Perry's Chemical Engineers' Handbook

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

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

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

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

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

DOW/DOW 9 0 2 1 0 9 8 7

ISBN 0-07-049841-5

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

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

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

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

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

ix

x

LIST OF CONTRIBUTORS

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

Ph.D.,

Research Scientist, Union Carbide Corporation

(Section 26,

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

LIST OF CONTRIBUTORS

xi

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

(Section

14, Gas Ab~orption

and

., Gas-Liquid

System

Design)

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

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

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

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

M.S.M.E.,

Consulting Engineer; Registered Professional

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

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

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

xii

LIST OF CONTRIBUTORS

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

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

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

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

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

E. Humphrey,

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

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

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

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

LIST OF CONTRIBUTORS

xiii

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

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

LIST OF CONTRIBUTORS

xv

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

xvi

LIST OF CONTRIBUTORS

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

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

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

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

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

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

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

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

Equipment

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

Process

(Section 18,

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

Ph.D.,

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

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

M.B.A.,

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

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

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

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

LIST OF CONTRIBUTORS

xvii

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

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

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

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

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

xviii

LIST OF CONTRIBUTORS

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

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

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

Preface to the Seventh Edition

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

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

Section

Conversion Factors and Mathematical Symbols

Jam£s 0. Maloney

1

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

2

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

3

Thermodynamics

4

Hendrick C. Van Ness, Michael M. Abbott

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

5

Fluid and Particle Dynamics

6

Reaction Kinetics

Jam£s N. Tilton

Stanley M. Walas

7

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

8

Process Economics

9

F A. Holland, J. K Wilkinson

Transport and Storage of Fluids

Meherwan 1'. Royce

10

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

11

Psychrometry, Evaporative Cooling, and Solids Drying Glenn W Baldwin

12

Distillation

Charles G. Moyers,

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

13

vii

viii

CONTENTS

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

james R. Fair; D. E. Steinmeyer;

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

14 Lanny A. Robbins,

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

Gas-Solid Operations and Equipment

Mel Pell, james B. Dunson

15 16 17

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

18

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

19

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

Richard L. Snow, Terry Allen,

Handling of Bulk Solids and Packaging of Solids and Liquids

Grantges I. Raymus

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

Stanley M. Walas

Biochemical Engineering

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

20 21

22 23 24

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

25

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

26

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

27

Materials of Construction

28

Oliver W Siebert, john G. Stoecker

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

Colin S. Howat

29 30

Section 1

Conversion Factors and Mathematical Symbols*

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

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

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

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

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

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

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

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

1-24 1-24

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

FIG. 1-1

1-2

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

TABLE 1-1

SI Base and Supplementary Quantities and Units

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

SI unit

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

meter kilogram second ampere kelvin mole* candela

m kg s A K mol cd

radian steradian

rad sr

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

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

Unit

Symbol

Formula

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

hertz newton pascal joule watt coulomb volt

Hz N Pa J W C V

l/s (kg⋅m)/s2 N/m2 N⋅m J/s A⋅s W/A

farad ohm siemens weber tesla henry lumen lux becquerel gray

F Ω S Wb T H lm lx Bq Gy

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

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

Unit

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

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

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

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

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

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

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

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

TABLE 1-3

Symbol

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

SI Prefixes

Multiplication factor

Prefix

Symbol

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

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

E P T G M k h da d c m µ n p f a

1 000 000 000 1 000 000 1 000 1

000 000 000 000 1

000 000 000 000 000 1

*Generally to be avoided.

1-3

TABLE 1-4

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

Quantity

SI unit

Alternate SI unit

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

Space,† time Length

naut mi mi chain link fathom yd ft

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

in in mil

km km m m m m m cm mm cm µm

Length/length

ft/mi

m/km

1.893 939 E − 01

Length/volume

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

m/m3 m/m3 m/m3

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

Area

mi2 section acre ha yd2 ft2 in2

km2 ha ha m2 m2 m2 mm2 cm2

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

Area/volume

ft2/in3 ft2/ft3

m2/cm3 m2/m3

5.699 291 E − 03 3.280 840 E + 00

Volume

cubem acre⋅ft

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

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

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

L L L L L L

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

Volume/length (linear displacement)

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

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

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

Plane angle

rad deg (°) min (′) sec (″)

rad rad rad rad

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

Solid angle

sr

sr

1

Time

year week h

a d s min s h ns

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

E + 00 E + 03 E + 01 E + 01 E − 02

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

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

min mµs Mass, amount of substance Mass

1-4

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

Mg Mg kg kg kg g g mg

t t

TABLE 1-4

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

Quantity Amount of substance

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

SI unit

Alternate SI unit

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

kmol kmol kmol

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

Btu/lbm cal/g cal/lbm

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

Caloric value, enthalpy (mole basis)

kcal/(g⋅mol) Btu/(lb⋅mol)

kJ/kmol kJ/kmol

Calorific value (volume basis—solids and liquids)

Btu/U.S. gal

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

kJ/dm3

Btu/U.K. gal Btu/ft3

cal/mL (ft⋅lbf)/U.S. gal

J/g J/g

2.326 000 2.326 000 6.461 112 4.184* 9.224 141

E − 03 E + 00 E − 04 E + 00 E + 00

4.184* E + 03 2.326 000 E + 00

kJ/dm3 kJ/dm3

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

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

Calorific value (volume basis—gases)

cal/mL kcal/m3 Btu/ft3

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

J/dm3 J/dm3 J/dm3

4.184* 4.184* 3.725 895 1.034 971

E + 03 E + 00 E + 01 E − 02

Specific entropy

Btu/(lbm⋅°R) cal/(g⋅K) kcal/(kg⋅°C)

kJ/(kg⋅K) kJ/(kg⋅K) kJ/(kg⋅K)

J/(g⋅K) J/(g⋅K) J/(g⋅K)

4.186 8* 4.184* 4.184*

E + 00 E + 00 E + 00

Specific-heat capacity (mass basis)

kWh/(kg⋅°C) Btu/(lbm⋅°F) kcal/(kg⋅°C)

kJ/(kg⋅K) kJ/(kg⋅K) kJ/(kg⋅K)

J/(g⋅K) J/(g⋅K) J/(g⋅K)

3.6* 4.186 8* 4.184*

E + 03 E + 00 E + 00

Specific-heat capacity (mole basis)

Btu/(lb⋅mol⋅°F) cal/(g⋅mol⋅°C)

kJ/(kmol⋅K) kJ/(kmol⋅K)

4.186 8* 4.184*

E + 00 E + 00

Temperature (absolute)

°R K

K K

5/9 1

Temperature (traditional)

°F

°C

5/9(°F − 32)

Temperature (difference)

°F

K, °C

5/9

Pressure

atm (760 mmHg at 0°C or 14,696 psi)

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

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

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

Vacuum, draft

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

kPa kPa kPa kPa kPa

3.376 85 2.490 82 2.488 4 1.333 224 9.806 38

E + 00 E − 01 E − 01 E − 01 E − 02

Liquid head

ft in

m mm cm

3.048* 2.54* 2.54*

E − 01 E + 01 E + 00

Pressure drop/length

psi/ft

kPa/m

2.262 059 E + 01

Temperature, pressure, vacuum

bar mmHg (0°C) = torr

1-5

TABLE 1-4

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

Quantity

SI unit

Alternate SI unit

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

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

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

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

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

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

6.242 796 6.242 796 6.242 796 1.002 242 8.345 404

E − 02 E − 05 E + 01 E + 01 E + 00

Specific volume (mole basis)

L/(g⋅mol) ft3/(lb⋅mol)

m3/kmol m3/kmol

1 6.242 796 E − 02

Specific volume

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

m3/t m3/t

1.752 535 E − 01 1.564 763 E − 01

Yield

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

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

Concentration (mass/mass)

wt % wt ppm

kg/kg g/kg mg/kg

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

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

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

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

Concentration (mole/volume)

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

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

Concentration (volume/mole)

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

dm3/kmol dm3/kmol

Throughput (mass basis)

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

Density

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

Specific volume

ft3/lbm 3

Concentration (mass/volume)

Concentration (volume/volume)

cm3/g cm3/g

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

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

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

L/kmol L/kmol

1.752 535 1.564 763 4.172 702 3.725 627

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

1.0* 1.0* 1

E − 02 E + 01

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

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

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

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

1.198 264 9.977 644 1.601 846 7.518 21

E + 02 E + 01 E + 01 E − 03

3.166 91 1.330 10

E + 00 E − 01

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

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

Facility throughput, capacity

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

1-6

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

TABLE 1-4

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

Quantity Throughput (volume basis)

bbl/day 3

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

SI unit

Alternate SI unit

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

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

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

kmol/h kmol/s

4.535 924 E − 01 1.259 979 E − 04

Throughput (mole basis)

(lbm⋅mol)/h

Flow rate (mass basis)

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

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

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

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

Flow rate (volume basis)

bbl/day

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

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

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

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

Flow rate (mole basis)

(lb⋅mol)/s (lb⋅mol)/h million scf/D

kmol/s kmol/s kmol/s

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

Flow rate/length (mass basis)

lbm/(s⋅ft) lbm/(h⋅ft)

kg/(s⋅m) kg/(s⋅m)

1.488 164 E + 00 4.133 789 E − 04

Flow rate/length (volume basis)

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

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

Flow rate/area (mass basis)

lbm/(s⋅ft2) lbm/(h⋅ft2)

kg/(s⋅m2) kg/(s⋅m2)

Flow rate/area (volume basis)

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

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

Flow rate

3

ft /day bbl/h ft3/h

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

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

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

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

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

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

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

1-7

TABLE 1-4

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

Quantity

SI unit

Alternate SI unit

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

Energy, work, power Energy, work

therm

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

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

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

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

Impact energy

kgf⋅m lbf⋅ft

J J

9.806 650* E + 00 1.355 818 E + 00

Surface energy

erg/cm2

mJ/m2

1.0*

Specific-impact energy

(kgf⋅m)/cm2 (lbf⋅ft)/in2

J/cm2 J/cm2

9.806 650* E − 02 2.101 522 E − 03

Power

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

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

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

Power/area

Btu/(s⋅ft2) cal/(h⋅cm2) Btu/(h⋅ft2)

kW/m2 kW/m2 kW/m2

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

Heat-release rate, mixing power

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

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

2.633 414 1.162 222 3.725 895 1.034 971

Cooling duty (machinery)

Btu/(bhp⋅h)

W/kW

Specific fuel consumption (mass basis)

lbm/(hp⋅h)

mg/J kg/kWh

kg/MJ

1.689 659 E − 01 6.082 774 E − 01

Specific fuel consumption (volume basis)

m3/kWh U.S. gal/(hp⋅h) U.K. pt/(hp⋅h)

dm3/MJ dm3/MJ dm3/MJ

mm3/J mm3/J mm3/J

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

Fuel consumption

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

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

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

2.824 807 2.352 146 4.251 437 3.540 064

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

1-8

E + 00

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

E + 01 E + 00 E + 01 E − 02

3.930 148 E − 01

E + 02 E + 02 E − 01 E − 01

TABLE 1-4

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

Quantity

SI unit

Alternate SI unit

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

in/s in/min

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

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

Corrosion rate

in/year (ipy) mil/year

mm/a mm/a

2.54* 2.54*

Rotational frequency

r/min

r/s rad/s

1.666 667 E − 02 1.047 198 E − 01

Acceleration (linear)

ft/s2

m/s2 cm/s2

3.048* 3.048*

Acceleration (rotational)

rpm/s

rad/s2

1.047 198 E − 01

Momentum

(lbm⋅ft)/s

(kg⋅m)/s

1.382 550 E − 01

Force

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

kN kN N N mN

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

Bending moment, torque

U.S. tonf⋅ft kgf⋅m lbf⋅ft lbf⋅in

kN⋅m N⋅m N⋅m N⋅m

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

Bending moment/length

(lbf⋅ft)/in (lbf⋅in)/in

(N⋅m)/m (N⋅m)/m

5.337 866 E + 01 4.448 222 E + 00

Moment of inertia

lbm⋅ft2

Velocity (linear), speed

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

MPa MPa MPa MPa kPa Pa

E − 01 E + 01

4.214 011 E − 02

kg⋅m2 2

E + 01 E − 02

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

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

Stress

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

Mass/length

lbm/ft

kg/m

1.488 164 E + 00

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

U.S. ton/ft2 lbm/ft2

Mg/m2 kg/m2

9.764 855 E + 00 4.882 428 E + 00

Diffusivity

ft2/s m2/s ft2/h

m2/s mm2/s m2/s

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

Thermal resistance

(°C⋅m2⋅h)/kcal (°F⋅ft2⋅h)/Btu

(K⋅m2)/kW (K⋅m2)/kW

8.604 208 E + 02 1.761 102 E + 02

Heat flux

Btu/(h⋅ft2)

kW/m2

3.154 591 E − 03

Thermal conductivity

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

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

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

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

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

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

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

Miscellaneous transport properties

2

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

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

1-9

TABLE 1-4

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

Quantity

SI unit

Alternate SI unit

kW/(m3⋅K) kW/(m3⋅K)

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

Volumetric heat-transfer coefficient

Btu/(s⋅ft3⋅°F) Btu/(h⋅ft3⋅°F)

6.706 611 E + 01 1.862 947 E − 02

Surface tension

dyn/cm

mN/m

Viscosity (dynamic)

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

Pa⋅s Pa⋅s Pa⋅s Pa⋅s Pa⋅s Pa⋅s Pa⋅s

Viscosity (kinematic)

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

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

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

Permeability

darcy millidarcy

µm2 µm2

9.869 233 E − 01 9.869 233 E − 04

Thermal flux

Btu/(h⋅ft2) Btu/(s⋅ft2) cal/(s⋅cm2)

W/m2 W/m2 W/m2

3.152 1.135 4.184

E + 00 E + 04 E + 04

Mass-transfer coefficient

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

m/s m/s

8.467 1.0

E − 05 E + 01

Admittance

S

S

1

Capacitance

µF

µF

1

Charge density

C/mm3

C/mm3

1

Conductance

S⍀

S S

1 1

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

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

Electricity, magnetism

(mho)

Conductivity

S/m ⍀ /m ⍀ m /m

S/m S/m mS/m

1 1 1

Current density

A/mm2

A/mm2

1

2

Displacement

C/cm

C/cm2

1

Electric charge

C

C

1

Electric current

A

A

1

Electric-dipole moment

C⋅m

C⋅m

1

Electric-field strength

V/m

V/m

1

Electric flux

C

C

1

Electric polarization

C/cm2

C/cm2

1

Electric potential

V mV

V mV

1 1

Electromagnetic moment

A⋅m2

A⋅m2

1

Electromotive force

V

V

1

Flux of displacement

C

C

1

Frequency

cycles/s

Hz

1

Impedance

Ω

Ω

1

Linear-current density

A/mm

A/mm

1

Magnetic-dipole moment

Wb⋅m

Wb⋅m

1

Magnetic-field strength

A/mm Oe gamma

A/mm A/m A/m

1 7.957 747 E + 01 7.957 747 E − 04

Magnetic flux

mWb

mWb

1

1-10

TABLE 1-4

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

Quantity Magnetic-flux density

mT G gamma

SI unit mT T nT

Alternate SI unit

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

Magnetic induction

mT

mT

1

Magnetic moment

A⋅m2

A⋅m2

1

Magnetic polarization

mT

mT

1

Magnetic potential difference

A

A

1

Magnetic-vector potential

Wb/mm

Wb/mm

1

Magnetization

A/mm

A/mm

1

Modulus of admittance

S

S

1

Modulus of impedance

Ω

Ω

1

Mutual inductance

H

H

1

Permeability

µH/m

µH/m

1

E − 04

Permeance

H

H

1

Permittivity

µF/m

µF/m

1

Potential difference

V

V

1

Quantity of electricity

C

C

1

Reactance

Ω

Ω

1

Reluctance

H−1

H−1

1

Resistance

Ω

Ω

1

Resistivity

Ω⋅cm Ω⋅m

Ω⋅cm Ω⋅m

1 1

Self-inductance

mH

mH

1

Surface density of change

mC/m2

mC/m2

1

Susceptance

S

S

1

Volume density of charge

C/mm3

C/mm3

1

Absorbed dose

rad

Gy

1.0*

Acoustical energy

J

J

1

Acoustical intensity

W/cm2

W/m2

1.0*

Acoustical power

W

W

1

Sound pressure

N/m2

N/m2

1.0*

Illuminance

fc

lx

1.076 391 E + 01

Illumination

fc

lx

1.076 391 E + 01

Irradiance

W/m2

W/m2

1

Light exposure

fc⋅s

lx⋅s

1.076 391 E + 01

Luminance

cd/m2

cd/m2

1

Acoustics, light, radiation

Luminous efficacy

lm/W

lm/W

1

Luminous exitance

lm/m2

lm/m2

1

Luminous flux

lm

lm

1

Luminous intensity

cd

cd

1

Radiance

W/m2⋅sr

W/m2⋅sr

1

Radiant energy

J

J

1

Radiant flux

W

W

1

Radiant intensity

W/sr

W/sr

1

Radiant power

W

W

1

E − 02 E + 04

1-11

TABLE 1-4

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

Quantity

SI unit

Alternate SI unit

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

Wavelength

Å

nm

1.0*

E − 01

Capture unit

10−3 cm−1

m−1

E + 01

m−1

m−1

1.0* 1 1

Ci

Bq

3.7*

E + 10

Radioactivity

10−3 cm−1

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

Name

Unit symbol

Name

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

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

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

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

NOTE:

1-12

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

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

Multiply by

To convert from

To

Multiply by

ampere coulomb farad henry mho ohm volt meter2 ampere

To

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

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

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

joule

+03 1.055 04

joule joule

+03 1.055 87 +03 1.054 350

joule joule meter3 meter meter joule

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

joule joule joule joule joule

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

joule joule

+03 4.190 02 +03 4.184

kilogram kelvin newton/meter2 newton/meter2 meter meter

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

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

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

meter2 meter3 coulomb

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

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

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

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

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

kilogram

−01 4.535 923

meter meter

+03 5.559 552 +03 5.556

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

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

Fahrenheit (temperature)

Celsius

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

farad coulomb

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

coulomb coulomb meter meter

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

1-13

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

1-14

To

Multiply by

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

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

kilogram

−01 4.535 923

kilogram

−01 3.732 417

newton meter3 meter3 joule/kilogram

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

kelvin 1/second-meter2

tK = (5/9)tR +10 1.00

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

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

To

Multiply by

second (ephemeris) second (mean solar)

To convert from

second second (ephemeris)

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

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

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

second

+07 3.155 692

TABLE 1-6

Alphabetical Listing of Common Conversions

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

To

Multiply by

To convert from

To

Multiply by

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

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

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

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

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

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

1-15

TABLE 1-6

Alphabetical Listing of Common Conversions (Concluded )

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

1-16

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

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

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

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

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

TABLE 1-6

Alphabetical Listing of Common Conversions (Concluded )

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

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

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

To convert from

To

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

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

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

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

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

1-17

TABLE 1-7 Mass (M)

Length (L)

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

Area (L2)

Volume (L3)

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

1 gallon Time (θ) 1 hour

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

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

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

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

TABLE 1-8

Kinematic-Viscosity Conversion Formulas

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

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

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

TABLE 1-9 Temp. scale

Values of the Gas-Law Constant Press. units

Vol. units

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

cm3 liters liters liters liters ft3 ft3

atm in Hg mm Hg lb/in2abs lb/ft2abs

ft3 ft3 ft3 ft3 ft3

Rankine

1-18

Wt. units

Energy units

R

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

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

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

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

冧 冧

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

TABLE 1-11

Temperature Conversion Formulas

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

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

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

冦

冧

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

冦

Volume Measure Solid:

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

1-19

TABLE 1-12

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

Lb per ft3 at 60°F wt in air

Sp gr 60°/ 60°

°Bé

104.33 102.38 100.47 98.58 96.73

4.9929 5.0346 5.0763 5.1180 5.1597

37.350 37.662 37.973 38.285 38.597

0.700 .705 .710 .715 .720

70.00 68.58 67.18 65.80 64.44

94.00 92.22 90.47 88.75 87.05

94.90 93.10 91.33 89.59 87.88

5.2014 5.2431 5.2848 5.3265 5.3682

39.910 39.222 39.534 39.845 40.157

.725 .730 .735 .740 .745

.650 .655 .660 .665 .670

85.38 83.74 82.12 80.53 78.96

86.19 84.53 82.89 81.28 79.69

5.4098 5.4515 5.4932 5.5349 5.5766

40.468 40.780 41.092 41.404 41.716

.675 .680 .685 .690 .695

77.41 75.88 74.38 72.90 71.44

78.13 76.59 75.07 73.57 72.10

5.6183 5.6600 5.7017 5.7434 5.7851

42.028 42.340 42.652 42.963 43.275

Lb per ft3 at 60°F wt in air

Sp gr 60°/ 60°

°Bé

Sp gr 60°/ 60°

°Bé

°API

0.600 .605 .610 .615 .620

103.33 101.40 99.51 97.64 95.81

.625 .630 .635 .640 .645

°API

Lb per gal at 60°F wt in air

Lb per ft3 at 60°F wt in air

Sp gr 60°/ 60°

°Bé

°API

70.64 69.21 67.80 66.40 65.03

5.8268 5.8685 5.9101 5.9518 5.9935

43.587 43.899 44.211 44.523 44.834

0.800 .805 .810 .815 .820

45.00 43.91 42.84 41.78 40.73

45.38 44.28 43.19 42.12 41.06

63.10 61.78 60.48 59.19 57.92

63.67 62.34 61.02 59.72 58.43

6.0352 6.0769 6.1186 6.1603 6.2020

45.146 45.458 45.770 46.082 46.394

.825 .830 .835 .840 .845

39.70 38.67 37.66 36.67 35.68

.750 .755 .760 .765 .770

56.67 55.43 54.21 53.01 51.82

57.17 55.92 54.68 53.47 52.27

6.2437 6.2854 6.3271 6.3688 6.4104

46.706 47.018 47.330 47.642 47.953

.850 .855 .860 .865 .870

.775 .780 .785 .790 .795

50.65 49.49 48.34 47.22 46.10

51.08 49.91 48.75 47.61 46.49

6.4521 6.4938 6.5355 6.5772 6.6189

47.265 48.577 48.889 49.201 49.513

.875 .880 .885 .890 .895

Lb per ft3 at 60°F wt in air

Sp gr 60°/ 60°

Lb per gal at 60°F wt in air

Lb per ft3 at 60°F wt in air

Sp gr 60°/ 60°

°Bé

°API

Lb per gal at 60°F wt in air

6.6606 6.7023 6.7440 6.7857 6.8274

49.825 50.137 50.448 50.760 51.072

0.900 .905 .910 .915 .920

25.56 24.70 23.85 23.01 22.17

25.72 24.85 23.99 23.14 22.30

7.4944 7.5361 7.5777 7.6194 7.6612

56.062 56.374 56.685 56.997 57.310

40.02 38.98 37.96 36.95 35.96

6.8691 6.9108 6.9525 6.9941 7.0358

51.384 51.696 52.008 52.320 52.632

.925 .930 .935 .940 .945

21.35 20.54 19.73 18.94 18.15

21.47 20.65 19.84 19.03 18.24

7.7029 7.7446 7.7863 7.8280 7.8697

57.622 57.934 58.246 58.557 58.869

34.71 33.74 32.79 31.85 30.92

34.97 34.00 33.03 32.08 31.14

7.0775 7.1192 7.1609 7.2026 7.2443

52.943 53.255 53.567 53.879 54.191

.950 .955 .960 .965 .970

17.37 16.60 15.83 15.08 14.33

17.45 16.67 15.90 15.13 14.38

7.9114 7.9531 7.9947 8.0364 8.0780

59.181 59.493 59.805 60.117 60.428

30.00 29.09 28.19 27.30 26.42

30.21 29.30 28.39 27.49 26.60

7.2860 7.3277 7.3694 7.4111 7.4528

54.503 54.815 55.127 55.438 55.750

.975 .980 .985 .990 .995 1.000

13.59 12.86 12.13 11.41 10.70 10.00

13.63 12.89 12.15 11.43 10.71 10.00

8.1197 8.1615 8.2032 8.2449 8.2866 8.3283

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

°Tw

Lb per gal at 60°F wt in air

°Tw

Lb per gal at 60°F wt in air

°Bé

°Tw

Lb per gal at 60°F wt in air

°Bé

°Tw

Lb per gal at 60°F wt in air

1.005 1.010 1.015 1.020 1.025

0.72 1.44 2.14 2.84 3.54

1 2 3 4 5

8.3700 8.4117 8.4534 8.4950 8.5367

62.612 62.924 63.236 63.547 63.859

1.255 1.260 1.265 1.270 1.275

29.46 29.92 30.38 30.83 31.27

51 52 53 54 55

10.4546 10.4963 10.5380 10.5797 10.6214

78.206 78.518 78.830 79.141 79.453

1.505 1.510 1.515 1.520 1.525

48.65 48.97 49.29 49.61 49.92

101 102 103 104 105

12.5392 12.5809 12.6226 12.6643 12.7060

93.800 94.112 94.424 94.735 95.047

1.755 1.760 1.765 1.770 1.775

62.38 62.61 62.85 63.08 63.31

151 152 153 154 155

14.6238 14.6655 14.7072 14.7489 14.7906

109.394 109.705 110.017 110.329 110.641

1.030 1.035 1.040 1.045 1.050

4.22 4.90 5.58 6.24 6.91

6 7 8 9 10

8.5784 8.6201 8.6618 8.7035 8.7452

64.171 64.483 64.795 65.107 65.419

1.280 1.285 1.290 1.295 1.300

31.72 32.16 32.60 33.03 33.46

56 57 58 59 60

10.6630 10.7047 10.7464 10.7881 10.8298

79.765 80.077 80.389 80.701 81.013

1.530 1.535 1.540 1.545 1.550

50.23 50.54 50.84 51.15 51.45

106 107 108 109 110

12.7477 12.7894 12.8310 12.8727 12.9144

95.359 95.671 95.983 96.295 96.606

1.780 1.785 1.790 1.795 1.800

63.54 63.77 63.99 64.22 64.44

156 157 158 159 160

14.8323 14.8740 14.9157 14.9574 14.9990

110.953 111.265 111.577 111.889 112.200

1.055 1.060 1.065 1.070 1.075

7.56 8.21 8.85 9.49 10.12

11 12 13 14 15

8.7869 8.8286 8.8703 8.9120 8.9537

65.731 66.042 66.354 66.666 66.978

1.305 1.310 1.315 1.320 1.325

33.89 34.31 34.73 35.15 35.57

61 62 63 64 65

10.8715 10.9132 10.9549 10.9966 11.0383

81.325 81.636 81.948 82.260 82.572

1.555 1.560 1.565 1.570 1.575

51.75 52.05 52.35 52.64 52.94

111 112 113 114 115

12.9561 12.9978 13.0395 13.0812 13.1229

96.918 97.230 97.542 97.854 98.166

1.805 1.810 1.815 1.820 1.825

64.67 64.89 65.11 65.33 65.55

161 162 163 164 165

15.0407 15.0824 15.1241 15.1658 15.2075

112.512 112.824 113.136 113.448 113.760

1.080 1.085 1.090 1.095 1.100

10.74 11.36 11.97 12.58 13.18

16 17 18 19 20

8.9954 9.0371 9.0787 9.1204 9.1621

67.290 67.602 67.914 68.226 68.537

1.330 1.335 1.340 1.345 1.350

35.98 36.39 36.79 37.19 37.59

66 67 68 69 70

11.0800 11.1217 11.1634 11.2051 11.2467

82.884 83.196 83.508 83.820 84.131

1.580 1.585 1.590 1.595 1.600

53.23 53.52 53.81 54.09 54.38

116 117 118 119 120

13.1646 13.2063 13.2480 13.2897 13.3313

98.478 98.790 99.102 99.414 99.725

1.830 1.835 1.840 1.845 1.850

65.77 65.98 66.20 66.41 66.62

166 167 168 169 170

15.2492 15.2909 15.3326 15.3743 15.4160

114.072 114.384 114.696 115.007 115.318

1.105 1.110 1.115 1.120 1.125

13.78 14.37 14.96 15.54 16.11

21 22 23 24 25

9.2038 9.2455 9.2872 9.3289 9.3706

68.849 69.161 69.473 69.785 70.097

1.355 1.360 1.365 1.370 1.375

37.99 38.38 38.77 39.16 39.55

71 72 73 74 75

11.2884 11.3301 11.3718 11.4135 11.4552

84.443 84.755 85.067 85.379 85.691

1.605 1.610 1.615 1.620 1.625

54.66 54.94 55.22 55.49 55.77

121 122 123 124 125

13.3730 13.4147 13.4564 13.4981 13.5398

100.037 100.349 100.661 100.973 101.285

1.855 1.860 1.865 1.870 1.875

66.83 67.04 67.25 67.46 67.67

171 172 173 174 175

15.4577 15.4993 15.5410 15.5827 15.6244

115.630 115.943 116.255 116.567 116.879

1.130 1.135 1.140 1.145 1.150

16.68 17.25 17.81 18.36 18.91

26 27 28 29 30

9.4123 9.4540 9.4957 9.5374 9.5790

70.409 70.721 71.032 71.344 71.656

1.380 1.385 1.390 1.395 1.400

39.93 40.31 40.68 41.06 41.43

76 77 78 79 80

11.4969 11.5386 11.5803 11.6220 11.6637

86.003 86.315 86.626 86.938 87.250

1.630 1.635 1.640 1.645 1.650

56.04 56.32 56.59 56.85 57.12

126 127 128 129 130

13.5815 13.6232 13.6649 13.7066 13.7483

101.597 101.909 102.220 102.532 102.844

1.880 1.885 1.890 1.895 1.900

67.87 68.08 68.28 68.48 68.68

176 177 178 179 180

15.6661 15.7078 15.7495 15.7912 15.8329

117.191 117.503 117.814 118.126 118.438

1.155 1.160 1.165 1.170 1.175

19.46 20.00 20.54 21.07 21.60

31 32 33 34 35

9.6207 9.6624 9.7041 9.7458 9.7875

71.968 72.280 72.592 72.904 73.216

1.405 1.410 1.415 1.420 1.425

41.80 42.16 42.53 42.89 43.25

81 82 83 84 85

11.7054 11.7471 11.7888 11.8304 11.8721

87.562 87.874 88.186 88.498 88.810

1.655 1.660 1.665 1.670 1.675

57.39 57.65 57.91 58.17 58.43

131 132 133 134 135

13.7900 13.8317 13.8734 13.9150 13.9567

103.156 103.468 103.780 104.092 104.404

1.905 1.910 1.915 1.920 1.925

68.88 69.08 69.28 69.48 69.68

181 182 183 184 185

15.8746 15.9163 15.9580 15.9996 16.0413

118.740 119.062 119.374 119.686 119.998

1.180 1.185 1.190 1.195 1.200

22.12 22.64 23.15 23.66 24.17

36 37 38 39 40

9.8292 9.8709 9.9126 9.9543 9.9960

73.528 73.840 74.151 74.463 74.775

1.430 1.435 1.440 1.445 1.450

43.60 43.95 44.31 44.65 45.00

86 87 88 89 90

11.9138 11.9555 11.9972 12.0389 12.0806

89.121 89.433 89.745 90.057 90.369

1.680 1.685 1.690 1.695 1.700

58.69 58.95 59.20 59.45 59.71

136 137 138 139 140

13.9984 14.0401 14.0818 14.1235 14.1652

104.715 105.027 105.339 105.651 105.963

1.930 1.935 1.940 1.945 1.950

69.87 70.06 70.26 70.45 70.64

186 187 188 189 190

16.0830 16.1247 16.1664 16.2081 16.2498

120.309 120.621 120.933 121.245 121.557

1.205 1.210 1.215 1.220 1.225

24.67 25.17 25.66 26.15 26.63

41 42 43 44 45

10.0377 10.0793 10.1210 10.1627 10.2044

75.087 75.399 75.711 76.022 76.334

1.455 1.460 1.465 1.470 1.475

45.34 45.68 46.02 46.36 46.69

91 92 93 94 95

12.1223 12.1640 12.2057 12.2473 12.2890

90.681 90.993 91.305 91.616 91.928

1.705 1.710 1.715 1.720 1.725

59.96 60.20 60.45 60.70 60.94

141 142 143 144 145

14.2069 14.2486 14.2903 14.3320 14.3737

106.275 106.587 106.899 107.210 107.522

1.955 1.960 1.965 1.970 1.975

70.83 71.02 71.21 71.40 71.58

191 192 193 194 195

16.2915 16.3332 16.3749 16.4166 16.4583

121.869 122.181 122.493 122.804 123.116

1.230 1.235 1.240 1.245 1.250

27.11 27.59 28.06 28.53 29.00

46 47 48 49 50

10.2461 10.2878 10.3295 10.3712 10.4129

76.646 76.958 77.270 77.582 77.894

1.480 1.485 1.490 1.495 1.500

47.03 47.36 47.68 48.01 48.33

96 97 98 99 100

12.3307 12.3724 12.4141 12.4558 12.4975

92.240 92.552 92.864 93.176 93.488

1.730 1.735 1.740 1.745 1.750

61.18 61.34 61.67 61.91 62.14

146 147 148 149 150

14.4153 14.4570 14.4987 14.5404 14.5821

107.834 108.146 108.458 108.770 109.082

1.980 1.985 1.990 1.995 2.000

71.77 71.95 72.14 72.32 72.50

196 197 198 199 200

16.5000 16.5417 16.5833 16.6250 16.6667

123.428 123.740 124.052 124.364 124.676

Sp gr 60°/ 60°

°Bé

Lb per ft3 at 60°F wt in air

Sp gr 60°/ 60°

Lb per ft3 at 60°F. wt. in air

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

1-20

TABLE 1-13

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

Gauge number

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

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

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

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

0000000 000000 00000 0000 000 00 0

— — — 0.460 .410 .365 .325

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

— — — 0.454 .425 .380 .340

1 2 3 4 5

.289 .258 .229 .204 .182

.2830 .2625 .2437 .2253 .2070

6 7 8 9 10

.162 .144 .128 .114 .102

11 12 13 14 15

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

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

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

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

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

Gauge number

Gauge number

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

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

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

Gauge number

— — — — — — —

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

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

0000000 000000 00000 0000 000 00 0

26 27 28 29 30

0.0159 .0142 .0126 .0113 .0100

0.0181 .0173 .0162 .0150 .0140

0.018 .016 .014 .013 .012

0.0188 .0172 .0156 .0141 .0125

0.0196 .0175 .0156 .0139 .0123

0.018 .0164 .0148 .0136 .0124

26 27 28 29 30

.300 .284 .259 .238 .220

— — 0.239 .224 .209

.3532 .3147 .2804 .2500 .2225

.300 .276 .252 .232 .212

1 2 3 4 5

31 32 33 34 35

.0089 .0080 .0071 .0063 .0056

.0132 .0128 .0118 .0104 .0095

.010 .009 .008 .007 .005

.0109 .0102 .0094 .0086 .0078

.0110 .0098 .0087 .0077 .0069

.0116 .0108 .0100 .0092 .0084

31 32 33 34 35

.1920 .1770 .1620 .1483 .1350

.203 .180 .165 .148 .134

.194 .179 .164 .150 .135

.1981 .1764 .1570 .1398 .1250

.192 .176 .160 .144 .128

6 7 8 9 10

36 37 38 39 40

.0050 .0045 .0040 .0035 .0031

.0090 .0085 .0080 .0075 .0070

.004 — — — —

.0070 .0066 .0062 — —

.0061 .0054 .0048 .0043 .0039

.0076 .0068 .0060 .0052 .0048

36 37 38 39 40

.091 .081 .072 .064 .057

.1205 .1055 .0915 .0800 .0720

.120 .109 .095 .083 .072

.120 .105 .090 .075 .067

.1113 .0991 .0882 .0785 .0699

.116 .104 .092 .080 .072

11 12 13 14 15

41 42 43 44 45

— — — — —

.0066 .0062 .0060 .0058 .0055

— — — — —

— — — — —

.0034 .0031 .0027 .0024 .0022

.0044 .0040 .0036 .0032 .0028

41 42 43 44 45

16 17 18 19 20

.051 .045 .040 .036 .032

.0625 .0540 .0475 .0410 .0348

.065 .058 .049 .042 .035

.060 .054 .0478 .0418 .0359

.0625 .0556 .0495 .0440 .0392

.064 .056 .048 .040 .036

16 17 18 19 20

46 47 48 49 50

— — — — —

.0052 .0050 .0048 .0046 .0044

— — — — —

— — — — —

.0019 .0017 .0015 .0014 .0012

.0024 .0020 .0016 .0012 .0010

46 47 48 49 50

21 22 23 24 25

.0285 .0253 .0226 .0201 .0179

.0317 .0286 .0258 .0230 .0204

.032 .028 .025 .022 .020

.0329 .0299 .0269 .0239 .0209

.0349 .0313 .0278 .0248 .0220

.032 .028 .024 .022 .020

21 22 23 24 25

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

1-21

TABLE 1-14

Fundamental Physical Constants

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

1-22

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

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

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

e = electronic charge

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

Definition of Btu: Btu = IT British Thermal Unit

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

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

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

(Difference due to rounding)

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

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

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

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

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

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

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

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

1-23

MATHEMATICAL SYMBOLS TABLE 1-15

Mathematical Signs, Symbols, and Abbreviations

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

冮

冕

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

b

integral between the limits a and b

a

y˙ y¨ ∆ or ∇2

δ

冖

1-24

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

冢

冣

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

Greek Alphabet

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

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

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

Section 2

Physical and Chemical Data*

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

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

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

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

2-11 2-7 2-28

2-12 2-13

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

2-76 2-76 2-76

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

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

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

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

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

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

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

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

2-90 2-90

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

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

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

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

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

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

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

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

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

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

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

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

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

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

2-3 2-183 2-183

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

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

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

2-186 2-186

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

2-184 2-184

2-186 2-187

2-195 2-199 2-200

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

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

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

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

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

2-224 2-225 2-225

2-303 2-304 2-305

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

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

2-346 2-347

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

2-271

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

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

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

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

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

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

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

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

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

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

2-6

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

PHYSICAL PROPERTIES OF PURE SUBSTANCES TABLE 2-1

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

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

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

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

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

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

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

Formula

Formula weight

Color, crystalline form and refractive index

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

26.97 204.10 162.07 266.72 374.82 143.91 133.34

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

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

241.44 101.99 294.05 77.99 375.14 81.96 101.94 101.94 121.95

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

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

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

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

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

25° 4

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

Boiling point, °C 2056 268 752mm

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

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

Hot water i. d.

Other reagents s. HCl, H2SO4, alk.

s. d.

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

v. s.

50 al.; s. et.

s.

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

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

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

TABLE 2-1

Physical Properties of the Elements and Inorganic Compounds (Continued )

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

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

Formula weight

Color, crystalline form and refractive index

Boiling point, °C

Solubility in 100 parts Cold water

mn., 1.590

2.9

d.

i.

556.49 362.21 209.96 524.29 342.12 906.64

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

2.56

1450 (1150)

2.90 2.61 2.71 1.64 20° 4

1000 1100 d. 770 93.5

i. s. sl. s. i. 31.30° 3.90°

956.72

gn. or vl., oct., 1.4842

1.72

964.40

vl., oct., 1.485

1.71

40

col., mn., 1.4564

1.76 26° 4

92

998.84

red or gn., cb., 1.4814

1.83

89

916.56

col., oct., 1.4388

17.03

col. gas, 1.325 (lq.)

77.08 337.33 79.06 97.96 114.11 157.11

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

272.22

wh.

53.50 444.05 373.14 367.52 152.09 44.06 252.10 392.21 37.04 57.05 63.06

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

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

51.11 35.05 196.03 1235.95 80.05 80.05

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

nitrite osmochloride oxalate oxalate, acid perchlorate persulfate phosphate, monobasic

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

64.05 439.02 142.12 125.08 117.50 228.20 115.04

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

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

−20H2O, 120; −24H2O, 200

100 d.

948.76

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

2-8

Melting point, °C

796.40

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

carbonate, sesqui-

Specific gravity

20° 4

1.675

−79°

0.817 0.5971 (A) 1.073 1.573 2.327 15° 4

1.53 3.065

2.4 1.91712° 0.79100° (A) 2.15

i. 89100° ∞ 100°

25°

124 −18H2O, 64.5

i. al.

50

106.4

0°

i. al. 45°

121.7

−77.7

−33.4

89.9

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

d.

1484° s. 11.90° 6810° 10015° 2515°

v. s. 2730° 145.6100°

2015°

5049°

d. 350 d. d.

subl. 520

d. 180 36 d. 185 d.

12° 2.21 12 1.266

114–116

2.27

d.

1.66 25° 4 1.725 25° 4

169.6

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

expl.

d.

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

d. 210 d. 210

0°

0°

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

i. al.

7.4

96°

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

6765°

77.3100° 1.25100° v. s.

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

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

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

53180°

s. al.

d.

i. al., NH3 i. al.

v. s. s. d. 4425° 118.30° 365.835°

241.830° 58080°

s.

d.

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

11.850°

sl. s. al.; i. NH3

46.9100° d. 173.2100°

220° al.; s. act.; i. et.

0°

d. d. d. 120

i. al.

∞93°

20

61

i. HCl d. a.

s. al.

25°

5.70°

Other reagents

s.

21.2

d. 17°

Hot water

2.5 s. 10.90° 58.20° 22.70°

s. al.

i. ac.

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

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

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

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

13115° s. 0.0315°

1.619 2.21 d.

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

2.01

i.

17.5°

s. alk.; i. al., HNO3

1380

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

87 17020° 3.0570° i.

73.4

220.2

601.60°

∞72°

1570

v. sl. s.

1.769 20° 4 1.78

132 235 d. 146.9 d.

1.41 2.03 12° 4 1.60 1.305 2.326 6.68425°

d. d. d. 149.6 d. 630.5

SbCl3

228.13

col., rhb., delq.

3.14 20° 4

Sb2O3 Sb2O3 Sb2S3

291.52 291.52 339.70

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

5.67 5.2 4.64

656 652 550

subl. d. 160

i. act.

490

d. 170

0.00017

55.5 35750° 103.3100°

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

60°

sl. s. 18°

d.

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

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

Sb2S5 Sb2Te3

403.82 626.35

golden gray

4.1200°

−2S, 135 629

i.

i.

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

333.94 371.58 683.13 39.94

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

2.60 4.89

−a H2O, 100 −189.2

−185.7

35.7100° d. d. 2.2350° cc

s. gly.; i. al.

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

As4 As4

299.64 299.64

met., hex. bk., amor.

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

5.268.7° d. i. 5.60° cc

81436atm.

subl. 615

i. i.

i. i.

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

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

As4 H3AsO4·a H2O HAsO3 H4As2O7 As2O5 As2S2

299.64 150.94 123.92 265.85 229.82 213.94

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

−H2O, 160

50 H3AsO4 H3AsO4 76.7100° d.

s. alk.

d. 565

16.7 d. to form d. to form 59.50° i.

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

As2S5

310.12

yel.

d. 500

0.0001360°

i.

s. HNO3, alk.

AsCl3 AsH3 As2O3 As2O3 As2O3

181.28 77.93 197.82 197.82 197.82

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

130 −55; d. 230

d. 20 cc sl. s. sl. s. 1.210°

d. sl. s. sl. s. sl. s. 2.9340°

Auric chloride

AuCl3·2H2O

339.60

or. cr.

v. s.

v. s.

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

Au(CN)3·6H2O AuCl AuCN

383.35 232.66 223.22

yel. cr. yel. cr.

7.4

v. s. d. i.

v. s. d. i.

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

Ba Ba(C2H3O2)2 Ba(C2H3O2)2·H2O BaBr2

137.36 255.45 273.46 297.19

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

3.5 2.468 2.19 4.781 24° 4

d. 58.80° 7530°(anh.) 980°

d. 75.0100° 7940°(anh.) 149100°

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

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

d. 290

850

1140

−H2O, 41 847

d.

5.1515° gly. 2425° cc al.

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

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

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

TABLE 2-1

Physical Properties of the Elements and Inorganic Compounds (Continued )

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

Formula

Formula weight

Color, crystalline form and refractive index

Specific gravity

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

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

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

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

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

169.36 313.49 331.35 233.35 602.04 448.68 279.42 233.42

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

BaS BaS3 BaS4·2H2O Be(Gl) Bi

169.42 233.54 301.63 9.02 209.00

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

528.03 279.91 315.37 485.10 305.02 466.00 466.00 466.00 260.46

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

6.86 4.86 4.75 2.82 4.92815° 8.9 8.55 8.20 7.7215°

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

3.69 4.29

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

2.9 4.16515° 4.116° 3.920° 4.27915° 4.49915°

Boiling point, °C d. d. 1450

1560 1560

2.988 1.816 9.8020°

Hot water v. s. 0.0065100°

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

0.002218° 20.350° s. 310°

0.0065100° 84.880° s. 59100°

s. a.; i. al.

76.8100° 101.480°

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

77.9 592

−8H2O, 550 d.

1923

2000

d. d. d.

tr. to mn. 1149

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

2767 1450

d. s. 4115° i. i.

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

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

i.

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

2.660°

40.2100°

−O, 800 −8H2O, 100

1580 d. d. 400 d. 200 1284 271

d. 300 447 −5H2O, 80 1900

Other reagents

v. s. 0.002218°

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

4.2515° 20°

Solubility in 100 parts Cold water

34.2100° 0.002424° 90.880°

0.09100° 0.00028530°

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

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

Boric acid

H3BO3

61.84

wh., tri.

1.43515°

Boron

B

10.82

2300

2550

i.

i.

B4C B2O3 B2O3·3H2O HBrO3 Br2

55.29 69.64 123.69 128.92 159.83

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

2.32

carbide oxide oxide (sassolite) Bromic acid Bromine

2.54 1.85 1.49

2450 577 d. d. 100 −7.2

>3500 >1500

i. 1.10° sl. s. v. s. 4.220°

i. 15.7100° s. d. 3.1330°

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

hydrate Cadmium acetate acetate carbonate

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

339.99 112.41 230.50 266.53 172.42

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

8.6520° 2.341 2.01 4.2584°

i.

wh., cb.

4.047 25° 4

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

chloride

CdCl2

183.32

3.11920°; 5.87 (A)

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

58.78 767 d.

960

s. i. v. s. v. s. i. 0°

90

i. 147100°

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

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

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

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

col., mn., 1.6513

3.327

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

4.79 15° 4

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

184.42 216.52 164.10 236.16 148.26 150.11 128.10 146.12 56.08

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

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

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

216.21 252.09 172.10 310.20 198.04 254.12 344.20 182.20 116.14

silicate (β) (wollastonite) sulfate (anhydrite)

CaSiO3 CaSO4

116.14 136.14

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

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

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

132

d. 900–1000 d. 1000 tr. 108 tr. 41.5

3.6720° 2.765

tr. 4 1750100atm 810 d. 1600 1551

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

760 d. 825 1339103atm. 772

1810

1.6817°

29.92 −2H2O, 130

−6H2O, 200 −4H2O, 185

1.7 3.18020° 2.015 1.7 2.2

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

subl. in N2, 980 1200  30

>1600

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

−8H2O, 100 −H2O, 100 d. 1670 975 1230

2850 expl. 275 d. 200

>1600 1540 tr. 1190 to α 1450(mn.)

180100°

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

60.8100° s. 127.660° s.

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

32659.5° i. i.

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

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

18°

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

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

tr. 1193 to rhb.

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

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

0.1619100°

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

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

Physical Properties of the Elements and Inorganic Compounds (Continued )

TABLE 2-1

Name Calcium (Cont.) sulfate (gypsum)

Formula

Formula weight

Color, crystalline form and refractive index

CaSO4·2H2O

172.17

col., mn., 1.5226

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

214.31 72.14 156.17 260.22 210.28 260.30 288.00

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

C C C CO2

12.01 12.01 12.01 44.01

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

disulfide

CS2

76.13

col. lq.

monoxide

CO

28.01

col., poisonous, odorless gas

oxychloride (phosgene) oxysulfide

COCl2 COS

98.92 60.07

poisonous gas gas

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

suboxide thionyl chloride Ceric hydroxide hydroxynitrate oxide sulfate Cerium

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

68.03 114.98 398.31 397.21 172.13 404.31 140.13

Cerous sulfate sulfate Cesium Chloric acid Chlorine

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

568.44 712.57 132.91 210.58 70.91

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

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

215.04 518.08 441.55 116.52 494.32 158.38 266.48 109.01 103.03

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

139.07 400.18 373.15 152.02 392.20 482.28 662.44 716.49

hydroxide nitrate nitrate oxide sulfate sulfate sulfate sulfate

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

Specific gravity 2.32 2.815°

Melting point, °C

Boiling point, °C

−1a H2O, 128

−2H2O, 163

d. 15 −2H2O, 100 d.

d. 650

Solubility in 100 parts Cold water

Hot water

0.2230°

0.25750°

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

v. s. d. 0.002790° 0.2285° v. s. d.

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

1.87316° 6.06

d.

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

>3500 >3500 >3500 −56.65.2atm.

4200 4200 4200 subl. −78.5

i. i. i. 179.70° cc

i. i. i. 90.120° cc

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

−108.6

46.3

0.20°

0.01450°

s. al.; et.

−207

−192

0.00440°; 3.50° cc

0.001850° 2.3220° cc

s. al., Cu2Cl2

−104 −138.2

8.2756mm −50.2760mm

v. s. sl. d. 1330° cc

d. 40.330° cc

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

−107

7761mm 73.5

d.

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

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

5.21 3.012 1.86717° 1.722°

1950 645

1400

d. i. s. d. i.

i.

0°

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

670 d. 40 −34.6

151.5765mm 1200–1500 d.

subl. 83 >1000

d.

−2H2O, 100 36.5 100 1900

d. 100 d.

100

−10H2O, 100 −12H2O, 100

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

Slowly oxidized 0.4100° 7.640°

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

s. a., al., NH3 0.57 ; 17730° cc 30°

v. s.

sl. s.

i. s. s. i. d. 67° d.

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

sulfide Chromium

Cr2S3 Cr

200.02 52.01

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

3.7719° 7.1

−S, 1350 1615

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

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

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

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

2.70 2.75

197 d.

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

1.7016 1.847 25° 1.819 25

CoO CoSO4 CoSO4·H2O

74.94 155.00 173.02

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

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

d. 3.97 1.92 8.920° 1.7318° 4.269 2.94

1550 −96.5 1480 51 subl.

5.18

d. 100 −1a H2O, 100 d. 900

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

−4H2O, 140 subl. 86 <100

5.68 3.71025° 3.13

d. 1800 d. 880 d.

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

96.8 >1100 1083

−7H2O, 420

115

240 d.

1.98

d. i.

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

206.7100° v. s.

i. 25.60° s.

i.

i. d.

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

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

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

s.

20

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

d. 110

33.80°

99.380°

s. a.

1.81 3.88

d. 150 d. 220

18.0521.5° i.

d. d.

i. 70.70°

d. 107.9100°

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

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

192.4100°

ammonium sulfate carbonate, basic (azurite)

CuSO4·4NH3·H2O 2CuCO3·Cu(OH)2

245.77 344.75

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

carbonate, basic (malachite) chloride (eriochalcite)

CuCO3·Cu(OH)2 CuCl2

221.17 134.48

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

3.9 3.054

d. 498

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

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

170.52 374.75 115.61 315.62 614.63 465.21 153.61 97.59 277.74 241.63 295.68

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

2.3922.4°

−2H2O, 110 −2H2O, 260 d. −2H2O, 100

277.51

1049 −6H2O, 110 d.

i. i.

3380° 0.0003818° i. s. 7.2 i.

ammonium chloride

dark gn., mn. gn.

117.6 2900 d. 52

20°

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

281.11 91.00 63.57 181.66 199.67 1013.83

2200

2.28618° 1.831 3.368

−H2O

2.0473.9° 2.074

114.5 −3H2O, 26.4

2300

Forms Cu2Cl2 993 d.

−HNO3, 170

i.

d. i. d. d. 45100° 66680° ∞

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

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

*Usual commercial form. †Also a soluble modification.

2-13

TABLE 2-1

Physical Properties of the Elements and Inorganic Compounds (Continued )

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

Formula

Specific gravity

249.71 95.63

tartate Cuprous ammonium iodide carbonate chloride (nantokite) cyanide

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

265.69 353.47 187.15 198.05 179.16

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

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

402.67 466.24 165.14 80.58 143.14 443.38 159.20 159.20 52.02

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

Fe(OH)(C2H3O2)2

190.95

brn., amor.

FeCl3 FeCl3·6H2O* Fe4[Fe(CN)6]3

162.22 270.32 859.27

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

2.80411°

282 37 d.

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

106.87 323.06 349.97 159.70

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

3.4 to 3.9

−1a H2O, 500

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

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

399.88 562.02 775.49 1662.70 231.55 303.61 392.15

chloride (lawrencite)

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

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

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

571.92 591.47 323.66 181.92 89.87 287.96 71.85

2-14

126.76

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

6.40 6.45

Melting point, °C

CuSO4·5H2O* CuS

hydroxide lactate nitrate oxide (hematite)

79.57 79.57 365.69 252.67 159.63

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

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

CuO CuO CuCl2·2CuO·4H2O Cu3P2 CuSO4

Formula weight

6.35 3.60615° 2.286 15.6° 4 4.6

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

4.4 3.53 2.9

d. 422 474.5

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

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

Hot water i. i. 75.4100°

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

Forms CuO, 650 −5H2O, 250 d. 220

24.3 0.00003318°

205100°

1.18° al. s. HNO3, KCN

0.1485°

1366 d.

0.0215° d. i. 1.5225° i.

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

d.

908 −a H2O, 360 1235

−O, 1800

1100 1130 −34.4

−20.5

subl. 1100

0°

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

i. i.

i. i.

i.

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

74.40° 2460° i.

535.8100° ∞ d.

i. v. s. 1500° i.

i. v. s. ∞ d. d. s.

5.2

d. 50 d. 180 1538 d.

sl. s. 440 s. i. i.

i.

s. d. h. HCl i. al.

1.864

d. d.

i. 180°

i. 10075°

s. a. i. al.

64.410°

105.7100°

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

v. s. i. i. sl. s. 0.00067 2000° i.

v. s.

20°

1.684 5.12

35 1560 d.

3.09718° 2.1

d. 480

2.7 2.714 d. 3.4 5.7

315 280

d.

delq.

d. lt. gn. cr. bk.

Boiling point, °C

60.5 1420

i. H2SO4, NH3 s. abs. al.

i. dil. a., al.

30025° i.

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

phosphate (vivianite)

Fe3(PO4)2·8H2O

501.64

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

FeSiO3 FeSO4·5H2O FeSO4·7H2O* FeS

131.91 241.99 278.02 87.91

HBF4 F2

87.83 38.00

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

H2SiF6 Gd GaBr3

144.08 156.9 309.47

Au Au

197.20 197.20

yel. met., cb. blue to vl.

19.3

1063

2600

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

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

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

12.1 0.1368 (A) 1.011 15° 4

>1700 <−272.2 1.4 128 −40

>3200(?) −268.9 113.5

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

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

80.92 116.96 36.47 36.47 72.50 90.51 27.03

Hydrofluoric acid Hydrofluoric acid Hydrogen

HF HF (35.35% in H2O) H2

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

peroxide selenide sulfide Hydroxylamine hydrochloride nitrate sulfate

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

20.01 20.01 2.016 34.02 81.22 34.08 33.03 69.50 96.05 82.07

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

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

1550

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

1.0321° 1.42

1.378 4.40° (A) 1.715°

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

0.697

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

i.

s. a.; i. ac.

64 1193

−5H2O, 300 −7H2O, 300 d.

s. 32.80° 0.00061618°

s. 14950°

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

−223

130 d. −187

∞ d.

∞

s. al.

s.

s.

s.

s.

i. s.

i.

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

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

1.0850° cc ∞

Absorbed by Pt s. al.

∞ v. s. v. s. v. s.

∞ al.; i. et. sl. s. al. s. al.

delq. cr. 20°

i.

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

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

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

−67

d. d. 26

∞ s. 82.30° ∞ ∞ ∞ ∞

19.4 120 −252.7

∞ 0° to 19.4° v. s. 2.10° cc

126 −85

760mm

151.4 −42 −59.6 56.522mm d. d. <100

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

27.6560° ∞ v. s.

130100°

s. 56.160°

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

v. s. 0.8580° cc

sl. s. Fe, Pd, Pt

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

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

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

TABLE 2-1

Physical Properties of the Elements and Inorganic Compounds (Continued )

Name

Formula

Hypobromous acid Illinium Indium Iodic acid

HBrO Il In HIO8

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

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

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

FeSi FeS2 FeS2 Fe7S8

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

Formula weight 96.92 146(?) 114.76 175.93

Color, crystalline form and refractive index

Specific gravity

Melting point, °C

yel.

Boiling point, °C

Solubility in 100 parts Cold water

Hot water

4050mm

s.

d.

soft, tet. met. col., rhb.

7.320° 4.6290°

155 110 d.

1450

i. 2860°

i. 576101°

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

4.9320° 4.799 25° 4

113.5 d. 300

184.35

0.0956660°

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

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

>4800

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

83.91 119.97 119.97 647.43

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

6.1 20° 4 4.87 5.0 4.6 20° 4

d. d.

i. 0.00049 0.0005 i.

i.

tr. 450 1171 d. >700

Kr La Pb

83.70 138.92 207.21

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

2.818 (A) 6.1520° 20° 11.337 20

−169 826 327.5

−151.8 1800 1620

11.050° cc d. i.

3.5760° cc

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

325.30 379.35 505.46 608.56 584.54

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

3.251 20° 4 2.55 1.689

280 −3H2O, 75 22

807.75

wh. nd.

489.06 347.13 453.03 676.24 291.26 367.05

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

4.46 5.94 6.4215° 15° 6.85 15 6.66

802 expl. 350 373

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

15°

d. 140 d. >200

3000

102.5760mm

−H2O, 280

918

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

i.

19.70° 45.6415° s. v. s. v. s.

22150° 200100° s.

5.55

18.2

d. i. d. i. i. 0.45540°

d. 0.05100° 4.75100°

sl. s.

PbCO3

267.22

wh., rhb., 2.0763

6.6

d. 315

0.0001120°

d.

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

775.67 278.12 323.22 546.43 297.25 687.65 331.23

6.14 5.80 6.12

d. 400 501 844

4.56 7.592 4.53

d. 190 −H2O, 130 d. 470

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

i. 3.34100° i. i. 18100° d.

oxide, suboxide, mono- (litharge)

Pb2O PbO

430.42 223.21

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

8.34 9.53

d. red heat 888

i. 0.006818°

i.

oxide, mono (massicotite)

PbO

223.21

yel., rhb., 2.61

8.0

2-16

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

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

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

954760mm d.

Other reagents

138.8100°

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

oxide, mono-

PbO

223.21

amor.

9.2 to 9.5

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

Pb3O4 Pb2O3 PbO2 PbSiO3 PbSO4

685.63 462.42 239.21 283.27 303.27

9.1

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

419.36 526.48 239.27 323.37 6.94 128.05 86.86

LiBr·2H2O Li2CO3 LiCl

122.89 73.89 42.40

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

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

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

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

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

MgCl2·NH4Cl·6H2O MgNH4PO4·6H2O

256.83 245.44

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

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

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

i.

i.

d. 500 d. 360 d. 290 766

i. i. i. i.

i. i. i.

1336  5

3.464 25° 4

547

1265

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

0.005640°

6.92 7.5 3.82 0.5320°

1170 d. 977 1120 d. 190 186

2.110° 2.068 25° 4

44 618 614

d. 1360

24620° 1.540° 670°

9.375 6.49 6.2

2.29521.5° 1.46 0.820 2.54 1.83 2.38 2.013 25° 4

d. 870 −H2O, 94 680 445 261 29.88

2.22 2.06 2.12313°

>100 837 100 d. 860 −H2O, 130 170.5

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

1.7420° 1.42 1.454 3.6

651 323 80 2135

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

1.456 1.715

−4H2O, 195 d. 100

2.461 2.53717.5° 1.645

1670 925 d. subl. <1000

1110

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

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

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

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

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

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

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

16.7 0.02310°

s. 0.019580°

29.9100° 35100°

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

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

360.62

col., mn.

1.72

>120

16.86

320.59 83.43 138.38 365.37

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

3.037 1.852 2.16

−3H2O, 110 d. 350 −H2O, 100 d.

4.525° (anh.) 0.0106 0.151819° 0.04

d. 0.011

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

95.23

col., hex., 1.675

2.32525°

712

52.80°

73100°

50 al.

203.33 58.34 100.98 40.32 223.23

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

1.56 2.4 3.65 2.6025°

118 d. d. d. 2800 d.

1412 d. 3600

0°

0°

281 0.000918° i. 0.00062 99.625°

100°

130 s.

100°

918 d. v. s.

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

Physical Properties of the Elements and Inorganic Compounds (Continued )

TABLE 2-1

Name

Formula

Formula weight

Color, crystalline form and refractive index

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

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

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

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

chloride (scacchite) chloride

MnCl2 MnCl2·4H2O*

125.84 197.91

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

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

196.76 88.95 175.88 287.04 70.93 157.86 86.93

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

MnSO4 MnSO4·H2O

150.99 169.01

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

sulfate (ous)

MnSO4·2H2O

187.02

2.52615° 15°

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

Specific gravity

Melting point, °C

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

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

2.66 1.68 7.220° 1.74 20° 4 1.589 3.125

1185 70 d. 1260

2.977 25° 4 2.01

650 58.0

3.25818° 3.258 1.8221° 5.18 4.81 5.026

d. d. 25.8 1650 −0, 1080 −0, >230

3.235 2.87

sulfate (ous)

MnSO4·7H2O

277.10

pink, mn. or rhb.

2.092

sulfate (ic)

Mn2(SO4)3

398.04

gn., delq. cr.

3.24

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

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

11.5 3.270 6.053

2300 (?) d. 237

5.44 4.42

277 expl. −H2O, 175 d. 100 d. 260

MnSO4·3H2O

205.04

sulfate (ous)

MnSO4·4H2O*

223.05

sulfate (ous)

MnSO4·5H2O

241.07

sulfate (ous)

MnSO4·6H2O

259.09

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

2-18

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

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

2.107 15°

2.103

11.14 7.93 6.47 6.44 7.307

d. d. subl. 345 d. 130

Solubility in 100 parts Cold water

Hot water

Other reagents

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

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

63.40° 1518°

123.8100° ∞

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

129.5

s. 0.00220° i. 4260° i. i. i.

s. i. i. ∞ i. i. i.

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

d. 850

530° 98.4748°

7350° 79.77100°

s. al.; i. et.

85.2735°

106.855°

1900

d.

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

sulfate (ous)

Boiling point, °C

1190 −H2O, 106; −4H2O, 200

5°

−4H2O, 450

74.22

99.3157°

13616°

16950°

5°

−7H2O, 280

322 304

s. 64.550°

200

2040°

2479°

176

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

i. al.

35°

142

0°

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

25114°

v. s.

d.

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

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

100100° 25100°

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

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

HgCl

236.07

wh., tet., 1.9733

7.150

302

383.7

0.00140°

0.000743°

iodide nitrate Mercurous oxide

HgI HgNO3·H2O Hg2O

327.53 280.63 417.22

yel., tet. wh. mn. bk.

7.70 4.7853.9° 9.8

290 d. 70 d. 100

subl. 140; 310d. expl.

2 × 10−8 v. s. i.

v. sl. s. d. 0.0007

sulfate Mercury† Molybdenum

Hg2SO4 Hg Mo

497.28 200.61 95.95

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

7.56 13.54620° 10.2

d. −38.87 2620  10

0.05516.5° i. i.

0.092100° i. i.

MoCl2

166.85

yel., amor.

3.714 25° 4

d.

i.

i.

3.578 25° 4

d.

chloride (calomel)

chloride, dichloride, tri-

MoCl3

202.32

dark red pd.

chloride, tetra-

MoCl4

237.78

brn., delq.

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

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

273.24 143.95 160.07 192.13 224.19 161.97 179.98 144.27 20.18

Np Ni

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

carbonyl chloride chloride

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

chloride, ammonia cyanide dimethylglyoxime

NiCl2·6NH3 Ni(CN)2·4H2O NiC8H14O4N4

231.80 182.79 288.91

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

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

184.76 109.71 97.21 290.80 286.87 74.69 258.97 154.75

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

170.73 129.60 237.70

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

2.928 25° 4 19.5°

4.50 4.80114°

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

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

i.

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

s.

d.

194

268

s.

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

subl.

−2H2O, 200 −245.9

d. 18°

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

Produced by Neutron bombardment of U 1452 2900 i.

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

1.798 1.645 1.923

d.

2.575 4.64 28° 4

16.6 15025° 2.53.5°

v. s. 39.288°

v. sl. s. (NH4)2SO4

d. d. −3H2O, 200

28 112.80° 1999° v. s.

156100° 316100° d.

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

d. d.

0.009325° i.

i. d.

s. a. s. a., NH4 salts

0.0189.8° 53.80° 180

i. 87.6100° v. s.

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

s. i. i.

d. i. i.

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

i. v. sl. s. ∞56.7°

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

1.837 3.715

gn. pl. scarlet red cr.

1.145° cc

d.

d.

8.9020

1.3117° 3.544

2.10679° i. s. i. sl. s. 2.1370°

i.

volt.

silv. met., cb.

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

356.9 3700

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

−25 subl.

43751mm 973

−4H2O, 200 subl. 250

d.

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

2.154 4.36 2.05

d. d. d. 56.7

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

7.45 1.87511° 3.68

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

136.7

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

i. 76.7100°

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

Physical Properties of the Elements and Inorganic Compounds (Continued )

TABLE 2-1

Name Nickel (Cont.) sulfate

Formula NiSO4·6H2O*

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

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

Nitrogen oxide, mono- (ous)

N2O

oxide, di- (ic)

NO or (NO)2

oxide, tri-

N2O3

oxide, tetra- (per- or di-)

NO2 or (NO2)2

Formula weight

Color, crystalline form and refractive index

262.85

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

280.86 63.02 81.03 117.06 127.08 28.02 44.02

col. gas

30.01 (60.02) 76.02

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

oxide, penta-

N2O5

46.01 (92.02) 108.02

oxybromide oxychloride

NOBr NOCl

109.92 65.47

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

Nitroxyl chloride Osmium chloride, dichloride, trichloride, tetraOxygen

NO2Cl Os OsCl2 OsCl3 OsCl4 O2

81.47 190.2 261.11 296.57 332.03 32.00

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

Ozone

O3

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

48.00

col. gas

Pd

106.70

silv. met., cb.

PdBr2 PdCl2 PdCl2·2H2O Pd(CN)2

266.53 177.61 213.65 158.74

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

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

214.41 211.68 100.46 118.48 136.50

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

PH4Cl

191.93 227.96 119.94 196.99 194.14 97.02 2365.88 34.00 70.47

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

Specific gravity 2.07 1.948 1.502

Melting point, °C

1.448

18°

Solubility in 100 parts Cold water

tr. 53.3

−6H2O, 280

13150°

98–100 −42 −38 −18.5 73 d. −209.86

−6H2O, 103 86

0°

Hot water 280100° 30°

117.8 ∞ ∞ ∞

−195.8

63.5 ∞ ∞ 263−20° d. 2.350° cc

−90.7

130.520° cc

−151

7.340° cc

60.8224° cc 0.0100°

3.5

s.

−9.3

21.3

d.

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

Boiling point, °C

1.5520° cc

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

1.63

30

47

s.

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

−55.5 −64.5

−2 −5.5

d. d.

<−30 2700

5 >5300

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

−218.4

−183

d i. s. d. sl. s. s. d. 4.890° cc

−251

−112

0.4940° cc

060° cc

s. oil turp., oil cinn.

1555

2200

i.

i.

i. s. s. i.

i. s. s. i.

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

d. 560–600

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

Forms HNO3 s. fuming H2SO4 i.

2.630° cc 1.7100° cc

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

d.

−90°

lq. 0.746 1.146 (A)

−112 50 −17.8

1618mm d. 200

d. 138 d. 110

subl. 110

<60 d. 78 −132.5 46atm

28

s. s. s. v. s.

−25H2O, 140 −85

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

subl.

d.

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

i.

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

Phosphoric acid, hypoPhosphoric acid, meta-

H4P2O6 HPO3

161.99 79.99

cr. vitreous, delq.

2.2–2.5

55 subl.

d. 70

s. s.

Phosphoric acid, orthoPhosphoric acid, pyro-

H3PO4† H4P2O7

98.00 177.99

col., rhb. wh. nd.

1.834

42.35 61

−a H2O, 213

2340 80028°

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

H3PO2 H3PO3 H4P2O6 P4 P4 P4

66.00 82.00 145.99 123.92 123.92 123.92

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

1.49318.8° 1.65121.2°

26.5 74 38

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

∞ 307.30° d. i. i. 0.0003

PCl3 PCl5

137.35 208.27

col., fuming lq. delq., tet.

−111.8 148 under pressure subl. 250 2

75.95760mm subl. 160

d. d.

21.4520° lq. 191755°

1755

4300

chloride, trichloride, pentaoxide, pentaoxychloride Phosphotungstic acid Platinum

P2O5 POCl2 P2O5·2WO3·42H2O Pt

141.96 153.35 3681.67 195.23

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

18.2°

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

59043atm 44.1; ign. 34

26°

Forms H3PO4 d. s. i.

107.2760mm

25°

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

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

v. s.

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

i.

PtCl4

337.06

brn.

d. 370

140

v. s.

chloride (ous)

PtCl2

266.14

brn.

5.8711°

d. 581

i.

i.

PtCl4·8H2O Pt(CN)2 Pu Pu K

481.19 247.27 238 239 39.10

red, mn. yel.-brn.

2.43

−4H2O, 100

v. s. i.

v. s. i.

d.

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

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

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

Forms KOH 39690°

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

silv. met., cb.

238

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

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

subl. 400

d. 1380 d. d. 400 1500

d. d. 1320 1330

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

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

i. i. sl. s.

chloride (ic)

chloride (ic) cyanide (ous) Plutonium Plutonium Potassium

i. lq. CO2

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

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

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

TABLE 2-1

Physical Properties of the Elements and Inorganic Compounds (Continued )

Formula

Formula weight

Color, crystalline form and refractive index

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

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

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

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

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

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

thiosulfate thiosulfate Praseodymium Radium bromide Radon (Niton)

K2S2O3 3K2S2O3·H2O Pr Ra RaBr2 Rn

190.31 588.95 140.92 226.05 385.88 222.0

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

Rhenium

Re

186.31

hex.

Rhodium chloride chloride Rubidium

Rh RhCl3 RhCl3·4H2O Rb

102.91 209.28 281.35 85.48

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

Ruthenium Ruthenium Samarium Scandium Selenic acid Selenic acid Selenium Selenium

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

101.70 101.70 150.43 45.10 144.98 162.99 631.68 631.68

bk., porous gray, hex.

Name

2-22

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

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

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

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

Boiling point, °C d. 225

d. 400 d. 350

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

1320 d. −3H2O, 300

−3H2O, 150 d.

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

d. 500

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

Hot water

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

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

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

96.10°

311.290°

>100 91.575° 278100° 6.1100° 21720°

d. 1140 subl. 900 −62

1955 d. 450

>2500 subl. 800

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

38.5 >1950 2450 >1300 1200 58 26 50 220

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

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

d. d. +H2 7020° 510° cc

s. 8.560° cc

3440 12.5

Other reagents

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

i. i.

700

i. i. v. s. d.

>2700

i. i.

i. i.

sl. s. aq. reg., a.

2400 260 205 688 688

130030° v. s. i. i.

∞60°

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

i. i.

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

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

Se8 H2SeO3 H2SiO3 H4SiO4 Si

631.68 128.98 78.08 96.09 28.06

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

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

217 d.

688

2600

i. 900° i. sl. s. i.

i. 40090° i. sl. s. i.

1420

2.0–2.5

2600

i.

i.

2 3.17 1.580° 1.50

>2700 −1 −70

2600 subl. 2200 144760mm 57.6

i. i. d. d.

i. i.

3.57 (A) lq. 0.68−185° 2.2 2.32

−95.7 −185 1600–1750 1710

−651810mm −112760mm subl. 1750 2230

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

Silicon, graphitic

Si

Silicon, amorphous carbide chloride, trichloride, tetra-

Si SiC Si2Cl6 SiCl4

28.06 40.07 268.86 169.89

SiF4 SiH4 SiO2·xH2O SiO2

104.06 32.09

SiO2 SiO2 SiO2 Ag AgBr

60.06 60.06 60.06 107.88 187.80

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

2.20 2.65020° 2.26 10.520° 6.473 25° 4

tr. <1425 tr. 1670 960.5 434

2230 2230 2230 1950 d. 700

i. i. i. i. 0.0000220°

i. 0.00037100°

carbonate chloride (cerargyrite)

Ag2CO3 AgCl

275.77 143.34

yel. pd. wh., cb., 2.071

6.077 5.56

218 d. 455

1550

0.00320° 0.00008910°

0.05100° 0.00217100°

cyanide nitrate (lunar caustic) Sodium

AgCN AgNO3 Na

133.90 169.89 22.997

wh., 1.685 col., rhb., 1.744 silv. met, cb.

3.95 4.352 19° 4 0.9720°

−(CN)2, 320 212 97.5

444 d. 880

acetate acetate aluminate amide Sodium ammonium phosphate antimonate, metaarsenate arsenate, acid (monobasic) arsenate, acid (dibasic) arsenate, acid (dibasic) arsenite, acid benzoate bicarbonate bifluoride bisulfate bisulfite borate, tetraborate, tetra

NaC2H3O2 NaC2H3O2·3H2O NaAlO2 NaNH2 NaNH4HPO4·4H2O 2NaSbO3·7H2O Na3AsO4·12H2O NaH2AsO4·H2O Na2HAsO4·7H2O* Na2HAsO4·12H2O Na2HAs03 NaC7H5O2 NaHCO3 NaHF2 NaHSO4 NaHSO3 Na2B4O7 Na2B4O7·5H2O

82.04 136.09 81.97 39.02 209.09 511.63 424.09 181.94 312.02 402.10 169.91 144.11 84.01 62.00 120.06 104.06 201.27 291.35

wh., mn., 1.464 wh., mn. amor. olive gn. col., mn. cb. hex., 1.4589 rhb., 1.5535 col., mn., 1.4658 mn., 1.4496 col. col. cr. wh., mn., 1.500 col. cr. col., tri. col., mn., 1.526

1.528 1.45

324 58 1650 210 79 d.

col., rhb., 1.461

2.742 1.48 2.367 1.815

Na2B4O7·10H2O*

381.43

wh., mn., 1.4694

1.73

fluoride hydride (silane) oxide, di- (opal) oxide, di- (cristobalite) oxide, di- (lechatelierite) oxide, di- (quartz) oxide, di- (tridymite) Silver bromide (bromyrite)

borate, tetra- (borax)

28.06

steel gray hex. amor., 1.41 amor. gray, cb., 3.736

60.06

1.574

−3H2O, 120 400

1.759 2.535 1.871 1.72 1.87

86.3 d. 100 125 28

2.20

−CO2, 270 d. >315 d. 741

d., −H2O

75

−10H2O, 200

17.5°

bromate bromide bromide

NaBrO3 NaBr NaBr·2H2O

150.91 102.91 138.95

col., cb. col., cb., 1.6412 col., mn.

3.339 3.20517.5° 2.176

381 755 50.7

carbonate (soda ash) carbonate

Na2CO3 Na2CO3·H2O

106.00 124.02

2.533 1.55

851 −H2O, 100

carbonate carbonate (sal soda)

Na2CO3·7H2O Na2CO3·10H2O

232.12 286.16

wh. pd., 1.535 wh., rhb., 1.506– 1.509 rhb. or trig. wh., mn., 1.425

1.51 1.46

d. 35.1

−7H2O, 100 −12H2O, 100

0.00002220° 1220° d., forms NaOH 46.520° v. s. s. d. 16.7 0.03112.8° 26.717° s. 6115° 5.590.1° v. s. 62.525° 6.90° 3.720° 500° sl. s. 1.30° 2262° (anh.)

i. i. i. i.

952100° 170100° v. s. v. s. 100

s. HF; i. alk. s. HF; i. alk. s. HF; i. alk. s. HNO3, h. H2SO4; i. alk. 0.5118° NH4OH; s. KCN, Na2S2O3 s. NH4OH, Na2S2O3; i. al. s. NH4OH, KCN; sl. s. HCl s. NH4OH, KCN, HNO3 s. gly.; v. sl. s. al. i. bz.; d. al. 2.118° al. 7.825° abs. al. i. al. d. al. i. al. sl. s. al., NH4 salts; i. ac. 1.67 al., 5015° gly. sl. s. al. sl. s. al. 2.325°, 8.378° al. i. al.

1390

27.5 9020° 79.50° (anh.)

d.

7.10° s. s. 21.50°

s. 23830°

0°

s. HNO3, al., et. i. al., et.; d. KOH s. HF, h. alk., fused CaCl2 s. HF; i. alk.

v. s. 140.730° 76.9100° 16.460° s. 100100° s. 8.7940° 52.3100° (anh.) 20.380° (anh.) 90.9100° 121100° 118.380° (anh.) 48.5104° s.

1.30.5 (anh).

i. CS2; s. H2SO4 v. s. al.; i. NH3 s. alk.; i. NH4Cl s. alk.; i. NH4Cl s. HNO3 + HF, Ag; sl. s. Pb, Zn; i. HF s. HNO3 + HF, fused alk.; i. HF. s. HF, KOH s. fused alk.; i. a. d. alk. d. conc. H2SO4, al.

d. al.; i. NH3 i. al., act. i. al. s. gly.; i. abs. al. i. al. sl. s. al. sl. s. al. i. al., et. s. gly.; i. al., et. i. al.

*Usual commercial form. 2-23

Physical Properties of the Elements and Inorganic Compounds (Continued )

TABLE 2-1

Name Sodium ammonium phosphate (Cont.) carbonate, sesqui- (trona) chlorate

Formula

Na3H(CO3)2·2H2O NaClO3

Formula weight

Specific gravity

NaCl Na2CrO4 Na2CrO4·10H2O 2Na3C6H5O7·11H2O NaCN Na2Cr2O7·2H2O

58.45 162.00 342.16 714.36 49.02 298.05

ferricyanide ferrocyanide

Na3Fe(CN)6·H2O Na4Fe(CN)6·10H2O

298.97 484.11

red, delq. yel., mn.

1.458

fluoride (villiaumite) formate hydride

NaF NaHCO2 NaH

tet., 1.3258 wh., mn. silv. nd., 1.470

2.79 1.919 0.92

992 253 d. 800

hydrosulfide hydrosulfide hydrosulfite hydroxide hydroxide hypochlorite iodide iodide lactate nitrate (soda niter) nitrite

NaSH·2H2O NaSH·3H2O Na2S2O4·2H2O NaOH NaOH·3a H2O NaOCl NaI* NaI·2H2O NaC3H5O3 NaNO3 NaNO2

2.130

d. 22 d. 318.4 15.5 d. 651

1300

2.257 2.1680°

d. 308 271

d. 380 d. 320

oxide

Na2O

2.27

subl.

2-24

NaBO3·H2O NaClO4 NaClO4·H2O Na2O2* Na2O2·8H2O NaH2PO4·H2O* NaH2PO4·2H2O Na2HPO4·7H2O Na2HPO4·12H2O Na3PO4 Na3PO4·12H2O* Na4P4O12 Na4P2O7* Na4P2O7·10H2O Na2H2P2O7 Na2H2P2O7·6H2O NaKC4H4O6·4H2O Na2SiO3 Na2SiO3·9H2O Na4SiO4 Na2SiF6 Na2SnO3·3H2O Na2SO4 Na2SO4

92.10 110.11 210.15 40.00 103.06 74.45 149.92 185.95 112.07 85.01 69.01 61.99 99.83 122.45 140.47 77.99 222.12 138.01 156.03 268.09 358.17 163.97 380.16 407.91 265.95 446.11 221.97 330.07 282.23 122.05 284.20 184.05 188.05 266.74 142.05 142.05

col., delq., nd. rhb. col. cr. wh., delq. mn. pa. yel., in soln. only col., cb., 1.7745 col., mn. col., amor. col., trig., 1.5874 pa. yel., rhb. wh., delq. wh. pd. rhb., 1.4617 hex. yel.-wh. pd. wh., hex. col., rhb., 1.4852 col., rhb., 1.4629 col., mn., 1.4424 col., mn., 1.4361 wh. wh., trig., 1.4458 col. wh. mn., 1.4525 col., mn., 1.510 col., mn., 1.4645 rhb., 1.493 col., rhb., 1.520 rhb. col., hex., 1.530 wh., hex., 1.312 hex. tablets col., rhb., 1.477 col., mn.

2.163 2.723 1.483 1.857 23.5° 4 18°

2.52

3.6670° 2.448

2.02 2.805 2.040 1.91 1.679 1.52 2.53717.5° 1.62 2.476 2.45 1.82 1.862 1.848 1.790

2.679 2.698

d. 248

Boiling point, °C

chloride chromate chromate citrate cyanide dichromate

42.00 68.01 24.005

2.112 2.49015°

Melting point, °C

wh., mn., 1.5073 wh., cb., or trig., 1.5151 col., cb., 1.5443 yel., rhb. yel., delq., mn. wh., rhb. wh., cb., 1.452 red, mn., 1.6994

perborate perchlorate perchlorate peroxide peroxide phosphate, monobasic phosphate, monobasic phosphate, dibasic phosphate, dibasic phosphate, tribasic phosphate, tribasic phosphate, metaphosphate, pyrophosphate, pyrophosphate (pyrodisodium) phosphate (pyrodisodium) potassium tartrate silicate, metaSodium silicate, metasilicate, orthosilicofluoride stannate sulfate (thenardite) sulfate

226.05 106.45

Color, crystalline form and refractive index

800.4 392 19.9 −11H2O, 150 563.7 −2H2O, 84.6; 356 (anh.)

d. 40 482 d. d. 130 d. d. 30 −H2O, 100 60 d. 34.6 1340 73.4 616 d. 988 d. d. 220 70 to 80 1088 47 1018 d. d. 140 tr. 100 to mn. tr. 500 to hex.

d. 1413 d. 1496 d. 400

Solubility in 100 parts Cold water

130° 790°

42100° 230100° 0°

1390

d. 200 −12H2O, 180 −11H2O, 100

−4H2O, 215 −6H2O, 100

100°

35.7 320° v. s. 9125° 4810° 2380°

39.8 126100° ∞ 250100° 8235° 50880°

18.90° 17.920° (anh.)

67100° 6398.5° (anh.) 5100° 160100°

40° 440° d. d.

Hot water

s. s. 2220° 420° s. 260° 158.70° v. s. v. s. 730° 72.10°

s. s. d. 347100° v. s. 15856° 302100° v. s. v. s. 180100° 163.2100°

Forms NaOH sl. s. 1700° 20915° s. d. s. d. 710° 91.10° 18540° 4.30° 4.50° 28.315° s. 2.260° 5.40° 4.50° 6.90° 260° s. v. s. s. 0.440° 500° 50° 48.840°

d. 320100° 28450° d. d. 39083° 30840° 2000100° 76.730° 77100° ∞ s. 4596° 93100° 2140° 3640° 6626° s. d. v. s. s. 2.45100° 6750° 42100° 42.5100°

Other reagents

s. al. sl. s. al.; i. conc. HCl sl. s. al. i. al. s. NH3; sl. s. al. i. al. i. al. v. sl. s. al. sl. s. al.; i. et. i. bz., CS2, CCl4, NH3; s. molten metal s. al.; d. a. s. al.; d. a. i. al. v. s. al., et., gly.; i. act. v. s. al., act. v. s. NH3 s. al.; i. et. s. NH3; sl. s. gly., al. 0.320° et.; 0.3 abs. al.; 4.420° m. al.; v. s. NH3 d. al. s. gly., alk. s. al.; 51 m. al.; 52 act.; i. et. s. al. s. dil. a. i. al. i. al. i. CS2 s. a., alk. d. a. i. al., NH3 sl. s. al. i. Na or K salts, al. 2918°, a N NaOH i. al. i. al., act. i. al. d. HI; s. H2SO4

sulfate sulfate sulfate (Glauber’s salt) sulfide, monosulfide, tetrasulfide, pentasulfite sulfite tartrate thiocyanate thiosulfate thiosulfate (hypo) tungstate tungstate tungstate, parauranate vanadate vanadate, pyroStannic chloride

Na2SO4 Na2SO4.7H2O Na2SO4.10H2O Na2S Na2S4 Na2S5 Na2SO3 Na2SO3.7H2O Na2C4H4O6.2H2O NaCNS Na2S2O3 Na2S2O3.5H2O* Na2WO4 Na2WO4.2H2O* Na6W7O24.16H2O Na2UO4 Na3VO4.16H2O Na4V2O7 SnCl4

142.05 268.17 322.21 78.05 174.23 206.29 126.05 252.17 230.10 81.08 158.11 248.19 293.91 329.95 2097.68 348.06 472.20 305.89 260.53

col., hex. tet. col., mn., 1.396 pink or wh., amor. yel., cb. yel. hex. pr., 1.565 mn. rhb. delq., rhb., 1.625 mn. mn. pr., 1.5079 wh., rhb. wh., rhb. wh., tri. yel. col. nd. hex. col., fuming lq.

884 1.464 1.856 2.633 15° 4 1.561 1.818

2.226

866 (anh.) 654 −30.2

7.0

1127

SnO2

150.70

wh., tet., 1.9968

346.85

col., delq., hex.

SnBr2 SnCl2 SnCl2.2H2O* SnSO4 Sr

278.53 189.61 225.65 214.76 87.63

yel., rhb. wh., rhb. wh., tri. wh. cr. silv. met.

2.71

acetate carbonate (strontianite) chloride chloride hydroxide hydroxide

Sr(C2H3O2)2 SrCO3 SrCl2 SrCl2.6H2O* Sr(OH)2 Sr(OH)2.8H2O*

205.72 147.64 158.54 266.64 121.65 265.77

wh. cr. wh., rhb., 1.664 wh., cb., 1.6499 wh., rhb., 1.5364 wh., delq. col., tet., 1.499

2.099 3.70 3.052 1.93317° 3.625 1.90

nitrate nitrate oxide (strontia)

Sr(NO3)2* Sr(NO3)2.4H2O SrO

211.65 283.71 103.63

col., cb., 1.5878 wh., mn. col., cb., 1.870

2.986 2.2 4.7

SrO2 SrO2.8H2O SrSO4 Sr(HSO4)2 NH2SO3H S S8 S8 S2Br2 S2Cl2 SCl2 SCl4 SO2

119.63 263.76 183.69 281.77 97.09 32.06 256.48 256.48 223.95 135.03 102.97 173.89 64.06

wh. pd. wh. cr. col., rhb., 1.6237 col., granular wh., rhb. pa. yel. pd., 2.0–2.9 pa. yel., mn. pa. yel., rhb. red, fuming lq. red-yel. lq. dark red fuming lq. yel.-brn. lq. col. gas

oxide, tri-(β) Sulfuric acid Sulfuric acid

SO3 (SO3)2 H2SO4* H2SO4.H2O

80.06 160.12 98.08 116.09

col. pr. col., silky, nd. col., viscous lq. pr. or lq.

d.

d. 48.0 692 −2H2O, 100 −16H2O, 300

Sn(SO4)2.2H2O

oxide, tri-(α)

275 251.8 d. −7H2O, 150 287

sulfate

peroxide peroxide sulfate (celestite) sulfate, acid Sulfamic acid Sulfur, amorphous Sulfur, monoclinic Sulfur, rhombic Sulfur bromide, monochloride, monochloride, dichloride, tetraoxide, di-

−10H2O, 100

1.667 1.685 4.179 3.245 3.98714°

oxide (cassiterite)

Stannous bromide chloride chloride (tin salt) sulfate Strontium

32.4

5.1217° 15.5°

2.6

3.96 2.03 12° 4 2.046 1.96 2.07 2.635 1.687 1.621 15° 15 lq., 1.4340°; 2.264 (A) lq., 1.923; 2.75 (A) 1.9720° 1.834 18° 4 1.842 15° 4

215.5 246.8 37.7 −SO2, 360 800 149760atm 873 −4H2O, 61 375 −7H2O in dry air 570

114.1

620 623 d. 1150 d. −CO2, 1350 −6H2O, 100

19.420° 44.90° 3615° 15.410° s. s. 13.90° 34.72° 296° 11010° 500° 74.70° 57.580° 880° 8 i. v. s. s. s. i.

i. d.

s. 83.90° 118.70° 1919° d.

d. 269.815° ∞ 18100° Forms Sr(OH)2 36.497° 0.065100° 100.8100° 19840° 21.83100° 47.7100°

36.90° 0.001118° 43.50° 1040° 0.410° 0.900°

444.6 444.6 444.6 540.18mm 138 59 d. > −20 −10.0

16.83

44.6

d.

50 10.49 8.62

d. 340 290

Forms H2SO4 ∞ ∞

d. −8H2O, 100 1580 d. d. 205 d. 120 119.0 112.8 −46 −80 −78 −30 −75.5

d.

d.

v. s.

400° 62.20° Forms Sr(OH)2 0.00820° 0.01820° 0.01130° d. 200° i. i. i. d. d. d. d. 22.80°

2430

45.360° 202.626° 41234° 57.390° s. s. 28.384° 67.818° 6643° 225100° 23180° 301.860° 97100° 123.5100° d. i. d.

i. al. sl. s. al.; i. et. s. al. s. al. i. al., NH i. al. i. al. v. s. al. s. NH3; v. sl. s. al. sl. s. NH3; i. a., al. s. alk. carb., dil. a. i. al. i. al. s. abs. al., act., NH3; s. ∞ CS2 s. conc. H2SO4; i. alk.; NH4OH, NH3 s. dil. H2SO4, HCl; d. abs. al. s. C6H5N s. alk., abs. al., et. s. tart. a., alk., al. s. H2SO4 s. al., a. 0.2615° m. al. s. a., NH4 salts, aq. CO2 v. sl. s. act., abs. al.; i. NH3 s. NH4Cl s. NH4Cl; i. act.

10089° 12420°

s. NH3; 0.012 abs. al. i. HNO3 sl. s. al.; i. et.

d. d. 0.011432°

s. al., NH4Cl; i. act. s. al.; i. NH4OH sl. s. a.; i. dil. H2SO4, al. 1470° H2SO4 sl. s. al., act.; i. et. sl. s. CS2 s. CS2, al. 240°, 18155° CS2

4070° i. i. i.

s. CS2, et., bz. d. al. 4.550°

s. H2SO4; al., ac. s. H2SO4

∞ ∞

s. H2SO4 d. al. d. al.

*Usual commercial form. 2-25

Physical Properties of the Elements and Inorganic Compounds (Concluded )

TABLE 2-1

Name

Formula

Formula weight

Color, crystalline form and refractive index

Specific gravity

Melting point, °C

Boiling point, °C

Solubility in 100 parts Cold water

Hot water

Sulfuric acid Sulfuric acid, pyroSulfuric oxychloride Sulfurous oxybromide oxychloride Tantalum

H2SO4·2H2O H2S2O7 SO2Cl2 SOBr2 SOCl2 Ta

134.11 178.14 134.97 207.89 118.97 180.88

col. lq. cr. col. lq. or.-yel. lq. col. lq. bk.-gray, cb.

1.650 40° 1.920° 1.667 20° 4 2.6818° 1.638 16.6

−38.9 35 −54.1 −50 −104.5 2850

167 d. 69.1760mm 6840mm 78.8 >4100

∞ d. d. d. d. i.

∞

Tellurium

Te

127.61

met., hex.

(α) 6.24; (β) 6.00

452

1390

i.

i.

Terbium Thallium acetate chloride, monochloride, sesquichloride, trichloride, trisulfate (ic) sulfate (ous) sulfate, acid Thio, cf. sulfo or sulfur Thorium

Tb Tl TlC2H3O2 TlCl Tl2Cl3 TlCl3 TlCl3·4H2O Tl2(SO4)3·7H2O Tl2SO4 TlHSO4

159.20 204.39 263.43 239.85 515.15 310.76 382.83 823.07 504.84 301.46

blue-wh., tet. silky nd. wh., cb. yel., hex. hex. pl. nd. lf. col., rhb., 1.8671 trimorphous

11.85 3.68 7.00 5.9

303.5 110 430 400–500 25 37 −6H2O, 200 632 115 d.

1650 806 d. d. −4H2O, 100 d. d.

i. v. s. 0.210° 0.2615° v. s. 86.217° d. 2.700°

Th

232.12

cb.

11.2

1845

>3000

i.

i.

oxide, di- (thorianite) sulfate sulfate Thulium Tin

ThO2 Th(SO4)2 Th(SO4)2·9H2O Tm Sn

264.12 424.24 586.38 169.40 118.70

wh., cb.

9.69 4.22517° 2.77

>2800

4400

silv. met., tet.

7.31

231.85

2260

i. 0.740° sl. s. i. i.

5.2250° sl. s. i. i.

Tin

Sn

118.70

gray, cb.

5.750

Stable −163 to +18

2260

i.

i.

Tin salts, cf. stannic and stannous Titanic acid

H2TiO3

97.92

wh. pd.

i.

i.

i. d.

d.

s. s. i.

s. d. i.

i.

i.

Titanium chloride, di-

mn. pr.

Ti TiCl2

47.90 118.81

dark gray, cb. bk., delq.

chloride, trichloride, tetraoxide, di- (anatase)

TiCl3 TiCl4* TiO2

154.27 189.73 79.90

oxide, di- (brookite)

TiO2

vl., delq. col. lq. brn. or bk., tet., 2.534–2.564 brn. or bk., rhb., 2.586 col. if pure, tet., 2.615 gray-bk., cb.

oxide, di- (rutile) Tungsten carbide carbide

TiO2 W

79.90 79.90 183.92

6.77

17.5°

4.50

lq., 1.726 3.84

136.4

s. HCl, H2SO; sl. s. HNO3; i. HF, alk. s. h. H2SO4; i. alk.

s. HCl, H2SO4, dil. HNO3 h. aq KOH s. a., h. alk. solns. s. alk.; v. sl. s. dil. a.; i. al. s. a. i. CS2, et., chl. s. dil. HCl sl. s. alk.

1640 d.

<3000

i.

i.

s. H2SO4, alk.

3370

5900

i.

i.

6000 6000

i. i.

i. i.

i. i.

i. sl. s.

s. h. conc. KOH; sl. s. NH3, HNO3, aq. reg. s. F2; i. a. s. h. HNO3; sl. s. HCl, H2SO4 s. alk.; i. a. s. HF, alk., NH3

i.

i.

s. a., alk. carb.; i. alk.

i. d. i.

i.

s. a.; i. alk. d. a. s. HNO3, conc. H2SO4

18°

15.7 16.0618°

2777 2877

oxide, triTungstic acid (tungstite)

WO3 H2WO4

231.92 249.94

yel., rhb. yel., rhb. 2.24

7.16 5.5

Uranic acid

H2UO4

304.09

yel. pd.

5.92615°

wh. cr. cr. bk., rhb.

18.485 13° 4

>2130 −a H2O, 100; 1473 −H2O, 250 to 300 1133 2400 2176

2-26

s. al., et. s. al., et. s. dil. H2SO4

19.3

gray pd., cb. iron gray

238.07 512.14 270.07

d. d. 18.45100°

s. HNO3, H2SO4; i. NH3 v. s. al. sl. s. HCl; i. al., NH4OH

4.26

195.93 379.85

U U2C3 UO2

>3000

4.17

WC W2C

Uranium carbide oxide, di- (uraninite)

i. 1.8100° 1.9100°

v. sl. s. dil. H2SO4

−9H2O, 400

1800 Unstable in air d. 440 −30

i.

Other reagents d. al. d. al. s. ac.; d. al. s. bz., CS2, CCl4; d. act. s. bz., chl. s. fused alk., HF; i. HCl, HNO3, H2SO4 s. H2SO4, HNO3, KCN, KOH, aq. reg.; i. CS2

11.28 10.9

3500

i.

oxide (pitchblende) sulfate (ous) Uranyl acetate carbonate (rutherfordine) nitrate sulfate Vanadic acid, metaVanadic acid, pyroVanadium chloride, dichloride, trichloride, tetraoxide, dioxide, trioxide, tetraoxide, pentaoxychloride, monoVanadyl chloride chloride, dichloride, triWater†

U3O8 U(SO4)2·4H2O UO2(C2H3O2)2·2H2O UO2CO3 UO2(NO3)2·6H2O UO2SO4·3H2O HVO3 H4V2O7 V VCl2 VCl3 VCl4 V2O2 V2O3 V2O4 V2O5 VOCl (VO)2Cl VOCl2 VOCl3 H2O

842.21 502.25 424.19 330.08 502.18 420.18 99.96 217.93 50.95 121.86 157.23 192.78 133.90 149.90 165.90 181.90 102.41 169.36 137.86 173.32 18.016

Water, heavy Xenon

D2O Xe

20.029 131.30

olive gn. gn., rhb. yel., rhb. tet. yel., rhb., 1.4967 yel. cr. yel. scales pa. yel., amor. lt. gray, cb. gn., hex., delq. pink, tabular, delq. red lq. lt. gray cr. bk. cr. blue cr. red-yel., rhb. brn. pd. yel. cr. gn., delq. yel. lq. col. lq., 1.3330020°; hex. solid, 1.309 col. lq., 1.3284420° col. gas

Ytterbium Yttrium Zinc acetate acetate bromide carbonate

Yb Y Zn Zn(C2H3O2)2 Zn(C2H3O2)2·2H2O* ZnBr2 ZnCO3

173.04 88.92 65.38 183.47 219.50 225.21 125.39

dark gray, hex. silv. met., hex. mn. wh., mn., 1.494 rhb. wh., trig., 1.818

ZnCl2

136.29

wh., delq., 1.687, uniaxial col., rhb. col., rhb. cb.

chloride cyanide hydroxide iodide

Zn(CN)2 Zn(OH)2 ZnI2

117.42 99.40 319.22

nitrate oxide (zincite) oxide peroxide phosphide silicate

Zn(NO3)2·6H2O ZnO ZnO ZnO2 Zn3P2 ZnSiO3

297.49 81.38 81.38 97.38 258.10 141.44

sulfate (zincosite)

ZnSO4

sulfate sulfate sulfate (goslarite) sulfide (α) (wurzite) sulfide (β) (sphalerite)

ZnSO4·H2O ZnSO4·6H2O ZnSO4·7H2O* ZnS ZnS

161.44 179.46 269.54 287.55 97.44 97.44

sulfide (blende) sulfite Zirconium oxide, di- (baddeleyite) oxide, di- (free from Hf)

ZnS ZnSO3·2a H2O Zr ZrO2 ZrO2

97.44 190.48 91.22 123.22 123.22

col., tet. wh., hex., 2.004 wh., amor. yel. steel gray, cb. hex. or rhb.; glass, 1.650 wh., rhb., 1.669 col. mn. rhb., 1.4801 wh., hex., 2.356 wh., cb.; glass (?) 2.18–2.25 wh., granular mn. cb., pd. ign. easily yel. or brn., mn., 2.19 wh., mn.

7.31 15°

2.89 5.6 2.807 3.2816.5° 5.96 3.2318° 3.0018° 1.81630° 3.64 4.87 18° 4 4.399 3.357 18° 4 2.824 3.64 2.8813° 1.829 1.004° (lq.); 0.9150° (ice) 1.10720° lq., 3.06−109.1 2.7−140° 4.53 (A)

d. −4H2O, 300 −2H2O, 110 60.2 d. 100

118

1710

3000

d. −109 ign. 1970 1967 800

148.5755mm

d. 1750

d. in air

i. 2311° 9.217°

i. 963° d.

s. HNO3, H2SO4 s. dil. a. s. al., act.

170.30° 18.913.2° i. i. i. s. s. s. d. i. sl. s. i. 0.820° i. i. d. s. d.

∞60° 23025°

v. s. ac., al., et.; i. dil., alk. 4 al.; s. a. s. a., alk.; i. NH3 s. a., alk., NH4OH s. HNO3, H2SO4; i. aq., alk. s. al., et. s. abs. al., et. s. abs. al., et., chl., ac. s. a. s. HNO3, HF, alk. s. a., alk. s. a., alk.; i. abs. al. v. s. HNO3 s. HNO3 s. abs. al., dil. HNO3 s. al., et., ∞Br2 ∞ al.; sl. s. et.

i. d. d. i. s. i.

<−15 0

127.19 100

3.82 −140

101.42 −109.1

∞ 24.20° cc

∞ 7.350° cc

∞ al.; sl. s. et.

5.51 7.140 1.840 1.735 4.2194° 4.42

1490 419.4 242 237 394 −CO2, 300

2500 907 subl. in vac. −2H2O, 100 650

sl. d. i. 3025° 4025° 3900° 0.00115°

d. i. 44.6100° 66.6100° 670100°

2.91 25° 4

283

732

43225°

615100°

v. s. dil. a., h. KOH s. a., ac., alk. 2.825°, 16679° al. v. s. al. v. s. NH4OH, al., et. s. a., alk., NH4 salts; i. act., NH3 10012.5° al.; v. s. et.; i. NH3 s. KCN, NH3, alk.; i. al. s. a., alk., NH4OH s. a., al., NH3, aq. (NH4)2CO3 v. s. al. s. a., alk., NH4Cl; i. NH3

3.053 4.666 14.2° 4

d. 80 d. 125 446

2.065 14° 4 5.606 5.47 1.571 4.55 13° 4 3.52

36.4 >1800 >1800 expl. 212 >420 1437

3.74 15° 4 3.28 15° 4 2.072 15° 4 1.96616.5° 4.087 4.102 25° 4

d. 740 d. 238 −5H2O, 70 tr. 39 1850150atm tr. 1020

4.04 6.4 5.49 5.73

−2a H2O, 100 1700 2700

18°

624 −6H2O, 105

1100

−7H2O, 280 subl. 1185

d. 200 >2900 4300

0.0005 0.0005218° 4300°

sl. s.

324.5 0.0004218° 0.0004218° 0.0022 i. i.

∞36.4°

420° s. s. 115.20° 0.0006918° i.

61100° 89.5100° s. 653.6100° i. i.

sl. s. al.; s. gly.

i. 0.16 i. i. i.

i. d. i. i. i.

v. s. a.; i. ac. s. H2SO3, NH4OH; i. al. s. HF, aq. reg.; sl. s. a. s. H2SO4, HF s. H2SO4, HF

510100°

i. NH4OH; d. a. s. dil. a.

sl. s. al.; i. act.; NH3 sl. s. al.; i. act.; NH3 v. s. a.; i. ac. s. a.

*Usual commercial form. †Cf. special tables on water and steam, Tables 2-3, 2-4, 2-5, 2-185, 2-186 and 2-351 through 2-357. NOTE: °F = 9⁄ 5 °C + 32.

2-27

TABLE 2-2

Physical Properties of Organic Compounds* Abbreviations Used in the Table

(A), density referred to air al., ethyl alcohol amor., amorphous aq., aqua, water brn., brown bz., benzene c., cubic cc., cubic centimeter chl., chloroform col., colorless

cr., crystalline d., decomposes d-, dextrorotatory dl-, dextro-laevorotatory et., ethyl ether expl., explodes gn., green h., hot hex., hexagonal

i-, iso-, containing the group (CH3)2CHi., insoluble ign., ignites l-, laevorotatory lf., leaflets lq., liquid m-, meta mn., monoclinic n-, normal

nd., needles o-, ortho or., orange p-, para pd., powder pet., petroleum ether pl., plates pr., prisms rhb., rhombic s., soluble

This table of the physical properties includes the organic compounds of most general interest. For the properties of other organic compounds, reference must be made to larger tables in Lange’s Handbook of Chemistry (Handbook Publishers), Handbook of Chemistry and Physics (Chemical Rubber Publishing Co.), Van Nostrand’s Chemical Annual, International Critical Tables (McGraw-Hill), and similar works. The molecular weights are based on the 1941 atomic weight values. The densities are given for the temperature indicated and are usually referred to water at 4°C, e.g., 1.02895/4 a density of 1.028 at 95°C Name Abietic acid Acenaphthene Acetal Acet-aldehyde -aldehyde, par-aldehyde ammonia -amide -anilide -phenetidide (o-) (m-) -toluidide (o-) (p-) Acetic acid anhydride nitrile Acetone Acetonyl urea Acetophenone benzoyl hydride Acetyl-chloride -phenylenediamine (-p) Acetylene dichloride (cis) (trans) Aconitic acid Acridine Acrolein ethylene aldehyde Acrylic acid nitrile Adipic acid amide nitrile Adrenaline (1-) (3,4,1) Alanine (α) (dl-) Aldol acetaldol Alizarin Allyl alcohol bromide chloride thiocyanate (i) thiourea Aluminum ethoxide Amino-anthraquinone (α) (β) -azobenzene -benzoic acid (m-) (p-)

2-28

Synonym sylvic acid, abietinic acid naphthylene ethylene acetaldehyde diethylacetal ethanal paraldehyde ethanamide antifebrin o-ethoxyacetanilide acetyl-m-phenetidine N-tolylacetamide N-tolylacetamide ethanoic acid, vinegar acid acetyl oxide, acetic oxide methyl cyanide propanone, dimethyl ketone dimethyl hydantoin methyl-phenyl ketone ethanoyl chloride amino-acetanilide (p) ethyne; ethine 1,2-dichloroethene dioform equisetic acid; citridic acid acrylic aldehyde, propenal propenoic acid vinyl cyanide hexandioc acid, adipinic acid tetramethylene 1-suprarenine 2-hydroxybutyraldehyde Anthraquinoic acid propen-1-ol-3,propenyl alcohol 3-bromo-propene-1 3-chloro-propene-1 mustard oil thiosinamide

aminodracylic acid

s-, sec-, secondary silv., silvery sl., slightly subl., sublimes sym., symmetrical t-, tertiary tet., tetragonal tri., triclinic uns., unsymmetrical v., very

v. s., very soluble v. sl. s., very slightly soluble wh., white yel., yellow (+), right rotation >, greater than <, less than ∞, infinitely

referred to water at 4°C, the 4 being omitted when it is not clear whether the reference is to water at 4°C or at the temperature indicated by the upper figure. The melting and boiling points given have been selected from available data as probably the most accurate. The solubility is given in grams of the substance in 100 g. of the solvent. In the case of gases, the solubility is often expressed in some manner as “510 cc.” which indicates that, at 10°C, 5 cc. of the gas are soluble in 100 g. of the solvent.

Formula

Formula weight

Form and color

C20H30O2 C10H6(CH2)2 CH3CH(OC2H5)2 CH3CHO (C2H4O)3 CH3CHOHNH2 CH3CONH2 C6H5NHCOCH3 CH3CONHC6H4OC2H5 CH3CONHC6H4OC2H5 CH3C6H4NHCOCH3 CH3C6H4NHCOCH3 CH3CO2H (CH3CO)2O CH3CN CH3COCH3 (CH3)2 CH3COC6H5 CH3COCl C2H3ONHC6H4NH2 HC⯗CH CHCl:CHCl CHCl:CHCl C3H3(CO2H)3 C6H4 < (CH)(N) > C6H4 CH2:CHCHO CH2:CHCO2H CH2:CHCN (CH2CH2CO2H)2 (CH2CH2CONH2)2 (CH2CH2CN)2 C6H3(OH)2(CHOHCH2NHCH3) CH3CH(NH2)CO2H CH3CH(OH)CH2CO2H C6H4(CO)2C6H2(OH)2 CH2:CHCH2OH CH2:CHCH2Br CH2:CHCH2Cl CH2:CHCH2NCS CH2:CHCH2NHCSNH2 Al(OCH2CH3)3 C6H4(CO)2C6H3NH2 C6H4(CO)2C6H3NH2 C6H5N:NC6H4NH2 H2NC6H4CO2H H2NC6H4CO2H

302.44 154.20 118.17 44.05 132.16 61.08 59.07 135.16 179.21 179.21 149.19 149.19 60.05 102.09 41.05 58.08 128.13 120.14 78.50 150.18 26.04 96.95 96.95 174.11 179.21 56.06 72.06 53.06 146.14 144.17 108.14 183.20 89.09 88.10 240.20 58.08 120.99 76.53 99.15 116.18 164.15 223.22 223.22 197.23 137.13 137.13

lf. rhb./al. lq. col. lq. col. cr. col. cr. col. cr. rhb./al. lf./al. lf./al. rhb. rhb. or mn. col. lq. col. lq. col. lq. col. lq. tri./al. lf. col. lq. nd./aq. col. gas col. lq. col. lq. cr./aq. rhb./aq. al. col. lq. col. lq. col. lq. mn. pr. cr. pd. col. oil col. pd. nd./aq. col. lq. red rhb. col. lq. lq. col. lq. col. oil col. pr. pd. red nd. red nd. yel. mn. nd./aq. mn. pr.

Specific gravity 1.06995/95 0.82122/4 0.78318/4 0.99420/4 1.159 1.214 1.16815 1.21215 1.04920/4 1.08220/4 0.78320/4 0.79220/4 1.03315/15 1.10520/4 (A) 0.906 1.29115/4 1.26515/4 0.84120/4 1.06216/4 0.81120 1.36025/4 19/19

0.951

1.10320/4 0.85420/4 1.39820/4 0.93820/4 1.01320/4 1.21920/20 1.14220/0

1.5114°

Melting point, °C 182 95 −123.5 10.5–12 97 81(69.4) 113–4 79 96–7 110 153 16.7 −73 −41 −94.6 175 20.5 −112.0 162 −81.5891 −80.5 −50 192 d. 110–1 −87.7 12–13 −82 151–3 226–7 1 d. 207–11 295 d. 289–90 −129 −119.4 −136.4 −80 77–8 150–60 256 302 126–7 173–4 187–8

Boiling point, °C 278–9 102.2 20.2 124.4752 100–10 d. 222 305 >250 296 306–7 118.1 139.6 81.6–2.0 56.5 subl. 202.3749 51–2 −84760 60.3 48.4 346 52.5 141–2 78–9 26510 295 subl. >200 8320 430 96.6 70–1753 44.6 152 200–510 subl. subl. 225120

Solubility in 100 parts Water

Alcohol

Ether

i. i. 625 ∞ 1213 v. s. s. 0.56 i. sl. s. 0.8619 0.0922 ∞ 12 c. ∞ ∞ s. i. d. s. h. 100 cc.18 0.3520 0.6320 3315 sl. s. h. 40 ∞ s. 1.415 0.412 v. sl. s. 0.0320 2217 ∞ 0.03100 ∞ i. <0.1 0.2 30 d. i. i. sl. s. h. v. sl. s. 0.313

v. s. s. h. ∞ ∞ ∞ v. s. s. 2120 s. s. s. 1025 ∞ ∞ ∞ ∞ s. s. d. v. s. 600 cc.18 ∞ ∞ sl. s. s. s. ∞

v. s. s. chl. ∞ ∞ ∞ sl. s. v. sl. s. 7 25

v. s.

0.615

s. v. sl. s. v. sl. s. ∞ v. s. ∞ ∞ ∞ ∞ s. i. s. s. s. h. 210 1110

v. sl. s. i. i. s. v. s. ∞ ∞ ∞ ∞ v. sl. s. v. sl. s. s. i. s. 1.86 8.26

s. s. ∞ ∞ ∞ ∞ s. s. ∞ v. s. ∞ ∞ v. sl. s. s. s.

Amino-diphenylamine (p-) -G-acid (2-)(6-,8-), Na2 salt -mono-potassium salt -sodium salt -J-acid (2-)(5-,7-) -mono-potassium salt -naphthol sulfonic (1-,2-,4-)(α-) (1-,8-,4-) -phenol (o-) (m-) (p-) -toluene sulfonic acid (1-,2-,3-) (1-,4-,2-) (1-,4-,3-) (1-,2-,5-) Amyl acetate (n-) (i-) (s-) (s-) (t-) alcohol (n-) fusel oil, (s-,n-) methyl-propyl carbinol, (prim.-,i-) isobutyl carbinol, (s-,i-) (t-) (d-) -amine (n-) (s-,n-) (i-) (t-)

aniline (i-) benzoate (i-) bromide (n-) (i-) (t-) n-butyrate (n-) (i-) (t-) i-butyrate (i-) chloride (n-) (s-) (s-) (i-) (s-,i-) (t-) i-cyanide (i-) formate (n-) (i-) iodide (n-) (i-) (s-,n-) (t-) mercaptan (n-) (n-) (i-) phenol (t-)(p-) propionate (n-) (i-) (act.) salicylate (n-) Amyl i-valerate (i) (t-)

2-aminophenol 3-aminophenol p-hydroxyaniline

common amyl acetate α-Me-Bu-acetate di Et-carbinol acetate pentanol-1 pentanol-2 2-methyl-butanol-4 2-methyl-butanol-3 2-methyl-butanol-2 active amyl alcohol

1-NH2-2-Me-butane 3-amino pentane 3-NH2-2-Me-butane 1-bromopentane 4-Br-2-Me-butane 2-Br-2-Me-butane

1-chloropentane 2-chloropentane 3-chloropentane 4-Cl-2-Me-butane 3-Cl-2-Me-butane 2-Cl-2-Me-butane 1-Cl-2-Me-butane iso-caproic iso-nitrile 1-iodopentane 4-I-2-Me-butane 2-iodopentane 2-I-2-Me-butane pentanthiol-1 pentanthiol-3 2-Me-butanthiol-4 pentaphen

H2NC6H4NHC6H5 C10H5(NH2)(SO3Na)2 C10H5(NH2)S2O6HK C10H5(NH2)S2O6HNa C10H5(NH2)(SO3H)2 C10H5(NH2)S2O6HK C10H5OHNH2SO3Ha H2O NH2(OH)C10H5SO3H H2NC6H4OH H2NC6H4OH H2NC6H4OH C6H3(CH3)(NH2)SO3H C6H3(CH3)(NH2)SO3HH2O C6H3(CH3)(NH2)SO3Ha H2O C6H3(CH3)(NH2)SO3HH2O CH3CO2CH2(CH2)3CH3 CH3CO2CH2CH2CH(CH3)2 CH3CO2CH2CH(CH3)C2H5 CH3CO2CH(CH3)CH2C2H5 CH3CO2CH(C2H5)2 CH3CO2C(CH3)2C2H5 CH3(CH2)3CH2OH C2H5CH2CH(OH)CH3 (CH3)2CHCH2CH2OH (C2H5)2CHOH (CH3)2CHCH(OH)CH3 (CH3)2C(OH)C2H5 (CH3)3CCH2OH C2H5CH(CH3)CH2OH CH3(CH2)4NH2 (C3H7)(CH3)CHNH2 (CH3)2CH(CH2)2NH2 (C2H5)(CH3)2CNH2 C2H5CH(CH3)CH2NH2 (C2H5)2CHNH2 (CH3)2CHCH(CH3)NH2 C6H5NHC5H11 C6H5CO2C5H11 CH3(CH2)3CH2Br (CH3)2CH(CH2)2Br (CH3)2C(Br)C2H5 C2H5CH2CO2(CH2)4CH3 C2H5CH2CO2C5H11 C3H7CO2C(CH3)2C2H5 (CH3)2CHCO2C5H11 CH3(CH2)3CH2Cl C2H5CH2CHClCH3 (C2H5)2CHCl (CH3)2CH(CH2)2Cl (CH3)CHCHClCH3 (CH3)2CClC2H5 (CH3)(C2H5)CHCH2Cl (CH3)2CH(CH2)2NC HCO2CH2(CH2)3CH3 HCO2CH2CH2CH(CH3)2 CH3(CH2)3CH2I (CH3)2CHCH2CH2I C2H5CH2CHICH3 (CH3)2CIC2H5 C2H5CH(CH3)CH2I CH3(CH2)3CH2SH (C2H5)2CHSH (CH3)2CH(CH2)2SH C5H11C6H4OH C2H5CO2(CH3)4CH3 C2H5CO2(CH2)2CH(CH3)2 C2H5CO2C5H11 HOC6H4CO2C5H11 C4H9CO2C5H11 C4H9CO2C5H11

184.23 347.28 341.39 325.29 303.30 341.39 248.25 239.24 109.12 109.12 109.12 187.21 205.23 196.22 205.23 130.18 130.18 130.18 130.18 130.18 130.18 88.15 88.15 88.15 88.15 88.15 88.15 88.15 88.15 87.16 87.16 87.16 87.16 87.16 87.16 87.16 163.25 192.25 151.05 151.05 151.05 158.23 158.23 158.23 158.23 106.60 106.60 106.60 106.60 106.60 106.60 106.60 97.16 116.16 116.16 198.06 198.06 198.06 198.06 198.06 104.21 104.21 104.21 164.24 144.21 144.21 144.21 208.25 172.26 172.26

nd./aq. al.

67

354

col. nd. pr. lf. nd. mn. nd. tri./aq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. cr. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. lq. col. lq. col. lq. col. lq. lq. col. lq. col. lq. col. lq. lq. col. lq. lq. col. lq. col. lq. lq. lq. lq. lq. lq. lq. lq. lq. lq. lq. lq. lq. col. lq. lq. cr. lq. col. lq. col. lq. lq. col. lq. col. lq.

173 122–3 184–6 d.

subl. subl.

d. 0.87920/20 0.87615/4 0.88013 0.9220 0.87120/4 0.87419 0.817 20/20 0.81020/20 0.81315/4 0.815 25/4 0.81919 0.809 20/4 0.81620/4 0.76619 0.749 20/4 0.75118/4 0.73125/4 0.75518 0.749 20/4 0.75718 0.92815/4 0.99214/14 1.21820/4 1.22017/15 1.21619/0 0.87115/4 0.86619/15 0.86515/0 0.8760/4 0.878 20/4 0.870 20/4 0.895 21 0.89320/4 0.8830 0.87120/4 0.88117.5 0.9020 0.882 20/4 1.51020/4 1.515 18/4 1.507 17/4 1.47119/15 1.524 20/4 0.857 20

−H2O, 120 −70.8

−78.5 −117.2 −11.9 52–3 −55 −105

−95 −73.2

−99

−72.9 −73.5 −93.5 −86

0.83520/4 0.87615/4 0.870 20/4 0.866 20/4 1.06515 0.85820/15 0.86114/0

93 −73.1

148.4737 142757 141–2 133.5 133 124.5 749 137.9 119.5 132.0 115.6 113–4 102 113–4 128 103–4 91–2 95 77–8 95–6 90–1 83–4 254.5 261746 129.7 120745 108765 186.4 178.6 164 168.8 108.4 96.7 97.3 99.7 758 91753 85.7 98–9 137–9 132 123.5 157.0 147 765 144–5 127 765 148 126767 105 120 265–7 168.7 160.2 5816 265 194 173–4

sl. s. v. sl. s. 12.820 2.718 10.0 20 3.418 v. s. v. sl. s. 1.7 0 2.6 0 1.10 0.97 11 0.5 20 0.47 311 v. sl. s. 0.315 v. sl. s. sl. s. sl. s. v. sl. s. 2.7 22 420 214 5.5 30 2.830 sl. s. sl. s. 3.630 s. ∞ ∞ ∞ ∞ ∞ ∞ i. i. i. 0.0216 i. 0.05 50 i. sl. s. i. i. i. i. i. i. i. i. i. v. sl. s. 0.322 i. i. i. i. i. i. i. i. sl. s. i. 0.125 v. sl. s. i. v. sl. s. sl. s.

s.

s.

4.30 s. 40

v. s. sl. s. i. bz.

i. i. ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ s. ∞ ∞ s. ∞ ∞ ∞ ∞ ∞ ∞ ∞ s. s. s. ∞ ∞ ∞ s. s. s. ∞ s. s. s. s. s. ∞ ∞ s. ∞ ∞ ∞ ∞ ∞ ∞ ∞ s. ∞ ∞ ∞ ∞ ∞ s.

∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ s. ∞ ∞ s. ∞ ∞ ∞ ∞ ∞ ∞ ∞ s. s. ∞ ∞ ∞ s. s. s. ∞ ∞ s. s. s. s. ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ s. ∞ ∞ ∞ ∞ ∞ s.

*By N. A. Lange, Ph.D., Handbook Publishers, Inc., Sandusky, Ohio. Abridged from table of Physical Constants of Organic Compounds in Lange’s “Handbook of Chemistry.” 2-29

TABLE 2-2

Physical Properties of Organic Compounds (Continued )

Name Amylene (n-)(α-) (i-) (α-) (-n)(β-) (i-)(β-) Anethole (p-) Anhydroformald-aniline Aniline hydrochloride nitrate sulfate Anisal-acetone (p-) Anisic acid (p-) aldehyde (p-) Anisidine (o-) (m-) (p-) Anisole Anthracene Anthramine (α) (β) Anthranil Anthranilic acid (o-) Anthrapurpurin (1-,2-,7-) Anthraquinone disulfonate Na2 (1-,5-) (1-,8-) (2-,6-) (2-,7-) sulfonate Na (1-) (2-) Anthrarufin (1-,5-) Antipyrene Apiole Arabinose (α)(d- or l-) (dl-) Arachidic acid Arsanilic acid (p-) Asparagine (l-) Aspirin (o-) Atropic acid Auramine Aurine, coralline (4-,4′-) Azo-anisole (2-,2′-) benzene Azoxybenzene Barbituric acid Benzal acetone Benzaldehyde Benzamide Benzanilide Benzene sulfinic acid sulfonic acid sulfonic amide sulfonic chloride Benzidine (4-,4′-) disulfonic acid (2-,2′-) (3-,3′-) Benzil Benzoic acid anhydride nitrile

2-30

Synonym pentene-1 2-methyl-butene-3 2-methyl-butene-1 pentene-2 2-methyl-butene-2 p-propenyl anisole methylene aniline amino benzene, phenyl amine, cyanol aniline salt, aniline chloride MeO-benzalacetone 2-amino-anisole MeO-aniline(m) 4-amino anisole methyl phenyl ether paranaphthalene, anthracin green oil α-amino-anthracene β-amino-anthracene

diphenyleneketone, dihydrodiketoanthracene ρ-anthraquinone disulfonate x-anthraquinone disulfonate

1-ph-2,3-diMepyrazolone-5 1-allyl-2, 5-diMeO-3,4 methylenedioxybenzene eicosanoic acid

α-phenyl acrylic acid 4,4′-dimethylaminobenzophenomide diMeO-azobenzene diphenyldiimide malonyl urea Me-cinnamyl ketone artificial almond oil benzol, phenyl hydride, cyclohexatriene benzene sulfonamide benzene sulfonyl chloride

dibenzoyl phenyl cyanide

Formula weight

Form and color

Specific gravity

C2H5CH2CH:CH2 (CH3)2CHCH:CH2 (C2H5)(CH3)C:CH2 C2H5CH:CHCH3 (CH3)2C:CHCH3 CH3CH:CHC6H4OCH3 (CH2NC6H5)3 C6H5NH2

70.13 70.13 70.13 70.13 70.13 148.20 315.40 93.12

lq. col. lq. col. lq. col. lq. col. lq. lf./al. pr./al. col. oil

0.644 20 0.63215 0.667 0/0 0.650 20/4 0.66319/4 0.99120/20

C6H5NH2HCl C6H5NH2HNO3 (C6H5NH2)2H2SO4 CH3OC6H4CH:CHCOCH3 CH3OC6H4CO2H CH3OC6H4CHO CH3OC6H4NH2 CH3OC6H4NH2 CH3OC6H4NH2 CH3OC6H5 C6H4:(CH)2:C6H4

129.59 156.14 284.32 176.22 152.14 136.14 123.15 123.15 123.15 108.13 178.22

cr. rhb. lf./al. lf./et. mn./aq. col. oil col. lq. oil pl./aq. col. lq. col. mn.

1.222 4 1.356 4 1.377 4

C6H4:(CH)2:C6H3NH2 C6H4:(CH)2:C6H3NH2 C6H4:(NH)CO H2NC6H4CO2H C14H5O2(OH)3 C6H4:(CO)2:C6H4

193.24 193.24 119.12 137.13 256.20 208.20

yel./al. yel./al. col. oil col. rhb. or. nd./al. yel. rhb.

C14H6O2(SO3Na)25H2O C14H6O2(SO3Na)24H2O C14H6O2(SO3Na)27H2O C14H6O2(SO3Na)24H2O C14H7O2SO3Na C14H7O2SO3Na C14H6O3(OH)2 C11H12ON2 C12H14O2

502.38 484.37 538.41 484.37 310.25 310.25 240.20 188.22 222.23

yel. lf. yel. pr. col. cr. cr. yel. lf. silv. lf. yel. lf. mn./aq. col. nd.

1.088113/4 1.0220/4

CH2OH(CHOH)3CHO CH2OH(CHOH)3CHO CH3(CH2)18CO2H H2NC6H4.AsO3H2 HO2CC2H3(NH2)CONH2 CH3CO2C6H4OH C6H5C(:CH2)CO2H [(CH3)2NC6H4]2C:NH

150.13 150.13 312.52 217.04 132.12 180.15 148.15 267.36

rhb. pr.

1.585 20/4

col. lf. nd./aq. rhb. nd./aq. nd./aq. col./al.

(HOC6H4)2C:C6H4:O (CH3OC6H4N:)2 C6H5N:NC6H5 (C6H5)2N2O CO:(NHCO)2:CH22H2O C6H5CH:CHCOCH3 C6H5CHO C6H5CONH2 C6H5CONHC6H5 C6H6

290.30 242.27 182.22 198.22 164.12 146.18 106.12 121.13 197.23 78.11

red or. pr. or. mn. yel. rhb. col./aq. pl. col. lq. col. pr. lf./al. col. lq.

C6H5SO2H C6H5SO3H C6H5SO2NH2 C6H5SO2Cl NH2C6H4C6H4NH2 (C6H3(NH2)SO3H)23H2O (C6H3(NH2)SO3H)2 C6H5COCOC6H5 C6H5CO2H (C6H5CO)2O C6H5CN

142.17 158.17 157.18 176.62 184.23 398.40 344.35 210.22 122.12 226.22 103.12

pr./aq. col. nd. mn./aq. cr. cr./aq. pr./aq.

1.38415/15

83–4 65–6 156 14.5 128–9 d. >175

pr. mn. pr. rhb./et. col. lq.

1.2315 1.26615/4 1.19915/4 1.00125/6

95 121.7 42 −12.9

Formula

1.02220/4

1.385 4 1.123 20/4 1.09815/15 1.096 20/4 1.089 55/55 0.990 22/4 1.25 27/4

1.18715/4 1.438 20/4

1.543

15/4

1.20320/4 1.248 20/20 1.035 20/20 1.046 20/4 1.341 1.314 0.879 20/4

Melting point, °C −135 −139 −124 22.5 143 −6.2

Boiling point, °C

Solubility in 100 parts Water

Alcohol

Ether

30–1 20.5 771 31–2758 36.4 37–8 235.3 185 184.4

i. i. i. v. sl. s. i. v. sl. s. i. 3.618

∞ ∞ ∞ ∞ s. s. sl. s. ∞

∞ ∞ ∞ ∞ ∞ ∞ s. ∞

245

s. s. sl. s. v. s. v. s. ∞ ∞ s. s. s. 1.520

i. sl. s. i. v. s. v. s. ∞ ∞ s. s. s.

198 d. 190 d. 73–4 184.2 2.5 5.2 <−12 57.2 −37.3 217–8

275–80 247–8 225 251 243 154–5 340–2

1815 s. 514 i. 0.0319 v. sl. s. v. sl. s. v. sl. s. s. h. i. i.

130 238 <−18 144–5 369 286

subl. d. >215 subl. 462 379–81

i. i. sl. s. h. 0.35 14 sl. s. h. i.

s. sl. s. s. 1110 v. s. h. 0.0518

sl. s. s. 167 sl. s. v. sl. s.

i.

i.

subl. 319174 294

v. s. sl. s. 3.920 30.5 20 0.5320 0.8425 i. 100 25 i

v. sl. s. i. i. sl. s. 100 s.

i. i. i. s. sl. s. s.

460 16.910 i. v. s. h. 3.128 137 0.1 c. i.

0.59°

i.

s. h. v. s. h. i. c. s. s. 720

v. s. i.

i. i. i. i. s. h. i. 0.3 1.35 25 i. 0.07 22

s. s. 4.220 11.415 sl. s. s. ∞ 1725 430 s.

s. s. ∞ sl. s. sl. s. ∞

v. s. h. v. s. 0.4316 i. 1 h. 0.09 25 v. sl. s. i. 0.217 i. 1100

v. s. v. s. v. s. v. s. 1 h. i.

v. s. i. v. s. s. 2 i.

v. s. 4615 s. ∞

v. s. 6615 s. ∞

280 113(109) 30 159.5 164.5 77 232 227–35 135–6 106–7 136 310 d. 153 68 36 d. 245 41–2 −26 130 163 5.5

328 d. 235 267 d.

297 d. 260–2 179 290 117–910 80.1 d. > 100 d. 251.5 400740 348 d. 249.2 360 190.7

520 s. 2.320 s. s.

Benzoin (dl-) Benzophenone Benzotrichloride Benzoyl-benzoic acid (o-) -chloride -peroxide Benzyl acetate alcohol amine aniline benzoate butyrate chloride ether formate propionate Berberonic acid (2-,4-,5-) Biuret Borneol (dl-) (d- or l-) (iso-) Bornyl acetate (d-) Bromo-aniline (p-) -benzene -camphor (3-)(d-) -diphenyl (p-) -naphthalene (α-) (β-) -phenol (o-) (m-) (p-) -styrene (ω)(1) (2) -toluene (o-) (m-) (p-) Bromoform Butadiene (1-,2-) (1-,3-) Butadienyl acetylene Butane (i-) Butyl acetate (n-) (s-) (i-) (tert-) alcohol (n-) (s-) (i-) (tert-) amine (n-) (s-) (i-) (t-) p-aminophenol (N)(n) (N)(i-) aniline (n-) (i-) arsonic acid (n-) benzoate (n-) (i-) bromide (n-) (s-) (i-) (t-) butyrate (n-)(n-) (n-)(i-) (i-)(i-) caproate carbamate (i-) cellosolve (n-)

diphenyl ketone phenyl chloroform

phenyl carbinol ω-amino toluene phenyl-benzylamine ω-chlorotoluene dibenzyl ether

allophanamide

phenyl bromide α-bromocamphor α-naphthyl bromide β-naphthyl bromide

o-tolyl bromide tribromo-methane methyl-allene erythrene diethyl trimethyl-methane

butanol-1 butanol-2 2-methyl-propanol-1 2-methyl-propanol-2

1-bromo-butane 2-bromo-butane 1-Br-2-Me-propane 2-Br-2-Me-propane

2-BuO-ethanol-1

C6H5COCHOHC6H5 C6H5COC6H5 C6H5CCl3 C6H5COC6H4CO2HH2O C6H5COCl (C6H5CO)2O2 CH3CO2CH2C6H5 C6H5CH2OH C6H5CH2NH2 C6H5CH2NHC6H5 C6H5CO2CH2C6H5 C2H5CH2CO2CH2C6H5 C6H5CH2Cl (C6H5CH2)2O HCO2CH2C6H5 C2H5CO2CH2C6H5 C5H2N(CO2H)32H2O NH(CONH2)2 C10H17OH C10H17OH C10H17OH CH3CO2C10H17 BrC6H4NH2 C6H5Br BrC10H15O BrC6H4C6H5 C10H7Br C10H7Br BrC6H4OH BrC6H4OH BrC6H4OH C6H5CH:CHBr C6H5CH:CHBr CH3C6H4Br CH3C6H4Br CH3C6H4Br CHBr3 CH3CH:C:CH2 CH2:CHCH:CH2 CH2:(CH)2:CHC⯗CH CH3CH2CH2CH3 (CH3)2CHCH3 CH3CO2(CH2)2C2H5 CH3CO2CH(CH3)C2H5 CH3CO2CH2CH(CH3)2 CH3CO2C(CH3)3 C2H5CH2CH2OH C2H5CH(OH)CH3 (CH3)2CHCH2OH (CH3)3COH C2H5CH2CH2NH2 C2H5CH(NH2)CH3 (CH3)2CHCH2NH2 (CH3)3CNH2 C4H9NHC6H4OH C4H9NHC6H4OH C4H9NHC6H5 C4H9NHC6H5 C4H9AsO(OH)2 C6H5CO2C4H9 C6H5CO2C4H9 C2H5CH2CH2Br C2H5CH(Br)CH3 (CH3)2CHCH2Br (CH3)3CBr C2H5CH2CO2CH2CH2C2H5 C2H5CH2CO2CH2CH(CH3)2 (CH3)2CHCO2CH2CH(CH3)2 CH3(CH2)4CO2C4H9 NH2CO2CH2CH(CH3)2 C4H9OCH2CH2OH

212.24 182.21 195.48 244.24 140.57 242.22 150.17 108.13 107.15 183.24 212.24 178.22 126.58 198.25 136.14 164.20 247.16 103.08 154.24 154.24 154.24 196.28 172.03 157.02 231.14 233.11 207.07 207.07 173.02 173.02 173.02 183.05 183.05 171.04 171.04 171.04 252.77 54.09 54.09 78.11 58.12 58.12 116.16 116.16 116.16 116.16 74.12 74.12 74.12 74.12 73.14 73.14 73.14 73.14 165.23 165.23 149.23 149.23 182.04 178.22 178.22 137.03 137.03 137.03 137.03 144.21 144.21 144.21 172.26 117.15 118.17

mn. col. rhb. col. lq. tri./aq. col. lq. rhb./et. col. lq. col. lq. lq. mn. pr. nd. col. lq. col. lq. lq. col. lq. lq. tri. nd./al. col. cr. col. cr. col. cr. rhb./pet. rhb. col. lq. cr. cr./al. col. oil lf./al. col. lq. cr. tet. cr. lq. lq. col. lq. col. lq. cr./al. col. lq. lq. col. gas col. lq. col. gas col. gas col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. lq. col. lq. col. lq. col. lq. col. lq. lq. oil col. lf. col. oil col. oil lq. lq. lq. lq. col. lq. col. lq. col. lq. col. lq. col. lf. col. lq.

1.08354 1.38014 1.21220/4 1.05717 1.04320/4 0.98220/4 1.065 25/25 1.1220/4 1.01616/18 1.100 20/20 1.03616 1.08123 1.03616/17 20/4

1.011 1.01120/4 0.99115 1.820 1.495 20/4 1.449 20/4 1.48220/4 1.605 0 1.55380 1.588 80 1.42220/4 1.427 20/4 1.42220/4 1.410 20/4 1.390 20/4 2.890 20/4 0.62120/4 0.773 20/4 0.600 0.600 0.882 20 0.865 25/4 0.87120/4 0.866 20/4 0.810 20/4 0.808 20/4 0.80517.5 0.779 26 0.739 25/4 0.724 20/4 0.73220/20 0.69818/4

133–7 48.5 −4.75 93(128) −0.5 108 d. −51.5 −15.3 37–8 21 238–40 −39 3.6 243 192–3 d. 210.5 208–9 212 29 63–4 −30.6 77–8 90–1 5–6 59 5.6 32–3 63.5 7 −7.5 −28 −39.8 28.5 8–9 −108.9 −135 −145 −76.3 −98.9 −79.9 −114.7 −108 25.5 −50 −104 −85 −67.5 71 79

0.940 20/4 1.005 25/25 0.997 25/25 1.277 20/4 1.25125/4 1.258 25/4 1.21120/4 0.87220/20 0.86318/4 0.875 0/4 0.8820/0 0.95676/4 0.90320/4

158–9 −22 −112.4 −112 −118.5 −16.2 −80.7 65

344768 305.4 220.7 197.2 expl. 213.5 204.7 184.5 306750 323–4 i. 179.4 295–8 202–3747 220–2 subl. 212–3 226–7 156.2 274 310 281.1 281–2 194–5 236–7 238 221 10826 181.8 183.7 184–5 150.5 18–9 −4.41 83–6 −0.6 −10 125 740 112744 118 95–6760 117 99.5 107–8 82.9 77.8 66772 68–9 45.2 235720 231–2 249–50 241.5 101.6 91.3 91.5 73.3 165.7736 156.9 148–9 204.3 206–7 171.2

v. sl. s. i. i. sl. s. d. i. i. 417 ∞ i. i. v. s. i. i. i. i. v. sl. s. 1.30 v. sl. s. v. sl. s. i. i. i. c. i. i. i. i. i. s. 1.415 i. i. i. i. i. 0.1 c. i. i. i. i. i. 0.7 i. 0.625 i. 915 12.520 1015 ∞ ∞ ∞ ∞ i. i. i. 0.0115 s. i. i. 0.0616 i. 0.0618 i. i. i. i. i. i. ∞

s. h. 6.515 s.

sl. s. 1513 s.

d. h. s. h. ∞ ∞ ∞

∞ s. ∞ ∞ ∞ s. ∞

∞ v. s. ∞ s. h. s.

∞ s. ∞

sl. s. h. s.

i.

v. s.

v. s.

s. v. s. s. 2026 s. s. 620 s. s. v. s. ∞ ∞ s. s. s. ∞ ∞ ∞

s. v. s. ∞ v. s. 34 25 ∞ v. s. ∞ s. v. s. ∞ ∞ ∞25 s. ∞25 ∞ ∞ ∞

s. s. ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞

s. s. ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞

v. s. v. s. s. s. ∞ ∞

v. s. v. s. i. s. ∞ ∞

∞ ∞ ∞ ∞ ∞

∞ ∞ ∞ ∞ ∞

s. ∞

s. ∞

2-31

TABLE 2-2

Physical Properties of Organic Compounds (Continued )

Name chloride (n-) (s-) (i-) (t-) dimethylbenzene (t-)(1-,3-,5-) formate (n-) (s-) (i-) furoate (n-) iodide (n-) (s-) (i-) (t-) lactate (n-) mercaptan (n-) (i-) (t-) methacrylate (n-) (i-) phenol (p-)(t-) propionate (n-) (s-) (i-) stearate (n-) (i-) iso-thiocyanate (n-) (i-) (s-)(d-) (t-) valerate (n-)(n-) (i-)(n-) (i-)(s-) (i-)(i-) Butylene (α-) (β-) Butyraldehyde (n-) (i-) Butyric acid (n-) (i-) amide (n-) (i-) anhydride (n-) (i-) anilide (n-) Caffeic acid (3-,4-) Caffeine Camphene (dl-) (d- or l-) Camphor (d-) Camphoric acid (d-) Cantharidine Capric acid Caproic acid (n-) (i-) Caprylic acid (n-) Carbazole Carbitol Carbon disulfide monoxide suboxide tetrabromide tetrachloride tetrafluoride Carbonyl sulfide Carminic acid Carvacrol (1-,2-,4-)

2-32

Synonym 1-chloro-butane 2-chloro-butane 1-Cl2-2-Me-propane 2-Cl2-2-Me-propane

1-iodo-butane 2-iodo-butane 1-iodo-2-Me-propane 2-iodo-2-Me-propane butanthiol-1 2-Me-propanthiol-1

butyl mustard oil iso-Bu mustard oil

butene-1 butene-2 2-Me-propanol butanoic acid 2-Me-propanoic acid n-butyramide iso-butyramide n-butyranilide

decanoic acid hexanoic acid 2-Me-pentanoic-5 acid octanoic acid diphenylenelimine, dibenzopyrrole diethylene glycol mono-Et ether

tetrabromomethane tetrachloromethane tetrafluoromethane

Formula C2H5CH2CH2Cl C2H5CHClCH3 (CH3)2CHCH2Cl (CH3)3CCl (CH3)3CC6H3:(CH3)2 HCO2CH2CH2C2H5 HCO2CH(CH3)C2H5 HCO2CH2CH(CH3)2 OC4H3CO2C4H9 C2H5CH2CH2I C2H5CHICH3 (CH3)2CHCH2I (CH3)3CI CH3CH(OH)CO2C4H3 C2H5CH2CH2SH (CH3)2CHCH2SH (CH3)3CSH CH2:C(CH3)CO2C4H9 CH2:C(CH3)CO2C4H9 (CH3)3CC6H4OH C2H5CO2C4H9 C2H5CO2C4H9 C2H5CO2C4H9 CH3(CH2)16CO2C4H9 CH3(CH2)16CO2C4H9 C2H5CH2CH2N:CS (CH3)2CHCH2N:CS C4H9N:CS (CH3)3CN:CS CH3(CH2)3CO2(CH2)3CH3 (CH3)2CHCH2CO2(CH2)3CH3 (CH3)2CHCH2CO2C4H9 C4H9CO2C4H9 C2H5CH:CH2 CH3CH:CHCH3 CH3CH2CH2CHO (CH3)2CHCHO C2H5CH2CO2H (CH3)2CHCO2H C2H5CH2CONH2 (CH3)2CHCONH2 (C2H5CH2CO)2O [(CH3)2CHCO]2O C3H7CONHC6H5 (HO)2C6H3C2H2CO2H C8H10O2N4H2O C10H16 C10H16 C10H16O C8H14(CO2H)2 C10H12O4 CH3(CH2)8CO2H CH3(CH2)4CO2H (CH3)2CH(CH2)2CO2H CH3(CH2)6CO2H (C6H4)2NH C2H5O(CH2)2O(CH2)2OH CS2 CO OC:C:CO CBr4 CCl4 CF4 COS C22H20O13 CH3C6H3(OH)CH(CH3)2

Formula weight

Form and color

Specific gravity

Melting point, °C

Boiling point, °C

92.57 92.57 92.57 92.57 162.26 102.13 102.13 102.13 168.19 184.03 184.03 184.03 184.03 146.18 90.18 90.18 90.18 142.19 142.19 150.21 130.18 130.18 130.18 340.57 340.57 115.19 115.19 115.19 115.19 158.23 158.23 158.23 158.23 56.10 56.10 72.10 72.10 88.10 88.10 87.12 87.12 158.19 158.19 163.21 180.15 212.21 136.23 136.23 152.23 200.23 196.20 172.26 116.16 116.16 144.21 167.20 134.17 76.13 28.01 68.03 331.67 153.84 88.01 60.07 492.40 150.21

col. lq. col. lq. col. lq. col. lq. col. lq. lq. lq. lq. col. lq. lq. lq. lq. lq. col. lq. col. lq. lq. lq. lq. lq. nd./aq. col. lq. col. lq. col. lq. col. lq. wax lq. lq. lq. lq. lq. lq. col. lq. col. lq. col. gas col. gas col. lq. col. lq. col. lq. col. lq. rhb. mn. pl. col. lq. col. lq. mn. pr. yel./aq. nd./al. cr. cr. trig. mn. cr. col. nd. oily lq. col. oil col. lf. lf. col. lq. col. lq. col. gas gas col. mn. col. lq. gas col. gas red pd. col. lq.

0.887 20 0.87120/4 0.88415 0.84715

−123.1 −131 −131.2 −26.5

77.9763 67.8767 68.9 51–2 200–2147 106.9 97 98.2 118–2025 129.9 118–9 120 99 75–66 97–8 88 65–7 155 155 236–8 146 132.5 136.8 220–525

0.9110 0.88220/4 0.885 20/4 1.056 20/4 1.617 20/4 1.595 20 1.606 20/4 1.370 19/15 0.968 0.837 25/4 0.836 20/4 0.889 15.6 0.889 15.6 0.908 112/4 0.88315 0.866 20/4 0.888 0/4 0.855 25/25 0.95611 0.96414/4 0.943 20/4 0.91910 0.87015/4 0.862 25/4 0.848 20/4 0.8740/4 0.69 0.817 20/4 0.79420/4 0.96420/4 0.949 20/4 1.032 1.013 0.968 20/20 0.950 25/4 1.134 1.2319 0.82278 0.845 50/4 0.999 9/9 1.186 0.889 87 0.922 20/4 0.925 20/4 0.910 20/4 0.990 20/20 1.263 20/4 0.81−195/4 1.1140 3.42 1.595 20/4 1.24−87 0.977 20/4

−95.3 −103.5 −104 −90.7 −34 −116 <−79

99 −89.55 −71.4 27.5 25

10.5 −93

−130 −127 −99 −65.9 −4.7 −47 115–6 129–30 −75 −53.5 92 195–213 237 50 42.7 178–9 187 212 31.5 −1.5 −35 16 244.8 −108.6 −207 −107 90.1(48) −22.6 −138.2 d. 136 0.5

165724 162 159–63 140770 186 168.8 163–4752 168.7 −5758 3746 75.7 64757 163.5757 154.5 216 216–20 199.5 181.5734 18915 d. subl. 160 159.6 209.1759 268–70 202761 207.7 237.5 354.8 201.9 46.3 −192 7761 189.5 76.8 −128 −50.2760 238

Solubility in 100 parts Water 0.0718 i. i. i. i. v. sl. s. sl. s. 1.122 i. i. i. i. i. sl. s. sl. s. v. sl. s. i. i. sl. s. i. i. i. 0.3 25 i. i. i. i. i. v. sl. s. i. i. i. i. 4 1120 ∞ 20 20 16.315 v. s. d. d. i. s. h. 2 i. i. 0.1 0.612 0.003 0.003 1.120 v. sl. s. 0.0715 i. ∞ 0.20 3.50 cc. d. 0.0230 0.0820 sl. s. 8014 cc. s. v. sl. s.

Alcohol

Ether

∞ ∞ ∞ ∞

∞ ∞ ∞ ∞

∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ v. s. s.

∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ v. s. s.

s. ∞ ∞ ∞ s.

s. ∞ ∞ ∞ s.

s. s. s. s. ∞ ∞ ∞ ∞ v. s.

s. s. s. s. ∞ ∞ ∞ ∞ v. s.

∞ ∞ ∞ ∞ s. s. d d s. s. 2 s. s. 12012 s.

∞ ∞ ∞ ∞ sl. s. sl. s. ∞ ∞ s. sl. s. 0.3 s. s. v. s.

s. s. s. s. 0.9214 v. s. ∞ s.

s. s. s. s. sl. s. s. ∞

s. ∞

s. s. ∞

s. s. ∞

s. v. sl. s. ∞

Carvacrylamine (2-,1-,4-) Carvone (d-) Cellosolve acetate Cellulose Cetyl acetate alcohol Chloral hydrate Chloranil Chloretone Chloro-acetanilide (p-) -acetic acid -acetone -acetophenone (ω-) -acetyl chloride -aniline (o-) (m-) (p-) -anthraquinone (1-) (2-) -benzaldehyde (o-) (m-) (p-) -benzene -benzoic acid (o-) (m-) (p-) -buta-1,3-diene (2-) (1-) -buta-1,2-diene (4-) -dimethylhydantoin -dinitrobenzene (α)(1-,2-)(4-) (α)(1-,3-)(4-) -diphenyl (o-) (m-) (p-) -hydroquinone -naphthalene (α-) (β-) -nitrobenzene (o-) (m-) (p-) -nitrotoluene (2-,4-) (2-,6-) -phenol (o-) (m-) (p-) -propionic acid (α)(dl-) -toluene (o-) (m-) (p-) Chloroform Chlorophyll (α-) Chloropicrin Cholesterol Chrysene Chrysoidine (2-,4-) Chrysophanic acid Cinchomeronic acid (3-,4-) Cineole, eucalyptole Cinnamic acid (cis-) (trans-) aldehyde Cinnamyl alcohol cinnamate Citraconic acid (cis-) Citral (α) Citric acid Citronellal (d-) Citronellol (d-) Coniine (d-)(2-)

H2NC6H3(CH3)C3H7 C10H14O C2H5O(CH2)2OH CH3CO2CH2CH2OC2H5 (C6H10O5)x CH3CO2(CH2)15CH3 CH3(CH2)14CH2OH CCl3CHO CCl3CH(OH)2 OC:(CClCCl)2:CO Cl3CC(OH)(CH3)2 CH3CO2NHC6H4Cl ClCH2CO2H CH3COCH2Cl C6H5COCH2Cl ClCH2COCl ClC6H4NH2 ClC6H4NH2 ClC6H4NH2 C6H4(CO)2C6H3Cl C6H4(CO)2C6H3Cl ClC6H4CHO ClC6H4CHO ClC6H4CHO C6H5Cl ClC6H4CO2H ClC6H4CO2H ClC6H4CO2H CH2:CClCH:CH2 CH2:CHCH:CHCl CH2:C:CHCH2Cl  C(CH3)2N(Cl)CON(Cl)CO ClC6H3(NO2)2 ClC6H3(NO2)2 C6H5C6H4Cl C6H5C6H4Cl C6H5C6H4Cl ClC6H3(OH)2 C10H7Cl C10H7Cl ClC6H4NO2 ClC6H4NO2 ClC6H4NO2 CH3C6H3(NO2)(Cl) CH3C6H3(NO2)(Cl) ClC6H4OH ClC6H4OH ClC6H4OH CH3CHClCO2H CH3C6H4Cl CH3C6H4Cl CH3C6H4Cl CHCl3 C55H72O5N4Mg Cl3CNO2 C27H45OHH2O C18H12 C6H5N:NC6H3(NH2)2 C14H5(OH)2(CH3)O2 C5H3N(CO2H)2 C10H18O C6H5CH:CHCO2H C6H5CH:CHCO2H C6H5CH:CHCHO C6H5CH:CHCH2OH C8H7CO2C9H9 CH3C(CO2H):CHCO2H C9H15CHO C3H4(OH)(CO2H)3 C9H17CHO C10H20O C3H7C5H10N

149.23 150.21 90.12 132.16 162.14 284.47 242.43 147.40 165.42 245.89 177.47 169.61 94.50 92.53 154.59 112.95 127.57 127.57 127.57 242.65 242.65 140.57 140.57 140.57 112.56 156.57 156.57 156.57 88.54 88.54 88.54 197.03 202.56 202.56 188.65 188.65 188.65 144.56 162.61 162.61 157.56 157.56 157.56 171.56 171.56 128.56 128.56 128.56 108.53 126.58 126.58 126.58 119.39 893.48 164.39 404.65 228.28 212.25 254.23 167.12 154.24 148.15 148.15 132.15 134.17 264.31 130.10 152.23 192.12 154.24 156.26 127.22

oil col. lq. col. lq. col. lq. amor. nd. lf. col. lq. mn. pr. yel./bz. col. cr. rhb. col. cr. col. lq. rhb. col. lq. lq. lq. rhb. yel. nd. nd./al. nd. pr. pr. col. lq. mn./aq. pr. tri. col. lq. col. lq. col. lq. cr./et. rhb./et. cr. cr. lf. mn. col. lq. lf./al. mn. nd. yel./al. mn. pr. cr. cr. col. lq. nd. nd. col. lq. col. lq. col. lq. col. lq. col. lq. lq. rhb./al. col. rhb. yel. cr. yel./al. cr./HCl col. oil mn. pr. mn. pr. lq. nd. nd. or pr. nd. col. oil cr. col. oil col. oil col. lq.

0.99420 0.96120/4 0.93120/4 0.975 20/4 1.3–1.4 0.858 20 0.818 50/4 1.505 25/4 1.619 50/4 1.385 22 1.58 20/20 1.16216 1.32415 1.498 20/20 1.21320/4 1.216 20/4 1.42719 1.298 1.25015 1.196 61 1.107 20/4 1.544 25/4 1.496 25/4 1.54124 0.958 20/20 0.965 20/20 0.99120/20 1.5 20/20 1.697 22

1.194 20/4 1.26616 1.305 80/4 1.34350/4 1.298 91 1.256 80 1.24118/15 1.268 25 1.306 20/4 1.306 9 1.082 20/4 1.07220/4 1.070 20/4 1.489 20 1.65123/4 1.067

0.927 20 1.2844 1.245 1.110 20/20 1.040 35/35 1.08516.5 1.617 0.89017/4 1.54220/4 0.85517.5 0.84820/4 0.84717

−16 −70 22–3 49–50 −57 51.7 290 97 175–6 61.2 −44.5 58–9 0 −10.4 70–1 162 208–9 11 17–8 47.8 −45.2 141–2 158 242–3

130 39(36) 53(43) 34 89 77.5 106 −20 56–7 32.5 44.4(24) 83–4 38.2 37.5 7(0) 32–3 41–3 <−20 −34 −47.8 7.5 −63.5 d. −64 149–51 253–4 117.5 195 258–9 d. 1.5 68 133 −7.5 33 44 92–3 153 −2

241 230766 135.1 156.3 i. 20015 189.515 97.6768 d. 98 subl. 167 189.5 121 245–7 105 210.5 230767 230–1 subl. 208748 213–4 213748 132.1 subl. 59.4 69 88 315 d. 315 d. 267–8 284–5 282 263 sl. d. 259.3 264751 245.5753 235.6 242761 240718 238 175–6 214 217 186 159.5 161.6 162.2 61.2 i. 112.3766 subl. 448 subl. subl. d. 176–7 12519 300 252 sl. d. 257.5 229 d. 204–8 224–5 166–7

v. sl. s. i. ∞ 22 i. i. i. v. s. 47417 i. 0.8 c. sl. s. v. s. ∞ 0.11 d. i. i. s. h. i. i. v. sl. s. v. sl. s. s. h. 0.049 20 0.20825 0.04125 0.00825 v. sl. s. v. sl. s. d. 0.2125 i. i. i. i. i. v. s. i. i. i. i. i. i. i. 2.8520 2.6020 2.7120 ∞ i. i. i. 0.8220 s. 0.1718 0.2620 i. sl. s. h. i. c. v. sl. s. 1.915

s. ∞ ∞ ∞ i. v. sl. s. c. s. ∞ v. s. i. c. 111 s. s. ∞ v. s. d.

0.0418 v. sl. s. sl. s. i. 36025 i. 207.725 v. sl. s. v. sl. s. 1.1

s. sl. s. h.

s. ∞ ∞ ∞ s. ∞ s. i. c. s. v. s. s. ∞ v. s. s. s. s.

v. s. v. s. v. s. ∞ s. s. s. ∞ ∞

v. s. v. s. v. s. ∞ s. s. s. ∞ ∞

v. s. h. s. h.

v. s. s.

v. s. s. v. s. s. h. v. s. h. v. s. h.

v. s. ∞ v. s. s. v. s. v. s.

s. s. v. s. ∞ s. s. s. ∞

s. s. v. s. ∞ ∞ ∞ ∞ ∞

s. 1.117 0.116 s. s. h. sl. s. ∞

s. 18 v. sl. s. s. sl. s. i. ∞

2420 s. v. s. 4 c. s. ∞ 7615 ∞ ∞ v. s.

v. s. ∞ v. s. 33 s. ∞ 215 ∞ ∞ v. s.

2-33

TABLE 2-2

Physical Properties of Organic Compounds (Continued ) Name

Coumaric acid (o-) (p-) Coumarin Coumarone Creatine Creatinine Creosol (3-,1-,4-) Cresidine (1-,2-,4-) Cresol (o-) (m-) (p-) Cresyl benzoate (o-) (m-) (p-) Crotonic acid (α-) acid (β-)(cis-) aldehyde (α) Cumene Cumic acid (p-) Cumidine (p-) Cyanamide Cyanic acid Cyanoacetic acid Cyanogen bromide chloride Cyanuric acid Cyclo-butane -heptane -hexane -hexanol -hexanone -hexene -hexyl acetate amine bromide chloride -pentadiene (1-,3-) -pentane -pentanone -propane Cymene (o-) (m-) (p-) Cystine (l-) Dambose Decahydronaphthalene (cis-) (trans-) Decane (n-) Decyl alcohol Dextrin Diacetone alcohol Diamino-benzophenone (4-,4′-) -diphenylamine (4-,4′-) -diphenylmethane (4-,4′-) -diphenylurea (4-,4′-) Diamyl-amine (i-) ether (n-) (i-) Diamyl ketone (i-) phthalate (n-) (i-) tartrate (i-) Dianisidine (o-)(4-,3-)2 Diazo-aminobenzene -aminotoluene (2-,2′-) -methane

2-34

Formula HOC6H4CH:CHCO2H HOC6H4CH:CHCO2H C9H6O2 C8H6O C4H9N3O2H2O C4H7N3O CH3OC6H3(CH3)OH CH3(NH2)C6H3OCH3 CH3C6H4OH CH3C6H4OH CH3C6H4OH C6H5CO2C6H4CH3 C6H5CO2C6H4CH3 C6H5CO2C6H4CH3 CH3CH:CHCO2H CH3CH:CHCO2H CH3CH:CHCHO C6H5CH(CH3)2 (CH3)2CHC6H4CO2H (CH3)2CHC6H4NH2 H2NCN HOCN or HNCO CH2(CN)CO2H (CN)2 BrCN ClCN C3H3O3N32H2O CH2 < (CH2)2 > CH2 CH2 < (CH2CH2CH2)2 > CH2 < (CH2CH2)2 > CH2 CH2 < (CH2CH2)2 > CHOH CH2 < (CH2CH2)2 > CO (CH2CH2CH:)2 CH3CO2C6H11 CH2 < (CH2CH2)2 > CHNH2 CH2 < (CH2CH2)2 > CHBr CH2 < (CH2CH2)2 > CHCl CH2 < (CH:CH)2 > CH2 < (CH2CH2)2 > < (CH2CH2)2 > CO < CH2CH2CH2 > CH3C6H3CH(CH3)2 CH3C6H4CH(CH3)2 CH3C6H4CH(CH3)2 [SCH2CH(NH2)CO2H]2 C6H6(OH)6 C10H18 C10H18 CH3(CH2)8CH3 CH3(CH2)8CH2OH (C6H10O5)x (CH3)2C(OH)CH2COCH3 H2NC6H4COC6H4NH2 H2NC6H4NHC6H4NH2 H2NC6H4CH2C6H4NH2 (H2NC6H4NH)2CO [(CH3)2CHCH2CH2]2NH (C2H5CH2CH2CH2)2O [(CH3)2CH(CH2)2]2O [(CH3)2CHCH2CH2]2CO C6H4(CO2C5H11)2 C6H4(CO2C5H11)2 (HOCHCO2C5H11)2 [NH2(OCH3)C6H3]2 C6H5N:NNHC6H5 C7H7N:NNHC7H7 CH2:N2

Formula weight

Form and color

164.15 164.15 146.14 118.13 149.15 113.12 138.16 137.18 108.13 108.13 108.13 212.24 212.24 212.24 86.09 86.09 70.09 120.19 164.20 135.20 42.04 43.03 85.06 52.04 105.93 61.48 165.11 56.10 98.18 84.16 100.16 98.14 82.14 142.19 99.17 163.06 118.61 66.10 70.13 84.11 42.08 134.21 134.21 134.21 240.29 180.16 138.24 138.24 142.28 158.28 162.14 116.16 212.24 199.25 198.26 242.28 157.29 158.28 158.28 170.29 306.39 306.39 290.35 244.28 197.23 225.28 42.04

nd./aq. cr./aq. rhb./et. oil mn./aq. mn. pr. nd./pet. cr. lq. pr. lq. cr. cr. col. mn. nd. col. lq. col. lq. tri. lq. col. nd. gas col. lq. col. gas nd. gas mn./aq. col. gas oil col. lq. col. nd. col. oil lq. oil col. lq. col. lq. col. lq. col. lq. col. oil col. oil col. gas col. lq. col. lq. col. lq. pl. mn./aq. lq. lq. col. lq. col. oil amor. lq. yel. nd. lf./aq. nd./aq. cr. col. lq. col. lq. col. lq. yel. oil col. lq. col. lq. lq. col. lf. yel. lf. or. cr. gas

Specific gravity

0.93520/4 1.07815/15 1.09220/20 1.04820/4 1.03420/4 1.03520/4

0.96479.7 1.03115/4 0.85320/20 0.86220/4 1.1624 0.953 1.07348/4 1.1400 0.86617 2.01520/4 1.2220 1.7680/4 0.7030/4 0.81020/4 0.77920/4 0.96220/4 0.94719/4 0.81020/4 0.9850/4 0.86520/0 1.32420/20 0.97718/4 0.80519/4 0.74520/4 0.94820 0.720−79 0.87520/4 0.86220 0.85720/4 1.752 0.89518/4 0.87220/4 0.7302 0.83020/4 1.038 0.93125

0.76721/4 0.77420/4 0.77720/4 0.82125/4

Melting point, °C

Boiling point, °C

207–8 206–7 d. 70 <−18 295 260 d. 5.5 93–4 30.8 10.9 35–6

subl.

55 71.5 72 15.5 −69 −96.9 116–7 <−20 44–5 −80 65–6 −34.4 52 −6.5 >360 −50 −12 6.5 23.9 −45 −103.7

−43.9 −85 −93.3 −58.2 −126.6 <−25 −73.5 d. 258–61 253 −51 −32 −29.7 7 −47 237–9 158 93–4 subl. 310 −44 −69 14.6

1.03 1.06315/4 131.5 96–8 51 −145

290–1 173–4 221–2765 235 190.8 202.8 202 308 314 316 189 170–1 d. 102.2 152.5 subl. 225761 14019 −640 1080.2 −21 61.3750 12.5–13 d. 11–12726 118–20 80–1 160–1 155–6 83.3 174750 134 165714 142 41–2 49–50 129–30 −34749 177 175–6 176–7 31915 193.3 185.3 174.0 232.9 167.9 d. 249–5315 188–90 190 173.4 228 204–611 22540 19516 expl. −23

Solubility in 100 parts Water sl. s. c. s. h. 0.3 c. i. 1.418 8.716 v. sl. s. v. sl. s. 2.5 0.5 1.8 i. i. i. 8.315 ∞25 18 i. 0.0225 i. v. s. sl. s. s. 45020 cc. s. 250020 cc. 0.2717 i. i. i. 3.620 s. v. sl. s. i. i. i. i. i. i. v. sl. s. i. i. i. i. 0.0119 212 i. i. i. i. s. ∞ sl. s. h. sl. s. sl. s. c. v. sl. s. sl. s. i. i. i. i. i. i. i. 0.05 d.

Alcohol

Ether

s. v. s. h. v. s.

v. sl. s. v. s. s. s. i.

0.0117 116 ∞ s. ∞30 ∞ ∞36

∞ s. ∞30 ∞ ∞36

s. ∞ ∞ s.

∞ ∞ s.

v. s. s. 230020 cc. s. v. s. 0.122 v. s. ∞ s. s. v. s. ∞ s. s. ∞ s. s. s. s. i. i. s. s. ∞ s. i. ∞ s. s. s.

v. s. s. s. 50020 cc. s. 500020 cc.

∞ s. s. v. s. ∞ s. ∞ ∞ s. s. s. s. i. s. s. ∞ i. ∞ s. s. s.

s. ∞ ∞ s.

∞ ∞ ∞ s.

s.

s.

s. s. h.

s. v. s. s.

Dibenzothiazyl-disulfide (2-,2′-) Dibensoyl methane Dibensyl-amine -aniline ketone phthalate (o-) succinate Dibromo-benzene (o-) (m-) (p-) -diphenyl (4-,4′-) Dibutyl-adipate (n-) (i-) -amine (n-) (i-) -p-aminophenol (s-) -aniline (n-) carbonate (n-) (i-) (s-) ether (n-) (i-) (s-) ketone (n-) (i-) malate (l-)(n-) oxalate (n-) phthalate (n-) tartrate (d-)(n-) (d-)(i-) Dichloro-acetic acid -acetone (αα-) -aniline (2-,5-) -anthraquinone (1-,3-) (1-,4-) (1-,5-) (1-,6-) (1-,8-) (2-,3-) (2-,6-) (2-,7-) -benzene (o-) (m-) (p-) -butane (n-)(1-,4-) -diphenyl (4-,4′-) -ethane (1-,2-) -naphthalene (β-)(1-,4-) (γ-)(1-,5-) -nitrobenzene (2-,5-) -pentane (1-,5-) -phenol (2-,4-) Dichloramine T (p-) Dicyandiamide Diethanolamine Diethyl adipate -amine -aminophenol (m-) -aniline sulfonic acid (m-) carbonate diethyl malonate Diethyl dimethyl malonate glutarate ketone malonate -malonic acid -naphthylamine (α-) (β-) oxalate phthalate (o-) sulfate sulfide

(C6H4NSC)2S2 (C6H5CO)2CH2 (C6H5CH2)2NH C6H5N(CH2C6H5)2 (C6H5CH2)2CO C6H4(CO2CH2C6H5)2 (CH2CO2CH2C6H5)2 C6H4Br2 C6H4Br2 C6H4Br2 BrC6H4C6H4Br (CH2CH2CO2C4H9)2 (CH2CH2CO2C4H9)2 (C2H5CH2CH2)2NH [(CH3)2CHCH2]2NH (C4H9)2NC6H4OH C6H5N(C4H9)2 CO(OC4H9)2 CO(OC4H9)2 CO(OC4H9)2 (C2H5CH2CH2)2O [(CH3)2CHCH2]2O [C2H5(CH3)CH]2O (C2H5CH2CH2)2CO [(CH3)2CHCH2]2CO C2H4O(CO2C4H9)2 (CO2C4H9)2 C6H4(CO2C4H9)2 (CHOHCO2C4H9)2 (CHOHCO2C4H9)2 Cl2CHCO2H Cl2CHCOCH3 Cl2C6H3NH2 C6H4:(CO)2:C6H2Cl2 C6H4:(CO)2:C6H2Cl2 C6H3Cl:(CO)2:C6H3Cl C6H3Cl:(CO)2:C6H3Cl C6H3Cl:(CO)2:C6H3Cl C6H4:(CO)2:C6H2Cl2 C6H3Cl:(CO)2:C6H3Cl C6H3Cl:(CO)2:C6H3Cl C6H4Cl2 C6H4Cl2 C6H4Cl2 ClCH2(CH2)2CH2Cl ClC6H4C6H4Cl ClCH2CH2Cl C10H6Cl2 C10H6Cl2 Cl2C6H3NO2 ClCH2(CH2)3CH2Cl Cl2C6H3OH CH3C6H4SO2NCl2 H2NC(:NH)NHCN HN(CH2CH2OH)2 (CH2CH2CO2C2H5)2 (C2H5)2NH (C2H5)2NC6H4OH (C2H5)2NC6H5 (C2H5)2NC6H4SO3H OC(OC2H5)2 (C2H5)2C(CO2C2H5)2 (CH3)2C(CO2C2H5)2 CH2(CH2CO2C2H5)2 (C2H5)2CO CH2(CO2C2H5)2 (C2H5)2C(CO2H)2 C10H7N(C2H5)2 C10H7N(C2H5)2 (CO2C2H5)2 C6H4(CO2C2H5)2 O2S(OC2H5)2 (C2H5)2S

232.46 224.25 197.27 273.36 210.26 346.36 298.32 235.92 235.92 235.92 312.02 258.35 258.35 129.24 129.24 221.33 205.33 174.23 174.23 174.23 130.22 130.22 130.22 142.23 142.23 246.30 202.24 278.34 262.30 262.30 128.95 126.98 162.02 277.10 277.10 277.10 277.10 277.10 277.10 277.10 277.10 147.01 147.01 147.01 127.02 223.10 98.97 197.06 197.06 192.01 141.04 163.01 240.11 84.08 105.14 202.24 73.14 165.23 149.23 229.29 118.13 216.27 188.22 188.22 86.13 160.17 160.17 199.28 199.28 146.14 222.23 154.18 90.18

cr. rhb./al. col. oil pr./al. cr. pr./al. lf./al. col. lq. col. lq. pl./al. mn. pr. col. lq. col. lq. col. lq. col. lq. lq. lq. col. lq. col. lq. col. lq. lq. lq. lq. lq. oil lq. col. lq. col. lq. pr. cr. lq. lq. nd. yel. nd. yel. nd. yel. nd. yel. nd. yel. nd. yel. nd. yel. nd. yel. nd. col. lq. col. lq. col. mn. lq. pr. col. lq. nd./al. lf./al. tri./al. col. lq. nd. cr. mn. pl. pr. col. lq. col. lq. rhb. oil cr. col. lq. col. lq. col. lq. syrup col. lq. col. lq. pr./aq. col. oil col. oil col. lq. col. lq. col. lq. col. lq.

1.50 1.028 25/25

20/4

1.956 1.952 20/4 2.26118 1.897 0.965 20/4 0.950 25 0.768 20/20 0.74125/4

180 78 −26 70–1 34–5 42–3 45–6 1.8 −6.9 87–8 164–5 −38 −20 −70

0.924 20/4 0.91915 0.769 20/20 0.76215 0.756 21 0.82718/4 0.805 21/4 1.038 20/4 0.986 20/4 1.045 21 1.09815 1.03175/4 1.560 25/25 1.23415

1.305 20/4 1.288 20/4 1.458 21 1.442 0/4 1.256 20/20 1.300 76/4 1.669 22 1.094 25/4 1.383 60/25 1.4014 1.09720/4 1.00920/4 0.712 15/15 0.934 20/4 20/4

0.975 0.985 20/4 0.994 25/25 1.025 21 0.816 19/4 1.055 20/4 1.005 1.026 1.079 20/4 1.121 25/25 1.172 25/4 0.837 20/4

−98 −5.9 −29.6 22–2.5 73–4 9.7(−4) 50 208–9 187.5 251 203–4 202–3 268–70 282 210–11 −17.6 −24.8 53 −38.7 148 −35.3 67–8 107 54.6 45 83 207–8 28 −21 −38.9 78 −34.4 270 d. −43 −24 −42 −49.8 125 −40.6 −25 −99.5

d. 219–2118 268–71250 >300 330.6 27412 23814 221–2 219755 218.6758 355–60 18314 278–80 159761 139–40 17010 262.8 207740 190 178–80 142.4 122.5 121 187.7 168.1 170–118 245.5 340 200–318 323–5 194.4 120 251

179 172766 174764 161–3 315–9 83.7 286–7 740 subl. 266 180–1 209–10 d. 270 748 239–41761 55.5759 276–80 216 759

126 230 196.7 237 101.7 198.9 d. 170–80 285–90 318 186 298–9 210 92–3 754

i. i. i. i. i. v. sl. s. i. i. i. i. i. i. i. ∞ v. sl. s. i. i. i. i.

4.420 s. v. s. h.

s. s. s.

s. s. s. s. 1.6 v. sl. s. h. ∞

s. s. s. s. 7125 ∞

∞ s.

∞ s.

∞ s.

∞

<0.05 i. i. i. <0.06 v. sl. s. i. 0.04 25 i. v. sl. s. ∞ v. sl. s. v. sl. s. i. i. i. i. i. i. i. i. i. i. i.

∞ ∞ ∞ s. ∞

∞ ∞ ∞ v. s. ∞

s. ∞

s. ∞

∞ s. s. i. v. sl. s. sl. s.

∞ s. s.

∞ s. v. s.

∞ s. v. s.

i. 0.90 i. i. i. i. 0.45 20 sl. s. 2.318 ∞ 0.4380 v. s. s. 1.412 s. i. i. i. 0.88 20 4.7 20 2.08 20 65 16 i. i. v. sl. s. i. i. 0.3120

v. sl. s. ∞ v. sl. s. s. v. s. h. s. v. s.

425 ∞

1.318 ∞ s. ∞

0.0118 v. sl. s. s. ∞

s.

s.

∞ ∞ ∞ v. s. ∞ ∞ v. s. ∞ ∞ ∞ ∞ s. ∞

∞ ∞ ∞ s. ∞ ∞ v. s. ∞ ∞ ∞ ∞ ∞ ∞

v. sl. s.

sl. s. sl. s.

s. s. v. s.

2-35

TABLE 2-2

Physical Properties of Organic Compounds (Continued ) Name

tartrate (d-) -toluidine (o-) (m-) (p-) Diethyleneglycol dinitrate Difluorodichloromethane Diglycerol Dihydroxy-dinaphthyl (α-) (-2,-2′,-1,-1′) -diphenyl (4-,4′-) -ethyl formal (β-) -naphthalene (1-,5-) (1-,8-) Dimethoxy-benzene (p-) -diphenylamine (4-,4′-) -ethyl adipate Dimethyl adipate -amine -aminoasobenzene (p-) -aminoethanol -aminophenol (m-) -aniline sulfonic acid (m-) (p-) carbonate ether -formamide fumarate glutarate glyoxime -naphthalene (1-,4-) (2-,3-) -naphthylamine (α-) (β-) oxalate phthalate (o-) sulfate sulfide tartrate (d-) -vinyl-ethenyl carbinol Dinaphthyl (αα-) -methane (αα′-) (β,β′-) Dinitro-anisole (1-)(2-,4-) -benzene (o-) (m-) (p-) sulfonic acid (2-,4-)(1-) -benzoic acid (2-,4-) (3-,5-) -benzophenone (4-,4′-) -diphenyl (4-,4′-) (2-,4′-) -naphthalene (1-,5-) (1-,8-) Dinitro-phenol (2-,3-) (2-,4-) (2-,6-) -salicylic acid (3-,5-) -stilbene (4-,4′-) -toluene (2-,4-) (3-,4-) (3-,5-) Dioxane Dipentene

2-36

Formula (CHOHCO2C2H5)2 CH3C6H4N(C2H5)2 CH3C6H4N(C2H5)2 CH3C6H4N(C2H5)2 O(CH2CH2ONO2)2 F2CCl2 [(HO)2C3H5]2O (HOC10H6)2 (HOC10H6)2 (HOC6H4)2 CH2(OCH2CH2OH)2 C10H6(OH)2 C10H6(OH)2 (CH3O)2C6H4 HN(C6H4OCH3)2 (CH2)4(CO2C2H4OCH3)2 [(CH2)2CO2CH3]2 (CH3)2NH C6H5N:NC6H4N(CH3)2 (CH3)2NCH2CH2OH (CH3)2NC6H4OH (CH3)2NC6H5 (CH3)2NC6H4SO3H (CH3)2NC6H4SO3HH2O OC(OCH3)2 CH3OCH3 HCON(CH3)2 (:CHCO2CH3)2 (CH2)3(CO2CH3)2 (CH3C:NOH)2 C10H6(CH3)2 C10H6(CH3)2 C10H7N(CH3)2 C10H7N(CH3)2 (CO2CH3)2 C6H4(CO2CH3)2 (CH3O)2SO2 (CH3)2S (CHOHCO2CH3)2 (CH3)2COHC⯗CCH:CH2 C10H7C10H7 (C10H7)2CH2 (C10H7)2CH2 CH3OC6H3(NO2)2 C6H4(NO2)2 C6H4(NO2)2 C6H4(NO2)2 (NO2)2C6H3SO3H3H2O (NO2)2C6H3CO2H (NO2)2C6H3CO2H (NO2C6H4)2CO (NO2C6H4)2 (NO2C6H4)2 C10H6(NO2)2 C10H6(NO2)2 (NO2)2C6H3OH (NO2)C6H3OH (NO2)C6H3OH (NO2)2C6H2(OH)CO2HH2O (NO2C6H4CH:)2 (NO2)2C6H3CH3 (NO2)2C6H3CH3 (NO2)2C6H3CH3 O < (CH2CH2)2 > O C13H10

Formula weight

Form and color

Specific gravity

Melting point, °C

Boiling point, °C

206.19 163.25 163.25 163.25 196.12 120.92 166.17 286.31 286.31 186.20 136.15 160.16 160.16 138.16 229.26 262.30 174.19 45.08 225.28 89.14 137.18 121.18 201.24 219.25 90.08 46.07 73.09 144.12 160.17 116.12 156.22 156.22 171.23 171.23 118.09 194.18 126.13 62.13 178.14 110.15 254.31 268.34 268.34 198.13 168.11 168.11 168.11 302.22 212.12 212.12 272.21 244.20 244.20 218.16 218.16 184.11 184.11 184.11 246.13 270.24 182.13 182.13 182.13 88.10 136.23

lq. lq. lq. lq. lq. gas lq. pl./al. nd./al. rhb./al. lq. pr./aq. nd. lf. cr. lq. col. lq. col. lq. yel./al. col. lq. nd. yel. lq. cr. pr. col. lq. gas lq. col. tri. lq. col. cr. lq. lf./al. col. oil col. cr. col. mn. col. lq. col. oil oil cr. lq. lf./al. pr./al. nd./al. col. mn. col. mn. col. rhb. col. mn. pr. cr./aq. mn. pr. col. nd. nd./al. mn. nd. rhb. yel. mn. yel. rhb. yel. rhb. pl./aq. yel. lf. nd. nd. mn. pr. col. lq. col. lq.

1.204 20/4

17

0.924 15.5 1.377 25/4 1.486 −30

−11.3 −155

280 208–9755 231–2 228–9

1.25 1.154 25 1.053

55/55

1.075 15.6 1.063 20/4 0.680 0/4 0.887 20/4 0.956 20/4 1.070 20/4 0.94525 1.08915.6 1.016 20/4 1.042 20 1.03970/70 1.14854 1.18925/25 1.3520/4 0.84621/4 1.32820/4 0.887 20/4

1.341 20 1.5918 1.575 20/4 1.625 18

1.445 1.474 1.681 20 1.683 24

1.321 71 1.259 111 1.277 111 1.033 20/4 0.865 18

300 218 270–2 −5.3 258–60 140 56 103 10–1 −96 116–7 85 2.5 d. 266 257 0.5 −138.5 −58.3 102 −37 240–6 <−18 104 46 54 −26.8 −83.2 61.5 160 109 92 94–5 117–8 89.8 173–4 106–8 179–80 204–5 189 233 93.5 216 170–2 144–5 114–5 63–4 173 d. 210–6 70 60–1 92–3 9.5–10.5

−29.2 220–3010 subl. subl. 264 d. 212.6 145–502 11518 7.4 d. 135756 265–8 193 89–90 −23.7 152.8 192 13050 264–6 265767 274.5711 304–5 163.3 280734 188.3 37.3 280 150 240–412 >360 319774 300–2 299777 subl.

subl. d. subl.

300 subl. 101.1 178

Solubility in 100 parts Water sl. s. i. i. i. i. 5.7 cc.26 s. h. i. i. sl. s. ∞ sl. s. sl. s. h. v. sl. s. i. 5 i. v. s. i. ∞ sl. s. h. i. s. s. h. i. 3700 cc.18 ∞ i. 20

0.06 i. i. i. i. 6 0.43 v. sl. s. i. s. 620 i. i. i. sl. s. h. 0.01 c. 0.399 0.18100 s. 1.8525 s h. i. i. i. i. i. sl. s. 0.5 c. s. h. s. c. i. 0.0322 i. sl. s. ∞ i.

Alcohol

Ether

∞ s. s.

∞ s. s.

s.

s. i. v. s. v. s. v. s.

s. s. v. s. s. v. s.

v. s. v. s. v. s.

s. s.

s. s.

s. s.

s. s.

v. sl. s. ∞ s.

v. sl. s. ∞ s.

sl. s.

sl. s.

v. s.

v. s.

sl. s. s. s. s.

s. s. s.

∞ s. 20015

∞ s.

s. h. 0.8 c. s. 1.520 1.921 320 0.1821 s. s. v. s.

s. v. s.

v. sl. s. sl. s.

1.520 v. s. h. 0.2 c. v. s. h. 420 s. h. v. s. v. sl. s. 1.215

v. s. v. s. h. s. v. s. v. sl. s. 916

s. h. s.

s. s.

Diphenyl -amine carbonate -chloroarsine -ethane ether guanidine -methane phenylenediamine (p-) succinate sulfide sulfone urea (uns.) Diphenylene oxide Dipropyl adipate (n-) -amine (n-) (i-) aniline (n-) carbonate (n-) ether (n-) (i-) ketone (n-) (i-) oxalate (n-) (i-) Disalicylal ethylenediamine Ditolyl guanidine (o-) Divinyl acetylene Docosane (n-) Dodecane (n-) Dulcitol Durene (1-,2-,4-,5-) Elaidic acid Eosine Ephedrine (l-) Epichlorhydrin (α-) Epidichlorohydrin (α-) Erythritol (dl-) tetranitrate Ethane Ethanol-amine formamide Ether Ethyl abietate acetate acetoacetate alcohol -amine hydrochloride aniline sulfonic acid (m-) anisate (p-) anthranilate (o-) benzene benzoate -benzyl-aniline bromide butyrate (n-) (i-) caprate (n-) Ethyl caproate (n-) caprylate (n-) chloride chloroacetate chlorocarbonate cinnamate (trans-) cyanoacetate formate furoate (α) heptoate hypochlorite iodide lactate

C6H5C6H5 C6H5NHC6H5 O(COC6H5)2 (C6H5)2AsCl (C6H5CH2)2 C6H5OC6H5 (C6H5NH)2C:NH (C6H5)2CH2 (C6H5NH)2C6H4 (CH2CO2C6H5)2 (C6H5)2S (C6H5)2SO2 (C6H5)2NCONH2 < (C6H4)2O (CH2CH2CO2C3H7)2 (C2H5CH2)2NH [(CH3)2CH]2NH C6H5N(C3H7)2 O(COCH2C2H5)2 (C2H5CH2)2O [(CH3)2CH]2O (C2H5CH2)2CO [(CH3)2CH]2CO (CO2CH2C2H5)2 [CO2CH(CH3)2]2 [HOC6H4CH:NCH2]2 (C7H7NH)2C:NH (H2C:CHC⯗)2 CH3(CH2)20CH3 CH3(CH2)10CH3 CH2OH(CHOH)4CH2OH (CH3)4C6H2 C8H17CH:CH(CH2)7CO2H C20H8O5Br4 C6H5CHOHCH(CH3)NHCH3 C2H3OCH2Cl CH2:CClCH2Cl CH2OH(CHOH)2CH2OH C4H6(ONO2)4 CH3CH3 HOCH2CH2NH2 HCONHCH2CH2OH (CH3CH2)2O C19H29CO2C2H5 CH3CO2C2H5 CH3COCH2CO2C2H5 CH3CH2OH C2H5NH2 C2H5NH2HCl C6H5NHC2H5 C2H5NHC6H4SO3H CH3OC6H4CO2C2H5 NH2C6H4CO2C2H5 C6H5C2H5 C6H5CO2C2H5 C6H5N(C2H5)CH2C6H5 C2H5Br C2H5CH2CO2C2H5 (CH3)2CHCO2C2H5 CH3(CH2)8CO2C2H5 CH3(CH2)4CO2C2H5 CH3(CH2)6CO2C2H5 CH3CH2Cl ClCH2CO2C2H5 ClCO2CH2CH3 C6H5CH:CHCO2H CH2(CN)CO2C2H5 HCO2CH2CH3 OC4H3CO2C2H5 CH3(CH2)5CO2C2H5 ClOCH2CH3 CH3CH2I CH3CH(OH)CO2C2H5

154.20 169.22 214.21 264.57 182.25 170.20 211.26 168.23 260.32 270.27 186.26 218.26 212.24 168.18 230.30 101.19 101.19 177.28 146.18 102.17 102.17 114.18 114.18 174.19 174.19 268.30 239.31 78.11 310.59 170.33 182.17 134.21 282.45 647.93 165.23 92.53 110.98 122.12 302.12 30.07 61.08 89.09 74.12 330.49 88.10 130.14 46.07 45.08 81.55 121.18 201.24 180.20 165.19 106.16 150.17 211.29 108.98 116.16 116.16 200.31 144.21 172.26 64.52 122.55 108.53 176.21 113.11 74.08 140.13 158.23 80.52 155.98 118.13

col. mn. col. mn. nd./al. rhb. col. pr. col. rhb. mn./al. col. pr. cr. lf./al. col. lq. nd./aq. rhb. lf./al. col. lq. col. lq. col. lq. yel. oil col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. cr. cr. lq. cr. lq. mn. mn. lf./al. col. cr. cr./et. lq. col. lq. tet. pr. lf./al. col. gas col. oil lq. col. lq. lq. col. lq. col. lq. col. lq. col. lq. mn. lq. nd./aq. lq. cr. col. lq. col. lq. yel. oil col. lq. col. lq. col. lq. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. lf. col. lq. yel. lq. col. lq. oil

0.992 73/4 1.160 20/20 1.272 14 1.583 40 0.978 50/50 1.073 20 1.001 26/4 1.119 15/15 1.248 25/4 1.276 0.979 20/4 0.739 20/4 0.722 22 0.910 20 0.968 22 0.744 21/0 0.725 21/0 0.822 20/4 0.806 20/4 1.038 0/0 1.34 1.1020/4 0.776 20/4 0.778 44/4 0.751 20/4 1.466 15 0.838 81/4 0.851 79/4 1.183 25/25 1.204 25 1.451 20/4 −88

0.546 1.022 20 1.169 25 0.708 25/4 1.020 20/20 0.901 20/4 1.025 20/4 0.789 20/4 0.689 15/15 1.216 0.963 20/4 1.103 25/25 1.117 20/4 0.867 20/4 1.052 15/15 1.034 18.5 1.431 20/4 0.879 20/4 0.871 20/4 0.859 28 0.873 20/20 0.878 17 0.917 6/6 1.159 20/4 1.138 20/4 1.049 20/4 1.062 20/4 0.923 20/4 1.117 21/4 0.872 20/20 1.013 −6/4 1.933 20/4 1.030 25/4

69–70 52.9 80 43–4 52–3 27 147–8 26–7 152 122–3 <−40 128–9 189 86–7 −20.3 −39.6 −61 −122 −60 −32.6 −51.7

254.9 302 302–6 d. 327 284 259 d. > 170 265 330 296–7 379 287–8 143–510 110–1 83.5743 245.4 168.2 91 69 144.2 123.7 213.5 190

125–6 178–9 44.5 −9.6 189 79–80 51–2 40 −25.6 126 61 −172 10.5 <−40 −116.3 −82.4 −45 −112 −80.6 108–9 −63.5 d. 294 7–8 13 −94.4 −34.6 −117.8 −93.3 −88.2 −20 −67.5 −45 −139 −26 −80.6 12 −22.5 −79 34 −66.1 expl. −105

85 224.515 214.5 290–53 193–5 288100 255 117756 94 329–31 expl. −88.6 171757 d. 34.6 2004 77.1 180755 78.4 16.6 204 269–70 266–8 136.2 211–2 28510 38.4 120–1 110–1 244.6758 165–6736 207–8753 13 144 94–5 271 208753 54760 195766 187–8 36752 72.4 155

i. 0.0325 i. 0.2 d. i. v. sl. s. v. sl. s. i. i. i. i. sl. s. h. v. sl. s. i. i. s. s. i. v. sl. s. sl. s. 0.2 0.43 v. sl. s. d. h. 0.0328 v. sl. s. i. i. i. 3.215 i. i. i. 5 <5 i. 60 i. c. 4.7 cc.20 ∞ ∞ 7.520 i. 8.515 1317 ∞ ∞ 24017 i. 2.1515 i. v. sl. s. 0.0115 0.0820 i. 1.060 0.6825 sl. s. 0.00220 i. i. 0.450 i. d. i. 225 1118 i. 0.02920 0.420 ∞

1020 5619.5 v. s. 20 s. s. 920 v. s.

6.620 s. s. s. v. s. ∞ sl. s. v. s. s. ∞

s. h. s. h. s. s. h. s. ∞ s. s.

s. v. s. s. ∞

∞ ∞ ∞ ∞

∞ ∞ ∞ ∞

s. h.

s.

4 h. v. s. v. sl. s. s. v. s. s. 500 ∞ ∞ sl. s. c. s. 150 cc. ∞

v. s. v. s. i. s. v. s.

s.

s. ∞ ∞ i. s. 1

∞ ∞ ∞ ∞ v. s. ∞ s. s. ∞ ∞ 18 ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞

∞ ∞ ∞ ∞ i. ∞ s. s. ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞

2-37

TABLE 2-2

Physical Properties of Organic Compounds (Continued ) Name

Formula

laurate mercaptan methacrylate naphthylamine (α-) naphthyl ether (α-) nitrate nitrite oleate palmitate pelargonate propionate salicylate (o-) stearate toluate (o-) (m-) toluene sulfonate (p-) toluidine (o-) (p-) urea valerate (n-) (i-) Ethylal Ethylene bromide bromohydrin chlorobromide chlorohydrin diamine oxide Ethylidene diacetate Eugenol (1-,4-,3-) (i-)(1-,3-,4-) Fenchyl alcohol (dl-) (d-)(α-) (i-)(l-) Ferric dimethyl-dithiocarbamate Fluorene Fluorescein Fluoro-dichloromethane -trichloromethane Formaldehyde (m-) (p-) Formamide Formanilide Formic acid Fructose Fuchsin Fulminic acid Fumaric acid (trans-) Furfural Furfuran Furfuryl acetate alcohol butyrate propionate Furoic acid G-acid, K salt (2-)(6-,8-) Na salt (2-)(6-,8-) Galactose (d-)(α-) Gallic acid (3-,4-,5-) Gamma acid (2-,8-,6-) Geraniol Glucose (d-)(α-) (d-)(β-) Glucuronic acid Glutam(in)ic acid (dl-)

CH3(CH2)10CO2C2H5 CH3CH2SH CH2:C(CH3)CO2C2H5 C10H7NHC2H5 C10H7OC2H5 C2H5ONO2 C2H5ONO C17H33CO2C2H5 CH3(CH2)14CO2C2H5 CH3(CH2)7CO2C2H5 CH3CH2CO2C2H5 HOC6H4CO2C2H5 CH3(CH2)16CO2C2H5 CH3C6H4CO2C2H5 CH3C6H4CO2C2H5 CH3C6H4SO3C2H5 CH3C6H4NHC2H5 CH3C6H4NHC2H5 C2H5NHCONH2 CH3(CH2)3CO2C2H5 (CH3)2CH(CH2)2CO2C2H5 CH2(OC2H5)2 H2C:CH2 BrCH2CH2Br BrCH2CH2OH ClCH2CH2Br ClCH2CH2OH H2NCH2CH2NH2 < (CH2)2 > O CH3CH(O2CCH3)2 C3H5C6H3(OH)OCH3 C3H5C6H3(OCH3)OH C10H17OH C10H17OH C10H17OH Fe[SSCN(CH3)2]3 (C6H4)2 > CH2 C20H12O5 FCHCl2 Cl3CF HCHO (CH2O)3 (CH2O)xxH2O HCONH2 HCONHC6H5 HCO2H CH2OH(CHOH)3COCH2OH C20H19N3HCl C:NOH HO2CCH:CHCO2H C4H3OCHO C4H4O CH3CO2CH2C4H3O C4H3OCH2OH C3H7CO2CH2C4H3O C2H5CO2CH2C4H3O C4H3OCO2H HOC10H5(SO3K)2 HOC10H5(SO3Na)2 C5H11O5CHO (HO)3C6H2CO2HH2O C10H5(NH2)(OH)SO3H C9H15CH2OH C5H11O5CHO C6H12O6H2O CHO(CHOH)4CO2H [CHNH2(CH2)2](CO2H)2

2-38

Formula weight

Form and color

Specific gravity

Melting point, °C

Boiling point, °C

228.36 62.13 114.14 171.23 172.22 91.07 75.07 310.50 284.47 186.29 102.13 166.17 312.52 164.20 164.20 200.25 135.20 135.20 88.11 130.18 130.18 104.15 28.05 187.88 124.98 143.43 80.52 60.10 44.05 146.14 164.20 164.20 154.24 154.24 154.24 416.47 166.21 332.30 102.93 137.38 30.03 90.08 (30.03) 45.04 121.13 46.03 180.16 337.84 43.03 116.07 96.08 68.07 140.13 98.10 168.19 154.16 112.08 380.46 348.26 180.16 188.13 239.24 154.24 180.16 198.17 194.14 147.13

oil lq. col. lq. oil cr. col. lq. lq. oil col. nd. col. lq. col. lq. col. lq. col. cr. lq. lq. pr./al. lq. lq. nd. col. lq. col. lq. lq. col. gas col. lq. col. lq. lq. col. lq. col. lq. lq. col. lq. oil oil col. cr. col. pr. col. cr. cr. cr./al. yel. red gas col. lq. gas wh. amor. lq. mn. col. lq. nd./aq. red

0.868 13/4 0.839 20/4 0.913 15.6 1.060 20/4 1.061 20/20 1.100 25/4 0.900 15.5 0.867 25 0.858 25/4 0.866 17.5 0.891 20/4 1.136 15/4 0.848 36.3 1.032 25/25 1.030 20/20 1.166 48/4 0.948 25/4 0.942 25/4 1.213 18 0.877 20 0.867 20/4 0.824 25/4 0.57−102/4 2.180 20/4 1.772 20/4 1.689 19 1.213 20/4 0.900 20/20 0.887 7/4 1.061 12 1.070 15/15 1.091 15/15 0.935 40 0.964 20/4 0.961

−10.7 −121

269 36–7 118 303723 276.4 87–8 17 216–815 19110 227–8757 99.1 233–4 20110 227 231750 221.3 215–6 217

col. pr. lq. col. lq. col. oil oil col. lq. col. lq. mn. pr. cr. cr. pr. mn./aq. cr. col. lq. rhb. cr. cr. cr./aq.

1.635 20/4 1.159 20/4 0.937 20/4 1.118 20/4 1.129 25/4 1.053 20/4 1.109 20/4

1.203 0/4 1.4260 1.494 17.2 0.815 −20 1.1765 1.139 20/4 1.147 15/15 1.220 20/4 1.669 17.5 1.22

5.5 −102 <−15 24–5 −44.5 −72.6 1.3 33.4(31) <−10 33–4 <−15 92 −91.2 −99.3 −66.5 −169 10 −16.6 −69 8.5 −111.3 18.85 10.3 −10 35 45–7 61–2 d. 100–30 115–6 d. > 290 −127 −92 64 150–60 2 47 8.6 95–105 d. >200 286–7 −38.7

133–4

1.694 4/4 0.883 15 1.544 25 1.562 18/4 1.460

145.5 135 89 −103.9 131.5 150.3 106.7 128.8 117.2 13.5747 168740 253.5 267.5 201 201–2 201–2 ign. >150 293–5 14.5 24.9 −21 114.5759 subl. 193 216120 100.8

290 161.7760 31–2756 175–7 169.5752 212–3 195–6 230–2

165.5 d. 220 <−15 146 150 154 199 d.

230 d.

Solubility in 100 parts Water

Alcohol

Ether

i. 1.5 i. i. i. 1.355 v. sl. s. i. i. i. 2.420 i. i. i. i. i. i. i. v. s. 0.2425 0.1720 918 26 cc.0 0.4380 sl. s. 0.6980 ∞ ∞ ∞ sl. s. v. sl. s. v. sl. s. sl. s. sl. s. i. v. sl. s. i. v. sl. s. h. i. i. v. s. 2125 20–3018 ∞ sl. s. ∞ v. s. 0.3

s. s. s. s. s. ∞ ∞ ∞ s. ∞ ∞ ∞ s. ∞ ∞ s.

∞ s. s. s. s. ∞ ∞ ∞ s. ∞ ∞ ∞ s. ∞ ∞ s.

80 ∞ ∞ ∞ 360 cc. ∞ s.

i. ∞ ∞ ∞ s. ∞

∞ ∞ ∞ ∞ ∞ ∞

∞ 0.3 v. s.

s.

s.

s. h. s. h. s. ∞ v. s. s. i. ∞ v. s. ∞ 818 s.

s.

0.717 9.113 i. i. ∞ v. sl. s. v. sl. s. 3.615 825 3420 10.30 113

5.830 ∞ s. s. s. s. s. s.

0.725 ∞ s. s. s. ∞ ∞ s.

0.640 2815

2.515

i. 8217.5 15415 v. s. 1.5 20

∞ sl. s.

∞ i.

v. sl. s.

v. sl. s.

∞ ∞

s. ∞ v. s. s. i. v. sl. s. s. i.

Glutaric acid Glycerol acetate (mono-) (di-) nitrate (mono-) (α-) (β-) dinitrate (1-,3-) Glyceryl triacetate tribenzoate tributyrate tricaprate tricaproate tricaprylate trilaurate trimyristate trinitrate trinitrite trioleate tripalmitate tristearate Glycide Glycine, Glycocoll Glycol diacetate dibenzoate dibutyrate dicaprylate diformate dilaurate dinitrate dinitrite dipalmitate dipropionate ether formal formate (mono-) Glycolic acid Guaiacol (o-) Guanidine H-acid, Na salt (1-,8-,3-,6-) Heptacosane (n-) Heptane (n-) (i-)

Heptoic acid aldehyde Heptyl acetate (n-) alcohol (n-) mercaptan Hexachloro-benzene -ethane Hexacosane (n-) Hexadecane (n-) Hexaethylbenzene Hexamethylbenzene Hexamethylene-diamine -diisocyanate -glycol tetramine Hexane (n-) (i-) (neo-)

CH2(CH2CO2H)2 CH2OHCHOHCH2OH C5H10O4 (CH3CO2)2C3H5OH CH2OHCHOHCH2NO3 CH2OHCHNO3CH2OH CHOH(CH2ONO2)2 (CH3CO2)3C3H5 (C6H5CO2)3C3H5 (C2H5CH2CO2)3C3H5 [CH3(CH2)8CO2]3C3H5 [CH3(CH2)4CO2]3C3H5 [CH3(CH2)6CO2]3C3H5 [CH3(CH2)10CO2]3C3H5 [CH3(CH2)12CO2]3C3H5 CH2NO3CHNO3CH2NO3 CH2NO2CHNO2CH2NO2 (C17H33CO2)3C3H5 [CH3(CH2)14CO2]3C3H5 [CH3(CH2)16CO2]3C3H5 C2H3OCH2OH NH2CH2CO2H CH2OHCH2OH (CH3CO2CH2)2 (C6H5CO2CH2)2 (C3H7CO2CH2)2 (C7H15CO2CH2)2 (HCO2CH2)2 (C11H23CO2CH2)2 (O2NOCH2)2 (ONOCH2)2 (C15H31CO2CH2)2 (C2H5CO2CH2)2 (HOCH2CH2)2O < OCH2CH2OCH2 > HCO2CH2CH2OH HOCH2CO2H CH3OC6H4OH NH:C(NH2)2 C10H8O7NS2Na1a H2O CH3(CH2)25CH3 CH3(CH2)5CH3 (CH3)2CH(CH2)3CH3 C3H7CH(CH3)C2H5 (CH3)3CCH2C2H5 [(CH3)2CH]2CH2 (CH3)2C(C2H5)2 (C2H5)3CH (CH3)3CCH(CH3)2 CH3(CH2)5CO2H CH3(CH2)5CHO CH3CO2CH2(CH2)5CH3 CH3(CH2)5CH2OH [(CH3)2CH]2CHOH (C2H5CH2CH2)2CHOH CH3CH(SH)C5H11 C6Cl6 CCl3CCl3 CH3(CH2)24CH3 CH3(CH2)14CH3 C6(C2H5)6 C6(CH3)6 NH2(CH2)6NH2 OCN(CH2)6NCO HO(CH2)6OH (CH2)6N4 CH3(CH2)4CH3 (CH3)2CH(CH2)2CH3 (CH3)3CC2H5 (CH3)2CHCH(CH3)2 (C2H5)2CHCH3

132.11 92.09 134.13 176.17 137.09 137.09 182.09 218.20 404.40 302.36 554.83 386.51 470.67 638.98 723.14 227.09 179.09 885.40 807.29 891.45 74.08 75.07 62.07 146.14 270.27 202.24 314.45 118.09 426.66 152.07 120.07 538.87 174.19 106.12 74.08 90.08 76.05 124.13 59.07 368.31 380.72 100.20 100.20 100.20 100.20 100.20 100.20 100.20 100.20 130.18 114.18 158.24 116.20 116.20 116.20 132.26 284.80 236.76 366.69 226.43 246.42 162.26 116.20 168.19 118.17 140.19 86.17 86.17 86.17 86.17 86.17

col. cr. col. lq. col. oil col. lq. col. pr. lf. oil col. lq. nd. col. lq. col. cr. col. lq. col. lq. col. nd. lf. yel. oil yel. lq. col. oil col. nd. col. pr. col. lq. mn. col. lq. col. lq. rhb./et. col. lq. lq. lq. amor. yel. lq. lq. nd. lq. lq. lq. lq. nd./aq. pr. col. cr. cr. col. cr. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. lq. lq. mn. rhb. cr. lf. pr./al. pl./al. lf. lq. nd./aq. col. rhb. col. lq. lq. lq. lq. lq.

1.429 15 1.260 50/4 1.20 20/4 1.178 15/15 1.4015 1.4015 1.4715 1.16117/4 1.228 12 1.032 20/4 0.921 40/4 0.987 20/4 0.954 20/4 0.894 60/4 0.885 60/6 1.601 15 1.291 10/16 0.915 15 0.866 80/4 0.862 80/4 1.114 16/16 1.161 1.113 19/4 1.109 14/4 1.024 0

97.5 17.9 40 58–9 54 <−30 −78 75–6 <−75 31(25) −25 8.3(−21) 45–6 56.5 13.3(2) −4 65.1 70.8(55)

20020 290 158165 175–640 155–60 155–60 146–815 258–9 d. 305–9

16015 150 sl. d. 24018 310–200.1 166 sl. d.

232–6 d. −15.6 −31 73–4

197.4 190.5 >360 240

22 1.482 21/2 1.216 0 1.045 25 1.118 20/20 1.060 20/4 1.199 15/4 1.140 15/15 0.780 60/4 0.684 20/4 0.679 20/4 0.687 20/4 0.674 20/4 0.675 20/4 0.693 20/4 0.698 20/4 0.690 20/4 0.918 20 0.850 20/ᐉ 0.874 16/16 0.824 20/4 0.829 20/4 0.820 20/4 0.835 20 2.044 24 2.091 20/4 0.779 57/4 0.774 20/4 0.831 130/4 1.0428 20/4

0.659 0.654 20/4 0.649 20/20 0.662 20/4 0.664 20/4

52–4 −20 <−15 71–2 −10.5 79(63) 28.3 50 59.5 −90.6 −118.2 −119.4 −125 −119.4 −135.0 −118.7 −25 −10 −42 34.6 −37 228–31 186–7 56.6 18.5 130 166 42 42 subl. −94 −153.7 −98.2 −129.8 −118

174 18820 expl. 114 96–8 2600.1 211–2 244.8 75–6 180 d. 205 27015 98.4760 90.0 91.8 79.1 80.8 86.0 93.5 80.8 221–2 155 191.5 759 175 756 140 156 174–5 765 309742 186777 26215 287.5 298.3 265 204–5 143–420 250 69 60.2 49.7 58.0760 63.2

63.920 ∞ v. s. s. 7015 7.1715 i. i. i. i. i. i. i. 0.1820 d. i. i. i. ∞ 23 c. ∞ 14.322 i. i. i. v. sl. s. i. 0.9225 i. i. sl. s. ∞ ∞ ∞ ∞ 1.715 v. s. 0.1720 i. 0.00515 i. i. i. i. i. i. i. 0.2515 0.0220 i. 0.18 25 v. sl. s. i. i. i. 0.00522 i. i. i. i. v. s. d. s. 8112 0.01415 i. i. i. i.

v. s. ∞ v. s. s. v. s. v. s. v. s. ∞ s. h. s. s. h. s. s. sl. s. c. s. 5020 d. sl. s. 0.00421 s. h. ∞ 0.1 c. ∞ ∞

v. s. i. sl. s. sl. s. v. sl. s. sl. s. v. s. ∞ s. s. v. s. s. s. v. s.

v. s.

∞ s. v. s. v. s. s. h. ∞ i. 1.0 ∞ s. v. s.

v. s. s. s. d. s. ∞ ∞

v. s. ∞ s. s. ∞ i.

9025 v. s. s.

v. s. v. s.

sl. s. s. s. s. s. s. s. s. s. ∞ s. ∞ ∞ s.

∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ s. ∞ s. ∞ ∞ s.

v. sl. s. h. v. s. v. sl. s. ∞ 0.7525 0.20 s. d. s. 3 5033

s. h. v. s. ∞ 825 v. s. sl. s. h. v. sl. s. ∞ s. s. s. s.

TABLE 2-2

Physical Properties of Organic Compounds (Continued ) Name

Hexyl acetate (n-) alcohol (n-) formate (n-) resorcinol (2-,4-) Hippuric acid Histidine (l-) Homophthalic acid (o-) Hydracrylic acid Hydro-cyanic acid -quinone (p-) Hydroxy-benzaldehyde (p-) -benzanilide (o-) -quinoline (2-)(α-) (8-)(o-) Indigo White Indole Indoxyl Iodo-benzene -phenol (p-) Iodoform Ionone (α-) (β-) Irone (β-) Isatin Isoprene Ketene Koch acid (1-)(3-,6-,8-) Lactic acid (dl-) anhydride Lactide (dl-) Lactose Lauric acid Laurone Lauryl alcohol Lead tetraethyl tetramethyl Lecithin (protagon) Lepidine (py-4) Leucine (l-) Levulinic acid Limonene (d- or l-) Linalool (d- or l-) Linalyl acetate Linoleic acid Maleic acid anhydride Malic acid (dl-) (d- or l-) Malonic acid Maltose Mandelic acid (dl-) Mannitol (d-) Mannose (d-) Margaric acid Mellitic acid Menthol (l-)(α-) Mercapto-benzothiazole (2-) -thiazoline (2-) Mercuric cyanide fulminate Mesityl oxide Mesitylene (1-,3-,5-) Metanilic acid (m-) Methane

2-40

Formula CH3CO2(CH2)5CH3 CH3(CH2)4CH2OH (CH3)2CHC(CH3)2OH (CH3)2COHCH2C2H5 HCO2CH2(CH2)4CH3 CH3(CH2)5C6H3(OH)2 C6H5CONHCH2CO2H C6H9O2N3 HO2CC6H4CH2CO2H HOCH2CH2CO2H HCN C6H4(OH)2 HOC6H4CHO HOC6H4CONHC6H5 C9H6NOH C9H6NOH [C6H4(CO)(NH)C:]2 C16H12O2N2 C8H7N C8H6NOH C6H5I IC6H4OH HCI3 C10H16:CHCOCH3 C10H16:CHCOCH3 C14H22O C6H4 < (CO)(N) > COH CH2:CHC(CH3):CH2 H2C:CO C10H4(NH2)S3O9HNa2 CH3CH(OH)CO2H C6H10O5 C6H8C4 C12H22O11H2O CH3(CH2)10CO2H [CH3(CH2)10]2CO CH3(CH2)10CH2OH Pb(CH2CH3)4 Pb(CH3)4 C42H48O9PN C9H6NCH3 (CH3)2CHCH2CH(NH2)CO2H CH3CO(CH2)2CO2H C10H16 C10H17OH CH3CO2C10H17 C17H31CO2H HO2CCH:CHCO2H < (CHCO)2 > O HO2CCH2CH(OH)CO2H HO2CCH2CH(OH)CO2H H2C(CO2H)2 C12H22O11H2O C6H5CH(OH)CO2H CH2OH(CHOH)4CH2OH CH2OH(CHOH)4CHO CH3(CH2)15CO2H C6(CO2H)6 C10H19OH < C6H4N:C(SH)S > < CH2N:C(SH)SCH2 > Hg(CN)2 Hg(ONC)2a H2O (CH3)2C:CHCOCH3 C6H3(CH3)3 H2NC6H4SO3H CH4

Formula weight

Form and color

Specific gravity

144.21 102.17 102.17 102.17 130.18 194.26 179.17 155.16 180.15 90.08 27.03 110.11 122.12 213.23 145.15 145.15 262.26 264.27 117.14 133.14 204.02 220.02 393.78 192.29 192.29 206.32 147.13 68.11 42.04 427.34 90.08 162.14 144.12 360.31 200.31 338.60 186.33 323.45 267.35 778.08 143.18 131.17 116.11 136.23 154.24 196.28 280.44 116.07 98.06 134.09 134.09 104.06 360.31 152.14 182.17 180.16 270.44 342.17 156.26 167.24 119.20 252.65 293.65 98.14 120.19 173.18 16.04

col. lq. col. lq. lq. lq. lq. col. nd. rhb. lf./aq. cr./aq. syrup lq. cr. nd./aq. pr./al. pr./al. pr. cr. gray lf./aq. yel. pr. col. lq. nd./aq. yel. hex. col. oil col. oil col. oil yel. red col. lq. col. gas cr. hyg. yel. oil tri./al. col. rhb. col. nd. pl. lf. col. lq. col. lq. wax lq. cr. lf. lq. col. oil col. lq. yel. oil mn. cr. col. cr. col. cr. col. tri. col. nd. rhb./aq. col. rhb. rhb. col. pl. nd./al. col. cr. nd. cr. cr. cr./aq. lq. col. lq. col. nd. gas

0.890 0/0 0.820 20/20 0.821 20/0 0.809 20/4 0.898 0 1.371 20/4

0.697 18 1.332 15 1.129 130

1.35

1.824 25/4 1.857 112 4.008 17 0.930 20 0.944 20 0.939 20

Melting point, °C −51.6 −14 −107 68–70 187–8 d. 287 175–80 −12 170.3 116–7 135 199–200 75–6 390–2 52 85 −28.5 93–4 119

0.681 20/4

200–1 −120 −151

1.249 15/4

16.8

10/4

0.862 1.525 20 0.869 50/4 0.809 69/4 0.831 24/4 1.659 18/4 1.995 20/4 1.086 20 1.29318 1.140 20/20 0.842 20/4 0.868 20 0.895 20 0.903 18/4 1.609 1.5 1.601 20/4 1.595 20/4 1.631 15 1.540 17 1.300 20/4 1.489 20/4 1.539 20/4 0.853 60 15/15

0.890 1.4220/4 1.50 4.003 22 4.4 0.858 20/4 0.865 20/4

0.415 −164

124.5 202 48(44) 69–70 24 −136 −27.5 150–200 d. 9–10 295 33.5 −96.9 −9.5 130.5 57–60 128–9 99–100 130–5 d. d. 118.1 166 132 60–1 286–8 42–3 179 106 d. 320 expl. −59 −45(−52) d. −182.6

Boiling point, °C 169.2 157.2 120–1 123762 153.6 1797 d. d. 25–6 285730 subl. d. subl. 266.6752 subl. 253–4 110 188.6 d. subl. 136.117 14018 14416 subl. 34 −56 12214 d. 250 255757 d. 225100 255–9 152291 110760 261–3 subl. 245–6 177 198–200 220762 d. 229–3016 135 d. 202 150 d. 140 d. d. 290–53 227100 d. 212 d.

130750 164.8 −161.4

Solubility in 100 parts Water i. 0.620 v. sl. s. v. sl. s. 0.05 0.420 s. s. h. ∞ 615 1.3831 v. sl. s. h. s. h. v. sl. s. c. i. i. s. h. s. 0.03420 sl. s. 0.0125 sl. s. sl. s. v. sl. s. s. h. i. d. 7.220 ∞ v. sl. s. v. sl. s. 1710 i. i. i. i. i. i. sl. s. 2.218 v. s. i. v. sl. s. v. sl. s. i. 7925 16.380 14426 v. s. 13816 10825 1620 1314 24817 i. v. s. 0.04 c. i. 1.660 12.515 0.0712 320 i. 215 0.420 cc.

Alcohol

Ether

v. s. ∞ ∞ ∞ ∞ v. s. s. h. v. sl. s. v. s.

v. s. ∞ ∞ ∞ ∞ s. 0.2518 i. sl. s.

∞ v. s.

∞ v. s.

s. v. s. s. i. s. s. h. s. s. v. s. 1.517 ∞ ∞ v. s. v. s. h. ∞ d.

s. v. s. sl. s. i. s. s. s. ∞ v. s. 13.625 ∞ ∞ v. s. sl. s. ∞ s.

∞ s. v. sl. s. c. i. s. i. c. s. sl. s. ∞ s. h. ∞

∞ s.

s. ∞ ∞ s. h. ∞

v. s. ∞ s. ∞ ∞ 7030

v. s. ∞ ∞ ∞ ∞ 825

v. s. v. s. 4225 v. sl. s. c. s. 0.0114 v. sl. s. 3228 v. s. v. s. s.

v. s. 8.415 815 i. s. i. i. v. s.

s. ∞ s. v. sl. s. 4720 cc.

i. s.

v. s. sl. s.

∞ ∞ v. sl. s. 10410 cc.

Methoxy-methoxyethanol Methyl acetate acrylic acid (α-) alcohol -amine -amine hydrochloride aniline anthracene (α-) (β-) anthranilate (o-) anthraquinone (2-) benzoate benzylaniline bromide butyrate (n-) (i-) caprate caproate (n-) caprylate cellosolve chloride chloroacetate chloroformate cinnamate cyclohexane ethyl carbonate ethyl ketone ethyl oxalate formate furoate glucamine glycolate heptoate hypochlorite iodide lactate laurate mercaptan methacrylate myristate naphthalene (α-) (β-) nitrate nitrite nonyl ketone (n-) oleate orange palmitate phosphine propionate propyl ketone (n-) salicylate (o-) stearate toluate (o-) (m-) (p-) Methyl toluidine (o-) (m-) (p-) valerate (n-) (i-) vinyl ketone Methylal Methylene-bis-(phenyl-4-isocyanate) bromide chloride dianiline iodide Michler’s hydrol (p-,p′-) ketone Morphine Mucic acid

CH3(OCH2)2CH2OH CH3CO2CH3 CH2:C(CH3)CO2H CH3OH CH3NH2 CH3NH2HCl C6H5NHCH3 C6H4:(CH)2:C6H3CH3 C6H4:(CH)2:C6H3CH3 NH2C6H4CO2CH3 C6H4:(CO)2:C6H3CH3 C6H5CO2CH3 C6H5N(CH3)CH2C6H5 CH3Br CH3(CH2)2CO2CH3 (CH3)2CHCO2CH3 CH3(CH2)8CO2CH3 CH3(CH2)4CO2CH3 CH3(CH2)6CO2CH3 CH3OCH2CH2OH CH3Cl ClCH2CO2CH3 ClCO2CH3 C6H5CH:CHCO2CH3 CH2 < (CH2CH2)2 > CHCH3 CH3OCOOC2H5 CH3.COC2H5 CH3OCOCO2C2H5 HCO2CH3 C4H3OCO2CH3 CH2OH(CHOH)4CH2NHCH3 HOCH2CO2CH3 CH3(CH2)5CO2CH3 ClOCH3 CH3I CH3CH(OH)CO2CH3 CH3(CH2)10CO2CH3 CH3SH CH2:C(CH3)CO2CH3 CH3(CH2)12CO2CH3 C10H7CH3 C10H7CH3 CH3ONO2 CH3ONO CH3(CH2)8COCH3 C17H33CO2CH3 (CH3)2NC6H4N2C6H4SO3Na CH3(CH2)14CO2CH3 CH3PH2 CH3CH2CO2CH3 CH3COCH2CH2CH3 HOC6H4CO2CH3 CH3(CH2)16CO2CH3 CH3C6H4CO2CH3 CH3C6H4CO2CH3 CH3C6H4CO2CH3 CH3C6H4NHCH3 CH3C6H4NHCH3 CH3C6H4NHCH3 CH3(CH2)3CO2CH3 (CH3)2CHCH2CO2CH3 CH3COCH:CH2 HCH(OCH3)2 (OCNC6H4)2CH2 CH2Br2 CH2Cl2 (C6H5NH)2CH2 CH2I2 [(CH3)2NC6H4]2CHOH [(CH3)2NC6H4]2CO C17H19O3NH2O (CHOHCHOHCO2H)2

106.12 74.08 86.09 32.04 31.06 67.52 107.15 192.25 192.25 151.16 222.23 136.14 197.27 94.95 102.13 102.13 186.29 130.18 158.23 76.09 50.49 108.53 94.50 162.18 98.18 104.10 72.10 132.11 60.05 126.11 195.21 90.08 144.21 66.49 141.95 104.10 214.34 48.10 100.11 242.39 142.19 142.19 77.04 61.04 170.29 296.48 327.33 270.44 48.03 88.10 86.13 152.14 298.49 150.17 150.17 150.17 121.18 121.18 121.18 116.16 116.16 70.09 76.09 250.25 173.86 84.94 198.26 267.87 270.36 268.35 303.35 210.14

lq. col. lq. pr. col. lq. col. gas pl./al. lq. lf./al. col. lf. col. lq. col. nd. col. lq. lq. gas col. lq. col. lq. lq. col. lq. col. lq. col. lq. gas col. lq. col. lq. cr. col. lq. lq. col. lq. lq. lq. col. lq. lq. lq. gas col. lq. lq. lq. gas lq. cr./al. oil mn. lq. gas col. oil oil red pd. col. cr. gas col. lq. col. lq. col. lq. col. cr. col. lq. col. lq. cr. lq. lq. lq. lq. col. lq. lq. col. lq. lq. col. lq. col. lq. cr. col. lq. gn. lf./al. pr./al. pd.

1.038 25 0.924 20/4 1.015 20/4 0.792 20/4 0.699 −11 1.23 0.989 20/4 1.047 99.4 1.181 0/4 1.168 19/4 1.087 25/25 1.732 0/0 0.898 20/4 0.891 20/4 0.904 0/0 0.887 18 0.965 20/4 0.952 0 1.236 20/4 1.236 15 1.042 36/0 0.769 20/4 1.002 27 0.805 20/4 1.156 0/0 0.974 20/4 1.179 21/4

<−70 −98.7 15–16 −97–8 −92.5 226–8 −57 86 207 24 176–7 −12.5 9.2 −93 <−95 −84.7 −18 −40 −97.7 −32.7 33.4 −126.3 −14.5 −85.9 −99.8

18

1.168 0.881 15/4 20/4

2.279 1.090 19

0.896 0 0.950 15.6 1.025 14/4 0.994 40/4 1.203 25 0.991 15 0.828 20/20 0.879 18

−64.4 5 −121 −48 18–9 −19 35–6 expl. 13.5 30–1

20/4

0.915 0.812 15/15 1.182 25/25 1.073 15 1.066 15 0.973 15 0.935 55/4 0.895 15/4 0.881 20/4 0.836 20/4 0.866 15/4 1.222 30 2.495 20/4 1.336 20/4 3.325 20/4 1.317

−87.5 −77.8 −8.3 38–9 <−50 33–4

−91 −104.8 −52.8 −96.7 65 5.7 96–7 174 254 d. 206–14

167.5 57.1 161–3 64.7 −6.7 758 23015 195.5 135.5 15 subl. 198–9 305–6 4.5 758 102.3 92.6 223–4 149.5 192–4 124–5 −24 130740 71–2 263 101 109.2 79.6 173.7 32 181.3 151.2 172–3 12726 42.4 144.8 14818 5.8752 100.3 295715 244.6 241–2 65 −12 228 190–110 19615 −14759 79.7 102 222.2 21515 213 215 217 206–7 206–7 211761 127.3 116.7764 81 42–3 210–213 98.5756 40–1 208–9 d. 180 d. >360 d.

∞ 3322 s. h. ∞ v. s. v. s. 0.0125 i. i. sl. s. i. 0.0230 i. v. sl. s. 1.7 v. sl. s. i. i. i. ∞ 28016 cc. v. sl. s. d. i. i. i. 3510 i. 3020 i.

∞ ∞ ∞ v. s. 23 h. s.

∞ ∞ ∞

v. sl. s. s. s. ∞ s. s. ∞ ∞ ∞ ∞ ∞ ∞ v. s. ∞ ∞ v. s. s. ∞ ∞ v. s. ∞ ∞

v. sl. s. s. s. ∞ s. s. ∞ ∞ ∞ ∞ ∞ ∞ v. s. ∞ ∞ v. s. s. ∞ ∞ v. s.

∞ s.

∞ s.

v. s.

v. s.

v. s. v. s. s. s. s. ∞

v. s. v. s. s. s. s. ∞

s. sl. s. ∞ ∞ ∞ s. ∞

∞ ∞ ∞ s. ∞

v. s. ∞ ∞ ∞ ∞ ∞

v. s. ∞ ∞ ∞ ∞ ∞

∞ d. ∞ ∞ s. ∞ s. h. sl. s. sl. s. i.

∞

i. ∞

∞

i. 1.815 ∞ i. s. i. i. i. i. sl. s. i. i. 0.2 c. i. i. 0.520 v. sl. s. 0.0730 i. i. i. i. i. i. i. v. sl. s. v. sl. s. >85 33 d. 1.170 220 i. 1.420 i. i. 0.0220 0.3314

s.

∞ ∞ s. ∞ s. v. sl. s. s. i.

TABLE 2-2

Physical Properties of Organic Compounds (Continued ) Name

Mustard gas Myricyl alcohol Myristic acid Myristyl alcohol Naphthalene disulfonic acid (1-,5-) (1-,6-) sulfonic acid (α-) (β-) Naphthasultam (1-,8-) disulfonate Na (1-,8-) (2-,4-) Naphthoic acid (α-) (β-) Naphthol (α-) (β-) sulfonic acid (α-)(1-,2-) (β-)(2-,6-) Naphthyl acetate (α-) (β-) amine (α-) (β-) amine hydrochloride (α-) (β-) amine sulfonic acid (1-,4-) (1-,5-) (1-,7-) (1-,8-) (2-,5-) (2-,6-) (2-,7-) isocyanate (α-) Nicotine Nicotinic acid (3-) (i-)(4-) Nitro-acetanilide (p-) -acetophenone (m-) -aminoanisole (4-,1-,2-) (5-,1-,2-) (3-,1-,4-) -aminophenol (4-,2-,1-) -aniline (o-) (m-) (p-) -anisole (o-) (p-) -anthraquinone (α-) -anthraquinone sulfonic acid (1-,5-) -benzal chloride (m-) -benzaldehyde (m-) Nitro-benzene -benzidine (2-) -benzoic acid (o-) (m-) (p-) -benzyl alcohol (m-) -benzyl bromide (p-) -chlorotoluene (1-,2-,6-) -cresol (1-,3-,4-) -cymene (1-,2-,4-) -dimethylaniline (o-) (m-) (p-) -diphenyl (o-) (p-) -diphenylamine (o-) -guanidine

2-42

Formula (ClCH2CH2)2S C31H63OH(?) CH3(CH2)12CO2H CH3(CH2)12CH2OH C10H8 C10H6(SO3H)2 C10H6(SO3H)2 C10H7SO3H2H2O C10H7SO2HH2O C10H7O2NS C10H5O8NS3Na22H2O C10H4O8NS3Na38a H2O C10H7CO2H C10H7CO2H C10H7OH C10H7OH HOC10H6SO3H HOC10H6SO3H CH3CO2C10H7 CH3CO2C10H7 C10H7NH2 C10H7NH2 C10H7NH2HCl C10H7NH2HCl NH2C10H6SO3H NH2C10H6SO3HH2O NH2C10H6SO3HH2O NH2C10H6SO3HH2O NH2C10H6SO3H NH2C10H6.SO3HH2O NH2C10H6SO3HH2O C10H7N:CO C10H14N2 C5H4NCO2H C5H4NCO2H CH3CONHC6H4NO2 CH3COC6H4NO2 NO2C6H3(OCH3)NH2 NO2C6H3(OCH3)NH2 NO2C6H3(OCH3)NH2 NO2C6H3(NH2)OH NO2C6H4NH2 NO2C6H4NH2 NO2C6H4NH2 CH3OC6H4NO2 CH3OC6H4NO2 C6H4:(CO)2:C6H3NO2 NO2C14H6O2SO3H NO2C6H4CHCl2 NO2C6H4CHO C6H5NO2 NH2C6H4C6H3(NH2)NO2 NO2C6H4CO2H NO2C6H4CO2H NO2C6H4CO2H NO2C6H4CH2OH NO2C6H4CH2Br CH3C6H3(NO2)Cl CH3C6H3(NO2)OH CH3C6H3(NO2)CH(CH3)2 NO2C6N4NHCH3 NO2C6H4NHCH3 NO2C6H4NHCH3 C6H5C6H4NO2 C6H5C6H4NO2 C6H5NHC6H4NO2 H2NC(NH)NHNO2

Formula weight

Form and color

Specific gravity

Melting point, °C

Boiling point, °C

159.08 452.82 228.36 214.38 128.16 288.28 288.28 244.26 226.24 205.22 445.35 584.45 172.17 172.17 144.16 144.16 224.22 224.22 186.20 186.20 143.18 143.18 179.65 179.65 223.24 241.26 241.26 241.26 223.24 241.26 241.26 169.17 162.23 123.11 123.11 180.16 165.14 168.15 168.15 168.15 154.12 138.12 138.12 138.12 153.13 153.13 253.20 333.26 206.03 151.12 123.11 229.23 167.12 167.12 167.12 153.13 216.04 171.58 153.13 179.21 166.18 166.18 166.18 199.20 199.20 214.22 104.07

oil cr. col. lf. cr. pl./al. lf. cr. cr. cr. nd. cr. lf. nd. mn. mn. mn. pl./aq. lf. nd./al. nd./al. rhb. lf./aq. nd. lf. nd. cr. cr. cr. cr. cr. cr. col. lq. oil nd./al. nd./aq. rhb. nd. red nd. yel. nd. red or. pr. yel. rhb. yel. rhb. yel. mn. col. cr. pr./al. nd. yel. cr. mn. nd./aq. yel. lq. red nd. tri./aq. mn. yel. mn. cr. nd./al. cr. yel. oil yel. oil red mn. yel. nd. rhb. nd./al. or. cr. nd./aq.

1.275 20/4 0.777 95 0.853 70/4 0.824 38/4 1.145 20/4

13–4 88 57–8 38 80.2 d. d. 125 90 125 177–8

217

1.077 100/4 1.224 4 1.217 4

1.123 25/25 1.061 98/4

160–1 184 96 122–3 >250 125 46–9 69–70 50 111–2

250.5100 16715 217.9

300 >300 278–80 285–6

300.8 306.1 subl.

d.

1.18 1.009 20/4

1.207 156 1.211 156 1.442 15 1.43 1.437 14 1.254 20/4 1.233 20

1.205

18/4

1.575 4/4 1.494 4/4 1.550 22/4

89/4

1.240 1.067 20/4 1.179 20/4 1.313 17 1.44

<−80 235.2 317 215–6 80–1 118 139–40 123 142–3 71.5 114 146–7 9.4 54 230 65 58 5.7 143 147.5 140–1 240–2 27 99–100 37.5 32 60–1 163–4 37 113–4 75–6 246–7

269–70 246730 subl. d. 202

284.1 306.4 331.7 272–3 274 2707 16423 210.9

subl. 175–803 238 12522 15215 151–380 280–5 320 340

Solubility in 100 parts Water 0.0725 i. i. <0.02 0.00325 10220 16420 v. s. 7730 s. h. v. s. v. s. v. sl. s. h. 0.00725 sl. s. h. 0.07425 v. s. h. v. s. sl. s. h. i. 0.17 c. v. s. h. 3.820 v. s. 0.2100 sl. s. 0.4625 0.42100 0.08 0.38100 0.28100 d. s. s. h. s. h. s. h. i. i.

Alcohol

Ether

s. v. sl. s. v. s. sl. s. 9.520 s. s. v. s.

s. v. s. v. s. s. v. s. i. i. sl. s.

sl. s.

s.

sl. s. s. h. s. v. s. v. s. v. s. s. s. v. s. s. s. v. s. i.

s. s. v. s. v. s. i. s. s. v. s. s. s. i.

s. ∞ s. h. sl. s. h. s. s. s.

s. ∞ v. sl. s. v. sl. s. s.

sl. s. sl. s. c. s. h. 0.1120 0.0819 0.1730 0.0630 i. s. i. 1.95112 0.1920 sl. s. h. 0.6520 0.24165 0.0215

s. v. s. v. s. 7.120 5.820 ∞ v. s. sl. s. i. v. s. h. v. s. h. v. s.

s. v. s. v. s. 7.920 6.120 ∞ v. s. v. sl. s. i. v. s. v. s. ∞

2811 3112 0.910

2211 2510 2.218

i. i. v. sl. s. i. v. sl. s. i. i. i. i.

219

v. s.

v. s.

v. s.

v. s. s. s. h. s. sl. s. c.

v. s. s.

sl. s.

v. sl. s.

9100

v. s. v. s.

-naphthalene (α-) (β-) -phenol (o-) (m-) (p-) -phenol sulfonic acid (1-,4-,2-) (1-,2-,4-) -phthalic acid (3-) (4-) -toluene (o-) (m-) (p-) -toluene sulfonic acid (1-,4-,2-) -toluidine (4-,1-,2-) (3-,1-,4-) Nitron Nitroso-dimethylaniline (p-) -naphthol (β-)(1-) Nonadecane (n-) Nonane (n-) Octadecane (n-) Octane (n-) (iso-) Octyl acetate (n-) (sec-) alcohol (n-) (sec-) Octylene (n-) Oleic acid Orcinol (1-,3-,5-) Oxalic acid Palmitic acid Pelargonic acid Penta-chloroethane -decane (n-) -erythritol Pentandiol Pentane (n-) (i-) (neo-) Phenacetin Phenanthrene Phenetidine (o-) (p-) Phenetole Phenol -phthalein -sulfonic acid (o-) Phenyl acetaldehyde acetic acid -acetylene aniline (o-) (p-) Phenyl-ethyl alcohol -glycine -hydrazine -hydrazine sulfonic acid (p-) isocyanate -methylpyrazolone (3-)(N-) -mustard oil naphthalene (α-) (β-) naphthylamine (α-) (β-) phenol (o-) (p-) propyl alcohol (γ-) quinoline (2-)(α-) (8-)(0-) salicylate, salol stearate urethane

C10H7NO2 C10H7NO2 NO2C6H4OH NO2C6H4OH NO2C6H4OH HOC6H3(NO2)SO3H3H2O HOC6H3(NO2)SO3H3H2O NO2C6H3(CO2H)2 NO2C6H3(CO2H)2 CH3C6H4NO2 CH3C6H4NO2 CH3C6H4NO2 CH3C6H3(NO2)SO3H2H2O NO2C6H3(CH3)NH2 NO2C6H3(CH3)NH2 C20H16N4 ONC6H4N(CH3)2 ONC10H6OH CH3(CH2)17CH3 CH3(CH2)7CH3 CH3(CH2)16CH3 CH3(CH2)6CH3 (CH3)3CCH2CH(CH3)2 CH3CO2CH2(CH2)6CH3 CH3CO2CH(CH3)C6H13 CH3(CH2)6CH2OH CH3(CH2)5CH(OH)CH3 CH3(CH2)5CH:CH2 C8H17CH:CH(CH2)7CO2H (HO)2C6H3CH3 HO2CCO2H2H2O CH3(CH2)14CO2H CH3(CH2)7CO2H CHCl2CCl3 CH3(CH2)13CH3 C(CH2OH)4 HOCH2(CH2)3CH2OH CH3(CH2)3CH3 (CH3)2CHCH2CH3 (CH3)2C(CH3)2 C2H5OC6H4NHCOCH3 < (C6H4CH)2 > C2H5OC6H4NH2 C2H5OC6H4NH2 C2H5OC6H5 C6H5OH C20H14O4 HOC6H4SO3He H2O C6H5CH2CHO C6H5CH2CO2H C6H5C:CH C6H5C6H4NH2 C6H5C6H4NH2 C6H5CH2CH2OH C6H5NHCH2CO2H C6H5NHNH2 H2NNHC6H4SO3H C6H5N:CO C4H5ON2C6H5 C6H5N:CS C10H7C6H5 C10H7C6H5 C10H7NHC6H5 C10H7NHC6H5 C6H5C6H4OH C6H5C6H4OH C6H5(CH2)3OH C6H5C9H6N C6H5C9H6N HOC6H4CO2C6H5 CH3(CH2)16CO2C6H5 C6H5NHCO2C2H5

173.16 173.16 139.11 139.11 139.11 273.22 273.22 211.13 211.13 137.13 137.13 137.13 253.23 152.15 152.15 312.36 150.18 173.16 268.51 128.25 254.48 114.22 114.22 172.26 172.26 130.22 130.22 112.21 282.45 124.13 126.07 256.42 158.23 202.31 212.41 136.15 104.15 72.15 72.15 72.15 179.21 178.22 137.18 137.18 122.16 94.11 318.31 187.68 120.14 136.14 102.13 169.22 169.22 122.16 151.16 108.14 188.20 119.12 174.20 135.18 204.26 204.26 219.27 219.27 170.20 170.20 136.19 205.25 205.25 214.21 360.56 165.19

yel./al. col./al. yel. mn. col. mn. yel. pr. nd. nd./aq. yel./aq. yel. cr. yel. lq. lq. rhb. pl./aq. yel. mn. red mn. yel. lf. gn. tri. brn. pr. cr. col. lq. cr. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. lq. col. nd. pr./bz. col. mn. col. pl. col. oil col. lq. col. lq. cr. lq. col. lq. col. lq. col. lq. col. mn. pl./al. oil lq. col. lq. col. nd. col. rhb. cr. lq. lf. col. lq. cr. lf. col. oil cr. yel. oil cr./al. lq. pr./aq. col. lq. waxy lf./al. pr./al. rhb. nd. nd. oil nd. lq. rhb./al. cr. pl./al.

1.223 62 45

1.295 1.485 20 1.479 20

1.163 20/4 1.160 18/4 1.139 55/55 1.365 15 1.312 17

0.777 32/4 0.718 20/4 0.775 28/4 0.703 20/4 0.692 20/4 0.885 0/4 0.863 14/4 0.827 20/4 0.822 20/4 0.721 18/4 0.854 78/4 1.290 4 1.653 19/4 0.849 70/4 0.906 20/4 1.671 25/4 0.770 20/4 0.994 20/4 0.630 18/4 0.621 19 0.613 20/4 1.179 25 1.061 15 0.967 20/4 1.071 25/4 1.299 25/4 1.025 20 1.081 80/4 0.930 20/4 1.023 18/4 1.097 23/4 1.096 20/4 1.138 15/15 1.17 1.18 1.008 20/4 1.250 20/4 1.106

30/4

59–60 79 44–5 96–7 113–4 d. 110 51.5 222 164–5 −4.1 15–16 51.9 130 105–7 116–7 189–90 d. 86–7 109.5 32 −53.7 28 −56.5 −107.4 −38.5 −16 −38.6 14 107–8 101.5 63–4 12.5 −22 10 262 −129.7 −160.0 −20 134–5 99–100 <−21 3–4 −30.2 42–3 261–2 50 d. 76–7 −43 45–6 50–2 127 19.6 286 128 −21 45 102.5 62 107–8 56–7 164–5 <−18 86 42–3 52 52–3

304 16515 214.5 19470 subl.

222.3 230–1 237.7

330 150.5759 317 125.7 99.3760 210 195 194–5 179–80 126 285–6100 287–90 subl. 271.5100 253–4 162 270.5 27630 239.4 36.3 27.95 9.5 d. 340 228–9 254–5 172 181.4 193–4 265.5 142–3 299760 302 219–21750 243.5 166769 19117 219–20 336–7 345–6 335258 399.5 275 305–8 235–7 363 283187 172–312 26715 237–8

i. i. 1.08100 1.3520 1.625 v. s. v. s. 2.0525 v. s. 0.0780 0.0580 0.0480 47.728 v. sl. s. sl. s. h. i. i. 0.120 i. i. i. 0.00216 i. i. i. 0.05425 0.09625 i. i. v. s. s. i. v. sl. s. 0.0520 i. 5.615 ∞ 0.03616 i. i. 0.720 i. i. i. i. 8.215 0.220 v. s. v. sl. s. 1.6620 i. v. sl. s. s. h. 1.620 s. sl. s. h. 0.612 d. 120 i. i. i. 0.0860 0.460 i. i. sl. s. sl. s. sl. s. 0.01525 i. i. c.

s. v. s. v. s. v. s. v. s. v. s. v. s. v. s. h. v. s. ∞ ∞ 8.615 v. s. s. s. s. h. s. 2.418 sl. s. sl. s. sl. s. sl. s. sl. s. s. s. ∞ ∞ ∞ ∞ v. s. s. 920 s. ∞ v. s. v. sl. s. ∞ ∞ s. 40 h. 10 h. s. s. ∞ ∞ 1025 v. s. ∞ v. s. ∞ s. s. s. s. ∞ sl. s. d. v. s. h. s. v. s. sl. s. s. v. s. h. s. s. ∞ s. h. s. v. s. s.

s. v. s. v. s. s. v. s. sl. s. sl. s. s. ∞ ∞ 80.815 v. s. s. v. sl. s. s. s. s. s. s. s. s. s. ∞ ∞ ∞ ∞ v. s. 1.3 s. s. ∞ v. s. i. ∞ ∞ s. 1.625 v. s. s. s. ∞ ∞ 5.9 c. ∞ v. s. ∞ s. s. ∞ sl. s. ∞ v. s. v. sl. s. s. v. s. sl. s. s. v. s. h. s. s. ∞ s. s. s. s.

2-43

TABLE 2-2

Physical Properties of Organic Compounds (Continued ) Name

Phenylene-diamine (o-) (m-) (p-) Phloroglucinol (1-,3-,5-) Phorone Phosgene Phthalic acid (o-) (m-)(iso-) anhydride (o-) nitrile (o-) Phthalide Phthalimide (o-) Picoline (α-) (β-) (γ-) Picramic acid (1-,2-,4-,6-) Picric acid (2-,4-,6-) Picryl chloride (2-,4-,6-) Pinacol Pinacoline Pinene (α-)(dl-) hydrochloride Pinol (dl-) Piperidine carboxylic acid (α-)(dl-) Piperidinium pentamethylene dithiocarbamate Propane Propionic acid aldehyde anhydride Propyl acetate (n-) (i-) alcohol (n-) (i-) amine (n-) (i-) aniline (n-) benzoate (n-) (i-) bromide (n-) (i-) n-butyrate (n-) i-butyrate (n-) n-butyrate (i-) i-butyrate (i-) chloride (n-) (i-) Propyl formate (n-) (i-) furoate (n-) lactate (n-) (i-) mercaptan (n-) (i-) propionate (n-) (i-) thiocyanate (i-) n-valerate (n-) i-valerate (n-) i-valerate (i-) Propylene bromide chlorohydrin chloride glycol oxide Protocatechuic acid (3-,4-)

2-44

Formula C6H4(NH2)2 C6H4(NH2)2 C6H4(NH2)2 C6H3(OH)32H2O [(CH3)2C:CH]2CO OCCl2 C6H4(CO2H)2 C6H4(CO2H)2 C6H4 < (CO)2 > O C6H4(CN)2 C6H4(CH2)(CO) > O C6H4 < (CO)2 > NH C5H4NCH3 C5H4NCH3 C5H4NCH3 HOC6H2(NH2)(NO2)2 HOC6H2(NO2)3 ClC6H2(NO2)3 [(CH3)2COH]2 CH3COC(CH3)3 C10H16 C10H17Cl C10H16O CH2 < (CH2CH2)2 > NH HO2CCH < (CH2CH2)2 > NH (CH2)5CS2HHN(CH2)5 CH3CH2CH3 CH3CH2CO2H CH3CH2CHO (CH3CH2CO)2O CH2CO2CH2CH2CH3 CH3CO2CH(CH3)2 CH3CH2CH2OH (CH3)2CHOH CH3CH2CH2NH2 (CH3)2CHNH2 C6H5NHCH2CH2CH3 C6H5CO2CH2CH2CH3 C6H5CO2CH(CH3)2 CH3CH2CH2Br (CH3)2CHBr C2H5CH2CO2CH2C2H5 (CH3)2CHCO2CH2C2H5 C2H5CH2CO2CH(CH3)2 (CH3)2CHCO2CH(CH3)2 CH3CH2CH2Cl (CH3)2CHCl HCO2CH2CH2CH3 HCO2CH(CH3)2 C4H3OCO2C3H7 CH3CH(OH)CO2CH2C2H5 CH3CH(OH)CO2CH(CH3)2 CH3CH2CH2SH (CH3)2CHSH C2H5CO2CH2C2H5 C2H5CO2CH(CH3)2 (CH3)2CHCNS CH3(CH2)3CO2CH2C2H5 (CH3)2CHCH2CO2C3H7 (CH3)2CHCH2CO2C3H7 CH3CH:CH2 CH3CHBrCH2Br CH3CHClCH2OH CH3CHClCH2Cl CH3CH(OH)CH2OH CH3(CHCH2)O (HO)2C6H3CO2HH2O

Formula weight

Form and color

108.14 108.14 108.14 162.14 138.20 98.92 166.13 166.13 148.11 128.13 134.13 147.13 93.12 93.12 93.12 199.12 229.11 247.56 118.17 100.16 136.23 172.69 152.23 85.15 129.16 232.41 44.09 74.08 58.08 130.14 102.13 102.13 60.09 60.09 59.11 59.11 135.20 164.20 164.20 123.00 123.00 130.18 130.18 130.18 130.18 78.54 78.54 88.10 88.10 154.16 132.16 132.16 76.15 76.15 116.16 116.16 101.16 144.21 144.21 144.21 42.08 201.91 94.54 112.99 6.09 58.08 172.13

lf./aq. rhb. mn. rhb. yel. pr. gas mn./aq. nd./aq. rhb. cr. nd./aq. cr./et. col. lq. col. lq. lq. red nd. yel. rhb. yel. mn. col. nd. col. lq. col. lq. lf. lq. lq. cr. cr. gas col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. lq. lq. col. lq. col. lq. lq. lq. col. lq. col. lq. gas col. lq. col. lq. col. lq. col. oil col. lq. nd./aq.

Specific gravity 1.139 15/15 0.885 20/4 1.392 19/4 1.593 20/4 1.527

4

1.164 99/4 0.950 15/4 0.961 15/4 0.957 15/4 1.763 20/4 1.797 20 0.967 15 0.800 16 0.878 20/4 0.953 20/20 0.860 20/4 1.13 0.585 −45/4 0.992 20/4 0.807 20/4 1.012 20/4 0.886 20/4 0.874 20/20 0.804 20/4 0.78920/4 0.718 20/20 0.694 15/4 0.949 18 1.021 25/25 1.010 25/25 1.353 20/4 1.310 20/4 0.879 15 0.884 0/4 0.865 18 0.869 0/4 0.890 20/4 0.859 20 0.901 20/4 0.873 20/4 1.075 26/4 0.836 25/4 0.809 25/4 0.883 20/4 0.893 0 0.963 20 0.874 15 0.863 20/4 0.854 17 0.609 −47/4 1.933 20/4 1.103 20 1.159 20/20 1.040 19.4 0.831 20/20 1.542 4/4

Melting point, °C

Boiling point, °C

103–4 62.8 140 117 28 −104 208 330 130.8 141 73(65) 238 −70

256–8 284–7 267 subl. 197.2743 8.2756 d. subl. 284.5

169 121.8 83 43(38) −52.5 −55 131–2 −9 264 175 −187.1 −22 −81 −45 −92.5 −73.4 −127 −85.8 −83 −101 −51.6 −109.9 −89 −95.2

−122.8 −117 −92.9

−112 −130.7 −76 −70.7 −185 −55.5 <−70 199 d.

290 subl. 128.8 143.5 143.1 expl. d. 171–2789 106.2 154–6 207–8 183–4 106 −42.2 141.1 49.5740 168.8780 101.6 88.4 97.8 82.5 49–50761 33–4 222 231 218.5 70.8 60 142.7 134–5 128 120.8 46.4 36.5 81.3 68-71751 211 122-3150 167.5 67-8 58-60 122-3 109-11750 152-3754 67.5 155.9 142756 −48749 141.6 133-4 96.8 188-9 35

Solubility in 100 parts Water 73381 35.125 669107 1.1325 0.150 v. sl. s. 0.7025 0.2100 v. sl. s. sl. s. c. v. sl. s. 0.0425 v. s. ∞ ∞ 0.1422 1.2320 0.01815 sl. s. c. 2.515 v. sl. s. i.

Alcohol

Ether

v. s. v. s. s. v. s. s.

v. s. s. s. v. s. s.

1218 s. s.

0.6815

s. 5 ∞ ∞ ∞ s. 620 4.817 v. s. s. s. 33 s. ∞

sl. s. s. h. ∞ ∞ ∞ sl. s. 113 717 v. s. s. ∞ s. s.

∞ s. 628 6.518 cc. ∞ 2020 d. 1.616 320 ∞ ∞ ∞ ∞ i. i. i. 0.2520 0.3220 0.1717 v. sl. s. v. sl. s. v. sl. s. 0.2720 0.3120 12.222 2.122 v. sl. s. s. s. v. sl. s. v. sl. s. 0.5625 0.625 i. i. i.

s. ∞ ∞ d. ∞ ∞ ∞ ∞ ∞ ∞ v. s. s. s. ∞ ∞ ∞

v. s. ∞ ∞

∞ ∞ ∞ ∞ s. s. s. s. ∞ ∞ ∞ ∞ ∞ ∞

∞ ∞ ∞ ∞ ∞ s. s. s. ∞ ∞ ∞ ∞ ∞ ∞

44.6 cc. 0.2520 s. 0.2720 ∞ 3320 1.8214

1200 cc. s. s. v. s. ∞ ∞ v. s.

v. s. s. v. s. 8 ∞ s.

∞ ∞ ∞ ∞ ∞ ∞ v. s. s. s. ∞ ∞ ∞

Pulegol (iso-)(d-) Pulegone Pyrazole Pyrazoline Pyrazolone Pyrene Pyridazine Pyridine Pyrocatechol (o-) Pyrogallol (1-,2-,3-) Pyrone Pyrrole Pyrrolidine Pyrroline Pyruvic acid Quercitrin Quinaldine (py-2) Quinoline (iso-) -diol (1-,3-) Quinone (p-) R-acid Ca salt (2-)(3-,6-) K salt Na salt Raffinose Resorcinol (m-) Retene Rhamnose (β-) Ricinoleic acid Rosaniline Rosolic acid Saccharin Safrole (1-,3-,4-) (iso-)(1-,3-,4-) Salicylic acid (o-) aldehyde (o-) Saligenin Schaeffer’s salt, Ca K Na Semicarbazide hydrochloride Skatole (3-) Sodium methylate Sorbitol Sorbose (d- or l-) Starch Stearic acid amide Styrene Suberic acid Succinic acid Sucrose Sulfanilic acid (p-) Sylvestrene (d-) Tartaric acid (meso-) (racemic) (d- or l-) Tartronic acid Terephthalic acid (p-) Terpin hydrate (cis-) Terpineol (α-)(d- or l-) (dl-) Terpinyl acetate (α-)(dl-) Tetrabromo-ethane (sym) (uns) Tetrachloro-ethane (sym) (uns) -ethylene Tetracosane (n-) Tetradecane (n-) Tetraethyl-thiuram disulfide

C10H17OH C10H16O NHN:CHCH:CH NHN:CHCH2CH2 NHCOCH2CH:N C16H10 N2 < (CHCH)2 > CH < (CHCH)2 > N C6H4(OH)2 C6H3(OH)3 CO < (CHCH)2 > O < (CH:CH)2 > NH < (CH2CH2)2 > NH < (CHCH2)2 > NH CH3COCO2H C21H20O112H2O CH3C9H6N C9H7N C9H7N C6H4CH:C(OH)N:C(OH) CO < (CHCH)2 > CO HOC10H5(SO3)2Ca HOC10H5(SO3K)2 HOC10H5(SO3Na)2 C18H32O165H2O C6H4(OH)2 C18H18 CH3(CHOH)4CHOH2O C17H32(OH)CO2H C20H21ON3 C20H16O3 C6H4(CO)(SO2) > NH CH2:CHCH2C6H3:O2CH2 CH3CH:CHC6H3:O2CH2 HOC6H4CO2H HOC6H4CHO HOC6H4CH2OH (HOC10H6SO3)2Ca5H2O HOC10H6SO3K HOC10H6SO3Na NH2CONHNH2 NH2CONHNH3Cl CH3C8H6N CH3ONa [CH2OH(CHOH)2]2 C6H12O6 (C6H10O5)x CH3(CH2)16CO2H CH3(CH2)16CONH2 C6H5CH:CH2 HO2C(CH2)6CO2H HO2C(CH2)2CO2H C12H22O11 H2NC6H4SO3H C10H16 (CHOHCO2H)2 (CHOHCO2H)2H2O (CHOHCO2H)2 CH(OH)(CO2H)2a H2O C6H4(CO2H)2 C10H20O2H2O C10H18O C10H18O CH3CO2C10H17 Br2CHCHBr2 Br3CCH2Br Cl2CHCHCl2 Cl3CCH2Cl Cl2C:CCl2 CH3(CH2)22CH3 CH3(CH2)12CH3 [(C2H5)2NCS]2S2

154.24 152.23 68.08 70.09 84.08 202.24 80.09 79.10 110.11 126.11 96.08 67.09 71.12 69.10 88.06 484.40 143.18 129.15 129.15 161.15 108.09 342.35 380.46 348.26 594.52 110.11 234.32 182.17 298.45 319.39 304.33 183.18 162.18 162.18 138.12 122.12 124.13 576.59 262.31 246.21 75.07 111.54 131.17 54.03 182.17 180.16 162.14 284.47 283.48 104.14 174.19 118.09 342.30 173.18 136.23 150.09 168.10 150.09 129.07 166.13 190.28 154.24 154.24 196.28 345.70 345.70 167.86 167.86 165.85 338.64 198.38 296.52

col. lq. col. lq. nd./et. lq. nd. yel. pr. lq. col. lq. nd./aq. nd. cr. lq. lq. lq. col. lq. yel. nd. lq. lq. pl. cr. yel. mn. cr. cr. cr. cr./aq. col. rhb. lf./al. col. mn. lq. col. nd. red lf. mn. col. mn. col. lq. mn. col. oil rhb./aq. cr. cr. cr. pr./al. pr. lf. pd. cr. rhb. amor. mn. col. cr. col. lq. nd./aq. col. mn. col. mn. col. cr. lq. cr. tri. mn. pr./aq. cr. rhb. col. cr. col. cr. lq. col. lq. col. lq. col. lq. lq. col. lq. cr. col. lq. cr.

0.911 20/4 0.932 20/20 70 1.277 0/4 1.107 20/4 0.982 20/4 1.344 4 1.453 4 1.190 40.3 0.948 20/4 0.852 22.5 0.910 20/4 1.267 20/4 1.059 20/4 1.095 20 1.099 21/4 1.318 20/4

1.465 0 1.272 15 1.1316 1.47120/4 0.954 16

1.100 20/4 1.122 20/4 1.443 20/4 1.153 25/4 1.161 25

1.654 15 1.5021 0.847 69.3 0.903 20/4 1.266 25/4 1.572 25/4 1.588 15 0.863 20/4 1.737 1.697 20/4 1.760 20/4 1.510 0.935 15 0.935 20/20 0.966 20/4 2.964 20/4 2.875 20/4 1.600 20/4 1.588 20/4 1.624 15/4 0.779 51/4 0.765 20/4 1.17

165 149-50 −8 −42 104-5 133-4 32.5

13.6 182-5 −1 −15 24.6 237 115.7

119 110.7 98-9 126 4-5 186 d. 308-10 d. 225-8 11.2 6-7 159 -7 86-7

96 173 d. 95 d. 300 110-2 165 d. 70-1 108-9 −31 140-4 189-90 170-86 d. d. > 280

86-9 10 224754 186-8 144 subl. d. >360 208 115-6 240-5 309 215-7 131 87-8 90-1 165 244-5750 237.1747 240.5763 subl.

d. 130 276.5 390-4 226-810 subl. 233-4 252-3 21120 196.5 subl.

755

265-6

291110 25112 145-6 279100 235 d. 176-7

159-60 205-6 168-70 d. 155-8 subl. 117 38-40 35 < −50 −1.0 0 −36 −19 51.1 5.5 70

d. subl. d. 219-21 218-9752 220 d. 15154 10413 146.3 129-30 120.8 324 252.5

v. sl. s. i. s. ∞ s. i. ∞ ∞ 45.120 40 13 v. sl. s. i. ∞ v. s. ∞ 0.04 20 v. sl. s. 6 sl. s. v. sl. s. sl. s. h. 30.6 25 29.5 25 25.2 25 14.3 20 14712 i. 60.8 21 i. v. sl. s. 0.1225 0.4 25 i. i. 0.223 1.7 86 6.615 4.7620 3.4625 6.2925 v. s. v. s. 0.05 c. d. v. s. 5517 i. 0.0325 i. v. sl. s. 0.1416 6.820 1790 0.810 12015 20.620 13920 v. s. 0.001 c. 0.415 i. i. i. i. 20

0.29 i. 0.0220 i. i.

∞ s. ∞ v. s. 3 h. s. ∞ v. s. s. s. s. ∞ ∞ ∞ s. ∞ s.

0.120 v. s. 69 h.

∞ s. sl. s. v. sl. s. v. s. s. s. v. s. s. v. s. s. ∞ ∞ ∞ sl. s. s. ∞ s. s.

∞ sl. s. v. s. h. 3.1 c. s. ∞ 4915 ∞ v. s.

v. s. v. s. h. i. ∞ i. sl. s. 1.05 c. ∞ ∞ 5115 ∞ v. s.

v. s. sl. s. s.

i. i. s.

v. s. h. sl. s. i. 220 s. h. ∞ s. 9.915 0.9 v. sl. s.

i. 6g s. h: ∞ 0.815 1.215 i. v. sl. s.

20 2515 v. s. sl. s. h. 1015 v. s. v. s. 20 ∞ s. ∞ ∞ ∞ v. s.

0.09 0.415 i. i. 115 v. s. v. s. ∞ ∞ ∞ ∞ s. v. s.

2-45

TABLE 2-2

Physical Properties of Organic Compounds (Concluded ) Name

Tetrafluoro-ethylene Tetrahydro-furan -furfuryl alcohol -pyran Tetralin Tetramethyl-thiuram disulfide Tetryl (2-,4-,6-) Theobromine Thio-acetic acid -aniline (4-, 4′-) -carbanilide -naphthol (β-) -phenol -salicylic acid (o-) -urea Thiophene Thymol (5-,2-,1-) Tolidine (0-)(3-,3′-,4-,4′-) Toluene sulfonic acid (o-) (p-) sulfonic amide (p-) sulfonic chloride (p-) Toluic acid (o-) (m-) (p-) Toluidine (o-) (m-) (p-) hydrochloride (o-) sulfonic acid (1-,2-,3-) Toluylenediamine (1-,2-,4-) Tolylene diisocyanate (1-,2-,4-) Trehalose Triamylamine (n-) (i-) Tributyl-amine (n-) phosphite Trichloro-acetic acid -benzene (s-)(1-,3-,5-) -ethane (1-,1-,1-) -ethylene -phenol Tricosane (n-) Tricresyl phosphate (o-) Tridecane (n-) Triethanol amine Triethyl-amine -benzene (1-,3-,5-) (1-,2-,4-) borate citrate Triethylene glycol Trifluoro-chloromethane chloroethylene -trichloroethane Trimethoxybutane (1-,3-,3-) Trimethylamine Trimethylene bromide chloride glycol Trinitro-benzene (1-,3-,5-) -benzoic acid (2-,4-,6-) -tert-butylxylene -naphthalene (α-)(1-,3-,5-) (β-)(1-,3-,8-) (γ-)(1-,4-,5-)

2-46

Formula F2C:CF2 CH2(CH2)2CH2O C4H7OCH2OH CH2(CH2)3CH2O C6H4CH2(CH2)2CH2 [(CH3)2NCS]2S2 (NO2)3C6H2N(CH3)NO2 C7H8O2N4 CH3COSH (NH2C6H4)2S (C6H5NH)2CS C10H7SH C6H5SH HSC6H4CO2H NH2CSNH2 < (CH:CH)2 > S (CH3)(C3H7)C6H3OH [CH3(NH2)C6H3]2 C6H5CH3 CH3C6H4SO3H2H2O CH3C6H4SO3HH2O CH3C6H4SO2NH2 CH3C6H4SO2Cl CH3C6H4CO2H CH3C6H4CO2H CH3C6H4CO2H CH3C6H4NH2 CH3C6H4NH2 CH3C6H4NH2 CH3C6H4NH3Cl CH3(NH2)C6H3SO3H CH3C6H3(NH2)2 CH3C6H3(NCO)2 C12H22O112H2O [CH3(CH2)3CH2]3N [(CH3)2CH(CH2)2]3N [CH3(CH2)2CH2]3N [CH3(CH2)3O]3P Cl3CCO2H C6H3Cl3 Cl3CCH3 Cl2C:CHCl Cl3C6H2OH CH3(CH2)21CH3 OP(OC6H4CH3)3 CH3(CH2)11CH3 (HOCH2CH2)3N (CH3CH2)3N (C2H5)3C6H3 (C2H5)3C6H3 B(OCH2CH3)3 HOC3H4(CO2C2H5)3 (CH2OCH2CH2OH)2 CF3Cl F2C:CFCl Cl2CFCClF2 CH2(OCH3)CH2C(OCH3)2CH3 (CH3)3N BrCH2CH2CH2Br ClCH2CH2CH2Cl HOCH2CH2CH2OH C6H3(NO2)3 (NO2)3C6H2CO2H (NO2)3C6(CH3)2C4H9 C10H5(NO2)3 C10H5(NO2)3 C10H5(NO2)3

Formula weight

Form and color

Specific gravity

Melting point, °C

Boiling point, °C

100.02 72.10 102.13 86.13 132.20 240.41 287.15 180.17 76.11 216.29 228.30 160.22 110.17 154.18 76.12 84.13 150.21 212.28 92.13 208.23 190.21 171.21 190.64 136.14 136.14 136.14 107.15 107.15 107.15 143.62 187.21 122.17 174.15 378.33 227.42 227.42 185.34 250.32 163.40 181.46 133.42 131.40 197.46 324.61 368.36 184.35 149.19 101.19 162.26 162.26 146.00 276.28 150.17 104.47 116.48 187.39 148.20 59.11 201.91 112.99 76.09 213.11 257.12 297.26 263.16 263.16 263.16

gas col. lq. col. lq. lq. col. lq. cr. yel. mn. rhb. yel. lq. nd./aq. rhb./al. cr./al. col. lq. yel. nd. rhb./al. col. lq. cr. lf. col. lq. cr. mn. mn. tri. cr./aq. pr./aq. cr./aq. col. lq. col. lq. cr. mn. pr. cr. rhb. lq. rhb./al. lq. col. lq. col. lq. lq. cr. nd. lq. col. lq. nd. lf. lq. col. lq. col. lq. col. oil lq. lq. lq. oil col. lq. gas gas lq. lq. gas lq. lq. oil col. rhb. rhb./aq. nd./al. rhb. cr./al. yel. cr.

1.58−78 0.88821/4 1.05020/4 0.88120/4 0.97318/4 1.29 1.5719

−142.5 −65

−76.3 65-6 177-8743 88 206764

1.07410 1.324 1.07423/4 1.40520/4 1.07015/4 0.97225/25 20/4

0.866

1.062115/4 1.054112/4 0.99920/4 0.98920/4 1.04620/4

1.2328

−31 155-6 130.5 330 < −17 108 154 81 164 180-2 −30 51.5 128-9 −95 d. 104-5 137 69 104-5 110-1 179-80 −16.3 −31.5 44-5 218-20 99

expl. 93 d. 286-8 168-9 subl. d. 84 232752 110.8 128.80 146-70 134.510 259751 263 274-5 199.7 203.3 200.3 242 283-5 134.520

97 0.78620/4 0.77820/20 0.92520/4 1.61746/15 1.32526/4 1.46620/20 1.49075/4 0.77948/4 0.75720/4 1.12620/20 0.72920/20 0.86120/4 0.88217/4 0.86420/20 1.13720/4 1.12520/20 1.726−130 1.57620/4 0.932 0.662 −5 1.987 15/4 1.201 15 1.060 20/4 1.688 20/4

58 63.5 −73 68-9 47.7 −6.2 20-1 −114.8

−5 −182 −157.5 −35 −124 −34.4 121 210-20 d. 110 122-3 218-9 148-9

240-5 235 216.5761 122-312 195.5754 208.5764 74.1 87.2 246 23415 234 277-9150 89.4 215 217-8755 120 294 290 −80 −27.9 47.6 63-525 3.5 167.5 123-5 214 d.

Solubility in 100 parts Water 0.0130 s. ∞ s. i. i. i. 0.0615 s. sl. s. h. i. v. sl. s. v. sl. s. sl. s. h. 9.213 i. 0.0919 v. sl. s. 0.0516 v. s. v. s. 0.29 i. 2.17100 1.6100 1.3100 1.525 sl. s. 0.7421 s. 0.9711 s. h. d. s. h. i. i. i. i. 12025 i. i. 0.125 0.0925 i. i. i. ∞ ∞ > 190 i. i. d. i. ∞ d. i. d. 4119 0.1730 0.2725 ∞ 0.0315 2.0524 i. i. 0.02100 i.

Alcohol

Ether

s. ∞

s. ∞

s.

s.

s. h. 0.06 c. ∞ s. v. s. v. s. v. s. s. s. s. v. s. s. s. s. s. 7.45 s. v. s. v. s. v. s. ∞ ∞ v. s. sl. s.

s. 0.03 h. ∞ s. v. s. v. s. ∞ sl. s. v. s. s. ∞

s. v. s. v. s. ∞ ∞ v. s.

s. d. sl. s. h.

i.

s.

s.

∞

s. sl. s. ∞ ∞ v. s.

∞ ∞ v. s.

v. s. ∞ ∞ s. s.

v. s. sl. s. ∞ s. s.

∞ ∞

∞ v. sl. s.

∞

∞

s. s. s. ∞ 1.918

s. s. s. 1.518

sl. s. s. 0.0523 0.1119

0.1315 0.419

s.

s.

-phenol (2-,3-,6-) -toluene (β-)(2-,3-,4-) (γ-)(2-,4-,5-) (α-)(2-,4-,6-) Trional Triphenyl-arsine carbinol guanidine (α-) methane methyl phosphate Tripropylamine (n-) Undecane (n-) Urea nitrate Uric acid Valeric acid (n-) (i-) aldehyde (n-) (i-) amide (n-) (i-) Vanillic acid (3-,4-,1-) alcohol (3-,4-,1-) hyl-thiuram disulfide Vanillin (3-,4-,1-) Veratrole (o-) Vinyl acetate (poly-) acetic acid acetylene alcohol (poly-) chloride propionate Xylene (o-) (m-) (p-) sulfonic acid (1-,4-,2-) Xylidine (1:2)(3-) (1:2)(4-) (1:3)(2-) (1:3)(4-) (1:3)(5-) (1:4)(2-) Xylose (l-)(+) Xylylene dichloride (p-) Zinc diethyl dimethyl dimethyl-dithiocarbamate NOTE:

(NO2)3C6H2OH CH3C6H2(NO2)3 CH3C6H2(NO2)3 CH3C6H3(NO2)3 C2H5(CH3)C(SO2C2H5)3 (C6H5)3As (C6H5)3COH C6H5N:C(NHC6H5)2 (C6H5)3CH (C6H5)3C . . . OP(OC6H5)3 (CH3CH2CH2)3N CH3(CH2)3CH3 H2NCONH2 CO(NH2)2HNO3 C5H4O3N4 C2H5CH2CH2CO2H (CH3)2CHCH2CO2H C2H5CH2CH2CHO (CH3)2CHCH2CHO C2H5CH2CH2CONH2 (CH3)2CHCH2CONH2 CH3O(OH)C6H3CO2H CH3O(OH)C6H3CH2OH [(C2H5)2NCS]2S2 CH3O(OH)C6H3CHO C6H4(OCH3)2 CH3CO2CH:CH2 (CH3CO2CH:CH2)x CH2:CHCH2CO2H CH2:CHC:CH CH2:CHOH (CH2:CHOH)x CH2:CHCl C2H5CO2CH:CH2 C6H4(CH3)2 C6H4(CH3)2 C6H4(CH3)2 (CH3)2C6H3SO3H2H2O (CH3)2C6H3NH2 (CH3)2C6H3NH2 (CH3)2C6H3NH2 (CH3)2C6H3NH2 (CH3)2C6H3NH2 (CH3)2C6H3NH2 CH2OH(CHOH)3CHO C6H4(CH2Cl)2 Zn(CH2CH3)2 Zn(CH3)2 Zn[S2CN(CH3)2]2

229.11 227.13 227.13 227.13 242.34 306.21 260.32 287.35 244.32 243.31 326.28 143.27 156.30 60.06 123.07 168.11 102.13 102.13 86.13 86.13 101.15 101.15 168.14 154.16 296.52 152.14 138.16 86.09 (86.09) 86.09 52.07 44.06 (44.06) 62.50 100.11 106.16 106.16 106.16 222.25 121.18 121.18 121.18 121.18 121.18 121.18 150.13 175.06 123.50 95.45 305.79

nd. cr. yel. pl. cr./al. pl./al. pl. cr. rhb./al. cr. col. cr. pr./al. col. lq. col. lq. col. pr. col. mn. cr. col. lq. col. lq. lq. col. lq. mn. pl. mn. nd./aq. mn./aq. cr. mn. cr. col. lq. col. lq. gas gas lq. col. lq. col. lq. col. lq. col. lf. lq. pr. lq. lq. oil oil nd. mn. col. lq. col. lq.

1.620 20/4 1.620 20/4 1.654 1.199 85/4 1.306 1.188 20/4 1.13 1.014 99/4 1.206 58/4 0.757 20/4 0.741 20/4 1.335 20/4 1.893 20 0.939 20/4 0.931 20/20 0.819 11 0.803 17 1.023 0.965 20/4 1.17 1.056 1.091 15/15 0.932 20/4 1.1920 1.013 15/15 0.705 1.5

117-8 112 104 80.8 76 59-60 162.5 144-5 93.4 145-7 49-50 −93.5 −25.6 132.7 152 d. d. −34.5 −37.6 −92 −51 106 135-7 207 115 70 81-2 22.5 < −60 100-25 −39

1.320 0.908 25/25

d. >200 −160

0.881 20/4 0.867 17/4 0.861 20/4

−25 −47.4 13.2 86 < −15 49-50

0.991 15 1.076 17.5 0.980 15 0.978 20/4 0.972 20/4 0.979 21/4 1.535 0 1.417 0 1.182 18 1.386 11 2.0040/4

15.5 153-4 100.5 −28 −40 248-50

expl. expl. expl. d. >360 >360 d. 359754 d. 24511 156.5 194.5 d. 187 176 103.4 92.5 232 subl. d. 285 207.1 72-3 163 5.5 −12 93-5 144 139.3 138.5 1490.1 223 224-6 216-7 213-4 221-2 215789 240-5 d. 118 46

s. h. i. i. 0.0120 0.315 i. i. i. i. i. i. v. sl. s. i. 10017 v. s. h. 0.06 h. 3.316 4.220 v. sl. s. sl. s. v. s. s. 0.1214 v. s. h. i. 114 v. sl. s. 220 i. s. 0.670.6 s. sl. s. v. sl. s. i. i. i. s. v. sl. s. v. sl. s. v. sl. s. v. sl. s. v. sl. s. v. sl. s. 11720 i. d. d. i.

v. s. sl. s. c. s. h. 1.522 50 s. v. s. 40 v. s. h. sl. s. h. 15525 ∞ ∞ 2020 s. i. ∞ ∞ s. s. v. s. s. v. s. v. s.

v. s. s. v. s. 533 6.615 v. s. v. s.

v. s. s. ∞

v. s. s. ∞

∞

∞

v. s. v. s. ∞ ∞ sl. s. i. ∞ ∞ s. s. v. s. s. v. s. v. s.

s.

v. s.

s. s. s.

∞ ∞ v. s.

s.

s.

v. sl. s. s. d. d.

i. v. sl. s.

°F = 9⁄5 °C + 32.

2-47

2-48

PHYSICAL AND CHEMICAL DATA

VAPOR PRESSURES OF PURE SUBSTANCES UNITS CONVERSIONS

ADDITIONAL REFERENCES

For this subsection, the following units conversions are applicable:

Additional compilations of vapor-pressure data include Boublik, Fried, and Hala, The Vapor Pressures of Pure Substances, Elsevier, Amsterdam, 1984. See also Hirata, Ohe, and Nagahama, Computer Aided Data Book of Vapor-Liquid Equilibria, Kodansha/Elsevier, Tokyo, 1975; Weishaupt, Landolt-Börnstein New Series Group IV, vol. 3; Thermodynamic Equilibria of Boiling Mixtures, Springer-Verlag, Berlin, 1975; Wichterle, Linek, and Hala, Vapor-Liquid Equilibrium Data Bibliography, Elsevier, Amsterdam, 1973; suppl. 1, 1976; suppl. 2, 1982.

°F = 9⁄5 °C + 32. To convert millimeters of mercury to pounds-force per square inch, multiply by 0.01934.

TABLE 2-3

Vapor Pressure of Water Ice from -15 to 0°C* mmHg

TABLE 2-4

Vapor Pressure of Liquid Water from -16 to 0°C* mmHg

t, °C

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

t, °C

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

−14 −13 −12 −11 −10

1.361 1.490 1.632 1.785 1.950

1.348 1.477 1.617 1.769 1.934

1.336 1.464 1.602 1.753 1.916

1.324 1.450 1.588 1.737 1.899

1.312 1.437 1.574 1.722 1.883

1.300 1.424 1.559 1.707 1.866

1.288 1.411 1.546 1.691 1.849

1.276 1.399 1.532 1.676 1.833

1.264 1.386 1.518 1.661 1.817

1.253 1.373 1.504 1.646 1.800

−15 −14 −13 −12 −11

1.436 1.560 1.691 1.834 1.987

1.425 1.547 1.678 1.819 1.971

1.414 1.534 1.665 1.804 1.955

1.402 1.522 1.651 1.790 1.939

1.390 1.511 1.637 1.776 1.924

1.379 1.497 1.624 1.761 1.909

1.368 1.485 1.611 1.748 1.893

1.356 1.472 1.599 1.734 1.878

1.345 1.460 1.585 1.720 1.863

1.334 1.449 1.572 1.705 1.848

−9 −8 −7 −6 −5

2.131 2.326 2.537 2.765 3.013

2.112 2.306 2.515 2.742 2.987

2.093 2.285 2.493 2.718 2.962

2.075 2.266 2.472 2.695 2.937

2.057 2.246 2.450 2.672 2.912

2.039 2.226 2.429 2.649 2.887

2.021 2.207 2.408 2.626 2.862

2.003 2.187 2.387 2.603 2.838

1.985 2.168 2.367 2.581 2.813

1.968 2.149 2.346 2.559 2.790

−10 −9 −8 −7 −6

2.149 2.326 2.514 2.715 2.931

2.134 2.307 2.495 2.695 2.909

2.116 2.289 2.475 2.674 2.887

2.099 2.271 2.456 2.654 2.866

2.084 2.254 2.437 2.633 2.843

2.067 2.236 2.418 2.613 2.822

2.050 2.219 2.399 2.593 2.800

2.034 2.201 2.380 2.572 2.778

2.018 2.184 2.362 2.553 2.757

2.001 2.167 2.343 2.533 2.736

−4 −3 −2 −1 −0

3.280 3.568 3.880 4.217 4.579

3.252 3.539 3.848 4.182 4.542

3.225 3.509 3.816 4.147 4.504

3.198 3.480 3.785 4.113 4.467

3.171 3.451 3.753 4.079 4.431

3.144 3.422 3.722 4.045 4.395

3.117 3.393 3.691 4.012 4.359

3.091 3.364 3.660 3.979 4.323

3.065 3.336 3.630 3.946 4.287

3.039 3.308 3.599 3.913 4.252

−5 −4 −3 −2 −1

3.163 3.410 3.673 3.956 4.258

3.139 3.384 3.647 3.927 4.227

3.115 3.359 3.620 3.898 4.196

3.092 3.334 3.593 3.871 4.165

3.069 3.309 3.567 3.841 4.135

3.046 3.284 3.540 3.813 4.105

3.022 3.259 3.514 3.785 4.075

3.000 3.235 3.487 3.757 4.045

2.976 3.211 3.461 3.730 4.016

2.955 3.187 3.436 3.702 3.986

*For data at 0(0.2)−30(2)−98°C see p. 2324, Handbook of Chemistry and Physics, 40th ed., Chemical Rubber Publishing Co.

−0

4.579 4.546 4.513 4.480 4.448 4.416 4.385 4.353 4.320 4.289

*Computed from the above table with the aid of the thermodynamic equation pw −1.1489t log10  =  − 1.330 × 10−5t2 + 9.084 × 10−8t3 273.1 + t pi

TABLE 2-5

Vapor Pressure of Liquid Water from 0 to 100°C* mmHg

t, °C

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 1 2 3 4

4.579 4.926 5.294 5.685 6.101

4.613 4.962 5.332 5.725 6.144

4.647 4.998 5.370 5.766 6.187

4.681 5.034 5.408 5.807 6.230

4.715 5.070 5.447 5.848 6.274

4.750 5.107 5.486 5.889 6.318

4.785 5.144 5.525 5.931 6.363

4.820 5.181 5.565 5.973 6.408

4.855 5.219 5.605 6.015 6.453

4.890 5.256 5.645 6.058 6.498

5 6 7 8 9

6.543 7.013 7.513 8.045 8.609

6.589 7.062 7.565 8.100 8.668

6.635 7.111 7.617 8.155 8.727

6.681 7.160 7.669 8.211 8.786

6.728 7.209 7.722 8.267 8.845

6.775 7.259 7.775 8.323 8.905

6.822 7.309 7.828 8.380 8.965

6.869 7.360 7.882 8.437 9.025

6.917 7.411 7.936 8.494 9.086

6.965 7.462 7.990 8.551 9.147

10 11 12 13 14

9.209 9.844 10.518 11.231 11.987

9.271 9.910 10.588 11.305 12.065

9.333 9.976 10.658 11.379 12.144

9.395 10.042 10.728 11.453 12.223

9.458 10.109 10.799 11.528 12.302

9.521 10.176 10.870 11.604 12.382

9.585 10.244 10.941 11.680 12.462

9.649 10.312 11.013 11.756 12.543

9.714 10.380 11.085 11.833 12.624

9.779 10.449 11.158 11.910 12.706

15 16 17 18 19

12.788 13.634 14.530 15.477 16.477

12.870 13.721 14.622 15.575 16.581

12.953 13.809 14.715 15.673 16.685

13.037 13.898 14.809 15.772 16.789

13.121 13.987 14.903 15.871 16.894

13.205 14.076 14.997 15.971 16.999

13.290 14.166 15.092 16.071 17.105

13.375 14.256 15.188 16.171 17.212

13.461 14.347 15.284 16.272 17.319

13.547 14.438 15.380 16.374 17.427

20 21 22 23 24

17.535 18.650 19.827 21.068 22.377

17.644 18.765 19.948 21.196 22.512

17.753 18.880 20.070 21.324 22.648

17.863 18.996 20.193 21.453 22.785

17.974 19.113 20.316 21.583 22.922

18.085 19.231 20.440 21.714 23.060

18.197 19.349 20.565 21.845 23.198

18.309 19.468 20.690 21.977 23.337

18.422 19.587 20.815 22.110 23.476

18.536 19.707 20.941 22.243 23.616

25 26 27 28 29

23.756 25.209 26.739 28.349 30.043

23.897 25.359 26.897 28.514 30.217

24.039 25.509 27.055 28.680 30.392

24.182 25.660 27.214 28.847 30.568

24.326 25.812 27.374 29.015 30.745

24.471 25.964 27.535 29.184 30.923

24.617 26.117 27.696 29.354 31.102

24.764 26.271 27.858 29.525 31.281

24.912 26.426 28.021 29.697 31.461

25.060 26.582 28.185 29.870 31.642

30 31 32 33 34

31.824 33.695 35.663 37.729 39.898

32.007 33.888 35.865 37.942 40.121

32.191 34.082 36.068 38.155 40.344

32.376 34.276 36.272 38.369 40.569

32.561 34.471 36.477 33.584 40.796

32.747 34.667 36.683 38.801 41.023

32.934 34.864 36.891 39.018 41.251

33.122 35.062 37.099 39.237 41.480

33.312 35.261 37.308 39.457 41.710

33.503 35.462 37.518 39.677 41.942

35 36 37 38 39

42.175 44.563 47.067 49.692 52.442

42.409 44.808 47.324 49.961 52.725

42.644 45.054 47.582 50.231 53.009

42.880 45.301 47.841 50.502 53.294

43.117 45.549 48.102 50.774 53.580

43.355 45.799 48.364 51.048 53.867

43.595 46.050 48.627 51.323 54.156

43.836 46.302 48.891 51.600 54.446

44.078 46.556 49.157 51.879 54.737

44.320 46.811 49.424 52.160 55.030

40 41 42 43 44

55.324 58.34 61.50 64.80 68.26

55.61 58.65 61.82 65.14 68.61

55.91 58.96 62.14 65.48 68.97

56.21 59.27 62.47 65.82 69.33

56.51 59.58 62.80 66.16 69.69

56.81 59.90 63.13 66.51 70.05

57.11 60.22 63.46 66.86 70.41

57.41 60.54 63.79 67.21 70.77

57.72 60.86 64.12 67.56 71.14

58.03 61.18 64.46 67.91 71.51

45 46 47 48 49

71.88 75.65 79.60 83.71 88.02

72.25 76.04 80.00 84.13 88.46

72.62 76.43 80.41 84.56 88.90

72.99 76.82 80.82 84.99 89.34

73.36 77.21 81.23 85.42 89.79

73.74 77.60 81.64 85.85 90.24

74.12 78.00 82.05 86.28 90.69

74.50 78.40 82.46 86.71 91.14

74.88 78.80 82.87 87.14 91.59

75.26 79.20 83.29 87.58 92.05

t, °C

0

1

2

3

4

5

6

7

8

9

50 60 70 80

92.51 149.38 233.7 355.1

97.20 156.43 243.9 369.7

102.09 163.77 254.6 384.9

107.20 171.38 265.7 400.6

112.51 179.31 277.2 416.8

118.04 187.54 289.1 433.6

123.80 196.09 301.4 450.9

129.82 204.96 314.1 468.7

136.08 214.17 327.3 487.1

142.60 223.73 341.0 506.1

90 91 92 93 94

525.76 546.05 566.99 588.60 610.90

527.76 548.11 569.12 590.80 613.17

529.77 550.18 571.26 593.00 615.44

531.78 552.26 573.40 595.21 617.72

533.80 554.35 575.55 597.43 620.01

535.82 556.44 577.71 599.66 622.31

537.86 558.53 579.87 601.89 624.61

539.90 560.64 582.04 604.13 626.92

541.95 562.75 584.22 606.38 629.24

544.00 564.87 586.41 608.64 631.57

95 96 97 98 99 100 101

633.90 657.62 682.07 707.27 733.24 760.00 787.57

636.24 660.03 684.55 709.83 735.88 762.72 790.37

638.59 662.45 687.04 712.40 738.53 765.45 793.18

640.94 664.88 689.54 714.98 741.18 768.19 796.00

643.30 667.31 692.05 717.56 743.85 770.93 796.82

645.67 669.75 694.57 720.15 746.52 773.68 801.66

648.05 672.20 697.10 722.75 749.20 776.44 804.50

650.43 674.66 699.63 725.36 751.89 779.22 807.35

652.82 677.12 702.17 727.98 754.58 782.00 810.21

655.22 679.69 704.71 730.61 757.29 784.78 813.06

*From the Physikalisch-technische Reichsanstalt, Holborn, Scheel, and Henning, Wärmetabellen, Friedrich Vieweg & Sohn, Brunswick, 1909. By permission. For data at 50(0.2)101.8°C, see Handbook of Chemistry and Physics, 40th ed., p. 2326, Chemical Rubber Publishing Co. For a tabulation of temperature for pressures 700(1)779 mm Hg, see Atack, Handbook of Chemical Data, p. 117, Reinhold, New York, 1957. For a tabulation of pressure for 105(5)200(10)370°C, see Atack, p. 134, and for 100(1)374°C, see Handbook of Chemistry and Physics, 40th ed., pp. 2328–2330, Chemical Rubber Publishing Co.

2-49

TABLE 2-6 Cmpd. no.

Vapor Pressure of Inorganic and Organic Liquids CAS no.

C1

C2

C3

C4

C5

Tmin, K

Ps at Tmin

Tmax, K

Ps at Tmax

1 2 3 4 5 6 7 8 9 10

Methane Ethane Propane n-Butane n-Pentane n-Hexane n-Heptane n-Octane n-Nonane n-Decane

CH4 C2H6 C3H8 C4H10 C5H12 C6H14 C7H16 C8H18 C9H20 C10H22

74828 74840 74986 106978 109660 110543 142825 111659 111842 124185

39.205 51.857 59.078 66.343 78.741 104.65 87.829 96.084 109.35 112.73

−1324.4 −2598.7 −3492.6 −4363.2 −5420.3 −6995.5 −6996.4 −7900.2 −9030.4 −9749.6

−3.4366 −5.1283 −6.0669 −7.046 −8.8253 −12.702 −9.8802 −11.003 −12.882 −13.245

3.1019E−05 1.4913E−05 1.0919E−05 9.4509E−06 9.6171E−06 1.2381E−05 7.2099E−06 7.1802E−06 7.8544E−06 7.1266E−06

2 2 2 2 2 2 2 2 2 2

90.69 90.35 85.47 134.86 143.42 177.83 182.57 216.38 219.66 243.51

1.1687E+04 1.1273E+00 1.6788E−04 6.7441E−01 6.8642E−02 9.0169E−01 1.8269E−01 2.1083E+00 4.3058E−01 1.3930E+00

190.56 305.32 369.83 425.12 469.7 507.6 540.2 568.7 594.6 617.7

4.5897E+06 4.8522E+06 4.2135E+06 3.7699E+06 3.3642E+06 3.0449E+06 2.7192E+06 2.4673E+06 2.3054E+06 2.0908E+06

11 12 13 14 15 16 17 18 19 20

n-Undecane n-Dodecane n-Tridecane n-Tetradecane n-Pentadecane n-Hexadecane n-Heptadecane n-Octadecane n-Nonadecane n-Eicosane

C11H24 C12H26 C13H28 C14H30 C15H32 C16H34 C17H36 C18H38 C19H40 C20H42

1120214 112403 629505 629594 629629 544763 629787 593453 629925 112958

131 137.47 137.45 140.47 135.57 156.06 156.95 157.68 182.54 203.66

−15.855 −16.698 −16.543 −16.859 −16.022 −18.941 −18.966 −18.954 −22.498 −25.525

8.1871E−06 8.0906E−06 7.1275E−06 6.5877E−06 5.6136E−06 6.8172E−06 6.4559E−06 5.9272E−06 7.4008E−06 8.8382E−06

2 2 2 2 2 2 2 2 2 2

247.57 263.57 267.76 279.01 283.07 291.31 295.13 301.31 305.04 309.58

4.0836E−01 6.1534E−01 2.5096E−01 2.5268E−01 1.2884E−01 9.2265E−02 4.6534E−02 3.3909E−02 1.5909E−02 9.2574E−03

639 658 675 693 708 723 736 747 758 768

1.9493E+06 1.8223E+06 1.6786E+06 1.5693E+06 1.4743E+06 1.4106E+06 1.3438E+06 1.2555E+06 1.2078E+06 1.1746E+06

21 22 23 24 25 26 27

2-Methylpropane 2-Methylbutane 2,3-Dimethylbutane 2-Methylpentane 2,3-Dimethylpentane 2,3,3-Trimethylpentane 2,2,4-Trimethylpentane

C4H10 C5H12 C6H14 C6H14 C7H16 C8H18 C8H18

75285 78784 79298 107835 565593 560214 540841

100.18 72.35 77.235 77.36 78.282 83.105 87.868

−4841.9 −5010.9 −5695.9 −5791.7 −6347 −6903.7 −6831.7

−13.541 −7.883 −8.5109 −8.4912 −8.502 −9.1858 −9.9783

2.0063E−02 8.9795E−06 8.0163E−06 7.7939E−06 6.4169E−06 6.4703E−06 7.7729E−06

1 2 2 2 2 2 2

113.54 113.25 145.19 119.55 160 172.22 165.78

1.4051E−02 1.1569E−04 1.5081E−02 9.2204E−06 1.2631E−02 1.6820E−02 1.6187E−02

408.14 460.43 499.98 497.5 537.35 573.5 543.96

3.6199E+06 3.3709E+06 3.1255E+06 3.0192E+06 2.8823E+06 2.8116E+06 2.5630E+06

28 29 30 31 32 33 34 35

Ethylene Propylene 1-Butene cis-2-Butene trans-2-Butene 1-Pentene 1-Hexene 1-Heptene

C2H4 C3H6 C4H8 C4H8 C4H8 C5H10 C6H12 C7H14

74851 115071 106989 590181 624646 109671 592416 592767

74.242 57.263 68.49 102.62 70.589 120.15 85.3 92.68

−2707.2 −3382.4 −4350.2 −5260.3 −4530.4 −6192.4 −6171.7 −7055.2

−9.8462 −5.7707 −7.4124 −13.764 −7.7229 −16.597 −9.702 −10.679

2.2457E−02 1.0431E−05 1.0503E−05 1.9183E−02 1.0928E−05 2.1922E−02 8.9604E−06 8.4459E−06

1 2 2 1 2 1 2 2

104 87.89 87.8 134.26 167.62 107.93 133.39 154.27

1.2361E+02 9.3867E−04 7.1809E−07 2.4051E−01 7.4729E+01 3.5210E−06 2.5272E−04 1.2810E−03

282.34 365.57 419.95 435.58 428.63 464.78 504.03 537.29

5.0296E+06 4.6346E+06 4.0391E+06 4.2388E+06 4.0811E+06 3.5557E+06 3.1397E+06 2.8225E+06

36 37 38 39 40 41 42 43 44

1-Octene 1-Nonene 1-Decene 2-Methylpropene 2-Methyl-1-butene 2-Methyl-2-butene 1,2-Butadiene 1,3-Butadiene 2-Methyl-1,3-butadiene

C8H16 C9H18 C10H20 C4H8 C5H10 C5H10 C4H6 C4H6 C5H8

111660 124118 872059 115117 563462 513359 590192 106990 78795

97.57 144.45 78.808 102.5 97.33 82.605 39.714 73.522 79.656

−7836 −9676.2 −8367.9 −5021.8 −5631.8 −5606.6 −3769.9 −4564.3 −5239.6

−11.272 −19.446 −7.9553 −13.88 −12.589 −9.4236 −2.6407 −8.1958 −9.4314

7.7267E−06 1.8031E−02 8.7442E−18 2.0296E−02 1.5395E−02 1.0512E−05 6.9379E−18 1.1580E−05 9.5850E−03

2 1 6 1 1 2 6 2 1

171.45 191.78 206.89 132.81 135.58 139.39 136.95 164.25 127.27

2.7570E−03 8.5514E−03 1.7308E−02 6.2213E−01 1.9687E−02 1.9447E−02 4.4720E−01 6.9110E+01 2.4768E−03

566.65 593.25 616.4 417.9 465 471 452 425.17 484

2.5735E+06 2.3308E+06 2.2092E+06 3.9760E+06 3.4544E+06 3.3769E+06 4.3613E+06 4.3041E+06 3.8509E+06

45 46 47 48 49 50

Acetylene Methylacetylene Dimethylacetylene 3-Methyl-1-butyne 1-Pentyne 2-Pentyne

C2H2 C3H4 C4H6 C5H8 C5H8 C5H8

74862 74997 503173 598232 627190 627214

172.06 119.42 66.592 69.459 82.805 137.29

−5318.5 −5364.5 −4999.8 −5250 −5683.8 −7447.1

−27.223 −16.81 −6.8387 −7.1125 −9.4301 −19.01

5.4619E−02 2.5523E−02 6.6793E−06 7.9289E−17 1.0767E−05 2.1415E−02

1 1 2 6 2 1

192.4 170.45 240.91 183.45 167.45 163.83

1.2603E+05 3.7264E+02 6.1212E+03 4.3551E+01 2.3990E+00 2.0462E−01

308.32 402.39 473.2 463.2 481.2 519

6.1467E+06 5.6206E+06 4.8699E+06 4.1986E+06 4.1701E+06 4.0198E+06

51 52 53 54 55 56

1-Hexyne 2-Hexyne 3−Hexyne 1-Heptyne 1-Octyne Vinylacetylene1

C6H10 C6H10 C6H10 C7H12 C8H14 C4H4

693027 764352 928494 628717 629050 689974

133.2 123.71 47.091 66.447 82.353 55.682

−7492.9 −7639 −5104 −6395.6 −7240.6 −4439.3

−18.405 −16.451 −3.6371 −6.3848 −9.1843 −5.0136

2.2062Ε−02 1.6495Ε−02 5.1621Ε−04 1.1250E−17 5.8038E−03 1.9650E−17

1 1 1 6 1 6

141.25 183.65 170.05 192.22 193.55 173.15

3.9157Ε−04 5.4026Ε−01 2.1950Ε−01 6.7026E−01 1.0092E−01 6.6899E+01

516.2 549 544 559 585 454

3.6352Ε+06 3.5301Ε+06 3.5397Ε+06 3.1343E+06 2.8202E+06 4.8874E+06

2-50

Name

Formula

−11143 −11976 −12549 −13231 −13478 −15015 −15557 −16093 −17897 −19441

57 58 59 60 61 62 63 64 65 66

Cyclopentane Methylcyclopentane Ethylcyclopentane Cyclohexane Methylcyclohexane 1,1-Dimethylcyclohexane Ethylcyclohexane Cyclopentene 1-Methylcyclopentene Cyclohexene

C5H10 C6H12 C7H14 C6H12 C7H14 C8H16 C8H16 C5H8 C6H10 C6H10

287923 96377 1640897 110827 108872 590669 1678917 142290 693890 110838

51.434 79.673 88.622 116.51 92.611 81.184 80.208 49.88 52.732 88.184

−4770.6 −6086.6 −7011 −7103.3 −7077.8 −6927 −7203.2 −4649.7 −5286.9 −6624.9

−4.3515 −8.7933 −10.038 −15.49 −10.684 −8.8498 −8.6023 −4.1191 −4.4509 −10.059

1.9605E−17 7.4046E−06 7.4481E−06 1.6959E−02 8.1239E−06 5.4580E−06 4.5901E−06 1.9564E−17 1.0883E−17 8.2566E−06

6 2 2 1 2 2 2 6 6 2

179.28 130.73 134.71 279.69 146.58 239.66 161.84 138.13 146.62 169.67

9.4420E+00 6.7059E−05 3.7061E−06 5.3802E+03 1.5256E−04 6.0584E+01 3.5747E−04 1.6884E−02 3.9787E−03 1.0377E−01

511.76 532.79 569.52 553.58 572.19 591.15 609.15 507 542 560.4

4.5028E+06 3.7808E+06 3.3970E+06 4.0958E+06 3.4828E+06 2.9387E+06 3.0411E+06 4.8062E+06 4.1303E+06 4.3922E+06

67 68 69 70 71 72 73

Benzene Toluene o-Xylene m-Xylene p-Xylene Ethylbenzene Propylbenzene

C6H6 C7H8 C8H10 C8H10 C8H10 C8H10 C9H12

71432 108883 95476 108383 106423 100414 103651

83.918 80.877 90.356 84.782 85.475 88.09 136.83

−6517.7 −6902.4 −7948.7 −7598.3 −7595.8 −7688.3 −9544.8

−9.3453 −8.7761 −10.081 −9.2612 −9.378 −9.7708 −18.190

7.1182E−06 5.8034E−06 5.9756E−06 5.5445E−06 5.6875E−06 5.8844E−06 1.6590E−02

2 2 2 2 2 2 1

278.68 178.18 247.98 225.3 286.41 178.15 324.18

4.7620E+03 4.2348E−02 2.1968E+01 3.2099E+00 5.8144E+02 4.0140E−03 2.0014E+03

562.16 591.8 630.33 617.05 616.23 617.2 638.32

4.8819E+06 4.1012E+06 3.7424E+06 3.5286E+06 3.4984E+06 3.5968E+06 3.2001E+06

74 75 76 77 78 79 80 81

1,2,4-Trimethylbenzene Isopropylbenzene 1,3,5-Trimethylbenzene p-Isopropyltoluene Naphthalene Biphenyl Styrene m-Terphenyl

C9H12 C9H12 C9H12 C10H14 C10H8 C12H10 C8H8 C18H14

95636 98828 108678 99876 91203 92524 100425 92068

60.658 143.62 48.603 107.71 62.447 76.811 105.93 88.044

−7260.4 −9687.7 −6545.2 −9402.7 −8109 −9878.5 −8685.9 −13367

−5.3772 −19.305 −3.6412 −12.545 −5.5571 −7.4384 −12.42 −8.6482

4.5816E−18 1.7703E−02 1.9307E−18 6.6661E−06 2.0800E−18 2.0436E−18 7.5583E−06 8.7874E−19

6 1 6 2 6 6 2 6

229.33 177.14 228.42 205.25 353.43 342.2 242.54 360

7.9735E−01 3.8034E−04 1.1889E+00 9.9261E−03 9.9229E+02 9.3752E+01 1.0613E+01 1.0112E+00

649.13 631.1 637.36 653.15 748.35 789.26 636 924.85

3.2533E+06 3.1837E+06 3.1119E+06 2.7957E+06 3.9941E+06 3.8615E+06 3.8234E+06 3.5297E+06

82 83 84 85 86 87 88

Methanol Ethanol 1-Propanol 1-Butanol 2-Butanol 2-Propanol 2-Methyl-2-propanol

CH4O C2H6O C3H8O C4H10O C4H10O C3H8O C4H10O

67561 64175 71238 71363 78922 67630 75650

81.768 74.475 88.134 93.173 152.54 76.964 172.31

−6876 −7164.3 −8498.6 −9185.9 −11111 −7623.8 −11590

−8.7078 −7.327 −9.0766 −9.7464 −19.025 −7.4924 −22.118

7.1926E−06 3.1340E−06 8.3303E−18 4.7796E−18 1.0426E−05 5.9436E−18 1.3709E−05

2 2 6 6 2 6 2

175.47 159.05 146.95 184.51 158.45 185.28 298.97

1.1147E−01 4.8459E−04 3.0828E−07 5.7220E−04 1.1323E−06 3.6606E−02 5.9356E+03

512.64 513.92 536.78 563.05 536.05 508.3 506.21

8.1402E+06 6.1171E+06 5.1214E+06 4.3392E+06 4.2014E+06 4.7908E+06 3.9910E+06

89 90 91 92 93 94 95 96

1-Pentanol 2-Methyl-1-butanol 3-Methyl-1-butanol 1-Hexanol 1-Heptanol Cyclohexanol Ethylene glycol 1,2-Propylene glycol

C5H12O C5H12O C5H12O C6H14O C7H16O C6H12O C2H6O2 C3H8O2

71410 137326 123513 111273 111706 108930 107211 57556

168.96 410.44 107.02 117.31 160.08 135.01 79.276 212.8

−12659 −20262 −10237 −11239 −14095 −12238 −10105 −15420

−21.366 −62.366 −11.695 −13.149 −19.211 −15.702 −7.521 −28.109

1.1591E−05 6.3353E−02 6.8003E−18 9.3676E−18 1.7043E−17 1.0349E−17 7.3408E−19 2.1564E−05

2 1 6 6 6 6 6 2

195.56 203 155.95 228.55 239.15 296.6 260.15 213.15

3.1816E−04 3.7992E−04 2.1036E−08 3.7401E−02 1.6990E−02 7.9382E+01 2.4834E−01 9.2894E−05

586.15 565 577.2 611.35 631.9 650 719.7 626

3.8657E+06 3.8749E+06 3.9013E+06 3.4557E+06 3.1810E+06 4.2456E+06 7.7100E+06 6.0413E+06

97 98 99 100

Phenol o-Cresol m-Cresol p-Cresol

C6H6O C7H8O C7H8O C7H8O

108952 95487 108394 106445

95.444 210.88 95.403 118.53

−10113 −13928 −10581 −11957

−10.09 −29.483 −10.004 −13.293

6.7603E−18 2.5182E−02 4.3032E−18 8.6988E−18

6 1 6 6

314.06 304.19 285.39 307.93

1.8798E+02 6.5326E+01 5.8624E+00 3.4466E+01

694.25 697.55 705.85 704.65

6.0585E+06 5.0583E+06 4.5221E+06 5.1507E+06

101 102 103 104 105 106 107 108 109 110 111 112

Dimethyl ether Methyl ethyl ether Methyl n-propyl ether Methyl isopropyl ether Methyl-n-butyl ether Methyl isobutyl ether Methyl tert-butyl ether Diethyl ether Ethyl propyl ether Ethyl isopropyl ether Methyl phenyl ether Diphenyl ether

C2H6O C3H8O C4H10O C4H10O C5H12O C5H12O C5H12O C4H10O C5H12O C5H12O C7H8O C12H10O

115106 540670 557175 598538 628284 625445 1634044 60297 628320 625547 100663 101848

44.704 205.79 50.83 55.096 102.04 58.165 55.875 136.9 143.11 57.723 128.06 59.969

−3.4444 −28.739 −4.1773 −4.8689 −12.278 −5.2568 −4.9604 −19.254 −18.751 −5.2136 −16.693 −5.1538

5.4574E−17 3.5317E−05 9.4076E−18 2.9518E−17 1.2131E−05 2.0194E−17 1.9123E−17 2.4508E−02 2.0620E−05 2.2998E−17 1.4919E−02 1.9983E−18

6 2 6 6 2 6 6 1 2 6 1 6

131.65 160 133.97 127.93 157.48 150 164.55 156.85 145.65 140 235.65 300.03

3.0496E+00 5.3423E−01 4.8875E−03 2.4971E−03 1.9430E−02 1.9801E−02 5.3566E−01 3.9545E−01 7.3931E−04 4.3092E−03 2.4466E+00 7.0874E+00

400.1 437.8 476.3 464.5 510 497 497.1 466.7 500.23 489 645.6 766.8

5.2735E+06 4.4658E+06 3.7721E+06 3.8892E+06 3.3089E+06 3.4130E+06 3.4106E+06 3.6412E+06 3.3729E+06 3.4145E+06 4.2731E+06 3.0971E+06

−3525.6 −9834.5 −4781.7 −4793.2 −6954.9 −5362.1 −5131.6 −6954.3 −8353.7 −5236.9 −9307.7 −8585.5

2-51

TABLE 2-6 Cmpd. no.

Vapor Pressure of Inorganic and Organic Liquids (Continued ) CAS no.

C1

C2

C3

C4

C5

Tmin, K

Ps at Tmin

Tmax, K

Ps at Tmax

113 114 115 116 117 118 119 120 121 122

Formaldehyde Acetaldehyde 1-Propanal 1-Butanal 1-Pentanal 1-Hexanal 1-Heptanal 1-Octanal 1-Nonanal 1-Decanal

CH2O C2H4O C3H6O C4H8O C5H10O C6H12O C7H14O C8H16O C9H18O C10H20O

50000 75070 123386 123728 110623 66251 111717 124130 124196 112312

101.51 193.69 80.581 99.33 149.58 81.507 107.17 250.25 337.71 201.64

−4917.2 −8036.7 −5896.1 −7083.6 −8890 −7776.8 −9070.3 −16162 −18506 −15133

−13.765 −29.502 −8.9301 −11.733 −20.697 −8.4516 −12.503 −33.927 −50.224 −26.264

2.2031E−02 4.3678E−02 8.2236E−06 1.0027E−05 2.2101E−02 1.5143E−17 7.4446E−06 2.2349E−05 4.7345E−02 1.4625E−05

1 1 2 2 1 6 2 2 1 2

181.15 150.15 170 176.75 182 217.15 229.8 246 255.15 267.15

8.8700E+02 3.2320E−01 1.3133E+00 3.1699E−01 5.2282E−02 1.2473E+00 1.1177E+00 4.1640E−01 3.4172E−01 4.8648E−01

408 466 504.4 537.2 566.1 591 617 638.1 658 674.2

6.5935E+06 5.5652E+06 4.9189E+06 4.3232E+06 3.9685E+06 3.4607E+06 3.1829E+06 2.9704E+06 2.7430E+06 2.5989E+06

123 124 125 126 127 128 129 130 131 132 133 134

Acetone Methyl ethyl ketone 2-Pentanone Methyl isopropyl ketone 2-Hexanone Methyl isobutyl ketone 3-Methyl-2-pentanone 3-Pentanone Ethyl isopropyl ketone Diisopropyl ketone Cyclohexanone Methyl phenyl ketone

C3H6O C4H8O C5H10O C5H10O C6H12O C6H12O C6H12O C5H10O C6H12O C7H14O C6H10O C8H8O

67641 78933 107879 563804 591786 108101 565617 96220 565695 565800 108941 98862

69.006 72.698 84.635 308.74 65.841 153.23 64.641 44.286 206.77 96.919 95.118 62.688

−5599.6 −6143.6 −7078.4 −13693 −7042 −10055 −6457.4 −5415.1 −12537 −8014.2 −8300.4 −8088.8

−7.0985 −7.5779 −9.3 −47.557 −6.1376 −19.848 −6.218 −3.0913 −27.894 −11.093 −10.796 −5.5434

6.2237E−06 5.6476E−06 6.2702E−06 5.7002E−02 7.2196E−18 1.6426E−05 3.4543E−06 1.8580E−18 2.2462E−05 7.3452E−06 6.5037E−06 2.0774E−18

2 2 2 1 6 2 2 6 2 2 2 6

178.45 186.48 196.29 181.15 217.35 189.15 167.15 234.18 200 204.81 242 292.81

2.7851E+00 1.3904E+00 7.5235E−01 2.2648E−02 1.5111E+00 3.3536E−02 3.2662E−03 7.3422E+01 6.0339E−02 3.9036E−01 6.9667E+00 3.5899E+01

508.2 535.5 561.08 553 587.05 571.4 573 560.95 567 576 653 709.5

4.7091E+06 4.1201E+06 3.7062E+06 3.8413E+06 3.3120E+06 3.2659E+06 3.3213E+06 3.6993E+06 3.3424E+06 3.0606E+06 4.0126E+06 3.8451E+06

135 136 137 138 139 140 141

Formic acid Acetic acid Propionic acid n-Butyric acid Isobutyric acid Benzoic acid Acetic anhydride

CH2O2 C2H4O2 C3H6O2 C4H8O2 C4H8O2 C7H6O2 C4H6O3

64186 64197 79094 107926 79312 65850 108247

50.323 53.27 54.552 93.815 110.38 88.513 100.95

−5378.2 −6304.5 −7149.4 −9942.2 −10540 −11829 −8873.2

−4.203 −4.2985 −4.2769 −9.8019 −12.262 −8.6826 −11.451

3.4697E−06 8.8865E−18 1.1843E−18 9.3124E−18 1.4310E−17 2.3248E−19 6.1316E−06

2 6 6 6 6 6 2

281.45 289.81 252.45 267.95 227.15 395.45 200.15

2.4024E+03 1.2769E+03 1.3142E+01 6.7754E+00 7.8244E−02 7.9550E+02 2.1999E−02

588 591.95 600.81 615.7 605 751 606

5.8074E+06 5.7390E+06 4.6080E+06 4.0705E+06 3.6834E+06 4.4691E+06 3.9702E+06

142 143 144 145 146 147 148

Methyl formate Methyl acetate Methyl propionate Methyl n-butyrate Ethyl formate Ethyl acetate Ethyl propionate

C2H4O2 C3H6O2 C4H8O2 C5H10O2 C3H6O2 C4H8O2 C5H10O2

107313 79209 554121 623427 109944 141786 105373

77.184 61.267 70.717 71.87 73.833 66.824 105.64

−5606.1 −5618.6 −6439.7 −6885.7 −5817 −6227.6 −8007

−8.392 −5.6473 −6.9845 −7.0944 −7.809 −6.41 −12.477

7.8468E−06 2.1080E−17 2.0129E−17 1.4903E−17 6.3200E−06 1.7914E−17 9.0000E−06

2 6 6 6 2 6 2

174.15 175.15 185.65 187.35 193.55 189.6 199.25

6.8808E+00 1.0170E+00 6.3409E−01 1.3435E−01 1.8119E+01 1.4318E+00 7.7988E−01

487.2 506.55 530.6 554.5 508.4 523.3 546

5.9829E+06 4.6948E+06 4.0278E+06 3.4797E+06 4.7080E+06 3.8502E+06 3.3365E+06

149 150 151 152 153 154 155

Ethyl n-butyrate n-Propyl formate n-Propyl acetate n-Butyl acetate Methyl benzoate Ethyl benzoate Vinyl acetate

C6H12O2 C4H8O2 C5H10O2 C6H12O2 C8H8O2 C9H10O2 C4H6O2

105544 110747 109604 123864 93583 93890 108054

57.661 104.08 115.16 71.34 82.976 53.024 57.406

−6346.5 −7535.9 −8433.9 −7285.8 −9226.1 −7676.8 −5702.8

−5.032 −12.348 −13.934 −6.9459 −8.4427 −4.1593 −5.0307

8.2534E−18 9.6020E−06 1.0346E−05 9.9895E−18 5.9115E−18 1.6850E−18 1.1042E−17

6 2 2 6 6 6 6

175.15 180.25 178.15 199.65 260.75 238.45 180.35

1.0390E−02 2.1101E−01 1.7113E−02 1.4347E−01 1.8653E+00 1.4385E−01 7.0586E−01

571 538 549.73 579.15 693 698 519.13

2.9352E+06 4.0310E+06 3.3657E+06 3.1097E+06 3.5896E+06 3.2190E+06 3.9298E+06

156 157 158 159 160 161

Methylamine Dimethylamine Trimethylamine Ethylamine Diethylamine Triethylamine

CH5N C2H7N C3H9N C2H7N C4H11N C6H15N

74895 124403 75503 75047 109897 121448

75.206 71.738 134.68 81.56 49.314 56.55

−5082.8 −5302 −6055.8 −5596.9 −4949 −5681.9

−8.0919 −7.3324 −19.415 −9.0779 −3.9256 −4.9815

8.1130E−06 6.4200E−17 2.8619E−02 8.7920E−06 9.1978E−18 1.2363E−17

2 6 1 2 6 6

179.69 180.96 156.08 192.15 223.35 158.45

1.7671E+02 7.5575E+01 9.9206E+00 1.5183E+02 3.7411E+02 1.0646E−02

430.05 437.2 433.25 456.15 496.6 535.15

7.4139E+06 5.2583E+06 4.1020E+06 5.5937E+06 3.6744E+06 3.0373E+06

162 163 164 165 166

n-Propylamine di-n-Propylamine Isopropylamine Diisopropylamine Aniline

C3H9N C6H15N C3H9N C6H15N C6H7N

107108 142847 75310 108189 62533

58.398 54 136.66 462.84 66.287

−5312.7 −6018.5 −7201.5 −18227 −8207.1

−5.2876 −4.4981 −18.934 −73.734 −6.0132

1.9913E−06 9.9684E−18 2.2255E−02 9.2794E−02 2.8414E−18

2 6 1 1 6

188.36 210.15 177.95 176.85 267.13

1.3004E+01 3.6942E+00 7.7251E+00 4.4724E−03 7.1322E+00

496.95 550 471.85 523.1 699

4.7381E+06 3.1113E+06 4.5404E+06 3.1987E+06 5.3514E+06

2-52

Name

Formula

167 168

N-Methylaniline N,N-Dimethylaniline

C7H9N C8H11N

100618 121697

70.843 51.352

−8517.5 −7160

−6.7007 −4.0127

5.6411E−18 8.1481E−07

6 2

216.15 275.6

1.0207E−02 1.7940E+01

701.55 687.15

5.1935E+06 3.6262E+06

169 170 171 172

Ethylene oxide Furan Thiophene Pyridine

C2H4O C4H4O C4H4S C5H5N

75218 110009 110021 110861

91.944 74.738 89.171 82.154

−5293.4 −5417 −6860.3 −7211.3

−11.682 −8.0636 −10.104 −8.8646

1.4902E−02 7.4700E−06 7.4769E−06 5.2528E−06

1 2 2 2

160.65 187.55 234.94 231.51

7.7879E+00 5.0026E+01 1.8538E+02 2.0535E+01

469.15 490.15 579.35 619.95

7.2553E+06 5.5497E+06 5.7145E+06 5.6356E+06

173 174 175 176 177 178 179 180

Formamide N,N-Dimethylformamide Acetamide N-Methylacetamide Acetonitrile Propionitrile n-Butyronitrile Benzonitrile

CH3NO C3H7NO C2H5NO C3H7NO C2H3N C3H5N C4H7N C7H5N

75127 68122 60355 79163 75058 107120 109740 100470

100.3 82.762 125.81 79.128 58.302 82.699 66.32 55.463

−10763 −7955.5 −12376 −9523.9 −5385.6 −6703.5 −6714.9 −7430.8

−10.946 −8.8038 −14.589 −7.7355 −5.4954 −9.1506 −6.3087 −4.548

3.8503E−06 4.2431E−06 5.0824E−06 3.1616E−18 5.3634E−06 7.5424E−06 1.3516E−17 1.7501E−18

2 2 2 6 2 2 6 6

275.6 212.72 353.33 301.15 229.32 180.26 161.25 260.4

1.0350E+00 1.9532E−01 3.3637E+02 2.8618E+01 1.8694E+02 1.6936E−01 6.1777E−04 5.1063E+00

771 649.6 761 718 545.5 564.4 582.25 699.35

7.7514E+06 4.3653E+06 6.5688E+06 4.9973E+06 4.8517E+06 4.1906E+06 3.7870E+06 4.2075E+06

181 182 183 184 185 186 187 188 189

Methyl mercaptan Ethyl mercaptan n-Propyl mercaptan n-Butyl mercaptan Isobutyl mercaptan sec-Butyl mercaptan Dimethyl sulfide Methyl ethyl sulfide Diethyl sulfide

CH4S C2H6S C3H8S C4H10S C4H10S C4H10S C2H6S C3H8S C4H10S

74931 75081 107039 109795 513440 513531 75183 624895 352932

54.15 65.551 62.165 65.382 61.736 60.649 83.485 79.07 60.867

−4337.7 −5027.4 −5624 −6262.4 −5909.2 −5785.9 −5711.7 −6114.1 −5969.6

−4.8127 −6.6853 −5.8595 −6.2585 −5.7554 −5.6113 −9.4999 −8.631 −5.5979

4.5000E−17 6.3208E−06 2.0597E−17 1.4943E−17 1.5119E−17 1.5877E−17 9.8449E−06 6.5333E−06 1.4530E−17

6 2 6 6 6 6 2 2 6

150.18 125.26 159.95 157.46 128.31 133.02 174.88 167.23 169.2

3.1479E+00 1.1384E−03 6.5102E−02 2.3532E−03 4.7502E−06 3.3990E−05 7.9009E+00 2.2456E−01 4.3401E−02

469.95 499.15 536.6 570.1 559 554 503.04 533 557.15

7.2309E+06 5.4918E+06 4.6272E+06 3.9730E+06 4.0603E+06 4.0598E+06 5.5324E+06 4.2610E+06 3.9629E+06

190 191 192 193 194 195 196 197

Fluoromethane Chloromethane Trichloromethane Tetrachloromethane Bromomethane Fluoroethane Chloroethane Bromoethane

CH3F CH3Cl CHCl3 CCl4 CH3Br C2H5F C2H5Cl C2H5Br

593533 74873 67663 56235 74839 353366 75003 74964

59.123 64.697 146.43 78.441 72.586 56.639 70.159 62.217

−3043.7 −4048.1 −7792.3 −6128.1 −4698.6 −3576.5 −4786.7 −5113.3

−6.1845 −6.8066 −20.614 −8.5766 −7.9966 −5.5801 −7.5387 −5.9761

1.6637E−05 1.0371E−05 2.4578E−02 6.8465E−06 1.1553E−05 9.8969E−06 9.3370E−06 4.7174E−17

2 2 1 2 2 2 2 6

131.35 175.43 207.15 250.33 179.47 129.95 134.8 154.55

4.3287E+02 8.7091E+02 5.2512E+01 1.1225E+03 1.9544E+02 8.3714E+00 1.1658E−01 3.7155E−01

317.42 416.25 536.4 556.35 467 375.31 460.35 503.8

5.8754E+06 6.6905E+06 5.5543E+06 4.5436E+06 7.9972E+06 5.0060E+06 5.4578E+06 6.2903E+06

198 199 200 201 202 203 204 205

1-Chloropropane 2-Chloropropane 1,1-Dichloropropane 1,2-Dichloropropane Vinyl chloride Fluorobenzene Chlorobenzene Bromobenzene

C3H7Cl C3H7Cl C3H6Cl2 C3H6Cl2 C2H3Cl C6H5F C6H5Cl C6H5Br

540545 75296 78999 78875 75014 462066 108907 108861

79.24 46.854 83.495 65.955 91.432 51.915 54.144 63.749

−5718.8 −4445.5 −6661.4 −6015.6 −5141.7 −5439 −6244.4 −7130.2

−8.789 −3.6533 −9.2386 −6.5509 −10.981 −4.2896 −4.5343 −5.879

8.4486E−06 1.3260E−17 6.7652E−06 4.3172E−06 1.4318E−05 8.7527E−18 4.7030E−18 5.2136E−18

2 6 2 2 2 6 6 6

150.35 155.97 200 172.71 119.36 230.94 227.95 242.43

6.9630E−02 9.0844E−01 4.5248E+00 8.2532E−02 1.9178E−02 1.5142E+02 8.4456E+00 7.8364E+00

503.15 489 560 572 432 560.09 632.35 670.15

4.5812E+06 4.5097E+06 4.2394E+06 4.2319E+06 5.7495E+06 4.5437E+06 4.5293E+06 4.5196E+06

206 207 208 209 210 211 212 213 214 215 216 217 218

Air3 Hydrogen Helium-44 Neon Argon Fluorine Chlorine Bromine Oxygen Nitrogen Ammonia Hydrazine Nitrous oxide

H2 He Ne Ar F2 Cl2 Br2 O2 N2 NH3 N2H4 N2O

132259100 1333740 7440597 7440019 7440371 7782414 7782505 7726956 7782447 7727379 7664417 302012 10024972

21.662 12.69 11.533 29.755 42.127 42.393 71.334 108.26 51.245 58.282 90.483 76.858 96.512

−692.39 −94.896 −8.99 −271.06 −1093.1 −1103.3 −3855 −6592 −1200.2 −1084.1 −4669.7 −7245.2 −4045

−0.39208 1.1125 0.6724 −2.6081 −4.1425 −4.1203 −8.5171 −14.16 −6.4361 −8.3144 −11.607 −8.22 −12.277

4.7574E−03 3.2915E−04 2.7430E−01 5.2700E−04 5.7254E−05 5.7815E−05 1.2378E−02 1.6043E−02 2.8405E−02 4.4127E−02 1.7194E−02 6.1557E−03 2.8860E−05

1 2 1 2 2 2 1 1 1 1 1 1 2

59.15 13.95 1.76 24.56 83.78 53.48 172.12 265.85 54.36 63.15 195.41 274.69 182.3

5.6421E+03 7.2116E+03 1.4625E+03 4.3800E+04 6.8721E+04 2.5272E+02 1.3660E+03 5.8534E+03 1.4754E+02 1.2508E+04 6.1111E+03 4.0847E+02 8.6908E+04

132.45 33.19 5.2 44.4 150.86 144.12 417.15 584.15 154.58 126.2 405.65 653.15 309.57

3.7934E+06 1.3154E+06 2.2845E+05 2.6652E+06 4.8963E+06 5.1674E+06 7.7930E+06 1.0276E+07 5.0206E+06 3.3906E+06 1.1301E+07 1.4731E+07 7.2782E+06

2-53

TABLE 2-6 Cmpd. no. 219 220 221 222 223 224 225 226 227 228 229 230 231

Vapor Pressure of Inorganic and Organic Liquids (Concluded ) Name Nitric oxide Cyanogen Carbon monoxide Carbon dioxide Carbon disulfide Hydrogen fluoride Hydrogen chloride Hydrogen bromide2 Hydrogen cyanide Hydrogen sulfide Sulfur dioxide Sulfur trioxide Water

Formula NO C2N2 CO CO2 CS2 HF HCl HBr HCN H2S SO2 SO3 H2O

CAS no.

C1

C2

C3

C4

C5

Tmin, K

Ps at Tmin

Tmax, K

Ps at Tmax

10102439 460195 630080 124389 75150 7664393 7647010 10035106 74908 7783064 7446095 7446119 7732185

72.974 88.589 45.698 140.54 67.114 59.544 104.27 29.315 36.75 85.584 47.365 180.99 73.649

−2650 −5059.9 −1076.6 −4735 −4820.4 −4143.8 −3731.2 −2424.5 −3927.1 −3839.9 −4084.5 −12060 −7258.2

−8.261 −10.483 −4.8814 −21.268 −7.5303 −6.1764 −15.047 −1.1354 −2.1245 −11.199 −3.6469 −22.839 −7.3037

9.7000E−15 1.5403E−05 7.5673E−05 4.0909E−02 9.1695E−03 1.4161E−05 3.1340E−02 2.3806E−18 3.8948E−17 1.8848E−02 1.7990E−17 7.2350E−17 4.1653E−06

6 2 2 1 1 2 1 6 6 1 6 6 2

109.5 245.25 68.15 216.58 161.11 189.79 158.97 185.15 259.83 187.68 197.67 289.95 273.16

2.1956E+04 7.3385E+04 1.5430E+04 5.1867E+05 1.4944E+00 3.3683E+02 1.3522E+04 2.9501E+04 1.8687E+04 2.2873E+04 1.6743E+03 2.0934E+04 6.1056E+02

180.15 400.15 132.92 304.21 552 461.15 324.65 363.15 456.65 373.53 430.75 490.85 647.13

6.5156E+06 5.9438E+06 3.4940E+06 7.3896E+06 8.0408E+06 6.4872E+06 8.3564E+06 8.4627E+06 5.3527E+06 8.9988E+06 7.8596E+06 8.1919E+06 2.1940E+07

All substances are listed in alphabetical order in Table 2-6a. Compiled from Daubert, T. E., R. P. Danner, H. M. Sibul, and C. C. Stebbins, DIPPR Data Compilation of Pure Compound Properties, Project 801 Sponsor Release, July, 1993, Design Institute for Physical Property Data, AIChE, New York, NY; and from Thermodynamics Research Center, “Selected Values of Properties of Hydrocarbons and Related Compounds,” Thermodynamics Research Center Hydrocarbon Project, Texas A&M University, College Station, Texas (extant 1994). Temperatures are in K; vapor pressures are in Pa. Pa × 9.869233E−06 = atm; Pa × 1.450377E−04 = psia; vapor pressure = exp [C1 + (C2/T) + C3 × ln (T) + C4 × T C5 ]. 1 Decomposes violently on heating. Forms explosive peroxides with air or oxygen. Polymerizes under pressure and heat. 2 Coefficients are hypothetical above the decomposition temperature. 3 At the bubble point. 4 Exhibits superfluid properties below 2.2 K.

2-54

VAPOR PRESSURES OF PURE SUBSTANCES

2-55

TABLE 2-6a Alphabetical Index to Substances in Tables 2-6, 2-30, 2-164, 2-193, 2-196, 2-198, and 2-221 Name Acetaldehyde Acetamide Acetic acid Acetic anhydride Acetone Acetonitrile Acetophenone Acetylene Air Ammonia Aniline Anisole Argon Benzene Benzoic acid Benzonitrile Biphenyl Bromine Bromobenzene Bromoethane Bromomethane 1,2-Butadiene 1,3-Butadiene n-Butane 1-Butanol 2-Butanol 1-Butene cis-2-Butene trans-2-Butene n-Butyl acetate n-Butyl mercaptan sec-Butyl mercaptan Butyraldehyde n-Butyric acid n-Butyronitrile Carbon dioxide Carbon disulfide Carbon monoxide Carbon tetrachloride Chlorine Chlorobenzene Chloroethane Chloroform Chloromethane 1-Chloropropane 2-Chloropropane o-Cresol m-Cresol p-Cresol Cumene Cyanogen Cyclohexane Cyclohexanol Cyclohexanone Cyclohexene Cyclopentane Cyclopentene p-Cymene 1-Decanal n-Decane 1-Decene 1,1-Dichloropropane 1,2-Dichloropropane Diethylamine Diethyl ether Diethyl sulfide Diisopropylamine Diisopropyl ketone Dimethylacetylene Dimethylamine N,N-Dimethylaniline 2,3-Dimethylbutane 1,1-Dimethylcyclohexane Dimethyl ether

Synonym Ethanal Ethanoic acid 2-Propanone Methyl cyanide Methyl phenyl ketone

Methyl phenyl ether

Phenyl cyanide 1,1′-Biphenyl Ethyl bromide Methyl bromide

sec-Butyl alcohol Z-2-Butene E-2-Butene 1-Butanethiol 2-Butanethiol Butanal Propyl cyanide

Tetrachloromethane Ethyl chloride Trichloromethane Methyl chloride Propyl chloride Isopropyl chloride 2-Methylphenol 3-Methylphenol 4-Methylphenol Isopropylbenzene Cyclohexyl alcohol Cyclohexyl ketone

p-Isopropyltoluene

Ethyl ether Ethyl sulfide 2,4-Dimethyl-3-pentanone 2-Butyne N,N-Dimethylbenzamine Diisopropyl Methyl ether

Cmpd. no.

Formula

114 175 136 141 123 177 134 45 206 216 166 111 210

C2H4O C2H5NO C2H4O2 C4H6O3 C3H6O C2H3N C8H8O C2H2

67 140 180 79 213 205 197 194 42 43 4 85 86 30 31 32 152 184 186 116 138 179

C6H6 C7H6O2 C7H5N C12H10 Br2 C6H5Br C2H5Br CH3Br C4H6 C4H6 C4H10 C4H10O C4H10O C4H8 C4H8 C4H8 C6H12O2 C4H10S C4H10S C4H8O C4H8O2 C4H7N

222 223 221 193 212 204 196 192 191 198 199 98 99 100 75 220 60 94 133 66 57 64 77

CO2 CS2 CO CCl4 Cl2 C6H5Cl C2H5Cl CHCl3 CH3Cl C3H7Cl C3H7Cl C7H8O C7H8O C7H8O C9H12 C2N2 C6H12 C6H12O C6H10O C6H10 C5H10 C5H8 C10H14

122 10 38 200 201 160 108 189 165 132 47 157 168 23 62 101

C10H20O C10H22 C10H20 C3H6Cl2 C3H6Cl2 C4H11N C4H10O C4H10S C6H15N C7H14O C4H6 C2H7N C8H11N C6H14 C8H16 C2H6O

NH3 C6H7N C7H8O Ar

Name

Synonym

1-Hexene 1-Hexyne 2-Hexyne 3-Hexyne Hydrazine Hydrogen Hydrogen bromide Hydrogen chloride Hydrogen cyanide Hydrogen fluoride Hydrogen sulfide

Cmpd. no.

Formula

34 51 52 53 217 207 226 225 227 224 228

C6H12 C6H10 C6H10 C6H10 N2H4 H2 HBr HCl HCN HF H2S

Isobutyl mercaptan Isobutyric acid Isooctane Isoprene Isopropylamine

2-Methyl-1-propanethiol 2-Methylpropanoic acid 2,2,4-Trimethylpentane 2-Methyl-1,3-butadiene

185 139 27 44 164

C4H10S C4H8O2 C8H18 C5H8 C3H9N

Mesitylene Methane Methanol N-Methylacetamide Methyl acetate Methylacetylene Methylamine N-Methylaniline Methyl benzoate 2-Methylbutane 2-Methyl-1-butanol 3-Methyl-1-butanol 2-Methyl-1-butene 2-Methyl-2-butene Methyl butyl ether 3-Methyl-1-butyne Methyl butyrate Methylcyclohexane Methylcyclopentane 1-Methylcyclopentene Methyl ethyl ether Methyl ethyl ketone Methyl ethyl sulfide Methyl formate Methyl isobutyl ether Methyl isobutyl ketone Methyl isopropyl ether Methyl isopropyl ketone Methyl mercaptan 2-Methylpentane 3-Methyl-2-pentanone 2-Methylpropane 2-Methyl-2-propanol 2-Methylpropene Methyl propionate Methyl propyl ether Methyl tert-butyl ether

1,3,5-Trimethylbenzene

76 1 82 176 143 46 156 167 153 22 90 91 40 41 105 48 145 61 58 65 102 124 188 142 106 128 104 126 181 24 129 21 88 39 144 103 107

C9H12 CH4 CH4O C3H7NO C3H6O2 C3H4 CH5N C7H9N C8H8O2 C5H12 C5H12O C5H12O C5H10 C5H10 C5H12O C5H8 C5H10O2 C7H14 C6H12 C6H10 C3H8O C4H8O C3H8S C2H4O2 C5H12O C6H12O C4H10O C5H10O CH4S C6H14 C6H12O C4H10 C4H10O C4H8 C4H8O2 C4H10O C5H12O

78 209 219 215 218 19 121 9 37

C10H8 Ne NO N2 N2O C19H40 C9H18O C9H20 C9H18

18 120 8 36 55 214

C18H38 C8H16O C8H18 C8H16 C8H14 O2

15 117 5 89

C15H32 C5H10O C5H12 C5H12O

Naphthalene Neon Nitric oxide Nitrogen Nitrous oxide n-Nonadecane 1-Nonanal n-Nonane 1-Nonene n-Octadecane 1-Octanal n-Octane 1-Octene 1-Octyne Oxygen n-Pentadecane 1-Pentanal n-Pentane 1-Pentanol

Methyl alcohol

Isopentane Isoamyl alcohol Amylene

2-Butanone 2-Thiabutane

Methanethiol Isohexane Methyl sec-butyl ketone Isobutane tert-Butyl alcohol Isobutene

n-Nonaldehyde

n-Octaldehyde

Valeraldehyde n-Amyl alcohol

2-56

PHYSICAL AND CHEMICAL DATA

TABLE 2-6a Alphabetical Index to Substances in Tables 2-6, 2-30, 2-164, 2-193, 2-196, 2-198, and 2-221 (Concluded ) Name N,N-Dimethylformamide 2,3-Dimethylpentane Dimethyl sulfide Diphenyl ether n-Dodecane n-Eicosane Ethane Ethanol Ethyl acetate Ethylamine Ethylbenzene Ethyl benzoate Ethyl butyrate Ethylcyclohexane Ethylcyclopentane Ethylene Ethylene glycol Ethylene oxide Ethyl formate Ethyl isopropyl ether Ethyl isopropyl ketone Ethyl mercaptan Ethyl propionate Ethyl propyl ether Fluorine Fluorobenzene Fluoroethane Fluoromethane Formaldehyde Formamide Formic acid Furan Helium-4 n-Heptadecane 1-Heptanal n-Heptane 1-Heptanol 1-Heptene 1-Heptyne n-Hexadecane 1-Hexanal n-Hexane 1-Hexanol 2-Hexanone

Synonym

Methyl sulfide

Ethyl alcohol Phenylethane

1,2-Ethanediol 1,2-Epoxyethane

Ethanethiol

Ethyl fluoride Methyl fluoride Methanal Methanoic acid

n-Heptaldehyde n-Heptyl alcohol

Caproaldehyde n-Hexyl alcohol Methyl n-butyl ketone

Cmpd. no.

Formula

174 25 187 112 12

C3H7NO C7H16 C2H6S C12H10O C12H26

20 2 83 147 159 72 154 149 63 59 28 95 169 146 110 131 182 148 109

C20H42 C2H6 C2H6O C4H8O2 C2H7N C8H10 C9H10O2 C6H12O2 C8H16 C7H14 C2H4 C2H6O2 C2H4O C3H6O2 C5H12O C6H12O C2H6S C5H10O2 C5H12O

211 203 195 190 113 173 135 170

F2 C6H5F C2H5F CH3F CH2O CH3NO CH2O2 C4H4O

208 17 119 7 93 35 54 16 118 6 92 127

He C17H36 C7H14O C7H16 C7H16O C7H14 C7H12 C16H34 C6H12O C6H14 C6H14O C6H12O

Name

Synonym

Cmpd. no.

Formula

Methyl n-propyl ketone Diethyl ketone

125 130 33 49 50 97 115 3 84 87 137 178 151 162 163 73 29 96 150 183 172

C5H10O C5H10O C5H10 C5H8 C5H8 C6H6O C3H6O C3H8 C3H8O C3H8O C3H6O2 C3H5N C5H10O2 C3H9N C6H15N C9H12 C3H6 C3H8O2 C4H8O2 C3H8S C5H5N

Styrene Sulfur dioxide Sulfur trioxide

80 229 230

C8H8 SO2 SO3

m-Terphenyl n-Tetradecane Thiophene Toluene n-Tridecane Triethylamine Trimethylamine 1,2,4-Trimethylbenzene 2,3,3-Trimethylpentane

81 14 171 68 13 161 158 74 26

C18H14 C14H30 C4H4S C7H8 C13H28 C6H15N C3H9N C9H12 C8H18

11

C11H24

Vinyl acetate Vinylacetylene Vinyl chloride

155 56 202

C4H6O2 C4H4 C2H3Cl

Water

231

H2O

69 70 71

C8H10 C8H10 C8H10

2-Pentanone 3-Pentanone 1-Pentene 1-Pentyne 2-Pentyne Phenol 1-Propanal n-Propane 1-Propanol 2-Propanol n-Propionic acid n-Propionitrile n-Propyl acetate n-Propylamine di-n-Propylamine n-Propylbenzene Propylene 1,2-Propylene glycol n-Propyl formate n-Propyl mercaptan Pyridine

Propionaldehyde n-Propyl alcohol Isopropyl alcohol Ethyl cyanide

1,2-Propanediol Propanethiol

n-Undecane

o-Xylene m-Xylene p-Xylene

1,2-Dimethylbenzene 1,3-Dimethylbenzene 1,4-Dimethylbenzene

VAPOR PRESSURES OF PURE SUBSTANCES TABLE 2-7

2-57

Vapor Pressures of Inorganic Compounds, up to 1 atm* Compound

Pressure, mm Hg 1

Name Aluminum borohydride bromide chloride fluoride iodide oxide Ammonia heavy Ammonium bromide carbamate chloride cyanide hydrogen sulfide iodide Antimony tribromide trichloride pentachloride triiodide trioxide Argon Arsenic Arsenic tribromide trichloride trifluoride pentafluoride trioxide Arsine Barium Beryllium borohydride bromide chloride iodide Bismuth tribromide trichloride Diborane hydrobromide Borine carbonyl triamine Boron hydrides dihydrodecaborane dihydrodiborane dihydropentaborane tetrahydropentaborane tetrahydrotetraborane Boron tribromide trichloride trifluoride Bromine pentafluoride Cadmium chloride fluoride iodide oxide Calcium Carbon (graphite) dioxide disulfide monoxide oxyselenide oxysulfide selenosulfide subsulfide tetrabromide tetrachloride tetrafluoride Cesium bromide chloride fluoride iodide

5

10

20

Al Al(BH4)3 AlBr3 Al2Cl6 AlF3 AlI3 Al2O3 NH3 ND3 NH4Br N2H6CO2 NH4Cl NH4CN NH4HS NH4I Sb SbBr3 SbCl3 SbCl5 SbI3 Sb4O6 A As AsBr3 AsCl3 AsF3 AsF5 As2O3 AsH3 Ba Be(BH4)2 BeBr2 BeCl2 BeI2 Bi BiBr3 BiCl3 B2H5Br BH3CO B3N3H6 B10H14 B2H6 B5H9 B5H11 B4H10 BBr3 BCl3 BF3 Br2 BrF5 Cd CdCl2 CdF2 CdI2 CdO Ca C CO2 CS2 CO COSe COS CSeS C3S2 CBr4 CCl4 CF4 Cs CsBr CsCl CsF CsI

40

60

100

200

400

760

Melting point, °C

1749 −3.9 176.1 152.0 1422 294.5 2665 −68.4 −67.4 320.0 26.7 271.5 −0.5 0.0 331.8 1223 203.5 143.3 114.1 303.5 957 −200.5 518 145.2 70.9 13.2 −84.3 332.5 −98.0 1301 58.6 405 411 411 1271 360 343 −29.0 −95.3 +4.0

1844 +11.2 199.8 161.8 1457 322.0 2766 −57.0 −57.0 345.3 37.2 293.2 +9.6 +10.5 355.8 1288 225.7 165.9

1947 28.1 227.0 171.6 1496 354.0 2874 −45.4 −45.4 370.8 48.0 316.5 20.5 21.8 381.0 1364 250.2 192.2

2056 45.9 256.3 180.2 1537 385.5 2977 −33.6 −33.4 396.0 58.3 337.8 31.7 33.3 404.9 1440 275.0 219.0

660 −64. 97. 192.4 1040

333.8 1085 −195.6 548 167.7 89.2 26.7 −75.5 370.0 −87.2 1403 69.0 427 435 435 1319 392 372 −15.4 −85.5 18.5

368.5 1242 −190.6 579 193.6 109.7 41.4 −64.0 412.2 −75.2 1518 79.7 451 461 461 1370 425 405 0.0 −74.8 34.3

401.0 1425 −185.6 610 220.0 130.4 56.3 −52.8 457.2 −62.1 1638 90.0 474 487 487 1420 461 441 +16.3 −64.0 50.6

142.3 −120.9 +9.6 20.1 −28.1 33.5 −32.4 −123.0 +9.3 −4.5 611 797 1486 640 1341 1207 4373 −100.2 −5.1 −205.7 −61.7 −85.9 28.3 109.9 119.7 23.0 −150.7 509 1072 1069 1025 1055

163.8 −111.2 24.6 34.8 −14.0 50.3 −18.9 −115.9 24.3 +9.9 658 847 1561 688 1409 1288 4516 −93.0 +10.4 −201.3 −49.8 −75.0 45.7 130.8 139.7 38.3 −143.6 561 1140 1139 1092 1124

−99.6 40.8 51.2 +0.8 70.0 −3.6 −108.3 41.0 25.7 711 908 1651 742 1484 1388 4660 −85.7 28.0 −196.3 −35.6 −62.7 65.2

−86.5 58.1 67.0 16.1 91.7 +12.7 −100.7 58.2 40.4 765 967 1751 796 1559 1487 4827 −78.2 46.5 −191.3 −21.9 −49.9 85.6

163.5 57.8 −135.5 624 1221 1217 1170 1200

189.5 76.7 −127.7 690 1300 1300 1251 1280

Temperature, °C

Formula 1284 81.3 100.0 1238 178.0 2148 −109.1

1421 −52.2 103.8 116.4 1298 207.7 2306 −97.5

1487 −42.9 118.0 123.8 1324 225.8 2385 −91.9

1555 −32.5 134.0 131.8 1350 244.2 2465 −85.8

1635 −20.9 150.6 139.9 1378 265.0 2549 −79.2

198.3 −26.1 160.4 −50.6 −51.1 210.9 886 93.9 49.2 22.7 163.6 574 −218.2 372 41.8 −11.4

234.5 −10.4 193.8 −35.7 −36.0 247.0 984 126.0 71.4 48.6 203.8 626 −213.9 416 70.6 +11.7

252.0 −2.9 209.8 −28.6 −28.7 263.5 1033 142.7 85.2 61.8 223.5 666 −210.9 437 85.2 +23.5

270.6 +5.3 226.1 −20.9 −20.8 282.8 1084 158.3 100.6 75.8 244.8 729 −207.9 459 101.3 36.0

−117.9 212.5 −142.6

−108.0 242.6 −130.8 984 19.8 325 328 322 1099 261 242 −75.3 −127.3 −45.0

−103.1 259.7 −124.7 1049 28.1 342 346 341 1136 282 264 −66.3 −121.1 −35.3

−98.0 279.2 −117.7 1120 36.8 361 365 361 1177 305 287 −56.4 −114.1 −25.0

290.0 14.0 245.0 −12.6 −12.3 302.8 1141 177.4 117.8 91.0 267.8 812 −204.9 483 118.7 50.0 −2.5 −92.4 299.2 −110.2 1195 46.2 379 384 382 1217 327 311 −45.4 −106.6 −13.2

1684 −13.4 161.7 145.4 1398 277.8 2599 −74.3 −74.0 303.8 19.6 256.2 −7.4 −7.0 316.0 1176 188.1 128.3 101.0 282.5 873 −202.9 498 130.0 58.7 +4.2 −88.5 310.3 −104.8 1240 51.7 390 395 394 1240 340 324 −38.2 −101.9 −5.8

3586 −134.3 −73.8 −222.0 −117.1 −132.4 −47.3 14.0

80.8 −149.5 −40.4 −29.9 −73.1 −20.4 −75.2 −145.4 −32.8 −51.0 455 618 1231 481 1100 926 3828 −124.4 −54.3 −217.2 −102.3 −119.8 −26.5 41.2

90.2 −144.3 −30.7 −19.9 −64.3 −10.1 −66.9 −141.3 −25.0 −41.9 484 656 1286 512 1149 983 3946 −119.5 −44.7 −215.0 −95.0 −113.3 −16.0 54.9

100.0 −138.5 −20.0 −9.2 −54.8 +1.5 −57.9 −136.4 −16.8 −32.0 516 695 1344 546 1200 1046 4069 −114.4 −34.3 −212.8 −86.3 −106.0 −4.4 69.3

−50.0 −184.6 279 748 744 712 738

−30.0 −174.1 341 838 837 798 828

−19.6 −169.3 375 887 884 844 873

−8.2 −164.3 409 938 934 893 923

117.4 −131.6 −8.0 +2.7 −44.3 14.0 −47.8 −131.0 −8.0 −21.0 553 736 1400 584 1257 1111 4196 −108.6 −22.5 −210.0 −76.4 −98.3 +8.6 85.6 96.3 +4.3 −158.8 449 993 989 947 976

127.8 −127.2 −0.4 10.2 −37.4 22.1 −41.2 −127.6 −0.6 −14.0 578 762 1436 608 1295 1152 4273 −104.8 −15.3 −208.1 −70.2 −93.0 17.0 96.0 106.3 12.3 −155.4 474 1026 1023 980 1009

+1.0 289 291 283 1021 −93.3 −139.2 −63.0 60.0 −159.7 −50.2 −90.9 −41.4 −91.5 −154.6 −48.7 −69.3 394 1112 416 1000

*Compiled from the extended tables published by D. R. Stull in Ind. Eng. Chem., 39, 517 (1947).

2050 −77.7 −74.0 520 36 630.5 96.6 73.4 2.8 167 656 −189.2 814 −18 −5.9 −79.8 312.8 −116.3 850 123 490 405 488 271 218 230 −104.2 −137.0 −58.2 99.6 −169 −47.0 −119.9 −45 −107 −126.8 −7.3 −61.4 320.9 568 520 385 851 −57.5 −110.8 −205.0 −138.8 −75.2 +0.4 90.1 −22.6 −183.7 28.5 636 646 683 621

2-58

PHYSICAL AND CHEMICAL DATA

TABLE 2-7

Vapor Pressures of Inorganic Compounds, up to 1 atm (Continued ) Compound

Name Chlorine fluoride trifluoride monoxide dioxide heptoxide Chlorosulfonic acid Chromium carbonyl oxychloride Cobalt chloride nitrosyl tricarbonyl Columbium fluoride Copper Cuprous bromide chloride iodide Cyanogen bromide chloride fluoride Deuterium cyanide Fluorine oxide Germanium bromide chloride hydride Trichlorogermane Tetramethylgermane Digermane Trigermane Gold Helium para-Hydrogen Hydrogen bromide chloride cyanide fluoride iodide oxide (water) sulfide disulfide selenide telluride Iodine heptafluoride Iron pentacarbonyl Ferric chloride Ferrous chloride Krypton Lead bromide chloride fluoride iodide oxide sulfide Lithium bromide chloride fluoride iodide Magnesium chloride Manganese chloride Mercury Mercuric bromide chloride iodide Molybdenum hexafluoride oxide

Pressure, mm Hg 1

5

10

20

−118.0 −98.5

−106.7 −143.4 −80.4 −81.6

−45.3 32.0 1616 36.0 −18.4

−23.8 53.5 1768 58.0 +3.2

−101.6 −139.0 −71.8 −73.1 −59.0 −13.2 64.0 1845 68.3 13.8

−93.3 −134.3 −62.3 −64.3 −51.2 −2.1 75.3 1928 79.5 25.7

86.3 1879 718 702 656 −76.8 −10.0 −53.8 −118.5 −46.7 −214.1 −182.3 56.8 −15.0 −145.3 −13.0 −45.2 −60.1 −0.9 2154 −271.3 −261.3 −121.8 −135.6 −47.7 −65.8 −102.3 11.2 −116.3 −15.2 −97.9 −75.4 73.2 −63.0 2039 +4.6 235.5 700 −187.2 1162 610 648 904 571 1085 975 881 888 932 1211 841 743 930 1505 778 184.0 179.8 180.2 204.5 3535 −40.8 814

−1.3 103.0 1970 777 766 716 −70.1 −1.0 −46.1 −112.8 −38.8 −211.0 −177.8 71.8 −4.1 −139.2 −3.0 −35.0 −49.9 +11.8 2256 −271.1 −260.4 −115.4 −130.0 −39.7 −56.0 −94.5 22.1 −109.7 −5.1 −91.8 −67.8 84.7 −54.5 2128 16.7 246.0 737 −182.9 1234 646 684 950 605 1134 1005 940 939 987 1270 883 789 988 1583 825 204.6 194.3 195.8 220.0 3690 −32.0 851

60

100

200

400

760

Melting point, °C

−71.7 −120.8 −34.7 −39.4 −29.4 29.1 105.3 2139 108.0 58.0 843 29.0 148.5 2207 951 960 907 −51.8 22.6 −24.9 −97.0 −17.5 −202.7 −165.8 113.2 27.5 −120.3 26.5 −6.3 −20.3 47.9 2521 −270.3 −257.9 −97.7 −114.0 −17.8 −28.2 −72.1 51.6 −91.6 22.0 −74.2 −45.7 116.5 −31.9 2360 50.3 272.5 842 −171.8 1421 745 784 1080 701 1265 1108 1097 1076 1129 1425 993 909 1142 1792 960 261.7 237.8 237.0 261.8 4109 −8.0 955

−60.2 −114.4 −20.7 −26.5 −17.8 44.6 120.0 2243 121.8 75.2 904 44.4 172.2 2325 1052 1077 1018 −42.6 33.8 −14.1 −89.2 −5.4 −198.3 −159.0 135.4 44.4 −111.2 41.6 +8.8 −4.7 67.0 2657 −269.8 −256.3 −88.1 −105.2 −5.3 −13.2 −60.3 66.5 −82.3 35.3 −65.2 −32.4 137.3 −20.7 2475 68.0 285.0 897 −165.9 1519 796 833 1144 750 1330 1160 1178 1147 1203 1503 1049 967 1223 1900 1028 290.7 262.7 256.5 291.0 4322 +4.1 1014

−47.3 −107.0 −4.9 −12.5 −4.0 62.2 136.1 2361 137.2 95.2 974 62.0 198.0 2465 1189 1249 1158 −33.0 46.0 −2.3 −80.5 +10.0 −193.2 −151.9 161.6 63.8 −100.2 58.3 26.0 +13.3 88.6 2807 −269.3 −254.5 −78.0 −95.3 +10.2 +2.5 −48.3 83.0 −71.8 49.6 −53.6 −17.2 159.8 −8.3 2605 86.1 298.0 961 −159.0 1630 856 893 1219 807 1402 1221 1273 1226 1290 1591 1110 1034 1316 2029 1108 323.0 290.0 275.5 324.2 4553 17.2 1082

−33.8 −100.5 +11.5 +2.2 +11.1 78.8 151.0 2482 151.0 117.1 1050 80.0 225.0 2595 1355 1490 1336 −21.0 61.5 +13.1 −72.6 26.2 −187.9 −144.6 189.0 84.0 −88.9 75.0 44.0 31.5 110.8 2966 −268.6 −252.5 −66.5 −84.8 25.9 19.7 −35.1 100.0 −60.4 64.0 −41.1 −2.0 183.0 +4.0 2735 105.0 319.0 1026 −152.0 1744 914 954 1293 872 1472 1281 1372 1310 1382 1681 1171 1107 1418 2151 1190 357.0 319.0 304.0 354.0 4804 36.0 1151

−100.7 −145 −83 −116 −59 −91 −80 1615

Temperature, °C

Formula Cl2 ClF ClF3 Cl2O ClO2 Cl2O7 HSO3Cl Cr Cr(CO)6 CrO2Cl2 CoCl2 Co(CO)3NO CbF5 Cu Cu2Br2 Cu2Cl2 Cu2I2 C2N2 CNBr CNCl CNF DCN F2 F2O GeBr4 GeCl4 GeH4 GeHCl3 Ge(CH3)4 Ge2H6 Ge3H8 Au He H2 HBr HCl HCN H2F2 HI H2O H2S HSSH H2Se H2Te I2 IF7 Fe Fe(CO)5 Fe2Cl6 FeCl2 Kr Pb PbBr2 PbCl2 PbF2 PbI2 PbO PbS Li LiBr LiCl LiF LiI Mg MgCl2 Mn MnCl2 Hg HgBr2 HgCl2 HgI2 Mo MoF6 MoO3

40

1628 572 546 −95.8 −35.7 −76.7 −134.4 −68.9 −223.0 −196.1 −45.0 −163.0 −41.3 −73.2 −88.7 −36.9 1869 −271.7 −263.3 −138.8 −150.8 −71.0 −123.3 −17.3 −134.3 −43.2 −115.3 −96.4 38.7 −87.0 1787 194.0 −199.3 973 513 547 479 943 852 723 748 783 1047 723 621 778 1292 126.2 136.5 136.2 157.5 3102 −65.5 734

1795 666 645 610 −83.2 −18.3 −61.4 −123.8 −54.0 −216.9 −186.6 43.3 −24.9 −151.0 −22.3 −54.6 −69.8 −12.8 2059 −271.5 −261.9 −127.4 −140.7 −55.3 −74.7 −109.6 +1.2 −122.4 −24.4 −103.4 −82.4 62.2 −70.7 1957 −6.5 221.8 −191.3 1099 578 615 861 540 1039 928 838 840 880 1156 802 702 877 1434 736 164.8 165.3 166.0 189.2 3393 −49.0 785

−84.5 −128.8 −51.3 −54.3 −42.8 +10.3 87.6 2013 91.2 38.5 770 +11.0 121.5 2067 844 838 786 −62.7 +8.6 −37.5 −106.4 −30.1 −207.7 −173.0 88.1 +8.0 −131.6 +8.8 −23.4 −38.2 26.3 2363 −270.7 −259.6 −108.3 −123.8 −30.9 −45.0 −85.6 34.0 −102.3 +6.0 −84.7 −59.1 97.5 −45.3 2224 30.3 256.8 779 −178.4 1309 686 725 1003 644 1189 1048 1003 994 1045 1333 927 838 1050 1666 879 228.8 211.5 212.5 238.2 3859 −22.1 892

−79.0 −125.3 −44.1 −48.0 −37.2 +18.2 95.2 2067 98.3 46.7 801 18.5 133.2 2127 887 886 836 −57.9 14.7 −32.1 −102.3 −24.7 −205.6 −170.0 98.8 16.2 −126.7 16.2 −16.2 −30.7 35.5 2431 −270.6 −258.9 −103.8 −119.6 −25.1 −37.9 −79.8 41.5 −97.9 12.8 −80.2 −53.7 105.4 −39.4 2283 39.1 263.7 805 −175.7 1358 711 750 1036 668 1222 1074 1042 1028 1081 1372 955 868 1088 1720 913 242.0 221.0 222.2 249.0 3964 −16.2 917

735 −11 75.5 1083 504 422 605 −34.4 58 −6.5 −12 −223 −223.9 26.1 −49.5 −165 −71.1 −88 −109 −105.6 1063 −259.1 −87.0 −114.3 −13.2 −83.7 −50.9 0.0 −85.5 −89.7 −64 −49.0 112.9 5.5 1535 −21 304 −156.7 327.5 373 501 855 402 890 1114 186 547 614 870 446 651 712 1260 650 −38.9 237 277 259 2622 17 795

VAPOR PRESSURES OF PURE SUBSTANCES TABLE 2-7

2-59

Vapor Pressures of Inorganic Compounds, up to 1 atm (Continued ) Compound

Name Neon Nickel carbonyl chloride Nitrogen Nitric oxide Nitrogen dioxide Nitrogen pentoxide Nitrous oxide Nitrosyl chloride fluoride Osmium tetroxide (yellow) (white) Oxygen Ozone Phosgene Phosphorus (yellow) (violet) tribromide trichloride pentachloride Phosphine Phosphonium bromide chloride iodide Phosphorus trioxide pentoxide oxychloride thiobromide thiochloride Platinum Potassium bromide chloride fluoride hydroxide iodide Radon Rhenium heptoxide Rubidium bromide chloride fluoride iodide Selenium dioxide hexafluoride oxychloride tetrachloride Silicon dioxide tetrachloride tetrafluoride Trichlorofluorosilane Iodosilane Diiodosilane Disiloxan Trisilane Trisilazane Tetrasilane Octachlorotrisilane Hexachlorodisiloxane Hexachlorodisilane Tribromosilane Trichlorosilane Trifluorosilane Dibromosilane Difluorosilane Monobromosilane Monochlorosilane Monofluorosilane Tribromofluorosilane Dichlorodifluorosilane Trifluorobromosilane

Pressure, mm Hg 1

5

10

20

40

−257.3 1810

−255.5 1979

−254.6 2057

−253.7 2143

671 −226.1 −184.5 −55.6 −36.8 −143.4

731 −221.3 −180.6 −42.7 −23.0 −133.4

759 −219.1 −178.2 −36.7 −16.7 −128.7

789 −216.8 −175.3 −30.4 −10.0 −124.0

−132.0 3.2 −5.6 −219.1 −180.4 −92.9 76.6 237 7.8 −51.6 55.5

−120.3 22.0 +15.6 −213.4 −168.6 −77.0 111.2 271 34.4 −31.5 74.0

−114.3 31.3 26.0 −210.6 −163.2 −69.3 128.0 287 47.8 −21.3 83.2

−107.8 41.0 37.4 −207.5 −157.2 −60.3 146.2 306 62.4 −10.2 92.5

−43.7 −91.0 −25.2 384

−28.5 −79.6 −9.0 39.7 424

50.0 −18.3 2730 341 795 821 885 719 745 −144.2 212.5 297 781 792 921 748 356 157.0 −118.6 34.8 74.0 1724

72.4 +4.6 3007 408 892 919 988 814 840 −132.4 237.5 358 876 887 982 839 413 187.7 −105.2 59.8 96.3 1835

−63.4 −144.0 −92.6

−44.1 −134.8 −76.4 −53.0 3.8 −95.8 −49.7 −49.9 −6.2 74.7 17.8 27.4 −8.0 −62.6 −142.7 −40.0 −136.0 −85.7 −104.3 −145.5 −25.4 −110.5

−21.2 −74.0 −1.1 53.0 442 2.0 83.6 16.1 3146 443 940 968 1039 863 887 −126.3 248.0 389 923 937 1016 884 442 202.5 −98.9 71.9 107.4 1888 1732 −34.4 −130.4 −68.3 −47.7 18.0 −88.2 −40.0 −40.4 +4.3 89.3 29.4 38.8 +3.4 −53.4 −138.2 −29.4 −130.4 −77.3 −97.7 −141.2 −15.1 −102.9

−13.3 −68.0 +7.3 67.8 462 13.6 95.5 29.0 3302 483 994 1020 1096 918 938 −119.2 261.0 422 975 990 1052 935 473 217.5 −92.3 84.2 118.1 1942 1798 −24.0 −125.9 −59.0 −33.4 34.1 −79.8 −29.0 −30.0 15.8 104.2 41.5 51.5 16.0 −43.8 −132.9 −18.0 −124.3 −68.3 −90.1 −136.3 −3.7 −94.5

−252.6 2234 −23.0 821 −214.0 −171.7 −23.9 −2.9 −118.3 −60.2 −100.3 51.7 50.5 −204.1 −150.7 −50.3 166.7 323 79.0 +2.3 102.5 −129.4 −5.0 −61.5 16.1 84.0 481 27.3 108.0 42.7 3469 524 1050 1078 1156 976 995 −111.3 272.0 459 1031 1047 1096 991 506 234.1 −84.7 98.0 130.1 2000 1867 −12.1 −120.8 −48.8 −21.8 52.6 −70.4 −16.9 −18.5 28.4 121.5 55.2 65.3 30.0 −32.9 −127.3 −5.2 −117.6 −57.8 −81.8 −130.8 +9.2 −85.0

100

200

400

760

Melting point, °C

−251.0 2364 −6.0 866 −209.7 −166.0 −14.7 7.4 −110.3 −46.3 −88.8 71.5 71.5 −198.8 −141.0 −35.6 197.3 349 103.6 21.0 117.0 −118.8 7.4 −52.0 29.3 108.3 510 47.4 126.3 63.8 3714 586 1137 1164 1245 1064 1080 −99.0 289.0 514 1114 1133 1168 1072 554 258.0 −73.9 118.0 147.5 2083 1969 +5.4 −113.3 −33.2 −4.4 79.4 −55.9 +1.6 −1.1 47.4 146.0 75.4 85.4 51.6 −16.4 −118.7 14.1 −107.3 −42.3 −68.5 −122.4 28.6 −70.3

−249.7 2473 +8.8 904 −205.6 −162.3 −5.0 15.6 −103.6 −34.0 −79.2 89.5 89.5 −194.0 −132.6 −22.3 222.7 370 125.2 37.6 131.3 −109.4 17.6 −44.0 39.9 129.0 532 65.0 141.8 82.0 3923 643 1212 1239 1323 1142 1152 −87.7 307.0 563 1186 1207 1239 1141 594 277.0 −64.8 134.6 161.0 2151 2053 21.0 −170.2 −19.3 +10.7 101.8 −43.5 17.8 +14.0 63.6 166.2 92.5 102.2 70.2 −1.8 −111.3 31.6 −98.3 −28.6 −57.0 −115.2 45.7 −58.0 −69.8

−248.1 2603 25.8 945 −200.9 −156.8 +8.0 24.4 −96.2 −20.3 −68.2 109.3 109.3 −188.8 −122.5 −7.6 251.0 391 149.7 56.9 147.2 −98.3 28.0 −35.4 51.6 150.3 556 84.3 157.8 102.3 4169 708 1297 1322 1411 1233 1238 −75.0 336.0 620 1267 1294 1322 1223 637 297.7 −55.2 151.7 176.4 2220 2141 38.4 −100.7 −4.0 27.9 125.5 −29.3 35.5 31.0 81.7 189.5 113.6 120.6 90.2 +14.5 −102.8 50.7 −87.6 −13.3 −44.5 −106.8 64.6 −45.0 −55.9

−246.0 2732 42.5 987 −195.8 −151.7 21.0 32.4 −85.5 −6.4 −56.0 130.0 130.0 −183.1 −111.1 +8.3 280.0 417 175.3 74.2 162.0 −87.5 38.3 −27.0 62.3 173.1 591 105.1 175.0 124.0 4407 774 1383 1407 1502 1327 1324 −61.8 362.4 679 1352 1381 1408 1304 680 317.0 −45.8 168.0 191.5 2287 2227 56.8 −94.8 +12.2 45.4 149.5 −15.4 53.1 48.7 100.0 211.4 135.6 139.0 111.8 31.8 −95.0 70.5 −77.8 +2.4 −30.4 −98.0 83.8 −31.8 −41.7

−248.7 1452 −25 1001 −210.0 −161 −9.3 30 −90.9 −64.5 −134 56 42 −218.7 −251 −104 44.1 590 −40 −111.8

Temperature, °C

Formula Ne Ni Ni(CO)4 NiCl2 N2 NO NO2 N2O5 N 2O NOCl NOF OsO4 OsO4 O2 O3 COCl2 P P PBr3 PCl3 PCl5 PH3 PH4Br PH4Cl PH4I P4O6 P4O10 POCl3 PSBr3 PSCl3 Pt K KBr KCl KF KOH KI Rn Re2O7 Rb RbBr RbCl RbF RbI Se SeO2 SeF6 SeOCl2 SeCl4 Si SiO2 SiCl4 SiF4 SiFCl3 SiH3I SiH2I2 (SiH3)2O Si3H8 (SiH3)3N Si4H10 Si3Cl3 (SiCl3)2O Si2Cl6 SiHBr3 SiHCl3 SiHF3 SiH2Br2 SiH2F2 SiH3Br SiH3Cl SiH3F SiFBr3 SiF2Cl2 SiF3Br

60

−112.5 −68.9 −68.7 −27.7 46.3 −5.0 +4.0 −30.5 −80.7 −152.0 −60.9 −146.7 −117.8 −153.0 −46.1 −124.7

−251.9 2289 −15.9 840 −212.3 −168.9 −19.9 +1.8 −114.9 −54.2 −95.7 59.4 59.4 −201.9 −146.7 −44.0 179.8 334 89.8 10.2 108.3 −125.0 +0.3 −57.3 21.9 94.2 493 35.8 116.0 51.8 3574 550 1087 1115 1193 1013 1030 −106.2 280.0 482 1066 1084 1123 1026 527 244.6 −80.0 106.5 137.8 2036 1911 −4.8 −117.5 −42.2 −14.3 64.0 −64.2 −9.0 −11.0 36.6 132.0 63.8 73.9 39.2 −25.8 −123.7 +3.2 −113.3 −51.1 −76.0 −127.2 17.4 −78.6

−132.5 −28.5 22.5 569 2 38 −36.2 1755 62.3 730 790 880 380 723 −71 296 38.5 682 715 760 642 217 340 −34.7 8.5 1420 1710 −68.8 −90 −120.8 −57.0 −1.0 −144.2 −117.2 −105.7 −93.6 −33.2 −1.2 −73.5 −126.6 −131.4 −70.2 −93.9 −82.5 −139.7 −70.5

2-60

PHYSICAL AND CHEMICAL DATA

TABLE 2-7

Vapor Pressures of Inorganic Compounds, up to 1 atm (Concluded ) Compound

Pressure, mm Hg

Name

Formula

Trifluorochlorosilane Hexafluorodisilane Dichlorofluorobromosilane Dibromochlorofluorosilane Silane Disilane Silver chloride iodide Sodium bromide chloride cyanide fluoride hydroxide iodide Strontium Strontium oxide Sulfur monochloride hexafluoride Sulfuryl chloride Sulfur dioxide trioxide (α) trioxide (β) trioxide (γ) Tellurium chloride fluoride Thallium Thallous bromide chloride iodide Thionyl bromide Thionyl chloride Tin Stannic bromide Stannous chloride Stannic chloride iodide hydride Tin tetramethyl trimethyl-ethyl trimethyl-propyl Titanium chloride Tungsten Tungsten hexafluoride Uranium hexafluoride Vanadyl trichloride Xenon Zinc chloride fluoride diethyl Zirconium bromide chloride iodide

SiF3Cl Si2F6 SiFCl2Br SiFClBr2 SiH4 Si2H6 Ag AgCl AgI Na NaBr NaCl NaCN NaF NaOH NaI Sr SrO S S2Cl2 SF6 SO2Cl2 SO2 SO3 SO3 SO3 Te TeCl4 TeF6 Tl TlBr TlCl TlI SOBr2 SOCl2 Sn SnBr4 SnCl2 SnCl4 SnI4 SnH4 Sn(CH3)4 Sn(CH3)3·C2H5 Sn(CH3)3·C3H7 TiCl4 W WF6 UF6 VOCl3 Xe Zn ZnCl2 ZnF2 Zn(C2H5)2 ZrBr4 ZrCl4 ZrI4

1

5

10

20

40

−144.0 −81.0 −86.5 −65.2 −179.3 −114.8 1357 912 820 439 806 865 817 1077 739 767

−133.0 −68.8 −68.4 −45.5 −168.6 −99.3 1500 1019 927 511 903 967 928 1186 843 857 847 2198 223.0 +15.7 −120.6 −35.1 −83.0 −23.7 −19.2 −2.0 605

−127.0 −63.1 −59.0 −35.6 −163.0 −91.4 1575 1074 983 549 952 1017 983 1240 897 903 898 2262 243.8 27.5 −114.7 −24.8 −76.8 −16.5 −12.3 +4.3 650 233 −92.4 983 522 517 531 31.0 −21.9 1703 72.7 391 +10.0 175.8 −118.5 −20.6 +3.8 21.8 21.3 4507 −49.2 −13.8 12.2 −152.8 593 508 1086 +11.7 250 230 311

−120.5 −57.0 −48.8 −24.5 −156.9 −82.7 1658 1134 1045 589 1005 1072 1046 1300 953 952 953 2333 264.7 40.0 −108.4 −13.4 −69.7 −9.1 −4.9 11.1 697 253 −86.0 1040 559 550 567 44.1 −10.5 1777 88.1 420 22.0 196.2 −111.2 −9.3 16.1 34.0 34.2 4690 −41.5 −5.2 26.6 −147.1 632 536 1129 24.2 266 243 329

−112.8 −50.6 −37.0 −12.0 −150.3 −72.8 1743 1200 1111 633 1063 1131 1115 1363 1017 1005 1018 2410 288.3 54.1 −101.5 −1.0 −60.5 −1.0 +3.2 17.9 753 273 −78.4 1103 598 589 607 58.8 +2.2 1855 105.5 450 35.2 218.8 −102.3 +3.5 30.0 48.5 48.4 4886 −33.0 +4.4 40.0 −141.2 673 566 1175 38.0 281 259 344

100

200

400

760

Melting point, °C

−108.2 −46.7 −29.0 −4.7 −146.3 −66.4 1795 1242 1152 662 1099 1169 1156 1403 1057 1039 1057

−101.7 −41.7 −19.5 +6.3 −140.5 −57.5 1865 1297 1210 701 1148 1220 1214 1455 1111 1083 1111

−91.7 −34.2 −3.2 23.0 −131.6 −44.6 1971 1379 1297 758 1220 1296 1302 1531 1192 1150 1192

−81.0 −26.4 +15.4 43.0 −122.0 −29.0 2090 1467 1400 823 1304 1379 1401 1617 1286 1225 1285

−70.0 −18.9 35.4 59.5 −111.5 −14.3 2212 1564 1506 892 1392 1465 1497 1704 1378 1304 1384

305.5 63.2 −96.8 +7.2 −54.6 +4.0 8.0 21.4 789 287 −73.8 1143 621 612 631 68.3 10.4 1903 116.2 467 43.5 234.2 −96.6 11.7 38.4 57.5 58.0 5007 −27.5 10.4 49.8 −137.7 700 584 1207 47.2 289 268 355

327.2 75.3 −90.9 17.8 −46.9 10.5 14.3 28.0 838 304 −67.9 1196 653 645 663 80.6 21.4 1968 131.0 493 54.7 254.2 −89.2 22.8 50.0 69.8 71.0 5168 −20.3 18.2 62.5 −132.8 736 610 1254 59.1 301 279 369

359.7 93.5 −82.3 33.7 −35.4 20.5 23.7 35.8 910 330 −57.3 1274 703 694 712 99.0 37.9 2063 152.8 533 72.0 283.5 −78.0 39.8 67.3 88.0 90.5 5403 −10.0 30.0 82.0 −125.4 788 648 1329 77.0 318 295 389

399.6 115.4 −72.6 51.3 −23.0 32.6 32.6 44.0 997 360 −48.2 1364 759 748 763 119.2 56.5 2169 177.7 577 92.1 315.5 −65.2 58.5 87.6 109.6 112.7 5666 +1.2 42.7 103.5 −117.1 844 689 1417 97.3 337 312 409

444.6 138.0 −63.5 69.2 −10.0 44.8 44.8 51.6 1087 392 −38.6 1457 819 807 823 139.5 75.4 2270 204.7 623 113.0 348.0 −52.3 78.0 108.8 131.7 136.0 5927 17.3 55.7 127.2 −108.0 907 732 1497 118.0 357 331 431

−142 −18.6 −112.3 −99.3 −185 −132.6 960.5 455 552 97.5 755 800 564 992 318 651 800 2430 112.8 −80 −50.2 −54.1 −73.2 16.8 32.3 62.1 452 224 −37.8 3035 460 430 440 −52.2 −104.5 231.9 31.0 246.8 −30.2 144.5 −149.9

60

Temperature, °C

2068 183.8 −7.4 −132.7 −95.5 −39.0 −34.0 −15.3 520 −111.3 825 440 −6.7 −52.9 1492 316 −22.7 −140.0 −51.3 −30.0 −12.0 −13.9 3990 −71.4 −38.8 −23.2 −168.5 487 428 970 −22.4 207 190 264

−98.8 931 490 487 502 +18.4 −32.4 1634 58.3 366 −1.0 156.0 −125.8 −31.0 −7.6 +10.7 +9.4 4337 −56.5 −22.0 +0.2 −158.2 558 481 1055 0.0 237 217 297

−30 3370 −0.5 69.2 −111.6 419.4 365 872 −28 450 437 499

VAPOR PRESSURES OF PURE SUBSTANCES TABLE 2-8

2-61

Vapor Pressures of Organic Compounds, up to 1 atm* Pressure, mm Hg Compound

1

Name

Formula

Acenaphthalene Acetal Acetaldehyde Acetamide Acetanilide Acetic acid anhydride Acetone Acetonitrile Acetophenone Acetyl chloride Acetylene Acridine Acrolein (2-propenal) Acrylic acid Adipic acid Allene (propadiene) Allyl alcohol (propen-1-ol-3) chloride (3-chloropropene) isopropyl ether isothiocyanate n-propyl ether 4-Allylveratrole iso-Amyl acetate n-Amyl alcohol iso-Amyl alcohol sec-Amyl alcohol (2-pentanol) tert-Amyl alcohol sec-Amylbenzene iso-Amyl benzoate bromide (1-bromo-3-methylbutane) n-butyrate formate iodide (1-iodo-3-methylbutane) isobutyrate Amyl isopropionate iso-Amyl isovalerate n-Amyl levulinate iso-Amyl levulinate nitrate 4-tert-Amylphenol Anethole Angelonitrile Aniline 2-Anilinoethanol Anisaldehyde o-Anisidine (2-methoxyaniline) Anthracene Anthraquinone Azelaic acid Azelaldehyde Azobenzene Benzal chloride (α,α-Dichlorotoluene) Benzaldehyde Benzanthrone Benzene Benzenesulfonylchloride Benzil Benzoic acid anhydride Benzoin Benzonitrile Benzophenone Benzotrichloride (α,α,α-Trichlorotoluene) Benzotrifluoride (α,α,α-Trifluorotoluene) Benzoyl bromide chloride nitrile Benzyl acetate alcohol

C12H10 C6H14O2 C2H4O C2H5NO C8H9NO C2H4O2 C4H6O3 C3H6O C2H3N C8H8O C2H3OCl C2H2 C13H9N C3H4O C3H4O2 C6H10O4 C3H4 C3H6O C3H5Cl C6H12O C4H5NS C6H12O C11H14O2 C7H14O2 C5H12O C5H12O C5H12O C5H12O C11H16 C12H16O2 C5H11Br C9H18O2 C6H12O2 C5H11I C9H18O2 C8H16O2 C10H20O2 C10H18O3 C10H18O3 C5H11NO3 C11H16O C10H12O C5H7N C6H7N C8H11NO C8H8O2 C7H9NO C14H10 C14H8O2 C9H16O4 C9H18O C12H10N2 C7H6Cl2 C7H6O C17H10O C6H6 C6H5ClO2S C14H10O2 C7H6O2 C14H10O3 C14H12O2 C7H5N C13H10O C7H5Cl3 C7H5F3 C7H5BrO C7H5ClO C8H5NO C9H10O2 C7H8O

5

10

20

114.8 −2.3 −65.1 92.0 146.6 +6.3 24.8 −40.5 −26.6 64.0 −35.0 −133.0 165.8 −46.0 27.3 191.0 −108.0 +0.2 −52.0 −23.1 +25.3 −18.2 113.9 +23.7 34.7 30.9 22.1 +7.2 55.8 104.5 +2.1 47.1 +5.4 +21.9 40.1 33.7 54.4 110.0 104.0 28.8 109.8 91.6 +15.0 57.9 134.3 102.6 88.0 173.5 219.4 210.4 58.4 135.7 64.0 50.1 274.5 −19.6 96.5 165.2 119.5 180.0 170.2 55.3 141.7 73.7 −10.3 75.4 59.1 71.7 73.4 80.8

131.2 +8.0 −56.8 105.0 162.0 17.5 36.0 −31.1 −16.3 78.0 −27.6 −128.2 184.0 −36.7 39.0 205.5 −101.0 10.5 −42.9 −12.9 38.3 −7.9 127.0 35.2 44.9 40.8 32.2 17.2 69.2 121.6 13.6 59.9 17.1 34.1 52.8 46.3 68.6 124.0 118.8 40.3 125.5 106.0 28.0 69.4 149.6 117.8 101.7 187.2 234.2 225.5 71.6 151.5 78.7 62.0 297.2 −11.5 112.0 183.0 132.1 198.0 188.1 69.2 157.6 87.6 −0.4 89.8 73.0 85.5 87.6 92.6

148.7 19.6 −47.8 120.0 180.0 29.9 48.3 −20.8 −5.0 92.4 −19.6 −122.8 203.5 −26.3 52.0 222.0 −93.4 21.7 −32.8 −1.8 52.1 +3.7 142.8 47.8 55.8 51.7 42.6 27.9 83.8 139.7 26.1 74.0 30.0 47.6 66.6 60.0 83.8 139.7 134.4 53.5 142.3 121.8 41.0 82.0 165.7 133.5 116.1 201.9 248.3 242.4 85.0 168.3 94.3 75.0 322.5 −2.6 129.0 202.8 146.7 218.0 207.0 83.4 175.8 102.7 12.2 105.4 87.6 100.2 102.3 105.8

40

60

100

200

400

760

197.5 50.1 −22.6 158.0 227.2 63.0 82.2 +7.7 27.0 133.6 +3.2 −107.9 256.0 +2.5 86.1 265.0 −72.5 50.0 −4.5 29.0 89.5 35.8 183.7 83.2 85.8 80.7 70.7 55.3 124.1 186.8 60.4 113.1 65.4 84.4 104.4 97.6 125.1 180.5 177.0 88.6 189.0 164.2 77.5 119.9 209.5 176.7 155.2 250.0 285.0 286.5 123.0 216.0 138.3 112.5 390.0 26.1 174.5 255.8 186.2 270.4 258.0 123.5 224.4 144.3 45.3 147.7 128.0 141.0 144.0 141.7

222.1 66.3 −10.0 178.3 250.5 80.0 100.0 22.7 43.7 154.2 16.1 −100.3 284.0 17.5 103.3 287.8 −61.3 64.5 10.4 44.3 108.0 52.6 204.0 101.3 102.0 95.8 85.7 69.7 145.2 210.2 78.7 133.2 83.2 103.8 124.2 117.3 146.1 203.1 198.1 106.7 213.0 186.1 96.3 140.1 230.6 199.0 175.3 279.0 314.6 309.6 142.1 240.0 160.7 131.7 426.5 42.2 198.0 283.5 205.8 299.1 284.4 144.1 249.8 165.6 62.5 169.2 149.5 161.3 165.5 160.0

250.0 84.0 +4.9 200.0 277.0 99.0 119.8 39.5 62.5 178.0 32.0 −92.0 314.3 34.5 122.0 312.5 −48.5 80.2 27.5 61.7 129.8 71.4 226.2 121.5 119.8 113.7 102.3 85.7 168.0 235.8 99.4 155.3 102.7 125.8 146.0 138.4 169.5 227.4 222.7 126.5 239.5 210.5 117.7 161.9 254.5 223.0 197.3 310.2 346.2 332.8 163.4 266.1 187.0 154.1

277.5 102.2 20.2 222.0 303.8 118.1 139.6 56.5 81.8 202.4 50.8 −84.0 346.0 52.5 141.0 337.5 −35.0 96.6 44.6 79.5 150.7 90.5 248.0 142.0 137.8 130.6 119.7 101.7 193.0 262.0 120.4 178.6 123.3 148.2 168.8 160.2 194.0 253.2 247.9 147.5 266.0 235.3 140.0 184.4 279.6 248.0 218.5 342.0 379.9 356.5 185.0 293.0 214.0 179.0

60.6 224.0 314.3 227.0 328.8 313.5 166.7 276.8 189.2 82.0 193.7 172.8 185.0 189.0 183.0

80.1 251.5 347.0 249.2 360.0 343.0 190.6 305.4 213.5 102.2 218.5 197.2 208.0 213.5 204.7

Temperature, °C −23.0 −81.5 65.0 114.0 −17.2 1.7 −59.4 −47.0 37.1 −50.0 −142.9 129.4 −64.5 +3.5 159.5 −120.6 −20.0 −70.0 −43.7 −2.0 −39.0 85.0 0.0 +13.6 +10.0 +1.5 −12.9 29.0 72.0 −20.4 21.2 −17.5 −2.5 14.8 +8.5 27.0 81.3 75.6 +5.2 62.6 −8.0 34.8 104.0 73.2 61.0 145.0 190.0 178.3 33.3 103.5 35.4 26.2 225.0 −36.7 65.9 128.4 96.0 143.8 135.6 28.2 108.2 45.8 −32.0 47.0 32.1 44.5 45.0 58.0

168.2 31.9 −37.8 135.8 199.6 43.0 62.1 −9.4 +7.7 109.4 −10.4 −116.7 224.2 −15.0 66.2 240.5 −85.2 33.4 −21.2 +10.9 67.4 16.4 158.3 62.1 68.0 63.4 54.1 38.8 100.0 158.3 39.8 90.0 44.0 62.3 81.8 75.5 100.6 155.8 151.7 67.6 160.3 139.3 55.8 96.7 183.7 150.5 132.0 217.5 264.3 260.0 100.2 187.9 112.1 90.1 350.0 +7.6 147.7 224.5 162.6 239.8 227.6 99.6 195.7 119.8 25.7 122.6 103.8 116.6 119.6 119.8

181.2 39.8 −31.4 145.8 211.8 51.7 70.8 −2.0 15.9 119.8 −4.5 −112.8 238.7 −7.5 75.0 251.0 −78.8 40.3 −14.1 18.7 76.2 25.0 169.6 71.0 75.5 71.0 61.5 46.0 110.4 171.4 48.7 99.8 53.3 71.9 91.7 85.2 110.3 165.2 162.6 76.3 172.6 149.8 65.2 106.0 194.0 161.7 142.1 231.8 273.3 271.8 110.0 199.8 123.4 99.6 368.8 15.4 158.2 238.2 172.8 252.7 241.7 109.8 208.2 130.0 34.0 133.4 114.7 127.0 129.8 129.3

Melting point, °C 95 −123.5 81 113.5 16.7 −73 −94.6 −41 20.5 −112.0 −81.5 110.5 −87.7 14 152 −136 −129 −136.4 −80

−117.2 −11.9

93 22.5 −6.2 2.5 5.2 217.5 286 106.5 68 −16.1 −26 174 +5.5 14.5 95 121.7 42 132 −12.9 48.5 −21.2 −29.3 0 −0.5 33.5 −51.5 −15.3

*Compiled from the extended tables published by D. R. Stull in Ind. Eng. Chem., 39, 517 (1947). For information on fuels see Hibbard, N.A.C.A. Research Mem. E56I21, 1956. For methane see Johnson (ed.), WADD-TR-60-56, 1960.

2-62

PHYSICAL AND CHEMICAL DATA

TABLE 2-8

Vapor Pressures of Organic Compounds, up to 1 atm (Continued ) Pressure, mm Hg Compound Name

Benzylamine Benzyl bromide (α-bromotoluene) chloride (α-chlorotoluene) cinnamate Benzyldichlorosilane Benzyl ethyl ether phenyl ether isothiocyanate Biphenyl 1-Biphenyloxy-2,3-epoxypropane d-Bornyl acetate Bornyl n-butyrate formate isobutyrate propionate Brassidic acid Bromoacetic acid 4-Bromoanisole Bromobenzene 4-Bromobiphenyl 1-Bromo-2-butanol 1-Bromo-2-butanone cis-1-Bromo-1-butene trans-1-Bromo-1-butene 2-Bromo-1-butene cis-2-Bromo-2-butene trans-2-Bromo-2-butene 1,4-Bromochlorobenzene 1-Bromo-1-chloroethane 1-Bromo-2-chloroethane 2-Bromo-4,6-dichlorophenol 1-Bromo-4-ethyl benzene (2-Bromoethyl)-benzene 2-Bromoethyl 2-chloroethyl ether (2-Bromoethyl)-cyclohexane 1-Bromoethylene Bromoform (tribromomethane) 1-Bromonaphthalene 2-Bromo-4-phenylphenol 3-Bromopyridine 2-Bromotoluene 3-Bromotoluene 4-Bromotoluene 3-Bromo-2,4,6-trichlorophenol 2-Bromo-1,4-xylene 1,2-Butadiene (methyl allene) 1,3-Butadiene n-Butane iso-Butane (2-methylpropane) 1,3-Butanediol 1,2,3-Butanetriol 1-Butene cis-2-Butene trans-2-Butene 3-Butenenitrile iso-Butyl acetate n-Butyl acrylate alcohol iso-Butyl alcohol sec-Butyl alcohol tert-Butyl alcohol iso-Butyl amine n-Butylbenzene iso-Butylbenzene sec-Butylbenzene tert-Butylbenzene iso-Butyl benzoate n-Butyl bromide (1-bromobutane) iso-Butyl n-butyrate carbamate Butyl carbitol (diethylene glycol butyl ether) n-Butyl chloride (1-chlorobutane) iso-Butyl chloride

1

5

10

20

29.0 32.2 22.0 173.8 45.3 26.0 95.4 79.5 70.6 135.3 46.9 74.0 47.0 70.0 64.6 209.6 54.7 48.8 +2.9 98.0 23.7 +6.2 −44.0 −38.4 −47.3 −39.0 −45.0 32.0 −36.0 −28.8 84.0 30.4 48.0 36.5 38.7 −95.4

70.0

54.8 59.6 47.8 206.3 70.2 52.0 127.7 107.8 101.8 169.9 75.7 103.4 74.8 99.8 93.7 241.7 81.6 77.8 27.8 133.7 45.4 30.0 −23.2 −17.0 −27.0 −17.9 −24.1 59.5 −18.0 −7.0 115.6 42.5 76.2 63.2 66.6 −77.8 22.0 117.5 135.4 42.0 49.7 50.8 47.5 146.2 65.0 −72.7 −87.6 −85.7 −94.1 67.5 132.0 −89.4 −81.1 −84.0 +2.9 +1.4 +23.5 +20.0 +11.6 +7.2 −3.0 −31.0 48.8 40.5 44.2 39.0 93.6 −11.2 30.0 83.7 95.7

67.7 73.4 60.8 221.5 83.2 65.0 144.0 121.8 117.0 187.2 90.2 118.0 89.3 114.0 108.0 256.0 94.1 91.9 40.0 150.6 55.8 41.8 −12.8 −6.4 −16.8 −7.2 −13.8 72.7 −9.4 +4.1 130.8 74.0 90.5 76.3 80.5 −68.8 34.0 133.6 152.3 55.2 62.3 64.0 61.1 163.2 78.8 −64.2 −79.7 −77.8 −86.4 85.3 146.0 −81.6 −73.4 −76.3 14.1 12.8 35.5 30.2 21.7 16.9 +5.5 −21.0 62.0 53.7 57.0 51.7 108.6 −0.3 42.2 96.4 107.8

81.8 88.3 75.0 239.3 96.7 79.6 160.7 137.0 134.2 205.8 106.0 133.8 104.0 130.0 123.7 272.9 108.2 107.8 53.8 169.8 67.2 54.2 −1.4 +5.4 −5.3 +4.6 −2.4 87.8 0.0 16.0 147.7 90.2 105.8 90.8 95.8 −58.8 48.0 150.2 171.8 69.1 76.0 78.1 75.2 181.8 94.0 −54.9 −71.0 −68.9 −77.9 100.0 161.0 −73.0 −64.6 −67.5 26.6 25.5 48.6 41.5 32.4 27.3 14.3 −10.3 76.3 67.8 70.6 65.6 124.2 +11.6 56.1 110.1 120.5

97.3 104.8 90.7 255.8 111.8 95.4 180.1 153.0 152.5 226.3 123.7 150.7 121.2 147.2 140.4 290.0 124.0 125.0 68.6 190.8 79.5 68.2 +11.5 18.4 +7.2 17.7 +10.5 103.8 +10.4 29.7 165.8 108.5 123.2 106.6 113.0 −48.1 63.6 170.2 193.8 84.1 91.0 93.9 91.8 200.5 110.6 −44.3 −61.3 −59.1 −68.4 117.4 178.0 −63.4 −54.7 −57.6 40.0 39.2 63.4 53.4 44.1 38.1 24.5 +1.3 92.4 83.3 86.2 80.8 141.8 24.8 71.7 125.3 135.5

C4H9Cl C4H9Cl

−49.0 −53.8

−28.9 −34.3

−18.6 −24.5

−7.4 −13.8

+5.0 −1.9

60

100

200

400

760

107.3 115.6 100.5 267.0 121.3 105.5 192.6 163.8 165.2 239.7 135.7 161.8 131.7 157.6 151.2 301.5 133.8 136.0 78.1 204.5 87.0 77.3 19.8 27.2 15.4 26.2 18.7 114.8 17.0 38.0 177.6 121.0 133.8 116.4 123.7 −41.2 73.4 183.5 207.0 94.1 100.0 104.1 102.3 213.0 121.6 −37.5 −55.1 −52.8 −62.4 127.5 188.0 −57.2 −48.4 −51.3 48.8 48.0 72.6 60.3 51.7 45.2 31.0 8.8 102.6 93.3 96.0 90.6 152.0 33.4 81.3 134.6 146.0

120.0 129.8 114.2 281.5 133.5 118.9 209.2 177.7 180.7 255.0 149.8 176.4 145.8 172.2 165.7 316.2 146.3 150.1 90.8 221.8 97.6 89.2 30.8 38.1 26.3 37.5 29.9 128.0 28.0 49.5 193.2 135.5 148.2 129.8 138.0 −31.9 85.9 198.8 224.5 107.8 112.0 117.8 116.4 229.3 135.7 −28.3 −46.8 −44.2 −54.1 141.2 202.5 −48.9 −39.8 −42.7 60.2 59.7 85.1 70.1 61.5 54.1 39.8 18.8 116.2 107.0 109.5 103.8 166.4 44.7 94.0 147.2 159.8

140.0 150.8 134.0 303.8 152.0 139.6 233.2 198.0 204.2 280.4 172.0 198.0 166.4 194.2 187.5 336.8 165.8 172.7 110.1 248.2 112.1 107.0 47.8 55.7 42.8 54.5 46.5 149.5 44.7 66.8 216.5 156.5 169.8 150.0 160.0 −17.2 106.1 224.2 251.0 127.7 133.6 138.0 137.4 253.0 156.4 −14.2 −33.9 −31.2 −41.5 161.0 222.0 −36.2 −26.8 −29.7 78.0 77.6 104.0 84.3 75.9 67.9 52.7 32.0 136.9 127.2 128.8 123.7 188.2 62.0 113.9 165.7 181.2

161.3 175.2 155.8 326.7 173.0 161.5 259.8 220.4 229.4 309.8 197.5 222.2 190.2 218.2 211.2 359.6 186.7 197.5 132.3 277.7 128.3 126.3 66.8 75.0 61.9 74.0 66.0 172.6 63.4 86.0 242.0 182.0 194.0 172.3 186.2 −1.1 127.9 252.0 280.2 150.0 157.3 160.0 160.2 278.0 181.0 +1.8 −19.3 −16.3 −27.1 183.8 243.5 −21.7 −12.0 −14.8 98.0 97.5 125.2 100.8 91.4 83.9 68.0 50.7 159.2 149.6 150.3 145.8 212.8 81.7 135.7 186.0 205.0

184.5 198.5 179.4 350.0 194.3 185.0 287.0 243.0 254.9 340.0 223.0 247.0 214.0 243.0 235.0 382.5 208.0 223.0 156.2 310.0 145.0 147.0 86.2 94.7 81.0 93.9 85.5 196.9 82.7 106.7 268.0 206.0 219.0 195.8 213.0 +15.8 150.5 281.1 311.0 173.4 181.8 183.7 184.5 305.8 206.7 18.5 −4.5 −0.5 −11.7 206.5 264.0 −6.3 +3.7 +0.9 119.0 118.0 147.4 117.5 108.0 99.5 82.9 68.6 183.1 172.8 173.5 168.5 237.0 101.6 156.9 206.5 231.2

13.0 +5.9

24.0 16.0

40.0 32.0

58.8 50.0

77.8 68.9

Temperature, °C

Formula C7H9N C7H7Br C7H7Cl C16H14O2 C7H8Cl2Si C9H12O C13H12O C8H7NS C12H10 C15H14O2 C12H20O2 C14H24O2 C11H18O2 C14H24O2 C13H22O2 C22H42O2 C2H3BrO2 C7H7BrO C6H5Br C12H9Br C4H9BrO C4H7BrO C4H7Br C4H7Br C4H7Br C4H7Br C4H7Br C6H4BrCl C2H4BrCl C2H4BrCl C6H3BrCl2O C8H9Br C8H9Br C4H8BrClO C8H15Br C2H3Br CHBr3 C10H7Br C12H9BrO C5H4BrN C7H7Br C7H7Br C7H7Br C6H2BrCl3O C8H9Br C4H6 C4H6 C4H10 C4H10 C4H10O2 C4H10O3 C4H8 C4H8 C4H8 C4H5N C6H12O2 C7H12O2 C4H10O C4H10O C4H10O C4H10O C4H11N C10H14 C10H14 C10H14 C10H14 C11H14O2 C4H9Br C8H16O2 C5H11NO2 C8H18O3

40

84.2 100.0 16.8 24.4 14.8 10.3 112.4 37.5 −89.0 −102.8 −101.5 −109.2 22.2 102.0 −104.8 −96.4 −99.4 −19.6 −21.2 −0.5 −1.2 −9.0 −12.2 −20.4 −50.0 22.7 14.1 18.6 13.0 64.0 −33.0 +4.6

Melting point, °C −4 −39 39

69.5 29

61.5 49.5 12.5 −30.7 90.5

−100.3 −133.4 −111.2 −114.6 16.6 −16.6 68 −45.0

−138 8.5 5.5 95 −28 39.8 28.5 +9.5 −108.9 −135 −145 77 −130 −138.9 −105.4 −98.9 −64.6 −79.9 −108 −114.7 25.3 −85.0 −88.0 −51.5 −75.5 −58 −112.4 65 −123.1 −131.2

VAPOR PRESSURES OF PURE SUBSTANCES TABLE 2-8

2-63

Vapor Pressures of Organic Compounds, up to 1 atm (Continued ) Pressure, mm Hg Compound Name

Formula

sec-Butyl chloride (2-Chlorobutane) tert-Butyl chloride sec-Butyl chloroacetate 2-tert-Butyl-4-cresol 4-tert-Butyl-2-cresol iso-Butyl dichloroacetate 2,3-Butylene glycol (2,3-butanediol) 2-Butyl-2-ethylbutane-1,3-diol 2-tert-Butyl-4-ethylphenol n-Butyl formate iso-Butyl formate sec-Butyl formate sec-Butyl glycolate iso-Butyl iodide (1-iodo-2-methylpropane) isobutyrate isovalerate levulinate naphthylketone (1-isovaleronaphthone) 2-sec-Butylphenol 2-tert-Butylphenol 4-iso-Butylphenol 4-sec-Butylphenol 4-tert-Butylphenol 2-(4-tert-Butylphenoxy)ethyl acetate 4-tert-Butylphenyl dichlorophosphate

C4H9Cl C4H9Cl C6H11ClO2 C11H16O C11H16O C6H10Cl2O2 C4H10O2 C10H22O2 C12H15O C5H10O2 C5H10O2 C5H10O2 C6H12O3 C4H9I C8H16O2 C9H18O2 C9H16O3 C15H16O C10H14O C10H14O C10H14O C10H14O C10H14O C14H20O3 C10H13Cl2 O2P C11H14O C7H14O2 C12H18O C12H18O C12H18O C12H18O C4H8O2 C4H8O2 C4H7N C11H14O C10H16 C10H16O2 C10H16O C10H19N C10H20O C10H20O2 C6H12O2 C6H12O2 C6H10O2 C6H11N C8H18O C8H16O C8H16O2 C8H15N C12H9N CO2 CS2 CO COSe COS CBr4 CCl4 CF4 C10H14O C10H14O C10H12O2 C2HCl3O C2H3Cl3O2

tert-Butyl phenyl ketone (pivalophenone) iso-Butyl propionate 4-tert-Butyl-2,5-xylenol 4-tert-Butyl-2,6-xylenol 6-tert-Butyl-2,4-xylenol 6-tert-Butyl-3,4-xylenol Butyric acid iso-Butyric acid Butyronitrile iso-Valerophenone Camphene Campholenic acid d-Camphor Camphylamine Capraldehyde Capric acid n-Caproic acid iso-Caproic acid iso-Caprolactone Capronitrile Capryl alcohol (2-octanol) Caprylaldehyde Caprylic acid (octanoic acid) Caprylonitrile Carbazole Carbon dioxide disulfide monoxide oxyselenide (carbonyl selenide) oxysulfide (carbonyl sulfide) tetrabromide tetrachloride tetrafluoride Carvacrol Carvone Chavibetol Chloral (trichloroacetaldehyde) hydrate (trichloroacetaldehyde hydrate) Chloranil Chloroacetic acid anhydride 2-Chloroaniline 3-Chloroaniline 4-Chloroaniline Chlorobenzene 2-Chlorobenzotrichloride (2-α,α,α-tetrachlorotoluene)

C6Cl4O2 C2H3ClO2 C4H4Cl2O3 C6H6ClN C6H6ClN C6H6ClN C6H5Cl C7H4Cl4

1

5

10

20

40

−60.2

−39.8

−29.2

−17.7

17.0 70.0 74.3 28.6 44.0 94.1 76.3 −26.4 −32.7 −34.4 28.3 −17.0 +4.1 16.0 65.0 136.0 57.4 56.6 72.1 71.4 70.0 118.0 96.0

41.8 98.0 103.7 54.3 68.4 122.6 106.2 −4.7 −11.4 −13.3 53.6 +5.8 28.0 41.2 92.1 167.9 86.0 84.2 100.9 100.5 99.2 150.0 129.6

54.6 112.0 118.0 67.5 80.3 136.8 121.0 +6.1 −0.8 −3.1 66.0 17.0 39.9 53.8 105.9 184.0 100.8 98.1 115.5 114.8 114.0 165.8 146.0

68.2 127.2 134.0 81.4 93.4 151.2 137.0 18.0 +11.0 +8.4 79.8 29.8 52.4 67.7 120.2 201.6 116.1 113.0 130.3 130.3 129.5 183.3 164.0

−5.0 −19.0 83.6 143.9 150.8 96.7 107.8 167.8 154.0 31.6 24.1 21.3 94.2 42.8 67.2 82.7 136.2 219.7 133.4 129.2 147.2 147.8 146.0 201.5 184.3

57.8 −2.3 88.2 74.0 70.3 83.9 25.5 14.7 −20.0 58.3

85.7 +20.9 119.8 103.9 100.2 113.6 49.8 39.3 +2.1 87.0

97.6 41.5 45.3 51.9 125.0 71.4 66.2 38.3 9.2 32.8 73.4 92.3 43.0

125.7 68.6 74.0 78.8 142.0 89.5 83.0 66.4 34.6 57.6 92.0 114.1 67.6

99.0 32.3 135.0 119.0 115.0 127.0 61.5 51.2 13.4 101.4 47.2 139.8 82.3 83.7 92.0 152.2 99.5 94.0 80.3 47.5 70.0 101.2 124.0 80.4

114.3 44.8 151.0 135.0 131.0 143.0 74.0 64.0 25.7 116.8 60.4 153.9 97.5 97.6 106.3 165.0 111.8 107.0 95.7 61.7 83.3 110.2 136.4 94.6

130.4 58.5 169.8 152.2 148.5 159.7 88.0 77.8 38.4 133.8 75.7 170.0 114.0 112.5 122.2 179.9 125.0 120.4 112.3 76.9 98.0 120.0 150.6 110.6

60

100

200

400

760

+3.4 −11.4 93.0 153.7 161.7 106.6 116.3 178.0 165.4 39.8 32.4 29.6 104.0 51.8 75.9 92.4 147.0 231.5 143.9 140.0 157.0 157.9 156.0 212.8 197.2

14.2 −1.0 105.5 167.0 176.2 119.8 127.8 191.9 179.0 51.0 43.4 40.2 116.4 63.5 88.0 105.2 160.2 246.7 157.3 153.5 171.2 172.4 170.2 228.0 214.3

31.5 +14.6 124.1 187.8 197.8 139.2 145.6 212.0 200.3 67.9 60.0 56.8 135.5 81.0 106.3 124.8 181.8 269.7 179.7 173.8 192.1 194.3 191.5 250.3 240.0

50.0 32.6 146.0 210.0 221.8 160.0 164.0 233.5 223.8 86.2 79.0 75.2 155.6 100.3 126.3 146.4 205.5 294.0 203.8 196.3 214.7 217.6 214.0 277.6 268.2

68.0 51.0 167.8 232.6 247.0 183.0 182.0 255.0 247.8 106.0 98.2 93.6 177.5 120.4 147.5 168.7 229.9 320.0 228.0 219.5 237.0 242.1 238.0 304.4 299.0

140.8 67.6 180.3 163.6 158.2 170.0 96.5 86.3 47.3 144.6 85.0 180.0 124.0 122.0 132.0 189.8 133.3 129.6 123.2 86.8 107.4 126.0 160.0 121.2 248.2 −104.8 −15.3 −208.1 −70.2 −93.0 106.3 12.3 −155.4 155.3 143.8 170.7 29.1 46.2

154.0 79.5 195.0 176.0 172.0 184.0 108.0 98.0 59.0 158.0 97.9 193.7 138.0 134.6 145.3 200.0 144.0 141.4 137.2 99.8 119.8 133.9 172.2 134.8 265.0 −100.2 −5.1 −205.7 −61.7 −85.9 119.7 23.0 −150.7 169.7 157.3 185.5 40.2 55.0

Temperature, °C

175.0 197.7 220.0 97.0 116.4 136.8 217.5 241.3 265.3 196.0 217.8 239.8 192.3 214.2 236.5 204.5 226.7 249.5 125.5 144.5 163.5 115.8 134.5 154.5 76.7 96.8 117.5 180.1 204.2 228.0 117.5 138.7 160.5 212.7 234.0 256.0 157.9 182.0 209.2 153.0 173.8 195.0 164.8 186.3 208.5 217.1 240.3 268.4 160.8 181.0 202.0 158.3 181.0 207.7 157.8 182.1 207.0 119.7 141.0 163.7 138.0 157.5 178.5 145.4 156.5 168.5 190.3 213.9 237.5 155.2 179.5 204.5 292.5 323.0 354.8 −93.0 −85.7 −78.2 +10.4 28.0 46.5 −201.3 −196.3 −191.3 −49.8 −35.6 −21.9 −75.0 −62.7 −49.9 139.7 163.5 189.5 38.3 57.8 76.7 −143.6 −135.5 −127.7 191.2 213.8 237.0 179.6 203.5 227.5 206.8 229.8 254.0 57.8 77.5 97.7 68.0 82.1 96.2

Melting point, °C −131.3 −26.5

22.5

−95.3 −90.7 −80.7

99

−71

−74 −47 50 178.5 31.5 −1.5 −35 −38.6 16 244.8 −57.5 −110.8 −205.0

−134.3 −73.8 −222.0 −117.1 −132.4

−124.4 −54.3 −217.2 −102.3 −119.8

−119.5 −44.7 −215.0 −95.0 −113.3

−114.4 −34.3 −212.8 −86.3 −106.0

−50.0 −184.6 70.0 57.4 83.6 −37.8 −9.8

−30.0 −174.1 98.4 86.1 113.3 −16.0 +10.0

−19.6 −169.3 113.2 100.4 127.0 −5.0 19.5

−8.2 −164.3 127.9 116.1 143.2 +7.2 29.2

−108.6 −22.5 −210.0 −76.4 −98.3 96.3 +4.3 −158.8 145.2 133.0 159.8 20.2 39.7

70.7 43.0 67.2 46.3 63.5 59.3 −13.0

89.3 68.3 94.1 72.3 89.8 87.9 +10.6

97.8 81.0 108.0 84.8 102.0 102.1 22.2

106.4 94.2 122.4 99.2 116.7 117.8 35.3

116.1 109.2 138.2 115.6 133.6 135.0 49.7

122.0 118.3 148.0 125.7 144.1 145.8 58.3

129.5 130.7 159.8 139.5 158.0 159.9 70.7

140.3 149.0 177.8 160.0 179.5 182.3 89.4

151.3 169.0 197.0 183.7 203.5 206.6 110.0

162.6 189.5 217.0 208.8 228.5 230.5 132.2

290 61.2 46 0 −10.4 70.5 −45.2

69.0

101.8

117.9

135.8

155.0

167.8

185.0

208.0

233.0

262.1

28.7

−138.8 90.1 −22.6 −183.7 +0.5 −57 51.7

2-64

PHYSICAL AND CHEMICAL DATA

TABLE 2-8

Vapor Pressures of Organic Compounds, up to 1 atm (Continued ) Pressure, mm Hg Compound Name

2-Chlorobenzotrifluoride (2-chloro-α,α,α-trifluorotoluene) 2-Chlorobiphenyl 4-Chlorobiphenyl α-Chlorocrotonic acid Chlorodifluoromethane Chlorodimethylphenylsilane 1-Chloro-2-ethoxybenzene 2-(2-Chloroethoxy) ethanol bis-2-Chloroethyl acetacetal 1-Chloro-2-ethylbenzene 1-Chloro-3-ethylbenzene 1-Chloro-4-ethylbenzene 2-Chloroethyl chloroacetate 2-Chloroethyl 2-chloroisopropyl ether 2-Chloroethyl 2-chloropropyl ether 2-Chloroethyl α-methylbenzyl ether Chloroform (trichloromethane) 1-Chloronaphthalene 4-Chlorophenethyl alcohol 2-Chlorophenol 3-Chlorophenol 4-Chlorophenol 2-Chloro-3-phenylphenol 2-Chloro-6-phenylphenol Chloropicrin (trichloronitromethane) 1-Chloropropene 2-Chloropyridine 3-Chlorostyrene 4-Chlorostyrene 1-Chlorotetradecane 2-Chlorotoluene 3-Chlorotoluene 4-Chlorotoluene Chlorotriethylsilane 1-Chloro-1,2,2-trifluoroethylene Chlorotrifluoromethane Chlorotrimethylsilane trans-Cinnamic acid Cinnamyl alcohol Cinnamylaldehyde Citraconic anhydride cis-α-Citral d-Citronellal Citronellic acid Citronellol Citronellyl acetate Coumarin o-Cresol (2-cresol; 2-methylphenol) m-Cresol (3-cresol; 3-methylphenol) p-Cresol (4-cresol; 4-methylphenol) cis-Crotonic acid trans-Crotonic acid cis-Crotononitrile trans-Crotononitrile Cumene 4-Cumidene Cuminal Cuminyl alcohol 2-Cyano-2-n-butyl acetate Cyanogen bromide chloride iodide Cyclobutane Cyclobutene Cyclohexane Cyclohexaneethanol Cyclohexanol Cyclohexanone 2-Cyclohexyl-4,6-dinitrophenol Cyclopentane Cyclopropane Cymene

1

5

10

20

60

100

200

400

760

88.3 197.0 212.5 155.9 −76.4 124.7 141.8 139.5 150.7 110.0 113.6 116.0 140.0 115.8 125.6 164.8 10.4 180.4 188.1 106.0 143.0 150.0 237.0 237.1 53.8 −15.1 104.6 121.2 122.0 215.5 94.7 96.3 96.6 82.3 −66.7 −111.7 +6.0 232.4 177.8 177.7 145.4 160.0 140.1 195.4 159.8 161.0 216.5 127.4 138.0 140.0 116.3 128.0 50.1 62.8 88.1 158.0 160.0 176.2 133.8 −51.8 22.6 −24.9 97.6 −32.8 −41.2 25.5 142.7 103.7 90.4 229.0 −1.3 −70.0 110.8

108.3 219.6 237.8 173.8 −65.8 145.5 162.0 157.2 169.8 130.2 133.8 137.0 159.8 135.7 146.3 186.3 25.9 204.2 210.0 126.4 164.8 172.0 261.3 261.6 71.8 +1.3 125.0 142.2 143.5 240.3 115.0 116.6 117.1 101.6 −55.0 −102.5 21.9 253.3 199.8 199.3 165.8 181.8 160.0 214.5 179.8 178.8 240.0 146.7 157.3 157.3 133.9 146.0 68.0 81.1 107.3 180.0 182.8 197.9 152.2 −42.6 33.8 −14.1 111.5 −18.9 −27.8 42.0 161.7 121.7 110.3 248.7 +13.8 −59.1 131.4

130.0 243.8 264.5 193.2 −53.6 168.6 185.5 176.5 190.5 152.2 156.7 159.8 182.2 156.5 169.8 210.8 42.7 230.8 234.5 149.8 188.7 196.0 289.4 289.5 91.8 18.0 147.7 165.7 166.0 267.5 137.1 139.7 139.8 123.6 −41.7 −92.7 39.4 276.7 224.6 222.4 189.8 205.0 183.8 236.6 201.0 197.8 264.7 168.4 179.0 179.4 152.2 165.5 88.0 101.5 129.2 203.2 206.7 221.7 173.4 −33.0 46.0 −2.3 126.1 −3.4 −12.2 60.8 183.5 141.4 132.5 269.8 31.0 −46.9 153.5

152.2 267.5 292.9 212.0 −40.8 193.5 208.0 196.0 212.6 177.6 181.1 184.3 205.0 180.0 194.1 235.0 61.3 259.3 259.3 174.5 214.0 220.0 317.5 317.0 111.9 37.0 170.2 190.0 191.0 296.0 159.3 162.3 162.3 146.3 −27.9 −81.2 57.9 300.0 250.0 246.0 213.5 228.0 206.5 257.0 221.5 217.0 291.0 190.8 202.8 201.8 171.9 185.0 108.0 122.8 152.4 227.0 232.0 246.6 195.2 −21.0 61.5 +13.1 141.1 +12.9 +2.4 80.7 205.4 161.0 155.6 291.5 49.3 −33.5 177.2

Temperature, °C

Formula C7H4ClF3 C12H9Cl C12H9Cl C4H5ClO2 CHClF2 C8H11ClSi C8H9ClO C4H9ClO2 C6H12Cl2O2 C8H9Cl C8H9Cl C8H9Cl C4H6Cl2O2 C5H10Cl2O C5H10Cl2O C10H13ClO CHCl3 C10H7Cl C8H9ClO C6H5ClO C6H5ClO C6H5ClO C12H9ClO C12H9ClO CCl3NO2 C3H5Cl C5H4ClN C8H7Cl C8H7Cl C14H29Cl C7H7Cl C7H7Cl C7H7Cl C6H15ClSi C2ClF3 CClF3 C3H9ClSi C9H8O2 C9H10O C9H8O C5H4O3 C10H16O C10H18O C10H18O2 C10H20O C12H22O2 C9H6O2 C7H8O C7H8O C7H8O C4H6O2 C4H6O2 C4H5N C4H5N C9H12 C9H13N C10H12O C10H14O C7H11NO2 C2N2 CBrN CClN CIN C4H8 C4H6 C6H12 C8H16O C6H12O C6H10O C12H14N2O5 C5H10 C3H6 C10H14

40

0.0 89.3 96.4 70.0 −122.8 29.8 45.8 53.0 56.2 17.2 18.6 19.2 46.0 24.7 29.8 62.3 −58.0 80.6 84.0 12.1 44.2 49.8 118.0 119.8 −25.5 −81.3 13.3 25.3 28.0 98.5 +5.4 +4.8 +5.5 −4.9 −116.0 −149.5 −62.8 127.5 72.6 76.1 47.1 61.7 44.0 99.5 66.4 74.7 106.0 38.2 52.0 53.0 33.5

24.7 109.8 129.8 95.6 −110.2 56.7 72.8 78.3 83.7 43.0 45.2 46.4 72.1 50.1 56.5 91.4 −39.1 104.8 114.3 38.2 72.0 78.2 152.2 153.7 −3.3 −63.4 38.8 51.3 54.5 131.8 30.6 30.3 31.0 +19.8 −102.5 −139.2 −43.6 157.8 102.5 105.8 74.8 90.0 71.4 127.3 93.6 100.2 137.8 64.0 76.0 76.5 57.4

−29.0 −19.5 +2.9 60.0 58.0 74.2 42.0 −95.8 −35.7 −76.7 25.2 −92.0 −99.1 −45.3 50.4 21.0 +1.4 132.8 −68.0 −116.8 17.3

−7.1 +3.5 26.8 88.2 87.3 103.7 68.7 −83.2 −18.3 −61.4 47.2 −76.0 −83.4 −25.4 77.2 44.0 26.4 161.8 −49.6 −104.2 43.9

37.1 134.7 146.0 108.0 −103.7 70.0 86.5 90.7 97.6 56.1 58.1 60.0 86.0 63.0 70.0 106.0 −29.7 118.6 129.0 51.2 86.1 92.2 169.7 170.7 +7.8 −54.1 51.7 65.2 67.5 148.2 43.2 43.2 43.8 32.0 −95.9 −134.1 −34.0 173.0 117.8 120.0 88.9 103.9 84.8 141.4 107.0 113.0 153.4 76.7 87.8 88.6 69.0 80.0 +4.0 15.0 38.3 102.2 102.0 118.0 82.0 −76.8 −10.0 −53.8 57.7 −67.9 −75.4 −15.9 90.0 56.0 38.7 175.9 −40.4 −97.5 57.0

50.6 151.2 164.0 121.2 −96.5 84.7 101.5 104.1 112.2 70.3 73.0 75.5 100.0 77.2 84.8 121.8 −19.0 134.4 145.0 65.9 101.7 108.1 186.7 189.8 20.0 −44.0 65.8 80.0 82.0 166.2 56.9 57.4 57.8 45.5 −88.2 −128.5 −23.2 189.5 133.7 135.7 103.8 119.4 99.8 155.6 121.5 126.0 170.0 90.5 101.4 102.3 82.0 93.0 16.4 27.8 51.5 117.8 117.9 133.8 96.2 −70.1 −1.0 −46.1 68.6 −58.7 −66.6 −5.0 104.0 68.8 52.5 191.2 −30.1 −90.3 71.1

65.9 169.9 183.8 135.6 −88.6 101.2 117.8 118.4 127.8 86.2 89.2 91.8 116.0 92.4 101.5 139.6 −7.1 153.2 162.0 82.0 118.0 125.0 207.4 208.2 33.8 −32.7 81.7 96.5 98.0 187.0 72.0 73.0 73.5 60.2 −79.7 −121.9 −11.4 207.1 151.0 152.2 120.3 135.9 116.1 171.9 137.2 140.5 189.0 105.8 116.0 117.7 96.0 107.8 30.0 41.8 66.1 134.2 135.2 150.3 111.8 −62.7 +8.6 −37.5 80.3 −48.4 −56.4 +6.7 119.8 83.0 67.8 206.7 −18.6 −82.3 87.0

75.4 182.1 196.0 144.4 −83.4 111.5 127.8 127.5 138.0 96.4 99.6 102.0 126.2 102.2 111.8 150.0 +0.5 165.6 173.5 92.0 129.4 136.1 219.6 220.0 42.3 −25.1 91.6 107.2 108.5 199.8 81.8 83.2 83.3 69.5 −74.1 −117.3 −4.0 217.8 162.0 163.7 131.3 146.3 126.2 182.1 147.2 149.7 200.5 115.5 125.8 127.0 104.5 116.7 38.5 50.9 75.4 145.0 146.0 161.7 121.5 −57.9 14.7 −32.1 88.0 −41.8 −50.0 14.7 129.8 91.8 77.5 216.0 −11.3 −77.0 97.2

Melting point, °C −6.0 34 75.5 −160

−80.2 −53.3 −62.6

−63.5 −20 7 32.5 42 +6 −64 −99.0 −15.0 +0.9 +7.3 −157.5 133 33 −7.5

70 30.8 10.9 35.5 15.5 72 −96.0

−34.4 58 −6.5 −50 +6.6 23.9 −45.0 −93.7 −126.6 −68.2

VAPOR PRESSURES OF PURE SUBSTANCES TABLE 2-8

2-65

Vapor Pressures of Organic Compounds, up to 1 atm (Continued ) Pressure, mm Hg Compound Name

cis-Decalin trans-Decalin Decane Decan-2-one 1-Decene Decyl alcohol Decyltrimethylsilane Dehydroacetic acid Desoxybenzoin Diacetamide Diacetylene (1,3-butadiyne) Diallyldichlorosilane Diallyl sulfide Diisoamyl ether oxalate sulfide Dibenzylamine Dibenzyl ketone (1,3-diphenyl2-propanone) 1,4-Dibromobenzene 1,2-Dibromobutane dl-2,3-Dibromobutane meso-2,3-Dibromobutane 1,2-Dibromodecane Di(2-bromoethyl) ether α,β-Dibromomaleic anhydride 1,2-Dibromo-2-methylpropane 1,3-Dibromo-2-methylpropane 1,2-Dibromopentane 1,2-Dibromopropane 1,3-Dibromopropane 2,3-Dibromopropene 2,3-Dibromo-1-propanol Diisobutylamine 2,6-Ditert-butyl-4-cresol 4,6-Ditert-butyl-2-cresol 4,6-Ditert-butyl-3-cresol 2,6-Ditert-butyl-4-ethylphenol 4,6-Ditert-butyl-3-ethylphenol Diisobutyl oxalate 2,4-Ditert-butylphenol Dibutyl phthalate sulfide Diisobutyl d-tartrate Dicarvacryl-mono-(6-chloro-2-xenyl) phosphate Dicarvacryl-2-tolyl phosphate Dichloroacetic acid 1,2-Dichlorobenzene 1,3-Dichlorobenzene 1,4-Dichlorobenzene 1,2-Dichlorobutane 2,3-Dichlorobutane 1,2-Dichloro-1,2-difluoroethylene Dichlorodifluoromethane Dichlorodiphenyl silane Dichlorodiisopropyl ether Di(2-chloroethoxy) methane Dichloroethoxymethylsilane 1,2-Dichloro-3-ethylbenzene 1,2-Dichloro-4-ethylbenzene 1,4-Dichloro-2-ethylbenzene cis-1,2-Dichloroethylene trans-1,2-Dichloro ethylene Di(2-chloroethyl) ether Dichlorofluoromethane 1,5-Dichlorohexamethyltrisiloxane Dichloromethylphenylsilane 1,1-Dichloro-2-methylpropane 1,2-Dichloro-2-methylpropane 1,3-Dichloro-2-methylpropane 2,4-Dichlorophenol 2,6-Dichlorophenol

1

5

10

20

22.5 −0.8 16.5 44.2 14.7 69.5 67.4 91.7 123.3 70.0 −82.5 +9.5 −9.5 18.6 85.4 43.0 118.3 125.5

50.1 +30.6 42.3 71.9 40.3 97.3 96.4 122.0 156.2 95.0 −68.0 34.8 +14.4 44.3 116.0 73.0 149.8 159.8

64.2 47.2 55.7 85.8 53.7 111.3 111.0 137.3 173.5 108.0 −61.2 47.4 26.6 57.0 131.4 87.6 165.6 177.6

79.8 65.3 69.8 100.7 67.8 125.8 126.5 153.0 192.0 122.6 −53.8 61.3 39.7 70.7 147.7 102.7 182.2 195.7

97.2 85.7 85.5 117.1 83.3 142.1 144.0 171.0 212.0 138.2 −45.9 76.4 54.2 86.3 165.7 120.0 200.2 216.6

C6H4Br2 61.0 C4H8Br2 7.5 C4H8Br2 +5.0 C4H8Br2 +1.5 C10H20Br2 95.7 C4H8Br2O 47.7 C4H2Br2O3 50.0 C4H8Br2 −28.8 C4H8Br2 14.0 C5H10Br2 19.8 C3H6Br2 −7.0 C3H6Br2 +9.7 C3H4Br2 −6.0 C3H6Br2O 57.0 C8H19N −5.1 C15H24O 85.8 C15H24O 86.2 C15H24O 103.7 C16H26O 89.1 C16H26O 111.5 C10H18O4 63.2 C14H22O 84.5 C16H22O4 148.2 C8H18S +21.7 C12H22O6 117.8 C32H34ClO4P 204.2

79.3 33.2 30.0 26.6 123.6 75.3 78.0 −3.0 40.0 45.4 +17.3 35.4 +17.9 84.5 +18.4 116.2 117.3 135.2 121.4 142.6 91.2 115.4 182.1 51.8 151.8 234.5

87.7 46.1 41.6 39.3 137.3 88.5 92.0 +10.5 53.0 58.0 29.4 48.0 30.0 98.2 30.6 131.0 132.4 150.0 137.0 157.4 105.3 130.0 198.2 66.4 169.0 249.3

103.6 60.0 56.4 53.2 151.0 103.6 106.7 25.7 67.5 72.0 42.3 62.1 43.2 113.5 43.7 147.0 149.0 167.0 154.0 174.0 120.3 146.0 216.2 80.5 188.0 264.5

C27H33O4P 180.2 C2H2Cl2O2 44.0 C6H4Cl2 20.0 C6H4Cl2 12.1 C6H4Cl2 C4H8Cl2 −23.6 C4H8Cl2 −25.2 C2Cl2F2 −82.0 CCl2F2 −118.5 C12H10Cl2Si 109.6 C6H12Cl2O 29.6 C5H10Cl2O2 53.0 C8H8Cl2OSi −33.8 C8H8Cl2 46.0 C8H8Cl2 47.0 C8H8Cl2 38.5 C2H2Cl2 −58.4 C2H2Cl2 −65.4 C4H8Cl2O 23.5 CHCl2F −91.3 C6H18Cl2 26.0 O2Si3 C7H8Cl2Si 35.7 C4H8Cl2 −31.0 C4H8Cl2 −25.8 C4H8Cl2 −3.0 C6H4Cl2O 53.0 C6H4Cl2O 59.5

209.3 69.8 46.0 39.0 −0.3 −3.0 −65.6 −104.6 142.4 55.2 80.4 −12.1 75.0 77.2 68.0 −39.2 −47.2 49.3 −75.5 52.0

221.8 82.6 59.1 52.0 54.8 +11.5 +8.5 −57.3 −97.8 158.0 68.2 94.0 −1.3 90.0 92.3 83.2 −29.9 −38.0 62.0 −67.5 65.1

63.5 −8.4 −4.2 +20.6 80.0 87.6

77.4 +2.6 +6.7 32.0 92.8 101.0

60

Melting point, °C

100

200

400

760

108.0 98.4 95.5 127.8 93.5 152.0 154.3 181.5 224.5 148.0 −41.0 86.3 63.7 96.0 177.0 130.6 212.2 229.4

123.2 114.6 108.6 142.0 106.5 165.8 169.5 197.5 241.3 160.6 −34.0 99.7 75.8 109.6 192.2 145.3 227.3 246.6

145.4 136.2 128.4 163.2 126.7 186.2 191.0 219.5 265.2 180.8 −20.9 119.4 94.8 129.0 215.0 166.4 249.8 272.3

169.9 160.1 150.6 186.7 149.2 208.8 215.5 244.5 293.0 202.0 −6.1 142.0 116.1 150.3 240.0 191.0 274.3 301.7

194.6 186.7 174.1 211.0 172.0 231.0 240.0 269.0 321.0 223.0 +9.7 165.3 138.6 173.4 265.0 216.0 300.0 330.5

−43.3 −30.7 −29.7 +3.5

120.8 76.0 72.0 68.0 167.4 119.8 123.5 42.3 83.5 87.4 57.2 77.8 57.8 129.8 57.8 164.1 167.4 185.3 172.1 192.3 137.5 164.3 235.8 96.0 208.5 280.5

131.6 86.0 82.0 78.0 177.5 130.0 133.8 53.7 93.7 97.4 66.4 87.8 67.0 140.0 67.0 175.2 179.0 196.1 183.9 204.4 147.8 175.8 247.8 105.8 221.6 290.7

146.5 99.8 95.3 91.7 190.2 144.0 147.7 68.8 107.4 110.1 78.7 101.3 79.5 153.0 79.2 190.0 194.0 211.0 198.0 218.0 161.8 190.0 263.7 118.6 239.5 304.9

168.5 120.2 115.7 111.8 209.6 165.0 168.0 92.1 117.8 130.2 97.8 121.7 98.0 173.8 97.6 212.8 217.5 233.0 220.0 241.7 183.5 212.5 287.0 138.0 264.7 323.8

192.5 143.5 138.0 134.2 229.8 188.0 192.0 119.8 150.6 151.8 118.5 144.1 119.5 196.0 118.0 237.6 243.4 257.1 244.0 264.6 205.8 237.0 313.5 159.0 294.0 342.0

218.6 166.3 160.5 157.3 250.4 212.5 215.0 149.0 174.6 175.0 141.6 167.5 141.2 219.0 139.5 262.5 269.3 282.0 268.6 290.0 229.5 260.8 340.0 182.0 324.0 361.0

87.5 −64.5

237.0 96.3 73.4 66.2 69.2 24.5 21.2 −48.3 −90.1 176.0 82.2 109.5 +11.3 105.9 109.6 99.8 −19.4 −28.0 76.0 −58.6 79.0

251.5 111.8 89.4 82.0 84.8 37.7 35.0 −38.2 −81.6 195.5 97.3 125.5 24.4 123.8 127.5 118.0 −7.9 −17.0 91.5 −48.8 94.8

260.3 121.5 99.5 92.2 95.2 47.8 43.9 −31.8 −76.1 207.5 106.9 135.8 32.6 135.0 139.0 129.0 −0.5 −10.0 101.5 −42.6 105.0

272.5 134.0 112.9 105.0 108.4 60.2 56.0 −23.0 −68.6 223.8 119.7 149.6 44.1 149.8 153.3 144.0 +9.5 −0.2 114.5 −33.9 118.2

290.0 152.3 133.4 125.9 128.3 79.7 74.0 −10.0 −57.0 248.0 139.0 170.0 61.0 172.0 176.0 166.2 24.6 +14.3 134.0 −20.9 138.3

309.8 173.7 155.8 149.0 150.2 100.8 94.2 +5.0 −43.9 275.5 159.8 192.0 80.3 197.0 201.7 191.5 41.0 30.8 155.4 −6.2 160.2

330.0 194.4 179.0 173.0 173.9 123.5 116.0 20.9 −29.8 304.0 182.7 215.0 100.6 222.1 226.6 216.3 59.0 47.8 178.5 +8.9 184.0

92.4 14.6 18.7 44.8 107.7 115.5

109.5 28.2 32.0 58.6 123.4 131.6

120.0 37.0 40.2 67.5 133.5 141.8

134.2 48.2 51.7 78.8 146.0 154.6

155.5 65.8 68.9 96.1 165.2 175.5

180.2 85.4 87.8 115.4 187.5 197.7

205.5 106.0 108.0 135.0 210.0 220.0

Temperature, °C

Formula C10H18 C10H18 C10H22 C10H20O C10H20 C10H22O C13H30Si C8H8O4 C14H12O C4H7NO2 C4H2 C6H10Cl2Si C6H10S C10H22O C12H22O4 C10H22S C14H15N C15H14O

40

+7 60 78.5 −34.9 −83

−26 34.5

−34.5

−70.3 −55.5 −34.4 −70

−79.7 73.5

9.7 −17.6 −24.2 53.0 −80.4 −112

−40.8 −76.4 −61.2 −80.5 −50.0 −135 −53.0

45.0

2-66

PHYSICAL AND CHEMICAL DATA

TABLE 2-8

Vapor Pressures of Organic Compounds, up to 1 atm (Continued ) Pressure, mm Hg Compound Name

α,α-Dichlorophenylacetonitrile Dichlorophenylarsine 1,2-Dichloropropane 2,3-Dichlorostyrene 2,4-Dichlorostyrene 2,5-Dichlorostyrene 2,6-Dichlorostyrene 3,4-Dichlorostyrene 3,5-Dichlorostyrene 1,2-Dichlorotetraethylbenzene 1,4-Dichlorotetraethylbenzene 1,2-Dichloro-1,1,2,2-tetrafluoroethane Dichloro-4-tolylsilane 3,4-Dichloro-α,α,α-trifluorotoluene Dicyclopentadiene Diethoxydimethylsilane Diethoxydiphenylsilane Diethyl adipate Diethylamine N-Diethylaniline Diethyl arsanilate 1,2-Diethylbenzene 1,3-Diethylbenzene 1,4-Diethylbenzene Diethyl carbonate cis-Diethyl citraconate Diethyl dioxosuccinate Diethylene glycol Diethyleneglycol-bis-chloroacetate Diethylene glycol dimethyl ether Di(2-methoxyethyl) ether glycol ethyl ether Diethyl ether ethylmalonate fumarate glutarate Diethylhexadecylamine Diethyl itaconate ketone (3-pentanone) malate maleate malonate mesaconate oxalate phthalate sebacate 2,5-Diethylstyrene Diethyl succinate isosuccinate sulfate sulfide sulfite d-Diethyl tartrate dl-Diethyl tartrate 3,5-Diethyltoluene Diethylzinc 1-Dihydrocarvone Dihydrocitronellol 1,4-Dihydroxyanthraquinone Dimethylacetylene (2-butyne) Dimethylamine N,N-Dimethylaniline Dimethyl arsanilate Di(α-methylbenzyl) ether 2,2-Dimethylbutane 2,3-Dimethylbutane Dimethyl citraconate 1,1-Dimethylcyclohexane cis-1,2-Dimethylcyclohexane trans-1,2-Dimethylcyclohexane trans-1,3-Dimethylcyclohexane cis-1,3-Dimethylcyclohexane cis-1,4-Dimethylcyclohexane trans-1,4-Dimethylcyclohexane

1

5

10

20

56.0 61.8 −38.5 61.0 53.5 55.5 47.8 57.2 53.5 105.6 91.7 −95.4 46.2 11.0 −19.1 111.5 74.0

84.0 100.0 −17.0 90.1 82.2 83.9 75.7 86.0 82.2 138.7 126.1 −80.0 71.7 38.3 34.1 +2.4 142.8 106.6

49.7

78.0

98.1 116.0 −6.1 104.6 97.4 98.2 90.0 100.4 97.4 155.0 143.8 −72.3 84.2 52.2 47.6 13.3 157.6 123.0 −33.0 91.9

113.8 133.1 +6.0 120.5 111.8 114.0 105.5 116.2 111.8 172.5 162.0 −63.5 97.8 67.3 62.0 25.3 174.3 138.3 −22.6 107.2

130.0 151.0 19.4 137.8 129.2 131.0 122.4 133.7 129.2 192.2 183.2 −53.7 113.2 84.0 77.9 38.0 193.2 154.6 −11.3 123.6

38.0 22.3 20.7 20.7 −10.1 59.8 70.0 91.8 148.3

62.6 48.7 46.8 47.1 +12.3 88.3 98.0 120.0 180.0

74.8 62.0 59.9 60.3 23.8 103.0 112.0 133.8 195.8

88.0 76.4 74.5 74.7 36.0 118.2 126.8 148.0 212.0

C6H14O3 C6H14O3 C4H10O C9H16O4 C8H12O4 C9H16O4 C20H43N C9H14O4 C5H10O C8H14O5 C8H12O4 C7H12O4 C9H14O4 C6H10O4 C12H14O4 C14H26O4 C12H16 C8H14O4 C8H14O4 C4H10O4S C4H10S C4H10O3S C8H14O6 C8H14O6 C11H16 C4H10Zn C10H16O C10H22O C14H8O4 C4H6 C2H7N C8H11N C8H12AsNO3 C16H18O C6H14 C6H14 C7H10O4 C8H16 C8H16 C8H16 C8H16 C8H16 C8H16 C8H16

13.0 45.3 −74.3 50.8 53.2 65.6 139.8 51.3 −12.7 80.7 57.3 40.0 62.8 47.4 108.8 125.3 49.7 54.6 39.8 47.0 −39.6 10.0 102.0 100.0 34.0 −22.4 46.6 68.0 196.7 −73.0 −87.7 29.5 15.0 96.7 −69.3 −63.6 50.8 −24.4 −15.9 −21.1 −19.4 −22.7 −20.0 −24.3

37.6 72.0 −56.9 77.8 81.2 94.7 175.8 80.2 +7.5 110.4 85.6 67.5 91.0 71.8 140.7 156.2 78.4 83.0 66.7 74.0 −18.6 34.2 133.0 131.7 61.5 0.0 75.5 91.7 239.8 −57.9 −72.2 56.3 39.6 128.3 −50.7 −44.5 78.2 −1.4 +7.3 +1.7 +3.4 0.0 +3.2 −1.7

50.0 85.8 −48.1 91.6 95.3 109.7 194.0 95.2 17.2 125.3 100.0 81.3 105.3 83.8 156.0 172.1 92.6 96.6 80.0 87.7 −8.0 46.4 148.0 147.2 75.3 +11.7 90.0 103.0 259.8 −50.5 −64.6 70.0 51.8 144.0 −41.5 −34.9 91.8 +10.3 18.4 13.0 14.9 +11.2 14.5 +10.1

63.0 100.3 −38.5 106.0 110.2 125.4 213.5 111.0 27.9 141.2 115.3 95.9 120.3 96.8 173.6 189.8 108.5 111.7 94.7 102.1 +3.5 59.7 164.2 163.8 90.2 24.2 106.0 115.0 282.0 −42.5 −56.0 84.8 65.0 160.3 −31.1 −24.1 106.5 23.0 31.1 25.6 27.4 23.6 27.1 22.6

60

100

200

400

760

141.0 163.2 28.0 149.0 140.0 142.0 133.3 144.6 140.0 204.8 195.8 −47.5 122.6 95.0 88.0 46.3 205.0 165.8 −4.0 133.8

154.5 178.9 39.4 163.5 153.8 155.8 147.6 158.2 153.8 220.7 212.0 −39.1 135.5 109.2 101.7 57.6 220.0 179.0 +6.0 147.3

176.2 202.8 57.0 185.7 176.0 178.0 169.0 181.5 176.0 245.6 238.5 −26.3 153.5 129.0 121.8 74.2 243.8 198.2 21.0 168.2

199.5 228.8 76.0 210.0 200.0 202.5 193.5 205.7 200.0 272.8 265.8 −12.0 175.2 150.5 144.2 93.2 259.7 219.1 38.0 192.4

223.5 256.5 96.8 235.0 225.0 227.0 217.0 230.0 225.0 302.0 296.5 +3.5 196.3 172.8 166.6 113.5 296.0 240.0 55.5 215.5

102.6 92.5 90.4 91.1 49.5 135.7 143.8 164.3 229.0

111.8 102.6 100.7 101.3 57.9 146.2 153.7 174.0 239.5

123.8 116.2 114.4 115.3 69.7 160.0 167.7 187.5 252.0

141.9 136.7 134.8 136.1 86.5 182.3 188.0 207.0 271.5

161.0 159.0 156.9 159.0 105.8 206.5 210.8 226.5 291.8

181.0 183.5 181.1 183.8 125.8 230.3 233.5 244.8 313.0

77.5 116.7 27.7 122.4 126.7 142.8 235.0 128.2 39.4 157.8 131.8 113.3 137.3 110.6 192.1 207.5 125.8 127.8 111.0 118.0 16.1 74.2 182.3 181.7 107.0 38.0 123.7 127.6 307.4 −33.9 −46.7 101.6 79.7 179.6 −19.5 −12.4 122.6 37.3 45.3 39.7 41.4 37.5 41.1 36.5

86.8 126.8 −21.8 132.4 137.7 153.2 248.5 139.9 46.7 169.0 142.4 123.0 147.9 119.7 204.1 218.4 136.8 138.2 121.4 128.6 24.2 83.8 194.0 193.2 117.7 47.2 134.7 136.7 323.3 −27.8 −40.7 111.9 88.6 191.5 −12.1 −4.9 132.7 45.7 54.4 48.7 50.4 46.4 50.1 45.4

99.5 140.3 −11.5 146.0 151.1 167.8 265.5 154.3 56.2 183.9 156.0 136.2 161.6 130.8 219.5 234.4 151.0 151.1 134.8 142.5 35.0 96.3 208.5 208.0 131.7 59.1 149.7 145.9 344.5 −18.8 −32.6 125.8 101.0 206.8 −2.0 +5.4 145.8 57.9 66.8 61.0 62.5 58.5 62.3 57.6

118.0 159.0 +2.2 166.0 172.2 189.5 292.8 177.5 70.6 205.3 177.8 155.5 183.2 147.9 243.0 255.8 173.2 171.7 155.1 162.5 51.3 115.8 230.4 230.0 152.4 77.0 171.8 160.2 377.8 −5.0 −20.4 146.5 119.8 229.7 +13.4 21.1 165.8 76.2 85.6 79.6 81.0 76.9 80.8 76.0

138.5 180.3 17.9 188.7 195.8 212.8 324.6 203.1 86.3 229.5 201.7 176.8 205.8 166.2 267.5 280.3 198.0 193.8 177.7 185.5 69.7 137.0 254.8 254.3 176.5 97.3 197.0 176.8 413.0 +10.6 −7.1 169.2 140.3 254.8 31.0 39.0 188.0 97.2 107.0 100.9 102.1 97.8 101.9 97.0

159.8 201.9 34.6 211.5 218.5 237.0 355.0 227.9 102.7 253.4 225.0 198.9 229.0 185.7 294.0 305.5 223.0 216.5 201.3 209.5 88.0 159.0 280.0 280.0 200.7 118.0 223.0 193.5 450.0 27.2 +7.4 193.1 160.5 281.0 49.7 58.0 210.5 119.5 129.7 123.4 124.4 120.1 124.3 119.3

Temperature, °C

Formula C8H5Cl2N C6H5AsCl2 C3H6Cl2 C8H6Cl2 C8H6Cl2 C8H6Cl2 C8H6Cl2 C8H6Cl2 C8H6Cl2 C14H20Cl2 C14H20Cl2 C2Cl2F4 C7H8Cl2Si C7H3Cl2F3 C10H8 C6H16O2Si C16H20O2Si C10H18O4 C4H11N C10H15N C10H16As NO3 C10H14 C10H14 C10H14 C5H10O3 C9H14O4 C8H10O6 C4H10O3 C8H12Cl2O5

40

Melting point, °C

−94 −12.1 32.9 −21 −38.9 −34.4 −31.4 −83.9 −43.2 −43

−116.3 +0.6

−42 −49.8 −40.6 1.3 −20.8 −25.0 −99.5 17 −28 194 −32.5 −96 +2.5 −99.8 −128.2 −34 −50.0 −88.0 −92.0 −76.2 −87.4 −36.9

VAPOR PRESSURES OF PURE SUBSTANCES TABLE 2-8

2-67

Vapor Pressures of Organic Compounds, up to 1 atm (Continued ) Pressure, mm Hg Compound Name

Dimethyl ether 2,2-Dimethylhexane 2,3-Dimethylhexane 2,4-Dimethylhexane 2,5-Dimethylhexane 3,3-Dimethylhexane 3,4-Dimethylhexane Dimethyl itaconate 1-Dimethyl malate Dimethyl maleate malonate trans-Dimethyl mesaconate 2,7-Dimethyloctane Dimethyl oxalate 2,2-Dimethylpentane 2,3-Dimethylpentane 2,4-Dimethylpentane 3,3-Dimethylpentane 2,3-Dimethylphenol (2,3-xylenol) 2,4-Dimethylphenol (2,4-xylenol) 2,5-Dimethylphenol (2,5-xylenol) 3,4-Dimethylphenol (3,4-xylenol) 3,5-Dimethylphenol (3,5-xylenol) Dimethylphenylsilane Dimethyl phthalate 3,5-Dimethyl-1,2-pyrone 4,6-Dimethylresorcinol Dimethyl sebacate 2,4-Dimethylstyrene 2,5-Dimethylstyrene α,α-Dimethylsuccinic anhydride Dimethyl sulfide d-Dimethyl tartrate dl-Dimethyl tartrate N,N-Dimethyl-2-toluidine N,N-Dimethyl-4-toluidine Di(nitrosomethyl) amine Diosphenol 1,4-Dioxane Dipentene Diphenylamine Diphenyl carbinol (benzhydrol) chlorophosphate disulfide 1,2-Diphenylethane (dibenzyl) Diphenyl ether 1,1-Diphenylethylene trans-Diphenylethylene 1,1-Diphenylhydrazine Diphenylmethane Diphenyl sulfide Diphenyl-2-tolyl thiophosphate 1,2-Dipropoxyethane 1,2-Diisopropylbenzene 1,3-Diisopropylbenzene Dipropylene glycol Dipropyleneglycol monobutyl ether isopropyl ether Di-n-propyl ether Diisopropyl ether Di-n-propyl ketone (4-heptanone) Di-n-propyl oxalate Diisopropyl oxalate Di-n-propyl succinate Di-n-propyl d-tartrate Diisopropyl d-tartrate Divinyl acetylene (1,5-hexadiene-3-yne) 1,3-Divinylbenzene Docosane n-Dodecane 1-Dodecene n-Dodecyl alcohol Dodecylamine Dodecyltrimethylsilane Elaidic acid

1

5

10

20

−115.7 −29.7 −23.0 −26.9 −26.7 −25.8 −22.1 69.3 75.4 45.7 35.0 46.8 +6.3 20.0 −49.0 −42.0 −48.0 −45.9 56.0 51.8 51.8 66.2 62.0 +5.3 100.3 78.6 49.0 104.0 34.2 29.0 61.4 −75.6 102.1 100.4 28.8 50.1 +3.2 66.7 −35.8 14.0 108.3 110.0 121.5 131.6 86.8 66.1 87.4 113.2 126.0 76.0 96.1 159.7 −38.8 40.0 34.7 73.8 64.7 46.0 −43.3 −57.0 23.0 53.4 43.2 77.5 115.6 103.7 −45.1 32.7 157.8 47.8 47.2 91.0 82.8 91.2 171.3

−101.1 −7.9 −1.1 −5.3 −5.5 −4.4 +0.2 94.0 104.0 73.0 59.8 74.0 30.5 44.0 −28.7 −20.8 −27.4 −25.0 83.8 78.0 78.0 93.8 89.2 30.3 131.8 107.6 76.8 139.8 61.9 55.9 88.1 −58.0 133.2 131.8 54.1 74.3 27.8 95.4 −12.8 40.4 141.7 145.0 160.5 164.0 119.8 97.8 119.6 145.8 159.3 107.4 129.0 179.8 −10.3 67.8 62.3 102.1 92.0 72.8 −22.3 −37.4 44.4 80.2 69.0 107.6 147.7 133.7 −24.4 60.0 195.4 75.8 74.0 120.2 111.8 122.1 206.7

−93.3 +3.1 +9.9 +5.2 +5.3 +6.1 11.3 106.6 118.3 86.4 72.0 87.8 42.3 56.0 −18.7 −10.3 −17.1 −14.4 97.6 91.3 91.3 107.7 102.4 42.6 147.6 122.0 90.7 156.2 75.8 69.0 102.0 −49.2 148.2 147.5 66.2 86.7 40.0 109.0 −1.2 53.8 157.0 162.0 182.0 180.0 136.0 114.0 135.0 161.0 176.1 122.8 145.0 201.6 +5.0 81.8 76.0 116.2 106.0 86.2 −11.8 −27.4 55.0 93.9 81.9 122.2 163.5 148.2 −14.0 73.8 213.0 90.0 87.8 134.7 127.8 137.7 223.5

−85.2 15.0 22.1 17.2 17.2 18.2 23.5 119.7 133.8 101.3 85.0 102.1 55.8 69.4 −7.5 +1.1 −5.9 −2.9 112.0 105.0 105.0 122.0 117.0 56.2 164.0 136.4 105.8 175.8 90.8 84.0 116.3 −39.4 164.3 164.0 80.2 100.0 53.7 124.0 +12.0 68.2 175.2 180.9 203.8 197.0 153.7 130.8 151.8 179.8 194.0 139.8 162.0 215.5 22.3 96.8 91.2 131.3 120.4 100.8 0.0 −16.7 66.2 108.6 95.6 138.0 180.4 164.0 −2.8 88.7 233.5 104.6 102.4 150.0 141.6 153.8 242.3

60

100

200

400

760

−62.7 48.2 56.0 50.6 50.5 52.5 57.7 153.7 175.1 140.4 121.9 141.5 93.9 104.8 23.9 33.3 25.4 29.3 152.2 143.0 143.0 161.0 156.0 94.2 210.0 177.5 147.3 222.6 132.3 124.7 155.3 −12.0 208.8 209.5 118.1 140.3 90.3 165.6 45.1 108.3 222.8 227.5 265.0 241.3 202.8 178.8 198.6 227.4 242.5 186.3 211.8 252.5 74.2 138.7 132.3 169.9 159.8 140.3 33.0 13.7 96.0 148.1 132.6 180.3 227.0 207.3 29.5 130.0 286.0 146.2 142.3 192.0 182.1 199.5 288.0

−50.9 65.7 73.8 68.1 68.0 70.0 75.6 171.0 196.3 160.0 140.0 161.0 114.0 123.3 40.3 50.1 41.8 46.2 173.0 161.5 161.5 181.5 176.2 114.2 232.7 198.0 167.8 245.0 153.2 145.6 175.8 +2.6 230.5 232.3 138.3 161.6 110.0 186.2 62.3 128.2 247.5 250.0 299.5 262.6 227.8 203.3 222.8 251.7 267.2 210.7 236.8 270.3 103.8 159.8 153.7 189.9 180.0 160.0 50.3 30.0 111.2 168.0 151.2 202.5 250.1 228.2 46.0 151.4 314.2 167.2 162.2 213.0 203.0 222.0 312.4

−37.8 85.6 94.1 88.2 87.9 90.4 96.0 189.8 219.5 182.2 159.8 183.5 136.0 143.3 59.2 69.4 60.6 65.5 196.0 184.2 184.2 203.6 197.8 136.4 257.8 221.0 192.0 269.6 177.5 168.7 197.5 18.7 255.0 257.4 161.5 185.4 131.3 209.5 81.8 150.5 274.1 275.6 337.2 285.8 255.0 230.7 249.8 278.3 294.0 237.5 263.9 290.0 140.0 184.3 177.6 210.5 203.8 183.1 69.5 48.2 127.3 190.3 171.8 226.5 275.6 251.8 64.4 175.2 343.5 191.0 185.5 235.7 225.0 248.0 337.0

−23.7 106.8 115.6 109.4 109.1 112.0 117.7 208.0 242.6 205.0 180.7 206.0 159.7 163.3 79.2 89.8 80.5 86.1 218.0 211.5 211.5 225.2 219.5 159.3 283.7 245.0 215.0 293.5 202.0 193.0 219.5 36.0 280.0 282.0 184.8 209.5 153.0 232.0 101.1 174.6 302.0 301.0 378.0 310.0 284.0 258.5 277.0 306.5 322.2 264.5 292.5 310.0 180.0 209.0 202.0 231.8 227.0 205.6 89.5 67.5 143.7 213.5 193.5 250.8 303.0 275.0 84.0 199.5 376.0 216.2 208.0 259.0 248.0 273.0 362.0

Temperature, °C

Formula C2H6O C8H18 C8H18 C8H18 C8H18 C8H18 C8H18 C7H10O4 C6H10O5 C6H8O4 C5H8O4 C7H10O4 C10H22 C4H6O4 C7H16 C7H16 C7H16 C7H16 C8H10O C8H10O C8H10O C8H10O C8H10O C8H12Si C10H10O4 C7H8O2 C8H10O2 C12H22O4 C10H12 C10H12 C6H8O3 C2H6S C6H10O6 C6H10O6 C9H13N C9H13N C2H5N3O2 C10H16O2 C4H8O2 C10H16 C12H11N C13H12O C12H10ClPO3 C12H10S2 C14H14 C12H10O C14H12 C14H12 C12H12N2 C13H12 C12H10S C18H17O3PS C8H18O2 C12H18 C12H18 C6H14O3 C10H22O3 C9H20O3 C6H14O C6H14O C7H14O C8H14O4 C8H14O4 C10H18O4 C10H18O6 C10H18O6 C6H6 C10H10 C22H46 C12H26 C12H24 C12H26O C12H27N C15H34Si C18H34O2

40 −76.2 28.2 35.6 30.5 30.4 31.7 37.1 133.7 150.1 117.2 100.0 118.0 71.2 83.6 +5.0 13.9 +6.5 +9.9 129.2 121.5 121.5 138.0 133.3 71.4 182.8 152.7 122.5 196.0 107.7 100.2 132.3 −28.4 182.4 182.4 95.0 116.3 68.2 141.2 25.2 84.3 194.3 200.0 227.9 214.8 173.7 150.0 170.8 199.0 213.5 157.8 182.8 230.6 42.3 114.0 107.9 147.4 136.3 117.0 +13.2 −4.5 78.1 124.6 110.5 154.8 199.7 181.8 +10.0 105.5 254.5 121.7 118.6 167.2 157.4 172.1 260.8

−70.4 36.7 44.2 39.0 38.9 40.4 45.8 142.6 160.4 127.1 109.7 127.8 80.8 92.8 13.0 22.1 14.5 18.1 139.5 131.0 131.0 148.0 143.5 81.3 194.0 163.8 133.2 208.0 118.0 110.7 142.4 −21.4 193.8 193.8 105.2 126.4 77.7 151.3 33.8 94.6 206.9 212.0 244.2 226.2 186.0 162.0 183.4 211.5 225.9 170.2 194.8 240.4 55.8 124.3 118.2 156.5 146.3 126.8 21.6 +3.4 85.8 134.8 120.0 166.0 211.7 192.6 18.1 116.0 268.3 132.1 128.5 177.8 168.0 184.2 273.0

Melting point, °C −138.5

−90.7 38 −62 −52.8 −123.7 −135 −119.5 −135.0 75 25.5 74.5 62.5 68 51.5 38

−83.2 61.5 89 −61

10 52.9 68.5 61 51.5 27 124 44 26.5

−105

−122 −60 −32.6

−66.9 44.5 −9.6 −31.5 24 51.5

2-68

PHYSICAL AND CHEMICAL DATA

TABLE 2-8

Vapor Pressures of Organic Compounds, up to 1 atm (Continued ) Pressure, mm Hg Compound

1

5

10

20

−16.5 −69.0 206.7 52.6 −159.5 36.3 −50.9 167.0 −43.4 28.5 −92.5 −29.5 47.0 −29.0 −31.3 −82.3 52.0 38.5 29.7 33.7 33.5 −9.8 44.0 107.6 −74.3 10.6 −18.4 −24.3 118.2 11.0 107.8 133.2 −89.8 +1.0 −5.1 +6.6 87.6 28.3 31.5 67.8 −14.5 −32.2 9.6 76.0 98.3 −168.3 112.0 −4.0 −11.0 −27.0 −44.5 53.0 −33.5

+5.6 −50.0 239.7 80.0 −148.5 63.1 −31.0 198.2 −23.5 54.0 −76.7 −8.7 70.7 −6.4 −12.0 −66.4 80.0 66.4 55.9 60.3 60.2 +13.9 72.0 136.4 −56.4 35.8 +4.0 −2.4 149.8 35.8 65.8 131.8 168.2 −73.9 25.4 +18.0 30.2 108.5 55.5 58.4 93.5 +9.2 −10.8 34.0 106.3 130.2 −158.3 142.4 +19.0 +10.5 +4.7 −24.0 79.7 −10.2

16.6 −40.3 254.5 93.7 −142.9 76.2 −20.7 213.5 −13.5 67.3 −68.7 +2.0 82.0 +5.0 −2.3 −58.3 93.8 80.6 69.0 73.9 73.9 25.9 86.0 150.3 −47.5 48.0 15.3 +8.4 165.0 48.0 77.8 143.7 186.0 −65.8 37.5 29.9 41.9 134.0 68.8 72.0 106.0 20.6 −0.1 46.3 121.7 146.0 −153.2 158.0 30.3 21.5 18.6 −13.6 92.1 +1.6

29.0 −29.5 270.6 108.4 −136.7 91.0 −9.8 230.0 −3.0 81.1 −59.9 13.0 94.4 17.7 +8.0 −48.6 109.0 96.0 83.1 88.5 88.5 38.6 101.4 166.8 −37.8 61.8 27.8 20.6 181.8 61.7 91.0 155.5 205.5 −56.8 50.4 42.0 54.3 150.3 83.6 86.7 119.8 33.4 +11.7 59.5 137.7 162.8 −147.6 173.5 42.5 33.0 32.7 −2.4 105.8 14.7

42.0 −17.3 289.1 124.6 −129.8 107.2 +3.7 247.0 +9.1 96.2 −50.0 26.0 108.1 31.8 19.0 −39.8 125.7 113.2 98.8 104.8 104.7 52.8 118.2 181.8 −26.7 77.0 41.5 33.8 199.8 76.3 105.6 168.8 226.5 −47.0 65.2 56.0 68.2 169.2 99.9 103.3 133.8 47.6 25.0 74.0 154.4 182.0 −141.3 191.0 56.0 45.8 48.0 +10.0 120.0 29.7

50.6 −9.7 300.2 135.2 −125.4 127.5 11.5 258.3 16.6 106.0 −43.4 33.5 116.7 40.6 26.0 −33.4 136.0 123.6 109.0 115.5 115.4 61.8 129.0 191.9 −19.5 86.7 50.1 42.3 211.5 85.8 114.8 177.3 239.8 −40.6 74.0 65.2 77.3 181.2 110.2 113.8 142.1 56.7 33.4 83.6 166.0 193.7 −137.3 201.8 64.1 53.8 57.9 18.1 129.5 39.0

62.0 +1.2 314.4 148.5 −119.3 131.4 22.1 273.5 27.0 118.5 −34.9 44.5 127.5 53.0 34.9 −25.1 149.8 137.3 122.3 129.2 128.4 74.1 143.2 205.0 −10.0 99.8 62.0 53.5 226.6 98.4 126.2 187.9 256.8 −32.0 86.0 76.6 89.3 196.0 124.3 127.2 152.8 69.0 45.0 96.1 180.3 209.8 −131.8 215.0 75.0 62.5 70.4 29.4 141.8 51.8

C4H10O2

−48.0

−26.2

−15.3

−3.0

+10.7

19.7

31.8

50.0

70.8

93.0

C3H8O2

−13.5

+10.2

22.0

34.3

47.8

56.4

68.0

85.3

104.3

124.4

−89.7 40.5 −117.0 −60.5 37.6 14.3 −20.0 50.0 −60.7 −112.5 −54.4 27.8 47.3 −76.7 26.5 −91.0

−73.8 67.3 −103.8 −42.2 63.8 38.8 +2.1 77.7 −41.9 −98.4 −34.3 57.3 74.0 −59.1 51.0 −75.6

−65.7 80.2 −97.7 −33.0 77.1 50.5 12.8 91.8 −32.3 −91.7 −24.3 72.1 87.3 −50.2 63.2 −67.8

−56.6 94.6 −90.0 −22.7 91.5 63.9 25.0 106.3 −21.9 −84.1 −13.1 88.0 101.8 −40.7 76.1 −59.1

−46.9 110.3 −81.8 −11.5 107.5 78.1 38.5 123.7 −10.2 −75.8 −0.9 106.0 117.7 −29.8 91.0 −49.4

−40.7 120.6 −76.4 −4.3 117.5 87.6 47.1 134.0 −2.9 −70.4 +7.2 117.8 127.6 −22.4 100.0 −43.3

−32.1 133.8 −69.3 −5.4 130.4 99.8 58.9 147.9 +7.2 −63.2 18.0 131.8 141.3 −13.0 112.0 −34.8

−19.5 153.2 −58.0 20.0 150.1 117.8 76.7 168.2 22.4 −52.0 34.1 149.8 160.2 +1.5 130.0 −22.0

−4.9 175.6 −45.5 37.1 172.5 138.0 97.0 192.2 39.8 −39.5 52.3 167.3 183.0 17.7 149.8 −7.8

+10.7 198.0 −32.0 54.3 195.0 158.2 118.5 216.0 57.4 −26.5 72.4 184.0 206.2 35.0 170.0 +7.5

Name

Formula

Epichlorohydrin 1,2-Epoxy-2-methylpropane Erucic acid Estragole (p-methoxy allyl benzene) Ethane Ethoxydimethylphenylsilane Ethoxytrimethylsilane Ethoxytriphenylsilane Ethyl acetate acetoacetate Ethylacetylene (1-butyne) Ethyl acrylate α-Ethylacrylic acid α-Ethylacrylonitrile Ethyl alcohol (ethanol) Ethylamine 4-Ethylaniline N-Ethylaniline 2-Ethylanisole 3-Ethylanisole 4-Ethylanisole Ethylbenzene Ethyl benzoate benzoylacetate bromide α-bromoisobutyrate n-butyrate isobutyrate Ethylcamphoronic anhydride Ethyl isocaproate carbamate carbanilate Ethylcetylamine Ethyl chloride chloroacetate chloroglyoxylate α-chloropropionate trans-cinnamate 3-Ethylcumene 4-Ethylcumene Ethyl cyanoacetate Ethylcyclohexane Ethylcyclopentane Ethyl dichloroacetate N,N-diethyloxamate N-Ethyldiphenylamine Ethylene Ethylene-bis-(chloroacetate) Ethylene chlorohydrin (2-chloroethanol) diamine (1,2-ethanediamine) dibromide (1,2-dibromethane) dichloride (1,2-dichloroethane) glycol (1,2-ethanediol) glycol diethyl ether (1,2-diethoxyethane) glycol dimethyl ether (1,2-dimethoxyethane) glycol monomethyl ether (2-methoxyethanol) oxide Ethyl α-ethylacetoacetate fluoride formate 2-furoate glycolate 3-Ethylhexane 2-Ethylhexyl acrylate Ethylidene chloride (1,1-dichloroethane) fluoride (1,1-difluoroethane) Ethyl iodide Ethyl l-leucinate Ethyl levulinate Ethyl mercaptan (ethanethiol) Ethyl methylcarbamate Ethyl methyl ether

C3H5ClO C4H8O C22H42O2 C10H12O C2H6 C10H16OSi C5H14OSi C20H20OSi C4H8O2 C6H10O3 C4H6 C5H8O2 C5H8O2 C5H7N C2H6O C2H7N C8H11N C8H11N C9H12O C9H12O C9H12O C8H10 C9H10O2 C11H12O3 C2H5Br C6H11BrO2 C6H12O2 C6H12O2 C11H16O5 C8H16O2 C3H7NO2 C9H11NO2 C18H39N C2H5Cl C4H7ClO2 C4H5ClO3 C5H9ClO2 C11H12O2 C11H16 C11H16 C5H7NO2 C8H16 C7H14 C4H6Cl2O2 C8H15NO3 C14H15N C2H4 C6H8Cl2O4 C2H5ClO C2H8N2 C2H4Br2 C2H4Cl2 C2H6O2 C6H14O2

C2H4O C8H14O3 C2H5F C3H6O2 C7H8O3 C4H8O3 C8H18 C11H20O2 C2H4Cl2 C2H4F2 C2H5I C8H17NO2 C7H12O3 C2H6S C4H9NO2 C3H8O

40

60

100

200

400

760

Temperature, °C 79.3 98.0 117.9 17.5 36.0 55.5 336.5 358.8 381.5 168.7 192.0 215.0 −110.2 −99.7 −88.6 151.5 175.0 199.5 38.1 56.3 75.7 295.0 319.5 344.0 42.0 59.3 77.1 138.0 158.2 180.8 −21.6 −6.9 +8.7 61.5 80.0 99.5 144.0 160.7 179.2 71.6 92.2 114.0 48.4 63.5 78.4 −12.3 +2.0 16.6 170.6 194.2 217.4 156.9 180.8 204.0 142.1 164.2 187.1 149.7 172.8 196.5 149.2 172.3 196.5 92.7 113.8 136.2 164.8 188.4 213.4 223.8 244.7 265.0 +4.5 21.0 38.4 119.7 141.2 163.6 79.8 100.0 121.0 71.0 90.0 110.0 248.5 272.8 298.0 117.8 139.2 160.4 144.2 164.0 184.0 203.8 220.0 237.0 283.3 313.0 342.0 −18.6 −3.9 +12.3 103.8 123.8 144.2 94.5 114.7 135.0 107.2 126.2 146.5 219.3 245.0 271.0 145.4 168.2 193.0 148.3 171.8 195.8 169.8 187.8 206.0 87.8 109.1 131.8 62.4 82.3 103.4 115.2 135.9 156.5 202.8 226.5 252.0 233.0 258.8 286.0 −123.4 −113.9 −103.7 237.3 259.5 283.5 91.8 110.0 128.8 81.0 99.0 117.2 89.8 110.1 131.5 45.7 64.0 82.4 158.5 178.5 197.3 71.8 94.1 119.5

Melting point, °C −25.6 33.5 −183.2

−82.4 −45 −130 −71.2 −112 −80.6 −4 −63.5

−94.9 −34.6 −117.8 −93.3 −88.2 49 52.5 −139 −26 12

−111.3 −138.6

−169 −69 8.5 10 −35.3 −15.6

−111.3 −79 34

−96.7 −117 −105 −121

VAPOR PRESSURES OF PURE SUBSTANCES TABLE 2-8

2-69

Vapor Pressures of Organic Compounds, up to 1 atm (Continued ) Pressure, mm Hg Compound Name

1-Ethylnaphthalene Ethyl α-naphthyl ketone (1-propionaphthone) Ethyl 3-nitrobenzoate 3-Ethylpentane 4-Ethylphenetole 2-Ethylphenol 3-Ethylphenol 4-Ethylphenol Ethyl phenyl ether (phenetole) Ethyl propionate Ethyl propyl ether Ethyl salicylate 3-Ethylstyrene 4-Ethylstyrene Ethylisothiocyanate 2-Ethyltoluene 3-Ethyltoluene 4-Ethyltoluene Ethyl trichloroacetate Ethyltrimethylsilane Ethyltrimethyltin Ethyl isovalerate 2-Ethyl-1,4-xylene 4-Ethyl-1,3-xylene 5-Ethyl-1,3-xylene Eugenol iso-Eugenol Eugenyl acetate Fencholic acid d-Fenchone dl-Fenchyl alcohol Fluorene Fluorobenzene 2-Fluorotoluene 3-Fluorotoluene 4-Fluorotoluene Formaldehyde Formamide Formic acid trans-Fumaryl chloride Furfural (2-furaldehyde) Furfuryl alcohol Geraniol Geranyl acetate Geranyl n-butyrate Geranyl isobutyrate Geranyl formate Glutaric acid Glutaric anhydride Glutaronitrile Glutaryl chloride Glycerol Glycerol dichlorohydrin (1,3-dichloro-2-propanol) Glycol diacetate Glycolide (1,4-dioxane-2,6-dione) Guaicol (2-methoxyphenol) Heneicosane Heptacosane Heptadecane Heptaldehyde (enanthaldehyde) n-Heptane Heptanoic acid (enanthic acid) 1-Heptanol Heptanoyl chloride (enanthyl chloride) 2-Heptene Heptylbenzene Heptyl cyanide (enanthonitrile) Hexachlorobenzene Hexachloroethane Hexacosane Hexadecane 1-Hexadecene n-Hexadecyl alcohol (cetyl alcohol)

1

5

10

20

70.0

101.4

116.8

133.8

152.0

C13H12O C9H9NO4 C7H16 C10H14O C8H10O C8H10O C8H10O C8H10O C5H10O2 C5H12O C9H10O3 C10H12 C10H12 C3H5NS C9H12 C9H12 C9H12 C4H5Cl3O2 C5H14Si C5H14Sn C7H14O2 C10H14 C10H14 C10H14 C10H12O2 C10H12O2 C12H14O3 C10H16O2 C10H16O C10H18O C13H10 C6H5F C7H7F C7H7F C7H7F CH2O CH3NO CH2O2 C4H2Cl2O2 C5H4O2 C5H6O2 C10H18O C12H20O2 C14H24O2 C14H24O2 C11H18O2 C5H8O4 C5H6O3 C5H6N2 C5H6Cl2O2 C3H8O3 C3H6Cl2O

124.0 108.1 −37.8 48.5 46.2 60.0 59.3 18.1 −28.0 −64.3 61.2 28.3 26.0 13.2 9.4 7.2 7.6 20.7 −60.6 −30.0 −6.1 25.7 26.3 22.1 78.4 86.3 101.6 101.7 28.0 45.8 −43.4 −24.2 −22.4 −21.8

155.5 140.2 −17.0 75.7 73.4 86.8 86.5 43.7 −7.2 −45.0 90.0 55.0 52.7 +10.6 34.8 32.3 32.7 45.5 −41.4 −7.6 +17.0 52.0 53.0 48.8 108.1 117.0 132.3 128.7 54.7 70.3 129.3 −22.8 −2.2 −0.3 +0.3

70.5 −20.0 +15.0 18.5 31.8 69.2 73.5 96.8 90.9 61.8 155.5 100.8 91.3 56.1 125.5 28.0

96.3 −5.0 38.5 42.6 56.0 96.8 102.7 125.2 119.6 90.3 183.8 133.3 123.7 84.0 153.8 52.2

171.0 155.0 −6.8 89.5 87.0 100.2 100.2 56.4 +3.4 −35.0 104.2 68.3 66.3 22.8 47.6 44.7 44.9 57.7 −31.8 +3.8 28.7 65.6 66.4 62.1 123.0 132.4 148.0 142.3 68.3 82.1 146.0 −12.4 +8.9 +11.0 11.8 −88.0 109.5 +2.1 51.8 54.8 68.0 110.0 117.9 139.0 133.0 104.3 196.0 149.5 140.0 97.8 167.2 64.7

188.1 173.6 +4.7 103.8 101.5 114.5 115.0 70.3 14.3 −24.0 119.3 82.8 80.8 36.1 61.2 58.2 58.5 70.6 −21.0 16.1 41.3 79.8 80.6 76.5 138.7 149.0 164.2 155.8 83.0 95.6 164.2 −1.2 21.4 23.4 24.0 −79.6 122.5 10.3 65.0 67.8 81.0 125.6 133.0 153.8 147.9 119.8 210.5 166.0 156.5 112.3 182.2 78.0

206.9 192.6 17.5 119.8 117.9 130.0 131.3 86.6 27.2 −12.0 136.7 99.2 97.3 50.8 76.4 73.3 73.6 85.5 −9.0 30.0 55.2 96.0 97.2 92.6 155.8 167.0 183.0 171.8 99.5 110.8 185.2 +11.5 34.7 37.0 37.8 −70.6 137.5 24.0 79.5 82.1 95.7 141.8 150.0 170.1 164.0 136.2 226.3 185.5 176.4 128.3 198.0 93.0

C6H10O4 C4H4O4 C7H8O2 C21H44 C27H56 C17H36 C7H14O C7H16 C7H14O2 C7H16O C7H13ClO C7H14 C13H20 C7H13N C6Cl6 C2Cl6 C26H54 C16H34 C16H32 C16H34O

38.3

64.1 103.0 79.1 188.0 248.6 145.2 32.7 −12.7 101.3 64.3 54.6 −14.1 94.6 47.8 149.3 49.8 240.0 135.2 131.7 158.3

77.1 116.6 92.0 205.4 266.8 160.0 43.0 −2.1 113.2 74.7 64.6 −3.5 110.0 61.6 166.4 73.5 257.4 149.8 146.2 177.8

90.8 132.0 106.0 223.2 284.6 177.7 54.0 +9.5 125.6 85.8 75.0 +8.3 126.0 76.3 185.7 87.6 275.8 164.7 162.0 197.8

106.1 148.6 121.6 243.4 305.7 195.8 66.3 22.3 139.5 99.8 86.4 21.5 144.0 92.6 206.0 102.3 295.2 181.3 178.8 219.8

100

200

400

760

Melting point, °C

164.1

180.0

204.6

230.8

258.1

−27

218.2 205.0 25.7 129.8 127.9 139.8 141.7 95.4 35.1 −4.0 147.6 109.6 107.6 59.8 86.0 82.9 83.2 94.4 −1.2 38.4 64.0 106.2 107.4 103.0 167.3 178.2 194.0 181.5 109.8 120.2 197.8 19.6 43.7 45.8 46.5 −65.0 147.0 32.4 89.0 91.5 104.0 151.5 160.3 180.2 174.0 147.2 235.5 196.2 189.5 139.1 208.0 102.0

233.5 220.3 36.9 143.5 141.8 152.0 154.2 108.4 45.2 +6.8 161.5 123.2 121.5 71.9 99.0 95.9 96.3 107.4 +9.2 50.0 75.9 120.0 121.2 116.5 182.2 194.0 209.7 194.0 123.6 132.3 214.7 30.4 55.3 57.5 58.1 −57.3 157.5 43.8 101.0 103.4 115.9 165.3 175.2 193.8 187.7 160.7 247.0 212.5 205.5 151.8 220.1 114.8

255.5 244.6 53.8 163.2 161.6 171.8 175.0 127.9 61.7 23.3 183.7 144.0 142.0 90.0 119.0 115.5 116.1 125.8 25.0 67.3 93.8 140.2 141.8 137.4 204.7 217.2 232.5 215.0 144.0 150.0 240.3 47.2 73.0 75.4 76.0 −46.0 175.5 61.4 120.0 121.8 133.1 185.6 196.3 214.0 207.6 182.6 265.0 236.5 230.0 172.4 240.0 133.3

280.2 270.6 73.0 185.7 184.5 193.3 197.4 149.8 79.8 41.6 207.0 167.2 165.0 110.1 141.4 137.8 136.4 146.0 42.8 87.6 114.0 163.1 164.4 159.6 228.3 242.3 257.4 237.8 166.8 173.2 268.6 65.7 92.8 95.4 96.1 −33.0 193.5 80.3 140.0 141.8 151.8 207.8 219.8 235.0 228.5 205.8 283.5 261.0 257.3 195.3 263.0 153.5

306.0 298.0 93.5 208.0 207.5 214.0 219.0 172.0 99.1 61.7 231.5 191.5 189.0 131.0 165.1 161.3 162.0 167.0 62.0 108.8 134.3 186.9 188.4 183.7 253.5 267.5 282.0 264.1 191.0 201.0 295.0 84.7 114.0 116.0 117.0 −19.5 210.5 100.6 160.0 161.8 170.0 230.0 243.3 257.4 251.0 230.0 303.0 287.0 286.2 217.0 290.0 174.3

115.8 158.2 131.0 255.3 318.3 207.3 74.0 30.6 148.5 108.0 93.5 30.0 154.8 103.0 219.0 112.0 307.8 193.2 190.8 234.3

128.0 173.2 144.0 272.0 333.5 223.0 84.0 41.8 160.0 119.5 102.7 41.3 170.2 116.8 235.5 124.2 323.2 208.5 205.3 251.7

147.8 194.0 162.7 296.5 359.4 247.8 102.0 58.7 179.5 136.6 116.3 58.6 193.3 137.7 258.5 143.1 348.4 231.7 226.8 280.2

168.3 217.0 184.1 323.8 385.0 274.5 125.5 78.0 199.6 155.6 130.7 78.1 217.8 160.0 283.5 163.8 374.6 258.3 250.0 312.7

190.5 240.0 205.0 350.5 410.6 303.0 155.0 98.4 221.5 175.8 145.0 98.5 244.0 184.6 309.4 185.6 399.8 287.5 274.0 344.0

60

Temperature, °C

Formula C12H12

40

52.4 152.6 211.7 115.0 12.0 −34.0 78.0 42.4 34.2 −35.8 64.0 21.0 114.4 32.7 204.0 105.3 101.6 122.7

47 −118.6 −45 −4 46.5 −30.2 −72.6 1.3 −5.9 −95.5

−99.3

−10 295 19 5 35 113 −42.1 −80 −110.8 −92 8.2

97.5

17.9 −31 97 28.3 40.4 59.5 22.5 −42 −90.6 −10 34.6

230 186.6 56.6 18.5 4 49.3

2-70

PHYSICAL AND CHEMICAL DATA

TABLE 2-8

Vapor Pressures of Organic Compounds, up to 1 atm (Continued ) Pressure, mm Hg Compound Name

Formula

n-Hexadecylamine (cetylamine) Hexaethylbenzene n-Hexane 1-Hexanol 2-Hexanol 3-Hexanol 1-Hexene n-Hexyl levulinate n-Hexyl phenyl ketone (enanthophenone) Hydrocinnamic acid Hydrogen cyanide (hydrocyanic acid) Hydroquinone 4-Hydroxybenzaldehyde α-Hydroxyisobutyric acid α-Hydroxybutyronitrile 4-Hydroxy-3-methyl-2-butanone 4-Hydroxy-4-methyl-2-pentanone 3-Hydroxypropionitrile Indene Iodobenzene Iodononane 2-Iodotoluene α-Ionone Isoprene Lauraldehyde Lauric acid Levulinaldehyde Levulinic acid d-Limonene Linalyl acetate Maleic anhydride Menthane 1-Menthol Menthyl acetate benzoate formate Mesityl oxide Methacrylic acid Methacrylonitrile Methane Methanethiol Methoxyacetic acid N-Methylacetanilide Methyl acetate acetylene (propyne) acrylate alcohol (methanol) Methylamine N-Methylaniline Methyl anthranilate benzoate 2-Methylbenzothiazole α-Methylbenzyl alcohol Methyl bromide 2-Methyl-1-butene 2-Methyl-2-butene Methyl isobutyl carbinol (2-methyl4-pentanol) n-butyl ketone (2-hexanone) isobutyl ketone (4-methyl-2-pentanone) n-butyrate isobutyrate caprate caproate caprylate chloride chloroacetate cinnamate α-Methylcinnamic acid Methylcyclohexane Methylcyclopentane Methylcyclopropane Methyl n-decyl ketone (n-dodecan-2-one) dichloroacetate N-Methyldiphenylamine

C16H35N C18H30 C6H14 C6H14O C6H14O C6H14O C6H12 C11H20O3 C13H18O C9H10O2 CHN C6H6O2 C7H6O2 C4H8O3 C5H9NO C5H10O2 C6H12O2 C3H5NO C9H8 C6H5I C9H19I C7H7I C13H20O C5H8 C12H24O C12H24O2 C5H8O2 C5H8O3 C10H16 C12H20O2 C4H2O3 C10H20 C10H20O C12H22O2 C17H24O2 C11H20O2 C6H10O C4H6O2 C4H5N CH4 CH4S C3H6O3 C9H11NO C3H6O2 C3H4 C4H6O2 CH4O CH5N C7H9N C8H9NO2 C8H8O2 C8H7NS C8H10O CH3Br C5H10 C5H10 C6H14O C6H12O C6H12O C5H10O2 C5H10O2 C11H22O2 C7H14O2 C9H18O2 CH3Cl C3H5ClO2 C10H10O2 C10H10O2 C7H14 C6H12 C4H8 C12H24O C3H4Cl2O2 C13H13N

1

5

10

20

40

60

100

123.6

157.8 134.3 −34.5 47.2 34.8 25.7 −38.0 120.0 130.3 133.5 −55.3 153.3 153.2 98.5 65.8 69.3 46.7 87.8 44.3 50.6 96.2 65.9 108.8 −62.3 108.4 150.6 54.9 128.1 40.4 82.5 63.4 35.7 83.2 85.8 154.2 75.8 +14.1 48.5 −23.3 −199.0 −75.3 79.3 103.8 −38.6 −97.5 −23.6 −25.3 −81.3 62.8 109.0 64.4 97.5 75.2 −80.6 −72.8 −57.0

176.0 150.3 −25.0 58.2 45.0 36.7 −28.1 134.7 145.5 148.7 −47.7 163.5 169.7 110.5 77.8 81.0 58.8 102.0 58.5 64.0 109.0 79.8 123.0 −53.3 123.7 166.0 68.0 141.8 53.8 96.0 78.7 48.3 96.0 100.0 170.0 90.0 26.0 60.0 −12.5 −195.5 −67.5 92.0 118.6 −29.3 −90.5 −13.5 −16.2 −73.8 76.2 124.2 77.3 111.2 88.0 −72.8 −64.3 −47.9

195.7 168.0 −14.1 70.3 55.9 49.0 −17.2 150.2 161.0 165.0 −39.7 174.6 186.8 123.8 90.7 94.0 72.0 117.9 73.9 78.3 123.0 95.6 139.0 −43.5 140.2 183.6 82.7 154.1 68.2 111.4 95.0 62.7 110.3 115.4 186.3 105.8 37.9 72.7 −0.6 −191.8 −58.8 106.5 135.1 −19.1 −82.9 −2.7 −6.0 −65.9 90.5 141.5 91.8 125.5 102.1 −64.0 −54.8 −37.9

215.7 187.7 −2.3 83.7 67.9 62.2 −5.0 167.8 178.9 183.3 −30.9 192.0 206.0 138.0 104.8 108.2 86.7 134.1 90.7 94.4 138.1 112.4 155.6 −32.6 157.8 201.4 98.3 169.5 84.3 127.7 111.8 78.3 126.1 132.1 204.3 123.0 51.7 86.4 +12.8 −187.7 −49.2 122.0 152.2 −7.9 −74.3 +9.2 +5.0 −56.9 106.0 159.7 107.8 141.2 117.8 −54.2 −44.1 −26.7

228.8 199.7 +5.4 92.0 76.0 70.7 +2.8 179.0 189.8 194.0 −25.1 203.0 217.5 146.4 113.9 117.4 96.0 144.7 100.8 105.0 147.7 123.8 166.3 −25.4 168.7 212.7 108.4 178.0 94.6 138.1 122.0 88.6 136.1 143.2 215.8 133.8 60.4 95.3 21.5 −185.1 −43.1 131.8 164.2 −0.5 −68.8 17.3 12.1 −51.3 115.8 172.0 117.4 150.4 127.4 −48.0 −37.3 −19.4

245.8 216.0 15.8 102.8 87.3 81.8 13.0 193.6 204.2 209.0 −17.8 216.5 233.5 157.7 125.0 129.0 108.2 157.7 114.7 118.3 159.8 138.1 181.2 −16.0 184.5 227.5 121.8 190.2 108.3 151.8 135.8 102.1 149.4 156.7 230.4 148.0 72.1 106.6 32.8 −181.4 −34.8 144.5 179.8 +9.4 −61.3 28.0 21.2 −43.7 129.8 187.8 130.8 163.9 140.3 −39.4 −28.0 −9.9

+22.1 28.8 +19.7 −5.5 −13.0 93.5 30.0 61.7 −99.5 19.0 108.1 155.0 −14.0 −33.8 −80.6 106.0 26.7 134.0

33.3 38.8 30.0 +5.0 −2.9 108.0 42.0 74.9 −92.4 30.0 123.0 169.8 −3.2 −23.7 −72.8 120.4 38.1 149.7

45.4 50.0 40.8 16.7 +8.4 123.0 55.4 89.0 −84.8 41.5 140.0 185.2 +8.7 −12.8 −64.0 136.0 50.7 165.8

58.2 62.0 52.8 29.6 21.0 139.0 70.0 105.3 −76.0 54.5 157.9 201.8 22.0 −0.6 −54.2 152.4 64.7 184.0

67.0 69.8 60.4 37.4 28.9 148.6 79.7 115.3 −70.4 63.0 170.0 212.0 30.5 +7.2 −48.0 163.8 73.6 195.4

78.0 79.8 70.4 48.0 39.6 161.5 91.4 128.0 −63.0 73.5 185.8 224.8 42.1 17.9 −39.3 177.5 85.4 210.1

200

400

760

Temperature, °C

−53.9 24.4 14.6 +2.5 −57.5 90.0 100.0 102.2 −71.0 132.4 121.2 73.5 41.0 44.6 22.0 58.7 16.4 24.1 70.0 37.2 79.5 −79.8 77.7 121.0 28.1 102.0 14.0 55.4 44.0 +9.7 56.0 57.4 123.2 47.3 −8.7 25.5 −44.5 −205.9 −90.7 52.5 −57.2 −111.0 −43.7 −44.0 −95.8 36.0 77.6 39.0 70.0 49.0 −96.3 −89.1 −75.4 −0.3 +7.7 −1.4 −26.8 −34.1 63.7 +5.0 34.2 −2.9 77.4 125.7 −35.9 −53.7 −96.0 77.1 3.2 103.5

272.2 300.4 330.0 241.7 268.5 298.3 31.6 49.6 68.7 119.6 138.0 157.0 103.7 121.8 139.9 98.3 117.0 135.5 29.0 46.8 66.0 215.7 241.0 266.8 225.0 248.3 271.3 230.8 255.0 279.8 −5.3 +10.2 25.9 238.0 262.5 286.2 256.8 282.6 310.0 175.2 193.8 212.0 142.0 159.8 178.8 146.5 165.5 185.0 126.8 147.5 167.9 178.0 200.0 221.0 135.6 157.8 181.6 139.8 163.9 188.6 179.0 199.3 219.5 160.0 185.7 211.0 202.5 225.2 250.0 −1.2 +15.4 32.6 207.8 231.8 257.0 249.8 273.8 299.2 142.0 164.0 187.0 208.3 227.4 245.8 128.5 151.4 175.0 173.3 196.2 220.0 155.9 179.5 202.0 122.7 146.0 169.5 168.3 190.2 212.0 178.8 202.8 227.0 253.2 277.1 301.0 169.8 194.2 219.0 90.0 109.8 130.0 123.9 142.5 161.0 50.0 70.3 90.3 −175.5 −168.8 −161.5 −22.1 −7.9 +6.8 163.5 184.2 204.0 202.3 227.4 253.0 24.0 40.0 57.8 −49.8 −37.2 −23.3 43.9 61.8 80.2 34.8 49.9 64.7 −32.4 −19.7 −6.3 149.3 172.0 195.5 212.4 238.5 266.5 151.4 174.7 199.5 183.2 204.5 225.5 159.0 180.7 204.0 −26.5 −11.9 +3.6 −13.8 +2.5 20.2 +4.9 21.6 38.5 94.9 94.3 85.6 64.3 55.7 181.6 109.8 148.1 −51.2 90.5 209.6 245.0 59.6 34.0 −26.0 199.0 103.2 232.8

113.5 111.0 102.0 83.1 73.6 202.9 129.8 170.0 −38.0 109.5 235.0 266.8 79.6 52.3 −11.3 222.5 122.6 257.0

131.7 127.5 119.0 102.3 92.6 224.0 150 193.0 −24.0 130.3 263.0 288.0 100.9 71.8 +4.5 246.5 143.0 282.0

Melting point, °C 130 −95.3 −51.6 −98.5 48.5 −13.2 170.3 115.5 79 −47 −2 −28.5

−146.7 44.5 48 33.5 −96.9 58 42.5 54.5 −59 15 −182.5 −121 102 −98.7 −102.7 −97.8 −93.5 −57 24 −12.5 15.4 −93 −135 −133 −56.9 −84.7 −84.7 −18 −40 −97.7 −31.9 33.4 −126.4 −142.4

−7.6

VAPOR PRESSURES OF PURE SUBSTANCES TABLE 2-8

2-71

Vapor Pressures of Organic Compounds, up to 1 atm* (Continued ) Pressure, mm Hg Compound Name

Formula

Methyl n-dodecyl ketone (2-tetradecanone) Methylene bromide (dibromomethane) chloride (dichloromethane) Methyl ethyl ketone (2-butanone) 2-Methyl-3-ethylpentane 3-Methyl-3-ethylpentane Methyl fluoride formate α-Methylglutaric anhydride Methyl glycolate 2-Methylheptadecane 2-Methylheptane 3-Methylheptane 4-Methylheptane 2-Methyl-2-heptene 6-Methyl-3-hepten-2-ol 6-Methyl-5-hepten-2-ol 2-Methylhexane 3-Methylhexane Methyl iodide laurate levulinate methacrylate myristate α-naphthyl ketone (1-acetonaphthone) β-naphthyl ketone (2-acetonaphthone) n-nonyl ketone (undecan-2-one) palmitate n-pentadecyl ketone (2-heptdecanone) 2-Methylpentane 3-Methylpentane 2-Methyl-1-pentanol 2-Methyl-2-pentanol Methyl n-pentyl ketone (2-heptanone) phenyl ether (anisole) 2-Methylpropene Methyl propionate 4-Methylpropiophenone 2-Methylpropionyl bromide Methyl propyl ether n-propyl ketone (2-pentanone) isopropyl ketone (3-Methyl-2-butanone) 2-Methylquinoline Methyl salicylate α-Methyl styrene 4-Methyl styrene Methyl n-tetradecyl ketone (2-hexadecanone) thiocyanate isothiocyanate undecyl ketone (2-tridecanone) isovalerate Monovinylacetylene (butenyne) Myrcene Myristaldehyde Myristic acid (tetradecanoic acid) Naphthalene 1-Naphthoic acid 2-Naphthoic acid 1-Naphthol 2-Naphthol 1-Naphthylamine 2-Naphthylamine Nicotine 2-Nitroaniline 3-Nitroaniline 4-Nitroaniline 2-Nitrobenzaldehyde 3-Nitrobenzaldehyde Nitrobenzene Nitroethane Nitroglycerin Nitromethane 2-Nitrophenol 2-Nitrophenyl acetate

C14H28O CH2Br2 CH2Cl2 C4H8O C8H18 C8H18 CH3F C2H4O2 C6H8O3 C3H6O3 C18H38 C8H18 C8H18 C8H18 C8H16 C8H16O C8H16O C7H16 C7H16 CH3I C13H26O2 C6H10O3 C5H8O2 C15H30O2 C12H10O C12H10O C11H22O C17H34O2 C17H34O C6H14 C6H14 C6H14O C6H14O C7H14O C7H8O C4H8 C4H8O2 C10H12O C4H7BrO C4H10O C5H10O C5H10O C10H9N C8H8O3 C9H10 C9H10 C16H32O C2H3NS C2H3NS C13H26O C6H12O2 C4H4 C10H16 C14H28O C14H28O2 C10H8 C11H8O2 C11H8O2 C10H8O C10H8O C10H9N C10H9N C10H14N2 C6H6N2O2 C6H6N2O2 C6H6N2O2 C7H5NO3 C7H5NO3 C6H5NO2 C2H5NO2 C3H5N3O9 CH3NO2 C6H5NO3 C8H7NO4

1

5

10

20

40

99.3 −35.1 −70.0 −48.3 −24.0 −23.9 −147.3 −74.2 93.8 +9.6 119.8 −21.0 −19.8 −20.4 −16.1 41.6 41.9 −40.4 −39.0

130.0 −13.2 −52.1 −28.0 −1.8 −1.4 −137.0 −57.0 125.4 33.7 152.0 +1.3 +2.6 +1.5 +6.7 65.0 66.0 −19.5 −18.1 −55.0 117.9 66.4 −10.0 145.7 146.3 152.3 95.5 166.8 161.6 −41.7 −39.8 38.0 +16.8 43.6 30.0 −96.5 −21.5 89.3 38.4 −54.3 +8.0 −1.0 104.0 81.6 34.0 42.0

145.5 −2.4 −43.3 −17.7 +9.5 +9.9 −131.6 −48.6 141.8 45.3 168.7 12.3 13.3 12.4 17.8 76.7 77.8 −9.1 −7.8 −45.8 133.2 79.7 +1.0 160.8 161.5 168.5 108.9 184.3 178.0 −32.1 −30.1 49.6 27.6 55.5 42.2 −81.9 −11.8 103.8 50.6 −45.4 17.9 +8.3 119.0 95.3 47.1 55.1

161.3 +9.7 −33.4 −6.5 21.7 22.3 −125.9 −39.2 157.7 58.1 186.0 24.4 25.4 24.5 30.4 89.3 90.4 +2.3 +3.6 −35.6 149.0 93.7 11.0 177.8 178.4 185.7 123.1 202.0 196.4 −21.4 −19.4 61.6 38.8 67.7 55.8 −73.4 −1.0 120.2 64.1 −35.4 28.5 18.3 134.0 110.0 61.8 69.2

179.8 23.3 −22.3 +6.0 35.2 36.2 −119.1 −28.7 177.5 72.3 204.8 37.9 38.9 38.0 44.0 102.7 104.0 14.9 16.4 −24.2 166.0 109.5 25.5 195.8 196.8 203.8 139.0

151.5 +9.8 −8.3 117.0 +2.9 −77.7 40.0 132.0 174.1 74.2 184.0 189.7 125.5 128.6 137.7 141.6 91.8 135.7 151.5 177.6 117.7 127.4 71.6 +1.5 167 −7.9 76.8 128.0

167.3 21.6 +5.4 131.8 14.0 −70.0 53.2 148.3 190.8 85.8 196.8 202.8 142.0 145.5 153.8 157.6 107.2 150.4 167.8 194.4 133.4 142.8 84.9 12.5 188 +2.8 90.4 142.0

184.6 34.5 20.4 147.8 26.4 −61.3 67.0 166.2 207.6 101.7 211.2 216.9 158.0 161.8 171.6 175.8 123.7 167.7 185.5 213.2 150.0 159.0 99.3 24.8 210 14.1 105.8 155.8

60

100

200

400

760

191.4 31.6 −15.7 14.0 43.9 45.0 −115.0 −21.9 189.9 81.8 216.3 46.6 47.6 46.6 52.8 111.5 112.8 23.0 24.5 −16.9 176.8 119.3 34.5 207.5 208.6 214.7 148.6

206.0 42.3 −6.3 25.0 55.7 57.1 −109.0 −12.9 205.0 93.7 231.5 58.3 59.4 58.3 64.6 122.6 123.8 34.1 35.6 −7.0 190.8 133.0 47.0 222.6 223.8 229.8 161.0

228.2 58.5 +8.0 41.6 73.6 75.3 −99.9 +0.8 229.1 111.8 254.5 76.0 77.1 76.1 82.3 139.5 140.0 50.8 52.4 +8.0

253.3 79.0 24.1 60.0 94.0 96.2 −89.5 16.0 255.5 131.7 279.8 96.2 97.4 96.3 102.2 156.6 156.6 69.8 71.6 25.3

278.0 98.6 40.7 79.6 115.6 118.3 −78.2 32.0 282.5 151.5 306.5 117.6 118.9 117.7 122.5 175.5 174.3 90.0 91.9 42.4

153.4 63.0 245.3 246.7 251.6 181.2

175.8 82.0 269.8 270.5 275.8 202.3

197.7 101.0 295.8 295.5 301.0 224.0

214.3 −9.7 −7.3 74.7 51.3 81.2 70.7 −63.8 +11.0 138.0 79.4 −24.3 39.8 29.6 150.8 126.2 77.8 85.0

226.7 −1.9 +0.1 83.4 58.8 89.8 80.1 −57.7 18.7 149.3 88.8 −17.4 47.3 36.2 161.7 136.7 88.3 95.0

242.0 +8.1 10.5 94.2 69.2 100.0 93.0 −49.3 29.0 164.2 101.6 −8.1 56.8 45.5 176.2 150.0 102.2 108.6

265.8 24.1 26.5 111.3 85.0 116.1 112.3 −36.7 44.2 187.4 120.5 +6.0 71.0 59.0 197.8 172.6 121.8 128.7

291.7 41.6 44.2 129.8 102.6 133.2 133.8 −22.2 61.8 212.7 141.7 22.5 86.8 73.8 211.7 197.5 143.0 151.2

319.5 60.3 63.3 147.9 121.2 150.2 155.5 −6.9 79.8 238.5 163.0 39.1 103.3 88.9 246.5 223.2 165.4 175.0

203.7 49.0 38.2 165.7 39.8 −51.7 82.6 186.0 223.5 119.3 225.0 231.5 177.8 181.7 191.5 195.7 142.1 186.0 204.2 234.2 168.8 177.7 115.4 38.0 235 27.5 122.1 172.8

215.0 58.1 47.5 176.6 48.2 −45.3 92.6 198.3 237.2 130.2 234.5 241.3 190.0 193.7 203.8 208.1 154.7 197.8 216.5 245.9 180.7 189.5 125.8 46.5 251 35.5 132.6 181.7

230.5 70.4 59.3 191.5 59.8 −37.1 106.0 214.5 250.5 145.5 245.8 252.7 206.0 209.8 220.0 224.3 169.5 213.0 232.1 261.8 196.2 204.3 139.9 57.8

254.4 89.8 77.5 214.0 77.3 −24.1 126.0 240.4 272.3 167.7 263.5 270.3 229.6 234.0 244.9 249.7 193.8 236.3 255.3 284.5 220.0 227.4 161.2 74.8

279.8 110.8 97.8 238.3 96.7 −10.1 148.3 267.9 294.6 193.2 281.4 289.5 255.8 260.6 272.2 277.4 219.8 260.0 280.2 310.2 246.8 252.1 185.8 94.0

307.0 132.9 119.0 262.5 116.7 +5.3 171.5 297.8 318.0 217.9 300.0 308.5 282.5 288.0 300.8 306.1 247.3 284.5 305.7 336.0 273.5 278.3 210.6 114.0

46.6 146.4 194.1

63.5 167.6 213.0

82.0 191.0 233.5

101.2 214.5 253.0

Temperature, °C

87.8 39.8 −30.5 115.0 115.6 120.2 68.2 134.3 129.6 −60.9 −59.0 15.4 −4.5 19.3 +5.4 −105.1 −42.0 59.6 13.5 −72.2 −12.0 −19.9 75.3 54.0 7.4 16.0 109.8 −14.0 −34.7 86.8 −19.2 −93.2 14.5 99.0 142.0 52.6 156.0 160.8 94.0 104.3 108.0 61.8 104.0 119.3 142.4 85.8 96.2 44.4 −21.0 127 −29.0 49.3 100.0

Melting point, °C −52.8 −96.7 −85.9 −114.5 −90 −99.8

−109.5 −120.8 −121.1

−118.2 −64.4 5 18.5 55.5 15 30 −154 −118 −103 −37.3 −140.3 −87.5

−77.8 −92 −1 −8.3 −23.2

−51 35.5 28.5

23.5 57.5 80.2 160.5 184 96 122.5 50 111.5 71.5 114 146.5 40.9 58 +5.7 −90 11 −29 45

2-72

PHYSICAL AND CHEMICAL DATA

TABLE 2-8

Vapor Pressures of Organic Compounds, up to 1 atm (Continued ) Pressure, mm Hg Compound Name

1-Nitropropane 2-Nitropropane 2-Nitrotoluene 3-Nitrotoluene 4-Nitrotoluene 4-Nitro-1,3-xylene (4-nitro-m-xylene) Nonacosane Nonadecane n-Nonane 1-Nonanol 2-Nonanone Octacosane Octadecane n-Octane n-Octanol (1-octanol) 2-Octanone n-Octyl acrylate iodide (1-Iodooctane) Oleic acid Palmitaldehyde Palmitic acid Palmitonitrile Pelargonic acid Pentachlorobenzene Pentachloroethane Pentachloroethylbenzene Pentachlorophenol Pentacosane Pentadecane 1,3-Pentadiene 1,4-Pentadiene Pentaethylbenzene Pentaethylchlorobenzene n-Pentane iso-Pentane (2-methylbutane) neo-Pentane (2,2-dimethylpropane) 2,3,4-Pentanetriol 1-Pentene α-Phellandrene Phenanthrene Phenethyl alcohol (phenyl cellosolve) 2-Phenetidine Phenol 2-Phenoxyethanol 2-Phenoxyethyl acetate Phenyl acetate Phenylacetic acid Phenylacetonitrile Phenylacetyl chloride Phenyl benzoate 4-Phenyl-3-buten-2-one Phenyl isocyanate isocyanide Phenylcyclohexane Phenyl dichlorophosphate m-Phenylene diamine (1,3-phenylenediamine) Phenylglyoxal Phenylhydrazine N-Phenyliminodiethanol 1-Phenyl-1,3-pentanedione 2-Phenylphenol 4-Phenylphenol 3-Phenyl-1-propanol Phenyl isothiocyanate Phorone iso-Phorone Phosgene (carbonyl chloride) Phthalic anhydride Phthalide Phthaloyl chloride 2-Picoline Pimelic acid α-Pinene β-Pinene

1

5

10

20

40

C3H7NO2 −9.6 C3H7NO2 −18.8 C7H7NO2 50.0 C7H7NO2 50.2 C7H7NO2 53.7 C8H9NO2 65.6 C29H60 234.2 C19H40 133.2 C9H20 +1.4 C9H20O 59.5 C9H18O 32.1 C28H58 226.5 C18H38 119.6 C8H18 −14.0 C8H18O 54.0 C8H16O 23.6 C11H20O2 58.5 C8H17I 45.8 C18H34O2 176.5 C16H32O 121.6 C16H32O2 153.6 C16H31N 134.3 C9H18O2 108.2 C6HCl5 98.6 C2HCl5 +1.0 C8H5Cl5 96.2 C6HCl5O C25H52 194.2 C15H32 91.6 C5H8 −71.8 C5H8 −83.5 C16H26 86.0 C16H25Cl 90.0 C5H12 −76.6 C5H12 −82.9 C5H12 −102.0 C5H12O3 155.0 C5H10 −80.4 C10H16 20.0 C14H10 118.2 C8H10O2 58.2 C8H11NO 67.0 C6H6O 40.1 C8H10O2 78.0 C10H12O3 82.6 C8H8O2 38.2 C8H8O2 97.0 C8H7N 60.0 C8H7ClO 48.0 C13H10O2 106.8 C10H10O 81.7 C7H5NO 10.6 C7H5N 12.0 C12H16 67.5 C6H5Cl2O2P 66.7

+13.5 +4.1 79.1 81.0 85.0 95.0 269.8 166.3 25.8 86.1 59.0 260.3 152.1 +8.3 76.5 48.4 87.7 74.8 208.5 154.6 188.1 168.3 126.0 129.7 27.2 130.0

25.3 15.8 93.8 96.0 100.5 109.8 286.4 183.5 38.0 99.7 72.3 277.4 169.6 19.2 88.3 60.9 102.0 90.0 223.0 171.8 205.8 185.8 137.4 144.3 39.8 148.0

230.0 121.0 −53.8 −66.2 120.0 123.8 −62.5 −65.8 −85.4 189.3 −63.3 45.7 154.3 85.9 94.7 62.5 106.6 113.5 64.8 127.0 89.0 75.3 141.5 112.2 36.0 37.0 96.5 95.9

248.2 135.4 −45.0 −57.1 135.8 140.7 −50.1 −57.0 −76.7 204.5 −54.5 58.0 173.0 100.0 108.6 73.8 121.2 128.0 78.0 141.3 103.5 89.0 157.8 127.4 48.5 49.7 111.3 110.0

37.9 28.2 109.6 112.8 117.7 125.8 303.6 200.8 51.2 113.8 87.2 295.4 187.5 31.5 101.0 74.3 117.8 105.9 240.0 190.0 223.8 204.2 149.8 160.0 53.9 166.0 192.2 266.1 150.2 −34.8 −47.7 152.4 158.1 −40.2 −47.3 −67.2 220.5 −46.0 72.1 193.7 114.8 123.7 86.0 136.0 144.5 92.3 156.0 119.4 103.6 177.0 143.8 62.5 63.4 126.4 125.9

51.8 41.8 126.3 130.7 136.0 143.3 323.2 220.0 66.0 129.0 103.4 314.2 207.4 45.1 115.2 89.8 135.6 123.8 257.2 210.0 244.4 223.8 163.7 178.5 69.9 186.2 211.2 285.6 167.7 −23.4 −37.0 171.9 178.2 −29.2 −36.5 −56.1 239.6 −34.1 87.8 215.8 130.5 139.9 100.1 152.2 162.3 108.1 173.6 136.3 119.8 197.6 161.3 77.7 78.3 144.0 143.4

C6H8N2 C8H6O2 C6H8N2 C10H15NO2 C11H12O2 C12H10O C12H10O C9H12O C7H5NS C9H14O C9H14O CCl2O C8H4O3 C8H6O2 C8H4Cl2O2 C6H7N C7H12O4 C10H16 C10H16

71.8 145.0 98.0 100.0

131.2 75.0 101.6 179.2 128.5 131.6

74.7 47.2 42.0 38.0 −92.9 96.5 95.5 86.3 −11.1 163.4 −1.0 +4.2

102.4 75.6 68.3 66.7 −77.0 121.3 127.7 118.3 +12.6 196.2 +24.6 30.0

147.0 87.8 115.8 195.8 144.0 146.2 176.2 116.0 89.8 81.5 81.2 −69.3 134.0 144.0 134.2 24.4 212.0 37.3 42.3

163.8 100.7 131.5 213.4 159.9 163.3 193.8 131.2 115.5 95.6 96.8 −60.3 151.7 161.3 151.0 37.4 229.3 51.4 58.1

182.5 115.5 148.2 233.0 178.0 180.3 213.0 147.4 122.5 111.3 114.5 −50.3 172.0 181.0 170.0 51.2 247.0 66.8 71.5

60

200

400

760

60.5 50.3 137.6 142.5 147.9 153.8 334.8 232.8 75.5 139.0 113.8 326.8 219.7 53.8 123.8 99.0 145.6 135.4 269.8 222.6 256.0 236.6 172.3 190.1 80.0 199.0 223.4 298.4 178.4 −16.5 −30.0 184.2 191.0 −22.2 −29.6 −49.0 249.8 −27.1 97.6 229.9 141.2 149.8 108.4 163.2 174.0 118.1 184.5 147.7 129.8 210.8 172.6 87.7 88.0 154.2 153.6

72.3 62.0 151.5 156.9 163.0 168.5 350.0 248.0 88.1 151.3 127.4 341.8 236.0 65.7 135.2 111.7 159.1 150.0 286.0 239.5 271.5 251.5 184.4 205.5 93.5 216.0 239.6 314.0 194.0 −6.7 −20.6 200.0 208.0 −12.6 −20.2 −39.1 263.5 −17.7 110.6 249.0 154.0 163.5 121.4 176.5 189.2 131.6 198.2 161.8 143.5 227.8 187.8 100.6 101.0 169.3 168.0

90.2 80.0 173.7 180.3 186.7 191.7 373.2 271.8 107.5 170.5 148.2 364.8 260.6 83.6 152.0 130.4 180.2 173.3 309.8 264.1 298.7 277.1 203.1 227.0 114.0 241.8 261.8 339.0 216.1 +8.0 −6.7 224.1 230.3 +1.9 −5.9 −23.7 284.5 −3.4 130.6 277.1 175.0 184.0 139.0 197.6 211.3 151.2 219.5 184.2 163.8 254.0 211.0 120.8 120.8 191.3 189.8

110.6 99.8 197.7 206.8 212.5 217.5 397.2 299.8 128.2 192.1 171.2 388.9 288.0 104.0 173.8 151.0 204.0 199.3 334.7 292.3 326.0 304.5 227.5 251.6 137.2 269.3 285.0 365.4 242.8 24.7 +8.3 250.2 257.2 18.5 +10.5 −7.1 307.0 +12.8 152.0 308.0 197.5 207.0 160.0 221.0 235.0 173.5 243.0 208.5 186.0 283.5 235.4 142.7 142.3 214.6 213.0

131.6 120.3 222.3 231.9 238.3 244.0 421.8 330.0 150.8 213.5 195.0 412.5 317.0 125.6 195.2 172.9 227.0 225.5 360.0 321.0 353.8 332.0 253.5 276.0 160.5 299.0 309.3 390.3 270.5 42.1 26.1 277.0 285.0 36.1 27.8 +9.5 327.2 30.1 175.0 340.2 219.5 228.0 181.9 245.3 259.7 195.9 265.5 233.5 210.0 314.0 261.0 165.6 165.0 240.0 239.5

−108 −93 −4.1 15.5 51.9 +2 63.8 32 −53.7 −5 −19 61.6 28 −56.8 −15.4 −16

194.0 124.2 158.7 245.3 189.8 192.2 225.3 156.8 133.3 121.4 125.6 −44.0 185.3 193.5 182.2 59.9 258.2 76.8 81.2

209.9 136.2 173.5 260.6 204.5 205.9 240.9 170.3 147.7 134.0 140.6 −35.6 202.3 210.0 197.8 71.4 272.0 90.1 94.0

233.0 153.8 195.4 284.5 226.7 227.9 263.2 191.2 169.6 153.5 163.3 −22.3 228.0 234.5 222.0 89.0 294.5 110.2 114.1

259.0 173.5 218.2 311.3 251.2 251.8 285.5 212.8 194.0 175.3 188.7 −7.6 256.8 261.8 248.3 108.4 318.5 132.3 136.1

285.5 193.5 243.5 337.8 276.5 275.0 308.0 235.0 218.5 197.2 215.2 +8.3 284.5 290.0 275.8 128.8 342.1 155.0 158.3

62.8 73 19.5

Temperature, °C

Formula

99.8

Melting point, °C

100

−45.9 14 34 64.0 31 12.5 85.5 −22 188.5 53.3 10

−129.7 −159.7 −16.6

99.5 40.6 11.6 −6.7 76.5 −23.8 70.5 41.5 +7.5

56.5 164.5 −21.0 28 −104 130.8 73 88.5 −70 103 −55

VAPOR PRESSURES OF PURE SUBSTANCES TABLE 2-8

2-73

Vapor Pressures of Organic Compounds, up to 1 atm (Continued ) Pressure, mm Hg Compound Name

Piperidine Piperonal Propane Propenylbenzene Propionamide Propionic acid anhydride Propionitrile Propiophenone n-Propyl acetate iso-Propyl acetate n-Propyl alcohol (1-propanol) iso-Propyl alcohol (2-propanol) n-Propylamine Propylbenzene Propyl benzoate n-Propyl bromide (1-bromopropane) iso-Propyl bromide (2-bromopropane) n-Propyl n-butyrate isobutyrate iso-Propyl isobutyrate Propyl carbamate n-Propyl chloride (1-chloropropane) iso-Propyl chloride (2-chloropropane) iso-Propyl chloroacetate Propyl chloroglyoxylate Propylene Propylene glycol (1,2-Propanediol) Propylene oxide n-Propyl formate iso-Propyl formate 4,4′-iso-Propylidenebisphenol n-Propyl iodide (1-iodopropane) iso-Propyl iodide (2-iodopropane) n-Propyl levulinate iso-Propyl levulinate Propyl mercaptan (1-propanethiol) 2-iso-Propylnaphthalene iso-Propyl β-naphthyl ketone (2-isobutyronaphthone) 2-iso-Propylphenol 3-iso-Propylphenol 4-iso-Propylphenol Propyl propionate 4-iso-Propylstyrene Propyl isovalerate Pulegone Pyridine Pyrocatechol Pyrocaltechol diacetate (1,2-phenylene diacetate) Pyrogallol Pyrotartaric anhydride Pyruvic acid Quinoline iso-Quinoline Resorcinol Safrole Salicylaldehyde Salicylic acid Sebacic acid Selenophene Skatole Stearaldehyde Stearic acid Stearyl alcohol (1-octadecanol) Styrene Styrene dibromide [(1,2-dibromoethyl) benzene] Suberic acid Succinic anhydride Succinimide Succinyl chloride α-Terpineol Terpenoline

1

5

10

20

87.0 −128.9 17.5 65.0 4.6 20.6 −35.0 50.0 −26.7 −38.3 −15.0 −26.1 −64.4 6.3 54.6 −53.0 −61.8 −1.6 −6.2 −16.3 52.4 −68.3 −78.8 +3.8 9.7 −131.9 45.5 −75.0 −43.0 −52.0 193.0 −36.0 −43.3 59.7 48.0 −56.0 76.0

−7.0 117.4 −115.4 43.8 91.0 28.0 45.3 −13.6 77.9 −5.4 −17.4 +5.0 −7.0 −46.3 31.3 83.8 −33.4 −42.5 +22.1 +16.8 +5.8 77.6 −50.0 −61.1 28.1 32.3 −120.7 70.8 −57.8 −22.7 −32.7 224.2 −13.5 −22.1 86.3 74.5 −36.3 107.9

+3.9 132.0 −108.5 57.0 105.0 39.7 57.7 −3.0 92.2 +5.0 −7.2 14.7 +2.4 −37.2 43.4 98.0 −23.3 −32.8 34.0 28.3 17.0 90.0 −41.0 −52.0 40.2 43.5 −112.1 83.2 −49.0 −12.6 −22.7 240.8 −2.4 −11.7 99.9 88.0 −26.3 123.4

15.8 148.0 −100.9 71.5 119.0 52.0 70.4 +8.8 107.6 16.0 +4.2 25.3 12.7 −27.1 56.8 114.3 −12.4 −22.0 47.0 40.6 29.0 103.2 −31.0 −42.0 53.9 55.6 −104.7 96.4 −39.3 −1.7 −12.1 255.5 +10.0 0.0 114.0 102.4 −15.4 140.3

29.2 165.7 −92.4 87.7 134.8 65.8 85.6 22.0 124.3 28.8 17.0 36.4 23.8 −16.0 71.6 131.8 −0.3 −10.1 61.5 54.3 42.4 117.7 −19.5 −31.0 68.7 68.8 −96.5 111.2 −28.4 +10.8 −0.2 273.0 23.6 +13.2 130.1 118.1 −3.2 159.0

60

100

200

400

760

37.7 177.0 −87.0 97.8 144.3 74.1 94.5 30.1 135.0 37.0 25.1 43.5 30.5 −9.0 81.1 143.3 +7.5 −2.5 70.3 63.0 51.4 126.5 −12.1 −23.5 78.0 77.2 −91.3 119.9 −21.3 18.8 +7.5 282.9 32.1 21.6 140.6 127.8 +4.6 171.4

49.0 191.7 −79.6 111.7 156.0 85.8 107.2 41.4 149.3 47.8 35.7 52.8 39.5 +0.5 94.0 157.4 18.0 +8.0 82.6 73.9 62.3 138.3 −2.5 −13.7 90.3 88.0 −84.1 132.0 −12.0 29.5 17.8 297.0 43.8 32.8 154.0 141.8 15.3 187.6

66.2 214.3 −68.4 132.0 174.2 102.5 127.8 58.2 170.2 64.0 51.7 66.8 53.0 15.0 113.5 180.1 34.0 23.8 101.0 91.8 80.2 155.8 +12.2 +1.3 108.8 104.7 −73.3 149.7 +2.1 45.3 33.6 317.5 61.8 50.0 175.6 161.6 31.5 211.8

85.7 238.5 −55.6 154.7 194.0 122.0 146.0 77.7 194.2 82.0 69.8 82.0 67.8 31.5 135.7 205.2 52.0 41.5 121.7 112.0 100.0 175.8 29.4 18.1 128.0 123.0 −60.9 168.1 17.8 62.6 50.5 339.0 81.8 69.5 198.0 185.2 49.2 238.5

106.0 263.0 −42.1 179.0 213.0 141.1 167.0 97.1 218.0 101.8 89.0 97.8 82.5 48.5 159.2 231.0 71.0 60.0 142.7 133.9 120.5 195.0 46.4 36.5 148.6 150.0 −47.7 188.2 34.5 81.3 68.3 360.5 102.5 89.5 221.2 208.2 67.4 266.0

Temperature, °C

Formula C5H11N C8H6O3 C3H8 C9H10 C3H7NO C3H6O2 C6H10O3 C3H5N C9H10O C5H10O2 C5H10O2 C3H8O C3H8O C3H9N C9H12 C10H12O2 C3H7Br C3H7Br C7H14O2 C7H14O2 C7H14O2 C4H9NO2 C3H7Cl C3H7Cl C5H9ClO2 C5H7ClO3 C3H6 C3H8O2 C3H6O C4H8O2 C4H8O2 C15H16O2 C3H7I C3H7I C8H14O3 C8H14O3 C3H8S C13H14

40

C14H14O C9H12O C9H12O C9H12O C6H12O2 C11H14 C8H16O2 C10H16O C5H5N C6H6O2

133.2 56.6 62.0 67.0 −14.2 34.7 +8.0 58.3 −18.9

165.4 83.8 90.3 94.7 +8.0 62.3 32.8 82.5 +2.5 104.0

181.0 97.0 104.1 108.0 19.4 76.0 45.1 94.0 13.2 118.3

197.7 111.7 119.8 123.4 31.6 91.2 58.0 106.8 24.8 134.0

215.6 127.5 136.2 139.8 45.0 108.0 72.8 121.7 38.0 150.6

227.0 137.7 146.6 149.7 53.8 118.4 82.3 130.2 46.8 161.7

242.3 150.3 160.2 163.3 65.2 132.8 95.0 143.1 57.8 176.0

264.0 170.1 182.0 184.0 82.7 153.9 113.9 162.5 75.0 197.7

288.2 192.6 205.0 206.1 102.0 178.0 135.0 189.8 95.6 221.5

313.0 214.5 228.0 228.2 122.4 202.5 155.9 221.0 115.4 245.5

C10H10O4 C6H6O3 C5H6O3 C3H4O3 C9H7N C9H7N C6H6O2 C10H10O2 C7H6O2 C7H6O3 C10H18O4 C4H4Se C9H9N C18H36O C18H36O2 C18H36O C8H8

98.0 69.7 21.4 59.7 63.5 108.4 63.8 33.0 113.7 183.0 −39.0 95.0 140.0 173.7 150.3 −7.0

129.8 151.7 99.7 45.8 89.6 92.7 138.0 93.0 60.1 136.0 215.7 −16.0 124.2 174.6 209.0 185.6 +18.0

145.7 167.7 114.2 57.9 103.8 107.8 152.1 107.6 73.8 146.2 232.0 −4.0 139.6 192.1 225.0 202.0 30.8

161.8 185.3 130.0 70.8 119.8 123.7 168.0 123.0 88.7 156.8 250.0 +9.1 154.3 210.6 243.4 220.0 44.6

179.8 204.2 147.8 85.3 136.7 141.6 185.3 140.1 105.2 172.2 268.2 24.1 171.9 230.8 263.3 240.4 59.8

191.6 216.3 158.6 94.1 148.1 152.0 195.8 150.3 115.7 182.0 279.8 33.8 183.6 244.2 275.5 252.7 69.5

206.5 232.0 173.8 106.5 163.2 167.6 209.8 165.1 129.4 193.4 294.5 47.0 197.4 260.0 291.0 269.4 82.0

228.7 255.3 196.1 124.7 186.2 190.0 230.8 186.2 150.0 210.0 313.2 66.7 218.8 285.0 316.5 293.5 101.3

253.3 281.5 221.0 144.7 212.3 214.5 253.4 210.0 173.7 230.5 332.8 89.8 242.5 313.8 343.0 320.3 122.5

278.0 309.0 247.4 165.0 237.7 240.5 276.5 233.0 196.5 256.0 352.3 114.3 266.2 342.5 370.0 349.5 145.2

C8H8Br2 C8H14O4 C4H4O3 C4H5NO2 C4H4Cl2O2 C10H18O C10H16

86.0 172.8 92.0 115.0 39.0 52.8 32.3

115.6 205.5 115.0 143.2 65.0 80.4 58.0

129.8 219.5 128.2 157.0 78.0 94.3 70.6

145.2 238.2 145.3 174.0 91.8 109.8 84.8

161.8 254.6 163.0 192.0 107.5 126.0 100.0

172.2 265.4 174.0 203.0 117.2 136.3 109.8

186.3 279.8 189.0 217.4 130.0 150.1 122.7

207.8 300.5 212.0 240.0 149.3 171.2 142.0

230.0 322.8 237.0 263.5 170.0 194.3 163.5

254.0 345.5 261.0 287.5 192.5 217.5 185.0

Melting point, °C −9 37 −187.1 −30.1 79 −22 −45 −91.9 21 −92.5 −127 −85.8 −83 −99.5 −51.6 −109.9 −89.0 −95.2

−122.8 −117 −185 −112.1 −92.9 −98.8 −90 −112

15.5 26 61 −76

−42 105 133 13.6 −15 24.6 110.7 11.2 −7 159 134.5 95 63.5 69.3 58.5 −30.6 142 119.6 125.5 17 35

2-74

PHYSICAL AND CHEMICAL DATA

TABLE 2-8

Vapor Pressures of Organic Compounds, up to 1 atm (Continued ) Pressure, mm Hg Compound Name

Formula

1,1,1,2-Tetrabromoethane 1,1,2,2-Tetrabromoethane Tetraisobutylene Tetracosane 1,2,3,4-Tetrachlorobenzene 1,2,3,5-Tetrachlorobenzene 1,2,4,5-Tetrachlorobenzene 1,1,2,2-Tetrachloro-1,2-difluoroethane 1,1,1,2-Tetrachloroethane 1,1,2,2-Tetrachloroethane 1,2,3,5-Tetrachloro-4-ethylbenzene Tetrachloroethylene 2,3,4,6-Tetrachlorophenol 3,4,5,6-Tetrachloro-1,2-xylene Tetradecane Tetradecylamine Tetradecyltrimethylsilane Tetraethoxysilane 1,2,3,4-Tetraethylbenzene Tetraethylene glycol Tetraethylene glycol chlorohydrin Tetraethyllead Tetraethylsilane Tetralin 1,2,3,4-Tetramethylbenzene 1,2,3,5-Tetramethylbenzene 1,2,4,5-Tetramethylbenzene 2,2,3,3-Tetramethylbutane Tetramethylene dibromide (1,4-dibromobutane) Tetramethyllead Tetramethyltin Tetrapropylene glycol monoisopropyl ether Thioacetic acid (mercaptoacetic acid) Thiodiglycol (2,2′-thiodiethanol) Thiophene Thiophenol (benzenethiol) α-Thujone Thymol Tiglaldehyde Tiglic acid Tiglonitrile Toluene Toluene-2,4-diamine 2-Toluic nitrile (2-tolunitrile) 4-Toluic nitrile (4-tolunitrile) 2-Toluidine 3-Toluidine 4-Toluidine 2-Tolyl isocyanide 4-Tolylhydrazine Tribromoacetaldehyde 1,1,2-Tribromobutane 1,2,2-Tribromobutane 2,2,3-Tribromobutane 1,1,2-Tribromoethane 1,2,3-Tribromopropane Triisobutylamine Triisobutylene 2,4,6-Tritertbutylphenol Trichloroacetic acid Trichloroacetic anhydride Trichloroacetyl bromide 2,4,6-Trichloroaniline 1,2,3-Trichlorobenzene 1,2,4-Trichlorobenzene 1,3,5-Trichlorobenzene 1,2,3-Trichlorobutane 1,1,1-Trichloroethane 1,1,2-Trichloroethane Trichloroethylene Trichlorofluoromethane 2,4,5-Trichlorophenol 2,4,6-Trichlorophenol

C2H2Br4 C2H2Br4 C16H32 C24H50 C6H2Cl4 C6H2Cl4 C6H2Cl4 C2Cl4F2 C2H2Cl4 C2H2Cl4 C8H6Cl4 C2Cl4 C6H2Cl4O C8H6Cl4 C14H30 C14H31N C17H38Si C8H20O4Si C14H22 C8H18O5 C8H17ClO4 C8H20Pb C8H20Si C10H12 C10H14 C10H14 C10H14 C8H18 C4H8Br2 C4H12Pb C4H12Sn C15H32O5 C2H4O2S C4H10O2S C4H4S C6H6S C10H16O C10H14O C5H8O C5H8O2 C5H7N C7H8 C7H10N2 C8H7N C8H7N C7H9N C7H9N C7H9N C8H7N C7H10N2 C2HBr3O C4H7Br3 C4H7Br3 C4H7Br3 C2H3Br3 C3H5Br3 C12H27N C12H24 C18H30O C2HCl3O2 C4Cl6O3 C2BrCl3O C6H4Cl3N C6H3Cl3 C6H3Cl3 C6H3Cl3 C4H7Cl3 C2H3Cl3 C2H3Cl3 C2HCl3 CCl3F C6H3Cl3O C6H3Cl3O

1

5

10

20

58.0 65.0 63.8 183.8 68.5 58.2

83.3 95.5 93.7 219.6 99.6 89.0

95.7 110.0 108.5 237.6 114.7 104.1

108.5 126.0 124.5 255.3 131.2 121.6

−37.5 −16.3 −3.8 77.0 −20.6 100.0 94.4 76.4 102.6 120.0 16.0 65.7 153.9 110.1 38.4 −1.0 38.0 42.6 40.6 45.0 −17.4

−16.0 +7.4 +20.7 110.0 +2.4 130.3 125.0 106.0 135.8 150.7 40.3 96.2 183.7 141.8 63.6 +23.9 65.3 68.7 65.8 65.0 +3.2

−5.0 19.3 33.0 126.0 13.8 145.3 140.3 120.7 152.0 166.2 52.6 111.6 197.1 156.1 74.8 36.3 79.0 81.8 77.8 74.6 13.5

32.0 −29.0 −51.3 116.6 60.0 42.0 −40.7 18.6 38.3 64.3 −25.0 52.0 −25.5 −26.7 106.5 36.7 42.5 44.0 41.0 42.0 25.2 82.4 18.5 45.0 41.0 38.2 32.6 47.5 32.3 18.0 95.2 51.0 56.2 −7.4 134.0 40.0 38.4

58.8 −6.8 −31.0 147.8 87.7 96.0 −20.8 43.7 65.7 92.8 −1.6 77.8 −2.4 −4.4 137.2 64.0 71.3 69.3 68.0 68.2 51.0 110.0 45.0 73.5 69.0 66.0 58.0 75.8 57.4 44.0 126.1 76.0 85.3 +16.7 157.8 70.0 67.3 63.8 27.2 −32.0 −2.0 −22.8 −67.6 102.1 105.9

72.4 +4.4 −20.6 163.0 101.5 128.0 −10.9 56.0 79.3 107.4 +10.0 90.2 +9.2 +6.4 151.7 77.9 85.8 81.4 82.0 81.8 64.0 123.8 58.0 87.8 83.2 79.8 70.6 90.0 69.8 56.5 142.0 88.2 99.6 29.3 170.0 85.6 81.7 78.0 40.0 −21.9 +8.3 −12.4 −59.0 117.3 120.2

40

60

100

200

400

760

Temperature, °C

+0.5 −52.0 −24.0 −43.8 −84.3 72.0 76.5

Melting point, °C

+6.7 32.1 46.2 143.7 26.3 161.0 156.0 135.6 170.0 183.5 65.8 127.7 212.3 172.6 88.0 50.0 93.8 95.8 91.0 88.0 24.6

123.2 144.0 142.2 276.3 149.2 140.0 146.0 19.8 46.7 60.8 162.1 40.1 179.1 174.2 152.7 189.0 201.5 81.1 145.8 228.0 190.0 102.4 65.3 110.4 111.5 105.8 104.2 36.8

132.0 155.1 152.6 288.4 160.0 152.0 157.7 28.1 56.0 70.0 175.0 49.2 190.0 185.8 164.0 200.2 213.3 90.7 156.7 237.8 200.5 111.7 74.8 121.3 121.8 115.4 114.8 44.5

144.0 170.0 167.5 305.2 175.7 168.0 173.5 38.6 68.0 83.2 191.6 61.3 205.2 200.5 178.5 215.7 227.8 103.6 172.4 250.0 214.7 123.8 88.0 135.3 135.7 128.3 128.1 54.8

161.5 192.5 190.0 330.5 198.0 193.7 196.0 55.0 87.2 102.2 215.3 79.8 227.2 223.0 201.8 239.8 250.0 123.5 196.0 268.4 236.5 142.0 108.0 157.2 155.7 149.9 149.5 70.2

181.0 217.5 214.6 358.0 225.5 220.0 220.5 73.1 108.2 124.0 243.0 100.0 250.4 248.3 226.8 264.6 275.0 146.2 221.4 288.0 258.2 161.8 130.2 181.8 180.0 173.7 172.1 87.4

200.0 243.5 240.0 386.4 254.0 246.0 245.0 92.0 130.5 145.9 270.0 120.8 275.0 273.5 252.5 291.2 300.0 168.5 248.0 307.8 281.5 183.0 153.0 207.2 204.4 197.9 195.9 106.3

87.6 16.6 −9.3 179.8 115.8 165.0 0.0 69.7 93.7 122.6 23.2 103.8 22.1 18.4 167.9 93.0 101.7 95.1 96.7 95.8 78.2 138.6 72.1 103.2 98.6 94.6 84.2 105.8 83.0 70.0 158.0 101.8 114.3 42.1 182.6 101.8 97.2 93.7 55.0 −10.8 21.6 −1.0 −49.7 134.0 135.8

104.0 30.3 +3.5 197.7 131.8 210.0 +12.5 84.2 110.0 139.8 37.0 119.0 36.7 31.8 185.7 110.0 109.5 110.0 113.5 111.5 94.0 154.1 87.8 120.2 116.0 111.8 100.0 122.8 97.8 86.7 177.4 116.3 131.2 57.2 195.8 119.8 114.8 110.8 71.5 +1.6 35.2 +11.9 −39.0 151.5 152.2

115.1 39.2 11.7 209.0 142.0 240.5 20.1 93.9 120.2 149.8 45.8 127.8 46.0 40.3 196.2 120.8 130.0 119.8 123.8 121.5 104.0 165.0 97.5 131.6 127.0 122.2 110.0 134.0 107.3 96.7 188.0 125.9 141.8 66.7 204.5 131.5 125.7 121.8 82.0 9.5 44.0 20.0 −32.3 162.5 163.5

128.7 50.8 22.8 223.3 154.0 285 30.5 106.6 134.0 164.1 57.7 140.5 58.2 51.9 211.5 135.0 145.2 133.0 136.7 133.7 117.7 178.0 110.2 146.0 141.8 136.3 123.5 148.0 119.7 110.0 203.0 137.8 155.2 79.5 214.6 146.0 140.0 136.0 96.2 20.0 55.7 31.4 −23.0 178.0 177.8

149.8 68.8 39.8 245.0

173.8 89.0 58.5 268.3

197.5 110.0 78.0 292.7

−20 −27.5

46.5 125.8 154.2 185.5 75.4 158.0 77.8 69.5 232.8 156.0 167.3 153.0 157.6 154.0 137.8 198.0 130.0 167.8 163.5 157.8 143.5 170.0 138.0 130.2 226.2 155.4 176.2 98.4 229.8 168.2 162.0 157.7 118.0 36.2 73.3 48.0 −9.1 201.5 199.0

64.7 146.7 177.8 209.2 95.5 179.2 99.7 89.5 256.0 180.0 193.0 176.2 180.6 176.9 159.9 219.5 151.6 192.0 188.0 182.2 165.4 195.0 157.8 153.0 250.6 175.2 199.8 120.2 246.4 193.5 187.7 183.0 143.0 54.6 93.0 67.0 +6.8 226.5 222.5

84.4 168.0 201.0 231.8 116.4 198.5 122.0 110.6 280.0 205.2 217.6 199.7 203.3 200.4 183.5 242.0 174.0 216.2 213.8 206.5 188.4 220.0 179.0 179.0 276.3 195.6 223.0 143.0 262.0 218.5 213.0 208.4 169.0 74.1 113.9 86.7 23.7 251.8 246.0

−38.3

51.1 46.5 54.5 139 26.5 −68.7 −36 −19.0 69.5 5.5

11.6 −136 −31.0 −6.2 −24.0 79.5 −102.2

−16.5

51.5 64.5 −95.0 99 −13 29.5 −16.3 −31.5 44.5 65.5

−26 16.5 −22 57 78 52.5 17 63.5 −30.6 −36.7 −73 62 68.5

VAPOR PRESSURES OF PURE SUBSTANCES TABLE 2-8

2-75

Vapor Pressures of Organic Compounds, up to 1 atm (Concluded ) Pressure, mm Hg Compound Name

Tri-2-chlorophenylthiophosphate 1,1,1-Trichloropropane 1,2,3-Trichloropropane 1,1,2-Trichloro-1,2,2-trifluoroethane Tricosane Tridecane Tridecanoic acid Triethoxymethylsilane Triethoxyphenylsilane 1,2,4-Triethylbenzene 1,3,4-Triethylbenzene Triethylborine Triethyl camphoronate citrate Triethyleneglycol Triethylheptylsilane Triethyloctylsilane Triethyl orthoformate phosphate Triethylthallium Trifluorophenylsilane Trimethallyl phosphate 2,3,5-Trimethylacetophenone Trimethylamine 2,4,5-Trimethylaniline 1,2,3-Trimethylbenzene 1,2,4-Trimethylbenzene 1,3,5-Trimethylbenzene 2,2,3-Trimethylbutane Trimethyl citrate Trimethyleneglycol (1,3-propanediol) 1,2,4-Trimethyl-5-ethylbenzene 1,3,5-Trimethyl-2-ethylbenzene 2,2,3-Trimethylpentane 2,2,4-Trimethylpentane 2,3,3-Trimethylpentane 2,3,4-Trimethylpentane 2,2,4-Trimethyl-3-pentanone Trimethyl phosphate 2,4,5-Trimethylstyrene 2,4,6-Trimethylstyrene Trimethylsuccinic anhydride Triphenylmethane Triphenylphosphate Tripropyleneglycol Tripropyleneglycol monobutyl ether Tripropyleneglycol monoisopropyl ether Tritolyl phosphate Undecane Undecanoic acid 10-Undecenoic acid Undecan-2-ol n-Valeric acid iso-Valeric acid γ-Valerolactone Valeronitrile Vanillin Vinyl acetate 2-Vinylanisole 3-Vinylanisole 4-Vinylanisole Vinyl chloride (1-chloroethylene) cyanide (acrylonitrile) fluoride (1-fluoroethylene) Vinylidene chloride (1,1-dichloroethene) 4-Vinylphenetole 2-Xenyl dichlorophosphate 2,4-Xyaldehyde 2-Xylene (2-xylene) 3-Xylene (3-xylene) 4-Xylene (4-xylene) 2,4-Xylidine 2,6-Xylidine

1

5

10

20

217.2

231.2

246.7

261.7

−7.0 33.7 −49.4 206.3 98.3 166.3 +22.8 98.8 74.2 76.0

+4.2 46.0 −40.3 223.0 104.0 181.0 34.6 112.6 88.5 90.2 −148.0 166.0 144.0 158.1 114.6 120.6 40.5 82.1 51.7 +0.8 149.8 122.3 −73.8 109.0 55.9 50.7 47.4 −18.8 160.4 100.6 84.6 80.5 +3.9 −4.3 +6.9 +7.1 46.4 67.8 91.6 79.7 97.4 197.0 249.8 140.5 147.0 127.3 198.0 73.9 149.0 156.3 112.8 79.8 71.3 79.8 30.0 154.0 −18.0 81.0 83.0 85.7 −83.7 −20.3 −132.2 −51.2 105.6 187.0 99.0 32.1 28.3 27.3 93.0 87.0

16.2 59.3 −30.0 242.0 120.2 195.8 47.2 127.2 104.0 105.8 −140.6 183.6 171.1 174.0 130.3 137.7 53.4 97.8 67.7 12.3 169.8 137.5 −65.0 123.7 69.9 64.5 61.0 −7.5 177.2 115.5 99.7 96.0 16.0 +7.5 19.2 19.3 57.6 83.0 107.1 94.8 113.8 206.8 269.7 155.8 161.8 143.7 213.2 85.6 166.0 172.0 127.5 93.1 84.0 95.2 43.3 170.5 −7.0 94.7 97.2 100.0 −75.7 −9.0 −125.4 −41.7 120.3 205.0 114.0 45.1 41.1 40.1 107.6 102.7

29.9 74.0 −18.5 261.3 137.7 212.4 61.7 143.5 121.7 122.6 −131.4 201.8 190.4 191.3 148.0 155.7 67.5 115.7 85.4 25.4 192.0 154.2 −55.2 139.8 85.4 79.8 76.1 +5.2 194.2 131.0 106.0 113.2 29.5 20.7 33.0 32.9 69.8 100.0 124.2 111.8 131.0 215.5 290.3 173.7 179.8 161.4 229.7 104.4 185.6 188.7 143.7 107.8 98.0 101.9 57.8 188.7 +5.3 110.0 112.5 116.0 −66.8 +3.8 −118.0 −31.1 136.3 223.8 129.7 59.5 55.3 54.4 123.8 120.2

60

100

200

400

760

271.5

283.8

302.8

322.0

341.3

38.3 83.6 −11.2 273.8 148.2 222.0 70.4 153.2 132.2 133.4 −125.2 213.5 202.5 201.5 158.2 168.0 76.0 126.3 95.7 33.2 207.0 165.7 −48.8 149.5 95.3 89.5 85.8 13.3 205.5 141.1 126.3 123.8 38.1 29.1 41.8 41.6 77.3 110.0 135.5 122.3 142.2 221.2 305.2 184.6 190.2 173.2 239.8 115.2 197.2 199.5 153.7 116.6 107.3 122.4 66.9 199.8 13.0 119.8 122.3 126.1 −61.1 11.8 −113.0 −24.0 146.4 236.0 139.8 68.8 64.4 63.5 133.7 131.5

50.0 96.1 −1.7 289.8 162.5 236.0 82.7 167.5 146.8 147.7 −116.0 228.6 217.8 214.6 174.0 184.3 88.0 141.6 112.1 44.2 225.7 179.7 −40.3 162.0 108.8 102.8 98.9 24.4 219.6 153.4 140.3 137.9 49.9 40.7 53.8 53.4 87.6 124.0 149.8 136.8 156.5 228.4 322.5 199.0 204.4 187.8 252.2 128.1 212.5 213.5 167.2 128.3 118.9 136.5 78.6 214.5 23.3 132.3 135.3 139.7 −53.2 22.8 −106.2 −15.0 159.8 251.5 152.2 81.3 76.8 75.9 146.8 146.0

67.7 115.6 +13.5 313.5 185.0 255.2 101.0 188.0 168.3 168.3 −101.0 250.8 242.2 235.2 196.0 208.0 106.0 163.7 136.0 60.1 255.0 201.3 −27.0 182.3 129.0 122.7 118.6 41.2 241.3 172.8 160.3 158.4 67.8 58.1 72.0 71.3 102.2 145.0 171.8 157.8 179.8 239.7 349.8 220.2 224.4 209.7 271.8 149.3 237.8 232.8 187.7 146.0 136.2 157.7 97.7 237.3 38.4 151.0 154.0 159.0 −41.3 38.7 −95.4 −1.0 180.0 275.3 172.3 100.2 95.5 94.6 166.4 168.0

87.5 137.0 30.2 339.8 209.4 276.5 121.8 210.5 193.7 193.2 −81.0 276.0 267.5 256.6 221.0 235.0 125.7 187.0 163.5 78.7 288.5 224.3 −12.5 203.7 152.0 145.4 141.0 60.4 264.2 193.8 184.5 183.5 88.2 78.0 92.7 91.8 118.4 167.8 196.1 182.3 205.5 249.8 379.2 244.3 247.0 232.8 292.7 171.9 262.8 254.0 209.8 165.0 155.2 182.3 118.7 260.0 55.5 172.1 175.8 182.0 −28.0 58.3 −84.0 +14.8 202.8 301.5 194.1 121.7 116.7 115.9 188.3 193.7

108.2 158.0 47.6 366.5 234.0 299.0 143.5 233.5 218.0 217.5 −56.2 301.0 294.0 278.3 247.0 262.0 146.0 211.0 192.1 98.3 324.0 247.5 +2.9 234.5 176.1 169.2 164.7 80.9 287.0 214.2 208.1 208.0 109.8 99.2 114.8 113.5 135.0 192.7 221.2 207.0 231.0 259.2 413.5 267.2 269.5 256.6 313.0 195.8 290.0 275.0 232.0 184.4 175.1 207.5 140.8 285.0 72.5 194.0 197.5 204.5 −13.8 78.5 −72.2 31.7 225.0 328.5 215.5 144.4 139.1 138.3 211.5 217.9

Temperature, °C

Formula C18H12Cl3O3 188.2 PS C3H5Cl3 −28.8 C3H5Cl3 +9.0 C2Cl3F3 −68.0 C23H48 170.0 C13H28 59.4 C13H26O2 137.8 C7H18O3Si −1.5 C12H20O3Si 71.0 C12H18 46.0 C12H18 47.9 C6H15B C15H26O6 C12H20O7 107.0 C6H14O4 114.0 C13H30Si 70.0 C14H32Si 73.7 C7H16O3 +5.5 C6H15O4P 39.6 C6H15Tl +9.3 C6H5F3Si −31.0 C12H21PO4 93.7 C11H14O 79.0 C3H9N −97.1 C9H13N 68.4 C9H12 16.8 C9H12 13.6 C9H12 9.6 C7H16 C9H14O7 106.2 C3H8O2 59.4 C11H16 43.7 C11H16 38.8 C8H18 −29.0 C8H18 −36.5 C8H18 −25.8 C8H18 −26.3 C8H16O 14.7 C3H9O4P 26.0 C11H14 48.1 C11H14 37.5 C7H10O3 53.5 C19H16 169.7 C18H15O4P 193.5 C9H20O4 96.0 C13H28O4 101.5 C12H26O4 82.4 C21H21O4P 154.6 C11H24 32.7 C11H22O2 101.4 C11H20O2 114.0 C11H24O 71.1 C5H10O2 42.2 C5H10O2 34.5 C5H8O2 37.5 C5H9N −6.0 C8H8O3 107.0 C4H6O2 −48.0 C9H10O 41.9 C9H10O 43.4 C9H10O 45.2 C2H3Cl −105.6 C3H3N −51.0 C2H3F −149.3 C2H2Cl2 −77.2 C10H12O 64.0 C12H9Cl2PO 138.2 C9H10O 59.0 C8H10 −3.8 C8H10 −6.9 C8H10 −8.1 C8H11N 52.6 C8H11N 44.0

40

150.2 138.7 144.0 99.8 104.8 29.2 67.8 37.6 −9.7 131.0 108.0 −81.7 95.9 42.9 38.3 34.7 146.2 87.2 71.2 67.0 −7.1 −15.0 −3.9 −4.1 36.0 53.7 77.0 65.7 82.6 188.4 230.4 125.7 131.6 112.4 184.2 59.7 133.1 142.8 99.0 67.7 59.6 65.8 +18.1 138.4 −28.0 68.0 69.9 72.0 −90.8 −30.7 −138.0 −60.0 91.7 171.1 85.9 +20.2 +16.8 +15.5 79.8 72.6

Melting point, °C

−77.7 −14.7 −35 47.7 −6.2 41

135

−63.0

−117.1 67 −25.5 −44.1 −44.8 −25.0 78.5

−112.3 −107.3 −101.5 −109.2

93.4 49.4

−25.6 29.5 24.5 −34.5 −37.6 81.5

−153.7 −82 −160.5 −122.5 75 −25.2 −47.9 +13.3

2-76

PHYSICAL AND CHEMICAL DATA

VAPOR PRESSURES OF SOLUTIONS UNITS CONVERSIONS

To convert cubic feet to cubic meters, multiply by 0.02832. To convert bars to pounds-force per square inch, multiply by 14.504. To convert bars to kilopascals, multiply by 1 × 102.

For this subsection, the following units conversions are applicable: °F = 9⁄5°C + 32. To convert millimeters of mercury to pounds-force per square inch, multiply by 0.01934.

TABLE 2-9

Partial Pressures of Water over Aqueous Solutions of HCl* log10 pmm = A − B/T, (T in K), which, however, agrees only approximately with the table. The table is more nearly correct. Partial pressure of H2O, mmHg, °C

% HCl

A

B

0°

5°

10°

15°

20°

25°

30°

35°

40°

45°

50°

60°

6 10 14 18 20

8.99156 8.99864 8.97075 8.98014 8.97877

2282 2295 2300 2323 2334

4.18 3.84 3.39 2.87 2.62

6.04 5.52 4.91 4.21 3.83

8.45 7.70 6.95 5.92 5.40

11.7 10.7 9.65 8.26 7.50

15.9 14.6 13.1 11.3 10.3

21.8 20.0 18.0 15.4 14.1

29.1 26.8 24.1 20.6 19.0

39.4 35.5 31.9 27.5 25.1

50.6 47.0 42.1 36.4 33.3

66.2 61.5 55.3 47.9 43.6

86.0 80.0 72.0 62.5 57.0

139 130 116 102 93.5

22 24 26 28 30

9.02708 8.96022 9.01511 8.97611 9.00117

2363 2356 2390 2395 2422

2.33 2.05 1.76 1.50 1.26

3.40 3.04 2.60 2.24 1.90

4.82 4.31 3.71 3.21 2.73

6.75 6.03 5.21 4.54 3.88

9.30 8.30 7.21 6.32 5.41

12.6 11.4 9.95 8.75 7.52

17.1 15.4 13.5 11.8 10.2

22.8 20.4 18.0 15.8 13.7

30.2 27.1 24.0 21.1 18.4

39.8 35.7 31.7 27.9 24.3

52.0 46.7 41.5 36.5 32.0

32 34 36 38 40 42

9.03317 9.07143 9.11815 9.20783 9.33923 9.44953

2453 2487 2526 2579 2647 2709

1.04 0.85 0.68 0.53 0.41 0.31

1.57 1.29 1.03 0.81 0.63 0.48

2.27 1.87 1.50 1.20 0.94 0.72

3.25 2.70 2.19 1.75 1.37 1.06

4.55 3.81 3.10 2.51 2.00 1.56

6.37 5.35 4.41 3.60 2.88 2.30

11.7 9.95 8.33 6.92 5.68 4.60

15.7 13.5 11.4 9.52 7.85 6.45

21.0 18.1 15.4 13.0 10.7 8.90

27.7 24.0 20.4 17.4 14.5 12.1

8.70 7.32 6.08 5.03 4.09 3.28

70°

80°

90°

100° 110°

220 204 185 162 150

333 310 273 248 230

492 463 425 374 345

715 677 625 550 510

960 892 783 729

85.6 77.0 69.0 60.7 53.5

138 124 112 99.0 87.5

211 194 173 154 136

317 290 261 234 207

467 426 387 349 310

670 611 555 499 444

46.5 40.5 34.8 29.6 25.0 21.2

76.5 66.5 57.0 49.1 42.1 35.8

120 104 90.0 77.5 67.3 57.2

184 161 140 120 105 89.2

275 243 212 182 158 135

396 355 311 266 230 195

*Accuracy, ca. 2 percent for solutions of 15 to 30 percent HCl between 0 and 100°; for solutions of > 30 percent HCl the accuracy is ca. 5 percent at the lower temperatures and ca. 15 percent at the higher temperatures. Below 15 percent HCl, the accuracy is ca. 5 percent at the lower temperatures and higher strengths to ca. 15 to 20 percent at the lower strengths and perhaps 15 to 20 percent at the higher temperatures and lower strengths.

TABLE 2-10

Partial Pressures of HCl over Aqueous Solutions of HCl*

log10 pmm = A − B/T, (T in K), which, however, agrees only approximately with the table. The table is more nearly correct. mmHg, °C

% HCl

A

2 4 6 8 10

11.8037 11.6400 11.2144 11.0406 10.9311

4736 0.0000117 0.000023 0.000044 0.000084 0.000151 0.000275 0.00047 0.00083 0.00140 0.00380 0.0100 0.0245 0.058 0.132 0.280 4471 0.000018 0.000036 .000069 .000131 .00024 .00044 .00077 .00134 .0023 .00385 .0064 .0165 .0405 .095 .21 .46 .93 4202 .000066 .000125 .000234 .000425 .00076 .00131 .00225 .0038 .0062 .0102 .0163 .040 .094 .206 .44 .92 1.78 4042 .000118 .000323 .000583 .00104 .00178 .0031 .00515 .0085 .0136 .022 .0344 .081 .183 .39 .82 1.64 3.10 3908 .00042 .00075 .00134 .00232 .00395 .0067 .0111 .0178 .0282 .045 .069 .157 .35 .73 1.48 2.9 5.4

12 14 16 18 20

10.7900 10.6954 10.6261 10.4957 10.3833

3765 3636 3516 3376 3245

22 24 26 28 30

10.3172 10.2185 10.1303 10.0115 9.8763

3125 .0734 2995 .175 2870 .41 2732 1.0 2593 2.4

.119 .277 .64 1.52 3.57

.187 .43 .98 2.27 5.23

.294 .66 1.47 3.36 7.60

.45 1.00 2.17 4.90 10.6

.68 1.49 3.20 7.05 15.1

1.02 2.17 4.56 9.90 21.0

32 34 36 38 40

9.7523 9.6061 9.5262 9.4670 9.2156

2457 2316 2229 2094 1939

5.7 13.1 29.0 63.0 130

8.3 18.8 41.0 87.0 176

11.8 26.4 56.4 117 233

16.8 36.8 78 158 307

23.5 50.5 105.5 210 399

32.5 68.5 142 277 515

44.5 92 188 360 627

42 44 46

8.9925 1800 253 8.8621 1681 510 940

332 655

430 840

560

709

900

B

0°

.00099 .0024 .0056 .0135 .0316

5°

.00175 .00415 .0095 .0225 .052

10°

.00305 .0071 .016 .037 .084

15°

.0052 .0118 .0265 .060 .132

20°

.0088 .0196 .0428 .095 .205

25°

.0145 .0316 .0685 .148 .32

30°

.0234 .050 .106 .228 .48

35°

.037 .078 .163 .345 .72

40°

45°

50°

60°

70°

80°

90°

100°

110°

.058 .121 .247 .515 1.06

.091 .185 .375 .77 1.55

.136 .275 .55 1.11 2.21

.305 .60 1.17 2.3 4.4

.66 1.25 2.40 4.55 8.5

1.34 2.50 4.66 8.6 15.6

2.65 4.8 8.8 15.7 28.1

5.1 9.0 16.1 28 49

9.3 16.0 28 48 83

1.50 3.14 6.50 13.8 28.6

2.18 4.5 9.2 19.1 39.4

3.14 6.4 12.7 26.4 53

4.42 8.9 17.5 35.7 71

8.6 16.9 32.5 64 124

16.3 31.0 58.5 112 208

29.3 54.5 100 188 340

52 94 169 309 542

90 157 276 493 845

146 253 436 760

60.0 122 246 465 830

81 161 322 598

107 211 416 758

141 273 535 955

238 450 860

390 720

623

970

*Accuracy, ca. 2 percent for solutions of 15 to 30 percent HCl between 0 and 100°; for solutions of > 30 percent HCl the accuracy is ca. 5 percent at the lower temperatures and ca. 15 percent at the higher temperatures. Below 15 percent HCl, the accuracy is ca. 5 percent at the lower temperatures and higher strengths to ca. 15 to 20 percent at the lower strengths and perhaps 15 to 20 percent at the higher temperatures and lower strengths.

VAPOR PRESSURES OF SOLUTIONS

FIG. 2-1 Vapor pressures of H3PO4 aqueous: partial pressure of H2O vapor. (Courtesy of Victor Chemical Works, Stauffer Chemical Company; measurements by W. H. Woodstock.)

TABLE 2-11 g SO2 / 100 g H2O

Temperature, ∞C 10

20

0.01 0.05 0.10 0.15 0.20

0.02 0.38 1.15 2.10 3.17

0.04 0.66 1.91 3.44 5.13

0.07 1.07 3.03 5.37 7.93

0.25 0.30 0.40 0.50 1.00

4.34 5.57 8.17 10.9 25.8

6.93 8.84 12.8 17.0 39.5

8.00 10.00 15.00 20.00

Vapor pressures of H3PO4 aqueous: weight of H2O in saturated air. (Courtesy of Victor Chemical Works, Stauffer Chemical Company; measurements by W. H. Woodstock.)

FIG. 2-2

Partial Pressures of H2O and SO2 over Aqueous Solutions of Sulfur Dioxide* Partial pressures of H2O and SO2, mmHg, °C 0

2.00 3.00 4.00 5.00 6.00

2-77

10.6 13.5 19.4 25.6 58.4

30

40

50

60

90

120

0.12 1.68 4.62 8.07 11.8

0.19 2.53 6.80 11.7 17.0

0.29 3.69 9.71 16.5 23.8

0.43 5.24 13.5 22.7 32.6

1.21 12.9 31.7 52.2 73.7

2.82 27.0 63.9 104 145

15.7 19.8 28.3 37.1 83.7

58.6 93.2 129 165 202

88.5 139 192 245 299

129 202 277 353 430

183 285 389 496 602

275 351 542 735

407 517 796

585 741

818

22.5 28.2 40.1 52.3 117

31.4 39.2 55.3 72.0 159

42.8 53.3 74.7 96.8 212

95.8 118 164 211 454

253 393 535 679 824

342 530 720

453 700

955

*Extracted with permission from J. Chem Eng. Data 8, 1963: 333–336. Copyright 1963 American Chemical Society.

186 229 316 404 856

2-78

PHYSICAL AND CHEMICAL DATA

TABLE 2-12 °C 0 10 20 30 40 50 60 70 80 90

Water Partial Pressure, bar, over Aqueous Sulfuric Acid Solutions* Weight percent, H2SO4

10.0 .582E−02 .117E−01 .223E−01 .404E−01 .703E−01 .117 .189 .296 .449 .664

20.0 .534E−02 .107E−01 .205E−01 .373E−01 .649E−01 .109 .175 .275 .417 .617

30.0 .448E−02 .909E−02 .174E−01 .319E−01 .558E−01 .939E−01 .152 .239 .365 .542

40.0 .326E−02 .670E−02 .130E−01 .241E−01 .427E−01 .725E−01 .119 .188 .290 .434

50.0

60.0

.193E−02 .405E−02 .802E−02 .151E−01 .272E−01 .470E−01 .782E−01 .126 .196 .298

.836E−03 .180E−02 .367E−02 .710E−02 .131E−01 .232E−01 .395E−01 .651E−01 .104 .161

75.0

80.0

85.0

.207E−03 .467E−03 .995E−03 .201E−02 .387E−02 .715E−02 .127E−01 .217E−01 .360E−01 .578E−01

70.0

.747E−04 .175E−03 .388E−03 .811E−03 .162E−02 .309E−02 .565E−02 .997E−02 .170E−01 .281E−01

.197E−04 .490E−04 .115E−04 .253E−03 .531E−03 .106E−02 .204E−02 .376E−02 .668E−02 .115E−01

.343E−05 .952E−05 .245E−04 .589E−04 .133E−03 .286E−03 .584E−03 .114E−02 .213E−02 .383E−02

.905E−01 .138 .206 .301 .481 .605 .837 1.138 1.525 2.017

.452E−01 .708E−01 .108 .162 .236 .339 .478 .662 .902 1.212

.192E−01 .312E−01 .493E−01 .760E−01 .115 .170 .246 .350 .489 .673

.666E−02 .112E−01 .183E−01 .291E−01 .451E−01 .682E−01 .101 .147 .208 .291

100 110 120 130 140 150 160 170 180 190

.957 1.349 1.863 2.524 3.361 4.404 5.685 7.236 9.093 11.289

.891 1.258 1.740 2.361 3.149 4.132 5.342 6.810 8.571 10.658

.786 1.113 1.544 2.101 2.810 3.697 4.793 6.127 7.731 9.640

.634 .904 1.264 1.732 2.333 3.090 4.031 5.185 6.584 8.259

.441 .638 .903 1.253 1.708 2.289 3.021 3.930 5.045 6.397

.244 .360 .519 .734 1.020 1.392 1.870 2.475 3.233 4.169

200 210 220 230 240 250 260 270 280 290

13.861 16.841 20.264 24.160 28.561 33.494 38.984 45.055 51.726 59.015

13.107 15.951 19.225 22.960 27.188 31.939 37.240 43.116 49.590 56.681

11.887 14.505 17.529 20.992 24.927 29.364 34.334 39.865 45.984 52.715

10.245 12.576 15.287 18.414 21.992 26.056 30.642 35.784 41.514 47.865

8.020 9.948 12.217 14.864 17.929 21.452 25.472 30.030 35.168 40.926

5.312 6.696 8.354 10.322 12.641 15.351 18.496 22.121 26.274 31.003

2.632 3.395 4.331 5.466 6.831 8.458 10.382 12.640 15.269 18.311

1.606 2.101 2.714 3.467 4.381 5.480 6.788 8.333 10.142 12.242

.913 1.220 1.609 2.096 2.699 3.435 4.326 5.395 6.663 8.155

.401 .542 .724 .952 1.237 1.587 2.012 2.525 3.136 3.857

300 66.934 310 75.495 320 84.705 330 94.567 340 105.083 350 116.251

64.407 72.781 81.816 91.518 101.894 112.946

60.081 68.100 76.792 86.172 96.252 107.043

54.868 62.553 70.947 80.077 89.969 100.646

47.346 54.470 62.337 70.988 80.463 90.802

36.360 42.395 49.164 56.721 65.123 74.426

21.808 25.804 30.343 35.473 41.240 47.692

14.665 17.438 20.591 24.153 28.154 32.622

9.897 11.912 14.227 16.867 19.855 23.217

4.701 5.680 6.806 8.093 9.551 11.193

*Vermeulen, Dong, Robinson, Nguyen, and Gmitro, AIChE meeting, Anaheim, Calif., 1982; and private communication from Prof. Theodore Vermeulen, Chemical Engineering Dept., University of California, Berkeley.

VAPOR PRESSURES OF SOLUTIONS TABLE 2-12

2-79

Water Partial Pressure, bar, over Aqueous Sulfuric Acid Solutions (Concluded ) Weight percent, H2SO4

°C

90.0

92.0

94.0

96.0

97.0

98.0

98.5

99.0

99.5

100.0

0 10 20 30 40 50 60 70 80 90

.518E−06 .159E−05 .448E−05 .117E−04 .285E−04 .652E−04 .141E−03 .290E−03 .569E−03 .107E−02

.242E−06 .762E−06 .220E−05 .587E−05 .146E−04 .341E−04 .754E−04 .158E−03 .316E−03 .606E−03

.107E−06 .344E−06 .101E−05 .275E−05 .696E−05 .166E−04 .372E−04 .795E−04 .162E−03 .315E−03

.401E−07 .130E−06 .390E−06 .108E−05 .278E−05 .672E−05 .154E−04 .334E−04 .691E−04 .137E−03

.218E−07 .713E−07 .215E−06 .598E−06 .155E−05 .379E−05 .875E−05 .192E−04 .400E−04 .801E−04

.980E−08 .323E−07 .978E−07 .275E−06 .720E−06 .177E−05 .413E−05 .912E−05 .192E−04 .388E−04

.569E−08 .188E−07 .572E−07 .161E−06 .424E−06 .105E−05 .245E−05 .544E−05 .115E−04 .234E−04

.268E−08 .888E−08 .271E−07 .766E−07 .202E−06 .503E−06 .118E−05 .263E−05 .559E−05 .114E−04

.775E−09 .258E−08 .789E−08 .224E−07 .595E−07 .149E−06 .350E−06 .784E−06 .168E−05 .343E−05

.196E−09 .655E−09 .201E−08 .575E−08 .153E−07 .384E−07 .910E−07 .205E−06 .439E−06 .903E−06

100 110 120 130 140 150 160 170 180 190

.194E−02 .338E−02 .571E−02 .938E−02 .150E−01 .233E−01 .354E−01 .526E−01 .766E−01 .110

.112E−02 .198E−02 .341E−02 .569E−02 .923E−02 .146E−01 .225E−01 .340E−01 .502E−01 .729E−01

.590E−03 .107E−02 .186E−02 .315E−02 .519E−02 .832E−02 .130E−01 .199E−01 .298E−01 .438E−01

.261E−03 .479E−03 .851E−03 .146E−02 .245E−02 .399E−02 .633E−02 .983E−02 .149E−01 .222E−01

.154E−03 .285E−03 .511E−03 .886E−03 .149E−02 .245E−02 .393E−02 .614E−02 .941E−02 .141E−01

.752E−04 .141E−03 .254E−03 .445E−03 .757E−03 .125E−02 .202E−02 .319E−02 .492E−02 .744E−02

.455E−04 .855E−04 .155E−03 .278E−03 .467E−03 .776E−03 .126E−02 .199E−02 .309E−02 .469E−02

.223E−04 .420E−04 .766E−04 .135E−03 .232E−03 .387E−03 .629E−03 .999E−03 .155E−02 .236E−02

.674E−05 .128E−04 .233E−04 .414E−04 .711E−04 .119E−03 .194E−03 .309E−03 .482E−03 .735E−03

.178E−05 .339E−05 .623E−05 .111E−04 .191E−04 .321E−04 .526E−04 .840E−04 .131E−03 .201E−03

.631E−01 .894E−01 .125 .171 .232 .310 .409 .534 .689 .880

.325E−01 .467E−01 .660E−01 .918E−01 .126 .170 .227 .300 .391 .505

.208E−01 .300E−01 .427E−01 .598E−01 .825E−01 .112 .151 .200 .263 .341

.110E−01 .161E−01 .230E−01 .325E−01 .451E−01 .618E−01 .835E−01 .111 .147 .192

.698E−02 .102E−01 .147E−01 .208E−01 .290E−01 .398E−01 .540E−01 .723E−01 .957E−01 .125

.352E−02 .516E−02 .743E−02 .105E−01 .147E−01 .202E−01 .274E−01 .366E−01 .485E−01 .634E−01

.110E−02 .161E−02 .232E−02 .329E−02 .460E−02 .633E−02 .858E−02 .115E−01 .152E−01 .199E−01

.300E−03 .442E−03 .638E−03 .906E−03 .127E−02 .174E−02 .237E−02 .317E−02 .420E−02 .548E−02

.248 .316 .400 .502 .624 .770

.162 .208 .264 .331 .413 .511

.820E−01 .105 .133 .167 .208 .256

.257E−01 .328E−01 .415E−01 .520E−01 .646E−01 .795E−01

.708E−02 .905E−02 .114E−01 .143E−01 .178E−01 .218E−01

200 210 220 230 240 250 260 270 280 290

.154 .213 .290 .389 .514 .673 .870 1.112 1.407 1.763

.104 .146 .201 .273 .366 .485 .635 .822 1.052 1.335

300 310 320 330 340 350

2.190 2.696 3.292 3.990 4.801 5.738

1.676 2.088 2.578 3.159 3.843 4.641

1.112 1.394 1.732 2.133 2.608 3.164

.646 .817 1.025 1.274 1.571 1.922

.437 .556 .701 .875 1.083 1.331

2-80

PHYSICAL AND CHEMICAL DATA

TABLE 2-13

Sulfur Trioxide Partial Pressure, bar, over Aqueous Sulfuric Acid Solutions* Weight percent, H2SO4

°C

10.0

20.0

30.0

40.0

50.0

60.0

70.0

75.0

80.0

85.0

0 10 20 30 40 50 60 70 80 90

.644E−29 .149E−27 .278E−26 .426E−25 .549E−24 .602E−23 .573E−22 .477E−21 .352E−20 .233E−19

.103E−27 .223E−26 .394E−25 .577E−24 .714E−23 .757E−22 .699E−21 .567E−20 .410E−19 .266E−18

.205E−26 .395E−25 .626E−24 .832E−23 .941E−22 .921E−21 .789E−20 .599E−19 .408E−18 .250E−17

.688E−25 .113E−23 .156E−22 .181E−21 .181E−20 .158E−19 .122E−18 .843E−18 .524E−17 .296E−16

.368E−23 .522E−22 .621E−21 .630E−20 .555E−19 .429E−18 .294E−17 .181E−16 .101E−15 .516E−15

.341E−21 .415E−20 .426E−19 .376E−18 .288E−17 .195E−16 .118E−15 .643E−15 .319E−14 .145E−13

.784E−19 .796E−18 .685E−17 .509E−16 .331E−15 .191E−14 .985E−14 .461E−13 .197E−12 .775E−12

.174E−17 .158E−16 .121E−15 .808E−15 .473E−14 .246E−13 .116E−12 .492E−12 .192E−11 .693E−11

.531E−16 .417E−15 .280E−14 .164E−13 .851E−13 .395E−12 .165E−11 .634E−11 .223E−10 .731E−10

.229E−14 .141E−13 .767E−13 .371E−12 .162E−11 .643E−11 .234E−10 .791E−10 .249E−09 .734E−09

100 110 120 130 140 150 160 170 180 190

.139E−18 .756E−18 .377E−17 .174E−16 .743E−16 .297E−15 .111E−14 .393E−14 .131E−13 .415E−13

.157E−17 .844E−17 .418E−16 .191E−15 .815E−15 .325E−14 .122E−13 .430E−13 .144E−12 .458E−12

.140E−16 .719E−16 .340E−15 .150E−14 .615E−14 .237E−13 .862E−13 .296E−12 .967E−12 .301E−11

.153E−15 .730E−15 .323E−14 .133E−13 .517E−13 .188E−12 .649E−12 .212E−11 .622E−11 .197E−10

.242E−14 .105E−13 .424E−13 .160E−12 .569E−12 .191E−11 .608E−11 .184E−10 .532E−10 .147E−09

.606E−13 .236E−12 .858E−12 .293E−11 .943E−11 .287E−10 .833E−10 .231E−09 .610E−09 .155E−08

.283E−11 .961E−11 .307E−10 .922E−10 .262E−09 .710E−09 .183E−08 .453E−08 .107E−07 .246E−07

.232E−10 .729E−10 .215E−09 .601E−09 .159E−08 .403E−08 .974E−08 .226E−07 .505E−07 .109E−06

.223E−09 .641E−09 .174E−08 .446E−08 .109E−07 .256E−07 .575E−07 .125E−06 .260E−06 .527E−06

.204E−08 .538E−08 .135E−07 .324E−07 .745E−07 .165E−06 .351E−06 .725E−06 .145E−05 .282E−05

200 210 220 230 240 250 260 270 280 290

.125E−12 .362E−12 .100E−11 .265E−11 .678E−11 .167E−10 .399E−10 .920E−10 .206E−09 .449E−09

.139E−11 .404E−11 .112E−10 .301E−10 .777E−10 .193E−09 .466E−09 .109E−08 .247E−08 .545E−08

.893E−11 .254E−10 .695E−10 .183E−09 .465E−09 .114E−08 .272E−08 .628E−08 .141E−07 .308E−07

.561E−10 .154E−09 .405E−09 .103E−08 .253E−08 .602E−08 .139E−07 .312E−07 .683E−07 .145E−06

.391E−09 .100E−08 .246E−08 .587E−08 .135E−07 .303E−07 .660E−07 .140E−06 .288E−06 .580E−06

.379E−08 .894E−08 .204E−07 .450E−07 .965E−07 .201E−06 .408E−06 .807E−06 .156E−05 .295E−05

.542E−07 .116E−06 .240E−06 .482E−06 .944E−06 .180E−05 .336E−05 .612E−05 .109E−04 .191E−04

.228E−06 .462E−06 .911E−06 .175E−05 .328E−05 .600E−05 .108E−04 .189E−04 .326E−04 .553E−04

.103E−05 .198E−05 .368E−05 .668E−05 .119E−04 .206E−04 .352E−04 .590E−04 .973E−04 .158E−03

.534E−05 .986E−05 .178E−04 .314E−04 .543E−04 .923E−04 .154E−03 .253E−03 .408E−03 .649E−03

300 310 320 330 340 350

.953E−09 .197E−08 .397E−08 .782E−08 .151E−07 .285E−07

.117E−07 .245E−07 .502E−07 .100E−06 .196E−06 .376E−06

.657E−07 .136E−06 .277E−06 .551E−06 .107E−05 .204E−05

.302E−06 .614E−06 .122E−05 .237E−05 .452E−05 .846E−05

.114E−05 .220E−05 .414E−05 .766E−05 .139E−04 .246E−04

.546E−05 .990E−05 .176E−04 .308E−04 .529E−04 .893E−04

.329E−04 .556E−04 .923E−04 .151E−03 .243E−03 .387E−03

.921E−04 .151E−03 .245E−03 .391E−03 .617E−03 .963E−03

.253E−03 .398E−03 .621E−03 .956E−03 .145E−02 .219E−02

.102E−02 .158E−02 .242E−02 .367E−02 .550E−02 .815E−02

*Vermeulen, Dong, Robinson, Nguyen, and Gmitro, AIChE meeting, Anaheim, Calif., 1982; and private communication from Prof. Theodore Vermeulen, Chemical Engineering Dept., University of California, Berkeley.

VAPOR PRESSURES OF SOLUTIONS TABLE 2-13

2-81

Sulfur Trioxide Partial Pressure, bar, over Aqueous Sulfuric Acid Solutions (Concluded ) Weight percent, H2SO4

°C

90.0

92.0

94.0

96.0

97.0

98.0

98.5

99.0

99.5

0 10 20 30 40 50 60 70 80 90

.671E−13 .345E−12 .159E−11 .664E−11 .254E−10 .897E−10 .294E−09 .904E−09 .261E−08 .712E−08

.216E−12 .107E−11 .475E−11 .192E−10 .709E−10 .242E−09 .771E−09 .230E−08 .643E−08 .171E−07

.677E−12 .326E−11 .141E−10 .557E−10 .201E−09 .669E−09 .207E−08 .602E−08 .165E−07 .426E−07

.240E−11 .114E−10 .482E−10 .186E−09 .655E−09 .214E−08 .647E−08 .184E−07 .492E−07 .124E−06

.500E−11 .234E−10 .986E−10 .376E−09 .131E−08 .424E−08 .127E−07 .357E−07 .946E−07 .237E−06

.124E−10 .578E−10 .241E−09 .911E−09 .315E−08 .101E−07 .299E−07 .833E−07 .218E−06 .541E−06

.224E−10 .104E−09 .433E−09 .163E−08 .562E−08 .179E−07 .528E−07 .146E−06 .381E−06 .940E−06

.502E−10 .232E−09 .961E−09 .360E−08 .123E−07 .391E−07 .115E−06 .316E−06 .820E−06 .201E−05

.182E−09 .839E−09 .346E−08 .129E−07 .440E−07 .139E−06 .405E−06 .111E−05 .286E−05 .698E−05

.755E−09 .347E−08 .142E−07 .528E−07 .179E−06 .560E−06 .163E−05 .444E−05 .114E−04 .276E−04

100 110 120 130 140 150 160 170 180 190

.184E−07 .456E−07 .108E−06 .244E−06 .533E−06 .112E−05 .229E−05 .453E−05 .870E−05 .163E−04

.430E−07 .103E−06 .238E−06 .526E−06 .112E−05 .230E−05 .459E−05 .886E−05 .166E−04 .304E−04

.105E−06 .247E−06 .555E−06 .120E−05 .250E−05 .504E−05 .983E−05 .186E−04 .343E−04 .615E−04

.300E−06 .689E−06 .152E−05 .321E−05 .656E−05 .129E−04 .247E−04 .459E−04 .829E−04 .146E−03

.565E−06 .128E−05 .280E−05 .586E−05 .118E−04 .231E−04 .438E−04 .806E−04 .144E−03 .252E−03

.127E−05 .287E−05 .619E−05 .128E−04 .257E−04 .497E−04 .932E−04 .170E−03 .301E−03 .520E−03

.220E−05 .494E−05 .106E−04 .219E−04 .435E−04 .837E−04 .156E−03 .283E−03 .499E−03 .859E−03

.470E−05 .105E−04 .224E−04 .459E−04 .910E−04 .174E−03 .324E−03 .586E−03 .103E−02 .177E−02

.162E−04 .359E−04 .764E−04 .156E−03 .308E−03 .588E−03 .109E−02 .196E−02 .343E−02 .587E−02

.638E−04 .141E−03 .298E−03 .606E−03 .119E−02 .226E−02 .416E−02 .746E−02 .130E−01 .222E−01

200 210 220 230 240 250 260 270 280 290

.297E−04 .528E−04 .919E−04 .157E−03 .261E−03 .428E−03 .690E−03 .109E−02 .170E−02 .261E−02

.543E−04 .946E−04 .161E−03 .269E−03 .441E−03 .708E−03 .112E−02 .174E−02 .266E−02 .401E−02

.108E−03 .185E−03 .309E−03 .508E−03 .819E−03 .130E−02 .202E−02 .309E−02 .466E−02 .694E−02

.251E−03 .422E−03 .694E−03 .112E−02 .178E−02 .276E−02 .423E−02 .638E−02 .948E−02 .139E−01

.429E−03 .714E−03 .117E−02 .187E−02 .293E−02 .453E−02 .688E−02 .103E−01 .152E−01 .221E−01

.878E−03 .145E−02 .235E−02 .373E−02 .582E−02 .891E−02 .134E−01 .200E−01 .293E−01 .423E−01

.144E−02 .237E−02 .383E−02 .605E−02 .939E−02 .143E−01 .215E−01 .319E−01 .465E−01 .670E−01

.296E−02 .486E−02 .781E−02 .123E−01 .191E−01 .291E−01 .437E−01 .646E−01 .943E−01 .136

.981E−02 .161E−01 .258E−01 .405E−01 .627E−01 .955E−01 .143 .212 .309 .444

.370E−01 .603E−01 .965E−01 .152 .234 .356 .532 .786 1.144 1.646

300 310 320 330 340 350

.395E−02 .589E−02 .868E−02 .126E−01 .181E−01 .258E−01

.595E−02 .873E−02 .126E−01 .181E−01 .255E−01 .357E−01

.102E−01 .148E−01 .211E−01 .299E−01 .418E−01 .578E−01

.201E−01 .287E−01 .405E−01 .565E−01 .780E−01 .107

.318E−01 .451E−01 .632E−01 .877E−01 .120 .164

.604E−01 .852E−01 .119 .164 .224 .303

.953E−01 .134 .186 .256 .348 .470

.193 .272 .378 .520 .708 .956

.632 .889 1.236 1.703 2.323 3.142

100.0

2.339 3.289 4.575 6.303 8.603 11.640

TABLE 2-14

Sulfuric Acid Partial Pressure, bar, over Aqueous Sulfuric Acid* Weight Percent, H2SO4

°C

10.0

20.0

30.0

40.0

50.0

60.0

70.0

75.0

80.0

85.0

0 10 20 30 40 50 60 70 80 90

.576E−21 .634E−20 .588E−19 .468E−18 .324E−17 .197E−16 .107E−15 .526E−15 .235E−14 .960E−14

.843E−20 .874E−19 .769E−18 .584E−17 .389E−16 .229E−15 .121E−14 .581E−14 .254E−13 .102E−12

.141E−18 .131E−17 .104E−16 .721E−16 .441E−15 .241E−14 .119E−13 .535E−13 .221E−12 .844E−12

.344E−17 .276E−16 .193E−15 .119E−14 .649E−14 .320E−13 .144E−12 .592E−12 .225E−11 .798E−11

.109E−15 .769E−15 .474E−14 .259E−13 .127E−12 .562E−12 .228E−11 .851E−11 .295E−10 .956E−10

.438E−14 .273E−13 .149E−12 .725E−12 .317E−11 .126E−10 .462E−10 .156E−09 .492E−09 .145E−08

.249E−12 .135E−11 .649E−11 .278E−10 .108E−09 .380E−09 .124E−08 .373E−08 .105E−07 .279E−07

.200E−11 .101E−10 .447E−10 .178E−09 .643E−09 .212E−08 .646E−08 .183E−07 .485E−07 .121E−06

.161E−10 .743E−10 .305E−09 .113E−08 .379E−08 .117E−07 .334E−07 .888E−07 .222E−06 .522E−06

.121E−09 .490E−09 .179E−08 .594E−08 .181E−07 .513E−07 .135E−06 .336E−06 .786E−06 .175E−05

100 110 120 130 140 150 160 170 180 190

.353E−13 .127E−12 .418E−12 .129E−11 .375E−11 .103E−10 .272E−10 .682E−10 .164E−09 .378E−09

.381E−12 .132E−11 .432E−11 .132E−10 .385E−10 .106E−09 .279E−09 .702E−09 .170E−08 .394E−08

.300E−11 .997E−11 .312E−10 .924E−10 .259E−09 .694E−09 .178E−08 .436E−08 .103E−07 .234E−07

.264E−10 .824E−10 .243E−09 .678E−09 .181E−08 .460E−08 .112E−07 .264E−07 .599E−07 .131E−06

.291E−09 .835E−09 .227E−08 .589E−08 .146E−07 .346E−07 .789E−07 .174E−06 .369E−06 .760E−06

.402E−08 .106E−07 .264E−07 .631E−07 .144E−06 .316E−06 .670E−06 .137E−05 .271E−05 .521E−05

.698E−07 .166E−06 .375E−06 .814E−06 .169E−05 .340E−05 .659E−05 .124E−04 .225E−04 .400E−04

.287E−06 .644E−06 .138E−05 .285E−05 .565E−05 .108E−04 .200E−04 .359E−04 .627E−04 .107E−03

.117E−05 .249E−05 .508E−05 .995E−05 .188E−04 .343E−04 .608E−04 .104E−03 .175E−03 .286E−03

.371E−05 .752E−05 .147E−04 .277E−04 .503E−04 .889E−04 .152E−03 .255E−03 .416E−03 .663E−03

200 210 220 230 240 250 260 270 280 290

.842E−09 .181E−08 .376E−08 .758E−08 .148E−07 .283E−07 .526E−07 .954E−07 .169E−06 .294E−06

.883E−08 .191E−07 .401E−07 .817E−07 .162E−06 .312E−06 .588E−06 .108E−05 .194E−05 .342E−05

.514E−07 .109E−06 .226E−06 .455E−06 .889E−06 .170E−05 .316E−05 .577E−05 .103E−04 .180E−04

.278E−06 .573E−06 .115E−05 .224E−05 .427E−05 .793E−05 .144E−04 .257E−04 .450E−04 .771E−04

.152E−05 .295E−05 .559E−05 .103E−04 .186E−04 .329E−04 .569E−04 .965E−04 .161E−03 .263E−03

.975E−05 .178E−04 .316E−04 .549E−04 .935E−04 .156E−03 .255E−03 .411E−03 .650E−03 .101E−02

.691E−04 .117E−03 .193E−03 .311E−03 .494E−03 .770E−03 .118E−02 .178E−02 .265E−02 .389E−02

.177E−03 .288E−03 .459E−03 .717E−03 .110E−02 .166E−02 .247E−02 .362E−02 .524E−02 .750E−02

.457E−03 .715E−03 .110E−02 .166E−02 .245E−02 .358E−02 .516E−02 .733E−02 .103E−01 .143E−01

.104E−02 .159E−02 .239E−02 .354E−02 .515E−02 .740E−02 .105E−01 .147E−01 .203E−01 .278E−01

300 310 320 330 340 350

.500E−06 .834E−06 .137E−05 .220E−05 .349E−05 .544E−05

.591E−05 .100E−04 .167E−04 .273E−04 .440E−04 .698E−04

.309E−04 .522E−04 .865E−04 .141E−03 .227E−03 .360E−03

.130E−03 .215E−03 .352E−03 .565E−03 .895E−03 .140E−02

.424E−03 .672E−03 .105E−02 .162E−02 .246E−02 .369E−02

.156E−02 .236E−02 .352E−02 .519E−02 .757E−02 .109E−01

.563E−02 .805E−02 .114E−01 .159E−01 .221E−01 .303E−01

.106E−01 .148E−01 .205E−01 .281E−01 .382E−01 .516E−01

.196E−01 .266E−01 .359E−01 .480E−01 .636E−01 .836E−01

.376E−01 .504E−01 .670E−01 .883E−01 .116 .150

°C

90.0

92.0

94.0

96.0

97.0

98.0

98.5

99.0

99.5

100.0

0 10 20 30 40 50 60 70 80 90

.534E−09 .200E−08 .677E−08 .211E−07 .607E−07 .163E−06 .411E−06 .976E−06 .220E−05 .473E−05

.803E−09 .296E−08 .993E−08 .306E−07 .870E−07 .231E−06 .575E−06 .135E−05 .302E−05 .642E−05

.112E−08 .409E−08 .136E−07 .415E−07 .117E−06 .309E−06 .765E−06 .179E−05 .396E−05 .835E−05

.148E−08 .540E−08 .179E−07 .543E−07 .153E−06 .400E−06 .985E−06 .229E−05 .504E−05 .106E−04

.167E−08 .609E−08 .201E−07 .611E−07 .171E−06 .449E−06 .110E−05 .256E−05 .562E−05 .118E−04

.187E−08 .679E−08 .224E−07 .680E−07 .191E−06 .498E−06 .122E−05 .283E−05 .622E−05 .130E−04

.196E−08 .714E−08 .236E−07 .714E−07 .200E−06 .523E−06 .128E−05 .297E−05 .652E−05 .136E−04

.206E−08 .750E−08 .247E−07 .749E−07 .210E−06 .548E−06 .134E−05 .310E−05 .681E−05 .143E−04

.217E−08 .788E−08 .260E−07 .786E−07 .220E−06 .574E−06 .140E−05 .325E−05 .712E−05 .149E−04

.228E−08 .827E−08 .273E−07 .824E−07 .230E−06 .600E−06 .147E−05 .339E−05 .743E−05 .155E−04

100 110 120 130 140 150 160 170 180 190

.973E−05 .192E−04 .366E−04 .672E−04 .120E−03 .207E−03 .348E−03 .572E−03 .917E−03 .144E−02

.131E−04 .256E−04 .482E−04 .879E−04 .155E−03 .266E−03 .444E−03 .723E−03 .115E−02 .179E−02

.169E−04 .328E−04 .614E−04 .111E−03 .195E−03 .332E−03 .550E−03 .889E−03 .140E−02 .217E−02

.213E−04 .412E−04 .767E−04 .138E−03 .241E−03 .408E−03 .673E−03 .108E−02 .170E−02 .262E−02

.237E−04 .457E−04 .849E−04 .153E−03 .266E−03 .449E−03 .740E−03 .119E−02 .186E−02 .286E−02

.261E−04 .503E−04 .935E−04 .168E−03 .292E−03 .493E−03 .810E−03 .130E−02 .204E−02 .312E−02

.274E−04 .527E−04 .977E−04 .175E−03 .304E−03 .514E−03 .844E−03 .135E−02 .212E−02 .325E−02

.285E−04 .549E−04 .102E−03 .182E−03 .316E−03 .534E−03 .876E−03 .140E−02 .220E−02 .336E−02

.298E−04 .572E−04 .106E−03 .190E−03 .329E−03 .554E−03 .909E−03 .145E−02 .227E−02 .348E−02

.310E−04 .595E−04 .110E−03 .197E−03 .341E−03 .574E−03 .941E−03 .150E−02 .235E−02 .359E−02

200 210 220 230 240 250 260 270 280 290

.221E−02 .333E−02 .494E−02 .719E−02 .103E−01 .146E−01 .203E−01 .279E−01 .380E−01 .510E−01

.273E−02 .408E−02 .601E−02 .869E−02 .124E−01 .174E−01 .240E−01 .329E−01 .444E−01 .592E−01

.329E−02 .490E−02 .715E−02 .103E−01 .146E−01 .203E−01 .279E−01 .380E−01 .510E−01 .676E−01

.395E−02 .585E−02 .850E−02 .122E−01 .171E−01 .238E−01 .326E−01 .441E−01 .589E−01 .778E−01

.431E−02 .637E−02 .924E−02 .132E−01 .186E−01 .257E−01 .352E−01 .475E−01 .633E−01 .835E−01

.470E−02 .693E−02 .100E−01 .143E−01 .201E−01 .278E−01 .380E−01 .513E−01 .683E−01 .900E−01

.488E−02 .720E−02 .104E−01 .149E−01 .209E−01 .289E−01 .394E−01 .531E−01 .706E−01 .930E−01

.505E−02 .744E−02 .108E−01 .153E−01 .215E−01 .297E−01 .405E−01 .545E−01 .725E−01 .954E−01

.522E−02 .768E−02 .111E−01 .158E−01 .221E−01 .305E−01 .416E−01 .560E−01 .744E−01 .978E−01

.538E−02 .791E−02 .114E−01 .162E−01 .227E−01 .314E−01 .427E−01 .574E−01 .762E−01 .100

300 310 320 330 340 350

.678E−01 .892E−01 .116 .150 .192 .243

.782E−01 .102 .132 .170 .216 .272

.888E−01 .115 .149 .190 .240 .301

.102 .132 .169 .214 .270 .337

.109 .141 .180 .228 .287 .358

.117 .151 .193 .245 .307 .383

.121 .156 .199 .252 .317 .394

.124 .160 .204 .258 .328 .402

.127 .164 .209 .263 .330 .410

.130 .167 .213 .269 .386 .417

Weight percent, H2SO4

*Vermeulen, Dong, Robinson, Nguyen, and Gmitro, AIChE meeting, Anaheim, CA, 1982; and private communication from Prof. Theodore Vermeulen, Chemical Engineering Dept., University of California, Berkeley. 2-82

TABLE 2-15

Total Pressure, bar, of Aqueous Sulfuric Acid Solutions* Weight percent, H2SO4

°C 0 10 20 30 40 50 60 70 80 90

10.0 .582E−02 .117E−01 .223E−01 .404E−01 .703E−01 .117 .189 .296 .449 .664

20.0 .534E−02 .107E−01 .205E−01 .373E−01 .649E−01 .109 .175 .275 .417 .617

30.0 .448E−02 .909E−02 .174E−01 .319E−01 .558E−01 .939E−01 .152 .239 .365 .542

40.0 .326E−02 .670E−02 .130E−01 .241E−01 .427E−01 .725E−01 .119 .188 .290 .434

50.0 .193E−02 .405E−02 .802E−02 .151E−01 .272E−01 .470E−01 .782E−01 .126 .196 .298

60.0 .836E−03 .180E−02 .367E−02 .710E−02 .131E−01 .232E−01 .395E−01 .651E−01 .104 .161

70.0

75.0

80.0

85.0

.207E−03 .467E−03 .995E−03 .201E−02 .387E−02 .715E−02 .127E−01 .217E−01 .360E−01 .578E−01

.747E−04 .175E−03 .388E−03 .811E−03 .162E−02 .309E−02 .565E−02 .997E−01 .170E−01 .281E−01

.197E−04 .490E−04 .115E−03 .253E−03 .531E−03 .106E−02 .204E−02 .376E−02 .668E−02 .115E−01

.343E−05 .952E−05 .245E−04 .589E−04 .134E−03 .286E−03 .584E−03 .114E−02 .213E−02 .383E−02

.905E−01 .138 .206 .301 .431 .605 .837 1.138 1.525 2.017

.452E−01 .708E−01 .108 .162 .236 .339 .478 .662 .902 1.212

.192E−01 .312E−01 .493E−01 .760E−01 .115 .170 .246 .350 .489 .673

.666E−02 .112E−01 .183E−01 .291E−01 .451E−01 .683E−01 .101 .147 .209 .292

100 110 120 130 140 150 160 170 180 190

.957 1.349 1.863 2.524 3.361 4.404 5.685 7.236 9.093 11.289

.891 1.258 1.740 2.361 3.149 4.132 5.342 6.810 8.571 10.658

.786 1.113 1.544 2.101 2.810 3.697 4.793 6.127 7.731 9.640

.634 .904 1.264 1.732 2.333 3.090 4.031 5.185 6.584 8.259

.441 .638 .903 1.253 1.708 2.289 3.021 3.930 5.045 6.397

.244 .360 .519 .734 1.020 1.392 1.870 2.475 3.233 4.169

200 210 220 230 240 250 260 270 280 290

13.861 16.841 20.264 24.160 28.561 33.494 38.984 45.055 51.726 59.015

13.107 15.951 19.225 22.960 27.188 31.939 37.240 43.116 49.590 56.681

11.887 14.505 17.529 20.992 24.927 29.364 34.334 39.865 45.984 52.715

10.245 12.576 15.287 18.414 21.992 26.056 30.642 35.784 41.514 47.866

8.020 9.948 12.217 14.864 17.929 21.452 25.472 30.030 35.168 40.926

5.312 6.696 8.354 10.322 12.641 15.351 18.496 22.122 26.275 31.004

2.633 3.396 4.331 5.466 6.832 8.459 10.384 12.642 15.272 18.315

1.606 2.101 2.715 3.468 4.382 5.481 6.791 8.337 10.147 12.250

.913 1.221 1.610 2.098 2.701 3.439 4.332 5.402 6.673 8.170

.402 .544 .726 .956 1.242 1.594 2.023 2.540 3.157 3.886

300 310 320 330 340 350

66.934 75.495 84.705 94.567 105.083 116.251

64.407 72.781 81.816 91.518 101.894 112.947

60.081 68.101 76.792 86.172 96.252 107.043

54.869 62.553 70.947 80.078 89.970 100.647

47.347 54.470 62.338 70.990 80.466 90.806

36.361 42.398 49.168 56.727 65.130 74.437

21.814 25.812 30.355 35.489 41.262 47.723

14.675 17.453 20.611 24.182 28.193 32.674

9.916 11.939 14.264 16.916 19.920 23.303

4.740 5.732 6.876 8.185 9.672 11.351

Weight percent, H2SO4 °C

90.0

92.0

94.0

96.0

97.0

98.0

98.5

99.0

99.5

100.0

0 10 20 30 40 50 60 70 80 90

.518E−06 .159E−05 .449E−05 .117E−04 .385E−04 .653E−04 .141E−03 .291E−03 .571E−03 .107E−02

.243E−06 .765E−06 .221E−05 .590E−05 .147E−04 .344E−04 .759E−04 .159E−03 .319E−03 .612E−03

.109E−06 .348E−06 .102E−05 .279E−05 .708E−05 .169E−04 .380E−04 .813E−04 .166E−03 .324E−03

.416E−07 .136E−06 .407E−06 .113E−05 .293E−05 .712E−05 .164E−04 .357E−04 .742E−04 .148E−03

.235E−07 .774E−07 .235E−06 .659E−06 .173E−05 .425E−05 .987E−05 .218E−04 .458E−04 .921E−04

.117E−07 .391E−07 .121E−06 .344E−06 .914E−06 .228E−05 .538E−05 .120E−04 .257E−04 .524E−04

.768E−08 .261E−07 .812E−07 .234E−06 .630E−06 .159E−05 .379E−05 .856E−05 .184E−04 .390E−04

.479E−08 .166E−07 .528E−07 .155E−06 .425E−06 .109E−05 .264E−05 .605E−05 .132E−04 .277E−04

.313E−08 .113E−07 .373E−07 .114E−06 .323E−06 .861E−06 .216E−05 .514E−05 .117E−04 .253E−04

.323E−08 .124E−07 .435E−07 .141E−06 .425E−06 .120E−05 .319E−05 .804E−05 .193E−04 .441E−04

100 110 120 130 140 150 160 170 180 190

.195E−02 .340E−02 .575E−02 .944E−02 .151E−01 .235E−01 .357E−01 .532E−01 .775E−01 .111

.113E−02 .201E−02 .346E−02 .578E−02 .939E−02 .149E−01 .230E−01 .347E−01 .514E−01 .747E−01

.607E−03 .110E−02 .192E−02 .327E−02 .539E−02 .866E−02 .136E−01 .208E−01 .312E−01 .460E−01

.283E−03 .521E−03 .929E−03 .161E−02 .270E−02 .441E−02 .703E−02 .110E−01 .167E−01 .250E−01

.178E−03 .332E−03 .598E−03 .104E−02 .177E−02 .293E−02 .471E−02 .741E−02 .114E−01 .172E−01

.103E−03 .194E−03 .354E−03 .626E−03 .107E−02 .180E−02 .293E−02 .466E−02 .726E−02 .111E−01

.751E−04 .143E−03 .263E−03 .470E−03 .815E−03 .137E−02 .226E−02 .363E−02 .571E−02 .880E−02

.555E−04 .107E−03 .201E−03 .363E−03 .639E−03 .109E−02 .183E−02 .299E−02 .478E−02 .749E−02

.527E−04 .106E−03 .206E−03 .387E−03 .708E−03 .126E−02 .219E−02 .372E−02 .619E−02 .101E−01

.966E−04 .204E−03 .414E−03 .314E−03 .155E−02 .287E−02 .516E−02 .905E−02 .155E−01 .260E−01

.665E−01 .944E−01 .132 .182 .247 .331 .439 .575 .744 .954

.367E−01 .530E−01 .752E−01 .105 .145 .197 .264 .351 .460 .597

.255E−01 .371E−01 .531E−01 .749E−01 .104 .143 .193 .258 .341 .446

.166E−01 .245E−01 .354E−01 .505E−01 .710E−01 .985E−01 .135 .153 .245 .324

.133E−01 .198E−01 .289E−01 .417E−01 .592E−01 .830E−01 .115 .157 .213 .285

.115E−01 .175E−01 .260E−01 .382E−01 .553E−01 .790E−01 .112 .156 .215 .295

.161E−01 .253E−01 .392E−01 .596E−01 .895E−01 .132 .193 .279 .398 .562

.427E−01 .687E−01 .109 .169 .258 .389 .577 .846 1.225 1.751

200 210 220 230 240 250 260 270 280 290

.156 .216 .295 .396 .525 .688 .881 1.141 1.447 1.817

.107 .150 .207 .282 .379 .503 .660 .856 1.099 1.398

300 310 320 330 340 350

2.261 2.791 3.417 4.153 5.011 6.006

1.761 2.199 2.723 3.347 4.084 4.949

1.211 1.524 1.901 2.353 2.889 3.523

.767 .977 1.234 1.545 1.919 2.366

.578 .742 .944 1.191 1.491 1.852

.425 .553 .713 .911 1.156 1.456

.379 .498 .649 .840 1.078 1.374

.399 .536 .714 .944 1.239 1.614

.785 1.085 1.486 2.018 2.718 3.631

2.476 3.465 4.800 6.586 8.957 12.079

*Vermeulen, Dong, Robinson, Nguyen, and Gmitro, AIChE meeting, Anaheim, Calif., 1982; and private communication from Prof. Theodore Vermeulen, Chemical Engineering Dept., University of California, Berkeley. 2-83

2-84

PHYSICAL AND CHEMICAL DATA

TABLE 2-16

Partial Pressures of HNO3 and H2O over Aqueous Solutions of HNO3 mmHg Percentages are weight % HNO3 in solution. 20%

°C

25%

HNO3

H2O

HNO3

30% H2O

HNO3

35% H2O

0 5 10 15 20

4.1 5.7 8.0 10.9 15.2

3.8 5.4 7.6 10.3 14.2

3.6 5.0 7.1 9.7 13.2

25 30 35 40 45

20.6 27.6 36.5 47.5 62

19.2 25.7 33.8 44 57.5

17.8 23.8 31.1 41 53

0.09

0.11 .17

40%

HNO3

H2O

45%

HNO3

H2O

3.3 4.6 6.5 8.9 12.0 16.2 21.7 28.3 37.7 48

0.09 .13 .20 .28

0.12 .17 .25 .36 .52

50%

HNO3

H2O

HNO3

3.0 4.2 5.8 8.0 10.8

H2O

0.10 .15

2.6 3.6 5.0 6.9 9.4

0.12 .18 .27

2.1 3.0 4.2 5.8 7.9

14.6 19.5 25.5 33.5 43

.23 .33 .48 .68 .96

12.7 16.9 22.3 29.3 38.0

.39 .56 .80 1.13 1.57

10.7 14.4 19.0 25.0 32.5

49.5 62.5 80 100 126

2.18 2.95 4.05 5.46 7.25

42.5 54 70 88 110

50 55 60 65 70

0.09 .13 .19 .27

80 100 128 162 200

.13 .18 .28 .40 .54

75 94 121 151 187

.25 .35 .51 .71 1.00

69 87 113 140 174

.42 .59 .85 1.18 1.63

63 79 102 127 159

.75 1.04 1.48 2.05 2.80

56 71 90 114 143

1.35 1.83 2.54 3.47 4.65

75 80 85 90 95

.38 .53 .74 1.01 1.37

250 307 378 458 555

.77 1.05 1.44 1.95 2.62

234 287 352 426 517

1.38 1.87 2.53 3.38 4.53

217 267 325 393 478

2.26 3.07 4.15 5.50 7.32

198 243 297 359 436

3.80 5.10 6.83 9.0 11.7

178 218 268 325 394

6.20 8.15 10.7 13.7 17.8

158 195 240 292 355

9.6 12.5 16.3 20.9 26.8

138 170 211 258 315

100 105 110 115 120

1.87 2.50

675 800

3.50 4.65

628 745

6.05 7.90

580 690

530 631 755

15.5 20.0 25.7 32.5

480 573 688 810

23.0 29.2 37.0 46

430 520 625 740

34.2 43.0 54.5 67 84

383 463 560 665 785

°C

HNO3

H2O

HNO3

H2O

HNO3

H2O

HNO3

H2O

0 5 10 15 20

0.14 .21 .31 .45

1.8 2.5 3.5 4.9 6.7

0.19 .28 .41 .59 .84

1.5 2.1 3.0 4.1 5.6

0.41 .60 .86 1.21 1.68

1.3 1.8 2.6 3.5 4.9

0.79 1.12 1.58 2.18 3.00

1.1 1.6 2.2 3.0 4.1

25 30 35 40 45

.66 .93 1.30 1.82 2.50

9.1 12.2 16.1 21.3 28.0

1.21 1.66 2.28 3.10 4.20

7.7 10.3 13.6 18.1 23.7

2.32 3.17 4.26 5.70 7.55

6.6 8.8 11.6 15.5 20.0

4.10 5.50 7.30 9.65 12.6

5.5 7.4 9.8 12.8 16.7

50 55 60 65 70

3.41 4.54 6.15 8.18 10.7

36.3 46 60 76 95

5.68 7.45 9.9 13.0 16.8

31 39 51 64 81

10.0 12.8 16.8 21.7 27.5

26.0 33.0 43.0 54.5 68

16.5 21.0 27.1 34.5 43.3

21.8 27.3 35.3 44.5 56

75 80 85 90 95

13.9 18.0 23.0 29.4 37.3

120 148 182 223 272

21.8 27.5 34.8 43.7 55.0

102 126 156 192 233

35.0 43.5 54.5 67.5 83.5

100 105 110 115 120 125

47 58.5 73 90 110

331 400 485 575 685

69.5 84.5 103 126 156 187

285 345 417 495 590 700

55%

60%

9.7 12.7 16.5

65%

103 124 152 181 218 260

70%

80% HNO3

90%

100%

H2O

HNO3

H2O

HNO3

2 3 4 6 8

1.2 1.7 2.4

5.5 8 11 15 20

10.5 14 18.5 24.5 32

3.2 4 5.5 7 9.5

27 36 47 62 80

1 1.3 1.8 2.4 3

57 77 102 133 170

11 15 22 30 42

41 52 67 85 106

12 15 20 25 31

103 127 157 192 232

4 5 6.5 8 10

215 262 320 385 460 540 625 720 820

86 106 131 160 195

54.5 67.5 83 103 125

70 86 107 130 158

130 158 192 230 278

38 48 60 73 89

282 338 405 480 570

13 16 20 24 29

238 288 345 410 490 580

152 183 221 262 312 372

192 231 278 330 393 469

330 392 465 545 640

108 129 155 185 219

675 790

35 42

VAPOR PRESSURES OF SOLUTIONS TABLE 2-17 Partial Pressures of H2O and HBr over Aqueous Solutions of HBr at 20 to 55°C mmHg 20°C

% HBr

HBr

32 34 36 38 40 42 44 46 48 50 52 54 56 58 60

25°C H2O

0.09 .23 .71 2.2 6.8 21

6.2 4.5 3.3 2.4 1.7 1.3

HBr

50°C H2O

0.0016 .0022 .0033 .0061 .011 .023 .048 .10 .13 .37 1.1 3.2 9.3 27

HBr

8.2 6.1 4.5 3.3 2.4 1.9

1.3 3.2 7.2 17 40 91

TABLE 2-20 Total Vapor Pressures of Aqueous Solutions of CH3COOH Percentages of weight % acetic acid in the solution mmHg

55°C

H2O

HBr

30.2 24.3 19.3 16.0 13.3 10.4

H2O

2.0 4.6 10.2 23.0 51 115 260

2-85

°C

25%

50%

75%

20 25 30 35 40

16.3 22.1 29.6 39.4 51.7

15.7 21.4 28.8 38.3 50.2

15.3 20.8 27.8 36.6 48.1

45 50 55 60 65

67.0 87.2 110 141 178

65.0 85.0 107 138 172

62.0 80.1 102 130 162

70 75 80 85 90

223 277 342 419 510

216 269 331 407 497

203 251 310 376 458

95 100

618 743

602 725

550 666

38 31 25 21 18 14 11.4

TABLE 2-18 Partial Pressures of HI over Aqueous Solutions of HI at 25°C mmHg %HI pHI

4 0.00064

46 0.0010

48 0.0022

50 0.0050

52 0.013

54 0.035

56 0.10

TABLE 2-19 Vapor Pressures of the System: Water-Sulfuric Acid-Nitric Acid For these data reference must be made to the graphs of International Critical Tables, vol. 3, pp. 306–308.

TABLE 2-21

Vapor pressure of aqueous diethylene glycol solutions. (Courtesy of Carbide and Carbon Chemicals Corp.)

FIG. 2-3

Partial Pressures of H2O over Aqueous Solutions of NH3* Pressures are in pounds per square inch absolute Molal concentration of ammonia in the solutions in percentages (Weight concentration of ammonia in the solution in percentages)

0

5

10

15

20

25

30

35

40

45

50

55

60

65

70

75

80

85

90

95

t, °F

(0)

32 40 50 60 70

0.09 0.084 .12 .115 .18 .17 .26 .24 .36 .34

0.079 .108 .16 .23 .32

0.074 .101 .15 .21 .30

0.070 .095 .14 .20 .28

0.065 .089 .13 .19 .26

0.060 .083 .12 .17 .25

0.056 .076 .11 .16 .23

0.051 .070 .10 .15 .21

0.047 .064 .094 .13 .19

0.042 .058 .085 .12 .17

0.038 .052 .076 .11 .15

0.034 .046 .068 .097 .14

0.030 .040 .059 .085 .12

0.025 .035 .051 .073 .10

0.021 .029 .042 .061 .086

0.017 .023 .034 .049 .069

0.013 .015 .025 .037 .052

0.008 .012 .017 .024 .034

0.004 .006 .008 .012 .017

80 90 100 110 120

.51 .48 .70 .66 .95 .90 1.27 1.20 1.69 1.60

.45 .63 .85 1.14 1.51

.42 .58 .79 1.07 1.42

.40 .55 .74 1.00 1.33

.37 .51 .69 .93 1.24

.34 .47 .64 .86 1.15

.32 .44 .59 .80 1.06

.29 .40 .55 .73 .97

.27 .37 .50 .67 .89

.24 .33 .45 .60 .80

.22 .30 .41 .54 .72

.19 .26 .36 .48 .64

.17 .23 .31 .42 .56

.14 .20 .27 .36 .48

.12 .16 .22 .30 .40

.096 .13 .18 .24 .32

.072 .10 .13 .18 .24

.048 .066 .090 .120 .160

.024 .033 .045 .061 .081

130 140 150 160 170

2.22 2.89 3.72 4.74 5.99

2.10 2.73 3.51 4.48 5.66

1.98 2.57 3.31 4.22 5.34

1.86 2.42 3.11 3.97 5.02

1.74 2.26 2.91 3.71 4.70

1.62 2.11 2.72 3.46 4.38

1.51 1.96 2.52 3.22 4.07

1.39 1.81 2.33 2.97 3.75

1.28 1.66 2.14 2.73 3.45

1.17 1.52 1.95 2.49 3.15

1.05 1.37 1.76 2.25 2.84

.95 1.23 1.59 2.02 2.56

.84 1.10 1.41 1.80 2.28

.74 .96 1.24 1.58 1.99

.63 .82 1.06 1.35 1.71

.53 .69 .88 1.12 1.42

.42 .55 .71 .90 1.13

.32 .41 .53 .67 1.85

.210 .270 .350 .450 .570

.100 .140 .180 .220 .300

180 7.51 7.10 190 9.34 8.83 200 11.53 10.90 210 14.12 13.35 220 17.19 16.25

6.69 8.32 10.27 12.58 15.32

6.30 7.82 9.65 11.82 14.39

5.89 7.32 9.04 11.07 13.48

5.49 6.83 8.43 10.32 12.57

5.10 6.34 7.83 9.59 11.67

4.71 5.86 7.23 8.86 10.78

4.33 5.38 6.64 8.13 9.90

3.94 4.91 6.06 7.42 9.03

3.57 4.44 5.48 6.71 8.17

3.21 3.99 4.93 6.04 7.31

2.85 3.55 4.38 5.34

2.50 3.10 3.81

2.14 2.65

1.77

1.42

1.06

230 20.78 19.64 240 24.97 23.60 250 29.83 28.20

18.51 22.25 26.58

17.40 20.91 25.00

16.29 19.58 23.39

15.19 18.26 21.82

14.11 16.95 20.25

13.03 15.66 18.71

11.97 14.38 17.18

10.91 13.12 15.67

9.87 11.86

(4.74) (9.50) (14.29) (19.10) (23.94) (28.81) (33.71) (38.64) (43.59) (48.57) (53.58) (58.62) (63.69) (68.79) (73.91) (79.07) (84.26) (89.47) (94.72)

*Wilson, Univ. Ill., Eng. Expt. Sta. Bull. 146.

TABLE 2-22

Mole Percentages of H2O over Aqueous Solutions of NH3* Molal concentration of ammonia in the solutions in percentages (Weight concentration of ammonia in the solutions in percentages)

t, °F

0 (0)

5 (4.74)

10 (9.50)

15 (14.29)

20 (19.10)

25 (23.94)

30 (28.81)

35 (33.71)

40 (38.64)

45 (43.59)

50 (48.57)

55 (53.58)

60 (58.62)

65 (63.69)

70 (68.79)

75 (73.91)

80 (79.07)

85 (84.26)

90 (89.47)

95 (94.72)

32 40 50 60 70

100 100 100 100 100

24.3 25.3 26.6 27.9 29.1

13.2 14.1 15.2 16.2 17.4

7.63 8.15 9.09 9.50 10.30

4.43 4.73 5.24 5.69 6.14

2.50 2.74 3.03 3.42 3.65

1.43 1.59 1.78 1.97 2.27

0.856 .943 1.060 1.210 1.390

0.514 .581 .652 .777 .873

0.335 .372 .434 .481 .569

0.216 .248 .290 .331 .383

0.151 .172 .202 .238 .266

0.109 .124 .148 .172 .205

0.0816 .0914 .1095 .1290 .1510

0.0585 .0706 .0838 .0986 .112

0.0457 .0533 .0630 .0754 .0882

0.0345 .0395 .0477 .0566 .0656

0.0249 .0243 .0332 .0406 .0474

0.0146 .0185 .0215 .0251 .0296

0.00689 .00879 .00959 .01125 .0135

80 90 100 110 120

100 100 100 100 100

31.6 32.7 34.4 35.9 37.5

18.5 20.0 21.0 22.2 23.4

11.20 12.00 12.90 13.80 14.70

6.89 7.40 7.92 8.59 9.22

4.08 4.47 4.85 5.29 5.75

2.45 2.73 3.00 3.30 3.63

1.550 1.730 1.890 2.110 2.320

.978 1.100 1.250 1.370 1.520

.659 .742 .834 .932 1.044

.444 .505 .574 .644 .714

.323 .366 .420 .466 .529

.230 .267 .307 .347 .395

.1750 .2020 .2290 .2640 .3020

.130 .157 .179 .208 .233

.103 .115 .135 .157 .180

.0772 .0884 .104 .118 .135

.0528 .0647 .0714 .0846 .0970

.0351 .0408 .0473 .0540 .0619

.0167 .0194 .0226 .0262 .0300

130 140 150 160 170

100 100 100 100 100

39.0 40.7 42.3 44.1 45.6

24.5 25.8 27.1 28.3 29.6

15.60 16.50 17.50 18.40 19.40

9.85 10.50 11.20 11.90 12.70

6.18 6.69 7.19 7.69 8.22

3.95 4.28 4.63 5.01 5.38

2.550 2.790 3.080 3.300 3.580

1.690 1.860 2.040 2.230 2.430

1.160 1.286 1.410 1.550 1.700

.811 .906 1.004 1.110 1.220

.596 .663 .741 .818 .904

.444 .501 .558 .617 .689

.3430 .3840 .4320 .4800 .5300

.263 .297 .334 .372 .414

.205 .232 .257 .287 .320

.154 .175 .197 .218 .242

.1117 .124 .140 .154 .174

.0703 .0786 .0892 .1005 .112

.0339 .0385 .0439 .0499 .0567

180 190 200 210 220

100 100 100 100 100

47.3 48.7 50.4 52.1 53.7

30.9 32.2 33.4 34.7 36.1

20.40 21.40 22.30 23.40 24.40

13.40 14.10 14.90 15.70 16.40

8.76 9.31 9.88 10.45 11.05

5.78 6.18 6.59 7.03 7.48

3.870 4.160 4.470 4.780 5.100

2.640 2.860 3.080 3.310 3.560

1.850 2.020 2.190 2.360 2.540

1.340 1.460 1.580 1.720 1.860

.994 1.087 1.187 1.272 1.390

.756 .830 .907 .983

.5860 .6420 .7010

.456 .501

.352

.268

.192

230 240 250

100 100 100

55.2 56.8 58.4

37.3 38.6 39.8

25.40 26.50 27.50

17.30 18.00 18.80

11.63 12.24 12.88

7.91 8.36 8.82

5.440 5.780 6.120

3.810 4.060 4.340

2.730 2.920 3.120

2.000 2.150

*Wilson, Univ. Ill., Eng. Expt. Sta. Bull. 146.

2-86

100 (100.00) 0.00

VAPOR PRESSURES OF SOLUTIONS TABLE 2-23

2-87

Partial Pressures of NH3 over Aqueous Solutions of NH3* Pressures are in pounds per square inch absolute Molal concentration of ammonia in the solutions in percentages (Weight concentration of ammonia in the solutions in percentages)

t, °F

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 (4.74) (9.50) (14.29) (19.10) (23.94) (28.81) (33.71) (38.64) (43.59) (48.57) (53.58) (58.62) (63.69) (68.79) (73.91) (79.07) (84.26) (89.47) (94.72)

32 40 50 60 70

0.26 .33 .47 .62 .83

0.52 .66 .89 1.19 1.52

0.90 1.14 1.50 2.00 2.60

1.51 1.92 2.53 3.21 4.28

2.67 3.16 4.16 5.36 6.87

4.27 5.13 6.63 8.48 10.76

6.54 7.98 10.24 13.06 16.33

8.93 11.98 15.24 19.15 23.84

14.13 17.14 21.56 26.92 33.20

19.36 23.33 29.17 36.14 44.25

25.12 30.15 37.46 46.12 56.29

31.13 37.15 45.86 56.22 68.32

36.74 43.69 53.79 65.81 79.42

42.69 49.56 60.82 73.99 89.26

45.92 54.40 66.63 80.90 97.42

49.26 58.31 71.26 86.44 104.01

52.13 61.62 75.22 91.04 109.55

54.89 64.77 79.05 95.67 114.83

58.01 68.31 83.40 100.65 120.61

80 90 100 110 120

1.04 1.36 1.72 2.14 2.67

1.98 2.52 3.20 4.00 4.95

3.34 4.25 5.34 6.65 8.21

5.45 6.88 8.60 10.64 13.09

8.69 10.89 13.53 16.65 20.30

13.52 16.76 20.68 25.21 30.54

20.29 25.04 30.57 37.01 44.56

29.40 35.94 43.57 52.43 62.62

40.69 49.45 59.49 71.20 84.44

53.84 64.99 77.85 92.59 109.40

67.97 81.61 97.27 115.16 135.48

82.36 98.35 116.81 137.62 161.44

95.52 113.79 134.70 158.42 185.14

107.06 127.22 150.23 176.18 205.81

116.42 138.18 162.94 190.85 222.28

124.20 147.02 173.22 203.02 236.05

130.57 154.46 181.97 212.71 247.14

136.35 161.74 190.13 222.22 258.24

143.70 169.73 199.17 232.79 270.02

130 140 150 160 170

3.28 3.97 4.78 5.68 6.75

6.09 7.41 8.92 10.70 12.67

10.05 12.21 14.70 17.57 20.85

15.93 19.23 23.09 27.45 32.41

24.58 29.43 35.09 41.56 48.89

36.74 43.77 51.91 61.03 71.48

53.16 62.97 74.28 86.91 101.09

74.27 87.53 102.51 119.37 138.30

99.69 116.72 136.15 157.71 181.95

128.45 149.93 173.64 200.45 230.36

158.45 184.17 212.91 244.98 280.54

188.16 218.18 251.24 288.38 329.42

215.14 248.70 286.00 327.82 373.61

238.70 275.33 316.24 361.75 411.59

257.87 297.12 340.82 389.08 442.28

272.88 314.45 360.39 411.30 466.67

286.08 328.99 376.57 429.73 487.85

298.46 342.93 392.45 447.35 507.63

311.80 358.46 409.62 466.38 528.50

180 7.90 190 9.23 200 10.70 210 12.26 220 14.02

14.96 17.55 20.45 23.68 27.15

24.56 28.78 33.49 38.76 44.61

38.13 44.49 51.58 59.65 68.43

57.19 66.49 76.90 88.48 101.24

83.07 96.22 110.85 126.83 144.74

116.97 134.89 154.58 176.24 200.46

159.37 182.72 208.56 236.97 268.30

208.66 238.39 270.94 307.08 346.07

263.43 299.86 340.02 383.99 431.43

319.89 363.11 410.17 462.36 518.19

374.25 424.15 478.62 537.56

424.10 479.40 539.79

466.26 526.15

500.63

528.08

551.24

230 15.95 240 17.92 250 20.12

31.09 35.40 40.09

51.06 58.00 65.74

78.14 89.02 100.69

115.45 130.94 147.66

164.17 185.79 209.37

226.67 255.26 286.89

302.53 339.72 380.42

389.29 435.78 486.73

483.53 540.44

*Wilson, Univ. Ill., Eng. Expt. Sta. Bull. 146.

TABLE 2-24

Total Vapor Pressures of Aqueous Solutions of NH3* Pressures are in pounds per square inch absolute Molal concentration of ammonia in the solutions in percentages (Weight concentration of ammonia in the solutions in percentages)

t, °F

0 (0)

5 (4.74)

10 (9.50)

15 (14.29)

20 (19.10)

25 (23.94)

30 (28.81)

35 (33.71)

40 (38.64)

45 (43.59)

50 (48.57)

55 (53.58)

60 (58.62)

65 (63.69)

70 (68.79)

75 (73.91)

80 (79.07)

85 (84.26)

90 (89.47)

95 (94.72)

100 (100.00)

32 40 50 60 70

0.09 .12 .18 .26 .36

0.34 .45 .64 .86 1.17

0.60 .77 1.05 1.42 1.84

0.97 1.24 1.65 2.21 2.90

1.58 2.01 2.67 3.51 4.56

2.60 3.25 4.29 5.55 7.13

4.20 5.21 6.75 8.65 11.01

6.54 8.06 10.35 13.22 16.56

9.93 12.05 15.34 19.30 24.05

14.18 17.20 21.65 27.05 33.39

19.40 23.39 29.26 36.26 44.42

25.16 30.20 37.54 46.23 56.44

31.16 37.20 45.93 56.32 68.46

36.77 43.73 53.85 65.90 79.54

42.72 49.60 60.87 74.06 89.36

45.94 54.43 66.67 80.96 97.51

49.28 58.33 71.29 86.49 104.08

52.14 61.64 75.25 91.08 109.60

54.90 64.78 79.07 95.69 114.86

58.01 68.32 83.41 100.66 120.63

62.29 73.32 89.19 107.6 128.8

80 90 100 110 120

.51 .70 .95 1.27 1.69

1.52 2.02 2.62 3.34 4.27

2.43 3.15 4.05 5.14 6.46

3.76 4.83 6.13 7.72 9.63

5.85 7.43 9.34 11.64 14.42

9.06 11.40 14.22 17.58 21.54

13.86 17.23 21.32 26.07 31.69

20.61 25.48 31.16 37.81 45.62

29.69 36.34 44.12 53.16 63.59

40.96 49.82 59.99 71.87 85.33

54.08 65.32 78.30 93.19 110.2

68.19 81.91 97.68 115.7 136.2

82.55 98.61 117.17 138.10 162.08

95.69 114.02 135.01 158.84 185.70

107.20 127.42 150.50 176.54 206.29

116.54 138.34 163.16 191.15 222.68

124.30 147.15 173.40 203.26 236.37

130.64 154.56 182.10 212.89 247.38

136.40 161.81 190.22 222.34 258.40

143.72 169.76 199.22 232.85 270.1

153.0 180.6 211.9 247.0 286.4

130 140 150 160 170

2.22 2.89 3.72 4.74 5.99

5.38 6.70 8.29 10.16 12.41

8.07 9.98 12.23 14.92 18.01

11.91 14.63 17.81 21.54 25.87

17.67 21.49 26.00 31.16 37.11

26.20 31.54 37.81 45.02 53.27

38.25 45.73 54.43 64.25 75.55

54.55 64.78 76.61 89.88 104.84

75.55 89.19 104.65 122.10 141.75

100.86 118.24 138.1 160.2 185.1

129.5 151.3 175.4 202.7 233.2

159.0 185.4 214.5 247.0 283.1

189.00 219.28 252.65 290.18 331.7

215.88 249.66 287.24 329.4 375.6

239.33 276.15 317.3 363.1 413.3

258.40 297.81 341.7 390.2 443.7

273.3 315.0 361.1 412.2 467.8

286.4 329.4 377.1 430.4 488.7

298.67 343.2 392.8 447.8 508.2

311.9 358.6 409.8 466.6 528.8

330.3 379.1 432.2 492.8 558.4

180 190 200 210 220

7.51 9.34 11.53 14.12 17.19

15.00 18.06 21.60 25.61 30.27

21.65 25.87 30.72 36.26 42.47

30.86 36.60 43.14 50.58 59.00

44.02 51.81 60.62 70.72 81.91

62.68 73.32 85.33 98.80 113.81

88.17 102.56 118.68 136.42 156.41

121.68 140.75 161.81 185.10 211.24

163.7 188.1 215.2 245.1 278.2

212.6 243.3 277.0 314.5 355.1

267.0 304.3 345.5 390.7 439.6

323.1 367.1 415.1 468.4 525.5

377.1 427.7 483.0 542.9

426.6 482.5 543.6

468.4 528.8

502.4

529.5

552.3

230 240 250

20.78 24.97 29.83

35.59 41.52 48.32

49.60 57.65 66.67

68.46 78.91 90.74

94.43 108.60 124.08

130.64 149.20 169.48

178.28 202.74 229.62

239.70 270.92 305.60

314.5 354.1 397.6

400.2 448.9 502.4

493.4 552.3

*Wilson, Univ. Ill., Eng. Expt. Sta. Bull. 146.

2-88

VAPOR PRESSURES OF SOLUTIONS TABLE 2-25 Partial Pressures of H2O over Aqueous Solutions of Sodium Carbonate mmHg %Na2CO3 t, °C

0

5

10

15

0 10 20 30 40 50 60 70 80 90 100

4.5 9.2 17.5 31.8 55.3 92.5 149.5 239.8 355.5 526.0 760.0

4.5 9.0 17.2 31.2 54.2 90.7 146.5 235 348 516 746

8.8 16.8 30.4 53.0 88.7 143.5 230.5 342 506 731

16.3 29.6 57.6 86.5 139.9 225 334 494 715

20

28.8 50.2 84.1 136.1 219 325 482 697

25

27.8 48.4 81.2 131.6 211.5 315 467 676

30

26.4 46.1 77.5 125.7 202.5 301 447 648

2-89

TABLE 2-26 Partial Pressures of H2O and CH3OH over Aqueous Solutions of Methyl Alcohol* 39.9°C

Mole fraction CH3OH

PH2O, mmHg

PCH3OH , mmHg

0 14.99 17.85 21.07 27.31 31.06 40.1 47.0 55.8 68.9 86.0 100.0

54.7 39.2 38.5 37.2 35.8 34.9 32.8 31.5 27.3 20.7 10.1 0

0 66.1 75.5 85.2 100.6 108.8 127.7 141.6 158.4 186.6 225.2 260.7

59.4°C

Mole fraction CH3OH

PH2O, mmHg

PCH3OH , mmHg

0 22.17 27.40 33.24 39.80 47.08 55.5 69.2 78.5 85.9 100.0

145.4 106.9 102.2 96.6 91.7 84.8 76.9 57.8 43.8 30.1 0

0 210.1 240.2 272.1 301.9 335.6 373.7 439.4 486.6 526.9 609.3

*International Critical Tables, vol. 3, McGraw-Hill, p. 290.

TABLE 2-27 Conc. g NaOH/ 100 g H2O 0 5 10 20 30 40 50 60 70 80 90 100 120 140 160 180 200 250 300 350 400 500 700 1000 2000 4000 8000

Partial Pressures of H2O over Aqueous Solutions of Sodium Hydroxide mmHg Temperature, °C 0

20

40

60

80

100

120

160

200

250

300

350

4.6 4.4 4.2 3.6 2.9 2.2

17.5 16.9 16.0 13.9 11.3 8.7 6.3 4.4 3.0 2.0 1.3 0.9

55.3 53.2 50.6 44.2 36.6 28.7 20.7 15.5 10.9 7.6 5.2 3.6 1.7

149.5 143.5 137.0 120.5 101.0 81.0 62.5 47.0 34.5 24.5 17.5 12.5 6.3 3.0 1.5

355.5 341.5 325.5 288.5 246.0 202.0 160.5 124.0 94.0 70.5 53.0 38.5 20.5 11.0 6.0 3.5 2.0 0.5 0.1

760.0 730.0 697.0 621.0 537.0 450.0 368.0 294.0 231.0 179.0 138.0 105.0 61.0 35.5 20.5 12.0 7.0 2.0 0.5

1,489 1,430 1,365 1,225 1,070 920 770 635 515 415 330 262 164 102 63 40 25 8 2.7 0.9

4,633 4,450 4,260 3,860 3,460 3,090 2,690 2,340 2,030 1,740 1,490 1,300 915 765 470 340 245 110 50 23 11

11,647 11,200 10,750 9,800 8,950 8,150 7,400 6,750 6,100 5,500 5,000 4,500 3,650 2,980 2,430 1,980 1,620 985 610 380 240 100

29,771 28,600 27,500 25,300 23,300 21,500 19,900 18,400 17,100 15,800 14,700 13,650 11,800 10,300 8,960 7,830 6,870 5,000 3,690 2,750 2,080 1,210 440

64,200 61,800 59,300 54,700 50,800 47,200 44,100 41,200 38,700 36,300 34,200 32,200 28,800 25,900 23,300 21,200 19,200 15,400 12,500 10,300 8,600 6,100 3,300 1,470 150

123,600 118,900 114,100 105,400 98,000 91,600 85,800 80,700 76,000 71,900 68,100 64,600 58,600 53,400 49,000 45,100 41,800 35,000 29,800 25,700 22,400 17,500 11,500 6,800 1,760 120 7

2-90

PHYSICAL AND CHEMICAL DATA

WATER-VAPOR CONTENT OF GASES CHART FOR GASES AT HIGH PRESSURES The accompanying figure is useful in determining the water-vapor content of air at high pressure in contact with liquid water.

Water content of air, °C = (°F − 32) × 5⁄ 9. (Landsbaum, Dadds, and Stutzman. Reprinted from vol. 47, January 1955 issue of Ind. Eng. Chem. [p. 192]. Copyright 1955 by the American Chemical Society and reproduced by permission of the copyright owner.)

FIG. 2-4

DENSITIES OF PURE SUBSTANCES

2-91

DENSITIES OF PURE SUBSTANCES UNITS CONVERSIONS For this subsection, the following units conversions are applicable:

To convert kilograms per cubic meter to pounds per cubic foot, multiply by 0.06243.

°F = 9⁄ 5 °C + 32.

TABLE 2-28

Density (kg/m3) of Water from 0 to 100°C* ρ, kg/m3

t, °C

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 1 2 3 4

999.839 999.898 999.940 999.964 999.972

999.846 999.903 999.943 999.966 999.972

999.852 999.908 999.946 999.967 999.972

999.859 999.913 999.949 999.968 999.971

999.865 999.917 999.952 999.969 999.971

999.871 999.921 999.954 999.970 999.970

999.877 999.925 999.956 999.971 999.969

999.882 999.929 999.959 999.971 999.968

999.888 999.933 999.961 999.972 999.967

999.893 999.936 999.962 999.972 999.965

5 6 7 8 9

999.964 999.940 999.901 999.848 999.781

999.962 999.937 999.897 999.842 999.773

999.960 999.934 999.892 999.836 999.765

999.958 999.930 999.887 999.829 999.758

999.956 999.926 999.882 999.823 999.750

999.954 999.923 999.877 999.816 999.742

999.951 999.919 999.871 999.809 999.734

999.949 999.915 999.866 999.802 999.725

999.946 999.910 999.860 999.795 999.717

999.943 999.906 999.854 999.788 999.708

10 11 12 13 14

999.699 999.605 999.497 999.377 999.244

999.691 999.595 999.486 999.364 999.230

999.682 999.584 999.474 999.351 999.216

999.672 999.574 999.462 999.338 999.202

999.663 999.563 999.451 999.325 999.188

999.654 999.553 999.439 999.312 999.173

999.644 999.542 999.426 999.299 999.159

999.635 999.531 999.414 999.285 999.144

999.625 999.520 999.402 999.272 999.129

999.615 999.509 999.389 999.258 999.114

15 16 17 18 19

999.099 998.943 998.775 998.595 998.405

999.084 998.926 998.757 998.577 998.385

999.069 998.910 998.740 998.558 998.366

999.054 999.894 998.722 998.539 998.346

999.038 998.877 998.704 998.520 998.326

999.022 998.860 998.686 998.502 998.306

999.007 998.843 998.668 998.482 998.286

998.991 998.826 998.650 998.463 998.265

998.975 998.809 998.632 998.444 998.245

998.958 998.792 998.614 998.425 998.224

20 21 22 23 24

998.204 997.992 997.770 997.538 997.296

998.183 997.971 997.747 997.515 997.272

998.162 997.949 997.725 997.491 997.247

998.141 997.927 997.702 997.467 997.222

998.120 997.905 997.679 997.443 997.197

998.099 997.883 997.656 997.419 997.172

998.078 997.860 997.632 997.394 997.146

998.057 997.838 997.609 997.370 997.121

998.035 997.816 997.585 997.345 997.096

998.014 997.793 997.562 997.321 997.070

25 26 27 28 29

997.045 996.783 996.513 996.233 995.945

997.019 996.757 996.485 996.205 995.915

996.993 996.730 996.458 996.176 995.886

996.967 996.703 996.430 996.148 995.856

996.941 996.676 996.402 996.119 995.827

996.915 996.649 996.374 996.090 995.797

996.889 996.622 996.346 996.061 995.767

996.863 996.595 996.318 996.032 995.737

996.836 996.568 996.290 996.003 995.707

996.810 996.540 996.262 995.974 995.677

30 31 32 33 34

995.647 995.341 995.026 994.703 994.371

995.617 995.310 994.997 994.670 994.338

995.586 995.278 994.962 994.637 994.304

995.556 995.247 994.930 994.604 994.270

995.526 995.216 994.898 994.571 994.236

995.495 995.184 994.865 994.538 994.202

995.464 995.153 994.833 994.505 994.168

995.433 995.121 994.801 994.472 994.134

995.403 995.090 994.768 994.438 994.100

995.372 995.058 994.735 994.405 994.066

35 36 37 38 39

994.032 993.684 993.328 992.965 992.594

993.997 993.648 993.292 992.928 992.557

993.963 993.613 993.256 992.891 992.519

993.928 993.578 993.220 992.855 992.481

993.893 993.543 993.184 992.818 992.444

993.859 993.507 993.148 992.780 992.406

993.824 993.471 993.111 992.743 992.368

993.789 993.436 993.075 992.706 992.330

993.754 993.400 993.038 992.669 992.292

993.719 993.364 993.002 992.631 992.254

40 41 42 43 44

992.215 991.830 991.436 991.036 990.628

992.177 991.791 991.396 990.995 990.587

992.139 991.751 991.357 990.955 990.546

992.100 991.712 991.317 990.914 990.504

992.062 992.673 991.277 990.873 990.463

992.023 991.634 991.237 990.833 990.421

991.985 991.594 991.197 990.792 990.380

991.946 991.555 991.157 990.751 990.338

991.907 991.515 991.116 990.710 990.297

992.868 991.476 991.076 990.669 990.255

45 46 47 48 49

990.213 989.792 989.363 988.928 988.485

990.171 989.749 989.320 988.884 988.441

990.129 989.706 989.276 988.840 988.396

990.087 989.664 989.233 988.796 988.352

990.045 989.621 989.190 988.752 988.307

990.003 989.578 989.146 988.707 988.262

989.961 989.535 989.103 988.663 988.217

989.919 989.492 989.059 988.619 988.172

989.876 989.449 989.015 988.574 988.127

989.834 989.406 988.971 988.530 988.082

*From “Water: Density at Atmospheric Pressure and Temperatures from 0 to 100°C,” Tables of Standard Handbook Data, Standartov, Moscow, 1978. To conserve space, only a few tables of density values are given. The reader is reminded that density values may be found as the reciprocal of the specific volume values tabulated in the “Thermodynamic Properties: Tables” subsection.

2-92

PHYSICAL AND CHEMICAL DATA

TABLE 2-28

Density (kg/m3) of Water from 0 to 100°C (Concluded ) ρ, kg/m3

t, °C

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

50 51 52 53 54

988.037 987.581 987.120 986.652 986.177

987.992 987.536 987.073 986.604 986.129

987.946 987.490 987.027 986.557 986.081

987.901 987.444 986.980 986.510 986.033

987.844 987.398 986.933 986.463 985.985

987.810 987.351 986.886 986.415 985.937

987.764 987.305 986.840 986.368 985.889

987.719 987.259 986.793 986.320 985.841

987.673 987.213 986.746 986.272 985.793

987.627 987.166 986.699 986.225 985.745

55 56 57 58 59

985.696 985.219 984.716 984.217 983.712

985.648 985.160 984.666 984.167 983.661

985.599 985.111 984.617 984.116 983.610

985.551 985.062 984.567 984.066 983.559

985.502 985.013 984.517 984.016 983.508

985.454 984.963 984.467 983.965 983.457

985.405 984.914 984.417 983.914 983.406

985.356 984.865 984.367 983.864 983.354

985.307 984.815 984.317 983.813 983.303

985.258 984.766 984.267 983.762 983.252

60 61 62 63 64

983.200 982.683 982.160 981.631 981.097

983.149 982.631 982.108 981.578 981.043

983.097 982.579 982.055 981.525 980.989

983.046 982.527 982.002 981.472 980.935

982.994 982.475 981.949 981.418 980.881

982.943 982.422 981.897 981.365 980.827

982.891 982.370 981.844 981.311 980.773

982.839 982.318 981.791 981.258 980.719

982.787 982.265 981.738 981.204 980.665

982.735 982.213 981.685 981.151 980.611

65 66 67 68 69

980.557 980.011 979.459 978.902 978.339

980.502 979.956 979.403 978.846 978.283

980.443 979.901 979.348 978.790 978.226

980.393 979.846 979.293 978.734 978.170

980.339 979.791 979.237 978.678 978.113

980.284 979.736 979.181 978.621 978.056

980.230 979.680 979.126 978.565 977.999

980.175 979.625 979.070 978.509 977.942

980.120 979.570 979.014 978.452 977.885

980.065 979.515 978.958 978.396 977.828

70 71 72 73 74

977.771 977.198 976.619 976.035 975.445

977.714 977.140 976.561 975.976 975.386

977.657 977.082 976.503 975.917 975.327

977.600 977.025 976.444 975.858 975.267

977.543 976.967 976.386 975.800 975.208

977.485 976.909 976.327 975.741 975.148

977.428 976.851 976.269 975.682 975.089

977.370 976.793 976.211 975.623 975.029

977.313 976.735 976.152 975.564 974.970

977.255 976.677 976.093 975.504 974.910

75 76 77 78 79

974.850 974.250 973.645 973.025 972.419

974.791 974.190 973.584 972.974 972.358

974.731 974.130 973.524 972.912 972.296

974.671 974.069 973.463 972.851 972.234

974.611 974.009 973.402 972.789 972.172

974.551 973.948 973.341 972.728 972.110

974.491 973.888 973.280 972.666 972.048

974.431 973.827 973.218 972.605 971.986

974.371 973.767 973.157 972.543 971.923

974.311 973.706 973.096 972.481 971.861

80 81 82 83 84

971.799 971.173 970.543 969.907 969.267

971.737 971.110 970.479 969.843 969.202

971.674 971.048 970.416 969.772 969.138

971.612 970.985 970.353 969.715 969.073

971.549 970.922 970.289 969.652 969.009

971.487 970.859 970.226 969.587 968.944

971.424 970.796 970.162 969.523 968.880

971.361 970.732 970.098 969.459 968.815

971.299 970.669 970.035 969.395 968.751

971.236 970.606 969.971 969.331 968.686

85 86 87 88 89

968.621 967.971 967.316 966.656 965.991

968.556 967.906 967.250 966.589 965.924

968.491 967.840 967.184 966.523 965.857

968.427 967.775 967.118 966.457 965.790

968.362 967.709 967.052 966.390 965.723

968.297 967.641 966.986 966.324 965.656

969.232 967.578 966.920 966.257 965.589

968.166 967.513 966.854 966.191 965.522

968.101 967.447 966.788 966.124 965.455

968.036 967.381 966.722 966.057 965.388

90 91 92 93 94

965.321 964.647 963.967 963.284 962.595

965.254 954.579 963.899 963.215 962.526

965.187 964.511 963.831 963.146 962.457

965.119 964.443 963.763 963.077 962.387

965.052 964.376 963.694 963.009 962.318

964.984 964.308 963.626 962.940 962.249

964.917 964.240 963.558 962.871 962.180

964.849 964.172 963.489 962.802 962.110

964.782 964.104 963.421 962.733 962.041

964.714 964.036 963.352 962.664 961.971

95 96 97 98 99

961.902 961.204 960.501 959.794 959.082

961.832 961.134 960.431 959.723 959.010

961.762 961.064 960.360 959.652 958.939

961.693 960.993 960.289 959.581 958.867

961.693 960.923 960.219 959.510 958.796

961.553 960.853 960.148 959.438 958.724

961.483 960.783 960.077 959.367 958.653

961.414 960.712 960.006 959.296 958.581

961.344 960.642 959.936 959.225 958.509

961.274 960.572 959.865 959.153 958.431

100

958.365

DENSITIES OF PURE SUBSTANCES TABLE 2-29

2-93

Density (kg/m3) of Mercury from 0 to 350°C* Density, kg /m3

t, °C

2

3

4

5

6

7

8

9

0 10 20 30 40

13595.08 13570.44 13545.87 13521.36 13496.92

0

13592.61 13567.98 13543.41 13518.91 13494.48

1

13590.14 13565.52 13540.96 13516.47 13492.04

13587.68 13563.06 13538.51 13514.02 13489.60

13585.21 13560.60 13536.06 13511.58 13487.16

13582.75 13558.14 13533.61 13509.13 13484.72

13580.29 13555.69 13531.16 13506.69 13482.29

13577.82 13553.23 13528.71 13504.25 13479.85

13575.36 13550.78 13526.26 13501.80 13477.41

13572.90 13548.32 13523.81 13499.36 13474.98

50 60 70 80 90

13472.54 13448.22 13423.96 13399.75 13375.59

13470.11 13445.80 13421.54 13397.34 13373.18

13467.67 13443.37 13419.12 13394.92 13370.77

13465.24 13440.94 13416.69 13392.50 13368.36

13462.81 13438.51 13414.27 13390.08 13365.94

13460.38 13436.09 13411.85 13387.67 13363.53

13457.94 13433.66 13409.43 13385.25 13361.12

13455.51 13431.23 13407.01 13382.84 13358.71

13453.08 13428.81 13404.59 13380.42 13356.30

13450.65 13426.39 13402.17 13378.01 13353.89

100 110 120 130 140

13351.5 13327.4 13303.4 13279.4 13255.4

13349.1 13325.0 13301.0 13277.0 13253.0

13346.7 13322.6 13298.6 13274.6 13250.6

13344.3 13320.2 13296.2 13272.2 13248.2

13341.9 13317.8 13293.8 13269.8 13245.8

13339.4 13315.4 13291.4 13267.4 13243.4

13337.0 13313.0 13288.9 13265.0 13241.0

13334.6 13310.6 13286.6 13262.6 13238.7

13332.2 13308.2 13284.2 13260.2 13236.3

13329.8 13305.8 13281.8 13257.8 13233.9

150 160 170 180 190

13231.5 13207.6 13183.7 13159.8 13136.0

13229.1 13205.2 13181.3 13157.4 13133.6

13226.7 13202.8 13178.9 13155.0 13131.2

13224.3 13200.4 13176.5 13152.6 13128.3

13221.9 13198.0 13174.1 13150.3 13126.4

13219.5 13195.6 13171.7 13147.9 13124.0

13217.1 13193.2 13169.4 13145.5 13121.7

13214.7 13190.8 13167.0 13143.1 13119.3

13212.4 13188.5 13164.6 13140.7 13116.9

13210.0 13186.1 13162.2 13138.3 13114.5

200 210 220 230 240

13112.1 13088.3 13064.5 13040.6 13016.8

13109.7 13085.9 13062.1 13038.3 13014.5

13107.4 13083.5 13059.7 13035.9 13012.1

13105.0 13081.1 13057.3 13033.5 13009.7

13102.6 13078.8 13054.9 13031.1 13007.3

13100.2 13076.4 13052.6 13028.7 13004.9

13097.8 13074.0 13050.2 13026.4 13002.5

13095.4 13071.6 13047.8 13024.0 13000.2

13093.1 13069.2 13045.4 13021.6 12997.8

13090.7 13066.8 13043.0 13019.2 12995.4

250 260 270 280 290

12993.0 12969.2 12945.4 12921.5 12897.7

12990.6 12966.8 12943.0 12919.1 12895.3

12988.3 12964.4 12940.6 12916.7 12892.9

12985.9 12962.0 12938.2 12914.4 12890.5

12983.5 12959.7 12935.8 12912.0 12888.1

12981.1 12957.3 12933.4 12909.6 12885.7

12978.7 12954.9 12931.1 12907.2 12883.3

12976.3 12952.5 12928.7 12904.8 12880.9

12974.0 12950.1 12926.3 12902.4 12878.5

12971.6 12947.7 12923.9 12900.0 12876.2

300 310 320 330 340

12873.8 12849.9 12825.9 12801.9 12777.8

12871.4 12847.5 12823.5 12799.5 12775.4

12869.0 12845.1 12821.1 12797.1 12773.0

12866.6 12842.7 12818.7 12794.7 12770.6

12864.2 12840.3 12816.3 12792.3 12768.2

12861.8 12837.9 12813.9 12789.9 12765.8

12859.4 12835.5 12811.5 12787.5 12763.4

12857.0 12833.1 12809.1 12785.1 12761.0

12854.6 12830.7 12806.7 12782.7 12758.6

12852.2 12828.3 12804.3 12780.2 12756.1

350

12753.7

*From “Mercury—Density and Thermal Expansion at Atmospheric Pressure and Temperatures from 0 to 350°C,” Tables of Standard Handbook Data, Standartov, Moscow, 1978. The density values obtainable from those cited for the specific volume of the saturated liquid in the “Thermodynamic Properties” subsection show minor differences. No attempt was made to adjust either set.

TABLE 2-30 Cmpd. no.

Densities of Inorganic and Organic Liquids Formula

CAS no.

Mol. wt.

C1

C2

C3

C4

Tmin, K

Density at Tmin

Tmax, K

Density at Tmax

1 2 3 4 5 6 7 8 9 10

Methane Ethane Propane n-Butane n-Pentane n-Hexane n-Heptane n-Octane n-Nonane n-Decane

CH4 C2H6 C3H8 C4H10 C5H12 C6H14 C7H16 C8H18 C9H20 C10H22

74828 74840 74986 106978 109660 110543 142825 111659 111842 124185

16.043 30.070 44.097 58.123 72.150 86.177 100.204 114.231 128.258 142.285

2.9214 1.9122 1.3757 1.0677 0.84947 0.70824 0.61259 0.53731 0.48387 0.42831

0.28976 0.27937 0.27453 0.27188 0.26726 0.26411 0.26211 0.26115 0.26147 0.25745

190.56 305.32 369.83 425.12 469.7 507.6 540.2 568.7 594.6 617.7

0.28881 0.29187 0.29359 0.28688 0.27789 0.27537 0.28141 0.28034 0.28281 0.28912

90.69 90.35 85.47 134.86 143.42 177.83 182.57 216.38 219.66 243.51

28.18 21.64 16.583 12.62 10.474 8.747 7.6998 6.6558 6.007 5.3811

190.56 305.32 369.83 425.12 469.7 507.6 540.2 568.7 594.6 617.7

10.082 6.845 5.011 3.927 3.178 2.682 2.337 2.058 1.851 1.664

11 12 13 14 15 16 17 18 19 20

n-Undecane n-Dodecane n-Tridecane n-Tetradecane n-Pentadecane n-Hexadecane n-Heptadecane n-Octadecane n-Nonadecane n-Eicosane

C11H24 C12H26 C13H28 C14H30 C15H32 C16H34 C17H36 C18H38 C19H40 C20H42

1120214 112403 629505 629594 629629 544763 629787 593453 629925 112958

156.312 170.338 184.365 198.392 212.419 226.446 240.473 254.500 268.527 282.553

0.39 0.35541 0.3216 0.30545 0.28445 0.26807 0.2545 0.23864 0.22451 0.21624

0.25678 0.25511 0.2504 0.2535 0.25269 0.25287 0.254 0.25272 0.25133 0.25287

639 658 675 693 708 723 736 747 758 768

0.2913 0.29368 0.3071 0.30538 0.30786 0.31143 0.31072 0.31104 0.3133 0.31613

247.57 263.57 267.76 279.01 283.07 291.31 295.13 301.31 305.04 309.58

4.9362 4.5132 4.2035 3.8924 3.6471 3.4187 3.2241 3.0466 2.8933 2.7496

21 22 23 24 25 26 27

2-Methylpropane 2-Methylbutane 2,3-Dimethylbutane 2-Methylpentane 2,3-Dimethylpentane 2,3,3-Trimethylpentane 2,2,4-Trimethylpentane

C4H10 C5H12 C6H14 C6H14 C7H16 C8H18 C8H18

75285 78784 79298 107835 565593 560214 540841

58.123 72.150 86.177 86.177 100.204 114.231 114.231

1.0463 0.9079 0.76929 0.73335 0.7229 0.6028 0.5886

0.27294 0.2761 0.27524 0.2687 0.28614 0.27446 0.27373

408.14 460.43 499.98 497.5 537.35 573.5 543.96

0.27301 0.28673 0.27691 0.28361 0.2713 0.2741 0.2846

113.54 113.25 145.19 119.55 160.00 172.22 165.78

28 29 30 31 32 33 34 35

Ethylene Propylene 1-Butene cis-2-Butene trans-2-Butene 1-Pentene 1-Hexene 1-Heptene

C2H4 C3H6 C4H8 C4H8 C4H8 C5H10 C6H12 C7H14

74851 115071 106989 590181 624646 109671 592416 592767

28.054 42.081 56.108 56.108 56.108 70.134 84.161 98.188

2.0961 1.4094 1.0972 1.1609 1.1426 0.9038 0.7389 0.63734

0.27657 0.26465 0.2649 0.27104 0.27095 0.26648 0.26147 0.26319

282.34 365.57 419.95 435.58 428.63 464.78 504.03 537.29

0.29147 0.295 0.29043 0.2816 0.2854 0.2905 0.2902 0.27375

36 37 38 39 40 41 42 43 44

1-Octene 1-Nonene 1-Decene 2-Methylpropene 2-Methyl-1-butene 2-Methyl-2-butene 1,2-Butadiene 1,3-Butadiene 2-Methyl-1,3-butadiene1

C8H16 C9H18 C10H20 C4H8 C5H10 C5H10 C4H6 C4H6 C5H8

111660 124118 872059 115117 563462 513359 590192 106990 78795

112.215 126.242 140.269 56.108 70.134 70.134 54.092 54.092 68.119

0.5871 0.4945 0.44244 1.1454 0.91619 0.93322 1.187 1.2384 0.95673

0.27005 0.26108 0.25838 0.2725 0.26752 0.27251 0.26114 0.2725 0.26488

566.65 593.25 616.4 417.9 465 471 452 425.17 484

45 46 47 48 49 50

Acetylene Methylacetylene Dimethylacetylene 3-Methyl-1-butyne 1-Pentyne 2-Pentyne

C2H2 C3H4 C4H6 C5H8 C5H8 C5H8

74862 74997 503173 598232 627190 627214

26.038 40.065 54.092 68.119 68.119 68.119

2.4091 1.6086 1.1717 0.94575 0.8491 0.92099

0.27223 0.26448 0.25895 0.26008 0.2352 0.25419

51 52 53

1-Hexyne 2-Hexyne 3-Hexyne1

C6H10 C6H10 C6H10

693027 764352 928494

82.145 82.145 82.145

0.84427 0.76277 0.78045

0.27185 0.25248 0.26065

2-94

Name

639 658 675 693 708 723 736 747 758 768

1.519 1.393 1.284 1.205 1.126 1.060 1.002 0.944 0.893 0.855

12.575 10.776 9.0343 9.2041 7.8746 7.0934 6.9163

408.14 460.43 499.98 497.5 537.35 573.5 543.96

3.833 3.288 2.795 2.729 2.526 2.196 2.150

104.00 87.89 87.80 134.26 167.62 107.93 133.39 154.27

23.326 18.143 14.326 13.895 13.1 11.543 9.6388 8.1759

282.34 365.57 419.95 435.58 428.63 464.78 504.03 537.29

7.579 5.326 4.142 4.283 4.217 3.392 2.826 2.422

0.27187 0.27319 0.28411 0.28186 0.28164 0.26031 0.3065 0.28813 0.28571

171.45 191.78 206.89 132.81 135.58 139.39 136.95 164.25 127.27

7.1247 6.333 5.7131 13.506 11.332 11.218 15.123 14.061 12.205

566.65 593.25 616.4 417.9 465 471 452 425.17 484

2.174 1.894 1.712 4.203 3.425 3.425 4.546 4.545 3.612

308.32 402.39 473.2 463.2 481.2 519

0.28477 0.279 0.27289 0.30807 0.353 0.31077

192.40 170.45 240.91 183.45 167.45 163.83

23.692 19.027 13.767 11.519 12.532 12.24

308.32 402.39 473.2 463.2 481.2 519

8.850 6.082 4.525 3.636 3.610 3.623

516.2 549 544

0.2771 0.31611 0.28571

141.25 183.65 170.05

10.23 10.133 10.021

516.2 549 544

3.106 3.021 2.994

54 55 56

1-Heptyne 1-Octyne Vinylacetylene2

C7H12 C8H14 C4H4

628717 629050 689974

96.172 110.199 52.076

0.67366 0.59229 1.2703

0.26003 0.26118 0.26041

559 585 454

0.29804 0.29357 0.297

192.22 193.55 173.15

8.4987 7.478 15.664

559 585 454

2.591 2.268 4.878

57 58 59 60 61 62

C5H10 C6H12 C7H14 C6H12 C7H14 C8H16

287923 96377 1640897 110827 108872 590669

70.134 84.161 98.188 84.161 98.188 112.215

1.124 0.84798 0.7193 0.8908 0.735 0.55873

0.28859 0.27042 0.26936 0.27396 0.27041 0.25143

511.76 532.79 569.52 553.58 572.19 591.15

0.2506 0.28276 0.2777 0.2851 0.2927 0.27758

179.28 130.73 134.71 279.69 146.58 239.66

11.883 10.492 9.018 9.3797 9.018 7.3417

511.76 532.79 569.52 553.58 572.19 591.15

3.895 3.136 2.670 3.252 2.718 2.222

63 64 65 66

Cyclopentane Methylcyclopentane Ethylcyclopentane Cyclohexane Methylcyclohexane 1,1-Dimethylcyclohexane Ethylcyclohexane Cyclopentene 1-Methylcyclopentene Cyclohexene

C8H16 C5H8 C6H10 C6H10

1678917 142290 693890 110838

112.215 68.119 82.145 82.145

0.61587 1.1035 0.88824 0.92997

0.26477 0.27035 0.26914 0.27056

609.15 507 542 560.4

0.28054 0.28699 0.27874 0.28943

161.84 138.13 146.62 169.67

7.8679 13.47 10.98 11.16

609.15 507 542 560.4

2.326 4.082 3.300 3.437

67 68 69 70 71 72 73

Benzene Toluene o-Xylene m-Xylene p-Xylene Ethylbenzene Propylbenzene

C6H6 C7H8 C8H10 C8H10 C8H10 C8H10 C9H12

71432 108883 95476 108383 106423 100414 103651

78.114 92.141 106.167 106.167 106.167 106.167 120.194

1.0162 0.8488 0.69883 0.69555 0.6816 0.6952 0.57695

0.2655 0.26655 0.26113 0.26204 0.25963 0.26037 0.25395

562.16 591.8 630.33 617.05 616.23 617.2 638.32

0.28212 0.2878 0.27429 0.27602 0.2768 0.2844 0.283

278.68 178.18 247.98 225.30 286.41 178.15 183.15

11.421 10.495 8.6285 8.6505 8.1616 9.0568 7.8942

562.16 591.8 630.33 617.05 616.23 617.2 638.32

3.828 3.184 2.676 2.654 2.625 2.670 2.272

74 75 76 77 78 79 80 81

1,2,4-Trimethylbenzene Isopropylbenzene 1,3,5-Trimethylbenzene p-Isopropyltoluene Naphthalene6 Biphenyl Styrene m-Terphenyl

C9H12 C9H12 C9H12 C10H14 C10H8 C12H10 C8H8 C18H14

95636 98828 108678 99876 91203 92524 100425 92068

120.194 120.194 120.194 134.221 128.174 154.211 104.152 230.309

0.60394 0.604 0.59879 0.51036 0.61674 0.5039 0.7397 0.30826

0.25955 0.25912 0.25916 0.25383 0.25473 0.25273 0.2603 0.23669

649.13 631.1 637.36 653.15 748.35 789.26 636 924.85

0.27716 0.2914 0.27968 0.28816 0.27355 0.281 0.3009 0.29678

229.33 177.14 228.42 205.25 333.15 342.20 242.54 360.00

7.6895 7.9496 7.6154 6.8779 7.7543 6.4395 9.1088 4.5223

649.13 631.1 637.36 653.15 748.35 789.26 636 924.85

2.327 2.331 2.311 2.011 2.421 1.994 2.842 1.302

82 83 84 85 86 87 88

Methanol Ethanol 1-Propanol 1-Butanol 2-Butanol 2-Propanol 2-Methyl-2-propanol

CH4O C2H6O C3H8O C4H10O C4H10O C3H8O C4H10O

67561 64175 71238 71363 78922 67630 75650

32.042 46.069 60.096 74.123 74.123 60.096 74.123

2.288 1.648 1.235 0.965 0.966 1.24 0.9212

0.2685 0.27627 0.27136 0.2666 0.26064 0.27342 0.2544

512.64 513.92 536.78 563.05 536.05 508.3 506.21

0.2453 0.2331 0.24 0.24419 0.2746 0.2353 0.276

175.47 159.05 146.95 184.51 158.45 185.28 298.97

27.912 19.413 15.231 12.016 12.57 14.547 10.555

512.64 513.92 536.78 563.05 536.05 508.3 506.21

8.521 5.965 4.551 3.620 3.706 4.535 3.621

89 90 91 92 93 94 95 96

1-Pentanol 2-Methyl-1-butanol 3-Methyl-1-butanol 1-Hexanol 1-Heptanol Cyclohexanol Ethylene glycol 1,2-Propylene glycol

C5H12O C5H12O C5H12O C6H14O C7H16O C6H12O C2H6O2 C3H8O2

71410 137326 123513 111273 111706 108930 107211 57556

88.150 88.150 88.150 102.177 116.203 100.161 62.068 76.095

0.8164 0.82046 0.837 0.70617 0.60481 0.8243 1.3151 1.0923

0.2673 0.26829 0.27375 0.26901 0.2632 0.26546 0.25125 0.26106

586.15 565 577.2 611.35 631.9 650 719.7 626

0.2506 0.2322 0.22951 0.2479 0.273 0.2848 0.2187 0.20459

195.56 203.00 155.95 228.55 239.15 296.60 260.15 213.15

10.057 10.017 10.204 8.4506 7.421 9.4693 18.31 14.363

586.15 565 577.2 611.35 631.9 650 719.7 626

3.054 3.058 3.058 2.625 2.298 3.105 5.234 4.184

97 98 99 100

Phenol o-Cresol m-Cresol p-Cresol

C6H6O C7H8O C7H8O C7H8O

108952 95487 108394 106445

94.113 108.140 108.140 108.140

1.3798 1.0861 0.9061 1.1503

0.31598 0.30624 0.28268 0.31861

694.25 697.55 705.85 704.65

0.32768 0.30587 0.2707 0.30104

314.06 304.19 285.39 307.93

11.244 9.5751 9.6115 9.4494

694.25 697.55 705.85 704.65

4.367 3.547 3.205 3.610

101 102 103 104 105 106 107 108

Dimethyl ether Methyl ethyl ether Methyl-n-propyl ether Methyl isopropyl ether Methyl-n-butyl ether Methyl isobutyl ether1 Methyl tert-butyl ether Diethyl ether

C2H6O C3H8O C4H10O C4H10O C5H12O C5H12O C5H12O C4H10O

115106 540670 557175 598538 628284 625445 1634044 60297

46.069 60.096 74.123 74.123 88.150 88.150 88.150 74.123

1.5693 1.2635 1.0124 1.0318 0.8281 0.8252 0.82157 0.9554

0.2679 0.27878 0.27942 0.28478 0.27245 0.27282 0.27032 0.26847

400.1 437.8 476.3 464.5 510 497 497.1 466.7

0.2882 0.2744 0.2555 0.2444 0.2827 0.2857 0.2829 0.2814

131.65 160.00 133.97 127.93 157.48 150.00 164.55 156.85

18.95 13.995 11.696 11.568 9.8068 9.7673 9.7682 11.487

400.1 437.8 476.3 464.5 510 497 497.1 466.7

5.858 4.532 3.623 3.623 3.040 3.025 3.039 3.559

2-95

TABLE 2-30 Cmpd. no.

Densities of Inorganic and Organic Liquids (Continued ) CAS no.

Mol. wt.

C1

C2

C3

C4

Tmin, K

Density at Tmin

Tmax, K

Density at Tmax

109 110 111 112

Ethyl propyl ether Ethyl isopropyl ether Methyl phenyl ether Diphenyl ether

C5H12O C5H12O C7H8O C12H10O

628320 625547 100663 101848

88.150 88.150 108.140 170.211

0.7908 0.82049 0.77488 0.52133

0.266 0.26994 0.26114 0.26218

500.23 489 645.6 766.8

0.292 0.30381 0.28234 0.31033

145.65 140.00 235.65 300.03

9.8474 9.9117 9.6675 6.2648

500.23 489 645.6 766.8

2.973 3.040 2.967 1.988

113 114 115 116 117 118 119 120 121 122

Formaldehyde3 Acetaldehyde 1-Propanal 1-Butanal 1-Pentanal 1-Hexanal 1-Heptanal 1-Octanal 1-Nonanal 1-Decanal

CH2O C2H4O C3H6O C4H8O C5H10O C6H12O C7H14O C8H16O C9H18O C10H20O

50000 75070 123386 123728 110623 66251 111717 124130 124196 112312

30.026 44.053 58.080 72.107 86.134 100.161 114.188 128.214 142.241 156.268

1.9415 1.6994 1.296 1.0361 0.83871 0.71899 0.62649 0.56833 0.49587 0.46802

0.22309 0.26167 0.26439 0.26731 0.26252 0.26531 0.26376 0.26939 0.26135 0.27146

408 466 504.4 537.2 566.1 591 617 638.1 658 674.2

0.28571 0.2913 0.29471 0.28397 0.29444 0.27628 0.29221 0.26975 0.30736 0.26869

181.15 150.15 170.00 176.75 182.00 217.15 229.80 246.00 255.15 267.15

30.945 21.499 15.929 12.589 10.534 8.7243 7.6002 6.6637 6.0165 5.3834

408 466 504.4 537.2 566.1 591 617 638.1 658 674.2

8.703 6.494 4.902 3.876 3.195 2.710 2.375 2.110 1.897 1.724

123 124 125 126

C3H6O C4H8O C5H10O C5H10O

67641 78933 107879 563804

58.080 72.107 86.134 86.134

1.2332 0.93767 0.90411 0.8374

0.25886 0.25035 0.27207 0.26204

508.2 535.5 561.08 553

0.2913 0.29964 0.30669 0.2857

178.45 186.48 196.29 181.15

15.683 12.663 10.398 10.565

508.2 535.5 561.08 553

4.764 3.745 3.323 3.196

127 128 129 130 131 132 133 134

Acetone Methyl ethyl ketone 2-Pentanone Methyl isopropyl ketone1 2-Hexanone Methyl isobutyl ketone 3-Methyl-2-pentanone1 3-Pentanone Ethyl isopropyl ketone Diisopropyl ketone Cyclohexanone Methyl phenyl ketone

C6H12O C6H12O C6H12O C5H10O C6H12O C7H14O C6H10O C8H8O

591786 108101 565617 96220 565695 565800 108941 98862

100.161 100.161 100.161 86.134 100.161 114.188 98.145 120.151

0.70659 0.71791 0.6969 0.71811 0.66469 0.56213 0.8663 0.64417

0.26073 0.26491 0.2587 0.24129 0.24527 0.23385 0.26941 0.24863

587.05 571.4 573 560.95 567 576 653 709.5

0.2963 0.28544 0.2857 0.27996 0.34305 0.2618 0.2977 0.28661

217.35 189.15 167.15 234.18 200.00 204.81 242.00 292.81

8.7505 8.8579 9.1722 10.102 9.0933 8.7779 10.081 8.5581

587.05 571.4 573 560.95 567 576 653 709.5

2.710 2.710 2.694 2.976 2.710 2.404 3.216 2.591

135 136 137 138 139 140 141

Formic acid Acetic acid Propionic acid n-Butyric acid Isobutyric acid Benzoic acid1 Acetic anhydride

CH2O2 C2H4O2 C3H6O2 C4H8O2 C4H8O2 C7H6O2 C4H6O3

64186 64197 79094 107926 79312 65850 108247

46.026 60.053 74.079 88.106 88.106 122.123 102.090

1.938 1.4486 1.1041 0.89213 0.88575 0.71587 0.86852

0.24225 0.25892 0.25659 0.25938 0.25736 0.24812 0.25187

588 591.95 600.81 615.7 605 751 606

0.24435 0.2529 0.26874 0.24909 0.26265 0.2857 0.31172

281.45 289.81 252.45 267.95 227.15 395.45 200.15

26.806 17.492 13.933 11.087 11.42 8.8935 11.643

588 591.95 600.81 615.7 605 751 606

8.000 5.595 4.303 3.440 3.442 2.885 3.448

142 143 144 145 146 147 148

Methyl formate Methyl acetate Methyl propionate Methyl n-butyrate Ethyl formate Ethyl acetate Ethyl propionate

C2H4O2 C3H6O2 C4H8O2 C5H10O2 C3H6O2 C4H8O2 C5H10O2

107313 79209 554121 623427 109944 141786 105373

60.053 74.079 88.106 102.133 74.079 88.106 102.133

1.525 1.13 0.9147 0.76983 1.1343 0.8996 0.7405

0.2634 0.2593 0.2594 0.26173 0.26168 0.25856 0.25563

487.2 506.55 530.6 554.5 508.4 523.3 546

0.2806 0.2764 0.2774 0.26879 0.2791 0.278 0.2795

174.15 175.15 185.65 187.35 193.55 189.60 199.25

18.811 14.475 11.678 9.7638 14.006 11.478 9.6317

487.2 506.55 530.6 554.5 508.4 523.3 546

5.790 4.358 3.526 2.941 4.335 3.479 2.897

149 150 151 152 153 154 155

Ethyl n-butyrate n-Propyl formate n-Propyl acetate n-Butyl acetate Methyl benzoate Ethyl benzoate Vinyl acetate

C6H12O2 C4H8O2 C5H10O2 C6H12O2 C8H8O2 C9H10O2 C4H6O2

105544 110747 109604 123864 93583 93890 108054

116.160 88.106 102.133 116.160 136.150 150.177 86.090

0.63566 0.915 0.73041 0.669 0.53944 0.4883 0.9591

0.25613 0.26134 0.25456 0.26028 0.23519 0.23878 0.2593

571 538 549.73 579.15 693 698 519.13

0.27829 0.28 0.27666 0.309 0.2676 0.28487 0.27448

175.15 180.25 178.15 199.65 260.75 238.45 180.35

8.4912 11.59 9.7941 8.3747 8.2133 7.2924 12.287

571 538 549.73 579.15 693 698 519.13

2.482 3.501 2.869 2.570 2.294 2.045 3.699

156 157 158 159 160

Methylamine Dimethylamine Trimethylamine Ethylamine Diethylamine

CH5N C2H7N C3H9N C2H7N C4H11N

74895 124403 75503 75047 109897

31.057 45.084 59.111 45.084 73.138

1.39 1.5436 1.0116 1.1477 0.85379

0.21405 0.27784 0.25683 0.23182 0.25675

430.05 437.2 433.25 456.15 496.6

0.2275 0.2572 0.2696 0.26053 0.27027

179.69 180.96 156.08 192.15 223.35

25.378 16.964 13.144 17.588 10.575

430.05 437.2 433.25 456.15 496.6

6.494 5.556 3.939 4.951 3.325

2-96

Name

Formula

161 162 163 164 165 166 167 168

Triethylamine n-Propylamine di-n-Propylamine Isopropylamine Diisopropylamine Aniline N-Methylaniline N,N-Dimethylaniline

C6H15N C3H9N C6H15N C3H9N C6H15N C6H7N C7H9N C8H11N

121448 107108 142847 75310 108189 62533 100618 121697

101.192 59.111 101.192 59.111 101.192 93.128 107.155 121.182

0.7035 0.9195 0.659 1.2801 0.6181 1.0405 0.6527 0.4923

0.27386 0.23878 0.26428 0.2828 0.25786 0.2807 0.24324 0.22868

535.15 496.95 550 471.85 523.1 699 701.55 687.15

0.2872 0.2461 0.2766 0.2972 0.271 0.29236 0.25374 0.2335

158.45 188.36 210.15 177.95 176.85 267.13 216.15 275.60

8.2843 13.764 7.9929 13.561 8.0541 11.176 9.7244 7.9705

535.15 496.95 550 471.85 523.1 699 701.55 687.15

2.569 3.851 2.494 4.527 2.397 3.707 2.683 2.153

169 170 171 172

Ethylene oxide Furan Thiophene Pyridine

C2H4O C4H4O C4H4S C5H5N

75218 110009 110021 110861

44.053 68.075 84.142 79.101

1.836 1.1339 1.2875 0.9815

0.26024 0.24741 0.28195 0.24957

469.15 490.15 579.35 619.95

0.2696 0.2612 0.3077 0.29295

160.65 187.55 234.94 231.51

23.477 15.702 13.431 13.193

469.15 490.15 579.35 619.95

7.055 4.583 4.566 3.933

173 174

CH3NO C3H7NO

75127 68122

45.041 73.095

1.2486 0.89615

0.20352 0.23478

771 649.6

0.25178 0.28091

275.60 212.72

25.488 13.954

771 649.6

6.135 3.817

175 176 177 178 179 180

Formamide5 N,N-Dimethylformamide Acetamide N-Methylacetamide Acetonitrile Propionitrile n-Butyronitrile Benzonitrile

C2H5NO C3H7NO C2H3N C3H5N C4H7N C7H5N

60355 79163 75058 107120 109740 100470

59.068 73.095 41.053 55.079 69.106 103.123

1.016 0.88268 1.3064 1.0224 0.87533 0.73136

0.21845 0.23568 0.22597 0.23452 0.24331 0.24793

761 718 545.5 564.4 582.25 699.35

0.26116 0.27379 0.28678 0.2804 0.28586 0.2841

353.33 301.15 229.32 180.26 161.25 260.40

16.936 13.012 20.628 16.027 13.047 10.009

761 718 545.5 564.4 582.25 699.35

4.651 3.745 5.781 4.360 3.598 2.950

181 182 183 184 185 186 187 188 189

Methyl mercaptan Ethyl mercaptan n-Propyl mercaptan n-Butyl mercaptan Isobutyl mercaptan sec-Butyl mercaptan Dimethyl sulfide Methyl ethyl sulfide Diethyl sulfide

CH4S C2H6S C3H8S C4H10S C4H10S C4H10S C2H6S C3H8S C4H10S

74931 75081 107039 109795 513440 513531 75183 624895 352932

48.109 62.136 76.163 90.189 90.189 90.189 62.136 76.163 90.189

1.9323 1.3047 1.0714 0.89458 0.88801 0.89137 1.4029 1.067 0.82413

0.28018 0.2694 0.27214 0.27463 0.27262 0.27365 0.27991 0.27101 0.26333

469.95 499.15 536.6 570.1 559 554 503.04 533 557.15

0.28523 0.27866 0.29481 0.28512 0.29522 0.2953 0.2741 0.29363 0.27445

150.18 125.26 159.95 157.46 128.31 133.02 174.88 167.23 169.20

21.564 16.242 12.716 10.585 10.851 10.761 15.556 12.672 10.476

469.95 499.15 536.6 570.1 559 554 503.04 533 557.15

6.897 4.843 3.937 3.257 3.257 3.257 5.012 3.937 3.130

190 191 192 193 194 195 196 197

Fluoromethane Chloromethane Trichloromethane Tetrachloromethane Bromomethane Fluoroethane Chloroethane Bromoethane

CH3F CH3Cl CHCl3 CCl4 CH3Br C2H5F C2H5Cl C2H5Br

593533 74873 67663 56235 74839 353366 75003 74964

34.033 50.488 119.377 153.822 94.939 48.060 64.514 108.966

2.1854 1.817 1.0841 0.99835 1.6762 1.6525 2.176 1.1908

0.24725 0.25877 0.2581 0.274 0.26141 0.27099 0.3377 0.25595

317.42 416.25 536.4 556.35 467 375.31 460.35 503.8

0.27558 0.2833 0.2741 0.287 0.28402 0.2442 0.3361 0.29152

131.35 175.43 209.63 250.33 179.47 129.95 134.80 154.55

29.526 22.347 13.702 10.843 20.64 19.785 16.934 15.833

317.42 416.25 536.4 556.35 467 375.31 460.35 503.8

8.839 7.022 4.200 3.644 6.412 6.098 6.444 4.653

198 199 200 201 202 203 204 205

1-Chloropropane 2-Chloropropane 1,1-Dichloropropane1 1,2-Dichloropropane Vinyl chloride Fluorobenzene Chlorobenzene Bromobenzene

C3H7Cl C3H7Cl C3H6Cl2 C3H6Cl2 C2H3Cl C6H5F C6H5Cl C6H5Br

540545 75296 78999 78875 75014 462066 108907 108861

78.541 78.541 112.986 112.986 62.499 96.104 112.558 157.010

1.087 1.1202 0.91064 0.89833 1.5115 1.0146 0.8711 0.8226

0.26832 0.27669 0.26561 0.26142 0.2707 0.27277 0.26805 0.26632

503.15 489 560 572 432 560.09 632.35 670.15

0.28055 0.27646 0.28571 0.2868 0.2716 0.28291 0.2799 0.2821

150.35 155.97 200.00 172.71 119.36 230.94 227.95 242.43

13.328 12.855 11.03 11.526 18.481 11.374 10.385 9.9087

503.15 489 560 572 432 560.09 632.35 670.15

4.051 4.049 3.429 3.436 5.584 3.720 3.250 3.089

206 207 208 209 210 211 212 213 214 215 216 217

Air Hydrogen Helium-44 Neon Argon Fluorine Chlorine Bromine Oxygen Nitrogen Ammonia Hydrazine

H2 He Ne Ar F2 Cl2 Br2 O2 N2 NH3 N2H4

132259100 1333740 7440597 7440019 7440371 7782414 7782505 7726956 7782447 7727379 7664417 302012

28.951 2.016 4.003 20.180 39.948 37.997 70.905 159.808 31.999 28.014 17.031 32.045

2.8963 5.414 7.2475 7.3718 3.8469 4.2895 2.23 2.1872 3.9143 3.2091 3.5383 1.0516

0.26733 0.34893 0.41865 0.3067 0.2881 0.28587 0.27645 0.29527 0.28772 0.2861 0.25443 0.16613

132.45 33.19 5.2 44.4 150.86 144.12 417.15 584.15 154.58 126.2 405.65 653.15

0.27341 0.2706 0.24096 0.2786 0.29783 0.28776 0.2926 0.3295 0.2924 0.2966 0.2888 0.1898

59.15 13.95 2.20 24.56 83.78 53.48 172.12 265.85 54.35 63.15 195.41 274.69

33.279 38.487 37.115 61.796 35.491 44.888 24.242 20.109 40.77 31.063 43.141 31.934

132.45 33.19 5.2 44.4 150.86 144.12 417.15 584.15 154.58 126.2 405.65 653.15

10.834 15.516 17.312 24.036 13.353 15.005 8.067 7.408 13.605 11.217 13.907 6.330 2-97

TABLE 2-30 Cmpd. no. 218 219 220 221 222 223 224 225 226 227 228 229 230 231

Densities of Inorganic and Organic Liquids (Concluded ) Name

Nitrous oxide Nitric oxide Cyanogen Carbon monoxide Carbon dioxide Carbon disulfide Hydrogen fluoride Hydrogen chloride Hydrogen bromide1 Hydrogen cyanide Hydrogen sulfide Sulfur dioxide Sulfur trioxide Water7

Formula N2O NO C2N2 CO CO2 CS2 HF HCl HBr HCN H2S SO2 SO3 H2O

CAS no.

Mol. wt.

C1

C2

C3

C4

Tmin, K

10024972 10102439 460195 630080 124389 75150 7664393 7647010 10035106 74908 7783064 7446095 7446119 7732185

44.013 30.006 52.036 28.010 44.010 76.143 20.006 36.461 80.912 27.026 34.082 64.065 80.064 18.015

2.781 5.246 1.0761 2.897 2.768 1.7968 2.5635 3.342 2.832 1.3413 2.7672 2.106 1.4969 5.459

0.27244 0.3044 0.20984 0.27532 0.26212 0.28749 0.1766 0.2729 0.2832 0.18589 0.27369 0.25842 0.19013 0.30542

309.57 180.15 400.15 132.92 304.21 552 461.15 324.65 363.15 456.65 373.53 430.75 490.85 647.13

0.2882 0.242 0.20635 0.2813 0.2908 0.3226 0.3733 0.3217 0.28571 0.28206 0.29015 0.2895 0.4359 0.081

182.30 109.50 245.25 68.15 216.58 161.11 189.79 158.97 185.15 259.83 187.68 197.67 289.95 273.16

Density at Tmin 27.928 44.487 18.513 30.18 26.828 19.064 60.203 34.854 27.985 27.202 29.13 25.298 24.241 55.583

Tmax, K

Density at Tmax

309.57 180.15 400.15 132.92 304.21 552 461.15 324.65 363.15 456.65 373.53 430.75 490.85 333.15

10.208 17.234 5.128 10.522 10.560 6.250 14.516 12.246 10.000 7.216 10.111 8.150 7.873 54.703

All substances are listed in alphabetical order in Table 2-6a. Compiled from Daubert, T. E., R. P. Danner, H. M. Sibul, and C. C. Stebbins, DIPPR Data Compilation of Pure Compound Properties, Project 801 Sponsor Release, July, 1993, Design Institute for Physical Property Data, AIChE, New York, NY; and from Thermodynamics Research Center, “Selected Values of Properties of Hydrocarbons and Related Compounds,” Thermodynamics Research Center Hydrocarbon Project, Texas A&M University, College Station, Texas (extant 1994). Temperatures are in kelvins. Liquid densities are in kmol/m3. Density formulas: kmol/m3 × (mol. wt./1E+03) = g/cm3; kmol/m3 × (mol. wt./1.601846E+01) = lb/ft3. C The liquid density equation is C1/C2[1 + (1 − T/C3) 4] unless otherwise noted. 1 The modified Rackett equation, density = (Pc /RTc)/ZRA1 + [1 − (T/Tc)]2/7, was used. See Spencer, C. F., and R. P. Danner, “Improved Equation for Prediction of Saturated Liquid Density,” J. Chem. Eng. Data 17, 236 (1972). 2 Decomposes violently on heating. Forms explosive peroxides with air or oxygen. Polymerizes under pressure and heat. 3 For the hypothetical pure liquid. 4 Exhibits superfluid properties below 2.2 K. 5 Coefficients are hypothetical above the decomposition temperature. 6 Lower limit is for the undercooled liquid. 7 For the temperature range 333.15 to 403.15 K, use the coefficients: C1 = 4.9669E+00, C2 = 2.7788E−01, C3 = 6.4713E+02, C4 = 1.8740E−01. For the temperature range 403.15 to 647.13 K, use C1 = 4.3910E+00, C2 = 2.4870E−01, C3 = 6.4713E+02, C4 = 2.5340E−01.

2-98

DENSITIES OF AQUEOUS INORGANIC SOLUTIONS

2-99

DENSITIES OF AQUEOUS INORGANIC SOLUTIONS UNITS AND UNITS CONVERSIONS Densities are given in grams per cubic centimeter. To convert to pounds per cubic foot, multiply by 62.43. °F = 9⁄ 5 °C + 32. ADDITIONAL REFERENCES For more detailed data on densities see International Critical Tables: tabular index, vol. 3, p. 1; abrasives, vol. 2, p. 87; air, moist, vol. 1, p. 71; building stones, vol. 2, p. 52; clays, vol. 2, p. 56; coals, vol. 2, p. 135; compounds, vol. 1. pp. 106, 176, 313, 341; elements, vol. 1, pp. 102, 340; fibers, vol. 2, p. 237; gases and vapors, vol. 3, pp. 3, 345; glass, vol. 2, p. 93; liquids and vitreous solids, vol. 3, p. 22; vol. 1, pp. 102, 340; vol. 2, pp. 456, 463; vol. 3, pp. 20, 35; liquid coolants and saturated TABLE 2-31

Aluminum Sulfate [Al2(SO4)3] d 15 4

%

d 15 4

1 2 4 8 12

1.0093 1.0195 1.0404 1.0837 1.1293

16 20 24 26

1.1770 1.2272 1.2803 1.3079

0°C

5°C

10°C

20°C

25°C

d 15 4

%

0.9943 0.9954 0.9959 0.9958 0.9955 0.9939 0.993 32 0.889 .9906 .9915 .9919 .9917 .9913 .9895 .988 36 .877 .9834 .9840 .9842 .9837 .9832 .9811 .980 40 .865 0.970 .9701 .9701 .9695 .9686 .9677 .9651 .964 45 .849 .958 .9576 .9571 .9561 .9548 .9534 .9501 .948 50 .832 .947 .9461 .9450 .9435 .9420 .9402 .9362 .934 60 .796 .9353 .9335 .9316 .9296 .9275 .9229 70 .755 .9249 .9226 .9202 .9179 .9155 .9101 80 .711 .9150 .9122 .9094 .9067 .9040 .8980 90 .665 .9101 .9070 .9040 .9012 .8983 .8920 100 .618

TABLE 2-33 Ammonium Acetate* (CH3COONH4)

TABLE 2-34 Ammonium Bichromate [(NH4)2Cr2O7]

%

d425

%

d412

1 2 4 8 12 16 20 24 28 30 35 40 45

0.9992 1.0013 1.0055 1.0136 1.0216 1.0294 1.0368 1.0439 1.0507 1.0540 1.0618 1.0691 1.0760

1 2 4 8 12 16 20

1.0051 1.0108 1.0223 1.0463 1.0715 1.0981 1.1263

*For data at 16°C for 3(1)52 percent see Atack Handbook of Chemical Data, p. 33, Reinhold, New York, 1957. TABLE 2-35

%

°C

d 4t

3.80 10.52 19.75 28.04

20 13 13.7 19.6

1.0219 1.0627 1.1189 1.1707

Ammonia (NH3)

% −15°C −10°C −5°C 1 2 4 8 12 16 20 24 28 30

TABLE 2-36 Ammonium Chromate [(NH4)2CrO4]

%

TABLE 2-32

vapors are available from WADC-TR-59-598, 1959; plastics are collected in the Handbook of Chemistry and Physics, Chemical Rubber Publishing Co.: solid helium, neon, argon, fluorine, and methane data are given by Johnson (ed.), WADD-TR-60-56, 1960; temperatures of maximum solubility, vol. 3, p. 107; metals, vol. 2, p. 463; oils, fats, and waxes, vol. 2, p. 201; orthobaric, vol. 3, pp. 202, 228, 237, 244; petroleums, vol. 2, pp. 137, 144; plastics, vol. 2, p. 296; porcelains, vol. 2, pp. 68, 75; refrigerating brines, vol. 2, p. 327; rubber, vol. 2, pp. 255, 259; soaps, vol. 5, p. 447; metallic solid solutions, vol. 2, p. 358; solids, vol. 3, pp. 43, 45; vol. 2, p. 456; vol. 3, p. 21; solutions and mixtures, vol. 3, pp. 17, 51, 95, 104, 107, 111, 125, 130; woods, vol. 2, p. 1. Also see the Handbook of Chemistry and Physics, Chemical Rubber Publishing Co., 40th ed., etc.

TABLE 2-37

Ammonium Nitrate (NH4NO3)

%

0°C

10°C

25°C

40°C

60°C

80°C

1.0 2.0 4.0 8.0 12.0 16.0 20.0 24.0 28.0 30.0 40.0 50.0

1.0043 1.0088 1.0178 1.0358 1.0539 1.0721 1.0905 1.1090 1.1277 1.1371 1.1862 1.2380

1.0039 1.0082 1.0168 1.0340 1.0515 1.0691 1.0870 1.1051 1.1234 1.1327 1.1810 1.2320

1.0011 1.0051 1.0132 1.0297 1.0464 1.0633 1.0806 1.0982 1.1161 1.1252 1.1727 1.2229

0.9961 1.0000 1.0079 1.0238 1.0400 1.0565 1.0734 1.0907 1.1082 1.1171 1.1640 1.2136

0.9870 .9908 .9985 1.0142 1.0301 1.0462 1.0627 1.0796 1.0968 1.1055 1.1515 1.2006

0.9755 .9793 .9869 1.0024 1.0181 1.0342 1.0506 1.0673 1.0844 1.0931 1.1385 1.1868

TABLE 2-38

Ammonium Sulfate [(NH4)2SO4]

%

0°C

20°C

40°C

80°C

100°C

1 2 4 8 12 16 20 24 28 35 40 50

1.0061 1.0124 1.0248 1.0495 1.0740 1.0980 1.1215 1.1448 1.1677 1.2072 1.2350 1.2899

1.0041 1.0101 1.0220 1.0456 1.0691 1.0924 1.1154 1.1383 1.1609 1.2800 1.2277 1.2825

0.9980 1.0039 1.0155 1.0387 1.0619 1.0849 1.1077 1.1304 1.1529 1.1919 1.2196 1.2745

0.9777 .9836 .9953 1.0187 1.0421 1.0653 1.0883 1.1111 1.1338 1.1731 1.2011 1.2568

0.9644 .9705 .9826 1.0066 1.0303 1.0539 1.0772 1.1003 1.1232 1.1629 1.1910 1.2466

Ammonium Chloride (NH4Cl)

%

0°C

10°C

20°C

30°C

50°C

80°C

100°C

1 2 4 8 12 16 20 24

1.0033 1.0067 1.0135 1.0266 1.0391 1.0510 1.0625 1.0736

1.0029 1.0062 1.0126 1.0251 1.0370 1.0485 1.0596 1.0705

1.0013 1.0045 1.0107 1.0227 1.0344 1.0457 1.0567 1.0674

0.9987 1.0018 1.0077 1.0195 1.0310 1.0422 1.0532 1.0641

0.9910 .9940 .9999 1.0116 1.0231 1.0343 1.0454 1.0564

0.9749 .9780 .9842 .9963 1.0081 1.0198 1.0312 1.0426

0.9617 .9651 .9718 .9849 .9975 1.0096 1.0213 1.0327

TABLE 2-39

Arsenic Acid (H3A3O4)

%

d 15 4

%

d 15 4

1 2 6 10 16

1.0057 1.0124 1.0398 1.0681 1.1128

20 30 40 50 60 70

1.1447 1.2331 1.3370 1.4602 1.6070 1.7811

2-100

PHYSICAL AND CHEMICAL DATA

TABLE 2-40

Barium Chloride (BaCl2)

TABLE 2-46

Chromic Acid (CrO3)

%

0°C

20°C

40°C

60°C

80°C

100°C

%

d 15 4

%

d 15 4

2 4 8 12 16 20 24 26

1.0181 1.0368 1.0760 1.1178 1.1627 1.2105

1.0159 1.0341 1.0721 1.1128 1.1564 1.2031 1.2531 1.2793

1.0096 1.0275 1.0648 1.1047 1.1478 1.1938 1.2430 1.2688

1.0004 1.0181 1.0551 1.0948 1.1373 1.1828 1.2316 1.2571

0.9890 1.0066 1.0434 1.0827 1.1249 1.1702 1.2186 1.2440

0.9755 .9931 1.0299 1.0692 1.1113 1.1563 1.2045 1.2298

1 2 6 10 16

1.006 1.014 1.045 1.076 1.127

20 26 30 40 50 60

1.163 1.220 1.260 1.371 1.505 1.663

TABLE 2-41

TABLE 2-47

d 18 4

%

d 18 4

2 4 8 12 16

1.0154 1.0326 1.0683 1.1061 1.1468

20 25 30 40 50

1.1904 1.2488 1.3124 1.4590 1.6356

TABLE 2-42 % −5°C 2 4 8 12 16 20 25 30 35 40

Cadmium Nitrate [Cd(NO3)2]

%

1.0708 1.1083 1.1471 1.1874

Chromium Chloride (CrCl3) d 18 4

%

Violet

Green

Equilibrium mixture of violet and green

1 2 4 8 12 14

1.0076 1.0166 1.0349 1.0724 1.1114 1.1316

1.0071 1.0157 1.0332 1.0691 1.1065

1.0075 1.0165 1.0347 1.0722 1.1111

Calcium Chloride (CaCl2)

0°C

20°C

30°C

40°C

60°C

1.0171 1.0346 1.0703 1.1072 1.1454 1.1853 1.2376 1.2922

1.0148 1.0316 1.0659 1.1015 1.1386 1.1775 1.2284 1.2816 1.3373 1.3957

1.0120 1.0286 1.0626 1.0978 1.1345 1.1730 1.2236 1.2764 1.3316 1.3895

1.0084 1.0249 1.0586 1.0937 1.1301 1.1684 1.2186 1.2709 1.3255 1.3826

0.9994 1.0158 1.0492 1.0840 1.1202 1.1581 1.2079 1.2597 1.3137 1.3700

80°C 100°C 120°C* 140°C 0.9881 1.0046 1.0382 1.0730 1.1092 1.1471 1.1965 1.2478 1.3013 1.3571

0.9748 .9915 1.0257 1.0610 1.0973 1.1352 1.1846 1.2359 1.2893 1.3450

0.9596 .9765 1.0111 1.0466 1.0835 1.1219

0.9428 .9601 .9954 1.0317 1.0691 1.1080

TABLE 2-48

Copper Nitrate [Cu(NO3)2]

%

d 20 4

%

d 20 4

1 2 4 8

1.007 1.015 1.032 1.069

12 16 20 25

1.107 1.147 1.189 1.248

*Corrected to atmospheric pressure.

TABLE 2-43 Calcium Hydroxide [Ca(OH)2]

TABLE 2-44 Calcium Hypochlorite* (CaOCl2)

TABLE 2-49 (CuSO4)

Copper Sulfate

TABLE 2-50 Cuprous Chloride (Cu2Cl2)

%

d 15 4

d 425

% total salt

d 15 4

%

0°C

20°C

40°C

%

0°C

20°C

40°C

0.05 .10 .15

0.99979 1.00044 1.00110

0.99773 .99838 .99904

2 4 6 8 10 12

1.0169 1.0345 1.0520 1.0697 1.0876 1.1060

1 4 8 12 16 18

1.0104 1.0429 1.0887 1.1379

1.0086 1.0401 1.084 1.1308 1.180 1.206

1.0024 1.0332 1.0764 1.1222

1 4 8 12 16 20

1.0095 1.0387 1.0788 1.1208 1.1653 1.2121

1.0072 1.036 1.0754 1.1165 1.1595 1.2052

1.002 1.0305 1.0682 1.107 1.151 1.1953

*CaOCl2 = 89.15% CaCl2 = 7.31% Ca(ClO3)2 = 0.26% Ca(OH)2 = 2.92%.

TABLE 2-45

Calcium Nitrate [Ca(NO3)2]

TABLE 2-51

Ferric Chloride (FeCl3)

%

6°C

18°C

25°C

30°C

%

0°C

10°C

20°C

30°C

2* 4 8 12 16 20 25 30 35 40 45 68*

1.0157 1.0316 1.0641 1.0979 1.1330 1.1694 1.2168

1.0137 1.0291 1.0608 1.0937 1.1279 1.1636 1.2106 1.260 1.311 1.365 1.422 1.747

1.0120 1.0272 1.0585 1.0911 1.1250 1.1602 1.2065

1.0105 1.0256 1.0565 1.0887 1.1224 1.1575 1.2032

1.741

1.736

1 2 4 8 12 16 20 25 30 35 40 45 50

1.0086 1.0174 1.0347 1.0703 1.1088 1.1475 1.1870 1.2400 1.2970 1.3605 1.4280

1.0084 1.0168 1.0341 1.0692 1.1071 1.1449 1.1847 1.2380 1.2950 1.3580 1.4235 1.4920 1.5610

1.0068 1.0152 1.0324 1.0669 1.1040 1.1418 1.1820 1.2340 1.2910 1.3530 1.4175 1.4850 1.5510

1.0040 1.0122 1.0292 1.0636 1.1006 1.1386 1.1786 1.2290 1.2850 1.3475 1.4115

*Supercooled tetrahydrate (m.p. 41.4°C).

DENSITIES OF AQUEOUS INORGANIC SOLUTIONS TABLE 2-52 [Fe2(SO4)3]

Ferric Sulfate

TABLE 2-53 [Fe(NO3)3]

17.5

%

d4

1 2 4 8 12 16 20 30 40 50 60

1.0072 1.0157 1.0327 1.0670 1.1028 1.1409 1.1811 1.3073 1.4487 1.6127 1.7983

TABLE 2-54 Ferrous Sulfate (FeSO4) % 0.2 0.4 0.8 1.0 4.0 8.0 12.0 16.0 20.0

Ferric Nitrate

d4

d4

%

d4

%

d4

1 2 4 8 12 16 20 25

1.0065 1.0144 1.0304 1.0636 1.0989 1.1359 1.1748 1.2281

5 10 20 30 40 50 60 70 80 90 95 100

1.020 1.040 1.080 1.119 1.159 1.198 1.235 1.258 1.259 1.178 1.089 1.0005

1.017 1.035 1.070 1.101 1.130 1.155

1 2 4 6 8 10 12 14 16 18 20 22 24

1.0022 1.0058 1.0131 1.0204 1.0277 1.0351 1.0425 1.0499 1.0574 1.0649 1.0725 1.0802 1.0880

26 28 30 35 40 45 50 55 60 70 80 90 100

1.0959 1.1040 1.1122 1.1327 1.1536 1.1749 1.1966 1.2188 1.2416 1.2897 1.3406 1.3931 1.4465

4

10

25

TABLE 2-60

20°C

%

d4

d4

d4

%

1.0090 1.0380 1.0790 1.1235 1.1690 1.2150

1.00068 1.00275 1.00645 1.0085 1.0375 1.0785 1.1220 1.1675 1.2135

1.0002 1.0022 1.0062 1.0082

1.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0 22.0 24.0 26.0 28.0 30.0 40.0 50.0 60.0 65.0

1.0073 1.0146 1.0295 1.0448 1.0604 1.0764 1.0928 1.1097 1.1272 1.1453 1.1640 1.1832 1.2030 1.2235 1.2446 1.2663 1.3877 1.5305 1.6950 1.7854

1.0068 1.0139 1.0285 1.0435 1.0589 1.0747 1.0910 1.1078 1.1251 1.1430 1.1615 1.1806 1.2003 1.2206 1.2415 1.2630 1.3838 1.5257 1.6892 1.7792

1.0041 1.0111 1.0255 1.0402 1.0552 1.0707 1.0867 1.1032 1.1202 1.1377 1.1557 1.1743 1.1935 1.2134 1.2340 1.2552 1.3736 1.5127 1.6731 1.7613

1 2 4 8 12

15

1 2 4 8 12 16 82 90 100

0.998 .996 .993 .984 .971 .956 .752 .724 .691

TABLE 2-57

18

%

18

18°C

d4

18

d4

15°C

%

0

%

TABLE 2-55 Hydrogen Bromide (HBr)

TABLE 2-56 Hydrogen Cyanide (HCN)

TABLE 2-59 Hydrogen Peroxide (H2O2)

TABLE 2-58 Hydrogen Fluoride (HF) 20

2-101

Hydrofluosilic Acid (H2SiF6) 17.5

d4

1.0080 1.0161 1.0324 1.0661 1.1011

TABLE 2-61

Hydrogen Chloride (HCl)

−5°C

0°C

10°C

20°C

40°C

60°C

80°C

100°C

1 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

1.0048 1.0104 1.0213 1.0321 1.0428 1.0536 1.0645 1.0754 1.0864 1.0975 1.1087 1.1200 1.1314 1.1426 1.1537 1.1648

1.0052 1.0106 1.0213 1.0319 1.0423 1.0528 1.0634 1.0741 1.0849 1.0958 1.1067 1.1177 1.1287 1.1396 1.1505 1.1613

1.0048 1.0100 1.0202 1.0303 1.0403 1.0504 1.0607 1.0711 1.0815 1.0920 1.1025 1.1131 1.1238 1.1344 1.1449 1.1553

1.0032 1.0082 1.0181 1.0279 1.0376 1.0474 1.0574 1.0675 1.0776 1.0878 1.0980 1.1083 1.1187 1.1290 1.1392 1.1493 1.1593 1.1691 1.1789 1.1885 1.1980

0.9970 1.0019 1.0116 1.0211 1.0305 1.0400 1.0497 1.0594 1.0692 1.0790 1.0888 1.0986 1.1085 1.1183 1.1280 1.1376

0.9881 .9930 1.0026 1.0121 1.0215 1.0310 1.0406 1.0502 1.0598 1.0694 1.0790 1.0886 1.0982 1.1076 1.1169 1.1260

0.9768 0.9819 0.9919 1.0016 1.0111 1.0206 1.0302 1.0398 1.0494 1.0590 1.0685 1.0780 1.0874 1.0967 1.1058 1.1149

0.9636 .9688 .9791 .9892 .9992 1.0090 1.0188 1.0286 1.0383 1.0479 1.0574 1.0668 1.0761 1.0853 1.0942 1.1030

d4

16 20 25 30 34

1.1373 1.1748 1.2235 1.2742 1.3162

Magnesium Chloride (MgCl2)

%

0°C

20°C

40°C

60°C

80°C

100°C

2 4 8 12 16 20 25 30

1.0168 1.0338 1.0683 1.1035 1.1395 1.1764 1.2246 1.2754

1.0146 1.0311 1.0646 1.0989 1.1342 1.1706 1.2184 1.2688

1.0084 1.0248 1.0580 1.0921 1.1272 1.1635 1.2111 1.2614

0.9995 1.0159 1.0493 1.0836 1.1188 1.1552 1.2031 1.2535

0.9883 1.0050 1.0388 1.0735 1.1092 1.1460 1.1942 1.2451

0.9753 .9923 1.0269 1.0622 1.0984 1.1359 1.1847 1.2360

TABLE 2-62

%

17.5

%

Magnesium Sulfate (MgSO4)

%

0°C

20°C

30°C

40°C

50°C

60°C

80°C

2 4 8 12 16 20 26

1.0210 1.0423 1.0858 1.1309 1.1777 1.2264 1.3032

1.0186 1.0392 1.0816 1.1256 1.1717 1.2198 1.2961

1.0158 1.0362 1.0782 1.1220 1.1679 1.2159 1.2922

1.0123 1.0326 1.0743 1.1179 1.1637 1.2117 1.2879

1.0081 1.0283 1.0700 1.1135 1.1592 1.2072 1.2836

1.0032 1.0234 1.0650 1.1083

0.9916 1.0118 1.0534 1.0968

TABLE 2-63 Nickel Chloride (NiCl2) 18

TABLE 2-64 Nickel Nitrate [Ni(NO3)2] 20

TABLE 2-65 Nickel Sulfate (NiSO4 ) 18

%

d4

%

d4

%

d4

1 2 4 8 12 16 20 30

1.0082 1.0179 1.0375 1.0785 1.1217 1.1674 1.2163 1.353

1 2 4 8 12 16 20 30 35

1.0065 1.0150 1.0325 1.0688 1.1070 1.1480 1.191 1.311 1.377

1 2 4 8 12 16 18

1.0091 1.0198 1.0415 1.0852 1.1325 1.1825 1.2090

2-102

PHYSICAL AND CHEMICAL DATA

TABLE 2-66 %

Nitric Acid (HNO3)

0°C

5°C

10°C

15°C

20°C

25°C

30°C

40°C

50°C

60°C

80°C

100°C

1 2 3 4

1.0058 1.0117 1.0176 1.0236

1.00572 1.01149 1.01730 1.02315

1.00534 1.01099 1.01668 1.02240

1.00464 1.01018 1.01576 1.02137

1.00364 1.00909 1.01457 1.02008

1.00241 1.00778 1.01318 1.01861

1.0009 1.0061 1.0114 1.0168

0.9973 1.0025 1.0077 1.0129

0.9931 .9982 1.0033 1.0084

0.9882 .9932 .9982 1.0033

0.9767 .9816 .9865 .9915

0.9632 .9681 .9730 .9779

5 6 7 8 9

1.0296 1.0357 1.0418 1.0480 1.0543

1.02904 1.03497 1.0410 1.0471 1.0532

1.02816 1.03397 1.0399 1.0458 1.0518

1.02702 1.03272 1.0385 1.0443 1.0502

1.02563 1.03122 1.0369 1.0427 1.0485

1.02408 1.02958 1.0352 1.0409 1.0466

1.0222 1.0277 1.0333 1.0389 1.0446

1.0182 1.0235 1.0289 1.0344 1.0399

1.0136 1.0188 1.0241 1.0295 1.0349

1.0084 1.0136 1.0188 1.0241 1.0294

.9965 1.0015 1.0066 1.0117 1.0169

.9829 .9879 .9929 .9980 1.0032

10 11 12 13 14

1.0606 1.0669 1.0733 1.0797 1.0862

1.0594 1.0656 1.0718 1.0781 1.0845

1.0578 1.0639 1.0700 1.0762 1.0824

1.0561 1.0621 1.0681 1.0742 1.0803

1.0543 1.0602 1.0661 1.0721 1.0781

1.0523 1.0581 1.0640 1.0699 1.0758

1.0503 1.0560 1.0618 1.0676 1.0735

1.0455 1.0511 1.0567 1.0624 1.0681

1.0403 1.0458 1.0513 1.0568 1.0624

1.0347 1.0401 1.0455 1.0509 1.0564

1.0221 1.0273 1.0326 1.0379 1.0432

1.0083 1.0134 1.0186 1.0238 1.0289

15 16 17 18 19

1.0927 1.0992 1.1057 1.1123 1.1189

1.0909 1.0973 1.1038 1.1103 1.1168

1.0887 1.0950 1.1014 1.1078 1.1142

1.0865 1.0927 1.0989 1.1052 1.1115

1.0842 1.0903 1.0964 1.1026 1.1088

1.0818 1.0879 1.0940 1.1001 1.1062

1.0794 1.0854 1.0914 1.0974 1.1034

1.0739 1.0797 1.0855 1.0913 1.0972

1.0680 1.0737 1.0794 1.0851 1.0908

1.0619 1.0675 1.0731 1.0787 1.0843

1.0485 1.0538 1.0592 1.0646 1.0700

1.0341 1.0393 1.0444 1.0496 1.0547

20 21 22 23 24

1.1255 1.1322 1.1389 1.1457 1.1525

1.1234 1.1300 1.1366 1.1433 1.1501

1.1206 1.1271 1.1336 1.1402 1.1469

1.1178 1.1242 1.1306 1.1371 1.1437

1.1150 1.1213 1.1276 1.1340 1.1404

1.1123 1.1185 1.1247 1.1310 1.1374

1.1094 1.1155 1.1217 1.1280 1.1343

1.1031 1.1090 1.1150 1.1210 1.1271

1.0966 1.1024 1.1083 1.1142 1.1201

1.0899 1.0956 1.1013 1.1070 1.1127

1.0754 1.0808 1.0862 1.0917 1.0972

1.0598 1.0650 1.0701 1.0753 1.0805

25 26 27 28 29

1.1594 1.1663 1.1733 1.1803 1.1874

1.1569 1.1638 1.1707 1.1777 1.1847

1.1536 1.1603 1.1670 1.1738 1.1807

1.1503 1.1569 1.1635 1.1702 1.1770

1.1469 1.1534 1.1600 1.1666 1.1733

1.1438 1.1502 1.1566 1.1631 1.1697

1.1406 1.1469 1.1533 1.1597 1.1662

1.1332 1.1394 1.1456 1.1519 1.1582

1.1260 1.1320 1.1381 1.1442 1.1503

1.1185 1.1244 1.1303 1.1362 1.1422

1.1027 1.1083 1.1139 1.1195 1.1251

1.0857 1.0910 1.0963 1.1016 1.1069

30 31 32 33 34

1.1945 1.2016 1.2088 1.2160 1.2233

1.1917 1.1988 1.2059 1.2131 1.2203

1.1876 1.1945 1.2014 1.2084 1.2155

1.1838 1.1906 1.1974 1.2043 1.2113

1.1800 1.1867 1.1934 1.2002 1.2071

1.1763 1.1829 1.1896 1.1963 1.2030

1.1727 1.1792 1.1857 1.1922 1.1988

1.1645 1.1708 1.1772 1.1836 1.1901

1.1564 1.1625 1.1687 1.1749 1.1812

1.1482 1.1542 1.1602 1.1662 1.1723

1.1307 1.1363 1.1419 1.1476 1.1533

1.1122 1.1175 1.1228 1.1281 1.1335

35 36 37 38 39

1.2306 1.2375 1.2444 1.2513 1.2581

1.2275 1.2344 1.2412 1.2479 1.2546

1.2227 1.2294 1.2361 1.2428 1.2494

1.2183 1.2249 1.2315 1.2381 1.2446

1.2140 1.2205 1.2270 1.2335 1.2399

1.2098 1.2163 1.2227 1.2291 1.2354

1.2055 1.2119 1.2182 1.2245 1.2308

1.1966 1.2028 1.2089 1.2150 1.2210

1.1876 1.1936 1.1995 1.2054 1.2112

1.1784 1.1842 1.1899 1.1956 1.2013

1.1591 1.1645 1.1699 1.1752 1.1805

1.1390 1.1440 1.1490 1.1540 1.1589

40 41 42 43 44

1.2649 1.2717 1.2786 1.2854 1.2922

1.2613 1.2680 1.2747 1.2814 1.2880

1.2560 1.2626 1.2692 1.2758 1.2824

1.2511 1.2576 1.2641 1.2706 1.2771

1.2463 1.2527 1.2591 1.2655 1.2719

1.2417 1.2480 1.2543 1.2606 1.2669

1.2370 1.2432 1.2494 1.2556 1.2618

1.2270 1.2330 1.2390 1.2450 1.2510

1.2170 1.2229 1.2287 1.2345 1.2403

1.2069 1.2126 1.2182 1.2238 1.2294

1.1858 1.1911 1.1963 1.2015 1.2067

1.1638 1.1687 1.1735 1.1783 1.1831

45 46 47 48 49

1.2990 1.3058 1.3126 1.3194 1.3263

1.2947 1.3014 1.3080 1.3147 1.3214

1.2890 1.2955 1.3021 1.3087 1.3153

1.2836 1.2901 1.2966 1.3031 1.3096

1.2783 1.2847 1.2911 1.2975 1.3040

1.2732 1.2795 1.2858 1.2921 1.2984

1.2680 1.2742 1.2804 1.2867 1.2929

1.2570 1.2630 1.2690 1.2750 1.2811

1.2461 1.2519 1.2577 1.2635 1.2693

1.2350 1.2406 1.2462 1.2518 1.2575

1.2119 1.2171 1.2223 1.2275 1.2328

1.1879 1.1927 1.1976 1.2024 1.2073

50 51 52 53 54

1.3327 1.3391 1.3454 1.3517 1.3579

1.3277 1.3339 1.3401 1.3462 1.3523

1.3215 1.3277 1.3338 1.3399 1.3459

1.3157 1.3218 1.3278 1.3338 1.3397

1.3100 1.3160 1.3219 1.3278 1.3336

1.3043 1.3102 1.3160 1.3218 1.3275

1.2987 1.3045 1.3102 1.3159 1.3215

1.2867 1.2923 1.2978 1.3033 1.3087

1.2748 1.2802 1.2856 1.2909 1.2961

1.2628 1.2680 1.2731 1.2782 1.2833

1.2377 1.2425 1.2473 1.2521 1.2568

1.2118 1.2163 1.2208 1.2252 1.2296

55 56 57 58 59

1.3640 1.3700 1.3759 1.3818 1.3875

1.3583 1.3642 1.3700 1.3757 1.3813

1.3518 1.3576 1.3634 1.3691 1.3747

1.3455 1.3512 1.3569 1.3625 1.3680

1.3393 1.3449 1.3505 1.3560 1.3614

1.3331 1.3386 1.3441 1.3495 1.3548

1.3270 1.3324 1.3377 1.3430 1.3482

1.3141 1.3194 1.3246 1.3298 1.3348

1.3013 1.3064 1.3114 1.3164 1.3213

1.2883 1.2932 1.2981 1.3029 1.3077

1.2615 1.2661 1.2706 1.2751 1.2795

1.2339 1.2382 1.2424 1.2466 1.2507

60 61 62 63 64

1.3931 1.3986 1.4039 1.4091

1.3868 1.3922 1.3975 1.4027 1.4078

1.3801 1.3855 1.3907 1.3958 1.4007

1.3734 1.3787 1.3838 1.3888 1.3936

1.3667 1.3719 1.3769 1.3818 1.3866

1.3600 1.3651 1.3700 1.3748 1.3795

1.3533 1.3583 1.3632 1.3679 1.3725

1.3398 1.3447 1.3494 1.3540

1.3261 1.3308 1.3354 1.3398

1.3124 1.3169 1.3213 1.3255

1.2839 1.2881 1.2922 1.2962

1.2547 1.2587 1.2625 1.2661

DENSITIES OF AQUEOUS INORGANIC SOLUTIONS Nitric Acid (HNO3) (Concluded )

TABLE 2-66 %

0°C

5°C

10°C

15°C

20°C

25°C

30°C

65 66 67 68 69

1.4128 1.4177 1.4224 1.4271 1.4317

1.4055 1.4103 1.4150 1.4196 1.4241

1.3984 1.4031 1.4077 1.4122 1.4166

1.3913 1.3959 1.4004 1.4048 1.4091

1.3841 1.3887 1.3932 1.3976 1.4019

1.3770 1.3814 1.3857 1.3900 1.3942

70 71 72 73 74

1.4362 1.4406 1.4449 1.4491 1.4532

1.4285 1.4328 1.4371 1.4413 1.4454

1.4210 1.4252 1.4294 1.4335 1.4376

1.4134 1.4176 1.4218 1.4258 1.4298

1.4061 1.4102 1.4142 1.4182 1.4221

1.3983 1.4023 1.4063 1.4103 1.4142

75 76 77 78 79

1.4573 1.4613 1.4652 1.4690 1.4727

1.4494 1.4533 1.4572 1.4610 1.4647

1.4415 1.4454 1.4492 1.4529 1.4565

1.4337 1.4375 1.4413 1.4450 1.4486

1.4259 1.4296 1.4333 1.4369 1.4404

1.4180 1.4217 1.4253 1.4288 1.4323

80 81 82 83 84

1.4764 1.4800 1.4835 1.4869 1.4903

1.4683 1.4718 1.4753 1.4787 1.4820

1.4601 1.4636 1.4670 1.4704 1.4737

1.4521 1.4555 1.4589 1.4622 1.4655

1.4439 1.4473 1.4507 1.4540 1.4572

1.4357 1.4391 1.4424 1.4456 1.4487

85 86 87 88 89

1.4936 1.4968 1.4999 1.5029 1.5058

1.4852 1.4883 1.4913 1.4942 1.4970

1.4769 1.4799 1.4829 1.4858 1.4885

1.4686 1.4716 1.4745 1.4773 1.4800

1.4603 1.4633 1.4662 1.4690 1.4716

1.4518 1.4548 1.4577 1.4605 1.4631

90 91 92 93 94

1.5085 1.5111 1.5136 1.5156 1.5177

1.4997 1.5023 1.5048 1.5068 1.5088

1.4911 1.4936 1.4960 1.4979 1.4999

1.4826 1.4850 1.4873 1.4892 1.4912

1.4741 1.4766 1.4789 1.4807 1.4826

1.4656 1.4681 1.4704 1.4722 1.4741

95 96 97 98 99 100

1.5198 1.5220 1.5244 1.5278 1.5327 1.5402

1.5109 1.5130 1.5152 1.5187 1.5235 1.5310

1.5019 1.5040 1.5062 1.5096 1.5144 1.5217

1.4932 1.4952 1.4974 1.5008 1.5056 1.5129

1.4846 1.4867 1.4889 1.4922 1.4969 1.5040

1.4761 1.4781 1.4802 1.4835 1.4881 1.4952

TABLE 2-67 15

%

d4

1 2 4 6 8 10 12 14 16 18 20 22 24 26

1.0050 1.0109 1.0228 1.0348 1.0471 1.0597 1.0726 1.0589 1.0995 1.1135 1.1279 1.1428 1.1581 1.1738

2-103

Perchloric Acid (HClO4) 20

d4

TABLE 2-69

40°C

50°C

d4

%

d4

d4

d4

°C

1%

2%

4%

1.0020 1.0070 1.0169 1.0270 1.0372 1.0475

0.9933 0.9986 0.9906 1.0205 1.0320 1.0440 1.0560 1.0680 1.0810 1.0940 1.1070 1.1205 1.1345 1.1490

28 30 32 34 36 38 40 45 50 55 60 65 70

1.1900 1.2067 1.2239 1.2418 1.2603 1.2794 1.2991 1.3521 1.4103 1.4733 1.5389 1.6059 1.6736

1.1851 1.2013 1.2183 1.2359 1.2542 1.2732 1.2927 1.3450 1.4018 1.4636 1.5298 1.5986 1.6680

1.1645 1.1800 1.1960 1.2130 1.2310 1.2490 1.2680 1.3180 1.3730 1.4320 1.4950 1.5620 1.6290

0 10 15 20 30 40 50 60 80 100

1.0066 1.0064 1.0058 1.0049 1.0024 0.9990 .9949 .9901 .9786 .9653

1.0134 1.0132 1.0125 1.0117 1.0092 1.0058 1.0017 0.9969 .9855 .9722

1.0270 1.0268 1.0260 1.0252 1.0228 1.0195 1.0154 1.0106 0.9993 .9860

1.1697

50

15

20

50

80°C

100°C

6%

8%

10%

1.0396

1.0534

1.0674

Potassium Bicarbonate (KHCO3)

d4

25

60°C

TABLE 2-70 Potassium Bromide (KBr) TABLE 2-68

Phosphoric Acid (H3PO4)

°C

2%

6%

14%

0 10 20 30 40

1.0113 1.0109 1.0092 1.0065 1.0029

1.0339 1.0330 1.0309 1.0279 1.0241

1.0811 1.0792 1.0764 1.0728 1.0685

20%

26%

35%

50%

75%

1.1192 1.1167 1.1567 1.221 1.341 1.1134 1.1529 1.216 1.335 1.579 1.1094 1.1484 1.211 1.329 1.572 1.1048

100%

1.870 1.862

%

d 20 4

1 2 6 12 20 30 40

1.0054 1.0127 1.0426 1.0903 1.1601 1.2593 1.3746

2-104

PHYSICAL AND CHEMICAL DATA

TABLE 2-71

Potassium Carbonate (K2CO3)

TABLE 2-77

Potassium Nitrate (KNO3)

%

0°C

10°C

20°C

40°C

60°C

80°C

100°C

%

0°C

10°C

20°C

40°C

60°C

80°C

100°C

1 2 4 8 12 16 20 24 28 30 35 40 45 50

1.0094 1.0189 1.0381 1.0768 1.1160 1.1562 1.1977 1.2405 1.2846 1.3071 1.3646 1.4244 1.4867 1.5517

1.0089 1.0182 1.0369 1.0746 1.1131 1.1530 1.1941 1.2366 1.2804 1.3028 1.3600 1.4195 1.4815 1.5462

1.0072 1.0163 1.0345 1.0715 1.1096 1.1490 1.1898 1.2320 1.2756 1.2979 1.3548 1.4141 1.4759 1.5404

1.0010 1.0098 1.0276 1.0640 1.1013 1.1399 1.1801 1.2219 1.2652 1.2873 1.3440 1.4029 1.4644 1.5285

0.9919 1.0005 1.0180 1.0538 1.0906 1.1290 1.1690 1.2106 1.2538 1.2759 1.3324 1.3913 1.4528 1.5169

0.9803 .9889 1.0063 1.0418 1.0786 1.1170 1.1570 1.1986 1.2418 1.2640 1.3206 1.3795 1.4408 1.5048

0.9670 .9756 .9951 1.0291 1.0663 1.1049 1.1451 1.1869 1.2301 1.2522 1.3089 1.3678 1.4290 1.4928

1 2 4 8 12 16 20 24

1.00654 1.01326 1.02677 1.05419 1.08221

1.00615 1.01262 1.02566 1.05226 1.07963

1.00447 1.01075 1.02344 1.04940 1.07620 1.10392 1.13261 1.16233

0.99825 1.00430 1.01652 1.04152 1.06740 1.09432 1.12240 1.15175

0.9890 .9949 1.0068 1.0313 1.0567 1.0831 1.1106 1.1391

0.9776 .9834 .9951 1.0192 1.0442 1.0703 1.0974 1.1256

0.9641 .9699 .9816 1.0056 1.0304 1.0562 1.0831 1.1110

TABLE 2-72 Potassium Chromate (K2Cr O4)

TABLE 2-73 (KClO3)

Potassium Chlorate

%

d4

d4

°C

1%

2%

3%

4%

1 2 4 8 12 16 20 24 28 30

1.0073 1.0155 1.0321 1.0659 1.1009

1.0066 1.0147 1.0311 1.0647 1.0999 1.1366 1.1748 1.2147 1.2566 1.2784

0 10 20 30 40 60 80 100

1.0061 1.0059 1.0045 1.0020 0.9986 .9895 .9781 .9646

1.0124 1.0122 1.0109 1.0085 1.0051 0.9959 .9845 .9709

1.0189 1.0187 1.0174 1.0151 1.0116 1.0024 0.9910 .9774

1.0256 1.0254 1.0241 1.0218 1.0183 1.0091 0.9977 .9840

15

TABLE 2-74

18

Potassium Chloride (KCl)

%

0°C

20°C

25°C

40°C

60°C

80°C

100°C

1.0 2.0 4.0 8.0 12.0 16.0 20.0 24.0 28.0

1.00661 1.01335 1.02690 1.05431 1.08222 1.11068 1.13973

1.00462 1.01103 1.02391 1.05003 1.07679 1.10434 1.13280 1.16226

1.00342 1.00977 1.02255 1.04847 1.07506 1.10245 1.13072 1.15995

0.99847 1.00471 1.01727 1.04278 1.06897 1.09600 1.12399 1.15299 1.18304

0.9894 .9956 1.0080 1.0333 1.0592 1.0861 1.1138 1.1425 1.1723

0.9780 .9842 .9966 1.0219 1.0478 1.0746 1.1024 1.1311 1.1609

0.9646 .9708 .9634 1.0888 1.0350 1.0619 1.0897 1.1185 1.1483

%

110°C

120°C

130°C

140°C

3.79 7.45 13.62

0.9733 .9978 1.0388

0.9663 .9899 1.0313

0.9583 .9827 1.0238

0.9502 .9745 1.0159

TABLE 2-75 Potassium Chrome Alum [K2Cr2(SO4)4]

TABLE 2-78 Potassium Dichromate (K2Cr2O7) 20

d 15 4

%

d 15 4

1 2 6 10 14 20 30 40 50

1.007 1.016 1.052 1.089 1.129 1.193 1.315 1.456 1.615

1.0 2.0 4.0 6.0 8.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 50.0 51.7

1.0083 1.0175 1.0359 1.0544 1.0730 1.0918 1.1396 1.1884 1.2387 1.2905 1.3440 1.3991 1.4558 1.5143 1.5355 (sat’d. soln.)

20

%

d4

%

d4

1 2 4 6 8 10

1.0052 1.0122 1.0264 1.0408 1.0554 1.0703

1 2 4 6 8 10

1.0063 1.0145 1.0310 1.0477 1.0646 1.0817

TABLE 2-80 Potassium Sulfite (K2SO3) 15

TABLE 2-81 Sodium Acetate (NaC2H3O2) 20

%

d4

%

d4

1 2 4 8 12 16 20 24 26

1.0073 1.0155 1.0322 1.0667 1.1026 1.1402 1.1793 1.2197 1.2404

1 2 4 8 12 18 20 26 28

1.0033 1.0084 1.0186 1.0392 1.0598 1.0807 1.1021 1.1351 1.1462

TABLE 2-82 Sodium Arsenate (Na3AsO4) 17

TABLE 2-83 Sodium Bichromate (Na2Cr2O7) 15

%

d4

%

d4

1 2 4 8 10 12

1.0097 1.0207 1.0431 1.0892 1.1130 1.1373

1 2 4 8 12 16 20 24 28 30 35 40 45 50

1.006 1.013 1.027 1.056 1.084 1.112 1.140 1.166 1.193 1.207 1.244 1.279 1.312 1.342

TABLE 2-76 Potassium Hydroxide (KOH)

%

TABLE 2-79 Potassium Sulfate (K2SO4)

TABLE 2-84 Sodium Bromide (NaBr) 17

TABLE 2-85 Sodium Formate (HCOONa) 25

%

d4

%

d4

1 2 4 8 10 12 20 30 40

1.0060 1.0139 1.0298 1.0631 1.0803 1.0981 1.1745 1.2841 1.4138

1 2 4 8 12 16 20 24 28 30 35 40

1.003 1.009 1.022 1.048 1.074 1.100 1.127 1.155 1.184 1.199 1.236 1.274

DENSITIES OF AQUEOUS INORGANIC SOLUTIONS TABLE 2-86

Sodium Carbonate (Na2CO3)

TABLE 2-90

2-105

Sodium Hydroxide (NaOH)

%

0°C

10°C

20°C

30°C

40°C

60°C

80°C

100°C

%

0°C

15°C

20°C

40°C

60°C

80°C

100°C

1 2 4 8 12 14 16 18 20 24 28 30

1.0109 1.0219 1.0439 1.0878 1.1319 1.1543

1.0103 1.0210 1.0423 1.0850 1.1284 1.1506

1.0086 1.0190 1.0398 1.0816 1.1244 1.1463

1.0058 1.0159 1.0363 1.0775 1.1200 1.1417 1.1636 1.1859 1.2086 1.2552 1.3031 1.3274

1.0022 1.0122 1.0323 1.0732 1.1150 1.1365

0.9929 1.0027 1.0223 1.0625 1.1039 1.1251

0.9814 .9910 1.0105 1.0503 1.0914 1.1125

0.9683 .9782 .9980 1.0380 1.0787 1.0996

1 2 4 8 12 16 20 24 28 32 36 40 44 48 50

1.0124 1.0244 1.0482 1.0943 1.1399 1.1849 1.2296 1.2741 1.3182 1.3614 1.4030 1.4435 1.4825 1.5210 1.5400

1.01065 1.02198 1.04441 1.08887 1.13327 1.17761 1.22183 1.26582 1.3094 1.3520 1.3933 1.4334 1.4720 1.5102 1.5290

1.0095 1.0207 1.0428 1.0869 1.1309 1.1751 1.2191 1.2629 1.3064 1.3490 1.3900 1.4300 1.4685 1.5065 1.5253

1.0033 1.0139 1.0352 1.0780 1.1210 1.1645 1.2079 1.2512 1.2942 1.3362 1.3768 1.4164 1.4545 1.4922 1.5109

0.9941 1.0045 1.0254 1.0676 1.1101 1.1531 1.1960 1.2388 1.2814 1.3232 1.3634 1.4027 1.4405 1.4781 1.4967

0.9824 .9929 1.0139 1.0560 1.0983 1.1408 1.1833 1.2259 1.2682 1.3097 1.3498 1.3889 1.4266 1.4641 1.4827

0.9693 .9797 1.0009 1.0432 1.0855 1.1277 1.1700 1.2124 1.2546 1.2960 1.3360 1.3750 1.4127 1.4503 1.4690

TABLE 2-87

Sodium Chlorate (NaClO3) 18

18

%

d4

%

d4

1 2 4 6 8 10 12 14 16

1.0053 1.0121 1.0258 1.0397 1.0538 1.0681 1.0827 1.0977 1.1131

18 20 22 24 26 28 30 32 34

1.1288 1.1449 1.1614 1.1782 1.1953 1.2128 1.2307 1.2491 1.2680

TABLE 2-88

TABLE 2-91

Sodium Chloride (NaCl)

%

0°C

10°C

25°C

40°C

60°C

80°C

100°C

1 2 4 8 12 16 20 24 26

1.00747 1.01509 1.03038 1.06121 1.09244 1.12419 1.15663 1.18999 1.20709

1.00707 1.01442 1.02920 1.05907 1.08946 1.12056 1.15254 1.18557 1.20254

1.00409 1.01112 1.02530 1.05412 1.08365 1.11401 1.14533 1.17776 1.19443

0.99908 1.00593 1.01977 1.04798 1.07699 1.10688 1.13774 1.16971 1.18614

0.9900 .9967 1.0103 1.0381 1.0667 1.0962 1.1268 1.1584 1.1747

0.9785 .9852 .9988 1.0264 1.0549 1.0842 1.1146 1.1463 1.1626

0.9651 .9719 .9855 1.0134 1.0420 1.0713 1.1017 1.1331 1.1492

Sodium Nitrate (NaNO3)

%

0°C

20°C

40°C

60°C

80°C

100°C

1 2 4 8 12 16 20 24 28 30 35 40 45

1.0071 1.0144 1.0290 1.0587 1.0891 1.1203 1.1526 1.1860 1.2204 1.2380 1.2834 1.3316

1.0049 1.0117 1.0254 1.0532 1.0819 1.1118 1.1429 1.1752 1.2085 1.2256 1.2701 1.3175 1.3683

0.9986 1.0050 1.0180 1.0447 1.0724 1.1013 1.1314 1.1629 1.1955 1.2122 1.2560 1.3027 1.3528

0.9894 .9956 1.0082 1.0340 1.0609 1.0892 1.1187 1.1496 1.1816 1.1980 1.2413 1.2875 1.3371

0.9779 .9840 .9964 1.0218 1.0481 1.0757 1.1048 1.1351 1.1667 1.1830 1.2258 1.2715 1.3206

0.9644 .9704 .9826 1.0078 1.0340 1.0614 1.0901 1.1200 1.1513 1.1674 1.2100 1.2555 1.3044

TABLE 2-89 Sodium Chromate (Na2CrO4)

TABLE 2-93

TABLE 2-92 Sodium Nitrite (NaNO2)

18

15

%

d4

%

d4

1 2 4 8 12 16 20 24 26

1.0074 1.0164 1.0344 1.0718 1.1110 1.1518 1.1942 1.2383 1.2611

1 2 4 8 12 16 20

1.0058 1.0125 1.0260 1.0535 1.0816 1.1103 1.1394

Sodium Silicates Concentration, % 1

2

4

8

10

14

Na2O/3.9SiO2 Na2O/3.36SiO2 Na2O/2.40SiO2 Na2O/2.44SiO2 Na2O/2.06SiO2 Na2O/1.69SiO2

20

24

30

36

40

45

50

1.445 1.450

1.520

1.594

20

Formula

d4 1.006 1.006 1.007

1.014 1.014 1.016

1.030 1.030 1.034

1.063 1.065 1.071

1.080 1.083 1.090

1.116 1.120 1.130

1.172 1.179

1.211 1.222

1.275 1.290

1.365

1.007 1.007

1.016 1.017

1.035 1.036

1.073 1.077

1.093 1.098

1.134 1.141

1.200 1.210

1.247 1.259

1.309 1.321 1.337

1.387 1.397 1.424

2-106

PHYSICAL AND CHEMICAL DATA TABLE 2-94 %

0°C

1 2 4 8 12 16 20 24

1.0094 1.0189 1.0381 1.0773 1.1174 1.1585 1.2008 1.2443

TABLE 2-95 Sodium Sulfide (Na2S)

Sodium Sulfate (Na2SO4) 20°C

30°C

40°C

60°C

80°C

100°C

1.0073 1.0164 1.0348 1.0724 1.1109 1.1586 1.1915 1.2336

1.0046 1.0135 1.0315 1.0682 1.1062 1.1456 1.1865 1.2292

1.0010 1.0098 1.0276 1.0639 1.1015 1.1406 1.1813 1.2237

0.9919 1.0007 1.0184 1.0544 1.0915 1.1299 1.1696

0.9805 .9892 1.0068 1.0426 1.0795 1.1176 1.1569

0.9671 .9758 .9934 1.0292 1.0661 1.1042

TABLE 2-96 Sodium Sulfite (Na2SO3)

TABLE 2-97 Sodium Thiosulfate (Na2S2O3)

%

d 18 4

%

d 19 4

%

d 20 4

1 2 4 8 12 16 18

1.0098 1.0211 1.0440 1.0907 1.1388 1.1885 1.2140

1 2 4 8 12 16 18

1.0078 1.0172 1.0363 1.0751 1.1146 1.1549 1.1755

1 2 4 8 12 16 20 24 28 30 35 40

1.0065 1.0148 1.0315 1.0654 1.1003 1.1365 1.1740 1.2128 1.2532 1.2739 1.3273 1.3827

TABLE 2-98 Sodium Thiosulfate Pentahydrate (Na2S2O3⋅5H2O) %

d 19 4

1 2 4 8 12 16 20 24 28 30 40 50

1.0052 1.0105 1.0211 1.0423 1.0639 1.0863 1.1087 1.1322 1.1558 1.1676 1.2297 1.2954

TABLE 2-99 Stannic Chloride (SnCl4) %

d 15 4

1 2 4 8 12 16 20 24 28 30 35 40 45 50 55 60 65 70

1.007 1.015 1.031 1.064 1.099 1.135 1.173 1.212 1.255 1.278 1.337 1.403 1.475 1.555 1.644 1.742 1.851 1.971

TABLE 2-100 Stannous Chloride (SnCl2) %

d 15 4

1 2 4 8 12 16 20 24 28 30 35 40 45 50 55 60 65

1.0068 1.0146 1.0306 1.0638 1.0986 1.1353 1.1743 1.2159 1.2603 1.2837 1.3461 1.4145 1.4897 1.5729 1.6656 1.7695 1.8865

DENSITIES OF AQUEOUS INORGANIC SOLUTIONS TABLE 2-101 %

2-107

Sulfuric Acid (H2SO4)

0°C

10°C

15°C

20°C

25°C

30°C

40°C

50°C

60°C

80°C

100°C

1 2 3 4

1.0074 1.0147 1.0219 1.0291

1.0068 1.0138 1.0206 1.0275

1.0060 1.0129 1.0197 1.0264

1.0051 1.0118 1.0184 1.0250

1.0038 1.0104 1.0169 1.0234

1.0022 1.0087 1.0152 1.0216

0.9986 1.0050 1.0113 1.0176

0.9944 1.0006 1.0067 1.0129

0.9895 .9956 1.0017 1.0078

0.9779 .9839 .9900 .9961

0.9645 .9705 .9766 .9827

5 6 7 8 9

1.0364 1.0437 1.0511 1.0585 1.0660

1.0344 1.0414 1.0485 1.0556 1.0628

1.0332 1.0400 1.0469 1.0539 1.0610

1.0317 1.0385 1.0453 1.0522 1.0591

1.0300 1.0367 1.0434 1.0502 1.0571

1.0281 1.0347 1.0414 1.0481 1.0549

1.0240 1.0305 1.0371 1.0437 1.0503

1.0192 1.0256 1.0321 1.0386 1.0451

1.0140 1.0203 1.0266 1.0330 1.0395

1.0022 1.0084 1.0146 1.0209 1.0273

.9888 .9950 1.0013 1.0076 1.0140

10 11 12 13 14

1.0735 1.0810 1.0886 1.0962 1.1039

1.0700 1.0773 1.0846 1.0920 1.0994

1.0681 1.0753 1.0825 1.0898 1.0971

1.0661 1.0731 1.0802 1.0874 1.0947

1.0640 1.0710 1.0780 1.0851 1.0922

1.0617 1.0686 1.0756 1.0826 1.0897

1.0570 1.0637 1.0705 1.0774 1.0844

1.0517 1.0584 1.0651 1.0719 1.0788

1.0460 1.0526 1.0593 1.0661 1.0729

1.0338 1.0403 1.0469 1.0536 1.0603

1.0204 1.0269 1.0335 1.0402 1.0469

15 16 17 18 19

1.1116 1.1194 1.1272 1.1351 1.1430

1.1069 1.1145 1.1221 1.1298 1.1375

1.1045 1.1120 1.1195 1.1271 1.1347

1.1020 1.1094 1.1168 1.1243 1.1318

1.0994 1.1067 1.1141 1.1215 1.1290

1.0968 1.1040 1.1113 1.1187 1.1261

1.0914 1.0985 1.1057 1.1129 1.1202

1.0857 1.0927 1.0998 1.1070 1.1142

1.0798 1.0868 1.0938 1.1009 1.1081

1.0671 1.0740 1.0809 1.0879 1.0950

1.0537 1.0605 1.0674 1.0744 1.0814

20 21 22 23 24

1.1510 1.1590 1.1670 1.1751 1.1832

1.1453 1.1531 1.1609 1.1688 1.1768

1.1424 1.1501 1.1579 1.1657 1.1736

1.1394 1.1471 1.1548 1.1626 1.1704

1.1365 1.1441 1.1517 1.1594 1.1672

1.1335 1.1410 1.1486 1.1563 1.1640

1.1275 1.1349 1.1424 1.1500 1.1576

1.1215 1.1288 1.1362 1.1437 1.1512

1.1153 1.1226 1.1299 1.1373 1.1448

1.1021 1.1093 1.1166 1.1239 1.1313

1.0885 1.0957 1.1029 1.1102 1.1176

25 26 27 28 29

1.1914 1.1996 1.2078 1.2160 1.2243

1.1848 1.1929 1.2010 1.2091 1.2173

1.1816 1.1896 1.1976 1.2057 1.2138

1.1783 1.1862 1.1942 1.2023 1.2104

1.1750 1.1829 1.1909 1.1989 1.2069

1.1718 1.1796 1.1875 1.1955 1.2035

1.1653 1.1730 1.1808 1.1887 1.1966

1.1588 1.1665 1.1742 1.1820 1.1898

1.1523 1.1599 1.1676 1.1753 1.1831

1.1388 1.1463 1.1539 1.1616 1.1693

1.1250 1.1325 1.1400 1.1476 1.1553

30 31 32 33 34

1.2326 1.2409 1.2493 1.2577 1.2661

1.2255 1.2338 1.2421 1.2504 1.2588

1.2220 1.2302 1.2385 1.2468 1.2552

1.2185 1.2267 1.2349 1.2432 1.2515

1.2150 1.2232 1.2314 1.2396 1.2479

1.2115 1.2196 1.2278 1.2360 1.2443

1.2046 1.2126 1.2207 1.2289 1.2371

1.1977 1.2057 1.2137 1.2218 1.2300

1.1909 1.1988 1.2068 1.2148 1.2229

1.1771 1.1849 1.1928 1.2008 1.2088

1.1630 1.1708 1.1787 1.1866 1.1946

35 36 37 38 39

1.2746 1.2831 1.2917 1.3004 1.3091

1.2672 1.2757 1.2843 1.2929 1.3016

1.2636 1.2720 1.2805 1.2891 1.2978

1.2599 1.2684 1.2769 1.2855 1.2941

1.2563 1.2647 1.2732 1.2818 1.2904

1.2526 1.2610 1.2695 1.2780 1.2866

1.2454 1.2538 1.2622 1.2707 1.2793

1.2383 1.2466 1.2550 1.2635 1.2720

1.2311 1.2394 1.2477 1.2561 1.2646

1.2169 1.2251 1.2334 1.2418 1.2503

1.2027 1.2109 1.2192 1.2276 1.2361

40 41 42 43 44

1.3179 1.3268 1.3357 1.3447 1.3538

1.3103 1.3191 1.3280 1.3370 1.3461

1.3065 1.3153 1.3242 1.3332 1.3423

1.3028 1.3116 1.3205 1.3294 1.3384

1.2991 1.3079 1.3167 1.3256 1.3346

1.2953 1.3041 1.3129 1.3218 1.3308

1.2880 1.2967 1.3055 1.3144 1.3234

1.2806 1.2893 1.2981 1.3070 1.3160

1.2732 1.2819 1.2907 1.2996 1.3086

1.2589 1.2675 1.2762 1.2850 1.2939

1.2446 1.2532 1.2619 1.2707 1.2796

45 46 47 48 49

1.3630 1.3724 1.3819 1.3915 1.4012

1.3553 1.3646 1.3740 1.3835 1.3931

1.3515 1.3608 1.3702 1.3797 1.3893

1.3476 1.3569 1.3663 1.3758 1.3854

1.3437 1.3530 1.3624 1.3719 1.3814

1.3399 1.3492 1.3586 1.3680 1.3775

1.3325 1.3417 1.3510 1.3604 1.3699

1.3251 1.3343 1.3435 1.3528 1.3623

1.3177 1.3269 1.3362 1.3455 1.3549

1.3029 1.3120 1.3212 1.3305 1.3399

1.2886 1.2976 1.3067 1.3159 1.3253

50 51 52 53 54

1.4110 1.4209 1.4310 1.4412 1.4515

1.4029 1.4128 1.4228 1.4329 1.4431

1.3990 1.4088 1.4188 1.4289 1.4391

1.3951 1.4049 1.4148 1.4248 1.4350

1.3911 1.4009 1.4109 1.4209 1.4310

1.3872 1.3970 1.4069 1.4169 1.4270

1.3795 1.3893 1.3991 1.4091 1.4191

1.3719 1.3816 1.3914 1.4013 1.4113

1.3644 1.3740 1.3837 1.3936 1.4036

1.3494 1.3590 1.3687 1.3785 1.3884

1.3348 1.3444 1.3540 1.3637 1.3735

55 56 57 58 59

1.4619 1.4724 1.4830 1.4937 1.5045

1.4535 1.4640 1.4746 1.4852 1.4959

1.4494 1.4598 1.4703 1.4809 1.4916

1.4453 1.4557 1.4662 1.4768 1.4875

1.4412 1.4516 1.4621 1.4726 1.4832

1.4372 1.4475 1.4580 1.4685 1.4791

1.4293 1.4396 1.4500 1.4604 1.4709

1.4214 1.4317 1.4420 1.4524 1.4629

1.4137 1.4239 1.4342 1.4446 1.4551

1.3984 1.4085 1.4187 1.4290 1.4393

1.3834 1.3934 1.4035 1.4137 1.4240

60 61 62 63 64

1.5154 1.5264 1.5375 1.5487 1.5600

1.5067 1.5177 1.5287 1.5398 1.5510

1.5024 1.5133 1.5243 1.5354 1.5465

1.4983 1.5091 1.5200 1.5310 1.5421

1.4940 1.5048 1.5157 1.5267 1.5378

1.4898 1.5006 1.5115 1.5225 1.5335

1.4816 1.4923 1.5031 1.5140 1.5250

1.4735 1.4842 1.4950 1.5058 1.5167

1.4656 1.4762 1.4869 1.4977 1.5086

1.4497 1.4602 1.4708 1.4815 1.4923

1.4344 1.4449 1.4554 1.4660 1.4766

2-108

PHYSICAL AND CHEMICAL DATA

TABLE 2-101

Sulfuric Acid (H2SO4) (Concluded )

%

0°C

10°C

15°C

20°C

25°C

30°C

40°C

50°C

60°C

80°C

100°C

65 66 67 68 69

1.5714 1.5828 1.5943 1.6059 1.6176

1.5623 1.5736 1.5850 1.5965 1.6081

1.5578 1.5691 1.5805 1.5920 1.6035

1.5533 1.5646 1.5760 1.5874 1.5989

1.5490 1.5602 1.5715 1.5829 1.5944

1.5446 1.5558 1.5671 1.5785 1.5899

1.5361 1.5472 1.5584 1.5697 1.5811

1.5277 1.5388 1.5499 1.5611 1.5724

1.5195 1.5305 1.5416 1.5528 1.5640

1.5031 1.5140 1.5249 1.5359 1.5470

1.4873 1.4981 1.5089 1.5198 1.5307

70 71 72 73 74

1.6293 1.6411 1.6529 1.6648 1.6768

1.6198 1.6315 1.6433 1.6551 1.6670

1.6151 1.6268 1.6385 1.6503 1.6622

1.6105 1.6221 1.6338 1.6456 1.6574

1.6059 1.6175 1.6292 1.6409 1.6526

1.6014 1.6130 1.6246 1.6363 1.6480

1.5925 1.6040 1.6155 1.6271 1.6387

1.5838 1.5952 1.6067 1.6182 1.6297

1.5753 1.5867 1.5981 1.6095 1.6209

1.5582 1.5694 1.5806 1.5919 1.6031

1.5417 1.5527 1.5637 1.5747 1.5857

75 76 77 78 79

1.6888 1.7008 1.7128 1.7247 1.7365

1.6789 1.6908 1.7026 1.7144 1.7261

1.6740 1.6858 1.6976 1.7093 1.7209

1.6692 1.6810 1.6927 1.7043 1.7158

1.6644 1.6761 1.6878 1.6994 1.7108

1.6597 1.6713 1.6829 1.6944 1.7058

1.6503 1.6619 1.6734 1.6847 1.6959

1.6412 1.6526 1.6640 1.6751 1.6862

1.6322 1.6435 1.6547 1.6657 1.6766

1.6142 1.6252 1.6361 1.6469 1.6575

1.5966 1.6074 1.6181 1.6286 1.6390

80 81 82 83 84

1.7482 1.7597 1.7709 1.7815 1.7916

1.7376 1.7489 1.7599 1.7704 1.7804

1.7323 1.7435 1.7544 1.7649 1.7748

1.7272 1.7383 1.7491 1.7594 1.7693

1.7221 1.7331 1.7437 1.7540 1.7639

1.7170 1.7279 1.7385 1.7487 1.7585

1.7069 1.7177 1.7281 1.7382 1.7479

1.6971 1.7077 1.7180 1.7279 1.7375

1.6873 1.6978 1.7080 1.7179 1.7274

1.6680 1.6782 1.6882 1.6979 1.7072

1.6493 1.6594 1.6692 1.6787 1.6878

85 86 87 88 89

1.8009 1.8095 1.8173 1.8243 1.8306

1.7897 1.7983 1.8061 1.8132 1.8195

1.7841 1.7927 1.8006 1.8077 1.8141

1.7786 1.7872 1.7951 1.8022 1.8087

1.7732 1.7818 1.7897 1.7968 1.8033

1.7678 1.7763 1.7842 1.7914 1.7979

1.7571 1.7657 1.7736 1.7809 1.7874

1.7466 1.7552 1.7632 1.7705 1.7770

1.7364 1.7449 1.7529 1.7602 1.7669

1.7161 1.7245 1.7324 1.7397 1.7464

1.6966 1.7050 1.7129 1.7202 1.7269

90 91 92 93 94

1.8361 1.8410 1.8453 1.8490 1.8520

1.8252 1.8302 1.8346 1.8384 1.8415

1.8198 1.8248 1.8293 1.8331 1.8363

1.8144 1.8195 1.8240 1.8279 1.8312

1.8091 1.8142 1.8188 1.8227 1.8260

1.8038 1.8090 1.8136 1.8176 1.8210

1.7933 1.7986 1.8033 1.8074 1.8109

1.7829 1.7883 1.7932 1.7974 1.8011

1.7729 1.7783 1.7832 1.7876 1.7914

1.7525 1.7581 1.7633 1.7681

1.7331 1.7388 1.7439 1.7485

95 96 97 98 99 100

1.8544 1.8560 1.8569 1.8567 1.8551 1.8517

1.8439 1.8457 1.8466 1.8463 1.8445 1.8409

1.8388 1.8406 1.8414 1.8411 1.8393 1.8357

1.8337 1.8355 1.8364 1.8361 1.8342 1.8305

1.8286 1.8305 1.8314 1.8310 1.8292 1.8255

1.8236 1.8255 1.8264 1.8261 1.8242 1.8205

1.8137 1.8157 1.8166 1.8163 1.8145 1.8107

1.8040 1.8060 1.8071 1.8068 1.8050 1.8013

1.7944 1.7965 1.7977 1.7976 1.7958 1.7922

%

d 5.96 4

%

d 13.00 4

d 18.00 4

0.005 .01 .02 .03 .04

1.000 0140 1.000 0576 1.000 1434 1.000 2276 1.000 3104

0.05 .1 .2 .3 .4

0.999 810 1.000 185 1.000 912 1.001 623 1.002 326

0.999 028 .999 400 1.000 119 1.000 820 1.001 512

.05 .06 .07 .08 .09

1.000 3920 1.000 4726 1.000 5523 1.000 6313 1.000 7098

.5 .6 .8 1.0 1.2

1.003 023 1.003 716 1.005 090 1.006 452 1.007 807

1.002 197 1.002 877 1.004 227 1.005 570 1.006 909

.10 .15 .20 .25 .30

1.000 7880 1.001 1732 1.001 5514 1.001 9254 1.002 2961

1.4 1.6 1.8 2.0 2.2

1.009 159 1.010 510 1.011 860 1.013 209 1.014 557

1.008 247 1.009 583 1.010 918 1.012 252 1.013 586

.35 .40 .45 .50

1.002 6639 1.003 0292 1.003 3923 1.003 7534

2.4

1.015 904

1.014 919

DENSITIES OF AQUEOUS ORGANIC SOLUTIONS TABLE 2-102

Zinc Bromide (ZnBr2)

%

0°C

20°C

40°C

60°C

80°C

100°C

2 4 8 12 16

1.0188 1.0381 1.0777 1.1186 1.1609

1.0167 1.0354 1.0738 1.1135 1.1544

1.0102 1.0285 1.0660 1.1046 1.1445

1.0008 1.0187 1.0554 1.0932 1.1320

0.9890 1.0065 1.0422 1.0789 1.1169

0.9751 0.9921 1.0270 1.0629 1.1000

20 30 40 50 60 65

1.2043 1.3288 1.477 1.661 1.891 2.026

1.1965 1.3170 1.462 1.643 1.869 2.002

1.1855 1.3030 1.445 1.623 1.845 1.976

1.1720 1.2868 1.427 1.602 1.822 1.951

1.1560 1.2688 1.406 1.579 1.797 1.924

1.1382 1.2489 1.385 1.555 1.771 1.898

TABLE 2-103

TABLE 2-104 [Zn(NO3)2]

Zinc Nitrate

TABLE 2-105 (ZnSO4)

2-109

Zinc Sulfate

%

18°C

%

18°C

%

20°C

2 4 6 8 10 12 14 16

1.0154 1.0322 1.0496 1.0675 1.0859 1.1048 1.1244 1.1445

18 20 25 30 35 40 45 50

1.1652 1.1865 1.2427 1.3029 1.3678 1.4378 1.5134 1.5944

2 4 6 8 10 12 14 16

1.019 1.0403 1.0620 1.0842 1.1071 1.1308 1.1553 1.1806

Zinc Chloride (ZnCl2)

%

0°C

20°C

40°C

60°C

80°C

100°C

2 4 8 12 16

1.0192 1.0384 1.0769 1.1159 1.1558

1.0167 1.0350 1.0715 1.1085 1.1468

1.0099 1.0274 1.0624 1.0980 1.1350

1.0003 1.0172 1.0508 1.0853 1.1212

0.9882 1.0044 1.0369 1.0704 1.1055

0.9739 .9894 1.0211 1.0541 1.0888

20 30 40 50 60 70

1.1970 1.3062 1.4329 1.5860

1.1866 1.2928 1.4173 1.5681 1.749 1.962

1.1736 1.2778 1.4003 1.5495

1.1590 1.2614 1.3824 1.5300

1.1428 1.2438 1.3637 1.5097

1.1255 1.2252 1.3441 1.4892

DENSITIES OF AQUEOUS ORGANIC SOLUTIONS* From International Critical Tables, vol. 3, pp. 115–129. All compositions are in weight percent in vacuo. All density values are d4t = g/mL in vacuo.

UNITS AND UNITS CONVERSIONS Unless otherwise noted, densities are given in grams per cubic centimeter. To convert to pounds per cubic foot, multiply by 62.43. °F = 9⁄ 5 °C + 32 TABLE 2-106

*For gasoline and aircraft fuels see Hibbard, NACA Res. Mem. E56I21 (declassified 1958).

Formic Acid (HCOOH)

%

0°C

15°C

20°C

30°C

%

0°C

15°C

20°C

30°C

%

0°C

15°C

20°C

30°C

%

0°C

15°C

20°C

30°C

0 1 2 3 4

0.9999 1.0028 1.0059 1.0090 1.0120

0.9991 1.0019 1.0045 1.0072 1.0100

0.9982 1.0019 1.0044 1.0070 1.0093

0.9957 0.9980 1.0004 1.0028 1.0053

25 26 27 28 29

1.0706 1.0733 1.0760 1.0787 1.0813

1.0627 1.0652 1.0678 1.0702 1.0726

1.0609 1.0633 1.0656 1.0681 1.0705

1.0540 1.0564 1.0587 1.0609 1.0632

50 51 52 53 54

1.1349 1.1374 1.1399 1.1424 1.1448

1.1225 1.1248 1.1271 1.1294 1.1318

1.1207 1.1223 1.1244 1.1269 1.1295

1.1098 1.1120 1.1142 1.1164 1.1186

75 76 77 78 79

1.1953 1.1976 1.1999 1.2021 1.2043

1.1794 1.1816 1.1837 1.1859 1.1881

1.1769 1.1785 1.1801 1.1818 1.1837

1.1636 1.1656 1.1676 1.1697 1.1717

5 6 7 8 9

1.0150 1.0179 1.0207 1.0237 1.0266

1.0124 1.0151 1.0177 1.0204 1.0230

1.0115 1.0141 1.0170 1.0196 1.0221

1.0075 1.0101 1.0125 1.0149 1.0173

30 31 32 33 34

1.0839 1.0866 1.0891 1.0916 1.0941

1.0750 1.0774 1.0798 1.0821 1.0844

1.0729 1.0753 1.0777 1.0800 1.0823

1.0654 1.0676 1.0699 1.0721 1.0743

55 56 57 58 59

1.1472 1.1497 1.1523 1.1548 1.1573

1.1341 1.1365 1.1388 1.1411 1.1434

1.1320 1.1342 1.1361 1.1381 1.1401

1.1208 1.1230 1.1253 1.1274 1.1295

80 81 82 83 84

1.2065 1.2088 1.2110 1.2132 1.2154

1.1902 1.1924 1.1944 1.1965 1.1985

1.1806 1.1876 1.1896 1.1914 1.1929

1.1737 1.1758 1.1778 1.1798 1.1817

10 11 12 13 14

1.0295 1.0324 1.0351 1.0379 1.0407

1.0256 1.0281 1.0306 1.0330 1.0355

1.0246 1.0271 1.0296 1.0321 1.0345

1.0197 1.0221 1.0244 1.0267 1.0290

35 36 37 38 39

1.0966 1.0993 1.1018 1.1043 1.1069

1.0867 1.0892 1.0916 1.0940 1.0964

1.0847 1.0871 1.0895 1.0919 1.0940

1.0766 1.0788 1.0810 1.0832 1.0854

60 61 62 63 64

1.1597 1.1621 1.1645 1.1669 1.1694

1.1458 1.1481 1.1504 1.1526 1.1549

1.1424 1.1448 1.1473 1.1493 1.1517

1.1317 1.1338 1.1360 1.1382 1.1403

85 86 87 88 89

1.2176 1.2196 1.2217 1.2237 1.2258

1.2005 1.2025 1.2045 1.2064 1.2084

1.1953 1.1976 1.1994 1.2012 1.2028

1.1837 1.1856 1.1875 1.1893 1.1910

15 16 17 18 19

1.0435 1.0463 1.0491 1.0518 1.0545

1.0380 1.0405 1.0430 1.0455 1.0480

1.0370 1.0393 1.0417 1.0441 1.0464

1.0313 1.0336 1.0358 1.0381 1.0404

40 41 42 43 44

1.1095 1.1122 1.1148 1.1174 1.1199

1.0988 1.1012 1.1036 1.1060 1.1084

1.0963 1.0990 1.1015 1.1038 1.1062

1.0876 1.0898 1.0920 1.0943 1.0965

65 66 67 68 69

1.1718 1.1742 1.1766 1.1790 1.1813

1.1572 1.1595 1.1618 1.1640 1.1663

1.1543 1.1565 1.1584 1.1604 1.1628

1.1425 1.1446 1.1467 1.1489 1.1510

90 91 92 93 94

1.2278 1.2297 1.2316 1.2335 1.2354

1.2102 1.2121 1.2139 1.2157 1.2174

1.2044 1.2059 1.2078 1.2099 1.2117

1.1927 1.1945 1.1961 1.1978 1.1994

20 21 22 23 24

1.0571 1.0598 1.0625 1.0652 1.0679

1.0505 1.0532 1.0556 1.0580 1.0604

1.0488 1.0512 1.0537 1.0561 1.0585

1.0427 1.0451 1.0473 1.0496 1.0518

45 46 47 48 49

1.1224 1.1249 1.1274 1.1299 1.1324

1.1109 1.1133 1.1156 1.1179 1.1202

1.1085 1.1108 1.1130 1.1157 1.1185

1.0987 1.1009 1.1031 1.1053 1.1076

70 71 72 73 74

1.1835 1.1858 1.1882 1.1906 1.1929

1.1685 1.1707 1.1729 1.1751 1.1773

1.1655 1.1677 1.1702 1.1728 1.1752

1.1531 1.1552 1.1573 1.1595 1.1615

95 96 97 98 99

1.2372 1.2390 1.2408 1.2425 1.2441

1.2191 1.2208 1.2224 1.2240 1.2257

1.2140 1.2158 1.2170 1.2183 1.2202

1.2008 1.2022 1.2036 1.2048 1.2061

100

1.2456

1.2273

1.2212

1.2073

2-110

PHYSICAL AND CHEMICAL DATA

TABLE 2-107 %

Acetic Acid (CH3COOH)

0°C

10°C

15°C

20°C

25°C

30°C

40°C

%

0°C

10°C

15°C

20°C

25°C

30°C

40°C

0 1 2 3 4

0.9999 1.0016 1.0033 1.0051 1.0070

0.9997 1.0013 1.0029 1.0044 1.0060

0.9991 1.0006 1.0021 1.0036 1.0051

0.9982 .9996 1.0012 1.0025 1.0040

0.9971 .9987 1.0000 1.0013 1.0027

0.9957 .9971 .9984 .9997 1.0011

0.9922 .9934 .9946 .9958 .9970

50 51 52 53 54

1.0729 1.0738 1.0748 1.0757 1.0765

1.0654 1.0663 1.0671 1.0679 1.0687

1.0613 1.0622 1.0629 1.0637 1.0644

1.0575 1.0582 1.0590 1.0597 1.0604

1.0534 1.0542 1.0549 1.0555 1.0562

1.0492 1.0499 1.0506 1.0512 1.0518

1.0408 1.0414 1.0421 1.0427 1.0432

5 6 7 8 9

1.0088 1.0106 1.0124 1.0142 1.0159

1.0076 1.0092 1.0108 1.0124 1.0140

1.0066 1.0081 1.0096 1.0111 1.0126

1.0055 1.0069 1.0083 1.0097 1.0111

1.0041 1.0055 1.0068 1.0081 1.0094

1.0024 1.0037 1.0050 1.0063 1.0076

.9982 .9994 1.0006 1.0018 1.0030

55 56 57 58 59

1.0774 1.0782 1.0790 1.0798 1.0805

1.0694 1.0701 1.0708 1.0715 1.0722

1.0651 1.0658 1.0665 1.0672 1.0678

1.0611 1.0618 1.0624 1.0631 1.0637

1.0568 1.0574 1.0580 1.0586 1.0592

1.0525 1.0531 1.0536 1.0542 1.0547

1.0438 1.0443 1.0448 1.0453 1.0458

10 11 12 13 14

1.0177 1.0194 1.0211 1.0228 1.0245

1.0156 1.0171 1.0187 1.0202 1.0217

1.0141 1.0155 1.0170 1.0184 1.0199

1.0125 1.0139 1.0154 1.0168 1.0182

1.0107 1.0120 1.0133 1.0146 1.0159

1.0089 1.0102 1.0115 1.0127 1.0139

1.0042 1.0054 1.0065 1.0077 1.0088

60 61 62 63 64

1.0813 1.0820 1.0826 1.0833 1.0838

1.0728 1.0734 1.0740 1.0746 1.0752

1.0684 1.0690 1.0696 1.0701 1.0706

1.0642 1.0648 1.0653 1.0658 1.0662

1.0597 1.0602 1.0607 1.0612 1.0616

1.0552 1.0557 1.0562 1.0566 1.0571

1.0462 1.0466 1.0470 1.0473 1.0477

15 16 17 18 19

1.0262 1.0278 1.0295 1.0311 1.0327

1.0232 1.0247 1.0262 1.0276 1.0291

1.0213 1.0227 1.0241 1.0255 1.0269

1.0195 1.0209 1.0223 1.0236 1.0250

1.0172 1.0185 1.0198 1.0210 1.0223

1.0151 1.0163 1.0175 1.0187 1.0198

1.0099 1.0110 1.0121 1.0132 1.0142

65 66 67 68 69

1.0844 1.0850 1.0856 1.0860 1.0865

1.0757 1.0762 1.0767 1.0771 1.0775

1.0711 1.0716 1.0720 1.0725 1.0729

1.0666 1.0671 1.0675 1.0678 1.0682

1.0621 1.0624 1.0628 1.0631 1.0634

1.0575 1.0578 1.0582 1.0585 1.0588

1.0480 1.0483 1.0486 1.0489 1.0491

20 21 22 23 24

1.0343 1.0358 1.0374 1.0389 1.0404

1.0305 1.0319 1.0333 1.0347 1.0361

1.0283 1.0297 1.0310 1.0323 1.0336

1.0263 1.0276 1.0288 1.0301 1.0313

1.0235 1.0248 1.0260 1.0272 1.0283

1.0210 1.0222 1.0233 1.0244 1.0256

1.0153 1.0164 1.0174 1.0185 1.0195

70 71 72 73 74

1.0869 1.0874 1.0877 1.0881 1.0884

1.0779 1.0783 1.0786 1.0789 1.0792

1.0732 1.0736 1.0738 1.0741 1.0743

1.0685 1.0687 1.0690 1.0693 1.0694

1.0637 1.0640 1.0642 1.0644 1.0645

1.0590 1.0592 1.0594 1.0595 1.0596

1.0493 1.0495 1.0496 1.0497 1.0498

25 26 27 28 29

1.0419 1.0434 1.0449 1.0463 1.0477

1.0375 1.0388 1.0401 1.0414 1.0427

1.0349 1.0362 1.0374 1.0386 1.0399

1.0326 1.0338 1.0349 1.0361 1.0372

1.0295 1.0307 1.0318 1.0329 1.0340

1.0267 1.0278 1.0289 1.0299 1.0310

1.0205 1.0215 1.0225 1.0234 1.0244

75 76 77 78 79

1.0887 1.0889 1.0891 1.0893 1.0894

1.0794 1.0796 1.0797 1.0798 1.0798

1.0745 1.0746 1.0747 1.0747 1.0747

1.0696 1.0698 1.0699 1.0700 1.0700

1.0647 1.0648 1.0648 1.0648 1.0648

1.0597 1.0598 1.0598 1.0598 1.0597

1.0499 1.0499 1.0499 1.0498 1.0497

30 31 32 33 34

1.0491 1.0505 1.0519 1.0532 1.0545

1.0440 1.0453 1.0465 1.0477 1.0489

1.0411 1.0423 1.0435 1.0446 1.0458

1.0384 1.0395 1.0406 1.0417 1.0428

1.0350 1.0361 1.0372 1.0382 1.0392

1.0320 1.0330 1.0341 1.0351 1.0361

1.0253 1.0262 1.0272 1.0281 1.0289

80 81 82 83 84

1.0895 1.0895 1.0895 1.0895 1.0893

1.0798 1.0797 1.0796 1.0795 1.0793

1.0747 1.0745 1.0743 1.0741 1.0738

1.0700 1.0699 1.0698 1.0696 1.0693

1.0647 1.0646 1.0644 1.0642 1.0638

1.0596 1.0594 1.0592 1.0589 1.0585

1.0495 1.0493 1.0490 1.0487 1.0483

35 36 37 38 39

1.0558 1.0571 1.0584 1.0596 1.0608

1.0501 1.0513 1.0524 1.0535 1.0546

1.0469 1.0480 1.0491 1.0501 1.0512

1.0438 1.0449 1.0459 1.0469 1.0479

1.0402 1.0412 1.0422 1.0432 1.0441

1.0371 1.0380 1.0390 1.0399 1.0408

1.0298 1.0306 1.0314 1.0322 1.0330

85 86 87 88 89

1.0891 1.0887 1.0883 1.0877 1.0872

1.0790 1.0787 1.0783 1.0778 1.0773

1.0735 1.0731 1.0726 1.0721 1.0715

1.0689 1.0685 1.0680 1.0675 1.0668

1.0635 1.0630 1.0626 1.0620 1.0613

1.0582 1.0576 1.0571 1.0564 1.0557

1.0479 1.0473 1.0467 1.0460 1.0453

40 41 42 43 44

1.0621 1.0633 1.0644 1.0656 1.0667

1.0557 1.0568 1.0578 1.0588 1.0598

1.0522 1.0532 1.0542 1.0551 1.0561

1.0488 1.0498 1.0507 1.0516 1.0525

1.0450 1.0460 1.0469 1.0477 1.0486

1.0416 1.0425 1.0433 1.0441 1.0449

1.0338 1.0346 1.0353 1.0361 1.0368

90 91 92 93 94

1.0865 1.0857 1.0848 1.0838 1.0826

1.0766 1.0758 1.0749 1.0739 1.0727

1.0708 1.0700 1.0690 1.0680 1.0667

1.0661 1.0652 1.0643 1.0632 1.0619

1.0605 1.0597 1.0587 1.0577 1.0564

1.0549 1.0541 1.0530 1.0518 1.0506

1.0445 1.0436 1.0426 1.0414 1.0401

45 46 47 48 49

1.0679 1.0689 1.0699 1.0709 1.0720

1.0608 1.0618 1.0627 1.0636 1.0645

1.0570 1.0579 1.0588 1.0597 1.0605

1.0534 1.0542 1.0551 1.0559 1.0567

1.0495 1.0503 1.0511 1.0518 1.0526

1.0456 1.0464 1.0471 1.0479 1.0486

1.0375 1.0382 1.0389 1.0395 1.0402

95 96 97 98 99

1.0813 1.0798 1.0780 1.0759 1.0730

1.0714

1.0652 1.0632 1.0611 1.0590 1.0567

1.0605 1.0588 1.0570 1.0549 1.0524

1.0551 1.0535 1.0516 1.0495 1.0468

1.0491 1.0473 1.0454 1.0431 1.0407

1.0386 1.0368 1.0348 1.0325 1.0299

100

1.0697

1.0545

1.0498

1.0440

1.0380

1.0271

DENSITIES OF AQUEOUS ORGANIC SOLUTIONS TABLE 2-108

TABLE 2-109

2-111

Oxalic Acid (H2C2O4)

%

d 17.5 4

%

d 17.5 4

1 2 4

1.0035 1.0070 1.0140

8 10 12

1.0280 1.0350 1.0420

Methyl Alcohol (CH3OH)*

%

0°C

10°C

20°C

15°C

%

0°C

10°C

20°C

15°C

%

0°C

10°C

20°C

15°C

0 1 2 3 4

0.9999 .9981 .9963 .9946 .9930

0.9997 .9980 .9962 .9945 .9929

15.56°C 0.9990 .9973 .9955 .9938 .9921

0.9982 .9965 .9948 .9931 .9914

0.99913 .99727 .99543 .99370 .99198

35 36 37 38 39

0.9534 .9520 .9505 .9490 .9475

0.9484 .9469 .9453 .9437 .9420

15.56°C 0.9456 .9440 .9422 .9405 .9387

0.9433 .9416 .9398 .9381 .9363

0.94570 .94404 .94237 .94067 .93894

70 71 72 73 74

0.8869 .8847 .8824 .8801 .8778

0.8794 .8770 .8747 .8724 .8699

15.56°C 0.8748 .8726 .8702 .8678 .8653

0.8715 .8690 .8665 .8641 .8616

0.87507 .87271 .87033 .86792 .86546

5 6 7 8 9

.9914 .9899 .9884 .9870 .9856

.9912 .9896 .9881 .9865 .9849

.9904 .9889 .9872 .9857 .9841

.9896 .9880 .9863 .9847 .9831

.99029 .98864 .98701 .98547 .98394

40 41 42 43 44

.9459 .9443 .9427 .9411 .9395

.9403 .9387 .9370 .9352 .9334

.9369 .9351 .9333 .9315 .9297

.9345 .9327 .9309 .9290 .9272

.93720 .93543 .93365 .93185 .93001

75 76 77 78 79

.8754 .8729 .8705 .8680 .8657

.8676 .8651 .8626 .8602 .8577

.8629 .8604 .8579 .8554 .8529

.8592 .8567 .8542 .8518 .8494

.86300 .86051 .85801 .85551 .85300

10 11 12 13 14

.9842 .9829 .9816 .9804 .9792

.9834 .9820 .9805 .9791 .9778

.9826 .9811 .9796 .9781 .9766

.9815 .9799 .9784 .9768 .9754

.98241 .98093 .97945 .97802 .97660

45 46 47 48 49

.9377 .9360 .9342 .9324 .9306

.9316 .9298 .9279 .9260 .9240

.9279 .9261 .9242 .9223 .9204

.9252 .9234 .9214 .9196 .9176

.92815 .92627 .92436 .92242 .92048

80 81 82 83 84

.8634 .8610 .8585 .8560 .8535

.8551 .8527 .8501 .8475 .8449

.8503 .8478 .8452 .8426 .8400

.8469 .8446 .8420 .8394 .8366

.85048 .84794 .84536 .84274 .84009

15 16 17 18 19

.9780 .9769 .9758 .9747 .9736

.9764 .9751 .9739 .9726 .9713

.9752 .9738 .9723 .9709 .9695

.9740 .9725 .9710 .9696 .9681

.97518 .97377 .97237 .97096 .96955

50 51 52 53 54

.9287 .9269 .9250 .9230 .9211

.9221 .9202 .9182 .9162 .9142

.9185 .9166 .9146 .9126 .9106

.9156 .9135 .9114 .9094 .9073

.91852 .91653 .91451 .91248 .91044

85 86 87 88 89

.8510 .8483 .8456 .8428 .8400

.8422 .8394 .8367 .8340 .8314

.8374 .8347 .8320 .8294 .8267

.8340 .8314 .8286 .8258 .8230

.83742 .83475 .83207 .82937 .82667

20 21 22 23 24

.9725 .9714 .9702 .9690 .9678

.9700 .9687 .9673 .9660 .9646

.9680 .9666 .9652 .9638 .9624

.9666 .9651 .9636 .9622 .9607

.96814 .96673 .96533 .96392 .96251

55 56 57 58 59

.9191 .9172 .9151 .9131 .9111

.9122 .9101 .9080 .9060 .9039

.9086 .9065 .9045 .9024 .9002

.9052 .9032 .9010 .8988 .8968

.90839 .90631 .90421 .90210 .89996

90 91 92 93 94

.8374 .8347 .8320 .8293 .8266

.8287 .8261 .8234 .8208 .8180

.8239 .8212 .8185 .8157 .8129

.8202 .8174 .8146 .8118 .8090

.82396 .82124 .81849 .81568 .81285

25 26 27 28 29

.9666 .9654 .9642 .9629 .9616

.9632 .9618 .9604 .9590 .9575

.9609 .9595 .9580 .9565 .9550

.9592 .9576 .9562 .9546 .9531

.96108 .95963 .95817 .95668 .95518

60 61 62 63 64

.9090 .9068 .9046 .9024 .9002

.9018 .8998 .8977 .8955 .8933

.8980 .8958 .8936 .8913 .8890

.8946 .8924 .8902 .8879 .8856

.89781 .89563 .89341 .89117 .88890

95 96 97 98 99

.8240 .8212 .8186 .8158 .8130

.8152 .8124 .8096 .8068 .8040

.8101 .8073 .8045 .8016 .7987

.8062 .8034 .8005 .7976 .7948

.80999 .80713 .80428 .80143 .79859

30 31 32 33 34

.9604 .9590 .9576 .9563 .9549

.9560 .9546 .9531 .9516 .9500

.9535 .9521 .9505 .9489 .9473

.9515 .9499 .9483 .9466 .9450

.95366 .95213 .95056 .94896 .94734

65 66 67 68 69

.8980 .8958 .8935 .8913 .8891

.8911 .8888 .8865 .8842 .8818

.8867 .8844 .8820 .8797 .8771

.8834 .8811 .8787 .8763 .8738

.88662 .88433 .88203 .87971 .87739

100

.8102

.8009

.7959

.7917

.79577

*It should be noted that the values for 100 percent do not agree with some data available elsewhere, e.g., American Institute of Physics Handbook, McGraw-Hill, New York, 1957. Also, see Atack, Handbook of Chemical Data, Reinhold, New York, 1957.

2-112

PHYSICAL AND CHEMICAL DATA

TABLE 2-110

Ethyl Alcohol (C2H5OH)*

%

10°C

15°C

20°C

25°C

30°C

35°C

40°C

%

10°C

15°C

20°C

25°C

30°C

35°C

40°C

0 1 2 3 4

0.99973 785 602 426 258

0.99913 725 542 365 195

0.99823 636 453 275 103

0.99708 520 336 157 .98984

0.99568 379 194 014 .98839

0.99406 217 031 .98849 672

0.99225 034 .98846 663 485

50 51 52 53 54

0.92126 .91943 723 502 279

0.91776 555 333 110 .90885

0.91384 160 .90936 711 485

0.90985 760 534 307 079

0.90580 353 125 .89896 667

0.90168 .89940 710 479 248

0.89750 519 288 056 .88823

5 6 7 8 9

098 .98946 801 660 524

032 .98877 729 584 442

.98938 780 627 478 331

817 656 500 346 193

670 507 347 189 031

501 335 172 009 .97846

311 142 .97975 808 641

55 56 57 58 59

055 .90831 607 381 154

659 433 207 .89980 752

258 031 .89803 574 344

.89850 621 392 162 .88931

437 206 .88975 744 512

016 .88784 552 319 085

589 356 122 .87888 653

10 11 12 13 14

393 267 145 026 .97911

304 171 041 .97914 790

187 047 .97910 775 643

043 .97897 753 611 472

.97875 723 573 424 278

685 527 371 216 063

475 312 150 .96989 829

60 61 62 63 64

.89927 698 468 237 006

523 293 062 .88830 597

113 .88882 650 417 183

699 446 233 .87998 763

278 044 .87809 574 337

.87851 615 379 142 .86905

417 180 .86943 705 466

15 16 17 18 19

800 692 583 473 363

669 552 433 313 191

514 387 259 129 .96997

334 199 062 .96923 782

133 .96990 844 697 547

.96911 760 607 452 294

670 512 352 189 023

65 66 67 68 69

.88774 541 308 074 .87839

364 130 .87895 660 424

.87948 713 477 241 004

527 291 054 .86817 579

100 .86863 625 387 148

667 429 190 .85950 710

227 .85987 747 407 266

20 21 22 23 24

252 139 024 .96907 787

068 .96944 818 689 558

864 729 592 453 312

639 495 348 199 048

395 242 087 .95929 769

134 .95973 809 643 476

.95856 687 516 343 168

70 71 72 73 74

602 365 127 .86888 648

187 .86949 710 470 229

.86766 527 287 047 .85806

340 100 .85859 618 376

.85908 667 426 184 .84941

470 228 .84986 743 500

025 .84783 540 297 053

25 26 27 28 29

665 539 406 268 125

424 287 144 .95996 844

168 020 .95867 710 548

.95895 738 576 410 241

607 442 272 098 .94922

306 133 .94955 774 590

.94991 810 625 438 248

75 76 77 78 79

408 168 .85927 685 442

.85988 747 505 262 018

564 322 079 .84835 590

134 .84891 647 403 158

698 455 211 .83966 720

257 013 .83768 523 277

.83809 564 319 074 .82827

30 31 32 33 34

.95977 823 665 502 334

686 524 357 186 011

382 212 038 .94860 679

067 .94890 709 525 337

741 557 370 180 .93986

403 214 021 .93825 626

055 .93860 662 461 257

80 81 82 83 84

197 .84950 702 453 203

.84772 525 277 028 .83777

344 096 .83848 599 348

.83911 664 415 164 .82913

473 224 .82974 724 473

029 .82780 530 279 027

578 329 079 .81828 576

35 36 37 38 39

162 .94986 805 620 431

.94832 650 464 273 079

494 306 114 .93919 720

146 .93952 756 556 353

790 591 390 186 .92979

425 221 016 .92808 597

051 .92843 634 422 208

85 86 87 88 89

.83951 697 441 181 .82919

525 271 014 .82754 492

095 .82840 583 323 062

660 405 148 .81888 626

220 .81965 708 448 186

.81774 519 262 003 .80742

322 067 .80811 552 291

40 41 42 43 44

238 042 .93842 639 433

.93882 682 478 271 062

518 314 107 .92897 685

148 .92940 729 516 301

770 558 344 128 .91910

385 170 .91952 733 513

.91992 774 554 332 108

90 91 92 93 94

654 386 114 .81839 561

227 .81959 688 413 134

.81797 529 257 .80983 705

362 094 .80823 549 272

.80922 655 384 111 .79835

478 211 .79941 669 393

028 .79761 491 220 .78947

45 46 47 48 49

226 017 .92806 593 379

.92852 640 426 211 .91995

472 257 041 .91823 604

085 .91868 649 429 208

692 472 250 028 .90805

291 069 .90845 621 396

.90884 660 434 207 .89979

95 96 97 98 99

278 .80991 698 399 094

.80852 566 274 .79975 670

424 138 .79846 547 243

.79991 706 415 117 .78814

555 271 .78981 684 382

114 .78831 542 247 .77946

670 388 100 .77806 507

100

.79784

360

.78934

506

075

641

203

*For data from −78° to 78°C, see p. 2-142, Table 2N-5, American Institute of Physics Handbook, McGraw-Hill, New York, 1957.

DENSITIES OF AQUEOUS ORGANIC SOLUTIONS TABLE 2-111 % alcohol by weight

2-113

Densities of Mixtures of C2H5OH and H2O at 20°C g/mL Tenths of %

0

1

2

3

4

5

6

7

0 1 2 3 4

0.99823 636 453 275 103

804 618 435 257 087

785 599 417 240 070

766 581 399 222 053

748 562 381 205 037

729 544 363 188 020

710 525 345 171 003

692 507 327 154 *987

5 6 7 8 9

.98938 780 627 478 331

922 765 612 463 316

906 749 597 449 301

890 734 582 434 287

874 718 567 419 273

859 703 553 404 258

843 688 538 389 244

827 673 523 374 229

10 11 12 13 14

187 047 .97910 775 643

172 033 896 761 630

158 019 883 748 617

144 006 869 735 604

130 *992 855 722 591

117 *978 842 709 578

103 *964 828 696 565

089 *951 815 683 552

15 16 17 18 19

514 387 259 129 .96997

501 374 246 116 984

488 361 233 103 971

475 349 220 089 957

462 336 207 076 944

450 323 194 063 931

438 310 181 050 917

425 297 168 037 904

412 284 155 024 891

20 21 22 23 24

864 729 592 453 312

850 716 578 439 297

837 702 564 425 283

823 688 551 411 269

810 675 537 396 254

796 661 523 382 240

783 647 509 368 225

769 634 495 354 211

756 620 481 340 196

25 26 27 28 29

168 020 .95867 710 548

153 005 851 694 532

139 *990 836 678 516

124 *975 820 662 499

109 *959 805 646 483

094 *944 789 630 466

080 *929 773 613 450

30 31 32 33 34

382 212 038 .94860 679

365 195 020 842 660

349 178 003 824 642

332 161 *985 806 624

315 143 *967 788 605

298 126 *950 770 587

35 36 37 38 39

494 306 114 .93919 720

475 287 095 899 700

456 268 075 879 680

438 249 056 859 660

419 230 036 840 640

40 41 42 43 44

518 314 107 .92897 685

498 294 086 876 664

478 273 065 855 642

458 253 044 834 621

45 46 47 48 49

472 257 041 .91823 604

450 236 019 801 582

429 214 *997 780 560

408 193 *976 758 538

Tenths of % 0

1

2

3

4

5

6

7

50 51 52 53 54

0.91384 160 .90936 711 485

361 138 914 689 463

339 116 891 666 440

317 093 869 644 417

295 071 846 621 395

272 049 824 598 372

250 026 801 576 349

228 206 183 004 *981 *959 779 756 734 553 531 508 327 304 281

796 642 493 345 201

55 56 57 58 59

258 031 .89803 574 344

236 008 780 551 321

213 *985 757 528 298

190 *962 734 505 275

167 *939 711 482 252

145 *917 688 459 229

122 *894 665 436 206

099 076 054 *871 *848 *825 643 620 597 413 390 367 183 160 137

075 061 *937 *923 801 788 670 657 539 526

60 61 62 63 64

113 .88882 650 417 183

090 859 626 393 160

067 836 603 370 136

044 812 580 347 113

021 789 557 323 089

*998 766 533 300 066

*975 743 510 277 042

*951 *928 *905 720 696 673 487 463 440 253 230 206 019 *995 *972

400 272 142 010 877

65 66 67 68 69

.87948 713 477 241 004

925 689 454 218 *981

901 666 430 194 *957

878 642 406 170 *933

854 619 383 147 *909

831 595 359 123 *885

807 572 336 099 *862

784 760 737 548 524 501 312 288 265 075 052 028 *838 *814 *790

742 606 467 326 182

70 71 72 73 74

.86766 527 287 047 .85806

742 503 263 022 781

718 479 239 *998 757

694 455 215 *974 733

671 431 191 *950 709

647 407 167 *926 685

623 383 143 *902 661

599 575 551 339 335 311 119 095 071 *878 *854 *830 636 612 588

065 *914 757 597 433

050 035 *898 *883 742 726 581 565 416 400

75 76 77 78 79

564 322 079 .84835 590

540 297 055 811 566

515 273 031 787 541

491 467 443 419 394 370 346 249 225 200 176 152 128 103 006 *982 *958 *933 *909 *884 *860 762 738 713 689 664 640 615 517 492 467 443 418 393 369

281 108 *932 752 568

264 091 *914 734 550

247 230 074 056 *896 *878 715 697 531 512

80 81 82 83 84

344 096 .83848 599 348

319 072 823 574 323

294 047 798 549 297

270 022 773 523 272

245 *997 748 498 247

220 *972 723 473 222

196 *947 698 448 196

171 146 121 *923 *898 *873 674 649 624 423 398 373 171 146 120

400 211 017 820 620

382 192 *997 800 599

363 172 *978 780 579

344 325 153 134 *958 *939 760 740 559 539

85 86 87 88 89

095 .82840 583 323 062

070 815 557 297 035

044 789 531 271 009

019 763 505 245 *983

*994 738 479 219 *956

*968 712 453 193 *930

*943 686 427 167 *903

*917 *892 *866 660 635 609 401 375 349 140 114 088 *877 *850 *824

437 232 023 812 600

417 212 002 791 579

396 191 *981 770 557

376 170 *960 749 536

356 335 149 129 *939 *918 728 707 515 493

90 91 92 93 94

.81797 529 257 .80983 705

770 502 230 955 677

744 475 203 928 649

717 448 175 900 621

690 421 148 872 593

664 394 120 844 565

637 366 093 817 537

386 171 *954 736 516

365 150 *932 714 494

343 128 *910 692 472

322 106 *889 670 450

300 279 085 063 *867 *845 648 626 428 406

95 96 97 98 99

424 138 .79846 547 243

395 109 816 517 213

367 080 787 487 182

338 051 757 456 151

310 281 253 224 195 166 022 *993 *963 *934 *905 *875 727 698 668 638 608 578 426 396 365 335 305 274 120 089 059 028 *997 *966

100

.78934

*Indicates change in the first two decimal places.

8

% alcohol by weight

9

673 655 489 471 310 292 137 120 *971 *954 811 658 508 360 215

610 339 066 789 509

8

583 312 038 761 480

9

556 285 010 733 452

2-114

PHYSICAL AND CHEMICAL DATA

TABLE 2-112 % alcohol by volume at 60°F 0 1 2 3 4

Specific Gravity (60°/60°F [(15.56°/15.56°C)]) of Mixtures by Volume of C2H5OH and H2O

Tenths of % 0

1

1.00000 *985 0.99850 835 703 688 559 545 419 405

2

3

4

5

6

7

8

9

*970 820 674 531 391

*955 806 659 516 378

*940 791 645 502 364

*925 776 630 488 350

*910 761 616 474 336

*895 747 602 460 323

*880 732 587 446 309

865 717 573 432 296

176 163 047 035 *923 *911 803 791 684 672

% alcohol by volume at 60°F

Tenths of % 0

1

2

3

4

5

6

50 51 52 53 54

0.93426 230 031 .92830 626

407 210 011 810 605

387 190 *991 789 585

368 171 *971 769 564

348 151 *951 749 544

328 131 *931 728 523

309 111 *911 708 502

55 56 57 58 59

419 210 .91999 784 565

398 189 978 762 543

377 168 956 741 521

357 147 935 719 499

336 126 914 697 477

315 105 892 675 455

294 084 871 653 433

7

8

9

289 270 250 091 071 051 *890 *870 *850 688 667 647 482 461 440

5 6 7 8 9

282 150 022 .98899 779

269 137 009 887 767

255 124 *997 875 755

242 111 *984 863 743

228 098 *972 851 731

215 085 *960 838 720

202 073 *947 826 708

189 060 *935 814 696

10 11 12 13 14

661 544 430 319 210

649 532 419 308 200

637 521 408 297 190

625 509 396 286 179

614 498 385 275 168

602 487 374 264 157

590 475 363 254 147

579 464 352 243 136

567 452 341 232 125

556 441 330 221 115

60 61 62 63 64

344 120 .90893 664 434

322 097 870 641 411

299 075 847 618 388

277 052 825 595 365

255 030 802 572 341

232 007 779 549 318

210 *984 756 526 295

15 16 17 18 19

104 .97998 895 794 694

093 988 885 784 684

083 977 875 774 674

072 967 864 764 664

062 956 854 754 654

051 946 844 744 645

040 936 834 734 635

030 925 824 724 625

019 915 814 714 615

009 905 804 704 605

65 66 67 68 69

202 .89967 729 489 245

179 943 705 465 220

155 920 681 441 196

132 896 657 416 171

108 872 633 392 147

085 848 609 368 122

061 825 585 343 098

038 801 561 319 073

014 *991 777 753 537 513 295 270 048 024

20 21 22 23 24

596 496 395 293 189

586 486 385 283 179

576 476 375 272 168

566 466 365 262 158

556 456 354 252 147

546 446 344 241 137

536 436 334 231 126

526 425 324 221 116

516 415 313 210 105

506 405 303 200 095

70 71 72 73 74

.88999 751 499 244 .87987

974 725 474 218 961

950 700 448 193 935

925 675 423 167 910

900 650 397 141 884

875 625 372 116 858

850 600 346 090 832

825 574 321 064 806

801 549 296 039 780

25 26 27 28 29

084 .96978 870 760 648

073 967 859 749 637

063 957 848 738 625

052 946 837 727 614

042 935 826 715 603

031 924 815 704 591

020 914 804 693 580

010 903 793 682 568

*999 *988 892 881 782 771 671 659 557 546

75 76 77 78 79

728 465 199 .86929 656

702 439 172 902 629

676 412 145 875 601

650 386 118 847 574

623 359 092 820 546

597 332 065 793 518

571 306 038 766 491

545 518 492 279 252 226 011 *984 *957 738 711 684 463 435 408

30 31 32 33 34

534 418 296 170 041

522 406 284 157 028

511 394 271 144 015

499 382 259 132 002

488 370 246 119 *988

476 358 234 106 *975

464 346 221 093 *962

453 334 209 080 *948

441 429 321 309 196 183 067 054 *935 *921

80 81 82 83 84

380 100 .85817 531 240

352 072 789 502 211

324 044 760 473 181

296 015 732 444 152

269 *987 703 415 122

241 *959 674 386 093

213 *931 646 357 063

185 157 129 *902 *874 *846 617 588 560 328 299 270 033 004 *974

35 36 37 38 39

.95908 770 628 482 332

894 756 614 467 317

881 742 599 452 302

867 728 585 437 286

854 714 570 423 271

840 700 556 408 256

826 685 541 393 240

812 671 526 378 225

784 643 497 347 194

85 86 87 88 89

.84944 642 336 025 .83707

914 612 305 *994 675

884 581 274 *962 643

854 551 243 *930 610

824 520 212 *899 578

794 490 181 *867 545

764 459 150 *835 513

734 703 673 428 398 367 119 088 056 *803 *771 *739 480 447 415

40 41 42 43 44

178 020 .94858 693 524

162 004 842 676 507

147 *988 825 660 490

131 *972 809 643 473

115 *956 792 626 455

100 *940 776 609 438

084 *923 759 592 421

068 *907 743 575 403

052 036 *891 *875 726 710 558 541 386 369

90 91 92 93 94

382 049 .82705 351 .81984

349 015 670 315 947

315 *981 635 279 909

282 *947 600 243 871

249 *913 565 206 834

216 *879 529 170 796

183 *845 494 133 757

150 116 083 *810 *776 *741 458 423 387 096 059 022 719 681 642

45 46 47 48 49

351 174 .93993 808 619

334 156 975 789 600

316 138 956 771 581

298 120 938 752 562

281 102 920 733 543

263 084 901 714 523

245 066 883 695 504

228 048 864 676 485

95 96 97 98 99

603 206 .80792 356 .79889

564 165 750 311 841

525 125 707 265 792

486 084 664 219 743

446 042 620 173 693

407 001 577 127 643

367 *960 533 080 593

327 287 247 *918 *876 *834 489 445 401 033 *985 *937 543 492 441

100

389

*Indicates change in first two decimal places.

798 657 512 362 209

210 030 845 657 465

192 011 827 638 446

273 062 849 631 410

252 041 827 610 388

231 020 806 588 366

188 165 143 *962 *939 *916 733 710 687 503 480 457 272 249 225

776 524 270 013 754

DENSITIES OF AQUEOUS ORGANIC SOLUTIONS

2-115

TABLE 2-113 n-Propyl Alcohol (C3H7OH) %

0°C

15°C

30°C

%

0°C

15°C

30°C

%

0°C

15°C

30°C

%

0°C

15°C

30°C

%

0°C

15°C

30°C

0 1 2 3 4

0.9999 .9982 .9967 .9952 .9939

0.9991 .9974 .9960 .9944 .9929

0.9957 .9940 .9924 .9908 .9893

20 21 22 23 24

0.9789 .9776 .9763 .9748 .9733

0.9723 .9705 .9688 .9670 .9651

0.9643 .9622 .9602 .9583 .9563

40 41 42 43 44

0.9430 .9411 .9391 .9371 .9352

0.9331 .9310 .9290 .9269 .9248

0.9226 .9205 .9184 .9164 .9143

60 61 62 63 64

0.9033 .9013 .8994 .8974 .8954

0.8922 .8902 .8882 .8861 .8841

0.8807 .8786 .8766 .8745 .8724

80 81 82 83 84

0.8634 .8614 .8594 .8574 .8554

0.8516 .8496 .8475 .8454 .8434

0.8394 .8373 .8352 .8332 .8311

5 6 7 8 9

.9926 .9914 .9904 .9894 .9883

.9915 .9902 .9890 .9877 .9864

.9877 .9862 .9848 .9834 .9819

25 26 27 28 29

.9717 .9700 .9682 .9664 .9646

.9633 .9614 .9594 .9576 .9556

.9543 .9522 .9501 .9481 .9460

45 46 47 48 49

.9332 .9311 .9291 .9272 .9252

.9228 .9207 .9186 .9165 .9145

.9122 .9100 .9079 .9057 .9036

65 66 67 68 69

.8934 .8913 .8894 .8874 .8854

.8820 .8800 .8779 .8759 .8739

.8703 .8682 .8662 .8641 .8620

85 86 87 88 89

.8534 .8513 .8492 .8471 .8450

.8413 .8393 .8372 .8351 .8330

.8290 .8269 .8248 .8227 .8206

10 11 12 13 14

.9874 .9865 .9857 .9849 .9841

.9852 .9840 .9828 .9817 .9806

.9804 .9790 .9775 .9760 .9746

30 31 32 33 34

.9627 .9608 .9589 .9570 .9550

.9535 .9516 .9495 .9474 .9454

.9439 .9418 .9396 .9375 .9354

50 51 52 53 54

.9232 .9213 .9192 .9173 .9153

.9124 .9104 .9084 .9064 .9044

.9015 .8994 .8973 .8952 .8931

70 71 72 73 74

.8835 .8815 .8795 .8776 .8756

.8719 .8700 .8680 .8659 .8639

.8600 .8580 .8559 .8539 .8518

90 91 92 93 94

.8429 .8408 .8387 .8364 .8342

.8308 .8287 .8266 .8244 .8221

.8185 .8164 .8142 .8120 .8098

15 16 17 18 19

.9833 .9825 .9817 .9808 .9800

.9793 .9780 .9768 .9752 .9739

.9730 .9714 .9698 .9680 .9661

35 36 37 38 39

.9530 .9511 .9491 .9471 .9450

.9434 .9413 .9392 .9372 .9351

.9333 .9312 .9289 .9269 .9247

55 56 57 58 59

.9132 .9112 .9093 .9073 .9053

.9023 .9003 .8983 .8963 .8942

.8911 .8890 .8869 .8849 .8828

75 76 77 78 79

.8736 .8716 .8695 .8675 .8655

.8618 .8598 .8577 .8556 .8536

.8497 .8477 .8456 .8435 .8414

95 96 97 98 99

.8320 .8296 .8272 .8248 .8222

.8199 .8176 .8153 .8128 .8104

.8077 .8054 .8031 .8008 .7984

100

.8194

.8077

.7958

TABLE 2-114

Isopropyl Alcohol (C3H7OH)

%

0°C

15°C*

15°C*

20°C

30°C

%

0°C

0 1 2 3 4

0.9999 .9980 .9962 .9946 .9930

0.9991 .9973 .9956 .9938 .9922

0.99913 .9972 .9954 .9936 .9920

0.9982 .9962 .9944 .9926 .9909

0.9957 .9939 .9921 .9904 .9887

35 36 37 38 39

0.9557 .9536 .9514 .9493 .9472

5 6 7 8 9

.9916 .9902 .9890 .9878 .9866

.9906 .9892 .9878 .9864 .9851

.9904 .9890 .9875 .9862 .9849

.9893 .9877 .9862 .9847 .9833

.9871 .9855 .9839 .9824 .9809

40 41 42 43 44

.9450 .9428 .9406 .9384 .9361

10 11 12 13 14

.9856 .9846 .9838 .9829 .9821

.9838 .9826 .9813 .9802 .9790

.98362 .9824 .9812 .9800 .9788

.9820 .9808 .9797 .9876 .9776

.9794 .9778 .9764 .9750 .9735

45 46 47 48 49

15 16 17 18 19

.9814 .9806 .9799 .9792 .9784

.9779 .9768 .9756 .9745 .9730

.9777 .9765 .9753 .9741 .9728

.9765 .9754 .9743 .9731 .9717

.9720 .9705 .9690 .9675 .9658

20 21 22 23 24

.9777 .9768 .9759 .9749 .9739

.9719 .9704 .9690 .9675 .9660

.97158 .9703 .9689 .9674 .9659

.9703 .9688 .9669 .9651 .9634

25 26 27 28 29

.9727 .9714 .9699 .9684 .9669

.9643 .9626 .9608 .9590 .9570

.9642 .9624 .9605 .9586 .9568

30 31 32 33 34

.9652 .9634 .9615 .9596 .9577

.9551

.95493 .9530 .9510 .9489 .9468

15°C*

15°C*

20°C

30°C

%

0°C

15°C*

15°C*

20°C

30°C

0.9446 .9424 .9401 .9379 .9356

0.9419 .9399 .9377 .9355 .9333

0.9338 .9315 .9292 .9269 .9246

70 71 72 73 74

0.8761 .8738 .8714 .8691 .8668

0.8639 .8615 .8592 .8568 .8545

0.86346 .8611 .8588 .8564 .8541

0.8584 .8560 .8537 .8513 .8489

0.8511 .8487 .8464 .8440 .8416

.93333 .9311 .9288 .9266 .9243

.9310 .9287 .9264 .9239 .9215

.9224 .9201 .9177 .9154 .9130

75 76 77 78 79

.8644 .8621 .8598 .8575 .8551

.8521 .8497 .8474 .8450 .8426

.8517 .8493 .8470 .8446 .8422

.8464 .8439 .8415 .8391 .8366

.8392 .8368 .8344 .8321 .8297

.9338 .9315 .9292 .9270 .9247

.9220 .9197 .9174 .9150 .9127

.9191 .9165 .9141 .9117 .9093

.9106 .9082 .9059 .9036 .9013

80 81 82 83 84

.8528 .8503 .8479 .8456 .8432

.8403 .8379 .8355 .8331 .8307

.83979 .8374 .8350 .8326 .8302

.8342 .8317 .8292 .8268 .8243

.8273 .8248 .8224 .8200 .8175

50 51 52 53 54

.9224 .9201 .9178 .9155 .9132

.91043 .9081 .9058 .9035 .9011

.9069 .9044 .9020 .8996 .8971

.8990 .8966 .8943 .8919 .8895

85 86 87 88 89

.8408 .8384 .8360 .8336 .8311

.8282 .8259 .8234 .8209 .8184

.8278 .8254 .8229 .8205 .8180

.8219 .8194 .8169 .8145 .8120

.8151 .8127 .8201 .8078 .8053

.9642 .9624 .9606 .9587 .9569

55 56 57 58 59

.9109 .9086 .9063 .9040 .9017

.8988 .8964 .8940 .8917 .8893

.8946 .8921 .8896 .8874 .8850

.8871 .8847 .8823 .8800 .8777

90 91 92 93 94

.8287 .8262 .8237 .8212 .8186

.8161 .8136 .8110 .8085 .8060

.81553 .8130 .8104 .8079 .8052

.8096 .8072 .8047 .8023 .7998

.8029 .8004 .7979 .7954 .7929

.9615 .9597 .9577 .9558 .9540

.9549 .9529 .9509 .9488 .9467

60 61 62 63 64

.8994 .8970 .8947 .8924 .8901

0.8829 .8805 .8781

.88690 .8845 .8821 .8798 .8775

.8825 .8800 .8776 .8751 .8727

.8752 .8728 .8704 .8680 .8656

95 96 97 98 99

.8160 .8133 .8106 .8078 .8048

.8034 .8008 .7981 .7954 .7926

.8026 .7999 .7972 .7945 .7918

.7973 .7949 .7925 .7901 .7877

.7904 .7878 .7852 .7826 .7799

.9520 .9500 .9481 .9460 .9440

.9446 .9426 .9405 .9383 .9361

65 66 67 68 69

.8878 .8854 .8831 .8807 .8784

.8757 .8733 .8710 .8686 .8662

.8752 .8728 .8705 .8682 .8658

.8702 .8679 .8656 .8632 .8609

.8631 .8607 .8583 .8559 .8535

100

.8016

.7896

.78913

.7854

.7770

*Two different observers; see International Critical Tables, vol. 3, p. 120.

2-116

PHYSICAL AND CHEMICAL DATA

TABLE 2-115

Glycerol* Density

Density

Glycerol, %

15°C

15.5°C

20°C

25°C

30°C

Glycerol, %

15°C

15.5°C

20°C

25°C

30°C

Glycerol, % 15°C

100 99 98 97 96

1.26415 1.26160 1.25900 1.25645 1.25385

1.26381 1.26125 1.25865 1.25610 1.25350

1.26108 1.25850 1.25590 1.25335 1.25080

1.15802 1.25545 1.25290 1.25030 1.24770

1.25495 1.25235 1.24975 1.24710 1.24450

65 64 63 62 61

1.17030 1.16755 1.16480 1.16200 1.15925

1.17000 1.16725 1.16445 1.16170 1.15895

1.16750 1.16475 1.16205 1.15930 1.15655

1.16475 1.16200 1.15925 1.15655 1.15380

1.16195 1.15925 1.15650 1.15375 1.15100

30 29 28 27 26

95 94 93 92 91

1.25130 1.24865 1.24600 1.24340 1.24075

1.25095 1.24830 1.24565 1.24305 1.24040

1.24825 1.24560 1.24300 1.24035 1.23770

1.24515 1.24250 1.23985 1.23725 1.23460

1.24190 1.23930 1.23670 1.23410 1.23150

60 59 58 57 56

1.15650 1.15370 1.15095 1.14815 1.14535

1.15615 1.15340 1.15065 1.14785 1.14510

1.15380 1.15105 1.14830 1.14555 1.14280

1.15105 1.14835 1.14560 1.14285 1.14015

1.14830 1.14555 1.14285 1.14010 1.13740

90 89 88 87 86

1.23810 1.23545 1.23280 1.23015 1.22750

1.23775 1.23510 1.23245 1.22980 1.22710

1.23510 1.23245 1.22975 1.22710 1.22445

1.23200 1.22935 1.22665 1.22400 1.22135

1.22890 1.22625 1.22360 1.22095 1.21830

55 54 53 52 51

1.14260 1.13980 1.13705 1.13425 1.13150

1.14230 1.13955 1.13680 1.13400 1.13125

1.14005 1.13730 1.13455 1.13180 1.12905

1.13740 1.13465 1.13195 1.12920 1.12650

85 84 83 82 81

1.22485 1.22220 1.21955 1.21690 1.21425

1.22445 1.22180 1.21915 1.21650 1.21385

1.22180 1.21915 1.21650 1.21380 1.21115

1.21870 1.21605 1.21340 1.21075 1.20810

1.21565 1.21300 1.21035 1.20770 1.20505

50 49 48 47 46

1.12870 1.12600 1.12325 1.12055 1.11780

1.12845 1.12575 1.12305 1.12030 1.11760

1.12630 1.12360 1.12090 1.11820 1.11550

80 79 78 77 76

1.21160 1.20885 1.20610 1.20335 1.20060

1.21120 1.20845 1.20570 1.20300 1.20025

1.20850 1.20575 1.20305 1.20030 1.19760

1.20545 1.20275 1.20005 1.19735 1.19465

1.20240 1.19970 1.19705 1.19435 1.19170

45 44 43 42 41

1.11510 1.11235 1.10960 1.10690 1.10415

1.11490 1.11215 1.10945 1.10670 1.10400

75 74 73 72 71

1.19785 1.19510 1.19235 1.18965 1.18690

1.19750 1.19480 1.19205 1.18930 1.18655

1.19485 1.19215 1.18940 1.18670 1.18395

1.19195 1.18925 1.18650 1.18380 1.18110

1.18900 1.18635 1.18365 1.18100 1.17830

40 39 38 37 36

1.10145 1.09875 1.09605 1.09340 1.09070

70 69 68 67 66

1.18415 1.18135 1.17860 1.17585 1.17305

1.18385 1.18105 1.17830 1.17555 1.17275

1.18125 1.17850 1.17575 1.17300 1.17025

1.17840 1.17565 1.17295 1.17020 1.16745

1.17565 1.17290 1.17020 1.16745 1.16470

35 34 33 32 31

1.08800 1.08530 1.08265 1.07995 1.07725

20°C

25°C

30°C

1.07455 1.07195 1.06935 1.06670 1.06410

1.07435 1.07175 1.06915 1.06655 1.06390

1.07270 1.07010 1.06755 1.06495 1.06240

1.07070 1.06815 1.06560 1.06305 1.06055

1.06855 1.06605 1.06355 1.06105 1.05855

25 24 23 22 21

1.06150 1.05885 1.05625 1.05365 1.05100

1.06130 1.05870 1.05610 1.05350 1.05090

1.05980 1.05720 1.05465 1.05205 1.04950

1.05800 1.05545 1.05290 1.05035 1.04780

1.05605 1.05350 1.05100 1.04850 1.04600

1.13470 1.13195 1.12925 1.12650 1.12380

20 19 18 17 16

1.04840 1.04590 1.04335 1.04085 1.03835

1.04825 1.04575 1.04325 1.04075 1.03825

1.04690 1.04440 1.04195 1.03945 1.03695

1.04525 1.04280 1.04035 1.03790 1.03545

1.04350 1.04105 1.03860 1.03615 1.03370

1.12375 1.12110 1.11840 1.11575 1.11310

1.12110 1.11845 1.11580 1.11320 1.11055

15 14 13 12 11

1.03580 1.03330 1.03080 1.02830 1.02575

1.03570 1.03320 1.03070 1.02820 1.02565

1.03450 1.03200 1.02955 1.02705 1.02455

1.03300 1.03055 1.02805 1.02560 1.02315

1.03130 1.02885 1.02640 1.02395 1.02150

1.11280 1.11010 1.10740 1.10470 1.10200

1.11040 1.10775 1.10510 1.10240 1.09975

1.10795 1.10530 1.10265 1.10005 1.09740

10 9 8 7 6

1.02325 1.02085 1.01840 1.01600 1.01360

1.02315 1.02075 1.01835 1.01590 1.01350

1.02210 1.01970 1.01730 1.01495 1.01255

1.02070 1.01835 1.01600 1.01360 1.01125

1.01905 1.01670 1.01440 1.01205 1.00970

1.10130 1.09860 1.09590 1.09320 1.09050

1.09930 1.09665 1.09400 1.09135 1.08865

1.09710 1.09445 1.09180 1.08915 1.08655

1.09475 1.09215 1.08955 1.08690 1.08430

5 4 3 2 1

1.01120 1.00875 1.00635 1.00395 1.00155

1.01110 1.00870 1.00630 1.00385 1.00145

1.01015 1.00780 1.00540 1.00300 1.00060

1.00890 1.00655 1.00415 1.00180 0.99945

1.00735 1.00505 1.00270 1.00035 0.99800

1.08780 1.08515 1.08245 1.07975 1.07705

1.08600 1.08335 1.08070 1.07800 1.07535

1.08390 1.08125 1.07860 1.07600 1.07335

1.08165 1.07905 1.07645 1.07380 1.07120

0

0.99913 0.99905 0.99823 0.99708 0.99568

*Bosart and Snoddy, Ind. Eng. Chem., 20, (1928): 1378.

TABLE 2-116

Density 15.5°C

Hydrazine (N2H4)

%

d 15 4

%

d 15 4

1 2 4 8 12 16 20 24 28

1.0002 1.0013 1.0034 1.0077 1.0121 1.0164 1.0207 1.0248 1.0286

30 40 50 60 70 80 90 100

1.0305 1.038 1.044 1.047 1.046 1.040 1.030 1.011

DENSITIES OF AQUEOUS ORGANIC SOLUTIONS TABLE 2-117

2-117

Densities of Aqueous Solutions of Miscellaneous Organic Compounds*

d (resp., dw, ds) = density of the solution (resp., water; resp., the pure liquid solute) in g/mL; ps (resp., pw) = wt % of solute (resp., water) in the solution; range = range of applicability of the equation. Section A Name

Formula

t, °C

Acetaldehyde Acetamide

C2H4O C2H5NO

Acetone

C3H6O

Acetonitrile Allyl alcohol Benzenepentacarboxylic acid Butyl alcohol (n-)

C2H3N C3H6O C11H6O10 C4H10O

Butyric acid (n-)

C4H8O2

Chloral hydrate

C2H3Cl3O2

Chloroacetic acid

C2H3ClO2

Citric acid (hydrate)

C6H3O7 + H2O

Dichloroacetic acid

C2H2Cl2O2

Diethylamine hydrochloride Ethylamine hydrochloride

C4H12ClN C2H8ClN

Ethylene glycol

C2H6O2

Ethyl ether tartrate Formaldehyde Formamide

C4H10O C8H14O6 CH2O CH3NO

Furfural Isoamyl alcohol

C5H4O2 C5H12O

Isobutyl alcohol

C4H10O




Isobutyric acid

C4H8O2




Isovaleric acid Lactic acid Maleic acid

C5H10O2 C3H6O C4H4O4

Malic acid

C4H6O5

Malonic acid Methyl acetate

C3H4O4 C3H6O2

glucoside (α-)

C7H14O6

Nicotine Nitrophenol (p-)

C10H14N2 C6H5NO3

Oxalic acid

C2H2O4

Phenol

C6H6O

Phenylglycolic acid Picoline (α-) (β-)

C8H8O3 C6H7N C6H7N

Propionic acid

C3H6O2

Pyridine Resorcinol Succinic acid

C5H5N C6H6O2 C4H6O4

Tartaric acid (d, l, or dl)

C4H6O6

*From “International Critical Tables,” vol. 3, pp. 111–114.





























18 15 0 4 15 20 25 15 0 25 20 18 25 0 15 30 20 25 18 20 25 21 21 0 15 20 25 15 15 25 20 25 20 15 20 15 18 25 25 25 25 20 25 20 20 0 30 20 15 0 15 17.5 20 25 15 80 25 25 25 18 25 25 18 25 15 17.5 20 30 40 50 60

d = d w + Aps + Bp s2 + Cp3s Range, ps

A

B

0– 30 0– 6 0–100 0–100 0–100 0–100 0–100 0– 16 0– 89 0– 0.6 0– 7.9 0– 10 0– 62 0– 70 0– 78 0– 90 0– 32 0– 86 0– 50 0– 30 0– 97 0– 36 0– 65 0–100 0– 6 0– 5 0– 4.5 0– 95 0– 40 22– 96 0– 8 0– 8 0– 2.5 0– 8 0– 8 0– 9 0– 9 0– 12 0– 5 0– 9 0– 40 0– 40 0– 40 0– 40 0– 20 26– 51 26– 51 0– 60 0– 1.5 0– 4 0– 4 0– 9 0– 4 0– 4 0– 5 0– 65 0– 11 0– 70 0– 60 0– 10 0– 40 0– 60 0– 52 0– 5.5 0– 15 0– 50 0– 50 0– 50 0– 50 0– 50 0– 50

+0.03255 +0.03639 −0.03856 −0.027648 −0.021009 −0.021233 −0.021171 −0.021175 −0.033729 +0.025615 −0.021651 +0.03414 +0.035135 +0.024489 +0.024455 +0.024401 +0.023648 +0.023602 +0.023824 +0.024427 +0.024427 +0.0334 +0.021193 +0.021483 +0.02133 −0.02221 −0.02221 +0.022367 +0.022518 +0.021217 +0.021827 +0.021664 +0.02155 −0.02146 −0.02169 +0.0352 +0.0345 +0.0337 +0.03253 +0.02231 +0.0234 +0.023933 +0.023736 +0.02389 +0.0340 +0.023336 +0.023151 +0.03642 +0.023216 +0.025898 +0.02494 +0.02494 +0.025264 +0.025108 +0.02111 +0.03462 +0.02207 −0.04386 −0.04683 +0.0395 +0.039245 +0.03229 +0.02201 +0.02304 +0.024482 +0.024455 +0.024432 +0.024335 +0.024265 +0.024205 +0.024155

−0.0516 +0.04171 −0.05449 −0.041193 −0.059682 −0.053529 −0.05904 −0.042024 −0.041232 −0.02117 +0.04285 +0.04131 −0.04166 +0.042802 +0.042198 +0.041887 +0.05302 +0.05552 +0.041141 +0.05537 +0.05537 +0.0676 −0.05307 +0.052992 −0.05108 +0.0448 +0.0435 +0.05358 −0.05658 +0.053199 +0.05366 +0.0421 +0.043 +0.056 +0.0438

−0.04282 +0.05186 +0.0575 +0.05957 +0.04175 +0.041066 −0.0574 +0.05996 +0.05975 +0.05454 −0.0455 −0.033185 −0.058 −0.058 −0.031996 −0.031607 −0.04283 −0.0686 +0.0423 −0.051405 −0.0513 −0.04172 −0.0599 −0.05204 +0.05519 +0.04185 +0.04185 +0.041837 +0.04185 +0.04185 +0.04185 +0.04185

C

−0.07588 +0.08272 −0.08624 −0.075327 −0.0856 +0.072984

+0.0611 −0.071291 +0.074366 +0.076549 +0.0722 +0.0717 +0.077534 +0.077534 −0.0747 −0.075248

−0.076005 +0.06542 −0.072529

+0.01544 +0.08978 −0.07687 +0.0441 +0.04254 +0.04208

−0.074167 +0.07361 −0.0828 −0.0819

2-118

PHYSICAL AND CHEMICAL DATA

TABLE 2-117

Densities of Aqueous Solutions of Miscellaneous Organic Compounds (Concluded ) d = d w + Aps + Bp s2 + Cp3s (Cont.)

Section A Name

t, °C

Formula

Tetraethyl ammonium chloride Thiourea

C8H20ClN CH4N2S

Trichloroacetic acid

C2HCl3O2

Triethylamine hydrochloride

C6H16ClN

Trimethyl carbinol

C4H10O

Urea

CH4N2O

Urethane Valeric acid (n-)

C3H7NO2 C5H10O2

Name







Formula C4H10O C4H8O2 C4H10O

Isobutyl alcohol

C4H10O

Isobutyric acid Nicotine Picoline (α-) (β-) Pyridine Trimethyl carbinol

C4H8O2 C10H14N2 C6H7N C6H7N C5H5N C4H10O




Name

Chloral hydrate

C2H3Cl3O2

Ethyl tartrate

C7H14O6

Furfural

C5H4O2

Pyridine

C5H5N

B

C

+0.056 +0.05374 +0.04153 +0.041387 +0.056119 +0.05558 −0.041908 −0.04176 −0.044802 +0.051552 +0.053712 −0.041817 −0.05245 −0.0427

+0.07122

t, °C

Range, pw

A

B

20 25 25 0 15 26 20 25 25 25 20

0–20 0–38 0– 1.1 0–14 0–16 0–80 0–40 0–30 0–40 0–40 0–20

+0.022103 +0.021854 +0.0234 +0.022437 +0.02224 +0.021808 +0.02199 +0.022715 +0.021925 +0.021157 +0.022287

−0.04113 −0.042314 +0.0336 −0.04285 −0.04129 −0.042358 −0.04331 −0.04393 −0.04352 −0.05536 +0.05275




+0.061038 −0.0869 +0.07957 +0.07887 +0.051216 +0.072573 −0.072285 +0.051379 −0.073437

d = ds + Apw + Bp 2w + Cp 3w

ds

C

+0.061253 +0.07315 +0.0625 −0.062

dt = do + At + Bt2

ps

do

Range, °C

A

B

76.60 80.95 2.00 10.00 5.00 10.00 25.00 4.62 5.69 6.56 9.34 21.20 29.50 40.40

0.9122 0.8614 1.0094 1.0476 1.0150 1.0270 1.0665 1.0125 1.0140 1.0155 1.0055 1.0115 1.0145 1.0182

0–45 0–43 7–80 7–80 15–80 15–80 15–80 22–74 22–74 22–74 11–73 14–73 12–72 9–74

−0.038 −0.037292 −0.042597 −0.047955 −0.032103 −0.032116 −0.03401 −0.03232 −0.03221 −0.03211 −0.03171 −0.03378 −0.03463 −0.03605

−0.0527 −0.0675 −0.054313 −0.054253 −0.052544 −0.062929 −0.0523 −0.05254 −0.05268 −0.05290 −0.053615 −0.05248 −0.05235 −0.05167

Formula C3H6O C4H10O

A +0.031884 +0.022995 +0.02499 +0.025053 +0.025051 +0.046 −0.02117 −0.021286 +0.023213 +0.022718 +0.022702 +0.022728 +0.021278 +0.0334

0.8097 0.9534 0.7077 0.8170 0.8055 0.9425 1.0093 0.9404 0.9515 0.9776 0.7856 Section C

Allyl alcohol Butyl alcohol (n-)

0– 63 0– 7 0– 61 10– 30 0– 94 0– 54 0–100 0–100 0– 12 0– 51 0– 35 0– 10 0– 56 0– 3




Section B Butyl alcohol (n-) Butyric acid (n-) Ethyl ether

Range, ps

21 15 12.5 20 25 21 20 25 14.8 18 20 25 20 25








DENSITIES OF MISCELLANEOUS MATERIALS

2-119

DENSITIES OF MISCELLANEOUS MATERIALS TABLE 2-118

Approximate Specific Gravities and Densities of Miscellaneous Solids and Liquids*

Water at 4°C and normal atmospheric pressure taken as unity. For more detailed data on any material, see the section dealing with the properties of that material.

Substance

Sp. gr.

Aver. weight lb/ft 3

Sp. gr.

Aver. weight lb/ft 3

Metals, Alloys, Ores Aluminum, cast-hammered bronze Brass, cast-rolled Bronze, 7.9 to 14% Sn phosphor

2.55–2.80 7.7 8.4–8.7 7.4–8.9 8.88

165 481 534 509 554

Timber, Air-dry Apple Ash, black white Birch, sweet, yellow Cedar, white, red

0.66–0.74 0.55 0.64–0.71 0.71–0.72 0.35

44 34 42 44 22

Copper, cast-rolled ore, pyrites German silver Gold, cast-hammered coin (U.S.)

8.8–8.95 4.1–4.3 8.58 19.25–19.35 17.18–17.2

556 262 536 1205 1073

Cherry, wild red Chestnut Cypress Elm, white Fir, Douglas

0.43 0.48 0.45–0.48 0.56 0.48–0.55

27 30 29 35 32

Iridium Iron, gray cast cast, pig wrought spiegeleisen

21.78–22.42 7.03–7.13 7.2 7.6–7.9 7.5

1383 442 450 485 468

balsam Hemlock Hickory Locust Mahogany

0.40 0.45–0.50 0.74–0.80 0.67–0.77 0.56–0.85

25 29 48 45 44

ferro-silicon ore, hematite ore, limonite ore, magnetite slag

6.7–7.3 5.2 3.6–4.0 4.9–5.2 2.5–3.0

437 325 237 315 172

Maple, sugar white Oak, chestnut live red, black

0.68 0.53 0.74 0.87 0.64–0.71

43 33 46 54 42

Lead ore, galena Manganese ore, pyrolusite Mercury

11.34 7.3–7.6 7.42 3.7–4.6 13.6

710 465 475 259 849

white Pine, Norway Oregon red Southern white

0.77 0.55 0.51 0.48 0.61–0.67 0.43

48 34 32 30 38–42 27

8.97 8.9 21.5 10.4–10.6 7.83 7.80 7.70–7.73 7.2–7.5 6.4–7.0 19.22

555 537 1330 656 489 487 481 459 418 1200

Poplar Redwood, California Spruce, white, red Teak, African Indian Walnut, black Willow

0.43 0.42 0.45 0.99 0.66–0.88 0.59 0.42–0.50

27 26 28 62 48 37 28

6.9–7.2 3.9–4.2

440 253

Various Solids Cereals, oats, bulk barley, bulk corn, rye, bulk wheat, bulk Cork

49 50 75 94 112

26 39 45 48 15

0.789 0.796 1.20 1.50 1.80

0.51 0.62 0.73 0.77 0.22–0.26

Various Liquids Alcohol, ethyl (100%) methyl (100%) Acid, muriatic, 40% nitric, 91% sulfuric, 87% Chloroform Ether Lye, soda, 66% Oils, vegetable mineral, lubricants

1.500 0.736 1.70 0.91–0.94 0.88–0.94

95 46 106 58 57

Cotton, flax, hemp Fats Flour, loose pressed Glass, common

1.47–1.50 0.90–0.97 0.40–0.50 0.70–0.80 2.40–2.80

93 58 28 47 162

2.45–2.72 2.90–3.00 3.2–4.7 0.32 0.86–1.02

161 184 247 20 59

54 62.428 59.830 56 8

plate or crown crystal dint Hay and straw, bales Leather

0.861–0.867 1.0 0.9584 0.88–0.92 0.125 1.02–1.03

64

Paper Potatoes, piled Rubber, caoutchouc goods Salt, granulated, piled

0.70–1.15 0.67 0.92–0.96 1.0–2.0 0.77

58 44 59 94 48

Ashlar Masonry Bluestone Granite, syenite, gneiss Limestone Marble Sandstone

2.3–2.6 2.4–2.7 2.1–2.8 2.4–2.8 2.0–2.6

153 159 153 162 143

Saltpeter Starch Sulfur Wool

1.07 1.53 1.93–2.07 1.32

67 96 125 82

Rubble Masonry Bluestone Granite, syenite, gneiss Limestone Marble Sandstone

2.2–2.5 2.3–2.6 2.0–2.7 2.3–2.7 1.9–2.5

147 153 147 156 137

Monel metal, rolled Nickel Platinum, cast-hammered Silver, cast-hammered Steel, cold-drawn machine tool Tin, cast-hammered cassiterite Tungsten Zinc, cast-rolled blende

Substance

Turpentine Water, 4°C max. density 100°C ice snow, fresh fallen sea water

*From Marks, Mechanical Engineers’ Handbook, McGraw-Hill.

Sp. gr.

Aver. weight lb/ft 3

Dry Rubble Masonry Granite, syenite, gneiss Limestone, marble Sandstone, bluestone

1.9–2.3 1.9–2.1 1.8–1.9

130 125 110

Brick Masonry Hard brick Medium brick Soft brick Sand-lime brick

1.8–2.3 1.6–2.0 1.4–1.9 1.4–2.2

128 112 103 112

Concrete Masonry Cement, stone, sand slag, etc. cinder, etc.

2.2–2.4 1.9–2.3 1.5–1.7

144 130 100

0.64–0.72 1.5 0.85–1.00 1.4–1.9 2.08–2.25

40–45 94 53–64 103 94–135

Portland cement Slags, bank slag bank screenings machine slag slag sand

3.1–3.2 1.1–1.2 1.5–1.9 1.5 0.8–0.9

196 67–72 98–117 96 49–55

Earth, etc., Excavated Clay, dry damp plastic and gravel, dry Earth, dry, loose dry, packed moist, loose moist, packed mud, flowing mud, packed Riprap, limestone

1.0 1.76 1.6 1.2 1.5 1.3 1.6 1.7 1.8 1.3–1.4

63 110 100 76 95 78 96 108 115 80–85

1.4 1.7 1.4–1.7 1.6–1.9 1.89–2.16

90 105 90–105 100–120 126

1.28 1.44 0.96 1.00 1.12 1.00

80 90 60 65 70 65

2.1–2.8 4.50 2.7–3.2 2.55 2.5–2.6

153 281 184 159 159

Borax Chalk Clay, marl Dolomite Feldspar, orthoclase

1.7–1.8 1.8–2.8 1.8–2.6 2.9 2.5–2.7

109 143 137 181 162

Gneiss Granite Greenstone, trap Gypsum, alabaster Hornblende Limestone Marble Magnesite Phosphate rock, apatite Porphyry

2.7–2.9 2.6–2.7 2.8–3.2 2.3–2.8 3.0 2.1–2.86 2.6–2.86 3.0 3.2 2.6–2.9

175 165 187 159 187 155 170 187 200 172

Substance

Various Building Materials Ashes, cinders Cement, Portland, loose Lime, gypsum, loose Mortar, lime, set Portland cement

Riprap, sandstone Riprap, shale Sand, gravel, dry, loose gravel, dry, packed gravel, wet Excavations in Water Clay River mud Sand or gravel and clay Soil Stone riprap Asbestos Barytes Basait Bauxite Bluestone

Minerals

2-120

PHYSICAL AND CHEMICAL DATA

TABLE 2-118

Approximate Specific Gravities and Densities of Miscellaneous Solids and Liquids (Concluded )

Water at 4°C and normal atmospheric pressure taken as unity. For more detailed data on any material, see the section dealing with the properties of that material.

Substance

Sp. gr.

Minerals (Cont.) Pumice, natural Quartz, flint Sandstone Serpentine Shale, slate Soapstone, talc Syenite Stone, Quarried, Piled Basalt, granite, gneiss Greenstone, hornblende Limestone, marble, quartz Sandstone Shale NOTE:

Aver. weight lb/ft3

0.37–0.90 2.5–2.8 2.0–2.6 2.7–2.8 2.6–2.9

40 165 143 171 172

2.6–2.8 2.6–2.7

169 165

1.5 1.7 1.5 1.3 1.5

96 107 95 82 92

Substance

Aver. weight lb/ft 3

Sp. gr.

Bituminous Substances Asphaltum Coal, anthracite bituminous lignite peat, turf, dry

1.1–1.5 1.4–1.8 1.2–1.5 1.1–1.4 0.65–0.85

81 97 84 78 47

charcoal, pine charcoal, oak coke Graphite Paraffin

0.28–0.44 0.47–0.57 1.0–1.4 1.64–2.7 0.87–0.91

23 33 75 135 56

Substance

Sp. gr.

Aver. weight lb/ft3

Bituminous Substances (Cont.) Petroleum refined (kerosene) benzine gasoline Pitch Tar, bituminous

0.87 0.78–0.82 0.73–0.75 0.70–0.75 1.07–1.15 1.20

54 50 46 45 69 75

Coal and Coke, Piled Coal, anthracite bituminous, lignite peat, turf charcoal coke

0.75–0.93 0.64–0.87 0.32–0.42 0.16–0.23 0.37–0.51

47–58 40–54 20–26 10–14 23–32

To convert pounds per cubic foot to kilograms per cubic meter, multiply by 16.02. °F = 9⁄ 5 °C + 32.

TABLE 2-119

Density (kg/m3) of Selected Elements as a Function of Temperature Element symbol

Temperature, K*

Al

Be†

Cr

Cu

Au

Ir

Fe

Pb

Mo

Ni

Pt

Ag

Zn†

50 100 150 200 250

2736 2732 2726 2719 2710

3650 3640 3630 3620 3610

7160 7155 7150 7145 7140

9019 9009 8992 8973 8951

19,490 19,460 19,420 19,380 19,340

22,600 22,580 22,560 22,540 22,520

7910 7900 7890 7880 7870

11,570 11,520 11,470 11,430 11,380

10,260 10,260 10,250 10,250 10,250

8960 8950 8940 8930 8910

21,570 21,550 21,530 21,500 21,470

10,620 10,600 10,575 10,550 10,520

7280 7260 7230 7200 7170

300 400 500 600 800

2701 2681 2661 2639 2591

3600 3580 3555 3530

7135 7120 7110 7080 7040

8930 8885 8837 8787 8686

19,300 19,210 19,130 19,040 18,860

22,500 22,450 22,410 22,360 22,250

7860 7830 7800 7760 7690

11,330 11,230 11,130 11,010 10,430

10,240 10,220 10,210 10,190 10,160

8900 8860 8820 8780 8690

21,450 21,380 21,330 21,270 21,140

10,490 10,430 10,360 10,300 10,160

7135 7070 7000 6935 6430

1000 1200 1400 1600 1800

2365 2305 2255

7000 6945 6890 6760 6700

8568 8458 7920 7750 7600

18,660 18,440 17,230 16,950

22,140 22,030 21,920 21,790 21,660

7650 7620 7520 7420 7320

10,190 9,940

10,120 10,080 10,040 10,000 9,950

8610 8510 8410 8320 7690

21,010 20,870 20,720 20,570 20,400

10,010 9,850 9,170 8,980

6260

21,510

7030

9,900

7450

20,220

2000

7460

NOTE:

Above the horizontal line the condensed phase is solid; below the line, it is liquid. *°R = 9⁄ 5 K. †Polycrystalline form tabulated. Similar tables for an additional 45 elements appear in the Handbook of Heat Transfer, 2d ed., McGraw-Hill, New York, 1984.

SOLUBILITIES UNITS CONVERSIONS For this subsection, the following units conversions are applicable: °F = 9⁄5 °C + 32. To convert cubic centimeters to cubic feet, multiply by 3.532 × 10−5.

To convert millimeters of mercury to pounds-force per square inch, multiply by 0.01934. To convert grams per liter to pounds per cubic foot, multiply by 6.243 × 10−2.

TABLE 2-120

Solubilities of Inorganic Compounds in Water at Various Temperatures*

This table shows the amount of anhydrous substance that is soluble in 100 g of water at the temperature in degrees Celsius as indicated; when the name is followed by †, the value is expressed in grams of substance in 100 cm3 of saturated solution. Solid phase gives the hydrated form in equilibrium with the saturated solution. Solid phase

Substance

Formula

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Aluminum chloride sulfate Ammonium aluminum sulfate bicarbonate bromide chloride chloroplatinate chromate chromium sulfate dichromate dihydrogen phosphite hydrogen phosphate iodide magnesium phosphate manganese phosphate nitrate oxalate perchlorate† persulfate sulfate thiocyanate vanadate (meta) Antimonious fluoride sulfide Arsenic oxide Arsenious sulfide

AlCl3 Al2(SO4)3 (NH4)2Al2(SO4)4 NH4HCO3 NH4Br NH4Cl (NH4)2PtCl6 (NH4)2CrO4 (NH4)2Cr2(SO4)4 (NH4)2Cr2O7 NH4H2PO3 (NH4)2HPO4 NH4I NH4MgPO4 NH4MnPO4 NH4NO3 (NH4)2C2O4 NH4ClO4† (NH4)2S2O8 (NH4)2SO4 NH4CNS NH4VO3 SbF3 Sb2S3 As2O5 As2S3

27 28 29

Barium acetate acetate carbonate

Ba(C2H3O2)2 Ba(C2H3O2)2 BaCO3

3H2O 1H2O

30 31 32 33 34 35 36 37 38

chlorate chloride chromate hydroxide iodide iodide nitrate nitrite oxalate

Ba(ClO3)2 BaCl2 BaCrO4 Ba(OH)2 BaI2 BaI2 Ba(NO3)2 Ba(NO2)2 BaC2O4

1H2O 2H2O

3H2O

39 40 41 42 43 44 45 46 47 48 49 50 51

perchlorate sulfate Beryllium sulfate sulfate sulfate Boric acid Boron oxide Bromine Cadmium chloride chloride chloride cyanide hydroxide

Ba(ClO4)2 BaSO4 BeSO4 BeSO4 BeSO4 H3BO3 B2O3 Br2 CdCl2 CdCl2 CdCl2 Cd(CN)2 Cd(OH)2

52 53 54

sulfate Calcium acetate acetate

CdSO4 Ca(C2H3O2)2 Ca(C2H3O2)2

6H2O 18H2O 24H2O

0°C 31.2 2.1 11.9 60.6 29.4

10°C 33.5 4.99 15.8 68 33.3 0.7

10.7825°

24II2O 171 6H2O 7H2O 1H2O

154.2 0.023 118.3 2.2 11.56 58.2 70.6 119.8

163.2

3.1 73.0 144

384.7

8H2O 6H2O 2H2O

20°C 69.8615° 36.4 7.74 21 75.5 37.2

59.5 5.17×10−5 at 18° 59

62.1 63 0.00168°

20.34 31.6 0.0002 1.67 170.2 5.0

1H2O

26.95 33.3 0.00028 2.48 185.7 7.0 0.00168°

205.8 1.15 × 10−4

2.0 × 10−4

19014.5° 13115 172.3 0.052 0 192 4.4 20.85 75.4 170 0.48 444.7 0.00017518° 65.8

4H2O 2aH2O 1H2O

2H2O 1H2O

3.57 1.5 3.4 125.1

50°C

60°C

70°C

80°C

90°C

100°C

46.1 14.88

52.2 20.10

59.2 26.70

66.1

73.0

80.8

89.0 109.796°

91.1 45.8

99.2 50.4

107.8 55.2

116.8 60.2

126 65.6

135.6 71.3

145.6 77.3 1.25

190.5 0.036 0 297.0 8.0 30.58

199.6 0.030

208.9 0.040 0 421.0

218.7 0.016 0.005 499.0

228.8 0.019 0.007 580.0

40.4 47.17 26031° 181.4 241.8 5.9 78.0 207.7 0.84 563.6

81.0

69.5

71.2

1.32

344.0 10.3

39.05

740.0

48.19

88.0 1.78

250.3 871.0 57.01

95.3

103.3

75.1

76.7

3.05 73.0

0.002218° 33.80 35.7 0.00037 3.89 203.1 9.2 67.5 0.002218° 289.1 2.4 × 10−4

5.04 2.2 3.20

75 0.0024 at 24.2° 41.70 38.2 0.00046 5.59 219.6 11.6

79

77

74

74

49.61 40.7

43.6

66.81 46.4

49.4

8.22

13.12

20.94

231.9 14.2

17.1

358.7

426.3

75 84.84 52.4

104.9 58.8

101.4

247.3 20.3

261.0 27.0 205.8

271.7 34.2 300

0.0024 at 24.2° 2.85 × 10−4 52 43.78 6.60

495.2

562.3

60.67 46.74 8.72 4.0

62 11.54

14.81 6.2

16.73

84.76 23.75 9.5

83 98 30.38

100 110 40.25 15.7

3.13 132.1

135.1

76.48 37.4

40.4 10.94 27 83.2 41.4

40°C

71

6H2O 4H2O 2H2O 2.66 1.1 4.22 97.59 90.01

30°C

76.00 36.0

134.5 1.715° 76.60 34.7

135.3

136.5

140.4

147.0

2.6 × 10−4 at 25° 33.8

78.54 33.2

83.68 32.7

63.13

60.77

31.1

29.7

33.5

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54

*By N. A. Lange. Abridged from “Table of Solubilities of Inorganic Compounds in Water at Various Temperatures” in Lange, Handbook of Chemistry, 10th ed., McGraw-Hill, New York, 1961. For tables of the solubility of gases in water at various temperatures, Atack (Handbook of Chemical Data, Reinhold, New York, 1957) gives values at closer temperature intervals, usually 1 or 5°C, than are tabulated here. For materials marked by ‡, additional data are given in tables subsequent to this one. For the solubility of various hydrocarbons in water at high pressures see J. Chem. Eng. Data, 4, 212 (1959).

2-121

TABLE 2-120

Solubilities of Inorganic Compounds in Water at Various Temperatures (Continued )

Substance

Formula

1 2 3 4 5 6 7 8 9 10 11

Calcium bicarbonate chloride chloride fluoride hydroxide nitrate nitrate nitrate nitrite nitrite oxalate

Ca(HCO3)2 CaCl2 CaCl2 CaF2 Ca(OH)2 Ca(NO3)2 Ca(NO3)2 Ca(NO3)2 Ca(NO2)2 Ca(NO2)2 CaC2O4

12 13 14 15 16 17 18 19 20 21 22 23 24

sulfate Carbon dioxide, 760 mm ‡ monoxide, 760 mm ‡ Cesium chloride nitrate sulfate Chlorine, 760 mm ‡ Chromic anhydride Cuprio chloride nitrate nitrate sulfate sulfide

CaSO4 CO2 CO CsCl CsNO3 Cs2SO4 Cl2 CrO3 CuCl2 Cu(NO3)2 Cu(NO3)2 CuSO4 CuS

25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62

Cuprous chloride Ferric chloride Ferrous chloride chloride nitrate sulfate sulfate Hydrobromic acid, 760 mm Hydrochloric acid, 760 mm Iodine Lead acetate bromide carbonate chloride chromate fluoride nitrate sulfate Magnesium bromide chloride hydroxide nitrate sulfate sulfate sulfate Manganous sulfate sulfate sulfate sulfate Mercurous chloride Molybdic oxide Nickel chloride nitrate nitrate sulfate sulfate Nitric oxide, 760 mm Nitrous oxide

CuCl FeCl3 FeCl2 FeCl2 Fe(NO3)2 FeSO4 FeSO4 HBr HCl I2 Pb(C2H3O2)2 PbBr2 PbCO3 PbCl2 PbCrO4 PbF2 Pb(NO3)2 PbSO4 MgBr2 MgCl2 Mg(OH)2 Mg(NO3)2 MgSO4 MgSO4 MgSO4 MnSO4 MnSO4 MnSO4 MnSO4 HgCl MoO3 NiCl2 Ni(NO3)2 Ni(NO3)2 NiSO4 NiSO4 NO N2O

2-122

Solid phase 6H2O 2H2O 4H2O 3H2O

0°C 16.15 59.5 0.185 102.0

10°C 65.0 0.176 115.3

20°C 16.60 74.5 0.001618° 0.165 129.3

30°C

40°C

50°C

17.05

60°C

70°C

17.50

80°C

90°C

17.95

18.40

102 0.001726° 0.153 152.6

136.8 0.141 195.9 237.5

0.128

0.116

141.7 0.106

147.0 0.094

152.7 0.085

62.07

2H2O

0.1759 0.3346 0.0044 161.4 9.33 167.1 1.46 164.9 70.7 81.8

2H2O 6H2O 3H2O 5H2O

6.7 × 10−4 at 13° 0.1928 0.2318 0.0035 174.7 14.9 173.1 0.980 73.76 95.28 17.4

74.4

81.9 64.5

6.8 × 10−4 at 25° 0.1688 0.0028 186.5 23.0 178.7 0.716 77.0 125.1 20.7 3.3 × 10−5 at 18° 1.5225° 91.8

9.5 × 10−4 at 50° 0.2090 0.1257 0.0024 197.3 33.9 184.1 0.562

132.6

80.34

14 × 10−4 at 95° 0.2097 0.0973 0.0021 208.0 47.2 189.9 0.451 174.0 83.8

25

159.8 28.5

33.3

77.3

315.1 82.5

73.0

0.0761 0.0018 218.5 64.4 194.9 0.386 182.1 87.44

0.2047 0.0576 0.0015 229.7 83.8 199.9 0.324

151.9

363.6 244.8

0.1966 0.0013 239.5 107.0 205.0 0.274

0.0010 250.0 134.0 210.3 0.219

91.2

99.2

178.8 40

207.8 55

88.7

525.8 100

0.0006 260.1 163.0 214.9 0.125 217.5

71.02 15.65

20.51

83.8 26.5

535.7

210.3

0.029 0.4554 0.6728

6H2O 6H2O 6H2O 7H2O 6H2O 1H2O 7H2O 5H2O 4H2O 1H2O

38.8 0.0028 91.0 52.8

0.060 48.3 0.0035 94.5 53.5

40.2

48.6

0.85 0.00011 0.99 7 × 10−6 0.064 56.5 0.0041 96.5 54.5 0.000918°

40.8 53.23

30.9 42.2 60.01 59.5

53.9 79.58

59.5

27.22

32

1.53

1.94

2.36

3.34

4.75

1.20

1.45

1.70

1.98

2.62

3.34

0.068 66 0.0049 99.2

40.8 45.3

62.9 64.5

67.76 66.44 0.264 68.9

75 0.0056 101.6 57.5

130 56.1

85

95

104.1

107.5 61.0

0.00757 0.1705

115

68.8 0.0007 0.476 73.3 122.2

38.8

50.4

53.5

59.5

64.2 62.9

69.0

74.0 68.3

72.6 58.17

55.0

52.0

48.0

42.5

34.0

0.687 78.3

1.206 82.2

2.055 85.2

113.7 66.0

84.74 45.6

120.2 73.0 137.0

163.1 0.00984

37.3

63.3 0.056

35.5 44.5

0.0002 0.138 64.2 96.31

43.6

67.3 0.04 55.0425° 1.15

171.5 59.6 0.078

66.55

0.00014 2H2O 6H2O 6H2O 3H2O 7H2O 6H2O

32.9

198

3H2O

105.8

165.6 50.9

221.2 82.3

0.1619 0 0 270.5 197.0 220.3 0 206.8 107.9 75.4

105.3 6H2O 7H2O 1H2O

0.077

76.68

14.3

4H2O

159

281.5 358.7

4H2O 2H2O

100°C

2.106 87.6

169.1

235.1

42.46 0.00618 0.1211

0.00517

0.00440

50.15 0.00376

54.80 0.00324

59.44 0.00267

63.17 0.00199

0.00114

76.7 0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64

Potassium acetate acetate alum bicarbonate bisulfate bitartrate carbonate chlorate chloride chromate dichromate ferricyanide hydroxide hydroxide nitrate nitrite perchlorate permanganate persulfate† sulfate thiocyanate Silver cyanide nitrate sulfate Sodium acetate acetate bicarbonate carbonate carbonate chlorate chloride chromate chromate chromate dichromate dichromate dihydrogen phosphate dihydrogen phosphate dihydrogen phosphate hydrogen arsenate hydrogen phosphate hydrogen phosphate hydrogen phosphate hydrogen phosphate hydroxide hydroxide hydroxide hydroxide nitrate nitrite oxalate phosphate, tripyrophosphate sulfate sulfate sulfate sulfide sulfide sulfide sulfite sulfite tetraborate tetraborate vanadate (meta)

KC2H3O2 KC2H3O2 K2SO4⋅Al2(SO4)3 KHCO3 KHSO4 KHC4H4O6 K2CO3 KClO3 KCl K2CrO4 K2Cr2O7 K3Fe(CN)6 KOH KOH KNO3 KNO2 KClO4 KMnO4 K2S2O8† K2SO4 KCNS AgCN AgNO3 Ag2SO4 NaC2H3O2 NaC2H3O2 NaHCO3 Na2CO3 Na2CO3 NaClO8 NaCl Na2CrO4 Na2CrO4 Na2CrO4 Na2Cr2O7 Na2Cr2O7 NaH2PO4 NaH2PO4 NaH2PO4 Na2HAsO4 Na2HPO4 Na2HPO4 Na2HPO4 Na2HPO4 NaOH NaOH NaOH NaOH NaNO3 NaNO2 Na2C2O4 Na3PO4 Na4P2O7 Na2SO4 Na2SO4 Na2SO4 Na2S Na2S Na2S Na2SO3 Na2SO3 Na2B4O7 Na2B4O7 NaVO8

1aH2O aH2O 24H2O

2H2O

2H2O 1H2O

†

3H2O 10H2O 1H2O 10H2O 4H2O

216.7 3.0 22.4 36.3 0.32 105.5 3.3 27.6 58.2 5 31 97

233.9 4.0 27.7 0.40 108 5 31.0 60.0 7 36 103

255.6 5.9 33.2 51.4 0.53 110.5 7.4 34.0 61.7 12 43 112

13.3 278.8 0.75 2.83 1.62 7.35 177.0

20.9

122 0.573 36.3 119 6.9 7

170 0.695 40.8 121 8.15 12.5

31.6 298.4 1.80 6.4 4.49 11.11 217.5 2.2 × 10−5 222 0.796 46.5 123.5 9.6 21.5

79 35.7 31.70

89 35.8 50.17

101 36.0 88.7

1.05 4.4 2.60 9.22

283.8 8.39 39.1 0.90 113.7 10.5 37.0 63.4 20 50 126 45.8 2.6 9.0 7.19 12.97 300 0.888 54.5 126 11.1 38.8 50.5 113 36.3 88.7

2H2O

163.0

2H2O 1H2O

57.9

323.3 11.70 45.4 67.3 1.32 116.9 14 40.0 65.2 26 60 63.9 334.9 4.4 12.56 9.89 14.76

337.3 17.00

350 24.75 60.0

1.83 121.2 19.3 42.6 66.8 34

2.46 126.8 24.5 45.5 68.6 43 66

140 85.5

110.0

364.8 40.0

133.1 48.3 70.4 52

138

380.1 71.0 4.6 139.8 38.5 51.1 72.1 61

169

396.3 109.0

147.5 54.0 73.9 70

202

6.5 16.89

9 22.2

11.8

14.8

18

16.50

18.17

19.75

21.4

22.8

376 0.979 65.5 129.5 12.7

455 1.08 83 134 14.45

525 1.15 139 139.5 16.4

1.22

669 1.30

48.5 126 36.6

140 37.0

46.4 155 37.3

104

114.6

95.96

177.8

244.8

146

153

172 37.8

45.8 189 38.4

123.0 316.7

124.8 376.2

1.36 161

39.0

121.6 6.95 155.7 57 56.7 75.6 80 82.6104 178 246 412.8 21.8 24.1 952 1.41 170 45.5 230 39.8 125.9 426.3

12H2O 12H2O 7H2O 2H2O

69.9

85.2

106.5

138.2

15.5 3.6

26.5 7.7

37 20.8

47

158.6 7.3 1.67

179.3 65

190.3

207.3 85

225.3

82.9

88.1

92.4

102.9

246.6

51.8 80.2

102.2 4H2O 3aH2O 1H2O

42 51.5 109

119

129

145

174

104 98.4

114 104.1

124

313

12H2O 10H2O 10H2O 7H2O 9H2O 5aH2O 6H2O 7H2O

73 72.1

80 78.0

1.5 3.16 5.0 19.5

13.9

96 91.6

4.1 3.95 9.0 30

88 84.5 3.7 11 6.23 19.4 44

15.42

18.8

22.5

20

26.9

36

20 9.95 40.8

31 13.50

43 17.45

55 21.83

81 30.04

347 180 163.2 6.33 108 40.26

48.8 28.5

46.7

45.3

43.7

42.5

39.82 36.4

42.69 39.1

28.2 10.5

28.8 20.3

28 10H2O 5H2O 2H2O

1.3

1.6

2.7 15.325°

3.9

148 132.6

45.73 43.31

68.4

59.23 57.28

28.3 24.4

30.2

51.40 49.14

31.5

41

52.5

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64

2-123

TABLE 2-120

Solubilities of Inorganic Compounds in Water at Various Temperatures (Concluded )

Substance 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Sodium vanadate (meta) Stannous chloride sulfate Strontium acetate acetate chloride chloride nitrate nitrate nitrate sulfate Sulfur dioxide, 760 mm † Thallium sulfate Thorium sulfate sulfate sulfate sulfate Zinc chlorate chlorate nitrate nitrate sulfate sulfate sulfate

2-124

Formula NaVO3 SnCl2 SnSO4 Sr(C2H3O2)2 Sr(C2H3O2)2 SrCl2 SrCl2 Sr(NO3)2 Sr(NO3)2 Sr(NO3)2 SrSO4 SO2 Tl2SO4 Th(SO4)2 Th(SO4)2 Th(SO4)2 Th(SO4)2 ZnClO3 ZnClO3 Zn(NO3)2 Zn(NO3)2 ZnSO4 ZnSO4 ZnSO4

Solid phase

0°C

10°C

9H2O 8H2O 6H2O 4H2O 6H2O 4H2O 6H2O 3H2O 7H2O 6H2O 1H2O

36.9 43.5

43.61 42.95 47.7

41.6 52.9

52.7 40.1

64.0 70.5

0.0113 22.83 2.70 0.74 1.0 1.50

0.0114 11.29 4.87 1.38 1.62 1.90

145.0

30°C

21.1025° 269.815° 19

83.9 4H2O aH2O 6H2O 2H2O 1H2O 4H2O

20°C

16.21 3.70 0.98 1.25

40°C

50°C

26.23

60°C

70°C

80°C

32.97

36.9

38.875°

36.24

36.10

85.9

90.5

90°C

100°C

18 39.5 58.7

88.6 0.0114 7.81 6.16 1.995

65.3

37.35 72.4

81.8

83.8

97.2

90.1

2.998

5.41

4.5 9.21 5.22

4.04

2.54

2.45

36.4 130.4

100.8 139

93.8

96

98

100

10.92

12.74

14.61

16.53

18.45

6.64 1.63

1.09

86.6

83.7

80.8

152.5 200.3 118.3

94.78

209.2

223.2

273.1

206.9 41.9

47

54.4 70.1

76.8

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

SOLUBILITIES The H in solubility tables (2-121 to 2-144) is the proportionality constant for the expression of Henry’s law, p = Hx, where x = mole fraction of the solute in the liquid phase; p = partial pressure of the solute in the gas phase, expressed in atmospheres; and H = a proportionality constant expressed in units of atmospheres of solute pressure in the gas phase per unit concentration of the solute in the liquid phase. (The unit of concentration of the solute in the liquid phase is moles solute per mole solution.)

TABLE 2-124

Ammonia (NH3)—Low Pressures

Weight NH3 per 100 weights H2O

0.105

0.244

0.32

0.38

0.576

0.751

1.02

Partial pressure NH3, mm. Hg, at 25°C

0.791

1.83

2.41

2.89

4.41

5.80

7.96

Weight NH3 per 100 weights H2O

1.31

1.53

1.71

1.98

2.11

2.58

2.75

10.31

11.91

13.46

15.75

16.94

20.86

22.38

Partial pressure NH3, mm. Hg, at 25°C TABLE 2-121

Acetylene (C2H2)

t, °C

0

5

10

15

20

25

30

10−3 × H*

0.72

0.84

0.96

1.08

1.21

1.33

1.46

2-125

“Landolt-Börnstein Physikalische-chemische Tabellen,” Eg. I, p. 303, 1927. Phase-equilibrium data for the binary system NH3-H2O are given by Clifford and Hunter, J. Phys. Chem., 37, 101 (1933).

International Critical Tables, vol. 3, p. 260, McGraw-Hill, 1928. *H. See footnote for Table 2-122. TABLE 2-125

TABLE 2-122

Carbon Dioxide (CO2)

Weight of CO2 per 100 weights of H2O* Total pressure, atm 12°C 18°C 25°C 31.04°C 35°C 40°C 50°C 75°C 100°C

Air

t, °C

0

5

10

15

20

25

30

35

10−4 × H*

4.32

4.88

5.49

6.07

6.64

7.20

7.71

8.23

t, °C

40

45

50

60

70

80

90

100

10−4 × H*

8.70

9.11

9.46

10.1

10.5

10.7

10.8

10.7

International Critical Tables, vol. 3, p. 257. *H is calculated from the absorption coefficients of O2 and N2, taking into consideration the correction for constant argon content.

25 50 75 100 150 200 300 400 500 700

7.03 7.18 7.27 7.59

3.86 6.33 6.69 6.72 7.07

7.86 8.12

7.35 7.77

5.38 6.17 6.28

7.54

2.80 4.77 5.80 5.97 6.25 6.48

2.56 4.39 5.51 5.76 6.03 6.29

2.30 4.02 5.10 5.50 5.81 6.28

7.27 7.65

7.06 7.51

6.89 7.26

1.92 3.41 4.45 5.07 5.47 5.76 6.20 6.58

1.35 2.49 3.37 4.07 4.86 5.27 5.83 6.30

1.06 2.01 2.82 3.49 4.49 5.08 5.84 6.40

7.58

7.43

7.61

*In the original, concentration is expressed in cubic centimeters of CO2 (reduced to 0°C and 1 atm) dissolved in 1 g of water. TABLE 2-123 Weight NH3 per 100 weights H2O 100 90 80 70 60 50 40 30 25 20 15 10 7.5 5 4 3 2.5 2 1.6 1.2 1.0 0.5

Ammonia (NH3) Partial pressure of NH3, mm. Hg 0°C

947 785 636 500 380 275 190 119 89.5 64 42.7 25.1 17.7 11.2

10°C

20°C

987 1450 780 1170 600 945 439 686 301 470 190 298 144 227 103.5 166 70.1 114 41.8 69.6 29.9 50.0 19.1 31.7 16.1 24.9 11.3 18.2 15.0 12.0

*Extrapolated values.

25°C 30°C

23.5 19.4 15.3 12.0 9.1 7.4 3.4

719 454 352 260 179 110 79.7 51.0 40.1 29.6 24.4 19.3 15.3 11.5

40°C

3300 2760 2130 1520 1065 692 534 395 273 167 120 76.5 60.8 45 (37.6)* (30.0) (24.1) (18.3) (15.4)

50°C 60°C

825 596 405 247 179 115 91.1 67.1 (55.7) (44.5) (35.5) (26.7) (22.2)

834 583 361 261 165 129.2 94.3 77.0 61.0 48.7 36.3 30.2

TABLE 2-126

Carbon Monoxide (CO) 10−4 × H

Partial pressure of CO, mm Hg

17.7°C

19.0°C

900 2000 3000 4000 5000 6000 7000 8000

4.77 4.77 4.77 4.78 4.80 4.82 4.86 4.88

4.88 4.91 4.93 4.95 4.97 4.98 5.02 5.08

International Critical Tables, vol. 3, p. 260.

TABLE 2-127

Carbonyl Sulfide (COS)

t °C

0

5

10

15

20

25

30

10−3 × H

0.92

1.17

1.48

1.82

2.19

2.59

3.04

International Critical Tables, vol. 3, p. 261.

2-126

PHYSICAL AND CHEMICAL DATA

TABLE 2-128 Partial pressure of Cl2, mm Hg

Chlorine (Cl2)

TABLE 2-129

Solubility, g of Cl2 per liter 10°C

20°C

30°C

40°C

50°C

5 10 30 50 100

0.488 .679 1.221 1.717 2.79

0.451 .603 1.024 1.354 2.08

0.438 .575 .937 1.210 1.773

0.424 .553 .873 1.106 1.573

0.412 .532 .821 1.025 1.424

0.398 .512 .781 .962 1.313

150 200 250 300 350

3.81 4.78 5.71

2.73 3.35 3.95 4.54 5.13

2.27 2.74 3.19 3.63 4.06

1.966 2.34 2.69 3.03 3.35

1.754 2.05 2.34 2.61 2.86

1.599 1.856 2.09 2.31 2.53

400 450 500 550 600

5.71 6.26 6.85 7.39 7.97

4.48 4.88 5.29 5.71 6.12

3.69 3.98 4.30 4.60 4.91

3.11 3.36 3.61 3.84 4.08

2.74 2.94 3.14 3.33 3.52

650 700 750 800 900

8.52 9.09 9.65 10.21

6.52 6.90 7.29 7.69 8.46

5.21 5.50 5.80 6.08 6.68

4.32 4.54 4.77 4.99 5.44

3.71 3.89 4.07 4.27 4.62

9.27 10.84 13.23 17.07 21.0

7.27 8.42 10.14 13.02 15.84

5.89 6.81 8.05 10.22 12.32

4.97 5.67 6.70 8.38 10.03

18.73 21.7 24.7 27.7 30.8

14.47 16.62 18.84 20.7 23.3

11.70 13.38 15.04 16.75 18.46

Cl2.8H2O2 separates

3000 3500 4000 4500 5000 Partial pressure of Cl2, mm Hg

70°C

80°C

90°C

1 3 5 7 10 11 12 13 14 15 16

100°C

110°C

5 10 30 50 100

0.383 .492 .743 .912 1.228

0.369 .470 .704 .863 1.149

0.351 .447 .671 .815 1.085

0.339 .431 .642 .781 1.034

0.326 .415 .627 .747 .987

0.316 .402 .598 .722 .950

150 200 250 300 350

1.482 1.706 1.914 2.10 2.28

1.382 1.580 1.764 1.932 2.10

1.294 1.479 1.642 1.793 1.940

1.227 1.396 1.553 1.700 1.831

1.174 1.333 1.480 1.610 1.736

1.137 1.276 1.413 1.542 1.661

400 450 500 550 600

2.47 2.64 2.80 2.97 3.13

2.25 2.41 2.55 2.69 2.83

2.08 2.22 2.35 2.47 2.59

1.965 2.09 2.21 2.32 2.43

1.854 1.972 2.08 2.19 2.29

1.773 1.880 1.986 2.09 2.19

650 700 750 800 900

3.29 3.44 3.59 3.75 4.04

2.97 3.10 3.23 3.37 3.63

2.72 2.84 2.96 3.08 3.30

2.55 2.66 2.76 2.87 3.08

2.41 2.50 2.60 2.69 2.89

2.28 2.37 2.47 2.56 2.74

1000 1200 1500 2000 2500

4.36 4.92 5.76 7.14 8.48

3.88 4.37 5.09 6.26 7.40

3.53 3.95 4.58 5.63 6.61

3.28 3.67 4.23 5.17 6.05

3.07 3.43 3.95 4.78 5.59

2.91 3.25 3.74 4.49 5.25

3000 3500 4000 4500 5000

9.83 11.22 12.54 13.88 15.26

8.52 9.65 10.76 11.91 13.01

7.54 8.53 9.52 10.46 11.42

6.92 7.79 8.65 9.49 10.35

6.38 7.16 7.94 8.72 9.48

5.97 6.72 7.42 8.13 8.84

0°C

5°C

10°C

15°C

20°C

30°C

40°C

2.00 6.00 10.0 14.0 20.0

1.50 4.7 7.8 10.9 15.5 17.0 18.6 20.3

1.25 3.85 6.30 8.95 12.8 14.0 15.3 16.6 18.0 19.2 20.3

1.00 3.20 5.25 7.35 10.5 11.7 12.8 13.8 14.9 16.0 17.0

0.90 2.70 4.30 6.15 8.80 9.70 10.55 11.5 12.3 13.2 14.2

0.60 1.95 3.20 4.40 6.30 7.00 7.50 8.20 8.80 9.50 10.1

0.46 1.30 2.25 3.20 4.50 5.00 5.45 5.85 6.35 6.80 7.20

Ishi, Chem. Eng. (Japan), 22, 153 (1958).

TABLE 2-130

Ethane (C2H6)

t, °C

0

5

10

15

20

25

30

35

10−4 × H

1.26

1.55

1.89

2.26

2.63

3.02

3.42

3.83

t, °C

40

45

50

60

70

80

90

100

10−4 × H

4.23

4.63

5.00

5.65

6.23

6.61

6.87

6.92

International Critical Tables, vol. 3, p. 261.

TABLE 2-131

Solubility, g of Cl2 per liter 60°C

Weight of ClO2, grams per liter of solution

Vol % of ClO2 in gas phase

0°C

1000 1200 1500 2000 2500

Chlorine Dioxide (ClO2)

Ethylene (C2H4)

t, °C

0

5

10

15

20

25

30

10−3 × H

5.52

6.53

7.68

8.95

10.2

11.4

12.7

International Critical Tables, vol. 3, p. 260.

TABLE 2-132

Helium (He)

t, °C

0

10

20

30

40

50

10−4 × H

12.9

12.6

12.5

12.4

12.1

11.5

See also Pray, Schweickert, and Minnich, Ind. Eng. Chem., 44, 1146 (1952).

TABLE 2-133

Hydrogen (H2)—Temperature

t, °C

0

5

10

15

20

25

30

35

10−4 × H

5.79

6.08

6.36

6.61

6.83

7.07

7.29

7.42

t, °C

40

45

50

60

70

80

90

100

10−4 × H

7.51

7.60

7.65

7.65

7.61

7.55

7.51

7.45

“International Critical Tables,” vol. 3, p. 256. See also Pray, Schweickert, and Minnich, Ind. Eng. Chem., 44, 1146 (1952).

SOLUBILITIES TABLE 2-134

Hydrogen (H2)—Pressure

TABLE 2-137

10−4 × H

Partial pressure H2, mm Hg

19.5°C

900 1100 2000 3000 4000 5000 6000 7000 8200 8250

23°C

7.42 7.75 7.76 7.77 7.81 7.89 8.00 8.16 8.41

7.42 7.43 7.47 7.56 7.70 7.87

2-127

Methane (CH4)

t, °C

0

5

10

15

20

25

30

35

10−4 × H

2.24

2.59

2.97

3.37

3.76

4.13

4.49

4.86

t, °C

40

45

50

60

70

80

90

100

10−4 × H

5.20

5.51

5.77

6.26

6.66

6.82

6.92

7.01

International Critical Tables, vol. 3, p. 260.

TABLE 2-138

8.17

International Critical Tables, vol. 3, p. 256.

Nitrogen (N2)—Temperature*

t, °C

0

5

10

15

20

25

30

35

10−4 × H

5.29

5.97

6.68

7.38

8.04

8.65

9.24

9.85

t, °C

40

45

50

60

70

80

90

100

10−4 × H

10.4

10.9

11.3

12.0

12.5

12.6

12.6

12.6

“International Critical Tables,” vol. 3, p. 256. See also Pray, Schweickert, and Minnich, Ind. Eng. Chem., 44, 1146 (1952). *Atmospheric nitrogen = 98.815 vol. % N2 + 1.185 vol. % A. TABLE 2-135

Hydrogen Chloride (HCl)

Weights of HCl per 100 weights of H2O

TABLE 2-139

Partial pressure of HCl, mm Hg 0°C

78.6 66.7 56.3 47.0 38.9 31.6 25.0 19.05 13.64 8.70 4.17 2.04

10°C

510 130 29.0 5.7 1.0 0.175 .0316 .0056 .00099 .000118 .000018

Weights of HCl per 100 weights of H2O

20°C

840 233 56.4 11.8 2.27 0.43 .084 .016 .00305 .000583 .000069 .0000117

30°C

399 105.5 23.5 4.90 1.00 0.205 .0428 .0088 .00178 .00024 .000044

Nitrogen (N2)—Pressure

627 188 44.5 9.90 2.17 0.48 .106 .0234 .00515 .00077 .000151

900 2000 3000 4000 5000 6000 7000 8100 8200

80°C

535 141 35.7 8.9 2.21 0.55 .136 .0344 .0064 .00140

110°C

623 188 54.5 15.6 4.66 1.34 0.39 .095 .0245

760 253 83 28 9.3 3.10 0.93 .280

Oxygen (O2)—Temperature

t, °C

0

5

10

15

20

25

30

35

10−4 × H

2.55

2.91

3.27

3.64

4.01

4.38

4.75

5.07

t, °C

40

45

50

60

70

80

90

100

10−4 × H

5.35

5.63

5.88

6.29

6.63

6.87

6.99

7.01

International Critical Tables, vol. 3, p. 257. Pray, Schweickert, and Minnich [Ind. Eng. Chem., 44, 1146 (1952)] give H = 4.46 × 10−4 at 25°C and other values up to 343°C.

Hydrogen Sulfide (H2S)

t, °C

0

5

10

15

20

25

30

35

10−2 × H

2.68

3.15

3.67

4.23

4.83

5.45

6.09

6.76

t, °C

40

45

50

60

70

80

90

100

7.45

9.08 9.15 9.25 9.38 9.49 9.62 9.75 9.91

TABLE 2-141

Oxygen (O2)—Pressure

Partial pressure of O2, mm Hg

10 × H

24.9°C

8.24 8.32 8.41 8.49 8.59 8.74 8.86 9.04

See also Goodman and Krase [Ind. Eng. Chem., 23, 401 (1931)] for values up to 169°C and 300 atm.

TABLE 2-140

Enthalpy and phase-equilibrium data for the binary system HCl-H2O are given by Van Nuys, Trans. Am. Inst. Chem. Engrs., 39, 663 (1943).

−2

19.4°C

Partial pressure of HCl, mm Hg 50°C

78.6 66.7 56.3 47.0 38.9 31.6 25.0 19.05 13.64 8.70 4.17 2.04

TABLE 2-136

10−4 × H

Partial pressure of N2, mm Hg

8.14

8.84

10.3

International Critical Tables, vol. 3, p. 259.

11.9

13.5

14.4

14.8

800 900 2000 3000 4000 5000 6000 7000 8150 8200

10−4 × H 23°C

25.9°C 4.79

4.58 4.59 4.60 4.68 4.73 4.80 4.88 4.98

4.80 4.83 4.88 4.92 4.98 5.05 5.16

International Critical Tables, vol. 3, p. 257. See also Trans. Am. Soc. Mech. Engrs., 76, 69 (1954) for solubility of O2 for 100°F < T < 650°F, 300 < P < 2000 lb/in2.

2-128

PHYSICAL AND CHEMICAL DATA

TABLE 2-142 t, °C

0

10−3 × H 1.94

Ozone (O3)

TABLE 2-143

5

10

15

20

25

30

35

2.18

2.48

2.88

3.76

4.57

5.98

8.18

40

50

12.0 27.4

International Critical Tables, vol. 3, p. 257.

TABLE 2-144

Propylene (C3H6)

t, °C

2

6

10

14

18

10−3 × H

3.04

3.84

4.46

5.06

5.69

International Critical Tables, vol. 3, p. 260.

Partial Vapor Pressure of Sulfur Dioxide over Water, mm Hg Temperature, °C

g SO2 / 100 g H2O

0

10

0.01 0.05 0.10 0.15 0.20

0.02 0.38 1.15 2.10 3.17

0.04 0.66 1.91 3.44 5.13

0.25 0.30 0.40 0.50 1.00

4.34 5.57 8.17 10.9 25.8

6.93 8.84 12.8 17.0 39.5

2.00 3.00 4.00 5.00 6.00 8.00 10.00 15.00 20.00

20 0.07 1.07 3.03 5.37 7.93 10.6 13.5 19.4 25.6 58.4

30

40

50

60

90

120

0.12 1.68 4.62 8.07 11.8

0.19 2.53 6.80 11.7 17.0

0.29 3.69 9.71 16.5 23.8

0.43 5.24 13.5 22.7 32.6

1.21 12.9 31.7 52.2 73.7

2.82 27.0 63.9 104 145

15.7 19.8 28.3 37.1 83.7

58.6 93.2 129 165 202

88.5 139 192 245 299

129 202 277 353 430

183 285 389 496 602

275 351 542 735

407 517 796

585 741

818

22.5 28.2 40.1 52.3 117

31.4 39.2 55.3 72.0 159

42.8 53.3 74.7 96.8 212

95.8 118 164 211 454

253 393 535 679 824

342 530 720

453 700

955

186 229 316 404 856

Condensed from Rabe, A. E. and Harris, J. F., J. Chem. Eng. Data, 8 (3), 333–336, 1963. Copyright © American Chemical Society and reproduced by permission of the copyright owner.

THERMAL EXPANSION UNITS CONVERSIONS For this subsection, the following units conversion is applicable:

2, p. 93; metals, vol. 2, p. 459; petroleums, vol. 2, p. 145; porcelains, vol. 2, pp. 70, 78; refractory materials, vol. 2, p. 83; solid insulators, vol. 2, p. 310.

°F = 9⁄ 5 °C + 32. THERMAL EXPANSION OF GASES ADDITIONAL REFERENCES The tables given under this subject are reprinted by permission from the Smithsonian Tables. For more detailed data on thermal expansion, see International Critical Tables: tabular index, vol. 3, p. 1; abrasives, vol. 2, p. 87; alloys, vol. 2, p. 463; building stones, vol. 2, p. 54; carbons, vol. 2, p. 303; elements, vol. 1, p. 102; enamels, vol. 2, p. 115; glass, vol.

No tables of the coefficients of thermal expansion of gases are given in this edition. The coefficient at constant pressure, 1/υ(∂υ/∂T)p, for an ideal gas is merely the reciprocal of the absolute temperature. For a real gas or liquid, both it and the coefficient at constant volume, 1/p (∂p/∂T)v, should be calculated either from the equation of state or from tabulated PVT data.

THERMAL EXPANSION TABLE 2-145

2-129

Linear Expansion of the Solid Elements*

C is the true expansion coefficient at the given temperature; M is the mean coefficient between given temperatures; where one temperature is given, the true coefficient at that temperature is indicated; α and β are coefficients in formula lt = l0(1 + αt + βt2); l0 is length at 0°C (unless otherwise indicated, when, if x is the reference temperature, lt = lx[1 + α(t − tx) + β(t − tx)2]; lt is length at t°C). Element

Temp. °C

C × 104

Aluminum Aluminum Antimony Arsenic Bismuth Cadmium Cadmium Carbon, diamond graphite Chromium Cobalt Copper Copper Gold Gold Indium Iodine Iridium Iridium Iron, soft cast wrought steel Lead (99.9)

20 300 20 20 20 0 0 50 50

0.224 0.284 0.136 0.05 0.014 0.54 0.20⊥ 0.012 0.06

20 20 200 20

0.123 0.162 0.170 0.140

40

0.417

20

0.065

40 20 20 20

0.1210 0.118 0.119 0.114

Magnesium

100 280 20

0.291 0.343 0.254

Manganese

20

0.233

Molybdenum†

20

0.053

Nickel

20

0.126

Osmium Palladium

40 20

0.066 0.1173

Platinum

20 20

0.0887 0.0893

40 40 0 40 20 20

0.0850 0.0963 0.439 0.0763 0.1846 0.195

Potassium Rhodium Ruthenium Selenium Silicon Silver Sodium Steel, 36.4Ni Tantalum†

20

Tellurium Thallium Tin

20 40 20 20 27 20‡ 20‡ 20

Tungsten† Zinc

0.065 0.016 0.302 0.214 0.305 0.0444 0.643 0.125⊥ 0.358

Temp. range, °C

M × 104

100 500 20

0.235 0.311 0.080⊥

20 −180, −140 −180, −140

0.103⊥ 0.59 0.117⊥

20, 100

0.068

100 300 17, 100 −191, 17

0.166 0.175 0.143 0.132

−190,

0.837

17

Temp. range, °C

α × 104

β × 106

0,

500

0.22

0.009

20, 20,

100 100

0.526 0.214⊥

20, 6, 0,

500 121 625

0.086 0.121 0.161

0.0064 0.0040

0,

520

0.142

0.0022

0.0636 0.0679

0.0032 0.0011

0, 80 1070, 1720 0, 100

20, 100 20, 200 −100, 20, 0, −190, 0, 25, 25, 0,

0, 6,

+ 20 100 100 0 100 100 500 100

0.11

0.291 0.300 0.240 0.260 0.228 0.159 0.052 0.049 0.055 0.130

50 21

0.83 0.0876

0, 100 −3, +18 0, 100

0.660 0.0249 0.197

−190, 20, 20, −78, 0,

−17 260 340 0 100 20

0.622 0.031 0.055 0.059 0.0655 0.272⊥

20 100 −100 100 100

0.154⊥ 0.045 0.656 0.639 0.141⊥

0, 0, 0, 100,

750 750 750 240

0.1158 0.1170 0.1118 0.269

0.0053 0.0053 0.0053 0.011

+ 20,

500

0.2480

0.0096

20, 300 −142, 19 19, +305

0.216 0.0515 0.0501

0.0121 0.0057 0.0014

−190, + 20 + 20, +300 500, 1000

0.1308 0.1236 0.1346

0.0166 0.0066 0.0033

−190, 0, −190, 0, 0,

+100 1000 −100 + 80 1000

0.1152 0.1167 0.0875 0.0890 0.0887

0.00517 0.0022 0.00314 0.00121 0.00132

−75, −112

0.0746

−75, −67 0, 875 20, 500 0, 50 260, 500 340, 500 20, 400

0.0182 0.1827 0.1939 0.72 0.144 0.136 0.0646

8, 0, −140, +20, +20,

0.00479 0.00295

0.0009

95

0.2033

0.0263

−105, +502 + 0, 400

0.0428 0.354

0.00058 0.010

*Smithsonian Tables. For more complete tabulations see Table 142, Smithsonian Physical Tables, 9th ed., 1954; Handbook of Chemistry and Physics, 40th ed., pp. 2239–2245. Chemical Rubber Publishing Co.; Goldsmith, and Waterman, WADC-TR-58-476, 1959; Johnson (ed.), WADD-TR-60-56, 1960, etc. †Molybdenum, 300° to 2500°C; lt = l300[1 + 5.00 × 10−6(t − 300) + 10.5 × 10−10(t − 300)2] Tantalum, 300° to 2800°C; lt = l300[1 + 6.60 × 10−6(t − 300) + 5.2 × 10−10(t − 300)2] Tungsten, 300° to 2700°C; lt = l300[1 + 4.44 × 10−6(t − 300) + 4.5 × 10−10(t − 300)2] Beryllium, 20° to 100°C; 12.3 × 10−6 per °C. Columbium, 0° to 100°C; 7.2 × 10−6 per °C. Tantalum, 20° to 100°C; 6.6 × 10−6 per °C. ‡Two errors in the data of zinc have been corrected. These values were taken from Grüneisen and Goens, Z. Physik., 29, 141 (1924).

2-130

PHYSICAL AND CHEMICAL DATA

TABLE 2-146

Linear Expansion of Miscellaneous Substances*

The coefficient of cubical expansion may be taken as three times the linear coefficient. In the following table, t is the temperature or range of temperature, and C, the coefficient of expansion. Substance

t°C

Amber

0−30 0−09 20−60

Bakelite, bleached Brass: Cast 0−100 Wire 0−100 Wire 0−100 71.5 Cu + 27.7 Zn + 0.3 Sn + 0.5 Pb 40 71 Cu + 29 Zn 0−100 Bronze: 3 Cu + 1 Sn 16.6−100 3 Cu + 1 Sn 16.6−350 3 Cu + 1 Sn 16.6−957 86.3 Cu + 9.7 Sn + 4 Zn 40 97.6 Cu + hard 0−80 2.2 Sn + soft 0−80 0.2 P Caoutchouc Caoutchouc 16.7−25.3 Celluloid 20−70 Constantan 4−29 Duralumin, 94Al 20−100 20−300 Ebonite 25.3−35.4 Fluorspar, CaF2 0−100 German silver 0−100 Gold-platinum, 2 Au + 1 Pt 0−100 Gold-copper, 2 Au + 1 Cu 0−100 Glass: Tube 0−100 Tube 0−100 Plate 0−100 Crown (mean) 0−100 Crown (mean) 50−60 Flint 50−60 Jena ther- 16III 0−100 mometer normal

{

}

C × 104 0.50 0.61 0.22 0.1875 0.1930 0.1783 to 0.193 0.1859 0.1906 0.1844 0.2116 0.1737 0.1782 0.1713 0.1708 0.657 to 0.686 0.770 1.00 0.1523 0.23 0.25 0.842 0.1950 0.1836 0.1523 0.1552 0.0833 0.0828 0.0891 0.0897 0.0954 0.0788 0.081

Substance

t°C

C × 104

Jena thermometer 59III Jena thermometer 59III Gutta percha Ice Iceland spar: Parallel to axis Perpendicular to axis Lead tin (solder) 2 Pb + 1 Sn Limestone Magnalium Manganin Marble Monel metal

0−100 −191 to +16 20 −20 to −1

0.058 0.424 1.983 0.51

0−80 0−80

0.2631 0.0544

0−100 25−100 12−39 15−100 25−100 25−600 0−16 16−38 38−49

0.2508 0.09 0.238 0.181 0.117 0.14 0.16 1.0662 1.3030 4.7707

40

0.0884

0−100 20−790 1000−1400

0.1523 0.0413 0.0553

0−80 −190 to + 16 0−80 −190 to + 16 16 to 500 16 to 1000 40 0 −160 0−100 25−100 25−600

0.0797 0.0521 0.1337 −0.0026 0.0057 0.0058 0.4040 0.691 0.300 0.1933 0.037 0.136

Paraffin Paraffin Paraffin Platinum-iridium, 10 Pt + 1 Ir Platinum-silver, 1 Pt + 2 Ag Porcelain Porcelain Bayeux Quartz: Parallel to axis Parallel to axis Perpend. to axis Quartz glass Quartz glass Quartz glass Rock salt Rubber, hard Rubber, hard Speculum metal Steel, 0.14 C, 34.5 Ni

Substance

t°C

Topas: Parallel to lesser horizontal axis 0−100 Parallel to greater horizontal axis 0−100 Parallel to vertical axis 0−100 Tourmaline: Parallel to longitudinal axis 0−100 Parallel to horizontal axis 0−100 Type metal 16.6−254 Vulcanite 0−18 Wedgwood ware 0−100 Wood: Parallel to fiber: Ash 0−100 Beech 2.34 Chestnut 2.34 Elm 2.34 Mahogany 2.34 Maple 2.34 Oak 2.34 Pine 2.34 Walnut 2.34 Across the fiber: Beech 2.34 Chestnut 2.34 Elm 2.34 Mahogany 2.34 Maple 2.34 Oak 2.34 Pine 2.34 Walnut 2.34 Wax white 10−26 Wax white 26−31 Wax white 31−43 Wax white 43−57

C × 104

0.0832 0.0836 0.0472 0.0937 0.0773 0.1952 0.6360 0.0890 0.0951 0.0257 0.0649 0.0565 0.0361 0.0638 0.0492 0.0541 0.0658 0.614 0.325 0.443 0.404 0.484 0.544 0.341 0.484 2.300 3.120 4.860 15.227

*Smithsonian Tables. For a more complete tabulation see Tables 143, 144. Smithsonian Physical Tables. 9th ed., 1954, also reprinted in American Institute of Physics Handbook, McGraw-Hill, New York, 1957; Handbook of Chemistry and Physics, 40th ed., pp. 2239–2245, Chemical Rubber Publishing Co. For data on many solids prior to 1926, see Gruneisen, Handbuch der Physik, vol. 10, pp. 1–52, 1926, translation available as N.A.S.A. RE 2-18-59W, 1959. For eight plastic solids below 300 K, see Scott, Cryogenic Engineering, p. 331, Van Nostrand, Princeton, NJ, 1959. For 11 other materials to 300 K, see Scott, loc. cit., p. 333. For quartz and silica, see Cook, Brit. J. Appl. Phys., 7, 285 (1956).

THERMAL EXPANSION TABLE 2-147

Cubical Expansion of Liquids*

TABLE 2-148

If V0 is the volume at 0°, then at t° the expansion formula is Vt = V0(1 + αt + βt2 + γ t3). The table gives values of α, β, and γ, and of C, the true coefficient of cubical expansion at 20° for some liquids and solutions. The temperature range of the observation is ∆t. Values for the coefficient of cubical expansion of liquids can be derived from the tables of specific volumes of the saturated liquid given as a function of temperature later in this section. Liquid

Range

α × 103

β × 106

Acetic acid Acetone Alcohol: Amyl Ethyl, 30% by volume Ethyl, 50% by volume Ethyl, 99.3% by volume Ethyl, 500 atm. pressure Ethyl, 3000 atm. pressure Methyl Benzene Bromine Calcium chloride: 5.8% solution 40.9% solution Carbon disulfide 500 atm. pressure 3000 atm. pressure Carbon tetrachloride Chloroform Ether Glycerin Hydrochloric acid, 33.2% solution Mercury Olive oil Pentane Potassium chloride, 24.3% solution Phenol Petroleum, 0.8467 density Sodium chloride, 20.6% solution Sodium sulfate, 24% solution Sulfuric acid: 10.9% solution 100.0% Turpentine Water

16−107 0−54

1.0630 1.3240

0.12636 3.8090

− 15−80 18−39 0−39

0.9001 0.2928 0.7450

0.6573 10.790 1.85

27−46

1.012

2.20

0−40

0.866

0−40 0−61 11−81 0−59

0.524 1.1342 1.17626 1.06218

18−25 17−24 −34−60 0−50 0−50 0−76 0−63 −15−38 0−33 0−100

γ × 108

C × 108 at 20°

1.0876 1.071 − 0.87983 1.487 1.18458 0.902 −11.87 0.730 1.12

1.3635 1.27776 1.87714

0.8741 1.199 0.80648 1.237 −0.30854 1.132

0.07878 0.42383 1.13980 0.940 0.581 1.18384 1.10715 1.51324 0.4853

4.2742 0.8571 1.37065

0.250 0.458 1.91225 1.218

0.89881 4.66473 2.35918 0.4895

1.35135 1.236 − 1.74328 1.273 4.00512 1.656 0.505

0−33

0.4460 0.18182 0.6821 1.4646

0.215 0.0078 1.1405 3.09319

− 0.539 1.6084

16−25 36−157

0.2695 0.8340

2.080 0.10732

0.4446

24−120

0.8994

1.396

0.955

0−29

0.3640

1.237

0.414

11−40

0.3599

1.258

0.410

0−30 0.2835 2.580 0−30 0.5758 −0.432 − 9−106 0.9003 1.9595 0−33 −0.06427 8.5053

0.455 0.18186 0.721 1.608 0.353 1.090

0.387 0.558 − 0.44998 0.973 − 6.7900 0.207

*Smithsonian Tables, Table 269. For a detailed discussion of mercury data, see Cook, Brit. J. Appl. Phys., 7, 285 (1956). For data on nitrogen and argon, see Johnson (ed.), WADD-TR-60-56, 1960. Bromoform1 7.7 − 50°C. Vt = 0.34204[1 + 0.00090411(t − 7.7) + 0.0000006766(t − 7.7)2] 0.34204 in the specific volume of bromoform at 7.7°C. Glycerin2 −62 to 0°C. Vt = V0(1 + 4.83 × 10−4t − 0.49 × 10−6t2) 0 − 80°C. Vt = V0(1 + 4.83 × 10−4t + 0.49 × 10−6t2) 3 Mercury 0 − 300°C. Vt − V0[1 + 10−8(18153.8t + 0.7548t2 + 0.001533t2 + 0.00000536t4)] 1 Sherman and Sherman, J. Am. Chem. Soc., 50, 1119 (1928). (An obvious error in their equation has been corrected.) 2 Samsoen, Ann. phys., (10) 9, 91 (1928). 3 Harlow, Phil. Mag., (7) 7, 674 (1929).

2-131

Cubical Expansion of Solids*

If v2 and v1 are the volumes at t2 and t1, respectively, then v2 = v1(1 + C∆t), C being the coefficient of cubical expansion and ∆t the temperature interval. Where only a single temperature is stated, C represents the true coefficient of cubical expansion at that temperature. Substance

t or ∆t

C × 104

Antimony Beryl Bismuth Copper† Diamond Emerald Galena Glass, common tube hard Jena, borosilicate 59 III pure silica Gold Ice Iron Lead† Paraffin Platinum Porcelain, Berlin chloride nitrate sulfate Quartz Rock salt Rubber Silver Sodium Stearic acid Sulfur, native Tin Zinc†

0−100 0−100 0−100 0−100 40 40 0−100 0−100 0−100 20−100 0−80 0−100 −20 to −1 0−100 0−100 20 0−100 20 0−100 0−100 20 0−100 50−60 20 0−100 20 33.8−45.4 13.2−50.3 0−100 0−100

0.3167 0.0105 0.3948 0.4998 0.0354 0.0168 0.558 0.276 0.214 0.156 0.0129 0.4411 1.1250 0.3550 0.8399 5.88 0.265 0.0814 1.094 1.967 1.0754 0.3840 1.2120 4.87 0.5831 2.13 8.1 2.23 0.6889 0.8928

*Smithsonian Tables, Table 268. †See additional data below. Aluminum1 100 − 530°C. V = V0(1 + 2.16 × 10−5t + 0.95 × 10−8t2) 1 Cadmium 130 − 270°C. V = V0(1 + 8.04 × 10−5t + 5.9 × 10−8t2) 1 Copper 110 − 300°C. V = V0(1 + 1.62 × 10−5t + 0.20 × 10−8t2) Colophony2 0 − 34°C. V = V0(1 + 2.21 × 10−4t + 0.31 × 10−6t2) 34 − 150°C. V = V34[1 + 7.40 × 10−4(t − 34) + 5.91 × 10−6(t − 34)2] 1 Lead 100 − 280°C. V = V0(1 + 1.60 × 10−5t + 3.2 × 10−8t2) 2 Shellac 0 − 46°C. V = V0(1 + 2.73 × 10−4t + 0.39 × 10−6t2) 46 − 100°C. V = V46[1 + 13.10 × 10−4(t − 46) + 0.62 × 10−6(t − 46)2] Silica (vitreous)3 0 − 300°C. Vt = V0[1 + 10−8(93.6t + 0.7776t2 − 0.003315t2 + 0.000005244t4) Sugar (cane, amorphous)2 0 − 67°C. Vt = V0(1 + 2.34 × 10−4t + 0.14 × 10−6t2) 67 − 160°C. Vt = V67[1 + 5.02 × 10−4(t − 67) + 0.43 × 10−6(t − 67)2] Zinc1 120 − 360°C. Vt = V0(1 + 8.50 × 10−5t + 3.9 × 10−8t2) 1 2 3

Uffelmann, Phil. Mag., (7) 10, 633 (1930). Samsoen, Ann. phys., (10) 9, 83 (1928). Harlow, Phil. Mag., (7) 7, 674 (1929).

2-132

PHYSICAL AND CHEMICAL DATA

JOULE-THOMSON EFFECT UNITS CONVERSIONS For this subsection, the following units conversions are applicable: To convert the Joule-Thomson coefficient, µ, in degrees Celsius per atmosphere to degrees Fahrenheit per atmosphere, multiply by 1.8.

TABLE 2-149

°F = 9⁄ 5 °C + 32; °R = 9⁄ 5 K To convert bars to pounds-force per square inch, multiply by 14.504; to convert bars to kilopascals, multiply by 1 × 102.

Additional References Available for the Joule-Thomson Coefficient Temp. range, °C

Pressure range, atm Gas Air Ammonia Argon Benzene Butane Carbon dioxide Carbon monoxide Deuterium Dowtherm A Ethane Ethylene Helium Hydrogen

>200

<0

0–10

10–50

50–200

12, 15, 16 19, 35 28 39 31 26 7, 8, 28 37 17

12, 15, 19 35

15, 19, 35

19, 35

39 31 26 7, 8, 37

39 31

39

7, 8, 37

7, 8, 37

1,* 22, 24 25

17 1,* 22, 24, 25

46 45 1, 38 24, 30

17 22, 24, 25 1* 46 45 1, 38 22, 24, 25 30 6

6 33 13, 40

Methane Mixtures Natural gas Nitrogen

13, 28, 40

13, 40

Nitrous oxide Pentane Propane Steam

26, 34, 44 41 28, 29, 42

34 43 29, 42, 47

38 24, 30

34 42, 47

1, 38 22, 24, 25 30 33 13

33 13, 40

0–300 12, 15, 16 19, 35 28 39 31 26 7, 8, 9, 10 37 17 46 45 9, 10 38 24 6 9, 11 33 9, 10, 13 28, 40 9, 10 26, 34, 44 43 28, 29, 42 45

>300

Unclassified 3, 4, 18 2, 3

31

46 48

13

19

29, 42, 47

29, 47

*See also 14 (generalized chart); 18 (review, to 1919); 20–22; 23 (review, to 1948); 27 (review, to 1905); 32, 36, 41, 50. REFERENCES: 1. Baehr. Z. Elektrochem., 60, 515 (1956). 2. Beattie, J. Math. Phys., 9, 11 (1930). 3. Beattie, Phys. Rev., 35, 643 (1930). 4. Bradley and Hale, Phys. Rev., 29, 258 (1909). 5. Brown and Dean, Bur. Stand. J. Res., 60, 161 (1958). 6. Budenholzer, Sage, et al., Ind. Eng. Chem., 29, 658 (1937). 7. Burnett, Phys. Rev., 22, 590 (1923). 8. Burnett, Univ. Wisconsin Bull. 9(6), 1926. 9. Charnley, Ph.D. thesis. University of Manchester, 1952. 10. Charnley, Isles, et al., Proc. R. Soc. (London), A217, 133 (1953). 11. Charnley, Rowlinson, et al., Proc. R. Soc. (London), A230, 354 (1955). 12. Dalton, Commun. Phys. Lab. Univ. Leiden, no. 109c, 1909. 13. Deming and Deming, Phys. Rev., 48, 448 (1935). 14. Edmister, Pet. Refiner, 28, 128 (1949). 15. Eucken, Clusius, et al., Z. Tech. Phys., 13, 267 (1932). 16. Eumorfopoulos and Rai, Phil. Mag., 7, 961 (1926). 17. Huang, Lin, et al., Z. Phys., 100, 594 (1936). 18. Hoxton, Phys. Rev., 13, 438 (1919). 19. Ishkin and Kaganev, J. Tech. Phys. U.S.S.R., 26, 2323 (1956). 20. Isles, Ph.D. thesis, Leeds University. 21. Jenkin and Pye, Phil. Trans. R. Soc. (London), A213, 67 (1914); A215, 353 (1915). 22. Johnston, J. Am. Chem. Soc., 68, 2362 (1946). 23. Johnston, Trans. Am. Soc. Mech. Eng., 70, 651 (1948). 24. Johnston, Bezman, et al., J. Am. Chem. Soc., 68, 2367 (1946). 25. Johnston, Swanson, et al., J. Am. Chem. Soc., 68, 2373 (1946). 26. Kennedy, Sage, et al., Ind. Eng. Chem., 28, 718 (1936). 27. Kester, Phys. Rev., 21, 260 (1905). 28. Keyes and Collins, Proc. Nat. Acad. Sci., 18, 328 (1932). 29. Kleinschmidt, Mech. Eng., 45, 165 (1923); 48, 155 (1926). 30. Koeppe, Kältetechnik, 8, 275 (1956). 31. Lindsay and Brown, Ind. Eng. Chem., 27, 817 (1935). 32. Noell, dissertation, Munich, 1914, Forschungsdienst, 184, p. 1, 1916. 33. Palienko, Tr. Inst. Ispol’ z. Gaza, Akad. Nauk Ukr. SSR, no. 4, p. 87, 1956. 34. Pattee and Brown, Ind. Eng. Chem., 26, 511, (1934). 35. Roebuck, Proc. Am. Acad. Arts Sci., 60, 537 (1925); 64, 287 (1930). 36. Roebuck, see 49 below, 37. Roebuck and Murrell, Phys. Rev., 55, 240 (1939). 38. Roebuck and Osterberg, Phys. Rev., 37, 110 (1931); 43, 60 (1933). 39. Roebuck and Osterberg, Phys. Rev., 46, 785 (1934). 40. Roebuck and Osterberg, Phys. Rev., 48, 450 (1935). 41. Roebuck, Murrell, et al., J. Am. Chem. Soc., 64, 400 (1942). 42. Sage, unpublished data, California Institute of Technology, 1959. 43. Sage and Lacy, Ind. Eng. Chem., 27, 1484 (1934). 44. Sage, Kennedy, et al., Ind. Eng. Chem., 28, 601 (1936). 45. Sage, Webster, et al., Ind. Eng. Chem., 29, 658 (1937). 46. Ullock, Gaffert, et al., Trans. Am. Inst. Chem. Eng., 32, 73 (1936). 47. Yang, Ind. Eng. Chem., 45, 786 (1953). 48. Zelmanov, J. Phys. U.S.S.R., 3, 43 (1940). 49. Roebuck, recalculated data. 50. Michels et al., van der Waals laboratory publications. Gunn, Cheuh, and Prausnitz, Cryogenics, 6, 324 (1966), review equations relating the inversion temperatures and pressures. The ability of various equations of state to relate these was also discussed by Miller, Ind. Eng. Chem. Fundam., 9, 585 (1970); and Juris and Wenzel, Am. Inst. Chem. Eng. J., 18, 684 (1972). Perhaps the most detailed review is that of Hendricks, Peller, and Baron. NASA Tech. Note D 6807, 1972.

JOULE-THOMSON EFFECT

2-133

TABLE 2-150 Approximate Inversion-Curve Locus in Reduced Coordinates (Tr = T/Tc; Pr = P/Pc)* Pr

0

0.5

1

1.5

2

2.5

3

4

TrL TrU

0.782 4.984

0.800 4.916

0.818 4.847

0.838 4.777

0.859 4.706

0.880 4.633

0.903 4.550

0.953 4.401

Pr

5

6

7

8

9

10

11

11.79

TrL TrU

1.01 4.23

1.08 4.06

1.16 3.88

1.25 3.68

1.35 3.45

1.50 3.18

1.73 2.86

2.24 2.24

*Calculated from the best three-constant equation recommended by Miller, Ind. Eng. Chem. Fundam., 9, 585 (1970). TrL refers to the lower curve, and TrU, to the upper curve.

TABLE 2-151

Joule-Thomson Data for Air* t, °C

P, atm 1 20 60 100 140 180 200

−150

−100

−75

−50

−25

0

25

50

75

100

150

200

250

0.0450 .0185 − .0070 − .0255 − .0330

0.5895 .5700 .4820 .2775 .1360 .0655 .0440

0.4795 .4555 .3835 .2880 .1855 .1136 .0855

0.3910 .3690 .3195 .2505 .1825 .1270 .1065

0.3225 .3010 .2610 .2130 .1650 .1240 .1090

0.2745 .2580 .2200 .1820 .1450 .1100 .0950

0.2320 .2173 .1852 .1550 .1249 .0959

0.1956 .1830 .1571 .1310 .1070 .0829

0.1614 .1508 .1293 .1087 .0889 .0707

0.1355 .1258 .1062 .0884 .0726 .0580

0.0961 .0883 .0732 .0600 .0482 .0376

0.0645 .0580 .0453 .0343 .0250 .0174

0.0409 .0356 .0254 .0165 .0092 .0027

*Free of water and CO2. Extracted from Table 261, Smithsonian Physical Tables, 9th rev. ed., Washington, DC, 1954. These data are corrected from earlier publications. µ in °C/atm.

TABLE 2-152 P, bar TL, K TU, K

Approximate Inversion-Curve Locus for Air

0 (112)* 653

25

50

75

100

125

150

175

200

225

114 641

117 629

120 617

124 606

128 594

132 582

137 568

143 555

149 541

P, bar

250

275

300

325

350

375

400

425

432

TL, K TU, K

156 526

164 509

173 491

184 470

197 445

212 417

230 386

265 345

300 300

*Hypothetical low-pressure limit.

TABLE 2-153

Joule-Thomson Data for Argon* Pressure, atm

t, °C

1

−150 −125 −100 −75 −50

20

60

100

140

180

200

1.812 1.112 0.8605 .7100 .5960

1.102 0.8485 .6895 .5720

−0.0025 .1250 .6900 .5910 .4963

−0.0277 .0415 .2820 .4225 .3970

−0.0403 .0090 .1137 .2480 .2840

−0.0595 −.0100 .0560 .1537 .2037

−0.0640 − .0165 .0395 .1215 .1860

−25 0 25 50 75

.5045 .4307 .3720 .3220 .2695

.4805 .4080 .3490 .3015 .2557

.4210 .3600 .3077 .2650 .2285

.3460 .3010 .2628 .2297 .1993

.2763 .2505 .2213 .1947 .1710

.2140 .2050 .1890 .1700 .1505

.1950 .1883 .1745 .1580 .1415

100 125 150 200 250

.2413 .2105 .1845 .1377 .0980

.2277 .1980 .1720 .1280 .0910

.1975 .1707 .1485 .1102 .0785

.1715 .1480 .1285 .0950 .0665

.1490 .1300 .1123 .0823 .0555

.1320 .1153 .0998 .0715 .0485

.1255 .1100 .0945 .0675 .0468

300

.0643

.0607

.0530

.0445

.0370

.0370

.0276

*Extracted from Table 263, Smithsonian Physical Tables, 9th rev. ed., Washington, DC, 1954. These data are corrected from an earlier publication. µ in °C/atm.

2-134

PHYSICAL AND CHEMICAL DATA TABLE 2-154

Approximate Inversion-Curve Locus for Argon

P, bar

0

25

50

75

100

125

150

175

200

225

TL, K TU, K

94 765

97 755

101 744

105 736

109 726

113 716

118 705

123 694

128 683

134 671

P, bar

250

275

300

325

350

375

400

425

450

475

TL, K TU, K

141 657

148 643

158 627

170 610

183 591

201 569

222 544

248 515

288 478

375 375

TABLE 2-155

Joule-Thomson Data for Carbon Dioxide* Pressure, atm

t, °C

1

20

60

73

100

140

180

200

−75 −50 0 50 100

2.4130 1.2900 0.8950 .6490

−0.0200 − .0140 1.4020 .8950 .6375

−0.0200 − .0150 .0370 .8800 .6080

−0.0232 − .0165 .0310 .8225 .5920

−0.0228 − .0160 .0215 .5570 .5405

−0.0240 − .0183 .0115 .1720 .4320

−0.0250 − .0228 .0085 .1025 .3000

−0.0290 − .0248 .0045 .0930 .2555

125 150 200 250 300

.5600 .4890 .3770 .3075 .2650

.5450 .4695 .3575 .2885 .2425

.5160 .4430 .3400 .2625 .2080

.5068 .4380 .3325 .2565 .2002

.4750 .4155 .3150 .2420 .1872

.4130 .3760 .2890 .2235 .1700

.3230 .3102 .2600 .2045 .1540

.2915 .2910 .2455 .1975 .1505

*Extracted from Table 266, Smithsonian Physical Tables, 9th rev. ed., Washington, DC, 1954. These data are corrected from an earlier publication. µ in °C/atm.

TABLE 2-156

Approximate Inversion-Curve Locus for Carbon Dioxide*

P, bar

50

100

150

200

250

300

350

400

450

TL, K TU, K

243 1290

251 1261

258 1233

266 1205

272 1175

283 1146

293 1111

302 1076

312 1045

P, bar

500

550

600

650

700

750

800

850

884

TL, K TU, K

325 1015

338 983

351 950

365 914

383 878

403 840

441 796

496 739

608 608

*Interpolated from Vukalovich and Altunin’s interpolation of data of Price, Ind. Eng. Chem., 47, 1691 (1955). TL = lower inversion temperature, and TU = upper inversion temperature.

TABLE 2-157 P, bar TL, K TU, K

Approximate Inversion-Curve Locus for Deuterium

0 (31)* 216

25

50

75

100

125

150

175

194

34 202

38 189

43 178

49 168

56 157

65 146

77 131

108 108

*Hypothetical low-pressure limit.

TABLE 2-158 P, bar

0

TL, K

Approximate Inversion-Curve Locus for Ethane 25

50

75

100

125

150

175

200

225

249

255

262

269

275

282

290

297

306

P, bar

250

275

300

325

350

375

400

425

450

475

TL, K

315

325

335

345

357

370

383

398

415

432

P, bar

500

525

550

575

600

TL, K

453

477

505

545

626

JOULE-THOMSON EFFECT TABLE 2-159

2-135

Joule-Thomson Data for Helium*

T, K

160

180

200

220

240

260

280

300

µ

−0.0574

−0.0587

−0.0594

−0.0601

−0.0608

−0.0614

−0.0619

−0.0625

T, K

320

340

360

380

400

420

440

460

µ

−0.0629

−0.0634

−0.0637

−0.0640

−0.0643

−0.0645

−0.0645

−0.0643

T, K

480

500

520

540

560

580

600

µ

−0.0640

−0.0636

−0.0630

−0.0622

−0.0611

−0.0587

−0.0540

*Interpolated and converted from data in Table 262, Smithsonian Physical Tables, 9th rev. ed., Washington, DC, 1954. These data are corrected from those in an earlier publication. µ is in °C/atm. Below about 200 atm, little change in the coefficient with pressure occurs.

TABLE 2-160

Approximate Inversion-Curve Locus for Normal Hydrogen

P, bar

0

25

50

75

100

125

150

164

TL, K TU, K

(28)* 202

32 193

38 183

44 171

52 157

61 141

73 119

92 92

*Hypothetical low-pressure limit.

TABLE 2-161 P, bar

Approximate Inversion-Curve Locus for Methane

25

TL, K

50

75

100

125

150

175

200

225

250

275

300

161

166

172

176

182

189

195

202

209

217

225

P, bar

325

350

375

400

425

450

475

500

525

534

TL, K TU, K

234

243

254

265

277

292

309 505

331 474

365 437

400 400

TABLE 2-162

Joule-Thomson Data for Nitrogen* Pressure, atm

t, °C

1

20

33.5

60

100

140

180

200

−150 −125 −100 −75 −50

1.2659 0.8557 .6490 .5033 .3968

1.1246 0.7948 .5958 .4671 .3734

0.1704 .7025 .5494 .4318 .3467

0.0601 .4940 .4506 .3712 .3059

0.0202 .1314 .2754 .2682 .2332

−0.0056 .0498 .1373 .1735 .1676

−0.0211 .0167 .0765 .1026 .1120

−0.0284 .0032 .0587 .0800 .0906

−25 0 25 50 75

.3224 .2656 .2217 .1855 .1555

.3013 .2494 .2060 .1709 .1421

.2854 .2377 .1961 .1621 .1336

.2528 .2088 .1729 .1449 .1191

.2001 .1679 .1400 .1164 .0941

.1506 .1316 .1105 .0915 .0740

.1101 .1015 .0874 .0732 .0583

.0932 .0891 .0779 .0666 .0543

100 125 150 200 250

.1292 .1070 .0868 .0558 .0331

.1173 .0973 .0776 .0472 .0256

.1100 .0904 .0734 .0430 .0230

.0975 .0786 .0628 .0372 .0160

.0768 .0621 .0482 .0262 .0071

.0582 .0459 .0348 .0168 .0009

.0462 .0347 .0248 .0094 −.0037

.0419 .0326 .0228 .0070 −.0058

300

.0140

.0096

.0050

−.0013

−.0075

−.0129

−.0160

−.0171

*Extracted from Table 264, Smithsonian Physical Tables, 9th rev. ed., Washington, DC, 1954. These data are corrected from an earlier publication. µ in °C/atm.

TABLE 2-163

Approximate Inversion-Curve Locus for Propane

P, bar

0

25

50

75

100

125

150

175

200

225

250

275

TL, K

(296)*

303

311

318

327

336

345

355

365

374

389

403

P, bar

300

325

350

375

400

425

450

475

500

525

541

TL, K

418

435

452

473

495

521

551

586

628

686

780

*Hypothetical low-pressure limit.

2-136

PHYSICAL AND CHEMICAL DATA

CRITICAL CONSTANTS ADDITIONAL REFERENCES Other data and estimation techniques for the elements are contained in Gates and Thodos, Am. Inst. Chem. Eng. J., 6 (1960):50–54; and Ohse and von Tippelskirch, High Temperatures—High Pressures, 9 TABLE 2-164 Cmpd. no.

(1977):367–385. For inorganic substances see Mathews, Chem. Rev., 72 (1972):71–100; for organics see Kudchaker, Alani, and Zwolinski, Chem. Rev., 68 (1968):659–735; and for fluorocarbons see Advances in Fluorine Chemistry, App. B, Butterworth. Washington, 1963, pp. 173–175.

Critical Constants and Acentric Factors of Inorganic and Organic Compounds Name

Formula

CAS no.

Mol. wt.

Tc, K

Pc × 1E-06 Pa

Vc, m3/Kmol

Zc

Acentric factor

1 2 3 4 5 6 7 8 9 10

Methane Ethane Propane n-Butane n-Pentane n-Hexane n-Heptane n-Octane n-Nonane n-Decane

CH4 C2H6 C3H8 C4H10 C5H12 C6H14 C7H16 C8H18 C9H20 C10H22

74828 74840 74986 106978 109660 110543 142825 111659 111842 124185

16.043 30.070 44.097 58.123 72.150 86.177 100.204 114.231 128.258 142.285

190.564 305.32 369.83 425.12 469.7 507.6 540.2 568.7 594.6 617.7

4.59 4.85 4.21 3.77 3.36 3.04 2.72 2.47 2.31 2.09

0.099 0.146 0.200 0.255 0.315 0.373 0.428 0.486 0.540 0.601

0.286 0.279 0.273 0.272 0.271 0.269 0.259 0.254 0.252 0.245

0.011 0.098 0.149 0.197 0.251 0.304 0.346 0.396 0.446 0.488

11 12 13 14 15 16 17 18 19 20

n-Undecane n-Dodecane n-Tridecane n-Tetradecane n-Pentadecane n-Hexadecane n-Heptadecane n-Octadecane n-Nonadecane n-Eicosane

C11H24 C12H26 C13H28 C14H30 C15H32 C16H34 C17H36 C18H38 C19H40 C20H42

1120214 112403 629505 629594 629629 544763 629787 593453 629925 112958

156.312 170.338 184.365 198.392 212.419 226.446 240.473 254.500 268.527 282.553

639 658 675 693 708 723 736 747 758 768

1.95 1.82 1.68 1.57 1.47 1.41 1.34 1.26 1.21 1.17

0.658 0.718 0.779 0.830 0.888 0.943 0.998 1.059 1.119 1.169

0.242 0.239 0.233 0.226 0.222 0.221 0.219 0.214 0.215 0.215

0.530 0.577 0.617 0.643 0.685 0.721 0.771 0.806 0.851 0.912

21 22 23 24 25 26 27

2-Methylpropane 2-Methylbutane 2,3-Dimethylbutane 2-Methylpentane 2,3-Dimethylpentane 2,3,3-Trimethylpentane 2,2,4-Trimethylpentane

C4H10 C5H12 C6H14 C6H14 C7H16 C8H18 C8H18

75285 78784 79298 107835 565593 560214 540841

58.123 72.150 86.177 86.177 100.204 114.231 114.231

408.14 460.43 499.98 497.5 537.35 573.5 543.96

3.62 3.37 3.13 3.02 2.88 2.81 2.56

0.261 0.304 0.358 0.366 0.396 0.455 0.465

0.278 0.268 0.269 0.267 0.255 0.268 0.264

0.177 0.226 0.246 0.279 0.292 0.289 0.301

28 29 30 31 32 33 34 35

Ethylene Propylene 1-Butene cis-2-Butene trans-2-Butene 1-Pentene 1-Hexene 1-Heptene

C2H4 C3H6 C4H8 C4H8 C4H8 C5H10 C6H12 C7H14

74851 115071 106989 590181 624646 109671 592416 592767

28.054 42.081 56.108 56.108 56.108 70.134 84.161 98.188

282.34 365.57 419.95 435.58 428.63 464.78 504.03 537.29

5.03 4.63 4.04 4.24 4.08 3.56 3.14 2.82

0.132 0.188 0.241 0.233 0.237 0.295 0.354 0.413

0.283 0.286 0.279 0.273 0.272 0.271 0.265 0.261

0.086 0.137 0.190 0.204 0.216 0.236 0.280 0.330

36 37 38 39 40 41 42 43 44

1-Octene 1-Nonene 1-Decene 2-Methylpropene 2-Methyl-1-butene 2-Methyl-2-butene 1,2-Butadiene 1,3-Butadiene 2-Methyl-1,3-butadiene

C8H16 C9H18 C10H20 C4H8 C5H10 C5H10 C4H6 C4H6 C5H8

111660 124118 872059 115117 563462 513359 590192 106990 78795

112.215 126.242 140.269 56.108 70.134 70.134 54.092 54.092 68.119

566.65 593.25 616.4 417.9 465 471 452 425.17 484

2.57 2.33 2.21 3.98 3.45 3.38 4.36 4.30 3.85

0.460 0.528 0.584 0.238 0.292 0.292 0.220 0.220 0.277

0.251 0.249 0.252 0.272 0.261 0.252 0.255 0.268 0.265

0.377 0.417 0.478 0.192 0.237 0.272 0.166 0.192 0.158

45 46 47 48 49 50

Acetylene Methylacetylene Dimethylacetylene 3-Methyl-1-butyne 1-Pentyne 2-Pentyne

C2H2 C3H4 C4H6 C5H8 C5H8 C5H8

74862 74997 503173 598232 627190 627214

26.038 40.065 54.092 68.119 68.119 68.119

308.32 402.39 473.2 463.2 481.2 519

6.15 5.62 4.87 4.20 4.17 4.02

0.113 0.164 0.221 0.275 0.277 0.276

0.271 0.276 0.274 0.300 0.289 0.257

0.188 0.216 0.239 0.308 0.290 0.174

51 52 53 54 55 56

1-Hexyne 2-Hexyne 3-Hexyne 1-Heptyne 1-Octyne Vinylacetylene

C6H10 C6H10 C6H10 C7H12 C8H14 C4H4

693027 764352 928494 628717 629050 689974

82.145 82.145 82.145 96.172 110.199 52.076

516.2 549 544 559 585 454

3.64 3.53 3.54 3.13 2.82 4.89

0.322 0.331 0.334 0.386 0.441 0.205

0.273 0.256 0.261 0.260 0.256 0.265

0.335 0.221 0.219 0.272 0.323 0.109

CRITICAL CONSTANTS TABLE 2-164

2-137

Critical Constants and Acentric Factors of Inorganic and Organic Compounds (Continued ) Mol. wt.

Tc, K

Pc × 1E-06 Pa

287923 96377 1640897 110827 108872 590669 1678917 142290 693890 110838

70.134 84.161 98.188 84.161 98.188 112.215 112.215 68.119 82.145 82.145

511.76 532.79 569.52 553.58 572.19 591.15 609.15 507 542 560.4

4.50 3.78 3.40 4.10 3.48 2.94 3.04 4.81 4.13 4.39

C6H6 C7H8 C8H10 C8H10 C8H10 C8H10 C9H12

71432 108883 95476 108383 106423 100414 103651

78.114 92.141 106.167 106.167 106.167 106.167 120.194

562.16 591.8 630.33 617.05 616.23 617.2 638.32

1,2,4-Trimethylbenzene Isopropylbenzene 1,3,5-Trimethylbenzene p-Isopropyltoluene Naphthalene Biphenyl Styrene m-Terphenyl

C9H12 C9H12 C9H12 C10H14 C10H8 C12H10 C8H8 C18H14

95636 98828 108678 99876 91203 92524 100425 92068

120.194 120.194 120.194 134.221 128.174 154.211 104.152 230.309

82 83 84 85 86 87 88

Methanol Ethanol 1-Propanol 1-Butanol 2-Butanol 2-Propanol 2-Methyl-2-propanol

CH4O C2H6O C3H8O C4H10O C4H10O C3H8O C4H10O

67561 64175 71238 71363 78922 67630 75650

89 90 91 92 93 94 95 96

1-Pentanol 2-Methyl-1-butanol 3-Methyl-1-butanol 1-Hexanol 1-Heptanol Cyclohexanol Ethylene glycol 1,2-Propylene glycol

C5H12O C5H12O C5H12O C6H14O C7H16O C6H12O C2H6O2 C3H8O2

97 98 99 100

Phenol o-Cresol m-Cresol p-Cresol

C6H6O C7H8O C7H8O C7H8O

101 102 103 104 105 106 107 108 109 110 111 112

Dimethyl ether Methyl ethyl ether Methyl n-propyl ether Methyl isopropyl ether Methyl n-butyl ether Methyl isobutyl ether Methyl tert-butyl ether Diethyl ether Ethyl propyl ether Ethyl isopropyl ether Methyl phenyl ether Diphenyl ether

113 114 115 116 117 118 119 120 121 122

Formaldehyde Acetaldehyde 1-Propanal 1-Butanal 1-Pentanal 1-Hexanal 1-Heptanal 1-Octanal 1-Nonanal 1-Decanal

Cmpd. no.

Name

Zc

Acentric factor

57 58 59 60 61 62 63 64 65 66

Cyclopentane Methylcyclopentane Ethylcyclopentane Cyclohexane Methylcyclohexane 1,1-Dimethylcyclohexane Ethylcyclohexane Cyclopentene 1-Methylcyclopentene Cyclohexene

C5H10 C6H12 C7H14 C6H12 C7H14 C8H16 C8H16 C5H8 C6H10 C6H10

0.257 0.319 0.374 0.308 0.368 0.450 0.430 0.245 0.303 0.291

0.272 0.272 0.269 0.274 0.269 0.269 0.258 0.279 0.278 0.274

0.196 0.230 0.271 0.212 0.236 0.233 0.246 0.196 0.232 0.216

67 68 69 70 71 72 73

Benzene Toluene o-Xylene m-Xylene p-Xylene Ethylbenzene Propylbenzene

4.88 4.10 3.74 3.53 3.50 3.60 3.20

0.261 0.314 0.374 0.377 0.381 0.375 0.440

0.273 0.262 0.267 0.259 0.260 0.263 0.265

0.209 0.262 0.311 0.325 0.320 0.301 0.344

74 75 76 77 78 79 80 81

649.13 631.1 637.36 653.15 748.35 789.26 636 924.85

3.25 3.18 3.11 2.80 3.99 3.86 3.82 3.53

0.430 0.429 0.433 0.497 0.413 0.502 0.352 0.768

0.259 0.260 0.254 0.256 0.265 0.295 0.254 0.352

0.380 0.322 0.397 0.366 0.296 0.367 0.295 0.561

32.042 46.069 60.096 74.123 74.123 60.096 74.123

512.64 513.92 536.78 563.05 536.05 508.3 506.21

8.14 6.12 5.12 4.34 4.20 4.79 3.99

0.117 0.168 0.220 0.276 0.270 0.221 0.276

0.224 0.240 0.252 0.256 0.254 0.250 0.262

0.566 0.643 0.617 0.585 0.574 0.670 0.613

71410 137326 123513 111273 111706 108930 107211 57556

88.150 88.150 88.150 102.177 116.203 100.161 62.068 76.095

586.15 565 577.2 611.35 631.9 650 719.7 626

3.87 3.87 3.90 3.46 3.18 4.25 7.71 6.04

0.327 0.327 0.327 0.381 0.435 0.322 0.191 0.239

0.260 0.270 0.266 0.259 0.263 0.253 0.246 0.277

0.592 0.678 0.586 0.572 0.592 0.371 0.487 1.102

108952 95487 108394 106445

94.113 108.140 108.140 108.140

694.25 697.55 705.85 704.65

6.06 5.06 4.52 5.15

0.229 0.282 0.312 0.277

0.240 0.246 0.240 0.244

0.438 0.438 0.444 0.507

C2H6O C3H8O C4H10O C4H10O C5H12O C5H12O C5H12O C4H10O C5H12O C5H12O C7H8O C12H10O

115106 540670 557175 598538 628284 625445 1634044 60297 628320 625547 100663 101848

46.069 60.096 74.123 74.123 88.150 88.150 88.150 74.123 88.150 88.150 108.140 170.211

400.1 437.8 476.3 464.5 510 497 497.1 466.7 500.23 489 645.6 766.8

5.27 4.47 3.77 3.89 3.31 3.41 3.41 3.64 3.37 3.41 4.27 3.10

0.171 0.221 0.276 0.276 0.329 0.331 0.329 0.281 0.336 0.329 0.337 0.503

0.271 0.271 0.263 0.278 0.257 0.273 0.272 0.264 0.273 0.276 0.268 0.244

0.192 0.229 0.264 0.280 0.335 0.310 0.264 0.281 0.347 0.306 0.353 0.441

CH2O C2H4O C3H6O C4H8O C5H10O C6H12O C7H14O C8H16O C9H18O C10H20O

50000 75070 123386 123728 110623 66251 111717 124130 124196 112312

30.026 44.053 58.080 72.107 86.134 100.161 114.188 128.214 142.241 156.268

408 466 504.4 537.2 566.1 591 617 638.1 658 674.2

6.59 5.57 4.92 4.32 3.97 3.46 3.18 2.97 2.74 2.60

0.115 0.154 0.204 0.258 0.313 0.369 0.421 0.474 0.527 0.580

0.223 0.221 0.239 0.250 0.264 0.260 0.261 0.265 0.264 0.269

0.282 0.292 0.256 0.278 0.347 0.387 0.427 0.474 0.514 0.582

Formula

CAS no.

Vc, m3/Kmol

2-138

PHYSICAL AND CHEMICAL DATA

TABLE 2-164

Critical Constants and Acentric Factors of Inorganic and Organic Compounds (Continued ) Mol. wt.

Tc, K

Pc × 1E-06 Pa

67641 78933 107879 563804 591786 108101 565617 96220 565695 565800 108941 98862

58.080 72.107 86.134 86.134 100.161 100.161 100.161 86.134 100.161 114.188 98.145 120.151

508.2 535.5 561.08 553 587.05 571.4 573 560.95 567 576 653 709.5

4.71 4.12 3.71 3.84 3.31 3.27 3.32 3.70 3.34 3.06 4.01 3.85

CH2O2 C2H4O2 C3H6O2 C4H8O2 C4H8O2 C7H6O2 C4H6O3

64186 64197 79094 107926 79312 65850 108247

46.026 60.053 74.079 88.106 88.106 122.123 102.090

588 591.95 600.81 615.7 605 751 606

Methyl formate Methyl acetate Methyl propionate Methyl n-butyrate Ethyl formate Ethyl acetate Ethyl propionate

C2H4O2 C3H6O2 C4H8O2 C5H10O2 C3H6O2 C4H8O2 C5H10O2

107313 79209 554121 623427 109944 141786 105373

60.053 74.079 88.106 102.133 74.079 88.106 102.133

149 150 151 152 153 154 155

Ethyl n-butyrate n-Propyl formate n-Propyl acetate n-Butyl acetate Methyl benzoate Ethyl benzoate Vinyl acetate

C6H12O2 C4H8O2 C5H10O2 C6H12O2 C8H8O2 C9H10O2 C4H6O2

105544 110747 109604 123864 93583 93890 108054

156 157 158 159 160 161

Methylamine Dimethylamine Trimethylamine Ethylamine Diethylamine Triethylamine

CH5N C2H7N C3H9N C2H7N C4H11N C6H15N

162 163 164 165 166 167 168

n-Propylamine di-n-Propylamine Isopropylamine Diisopropylamine Aniline N-Methylaniline N,N-Dimethylaniline

169 170 171 172

Cmpd. no.

Name

Formula

Zc

Acentric factor

123 124 125 126 127 128 129 130 131 132 133 134

Acetone Methyl ethyl ketone 2-Pentanone Methyl isopropyl ketone 2-Hexanone Methyl isobutyl ketone 3-Methyl-2-pentanone 3-Pentanone Ethyl isopropyl ketone Diisopropyl ketone Cyclohexanone Methyl phenyl ketone

C3H6O C4H8O C5H10O C5H10O C6H12O C6H12O C6H12O C5H10O C6H12O C7H14O C6H10O C8H8O

0.210 0.267 0.301 0.313 0.369 0.369 0.371 0.336 0.369 0.416 0.311 0.386

0.234 0.247 0.239 0.261 0.250 0.254 0.259 0.267 0.262 0.266 0.230 0.252

0.307 0.320 0.345 0.349 0.395 0.389 0.386 0.340 0.394 0.411 0.308 0.365

135 136 137 138 139 140 141

Formic acid Acetic acid Propionic acid n-Butyric acid Isobutyric acid Benzoic acid Acetic anhydride

5.81 5.74 4.61 4.07 3.68 4.47 3.97

0.125 0.179 0.232 0.291 0.291 0.347 0.290

0.148 0.208 0.214 0.231 0.213 0.248 0.229

0.317 0.463 0.574 0.682 0.612 0.603 0.450

142 143 144 145 146 147 148

487.2 506.55 530.6 554.5 508.4 523.3 546

5.98 4.69 4.03 3.48 4.71 3.85 3.34

0.173 0.229 0.284 0.340 0.231 0.287 0.345

0.255 0.256 0.259 0.257 0.257 0.254 0.254

0.254 0.326 0.349 0.378 0.282 0.363 0.391

116.160 88.106 102.133 116.160 136.150 150.177 86.090

571 538 549.73 579.15 693 698 519.13

2.94 4.03 3.37 3.11 3.59 3.22 3.93

0.403 0.286 0.349 0.389 0.436 0.489 0.270

0.249 0.257 0.257 0.251 0.272 0.271 0.246

0.399 0.310 0.390 0.410 0.421 0.477 0.348

74895 124403 75503 75047 109897 121448

31.057 45.084 59.111 45.084 73.138 101.192

430.05 437.2 433.25 456.15 496.6 535.15

7.41 5.26 4.10 5.59 3.67 3.04

0.154 0.180 0.254 0.202 0.301 0.389

0.319 0.260 0.289 0.298 0.268 0.266

0.279 0.293 0.210 0.283 0.300 0.316

C3H9N C6H15N C3H9N C6H15N C6H7N C7H9N C8H11N

107108 142847 75310 108189 62533 100618 121697

59.111 101.192 59.111 101.192 93.128 107.155 121.182

496.95 550 471.85 523.1 699 701.55 687.15

4.74 3.11 4.54 3.20 5.35 5.19 3.63

0.260 0.401 0.221 0.417 0.270 0.373 0.465

0.298 0.273 0.256 0.307 0.248 0.332 0.295

0.280 0.446 0.276 0.388 0.381 0.480 0.403

Ethylene oxide Furan Thiophene Pyridine

C2H4O C4H4O C4H4S C5H5N

75218 110009 110021 110861

44.053 68.075 84.142 79.101

469.15 490.15 579.35 619.95

7.26 5.55 5.71 5.64

0.142 0.218 0.219 0.254

0.264 0.297 0.260 0.278

0.201 0.205 0.195 0.239

173 174 175 176 177 178 179 180

Formamide N,N-Dimethylformamide Acetamide N-Methylacetamide Acetonitrile Propionitrile n-Butyronitrile Benzonitrile

CH3NO C3H7NO C2H5NO C3H7NO C2H3N C3H5N C4H7N C7H5N

75127 68122 60355 79163 75058 107120 109740 100470

45.041 73.095 59.068 73.095 41.053 55.079 69.106 103.123

771 649.6 761 718 545.5 564.4 582.25 699.35

7.75 4.37 6.57 5.00 4.85 4.19 3.79 4.21

0.163 0.262 0.215 0.267 0.173 0.229 0.278 0.339

0.197 0.212 0.223 0.224 0.185 0.205 0.217 0.245

0.410 0.312 0.419 0.437 0.340 0.325 0.371 0.352

181 182 183 184 185 186 187 188 189

Methyl mercaptan Ethyl mercaptan n-Propyl mercaptan n-Butyl mercaptan Isobutyl mercaptan sec-Butyl mercaptan Dimethyl sulfide Methyl ethyl sulfide Diethyl sulfide

CH4S C2H6S C3H8S C4H10S C4H10S C4H10S C2H6S C3H8S C4H10S

74931 75081 107039 109795 513440 513531 75183 624895 352932

48.109 62.136 76.163 90.189 90.189 90.189 62.136 76.163 90.189

469.95 499.15 536.6 570.1 559 554 503.04 533 557.15

7.23 5.49 4.63 3.97 4.06 4.06 5.53 4.26 3.96

0.145 0.206 0.254 0.307 0.307 0.307 0.200 0.254 0.320

0.268 0.273 0.263 0.257 0.268 0.271 0.264 0.244 0.273

0.158 0.188 0.232 0.272 0.253 0.251 0.194 0.209 0.294

CAS no.

Vc, m3/Kmol

CRITICAL CONSTANTS TABLE 2-164 Cmpd. no.

2-139

Critical Constants and Acentric Factors of Inorganic and Organic Compounds (Concluded ) Name

Formula

CAS no.

Mol. wt.

Tc, K

Pc × 1E-06 Pa

Vc, m3/Kmol

Zc

Acentric factor

190 191 192 193 194 195 196 197

Fluoromethane Chloromethane Trichloromethane Tetrachloromethane Bromomethane Fluoroethane Chloroethane Bromoethane

CH3F CH3Cl CHCl3 CCl4 CH3Br C2H5F C2H5Cl C2H5Br

593533 74873 67663 56235 74839 353366 75003 74964

34.033 50.488 119.377 153.822 94.939 48.060 64.514 108.966

317.42 416.25 536.4 556.35 467 375.31 460.35 503.8

5.88 6.69 5.55 4.54 8.00 5.01 5.46 6.29

0.113 0.142 0.238 0.274 0.156 0.164 0.155 0.215

0.252 0.275 0.296 0.270 0.321 0.263 0.221 0.323

0.198 0.154 0.228 0.191 0.192 0.218 0.206 0.259

198 199 200 201 202 203 204 205

1-Chloropropane 2-Chloropropane 1,1-Dichloropropane 1,2-Dichloropropane Vinyl chloride Fluorobenzene Chlorobenzene Bromobenzene

C3H7Cl C3H7Cl C3H6Cl2 C3H6Cl2 C2H3Cl C6H5F C6H5Cl C6H5Br

540545 75296 78999 78875 75014 462066 108907 108861

78.541 78.541 112.986 112.986 62.499 96.104 112.558 157.010

503.15 489 560 572 432 560.09 632.35 670.15

4.58 4.51 4.24 4.23 5.75 4.54 4.53 4.52

0.247 0.247 0.292 0.291 0.179 0.269 0.308 0.324

0.270 0.274 0.266 0.259 0.287 0.262 0.265 0.263

0.228 0.196 0.253 0.256 0.106 0.247 0.251 0.251

206 207 208 209 210 211 212 213 214 215 216 217 218

Air Hydrogen Helium-4 Neon Argon Fluorine Chlorine Bromine Oxygen Nitrogen Ammonia Hydrazine Nitrous oxide

H2 He Ne Ar F2 Cl2 Br2 O2 N2 NH3 N2H4 N2O

132259100 1333740 7440597 7440019 7440371 7782414 7782505 7726956 7782447 7727379 7664417 302012 10024972

28.951 2.016 4.003 20.180 39.948 37.997 70.905 159.808 31.999 28.014 17.031 32.045 44.013

132.45 33.19 5.2 44.4 150.86 144.12 417.15 584.15 154.58 126.2 405.65 653.15 309.57

3.79 1.32 0.23 2.67 4.90 5.17 7.79 10.28 5.02 3.39 11.30 14.73 7.28

0.092 0.064 0.058 0.042 0.075 0.067 0.124 0.135 0.074 0.089 0.072 0.158 0.098

0.318 0.307 0.305 0.300 0.292 0.287 0.279 0.286 0.287 0.288 0.241 0.429 0.277

0.000 −0.215 −0.388 −0.038 0.000 0.053 0.073 0.128 0.020 0.037 0.253 0.315 0.143

219 220 221 222 223 224 225 226 227 228 229 230 231

Nitric oxide Cyanogen Carbon monoxide Carbon dioxide Carbon disulfide Hydrogen fluoride Hydrogen chloride Hydrogen bromide Hydrogen cyanide Hydrogen sulfide Sulfur dioxide Sulfur trioxide Water

NO C2N2 CO CO2 CS2 HF HCl HBr HCN H2S SO2 SO3 H2O

10102439 460195 630080 124389 75150 7664393 7647010 10035106 74908 7783064 7446095 7446119 7732185

30.006 52.036 28.010 44.010 76.143 20.006 36.461 80.912 27.026 34.082 64.065 80.064 18.015

180.15 400.15 132.92 304.21 552 461.15 324.65 363.15 456.65 373.53 430.75 490.85 647.13

6.52 5.94 3.49 7.39 8.04 6.49 8.36 8.46 5.35 9.00 7.86 8.19 21.94

0.058 0.195 0.095 0.095 0.160 0.069 0.082 0.100 0.139 0.099 0.123 0.127 0.056

0.252 0.348 0.300 0.277 0.280 0.117 0.253 0.280 0.195 0.287 0.269 0.255 0.228

0.585 0.276 0.048 0.224 0.118 0.383 0.134 0.069 0.407 0.096 0.244 0.423 0.343

All substances are listed in alphabetical order in Table 2-6a. Compiled from Daubert, T. E., R. P. Danner, H. M. Sibul, and C. C. Stebbins, DIPPR Data Compilation of Pure Compound Properties, Project 801 Sponsor Release, July, 1993, Design Institute for Physical Property Data, AIChE, New York, NY; and from Ambrose, D. “Vapour-Liquid Critical Properties”, Report Chem 107, National Physical Laboratory, Teddington, UK, October, 1979. In order to ensure thermodynamic consistency, in almost all cases these properties are calculated from Tc and the vapor pressure and liquid density correlation coefficients listed in those tables. This means that there will be slight differences between the values listed here and those in the DIPPR tables. Most of the differences are less than 1%, and almost all the rest are less than the estimated accuracy of the quantity in question. The atomic weights used, taken from J. Phys. Chem. Ref. Data 22(6), 1993, are C = 12.011, H = 1.00794, O = 15.9994, N = 14.00674, S = 32.066, F = 18.9984, Cl = 35.4527, Br = 79.904, and I = 126.90447. The value of the gas constant, R, used here is 8314.51 J/(kmol·K), as given by E. R. Cohen and B. N. Taylor in J. Phys. Chem. Ref. Data 17, 1988. K − 273.15 = °C; 1.8 × K − 459.67 = °F; Pa × 9.869233E-06 = atm; Pa × 1.450377E-04 = psia j; m3/kmol × (1E + 03/mol. wt.) = cm3/g; m3/kmol × (1.601846E + 01/mol wt) = ft3/lb.

2-140

PHYSICAL AND CHEMICAL DATA

COMPRESSIBILITIES reminded that compressibilities can be calculated from the pressure—volume (or density)—temperature tables of the subsection “Thermodynamic Properties.”

INTRODUCTION The increasing ranges of pressure and temperature of interest to technology for an ever-increasing number of substances would necessitate additional tables in this subsection as well as in the subsection “Thermodynamic Properties.” Space restrictions preclude this. Hence, in the present revision, an attempt was made to update the fluidcompressibility tables for selected fluids and to omit tables for other fluids. The reader is thus referred to the fourth edition for tables on miscellaneous gases at 0°C, acetylene, ammonia, ethane, ethylene, hydrogen-nitrogen mixtures, and methyl chloride. The reader is also

TABLE 2-165

UNITS CONVERSIONS For this subsection, the following units conversions are applicable: °R = 9⁄5 K. To convert bars to pounds-force per cubic inch, multiply by 14.504. To convert bars to kilopascals, multiply by 1 × 102.

Compressibility Factors for Air* Pressure, bar

Temp., K

1

5

10

20

40

60

80

100

150

200

250

300

400

500

75 80 90 100 120

0.0052 0.9764 0.9797 0.9880

0.0260 0.0250 0.0236 0.8872 0.9373

0.0519 0.0499 0.0471 0.0453 0.8660

0.1036 0.0995 0.0940 0.0900 0.6730

0.2063 0.1981 0.1866 0.1782 0.1778

0.3082 0.2958 0.2781 0.2635 0.2557

0.4094 0.3927 0.3686 0.3498 0.3371

0.5099 0.4887 0.4581 0.4337 0.4132

0.7581 0.7258 0.6779 0.6386 0.5964

1.0025 0.9588 0.8929 0.8377 0.7720

1.1931 1.1098 1.0395 0.9530

1.4139 1.3110 1.2227 1.1076

1.7161 1.5937 1.5091

2.1105 1.9536 1.7366

140 160 180 200 250

0.9927 0.9951 0.9967 0.9978 0.9992

0.9614 0.9748 0.9832 0.9886 0.9957

0.9205 0.9489 0.9660 0.9767 0.9911

0.8297 0.8954 0.9314 0.9539 0.9822

0.5856 0.7803 0.8625 0.9100 0.9671

0.3313 0.6603 0.7977 0.8701 0.9549

0.3737 0.5696 0.7432 0.8374 0.9463

0.4340 0.5489 0.7084 0.8142 0.9411

0.5909 0.6340 0.7180 0.8061 0.9450

0.7699 0.7564 0.7986 0.8549 0.9713

0.9114 0.8840 0.9000 0.9311 1.0152

1.0393 1.0105 1.0068 1.0185 1.0702

1.3202 1.2585 1.2232 1.2054 1.1990

1.5903 1.4970 1.4361 1.3944 1.3392

300 350 400 450 500

0.9999 1.0000 1.0002 1.0003 1.0003

0.9987 1.0002 1.0012 1.0016 1.0020

0.9974 1.0004 1.0025 1.0034 1.0034

0.9950 1.0014 1.0046 1.0063 1.0074

0.9917 1.0038 1.0100 1.0133 1.0151

0.9901 1.0075 1.0159 1.0210 1.0234

0.9903 1.0121 1.0229 1.0287 1.0323

0.9930 1.0183 1.0312 1.0374 1.0410

1.0074 1.0377 1.0533 1.0614 1.0650

1.0326 1.0635 1.0795 1.0913 1.0913

1.0669 1.0947 1.1087 1.1183 1.1183

1.1089 1.1303 1.1411 1.1463 1.1463

1.2073 1.2116 1.2117 1.2090 1.2051

1.3163 1.3015 1.2890 1.2778 1.2667

600 800 1000

1.0004 1.0004 1.0004

1.0022 1.0020 1.0018

1.0039 1.0038 1.0037

1.0081 1.0077 1.0068

1.0164 1.0157 1.0142

1.0253 1.0240 1.0215

1.0340 1.0321 1.0290

1.0434 1.0408 1.0365

1.0678 1.0621 1.0556

1.0920 1.0844 1.0744

1.1172 1.1061 1.0948

1.1427 1.1283 1.1131

1.1947 1.1720 1.1515

1.2475 1.2150 1.1889

*Calculated from values of pressure, volume (or density), and temperature in Vasserman, Kazavchinskii, and Rabinovich, Thermophysical Properties of Air and Air Components, Moscow, Nauka, 1966, and NBS-NSF Trans. TT 70-50095, 1971; and Vasserman and Rabinovich, Thermophysical Properties of Liquid Air and Its Components, Moscow, 1968, and NBS-NSF Trans. 69-55092, 1970.

TABLE 2-166

Compressibility Factors for Argon*

Temp., K

1

5

10

20

40

60

80

100

200

300

400

500

100 150 200 250 300

0.9773 0.9932 0.9972 0.9988 0.9995

0.0183 0.9647 0.9857 0.9935 0.9969

0.0366 0.9273 0.9713 0.9869 0.9941

0.0729 0.8447 0.9419 0.9741 0.9884

0.1449 0.6101 0.8810 0.9494 0.9777

0.2162 0.2249 0.8208 0.9263 0.9686

0.2867 0.2781 0.7624 0.9056 0.9611

0.3567 0.3324 0.7121 0.8877 0.9552

0.6975 0.5934 0.6870 0.8590 0.9533

1.0267 0.8387 0.8360 0.9207 0.9950

1.3470 1.0732 1.0051 1.0262 1.0673

1.6932 1.2995 1.1982 1.1479 1.1786

400 500 600 800 1000

1.0001 1.0002 1.0003 1.0003 1.0002

0.9997 1.0007 1.0012 1.0012 1.0013

0.9998 1.0012 1.0025 1.0023 1.0022

0.9999 1.0034 1.0046 1.0050 1.0050

1.0004 1.0071 1.0094 1.0102 1.0096

1.0018 1.0113 1.0143 1.0151 1.0142

1.0031 1.0154 1.0198 1.0205 1.0193

1.0056 1.0205 1.0250 1.0258 1.0239

1.0280 1.0501 1.0553 1.0532 1.0484

1.0656 1.0874 1.0904 1.0830 1.0736

1.1157 1.1301 1.1291 1.1147 1.0999

1.1976 1.1997 1.1933 1.1707 1.1497

Pressure, bar

*Calculated from PVT values tabulated in Rabinovich (ed.), Thermophysical Properties of Neon, Argon, Krypton and Xenon, Standard Press, Moscow, 1976. This book was published in English translation by Hemisphere, New York, 1988 (604 pp.).

COMPRESSIBILITIES TABLE 2-167

2-141

Compressibility Factors for Carbon Dioxide* Pressure, bar

Temp., °C

1

5

10

20

40

60

80

100

200

300

400

500

0 50 100 150 200

0.9933 0.9964 0.9977 0.9985 0.9991

0.9658 0.9805 0.9883 0.9927 0.9953

0.9294 0.9607 0.9764 0.9853 0.9908

0.8496 0.9195 0.9524 0.9705 0.9818

0.8300 0.9034 0.9416 0.9640

0.7264 0.8533 0.9131 0.9473

0.5981 0.8022 0.8854 0.9313

0.4239 0.7514 0.8590 0.9170

0.5891 0.7651 0.8649

0.6420 0.7623 0.8619

0.8235 0.8995

0.9098 0.9621

250 300 350 400 450

0.9994 0.9996 0.9998 0.9999 1.0000

0.9971 0.9982 0.9991 0.9997 1.0000

0.9943 0.9967 0.9983 0.9994 1.0003

0.9886 0.9936 0.9964 0.9989 1.0005

0.9783 0.9875 0.9938 0.9982 1.0013

0.9684 0.9822 0.9914 0.9979 1.0023

0.9593 0.9773 0.9896 0.9979 1.0038

0.9511 0.9733 0.9882 0.9984 1.0056

0.9253 0.9640 0.9895 1.0073 1.0070

0.9294 0.9746 1.0053 1.0266 1.0412

0.9508 1.0030 1.0340 1.0559 1.0709

1.0096 1.0464 1.0734 1.0928 1.1067

500 600 700 800 900

1.0000 1.0000 1.0003 1.0002 1.0002

1.0004 1.0007 1.0010 1.0009 1.0009

1.0008 1.0013 1.0017 1.0019 1.0020

1.0015 1.0030 1.0036 1.0040 1.0041

1.0035 1.0062 1.0073 1.0082 1.0083

1.0056 1.0093 1.0161 1.0122 1.0128

1.0079 1.0129 1.0155 1.0168 1.0171

1.0107 1.0168 1.0198 1.0212 1.0221

1.0282 1.0386 1.0436 1.0458 1.0463

1.0522 1.0648 1.0707 1.0731 1.0726

1.0820 1.0948 1.1000 1.1016 1.1012

1.1165 1.1277 1.1318 1.1324 1.1303

1000

1.0002

1.0009

1.0021

1.0042

1.0084

1.0128

1.0172

1.0218

1.0460

1.0725

1.0725

1.1274

*Calculated from density-pressure-temperature data in Vukalovitch and Altunin, Thermophysical Properties of Carbon Dioxide, Atomizdat, Moscow, 1965, and Collet’s, London, 1968, translation.

TABLE 2-168

Compressibility Factors for Carbon Monoxide*

Temp., K

1

4

7

10

40

70

100

200 250 300 350 400

0.9973 0.9989 0.9997 1.0000 1.0002

0.9893 0.9957 0.9987 1.0002 1.0010

0.9813 0.9926 0.9977 1.0003 1.0017

0.9734 0.9896 0.9968 1.0005 1.0025

0.9632 0.9907 1.0042 1.0042

0.9896 1.0112 1.0112

0.9935 1.0216 1.0216

450 500 600 700 800

1.0003 1.0004 1.0005 1.0005 1.0004

1.0014 1.0016 1.0018 1.0018 1.0017

1.0025 1.0029 1.0032 1.0032 1.0030

1.0035 1.0041 1.0045 1.0045 1.0044

1.0152 1.0172 1.0186 1.0183 1.0175

1.0285 1.0314 1.0332 1.0325 1.0309

1.0433 1.0469 1.0485 1.0470 1.0445

900 1000 1500 2000 2500

1.0004 1.0004 1.0003 1.0002 1.0002

1.0017 1.0016 1.0012 1.0009 1.0007

1.0029 1.0027 1.0021 1.0016 1.0013

1.0041 1.0039 1.0029 1.0022 1.0018

1.0166 1.0156 1.0115 1.0088 1.0071

1.0291 1.0273 1.0200 1.0155 1.0124

1.0418 1.0391 1.0286 1.0221 1.0178

3000

1.0002

1.0006

1.0010

1.0015

1.0059

1.0104

1.0148

Pressure, atm

*From Hilsenrath et al., N.B.S. Circ. 564, 1955. Some of the above values have been rounded to four decimal places. Values at 10-K increments below 1000 K and at 50 K increments for higher temperatures appear in the original, also for pressures below atmospheric.

TABLE 2-169

Compressibility Factors for Ethanol

Temp., K

0.1

0.5

1.013

10

20

50

100

250

500

300 350 400 450 500

0.0022

0.0023

0.0024

0.999 1.000 1.000

0.993 0.997 0.997

0.986 0.991 0.994

0.0229 0.0215 0.0204 0.908 0.941

0.0458 0.0411 0.0408 0.874

0.114 0.107 0.101 0.101 0.122

0.228 0.208 0.201 0.198 0.214

0.565 0.509 0.490 0.472 0.473

1.11 1.03 0.95 0.898 0.868

600 700 800 900 1000

1.000 1.000 1.000 1.000 1.000

0.998 0.999 1.000 1.000 1.000

0.997 0.999 0.999 1.000 1.000

0.972 0.985 0.992 0.996 0.998

0.943 0.971 0.984 0.992 0.997

0.948 0.973 0.988 0.993

0.672 0.902 0.953 0.981 0.990

0.470 0.760 0.890 0.962 1.002

0.868 0.921 0.988 1.04 1.08

Pressure, bar

Rounded and interpolated from Thermodynamics Research Center tables, Texas A&M University.

2-142

PHYSICAL AND CHEMICAL DATA TABLE 2-170

Compressibility Factors for Ethylene Temperature, K

Pressure, bar

110

150

200

250

300

350

400

450

500

1 5 10 15 20

0.0047 0.0237 0.0472 0.0710 0.0946

0.0038 0.0189 0.0378 0.0566 0.0754

0.9808 0.0162 0.0323 0.0484 0.0644

0.9902 0.9495 0.8946 0.8320 0.7578

0.9944 0.9717 0.9425 0.9121 0.8804

0.9966 0.9828 0.9659 0.9479 0.9299

0.9979 0.9894 0.9785 0.9679 0.9574

0.9986 0.9935 0.9867 0.9749 0.9734

0.9991 0.9959 0.9919 0.9876 0.9833

30 40 60 80 100

0.1418 0.1889 0.2831 0.3767 0.4702

0.1129 0.1504 0.2251 0.2994 0.3734

0.0963 0.1280 0.1910 0.2533 0.3150

0.0950 0.1251 0.1838 0.2410 0.2968

0.8122 0.7342 0.5235 0.3302 0.3480

0.8936 0.8560 0.7791 0.7023 0.6359

0.9357 0.9144 0.8730 0.9056 0.9220

0.9603 0.9477 0.9231 0.9009 0.8825

0.9754 0.9677 0.9541 0.9428 0.9321

150 200 250 300 400

0.7030 0.9337 1.1636 1.3917 1.8441

0.5567 0.7382 0.9179 1.0960 1.4475

0.4671 0.6161 0.7630 0.9075 1.1910

0.4324 0.5630 0.6904 0.8148 1.0565

0.4528 0.5641 0.6740 0.7816 0.9909

0.5842 0.6347 0.7110 0.7969 0.9726

0.7483 0.7499 0.7895 0.8479 0.9849

0.8523 0.8494 0.8710 0.9095 1.0142

0.9167 0.9184 0.9343 0.9631 1.0450

500

solid

1.7934

1.4679

1.2908

1.1932

1.1468

1.1304

1.1341

1.1436

Calculated from Jacobsen, R.T., M. Jahangiri, et al., Ethylene, Blackwell Sci. Publs., Oxford, 1988 (299 pp.).

TABLE 2-171

Compressibility Factors for Normal Hydrogen* Pressure, bar

Temp., K

1

10

20

40

60

80

100

200

400

600

800

1000

20 40 60 80 100

0.0169 0.9848 0.9955 0.9986 0.9998

0.1680 0.8340 0.9562 0.9776 0.9979

0.3302 0.6311 0.9169 0.9763 0.9976

0.6430 0.5240 0.8608 0.9655 1.0022

0.9434 0.6627 0.8498 0.9676 1.0133

1.2346 0.8118 0.8832 0.9842 1.0280

1.5166 0.9590 0.9432 1.0138 1.0528

2.844 1.650 1.347 1.257 1.225

2.878 2.158 1.834 1.659

3.993 2.902 2.389 2.095

5.034 3.598 2.907 2.512

6.019 4.263 3.404 2.902

200 300 400 500 600

1.0007 1.0005 1.0004 1.0004 1.0003

1.0066 1.0059 1.0048 1.0040 1.0034

1.0134 1.0117 1.0096 1.0080 1.0068

1.0275 1.0236 1.0192 1.0160 1.0136

1.0422 1.0357 1.0289 1.0240 1.0204

1.0575 1.0479 1.0386 1.0320 1.0272

1.0734 1.0603 1.0484 1.0400 1.0340

1.163 1.124 1.098 1.080 1.068

1.355 1.253 1.196 1.159 1.133

1.555 1.383 1.293 1.236 1.197

1.753 1.510 1.388 1.311 1.259

1.936 1.636 1.481 1.385 1.320

800 1000 2000

1.0002 1.0002 1.0009

1.0026 1.0021 1.0013

1.0052 1.0042 1.0023

1.0104 1.0084 1.0044

1.0156 1.0126 1.0065

1.0208 1.0168 1.0086

1.0259 1.0209 1.0107

1.051 1.041 1.021

1.100 1.080 1.040

1.147 1.117 1.057

1.193 1.153 1.073

1.237 1.187 1.088

*Calculated from PVT tables of McCarty, Hord, and Roder, NBS Monogr. 168, 1981.

TABLE 2-172

Compressibility Factors for KLEA 60

Temp., K

1

Pressure, bar 5

10

15

20

25

30

250 260 270 280 290

0.9687 0.9780 0.9803 0.9824 0.9848

0.9099 0.9199

300 310 320 330 340

0.9867 0.9872 0.9884 0.9894 0.9905

0.9284 0.9359 0.9425 0.9484 0.9537

0.8459 0.8637 0.8790 0.8908 0.9026

0.7800 0.8066 0.8299 0.8488

0.7577 0.7888

0.6700 0.7184

0.6305

350

0.9920

0.9582

0.9139

0.8663

0.8145

0.7570

0.6908

Zsat Tsat

0.9712 234.0

0.9022 273.1

0.8361 295.0

0.7777 309.5

0.7224 320.7

0.6677 329.8

0.6118 337.6

Zsat

Psat

0.9494 0.9315 0.9098 0.8839 0.8538

2.08 3.11 4.49 6.30 8.62

0.8175 0.7756 0.7261 0.6666

11.55 15.19 19.66 25.10

Converted and interpolated from “Thermodynamic Properties of KLEA 60,” British units, © ICI Chemicals and Polymers, 1993 (20 pp.). Reproduced by permission. KLEA 60 is R32/125/134a (20/40/40 wt %).

COMPRESSIBILITIES TABLE 2-173

Compressibility Factors for KLEA 61

Temp., K

1

2-143

Pressure, bar 5

10

15

20

25

30

250 260 270 280 290

0.9746 0.9773 0.9798 0.9787 0.9838

0.9067 0.9185

300 310 320 330 340

0.9854 0.9868 0.9881 0.9892 0.9903

0.9270 0.9348 0.9416 0.9481 0.9529

0.8431 0.8615 0.8772 0.8909 0.9027

0.7755 0.8042 0.8280 0.8484

0.7148 0.7518 0.7934

0.6659 0.7174

0.6312

350

0.9917

0.9577

0.9131

0.8653

0.8134

0.7565

0.6916

Zsat Tsat

0.9686 230.0

0.8944 269.0

0.8237 290.9

0.7602 305.5

0.7003 316.7

0.6399 325.9

0.5780 333.7

Zsat

Psat

0.9381 0.9172 0.8920 0.8622 0.8272

2.46 3.63 5.18 7.19 9.75

0.7868 0.7377 0.6801 0.6087

12.21 16.88 21.68 27.50

Converted and interpolated from “Thermodynamic Properties of KLEA 61,” British units, © ICI Chemicals and Polymers, 1993 (23 pp.). Reproduced by permission. KLEA 61 is R32/125/134a (10/70/20 wt %).

TABLE 2-174

Compressibility Factors for KLEA 66

Temp., K

1

Pressure, bar 5

10

15

20

25

30

250 260 270 280 290

0.974 0.9772 0.9796 0.9838 0.9858

0.9089 0.9209

300 310 320 330 340

0.9872 0.9883 0.9896 0.9907 0.9917

0.9287 0.9359 0.9431 0.9490 0.9540

0.8461 0.8663 0.8786 0.8910 0.9035

0.8056 0.8292 0.8492

0.7551 0.7878

0.7147

350

0.9926

0.9588

0.9137

0.8659

0.8127

0.7542

0.6843

Zsat Tsat

0.9719 236.1

0.9044 275.5

0.8397 297.6

0.7827 312.1

0.7289 323.5

0.6759 332.6

0.6220 340.4

Zsat

Psat

0.9541 0.9374 0.9172 0.8931 0.8645

1.89 2.84 4.12 5.81 7.98

0.8328 0.7920 0.7462 0.6918 0.6255

10.73 14.15 18.37 23.50 29.73

Converted and interpolated from “Thermodynamic properties of KLEA 66,” British units, © ICI Chemicals and Polymers, 1993 (20 pp.). Reproduced by permission. KLEA 66 is R32/125/134a (23/25/52 wt %).

TABLE 2-175

Compressibility Factors for Krypton*

Temp., K

1

5

10

20

40

60

80

100

200

300

400

500

150 200 250 300 350

0.9837 0.9933 0.9966 0.9982 0.9989

0.9155 0.9648 0.9841 0.9899 0.9949

0.0310 0.9278 0.9635 0.9800 0.9897

0.0618 0.8459 0.9265 0.9595 0.9793

0.1227 0.6039 0.8468 0.9197 0.9522

0.1829 0.1870 0.7605 0.8807 0.9415

0.2423 0.2393 0.6680 0.8437 0.9250

0.3012 0.2903 0.5810 0.8097 0.9110

0.5875 0.5313 0.5785 0.7337 0.8774

0.8636 0.7568 0.7461 0.7954 0.8992

1.1315 0.9730 0.9197 0.9302 0.9799

1.3932 1.1820 1.0891 1.0627 1.0664

400 450 500 600 800

0.9993 0.9998 0.9998 1.0000 1.0002

0.9967 0.9985 0.9992 1.0003 1.0010

0.9933 0.9969 0.9984 1.0005 1.0020

0.9867 0.9939 0.9970 1.0012 1.0041

0.9746 0.9886 0.9942 1.0025 1.0079

0.9635 0.9838 0.9921 1.0043 1.0122

0.9539 0.9800 0.9910 1.0064 1.0170

0.9459 0.9774 0.9906 1.0091 1.0214

0.9323 0.9663 1.0019 1.0301 1.0475

0.9570 1.0011 1.0311 1.0618 1.0779

1.0150 1.0543 1.0732 1.1000 1.1112

1.0910 1.1142 1.1258 1.1431 1.1147

1000

1.0002

1.0013

1.0023

1.0045

1.0091

1.0135

1.0184

1.0230

1.0486

1.0767

1.1063

1.1369

Pressure, bar

*Calculated from PVT values tabulated in Rabinovich (ed.), Thermophysical Properties of Neon, Argon, Krypton and Xenon, Standards Press, Moscow, 1976. This book was published in English translation by Hemisphere, New York, 1988 (604 pp.).

2-144

PHYSICAL AND CHEMICAL DATA

TABLE 2-176

Compressibility Factors for Methane (R50)*

Temp., K

1

5

10

20

40

60

80

100

200

300

400

500

150 200 250 300 350

0.9854 0.9936 0.9965 0.9983 0.9991

0.9225 0.9676 0.9838 0.9915 0.9954

0.8275 0.9339 0.9680 0.9830 0.9911

0.0714 0.8599 0.9352 0.9667 0.9825

0.1411 0.6784 0.8682 0.9343 0.9662

0.2093 0.3559 0.8020 0.9047 0.9520

0.2763 0.3172 0.7386 0.8783 0.9401

0.3423 0.3618 0.6854 0.8556 0.9306

0.6599 0.6141 0.6899 0.8280 0.9227

0.9623 0.8568 0.8554 0.9154 0.9800

1.2537 1.0887 1.0359 1.0432 1.0723

1.5363 1.3122 1.2155 1.1829 1.1804

400 450 500 600 800

0.9995 0.9997 0.9999 1.0000 1.0003

0.9977 0.9989 0.9997 1.0009 1.0017

0.9953 0.9979 0.9995 1.0020 1.0034

0.9912 0.9963 0.9995 1.0039 1.0068

0.9835 0.9935 0.9996 1.0081 1.0130

0.9772 0.9917 1.0005 1.0125 1.0197

0.9726 0.9911 1.0022 1.0171 1.0263

0.9696 0.9916 1.0048 1.0217 1.0330

0.9779 1.0098 1.0285 1.0540 1.0678

1.0245 1.0528 1.0699 1.0969 1.1068

1.0986 1.1152 1.1248 1.1470 1.1496

1.1859 1.1899 1.1899 1.2019 1.1951

1000

1.0004

1.0014

1.0035

1.0071

1.0141

1.0207

1.0274

1.0342

1.0678

1.1033

1.1400

1.1790

Pressure, bar

*Calculated from PVT values tabulated in Goodwin, NBS Tech. Note 653, 1974, for temperatures up to 500 K, and from PVT values tabulated in Zhuravlev. Thermophysical Properties of Gaseous and Liquid Methane, Standartov, Moscow, 1969, and NBS-NSF transl. TT 70-50097, 1970.

TABLE 2-177

Compressibility Factors for Methanol Pressure, bar

Temp., K

0.1

0.5

1.0133

10

20

50

100

150

200

250

300

400

500

200 250 300 350 400

0.0002 0.0002 0.9792 0.9844 0.9872

0.0011 0.0009 0.0008 0.9713 0.9795

0.0022 0.0019 0.0017 0.9551 0.9722

0.0219 0.0185 0.0164 0.0150 0.0142

0.0438 0.0370 0.0327 0.0298 0.0283

0.1091 0.0923 0.0813 0.0742 0.0702

0.2174 0.1837 0.1617 0.1473 0.1386

0.3250 0.2743 0.2413 0.2193 0.2056

0.4319 0.3643 0.3201 0.2904 0.2714

0.5381 0.4535 0.3981 0.3606 0.3362

0.6437 0.5422 0.4755 0.4301 0.4000

0.8531 0.7176 0.6284 0.5671 0.5253

1.6030 0.8909 0.7791 0.7016 0.6478

450 500 600 700 800

0.9890 0.9903 0.9922 0.9934 0.9964

0.9835 0.9859 0.9889 0.9907 0.9920

0.9792 0.9828 0.9867 0.9889 0.9904

0.9145 0.9525 0.9756 0.9816 0.9838

0.7989 0.9081 0.9643 0.9778 0.9818

0.0701 0.6799 0.9042 0.9541 0.9711

0.1366 0.1505 0.7629 0.8932 0.9411

0.2007 0.2110 0.6275 0.8392 0.9156

0.2629 0.2699 0.5255 0.8027 0.9025

0.3238 0.3271 0.4921 0.7797 0.8994

0.3834 0.3829 0.5010 0.7675 0.9026

0.4997 0.4912 0.5606 0.7713 0.9205

0.6128 0.5959 0.6358 0.7993 0.9485

Goodwin, R.D., J. Phys. Chem. Ref. Data, 16 (4), 799, 1987.

TABLE 2-178

Compressibility Factors for Neon*

Temp., K

1

5

10

20

40

60

80

100

200

300

400

500

50 100 150 200 250

0.9913 0.9993 1.0002 1.0003 1.0001

0.9472 0.9970 1.0017 1.0023 1.0022

0.9083 0.9949 1.0036 1.0049 1.0045

0.8013 0.9913 1.0078 1.0100 1.0097

0.3810 0.9854 1.0162 1.0204 1.0198

0.4398 0.9245 1.0262 1.0318 1.0295

0.4984 0.9864 1.0375 1.0427 1.0403

0.5850 0.9930 1.0497 1.0551 1.0502

0.9864 1.0796 1.1236 1.1191 1.1057

1.3659 1.2197 1.2131 1.1909 1.1633

1.7289 1.3796 1.3113 1.2655 1.2223

2.0794 1.5473 1.4150 1.3422 1.2822

300 400 500 600 800

1.0000 1.0000 1.0000 1.0000 1.0000

1.0020 1.0017 1.0014 1.0012 1.0009

1.0041 1.0036 1.0029 1.0024 1.0018

1.0091 1.0074 1.0058 1.0049 1.0043

1.0181 1.0151 1.0124 1.0107 1.0081

1.0277 1.0216 1.0188 1.0160 1.0123

1.0369 1.0301 1.0252 1.0214 1.0163

1.0469 1.0376 1.0316 1.0267 1.0206

1.0961 1.0771 1.0641 1.0542 1.0413

1.1476 1.1172 1.0963 1.0814 1.0622

1.1997 1.1575 1.1291 1.1091 1.0829

1.2520 1.1981 1.1621 1.1369 1.1039

1000

1.0000

1.0007

1.0014

1.0034

1.0068

1.0098

1.0132

1.0165

1.0330

1.0500

1.0670

1.0836

Pressure, bar

*Calculated from PVT values tabulated in Rabinovich (ed.), Thermophysical Properties of Neon, Argon, Krypton and Xenon, Standards Press, Moscow, 1976. This book was published in English translation by Hemisphere, New York, 1988 (604 pp.).

COMPRESSIBILITIES

2-145

TABLE 2-179

Compressibility Factors for Nitrogen*

Temp., K

1

5

10

20

40

60

80

100

200

300

400

500

70 80 90 100 120

0.0057 0.9593 0.9722 0.9798 0.9883

0.0287 0.0264 0.0251 0.8910 0.9397

0.0573 0.0528 0.0500 0.0487 0.8732

0.1143 0.1053 0.0996 0.0966 0.7059

0.2277 0.2093 0.1975 0.1905 0.1975

0.3400 0.3122 0.2938 0.2823 0.2822

0.4516 0.4140 0.3888 0.3720 0.3641

0.5623 0.5148 0.4826 0.4605 0.4438

1.1044 1.0061 0.9362 0.8840 0.8188

1.6308 1.4797 1.3700 1.2852 1.1684

Solid 1.9396 1.7890 1.6707 1.5015

Solid 2.3879 2.1962 2.0441 1.8223

140 160 180 200 250

0.9927 0.9952 0.9967 0.9978 0.9992

0.9635 0.9766 0.9846 0.9897 0.9960

0.9253 0.9529 0.9690 0.9791 0.9924

0.8433 0.9042 0.9381 0.9592 0.9857

0.6376 0.8031 0.8782 0.9212 0.9741

0.4251 0.7017 0.8125 0.8882 0.9655

0.4278 0.6304 0.7784 0.8621 0.9604

0.4799 0.6134 0.7530 0.8455 0.9589

0.7942 0.8107 0.8550 0.9067 1.0048

1.0996 1.0708 1.0669 1.0760 1.1143

1.3920 1.3275 1.2893 1.2683 1.2501

1.6726 1.5762 1.5105 1.4631 1.3962

300 350 400 450 500

0.9998 1.0001 1.0002 1.0003 1.0004

0.9990 1.0007 1.0011 1.0018 1.0020

0.9983 1.0011 1.0024 1.0033 1.0040

0.9971 1.0029 1.0057 1.0073 1.0081

0.9964 1.0069 1.0125 1.0153 1.0167

0.9973 1.0125 1.0199 1.0238 1.0257

1.0000 1.0189 1.0283 1.0332 1.0350

1.0052 1.0271 1.0377 1.0430 1.0451

1.0559 1.0810 1.0926 1.0973 1.0984

1.1422 1.1560 1.1609 1.1606 1.1575

1.2480 1.2445 1.2382 1.2303 1.2213

1.3629 1.3405 1.3216 1.3043 1.2881

600 800 1000

1.0004 1.0004 1.0003

1.0021 1.0017 1.0015

1.0040 1.0036 1.0034

1.0084 1.0074 1.0067

1.0173 1.0157 1.0136

1.0263 1.0237 1.0205

1.0355 1.0320 1.0275

1.0450 1.0402 1.0347

1.0951 1.0832 1.0714

1.1540 1.1264 1.1078

1.2028 1.1701 1.1449

1.2657 1.2140 1.1814

Pressure, bar

*Computed from pressure-volume-temperature tables in the Vasserman monographs referenced under Table 2-165. TABLE 2-180

Compressibility Factors for Oxygen* Pressure, bar

Temp., K

1

5

10

20

40

60

80

100

200

300

400

500

75 80 90 100 120

0.0043 0.0041 0.0038 0.9757 0.9855

0.0213 0.0203 0.0188 0.0177 0.9246

0.0425 0.0406 0.0376 0.0354 0.8367

0.0849 0.0811 0.0750 0.0705 0.0660

0.1693 0.1616 0.1494 0.1404 0.1302

0.2533 0.2418 0.2233 0.2096 0.1935

0.3368 0.3214 0.2966 0.2783 0.2558

0.4200 0.4007 0.3696 0.3464 0.3173

0.8301 0.7912 0.7281 0.6798 0.6148

1.2322 1.1738 1.0780 1.0040 0.8999

1.6278 1.5495 1.4211 1.3206 1.1762

2.0175 1.9196 1.7580 1.6309 1.4456

140 160 180 200 250

0.9911 0.9939 0.9960 0.9970 0.9987

0.9535 0.9697 0.9793 0.9853 0.9938

0.9034 0.9379 0.9579 0.9705 0.9870

0.7852 0.8689 0.9134 0.9399 0.9736

0.1334 0.6991 0.8167 0.8768 0.9477

0.1940 0.3725 0.7696 0.8140 0.9237

0.2527 0.2969 0.5954 0.7534 0.9030

0.3099 0.3378 0.5106 0.6997 0.8858

0.5815 0.5766 0.6043 0.6720 0.8563

0.8374 0.8058 0.8025 0.8204 0.9172

1.0832 1.0249 0.9990 0.9907 1.0222

1.3214 1.2364 1.1888 1.1623 1.1431

300 350 400 450 500

0.9994 0.9998 1.0000 1.0002 1.0002

0.9968 0.9990 1.0000 1.0007 1.0011

0.9941 0.9979 1.0000 1.0015 1.0022

0.9884 0.9961 1.0000 1.0024 1.0038

0.9771 0.9919 1.0003 1.0048 1.0075

0.9676 0.9890 1.0011 1.0074 1.0115

0.9597 0.9870 1.0022 1.0106 1.0161

0.9542 0.9870 1.0045 1.0152 1.0207

0.9560 1.0049 1.0305 1.0445 1.0523

0.9972 1.0451 1.0718 1.0859 1.0927

1.0689 1.1023 1.1227 1.1334 1.1380

1.1572 1.1722 1.1816 1.1859 1.1866

600 800 1000

1.0003 1.0003 1.0003

1.0014 1.0014 1.0013

1.0024 1.0026 1.0026

1.0052 1.0055 1.0053

1.0102 1.0109 1.0101

1.0153 1.0164 1.0149

1.0207 1.0219 1.0198

1.0266 1.0271 1.0253

1.0582 1.0565 1.0507

1.0961 1.0888 1.0783

1.1374 1.1231 1.1072

1.1803 1.1582 1.1369

*Calculated from pressure-volume-temperature tables in the Vasserman monographs listed under Table 2-165. TABLE 2-181

Compressibility Factors for Refrigerant 32* Pressure, bar

Temp., K

1

5

10

15

20

25

30

40

50

230 240 250 260 270

0.9656 0.9711 0.9755 0.9791 0.9819

0.8865 0.9036

280 290 300 310 320

0.9844 0.9864 0.9880 0.9894 0.9904

0.9180 0.9285 0.9376 0.9453 0.9518

0.8210 0.8476 0.8686 0.8358 0.8998

0.7899 0.8197 0.8436

0.7439 0.7812

0.7089

330 340 350

0.9914 0.9923 0.9932

0.9573 0.9619 0.9655

0.9118 0.9203 0.9296

0.8628 0.8790 0.8932

0.8102 0.8338 0.8534

0.7518 0.7846 0.8115

0.6851 0.7316 0.7671

0.6021 0.6675

0.5312

0.9595 221.2

0.8843 258.8

0.8202 279.8

0.7670 293.8

0.7191 304.6

0.6722 313.5

0.6303 321.1

0.5427 333.9

0.4467 344.3

Zsat Tsat

Zsat

Psat

0.9453 0.9278 0.9062 0.8811 0.8522

1.54 2.40 3.60 5.22 7.34

0.8194 0.7822 0.7401 0.6922 0.6370

10.07 13.51 17.76 22.95 29.21

0.5719 0.4905 0.3702

36.72 45.66 56.35

*Converted and interpolated from British units shown in Thermodynamic properties of KLEA 32, ICI Chemicals and Polymers, 1993. Reproduced by permission.

2-146

PHYSICAL AND CHEMICAL DATA

TABLE 2-182

Compressibility Factors for Refrigerant 123

Temp., °C

1

40 50 60 70 80

0.9639 0.9682 0.9717 0.9745 0.9766

0.9248 0.9327 0.9401

100 120 140 160 180

0.9804 0.9839 0.9861 0.9886 0.9908

0.9501 0.9591 0.9650 0.9714 0.9762

0.9197 0.9146 0.9282 0.9406 0.9518

0.8355 0.8667 0.8915 0.9077 0.9254

0.8140 0.8503 0.8747 0.8970

0.8023 0.8398 0.8709

0.7479 0.8026 0.8402

0.6916 0.7600 0.8072

0.7134 0.7712

0.6553 0.7346

200 225 250

0.9924 0.9938 0.9954

0.9806 0.9846 0.9885

0.9602 0.9692 0.9

0.9388 0.9526 0.9651

0.9174 0.9378 0.9528

0.8931 0.9170 0.9382

0.8688 0.8972 0.9229

0.8422 0.8800 0.9101

0.8163 0.8566 0.8930

Zsat T

0.9575 27.5

0.9210 55.4

0.9730 80.8

0.8292 97.9

0.7947 111.1

0.7654 122.0

0.7229 131.4

0.7110 139.7

Pressure, bar 2.5

5

7.5

10

12.5

15

17.5

20

22.5

25

Zsat

Psat

0.9427 0.9294 0.9134 0.8950 0.8727

1.54 2.13 2.96 3.78 4.90

0.6841

0.8262 0.7640 0.6890 0.5820 0.3926

7.87 12.01 17.59 24.92 34.54

0.7882 0.8401 0.8771

0.7539 0.8157 0.8581

— — —

— — —

0.6564 147.2

0.6206 154.0

0.5821 160.2

— —

— —

20

22.5

25

Dashes indicate inaccessible states; blanks indicate no available data. TABLE 2-183

Compressibility Factors for Refrigerant 124

Temp., °C

1

Pressure, bar 2.5

−20 −10 0 10 20

0.9573 0.9641 0.9693

30 40 50 60 80

0.9736 0.9675 0.9798 0.9820 0.9854

0.9313 0.9396 0.9473 0.9534 0.9633

100 120 140 160 180

0.9880 0.9899 0.9917

0.9700 0.9749 0.9794 0.9825

5

7.5

0.9468 −12.4

0.9071 11.9

12.5

15

17.5

0.8728 0.8889 0.9017 0.9226

0.8229 0.8462 0.8820

0.8366

0.9370 0.9478 0.9575 0.9645 0.9690

0.9040 0.9206 0.9357 0.9464 0.9536

0.8710 0.8935 0.9138 0.9247 0.9382

0.8314 0.8634 0.8884 0.9061 0.9213

0.7918 0.8329 0.8641 0.8868 0.9056

0.7463 0.8022 0.8391 0.8644 0.8857

0.6950 0.7682 0.8105 0.8489 0.8634

0.6380 0.7285 0.7896 0.8285 0.8574

0.6878 0.7647 0.7868 0.8379

0.9601

0.9471 0.9589 0.9650

0.9338 0.9488 0.9573

0.9211 0.9391 0.9443

0.9042 0.9223 0.9412

0.8951 0.9160 0.9333

0.8783 0.9040 0.9252

0.8647 0.8947 0.9174

0.8185 48.7

0.7830 60.2

0.7488 69.6

0.7157 77.8

0.6825 85.0

0.6484 91.4

0.6279 96.3

0.5788 102.6

200 225 250 Zsat T

10

0.8605 34.0

0.7251

Zsat

Psat

0.9562 0.9431 0.9284 0.9243 0.8920

0.72 1.10 1.63 2.34 3.27

0.8828 0.8427 0.8151 0.7803 0.7024

4.45 5.93 7.75 9.96 15.74

0.5955 0.3912

23.75 34.70

Dashes indicate inaccessible states; blanks indicate no available data. TABLE 2-184

Compressibility Factors for Refrigerant 134a Pressure, bar

Temp., °C

1

−10 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 satn. sat. T

0.9622 0.9710 0.9752 0.9778 0.9817 0.9839 0.9857 0.9872 0.9886 0.9897 0.9908 0.9916 0.9920 0.9924 0.9927 0.9929 0.9931 0.9567 −26.37

5

0.8819 0.8973 0.9098 0.9206 0.9296 0.9376 0.9442 0.9495 0.9543 0.9592 0.9638 0.9673 0.9691 0.9727 0.8741 15.74

10

0.8005 0.8280 0.8449 0.8678 0.8828 0.8954 0.9062 0.9151 0.9235 0.9308 0.9370 0.9428 0.7989 39.39

15

0.7361 0.7917 0.8137 0.8390 0.8555 0.8630 0.8802 0.8949 0.9040 0.8877 0.7017 55.23

Dashes indicate inaccessible states; blanks indicate no available data.

20

0.6853 0.7327 0.7682 0.7965 0.8144 0.8386 0.8553 0.8694 0.8817 0.6704 67.49

25

0.6290 0.6860 0.7335 0.7630 0.7915 0.8165 0.8350 0.8495 0.6094 77.57

30

0.5832 0.6557 0.7046 0.7418 0.7716 0.7964 0.8173 0.5415 86.20

40

0.5732 0.6249 0.6771 0.7169 0.7489 0.4442 100.35

50

Zsat

Psat

0.4530 0.4885 0.5645 0.6303 0.6783 — —

0.9316 0.9119 0.8888 0.8621 0.8314 0.7963 0.7560 0.7098 0.6562 0.5911 0.5054 0.3462 — — — — — — —

2.005 2.926 4.144 5.716 7.701 10.17 13.18 16.82 21.17 26.38 32.45 39.72 — — — — — — —

TABLE 2-185 Compressibility Factors for Water Substance (fps units)* Pressure, lb/in2 abs.

Temp., °F 400

600

800

1000

1200

1400

1600

1800

2000

2200

2400

2600

2800

3000

3200

3400

3600

3800

4000

10 15 20 40 60

0.9965 0.9943 0.9930 0.9861 0.9788

0.9989 0.9972 0.9970 0.9940 0.9910

0.9992 0.9986 0.9981 0.9967 0.9951

0.9995 0.9993 0.9991 0.9981 0.9973

0.9999 0.9997 0.9995 0.9990 0.9984

0.9999 0.9998 0.9996 0.9994 0.9991

0.9999 0.9999 0.9998 0.9996 0.9994

1.0000 0.9999 0.9999 0.9998 0.9997

1.0000 1.0000 1.0000 0.9999 0.9999

1.0000 1.0000 1.0000 0.9999 0.9999

1.0001 1.0001 1.0001 1.0001 1.0001

1.0006 1.0004 1.0003 1.0003 1.0003

1.0012 1.0012 1.0011 1.0010 1.0009

1.0024 1.0022 1.0020 1.0018 1.0018

1.0053 1.0042 1.0036 1.0028 1.0024

1.0084 1.0072 1.0065 1.0054 1.0048

1.0145 1.0124 1.0112 1.0090 1.0080

1.0211 1.0188 1.0173 1.0139 1.0120

1.0332 1.0295 1.0269 1.0214 1.0186

80 100 150 200 400

0.9714 0.9469 0.9435 0.9216

0.9878 0.9848 0.9770 0.9690 0.9356

0.9935 0.9919 0.9879 0.9839 0.9675

0.9963 0.9954 0.9931 0.9908 0.9817

0.9979 0.9974 0.9960 0.9947 0.9893

0.9987 0.9985 9.9976 0.9968 0.9935

0.9992 0.9990 0.9985 0.9980 0.9960

0.9996 0.9995 0.9993 0.9991 0.9982

0.9998 0.9998 0.9997 0.9996 0.9992

0.9999 0.9999 0.9998 0.9998 0.9998

1.0001 1.0001 1.0001 1.0001 1.0002

1.0003 1.0004 1.0004 1.0005 1.0007

1.0008 1.0007 1.0006 1.0007 1.0011

1.0016 1.0015 1.0014 1.0015 1.0017

1.0023 1.0022 1.0021 1.0021 1.0023

1.0044 1.0042 1.0039 1.0037 1.0033

1.0073 1.0067 1.0059 1.0055 1.0049

1.0108 1.0099 1.0087 1.0080 1.0070

1.0170 1.0157 1.0137 1.0126 1.0105

0.8989 0.8586 0.8138 0.6702

0.9509 0.9336 0.9162 0.8695 0.8188

0.9725 0.9633 0.9540 0.9305 0.9067

0.9839 0.9790 0.9733 0.9600 0.9468

0.9904 0.9872 0.9841 0.9764 0.9687

0.9942 0.9925 0.9905 0.9859 0.9813

0.9973 0.9964 0.9955 0.9932 0.9900

0.9988 0.9985 0.9981 0.9971 0.9958

0.9997 0.9996 0.9994 0.9992 0.9990

1.0002 1.0003 1.0004 1.0007 1.0010

1.0008 1.0010 1.0012 1.0017 1.0023

1.0014 1.0016 1.0019 1.0026 1.0034

1.0019 1.0022 1.0025 1.0033 1.0042

1.0026 1.0029 1.0032 1.0040 1.0049

1.0034 1.0036 1.0039 1.0048 1.0058

1.0048 1.0049 1.0052 1.0059 1.0068

1.0066 1.0065 1.0066 1.0072 1.0082

1.0097 1.0094 1.0092 1.0096 1.0104

0.5608

0.8060 0.7042 0.6185 0.5699

0.8942 0.8442 0.8003 0.7657

0.9392 0.9121 0.8883 0.8693

0.9647 0.9497 0.9371 0.9274

0.9836 0.9771 0.9714 0.9668

0.9930 0.9907 0.9895 0.9890

0.9989 0.9991 1.0004 1.0025

1.0024 1.0048 1.0075 1.0105

1.0050 1.0081 1.0118 1.0158

1.0069 1.0110 1.0152 1.0196

1.0082 1.0128 1.0172 1.0220

1.0093 1.0139 1.0188 1.0240

1.0106 1.0152 1.0204 1.0258

1.0118 1.0165 1.0216 1.0271

1.0132 1.0179 1.0229 1.0284

1.0149 1.0195 1.0242 1.0298

600 800 1,000 1,500 2,000 4,000 6,000 8,000 10,000

*Calculated by P. E. Liley from various steam tables for the lower temperatures and from Paper B-11 by P. H. Kesselman and Yu. I. Blank, 7th. Int. Conf. Properties of Steam, Tokyo, 1968, for the higher temperatures.

2-147

TABLE 2-186 Temperature, K

Compressibility Factors of Water Substance (SI units)* Pressure, bar 1

5

10

15

20

25

30

40

50

60

80

100

150

200

250

300

400

500

600

800

400 450 500 550 600 650

0.990 0.993 0.996 0.997 0.998 0.999

0.003 0.003 0.980 0.985 0.990 0.992

0.006 0.006 0.958 0.969 0.979 0.984

0.009 0.009 0.930 0.956 0.970 0.977

0.012 0.012 0.901 0.939 0.961 0.968

0.014 0.014 0.878 0.922 0.948 0.959

0.017 0.016 0.016 0.904 0.935 0.958

0.023 0.022 0.021 0.865 0.910 0.937

0.029 0.027 0.026 0.822 0.885 0.919

0.035 0.033 0.031 0.773 0.858 0.902

0.046 0.043 0.042 0.042 0.798 0.864

0.058 0.054 0.052 0.052 0.726 0.824

0.086 0.080 0.077 0.077 0.082 0.702

0.114 0.107 0.102 0.102 0.107 0.514

0.143 0.134 0.127 0.126 0.131 0.177

0.171 0.159 0.152 0.150 0.155 0.183

0.227 0.206 0.201 0.181 0.201 0.221

0.282 0.255 0.249 0.198 0.246 0.260

0.336 0.304 0.297 0.289 0.290 0.303

0.445 0.402 0.390 0.378 0.375 0.383

1000 0.552 0.498 0.482 0.464 0.457 0.460

700 750 800 850 900 950

1.000 1.000 1.000 1.000 1.000 1.000

0.994 0.996 0.997 0.997 0.998 0.998

0.988 0.991 0.993 0.995 0.997 0.997

0.984 0.988 0.991 0.992 0.993 0.994

0.976 0.981 0.985 0.989 0.992 0.994

0.967 0.975 0.982 0.984 0.989 0.993

0.966 0.971 0.976 0.981 0.986 0.991

0.952 0.961 0.970 0.977 0.982 0.985

0.941 0.955 0.966 0.973 0.979 0.983

0.929 0.945 0.957 0.967 0.974 0.980

0.900 0.927 0.945 0.957 0.965 0.973

0.876 0.907 0.929 0.946 0.958 0.967

0.800 0.856 0.892 0.917 0.936 0.950

0.716 0.801 0.853 0.889 0.915 0.933

0.618 0.743 0.813 0.860 0.893 0.916

0.503 0.682 0.773 0.831 0.872 0.901

0.326 0.557 0.693 0.775 0.830 0.867

0.316 0.465 0.620 0.715 0.792 0.839

0.340 0.435 0.568 0.679 0.760 0.816

0.406 0.456 0.538 0.631 0.714 0.780

0.476 0.509 0.561 0.629 0.700 0.761

1000 1200 1400 1600 1800 2000

1.000 1.000 1.000 1.000 1.001 1.003

0.999 1.000 1.000 1.000 1.001 1.002

0.998 0.999 1.000 1.000 1.001 1.002

0.995 0.998 1.000 1.000 1.000 1.002

0.995 0.998 1.000 1.000 1.000 1.002

0.994 0.997 1.000 1.000 1.000 1.002

0.993 0.997 1.000 1.000 1.000 1.002

0.990 0.995 0.999 1.000 1.000 1.002

0.987 0.994 0.998 1.000 1.000 1.002

0.985 0.994 0.998 1.000 1.001 1.003

0.978 0.992 0.998 1.000 1.002 1.003

0.973 0.990 0.997 1.000 1.003 1.004

0.960 0.986 0.996 1.001 1.003 1.004

0.948 0.982 0.995 1.002 1.004 1.006

0.935 0.975 0.995 1.002 1.005 1.008

0.923 0.968 0.994 1.004 1.008 1.011

0.900 0.961 0.993 1.006 1.011 1.014

0.878 0.957 0.992 1.009 1.014 1.018

0.859 0.949 0.994 1.012 1.017 1.021

0.831 0.942 0.996 1.015 1.021 1.032

0.816 0.937 0.998 1.020 1.031 1.043

*Calculated by P. E. Liley from various steam tables for the lower temperatures and from Pap. B-11 by P. H. Kesselman and Yu. I. Blank, 7th Internal Conference on the Properties of Steam, Tokyo, 1968, for the higher temperatures.

2-148

COMPRESSIBILITIES TABLE 2-187 Temperature, K

2-149

Compressibility Factors for Xenon* Pressure, bar 1

5

10

20

40

60

80

100

200

300

400

500

200 250 300 350 400

0.9831 0.9911 0.9949 0.9967 0.9977

0.9088 0.9545 0.9736 0.9834 0.9892

0.0293 0.9052 0.9465 0.9669 0.9183

0.0584 0.7887 0.8885 0.9322 0.9562

0.1162 0.1114 0.7517 0.8473 0.9128

0.1733 0.1642 0.5492 0.7840 0.8696

0.2300 0.2158 0.2794 0.7039 0.8278

0.2861 0.2663 0.3016 0.6249 0.7888

0.5601 0.5074 0.5021 0.5645 0.6916

0.8253 0.7355 0.6997 0.7124 0.7642

1.0833 0.9546 0.8886 0.8706 0.8850

1.3356 1.1670 1.0707 1.0269 1.0148

450 500 600 800 1000

0.9989 0.9982 0.9996 1.0000 1.0000

0.9928 0.9951 0.9979 0.9998 1.0004

0.9856 0.9902 0.9957 1.0002 1.0015

0.9714 0.9810 0.9917 1.0004 1.0031

0.9429 0.9623 0.9841 1.0012 1.0144

0.9163 0.9452 0.9772 1.0020 1.0101

0.8911 0.9293 0.9715 1.0034 1.0133

0.8679 0.9156 0.9667 1.0054 1.0172

0.7335 0.8774 0.9596 1.0213 1.0394

0.8331 0.8953 0.9791 1.0476 1.0669

0.9187 0.9572 1.0211 1.0818 1.0979

1.0224 1.0412 1.0799 1.1222 1.1331

*Calculated from PVT values tabulated in Rabinovich (ed.), Thermophysical Properties of Neon, Argon, Krypton and Xenon, Standards Press, Moscow, 1976. This book was published in English translation by Hemisphere, New York, 1988 (604 pp.).

TABLE 2-188

Compressibilities of Liquids*

At the constant temperature T, the compressibility β = (1/V 0)(dV/dP). In general as P increases, β decreases rapidly at first and then slowly; the change of β with T is large at low pressures but very small at pressures above 1000 to 2000 megabars. 1 megabar = 0.987 atm. = 106 dynes/cm2 based upon the older usage, 1 bar = 1 dyne/cm2. The use of the bar as a pressure unit is not encouraged.

Substance

Temp., °C

Pressure, megabars

Acetone Acetone Acetone Acetone Amyl alcohol alcohol, iso. alcohol, iso. alcohol, n alcohol, n alcohol, n alcohol, n Benzene Benzene Benzene Bromine Bromine Butyl alcohol, iso alcohol, iso alcohol, iso alcohol, iso alcohol, iso alcohol, iso Carbon bisulfide bisulfide bisulfide bisulfide tetrachloride tetrachloride Chloroform Chloroform Dichloroethylsulfide Dichloroethylsulfide Ethyl acetate acetate

14 20 20 40 14 20 20 20 20 20 40 17 20 20 20 20 18 20 20 20 20 20 16 20 20 20 20 20 20 20 32 32 13 20

23 500 1,000 12,000 23 200 400 500 1,000 12,000 12,000 5 200 400 200 400 8 200 400 500 1,000 12,000 21 500 1,000 12,000 200 400 200 400 1,000 2,000 23 200

Compressibility per megabar β × 106

Substance

111 61 52 9 88 84 70 61 46 8 8 89 77 67 56 51 97 81 64 56 46 8 86 57 48 6 86 73 83 70 34 24 103 90

Ethyl acetate alcohol alcohol alcohol alcohol bromide bromide bromide bromide bromide chloride chloride chloride chloride ether ether ether ether iodide iodide iodide iodide iodide Gallium Glycerol Hexane Hexane Kerosene Kerosene Kerosene Mercury Mercury Mercury Mercury

Temp., °C

Pressure, megabars

Compressibility per megabar β × 106

Substance

20 14 20 20 20 20 20 20 20 20 15 20 20 20 25 20 20 20 20 20 20 20 20 30 15 20 20 20 20 20 20 22 22 22

400 23 500 1,000 12,000 200 400 500 1,000 12,000 23 500 1,000 12,000 23 500 1,000 12,000 200 400 500 1,000 12,000 300 5 200 400 500 1,000 12,000 300 500 1,000 12,000

75 100 63 54 8 100 82 70 54 8 151 102 66 8 188 84 61 10 81 69 64 50 8 3.97 22 117 91 55 45 8 3.95 3.97 3.91 2.37

Methyl alcohol alcohol alcohol alcohol alcohol alcohol Nitric acid Oils: Almond Castor Linseed Olive Rapeseed Phosphorus trichloride trichloride trichloride trichloride Propyl alcohol (n) alcohol (n) alcohol (n?) alcohol (n?) alcohol (n?) Toluene Toluene Turpentine Water Water Water Water Water Water Water Xylene, meta meta

Temp., °C

Pressure, megabars

Compressibility per megabar β × 106

15 20 20 20 20 20 0

23 200 400 500 1,000 12,000 17

103 95 80 65 54 8 32

15 15 15 15 20 10 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 40 40 40 20 20

5 5 5 5

53 46 51 55 59 71 63 47 8 77 67 65 47 7 74 64 74 49 43 41 39 38 33 9 69 60

250 500 1,000 12,000 200 400 500 1,000 12,000 200 400 13 200 400 500 500 1,000 12,000 200 400

* Smithsonian Tables, Table 106. Scott (Cryogenic Engineering, Van Nostrand, Princeton, NJ, 1959) gives data for liquid nitrogen (p. 283), oxygen (p. 276), and hydrogen (p. 303). For a convenient index to the high-pressure work of Bridgman, see American Institute of Physics Handbook, p. 2-163, McGraw-Hill, New York, 1957.

2-150

PHYSICAL AND CHEMICAL DATA TABLE 2-189

Compressibilities of Solids

Many data on the compressibility of solids obtained prior to 1926 are contained in Gruneisen, Handbuch der Physik, vol. 10, Springer, Berlin, 1926, pp. 1–52; also available as translation, NASA RE 2-18-59W, 1959. See also Tables 271, 273, 276, 278, and other material in Smithsonian Physical Tables, 9th ed., 1954. For a review of high-pressure work to 1946, see Bridgman, Rev. Mod. Phys., 18, 1 (1946).

LATENT HEATS UNITS CONVERSIONS For this subsection, the following units conversions are applicable: °F = 9⁄ 5 °C + 32. To convert calories per gram-mole to British thermal units per

pound-mole, multiply by 1.799; to convert calories per gram to British thermal units per pound, multiply by 1.799. To convert millimeters of mercury to pounds-force per square inch, multiply by 1.934 × 10−2.

LATENT HEATS TABLE 2-190

2-151

Heats of Fusion and Vaporization of the Elements and Inorganic Compounds*

Unless stated otherwise, the values have been taken from the compilations by K. K. Kelley on “Heats of Fusion of Inorganic Compounds,” U.S. Bur. Mines Bull. 393 (1936), and “The Free Energies of Vaporization and Vapor Pressures of Inorganic Substances,” U.S. Bur. Mines Bull. 383 (1935).

Substance Aluminum Al Al2Br6 Al2Cl6 AlF3·3NaF Al2I6 Al2O3 Antimony Sb SbBr3 SbCl3 SbCl5 Sb4O6 Sb4S6 Argon A Arsenic As AsBr3 AsCl3 AsF5 As4O6 Barium Ba BaBr2 BaCl2 BaF2 Ba(NO3)2 Ba3(PO4)2 BaSO4 Beryllium Be Bismuth Bi BiBr3 BiCl3 Bi2O3 Bi2S5 Boron BBr3 BCl3 BF3 B2H6 B3H10 B5H9 B5H11 B10H14 B2H5Br B3N3H6 Bromine Br2 BrF5 Cadmium Cd CdBr2 CdCl2 CdF2 CdI2 CdO CdSO4 Calcium Ca CaBr2 CaCO3 CaCl2 CaF2 Ca(NO3)2 CaO CaO·Al2O3·2SiO2 CaO·MgO·2SiO2 CaO·SiO2 CaSO4 Carbon C (graphite) CBr4 CCl4 CF4 CH4 C2N2 CNBr CNCl

mp, °C

Heat of fusion,a,b cal/mole

660.0 97.5 192.5 1000 191.0 2045

2,550 5,420 16,960 16,380 7,960 (26,000)

2057 256.4 180.2c

61,020 10,920 26,750c

385.5 3000

15,360

630.5 97 73.4 4 655 546

4,770 3,510 3,030 2,400 (27,000) 11,200

1440

46,670

219 172d 1425

10,360 11,570 17,820

−185.8

1,590

610c

31,000c

122 −52.8 457.2

7,570 4,980 14,300

−189.3 814 31 −16 −80.7 313

290 (6,620) 2,810 2,420 2,800 8,000

704 847 960 1287 595 1730 1350

(1,400)e 6,000 5,370 3,000 (5,980) 18,600 9,700

1280

2,500e

271.3

2,505

224 817 747

2,600 6,800 8,900

−128 −165.5 −119.8 −46.9

480

99.7 −104 −58

7,800

−7.2 −61.3

2,580 1,355

320.9 568 568 1110 387

1,460 (5,000) 5,300 (5,400) 3,660

1000

4,790

851 730 1282 782 1392 561 2707 1550 1392 1512 1297

2,230 4,180 (12,700) 6,100 4,100 5,120 (12,240) 29,400 (18,200) 13,400 6,700

bp at 1 atm, °C

1638

35,670

1420 461 441

18,020 17,350

91.3 12.5 −100.9 −92.4 16 58 67 f 16 50.4

7,300 5,680 4,620 3,685 6,470 7,700 8,500 11,600 6,230 7,670

58.0 40.4

7,420 7,470

765

23,870

967

29,860

796 1559c

25,400 53,820c

1487

36,580

3600 90 −24.0

11,000e 1,050 644

−182.5 −27.8 52 −5

224 1,938u

77 −127.9 −161.4 −21.1

2,240

13

*See also subsection “Thermodynamic Properties.”

Heat of vaporization,a,b cal/mole

7,280 3,110 2,040 5,576u 11,010c 6,300

Substance Carbon (Cont.) CNF CNI CO CO2 COS COCl2 CS2 Cerium Ce Cesium Cs CsBr CsCl CsF CsI CsNO3 Chlorine Cl2 ClF ClF3 Cl2O ClO2 Cl2O7 Chromium Cr CrO2Cl2 Cobalt Co CoCl2 Copper Cu Cu2Br2 Cu2Cl2 CuI Cu2(CN)2 Cu2O CuO Cu2S Fluorine F2 F2O Gallium Ga Germanium Ge GeH4 Ge2H6 Ge3H8 GeHCl3 GeBr4 GeCl4 Ge(CH3)4 Gold Au Helium He Hydrogen H2 HBr HCl HCN HF (HF)6 HI H2O H22O (= D2O) H2O2 HNO3 H3PO2 H3PO3 H3PO4 H4P2O6 H2S H2S2 H2SO4 H2Se H2SeO4 H2Te Indium In

mp, °C

Heat of fusion,a,b cal/mole

−205.0 −57.5 −138.8

200 1,900 1,129 k

−112.0

1,049 l

775

2,120

28.4

500

642 715

3,600 (2,450)

407

3,250

−101.0

1,531m

1550

3,930

1490 727

3,660 7,390

1083.0

3,110

430

4,890

473 1230 1447 1127

(5,400) (13,400) 2,820 5,500

−223 29.8

bp at 1 atm, °C −72.8 141 −191.5 −78.4c −50.2 8.0

5,780c 13,980c 1,444 6,030 c, r 4,423 k 5,990

690 1300 1300 1251 1280

16,320 35,990 35,690 34,330 35,930

−34.1 −101 11.3 2.0 10.9 79

5,890 6,280 7,100 8,480

2475 117

8,250

1050

27,170

2595 1355 1490 1336

72,810 16,310 11,920 15,940

−188.2 −144.8 1,336

959 −165 −109 −105.6 −71 26.1 −49.5 −88

(8,300)

1063.0

3,030

−271.4 −259.2 −86.9 −114.2 −13.2 −83.0

28 575 476 2,009i 1,094

−50.8 0.0 3.8 −2 −47 17.4 74 42.4 55 −85.5 −87.6 10.5

686 1,436 1,501s 2,520c 600 2,310 3,070 2,520 8,300 568t 1,805 2,360

58 −48.9

3,450 1,670

156.4

781

Heat of vaporization,a,b cal/mole

4,878 m

1,640 2,650

2071 −89.1 31.4 110.6 75g 189 84 44 2966

3,580 5,900 7,550 8,000 8,560 7,030 6,460 81,800

−268.4

22

−252.7 −66.7 −85.0 25.7 33.3 51.2

216 4,210 3,860 6,027i 7,460 5,020

100.0 101.4 158

9,729 h,q 9,945 r,q 10,270

−60.3

4,463 t

−41.3

4,880

−2.2

5,650

2-152

PHYSICAL AND CHEMICAL DATA

TABLE 2-190

Substance Iodine I2 ICl(α) ICl(β) IF7 Iron Fe FeCl2 Fe2Cl6 Fe(CO)5 FeO FeS Krypton Kr Lead Pb PbBr2 PbCl2 PbF2 PbI2 PbMoO4 PbO PbS PbSO4 PbWO4 Lithium Li LiBO2 LiBr LiCl LiF LiI LiOH Li2MoO4 LiNO3 Li2SiO3 Li4SiO4 Li2SO4 Li2WO4 Magnesium Mg MgBr2 MgCl2 MgF2 MgO Mg3(PO4)2 MgSiO3 MgSO4 MgZn2 Manganese Mn MnCl2 MnSiO3 MnTiO3 Mercury Hg HgBr2 HgCl2 HgI2 HgSO4 Molybdenum Mo MoF6 MoO3 Neon Ne Nickel Ni NiCl2 Ni(CO)4 Ni2S Ni3S2 Nitrogen N2 NF3 NH3 NH4CNS NH4NO3 N2O NO N2O4 N2O5 NOCl Osmium OsF8 OsO4 (yellow) OsO4 (white) Oxygen O2 O3

Heats of Fusion and Vaporization of the Elements and Inorganic Compounds (Continued ) mp, °C 113.0 17.2 13.9

Heat of fusion,a,b cal/mole

bp at 1 atm, °C

Heat of vaporization,a,b cal/mole

3,650 2,660 2,270

183 4c

7,460c

1530 677 304 −21 1380 1195

3,560 7,800 20,590 3,250 (7,700) 5,000

2735 1026 319 105

84,600 30,210 12,040 9,000

−157

360e

152.9

10,390

2,310e

327.4 488 498 824 412 1065 890 1114 1087 1123

1,224 4,290 5,650 1,860 5,970 (25,800) 2,820 4,150 9,600 (15,200)

1744 914 954 1293 872

42,060 27,700 29,600 38,300 24,850

1472 1281

51,310 (50,000)

179 845 552 614 847 440 462 705

1,100 (5,570) 2,900 3,200 (2,360) (1,420) 2,480 4,200

1372

32,250

1310 1382 1681 1171

35,420 35,960 50,970 40,770

1177 1249 857 742

7,210 7,430 3,040 (6,700)

650 711 712 1221 2642 1184 1524 1127 589

2,160 8,300 8,100 5,900 18,500 (11,300) 14,700 3,500 (8,270)

1220 650 1274 1404

3,450 7,340 (8,200) (7,960)

2152 1190

557 3,960 4,150 4,500 (1,440)

361 319 304 354

13,980 14,080 14,080 14,260

(6,660) 2,500 (2,500)

(4800) 36 1151

(128,000) 6,000

−38.9 241 277 250 850 2622 17 745 −248.5

1107 1418

32,520 32,690

55,150 29,630

77

−246.0

440e

1455

4,200

2730 987c 42.5

87,300 48,360c 7,000

645 790

(2,980) 5,800 −195.8 −129.0 −33.4

1,336 3,000 5,581n

−88.5 −151.7 30 32.4 −6.4

3,950 3,307 7,040 13,800c 6,140

−210.0 −77.7 146 169.6 −90.8 −163.6 −13

56 42 −218.9

172 1,352n (4,700) 1,460 1,563 550 5,540

4,060 2,340 106

47.4 130

6,840 9,450

−183.0 −111

1,629 2,880

Substance Palladium Pd Phosphorus P4 (yellow) P4 (violet) P4 (black) PCl3 PH3 P4O6 P4O10(α) P4O10(β) POCl3 P2S3 Platinum Pt Potassium K KBO2 KBr KCl KCN KCNS K2CO3 K2CrO4 K2Cr2O7 KF KI K2MoO4 KNO3 KOH KPO3 K3PO4 K4P2O7 K2SO4 K2TiO3 K2WO4 Praseodymium Pr Radon Rn Rhenium Re Re2O7 Re2O8 Rubidium Rb RbBr RbCl RbF RbI RbNO3 Selenium Se2 Se6 SeF6 SeO2 SeOCl2 Silicon Si SiCl4 Si2Cl6 Si3Cl8 (SiCl3)2O SiF4 Si2F6 SiF3Cl SiF2Cl2 SiH4 Si2H6 Si3H8 Si4H10 SiH3Br SiH2Br2 SiHCl3 (SiH3)3N (SiH3)2O SiO2 (quartz) SiO2 (cristobalite) Silver Ag AgBr AgCl AgCN AgI AgNO3 Ag2S Ag2SO4 Sodium Na NaBO2

mp, °C

Heat of fusion,a,b cal/mole

1554

4,120

bp at 1 atm, °C

44.2

615

−133.8 23.8 569

270o 3,360 17,080

1.1

3,110

1773.5

4,700

(4400)

(107,000)

63.5 947 742 770 623 179 897 984 398 857 682 922 338 360 817 1340 1092 1074 810 927

574 (5,700) 5,000 6,410 (3,500) 2,250 7,800 6,920 8,770 6,500 4,100 (4,000) 2,840 (2,000) 2,110 8,900 14,000 8,100 (10,600) (4,400)

776

18,920

1383 1407

37,060 38,840

1324

34,690

1327

30,850

932

2,700

−71 (3000) 296 147

15,340 3,800

39.1 677 717 833 638 305

525 3,700 4,400 4,130 2,990 1,340

217

1,220

10

1,010

1427 −67.6 −1

9,470 1,845

−33

280 417c 453c 74.2 −87.7 174 591 358c 105.1 508

Heat of vaporization,a,b cal/mole

12,520 25,600c 33,100 7,280 3,489 o 10,380 20,670 8,380

−61.8

4,010

362.4

18,060

679 1352 1381 1408 1304 753 736 −45.8c 317c 168 2290 56.8 139 211.4 135.6 −94.8c −18.9c −70.1 −31.5 −111.6 −14.3 53.1 100 2.4 70.5 31.8 48.7 −15.4 2230

18,110 37,120 36,920 39,510 35,960 25,490 20,600 6,350c 20,900

6,860 12,340 8,820 6,130c 10,400c 4,460 5,080 2,960 5,110 6,780 8,890 5,650 6,840 6,360 6,850 5,350

−18.5 −138 −144 −185 −132.5 −117 −93.5 −93.8 −70.0 −126.5 −105.6 −144 1470 1700

3,900

960.5 430 455 350 557 209 842 657

2,700 2,180 3,155 2,750 2,250 2,755 3,360 (4,300)

2212

60,720

1564

42,520

1506

34,450

97.7 966

630 8,660

914

23,120

3,400 2,100

LATENT HEATS TABLE 2-190

2-153

Heats of Fusion and Vaporization of the Elements and Inorganic Compounds (Concluded )

Substance Sodium (Cont.) NaBr NaCl NaClO3 NaCN NaCNS Na2CO3 NaF NaI Na2MoO4 NaNO3 NaOH aNa2O·aAl2O3·3SiO2 NaPO3 Na4P2O7 Na2S Na2SiO3 Na2Si2O5 Na2SO4 Na2WO4 Strontium Sr SrBr2 SrCl2 SrF2 Sr3(PO4)2 Sulfur S (rhombic) S (monoclinic) S2Cl2 SF6 SO2 SO3(α) SO3(β) SO3(γ) SOBr2 SOCl2 SO2Cl2 Tellurium Te TeCl4 TeF6

mp, °C

Heat of fusion,a,b cal/mole

bp at 1 atm, °C

Heat of vaporization,a,b cal/mole

747 800 255 562 323 854 992 662 687 310 322 1107 988 970 920 1087 884 884 702

6,140 7,220 5,290 (4,400) 4,450 7,000 7,000 5,240 3,600 3,760 2,000 13,150 (5,000) (13,700) (1,200) 10,300 8,460 5,830 5,800

1392 1465

37,950 40,810

1500

37,280

757 643 872 1400 1770

2,190 4,780 4,100 4,260 18,500

1384

112.8 119.2 −75.5 17 32.4 62.2

453

1,769p 2,060 2,890 6,310

3,230

1704

53,260

1378

33,610

444.6

2,200

138 −63.5c −5.0 44.8

8,720 5,600c 5,960p 10,190

139.5 75.4 69.2

9,920 7,600 7,760

1090 392 −38.6c

16,830 6,700c

Substance Thallium Tl TlBr TlCl Tl2CO3 TlI TlNO3 Tl2S Tl2SO4 Tin Sn4 SnBr2 SnBr4 SnCl2 SnCl4 Sn(CH3)4 SnH4 SnI4 Titanium TiBr4 TiCl4 TiO2 Tungsten W WF6 Uranium UF6 Xenon Xe Zinc Zn ZnCl2 Zn(C2H5)2 ZnO ZnS Zirconium ZrBr4 ZrCl4 ZrI4 ZrO2

a

k

b

l

c

m

Values in parentheses are uncertain. For the freezing point or the normal boiling point unless otherwise stated. Sublimation. d Decomposes at about 75°C; value obtained by extrapolation. e Bichowsky and Rossini, “Thermochemistry of the Chemical Substances,” Reinhold, New York (1936). f Decomposes before the normal boiling point is reached. g Decomposes at about 40°C; value obtained by extrapolation. h See also pp. 2-304 through 2-307 on steam table. i Giauque and Ruehrwein, J. Am. Chem. Soc., 61 (1939): 2626. j Giauque and Egan, J. Chem. Phys., 5 (1937): 45.

TABLE 2-191

mp, °C

Alloys 30.5 Pb + 69.5 Sn 36.9 Pb + 63.1 Sn 63.7 Pb + 36.3 Sn 77.8 Pb + 22.2 Sn 1 Pb + 9 Sn 24 Pb + 27.3 Sn + 48.7 Bi 25.8 Pb + 14.7 Sn + 52.4 Bi + 7 Cd Silicates Anorthite (CaAl2Si2O8) Orthoclase (KAlSi2O8) Microcline (KAlSi3O8) Wollastonite (CaSiO8) Malacolite (Ca8MgSi4O12) Diopside (CaMgSi2O4) Olivine (Mg2SiO4) Fayalite (Fe2SiO4) Spermaceti Wax (bees’)

bp at 1 atm, °C

Heat of vaporization,a,b cal/mole

302.5 460 427 273 440 207 449 632

1,030 5,990 4,260 4,400 3,125 2,290 3,000 5,500

1457 819 807

38,810 23,800 24,420

823

25,030

231.8 232 30 247 −33.2

1,720 (1,700) 3,000 3,050 2,190

2270

68,000

−149.8 143.5

(4,300)

38.2 −23 1825

(2,060) 2,240 (11,400)

136

3390 −0.4

(8,400) 1,800

(5900) 17.3

(176,000) 6,350

55.1c

9,990c

−111.5 419.5 283 1975 1645

2715

740 1,595 (5,500)

623 113 78.3 −52.3

20,740 8,330 7,320 4,420

8,350

−108.0

3,110

907 732 118

27,430 28,710 8,960

357c 311c 431c

25,800c 25,290c 29,030c

4,470 (9,000)

20,800

Kemp and Giauque, J. Am. Chem. Soc., 59 (1937): 79. Brown and Manov, J. Am. Chem. Soc., 59 (1937): 500. Giauque and Powell, J. Am. Chem. Soc. 61 (1939): 1970. n Overstreet and Giauque, J. Am. Chem. Soc 59 (1937): 254. o Stephenson and Giauque, J. Chem. Phys., 5 (1937): 149. p Giauque and Stephenson, J. Am. Chem. Soc., 60 (1938): 1389. q Osborne, Stimson, and Ginnings, Bur. Standards J. Research, 23, 197 (1939): 261. r Miles and Menzies, J. Am. Chem. Soc., 58 (1936): 1067. s Long and Kemp, J. Am. Chem. Soc., 58 (1936): 1829. t Giauque and Blue, J. Am. Chem. Soc., 58 (1936): 831. u Ruehrwein and Giauque, J. Am. Chem. Soc., 61 (1939): 2940.

Heats of Fusion of Miscellaneous Materials

Material

Heat of fusion,a,b cal/mole

mp, °C

Heat of fusion, cal/g

183 179 177.5 176.5 236 98.8 75.5

17 15.5 11.6 9.54 28 6.85 8.4

43.9 61.8

100 100 83 100 94 100 130 85 37.0 42.3

2-154

PHYSICAL AND CHEMICAL DATA

TABLE 2-192

Heats of Fusion of Organic Compounds

The values for the hydrocarbons are from the tables of the American Petroleum Institute Research Project 44 at the National Bureau of Standards, with some from Parks and Huffman, Ind. Eng. Chem., 23, 1138 (1931). The values for the nonhydrocarbon compounds were recalculated from data in International Critical Tables, vol. 5. Hydrocarbon compounds Paraffins Methane Ethane Propane n-Butane 2-Methylpropane n-Pentane 2-Methylbutane 2,2-Dimethylpropane n-Hexane 2-Methylpentane 2,2-Dimethylbutane 2,3-Dimethylbutane n-Heptane 2-Methylhexane 3-Ethylpentane 2,2-Dimethylpentane 2,4-Dimethylpentane 3,3-Dimethylpentane 2,2,3-Trimethylbutane n-Octane 2-Methylheptane 3-Methylpentane 4-Methylheptane 2,2-Dimethylhexane 2,5-Dimethylhexane 3,3-Dimethylhexane 2-Methyl-3-ethylpentane 3-Methyl-3-ethylpentane 2,2,3-Trimethylpentane 2,2,4-Trimethylpentane 2,3,3-Trimethylpentane 2,3,4-Trimethylpentane 2,2,3,3-Tetramethylbutane n-Nonane n-Decane n-Undecane n-Dodecane Eicosane Pentacosane Tritriacontane Aromatics Benzene Methylbenzene (Toluene) Ethylbenzene o-Xylene m-Xylene p-Xylene n-Propylbenzene Isopropylbenzene 1-Methyl-2-ethylbenzene

Formula

mp, °C

Heat of fusion, cal/g

CH4 C2H6 C3H8 C4H10 C4H10 C5H12 C5H12 C5H12 C6H14 C6H14 C6H14 C6H14 C7H16 C7H16 C7H16 C7H16 C7H16 C7H16 C7H16 C8H18 C8H18 C8H18 C8H18 C8H18 C8H18 C8H18 C8H18 C8H18 C8H18 C8H18 C8H18 C8H18 C8H18 C9H20 C10H22 C11H24 C12H26 C20H42 C25H52 C33H68

−182.48 −183.23 −187.65 −138.33 −159.60 −129.723 −159.890 −16.6 −95.320 −153.680 −99.73 −128.41 −90.595 −118.270 −118.593 −123.790 −119.230 −134.46 −24.96 −56.798 −109.04 −120.50 −120.955 −121.18 −91.200 −126.10 −114.960 −90.870 −112.27 −107.365 −100.70 −109.210 +100.69 −53.9 −30.0 −25.9 −9.6 +36.4 +53.3 +71.1

14.03 22.712 19.100 19.167 18.668 27.874 17.076 10.786 36.138 17.407 1.607 2.251 33.513 21.158 22.555 13.982 15.968 16.856 5.250 43.169 21.458 23.795 22.692 24.226 26.903 14.9 23.690 22.657 18.061 19.278 3.204 19.392 14.900 41.2 48.3 34.1 51.3 52.0 53.6 54.0

C6H6 C7H8 C8H10 C8H10 C8H10 C8H10 C9H12 C9H12 C9H12

+5.533 −94.991 −94.950 −25.187 −47.872 +13.263 −99.500 −96.028 −80.833

30.100 17.171 20.629 30.614 26.045 38.526 16.97 19.22 21.13 Heat of fusion, cal/g

Formula

mp, °C

Acetic acid Acetone Acrylic acid Allo-cinnamic acid Aminobenzoic acid (o-) (m-) (p-) Amyl alcohol Anethole Aniline Anthraquinone Apiol Azobenzene Azoxybenzene

C2H4O2 C3H6O C3H4O2 C9H8O2 C7H7NO2 C7H7NO2 C7H7NO2 C5H12O C10H12O C6H5NH2 C14H8O2 C12H14O4 C12H10N2 C12H10N2O

16.7 −95.5 12.3 68 145 179.5 188.5 −78.9 22.5 −6.3 284.8 29.5 67.1 36

46.68 23.42 37.03 27.35 35.48 38.03 36.46 26.65 25.80 27.09 37.48 25.80 28.91 21.62

Benzil Benzoic acid Benzophenone Benzylaniline Bromocamphor Bromochlorbenzene (o-) (m-) (p-) Bromoiodobenzene (o-) (m-) (p-) Bromol hydrate Bromophenol (p-) Bromotoluene (p-)

C14H10O2 C7H8O2 C13H10O C13H13N C10H15BrO C6H4BrCl C6H4BrCl C6H4BrCl C6H4BrI C6H4BrI C6H4BrI C2H3Br3O2 C6H5BrO C7H7Br

95.2 122.45 47.85 32.37 78 −12.6 −21.2 64.6 21 9.3 90.1 46 63.5 28

22.15 33.90 23.53 21.86 41.57 15.41 15.29 23.41 12.18 10.27 16.60 16.90 20.50 20.86

Nonhydrocarbon compounds

Hydrocarbon compounds Aromatics—(Cont.) 1-Methyl-3-ethylbenzene 1-Methyl-4-ethylbenzene 1,2,3-Trimethylbenzene 1,2,4-Trimethylbenzene 1,3,5-Trimethylbenzene Naphthalene Camphene Durene Isodurene Prehnitene p-Cymene n-Butyl benzene tert-Butyl benzene β-Methyl naphthalene Diphenyl Hexamethyl benzene Diphenyl methane Anthracene Phenanthrene Tolane Stilbene Dibenzil Triphenyl methane Alkyl cyclohexanes Cyclohexane Methylcyclohexane Alkyl cyclopentanes Cyclopentane Methylcyclopentane Ethylcyclopentane 1,1-Dimethylcyclopentane cis-1,2-Dimethylcyclopentane trans-1,2-Dimethylcyclopentane trans-1,3-Dimethylcyclopentane Monoolefins Ethene (Ethylene) Propene (Propylene) 1-Butene cis-2-Butene trans-2-Butene 2-Methylpropene (isobutene) 1-Pentene cis-2-pentene trans-2-pentene 2-Methyl-1-butene 3-Methyl-1-butene 2-Methyl-2-butene Acetylenes Acetylene 2-Butyne (dimethylacetylene) Nonhydrocarbon compounds

mp, °C

Heat of fusion, cal/g

C9H12 C9H12 C9H12 C9H12 C9H12 C10H8 C10H12 C10H14 C10H14 C10H14 C10H14 C10H14 C10H14 C11H10 C12H10 C12H18 C13H12 C14H10 C14H10 C14H10 C14H12 C14H14 C19H16

−95.55 −62.350 −25.375 −43.80 −44.720 +80.0 +51 +79.3 −24.0 −7.7 −68.9 −88.5 −58.1 +34.1 +68.6 +165.5 +25.2 +216.5 +96.3 +60 +124 +51.4 +92.1

15.14 25.29 16.64 24.54 18.97 36.0 57 37.4 23.0 20.0 17.1 19.5 14.9 20.1 28.8 30.4 26.4 38.7 25.0 28.7 40.0 30.7 21.1

C6H12 C7H14

+6.67 −126.58

7.569 16.429

C5H10 C6H12 C7H14 C7H14 C7H14 C7H14 C7H14

−93.80 −142.445 −138.435 −69.73 −53.85 −117.57 −133.680

2.068 19.68 11.10 3.36 3.87 15.68 17.93

C2H4 C3H6 C4H8 C4H8 C4H8 C4H8 C5H10 C5H10 C5H10 C5H10 C5H10 C5H10

−169.15 −185.25 −185.35 −138.91 −105.55 −140.35 −165.27 −151.363 −140.235 −137.560 −168.500 −133.780

28.547 17.054 16.393 31.135 41.564 25.265 16.82 24.239 26.536 26.879 18.009 25.738

C2H2 C4H6

−81.5 −132.23

23.04 40.808

Formula

Formula

mp, °C

Heat of fusion, cal/g

Butyl alcohol (n-) (t-) Butyric acid (n-)

C4H10O C4H10O C4H8O2

−89.2 25.4 −5.7

29.93 21.88 30.04

Capric acid (n-) Caprylic acid (n-) Carbazole Carbon tetrachloride Carvoxime (d-) (l-) (dl-) Cetyl alcohol Chloracetic acid (α-) (β-) Chloral alcoholate hydrate Chloroaniline (p-) Chlorobenzoic acid (o-) (m-) (p-) Chloronitrobenzene (m-) (p-) Cinnamic acid anhydride Cresol (p-) Crotonic acid (α-) (cis-) Cyanamide Cyclohexanol

C10H20O2 C8H16O2 C12H9N CCl4 C10H15NO C10H15NO C10H15NO C16H34O C2H3ClO2 C2H3ClO2 C4H7Cl3O2 C2H3Cl3O2 C6H6ClN C7H5ClO2 C7H5ClO2 C7H5ClO2 C6H4ClNO2 C6H4ClNO2 C9H8O2 C18H14O3 C7H8O C4H6O2 C4H6O2 CH2N2 C6H12O

31.99 16.3 243 −22.8 71.5 71 91 49.27 61.2 56 9 47.4 71 140.2 154.25 239.7 44.4 83.5 133 48 34.6 72 71.2 44 25.46

38.87 35.40 42.05 41.57 23.29 23.41 24.61 33.80 31.06 35.12 24.03 33.18 37.15 39.30 36.41 49.21 29.38 31.51 36.50 28.14 26.28 25.32 34.90 49.81 4.19

LATENT HEATS TABLE 2-192

2-155

Heats of Fusion of Organic Compounds (Concluded ) Heat of fusion, cal/g

Formula

mp, °C

Dibromobenzene (o-) (m-) (p-) Dibromophenol (2, 4-) Dichloroacetic acid Dichlorobenzene (o-) (m-) ( p-) Dihydroxybenzene (o-) (m-) (p-) Di-iodobenzene (o-) (m-) (p-) Dimethyl tartrate (dl-) (d-) pyrone Dinitrobenzene (o-) (m-) ( p-) Dinitrotoluene (2, 4-) Dioxane Diphenyl amine

C6H4Br2 C6H4Br2 C6H4Br2 C6H4Br2O C2H2Cl2O2 C6H4Cl2 C6H4Cl2 C6H4Cl2 C6H6O2 C6H6O2 C6H6O2 C6H4I2 C6H4I2 C6H4I2 C6H10O6 C6H10O6 C7H8O2 C6H4N2O4 C6H4N2O4 C6H4N2O4 C7H6N2O4 C4H8O2 C12H11N

1.8 −6.9 86 12 −4(?) −16.7 −24.8 53.13 104.3 109.65 172.3 23.4 34.2 129 87 49 132 116.93 89.7 173.5 70.14 11.0 52.98

12.78 13.38 20.55 13.97 14.21 21.02 20.55 29.67 49.40 46.20 58.77 10.15 11.54 16.20 35.12 21.50 56.14 32.25 24.70 39.99 26.40 34.85 25.23

Elaidic acid Ethyl acetate alcohol Ethylene dibromide Ethyl ether

C18H34O2 C4H8O2 C2H6O C2H4Br2 C4H10O

44.4 83.8 −114.4 10.012 −116.3

52.08 28.43 25.76 13.52 23.54

Formic acid

CH2O2

Glutaric acid Glycerol Glycol, ethylene

Nonhydrocarbon compounds

8.40

58.89

C6H8O4 C3H8O3 C2H6O2

97.5 18.07 −11.5

37.39 47.49 43.26

Hydrazo benzene Hydrocinnamic acid Hydroxyacetanilide

C12H12N2 C9H10O2 C8H9NO2

134 48 91.3

22.89 28.14 33.59

Iodotoluene (p-) Isopropyl alcohol ether

C7H7I C3H8O C6H14O

34 −88.5 −86.8

18.75 21.08 25.79

Lauric acid (n-) Levulinic acid

C12H24O2 C5H8O3

43.22 33

43.72 18.97

Menthol (l-) (α) Methyl alcohol Myristic acid Methyl cinnamate fumarate oxalate phenylpropiolate succinate

C10H20O CH4O C14H28O2 C10H10O2 C6H8O4 C4H6O4 C10H8O2 C6H10O4

43.5 −97.8 53.86 36 102 54.35 18 19.5

18.63 23.7 47.49 26.53 57.93 42.64 22.86 35.72

Formula

mp, °C

Heat of fusion, cal/g

Naphthol (α-) (β-) Naphthylamine (α-) Nitroaniline (o-) (m-) (p-) Nitrobenzene Nitrobenzoic acid (o-) (m-) (p-) Nitronaphthalene Nitrophenol (o-)

C10H8O C10H8O C10H9N C6H6N2O2 C6H6N2O2 C6H6N2O2 C6H5NO2 C7H5NO4 C7H5NO4 C7H5NO4 C10H7NO2 C6H5NO3

95.0 120.6 50 71.2 114.0 147.3 5.85 145.8 141.1 239.2 56.7 45.13

38.94 31.30 22.34 27.88 40.97 36.46 22.52 40.06 27.59 52.80 25.44 26.76

Palmitic acid Paraldehyde Pelargic acid (n-) (β-) Pelargonic acid (n-) (α-) Phenol Phenylacetic acid Phenylhydrazine Propyl ether (n)

C16H32O2 C6H12O3 C9H18O2 C9H18O2 C6H6O C8H8O2 C6H8N2 C6H14O

61.82 10.5 12.35 40.92 76.7 19.6 −126.1

39.18 25.02 39.04 30.63 29.03 25.44 36.31 20.66

Quinone

C6H4O2

115.7

40.85

Stearic acid Succinic anhydride Succinonitrile

C18H30O2 C4H4O3 C4H4N2

68.82 119 54.5

47.54 48.74 11.71

Tetrachloroxylene (o-) (p-) Thiophene Thiosinamine Thymol Toluic acid (o-) (m-) (p-) Toluidine (p-) Tribromophenol (2, 4, 6-) Trichloroacetic acid Trinitroglycerol Trinitrotoluene (2, 4, 6-) Tristearin

C8H6Cl4 C8H6Cl4 C4H4S C4H8N2S C10H14O C8H8O2 C8H8O2 C8H8O2 C7H9N C6H3Br3O C2HCl3O2 C3H5N3O9 C7H5N3O6 C57H110O6

86 95 −39.4 77 51.5 103.7 108.75 179.6 43.3 93 57.5 12.3 80.83 70.8, 54.5

21.02 22.10 14.11 33.45 27.47 35.40 27.59 39.90 39.90 13.38 8.60 23.02 22.34 45.63

Undecylic acid (α-) (n-) (β-) (n-) Urethane

C11H22O2 C11H22O2 C3H7NO2

28.25 48.7

32.20 42.91 40.85

Veratrol

C8H10O2

22.5

27.45

Xylene dibromide (o-) (m-) dichloride (o-) (m-) (p-)

C8H8Br2 C8H8Br2 C8H8Cl2 C8H8Cl2 C8H8Cl2

Nonhydrocarbon compounds

95 77 55 34 100

24.25 21.45 29.03 26.64 32.73

TABLE 2-193 Cmpd. no.

Heats of Vaporization of Inorganic and Organic Compounds Name

Formula

CAS no.

Mol wt

C1 × 1E−07

C2

C3

C4

Tmin, K

∆Hv at Tmin × 1E−07

Tmax, K

∆Hv at Tmax

1 2 3 4 5 6 7 8 9 10

Methane Ethane Propane n-Butane n-Pentane n-Hexane n-Heptane n-Octane n-Nonane n-Decane

CH4 C2H6 C3H8 C4H10 C5H12 C6H14 C7H16 C8H18 C9H20 C10H22

74828 74840 74986 106978 109660 110543 142825 111659 111842 124185

16.043 30.070 44.097 58.123 72.150 86.177 100.204 114.231 128.258 142.285

1.0194 2.1091 2.9209 3.6238 3.9109 4.4544 5.0014 5.5180 6.0370 6.6126

0.26087 0.60646 0.78237 0.8337 0.38681 0.39002 0.38795 0.38467 0.38522 0.39797

−0.14694 −0.55492 −0.77319 −0.82274 0 0 0 0 0 0

0.22154 0.32799 0.39246 0.39613 0 0 0 0 0 0

90.69 90.35 85.47 134.86 143.42 177.83 182.57 216.38 219.66 243.51

0.8724 1.7879 2.4787 2.8684 3.3968 3.7647 4.2619 4.5898 5.0545 5.4168

190.56 305.32 369.83 425.12 469.7 507.6 540.2 568.7 594.6 617.7

0 0 0 0 0 0 0 0 0 0

11 12 13 14 15 16 17 18 19 20

n-Undecane n-Dodecane n-Tridecane n-Tetradecane n-Pentadecane n-Hexadecane n-Heptadecane n-Octadecane n-Nonadecane n-Eicosane

C11H24 C12H26 C13H28 C14H30 C15H32 C16H34 C17H36 C18H38 C19H40 C20H42

1120214 112403 629505 629594 629629 544763 629787 593453 629925 112958

156.312 170.338 184.365 198.392 212.419 226.446 240.473 254.500 268.527 282.553

7.2284 7.7337 8.4339 9.0539 9.6741 10.1560 10.4730 10.9690 11.6740 12.8600

0.40607 0.40681 0.4257 0.44467 0.45399 0.45726 0.4374 0.44327 0.45865 0.50351

0 0 0 0 0 0 0 0 0 0.32986

0 0 0 0 0 0 0 0 0 −0.42184

247.57 263.57 267.76 279.01 283.07 291.31 295.13 301.31 305.04 309.58

5.9240 6.2802 6.8015 7.2002 7.6728 8.0225 8.3699 8.7246 9.2185 9.5933

639 658 675 693 708 723 736 747 758 768

0 0 0 0 0 0 0 0 0 0

21 22 23 24 25 26 27

2-Methylpropane 2-Methylbutane 2,3-Dimethylbutane 2-Methylpentane 2,3-Dimethylpentane 2,3,3-Trimethylpentane 2,2,4-Trimethylpentane

C4H10 C5H12 C6H14 C6H14 C7H16 C8H18 C8H18

75285 78784 79298 107835 565593 560214 540841

58.123 72.150 86.177 86.177 100.204 114.231 114.231

3.1667 3.7700 4.1404 4.2780 4.6536 4.9910 4.7721

0.3855 0.3952 0.38124 0.384 0.37579 0.383 0.37992

0 0 0 0 0 0 0

0 0 0 0 0 0 0

113.54 113.25 145.19 119.55 160 172.22 165.78

2.7927 3.3720 3.6328 3.8495 4.0747 4.3530 4.1565

408.14 460.43 499.98 497.5 537.35 573.5 543.96

0 0 0 0 0 0 0

28 29 30 31 32 33 34 35

Ethylene Propylene 1-Butene cis-2-Butene trans-2-Butene 1-Pentene 1-Hexene 1-Heptene

C2H4 C3H6 C4H8 C4H8 C4H8 C5H10 C6H12 C7H14

74851 115071 106989 590181 624646 109671 592416 592767

28.054 42.081 56.108 56.108 56.108 70.134 84.161 98.188

2.8694 3.2300 3.4190 3.3320 3.7740 4.3236 4.8120

0.3746 0.8375 0.3747 0.3754 0.3736 0.37647 0.3788 0.3685

0 −0.9216 0 0 0 0 0 0

0 0.5012 0 0 0 0 0 0

104 87.89 87.8 134.26 167.62 107.93 133.39 154.27

1.6025 2.4031 2.9582 2.9773 2.7684 3.4166 3.8483 4.2478

282.34 365.57 419.95 435.58 428.63 464.78 504.03 537.29

0 0 0 0 0 0 0 0

36 37 38 39 40 41 42 43 44

1-Octene 1-Nonene 1-Decene 2-Methylpropene 2-Methyl-1-butene 2-Methyl-2-butene 1,2-Butadiene 1,3-Butadiene 2-Methyl-1,3-butadiene

C8H16 C9H18 C10H20 C4H8 C5H10 C5H10 C4H6 C4H6 C5H8

111660 124118 872059 115117 563462 513359 590192 106990 78795

112.215 126.242 140.269 56.108 70.134 70.134 54.092 54.092 68.119

5.3980 5.9940 6.4898 3.2720 3.9091 3.9121 3.5220 3.2580 3.9310

0.3835 0.3953 0.39187 0.383 0.39866 0.3634 0.395 0.373 0.425

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

171.45 191.78 206.89 132.81 135.58 139.39 136.95 164.25 127.27

4.7013 5.1366 5.5289 2.8262 3.4072 3.4437 3.0540 2.7155 3.4529

566.65 593.25 616.4 417.9 465 471 452 425.17 484

0 0 0 0 0 0 0 0 0

45 46 47 48 49 50

Acetylene Methylacetylene Dimethylacetylene 3-Methyl-1-butyne 1-Pentyne 2-Pentyne

C2H2 C3H4 C4H6 C5H8 C5H8 C5H8

74862 74997 503173 598232 627190 627214

26.038 40.065 54.092 68.119 68.119 68.119

2.3795 3.2775 3.8560 3.7920 3.9540 4.4158

0.375 0.3997 0.3737 0.3565 0.3512 0.44347

0 0 0 0 0 0

0 0 0 0 0 0

192.4 170.45 240.91 183.45 167.45 163.83

1.6488 2.6297 2.9557 3.1681 3.4025 3.7321

308.32 402.39 473.2 463.2 481.2 519

0 0 0 0 0 0

51 52

1-Hexyne 2-Hexyne

C6H10 C6H10

693027 764352

82.145 82.145

4.5740 4.9110

0.3698 0.4392

0 0

0 0

141.25 183.65

4.0640 4.1067

516.2 549

0 0

2-156

53 54 55 56

3-Hexyne 1-Heptyne 1-Octyne Vinylacetylene1

C6H10 C7H12 C8H14 C4H4

928494 628717 629050 689974

82.145 96.172 110.199 52.076

4.8080 5.0514 5.6306 3.6490

0.436 0.41163 0.4148 0.4

0 0 0 0.043

0 0 0 0

170.05 192.22 193.55 173.15

4.0831 4.2470 4.7663 2.9876

544 559 585 454

0 0 0 0

57 58 59 60 61 62 63 64 65 66

Cyclopentane Methylcyclopentane Ethylcyclopentane Cyclohexane Methylcyclohexane 1,1-Dimethylcyclohexane Ethylcyclohexane Cyclopentene 1-Methylcyclopentene Cyclohexene

C5H10 C6H12 C7H14 C6H12 C7H14 C8H16 C8H16 C5H8 C6H10 C6H10

287923 96377 1640897 110827 108872 590669 1678917 142290 693890 110838

70.134 84.161 98.188 84.161 98.188 112.215 112.215 68.119 82.145 82.145

3.8900 4.3600 4.8288 4.4940 4.7534 5.0402 5.3832 3.8107 4.3541 4.4405

0.361 0.38531 0.37809 0.3974 0.39461 0.4036 0.41763 0.3543 0.36805 0.37479

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

179.28 130.73 134.71 279.69 146.58 239.66 161.84 138.13 146.62 169.67

3.3292 3.9118 4.3604 3.3977 4.2295 4.0862 4.7318 3.4046 3.8769 3.8791

511.76 532.79 569.52 553.58 572.19 591.15 609.15 507 542 560.4

0 0 0 0 0 0 0 0 0 0

67 68 69 70 71 72 73

Benzene Toluene o-Xylene m-Xylene p-Xylene Ethylbenzene Propylbenzene

C6H6 C7H8 C8H10 C8H10 C8H10 C8H10 C9H12

71432 108883 95476 108383 106423 100414 103651

78.114 92.141 106.167 106.167 106.167 106.167 120.194

4.7500 5.0144 5.5330 5.4600 5.3740 5.4640 5.7663

0.45238 0.3859 0.377 0.3726 0.3656 0.392 0.3956

0.0534 0 0 0 0 0 −8.9129E−03

−0.1181 0 0 0 0 0 0

278.68 178.18 247.98 225.3 286.41 178.15 215.03

3.4909 4.3670 4.5826 4.6097 4.2761 4.7811 5.0574

562.16 591.8 630.33 617.05 616.23 617.2 574.54

0 0 0 0 0 0 2.4695E+07

74 75 76 77 78 79 80 81

1,2,4-Trimethylbenzene Isopropylbenzene 1,3,5-Trimethylbenzene p-Isopropyltoluene Naphthalene Biphenyl Styrene m-Terphenyl

C9H12 C9H12 C9H12 C10H14 C10H8 C12H10 C8H8 C18H14

95636 98828 108678 99876 91203 92524 100425 92068

120.194 120.194 120.194 134.221 128.174 154.211 104.152 230.309

5.9126 5.7950 6.0380 6.3314 7.0510 7.5736 5.7260 10.1230

0.35632 0.3956 0.37999 0.40289 0.4612 0.3975 0.4055 0.3767

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

229.33 177.14 228.42 205.25 353.43 342.2 242.54 360

5.0621 5.0869 5.1010 5.4387 5.2508 6.0420 4.7128 8.4070

649.13 631.1 637.36 653.15 748.35 789.26 636 924.85

0 0 0 0 0 0 0 0

82 83 84 85 86 87 88

Methanol Ethanol 1-Propanol 1-Butanol 2-Butanol 2-Propanol 2-Methyl-2-propanol

CH4O C2H6O C3H8O C4H10O C4H10O C3H8O C4H10O

67561 64175 71238 71363 78922 67630 75650

32.042 46.069 60.096 74.123 74.123 60.096 74.123

5.2390 5.6900 6.3300 6.7390 7.2560 6.3080 7.7320

0.3682 0.3359 0.3575 0.173 0.4774 0.3921 0.5645

0 0 0 0.2915 0 0 0

0 0 0 0 0 0 0

175.47 159.05 146.95 184.51 158.45 185.28 298.97

4.4900 5.0245 5.6460 6.0575 6.1383 5.2807 4.6703

512.64 513.92 536.78 563.05 536.05 508.3 506.21

0 0 0 0 0 0 0

89 90 91 92 93 94 95 96

1-Pentanol 2-Methyl-1-butanol 3-Methyl-1-butanol 1-Hexanol 1-Heptanol Cyclohexanol Ethylene glycol 1,2-Propylene glycol

C5H12O C5H12O C5H12O C6H14O C7H16O C6H12O C2H6O2 C3H8O2

71410 137326 123513 111273 111706 108930 107211 57556

88.150 88.150 88.150 102.177 116.203 100.161 62.068 76.095

8.3100 7.7839 8.0815 8.5980 9.6900 9.2440 8.2900 8.0700

0.511 0.45313 0.50185 0.513 0.572 0.64825 0.4266 0.295

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

195.56 203 155.95 228.55 239.15 296.6 260.15 213.15

6.7533 6.3619 6.8999 6.7623 7.3822 6.2273 6.8461 7.1374

586.15 565 577.2 611.35 631.9 650 719.7 626

0 0 0 0 0 0 0 0

97 98 99 100

Phenol o-Cresol m-Cresol p-Cresol

C6H6O C7H8O C7H8O C7H8O

108952 95487 108394 106445

94.113 108.140 108.140 108.140

7.3060 7.1979 8.0082 8.4942

0.4246 0.40317 0.45314 0.50234

0 0 0 0

0 0 0 0

314.06 304.19 285.39 307.93

5.6577 5.7135 6.3326 6.3649

694.25 697.55 705.85 704.65

0 0 0 0

101 102 103 104 105 106 107

Dimethyl ether Methyl ethyl ether Methyl n-propyl ether Methyl isopropyl ether Methyl-n-butyl ether Methyl isobutyl ether Methyl tert-butyl ether

C2H6O C3H8O C4H10O C4H10O C5H12O C5H12O C5H12O

115106 540670 557175 598538 628284 625445 1634044

46.069 60.096 74.123 74.123 88.150 88.150 88.150

2.9940 3.5300 3.9795 3.9305 4.5328 4.2678 4.2024

0.3505 0.376 0.3729 0.3711 0.3824 0.37995 0.37826

0 0 0 0 0 0 0

0 0 0 0 0 0 0

131.65 160 133.97 127.93 157.48 150 164.55

2.6032 2.9751 3.5184 3.4876 3.9358 3.7232 3.6096

400.1 437.8 476.3 464.5 510 497 497.1

0 0 0 0 0 0 0 2-157

TABLE 2-193 Cmpd. no.

Heats of Vaporization of Inorganic and Organic Compounds (Continued ) Name

Formula

CAS no.

Mol wt

C1 × 1E−07

C2

C3

C4

Tmin, K

∆Hv at Tmin × 1E−07

Tmax, K

∆Hv at Tmax

108 109 110 111 112

Diethyl ether Ethyl propyl ether Ethyl isopropyl ether Methyl phenyl ether Diphenyl ether

C4H10O C5H12O C5H12O C7H8O C12H10O

60297 628320 625547 100663 101848

74.123 88.150 88.150 108.140 170.211

4.0600 5.4380 4.2580 5.8662 6.8243

0.3868 0.60624 0.37221 0.37127 0.30877

0 0 0 0 0

0 0 0 0 0

156.85 145.65 140 235.65 300.03

3.4651 4.4140 3.7556 4.9560 5.8546

466.7 500.23 489 645.6 766.8

0 0 0 0 0

113 114 115 116 117 118 119 120 121 122

Formaldehyde Acetaldehyde 1-Propanal 1-Butanal 1-Pentanal 1-Hexanal 1-Heptanal 1-Octanal 1-Nonanal 1-Decanal

CH2O C2H4O C3H6O C4H8O C5H10O C6H12O C7H14O C8H16O C9H18O C10H20O

50000 75070 123386 123728 110623 66251 111717 124130 124196 112312

30.026 44.053 58.080 72.107 86.134 100.161 114.188 128.214 142.241 156.268

3.0760 4.6070 4.1492 4.6403 5.1478 5.6661 6.1299 6.8347 7.3363 7.9073

0.2954 0.62 0.36751 0.3849 0.37541 0.38533 0.37999 0.41039 0.41735 0.4129

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

181.15 150.15 170 176.75 182 217.15 229.8 246 255.15 267.15

2.5863 3.6199 3.5675 3.9797 4.4502 4.7495 5.1353 5.5966 5.9779 6.4201

408 466 504.4 537.2 566.1 591 617 638.1 658 674.2

0 0 0 0 0 0 0 0 0 0

123 124 125 126 127 128 129 130 131 132 133 134

Acetone Methyl ethyl ketone 2-Pentanone Methyl isopropyl ketone 2-Hexanone Methyl isobutyl ketone 3-Methyl-2-pentanone 3-Pentanone Ethyl isopropyl ketone Diisopropyl ketone Cyclohexanone Methyl phenyl ketone

C3H6O C4H8O C5H10O C5H10O C6H12O C6H12O C6H12O C5H10O C6H12O C7H14O C6H10O C8H8O

67641 78933 107879 563804 591786 108101 565617 96220 565695 565800 108941 98862

58.080 72.107 86.134 86.134 100.161 100.161 100.161 86.134 100.161 114.188 98.145 120.151

4.2150 4.6220 5.1740 5.1400 5.6770 5.4000 5.1130 5.2359 5.3880 5.5980 5.5500 6.6104

0.3397 0.355 0.39422 0.3858 0.3817 0.383 0.3395 0.40465 0.40616 0.3774 0.3538 0.37425

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

178.45 186.48 196.29 250 217.35 189.15 167.15 234.18 200 204.81 242 292.81

3.6390 3.9704 4.3663 4.0753 4.7584 4.6294 4.5480 4.2075 4.5154 4.7426 4.7114 5.4166

508.2 535.5 561.08 553 587.05 571.4 573 560.95 567 576 653 709.5

0 0 0 0 0 0 0 0 0 0 0 0

135 136 137 138 139 140 141

Formic acid Acetic acid Propionic acid n-Butyric acid Isobutyric acid Benzoic acid2 Acetic anhydride

CH2O2 C2H4O2 C3H6O2 C4H8O2 C4H8O2 C7H6O2 C4H6O3

64186 64197 79094 107926 79312 65850 108247

46.026 60.053 74.079 88.106 88.106 122.123 102.090

2.3700 2.0265 2.7290 7.4996 4.4967 10.1900 6.3520

1.999 0.11911 0.06954 2.333 1.1615 0.478 0.3986

−5.1503 −1.3487 −1.0423 −3.8644 −2.4573 0 0

3.331 1.4227 1.1152 2.016 1.5823 0 0

281.45 289.81 252.45 267.95 227.15 395.45 200.15

1.9532 2.3185 2.9964 4.1566 3.6179 7.1277 5.4139

588 591.95 600.81 615.7 605 751 606

0 0 0 0 0 0 0

142 143 144 145 146 147 148

Methyl formate Methyl acetate Methyl propionate Methyl n-butyrate Ethyl formate Ethyl acetate Ethyl propionate

C2H4O2 C3H6O2 C4H8O2 C5H10O2 C3H6O2 C4H8O2 C5H10O2

107313 79209 554121 623427 109944 141786 105373

60.053 74.079 88.106 102.133 74.079 88.106 102.133

4.1030 4.4920 5.0080 5.3781 4.5909 4.9330 5.3325

0.3825 0.3685 0.3959 0.39523 0.4123 0.3847 0.401

0 0 0 0 0 0 0

0 0 0 0 0 0 0

174.15 175.15 185.65 187.35 193.55 189.6 199.25

3.4644 3.8418 4.2231 4.5694 3.7679 4.1490 4.4449

487.2 506.55 530.6 554.5 508.4 523.3 546

0 0 0 0 0 0 0

149 150 151 152 153 154 155

Ethyl n-butyrate n-Propyl formate n-Propyl acetate n-Butyl acetate Methyl benzoate Ethyl benzoate Vinyl acetate

C6H12O2 C4H8O2 C5H10O2 C6H12O2 C8H8O2 C9H10O2 C4H6O2

105544 110747 109604 123864 93583 93890 108054

116.160 88.106 102.133 116.160 136.150 150.177 86.090

5.6419 4.9687 5.4327 5.7800 6.9650 6.3400 4.7700

0.37985 0.4025 0.407 0.3935 0.4061 0.2911 0.3765

0 0 0 0 0 0 0

0 0 0 0 0 0 0

175.15 180.25 178.15 199.65 260.75 238.45 180.35

4.9090 4.2162 4.6322 4.8943 5.7500 5.6137 4.0619

571 538 549.73 579.15 693 698 519.13

0 0 0 0 0 0 0

156 157 158

Methylamine Dimethylamine Trimethylamine

CH5N C2H7N C3H9N

74895 124403 75503

31.057 45.084 59.111

3.8580 4.0900 3.3050

0.404 0.42005 0.354

0 0 0

0 0 0

179.69 180.96 156.08

3.1006 3.2678 2.8216

430.05 437.2 433.25

0 0 0

2-158

159 160 161

Ethylamine Diethylamine Triethylamine

C2H7N C4H11N C6H15N

75047 109897 121448

45.084 73.138 101.192

4.2750 4.6133 4.6640

0.5857 0.42628 0.3663

−0.332 0 0

0.169 0 0

192.15 223.35 158.45

3.2955 3.5761 4.1011

456.15 496.6 535.15

0 0 0

162 163 164 165 166 167 168

n-Propylamine di-n-Propylamine Isopropylamine Diisopropylamine Aniline N-Methylaniline N,N-Dimethylaniline

C3H9N C6H15N C3H9N C6H15N C6H7N C7H9N C8H11N

107108 142847 75310 108189 62533 100618 121697

59.111 101.192 59.111 101.192 93.128 107.155 121.182

4.4488 5.4280 4.4041 5.0070 7.1950 6.3860 6.7900

0.39494 0.3665 0.43325 0.4362 0.458 0.3104 0.4053

0 0 0 0 0 0 0

0 0 0 0 0 0 0

188.36 210.15 177.95 176.85 267.13 216.15 275.6

3.6857 4.5500 3.5874 4.1823 5.7710 5.6961 5.5162

496.95 550 471.85 523.1 699 701.55 687.15

0 0 0 0 0 0 0

169 170 171 172

Ethylene oxide Furan Thiophene Pyridine

C2H4O C4H4O C4H4S C5H5N

75218 110009 110021 110861

44.053 68.075 84.142 79.101

3.6652 4.0050 4.5793 5.1740

0.37878 0.3995 0.38557 0.38865

0 0 0 0

0 0 0 0

160.65 196.29 234.94 231.51

3.1271 3.2647 3.7472 4.3144

469.15 490.15 579.35 619.95

0 0 0 0

173 174 175 176 177 178 179 180

Formamide3 N,N-Dimethylformamide Acetamide N-Methylacetamide Acetonitrile Propionitrile n-Butyronitrile Benzonitrile

CH3NO C3H7NO C2H5NO C3H7NO C2H3N C3H5N C4H7N C7H5N

75127 68122 60355 79163 75058 107120 109740 100470

45.041 73.095 59.068 73.095 41.053 55.079 69.106 103.123

7.3580 5.9217 8.1070 7.3402 4.3511 4.9348 5.2200 6.2615

0.3564 0.37996 0.42 0.38974 0.34765 0.41873 0.165 0.35427

0 0 0 0 0 0 0.6692 0

0 0 0 0 0 0 −0.539 0

275.7 212.72 353.15 301.15 229.32 180.26 161.25 260.4

6.2844 5.0931 6.2386 5.9384 3.5996 4.2005 4.7223 5.3091

771 649.6 761 718 545.5 564.4 582.25 699.35

0 0 0 0 0 0 0 0

181 182 183 184 185 186 187 188 189

Methyl mercaptan Ethyl mercaptan n-Propyl mercaptan n-Butyl mercaptan Isobutyl mercaptan sec-Butyl mercaptan Dimethyl sulfide Methyl ethyl sulfide Diethyl sulfide

CH4S C2H6S C3H8S C4H10S C4H10S C4H10S C2H6S C3H8S C4H10S

74931 75081 107039 109795 513440 513531 75183 624895 352932

48.109 62.136 76.163 90.189 90.189 90.189 62.136 76.163 90.189

3.4448 3.8440 4.4782 4.9702 4.7420 4.6432 3.8690 4.4740 4.7182

0.37427 0.37534 0.41073 0.41199 0.40535 0.399 0.3694 0.4097 0.3643

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

150.18 125.26 159.95 157.46 128.31 133.02 174.88 167.23 169.2

2.9825 3.4489 3.8723 4.3505 4.2664 4.1614 3.3042 3.8344 4.1353

469.95 499.15 536.6 570.1 559 554 503.04 533 557.15

0 0 0 0 0 0 0 0 0

190 191 192 193 194 195 196 197

Fluoromethane Chloromethane Trichloromethane Tetrachloromethane Bromomethane Fluoroethane Chloroethane Bromoethane

CH3F CH3Cl CHCl3 CCl4 CH3Br C2H5F C2H5Cl C2H5Br

593533 74873 67663 56235 74839 353366 75003 74964

34.033 50.488 119.377 153.822 94.939 48.060 64.514 108.966

2.4708 2.9745 4.1860 4.3252 3.1690 2.7617 3.5240 3.9004

0.37014 0.353 0.3584 0.37688 0.3015 0.32162 0.3652 0.38012

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

131.35 175.43 209.63 250.33 179.47 129.95 134.8 154.55

2.0276 2.4520 3.5047 3.4528 2.7379 2.4089 3.1052 3.3933

317.42 416.25 536.4 556.35 467 375.31 460.35 503.8

0 0 0 0 0 0 0 0

198 199 200 201 202 203 204 205

1-Chloropropane 2-Chloropropane 1,1-Dichloropropane 1,2-Dichloropropane Vinyl chloride Fluorobenzene Chlorobenzene Bromobenzene

C3H7Cl C3H7Cl C3H6Cl2 C3H6Cl2 C2H3Cl C6H5F C6H5Cl C6H5Br

540545 75296 78999 78875 75014 462066 108907 108861

78.541 78.541 112.986 112.986 62.499 96.104 112.558 157.010

3.9890 3.8871 4.7740 4.6750 3.4125 4.5820 5.1480 5.5520

0.37956 0.38043 0.39204 0.36529 0.4513 0.3717 0.36614 0.37694

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

150.35 155.97 200 172.71 119.36 230.94 227.95 242.43

3.4862 3.3586 4.0147 4.0997 2.9491 3.7605 4.3707 4.6875

503.15 489 560 572 432 560.09 632.35 670.15

0 0 0 0 0 0 0 0

206 207 208 209 210 211 212 213

Air Hydrogen Helium-4 Neon Argon Fluorine Chlorine Bromine

H2 He Ne Ar F2 Cl2 Br2

132259100 1333740 7440597 7440019 7440371 7782414 7782505 7726956

28.951 2.016 4.003 20.180 39.948 37.997 70.905 159.808

0.8474 0.1013 0.0125 0.2389 0.8731 0.8876 3.0680 4.0000

0.3822 0.698 1.3038 0.3494 0.3526 0.34072 0.8458 0.351

0 −1.817 −2.6954 0 0 0 −0.9001 0

0 1.447 1.7098 0 0 0 0.453 0

59.15 13.95 2.2 24.56 83.78 53.48 172.12 265.85

0.6759 0.0913 0.0097 0.1803 0.6561 0.7578 2.2878 3.2323

132.45 33.19 5.2 44.4 150.86 144.12 417.15 584.15

0 0 0 0 0 0 0 0 2-159

TABLE 2-193 Cmpd. no.

Heats of Vaporization of Inorganic and Organic Compounds (Concluded ) Name

Formula

CAS no.

Mol wt

C1 × 1E−07

C2

C3

C4

Tmin, K

∆Hv at Tmin × 1E−07

Tmax, K

∆Hv at Tmax

214 215 216 217 218

Oxygen Nitrogen Ammonia Hydrazine Nitrous oxide

O2 N2 NH3 N2H4 N2O

7782447 7727379 7664417 302012 10024972

31.999 28.014 17.031 32.045 44.013

0.9008 0.7491 3.1523 5.9794 2.3215

0.4542 0.40406 0.3914 0.9424 0.384

−0.4096 −0.317 −0.2289 −1.398 0

0.3183 0.27343 0.2309 0.8862 0

54.36 63.15 195.41 274.69 182.3

0.7742 0.6024 2.5298 4.5238 1.6502

154.58 126.2 405.65 653.15 309.57

0 0 0 0 0

219 220 221 222 223 224 225 226 227 228 229 230 231

Nitric oxide Cyanogen Carbon monoxide Carbon dioxide Carbon disulfide Hydrogen fluoride Hydrogen chloride Hydrogen bromide Hydrogen cyanide2 Hydrogen sulfide Sulfur dioxide Sulfur trioxide Water

NO C2N2 CO CO2 CS2 HF HCl HBr HCN H2S SO2 SO3 H2O

10102439 460195 630080 124389 75150 7664393 7647010 10035106 74908 7783064 7446095 7446119 7732185

30.006 52.036 28.010 44.010 76.143 20.006 36.461 80.912 27.026 34.082 64.065 80.064 18.015

2.1310 3.3840 0.8585 2.1730 3.4960 13.4510 2.2093 2.4850 3.3490 2.5676 3.6760 7.3370 5.2053

0.4056 0.3707 0.4921 0.382 0.2986 13.36 0.3466 0.39 0.2053 0.37358 0.4 0.5647 0.3199

0 0 −0.326 −0.4339 0 −23.383 0 0 0 0 0 0 −0.212

0 0 0.2231 0.42213 0 10.785 0 0 0 0 0 0 0.25795

109.5 245.25 68.13 216.58 161.11 277.56 158.97 185.15 259.83 187.68 197.67 289.95 273.16

1.4578 2.3803 0.6517 1.5202 3.1537 0.7104 1.7498 1.8817 2.8176 1.9782 2.8753 4.4303 4.4733

180.15 400.15 132.5 304.21 552 461.15 324.65 363.15 456.65 373.53 430.75 490.85 647.13

0 0 915280 0 0 0 0 0 0 0 0 0 0

All substances are listed in alphabetical order in Table 2-6a. Compiled from Daubert, T. E., R. P. Danner, H. M. Sibul, and C. C. Stebbins, DIPPR Data Compilation of Pure Compound Properties, Project 801 Sponsor Release, July, 1993, Design Institute for Physical Property Data, AIChE, New York, NY; and from Thermodynamics Research Center, “Selected Values of Properties of Hydrocarbons and Related Compounds,” Thermodynamics Research Center Hydrocarbon Project, Texas A&M University, College Station, Texas (extant 1994). Temperatures are expressed in kelvins; heats of vaporization, in J/kmol. J/ kmol × 2.390E−04 = cal/gmol; J/ kmol × 4.302106E−04 = Btu/lbmol. The heat of vaporization equation used is ∆Hv = C1 × (1 − Tr)C2 + C3 × Tr + C4 × Tr × Tr. Tr is the reduced temperature, T/Tc. 1 Coefficients are hypothetical; compound decomposes violently on heating. 2 For the monomer. 3 Equation coefficients are hypothetical above the decomposition temperature. 2-160

SPECIFIC HEATS OF PURE COMPOUNDS

2-161

SPECIFIC HEATS OF PURE COMPOUNDS To convert kilojoules per kilogram-kelvin to British thermal units per pound-degree Rankine, multiply by 0.2388.

UNITS CONVERSIONS For this subsection, the following units conversions are applicable: °F = 9⁄ 5 °C + 32 °F = 1.8 K

ADDITIONAL REFERENCES

To convert calories per gram-kelvin to British thermal units per pound-degree Rankine, multiply by 1.0; to convert calories per grammole-kelvin to British thermal units per pound-mole-degree Rankine, multiply by 1.0. TABLE 2-194

Heat Capacities of the Elements and Inorganic Compounds* Heat capacity at constant pressure (T = K; 0°C = 273.1 K), cal/deg mol

Range of temperature, K

Uncertainty, %

c l c l c l c c c c l c l c c, sillimanite c, disthene c, andalusite c, mullite c c c

4.80 + 0.00322T 7.00 18.74 + 0.01866T 29.5 13.25 + 0.02800T 31.2 76 19.3 50.5 38.63 + 0.04760T − 449200/T 2 142 16.88 + 0.02266T 28.8 22.08 + 0.008971T − 522500/T 2 40.79 + 0.004763T − 992800/T 2 41.81 + 0.005283T − 1211000/T 2 43.96 + 0.001923T − 1086000/T 2 59.65 + 0.0670T 113.2 + 0.0652T 63.5 235

273–931 931–1273 273–370 370–407 273–465 465–504 288–327 288–326 288–326 273–1273 1273–1373 273–464 464–480 273–1973 273–1573 273–1673 273–1573 273–576 273–575 273–373 288–325

1 5 3 5 3 3 ? ? ? 2 ? 3 5 3 3 2 3 5 3 ? ?

c l c c c c c

5.51 + 0.00178T 7.15 17.2 + 0.0293T 10.3 + 0.0511T 19.1 + 0.0171T 22.6 + 0.0162T 24.2 + 0.0132T

273–903 903–1273 273–370 273–346 273–929 273–1198 273–821

2 5 ? ? ? ? ?

Substance Aluminum1 Al AlBr3 AlCl3 AlCl3·6H2O AlF3 AlF3·3aH2O AlF3·3NaF AlI3 Al2O3 Al2O3·SiO2 3Al2O3·2SiO2 4Al2O3·3SiO2 Al2(SO4)3 Al2(SO4)3·18H2O Antimony Sb SbBr3 SbCl3 Sb2O3 Sb2O4 Sb2S3 Argon2 A Arsenic As AsCl3 As2O3 As2S3 Barium BaCl2 BaCl2·H2O BaCl2·2H2O Ba(ClO3)2·H2O BaCO3 BaMoO4 Ba(NO3)2 BaSO4 Beryllium3,4 Be BeO BeO·Al2O3 BeSO4

Additional data are contained in the subsection “Thermodynamic Properties.” Data on water are also contained in that subsection. Additional tables for water are found in Eng. Sci. Data Item 68008, 251 Regent Street, London, England, which contains about 5000 values from 1 to 1000 bar, 0 to 1500°C.

State†

g

4.97

c l c c

5.17 + 0.00234T 31.9 8.37 + 0.0486T 25.8

273–1168 286–371 273–548 293–373

All

5 ? ? ?

0

c c c c c, α c, β c c c

17.0 + 0.00334T 28.2 37.3 51 17.26 + 0.0131T 30.0 34 39.8 21.35 + 0.0141T

273–1198 273–307 273–307 289–320 273–1083 1083–1255 273–297 285–371 273–1323

? ? ? ? 5 15 ? ? 5

c c c c

4.698 + 0.001555T − 121000/T 2 8.69 + 0.00365T − 313000/T 2 25.4 20.8

273–1173 273–1175 273–373 273–373

1 5 ? ?

*From Kelley, U.S. Bur. Mines Bull. 371, 1934. For a revision see Kelley, U.S. Bur. Mines Bull. 477, 1948. Data for many elements and compounds are given by Johnson (ed.), WADD-TR-60-56, 1960, for cryogenic temperatures. Tabulated data for gases can be obtained from many of the references cited in the “Thermodynamic Properties” subsection and other tables in this section. Thinh, Duran, et al., Hydrocarbon Process., 50, 98 (January 1971), review previous equation fits and give newer fits for 408 hydrocarbons and related compounds. Later publications include Duran, Thinh, et al., Hydrocarbon Process., 55, 153 (August 1976); Thompson, J. Chem. Eng. Data, 22(4), 431 (1977); and Passut and Danner, Ind. Eng. Chem. Process Des. Dev., 11, 543 (1972); 13, 193 (1974). † The symbols in this column have the following meaning; c, crystal; l, liquid; g, gas; gls, glass.

TABLE 2-194

Heat Capacities of the Elements and Inorganic Compounds (Continued )

Substance Bismuth4 Bi Bi2O3 Bi2S3 Boron B B2O3 BN Bromine Br2 Cadmium Cd CdO CdS CdSO4·8/3H2O Calcium Ca CaCl2 CaCO3 CaF2 CaMg(CO3)2 CaMoO4 CaO Ca(OH)2 CaO·Al2O3·2SiO2 CaO·MgO·2SiO2 CaO·SiO2 CaP2O6 CaSO4 CaSO4·2H2O CaWO4 Carbon5 C CH4 CO6 CO2 CS2 Cerium Ce CeO2 Ce2(MoO4)3 Ce2(SO4)3 Ce2(SO4)3·5H2O Cesium Cs CsBr CsCl CsF CsI Chlorine Cl2 Chromium4 Cr CrCl3 Cr2O3 CrSb CrSb2 Cr2(SO4)3 Cobalt4 Co CoAs2·CoS2 CoSb Co2Sn CoS CoSO4·7H2O 2-162

State†

Heat capacity at constant pressure (T = K; 0°C = 273.1 K), cal/deg mol

Range of temperature, K

Uncertainty, %

c l c c

5.38 + 0.00260T 7.60 23.27 + 0.01105T 30.4

273–544 544–1273 273–777 284–372

3 3 2 ?

c gls gls c

1.54 + 0.00440T 5.14 + 0.0320T 30.4 1.61 + 0.00400T

273–1174 273–513 513–623 273–1173

5 3 3 5

g

9.00

300–2000

5

c l c c c

5.46 + 0.002466T 7.13 9.65 + 0.00208T 12.9 + 0.00090T 51.3

273–594 594–973 273–2086 273–1273 293

1 5 ? ? ?

c c c c c c c c c c, anorthite gls c, diopside gls c, wollastonite c, pseudowollastonite gls c c c c

5.31 + 0.00333T 6.29 + 0.00140T 16.9 + 0.00386T 19.68 + 0.01189T − 307600/T 2 14.7 + 0.00380T 40.1 33 10.00 + 0.00484T − 108000/T 2 21.4 63.13 + 0.01500T − 1537000/T 2 67.41 + 0.01048T − 1874000/T 2 54.46 + 0.005746T − 1500000/T 2 51.68 + 0.009724T − 1308000/T 2 27.95 + 0.002056T − 745600/T 2 25.48 + 0.004132T − 488100/T 2 23.16 + 0.009672T − 487100/T 2 39.5 18.52 + 0.02197T − 156800/T 2 46.8 27.9

273–673 673–873 273–1055 273–1033 273–1651 299–372 273–297 273–1173 276–373 273–1673 273–973 273–1573 273–973 273–1573 273–1673 273–973 287–371 273–1373 282–373 292–322

2 2 ? 3 ? ? ? 2 ? 1 1 1 1 1 1 1 ? 5 ? ?

c, graphite c, diamond g g g l

2.673 + 0.002617T − 116900/T 2 2.162 + 0.003059T − 130300/T 2 5.34 + 0.0115T 6.60 + 0.00120T 10.34 + 0.00274T − 195500/T 2 18.4

273–1373 273–1313 273–1200 273–2500 273–1200 293

2 3 2 1a 1a ?

c c c c c

5.88 + 0.00123T 15.1 96 66.4 131.6

273–908 273–373 273–297 273–373 273–319

? ? ? ? ?

c l g c c c c

1.96 + 0.0182T 8.00 4.97 12.6 + 0.00259T 11.7 + 0.00309T 11.3 + 0.00285T 11.6 + 0.00268T

273–301 302 All 273–909 273–752 273–957 273–894

3 3 0 ? ? ? ?

g

8.28 + 0.00056T

273–2000

1a

c l c c c c c

4.84 + 0.00295T 9.70 23 26.0 + 0.00400T 12.3 + 0.00120T 19.2 + 0.00184T 67.4

273–1823 1823–1923 286–319 273–2263 273–1383 273–949 273–373

5 10 ? ? ? ? ?

c l c c c c c

5.12 + 0.00333T 8.40 32.9 11.7 + 0.00156T 15.83 + 0.00950T 10.6 + 0.00251T 96

273–1763 1763–1873 283–373 273–1464 273–903 273–1373 286–303

5 5 ? ? 2 ? ?

TABLE 2-194

Heat Capacities of the Elements and Inorganic Compounds (Continued )

Substance Copper7 Cu CuAl CuAl2 Cu3Al CuI CuI2 CuO CuO·SiO2·H2O CuS Cu2S CuS·FeS Cu2Sb Cu2Sb Cu2Se Cu3Si CuSO4 CuSO4·H2O CuSO4·3H2O CuSO4·5H2O Fluorine8 F2 Gallium Ga2O3 Ga2(SO4)3 Germanium4 Ge Gold Au AuSb2 Helium9 He Hydrogen10 H H2 HBr HCl HI H2O H2S H2S2O7 Indium In Iodine I2 Iridium Ir Iron4 Fe

FeAs2 Fe3C FeCO3 FeO Fe2O3 Fe3O4 Fe2O3·3H2O FeS FeS2 FeSi Fe2SiO4 FeSO4 Fe2(SO4)3 FeSO4·4H2O FeSO4·7H2O Krypton Kr

State†

Heat capacity at constant pressure (T = K; 0°C = 273.1 K), cal/deg mol

Range of temperature, K

Uncertainty, %

273–1357 1357–1573 273–733 273–773 273–775 273–675 274–328 273–810 293–323 273–1273 273–376 376–1173 292–321 273–573 273–693 273–383 383–488 273–1135 282 282 282 282

1 3 2 2 2 ? ? 2 ? ? 3 2 ? 2 2 5 5 ? ? ? ? ?

c l c c c c c c c c c, α c, β c c c c, α c, β c c c c c

5.44 + 0.001462T 7.50 9.88 + 0.00500T 16.78 + 0.00366T 19.61 + 0.01054T 12.1 + 0.00286T 20.1 10.87 + 0.003576T − 150600/T 2 29 10.6 + 0.00264T 9.38 + 0.0312T 20.9 24 13.73 + 0.01350T 21.79 + 0.00900T 20.85 20.35 20.3 + 0.00587T 24.1 31.3 49.0 67.2

g

6.50 + 0.00100T

300–3000

5

c c

18.2 + 0.0252T 62.4

273–923 273–373

? ?

273–1336 1336–1573 273–628 628–713

2 5 1 ?

c c l c, α c, βγ

5.61 + 0.00144T 7.00 17.12 + 0.00465T 11.47 + 0.01756T

g

4.97

g g g g g l g g c l

4.97 6.62 + 0.00081T 6.80 + 0.00084T 6.70 + 0.00084T 6.93 + 0.00083T See Tables 2-355 through 2-357 8.22 + 0.00015T + 0.00000134T 2 7.20 + 0.00360T 27 58

All 273–2500 273–2000 273–2000 273–2000

0 2 2 1a 2

300–2500 300–600 281 308

? 8 ? ?

g

9.00

300–2000

5

c

5.50 + 0.00148T

273–1873

1

c, α c, β c, γ c, δ l c c c c c c c c, α c, β c c c c c c c

4.13 + 0.00638T 6.12 + 0.00336T 8.40 10.0 8.15 17.8 25.17 + 0.00223T 22.7 12.62 + 0.001492T − 76200/T 2 24.72 + 0.01604T − 423400/T 2 41.17 + 0.01882T − 979500/T 2 47.8 2.03 + 0.0390T 12.05 + 0.00273T 10.7 + 0.01336T 10.54 + 0.00458T 33.57 + 0.01907T − 879700/T 2 22 66.2 63.6 96

273–1041 1041–1179 1179–1674 1674–1803 1803–1873 283–373 273–1173 293–368 273–1173 273–1097 273–1065 286–373 273–411 411–1468 273–773 273–903 273–1161 293–373 273–373 282 291–319

3 3 5 5 5 ? 10 ? 2 2 2 ? 5 3 ? 2 2 ? ? ? ?

g

4.97

All

0

c

All

0 2-163

2-164

PHYSICAL AND CHEMICAL DATA TABLE 2-194

Heat Capacities of the Elements and Inorganic Compounds (Continued )

Substance Lanthanum La La2O3 La2(MoO4)3 La2(SO4)3 La2(SO4)3·9H2O Lead4 Pb Pb3(AsO4)2 PbB2O4 PbB4O7 PbBr2 PbCl2 2PbCl2·NH4Cl PbCO3 PbCrO4 PbF2 PbI2 PbMoO4 Pb(NO3)2 PbO PbO2 Pb2P2O7 PbS PbSO4 PbS2O3 PbWO4 Lithium Li LiBr LiBr·H2O LiCl LiCl·H2O LiF LiI LiI·H2O LiI·2H2O LiI·3H2O LiNO3 Magnesium4 Mg MgAg Mg4Al3 MgAu Mg2Au Mg3Au MgCl2 MgCl2·6H2O MgCO3 MgCu2 Mg2Cu MgNi2 MgO MgO·Al2O3 MgO·SiO2 6MgO·MgCl2·8B2O3 Mg(OH)2 Mg3Sb2 Mg2Si MgSO4 MgSO4·H2O MgSO4·6H2O MgSO4·7H2O

State†

Heat capacity at constant pressure (T = K; 0°C = 273.1 K), cal/deg mol

Range of temperature, K

Uncertainty, %

c c c c c

5.91 + 0.00100T 22.6 + 0.00544T 86 66.9 152

273–1009 273–2273 273–307 273–373 273–319

? ? ? ? ?

c l c c c c l c l c c c c c l c c c c c c c c c

5.77 + 0.00202T 6.8 65.5 26.5 41.4 18.13 + 0.00310T 27.4 15.88 + 0.00835T 27.2 53.1 21.1 29.1 16.5 + 0.00412T 18.66 + 0.00293T 32.3 30.4 36.4 10.33 + 0.00318T 12.7 + 0.00780T 48.3 10.63 + 0.00401T 26.4 29 35

273–600 600–1273 286–370 288–371 289–371 273–761 761–860 273–771 771–851 293 286–320 292–323 273–1091 273–648 648–776 292–322 286–320 273–544 273–? 284–371 273–873 293–372 293–373 273–297

2 5 ? ? ? 2 10 2 10 ? ? ? ? 2 20 ? ? 2 ? ? 3 ? ? ?

c g c c c c c c c c c c l

0.68 + 0.0180T 4.97 11.5 + 0.00302T 22.6 11.0 + 0.00339T 23.6 8.20 + 0.00520T 12.5 + 0.00208T 23.6 32.9 43.2 9.17 + 0.0360T 26.8

273–459 All 273–825 278–318 273–887 279–360 273–1117 273–723 277–359 277–345 277–347 273–523 523–575

10 0 ? ? ? ? ? ? ? ? ? 5 5

c l c c c c c c c c c c c c c c, amphibole c, pyroxene gls c, α c, β c c c c c c c

6.20 + 0.00133T − 67800/T 2 7.4 10.58 + 0.00412T 34.4 + 0.0198T 11.3 + 0.00189T 16.2 + 0.00451T 21.2 + 0.00614T 17.3 + 0.00377T 77.1 16.9 14.96 + 0.00776T 15.5 + 0.00652T 15.87 + 0.00692T 10.86 + 0.001197T − 208700/T 2 28 25.60 + 0.004380T − 674200/T 2 23.35 + 0.008062T − 558800/T 2 23.30 + 0.007734T − 542000/T 2 58.7 + 0.408T 107.2 + 0.2876T 18.2 28.2 + 0.00560T 15.4 + 0.00415T 26.7 33 80 89

273–923 923–1048 273–905 273–736 273–1433 273–1073 273–1103 273–991 292–342 290 273–903 273–843 273–903 273–2073 288–319 273–1373 273–773 273–973 273–538 538–623 292–323 273–1234 273–1343 296–372 282 282 291–319

1 10 2 ? ? ? ? ? ? ? 3 ? 2 2 ? 1 1 1 5 5 ? ? ? ? ? ? ?

SPECIFIC HEATS OF PURE COMPOUNDS TABLE 2-194

Heat Capacities of the Elements and Inorganic Compounds (Continued ) Heat capacity at constant pressure (T = K; 0°C = 273.1 K), cal/deg mol

Range of temperature, K

Uncertainty, %

c, α c, β c, γ l c c c c c c c c c c

3.76 + 0.00747T 5.06 + 0.00395T 4.80 + 0.00422T 11.0 16.2 + 0.00520T 7.79 + 0.0421T + 0.0000090T 2 7.43 + 0.01038T − 0.00000362T 2 10.33 + 0.0530T − 0.0000257T 2 19.25 + 0.0538T − 0.0000209T 2 1.92 + 0.0471T − 0.0000297T 2 31 10.21 + 0.00656T − 0.00000242T 2 27.5 78

273–1108 1108–1317 1317–1493 1493–1673 273–923 273–773 273–1923 273–1173 273–1773 273–773 291–322 273–1883 293–373 290–319

5 5 5 10 ? ? ? ? ? ? ? ? ? ?

l g g c c c c c, α c, β c c c

6.61 4.97 9.00 11.05 + 0.00370T 15.3 + 0.0103T 25 11.4 + 0.00461T 17.4 + 0.004001T 20.2 11.5 10.9 + 0.00365T 31.0

273–630 All 300–2000 273–798 273–553 285–319 273–563 273–403 403–523 278–371 273–853 273–307

1 0 5 ? ? ? ? 3 3 ? ? ?

c c c

5.69 + 0.00188T − 50300/T 2 15.1 + 0.0121T 19.7 + 0.00315T

273–1773 273–1068 273–729

5 ? ?

g

4.97

c, α c, β l c c c c c c c c

4.26 + 0.00640T 6.99 + 0.000905T 8.55 11.3 + 0.00215T 9.25 + 0.00640T 15.8 + 0.00329T 10.0 + 0.00312T 20.78 + 0.0102T 33.4 82 11.00 + 0.00433T

g g c c, α c, β c c c g

Substance Manganese Mn

MnCl2 MnCO3 MnO Mn2O3 Mn3O4 MnO2 Mn2O3·H2O MnS MnSO4 MnSO4·5H2O Mercury11 Hg Hg2 HgCl HgCl2 Hg(CN)2 HgI HgI2 HgO HgS Hg2SO4 Molybdenum Mo MoO3 MoS2 Neon12 Ne Nickel4 Ni NiO NiS Ni2Si NiSi Ni3Sn NiSO4 NiSO4·6H2O NiTe Nitrogen13 N2 NH3 NH4Br NH4Cl NH4I NH4NO3 (NH4)2SO4 NO Osmium Os Oxygen14 O2 Palladium Pd Phosphorus P PCl3 P4O10 Platinum4 Pt Potassium K

State†

All

0

273–626 626–1725 1725–1903 273–1273 273–597 273–1582 273–1273 273–904 293–373 291–325 273–700

2 5 10 ? 3 ? ? 2 ? ? 2

6.50 + 0.00100T 6.70 + 0.00630T 22.8 9.80 + 0.0368T 5.0 + 0.0340T 17.8 31.8 51.6 8.05 + 0.000233T − 156300/T 2

300–3000 300–800 274–328 273–457 457–523 273–328 273–293 275–328 300–5000

3 1a ? 5 5 ? ? ? 2

c

5.686 + 0.000875T

273–1877

1

g

8.27 + 0.000258T − 187700/T

300–5000

1

2

c

5.41 + 0.00184T

273–1822

2

c, yellow c, red l l c g

5.50 0.21 + 0.0180T 6.6 28.7 15.72 + 0.1092T 73.6

273–317 273–472 317–373 284–371 273–631 631–1371

5 10 10 ? 2 3

c

5.92 + 0.00116T

273–1873

1

c l

5.24 + 0.00555T 7.7

273–336 336–373

5 5

2-165

2-166

PHYSICAL AND CHEMICAL DATA TABLE 2-194

Heat Capacities of the Elements and Inorganic Compounds (Continued )

Substance Potassium—(Cont.) K K2 KAsO3 KBO2 K2B4O7 KBr KCl KClO3 KClO4 2KCl·CuCl2.2H2O 2KCl·PtCl4 2KCl·SnCl4 2KCl·ZnCl2 2KCN·Zn(CN)2 K2CO3 K2CrO4 K2Cr2O7 KF K4Fe(CN)6 K4Fe(CN)6·3H2O KH2AsO4 KH2PO4 KHSO4 KMnO4 KNO3 K2O·Al2O3·3SiO2

K4P2O7 K2SO4 K2S2O3 K2SO4·Al2(SO4)3·24H2O K2SO4·Cr2(SO4)3·24H2O K2SO4·MgSO4·6H2O K2SO4·NiSO4·6H2O K2SO4·ZnSO4·6H2O Prometheum Pr Radon Rn Rhenium Re Rhodium Rh Rubidium Rb RbBr RbCl Rb2CO3 RbF RbI Scandium Sc2O3 Sc2(SO4)3 Selenium Se Silicon Si SiC SiCl4 SiO2

Silver4 Ag

State† g g c c c c c c c c c c c c c c c l c c c c c c c c c l c, orthoclase gls, orthoclase c, microcline gls, microcline c c c c c c c c

Heat capacity at constant pressure (T = K; 0°C = 273.1 K), cal/deg mol 4.97 9.00 25.3 12.6 + 0.0126T 51.3 11.49 + 0.00360T 10.93 + 0.00376T 25.7 26.3 63 55 54.5 43.4 57.4 29.9 35.9 42.80 + 0.0410T 96.9 10.8 + 0.00284T 80.1 114.5 32 28.3 30 28 6.42 + 0.0530T 28.8 29.5 69.26 + 0.00821T − 2331000/T 2 69.81 + 0.01053 − 2403000/T 2 65.65 + 0.01102T − 1748000/T 2 64.83 + 0.01438T − 1641000/T 2 63.1 33.1 37 352 324 106 107 120

Range of temperature, K

Uncertainty, %

All 300–2000 290–372 273–1220 290–372 273–543 273–1043 289–371 287–318 292–323 286–319 292–323 279–319 277–319 296–372 289–371 273–671 671–757 273–1129 273–319 273–310 289–319 290–320 292–324 287–318 273–401 401–611 611–683 273–1373 273–1373 273–1373 273–1373 290–371 287–371 293–373 292–322 292–324 292–323 289–319 293–317

0 5 ? ? ? 2 2 ? ? ? ? ? ? ? ? ? 5 5 ? ? ? ? ? ? ? 10 5 10 1a 1a 1a 1a ? ? ? ? ? ? ? ?

c g

4.97

c

6.30 + 0.00053T

273–2273

All

0 ?

c

5.40 + 0.00219T

273–1877

2

c l c c c c c

3.27 + 0.0131T 7.85 11.6 + 0.00255T 11.5 + 0.00249T 28.4 11.3 + 0.00256T 11.6 + 0.00263T

273–312 312–373 273–954 273–987 291–320 273–1048 273–913

2 5 ? ? ? ? ?

c c

21.1 62.0

273–373 273–373

? ?

c l

4.53 + 0.00550T 8.35

273–490 490–570

2 3

c c l c, quartz, α c, quartz, β c, cristobalite, α c, cristobalite, β gls

5.74 + 0.000617T − 101000/T 2 8.89 + 0.00291T − 284000/T 2 32.4 10.87 + 0.008712T − 241200/T 2 10.95 + 0.00550T 3.65 + 0.0240T 17.09 + 0.000454T − 897200/T 2 12.80 + 0.00447T − 302000/T 2

273–1174 273–1629 293–373 273–848 848–1873 273–523 523–1973 273–1973

2 2 ? 1 3a 2a 2 3a

c l

5.60 + 0.00150T 8.2

273–1234 1234–1573

1 3

SPECIFIC HEATS OF PURE COMPOUNDS TABLE 2-194

Heat Capacities of the Elements and Inorganic Compounds (Continued )

Substance Silver—(Cont.) Ag3Al Ag2Al AgAl12 AgBr AgCl AgCNO AgI AgNO3 Ag3PO4 Ag2S Ag3Sb Ag2Se Sodium15 Na NaBO2 Na2B4O7 Na2B4O7·10H2O NaBr NaCl NaClO3 NaCNO Na2CO3 NaF Na2HPO4·7H2O Na2HPO4·12H2O NaI NaNO3 Na2O·Al2O3·3SiO2 NaPO3 Na4P2O7 Na2SO4 Na2S2O3 Na2S2O3·5H2O Sodium-potassium alloys15 Strontium SrBr2 SrBr2·H2O SrBr2·6H2O SrCl2 SrCl2·H2O SrCl2·2H2O SrCO3 SrI2 SrI2·H2O SrI2·2H2O SrI2·6H2O SrMoO4 Sr(NO3)2 SrSO4 Sulfur16 S S2 S2Cl2 SO2 Tantalum Ta Tellurium Te Thallium Tl

State†

Heat capacity at constant pressure (T = K; 0°C = 273.1 K), cal/deg mol

Range of temperature, K

Uncertainty, %

273–902 273–903 273–768 273–703 703–836 273–728 728–806 273–353 273–423 273–433 433–482 482–541 293–325 273–448 448–597 273–694 273–406 406–460

2 2 5 6 5 2 5 ? 6 2 5 5 ? 5 5 5 5 5

273–371 371–451 All 273–1239 289–371 292–323 273–543 273–1074 1073–1205 273–528 528–572 273–353 288–371 273–1261 275–307 275–307 273–936 273–583 583–703 273–1373 273–1173 290–319 290–371 289–371 273–307 273–307

1a 2 0 ? ? ? 2 2 3 3 5 ? ? ? ? ? ? 5 10 1 1 ? ? ? ? ?

c c c c l c l c c, α c, α c, β l c c, α c, β c c, α c, β

22.56 + 0.00570T 16.85 + 0.00450T 58.62 + 0.0575T 8.58 + 0.0141T 14.9 9.60 + 0.00929T 14.05 18.7 8.58 + 0.0141T 18.83 + 0.0160T 25.7 30.2 37.5 18.8 21.8 19.53 + 0.0160T 20.2 20.4

c l g c c c c c l c l c c c c c c c l c, albite gls c c c c c l

5.01 + 0.00536T 7.50 4.97 10.4 + 0.0199T 47.9 147 11.74 + 0.00233T 10.79 + 0.00420T 15.9 9.48 + 0.0468T 31.8 13.1 28.9 10.4 + 0.00289T 86.6 133.4 12.5 + 0.00162T 4.56 + 0.0580T 37.2 63.78 + 0.01171T − 1678000/T 2 61.25 + 0.01768T − 1545000/T 2 22.1 60.7 32.8 34.9 86.2

c c c c c c c c c c c c c c

18.1 + 0.00311T 28.9 82.1 18.2 + 0.00244T 28.7 38.3 21.8 18.6 + 0.00304T 28.5 39.1 84.9 37 38.3 26.2

273–923 277–370 276–327 273–1143 276–365 277–366 281–371 273–783 276–363 275–336 275–333 273–297 290–320 293–369

? ? ? ? ? ? ? ? ? ? ? ? ? ?

c, rhombic c, monoclinic g l g

3.63 + 0.00640T 4.38 + 0.00440T 8.58 + 0.00030T 27.5 7.70 + 0.00530T − 0.00000083T 2

273–368 368–392 300–2500 273–332 300–2500

3 3 5 ? 2a

c

5.91 + 0.00099T

273–1173

2

c

5.19 + 0.00250T

273–600

3

c, α c, β

5.32 + 0.00385T 8.12

273–500 500–576

1 1

2-167

2-168

PHYSICAL AND CHEMICAL DATA TABLE 2-194

Heat Capacities of the Elements and Inorganic Compounds (Concluded )

Substance Thallium—(Cont.) Tl TlBr TlCl Thorium Th ThO2 Th(SO4)2 Tin4 Sn SnAu SnCl2 SnCl4 SnO SnO2 SnPt SnS SnS2 Titanium Ti TiCl4 TiO2 Tungsten W WO3 Uranium U U3O8 Vanadium V Xenon Xe Zinc4 Zn ZnCl2 ZnO ZnS ZnSb ZnSO4 ZnSO4·H2O ZnSO4·6H2O ZnSO4·7H2O Zirconium ZrO2 ZrO2·SiO2 1

State†

Heat capacity at constant pressure (T = K; 0°C = 273.1 K), cal/deg mol

Range of temperature, K

Uncertainty, %

l c l c l

7.12 12.53 + 0.00100T 16.0 12.56 + 0.00088T 14.2

576–773 273–733 733–800 273–700 700–803

3 10 10 5 10

c c c

6.40 14.6 + 0.00507T 41.2

273–373 273–1273 273–373

? ? ?

c l c c l c c c c c

5.05 + 0.00480T 6.6 11.79 + 0.00233T 16.2 + 0.00926T 38.4 9.40 + 0.00362T 13.94 + 0.00565T − 252000/T 2 11.49 + 0.00190T 12.1 + 0.00165T 20.5 + 0.00400T

273–504 504–1273 273–581 273–520 286–371 273–1273 273–1373 273–1318 273–1153 273–873

2 10 1 ? ? ? ? 1 ? ?

c l c

8.91 + 0.00114T − 433000/T 2 35.7 11.81 + 0.00754T − 41900/T 2

273–713 285–372 273–713

3 ? 3

c c

5.65 + 0.00866 16.0 + 0.00774T

273–2073 273–1550

1 ?

c c

6.64 59.8

273–372 276–314

? ?

c

5.57 + 0.00097T

273–1993

?

g

4.97

c l c c c c c c c c

5.25 + 0.00270T 7.59 + 0.00055T 15.9 + 0.00800T 11.40 + 0.00145T − 182400/T 2 12.81 + 0.00095T − 194600/T 2 11.5 + 0.00313T 28 34.7 80.8 100.2

273–692 692–1122 273–638 273–1573 273–1173 273–810 293–373 282 282 273–307

1 3 ? 1 5 ? ? ? ? ?

c c

11.62 + 0.01046T − 177700/T 2 26.7

273–1673 297–372

5 ?

All

0

See also Table 2-195. Data to 298 K are also given by Scott, Cryogenic Engineering, Van Nostrand, Princeton, N.J., 1959. For liquid and gas data, see Johnson (ed.), WADD-TR-60-56, 1960. 3 Stalder, NACA Tech. Note 4141, 1957 (Fig. 5), gives data from 400 to 2600°R. 4 See also Table 2-195. 5 For data from 400 to 5500°R see Stalder, NACA Tech. Note 4141, 1975 (Fig. 4). 6 For solid, liquid, and gas data, see Johnson (ed.), WADD-TR-60-56, 1960. 7 For data from 400 to 2350°R see Stalder, NACA Tech. Note 4141, 1957. 8 For solid, liquid, and gas data, see Johnson (ed.), WADD-TR-60-56, 1960. 9 For liquid and gas data, see Johnson (ed.), WADD-TR-60-56, 1960. 10 For solid, liquid, and gas data, see Johnson (ed.), WADD-TR-60-56, 1960. 11 See also Table 2-195; Douglas, Ball, et al., Bur. Stand. J. Res., 46 (1951): 334; Busey and Giaque, J. Am. Chem. Soc., 75 (1953): 806; Sheldon, ASME Pap. 49-A-30, 1949. 12 For solid, liquid, and gas data, see Johnson (ed.), WADD-TR-56-60, 1960. 13 For solid, liquid, and gas data, see Johnson (ed.), WADD-TR-56-60, 1960. 14 For solid, liquid, and gas data, see Johnson (ed.), WADD-TR-56-60, 1960. Ozone: For liquid see Brabets and Waterman, J. Chem. Phys., 28 (1958): 1212. 15 For data on liquid Na-K alloys to 1500°F and for liquid Na to 1460°F, see Lubarsky and Kaufman, NACA Rep. 1270, 1956. 16 See also Evans and Wagman, Bur. Stand. J. Res. 49 (1952): 141; Gratch, OTS PB 124957, 1950; Guthrie, Scott, et al., J. Am. Chem. Soc., 76 (1954): 1488. 2

SPECIFIC HEATS OF PURE COMPOUNDS TABLE 2-195

2-169

Specific Heat [kJ/(kg·K)] of Selected Elements Temperature, K

Symbol

4

6

8

10

20

40

60

80

100

200

250

300

400

600

800

Al Be Bi Cr Co

0.00026 0.00008 0.00054 0.00016 0.00036

0.00050

0.00088

0.214

0.357

0.00541 0.00050 0.00085

0.0089 0.0014 0.0340 0.0021 0.0048

0.0775

0.00220 0.00029 0.00059

0.00140 0.00028 0.01040 0.00081 0.00121

0.0729 0.0107 0.0404

0.092 0.059 0.110

0.102 0.127 0.184

0.481 0.195 0.109 0.190 0.234

0.797 1.109 0.120 0.382 0.376

0.859 1.537 0.121 0.424 0.406

0.902 1.840 0.122 0.450 0.426

0.949 2.191 0.123 0.501 0.451

1.042 2.605 0.142 0.565 0.509

1.134 2.823 0.136 0.611 0.543

Cu Ge Au Ir Fe

0.00011

0.00024

0.00018

0.00047

0.00048 0.00037 0.00126

0.137 0.108 0.084

0.203 0.153 0.100

0.00061

0.00090

0.0076 0.0129 0.0163 0.0021 0.0039

0.059 0.0619 0.0569

0.00038

0.00086 0.00081 0.00255 0.00032 0.00127

0.0276

0.086

0.154

0.254 0.192 0.109 0.090 0.216

0.357 0.286 0.124 0.122 0.384

0.377 0.305 0.127 0.128 0.422

0.386 0.323 0.129 0.131 0.450

0.396 0.343 0.131 0.133 0.491

0.431 0.364 0.136 0.140 0.555

0.448 0.377 0.141 0.146 0.692

Pb Mg Hg Mo Ni

0.00075 0.00034 0.00417 0.00011 0.00054

0.00242 0.00080 0.01420 0.00019 0.00086

0.00747 0.00155 0.01820 0.00032 0.00121

0.01350 0.00172 0.02250 0.00050 0.00178

0.0531 0.0148 0.0515 0.0029 0.0058

0.0944 0.138 0.0895 0.0236 0.0380

0.108 0.336 0.107 0.061 0.103

0.114 0.513 0.116 0.105 0.173

0.118 0.648 0.121 0.140 0.232

0.125 0.929 0.136 0.223 0.383

0.127 0.985 0.141 0.241 0.416

1.129 1.005 0.139 0.248 0.444

0.132 1.082 0.136 0.261 0.490

0.142 1.177 0.135 0.280 0.590

1.263 0.104 0.292 0.530

Pt Ag Sn Zn

0.00019 0.00016 0.00024 0.00011

0.00028 0.00035 0.00127 0.00029

0.00067 0.00093 0.00423 0.00096

0.00112 0.00186 0.00776 0.00250

0.0077 0.0159 0.0400 0.0269

0.0382 0.0778 0.108 0.123

0.069 0.133 0.149 0.205

0.088 0.166 0.173 0.258

0.101 0.187 0.189 0.295

0.127 0.225 0.214 0.366

0.132 0.232 0.220 0.380

0.134 0.236 0.222 0.389

0.136 0.240 0.245 0.404

0.140 0.251 0.257 0.435

0.146 0.264 0.257 0.479

TABLE 2-196 Cmpd. no.

Heat Capacities of Inorganic and Organic Liquids Name

Formula

CAS no.

Mol wt

C1

C2

C3

C4

C5

Tmin, K

Cp at Tmin × 1E−05

Tmax, K

Cp at Tmax × 1E−05

1 2 3 4 5 6 7 8 9 10

Methane (eqn. 2) Ethane (eqn. 2) Propane (eqn. 2) n-Butane (eqn. 2) n-Pentane n-Hexane n-Heptane (eqn. 2) n-Octane n-Nonane n-Decane

CH4 C2H6 C3H8 C4H10 C5H12 C6H14 C7H16 C8H18 C9H20 C10H22

74828 74840 74986 106978 109660 110543 142825 111659 111842 124185

16.043 30.070 44.097 58.123 72.150 86.177 100.204 114.231 128.258 142.285

6.5708E+01 4.4009E+01 6.2983E+01 6.4730E+01 1.5908E+05 1.7212E+05 6.1260E+01 2.2483E+05 3.8308E+05 2.7862E+05

3.8883E+04 8.9718E+04 1.1363E+05 1.6184E+05 −2.7050E+02 −1.8378E+02 3.1441E+05 −1.8663E+02 −1.1398E+03 −1.9791E+02

−2.5795E+02 9.1877E+02 6.3321E+02 9.8341E+02 9.9537E−01 8.8734E−01 1.8246E+03 9.5891E−01 2.7101E+00 1.0737E+00

6.1407E+02 −1.8860E+03 −8.7346E+02 −1.4315E+03 0 0 −2.5479E+03 0 0 0

0 0 0 0 0 0 0 0 0 0

90.69 92 85.47 134.86 143.42 177.83 182.57 216.38 219.66 243.51

0.5361 0.6855 0.8488 1.1380 1.4076 1.6750 1.9989 2.2934 2.6348 2.9409

190 290 360 420 390 460 520 460 325 460

11 12 13 14 15 16 17 18 19 20

n-Undecane n-Dodecane n-Tridecane n-Tetradecane n-Pentadecane n-Hexadecane n-Heptadecane n-Octadecane n-Nonadecane n-Eicosane

C11H24 C12H26 C13H28 C14H30 C15H32 C16H34 C17H36 C18H38 C19H40 C20H42

1120214 112403 629505 629594 629629 544763 629787 593453 629925 112958

156.312 170.338 184.365 198.392 212.419 226.446 240.473 254.500 268.527 282.553

2.9398E+05 5.0821E+05 3.5018E+05 3.5314E+05 3.4691E+05 3.7035E+05 3.7697E+05 3.9943E+05 3.4257E+05 3.5272E+05

−1.1498E+02 −1.3687E+03 −1.0470E+02 2.9130E+01 2.1954E+02 2.3147E+02 3.4782E+02 3.7464E+02 7.6208E+02 8.0732E+02

9.6936E−01 3.1015E+00 1.0022E+00 8.6116E−01 6.5632E−01 6.8632E−01 5.7895E−01 5.8156E−01 2.0481E−01 2.1220E−01

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

247.57 263.57 267.76 279.01 283.07 291.31 295.13 301.31 305.04 309.58

3.2493 3.6292 3.9400 4.2831 4.6165 4.9602 5.3005 5.6511 5.9409 6.2299

433.42 330 508.62 526.73 543.84 560.01 575.3 589.86 603.05 616.93

4.2624 3.9429 5.5619 6.0741 6.6042 7.1521 7.6869 8.2276 8.7663 9.3154

21 22 23 24 25 26 27

2-Methylpropane 2-Methylbutane 2,3-Dimethylbutane 2-Methylpentane 2,3-Dimethylpentane1 2,3,3-Trimethylpentane 2,2,4-Trimethylpentane

C4H10 C5H12 C6H14 C6H14 C7H16 C8H18 C8H18

75285 78784 79298 107835 565593 560214 540841

58.123 72.150 86.177 86.177 100.204 114.231 114.231

1.7237E+05 1.0830E+05 1.2945E+05 1.4222E+05 1.4642E+05 3.8862E+05 9.5275E+04

−1.7839E+03 1.4600E+02 1.8500E+01 −4.7830E+01 5.9200E+01 −1.4395E+03 6.9670E+02

1.4759E+01 −2.9200E−01 6.0800E−01 7.3900E−01 6.0400E−01 3.2187E+00 −1.3765E+00

−4.7909E−02 1.5100E−03 0 0 0 0 2.1734E−03

5.8050E−05 0 0 0 0 0 0

113.54 113.25 145.19 119.55 90 280 165.78

0.9961 1.2328 1.4495 1.4706 1.5664 2.3791 1.8285

380 310 331.13 333.41 380 320 520

2.0725 1.7048 2.0224 2.0842 2.5613 2.5757 3.9095

28 29 30 31 32 33 34 35

Ethylene Propylene 1-Butene cis-2-Butene trans-2-Butene 1-Pentene 1-Hexene 1-Heptene

C2H4 C3H6 C4H8 C4H8 C4H8 C5H10 C6H12 C7H14

74851 115071 106989 590181 624646 109671 592416 592767

28.054 42.081 56.108 56.108 56.108 70.134 84.161 98.188

2.4739E+05 1.1720E+05 1.3589E+05 1.2668E+05 1.1276E+05 1.5467E+05 1.9263E+05 1.8997E+05

−4.4280E+03 −3.8632E+02 −4.7739E+02 −6.5470E+01 −1.0470E+02 −4.2600E+02 −5.7116E+02 −1.5670E+02

4.0936E+01 1.2348E+00 2.1835E+00 −6.4000E−01 5.2140E−01 1.9640E+00 2.4004E+00 3.4300E−01

−1.6970E−01 0 −2.2230E−03 2.9120E−03 0 −1.8038E−03 −1.9758E−03 1.5222E−03

2.6816E−04 0 0 0 0 0 0 0

103.97 87.89 87.8 134.26 167.62 107.93 133.39 154.27

0.7013 0.9279 1.0930 1.1340 1.0986 1.2930 1.5446 1.7955

252.7 298.15 300 350 274.03 310 336.63 330

0.9758 1.1178 1.2917 1.5022 1.2322 1.5761 1.9700 2.3032

36 37 38 39 40 41 42 43 44

1-Octene 1-Nonene 1-Decene 2-Methylpropene 2-Methyl-1-butene2 2-Methyl-2-butene2 1,2-Butadiene 1,3-Butadiene 2-Methyl-1,3-butadiene

C8H16 C9H18 C10H20 C4H8 C5H10 C5H10 C4H6 C4H6 C5H8

111660 124118 872059 115117 563462 513359 590192 106990 78795

112.215 126.242 140.269 56.108 70.134 70.134 54.092 54.092 68.119

3.7930E+05 2.5875E+05 3.1950E+05 8.7680E+04 1.4951E+05 1.5160E+05 1.3515E+05 1.2886E+05 1.4148E+05

−2.1175E+03 −3.5450E+02 −5.7621E+02 2.1710E+02 −2.4763E+02 −2.6672E+02 −3.1114E+02 −3.2310E+02 −2.8870E+02

8.2362E+00 1.3126E+00 1.7087E+00 −9.1530E−01 9.1849E−01 9.0847E−01 9.7007E−01 1.0150E+00 1.0910E+00

−9.0093E−03 0 0 2.2660E−03 0 0 −1.5230E−04 3.2000E−05 0

0 0 0 0 0 0 0 0 0

171.45 191.78 206.89 132.81 135.58 139.39 136.95 165 130.32

2.1295 2.3904 2.7343 1.0568 1.3282 1.3207 1.1034 1.0333 1.2239

315 420.02 443.75 343.15 304.31 311.71 290 350 307.2

2.4793 3.4142 4.0027 1.4596 1.5921 1.5673 1.2279 1.4148 1.5575

45 46 47 48 49 50

Acetylene Methylacetylene Dimethylacetylene 3-Methyl-1-butyne 1-Pentyne 2-Pentyne

C2H2 C3H4 C4H6 C5H8 C5H8 C5H8

74862 74997 503173 598232 627190 627214

26.038 40.065 54.092 68.119 68.119 68.119

2.0011E+05 7.9791E+04 8.8153E+04 1.0520E+05 8.6200E+04 6.8671E+04

−1.1988E+03 8.9490E+01 1.2416E+02 1.9110E+02 2.5660E+02 2.4666E+02

3.0027E+00 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

192.4 200 240.91 200 200 200

0.8061 0.9769 1.1806 1.4342 1.3752 1.1800

250 249.94 300.13 299.49 313.33 329.27

0.8808 1.0216 1.2542 1.6243 1.6660 1.4989

2-170

14.9780 1.2444 2.6079 5.0822 2.0498 2.7534 4.0657 3.4189 2.9890 4.1478

51 52 53 54 55 56

1-Hexyne 2-Hexyne 3-Hexyne 1-Heptyne 1-Octyne Vinylacetylene3

C6H10 C6H10 C6H10 C7H12 C8H14 C4H4

693027 764352 928494 628717 629050 689974

82.145 82.145 82.145 96.172 110.199 52.076

9.3000E+04 9.4860E+04 8.2795E+04 8.5122E+04 9.1748E+04 6.8720E+04

3.2600E+02 2.5415E+02 2.8340E+02 4.0247E+02 4.7140E+02 1.3500E+02

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

200 300 300 192.22 193.55 200

1.5820 1.7110 1.6781 1.6248 1.8299 0.9572

344.48 357.67 354.35 372.93 399.35 278.25

2.0530 1.8576 1.8322 2.3522 2.8000 1.0628

57 58 59 60 61 62 63 64 65 66

Cyclopentane Methylcyclopentane Ethylcyclopentane Cyclohexane Methylcyclohexane 1,1-Dimethylcyclohexane Ethylcyclohexane Cyclopentene 1-Methylcyclopentene Cyclohexene

C5H10 C6H12 C7H14 C6H12 C7H14 C8H16 C8H16 C5H8 C6H10 C6H10

287923 96377 1640897 110827 108872 590669 1678917 142290 693890 110838

70.134 84.161 98.188 84.161 98.188 112.215 112.215 68.119 82.145 82.145

1.2253E+05 1.5592E+05 1.7852E+05 −2.2060E+05 1.3134E+05 1.3450E+05 1.3236E+05 1.2538E+05 5.3271E+04 1.0585E+05

−4.0380E+02 −4.9000E+02 −5.1835E+02 3.1183E+03 −6.3100E+01 8.7650E+00 7.2740E+01 −3.4970E+02 3.2792E+02 −6.0000E+01

1.7344E+00 2.1383E+00 2.3255E+00 −9.4216E+00 8.1250E−01 8.1151E−01 6.4738E−01 1.1430E+00 0 6.8000E−01

−1.0975E−03 −1.5585E−03 −1.6818E−03 1.0687E−02 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

179.28 130.73 134.71 279.69 146.58 239.66 161.84 138.13 200 169.67

0.9956 1.2492 1.4678 1.4836 1.3955 1.8321 1.6109 0.9888 1.1885 1.1525

322.4 366.48 301.82 400 320 392.7 404.95 317.38 348.64 356.12

1.3584 1.8682 1.8767 2.0323 1.9435 2.6309 2.6798 1.2953 1.6760 1.7072

67 68 69 70 71 72 73

Benzene Toluene o-Xylene m-Xylene p-Xylene Ethylbenzene Propylbenzene

C6H6 C7H8 C8H10 C8H10 C8H10 C8H10 C9H12

71432 108883 95476 108383 106423 100414 103651

78.114 92.141 106.167 106.167 106.167 106.167 120.194

1.2944E+05 1.4014E+05 3.6500E+04 1.7555E+05 −3.5500E+04 1.3316E+05 2.3477E+05

−1.6950E+02 −1.5230E+02 1.0175E+03 −2.9950E+02 1.2872E+03 4.4507E+01 −8.0022E+02

6.4781E−01 6.9500E−01 −2.6300E+00 1.0880E+00 −2.5990E+00 3.9645E−01 3.4037

0 0 3.0200E−03 0 2.4260E−03 0 −3.1739E−03

0 0 0 0 0 0 0

278.68 178.18 248 225.3 286.41 178.15 173.59

1.3251 1.3507 1.7315 1.6330 1.7697 1.5367 1.8182

353.24 500 415 360 600 409.35 370

1.5040 2.3774 2.2166 2.0873 3.2520 2.1781 2.4389

74 75 76 77 78 79 80 81

1,2,4-Trimethylbenzene Isopropylbenzene 1,3,5-Trimethylbenzene p-Isopropyltoluene Naphthalene Biphenyl Styrene m-Terphenyl

C9H12 C9H12 C9H12 C10H14 C10H8 C12H10 C8H8 C18H14

95636 98828 108678 99876 91203 92524 100425 92068

120.194 120.194 120.194 134.221 128.174 154.211 104.152 230.309

1.7880E+05 1.8290E+05 1.4805E+05 1.4560E+05 2.9800E+04 1.2177E+05 1.1334E+05 1.9567E+05

−1.2847E+02 −1.7400E+02 1.9700E+01 2.4870E+02 5.2750E+02 4.2930E+02 2.9020E+02 5.9407E+02

8.3741E−01 9.1200E−01 6.2260E−01 1.8700E−01 0 0 −6.0510E−01 0

0 0 0 0 0 0 1.3567E−03 0

0 0 0 0 0 0 0 0

229.33 177.14 228.42 205.25 353.43 342.2 242.54 360

1.9338 1.8069 1.8503 2.0452 2.1623 2.6868 1.6749 4.0954

350 500 350 450.28 491.14 533.37 418.31 650

2.3642 3.2390 2.3121 2.9550 2.8888 3.5075 2.2816 5.8182

82 83 84 85 86 87 88

Methanol Ethanol 1-Propanol 1-Butanol 2-Butanol 2-Propanol 2-Methyl-2-propanol

CH4O C2H6O C3H8O C4H10O C4H10O C3H8O C4H10O

67561 64175 71238 71363 78922 67630 75650

32.042 46.069 60.096 74.123 74.123 60.096 74.123

1.0580E+05 1.0264E+05 1.5876E+05 1.9120E+05 2.0670E+05 7.2355E+05 −9.2546E+05

−3.6223E+02 −1.3963E+02 −6,3500E+02 −7.3040E+02 −1.0204E+03 −8.0950E+03 7.8949E+03

9.3790E−01 −3.0341E−02 1.9690E+00 2.2998E+00 3.2900E+00 3.6662E+01 −1.7661E+01

0 2.0386E−03 0 0 0 −6.6395E−02 1.3617E−02

0 0 0 0 0 4.4064E−05 0

175.47 159.05 146.95 184.51 158.45 185.28 298.96

0.7112 0.8787 1.0797 1.3473 1.2762 1.1189 2.2016

400 390 400 390.81 372.7 480 460

1.1097 1.6450 2.1980 2.5701 2.8340 2.8122 2.9455

89 90 91 92 93 94 95 96

1-Pentanol 2-Methyl-1-butanol 3-Methyl-1-butanol 1-Hexanol 1-Heptanol Cyclohexanol Ethylene glycol 1,2-Propylene glycol

C5H12O C5H12O C5H12O C6H14O C7H16O C6H12O C2H6O2 C3H8O2

71410 137326 123513 111273 111706 108930 107211 57556

88.150 88.150 88.150 102.177 116.203 100.161 62.068 76.095

2.0120E+05 8.2937E+04 −5.3777E+04 4.8466E+05 4.3790E+05 −4.0000E+04 3.5540E+04 5.8080E+04

−6.5130E+02 4.5998E+02 8.8342E+02 −2.7613E+03 −2.0947E+03 8.5300E+02 4.3678E+02 4.4520E+02

2.2750E+00 0.0000E+00 0 6.5555E+00 5.2090E+00 0 −1.8486E−01 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

200.14 250 295.52 228.55 239.15 296.6 260.15 213.15

1.6198 1.9793 2.0729 1.9599 2.3487 2.1300 1.3666 1.5297

389.15 401.85 350 320 370 434 493.15 460.75

2.9227 2.6778 2.5542 2.7233 3.7597 3.3020 2.0598 2.6321

Phenol o-Cresol m-Cresol p-Cresol

C6H6O C7H8O C7H8O C7H8O

108952 95487 108394 106445

94.113 108.140 108.140 108.140

1.0172E+05 −1.8515E+05 −2.4670E+05 2.5998E+05

3.1761E+02 3.1480E+03 3.2568E+03 −1.1123E+03

0 −8.0367E+00 −7.4202E+00 4.9427E+00

0 7.2540E−03 6.0467E−03 −5.4367E−03

0 0 0 0

314.06 304.2 285.39 307.93

2.0147 2.3297 2.1895 2.2740

425 400 400 400

2.3670 2.5243 2.5578 2.5794

97 98 99 100

2-171

TABLE 2-196

Heat Capacities of Inorganic and Organic Liquids (Continued )

Cmpd. no.

Name

Formula

CAS no.

Mol wt

C1

C2

C3

101 102 103 104 105 106 107 108 109 110 111 112

Dimethyl ether Methyl ethyl ether Methyl-n-propyl ether Methyl isopropyl ether Methyl-n-butyl ether Methyl isobutyl ether Methyl tert-butyl ether Diethyl ether Ethyl propyl ether Ethyl isopropyl ether Methyl phenyl ether Diphenyl ether

C2H6O C3H8O C4H10O C4H10O C5H12O C5H12O C5H12O C4H10O C5H12O C5H12O C7H8O C12H10O

115106 540670 557175 598538 628284 625445 1634044 60297 628320 625547 100663 101848

46.069 60.096 74.123 74.123 88.150 88.150 88.150 74.123 88.150 88.150 108.140 170.211

1.1010E+05 1.2977E+05 1.4411E+05 1.4344E+05 1.7785E+05 5.1380E+04 1.4012E+05 4.4400E+04 1.0368E+05 1.0625E+05 1.5094E+05 1.3416E+05

−1.5747E+02 −3.3196E+02 −1.0209E+02 −1.5407E+02 −1.7157E+02 4.5040E+02 −9.0000E+00 1.3010E+03 7.2630E+02 2.9215E+02 9.3455E+01 4.4767E+02

5.1853E−01 1.3869E+00 5.8113E−01 7.2550E−01 7.4379E−01 0 5.6300E−01 −5.5000E+00 −2.6047E+00 0 2.3602E−01 0

113 114 115 116 117 118 119 120 121 122

Formaldehyde4 Acetaldehyde 1-Propanal 1-Butanal 1-Pentanal 1-Hexanal 1-Heptanal 1-Octanal 1-Nonanal 1-Decanal

CH2O C2H4O C3H6O C4H8O C5H10O C6H12O C7H14O C8H16O C9H18O C10H20O

50000 75070 123386 123728 110623 66251 111717 124130 124196 112312

30.026 44.053 58.080 72.107 86.134 100.161 114.188 128.214 142.241 156.268

6.1900E+04 1.1510E+05 9.9306E+04 6.5682E+04 1.1205E+05 1.1770E+05 2.2236E+05 1.3065E+05 1.3682E+05 1.5046E+05

2.8300E+01 −4.3300E+02 1.1573E+02 1.3291E+03 2.5778E+02 3.2952E+02 −1.0517E+02 4.6361E+02 5.3129E+02 5.8663E+02

123 124 125 126 127 128 129 130 131 132 133 134

Acetone Methyl ethyl ketone 2-Pentanone Methyl isopropyl ketone 2-Hexanone Methyl isobutyl ketone 3-Methyl-2-pentanone 3-Pentanone Ethyl isopropyl ketone Diisopropyl ketone Cyclohexanone Methyl phenyl ketone

C3H6O C4H8O C5H10O C5H10O C6H12O C6H12O C6H12O C5H10O C6H12O C7H14O C6H10O C8H8O

67641 78933 107879 563804 591786 108101 565617 96220 565695 565800 108941 98862

58.080 72.107 86.134 86.134 100.161 100.161 100.161 86.134 100.161 114.188 98.145 120.151

1.3560E+05 1.3230E+05 1.9459E+05 1.8361E+05 2.7249E+05 1.2492E+05 9.9815E+04 1.9302E+05 8.3630E+04 1.7927E+05 1.0980E+05 7.2692E+04

135 136 137 138 139 140 141

Formic acid Acetic acid Propionic acid n-Butyric acid Isobutyric acid Benzoic acid Acetic anhydride

CH2O2 C2H4O2 C3H6O2 C4H8O2 C4H8O2 C7H6O2 C4H6O3

64186 64197 79094 107926 79312 65850 108247

46.026 60.053 74.079 88.106 88.106 122.123 102.090

142 143 144 145 146 147 148

Methyl formate Methyl acetate Methyl propionate Methyl-n-butyrate Ethyl formate Ethyl acetate Ethyl propionate

C2H4O2 C3H6O2 C4H8O2 C5H10O2 C3H6O2 C4H8O2 C5H10O2

107313 79209 554121 623427 109944 141786 105373

60.053 74.079 88.106 102.133 74.079 88.106 102.133

2-172

C5

Tmin, K

Cp at Tmin × 1E−05

Tmax, K

Cp at Tmax × 1E−05

0 0 0 0 0 0 0 8.7630E−03 4.0957E−03 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

131.65 218.9 133.97 127.93 157.48 300 164.55 156.92 145.65 298.15 298.15 300.03

0.9836 1.2356 1.4086 1.3560 1.6928 1.8650 1.5388 1.4698 1.6686 1.9335 1.9978 2.6847

250 328.35 312.2 310 343.35 370 328.35 460 320 326.15 484.2 570

1.0314 1.7030 1.6888 1.6540 2.0663 2.1803 1.9786 3.3202 2.0358 2.0153 2.5153 3.8933

0 1.4250E+00 0 −7.1579E+00 0 0 6.5074E−01 0 0 0

0 0 0 1.2755E−02 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

204 150.15 200 176.75 200 217.15 229.8 246 255.15 267.15

0.6767 0.8221 1.2245 1.4741 1.6361 1.8926 2.3256 2.4470 2.7238 3.0718

234 294 328.75 300 376.15 401.45 381.25 447.15 468.15 488.15

0.6852 1.1097 1.3735 1.6459 2.0901 2.4999 2.7685 3.3795 3.8554 4.3682

−1.7700E+02 2.0087E+02 −2.6386E+02 −2.6885E+02 −7.9070E+02 3.0410E+02 3.4672E+02 −1.7643E+02 3.9900E+02 2.8370E+01 2.6150E+02 3.3783E+02

2.8370E−01 −9.5970E−01 7.6808E−01 8.6080E−01 2.5834E+00 0 0 5.6690E−01 0 5.3750E−01 0 3.5572E−01

6.8900E−04 1.9533E−03 0 0 −2.0040E−03 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

178.45 186.48 196.29 181.15 220.87 298.15 298.15 234.18 298.15 204.81 290 298.2

1.1696 1.4905 1.7239 1.6316 2.0228 2.1559 2.0319 1.8279 2.0259 2.0763 1.8563 2.0506

329.44 373.15 375.46 367.55 382.62 390 390.55 375.14 425 410 486.5 532.12

1.3271 1.7511 2.0380 2.0108 2.3590 2.4352 2.3523 2.0661 2.5320 2.8126 2.3702 3.5318

7.8060E+04 1.3964E+05 2.1366E+05 2.3770E+05 1.2754E+05 −5.4800E+03 3.6600E+04

7.1540E+01 −3.2080E+02 −7.0270E+02 −7.4640E+02 −6.5350E+01 6.4712E+02 5.1100E+02

0 8.9850E−01 1.6605E+00 1.8290E+00 8.2867E−01 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

281.45 289.81 252.45 267.95 270 395.45 250

0.9820 1.2213 1.4209 1.6902 1.7031 2.5042 1.6435

380 391.05 414.32 436.42 427.65 450 350

1.0525 1.5159 2.0756 2.6031 2.5114 2.8572 2.1545

1.3020E+05 6.1260E+04 7.1140E+04 1.0293E+05 8.0000E+04 2.2623E+05 7.6330E+04

−3.9600E+02 2.7090E+02 3.3550E+02 1.2910E+02 2.2360E+02 −6.2480E+02 4.0010E+02

1.2100E+00 0 0 6.2516E−01 0 1.4720E+00 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

174.15 253.4 300 277.25 254.2 189.6 298.15

1.2991 1.7179 1.8678 1.3684 1.6068 1.9562

304.9 373.4 390 415.87 374.2 350.21 410

1.2195 1.6241 2.0198 2.6474 1.6367 1.8796 2.4037

C4

149 150 151 152 153 154 155

Ethyl-n-butyrate n-Propyl formate n-Propyl acetate n-Butyl acetate Methyl benzoate Ethyl benzoate Vinyl acetate

C6H12O2 C4H8O2 C5H10O2 C6H12O2 C8H8O2 C9H10O2 C4H6O2

105544 110747 109604 123864 93583 93890 108054

116.160 88.106 102.133 116.160 136.150 150.177 86.090

8.2434E+04 7.5700E+04 8.3400E+04 1.1730E+05 1.1950E+05 1.2450E+05 1.3630E+05

4.2245E+02 3.2610E+02 3.8410E+02 3.5220E+02 2.9400E+02 3.7060E+02 −1.0617E+02

2.0992E−01 0 0 0 0 0 7.5175E−01

156 157 158 159 160 161

Methylamine Dimethylamine Trimethylamine Ethylamine Diethylamine Triethylamine

CH5N C2H7N C3H9N C2H7N C4H11N C6H15N

74895 124403 75503 75047 109897 121448

31.057 45.084 59.111 45.084 73.138 101.192

9.2520E+04 −2.1487E+05 1.3605E+05 1.2170E+05 1.0133E+05 1.1148E+05

3.7450E+01 3.7872E+03 −2.8800E+02 3.8993E+01 2.4318E+02 3.6813E+02

0 −1.3781E+01 9.9130E−01 0 0 0

162 163 164 165 166 167 168

n-Propylamine di-n-Propylamine Isopropylamine Diisopropylamine Aniline N-Methylaniline N,N-Dimethylaniline

C3H9N C6H15N C3H9N C6H15N C6H7N C7H9N C8H11N

107108 142847 75310 108189 62533 100618 121697

59.111 101.192 59.111 101.192 93.128 107.155 121.182

1.3953E+05 4.9120E+04 −3.2469E+04 9.8434E+04 1.4150E+05 1.2850E+05 4.1860E+04

7.8000E+01 5.6224E+02 1.9771E+03 4.2904E+02 1.7120E+02 1.0020E+02 5.2750E+02

169 170 171 172

Ethylene oxide Furan Thiophene Pyridine

C2H4O C4H4O C4H4S C5H5N

75218 110009 110021 110861

44.053 68.075 84.142 79.101

1.4471E+05 1.1437E+05 8.1350E+04 1.0785E+05

173 174 175 176 177 178 179 180

Formamide5 N,N-Dimethylformamide Acetamide N-Methylacetamide Acetonitrile Propionitrile n-Butyronitrile Benzonitrile

CH3NO C3H7NO C2H5NO C3H7NO C2H3N C3H5N C4H7N C7H5N

75127 68122 60355 79163 75058 107120 109740 100470

45.041 73.095 59.068 73.095 41.053 55.079 69.106 103.123

181 182 183 184 185 186 187 188 189

Methyl mercaptan Ethyl mercaptan n-Propyl mercaptan n-Butyl mercaptan Isobutyl mercaptan sec-Butyl mercaptan2 Dimethyl sulfide Methyl ethyl sulfide Diethyl sulfide

CH4S C2H6S C3H8S C4H10S C4H10S C4H10S C2H6S C3H8S C4H10S

74931 75081 107039 109795 513440 513531 75183 624895 352932

190 191 192 193 194 195 196 197

Fluoromethane2 Chloromethane Trichloromethane Tetrachloromethane Bromomethane Fluoroethane Chloroethane Bromoethane

CH3F CH3Cl CHCl3 CCl4 CH3Br C2H5F C2H5Cl C2H5Br

593533 74873 67663 56235 74839 353366 75003 74964

0 0 0 0 0 0 0

0 0 0 0 0 0 0

285.5 298.15 274.7 289.58 260.75 238.45 259.56

2.2015 1.7293 1.8891 2.1929 1.9616 2.1287 1.5939

428.25 398.15 404.7 429.58 472.65 486.55 389.35

3.0185 2.0554 2.3885 2.6860 2.5846 3.0482 2.0892

0 1.6924E−02 0 0 0 0

0 0 0 0 0 0

179.69 180.96 156.08 192.15 223.35 200

0.9925 1.1947 1.1525 1.2919 1.5564 1.8511

266.82 298.15 276.02 289.73 328.6 361.92

1.0251 1.3779 1.3208 1.3300 1.8124 2.4471

0 0 −7.0145E+00 0 0 3.7400E−01 0

0 0 8.6913E−03 0 0 0 0

0 0 0 0 0 0 0

188.36 277.9 177.95 275 267.13 216.15 343.58

1.5422 2.0537 1.4621 2.1642 1.8723 1.6763 2.2310

340 407.9 320 357.05 457.15 469.02 513.58

1.6605 2.7846 1.6671 2.5162 2.1976 2.5777 3.1277

−7.5887E+02 −2.1569E+02 1.2980E+02 −3.4787E+01

2.8261E+00 7.2691E−01 −3.9000E−03 3.9565E−01

−3.0640E−03 0 0 0

0 0 0 0

160.65 187.55 234.94 231.51

0.8303 0.9949 1.1163 1.2100

283.85 304.5 357.31 388.41

0.8693 1.1609 1.2723 1.5403

6.3400E+04 1.4790E+05 1.0230E+05 6.2600E+04 9.7582E+04 1.1819E+05 1.0400E+05 7.6900E+04

1.5060E+02 −1.0600E+02 1.2870E+02 2.4340E+02 −1.2220E+02 −1.2098E+02 1.7400E+02 3.1420E+02

0 3.8400E−01 0 0 3.4085E−01 4.2075E−01 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

292 273.82 354.15 359 229.32 180.26 161.25 260.4

1.0738 1.4767 1.4788 1.4998 0.8748 1.1005 1.3206 1.5872

493 466.44 571 538.5 354.75 370.5 390.75 464.15

1.3765 1.8200 1.7579 1.9367 0.9713 1.3112 1.7199 2.2274

48.109 62.136 76.163 90.189 90.189 90.189 62.136 76.163 90.189

1.1530E+05 1.3467E+05 1.6733E+05 2.3219E+05 1.7336E+05 1.9789E+05 1.4695E+05 1.6124E+05 2.3852E+05

−2.6323E+02 −2.3439E+02 −3.1910E+02 −8.0435E+02 −2.1732E+02 −4.9154E+02 −3.8006E+02 −2.8861E+02 −1.0384E+03

6.0412E−01 5.9656E−01 8.1270E−01 2.7063E+00 7.0933E−01 1.7219E+00 1.2035E+00 7.8179E−01 4.0587E+00

0 0 0 −2.3017E−03 0 −1.2499E−03 −8.4787E−04 0 −4.4691E−03

0 0 0 0 0 0 0 0 0

150.18 125.26 159.95 157.46 128.31 133.02 174.88 167.23 181.95

0.8939 1.1467 1.3708 1.6365 1.5715 1.6003 1.1276 1.3484 1.5703

298.15 315.25 340.87 390 361.64 370 310.48 339.8 322.08

0.9052 1.2007 1.5299 1.9359 1.8754 1.8844 1.1959 1.5344 1.7579

34.033 50.488 119.377 153.822 94.939 48.060 64.514 108.966

7.4746E+04 9.6910E+04 1.2485E+05 −7.5270E+05 1.2973E+05 8.3303E+04 1.2790E+05 9.4364E+04

−1.3232E+02 −2.0790E+02 −1.6634E+02 8.9661E+03 −5.9654E+02 6.5454E+01 −3.4515E+02 −1.0912E+02

5.3772E−01 3.7456E−01 4.3209E−01 −3.0394E+01 2.1600E+00 0 9.1500E−01 4.4032E−01

0 4.8800E−04 0 3.4455E−02 −2.4234E−03 0 0 0

0 0 0 0 0 0 0 0

140 175.43 233.15 250.33 184.45 200 134.8 160

0.6676 0.7460 1.0956 1.2763 0.7798 0.9639 0.9800 0.8818

220 373.15 366.48 388.71 276.71 281.48 340 320

0.7166 0.9684 1.2192 1.6374 0.7870 1.0173 1.1632 1.0453

2-173

TABLE 2-196 Cmpd. no.

Heat Capacities of Inorganic and Organic Liquids (Concluded ) Name

Formula

CAS no.

Mol wt

C1

C2

C3

540545 75296 78999 78875 75014 462066 108907 108861

78.541 78.541 112.986 112.986 62.499 96.104 112.558 157.010

9.6344E+04 6.9362E+04 7.0010E+04 1.1094E+05 −1.0320E+04 −9.9120E+05 −1.3075E+06 1.2160E+05

1.1752E+02 2.1501E+02 2.6660E+02 8.3496E+00 3.2280E+02 1.1734E+04 1.5338E+04 −9.4500E+00

0 0 0 4.7218E−01 0 −4.0669E+01 −5.3974E+01 3.5800E−01

C5

Tmin, K

Cp at Tmin × 1E−05

Tmax, K

Cp at Tmax × 1E−05

0 0 0 0 0 4.7333E−02 6.3483E−02 0

0 0 0 0 0 0 0 0

230 200 280 286 200 239.99 227.95 293.15

1.2337 1.1236 1.4466 1.5195 0.5424 1.3675 1.3617 1.4960

319.67 308.85 420 429 400 319.99 360 495.08

1.3391 1.3577 1.8198 2.0142 1.1880 1.5018 1.8101 2.0467

C4

198 199 200 201 202 203 204 205

1-Chloropropane 2-Chloropropane 1,1-Dichloropropane 1,2-Dichloropropane Vinyl chloride Fluorobenzene Chlorobenzene Bromobenzene

C3H7Cl C3H7Cl C3H6Cl2 C3H6Cl2 C2H3Cl C6H5F C6H5Cl C6H5Br

206 207 208 209 210 211 212 213 214 215 216 217 218

Air Hydrogen (eqn. 2) Helium-46 Neon Argon Fluorine Chlorine Bromine Oxygen Nitrogen Ammonia (eqn. 2) Hydrazine Nitrous oxide

H2 He Ne Ar F2 Cl2 Br2 O2 N2 NH3 N2H4 N 2O

132259100 1333740 7440597 7440019 7440371 7782414 7782505 7726956 7782447 7727379 7664417 302012 10024972

28.951 2.016 4.003 20.180 39.948 37.997 70.905 159.808 31.999 28.014 17.031 32.045 44.013

−2.1446E+05 6.6653E+01 3.8722E+05 1.0341E+06 1.3439E+05 −9.4585E+04 6.3936E+04 3.7570E+04 1.7543E+05 2.8197E+05 6.1289E+01 7.9815E+04 6.7556E+04

9.1851E+03 6.7659E+03 −4.6557E+05 −1.3877E+05 −1.9894E+03 7.5299E+03 4.6350E+01 3.2850E+02 −6.1523E+03 −1.2281E+04 8.0925E+04 5.0929E+01 5.4373E+01

−1.0612E+02 −1.2363E+02 2.1180E+05 7.1540E+03 1.1043E+01 −1.3960E+02 −1.6230E−01 −6.7000E−01 1.1392E+02 2.4800E+02 7.9940E+02 4.3379E−02 0

4.1616E−01 4.7827E+02 −4.2494E+04 −1.6255E+02 0 1.1301E+00 0 0 −9.2382E−01 −2.2182E+00 −2.6510E+03 0 0

0 0 3.2129E+03 1.3841E+00 0 −3.3241E−03 0 0 2.7963E−03 7.4902E−03 0 0 0

75 13.95 2.2 24.56 83.78 58 172.12 265.9 54.36 63.15 203.15 274.69 182.3

0.5307 0.1262 0.1087 0.3666 0.4523 0.5541 0.6711 0.7755 0.5365 0.5593 0.7575 0.9708 0.7747

115 32 4.6 40 135 98 239.12 305.37 142 112 401.15 653.15 200

0.7132 1.3122 0.2965 0.6980 0.6708 0.5966 0.6574 0.7541 0.9066 0.7960 4.1847 1.3158 0.7843

219 220 221 222 223 224 225 226 227 228 229 230 231

Nitric oxide Cyanogen Carbon monoxide (eqn. 2) Carbon dioxide Carbon disulfide Hydrogen fluoride Hydrogen chloride Hydrogen bromide Hydrogen cyanide Hydrogen sulfide (eqn. 2) Sulfur dioxide Sulfur trioxide Water

NO C2N2 CO CO2 CS2 HF HCl HBr HCN H 2S SO2 SO3 H 2O

10102439 460195 630080 124389 75150 7664393 7647010 10035106 74908 7783064 7446095 7446119 7732185

30.006 52.036 28.010 44.010 76.143 20.006 36.461 80.912 27.026 34.082 64.065 80.064 18.015

−2.9796E+06 3.1322E+06 6.5429E+01 −8.3043E+06 8.5600E+04 6.2520E+04 4.7300E+04 5.7720E+04 9.5398E+04 6.4666E+01 8.5743E+04 2.5809E+05 2.7637E+05

7.6602E+04 −2.4320E+04 2.8723E+04 1.0437E+05 −1.2200E+02 −2.2302E+02 9.0000E+01 9.9000E+00 −1.9752E+02 4.9354E+04 5.7443E+00 0.0000E+00 −2.0901E+03

−6.5259E+02 4.8844E+01 −8.4739E+02 −4.3333E+02 5.6050E−01 6.2970E−01 0 0 3.8830E−01 2.2493E+01 0 0 8.1250E+00

1.8879E+00 0 1.9596E+03 6.0052E−01 −1.4520E−03 0 0 0 0 −1.6230E+03 0 0 −1.4116E−02

0 0 0 0 2.0080E−06 0 0 0 0 0 0 0 9.3701E−06

109.5 245.25 68.15 220 161.11 189.79 165 185.15 259.83 187.68 197.67 303.15 273.16

0.6229 1.0557 0.5912 0.7827 0.7577 0.4288 0.6215 0.5955 0.7029 0.6733 0.8688 2.5809 0.7615

150 300 132 290 552 292.67 185 206.45 298.85 370 350 303.15 533.15

1.9909 2.3216 6.4799 1.6603 1.3125 0.5119 0.6395 0.5976 0.7105 4.9183 0.8775 2.5809 0.8939

All substances are listed in alphabetical order in Table 2-6a. Compiled from Daubert, T. E., R. P. Danner, H. M. Sibul, and C. C. Stebbins, DIPPR Data Compilation of Pure Compound Properties, Project 801 Sponsor Release, July, 1993, Design Institute for Physical Property Data, AIChE, New York, NY; and from Thermodynamics Research Center, “Selected Values of Properties of Hydrocarbons and Related Compounds,” Thermodynamics Research Center Hydrocarbon Project, Texas A&M University, College Station, Texas (extant 1994). Temperatures are expressed in kelvins; liquid heat capacities are in J/kmol-K. J/(kmol·K) × 2.390E−04 = cal/(gmol·°C); J/(kmol·K) × 2.390059E−04 = Btu/(lbmol·°F). Equation 1, heat capacity = C1 + C2 × T + C3 × T 2 + C4 × T 3 + C5 × T 4, should be used except as otherwise specified. Equation 2 is heat capacity = C12/t + C2 − (2 × C1 × C3)t − (C1 × C4)t2 − (C32/3)t3 − (C3 × C4/2)t4 − (C42/5)t5. t = (1 − Tr) and Tr is the reduced temperature, T/Tc. 1 Coefficients are for the monomer and are hypothetical above 473 K. 2 For the saturated heat capacity. 3 Coefficients are hypothetical; compound decomposes violently on heating. 4 Coefficients are hypothetical and are based on predicted data. 5 Coefficients are hypothetical. 6 Exhibits superfluid properties below 2.2 K. 2-174

SPECIFIC HEATS OF PURE COMPOUNDS TABLE 2-197

Specific Heats of Organic Solids Recalculated from International Critical Tables, vol. 5, pp. 101–105

Compound

Formula

Acetic acid Acetone Aminobenzoic acid (o-) (m-) (p-) Aniline Anthracene

C2H4O2 C3H6O C7H7NO2 C7H7NO2 C7H7NO2 C6H7N C14H10

Anthraquinone Apiol Azobenzene

C14H8O2 C12H14O4 C12H10N2

Benzene

C6H6

Benzoic acid Benzophenone

C7H6O2 C13H10O

Betol

C17H12O3

Bromoiodobenzene (o-) (m-) (p-) Bromonaphthalene (β-) Bromophenol

C6H4BrI C6H4BrI C6H4BrI C10H7Br C6H5BrO

Camphene Capric acid Caprylic acid Carbon tetrachloride

C10H16 C10H20O2 C8H16O2 CCl4

Cerotic acid Chloral alcoholate hydrate Chloroacetic acid Chlorobenzoic acid (o-) (m-) (p-) Chlorobromobenzene (o-) (m-) (p-) Crotonic acid Cyamelide Cyanamide Cyanuric acid

C27H54O2 C4H7Cl3O2 C2H3Cl3O2 C2H3ClO2 C7H5ClO2 C7H5ClO2 C7H5ClO2 C6H4BrCl C6H4BrCl C6H4BrCl C4H6O2 C3H3N3O3 CH2N2 C3H3N3O3

Dextrin Dextrose

(C6H10O5)x C6H12O6

Dibenzyl Dibromobenzene (o-) (m-) (p-) Dichloroacetic acid Dichlorobenzene (o-) (m-) (p-) Dicyandiamide

C14H14 C6H4Br2 C6H4Br2 C6H4Br2 C2H2Cl2O2 C6H4Cl2 C6H4Cl2 C6H4Cl2 C2H4N4

Temperature, °C −200 to +25 −210 to −80 85 to mp 120 to mp 128 to mp

sp ht, cal/g °C

50 100 150 0 to 270 10 28

0.330 + 0.00080t 0.540 + 0.0156t 0.254 + 0.00136t 0.253 + 0.00122t 0.287 + 0.00088t 0.741 0.308 0.350 0.382 0.258 + 0.00069t 0.299 0.330

−250 −225 −200 −150 −100 −50 0 20 to mp −150 −100 −50 0 +20 −150 −100 0 +50 −50 to 0 −75 to −15 −40 to 50 41 32

0.0399 0.0908 0.124 0.170 0.227 0.299 0.375 0.287 + 0.00050t 0.115 0.172 0.220 0.275 0.303 0.129 0.167 0.248 0.308 0.143 + 0.00025t 0.143 0.116 + 0.00032t 0.260 0.263

35 8 −2 −240 −200 −160 −120 −80 −40 15 78 32 60 80 to mp 94 to mp 180 to mp −34 −52 −40 38 to 70 40 20 40

0.380 0.695 0.628 0.013 0.081 0.131 0.162 0.182 0.201 0.387 0.509 0.213 0.363 0.228 + 0.00084t 0.232 + 0.00073t 0.242 + 0.00055t 0.192 0.150 0.150 0.520 + 0.00020t 0.263 0.547 0.318

0 to 90 −250 −200 −100 0 20 28 −36 −25 −50 to +50

0.291 + 0.00096t 0.016 0.077 0.160 0.277 0.300 0.363 0.248 0.134 0.139 + 0.00038t 0.406 0.185 0.186 0.219 + 0.0021t 0.456

−48.5 −52 −50 to +53 0 to 204

2-175

2-176

PHYSICAL AND CHEMICAL DATA TABLE 2-197

Specific Heats of Organic Solids (Continued ) Recalculated from International Critical Tables, vol. 5, pp. 101–105

Compound

Formula

Dihydroxybenzene (o-) (m-) (p-)

C6H6O2 C6H6O2 C6H6O2

Di-iodobenzene (o-) (m-) (p-) Dimethyl oxalate Dimethylpyrene Dinitrobenzene (o-) (m-) (p-) Diphenyl Diphenylamine Dulcitol

C6H4I2 C6H4I2 C6H4I2 C4H6O4 C7H8O2 C6H4N2O4 C6H4N2O4 C6H4N2O4 C12H10 C12H11N C6H14O6

Erythritol Ethyl alcohol

C4H10O4 C2H6O (crystalline)

(vitreous)

Temperature, °C

sp ht, cal/g °C

−163 to mp −160 to mp −250 −240 −220 −200 −150 to mp −50 to +15 −52 to −42 −50 to +80 10 to 50 50 −160 to mp −160 to mp 119 to mp 40 26 20

0.278 + 0.00098t 0.269 + 0.00118t 0.025 0.038 0.061 0.081 0.268 + 0.00093t 0.109 + 0.00026t 0.100 + 0.00026t 0.101 + 0.00026t 0.212 + 0.0044t 0.368 0.252 + 0.00083t 0.248 + 0.00077t 0.259 + 0.00057t 0.385 0.337 0.282

60 −190 −180 −160 −140 −130 −190 −180 −175 −170 −190 to −40

0.351 0.232 0.248 0.282 0.318 0.376 0.260 0.296 0.380 0.399 0.366 + 0.00110t

Ethylene glycol

C2H6O2

Formic acid

CH2O2

−22 0

0.387 0.430

Glutaric acid Glycerol

C5H8O4 C3H8O3

20 −265 −260 −250 −220 −200 −100 0

0.299 0.009 0.022 0.047 0.085 0.115 0.217 0.330

Hexachloroethane Hexadecane Hydroxyacetanilide

C2Cl6 C16H34 C8H9NO2

Iodobenzene Isopropyl alcohol

C6H5I C3H8O

Lactose Lauric acid Levoglucosane Levulose

C12H22O11 C12H22O11·H2O C12H24O2 C6H10O5 C6H12O6

Malonic acid Maltose Mannitol Melamine Myristic acid Naphthalene Naphthol (α-) (β-) Naphthylamine (α-) Nitroaniline (o-) (m-) (p-) Nitrobenzoic acid (o-) (m-) (p-) Nitronaphthalene

25 41 to mp

0.174 0.495 0.249 + 0.00154t

40 −200 to −160

0.191 0.051 + 0.00165t

20 20 −30 to +40 40 20

0.287 0.299 0.430 + 0.000027t 0.607 0.275

C3H4O4 C12H22O11 C6H14O6 C3H6N6 C14H28O2

20 20 0 to 100 40 0 to 35

0.275 0.320 0.313 + 0.00025t 0.351 0.381 + 0.00545t

C10H8 C10H8O C10H8O C10H9N C6H6N2O2 C6H6N2O2 C6H6N2O2 C7H5NO4 C7H5NO4 C7H5NO4 C10H7NO2

−130 to mp 50 to mp 61 to mp 0 to 50 −160 to mp −160 to mp −160 to mp −163 to mp 66 to mp −160 to mp 0 to 55

0.281 + 0.00111t 0.240 + 0.00147t 0.252 + 0.00128t 0.270 + 0.0031t 0.269 + 0.000920t 0.275 + 0.000946t 0.276 + 0.001000t 0.256 + 0.00085t 0.258 + 0.00091t 0.247 + 0.00077t 0.236 + 0.00215t

SPECIFIC HEATS OF PURE COMPOUNDS TABLE 2-197

Specific Heats of Organic Solids (Concluded ) Recalculated from International Critical Tables, vol. 5, pp. 101–105

Compound

Formula

Temperature, °C

sp ht, cal/g °C

Oxalic acid

C2H2O4 C2H2O4.2H2O

−200 to +50 −200 −100 0 +50 100

0.259 + 0.00076t 0.117 0.239 0.338 0.385 0.416

Palmitic acid

C16H32O2

Phenol Phthalic acid Picric acid

C6H6O C8H6O4 C6H3N3O7

Propionic acid Propyl alcohol (n-)

C3H6O2 C3H8O

Pyrotartaric acid

C6H8O4

−180 −140 −100 −50 0 +20 14 to 26 20 −100 0 +50 100 120 −33 −200 −175 −150 −130 20

0.167 0.208 0.251 0.306 0.382 0.430 0.561 0.232 0.165 0.240 0.263 0.297 0.332 0.726 0.170 0.363 0.471 0.497 0.301

Quinhydrone

C12H10O4

Quinone

C6H4O2

−250 −225 −200 −100 0 −250 −225 −200 −150 to mp

0.017 0.061 0.098 0.191 0.256 0.031 0.082 0.113 0.282 + 0.00083t

Salol Stearic acid Succinic acid Sucrose Sugar (cane)

C13H10O3 C18H36O2 C4H6O4 C12H22O11 C12H22O11

32 15 0 to 160 20 22 to 51

0.289 0.399 0.248 + 0.00153t 0.299 0.301

Tartaric acid Tartaric acid

C4H6O6 C4H6O6·H2O

Tetrachloroethylene Tetryl

C2Cl4 C7H5N5O8

1 Tetryl + 1 picric acid 1 Tetryl + 2 TNT

C13H8N8O15 C21H15N11O20

Thymol Toluic acid (o-) (m-) (p-) Toluidine (p-)

C10H14O C8H8O2 C8H8O2 C8H8O2 C7H9N

Trichloroacetic acid Trimethyl carbinol Trinitrotoluene

C2HCl3O2 C4H10O C7H5N3O6

36 −150 −100 −50 0 +50 −40 to 0 −100 −50 0 +100 −100 to +100 −100 0 +50 0 to 49 54 to mp 54 to mp 130 to mp 0 20 40 solid −4 −100 −50 0 +100 −185 to +23 20 to 50 0 to 91

0.287 0.112 0.170 0.231 0.308 0.366 0.198 + 0.00018t 0.182 0.199 0.212 0.236 0.253 + 0.00072t 0.172 0.280 0.325 0.315 + 0.0031t 0.277 + 0.00120t 0.239 + 0.00195t 0.271 + 0.00106t 0.337 0.387 0.440 0.459 0.559 0.170 0.253 0.311 0.385 0.241 0.423 0.189 + 0.0027t

Trinitroxylene

C8H7N3O6

Triphenylmethane

C19H16

Urea

CH4N2O

20

0.320

2-177

TABLE 2-198 Cmpd. no.

Heat Capacities of Inorganic and Organic Compounds in the Ideal Gas State Name

Formula

CAS no.

Mol wt.

C1 × 1E−05

C2 × 1E−05

C3 × 1E−03

C4 × 1E−05

C5

Tmin, K

Cp at Tmin × 1E−05

Tmax, K

Cp at Tmax × 1E−05

1 2 3 4 5 6 7 8 9 10

Methane Ethane Propane n-Butane n-Pentane n-Hexane n-Heptane n-Octane n-Nonane n-Decane

CH4 C2H6 C3H8 C4H10 C5H12 C6H14 C7H16 C8H18 C9H20 C10H22

74828 74840 74986 106978 109660 110543 142825 111659 111842 124185

16.043 30.070 44.097 58.123 72.150 86.177 100.204 114.231 128.258 142.285

0.3330 0.4033 0.5192 0.7134 0.8805 1.0440 1.2015 1.3554 1.5175 1.6720

0.7993 1.3422 1.9245 2.4300 3.0110 3.5230 4.0010 4.4310 4.9150 5.3530

2.0869 1.6555 1.6265 1.6300 1.6502 1.6946 1.6766 1.6356 1.6448 1.6141

0.4160 0.7322 1.1680 1.5033 1.8920 2.3690 2.7400 3.0540 3.4700 3.7820

991.96 752.87 723.6 730.42 747.6 761.6 756.4 746.4 749.6 742

50 200 200 200 200 200 200 200 200 200

0.3330 0.4256 0.5632 0.7673 0.9404 1.1117 1.2828 1.4529 1.6257 1.7967

1500 1500 1500 1500 1500 1500 1500 1500 1500 1500

0.8890 1.4562 2.0556 2.6602 3.2927 3.8620 4.4283 4.9764 5.5407 6.0932

11 12 13 14 15 16 17 18 19 20

n-Undecane n-Dodecane n-Tridecane n-Tetradecane n-Pentadecane n-Hexadecane n-Heptadecane n-Octadecane n-Nonadecane n-Eicosane

C11H24 C12H26 C13H28 C14H30 C15H32 C16H34 C17H36 C18H38 C19H40 C20H42

1120214 112403 629505 629594 629629 544763 629787 593453 629925 112958

156.312 170.338 184.365 198.392 212.419 226.446 240.473 254.500 268.527 282.553

1.9529 2.1295 2.1496 2.3082 2.4679 2.6283 2.7878 2.9502 3.1062 3.2481

6.0998 6.6330 7.3045 7.8678 8.4212 8.9733 9.5247 10.0340 10.5750 11.0900

1.7087 1.7155 1.6695 1.6823 1.6865 1.6912 1.6935 0.7711 0.7679 1.6360

4.1302 4.5161 4.9998 5.4486 5.8537 6.2640 6.6651 −4.3012 −4.5661 7.4500

775.4 777.5 741.02 743.1 743.6 744.41 744.57 916.73 −912.03 −726.27

200 200 200 200 200 200 200 200 200 200

2.0594 2.2442 2.3156 2.4864 2.6586 2.8312 3.0034 3.1800 3.3533 3.5235

1500 1500 1500 1500 1500 1500 1500 1500 1500 1500

6.8342 7.4325 8.0251 8.6225 9.2209 9.8182 10.4160 11.0160 11.6130 12.2110

21 22 23 24 25 26 27

2-Methylpropane 2-Methylbutane 2,3-Dimethylbutane 2-Methylpentane 2,3-Dimethylpentane 2,3,3-Trimethylpentane 2,2,4-Trimethylpentane

C4H10 C5H12 C6H14 C6H14 C7H16 C8H18 C8H18

75285 78784 79298 107835 565593 560214 540841

58.123 72.150 86.177 86.177 100.204 114.231 114.231

0.6549 0.7460 0.7772 0.9030 0.8544 0.9820 1.1390

2.4776 3.2650 4.0320 3.8010 4.5772 5.4020 5.2860

1.5870 1.5450 1.5440 1.6020 1.5181 1.5310 1.5940

1.5750 1.9230 2.5080 2.4530 2.9740 3.4930 3.3510

−706.99 666.7 −649.95 −691.6 641.01 639.9 677.94

200 200 200 200 200 200 200

0.7218 0.8546 0.9363 1.0192 1.0550 1.2194 1.3139

1500 1500 1500 1500 1500 1500 1500

2.6656 3.3792 4.0353 3.9617 4.5983 5.3754 5.3769

28 29 30 31 32 33 34 35

Ethylene Propylene 1-Butene cis-2-Butene trans-2-Butene 1-Pentene 1-Hexene 1-Heptene

C2H4 C3H6 C4H8 C4H8 C4H8 C5H10 C6H12 C7H14

74851 115071 106989 590181 624646 109671 592416 592767

28.054 42.081 56.108 56.108 56.108 70.134 84.161 98.188

0.3338 0.4339 0.5998 0.5765 0.6592 0.7595 0.9180 1.0775

0.9479 1.5200 2.0846 2.1150 2.0700 2.5525 3.0220 3.4900

1.5960 1.4250 1.5884 1.6299 1.6733 1.5820 1.5742 1.5705

0.5510 0.7860 1.2940 1.2872 1.2510 1.6660 2.0320 2.4030

740.8 623.9 707.3 739.1 742.2 713 715 717.4

60 130 200 200 200 200 200 200

0.3338 0.4388 0.6547 0.6199 0.7004 0.8273 0.9995 1.1723

1500 1500 1500 1500 1500 1500 1500 1500

1.0987 1.6836 2.2853 2.2715 2.2904 2.8467 3.4088 3.9706

36 37 38 39 40 41 42 43 44

1-Octene 1-Nonene 1-Decene 2-Methylpropene 2-Methyl-1-butene 2-Methyl-2-butene 1,2-Butadiene 1,3-Butadiene 2-Methyl-1,3-butadiene

C8H16 C9H18 C10H20 C4H8 C5H10 C5H10 C4H6 C4H6 C5H8

111660 124118 872059 115117 563462 513359 590192 106990 78795

112.215 126.242 140.269 56.108 70.134 70.134 54.092 54.092 68.119

1.2355 1.3950 1.7573 0.6125 0.8703 0.8192 0.5750 0.5095 0.6527

3.9570 4.4255 5.1710 2.0660 2.5556 2.6038 1.6476 1.7050 2.2993

1.5640 1.5624 1.7664 1.5450 1.7757 1.7593 1.5270 1.5324 1.4943

2.7669 3.1370 3.6210 1.2057 1.7636 1.7195 0.9900 1.3370 1.5164

718.17 719.6 803.02 676 807.82 800.93 677.3 685.6 −647.15

200 200 200 200 200 200 200 200 200

1.3440 1.5168 1.8333 0.6763 0.9060 0.8559 0.6269 0.5756 0.7508

1500 1500 1500 1500 1500 1500 1500 1500 1500

4.5322 5.0938 5.8682 2.2814 2.8923 2.8709 1.9202 1.9555 2.5571

45 46 47 48 49 50

Acetylene Methylacetylene Dimethylacetylene 3-Methyl-1-butyne 1-Pentyne 2-Pentyne

C2H2 C3H4 C4H6 C5H8 C5H8 C5H8

74862 74997 503173 598232 627190 627214

26.038 40.065 54.092 68.119 68.119 68.119

0.3199 0.4478 0.6534 0.8274 0.7530 0.7074

0.5424 1.0917 1.6179 2.1377 2.0905 2.2229

1.5940 1.5508 1.7837 1.7550 1.5307 1.5570

0.4325 0.6750 1.0242 1.5149 1.3780 1.3125

607.1 658.2 821.4 782 672.8 690.78

200 200 200 200 200 200

0.3566 0.4882 0.6721 0.8646 0.8276 0.7700

1500 1500 1500 1500 1500 1500

0.7575 1.3293 1.9148 2.5255 2.4754 2.5052

51 52 53 54

1-Hexyne 2-Hexyne 3-Hexyne 1-Heptyne

C6H10 C6H10 C6H10 C7H12

693027 764352 928494 628717

82.145 82.145 82.145 96.172

0.9129 1.0360 0.9376 1.0712

2.5577 3.0090 3.0150 3.0258

1.5290 2.1160 1.9057 1.5273

1.7370 2.1060 1.9860 2.0975

683 902.4 817 689.62

200 300 300 200

1.0004 1.2215 1.1909 1.1721

1500 1500 1500 1500

3.0371 3.1894 3.1889 3.5985

2-178

55 56

1-Octyne Vinylacetylene

C8H14 C4H4

629050 689974

110.199 52.076

1.2307 0.5598

3.4942 1.2141

1.5280 1.6102

2.4617 0.8908

694.81 −710.4

200 200

1.3448 0.5967

1500 1500

4.1604 1.5590

57 58 59 60 61 62 63 64 65 66

Cyclopentane Methylcyclopentane Ethylcyclopentane Cyclohexane Methylcyclohexane 1,1-Dimethylcyclohexane Ethylcyclohexane Cyclopentene 1-Methylcyclopentene Cyclohexene

C5H10 C6H12 C7H14 C6H12 C7H14 C8H16 C8H16 C5H8 C6H10 C6H10

287923 96377 1640897 110827 108872 590669 1678917 142290 693890 110838

70.134 84.161 98.188 84.161 98.188 112.215 112.215 68.119 82.145 82.145

0.4160 0.6646 0.8205 0.4320 0.9227 1.0776 1.1059 0.4807 0.6941 0.5817

3.0140 3.5070 4.0342 3.7350 4.1150 4.6718 4.6306 2.5159 3.0209 3.1717

1.4617 1.5892 1.5670 1.1920 1.6504 1.6540 1.6628 1.5803 1.6903 1.5435

1.8095 2.3526 2.6697 1.6350 2.9006 3.3397 3.2990 1.7454 2.1209 2.1273

−668.8 727.13 715.52 −530.1 779.48 792.5 781.1 718.37 781.56 701.62

100 200 200 100 200 200 200 150 200 150

0.4165 0.7510 0.9272 0.4366 0.9953 1.1535 1.1875 0.4918 0.7464 0.5978

1500 1500 1500 1500 1500 1500 1500 1500 1500 1500

2.9298 3.5495 4.1472 3.6516 4.3180 4.9543 4.9184 2.5619 3.1496 3.2132

67 68 69 70 71 72 73

Benzene Toluene o-Xylene m-Xylene p-Xylene Ethylbenzene Propylbenzene (eqn. 3)

C6H6 C7H8 C8H10 C8H10 C8H10 C8H10 C9H12

71432 108883 95476 108383 106423 100414 103651

78.114 92.141 106.167 106.167 106.167 106.167 120.194

0.4442 0.5814 0.8521 0.7568 0.7512 0.7844 −21.4827

2.3205 2.8630 3.2954 3.3924 3.3970 3.3990 3.8070

1.4946 1.4406 1.4944 1.4960 1.4928 1.5590 54701

1.7213 1.8980 2.1150 2.2470 2.2470 2.4260 −0.001713

−678.15 −650.43 -675.8 −675.9 −675.1 −702 0

200 200 200 200 200 200 200

0.5340 0.7016 0.9643 0.8759 0.8710 0.8912 1.0802

1500 1500 1500 1500 1500 1500 1500

2.4169 3.0029 3.5965 3.5920 3.5923 3.6147 4.1537

74 75 76 77 78 79 80 81

1,2,4-Trimethylbenzene Isopropylbenzene 1,3,5-Trimethylbenzene p-Isopropyltoluene Naphthalene Biphenyl Styrene m-Terphenyl

C9H12 C9H12 C9H12 C10H14 C10H8 C12H10 C8H8 C18H14

95636 98828 108678 99876 91203 92524 100425 92068

120.194 120.194 120.194 134.221 128.174 154.211 104.152 230.309

1.0106 1.0810 0.9154 1.3186 0.6805 0.9060 0.8930 1.6397

3.8314 3.7932 3.9270 4.3036 3.5494 4.2634 2.1503 6.0125

1.5010 1.7505 1.4980 1.7734 1.4262 1.4553 0.7720 1.6902

2.3950 3.0027 2.5090 3.2570 2.5984 3.1550 0.9990 5.1314

678.3 794.8 676.9 811.9 650.1 661.2 2442 757.5

200 200 200 200 200 200 100 298.15

1.1354 1.1480 1.0474 1.3825 0.8454 1.0913 0.8931 2.4618

1500 1500 1500 1500 1500 1500 1500 1500

4.1854 4.1808 4.1807 4.7952 3.7359 4.5581 3.2416 6.6678

82 83 84 85 86 87 88

Methanol Ethanol 1-Propanol 1-Butanol 2-Butanol 2-Propanol 2-Methyl-2-propanol

CH4O C2H6O C3H8O C4H10O C4H10O C3H8O C4H10O

67561 64175 71238 71363 78922 67630 75650

32.042 46.069 60.096 74.123 74.123 60.096 74.123

0.3925 0.4920 0.6190 0.7454 0.8202 0.5723 0.7704

0.8790 1.4577 2.0213 2.5907 2.5220 1.9100 2.5390

1.9165 1.6628 1.6293 1.6073 1.6010 1.4210 1.5502

0.5365 0.9390 1.2956 1.7320 1.5864 1.2155 1.6690

896.7 744.7 727.4 712.4 −704.15 626 −679.3

200 200 200 200 200 150 200

0.3980 0.5224 0.6665 0.8162 0.8890 0.5924 0.8567

1500 1500 1500 1500 1500 1500 1500

1.0533 1.6576 2.2458 2.8509 2.8513 2.1792 2.8508

89 90 91 92 93 94 95 96

1-Pentanol 2-Methyl-1-butanol 3-Methyl-1-butanol 1-Hexanol 1-Heptanol Cyclohexanol Ethylene glycol 1,2-Propylene glycol

C5H12O C5H12O C5H12O C6H14O C7H16O C6H12O C2H6O2 C3H8O2

71410 137326 123513 111273 111706 108930 107211 57556

88.150 88.150 88.150 102.177 116.203 100.161 62.068 76.095

0.9060 1.0890 1.1060 1.0625 1.2215 0.9043 0.8200 2.0114

3.0620 2.1850 2.2100 3.5210 3.9910 2.5771 1.2780 0.8082

1.6054 0.8530 0.8760 1.5835 1.5800 0.7882 1.6980 1.8656

2.1150 1.4000 1.2200 2.4620 2.8350 1.3068 0.9290 −2.4404

−717.97 2906 2940 715.75 717.7 1952.2 −754 279.98

200 298.15 298.15 200 200 200 200 298.15

0.9890 1.3247 1.3213 1.1607 1.3330 0.9648 0.8481 1.0218

1500 1500.1 1200.15 1500 1500 1500 1500 1000.15

3.4133 3.4718 3.1770 3.9726 4.5346 3.8251 1.8521 2.1175

97 98 99 100

Phenol o-Cresol m-Cresol p-Cresol

C6H6O C7H8O C7H8O C7H8O

108952 95487 108394 106445

94.113 108.140 108.140 108.140

0.4340 0.7988 0.7515 0.7384

2.4450 2.8530 2.0900 2.9080

1.1520 1.4765 0.6666 1.4559

1.5120 2.0420 1.2120 2.0910

−507 −664.7 2214 −650.42

100 200 200 200

0.4401 0.9158 0.8701 0.8707

1500 1500 1500 1500

2.6045 3.2163 3.2075 3.2102

101 102 103 104 105 106

Dimethyl ether Methyl ethyl ether Methyl-n-propyl ether Methyl isopropyl ether Methyl-n-butyl ether Methyl isobutyl ether

C2H6O C3H8O C4H10O C4H10O C5H12O C5H12O

115106 540670 557175 598538 628284 625445

46.069 60.096 74.123 74.123 88.150 88.150

0.5148 0.6868 0.9215 0.8923 0.8205 0.7284

1.4420 1.9959 2.3943 2.4765 3.0869 3.1713

1.6034 1.5534 1.6936 1.6960 1.3864 1.3520

0.7747 1.1168 1.4896 1.5598 1.7886 1.8948

725.4 692.04 797.79 791.4 613.87 585.14

200 200 298 200 300 300

0.5436 0.7396 1.1251 0.9280 1.3300 1.3200

1500 1500 1200 1500 1200 1200

1.6581 2.2931 2.6391 2.8696 3.1994 3.1987 2-179

TABLE 2-198 Cmpd. no.

Heat Capacities of Inorganic and Organic Compounds in the Ideal Gas State (Continued ) Mol wt.

C1 × 1E−05

C2 × 1E−05

C3 × 1E−03

C4 × 1E−05

C5

1634044 60297 628320 625547 100663 101848

88.150 74.123 88.150 88.150 108.140 170.211

0.9933 0.8621 1.1320 1.0953 0.7637 1.0985

3.0667 2.5510 2.9400 3.0032 2.9377 4.3412

1.7426 1.5413 1.8270 1.7988 1.6051 1.6222

2.0764 1.4370 2.0550 2.1311 2.1700 3.6455

795.59 −688.9 −852 817.35 751.2 743.62

CH2O C2H4O C3H6O C4H8O C5H10O C6H12O C7H14O C8H16O C9H18O C10H20O

50000 75070 123386 123728 110623 66251 111717 124130 124196 112312

30.026 44.053 58.080 72.107 86.134 100.161 114.188 128.214 142.241 156.268

0.3327 0.4451 0.7174 0.8966 1.0743 1.2320 1.4040 1.6088 1.7347 1.9641

0.4954 1.0687 1.9140 2.3731 2.8363 2.2146 2.5907 4.2180 4.5115 5.1412

1.8666 1.6141 2.0144 1.9754 1.9549 0.8400 0.8315 1.9126 1.7120 1.8989

0.2808 0.6135 1.1708 1.5866 2.0146 1.2190 1.3120 3.2780 3.3256 4.1278

Acetone Methyl ethyl ketone 2-Pentanone Methyl isopropyl ketone 2-Hexanone Methyl isobutyl ketone 3-Methyl-2-pentanone 3-Pentanone Ethyl isopropyl ketone Diisopropyl ketone Cyclohexanone Methyl phenyl ketone

C3H6O C4H8O C5H10O C5H10O C6H12O C6H12O C6H12O C5H10O C6H12O C7H14O C6H10O C8H8O

67641 78933 107879 563804 591786 108101 565617 96220 565695 565800 108941 98862

58.080 72.107 86.134 86.134 100.161 100.161 100.161 86.134 100.161 114.188 98.145 120.151

0.5704 0.7840 0.9005 1.5914 1.0940 1.2270 1.0028 0.9690 1.2400 1.0869 0.5776 0.8540

1.6320 2.1032 2.7085 1.7640 1.8070 2.1950 3.3169 2.4907 3.2000 4.0540 3.3535 2.3340

1.6070 1.5488 1.6592 1.2076 0.6890 0.8420 1.6900 1.4177 1.9670 1.7802 1.2202 0.8310

135 136 137 138 139 140 141

Formic acid1 Acetic acid2 Propionic acid2 n-Butyric acid2 Isobutyric acid2 Benzoic acid Acetic anhydride

CH2O2 C2H4O2 C3H6O2 C4H8O2 C4H8O2 C7H6O2 C4H6O3

64186 64197 79094 107926 79312 65850 108247

46.026 60.053 74.079 88.106 88.106 122.123 102.090

0.3381 0.4020 0.6959 1.4880 0.7469 0.7759 0.7130

0.7593 1.3675 1.7778 1.3522 2.4356 2.6455 2.2220

142 143 144 145 146 147 148

Methyl formate Methyl acetate Methyl propionate Methyl n-butyrate Ethyl formate Ethyl acetate Ethyl propionate

C2H4O2 C3H6O2 C4H8O2 C5H10O2 C3H6O2 C4H8O2 C5H10O2

107313 79209 554121 623427 109944 141786 105373

60.053 74.079 88.106 102.133 74.079 88.106 102.133

0.5060 0.5550 0.7765 0.8940 0.5370 0.9981 0.9370

149 150 151 152 153 154 155

Ethyl n-butyrate n-Propyl formate n-Propyl acetate n-Butyl acetate Methyl benzoate Ethyl benzoate Vinyl acetate

C6H12O2 C4H8O2 C5H10O2 C6H12O2 C8H8O2 C9H10O2 C4H6O2

105544 110747 109604 123864 93583 93890 108054

116.160 88.106 102.133 116.160 136.150 150.177 86.090

156 157 158 159

Methylamine Dimethylamine Trimethylamine Ethylamine

CH5N C2H7N C3H9N C2H7N

74895 124403 75503 75047

31.057 45.084 59.111 45.084

Name

Formula

107 108 109 110 111 112

Methyl tert-butyl ether Diethyl ether Ethyl propyl ether Ethyl isopropyl ether Methyl phenyl ether Diphenyl ether

C5H12O C4H10O C5H12O C5H12O C7H8O C12H10O

113 114 115 116 117 118 119 120 121 122

Formaldehyde Acetaldehyde 1-Propanal 1-Butanal 1-Pentanal 1-Hexanal 1-Heptanal 1-Octanal 1-Nonanal 1-Decanal

123 124 125 126 127 128 129 130 131 132 133 134

2-180

CAS no.

Cp at Tmin × 1E−05

Tmax, K

Cp at Tmax × 1E−05

200 200 298.15 298.15 300 300

1.0394 0.9316 1.3538 1.3620 1.1302 1.7298

1500 1500 1500 1200 1200 1200

3.4321 2.9244 3.4535 3.2289 3.0226 4.5143

934.9 737.8 930.6 904.13 890.44 2205 2201 869 810.96 862.51

50 200 200 200 200 200 200 200 200 200

0.3327 0.4660 0.7266 0.9119 1.0960 1.2672 1.4479 1.6504 1.8005 2.0192

1500 1500 1500 1500 1500 1500 1500 1500 1500 1500

0.7113 1.2994 2.1149 2.6775 3.2404 3.7314 4.2863 4.9286 5.4439 6.0539

0.9680 1.1855 1.8012 −407.4000 1.4740 1.1910 2.3000 1.3010 2.3460 2.9786 1.5700 0.7730

731.5 693 743.96 10.503 1772 2460 770.7 646.7 896 791.6 586.92 2227

200 200 200 300 200 298.15 300 200 298.15 300 200 298.15

0.6049 0.8397 0.9591 1.1291 1.1815 1.4755 1.3604 1.0536 1.4479 1.5102 0.7321 1.1313

1500 1500 1500 1500 1200 1500.15 1200 1500 1200 1500 1500 1500

1.8820 2.4816 3.0797 2.9991 3.3207 3.6532 3.4275 3.0358 3.4234 4.3093 3.4870 3.2797

1.1925 1.2620 1.7098 1.1460 1.7150 1.7925 1.6203

0.3180 0.7003 1.2654 −678.0000 1.8484 2.2382 1.6760

550 569.7 −763.78 6.98 757.75 835.9 746.5

50 50 298.15 298.15 298.15 200 200

0.3381 0.4020 0.8938 1.1533 1.0427 0.8126 0.7665

1500 1500 1500 1200.1 1200 1500 1500

0.9933 1.5756 2.1248 2.4716 2.5383 2.9712 2.5675

1.2190 1.7820 2.4420 2.9100 1.8860 2.0931 2.8290

1.6370 1.2600 1.7140 1.5700 1.2070 2.0226 1.6480

0.8940 0.8530 1.8180 2.0730 0.8640 1.8030 2.1550

743 562 716 678.3 496 928.05 724.7

250 298 300 298 100 200 300

0.5888 0.8489 1.1242 1.3461 0.5412 1.0126 1.3377

1500 1500 1200 1200 1500 1500 1200

1.5109 2.0754 2.5276 3.0766 2.1485 2.6594 3.0569

1.1150 0.8710 1.7994 1.1684 0.9396 1.0944 0.5360

3.3910 2.4470 1.7530 3.7690 2.5590 4.1794 2.1190

1.6705 1.9254 1.1960 1.9560 0.8250 0.8838 1.1980

2.5180 1.8880 −4.1200 2.8180 1.3600 −1.6090 1.1470

298 298.15 298.15 300 300 300 100

1.5583 1.1022 1.3594 1.5358 1.2586 1.4598 0.5404

1200 1500 1500 1200 1200 1500 1500

3.6213 2.7484 3.2024 3.6724 3.3569 4.2540 2.3750

0.4100 0.5565 0.7107 0.5940

1.0578 1.6384 1.5051 1.6180

1.7080 1.7341 0.7966 1.8120

0.6836 1.0899 0.8454 1.0780

150 200 200 200

0.4136 0.5812 0.7439 0.6139

1500 1500 1500 1500

1.2388 1.8585 2.4322 1.8528

733.6 −821.3 108.2 811.2 3000 −1183.1 510 735 793.04 2187.6 820

Tmin, K

160 161

Diethylamine Triethylamine

C4H11N C6H15N

109897 121448

73.138 101.192

0.9102 1.2766

2.6740 2.5559

1.7190 0.8094

1.7926 1.4829

794.94 2231.7

200 200

0.9502 1.3278

1500 1500

3.0519 4.2046

162 163 164 165 166 167 168

n-Propylamine di-n-Propylamine Isopropylamine Diisopropylamine Aniline N-Methylaniline N,N-Dimethylaniline

C3H9N C6H15N C3H9N C6H15N C6H7N C7H9N C8H11N

107108 142847 75310 108189 62533 100618 121697

59.111 101.192 59.111 101.192 93.128 107.155 121.182

0.7608 1.2114 0.6855 1.1384 0.6533 0.7796 0.8742

2.1049 2.6127 2.1876 2.5747 2.5192 3.0280 2.7204

1.7256 0.7896 1.5831 0.7384 1.4608 1.5203 0.7242

1.3936 1.6903 1.3855 1.6200 1.8870 2.3280 1.1300

789.03 2394.4 691.76 2143 −653.1 699.8 1949

200 300 200 300 200 300 300

1500 1500 1500 1500 1500 1500 1500

2.4353 4.2484 2.4540 4.1941 2.8047 3.3641 3.8844

169 170 171 172

Ethylene oxide Furan Thiophene Pyridine

C2H4O C4H4O C4H4S C5H5N

75218 110009 110021 110861

44.053 68.075 84.142 79.101

0.3346 0.3727 0.4040 0.4413

1.2116 1.6606 1.6270 2.0830

1.6084 1.5112 1.4564 1.4783

0.8241 1.3145 1.3212 1.5330

737.3 686 649 676.8

50 200 200 200

0.7933 1.5900 0.7510 1.5995 0.7705 1.2602 1.3903 0.0000 0.3346 0.4376 0.4884 0.5220

1500 1500 1500 1500

1.3297 1.7940 1.8097 2.2194

173 174 175 176 177 178 179 180

Formamide N,N-Dimethylformamide Acetamide N-Methylacetamide Acetonitrile Propionitrile n-Butyronitrile Benzonitrile

CH3NO C3H7NO C2H5NO C3H7NO C2H3N C3H5N C4H7N C7H5N

75127 68122 60355 79163 75058 107120 109740 100470

45.041 73.095 59.068 73.095 41.053 55.079 69.106 103.123

0.3822 0.7220 0.3420 0.6116 0.4191 0.5357 0.6906 0.7186

0.9300 1.7830 1.2940 2.0290 0.8876 1.4617 1.9996 2.2700

1.8450 1.5320 1.0750 1.7683 1.5818 1.5530 1.5494 1.4669

0.6900 1.3100 0.6400 1.3302 0.5032 0.9120 1.3146 1.6930

850 762 502 835.5 699.8 678.2 675 −680.77

150 200 100 300 100 200 200 200

0.3833 0.7594 0.3448 0.7698 0.4192 0.5832 0.7607 0.8053

1500 1500 1500 1500 1500 1500 1500 1500

1.1203 2.2596 1.4997 2.2209 1.1285 1.7235 2.3273 2.6706

181 182 183 184 185 186 187 188 189

Methyl mercaptan Ethyl mercaptan n-Propyl mercaptan n-Butyl mercaptan Isobutyl mercaptan sec-Butyl mercaptan Dimethyl sulfide Methyl ethyl sulfide Diethyl sulfide

CH4S C2H6S C3H8S C4H10S C4H10S C4H10S C2H6S C3H8S C4H10S

74931 75081 107039 109795 513440 513531 75183 624895 352932

48.109 62.136 76.163 90.189 90.189 90.189 62.136 76.163 90.189

0.4146 0.5576 0.7474 0.9248 0.9142 0.9237 0.6037 0.7508 0.9429

0.8307 1.3617 1.9523 2.7795 2.4513 2.5166 1.3747 1.9577 2.6863

1.5890 1.5221 1.6310 1.6837 1.6265 1.6109 1.6410 1.6424 1.7624

0.4612 0.8073 1.2112 1.5974 1.6157 1.5641 0.7988 1.1949 1.6752

716.7 687.5 750.92 758.68 745.8 739.2 −743.5 749.19 −798.3

200 200 200 200 200 200 200 273.16 200

0.4329 0.5970 0.7848 0.9714 0.9660 0.9763 0.6298 0.9004 0.9794

1500 1500 1500 1500 1500 1500 1500 1500 1500

1.0781 1.6729 2.3216 3.1008 2.9095 2.9615 1.6949 2.3178 3.0338

190 191 192 193 194 195 196 197

Fluoromethane Chloromethane Trichloromethane Tetrachloromethane Bromomethane Fluoroethane Chloroethane Bromoethane

CH3F CH3Cl CHCl3 CCl4 CH3Br C2H5F C2H5Cl C2H5Br

593533 74873 67663 56235 74839 353366 75003 74964

34.033 50.488 119.377 153.822 94.939 48.060 64.514 108.966

0.3329 0.3409 0.3942 0.3758 0.3377 0.4437 0.4568 0.4719

0.7399 0.7246 0.6573 0.7054 0.7150 1.3119 1.2967 1.2787

1.8639 1.7230 0.9280 0.5121 1.5780 1.6422 1.5992 1.5957

0.4608 0.4480 0.4930 0.4850 0.4175 0.8544 0.8590 0.8517

891.16 780.5 399.6 236.1 691.4 738.77 708.8 703.87

50 150 100 100 100 200 100 200

0.3329 0.3424 0.4048 0.4730 0.3378 0.4726 0.4569 0.5089

1500 1500 1500 1500 1500 1500 1500 1500

0.9024 0.9097 1.0063 1.0662 0.9107 1.5008 1.5112 1.5121

198 199 200 201 202 203 204 205

1-Chloropropane 2-Chloropropane 1,1-Dichloropropane 1,2-Dichloropropane Vinyl chloride Fluorobenzene Chlorobenzene Bromobenzene

C3H7Cl C3H7Cl C3H6Cl2 C3H6Cl2 C2H3Cl C6H5F C6H5Cl C6H5Br

540545 75296 78999 78875 75014 462066 108907 108861

78.541 78.541 112.986 112.986 62.499 96.104 112.558 157.010

0.6210 0.6181 0.7145 0.7866 0.4236 0.6265 0.8011 0.7210

1.8430 1.8023 1.7344 1.7429 0.8735 2.1646 2.3100 2.0640

1.6290 1.5438 1.5240 1.7157 1.6492 1.5640 2.1570 1.6504

1.2337 1.1893 1.2230 1.2627 0.6556 1.7278 2.0460 1.6870

724 685.93 674.2 765.1 739.07 −724.29 −897.6 765.3

200 200 150 200 200 200 200 200

0.6674 0.6768 0.7268 0.8217 0.4457 0.6914 0.8219 0.7679

1500 1500 1500 1500 1500 1500 1500 1500

2.1126 2.1023 2.1609 2.1894 1.1423 2.4736 2.5327 2.4628

206 207 208 209 210 211

Air Hydrogen3 Helium-4 (eqn 2) Neon Argon Fluorine

H2 He Ne Ar F2

132259100 1333740 7440597 7440019 7440371 7782414

28.951 2.016 4.003 20.180 39.948 37.997

0.2896 0.2762 0.2079 0.2079 0.2079 0.2912

0.0939 0.0956 0 0 0 0.1013

3.0120 2.4660 0 0 0 1.4530

0.0758 0.0376 0 0 0 0.0941

1484 567.6 0 0 0 662.91

50 250 100 100 100 50

0.2896 0.2843 0.2079 0.2079 0.2079 0.2912

1500 1500 1500 1500 1500 1500

0.3496 0.3225 0.2079 0.2079 0.2079 0.3812 2-181

TABLE 2-198 Cmpd. no.

Heat Capacities of Inorganic and Organic Compounds in the Ideal Gas State (Concluded ) Name

Formula

CAS no.

Mol wt.

C1 × 1E−05

C2 × 1E−05

C3 × 1E−03

C4 × 1E−05

C5

Tmin, K

Cp at Tmin × 1E−05

Tmax, K

Cp at Tmax × 1E−05

212 213 214 215 216 217 218

Chlorine Bromine Oxygen Nitrogen Ammonia Hydrazine Nitrous oxide

Cl2 Br2 O2 N2 NH3 N2H4 N2O

7782505 7726956 7782447 7727379 7664417 302012 10024972

70.905 159.808 31.999 28.014 17.031 32.045 44.013

0.2914 0.3011 0.2910 0.2911 0.3343 0.3871 0.2934

0.0918 0.0801 0.1004 0.0861 0.4898 0.8576 0.3236

0.9490 0.7514 2.5265 1.7016 2.0360 1.7228 1.1238

0.1003 0.1078 0.0936 0.0010 0.2256 0.5664 0.2177

425 314.6 1153.8 909.79 882 733.53 479.4

50 100 50 50 100 200 100

0.2914 0.3090 0.2910 0.2911 0.3343 0.4070 0.2948

1500 1500 1500 1500 1500 1500 1500

0.3793 0.3794 0.3653 0.3484 0.6647 1.0571 0.5828

219 220 221 222 223 224 225 226 227 228 229 230 231

Nitric oxide (eqn 2) Cyanogen Carbon monoxide Carbon dioxide Carbon disulfide Hydrogen fluoride Hydrogen chloride Hydrogen bromide Hydrogen cyanide Hydrogen sulfide Sulfur dioxide Sulfur trioxide Water

NO C2N2 CO CO2 CS2 HF HCl HBr HCN H2S SO2 SO3 H2O

10102439 460195 630080 124389 75150 7664393 7647010 10035106 74908 7783064 7446095 7446119 7732185

30.006 52.036 28.010 44.010 76.143 20.006 36.461 80.912 27.026 34.082 64.065 80.064 18.015

0.3498 0.3545 0.2911 0.2937 0.3010 0.2913 0.2916 0.2912 0.3013 0.3329 0.3338 0.3341 0.3336

−3.5320E−04 0.5015 0.0877 0.3454 0.3338 0.0933 0.0905 0.0953 0.3171 0.2609 0.2586 0.4968 0.2679

7.7290E−05 1.0570 3.0851 1.4280 0.8960 2.9050 2.0938 2.1420 1.6102 0.9134 0.9328 0.8732 2.6105

−5.7357E−10 0.4520 0.0846 0.2640 0.2893 0.0020 −0.0011 0.0157 0.2179 −0.1798 0.1088 0.2856 0.0890

1.4526E−08 −396 1538.2 588 374.7 1326 120 1400 626 949.4 423.7 393.74 1169

100 100 60 50 100 50 50 50 100 100 100 100 100

0.3217 0.3648 0.2911 0.2937 0.3100 0.2913 0.2914 0.2912 0.3014 0.3329 0.3354 0.3408 0.3336

1500 1500 1500 5000 1500 1500 1500 1500 1500 1500 1500 1500 2273.15

0.3586 0.8100 0.3521 0.6335 0.6148 0.3224 0.3406 0.3479 0.5522 0.5143 0.5695 0.7967 0.5276

All substances are listed in alphabetical order in Table 2-6a. Compiled from Daubert, T. E., R. P. Danner, H. M. Sibul, and C. C. Stebbins, DIPPR Data Compilation of Pure Compound Properties, Project 801 Sponsor Release, July, 1993, Design Institute for Physical Property Data, AIChE, New York, NY; and from Thermodynamics Research Center, “Selected Values of Properties of Hydrocarbons and Related Compounds,” Thermodynamics Research Center Hydrocarbon Project, Texas A&M University, College Station, Texas (extant 1994). Temperatures are expressed in kelvins; heat capacities, in J/kmol-K. J/(kmol·K) × 2.390E−04 = cal/(gmol·°C); J/(kmol·K) × 2.390059E−04 = Btu/(lbmol·°F). cosh    unless otherwise specified.     + C4 T  T

C3 C3 Use heat capacity = C1 + C2  sinh  T T

2

C5

C5

2

Equation 2 is heat capacity = C1 + C2 × T + C3 × T 2 + C4 × T 3 + C5 × T 4. Equation 3 is heat capacity = C1 + C2 × ln T + C3/T + C4 × T. 1 For the monomer. Monomer and dimer are in equilibrium below 600 K. 2 For the monomer. 3 For equilibrium mixture of ortho and para hydrogen. 2-182

SPECIFIC HEATS OF PURE COMPOUNDS

2-183

TABLE 2-199 Cp /Cv : Ratios of Specific Heats of Gases at 1-atm Pressure*

Compound

Formula

Acetaldehyde Acetic acid Acetylene

C2H4O C2H4O2 C2H2

Air

Ammonia Argon

NH3 A

Temperature, °C

Ratio of specific heats, (γ) = Cp /Cv

30 136 15 −71 925 17 −78 −118 15 15 −180 0–100

1.14 1.15 1.26 1.31 1.36 1.403 1.408 1.415 1.310 1.668 1.76 (?) 1.67

Benzene Bromine

C6H6 Br2

90 20–350

1.10 1.32

Carbon dioxide

CO2

disulfide monoxide

CS2 CO

1.304 1.37 1.21 1.404 1.41 1.355 1.15 1.256 1.08

Chlorine Chloroform Cyanogen Cyclohexane

Cl2 CHCl3 (CN)2 C6H12

15 −75 100 15 −180 15 100 15 80

Dichlorodifluormethane

CCl2F2

25

1.139

Ethane

C2H6

Ethyl alcohol ether

C2H6O C4H10O

Ethylene

C2H4

100 15 −82 90 35 80 100 15 −91

1.19 1.22 1.28 1.13 1.08 1.086 1.18 1.255 1.35

Helium Hexane (n-) Hydrogen

He C6H14 H2

bromide chloride

HBr HCl

cyanide

HCN

−180 80 15 −76 −181 20 15 100 65 140 210

1.660 1.08 1.410 1.453 1.597 1.42 1.41 1.40 1.31 1.28 1.24

Compound

Formula

Hydrogen (Cont.) iodide sulfide

HI H2S

Iodine Isobutane

I2 C4H10

Krypton

Kr

Mercury Methane

Hg CH4

Methyl acetate alcohol ether Methylal

C3H6O2 CH4O C2H6O C3H8O2

Neon Nitric oxide

Ne NO

Nitrogen

N2

Nitrous oxide

N2O

Oxygen

O2

Pentane (n-) Phosphorus Potassium

C5H12 P K

Sodium Sulfur dioxide

Na SO2

Xenon

Xe

Temperature, °C

Ratio of specific heats, (γ) = Cp /Cv

20–100 15 −45 −57

1.40 1.32 1.30 1.29

185 15

1.30 1.11

19

1.68

360 600 300 15 −80 −115 15 77 6–30 13 40

1.67 1.113 1.16 1.31 1.34 1.41 1.14 1.203 1.11 1.06 1.09

19 15 −45 −80 15 −181 100 15 −30 −70

1.64 1.400 1.39 1.38 1.404 1.47 1.28 1.303 1.31 1.34

15 −76 −181

1.401 1.415 1.45

86 300 850

1.086 1.17 1.77

750–920 15

1.68 1.29

19

1.66

*From International Critical Tables, vol. 5, pp. 80–82.

TABLE 2-200

Specific Heat Ratio, Cp /Cv , for Air Pressure, bar

Temperature, K

1

10

20

40

60

80

100

150

200

250

300

400

500

600

800

1000

150 200 250 300 350

1.410 1.406 1.403 1.402 1.399

1.510 1.452 1.429 1.418 1.411

1.668 1.505 1.457 1.436 1.422

2.333 1.630 1.517 1.470 1.446

4.120 1.781 1.577 1.505 1.467

3.973 1.943 1.640 1.537 1.488

3.202 2.093 1.699 1.570 1.509

2.507 2.274 1.816 1.640 1.553

2.243 2.236 1.877 1.687 1.589

2.091 2.140 1.896 1.716 1.612

1.988 2.050 1.885 1.730 1.627

1.851 1.920 1.836 1.727 1.640

1.768 1.832 1.782 1.707 1.638

1.712 1.771 1.743 1.683 1.629

1.654 1.682 1.681 1.645 1.605

1.639 1.619 1.636 1.619 1.585

400 450 500 600 800

1.395 1.392 1.387 1.377 1.353

1.404 1.397 1.391 1.378 1.355

1.412 1.404 1.395 1.382 1.357

1.429 1.416 1.406 1.386 1.359

1.444 1.428 1.414 1.392 1.361

1.460 1.438 1.421 1.398 1.365

1.472 1.449 1.430 1.403 1.366

1.505 1.471 1.448 1.413 1.372

1.529 1.490 1.463 1.423 1.375

1.548 1.505 1.474 1.432 1.381

1.563 1.518 1.484 1.439 1.384

1.579 1.533 1.499 1.448 1.392

1.584 1.541 1.507 1.457 1.397

1.580 1.542 1.510 1.461 1.401

1.567 1.537 1.510 1.465 1.406

1.555 1.528 1.504 1.466 1.409

1000

1.336

1.337

1.338

1.339

1.342

1.343

1.343

1.345

1.348

1.350

1.354

1.358

1.361

1.365

1.368

1.372

Calculated from Cp, Cv values of Sychev, V. V., A. A. Vasserman, et al., “Thermodynamic Properties of Air,” Standartov, Moscow, 1978 and Hemisphere, New York, 1988 (276 pp.).

2-184

PHYSICAL AND CHEMICAL DATA

SPECIFIC HEATS OF AQUEOUS SOLUTIONS UNITS CONVERSIONS

ADDITIONAL REFERENCES

For this subsection, the following units conversions are applicable: °F = 9⁄5 °C + 32. To convert calories per gram-degree Celsius to British thermal units per pound-degree Fahrenheit, multiply by 1.0.

For additional data, see International Critical Tables, vol. 5, pp. 115–116, 122–125.

TABLE 2-201

TABLE 2-208

Acetic Acid (at 38°C)

Mole % acetic acid Cal/g °C

TABLE 2-202

0 1.0

6.98 0.911

30.9 0.73

54.5 0.631

100 0.535

Ammonia Specific heat, cal/g °C

Mole % NH3

2.4°C

20.6°C

41°C

61°C

0 10.5 20.9 31.2 41.4

1.01 0.98 .96 .956 .985

1.0 0.995 .99 1.0

0.995 1.06 1.03

1.0 1.02

TABLE 2-203

TABLE 2-204

100 0.497

5°C

20°C

40°C

5.88 12.3 27.3 45.8 69.6 100

1.02 0.975 .877 .776 .681 .576

1.0 0.982 .917 .811 .708 .60

0.995 .98 .92 .83 .726 .617

TABLE 2-209

95 0.52

90.5 0.53

82.3 0.56

75.2 0.581

Copper Sulfate

Composition CuSO4 + 50H2O CuSO4 + 200H2O CuSO4 + 400H2O TABLE 2-205

Specific heat, cal/g °C Mole % CH3OH

Aniline (at 20°C)

Mol % aniline Cal/g °C

Methyl Alcohol

Temperature

Specific heat, cal/g °C

12° to 15°C 12° to 14°C 13° to 17°C

0.848 .951 .975

Nitric Acid

% HNO3 by Weight

Specific Heat at 20°C, cal/g °C

0 10 20 30 40 50 60 70 80 90

1.000 0.900 .810 .730 .675 .650 .640 .615 .575 .515

Ethyl Alcohol Specific heat, cal/g °C

Mole % C2H5OH 4.16 11.5 37.0 61.0 100.0 TABLE 2-206

3°C

23°C

41°C

1.05 1.02 0.805 .67 .54

1.02 1.03 0.86 .727 .577

1.02 1.03 0.875 .748 .621

Glycerol Specific heat, cal/g °C

Mole % C3H5(OH)3

15°C

32°C

2.12 4.66 11.5 22.7 43.9 100.0

0.961 .929 .851 .765 .67 .555

0.960 .924 .841 .758 .672 .576

TABLE 2-207

Hydrochloric Acid Specific heat, cal/g °C

Mole % HCl 0.0 9.09 16.7 20.0 25.9

0°C

10°C

20°C

40°C

60°C

1.00 0.72 .61 .58 .55

0.72 .605 .575

0.74 .631 .591

0.75 .645 .615

0.78 .67 .638 .61

TABLE 2-210

Phosphoric Acid*

%H2PO4

Cp at 21.3°C cal/g °C

%H3PO4

Cp at 21.3°C cal/g °C

2.50 3.80 5.33 8.81 10.27 14.39 16.23 19.99 22.10 24.56 25.98 28.15 29.96 32.09 33.95 36.26 38.10 40.10 42.08 44.11 46.22 48.16 49.79

0.9903 .9970 .9669 .9389 .9293 .8958 .8796 .8489 .8300 .8125 .8004 .7856 .7735 .7590 .7432 .7270 .7160 .7024 .6877 .6748 .6607 .6475 .6370

50.00 52.19 53.72 56.04 58.06 60.23 62.10 64.14 66.13 68.14 69.97 69.50 71.88 73.71 75.79 77.69 79.54 80.00 82.00 84.00 85.98 88.01 89.72

0.6350 .6220 .6113 .5972 .5831 .5704 .5603 .5460 .5349 .5242 .5157 .5160 .5046 .4940 .4847 .4786 .4680 .4686 .4593 .4500 .4419 .4359 .4206

*Z. Physik. Chem., A167, 42 (1933).

SPECIFIC HEATS OF AQUEOUS SOLUTIONS TABLE 2-211

Potassium Chloride

TABLE 2-215

Sodium Chloride Specific heat, cal/g °C

Specific heat, cal/g °C Mole % KCl

6°C

20°C

33°C

40°C

Mole % NaCl

0.99 3.85 5.66 7.41

0.945 .828 .77

0.947 .831 .775 .727

0.947 .835 .778

0.947 .837 .775

0.249 .99 2.44 9.09

TABLE 2-212

Potassium Hydroxide (at 19°C)

Mole % KOH Cal/g °C

TABLE 2-213

0 1.0

0.497 0.975

1.64 0.93

TABLE 2-216 4.76 0.814

9.09 0.75

Normal Propyl Alcohol

Mole % NaOH Cal/g °C

TABLE 2-217

Mole % C3H7OH

5°C

20°C

40°C

1.55 5.03 11.4 23.1 41.2 73.0 100.0

1.03 1.07 1.035 0.877 .75 .612 .534

1.02 1.06 1.032 0.90 .78 .645 .57

1.01 1.03 0.99 .91 .815 .708 .621

% Na2CO3 by weight 0.000 1.498 2.000 2.901 4.000 5.000 6.000 8.000 10.000 13.790 13.840 20.000 25.000

Sodium Carbonate* Temperature, °C 17.6

30.0

76.6

98.0

0.9992 .9807

0.9986

1.0098

1.0084

.9786 .9597

20°C

33°C

57°C

0.96 .91 .805

0.99 .97 .915 .81

0.97 .915 .81

0.923 .82

0 1.0

0.5 0.985

1.0 0.97

9.09 0.835

16.7 0.80

28.6 0.784

37.5 0.782

Sulfuric Acid*

%H2SO4

Cp at 20°C, cal/g °C

%H2SO4

Cp at 20°C, cal/g °C

0.34 0.68 1.34 2.65 3.50 5.16 9.82 15.36 21.40 22.27 23.22 24.25 25.39 26.63 28.00 29.52 30.34 31.20 33.11

0.9968 .9937 .9877 .9762 .9688 .9549 .9177 .8767 .8339 .8275 .8205 .8127 .8041 .7945 .7837 .7717 .7647 .7579 .7422

35.25 37.69 40.49 43.75 47.57 52.13 57.65 64.47 73.13 77.91 81.33 82.49 84.48 85.48 89.36 91.81 94.82 97.44 100.00

0.7238 .7023 .6770 .6476 .6153 .5801 .5420 .5012 .4628 .4518 .4481 .4467 .4408 .4346 .4016 .3787 .3554 .3404 .3352

*Vinal and Craig, Bur. Standards J. Research, 24, 475 (1940).

.9594 .9428

6°C

Sodium Hydroxide (at 20°C)

Specific heat, cal/g °C

TABLE 2-214

2-185

0.9761 .9392

.9183 .9086 .8924

*J. Chem. Soc. 3062–3079 (1931).

.9452 .8881 .8631

TABLE 2-218 .8936 .8615

0.8911

Zinc Sulfate

Composition

Temperature

Specific heat, cal/g °C

ZnSO4 + 50H2O ZnSO4 + 200H2O

20° to 52°C 20° to 52°C

0.842 .952

2-186

PHYSICAL AND CHEMICAL DATA

SPECIFIC HEATS OF MISCELLANEOUS MATERIALS TABLE 2-219 and Solids

Specific Heats of Miscellaneous Liquids

Material Alumina Alundum Asbestos Asphalt Bakelite Brickwork Carbon (gas retort) (see under Graphite) Cellulose Cement, Portland Clinker Charcoal (wood) Chrome brick Clay Coal tar oils Coal tars Coke Concrete Cryolite Diamond Fireclay brick Fluorspar Gasoline Glass (crown) (flint) (pyrex) (silicate) wool Granite Graphite Gypsum Kerosene Limestone Litharge Magnesia Magnesite brick Marble Porcelain, fired Berlin Porcelain, green Berlin Porcelain, fired earthenware Porcelain, green earthenware

Specific heat, cal/g °C 0.2 (100°C); 0.274 (1500°C) 0.186 (100°C) 0.25 0.22 0.3 to 0.4 About 0.2 0.168 (26° to 76°C) 0.314 (40° to 892°C) 0.387 (56° to 1450°C) 0.204 0.32 0.186 0.242 0.17 0.224 0.26 to 0.37 0.34 (15° to 90°C) 0.35 (40°C); 0.45 (200°C) 0.265 (21° to 400°C) 0.359 (21° to 800°C) 0.403 (21° to 1300°C) 0.156 (70° to 312°F); 0.219 (72° to 1472°F) 0.253 (16° to 55°C) 0.147 0.198 (100°C); 0.298 (1500°C) 0.21 (30°C) 0.53 0.16 to 0.20 0.117 0.20 0.188 to 0.204 (0 to 100°C) 0.24 to 0.26 (0 to 700°C) 0.157 0.20 (20° to 100°C) 0.165 (26° to 76°C); 0.390 (56° to 1450°C) 0.259 (16° to 46°C) 0.47 0.217 0.055 0.234 (100°C); 0.188 (1500°C) 0.222 (100°C); 0.195 (1500°C) 0.21 (18°C) 0.189 (60°C) 0.185 (60°C) 0.186 (60°C) 0.181 (60°C)

TABLE 2-219 Specific Heats of Miscellaneous Liquids and Solids (Concluded ) Specific heat, cal/g °C

Material Pyrex glass Pyrites (copper) Pyrites (iron) Pyroxylin plastics Quartz Rubber (vulcanized) Sand Silica Silica brick Silicon carbide brick Silk Steel Stone Stoneware (common) Turpentine Wood (Oak) Woods, miscellaneous Wool Zirconium oxide

0.20 0.131 (30°C) 0.136 (30°C) 0.34 to 0.38 0.17 (0°C); 0.28 (350°C) 0.415 0.191 0.316 0.202 (100°C); 0.195 (1500°C) 0.202 (100°C) 0.33 0.12 about 0.2 0.188 (60°C) 0.42 (18°C) 0.570 0.45 to 0.65 0.325 0.11 (100°C); 0.179 (1500°C)

TABLE 2-219a Oils (Animal, Vegetable, Mineral Oils) 15 Cp[cal/(g ⋅ °C) = A/  d 4 + B(t − 15)

where d = density, g/cm3. °F = 9⁄5 °C + 32; to convert calories per gram-degree Celsius to British thermal units per pound-degree Fahrenheit, multiply by 1.0; to convert grams per cubic centimeter to pounds per cubic foot, multiply by 62.43. Oils

A

Castor Citron Fatty drying non-drying semidrying oils (except castor) Naphthene base Olive Paraffin base Petroleum oils

0.500

HEATS AND FREE ENERGIES OF FORMATION UNITS CONVERSIONS °F = 9⁄5 °C + 32; to convert kilocalories per gram-mole to British thermal units per pound-mole, multiply by 1.799 × 10−3.

B

0.0007 (0.438 at 54°C) 0.440 0.0007 0.450 0.0007 0.445 0.0007 0.450 0.0007 0.405 0.0009 (0.47 at 7°C) 0.425 0.0009 0.415 0.0009

HEATS AND FREE ENERGIES OF FORMATION TABLE 2-220

2-187

Heats and Free Energies of Formation of Inorganic Compounds

The values given in the following table for the heats and free energies of formation of inorganic compounds are derived from (a) Bichowsky and Rossini, “Thermochemistry of the Chemical Substances,” Reinhold, New York, 1936; (b) Latimer, “Oxidation States of the Elements and Their Potentials in Aqueous Solution,” PrenticeHall, New York, 1938; (c) the tables of the American Petroleum Institute Research Project 44 at the National Bureau of Standards; and (d) the tables of Selected Values of Chemical Thermodynamic Properties of the National Bureau of Standards. The reader is referred to the preceding books and tables for additional details as to methods of calculation, standard states, and so on.

Compound Aluminum Al AlBr3 Al4C3 AlCl3 AlF3 AlI3 AlN Al(NH4)(SO4)2 Al(NH4)(SO4)2·12H2O Al(NO3)3·6H2O Al(NO3)3·9H2O Al2O3 Al(OH)3 Al2O3·SiO2 Al2O3·SiO2 Al2O3·SiO2 3Al2O3·2SiO2 Al2S3 Al2(SO4)3 Al2(SO4)3·6H2O Al2(SO4)3·18H2O Antimony Sb SbBr3 SbCl3 SbCl5 SbF3 SbI3 Sb2O3 Sb2O4 Sb2O5 Sb2S3 Arsenic As AsBr3 AsCl3 AsF3 AsH3 AsI3 As2O3 As2O5 As2S3 Barium Ba BaBr2 BaCl2 Ba(ClO3)2 Ba(ClO4)2 Ba(CN)2 Ba(CNO)2 BaCN2 BaCO3 BaCrO4

State† c c aq c c aq, 600 c aq c aq c c c c c c, corundum c c, sillimanite c, disthene c, andalusite c, mullite c c aq c c c c c l c c c, I, orthorhombic c, II, octahedral c c c, black

Heat of formation‡§ ∆H (formation) at 25°C, kcal/mole 0.00 −123.4 −209.5 −30.8 −163.8 −243.9 −329 −360.8 −72.8 −163.4 −57.7 −561.19 −1419.36 −680.89 −897.59 −399.09 −304.8 −648.7 −642.4 −642.0 −1874 −121.6 −820.99 −893.9 −1268.15 −2120 0.00 −59.9 −91.3 −104.8 −216.6 −22.8 −165.4 −166.6 −213.0 −230.0 −38.2

c c l l g c c c c amorphous

0.00 −45.9 −80.2 −223.76 43.6 −13.6 −154.1 −217.9 −20 −34.76

c c aq, 400 c aq, 300 c aq, 1600 c aq, 800 c c aq c c, witherite c

0.00 −180.38 −185.67 −205.25 −207.92 −176.6 −170.0 −210.2

*For footnotes see end of table.

−48 −212.1 −63.6 −284.2 −342.2

Free energy of formation¶ ∆F (formation) at 25°C, kcal/mole 0.00 −189.2 −29.0 −209.5 −312.6 −152.5 −50.4 −486.17 −1179.26 −526.32 −376.87 −272.9

−739.53 −759.3 −1103.39

Compound Barium (Cont.) BaF2 BaH2 Ba(HCO3)2 BaI2 Ba(IO3)2 BaMoO4 Ba3N2 Ba(NO2)2 Ba(NO3)2 BaO Ba(OH)2 BaO·SiO2 Ba3(PO4)2 BaPtCl6 BaS BaSO3 BaSO4 BaWO4 Beryllium Be BeBr2

0.00 −77.8

BeCl2 BeI2

−146.0 −186.6 −196.1 −36.9 0.00 −70.5 −212.27 37.7 −134.8 −183.9 −20 0.00 −183.0 −196.5 −134.4 −155.3 −180.7 −271.4

Be3N2 BeO Be(OH)2 BeS BeSO4 Bismuth Bi BiCl3 BiI3 BiO Bi2O3 Bi(OH)3 Bi2S3 Bi2(SO4)3 Boron B BBr3 BCl3 BF3 B2H6 BN B2O3 B(OH)3 B2S3 Bromine Br2 BrCl

State†

Heat of formation‡§ ∆H (formation) at 25°C, kcal/mole

c aq, 1600 c aq c aq, 400 c aq c c c aq c aq, 600 c c aq, 400 c c c c c c c

−287.9 −284.6 −40.8 −459 −144.6 −155.17 −264.5 −237.50 −370 −90.7 −184.5 −179.05 −236.99 −227.74 −133.0 −225.9 −237.76 −363 −992 −284.9 −111.2 −282.5 −340.2 −402

c c aq c aq c aq c c c c c aq

0.00 −79.4 −142 −112.6 −163.9 −39.4 −112 −134.5 −145.3 −215.6 −56.1 −281

c c aq c aq c c c c c

0.00 −90.5 −101.6 −24 −27 −49.5 −137.1 −171.1 −43.9 −607.1

c l g g g g c c gls c c

0.00 −52.7 −44.6 −94.5 −265.2 7.5 −32.1 −302.0 −297.6 −260.0 −56.6

l g g

0.00 7.47 3.06

Free energy of formation¶ ∆F (formation) at 25°C, kcal/mole −265.3 −31.5 −414.4 −158.52 −198.35

−150.75 −189.94

−209.02

−313.4 0.00 −127.9 −141.4 −103.4 −122.4 −138.3

−254.8 0.00 −76.4

−43.2 −117.9 −39.1 0.00 −50.9 −90.8 −261.0 19.9 −27.2 −282.9 −280.3 −229.4 0.00 0.931 −0.63

2-188

PHYSICAL AND CHEMICAL DATA

TABLE 2-220

Heats and Free Energies of Formation of Inorganic Compounds (Continued )

Compound Cadmium Cd CdBr2 CdCl2 Cd(CN)2 CdCO3 CdI2 Cd3N2 Cd(NO3)2 CdO Cd(OH)2 CdS CdSO4 Calcium Ca CaBr2 CaC2 CaCl2 CaCN2 Ca(CN)2 CaCO3 CaCO3·MgCO3 CaC2O4 Ca(C2H3O2)2 CaF2 CaH2 CaI2 Ca3N2 Ca(NO3)2 Ca(NO3)2·2H2O Ca(NO3)2·3H2O Ca(NO3)2·4H2O CaO Ca(OH)2 CaO·SiO2 CaS CaSO4 CaSO4·aH2O CaSO4·2H2O CaWO4 Carbon C CO CO2 Cerium Ce CeN Cesium Cs CsBr CsCl

State†

Heat of formation‡§ ∆H (formation) at 25°C, kcal/mole

c c aq, 400 c aq, 400 c c c aq, 400 c aq, 400 c c c c aq, 400

0.00 −75.8 −76.6 −92.149 −96.44 36.2 −178.2 −48.40 −47.46 39.8 −115.67 −62.35 −135.0 −34.5 −222.23 −232.635

c c aq, 400 c c aq c c aq c, calcite c, aragonite c c c aq c aq c c aq, 400 c c aq, 400 c c c c c aq, 800 c, II, wollastonite c, I, pseudowollastonite c c, insoluble form c, soluble form α c, soluble form β c c c

0.00 −162.20 −187.19 −14.8 −190.6 −209.15 −85 −43.3 −289.5 −289.54 −558.8 −332.2 −356.3 −364.1 −290.2 −286.5 −46 −128.49 −156.63 −103.2 −224.05 −228.29 −367.95 −439.05 −509.43 −151.7 −235.58 −239.2 −377.9 −376.6 −114.3 −338.73 −336.58 −335.52 −376.13 −479.33 −387

Free energy of formation¶ ∆F (formation) at 25°C, kcal/mole 0.00 −70.7 −67.6 −81.889 −81.2 −163.2 −43.22 −71.05 −55.28 −113.7 −33.6 −194.65 0.00 −181.86 −16.0 −179.8 −195.36 −54.0 −270.8 −270.57

−311.3 −264.1 −35.7 −157.37 −88.2 −177.38 −293.57 −351.58 −409.32 −144.3 −213.9 −207.9 −357.5 −356.6 −113.1 −311.9 −309.8 −308.8 −425.47

c, graphite c, diamond g g

0.00 0.453 −26.416 −94.052

0.00 0.685 −32.808 −94.260

c c

0.00 −78.2

0.00 −70.8

0.00 −97.64 −91.39 −106.31 −102.01

0.00

c c aq, 500 c aq, 400

−94.86 −101.61

Compound Cesium (Cont.) Cs2CO3 CsF CsH CsHCO3 CsI CsNH2 CsNO3 Cs2O CsOH Cs2S Cs2SO4 Chlorine Cl2 ClF ClO ClO2 ClO3 Cl2O Cl2O7 Chromium Cr CrBr3 Cr3C2 Cr4C CrCl2 CrF2 CrF3 CrI2 CrO3 Cr2O3 Cr2(SO4)3 Cobalt Co CoBr2 Co3C CoCl2 CoCO3 CoF2 CoI2 Co(NO3)2 CoO Co3O4 Co(OH)2 Co(OH)3 CoS Co2S3 CoSO4 Columbium Cb Cb2O5 Copper Cu CuBr CuBr2 CuCl CuCl2

State† c c aq, 400 c c aq, 2000 c aq, 400 c c aq, 400 c c aq, 200 c c aq g g g g g g g c aq c c c aq c c c aq c c aq

Heat of formation‡§ ∆H (formation) at 25°C, kcal/mole −271.88 −131.67 −140.48 −12 −230.6 −226.6 −83.91 −75.74 −28.2 −121.14 −111.54 −82.1 −100.2 −117.0 −87 −344.86 −340.12 0.00 −25.7 33 24.7 37 18.20 63 0.00 −21.008 −16.378 −103.1 −152 −231 −63.7 −139.3 −268.8

Free energy of formation¶ ∆F (formation) at 25°C, kcal/mole

−135.98 −7.30 −210.56 −82.61 −96.53 −107.87 −316.66 0.00 29.5 22.40 0.00 −122.7 −21.20 −16.74 −93.8 −102.1

−64.1 −249.3 −626.3

c c aq c c aq, 400 c aq c aq c aq c c c c c c c aq, 400

0.00 −55.0 −73.61 9.49 −76.9 −95.58 −172.39 −172.98 −24.2 −43.15 −102.8 −114.9 −57.5 −196.5 −131.5 −177.0 −22.3 −40.0 −216.6

c c

0.00 −462.96

0.00

0.00 −26.7 −34.0 −42.4 −31.4 −48.83 −64.7

0.00 −23.8

c c c aq c c aq, 400

0.00 −61.96 7.08 −66.6 −75.46 −155.36 −144.2 −37.4 −65.3 −108.9 −142.0 −19.8 −188.9

−33.25 −24.13

HEATS AND FREE ENERGIES OF FORMATION TABLE 2-220

Heats and Free Energies of Formation of Inorganic Compounds (Continued )

Compound Copper (Cont.) CuClO4 Cu(ClO3)2 Cu(ClO4)2 CuI CuI2 Cu3N Cu(NO3)2 CuO Cu2O Cu(OH)2 CuS Cu2S CuSO4 Cu2SO4 Erbium Er Er(OH)3 Fluorine F2 F2O Gallium Ga GaBr3 GaCl3 GaN Ga2O Ga2O3 Germanium Ge Ge3N4 GeO2 Gold Au AuBr AuBr3 AuCl AuCl3 AuI Au2O3 Au(OH)3 Hafnium Hf HfO2 Hydrogen H3AsO3 H3AsO4 HBr HBrO HBrO3 HCl HCN HClO HClO3 HClO4 HC2H3O2 H2C2O4 HCOOH

2-189

State† aq aq, 400 aq c c aq c c aq, 200 c c c c c c aq, 800 c aq

Heat of formation‡§ ∆H (formation) at 25°C, kcal/mole −28.3 −17.8 −4.8 −11.9 17.78 −73.1 −83.6 −38.5 −43.00 −108.9 −11.6 −18.97 −184.7 −200.78 −179.6

Free energy of formation¶ ∆F (formation) at 25°C, kcal/mole 1.34 15.4 −5.5 −16.66 −8.76 −36.6 −31.9 −38.13 −85.5 −11.69 −20.56 −158.3 −160.19 −152.0

Compound Hydrogen (Cont.) H2CO3 HF HI HIO HIO3 HN3 HNO3 HNO3·H2O HNO3·3H2O H2O H2O2 H3PO2

c c

0.00 −326.8

0.00

g g

0.00 5.5

0.00 9.7

H3PO3 H3PO4 H2S

c c c c c c

0.00 −92.4 −125.4 −26.2 −84.3 −259.9

0.00

c c c

0.00 −15.7 −128.6

0.00

c c c aq c c aq c c c

0.00 −3.4 −14.5 −11.0 −8.3 −28.3 −32.96 0.2 11.0 −100.6

0.00

c c

0.00 −271.1

0.00 −258.2

aq c aq g aq, 400 aq aq g aq, 400 g aq, 100 aq, 400 aq aq, 660 l aq, 400 c aq, 300 l aq, 200

−175.6 −214.9 −214.8 −8.66 −28.80 −25.4 −11.51 −22.063 −39.85 31.1 24.2 −28.18 −23.4 −31.4 −116.2 −116.74 −196.7 −194.6 −97.8 −98.0

−153.04

H2S2 H2SO3 H2SO4 H2Se H2SeO3 H2SeO4

24.47

H2SiO3 H4SiO4 H2Te H2TeO3

4.21 −0.76 18.71

H2TeO4 Indium In InBr3 InCl3

−183.93 −12.72 −24.58 −19.90 5.00 −22.778 −31.330 27.94 26.55 −19.11 −0.25 −10.70 −93.56 −96.8 −165.64 −82.7 −85.1

InI3 InN In2O3 Iodine I2 IBr ICl ICl3 I2O5 Iridium Ir IrCl IrCl2 IrCl3 IrF6 IrO2 Iron Fe FeBr2

State†

Heat of formation‡§ ∆H (formation) at 25°C, kcal/mole

Free energy of formation¶ ∆F (formation) at 25°C, kcal/mole

aq g aq, 200 g aq, 400 aq c aq g g l aq, 400 l l g l l aq, 200 c aq c aq c aq, 400 g aq, 2000 l aq, 200 l aq, 400 g aq c aq c aq, 400 c c g c aq aq

−167.19 −64.2 −75.75 6.27 −13.47 −38 −56.77 −54.8 70.3 −31.99 −41.35 −49.210 −112.91 −252.15 −57.7979 −68.3174 −45.16 −45.80 −145.5 −145.6 −232.2 −232.2 −306.2 −309.32 −4.77 −9.38 −3.6 −146.88 −193.69 −212.03 20.5 18.1 −126.5 −122.4 −130.23 −143.4 −267.8 −340.6 36.9 −145.0 −145.0 −165.6

c c aq c aq c aq c c

0.00 −97.2 −112.9 −128.5 −145.6 −56.5 −67.2 −4.8 −222.47

c g g g c c

0.00 14.88 10.05 4.20 −21.8 −42.5

0.00 4.63 1.24 −1.32 −6.05

c c c c l c

0.00 −20.5 −40.6 −60.5 −130 −40.14

0.00 −16.9 −32.0 −46.5

c, α c aq, 540

0.00 −57.15 −78.7

−149.0 −64.7 0.365 −12.35 −23.33 −32.25 78.50 −17.57 −19.05 −78.36 −193.70 −54.6351 −56.6899 −28.23 −31.47 −120.0 −204.0 −270.0 −7.85 −128.54 17.0 18.4 −101.36 −247.9 33.1 −115.7

0.00 −97.2 −117.5 −60.5

0.00 −69.47

2-190

PHYSICAL AND CHEMICAL DATA

TABLE 2-220

Heats and Free Energies of Formation of Inorganic Compounds (Continued )

Compound Iron (Cont.) FeBr3 Fe3C Fe(CO)5 FeCO3 FeCl2 FeCl3 FeF2 FeI2 FeI3 Fe4N Fe(NO3)2 Fe(NO3)3 FeO Fe2O3 Fe3O4 Fe(OH)2 Fe(OH)3 FeO·SiO2 Fe2P FeSi FeS FeS2 FeSO4 Fe2(SO4)3 FeTiO3 Lanthanum La LaCl3 La3H8 LaN La2O3 LaS2 La2S3 La2(SO4)3 Lead Pb PbBr2 PbCO3 Pb(C2H3O2)2 PbC2O4 PbCl2 PbF2 PbI2 Pb(NO3)2 PbO PbO2 Pb3O4 Pb(OH)2 PbS PbSO4 Lithium Li LiBr LiBrO3 Li2C2 LiCN LiCNO

State† aq c l c, siderite c aq c aq, 2000 aq, 1200 c aq aq c aq aq, 800 c c c c c c c c c c, pyrites c, marcasite c aq, 400 aq, 400 c, ilmenite

Heat of formation‡§ ∆H (formation) at 25°C, kcal/mole −95.5 5.69 −187.6 −172.4 −81.9 −100.0 −96.4 −128.5 −177.2 −24.2 −47.7 −49.7 −2.55 −118.9 −156.5 −64.62 −198.5 −266.9 −135.9 −197.3 −273.5 −13 −19.0 −22.64 −38.62 −33.0 −221.3 −236.2 −653.3 −295.51

Free energy of formation¶ ∆F (formation) at 25°C, kcal/mole −76.26 4.24 −154.8 −72.6 −83.0 −96.5 −151.7 −45 −39.5 0.862 −72.8 −81.3 −59.38 −179.1 −242.3 −115.7 −166.3

Compound Lithium (Cont.) LiC2H3O2 Li2CO3 LiCl LiClO3 LiClO4 LiF LiH LiHCO3 LiI LiIO3 Li3N LiNO3 Li2O Li2O2 LiOH

−23.23 −35.93 −195.5 −196.4 −533.4 −277.06

c c aq c c c c c aq

0.00 −253.1 −284.7 −160 −72.0 −539 −148.3 −351.4 −972

0.00

c c aq c, cerussite c aq, 400 c c aq c c c aq, 400 c, red c, yellow c c c c c

0.00 −66.24 −56.4 −167.6 −232.6 −234.2 −205.3 −85.68 −82.5 −159.5 −41.77 −106.88 −99.46 −51.72 −50.86 −65.0 −172.4 −123.0 −22.38 −218.5

0.00 −62.06 −54.97 −150.0

c c aq, 400 aq c aq aq

0.00 −83.75 −95.40 −77.9 −13.0 −31.4 −101.2

0.00

LiOH·H2O Li2O·SiO2 Li2Se Li2SO4 Li2SO4·H2O Magnesium Mg Mg(AsO4)2 MgBr2

−64.6

−184.40 −75.04 −68.47 −148.1 −41.47 −58.3 −45.53 −43.88 −52.0 −142.2 −102.2 −21.98 −192.9

−95.28 −65.70 −31.35 −94.12

Mg(CN)2 MgCN2 Mg(C2H3O2)2 MgCO3 MgCl2 MgCl2·H2O MgCl2·2H2O MgCl2·4H2O MgCl2·6H2O MgF2 MgI2 MgMoO4 Mg3N2 Mg(NO3)2 Mg(NO3)2·2H2O Mg(NO3)2·6H2O MgO MgO·SiO2 Mg(OH)2 MgS MgSO4 MgTe MgWO4 Manganese Mn MnBr2 Mn3C

State†

Heat of formation‡§ ∆H (formation) at 25°C, kcal/mole

aq c aq, 1900 c aq, 278 aq aq c aq, 400 c aq, 2000 c aq, 400 aq c c aq, 400 c c aq c aq, 400 c gls c aq c aq, 400 c

−183.9 −289.7 −293.1 −97.63 −106.45 −87.5 −106.3 −145.57 −144.85 −22.9 −231.1 −65.07 −80.09 −121.3 −47.45 −115.350 −115.88 −142.3 −151.9 −159 −116.58 −121.47 −188.92 −374 −84.9 −95.5 −340.23 −347.02 −411.57

c c aq c aq, 400 aq c aq c c aq, 400 c c c c c c aq, 400 c c c aq, 400 c c c c c, ppt. c, brucite c aq c aq, 400 c c

0.00 −731.3 −749 −123.9 −167.33 −39.7 −61 −344.6 −261.7 −153.220 −189.76 −230.970 −305.810 −453.820 −597.240 −263.8 −86.8 −136.79 −329.9 −115.2 −188.770 −209.927 −336.625 −624.48 −143.84 −347.5 −221.90 −223.9 −84.2 −108 −304.94 −325.4 −25 −345.2

c, α c aq c

0.00 −91 −106 1.1

Free energy of formation¶ ∆F (formation) at 25°C, kcal/mole −160.00 −269.8 −267.58 −102.03 −70.95 −81.4 −136.40 −210.98 −83.03 −102.95 −37.33 −96.95 −138.0 −106.44 −108.29

−105.64 −314.66 −375.07 0.00 −630.14 −156.94 −29.08 −286.38 −241.7 −143.77 −205.93 −267.20 −387.98 −505.45 −132.45 −100.8 −140.66 −160.28 −496.03 −136.17 −326.7 −200.17 −193.3 −277.7 −283.88

0.00 −97.8 1.26

HEATS AND FREE ENERGIES OF FORMATION TABLE 2-220

Heats and Free Energies of Formation of Inorganic Compounds (Continued )

Compound Manganese (Cont.) Mn(C2H3O2)2 MnCO3 MnC2O4 MnCl2 MnF2 MnI2 Mn5N2 Mn(NO3)2 Mn(NO3)2.6H2O MnO MnO2 Mn2O3 Mn3O4 MnO.SiO2 Mn(OH)2 Mn(OH)3 Mn3(PO4)2 MnSe MnS MnSO4 Mn2(SO4)3 Mercury Hg HgBr HgBr2 Hg(C2H3O2)2 HgCl2 HgCl Hg2Cl2 Hg(CN)2 HgC2O4 HgH HgI2 HgI Hg2I2 Hg(NO3)2 Hg2(NO3)2 HgO Hg2O HgS HgSO4 Hg2SO4 Molybdenum Mo Mo2C Mo2N MoO2 MoO3 MoS2 MoS3 Nickel Ni NiBr2 Ni3C Ni(C2H3O2)2 Ni(CN)2 NiCl2

2-191

State†

Heat of formation‡§ ∆H (formation) at 25°C, kcal/mole

c aq c c c aq, 400 aq, 1200 c aq c c aq, 400 c c c c c c c c c c c, green c aq, 400 c aq

−270.3 −282.7 −211 −240.9 −112.0 −128.9 −206.1 −49.8 −76.2 −57.77 −134.9 −148.0 −557.07 −92.04 −124.58 −229.5 −331.65 −301.3 −163.4 −221 −736 −26.3 −47.0 −254.18 −265.2 −635 −657

l g c aq c aq c aq g c c aq, 1110 c g c, red g c aq aq c, red c, yellow ppt. c c, black c c

0.00 23 −40.68 −38.4 −196.3 −192.5 −53.4 −50.3 19 −63.13 62.8 66.25 −159.3 57.1 −25.3 33 −28.88 −56.8 −58.5 −21.6 −20.8 −21.6 −10.7 −166.6 −177.34

c c c c c c c

0.00 4.36 −8.3 −130 −180.39 −56.27 −61.48

c c aq c aq aq c

0.00 −53.4 −72.6 9.2 −249.6 230.9 −75.0

Free energy of formation¶ ∆F (formation) at 25°C, kcal/mole

Heat of formation‡§ ∆H (formation) at 25°C, kcal/mole

Free energy of formation¶ ∆F (formation) at 25°C, kcal/mole

aq, 400 c aq c aq c aq, 200 c c c c c aq, 200

−94.34 −157.5 −171.6 −22.4 −42.0 −101.5 −113.5 −58.4 −129.8 −163.2 −20.4 −216 −231.3

−74.19

g g g aq, 200 c aq c aq, 400 c aq c aq aq c aq c aq, 400 c aq c aq c aq c aq c aq, 500 aq aq, 400 c aq, 400 l l c g g g g c l g

0.00 −27 −10.96 −19.27 −64.57 −60.27 −148.1 −148.58 −0.7 3.6 −17.8 −12.3 −223.4 −266.3 −260.6 −75.23 −71.20 −69.4 −63.2 −276.9 −271.3 −111.6 −110.2 −48.43 −44.97 −87.40 −80.89 −87.59 −55.21 −281.74 −279.33 12.06 −57.96 −232.2 19.55 21.600 7.96 2.23 −10.0 11.6 12.8

Compound

State†

Nickel (Cont.) −227.2 −192.5 −102.2 −180.0 −73.3 −46.49 −101.1 −441.2 −86.77 −111.49 −209.9 −306.22 −282.1 −143.1 −190 −27.5 −48.0 −228.41

NiF2 NiI2 Ni(NO3)2 NiO Ni(OH)2 Ni(OH)3 NiS NiSO4 Nitrogen N2 NF3 NH3 NH4Br NH4C2H3O2 NH4CN NH4CNS

0.00 18 −38.8 −9.74 −139.2 −42.2 −23.25 14

(NH4)2CO3 (NH4)2C2O4 NH4Cl NH4ClO4 (NH4)2CrO4 NH4F NH4I NH4NO3

52.25 −24.0 23 −26.53 −13.09 −15.65 −13.94 −12.80 −8.80 −149.12 0.00 2.91 −118.0 −162.01 −54.19 −57.38 0.00 −60.7 8.88 −190.1 66.3

NH4OH (NH4)2S (NH4)2SO4 N2H4 N2H4·H2O N2H4·H2SO4 N2O NO NO2 N2O4 N2O5 NOBr NOCl Osmium Os OsO4 Oxygen O2 O3 Palladium Pd PdO Phosphorus P P

−142.9 −36.2 −64.0 −51.7 −105.6

−187.6 0.00 −3.903 −43.54 −108.26 20.4 4.4 −164.1 −196.2 −48.59 −21.1 −209.3 −84.7 −31.3

−14.50 −215.06 −214.02

24.82 20.719 12.26 23.41 19.26 16.1

c c g

0.00 −93.6 −80.1

0.00 −70.9 −68.1

g g

0.00 33.88

0.00 38.86

c c

0.00 −20.40

0.00

c, white (“yellow”) c, red (“violet”) g

0.00 −4.22 150.35

0.00 −1.80 141.88

2-192

PHYSICAL AND CHEMICAL DATA

TABLE 2-220

Heats and Free Energies of Formation of Inorganic Compounds (Continued )

Compound Phosphorus (Cont.) P2 P4 PBr3 PBr5 PCl3 PCl5 PH3 PI3 P2O5 POCl3 Platinum Pt PtBr4 PtCl2 PtCl4 PtI4 Pt(OH)2 PtS PtS2 Potassium K K3AsO3 K3AsO4 KH2AsO4 KBr KBrO3 KC2H3O2 KCl KClO3 KClO4 KCN KCNO KCNS K2CO3 K2C2O4 K2CrO4 K2Cr2O7 KF K3Fe(CN)6 K4Fe(CN)6 KH KHCO3 KI KIO3 KIO4 KMnO4 K2MoO4

State† g g l c g l g g c c g c c aq c c aq c c c c c aq aq c c aq, 400 c aq, 1667 c aq, 400 c aq, 400 c aq, 400 c aq, 400 c aq, 400 c aq c aq, 400 c aq, 400 c aq, 400 c aq, 400 c aq, 400 c aq, 180 c aq c aq c c aq, 2000 c aq, 500 c aq, 400 aq c aq, 400 aq, 880

Heat of formation‡§ ∆H (formation) at 25°C, kcal/mole 33.82 13.2 −45 −60.6 −70.0 −76.8 −91.0 2.21 −10.9 −360.0 −138.4 0.00 −40.6 −50.7 −34 −62.6 −82.3 −18 −87.5 −20.18 −26.64 0.00 −323.0 −390.3 −271.2 −94.06 −89.19 −81.58 −71.68 −173.80 −177.38 −104.348 −100.164 −93.5 −81.34 −103.8 −101.14 −28.1 −25.3 −99.6 −94.5 −47.0 −41.07 −274.01 −280.90 −319.9 −315.5 −333.4 −328.2 −488.5 −472.1 −134.50 −138.36 −48.4 −34.5 −131.8 −119.9 −10 −229.8 −224.85 −78.88 −73.95 −121.69 −115.18 −98.1 −192.9 −182.5 −364.2

Free energy of formation¶ ∆F (formation) at 25°C, kcal/mole 24.60 5.89 −65.2 −63.3 −73.2 −1.45 −127.2 0.00

Compound Potassium (Cont.) KNH2 KNO2 KNO3 K2O K2O·Al2O3·SiO2 K2O·Al2O3·SiO2 KOH K3PO3 K3PO4 KH2PO4 K2PtCl4 K2PtCl6

−67.9 −18.55 −24.28 0.00 −355.7 −236.7 −90.8 −92.0 −60.30 −156.73 −97.76 −98.76 −69.30 −72.86 −28.08 −90.85 −44.08

K2Se K2SeO4 K2S K2SO3 K2SO4 K2SO4·Al2(SO4)3 K2SO4·Al2(SO4)3· 24H2O K2S2O6 Rhenium Re ReF6 Rhodium Rh RhO Rh2O Rh2O3 Rubidium Rb RbBr

−264.04

RbCN Rb2CO3

−293.1

RbCl

−306.3 −440.9 −133.13

−5.3 −207.71 −77.37 −79.76 −101.87 −99.68 −169.1 −168.0 −342.9

RbF RbHCO3 RbI RbNH2 RbNO3 Rb2O Rb2O2 RbOH Ruthenium Ru RuS2 Selenium Se

State†

Heat of formation‡§ ∆H (formation) at 25°C, kcal/mole

Free energy of formation¶ ∆F (formation) at 25°C, kcal/mole

c aq c aq, 400 c c, leucite gls c, adularia c, microcline gls c aq, 400 aq aq c c aq c aq, 9400 c aq aq c aq, 400 c aq c aq, 400 c

−28.25 −86.0 −118.08 −109.79 −86.2 −1379.6 −1368.2 −1784.5 −1784.5 −1747 −102.02 −114.96 −397.5 −478.7 −362.7 −254.7 −242.6 −299.5 −286.1 −74.4 −83.4 −267.1 −121.5 −110.75 −267.7 −269.7 −342.65 −336.48 −1178.38

c c

−2895.44 −418.62

−2455.68

c g

0.00 −274

0.00

c c c c

0.00 −21.7 −22.7 −68.3

0.00

0.00 −95.82 −45.0 −90.54 −25.9 −273.22 −282.61 −105.06 −53.6 −101.06 −133.23 −139.31 −230.01 −225.59 −81.04 −31.2 −74.57 −27.74 −119.22 −110.52 −82.9 −107 −101.3 −115.8

0.00

c c g aq, 500 aq c aq, 220 c g aq, ∞ c aq, 400 c aq, 2000 c g aq, 400 c c aq, 400 c c c aq, 200 c c c, I, hexagonal

−75.9 −94.29 −93.68

−105.0 −443.3 −326.1 −226.5 −263.6 −99.10 −240.0 −111.44 −251.3 −314.62 −310.96 −1068.48

−52.50 −93.38 −263.78 −98.48 −57.9 −100.13 −134.5 −209.07 −40.5 −81.13 −95.05

−106.39

0.00 −46.99

0.00 −44.11

0.00

0.00

HEATS AND FREE ENERGIES OF FORMATION TABLE 2-220

Compound

Heats and Free Energies of Formation of Inorganic Compounds (Continued )

State†

Heat of formation‡§ ∆H (formation) at 25°C, kcal/mole

Free energy of formation¶ ∆F (formation) at 25°C, kcal/mole

SiF4 SiH4 SiI4 Si3N4 SiO2

Silver Ag AgBr Ag2C2 AgC2H3O2 AgCN Ag2CO3 Ag2C2O4 AgCl AgF AgI AgIO3 AgNO2 AgNO3 Ag2O Ag2S Ag2SO4 Sodium Na Na3AsO3 Na3AsO4 NaBr NaBrO NaBrO3 NaC2H3O2 NaCN NaCNO NaCNS Na2CO3 NaCO2NH2 Na2C2O4 NaCl NaClO3 NaClO4

Compound

State†

Sodium (Cont.)

Selenium (Cont.) Se2Cl2 SeF6 SeO2 Silicon Si SiBr4 SiC SiCl4

2-193

c, II, red, monoclinic l g c

0.2 −22.06 −246 −56.33

−13.73 −222

c l c l g g g c c c, cristobalite, 1600° form c, cristobalite, 1100° form c, quartz c, tridymite

0.00 −93.0 −28 −150.0 −142.5 −370 −14.8 −29.8 −179.25 −202.62

0.00

c c c c aq c c c c c aq, 400 c c c aq c aq, 6500 c c c aq

0.00 −23.90 84.5 −95.9 −91.7 33.8 −119.5 −158.7 −30.11 −48.7 −53.1 −15.14 −42.02 −11.6 −2.9 −29.4 −24.02 −6.95 −5.5 −170.1 −165.8

c aq, 500 c aq, 500 c aq, 400 aq aq, 400 c aq, 400 c aq, 200 c aq c aq, 400 c aq, 1000 c c aq, 600 c aq, 400 c aq, 400 c

0.00 −314.61 −366 −381.97 −86.72 −86.33 −78.9 −68.89 −170.45 −175.450 −22.47 −22.29 −96.3 −91.7 −39.94 −38.23 −269.46 −275.13 −142.17 −313.8 −309.92 −98.321 −97.324 −83.59 −78.42 −101.12

−27.4 −133.9 −133.0 −360 −9.4 −154.74

Na2Cr2O7 NaF NaH NaHCO3 NaI NaIO3 Na2MoO4 NaNO2 NaNO3

−202.46 −203.35 −203.23

Na2CrO4

−190.4 0.00 −23.02 −70.86 38.70 −103.0 −25.98 −47.26 −16.17 −24.08 3.76 9.99 −7.66 −7.81 −2.23 −7.6 −146.8 −139.22 0.00 −341.17 −87.17 −57.59 −152.31 −23.24 −86.00 −39.24 −249.55 −251.36 −283.42 −91.894 −93.92 −62.84

Na2O Na2O2 Na2O·SiO2 Na2O·Al2O3·3SiO2 Na2O·Al2O3·4SiO2 NaOH Na3PO3 Na3PO4 Na2PtCl4 Na2PtCl6 Na2Se Na2SeO4 Na2S Na2SO3 Na2SO4 Na2SO4·10H2O Na2WO4 Strontium Sr SrBr2 Sr(C2H3O2)2 Sr(CN)2 SrCO3 SrCl2 SrF2 Sr(HCO3)2 SrI2 Sr3N2 Sr(NO3)2 SrO SrO·SiO2 SrO2 Sr2O Sr(OH)2 Sr3(PO4)2 SrS

aq, 476 c aq, 800 aq, 1200 c aq, 400 c c aq c aq, ∞ aq, 400 c aq c aq c aq, 400 c c c c, natrolite c c aq, 400 aq, 1000 c aq, 400 aq c aq c aq, 440 c aq, 800 c aq, 400 c aq, 800 c aq, 1100 c c aq c c aq, 400 c aq aq c c aq, 400 c aq c aq, 400 c c aq, 400 c gls c c c aq, 800 c aq c

Heat of formation‡§ ∆H (formation) at 25°C, kcal/mole

Free energy of formation¶ ∆F (formation) at 25°C, kcal/mole

−97.66 −319.8 −323.0 −465.9 −135.94 −135.711 −14 −226.0 −222.1 −69.28 −71.10 −112.300 −364 −358.7 −86.6 −83.1 −111.71 −106.880 −99.45 −119.2 −383.91 −1180 −1366 −101.96 −112.193 −389.1 −457 −471.9 −237.2 −272.1 −280.9 −59.1 −78.1 −254 −261.5 −89.8 −105.17 −261.2 −264.1 −330.50 −330.82 −1033.85 −391 −381.5

−73.29

0.00 −171.0 −187.24 −358.0 −364.4 −59.5 −290.9 −197.84 −209.20 −289.0 −459.1 −136.1 −156.70 −91.4 −233.2 −228.73 −140.8 −364 −153.3 −153.6 −228.7 −239.4 −980 −985 −113.1

−296.58 −431.18 −129.0 −128.29 −9.30 −202.66 −202.87 −74.92 −94.84 −333.18 −71.04 −87.62 −88.84 −90.06 −105.0 −361.49 −90.60 −100.18 −428.74 −216.78

−89.42 −230.30 −101.76 −240.14 −241.58 −302.38 −301.28 −870.52 −345.18 0.00 −182.36 −311.80 −54.50 −271.9 −195.86 −413.76 −157.87 −76.5 −185.70 −133.7 −139.0 −208.27 −881.54

2-194

PHYSICAL AND CHEMICAL DATA

TABLE 2-220

Heats and Free Energies of Formation of Inorganic Compounds (Continued )

Compound

State†

Strontium (Cont.) SrSO4 SrWO4 Sulfur S

S2 S6 S8 S2Br2 SCl4 S2Cl2 S2Cl4 SF6 SO SO2 SO3

SO2Cl2 Tantalum Ta TaN Ta2O5 Tellurium Te TeBr4 TeCl4 TeF6 TeO2 Thallium Tl TlBr TlCl TlCl3 TlF TlI TlNO3 Tl2O Tl2O3 TlOH Tl2S Tl2SO4 Thorium Th ThBr4 ThC2 ThCl4 ThI4 Th3N4 ThO2 Th(OH)4 Th(SO4)2

aq c aq, 400 c

Heat of formation‡§ ∆H (formation) at 25°C, kcal/mole −120.4 −345.3 −345.0 −393

Free energy of formation¶ ∆F (formation) at 25°C, kcal/mole −109.78 −309.30

0.00 −0.071 0.257

0.00 0.023 0.072 0.071 43.57 19.36 13.97 12.770

53.25 31.02 27.78 27.090 −4 −13.7 −14.2 −24.1 −262 19.02 −70.94 −94.39 −103.03 −105.09 −105.92 −109.34 −82.04 −89.80

−237 12.75 −71.68 −88.59 −88.28 −88.22 −88.34 −88.98 −74.06 −75.06

c c c

0.00 −51.2 −486.0

0.00 −45.11 −453.7

c c c g c

0.00 −49.3 −77.4 −315 −77.56

0.00

c c aq c aq c aq aq c aq c aq c c c aq c c aq, 800

0.00 −41.5 −28.0 −49.37 −38.4 −82.4 −91.0 −77.6 −31.1 −12.7 −58.2 −48.4 −43.18 −120 −57.44 −53.9 −22 −222.8 −214.1

c c aq c c aq aq c c c, “soluble” c aq

0.00 −281.5 −352.0 −45.1 −335 −392 −292.0 −309.0 −291.6 −336.1 −632 −668.1

−5.90

−57.4 −292 −64.66 0.00 −39.43 −32.34 −44.46 −39.09 −44.25 −73.46 −31.3 −20.09 −36.32 −34.01 −45.54 −45.35 −197.79 −191.62 0.00

SnBr2

SnCl2 SnCl4 SnI2 SnO SnO2 Sn(OH)2 Sn(OH)4 SnS Titanium Ti TiC TiCl4 TiN TiO2 Tungsten W WO2 WO3 WS2 Uranium U UC2 UCl3 UCl4 U3N4 UO2 UO2(NO3)2·6H2O UO3 U3O8 Vanadium V VCl2 VCl3 VCl4 VN V2O2 V2O3 V2O4 V2O5 Zinc Zn ZnSb ZnBr2 Zn(C2H3O2)2 Zn(CN)2 ZnCO3 ZnCl2

−295.31

ZnF2 ZnI2

−322.32 −246.33 −282.3 −280.1

Zn(NO3)2 ZnO ZnO·SiO2 Zn(OH)2 ZnS ZnSO4

−549.2

c, II, tetragonal c, III, “gray,” cubic c aq c aq c aq l aq c aq c c c c c

0.00 0.6 −61.4 −60.0 −94.8 −110.6 −83.6 −81.7 −127.3 −157.6 −38.9 −33.3 −67.7 −138.1 −136.2 −268.9 −18.61

−68.94 −110.4 −124.67

c c l c c, III, rutil amorphous

0.00 −110 −181.4 −80.0 −225.0 −214.1

0.00 −109.2 −165.5 −73.17 −211.9 −201.4

c c c c

0.00 −130.5 −195.7 −84

0.00 −118.3 −177.3

c c c c c c c c c

0.00 −29 −213 −251 −274 −256.6 −756.8 −291.6 −845.1

0.00

c c l l c c c c c

0.00 −147 −187 −165 −41.43 −195 −296 −342 −373

c c c aq, 400 c aq, 400 c c c aq, 400 aq c aq aq, 400 c, hexagonal c c, rhombic c, wurtzite c aq, 400

0.00 −3.6 −77.0 −93.6 −259.4 −269.4 17.06 −192.9 −99.9 −115.44 −192.9 −50.50 −61.6 −134.9 −83.36 −282.6 −153.66 −45.3 −233.4 −252.12

Compound Tin Sn

SnBr4 c, rhombic c, monoclinic l, λ l, λµ equilibrium g g g g l l l l g g g g l c, α c, β c, γ g l

State†

Heat of formation‡§ ∆H (formation) at 25°C, kcal/mole

Free energy of formation¶ ∆F (formation) at 25°C, kcal/mole 0.00 1.1 −55.43 −97.66

−30.95 −60.75 −123.6 −115.95 −226.00

−249.6 −242.2 −617.8

0.00

−35.08 −277 −316 −342 0.00 −3.88 −72.9 −214.4 −173.5 −88.8 −166.6 −49.93 −87.7 −76.19 −44.2 −211.28

HEATS OF COMBUSTION TABLE 2-220

2-195

Heats and Free Energies of Formation of Inorganic Compounds (Concluded )

Compound

State†

Zirconium Zr ZrC ZrCl4 ZrN

c c c c

Heat of formation‡§ ∆H (formation) at 25°C, kcal/mole 0.00 −29.8 −268.9 −82.5

Free energy of formation¶ ∆F (formation) at 25°C, kcal/mole 0.00 −34.6 −75.9

Compound Zirconium (Cont.) ZrO2 Zr(OH)4 ZrO(OH)2

Heat of formation‡§ ∆H (formation) at 25°C, kcal/mole

State† c, monoclinic c c

−258.5 −411.0 −337

Free energy of formation¶ ∆F (formation) at 25°C, kcal/mole −244.6 −307.6

† The physical state is indicated as follows: c, crystal (solid); l, liquid; g, gas; gls, glass or solid supercooled liquid; aq, in aqueous solution. A number following the symbol aq applies only to the values of the heats of formation (not to those of free energies of formation); and indicates the number of moles of water per mole of solute; when no number is given, the solution is understood to be dilute. For the free energy of formation of a substance in aqueous solution, the concentration is always that of the hypothetical solution of unit molality. ‡ The increment in heat content, ∆H, in the reaction of forming the given substance from its elements in their standard states. When ∆H is negative, heat is evolved in the process, and, when positive, heat is absorbed. § The heat of solution in water of a given solid, liquid, or gaseous compound is given by the difference in the value for the heat of formation of the given compound in the solid, liquid, or gaseous state and its heat of formation in aqueous solution. The following two examples serve as an illustration of the procedure: (1) For NaCl(c) and NaCl(aq, 400H2O), the values of ∆H(formation) are, respectively, −98.321 and −97.324 kg-cal per mole. Subtraction of the first value from the second gives ∆H = 0.998 kg-cal per mole for the reaction of dissolving crystalline sodium chloride in 400 moles of water. When this process occurs at a constant pressure of 1 atm, 0.998 kg-cal of energy are absorbed. (2) For HCl(g) and HCl(aq, 400H2O), the values for ∆H(formation) are, respectively, −22.06 and −39.85 kg-cal per mole. Subtraction of the first from the second gives ∆H = −17.79 kg-cal per mole for the reaction of dissolving gaseous hydrogen chloride in 400 moles of water. At a constant pressure of 1 atm, 17.79 kg-cal of energy are evolved in this process.  The increment in the free energy, ∆F, in the reaction of forming the given substance in its standard state from its elements in their standard states. The standard states are: for a gas, fugacity (approximately equal to the pressure) of 1 atm; for a pure liquid or solid, the substance at a pressure of 1 atm; for a substance in aqueous solution, the hypothetical solution of unit molality, which has all the properties of the infinitely dilute solution except the property of concentration. ¶ The free energy of solution of a given substance from its normal standard state as a solid, liquid, or gas to the hypothetical one molal state in aqueous solution may be calculated in a manner similar to that described in footnote § for calculating the heat of solution.

HEATS OF COMBUSTION TABLE 2-221 Enthalpies and Gibbs Energies of Formation, Entropies, and Net Enthalpies of Combustion of Inorganic and Organic Compounds at 298.15 K

Mol wt

Ideal gas enthalpy of formation, J/kmol × 1E-07

Methane Ethane Propane n-Butane n-Pentane n-Hexane n-Heptane n-Octane n-Nonane

CH4 C2H6 C3H8 C4H10 C5H12 C6H14 C7H16 C8H18 C9H20

74828 74840 74986 106978 109660 110543 142825 111659 111842

16.043 30.070 44.097 58.123 72.150 86.177 100.204 114.231 128.258

−7.4520 −8.3820 −10.4680 −12.5790 −14.6760 −16.6940 −18.7650 −20.8750 −22.8740

−5.0490 −3.1920 −2.4390 −1.6700 −0.8813 −0.0066 0.8165 1.6000 2.4980

1.8627 2.2912 2.7020 3.0991 3.4945 3.8874 4.2798 4.6723 5.0640

−0.8026 −1.4286 −2.0431 −2.6573 −3.2449 −3.8551 −4.4647 −5.0742 −5.6846

10 11 12 13 14 15 16 17 18 19

n-Decane n-Undecane n-Dodecane n-Tridecane n-Tetradecane n-Pentadecane n-Hexadecane n-Heptadecane n-Octadecane n-Nonadecane

C10H22 C11H24 C12H26 C13H28 C14H30 C15H32 C16H34 C17H36 C18H38 C19H40

124185 1120214 112403 629505 629594 629629 544763 629787 593453 629925

142.285 156.312 170.338 184.365 198.392 212.419 226.446 240.473 254.500 268.527

−24.9460 −27.0430 −29.0720 −31.1770 −33.2440 −35.3110 −37.4170 −39.4450 −41.5120 −43.5790

3.3180 4.1160 4.9810 5.7710 6.5990 7.4260 8.2160 9.0830 9.9100 10.7400

5.4570 5.8493 6.2415 6.6337 7.0259 7.4181 7.8102 8.2023 8.5945 8.9866

−6.2942 −6.9036 −7.5137 −8.1229 −8.7328 −9.3424 −9.9515 −10.5618 −11.1715 −11.7812

20 21 22 23 24 25 26 27 28 29

n-Eicosane 2-Methylpropane 2-Methylbutane 2,3-Dimethylbutane 2-Methylpentane 2,3-Dimethylpentane 2,3,3-Trimethylpentane 2,2,4-Trimethylpentane Ethylene Propylene

C20H42 C4H10 C5H12 C6H14 C6H14 C7H16 C8H18 C8H18 C2H4 C3H6

112958 75285 78784 79298 107835 565593 560214 540841 74851 115071

282.553 58.123 72.150 86.177 86.177 100.204 114.231 114.231 28.054 42.081

−45.6460 −13.4180 −15.3700 −17.6800 −17.4550 −19.4100 −21.8450 −22.4010 5.2510 1.9710

11.5700 −2.0760 −1.4050 −0.3125 −0.5338 0.5717 1.8280 1.3940 6.8440 6.2150

9.3787 2.9539 3.4374 3.6592 3.8089 4.1455 4.2702 4.2296 2.1920 2.6660

−12.3908 −2.6490 −3.2395 −3.8476 −3.8492 −4.4608 −5.0688 −5.0653 −1.3230 −1.9257

Cmpd. no. 1 2 3 4 5 6 7 8 9

Name

Formula

CAS no.

Ideal gas Gibbs energy Ideal gas Standard net enthalpy of formation, entropy, of combustion, J/kmol × 1E-07 J/(kmol·K) × 1E-05 J/kmol × 1E-09

2-196

PHYSICAL AND CHEMICAL DATA

TABLE 2-221 Enthalpies and Gibbs Energies of Formation, Entropies, and Net Enthalpies of Combustion of Inorganic and Organic Compounds (Continued ) Cmpd. no.

Name

Formula

CAS no.

Mol wt

Ideal gas enthalpy of formation, J/kmol × 1E-07

Ideal gas Gibbs energy Ideal gas Standard net enthalpy of formation, entropy, of combustion, J/kmol × 1E-07 J/(kmol·K) × 1E-05 J/kmol × 1E-09

30 31 32 33 34 35 36 37 38 39

1-Butene cis-2-Butene trans-2-Butene 1-Pentene 1-Hexene 1-Heptene 1-Octene 1-Nonene 1-Decene 2-Methylpropene

C4H8 C4H8 C4H8 C5H10 C6H12 C7H14 C8H16 C9H18 C10H20 C4H8

106989 590181 624646 109671 592416 592767 111660 124118 872059 115117

56.108 56.108 56.108 70.134 84.161 98.188 112.215 126.242 140.269 56.108

−0.0540 −0.7400 −1.1000 −2.1300 −4.2000 −6.2800 −8.3600 −10.4000 −12.4700 −1.7100

7.0270 6.5360 6.3160 7.8450 8.7390 9.4830 10.3000 11.1500 11.9800 5.8080

3.0775 3.0120 2.9650 3.4699 3.8389 4.2549 4.6469 5.0399 5.4319 2.9309

−2.5408 −2.5339 −2.5303 −3.1296 −3.7394 −4.3489 −4.9606 −5.5684 −6.1781 −2.5242

40 41 42 43 44 45 46 47 48 49

2-Methyl-1-butene 2-Methyl-2-butene 1,2-Butadiene 1,3-Butadiene 2-Methyl-1,3-butadiene Acetylene Methylacetylene Dimethylacetylene 3-Methyl-1-butyne 1-Pentyne

C5H10 C5H10 C4H6 C4H6 C5H8 C2H2 C3H4 C4H6 C5H8 C5H8

563462 513359 590192 106990 78795 74862 74997 503173 598232 627190

70.134 70.134 54.092 54.092 68.119 26.038 40.065 54.092 68.119 68.119

−3.5300 −4.1800 16.2300 10.9240 7.5730 22.8200 18.4900 14.5700 13.8000 14.4400

6.6680 6.0450 19.8600 14.9720 14.5896 21.0680 19.3840 18.4900 20.7200 21.0300

3.3950 3.3860 2.9300 2.7889 3.1564 2.0081 2.4836 2.8330 3.1890 3.2980

−3.1159 −3.1088 −2.4617 −2.4090 −2.9842 −1.2570 −1.8487 −2.4189 −3.0460 −3.0510

50 51 52 53 54 55 56 57 58 59

2-Pentyne 1-Hexyne 2-Hexyne 3-Hexyne 1-Heptyne 1-Octyne Vinylacetylene Cyclopentane Methylcyclopentane Ethylcyclopentane

C5H8 C6H10 C6H10 C6H10 C7H12 C8H14 C4H4 C5H10 C6H12 C7H14

627214 693027 764352 928494 628717 629050 689974 287923 96377 1640897

68.119 82.145 82.145 82.145 96.172 110.199 52.076 70.134 84.161 98.188

12.5100 12.3700 10.5000 10.6000 10.3000 8.2300 30.4600 −7.7030 −10.6200 −12.6900

19.0700 21.8500 19.9000 19.9000 22.7000 23.5000 30.6000 3.8850 3.6300 4.4800

3.3084 3.6940 3.7200 3.7600 4.0850 4.4780 2.7940 2.9290 3.3990 3.7830

−3.0291 −3.6610 −3.6400 −3.6400 −4.2717 −4.8815 −2.3620 −3.0709 −3.6741 −4.2839

60 61 62 63 64 65 66 67 68 69

Cyclohexane Methylcyclohexane 1,1-Dimethylcyclohexane Ethylcyclohexane Cyclopentene 1-Methylcyclopentene Cyclohexene Benzene Toluene o-Xylene

C6H12 C7H14 C8H16 C8H16 C5H8 C6H10 C6H10 C6H6 C7H8 C8H10

110827 108872 590669 1678917 142290 693890 110838 71432 108883 95476

84.161 98.188 112.215 112.215 68.119 82.145 82.145 78.114 92.141 106.167

−12.3300 −15.4800 −18.1000 −17.1500 3.3100 −0.3800 −0.4600 8.2880 5.0170 1.9080

3.1910 2.7330 3.5229 3.9550 11.0500 10.3800 10.7700 12.9600 12.2200 12.2000

2.9728 3.4330 3.6501 3.8260 2.9127 3.2640 3.1052 2.6930 3.2099 3.5383

−3.6560 −4.2571 −4.8639 −4.8705 −2.9393 −3.5340 −3.5320 −3.1360 −3.7340 −4.3330

70 71 72 73 74 75 76 77 78 79

m-Xylene p-Xylene Ethylbenzene Propylbenzene 1,2,4-Trimethylbenzene Isopropylbenzene 1,3,5-Trimethylbenzene p-Isopropyltoluene Naphthalene Biphenyl

C8H10 C8H10 C8H10 C9H12 C9H12 C9H12 C9H12 C10H14 C10H8 C12H10

108383 106423 100414 103651 95636 98828 108678 99876 91203 92524

106.167 106.167 106.167 120.194 120.194 120.194 120.194 134.221 128.174 154.211

1.7320 1.8030 2.9920 0.7910 −1.3800 0.4000 −1.5900 −2.9000 15.0580 18.2420

11.8760 12.1400 13.0730 13.8090 11.7100 13.7900 11.8100 13.3520 22.4080 28.0230

3.5854 3.5223 3.6063 3.9843 3.9610 3.8600 3.8560 4.2630 3.3315 3.9367

−4.3318 −4.3330 −4.3450 −4.9542 −4.9307 −4.9510 −4.9291 −5.5498 −4.9809 −6.0317

80 81 82 83 84 85 86 87 88 89

Styrene m-Terphenyl Methanol Ethanol 1-Propanol 1-Butanol 2-Butanol 2-Propanol 2-Methyl-2-propanol 1-Pentanol

C8H8 C18H14 CH4O C2H6O C3H8O C4H10O C4H10O C3H8O C4H10O C5H12O

100425 92068 67561 64175 71238 71363 78922 67630 75650 71410

104.152 230.309 32.042 46.069 60.096 74.123 74.123 60.096 74.123 88.150

14.7400 27.6600 −20.0940 −23.4950 −25.5200 −27.4600 −29.2900 −27.2700 −31.2400 −29.8737

21.3900 42.3000 −16.2320 −16.7850 −15.9900 −15.0300 −16.9600 −17.3470 −17.7600 −14.6022

3.4510 5.2630 2.3988 2.8064 3.2247 3.6148 3.6469 3.0920 3.2630 4.0250

−4.2190 −9.0530 −0.6382 −1.2350 −1.8438 −2.4560 −2.4408 −1.8300 −2.4239 −3.0605

90 91 92 93 94 95 96

2-Methyl-1-butanol 3-Methyl-1-butanol 1-Hexanol 1-Heptanol Cyclohexanol Ethylene glycol 1,2-Propylene glycol

C5H12O C5H12O C6H14O C7H16O C6H12O C2H6O2 C3H8O2

137326 123513 111273 111706 108930 107211 57556

88.150 88.150 102.177 116.203 100.161 62.068 76.095

−30.2085 −30.2100 −31.6500 −33.6400 −28.6200 −38.7500 −42.1500

−14.6709 −14.5000 −13.4400 −12.5300 −10.9500 −30.2600 −30.4000

3.9351 3.8770 4.4010 4.7919 3.2770 3.2350 3.5200

−3.0620 −3.0623 −3.6766 −4.2860 −3.4639 −1.0590 −1.6476

HEATS OF COMBUSTION

2-197

TABLE 2-221 Enthalpies and Gibbs Energies of Formation, Entropies, and Net Enthalpies of Combustion of Inorganic and Organic Compounds (Continued ) Cmpd. no.

Name

Formula

CAS no.

Mol wt

Ideal gas enthalpy of formation, J/kmol × 1E-07

Ideal gas Gibbs energy Ideal gas Standard net enthalpy of formation, entropy, of combustion, J/kmol × 1E-07 J/(kmol·K) × 1E-05 J/kmol × 1E-09

97 98 99

Phenol o-Cresol m-Cresol

C6H6O C7H8O C7H8O

108952 95487 108394

94.113 108.140 108.140

−9.6399 −12.8570 −13.2300

−3.2637 −3.5430 −4.0190

3.1481 3.5259 3.5604

−2.9210 −3.5280 −3.5278

100 101 102 103 104 105 106 107 108 109

p-Cresol Dimethyl ether Methyl ethyl ether Methyl n-propyl ether Methyl isopropyl ether Methyl n-butyl ether Methyl isobutyl ether Methyl tert-butyl ether Diethyl ether Ethyl propyl ether

C7H8O C2H6O C3H8O C4H10O C4H10O C5H12O C5H12O C5H12O C4H10O C5H12O

106445 115106 540670 557175 598538 628284 625445 1634044 60297 628320

108.140 46.069 60.096 74.123 74.123 88.150 88.150 88.150 74.123 88.150

−12.5350 −18.4100 −21.6400 −23.8200 −25.2000 −25.8100 −26.6000 −28.3500 −25.2100 −27.2200

−3.1660 −11.2800 −11.7100 −11.1000 −12.1800 −10.1700 −10.7000 −11.7500 −12.2100 −11.5200

3.5075 2.6670 3.0881 3.5200 3.4160 3.9010 3.8100 3.5780 3.4230 3.8810

−3.5226 −1.3284 −1.9314 −2.5174 −2.5311 −3.1282 −3.1220 −3.1049 −2.5035 −3.1200

110 111 112 113 114 115 116 117 118 119

Ethyl isopropyl ether Methyl phenyl ether Diphenyl ether Formaldehyde Acetaldehyde 1-Propanal 1-Butanal 1-Pentanal 1-Hexanal 1-Heptanal

C5H12O C7H8O C12H10O CH2O C2H4O C3H6O C4H8O C5H10O C6H12O C7H14O

625547 100663 101848 50000 75070 123386 123728 110623 66251 111717

88.150 108.140 170.211 30.026 44.053 58.080 72.107 86.134 100.161 114.188

−28.5800 −6.7900 5.2000 −10.8600 −16.6200 −18.6300 −20.7000 −22.7800 −24.8600 −26.9400

−12.6400 2.2700 17.5000 −10.2600 −13.3100 −12.4600 −11.6300 −10.7100 −10.0050 −9.1910

3.8000 3.6100 4.1300 2.1866 2.6420 3.0440 3.4365 3.8289 4.2214 4.6138

−3.1030 −3.6072 −5.8939 −0.5268 −1.1045 −1.6857 −2.3035 −2.9100 −3.5200 −4.1360

120 121 122 123 124 125 126 127 128 129

1-Octanal 1-Nonanal 1-Decanal Acetone Methyl ethyl ketone 2-Pentanone Methyl isopropyl ketone 2-Hexanone Methyl isobutyl ketone 3-Methyl-2-pentanone

C8H16O C9H18O C10H20O C3H6O C4H8O C5H10O C5H10O C6H12O C6H12O C6H12O

124130 124196 112312 67641 78933 107879 563804 591786 108101 565617

128.214 142.241 156.268 58.080 72.107 86.134 86.134 100.161 100.161 100.161

−29.0200 −31.0900 −33.1700 −21.5700 −23.9000 −25.9200 −26.2400 −27.9826 −28.8000 −28.1000

−8.3770 −7.5530 −6.7390 −15.1300 −14.7000 −13.8300 −13.9000 −13.0081 −13.5000 −12.9000

5.0063 5.3988 5.7912 2.9540 3.3940 3.7860 3.6990 4.1786 4.0700 4.1200

−4.7400 −5.3500 −5.9590 −1.6590 −2.2680 −2.8796 −2.8770 −3.4900 −3.4900 −3.4900

130 131 132 133 134 135 136 137 138 139

3-Pentanone Ethyl isopropyl ketone Diisopropyl ketone Cyclohexanone Methyl phenyl ketone Formic acid Acetic acid Propionic acid n-Butyric acid Isobutyric acid

C5H10O C6H12O C7H14O C6H10O C8H8O CH2O2 C2H4O2 C3H6O2 C4H8O2 C4H8O2

96220 565695 565800 108941 98862 64186 64197 79094 107926 79312

86.134 100.161 114.188 98.145 120.151 46.026 60.053 74.079 88.106 88.106

−25.7900 −28.6100 −31.1400 −22.6100 −8.6700 −37.8600 −43.2800 −45.3500 −47.5800 −48.4100

−13.4400 −13.3000 −13.2000 −8.6620 −0.1364 −35.1000 −37.4600 −36.6700 −36.0000 −36.2100

3.7000 4.0690 4.5700 3.2200 3.8450 2.4870 2.8250 3.2300 3.6200 3.4120

−2.8804 −3.4860 −4.0950 −3.2990 −3.9730 −0.2115 −0.8146 −1.3950 −2.0077 −2.0004

140 141 142 143 144 145 146 147 148 149

Benzoic acid Acetic anhydride Methyl formate Methyl acetate Methyl propionate Methyl n-butyrate Ethyl formate Ethyl acetate Ethyl propionate Ethyl n-butyrate

C7H6O2 C4H6O3 C2H4O2 C3H6O2 C4H8O2 C5H10O2 C3H6O2 C4H8O2 C5H10O2 C6H12O2

65850 108247 107313 79209 554121 623427 109944 141786 105373 105544

122.123 102.090 60.053 74.079 88.106 102.133 74.079 88.106 102.133 116.160

−29.4100 −57.2500 −35.2400 −41.1900 −42.7500 −45.0700 −38.8300 −44.4500 −46.3600 −48.5500

−21.4200 −47.3400 −29.5000 −32.4200 −31.1000 −30.5300 −30.3100 −32.8000 −31.9300 −31.2200

3.6900 3.8990 2.8520 3.1980 3.5960 3.9880 3.2820 3.5970 4.0250 4.4170

−3.0951 −1.6750 −0.8924 −1.4610 −2.0780 −2.6860 −1.5070 −2.0610 −2.6740 −3.2840

150 151 152 153 154 155 156 157 158 159

n-Propyl formate n-Propyl acetate n-Butyl acetate Methyl benzoate Ethyl benzoate Vinyl acetate Methylamine Dimethylamine Trimethylamine Ethylamine

C4H8O2 C5H10O2 C6H12O2 C8H8O2 C9H10O2 C4H6O2 CH5N C2H7N C3H9N C2H7N

110747 109604 123864 93583 93890 108054 74895 124403 75503 75047

88.106 102.133 116.160 136.150 150.177 86.090 31.057 45.084 59.111 45.084

−40.7600 −46.4800 −48.5600 −28.7900 −32.6000 −31.4900 −2.2970 −1.8450 −2.4310 −4.7150

−29.3600 −32.0400 −31.2600 −18.1000 −19.0500 −22.7900 3.2070 6.8390 9.8990 3.6160

3.6780 4.0230 4.4250 4.1400 4.5500 3.2800 2.4330 2.7296 2.8700 2.8480

−2.0410 −2.6720 −3.2800 −3.7720 −4.4100 −1.9500 −0.9751 −1.6146 −2.2449 −1.5874

160 161 162

Diethylamine Triethylamine n-Propylamine

C4H11N C6H15N C3H9N

109897 121448 107108

73.138 101.192 59.111

−7.1420 −9.5800 −7.0500

7.3080 11.4100 4.1700

3.5220 4.0540 3.2420

−2.8003 −4.0405 −2.1650

2-198

PHYSICAL AND CHEMICAL DATA

TABLE 2-221 Enthalpies and Gibbs Energies of Formation, Entropies, and Net Enthalpies of Combustion of Inorganic and Organic Compounds (Continued ) Cmpd. no.

Name

Formula

CAS no.

Mol wt

Ideal gas enthalpy of formation, J/kmol × 1E-07

Ideal gas Gibbs energy Ideal gas Standard net enthalpy of formation, entropy, of combustion, J/kmol × 1E-07 J/(kmol·K) × 1E-05 J/kmol × 1E-09

163 164 165 166 167 168 169

di-n-Propylamine Isopropylamine Diisopropylamine Aniline N-Methylaniline N,N-Dimethylaniline Ethylene oxide

C6H15N C3H9N C6H15N C6H7N C7H9N C8H11N C2H4O

142847 75310 108189 62533 100618 121697 75218

101.192 59.111 101.192 93.128 107.155 121.182 44.053

−11.6000 −8.3800 −15.0000 8.7100 8.8000 10.0500 −5.2630

8.6800 3.1920 5.7900 16.6800 20.2000 24.7728 −1.3230

4.2900 3.1240 4.1200 3.1980 3.4100 3.6600 2.4299

−4.0189 −2.1566 −3.9900 −3.2390 −3.9000 −4.5250 −1.2180

170 171 172 173 174 175 176 177 178 179

Furan Thiophene Pyridine Formamide N,N-Dimethylformamide Acetamide N-Methylacetamide Acetonitrile Propionitrile n-Butyronitrile

C4H4O C4H4S C5H5N CH3NO C3H7NO C2H5NO C3H7NO C2H3N C3H5N C4H7N

110009 110021 110861 75127 68122 60355 79163 75058 107120 109740

68.075 84.142 79.101 45.041 73.095 59.068 73.095 41.053 55.079 69.106

−3.4800 11.5440 14.0370 −19.2200 −19.1700 −23.8300 −24.0000 7.4040 5.1800 3.4058

0.0823 12.6620 19.0490 −14.7100 −8.8400 −15.9600 −13.5000 9.1868 9.7495 10.8658

2.6714 2.7865 2.8278 2.4857 3.2600 2.7220 3.2000 2.4329 2.8614 3.2543

−1.9959 −2.4352 −2.6721 −0.5021 −1.7887 −1.0741 −1.7100 −1.1904 −1.8007 −2.4148

180 181 182 183 184 185 186 187 188 189

Benzonitrile Methyl mercaptan Ethyl mercaptan n-Propyl mercaptan n-Butyl mercaptan Isobutyl mercaptan sec-Butyl mercaptan Dimethyl sulfide Methyl ethyl sulfide Diethyl sulfide

C7H5N CH4S C2H6S C3H8S C4H10S C4H10S C4H10S C2H6S C3H8S C4H10S

100470 74931 75081 107039 109795 513440 513531 75183 624895 352932

103.123 48.109 62.136 76.163 90.189 90.189 90.189 62.136 76.163 90.189

21.8823 −2.2900 −4.6300 −6.7500 −8.7800 −9.6900 −9.6600 −3.7240 −5.9600 −8.3470

26.0872 −0.9800 −0.4814 0.2583 1.1390 0.5982 0.5120 0.7302 1.1470 1.7780

3.2104 2.5500 2.9610 3.3650 3.7520 3.6280 3.6670 2.8585 3.3320 3.6800

−3.5224 −1.1517 −1.7366 −2.3458 −2.9554 −2.9490 −2.9490 −1.7449 −2.3531 −2.9607

190 191 192 193 194 195 196 197 198 199

Fluoromethane Chloromethane Trichloromethane Tetrachloromethane Bromomethane Fluoroethane Chloroethane Bromoethane 1-Chloropropane 2-Chloropropane

CH3F CH3Cl CHCl3 CCl4 CH3Br C2H5F C2H5Cl C2H5Br C3H7Cl C3H7Cl

593533 74873 67663 56235 74839 353366 75003 74964 540545 75296

34.033 50.488 119.377 153.822 94.939 48.060 64.514 108.966 78.541 78.541

−23.4300 −8.1960 −10.2900 −9.5810 −3.7700 −26.4400 −11.2260 −6.3600 −13.3180 −14.4770

−21.0400 −5.8440 −7.0100 −5.3540 −2.8190 −21.2300 −6.0499 −2.5820 −5.2610 −6.1360

2.2273 2.3418 2.9560 3.0991 2.4580 2.6440 2.7578 2.8730 3.1547 3.0594

−0.5219 −0.6754 −0.3800 −0.2653 −0.7054 −1.1270 −1.2849 −1.2850 −1.8670 −1.8630

200 201 202 203 204 205 206 207 208 209

1,1-Dichloropropane 1,2-Dichloropropane Vinyl chloride Fluorobenzene Chlorobenzene Bromobenzene Air Hydrogen Helium-4 Neon

C3H6Cl2 C3H6Cl2 C2H3Cl C6H5F C6H5Cl C6H5Br H2 He Ne

78999 78875 75014 462066 108907 108861 132259100 1333740 7440597 7440019

112.986 112.986 62.499 96.104 112.558 157.010 28.951 2.016 4.003 20.180

−15.0800 −16.2800 2.8450 −11.6566 5.1090 10.5018 0 0 0 0

−6.5200 −8.0180 4.1950 −6.9036 9.8290 13.8532 0 0 0 0

3.4480 3.5480 2.7354 3.0263 3.1403 3.2439 1.9900 1.3057 1.2604 1.4622

−1.7200 −1.7070 −1.1780 −2.8145 −2.9760 −3.0192 0 −0.2418 0 0

210 211 212 213 214 215 216 217 218 219

Argon Fluorine Chlorine Bromine Oxygen Nitrogen Ammonia Hydrazine Nitrous oxide Nitric oxide

Ar F2 Cl2 Br2 O2 N2 NH3 N2H4 N2O NO

7440371 7782414 7782505 7726956 7782447 7727379 7664417 302012 10024972 10102439

39.948 37.997 70.905 159.808 31.999 28.014 17.031 32.045 44.013 30.006

0 0 0 3.0910 0 0 −4.5898 9.5353 8.2050 9.0250

0 0 0 0.3140 0 0 −1.6400 15.9170 10.4160 8.6570

1.5474 2.0268 2.2297 2.4535 2.0504 1.9150 1.9266 2.3861 2.1985 2.1060

0 0 0 0 0 0 −0.3168 −5.3420 −0.0820 −0.0902

220 221 222 223 224 225 226 227

Cyanogen Carbon monoxide Carbon dioxide Carbon disulfide Hydrogen fluoride Hydrogen chloride Hydrogen bromide Hydrogen cyanide

C2N2 CO CO2 CS2 HF HCl HBr HCN

460195 630080 124389 75150 7664393 7647010 10035106 74908

52.036 28.010 44.010 76.143 20.006 36.461 80.912 27.026

30.9072 −11.0530 −39.3510 11.6900 −27.3300 −9.2310 −3.6290 13.5143

29.7553 −13.7150 −39.4370 6.6800 −27.5400 −9.5300 −5.3340 12.4725

2.4146 1.9756 2.1368 2.3790 1.7367 1.8679 1.9859 2.0172

−1.0961 −0.2830 0 −1.0769 0.1524 −0.0286 −0.0690 −0.6233

HEATS OF COMBUSTION

2-199

TABLE 2-221 Enthalpies and Gibbs Energies of Formation, Entropies, and Net Enthalpies of Combustion of Inorganic and Organic Compounds (Concluded ) Cmpd. no. 228 229 230 231

Name

Formula

Hydrogen sulfide Sulfur dioxide Sulfur trioxide Water

H2S SO2 SO3 H2O

CAS no. 7783064 7446095 7446119 7732185

Mol wt

Ideal gas enthalpy of formation, J/kmol × 1E-07

34.082 64.065 80.064 18.015

−2.0630 −29.6840 −39.5720 −24.1814

Ideal gas Gibbs energy Ideal gas Standard net enthalpy of formation, entropy, of combustion, J/kmol × 1E-07 J/(kmol·K) × 1E-05 J/kmol × 1E-09 −3.3440 −30.0120 −37.0950 −22.8590

−0.5180 0 0.0989 0

2.0560 2.4810 2.5651 1.8872

All substances are listed in alphabetical order in Table 2-6a. Compiled from Daubert, T. E., R. P. Danner, H. M. Sibul, and C. C. Stebbins, DIPPR Data Compilation of Pure Compound Properties, Project 801 Sponsor Release, July, 1993, Design Institute for Physical Property Data, AIChE, New York, NY; and from Thermodynamics Research Center, “Selected Values of Properties of Hydrocarbons and Related Compounds,” Thermodynamics Research Center Hydrocarbon Project, Texas A&M University, College Station, Texas (extant 1994). The compounds are considered to be formed from the elements in their standard states at 298.15 K and 101,325 Pa. These include C (graphite) and S (rhombic). Enthalpy of combustion is the net value for the compound in its standard state at 298.15K and 101,325 Pa. Products of combustion are taken to be CO2 (gas), H2O (gas), F2 (gas), Cl2 (gas), Br2 (gas), I2 (gas), SO2 (gas), N2 (gas), H3PO4 (solid), and SiO2 (crystobalite). J/kmol × 2.390E-04 = cal/gmol; J/kmol × 4.302106E-04 = Btu/lbmol. J/(kmol·K) × 2.390E-04 = cal/(gmol·°C); J/(kmol·K) × 2.390059E-04 = Btu/(lbmol·°F). TABLE 2-222

Ideal Gas Sensible Enthalpies, hT - h298 (kJ/kgmol), of Combustion Products

Temperature, K

CO

CO2

H

OH

H2

N

NO

NO2

N2

N2O

O

O2

SO2

H2O

200 240 260 280 298.15

−2858 −1692 −1110 −529 0

−3414 −2079 −1383 −665 0

−2040 −1209 −793 −377 0

−2976 −1756 −1150 −546 0

−2774 −1656 −1091 −522 0

−2040 −1209 −793 −378 0

−2951 −1743 −1142 −543 0

−3495 −2104 −1392 −672 0

−2857 −1692 −1110 −528 0

−3553 −2164 −1438 −692 0

−2186 −1285 −840 −398 0

−2868 −1703 −1118 −533 0

−3736 −2258 −1496 −718 0

−3282 −1948 −1279 −609 0

300 320 340 360 380

54 638 1221 1805 2389

69 823 1594 2382 3184

38 454 870 1285 1701

55 654 1251 1847 2442

53 630 1209 1791 2373

38 454 870 1286 1701

55 652 1248 1845 2442

68 816 1571 2347 3130

54 636 1219 1802 2386

72 854 1654 2470 3302

41 478 913 1346 1777

54 643 1234 1828 2425

74 881 1702 2538 3387

62 735 1410 2088 2769

400 420 440 460 480

2975 3563 4153 4643 5335

4003 4835 5683 6544 7416

2117 2532 2948 3364 3779

3035 3627 4219 4810 5401

2959 3544 4131 4715 5298

2117 2533 2949 3364 3780

3040 3638 4240 4844 5450

3927 4735 5557 6392 7239

2971 3557 4143 4731 5320

4149 5010 5884 6771 7670

2207 2635 3063 3490 3918

3025 3629 4236 4847 5463

4250 5126 6015 6917 7831

3452 4139 4829 5523 6222

500 550 600 650 700

5931 7428 8942 10477 12023

8305 10572 12907 15303 17754

4196 5235 6274 7314 8353

5992 7385 8943 10423 11902

5882 6760 8811 10278 11749

4196 5235 6274 7314 8353

6059 7592 9144 10716 12307

8099 10340 12555 14882 17250

5911 7395 8894 10407 11937

8580 10897 13295 15744 18243

4343 5402 6462 7515 8570

6084 7653 9244 10859 12499

8758 11123 13544 16022 18548

6925 8699 10501 12321 14192

750 800 850 900 950

13592 15177 16781 18401 20031

20260 22806 25398 28030 30689

9392 10431 11471 12510 13550

13391 14880 16384 17888 19412

13223 14702 16186 17676 19175

9329 10431 11471 12510 13550

13919 15548 17195 18858 20537

19671 22136 24641 27179 29749

13481 15046 16624 18223 19834

20791 23383 26014 28681 31381

9620 10671 11718 12767 13812

14158 15835 17531 19241 20965

21117 23721 26369 29023 31714

16082 18002 19954 21938 23954

1000 1100 1200 1300 1400

21690 25035 28430 31868 35343

33397 38884 44473 50148 55896

14589 16667 18746 20824 22903

20935 24024 27160 30342 33569

20680 23719 26797 29918 33082

14589 16667 18746 20824 22903

22229 25653 29120 32626 36164

32344 37605 42946 48351 53808

21463 24760 28109 31503 34936

34110 39647 45274 50976 56740

14860 16950 19039 21126 23212

22703 26212 29761 33344 36957

34428 39914 45464 51069 56718

26000 30191 34506 38942 43493

1500 1600 1700 1800 1900

38850 42385 45945 49526 53126

61705 67569 73480 79431 85419

24982 27060 29139 31217 33296

36839 40151 43502 46889 50310

36290 39541 42835 46169 49541

24982 27060 29139 31218 33296

39729 43319 46929 50557 54201

59309 64846 70414 76007 81624

38405 41904 45429 48978 52548

62557 68420 74320 80254 86216

25296 27381 29464 31547 33630

40599 44266 47958 51673 55413

62404 68123 73870 79642 85436

48151 52908 57758 62693 67706

2000 2100 2200 2300 2400

56744 60376 64021 67683 71324

91439 97488 103562 109660 115779

35375 37453 39532 41610 43689

53762 57243 60752 64285 67841

52951 56397 59876 63387 66928

35375 37454 39534 41614 43695

57859 61530 65212 68904 72606

87259 92911 98577 104257 109947

56137 59742 63361 66995 70640

92203 98212 104240 110284 116344

35713 37796 39878 41962 44045

59175 62961 66769 70600 74453

91250 97081 102929 108792 114669

72790 77941 83153 88421 93741

2500 2600 2700 2800 2900

74985 78673 82369 86074 89786

121917 128073 134246 140433 146636

45768 47846 49925 52004 54082

71419 75017 78633 82267 85918

70498 74096 77720 81369 85043

45777 47860 49945 52033 54124

76316 80034 83759 87491 91229

115648 121357 127075 132799 138530

74296 77963 81639 85323 89015

122417 128501 134596 140701 146814

46130 48216 50303 52391 54481

78328 82224 86141 90079 94036

120559 126462 132376 138302 144238

99108 104520 109973 115464 120990

3000 3500 4000 4500 5000

93504 112185 130989 149895 168890

152852 184109 215622 247354 279283

56161 66554 75947 87340 97733

89584 108119 126939 145991 165246

88740 107555 126874 146660 166876

56218 66769 77532 88614 100111

94973 113768 132671 151662 170730

144267 173020 201859 230756 259692

92715 111306 130027 148850 167763

152935 183636 214453 245348 276299

56574 67079 77675 88386 99222

98013 118165 188705 159572 180749

150184 180057 210145 240427 270893

126549 154768 183552 212764 242313

Converted and usually rounded off from JANAF Thermochemical Tables, NSRDS-NBS-37, 1971 (1141 pp.)

2-200

PHYSICAL AND CHEMICAL DATA

TABLE 2-223 Temperature, K

Ideal Gas Entropies, s°, kJ/kgmol·K, of Combustion Products CO

CO2

H

OH

H2

N

NO

NO2

N2

N2O

O

O2

SO2

H2O

200 240 260 280 298.15

186.0 191.3 193.7 195.3 197.7

200.0 206.0 208.8 211.5 213.8

106.4 110.1 111.8 113.3 114.7

171.6 177.1 179.5 181.8 183.7

119.4 124.5 126.8 129.2 130.7

145.0 148.7 150.4 151.9 153.3

198.7 204.1 206.6 208.8 210.8

225.9 232.2 235.0 237.7 240.0

180.0 185.2 187.6 189.8 191.6

205.6 211.9 214.8 217.5 220.0

152.2 156.2 158.0 159.7 161.1

193.5 198.7 201.1 203.3 205.1

233.0 239.9 242.8 245.8 248.2

175.5 181.4 184.1 186.6 188.8

300 320 340 360 380

197.8 199.7 201.5 203.2 204.7

214.0 216.5 218.8 221.0 223.2

114.8 116.2 117.4 118.6 119.7

183.9 185.9 187.7 189.4 191.0

130.9 132.8 134.5 136.2 137.7

153.4 154.8 156.0 157.2 158.3

210.9 212.9 214.7 216.4 218.0

240.3 242.7 245.0 247.2 249.3

191.8 193.7 195.5 197.2 198.7

220.2 222.7 225.2 227.5 229.7

161.2 162.6 163.9 165.2 166.3

205.3 207.2 209.0 210.7 212.5

248.5 251.1 253.6 256.0 258.2

189.0 191.2 193.3 195.2 197.1

400 420 440 460 480

206.2 207.7 209.0 210.4 211.6

225.3 227.3 229.3 231.2 233.1

120.8 121.8 122.8 123.7 124.6

192.5 194.0 195.3 196.6 197.9

139.2 140.6 141.9 143.2 144.5

159.4 160.4 161.4 162.3 163.1

219.5 221.0 222.3 223.7 225.0

251.3 253.2 255.1 257.0 258.8

200.2 201.5 202.9 204.2 205.5

231.9 234.0 236.0 238.0 239.9

167.4 168.4 169.4 170.4 171.3

213.8 215.3 216.7 218.0 219.4

260.4 262.5 264.6 266.6 268.5

198.8 200.5 202.0 203.6 205.1

500 550 600 650 700

212.8 215.7 218.3 220.8 223.1

234.9 239.2 243.3 247.1 250.8

125.5 127.5 129.3 131.0 132.5

199.1 201.8 204.4 206.8 209.0

145.7 148.6 151.1 153.4 155.6

164.0 166.0 167.8 169.4 171.0

226.3 229.1 231.9 234.4 236.8

260.6 264.7 268.8 272.6 276.0

206.7 209.4 212.2 214.6 216.9

241.8 246.2 250.4 254.3 258.0

172.2 174.2 176.1 177.7 179.3

220.7 223.7 226.5 229.1 231.5

270.5 274.9 279.2 283.1 286.9

206.5 210.5 213.1 215.9 218.7

750 800 850 900 950

225.2 227.3 229.2 231.1 232.8

255.4 257.5 260.6 263.6 266.5

133.9 135.2 136.4 137.7 138.8

211.1 213.0 214.8 216.5 218.1

157.6 159.5 161.4 163.1 164.7

172.5 173.8 175.1 176.3 177.4

239.0 241.1 243.0 245.0 246.8

279.3 282.5 285.5 288.4 291.3

219.0 221.0 223.0 224.8 226.5

261.5 264.8 268.0 271.1 274.0

180.7 182.1 183.4 184.6 185.7

233.7 235.9 237.9 239.9 241.8

290.4 293.8 297.0 300.1 303.0

221.3 223.8 226.2 228.5 230.6

1000 1100 1200 1300 1400

234.5 237.7 240.7 243.4 246.0

269.3 274.5 279.4 283.9 288.2

139.9 141.9 143.7 145.3 146.9

219.7 222.7 225.4 228.0 230.3

166.2 169.1 171.8 174.3 176.6

178.5 180.4 182.2 183.9 185.4

248.4 251.8 254.8 257.6 260.2

293.9 298.9 303.6 307.9 311.9

228.2 231.3 234.2 236.9 239.5

276.8 282.1 287.0 291.5 295.8

186.8 188.8 190.6 192.3 193.8

243.6 246.9 250.0 252.9 255.6

305.8 311.0 315.8 320.3 324.5

232.7 236.7 240.5 244.0 247.4

1500 1600 1700 1800 1900

248.4 250.7 252.9 254.9 256.8

292.2 296.0 299.6 303.0 306.2

148.3 149.6 150.9 152.1 153.2

232.6 234.7 236.8 238.7 240.6

178.8 180.9 182.9 184.8 186.7

186.9 188.2 189.5 190.7 191.8

262.7 265.0 267.2 269.3 271.3

315.7 319.3 322.7 325.9 328.9

241.9 244.1 246.3 248.3 250.2

299.8 303.6 307.2 310.6 313.8

195.3 196.6 197.9 199.1 200.2

258.1 260.4 262.7 264.8 266.8

328.4 332.1 335.6 338.9 342.0

250.6 253.7 256.6 259.5 262.2

2000 2100 2200 2300 2400

258.7 260.5 262.2 263.8 265.4

309.3 312.2 315.1 317.8 320.4

154.3 155.3 156.3 157.2 158.1

242.3 244.0 245.7 247.2 248.7

188.4 190.1 191.7 193.3 194.8

192.9 193.9 194.8 195.8 196.7

273.1 274.9 276.6 278.3 279.8

331.8 334.5 337.2 339.7 342.1

252.1 253.8 255.5 257.1 258.7

316.9 319.8 322.6 325.3 327.9

201.3 202.3 203.2 204.2 205.0

268.7 270.6 272.4 274.1 275.7

345.0 347.9 350.6 353.2 355.7

264.8 267.3 269.7 272.0 274.3

2500 2600 2700 2800 2900

266.9 268.3 269.7 271.0 272.3

322.9 325.3 327.6 329.9 332.1

158.9 159.7 160.5 161.3 162.0

250.2 251.6 253.0 254.3 255.6

196.2 197.7 199.0 200.3 201.6

197.5 198.3 199.1 199.9 200.6

281.4 282.8 284.2 285.6 286.9

344.5 346.7 348.9 350.9 352.9

260.2 261.6 263.0 264.3 265.6

330.4 332.7 335.0 337.3 339.4

205.9 206.7 207.5 208.3 209.0

277.3 278.8 280.3 281.7 283.1

358.1 360.4 362.6 364.8 366.9

276.5 278.6 380.7 282.7 284.6

3000 3500 4000 4500 5000

273.6 279.4 284.4 288.8 292.8

334.2 343.8 352.2 359.7 366.4

162.7 165.9 168.7 171.1 173.3

256.8 262.5 267.6 272.1 276.1

202.9 208.7 213.8 218.5 222.8

201.3 204.6 207.4 210.1 212.5

288.2 294.0 299.0 303.5 307.5

354.9 363.8 371.5 378.3 384.4

266.9 272.6 277.6 282.1 286.0

341.5 350.9 359.2 366.5 373.0

209.7 212.9 215.8 218.3 220.6

284.4 290.7 296.2 301.1 305.5

368.9 378.1 386.1 393.3 399.7

286.5 295.2 302.9 309.8 316.0

Usually rounded off from JANAF Thermochemical Tables, NSRDS-NBS-37, 1971 (1141 pp.). Equilibrium constants can be calculated by combining ∆h°f values from Table 2-221, hT − h298 from Table 2-222, and s° values from the above, using the formula ln kp = −∆G/(RT), where ∆G = ∆h°f + (hT − h298) − T °. s

HEATS OF SOLUTION

2-201

HEATS OF SOLUTION TABLE 2-224

Heats of Solution of Inorganic Compounds in Water

Heat evolved, in kilogram-calories per gram formula weight, on solution in water at 18°C. Computed from data in Bichowsky and Rossini, Thermochemistry of Chemical Substances, Reinhold, New York, 1936.

Substance

Dilution*

Formula

Heat, kg-cal/ g-mole

Aluminum bromide chloride

aq 600 600 aq aq aq aq aq aq aq aq ∞ aq 600 aq ∞ aq ∞ 800 aq aq aq aq aq

AlBr3 AlCl3 AlCl3·6H2O AlF3 AlF3·aH2O AlF3·3aH2O AlI3 Al2(SO4)3 Al2(SO4)3·6H2O Al2(SO4)3·18H2O NH4Br NH4Cl (NH4)2CrO4 (NH4)2Cr2O7 NH4I NH4NO3 NH4BO3·H2O (NH4)2SO4 NH4HSO4 (NH4)2SO3 (NH4)2SO3·H2O SbF3 SbI3 H3AsO4

+85.3 +77.9 +13.2 +31 +19.0 −1.7 +89.0 +126 +56.2 +6.7 −4.45 −3.82 −5.82 −12.9 −3.56 −6.47 −9.0 −2.75 +0.56 −1.2 −4.13 −1.7 −0.8 −0.4

∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ aq aq aq ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ aq aq aq aq aq aq aq aq aq

Ba(BrO3)2·H2O BaBr2 BaBr2·H2O BaBr2·2H2O Ba(ClO3)2 Ba(ClO3)2·H2O BaCl2 BaCl2·H2O BaCl2.2H2O Ba(CN)2 Ba(CN)2·H2O Ba(CN)2·2H2O Ba(IO3)2 Ba(IO3)2·H2O BaI2 BaI2·H2O BaI2·2H2O BaI2·2aH2O BaI2·7H2O Ba(NO3)2 Ba(ClO4)2 Ba(ClO4)2·3H2O BaS BeBr2 BeCl2 BeI2 BeSO4 BeSO4·H2O BeSO4·2H2O BeSO4·4H2O BiI3 H3BO3

−15.9 +5.3 −0.8 −3.87 −6.7 −10.6 +2.4 −2.17 −4.5 +1.5 −2.4 −4.9 −9.1 −11.3 +10.5 +2.7 +0.14 −0.58 −6.61 −10.2 −2.8 −10.5 +7.2 +62.6 +51.1 +72.6 +18.1 +13.5 +7.9 +1.1 +3 −5.4

400 400 400 400 400 400 400 400 400 400 ∞ ∞

CdBr2 CdBr2·4H2O CdCl2 CdCl2·H2O CdCl2·2aH2O Cd(NO3)2·H2O Cd(NO3)2·4H2O CdSO4 CdSO4·H2O CdSO4·2wH2O Ca(C2H3O2)2 Ca(C2H3O2)2·H2O

+0.4 −7.3 +3.1 +0.6 −3.00 +4.17 −5.08 +10.69 +6.05 +2.51 +7.6 +6.5

fluoride iodide sulfate Ammonium bromide chloride chromate dichromate iodide nitrate perborate sulfate sulfate, acid sulfite Antimony fluoride iodide Arsenic acid Barium bromate bromide chlorate chloride cyanide iodate iodide

nitrate perchlorate sulfide Beryllium bromide chloride iodide sulfate

Bismuth iodide Boric acid Cadmium bromide chloride nitrate sulfate Calcium acetate

Substance Calcium—(Cont.) bromide

Dilution*

Formula

Heat, kg-cal/ g-mole

∞ ∞ ∞ ∞ ∞ ∞ ∞ 400 ∞ ∞ ∞ ∞ ∞ ∞ ∞ aq aq ∞ ∞ ∞ aq

+24.86 −0.9 +4.9 +12.3 +12.5 +2.4 −4.11 +0.7 +28.0 +1.8 +4.1 +0.7 −3.2 −4.2 −7.99 −0.6 −1 +5.1 +3.6 −0.18 +18.6 +5.3 +2.0 +5.7 +18.4 −1.25 +18.5 +9.8 −2.9 +18.8 +15.0 −1.4 −3.6 +2.4 +0.5 +10.3 −2.6 −10.7 +15.9 +9.3 +3.65 −2.85 +11.6

Cuprous sulfate

aq

CaBr2 CaBr2·6H2O CaCl2 CaCl2·H2O CaCl2·2H2O CaCl2·4H2O CaCl2·6H2O Ca(CHO2)2 CaI2 CaI2·8H2O Ca(NO3)2 Ca(NO3)2·H2O Ca(NO3)2·2H2O Ca(NO3)2·3H2O Ca(NO3)2·4H2O Ca(H2PO4)2·H2O CaHPO4·2H2O CaSO4 CaSO4·aH2O CaSO4·2H2O CrCl2 CrCl2·3H2O CrCl2·4H2O CrI2 CoBr2 CoBr2·6H2O CoCl2 CoCl2·2H2O CoCl2·6H2O CoI2 CoSO4 CoSO4·6H2O CoSO4·7H2O Cu(C2H3O2)2 Cu(CHO2)2 Cu(NO3)2 Cu(NO3)2·3H2O Cu(NO3)2·6H2O CuSO4 CuSO4·H2O CuSO4·3H2O CuSO4·5H2O Cu2SO4

Ferric chloride

1000 1000 1000 800 aq 400 400 400 aq 400 400 400 400

FeCl3 FeCl3·2aH2O FeCl3·6H2O Fe(NO3)3·9H2O FeBr2 FeCl2 FeCl2·2H2O FeCl2·4H2O FeI2 FeSO4 FeSO4·H2O FeSO4·4H2O FeSO4·7H2O

+31.7 +21.0 +5.6 −9.1 +18.0 +17.9 +8.7 +2.7 +23.3 +14.7 +7.35 +1.4 −4.4

400 400 aq aq aq 400 ∞ ∞ ∞ ∞ ∞

Pb(C2H3O2)2 Pb(C2H3O2)2·3H2O PbBr2 PbCl2 Pb(CHO2)2 Pb(NO3)2 LiBr LiBr·H2O LiBr·2H2O LiBr·3H2O LiCl

+1.4 −5.9 −10.1 −3.4 −6.9 −7.61 +11.54 +5.30 +2.05 −1.59 +8.66

chloride

formate iodide nitrate

phosphate, monodibasic sulfate Chromous chloride iodide Cobaltous bromide chloride iodide sulfate Cupric acetate formate nitrate sulfate

nitrate Ferrous bromide chloride iodide sulfate

Lead acetate bromide chloride formate nitrate Lithium bromide

chloride

aq aq aq 400 400 400 aq 400 400 400 aq aq 200 200 200 800

*The numbers represent moles of water used to dissolve 1 g formula weight of substance; ∞ means “infinite dilution”; and aq means “aqueous solution of unspecified dilution.”

2-202

PHYSICAL AND CHEMICAL DATA

TABLE 2-224

Heats of Solution of Inorganic Compounds in Water (Continued )

Substance Lithium—(Cont.)

fluoride hydroxide iodide

nitrate sulfate Magnesium bromide chloride

iodide nitrate phosphate sulfate

sulfide Manganic nitrate sulfate Manganous acetate bromide chloride formate iodide

sulfate Mercuric acetate bromide chloride nitrate Mercurous nitrate Nickel bromide Nickel chloride

iodide nitrate sulfate

Dilution*

Formula

Heat, kg-cal/ g-mole

∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞

LiCl·H2O LiCl·2H2O LiCl·3H2O LiF LiOH LiOH·fH2O LiOH·H2O LiI LiI·aH2O LiI·H2O LiI·2H2O LiI·3H2O LiNO3 LiNO3·3H2O Li2SO4 Li2SO4·H2O

+4.45 +1.07 −1.98 −0.74 +4.74 +4.39 +9.6 +14.92 +10.08 +6.93 +3.43 −0.17 +0.466 −7.87 +6.71 +3.77

∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ aq ∞ ∞ ∞ ∞ ∞ ∞ aq 400 400 400 aq aq aq aq aq aq 400 400 400 aq aq aq aq aq aq aq 400 400 400 aq aq aq aq aq

MgBr2 MgBr2·H2O MgBr2·6H2O MgCl2 MgCl2·2H2O MgCl2·4H2O MgCl2·6H2O MgI2 Mg(NO3)2·6H2O Mg3(PO4)2 MgSO4 MgSO4·H2O MgSO4·2H2O MgSO4·4H2O MgSO4·6H2O MgSO4·7H2O MgS Mn(NO3)2 Mn(NO3)2·3H2O Mn(NO3)2·6H2O Mn2(SO4)3 Mn(C2H3O2)2 Mn(C2H3O2)2·4H2O MnBr2 MnBr2·H2O MnBr2·4H2O MnCl2 MnCl2·2H2O MnCl2·4H2O Mn(CHO2)2 Mn(CHO2)2·2H2O MnI2 MnI2·H2O MnI2·2H2O MnI2·4H2O MnI2·6H2O MnSO4 MnSO4·H2O MnSO4·7H2O Hg(C2H3O2)2 HgBr2 HgCl2 Hg(NO3)2·aH2O Hg2(NO3)2·2H2O

+43.7 +35.9 +19.8 +36.3 +20.8 +10.5 +3.4 +50.2 −3.7 +10.2 +21.1 +14.0 +11.7 +4.9 +0.55 −3.18 +25.8 +12.9 −3.9 −6.2 +22 +12.2 +1.6 +15 +14.4 +16.1 +16.0 +8.2 +1.5 +4.3 −2.9 +26.2 +24.1 +22.7 +19.9 +21.2 +13.8 +11.9 −1.7 −4.0 −2.4 −3.3 −0.7 −11.5

aq aq 800 800 800 800 aq 200 200 200 200

NiBr2 NiBr2·3H2O NiCl2 NiCl2·2H2O NiCl2·4H2O NiCl2·6H2O NiI2 Ni(NO3)2 Ni(NO3)2·6H2O NiSO4 NiSO4·7H2O

+19.0 +0.2 +19.23 +10.4 +4.2 −1.15 +19.4 +11.8 −7.5 +15.1 −4.2

Substance

Dilution*

Phosphoric acid, orthopyroPotassium acetate aluminum sulfate

∞ 400 aq aq aq ∞ 800 ∞ aq aq ∞ aq ∞

+2.79 −0.1 +25.9 +4.65 +3.55 +48.5 +26.6 −10.1 −5.1 −10.13 −5.13 +6.58 +4.25 −0.43 −10.31 −4.404 −4.9 +55 +42 +33 +7 −9.5 −3.0 −17.8 +3.96 −1.85 −6.05 +0.86 +1.21 +12.91 +4.27 +3.48 +0.86 −6.93 −5.23 −8.633 −4.6 −7.5 −12.94 −10.4 +4.7 −11.0 −10.22 −6.32 −3.10 −11.0 +1.8 +1.37 −6.08 −13.0 −4.5

aq 200 ∞ ∞ 500 500 1800 900 900 ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ 800 800 800 200 200

AgC2H3O2 AgNO3 NaC2H3O2 NaC2H3O2·3H2O Na3AsO4 Na3AsO4·12H2O NaHCO3 Na2B4O7 Na2B4O7·10H2O NaBr NaBr·2H2O Na2CO3 Na2CO3·H2O Na2CO3·7H2O Na2CO3·10H2O NaClO3 NaCl Na2CrO4 Na2CrO4·4H2O Na2CrO4·10H2O NaCN NaCN·aH2O

−5.4 −4.4 +4.085 −4.665 +15.6 −12.61 −4.1 +10.0 −16.8 −0.58 −4.57 +5.57 +2.19 −10.81 −16.22 −5.37 −1.164 +2.50 −7.52 −16.0 −0.37 −0.92

2000 ∞ ∞ ∞

chlorate chloride chromate chrome sulfate

∞ ∞ 2185 600

cyanide dichromate fluoride

200 1600 ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ 400

hydroxide

iodate iodide nitrate oxalate perchlorate permanganate phosphate, dihydrogen pyrosulfite sulfate sulfate, acid sulfide sulfite thiocyanate thionate, dithiosulfate Silver acetate nitrate Sodium acetate arsenate bicarbonate borate, tetrabromide carbonate

chlorate chloride chromate cyanide

Heat, kg-cal/ g-mole

H3PO4 H3PO4·aH2O H4P2O7 H4P2O7·1aH2O KC2H3O2 KAl(SO4)2 KAl(SO4)2·3H2O KAl(SO4)2·12H2O KHCO3 KBrO3 KBr K2CO3 K2CO3·aH2O K2CO3·1aH2O KClO3 KCl K2CrO4 KCr(SO4)2 KCr(SO4)2·H2O KCr(SO4)2·2H2O KCr(SO4)2·6H2O KCr(SO4)2·12H2O KCN K2Cr2O7 KF KF·2H2O KF·4H2O KHS KHS·dH2O KOH KOH·eH2O KOH·H2O KOH·7H2O KIO3 KI KNO3 K2C2O4 K2C2O4·H2O KClO4 KMnO4 KH2PO4 K2S2O5 K2S2O5·aH2O K2SO4 KHSO4 K2S K2SO3 K2SO3·H2O KCNS K2S2O6 K2S2O3

400 400 aq aq ∞ 600 600

bicarbonate bromate bromide carbonate

hydrosulfide

Formula

HEATS OF SOLUTION TABLE 2-224

Heats of Solution of Inorganic Compounds in Water (Concluded )

Substance

Dilution*

Formula

200 ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ 600 ∞ aq ∞ 1600 1600 1600 1600 1600 1600 600 600 800 800 1600 1600 1200 1200 ∞ ∞ 800 800 ∞ ∞ ∞ ∞ ∞ ∞ ∞

NaCN·2H2O NaF NaHS NaHS·2H2O NaOH NaOH·aH2O NaOH·wH2O NaOH·eH2O NaOH·H2O NaI NaI·2H2O NaPO3 NaNO3 NaNO2 NaClO4 Na2HPO4 Na3PO4 Na3PO4·12H2O Na2HPO4·2H2O Na2HPO4·7H2O Na2HPO4·12H2O NaH2PO3 NaH2PO3·2aH2O Na2HPO3 Na2HPO3·5H2O Na4P2O7 Na4P2O7·10H2O Na2H2P2O7 Na2H2P2O7·6H2O Na2SO4 Na2SO4·10H2O NaHSO4 NaHSO4·H2O Na2S Na2S·4aH2O Na2S·5H2O Na2S·9H2O Na2SO3 Na2SO3·7H2O NaCNS

Sodium—(Cont.) fluoride hydrosulfide Sodium hydroxide

iodide metaphosphate nitrate nitrite perchlorate phosphate di triphosphate di diphosphite, monodipyrophosphate disulfate sulfate, acid sulfide

sulfite thiocyanate NOTE:

2-203

Heat, kg-cal/ g-mole −4.41 −0.27 +4.62 −1.49 +10.18 +8.17 +7.08 +6.48 +5.17 +1.57 −3.89 +3.97 −5.05 −3.6 −4.15 +5.21 +13 −15.3 −0.82 −12.04 −23.18 +0.90 −5.29 +9.30 −4.54 +11.9 −11.7 −2.2 −14.0 +0.28 −18.74 +1.74 +0.15 +15.2 +0.09 −6.54 −16.65 +2.8 −11.1 −1.83

Substance Sodium—(Cont.) thionate, diSodium thiosulfate Stannic bromide Stannous bromide iodide Strontium acetate bromide

chloride

iodide

nitrate sulfate Sulfuric acid, pyroZinc acetate bromide chloride iodide nitrate sulfate

To convert kilocalories per gram-mole to British thermal units per pound-mole, multiply by 1.799 × 10−3.

Dilution*

Formula

Heat, kg-cal/ g-mole

aq aq aq aq aq aq aq ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞

Na2S2O6 Na2S2O6·2H2O Na2S2O3 Na2S2O3·5H2O SnBr4 SnBr2 SnI2 Sr(C2H3O2)2 Sr(C2H3O2)2·aH2O SrBr2 SrBr2·H2O SrBr2·2H2O SrBr2·4H2O SrBr2·6H2O SrCl2 SrCl2·H2O SrCl2·2H2O SrCl2·6H2O SrI2 SrI2·H2O SrI2·2H2O SrI2·6H2O Sr(NO3)2 Sr(NO3)2·4H2O SrSO4 H2S2O7

−5.80 −11.86 +2.0 −11.30 +15.5 −1.6 −5.8 +6.2 +5.9 +16.4 +9.25 +6.5 +0.4 −6.1 +11.54 +6.4 +2.95 −7.1 +20.7 +12.65 +10.4 −4.5 −4.8 −12.4 +0.5 −18.08

400 400 400 400 400 aq 400 400 400 400 400 400

Zn(C2H3O2)2 Zn(C2H3O2)2·H2O Zn(C2H3O2)2·2H2O ZnBr2 ZnCl2 ZnI2 Zn(NO3)2·3H2O Zn(NO3)2·6H2O ZnSO4 ZnSO4·H2O ZnSO4·6H2O ZnSO4·7H2O

+9.8 +7.0 +3.9 +15.0 +15.72 +11.6 −5 −6.0 +18.5 +10.0 −0.8 −4.3

2-204

PHYSICAL AND CHEMICAL DATA TABLE 2-225 Heats of Solution of Organic Compounds in Water (at Infinite Dilution and Approximately Room Temperature) Recalculated and rearranged from International Critical Tables, vol. 5, pp. 148–150. (g⋅cal)/(g⋅mol) = Btu/(lb⋅mol) × 1.799.

Solute Acetic acid (solid), C2H4O2 Acetylacetone, C5H8O2 Acetylurea, C3H6N2O2 Aconitic acid, C6H6O6 Ammonium benzoate, C7H9NO2 picrate succinate (n-) Aniline, hydrochloride, C6H8ClN Barium picrate Benzoic acid, C7H6O2 Camphoric acid, C10H16O4 Citric acid, C6H8O7 Dextrin, C12H20O10 Fumaric acid, C4H4O4 Hexamethylenetetramine, C6H12N4 Hydroxybenzamide (m-), C7H7NO2 (m-), (HCl) (o-), C7H7NO2 (p-) Hydroxybenzoic acid (o-), C7H6O3 (p-), C7H6O3 Hydroxybenzyl alcohol (o-), C7H8O2 Inulin, C36H62O31 Isosuccinic acid, C4H6O4 Itaconic acid, C5H6O4 Lactose, C12H22O11·H2O Lead picrate (2H2O) Magnesium picrate (8H2O) Maleic acid, C4H4O4 Malic acid, C4H6O5 Malonic acid, C3H4O4 Mandelic acid, C8H2O3 Mannitol, C6H14O6 Menthol, C10H20O Nicotine dihydrochloride, C10H16Cl2N2 Nitrobenzoic acid (m-), C7H5NO4 (o-), C7H5NO4 (p-), C7H5NO4 Nitrophenol (m-), C6H5NO3 (o-), C6H5NO3 (p-), C6H5NO3

Heat of Solution, G-cal/g-mole Solute* −2,251 −641 −6,812 −4,206 −2,700 −8,700 −3,489 −2,732 −4,708 −6,501 −502 −5,401 268 −5,903 4,780 −4,161 −7,003 −4,340 −5,392 −6,350 −5,781 −3,203 −96 −3,420 −5,922 −3,705 −7,098 −13,193 14,699 −15,894 −4,441 −3,150 −4,493 −3,090 −5,260 0 6,561 −5,593 −5,306 −8,891 −5,210 −6,310 −4,493

Solute Oxalic acid, C2H2O4 (2H2O) Phenol (solid), C6H6O Phthalic acid, C8H6O4 Picric acid, C6H3N3O7 Piperic acid, C12H10O4 Piperonylic acid, C8H6O4 Potassium benzoate citrate tartrate (n-) (0.5 H2O) Pyrogallol, C6H6O3 Pyrotartaric acid Quinone Raffinose, C18H32O16 (5H2O) Resorcinol, C6H6O2 Silver malonate (n-) Sodium citrate (tri-) picrate potassium tartrate (4H2O) succinate (n-) (6H2O) tartrate (n-) (2H2O) Strontium picrate (6H2O) Succinic acid, C4H6O4 Succinimide, C4H5NO2 Sucrose, C12H22O11 Tartaric acid (d-) Thiourea, CH4N2S Urea, CH4N2O acetate formate nitrate oxalate Vanillic acid Vanillin Zinc picrate (8H2O)

Heat of Solution, G-cal/g-mole Solute* −2,290 −8,485 −2,605 −4,871 −7,098 −10,492 −9,106 −1,506 2,820 −5,562 −3,705 −5,019 −3,991 −9,703 −3,960 −9,799 5,270 −6,441 −1,817 −12,342 2,390 −10,994 −1,121 −5,882 7,887 −14,412 −6,405 −4,302 −1,319 −3,451 −5,330 −3,609 −8,795 −7,194 −10,803 −17,806 −5,160 −5,210 −11,496 −15,894

* + denotes heat evolved, and − denotes heat absorbed. All values are positive unless otherwise noted. The data in the International Critical Tables were calculated by E. Anderson.

THERMODYNAMIC PROPERTIES

2-205

THERMODYNAMIC PROPERTIES EXPLANATION OF TABLES The following subsection presents information on the thermodynamic properties of a number of fluids. In some cases transport properties are also included. Notation cp = specific heat e = specific internal energy h = enthalpy k = thermal conductivity p = pressure s = specific entropy t = temperature T = absolute temperature u = specific internal energy µ = viscosity v = specific volume f = subscript denoting saturated liquid g = subscript denoting saturated vapor UNITS CONVERSIONS For this subsection, the following units conversions are applicable: cp, specific heat: To convert kilojoules per kilogram-kelvin to British thermal units per pound–degree Fahrenheit, multiply by 0.23885. e, internal energy: To convert kilojoules per kilogram to British thermal units per pound, multiply by 0.42992. g, gravity acceleration: To convert meters per second squared to feet per second squared, multiply by 3.2808. h, enthalpy: To convert kilojoules per kilogram to British thermal units per pound, multiply by 0.42992. k, thermal conductivity: To convert watts per meter-kelvin to British thermal unit–feet per hour–square foot–degree Fahrenheit, multiply by 0.57779. p, pressure: To convert bars to kilopascals, multiply by 1 × 102; to convert bars to pounds-force per square inch, multiply by 14.504; and to convert millimeters of mercury to pounds-force per square inch, multiply by 0.01934. s, entropy: to convert kilojoules per kilogram-kelvin to British thermal units per pound–degree Rankine, multiply by 0.23885.

t, temperature: °F = 9⁄5 °C + 32. T, absolute temperature: °R = 9⁄ 5 K. u, internal energy: to convert kilojoules per kilogram to British thermal units per pound, multiply by 0.42992. µ, viscosity: to convert pascal-seconds to pound-force–seconds per square foot, multiply by 0.020885; to convert pascal-seconds to cp, multiply by 1000. v, specific volume: to convert cubic meters per kilogram to cubic feet per pound, multiply by 16.018. ρ, density: to convert kilograms per cubic meter to pounds per cubic foot, multiply by 0.062428. ADDITIONAL REFERENCES Bretsznajder, Prediction of Transport and Other Physical Properties of Fluids, Pergamon, New York, 1971. D’Ans and Lax, Handbook for Chemists and Physicists (in German), 3 vols., Springer-Verlag, Berlin. Engineering Data Book, Natural Gas Processors Suppliers Association, Tulsa, Okla. Ganic, Hartnett, and Rohsenow, Handbook of Heat Transfer, 2d ed., McGraw-Hill, New York, 1984. Gray, American Institute of Physics Handbook, 3d ed., McGraw-Hill, New York, 1972. Kay and Laby, Tables of Physical and Chemical Constants, Longman, London, various editions and dates. Landolt-Börnstein Tables, many volumes and dates, Springer-Verlag, Berlin. Lange, Handbook of Chemistry, McGraw-Hill, New York, various editions and dates. Partington, Advanced Treatise on Physical Chemistry, 5 vols., Longman, London, 1950. Raznjevic, Handbook of Thermodynamic Tables and Charts, McGraw-Hill, New York, 1976 and other editions. Reynolds, Thermodynamic Properties in SI, Department of Mechanical Engineering, Stanford University, 1979. Stephan and Lucas, Viscosity of Dense Fluids, Plenum, New York and London, 1979. Selected Values of Properties of Chemical Compounds and Selected Values of the Properties of Hydrocarbons and Related Compounds, Thermodynamics Research Center, Texas A&M University, College Station, looseleaf, intermittent publication. Vargaftik, Tables of the Thermophysical Properties of Gases and Liquids, Wiley, New York, 1975. Vargaftik, Filippov, Tarzimanov, and Totskiy, Thermal Conductivity of Liquids and Gases (in Russian), Standartov, Moscow, 1978. Weast, Handbook of Chemistry and Physics, Chemical Rubber Co., Boca Raton, FL, annually.

2-206

PHYSICAL AND CHEMICAL DATA

TABLE 2-226

Thermophysical Properties of Saturated Acetone

Temperature, K

Pressure, bar

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/kg⋅K

sg, kJ/kg⋅K

c pf, kJ/kg⋅K

µ f, 10−6 Pa⋅s

k f, W/m⋅K

Pr

300 310 320 329.3 b 330

0.318 0.482 0.710 1.013 1.040

0.001 0.001 0.001 0.001 0.001

261 285 309 333 335

1.415 0.942 0.645 0.456 0.448

−67 −46 −22 0 2

466 476 490 506 506

−0.213 −0.144 −0.068 0 0.003

1.561 1.540 1.531 1.537 1.521

2.29 2.29

232 231

0.141 0.141

3.77 3.75

340 350 360 370 380

1.52 2.04 2.74 3.60 4.52

0.001 0.001 0.001 0.001 0.001

359 383 408 435 464

0.311 0.237 0.179 0.138 0.110

25 51 78 103 127

509 529 543 554 566

0.075 0.150

1.514 1.516

2.33 2.38 2.43 2.48 2.53

212 200 187 176 165

0.137 0.132 0.128 0.124 0.119

3.61 3.61 3.55 3.52 3.51

390 400 410 420 430

5.87 7.31 8.94 10.82 13.64

0.001 0.001 0.001 0.001 0.001

495 528 564 604 647

0.0854 0.0684 0.0556 0.0454 0.0356

151 184 207 231 256

577 588 598 608 618

2.59 2.65 2.73 2.82 2.92

153 141 130 119 109

0.115 0.111 0.107 0.103 0.099

3.45 3.37 3.32 3.26 3.21

440 450 460 470 480

16.37 19.42 22.79 27.52 32.52

0.001 0.001 0.001 0.001 0.001

695 748 81 88 98

0.0292 0.0240 0.0199 0.0159 0.0130

281 308 337 365 396

625 632 637 641 638

3.03 3.15 3.29 3.45 3.76

99 90 80 71 64

0.095 0.092 0.088 0.083 0.077

3.16 3.08 2.99 2.95 3.13

490 500 508.2c

37.73 43.08 47.61

0.002 15 0.002 46 0.003 67

vf, m3/kg

0.0091 0.0063 0.0037

b = normal boiling point; c = critical point P, v, h, and s interpolated and converted from Heat Exchanger Design Handbook, vol. 5, Hemisphere, Washington, DC, 1983 and reproduced in Beaton, C. F. and G. F. Hewitt, Physical Property Data for the Design Engineer, Hemisphere, New York, 1989 (394 pp.). Other values compiled by P. E. Liley An enthalpy-pressure diagram to 1000 psia, 250–500 °F appears in J. Chem. Eng. Data 7, 1 (1962): 75–78.

TABLE 2-227

Saturated Acetylene*

Temperature, K

Pressure, bar

162.0 169.3 173.9 180.0 184.3

vcond, m3/kg

vg, m3/kg

hcond, kJ/kg

hg, kJ/kg

scond, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

0.101 0.203 0.304 0.507 0.709

5.081 2.644 1.805 1.116 0.810

158 173 182 194 203

983 994 999 1007 1011

2.967 3.039 3.095 3.161 3.216

8.062 7.889 7.797 7.672 7.596

189.1 192.4t

1.013 1.283

0.5780 0.4617

214 221

1015 1018

3.272 3.312

7.511 7.455

192.4t 200.9 209.4

1.283 2.027 3.040

0.00164 0.00165 0.00169

0.4617 0.3011 0.2074

378 411 445

1018 1027 1035

4.127 4.296 4.461

7.455 7.362 7.280

221.5 230.4 240.7 253.2 263.0

5.066 7.093 10.13 15.20 20.27

0.00174 0.00179 0.00186 0.00195 0.00204

0.1264 0.0907 0.0635 0.0420 0.0309

493 528 565 602 628

1046 1052 1058 1061 1061

4.684 4.837 4.990 5.133 5.231

7.180 7.111 7.037 6.947 6.878

271.6 278.9 284.9 290.4 300.0

25.33 30.40 35.46 40.53 50.66

0.00213 0.00223 0.00232 0.00242 0.00270

0.0240 0.0193 0.0159 0.0133 0.0093

654 680 704 727 778

1060 1057 1051 1041 1017

5.326 5.414 5.494 5.576 5.737

6.822 6.767 6.716 6.658 6.534

307.8 308.7c

60.80 62.47

0.00335 0.00434

0.0061 0.0043

850 908

968 908

5.965 6.158

6.351 6.158

*Values recalculated into SI units from those of Din. Thermodynamic Functions of Gases, vol. 2, Butterworth, London, 1956. Above the solid line the condensed phase is solid; below the line it is liquid. t = triple point; c = critical point.

THERMODYNAMIC PROPERTIES TABLE 2-228 T, K

2-207

Saturated Air*

Pf, bar

Pg, bar

vf, m3/kg

vg, m3/kg

h f, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

5.55 3.73 2.57 1.82 1.313

−159.2 −155.2 −151.4 −147.8 −144.2

59.7 61.7 63.6 65.5 67.4

2.528 2.585 2.641 2.696 2.747

6.255 6.164 6.080 6.002 5.929

cpf, kJ/(kg⋅K)

µ f, 10−4 Pa⋅s

k f, W/(m⋅K)

3.25 2.98 2.75 2.54 2.36

0.180 0.176 0.173 0.169 0.166

60 62 64 66 68

0.123 0.174 0.239

0.071 0.104 0.147

1.040.–3 1.050.–3 1.060.–3 1.070.–3 1.080.–3

70 72 74 76 78

0.323 0.429 0.560 0.721 0.915

0.205 0.280 0.376 0.495 0.644

1.089.–3 1.101.–3 1.113.–3 1.125.–3 1.136.–3

0.968 0.728 0.556 0.431 0.339

−140.6 −137.1 −133.5 −129.9 −126.3

69.2 71.0 72.8 74.5 76.2

2.797 2.847 2.895 2.941 2.988

5.862 5.799 5.740 5.685 5.634

1.817 1.827 1.838 1.849 1.861

2.21 2.07 1.95 1.84 1.74

0.163 0.160 0.156 0.152 0.148

80 82 84 86 88

1.146 1.420 1.741 2.114 2.544

0.825 1.043 1.305 1.614 1.976

1.146.–3 1.160.–3 1.173.–3 1.187.–3 1.201.–3

0.270 0.217 0.177 0.145 0.120

−122.6 −118.8 −115.0 −111.2 −107.4

77.8 79.4 80.9 82.3 83.6

3.034 3.079 3.123 3.167 3.209

5.585 5.540 5.496 5.454 5.414

1.873 1.885 1.898 1.912 1.927

1.65 1.58 1.51 1.44 1.38

0.145 0.142 0.139 0.135 0.132

90 92 94 96 98

3.036 3.596 4.229 4.940 5.736

2.397 2.884 3.441 4.075 4.792

1.216.–3 1.231.–3 1.247.–3 1.265.–3 1.283.–3

0.1002 0.0843 0.0713 0.0607 0.0520

−103.5 −99.5 −95.5 −91.5 −87.5

84.8 85.9 87.0 87.9 88.7

3.251 3.293 3.335 3.376 3.416

5.376 5.340 5.304 5.270 5.236

1.944 1.962 1.982 2.003 2.027

1.32 1.27 1.23 1.18 1.14

0.128 0.125 0.121 0.117 0.114

100 105 110 115 120

6.621 9.265 12.59 16.68 21.61

5.599 8.056 11.22 15.21 20.14

1.302.–3 1.355.–3 1.418.–3 1.495.–3 1.596.–3

0.0447 0.0312 0.0222 0.0159 0.0115

−83.3 −72.8 −61.9 −50.3 −37.5

89.3 90.2 90.1 88.4 84.8

3.456 3.553 3.649 3.747 3.850

5.204 5.124 5.045 4.964 4.877

2.053 2.137 2.264 2.477 2.916

1.10 1.02 0.95 0.87 0.75

0.110 0.102 0.093 0.084 0.076

125 130 132.55c

27.43 34.16

26.14 33.32 37.69

1.757.–3 2.075.–3 3.196.–3

0.0081 0.0054 0.0032

−22.0 0.4 37.4

78.2 66.1 37.4

3.969 4.136 4.410

4.776 4.644 4.410

4.585

0.42

∞

0.067 ∞

*Liquid properties extracted or converted from Vasserman and Rabinovich, Thermophysical Properties of Liquid Air and Its Components, Moscow, 1968, and NBSNSF transl. TT 69-55092, 1970. Copyrighted material. Reproduced by permission. Vapor properties extracted or converted from Vasserman, Kazavchinskii, and Rabinovich, Thermophysical Properties of Air and Its Components, Nauka, Moscow, 1966, and NBS-NSF transl. TT 70-50095, 1971. Copyrighted material. Reproduced by permission. Note that on pages 150–151 of the TT 69-55092 publication certain values of TT 70-50095 were adjusted. As a complete retabulation was not given, the tables here are based upon the two separate publications, as indicated. See also Table 2-235 for the argon-oxygen-nitrogen equilibrium data. c = critical point. The notation 1.040.–3 signifies 1.040 × 10−3.

2-208

PHYSICAL AND CHEMICAL DATA

TABLE 2-229 Pressure, bar 1v h s Cp µ k

Thermophysical Properties of Compressed Air* Temperature, K 80

Mix

90

100

120

140

160

180

200

220

240

260

280

300

0.251 87.9 5.650 1.044 0.064 0.0084

0.281 98.3 5.759 1.032 0.071 0.0093

0.340 118.8 5.946 1.020 0.085 0.0112

0.399 139.1 6.103 1.014 0.097 0.0129

0.457 159.3 6.238 1.010 0.109 0.0147

0.515 179.5 6.357 1.008 0.121 0.0164

0.537 199.7 6.463 1.007 0.133 0.0181

0.631 219.8 6.559 1.006 0.144 0.0198

0.688 239.9 6.647 1.006 0.154 0.0214

0.746 260.0 6.727 1.006 0.165 0.0231

0.803 280.2 6.802 1.006 0.175 0.0247

0.861 300.3 6.871 1.007 0.185 0.0263

5v h s Cp µ k

0.00115 −122.3 3.031 1.868 1.794 0.146

0.00122 −103.3 3.250 1.941 1.163 0.128

0.0509 90.6 5.246 1.212 0.077 0.0103

0.0646 113.6 5.455 1.107 0.087 0.0119

0.0773 135.3 5.623 1.065 0.098 0.0135

0.0895 156.4 5.763 1.045 0.110 0.0151

0.102 177.1 5.885 1.033 0.122 0.0168

0.114 197.7 5.994 1.025 0.134 0.0185

0.125 218.1 6.092 1.020 0.145 0.0201

0.137 238.5 6.180 1.017 0.155 0.0217

0.149 258.8 6.262 1.015 0.165 0.0234

0.160 279.1 6.337 1.013 0.175 0.0250

0.172 299.4 6.406 1.013 0.185 0.0265

10 v h s Cp µ k

0.00115 −122.0 3.028 1.863 1.816 0.146

0.00121 −103.1 3.246 1.932 1.177 0.128

0.00130 −83.2 3.452 2.041 0.838 0.111

0.0298 106.2 5.214 1.270 0.089 0.0126

0.0370 130.2 5.398 1.146 0.101 0.0141

0.0436 152.5 5.548 1.093 0.112 0.0157

0.0499 174.1 5.675 1.065 0.124 0.0173

0.0561 195.2 5.786 1.049 0.135 0.0189

0.0621 216.1 5.885 1.038 0.146 0.0205

0.0681 236.7 5.975 1.031 0.156 0.0221

0.0741 257.3 6.058 1.026 0.166 0.0237

0.0800 277.8 6.134 1.023 0.176 0.0253

0.0859 298.3 6.204 1.201 0.186 0.0268

20 v h s Cp µ k

0.00114 −121.3 3.022 1.853 1.859 0.147

0.00121 −102.5 3.239 1.916 1.205 0.130

0.00129 −82.9 3.442 2.010 0.857 0.112

0.0116 85.2 4.882 2.237 0.098 0.0152

0.0167 118.5 5.140 1.390 0.106 0.0157

0.0206 144.3 5.312 1.215 0.116 0.0169

0.0241 167.7 5.450 1.141 0.127 0.0182

0.0274 190.1 5.568 1.101 0.137 0.0197

0.0306 211.9 5.672 1.076 0.148 0.0212

0.0337 233.2 5.765 1.061 0.158 0.0228

0.0368 254.3 5.849 1.050 0.168 0.0243

0.0398 275.2 5.927 1.042 0.178 0.0258

0.0428 296.0 5.998 1.037 0.187 0.0273

40 v h s Cp µ k

0.00114 −120.0 3.011 1.834 1.943 0.149

0.00120 −101.4 3.225 1.886 1.261 0.132

0.00128 −82.2 3.424 1.958 0.896 0.115

0.00153 −39.8 3.807 2.432 0.516 0.0814

0.0058 83.6 4.745 3.193 0.132 0.0460

0.0090 125.3 5.025 1.610 0.129 0.0201

0.0114 154.3 5.196 1.335 0.135 0.0206

0.0131 179.7 5.330 1.221 0.144 0.0217

0.0148 203.5 5.444 1.159 0.154 0.0229

0.0165 226.3 5.543 1.122 0.163 0.0242

0.0182 248.5 5.632 1.097 0.172 0.0256

0.0198 270.2 5.712 1.081 0.182 0.0270

0.0214 291.7 5.786 1.068 0.191 0.0284

60 v h s Cp µ k

0.00113 −118.6 3.000 1.818 2.028 0.150

0.00119 −100.3 3.211 1.860 1.318 0.134

0.00126 −81.4 3.407 1.915 0.936 0.117

0.00147 −40.8 3.773 2.205 0.559 0.0861

0.00222 22.8 4.260 4.808 0.277 0.0480

0.00505 90.0 4.798 2.338 0.153 0.0360

0.00687 132.6 5.020 1.594 0.149 0.0238

0.00833 163.9 5.174 1.361 0.154 0.0240

0.00963 191.1 5.298 1.249 0.161 0.0248

0.0108 216.1 5.404 1.186 0.169 0.0258

0.0120 240.0 5.497 1.146 0.178 0.0270

0.0131 263.1 5.581 1.119 0.186 0.0283

0.0142 285.6 5.657 1.100 0.195 0.0296

80 v h s Cp µ k

0.00113 −117.2 2.989 1.802 2.12 0.152

0.00119 −99.1 3.198 1.838 1.38 0.134

0.00126 −80.4 3.391 1.881 0.977 0.120

0.00145 −41.3 3.745 2.078 0.597 0.0901

0.00188 9.0 4.138 2.992 0.356 0.0599

0.00327 78.4 4.597 3.029 0.194 0.0420

0.00480 125.3 4.875 1.887 0.167 0.0278

0.00601 158.7 5.051 1.510 0.166 0.0268

0.00706 187.1 5.186 1.342 0.170 0.0269

0.00803 212.9 5.299 1.250 0.177 0.0276

0.00894 237.3 5.396 1.194 0.184 0.0286

0.00981 260.8 5.484 1.156 0.191 0.0296

0.0107 283.7 5.562 1.130 0.200 0.0308

100 v h s Cp µ k

0.00112 −115.8 2.978 1.789 2.21 0.154

0.00118 −97.8 3.186 1.818 1.44 0.137

0.00125 −79.4 3.376 1.852 1.02 0.122

0.00142 −41.3 3.721 1.992 0.631 0.0936

0.00174 3.9 4.076 2.506 0.405 0.0669

0.00252 61.7 4.457 2.874 0.249 0.0500

0.00366 111.8 4.753 2.114 0.193 0.0327

0.00467 148.8 4.949 1.650 0.181 0.0299

0.00556 179.4 5.095 1.431 0.181 0.0293

0.00637 206.7 5.214 1.311 0.185 0.0295

0.00713 232.2 5.315 1.239 0.191 0.0302

0.00785 256.4 5.406 1.191 0.198 0.0311

0.00855 279.9 5.486 1.158 0.205 0.0320

150 v h s Cp µ k

0.00111 −112.2 2.954 1.789 2.44 0.157

0.00116 −94.5 3.157 1.818 1.60 1.142

0.00122 −76.6 3.342 1.852 1.13 0.127

0.00137 −40.1 3.673 1.992 0.709 0.101

0.00158 0.5 3.988 2.506 0.490 0.0785

0.00194 45.2 4.287 2.874 0.349 0.0588

0.00247 89.5 4.548 2.114 0.266 0.0455

0.00309 129.2 4.757 1.650 0.229 0.0389

0.00369 163.2 4.919 1.431 0.215 0.0360

0.00425 193.4 5.051 1.311 0.211 0.0348

0.00478 221.0 5.161 1.239 0.212 0.0346

0.00529 247.0 5.257 1.267 0.215 0.0349

0.00578 271.8 5.343 1.220 0.220 0.0354

200 v h s Cp µ k

0.00110 −108.5 2.930 1.733 2.70 0.161

0.00115 −91.2 3.130 1.747 1.78 0.146

0.00120 −73.6 3.312 1.761 1.25 0.132

0.00133 −38.0 3.634 1.809 0.782 0.107

0.00150 0.2 3.931 1.905 0.561 0.0868

0.00174 40.2 4.198 1.988 0.420 0.0691

0.00206 79.8 4.432 1.953 0.331 0.0559

0.00245 117.6 4.631 1.814 0.279 0.0476

0.00287 152.2 4.796 1.643 0.253 0.0429

0.00328 183.6 4.932 1.501 0.241 0.0405

0.00368 212.5 5.048 1.396 0.236 0.0393

0.00407 239.6 5.149 1.321 0.235 0.0389

0.00446 265.5 5.238 1.266 0.237 0.0389

250 v h s Cp µ k

0.00109 −104.8 2.909 1.712 2.96 0.165

0.00114 −87.6 3.106 1.722 1.97 0.150

0.00119 −70.3 3.285 1.733 1.39 0.137

0.00130 −35.4 3.601 1.767 0.855 0.113

0.00144 1.3 3.886 1.824 0.625 0.0935

0.00162 38.9 4.138 1.854 0.476 0.0769

0.00186 75.8 4.355 1.831 0.385 0.0641

0.00214 111.7 4.544 1.748 0.327 0.0552

0.00244 145.6 4.706 1.635 0.292 0.0495

0.00276 177.1 4.843 1.522 0.272 0.0460

0.00307 206.6 4.961 1.427 0.262 0.0441

0.00338 234.3 5.064 1.353 0.257 0.0430

0.00368 260.8 5.155 1.297 0.256 0.0426

*For sources, units, and remarks, see Table 2-228. v = specific volume, m3/kg; h = specific enthalpy, kJ/kg; s = specific entropy, kJ/(kg⋅K); cp = specific heat at constant pressure, kJ/(kg⋅K); µ = viscosity, 10−4 Pa⋅s; and k = thermal conductivity, W/(m⋅K). For specific heat ratio, see Table 2-200; for Prandtl number, see Table 2-369.

THERMODYNAMIC PROPERTIES

2-209

Temperature, K 350

400

450

500

600

800

1000

1200

1400

1600

1800

2000

2500

1.005 350.7 7.026 1.009 0.208 0.0301

1.148 401.2 7.161 1.014 0.230 0.0336

1.292 452.1 7.282 1.021 0.251 0.0371

1.436 503.4 7.389 1.030 0.270 0.0404

1.723 607.5 7.579 1.051 0.306 0.0466

2.297 822.5 7.888 1.099 0.370 0.0577

2.872 1046.8 8.138 1.141 0.424 0.0681

3.446 1278 8.349 1.175 0.473 0.0783

4.020 1515 8.531 1.207 0.527 0.0927

4.594 1764 8.695 1.248 0.584 0.106

5.168 2017 8.844 1.286 0.637 0.120

5.743 2279 8.983 1.337 0.689 0.137

7.200 3011 9.308 1.665 0.818 0.222

0.201 350.0 6.563 1.014 0.208 0.0303

0.230 400.8 6.698 1.017 0.230 0.0338

0.259 451.8 6.818 1.024 0.251 0.0372

0.288 503.2 6.927 1.032 0.270 0.0405

0.345 607.4 7.116 1.053 0.306 0.0467

0.460 822.6 7.426 1.100 0.370 0.0578

0.575 1046.9 7.676 1.142 0.425 0.0681

0.690 1279 7.887 1.175 0.473 0.0783

0.805 1516 8.069 1.208 0.527 0.0927

0.920 1764 8.233 1.248 0.584 0.106

1.034 2017 8.382 1.285 0.637 0.120

1.149 2278 8.520 1.326 0.689 0.136

1.438 2981 8.832 1.516 0.818 0.195

0.101 349.2 6.361 1.019 0.209 0.0305

0.115 400.2 6.497 1.021 0.231 0.0340

0.130 451.4 6.618 1.027 0.252 0.0374

0.144 502.9 6.727 1.034 0.271 0.0407

0.173 607.3 6.917 1.055 0.306 0.0469

0.231 822.7 7.226 1.100 0.370 0.0579

0.288 1047.2 7.477 1.142 0.425 0.0682

0.345 1279 7.688 1.175 0.473 0.0784

0.403 1516 7.870 1.208 0.527 0.0927

0.460 1765 8.034 1.248 0.584 0.106

0.518 2018 8.183 1.284 0.637 0.120

0.575 2279 8.321 1.324 0.689 0.135

0.720 2974 8.630 1.481 0.817 0.187

0.0503 347.7 6.158 1.030 0.210 0.0309

0.0577 399.1 6.295 1.029 0.232 0.0344

0.0650 450.7 6.417 1.033 0.253 0.0377

0.0723 502.4 6.526 1.039 0.272 0.0410

0.0868 607.2 6.716 1.057 0.307 0.0471

0.116 823.0 7.027 1.102 0.371 0.0581

0.145 1047.7 0.277 1.143 0.425 0.0685

0.173 1280 7.489 1.176 0.474 0.0787

0.202 1517 7.671 1.209 0.527 0.0928

0.231 1766 7.835 1.249 0.584 0.106

0.260 2019 7.984 1.284 0.637 0.120

0.288 2279 8.121 1.322 0.689 0.135

0.360 2970 8.428 1.456 0.817 0.181

0.0252 344.6 5.950 1.051 0.213 0.0318

0.0290 397.0 6.090 1.044 0.235 0.0351

0.0327 449.2 6.212 1.044 0.255 0.0384

0.0364 501.5 6.323 1.049 0.274 0.0416

0.0438 606.9 6.515 1.063 0.309 0.0476

0.0583 823.7 6.826 0.105 0.372 0.0584

0.0728 1048.8 7.077 1.145 0.426 0.0687

0.0872 1281 7.289 1.177 0.474 0.0789

0.102 1519 7.473 1.210 0.527 0.0928

0.116 1768 7.636 1.249 0.584 0.106

0.130 2021 7.785 1.284 0.637 0.120

0.145 2281 7.922 1.322 0.689 0.135

0.181 2969 8.229 1.438 0.817 0.177

0.0169 340.4 5.824 1.072 0.217 0.0328

0.0194 394.0 5.967 1.059 0.237 0.0359

0.0220 447.1 6.091 1.055 0.257 0.0391

0.0245 500.6 6.202 1.057 0.275 0.0422

0.0294 606.8 6.396 1.069 0.310 0.0481

0.0392 824.3 6.708 1.108 0.373 0.0588

0.0489 1050.0 6.960 1.147 0.427 0.0690

0.0585 1283 7.172 1.178 0.475 0.0790

0.0681 1521 7.355 1.210 0.527 0.0929

0.0776 1770 7.520 1.249 0.584 0.106

0.0872 2023 7.669 1.286 0.637 0.120

0.0968 2284 7.806 1.322 0.689 0.134

0.1207 2969 8.112 1.430 0.817 0.176

0.0127 339.0 5.733 1.091 0.220 0.0337

0.0147 393.1 5.878 1.073 0.240 0.0368

0.0166 446.5 6.004 1.066 0.259 0.0398

0.0185 499.8 6.116 1.065 0.278 0.0428

0.0223 606.7 6.311 1.075 0.312 0.0486

0.0296 825.1 6.624 1.111 0.374 0.0592

0.0369 1051.1 6.877 1.149 0.428 0.0693

0.0442 1284 7.089 1.180 0.475 0.0793

0.0513 1522 7.273 1.210 0.527 0.0929

0.0585 1772 7.437 1.249 0.584 0.106

0.0657 2025 7.586 1.286 0.637 0.120

0.0729 2285 7.723 1.322 0.689 0.134

0.0908 2971 8.029 1.426 0.817 0.175

0.0102 336.5 5.661 1.110 0.224 0.0347

0.0118 391.3 5.807 1.087 0.243 0.0376

0.0134 445.3 5.935 1.076 0.262 0.0405

0.0149 499.0 6.048 1.073 0.280 0.0434

0.0180 606.6 6.244 1.080 0.314 0.0491

0.0239 825.8 6.559 1.114 0.375 0.0595

0.0298 1052.4 6.812 1.151 0.429 0.0696

0.0356 1286 7.024 1.181 0.477 0.0795

0.0413 1524 7.208 1.211 0.527 0.0930

0.0470 1774 7.373 1.250 0.584 0.106

0.0528 2027 7.522 1.288 0.637 0.120

0.0584 2288 7.659 1.323 0.689 0.134

0.0729 2972 7.964 1.423 0.817 0.175

0.00695 330.9 5.525 1.151 0.235 0.0374

0.00806 387.5 5.677 1.117 0.252 0.0398

0.00914 442.9 5.807 1.099 0.270 0.0424

0.0102 497.5 5.922 1.092 0.286 0.0451

0.0123 606.6 6.121 1.093 0.318 0.0504

0.0163 827.8 6.439 1.121 0.379 0.0605

0.0202 1055.5 6.693 1.155 0.431 0.0703

0.0241 1290 6.906 1.184 0.478 0.0801

0.0279 1529 7.092 1.213 0.527 0.0932

0.0317 1779 7.256 1.252 0.584 0.106

0.0356 2033 7.405 1.290 0.637 0.120

0.0394 2294 7.543 1.325 0.689 0.133

0.0490 2977 7.848 1.418

0.00534 326.5 5.426 1.184 0.248 0.0400

0.00620 384.5 5.581 1.141 0.262 0.0420

0.00702 440.9 5.715 1.119 0.278 0.0423

0.00783 496.6 5.831 1.108 0.293 0.0467

0.00940 607.0 6.033 1.104 0.324 0.0517

0.0125 829.9 6.353 1.128 0.382 0.0614

0.0154 1058.7 6.608 1.160 0.434 0.0711

0.0184 1294 6.822 1.187 0.481 0.0808

0.0212 1533 7.009 1.214 0.528 0.0934

0.0241 1783 7.173 1.254 0.585 0.106

0.0269 2038 7.323 1.292 0.638 0.120

0.0298 2299 7.460 1.326

0.0370 2982 7.765 1.415

0.00440 323.2 5.348 1.208 0.262 0.0429

0.00509 382.3 5.506 1.161 0.273 0.0443

0.00576 439.6 5.641 1.135 0.286 0.0462

0.00642 496.0 5.760 1.121 0.301 0.0484

0.00770 607.6 5.963 1.115 0.329 0.0531

0.0102 832.2 6.286 1.135 0.386 0.0624

0.0126 1062.0 6.542 1.164 0.437 0.0718

0.0149 1298 6.757 1.190 0.483 0.0814

0.0172 1538 6.944 1.216 0.528 0.0937

0.0195 1789 7.108 1.256 0.585 0.106

0.0218 2043 7.258 1.294

0.0241 2304 7.396 1.328

0.0298 2988 7.701 1.414

2-210

PHYSICAL AND CHEMICAL DATA

TABLE 2-229 Pressure, bar

Thermophysical Properties of Compressed Air (Concluded ) Temperature, K 90

100

120

140

160

180

200

220

240

260

280

300

0.00112 −84.0 3.083 1.703 2.18 0.154

0.00117 −67.0 3.260 1.713 1.53 0.141

0.00127 −32.4 3.572 1.740 0.932 0.118

0.00139 3.1 3.849 1.769 0.687 0.0996

0.00155 39.2 4.090 1.777 0.529 0.0836

0.00173 74.5 4.298 1.751 0.433 0.0710

0.00195 109.0 4.480 1.689 0.370 0.0619

0.00219 142.0 4.637 1.607 0.329 0.0555

0.00243 173.2 4.773 1.518 0.303 0.0514

0.00269 202.7 4.891 1.438 0.288 0.0487

0.00294 230.8 4.995 1.370 0.280 0.0471

0.00318 257.7 5.088 1.316 0.276 0.0462

400 v h s Cp µ k

0.00110 −76.6 3.042 1.674 2.63 0.161

0.00114 −59.8 3.216 1.686 1.86 0.149

0.00123 −25.9 3.523 1.704 1.10 0.127

0.00133 8.3 3.788 1.702 0.802 0.110

0.00145 42.4 4.016 1.685 0.631 0.0946

0.00158 75.8 4.214 1.654 0.500 0.0823

0.00173 108.5 4.386 1.607 0.446 0.0729

0.00189 140.1 4.537 1.550 0.397 0.0660

0.00206 170.5 4.669 1.490 0.364 0.0610

0.00224 199.7 4.786 1.431 0.341 0.0574

0.00242 227.8 4.890 1.378 0.325 0.0550

0.00260 254.8 4.983 1.331 0.316 0.0533

500 v h s Cp µ k

0.00109 −69.0 3.005 1.655 3.13 0.167

0.00112 −52.3 3.177 1.670 2.24 0.156

0.00120 −18.7 3.482 1.686 1.31 0.135

0.00128 14.4 3.743 1.667 0.924 0.119

0.00138 47.4 3.966 1.644 0.710 0.104

0.00148 79.8 4.151 1.598 0.0560 0.0916

0.00160 111.4 4.317 1.557 0.512 0.0822

0.00173 142.0 4.463 1.509 0.459 0.0749

0.00186 171.7 4.593 1.461 0.420 0.0694

0.00199 200.5 4.708 1.415 0.391 0.0653

0.00213 228.4 4.811 1.371 0.370 0.0622

0.00227 255.4 4.905 1.331 0.356 0.0599

0.00151 116.0 2.263 1.525

0.00161 146.1 4.406 1.480 0.516 0.0828

0.00172 175.3 4.533 1.438 0.472 0.0769

0.00183 203.6 4.646 1.398 0.439 0.0724

0.00194 231.2 4.749 1.361 0.414 0.0689

0.00205 258.1 4.842 1.327 0.396 0.0662

0.00147 157.4 4.318 1.445

0.00155 185.9 4.442 1.406

0.0964

0.0901

0.00163 213.7 4.553 1.372 0.529 0.0850

0.00171 240.3 4.653 1.342 0.497 0.0809

0.00179 267.3 4.745 1.314 0.473 0.0776

0.00151 226.4 4.482 1.355

0.00157 253.2 4.582 1.327

0.0961

0.0916

0.00163 279.5 4.672 1.303 0.546 0.0878

300 v h s Cp µ k

600 v h s Cp µ k 800 v h s Cp µ k 1000 v h s Cp µ k

80 0.00108 −101.0 2.888 1.694 3.24 0.168

0.0903

THERMODYNAMIC PROPERTIES

2-211

Temperature, K 350

400

450

500

600

800

1000

1200

1400

1600

1800

2000

2500

0.00379 320.9 5.283 1.226 0.276 0.0457

0.00437 380.9 5.443 1.176 0.284 0.0466

0.00493 438.9 5.580 1.148 0.296 0.0481

0.00548 495.9 5.700 1.133 0.308 0.0501

0.00656 608.5 5.906 1.124 0.335 0.0544

0.00864 834.5 6.230 1.140 0.390 0.0634

0.0107 1065.3 6.488 1.168 0.440 0.0726

0.0126 1302 6.703 1.193 0.485 0.0820

0.0145 1542 6.891 1.217 0.529 0.0940

0.0164 1794 7.056 1.257

0.0183 2049 7.206 1.298

0.0202 2310 7.344 1.330

0.0250 2993 7.648 1.413

0.00304 319.1 5.181 1.246 0.307 0.0513

0.00348 380.0 5.344 1.195 0.308 0.0512

0.00390 439.0 5.483 1.166 0.315 0.0521

0.00432 496.8 5.605 1.149 0.325 0.0535

0.00514 611.0 5.813 1.138 0.348 0.0571

0.00673 839.4 6.142 1.151 0.398 0.0653

0.00826 1072.0 6.401 1.176 0.446 0.0740

0.00977 1310 6.618 1.199 0.490 0.0832

0.0111 1552 6.808 1.222

0.0126 1804 6.972 1.258

0.0140 2059 7.123 1.301

0.0155 2321 7.261 1.333

0.0190 3004 7.566 1.412

0.00262 319.9 5.103 1.255 0.338 0.0568

0.00296 381.3 5.267 1.206 0.333 0.0557

0.00330 440.8 5.408 1.176 0.336 0.0560

0.00364 499.1 5.531 1.159 0.343 0.0569

0.00430 614.3 5.741 1.148 0.361 0.0598

0.00558 844.6 6.072 1.159 0.407 0.0672

0.00683 1078.8 6.333 1.183 0.452 0.0755

0.00804 1318 6.550 1.205 0.495 0.0844

0.00911 1561 6.743 1.226

0.0103 1814 6.907 1.265

0.0114 2070 7.058 1.306

0.0126 2332 7.196 1.337

0.0154 3015 7.501 1.412

0.00234 322.6 5.041 1.258 0.370 0.0620

0.00262 384.2 5.205 1.211 0.359 0.0602

0.00290 444.0 5.346 1.182 0.358 0.0598

0.00318 502.6 5.470 1.166 0.361 0.0603

0.00374 618.5 5.681 1.154 0.375 0.0625

0.00481 850.1 6.014 1.166 0.416 0.0691

0.00586 1085.5 6.277 1.189 0.459 0.0770

0.00689 1326 6.495 1.210 0.501 0.0857

0.00776 1570 6.690 1.231

0.00873 1824 6.854 1.267

0.00970 2080 7.005 1.310

0.0107 2343 7.144 1.341

0.0130 3026 7.449 1.412

0.00200 331.6 4.943 1.257 0.432 0.0718

0.00221 393.8 5.108 1.216 0.411 0.0688

0.00242 453.4 5.250 1.188 0.402 0.0673

0.00263 512.3 5.374 1.172 0.399 0.0669

0.00304 625.8 5.586 1.161 0.405 0.0679

0.00385 862.0 5.922 1.175 0.436 0.0730

0.00465 1099.3 6.136 1.198 0.474 0.0800

0.00544 1341 6.407 1.219 0.512 0.0881

0.00608 1588 6.605 1.240

0.00681 1844 6.769 1.275

0.00754 2101 6.921 1.318

0.00826 2365 7.060 1.347

0.0101 3049 7.366 1.412

0.00180 343.4 4.869 1.254 0.494 0.0810

0.00196 405.1 5.034 1.217 0.463 0.0768

0.00213 465.3 5.176 1.192 0.446 0.0744

0.00230 524.4 5.300 1.175 0.438 0.0733

0.00262 641.2 5.513 1.164 0.435 0.0732

0.00328 875.1 5.850 1.179 0.456 0.0768

0.00392 1113.3 6.115 1.204 0.489 0.0830

0.00455 1356 6.337 1.225 0.524 0.0906

0.00507 1606 6.539 1.248

0.00565 1863 6.703 1.283

0.00624 2121 6.856 1.325

0.00681 2386 6.995 1.354

0.00825 3071 7.302 1.413

2-212

PHYSICAL AND CHEMICAL DATA TABLE 2-230

Enthalpy and Psi Functions for Ideal-Gas Air*

T, K

h, kJ/kg

Ψ

T, K

h, kJ/kg

Ψ

T, K

h, kJ/kg

Ψ

200 210 220 230 240

200.0 210.0 220.0 230.1 240.1

−0.473 −0.400 −0.329 −0.262 −0.197

650 660 670 680 690

659.8 670.5 681.1 691.8 702.5

1.339 1.364 1.388 1.412 1.436

1200 1220 1240 1260 1280

1278 1301 1325 1349 1372

2.376 2.406 2.435 2.463 2.491

250 260 270 280 290

250.1 260.1 270.1 280.1 290.2

−0.135 −0.076 −0.018 0.037 0.090

700 710 720 730 740

713.3 724.0 734.8 745.6 756.4

1.459 1.482 1.505 1.528 1.550

1300 1320 1340 1360 1380

1396 1420 1444 1467 1491

2.519 2.547 2.574 2.601 2.627

300 310 320 330 340

300.2 310.3 320.3 330.4 340.4

0.142 0.191 0.240 0.286 0.332

750 760 770 780 790

767.3 778.2 789.1 800.0 811.0

1.572 1.594 1.615 1.637 1.658

1400 1420 1440 1460 1480

1515 1539 1563 1587 1612

2.653 2.679 2.705 2.730 2.755

350 360 370 380 390

350.5 360.6 370.7 380.8 390.9

0.376 0.419 0.461 0.502 0.541

800 810 820 830 840

821.9 832.9 844.0 855.0 866.1

1.679 1.699 1.720 1.740 1.760

1500 1520 1540 1560 1580

1636 1660 1684 1709 1738

2.779 2.803 2.827 2.851 2.875

400 410 420 430 440

401.0 411.2 421.3 431.5 441.7

0.580 0.618 0.655 0.691 0.727

850 860 870 880 890

877.2 888.3 899.4 910.6 921.8

1.780 1.800 1.819 1.838 1.857

1600 1620 1640 1660 1680

1758 1782 1806 1831 1855

2.898 2.921 2.944 2.966 2.988

450 460 470 480 490

451.8 462.1 472.3 482.5 492.8

0.761 0.795 0.829 0.861 0.893

900 910 920 930 940

933.0 944.2 955.4 966.7 978.0

1.876 1.895 1.914 1.932 1.950

1700 1720 1740 1760 1780

1880 1905 1929 1954 1979

3.010 3.032 3.054 3.075 3.096

500 510 520 530 540

503.1 513.4 523.7 534.0 544.4

0.925 0.956 0.986 1.016 1.045

950 960 970 980 990

989.3 1000.6 1011.9 1023.3 1034.7

1.969 1.987 2.004 2.022 2.039

1800 1820 1840 1860 1880

2003 2028 2053 2078 2102

3.117 3.138 3.158 3.178 3.198

550 560 570 580 590

554.8 565.2 575.6 586.1 596.5

1.074 1.102 1.130 1.158 1.185

1000 1020 1040 1060 1080

1046.1 1068.9 1091.9 1114.9 1138.0

2.057 2.091 2.125 2.158 2.190

1900 1920 1940 1960 1980

2127 2152 2177 2202 2227

3.218 3.238 3.258 3.277 3.296

600 610 620 630 640

607.0 617.5 628.1 638.6 649.2

1.211 1.238 1.264 1.289 1.314

1100 1120 1140 1160 1180

1161.1 1184.3 1207.6 1230.9 1254.3

2.223 2.254 2.285 2.316 2.346

2000 2050 2100 2150 2200

2252 2315 2377 2440 2504

3.215 3.362 3.408 3.453 3.496

*Values rounded off from Chappell and Cockshutt, Nat. Res. Counc. Can. Rep. NRC LR 759 (NRC No. 14300), 1974. This source tabulates values of seven thermodynamic functions at 1-K increments from 200 to 2200 K in SI units and at other increments for two other unit systems. An earlier report (NRC LR 381, 1963) gives a more detailed description of an earlier fitting from 200 to 1400 K. In the above table h = specific enthalpy, kJ/kg, and Ψ2 − Ψ1 = log10 (P2 /P1)s for an isentrope. In terms of the Keenan and Kaye function φ, Ψ = (log10 e/R) ⋅ φ.

THERMODYNAMIC PROPERTIES

FIG. 2-5 Temperature-entropy diagram for air. [Landsbaum, Dadds, Stevens, et al., Am. Inst. Chem. Eng. J., 1(3), 303 (1955). Reproduced by permission of the authors and of the editor, American Institute of Chemical Engineers.]

2-213

2-214

PHYSICAL AND CHEMICAL DATA

TABLE 2-231

Air

Other tables include Stewart, R. B., S. G. Penoncello, et al., University of Idaho CATS report, 85-5, 1985 (0.1–700 bar, 85–750 K), and a revision is in process of publication. Tables including reactions with hydrocarbons include Gordon, S., NASA Techn. Paper 1907, 4 vols., 1982. See also Gupta, R. N., K-P. Lee, et al., NASA RP 1232, 1990 (89 pp.) and RP 1260, 1991 (75 pp.). Analytic expressions for high temperatures were given by Matsuzaki, R., Jap. J. Appl. Phys., 21, 7 (1982): 1009–1013 and Japanese National Aerospace Laboratory report NAL TR 671, 1981 (45 pp.). Functions from 1500 to 15000 K were tabulated by Hilsenrath, J. and M. Klein, AEDC-TR-65-58 = AD 612 301, 1965 (333 pp.). Tables from 10000 to 10,000,000 K were authored by Gilmore, F. R., Lockheed rept. 3-27-67-1, vol 1., 1967 (340 pp.), also published as Radiative Properties of Air, IFI/Plenum, New York, 1969 (648 pp.). Saturation and superheat tables and a chart to 7000 psia, 660°R appear in Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat, thermal conductivity, and viscosity see Thermophysical Properties of Refrigerants, ASHRAE, 1993. AIR, MOIST An ASHRAE publication, Thermodynamic Properties of Dry Air and Water and S. I. Psychrometric Charts, 1983 (360 pp.), extensively reviews moist air properties. Gandiduson, P., Chem. Eng., Oct. 29, 1984 gives on page 118 a nomograph from 50 to 120°F, while equations in SI units were given by Nelson, B., Chem. Eng. Progr. 76, 5 (May 1980): 83–85. Liley, P. E., 2000 Solved Problems in M.E. Thermodynamics, McGraw-Hill, New York, 1989, gives four simple equations with which most calculations can be made. Devres, Y.O., Appl. Energy 48 (1994): 1–18 gives equations with which three known properties can be used to determine four others. Klappert, M. T. and G. F. Schilling, Rand RM-4244-PR = AD 604 856, 1984 (40 pp.) gives tables from 100 to 270 K, while programs from −60 to 2°F are given by Sando, F. A., ASHRAE Trans., 96, 2 (1990): 299–308. Viscosity references include Kestin, J. and J. H. Whitelaw, Int. J. Ht. Mass Transf. 7, 11 (1964): 1245–1255; Studnokov, E. L., Inz.-Fiz. Zhur. 19, 2 (1970): 338–340; Hochramer, D. and F. Munczak, Setzb. Ost. Acad. Wiss II 175, 10 (1966): 540–550. For thermal conductivity see, for instance, Mason, E. A. and L. Monchick, Humidity and Moisture Control in Science and Industry, Reinhold, New York, 1965 (257–272).

TABLE 2-232

Saturated Ammonia*

P, bar

vf, m3/kg

vg, m3/kg

h f, kJ/kg

h g, kJ/kg

s f, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

c pf, kJ/(kg⋅K)

µ f, 10−4 Pa⋅s

k f, W/(m⋅K)

195.5t 200 210 220 230

0.0608 0.0865 0.1775 0.3381 0.6044

1.327.–3 1.372.–3 1.394.–3 1.417.–3 1.442.–3

15.648 11.237 5.729 3.135 1.822

−1110.1 −1088.8 −1044.1 −1000.6 −957.0

380.1 388.5 406.7 424.1 440.7

4.203 4.311 4.529 4.731 4.925

11.827 11.698 11.438 11.207 11.002

4.73 4.61 4.38 4.35 4.38

4.25 4.07 3.69 3.34 3.02

0.715 0.709 0.685 0.661 0.638

240 250 260 270 280

1.0226 1.6496 2.5529 3.8100 5.5077

1.468.–3 1.495.–3 1.524.–3 1.551.–3 1.589.–3

1.115 0.712 0.472 0.324 0.228

−912.9 −868.2 −823.1 −777.3 −730.9

456.2 470.6 483.8 495.6 506.0

5.113 5.294 5.471 5.643 5.811

10.817 10.650 10.498 10.358 10.228

4.43 4.48 4.54 4.60 4.66

2.73 2.45 2.20 1.97 1.76

0.615 0.592 0.569 0.546 0.523

T, K

290 300 310 320 330

7.741 10.61 14.24 18.72 24.20

1.626.–3 1.666.–3 1.710.–3 1.760.–3 1.815.–3

0.165 0.121 0.091 0.069 0.053

−683.8 −636.0 −587.2 −537.5 −486.7

514.7 521.5 526.1 528.2 527.5

5.975 6.135 6.293 6.448 6.602

10.108 9.994 9.885 9.779 9.675

4.73 4.82 4.91 5.02 5.17

1.58 1.41 1.26 1.13 1.02

0.500 0.477 0.454 0.431 0.408

340 350 360 370 380

30.79 38.64 47.90 58.74 71.35

1.878.–3 1.952.–3 2.039.–3 2.148.–3 2.291.–3

0.0410 0.0319 0.0249 0.0194 0.0149

−434.3 −380.0 −323.2 −262.6 −196.5

523.3 515.1 501.8 481.9 452.7

6.755 6.908 7.063 7.222 7.391

9.571 9.465 9.354 9.235 9.100

5.37 5.64 6.04 6.68 7.80

0.92 0.83 0.75 0.69 0.61

0.385 0.361 0.337 0.313 0.286

85.98 103.0 113.0

2.499.–3 2.882.–3 4.255.–3

0.0113 0.0077 0.0043

−120.9 −23.5 142.7

408.1 329.0 142.7

7.578 7.813 8.216

8.935 8.694 8.216

0.50 0.39 0.25

0.254 0.21 ∞

390 400 405.4c

10.3 21. ∞

*P, v, h, and s values condensed from ASHRAE Handbook, 1981: Fundamentals. Copyright 1981 by the American Society of Heating, Refrigerating and Air Conditioning Engineers, Inc., and reproduced by permission of the copyright owner. cp, µ, and k values are interpolated and converted from Thermophysical Properties of Refrigerants, ASHRAE, New York, 1976. t = triple point; c = critical point. The notation 1.327.–3 signifies 1.327 × 10−3. At 195.5 K, the viscosity of the saturated liquid is 4.25 × 10−4 Pa⋅s. Most recent tabulations of ammonia properties are based upon the extensive tabulation to 5000 bar, 750 K of Haar, L. and J. S. Gallagher, J. Phys. Chem. Ref. Data, 7, 3 (1978): 635–792, which does, however, neglect dissociation. For tables to 70,000 psia, 920°F, see Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). A chart in fps units corresponding with these tables appears on page 17.34 of the ASHRAE 1989 Fundamentals Handbook. Simmons, A. L., C. E. Miller III, et al., Tables and Charts of Equilibrium Thermodynamic Properties of Ammonia for Temperatures from 500 to 50000 K, NASA SP 3099, 1976 (255 pp.), tabulates ρ, h, s, cp, cv, Z, and so on, from 0.01 to 400 bar and also 18 species of decomposition products. The 1993 ASHRAE Handbook—Fundamentals (SI ed.) gives material for integral degrees Celsius with temperatures on the ITS 90 scale for saturation temperatures from −77.66 to 132.22 °C. The same diagram reproduced here appears in that source.

THERMODYNAMIC PROPERTIES

FIG. 2-6 Enthalpy–log-pressure diagram for ammonia. 1 MPa = 10 bar. (Copyright 1981 by the American Society of Heating, Refrigerating and Air-Conditioning Engineers and reproduced by permission of the copyright owner.)

2-215

2-216

FIG. 2-7 Enthalpy-concentration diagram for aqueous ammonia. From Thermodynamic and Physical Properties NH3 –H2O, Int. Inst. Refrigeration, Paris, France, 1994 (88 pp.). Reproduced by permission. In order to determine equilibrium compositions, draw a vertical from any liquid composition on any boiling line (the lowest plots) to intersect the appropriate auxiliarly curve (the intermediate curves). A horizontal then drawn from this point to the appropriate dew line (the upper curves) will establish the vapor composition. The Int. Inst. Refrigeration publication also gives extensive P-v-x tables from −50 to 316°C. Other sources include Park, Y. M. and Sonntag, R. E., ASHRAE Trans., 96, 1 (1990): 150–159 (x, h, s, tables, 360 to 640 K); Ibrahim, O. M. and S. A. Klein, ASHRAE Trans., 99, 1 (1993): 1495–1502 (Eqs., 0.2 to 110 bar, 293 to 413 K); Smolen, T. M., D. B. Manley, et al., J. Chem. Eng. Data, 36 (1991): 202–208 (p-x correlation, 0.9 to 450 psia, 293–413 K); Ruiter, J. P., Int. J. Refrig., 13 (1990): 223–236 gives ten subroutines for computer calculations.

TABLE 2-233 T, K

Saturated Argon (R740)* vf, m3/kg

P, bar

10 20 30 40 50

vg, m3/kg

5.646.–4 5.666.–4 5.707.–4 5.763.–4 5.831.–4

hf, kJ/kg

hg, kJ/kg

0.20 2.20 6.12 11.30 17.26

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

0.0266 0.1559 0.3129 0.4610 0.5937

µf, 10−4 Pa⋅s

kf, W/(m⋅K)

0.083 0.306 0.466 0.560 0.627

60 70 80 83.8t 83.8t

0.082 0.406 0.687 0.687

5.912.–4 6.008.–4 6.125.–4 6.178.–4 7.068.–4

2.1800 0.3918 0.2434 0.2434

23.85 31.08 39.07 42.34 71.88

229.08 232.88 235.06 235.06

0.7138 0.8250 0.9316 0.9720 1.333

3.415 3.364 3.280 3.280

0.687 0.752 0.836 0.877 1.050

2.93

0.134

85 87.3 90 95 100

0.790 1.013 1.338 2.137 3.247

7.107.–4 7.174.–4 7.269.–4 7.440.–4 7.628.–4

0.2145 0.1710 0.1327 0.0864 0.0588

73.16 75.61 78.55 84.15 89.85

235.55 236.39 237.37 238.91 240.20

1.348 1.375 1.403 1.462 1.520

3.258 3.216 3.168 3.091 3.023

1.058 1.073 1.091 1.124 1.158

2.81 2.60 2.40 2.08 1.82

0.132 0.128 0.124 0.116 0.109

110 115 120 125 130

6.665 9.107 12.13 15.81 20.23

8.064.–4 8.322.–4 8.618.–4 8.965.–4 9.620.–4

0.0299 0.0221 0.0166 0.0126 0.0096

101.83 108.11 114.62 121.50 128.79

241.66 241.78 241.33 240.30 238.41

1.632 1.685 1.738 1.792 1.846

2.903 2.848 2.794 2.743 2.690

1.229 1.274 1.336 1.427 1.550

1.46 1.32 1.21 1.12 1.01

0.096 0.090 0.084 0.078 0.072

135 140 145 150 150.9

25.49 31.68 38.93 47.39 48.98

9.906.–4 1.061.–3 1.172.–3 1.468.–3 1.867.–3

0.0074 0.0056 0.0041 0.0026 0.0019

136.76 145.58 155.73 174.64 189.94

234.60 230.74 223.09 204.35 189.94

1.902 1.961 2.026 2.133 2.201

2.633 2.570 2.490 2.331 2.201

1.752

0.89 0.75 0.60 0.45 0.28

0.066 0.060 0.054 ∞

*Values extracted and in some cases rounded off from those cited in Rabinovich (ed.), Thermophysical Properties of Neon, Argon, Krypton and Xenon, Standards Press, Moscow, 1976. This source contains values for the compressed state for pressures up to 1000 bar, etc. t = triple point. Above the solid line the condensed phase is solid; below it, it is liquid. The notation 5.646.–4 signifies 5.646 × 10−4. At 83.8 K, the viscosity of the saturated liquid is 2.93 × 10−4 Pa⋅s = 0.000293 Ns/m2. This book was published in English translation by Hemisphere, New York, 1988 (604 pp.). TABLE 2-234

Thermodynamic Properties of Compressed Argon* Pressure, bar

T, K

1

100

200

300

400

500

600

700

800

900

v 100 h s

0.2035 243.4 3.299

7.420.–4 93.6 1.494

7.255.–4 97.9 1.464

7.120.–4 102.5 1.438

7.006.–4 107.2 1.414

6.907.–4 112.0 1.393

6.819.–4 116.8 1.372

6.050.–4 91.1 1.037

6.009.–4 96.1 1.026

5.976.–4 101.0 1.016

5.935.–4 106.0 1.007

1000

v 200 h s

0.4151 296.4 3.667

2.96.–3 250.2 2.538

1.430.–5 217.1 2.276

1.159.–3 209.1 2.173

1.045.–3 207.9 2.112

9.778.–4 209.2 2.068

9.312.–4 211.9 2.033

8.962.–4 215.1 2.004

8.683.–4 218.9 1.979

8.454.–4 223.0 1.957

8.260.–4 227.4 1.936

v 300 h s

0.6241 348.6 3.879

5.96.–3 330.9 2.872

2.976.–3 316.3 2.686

2.071.–3 306.6 2.572

1.666.–3 301.4 2.493

1.443.–3 299.3 2.435

1.304.–3 299.2 2.389

1.207.–3 300.5 2.352

1.136.–3 302.7 2.320

1.081.–3 305.6 2.293

1.037.–3 310.0 2.269

v 400 h s

0.8326 400.7 4.028

8.37.–3 391.3 3.048

4.279.–3 383.6 2.881

2.957.–3 378.4 2.780

2.322.–3 375.2 2.707

1.955.–3 373.8 2.651

1.719.–3 373.8 2.603

1.557.–3 374.8 2.565

1.435.–3 376.6 2.533

1.344.–3 379.2 2.505

1.271.–3 382.0 3.480

v 500 h s

1.0409 452.8 4.145

1.062.–2 447.7 3.174

5.464.–3 444.3 3.018

3.772.–3 442.0 2.924

2.940.–3 440.9 2.854

2.448.–3 440.6 2.801

2.124.–3 441.4 2.755

1.899.–3 422.9 2.718

1.730.–3 444.7 2.685

1.607.–3 447.1 2.658

1.506.–3 449.9 2.633

v 600 h s

1.2489 504.9 4.240

1.280.–2 502.4 3.274

6.589.–3 501.6 3.122

4.539.–3 501.4 3.031

3.525.–3 501.8 2.966

2.922.–3 503.0 2.914

2.522.–3 504.6 2.870

2.238.–3 506.6 2.834

2.023.–3 508.7 2.801

1.866.–3 511.2 2.774

1.736.–3 513.9 2.750

v 700 h s

1.4569 556.9 4.320

1.495.–2 556.5 3.356

7.686.–3 556.9 3.207

5.281.–3 558.0 3.118

4.088.–3 559.8 3.054

3.377.–3 561.8 3.005

2.906.–3 564.2 2.963

2.570.–3 566.9 2.928

2.317.–3 569.6 2.897

2.123.–3 527.5 2.870

1.966.–3 575.3 2.845

v 800 h s

1.6659 609.9 4.389

1.708.–2 609.8 3.427

8.768.–3 611.0 3.279

6.011.–3 612.9 3.191

4.640.–3 615.2 3.129

3.822.–3 618.1 3.081

3.280.–3 621.2 3.039

2.893.–3 624.5 3.005

2.603.–3 627.8 2.975

2.376.–3 631.3 2.948

2.196.–3 634.8 2.924

v 900 h s

1.8739 661.0 4.451

1.920.–2 662.7 3.490

9.841.–3 664.6 3.342

6.732.–3 667.2 3.255

5.183.–3 670.1 3.193

4.259.–3 673.3 3.145

3.646.–3 676.8 3.105

3.209.–3 680.7 3.071

2.881.–3 684.4 3.042

2.626.–3 688.3 3.016

2.423.–3 692.3 2.992

v 1000 h s

2.0819 713.1 4.506

2.131.–2 715.4 3.545

1.091.–2 717.9 3.398

7.448.–3 720.9 3.312

5.723.–3 724.3 3.250

4.692.–3 727.8 3.203

4.008.–3 731.5 3.163

3.520.–3 735.6 3.129

3.156.–3 739.8 3.100

2.872.–3 744.1 3.074

2.645.–3 748.5 3.051

*Values extracted and in some cases rounded off from those cited in Rabinovich (ed.), Thermophysical Properties of Neon, Argon, Krypton and Xenon, Standards Press, Moscow, 1976. v = specific volume, m3/kg; h = specific enthalpy, kJ/kg; s = specific entropy, kJ/(kg⋅K). This source contains an exhaustive tabulation of values. The notation 7.420.–4 signifies 7.420 × 10−4. This book was published in English translation by Hemisphere, New York, 1988 (604 pp.). The 1993 ASHRAE Handbook—Fundamentals (SI ed.) has a thermodynamic chart for pressures from 1 to 2000 bar, temperatures from 90 to 700 K. Saturation and superheat tables and a chart to 50,000 psia, 1220 °R appear in Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat, thermal conductivity, and viscosity see Thermophysical Properties of Refrigerants, ASHRAE, 1993. Extensive tables for 10 properties from 0.9–100 bar, 86–400 K are given by Jacques, A., Fermi Accelerator Lab., Batavia, IL, rept TM 1517, 1988 (201 pp.). In Hilsenrath, J., C. G. Messina, et al., AEDC-TR-66-248 = AD 644 081, 1966 (121 pp.), thermodynamic properties and chemical composition from 2400 to 35,000 K are tabulated. See also Drellishak, K. S. et al., AEDC-TDR-63-146, 1963; AEDC-TDR-64-12 = AD 427839, 1964. 2-217

2-218

PHYSICAL AND CHEMICAL DATA

TABLE 2-235

Liquid-Vapor Equilibrium Data for the Argon-Nitrogen-Oxygen System*

Liquid mole fraction N2 /N2 + O2

Vapor mole fraction Ar

N2

Ar

O2

Temperature,°R

Relative volatility N2 /Ar

N2 /O2

Ar/O2

Pressure activity coefficient

Enthalpy, Btu/ (lb·mol)

Heat capacity, Btu/(lb·mol·°R)

N2

Ar

O2

Liquid

Vapor

Liquid

Vapor

Pressure, 1 atm 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.

0. 0.01 0.02 0.03 0.04 0.05 0.07 0.10 0.20 0.40 0.60 0.80 0.90

0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.

0. 0.0154 0.0306 0.0456 0.0603 0.0748 0.1031 0.1439 0.2687 0.4796 0.6605 0.8293 0.9136

1.0000 0.9845 0.9694 0.9544 0.9397 0.9253 0.8970 0.8561 0.7313 0.5204 0.3395 0.1707 0.0865

162.4 162.3 162.2 162.1 162.0 161.9 161.7 161.5 160.7 159.4 158.5 157.7 157.5

2.575 2.581 2.586 2.592 2.597 2.602 2.613 2.629 2.682 2.786 2.888 2.991 3.042

4.010 4.007 4.004 4.001 3.998 3.995 3.989 3.979 3.941 3.852 3.746 3.632 3.572

1.557 1.553 1.548 1.544 1.540 1.535 1.526 1.513 1.469 1.382 1.297 1.214 1.174

1.118 1.117 1.115 1.113 1.112 1.110 1.107 1.103 1.091 1.076 1.075 1.087 1.099

1.165 1.161 1.158 1.155 1.151 1.148 1.142 1.132 1.104 1.058 1.026 1.008 1.003

0.999 1.000 1.000 1.000 1.001 1.001 1.002 1.003 1.010 1.034 1.072 1.127 1.162

−1841. −1844. −1847. −1850. −1852. −1855. −1860. −1868. −1893. −1938. −1978. −2015. −2032.

1093. 1087. 1082. 1076. 1071. 1066. 1056. 1041. 997. 924. 862. 807. 779.

13.2 13.1 13.1 13.1 13.0 13.0 12.9 12.9 12.6 11.9 11.3 10.7 10.4

7.406 7.374 7.342 7.311 7.281 7.251 7.192 7.107 6.847 6.406 6.026 5.669 5.491

0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10

0. 0.01 0.02 0.03 0.04 0.05 0.07 0.10 0.20 0.40 0.60 0.80 0.90

0.3135 0.3095 0.3056 0.3017 0.2978 0.2939 0.2863 0.2752 0.2399 0.1759 0.1169 0.0595 0.0303

0. 0.0119 0.0237 0.0354 0.0470 0.0585 0.0812 0.1145 0.2207 0.4170 0.6036 0.7933 0.8937

0.6865 0.6786 0.6707 0.6630 0.6553 0.6476 0.6325 0.6103 0.5394 0.4072 0.2795 0.1471 0.0762

157.7 157.6 157.6 157.6 157.6 157.5 157.5 157.4 157.2 157.0 156.9 156.9 157.1

2.621 2.626 2.631 2.636 2.641 2.645 2.655 2.669 2.717 2.812 2.906 3.001 3.048

4.111 4.106 4.100 4.095 4.090 4.085 4.074 4.058 4.003 3.887 3.766 3.640 3.576

1.568 1.563 1.558 1.554 1.549 1.544 1.534 1.520 1.473 1.382 1.296 1.213 1.173

1.103 1.102 1.100 1.099 1.098 1.096 1.094 1.090 1.080 1.070 1.072 1.086 1.099

1.168 1.164 1.161 1.157 1.154 1.151 1.144 1.135 1.106 1.061 1.029 1.009 1.004

1.012 1.012 1.012 1.013 1.013 1.013 1.014 1.015 1.022 1.045 1.082 1.134 1.166

−1834. −1837. −1839. −1842. −1844. −1846. −1851. −1858. −1882. −1926. −1969. −2009. −2029.

1060. 1057. 1053. 1049. 1045. 1042. 1034. 1024. 990. 928. 871. 813. 783.

13.2 13.1 13.1 13.1 13.1 13.0 13.0 12.9 12.5 11.9 11.3 10.7 10.4

7.410 7.386 7.361 7.337 7.313 7.289 7.242 7.173 6.951 6.540 6.147 5.746 5.534

0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20

0. 0.01 0.02 0.03 0.04 0.05 0.07 0.10 0.20 0.40 0.60 0.80 0.90

0.5095 0.5042 0.4990 0.4938 0.4886 0.4834 0.4732 0.4580 0.4083 0.3123 0.2162 0.1144 0.0593

0. 0.0096 0.0192 0.0288 0.0383 0.0477 0.0666 0.0946 0.1866 0.3680 0.5550 0.7602 0.8744

0.4905 0.4861 0.4818 0.4775 0.4731 0.4688 0.4602 0.4474 0.4051 0.3197 0.2288 0.1254 0.0663

154.0 154.0 154.0 154.1 154.1 154.1 154.1 154.2 154.3 154.8 155.4 156.2 156.7

2.641 2.646 2.651 2.655 2.660 2.665 2.674 2.688 2.735 2.829 2.921 3.009 3.052

4.155 4.149 4.143 4.137 4.131 4.125 4.112 4.094 4.032 3.907 3.779 3.647 3.580

1.573 1.568 1.563 1.558 1.553 1.548 1.538 1.523 1.474 1.381 1.294 1.212 1.173

1.085 1.084 1.083 1.082 1.081 1.080 1.078 1.075 1.068 1.062 1.068 1.086 1.099

1.171 1.168 1.164 1.161 1.158 1.154 1.148 1.139 1.110 1.064 1.032 1.011 1.006

1.026 1.026 1.026 1.027 1.027 1.027 1.028 1.030 1.036 1.058 1.093 1.140 1.169

−1814. −1816. −1819. −1821. −1824. −1826. −1831. −1839. −1863. −1911. −1958. −2004. −2026.

1035. 1032. 1029. 1027. 1024. 1021. 1016. 1008. 981. 930. 877. 819. 787.

13.2 13.2 13.1 13.1 13.1 13.0 13.0 12.9 12.6 11.9 11.3 10.7 10.4

7.422 7.402 7.382 7.362 7.342 7.322 7.283 7.224 7.031 6.648 6.252 5.817 5.574

0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40

0. 0.01 0.02 0.03 0.04 0.05 0.07 0.10 0.20 0.40 0.60 0.80 0.90

0.7333 0.7279 0.7226 0.7172 0.7118 0.7064 0.6956 0.6794 0.6244 0.5075 0.3743 0.2121 0.1141

0. 0.0070 0.0140 0.0210 0.0280 0.0350 0.0491 0.0703 0.1426 0.2977 0.4776 0.7010 0.8382

0.2667 0.2651 0.2635 0.2619 0.2602 0.2586 0.2553 0.2503 0.2331 0.1948 0.1482 0.0870 0.0477

148.7 148.8 148.8 148.9 148.9 149.0 149.1 149.3 149.9 151.2 152.8 154.8 155.9

2.629 2.634 2.640 2.645 2.650 2.656 2.667 2.683 2.737 2.841 2.939 3.025 3.063

4.124 4.119 4.114 4.108 4.103 4.098 4.087 4.072 4.018 3.907 3.788 3.657 3.586

1.569 1.564 1.558 1.553 1.548 1.543 1.533 1.517 1.468 1.375 1.289 1.209 1.171

1.050 1.049 1.049 1.048 1.048 1.047 1.047 1.046 1.044 1.048 1.062 1.084 1.099

1.187 1.183 1.179 1.176 1.172 1.169 1.162 1.152 1.122 1.074 1.039 1.016 1.008

1.065 1.065 1.065 1.066 1.066 1.066 1.066 1.067 1.071 1.088 1.116 1.154 1.177

−1748. −1751. −1754. −1757. −1760. −1763. −1770. −1779. −1810. −1871. −1932. −1991. −2020.

997. 996. 994. 992. 991. 989. 986. 981. 964. 928. 885. 829. 794.

13.3 13.3 13.2 13.2 13.2 13.1 13.1 13.0 12.6 11.9 11.3 10.7 10.4

7.452 7.437 7.422 7.407 7.392 7.377 7.347 7.301 7.145 6.811 6.423 5.944 5.651

0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60

0. 0.01 0.02 0.03 0.04 0.05 0.07 0.10 0.20 0.40 0.60 0.80 0.90

0.8569 0.8521 0.8472 0.8424 0.8375 0.8326 0.8227 0.8076 0.7557 0.6391 0.4938 0.2961 0.1647

0. 0.0056 0.0111 0.0167 0.0224 0.0280 0.0394 0.0566 0.1163 0.2507 0.4191 0.6500 0.8047

0.1431 0.1424 0.1416 0.1409 0.1402 0.1395 0.1380 0.1357 0.1280 0.1102 0.0871 0.0539 0.0306

144.9 145.0 145.1 145.1 145.2 145.3 145.5 145.7 146.5 148.4 150.6 153.5 155.2

2.575 2.582 2.589 2.595 2.602 2.609 2.622 2.642 2.707 2.833 2.945 3.037 3.071

3.993 3.991 3.988 3.985 3.983 3.980 3.975 3.966 3.937 3.867 3.777 3.662 3.592

1.551 1.546 1.541 1.536 1.531 1.526 1.516 1.501 1.454 1.365 1.283 1.206 1.170

1.024 1.024 1.024 1.024 1.024 1.023 1.023 1.023 1.025 1.036 1.056 1.083 1.098

1.218 1.214 1.210 1.206 1.202 1.198 1.190 1.179 1.145 1.090 1.049 1.021 1.010

1.126 1.125 1.125 1.124 1.124 1.123 1.122 1.121 1.120 1.126 1.143 1.169 1.185

−1663. −1667. −1672. −1676. −1680. −1684. −1692. −1704. −1744. −1824. −1903. −1978. −2014.

970. 969. 968. 967. 966. 965. 963. 960. 948. 923. 888. 837. 801.

13.6 13.5 13.5 13.4 13.4 13.4 13.3 13.2 12.8 12.0 11.3 10.7 10.4

7.483 7.471 7.459 7.446 7.434 7.421 7.396 7.357 7.224 6.926 6.556 6.055 5.723

0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80

0. 0.01 0.02 0.03 0.04 0.05 0.07 0.10 0.20 0.40 0.60 0.80 0.90

0.9384 0.9340 0.9296 0.9252 0.9207 0.9162 0.9071 0.8934 0.8452 0.7339 0.5869 0.3690 0.2117

0. 0.0047 0.0094 0.0142 0.0189 0.0237 0.0334 0.0481 0.0994 0.2177 0.3740 0.6059 0.7736

0.0616 0.0613 0.0610 0.0607 0.0604 0.0601 0.0595 0.0586 0.0554 0.0483 0.0391 0.0252 0.0147

142.0 142.0 142.1 142.2 142.3 142.4 142.6 142.8 143.8 146.0 148.7 152.2 154.5

2.501 2.509 2.517 2.525 2.533 2.540 2.556 2.580 2.658 2.809 2.943 3.045 3.078

3.811 3.811 3.812 3.812 3.813 3.813 3.814 3.814 3.814 3.797 3.751 3.662 3.595

1.524 1.519 1.515 1.510 1.506 1.501 1.492 1.478 1.435 1.352 1.275 1.202 1.168

1.013 1.013 1.013 1.013 1.013 1.013 1.012 1.013 1.015 1.028 1.051 1.082 1.098

1.273 1.268 1.263 1.257 1.258 1.247 1.238 1.224 1.181 1.112 1.062 1.026 1.013

1.214 1.212 1.210 1.209 1.207 1.205 1.202 1.197 1.185 1.173 1.173 1.184 1.193

−1570. −1575. −1580. −1585. −1590. −1595. −1606. −1621. −1671. −1772. −1870. −1964. −2007.

949. 948. 947. 947. 946. 945. 944. 942. 934. 916. 889. 843. 806.

14.0 13.9 13.9 13.8 13.8 13.7 13.6 13.5 13.0 12.2 11.4 10.7 10.3

7.514 7.503 7.492 7.481 7.470 7.459 7.437 7.403 7.284 7.013 6.662 6.153 5.790

*Calculated values, from Wilson, Silverberg, and Zellner, USAF Aero Propulsion Laboratory. Rep. APL TDR 64-64 (AD 603 151), 1964. Relative volatility = α i − j = (yi /xi)(x j /yj), where x = liquid composition, y = vapor composition. Pressure activity coefficient = yi P/x i p0, where p0 = vapor pressure. These data were confirmed by the analyses of Bender, Cryogenics, 13, 11 (1973); and by Elshayal and Lu, J. Chem. Eng. Data, 16, 31 (1971). See also Armstrong, G. T., J. M. Goldstein, et al., J. Res. N.B.S., 55, 5 (1955): 265–277; Bender, E., Cryogenics, 13, 1 (1973): 11–18; Elshayal, I. M. and B. C-y Lu, J. Chem. Eng. Data, 16, 1 (1971): 31–37; Funada, I., S. Yoshimura, et al., Advan. Cryog. Engng., 7 (1982): 893–901; and Hwang, S-C., Fluid Phase Equila., 37 (1987): 153–167.

THERMODYNAMIC PROPERTIES TABLE 2-235

Liquid-Vapor Equilibrium Data for the Argon-Nitrogen-Oxygen System (Continued )

Liquid mole fraction N2 /N2 + O2

2-219

Vapor mole fraction Ar

N2

Ar

O2

Temperature,°R

Relative volatility N2 /Ar

N2 /O2

Pressure activity coefficient

Ar/O2

Enthalpy, Btu/ (lb·mol)

Heat capacity, Btu/(lb·mol·°R)

N2

Ar

O2

Liquid

Vapor

Liquid

Vapor

Pressure, 1 atm (Cont.) 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90

0. 0.01 0.02 0.03 0.04 0.05 0.07 0.10 0.20 0.40 0.60 0.80 0.90

0.9709 0.9667 0.9624 0.9581 0.9538 0.9494 0.9407 0.9274 0.8808 0.7723 0.6262 0.4018 0.2339

0. 0.0044 0.0088 0.0133 0.0177 0.0222 0.0312 0.0450 0.0931 0.2048 0.3552 0.5860 0.7589

0.0291 0.0289 0.0288 0.0286 0.0285 0.0284 0.0281 0.0276 0.0261 0.0229 0.0186 0.0122 0.0072

140.6 140.7 140.8 140.9 141.0 141.1 141.3 141.6 142.6 144.9 147.8 151.7 154.2

2.459 2.468 2.476 2.485 2.493 2.502 2.519 2.545 2.629 2.793 2.938 3.048 3.082

3.710 3.712 3.714 3.716 3.718 3.720 3.724 3.729 3.743 3.755 3.733 3.660 3.597

1.509 1.504 1.500 1.496 1.491 1.487 1.478 1.465 1.424 1.344 1.270 1.201 1.167

1.015 1.014 1.014 1.013 1.013 1.013 1.013 1.012 1.014 1.026 1.050 1.081 1.098

1.311 1.305 1.299 1.293 1.287 1.281 1.270 1.254 1.204 1.126 1.069 1.029 1.014

1.271 1.268 1.265 1.263 1.260 1.257 1.252 1.245 1.226 1.200 1.190 1.192 1.197

−1522. −1527. −1533. −1538. −1544. −1550. −1561. −1578. −1633. −1745. −1853. −1956. −2004.

939. 938. 938. 937. 937. 936. 935. 934. 928. 912. 889. 846. 809.

14.2 14.2 14.1 14.1 14.0 14.0 13.8 13.7 13.2 12.3 11.4 10.7 10.3

7.530 7.519 7.509 7.498 7.488 7.477 7.456 7.423 7.310 7.050 6.708 6.197 5.822

0.97 0.97 0.97 0.97 0.97 0.97 0.97 0.97 0.97 0.97 0.97 0.97 0.97

0. 0.01 0.02 0.03 0.04 0.05 0.07 0.10 0.20 0.40 0.60 0.80 0.90

0.9916 0.9874 0.9832 0.9790 0.9748 0.9705 0.9619 0.9488 0.9032 0.7965 0.6513 0.4236 0.2489

0. 0.0042 0.0085 0.0127 0.0170 0.0213 0.0300 0.0432 0.0893 0.1969 0.3433 0.5728 0.7489

0.0084 0.0084 0.0083 0.0083 0.0083 0.0082 0.0081 0.0080 0.0076 0.0066 0.0054 0.0036 0.0021

139.8 139.9 140.0 140.1 140.2 140.3 140.5 140.8 141.8 144.2 147.2 151.3 153.9

2.429 2.438 2.447 2.456 2.465 2.474 2.492 2.519 2.608 2.780 2.934 3.050 3.084

3.638 3.641 3.644 3.647 3.650 3.653 3.658 3.667 3.691 3.722 3.718 3.658 3.597

1.498 1.494 1.489 1.485 1.481 1.477 1.468 1.456 1.415 1.339 1.267 1.199 1.167

1.018 1.018 1.017 1.017 1.016 1.016 1.015 1.014 1.014 1.025 1.049 1.081 1.098

1.342 1.335 1.329 1.322 1.315 1.309 1.296 1.278 1.224 1.137 1.075 1.031 1.015

1.316 1.313 1.309 1.306 1.302 1.299 1.293 1.284 1.257 1.220 1.202 1.198 1.200

−1488. −1494. −1500. −1505. −1511. −1517. −1529. −1547. −1606. −1725. −1841. −1951. −2002.

933. 932. 932. 931. 931. 930. 929. 928. 923. 910. 889. 847. 810.

14.4 14.4 14.3 14.2 14.2 14.1 14.0 13.9 13.3 12.3 11.4 10.7 10.3

7.541 7.531 7.520 7.510 7.500 7.489 7.469 7.437 7.327 7.073 6.737 6.226 5.844

1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

0. 0.01 0.02 0.03 0.04 0.05 0.07 0.10 0.20 0.40 0.60 0.80 0.90

1.0000 0.9959 0.9917 0.9875 0.9833 0.9791 0.9705 0.9576 0.9122 0.8063 0.6616 0.4327 0.2553

0. 0.0041 0.0083 0.0125 0.0167 0.0209 0.0295 0.0424 0.0878 0.1937 0.3384 0.5674 0.7447

0. 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0. 0. 0.

139.4 139.5 139.6 139.7 139.8 139.9 140.1 140.4 141.5 143.9 147.0 151.1 153.8

2.416 2.425 2.434 2.443 2.452 2.462 2.480 2.507 2.598 2.774 2.932 3.050 3.085

3.607 3.611 3.614 3.617 3.621 3.624 3.630 3.640 3.668 3.708 3.712 3.657 3.598

1.493 1.489 1.485 1.480 1.476 1.472 1.464 1.452 1.412 1.337 1.266 1.199 1.166

1.021 1.020 1.019 1.019 1.018 1.018 1.017 1.016 1.015 1.025 1.048 1.080 1.098

1.357 1.350 1.343 1.336 1.329 1.322 1.309 1.290 1.232 1.142 1.077 1.032 1.016

1.338 1.334 1.330 1.326 1.322 1.318 1.311 1.301 1.271 1.230 1.208 1.201 1.201

−1473. −1479. −1485. −1491. −1497. −1503. −1516. −1534. −1595. −1716. −1836. −1949. −2001.

930. 929. 929. 929. 928. 928. 927. 926. 921. 909. 888. 848. 811.

14.5 14.4 14.4 14.3 14.3 14.2 14.1 13.9 13.4 12.4 11.4 10.7 10.3

7.546 7.535 7.525 7.515 7.505 7.495 7.474 7.443 7.333 7.082 6.749 6.838 5.853

0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.

0. 0.01 0.02 0.03 0.04 0.05 0.07 0.10 0.20 0.40 0.60 0.80 0.90

0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.

0. 0.0137 0.0272 0.0405 0.0537 0.0668 0.0924 0.1299 0.2471 0.4548 0.6410 0.8190 0.9084

1.0000 0.9863 0.9728 0.9595 0.9463 0.9332 0.9076 0.8701 0.7529 0.5452 0.3590 0.1811 0.0916

190.6 190.5 190.4 190.3 190.2 190.1 189.9 189.6 188.8 187.5 186.5 185.8 185.5

2.020 2.023 2.026 2.030 2.033 2.037 2.043 2.053 2.086 2.148 2.207 2.264 2.292

2.776 2.775 2.773 2.772 2.770 2.768 2.765 2.759 2.738 2.688 2.627 2.560 2.524

1.375 1.372 1.369 1.366 1.362 1.359 1.353 1.344 1.313 1.251 1.190 1.131 1.101

0.987 0.986 0.985 0.985 0.984 0.983 0.982 0.980 0.974 0.967 0.967 0.976 0.983

1.111 1.109 1.106 1.104 1.102 1.100 1.096 1.089 1.070 1.038 1.016 1.003 1.000

1.000 1.001 1.001 1.002 1.002 1.003 1.004 1.005 1.013 1.034 1.066 1.110 1.137

−1466. −1470. −1473. −1477. −1480. −1484. −1490. −1500. −1533. −1593. −1647. −1698. −1723.

1224. 1218. 1212. 1207. 1201. 1196. 1185. 1169. 1121. 1037. 964. 896. 862.

13.4 13.4 13.4 13.4 13.4 13.3 13.3 13.3 13.1 12.7 12.3 11.9 11.8

8.122 8.094 8.065 8.037 8.010 7.982 7.929 7.850 7.605 7.166 6.771 6.389 6.196

0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10

0. 0.01 0.02 0.03 0.04 0.05 0.07 0.10 0.20 0.40 0.60 0.80 0.90

0.2393 0.2363 0.2334 0.2305 0.2277 0.2248 0.2192 0.2109 0.1843 0.1353 0.0896 0.0452 0.0229

0. 0.0116 0.0230 0.0344 0.0457 0.0569 0.0792 0.1120 0.2174 0.4150 0.6045 0.7961 0.8957

0.7607 0.7521 0.7436 0.7351 0.7267 0.7183 0.7017 0.6772 0.5983 0.4497 0.3058 0.1588 0.0814

186.2 186.1 186.1 186.0 186.0 186.0 185.9 185.8 185.5 185.2 185.0 185.1 185.2

2.063 2.066 2.069 2.072 2.075 2.078 2.083 2.092 2.119 2.173 2.224 2.273 2.296

2.831 2.828 2.825 2.822 2.820 2.817 2.811 2.802 2.772 2.707 2.638 2.564 2.526

1.372 1.369 1.365 1.362 1.359 1.356 1.349 1.340 1.308 1.246 1.186 1.128 1.100

0.994 0.994 0.993 0.992 0.991 0.990 0.988 0.986 0.979 0.971 0.971 0.978 0.985

1.120 1.117 1.115 1.113 1.110 1.108 1.104 1.097 1.077 1.044 1.020 1.006 1.002

1.020 1.021 1.021 1.021 1.022 1.022 1.023 1.024 1.030 1.049 1.078 1.117 1.141

−1452. −1455. −1459. −1462. −1465. −1469. −1475. −1485. −1517. −1578. −1637. −1693. −1720.

1193. 1188. 1184. 1180. 1175. 1171. 1163. 1150. 1110. 1037. 968. 900. 865.

13.7 13.6 13.6 13.6 13.6 13.5 13.5 13.4 13.2 12.8 12.4 12.0 11.8

8.150 8.126 8.102 8.078 8.054 8.031 7.984 7.915 7.691 7.269 6.860 6.444 6.226

0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20

0. 0.01 0.02 0.03 0.04 0.05 0.07 0.10 0.20 0.40 0.60 0.80 0.90

0.4170 0.4126 0.4082 0.4038 0.3994 0.3951 0.3864 0.3735 0.3316 0.2506 0.1706 0.0883 0.0452

0. 0.0099 0.0198 0.0297 0.0395 0.0493 0.0688 0.0979 0.1932 0.3808 0.5714 0.7742 0.8834

0.5831 0.5776 0.5721 0.5666 0.5611 0.5557 0.5448 0.5286 0.4752 0.3686 0.2580 0.1376 0.0715

182.4 182.4 182.4 182.4 182.4 182.4 182.4 182.5 182.6 183.0 183.6 184.3 184.8

2.094 2.097 2.100 2.102 2.105 2.107 2.112 2.120 2.145 2.194 2.239 2.281 2.301

2.861 2.857 2.854 2.851 2.847 2.844 2.837 2.827 2.792 2.720 2.645 2.568 2.528

1.366 1.363 1.359 1.356 1.353 1.349 1.343 1.333 1.301 1.240 1.181 1.126 1.099

0.997 0.996 0.995 0.994 0.993 0.992 0.990 0.988 0.982 0.974 0.974 0.980 0.986

1.128 1.125 1.123 1.120 1.118 1.116 1.111 1.104 1.083 1.049 1.024 1.008 1.003

1.041 1.041 1.042 1.042 1.042 1.042 1.043 1.044 1.049 1.065 1.090 1.124 1.145

−1431. −1434. −1438. −1441. −1445. −1448. −1455. −1465. −1498. −1562. −1625. −1687. −1717.

1166. 1162. 1159. 1155. 1152. 1149. 1142. 1132. 1099. 1035. 971. 904. 867.

13.8 13.8 13.8 13.8 13.7 13.7 13.7 13.6 13.3 12.9 12.4 12.0 11.8

8.189 8.167 8.146 8.125 8.104 8.083 8.041 7.978 7.771 7.363 6.944 6.497 6.255

Pressure, 4 atm

2-220

PHYSICAL AND CHEMICAL DATA

TABLE 2-235

Liquid-Vapor Equilibrium Data for the Argon-Nitrogen-Oxygen System (Concluded )

Liquid mole fraction N2 /N2 + O2

Vapor mole fraction Ar

N2

Ar

O2

Temperature,°R

Relative volatility N2 /Ar

N2 /O2

Pressure activity coefficient

Ar/O2

Enthalpy, Btu/ (lb·mol)

Heat capacity, Btu/(lb·mol·°R)

N2

Ar

O2

Liquid

Vapor

Liquid

Vapor

Pressure, 4 atm (Cont.) 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40

0. 0.01 0.02 0.03 0.04 0.05 0.07 0.10 0.20 0.40 0.60 0.80 0.90

0.6560 0.6506 0.6453 0.6400 0.6347 0.6293 0.6186 0.6025 0.5482 0.4348 0.3104 0.1684 0.0882

0. 0.0077 0.0155 0.0232 0.0310 0.0387 0.0543 0.0778 0.1575 0.3259 0.5141 0.7334 0.8595

0.3441 0.3417 0.3393 0.3369 0.3345 0.3320 0.3271 0.3197 0.2943 0.2393 0.1756 0.0982 0.0523

176.3 176.4 176.4 176.5 176.6 176.6 176.8 177.0 177.7 179.2 180.9 182.9 184.1

2.124 2.127 2.129 2.132 2.135 2.137 2.143 2.151 2.176 2.223 2.264 2.297 2.309

2.859 2.856 2.853 2.850 2.846 2.843 2.837 2.827 2.794 2.725 2.652 2.573 2.531

1.346 1.343 1.340 1.337 1.333 1.330 1.324 1.315 1.284 1.226 1.171 1.120 1.096

0.992 0.991 0.991 0.990 0.990 0.989 0.988 0.986 0.982 0.978 0.979 0.985 0.989

1.146 1.143 1.140 1.138 1.135 1.133 1.128 1.120 1.098 1.061 1.033 1.013 1.006

1.090 1.090 1.090 1.090 1.090 1.090 1.090 1.090 1.091 1.100 1.116 1.139 1.153

−1372. −1376. −1380. −1384. −1387. −1391. −1399. −1410. −1449. −1525. −1600. −1674. −1710.

1121. 1119. 1117. 1115. 1112. 1110. 1106. 1099. 1077. 1029. 975. 910. 872.

14.1 14.0 14.0 14.0 13.9 13.9 13.9 13.8 13.5 13.0 12.5 12.0 11.8

8.277 8.260 8.242 8.224 8.206 8.188 8.153 8.098 7.915 7.527 7.095 6.597 6.312

0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60

0. 0.01 0.02 0.03 0.04 0.05 0.07 0.10 0.20 0.40 0.60 0.80 0.90

0.8076 0.8023 0.7971 0.7918 0.7865 0.7812 0.7704 0.7542 0.6980 0.5739 0.4261 0.2413 0.1293

0. 0.0064 0.0127 0.0192 0.0256 0.0321 0.0451 0.0649 0.1332 0.2848 0.4667 0.6963 0.8367

0.1924 0.1913 0.1902 0.1891 0.1879 0.1868 0.1845 0.1810 0.1688 0.1413 0.1073 0.0625 0.0340

171.7 171.8 171.9 172.0 172.1 172.2 172.4 172.6 173.7 175.9 178.5 181.6 183.4

2.120 2.123 2.127 2.130 2.133 2.137 2.143 2.153 2.184 2.239 2.283 2.311 2.317

2.798 2.796 2.794 2.792 2.790 2.788 2.784 2.778 2.757 2.708 2.648 2.576 2.533

1.320 1.317 1.314 1.311 1.308 1.305 1.299 1.290 1.262 1.209 1.160 1.115 1.093

0.986 0.986 0.985 0.985 0.985 0.984 0.984 0.983 0.981 0.980 0.984 0.989 0.992

1.173 1.170 1.167 1.164 1.161 1.159 1.153 1.144 1.119 1.076 1.043 1.019 1.009

1.154 1.153 1.152 1.152 1.151 1.150 1.149 1.148 1.143 1.141 1.145 1.155 1.161

−1296. −1301. −1305. −1310. −1315. −1319. −1329. −1343. −1389. −1482. −1573. −1661. −1704.

1086. 1084. 1083. 1082. 1080. 1079. 1076. 1072. 1056. 1021. 976. 916. 876.

14.2 14.2 14.1 14.1 14.1 14.0 14.0 13.9 13.6 13.0 12.5 12.0 11.8

8.369 8.354 8.338 8.322 8.306 8.289 8.257 8.208 8.038 7.666 7.228 6.690 6.367

0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80

0. 0.01 0.02 0.03 0.04 0.05 0.07 0.10 0.20 0.40 0.60 0.80 0.90

0.9152 0.9102 0.9052 0.9001 0.8950 0.8899 0.8795 0.8638 0.8087 0.6826 0.5231 0.3077 0.1684

0. 0.0055 0.0110 0.0165 0.0221 0.0277 0.0390 0.0562 0.1162 0.2535 0.4274 0.6624 0.8150

0.0848 0.0843 0.0839 0.0834 0.0829 0.0825 0.0815 0.0801 0.0751 0.0638 0.0496 0.0299 0.0166

168.0 168.1 168.2 168.3 168.4 168.5 168.7 169.1 170.3 173.1 176.4 180.3 182.7

2.095 2.099 2.103 2.107 2.112 2.116 2.124 2.136 2.175 2.244 2.295 2.323 2.325

2.699 2.699 2.699 2.698 2.698 2.698 2.698 2.697 2.693 2.674 2.636 2.576 2.535

1.288 1.286 1.283 1.280 1.278 1.275 1.270 1.263 1.238 1.192 1.149 1.109 1.090

0.986 0.986 0.986 0.985 0.985 0.984 0.984 0.983 0.981 0.983 0.988 0.993 0.994

1.216 1.212 1.209 1.205 1.201 1.198 1.190 1.180 1.148 1.095 1.055 1.024 1.012

1.239 1.237 1.235 1.234 1.232 1.230 1.227 1.222 1.208 1.188 1.176 1.171 1.169

−1209. −1215. −1220. −1226. −1232. −1237. −1249. −1266. −1322. −1434. −1543. −1647. −1698.

1057. 1056. 1055. 1054. 1053. 1052. 1050. 1047. 1037. 1012. 976. 920. 880.

14.3 14.2 14.2 14.2 14.1 14.1 14.0 14.0 13.7 13.1 12.5 12.0 11.8

8.462 8.447 8.432 8.417 8.402 8.387 8.356 8.309 8.148 7.786 7.347 6.777 6.420

0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90

0. 0.01 0.02 0.03 0.04 0.05 0.07 0.10 0.20 0.40 0.60 0.80 0.90

0.9596 0.9547 0.9497 0.9448 0.9397 0.9347 0.9245 0.9090 0.8546 0.7287 0.5659 0.3387 0.1873

0. 0.0051 0.0103 0.0155 0.0208 0.0260 0.0367 0.0529 0.1096 0.2407 0.4102 0.6467 0.8045

0.0404 0.0402 0.0399 0.0397 0.0395 0.0393 0.0388 0.0381 0.0358 0.0305 0.0239 0.0146 0.0082

166.3 166.4 166.5 166.7 166.8 166.9 167.2 167.5 168.8 171.8 175.3 179.7 182.4

2.077 2.081 2.086 2.091 2.095 2.100 2.109 2.122 2.166 2.242 2.299 2.328 2.329

2.641 2.641 2.642 2.643 2.644 2.645 2.646 2.649 2.653 2.651 2.627 2.575 2.536

1.271 1.269 1.267 1.264 1.262 1.260 1.255 1.248 1.225 1.182 1.143 1.106 1.089

0.990 0.990 0.989 0.989 0.988 0.988 0.987 0.986 0.984 0.985 0.990 0.995 0.995

1.245 1.241 1.236 1.232 1.228 1.224 1.215 1.203 1.167 1.107 1.062 1.027 1.013

1.292 1.289 1.287 1.284 1.281 1.279 1.274 1.267 1.246 1.215 1.193 1.179 1.173

−1163. −1169. −1175. −1181. −1188. −1194. −1206. −1225. −1287. −1409. −1527. −1640. −1694.

1044. 1043. 1042. 1042. 1041. 1040. 1039. 1036. 1028. 1007. 975. 922. 882.

14.3 14.3 14.2 14.2 14.2 14.1 14.1 14.0 13.7 13.1 12.5 12.0 11.8

8.509 8.494 8.479 8.464 8.449 8.434 8.404 8.358 8.198 7.840 7.400 6.818 6.446

0.97 0.97 0.97 0.97 0.97 0.97 0.97 0.97 0.97 0.97 0.97 0.97 0.97 0.97

0. 0.01 0.02 0.03 0.04 0.05 0.07 0.10 0.10 0.20 0.40 0.60 0.80 0.90

0.9882 0.9834 0.9784 0.9735 0.9685 0.9635 0.9534 0.9380 0.9380 0.8840 0.7584 0.5940 0.3597 0.2003

0. 0.0050 0.0099 0.0149 0.0200 0.0250 0.0353 0.0509 0.0509 0.1056 0.2327 0.3990 0.6361 0.7972

0.0118 0.0117 0.0116 0.0116 0.0115 0.0114 0.0113 0.0111 0.0111 0.0104 0.0089 0.0070 0.0043 0.0024

165.2 165.3 165.5 165.6 165.7 165.8 166.1 166.5 166.5 167.9 170.9 174.6 179.3 182.1

2.062 2.067 2.072 2.077 2.082 2.087 2.097 2.111 2.111 2.158 2.240 2.302 2.332 2.331

2.598 2.599 2.601 2.602 2.604 2.605 2.608 2.612 2.612 2.624 2.633 2.620 2.574 2.537

1.260 1.257 1.255 1.253 1.251 1.248 1.244 1.237 1.237 1.216 1.176 1.138 1.104 1.088

0.995 0.994 0.994 0.993 0.992 0.992 0.991 0.989 0.989 0.987 0.986 0.991 0.996 0.996

1.269 1.264 1.259 1.254 1.250 1.245 1.236 1.222 1.222 1.182 1.116 1.067 1.030 1.014

1.334 1.330 1.327 1.324 1.321 1.317 1.311 1.302 1.302 1.276 1.235 1.206 1.185 1.176

−1130. −1136. −1143. −1149. −1156. −1163. −1176. −1195. −1195. −1261. −1390. −1516. −1635. −1692.

1035. 1035. 1034. 1033. 1033. 1032. 1031. 1029. 1029. 1022. 1003. 974. 923. 883.

14.3 14.3 14.2 14.2 14.2 14.2 14.1 14.0 14.0 13.7 13.1 12.5 12.0 11.8

8.541 8.527 8.512 8.497 8.482 8.467 8.437 8.391 8.391 8.233 7.877 7.436 6.846 6.464

1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

0. 0.01 0.02 0.03 0.04 0.05 0.07 0.10 0.20 0.40 0.60 0.80 0.90

1.0000 0.9951 0.9902 0.9853 0.9803 0.9753 0.9653 0.9499 0.8960 0.7706 0.6055 0.3684 0.2058

0. 0.0049 0.0098 0.0147 0.0197 0.0247 0.0347 0.0501 0.1040 0.2294 0.3945 0.6316 0.7942

0. 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0. 0. 0.

164.7 164.9 165.0 165.1 165.3 165.4 165.7 166.1 167.4 170.6 174.4 179.1 182.0

2.057 2.061 2.066 2.071 2.076 2.081 2.091 2.106 2.154 2.239 2.303 2.333 2.332

2.580 2.581 2.583 2.584 2.586 2.588 2.591 2.596 2.610 2.626 2.617 2.574 2.537

1.254 1.252 1.250 1.248 1.246 1.243 1.239 1.233 1.212 1.173 1.136 1.103 1.088

0.997 0.997 0.996 0.995 0.995 0.994 0.993 0.991 0.988 0.987 0.992 0.996 0.996

1.280 1.275 1.270 1.265 1.260 1.255 1.245 1.231 1.189 1.120 1.069 1.030 1.015

1.352 1.349 1.346 1.342 1.338 1.335 1.328 1.319 1.289 1.244 1.211 1.188 1.177

−1115. −1122. −1129. −1136. −1142. −1149. −1163. −1183. −1250. −1382. −1511. −1633. −1691.

1032. 1031. 1031. 1030. 1029. 1029. 1028. 1026. 1019. 1002. 973. 923. 884.

14.3 14.3 14.3 14.2 14.2 14.2 14.1 14.0 13.7 13.1 12.5 12.0 11.8

8.555 8.541 8.526 8.511 8.496 8.481 8.451 8.405 8.247 7.892 7.451 6.858 6.471

THERMODYNAMIC PROPERTIES TABLE 2-236

2-221

Thermodynamic Properties of the International Standard Atmosphere* T, K

P, bar

ρ, kg/m3

g, m/s2

M

a, m/s

µ, Pa⋅s

k, W/(m⋅K)

λ, m

0 1,000 2,000 3,000 4,000

288.15 281.65 275.15 268.66 262.17

1.01325 0.89876 0.79501 0.70121 0.61660

1.2250 1.1117 1.0066 0.90925 0.81935

9.80665 9.8036 9.8005 9.7974 9.7943

28.964 28.964 28.964 28.964 28.964

340.29 336.43 332.53 328.58 324.59

1.79.−5 1.76.−5 1.73.−5 1.69.−5 1.66.−5

2.54.−5 2.49.−5 2.43.−5 2.38.−5 2.33.−5

6.63.−8 7.31.−8 8.07.−8 8.94.−8 9.92.−8

0 1,000 2,999 2,999 3,997

5,000 6,000 7,000 8,000 9,000

255.68 249.19 242.70 236.22 229.73

0.54048 0.47217 0.41105 0.35651 0.30800

0.73643 0.66011 0.59002 0.52579 0.46706

9.7912 9.7882 9.7851 9.7820 9.7789

28.964 28.964 28.964 28.964 28.964

320.55 316.45 312.31 308.11 303.85

1.63.−5 1.59.−5 1.56.−5 1.53.−5 1.49.−5

2.28.−5 2.22.−5 2.17.−5 2.12.−5 2.06.−5

1.10.−7 1.23.−7 1.38.−7 1.55.−7 1.74.−7

4,996 5,994 6,992 7,990 8,987

10,000 15,000 20,000 25,000 30,000

223.25 216.65 216.65 221.55 226.51

0.26499 0.12111 0.05529 0.02549 0.01197

0.41351 0.19476 0.08891 0.04008 0.01841

9.7759 9.7605 9.7452 9.7300 9.7147

28.964 28.964 28.964 28.964 28.964

299.53 295.07 295.07 298.39 301.71

1.46.−5 1.42.−5 1.42.−5 1.45.−5 1.48.−5

2.01.−5 1.95.−5 1.95.−5 1.99.−5 2.04.−5

1.97.−7 4.17.−7 9.14.−7 2.03.−6 4.42.−6

9,984 14,965 19,937 24,902 29,859

40,000 50,000 60,000 70,000 80,000

250.35 270.65 247.02 219.59 198.64

2.87.−3 8.00.−4 2.20.−4 5.22.−5 1.05.−5

4.00.−3 1.03.−3 3.10.−4 8.28.−5 1.85.−5

9.6844 9.6542 9.6241 9.5942 9.5644

28.964 28.964 28.964 28.964 28.964

317.19 329.80 315.07 297.06 282.54

1.60.−5 1.70.−5 1.58.−5 1.44.−5 1.32.−5

2.23.−5 2.40.−5 2.21.−5 1.98.−5 1.80.−5

2.03.−5 7.91.−5 2.62.−4 9.81.−4 4.40.−3

39,750 49,610 59,439 69,238 79,006

90,000 100,000 150,000 200,000 250,000

186.87 195.08 634.39 854.56 941.33

1.84.−6 3.20.−7 4.54.−9 8.47.−10 2.48.−10

3.43.−6 5.60.−7 2.08.−9 2.54.−10 6.07.−11

9.5348 9.5052 9.3597 9.2175 9.0785

28.95 28.40 24.10 21.30 19.19

2.37.−2 0.142 33 240 890

88,744 98,451 146,542 193,899 240,540

300,000 400,000 500,000 600,000 800,000

976.01 995.83 999.24 999.85 999.99

8.77.−11 1.45.−11 3.02.−12 8.21.−13 1.70.−13

1.92.−11 2.80.−12 5.22.−13 1.14.−13 1.14.−14

8.9427 8.6799 8.4286 8.1880 7.7368

17.73 15.98 14.33 11.51 5.54

2600 1.6.+4 7.7.+4 2.8.+5 1.4.+6

286,480 376,320 463,540 548,252 710,574

1,000,000

1000.00

7.51.−14

3.56.−15

7.3218

3.94

3.1.+6

864,071

Z, m

H, m

*Extracted from U.S. Standard Atmosphere, 1976, National Oceanic and Atmospheric Administration, National Aeronautics and Space Administration and the U.S. Air Force, Washington, 1976. Z = geometric altitude, T = temperature, P = pressure, g = acceleration of gravity, M = molecular weight, a = velocity of sound, µ = viscosity, k = thermal conductivity, λ = mean free path, ρ = density, and H = geopotential altitude. The notation 1.79.−5 signifies 1.79 × 10−5. TABLE 2-237 T, K

Saturated Benzene*

P, bar

vf, m3/kg –3

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µf, 10−4 Pa⋅s

kf, W/(m⋅K)

290 300 310 320 330

0.0860 0.1382 0.2139 0.3206 0.4665

1.133.−10 1.147.−10–3 1.162.−10–3 1.176.−10–3 1.192.−10–3

3.569.−10 2.292.−10 1.525.−10 1.046.−10 7.379.−10–1

371.1 388.3 405.9 423.8 442.1

810.3 820.4 830.8 841.5 852.4

2.172 2.229 2.286 2.344 2.400

3.686 3.670 3.657 3.650 3.643

1.719 1.746 1.774 1.804 1.836

6.75 5.80 5.14 4.52 3.95

0.147 0.144 0.141 0.138 0.135

340 350 360 370 380

0.6615 0.9162 1.2419 1.6517 2.1588

1.207.−10–3 1.224.−10–3 1.241.−10–3 1.259.−10–3 1.277.−10–3

5.332.−10–1 3.938.−10–1 2.965.−10–1 2.233.−10–1 1.767.−10–1

460.8 479.6 498.7 518.1 537.7

863.6 875.0 886.7 898.6 910.6

2.455 2.510 2.564 2.617 2.669

3.641 3.641 3.642 3.646 3.651

1.868 1.890 1.920 1.950 1.989

3.55 3.23 2.99 2.72 2.46

0.132 0.129 0.126 0.123 0.120

390 400 410 420 430

2.7774 3.5228 4.4091 5.4540 6.6739

1.297.−10–3 1.318.−10–3 1.340.−10–3 1.363.−10–3 1.388.−10–3

1.393.−10–1 1.112.−10–1 8.972.−10–2 7.309.−10–2 6.003.−10–2

557.6 577.9 598.6 619.7 641.3

922.9 935.2 947.8 960.4 973.0

2.592 2.644 2.823 2.873 2.924

3.657 3.665 3.674 3.684 3.695

2.030 2.070 2.110 2.160 2.210

2.24 2.05 1.88 1.73 1.60

0.117 0.114 0.111 0.107 0.104

440 450 460 470 480

8.0861 9.7088 11.451 13.660 16.028

1.415.−10–3 1.444.−10–3 1.475.−10–3 1.510.−10–3 1.548.−10–3

4.965.−10–2 4.131.−10–2 3.455.−10–2 2.901.−10–2 2.441.−10–2

663.5 686.3 709.7 733.8 758.6

985.6 998.2 1010.7 1022.9 1034.9

2.974 3.025 3.075 3.126 3.179

3.706 3.718 3.730 3.742 3.753

2.260 2.320 2.380 2.450 2.519

1.48 1.37 1.28 1.10 1.12

0.101 0.098 0.095 0.092 0.089

490 500 510 520 530

18.685 21.651 24.952 28.613 32.669

1.591.−10–3 1.640.−10–3 1.697.−10–3 1.765.−10–3 1.849.−10–3

2.059.−10–2 1.736.−10–2 1.462.−10–2 1.226.−10–2 1.020.−10–2

784.3 810.9 838.5 867.2 897.2

1046.4 1057.3 1067.5 1076.6 1084.3

3.230 3.284 3.336 3.391 3.446

3.765 3.777 3.785 3.794 3.800

2.590 2.670 2.750 2.839 2.941

1.05 0.98 0.91 0.84 0.77

0.086 0.083

540 550 560 562.2

37.161 42.144 47.696 48.979

2.126.−10–3 2.258.−10–3 2.512.−10–3 3.290.−10–3

8.349.−10–3 6.616.−10–3 4.696.−10–3 3.290.−10–3

928.8 963.2 1007.3 1043.0

1089.5 1090.4 1077.6 1043.0

3.504 3.565 3.642 3.706

3.802 3.797 3.769 3.706

0.70 0.65 0.60

*Converted from a tabulation by Counsell, Lawrenson, and Lees, Nat. Phys. Lab. Teddington (U.K.) Rep. Chem. 52, 1976. Another tabulation by Kesselman et al., in Vargaftik (ed.), Tables on the Thermophysical Properties of Liquids and Gases, Hemisphere, Washington and London, 1975, shows some differences. The notation 1.133.−6 signifies 1.133 × 10−6. Other tables are given by Goodwin, R. D., J. Phys. Chem. Ref. Data, 17, 4 (1988): 1541–1636.

2-222

PHYSICAL AND CHEMICAL DATA

TABLE 2-238

Saturated Bromine* vf, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µf, 10−4 Pa⋅s

kf, W/(m⋅K)

260 280 300 320 340

0.042 0.124 0.310 0.680 1.330

3.106.−4 3.168.−4 3.232.−4 3.311.−4 3.385.−4

3.195 1.169 0.5002 0.2425 0.1309

−147.2 −138.9 −131.6 −124.2 −112.3

51.8 56.2 60.6 64.8 71.1

0.903 0.933 0.956 0.978 1.004

1.669 1.629 1.597 1.570 1.539

0.486 0.479 0.475 0.473 0.471

13.4 11.5 9.3 7.8 6.7

0.131 0.127 0.122 0.118 0.114

360 380 400 420 440

2.384 4.010 6.390 9.730 14.25

3.464.−4 3.550.−4 3.647.−4 3.752.−4 3.885.−4

0.0767 0.0477 0.0311 0.0211 0.0148

−108.6 −100.6 −93.4 −85.8 −77.7

73.1 76.9 80.6 84.0 87.1

1.026 1.048 1.063 1.084 1.103

1.531 1.515 1.501 1.488 1.477

0.470 0.471 0.475 0.480 0.489

5.7 5.0 4.5 4.0 3.7

0.109 0.104 0.099 0.094 0.089

460 480 500 520 540

20.17 27.75 37.21 48.81 62.80

4.023.−4 4.179.−4 4.378.−4 4.623.−4 4.938.−4

0.0107 0.00786 0.00589 0.00445 0.00337

−69.0 −59.7 −49.3 −37.7 −24.0

89.9 92.2 94.0 95.0 94.8

1.122 1.142 1.161 1.183 1.207

1.467 1.457 1.448 1.438 1.428

0.503 0.527 0.595 0.710 0.860

3.3 3.1 2.8 2.6 2.5

0.084 0.079 0.073 0.066 0.059

79.41 98.90 103.4

5.368.−4 6.250.−4 8.475.−4

0.00251 0.00167 0.00085

−7.1 18.8 64.8

92.5 82.5 64.8

1.237 1.280 1.356

1.414 1.390 1.356

1.063 2.31 ∞

2.3 2.2 2.1

0.050 0.035 ∞

T, K

P, bar

560 580 584.2c

*Reproduced or converted from a tabulation by Seshadri, Viswanath, and Kuloor, Ind. J. Technol., 6 (1970): 191–198. c = critical point.

TABLE 2-239

Saturated Normal Butane (R600)* hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µf, 10−4 Pa⋅s

kf, W/(m⋅K)

0.00 9.95 29.44 49.10 68.94

494.21 499.96 511.39 523.13 535.16

2.3056 2.3778 2.5121 2.6389 2.7592

5.9702 5.8779 5.7251 5.6016 5.5017

1.946 1.953 1.970 1.985 2.001

15.8 14.4 12.0 9.94 8.26

0.181 0.179 0.175 0.171 0.167

88.97 109.22 129.71 150.45 171.49

547.48 560.07 572.93 586.06 599.42

2.8738 2.9835 3.0887 3.1900 3.2879

5.4211 5.3564 5.3048 5.2643 5.2331

2.018 2.035 2.055 2.077 2.101

6.87 5.71 4.83 4.15 3.61

0.163 0.160 0.156 0.152 0.148

2.31 1.40 0.893 0.592 0.406

192.83 214.50 236.52 258.92 281.72

613.02 626.83 640.82 654.97 669.24

3.3828 3.4749 3.5647 3.6523 3.7380

5.2097 5.1929 5.1818 5.1755 5.1732

2.128 2.158 2.192 2.231 2.274

3.18 2.83 2.55 2.31 2.10

0.144 0.140 0.136 0.132 0.128

1.686.−3 1.718.−3 1.752.−3 1.790.−3 1.830.−3

0.286 0.207 0.1533 0.1156 0.0885

309.94 328.62 352.77 377.46 402.71

683.60 697.99 712.36 726.67 740.84

3.8220 3.9046 3.9860 4.0663 4.1458

5.1744 5.1783 5.1846 5.1928 5.2025

2.323 2.377 2.437 2.503 2.577

1.93 1.77 1.62 1.47 1.34

0.124 0.120 0.116 0.113 0.109

5.9179 7.5354 9.4573 11.72 14.35

1.874.−3 1.923.−3 1.978.−3 2.041.−3 2.114.−3

0.0687 0.0539 0.0427 0.0340 0.0272

428.61 455.25 482.74 511.22 540.88

754.80 768.49 781.79 794.60 806.72

4.2248 4.3035 4.3822 4.4613 4.5412

5.2132 5.2248 5.2367 5.2485 5.2597

2.657 2.746 2.842 2.947 3.062

1.21 1.08 0.97 0.87 0.78

0.105 0.101 0.097 0.093 0.089

380 390 400 410 420

17.40 20.90 24.92 29.54 34.86

2.200.−3 2.307.−3 2.447.−3 2.652.−3 3.048.−3

0.0218 0.0174 0.0138 0.0106 0.0075

571.94 604.76 639.85 678.30 723.89

817.86 827.56 834.95 838.10 830.34

4.6225 4.7058 4.7922 4.8842 4.9903

5.2696 5.2771 5.2800 5.2740 5.2437

3.20 3.34 3.50 3.69 3.84

0.69 0.62 0.55 0.49 0.44

0.085 0.081 0.077 0.074 0.072

425.2c

37.96

4.405.−3

0.0044

783.50

783.50

5.1290

5.1290

∞

P, bar

vf, m3/kg

134.9 140 150 160 170

6.7.−6 1.7.−5 8.7.−5 3.5.−4 1.17.−3

1.360.−3 1.369.−3 1.387.−3 1.405.−3 1.424.−3

180 190 200 210 220

3.37.−3 8.53.−3 1.94.−2 4.05.−2 7.81.−2

1.443.−3 1.463.−3 1.484.−3 1.505.−3 1.528.−3

230 240 250 260 270

0.1411 0.2408 0.3915 0.6100 0.9155

1.551.−3 1.575.−3 1.601.−3 1.628.−3 1.656.−3

280 290 300 310 320

1.3297 1.8765 2.5811 3.4706 4.5731

330 340 350 360 370

T, K t

vg, m3/kg 28630 11635 2470 654 207 76.4 31.8 14.7 7.39 4.00

∞

*Values rounded and reproduced or converted from Goodwin, NBSIR 79-1621, 1979. t = triple point; c = critical point. The notation 6.7.−6 signifies 6.7 × 10−6.

THERMODYNAMIC PROPERTIES TABLE 2-240

2-223

Superheated Normal Butane* Temperature, K

P, bar

150

200

250

300

350

400

450

500

600

700

v 1.013 h s

0.00139 29.6 2.512

0.00148 129.8 3.088

0.00160 236.6 3.564

0.4106 718.9 5.334

0.4847 810.7 5.616

0.5575 913.1 5.889

0.6297 1026.0 6.155

0.7013 1149.0 6.414

0.8440 1423 6.913

0.9861 1730 7.386

v 5h s

0.00139 30.0 2.511

0.00148 130.2 3.088

0.00160 237.0 3.563

0.00175 352.9 3.985

0.0909 798.5 5.363

0.1078 904.3 5.645

0.1238 1019.3 5.916

0.1393 1143.7 6.178

0.1693 1420 6.680

0.1988 1728 7.155

v 10 h s

0.00139 30.6 2.510

0.00148 130.8 3.087

0.00160 237.4 3.562

0.00175 353.3 3.983

0.00198 482.7 4.382

0.0502 891.9 5.524

0.0593 1010.3 5.803

0.0677 1136.8 6.069

0.0835 1415 6.575

0.0987 1725 7.052

v 20 h s

0.00138 31.7 2.509

0.00148 131.8 3.085

0.00160 238.4 3.560

0.00174 354.0 3.980

0.00196 482.6 4.376

0.0205 860.0 5.364

0.0268 990.1 5.670

0.0318 1122.0 5.948

0.0406 1406 6.464

0.0487 1718 6.945

v 30 h s

0.00138 32.8 2.507

0.00148 132.9 3.082

0.00159 239.3 3.557

0.00174 354.7 3.976

0.00195 482.6 4.370

0.00240 637.3 4.783

0.0156 965.5 5.570

0.0198 1105.9 5.866

0.0263 1396 6.394

0.0320 1711 6.880

v 40 h s

0.00138 33.9 2.505

0.00148 134.0 3.080

0.00159 240.3 3.555

0.00173 355.4 3.973

0.00194 482.7 4.365

0.00234 633.6 4.768

0.0097 932.2 5.468

0.0137 1088.1 5.797

0.0192 1387 6.341

0.0237 1705 6.832

v 50 h s

0.00138 35.0 2.503

0.00148 135.0 3.078

0.00159 241.3 3.552

0.00173 356.2 3.970

0.00193 428.8 4.360

0.00229 631.0 4.755

0.00549 877.0 5.329

0.0101 1068.2 5.734

0.0149 1377 6.297

0.0188 1699 6.792

v 60 h s

0.00138 36.2 2.501

0.00148 136.1 3.076

0.00159 242.3 3.550

0.00172 356.9 3.967

0.00192 483.1 4.355

0.00255 629.1 4.745

0.00352 825.1 5.204

0.00764 1046.4 5.673

0.0121 1367 6.258

0.0155 1692 6.759

v 80 h s

0.00138 38.4 2.498

0.00147 138.3 3.072

0.00158 244.2 3.545

0.00172 358.5 3.960

0.00190 483.7 4.346

0.00219 626.5 4.727

0.00286 798.1 5.130

0.00482 1001.5 5.559

0.00868 1347 6.191

0.0114 1680 6.704

v 100 h s

0.00138 40.6 2.495

0.00147 140.4 3.069

0.00158 246.2 3.540

0.00171 360.1 3.954

0.00188 484.5 4.337

0.00214 624.9 4.712

0.00264 787.9 5.095

0.00368 971.3 5.310

0.00669 1329 6.134

0.00901 1668 6.658

v 200 h s

0.00137 51.9 2.478

0.00146 151.3 3.049

0.00156 257.9 3.518

0.00167 368.8 3.927

0.00178 490.3 4.301

0.00200 624.4 4.660

0.00225 773.3 5.010

0.00258 933.7 5.348

0.00349 1270 5.960

0.00460 1623 6.849

v 300 h s

0.00136 63.2 2.462

0.00145 162.2 3.032

0.00154 266.7 3.498

0.00164 378.3 3.903

0.00176 498.0 4.273

0.00191 629.2 4.623

0.00209 773.4 4.962

0.00231 928.4 5.288

0.00284 1255 5.884

0.00345 1603 6.419

v 400 h s

0.00136 74.5 2.447

0.00144 173.3 3.015

0.00152 277.4 3.479

0.00162 388.2 3.882

0.00173 506.8 4.248

0.00185 636.2 4.593

0.00200 778.0 4.927

0.00217 930.2 5.247

0.00255 1253 5.836

0.00298 1600 6.366

v 500 h s

0.00136 85.8 2.432

0.00143 184.4 2.999

0.00151 288.1 3.461

0.00160 398.4 3.863

0.00170 516.3 4.226

0.00181 644.5 4.569

0.00193 784.8 4.898

0.00207 935.3 5.215

0.00240 1256 5.799

0.00272 1599 6.328

*Converted and rounded from tables of Goodwin, NBSIR 79-1621, 1979. Saturation and superheat tables and a diagram to 100 bar, 580 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (173 pp.). For material to 10,000 psia, 640°F, see Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat, thermal conductivity, and viscosity, see Thermophysical Properties of Refrigerants, ASHRAE, 1993.

2-224

PHYSICAL AND CHEMICAL DATA

TABLE 2-241

Saturated Carbon Dioxide*

T, K

P, bar

vf, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

216.6 220 225 230 235

5.180 5.996 7.357 8.935 10.75

8.484.−4 8.574.−4 8.710.−4 8.856.−4 9.011.−4

0.0712 0.0624 0.0515 0.0428 0.0357

386.3 392.6 401.8 411.1 420.5

731.5 733.1 735.1 736.7 737.9

2.656 2.684 2.723 2.763 2.802

4.250 4.232 4.204 4.178 4.152

1.707 1.761

240 245 250 255 260

12.83 15.19 17.86 20.85 24.19

9.178.−4 9.358.−4 9.554.−4 9.768.−4 1.000.−3

0.0300 0.0253 0.0214 0.0182 0.0155

430.2 440.1 450.3 460.8 471.6

738.9 739.4 739.6 739.4 738.7

2.842 2.882 2.923 2.964 3.005

4.128 4.103 4.079 4.056 4.032

270 275 280 290 300

32.03 36.59 41.60 53.15 67.10

1.056.−3 1.091.−3 1.130.−3 1.241.−3 1.470.−3

0.0113 0.0097 0.0082 0.0058 0.0037

494.4 506.5 519.2 547.6 585.4

735.6 732.8 729.1 716.9 690.2

3.089 3.132 3.176 3.271 3.393

3.981 3.954 3.925 3.854 3.742

304.2c

73.83

2.145.−3

0.0021

636.6

636.6

3.558

3.558

1.879

µf, 10−4 Pa⋅s

1.64

kf, W/(m⋅K) 0.182 0.178 0.171 0.164 0.160

1.933

1.45

1.992

1.28

0.156 0.148 0.140 0.134 0.128

2.125

1.14

2.410

1.02

2.887 3.724

0.91 0.79 0.60

0.116 0.109 0.102 0.088 0.074

∞

0.31

∞

*c = critical point. The notation 8.484.−4 signifies 8.484 × 10−4. The 1993 ASHRAE Handbook—Fundamentals (SI ed.) gives material for integral degrees Celsius with temperatures on the LPTS 68 scale for saturation temperatures from −56.57 to 30.98 degrees Celsius. The thermodynamic diagram from 4 to 1000 bar extends to 420°C. Saturation and superheat tables and a chart to 15,000 psia, 840°F appear in Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat, thermal conductivity, and viscosity, see ASHRAE Thermophysical Properties of Refrigerants, 1993. Saturation and superheat tables and a diagram to 200 bar, 1000 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (173 pp.). Holste, J. C., D. M. Bailey, et al., Energy Progr., 6, 2 (1986): 125–130, give properties mainly in the range 0–100 bar, 200–450 K for the superheated vapor. Compare these with Angus, S., B. Armstrong, et al., International Tables of the Fluid State—Carbon Dioxide, Pergamon, Oxford, 1976 (377 pp.). In Miller, C. E. III and S. E. Wilder, NASA SP 3097, 1976 (489 pp.), many properties and decomposition products are tabulated for pressures from 10−7 to 104 atm., 100–25,000 K. For the range to 50 kb, 400–2100 K, see Bottinga, Y. and P. Richet, Amer. J. Sci., 281 (1981): 615–660.

THERMODYNAMIC PROPERTIES TABLE 2-242

2-225

Superheated Carbon Dioxide* Temperature, K

P, bar

300

350

400

450

500

600

700

800

900

1000

v 1h s

0.5639 809.3 4.860

0.6595 853.1 4.996

0.7543 899.1 5.118

0.8494 947.1 5.231

0.9439 997.0 5.337

1.1333 1102 5.527

1.3324 1212 5.697

1.5115 1327 5.850

1.7005 1445 5.990

1.8894 1567 6.120

v 5h s

0.1106 805.5 4.548

0.1304 850.3 4.686

0.1498 897.0 4.810

0.1691 945.5 4.925

0.1882 995.8 50.31

0.2264 1101 5.222

0.2645 1211 5.392

0.3024 1326 5.546

0.3403 1445 5.685

0.3782 1567 5.814

v 10 h s

0.0539 800.7 4.405

0.0642 846.9 4.548

0.0742 894.4 4.674

0.0841 943.5 4.790

0.0938 994.1 4.897

0.1131 1100 5.089

0.1322 1211 5.260

0.1513 1326 5.414

0.1703 1445 5.555

0.1893 1567 5.683

v 20 h s

0.0255 790.2 4.249

0.0311 839.8 4.402

0.0364 889.3 4.534

0.0416 939.4 4.653

0.0466 990.8 4.762

0.0564 1098 4.955

0.0661 1209 5.127

0.0757 1325 5.282

0.0853 1444 5.423

0.0948 1567 5.551

v 30 h s

0.0159 778.5 4.144

0.0201 832.4 4.341

0.0238 883.8 4.447

0.0274 935.2 4.569

0.0309 987.3 4.679

0.0375 1096 4.876

0.0441 1208 5.049

0.0505 1324 5.204

0.0570 1444 5.346

0.0633 1566 5.474

v 40 h s

0.0110 764.9 4.055

0.0146 824.6 4.239

0.0175 878.3 4.380

0.0203 931.1 4.507

0.0230 984.3 4.619

0.0281 1094 4.818

0.0331 1205 4.993

0.0379 1323 5.148

0.0428 1443 5.291

0.0476 1566 5.419

v 50 h s

0.0080 748.2 3.968

0.0112 816.3 4.179

0.0138 872.6 4.330

0.0161 926.9 4.457

0.0183 981.1 4.572

0.0224 1091 4.773

0.0265 1205 4.948

0.0304 1322 5.104

0.0343 1443 5.247

0.0382 1566 5.377

v 60 h s

0.0058 726.9 3.878

0.0090 807.7 4.126

0.0113 866.9 4.314

0.0133 922.7 4.416

0.0151 977.8 4.532

0.0187 1089 4.736

0.0221 1204 4.912

0.0254 1321 5.069

0.0286 1442 5.212

0.0318 1565 5.341

v 80 h s

0.0062 788.4 4.029

0.0081 855.1 4.208

0.0097 914.2 4.347

0.0112 971.3 4.468

0.0140 1085 4.675

0.0166 1201 4.854

0.0191 1320 5.011

0.0216 1441 5.155

0.0240 1565 5.286

v 100 h s

0.0045 766.2 3.936

0.0062 843.0 4.144

0.0076 905.7 4.290

0.0089 964.9 4.417

0.0111 1081 4.627

0.0133 1198 4.808

0.0153 1318 4.967

0.0173 1440 5.111

0.0193 1564 5.241

v 150 h s

0.0023 704.5 3.716

0.0038 811.9 4.005

0.0049 884.8 4.177

0.0058 949.4 4.313

0.0074 1072 4.536

0.0089 1192 4.722

0.0103 1314 4.884

0.0117 1437 5.030

0.0130 1562 5.162

v 200 h s

0.0017 670.0 3.591

0.0027 783.2 3.894

0.0035 865.2 4.088

0.0043 934.9 4.234

0.0056 1063 4.468

0.0067 1186 4.668

0.0078 1310 4.824

0.0088 1435 4.970

0.0099 1561 5.104

v 300 h s

0.0017 745.3 3.747

0.0023 834.0 3.956

0.0029 910.6 4.118

0.0038 1047 4.367

0.0046 1176 4.573

0.0053 1303 4.743

0.0060 1431 4.886

0.0067 1559 5.021

v 400 h s

0.0015 728.1 3.663

0.0018 814.6 3.867

0.0022 893.3 4.033

0.0029 1035 4.292

0.0035 1168 4.497

0.0041 1298 4.671

0.0047 1428 4.824

0.0052 1558 4.960

0.0016 803.5 3.805

0.0018 881.9 3.970

0.0024 1027 4.234

0.0029 1162 4.443

0.0034 1294 4.620

0.0038 1426 4.774

0.0043 1557 4.913

v 500 h s

*Interpolated and rounded from Vukalovich and Altunin, Thermophysical Properties of Carbon Dioxide, Atomizdat, Moscow, 1965; and Collett, England, 1968. TABLE 2-243

Saturated Carbon Monoxide*

T, K

P, bar

vf, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

81.62 83.36 88.25 96.16 101.51

1.01 1.52 2.03 4.05 6.08

1.268.−3 1.295.−3 1.317.−3 1.385.−3 1.440.−3

0.0666 0.0631 0.0606 0.0547 0.0513

150.25 158.56 165.00 182.76 195.0

365.30 368.07 370.00 374.21 375.98

3.005 3.104 3.178 3.368 3.489

5.640 5.559 5.501 5.359 5.271

105.69 109.17 116.08 121.48 125.97

8.12 10.13 15.20 20.27 25.33

1.489.−3 1.535.−3 1.651.−3 1.778.−3 1.936.−3

0.0318 0.0253 0.0163 0.0116 0.0085

204.8 213.2 231.0 246.3 261.2

376.6 376.6 374.5 370.2 363.6

3.580 3.656 3.807 3.918 4.041

5.206 5.152 5.043 4.948 4.854

129.84 132.91c

30.40 34.96

2.168.−3 3.337.−3

0.0063 0.0033

277.6

313.15

4.161

4.747

*Pressure and volume values converted, and enthalpy and entropy values reproduced, from Hust and Stewart, NBS Tech. Note 202, 1963. This source gives values at and above 72.373 K at closer pressure intervals. c = critical point. The notation 1.268.−3 signifies 1.268 × 10−3. Goodwin, R. D., J. Phys. Chem. Ref. Data, 14, 4 (1985): 849–932, gives properties to 1000 bar, 68–1000 K.

2-226

PHYSICAL AND CHEMICAL DATA

FIG. 2-8 Temperature-entropy diagram for carbon monoxide. Pressure P, in atmospheres; density ρ, in grams per cubic centimeter; enthalpy H, in joules per gram. (From Hust and Stewart, NBS Tech. Note 202, 1963.)

THERMODYNAMIC PROPERTIES TABLE 2-244

2-227

Thermophysical Properties of Saturated Carbon Tetrachloride hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µf , 10−6 Pa·s

kf , W/(m⋅K)

Pr

280 290 300 310 320

0.064 0.105 0.165 0.251 0.370

0.000 0.000 0.000 0.000 0.000

619 625 633 641 649

2.414 1.495 0.971 0.669 0.463

205.5 212.9 220.9 228.8 236.9

420.7 425.7 430.9 436.1 441.3

1.018 1.042 1.068 1.095 1.121

1.787 1.775 1.768 1.764 1.760

0.835 0.844 0.853 0.863 0.874

1042 892 774 679 603

0.1043 0.1020 0.0998 0.0975 0.0952

8.34 7.38 6.62 6.01 5.54

330 340 350 360 370

0.531 0.743 1.017 1.361 1.795

0.000 0.000 0.000 0.000 0.000

657 666 674 684 694

0.3306 0.2407 0.1802 0.1370 0.1053

246.0 254.5 263.1 271.8 280.8

446.4 451.5 456.6 461.7 466.6

1.149 1.174 1.199 1.224 1.248

1.756 1.754 1.752 1.751 1.751

0.885 0.897 0.910 0.924 0.939

539 486 441 402 368

0.0930 0.0907 0.0884 0.0861 0.0839

5.13 4.81 4.54 4.31 4.12

380 390 400 410 420

2.327 2.970 3.735 4.642 5.700

0.000 0.000 0.000 0.000 0.000

704 715 727 739 753

0.0820 0.0651 0.0525 0.0426 0.0350

289.7 298.1 307.9 317.1 326.0

471.5 475.8 481.2 485.8 490.4

1.272 1.295 1.319 1.341 1.363

1.750 1.751 1.752 1.753 1.754

0.954 0.970 0.987 1.010 1.034

338 311 287 265 246

0.0816 0.0794 0.0771 0.0749 0.0726

3.95 3.80 3.67 3.57 3.50

430 440 450 460 470

6.927 8.342 9.958 11.792 13.869

0.000 0.000 0.000 0.000 0.000

766 780 796 801 834

0.02899 0.02413 0.02020 0.01692 0.01425

335.2 344.3 353.6 363.1 372.8

494.9 499.2 503.4 507.3 511.1

1.384 1.405 1.426 1.446 1.467

1.756 1.757 1.759 1.760 1.761

1.060 1.094 1.141 1.207 1.240

227 211 195 180 167

0.0704 0.0682 0.0660 0.0638 0.0666

3.42 3.38 3.37 3.36 3.36

480 490 500 510 520

16.21 18.83 21.77 25.02 28.68

0.000 0.000 0.000 0.000 0.000

856 880 858 945 987

0.01205 0.01011 0.00858 0.00722 0.00607

382.6 392.0 402.5 412.9 424.3

514.6 517.5 520.2 522.6 524.2

1.487 1.507 1.526 1.546 1.568

1.762 1.763 1.762 1.761 1.760

1.278 1.320 1.375 1.44 1.52

156 145 133

0.0594 0.0511 0.0549

3.36 3.35 3.35

530 540 550 556.4c

32.71 37.18 44.12 45.60

0.001 0.001 0.001 0.001

041 121 248 792

0.00500 0.00400 0.00309 0.00179

436.4 448.3 463.4 494.4

524.5 522.7 518.2 494.4

1.590 1.614 1.638 1.692

1.756 1.749 1.738 1.692

T, K

vf, m3/kg

P, bar

vg, m3/kg

c = critical point. Base points: hf = 200 at 273.15 K = 0°C = hA − 300 kJ/kg; sf = 1.000 at 273.15 K = 0°C = sA − 4.000 kJ/(kg⋅K). Values mostly rounded and converted from Altunin, V. V., V. Z. Geller, et al., Thermophysical Properties of Freons, vol. 9, Hemisphere, Washington, DC, 1987 (243 pp.). Some irregularities exist in these data.

TABLE 2-245

Saturated Carbon Tetrafluoride (R14)*

P, bar

vf, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µf, 10−4 Pa⋅s

kf, W/(m⋅K)

100 110 120 130 140

0.0089 0.0286 0.0924 0.2986 0.6901

5.370.−4 5.515.−4 5.668.−4 5.834.−4 6.018.−4

10.77 3.648 1.228 0.4051 0.1855

495.8 502.7 510.4 518.8 527.7

648.4 652.9 657.1 661.1 664.8

5.487 5.556 5.624 5.691 5.757

7.003 6.919 6.847 6.786 6.736

0.887 0.887 0.890 0.896 0.904

3.56

0.136 0.128 0.119 0.111 0.104

150 160 170 180 190

1.4074 2.598 4.426 7.067 10.702

6.225.−4 6.460.−4 6.733.−4 7.055.−4 7.449.−4

0.0951 0.0532 0.0318 0.0200 0.0131

537.2 549.4 557.6 568.2 579.3

668.3 671.4 674.0 676.1 677.4

5.822 5.885 5.947 6.007 6.066

6.696 6.662 6.629 6.607 6.583

0.922 0.975 1.031 1.104 1.203

3.28 3.03 2.80 2.59 2.39

0.097 0.089 0.081 0.072 0.064

200 210 220 227.5c

15.531 21.794 29.269 37.45

7.957.−4 8.674.−4 9.931.−4 1.598.−3

0.0087 0.0058 0.0036 0.0016

591.0 603.5 618.5 646.9

677.8 676.4 671.4 646.9

6.124 6.182 6.233 6.371

6.558 6.536 6.490 6.371

1.334 1.506 1.73 ∞

2.19 2.01 1.85

0.057 0.049 0.042 ∞

T, K

*P, v, h, and s values interpolated, extrapolated, and converted from Oguchi, Reito, 52 (1977): 869–889. c = critical point. The notation 5.370.−4 signifies 5.370 × 10−4. Equations and constants approximated to ASHRAE tables are given by Mecaryk, K. and M. Masaryk, Heat Recovery Systems and CHP, 11, 2–3 (1991). The 1993 ASHRAE Handbook—Fundamentals (S.I. ed.) contains a saturation table from −140 to −45.65°C and an enthalpy–log-pressure diagram from 0.1 to 300 bar, −140 to 300°C. For properties to 1000 bar from 90 to 420 K, see Rublo, R. G., J. A. Zollweg, et al., J. Chem. Eng. Data, 36 (1991): 171–184. Saturation and superheat tables and a diagram to 80 bar, 600 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (173 pp.). Chari, Ph.D. thesis, University of Michigan, 1960, presents saturation-temperature tables in fps units for 1°F increments from −270 to −51°F. Thermodynamic and transport properties, equations, and computer code and tables at constant entropy from 89 to 845 K are given by Hunt, J. L. and Boney, L. R., NASA TN D-7181, 1973 (105 pp.), largely based upon the Chari data.

2-228

PHYSICAL AND CHEMICAL DATA

TABLE 2-246 T, K 301.6m 400 500 600 700 800 900 1000 1200 1500

Saturated Cesium* P, bar

vf, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

2.66.−9 3.83.−6 3.11.−4 5.65.−3 0.0440

5.444.−4 5.615.−4 5.800.−4 5.999.−4 6.215.−4

7.01.+7 6.54.+4 1001 65.63 9.671

74.6 98.5 122.0 144.9 167.0

637.6 651.9 666.1 678.4 688.9

0.696 0.765 0.817 0.859 0.893

2.563 2.148 1.905 1.748 1.638

0.245 0.240 0.232 0.224 0.219

0.2029 0.6620 1.693 6.790 27.6

6.443.−4 6.689.−4 6.954.−4 7.628.−4 8.84.−4

188.7 210.6 233.2 281.1 358.8

698.3 707.3 716.4 736.1 772.2

0.922 0.975 0.972 1.015 1.072

1.559 1.500 1.455 1.394 1.345

0.217 0.222 0.231 0.248 0.275

2.353 0.796 0.335 0.097 0.029

*Converted from tables in Vargaftik, Tables of the Thermophysical Properties of Liquids and Gases, Nauka, Moscow, 1972, and Hemisphere, Washington, 1975. m = melting point. The notation 2.66.−9 signifies 2.66 × 109. Many of the Vargaftik values also appear in Ohse, R. W., Handbook of Thermodynamic and Transport Properties of Alkali Metals, Blackwell Sci. Pubs., Oxford, 1985 (1020 pp.). This source contains superheat data. Saturation and superheat tables and a diagram to 30 bar, 1550 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (173 pp.). For a Mollier diagram from 0.1 to 327 psia, 1300−2700°R, see Weatherford, W. D., J. C. Tyler, et al., WADD-TR-61-96, 1961. An extensive review of properties of the solid and the saturated liquid was given by Alcock, C. B., M. W. Chase, et al., J. Phys. Chem. Ref. Data, 23, 3 (1994): 385–497.

TABLE 2-247

Thermophysical Properties of Saturated Chlorine

T, °C

P, bar

vf , m3/kg

vg , m3/kg

hf , kJ/ kg

hg , kJ/kg

sf , kJ/(kg·K)

sg , kJ/(kg·K)

cpf , kJ/(kg·K)

cpg , kJ/(kg·K)

µf , 10−6 Pa· s

µg , 10−6 Pa·s

kf ,W/(m ·K)

kg,W/(m· K)

Prf

Prg

−50 −40 −30 −20 −10

0.475 0.773 1.203 1.802 2.608

0.000 0.000 0.000 0.000 0.000

623 634 645 656 668

0.5448 0.3481 0.2314 0.1593 0.1134

221.5 231.0 240.6 250.3 260.0

518.2 522.2 526.1 529.9 533.9

1.7650 1.8074 1.8480 1.8869 1.9243

3.0946 3.0562 3.0223 2.9921 2.9649

0.9454 0.9474 0.9496 0.9520 0.9547

0.476 0.484 0.497 0.513 0.532

565 520 483 452 422

10.3 10.8 11.4 11.9 12.4

0.1684 0.1650 0.1613 0.1573 0.1527

0.0061 0.0065 0.0069 0.0074 0.0078

3.17 2.99 2.85 2.74 2.64

0.809 0.815 0.820 0.826 0.841

0 10 20 30 40

3.664 5.014 6.702 8.774 11.27

0.000 0.000 0.000 0.000 0.000

681 695 710 726 744

0.0829 0.0619 0.0471 0.0364 0.0286

269.7 279.4 289.2 299.0 308.8

537.4 540.5 543.3 545.7 548.0

1.9604 1.9953 2.0291 2.0622 2.0946

2.9402 2.9177 2.8924 2.8777 2.8593

0.9579 0.9618 0.9667 0.9728 0.9816

0.554 0.579 0.607 0.638 0.674

393 368 348 333 318

13.0 13.5 14.1 14.7 15.2

0.1478 0.1427 0.1378 0.1327 0.1282

0.0083 0.0088 0.0093 0.0099 0.0104

2.55 2.48 2.45 2.44 2.43

0.864 0.888 0.918 0.950 0.985

50 60 70 80 90

14.25 17.76 21.85 26.65 32.17

0.000 0.000 0.000 0.000 0.000

763 784 808 834 865

0.02276 0.01827 0.01481 0.01202 0.00972

318.6 329.1 340.0 351.4 364.1

549.8 551.2 552.1 552.5 552.4

2.1264 2.1578 2.1892 2.2207 2.2528

2.8417 2.8245 2.8074 2.7900 2.7714

0.9968 1.022 1.054 1.124 1.253

0.720 0.786 0.885 1.017 1.205

304 290 278 267 256

15.8 16.4 17.1 17.9 18.7

0.1230 0.1171 0.1122 0.1050 0.0986

0.0110 0.0117 0.0126 0.0137 0.0149

2.46 2.53 2.61 2.85 3.26

1.034 1.107 1.201 1.331 1.510

100 110 120 130 140

38.44 45.54 53.57 62.68 72.84

0.000 0.000 0.001 0.001 0.001

901 956 016 121 335

0.00789 0.00639 0.00508 0.00392 0.00282

377.8 391.3 407.1 426.1 451.1

551.0 548.8 543.7 535.0 517.3

2.2860 2.3207 2.3590 2.4032 2.4595

2.7502 2.7317 2.7064 2.6733 2.6198

1.418 1.632 1.891

1.434 1.696 1.960

247 238 230

19.5 20.6 22.2

0.0916 0.0850 0.0775

0.0163 0.0178 0.0195

3.82 4.57 5.61

1.700 1.96 2.23

144c

77.10

0.001 77

0.00177

483.1

483.1

2.5365

2.5365

c = critical point. Values interpolated and converted from Martin, J. J., 1977 (private communication), and from Heat Exchanger Design Handbook, vol. 5, Hemisphere, Washington, DC, 1983. Values of Ziegler, Chem.-Ing.Tech., 22 (1950): 229, apparently were also used in Landolt-Bornstein, IVa, (1967): 238–239, and in Ullmans Enzyklopädie der technische Chemie, 9, Verlag Chemie, Weinheim, 1975 (317–372). 2-229

PHYSICAL AND CHEMICAL DATA 10 Chlorine

8 6

200°C 180

4

2.

6

160

140 2. 7

2 120

100°C 2. 8

10 8 80 6

9

2.

0 3.

4

Satu rate d va por

Pressure, bar

2-230

2

kJ

/kg

•

K

60

0

3. t En

p ro

y

40

20

1

3.

1 2

3.

0.8 0°C 0.6 –20

3.3

0.4

–40 3.4

0.2 –60

0.1

FIG. 2-9

500 Enthalpy–log-pressure diagram for chlorine.

3.5

550 Enthalpy, kJ/kg

600

THERMODYNAMIC PROPERTIES TABLE 2-248 T, K

2-231

Saturated Chloroform (R20) vf, m3/kg

P, bar

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg·K)

sg, kJ/(kg·K)

280 300 320 340 360

0.115 0.293 0.620 1.224 2.255

0.000 0.000 0.000 0.000 0.000

660 678 695 715 739

1.689 0.714 0.358 0.190 0.107

−46.0 −32.6 −13.4 5.2 23.3

219.5 230.6 241.1 252.1 263.0

−0.165 −0.105 −0.041 0.015 0.065

0.798 0.773 0.754 0.741 0.731

380 400 420 440 460

3.830 6.039 9.058 13.39 18.80

0.000 0.000 0.000 0.000 0.000

765 795 822 871 921

0.0653 0.0425 0.0288 0.0195 0.0137

41.7 61.4 82.8 106.1 131.6

273.7 284.2 294.2 303.6 311.2

0.114 0.165 0.217 0.270 0.325

480 500 520 530 536.6c

26.00 34.66 44.68 50.44 54.72

0.000 0.001 0.001 0.001 0.002

980 059 193 328 00

0.00962 0.00673 0.00467 0.00359 0.00200

157.4 186.2 219.6 242.7 284.1

316.5 320.8 321.3 315.7 284.1

0.380 0.436 0.499 0.540 0.602

cpf, kJ/(kg·K)

µf, 10−6 Pa·s

kf, W/(m·K)

Prf

1.03

748 587 468 381 319

0.120 0.114 0.109 0.103 0.095

3.35

0.725 0.722 0.721 0.719 0.716

1.07 1.11 1.15 1.21 1.32

273 237 206 177 155

0.0921 0.0863 0.0808 0.0750 0.0694

3.17 3.04 2.93 2.86 2.95

0.711 0.706 0.694 0.678 0.602

1.43 1.59

129.6 105.5 81.2 67.7

0.0641 0.0584 0.0518 0.0461

2.89 2.87

c = critical point. hf = sf = 0 at n.b.p., 334.5 K. P, v, h, and s interpolated from Altunin, V. V., V. Z. Geller, et al., Thermophysical Properties of Freons, U.S.S.R. N.S.R.D.S. series, vol. 9., Hemisphere.

TABLE 2-249

Saturated Decane*

P, bar

vf, m3/kg

243.5 260 280 300 320

0.00001 0.00006 0.00042 0.00197 0.00720

1.319.−3 1.334.−3 1.356.−3 1.381.−3 1.410.−3

340 360 380 400 420

0.02155 0.05522 0.1248 0.2549 0.4789

1.442.−3 1.478.−3 1.515.−3 1.552.−3 1.591.−3

440 447.3 460 480 500

0.8387 1.0133 1.3852 2.1745 3.2690

T, K m

vg, m3/kg

µf, 10−4 Pa·s

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg·K)

sg, kJ/(kg·K)

cpf, kJ/(kg·K)

418.1 452.7 495.3 539.0 584.0

812.5 836.3 866.9 899.2 933.2

2.561 2.699 2.856 3.007 3.153

4.092 4.120 4.158 4.200 4.246

2.119 2.109 2.155 2.217 2.286

8.883 3.763 1.750 0.892 0.490

631.1 680.1 730.7 782.0 835.6

968.9 1006.2 1045.0 1085.0 1126.2

3.303 3.443 3.581 3.712 3.842

4.296 4.350 4.408 4.469 4.534

5.2 4.16 3.52 2.98 2.54

1.632.−3 1.650.−3 1.682.−3 1.735.−3 1.797.−3

0.290 0.243 0.178 0.115 0.0759

889.6 909.4 944.5 1002.6 1062.7

1168.4 1184.0 1211.4 1255.2 1299.4

3.968 4.014 4.089 4.213 4.335

4.602 4.627 4.670 4.739 4.808

2.23 2.09

20750 3300. 443. 88.74 22.73

520 540 560 580 600

4.733 6.633 9.062 12.16 16.12

1.868.−3 1.952.−3 2.067.−3 2.255.−3 2.588.−3

0.0525 0.0369 0.0248 0.0154 0.0093

1124.5 1190.1 1256.1 1318.5 1384.5

1344.4 1389.5 1432.2 1468.1 1495.6

4.456 4.573 4.698 4.802 4.913

4.879 4.949 5.011 5.060 5.098

617.5c

20.97

4.238.−3

0.0042

1483.2

1483.2

5.073

5.073

kf, W/(m · K)

25.0 16.6 11.3 8.2 6.5

*Values converted from Das and Kuloor, Ind. J. Technol., 5 (1967): 75. m = melting point; c = critical point. The notation 1.319.−3 signifies 1.319 × 10−3.

0.149 0.144 0.139 0.134 0.129 0.124 0.119 0.116 0.110

2-232

PHYSICAL AND CHEMICAL DATA TABLE 2-250

Saturated Normal Deuterium*

T, K

P, bar

vf, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

18.71 19 20 21 22

0.1709 0.1944 0.2944 0.4297 0.6072

0.005752 0.005771 0.005840 0.005914 0.005993

2.232 1.988 1.365 0.968 0.705

−161.1 −160.0 −152.8 −145.9 −138.7

158.6 159.1 163.9 167.6 170.6

4.54 4.68 4.97 5.30 5.63

21.62 21.48 20.81 20.23 19.69

23 24 25 26 27

0.8344 1.1192 1.4694 1.8932 2.3989

0.00608 0.00617 0.00627 0.00638 0.00650

0.5256 0.3995 0.3088 0.2421 0.1921

−131.4 −123.8 −116.1 −108.2 −100.2

173.0 174.6 175.5 175.7 175.1

5.95 6.26 6.57 6.87 7.16

19.18 18.70 18.23 17.79 17.36

28 29 30 31 32

2.995 3.690 4.493 5.412 6.457

0.00663 0.00678 0.00694 0.00713 0.00735

0.1540 0.1246 0.1015 0.0831 0.0683

−92.0 −83.6 −74.9 −65.9 −56.5

173.8 171.7 168.7 165.0 160.3

7.44 7.72 8.00 8.27 8.54

16.94 16.52 16.12 15.72 15.32

33 34 35 36 37

7.455 8.962 10.44 12.09 13.91

0.00761 0.00793 0.00834 0.00890 0.00976

0.0563 0.0465 0.0382 0.0311 0.0249

−46.4 −35.5 −23.2 −8.6 10.0

154.7 148.0 140.0 130.1 117.1

8.83 9.12 9.45 9.82 10.28

14.92 14.52 14.11 13.67 13.17

38 38.34c

15.92 16.65

0.01158 0.01433

0.0185 0.0143

39.7 69.2

95.0 69.2

11.01 11.76

12.47 11.76

*Condensed and converted from tables of Prydz, NBS Rep. 9276, 1967. c = critical point. For equations and T-s and Z charts from 0.1 to 100 atm, 20–300 K, see also Prydz, R. and K. D. Timmerhaus, Advan. Cryog. Eng., 13 (1968): 384–396. TABLE 2-251

Saturated Deuterium Oxide*

P, bar

vf, m3/kg

vg, m3/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

277.0 278.2 283.2 288.2 293.2

0.00668 0.00720 0.01030 0.01449 0.02011

9.047.−4 9.045.−4 9.042.−4 9.043.−4 9.047.−4

172.2 160.4 114.1 82.48 60.45

0.0 5.0 25.9 46.9 67.8

2320.9 2322.5 2330.9 2339.3 2347.6

0.000 0.0188 0.0920 0.166 0.239

8.380 8.351 8.233 8.122 8.016

298.2 303.2 308.2 313.2 318.2

0.02758 0.03730 0.04990 0.06598 0.08638

9.054.−4 9.063.−4 9.075.−4 9.091.−4 9.108.−4

44.88 33.71 25.59 19.66 15.24

88.7 109.6 130.5 151.5 172.4

2356.0 2364.0 2372.3 2380.7 2388.6

0.311 0.382 0.450 0.518 0.585

7.915 7.818 7.725 7.637 7.550

323.2 333.2 353.2 373.2 398.2

0.1120 0.1831 0.4439 0.9646 2.2427

9.127.−4 9.170.−4 9.274.−4 9.403.−4 9.599.−4

11.93 7.52 3.27 1.58 0.72

193.3 234.7 318.4 402.0 507.5

2396.6 2413.3 2445.1 2474.8 2509.6

0.650 0.776 1.020 1.253 1.527

7.468 7.315 7.042 6.807 6.555

T, K t

423.2 448.2 473.2 498.2 523.2 548.2 573.2 598.2 623.2 644.7c

4.653 8.806 15.46 25.52 39.99 60.04 86.97 122.4 168.3 218.4

hf, kJ/kg

9.835.−4 1.012.−3 1.044.−3 1.082.−3 1.133.−3

0.362 0.198 0.115 0.0704 0.0447

612.5 718.8 826.8 938.5 1055.2

2541.8 2569.4 2585.7 2597.0 2598.7

1.781 2.020 2.256 2.483 2.707

6.341 6.149 5.973 5.812 5.658

1.200.−3 1.276.−3 1.392.−3 1.596.−3 2.950.−3

0.0290 0.0191 0.0124 0.0075 0.0030

1177.4 1306.7 1445.6 1607.1

2587.0 2555.6 2492.4 2366.5

2.930 3.153 3.356 3.631

5.501 5.332 5.132 4.850

*Extracted or converted from values in Kazavchinskii, Kesselman, et al., Thermophysical Properties of Heavy Water, Moscow and Leningrad, 1963; NBS-NSF transl. 70-50094, 1971. t = triple point; c = critical point. The notation 9.047.−4 signifies 9.047 × 10−4. Hill, P. G., MacMillan, R. D. and others give extensive tables for 0–1000 bar, 4–800°C in Atomic Energy of Canada, Chalk River rept. AECL-7531, 1981 (196 pp.). See also J. Phys. Chem. Ref. Data, 11, 1 (1982): 1–14; 19, 5 (1990): 1233–1274. TABLE 2-252

Deuterium Oxide Gas at 1-kg/cm3 Pressure

T, K

400

450

500

550

600

650

700

750

v, m3/ kg h, kJ/ kg s, kJ/(kg⋅K)

1.676 2525 6.931

1.895 2619 7.151

2.112 2712 7.349

2.322 2807 7.529

2.535 2904 7.697

2.747 3002 7.855

2.960 3102 8.003

3.172 3205 8.153

THERMODYNAMIC PROPERTIES TABLE 2-253

2-233

Saturated Diphenyl* hg, kJ/ kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µf, 10−4 Pa⋅s

kf, W/(m⋅K)

0.0 13.0 30.0 47.2 65.0

444.2 444.2 446.7 449.7 454.5

0.000 0.036 0.084 0.130 0.178

1.298 1.266 1.236 1.213 1.200

1.760 1.782 1.813 1.844 1.875

15.0 13.5 11.7 10.3 9.1

0.139 0.138 0.136 0.135 0.133

82.7 99.3 139.9 180.3 222.7

462.7 461.2 499.0 532.4 569.7

0.224 0.273 0.358 0.451 0.545

1.194 1.202 1.228 1.267 1.378

1.906 1.936 1.998 2.060 2.122

8.1 7.3 6.0 5.0 4.3

0.132 0.130 0.127 0.125 0.122

0.9594 0.4452 0.3652 0.2261 0.1447

267.6 314.9 361.5 404.5 457.2

611.6 651.8 687.8 723.8 762.7

0.652 0.746 0.824 0.915 1.032

1.367 1.424 1.477 1.529 1.582

2.184 2.246 2.308 2.370 2.432

3.7 3.3 2.7 2.4 2.2

0.119 0.116 0.113 0.110 0.107

1.258.−3 1.291.−3 1.326.−3 1.366.−3 1.412.−3

0.0977 0.0685 0.0504 0.0381 0.0301

522.3 563.7 630.4 689.1 745.9

801.7 842.4 886.4 930.9 977.1

1.125 1.223 1.316 1.375 1.457

1.635 1.688 1.740 1.748 1.791

2.494 2.556 2.618 2.680 2.741

1.90 1.71 1.54 1.39 1.24

0.105 0.102 0.099 0.096 0.093

2.803 2.865 2.93 3.00

1.10 0.97

0.090 0.087

T, K

P, bar

vf, m3/kg

vg, m3/kg

343 350 360 370 380

0.0010 0.0016 0.0029 0.0049 0.0064

1.010.−3 1.014.−3 1.021.−3 1.030.−3 1.037.−3

252.5 156.1 85.0 49.9 29.9

390 400 420 440 460

0.0129 0.0200 0.0432 0.0879 0.1694

1.046.−3 1.054.−3 1.072.−3 1.092.−3 1.112.−3

480 500 520 540 560

0.3112 0.5218 0.8375 1.290 1.941

1.132.−3 1.154.−3 1.177.−3 1.204.−3 1.230.−3

580 600 620 640 660

2.818 3.926 5.408 7.328 9.572

18.3 11.7 5.84 3.021 1.652

hf, kJ/kg

680 700 720 740 760

12.05 15.21 19.14 23.93 28.71

1.465.−3 1.529.−3 1.56.−3 1.70.−3 1.95.−3

0.0236 0.0186 0.0147 0.0113 0.0085

802.8 860.1 917.5 975.2 1033.1

1024.9 1073.1 1116.7 1152.8 1182.5

1.585 1.663 1.746 1.822 1.901

1.856 1.951 2.003 2.058 2.099

780 800

34.83 42.46

2.16.−3 3.18.−3

0.0058 0.0032

1091.2 1148.4

1163.0 1148.4

1.977 2.047

2.107 2.047

*Interpolated by P. E. Liley from the Landolt-Börnstein band IVa, p. 557, 1967 tables based on Technical Data on Fuel, British National Committee, World Energy Conference, London.

TABLE 2-254

Saturated Ethane (R170)* hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µf, 10−4 Pa⋅s

kf, W/(m⋅K)

176.8 198.7 221.5 244.4 267.4

769.4 782.4 795.0 807.2 819.3

2.560 2.790 3.008 3.207 3.391

9.113 8.627 8.222 7.897 7.637

2.260 2.274 2.284 2.292 2.302

14.19 9.37 6.57 4.89 3.81

0.215 0.208 0.201 0.194 0.187

10.08 4.263 2.039 1.075 0.6139

290.5 313.7 337.2 360.9 384.9

831.4 843.5 855.6 867.6 879.4

3.562 3.722 3.873 4.017 4.154

7.426 7.254 7.113 6.998 6.901

2.316 2.333 2.355 2.383 2.417

3.07 2.55 2.17 1.88 1.65

0.180 0.174 0.167 0.160 0.153

1.862 1.908 1.958 2.014 2.076

0.3738 0.2395 0.1602 0.1109 0.0789

409.3 434.2 459.7 485.9 512.8

890.8 901.7 911.9 921.4 929.6

4.285 4.412 4.535 4.655 4.773

6.819 6.750 6.689 6.635 6.585

2.458 2.508 2.568 2.640 2.730

1.47 1.33 1.21 1.11 1.03

0.147 0.140 0.133 0.126 0.119

9.670 13.01 17.12 22.10 28.06

2.148 2.231 2.330 2.452 2.613

0.0573 0.0423 0.0316 0.0237 0.0177

540.8 569.9 600.7 633.6 669.3

936.6 941.9 945.4 946.4 943.6

4.890 5.006 5.123 5.233 5.370

6.539 6.493 6.449 6.392 6.350

2.843 2.991 3.214 3.511 4.011

0.96 0.82 0.73 0.64 0.55

0.112 0.106 0.099 0.092 0.085

35.14 43.54 48.71

2.847 3.295 4.891

0.0129 0.0087 0.0048

709.8 761.6 841.2

934.7 910.8 841.2

5.502 5.669 5.919

6.278 6.166 5.919

5.089 9.919 ∞

0.44 0.31

0.078 0.067

P, bar

vf, m3/kg

90.4 100 110 120 130

1.131.−5 1.110.−4 7.467.−3 3.545.−3 1.291.−2

1.534.−3 1.546 1.573 1.615 1.644

140 150 160 170 180

3.831.−2 9.672.−2 0.2146 0.4290 0.7874

1.675 1.708 1.743 1.780 1.819

190 200 210 220 230

1.347 2.174 3.340 4.922 7.004

240 250 260 270 280 290 300 305.3c

T, K t

vg, m3/kg 21945 2484.5 407.0 93.61 27.83

*Values reproduced or converted from Goodwin, Roder, and Straty, NBS Tech. Note 684, 1976. t = triple point; c = critical point. The notation 1.131.−5 signifies 1.131 × 10−5.

2-234

PHYSICAL AND CHEMICAL DATA

TABLE 2-255

Superheated Ethane* Temperature, K

P, bar

100

150

200

250

300

350

400

450

500

600

700

v 1.013 h s

0.00156 198.9 2.790

0.00171 313.8 3.722

0.5310 909.3 6.993

0.6725 984.7 7.330

0.8118 1068.3 7.634

0.9500 1161.5 7.921

1.0877 1265.3 8.198

1.2250 1379.8 8.467

1.3622 1504.6 8.730

1.6360 1783 9.237

1.9096 2097 9.720

v 5h s

0.00156 199.4 2.789

0.00171 314.3 3.720

0.00191 434.5 4.411

0.1288 973.3 6.858

0.1595 1060.3 7.175

0.1890 1155.6 7.468

0.2178 1260.7 7.748

0.2464 1376.1 8.020

0.2747 1501.5 8.284

0.3308 1781 8.793

0.3867 2096 9.227

v 10 h s

0.00156 200.0 2.788

0.00171 314.9 3.719

0.00190 435.0 4.408

0.0590 956.5 6.618

0.0765 1050.0 6.959

0.0923 1148.2 7.262

0.1073 1255.0 7.547

0.1220 1371.5 7.821

0.1365 1497.9 8.087

0.1650 1777 8.598

0.1933 2094 9.083

v 20 h s

0.00156 201.3 2.785

0.00170 316.1 3.715

0.00190 435.9 4.404

0.00222 569.8 4.999

0.0346 1026.1 6.710

0.0438 1132.3 7.038

0.0521 1243.3 7.334

0.0599 1362.4 7.614

0.0674 1490.5 7.884

0.0822 1774 8.399

0.0966 2090 8.886

v 40 h s

0.00155 203.9 2.780

0.00170 318.5 3.709

0.00189 437.9 4.394

0.00219 569.9 4.982

0.0118 947.9 6.309

0.0193 1096.2 6.770

0.0244 1218.6 7.097

0.0288 1343.8 7.391

0.0329 1475.9 7.670

0.0407 1764 8.194

0.0482 2083 8.686

v 60 h s

0.00155 206.5 2.775

0.00170 321.0 3.702

0.00188 439.8 4.385

0.00217 570.3 4.966

0.00290 738.1 5.574

0.0109 1050.9 6.557

0.0132 1192.0 6.934

0.0185 1324.8 7.247

0.0215 1461.2 7.535

0.0270 1754 8.068

0.0321 2077 8.564

v 80 h s

0.00155 209.1 2.769

0.00169 323.4 3.696

0.00188 441.9 4.377

0.00215 570.9 4.951

0.00273 728.1 5.522

0.00667 993.8 6.345

0.0106 1163.6 6.800

0.0134 1305.5 7.135

0.0158 1446.7 7.432

0.0201 1745 7.975

0.0459 2070 8.476

v 100 h s

0.00155 211.7 2.764

0.00169 325.8 3.690

0.00187 443.9 4.368

0.00213 571.8 4.938

0.00263 722.7 5.486

0.00465 924.4 6.166

0.00791 1134.7 6.682

0.0104 1286.3 7.040

0.0124 1432.4 7.348

0.0160 1736 7.900

0.0193 2064 8.406

v 150 h s

0.00155 218.1 2.752

0.00168 332.0 3.674

0.00185 449.2 4.348

0.00209 574.6 4.907

0.00247 716.4 5.423

0.00328 887.4 5.955

0.00488 1075.2 6.457

0.00655 1242.3 6.851

0.00805 1399.3 7.182

0.0107 1715 7.758

0.0130 2050 8.274

v 200 h s

0.00154 224.6 2.738

0.00167 338.2 3.660

0.00184 454.7 4.329

0.00205 578.2 4.880

0.00237 714.8 5.377

0.00291 870.5 5.863

0.00383 1041.7 6.320

0.00495 1210.2 6.717

0.00605 1327.3 7.059

0.00806 1697 7.651

0.00986 2038 8.176

v 300 h s

0.00153 237.6 2.715

0.00166 350.6 3.632

0.00181 465.9 4.294

0.00200 586.8 4.833

0.00225 715.9 5.309

0.00259 860.9 5.757

0.00307 1014.9 6.168

0.00367 1175.5 6.547

0.00433 1338.7 6.891

0.00563 1671 7.496

0.00686 2019 8.032

v 400 h s

0.00153 250.6 2.692

0.00165 363.2 3.605

0.00179 477.6 4.262

0.00195 596.6 4.793

0.00216 723.7 5.257

0.00244 861.6 5.688

0.00276 1008.3 6.080

0.00316 62.5 6.443

0.00361 1322.7 6.780

0.00454 1657 7.388

0.00545 2008 7.930

v 500 h s

0.00152 263.5 2.670

0.00163 375.8 3.580

0.00176 489.3 4.234

0.00192 607.1 4.758

0.00210 732.0 5.213

0.00232 866.5 5.634

0.00258 1009.3 6.015

0.00288 1159.3 6.369

0.00322 1316.9 6.00

0.00392 1650 7.306

0.00465 2003 7.851

*Converted and rounded off from the tables of Goodwin, Roder, and Straty, NBS Tech. Note 684, 1976. v = specific volume, m3/kg; h = specific enthalpy, kJ/ kg; s = specific entropy, kJ/(kg⋅K). Saturation and superheat tables and a diagram to 300 bar, 580 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (173 pp.). Saturation and superheat tables and a chart to 10,000 psia, 640°F appear in Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat, thermal conductivity, and viscosity, see Thermophysical Properties of Refrigerants, ASHRAE, 1993. The 1993 ASHRAE Handbook—Fundamentals (SI ed.) contains a thermodynamic diagram from 0.1 to 700 bar for temperatures to 600 K.

THERMODYNAMIC PROPERTIES TABLE 2-256 T, K

P, bar

2-235

Saturated Ethanol vf, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µf, 10−6 Pa·s

kf, W/(m·K)

Prf

250 260 270 280 290

0.0027 0.0059 0.0128 0.025 0.048

0.001 0.001 0.001 0.001 0.001

184 196 208 220 233

2.113 2.167 2.227 2.294 2.369

295 229 193 156 127

0.177 0.175 0.173 0.171 0.170

35.2 28.4 24.8 20.9 17.7

300 310 320 330 340

0.088 0.151 0.253 0.406 0.632

0.001 0.001 0.001 0.001 0.001

246 260 274 288 304

2.45 2.54 2.64 2.75 2.86

104 86 72 61 52

0.168 0.165 0.162 0.159 0.157

15.2 13.2 11.7 10.6 9.5

350 360 370 380 390

0.956 1.409 2.023 2.837 3.897

0.001 0.001 0.001 0.001 0.001

318 337 357 379 403

0.7656 0.5052 0.3555 0.2556 0.1873

199.9 230.1 262.2 295.1 329.1

1161.9 1178.4 1193.9 1208.4 1221.5

2.99 3.12 3.27 3.42 3.58

45.0 39.0 34.2 30.0 26.1

0.155 0.153 0.151 0.149 0.147

8.7 8.0 7.4 6.9 6.3

400 410 420 430 440

5.251 6.954 9.063 11.64 14.72

0.001 0.001 0.001 0.001 0.001

430 461 495 532 574

0.1398 0.1058 0.0812 0.0631 0.0493

364.2 400.8 435.7 472.2 512.7

1233.6 1244.2 1254.2 1262.3 1269.2

3.74 3.99 4.26 4.55 4.88

22.7 20.0 17.6 15.3 13.9

0.145 0.144 0.142 0.140 0.139

5.9 5.5 5.3 5.0 4.9

450 460 470 480 490

18.33 22.61 27.66 33.55 40.39

0.001 0.001 0.001 0.001 0.001

623 682 752 832 950

0.0389 0.0308 0.0243 0.0193 0.0148

557.2 605.0 653.7 704.5 757.7

1274.2 1275.5 1271.1 1262.3 1250.2

5.23

12.5

0.137

4.8

500 510 516.3c

48.28 57.32 63.90

0.002 091

0.0110

818.9

1232.7

c = critical point. Values interpolated and converted from Heat Exchanger Design Handbook, vol. 5, Hemisphere, Washington, DC, 1983, and from various literature sources.

FIG. 2-10 Enthalpy-concentration diagram for aqueous ethyl alcohol. Reference states: Enthalpies of liquid water and ethyl alcohol at 0°C are zero. NOTE: In order to interpolate equilibrium compositions, a vertical may be erected from any liquid composition on the boiling line and its intersection with the auxiliary line determined. A horizontal from this intersection will establish the equilibrium vapor composition on the dew line. (Bosnjakovic, Technische Thermodynamik, T. Steinkopff, Leipzig, 1935.)

2-236

PHYSICAL AND CHEMICAL DATA

TABLE 2-257

Saturated Ethylene (Ethene—R1150)

Temperature, K

Pressure, bar

104.0t 110 120 130 140

0.00123 0.00334 0.01380 0.04456 0.1191

0.001 0.001 0.001 0.001 0.001

527 545 576 609 644

150 160 170 180 190

0.2747 0.5636 1.0526 1.8207 2.9574

0.001 0.001 0.001 0.001 0.001

681 721 763 810 861

vf, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

251.36 97.57 25.75 8.62 3.46

−323.81 −309.54 −284.17 −259.13 −234.80

244.36 251.47 263.23 274.87 286.28

−1.9901 −1.8571 −1.6362 −1.4358 −1.2554

3.4730 3.2431 2.9255 2.6717 2.4663

2.497 2.500 2.539 2.465 2.405

1.5977 0.8232 0.4625 0.2784 0.1770

−210.90 −187.12 −163.23 −139.05 −114.46

297.37 308.00 318.04 327.35 335.79

−1.0908 −0.9378 −0.7935 −0.6559 −0.5244

2.2977 2.1566 2.0375 1.9352 1.7812

2.377 2.377 2.395 2.427 2.472

200 210 220 230 240

4.560 6.730 9.575 13.206 17.742

0.001 0.001 0.002 0.002 0.002

918 981 054 139 241

0.1177 0.0810 0.0573 0.0413 0.0302

−89.33 −63.52 −36.84 −9.04 20.23

343.21 349.41 354.18 357.17 357.90

−0.3967 −0.2730 −0.1515 −0.0314 0.0088

1.7659 1.6932 1.6258 1.5609 1.4957

2.531 2.608 2.711 2.852 3.055

250 260 270 280 282.3c

23.307 30.046 38.132 47.834 50.403

0.002 0.002 0.002 0.003 0.004

369 541 804 442 669

0.02222 0.01624 0.01152 0.00720 0.00467

51.55 85.91 125.79 183.40 234.55

355.37 348.68 333.71 292.83 234.55

0.2114 0.3397 0.4819 0.6803 0.8585

1.4276 1.3503 1.3054 1.0711 0.8585

3.372 3.945 5.40 20.0

t = triple point; c = critical point. hf = sf = 0 at 233.15 K = −40°C. Converted from Jacobsen, R. T., M. Jahangiri, et al., Ethylene—Intl. Thermodyn. Tables of the Fluid State—10, Blackwell Sci. Publ., Oxford, U.K., 1988 (299 pp.). Saturation and superheat tables and a diagram to 100 bar, 460 K are given by Reynolds, W. C., Thermodynamic properties in S.I., Stanford Univ. publ., 1979 (173 pp.). Saturation and superheat tables and a chart to 6000 psia, 360°F appear in Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat, thermal conductivity, and viscosity, see Thermophysical Properties of Refrigerants, ASHRAE, 1993. The 1993 ASHRAE Handbook—Fundamentals (SI ed.) has a thermodynamic chart for pressures from 0.1 to 400 bar and temperatures up to 460 K.

THERMODYNAMIC PROPERTIES TABLE 2-258

Compressed Ethylene Temperature, K

Pressure, bar

110

125

150

175

200

225

250

275

v (m3/kg) 1 h (kJ/kg) s (kJ/kg⋅K)

0.001 545 −309.4 −1.858

0.001 592 −271.4 −1.534

0.001 681 −210.8 −1.091

0.5036 324.8 2.091

0.5814 357.0 2.264

0.6580 389.9 2.419

0.7337 424.0 2.562

0.8091 459.7 2.698

v (m3/kg) 5 h (kJ/kg) s (kJ/kg⋅K)

0.001 544 −308.9 −1.859

0.001 591 −271.0 −1.535

0.001 680 −210.4 −1.093

0.001 785 −150.8 −0.726

0.001 917 −89.3 −0.397

0.1240 378.4 1.907

0.1407 415.0 2.061

0.1569 452.3 2.203

v (m3/kg) 10 h (kJ/kg) s (kJ/kg⋅K)

0.001 543 −308.3 −1.860

0.001 591 −270.4 −1.537

0.001 679 −209.8 −1.095

0.001 783 −150.3 −0.728

0.001 914 −89.0 −0.400

0.05643 361.2 1.646

0.06672 402.4 1.820

0.07525 442.3 1.973

v (m3/kg) 20 h (kJ/kg) s (kJ/kg⋅K)

0.001 542 −307.1 −1.863

0.001 589 −269.2 −1.540

0.001 676 −208.7 −1.098

0.001 780 −149.4 −0.733

0.001 908 −88.2 −0.406

0.002 084 −23.0 −0.099

0.02810 370.3 1.520

0.03405 419.7 1.708

v (m3/kg) 30 h (kJ/kg) s (kJ/kg⋅K)

0.001 541 −305.9 −1.866

0.001 588 −268.0 −1.543

0.001 674 −207.6 −1.102

0.001 776 −148.4 −0.737

0.001 903 −87.5 −0.412

0.002 072 −22.8 −0.107

0.002 347 50.5 0.201

0.01978 390.7 1.508

v (m3/kg) 40 h (kJ/kg) s (kJ/kg⋅K)

0.001 540 −304.7 −1.869

0.001 587 −266.8 −1.546

0.001 672 −206.5 −1.106

0.001 773 −147.4 −0.741

0.001 897 −86.7 −0.418

0.002 062 −22.5 −0.115

0.002 318 49.1 0.186

0.01163 344.7 1.284

v (m3/kg) 50 h (kJ/kg) s (kJ/kg⋅K)

0.001 539 −303.5 −1.872

0.001 585 −265.7 −1.550

0.001 670 −205.4 −1.110

0.001 770 −146.4 −0.746

0.001 892 −85.9 −0.423

0.002 052 −22.2 −0.123

0.002 293 48.1 0.173

0.002 846 139.8 0.521

v (m3/kg) 60 h (kJ/kg) s (kJ/kg⋅K)

0.001 538 −302.3 −1.875

0.001 584 −264.5 −1.553

0.001 668 −204.2 −1.113

0.001 767 −145.4 −0.750

0.001 887 −85.1 −0.428

0.002 043 −21.8 −0.130

0.002 270 47.4 0.161

0.002 723 132.3 0.484

v (m3/kg) 80 h (kJ/kg) s (kJ/kg⋅K)

0.001 535 −299.8 −1.881

0.001 581 −262.1 −1.559

0.001 664 −202.0 −1.120

0.001 761 −143.4 −0.759

0.001 877 −83.5 −0.439

0.002 025 −20.9 −0.145

0.002 232 46.5 0.139

0.002 585 124.1 0.434

v (m3/kg) 100 h (kJ/kg) s (kJ/kg⋅K)

0.001 533 −297.4 −1.887

0.001 579 −259.7 −1.565

0.001 660 −199.7 −1.127

0.001 754 −141.2 −0.767

0.001 867 −81.8 −0.449

0.002 009 −19.9 −0.158

0.002 199 46.1 0.120

0.002 495 119.6 0.400

v (m3/kg) 150 h (kJ/kg) s (kJ/kg⋅K)

0.001 528 −291.3 −1.901

0.001 571 −253.7 −1.580

0.001 650 −194.0 −1.145

0.001 740 −136.0 −0.787

0.001 846 −77.3 −0.473

0.001 973 −16.7 −0.188

0.002 136 46.6 0.079

0.002 356 114.4 0.337

v (m3/kg) 200 h (kJ/kg) s (kJ/kg⋅K)

0.001 522 −285.3 −1.914

0.001 565 −247.7 −1.595

0.001 641 −188.3 −1.161

0.001 727 −130.7 −0.806

0.001 826 −72.5 −0.495

0.001 943 −13.0 −0.215

0.002 086 48.6 0.045

0.002 268 113.2 0.291

v (m3/kg) 250 h (kJ/kg) s (kJ/kg⋅K)

0.001 517 −279.2 −1.928

0.001 559 −241.7 −1.610

0.001 633 −182.5 −1.177

0.001 715 −125.2 −0.824

0.001 809 −67.6 −0.516

0.001 918 −8.9 −0.240

0.002 046 51.4 0.015

0.002 203 113.9 0.253

v (m3/kg) 300 h (kJ/kg) s (kJ/kg⋅K)

0.001 512 −273.0 −1.942

0.001 552 −235.7 −1.623

0.001 625 −174.5 −1.192

0.001 704 −119.6 −0.841

0.001 793 −62.5 −0.536

0.001 895 −4.4 −0.262

0.002 012 54.9 −0.012

0.002 151 115.9 0.220

v (m3/kg) 400 h (kJ/kg) s (kJ/kg⋅K)

0.001 503 −260.8 −1.968

0.001 542 −223.6 −1.650

0.001 609 −164.8 −1.221

0.001 683 −108.3 −0.873

0.001 765 −51.9 −0.572

0.001 855 5.1 −0.303

0.001 957 63.0 −0.059

0.002 072 122.1 0.166

v (m3/kg) 500 h (kJ/kg) s (kJ/kg⋅K)

0.001 499 −246.9 −1.978

0.001 531 −211.4 −1.676

0.001 596 −152.9 −1.249

0.001 665 −96.8 −0.906

0.001 740 −40.9 −0.605

0.001 823 15.3 −0.339

0.001 913 72.3 −0.099

0.002 01 130.1 0.121

Converted from Jacobsen, R. T., M. Jahangiri, et al., Ethylene—Intl. Thermodyn. Tables of the Fluid State—10, Blackwell Sci. Publ., Oxford, 1988 (299 pp.). sf = hf = 0 at 233.15 K = −40°C.

2-237

2-238

PHYSICAL AND CHEMICAL DATA TABLE 2-258

Compressed Ethylene (Concluded ) Temperature, K

Pressure, bar

300

325

350

375

400

425

450

v (m3/kg) 1 h (kJ/kg) s (kJ/kg⋅K)

0.8842 497.3 2.829

0.9591 536.9 2.956

1.0339 578.8 3.079

1.1084 622.8 3.201

1.1830 668.9 3.320

1.2575 717.2 3.437

1.3319 767.9 3.553

v (m3/kg) 5 h (kJ/kg) s (kJ/kg⋅K)

0.1728 491.0 2.338

0.1884 531.5 2.467

0.2039 574.1 2.593

0.2193 618.6 2.716

0.2346 665.2 2.836

0.2499 713.9 2.954

0.2650 764.9 3.071

v (m3/kg) 10 h (kJ/kg) s (kJ/kg⋅K)

0.08380 482.8 2.113

0.09207 542.5 2.247

0.1002 568.0 2.375

0.1081 613.3 2.500

0.1160 660.5 2.622

0.1238 709.7 2.742

0.1316 761.1 2.859

v (m3/kg) 20 h (kJ/kg) s (kJ/kg⋅K)

0.03914 465.0 1.866

0.04379 509.8 2.009

0.04823 555.5 2.144

0.05257 602.4 2.274

0.05675 650.9 2.399

0.06088 701.2 2.521

0.06491 753.5 2.640

v (m3/kg) 30 h (kJ/kg) s (kJ/kg⋅K)

0.02404 444.7 1.696

0.02763 493.8 1.853

0.03090 542.3 1.996

0.03400 591.2 2.131

0.03700 641.2 2.261

0.03990 692.6 2.387

0.04270 745.9 2.508

v (m3/kg) 40 h (kJ/kg) s (kJ/kg⋅K)

0.01630 420.6 1.550

0.01947 476.3 1.728

0.02220 528.3 1.882

0.02473 579.5 2.023

0.02710 631.2 2.157

0.02938 688.9 2.286

0.03160 738.2 2.409

v (m3/kg) 50 h (kJ/kg) s (kJ/kg⋅K)

0.01140 390.4 1.404

0.01451 456.9 1.617

0.01697 513.4 1.784

0.01916 567.5 1.933

0.02119 621.1 2.072

0.02311 675.1 2.207

0.02495 730.5 2.330

v (m3/kg) 60 h (kJ/kg) s (kJ/kg⋅K)

0.007 757 347.8 1.230

0.01116 435.1 1.510

0.01347 497.7 1.696

0.01546 555.2 1.854

0.01725 610.9 1.999

0.01892 666.3 2.135

0.02052 722.9 2.263

v (m3/kg) 80 h (kJ/kg) s (kJ/kg⋅K)

0.003 672 238.7 0.832

0.006 864 382.8 1.295

0.009 136 463.5 1.534

0.01085 529.4 1.717

0.01237 590.1 1.874

0.01374 648.5 2.016

0.01502 707.6 2.151

v (m3/kg) 100 h (kJ/kg) s (kJ/kg⋅K)

0.003 094 210.2 0.715

0.004 698 330.2 1.098

0.006 596 427.5 1.387

0.008 163 503.1 1.596

0.009 492 569.3 1.768

0.01068 630.9 1.918

0.01177 692.7 2.059

v (m3/kg) 150 h (kJ/kg) s (kJ/kg⋅K)

0.002 684 188.8 0.596

0.003 223 272.7 0.864

0.004 040 361.4 1.126

0.004 983 445.5 1.359

0.005 914 521.6 1.556

0.006 765 592.1 1.722

0.007 578 658.1 1.878

v (m3/kg) 200 h (kJ/kg) s (kJ/kg⋅K)

0.002 508 181.7 0.529

0.002 840 255.0 0.763

0.003 292 332.4 0.992

0.003 838 410.6 1.208

0.004 445 487.0 1.406

0.005 058 560.1 1.580

0.005 664 629.5 1.742

v (m3/kg) 250 h (kJ/kg) s (kJ/kg⋅K)

0.002 397 179.2 0.480

0.002 644 247.7 0.698

0.003 024 319.1 0.910

0.003 327 392.1 1.111

0.003 743 465.5 1.301

0.004 190 538.2 1.476

0.004 648 608.5 1.639

v (m3/kg) 300 h (kJ/kg) s (kJ/kg⋅K)

0.002 317 179.1 0.440

0.002 517 244.6 0.670

0.002 578 312.6 0.850

0.003 037 382.2 1.043

0.003 351 452.9 1.226

0.003 690 524.1 1.398

0.004 042 593.9 1.558

v (m3/kg) 400 h (kJ/kg) s (kJ/kg⋅K)

0.002 203 182.7 0.377

0.002 352 244.9 0.576

0.002 522 308.8 0.764

0.002 711 374.3 0.946

0.002 919 441.3 1.119

0.003 413 509.8 1.285

0.003 382 578.3 1.442

v (m3/kg) 500 h (kJ/kg) s (kJ/kg⋅K)

0.002 122 189.1 0.326

0.002 245 250.7 0.523

0.002 379 312.8 0.707

0.002 524 376.1 0.882

0.002 678 440.4 1.048

0.002 847 505.8 1.206

0.003 022 572.0 1.358

THERMODYNAMIC PROPERTIES TABLE 2-259

2-239

Saturated Fluorine* hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µf, 10−4 Pa⋅s

kf, W/(m⋅K)

−158.6 −153.5 −149.1 −141.8 −134.4

40.9 42.0 45.8 49.6 53.2

1.602 1.642 1.768 1.885 1.995

5.314 5.235 5.004 4.816 4.666

1.446 1.442 1.437 1.442 1.450

8.8 8.0 6.0 4.7 3.8

0.186 0.184 0.177 0.170 0.162

0.583 0.309 0.176 0.108 0.069

−127.0 −119.5 −111.9 −104.3 −96.5

56.8 60.1 63.3 66.1 68.6

2.097 2.194 2.285 2.372 2.455

4.540 4.433 4.342 4.262 4.191

1.460 1.474 1.498 1.535 1.555

3.19 2.71 2.33 2.00 1.76

0.154 0.146 0.137 0.129 0.120

7.193.−4 7.412.−4 7.659.−4 7.948.−4 8.283.−4

0.0466 0.0323 0.0231 0.0168 0.0125

−88.6 −80.5 −72.2 −63.6 −54.5

70.7 72.4 73.6 74.1 73.9

2.535 2.612 2.688 2.763 2.837

4.127 4.068 4.012 3.959 3.906

1.585 1.630 1.692 1.782 1.888

1.53 1.36 1.21 1.08 0.96

0.112 0.103 0.095 0.087 0.080

8.696.−4 9.223.−4 9.963.−4 1.119.−3 1.743.−3

0.0093 0.0069 0.0051 0.0036 0.0017

−44.9 −34.5 −22.7 −8.4 23.9

72.7 70.2 65.6 56.9 23.9

2.912 2.989 3.073 3.170 3.388

3.864 3.795 3.727 3.636 3.388

2.05 2.33 2.90 3.64 ∞

0.86 0.74 0.63 0.49

0.073 0.066 0.070 0.105 ∞

T, K

P, bar

vf, m3/kg

vg, m3/kg

53.5t 55 60 65 70

0.0025 0.0041 0.0155 0.0477 0.1230

5.866.−4 5.898.−4 6.005.−4 6.119.−4 6.240.−4

46.2 17.1 8.46 2.93 1.24

75 80 85 90 95

0.276 0.555 1.019 1.740 2.802

6.369.−4 6.508.−4 6.657.−4 6.819.−4 6.997.−4

100 105 110 115 120

4.280 6.280 8.885 12.20 16.33

125 130 135 140 144.3c

21.37 27.48 34.72 43.47 52.15

*Values reproduced or converted from Prydz and Straty, NBS Tech. Note 392, rev., September 1973. t = triple point; c = critical point. The notation 5.866.−4 signifies 5.866 × 10−4. TABLE 2-260

Fluorine Gas at Atmospheric Pressure*

T, K

84.95

90

100

120

140

160

180

200

220

240

260

280

300

v, m3/kg h, kJ/kg s, kJ/(kg⋅K)

0.1776 63.22 4.342

0.1892 67.30 4.390

0.2118 75.27 4.474

0.2562 90.96 4.616

0.3002 106.53 4.737

0.3439 122.06 4.840

0.3874 137.62 4.932

0.4309 153.2 5.014

0.4744 169.0 5.090

0.5176 184.9 5.158

0.5610 201.0 5.221

0.6043 217.2 5.282

0.6476 233.7 5.340

*Extracted from Prydz and Straty, NBS Tech. Note 392, 1970. This source is recommended for other pressures and temperatures. Other information is contained in J. Chem. Phys., 53 (1970): 2359; and J. Res. NBS, 74A (1970): 499, 661, 747. TABLE 2-261

TABLE 2-262

Flutec

Proprietary name for a series of fluorocarbons produced by the Imperial Smelting Corp., Avonmouth, Bristol, UK. Bulletins of thermodynamic properties include PP1 (C6F14), PP2 (C7F14), PP3 (C8F16), PP5 (C10F18), PP9 (C11F20), and PP50, usually for 0.1–100 kg/m2, 0–500°C. See also Green, S. W., Chem. & Ind. (1969): 63–67.

TABLE 2-263

Halon

A series of fire-extinguishing fluids. Halon 1211 is produced by ICI, and Halon 1301, by duPont, the latter issuing a bulletin with thermodynamic properties and a diagram for the range 0.6–600 psia, −160–460°F.

Saturated Helium3*

T, K

P, bar

vf, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

1.0 1.1 1.2 1.3 1.4

0.0122 0.0182 0.0274 0.0370 0.0517

0.01222 0.01224 0.01227 0.01231 0.01236

1.72 1.33 1.02 0.805 0.649

−5.69 −5.49 −5.26 −5.01 −4.75

6.75 7.40 8.03 8.65 9.27

2.28 2.65 2.95 3.20 3.40

14.72 14.34 14.02 13.70 13.41

1.5 1.6 1.7 1.8 1.9

0.0659 0.0871 0.107 0.137 0.163

0.01241 0.01247 0.01254 0.01262 0.01271

0.526 0.437 0.363 0.308 0.260

−4.47 −4.17 −3.84 −3.47 −3.07

9.88 10.46 11.04 11.60 12.15

3.60 3.80 3.91 4.01 4.13

13.13 12.88 12.53 12.38 12.14

2.0 2.1 2.2 2.3 2.4

0.202 0.237 0.284 0.326 0.385

0.01282 0.01294 0.01308 0.01324 0.01343

0.222 0.189 0.164 0.142 0.124

−2.64 −2.17 −1.55 −0.99 −0.34

12.68 13.19 13.67 14.13 14.57

4.26 4.40 4.55 4.71 4.87

11.91 11.69 11.47 11.25 11.04

2.5 2.6 2.7 2.8 2.9

0.438 0.508 0.576 0.653 0.732

0.01365 0.01390 0.01419 0.01456 0.01497

0.109 0.096 0.085 0.074 0.064

0.36 1.16 2.01 2.96 4.01

14.98 15.37 15.89 16.40 16.37

5.03 5.20 5.38 5.57 5.77

10.84 10.64 10.41 10.17 9.92

3.0 3.1 3.2 3.3 3.32c

0.803 0.907 1.023 1.128 1.165

0.01549 0.01614 0.01720 0.01902 0.02394

0.055 0.047 0.039 0.028 0.024

5.28 6.70 8.44 10.66 13.25

16.32 16.20 15.98 14.50 13.25

6.00 6.24 6.54 6.96 7.50

9.66 9.34 8.90 8.35 7.50

*Converted and smoothed from a tabulation of Gibbons and Nathan, USAF Rep. AFML-TR-67-175, 1967. c = critical point. Kelly, D. P. and W. K. Haubach, in AEC R&D rept. MLM 1161, 1963 (56 pp.), give a comprehensive graphical comparison of the properties of He3 and He4.

2-240

PHYSICAL AND CHEMICAL DATA

TABLE 2-264

Saturated Helium4

Temperature, K

Pressure, bar

vf, m3/kg

0.8 0.9 1.0 1.1 1.2

1.475.−5 5.379.−5 1.557.−4 3.800.−4 8.148.−4

0.00689 0.00689 0.00689 0.00689 0.00689

1.3 1.4 1.5 1.6 1.7

0.00158 0.00282 0.00472 0.00746 0.01128

0.00689 0.00689 0.00689 0.00688 0.00688

1.8 1.9 2.0 2.1 2.2

0.01638 0.02299 0.03129 0.04141 0.05335

0.00688 0.00687 0.00686 0.00685 0.00684

2.3 2.4 2.5 2.6 2.7

0.06730 0.08354 0.01023 0.1237 0.1481

2.8 2.9 3.0 3.1 3.2

vg, m3/kg

hf, kJ/kg

h g, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf , kJ/(kg⋅K)

cpg, kJ/(kg⋅K)

0.0019 0.0054 0.0127 0.0268 0.0518

19.42 19.94 20.44 20.95 21.44

0.0047 0.0087 0.0163 0.0296 0.0510

23.94 21.86 20.44 18.82 17.67

0.022 0.050 0.100 0.185 0.318

5.210 5.230 5.262 5.305 5.360

0.0932 0.1579 0.2543 0.3923 0.5836

21.92 22.40 22.87 23.32 23.77

0.0836 0.1308 0.1962 0.2839 0.3981

16.70 15.86 15.13 14.49 13.93

0.511 0.780 1.138 1.602 2.193

5.424 5.496 5.574 5.654 5.736

2.1993 1.6420 1.2601 0.9921 0.7994

0.8422 1.186 1.642 2.261 3.090

24.20 24.63 25.04 25.45 25.85

0.5437 0.7270 0.9578 1.256 1.638

13.43 12.98 12.58 12.23 11.92

2.938 3.893 5.187 7.244 4.222

5.818 5.898 5.975 6.046 6.111

0.00685 0.00687 0.00690 0.00693 0.00695

0.6566 0.5470 0.4608 0.3923 0.3367

3.418 3.678 3.922 4.161 4.408

26.24 26.63 27.00 27.37 27.72

1.780 1.886 1.980 2.068 2.155

11.65 11.40 11.17 10.96 10.76

2.685 2.375 2.284 2.320 2.351

6.170 6.228 6.285 6.344 6.406

0.1755 0.2063 0.2405 0.2784 0.3201

0.00699 0.00703 0.00707 0.00713 0.00717

0.2913 0.2537 0.2223 0.1958 0.1728

4.662 4.923 5.195 5.483 5.787

28.06 28.38 28.69 28.98 29.26

2.240 2.324 2.408 2.494 2.581

10.57 10.39 10.22 10.05 9.90

2.403 2.486 2.597 2.740 2.896

6.470 6.540 6.616 6.700 6.792

3.3 3.4 3.5 3.6 3.7

0.3659 0.4159 0.4704 0.5296 0.5935

0.00723 0.00728 0.00735 0.00742 0.00749

0.1542 0.1376 0.1232 0.1107 0.0997

6.108 6.448 6.806 7.184 7.581

29.52 29.76 29.97 30.17 30.34

2.670 2.780 2.852 2.946 3.042

9.747 9.600 9.458 9.318 9.181

3.061 3.273 3.413 3.601 3.801

6.897 7.015 7.150 7.305 7.484

3.8 3.9 4.0 4.1 4.2

0.6625 0.7366 0.8162 0.9014 0.9923

0.00758 0.00766 0.00776 0.00786 0.00797

0.0900 0.0814 0.0738 0.0669 0.0606

7.998 8.437 8.899 9.387 9.901

30.48 30.60 30.68 30.73 30.74

3.140 3.239 3.341 3.444 3.551

9.046 8.911 8.776 8.641 8.504

4.017 4.254 4.519 4.820 5.170

7.694 7.942 8.238 8.641 9.033

4.3 4.4 4.5 4.6 4.7

1.089 1.193 1.303 1.419 1.543

0.00810 0.00824 0.00841 0.00860 0.00881

0.0550 0.0499 0.0452 0.0408 0.0367

10.45 11.02 11.64 12.31 13.04

30.71 30.62 30.47 30.24 29.91

3.661 3.775 3.893 4.018 4.151

8.363 8.218 8.067 7.906 7.732

5.587 6.097 6.742 7.590 8.763

4.8 4.9 5.0 5.1 5.195c

1.674 1.813 1.960 2.116 2.275

0.00907 0.00941 0.00986 0.01056 0.01436

0.0329 0.0291 0.0252 0.0207 0.0145

13.85 14.76 15.85 17.26

29.45 28.80 27.83 26.08

4.296 4.458 4.649 4.898

7.539 7.327 7.041 6.624

1125.9 347.1 133.0 59.8 30.4 16.93 10.17 6.49 4.35 3.04

c = critical pt. From Arp, V. D. and R. D. McCarty, N.I.S.T. TN 1334, 1989 (142 pp.).

10.51 13.38 19.02 34.60

9.58 10.29 11.22 12.50 14.37 17.32 22.64 34.93 95.84

THERMODYNAMIC PROPERTIES TABLE 2-265

Superheated Helium*

P, bars

0

2-241

Temp., °C 1 v h s

5.677 0.327 0.0116

100

200

300

400

500

600

800

1000

7.754 519.6 1.620

9.831 1039 2.853

11.908 1558 3.849

13.985 2078 4.684

16.063 2597 5.403

18.140 3116 6.035

22.294 4155 7.106

26.448 5193 7.993

5 v h s

1.138 1.636 −3.343

1.553 520.9 −1.723

1.968 1040 −0.490

2.384 1560 0.506

2.799 2079 1.341

3.215 2598 2.060

3.630 3117 2.692

4.461 4156 3.763

5.291 5194 4.650

10 v h s

0.5704 3.272 −4.782

0.780 522.5 −3.162

0.986 1042 −1.929

1.193 1561 −0.934

1.401 2080 −0.098

1.609 2600 0.621

1.816 3119 1.252

2.232 4157 2.323

2.647 5196 3.211

20 v h s

0.2867 6.544 −6.221

0.3904 525.8 −4.601

0.4942 1045 −3.368

0.5979 1564 −2.373

0.7017 2083 −1.537

0.9093 2603 −0.818

1.1169 3122 −0.187

1.3245 4160 0.884

1.8435 5199 1.771

50 v h s

0.1164 16.360 −8.121

0.1579 535.5 −6.501

0.1993 1055 −5.268

0.2408 1574 −4.273

0.2822 2093 −3.438

0.3257 2612 −2.719

0.3652 3131 −2.088

0.4481 4169 −1.017

0.5311 5207 −0.130

100 v h s

0.0597 37.720 −9.555

0.0803 551.7 −7.936

0.1010 1071 −6.703

0.1217 1590 −5.709

0.1424 2108 −4.874

0.1631 2627 −4.155

0.1838 3146 −3.524

0.2252 4184 −2.454

0.2666 5222 −1.567

150 v h s

0.0407 49.080 −10.391

0.0545 567.9 −8.773

0.0682 1087 −7.541

0.0820 1605 −6.546

0.0958 2124 −5.712

0.1095 2643 −4.994

0.1233 3161 −4.363

0.1509 4199 −3.293

0.1785 5236 −2.407

200 v h s

0.0312 65.440 −10.983

0.0416 584.1 −9.635

0.0518 1103 −8.134

0.0622 1621 −7.139

0.0725 2140 −6.306

0.0828 2658 −5.588

0.0931 3176 −4.957

0.1137 4213 −3.888

0.1344 5250 −3.002

*Extracted from Tsederberg, Popov, et al., Thermodynamic and Thermophysical Properties of Helium, Atomizdat, Moscow, 1969, and NBS-NSF TT 50096, 1971. Copyright material. Reproduced by permission. This source contains entries for many more temperatures and pressures than can be reproduced here. v = volume, m3/kg; h = enthalpy, kJ/ kg; s = entropy, kJ/(kg·K). The 1993 ASHRAE Handbook—Fundamentals (SI ed.) has a thermodynamic chart for pressures from 0.1 to 50 bar and temperatures from 2.5 to 15 K. Saturation and superheat tables to 9000 psia, 800°R; and a chart to 700 psia, 40°R appear in Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat, thermal conductivity, and viscosity, see Thermophysical Properties of Refrigerants, ASHRAE, 1993. A useful compilation of properties is given by Betts, D. S., Cryogenics, 16, 1 (1976): 3–16. A 32-term equation of state for the range up to 20,000 bar, 2–1500 K is given by McCarty, R. D. and V. D. Arp, Advan. Cryog. Engng., 35 (1990): 1465–1475.

TABLE 2-266

Helium4 Gas at Atmospheric Pressure*

T, K

4.224

5

10

20

30

40

50

75

100

200

300

400

500

600

800

1000

v, m3/kg h, kJ/kg s, kJ/(kg⋅K)

0.0591 30.30 8.327

0.0834 36.18 9.614

0.1612 64.91 13.369

0.4094 117.95 17.321

0.6161 170.24 19.442

0.8218 222.4 20.94

1.0273 274.4 22.10

1.5403 404.4 24.21

2.053 534.2 25.71

4.102 1054 29.30

6.154 1573 31.41

8.191 2092 32.90

10.24 2612 34.06

12.31 3131 35.01

16.40 4170 36.50

20.50 5208 37.66

*From McCarty, NBS Rep. 9762, 1970. Reproduced by permission. The source contains values for further temperatures and for other functions, usually to additional significant figures.

2-242

PHYSICAL AND CHEMICAL DATA

TABLE 2-267

Saturated n-Heptane*

P, bar

vf, m3/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µ f, 10−4 Pa⋅s

k f, W/(m⋅K)

182.6t 200 220 240 250

0.00002 0.00019 0.00133 0.00303

1.292.−3 1.316.−3 1.344.−3 1.374.−3 1.389.−3

284.1 319.4 359.7 400.5 421.3

722.6 757.1 791.4 808.3

2.260 2.441 2.636 2.814 2.899

4.457 4.442 4.443 4.447

2.025 2.011 2.026 2.063 2.088

39.4 21.0 12.6 8.52 7.23

0.150 0.148 0.145 0.142 0.140

260 270 280 290 300

0.00635 0.01316 0.02347 0.03997 0.06674

1.405.−3 1.422.−3 1.440.−3 1.457.−3 1.475.−3

3.744

442.3 463.6 485.2 507.2 529.6

824.9 841.2 857.8 874.8 891.9

2.981 3.061 3.140 3.217 3.293

4.453 4.460 4.471 4.485 4.501

2.117 2.147 2.180 2.216 2.252

6.52 5.46 4.83 4.29 3.85

0.137 0.135 0.132 0.129 0.126

310 320 330 340 350

0.1070 0.1656 0.2461 0.3614 0.5130

1.494.−3 1.514.−3 1.534.−3 1.555.−3 1.578.−3

2.412 1.596 1.101 0.7650 0.5510

552.3 575.4 598.8 622.8 647.0

908.9 926.0 943.3 961.2 979.1

3.367 3.441 3.513 3.584 3.655

4.517 4.537 4.557 4.579 4.604

2.291 2.329 2.370 2.412 2.454

3.48 3.17 2.89 2.66 2.45

0.123 0.121 0.119 0.116 0.114

360 370 371.6 380 390

0.712 0.967 1.013 1.289 1.689

1.601.−3 1.625.−3 1.629.−3 1.651.−3 1.678.−3

0.4058 0.3036 0.2904 0.2308 0.1781

671.9 697.1 701.9 723.9 750.4

997.5 1016.1 1019.8 1035.4 1054.2

3.725 3.794 3.805 3.864 3.932

4.629 4.656 4.660 4.684 4.711

2.500 2.548 2.556 2.60 2.65

2.24 2.04 2.01 1.86 1.71

0.111 0.109 0.108 0.107 0.105

400 420 440 460 480

2.180 3.471 5.268 7.691 10.92

1.708.−3 1.775.−3 1.853.−3 1.954.−3 2.065.−3

0.1388 0.0734 0.0576 0.0389 0.0265

777.2

1073.2

4.000

4.740

2.70 2.81 2.93 3.05 3.19

1.58 1.35 1.15 0.97 0.82

0.103 0.099 0.095 0.091 0.087

500 520 540.1c

15.10 20.43 27.35

2.235.−3 2.52.−3 4.3.−3

3.38 3.7

0.67

0.080

0.0043

T, K

vg, m3/kg

h f, kJ/kg

0.0178

*Values of P and v interpolated and converted from tables in Vargaftik, Handbook of Thermophysical Properties of Gases and Liquids, Hemisphere, Washington, and McGraw-Hill, New York, 1975. Values of h and s calculated from API tables published by the Thermodynamics Research Center, Texas A&M University, College Station. t = triple point; c = critical point. Saturation and superheat tables and a diagram to 200 bar, 680 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (173 pp.).

TABLE 2-268

Hexane

Saturation and superheat tables and a diagram to 100 bar, 680 K, are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (173 pp.).

THERMODYNAMIC PROPERTIES TABLE 2-269

2-243

Saturated Hydrazine

Temperature, K

vf, m3/kg

vg, m3/kg

h f, kJ/kg

hg, kJ/kg

s f, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

1.013 1.135 1.560 2.102 2.786

0.001 053 0.001 060 0.001 081 0.001 104 0.001 127

0.9833 0.8850 0.6579 0.4994 0.3850

−105.9 −104.8 −101.4 −97.6 −93.9

65.4 66.0 68.2 70.6 73.0

0.5994 0.6029 0.6120 0.6211 0.6300

1.0426 1.0409 1.0360 1.0314 1.0275

4.732 7.610 11.76 17.42 29.59

0.001 178 0.001 235 0.001 299 0.001 374 0.001 460

0.2355 0.1500 0.1005 0.0690 0.0407

−86.1 −76.9 −67.1 −57.3 −47.8

77.6 82.1 86.6 90.8 94.6

0.6492 0.6707 0.6916 0.7124 0.7320

1.0212 1.0163 1.0118 1.0086 1.0058

Pressure, bar

386.6 390 400 410 420 440 460 480 500 520 540 560 580 600 620

34.75 47.09 62.44 81.17 102.7

0.001 563 0.001 681 0.001 835 0.002 045 0.002 320

0.0353 0.0263 0.0196 0.0142 0.0106

−36.0 −25.2 −12.4 5.2 23.2

97.7 101.2 103.6 104.2 103.6

0.7566 0.7762 0.8002 0.8335 0.8671

1.0042 1.0020 1.0002 0.9988 0.9967

640 653 c

128.1 146.9

0.002 86 0.004 33

0.0074 0.0043

45.9 83.7

98.1 83.7

0.9035 0.9715

0.9906 0.9715

Converted from E. F. Fricke, Republic Aviation Co. rept. F-5028-101. c = critical point.

TABLE 2-270

Saturated n-Hydrogen*

T, K

P, bar

vf, m3/kg

v g, m3/kg

hf, kJ/kg

h g, kJ/kg

s f, kJ/(kg⋅K)

s g, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µ f, 10−4 Pa⋅s

k f, W/(m⋅K)

13.95t 14 15 16 17

0.072 0.074 0.127 0.204 0.314

0.01298 0.01301 0.01316 0.01332 0.01348

7.974 7.205 4.488 2.954 2.032

218.3 219.6 226.4 233.8 241.6

667.4 669.3 678.2 686.7 694.7

14.079 14.173 14.640 15.104 15.568

46.635 46.301 44.763 43.418 42.227

6.36 6.47 6.91 7.36 7.88

0.255 0.248 0.218 0.194 0.175

0.073 0.075 0.083 0.089 0.093

18 19 20 21 22

0.461 0.654 0.901 1.208 1.585

0.01366 0.01387 0.01407 0.01430 0.01455

1.449 1.064 0.8017 0.6177 0.4828

249.9 258.8 268.3 278.4 289.2

702.1 708.8 714.8 720.2 724.4

16.032 16.498 16.966 17.440 17.919

41.158 40.188 39.299 38.485 37.710

8.42 8.93 9.45 10.13 10.82

0.159 0.146 0.135 0.125 0.116

0.095 0.097 0.098 0.100 0.101

23 24 25 26 27

2.039 2.579 3.213 3.950 4.800

0.01483 0.01515 0.01551 0.01592 0.01639

0.3829 0.3072 0.2489 0.2032 0.1667

300.8 313.3 326.7 341.2 357.0

727.6 729.8 730.7 730.2 728.0

18.405 18.901 19.408 19.929 20.473

36.973 36.266 35.579 34.900 34.221

11.69 12.52 13.44 14.80 16.17

0.108 0.101 0.094 0.088 0.082

0.101 0.101 0.100 0.098 0.096

28 29 30 31 32

5.770 6.872 8.116 9.510 11.07

0.01696 0.01765 0.01854 0.01977 0.02174

0.1370 0.1125 0.0919 0.0738 0.0571

374.3 393.6 415.4 441.3 474.7

723.7 716.6 705.9 689.7 663.2

21.041 21.650 22.315 23.075 24.032

33.524 32.795 32.002 31.091 29.926

18.48 22.05 26.59 36.55 65.37

0.076 0.070 0.065 0.058 0.051

0.094 0.091 0.087 0.086 0.092

33.18c

13.13

0.03182

0.0318

565.4

565.4

26.680

26.680

∞

*Values extracted and occasionally rounded off from McCarty, Hord, and Roder, NBS Monogr. 168, 1981. t = triple point; c = critical point.

∞

2-244

PHYSICAL AND CHEMICAL DATA

TABLE 2-271

Compressed n-Hydrogen* Temperature, K

Pressure, bar

15

20

30

40

50

60

80

100

150

200

6.076 679.2 46.02

8.176 731.6 49.04

12.333 835.5 53.25

16.473 938.9 56.23

20.606 1042.3 58.53

24.736 1146 60.43

32.991 1356 63.45

41.244 1575 65.89

61.870 2172 70.68

82.495 2826 74.46

1 v h s

0.0131 227.3 14.62

0.0141 268.3 16.96

1.196 826.0 43.56

1.625 932.7 46.63

2.046 1037.9 48.98

2.463 1143 50.89

3.295 1354 53.93

4.123 1574 56.38

6.190 2172 61.17

8.254 2826 64.96

5 v h s

0.0131 231.7 14.57

0.0140 272.1 16.88

0.2006 775.0 35.80

0.3039 903.4 39.52

0.3958 1017.6 42.07

0.4839 1128 44.07

0.6553 1345 47.20

0.8238 1568 49.68

1.241 2170 54.66

1.655 2826 58.31

10 v h s

0.0130 237.2 14.50

0.0138 277.0 16.77

0.0181 412.1 22.09

0.1376 861.8 35.95

0.1895 991.1 38.85

0.2366 1109 40.99

0.3255 1334 44.23

0.4116 1560 46.75

0.6221 2167 51.63

0.8303 2826 55.44

20 v h s

0.0129 248.2 14.37

0.0136 286.9 16.58

0.0167 406.5 21.33

0.0521 752.0 31.07

0.0866 934.7 35.19

0.1135 1070 37.67

0.1611 1312 41.15

0.2057 1546 43.76

0.3129 2163 48.71

0.4179 2826 52.55

40 v h s

0.0133 307.3 16.26

0.0155 413.5 20.50

0.0216 589.3 25.49

0.0376 823.5 30.73

0.0533 997 33.91

0.0796 1271 37.87

0.1033 1521 40.65

0.1586 2155 45.75

0.2119 2826 49.64

60 v h s

0.0130 328.0 15.98

0.0147 427.2 19.95

0.0182 570.1 24.03

0.0254 757.0 28.19

0.0351 940 31.54

0.0532 1237 35.82

0.0697 1499 38.76

0.1073 2149 43.99

0.1433 2828 47.92

80 v h s

0.0127 348.9 15.74

0.0142 443.5 19.53

0.0167 572.3 23.21

0.0211 732.8 26.78

0.0273 905 29.93

0.0406 1210 34.34

0.0531 1482 37.37

0.0818 2146 42.72

0.1090 2831 46.69

100 v h s

0.0125 369.8 15.53

0.0138 461.1 19.19

0.0158 581.5 22.63

0.0190 727.4 25.88

0.0233 888 28.80

0.0335 1192 33.19

0.0434 1469 36.28

0.0666 2144 41.73

0.0885 2835 45.73

200 v h s

0.0117 474.4 14.71

0.0125 556.1 17.99

0.0136 658.7 20.93

0.0150 776.9 23.56

0.0167 908 25.94

0.0207 1182 29.88

0.0253 1458 32.97

0.0368 2156 38.59

0.0480 2869 42.72

400 v h s

0.0113 751.0 16.59

0.0119 841.9 19.20

0.0126 945.4 21.50

0.0134 1059 23.58

0.0151 1303 27.07

0.0171 1560 29.94

0.0225 2249 35.48

0.0279 2973 39.67

600 v h s

0.0106 941.5 15.68

0.0110 1027 18.14

0.0115 1124 20.29

0.0120 1231 22.24

0.0131 1463 25.57

0.0144 1709 28.31

0.0178 2385 33.74

0.0214 3107 37.92

800 v h s

0.0104 1209 17.35

0.0107 1302 19.43

0.0111 1405 21.30

0.0120 1628 24.50

0.0130 1870 27.20

0.0155 2535 32.54

0.0181 3255 36.70

1000 v h s

0.0099 1387 16.72

0.0102 1478 18.75

0.0106 1578 20.58

0.0112 1796 23.70

0.0120 2032 26.33

0.0140 2692 31.63

0.0160 3403 35.75

0.1 v h s

*Values extracted and sometimes rounded off from the tables of McCarty, Hord, and Roder, NBS Monogr. 168, 1981. This source contains an exhaustive tabulation of property values for both the normal and the para forms of hydrogen. v = specific volume, m3/kg; h = specific enthalpy, kJ/kg; s = specific entropy, kJ/(kg⋅K). The 1993 ASHRAE Handbook—Fundamentals (SI ed.) has a thermodynamic diagram for 0.1 to 500 bar for temperatures up to 100 K. Tables and a Mollier chart from 10−4 to 1000 atm, 300–20,000 K are given by Kubin, R. F. and L. L. Presley, NASA SP 3002, 1964. Liebenberg, D. H., R. L. Mills, and others, in LA-6645-MS, 1977 (26 pp.), give properties from 75 to 307 K for pressures from 2 to 20 kbar. See also Baker, J. R. and H. F. Swift, J. Appl. Phys., 43, 3 (1972): 950–953. An extensive collection of data for H2, D2, T2, and so on below 30 K is given by Souers, P. C., UCRL 52628, 1979 (91 pp.); and for temperatures below 40 K by Roder, H. M., G. E. Childs, et al., NBS TN 641, 1973 (114 pp.). Saturation and superheat tables to 10,000 psia, 900°R and a chart to 180°R appear in Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For viscosity, thermal conductivity, and specific heat, see Thermophysical Properties of Refrigerants, ASHRAE, 1993.

THERMODYNAMIC PROPERTIES

2-245

Temperature, K 250

300

350

400

450

500

600

700

800

900

1000

103.12 3517 77.53

123.23 4227 80.13

144.35 4945 82.34

164.97 5668 84.27

185.60 6393 85.98

206.22 7118 87.51

247.46 8571 90.15

288.70 10028 92.40

329.94 11493 94.36

371.18 12969 96.10

412.43 14458 97.66

10.32 3517 68.03

12.38 4227 70.63

14.44 4946 72.85

16.50 5669 74.78

18.57 6393 76.48

20.63 7118 78.01

24.75 8571 80.66

28.88 10029 82.91

33.00 11494 84.86

37.13 12969 86.60

41.25 14459 88.17

2.069 3518 61.39

2.482 4229 63.99

2.895 4948 66.21

3.307 5671 68.14

3.720 6396 69.84

4.132 7121 71.37

4.957 8574 74.02

5.782 10032 76.27

6.607 11497 78.23

7.432 12973 79.96

8.257 14462 81.53

1.038 3519 58.52

1.245 4231 61.12

1.451 4951 63.34

1.658 5674 65.28

1.864 6399 66.98

2.070 7125 68.51

2.483 8578 71.16

2.896 10036 73.41

3.308 11501 75.37

3.720 12977 77.10

4.133 14467 78.67

0.522 3522 55.65

0.6259 4235 58.26

0.7294 4956 60.48

0.8328 5680 62.41

0.9361 6406 64.12

1.040 7132 65.65

1.246 8586 68.30

1.452 10044 70.55

1.658 11509 72.51

1.865 12985 74.24

2.071 14475 75.81

0.2644 3527 52.76

0.3166 4244 55.38

0.3685 4967 57.61

0.4204 5692 59.55

0.4721 6419 61.26

0.5238 7146 62.79

0.6271 8601 65.44

0.7303 10059 67.69

0.8335 11525 69.65

0.9366 13002 71.39

1.040 14492 72.95

0.1786 3533 51.05

0.2136 4253 53.69

0.2483 4978 55.92

0.2829 5705 57.86

0.3174 6432 59.58

0.3519 7160 61.11

0.4209 8616 63.76

0.4897 10075 66.02

0.5585 11542 67.97

0.6273 13018 70.51

0.6961 14508 71.28

0.1357 3540 49.84

0.1621 4263 52.49

0.1882 4989 54.73

0.2142 5718 56.67

0.2401 6446 58.39

0.2660 7174 59.92

0.3177 8631 62.57

0.3694 10091 64.83

0.4120 11558 66.79

0.4726 13035 68.52

0.5242 14525 70.09

0.1099 3547 48.89

0.1312 4273 51.55

0.1521 5001 53.79

0.1730 5731 55.74

0.1937 6460 57.46

0.2145 7189 59.00

0.2559 8647 61.65

0.2972 10107 63.90

0.3385 11574 65.87

0.3798 13051 67.60

0.4211 14542 69.17

0.0588 3594 45.94

0.0695 4329 48.62

0.0801 5064 50.89

0.0905 5798 52.85

0.1001 6531 54.58

0.1114 7263 56.12

0.1321 8724 58.78

0.1528 10187 61.04

0.1734 11656 63.00

0.1941 13134 64.74

0.2147 14625 66.31

0.0334 3716 42.98

0.0388 4458 45.68

0.0441 5202 47.97

0.0493 5943 49.95

0.0545 6681 51.69

0.0597 7416 53.24

0.0701 8883 55.91

0.0804 10349 58.17

0.0908 11820 60.14

0.1011 13300 61.88

0.1114 14792 63.45

0.0249 3854 41.24

0.0285 4600 43.95

0.0321 5349 46.26

0.0355 6095 48.26

0.0390 6836 50.00

0.0425 7574 51.56

0.0494 9045 54.24

0.0562 10513 56.50

0.0631 11985 58.47

0.0700 13466 60.21

0.0768 14958 61.78

0.0207 4003 40.03

0.0234 4748 42.73

0.0260 5501 45.05

0.0286 6249 47.05

0.0312 6993 48.81

0.0338 7734 50.37

0.0390 9207 53.05

0.0441 10677 55.32

0.0492 12150 57.29

0.0543 13631 59.03

0.0594 15124 60.60

0.0181 4156 39.10

0.0202 4898 41.79

0.0223 5654 44.12

0.0244 6405 46.12

0.0265 7151 47.88

0.0286 7893 49.45

0.0327 9370 52.14

0.0367 10842 54.41

0.0408 12316 56.38

0.0449 13797 58.12

0.0490 15289 59.69

2-246

PHYSICAL AND CHEMICAL DATA

TABLE 2-272

Saturated para-Hydrogen*

T, K

P, bar

vf, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf , kJ/(kg⋅K)

µf, 10−4 Pa⋅s

kf, W/(m⋅K)

13.8t 14 15 16 17

0.070 0.079 0.134 0.216 0.329

0.0130 0.0130 0.0132 0.0133 0.0135

7.97 7.20 4.49 2.96 2.03

−308.9 −307.6 −300.9 −293.4 −285.6

140.3 142.1 151.1 159.6 167.6

4.97 5.06 5.53 5.99 6.45

37.52 37.19 36.65 34.31 33.11

6.37 6.47 6.91 7.36 7.88

0.255 0.248 0.218 0.194 0.175

0.073 0.075 0.082 0.089 0.092

18 19 20 21 22

0.482 0.682 0.935 1.250 1.634

0.0137 0.0139 0.0141 0.0143 0.0146

1.45 1.07 0.802 0.618 0.483

−277.3 −268.4 −258.9 −248.8 −237.9

175.0 181.7 187.7 193.0 197.3

6.92 7.38 7.85 8.32 8.80

32.05 31.08 30.19 29.37 28.60

8.42 8.93 9.45 10.13 10.82

0.159 0.146 0.135 0.125 0.116

0.095 0.097 0.098 0.100 0.101

23 24 25 26 27

2.096 2.645 3.288 4.035 4.892

0.0148 0.0152 0.0155 0.0159 0.0164

0.383 0.307 0.249 0.203 0.167

−226.3 −213.9 −200.4 −185.9 −170.2

200.5 202.7 203.6 203.1 200.9

9.29 9.78 10.29 10.81 11.36

27.86 27.15 26.46 25.79 25.11

11.69 12.52 13.44 14.81 16.18

0.108 0.101 0.094 0.088 0.082

0.101 0.100 0.099 0.098 0.096

0.076 0.070 0.065 0.058 0.051

0.094 0.091 0.087 0.088 0.092

28 29 30 31 32

5.88 6.98 8.23 9.63 11.20

0.0170 0.0177 0.0185 0.0198 0.0217

0.137 0.113 0.092 0.074 0.057

−152.9 −133.6 −111.7 −85.8 −52.4

196.5 189.5 178.8 162.6 136.1

11.93 12.54 13.20 13.96 14.92

24.41 23.68 22.89 21.98 20.81

18.5 22.1 26.6 36.6 65.4

33c

12.93

0.0318

0.032

38.3

38.3

17.56

17.56

∞

∞

*Values extracted and occasionally rounded off from McCarty, Hord, and Roder, NBS Monogr. 168, 1981. t = triple point; c = critical point. Saturation and superheat tables to 12,000 psia, 900°R and a chart to 180°R appear in Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat, thermal conductivity, and viscosity, see Thermophysical Properties of Refrigerants, ASHRAE, 1993. The 1993 ASHRAE Handbook—Fundamentals (SI ed.) has a thermodynamic chart for pressures from 0.1 to 1000 bar for temperatures up to 100 K.

TABLE 2-273 T, K 273 300 350 400 450

Saturated Hydrogen Peroxide*

P, bar

vf, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µf, 10−4 Pa⋅s

kf, W/(m⋅K)

0.0004 0.0031 0.0564 0.4521 2.143

0.00068 0.00069 0.00072 0.00076 0.00081

1672 235 15.1 2.12 0.487

−5577 −5510 −5376 −5238 −5091

−4027 −3995 −3933 −3878 −3820

2.990 3.224 3.631 4.032 4.346

8.662 8.269 7.758 7.440 7.172

1.45 1.48 1.54 1.61 1.68

18.0 11.3 4.3 2.2 1.3

0.483 0.481 0.474 0.464 0.453

1.75 1.82 1.90

500 550 600 650 700

7.126 18.56 40.75 79.27 141.7

0.00088 0.00095 0.00107 0.00125 0.00171

0.155 0.0605 0.0268 0.0125 0.0048

−4945 −4794 −4635 −4463 −4195

−3777 −3745 −3731 −3746 −3860

4.656 4.941 5.209 5.485 5.682

6.992 6.846 6.720 6.582 6.339

708.5c

155.3

0.00284

0.0028

−4012

−4012

5.732

5.732

0.89 0.65 0.50

0.443 0.431 0.416

*Values reproduced or converted from a tabulation by Tsykalo and Tabachnikov in V. A. Rabinovich (ed.), Thermophysical Properties of Gases and Liquids, Standartov, Moscow, 1968; NBS-NSF transl. TT 69-55091, 1970. The reader may be reminded that very pure hydrogen peroxide is very difficult to obtain owing to its decomposition or instability. c = critical point. The FMC Corp., Philadelphia, PA tech. bull. 67, 1969 (100 pp.) contains an enthalpy-pressure diagram to 3000 psia, 1100 K.

TABLE 2-274

Hydrogen Sulfide

West, J. R., Chem. Eng. Progr., 44, 4 (1948): 207–292 gives tables and a chart for the range 1–90 atm., −76 to 1300°F while properties from 10 to 330 bar, 300 to 500 K were tabulated by Lui, C-H., D. M. Bailey, et al., Hydroc. Proc., 65, 7 ( July 1986): 41–43.

THERMODYNAMIC PROPERTIES

Enthalpy-concentration diagram for aqueous hydrogen chloride at 1 atm. Reference states: enthalpy of liquid water at 0°C is zero; enthalpy of pure saturated HCl vapor at 1 atm (−85.03°C) is 8000 kcal/mol. NOTE: It should be observed that the weight basis includes the vapor, which is particularly important in the two-phase region. Saturation values may be read at the ends of the tie lines. [Van Nuys, Trans. Am. Inst. Chem. Eng., 39, 663 (1943).]

FIG. 2-11

2-247

2-248

PHYSICAL AND CHEMICAL DATA

TABLE 2-275

Saturated Isobutane (R600a)* µf , 10−4 Pa⋅s

kf, W/(m⋅K)

1.78 1.87 1.93 1.99

9.46

0.163 0.158 0.149

4.984 4.916 4.878 4.863 4.861

2.05 2.12 2.19 2.28 2.33

6.06 4.21 3.11 2.40 2.14

0.142 0.134 0.127 0.120 0.117

3.617 3.700 3.783 3.865 3.946

4.863 4.867 4.874 4.882 4.891

2.39 2.46 2.53 2.61 2.70

1.93 1.75 1.59 1.46 1.35

0.113 0.110 0.106 0.102 0.099

T, K

P, bar

vf, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

113.6t 120 140 160 180

1.9.−7 9.3.−7 4.8.−5 8.2.−4 0.0070

1.349.−3 1.360.−3 1.396.−3 1.435.−3 1.476.−3

8.60.+6 1.84.+6 4210 278.2 36.66

0.0 11.0 46.0 82.1 119.5

485.3 491.1 510.1 530.8 533.0

1.863 1.957 2.226 2.467 2.688

6.136 5.957 5.541 5.272 5.097

200 220 240 260 270

0.0369 0.1374 0.3989 0.9600 1.4081

1.520.−3 1.568.−3 1.621.−3 1.680.−3 1.712.−3

7.723 2.265 0.8432 0.3738 0.2617

158.5 199.0 241.4 285.8 308.8

576.7 601.5 627.4 654.2 667.7

2.893 3.086 3.270 3.446 3.532

280 290 300 310 320

2.0020 2.7686 3.7365 4.934 6.392

1.746.−3 1.784.−3 1.824.−3 1.868.−3 1.916.−3

0.1882 0.1385 0.1040 0.0794 0.0614

332.3 356.4 381.1 406.4 432.4

681.3 694.9 708.4 721.7 734.8

330 340 350 360 370

8.140 10.21 12.64 15.46 18.72

1.971.−3 2.032.−3 2.103.−3 2.187.−3 2.289.−3

0.0481 0.0380 0.0301 0.0240 0.0190

459.2 486.9 515.7 545.6 577.1

747.7 760.0 771.8 782.7 792.3

4.028 4.109 4.191 4.273 4.357

4.902 4.912 4.923 4.932 4.939

2.81 2.92 3.04 3.17 3.31

1.25 1.15 1.05 0.95 0.85

0.095 0.092 0.088 0.083 0.080

380 390 400 408.0c

22.48 26.82 31.86 36.55

2.420.−3 2.604.−3 2.920.−3 4.464.−3

0.0150 0.0115 0.0083 0.0045

610.6 647.1 689.6 752.5

799.8 803.7 799.6 752.5

4.444 4.536 4.639 4.791

4.942 4.937 4.915 4.791

3.45 3.62 3.85 ∞

0.75 0.63 0.51

0.076 0.071 0.065 ∞

*Values reproduced or converted from Goodwin, NBSIR 79-1612, 1979. t = triple point; c = critical point. The notation 1.9.−7 signifies 1.9 × 10−7. Slightly different values for the range 0.5 to 34.5 bar, 250–404 K appear in Waxman, M. and J. S. Gallagher, J. Chem. Eng. Data, 28, (1983): 224–241. This source also contains superheat tables for 1–400 bar, 250–600 K. Saturation and superheat tables and a diagram to 200 bar, 600 K are given by Reynolds, W. C., Thermodynamic properties in S.I., Stanford Univ. publ., 1979 (173 pp.). Saturation and superheat tables and a chart to 10,000 psia, 640°F appear in Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat, thermal conductivity, and viscosity, see Thermophysical Properties of Refrigerants, ASHRAE, 1993. Equations and data for thermal conductivity and viscosity are given by Nieuwoldt, J. C., B. LeNeindre, et al., J. Chem. Eng. Data, 32, (1987): 1–8.

TABLE 2-276 T, K

Saturated Krypton*

P, bar

vf, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µf, 10−4 Pa⋅s

kf, W/(m⋅K)

10 20 30 40 50

3.235.−4 3.246.−4 3.265.−4 3.288.−4 3.313.−4

0.22 1.59 3.84 6.49 9.37

0.0256 0.1141 0.2034 0.2791 0.3431

0.070 0.188 0.247 0.276 0.295

60 70 80 90 100

3.341.−4 3.372.−4 3.407.−4 3.446.−4 3.492.−4

12.40 15.57 18.97 22.58 26.42

0.3982 0.4471 0.4925 0.5353 0.5765

0.311 0.327 0.345 0.366 0.389

110 115.76 115.76 119.76 120

0.732 1.013 1.032

3.544.−4 3.579.−4 4.090.−4 4.143.−4 4.146.−4

0.1529 0.1136 0.1116

30.52 33.18 52.78 54.99 55.09

161.8 162.6 162.6

0.6165 0.6390 0.8095 0.8279 0.8291

1.751 1.726 1.724

0.414 0.427 0.547 0.545 0.544

3.72

0.0900

130 140 150 160 170

2.112 3.878 6.552 10.37 15.57

4.284.−4 4.440.−4 4.619.−4 4.831.−4 5.091.−4

0.0578 0.0330 0.0201 0.0130 0.0086

60.55 66.02 71.58 77.34 83.48

164.1 165.3 166.1 166.4 166.0

0.8724 0.9124 0.9499 0.9859 1.022

1.669 1.622 1.580 1.543 1.507

0.542 0.546 0.559 0.587 0.641

3.16 2.64 2.20 1.87 1.54

0.0828 0.0756 0.0688 0.0625 0.0558

180 190 200 209.39

22.41 31.20 42.23 54.96

5.423.−4 5.882.−4 6.641.−4 1.098.−3

0.0059 0.0040 0.0026 0.0011

90.26 98.19 108.40 133.90

164.6 161.8 156.0 133.9

1.058 1.098 1.147 1.262

1.472 1.433 1.386 1.262

0.734 0.905 1.515 ∞

1.28 1.05 0.80

0.0494 0.0433 0.0348 ∞

*Values extracted and in some cases rounded off from those cited in Rabinovich (ed.), Thermophysical Properties of Neon, Argon, Krypton and Xenon, Standards Press, Moscow, 1976. This source contains values for the compressed state for pressures up to 1000 bar, etc. The notation 3.235.−4 signifies 3.235 × 10−4. This book was published in English translation by Hemisphere, New York, 1988 (604 pp.).

THERMODYNAMIC PROPERTIES TABLE 2-277 Temperature, K

2-249

Compressed Krypton* Pressure, bar 1

10

20

40

60

80

100

200

400

600

800

1000

100

v h s

3.49.−4 26.42 0.5765

3.49.−4 26.69 0.5760

3.49.−4 26.99 0.5755

3.48.−4 27.59 0.5745

3.48.−4 28.18 0.5735

3.47.−4 28.78 0.5724

3.47.−4 29.38 0.5714

3.45.−4 32.38 0.5667

3.42.−4 38.42 0.5580

3.39.−4 44.47 0.5503

3.36.−4 50.52 0.5432

3.33.−4 56.57 0.5366

200

v h s

0.1971 183.1 1.859

0.0184 179.3 1.618

8.39.−3 174.5 1.533

3.00.−3 159.4 1.405

6.19.−4 105.6 1.129

5.94.−4 104.4 1.116

5.76.−4 103.6 1.106

5.27.−4 102.7 1.073

4.83.−4 105.3 1.037

4.58.−4 109.7 1.013

4.41.−4 114.9 0.993

4.28.−4 120.3 0.977

v 300 h s

0.2971 208.1 1.961

0.0292 206.3 1.728

0.0143 204.2 1.654

6.84.−3 200.0 1.575

4.37.−3 195.7 1.525

3.14.−3 191.2 1.485

2.41.−3 186.6 1.451

1.09.−3 166.8 1.333

6.92.−4 155.1 1.239

5.94.−4 155.2 1.196

5.44.−4 158.2 1.169

5.13.−4 162.3 1.149

400

v h s

0.3966 233.0 2.032

0.0394 231.9 1.802

0.0196 230.7 1.730

9.67.−3 228.3 1.657

6.37.−3 225.9 1.612

4.73.−3 223.6 1.579

3.75.−3 221.3 1.552

1.85.−3 211.4 1.463

1.01.−3 199.7 1.368

7.79.−4 196.8 1.317

6.76.−4 197.9 1.284

6.14.−4 200.8 1.259

500

v h s

0.4960 257.8 2.088

0.0495 257.1 1.858

0.0247 256.3 1.788

0.0123 254.9 1.716

8.20.−3 253.3 1.673

6.15.−3 251.9 1.642

4.91.−3 250.5 1.617

2.49.−3 244.5 1.537

1.33.−3 236.9 1.451

9.81.−4 234.2 1.400

8.22.−4 234.7 1.365

7.29.−4 237.4 1.340

600

v h s

0.5953 282.7 2.133

0.0596 282.2 1.904

0.0298 281.7 1.834

0.0149 280.7 1.763

9.96.−3 279.7 1.721

7.49.−3 278.8 1.691

6.01.−3 277.9 1.667

3.07.−3 274.2 1.591

1.64.−3 269.6 1.511

1.18.−3 268.1 1.462

9.67.−4 269.1 1.428

8.44.−4 271.7 1.403

700

v h s

0.6946 307.5 2.171

0.0696 307.2 1.942

0.0348 306.9 1.873

0.0175 306.2 1.803

0.0117 305.6 1.761

8.80.−3 305.1 1.732

7.07.−3 304.5 1.708

3.62.−3 302.2 1.634

1.93.−3 299.8 1.557

1.38.−3 299.6 1.511

1.11.−3 301.1 1.478

9.56.−4 304.0 1.453

800

v h s

0.7939 332.3 2.204

0.0795 332.2 1.975

0.0399 331.9 1.906

0.0200 331.6 1.837

0.0134 331.2 1.795

0.0101 330.9 1.766

8.11.−3 330.5 1.743

4.16.−3 329.3 1.671

2.21.−3 328.6 1.596

1.57.−3 329.4 1.551

1.25.−3 331.5 1.518

1.07.−3 334.6 1.494

900

v h s

0.8931 357.1 2.233

0.0895 357.0 2.005

0.0448 356.9 1.936

0.0225 356.8 1.866

0.0151 356.6 1.825

0.0114 356.4 1.796

9.13.−3 356.3 1.773

4.68.−3 355.8 1.702

2.48.−3 356.3 1.628

1.75.−3 358.0 1.584

1.39.−3 360.7 1.553

1.18.−3 364.2 1.528

1000

v h s

0.9924 381.9 2.260

0.0994 381.9 2.031

0.0498 381.9 1.962

0.0250 381.8 1.893

0.0168 381.8 1.852

0.0126 381.8 1.823

0.0102 381.8 1.800

5.20.−3 381.9 1.729

2.74.−3 383.4 1.657

1.93.−3 385.9 1.614

1.53.−3 389.0 1.583

1.29.−3 392.8 1.559

*Values extracted and in some cases rounded off from those cited in Rabinovich (ed.), Thermophysical Properties of Neon, Argon, Krypton and Xenon, Standards Press, Moscow, 1976. This source contains an exhaustive tabulation of values. v = specific volume, m3/kg; h = specific enthalpy, kJ/kg; s = specific entropy, kJ/(kg⋅K). The notation 3.49.−4 signifies 3.49 × 10−4. This book was published in English translation by Hemisphere, New York, 1988 (604 pp.).

TABLE 2-278 T, K 453.7m 500 600 700 800 900 1000 1200 1400 1500

Saturated Lithium*

P, bar

vf, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

1.78.−13 8.21.−12 4.18.−9 3.51.−7 9.57.−6

1.912.−3 1.946.−3 1.988.−3 2.028.−3 2.070.−3

2.40.+7 9.94.+5

1703 1905 2334 2697 3174

24259 24390 24674 24869 25162

6.776 7.199 7.983 8.633 9.192

56.492 52.169 45.216 40.307 36.678

4.30 4.34 4.23 4.19 4.17

1.24.−4 9.60.−4 0.0204 0.1794 0.4269

2.114.−3 2.160.−3 2.262.−3 2.370.−3 2.433.−3

8.55.+4 1.22.+4 669.3 86.06 38.17

3590 4006 4835 5668 6088

25341 25477 25654 25778 25845

9.682 10.120 10.876 11.518 11.808

33.850 31.591 28.225 25.882 24.979

4.16 4.16 4.14 4.19 4.20

*Converted from tables in Vargaftik, Tables of the Thermophysical Properties of Liquids and Gases, Nauka, Moscow, 1972, and Hemisphere, Washington, 1975. m = melting point. The notation 1.78.−13 signifies 1.78 × 10−13. Many of the Vargaftik values also appear in Ohse, R. W., Handbook of Thermodynamic and Transport Properties of Alkali Metals, Blackwell Sci. Pubs., Oxford, 1985 (1020 pp.). This source contains superheat data. Saturation and superheat tables and a diagram to 14 bar, 2200 K are given by Reynolds, W. C., Thermodynamic properties in S.I., Stanford Univ. publ., 1979 (173 pp.). For a Mollier diagram from 0.1 to 140 psia, 2100–3600°R, see Weatherford, P. M., J. C. Tyler, et al., WADD-TR-61-96, 1961. An extensive review of properties of the solid and the saturated liquid was given by Alcock, C. B., M. W. Chase, et al., J. Phys. Chem. Ref. Data, 23, 3 (1994): 385–497.

TABLE 2-279

Lithium Bromide—Water Solutions

Ruiter, J. P., Rev. Int. Froid = Int. J. Refrig., 13 (1990): 223–236 gives subroutines for computer calculations. See also ASHRAE Handbook—Fundamentals.

2-250

PHYSICAL AND CHEMICAL DATA

TABLE 2-280

Saturated Mercury*

T, K

P, bar

vf × 105, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

hfg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

203.15 213.15 223.15 233.15 243.15

2.298 ⋅ 10−11 1.288 ⋅ 10−10 6.169 ⋅ 10−10 2.580 ⋅ 10−9 9.573 ⋅ 10−9

7.26239 7.27570 7.28900 7.30231 7.31563

3.665 ⋅ 109 6.862 ⋅ 108 1.499 ⋅ 108 3.746 ⋅ 107 1.053 ⋅ 107

33.131 34.567 35.997 37.422 38.842

342.637 343.674 344.710 345.746 346.782

309.506 309.107 308.713 308.324 307.940

0.32434 0.33124 0.33780 0.34404 0.35001

1.84787 1.78142 1.72123 1.66647 1.61647

253.15 263.15 273.15 283.15 293.15

3.198 ⋅ 10−8 9.736 ⋅ 10−8 2.728 ⋅ 10−7 7.101 ⋅ 10−7 1.729 ⋅ 10−6

7.32896 7.34229 7.35563 7.36898 7.38234

3.281 ⋅ 106 1.120 ⋅ 106 4.150 ⋅ 105 1.653 ⋅ 105 7.026 ⋅ 104

40.258 41.668 43.074 44.476 45.874

347.819 348.855 349.891 350.927 351.964

307.561 307.187 306.817 306.451 306.090

0.35571 0.36118 0.36642 0.37146 0.37631

1.57065 1.52852 1.48967 1.45375 1.42045

303.15 313.15 323.15 333.15 343.15

3.968 ⋅ 10−6 8.626 ⋅ 10−6 1.786 ⋅ 10−5 3.356 ⋅ 10−5 6.724 ⋅ 10−5

7.39572 7.40911 7.42252 7.43594 7.44938

3.167 ⋅ 104 1.505 ⋅ 104 7.501 ⋅ 103 3.905 ⋅ 103 2.115 ⋅ 103

47.268 48.659 50.046 51.430 52.810

353.000 354.036 355.072 356.108 357.145

305.732 305.377 305.026 304.678 304.335

0.38099 0.38550 0.38986 0.39408 0.39816

1.38951 1.36068 1.33378 1.30862 1.28505

353.15 363.15 373.15 383.15

1.232 ⋅ 10−4 2.182 ⋅ 10−4 3.745 ⋅ 10−4 6.247 ⋅ 10−4

7.46285 7.47633 7.48984 7.50337

1.188 ⋅ 103 6.899 ⋅ 102 413.0 254.2

54.188 55.563 56.936 58.306

358.181 359.217 360.253 361.289

303.993 303.654 303.317 302.983

0.40212 0.40596 0.40969 0.41331

1.26292 1.24213 1.22255 1.20408

393.15 403.15 413.15 423.15 433.15

1.015 ⋅ 10−3 1.608 ⋅ 10−3 2.491 ⋅ 10−3 3.778 ⋅ 10−3 5.618 ⋅ 10−3

7.51693 7.53052 7.55415 7.55780 7.57148

153.6 103.9 68.75 46.43 31.96

59.674 61.039 62.403 63.765 65.125

362.326 363.362 364.397 365.433 366.469

302.652 302.323 301.994 301.668 301.344

0.41684 0.42027 0.42361 0.42687 0.43004

1.18665 1.17017 1.15456 1.13978 1.12575

443.15 453.15 463.15 473.15 483.15

8.204 ⋅ 10−3 1.178 ⋅ 10−2 1.664 ⋅ 10−2 2.315 ⋅ 10−2 3.177 ⋅ 10−2

7.58520 7.59897 7.61277 7.62662 7.64051

66.484 67.842 69.198 70.553 71.908

367.504 368.539 369.574 370.609 371.642

301.020 300.697 300.376 300.056 299.734

0.43314 0.43617 0.43913 0.44203 0.44486

1.11242 1.09975 1.08768 1.07619 1.06524

493.15 503.15 513.15 523.15 533.15

4.304 ⋅ 10−2 5.758 ⋅ 10−2 7.614 ⋅ 10−2 9.959 ⋅ 10−2 0.12892

7.65444 7.66843 7.68247 7.69656 7.71071

4.748 3.621 2.793 2.176 1.7132

73.261 74.614 75.967 77.319 78.671

372.676 373.708 374.740 375.771 376.800

299.415 299.094 298.773 298.452 298.129

0.44763 0.45035 0.45301 0.45562 0.45818

1.05478 1.04479 1.03525 1.02611 1.01737

543.15 553.15 563.15 573.15 583.15

0.16527 0.20993 0.26435 0.33015 0.40910

7.72491 7.73918 7.75351 7.7679 7.7823

1.3613 1.0912 0.88213 0.71874 0.59002

80.023 81.375 82.728 84.080 85.434

377.829 378.855 379.880 380.904 381.925

297.806 297.480 297.152 296.824 296.491

0.46069 0.46316 0.46558 0.46796 0.47030

1.00899 1.00095 0.99324 0.98584 0.97893

593.15 603.15 613.15 623.15 633.15

0.50320 0.61460 0.74567 0.89896 1.0772

7.7969 7.8115 7.8262 7.8409 7.8558

0.48779 0.40600 0.34008 0.28660 0.24291

86.788 88.143 89.499 90.856 92.215

382.944 383.960 384.973 385.984 386.991

296.156 295.817 295.474 295.128 294.776

0.47260 0.47487 0.47709 0.47929 0.48145

0.97190 0.96532 0.95899 0.95289 0.94702

643.15 653.15 663.15 673.15 683.15

1.2834 1.5207 1.9725 2.1024 2.454

7.8707 7.8858 7.9008 7.9160 7.9313

0.20702 0.17735 0.15269 0.13207 0.11476

93.575 94.937 96.300 97.666 99.033

387.994 388.994 389.989 390.980 391.966

294.419 294.057 293.689 293.314 292.933

0.48358 0.48568 0.48774 0.48978 0.49180

0.94135 0.93589 0.93061 0.92552 0.92059

693.15 703.15 713.15 723.15 733.15

2.852 3.299 3.801 4.362 4.986

7.9467 7.9622 7.9778 7.9935 8.0094

0.10014 0.08775 0.07719 0.06815 0.06039

100.403 101.775 103.150 104.528 105.908

392.947 393.923 394.893 395.858 396.816

292.544 292.148 291.743 291.330 290.908

0.49378 0.49574 0.49768 0.49959 0.50148

0.91583 0.91123 0.90677 0.90245 0.89827

743.15 753.15 763.15 773.15 783.15

5.679 6.446 7.292 8.222 9.242

8.0252 8.0413 8.0574 8.074 8.090

0.05369 0.04789 0.04285 0.03846 0.03462

107.292 108.679 110.069 111.463 112.861

397.767 398.711 399.649 400.579 401.501

290.475 290.032 289.580 289.116 288.640

0.50335 0.50519 0.50702 0.50882 0.51061

0.89422 0.89029 0.88647 0.88277 0.87917

793.15 803.15 813.15 823.15 833.15

10.358 11.576 12.901 14.340 15.899

8.106 8.123 8.140 8.157 8.174

0.03124 0.02827 0.02565 0.02333 0.02126

114.262 115.668 117.078 118.492 119.911

402.415 403.321 404.218 405.106 405.985

288.153 287.653 287.140 286.614 286.074

0.51238 0.51412 0.51586 0.51757 0.51927

0.87568 0.87228 0.86898 0.86576 0.86263

22.39 15.95 11.54 8.469 6.301

*From Vukalovich, Ivanov, Fokin, and Yakovlev, Thermophysical Properties of Mercury, Standartov, Moscow, 1971. For the saturated liquid the specific volume at 203.15 K is 7.26239 × 10−5 m3/kg, etc. All the tabular values for 203.15 K, 213.15 K, 223.15 K, and 233.15 K represent a metastable equilibrium between the subcooled liquid and the saturated vapor. Saturation and superheat tables and a diagram to 100 bar, 1600 K are given by Reynolds, W. C., Thermodynamic properties in S.I., Stanford Univ. publ., 1979 (173 pp.). For a Mollier diagram from 1 to 8200 psia and 2700°R, see Weatherford, W. D., J. C. Tyler, et al., WADD-TR-61-96, 1961.

THERMODYNAMIC PROPERTIES

2-251

Saturated Mercury* (Concluded )

TABLE 2-280 T, K

P, bar

vf × 105, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

hfg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

843.15 853.15 863.15 873.15 883.15

17.584 19.403 21.36 23.46 25.72

8.191 8.209 8.226 8.244 8.262

0.019426 0.017785 0.016317 0.015000 0.013815

121.335 122.763 124.197 125.636 127.080

406.855 407.715 408.565 409.405 410.235

285.520 284.952 284.368 283.769 283.155

0.52095 0.52262 0.52427 0.52591 0.52753

0.85959 0.85662 0.85372 0.85090 0.84815

893.15 903.15 913.15 923.15 933.15

28.14 30.72 33.47 36.41 39.53

8.280 8.298 8.316 8.335 8.353

0.012748 0.011784 0.010911 0.010120 0.009401

128.530 129.986 131.448 132.915 134.389

411.054 411.861 412.658 413.444 414.218

282.524 281.875 281.210 280.529 279.829

0.52914 0.53074 0.53232 0.53389 0.53545

0.84546 0.84284 0.84028 0.83777 0.83533

943.15 953.15 963.15 973.15 983.15

42.85 46.36 50.09 54.03 58.20

8.372 8.391 8.410 8.430 8.450

0.008746 0.008150 0.007604 0.007105 0.006648

135.869 137.356 138.850 140.350 141.858

414.980 415.731 416.469 417.195 417.909

279.111 278.375 277.619 276.845 276.051

0.53700 0.53854 0.54006 0.54158 0.54308

0.83294 0.83060 0.82831 0.82606 0.82387

993.15 1003.15 1013.15 1023.15 1033.15

62.59 67.22 72.10 77.22 82.60

8.468 8.488 8.508 8.529 8.550

0.006228 0.005842 0.005487 0.005159 0.004856

143.372 144.894 146.424 147.961 149.506

418.610 419.298 419.974 420.636 421.286

275.238 274.404 273.550 272.675 271.780

0.54458 0.54607 0.54754 0.54901 0.55047

0.82172 0.81961 0.81754 0.81552 0.81353

1043.15 1053.15 1063.15 1073.15

88.25 94.17 100.37 106.85

8.570 8.590 8.612 8.632

0.004576 0.004317 0.004077 0.003854

151.059 152.619 154.188 155.766

421.923 422.546 423.156 423.752

270.864 269.927 268.968 267.986

0.55192 0.55336 0.55479 0.55621

0.81158 0.80966 0.80778 0.80593

TABLE 2-281

Saturated Methane*

P, bar

vf, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µf, 10−4 Pa⋅s

kf, W/(m⋅K)

90.7 95 100 105 110

0.117 0.198 0.345 0.565 0.884

2.215.−3 2.244.−3 2.278.−3 2.316.−3 2.353.−3

3.976 2.463 1.479 0.940 0.625

216.4 232.5 246.3 263.2 280.1

759.9 769.0 776.9 785.7 794.5

4.231 4.406 4.556 4.719 4.882

10.225 10.034 9.862 9.710 9.558

3.288 3.318 3.369 3.425 3.478

2.02 1.71 1.56 1.33 1.22

0.225 0.215 0.206 0.197 0.189

115 120 125 130 135

1.325 1.919 2.693 3.681 4.912

2.396.−3 2.438.−3 2.487.−3 2.536.−3 2.594.−3

0.430 0.306 0.223 0.167 0.127

297.7 315.3 333.5 351.7 370.6

802.5 810.4 817.3 824.1 829.5

5.035 5.188 5.332 5.476 5.614

9.436 9.314 9.062 8.810 8.871

3.525 3.570 3.620 3.679 3.755

1.09 0.98 0.89 0.81 0.73

0.181 0.173 0.165 0.158 0.150

140 145 150 155 160

6.422 8.246 10.41 12.97 15.94

2.652.−3 2.722.−3 2.792.−3 2.882.−3 2.971.−3

0.098 0.077 0.061 0.049 0.039

389.5 409.5 429.4 450.8 472.1

834.8 844.4 853.9 848.5 843.0

5.751 5.885 6.019 6.151 6.283

8.932 8.891 8.849 8.725 8.601

3.849 3.965 4.101 4.27 4.47

0.66 0.61 0.56 0.51 0.46

0.143 0.136 0.129 0.122 0.115

165 170 175 180 185

19.39 23.81 27.81 32.86 38.59

3.095.−3 3.218.−3 3.419.−3 3.619.−3 3.979.−3

0.032 0.026 0.020 0.016 0.012

495.4 518.6 545.8 572.9 605.4

840.0 837.0 827.6 818.1 797.7

6.417 6.551 6.697 6.843 7.017

8.513 8.424 8.315 8.205 8.049

4.75 5.16 5.89 7.27 11.1

0.42 0.38 0.34 0.30 0.25

0.108 0.101 0.094 0.088 0.085

190 190.6c

45.20 45.99

4.900.−3 6.233.−3

0.008 0.006

661.6 704.4

750.7 704.4

7.293 7.516

7.762 7.516

70. ∞

0.19 0.17

0.090 ∞

T, K t

*Values reproduced or converted from Goodwin, NBS Tech. Note 653, 1974. t = triple point; c = critical point. The notation 2.215.−3 signifies 2.215 × 10−3.

2-252

PHYSICAL AND CHEMICAL DATA

FIG. 2-12

2-280.)

Enthalpy-log-pressure diagram for mercury. (Drawn from tabular data in footnote reference to Table

THERMODYNAMIC PROPERTIES TABLE 2-282

2-253

Superheated Methane* Temperature, K

P, bar

100

150

200

250

300

350

400

450

500

1

v h s

0.00228 246.4 4.555

0.7661 879.0 10.152

1.0299 984.3 10.757

1.2915 1090.4 11.230

1.5521 1199.8 11.629

1.8122 1314.8 11.983

2.0719 1437.4 12.310

2.3669 1568.8 12.618

2.5911 1708.9 12.914

5

v h s

0.00228 247.0 4.553

0.1434 865.0 9.256

0.2006 976.1 9.896

0.2549 1084.7 10.381

0.3083 1195.5 10.785

0.3611 1311.5 11.142

0.4136 1434.7 11.471

0.4657 1566.6 11.781

0.5181 1706.9 12.066

10

v h s

0.00227 247.8 4.549

0.0643 843.6 8.797

0.0968 965.5 9.501

0.1254 1077.9 10.002

0.1528 1190.6 10.414

0.1798 1307.9 10.775

0.2063 1432.0 11.106

0.2327 1564.1 11.417

0.2590 1705.3 11.715

20

v h s

0.00227 249.4 4.542

0.00277 429.8 6.003

0.0446 941.9 9.059

0.0606 1063.6 9.603

0.0751 1180.7 10.030

0.0891 1300.6 10.400

0.1027 1426.5 10.736

0.1162 1560.3 11.050

0.1295 1702.1 11.349

40

v h s

0.00226 252.5 4.528

0.00274 430.8 5.973

0.0176 879.3 8.465

0.0281 1032.9 9.155

0.0363 1160.5 9.621

0.0438 1286.0 10.008

0.0510 1415.7 10.354

0.0579 1552.1 10.674

0.0648 1696.0 10.978

60

v h s

0.00226 255.7 4.515

0.00271 432.2 5.946

0.00615 734.0 7.623

0.0173 999.8 8.847

0.0234 1140.0 9.359

0.0287 1271.7 9.765

0.0338 1405.1 10.121

0.0386 1544.2 10.440

0.0432 1690.0 10.756

80

v h s

0.00225 258.9 4.502

0.00268 433.8 5.920

0.00411 660.5 7.209

0.0119 964.4 8.590

0.0171 1119.7 9.158

0.0213 1257.7 9.584

0.0252 1394.9 9.951

0.0289 1536.6 10.283

0.0324 1684.4 10.595

100

v h s

0.00224 262.1 4.489

0.00266 435.5 5.897

0.00375 644.5 7.090

0.00888 928.5 8.364

0.0133 1099.6 8.991

0.0169 1244.2 9.437

0.0201 1385.2 9.814

0.0231 1529.4 10.153

0.0260 1679.0 10.469

150

v h s

0.00223 270.2 4.458

0.00261 440.7 5.843

0.00337 630.2 6.930

0.00555 860.0 7.953

0.00852 1054.1 8.664

0.0111 1213.1 9.155

0.0134 1362.8 9.555

0.0155 1513.0 9.907

0.0175 1667.0 10.233

200

v h s

0.00221 278.3 4.429

0.00256 446.5 5.796

0.00318 626.5 6.829

0.00447 825.0 7.719

0.00644 1019.8 8.426

0.00837 1187.2 8.944

0.0101 1343.8 9.362

0.0118 1498.9 9.727

0.0133 1656.9 10.060

300

v h s

0.00218 294.7 4.373

0.00249 459.6 5.714

0.00296 629.2 6.690

0.00369 804.4 7.471

0.00474 982.9 8.122

0.00593 1153.6 8.649

0.00708 1316.8 9.085

0.00818 1478.5 9.465

0.00924 1642.2 9.811

400

v h s

0.00244 473.8 5.645

0.00282 637.7 6.588

0.00336 802.4 7.323

0.00406 970.1 7.935

0.00486 1137.8 8.451

0.00569 1303.0 8.893

0.00560 1467.7 9.280

0.00729 1634.7 9.633

500

v h s

0.00239 488.8 5.584

0.00272 648.9 6.507

0.00315 807.7 7.215

0.00368 969.0 7.802

0.00428 1132.8 8.307

0.00492 1297.8 8.748

0.00555 1464.2 9.139

0.00616 1633.2 9.496

*Converted and rounded off from the tables of Goodwin, NBS Tech. Note 654, 1974. v = specific volume, m3/kg; h = specific enthalpy, kJ/kg; s = specific entropy, kJ/(kg⋅K). For a thermodynamic diagram from 0.1 to 400 bar and 620°C, see the 1993 ASHRAE Handbook—Fundamentals (SI ed.). Saturation and superheat tables and a chart to 6000 psia, 680°F appear in Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat, thermal conductivity, and viscosity, see Thermophysical Properties of Refrigerants, ASHRAE, 1993. See also Friend, D. G., J. F. Ely, et al., J. Phys. Chem. Ref. Data. 18, 2 (1989): 583–638.

2-254

PHYSICAL AND CHEMICAL DATA

TABLE 2-283 Pressure, bar

Thermophysical Properties of Saturated Methanol kf, W/(m⋅K)

Prf

2.531 2.554 2.669 2.777

625 525 401 329

0.204 0.196 0.193 0.189

7.75 6.84 5.55 4.83

7.5379 7.4836 7.4398 7.4051 7.3474

2.845 2.894 2.946 2.984 3.050

288 268 242 227 204

0.186 0.184 0.182 0.181 0.179

4.41 4.22 3.92 3.74 3.48

4.7307 4.7836 4.8678 4.9366 5.0708

7.2992 7.2624 7.1988 7.1471 7.0461

3.117 3.176 3.265 3.349 3.540

187 174 156 141 117

0.178 0.177 0.175 0.173 0.171

3.27 3.12 2.91 2.73 2.42

1553.8 1486.4 1474.7 1450.1 1423.2

5.1744 5.2605 5.3355 5.4650 5.5793

6.9677 6.9017 6.8435 6.7409 6.6475

3.72 3.91 4.12 4.67 5.55

102 92 84 72 63

0.169 0.167 0.165 0.160 0.154

2.25 2.15 2.10 2.10 2.27

1391.8 1318.7 1186.8

5.6889 5.8803 6.0979

6.5543 6.3791 6.0979

hf, kJ/kg

hg, kJ/kg

0.001 0.001 0.001 0.001 0.001

057 257 276 307 336

1700000 7.309 3.801 1.599 0.819

0.0 261.0 293.9 345.0 391.7

1303.1 1440.3 1455.4 1476.2 1492.1

2.8114 3.9383 4.0493 4.2117 4.3516

10.2328 8.0281 7.9032 7.7386 7.6104

1.5 2.0 2.5 3.0 4.0

348.0 356.0 362.5 368.0 377.1

0.001 0.001 0.001 0.001 0.001

356 371 385 396 417

0.5632 0.4276 0.3443 0.2893 0.2188

421.0 444.2 463.6 479.8 507.8

1500.3 1505.8 1509.8 1512.4 1515.9

4.4361 4.5014 4.5536 4.5992 4.6728

5 6 8 10 15

384.5 390.8 401.3 409.8 426.3

0.001 0.001 0.001 0.001 0.001

434 450 479 504 560

0.17569 0.14683 0.11015 0.08783 0.05761

529.7 549.6 582.7 610.3 665.8

1517.4 1518.4 1518.0 1516.1 1507.9

20 25 30 40 50

438.9 449.3 458.2 472.9 484.9

0.001 0.001 0.001 0.001 0.001

611 666 710 814 934

0.04224 0.03290 0.02661 0.01863 0.01373

710.5 749.0 783.8 846.7 905.2

60 80 80.95c

495.1 508.1 512.6

0.002 086 0.002 507 0.003 715

0.01032 0.00642 0.00372

963.3 1065.3 1186.8

t

Temp., K

vf, m3/kg

µf, 10−6 Pa·s

vg, m3/kg

175.6 288.4 301.7 320.7 337.7

4 × 10−6 0.1 0.2 0.5 1.013

sf, kJ/(kg⋅K) sg, kJ/(kg⋅K) cpf, kJ/(kg⋅K)

t = triple point; c = critical point. v, h, s, and cp interpolated and converted from Goodwin, R. D., J. Phys. Chem. Ref. Data, 16, 4 (1987): 799–891.

THERMODYNAMIC PROPERTIES TABLE 2-284

2-255

Thermodynamic Properties of Compressed Methanol Temperature, K

Pressure, bar

200

250

300

350

v (m3/ kg) 0.1 h (kJ/ kg) s (kJ/ kg·K)

0.001137 57.5 3.096

400

450

500

550

600

0.001203 169.8 3.597

7.630 1456.7 8.081

8.942 1529.5 8.305

10.23 1607.5 8.514

11.56 1691.5 8.711

12.84 1781.7 8.901

1878.1 9.085

15.45 1980.2 9.263

v (m3/ kg) 0.5 h (kJ/ kg) s (kJ/ kg·K)

0.001137 57.5 3.096

0.001202 169.9 3.597

0.001274 290.5 4.038

1.764 1522.7 7.877

2.033 1603.0 8.091

2.296 1687.9 8.291

2.558 1778.9 8.482

2.818 1875.3 8.666

3.078 1977.4 8.844

v (m3/ kg) 1.013 h (kJ/ kg) s (kJ/ kg·K)

0.001137 57.6 3.096

0.001202 169.9 3.597

0.001274 290.5 4.038

0.8560 1514.0 7.675

0.9958 1598.7 7.902

1.1283 1685.1 8.105

1.2843 1795.4 8.117

1.3870 1873.5 8.482

1.5157 1975.8 8.660

v (m3/ kg) 10 h (kJ/ kg) s (kJ/ kg·K)

0.001136 58.4 3.095

0.001201 170.7 3.596

0.001272 291.2 4.036

0.001357 427.4 4.451

0.001474 578.8 4.857

0.1068 1638.1 7.427

0.1236 1751.5 7.667

0.1381 1858.0 7.870

0.1519 1965.2 8.056

v (m3/ kg) 15 h (kJ/ kg) s (kJ/ kg·K)

0.001136 58.8 3.094

0.001201 171.1 3.595

0.001272 291.6 4.035

0.001356 427.7 4.450

0.001472 578.9 4.856

0.0673 1601.9 7.253

0.0806 1735.6 7.536

0.0911 1849.4 7.752

0.1007 1960.2 7.946

v (m3/ kg) 20 h (kJ/ kg) s (kJ/ kg·K)

0.001135 59.2 3.094

0.001200 171.6 3.595

0.001271 292.0 4.035

0.001355 428.1 4.449

0.001469 579.0 4.854

0.0466 1565.3 7.087

0.0589 1717.7 7.431

0.0675 1840.0 7.664

0.0751 1954.8 7.864

v (m3/ kg) 30 h (kJ/ kg) s (kJ/ kg·K)

0.001134 60.1 3.092

0.001199 172.4 3.593

0.001269 292.9 4.036

0.001355 428.8 4.447

0.001465 579.4 4.851

0.001659 751.3 5.264

0.0367 1675.4 7.253

0.0436 1818.7 7.526

0.0492 1942.7 7.743

v (m3/ kg) 40 h (kJ/ kg) s (kJ/ kg·K)

0.001133 61.0 3.091

0.001198 173.3 3.592

0.001268 293.7 4.032

0.001350 429.5 4.445

0.001461 579.8 4.849

0.001649 750.4 5.258

0.0251 1623.0 7.088

0.0314 1794.2 7.414

0.0361 1928.8 7.650

v (m3/ kg) 50 h (kJ/ kg) s (kJ/ kg·K)

0.001133 61.9 3.090

0.001197 174.2 3.591

0.001266 294.5 4.030

0.001348 430.2 4.443

0.001457 580.2 4.846

0.001637 749.7 5.579

0.0176 1556.4 6.912

0.0239 1766.7 7.314

0.0282 1913.4 7.570

v (m3/ kg) 60 h (kJ/ kg) s (kJ/ kg·K)

0.001131 62.8 3.089

0.001196 175.0 3.589

0.001265 295.3 4.029

0.001346 430.9 4.442

0.001453 580.6 4.843

0.001628 749.1 5.248

0.0120 1461.8 6.692

0.0188 1736.1 7.220

0.0228 1896.6 7.500

v (m3/ kg) 75 h (kJ/ kg) s (kJ/ kg·K)

0.001130 64.1 3.087

0.001194 176.3 3.587

0.001263 296.6 4.027

0.001343 431.9 4.439

0.001448 581.2 4.839

0.001614 748.3 5.241

0.002084 982.1 5.718

0.01359 1683.9 7.081

0.0174 1869.1 7.405

v (m3/ kg) 100 h (kJ/ kg) s (kJ/ kg·K)

0.001128 66.3 3.084

0.001191 178.5 3.584

0.001259 298.6 4.023

0.001337 433.8 4.435

0.001439 582.4 4.833

0.001595 747.5 5.230

0.001952 964.8 5.673

0. 1572.9 6.829

0.01188 1818.8 7.261

v (m3/ kg) 150 h (kJ/ kg) s (kJ/ kg·K)

0.001125 70.7 3.078

0.001186 182.8 3.578

0.001252 302.8 4.016

0.001328 437.4 4.426

0.001423 584.9 4.822

0.001562 746.8 5.211

0.001825 948.4 5.622

1248.8 6.302

0.006513 1704.3 6.997

v (m3/ kg) 200 h (kJ/ kg) s (kJ/ kg·K)

0.001121 75.1 3.071

0.001182 187.2 3.571

0.001246 307.0 4.009

0.001317 441.2 4.418

0.001408 587.8 4.811

0.001535 747.0 5.194

0.001751 939.9 5.587

0.002314 1223.5 6.125

0.004091 1583.5 6.752

v (m3/ kg) 300 h (kJ/ kg) s (kJ/ kg·K)

0.001113 83.9 3.060

0.001172 195.9 3.559

0.001234 315.4 3.996

0.001302 448.9 4.403

0.001384 593.8 4.791

0.001492 749.4 5.166

0.001656 932.0 5.537

0.001957 1173.4 5.996

0.002600 1443.5 6.466

v (m3/ kg) 400 h (kJ/ kg) s (kJ/ kg·K)

0.001107 92.7 3.048

0.001164 204.7 3.548

0.001223 324.0 3.983

0.001288 456.9 4.388

0.001363 600.5 4.774

0.001459 753.4 5.142

0.001593 929.6 5.500

0.001808 1154.4 5.926

0.002182 1388.1 6.335

v (m3/ kg) 500 h (kJ/ kg) s (kJ/ kg·K)

0.001101 101.5 3.037

0.001156 213.4 3.536

0.001213 332.6 3.971

0.001274 465.1 4.375

0.001345 607.7 4.757

0.001431 758.4 5.121

0.001546 930.4 5.470

0.001716 1145.6 5.880

0.001980 1360.6 6.254

Converted and interpolated from Goodwin, R. D., J. Phys. Chem. Ref. Data, 16, 4 (1987): 799–891. These extensive tables extend to 700 bar for temperatures from 175.6 to 800 K. Another extensive compilation is by deReuck, K. M. and R. J. B. Craven, Methanol, C.R.C. Press, 1993 (320 pp.). Equations and diagrams to 30 bar, 200°C are given by Eicholz, H. D., S. Schulz, et al., Kalte u Klim., no. 9, (1981) 322–331. For pressures to 1040 bar, 298–489 K, see Machado, J. R. S. and W. B. Street, J.Chem.Eng.Data, 28 (1983): 218–223; to 2800 bar from 273 to 333 K, see Sun, T., S. N. Biswas, et al., J. Chem. Eng. Data, 33 (1988): 395–398. Dissociation was considered by Yerlett, T. K. and C. J. Wormald, J. Chem. Thermo., 18 (1986): 719–726, and by Kazarnovskii, Ya. S. and E. V. Pavlova, Russ. J. Phys. Chem., 56, 6 (1982): 847–851.

TABLE 2-285

Saturated Methyl Chloride* µf, 10−4 Pa⋅s

kf, W/(m⋅K)

1.483 1.486 1.489 1.492 1.496

4.44 4.27 4.11 3.96 3.82

0.241 0.236 0.232 0.228 0.224

5.896 5.866 5.845 5.822 5.786

1.500 1.504 1.508 1.513 1.518

3.69 3.57 3.46 3.35 3.25

0.219 0.215 0.211 0.207 0.202

4.050 4.080 4.110 4.139 4.168

5.762 5.740 5.720 5.699 5.680

1.523 1.528 1.533 1.539 1.546

3.16 3.08 3.00 2.92 2.85

0.198 0.194 0.190 0.186 0.182

824.4 826.8 829.0 831.2 833.2

4.197 4.225 4.253 4.280 4.308

5.662 5.644 5.628 5.612 5.597

1.554 1.565 1.574 1.583 1.594

2.78 2.72 2.66 2.61 2.56

0.177 0.173 0.169 0.165 0.160

461.2 469.3 477.4 485.6 493.8

835.2 837.0 838.8 840.5 841.9

4.334 4.361 4.388 4.414 4.440

5.581 5.567 5.553 5.540 5.527

1.605 1.617 1.631 1.644 1.658

2.51 2.46 2.42 2.37 2.33

0.156 0.152 0.148 0.143 0.139

0.0386 0.0343 0.0282 0.0228 0.0186

502.1 510.4 518.8 538.3 562.9

843.3 844.5 846.4 847.5 847.6

4.465 4.491 4.542 4.592 4.643

5.516 5.504 5.481 5.457 5.434

2.30 2.27 2.12 1.99 1.87

0.135 0.131 0.124 0.117 0.110

13.47.−4 14.11.−4 14.67.−4 15.66.−4 16.48.−4

0.0151 0.0117 0.0096 0.0075 0.0063

581.6 602.8 622.9 643.6 663.2

845.9 842.6 837.4 826.4 819.1

4.694 4.747 4.805 4.870 4.904

5.398 5.382 5.358 5.323 5.289

1.77 1.67 1.59 1.51

0.103 0.095 0.086 0.075

17.97.−4 21.10.−4 27.40.−4

0.0052 0.0038 0.0027

677.3 714.1 749.3

807.1 778.6 749.3

4.954 5.025 5.116

5.256 5.200 5.116

T, K

P, bar

vf, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

175 180 185 190 195

0.0117 0.0165 0.0233 0.0327 0.0462

8.84.−4 8.91.−4 8.97.−4 9.04.−4 9.10.−4

27.90 19.85 14.12 10.12 7.208

274.5 280.9 287.5 294.5 301.7

764.3 767.7 771.0 774.3 777.5

3.529 3.570 3.603 3.647 3.684

6.328 6.274 6.222 6.172 6.124

1.469 1.472 1.475 1.477 1.480

200 205 210 215 220

0.0653 0.0919 0.1315 0.181 0.243

9.17.−4 9.25.−4 9.33.−4 9.40.−4 9.48.−4

5.137 3.835 2.656 1.975 1.505

309.0 316.3 323.7 331.0 338.4

780.7 783.9 787.0 790.1 793.2

3.722 3.756 3.791 3.825 3.859

6.080 6.038 5.998 5.961 5.928

225 230 235 240 245

0.319 0.417 0.539 0.688 0.866

9.56.−4 9.65.−4 9.73.−4 9.81.−4 9.89.−4

1.168 0.911 0.718 0.572 0.462

345.7 353.1 360.5 368.0 375.6

796.3 799.3 802.3 805.3 808.2

3.892 3.925 3.957 3.988 4.019

250 255 260 265 270

1.076 1.328 1.627 1.970 2.364

9.98.−4 10.08.−4 10.18.−4 10.27.−4 10.36.−4

0.377 0.311 0.257 0.215 0.1807

383.2 390.7 398.3 406.0 413.7

811.1 814.0 816.8 819.4 822.0

275 280 285 290 295

2.830 3.347 3.936 4.612 5.361

10.46.−4 10.57.−4 10.68.−4 10.79.−4 10.91.−4

0.1524 0.1301 0.1115 0.0960 0.0830

421.5 429.4 437.3 445.2 453.2

300 305 310 315 320

6.189 7.110 8.111 9.243 10.47

11.03.−4 11.15.−4 11.27.−4 11.40.−4 11.55.−4

0.0723 0.0632 0.0556 0.0489 0.0433

325 330 340 350 360

11.78 13.27 16.52 20.53 25.29

11.70.−4 11.86.−4 12.17.−4 12.54.−4 12.97.−4

370 380 390 400 405

30.74 36.99 44.05 52.29 56.6

410 415 416c

61.5 67.4 69.0

*Interpolated by P. E. Liley from the Landolt-Börnstein band IVa, p. 677, 1967 tables by Steinle/Dienemann. c = critical point. The notation 8.84.−4 signifies 8.84 × 10−4. TABLE 2-286

Saturated Neon*

P, bar

vf, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µf, 10−4 Pa⋅s

kf, W/(m⋅K)

10 20 24.6m 24.6m 26

0.434 0.718

6.654.−4 6.823.−4 6.696.−4 8.012.−4 8.172.−4

0.2266 0.1429

0.75 6.78 11.96 28.22 30.90

117.0 118.1

0.0992 0.4906 0.7257 1.388 1.494

5.006 4.846

0.278 0.945 1.345 1.802 1.868

1.57 1.37

0.146 0.132

28 30 32 34 36

1.321 2.238 3.552 5.352 7.728

8.413.−4 8.687.−4 9.001.−4 9.370.−4 9.820.−4

0.0817 0.0501 0.0323 0.0217 0.0149

34.75 38.80 43.06 47.57 52.34

119.3 120.1 120.6 120.6 119.9

1.634 1.771 1.905 2.036 2.166

4.653 4.483 4.329 4.184 4.043

1.955 2.052 2.163 2.302 2.506

1.16 1.00 0.84 0.71 0.59

0.124 0.115 0.106 0.097 0.088

1.039.−3 1.116.−3 1.232.−3 1.538.−3 2.070.−3

0.0104 0.0073 0.0050 0.0031 0.0021

57.52 63.33 69.82 80.83 92.50

118.4 115.8 111.8 103.0 92.5

2.297 2.435 2.582 2.812 3.062

3.900 3.749 3.582 3.316 3.062

2.825 3.436 5.26 25.0 ∞

0.48 0.38 0.31 0.25

0.078 0.069 0.059

T, K

38 40 42 44 44.4c

10.78 14.62 19.39 25.22 26.53

∞

*Values extracted and in some cases rounded off from those cited in Rabinovich (ed.), Thermophysical Properties of Neon, Argon, Krypton and Xenon, Standards Press, Moscow, 1976. m = melting point; c = critical point. The notation 6.654.−4 signifies 6.654 × 10−4. This source contains values for the compressed state up to 1000 bar, etc. This book was published in English translation by Hemisphere, New York 1988 (604 pp.). Saturation and superheat tables and a diagram to 200 bar, 320 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (173 pp.). Saturation and superheat tables to 60,000 psia, 900°R and a chart to 4000 psia, 560°R appear in Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat, thermal conductivity, and viscosity, see Thermophysical Properties of Refrigerants, ASHRAE, 1993. 2-256

THERMODYNAMIC PROPERTIES TABLE 2-287

2-257

Compressed Neon* Pressure, bar

Temperature, K

1

10

20

40

60

80

100

200

400

600

800

1000

v 100 h s

0.4117 195.4 6.129

0.0410 194.0 5.168

0.0204 192.4 4.869

0.0102 189.4 4.556

6.76.−3 186.6 4.363

5.08.−3 184.0 4.221

4.09.−3 181.6 4.106

2.22.−3 174.1 3.739

1.42.−3 173.3 3.386

1.18.−3 180.0 3.197

1.06.−3 189.2 3.066

9.74.−4 199.2 2.964

v 200 h s

0.8243 298.5 6.844

0.0828 298.4 5.893

0.0416 298.4 5.605

0.0210 298.2 5.315

0.0142 298.2 5.143

0.0107 298.2 5.020

8.69.−3 298.3 4.924

4.61.−3 299.4 4.620

2.61.−3 304.6 4.308

1.95.−3 312.4 4.124

1.63.−3 321.8 3.994

1.43.−3 332.1 3.893

v 300 h s

1.236 401.6 7.262

0.1241 401.8 6.312

0.0624 402.2 6.026

0.0315 402.8 5.739

0.0212 403.5 5.570

0.0160 404.1 5.450

0.0129 404.9 5.357

6.77.−3 408.8 5.065

3.71.−3 417.8 4.769

2.69.−3 427.8 4.593

2.18.−3 438.5 4.469

1.87.−3 449.7 4.372

v 400 h s

1.648 504.6 7.558

0.1654 505.0 6.609

0.0830 505.4 6.323

0.0418 506.4 6.037

0.0281 507.4 5.896

0.0212 508.3 5.750

0.0171 509.3 5.657

8.88.−3 514.4 5.369

4.77.−3 525.3 5.078

3.40.−3 536.7 4.907

2.72.−3 548.4 4.785

2.30.−3 560.2 4.690

v 500 h s

2.060 607.6 7.788

0.2066 608.1 6.839

0.1036 608.6 6.553

0.0521 609.7 6.267

0.0350 610.8 6.100

0.0264 611.9 5.981

0.0213 613.0 5.889

0.0110 618.8 5.601

5.82.−3 630.7 5.313

4.10.−3 642.9 5.144

3.24.−3 655.2 5.023

2.73.−3 667.5 4.929

v 600 h s

2.472 710.6 7.975

0.2478 711.1 7.027

0.1242 711.7 6.741

0.0625 712.9 6.455

0.0419 714.1 6.288

0.0316 715.3 6.169

0.0254 716.5 6.077

0.0130 722.5 5.791

6.85.−3 735.0 5.504

4.80.−3 747.8 5.335

3.77.−3 760.5 5.215

3.15.−3 773.2 5.122

v 700 h s

2.884 813.5 8.134

0.2890 814.1 7.186

0.1449 814.7 6.900

0.0728 816.0 6.614

0.0487 817.2 6.447

0.0367 818.5 6.328

0.0295 819.7 6.236

0.0151 826.0 5.950

7.89.−3 838.9 5.664

5.49.−3 851.9 5.496

4.29.−3 865.0 5.376

3.57.−3 878.1 5.284

v 800 h s

3.296 916.5 8.272

0.3302 917.1 7.323

0.1655 917.7 7.038

0.0831 919.0 6.752

0.0556 920.3 6.585

0.0419 921.6 6.466

0.0336 922.9 6.374

0.0172 929.3 6.088

8.92.−3 942.4 5.802

6.18.−3 955.7 5.634

4.81.−3 969.0 5.515

3.98.−3 982.2 5.423

v 900 h s

3.708 1020 8.393

0.3714 1020 7.444

0.1861 1021 7.159

0.0934 1022 6.873

0.0625 1023 6.706

0.0470 1025 6.588

0.0378 1026 6.496

0.0192 1033 6.210

9.96.−3 1046 5.924

6.87.−3 1059 5.756

5.32.−3 1073 5.637

4.40.−3 1086 5.545

v 1000 h s

4.120 1123 8.502

0.4126 1123 7.553

0.2067 1124 7.267

0.1037 1125 6.982

0.0693 1126 6.815

0.0522 1128 6.696

0.0419 1129 6.604

0.0213 1136 6.318

0.0110 1149 6.032

7.56.−3 1163 5.856

5.84.−3 1176 5.746

4.81.−3 1190 5.654

*Values extracted and in some cases rounded off from those cited in Rabinovich (ed.), Thermophysical Properties of Neon, Argon, Krypton and Xenon, Standards Press, Moscow, 1976. This source contains an exhaustive tabulation of values. v = specific volume, m3/kg; h = specific enthalpy, kJ/kg; s = specific entropy, kJ/(kg⋅K). The notation 6.76.−3 signifies 6.76 × 10−3. This book was published in English translation by Hemisphere, New York, 1988 (604 pp.). TABLE 2-288

Saturated Nitrogen (R728)*

T, K

P, bar

vf, 10–3 m3/kg

63.15t 65 70 75 77.35

0.1253 0.1743 0.3859 0.7609 1.0133

1.155 1.165 1.193 1.224 1.239

80 85 90 95 100

1.369 2.287 3.600 5.398 7.775

1.258 1.297 1.340 1.390 1.447

vg, m3/kg 1477 1091 525.6 281.8 216.9 164.0 101.7 66.28 44.87 31.26

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µf, 10−4 Pa⋅s

kf, W/(m⋅K)

−148.5 −144.9 −135.2 −125.4 −120.8

64.1 65.8 70.5 74.9 76.8

2.459 2.516 2.657 2.789 2.849

5.826 5.757 5.595 5.460 5.404

1.928 1.930 1.937 1.948 1.955

2.74 2.17 1.77 1.60

0.170 0.160 0.151 0.141 0.136

−115.6 −105.7 −95.6 −85.2 −74.5

78.9 82.3 85.0 86.8 87.7

2.913 3.032 3.147 3.256 3.363

5.345 5.244 5.152 5.067 4.985

1.964 1.989 2.028 2.086 2.176

1.48 1.27 1.10 0.97 0.87

0.132 0.123 0.114 0.105 0.097

0.79 0.71 0.60 0.48 0.32

0.088 0.080 0.071 0.063 0.052

105 110 115 120 125

10.83 14.67 19.40 25.15 32.05

1.514 1.597 1.714 1.892 2.324

22.23 15.98 11.47 8.031 5.016

−63.8 −51.4 −38.1 −21.4 5.1

87.4 85.6 81.8 74.3 57.2

3.469 3.575 3.687 3.821 4.024

4.904 4.820 4.729 4.619 4.444

2.319 2.566 3.063

126.25c

33.96

3.289

3.289

34.8

34.8

4.252

4.252

∞

∞

*Reproduced and converted from Vasserman and Rabinovich, Thermophysical Properties of Liquid Air and Its Components, Standartov, Moscow, 1968; and Israel Program for Scientific Translations, TT 69-55092, 1970. t = triple point; c = critical point. Other extensive tables are given by Angus, S., International Thermodynamic Tables of the Fluid State—6. Nitrogen, Pergamon, 1977 (244 pp.); Hanley, H. J. M., R. D. McCarty, et al., J. Phys. Chem. Ref. Data, 3 (1974): 979–1019. Saturation and superheat tables to 30,000 psia and a chart to 10,000 psia, all to 860°R, appear in Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat, thermal conductivity, and viscosity, see Thermophysical Properties of Refrigerants, ASHRAE, 1993. The 1993 ASHRAE Handbook—Fundamentals (SI ed.) has a thermodynamic chart for pressures from 0.1 to 800 bar and temperatures from 80 to 500 K.

2-258

FIG. 2-13

4729.)

PHYSICAL AND CHEMICAL DATA

Temperature-entropy diagram for nitrogen. Section of T-S diagram for nitrogen by E. S. Burnett, 1950. (Reprinted from U.S. Bur. Mines Rep. Invest.

TABLE 2-289

Thermophysical Properties of Nitrogen (R728) at Atmospheric Pressure

T (K) v (m3/ kg) h (kJ/ kg) s (kJ/ kg·K) cp (kJ/ kg·K) Z vs (m/s) η (10−6 Pa·s) k (W/m·K) NPr

77.4b 0.2164 76.7 5.403 1.341 0.9545 172 5.0 0.0074 0.913

80 0.2252 80.0 5.446 1.196 0.9610 177 5.2 0.0077 0.811

100 0.2871 101.9 5.690 1.067 0.9801 202 6.7 0.0098 0.728

120 0.3474 123.1 5.884 1.056 0.9883 222 8.0 0.0117 0.727

140 0.4071 144.2 6.046 1.050 0.9927 240 9.3 0.0136 0.723

160 0.4664 165.2 6.186 1.047 0.9952 257 10.6 0.0154 0.721

180 0.5255 186.1 6.309 1.045 0.9967 273 11.8 0.0171 0.720

200 0.5845 207.0 6.419 1.043 0.9977 288 12.9 0.0187 0.719

220 0.6434 227.8 6.519 1.043 0.9984 302 14.0 0.0203 0.718

240 0.7023 248.7 6.609 1.042 0.9990 316 15.0 0.0218 0.717

T (K) v (m3/ kg) h (kJ/ kg) s (kJ/ kg·K) cp (kJ/ kg·K) Z vs (m/s) η (10−6 Pa·s) k (W/m·K) NPr

260 0.7611 269.5 6.693 1.042 0.9994 329 16.0 0.0232 0.717

280 0.8199 290.3 6.770 1.041 0.9997 341 17.0 0.0247 0.716

300 0.8786 311.2 6.842 1.041 0.9998 359 17.9 0.0260 0.716

320 0.9371 332.0 6.909 1.042 0.9999 365 18.8 0.0273 0.717

340 0.9960 352.8 6.972 1.042 1.0000 376 19.7 0.0286 0.717

360 1.0546 373.7 7.032 1.043 1.0001 387 20.5 0.0299 0.717

380 1.1134 394.5 7.088 1.044 1.0002 397 21.4 0.0311 0.717

400 1.1719 411.5 7.142 1.045 1.0002 408 22.2 0.0324 0.717

420 1.2305 436.3 7.193 1.047 1.0002 417 23.0 0.0336 0.717

440 1.2892 457.3 7.242 1.048 1.0003 427 23.8 0.0347 0.717

T (K) v (m3/ kg) h (kJ/ kg) s (kJ/ kg·K) cp (kJ/ kg·K) Z vs (m/s) η (10−6 Pa·s) k (W/m·K) NPr

460 1.3481 478.3 7.288 1.051 1.0003 437 24.5 0.0359 0.718

480 1.4065 499.3 7.333 1.053 1.0004 446 25.3 0.0371 0.718

500 1.4654 520.4 7.376 1.056 1.0004 455 26.0 0.0383 0.718

600 1.758 626.9 7.570 1.075 1.000 496 29.5 0.0440 0.722

700 2.052 735.6 7.738 1.098 1.000 534 32.8 0.0496 0.726

800 2.344 846.6 7.886 1.122 1.000 568 35.9 0.0551 0.730

900 2.636 960.0 8.019 1.146 1.000 601 38.8 0.0606 0.734

1000 2.931 1075.7 8.141 1.167 1.001 631 41.6 0.658 0.737

1500 4.396 1680.5 8.630 1.244 1.001 765

2000 5.862 2313.5 8.995 1.284 1.001 879

b = normal boiling point. TABLE 2-290

Saturated Nitrogen Tetroxide vg, m3/kg

Mf

Mg

299.32 309.57 326.66 337.43 345.45

0.000 0.000 0.000 0.000 0.000

694 711 733 749 762

0.2996 0.1630 0.0876 0.0608 0.0469

91.857 91.886 91.766 91.625 91.488

79.157 76.503 73.538 71.748 70.480

10 15 20 30 40

351.88 364.09 373.17 386.57 396.52

0.000 0.000 0.000 0.000 0.000

774 800 822 863 903

0.0382 0.0262 0.0199 0.0133 0.0098

91.346 90.979 90.601 89.823 89.018

69.483 67.742 66.547 64.997 64.099

50 60 80 100

404.50 411.20 422.07 430.76

0.000 0.000 0.001 0.001

945 993 129 577

0.00761 0.00607 0.00394 0.00209

88.191 87.344 85.602 83.817

63.532 63.181 62.959 63.366

Pressure, bar

Temperature, K

1.0133 2 4 6 8

vf, m3/kg

Condensed from McCarty, R. D., H-U. Steurer, et al., NBS IR 86 - 3054, 1986 (106 pp.). M = mol wt for the reaction N2O4 A 2NO2 A 2NO + O2. No derived thermodynamic functions were tabulated due to unduly large differences in literature values, but 92 references are given. TABLE 2-291

Saturated Nitrous Oxide

Temp., °F

Pressure, psia

vf, ft3/lbm

vg, ft3/lbm

h f, Btu/lbm

h g, Btu/lbm

s f, Btu/lbm°R

sg, Btu/lbm°R

−127.2 −100 −80 −60 −40

14.70 33.68 56.79 90.29 136.68

0.01310 0.01358 0.01398 0.01444 0.01495

5.069 2.374 1.463 0.939 0.648

0.0 11.7 20.8 30.2 40.3

161.7 165.9 168.8 171.5 173.7

0.0000 0.0304 0.0534 0.0782 0.1044

0.4864 0.4591 0.4433 0.4315 0.4222

−20 0 20 40 60

198.62 278.97 380.88 507.51 662.69

0.01555 0.01625 0.01711 0.01819 0.01968

0.450 0.316 0.227 0.164 0.117

50.6 60.5 70.2 80.3 91.9

175.3 176.2 176.2 175.0 172.1

0.1296 0.1518 0.1718 0.1920 0.2145

0.4133 0.4036 0.3928 0.3815 0.3687

0.0222 0.0247 0.0354

0.0792 0.0611 0.0354

105.7 114.7 136.4

165.0 157.5 136.4

0.2382 0.2523 0.2890

0.3480 0.3302 0.2890

80 90 97.6c

851.5 961.0 1052.2

Rounded and condensed from Couch, E. J. and K. A. Kobe, Univ. Texas Rep., Cont. DAI-23-072-ORD-685, June 1, 1956. c = critical point. 2-259

2-260

PHYSICAL AND CHEMICAL DATA

Mollier diagram for nitrous oxide. (Fig. 9, Univ. Texas Rep., Cont. DAI-23-072-ORD685, June 1, 1956, by Couch and Kobe. Reproduced by permission.) Some irregularity in the compressibility factors from 80 to 160 atm, 50 to 100°C exists (Couch, private communication, 1967). See Couch et al., J. Chem. Eng. Data, 6, (1961) for PVT data.

FIG. 2-14

THERMODYNAMIC PROPERTIES TABLE 2-292

2-261

Nonane* h g, kJ/kg

s f, kJ/(kg⋅K)

s g, kJ/(kg⋅K)

c pf, kJ/(kg⋅K)

µ f, 10−4 Pa⋅s

k f, W/(m⋅K)

358.4 359.2 400.6 442.2 484.8

828.7 859.4

2.424 2.427 2.607 2.774 2.932

4.210 4.243

2.07 2.07 2.08 2.10 2.16

33.5 33.0 17.9 12.1 8.7

0.150 0.150 0.145 0.140 0.134

30.35 10.19 4.00 1.80 0.894

528.6 573.8 622.0 671.3 722.5

891.7 925.6 961.1 998.2 1036.5

3.083 3.229 3.370 3.511 3.650

4.282 4.324 4.368 4.419 4.476

1.591.−3 1.637.−3 1.690.−3 1.748.−3 1.815.−3

0.485 0.286 0.161 0.104 0.069

776.7 833.3 890.2 950.3 1012.1

1076.0 1116.6 1157.1 1199.2 1241.3

3.788 3.927 4.053 4.186 4.316

4.536 4.601 4.660 4.727 4.794

5.309 7.437 10.20 13.76 18.02

1.895.−3 2.00.−3 2.13.−3 2.35.−3 2.78.−3

0.045 0.030 0.021 0.013 0.008

1076.2 1141.3 1207.7 1275.4 1342.9

1282.9 1324.5 1363.8 1338.7 1318.1

4.444 4.569 4.691 4.811 4.927

4.857 4.921 3.980 5.029 5.056

22.90

4.23.−3

0.004

1305.2

1305.2

5.032

5.032

vf, m3/kg

T, K

P, bar

219.7t 220 240 260 280

2.6.−6 2.7.−6 3.74.−5 2.97.−4 1.61.−3

300 320 340 360 380

6.40.−3 0.0203 0.0547 0.1279 0.2678

1.404.−3 1.436.−3 1.471.−3 1.508.−3 1.548.−3

400 420 440 460 480

0.513 0.911 1.521 2.401 3.639

500 520 540 560 580 594.6c

vg, m3/kg

h f, kJ/kg

2.22 2.30

6.53 5.13 4.16 3.44 2.91

0.129 0.123 0.118 0.112 0.107

2.50 2.18

0.101 0.096 0.092 0.089 0.085 0.082

*Values of p and v interpolated and converted from tables in Vargaftik, Handbook of Thermophysical Properties of Gases and Liquids, Hemisphere, Washington, and McGraw-Hill, New York, 1975. Values of h and s calculated from API tables published by Texas A&M University, College Station. t = triple point; c = critical point.

TABLE 2-293

Octane* h g, kJ/kg

s f, kJ/(kg⋅K)

s g, kJ/(kg⋅K)

c pf, kJ/(kg⋅K)

µ f, 10−3 Pa⋅s

k f, W/(m⋅K)

365.9 373.2 414.1 455.8 498.4

811.4 842.1 873.5

2.487 2.520 2.698 2.865 3.023

4.207 4.259 4.312

2.033 2.035 2.059 2.105 2.165

2.25 2.01 1.24 0.87 0.65

0.149 0.148 0.143 0.138 0.133

10.7 4.01 1.752 0.844 0.448

542.4 589.8 637.9 687.1 737.7

906.2 939.8 974.6 1010.4 1047.3

3.175 3.325 3.471 3.611 3.747

4.366 4.419 4.461 4.509 4.562

2.231

0.504 0.405 0.334 0.282 0.244

0.128 0.123 0.118 0.112 0.107

1.632.−3 1.685.−3 1.747.−3 1.818.−3 1.904.−3

0.252 0.155 0.100 0.066 0.045

790.1 843.1 897.5 954.8 1013.5

1084.8 1123.6 1162.5 1202.0 1241.8

3.881 4.010 4.137 4.264 4.388

4.617 4.677 4.740 4.802 4.864

0.200 0.167 0.143 0.121 0.103

0.102 0.099 0.095 0.091 0.087

2.013.−3 2.16.−3 2.37.−3 2.81.−3 4.26.−3

0.031 0.021 0.014 0.008 0.004

1072.8 1136.0 1201.5 1276.7 1331.7

1281.2 1318.6 1352.4 1370.4 1331.7

4.508 4.629 4.749 4.880 4.977

4.924 4.980 5.028 5.048 4.977

0.086 0.072 0.058 0.044

0.083

vf, m3/kg

T, K

P, bar

216.4t 220 240 260 280

1.49.−5 2.41.−5 2.18.−4 0.0014 0.0061

1.353.−3 1.368.−3 1.384.−3

300 320 340 360 380

0.0207 0.0575 0.1384 0.3000 0.5856

1.420.−3 1.457.−3 1.495.−3 1.536.−3 1.582.−3

400 420 440 460 480

1.0507 1.758 2.797 4.246 6.201

500 520 540 560 568.8c

8.785 12.15 16.46 21.98 24.97

vg, m3/kg

700 125 31.9

h f, kJ/kg

*Values of p and v interpolated and converted from tables in Vargaftik, Handbook of Thermophysical Properties of Gases and Liquids, Hemisphere, Washington, and McGraw-Hill, New York, 1975. Values of h and s calculated from API tables published by Texas A&M University, College Station. t = triple point; c = critical point. Saturation and superheat tables and a diagram to 100 bar, 680 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (173 pp.).

2-262

PHYSICAL AND CHEMICAL DATA

TABLE 2-294

Saturated Oxygen (R732)*

T, K

P, bar

vf, 10–3 m3/kg

54.35 t 55 60 65 70

0.0015 0.0018 0.0073 0.0233 0.0624

0.776 0.778 0.790 0.802 0.816

75 80 85 90 90.18

0.1448 0.3003 0.5677 0.9943 1.0133

0.827 0.845 0.862 0.880 0.881

1.634 2.547 3.794 5.443 7.559

0.899 0.920 0.944 0.970 0.998

95 100 105 110 115

vg, 10–3 m3/kg 93980 77920 21240 7200 2894 1330 680.7 379.7 227.1 223.2 143.9 95.46 65.81 46.81 34.15

cpf, kJ/(kg⋅K)

µ f, 10−4 Pa⋅s

kf , W/(m⋅K)

5.642 5.510 5.397 5.301 5.297

1.570 1.589 1.607 1.625 1.626

3.04 2.54 2.16 1.88 1.87

0.170 0.164 0.157 0.151 0.151

3.045 3.113 3.196 3.276 3.354

5.216 5.141 5.073 5.009 4.950

1.645 1.672 1.706 1.752 1.814

1.66 1.51 1.34 1.20 1.07

0.144 0.138 0.131 0.125 0.118

h f, kJ/kg

hg, kJ/kg

s f, kJ/(kg⋅K)

s g, kJ/(kg⋅K)

−189.8 −188.9 −181.1 −173.3 −165.5

48.9 49.5 53.8 58.1 62.4

2.156 2.172 2.308 2.432 2.545

6.548 6.507 6.223 5.992 5.801

−159.2 −149.7 −141.7 −133.7 −133.4

66.6 70.8 74.9 78.8 78.9

2.631 2.754 2.849 2.940 2.943

−125.4 −117.1 −108.6 −99.9 −90.0

82.4 85.7 88.5 90.8 92.6

120 125 130 135 140

10.21 13.48 17.44 22.19 27.82

1.031 1.070 1.116 1.170 1.237

25.42 19.21 14.67 11.25 8.612

−81.6 −71.8 −61.5 −50.6 −38.9

93.6 93.9 93.3 91.6 88.4

3.432 3.510 3.588 3.667 3.748

4.892 4.836 4.779 4.720 4.657

1.896 2.004 2.148 2.341 2.629

0.97 0.86 0.78 0.70 0.60

0.111 0.103 0.096 0.088 0.080

145 150 154.77c

34.45 42.23 50.87

1.332 1.487 2.464

6.499 4.705 2.464

−25.9 −10.8 35.2

82.9 73.1 35.2

3.833 3.928 4.219

4.583 4.487 4.219

3.141 3.935 ∞

0.52

0.072 ∞

*Reproduced and converted from Vasserman and Rabinovich, Thermophysical Properties of Liquid Air and Its Components, Standartov, Moscow, 1968; and Israel Program for Scientific Translations, TT 69-55092, 1970. t = triple point; c = critical point. Other tables are given by Sytchev, V. V., A. A. Vasserman, et al., Thermodynamic Properties of Oxygen, Hemisphere, New York, 1987 (307 pp.); Stewart, R. B., R. T. Jacobsen, et al., J. Phys. Chem. Ref. Data, 20, 5 (1991): 917–1021; For fps units, see Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat, thermal conductivity, and viscosity, see Thermophysical Properties of Refrigerants, ASHRAE, 1993. See also Roder, H. M., Transport Properties of Oxygen, NASA Ref. Publ. 1102, 1983 (87 pp.); Laesecke, A., K. Krauss, et al., J. Phys. Chem. Ref. Data, 19, 5 (1990): 1089–1122.

THERMODYNAMIC PROPERTIES

FIG. 2-15 Temperature-entropy chart for oxygen. Pressure P, atm; density ρ, (g⋅mol)/L; temperature, K; enthalpy H, J/(g⋅mol); entropy, J/(g⋅mol⋅K). (NBS Chart D-56. Reproduced by permission.)

2-263

2-264

PHYSICAL AND CHEMICAL DATA

Enthalpy-concentration diagram for oxygen-nitrogen mixture at 1 atm. Reference states: Enthalpies of liquid oxygen and liquid nitrogen at the normal boiling point of nitrogen are zero. (Dodge, Chemical Engineering Thermodynamics, McGraw-Hill, New York, 1944.) Wilson, Silverberg, and Zellner, AFAPL TDR 64-64 (AD 603151), 1964, p. 314, present extensive vapor-liquid equilibrium data for the three-component system argon-nitrogen-oxygen as well as for binary systems including oxygen-nitrogen.

FIG. 2-16

TABLE 2-295

Pentane

Canjar and Manning (Thermodynamic Properties and Reduced Correlations for Gases, Gulf, Houston, 1967) give extensive tables and an enthalpy–log-pressure diagram, based upon Brydon, Walen, and Canjar [Chem. Eng. Prog. Symp. Ser., 49, 7, (1951): 151–157]. For isopentane, Arnold, Liou, and Eldridge [ J. Chem. Eng. Data, 10, 88 (1965)] used the Benedict-WebbRubin equation to generate information to 600°F and 60 atm. Das and Kuloor used the same equation in Ind. J. Technol., 5, 46 (1967) to calculate information up to 1500 K and 1000 atm. Saturation and superheat tables and a diagram to 200 bar, 600 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (173 pp.). For equations, see Grigoryev, B. A., Yu. L. Rastorguyev, et al., Int. J. Thermophys., 11, 3 (1990): 487–502.

TABLE 2-296 T, K m

336.4 400 500 600 700 800 1000 1200 1400 1500

Saturated Potassium*

P, bar

vf, m3/kg

1.37.−9 1.84.−7 3.13.−5 9.26.−4 0.01022

0.001208 0.001229 0.001266 0.001304 0.001346

0.06116 0.7322 3.913 12.44 20.0

0.001389 0.001488 0.001605 0.001742 0.001816

vg, m3/kg

h f, kJ/kg

h g, kJ/kg

sf, kJ/(kg⋅K)

s g, kJ/(kg⋅K)

c pf, kJ/(kg⋅K)

4.64.+6 3.39.+4 3164 142.3

93.8 145.5 225.1 302.7 379.4

2327 2342 2390 2433 2468

1.928 2.068 2.246 2.388 2.506

8.567 7.559 6.576 5.937 5.490

0.822 0.805 0.785 0.771 0.762

455.5 609.7 773.5 948.0 1040.0

2498 2552 2610 2679 2718

2.608 2.780 2.929 3.063 3.123

5.161 4.722 4.459 4.299 4.209

0.761 0.792 0.846 0.899 0.924

26.75 2.691 0.584 0.207 0.132

*Converted from tables in Vargaftik, Tables of the Thermophysical Properties of Liquids and Gases, Nauka, Moscow, 1972; and Hemisphere, Washington, 1975. m = melting point. The notation 1.37.−9 signifies 1.37 × 10−9. Many of the Vargaftik values also appear in Ohse, R. W., Handbook of Thermodynamic and Transport Properties of Alkali Metals, Blackwell Sci. Pubs., Oxford, 1985 (1020 pp.). This source contains superheat data. Saturation and superheat tables and a diagram to 30 bar, 1650 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (173 pp.). For a Mollier diagram from 0.1 to 250 psia, 1300 to 2700°R, see Weatherford, W. D., J. C. Tyler, et al., WADD-TR61-96, 1961. An extensive review of properties of the solid and the saturated liquid is given by Alcock, C. B., M. W. Chase, et al., J. Phys. Chem. Ref. Data, 23, 3 (1994): 385–497.

THERMODYNAMIC PROPERTIES

2-265

FIG. 2-17 Mollier diagram for potassium. Basis: enthalpy = 0.0 cal/g atom at 298 K; entropy = 15.8 cal/(g atom·K) at 298 K. (Aerojet-General Rep. AGN8194, vol. 2, 1967. Reproduced by permission.)

2-266

PHYSICAL AND CHEMICAL DATA

TABLE 2-297

Saturated Propane (R290)* vg, m3/kg

µ f, 10−4 Pa⋅s

k f, W/(m⋅K)

1.96 1.98 2.00 2.02 2.04

6.61 5.54 4.67

0.191 0.183 0.175

5.9862 5.9114 5.8502 5.8005 5.7603

2.07 2.10 2.13 2.16 2.20

3.97 3.27 2.98 2.65 2.36

0.166 0.158 0.150 0.143 0.136

3.8631 3.9605 4.0563 4.1505 4.2433

5.7280 5.7022 5.6817 5.6656 5.6528

2.25 2.29 2.34 2.41 2.48

2.07 1.86 1.69 1.53 1.40

0.129 0.123 0.117 0.111 0.106

906.03 916.54 926.41 935.45 943.38

4.3349 4.4257 4.5160 4.6062 4.6971

5.6426 5.6343 5.6270 5.6200 5.6124

2.56 2.65 2.76 2.89 3.06

1.29 1.19 1.10 0.93 0.82

0.100 0.096 0.091 0.086 0.082

949.79 953.92 954.23 946.56 879.20

4.7896 4.8850 4.9861 5.0997 5.3300

5.6030 5.5896 5.5681 5.5277 5.3300

3.28 3.62 4.23 5.98 ∞

0.72 0.62 0.52 0.40 0.29

0.078 0.073 0.069 0.066 ∞

T, K

P, bar

v f, m3/kg

h f, kJ/kg

h g, kJ/kg

s f, kJ/(kg⋅K)

s g, kJ/(kg⋅K)

c pf, kJ/(kg⋅K)

85.5t 90 100 110 120

3.0.−9 1.5.−8 3.2.−7 3.9.−6 3.1.−5

1.364.−3 1.373.−3 1.392.−3 1.412.−3 1.432.−3

5.37.+7 1.12.+7 5.85.+5 53275 7350

124.92 133.56 152.74 172.03 191.46

690.02 693.58 702.23 711.71 721.78

1.8738 1.9723 2.1743 2.3581 2.5271

8.3548 8.0953 7.6163 7.2377 6.9343

1.92 1.92 1.93 1.94 1.95

130 140 150 160 170

1.8.−4 7.7.−4 2.74.−3 8.22.−3 0.0214

1.453.−3 1.475.−3 1.497.−3 1.521.−3 1.545.−3

1400 344 103 36.8 15.0

211.03 230.77 250.67 270.78 291.10

732.27 743.07 754.12 765.37 776.80

2.6838 2.8300 2.9674 3.0971 3.2202

6.6885 6.4881 6.3237 6.1886 6.0775

180 190 200 210 220

0.0495 0.1035 0.1993 0.3574 0.6031

1.570.−3 1.597.−3 1.625.−3 1.654.−3 1.686.−3

6.84 3.43 1.868 1.087 0.669

311.66 332.48 353.61 375.07 396.90

788.40 800.15 812.03 824.01 836.04

3.3377 3.4503 3.5586 3.6631 3.7645

230 240 250 260 270

0.9661 1.4800 2.1819 3.1118 4.3120

1.719.−3 1.754.−3 1.792.−3 1.833.−3 1.878.−3

0.432 0.290 0.2020 0.1445 0.1059

419.16 442.07 465.58 489.70 514.45

848.08 860.07 871.94 883.62 895.02

280 290 300 310 320

5.8278 7.7063 9.9973 12.75 16.03

1.927.−3 1.982.−3 2.044.−3 2.115.−3 2.200.−3

0.0791 0.0600 0.0461 0.0357 0.0279

539.88 566.06 593.11 621.18 650.49

330 340 350 360 369.8c

19.88 24.36 29.56 35.55 42.42

2.301.−3 2.430.−3 2.607.−3 2.896.−3 4.566.−3

0.0218 0.0170 0.0130 0.0095 0.0046

681.37 714.38 750.52 792.50 879.20

*Values converted and mostly rounded off from those of Goodwin, NBSIR 77-860, 1977. t = triple point; c = critical point. The notation 3.0.−9 signifies 3.0 × 10−9. Later tables for the same temperature range for saturation and for the superheat state from 0.1 to 1000 bar, 85.5 to 600 K, were published by Younglove, B. A. and J. F. Ely, J. Phys. Chem. Ref. Data, 16, 4 (1987): 685–721, but the lower temperature saturation tables contain some errors. Saturation and superheat tables and a chart to 10,000 psia, 800°F appear in Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For thermodynamic properties for 0.1 to 1000 bar, 100 to 700 K, see Sychev, V. V., A. A. Vasserman, et al., Thermodynamic Properties of Propane, Hemisphere, New York, NY, 1991 (275 pp.). Saturation and superheat tables and a diagram to 200 bar, 600 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (173 pp.). For specific heat, thermal conductivity, and viscosity, see Thermophysical Properties of Refrigerants, ASHRAE, 1993.

THERMODYNAMIC PROPERTIES TABLE 2-298

2-267

Saturated Propylene (Propene, R1270)

T, K

P, bar

v f, m3/kg

vg, m3/kg

h f, kJ/kg

hg, kJ/ kg

s f, kJ/(kg⋅K)

s g, kJ/(kg⋅K)

c pf, kJ/(kg⋅K)

87.9t 90 100 110 120

9.54.−9 2.05.−8 4.81.−7 6.08.−6 4.88.−5

0.001 301 0.001 305 0.001 325 0.001 346 0.001 367

1.82.+7 8.66.+6 411 165 35 753 4 856

−290.1 −285.1 −265.4 −247.7 −229.8

279.2 281.1 290.2 299.6 309.3

−1.923 −1.867 −1.659 −1.490 −1.335

4.554 4.424 3.897 3.488 3.158

1.695 1.760 1.820

130 140 150 160 170

2.77.−4 1.20.−3 4.17.−3 0.0122 0.0309

0.001 389 0.001 411 0.001 434 0.001 458 0.001 483

927.0 230.91 71.043 25.903 10.842

−211.4 −192.4 −172.9 −153.1 −133.1

319.3 329.4 339.8 350.4 361.2

−1.187 −1.046 −0.912 −0.784 −0.663

2.895 2.681 2.506 2.363 2.245

180 190 200 210 220

0.0697 0.1425 0.2686 0.4727 0.7849

0.001 508 0.001 535 0.001 563 0.001 593 0.001 624

5.080 2.613 1.452 0.860 0.538

−112.7 −92.2 −71.4 −50.3 −28.8

372.1 383.1 394.2 405.3 416.3

−0.547 −0.436 −0.329 −0.226 −0.127

225.5 b 230 240 250 260

1.0133 1.2401 1.8775 2.7401 3.8737

0.001 642 0.001 657 0.001 693 0.001 732 0.001 774

0.4241 0.3515 0.2388 0.1674 0.1206

−16.9 −7.0 15.3 38.0 61.3

422.2 427.1 437.8 448.2 458.2

270 280 290 300 310

5.3269 7.1499 9.3954 12.12 15.38

0.001 820 0.001 872 0.001 929 0.001 995 0.002 071

0.0888 0.0666 0.0507 0.0390 0.0303

85.2 109.9 135.3 161.6 189.0

320 330 340 350 360

19.23 23.75 29.01 35.12 42.20

0.002 162 0.002 273 0.002 418 0.002 628 0.003 038

0.0236 0.0184 0.0142 0.0107 0.0075

365.6 c

46.65

0.004 476

0.0045

µ f, 10−6 Pa·s

k f, W/(m⋅K)

Pr

2017 1526 1185

0.214 0.209 0.204

15.98 12.85 10.57

1.875 1.923 1.964 1.996 2.020

941 735 587 478 397

0.198 0.193 0.188 0.183 0.178

8.91 7.32 6.13 5.21 4.50

2.147 2.066 1.999 1.943 1.896

2.044 2.067 2.094 2.128 2.162

334.5 286.1 244.9 212.7 187.0

0.173 0.168 0.162 0.157 0.152

3.95 3.52 3.17 2.88 2.66

−0.073 −0.030 0.064 0.157 0.247

1.874 1.857 1.825 1.797 1.774

2.182 2.199 2.243 2.298 2.369

175.0 166.2 149.2 135.0 123.0

0.149 0.147 0.142 0.137 0.131

2.56 2.49 2.36 2.26 2.22

467.8 476.9 485.3 492.8 499.3

0.336 0.425 0.512 0.600 0.688

1.753 1.735 1.719 1.704 1.688

2.418 2.494 2.584 2.693 2.842

112.9 106.6 100.0 93.0 85.6

0.126 0.121 0.116 0.112 0.109

2.17 2.20 2.23 2.24 2.23

217.7 248.2 280.9 317.6 364.1

504.3 507.4 507.6 502.8 486.0

0.776 0.867 0.961 1.062 1.188

1.672 1.652 1.627 1.592 1.527

3.007 3.335 3.723 4.669

77.8 69.6 61.0

0.104 0.097 0.090 0.082

2.25 2.39 2.52

433.3

433.3

1.374

1.374

t = triple point; b = normal boiling point; c = critical point. The notation 9.54.−9 signifies 9.54 × 10−9. hf = sf = 0 at 233.15 K = −40°C. Converted from Angus, S., B. Armstrong, et al., Intnl. Thermodynamic Properties of the Fluid State—7. Propylene (Propene) R1270, Pergamon Press, Oxford, 1980 (401 pp.).

2-268

PHYSICAL AND CHEMICAL DATA

TABLE 2-299

Compressed Propylene (Propene, R1270) Temperature, K

Pressure, bar

225

250

275

300

325

350

375

400

425

450

v (m3/ kg) 1 h (kJ/ kg) s (kJ/ kg⋅K)

0.00164 −17.9 −0.0779

0.4817 455.5 2.0169

0.5334 491.2 2.1530

0.5846 529.1 2.2847

0.6354 569.1 2.4128

0.6858 611.4 2.5380

0.7361 656.0 2.6610

0.7861 702.8 2.7821

0.8361 751.9 2.9010

0.8859 803.2 3.0183

v (m3/ kg) 10 h (kJ/ kg) s (kJ/ kg⋅K)

0.00164 −17.2 −0.0810

0.00173 38.5 0.1535

0.00184 97.6 0.3788

0.04986 501.2 1.7631

0.05670 547.0 1.9653

0.06295 593.2 2.0466

0.06889 640.8 2.1774

0.07460 689.7 2.3042

0.08014 740.6 2.4273

0.08557 793.3 2.5480

v (m3/ kg) 20 h (kJ/ kg) s (kJ/ kg⋅K)

0.00163 −16.3 −0.0841

0.00172 39.3 0.1497

0.00184 98.1 0.3736

0.00198 161.5 0.5941

0.02324 512.7 1.6920

0.02773 568.2 1.8567

0.03149 621.5 2.0024

0.03489 673.5 2.1381

0.03803 726.5 2.2674

0.04104 781.5 2.3923

v (m3/ kg) 40 h (kJ/ kg) s (kJ/ kg⋅K)

0.00163 −14.4 −0.0908

0.00172 40.8 0.1419

0.00182 99.0 0.3638

0.00196 161.3 0.5806

0.00216 230.0 0.8001

0.00256 313.3 1.0466

0.01465 633.6 1.9256

0.01682 695.4 2.0758

0.01872 755.6 2.2131

v (m3/ kg) 60 h (kJ/ kg) s (kJ/ kg⋅K)

0.00162 −12.6 −0.0970

0.00171 42.3 0.1345

0.00181 100.1 0.3546

0.00194 161.5 0.5684

0.00211 228.2 0.7816

0.00240 303.8 1.0055

0.00743 575.4 1.7272

0.00944 656.7 1.9250

0.01126 726.1 2.0832

v (m3/ kg) 80 h (kJ/ kg) s (kJ/ kg⋅K)

0.00162 −10.7 −0.1031

0.00170 44.0 0.1274

0.00180 101.3 0.3458

0.00192 162.0 0.5570

0.00208 227.2 0.7657

0.00231 299.0 0.9781

0.00402 499.7 1.5107

0.00605 607.7 1.7795

0.00757 693.3 1.9693

v (m3/ kg) 100 h (kJ/ kg) s (kJ/ kg⋅K)

0.00161 −8.8 −0.1091

0.00169 45.6 0.1202

0.00179 102.6 0.3374

0.00190 162.7 0.5466

0.00204 226.7 0.7514

0.00224 296.1 0.9570

0.00256 373.5 1.1704

0.00316 464.9 1.4061

0.00426 567.8 1.6456

0.00551 660.1 1.8669

v (m3/ kg) 150 h (kJ/ kg) s (kJ/ kg⋅K)

0.00160 −4.1 −0.1236

0.00167 49.9 0.1038

0.00176 106.1 0.3180

0.00186 165.0 0.5228

0.00198 227.0 0.7214

0.00214 292.8 0.9163

0.00234 362.9 1.1100

0.00262 438.5 1.3049

0.00300 519.5 1.5021

0.00354 604.6 1.6958

v (m3/ kg) 200 h (kJ/ kg) s (kJ/ kg⋅K)

0.00159 0.8 −0.1371

0.00166 54.4 0.0884

0.00174 110.0 0.3004

0.00183 167.9 0.5021

0.00194 228.7 0.6963

0.00207 292.3 0.8852

0.00222 359.2 1.0701

0.00242 429.9 1.2521

0.00266 504.4 1.4319

0.00296 581.4 1.6086

v (m3/ kg) 300 h (kJ/ kg) s (kJ/ kg⋅K)

0.00157 10.9 −0.1625

0.00163 63.8 0.0601

0.00170 118.5 0.2688

0.00178 175.2 0.4660

0.00187 234.2 0.6549

0.00197 295.5 0.8367

0.00208 359.3 1.0127

0.00221 425.7 1.1839

0.00236 494.4 1.3507

0.00253 565.6 1.5133

v (m3/ kg) 400 h (kJ/ kg) s (kJ/ kg⋅K)

0.00155 21.2 −0.1863

0.00161 73.6 0.0347

0.00167 127.7 0.2407

0.00174 183.6 0.4351

0.00182 241.5 0.6207

0.00190 301.5 0.7985

0.00199 363.7 0.9705

0.00209 428.0 1.1362

0.00220 494.4 1.2972

0.00232 562.9 1.4536

v (m3/ kg) 500 h (kJ/ kg) s (kJ/ kg⋅K)

0.00153 31.6 −0.2082

0.00159 83.7 0.0112

0.00165 137.3 0.2155

0.00171 192.6 0.4080

0.00178 249.8 0.5910

0.00185 309.0 0.7664

0.00193 370.1 0.9351

0.00201 433.3 1.0981

0.00210 498.7 1.2559

0.00220 565.4 1.4092

Converted and interpolated from Angus, S., B. Armstrong, et al., International Thermodynamic Tables of the Fluid State—7. Propylene, Pergamon, Oxford, 1980 (401 pp.). The 1993 ASHRAE Handbook—Fundamentals (SI ed.) has a thermodynamic chart for pressures from 0.1 to 1000 bar for temperatures up to 580 K. Saturation and superheat tables and a diagram to 30,000 psia, 580°F appear in Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat, thermal conductivity, and viscosity, see Thermophysical Properties of Refrigerants, ASHRAE, 1993.

THERMODYNAMIC PROPERTIES TABLE 2-300

2-269

Saturated Refrigerant 11*

P, bar

vf, m3/kg

vg, m3/kg

h f, kJ/kg

h g, kJ/kg

s f, kJ/(kg⋅K)

s g, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µ f, 10−4 Pa⋅s

k f, W/(m⋅K)

200 220 240 260 270

0.0043 0.0417 0.0768 0.2215 0.3514

5.901.−4 6.061.−4 6.225.−4 6.398.−4 6.491.−4

28.06 6.272 1.882 0.703 0.458

−14.37 −8.20 4.97 21.01 29.53

186.30 195.89 205.85 216.06 221.23

−0.0651 −0.0361 0.0210 0.0851 0.1172

0.9431 0.8925 0.8581 0.8353 0.8272

0.815 0.828 0.842 0.856 0.863

1.674 1.142 0.831 0.635 0.563

0.115 0.110 0.104 0.098 0.095

280 290 300 310 320

0.5364 0.7917 1.1341 1.5821 2.1556

6.587.−4 6.688.−4 6.794.−4 6.908.−4 7.027.−4

0.309 0.216 0.154 0.113 0.0847

38.25 47.10 56.06 65.10 74.22

226.40 231.58 236.73 241.83 246.88

0.1489 0.1799 0.2102 0.2397 0.2686

0.8209 0.8160 0.8124 0.8099 0.8081

0.870 0.878 0.887 0.897 0.907

0.504 0.454 0.413 0.377 0.346

0.093 0.090 0.087 0.084 0.081

330 340 350 360 380

2.876 3.764 4.845 6.142 9.487

7.156.−4 7.293.−4 7.442.−4 7.603.−4 7.974.−4

0.0645 0.0500 0.0392 0.0311 0.0201

83.42 92.72 102.12 111.64 131.12

251.84 256.69 261.40 265.95 274.40

0.2967 0.3243 0.3513 0.3778 0.4298

0.8071 0.8065 0.8064 0.8065 0.8069

0.917 0.928 0.939 0.950 0.975

0.320 0.297 0.276 0.259 0.229

0.079 0.076 0.073 0.070 0.065

8.435.−4 9.042.−4 9.930.−4 1.167.−3 1.799.−3

0.0134 0.0090 0.0059 0.0036 0.0018

151.38 172.76 196.01 223.85 258.70

281.69 287.20 289.72 285.36 258.70

0.4808 0.5317 0.5840 0.6435 0.7162

0.8066 0.8041 0.7970 0.7773 0.7162

1.004 1.04 1.09 1.19 ∞

0.203 0.169 0.131 0.084 0.033

0.059 0.053 0.048 0.037 ∞

T, K

400 420 440 460 471.2c

14.02 19.98 27.65 37.36 44.09

*Values reproduced or converted from Table 1, p. 17.75, ASHRAE Handbook, 1981: Fundamentals, American Society of Heating, Refrigerating and AirConditioning Engineers, Atlanta, 1981. Copyright material. Reproduced by permission of the copyright owner. c = critical point. The notation 5.901.−4 signifies 5.901 × 10−4. The 1993 ASHRAE Handbook—Fundamentals (SI ed.) gives material for integral degrees Celsius with temperatures on the ITS 90 scale. For experimental isochores for the compressed liquid from 12 to 301 bar, 254 to 453 K, see Blanke, W. and R. Weiss, PTB Bericht W 30, Braunschweig, Germany, 1992 (54 pp.). Equations and constants approximated to 1985 ASHRAE tables are given by Mecarik, K. and M. Masaryk, Heat Recovery Systems and CHP, 11, 2/3 (1991): 193–197. For tables and a chart to 3000 psia, 460°F, see Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For similar material to 80 bar, 650 K, see Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (173 pp.). For specific heat at constant pressure, thermal conductivity, and viscosity in both SI and fps units, see Liley, P. E., Thermophysical Properties of Refrigerants, ASHRAE, Atlanta, GA, 1993.

FIG. 2-18 Enthalpy–log-pressure diagram for Refrigerant 11. 1 MPa = 10 bar. (Copyright 1981 by the American Society of Heating, Refrigerating and Air-Conditioning Engineers and reproduced by permission of the copyright owner.) This chart, redrawn with a different zero point and temperatures in Celsius, appears in ASHRAE Handbook—Fundamentals (SI ed.), Atlanta, GA, 1993.

2-270

PHYSICAL AND CHEMICAL DATA

TABLE 2-301

Saturated Refrigerant 12* h f, kJ/kg

h g, kJ/kg

s f, kJ/(kg⋅K)

s g, kJ/(kg⋅K)

c pf, kJ/(kg⋅K)

µ f, 10−4 Pa⋅s

k f, W/(m⋅K)

294.6 302.3 310.3 318.3 326.5

496.0 500.2 504.5 508.9 513.5

3.492 3.543 3.591 3.637 3.681

4.835 4.780 4.734 4.696 4.665

0.808 0.817 0.827 0.836 0.845

18.9 15.1 12.1 9.69 7.94

0.123 0.119 0.116 0.113 0.109

1.370 0.7589 0.4476 0.2784 0.1811

334.8 343.2 351.8 360.6 369.5

518.1 522.7 527.4 531.1 536.8

3.724 3.765 3.805 3.844 3.881

4.640 4.620 4.603 4.590 4.579

0.855 0.864 0.873 0.882 0.891

6.64 5.65 4.88 4.26 3.77

0.105 0.102 0.098 0.094 0.090

6.810.−4 6.970.−4 7.112.−4 7.282.−4 7.470.−4

0.1225 0.08559 0.06147 0.04543 0.03888

378.0 387.7 397.0 406.5 416.1

541.5 546.1 550.7 555.1 559.4

3.918 3.954 3.989 4.023 4.057

4.570 4.563 4.558 4.554 4.551

0.902 0.913 0.926 0.942 0.959

3.37 3.03 2.75 2.52 2.31

0.087 0.083 0.080 0.076 0.072

T, K

P, bar

vf, m3/kg

v g, m3/kg

150 160 170 180 190

0.00091 0.00305 0.00871 0.02178 0.04877

5.767.−4 5.849.−4 5.926.−4 6.024.−4 6.118.−4

179.12 36.05 13.40 5.666 2.665

200 210 220 230 240

0.0996 0.1879 0.3317 0.5531 0.8781

6.217.−4 6.139.−4 6.431.−4 6.549.−4 6.675.−4

250 260 270 280 290

1.3359 1.959 2.784 3.825 5.184

300 310 320 330 340

6.840 8.860 11.29 14.17 17.58

7.678.−4 7.912.−4 8.173.−4 8.478.−4 8.840.−4

0.02582 0.01992 0.01553 0.01218 0.00957

426.0 436.0 446.2 456.8 467.8

563.5 567.3 570.9 574.0 576.5

4.090 4.122 4.154 4.186 4.218

4.548 4.546 4.543 4.541 4.538

0.979 1.005 1.041 1.093 1.166

2.14 2.00 1.86 1.74 1.60

0.069 0.065 0.061 0.058 0.054

350 360 370 380 385c

21.57 26.19 31.56 37.76 41.31

9.286.−4 9.868.−4 1.072.−3 1.237.−3 1.876.−3

0.00750 0.00582 0.00439 0.00305 0.00188

479.4 492.1 506.4 524.7 551.1

578.2 578.7 577.2 571.2 551.1

4.250 4.285 4.322 4.369 4.437

4.533 4.525 4.514 4.900 4.437

1.264 1.39 1.55

1.45 1.28 1.06 0.75 0.31

0.050 0.046 0.041

∞

∞

*P, v, h, and s data interpolated from Perelshteyn (ed.), Tables and Diagrams of the Thermodynamic Properties of Refrigerants 12, 13, and 22, Moscow, 1971. cp, µ, and k data interpolated and converted from Thermophysical Properties of Refrigerants, American Society of Heating, Refrigerating and Air-Conditioning Engineers, New York, 1976. c = critical point. The notation 5.767.−4 signifies 5.767 × 10−4.

TABLE 2-302

Saturated Refrigerant 13*

P, bar

vf, m3/kg

91 100 110 120 130

3.817.−6 3.418.−5 2.563.−4 0.00137 0.00571

5.367.−4 5.448.−4 5.538.−4 5.635.−4 5.739.−4

140 150 160 170 180

0.01895 0.05250 0.1258 0.2680 0.5186

5.850.−4 5.969.−4 6.095.−4 6.231.−4 6.380.−4

190 200 210 220 230

0.9269 1.5507 2.456 3.712 5.396

T, K

vg, m3/kg

cpf, kJ/(kg⋅K)

µf, 10−4 Pa⋅s

kf, W/(m⋅K)

4.610 4.558 4.516 4.482 4.454

0.826 0.845 0.865 0.884

6.83 5.60 4.59 3.83

0.114 0.109 0.104 0.099

3.642 3.688 3.732 3.777 3.820

4.431 4.413 4.397 4.385 4.374

0.898 0.910 0.924 0.943 0.972

3.26 2.82 2.48 2.20 1.97

0.093 0.088 0.083 0.078 0.072

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

238.1 243.8 251.1 258.2 265.8

424.9 429.0 433.1 437.2 441.3

3.080 3.140 3.205 3.267 3.327

5.133 4.990 4.860 4.759 4.677

5.865 2.2617 1.0019 0.4962 0.2689

273.7 281.7 290.0 298.4 307.1

455.3 449.3 453.5 457.6 461.8

3.385 3.441 3.494 3.545 3.594

6.536.−4 6.709.−4 6.899.−4 7.110.−4 7.346.−4

0.1567 9.69.−2 6.28.−2 4.24.−2 2.95.−2

315.9 325.0 334.3 343.8 353.6

465.9 469.9 473.8 477.5 481.0

19557 2392 347.0 70.25 18.15

240 250 260 270 280

7.589 10.37 13.85 18.13 23.32

7.615.−4 7.928.−4 8.302.−4 8.769.−4 9.320.−4

2.11.−2 1.53.−2 1.13.−2 8.28.−3 6.10.−3

363.5 373.9 384.7 396.2 408.8

484.1 486.1 489.1 490.5 490.6

3.862 3.903 3.944 3.986 4.029

4.364 4.355 4.346 4.336 4.323

1.014 1.072 1.151 1.255 1.386

1.79 1.63 1.50 1.34 1.14

0.067 0.062 0.057 0.051 0.045

290 300 302.0c

29.57 37.05 38.70

1.035.−3 1.284.−3 1.808.−3

4.34.−3 2.60.−3 1.81.−3

423.6 445.3 463.1

488.3 477.5 463.1

4.080 4.151 4.209

4.303 4.257 4.209

1.549 1.75 ∞

0.87 0.52 0.29

0.038 ∞

*P, v, h, and s data interpolated from Perelshteyn (ed.), Tables and Diagrams of the Thermodynamic Properties of Refrigerants 12, 13 and 22, Moscow, 1971. cp, µ, and k data interpolated and converted from Thermophysical Properties of Refrigerants, American Society of Heating, Refrigerating and Air-Conditioning Engineers, New York, 1976. c = critical point. The notation 3.817.−6 signifies 3.817 × 10−6. The 1993 ASHRAE Handbook—Fundamentals (SI ed.) contains a table at closer temperature increments and also an enthalpy–log-pressure diagram from 0.1 to 70 bar, −100 to 240°C. Equations and constants approximated to 1985 ASHRAE tables are given by Mecarik, K. and M. Masaryk, Heat Recovery Systems and CHP, 11, 2/3 (1991): 193–197. Saturation and superheat tables and a diagram to 60 bar, 600 K are given by Reynolds, W. C., Thermodynamic Properties in SI, Stanford Univ. publ., 1979 (173 pp.). For tables and a chart to 1000 psia, 520°F, see Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat, thermal conductivity, and viscosity, see Thermophysical Properties of Refrigerants, ASHRAE, 1993.

THERMODYNAMIC PROPERTIES

2-271

Enthalpy–log-pressure diagram for Refrigerant 12. 1 MPa = 10 bar. (Copyright 1981 by the American Society of Heating, Refrigerating and AirConditioning Engineers and reproduced by permission of the copyright owner.) This chart, redrawn for integral Celsius temperatures with a different zero point, appears on p. 17.4 of the 1993 ASHRAE Handbook—Fundamentals (SI ed.). This handbook gives material for integral degrees Celsius with temperatures on the ITS 90 scale. For experimental isochores for the compressed liquid from 10 to 302 bar, 122 to 462 K, see Blanke, W. and R. Weiss, PTB Bericht W30, Braunschweig, Germany, 1992 (54 pp.). Equations and constants approximated to 1985 ASHRAE tables are given by Mecarik, K. and M. Masaryk, Heat Recovery Systems and CHP, 11, 2/3 (1991): 193–197. Tables at 2°C increments to 240°C, 50 bar are given by Watson, J. T. R., Thermophysical Properties of Refrigerant 12, H.M.S.O., Edinburgh, Scotland, 1975 (183 pp.). Saturation and superheat tables and a diagram to 40 bar, 620 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ., 1979 (173 pp.). Tables and a chart to 1100 psia, 480°F are given by Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA 1986 (521 pp.). For specific heat, thermal conductivity, and viscosity, see Thermophysical Properties of Refrigerants, ASHRAE, 1993.

FIG. 2-19

2-272

PHYSICAL AND CHEMICAL DATA

TABLE 2-303

Saturated Refrigerant 13B1*

P, bar

vf, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µf, 10−4 Pa⋅s

kf, W/(m⋅K)

170 180 190 200 210

0.059 0.127 0.250 0.455 0.777

4.594.−4 4.677.−4 4.765.−4 4.860.−4 4.961.−4

1.6015 0.7840 0.4190 0.2407 0.1467

−40.90 −34.75 −28.51 −22.17 −15.68

90.95 94.37 97.83 101.32 104.82

−0.2033 −0.1682 −0.1345 −0.1020 −0.0704

0.5723 0.5491 0.5305 0.5154 0.5033

0.597 0.618 0.634 0.648 0.663

9.54 7.60 6.20 5.13 4.33

0.101 0.096 0.091 0.086 0.082

215.4 220 230 240 250

1.013 1.254 1.933 2.863 4.096

5.020.−4 5.071.−4 5.190.−4 5.321.−4 5.466.−4

0.1147 0.0940 0.0628 0.0433 0.0308

−12.09 −9.02 −2.19 4.83 12.03

106.70 108.28 111.68 114.99 118.16

−0.0536 −0.0396 −0.0094 0.0202 0.0494

0.4978 0.4936 0.4857 0.4793 0.4739

0.670 0.676 0.690 0.703 0.721

3.97 3.71 3.22 2.83 2.51

0.079 0.077 0.073 0.068 0.063

260 270 280 290 300

5.690 7.703 10.20 13.25 16.91

5.627.−4 5.809.−4 6.018.−4 6.264.−4 6.562.−4

0.0224 0.0166 0.0124 0.0094 0.0072

19.44 27.06 34.94 43.11 51.68

121.16 123.93 126.41 128.51 130.09

0.0781 0.1064 0.1345 0.1625 0.1908

0.4693 0.4652 0.4612 0.4570 0.4522

0.742 0.767 0.800 0.842 0.891

2.25 2.04 1.84 1.69 1.57

0.059 0.054 0.049 0.045 0.040

310 320 330 340.2c

21.28 26.44 32.48 39.64

6.940.−4 7.458.−4 8.295.−4 1.344.−3

0.0055 0.0041 0.0030 0.0013

60.81 70.80 82.42 108.70

130.97 130.76 128.59 108.70

0.2197 0.2503 0.2845 0.3605

0.4460 0.4376 0.4245 0.3605

0.951 1.09 1.29 ∞

1.45 1.26 0.99 0.35

0.035 0.030 0.026 ∞

T, K

*Values reproduced or converted from Table 4, p. 17.83, ASHRAE Handbook, 1981: Fundamentals, American Society of Heating, Refrigerating and Air-Conditioning Engineers, Atlanta, 1981. Copyright material. Reproduced by permission of the copyright owner. c = critical point. The notation 4.594.−4 signifies 4.594 × 10−4. The 1993 ASHRAE Handbook—Fundamentals (SI ed.) contains a table at closer temperature increments and also an enthalpy–log-pressure diagram from 0.1 to 35 bar, −80 to 220°C. For tables and a chart to 500 psia, 480°F, see Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat, thermal conductivity, and viscosity, see Thermophysical Properties of Refrigerants, ASHRAE, 1993.

Refrigerant 14 (tetrafluoromethane) See Carbon Tetrafluoride (Table 2-245).

TABLE 2-304

Refrigerant 20 See Chloroform (Table 2-248).

Saturated Refrigerant 21

Temperature, K

Pressure, bar

250 260 270 280 290

0.2415 0.3953 0.6200 0.9364 1.3682

0.000 0.000 0.000 0.000 0.000

300 310 320 330 340

1.9417 2.6849 3.6279 4.8022 6.2409

vf, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

677 687 698 709 722

0.8292 0.5247 0.3455 0.2355 0.1654

16.6 26.5 36.6 46.7 57.1

274.8 279.9 284.9 290.0 295.0

0.0687 0.1076 0.1454 0.1824 0.2186

1.1015 1.0820 1.0653 1.0511 1.0389

0.000 0.000 0.000 0.000 0.000

735 748 763 778 794

0.1192 0.0879 0.0661 0.0505 0.0391

67.7 78.4 89.5 100.7 112.3

300.0 304.8 309.5 314.1 318.4

0.2543 0.2894 0.3242 0.3586 0.3927

1.0286 1.0196 1.0119 1.0051 0.9989

sg, kJ⋅(kg⋅K)

350 360 370 380 390

7.978 10.049 12.489 15.337 18.630

0.000 0.000 0.000 0.000 0.000

812 830 850 870 893

0.0307 0.0243 0.0194 0.0155 0.0125

124.1 136.2 148.6 161.2 173.9

322.4 326.1 329.3 331.9 333.8

0.4266 0.4602 0.4935 0.5264 0.5587

0.9932 0.9877 0.9820 0.9758 0.9688

400 410 420 430 440

22.41 26.72 31.60 37.10 43.26

0.000 0.000 0.000 0.001 0.001

918 944 972 002 034

0.01011 0.00820 0.00672 0.00564 0.00491

186.4 198.3 208.7 216.4 221.1

334.8 334.7 333.7 332.4 332.3

0.5896 0.6180 0.6418 0.6587 0.6682

0.9605 0.9506 0.9394 0.9286 0.9208

Reproduced and rounded from unpublished Center for Applied Thermodynamic Studies, Moscow ID report, 1981. For a thermodynamic diagram to 350 bar, 370°C, see Rombusch, U. K., Allgem. Warme., 11, 3 (1962).

TABLE 2-305

Saturated Refrigerant 22*

P, bar

vf, m3/kg

vg, m3/kg

150 160 170 180 190

0.0017 0.0054 0.0150 0.0369 0.0821

6.209.−4 6.293.−4 6.381.−4 6.474.−4 6.573.−4

83.40 28.20 10.85 4.673 2.225

200 210 220 230 240

0.1662 0.3116 0.5470 0.9076 1.4346

6.680.−4 6.794.−4 6.917.−4 7.050.−4 7.195.−4

250 260 270 280 290

2.174 3.177 4.497 6.192 8.324

300 310 320 330 340 350 360 369.3c

T, K

µf, 10−4 Pa⋅s

kf, W/(m⋅K)

1.059 1.058 1.057 1.058 1.060

0.770 0.647 0.554

0.161 0.156 0.151 0.146 0.141

4.921 4.885 4.854 4.828 4.805

1.065 1.071 1.080 1.091 1.105

0.481 0.424 0.378 0.340 0.309

0.136 0.131 0.126 0.121 0.117

3.898 3.942 3.986 4.029 4.071

4.785 4.768 4.752 4.738 4.725

1.122 1.143 1.169 1.193 1.220

0.282 0.260 0.241 0.225 0.211

0.112 0.107 0.102 0.097 0.092

612.8 615.1 616.7 617.3 616.5

5.113 4.153 4.194 4.235 4.278

4.713 4.701 4.688 4.674 4.658

1.257 1.305 1.372 1.460 1.573

0.198 0.186 0.176 0.167 0.151

0.087 0.082 0.077 0.072 0.067

613.3 605.5 570.0

4.324 4.378 4.501

4.637 4.605 4.501

1.718 1.897 ∞

0.130 0.106

0.062

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

268.2 278.2 288.3 298.7 308.6

547.3 552.1 557.0 561.9 566.8

3.355 3.430 3.494 3.551 3.605

5.215 5.141 5.075 5.013 4.963

1.145 0.6370 0.3772 0.2352 0.1532

318.8 329.1 339.7 350.6 361.7

571.6 576.5 581.2 585.9 590.5

3.657 3.707 3.756 3.804 3.852

7.351.−4 7.523.−4 7.733.−4 7.923.−4 8.158.−4

0.1037 0.07237 0.05187 0.03803 0.02838

373.0 384.5 396.3 408.2 420.4

594.9 599.0 603.0 606.6 610.0

10.956 14.17 18.02 22.61 28.03

8.426.−4 8.734.−4 9.096.−4 9.535.−4 1.010.−3

0.02148 0.01643 0.01265 9.753.−3 7.479.−3

432.7 445.5 458.6 472.4 487.2

34.41 41.86 49.89

1.086.−3 1.212.−3 2.015.−3

5.613.−3 4.036.−3 2.015.−3

503.7 523.7 570.0

*P, v, h, and s data interpolated from Perelshteyn (ed.), Tables and Diagrams of the Thermodynamic Properties of Refrigerants 12, 13 and 22, Moscow, 1971. cp, µ, and k data interpolated and converted from Thermophysical Properties of Refrigerants, American Society of Heating, Refrigerating and Air-Conditioning Engineers, New York, 1976. c = critical point. The notation 6.209.−4 signifies 6.209 × 10−4. The 1993 ASHRAE Handbook—Fundamentals (SI ed.) gives a saturation table from −150 to 96.14°C and an enthalpy–log-pressure diagram from 0.1 to 150 bar, −60 to 200°C. For experimental isochores for the compressed liquid from 12 to 297 bar, 120 to 378 K, see Blanke, W. and R. Weiss, PTB Bericht W 30, Braunschweig, Germany, 1992 (54 pp.). Equations and constants approximated to 1985 ASHRAE tables are given by Mecarik, K. and M. Masaryk, Heat Recovery Systems and CHP, 11 2/3 (1991): 193–197. Saturation and superheat tables and a diagram to 100 bar, 620 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (179 pp.). For tables and a chart to 2000 psia, 480°F, see Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat, thermal conductivity, and viscosity, see Thermophysical Properties of Refrigerants, ASHRAE, 1993.

FIG. 2-20 Enthalpy–log-pressure diagram for Refrigerant 22. 1 MPa = 10 bar. (Copyright 1981 by the American Society of Heating, Refrigerating and Air-Conditioning Engineers and reproduced by permission of the copyright owner.)

2-273

2-274

PHYSICAL AND CHEMICAL DATA

TABLE 2-306

Thermophysical Properties of Compressed R22 Temperature, K

Pressure, bar

Property

275

300

325

350

375

400

425

450

1

cp (kJ/ kg⋅K) µ (10−6 Pa·s) k (W/m·K) Pr

0.639 11.8 0.0091 0.829

0.653 12.8 0.0106 0.793

0.689 13.9 0.0121 0.787

0.714 14.9 0.0136 0.782

0.739 15.8 0.0151 0.773

0.758 16.7 0.0166 0.762

0.781 17.7 0.0181 0.765

0.806 18.7 0.0196 0.769

5

cp (kJ/ kg⋅K) µ (10−6 Pa·s) k (W/m·K) Pr

0.725 11.8 0.0096 0.887

0.728 12.8 0.0107 0.871

0.744 13.8 0.0123 0.852

0.759 15.0 0.0138 0.839

0.766 16.2 0.0153 0.803

0.775 17.0 0.0170 0.775

0.791 18.0 0.0184 0.773

0.816 18.8 0.0199 0.771

10

cp (kJ/ kg⋅K) µ (10−6 Pa·s) k (W/m·K) Pr

1.166 211 0.0954 2.58

0.847 13.7 0.0121 0.959

0.810 14.4 0.0128 0.901

0.799 15.1 0.0144 0.838

0.797 16.1 0.0160 0.802

0.803 17.1 0.0175 0.785

0.814 18.1 0.0190 0.775

0.828 19.0 0.0205 0.767

20

cp (kJ/ kg⋅K) µ (10−6 Pa·s) k (W/m·K) Pr

0.164 211 0.0963 2.55

1.237 159 0.0849 2.32

0.949 16.5 0.0157 0.997

0.889 17.3 0.0172 0.894

0.865 18.0 0.0184 0.846

0.858 18.8 0.0199 0.811

0.859 19.6 0.0214 0.787

40

cp (kJ/ kg⋅K) µ (10−6 Pa·s) k (W/m·K) Pr

1.152 218 0.0980 2.56

1.217 164 0.0872 2.29

1.359 123 0.0767 2.18

1.373 20.7 0.0219 1.30

1.089 20.5 0.0210 1.063

0.996 20.7 0.0217 0.950

0.956 21.2 0.0233 0.870

60

cp (kJ/ kg⋅K) µ (10−6 Pa·s) k (W/m·K) Pr

1.142 221 0.0993 2.54

1.191 170 0.0889 2.28

1.311 128 0.0786 2.14

1.460 94.6

1.767 24.7 0.0305 1.431

1.221 24.2 0.0287 1.030

1.089 23.9 0.0268 0.971

80

cp (kJ/ kg⋅K) µ (10−6 Pa·s) k (W/m·K) Pr

1.132 226 0.1003 2.55

1.177 175 0.0904 2.28

1.277 133 0.0803 2.12

1.444 101 0.0690 2.12

1.861 73.6 0.0523 2.62

1.396 29.9 0.0374 1.12

1.262 27.7 0.0337 1.04

100

cp (kJ/ kg⋅K) µ (10−6 Pa·s) k (W/m·K) Pr

1.122 230 0.1013 2.55

1.154 179 0.0916 2.26

1.247 138 0.0817 2.11

1.361 108 0.0716 2.04

1.564 83.2 0.0607 2.14

1.923 37.6 0.0421 1.72

1.471 32.4 0.0378 1.26

2.073 55.5 0.0504 2.28

Some values are approximate as significant differences exist in the literature.

TABLE 2-307

Saturated Refrigerant 23

Temp., K

Pressure, bar

180 190 191.1b 200 210

0.510 0.950 1.013 1.652 2.709

0.000 0.000 0.000 0.000 0.000

220 230 240 250 260

4.298 6.312 9.091 12.69 17.25

270 280 290 299.1c

22.94 29.98 38.68 48.36

vf, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

678 693 695 710 729

0.4088 0.2279 0.2139 0.1353 0.0845

−66.0 −54.4 −53.1 −42.6 −30.3

181.1 185.3 185.7 189.1 192.4

−0.3179 −0.2554 −0.2485 −0.1948 −0.1353

1.0549 1.0062 1.0011 0.9635 0.9254

0.000 0.000 0.000 0.000 0.000

751 777 807 844 889

0.0551 0.0372 0.0259 0.0183 0.0132

−17.5 −4.3 9.4 23.6 38.1

195.4 197.8 199.6 200.7 200.9

−0.0764 −0.0182 0.0392 0.0957 0.1512

0.8913 0.8602 0.8314 0.8042 0.7773

0.000 0.001 0.001 0.001

948 031 169 905

0.0095 0.0068 0.0046 0.0019

53.5 70.5 92.0 143.0

199.8 196.4 188.1 143.0

0.2071 0.2665 0.3387 0.5062

0.7493 0.7162 0.6698 0.5062

cpf, kJ/(kg⋅K)

µf, 10−6 Pa·s

kf, W/(m⋅K)

Pr

0.710 1.043 1.289 1.497

170.1 150.4 131.2 113.0

0.105 0.098 0.091 0.084

1.15 1.56 1.85 2.00

b = normal boiling point; c = critical point. hf = sf = 0 at 233.15 K = −40°C. Interpolated and converted from ASHRAE Handbook—Fundamentals, 1993. Experimental P−ρ−T data from 95 to 413 K reported in J. Phys. Chem., 89 (1985): 4637–4646 were used by Rubio, R. G., J. A. Zollweg, et al., J. Chem. Eng. Data, 36, (1991): 171–184, to calculate properties up to 1000 bar from 126 to 332° K. The 1993 ASHRAE Handbook—Fundamentals (SI ed.) gives a saturation table from −100 to 25.92°C and an enthalpy–log-pressure diagram from 0.1 to 80 bar, −100 to 280°C. Equations and constants approximated to the 1985 ASHRAE tables are given by Mecaryk, K. and M. Masaryk, Heat Recovery Systems and CHP, 11, 2/3 (1991): 193–197. For an enthalpy–log-pressure diagram from 0.005 to 200 bar, −140 to 180°C, see Morsy, T. E., Kaltetechnik—Klimat., 18, 9 (1966): 347–349. Saturation and superheat tables and a diagram to 100 bar, 600 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (173 pp.). For tables and a chart to 1000 psia, 560°F, see Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (21 pp.). For specific heat, thermal conductivity, and viscosity, see Thermophysical Properties of Refrigerants, ASHRAE, 1993.

TABLE 2-308 Temp., K

Thermophysical Properties of Saturated Difluoromethane (R32)

Pressure, bar

vf , m3/kg

vg , m3/kg

hf , kJ/kg

hg , kJ/kg

sf , kJ/(kg⋅K)

sg , kJ/(kg⋅K)

cpf , kJ/(kg⋅K)

cpg , kJ/(kg⋅K)

µf , 10−6 Pa·s

µg , 10−6 Pa·s

200 210 220 230 240

0.2960 0.5440 0.9384 1.5345 2.3963

7.845.−4 8.025.−4 8.208.−4 8.402.−4 8.611.−4

1.0580 0.5990 0.3593 0.2261 0.1483

−52.340 −37.750 −20.690 −4.984 10.947

351.160 356.880 361.970 366.773 371.129

−0.2418 −0.1652 −0.0909 −0.0215 0.0459

1.7757 1.7098 1.6485 1.5948 1.5468

1.557 1.580 1.613

0.799 0.839 0.894

283.8 247.8 220.4

10.30 10.37 10.46

250 260 270 280 290

3.5966 5.2160 7.3423 10.070 13.502

8.842.−4 9.096.−4 9.376.−4 9.696.−4 1.006.−3

0.1005 0.07020 0.05009 0.03643 0.02687

374.971 378.224 380.786 382.525 383.262

0.1117 0.1763 0.2397 0.3029 0.3654

1.5030 1.4624 1.4247 1.3886 1.3534

1.642 1.682 1.730 1.786 1.863

0.963 1.043 1.138 1.244 1.375

198.1 177.9 159.1 141.8 126.1

300 310 320 330 340

17.749 22.931 29.186 36.675 45.603

1.049.−3 1.100.−3 1.166.−3 1.243.−3 1.394.−3

0.02001 0.01497 0.01117 0.00822 0.00581

115.754 135.801 157.212 180.724 208.262

382.737 380.576 376.163 368.357 354.460

0.4283 0.4919 0.5574 0.6264 0.7047

1.3182 1.2815 1.2415 1.1950 1.1347

1.955 2.084 2.282 2.620 3.560

1.560 1.810 2.16 2.62 4.21

112.0 98.8 86.1 75.1 65.4

350 351.4c

56.336 57.927

0.00237

0.00317 0.00237

274.640 286.675

337.933 286.675

0.8927 0.9269

1.0735 0.9269

27.1778 43.786 60.849 78.456 96.713

kf , W/(m⋅K)

kg , W/(m⋅K)

Prf

Prg

10.66 10.95 11.31 11.70 12.21

0.1646 0.1562 0.1487 0.1403 0.1308

0.0097 0.0106 0.0115 0.0125 0.0136

1.98 1.92 1.85 1.81 1.80

1.06 1.08 1.12 1.16 1.23

12.82 13.71 14.4 15.3 17.3

0.1228 0.1155 0.1073 0.0990

0.0149 0.0165 0.0184 0.0205 0.0236

1.78 1.78 1.83 1.99

1.34 1.50 1.69 1.96 3.10

c = critical point. The notation 7.845.−4 signifies 7.845 × 10−4. P, v, T, h, s, and cp converted and extrapolated from Defibaugh, D. R., G. Morrison, et al., J. Chem. Eng. Data, 39 (1994): 333–340. Saturated liquid and vapor viscosities from smooth curve fits of Oliveira, C. M. B. P. and W. A. Wakeham, Int. J. Thermophys., 14, 6 (1993): 1131–1143. Thermal conductivity values based upon papers by Geller, V. Z. and M. E. Perlaitis, and by Gross, Proc. 10th Symp. Thermophys. Props., Boulder, CO, 1994. The 1993 ASHRAE Handbook—Fundamentals (SI ed.) gives a saturation table to 78.41°C and a diagram to 200 bar, 200°C.

2-275

2-276

PHYSICAL AND CHEMICAL DATA

TABLE 2-309

Specific Heat at Constant Pressure, Thermal Conductivity, Viscosity, and Prandtl Number of R32 Gas P, bar

Temp., K 250

260

270

280

290

300

5

P, bar

Property

1

10

cp (kJ/ kg⋅K) µ (10−6 Pa·s) k (W/m⋅K) Pr

0.805 10.55 0.0094 0.908

cp (kJ/ kg⋅K) µ (10−6 Pa·s) k (W/m⋅K) Pr

0.810 11.00 0.0100 0.890

1.025 10.96 0.0104 1.080

cp (kJ/ kg⋅K) µ (10−6 Pa·s) k (W/m⋅K) Pr

0.818 11.42 0.0107 0.873

0.991 11.37 0.0111 1.015

cp (kJ/ kg⋅K) µ (10−6 Pa·s) k (W/m⋅K) Pr

0.825 11.82 0.0116 0.860

0.969 11.77 0.0118 0.967

1.238 11.68 0.0125 1.157

cp (kJ/ kg⋅K) µ (10−6 Pa·s) k (W/m⋅K) Pr

0.837 12.28 0.0121 0.849

0.959 12.22 0.0125 0.938

1.161 12.17 0.0131 1.079

cp (kJ/ kg⋅K) µ (10−6 Pa·s) k (W/m⋅K) Pr

0.849 12.70 0.0128 0.842

0.951 12.69 0.0132 0.914

1.118 12.66 0.0138 1.026

15

Temp., K 310

320

330

340

350

1

5

10

15

20

0.861 13.12 0.0135 0.878

0.945 13.10 0.0139 0.891

1.084 13.08 0.0144 0.985

1.279 13.07 0.0150 1.114

1.560 13.09 0.0159 1.284

25

30

40

0.873 13.54 0.0142 0.836

0.944 13.54 0.0146 0.875

1.059 13.55 0.0150 0.957

1.207 13.56 0.0156 1.049

1.400 13.60 0.0164 1.161

1.704 13.75 0.0173 1.354

0.885 13.96 0.0148 0.834

0.942 13.96 0.0152 0.865

1.038 13.98 0.0156 0.930

1.158 14.01 0.0162 1.001

1.301 14.15 0.0169 1.089

1.508 14.28 0.0177 1.217

1.837 14.52 0.0187 1.426

0.897 14.38 0.0155 0.832

0.937 14.40 0.0159 0.849

1.020 14.43 0.0163 0.903

1.135 14.47 0.0168 0.978

1.242 14.53 0.0175 1.031

1.388 14.65 0.0182 1.117

1.612 14.85 0.0190 1.260

2.488 16.00 0.0217 1.834

0.910 14.80 0.0162 0.831

0.934 14.82 0.0165 0.839

1.004 14.84 0.0169 0.882

1.118 14.87 0.0174 0.955

1.200 14.92 0.0180 0.995

1.308 15.06 0.0186 1.060

1.440 15.21 0.0194 1.130

1.914 16.16 0.0216 1.432

1.370 12.62 0.0144 1.201

Some values read from charts may be approximate. cp values interpolated and converted from Thermodynamic Properties of KLEA 32, I.C.I., 1993 (47 pp.). Viscosity interpolated from Takahashi, M., C. Yokoyama, et al., Proc. 14th Symp. Thermophys. Props., Japan, 1993 (pp. 427–430). Thermal conductivities are taken from Geller, V. Z. and M. E. Perlaitis, and from Gross, Proc. 10th Symp. Thermophys. Props., Boulder, CO, 1994.

TABLE 2-310 Temp., °C

Saturated SUVA MP 39

Pf, bar

Pg, bar

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µf, 10−6 Pa·s

kf , W/(m·K)

Prf

−40 −30 −20 −10 0

0.733 1.155 1.748 2.553 3.615

0.533 0.871 1.361 2.043 2.965

0.000 0.000 0.000 0.000 0.000

712 728 744 762 781

0.3778 0.2391 0.1576 0.1075 0.0755

154.0 164.9 176.2 188.6 200.0

385.0 390.6 396.3 401.8 407.3

0.8188 0.8647 0.9099 0.9577 1.0000

1.8244 1.8059 1.7907 1.7781 1.7675

1.078 1.109 1.137 1.165 1.197

351 323 291 266 241

0.1209 0.1154 0.1107 0.1057 0.1012

3.13 3.06 2.99 2.93 2.85

10 20 30 40 50

4.984 6.712 8.857 11.475 14.628

4.177 5.733 7.697 10.133 13.112

0.000 0.000 0.000 0.000 0.000

803 826 851 878 909

0.0544 0.0399 0.0298 0.0225 0.0172

212.7 225.3 238.3 252.0 266.4

412.6 417.6 422.2 426.5 430.1

1.0454 1.0884 1.1316 1.1752 1.2194

1.7587 1.7510 1.7439 1.7372 1.7304

1.233 1.277 1.329 1.392 1.468

221 202 186 170 157

0.0967 0.0922 0.0877 0.0830 0.0781

2.82 2.80 2.83 2.85 2.95

60 70 80 90 100

18.378 22.79 27.92 33.83 40.53

16.711 21.01 26.12 32.13 39.22

0.000 0.000 0.001 0.001 0.001

944 988 028 084 140

0.01313 0.01005 0.00764 0.00570 0.00403

281.6 297.9 315.9 336.2 361.4

433.0 434.9 435.4 433.5 426.9

1.2647 1.3118 1.3616 1.4163 1.4820

1.7228 1.7138 1.7022 1.6858 1.6584

1.564 1.652 1.802 1.958 2.16

143 131 122 115 110

0.0737 0.0684 0.0631 0.0577 0.0533

3.04 3.16 3.48 3.90 4.46

108.0c

46.04

46.04

0.001 96

0.00196

397

397

vf, m3/kg

c = critical point. SUVA MP 39 = R401A = CHClF2 (R22) 53% wt + CH3CHF2 (R 152a) 13% wt + CHClFCF3 (R124) 34% wt, near-azeotropic blend. Some values read from charts are approximate. Material used by permission of DuPont Fluoroproducts.

TABLE 2-311

SUVA MP 39 at Atmospheric Pressure

Temp., °C

−27.01

−20

0

20

40

60

80

100

120

140

v (m3/ kg) h (kJ/ kg) s (kJ/ kg⋅K) cp (kJ/ kg⋅K) µ (10−6 Pa·s) k (W/m⋅K) Pr Z

0.2102 351.7 1.8009 0.648 10.17 0.00878 0.750 0.9829

0.2167 396.9 1.8193 0.669 10.43 0.00921 0.758 0.9852

0.2351 410.4 1.8706 0.698 11.18 0.01041 0.750 0.9906

0.2534 424.5 1.9204 0.727 11.93 0.01161 0.749 0.9949

0.2715 439.2 1.9689 0.757 12.68 0.01282 0.749 0.9979

0.2896 454.4 2.0161 0.787 13.42 0.01404 0.748 1.0005

0.3076 470.3 2.0623 0.811 14.17 0.01536 0.748 1.0025

0.3256 486.6 2.1073 0.836 14.89 0.01668 0.748 1.0043

0.3435 503.5 2.1513 0.859 15.61 0.01796 0.747 1.0056

0.3613 521.2 2.1943 0.883 16.32 0.01929 0.747 1.0060

For composition see footnote to Table 2-310. Some values read from charts are approximate. Material used by permission of DuPont Fluoroproducts.

50 R 32

20

1.

40

30

0

1.3

350 20

g

.40

•

340

K

y1

p tro

0.95

0.90

k kJ/

330 320

En

310 300 K

0

1.5

Satu rated vapor

Pressure, bar

10 8

6

290

1.6

0

280 270

4

0

1.7

260

2

0 1.8 250 240 230

1

FIG. 2-21

360

380

400 Enthalpy, kJ/kg

420

440

Enthalpy–log-pressure diagram for Refrigerant 32. 2-277

2-278

PHYSICAL AND CHEMICAL DATA TABLE 2-312 Pressure, bar

Thermodynamic Properties of Saturated KLEA 60 Tf , K

Tg , K

vg , m3/kg

hf , kJ/kg

hg , kJ/kg

227.3 236.1 242.8 248.3 253.0

234.0 242.5 249.1 254.5 259.1

0.000 0.000 0.000 0.000 0.000

7118 7263 7381 7483 7573

0.2097 0.1433 0.1093 0.0885 0.0744

−7.80 3.92 12.89 20.27 26.57

229.64 235.02 239.07 242.35 245.08

0.9965 0.9833 0.9744 0.9679 0.9629

4 5 6 8 10

260.7 267.3 272.9 282.1 289.8

266.8 273.1 278.5 287.5 295.0

0.000 0.000 0.000 0.000 0.000

7735 7880 8012 8254 8480

0.0564 0.0442 0.0384 0.0286 0.0228

37.23 46.12 53.84 67.02 78.23

249.54 253.07 255.24 260.70 263.86

0.9552 0.9496 0.9450 0.9378 0.9318

12.5 15 17.5 20 22.5

297.9 304.8 311.0 316.5 321.4

302.8 309.5 315.4 320.7 325.5

0.000 0.000 0.000 0.000 0.000

8750 9017 9290 9613 9884

0.01802 0.01481 0.01247 0.01069 0.00928

90.50 101.51 111.64 121.18 130.31

266.95 269.12 270.58 271.46 271.79

0.9257 0.9190 0.9128 0.9065 0.8999

25 27.5 30

326.1 330.4 334.5

329.8 333.9 337.6

0.001 023 0.001 063 0.001 115

0.00828 0.00717 0.00635

139.17 147.89 156.58

271.63 270.97 269.81

0.8927 0.8850 0.8765

1 1.5 2 2.5 3

vf , m3/kg

sf , kJ/(kg⋅K)

sg , kJ/(kg⋅K)

hf = sf = 0 at 233.15 K = −40°C. Converted and interpolated from Thermodynamic Properties of Klea 60 (British units, 20 pp.), copyright ICI Chemicals and Polymers Limited, 1993. Reproduced by permission. Tf = bubble point temperature; Tg = dew point temperature.

TABLE 2-313 Pressure, bar

Thermodynamic Properties of Saturated KLEA 61 vf , m3/kg

vg , m3/kg

hf , kJ/kg

hg , kJ/kg

6852 6994 7110 7211 7301

0.1800 0.1230 0.0937 0.0758 0.0637

−9.45 2.52 9.72 16.59 22.47

191.64 196.90 200.88 204.10 206.80

0.8433 0.8341 0.8282 0.8245 0.8215

0.000 0.000 0.000 0.000 0.000

7463 7607 7740 7985 8214

0.04831 0.03888 0.03249 0.02435 0.01936

32.43 40.76 48.00 59.82 70.98

211.22 214.74 217.65 222.21 225.63

0.8172 0.8141 0.8123 0.8080 0.8048

298.7 305.5 311.4 316.7 321.5

0.000 0.000 0.000 0.000 0.000

8491 8768 9053 9353 9680

0.01528 0.01251 0.01049 0.00896 0.00774

82.59 93.02 102.67 111.79 120.55

228.80 231.08 232.64 233.60 233.99

0.8010 0.7971 0.7929 0.7882 0.7829

325.9 330.0 333.7

0.001 005 0.001 048 0.001 102

0.00674 0.00590 0.00518

129.11 137.62 146.21

233.85 233.16 231.84

0.7769 0.7700 0.7619

Tf , K

Tg , K

225.6 234.3 241.8 246.4 251.1

230.0 238.5 245.0 250.4 254.9

0.000 0.000 0.000 0.000 0.000

4 5 6 8 10

258.9 265.4 270.9 280.2 287.8

262.6 269.0 274.4 283.4 290.9

12.5 15 17.5 20 22.5

295.8 302.8 308.8 314.3 319.3

25 27.5 30

323.9 328.1 332.1

1 1.5 2 2.5 3

sf , kJ/(kg⋅K)

sg , kJ/(kg⋅K)

Converted and interpolated from Thermodynamic Properties of Klea 61 (British units, 20 pp.), copyright ICI Chemicals and Polymers Limited, 1993. Reproduced by permission. Tf = bubble-point temperature; Tg = dew-point temperature. hf = sf = 0 at 233.15 K = −40 °C.

THERMODYNAMIC PROPERTIES

2-279

30 0.9 0

5

0.70 0.75

0 .9

20

R 60

0.80

0

425

1.0

15

0.90

400

1.1 0

10

1.0

5

kJ/ kg

•

K

0.85

8

5

375

350

1.2

6

0

1.1

Pressure, bar

0.95

1.3

0

325

1.2

5

4

1.

35

300 K

40

2

5

1.

275

1

FIG. 2-22

1.4

1.5

250

Enthalpy–log-pressure diagram for KLEA 60.

300 Enthalpy (h), kJ/kg

350

400

2-280

PHYSICAL AND CHEMICAL DATA

30 0.80

0.70 0.75

0.8 5

0.80

20

450

0.9 0

0.85 0.90 15

0.9

5

0.95

425

1.0 0k J/k g

•

K

400

10

5 1.0

rated

0

375

350

1.1

6

Satu

Pressure, bar

vapo

r

8

1.1

5

4

1.2

0

325

1.2

5

300

30

2

1.3

5

1.

275 1.5 250 K

1 200 FIG. 2-23

250 Enthalpy–log-pressure diagram for KLEA 61.

300 Enthalpy (h), kJ/kg

350

THERMODYNAMIC PROPERTIES TABLE 2-314 Temp., °C

2-281

Saturated SUVA HP 62

Pf, bar

Pg, bar

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µf, 10−6 Pa·s

kf, W/(m⋅K)

−50 −40 −30 −20 −10

0.852 1.367 2.095 3.087 4.404

0.821 1.325 2.041 3.018 4.321

0.000 0.000 0.000 0.000 0.000

761 779 799 820 843

0.2244 0.1434 0.0953 0.0656 0.0463

133.1 145.6 159.9 172.8 186.1

337.3 343.8 350.3 356.5 362.6

0.7318 0.7862 0.8460 0.8975 0.9487

1.6487 1.6380 1.6301 1.6245 1.6202

0.0970

1.220 1.260 1.302

370 318 276 238 207

0.0868 0.0834 0.0801

3.88 3.60 3.37

0 10 20 30 40

6.111 8.278 10.977 14.287 18.292

6.013 8.165 10.851 14.150 18.148

0.000 0.000 0.000 0.000 0.001

868 898 933 977 037

0.03338 0.02444 0.01809 0.01348 0.01003

200.0 214.5 229.9 246.2 263.8

368.3 373.6 378.3 382.2 385.0

1.0000 1.0515 1.1038 1.1574 1.2130

1.6188 1.6138 1.6106 1.6065 1.6005

1.351 1.412 1.489 1.592 1.753

181 158 138 122 106

0.0767 0.0733 0.0698 0.0663 0.0624

3.19 3.04 2.94 2.93 2.98

50 60 70 72.1c

23.08 28.75 35.58 37.32

22.94 28.63

0.001 122 0.001 261

386.1 384.2 375.9 361

1.5910 1.5742

2.09

91 76 61

0.0583 0.0535

3.26

0.002 06

283.2 305.8 339.8 361

1.2723 1.3389

37.32

0.00739 0.00527 0.00285 0.00206

vf, m3/kg

Prf

c = critical point. SUVA HP 62 = CHF2CF3 (R125) 44% wt + CH3CF3 (R143a) 52% wt + CH2FCF3 (R134a) 4% wt, near-azeotropic blend. Material used by permission of DuPont Fluoroproducts. Some values read from charts may be approximate.

TABLE 2-315

SUVA HP 62 at Atmospheric Pressure

Temp., °C

−45.63

−40

−20

0

20

40

60

80

100

120

v (m3/ kg) h (kJ/ kg) s (kJ/ kg⋅K) cp (kJ/ kg⋅K) µ (10−6 Pa⋅s) k (W/m⋅K) Pr Z

0.1866 336.0 1.6599 0.732 9.47 0.00860 0.806 0.9755

0.1921 344.4 1.6636 0.738 9.68 0.00932 0.767 0.9800

0.2100 359.9 1.7274 0.781 10.45 0.01059 0.771 0.9867

0.2278 376.2 1.7891 0.821 11.22 0.01186 0.777 0.9919

0.2455 393.1 1.8491 0.860 11.99 0.01313 0.785 0.9961

0.2630 410.9 1.9076 0.897 12.76 0.01440 0.795 0.9989

0.2805 429.3 1.9646 0.933 13.53 0.01568 0.805 1.0014

0.2980 448.4 2.0203 0.967 14.30 0.01695 0.816 1.0037

0.3153 468.2 2.0747 1.000 15.07 0.01827 0.827 1.0050

0.3325 488.7 2.1278 1.032 15.84 0.01949 0.839 1.0060

v,h, and s from DuPont bull. T—HP62—SI, June 1993 (17 pp.). cp and k from DuPont bull. ART 18, June 1993 (37 pp.). Some values read from charts may be approximate. Material used by permission of DuPont Fluoroproducts.

TABLE 2-316

Thermodynamic Properties of Saturated KLEA 66

Pressure, bar

Tf, K

Tg, K

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

0.69 1 1.5 2 2.5

221.46 228.89 237.69 244.45 249.99

228.77 236.05 244.69 251.33 256.76

0.000 0.000 0.000 0.000 0.000

7122 7237 7382 7501 7600

0.31325 0.22131 0.15140 0.11537 0.09104

−16.16 −5.89 6.22 15.51 23.12

241.25 245.91 251.38 255.52 258.95

1.0729 1.0580 1.0430 1.0330 1.0258

3 4 5 6 8

254.36 262.60 269.12 274.70 284.03

261.39 269.14 275.51 280.98 290.08

0.000 0.000 0.000 0.000 0.000

7695 7857 8001 8133 8375

0.07855 0.05964 0.04806 0.04021 0.03022

29.62 40.60 49.74 57.69 71.22

261.63 266.16 269.74 272.68 277.25

1.0201 1.0114 1.0055 0.9993 0.9913

10 12.5 15 17.5 20

291.74 299.87 306.87 313.05 318.60

297.56 305.44 312.18 318.10 323.40

0.000 0.000 0.000 0.000 0.000

8599 8867 9131 9400 9680

0.02410 0.01910 0.01571 0.01324 0.01137

82.73 95.32 106.59 116.97 126.73

280.64 283.74 285.93 287.28 288.26

0.9834 0.9770 0.9701 0.9633 0.9564

22.5 25 27.5 30

323.7 328.3 332.7 336.7

328.2 332.5 336.6 340.4

0.000 0.001 0.001 0.001

9981 032 072 125

0.00988 0.00883 0.00766 0.00703

136.0 145.1 153.9 162.7

288.6 288.4 287.8 286.6

0.9493 0.9418 0.9338 0.9251

vf, m3/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

Converted and interpolated from Thermodynamic Properties of Klea 66 (British units, 22 pp.), copyright ICI Chemicals and Polymers Limited, 1993. Reproduced by permission. Tf = bubble-point temperature; Tg = dew-point temperature. hf = sf = 0 at 233.15 K = −40°C.

2-282

PHYSICAL AND CHEMICAL DATA

0 1 .2

5 1 .1

0

450 1 .1

1.0 5

0.9

5

0.75

1.0 0

30

K

0.80 20

1.2 5

15

kJ

425

0.90

/kg

•

0.85

Satu ra

8

1 .3

375

1. 35

Pressure, bar

400 K ted v apor

10

0

Qua

lity

0.95

6

350

py En

tro

325

1.

40

4

1.4

5

300

2 275

1.5

0

1.5 250 K 1 250 FIG. 2-24

300 Enthalpy–log-pressure diagram for KLEA 66.

350 Enthalpy (h), kJ/kg

400

THERMODYNAMIC PROPERTIES TABLE 2-317 Temp., °C

2-283

Saturated SUVA MP 66

Pf, bar

Pg, bar

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µf, 10−6 Pa·s

kf, W/(m⋅K)

Prf

−40 −30 −20 −10 0

0.788 1.239 1.872 2.726 3.850

0.585 0.952 1.479 2.212 3.198

0.000 0.000 0.000 0.000 0.000

710 725 740 758 778

0.3498 0.2224 0.1471 0.1008 0.0710

153.8 164.8 176.0 188.6 200.0

386.0 391.6 397.1 402.6 407.8

0.8184 0.8643 0.9095 0.9577 1.0000

1.8291 1.8100 1.7940 1.7807 1.7694

1.078 1.109 1.137 1.165 1.197

349 313 282 257 236

0.1209 0.1154 0.1106 0.1057 0.1012

3.11 3.01 2.90 2.83 2.79

10 20 30 40 50

5.297 7.120 9.379 12.133 15.444

4.491 6.146 8.229 10.808 13.955

0.000 0.000 0.000 0.000 0.000

801 827 858 895 939

0.05124 0.03771 0.02818 0.02131 0.01625

212.6 225.1 238.2 251.9 266.3

412.9 417.7 422.1 426.1 429.4

1.0450 1.0879 1.1311 1.1747 1.2190

1.7598 1.7512 1.7433 1.7357 1.7278

1.233 1.277 1.329 1.392 1.468

217 198 181 168 151

0.0967 0.0922 0.0877 0.0830 0.0781

2.77 2.74 2.74 2.82 2.84

60 70 80 90 100

19.378 24.00 29.37 35.55 42.30

17.750 22.28 27.64 33.96

0.000 0.001 0.001 0.001

994 066 164 313

0.01244 0.00951 0.00721 0.00534

281.6 298.1 316.3 337.2

431.9 433.4 433.2 430.4

1.2645 1.3120 1.3625 1.4187

1.7191 1.7088 1.6956 1.6768

1.564 1.652 1.802

139 127 116

0.0737 0.0684 0.0631 0.0577 0.0533

2.95 3.07 3.31

106.1c

46.82

46.82

0.001 95

0.00195

389

389

vf, m3/kg

c = critical point. SUVA MP 66 = R401 = CHClF2 (R22) 61% wt + CH3CHF2 (R152a) 11% wt + CHClFCF3 (R124) 28% wt, near-azeotropic blend. Material used by permission of DuPont Fluoroproducts. Some values read from charts are approximate.

TABLE 2-318

SUVA MP 66 at Atmospheric Pressure

Temp., °C

−28.63b

−20

0

20

40

60

80

100

120

140

v (m3/ kg) h (kJ/ kg) s (kJ/ kg⋅K) cp (kJ/ kg⋅K) µ (10−6 Pa·s) k (W/m⋅K) Pr Z

0.2086 392.2 1.8081 0.641 9.78 0.00817 0.767 0.9652

0.2177 397.9 1.8299 0.652 10.43 0.00921 0.738 0.9730

0.2362 411.2 1.8804 0.688 11.18 0.01041 0.737 0.9783

0.2545 425.1 1.9295 0.716 11.93 0.01161 0.736 0.9822

0.2727 439.6 1.9772 0.744 12.68 0.01282 0.735 0.9852

0.2908 454.6 2.0237 0.771 13.42 0.01404 0.735 0.9876

0.3089 470.1 2.0690 0.796 14.17 0.01536 0.734 0.9896

0.3269 486.2 2.1132 0.822 14.89 0.01668 0.734 0.9912

0.3449 502.7 2.1564 0.844 15.61 0.01796 0.733 0.9925

0.3629 519.4 2.1986 0.866 16.32 0.01929 0.733 0.9937

v, h, and s from DuPont bull. T—MP 66—SI, Jan. 1993 (17 pp.). cp, µ, and k from DuPont bull. ART 10, Jan. 1993 (27 pp.). Some values read from charts may be approximate. Material used by permission of DuPont Fluoroproducts. b = normal boiling point.

TABLE 2-319 Temp., °C

Saturated SUVA HP 80 vf, m3/kg

cpf, kJ/(kg⋅K)

µf, 10−6 Pa·s

kf, W/(m⋅K)

1.6327 1.6206 1.6110 1.6034 1.5972

0.0970

1.193 1.217 1.236

377 317 283 247 215

0.0880 0.0849 0.0813

3.84 3.54 3.27

1.0000 1.0461 1.0927 1.1403 1.1897

1.5919 1.5870 1.5820 1.5762 1.5690

1.253 1.286 1.340 1.412 1.512

188 165 146 128 113

0.0778 0.0743 0.0708 0.0672 0.0634

3.03 2.86 2.76 2.69 2.70

1.2420 1.2998

1.5589 1.5433

1.64 1.81

98 83 68

0.0593 0.0551

2.71 2.79

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

679 695 713 733 757

0.2033 0.1303 0.0869 0.0598 0.0423

139.6 150.8 163.1 174.9 187.6

334.1 339.9 345.6 351.1 356.4

0.7578 0.8070 0.8584 0.9053 0.9541

785 819 860 911 977

0.03060 0.02248 0.01671 0.01250 0.00936

200.0 213.0 226.7 241.2 256.8

361.3 365.9 369.8 373.1 375.4

24.04 29.97

0.001 070 0.001 212

0.00696 0.00505

273.9 293.6

376.2 374.6

41.35

0.001 850

0.00185

340

340

Pf, bar

Pg, bar

−50 −40 −30 −20 −10

0.962 1.520 2.305 3.370 4.776

0.872 1.403 2.156 3.188 4.560

0.000 0.000 0.000 0.000 0.000

0 10 20 30 40

6.588 8.877 11.720 15.195 19.388

6.336 8.592 11.404 14.855 19.034

0.000 0.000 0.000 0.000 0.000

50 60 70 75.5c

24.39 30.30 41.35

Prf

c = critical point. SUVA HP 80 = R402 = CHF2CF3 (R125) 60% wt + CH3CH2CH3 (R290) 2% wt + CHClF2 (R22) 38% wt, near-azeotropic blend. Material used by permission of DuPont Fluoroproducts. Some values, read from charts, may be approximate.

2-284

PHYSICAL AND CHEMICAL DATA

TABLE 2-320

SUVA HP 80 at Atmospheric Pressure

Temp., °C

−46.95b

−40

−20

0

20

40

60

80

100

120

v (m3/ kg) h (kJ/ kg) s (kJ/ kg⋅K) cp (kJ/ kg⋅K) µ (10−6 Pa·s) k (W/m⋅K) Pr Z

0.1768 335.9 1.6286 0.648 9.42 0.00888 0.687 0.9673

0.1827 340.5 1.6490 0.654 9.69 0.00932 0.680 0.9697

0.1996 354.3 1.7055 0.687 10.45 0.01059 0.678 0.9758

0.2164 368.6 1.7599 0.721 11.22 0.01186 0.681 0.9804

0.2331 383.5 1.8124 0.749 11.99 0.01313 0.685 0.9840

0.2497 398.7 1.8633 0.779 12.75 0.01440 0.690 0.9868

0.2663 414.9 1.9128 0.807 13.52 0.01568 0.696 0.9892

0.2828 431.4 1.9610 0.836 14.29 0.01695 0.703 0.9910

0.2992 448.5 2.0081 0.863 15.06 0.01822 0.713 0.9923

0.3155 466.1 2.0541 0.890 15.82 0.01949 0.722 0.9932

b = normal boiling pt. v, h, and s from DuPont bull. T—HP 80—SI, Jan. 1993 (17 pp.). cp, µ, and k from DuPont bull. ART 18, June 1993 (37 pp.). Some values read from charts may be approximate. Material used by permission of DuPont Fluoroproducts.

TABLE 2-321 Temp., °C

Saturated SUVA HP 81 vf, m3/kg

cpf, kJ/(kg⋅K)

µf, 10−6 Pa·s

kf, W/(m⋅K)

Prf

1.7122 1.6957 1.6820 1.6706 1.6611

1.178 1.191 1.204

383 333 290 253 223

0.1031 0.0983 0.0941 0.0900 0.0863

3.63 3.35 3.11

1.0000 1.0450 1.0905 1.1367 1.1842 1.2339

1.6528 1.6451 1.6376 1.6299 1.6211 1.6104

1.221 1.288 1.313 1.37 1.75 2.07

195 173 151 137 122 106

0.0818 0.0790 0.0753 0.0715 0.0676 0.0633

2.91 2.82 2.63 2.49 3.16 3.47

1.2873 1.3164

1.5961 1.5866

91 75

0.0586 0.0544

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

687 702 719 739 761

0.2425 0.1548 0.1028 0.0707 0.0499

140.3 151.4 163.3 174.9 187.8

351.7 357.2 362.7 368.0 373.0

0.7606 0.8092 0.8589 0.9054 0.9550

787 817 854 899 955 030

0.03610 0.02656 0.01980 0.01490 0.01125 0.00848

200.0 212.7 226.0 240.1 255.1 271.4

377.8 382.2 386.0 389.3 391.5 392.8

28.03 34.60

0.001 136 0.001 307

0.00632 0.00456

289.5 299.6

392.2 390.9

44.45

0.001 88

0.00188

351

351

Pf, bar

Pg, bar

−50 −40 −30 −20 −10

0.883 1.403 2.135 3.132 4.451

0.787 1.273 1.967 2.923 4.198

0.000 0.000 0.000 0.000 0.000

0 10 20 30 40 50

6.153 8.307 10.984 14.261 18.216 22.93

5.852 7.959 10.591 13.827 17.750 22.45

0.000 0.000 0.000 0.000 0.000 0.001

60 70 80 82.6c

28.50 35.01 44.45

c = critical point. SUVA HP 81 = R402 (38/2/60) = CHF2CF3 (R125) 38% wt + CH3CH2CH3 (R290) 2% wt + CHClF2 (R22) 60% wt, near-azeotropic blend. Material used by permission of DuPont Fluoroproducts. Some values read from charts may be approximate.

TABLE 2-322

SUVA HP 81 at Atmospheric Pressure

Temp., °C

−44.87b

−40

−20

0

20

40

60

80

100

120

v (m3/ kg) h (kJ/ kg) s (kJ/ kg⋅K) cp (kJ/ kg⋅K) µ (10−6 Pa·s) k (W/m⋅K) Pr Z

0.1903 354.7 1.7032 1.187 10.16 0.00739 1.632 0.9622

0.1960 357.7 1.7169 1.177 10.33 0.00768 1.583 0.9703

0.2142 370.8 1.7711 1.169 11.10 0.00902 1.439 0.9766

0.2322 384.6 1.8232 1.159 11.86 0.01036 1.327 0.9811

0.2500 398.8 1.8735 1.149 12.62 0.01170 1.239 0.9843

0.2678 413.6 1.9222 1.143 13.39 0.01304 1.174 0.9870

0.2856 428.9 1.9696 1.134 14.15 0.01438 1.124 0.9894

0.3032 444.7 2.0158 1.128 14.78 0.01572 1.061 0.9909

0.3209 461.0 2.0607 1.124 15.54 0.01706 1.024 0.9926

0.3386 477.7 2.1047 1.120 16.30 0.01840 0.992 0.9940

b = normal boiling point. v, h, and s from DuPont bull. T—HP 81—SI, Jan. 1993 (17 pp.). cp, µ, and k from DuPont bull. ART 18, June 1993 (37 pp.). Some values, read from charts, may be approximate. Material used by permission of DuPont Fluoroproducts.

THERMODYNAMIC PROPERTIES TABLE 2-323

2-285

Saturated Refrigerant 113*

P, bar

v f, m3/kg

vg, m3/kg

h f, kJ/kg

hg, kJ/kg

s f, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

c pf, kJ/(kg⋅K)

µ f, 10−4 Pa⋅s

k f, W/(m⋅K)

240 250 260 270 280

0.0233 0.0435 0.0767 0.1290 0.2076

5.908.−4 5.986.−4 6.066.−4 6.150.−4 6.237.−4

4.548 2.537 1.492 0.9189 0.5893

5.70 14.19 22.83 31.65 40.63

171.97 178.06 184.22 190.46 196.75

0.0241 0.0587 0.0926 0.1259 0.1585

0.7169 0.7142 0.7134 0.7141 0.7161

0.845 0.877 0.895 0.916 0.933

17.9 14.8 12.3 10.4 8.9

0.087 0.084 0.083 0.081 0.079

290 300 310 320 330

0.3217 0.4817 0.6999 0.9897 1.3657

6.328.−4 6.422.−4 6.522.−4 6.626.−4 6.737.−4

0.3917 0.2687 0.1895 0.1370 0.1012

49.77 59.07 68.51 78.09 87.80

203.08 209.44 215.80 222.17 228.53

0.1906 0.2221 0.2530 0.2833 0.3131

0.7192 0.7233 0.7281 0.7336 0.7396

0.946 0.958 0.971 0.983 0.992

7.6 6.6 5.9 5.2 4.7

0.077 0.075 0.073 0.071 0.069

340 350 360 370 380

1.8347 2.4406 3.174 4.062 5.123

6.854.−4 6.979.−4 7.112.−4 7.255.−4 7.411.−4

0.0762 0.0584 0.0454 0.0357 0.0284

97.64 107.58 117.65 127.82 138.11

234.86 241.16 247.41 253.59 259.70

0.3424 0.3711 0.3993 0.4270 0.4542

0.7460 0.7528 0.7598 0.7669 0.7742

1.000 1.013 1.029 1.042 1.059

4.2 3.8 3.4 3.2 2.9

0.066 0.065 0.062 0.060 0.058

T, K

390 400 410 420 430

6.379 7.849 9.556 11.52 13.78

7.580.−4 7.767.−4 7.975.−4 8.211.−4 8.483.−4

0.0229 0.0185 0.0151 0.0124 0.0102

148.52 159.07 169.78 180.69 191.85

265.71 271.59 277.31 282.83 288.09

0.4810 0.5075 0.5336 0.5595 0.5853

0.7815 0.7888 0.7958 0.8027 0.8091

1.084 1.109 1.14 1.18 1.22

2.7 2.46 2.28 2.10 1.93

0.056 0.054 0.052 0.050 0.047

440 450 460 470 480

16.35 19.26 22.56 26.29 30.52

8.806.−4 9.201.−4 9.713.−4 1.044.−3 1.174.−3

0.0083 0.0068 0.0055 0.0044 0.0032

203.35 215.31 227.97 241.79 258.16

292.98 297.38 301.03 303.41 303.00

0.6112 0.6375 0.6645 0.6933 0.7264

0.8149 0.8198 0.8234 0.8244 0.8198

1.27 1.32 1.38 1.45 1.54

1.75 1.58 1.33 1.07 0.77

0.045 0.042 0.039 0.035 0.031

487.5c

34.11

1.754.−3

0.0018

288.10

288.10

0.7828

0.7828

∞

0.30

∞

*Values reproduced or converted from Table 8, p. 17.91, ASHRAE Handbook, 1981: Fundamentals, American Society of Heating, Refrigerating and AirConditioning Engineers, Atlanta, 1981. Copyright material. Reproduced by permission of the copyright owner. c = critical point. The notation 5.908.−4 signifies 5.908 × 10−4. The 1993 ASHRAE Handbook—Fundamentals (SI ed.) gives a saturation table from −30 to 214.4°C and an enthalpy–log-pressure diagram from 0.1 to 60 bar, 0 to 260°C. Equations and constants approximated to the 1985 ASHRAE tables were given by Mecaryk, K. and M. Masaryk, Heat Recovery Systems and CHP, 11, 2/3 (1991): 193–197. For experimental isochores for the compressed liquid from 21 to 304 bar, 266 to 453 K, see Blanke, W. and R. Weiss, PTB Bericht W 30, Braunschweig, Germany, 1992 (54 pp.). For tables to 300 bar, 460 K, see Geller, V. Z. and V. A. Rabinovich (ed.), Thermophysical Properties of Substances and Materials, Standartov, Moscow, 7 (1973): 135–154. Mastroianni, M. J., R. F. Stahl, et al., J. Chem. Eng. Data, 23, 2 (1978): 113–118 give a diagram to 1000 psia, 600°F. Tables and a diagram to 800 psia, 520°F are given by Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat, thermal conductivity, and viscosity, see Thermophysical Properties of Refrigerants, ASHRAE, 1993.

2-286

PHYSICAL AND CHEMICAL DATA

TABLE 2-324

Saturated Refrigerant 114*

P, bar

vf, m3/kg

vg, m3/kg

h f, kJ/kg

h g, kJ/kg

s f, kJ/(kg⋅K)

s g, kJ/(kg⋅K)

c pf, kJ/(kg⋅K)

µ f , 10−4 Pa⋅s

k f, W/(m⋅K)

190 200 210 220 230

0.0058 0.0137 0.029 0.059 0.109

6.326.−4 6.344.−4 6.366.−4 6.391.−4 6.421.−4

15.823 7.094 3.465 1.822 1.021

−42.58 −31.87 −21.48 −11.37 −1.50

125.78 131.01 136.41 141.95 147.61

−0.2091 −0.1542 −0.1035 −0.0565 −0.0126

0.6794 0.6648 0.6541 0.6466 0.6419

0.765 0.787 0.810 0.831 0.854

23.9 18.2 14.3 11.5 9.4

0.093 0.090 0.088 0.085 0.082

240 250 260 270 280

0.190 0.317 0.505 0.773 1.143

6.457.−4 6.500.−4 6.554.−4 6.619.−4 6.700.−4

0.604 0.375 0.2431 0.1633 0.1132

8.18 17.74 27.22 36.71 46.27

153.36 159.18 165.05 170.95 176.85

0.0286 0.0676 0.1047 0.1405 0.1751

0.6393 0.6387 0.6396 0.6418 0.6452

0.877 0.900 0.923 0.946 0.967

7.9 6.61 5.66 4.96 4.30

0.080 0.077 0.075 0.072 0.069

290 300 310 320 330

1.636 2.279 3.096 4.116 5.366

6.799.−4 6.918.−4 7.060.−4 7.224.−4 7.412.−4

0.0807 0.0590 0.0440 0.0334 0.0257

55.95 65.79 75.79 85.92 96.16

182.75 188.61 194.44 200.19 205.84

0.2090 0.2422 0.2748 0.3067 0.3379

0.6494 0.6543 0.6598 0.6657 0.6719

0.991 1.015 1.038 1.062 1.087

3.80 3.35 3.02 2.69 2.48

0.067 0.064 0.061 0.059 0.056

340 350 360 370 380

6.877 8.683 10.82 13.32 16.24

7.624.−4 7.863.−4 8.135.−4 8.453.−4 8.836.−4

0.0201 0.0158 0.0125 0.0099 0.0079

106.49 116.96 127.63 138.60 149.99

211.37 216.71 221.82 226.57 230.84

0.3685 0.3984 0.4280 0.4575 0.4872

0.6781 0.6843 0.6903 0.6957 0.7002

1.111 1.136 1.160 1.185 1.210

2.27 2.07 1.91 1.76 1.59

0.054 0.051 0.048 0.045 0.042

390 400 410 419.0c

19.62 23.52 28.00 32.61

9.324.−4 1.001.−3 1.118.−3 1.795.−3

0.0062 0.0048 0.0035 0.0018

162.01 175.03 190.13 219.90

234.36 236.61 236.20 219.90

0.5176 0.5496 0.5857 0.6559

0.7032 0.7036 0.6980 0.6559

1.236 1.261 1.5 ∞

1.39 1.17 0.87 0.34

0.038 0.034 0.030 ∞

T, K

*Values reproduced or converted from Table 9, p. 17.93, ASHRAE Handbook, 1981: Fundamentals, American Society of Heating, Refrigerating and AirConditioning Engineers, Atlanta, 1981. Copyright material. Reproduced by permission of the copyright owner. c = critical point. The notation 6.326.−4 signifies 6.326 × 10−4. The 1993 ASHRAE Handbook—Fundamentals (SI ed.) gives a saturation table from −80 to 145.88°C and an enthalpy–log-pressure diagram from 0.1 to 100 bar, −20 to 220°C. Equations and constants approximated to the 1985 ASHRAE tables were given by Mecaryk, K. and M. Masaryk, Heat Recovery Systems and CHP., 11, 2/3 (1991): 193–197. Saturation and superheat tables and a diagram to 60 bar, 540 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (179 pp.). Tables and a chart to 1500 psia, 480°F are given by Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat, thermal conductivity, and viscosity, see Thermophysical Properties of Refrigerants, ASHRAE, 1993.

TABLE 2-325 Temp., °F −100 −80 −60 −40 −20

Saturated Refrigerant 115* Volume, ft3/lb

Enthalpy, Btu/lb

Entropy, Btu/(lb)(°F)

Pressure, lb/in2 abs.

Liquid

Vapor

Liquid

Vapor

Liquid

Vapor

2.327 4.573 8.306 14.13 22.74

0.00966 0.00986 0.01009 0.01033 0.01060

10.57 5.624 3.218 1.953 1.245

−13.07 −8.78 −4.43 0.00 4.50

45.83 48.39 50.96 53.53 56.07

−0.0335 −0.0219 −0.0108 0.0000 0.0104

0.1302 0.1286 0.1278 0.1275 0.1277

0 20 40 60 80

34.94 51.59 73.65 102.1 138.1

0.01090 0.01123 0.01161 0.01204 0.01255

0.8257 0.5657 0.3979 0.2857 0.2081

9.09 13.76 18.54 23.45 28.54

58.56 61.00 63.35 65.60 67.71

0.0206 0.0305 0.0401 0.0496 0.0591

0.1282 0.1290 0.1298 0.1308 0.1317

100 120 140 160 170

182.7 237.3 303.2 382.0 427.0

0.01316 0.01393 0.01496 0.01664 0.01838

0.1530 0.1125 0.0817 0.0567 0.0444

33.85 39.50 45.67 52.76 56.56

69.63 71.24 72.36 72.42 71.33

0.0686 0.0782 0.0884 0.0996 0.1055

0.1325 0.1330 0.1329 0.1314 0.1290

175.89 c

457.6

0.0261

0.0261

64.30

64.30

0.1175

0.1175

*Unpublished data of General Chemicals Division, Allied Chemical Company. Used by permission. c = critical temperature. No material in SI units appears in the 1993 ASHRAE Handbook—Fundamentals (SI ed.). Tables and a chart to 50 ata, 200°C are given by Mathias, H. and H. J. Loffler, Techn. Univ. Berlin rept., 1966 (42 pp.). A chart to 1500 psia, 500°F was given by Mears, W. H., E. Rosenthal, et al., J. Chem. Eng. Data, 11, 3 (1966): 338–343.

THERMODYNAMIC PROPERTIES TABLE 2-326

2-287

Thermodynamic Properties of Refrigerant 123

Pressure, bar

Temp., K

vg, m3/kg

h f, kJ/kg

h g, kJ/kg

s f, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

c pf, kJ/(kg⋅K)

k f, W/(m⋅K)

µ f, 10−6 Pa·s

Prf

0.1 0.5 1.0 1.013 1.5

249.49 282.87 300.62 300.99 312.25

0.000 0.000 0.000 0.000 0.000

6315 6664 6862 6868 7008

1.3430 0.2993 0.1567 0.1546 0.1070

13.25 41.72 58.62 58.99 70.51

198.51 218.53 229.20 229.43 236.52

0.0548 0.1610 0.2195 0.2208 0.2582

0.7977 0.7863 0.7869 0.7870 0.7892

0.849 0.923 1.000 1.001 1.038

0.0908 0.0811 0.0759 0.0758 0.0726

798.7 503.7 409.8 408.1 361.5

7.46 5.73 5.40 5.39 5.17

2.0 2.5 3.0 4.0 5.0

321.18 328.50 334.79 345.29 353.95

0.000 0.000 0.000 0.000 0.000

7126 7230 7323 7490 7640

0.08139 0.06546 0.05525 0.03836 0.03358

79.90 87.76 94.59 106.16 115.83

241.76 246.20 249.96 256.17 261.17

0.2877 0.3118 0.3323 0.3661 0.3935

0.7917 0.7942 0.7965 0.8006 0.8042

1.063 1.079 1.091 1.108 1.120

0.0696 0.0678 0.0660 0.0630 0.0605

329.6 306.1 287.4 259.7 239.1

5.03 4.87 4.75 4.57 4.43

6 8 10 15 20

361.41 373.92 384.19 404.54 420.30

0.000 0.000 0.000 0.000 0.000

7779 8038 8280 8874 9512

0.02799 0.02090 0.01675 0.01062 0.00751

124.23 138.48 150.35 174.49 194.19

265.34 272.04 277.18 286.01 291.01

0.4168 0.4551 0.4860 0.5462 0.5928

0.8073 0.8124 0.8162 0.8218 0.8232

1.130 1.148 1.168 1.234 1.345

25 30 36.68*

433.33 444.10 456.83

0.001 030 0.001 136 0.001 818

0.00549 0.00408 0.00182

212.00 228.26 264.54

293.05 291.27 264.54

0.6334 0.6692 0.7393

0.8203 0.8112 0.7393

1.559 2.005

vf, m3/kg

h f = s f = 0 at −40°C = 233.15 K. s f, s g, cp,f units: kJ/kg⋅K. Interpolated and converted from 1993 ASHRAE Handbook—Fundamentals (SI ed.) saturation table from −40 to 183.68°C. This source also contains an enthalpy–log-pressure diagram from 0.1 to 200 bar, −40 to 320°C.

TABLE 2-327 Temp., °C

Saturated Refrigerant 124 Pressure, bar

vf, m3/kg

vg, m3/kg

h f, kJ/kg

h g, kJ/kg

s f, kJ/(kg⋅K)

s g, kJ/(kg⋅K)

−40 −30 −20 −10 0

0.2680 0.4499 0.7197 1.1044 1.6348

0.000 644 0.000 655 0.000 668 0.000 681 0.000 696

0.5173 0.3185 0.2049 0.1369 0.0945

159.1 169.3 179.5 189.7 200.0

334.9 340.6 346.2 351.8 357.4

0.8384 0.8813 0.9222 0.9616 1.0000

1.5927 1.5856 1.5808 1.5777 1.5762

10 20 30 40 50

2.3447 3.2710 4.4529 5.9320 7.7521

0.000 711 0.000 728 0.000 747 0.000 768 0.000 791

0.06703 0.04867 0.03604 0.02713 0.02069

210.5 221.3 282.3 243.7 255.4

363.0 368.5 373.9 379.2 384.4

1.0376 1.0747 1.1115 1.1480 1.1843

1.5760 1.5768 1.5785 1.5808 1.5836

60 70 80 90 100

9.9599 12.605 15.742 19.432 23.749

0.000 818 0.000 849 0.000 887 0.000 935 0.000 999

0.01594 0.01236 0.00961 0.00744 0.00569

267.5 280.1 293.2 307.0 321.9

389.3 393.9 398.0 401.3 403.4

1.2207 1.2572 1.2942 1.3318 1.3710

1.5864 1.5890 1.5909 1.5915 1.5894

110 120 122.5c

28.787 34.702 36.340

0.001 098 0.001 338 0.001 810

0.00420 0.00269 0.00181

338.4 360.6 378.5

403.0 394.9 378.5

1.4133 1.4685

1.5820 1.5558

c = critical point. Bull. T—124—SI, Jan. 1993 (28 pp.). Used by permission of DuPont Fluoroproducts. The 1993 ASHRAE Handbook— Fundamentals (SI ed.) gives a saturation table to 122.47°C and a diagram to 200 bar, 320°C.

FIG. 2-25

2-288

Enthalpy–log-pressure diagram for Refrigerant 123.

THERMODYNAMIC PROPERTIES TABLE 2-328

2-289

Thermophysical Properties of Saturated Refrigerant 125 µf , 10−6 Pa·s

Temp., K

Pressure, bar

vf, m3/kg

vg , m3/kg

h f, kJ/kg

h g , kJ/kg

s f, kJ/(kg⋅K)

s g, kJ/(kg·K)

172.5* 180 190 200 210

0.035 0.064 0.133 0.257 0.465

0.000591 0.000599 0.000611 0.000624 0.000638

3.48 1.958 0.986 0.5312 0.3057

−30.5 −23.3

140.8 146.7

−0.1386 −0.1024

0.7183 0.7067

220 224.9† 230 240 250

0.794 1.013 1.290 2.005 3.000

0.000653 0.000660 0.000669 0.000686 0.000705

0.1854 0.1475 0.1175 0.0775 0.0527

−13.9 −8.8 −3.4 7.7 19.3

152.6 155.5 158.4 164.2 170.0

−0.0604 −0.0386 −0.0147 0.0324 0.0800

0.6981 0.6948 0.6919 0.6875 0.6847

1.077 1.139 1.184

445.8 411.2 379.6 326.6 282.8

260 270 280 290 300

4.336 6.078 8.298 11.068 14.476

0.000725 0.000749 0.000776 0.000809 0.000848

0.0369 0.0264 0.0193 0.0143 0.0106

31.3 43.7 56.5 69.7 83.5

175.3 180.5 185.4 189.9 194.2

0.1274 0.1743 0.2206 0.2666 0.3126

0.6831 0.6822 0.6819 0.6815 0.6805

1.221 1.257 1.299 1.356 1.437

245.8 213.6 185.3 159.7 136.3

310 320 330 339.4‡

18.62 23.63 29.65 35.95

0.000898 0.000969 0.001088 0.00175

0.0079 0.0059 0.0041 0.0018

98.1 113.9 132.2 169.0

196.9 198.5 197.4 169.0

0.3597 0.4079 0.4621 0.5699

0.6774 0.6726 0.6639 0.5699

1.57 1.82

115.0 95.4

cpf, kJ/(kg⋅K)

644 531

* = triple point; † = normal boiling point; ‡ = critical point. Converted, extrapolated and interpolated from 1993 ASHRAE Handbook—Fundamentals (SI ed.) hf = sf = 0 at 233.15 K = −40°C. This source also contains an enthalpy–log-pressure diagram from 0.3 to 100 bar, −65 to 175°C. An apparently identical diagram but a different saturation table is contained in Duarte-Garza, H.A., Hwang, C.A. et al., ASHRAE Trans., 99, 2 (1993): 649–664. R124: The 1993 ASHRAE Handbook—Fundamentals (SI ed.) contains a saturation table from −60 to 122.47°C.

TABLE 2-329

Thermophysical Properties of Refrigerant 134a Temp., K

vf, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

s f, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf , kJ/(kg·K)

0.0039t 0.5 0.6 0.8 1.0

169.85 232.69 236.22 242.04 246.80

0.0006285 0.0007062 0.0007113 0.0007199 0.0007272

35.263 0.3692 0.3015 0.2375 0.1924

−76.68 −0.57 3.85 11.15 17.14

186.50 225.27 227.52 231.19 234.15

−0.3830 −0.0025 0.0161 0.0467 0.0713

1.1665 0.9669 0.9636 0.9560 0.9507

1.147 1.242 1.248 1.258 1.267

1.013 1.5 2.0 2.5 3.0

247.03 256.03 263.09 268.88 273.82

0.0007276 0.0007421 0.0007543 0.0007648 0.0007743

0.1902 0.1312 0.0999 0.0806 0.0677

17.50 28.96 38.13 45.75 52.33

234.33 239.86 244.14 247.60 250.50

0.0728 0.1181 0.1533 0.1819 0.2059

0.9503 0.9419 0.9364 0.9326 0.9297

1.268 1.288 1.306 1.322 1.337

4.0 5 6 8 10

282.08 288.89 294.72 304.47 312.53

0.0007912 0.0008063 0.0008203 0.0008460 0.0008703

0.0512 0.04116 0.03434 0.02565 0.02035

63.50 72.87 81.04 95.00 106.86

255.22 258.99 262.09 267.01 270.74

0.2458 0.2784 0.3062 0.3522 0.3901

0.9256 0.9232 0.9208 0.9171 0.9144

12 14 16 18 20

319.47 325.57 330.11 336.04 340.63

0.0008938 0.0009170 0.0009362 0.0009555 0.0009894

0.01675 0.01414 0.01247 0.01059 0.00931

117.34 126.80 134.00 143.68 151.39

273.65 275.92 277.40 279.01 279.95

0.4227 0.4515 0.4729 0.5013 0.5236

25 30 35 40 40.56c

350.73 359.37 366.89 373.50 374.18

0.0010585 0.001144 0.001270 0.001606 0.001948

0.00695 0.00528 0.00399 0.00255 0.00195

169.30 185.05 203.19 229.24 241.22

280.64 278.32 273.52 257.12 241.22

0.5738 0.6212 0.6657 0.7292 0.7620

Pressure, bar

µ f, 10−6 Pa·s

k f, W/(m⋅K)

Prf

0.1121 0.1105 0.1078 0.1056

5.61 5.42 5.12 4.90

406 358.7 326.6 303.2 285.1

0.1054 0.1013 0.0980 0.0954 0.0931

4.89 4.56 4.35 4.20 4.09

1.363 1.387 1.410 1.454 1.497

257.7 237.5 221.6 197.6 179.5

0.0893 0.0861 0.0835 0.0790 0.0753

3.93 3.83 3.74 3.64 3.57

0.9120 0.9095 0.9073 0.9041 0.9010

1.541 1.589 1.631 1.698 1.764

165.1 153.0 144.3 133.2 124.8

0.0721 0.0693 0.0672 0.0645 0.0623

3.53 3.51 3.50 3.51 3.53

0.8913 0.8807 0.8574 0.8038 0.7620

1.987 2.418

106.6 90.4

0.0577 0.0538

3.67 4.06

2187 506 480 438 408

t = triple point, c = critical point. hf = sf = 0 at −40°C = 233.15 K. T, v, h, and s interpolated and converted from Refrigerant 134a—Thermodynamic and Physical Properties, Int. Inst. Refrig., Paris, France, 1992 (28 pp.). Other properties from this source and from Oliveira, C. M. B. P. and W. A. Wakeham, Int. J. Thermophys., 14, 1 (1993): 33–44; Krauss, R., J. Luettmer-Strathmann, et al., Int. J. Thermophys., 14, 4 (1993): 951–988; ASHRAE Handbook—Fundamentals, Atlanta, GA, 1993; ICI KLEA 134a bulletin, 1993 (43 pp.); and R134a— Thermodynamic and Physical Properties, Int. Inst. Refrig., Paris, France, 1992 (28 pp.). Papers giving polynomial curve fits and similar simple equations include Cleland, A. C., Rev. Int. Froid = Int. J. Refrig., 17, 4 (1994): 245–249; Dobrokhotov, A., A. Grebenkov, et al., Proc. 14th Japan Symp. Thermophys. Props., (1993): 271–274; Huber, M. L. and J. F. Ely, Rev. Int. Froid = Int. J. Refrig., 17, 1 (1994): 18–31 (includes extensive list of vapor pressure and liquid density sources for many refrigerants); Modic, J., Proc. 11th Int. Symp. Htg, Refrig and Air-Condg., Zagreb, (1991): 174–185; and Kabelac, S., Int. J. Refrig., 14, (1991): 217–222. The 1993 ASHRAE Handbook—Fundamentals (SI ed.) gives saturation data for integral degrees Celsius with temperatures on the ITS 90 scale from −103.03°C to 101.03°C. The thermodynamic diagram from 0.1 to 200 bar extends to 320°C.

FIG. 2-26

2-290

Enthalpy–log-pressure diagram for Refrigerant 125.

THERMODYNAMIC PROPERTIES TABLE 2-330

2-291

Thermophysical Properties of Compressed Gaseous Refrigerant 134a Pressure, bar

Temp., K

0

1

2

3

4

5

6

cp (kJ/ kg⋅K) µ (10−6 Pa·s) 230 k (W/m⋅K) Pr

— — — —

— — — —

— — — —

— — — —

— — — —

— — — —

cp (kJ/ kg⋅K) −6 240 µ (10 Pa·s) k (W/m⋅K) Pr

— — — —

— — — —

— — — —

— — — —

— — — —

— — — —

cp (kJ/ kg⋅K) −6 250 µ (10 Pa·s) k (W/m⋅K) Pr

0.7437 10.11 0.0096 0.783

0.7953 10.15 0.0097 0.797

— — — —

— — — —

— — — —

— — — —

— — — —

cp (kJ/ kg⋅K) −6 260 µ (10 Pa·s) k (W/m⋅K) Pr

0.7627 10.47 0.0105 0.761

0.8048 10.51 0.0107 0.790

— — — —

— — — —

— — — —

— — — —

— — — —

cp (kJ/ kg) −6 270 µ (10 Pa·s) k (W/m⋅K) Pr

0.7813 10.84 0.0117 0.724

0.8158 10.88 0.0118 0.761

0.8557 10.94 0.0118 0.793

— — — —

— — — —

— — — —

— — — —

cp (kJ/ kg⋅K) −6 280 µ (10 Pa·s) k (W/m⋅K) Pr

0.7996 11.22 0.0122 0.735

0.8283 11.26 0.0123 0.757

0.8604 11.29 0.0123 0.790

— — — —

— — — —

— — — —

cp (kJ/ kg⋅K) −6 290 µ (10 Pa·s) k (W/m⋅K) Pr

0.8176 11.62 0.0130 0.731

0.8412 11.65 0.0131 0.748

0.8673 11.68 0.0131 0.773

0.8938 11.71 0.0133 0.787

0.9335 11.74

cp (kJ/ kg⋅K) −6 300 µ (10 Pa·s) k (W/m⋅K) Pr

0.8354 12.05 0.0139 0.730

0.8556 12.06 0.0139 0.742

0.8771 12.08 0.0140 0.757

0.8972 12.10 0.0141 0.770

0.9277 12.15 0.0142 0.792

0.0143 0.816

cp (kJ/ kg⋅K) −6 310 µ (10 Pa·s) k (W/m⋅K) Pr

0.8530 12.44 0.0145 0.730

0.8703 12.45 0.0146 0.742

0.8875 12.47 0.0147 0.753

0.9046 12.49 0.0148 0.763

0.9292 12.52 0.0149 0.781

0.9546 12.54 0.0150 0.798

cp (kJ/ kg⋅K) −6 320 µ (10 Pa·s) k (W/m⋅K) Pr

0.8703 12.83 0.0153 0.730

0.8843 12.84 0.0153 0.740

0.8993 12.86 0.0154 0.751

0.9163 12.88 0.0155 0.761

0.9356 12.90 0.0156 0.774

0.9548 12.93 0.0157 0.786

0.9750 12.97 0.0158 0.800

cp (kJ/ kg⋅K) −6 330 µ (10 Pa·s) k (W/m⋅K) Pr

0.8874 13.22 0.0160 0.729

0.8996 13.23 0.0161 0.739

0.9114 13.25 0.0161 0.750

0.9268 13.27 0.0162 0.759

0.9398 13.29 0.0163 0.766

0.9569 13.32 0.0164 0.777

0.9750 13.35 0.0165 0.789

cp (kJ/ kg⋅K) −6 340 µ (10 Pa·s) k (W/m⋅K) Pr

0.9042 13.61 0.0169 0.728

0.9152 13.62 0.0169 0.738

0.9262 13.64 0.0169 0.748

0.9372 13.66 0.0170 0.755

0.9502 13.68 0.0170 0.765

0.9632 13.70 0.0171 0.772

0.9770 13.73 0.0171 0.780

cp (kJ/ kg⋅K) −6 350 µ (10 Pa·s) k (W/m⋅K) Pr

0.9208 13.98 0.0175 0.730

0.9307 13.99 0.0176 0.740

0.9406 14.01 0.0176 0.749

0.9505 14.03 0.0177 0.754

0.9607 14.05 0.0177 0.763

0.9695 14.07 0.0178 0.767

0.9830 14.10 0.0179 0.774

— — — — 0.9606

0.9976

0.9827 0.0152

Dashes indicate unavailable states; blanks indicate no data. Note that “profound differences” presently exist in the transport properties of R134a according to Chemistry International, 16(6), 233, Nov. 1994, and 18(2) 44–47, 1996.

2-292

PHYSICAL AND CHEMICAL DATA

TABLE 2-330

Thermophysical Properties of Compressed Gaseous R134a (Concluded ) Pressure, bar

Temp., K

8

10

12.5

15

17.5

20

22.5

cp (kJ/ kg⋅K) 300 µ (10−6 Pa·s) k (W/m⋅K) Pr

— — — —

— — — —

— — — —

— — — —

— — — —

— — — —

— — — —

cp (kJ/ kg⋅K) 310 µ (10−6 Pa·s) k (W/m⋅K) Pr

1.053

— — — —

— — — —

— — — —

— — — —

— — — —

— — — —

cp (kJ/ kg⋅K) 320 µ (10−6 Pa·s) k (W/m⋅K) Pr

1.028 13.05 0.0161 0.833

1.097 13.13

— — — —

— — — —

— — — —

— — — —

— — — —

cp (kJ/ kg⋅K) 330 µ (10−6 Pa·s) k (W/m⋅K) Pr

1.015 13.41 0.0168 0.810

1.065 13.49 0.0171 0.840

1.151 13.64 0.0177 0.887

1.276 13.86 0.0184 0.961

— — — —

— — — —

— — — —

cp (kJ/ kg⋅K) 340 µ (10−6 Pa·s) k (W/m⋅K) Pr

1.008 13.79 0.0174 0.799

1.049 13.86 0.0177 0.821

1.107 13.98 0.181 0.855

1.187 14.17 0.0187 0.899

1.319

— — — —

— — — —

cp (kJ/ kg⋅K) 350 µ (10−6 Pa·s) k (W/m⋅K) Pr

1.008 14.15 0.0181 0.788

1.040 14.22 0.0183 0.828

1.086 14.34 0.0186 0.837

1.148 14.49 0.0192 0.866

1.225

1.340 14.97 0.0205

0.0155

TABLE 2-331

0.0198

Refrigerant 141b

The 1993 ASHRAE Handbook—Fundamentals (SI ed.) gives saturation data to 150°C and a diagram to 20 bar, 150°C. For equation of state including decomposition, see Weber, L. A., paper 69, Proc. 18th Int. Congr. Refrig., Montreal, 1991.

1.525 0.0215

THERMODYNAMIC PROPERTIES

2-293

40 410

5

5

0.

6

400 K

1 .7

0.

0.

8

Q ua

lit

y

1 .8

=

20

0

0.

7

25

1 .7

30

0

0.4

1. 25

390

90

0

10 9 8 7

1.

1.

6 95

py

5

tro

1.

En

Pressure (P ), bar

0. 9

1.

kJ /k g

•

85

K

15

4

00

2.

380

3

370 360

2.5 340 330

2 310 1.5

290 260

1

270

320

05

2.

300 K

280 10

2.

250 400

450 Enthalpy (h), kJ/kg

FIG. 2-27

350 K

Enthalpy–log-pressure diagram for Refrigerant 134a.

500

2-294

PHYSICAL AND CHEMICAL DATA

TABLE 2-332

Refrigerant 142b* µ f, 10−4 Pa⋅s

k f, W/(m⋅K)

1.15 1.17 1.18 1.19

0.517

0.123 0.118 0.114 0.111 0.109

0.9341 0.9264 0.9204 0.9155 0.9116

1.21 1.22 1.24 1.26 1.28

0.466 0.422 0.385 0.355 0.329

0.103 0.099 0.095 0.091 0.088

1.30 1.32 1.34

0.305 0.285 0.267 0.241 0.216

0.084 0.080 0.075 0.072 0.068

0.192

0.064 0.060 0.056 0.052 0.048

T, K

P, bar

vf, m3/kg

vg, m3/kg

h f, kJ/kg

h g, kJ/kg

s f, kJ/(kg⋅K)

s g, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

200 210 220 230 240

0.0380 0.0728 0.1314 0.2252 0.3691

7.505.−4 7.626.−4 7.751.−4 7.883.−4 8.019.−4

4.337 2.374 1.373 0.833 0.527

−24.36 −17.48 −10.21 −2.52 9.92

200.49 206.82 213.28 219.82 229.74

−0.1123 −0.0788 −0.0450 −0.0109 0.0414

1.0119 0.9893 0.9708 0.9558 0.9387

250 260 270 280 290

0.5815 0.8846 1.3046 1.8714 2.6184

8.164.−4 8.317.−4 8.480.−4 8.653.−4 8.843.−4

0.346 0.234 0.162 0.115 0.0838

14.32 23.54 33.32 43.68 54.60

233.05 239.66 246.18 252.57 258.77

0.0592 0.0952 0.1320 0.1695 0.2076

300 310 320 330 340

3.583 4.803 6.324 8.187 10.44

9.047.−4 9.273.−4 9.525.−4 9.810.−4 1.014.−3

0.0619 0.0464 0.0353 0.0271 0.0210

66.07 78.07 90.55 103.45 116.71

264.69 270.26 275.40 280.01 283.99

0.2462 0.2851 0.3243 0.3634 0.4024

0.9082 0.9051 0.9020 0.8985 0.8943

350 360 370 380 390

13.13 16.30 20.01 24.29 29.20

1.052.−3 1.099.−3 1.157.−3 1.235.−3 1.348.−3

0.0164 0.0129 0.0102 0.0080 0.0062

130.30 144.18 158.45 173.45 190.16

287.23 289.61 291.01 291.22 289.77

0.4409 0.4791 0.5170 0.5557 0.5974

0.8893 0.8831 0.8753 0.8656 0.8528

400 410c

34.78 41.5

1.541.−3 2.300.−3

0.0046 0.0023

212.57 255.00

284.04 255.00

0.6521

0.8307

0.044 ∞

*Values reproduced and converted from Table 10, p. 17.95, ASHRAE Handbook, 1981: Fundamentals, American Society of Heating, Refrigerating and AirConditioning Engineers, Atlanta, 1981. Copyright material. Reproduced by permission of the copyright owner. c = critical point. The notation 7.505.−4 signifies 7.505 × 10−4. The 1993 ASHRAE Handbook—Fundamentals (SI ed.) gives material for integral degrees Celsius with temperatures on the IPTS 68 scale from −50 to 125°C. The thermodynamic diagram from 0.1 to 35 bar extends to 180°C. For experimental isochores for the compressed liquid from 6 to 298 bar, 147 to 432 K, see Blanke, W. and R. Weiss, PTB Bericht W 30, Braunschweig, Germany, 1992 (54 pp.). Tables and a diagram to 500 psia, 400°F are given in Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat, thermal conductivity, and viscosity, see Thermophysical Properties of Refrigerants, ASHRAE, 1993.

TABLE 2-333 Temp., K t

161.82 170 180 190 200 210 220 225.92 230 240 250 260 270 280 290 300 310 320 330 340 346.75c

Saturated Refrigerant R143a*

Pressure, bar

vf, m3/kg

vg, m3/kg

hf, kJ/kg

h g, kJ/kg

s f , kJ/(kg⋅K)

s g, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µ f , 10−4 Pa⋅s

k f, W/(m⋅K)

0.01124 0.02497 0.05914 0.126 0.2458 0.4455 0.7586 1.01325 1.225 1.89 2.806 4.027 5.613 7.629 10.14 13.23 16.98 21.48 26.85 33.25 38.32

0.000752 0.000764 0.000778 0.000793 0.000809 0.000826 0.000845 0.000856 0.000865 0.000886 0.000910 0.000936 0.000966 0.000999 0.001038 0.001084 0.001140 0.001214 0.001321 0.001514 0.002311

14.22 6.709 2.991 1.474 0.7898 0.4532 0.2754 0.2098 0.1755 0.1164 0.07975 0.05617 0.04045 0.02964 0.02200 0.01646 0.01234 0.009182 0.006678 0.004520 0.002311

53.2 63.0 75.3 87.7 100.3 113.1 126.1 133.9 139.3 152.8 166.6 180.8 195.3 210.3 225.9 242.1 259.2 277.4 297.5 321.8 360.6

320.5 325.5 331.8 338.1 344.5 350.8 357.1 360.8 363.3 369.4 375.3 380.9 386.2 391.1 395.4 399.0 401.6 402.7 401.0 393.4 360.6

0.3181 0.3774 0.4474 0.5147 0.5792 0.6415 0.7018 0.7367 0.7604 0.8176 0.8736 0.9287 0.983 1.037 1.091 1.144 1.199 1.255 1.315 1.385 1.471

1.970 1.922 1.872 1.832 1.800 1.774 1.752 1.741 1.734 1.720 1.708 1.698 1.690 1.682 1.675 1.668 1.659 1.647 1.629 1.595 1.471

1.188 1.215 1.235 1.252 1.268 1.287 1.308 1.323 1.333 1.362 1.394 1.431 1.475 1.527 1.593 1.679 1.804 2.006 2.421 4.021 —

6.011 5.366 4.692 4.121 3.636 3.221 2.864 2.676 2.556 2.288 2.055 1.852 1.673 1.496 1.334 1.192 1.067 0.9569 0.8321 0.6560 —

0.1416 0.1403 0.1372 0.1331 0.1281 0.1226 0.1168 0.1133 0.1108 0.1047 0.09862 0.09272 0.08682 0.08098 0.07521 0.06951 0.06381 0.05803 0.05202 0.04480 —

*Values calculated from NIST Thermodynamic Properties of Refrigerants and Refrigerant Mixtures Database (REFPROP, Version 5). Thermodynamic properties are from 32-term MBWR equation of state; transport properties are from extended corresponding states model. t = triple point; c = critical point.

THERMODYNAMIC PROPERTIES TABLE 2-334

2-295

Saturated Refrigerant R152a*

Temp., K

Pressure, bar

vf, m3/kg

vg, m3/kg

hf, kJ/kg

h g, kJ/kg

s f , kJ/(kg⋅K)

s g, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µ f , 10−4 Pa⋅s

k f, W/(m⋅K)

154.56t 160 170 180 190 200 210 220 230 240 249.12 250 260 270 280 290 300 310 320 330 340 350 360 370 380 386.41c

0.000641 0.001297 0.004145 0.01141 0.02775 0.06088 0.1224 0.2284 0.4004 0.6647 1.01325 1.053 1.603 2.354 3.354 4.650 6.297 8.351 10.87 13.92 17.57 21.90 27.00 32.97 39.97 45.17

0.000839 0.000846 0.000859 0.000873 0.000887 0.000902 0.000918 0.000935 0.000952 0.000971 0.000989 0.000991 0.001012 0.001035 0.001060 0.001087 0.001118 0.001152 0.001190 0.001235 0.001289 0.001355 0.001440 0.001563 0.001785 0.002717

303.6 155.2 51.59 19.82 8.588 4.110 2.138 1.193 0.7064 0.4397 0.2961 0.2855 0.1922 0.1334 0.09500 0.06916 0.05126 0.03857 0.02935 0.02251 0.01735 0.01335 0.01020 0.007603 0.005274 0.002717

14.0 22.2 37.2 52.4 67.6 82.9 98.3 113.8 129.5 145.5 160.2 161.6 178.0 194.7 211.7 229.1 246.9 265.2 284.1 303.7 324.2 345.7 368.8 394.3 425.4 477.3

419.8 423.5 430.6 437.9 445.3 452.8 460.4 468.0 475.6 483.1 489.8 490.5 497.7 504.7 511.4 517.8 523.8 529.3 534.1 538.2 541.3 542.9 542.5 538.5 526.2 477.3

0.1130 0.1647 0.2560 0.3425 0.4247 0.5032 0.5784 0.6507 0.7205 0.7881 0.8481 0.8538 0.9178 0.9805 1.042 1.103 1.162 1.222 1.281 1.340 1.400 1.460 1.523 1.591 1.671 1.778

2.738 2.673 2.570 2.484 2.413 2.353 2.303 2.261 2.225 2.195 2.171 2.169 2.147 2.129 2.112 2.098 2.085 2.073 2.062 2.051 2.038 2.024 2.006 1.980 1.936 1.778

1.492 1.500 1.510 1.517 1.525 1.535 1.547 1.562 1.580 1.600 1.622 1.624 1.651 1.681 1.716 1.756 1.803 1.859 1.928 2.015 2.131 2.299 2.573 3.143 5.407 —

10.85 9.614 8.058 6.940 6.012 5.236 4.582 4.028 3.556 3.153 2.834 2.805 2.500 2.231 1.995 1.789 1.607 1.447 1.305 1.180 1.069 1.005 0.9225 0.8191 0.6638 —

0.1932 0.1894 0.1822 0.1753 0.1685 0.1618 0.1552 0.1487 0.1423 0.1361 0.1304 0.1299 0.1239 0.1179 0.1121 0.1064 0.1008 0.09526 0.08986 0.08457 0.07939 0.07336 0.06666 0.06032 0.05176 —

*Values calculated from NIST Thermodynamic Properties of Refrigerants and Refrigerant Mixtures Database (REFPROP, Version 5). Thermodynamic properties are from 32-term MBWR equation of state; transport properties are from extended corresponding states model. t = triple point; c = critical point.

TABLE 2-335 Temp., °F

Saturated Refrigerant 216* Pressure, lb/in2 abs.

Volume, ft3/lb

Enthalpy, Btu/lb

Entropy, Btu/(lb)(°F)

Liquid

Vapor

Liquid

Vapor

Liquid

Vapor

−40 −20 0 20 40

0.339 0.713 1.382 2.497 4.247

0.00927 0.00942 0.00958 0.00974 0.00992

59.957 29.749 15.986 9.184 5.582

0.000 4.778 9.541 14.298 19.056

62.415 65.276 68.208 71.199 74.239

0.0000 0.0111 0.0217 0.0318 0.0415

0.1487 0.1487 0.1493 0.1504 0.1520

60 80 100 120 140

6.862 10.612 15.797 22.753 31.845

0.01010 0.01030 0.01050 0.01073 0.01097

3.558 2.361 1.6215 1.1462 0.8304

23.821 28.598 33.391 38.205 43.049

77.319 80.429 83.559 86.701 89.845

0.0509 0.0599 0.0686 0.0770 0.0852

0.1538 0.1559 0.1582 0.1607 0.1632

160 180 200 220 240

43.468 58.046 76.033 97.913 124.21

0.01124 0.01153 0.01186 0.01223 0.01266

0.6142 0.4623 0.3529 0.2725 0.2121

47.930 52.861 57.857 62.939 68.132

92.981 96.099 99.186 102.225 105.196

0.0931 0.1009 0.1085 0.1161 0.1235

0.1658 0.1685 0.1712 0.1739 0.1765

260 280 300 320 340

155.50 192.40 235.63 286.03 344.81

0.01317 0.01378 0.01458 0.01570 0.01764

0.1660 0.1300 0.1013 0.0776 0.0565

73.474 79.015 84.835 91.089 98.234

108.066 110.789 113.282 115.373 116.538

0.1309 0.1384 0.1460 0.1539 0.1628

0.1790 0.1813 0.1834 0.1851 0.1856

355.98c

399.45

0.02771

0.0277

110.248

110.248

0.1773

0.1773

*From published data, Chemicals Division, Union Carbide Corporation. Used by permission. The paper describing these data is by Shank, ASHRAE J., 7 (1965): 94–101. c = critical temperature. The 1993 ASHRAE Handbook—Fundamentals (SI ed.) gives material for integral degrees Celsius with temperatures on the ITS 90 scale from −118.59 to 113.26°C. The thermodynamic diagram from 0.1 to 30 bar extends to 180°C. For tables and a diagram to 400 psia, 360°F, see Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat and viscosity, see Thermophysical Properties of Refrigerants, ASHRAE, 1993. Thermal conductivity data as a function of pressure and temperature are reported by Krauss, R. and K. Stephan, Proc. 12th Symp. Thermophys. Props., Boulder, CO, 1994.

2-296

PHYSICAL AND CHEMICAL DATA TABLE 2-336

Saturated Refrigerant 245*

P, bar

v f, m3/kg

vg, m3/kg

172 180 190 200 210

0.0034 0.0076 0.0190 0.0425 0.0870

6.46.−4 6.57.−4 6.70.−4 6.83.−4 6.97.−4

31.49 14.63 6.20 2.91 1.48

220 230 240 250 260

0.1654 0.2946 0.4958 0.7946 1.2204

7.11.−4 7.25.−4 7.40.−4 7.55.−4 7.72.−4

270 280 290 300 310

1.806 2.584 3.600 4.888 6.491

320 330 340 350 360 370 375 380.1c

T, K

h f, kJ/kg

h g, kJ/kg

s f, kJ/(kg⋅K)

s g, kJ/(kg⋅K)

−63.4 −55.9 −46.2 −36.0 −25.7

133.8 138.7 145.1 151.7 158.5

−0.3131 −0.2707 −0.2182 −0.1666 −0.1157

0.8327 0.8099 0.7885 0.7725 0.7612

0.822 0.475 0.292 0.192 0.125

−14.8 −3.6 8.0 19.9 32.3

165.4 172.5 179.6 186.8 194.0

−0.0654 −0.0156 0.0337 0.0824 0.1305

0.7539 0.7500 0.7487 0.7497 0.7525

7.89.−4 8.08.−4 8.30.−4 8.53.−4 8.80.−4

0.0862 0.0611 0.0443 0.0327 0.0246

44.9 57.9 71.1 84.6 98.4

201.1 208.3 215.3 222.2 228.9

0.1781 0.2249 0.2711 0.3161 0.3614

0.7567 0.7621 0.7683 0.7751 0.7822

8.456 10.83 13.67 17.04 21.02

9.11.−4 9.48.−4 9.93.−4 0.00105 0.00113

0.0186 0.0143 0.0111 0.0084 0.0063

112.6 127.1 142.1 157.2 174.7

235.3 241.4 246.9 251.5 254.8

0.4057 0.4497 0.4937 0.5382 0.5844

0.7893 0.7960 0.8018 0.8060 0.8071

25.71 28.46 31.37

0.00125 0.00137 0.00204

0.0045 0.0036 0.0020

193.6 205.2 231.8

255.2 252.5 231.8

0.6349 0.6649 0.7341

0.8013 0.7953 0.7341

*Values converted from tables of Shank, Thermodynamic Properties of UCON 245 Refrigerant, Union Carbide Corporation, New York, 1966. See also Shank, J. Chem. Eng. Data, 12, 474–480 (1967). c = critical point. The notation 6.46.−4 signifies 6.46 × 10−4.

TABLE 2-337

Refrigerant C 318*

P, bar

vf, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µf, 10−4 Pa⋅s

kf, W/(m⋅K)

200 210 220 230 240

0.0216 0.0449 0.0875 0.1608 0.2810

5.507.−4 5.593.−4 5.683.−4 5.778.−4 5.879.−4

3.810 1.931 1.038 0.588 0.349

353.5 361.0 369.2 377.6 386.4

498.0 500.1 502.2 504.4 510.9

3.909 3.947 3.984 4.022 4.060

4.560 4.564 4.569 4.574 4.578

0.98 1.00

11.7 9.55

0.088 0.085

250 260 270 280 290

0.466 0.741 1.133 1.672 2.392

5.988.−4 6.106.−4 6.234.−4 6.375.−4 6.529.−4

0.2166 0.1401 0.0938 0.0647 0.0458

395.6 405.2 415.1 425.8 436.2

517.4 524.0 530.7 537.3 543.9

4.097 4.133 4.172 4.210 4.247

4.584 4.592 4.599 4.609 4.618

1.02 1.03 1.05 1.07 1.09

7.90 6.63 5.64 4.85 4.22

0.082 0.078 0.075 0.071 0.068

300 310 320 330 340

3.325 4.522 6.007 7.826 10.018

6.694.−4 6.893.−4 7.115.−4 7.365.−4 7.666.−4

0.0332 0.0245 0.0184 0.0139 0.0106

447.3 458.7 470.5 482.7 495.2

550.4 556.9 563.3 569.4 575.4

4.284 4.322 4.359 4.396 4.433

4.626 4.638 4.648 4.659 4.669

1.12 1.15 1.18 1.23 1.27

3.70 3.20 2.94 2.66 2.33

0.065 0.061 0.058 0.054 0.051

350 360 370 380 388.5c

12.632 15.71 19.33 23.59 27.83

8.034.−4 8.508.−4 9.172.−4 1.031.−3 1.613.−3

0.0082 0.0062 0.0047 0.0033 0.0016

508.1 521.5 535.6 551.4 577.2

581.0 585.8 589.9 591.5 577.2

4.469 4.507 4.544 4.585 4.651

4.678 4.685 4.691 4.691 4.651

1.32 1.39

2.00

0.048

T, K

*Values of P, v, h, and s interpolated, extrapolated, and converted from tables of Oguchi, Reito, 52 (1977): 869–889. Values of cp, µ, and k interpolated and converted from tables in Thermophysical Properties of Refrigerants, American Society of Heating, Refrigerating and Air-Conditioning Engineers, New York, 1976. c = critical point. Saturation and superheat tables and a diagram to 80 bar, 580 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (173 pp.). For equations, see Cipollone, R., ASHRAE Trans., 97, 2 (1991): 262–267.

THERMODYNAMIC PROPERTIES TABLE 2-338

2-297

Saturated Refrigerant 500*

P, bar

vf, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µf, 10−4 Pa⋅s

kf, W/(m⋅K)

200 210 220 230 240

0.1219 0.2258 0.3936 0.6511 1.0291

6.966.−4 7.090.−4 7.222.−4 7.361.−4 7.509.−4

1.360 0.766 0.457 0.286 0.187

−29.56 −21.03 −12.17 −2.97 6.58

185.87 191.25 196.63 201.96 207.23

−0.1363 −0.0948 −0.0536 −0.0130 0.0277

0.9408 0.9161 0.8955 0.8782 0.8638

1.044 1.018 0.997 0.987 0.987

6.11 5.15 4.42 3.85 3.42

0.113 0.109 0.106 0.102 0.098

250 260 270 280 290

1.5632 2.2932 3.2624 4.5172 6.1064

7.668.−4 7.839.−4 8.024.−4 8.226.−4 8.450.−4

0.1261 0.0879 0.0628 0.0459 0.0342

16.50 26.78 37.44 48.48 59.91

212.40 217.45 222.35 227.06 231.56

0.0680 0.1082 0.1481 0.1878 0.2275

0.8517 0.8415 0.8329 0.8257 0.8194

0.997 1.017 1.048 1.089 1.140

3.04 2.74 2.48 2.26 2.08

0.094 0.090 0.086 0.082 0.078

300 310 320 330 340

8.0809 10.49 13.40 16.86 20.93

8.699.−4 8.981.−4 9.306.−4 9.690.−4 1.016.−3

0.0259 0.0198 0.0154 0.0119 0.0093

71.76 84.05 96.83 110.17 124.20

235.79 239.69 243.19 246.14 248.36

0.2671 0.3067 0.3464 0.3864 0.4271

0.8139 0.8088 0.8038 0.7985 0.7922

1.201 1.273 1.355 1.447 1.550

1.92 1.77 1.63 1.48 1.34

0.074 0.070 0.066 0.062 0.058

350 360 370 378.6c

25.70 31.25 37.72 44.26

1.077.−3 1.162.−3 1.307.−3 2.012.−3

0.0072 0.0055 0.0040 0.0020

139.18 155.66 175.59 219.50

249.47 248.71 244.26 219.50

0.4689 0.5135 0.5650 0.6729

0.7841 0.7721 0.7509 0.6729

1.663 1.919 2.07 ∞

T, K

*Values reproduced and converted from Table 12, p. 17.99, ASHRAE Handbook, 1981: Fundamentals, American Society of Heating, Refrigerating and Air-Conditioning Engineers, Atlanta, 1981. Copyright material. Reproduced by permission of the copyright owner. c = critical point. The notation 6.966.−4 signifies 6.966 × 10−4. The 1993 ASHRAE Handbook—Fundamentals (SI ed.) gives material for integral degrees Celsius with temperatures on the IPTS 68 scale from −70 to 105.60°C. The thermodynamic diagram from 0.1 to 70 bar extends to 240°C. Equations and constants approximated to the 1985 ASHRAE tables were given by Mecaryk, K. and M. Masaryk, Heat Recovery Systems and CHP, 11, 2/3 (1991): 193–197. Saturation and superheat tables and a diagram to 80 bar, 560 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (173 pp.). Tables and a chart to 1000 psia, 480°F are given by Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). Specific heat and viscosity appear in Thermophysical Properties of Refrigerants, ASHRAE, 1993.

TABLE 2-339

Saturated Refrigerant 502*

P, bar

vf, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µf, 10−4 Pa⋅s

kf, W/(m⋅K)

200 210 220 230 240

0.2274 0.4098 0.6965 1.1251 1.7392

6.381.−4 6.507.−4 6.640.−4 6.783.−4 6.938.−4

0.646 0.374 0.228 0.146 0.0969

−29.04 −20.83 −12.15 −2.99 6.66

153.34 158.42 163.49 168.50 173.42

−0.1337 −0.0937 −0.0534 −0.0128 0.0280

0.7782 0.7599 0.7449 0.7328 0.7228

1.018 1.036 1.055 1.075 1.097

5.72 4.88 4.23 3.71 3.28

0.103 0.099 0.095 0.091 0.087

250 260 270 280 290

2.5867 3.7188 5.1893 7.0530 9.3660

7.105.−4 7.289.−4 7.492.−4 7.720.−4 7.979.−4

0.0665 0.0470 0.0340 0.0251 0.0188

16.78 27.36 38.36 49.77 61.55

178.20 182.81 187.21 191.35 195.16

0.0691 0.1102 0.1514 0.1923 0.2330

0.7148 0.7082 0.7027 0.6980 0.6937

1.120 1.144 1.170 1.197 1.225

2.94 2.65 2.41 2.18 1.99

0.083 0.079 0.075 0.072 0.068

T, K

300 310 320 330 340

12.19 15.57 19.60 24.35 29.95

8.280.−4 8.637.−4 9.081.−4 9.666.−4 1.053.−3

0.0143 0.0109 0.0084 0.0064 0.0048

73.68 86.17 99.06 112.53 127.13

198.56 201.43 203.57 204.62 203.71

0.2734 0.3134 0.3532 0.3933 0.4351

0.6896 0.6852 0.6798 0.6723 0.6604

1.254 1.285 1.317 1.351 1.386

1.79 1.59 1.40 1.23 1.07

0.064 0.060 0.056 0.052 0.048

350 355.3c

36.62 40.75

1.220.−3 1.786.−3

0.0033 0.0018

145.44 174.00

197.82 174.00

0.4859 0.5634

0.6355 0.5634

1.422

0.93

0.044

*Values reproduced and converted from Table 13, p. 17.101, ASHRAE Handbook, 1981: Fundamentals, American Society of Heating, Refrigerating and AirConditioning Engineers, Atlanta, 1981. Copyright material. Reproduced by permission of the copyright owner. c = critical point. The notation 6.381.−4 signifies 6.381 × 10−4. The 1993 ASHRAE Handbook—Fundamentals (SI ed.) gives material for integral degrees Celsius with temperatures on the IPTS 68 scale from −70 to 82.2°C. The thermodynamic diagram from 0.1 to 80 bar extends to 180°C. Equations and constants approximated to 1985 ASHRAE tables are given by Mecaryk, K., and M. Masaryk, Heat Recovery Systems and CHP, 11, 2/3 (1991): 193–197. Saturation and superheat tables and a diagram to 20 bar, 515 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (173 pp.). Tables and a chart to 1000 psia, 400°F appear in Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat and viscosity, see Thermophysical Properties of Refrigerants, ASHRAE, 1993.

2-298

PHYSICAL AND CHEMICAL DATA

TABLE 2-340

Saturated Refrigerant 503*

P, bar

vf, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µf, 10−4 Pa⋅s

kf, W/(m⋅K)

150 160 170 180 190

0.0750 0.1798 0.3828 0.7395 1.3187

6.384.−4 6.478.−4 6.585.−4 6.700.−4 6.850.−4

1.894 0.837 0.414 0.224 0.130

−89.60 −79.73 −69.55 −59.08 −48.36

111.02 115.40 119.70 123.84 127.77

−0.4694 −0.4057 −0.3441 −0.2844 −0.2267

0.8681 0.8139 0.7691 0.7318 0.7003

0.482 0.554 0.620 0.682 0.747

6.12 5.05 4.16 3.43 2.94

0.128 0.123 0.116 0.111 0.105

200 210 220 230 240

2.1999 3.4713 5.2281 7.5713 10.61

7.014.−4 7.204.−4 7.426.−4 7.687.−4 8.001.−4

0.0803 0.0520 0.0350 0.0242 0.0172

−37.45 −26.36 −15.10 − 3.65 8.07

131.45 134.84 137.87 140.49 142.58

−0.1710 −0.1173 −0.0656 −0.0155 0.0334

0.6735 0.6503 0.6298 0.6112 0.5939

0.817 0.896 0.988 1.017 1.227

2.56 2.25 1.98 1.73 1.52

0.099 0.094 0.088 0.082 0.076

250 260 270 280 290

14.46 19.25 25.13 32.27 40.87

8.386.−4 8.874.−4 9.526.−4 1.050.−3 1.264.−3

0.0124 0.0090 0.0064 0.0045 0.0028

20.22 33.10 47.22 63.64 86.41

143.98 144.38 143.23 139.25 127.51

0.0817 0.1305 0.1816 0.2384 0.3131

0.5767 0.5585 0.5373 0.5085 0.4548

1.382 1.57 1.79 2.03 2.35

1.33 1.17 1.03 0.91

0.070 0.065 0.059 0.054

292.6c

43.57

1.773.−3

0.0018

110.20

110.20

0.3864

0.3864

∞

T, K

∞

*P, v, h, and s values reproduced and converted from Table 14, p. 17.103, ASHRAE Handbook, 1981: Fundamentals, American Society of Heating, Refrigerating and Air-Conditioning Engineers, Atlanta, 1981. Copyright material. Reproduced by permission of the copyright owner. cp, µ, and k values interpolated and converted from Thermophysical Properties of Refrigerants, American Society of Heating, Refrigerating and Air-Conditioning Engineers, New York, 1976. c = critical point. The notation 6.384.−4 signifies 6.384 × 10−4. Saturation and superheat tables and a diagram to 80 bar, 600 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (173 pp.). Tables and a chart to 1000 psia, 460°F are given by Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat and viscosity see Thermophysical Properties of Refrigerants, ASHRAE, 1993. The 1993 ASHRAE Handbook—Fundamentals (SI ed.) gives material for integral degrees Celsius with temperatures on the IPTS 68 scale for saturation conditions from −125 to 19.50°C. The thermodynamic diagram from 0.1 to 80 bar extends to 220°C. TABLE 2-341

Saturated Refrigerant 504* Volume, ft3/lb

Enthalpy, Btu/lb

Entropy, Btu/(lb)(°F)

Temp., °F

Pressure, lb/in2 abs.

Liquid

Vapor

Liquid

Vapor

Liquid

Vapor

−120 −100 −80 −60 −40

2.964 6.042 11.34 19.85 32.76

0.01095 0.01119 0.01146 0.01175 0.01206

15.31 7.874 4.372 2.585 1.609

−21.48 −16.39 −11.12 −5.65 0.00

86.69 89.31 91.84 94.25 96.50

−0.0565 −0.0420 −0.0277 −0.0137 0.0000

0.2609 0.2519 0.2435 0.2362 0.2299

−20 0 20 40 60

51.44 77.41 112.3 158.0 216.2

0.01242 0.01282 0.01328 0.01379 0.01443

1.045 0.7029 0.4859 0.3431 0.2458

5.85 11.91 18.22 24.81 31.78

98.58 100.45 102.09 103.44 104.41

0.0135 0.0269 0.0401 0.0533 0.0667

0.2244 0.2195 0.2150 0.2107 0.2065

80 100 120 140 150

289.2 379.1 488.3 618.1 692.2

0.01522 0.01629 0.01783 0.02083 0.02597

0.1773 0.1274 0.0893 0.0578 0.0394

39.25 47.43 56.78 69.97 76.96

104.85 104.49 102.72 97.70 89.76

0.0804 0.0948 0.1107 0.1322 0.1432

0.2020 0.1968 0.1899 0.1784 0.1642

*Unpublished data of Allied Chemical Company, 1970. Used by permission. TABLE 2-342

Thermodynamic Properties of Refrigerant 507*

Temp., K

Pressure, bar

230.5 240 250 260 270 280

1.013 1.59 2.42 3.54 4.95 6.70

vf, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

0.000 0.000 0.000 0.000 0.000 0.000

574 602 627 658 695 738

0.1280 0.0826 0.0546 0.0377 0.0270 0.0198

−3.1 10.3 22.6 37.6 51.6 64.7

143.3 150.2 154.5 159.0 163.8 169.0

−0.015 0.042 0.092 0.149 0.202 0.250

0.620 0.623 0.619 0.617 0.618 0.620

290 300 310 320 330

8.85 11.52 14.74 18.76 23.65

0.000 0.000 0.000 0.001 0.001

787 839 903 006 221

0.0148 0.0112 0.0084 0.0062 0.0042

77.2 89.4 101.6 115.7 135.5

174.6 180.3 185.4 188.6 189.3

0.295 0.336 0.378 0.422 0.481

0.634 0.640 0.648 0.649 0.641

340 341.5c

29.57 32.67

0.001 618 0.001 97

0.0025 0.0020

161.7 172.7

179.9 172.7

0.557 0.590

0.611 0.590

*Azeotropic mixture of R152a and R218. hf = sf = 0 at 233.15 K = −40°C. Interpolated, extrapolated and converted from Lavrenchenko, G. K., M. G. Khmelnuk, et al., Int. J. Refrig., 17, 7 (1994): 461. Some values are tentative. This source also gives a ln P–h diagram from 0.6 to 30 bar, −50 to 70°C. Differences exist between the published diagram and tables. c = critical point.

THERMODYNAMIC PROPERTIES TABLE 2-343

Saturated Rubidium*

T, K

P, bar

vf, m3/kg

312.7m 400 500 600 700

2.46.−9 1.69.−6 1.73.−4 0.0037 0.0317

6.75.−4 6.98.−4 7.22.−4 7.46.−4 7.73.−4

0.1584 1.467 6.466 18.6 28.5

8.10.−4 8.65.−4 9.40.−4 1.03.−3 1.08.−3

800 1000 1200 1400 1500

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

2.3.+5 2790 156.6 20.75

118.7 151.6 188.8 225.4 261.3

1036 1057 1078 1096 1111

0.998 1.091 1.174 1.241 1.296

3.932 3.355 2.953 2.692 2.511

0.379 0.375 0.369 0.362 0.357

4.662 0.605 0.159

296.8 367.6 440.1

1124 1150 1179

1.343 1.422 1.490

2.378 2.205 2.104

0.353 0.360 0.385

*Converted from tables in Vargaftik, Tables of the Thermophysical Properties of Liquids and Gases, Nauka, Moscow, 1972, and Hemisphere, Washington, 1975. m = melting point. The notation 2.46.−9 signifies 2.46 × 10−9. Many of the Vargaftik values also appear in Ohse, R. W., Handbook of Thermodynamic and Transport Properties of Alkali Metals, Blackwell Sci. Pubs., Oxford, 1985 (1020 pp.). This source contains superheat data. Saturation and superheat tables and a diagram to 40 bar, 1600 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (173 pp.). For a Mollier diagram from 0.1 to 320 psia, 1200 to 2700°R, see Weatherford, W. D., J. C. Tyler, et al., WADD-TR-61-96, 1961. An extensive review of properties of the solid and the saturated liquid was given by Alcock, C. B., M. W. Chase, et al., J. Phys. Chem. Ref. Data, 23, 3 (1994): 385–497.

TABLE 2-344 Temp., °C

Thermophysical Properties of Saturated Seawater

Pressure, bar

v, (m3/kg)103

cp, kJ/(kg⋅K)

µ, Ns/m2

k, W/(m⋅K)

NPr

105κ, 1/bar

0 1 2 3 4

0.005993 0.006438 0.006916 0.007427 0.007970

1.000158 1.000099 1.000057 1.000033 1.000025

4.000 4.000 4.000 4.000 4.001

0.001884 0.001827 0.001772 0.001720 0.001669

0.560 0.563 0.565 0.567 0.569

13.46 12.98 12.55 12.13 11.74

5.06 5.02 4.98 4.95 4.92

5 6 7 8 9

0.008548 0.009163 0.009816 0.010511 0.011248

1.000033 1.000057 1.000096 1.000149 1.000261

4.001 4.001 4.002 4.002 4.002

0.001620 0.001574 0.001529 0.001486 0.001445

0.571 0.574 0.576 0.578 0.580

11.35 10.97 10.62 10.29 9.97

4.89 4.86 4.83 4.80 4.78

10 11 12 13 14

0.01203 0.01286 0.01374 0.01467 0.01566

1.000298 1.000392 1.000500 1.000620 1.000727

4.003 4.003 4.003 4.004 4.004

0.001405 0.001367 0.001330 0.001294 0.001259

0.582 0.584 0.586 0.588 0.590

9.70 9.37 9.09 8.81 8.54

4.76 4.74 4.72 4.70 4.68

15 16 17 18 19

0.01671 0.01781 0.01898 0.02022 0.02153

1.000899 1.001055 1.001224 1.001404 1.001595

4.005 4.005 4.006 4.006 4.007

0.001226 0.001195 0.001165 0.001136 0.001107

0.592 0.594 0.595 0.597 0.599

8.29 8.06 7.82 7.62 7.41

4.66 4.65 4.63 4.62 4.60

20 21 22 23 24

0.02291 0.02437 0.02591 0.02753 0.02924

1.001796 1.002009 1.002232 1.002465 1.002708

4.007 4.007 4.008 4.008 4.009

0.001080 0.001054 0.001029 0.001005 0.000981

0.600 0.602 0.604 0.605 0.607

7.21 7.02 6.82 6.66 6.48

4.59 4.57 4.56 4.55 4.54

25 26 27 28 29

0.03104 0.03294 0.03494 0.03705 0.03926

1.002961 1.003224 1.003496 1.003778 1.004069

4.009 4.009 4.010 4.010 4.011

0.000958 0.000936 0.000915 0.000895 0.000875

0.608 0.609 0.611 0.612 0.614

6.31 6.16 6.01 5.86 5.72

4.53 4.52 4.51 4.50 4.49

30

0.04159

1.004369

4.011

0.000855

0.615

5.58

4.48

κ = (−1/V)(∂v/∂p)T ⋅ 105. Thus, at 0°C, the compressibility is 5.06 × 10−5/bar. For further information see, for instance, Bromley, LeR. A., J. Chem. Eng. Data, 12, 2 (1967): 202–206; 13, 1 (1968): 60–62 and 13, 3: 399–402; 15, 2 (1970): 246–253; and A.I.Ch.E.J., 20, 2 (1974): 326–335. Thermal conductivity data sources include Castelli, V. J., E. M. Stanley, et al., Deep Sea Res., 211 (1974): 311–318; Levy, F. L., Int. J. Refrig., 5, 3 (1982): 155–159. For velocity of sound, see, for instance, U.S. Naval Oceanographic Office SP 58, 1962 (50 pp.). More recent information is contained in UNESCO technical papers. See Marine Science No. 38, 1981 (6 pp.) and No. 44, 1983 (53 pp.). For sea ice properties, see Fukusako, S., Int. J. Thermophys., 11, 2 (1990): 353–372.

2-299

TABLE 2-345 Saturated Sodium Temp., K

Pressure, bar

vf, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

cpg, kJ/(kg⋅K)

µf, 10−6 Pa·s

µg, 10−6 Pa·s

kf, W/(m⋅K)

kg, W/(m⋅K)

Prg

1.59.−10 1.80.−9 8.99.−7 5.57.−5 0.00105

0.001 0.001 0.001 0.001 0.001

078 088 115 144 174

8.54.+9 8.08.+8 1.99.+6 38022 2320

207 247 382 514 642

4739 4757 4817 4872 4921

2.259 2.920 3.222 3.462 3.661

14.475 14.195 12.092 10.745 10.631

1.383 1.372 1.334 1.301 1.277

0.86 1.25 1.80 2.28

688 599 415 321 264

800 900 1000 1100 1154.7

0.00941 0.05147 0.1995 0.6016 1.013

0.001 0.001 0.001 0.001 0.001

208 242 280 323 347

291.5 58.8 16.6 5.95 3.89

769 895 1020 1146 1215

4966 5007 5044 5079 5097

3.830 3.978 4.110 4.230 4.290

9.076 8.547 8.134 7.805 7.652

1.260 1.252 1.252 1.261 1.271

2.59 2.72 2.70 2.62 2.56

227 201 181 166 159

19.6 20.6 23.0 25.3 26.5

62.9 58.3 54.2 50.5 48.7

0.0343 0.0406 0.0455 0.0492 0.0522

0.0045 0.0043 0.0042 0.0042 0.0041

1.48 1.38 1.36 1.35 1.30

27.5 29.9 32.2 34.6 37.1

47.2 44.0 41.1 38.2 35.4

0.0547 0.0570 0.0592

0.0041 0.0042 0.0044 0.0046 0.0050

1.26 1.27 1.30

1200 1300 1400 1500 1600

1.50 3.26 6.30 11.13 18.28

0.001 0.001 0.001 0.001 0.001

366 416 471 531 597

2.54 1.24 0.676 0.400 0.253

1273 1402 1534 1671 1812

5111 5140 5168 5193 5217

4.340 4.444 4.542 4.636 4.727

7.538 7.319 7.138 6.984 6.855

1.279 1.305 1.340 1.384 1.437

2.51 2.43 2.39 2.36 2.34

153 143 135 128 122

1700 1800 1900 2000 2100

28.28 41.61 58.70 79.91 105.5

0.001 0.001 0.001 0.001 0.002

675 761 862 984 174

0.168 0.117 0.084 0.063 0.0472

1959 2113 2274 2444 2625

5238 5256 5268 5273 5265

4.816 4.904 4.992 5.079

6.745 6.650 6.568 6.494

1.500 1.574 1.661 1.764 1.926

2.41 2.46 2.53 2.66 2.91

117 112 108 104

2200 2300 2400 2500 2503.7c

135.7 170.6 210.3 254.7 256.4

0.002 0.002 0.002 0.004 0.004

320 584 985 19 57

0.0361 0.0275 0.0203 0.0098 0.0046

2822 3047 3331 3965 4294

5241 5188 5078 4617 4294

2.190 2.690 4.012 39.3

3.40 4.47 8.03 417.

89.4 87.2 80.1 73.7 68.0

Prf

371 400 500 600 700

32.6 29.7 26.6 23.2

0.0106 0.0094 0.0069 0.0057 0.0050

0.0054 0.0059 0.0067 0.0079

c = critical point. sf values converted from Cordfunke, E. H. P. and R. J. M. Konings, Thermochemical Data for Reactor Materials and Fission Products, North Holland Elsevier, NY, 1990. sg determined as sf + (hg − hf)/T. µg and kg values estimated by P. E. Liley. All other values are from Fink, J. K. and L. Leibowitz, Argonne Nat. Lab Rept. ANL/RE-95-2, 1995. The Fink and Leibowitz work also appeared in High Temp. Materials Sci., 35, 65–103, 1996. Saturation and superheat tables and a diagram to 14 bar, 1700 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (173 pp.). For a Mollier diagram for 0.1–150 psia, 1500–2700°R, see Weatherford, P. M., J. C. Tyler, et al., WADD-TR-61-96, 1961.

2-300

THERMODYNAMIC PROPERTIES

2-301

2500 K 2400 6500

2300 2200 2100 2000 1900 1800

ar

1700

5

10 b

1600

1 ba

r

2

h, kJ/kg

6000

0.5

1500

0.2

1400

0.1

1300

5500

0.0

5

1200

0.0

2

1100

0.0 1

1000

5000 7.0

800

r

8.0

9.0 Entropy (s), kJ/kg • K

bar

vapo

01

te d

0.0

tura

0.0 02

Sa

0.0 05

900

700

10.0

FIG. 2-28 Mollier Diagram for Sodium. Drawn from the Vargaftik et al. values in Ohse, R. W., Handbook of Thermodynamic and Transport Properties of Alkali Metals, Blackwell Sci. Pubs., Oxford, UK, 1985. These values are identical with those of Vargaftik, N. B., Handbook of Thermophysical Properties of Gases and Liquids, Moscow, 1972, and the Hemisphere translation, pp. 19. An apparent discontinuity exists between the superheat values and the saturation values, not reproduced here. For a Mollier diagram in f.p.s. units from 0.1 to 150 psia, 1500 to 2700°R, see Fig. 3-36, p. 3-232 of the 6th edition of this handbook. An extensive review of properties of the solid and the saturated liquid was given by Alcock, C. B., Chase, M. W. et al., J. Phys. Chem. Ref. Data, 23(3), 385–497, 1994.

2-302

PHYSICAL AND CHEMICAL DATA

FIG. 2-29 Enthalpy-concentration diagram for aqueous sodium hydroxide at 1 atm. Reference states: enthalpy of liquid water at 32°F and vapor pressure is zero; partial molal enthalpy of infinitely dilute NaOH solution at 64°F and 1 atm is zero. [McCabe, Trans. Am. Inst. Chem. Eng., 31, 129 (1935).]

TABLE 2-346

Saturated Sulfur Dioxide*

P, bar

vf, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µf, 10−4 Pa⋅s

200 210 220 230 240

0.02056 0.04569 0.09997 0.1844 0.3202

6.189.−4 6.284.−4 6.384.−4 6.488.−4 6.596.−4

12.602 5.946 2.876 1.605 0.9602

7.4 9.1 28.6 43.5 56.5

433.3 446.1 453.8 459.5 464.5

0.033 0.041 0.123 0.198 0.254

2.212 2.159 2.075 2.001 1.952

1.280 1.284 1.288 1.293 1.299

12.3 10.6 8.37 7.03 5.97

250 260 270 280 290

0.5430 0.8778 1.3634 2.0402 2.9574

6.707.−4 6.819.−4 6.938.−4 7.057.−4 7.184.−4

0.5864 0.3745 0.2479 0.1699 0.1197

70.0 85.1 99.8 114.8 129.2

469.7 474.5 479.3 484.3 488.5

0.308 0.363 0.425 0.473 0.523

1.906 1.865 1.827 1.793 1.763

1.308 1.317 1.328 1.343 1.363

5.11 4.39 3.78 3.30 2.87

0.262 0.243 0.224 0.206 0.190

300 310 320 330 340

4.1675 5.7372 7.8226 10.301 13.229

7.312.−4 7.447.−4 7.590.−4 7.847.−4 8.066.−4

0.08647 0.06366 0.04707 0.03572 0.02792

143.1 157.1 170.1 183.0 196.0

492.5 496.3 498.9 501.2 502.5

0.568 0.612 0.649 0.690 0.731

1.732 1.706 1.678 1.654 1.633

1.389 1.422 1.459 1.499 1.546

2.51 2.19 1.91 1.67 1.46

0.174 0.162 0.151 0.139 0.128

350 360 370 380 390

16.759 21.01 26.01 31.92 38.76

8.303.−4 8.571.−4 8.877.−4 9.236.−4 9.671.−4

0.02209 0.01755 0.01399 0.01110 0.00877

211.2 223.7 239.9 257.9 277.7

502.9 503.1 502.9 502.7 500.7

0.781 0.817 0.862 0.910 0.962

1.614 1.593 1.573 1.555 1.534

1.603 1.68 1.75 1.84 1.97

1.27 1.11 0.96 0.84 0.73

0.117 0.108 0.098 0.089 0.081

400 410 420 425.1c

46.67 55.80 66.19 78.81

1.023.−3 1.098.−3 1.235.−3 1.906.−3

0.00685 0.00559 0.00387 0.00191

300.2 326.2 355.6 423.6

496.7 489.5 474.1 423.6

1.020 1.083 1.155 1.304

1.511 1.481 1.436 1.304

2.12

0.63 0.53 0.44

0.072 0.064 0.055

T, K

kf, W/(m⋅K)

*Values interpolated and converted from tables of Kang, McKetta, et al., Bur. Eng. Res. Repr. 59, University of Texas, Austin, 1961. See also J. Chem. Eng. Data, 6 (1961): 220–227; and Am. Inst. Chem. Eng. J., 7 (1961): 418. c = critical point. The notation 6.189.−4 signifies 6.189 × 10−4. The AIChE publication contains a Mollier diagram to 4500 psia, 480°F, while the reprint contains saturation and superheat tables.

THERMODYNAMIC PROPERTIES TABLE 2-347

2-303

Thermodynamic Properties of Saturated Sulfur Hexafluoride (SF6)*

Temp., K

Pressure, bar

vf, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

222.4 225 230 235 240 245 250 255 260 265 270 275 280 285 290 295 300 305 310 315 318.7

2.200 2.470 3.045 3.710 4.475 5.346 6.332 7.442 8.684 10.07 11.60 13.30 15.18 17.25 19.52 22.01 24.75 27.76 31.05 34.67 37.79

0.0005389 0.0005429 0.0005507 0.0005588 0.0005675 0.0005768 0.0005866 0.0005971 0.0006085 0.0006207 0.0006341 0.0006488 0.0006652 0.0006836 0.0007047 0.0007295 0.0007594 0.000798 0.000851 0.000949 0.001372

0.05428 0.04861 0.03978 0.03286 0.02737 0.02296 0.01939 0.01647 0.01406 0.01205 0.01035 0.00892 0.00769 0.00663 0.00571 0.00490 0.00418 0.00352 0.00291 0.00228 0.00137

−57.55 −54.41 −48.51 −42.67 −36.87 −31.13 −25.47 −19.87 −14.33 −8.85 −3.41 2.00 7.42 12.88 18.45 24.20 30.22 36.75 44.05 53.98 71.74

59.08 60.49 63.12 65.68 68.18 70.61 72.96 75.22 77.39 79.44 81.38 83.19 84.84 86.30 87.52 88.45 89.00 88.97 88.06 85.22 71.74

−0.2310 −0.2171 −0.1913 −0.1663 −0.1421 −0.1186 −0.0960 −0.0741 −0.0528 −0.0323 −0.0123 0.0071 0.0262 0.0451 0.0639 0.0829 0.1025 0.1233 0.1462 0.1769 0.2317

0.2935 0.2936 0.2940 0.2947 0.2956 0.2966 0.2977 0.2988 0.2999 0.3009 0.3017 0.3024 0.3027 0.3027 0.3021 0.3008 0.2984 0.2945 0.2881 0.2761 0.2317

cpf, kJ/(kg⋅K)

cpg, kJ/(kg⋅K)

0.409 0.631 0.870 1.17 1.63 2.48 ∞

0.579 0.583 0.592 0.602 0.613 0.626 0.640 0.656 0.674 0.695 0.720 0.748 0.783 0.827 0.882 0.941 1.070 1.26 1.63 2.40 ∞

*See also Oda, A., M. Uematsu, et al., Bull. JSME, 26, 219 (1983): 1590–1596. Ulybin, S.A., Thermodynamic Properties of Sulfur Hexafluoride, Moscow, 1977 (53 pp.). For thermal conductivity to 500 bar, see Rastorguev, Yu L., B. A. Grigorev, et al., Teploenergetika 24, 6 (1977): 78–81 and Bakulin, S. S. and S. A. Ulybin, Teplofiz. Vysok. Temp., 16, 1 (1978): 59–66. For viscosity to 400 bar, see Grigorev, B. A., A. S. Keramidi, et al., Teploenergetika, 24, 9 (1977): 85–87; and Ulybin, S. A. and V. I. Makanushkin, Teplofiz. Vysok. Temp., 15, 6 (1977): 1195–1201.

FIG. 2-30 Enthalpy-concentration diagram for aqueous sulfuric acid at 1 atm. Reference states: enthalpies of pure-liquid components at 32°F and vapor pressures are zero. NOTE: It should be observed that the weight basis includes the vapor, which is particularly important in the two-phase region. The upper ends of the tie lines in this region are assumed to be pure water. (Hougen and Watson, Chemical Process Principles, part I, Wiley, New York, 1943.)

FIG. 2-31

Enthalpy–log-pressure diagram for sulfur hexafluoride.

TABLE 2-348

Saturated SUVA AC 9000

DuPont bulletin T–AC–9000–SI, 1994 (16 pp.) gives tables and a chart to 100 bar, 235°C. With a stated composition of 23% wt CH2F2 (R23), 25% wt CHF2CF3 (R125), and 52% wt CH2FCH3 (R134a) this is apparently identical to KLEA 66, to which the reader is referred. TABLE 2-349

Saturated Toluene* hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µf, 10−4 Pa⋅s

kf, W/(m⋅K)

316.7 333.0 349.6 366.5 383.7

745.7 756.1 766.8 777.8 789.2

2.236 2.295 2.353 2.410 2.467

3.825 3.806 3.792 3.782 3.776

1.64 1.66 1.68 1.71 1.74

8.02 6.96 6.10 5.41 4.83

0.141 0.138 0.136 0.133 0.131

2.67 1.80 1.25 0.891 0.698

401.3 419.6 437.4 456.0 475.1

800.9 812.9 825.2 837.8 850.7

2.522 2.577 2.632 2.686 2.739

3.771 3.771 3.772 3.777 3.783

1.78 1.81 1.84 1.88 1.92

4.34 3.93 3.58 3.28 3.01

0.128 0.126 0.124 0.121 0.119

1.261.−3 1.277.−3 1.294.−3 1.312.−3 1.350.−3

0.481 0.364 0.279 0.218 0.137

494.6 514.4 534.7 555.4 598.1

863.8 877.2 890.9 904.8 933.1

2.792 2.846 2.898 2.950 3.054

3.791 3.801 3.811 3.824 3.852

1.96 2.01 2.05 2.09 2.17

2.78 2.56 2.37 2.19 1.89

0.117 0.114 0.112 0.110 0.105

1.64

0.101 0.096 0.091 0.086 0.082

P, bar

vf, m3/kg

vg, m3/kg

270 280 290 300 310

0.0076 0.0139 0.0246 0.0418 0.0682

1.127.−3 1.138.−3 1.150.−3 1.162.−3 1.175.−3

34.9 19.1 10.6 6.46 4.08

320 330 340 350 360

0.1072 0.1633 0.2416 0.3480 0.4894

1.188.−3 1.201.−3 1.215.−3 1.230.−3 1.245.−3

370 380 390 400 420

0.6736 0.9090 1.2049 1.5713 2.5589

T, K

440 460 480 500 520

3.965 5.892 8.451 11.76 15.96

1.393.−3 1.443.−3 1.499.−3 1.567.−3 1.651.−3

9.00.−2 6.11.−2 4.26.−2 3.03.−2 2.19.−2

642.3 688.1 735.5 784.4 834.9

962.0 991.3 1021.1 1051.3 1081.4

3.156 3.258 3.358 3.457 3.554

3.883 3.917 3.953 3.989 4.027

2.24 2.31 2.38 2.45 2.53

540 560 580 590 591.8c

21.99 27.65 35.56 40.16 41.04

1.761.−3 1.919.−3 2.213.−3 2.650.−3 3.432.−3

1.58.−2 1.13.−2 7.59.−3 5.28.−3 3.43.−3

887.3 942.8 1005.6 1050.2 1084.9

1109.6 1132.1 1142.3 1128.1 1084.9

3.651 3.750 3.857 3.932 3.989

4.062 4.088 4.093 4.063 3.989

2.65 2.82

0.078 0.074

*Values converted and mostly rounded off from the tables of Counsell, Lawrenson, and Lees, Nat. Phys. Lab., Teddington (U.K.) Rep. Chem. 52, 1976. c = critical point. The notation 1.127.−6 signifies 1.127 × 10−6. For other tables, see Goodwin, R. D., J. Phys. Chem. Ref. Data, 18, 4 (1989): 1565–1636. TABLE 2-350

Saturated Solid/Vapor Water* Volume, ft3/lb

Enthalpy, Btu/lb

Entropy, Btu/(lb)(°F)

Temp., °F

Pressure, lb/in2 abs.

Solid

Vapor

Solid

Vapor

Solid

Vapor

−160 −150 −140 −130 −120

4.949.−8 1.620.−7 4.928.−7 1.403.−6 3.757.−6

0.01722 0.01723 0.01724 0.01725 0.01726

3.607.+9 1.139.+9 3.864.+8 1.400.+8 5.386.+7

−222.05 −218.82 −215.49 −212.08 −208.58

990.38 994.80 999.21 1003.63 1008.05

−0.4907 −0.4801 −0.4695 −0.4590 −0.4485

3.5549 3.4387 3.3301 3.2284 3.1330

−110 −100 −90 −80 −70

9.517.−6 2.291.−5 5.260.−5 1.157.−4 2.443.−4

0.01728 0.01729 0.01730 0.01731 0.01732

2.189.+7 9.352.+6 4.186.+6 1.955.+6 9.501.+5

−204.98 −201.28 −197.49 −193.60 −189.61

1012.47 1016.89 1021.31 1025.73 1030.15

−0.4381 −0.4277 −0.4173 −0.4069 −0.3965

3.0434 2.9591 2.8796 2.8045 2.7336

−60 −50 −45 −40 −35

4.972.−4 9.776.−4 1.354.−3 1.861.−3 2.540.−3

0.01734 0.01735 0.01736 0.01737 0.01737

4.788.+5 2.496.+5 1.824.+5 1.343.+5 9.961.+4

−185.52 −181.34 −179.21 −177.06 −174.88

1034.58 1039.00 1041.21 1043.42 1045.63

−0.3862 −0.3758 −0.3707 −0.3655 −0.3604

2.6664 2.6028 2.5723 2.5425 2.5135

−30 −25 −20 −15 −10

3.440.−3 4.627.−3 6.181.−3 8.204.−3 1.082.−2

0.01738 0.01739 0.01739 0.01740 0.01741

7.441.+4 5.596.+4 4.237.+4 3.228.+4 2.475.+4

−172.68 −170.46 −168.21 −165.94 −163.65

1047.84 1050.05 1052.26 1054.47 1056.67

−0.3552 −0.3501 −0.3449 −0.3398 −0.3347

2.4853 2.4577 2.4308 2.4046 2.3791

−5 0 5 10 15

1.419.−2 1.849.−2 2.396.−2 3.087.−2 3.957.−2

0.01741 0.01742 0.01743 0.01744 0.01744

1.909.+4 1.481.+4 1.155.+4 9.060.+3 7.144.+3

−161.33 −158.98 −156.61 −154.22 −151.80

1058.88 1061.09 1063.29 1065.50 1067.70

−0.3295 −0.3244 −0.3193 −0.3142 −0.3090

2.3541 2.3297 2.3039 2.2827 2.2600

16 18 20 22 24

4.156.−2 4.581.−2 5.045.−2 5.552.−2 6.105.−2

0.01745 0.01745 0.01745 0.01746 0.01746

6.817.+3 6.210.+3 5.662.+3 5.166.+3 4.717.+3

−151.32 −150.34 −149.36 −148.38 −147.39

1068.14 1069.02 1069.90 1070.38 1071.66

−0.3080 −0.3060 −0.3039 −0.3019 −0.2998

2.2555 2.2466 2.2378 2.2291 2.2205

26 28 30 31 32

6.708.−2 7.365.−2 8.080.−2 8.461.−2 8.858.−2

0.01746 0.01746 0.01747 0.01747 0.01747

4.311.+3 3.943.+3 3.608.+3 3.453.+3 3.305.+3

−146.40 −145.40 −144.40 −143.90 −143.40

1072.53 1073.41 1074.29 1074.73 1075.16

−0.2978 −0.2957 −0.2937 −0.2927 −0.2916

2.2119 2.2034 2.1950 2.1908 2.1867

*Condensed from Fundamentals, American Society of Heating, Refrigerating and Air-Conditioning Engineers, 1967 and 1972. Reproduced by permission. The validity of many standard reference tables has been critically reviewed by Jancso, Pupezin, and van Hook, J. Phys. Chem., 74 (1970): 2984. This source is recommended for further study. The notation 4.949.−8, 3.607.+9, etc., means 4.949 × 10−8, 3.607 × 109, etc. 2-304

THERMODYNAMIC PROPERTIES TABLE 2-351 Temp., °F

Saturated Water Substance—Temperature (fps units)* Pressure, lb/in2 abs.

Volume, ft3/lb Liquid

Enthalpy, Btu/lb Vapor

Entropy, Btu/(lb)(°F)

Liquid

Vapor

Liquid

Vapor

32.018 35 40 45 50

0.08865 0.09991 0.12163 0.14744 0.17796

0.016022 0.016020 0.016019 0.016020 0.016023

3302.4 2948.1 2445.8 2037.8 1704.8

0.000 3.002 8.027 13.044 18.054

1075.5 1076.8 1079.0 1081.2 1083.4

0.0000 0.0061 0.0162 0.0262 0.0361

2.1872 2.1767 2.1594 2.1426 2.1262

55 60 65 70 75

0.21392 0.25611 0.30545 0.36292 0.42964

0.016027 0.016033 0.016041 0.016050 0.016060

1432.0 1207.6 1022.1 868.4 740.3

23.059 28.060 33.057 38.052 43.045

1085.6 1087.7 1089.9 1092.1 1094.3

0.0458 0.0555 0.0651 0.0745 0.0839

2.1102 2.0946 2.0794 2.0645 2.0500

80 85 90 95 100

0.50683 0.59583 0.69813 0.81534 0.94294

0.016072 0.016085 0.016099 0.016114 0.016130

633.3 543.6 468.1 404.4 350.4

48.037 53.027 58.018 63.008 67.999

1096.4 1098.6 1100.8 1102.9 1105.1

0.0932 0.1024 0.1115 0.1206 0.1295

2.0359 2.0221 2.0086 1.9954 1.9825

110 120 130 140 150

1.2750 1.6927 2.2230 2.8892 3.7184

0.016165 0.016204 0.016247 0.016293 0.016343

265.39 203.26 157.33 122.98 97.07

77.98 87.97 97.96 107.89 117.95

1109.3 1113.6 1117.8 1122.0 1126.1

0.1472 0.1646 0.1817 0.1985 0.2150

1.9577 1.9339 1.9112 1.8895 1.8686

160 170 180 190 200

4.7414 5.9926 7.5110 9.340 11.526

0.016395 0.016451 0.016510 0.016572 0.016637

77.27 62.06 50.225 40.957 33.639

127.96 137.97 148.00 158.04 168.09

1130.2 1134.2 1138.2 1142.1 1146.0

0.2313 0.2473 0.2631 0.2787 0.2940

1.8487 1.8295 1.8111 1.7934 1.7764

210 212 220 230 240

14.123 14.696 17.186 20.779 24.968

0.016705 0.016719 0.016775 0.016849 0.016926

27.816 26.799 23.148 19.381 16.321

178.15 180.17 188.23 198.33 208.45

1149.7 1150.5 1153.4 1157.1 1160.6

0.3091 0.3121 0.3241 0.3388 0.3533

1.7600 1.7568 1.7442 1.7290 1.7142

250 260 270 280 290

29.825 35.427 41.856 49.200 57.550

0.017066 0.017089 0.017175 0.017264 0.01736

13.819 11.762 10.060 8.644 7.4603

218.59 228.76 238.95 249.17 259.4

1164.0 1167.4 1170.6 1173.8 1167.8

0.3677 0.3819 0.3960 0.4098 0.4236

1.7000 1.6862 1.6729 1.6599 1.6473

300 320 340 360 380

67.005 89.643 117.992 153.01 195.73

0.01745 0.01766 0.01787 0.01811 0.01836

6.4658 4.9138 3.7878 2.9573 2.3353

269.7 290.4 311.3 332.3 353.6

1179.7 1185.2 1190.1 1194.4 1198.0

0.4372 0.4640 0.4902 0.5161 0.5416

1.6351 1.6116 1.5892 1.5678 1.5473

400 420 440 460 480

247.26 308.78 381.54 466.87 566.15

0.01864 0.01894 0.01926 0.01961 0.02000

1.8630 1.4997 1.2169 0.99424 0.81717

375.1 396.9 419.0 441.5 464.5

1201.0 1203.1 1204.4 1204.8 1204.1

0.5667 0.5915 0.6161 0.6405 0.6648

1.5274 1.5080 1.4890 1.4704 1.4518

500 520 540 560 580

680.86 812.53 962.79 1133.38 1326.17

0.02043 0.02091 0.02146 0.02207 0.02279

0.67492 0.55956 0.46513 0.38714 0.32216

487.9 512.0 536.8 562.4 589.1

1202.2 1199.0 1194.3 1187.7 1179.0

0.6890 0.7133 0.7378 0.7625 0.7876

1.4333 1.4146 1.3954 1.3757 1.3550

600 620 640 660 680

1543.2 1786.9 2059.9 2365.7 2708.6

0.02364 0.02466 0.02595 0.02768 0.03037

0.26747 0.22081 0.18021 0.14431 0.11117

617.1 646.9 679.1 714.9 758.5

1167.7 1153.2 1133.7 1107.0 1068.5

0.8134 0.8403 0.8686 0.8995 0.9365

1.3330 1.3092 1.2821 1.2498 1.2086

700 702 704 705.47

3094.3 3135.5 3177.2 3208.2

0.03662 0.03824 0.04108 0.05078

0.07519 0.06997 0.06300 0.05078

825.2 835.0 854.2 906.0

991.7 979.7 956.2 906.0

0.9924 1.0006 1.0169 1.0612

1.1359 1.1210 1.1046 1.0612

*Extracted and condensed from 1967 ASME Steam Tables. Copyright reserved. Reproduced by permission.

2-305

TABLE 2-352

Temp., K

Saturated Water Substance—Temperature (SI units)

Pressure, bar*

Volume, m3/ kg Vapor

Entropy, kJ/(kg⋅K)

Condensed† Vapor Condensed†

Specific heat, Cp, kJ/(kg⋅K) Vapor

Viscosity, Ns/m2 Condensed†

Vapor

Thermal conductivity, W/(m⋅K) Condensed†

Vapor

Prandtl no. Condensed†

Vapor

Surface tension, N/m Temp., Condensed† K

Vapor

Condensed†

150

6.30.−11

1.073.−3

9.55.+9

−539.6

2273

−2.187

16.54

1.155

3.73

150

160 170 180 190 200

7.72.−10 7.29.−9 5.38.−8 3.23.−7 1.62.−6

1.074.−3 1.076.−3 1.077.−3 1.078.−3 1.079.−3

9.62.+8 1.08.+8 1.55.+7 2.72.+6 5.69.+5

−525.7 −511.7 −497.8 −483.8 −467.5

2291 2310 2328 2347 2366

−2.106 −2.026 −1.947 −1.868 −1.789

15.49 14.57 13.76 13.03 12.38

1.233 1.311 1.389 1.467 1.545

3.52 3.34 3.18 3.04 2.91

160 170 180 190 200

210 220 230 240 250

7.01.−6 2.65.−5 8.91.−5 3.72.−4 7.59.−4

1.081.−3 1.082.−3 1.084.−3 1.085.−3 1.087.−3

1.39.+5 3.83.+4 1.18.+4 4.07.+3 1.52.+3

−451.2 −435.0 −416.3 −400.1 −381.5

2384 2403 2421 2440 2459

−1.711 −1.633 −1.555 −1.478 −1.400

11.79 11.20 10.79 10.35 9.954

1.623 1.701 1.779 1.857 1.935

2.79 2.69 2.59 2.50 2.42

210 220 230 240 250

255 260 265 270 273.15

1.23.−3 1.96.−3 3.06.−3 4.69.−3 6.11.−3

1.087.−3 1.088.−3 1.089.−3 1.090.−3 1.091.−3

956.4 612.2 400.4 265.4 206.3

−369.8 −360.5 −351.2 −339.6 −333.5

2468 2477 2486 2496 2502

−1.361 −1.323 −1.281 −1.296 −1.221

9.768 9.590 9.461 9.255 9.158

1.974 2.013 2.052 2.091 2.116

2.38 2.35 2.31 2.27 2.26

255 260 265 270 273.15

273.15 275 280 285 290

0.00611 0.00697 0.00990 0.01387 0.01917

1.000.−3 1.000.−3 1.000.−3 1.000.−3 1.001.−3

206.3 181.7 130.4 99.4 69.7

0.0 7.8 28.8 49.8 70.7

2502 2505 2514 2523 2532

0.000 0.028 0.104 0.178 0.251

9.158 9.109 8.980 8.857 8.740

4.217 4.211 4.198 4.189 4.184

1.854 1.855 1.858 1.861 1.864

1750.−6 1652.−6 1422.−6 1225.−6 1080.−6

8.02.−6 8.09.−6 8.29.−6 8.49.−6 8.69.−6

0.569 0.574 0.582 0.590 0.598

0.0182 0.0183 0.0186 0.0189 0.0193

12.99 12.22 10.26 8.81 7.56

0.815 0.817 0.825 0.833 0.841

0.0755 0.0753 0.0748 0.0743 0.0737

273.15 275 280 285 290

295 300 305 310 315

0.02617 0.03531 0.04712 0.06221 0.08132

1.002.−3 1.003.−3 1.005.−3 1.007.−3 1.009.−3

51.94 39.13 27.90 22.93 17.82

91.6 112.5 133.4 154.3 175.2

2541 2550 2559 2568 2577

0.323 0.393 0.462 0.530 0.597

8.627 8.520 8.417 8.318 8.224

4.181 4.179 4.178 4.178 4.179

1.868 1.872 1.877 1.882 1.888

959.−6 855.−6 769.−6 695.−6 631.−6

8.89.−6 9.09.−6 9.29.−6 9.49.−6 9.69.−6

0.606 0.613 0.620 0.628 0.634

0.0195 0.0196 0.0201 0.0204 0.0207

6.62 5.83 5.20 4.62 4.16

0.849 0.857 0.865 0.873 0.883

0.0727 0.0717 0.0709 0.0700 0.0692

295 300 305 310 315

320 325 330 335 340

0.1053 0.1351 0.1719 0.2167 0.2713

1.011.−3 1.013.−3 1.016.−3 1.018.−3 1.021.−3

13.98 11.06 8.82 7.09 5.74

196.1 217.0 237.9 258.8 279.8

2586 2595 2604 2613 2622

0.649 0.727 0.791 0.854 0.916

8.151 8.046 7.962 7.881 7.804

4.180 4.182 4.184 4.186 4.188

1.895 1.903 1.911 1.920 1.930

577.−6 528.−6 489.−6 453.−6 420.−6

9.89.−6 10.09.−6 10.29.−6 10.49.−6 10.69.−6

0.640 0.645 0.650 0.655 0.660

0.0210 0.0213 0.0217 0.0220 0.0223

3.77 3.42 3.15 2.88 2.66

0.894 0.901 0.908 0.916 0.925

0.0683 0.0675 0.0666 0.0658 0.0649

320 325 330 335 340

345 350 355 360 365

0.3372 0.4163 0.5100 0.6209 0.7514

1.024.−3 1.027.−3 1.030.−3 1.034.−3 1.038.−3

4.683 3.846 3.180 2.645 2.212

300.7 321.7 342.7 363.7 384.7

2630 2639 2647 2655 2663

0.977 1.038 1.097 1.156 1.214

7.729 7.657 7.588 7.521 7.456

4.191 4.195 4.199 4.203 4.209

1.941 1.954 1.968 1.983 1.999

389.−6 365.−6 343.−6 324.−6 306.−6

10.89.−6 11.09.−6 11.29.−6 11.49.−6 11.69.−6

0.665 0.668 0.671 0.674 0.677

0.0226 0.0230 0.0233 0.0237 0.0241

2.45 2.29 2.14 2.02 1.91

0.933 0.942 0.951 0.960 0.969

0.0641 0.0632 0.0623 0.0614 0.0605

345 350 355 360 365

370 373.15 375 380 385

0.9040 1.0133 1.0815 1.2869 1.5233

1.041.−3 1.044.−3 1.045.−3 1.049.−3 1.053.−3

1.861 1.679 1.574 1.337 1.142

405.8 419.1 426.8 448.0 469.2

2671 2676 2679 2687 2694

1.271 1.307 1.328 1.384 1.439

7.394 7.356 7.333 7.275 7.218

4.214 4.217 4.220 4.226 4.232

2.017 2.029 2.036 2.057 2.080

289.−6 279.−6 274.−6 260.−6 248.−6

11.89.−6 12.02.−6 12.09.−6 12.29.−6 12.49.−6

0.679 0.680 0.681 0.683 0.685

0.0245 0.0248 0.0249 0.0254 0.0258

1.80 1.76 1.70 1.61 1.53

0.978 0.984 0.987 0.995 1.004

0.0595 0.0589 0.0586 0.0576 0.0566

370 373.15 375 380 385

390 400 410 420 430

1.794 2.455 3.302 4.370 5.699

1.058.−3 1.067.−3 1.077.−3 1.088.−3 1.099.−3

0.980 0.731 0.553 0.425 0.331

490.4 532.9 575.6 618.6 661.8

2702 2716 2729 2742 2753

1.494 1.605 1.708 1.810 1.911

7.163 7.058 6.959 6.865 6.775

4.239 4.256 4.278 4.302 4.331

2.104 2.158 2.221 2.291 2.369

237.−6 217.−6 200.−6 185.−6 173.−6

12.69.−6 13.05.−6 13.42.−6 13.79.−6 14.14.−6

0.686 0.688 0.688 0.688 0.685

0.0263 0.0272 0.0282 0.0293 0.0304

1.47 1.34 1.24 1.16 1.09

1.013 1.033 1.054 1.075 1.10

0.0556 0.0536 0.0515 0.0494 0.0472

390 400 410 420 430

2-306

Condensed†

Enthalpy, kJ/kg

440 450 460 470 480

7.333 9.319 11.71 14.55 17.90

1.110.−3 1.123.−3 1.137.−3 1.152.−3 1.167.−3

0.261 0.208 0.167 0.136 0.111

705.3 749.2 793.5 838.2 883.4

2764 2773 2782 2789 2795

2.011 2.109 2.205 2.301 2.395

6.689 6.607 6.528 6.451 6.377

4.36 4.40 4.44 4.48 4.53

2.46 2.56 2.68 2.79 2.94

162.−6 152.−6 143.−6 136.−6 129.−6

14.50.−6 14.85.−6 15.19.−6 15.54.−6 15.88.−6

0.682 0.678 0.673 0.667 0.660

0.0317 0.0331 0.0346 0.0363 0.0381

1.04 0.99 0.95 0.92 0.89

1.12 1.14 1.17 1.20 1.23

0.0451 0.0429 0.0407 0.0385 0.0362

440 450 460 470 480

490 500 510 520 530

21.83 26.40 31.66 37.70 44.58

1.184.−3 1.203.−3 1.222.−3 1.244.−3 1.268.−3

0.0922 0.0766 0.0631 0.0525 0.0445

929.1 975.6 1023 1071 1119

2799 2801 2802 2801 2798

2.479 2.581 2.673 2.765 2.856

6.312 6.233 6.163 6.093 6.023

4.59 4.66 4.74 4.84 4.95

3.10 3.27 3.47 3.70 3.96

124.−6 118.−6 113.−6 108.−6 104.−6

16.23.−6 16.59.−6 16.95.−6 17.33.−6 17.72.−6

0.651 0.642 0.631 0.621 0.608

0.0401 0.0423 0.0447 0.0475 0.0506

0.87 0.86 0.85 0.84 0.85

1.25 1.28 1.31 1.35 1.39

0.0339 0.0316 0.0293 0.0269 0.0245

490 500 510 520 530

540 550 560 570 580

52.38 61.19 71.08 82.16 94.51

1.294.−3 1.323.−3 1.355.−3 1.392.−3 1.433.−3

0.0375 0.0317 0.0269 0.0228 0.0193

1170 1220 1273 1328 1384

2792 2784 2772 2757 2737

2.948 3.039 3.132 3.225 3.321

5.953 5.882 5.808 5.733 5.654

5.08 5.24 5.43 5.68 6.00

4.27 4.64 5.09 5.67 6.40

101.−6 97.−6 94.−6 91.−6 88.−6

18.1.−6 18.6.−6 19.1.−6 19.7.−6 20.4.−6

0.594 0.580 0.563 0.548 0.528

0.0540 0.0583 0.0637 0.0698 0.0767

0.86 0.87 0.90 0.94 0.99

1.43 1.47 1.52 1.59 1.68

0.0221 0.0197 0.0173 0.0150 0.0128

540 550 560 570 580

1.05 1.14 1.30 1.52 1.65

1.84 2.15 2.60 3.46 4.20

0.0105 0.0084 0.0063 0.0045 0.0035

590 600 610 620 625

0.0026 0.0015 0.0008 0.0001 0.0000

630 635 640 645 647.3‡

590 600 610 620 625

108.3 123.5 137.3 159.1 169.1

1.482.−3 1.541.−3 1.612.−3 1.705.−3 1.778.−3

0.0163 0.0137 0.0115 0.0094 0.0085

1443 1506 1573 1647 1697

2717 2682 2641 2588 2555

3.419 3.520 3.627 3.741 3.805

5.569 5.480 5.318 5.259 5.191

6.41 7.00 7.85 9.35 10.6

7.35 8.75 11.1 15.4 18.3

84.−6 81.−6 77.−6 72.−6 70.−6

21.5.−6 22.7.−6 24.1.−6 25.9.−6 27.0.−6

0.513 0.497 0.467 0.444 0.430

0.0841 0.0929 0.103 0.114 0.121

630 635 640 645 647.3‡

179.7 190.9 202.7 215.2 221.2

1.856.−3 1.935.−3 2.075.−3 2.351.−3 3.170.−3

0.0075 0.0066 0.0057 0.0045 0.0032

1734 1783 1841 1931 2107

2515 2466 2401 2292 2107

3.875 3.950 4.037 4.223 4.443

5.115 5.025 4.912 4.732 4.443

12.6 16.4 26 90 ∞

22.1 27.6 42

67.−6 64.−6 59.−6 54.−6 45.−6

28.0.−6 30.0.−6 32.0.−6 37.0.−6 45.0.−6

0.412 0.392 0.367 0.331 0.238

0.130 0.141 0.155 0.178 0.238

∞

2.0 2.7 4.2 12 ∞

4.8 6.0 9.6 26 ∞

*1 bar = 105 N/m2. †Above the solid line, the condensed phase is solid; below it, liquid. ‡Critical temperature. −11 −3 9 NOTE: The notations 6.30.−11, 1.073.−3, 9.55.+9, etc. signify 6.30 × 10 , 1.073 × 10 , 955 × 10 , etc. Tables 2-351 and 2-352 are provided for general use. Tables to higher precision are available over certain ranges and for various properties. The most current internationally accepted tables are found in Haar, L., J. S. Gallagher, and G. S. Kell, NBS/NRC Steam Tables, Hemisphere, Washington, DC, 1984 (320 pp.). These do not tabulate certain properties at saturation states. A revised release on the IAPWS Skeleton Tables 1985 for the thermodynamic properties of ordinary water substance, Sept. 1993 (15 pp), is apparently the latest international publication. In J. Phys. Chem. Ref. Data 17, 4 (1988): 1439–1540, H. Sato, M. Uematsu, and others review existing steam tables and present the 1985 formulation of skeleton tables. Property codes and programs include Cheng, S. C. and C. Nguyen, Modeling and Simulation on Microcomputers 1989 (R. W. Allen, ed.), S.C.S. Intl., San Diego, 1989 (pp. 138–141); Garland, W. J. and B. J. Hand, Nucl. Engng. & Des., 113, (1989): 21–34; Dickey, D. S., Chem. Eng. 98, 9 (1991): 207–8 and 98, 11: 235–6; Muneer, T. and S. M. Scott, Proc. Inst. Mech. Eng., 205, (1991): 25–29; and Energy Convsn. Mgmt., 31, 4 (1991): 315–325. Useful pictorial representations of 20 properties as a function of both temperature (to 800°C) and pressure (to 1000 bar) are given by Grigull, U., J. Bach, et al., Warme- u. Stoff., 1 (1968): 202–213. Property equations for the saturated liquid for the range 0–300°C are given by Charters, W. W. S. and H. A. Sadafi, Rev. Int. Froid, 10, (Mar. 1987): 105–6. Gordon, S., NASA Tech. Paper 1906, 1982 gives detailed tables for ice from 0 K. Ice and snow properties are reviewed by Fukusako, S., Int. J. Thermophys., 11, 2 (1990): 353–372. See also Wagner, W., A. Saul, et al., J. Phys. Chem. Ref. Data, 23, 3 (1994): 515–525, and Table 2-358.

2-307

2-308

PHYSICAL AND CHEMICAL DATA

TABLE 2-353

Saturated Liquid Water—Miscellaneous Properties

Temperature, °C

104 β

104 kT / bar

104 ks / bar

vs, m/s

µf , 10−6 Pa⋅s

cp, kJ/kg⋅K

k, W/m⋅K

Pr, bar

0 1 2 3 4

−0.681 −0.501 −0.327 −0.160 0.003

0.50885 0.50509 0.50151 0.49808 0.49481

0.50855 0.50493 0.50143 0.49806 0.49481

1402.4 1407.4 1412.2 1417.0 1421.6

1.793 1.732 1.675 1.621 1.569

4.2176 4.2140 4.2107 4.2077 4.2048

0.567 0.569 0.570 0.572 0.573

13.32 12.83 12.37 11.93 11.51

σ, N/m 0.07565 0.07551 0.07537 0.07522 0.07508

5 6 7 8 9

0.160 0.312 0.460 0.604 0.744

0.49169 0.48871 0.48587 0.48315 0.48056

0.49167 0.48865 0.48573 0.48291 0.48019

1426.2 1430.6 1434.9 1439.1 1443.3

1.520 1.474 1.429 1.387 1.346

4.2022 4.1999 4.1977 4.1956 4.1938

0.575 0.577 0.578 0.580 0.581

11.11 10.73 10.38 10.04 9.72

0.07494 0.07480 0.07465 0.07451 0.07436

10 11 12 13 14

0.880 1.012 1.141 1.267 1.389

0.47809 0.47573 0.47347 0.47133 0.46928

0.47757 0.47504 0.47260 0.47024 0.46797

1447.3 1451.2 1455.0 1458.7 1462.4

1.308 1.271 1.236 1.202 1.170

4.1921 4.1906 4.1892 4.1879 4.1867

0.5828 0.5844 0.5859 0.5875 0.5891

9.41 9.11 8.84 8.57 8.32

0.07422 0.07407 0.07393 0.07378 0.07364

15 16 17 18 19

1.509 1.626 1.740 1.852 1.961

0.46733 0.46548 0.46371 0.46203 0.46043

0.46578 0.46366 0.46162 0.45966 0.45776

1465.9 1469.4 1472.7 1476.0 1479.2

1.139 1.110 1.081 1.054 1.028

4.1856 4.1847 4.1838 4.1830 4.1823

0.5906 0.5922 0.5937 0.5953 0.5968

8.07 7.84 7.62 7.41 7.20

0.07349 0.07334 0.07319 0.07304 0.07289

20 21 22 23 24

2.068 2.173 2.275 2.376 2.475

0.45892 0.45748 0.45612 0.45484 0.45362

0.45593 0.45417 0.45248 0.45084 0.44927

1482.3 1485.3 1488.3 1491.2 1493.9

1.003 0.979 0.955 0.933 0.911

4.1817 4.1812 4.1807 4.1802 4.1798

0.5983 0.5999 0.6014 0.6029 0.6044

7.01 6.82 6.64 6.47 6.30

0.07274 0.07259 0.07244 0.07228 0.07213

25 26 27 28 30

2.572 2.667 2.761 2.852 3.032

0.45247 0.45139 0.45038 0.44943 0.44771

0.44776 0.44630 0.44490 0.44355 0.44102

1496.7 1499.3 1501.9 1504.3 1509.1

0.891 0.871 0.852 0.833 0.798

4.1795 4.1792 4.1790 4.1788 4.1785

0.6059 0.6074 0.6089 0.6104 0.6133

6.15 5.99 5.85 5.70 5.44

0.07198 0.07182 0.07167 0.07151 0.07120

32 34 36 38 40

3.206 3.375 3.539 3.698 3.853

0.44622 0.44496 0.44390 0.44305 0.44239

0.43869 0.43655 0.43459 0.43280 0.43118

1513.6 1517.8 1521.7 1525.4 1528.9

0.765 0.734 0.705 0.679 0.653

4.1783 4.1782 4.1783 4.1784 4.1786

0.6162 0.6190 0.6218 0.6246 0.6273

5.19 4.95 4.74 4.54 4.35

0.07089 0.07058 0.07025 0.06992 0.06960

42 44 46 48 50

4.004 4.152 4.296 4.438 4.576

0.44192 0.44162 0.44149 0.44153 0.44173

0.42972 0.42842 0.42726 0.42624 0.42535

1532.1 1535.0 1537.7 1540.3 1542.6

0.629 0.607 0.586 0.566 0.547

4.1789 4.1792 4.1797 4.1801 4.1807

0.6299 0.6315 0.6351 0.6375 0.6400

4.17 4.02 3.86 3.71 3.57

0.06927 0.06894 0.06861 0.06828 0.06795

55 60 65 70 75

4.910 5.231 5.539 5.837 6.128

0.44290 0.44496 0.44788 0.45162 0.45614

0.42370 0.42281 0.42262 0.42309 0.42418

1547.4 1551.0 1553.4 1554.8 1555.1

0.5043 0.4668 0.4338 0.4045 0.3784

4.1824 4.1844 4.1869 4.1897 4.1929

0.6457 0.6511 0.6561 0.6607 0.6649

3.267 3.000 2.768 2.565 2.386

0.06710 0.06624 0.06537 0.06449 0.06359

80 85 90 95 100

6.411 6.689 6.962 7.233 7.501

0.46143 0.46748 0.47429 0.48185 0.49019

0.42587 0.42812 0.43093 0.43429 0.43819

1554.4 1552.9 1550.5 1547.2 1543.1

0.3550 0.3340 0.3150 0.2979 0.2823

4.1965 4.2005 4.2050 4.2102 4.2164

0.6686 0.6721 0.6753 0.6779 0.6800

2.228 2.088 1.962 1.850 1.756

0.06268 0.06176 0.06083 0.05988 0.05892

Values mostly from Aleksandrov, A. A. and M. S. Trakhtenhertz, Thermophysical Properties of Water at Atmospheric Pressure, Standartov, Moscow, 1977 (99 pp.).

THERMODYNAMIC PROPERTIES TABLE 2-354 Temperature, K

2-309

Thermodynamic Properties of Compressed Steam* Pressure, bar 0.1

0.5

1

5

10

20

40

60

80

350

v h s

16.12 2644 8.327

1.027.−3 321.7 1.037

1.027.−3 231.8 1.037

1.027.−3 322.1 1.037

1.027.−3 322.5 1.037

1.026.−3 323.3 1.036

1.025.−3 324.9 1.035

1.024.−3 326.4 1.034

1.023.−3 328.1 1.032

1.023.−3 329.7 1.031

400

v h s

18.44 2739 8.581

3.67 2735 7.831

1.827 2730 7.502

1.067.−3 533.1 1.601

1.067.−3 533.4 1.600

1.066.−3 534.1 1.599

1.065.−3 535.4 1.597

1.064.−3 536.8 1.595

1.063.−3 538.2 1.593

1.061.−3 539.6 1.592

450

v h s

20.75 2835 8.811

4.14 2833 8.061

2.063 2830 7.736

0.410 2804 6.949

1.124.−3 749.0 2.110

1.123.−3 749.8 2.107

1.121.−3 750.8 2.105

1.119.−3 751.9 2.102

1.118.−3 753.0 2.099

1.116.−3 754.1 2.097

500

v h s

23.07 2932 9.012

4.61 2931 8.261

2.298 2929 7.944

0.452 2912.4 7.177

0.221 2891.2 6.823

0.104 2839.4 6.422

1.201.−3 975.9 2.578

1.198.−3 976.3 2.575

1.196.−3 976.8 2.571

1.193.−3 977.3 2.567

600

v h s

27.7 3131 9.374

5.53 3130 8.630

2.76 3129 8.309

0.548 3120 7.560

0.271 3109 7.223

0.133 3087 6.875

0.0630 3036 6.590

0.0396 2976 6.224

0.0276 2906 5.997

0.0201 2820 5.775

700

v h s

32.3 3335 9.692

6.46 3335 8.946

3.23 3334 8.625

0.643 3328 7.877

0.319 3322 7.550

0.158 3307 7.215

0.0769 3278 6.864

0.0500 3247 6.644

0.0346 3214 6.431

0.0283 3179 6.334

800

v h s

36.9 3547 9.971

7.38 3546 9.228

3.69 3546 8.908

0.736 3542 8.161

0.367 3537 7.837

0.182 3526 7.507

0.0889 3506 7.151

0.0589 3485 6.965

0.0436 3464 6.809

0.0343 3442 6.685

900

v h s

41.5 3765 10.228

8.31 3765 9.485

4.15 3764 9.165

0.829 3761 8.420

0.414 3757 8.097

0.206 3750 7.770

0.102 3737 7.462

0.0674 3719 7.237

0.0501 3704 7.092

0.0398 3688 6.975

1000

v h s

46.2 3990 10.466

9.23 3990 9.723

4.615 3990 9.402

0.921 3987 8.659

0.460 3984 8.336

0.229 3978 8.011

0.114 3967 7.682

0.0758 3955 7.486

0.0564 3944 7.345

0.0449 3935 7.233

1500

v h s

69.2 5231 11.47

13.9 5228 10.77

6.92 5227 10.40

1.385 5225 9.66

0.692 5224 9.34

0.341 5221 9.015

0.1730 5217 8.693

0.1153 5212 8.503

0.0865 5207 8.368

0.0692 5203 8.262

2000

v h s

93.0 6832 12.38

18.6 6734 11.58

9.26 6706 11.25

1.850 6662 10.48

0.925 6649 10.15

0.462 6639 9.828

0.231 6629 9.503

0.1543 6623 9.313

0.1157 6619 9.178

0.0926 6616 9.073

2500

v h s

123.7 10417 13.95

24.0 9330 12.73

11.90 9046 12.28

2.35 8621 11.35

1.171 8504 10.80

0.583 8413 10.62

0.291 8342 10.26

0.1942 8307 10.06

0.1457 8285 9.920

0.1166 8269 9.810

*v = specific volume, m3/kg; h = specific enthalpy, kJ/kg; s = specific entropy, kJ/(kg⋅K). The notation 1.027.−3 signifies 1.027 × 10−3.

100

2-310

PHYSICAL AND CHEMICAL DATA

TABLE 2-354

Thermodynamic Properties of Compressed Steam (Concluded ) Pressure, bar

Temperature, K

150

200

250

300

400

500

600

700

800

900

1000

v 350 h s

1.020.−3 333.7 1.028

1.018.−3 337.7 1.025

1.016.−3 341.7 1.022

1.014.−3 344.7 1.019

1.009.−3 353.8 1.013

1.005.−3 361.8 1.007

1.002.−3 369.7 1.001

9.977.−4 377.7 0.996

9.937.−4 385.7 0.991

9.900.−4 393.7 0.985

9.865.−4 401.7 0.979

400

v h s

1.059.−3 543.1 1.587

1.056.−3 546.5 1.583

1.053.−3 550.1 1.578

1.050.−3 553.5 1.574

1.045.−3 560.6 1.565

1.041.−3 567.8 1.557

1.035.−3 574.9 1.549

1.031.−3 582.1 1.541

1.027.−3 589.3 1.533

1.022.−3 596.5 1.526

1.018.−3 603.8 1.518

450

v h s

1.112.−3 756.8 2.088

1.108.−3 759.5 2.082

1.105.−3 762.3 2.076

1.101.−3 765.2 2.070

1.094.−3 771.0 2.060

1.088.−3 776.9 2.049

1.082.−3 783.0 2.039

1.076.−3 789.6 2.029

1.070.−3 795.3 2.019

1.065.−3 801.6 2.010

1.059.−3 807.9 2.002

500

v h s

1.187.−3 978.8 2.558

1.181.−3 980.3 2.549

1.175.−3 981.9 2.541

1.170.−3 983.7 2.533

1.160.−3 987.4 2.517

1.151.−3 991.5 2.502

1.142.−3 995.9 2.488

1.134.−3 1000.5 2.474

1.126.−3 1005.3 2.461

1.119.−3 1010.3 2.449

1.112.−3 1015.4 2.437

600

v h s

1.519.−3 1499 3.501

1.483.−3 1489 3.469

1.454.−3 1479 3.443

1.428.−3 1472 3.419

1.392.−3 1462 3.379

1.362.−3 1456 3.346

1.337.−3 1452 3.316

1.315.−3 1449 3.290

1.296.−3 1447 3.266

1.280.−3 1447 3.244

1.265.−3 1447 3.223

700

v h s

1.724.−2 3082 6.037

1.157.−2 2965 5.770

7.986.−3 2821 5.494

5.416.−3 2635 5.179

2.630.−3 2233 4.554

2.038.−3 2084 4.308

1.831.−3 2021 4.192

1.716.−3 1986 4.116

1.639.−3 1962 4.058

1.589.−3 1946 4.012

1.536.−3 1931 3.972

800

v h s

2.195.−2 3386 6.444

1.575.−2 3325 6.252

1.201.−2 3261 6.086

9.512.−3 3193 5.934

6.391.−3 3047 5.654

4.576.−3 2895 5.397

3.496.−3 2734 5.175

2.866.−3 2648 4.998

2.484.−3 2567 4.864

2.239.−3 2508 4.761

2.072.−3 2465 4.701

900

v h s

2.590.−2 3649 6.755

1.899.−2 3609 6.587

1.483.−2 3568 6.449

1.207.−2 3526 6.327

8.619.−3 3440 6.119

6.581.−3 3354 5.940

5.257.−3 3269 5.780

4.348.−3 3188 5.637

3.704.−3 3113 5.510

3.454.−3 3049 5.399

2.907.−3 2995 5.305

1000

v h s

2.954.−2 3904 7.023

2.186.−2 3874 6.867

1.726.−2 3845 6.741

1.420.−2 3816 6.633

1.038.−2 3756 6.453

8.102.−3 3697 6.302

6.605.−3 3640 6.172

5.557.−3 3584 6.055

4.792.−3 3532 5.951

4.212.−3 3482 5.856

3.763.−3 3435 5.727

1500

v h s

0.0461 5202 8.074

0.0346 5198 7.936

0.0277 5186 7.827

0.0231 5180 7.738

0.0173 5171 7.597

0.0139 5157 7.484

0.0116 5144 7.391

0.00993 5133 7.310

0.00871 5120 7.239

0.00776 5108 7.176

0.00700 5095 7.118

2000

v h s

0.0619 6613 8.883

0.0465 6610 8.748

0.0372 6608 8.642

0.0311 6605 8.555

0.0234 6599 8.418

0.0188 6595 8.310

0.0157 6590 8.222

0.0135 6585 8.147

0.0119 6581 8.082

0.0106 6577 8.024

0.0096 6574 7.971

2500

v h s

0.0778 8269 9.610

0.0584 8269 9.468

0.0468 8269 9.358

0.0391 8268 9.270

0.0294 8267 9.129

0.0236 8265 9.020

0.0197 8261 8.930

0.0170 2856 8.854

0.0149 8250 8.788

0.0133 8244 8.730

0.0120 8240 8.677

TABLE 2-355 Density, Specific Heats at Constant Pressure and at Constant Volume and Velocity of Sound for Compressed Water, 1–1000 bar, 0–150°C 0°C (ITS-90)

10°C (ITS-90)

20°C (ITS-90)

30°C (ITS-90)

P, bar

density, kg/m3

Cp, kJ/ (kg⋅K)

Cv, kJ/ (kg⋅K)

w, m/s

density, kg/m3

Cp, kJ/(kg⋅K)

Cv, kJ/(kg⋅K)

w, m/s

density, kg/m3

Cp, kJ/(kg⋅K)

Cv, kJ/(kg⋅K)

w, m/s

density, kg/m3

Cp, kJ/(kg⋅K)

Cv, kJ/(kg⋅K)

w, m/s

1 50 100 150 200 250 300 400 500 600 800 1000

999.702 1002.03 1004.38 1006.71 1009.01 1011.28 1013.53 1017.97 1022.31 1026.57 1034.85 1042.83

4.1923 4.174 4.156 4.139 4.123 4.108 4.093 4.065 4.040 4.018 3.979 3.948

4.1877 4.168 4.149 4.130 4.112 4.095 4.078 4.046 4.016 3.988 3.937 3.892

1447.3 1455 1464 1472 1480 1489 1497 1513 1529 1545 1577 1609

998.207 1000.44 1002.69 1004.93 1007.13 1009.32 1011.48 1015.74 1019.92 1024.02 1031.99 1039.68

4.1812 4.166 4.151 4.137 4.124 4.110 4.098 4.074 4.052 4.032 3.996 3.967

4.1538 4.137 4.119 4.103 4.087 4.071 4.056 4.027 3.999 3.974 3.926 3.884

1482.3 1491 1499 1507 1516 1524 1532 1548 1565 1581 1613 1644

995.650 997.82 1000.02 1002.19 1004.34 1006.47 1008.57 1012.72 1016.79 1020.79 1028.56 1036.06

4.1774 4.164 4.151 4.139 4.127 4.115 4.104 4.083 4.063 4.044 4.011 3.982

4.1148 4.099 4.084 4.069 4.055 4.041 4.027 4.001 3.976 3.952 3.908 3.869

1509.1 1517 1526 1534 1543 1551 1559 1576 1592 1608 1640 1671

992.217 994.36 996.52 998.66 1000.77 1002.87 1004.94 1009.03 1013.03 1016.97 1024.62 1032.00

4.1775 4.166 4.154 4.142 4.131 4.121 4.110 4.091 4.072 4.055 4.023 3.995

4.0715 4.058 4.044 4.031 4.018 4.005 3.993 3.969 3.946 3.924 3.884 3.847

1528.9 1537 1546 1554 1563 1571 1579 1596 1612 1628 1660 1692

P, bar

density, kg/m3

Cp, kJ/ (kg⋅K)

Cv, kJ/ (kg⋅K)

w, m/s

density, kg/m3

Cp, kJ/(kg⋅K)

Cv, kJ/(kg⋅K)

w, m/s

density, kg/m3

Cp, kJ/(kg⋅K)

Cv, kJ/(kg⋅K)

w, m/s

density, kg/m3

Cp, kJ/ (kg⋅K)

Cv, kJ/(kg⋅K)

w, m/s

1 50 100 150 200 250 300 400 500 600 800 1000

988.036 990.16 992.31 994.44 996.54 998.62 1000.68 1004.74 1008.72 1012.62 1020.21 1027.53

4.1799 4.169 4.158 4.147 4.137 4.126 4.117 4.098 4.080 4.063 4.033 4.005

4.0248 4.012 4.000 3.988 3.976 3.965 3.954 3.932 3.911 3.892 3.854 3.820

1542.6 1551 1560 1568 1577 1585 1594 1610 1627 1643 1676 1707

983.197 985.33 987.48 989.61 991.71 993.80 995.86 999.92 1003.90 1007.80 1015.38 1022.69

4.1840 4.173 4.163 4.152 4.142 4.132 4.123 4.105 4.087 4.071 4.041 4.013

3.9755 3.964 3.953 3.943 3.932 3.922 3.911 3.892 3.873 3.855 3.821 3.789

1551.0 1560 1568 1577 1586 1594 1603 1620 1637 1653 1686 1718

977.766 979.92 982.09 984.23 986.36 988.46 990.53 994.62 998.62 1002.54 1010.15 1017.48

4.1896 4.179 4.169 4.158 4.148 4.139 4.129 4.111 4.094 4.078 4.048 4.020

3.9246 3.915 3.905 3.895 3.885 3.876 3.867 3.849 3.832 3.815 3.784 3.754

1554.8 1564 1573 1582 1590 1599 1608 1625 1642 1659 1693 1726

971.791 973.98 976.18 978.35 980.51 982.63 984.74 988.87 992.92 996.88 1004.56 1011.94

4.1967 4.186 4.176 4.165 4.155 4.146 4.136 4.118 4.101 4.085 4.054 4.027

3.8727 3.864 3.855 3.846 3.838 3.829 3.821 3.805 3.789 3.774 3.745 3.717

1554.5 1564 1573 1582 1591 1600 1609 1627 1644 1662 1696 1730

50°C (ITS-90)

40°C (ITS-90)

70°C (ITS-90)

60°C (ITS-90)

Prepared by H. Sato, Keio University, Japan, Oct. 1994. Based upon “An equation of state for the thermodynamic properties of water in the liquid phase including the metastable state,” from Properties of Water and Steam,” Proc. 11th Int. Conf. Props. Steam (M. Pichal and O. Sifner, eds.), Hemisphere, New York, 1990 (551 pp.).

2-311

TABLE 2-355 Density, Specific Heats at Constant Pressure and at Constant Volume and Velocity of Sound for Compressed Water, 1–1000 bar, 0–150°C (Concluded ) 80°C (ITS-90)

90°C (ITS-90)

100°C (ITS-90)

110°C (ITS-90)

P, bar

density, kg/m3

Cp, kJ/ (kg⋅K)

Cv, kJ/ (kg⋅K)

w, m/s

density, kg/m3

Cp, kJ/(kg⋅K)

Cv, kJ/(kg⋅K)

w, m/s

density, kg/m3

Cp, kJ/(kg⋅K)

Cv, kJ/(kg⋅K)

w, m/s

density, kg/m3

Cp, kJ/(kg⋅K)

Cv, kJ/(kg⋅K)

w, m/s

1 50 100 150 200 250 300 400 500 600 800 1000

965.309 967.54 969.79 972.00 974.20 976.36 978.50 982.71 986.82 990.83 998.62 1006.08

4.2056 4.195 4.184 4.174 4.164 4.154 4.144 4.126 4.108 4.092 4.061 4.033

3.8206 3.813 3.805 3.797 3.789 3.782 3.774 3.759 3.745 3.731 3.704 3.678

1550.5 1560 1569 1579 1588 1597 1607 1625 1643 1661 1696 1731

958.348 960.64 962.94 965.21 967.45 969.67 971.85 976.15 980.34 984.43 992.34 999.92

4.2164 4.205 4.194 4.184 4.173 4.163 4.153 4.135 4.117 4.100 4.068 4.039

3.7689 3.762 3.755 3.748 3.741 3.734 3.727 3.714 3.701 3.688 3.663 3.639

1543.1 1553 1563 1572 1582 1592 1601 1620 1639 1658 1694 1730

950.927 953.28 955.65 957.99 960.30 962.57 964.82 969.21 973.50 977.69 985.76 993.47

4.2296 4.218 4.206 4.195 4.184 4.174 4.164 4.144 4.126 4.108 4.076 4.046

3.7181 3.712 3.706 3.699 3.693 3.687 3.681 3.669 3.657 3.645 3.621 3.598

1532.5 1543 1553 1563 1573 1583 1593 1613 1632 1652 1690 1727

943.059 945.50 947.95 950.36 952.74 955.09 957.40 961.92 966.32 970.61 978.87 986.73

4.2453 4.233 4.221 4.209 4.198 4.187 4.176 4.155 4.136 4.118 4.084 4.053

3.6684 3.663 3.657 3.652 3.646 3.641 3.635 3.624 3.613 3.602 3.579 3.556

1519.0 1530 1540 1551 1561 1572 1582 1603 1623 1644 1683 1723

P, bar

density, kg/m3

Cp, kJ/ (kg⋅K)

Cv, kJ/ (kg⋅K)

w, m/s

density, kg/m3

Cp, kJ/(kg⋅K)

Cv, kJ/(kg⋅K)

w, m/s

density, kg/m3

Cp, kJ/(kg⋅K)

Cv, kJ/(kg⋅K)

w, m/s

density, kg/m3

Cp, kJ/(kg⋅K)

Cv, kJ/(kg⋅K)

w, m/s

1 50 100 150 200 250 300 400 500 600 800 1000

934.749 937.28 939.83 942.33 944.79 947.22 949.61 954.27 958.81 963.22 971.68 979.72

4.2639 4.251 4.238 4.225 4.213 4.201 4.190 4.168 4.148 4.128 4.092 4.059

3.6201 3.615 3.610 3.605 3.600 3.595 3.590 3.580 3.569 3.558 3.536 3.512

1502.8 1514 1525 1536 1547 1558 1569 1591 1612 1634 1676 1717

925.997 928.64 931.29 933.90 936.46 938.97 941.45 946.28 950.96 955.51 964.20 972.44

4.2859 4.271 4.257 4.244 4.231 4.218 4.206 4.183 4.161 4.140 4.101 4.066

3.5733 3.569 3.564 3.560 3.555 3.550 3.545 3.535 3.525 3.514 3.491 3.466

1484.1 1496 1508 1519 1531 1543 1554 1577 1600 1623 1667 1710

916.797 919.57 922.34 925.06 927.73 930.35 932.92 937.94 942.79 947.48 956.43 964.88

4.3114 4.296 4.280 4.266 4.251 4.238 4.224 4.199 4.176 4.153 4.112 4.073

3.5279 3.524 3.520 3.516 3.511 3.506 3.501 3.491 3.480 3.469 3.444 3.417

1463.0 1475 1488 1501 1513 1525 1538 1562 1586 1610 1658 1704

907.143 910.06 912.97 915.82 918.61 921.34 924.03 929.24 934.27 939.13 948.36 957.04

4.3408 4.324 4.307 4.291 4.276 4.261 4.246 4.219 4.194 4.169 4.124 4.081

3.4848 3.481 3.477 3.473 3.469 3.464 3.459 3.448 3.436 3.423 3.395 3.364

1439.8 1453 1467 1480 1493 1507 1520 1546 1572 1598 1648 1698

120°C (ITS-90)

2-312

140°C (ITS-90)

130°C (ITS-90)

150°C (ITS-90)

TABLE 2-356

Specific Heat and Other Thermophysical Properties of Water Substance* Temperature, K

Pressure, bar

300

350

400

450

500

600

700

800

900

1000

1200

1400

1600

1800

µ 1 cp k Pr

8.57.−4 4.18 0.614 5.81

3.70.−4 4.19 0.668 2.32

1.32.−5 1.99 0.0268 0.980

1.52.−5 1.97 0.0311 0.967

1.73.−5 1.98 0.0358 0.955

2.15.−5 2.02 0.0464 0.936

2.57.−5 2.09 0.0581 0.920

2.98.−5 2.15 0.0710 0.906

3.39.−5 2.22 0.0843 0.891

3.78.−5 2.29 0.0981 0.881

4.48.−5 2.43 0.13 0.83

5.06.−5 2.58 0.16 0.80

5.65.−5 2.73 0.21 0.75

6.19.−5 3.02 0.33 0.57

6.70.−5 3.79 0.57 0.45

5

µ cp k Pr

8.57.−4 4.18 0.614 5.82

3.70.−4 4.19 0.668 2.32

2.17.−4 4.26 0.689 1.34

1.49.−5 2.21 0.0335 0.983

1.72.−5 2.10 0.0369 0.973

2.15.−5 2.07 0.0469 0.947

2.57.−5 2.11 0.0585 0.925

2.98.−5 2.16 0.0713 0.907

3.39.−5 2.23 0.0846 0.892

3.78.−5 2.29 0.0984 0.881

4.45.−5 2.43 0.13 0.83

5.06.−5 2.58 0.16 0.81

5.65.−5 2.73 0.20 0.77

6.19.−5 2.98 0.28 0.65

6.70.−5 3.40 0.43 0.53

10

µ cp k Pr

8.57.−4 4.18 0.615 5.82

3.70.−4 4.19 0.668 2.32

2.17.−4 4.25 0.689 1.34

1.51.−4 4.39 0.677 0.981

1.71.−5 2.29 0.0380 1.028

2.15.−5 2.13 0.0474 0.963

2.58.−5 2.13 0.0590 0.931

2.99.−5 2.18 0.0717 0.908

3.39.−5 2.24 0.0851 0.892

3.78.−5 2.30 0.0988 0.881

4.45.−5 2.44 0.13 0.84

5.06.−5 2.58 0.16 0.82

5.65.−5 2.73 0.20 0.78

6.19.−5 2.95 0.26 0.70

6.70.−5 3.29 0.39 0.57

20

µ cp k Pr

8.56.−4 4.17 0.616 5.80

3.71.−4 4.19 0.669 2.32

2.18.−4 4.25 0.689 1.34

1.51.−4 4.39 0.679 0.979

1.68.−5 2.84 0.0402 1.19

2.15.−5 2.26 0.0485 0.999

2.59.−5 2.19 0.0599 0.946

3.00.−5 2.21 0.0726 0.912

3.40.−5 2.26 0.0859 0.893

3.79.−5 2.32 0.0996 0.881

4.46.−5 2.45 0.13 0.84

5.06.−5 2.59 0.16 0.82

5.65.−5 2.73 0.20 0.79

6.19.−5 2.92 0.25 0.72

6.70.−5 3.21 0.36 0.60

40

µ cp k Pr

8.55.−4 4.17 0.617 5.78

3.71.−4 4.19 0.671 2.31

2.18.−4 4.25 0.690 1.34

1.52.−4 4.38 0.680 0.977

1.19.−4 4.65 0.644 0.862

2.15.−5 2.60 0.516 1.08

2.61.−5 2.32 0.0620 0.975

3.02.−5 2.28 0.0744 0.924

3.42.−5 2.30 0.0877 0.895

3.80.−5 2.34 0.101 0.881

4.47.−5 2.46 0.13 0.84

5.07.−5 2.59 0.16 0.82

5.65.−5 2.73 0.19 0.80

6.19.−5 2.90 0.24 0.73

6.70.−5 3.14 0.33 0.63

60

µ cp k Pr

8.54.−4 4.16 0.619 5.74

3.72.−4 4.18 0.672 2.31

2.19.−4 4.24 0.692 1.34

1.53.−4 4.37 0.682 0.976

1.20.−4 4.63 0.646 0.859

2.14.−5 3.11 0.0561 1.19

2.63.−5 2.47 0.0645 1.008

3.04.−5 2.35 0.0764 0.934

3.43.−5 2.34 0.0895 0.899

3.82.−5 2.37 0.103 0.879

4.48.−5 2.48 0.13 0.84

5.07.−5 2.60 0.16 0.82

5.66.−5 2.73 0.19 0.81

6.19.−5 2.89 0.24 0.74

6.70.−5 3.11 0.32 0.65

80

µ cp k Pr

8.53.−4 4.16 0.620 5.72

3.72.−4 4.18 0.674 2.31

2.19.−4 4.24 0.693 1.34

1.53.−4 4.36 0.684 0.975

1.20.−4 4.62 0.648 0.856

2.14.−5 3.88 0.0628 1.33

2.66.−5 2.65 0.0672 1.046

3.06.−5 2.43 0.0785 0.946

3.45.−5 2.39 0.0914 0.902

3.83.−5 2.40 0.105 0.877

4.48.−5 2.49 0.13 0.84

5.08.−5 2.61 0.16 0.83

5.66.−5 2.73 0.10 0.81

6.19.−5 2.88 0.24 0.74

6.70.−5 3.09 0.31 0.66

100

µ cp k Pr

8.52.−4 4.15 0.622 5.69

3.73.−4 4.17 0.675 2.31

2.20.−4 4.23 0.694 1.34

1.53.−4 4.35 0.685 0.975

1.21.−4 4.60 0.651 0.853

2.14.−5 5.22 0.0730 1.74

2.69.−5 2.85 0.0704 1.088

3.08.−5 2.52 0.0807 0.960

3.47.−5 2.44 0.0934 0.905

3.85.−5 2.44 0.107 0.876

4.49.−5 2.50 0.13 0.84

5.08.−5 2.62 0.16 0.83

5.66.−5 2.73 0.19 0.81

6.19.−5 2.88 0.24 0.74

6.70.−5 3.08 0.31 0.67

µ cp k Pr

8.51.−4 4.14 0.624 5.64

3.74.−4 4.16 0.678 2.30

2.22.−4 4.22 0.699 1.34

1.56.−4 4.34 0.693 0.974

1.22.−4 4.54 0.657 0.842

8.22.−5

150

2.72.−5 3.55 0.079 1.22

3.12.−5 2.74 0.086 0.994

3.51.−5 2.57 0.098 0.916

3.89.−5 2.53 0.110 0.891

4.52.−5 2.54 0.14 0.84

5.09.−5 2.65 0.16 0.83

5.67.−5 2.75 0.19 0.82

6.19.−5 2.88 0.23 0.76

6.70.−5 3.06

µ cp k Pr

8.50.−4 4.12 0.626 5.59

3.75.−4 4.15 0.681 2.29

2.24.−4 4.21 0.702 1.34

1.57.−4 4.32 0.697 0.974

1.23.−4 4.51 0.661 0.833

2.80.−5 4.67 0.095 1.38

3.17.−5 3.04 0.095 1.014

3.54.−5 2.71 0.104 0.925

3.93.−5 2.62 0.113 0.903

4.54.−5 2.57 0.14 0.84

5.11.−5 2.67 0.16 0.83

5.67.−5 2.76 0.19 0.82

2.88

3.05

250

µ cp k Pr

8.49.−4 4.12 0.627 5.57

3.76.−4 4.14 0.683 2.28

2.26.−4 4.20 0.705 1.34

1.59.−4 4.30 0.701 0.974

1.23.−4 4.49 0.672 0.826

8.41.−5 5.90 0.537 0.924

2.89.−5 6.16 0.112 1.590

3.24.−5 3.40 0.103 1.070

3.59.−5 2.86 0.110 0.940

3.98.−5 2.71 0.119 0.910

4.56.−5 2.61 0.136 0.85

5.12.−5 2.69 0.16 0.84

5.68.−5 2.77

2.89

3.04

300

µ cp k Pr

8.49.−4 4.10 0.629 5.53

3.77.−4 4.13 0.685 2.27

2.28.−4 4.19 0.708 1.34

1.60.−4 4.29 0.704 0.973

1.24.−4 4.44 0.675 0.820

8.50.−5 5.60 0.548 0.859

3.7.−5 10.20 0.173 2.18

3.4.−5 3.82 0.113 1.149

3.64.−5 3.03 0.113 0.976

4.02.−5 2.81 0.123 0.917

4.59.−5 2.65 0.14 0.87

5.14.−5 2.72

5.68.−5 2.78

2.90

3.04

400

µ cp k Pr

8.49.−4 4.08 0.631 5.49

3.80.−4 4.12 0.689 2.26

2.30.−4 4.16 0.714 1.34

1.62.−4 4.26 0.710 0.971

1.26.−4 4.42 0.676 0.817

8.64.−5 5.31 0.567 0.799

5.3.−5 13.20 0.327 2.14

3.6.−5 4.86 0.145 1.207

3.8.−5 3.39 0.129 0.999

4.1.−5 3.01 0.134 0.926

4.6.−5 2.70 0.15

5.17.−5 2.77

2.81

2.91

3.04

500

µ cp k Pr

8.50.−4 4.06 0.634 5.44

3.82.−4 4.10 0.695 2.25

2.31.−4 4.15 0.719 1.33

1.64.−4 4.23 0.717 0.971

1.28.−4 4.38 0.693 0.814

8.83.−5 5.08 0.583 0.773

5.8.−5 8.44 0.378 1.30

4.0.−5 5.70 0.186 1.225

4.0.−5 3.90 0.147 1.061

4.2.−5 3.21 0.145 0.932

4.7.−5 2.77

2.81

2.84

2.92

3.04

200

0.520 8.32.−5 0.525

2000

2-313

TABLE 2-356

Specific Heat and Other Thermophysical Properties of Water Substance (Concluded ) Temperature, K

Pressure, bar

300

350

400

450

500

600

700

800

900

1000

µ 600 cp k Pr

8.51.−4 4.04 0.639 5.38

3.85.−4 4.08 0.699 2.24

2.32.−4 4.13 0.725 1.32

1.66.−4 4.20 0.725 0.970

1.30.−4 4.33 0.700 0.812

9.17.−5 4.92 0.597 0.755

6.5.−5 6.93 0.420 1.073

4.4.−5 6.83 0.239 1.175

4.2.−5 4.19 0.170 1.035

4.4.−5 3.38 0.159 0.935

2.87

2.86

2.86

2.92

3.04

700

µ cp k Pr

8.52.−4 4.01 0.644 5.33

3.87.−4 4.07 0.706 2.23

2.33.−4 4.12 0.730 1.32

1.69.−4 4.17 0.732 0.970

1.33.−4 4.29 0.707 0.810

9.50.−5 4.78 0.614 0.739

6.9.−5 6.12 0.442 1.047

4.9.−5 6.26 0.279 1.098

4.5.−5 4.62 0.198 1.010

4.6.−5 3.59 0.177 0.935

2.94

2.91

2.88

2.93

3.05

800

µ cp k Pr

8.53.−4 3.99 0.648 5.28

3.90.−4 4.05 0.709 2.23

2.34.−4 4.10 0.735 1.31

1.72.−4 4.15 0.736 0.970

1.36.−4 4.26 0.714 0.808

9.82.−5 4.67 0.625 0.725

7.3.−5 5.60 0.478 0.855

5.4.−5 6.09 0.320 1.028

4.8.−5 4.77 0.228 1.003

4.8.−5 3.75 0.193 0.933

3.01

2.96

2.91

2.95

3.05

900

µ cp k Pr

8.54.−4 3.98 0.651 5.23

3.93.−4 4.03 0.713 2.22

2.35.−4 4.08 0.738 1.30

1.74.−4 4.13 0.742 0.969

1.38.−4 4.23 0.724 0.806

1.00.−4 4.57 0.636 0.712

7.6.−5 5.29 0.496 0.810

5.8.−5 5.86 0.351 0.968

5.1.−5 4.85 0.260 0.950

5.0.−5 3.86 0.210 0.919

3.08

3.00

2.94

2.97

3.06

1000

µ cp k Pr

8.56.−4 3.97 0.653 5.19

3.96.−4 4.02 0.717 2.22

2.36.−4 4.06 0.743 1.30

1.76.−4 4.11 0.747 0.968

1.40.−4 4.20 0.731 0.804

1.02.−4 4.47 0.650 0.701

7.9.−5 5.08 0.516 0.778

6.2.−5 5.51 0.372 0.918

5.4.−5 4.88 0.288 0.900

5.1.−5 3.96 0.228 0.886

3.16

3.05

2.97

2.98

3.07

*µ = viscosity, Ns/m2; cp = specific heat at constant pressure, kJ/(kg⋅K); k = thermal conductivity, W/(m⋅K); Pr = Prandtl number.

2-314

1200

1400

1600

1800

2000

THERMODYNAMIC PROPERTIES TABLE 2-357

Thermodynamic Properties of Water Substance along the Melting Line T, °C

103 v f , m3/kg

h f , kJ/kg

s f, kJ/kg⋅K

cpf, kJ/kg⋅K

cmelt, kJ/kg⋅K

106α f, K−1

106K f,T bar−1

6.117.−5t 1.01325 50 100 150

0.0100 0.0026 −0.3618 −0.7410 −1.1249

1.00021 1.00016 0.99770 0.99523 0.99278

0 0.0719 3.5140 6.9794 10.3964

0 −0.0001 −0.0054 −0.0110 −0.0167

4.219 4.218 4.196 4.174 4.152

3.969 3.970 3.997 4.023 4.047

−67.42 −67.17 −54.92 −42.52 −30.24

50.90 50.88 50.30 49.73 49.17

200 250 300 400 500

−1.5166 −1.9151 −2.3206 −3.1532 −4.0156

0.99037 0.98798 0.98562 0.98098 0.97643

13.7648 17.0843 20.3547 26.7472 32.9403

−0.0225 −0.0285 −0.0347 −0.0474 −0.0607

4.132 4.112 4.092 4.056 4.022

4.070 4.092 4.113 4.150 4.184

−18.05 −5.93 6.12 30.09 53.97

48.63 48.11 47.59 46.61 45.68

600 800 1000

−4.909 −6.790 −8.803

0.97196 0.96326 0.95493

38.932 50.300 60.836

−0.0747 −0.1046 −0.1371

3.992 3.937 3.893

4.215 4.270 4.320

77.87 126.18 175.98

44.80 43.19 41.74

P, bar

2-315

Condensed from U. Grigull, Private communication, January 18, 1995. Materials prepared at Technical University München, Germany by U. Grigull and S. Marek. For a table as a function of temperature, see Grigull, U. and S. Marek, Warme u. Stoff., 30 (1994): 1–8. t = the triple point (at 6.117 × 10−5 bar, 0.01°C); vf = 0.0010021 m3/kg: α f = −67.42 × 10−6/K. Other equations for properties are given by Jones, F. E. and G. L. Harris, J. Res. N.I.S.T., 97, 3 (1992): 335–340, and by Wagner, W. and A. Pruss, J. Phys. Chem. Ref. Data, 22, 3 (1993): 783–787. Steam tables include Walker, W. A., U.S. Naval Ordn. Lab. rept. NOLTR NOLTR-66-217 = AD 651105 (0–1000 bar, 0–150°C), 1967 (72 pp.); Grigull, U., J. Straub, et al., Steam Tables in S.I. Units (0.01–1000 bar, 0–1000°C), Springer-Verlag, Berlin, 1990 (133 pp.); Tseng, C. M., T. A. Hamp, et al., Atomic Energy of Canada rept. (30 props, sat liq & vap., 1–220 bar), AECL-5910 1977 (90 pp.). For dissociation, see e.g., Knonicek, V., Rozpr. Cesko Acad Ved., Rada techn ved (0.01–100 bar, 1000–5000 K). 77, 1 (1967). The proceedings of the 10th international conference on the properties of steam were edited by Sytchev, V. V. and A. A. Aleksandrov, Plenum, NY, 1984; and for the 11th conference by Pichal, M. and O. Sifner, Hemisphere, 1989 (550 pp.). For electrical conductivity, see e.g., Marshall, W. L., J. Chem. Eng. Data, 32 (1987): 221–226.

TABLE 2-358 T, K

Saturated Xenon*

P, bar

v f, m3/kg

vg, m3/kg

h f, kJ/kg

hg, kJ/kg

s f, kJ/(kg⋅K)

s g, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µ f, 10−4 Pa⋅s

k f, W/(m⋅K)

10 20 30 40 50

2.642.−4 2.650.−4 2.661.−4 2.675.−4 2.689.−4

0.19 1.21 2.74 4.47 6.31

0.0236 0.0901 0.1510 0.2003 0.2410

0.058 0.133 0.164 0.178 0.186

60 80 100 120 140

2.704.−4 2.737.−4 2.776.−4 2.820.−4 2.874.−4

8.21 12.14 16.30 20.81 25.67

0.2755 0.3319 0.3783 0.4197 0.4581

0.191 0.202 0.214 0.231 0.251

160 161.4m 161.4m 170 180

0.816 1.336 2.218

2.941.−4 2.946.−4 3.372.−4 3.439.−4 3.523.−4

0.1219 0.0776 0.0487

30.94 31.30 48.98 52.01 55.52

145.5 146.5 147.5

0.4946 0.4969 0.6072 0.6253 0.6452

1.206 1.181 1.156

0.270 0.271 0.350 0.349 0.349

4.50 3.99

0.0707 0.0663

190 200 210 220 230

3.480 5.212 7.504 10.45 14.16

3.615.−4 3.715.−4 3.828.−4 3.955.−4 4.100.−4

0.0321 0.0220 0.0156 0.0113 0.0084

59.04 62.61 66.25 70.00 73.91

148.3 148.9 149.2 149.4 149.2

0.6641 0.6820 0.6994 0.7163 0.7330

1.134 1.113 1.095 1.077 1.060

0.352 0.357 0.365 0.379 0.400

3.51 3.09 2.71 2.39 2.09

0.0622 0.0582 0.0542 0.0506 0.0468

240 250 260 270 280

18.72 24.25 30.87 38.69 47.86

4.271.−4 4.476.−4 4.730.−4 5.079.−4 5.689.−4

0.0063 0.0047 0.0036 0.0027 0.0019

78.05 82.54 87.52 93.30 100.6

148.5 147.5 145.7 142.8 138.0

0.7498 0.7671 0.7855 0.8058 0.8308

1.044 1.027 1.009 0.989 0.964

0.432 0.482 0.560 0.685 0.995

1.83 1.60 1.38 1.18 0.95

0.0429 0.0393 0.0355 0.0313 0.0275

289.7 c

58.21

9.091.−4

0.0009

120.0

120.0

0.8962

0.896

∞

∞

*Values extracted and in some cases rounded off from those cited in Rabinovich (ed.), Thermophysical Properties of Neon, Argon, Krypton and Xenon, Standards Press, Moscow, 1976. This source contains values for the compressed state for pressures up to 1000 bar, etc. m = melting point; c = critical point. The notation 2.642.−4 signifies 2.642 × 10−4. This book was published in English translation by Hemisphere, New York, 1988 (604 pp.).

2-316

PHYSICAL AND CHEMICAL DATA

TABLE 2-359

Compressed Xenon* Pressure, bar

T, K 100 v h s

1 2.776.−4 16.30 0.3783

100

200

300

400

500

600

700

2.764.−4 18.84 0.3762

2.752.−4 21.40 0.3742

2.742.−4 23.95 0.3723

2.731.−4 26.50 0.3704

2.721.−4 29.05 0.3686

2.711.−4 31.59 0.3669

2.702.−4 34.13 0.3652

2.693.−4 36.67 0.3636

800

2.684.−4 39.21 0.3621

900

2.675.−4 41.74 0.3802

1000

200 v h s

0.1245 151.8 1.228

3.623.−4 64.22 0.6727

3.547.−4 66.14 0.6643

3.484.−4 68.19 0.6570

3.430.−4 70.34 0.6505

3.383.−4 72.56 0.6446

3.342.−4 74.83 0.6391

3.304.−4 77.13 0.6340

3.270.−4 79.46 0.6292

3.240.−4 81.81 0.6247

3.211.−4 84.18 0.6204

300 v h s

0.1890 168.0 1.294

5.729.−4 106.4 0.8401

4.769.−4 101.6 0.8073

4.431.−4 101.3 0.7908

4.220.−4 102.0 0.7789

4.068.−4 103.3 0.7691

3.955.−4 104.9 0.7608

3.862.−4 106.7 0.7540

3.783.−4 108.5 0.7477

3.716.−4 110.6 0.7424

3.657.−4 112.8 0.7370

400 v h s

0.2527 183.9 1.340

1.998.−3 164.2 1.012

8.759.−4 145.4 0.9330

6.452.−4 137.4 0.8945

5.604.−4 134.7 0.8730

5.141.−4 134.1 0.8581

4.839.−4 134.5 0.8467

4.622.−4 135.5 0.8373

4.457.−4 136.8 0.8292

4.325.−4 138.3 0.8220

4.217.−4 140.0 0.8162

500 v h s

0.3163 199.8 1.375

2.899.−3 187.8 1.065

1.389.−3 177.1 1.004

9.449.−4 169.4 0.9664

7.577.−4 165.1 0.9409

6.593.−4 163.0 0.9228

5.986.−4 162.3 0.9088

5.570.−4 162.4 0.8975

5.268.−4 163.1 0.8881

5.038.−4 164.3 0.8801

4.859.−4 165.7 0.8731

600 v h s

0.3798 215.7 1.404

3.673.−3 207.4 1.101

1.823.−3 200.3 1.047

1.240.−3 194.8 1.013

9.699.−4 191.1 0.9885

8.206.−4 188.9 0.9700

7.273.−4 187.9 0.9555

6.636.−4 187.6 0.9435

6.172.−4 188.0 0.9334

5.820.−4 188.8 0.9247

5.545.−4 189.9 0.9172

700 v h s

0.4432 231.5 1.428

4.397.−3 225.6 1.129

2.217.−3 220.6 1.078

1.513.−3 216.7 1.047

1.175.−3 213.8 1.023

9.815.−4 212.2 1.006

8.583.−4 211.3 0.9916

7.734.−4 211.1 0.9797

7.115.−4 211.3 0.9695

6.642.−4 212.0 0.9606

6.268.−4 213.1 0.9528

800 v h s

0.5066 247.4 1.450

5.093.−3 243.0 1.152

2.587.−3 239.5 1.103

1.769.−3 236.7 1.073

1.370.−3 234.8 1.052

1.137.−3 233.6 1.035

9.870.−4 233.0 1.021

8.824.−4 232.9 1.009

8.057.−4 233.3 0.9988

7.469.−4 234.0 0.9901

7.005.−4 235.0 0.9823

900 v h s

0.5700 263.2 1.468

5.773.−3 260.1 1.172

2.944.−3 257.5 1.125

2.014.−3 255.7 1.096

1.557.−3 254.4 1.075

1.288.−3 253.6 1.058

1.112.−3 253.4 1.045

9.893.−4 253.7 1.033

8.989.−4 254.2 1.023

8.289.−4 254.9 1.015

7.737.−4 256.1 1.007

1000 v h s

0.6333 279.1 1.485

6.441.−3 276.8 1.190

3.291.−3 275.1 1.143

2.252.−3 273.9 1.115

1.738.−3 273.2 1.095

1.435.−3 272.9 1.079

1.235.−3 273.0 1.065

1.094.−3 273.4 1.054

9.899.−4 274.1 1.044

9.097.−4 275.1 1.036

8.461.−4 276.2 1.028

*Values extracted and in some cases rounded off from those cited in Rabinovich (ed.), Thermophysical Properties of Neon, Argon, Krypton and Xenon. Standards Press, Moscow, 1976. This source contains an exhaustive tabulation of values. v = specific volume, m3/kg; h = specific enthalpy, kJ/kg; s = specific entropy, kJ/(kg⋅K). The notation 2.776.−4 signifies 2.776 × 10−4. This book was published in English translation by Hemisphere, New York, 1988 (604 pp.).

THERMODYNAMIC PROPERTIES

2-317

TABLE 2-360

Surface Tension (N/m) of Saturated Liquid Refrigerants*

R no.

−50

−25

0

25

50

75

100

125

150

0.0279 0.0188 0.0092 0.0197 0.0115

0.0244 0.0152 0.0056 0.0156 0.0065

0.0210 0.0118 0.0025 0.0117 0.0025

0.0178 0.0085 0.0002 0.0081

0.0146 0.0055 — 0.0047 —

0.0116 0.0029 — 0.0018 —

0.0087 0.0007 — — —

0.0060 — — — —

0.0036 — — — —

—

0.0231 0.0154

0.0069 0.0172 0.0109 0.0047 0.0082

0.0032 0.0144 0.0082 0.0022 0.0050

0.0002 0.0118 0.0056

0.0192

0.0201 0.0138 0.0075 0.0117

0.0021

— 0.0092 0.0033 — 0.0000

— 0.0067 0.0012 — —

— 0.0045 — — —

142b 152a 170 290 C318

0.0213 0.0201 0.0100

0.0178 0.0166 0.0051

—

0.0143

0.0145 0.0132 0.0032 0.0101 0.0113

0.0113 0.0100 0.0005 0.0082 0.0085

0.0083 0.0068 — 0.0041 0.0048

0.0055 0.0038 — 0.0016 0.0033

0.0029 0.0011 — — 0.0011

— — — —

— — — — —

502 503 600 600a 718

0.0159 0.0094

0.0121 0.0053 0.0180

0.0054 — 0.0122 0.0101 0.0720

0.0026 — 0.0094 0.0073 0.0680

— 0.0068 0.0047 0.0636

— — 0.0043 0.0024 0.0590

— — 0.0020 0.0005 0.0540

— — 0.0001 — 0.0488

0.0005

—

—

—

—

—

0.0070

0.0041

0.0014

Temperature, °C 11 12 13 22 23 32 113 114 115 134a

744 1150 1270

—

—

0.0086 0.0018 0.0150 0.0132 0.0755

0.0100 0.0171

0.0096 0.0055 0.0136

0.0044 0.0013 0.0102

*Dashes indicate inaccessible states; blanks indicate no available data. Values and equations were given by Srinivasan, K., Can. J. Chem. Eng. (27 liquids), 68 (1990): 493; Lielmezs, J. and T. A. Herrick, Chem. Eng. J. (34 liquids), 32 (1986): 165–169. Somayajulu, G. R., Int. J. Thermophys. (64 liquids); 9, 4 (1988): 559–566; Ibrahim, N. and S. Murad, Chem. Eng. Commun. (29 polar liquids), 79 (1979): 165–174; Yaws, C. L.; Morachevsky, A. G. and I. B. Sladkov, Physico-Chemical Properties of Molecular Inorganic Compounds (200 compounds), Khimiya, Leningrad, 1987, Jasper, J., J. Phys. Chem. Ref. Data (2200 compounds), 1, (1972): 841–1009; and Vargaftik, N. B., B. N. Volkov, et al., J. Phys. Chem. Ref. Data (water), 12, 3 (1983): 817–820. See also Escobedo, J. and Mansoori, G. R., AIChE J., 42(5), May 1996: 1425–1433.

TABLE 2-361

Velocity of Sound (m/s) in Gaseous Refrigerants at Atmospheric Pressure* Temperature, °C

−50

−25

0

25

50

75

100

125

150

11 12 13 14 22

— — 142 158 —

— 136 150 166 166

— 143 157 173 174

141 150 164 180 182

147 156 170 187 189

153 162 176 194 196

158 168 182 200 202

163 173 188 206 208

168 179 193 212 215

23 32 113 114 134a

179

188

197

205

212

220

227

234

240

— —

— — 146

— — 154

— 120 162

121 126 169

126 131 175

131 136 180

135 141 186

140 146 192

170 290 600 600a 718

272 — — — —

286 227 — — —

299 238 200 201 —

311 249 210 211 —

323 258 220 221 —

334 268 228 229 —

344 277 237 237 473

355 286 245 246 490

364 294 252 253 505

290 —

248 305 235

258 318 246

269 330 257

279 341 267

288 352 277

297 363 286

307 373 295

316 384 303

R. no.

744 1150 1270

*Dashes indicate inaccessible states; blanks indicate no available data.

2-318

PHYSICAL AND CHEMICAL DATA TABLE 3-362

Velocity of Sound (m/s) in Saturated Liquid Refrigerants* Temperature, °C

R. no.

−50

−25

0

25

50

75

100

125

150

11 12 13 14 22

933 829 602 182 899

843 695 444 — 790

772 564 302 — 682

705 434

639

569

493

— 571

— — 446

— — 319

— — —

408 — — — —

323 — — — —

538

348

191

—

—

— —

— —

— —

— 853

871 726

786 623 454

700 540 346

633 453 255

371

284 —

183 —

— —

858 1210 1290 1205 —

743 982 1163 1078 —

626 884 1031 947 1402

517 719 896 812 1495

387 551 759 661 1542

262 367 609 528 1554

105 — 477 378 1543

— — 325 208 1514

— — 142 — 1468

751 644 1022

525 372 859

272

—

—

—

—

—

874 1184

694

524

335

23 32 113 114 115 134a 290 600 600a 718 744 1150 1270

*Dashes indicate inaccessible states; blanks indicate no available data.

TRANSPORT PROPERTIES INTRODUCTION Extensive tables of the viscosity and thermal conductivity of air and of water or steam for various pressures and temperatures are given with the thermodynamic-property tables. The thermal conductivity and the viscosity for the saturated-liquid state are also tabulated for many fluids along with the thermodynamic-property tables earlier in this section. UNITS CONVERSIONS For this subsection the following units conversions are applicable: Diffusivity: to convert square centimeters per second to square feet per hour, multiply by 3.8750; to convert square meters per second to square feet per hour, multiply by 38,750. Pressure: to convert bars to pounds-force per square inch, multiply by 14.504. Temperature: °F = 9⁄5 °C + 32; °R = 9⁄5 K. Thermal conductivity: to convert watts per meter-kelvin to British thermal unit–feet per hour–square foot–degree Fahrenheit, multiply by 0.57779; and to convert British thermal unit–feet per hour–square foot–degree Fahrenheit to watts per meter-kelvin, multiply by 1.7307. Viscosity: to convert pascal-seconds to centipoises, multiply by 1000. ADDITIONAL REFERENCES An extensive coverage of the general pressure and temperature variation of thermal conductivity is given in the monograph by Vargaftik,

Filippov, Tarzimanov, and Totskiy, Thermal Conductivity of Liquids and Gases (in Russian), Standartov, Moscow, 1978, now published in English translation by CRC Press, Miami, FL. For a similar work on viscosity, see Stephan and Lucas, Viscosity of Dense Fluids, Plenum, New York and London, 1979. Tables and polynomial fits for refrigerants in both the gaseous and the liquid state are contained in ASHRAE Thermophysical Properties of Refrigerants, American Society of Heating, Refrigerating and Ventilating Engineers, Atlanta, GA, 1993. Other sources for viscosity include Fischer & Porter Co. catalog 10-A-94, “Fluid densities and viscosities,” 1953 (200 industrial fluids in 48 pp.) and van Velzen, D., R. L. Cardozo et al., EURATOM Ispra, Italy rept. 4735 e, 1972 (160 pp.). Liquid viscosity, 314 cpds, is summarized in I&EC Fundtls., 11 (1972): 20–26. Five hundred forty-nine binary and ternary systems are discussed in Skubla, P., Coll. Czech. Chem. Commun., 46 (1981): 303–339. See also Duhne, C. R., Chem. Eng. (NY), 86, 15 (July 16, 1979): 83–91 (equations and 326 liquids); and Rao, K. V. K., Chem. Eng. (NY), 90, 11 (May 30, 1983): 90–91 (nomograph, 87 liquids). For rheology, non-Newtonian behavior, and the like, see, for instance, Barnes, H., The Chem. Engr. (UK), (June 24, 1993): 17–23; Hyman, W. A., I&EC Fundtls., 16 (1976): 215–218; and Ferguson, J. and Z. Kemblowski, Applied Fluid Rheology, Elsevier, 1991 (325 pp.). Other sources for thermal conductivity include Ho, C. Y., R. W. Powell et al., J. Phys. Chem. Ref. Data, 1 (1972) and 3, suppl. 1 (1974); Childs, Ericks et al., N.B.S. Monogr. 131, 1973; Jamieson, D. T., J. B. Irving et al., Liquid Thermal Conductivity, H.M.S.O., Edinburgh, Scotland, 1975 (220 pp.).

TRANSPORT PROPERTIES TABLE 2-363

2-319

Transport Properties of Selected Gases at Atmospheric Pressure* Viscosity, 10−4 Pa⋅s Temperature, K

Thermal conductivity, W/(m⋅K) Temperature, K Substance

250

300

400

500

Acetone Acetylene Ammonia Argon Benzene

0.0080 0.0162 0.0197 0.0152 0.0077

0.0115 0.0213 0.0246 0.0177 0.0104

0.0201 0.0332 0.0364 0.0223 0.0195

0.0310 0.0452 0.0506 0.0264 0.0335

600

Bromine Butane CO2 CCl4 Chlorine

0.0038 0.0117 0.0129 0.0053 0.0071

0.0048 0.0160 0.0166 0.0067 0.0089

0.0067 0.0264 0.0244 0.0099 0.0124

0.0377 0.0323 0.0126 0.0156

Deuterium Ethane Ethylene Helium Heptane

0.122 0.0156 0.0152 0.134 0.0082

0.141 0.0218 0.0214 0.150 0.0120

0.176 0.0360 0.0342 0.180 0.0214

0.0516 0.0491 0.211 0.0325

0.0685 0.0653 0.247 0.0447

Hydrogen Methane Nitrogen Oxygen Pentane

0.156 0.0277 0.0222 0.0225 0.0107

0.182 0.0343 0.0260 0.0267 0.0152

0.221 0.0484 0.0325 0.0343 0.0250

0.256 0.0671 0.0386 0.0412 0.0362

0.291 0.0948 0.0441 0.0480

Propane Propylene R 11 R 12 R 13

0.0129 0.0114

0.0295 0.0226 0.0119 0.0151 0.0185

0.0417 0.0430

0.0072 0.0091

0.0183 0.0168 0.0078 0.0097 0.0121

0.0208 0.0248

R 21 R 22 SO2

0.0080 0.0078

0.0088 0.0109 0.0096

0.0135 0.0170 0.0143

0.0181 0.0230 0.0200

250

0.0561 0.0656 0.0301 0.0524

0.085 0.195

0.0403

0.126

0.0190

0.0580

0.0290 0.0256

Prandtl number, dimensionless Temperature, K

300

400

500

600

0.077 0.104 0.102 0.229 0.076

0.101 0.135 0.139 0.289 0.101

0.128 0.164 0.175 0.343 0.127

0.156

250

300

400

500

0.211 0.390 0.154

0.669

0.91 0.668

0.87 0.666

0.86 0.663

0.076 0.150 0.101 0.136

0.203 0.101 0.196 0.131 0.178

0.260 0.125 0.239 0.162 0.218

0.291 0.151 0.278 0.191 0.259

0.793

0.805 0.778

0.820 0.752

0.734

0.817 0.812 0.671

0.773 0.796 0.668

0.746 0.769 0.663

0.746 0.750 0.661

0.71 0.742 0.721

0.71 0.739 0.714

0.71 0.737 0.708

0.71 0.736 0.707

0.810 0.860

0.788 0.762 0.761 0.745 0.759

0.826

0.827 0.796

0.774 0.797 0.814 0.781 0.766

0.820

0.779 0.771

0.773 0.760

0.111 0.079 0.087 0.176

0.126 0.094 0.103 0.199

0.153 0.123 0.135 0.243 0.080

0.178 0.148 0.162 0.284 0.099

0.201 0.171 0.187 0.322 1.116

0.080 0.095 0.156 0.179

0.090 0.112 0.180 0.207

0.109 0.142 0.223 0.258

0.126 0.170 0.261 0.306

0.143 0.195 0.295 0.348

0.069 0.073 0.094 0.108 0.123

0.082 0.087 0.110 0.126 0.145

0.108 0.115 0.144 0.162 0.190

0.131 0.141

0.100 0.109

0.115 0.129 0.129

0.154 0.168 0.175

0.217

0.708 0.757

0.256

*An approximate interpolation scheme is to plot the logarithm of the viscosity or the thermal conductivity versus the logarithm of the absolute temperature. At 250 K the viscosity of gaseous argon is to be read as 1.95 × 10−5 Pa⋅s = 0.0000195 Ns/m2.

2-320

PHYSICAL AND CHEMICAL DATA

TABLE 2-364

Viscosities of Gases: Coordinates for Use with Fig. 2-32* µ × 107 p

Gas

X

Y

Acetic acid Acetone Acetylene Air Ammonia Amylene (β) Argon Arsine Benzene Bromine Butane (n) Butane (iso) Butyl acetate (iso) Butylene (α) Butylene (β) Butylene (iso) Butyl formate (iso) Cadmium Carbon dioxide Carbon disulfide Carbon monoxide Carbon oxysulfide Carbon tetrachloride Chlorine Chloroform Cyanogen Cyclohexane Cyclopropane Deuterium Diethyl ether Dimethyl ether Diphenyl ether Diphenyl methane Ethane Ethanol Ethyl acetate Ethyl chloride Ethylene Ethyl propionate Fluorine Freon-11 Freon-12 Freon-14 Freon-21 Freon-22 Freon-113 Freon-114 Helium Heptane (n) Hexane (n) Hydrogen Hydrogen-helium 10% H2, 90% He 25% H2, 75% He 40% H2, 60% He 60% H2, 40% He 81% H2, 19% He

7.0 8.4 9.3 10.4 8.4 8.6 9.7 8.6 8.7 8.8 8.6 8.6 5.7 8.4 8.7 8.3 6.6 7.8 8.9 8.5 10.5 8.2 8.0 8.8 8.8 8.2 9.0 8.3 11.0 8.8 9.0 8.6 8.0 9.0 8.2 8.4 8.5 9.5 12.0 7.3 8.6 9.0 9.5 9.0 9.0 11.0 9.4 11.3 8.6 8.4 11.3

14.6 13.2 15.5 20.4 16.0 12.2 22.6 20.0 13.2 19.4 13.2 13.2 16.3 13.5 13.1 13.9 16.0 22.5 19.1 15.8 20.0 17.9 15.3 18.3 15.7 16.2 12.2 14.7 16.2 12.7 15.0 10.4 10.3 14.5 14.5 13.4 15.6 15.2 12.4 23.8 16.2 17.4 20.4 16.7 17.7 14.0 16.4 20.8 10.6 12.0 12.4

825 (50°C) 735 1017 1812 1000 676 2215 1576 746 1495 735 744 778 761 746 786 840 5690 (500) 1463 990 1749 1220 966 1335 1000 1002 701 870 1240 730 925 610 (50) 605 (50) 915 835 743 978 1010 890 2250 1298 (93) 1496 (93) 1716 1389 (93) 1554 (93) 1166 (93) 1364 (93) 1946 618 (50) 644 880

11.0 11.0 10.7 10.8 10.5

20.5 19.4 18.4 16.7 15.0

1780 (0) 1603 (0) 1431 (0) 1227 (0) 1016 (0)

Ref.

Gas

X

Y

µ × 107 p

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 3 3 5 3 3 3 3 1 1 1 1 1

Hydrogen–sulfur dioxide 10% H2, 90% SO2 20% H2, 80% SO2 50% H2, 50% SO2 80% H2, 20% SO2 Hydrogen bromide Hydrogen chloride Hydrogen cyanide Hydrogen iodide Hydrogen sulfide Iodine Krypton Mercury Mercuric bromide Mercuric chloride Mercuric iodide Mesitylene Methane Methane (deuterated) Methanol Methyl acetate Methyl acetylene 3-Methyl-1-butene Methyl butyrate (iso) Methyl bromide Methyl chloride 3-Methylene-1-butene Methylene chloride Methyl formate Neon Nitric oxide Nitrogen Nitrous oxide Nonane (n) Octane (n) Oxygen Pentane (n) Pentane (iso) Phosphene Propane Propanol (n) Propanol (iso) Propyl acetate Propylene Pyridine Silane Stannic chloride Stannic bromide Sulfur dioxide Thiazole Thiophene Toluene 2,2,3-Trimethylbutane Trimethylethane Water Xenon Zinc

8.7 8.6 8.9 9.7 8.4 8.5 7.1 8.5 8.4 8.7 9.4 7.4 8.5 7.7 8.4 9.5 9.5 9.5 8.3 8.4 8.9 8.0 6.6 8.1 8.5 8.0 8.5 5.1 11.1 10.4 10.6 9.0 9.2 8.8 10.2 8.5 8.9 8.8 8.9 8.4 8.4 8.0 8.5 8.6 9.0 9.1 9.0 8.4 10.0 8.3 8.6 10.0 8.0 8.0 9.3 8.0

18.1 18.2 18.3 17.7 21.6 19.2 14.5 21.5 18.0 18.7 24.0 24.9 19.0 18.7 18.0 10.2 15.8 17.6 15.6 14.0 14.3 13.3 15.8 18.7 16.5 13.3 15.8 18.0 25.8 20.8 20.0 19.0 8.9 9.8 21.6 12.3 12.1 17.0 13.5 13.5 13.6 14.3 14.4 13.3 16.8 16.0 16.7 18.2 14.4 14.2 12.5 10.4 13.0 16.0 23.0 22.0

1259 (17) 1277 (17) 1332 (17) 1306 (17) 1843 1425 737 1830 1265 1730 (100) 2480 4500 (200) 2253 2200 (200) 2045 (200) 660 (50) 1092 1290 935 870 (50) 867 716 824 1327 1062 716 989 923 3113 1899 1766 1460 554 (50) 586 (50) 2026 668 685 1150 800 770 774 797 840 830 (50) 1148 1330 (100) 142 (100) 1250 958 901 (50) 686 691 (50) 686 1250 (100) 2255 5250 (500)

Ref. 4

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 6 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

*Viscosity at 20°C unless otherwise indicated. From Beerman, Meas. Control (June 1982): 154–157. References: 1. I. F. Golubev, Viscosity of Gases and Gas Mixtures, Moscow 1959; transl. U.S. Department of Commerce, Clearinghouse for Federal Scientific and Technical Information, Springfield, Va., TT 70-50022, ISPT Cat. No. 5680, Table 4, Jerusalem 1970. 2. R. H. Perry and C. H. Chilton, Chemical Engineers’ Handbook, 5th ed., McGraw-Hill, New York, 1973, pp. 3-210, 3-211. 3. Ibid., Table 3-282, p. 3-210. 4. By interpolation of data in Ref. 1. 5. Thermophysical Properties of Refrigerants, American Society of Heating, Refrigerating and Air-Conditioning Engineers, New York. 6. N. A. Lange, Handbook of Chemistry, 4th ed., Handbook Publishers, Sandusky, Ohio, 1941. For another alignment chart for 165 hydrocarbons from −100 to 500°C, see Sastry, R. C. and A. Satyanarayan, Chem. Industry Devs. (July 1978): 11–14.

TRANSPORT PROPERTIES

Nomograph for determining (a) absolute viscosity of a gas as a function of temperature near ambient pressure and (b) relative viscosity of a gas compared with air. For coordinates see Table 2-364. To convert poises to pascal-seconds, multiply by 0.1. [From Beerman, Meas. Control, 154–157 (June 1982).]

FIG. 2-32

2-321

2-322

PHYSICAL AND CHEMICAL DATA

TABLE 2-365

Viscosities of Liquids: Coordinates for Use with Fig. 2-33

Liquid

X

Y

Liquid

X

Y

Acetaldehyde Acetic acid, 100% Acetic acid, 70% Acetic anhydride Acetone, 100% Acetone, 35% Acetonitrile Acrylic acid Allyl alcohol Allyl bromide Allyl iodide Ammonia, 100% Ammonia, 26% Amyl acetate Amyl alcohol Aniline Anisole Arsenic trichloride Benzene Brine, CaCl2, 25% Brine, NaCl, 25% Bromine Bromotoluene Butyl acetate Butyl acrylate Butyl alcohol Butyric acid Carbon dioxide Carbon disulfide Carbon tetrachloride Chlorobenzene Chloroform Chlorosulfonic acid Chlorotoluene, ortho Chlorotoluene, meta Chlorotoluene, para Cresol, meta Cyclohexanol Cyclohexane Dibromomethane Dichloroethane Dichloromethane Diethyl ketone Diethyl oxalate Diethylene glycol Diphenyl Dipropyl ether Dipropyl oxalate Ethyl acetate Ethyl acrylate Ethyl alcohol, 100% Ethyl alcohol, 95% Ethyl alcohol, 40% Ethyl benzene Ethyl bromide 2-Ethyl butyl acrylate Ethyl chloride Ethyl ether Ethyl formate 2-Ethyl hexyl acrylate Ethyl iodide Ethyl propionate Ethyl propyl ether Ethyl sulfide Ethylene bromide Ethylene chloride Ethylene glycol Ethylidene chloride Fluorobenzene Formic acid Freon-11 Freon-12 Freon-21 Freon-22

15.2 12.1 9.5 12.7 14.5 7.9 14.4 12.3 10.2 14.4 14.0 12.6 10.1 11.8 7.5 8.1 12.3 13.9 12.5 6.6 10.2 14.2 20.0 12.3 11.5 8.6 12.1 11.6 16.1 12.7 12.3 14.4 11.2 13.0 13.3 13.3 2.5 2.9 9.8 12.7 13.2 14.6 13.5 11.0 5.0 12.0 13.2 10.3 13.7 12.7 10.5 9.8 6.5 13.2 14.5 11.2 14.8 14.5 14.2 9.0 14.7 13.2 14.0 13.8 11.9 12.7 6.0 14.1 13.7 10.7 14.4 16.8 15.7 17.2

4.8 14.2 17.0 12.8 7.2 15.0 7.4 13.9 14.3 9.6 11.7 2.0 13.9 12.5 18.4 18.7 13.5 14.5 10.9 15.9 16.6 13.2 15.9 11.0 12.6 17.2 15.3 0.3 7.5 13.1 12.4 10.2 18.1 13.3 12.5 12.5 20.8 24.3 12.9 15.8 12.2 8.9 9.2 16.4 24.7 18.3 8.6 17.7 9.1 10.4 13.8 14.3 16.6 11.5 8.1 14.0 6.0 5.3 8.4 15.0 10.3 9.9 7.0 8.9 15.7 12.2 23.6 8.7 10.4 15.8 9.0 5.6 7.5 4.7

Freon-113 Glycerol, 100% Glycerol, 50% Heptane Hexane Hydrochloric acid, 31.5% Iodobenzene Isobutyl alcohol Isobutyric acid Isopropyl alcohol Isopropyl bromide Isopropyl chloride Isopropyl iodide Kerosene Linseed oil, raw Mercury Methanol, 100% Methanol, 90% Methanol, 40% Methyl acetate Methyl acrylate Methyl i-butyrate Methyl n-butyrate Methyl chloride Methyl ethyl ketone Methyl formate Methyl iodide Methyl propionate Methyl propyl ketone Methyl sulfide Napthalene Nitric acid, 95% Nitric acid, 60% Nitrobenzene Nitrogen dioxide Nitrotoluene Octane Octyl alcohol Pentachloroethane Pentane Phenol Phosphorus tribromide Phosphorus trichloride Propionic acid Propyl acetate Propyl alcohol Propyl bromide Propyl chloride Propyl formate Propyl iodide Sodium Sodium hydroxide, 50% Stannic chloride Succinonitrile Sulfur dioxide Sulfuric acid, 110% Sulfuric acid, 100% Sulfuric acid, 98% Sulfuric acid, 60% Sulfuryl chloride Tetrachloroethane Thiophene Titanium tetrachloride Toluene Trichloroethylene Triethylene glycol Turpentine Vinyl acetate Vinyl toluene Water Xylene, ortho Xylene, meta Xylene, para

12.5 2.0 6.9 14.1 14.7 13.0 12.8 7.1 12.2 8.2 14.1 13.9 13.7 10.2 7.5 18.4 12.4 12.3 7.8 14.2 13.0 12.3 13.2 15.0 13.9 14.2 14.3 13.5 14.3 15.3 7.9 12.8 10.8 10.6 12.9 11.0 13.7 6.6 10.9 14.9 6.9 13.8 16.2 12.8 13.1 9.1 14.5 14.4 13.1 14.1 16.4 3.2 13.5 10.1 15.2 7.2 8.0 7.0 10.2 15.2 11.9 13.2 14.4 13.7 14.8 4.7 11.5 14.0 13.4 10.2 13.5 13.9 13.9

11.4 30.0 19.6 8.4 7.0 16.6 15.9 18.0 14.4 16.0 9.2 7.1 11.2 16.9 27.2 16.4 10.5 11.8 15.5 8.2 9.5 9.7 10.3 3.8 8.6 7.5 9.3 9.0 9.5 6.4 18.1 13.8 17.0 16.2 8.6 17.0 10.0 21.1 17.3 5.2 20.8 16.7 10.9 13.8 10.3 16.5 9.6 7.5 9.7 11.6 13.9 25.8 12.8 20.8 7.1 27.4 25.1 24.8 21.3 12.4 15.7 11.0 12.3 10.4 10.5 24.8 14.9 8.8 12.0 13.0 12.1 10.6 10.9

TRANSPORT PROPERTIES

Nomograph for viscosities of liquids at 1 atm. For coordinates see Table 2-365. To convert centipoises to pascalseconds, multiply by 0.001.

FIG. 2-33

2-323

2-324

PHYSICAL AND CHEMICAL DATA

TABLE 2-367

TABLE 2-366

Viscosity of Sucrose Solutions* Viscosity in centipoises

Percentage sucrose by weight

Temp., °C

20

40

0 5 10 15 20 25 30 35 40 45

3.818 3.166 2.662 2.275 1.967 1.710 1.510 1.336 1.197 1.074

14.82 11.60 9.830 7.496 6.223 5.206 4.398 3.776 3.261 2.858

60

Temp., °C

113.9 74.9 56.7 44.02 34.01 26.62 21.30 17.24

50 55 60 65 70 75 80 85 90 95

Percentage sucrose by weight 20

40

60

0.974 0.887 0.811 0.745 0.688 0.637 0.592 0.552

2.506 2.227 1.989 1.785 1.614 1.467 1.339 1.226 1.127 1.041

14.06 11.71 9.87 8.37 7.18 6.22 5.42 4.75 4.17 3.73

*International Critical Tables, vol. 5, p. 23. Bingham and Jackson, Bur. Standards Bull. 14 (1919): 59.

FIG. 2-34 and TABLE 2-367

Nomograph (right) for thermal conductivity of organic liquids.

TRANSPORT PROPERTIES TABLE 2-368

2-325

Prandtl Number of Air* Pressure, bar

Temperature, K

1

5

10

20

30

40

50

60

70

80

90

100

80 90 100 120 140

mix 0.796 0.786 0.773 0.763

2.31 1.76 0.872 0.813 0.782

2.32 1.77 1.54 0.89 0.82

2.35 1.78 1.53 1.44 0.94

2.37 1.79 1.53 1.65 1.20

2.40 1.81 1.53 1.54 1.59

2.42 1.82 1.53 1.48 2.14

2.45 1.83 1.53 1.43 2.43

2.48 1.85 1.53 1.40 2.07

2.51 1.87 1.54 1.38 1.78

2.54 1.89 1.54 1.36 1.62

2.57 1.91 1.55 1.34 1.52

160 180 200 240 280

0.754 0.745 0.738 0.724 0.710

0.765 0.754 0.743 0.727 0.711

0.78 0.763 0.749 0.729 0.713

0.84 0.792 0.766 0.737 0.717

0.92 0.830 0.788 0.746 0.721

1.03 0.876 0.812 0.756 0.726

1.13 0.932 0.841 0.767 0.731

1.25 1.00 0.87 0.78 0.737

1.37 1.07 0.90 0.80 0.742

1.65 1.14 0.95 0.81 0.75

1.83 1.20 0.97 0.81 0.75

1.72 1.25 1.00 0.82 0.76

300 350 400 450 500

0.705 0.699 0.694 0.691 0.689

0.707 0.699 0.694 0.691 0.689

0.708 0.699 0.694 0.691 0.689

0.712 0.701 0.695 0.691 0.689

0.715 0.703 0.696 0.692 0.689

0.717 0.705 0.697 0.692 0.690

0.721 0.707 0.698 0.693 0.690

0.725 0.709 0.699 0.693 0.690

0.728 0.711 0.700 0.694 0.690

0.732 0.712 0.701 0.695 0.691

0.737 0.714 0.703 0.695 0.691

0.742 0.716 0.704 0.696 0.691

600 700 800 900 1000

0.690 0.696 0.705 0.709 0.711

0.690 0.696 0.704 0.709 0.711

0.690 0.695 0.704 0.708 0.711

0.689 0.695 0.704 0.708 0.711

0.689 0.695 0.704 0.708 0.711

0.689 0.695 0.703 0.708 0.710

0.689 0.695 0.703 0.708 0.710

0.689 0.695 0.703 0.708 0.710

0.689 0.695 0.703 0.708 0.710

0.690 0.695 0.702 0.708 0.709

0.690 0.695 0.702 0.708 0.709

0.690 0.695 0.702 0.708 0.709

*Compiled by P. E. Liley from tables of specific heat at constant pressure, thermal conductivity, and viscosity given in SI units for integral kelvin temperatures and pressures in bars by Vasserman. Thermophysical Properties of Air and Its Components and Thermophysical Properties of Liquid Air and Its Components. Nauka, Moscow, and in translated form by the National Bureau of Standards, Washington. The number of significant figures given above reflects the similar numbers appearing for the constituent properties in the source references. While reasonable agreement occurs for atmospheric pressure with some other works, the fragmentary data available for the saturated, etc., states show large deviations.

TABLE 2-369

Prandtl Number of Liquid Refrigerants* Temperature, K Refrigerant

Trichlorofluoromethane Dichlorodifluoromethane Chlorotrifluoromethane Bromotrifluoromethane Dichlorofluoromethane Chlorodifluoromethane Methyl chloride Trichlorotrifluoroethane Dichlorotetrafluoroethane Chloropentafluoroethane Ethane Propane Octafluorocyclobutane Dichlorodifluoromethane/difluoroethane Chlorodifluoromethane/chloropentafluoroethane Trifluoromethane/chlorotrifluoromethane Methylene fluoride/chloropentafluoroethane Butane Isobutane (2-methyl propane) Ammonia Water Ethylene Propylene

No. 11 12 13 13B1 21 22 40 113 114 115 170 290 C318 500 502 503 504 600 600a 717 718 1150 1270

180

200

220

240

260

280

300

11.9 5.25 2.96 3.75

8.64 4.27 2.67 3.27 5.72

6.73 3.65 2.69 2.94 4.50

5.33 3.27 3.05 2.83 3.87

4.74 3.08 3.57 3.03 3.48

4.18 3.04 3.61 3.25

2.93 2.42

2.79 2.40

2.77 2.45

2.87 2.60

— 15.13 7.85

3.23 2.53 — 11.18 6.16

8.59 5.21

6.94 4.67

5.77 4.40

5.06 4.46

2.55 5.28 —

2.29 4.46 — 5.78 5.73

2.22 3.88 — 4.23 4.71

2.40 3.44 11.2 3.40 4.13

2.70 3.16 8.74 3.13 3.81

—

—

3.02 7.35 3.01

3.16 6.37 3.13

5.87 3.35

5.96 3.72

2.10

2.09 4.90 6.19 8.26

2.24 3.60 5.20 6.36

2.43 3.04 4.44 5.18 1.97

2.89 2.79 3.83 4.49 1.76

2.69 3.44 3.93 1.54

— 2.85 3.22 3.66 1.40

— 3.30 3.07 3.53 1.29

— — 3.02 3.53 1.24

— 1.74 2.24

— 1.78 1.88

— 2.07 1.71

— 2.70 1.71

10.3 4.4 1.88

5.69 — 2.24

3.65 — 3.91

2.60 — 4.73

7.00 4.80 4.68 — 25.7 —

8.35

3.76

— — 1.85 3.80

320

340

360

380

3.19 — 4.52 3.16

3.44 —

4.00 — —

— —

3.18 2.85 7.04 4.78 4.90

3.54

3.17

6.23 4.82

— 5.61

5.18

—

—

—

—

—

— — —

— —

— —

3.77 1.25

4.68 1.34

1.99 —

1.59 — —

*Dashes indicate inaccessible states. Average uncertainty is about 20 percent. Values derived from formulations for thermal conductivity, specific heat at constant pressure, and viscosity contained in Thermophysical Properties of Refrigerants. American Society of Heating, Refrigerating and Air-Conditioning Engineers, New York, 1976. For further details see M. W. Johnson, M.S.M.E. thesis, Purdue University, West Lafayette, Ind., 1976.

2-326

PHYSICAL AND CHEMICAL DATA

TABLE 2-370

Thermophysical Properties of Miscellaneous Saturated Liquids Temperature, °C

Substance

Property

Acetaldehyde ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr Acetic acid

−50

−40

−30

−20

−10

0

863 2.05 460 0.211 4.47

852 2.08 404 0.206 4.08

840 2.11 358 0.200 3.78

828 2.14 321 0.195 3.52

816 2.17 290 0.189 3.33

804 2.20 263 0.184 3.14

10

20

794 2.24 241 0.182 2.97

783 2.28 222 0.180 2.81

ρ (kg/m3) cp (kJ/ kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr

1049 2.031 1210 0.173 14.2

30

40

50

60

1039

1028

1018

1006

1102 0.170

1010 0.168

795 0.167

600 0.165

70

80

90

100

995

984

972

960

0.163

0.161

Aniline

ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr

— — — — —

— — — — —

— — — — —

— — — — —

— — — — —

1039 2.024 10200 0.186 111

1030 2.047 6500 0.184 72

1022 2.071 4400 0.182 50

1013 2.093 3160 0.180 36.7

1005 2.113 2370 0.177 28.3

996 2.132 1850 0.174 22.7

987 2.17 1510 0.171 19.2

978 2.20 1270 0.169 16.5

969 2.23 1090 0.168 14.5

960 2.27 935 0.167 12.7

951 2.32 825 0.167 11.5

Butanol

ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr

845 1.947 34700 0.175 3860

841 1.996 22400 0.174 2570

837 2.046 14700 0.173 1740

833 2.100 10300 0.172 1260

829 2.153 7400 0.171 930

825 2.202 5190 0.170 670

817 2.262 3870 0.168 120

810 2.345 2950 0.167 41

803 2.437 2300 0.166 33.8

797 2.524 1780 0.165 27.2

791 2.621 1410 0.164 22.5

784

776

768

760

753

1140 0.163

930 0.162

760 630 535 0.161 0.160 0.159

ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr

1362 0.988 630 0.194 3.21

1348 0.989 580 0.190 3.02

1334 0.990 535 0.186 2.85

1320 0.991 496 0.182 2.70

1306 0.993 463 0.178 2.58

1292 0.996 435 0.174 2.49

1278 1.004 405 0.170 2.39

1263 1.017 375 0.166 2.30

350 0.161

330 0.158

0.156

0.154

0.152

0.150

ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr

— — — — —

— — — — —

— — — — —

— — — — —

— — — — —

— — — — —

789 2.068 1175 0.122 19.9

779 2.081 980 0.120 17.0

769 2.094 820 0.119 14.4

759 2.106 710 0.118 12.7

750 2.119 605 0.117 11.0

740

731

721

540 0.116

0.114

0.112

ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr

2.01 6400 0.188 68.4

2.04 4790 0.186 52.5

2.08 3650 0.184 41.3

2.13 2825 0.181 33.2

2.19 2220 0.179 27.2

806 2.27 1770 0.177 22.7

798 2.35 1470 0.175 19.7

789 2.43 1200 0.173 16.9

781 2.52 1000 0.171 14.7

776 2.62 835 0.168 13.0

763 2.73 700 0.165 11.6

754 2.83 590 0.162 10.3

745 2.93 500 0.159 9.2

735 3.03 435 0.156 8.4

725 3.19 370 0.153 7.7

716 3.30 314 0.151 6.9

947

935

924

912

888

876

863

851

838

825

811

797

580

510

901 2.01 455 0.145 6.3

400 0.142

370 0.139

345 0.136

310 0.133

280 0.130

250 230 220 0.127 0.123 0.119

Carbon disulfide

Cyclohexane

Ethanol

Ethyl acetate

ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) 1090 k (W/m⋅K) Pr

Ethylamine

ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr

761 2.95 580 0.204 8.39

750 2.97 500 0.201 7.39

739 2.98 435 0.199 6.51

729 3.00 390 0.196 5.97

718 3.01 350 0.194 5.43

707 3.03 320 0.191 5.08

695

683

671

658

646

633

620

607

Ethyl ether

ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr

790 2.135 550 0.159 7.39

780 2.156 470 0.155 6.54

769 2.179 410 0.151 5.92

758 2.205 365 0.147 5.48

747 2.233 330 0.144 5.12

736 2.265 290 0.140 4.69

725 2.299 265 0.139 4.38

714 2.332 233 0.134 4.05

702 2.36 214 0.129 3.92

689 2.39 197 0.125 3.77

676 2.43 181 0.120 3.67

666 2.47 166 0.116 3.54

653 2.51 153 0.112 3.43

640

625

611

140

129

118

Ethyl iodide

ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr

0.656 0.663 0.670 0.677 730 0.092 5.37

0.684 655 0.090 4.98

0.691 590 0.088 4.63

0.698 539 0.086 4.30

0.705 495 0.085 4.11

0.712 455 0.083 3.90

0.718 420 0.081 3.72

0.724 390 0.080 3.53

Ethylene glycol

ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr

1127 2.272 57000 0.254 510

1120 2.327 33300 0.255 305

1113 2.381 20200 0.256 190

1106 2.431 13400 0.258 126

1099 2.484 9100 0.259 87.3

1092 2.536 7070 0.260 69.0

1085 2.586 4000

1077 2.636 3450

1070 1063 1056 2.685 2.734 2.779 3000 2440 2000

Formic acid

ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr

1241

1231

1220

1209

1196

1184

1170

1156

1140

0.265

2260 0.261

1800 0.257

1470 0.257

1220 0.253

1030 0.250

890 0.246

780 0.243

680 615 550 0.240 0.236 0.232

1124

1108

TRANSPORT PROPERTIES TABLE 2-370

2-327

Thermophysical Properties of Miscellaneous Saturated Liquids (Concluded ) Temperature, °C

Substance Gasoline

Glycerol

Kerosine

Property

−50

ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) 1710 k (W/m⋅K) 0.131 Pr ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr

—

−40

1400 0.128 —

−30

−20

784 1.88 1170 990 0.125 0.123 15.1 —

—

−10

0

775 1.92 850 0.121 13.5 —

10

20

30

40

50

60

70

80

90

100

767 1.97 735 0.120 12.1

759 2.02 645 0.118 11.0

751 2.06 530 0.116 9.41

743 2.11 464 0.114 8.59

735 2.15 410 0.112 7.87

721 2.20 367 0.110 7.34

717 2.25 330 0.108 6.88

708 2.30 298 0.106 6.47

699 2.35 270 0.104 6.10

690 2.41 246 0.102 5.81

681 2.46 225 0.100 5.54

1276

1270

1260 2.393 1.5.+6 0.284 12650

1254 2.406

1248 2.457

1242 2.504

2.548

2.588

2.625 2.657 2.686

0.285

0.287

0.288

0.289

0.291

0.293 0.294 0.295

2.28 73

2.32 66

2.35 60

2.38 55

1.2.+7 4.0.+6

ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) 1150 k (W/m⋅K) Pr

725

500

360

275

781 1.91 215 0.140 2.93

774 1.96 173 0.139 2.44

767 2.02 149 0.139 2.17

760 2.07 126 0.138 1.89

754 2.13 108 0.138 1.67

748 2.18 95 0.137 1.51

742 2.23 83 0.137 1.35

ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) k(W/m⋅K) Pr

2.30 2305 0.225 23.6

2.32 1800 0.222 18.8

2.35 1410 0.219 15.1

2.37 1170 0.216 12.9

2.40 975 0.212 11.0

2.42 820 0.209 9.53

2.45 692 0.206 8.23

2.47 590 0.203 7.18

783 2.49 510 0.199 6.38

774 2.52 455 0.195 5.88

766 2.55 400 0.192 5.31

756 2.65 355 0.189 4.98

746 2.78 315 0.187 4.68

736 2.94 271 0.184 4.34

725 3.13 240 0.182 4.13

711 3.30 218 0.180 3.99

Methyl formate

ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr

1069 1.84 830 0.217 7.04

1056 1.86 711 0.213 6.21

1043 1.88 618 0.209 5.56

1030 1.90 544 0.205 5.04

1017 1.92 481 0.200 4.62

1003 1.95 430 0.195 4.30

989 1.99 380 0.191 3.96

975 2.03 345 0.186 3.77

960 2.08 315 0.180 3.64

944

929

913

897

880

863

845

Oil, castor

ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr

Methanol

Oil, olive

Pentane

Propanol

Sulfuric acid

2,420,000 986,000 451,000 231,000 125,000 74,000 43,000 0.182 0.181 0.180 0.179 0.178 0.177 0.176 0.175 0.174 0.17

ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr

138,000 0.170

914 1.633 84,000 0.169 810

52,000 0.168

36,300 0.167

24,500 0.166

17,000 12,400 0.166 0.165 0.165 0.164 0.164

616

606

596

585

574

562

209 0.115

190 0.112

175 0.108

161 0.105

148 0.101

137 124 113 0.098 0.095 0.091

ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr

693 2.060 489 0.142 7.14

665 2.137 339 0.132 5.49

656 2.167 307 0.128 5.20

646 2.206 279 0.125 4.92

636 2.239 254 0.122 4.66

ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr

849 1.955 20,200 13,500 9500 6900 0.167 0.166 0.165 236

819 2.219 3900

811

814

796

788

779

770

761

752

5110

2900

2245

1720 0.171

1400 0.169

1130 0.168

921 0.167

760 0.165

630 508 447 0.164 0.163 0.162

1834 1.382 25,400

15,700

11,500

8820

7220

6090

5190

829 1.80 380 0.124 5.5

820 1.83 355 0.122 5.3

810 1.87 325 0.119 5.1

820

730

675

684 2.084 428 0.139 6.42

674 2.110 379 0.136 5.88

ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr

Toluene

ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr

Turpentine

ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr

48,400 35,200 0.314 932 1.514 2120 0.152 21.1

923 1.535 1670 0.149 17.8

913 1.556 1345 0.147 14.2

904 1.579 1100 0.144 12.1

895 1.602 915 0.142 10.3

626 2.273 234 0.119 4.47

886 1.633 770 0.139 9.0

876 1.652 670 0.137 8.1

867 1.675 590 0.134 7.4

858 1.701 520 0.132 6.7

848 1.73 470 0.129 6.3

839 1.76 420 0.126 5.9

1.72 2250 0.130 29.8

1.76 1780 0.129 24.3

1.80 1490 0.128 20.9

1270 0.127 18.4

1070 0.126 16.1

1.93 925 0.125 14.3

550

747

800 1.92 295 0.117 4.8

538

743

790 1.97 270 0.114 4.7

2-328

PHYSICAL AND CHEMICAL DATA

TABLE 2-371

Diffusivities of Pairs of Gases and Vapors (1 atm) Dv in cm2/s

Substance Acetic acid Acetone n-Amyl alcohol sec-Amyl alcohol Amyl butyrate Amyl formate i-Amyl formate Amyl isobutyrate Amyl propionate Aniline Anthracene Argon Benzene Benzidine Benzyl chloride n-Butyl acetate i-Butyl acetate n-Butyl alcohol i-Butyl alcohol Butyl amine i-Butyl amine i-Butyl butyrate i-Butyl formate i-Butyl isobutyrate i-Butyl proprionate i-Butyl valerate Butyric acid i-Butyric acid Cadmium Caproic acid i-Caproic acid Carbon dioxide

Carbon disulfide Carbon monoxide Carbon tetrachloride Chlorobenzene Chloroform Chloropicrin m-Chlorotoluene o-Chlorotoluene p-Chlorotoluene Cyanogen chloride Cyclohexane n-Decane Diethylamine 2,3-Dimethyl butane Diphenyl n-Dodecane Ethane Ethanol Ether (diethyl) Ethyl acetate Ethyl alcohol Ethyl benzene Ethyl n-butyrate Ethyl i-butyrate Ethylene Ethyl formate Ethyl propionate Ethyl valerate Eugenol Formic acid Helium n-Heptane n-Hexane Hexyl alcohol Hydrogen

Temp., °C 0 0 0 30 0 0 0 0 0 0 30 0 20 0 0 0 0 0 0 30 0 0 0 0 0 0 0 0 0 0 0 0 0 0 20 25 500‡ 0 0 450‡ 0 30 0 25 0 0 0 0 15 45 90 0 15 0 126 0 0 0 0 30 0 0 0 0 0 0 0 0 0 0 0 20 38 15 0 0 25 500

Air

A

H2

0.1064 .109 .0589 .072 .040 .0543 .058 .0419 .046 .0610 .075 .0421

0.416 .361 .235

.077 .0298 .066 .058 .0612 .0703 .088 .0727 .0821 .0853 .0468 .0705 .0457 .0529 .0424 .067 .0679

.306

O2

N2

CO2

N2O

CH4

C2H6

C2H4

n-C4H10

i-C4H10

0.0716 .0422

.171 .1914

.0347

0.194 0.0797

.0528

.2364 .2716

.0425 .0476

.2771

.0483

.185

.0327

.191 .203 .173 .264 .271

.0364 .0366 .0308 .0476 .0471 .17

.050 .0513 .138

.550

.139

0.096

0.153

.163 .0996*

.00215†

.9 .0892

.369 .651 .293

.185 1.0 0.0636

.319

.0744

.063 .137

0.116

.075 .091 .088 .054 .059 .051 .111 0.0719

.0760

.086 .306

.0841

.0884 .0657

.301

.0753

.0751

.0610 .308 .459 .377 .298 .273

.0778 .0715 .089 .102 .0658 .0579 .0591 .0840 .068 .0512 .0377 .1308

.0813 .0686 .0546 .0487

.375

.0685

.224 .229 .486 .337 .236 .205

.0407 .0413 .0573 .0450 .0367

.510

Ref. 8 6, 16 8 5 8 8 8 8 8 8 5 8 18 8, 15 8 8 8 8 8 5 8 8 8 8 8 8 8 8 8 8 13 8 8 8 19 1, 9 18 8 8 18 16, 17 5 6 10 8 8 8 10 3 6 3 8 3 8 3 8 20 7, 8 8 5 8 8 8 8 8 8 4, 8 8 8 8 8 19

.0874

.641 .705 .066§ .0663 .0499 .611

.290 .200

.0753 .697 4.2

.0757 .674

.0351 .550 .646

.535

.625

0.459 .537

0.486 .726

0.272

0.277

3 8 8 2 18

TRANSPORT PROPERTIES TABLE 2-371

2-329

Diffusivities of Pairs of Gases and Vapors (1 atm) (Concluded ) Dv in cm2/s

Substance

Temp., °C

Air

Hydrogen cyanide Hydrogen peroxide Iodine Mercury Mesitylene Methane Methyl acetate Methyl alcohol Methyl butyrate Methyl i-butyrate Methyl cyclopentane Methyl formate Methyl propionate Methyl valerate Naphthalene Nitrogen

0 60 0 0 0 500 0 0 0 0 15 0 0 0 0 0 25 0 0 30 0 0 0 0 0 0 30 0 0 0 0 0 0 0 0 0 0 0 0 0 0 25 0 30 0

.173 .188 .07 .112 .056

Nitrous oxide n-Octane Oxygen Phosgene Propionic acid Propyl acetate n-Propyl alcohol i-Propyl alcohol n-Propyl benzene i-Propyl benzene n-Propyl bromide i-Propyl bromide Propyl butyrate Propyl formate n-Propyl iodide i-Propyl iodide n-Propyl isobutyrate i-Propyl isobutyrate Propyl propionate Propyl valerate Safrol i-Safrol Sulfur hexafluoride Toluene Trimethyl carbinol 2,2,4-Trimethyl pentane 2,2,3-Trimethyl heptane n-Valeric acid i-Valeric acid Water

H2

O2

N2

CO2

N2O

CH4

C2H6

C2H4

n-C4H10

i-C4H10

.070 .13

.53 1.1

.084 .132 .0633 .0639 .0731 .0872 .0735 0.0569 .0513

.333 .506 .242 .257 .318

.0567 .0879 .0446 .0451 0.0742

.0758

.295

.0528 0.181 0.165 .096

0.535 .0505 0.0642 .178 .095 .0829 .067 .085 .0818 .101 .0481 .0489 .085 .0902 .0530 .0712 .079 .0802 .0549 .059 .057 .0466 .0434 .0455

.271 .697

.0705

0.0710 .181

.139

.330

.0588

.315

.0577

.206 .281

.0364 .0490

.212

.0388

.212 .189

.0395 .0341

.418 .076 .088 .087

30 90 0 0 0 450

A

.071

.0618

.288

.0688

.270 .050 .0544 .220

.212 .75

0.148

0.163

0.0960

0.0908

Ref. 10 11 8, 12, 14 8, 12, 13 8 18 8 8 8 8 3 8 8 8 8 8 2 8 8 3 8 10 8 8 8 8 5 8 8 8 8 8 8 8 8 8 8 8 8 8 8 2 4, 8 5 8

.0705

3

.0684

3 8 8 8, 20 18

.0376 .138 1.3

* 320 mm Hg. † 40 atm. ‡ Also at other temperatures. § Strong function of concentration. References 1 Amdur, Irvine, Mason, and Ross, J. Chem. Phys., 20, 436 (1952). 2 Boyd, Stein, Steingrimsson, and Rumpel, J. Chem. Phys., 19, 548 (1951). 3 Cummings and Ubbelohde, J. Chem. Soc. (London), 1953, p. 3751. 4 Fairbanks and Wilke, Ind. Eng. Chem., 42, 471 (1950). 5 Gilliland, Ind. Eng. Chem., 26, 681 (1934). 6 Gorynnova and Kuvskinskii, Zhur. Tekh. Fiz., 18, 1421 (1948). 7 Hansen, Dissertation, Jena, 1907. 8 “International Critical Tables,” vol. 5, p. 62. 9 Jeffries and Drickamer, J. Chem. Phys., 22, 436 (1954). 10 Klotz and Miller, J. Am. Chem. Soc., 69, 2557 (1947). 11 McMurtrie and Keyes, J. Am. Chem. Soc., 70, 3755 (1948). 12 Mullaly and Jacques, Phil. Mag., 48, 6, 1105 (1924). 13 Spier, Physica, 6 (1939): 453; 7, 381 (1940). 14 Topley and Whytlaw-Gray, Phil. Mag., 4, 873 (1927). 15 Trautz and Ludwig, Ann. Physik, 5, 5, 887 (1930). 16 Trautz and Muller, Ann. Physik, 22, 353 (1935). 17 Trautz and Ries, Ann. Physik, 8, 163 (1931). 18 Walker and Westenberg, J. Chem. Phys., 32, 136 (1960). 19 Westenberg and Walker, J. Chem. Phys., 26, 1753 (1957). 20 Winkelmann, Wied. Ann., 22, 152 (1884); 23, 203 (1884); 26, 105 (1885); 33, 445 (1888); 36, 92 (1889).

2-330

PHYSICAL AND CHEMICAL DATA

In this table are a representative selection of diffusion coefficients. The subsection “Prediction and Correlation of Physical Properties” should be consulted for estimation techniques. As general references, the works by Hirschfelder, Curtiss, and Bird, Molecular Theory of Gases and Liquids, Wiley, New York, 1964; Chapman and Cowling, The Mathematical Theory of Non-Uniform Gases, Cambridge, New York, 1970; Reid and Sherwood, The Properties of Gases and Liquids, TABLE 2-372

McGraw-Hill, New York, 1964; and Bretsznajder, Prediction of Transport and Other Physical Properties of Fluids, Pergamon, New York, 1971, may be found useful. The most exhaustive recent compilation for gases is by Mason and Marrero, J. Phys. Chem. Ref. Data, 1 (1972). Unfortunately, the Mason and Marrero work cites only equations and equation constants and not direct tabulations. For these, the LandoltBörnstein series is suggested.

Diffusivities in Liquids (25°C)

Dilute solutions and 1 atm unless otherwise noted; use DLµ/T = constant to estimate effect of temperature; * indicates that reference gives effect of concentration.

Solute

Solvent

DL × 105, sq cm/sec

Acetal* Acetamide* Acetamide* Acetic acid Acetic acid Acetic acid Acetic acid Acetic acid Acetic acid* Acetonitrile Acetylene Allyl alcohol* Allyl alcohol Ammonia* i-Amyl alcohol* i-Amyl alcohol Benzene Benzene (50 mole %) Benzene (50 mole %) Benzene (50 mole %) Benzene (50 mole %) Benzene (50 mole %) Benzene (50 mole %) Benzoic acid Benzoic acid Benzoic acid Benzoic acid Benzoic acid Bromine Bromine Bromine Bromobenzene Bromoform* Bromoform Bromoform Bromoform* Bromoform Bromoform n-Butanol Caffeine Carbon dioxide Carbon dioxide Carbon disulfide (50 mole %, 200 atm.) Carbon disulfide (50 mole %, 200 atm.) Carbon disulfide (50 mole %, 218 atm.) Carbon disulfide (50 mole %, 200 atm.) Carbon disulfide (50 mole %, 100 atm.) Carbon disulfide (50 mole %, 50 atm.) Carbon disulfide (50 mole %, 200 atm.) Carbon disulfide (50 mole %) Carbon tetrachloride Carbon tetrachloride* Carbon tetrachloride Carbon tetrachloride Carbon tetrachloride* Carbon tetrachloride Carbon tetrachloride Carbon tetrachloride Carbon tetrachloride Carbon tetrachloride Chloral* Chloral hydrate

Ethanol Ethanol Water Acetone Benzene Carbon tetrachloride Ethylene glycol Toluene Water Water Water Ethanol Water Water Ethanol Water Carbon tetrachloride n-Decane 2,4-Dimethyl pentane n-Dodecane n-Heptane n-Hexadecane n-Octadecane Acetone Benzene Carbon tetrachloride Ethylene glycol Toluene Benzene Carbon disulfide Water Benzene Acetone i-Amyl alcohol Ethanol Ethyl ether Methanol n-Propanol Water Water Ethanol Water n-Butanol i-Butanol Chlorobenzene 2,4-Dimethyl pentane n-Heptane Methyl cyclohexane n-Octane Toluene Benzene Cyclohexane Decalin Dioxane Ethanol n-Heptane Kerosene Methanol i-Octane Tetralin Ethanol Water

1.25 0.68 1.19 3.31 2.11 1.49 0.13 2.26 1.24 1.66 1.78, 2.11 1.06 1.19 1.7, 2.0, 2.3 0.87 1.0 1.53 1.72 2.49 1.40 2.47 0.96 0.86 2.62 1.38 0.91 0.043 1.49 2.7 4.1 1.3 2.30 2.90 0.53 1.08 3.62 2.20 0.94 0.96 0.63 4.0 1.96 3.57 2.42 3.00 3.63 3.0 3.5 3.10 2.06 2.04 1.49 0.776 1.02 1.50 3.17 0.961 2.30 2.57 0.735 0.68 0.77

Estimated possible, error,  %1 5 5 3

3 5 5 6 5 8

5

5 6 6 1

3 2 2 2 2 2 2 2 2 2 5 7

Ref. 11 11 11 4 1, 4 4 4 4 11 11 1, 24 11 11 1, 11 11 11, 25 7 26 26 26 26 26 26 4 4 4 4 4 11 11 11 25 11 11 11 11 23 11 1, 11, 18, 25 11 11 1, 3, 5, 20, 24, 28 14 14 14 14 14 14 14 14 7, 9 9, 10* 9 9 9, 10* 9 9 9 9 9 11 11

TRANSPORT PROPERTIES TABLE 2-372

Diffusivities in Liquids (25°C) (Continued )

Dilute solutions and 1 atm unless otherwise noted; use DL µ /T = constant to estimate effect of temperature; * indicates that reference gives effect of concentration.

Solute Chlorine Chlorobenzene Chloroform Chloroform Cinnamic acid Cinnamic acid Cinnamic acid Cinnamic acid 1,1′-Dichloropropanol Dicyanodiamide* Diethyl ether Diethyl ether 2,4-Dimethyl pentane (50 mole %) 2,4-Dimethyl pentane (50 mole %) Ethanol* Ethyl acetate Ethylene dichloride Formic acid Formic acid Formic acid Formic acid Formic acid Formic acid Glucose Glycerol Glycerol Glycerol* n-Heptane (50 mole %) n-Heptane (50 mole %) n-Heptane (50 mole %) n-Heptane (50 mole %) Hexamethylene tetramine Hydrogen chloride* Hydrogen Hydrogen sulfide Hydroquinone* Hydroquinone* Iodine Iodine Iodine Iodine Iodine Iodine Iodine Iodine Iodine Iodine* Iodine Iodine Iodine Iodine Iodine Iodine Iodine Iodine Iodine Iodine Iodine Iodine Iodine Iodobenzene Lactose* Maltose* Mannitol* Methanol Nicotine* Nitric acid* Nitrobenzene Nitrogen Nitrous oxide Oxalic acid*

Solvent

DL × 105, sq cm/sec

Water Benzene Benzene Ethanol Acetone Benzene Carbon tetrachloride Toluene Water Water Benzene Water n-Dodecane n-Hexadecane Water Ethyl benzoate Benzene Acetone Benzene Carbon tetrachloride Ethylene glycol Toluene Water Water i-Amyl alcohol Ethanol Water n-Dodecane n-Hexadecane n-Octadecane n-Tetradecane Water Water Water Water Ethanol Water Acetic acid Anisole Benzene Bromobenzene Carbon disulfide Carbon tetrachloride Chloroform Cyclohexane Dioxane Ethanol Ethyl acetate Ethyl ether Ethylene bromide n-Heptane n-Hexane Mesitylene Methanol Methyl cyclohexane n-Octane Tetrabromoethane n-Tetradecane Toluene m-Xylene Ethanol Water Water Water Water Water Water Carbon tetrachloride Water Water Water

1.44 2.66 2.50 1.38 2.41 1.12 0.76 2.41 1.0 1.18 2.73 0.85 1.44 0.88 1.28 0.94 2.8 3.77 2.28 1.89 0.094 2.65 1.37 0.69 0.12 0.56 0.94 1.58 1.00 0.92 1.29 0.67 3.10 5.85 (4.4) 1.61 0.53 0.88, 1.12 1.13 1.25 1.98 1.25 3.2 1.45 2.30 1.80 1.07 1.30 2.2 3.61 0.93 3.4, 2.5 4.15 1.49 1.74 2.1 2.76 2.0 0.96 2.1 1.82 1.09 0.49 0.48 0.65 1.6 0.60 2.98 1.00 1.9 1.8 1.61

Estimated possible, error,  %1 4 6 3

6 4

4

10 6 6

3 5

10 8 3

3 5 5 5 8 2

2

Ref. 1, 28 25 1, 25 11 4 4 4 4 11 11 25 2 26 26 1, 7, 9,* 11,* 22 6 1, 25 4 4 4 4 11 11 11 11 1, 11* 26 26 26 26 11 4, 11,* 12* 1, 11, 24(?) 1 11 2, 11* 11 11 9, 19, 23 4, 11, 19 11, 19, 23 9, 11, 19 11, 23 4 9 4, 11* 11, 19 11 11 9, 11, 19 4, 9 9 19 4 4 11 4 11 9, 11 11 11 11 11 1, 7, 11 11 11 7 1, 24 1, 11 11

2-331

2-332

PHYSICAL AND CHEMICAL DATA TABLE 2-372

Diffusivities in Liquids (25°C) (Concluded )

Dilute solutions and 1 atm unless otherwise noted; use DL µ /T = constant to estimate effect of temperature; * indicates that reference gives effect of concentration.

Solute Oxygen Oxygen Oxygen Pentaerythritol* Phenol Phenol Phenol Phenol Phenol Phenol n-Propanol Pyridine* Pyridine Pyrogallol Raffinose* Resorcinol* Resorcinol* Saccharose* Stearic acid* Succinic acid* Sucrose Sulfur dioxide Sulfuric acid* Tartaric acid* 1,1,2,2-Tetrabromoethane Toluene Toluene Toluene Toluene Toluene Urea Urea Urethane Water

Solvent Glycerol*-water (106 poise) Sucrose*-water (125 poise) Water Water i-Amyl alcohol Benzene Carbon disulfide Chloroform Ethanol Ethyl ether Water Ethanol Water Water Water Ethanol Water Water Ethanol Water Water Water Water Water 1,1,2,2-Tetrachloroethane n-Decane n-Dodecane n-Heptane n-Hexane n-Tetradecane Ethanol Water Water Glycerol

DL × 105, sq cm/sec

Estimated possible, error,  %1

Ref.

0.24

13

0.25

13

2.5 0.77 0.2 1.68 3.7 2.0 0.89 3.9 1.1 1.24 0.76 0.74 0.41 0.46 0.87 0.49 0.65 0.94 0.56 1.7 1.97 0.80 0.61 2.09 1.38 3.72 4.21 1.02 0.73 1.37 1.06 0.021

References 1 Arnold, J. Am. Chem. Soc., 52, 3937 (1930). 2 Calvet, J. Chim. Phys., 44, 47 (1947). 3 Carlson, J. Am. Chem. Soc., 33, 1027 (1911). 4 Chang and Wilke, J. Phys. Chem., 59, 592 (1955). 5 Davidson and Cullen, Trans. Inst. Chem. Eng., 35, 51 (1957). 6 Dummer, Z. Anorg. Chem., 109, 31 (1949). 7 Gerlach, Ann. Phys. (Leipzig), 10, 437 (1931). 8 Gosting and Akeley, J. Am. Chem. Soc., 74, 2058 (1952). 9 Hammond and Stokes, Trans. Faraday Soc., 49, 890 (1953); 49, 886 (1953). 10 Hammond and Stokes, Trans. Faraday Soc., 52, 781 (1956). 11 International Critical Tables, vol. 5, p. 63. 12 James, Hollingshead, and Gordon, J. Chem. Phys., 7, 89 (1939); 7, 836 (1939). 13 Jordon, Ackermann, and Berger, J. Am. Chem. Soc., 78, 2979 (1956). 14 Koeller and Drickamer, J. Chem. Phys., 21, 575 (1953). 15 Kolthoff and Miller, J. Am. Chem. Soc., 63, 1013 (1941).

20 4

3 7 7 4 5 4 4 5 6 3 10 4

2

1, 3, 15, 21, 24 11 11 1 11 11 11 11 1, 7, 11 11 11 11 11 11 11 11 11 11 2, 27 15, 17 11 11 11 4 4 4 4 4 11 8, 11 11, 25 16

TRANSPORT PROPERTIES TABLE 2-373

2-333

Thermal Conductivities of Some Building and Insulating Materials* k = Btu/(h⋅ft2)(°F/ft)

Material

Apparent density ρ, lb/ft3 at room temperature

Aerogel, silica, opacified

8.5

Asbestos-cement boards Asbestos sheets Asbestos slate

120 55.5 112 112 29.3 29.3 36 36 36 36 43.5 43.5 0.2

Asbestos

Aluminum foil (7 air spaces per 2.5 in.) Ashes, wood Asphalt Boiler scale (Note 1) Bricks: Alumina (92–99% Al2O3 by wt.) fused Alumina (64–65% Al2O3 by wt.) (See also Bricks, fire clay) Building brick work Carbon Chrome brick (32% Cr2O3 by wt.) Diatomaceous earth, natural, across strata (Note 2) Diatomaceous, natural, parallel to strata (Note 2) Diatomaceous earth, molded and fired (Note 2) Diatomaceous earth and clay, molded and fired (Note 2) Diatomaceous earth, high burn, large pores (Note 3)

132

Kaolin insulating firebrick (Note 4) Magnesite (86.8% MgO, 6.3% Fe2O3, 3% CaO, 2.6% SiO2 by wt.)

Silicon carbide brick, recrystallized (Note 3)

Calcium carbonate, natural White marble Chalk Calcium sulfate (4H2O), artificial plaster (artificial) (building) Cambric (varnished) Carbon, gas Carbon stock Cardboard, corrugated

k

120 290 20 51 0 60 −200 0 0 100 200 400 −200 0 38 177 0–100 20

0.013 .026 .43 .096 .087 .114 .043 .090 .087 .111 .120 .129 .090 .135 .025 .038 .041 .43

427 1315 800 1100 20

96.7 200 200 200

200 650 1315

1.8 2.7 0.62 .63 .4 3.0 .67 .85 1.0

27.7 27.7

204 871

0.051 .077

27.7 27.7 38 38

204 871 204 871

.081 .106 .14 .18

42.3 42.3

204 871

.14 .19

27 27 19 19

200 1000 200 600 1000 1400 500 1150 200 760

.13 .34 .58 .85 .95 1.02 0.15 .26 .050 .113

158 158 158

204 650 1200

2.2 1.6 1.1

129 129 129 129 129 162

600 800 1000 1200 1400 30

10.7 9.2 8.0 7.0 6.3 1.3 1.7 0.4 .22 .43 .25 .091 2.0 0.55 3.6 0.037

115 115

37 37

Fire clay (Missouri)

Kaolin insulating brick (Note 3)

t, °C

96 84.6 132 77.9 94

40 75 25 38 0–100 −184 0

Material Cotton wool Cork board Cork (regranulated) (ground) Diatomaceous earth powder, coarse (Note 2) fine (Note 2) molded pipe covering (Note 2) 4 vol. calcined earth and 1 vol. cement, poured and fired (Note 2) Dolomite Ebonite Enamel, silicate Felt, wool Fiber insulating board Fiber, red (with binder, baked) Gas carbon Glass Borosilicate type Window glass Soda glass Granite Graphite, longitudinal powdered, through 100 mesh Gypsum (molded and dry) Hair felt (perpendicular to fibers) Ice Infusorial earth, see diatomaceous earth Kapok Lampblack Lava Leather, sole Limestone (15.3 vol. % H2O) Linen Magnesia (powdered) Magnesia (light carbonate) Magnesium oxide (compressed) Marble Mica (perpendicular to planes) Mill shavings Mineral wool

Apparent density ρ, lb/ft3 at room temperature

t, °C

k

5 10 8.1 9.4 20.0 20.0 17.2 17.2 26.0 26.0

30 30 30 30 38 871 204 871 204 871

0.024 .025 .026 .025 .036 .082 .040 .074 .051 .088

61.8 61.8 167

204 871 50

38 20.6 14.8 80.5

139

30 78 17 57.5 0.88 10 62.4 103

20 40

0 21 21 20 40 21

0.020 .038 .49 .092 .54 .05 .35 0.034 .32 1.2–1.7 0.25 0.033–0.05 0.0225 .024 .075 .14 3.4 2.9 0.88 .17 .14 .075 .087 .109 0.075–0.092 0.19 1.06 0.03

38 24–127 30 94 0 100 21 21 30 30

.026 .096 .064 .022 .86 .27 0.09–0.097 0.16 .028 .04 .034

49.7 13 49.9

24 30 47 21 20

9.4 19.7

30 30

50

Paper Paraffin wax Petroleum coke Porcelain Portland cement, see concrete Pumice stone Pyroxylin plastics Rubber (hard) (para) (soft) Sand (dry) Sandstone Sawdust Scale (Note 1) Silk varnished Slag, blast furnace Slag wool Slate Snow Sulfur (monoclinic) (rhombic) Wall board, insulating type Wall board, stiff paste board Wood shavings

.16 .23 1.0 0.10 0.5–0.75 30 0.03 21 .028 20 .27 20–97 .097 0–100 2.0 0.2–0.73 30–75 0.63 0.3–0.61 0.3–0.44 1.0–2.3 20 95 40 0.104 20 .25 30 .021 0 1.3

0 100 500 200 90 21–66 74.8 94.6 140 12 6.3 12 34.7 14.8 43 8.8

2-334

PHYSICAL AND CHEMICAL DATA

TABLE 2-373

Thermal Conductivities of Some Building and Insulating Materials* (Concluded ) k = Btu/(h⋅ft2)(°F/ft) Apparent density ρ, lb/ft3 at room temperature

Material Celluloid Charcoal flakes

87.3 11.9 15

Clinker (granular) Coke, petroleum Coke, petroleum (20–100 mesh) Coke (powdered) Concrete (cinder) (stone) (1:4 dry)

62

t, °C

k

30 80 80 0–700 100 500 400 0–100

.12 .043 .051 .27 3.4 2.9 0.55 .11 .20 .54 .44

Material Wood (across grain): Balsa Oak Maple Pine, white Teak White fir Wood (parallel to grain): Pine Wool, animal

Apparent density ρ, lb/ft3 at room temperature

t, °C

k

7–8 51.5 44.7 34.0 40.0 28.1

30 15 50 15 15 60

0.025–0.03 0.12 .11 .087 .10 .062

34.4 6.9

21 30

.20 .021

*Marks, Mechanical Engineers’ Handbook, 4th ed., McGraw-Hill, New York, 1941. International Critical Tables, McGraw-Hill, 1929, and other sources. Note 1: B. Kamp [Z. tech. Physik, 12, 30 (1931)] shows the effect of increased porosity in decreasing thermal conductivity of boiler scale. Partridge [University of Michigan, Eng. Research Bull., 15, 1930] has published a 170-page treatise on Formation and Properties of Boiler Scale. Note 2: Townshend and Williams, Chem. & Met., 39, 219 (1932). Note 3: Norton, Refractories, 2d ed., McGraw-Hill, New York, 1942. Note 4: Norton, private communication. TABLE 2-374

Thermal-Conductivity-Temperature Table for Metals* Thermal conductivities tabulated in watts per meter-kelvin Temperature, K

Substance

10

20

40

60

80

100

200

300

400

500

7 38,000 470 47 240

32 13,500 230 196 100

121 2,300 110 810 45

174 850 80 1,400 31

160 380 60 1,650 24

125 300 48 1,490 22

55 237 32 480 18

36 273 26 272 16

26 240 22 196 14

20 237 20 146 12

165 900 400 250 4

305 250 570 450 9

400 150 450 380 16

327 120 250 250 18

230 110 180 190 19

170 110 158 160 20

45 105 111 120 23

25 104 90 100 25

15 101 87 85 27

Copper Gallium Gold Graphite† Graphite‡

19,000 2,200 2,800 27 81

10,700 640 1,500 108 420

2,100 250 520 135 1,630

850 200 380 81 2,980

570 170 350 54 4,290

483 140 345 39 4,980

413 100 327 15 3,250

398 85 315 10 2,000

Hastelloy Inconel Iridium Iron Lead

1 2 1,300 710 175

3 4 1,900 1,000 57

4 8 750 560 43

5 10 360 270 42

6 11 230 170 41

7 11 172 132 40

9 14 147 94 37

10 15 145 80 35

11

13

143 69 34

140 61 33

Magnesium Magnesium oxide Manganese Manganin Mercury

1,200 1,100 2 2 54

1,300 3,100 2 4 40

620 2,200 4 9 35

290 950 5 11 33

190 460 5 13 33

169 260 6 13 32

159 75 7 17 32

156 48 8 22 8

153 36 9 28 10

Molybdenum Nickel Nylon Palladium Platinum

150 2,600 0.04 1,200 1,200

280 1,700 0.10 610 490

350 570 0.17 160 130

250 290 0.20 100 92

210 200 0.23 88 82

179 158 0.25 80 79

143 106 0.28 78 75

138 91 0.30 78 73

PTFE§ Pyrex Quartz Rhodium Rubber

0.94 0.12 1,200 2,900

1.43 0.20 480 3,900

1.94 0.33 82 1,000 0.13

2.1 0.42 40 370 0.15

2.15 0.51 30 250 0.16

2.16 0.57

2.20 0.88

2.25 1.1

190 0.17

160 0.20

Alumina Aluminum Antimony Beryllium oxide Bismuth Boron Cadmium Chromium Cobalt Constantan

Selenium (axis) Silica Silver Tantalum Tellurium Tin Titanium Tungsten Uranium Zinc Zirconium

1200

1400

16 232

10 220

8 93

7 99

6 105

111

70

47

33

25

12 99 85 70 30

81

71

65

62

61

392

388

383

371

357

342

312 7 1,460

309 5 1,140

304 4 930

292 3 680

278 3 530

262 2 440

2 370

55 31

43 19

33 22

28 24

31 26

151 27 9 34 11

149 21

146 13

84 10

98 8

112 7

40 12

13

14

134 80

130 72

126 66

118 67

112 72

105 76

100 80

78 72

80 72

72

73

78

78

81

358 61

62

2.3 1.6

2.5 2.1

150 0.22

145 0.24

140 0.25

140

57

25

15

10

8

6

16,500 108 300

5,200 146 93

1,100 88 29

630 68 17

500 62 13

430 59 11

425 58 6

4 1.34 424 57 4

3 1.52 420 58 3

2 1.70 413 58 3

14

320 28

130 39 880

100

110

59

101 37 330 20 150 42

90 33 310 22 135 38

84 31 280 23 130 34

72 26 190 26 123 25

67 21 180 28 120 23

62 20 170 30 116 22

60 20 150 32 110 21

600

1.87 405 59

800

2.22 389 59

1000

2.60 374 60

19 140 110 21

* Especially at low temperatures, the thermal conductivity can often be markedly reduced by even small traces of impurities. This table, for the highest-purity specimens available, should thus be used with caution in applications with commercial materials. From Perry, Engineering Manual, 3d ed., McGraw-Hill, New York, 1976. A more detailed table appears as Section 5.5.6 in the Heat Exchanger Design Handbook, Hemisphere Pub. Corp., Washington, DC, 1983. † Parallel to basal plane. ‡ Perpendicular to basal plane. § Also known as Teflon, etc.

TRANSPORT PROPERTIES TABLE 2-375

Thermal Conductivity of Chromium Alloys* k = Btu/(h⋅ft2)(°F/ft)

American Iron and Steel Institute Type No.

k at 212°F

k at 932°F

9.4 8.8 8.0 9.3 14.4 15.1 12.5 21.2

12.4 12.5 10.8 12.8 16.6 15.2 14.2 19.5

301, 302, 302B, 303, 304, 316† 308 309, 310 321, 347 403, 406, 410, 414, 416† 430, 430F† 442 501, 502†

* Table 3-322 is based on information from manufacturers. † Shelton and Swanger (National Bureau of Standards), Trans. Am. Soc. Steel Treat., 21, 1061–1078 (1933).

TABLE 2-376 Thermal Conductivity of Some Alloys at High Temperature*

TABLE 2-377 Thermal Conductivities of Some Materials for Refrigeration and Building Insulation* k = Btu/(h⋅ft2)(°F/ft) at approximately room temperature

Material Soft flexible materials in sheet form: Chemically treated wood fiber Eel grass between paper Felted cattle hair Flax fibers between paper Hair and asbestos fibers, felted Insulating hair, and jute Jute and asbestos fibers, felted Loose materials: Cork, regranulated, fine particles Charcoal, 6 mesh Diatomaceous earth, powdered Glass wool, curled Gypsum in powdered form Mineral wool, fibrous

Thermal conductivity, Btu/(ft)(hr)(°R) °R

Kovar

500 600 700 800 900

2-335

Advance

Monel

Hastelloy A

Inconel

Nichrome V

7.8 8.3 8.6 8.7 8.7

11.4 12.6 13.9 15.1

9.0 10.2 11.2 12.3 13.4

5.6 6.2 6.8 7.3 7.8

6.0 6.5 7.0 7.6 8.1

5.5 6.1 6.7 7.3 7.8

Sawdust Wood shavings, from planer Semiflexible materials in sheet form: Flax fiber Semirigid materials in board form: Corkboard

1000 1100 1200 1300 1400

8.9 9.2 9.5 9.8 10.2

16.4 17.6 18.8 20.0 21.2

14.4 15.4 16.5 17.6 18.7

8.4 9.0 9.5 10.1 10.7

8.6 9.1 9.7 10.2 10.8

8.4 9.0 9.5 10.1 10.7

Mineral wool, block, with binder Stiff fibrous materials in sheet form: Wood pulp Sugar-cane fiber Cellular gypsum

1500 1600 1700 1800 1900

10.5 10.8 11.1 11.3 11.5

22.5 23.8 25.0 26.2 27.4

19.8 20.8 21.9 23.0 24.0

11.3 11.8 12.3 12.9 13.4

11.3 11.8 12.4 13.0 13.6

11.3 11.9 12.4 13.0 13.5

2000 2100 2200

11.8 12.1 12.3

28.7 30.0

25.1 26.1 27.2

14.0 14.6 15.1

14.0 14.5 15.0

14.1 14.7 15.3

Apparent density, lb/cu ft room temp.

k

2.2 3.4–4.6 11–13 4.9 7.8 6.1–6.3 10.0

0.023 0.021–0.022 0.022 .023 .023 0.022–0.023 0.031

8–9 15.2 10.6 4–10 26–34 6 10 14 18 12 8.8

.025 .031 .026 .024 0.043–0.05 0.0217 .0225 .0233 .0242 .034 .034

13.0

.026

7.0 10.6 16.7

.0225 .025 .031

16.2–16.9 13.2–14.8 8 12 18 24 30

.028 .028 .029 .037 .049 .064 .083

*Abstracted from U.S. Bur. Standards Letter Circ. 227, Apr. 19, 1927.

*Silverman, J. Metals, 5, 631 (1953). Copyright American Institute of Mining, Metallurgical and Petroleum Engineers, Inc.

TABLE 2-378

Thermal Conductivities of Insulating Materials at High Temperatures* k = Btu/(h⋅ft2)(°F/ft)

Material Laminated asbestos felt (approx. 40 laminations per in) Laminated asbestos felt (approx. 20 laminations per in) Corrugated asbestos (4 plies per in) 85% magnesia (density, 13 lb/ft3) Diatomaceous earth, asbestos and bonding material Diatomaceous earth brick Diatomaceous earth brick Diatomaceous earth brick Diatomaceous earth powder (density, 18 lb/ft3) Rock wool

For temperatures, °F up to 700 500 300 600 1600 1600 2000 2500

Mean temperatures, °F 100

200

300

400

500

0.033 .045 .050 .034 .045 .054 .127 .128 .039 .030

0.037 .050 .058 .036 .047 .056 .130 .131 .042 .034

0.040 .055 .069 .038 .049 .058 .133 .135 .044 .039

0.044 .060

0.048 .065

.040 .050 .060 .137 .139 .048 .044

.053 .063 .140 .143 .051 .050

Asbestos cement, 1.2; 85% magnesia cement, 0.05; asbestos and rock wool cement, 0.075 approx. *Marks, “Mechanical Engineers’ Handbook,” 4th ed., McGraw-Hill, New York, 1941.

600

800

1000

0.055 .065 .143 .148 .054 .057

0.060 .069 .150 .155 .061

0.065 .073 .158 .163 .068

1500

2000

0.176 .183

0.203

2-336

PHYSICAL AND CHEMICAL DATA

TABLE 2-379 Thermal Conductivities of Insulating Materials at Moderate Temperatures (Nusselt)* k = Btu/(h⋅ft2)(°F/ft) Weight, lb/cu ft

Material Asbestos Burned infusorial earth for pipe coverings Insulating composition (loose) Cotton Silk hair Silk Wool Pulverized cork Infusorial earth (loose)

TABLE 2-380 Thermal Conductivities of Insulating Materials at Low Temperatures (Gröber)* k = Btu/(h⋅ft2)(°F/ft)

Temperatures, °F 32

100

200

300

400

600

800

36.0

0.087 0.097 0.110 0.117 0.121 0.125 0.130

12.5

.043 .046 .052 .057 .062 .073 .085

25.0 5.0 9.1 6.3 8.5 10.0 22.0

.040 .032 .026 .025 .022 .021 .035

.046 .035 .030 .028 .027 .026 .039

.050 .053 .055 .039 .034 .034 .033 .032 .045 .047 .050 .053

Temperatures, °F

Material

Weight, lb/cu ft

32

−50

−100

−200

−300

Asbestos Asbestos Cotton Silk

44.0 29.0 5.0 6.3

0.135 .0894 .0325 .0290

0.132 .0860 .0302 .0256

0.130 .0820 .0276 .0235

0.125 .0720 .0235 .0196

0.100 .0545 .0198 .0155

*Marks, Mechanical Engineers’ Handbook, 4th ed., McGraw-Hill, New York, 1941.

*Marks, Mechanical Engineers’ Handbook, 4th ed., McGraw-Hill, New York, 1941.

TABLE 2-381

Thermal Diffusivity (m2/s) of Selected Elements* Temperature, K

Element

20

40

60

80

100

Aluminum Beryllium Chromium Copper Gold

0.50

200

400

600

800

1000

0.012

0.0014

4.4. − 4

0.038 0.16 0.005

0.0037 0.0040 4.5. − 4

5.9. − 4 6.9. − 4 2.3. − 4

2.0. − 4 3.1. − 4 1.8. − 4

2.3. − 4 0.0036 1.2. − 4 2.2. − 4 1.5. − 4

1.1. − 4 1.5. − 4 4.1. − 5 1.3. − 4 1.3. − 4

9.4. − 5 4.0. − 5 2.6. − 5 1.1. − 4 1.2. − 4

8.4. − 5 2.6. − 5 2.0. − 5 1.0. − 4 1.2. − 4

7.4. − 5 2.1. − 5 1.7. − 5 9.0. − 5 1.1. − 4

6.6. − 5 1.7. − 5 1.4. − 5 9.0. − 5 9.8. − 5

Iridium Iron Lead Molybdenum Nickel

0.046 0.043 9.3. − 5 0.0095 0.033

3.2. − 3 3.9. − 5 0.0014 0.0017

4.9. − 4 3.3. − 5 4.0. − 4 3.1. − 4

1.6. − 4 3.1. − 5 2.0. − 4 1.3. − 4

8.4. − 5 8.2. − 5 2.9. − 5 1.3. − 4 8.0. − 5

5.6. − 5 3.1. − 5 2.6. − 5 6.3. − 5 3.1. − 5

4.8. − 5 1.8. − 5 2.3. − 5 5.1. − 5 1.9. − 5

4.4. − 5 1.3. − 5 2.0. − 5 4.5. − 5 1.3. − 5

4.1. − 5 1.1. − 5 1.3. − 5 4.2. − 5 1.4. − 5

3.5. − 5 1.0. − 5 1.5. − 5 3.8. − 5 1.5. − 5

Platinum Silver Zinc

0.0029 0.031 0.0046

1.6. − 4 0.0013 3.1. − 4

6.3. − 5 4.5. − 4 1.0. − 4

4.3. − 5 2.8. − 4 7.0. − 5

3.6. − 5 2.3. − 4 5.5. − 5

2.7. − 5 1.8. − 4 4.7. − 5

2.5. − 5 1.7. − 4 3.9. − 5

2.5. − 5 1.6. − 4 3.4. − 5

2.5. − 5 1.5. − 4 1.8. − 5

2.5. − 5 1.4. − 4 2.2. − 5

*Tables for up to 24 temperatures for 47 elements appear in the Handbook of Heat Transfer, 2d ed., McGraw-Hill, New York, 1984. The notation 3.2. − 4 signifies 2.3 × 10−4.

PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES TABLE 2-382

2-337

Thermophysical Properties of Selected Nonmetallic Solid Substances Density, kg/m3

Material Alumina Asphalt Bakelite Beryllia Brick

3975 2110 1300 3000 1925

Brick, fireclay Carbon, amorphous Clay Coal Cotton

2640 1950 1460 1350 80

Diamond Granite Hardboard Magnesite Magnesia

3500 2630 1000 3025 3635

Emissivity

Specific heat, kJ/(kg⋅K)

Thermal conductivity, W/(m⋅K)

0.82 0.93

0.765 0.920 1.465 1.030 0.835

36 0.06 1.4 270 0.72

11.9 0.03 0.74 88 0.45

0.960 0.724 0.880 1.26 1.30

1.0 1.6 1.3 0.26 0.06

0.39 1.13 1.01 0.15 0.58

0.509 0.775 1.38 1.13 0.943

2300 2.79 0.15 4.0 48

1290 1.37 0.11 1.2 14

0.93 0.86 0.91 0.80

0.38 0.72

Oak Paper Pine Plaster board Plywood

770 930 525 800 540

0.90 0.83 0.84 0.91

2.38 1.34 2.75

Pyrex Rubber Rubber, foam Salt Sandstone

2250 1150 70

0.835 2.00

2150

0.92 0.92 0.90 0.34 0.59

Silica Sapphire Silicon carbide Soil

3975 3160 2050

Teflon Thoria Urethane foam Vermiculite

2200 4160 70 120

Thermal diffusivity, m2/s × 106

0.18 0.011 0.12 0.17 0.12

0.10 0.01 0.54

0.74 0.09

0.854 0.745

1.4 0.2 0.03 7.1 2.9

0.79 0.48 0.86 0.38

0.743 0.765 0.675 1.84

1.3 46 490 0.52

15 230 0.14

0.92 0.28

0.35 0.71 1.05 0.84

0.26 14 0.03 0.06

0.34 4.7 0.36 0.60

1.22

0.18

1.8

NOTE: Difficulties of accurately characterizing many of the specimens mean that many of the values presented here must be regarded as being of order of magnitude only. For some materials, actual measurement may be the only way to obtain data of the required accuracy. To convert kilograms per cubic meter to pounds per cubic foot, multiply by 0.062428; to convert kilojoules per kilogram-kelvin to British thermal units per pound-degree Fahrenheit, multiply by 0.23885.

PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES INTRODUCTION In the absence of reliable experimental data, the methods presented here provide physical property estimates that are sufficiently accurate for many engineering applications. These techniques have been selected on the basis of accuracy, generality, and, in most cases, simplicity; they are divided into 11 categories: (1) pure component constants: critical properties, normal freezing and boiling temperatures, acentric factor, radius of gyration, dipole moment, and van der Waals area and volume; (2) vapor pressure; (3) ideal gas thermal properties: heat capacity and enthalpy, Gibbs energy, and entropy of formation; (4) enthalpy of vaporization and fusion; (5) solid and liquid heat capacity; (6) vapor, liquid, and solid density; (7) vapor and liquid viscosity; (8) vapor and liquid thermal conductivity; (9) vapor and liquid diffusivity; (10) surface tension; and (11) flammability properties: flash point, flammability limits, and autoignition temperature. The definition of the property and limitations and accuracy of each method of correla-

tion or prediction are given for each property. Numerical examples are included for many of the methods. Equation symbols are listed under “Nomenclature,” and literature citations, indicated by superscript numbers, follow the nomenclature under “References.” Essentially all of the methods are derived from work on the American Petroleum Institute Technical Data Book 23 (hydrocarbon compounds and their mixtures), the AIChE Design Institute for Physical Property Data (DIPPR) Data Prediction Manual 22 (nonhydrocarbon compounds and their mixtures), and the DIPPR Data Compilation Project 24. UNITS Applicable dimensional units are shown individually with each equation. The International Metric System (SI) is used when feasible; otherwise commonly used U.S. engineering units are employed. The reader is referred to Sec. 1 for unit conversion factors.

2-338

PHYSICAL AND CHEMICAL DATA Nomenclature Symbol

Definition

SI units

U.S. customary units

B Cp Cv d D F k K ln M MeABP N NA P P sat [P] rel den R R  S T V x y Z

Second virial coefficient Heat capacity at constant pressure Heat capacity at constant volume Mass density Diffusivity Conductivity factor in Eq. (2-136) Thermal conductivity Watson/UOP characterization factor = (1.8Tb)1/3/rel den Denotes natural logarithm Molecular weight Mean average boiling point Carbon number Avogadro’s number Pressure Vapor pressure Parachor Relative density at 15°C and 0.1 MPa Universal gas constant; e.g., 8314 Pa·m3/kmol K Radius of gyration Absolute entropy Absolute temperature Molar volume Mole fraction of component in liquid phase Mole fraction of component in vapor phase Compressibility factor = PV/RT

m3/kmol J/(kmol·K) kg/m3 m2/s

ft3/lbmol Btu/(lbm °F) lbm/ft3 ft2/s

W/(m·K)

Btu/(h·ft °F)

Pa Pa

lbf/in2 lbf/in2

J/kg K m3/kmol

Btu/lbm R ft3/lbmole

∆Gf ∆Hf ∆Hfus ∆HV ∆Sf ∆Sfus ∆ZV

J/kg J/kg J/kg J/kg J/(kg·K) J/(kg·K)

Btu/lbm Btu/lbm Btu/lbm Btu/lbm Btu/lbm Btu/lbm

Θ λ µ ν ρ σ φ ω

Gibbs energy of formation Enthalpy (heat) of formation Enthalpy (heat) of fusion Enthalpy (heat) of vaporization Entropy of formation Entropy of fusion Difference of vapor and liquid compressibility factors defined in Eq. (2-55) Tb /Tc Dipole moment Absolute viscosity Kinematic viscosity Molar density Surface tension Volume fraction Acentric factor

Pa·s m2/s kmol/m3 N/m

lbm/(ft·s) ft2/s lb·mole/ft3 dyne/cm

r (0) (1) ° ′

At reference condition Simple spherical molecule, corresponding states Correction factor, corresponding states Of the ideal gas At atmospheric pressure

b bp c G HI i j L m mc mlt o pc r RA sat S V w

At normal boiling temperature At bubble point At critical point Of the gas/vapor Upper limit Component index, of the ith component Component index, of the jth component Of the liquid Of the mixture Mixture correspondence At melting temperature Of organic component Pseudocritical quantity Reduced quantity Rackett parameter Saturated Of the solid Of the vapor Of water

Greek symbols

Superscripts

Subscripts

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2-339

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Basic pure component constants required to characterize components or mixtures for calculation of other properties include the melting point, normal boiling point, critical temperature, critical pressure, critical volume, critical compressibility factor, acentric factor, and several other characterization properties. This section details for each property the method of calculation for an accurate technique of prediction for each category of compound, and it references other accurate techniques for which space is not available for inclusion. Critical Temperature The critical temperature of a compound is the temperature above which a liquid phase cannot be formed, no matter what the pressure on the system. The critical temperature is important in determining the phase boundaries of any compound and is a required input parameter for most phase equilibrium thermal property or volumetric property calculations using analytic equations of state or the theorem of corresponding states. Critical temperatures are predicted by various empirical methods according to the type of compound or mixture being considered. For pure hydrocarbons, the method of Ambrose2 is the most accurate and will also be useful for predicting critical pressure and volume. Equation (2-1) requires only the normal boiling point, Tb, and the molecular structure of the compound.



log Tc = A + B log10 (rel den) + C log Tb



(2-1)

Tc and Tb are the critical and normal boiling temperatures, respectively, expressed in kelvins. Values of ∆T from Table 2-383 are summed for each part of the molecule to yield ∆T (e.g., for isobutane, 3 × CH3 + 1 × CH). ∆(Platt no.) is equal to the Platt number of any alkyl chains in the molecule minus the Platt number of the n-alkane with the same number of carbon atoms. The Platt number is defined as the number of pairs of carbon atoms that are separated by three carbon-carbon bonds and is an indicator of branching (e.g., for 2,2,3trimethylpentane, the Platt number is 5). The Platt number of an n-alkane is the number of carbon atoms minus three (e.g., for n-octane, the Platt number is five). Errors in Tc average about 4 K for paraffins to C20 and other hydrocarbons to C14.

(2-2)

Tc and Tb are the critical and normal boiling temperatures, respectively, expressed in kelvins. The relative density (rel den) of the liquid at 15°C is 0.1 MPa. The regression constants A, B, and C are tabulated by family in Table 2-384. Errors average about 3 K. For pure nonhydrocarbon organics, the most accurate method for prediction of critical temperature for all compound groups is also the Ambrose2 method. Equation (2-1) applies to all nonhydrocarbon compounds except perfluorocarbons, where the constant 1.242 is replaced by 1.570. For compounds containing any of C, H, O, N, S, or halogens up to C13 and ranging in critical temperature from 228– 790 K, the average error is about 6 K. Alternate methods for nonhydrocarbon organics are the first order method of Lydersen63 with an average error of 9 K although the method of Ambrose is considerably better for alcohols and ketones. Equation (2-3) is the Lydersen equation for critical temperature and requires only the normal boiling point and the molecular structure for solution. Tb Tc =  [0.567 + ∆T − ( ∆T)2]

(2-3)

Contributions to ∆T are given in Table 2-385. For pure inorganic compounds, the method of Gambill31 was modified to yield Eq. (2-4) and only requires the normal boiling point as input. Tc = 1.64Tb

PURE COMPONENT CONSTANTS

1 Tc = Tb 1 +  1.242 + ∆T − 0.023∆(Platt no.)

Equation (2-2), another somewhat simpler method for estimating the critical temperature of pure hydrocarbons only, is the method of Nokay76 and requires the normal boiling point, the relative density, and the compound family.

(2-4)

Although this equation was tested with available experimental data (38 compounds), it can only be considered a rough approximation. Inorganic-organic and inorganic-halide compounds are predicted better by replacing the constant 1.64 in Eq. (2-4) by 1.55. For both hydrocarbons and nonhydrocarbon organic defined mixtures, the method of Li60 is used with a relatively simple volumetric average mixing rule as shown in Eq. (2-5) to calculate the true critical temperature. yjVc j Tcm =  Tc j (2-5) j

yiVc i



i



Tcm is the mixture critical temperature in K. Vc is the critical volume of a component, m3/kmole. The mole fraction of a component is y. The mixture contains i components. For hydrocarbon systems, the average error is about 3 K, while for systems containing nonhydrocarbons, the average error is 15 K with the highest errors occurring where simple gases are present. The method of Chueh and Prausnitz18 yields average errors slightly lower than the method of Li, but it is computationally more complex. Critical Pressure The critical pressure of a compound is the vapor pressure of the compound at the critical temperature. Below the critical temperature, any compound above its vapor pressure will be a liquid. The critical pressure is required for calculations discussed in the part of the section on critical temperature. For pure hydrocarbons, the method of Ambrose2 is the most accurate. Equation (2-6) requires only the molecular weight (M) and the molecular structure of the compound. 0.101325M Pc =  [0.339 + ∆P − 0.026∆(Platt no.)]2

(2-6)

Pc is the critical pressure, MPa. Values of ∆P from Table 2-383 are summed for each part of the molecule to yield ∆P. Calculation of the Platt number is discussed under “Critical Temperature.” Errors in Pc average 0.07 MPa and are less reliable for compounds with 12 or more carbon atoms.

TABLE 2-383

Group Increments for the Ambrose Method ∆T

∆P

∆V

Group description

CH3

0.138

0.2260

55.1

Aromatic Compounds (Cont.)

CH2

0.138

0.2260

55.1

CH

0.095

0.2200

47.1

C

0.018

0.1960

38.1

CH2

0.113

0.1935

45.1

CH

0.113

0.1935

45.1

C

0.070

0.1875

37.1

C

0.088

0.1610

35.1

CH

0.038

0.1410

35.1

C

0.038

0.1410

35.1

Group description

Ring Increments CH2

0.090

0.1820

44.5

CH

0.090

0.1820

44.5

CH

in fused ring *

0.030

0.1820

44.5

C

0.090

0.1820

44.5

CH

0.075

0.1495

37.0

C

0.075

0.1495

37.0

C

0.060

0.1170

29.5

Aromatic Compounds 0.458

0.448

0.488

0.9240

0.8940

0.9440

222

222

222

0.488

0.9440

222

0.438

0.8640

222

0.478

0.9140

0.8340

∆P

∆V

0.468

0.8840

222

0.468

0.8840

222

0.418

0.8040

222

0.368

0.7240

222

0.220

0.5150

148

in fused ring* Nonring Increments OH

Use Eq. (a), below Use Eq. (b), below

O

0.138

0.160

CO

0.220

0.282

0.220

0.220

0.578

0.450

1.156

0.900

CHO COOH CO

O

CO

O

OC

0.330

0.470

NO2

0.370

0.420

NH2

0.208

0.095

NH

0.208

0.135

N

0.088

0.170

CN

0.423

0.360

S

0.105

0.270

SH

0.090

0.270

Si

0.138

0.461

SiH

0.371

0.507

SiH3

0.195

—

0.159

0.725

0.131

0.663

222 Si

0.428

∆T

222

Si

O

O

cyclic

2-341

2-342

PHYSICAL AND CHEMICAL DATA

TABLE 2-383

Group Increments for the Ambrose Method (Concluded ) ∆T

∆P

−0.050 −0.200 0.180 (1st) 0.055 0.110 (1st) 0.055 0.110 (1st) 0.055

−0.065 −0.170 0.223 0.318 0.500

0.448 0.448 0.198 0.220 0.080

0.924 0.850 −0.025 0.515 0.183

Group description Double Bond Triple Bond F Cl Br Aromatic Compounds Corrections benzene pyridine OH C4H4 (fused ring) F *Group contributions for

CH and

∆V

∆T

∆P

0.080 0.080

0.318 0.600 0.850

0.010

0

0.030

0.020

−0.040 −0.080

−0.050 0

Group description Cl Br I The single or first substituent on an aromatic ring The second or subsequent ring substituents Each pair of ring substituents in ortho positions with respect to each other If one of the ortho pair is OH

∆V

in fused rings have been calculated from minimal data and may be less reliable than the other values.

Substituent increments do not apply to halogens. Eq. (a): ∆T (OH) = 0.87 − 0.11n + 0.003n2 Eq. (b): ∆p (OH) = 0.100 − 0.013n n = carbon number of compound. For branched alcohols, an effective carbon number can be determined by interpolation between the normal alcohol of the same carbon number and the immediately lower normal alcohol using normal boiling points. (See Example 1.)

TABLE 2-384

∆T = 2(.138) + .138 + .095 + .558 = 1.067 1 Tc = 372.7 1 +  1.242 + 1.067 − 0

Nokay Equation

Type compound

A

B

C

Paraffin Naphthene Olefin Acetylene Diolefin Aromatic

1.35940 0.65812 1.09534 0.74673 0.1384 1.0615

0.43684 −0.07165 0.27749 0.30381 −0.39618 0.22732

0.56224 0.81196 0.65563 0.79987 0.99481 0.66929





Tc = 534.1 K An accurate experimental value is 536.05 K.

∆P = 2(.226) + .226 + .220 + .0597 = 0.9577 .101325(74.12) Pc = 2 (.339 + .9577 − 0)

Example 1 Estimate the critical temperature and critical pressure of 2-butanol using the Ambrose method, Eqs. (2-1) and (2-6). The experimental normal boiling point is 372.7 K. 1 Tc = Tb 1 +  1.242 + ∆ T − 0.023∆ (Platt no.)





0.101325M Pc =  [0.339 + ∆P − 0.026∆ (Platt no.)]2 Determine group contributions from Table 2-383 for a structure CH3CHOHCH2CH3: Number of groups

∆T

∆P

CH3

2

0.138

0.2260

CH2

1

0.138

0.2260

CH–

1

0.095

0.2200

OH

1

(a)

(b)

Group

Pc = 4.467 MPa An accurate experimental value is 4.179 MPa. For pure nonhydrocarbon organics, the simplest accurate method for prediction of critical pressure is the method of Lydersen.63 Equation (2-7) requires the molecular weight (M) and the molecular structure of the compound. 0.101325M Pc =  (2-7) (0.34 + ∆P)2 Pc is the critical pressure, MPa. Values of ∆P from Table 2-385 are summed to yield ∆P. The average error in Pc is about 0.2 MPa when tested on compounds ranging in carbon number from C1 to C10.

Example 2 Estimate the critical temperature and critical pressure of 2-butanol, which has an experimental normal boiling point of 372.7 K. Use the Lydersen method, Eqs. (2-3) and (2-7). The structure of 2-butanol is CH3CHOHCH2CH3. Determine group contributions from Table 2-385. Group

Number of groups

∆T

∆P

CH3

2

0.020

0.227

CH2

1

0.020

0.227

To determine n, the normal boiling points of 1-propanol (370.3 K) and 1-butanol (390.9 K) are required:

CH

1

0.012

0.210

372.7 − 370.3 n = 3 +  (4 − 3) = 3.1 390.9 − 370.3

OH

1

0.082

0.06

∆T°(OH) = 0.87 − 0.11(n) + 0.003 n = 0.558 ∆p°(OH) = 0.100 − 0.013(n) = 0.0597 2





The Platt number of the compound is determined by substituting a CH3 group for the OH. CH3CH(CH3) − CH2CH3, Platt no. = 2 The Platt number of the n-alkane of the same carbon number is 5 − 3 = 2. Thus, ∆(Platt no.) = 2 − 2 = 0. Therefore:

0.154 Tb 372.7 Tc = 2 = 2 0.567 + ∆T − ( ∆T) .567 + .154 − (.154) Tc = 534.5 K

0.951

PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES TABLE 2-385

Group Increments for the Lydersen Method Incremental contributions

Group description Nonring Elements CH3, CH2 CH

∆T

∆P

∆V

Incremental contributions ∆P

∆V

0.085

(0.4)

0.080

0.047

0.47

0.080

(0.02)

(0.12)

(0.011)

0.018 0.017 0.010 0.012

0.224 0.320 (0.50) (0.83)

0.018 0.049 (0.070) (0.095)

0.031 0.031

0.095 0.135

0.028 (0.037)

(0.024)

(0.09)

(0.027)

0.014

0.17

(0.042)

(0.007)

(0.13)

(0.032)

CN

(0.060)

(0.36)

(0.080)

NO2

(0.055)

(0.42)

(0.078)

0.015 0.015 (0.008) (0.003)

0.27 0.27 (0.24) (0.24)

0.055 0.055 (0.045) (0.047)

Si

0.026

0.468

—

SiH

0.040

0.513

—

SiH3

0.027

—

—

0.025

0.730

—

0.027

0.668

—

Group description

C

0.020

0.227

0.055

0.012

0.210

0.051

0.00

C CH,

0.210

0.041

CH

0.018

0.198

0.045

C

0.00

0.198

0.036

0.005

0.153

(0.036)

0.013

0.184

0.0445

, C

Ring Increments CH2

∆T

Oxygen Increments (Cont.) COOH (acid) COO

CH2,

2-343

(ester)

O (except for combinations above) Halogen Increments F Cl Br I Nitrogen Increments NH2 NH (nonring) NH (ring)

0.012

CH

0.192

0.046 N

(−0.007)

C

0.154

(0.031) N

C

,

C

CH

0.011

0.154

0.036

0.011

0.154

0.037

0.066

0.924

Organometallic Increments

OH (alcohols)

0.082

0.06

(0.018)

OH (phenols)

0.031

(−0.02)

(0.003)

O

(nonring)

0.021

0.16

0.020

O

(ring)

(0.014)

(0.12)

(0.008)

O (nonring)

C

O (ring)

HC

O (aldehyde)

(ring)

Sulfur Increments SH S (nonring) S (ring) 苷S

Oxygen Increments

C

(nonring)

0.040

0.29

0.060

(0.033)

(0.2)

(0.050)

0.048

0.33

0.073

Values in parentheses are based on too few experimental points to be reliable.

Si

O

Si

O

cyclic

2-344

PHYSICAL AND CHEMICAL DATA TABLE 2-386 Atomic and Structural Contributions for the Fedors Method

The accurate experimental critical temperature is 536.05 K. 0.101325M (0.101325)(74.12) =  Pc =  (0.34 + ∆P)2 (0.34 + 0.951)2

Atomic increments

Pc = 4.506 MPa The accurate experimental critical pressure is 4.179 MPa. No known method is available to predict the critical pressure of inorganic compounds. For both hydrocarbon and nonhydrocarbon organic defined mixtures, the method of Kreglewski and Kay52 is recommended. The critical temperature, critical pressure, and acentric factor of each compound and the critical temperature of the mixture must be known or predicted from the methods of this section.



Tcm − Tpc

 x ω   T

Pcm = Ppc + Ppc 5.808 + 4.93

n

i

i

i=1

∆V

Feature

∆V

C H O O (alcohols) N N (amines) F

0.034426 0.009172 0.020291 0.018000 0.048855 0.047422 0.022242

Si Sisiloxane Sicyclic siloxane

0.086174 0.126483 0.126483

Cl Br I S 3-membered ring 4-membered ring 5-membered ring 6-membered ring double bond triple bond ring attached directly to another ring

0.052801 0.071774 0.096402 0.050866 −0.105824 −0.017247 −0.039126 −0.039508 +0.005028 +0.000797 +0.035524

(2-8)

pc

Use of Eq. (2-8) requires the pseudocritical properties defined by Eqs. (2-9) and (2-10)

Structural increments

Atom

n

Tpc = xiTc i

(2-9)

i=1 n

Ppc = xiPc i

(2-10)

i=1

Each component i of the mixture must have available its Tc, Pc, and ω. Tcm can be predicted from Eq. (2-5). For hydrocarbon systems, average errors in predicted critical pressures are about 0.2 MPa, except when organic gases are present and errors are unacceptably large. Errors for nonhydrocarbon organics not including inorganic gases average 0.5 MPa.

Critical Volume The critical volume of a compound is the volume occupied by a set mass of a compound at its critical temperature and pressure. While useful in itself, the critical volume is extensively used in equations for estimating volumetric fractions. For pure hydrocarbons, two methods are quite accurate. The Ambrose2 method used for Tc and Pc is also used for critical volume. Eq. (2-11) only requires the molecular structure of the compound. Vc = 10−3(40 + ∆V)

(2-11)

Vc is the critical volume, m3/kmole. Values of ∆V are given in Table 2-383. The average error for hydrocarbons of twelve or less carbon atoms is about 0.01 m3/kmole. The Riedel method90 requires the critical temperature (Tc), critical pressure (Pc), and acentric factor (ω) of the compound as given by Eqs. (2-12) and (2-13). If the gas constant is in Pa·m3/kmole·K, the critical volume will be in m3/kmole. RTc Vc =  (2-12) Pc[3.72 + 0.26(α − 7.00)] α = 5.811 + 4.919ω

(2-13)

The average error for paraffins up to C18 and other hydrocarbon families up to C11 is about 0.015 m3/kmole. If estimated values of Tc and Pc are used, errors may be higher. For pure nonhydrocarbon organics, the method of Fedors,29 which requires only molecular structure, is the most accurate. Equation (2-14) shows the method to depend only on the molecular structure. The resulting Vc will be in m3/kmole. Vc = 0.0266 + ∆V

(2-14)

Values for ∆V are given in Table 2-386. The average error for compounds up to C7 is about 0.007 m3/kmole, although a maximum error of 0.03 m3/kmole has been noted. No experimental data above C7 are available for comparison. Example 3 Estimate the critical volume of 2-butanol. Use the method of Fedors, Eq. (2-14). Vc = 0.0266 + ∆V The molecular formula is C4H10O. Using the atomic contribution values from Table 2-386.

∆V = 4(.034426) + 10(.009172) + 1(.018000)

∆V = 0.2474 Vc = 0.0266 + 0.2474 = 0.2740 m3/kmole The accurate experimental critical volume is 0.2690 m3/kmole.

The method of Lydersen63 may also be used for prediction of critical volume, but it is not so accurate as the method of Fedors. Equation (2-15) depends only on molecular structure and gives a critical volume in m3/kmole. Vc = 0.040 + ∆V

(2-15)

Group contributions for ∆V are given in Table 2-385. Errors average about 0.01 m3/kmole. There is no known method for predicting the critical volume of inorganic compounds. For both hydrocarbon and nonhydrocarbon organic defined mixtures, the method of Chueh and Prausnitz19 is useful. For hydrocarbon systems, the mixing rule is shown by Eq. (2-16) for binaries and by Eq. (2-17) for multicomponents. Equations (2-18) through (2-20) give the input parameters. The mixture critical volume Vcm is a function of the pure component critical volumes. The constant C is zero for hydrocarbon systems and 0.1559 for systems containing a nonhydrocarbon gas. Vcm = φ1Vc1 + φ2Vc2 + 2φ1φ2ν12 n

n

i

j

Vcm = φiφ jνi j

(i ≠ j)

2/3 cj

(2-16) (2-17)

xjV φj =  n

xiV 2/3ci

(2-18)

Vij(Vci + Vcj) νij =  2.0

(2-19)

i=1

Vci − Vcj (2-20) Vij = −1.4684  + C Vci + Vcj Errors average about 10 percent for systems containing hydrocarbons. Systems containing only organics or organics and gases may give very high errors. A specialized modification of the method (Chueh and Prausnitz18) is available for binary mixtures containing organics but may give errors over 20 percent.





Critical Compressibility Factor The critical compressibility factor of a compound is calculated from the experimental or predicted values of the critical properties by the definition, Eq. (2-21). PcVc Zc =  (2-21) RTc Critical compressibility factors are used as characterization parameters in corresponding states methods (especially those of Lydersen) to predict volumetric and thermal properties. The factor varies from about 0.23 for water to 0.26–0.28 for most hydrocarbons to slightly above 0.30 for light gases. Normal Freezing Temperature (Melting Point) The melting point is the temperature at which melting occurs at atmospheric pressure. In most cases, measurements are made in air, making values slightly lower than if the measurements were made in vacuum. Impurities can cause a substantial decrease in the measured melting point. The melting point is very slightly higher than the triple point temperature—the temperature at which equilibrium exists between solid, liquid, and vapor—for a pure compound. For practical purposes, the two temperatures are equal. Reliable methods for predicting melting points have not until recently been advanced. Constantinou and

PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES Gani20 derived a group contribution method that shows promise. Initial evaluations show average errors of 5 to 10 percent (10–30 K) on a wide variety of compounds, but larger errors can occur. It is recommended that several compounds of known melting point in the same or a similar family be predicted in order to estimate the probable error. Normal Boiling Temperature The normal boiling temperature (point) is the temperature at which the vapor pressure equals exactly 101,325 Pa (1 atmosphere). Caution should be taken in using values from older references, where the temperature may be reported for the prevailing pressure (0.95–0.97 atm) rather than at 1 atmosphere. If at least two values of vapor pressure very close to 1 atmosphere are available, the normal boiling point can be interpolated or extrapolated on a plot of log P sat vs. 1/T. The section on vapor pressure discusses this in more detail. Various methods are available for estimation of the normal boiling point of organic compounds. Lyman et al.64 review and give calculational procedures for the methods of Meissner, Miller, and Lydersen/ Forman-Thodos. A more recent method that has been determined to be more accurate is the method of Pailhes,80 which requires one experimental vapor pressure point and Lydersen group contributions for critical temperature and critical pressure (Table 2-385). log Pc + (1 − Θ) log (1/p) Tb = T  (2-22) log Pc where T and p = the one low pressure vapor pressure point, in K and atm, respectively Θ = Tc /Tb calculated from Eq. (2-3) using Lydersen contributions Pc = critical pressure calculated from Eq. (2-7) using Lydersen contributions A recent study of the method on a wide variety of complex organics shows an overall average error of less than 2 percent (~10 K). If no vapor pressure point is available, the new group contribution method of Constantinou and Gani20 discussed under the section on melting point gives an overall average error of about 4 percent (~20 K) and may be useful. The method of Miller (Lyman et al.64), which requires only the molecular structure, has also been found to be relatively accurate for organics. Example 4 Estimate the normal boiling point of 2-butanol. One vapor pressure point of 0.802 psia at 100°F is available. Use the Pailhes method, Eq. (2-22). log Pc + (1 − θ) log (1/p) TB = T  log Pc From Example 2, Pc = 4.506 MPa; θ = Tc /Tb = 1.434; T = 100°F = 310.9 K; and p = 0.802 psia = 5528 Pa. log (4.506 × 106) + (1 − 1.434) log (1/5528) Tb = (310.9)  log (4.506 × 106) Tb = (310.9)(1.244) = 386.8 K An accepted experimental normal boiling point is 372.7 K. Note the error here is 3.8 percent, a value above the average. If the vapor pressure point available would have been closer to one atmosphere, the error would have been much lower.

Acentric Factor The acentric factor of a compound (ω) is primarily a measure of the shape of a molecule, though it also measures a molecule’s polarity. It is calculated from the reduced vapor pressure (Prsat ) at a reduced temperature of 0.7 by the definition, Eq. (2-23). ω = −log (Prsat)Tr = 0.7 − 1.000

(2-23)

Critical temperature and pressure are required and can be estimated from the methods of this section. Vapor pressure is predicted by the methods of the next section. Experimental values should be used if available. The acentric factor is used as a third parameter with Tc and Pc in Pitzer-type corresponding states methods to predict volumetric properties and in cubic equations of state such as the Redlich-KwongSoave and Peng-Robinson equations. For simple spherical molecules, the acentric factor is essentially zero, rising as branching and molecu-

2-345

lar weight increases. For compounds of similar size and shape the acentric factor increases slightly with increasing polarity. For mixtures, the acentric factor is usually taken as a simple molar average value of the n components of the mixture. n

ω = xiω i

(2-24)

i=1

Miscellaneous Characterizing Constants The radius of gyration (R ) is a simultaneous size-shape factor varying with the manner in which mass is distributed about the center of gravity of the molecule. For planar molecules, the radius of gyration is R= 

(AB) N 

 M 1/2

A

(2-25)

For three-dimensional molecules, it is R= 

(ABC) 2πN 

 M 1/3

A

(2-26)

AB and ABC are the products of the principal moments of inertia. Moments of inertia are calculated from bond angles and bond lengths. Many values are given by Landolt-Bornstein.53 NA is Avogadro’s number, and M is the molecular weight of the molecule. Stuper et al.105 give a computerized method for prediction of the radius of gyration. The dipole moment (λ) of a molecule is the first moment of the electric charge density of a molecule. Paraffins have dipole moments of zero, while dipole moments of almost all hydrocarbons are small. McClellan68 lists many dipole moments. The computer method of Dixon and Jurs27 is the most useful method for predicting dipole moments. Lyman et al.64 give other methods of calculation. The van der Waals volume and area are characterizing parameters relating molecular configurations. Bondi8 describes group contribution methods for their calculation. VAPOR PRESSURE Vapor pressure is the most important of the basic thermodynamic properties affecting liquids and vapors. The vapor pressure is the pressure exerted by a pure component at equilibrium at any temperature when both liquid and vapor phases exist and thus extends from a minimum at the triple point temperature to a maximum at the critical temperature, the critical pressure. This section briefly reviews methods for both correlating vapor pressure data and for predicting vapor pressure of pure compounds. Except at very high total pressures (above about 10 MPa), there is no effect of total pressure on vapor pressure. If such an effect is present, a correction, the Poynting correction, can be applied. The pressure exerted above a solid-vapor mixture may also be called vapor pressure but is normally only available as experimental data for common compounds that sublime. Correlation Methods Vapor pressure is correlated as a function of temperature by numerous methods mainly derived from the Clapeyron equation discussed in the section on enthalpy of vaporization. The classic simple equation used for correlation of low to moderate vapor pressures is the Antoine4 equation (2-27). B (2-27) ln Psat = A +  T+C A, B, and C are regression constants for the specific compound. The Antoine equation does not fit data accurately much above the normal boiling point. Thus, as regression by computer is now standard, more accurate expressions applicable to the critical point have become usable. The entire DIPPR Compilation24 is regressed with the modified Riedel89 equation (2-28) with constants available for over 1500 compounds. B ln P sat = A +  + C ln T + DT E (2-28) T A, B, C, and D are regression constants and E is an exponent equal to 1, 2, or 6 depending on which regression gives the most accurate fit of the data.

2-346

PHYSICAL AND CHEMICAL DATA

For purposes of the API Technical Data Book (Daubert and Danner23), another modified Riedel equation (2-29) was chosen and found to fit hydrocarbon data well over the entire pressure range. Coefficients are given for several hundred hydrocarbons. B E ln P sat = A +  + C ln T + DT 2 + 2 (2-29) T T Both equations (2-28) and (2-29) are also extrapolatable above the critical temperature where necessary for thermodynamic calculations. The other modern equation used for correlation is the modified and linearized Wagner124 equation (2-30), which has the advantage that it will match critical data exactly, although it cannot be extrapolated above the critical point. The equation is also included with coefficients to several hundred compounds in the Technical Data Book—Petroleum Refining. ln Pr = aX1 + bX2 + cX3 + dX4 where

(2-30)

1−T (1 − Tr)1.5 (1 − Tr)2.6 (1 − Tr)5 X1 = r , X2 =  , X3 =  , X4 =  Tr Tr Tr Tr Pr = P /Pc sat

Tr = T / Tc sat

(2-31)

(ln P ) = 5.92714 − 6.09648/Tr − 1.28862 ln Tr + 0.169347T 6r (2-32) sat (0) r

(1) (ln P sat = 15.2518 − 15.6875/Tr − 13.4721 ln Tr + 0.43577T 6r r ) (2-33)

The method is applicable at reduced temperatures above 0.30 or the freezing point, whichever is higher, and below the critical point. The method is most reliable when 0.5 < Tr < 0.95, where errors in prediction average 3.5 percent when experimental critical properties are known. Errors are higher for predicted criticals. The method is useful when solved iteratively with Eq. (2-23) to predict the acentric factor. Example 5 Estimate the vapor pressure of 1-butene at 98°C. Use Eq. (2-31): (1) sat (0) ln P sat + ω(ln P sat r = (ln P r ) r )

Pure component properties of 1-butene are Tc = 146.4°C, Pc = 4.02 MPa, and ω = 0.1867. 371.1 Tr =  = 0.885 419.5 From Eq. (2-32):

(ln Prsat)(0) = −0.7227

From Eq. (2-33):

(ln Prsat)(1) = −0.6190

3000.538X − 6.761560 log Psat =  43X − 0.987672

ln Prsat = −.7227 + (.1867)(−0.6190) ln Prsat = −0.8383 Prsat = 0.4325 P sat = PrsatPc = (.4325)(4.02) = 1.74 MPa An experimental value is 1.72 MPa. When criticals cannot be estimated with reasonable accuracy, the method of Maxwell and Bonnell67 is recommended. The normal boiling point and the specific gravity at 60°F (15.5°C) are required inputs. According to what vapor pressure range is expected, the vapor pressure is calculated from Eqs. (2-34), (2-35), or (2-36). If the wrong range is selected, the procedure will need to be repeated.

(2-34)

For 0.0013 ≤ X ≤ 0.0022 (2 mm Hg ≤ P ≤ 760 mm Hg): sat

2663.129X − 5.994296 log Psat =  95.76X − 0.972546

(2-35)

For X < 0.0013 (P sat > 760 mm Hg): 2770.085X − 6.412631 log Psat =  36X − 0.989679

(2-36)

X is calculated from Eq. (2-37) and T′b is calculated from Eq. (2-38). Iterative calculation may be required. T ′b  − 0.0002867(T′b) T  X = 748.1 − 0.2145(T′b) (2-37) Psat Tb − T′b = 2.5f(K − 12) log  760 where

Both Riedel and Wagner regressions usually fit data within a few tenths of a percent over the entire range between the triple point and the critical point. Prediction Methods Two methods have gained almost universal acceptance for prediction of the vapor pressure of pure hydrocarbons. The method of Lee and Kesler55 is the preferred method if the critical temperature and the critical pressure of the hydrocarbon is known or can be reasonably predicted by the methods of the first section. The corresponding states method is shown in equation (2-31) with the simple fluid and correction terms to be calculated from equations (2-32) and (2-33), respectively, for any Tr. (1) sat (0) ln P sat + ω(ln P sat r = (ln P r ) r )

For X > 0.0022 (Psat < 2 mm Hg):

(2-38)

Tb = normal boiling point, °R T ′b = normal boiling point corrected to K = 12, °R T = absolute temperature, °R f = correction factor. For all subatmospheric vapor pressures and for all substances having normal boiling points greater than 400°F, f = 1. For substances having normal boiling points less than 200°F, f = 0. For superatmospheric vapor pressures of substances having normal boiling points between 200°F and 400°F, f is given by (Tb − 659.7)/200 K = Watson characterization factor, T b1/3/sp gr

Evaluation of the method for pure hydrocarbons shows errors averaging 8 percent for vapor pressures above 1 mm Hg and 30 percent below 1 mm Hg. The method is also usable for narrow boiling (range up to 50°F) undefined hydrocarbon mixtures with the only change being that the mean average boiling point replaces the normal boiling point in all calculations.

Example 6 Estimate the vapor pressure of tetralin at 150°C (302°F). Its normal boiling point is 207.6°C (405.7°F) and its Watson characterization factor is 9.78. Use the Maxwell-Bonnell method, Eq. (2-35). 2663.129X − 5.994296 log Psat =  95.76X − 0.972546 At 150°C with a normal boiling point of 207.6°C, a vapor pressure between 2 and 760 mm Hg would be expected. Assume T b′ = Tb = 405.7°F as a first trial. From Eq. (2-37): (865.7/762) − 0.0002867(865.7) X =  748.1 − 0.2145(865.7) X = 0.001579 2663.129(0.001579) − 5.994296 log Psat =  = 2.179 95.76(0.001579) − 0.972546 Psat = 151.0 mm Hg Use Eq. (2-38) to calculate the correction. 151 Psat Tb − T′b = 2.5f(K − 12) log  = 2.5(1)(9.48 − 12) log  760 760 Tb − T′b = 4.4° Thus, for the second trial, T′b = 405.7 − 4.4 = 401.3°F. Using Eq. (2-37), recalculate X. (861.3/762) − 0.0002867(861.3) X =  = 0.001568 748.1 − 0.2145(861.3) From Eq. (2-35), log Psat = 2.214; Psat = 163.7 mm Hg. Use Eq. (2-38) to recalculate the correction. 163.7 Tb − T′b = 2.5(1)(9.48 − 12) log  = 4.2° 760 Thus, T′b = 401.5°F. If greater accuracy is desired, carry out a third trial. (An experimental vapor pressure is 161.8 mm Hg.) For nonhydrocarbon organics, vapor pressures above 15 kPa for com-

PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES pounds of known or estimable normal boiling point are predicted using the method of Riedel89 given by Eq. (2-39). log P°r = φ(Tr) − (α − 7)ψ(Tr)

Use Eq. (2-44), determining parameters from Eqs. (2-45), (2-46), and (2-47). log Psat = m log Pwsat + C

(2-39)

at T1 = 256.55 K, P1sat = 2.00 × 104 Pa, log P1sat = 4.3010

Correlation functions φ(Tr), ψ(Tr), and ζ(Tr) are given by Eqs. (2-40), (2-41), and (2-42), respectively.

at T2 = 294.95 K, P2sat = 1.067 × 105 Pa, log P2sat = 5.0282 Use Eq. (2-45) to calculate the vapor pressure of water at T1 and T2.

φ(Tr) = 0.118ζ(Tr) − 7 log10 Tr

(2-40)

ψ(Tr) = 0.0364ζ(Tr) − log10 Tr

(2-41)

log Pωsat1 = 2.2282,

Pωsat1 = 169.12 Pa

ζ(Tr) = 36/Tr + 96.7 log10 Tr − 35 − Tr6

(2-42)

log Pωsat2 = 3.4213,

Pωsat2 = 2637.9 Pa

From Eq. (2-46): log P1sat − log P sat 4.3010 − 5.0282 2 m =  =  = 0.6093 log P wsat1 − log P wsat2 2.2282 − 3.4213 From Eq. (2-47):

The Riedel α is calculated from Eq. (2-43). 0.136ζ(Tr ) + log10 Pc − 5.01 α =  0.0364ζ(Tr ) − log10 Tr b

b

(2-43)

b

Critical properties, if not available, can be estimated from the methods of the previous section. Tr is the reduced temperature at the temperature of interest, while Tr is the reduced temperature at the normal boiling point. The method is accurate within 2 to 3 percent above 15 kPa, while errors increase to 10–30 percent at lower pressures. Care should be taken not to use the method below the freezing point temperature.

2-347

C = log P1sat − m log P wsat1 = 4.3010 − 0.6093(2.2282) C = 2.9434

b

T = 273.15 K (0°C) 3129.8 log P = 31.51 −  − 7.1385 log 273.15 + 1.757 × 10−6(273.15)2 273.15 log P sat w = 2.7907

at

sat w

Example 7 Estimate the vapor pressure of thiophene at 500 K. Pure component properties are Tc = 579.4 K, Pc = 5.694 MPa, and Tb = 357.5 K. Use the Riedel method, Eq. (2-39).

log Psat = (0.6093)(2.7907) + 2.9434 log Psat = 4.6438

log P sat r = −φ(Tr) − (α − 7) ψ (Tr) 500 357.5 Tr =  = 0.8629 Tr =  = 0.6170 579.4 579.4 From Eq. (2-42): (36) ζ(Tr ) =  + 96.7 log (0.6170) − 35 − (0.6170)6 = 3.01 (0.6170) From Eq. (2-43): 0.136(3.01) + log (5.694 × 106) − 5.01 α =  (0.0364)(3.01) − log (0.6170)

Psat = 44,030 Pa = 44.0 kPa

b

IDEAL GAS THERMAL PROPERTIES

b

α = 6.749 From Eq. (2-42), calculate ζ(Tr) and then calculate φ(Tr) and ψ(Tr) from Eqs. (2-40) and (2-41). log Prsat = −0.461 − (6.749 − 7)(0.068) log Prsat = −0.444 Prsat = 0.3598 Psat = PrsatPc = (.3598)(5.694) = 2.049 MPa An experimental value is 2.037 MPa. For nonhydrocarbon organics for which normal boiling points are unknown or expected vapor pressures are below 15 kPa, the reference substance method of Othmer and Yu78 as given by Eq. (2-44) is recommended. log Psat = m log P wsat + C

(2-44)

The vapor pressure of water P wsat may be calculated by Eq. (2-45). 3.1298 × 103 log Pwsat = 31.51 −  − 7.1385 log T + 1.757 × 10−6T 2 (2-45) T with temperatures in K and vapor pressures in Pa. Values of the compound specific constants m and c were originally derived by Othmer et al. and greatly expanded to over 600 common organics by Danner and Daubert.22 If constants are not available but any two vapor pressure data points are available, the constants m and C can be calculated using Eqs. (2-46) and (2-47). sat log P sat 1 − log P 2 m =  (2-46) log P wsat1 − log P wsat2 sat C = log P sat 1 − m log P w1

(2-47)

where the subscripts 1 and 2 refer to the two reference temperatures T1 and T2. Average errors at low pressures for compounds with tabulated m and C are within a few percent. When values of m and C are calculated from only two vapor pressure points, the method should be used only for interpolation and limited extrapolation. The method is usable from about 220 K (so long as it is above the freezing point of the compound) to the critical point of water (about 647 K).

Example 8 Estimate the vapor pressure of acetaldehyde at 0°C. Two vapor pressure points are 20.0 kPa at 256.55 K and 107.6 kPa at 294.85 K.

A substance is in the ideal gas state when the volume of its molecules is a zero fraction of the total volume taken up by the substance and when the individual molecules are far enough apart from each other so that there is no interaction between them. Although this only occurs at infinite volume and zero pressure, in practice, ideal gas properties can be used for gases up to a pressure of two atmospheres with little loss of accuracy. Thermal properties of ideal gas mixtures may be obtained by mole-fraction averaging the pure component values. Heat Capacity, Cpo Heat capacity is defined as the amount of energy required to change the temperature of a unit mass or mole one degree; typical units are J/kg·K or J/kmol·K. There are many sources of ideal gas heat capacities in the literature; e.g., Daubert et al.,24 Daubert and Danner,23 JANAF thermochemical tables,15 TRC thermodynamic tables,115,116 and Stull et al.104 If Cpo values are not in the preceding sources, there are several estimation techniques that require only the molecular structure. The methods of Thinh et al.113 and Benson et al.6,7 are the most accurate but are also somewhat complicated to use. The equation of Harrison and Seaton36 for Cpo between 300 and 1500 K is almost as accurate and easy to use: Cpo = a1 + a2C + a3H + a4O + a5N + a6S + a7F + a8Cl + a9I + a10Br + a11Si + a12Al + a13B + a14P + a15E where

(2-48)

C = ideal gas heat capacity, J/mol K a1–a15 = constant parameters obtained from Table 2-387 as a function of temperature C = number of carbon atoms in the molecule H = number of hydrogen atoms in the molecule O = number of oxygen atoms in the molecule N = number of nitrogen atoms in the molecule S = number of sulfur atoms in the molecule F = number of fluorine atoms in the molecule Cl = number of chlorine atoms in the molecule I = number of iodine atoms in the molecule Br = number of bromine atoms in the molecule Si = number of silicon atoms in the molecule Al = number of aluminum atoms in the molecule B = number of boron atoms in the molecule P = number of phosphorus atoms in the molecule E = number of atoms in the molecule excluding the 13 atom-types listed above o p

2-348

PHYSICAL AND CHEMICAL DATA

TABLE 2-387

Values of the Constant Parameters a1–a15 in Eq. (2-48) at Different Temperatures

Temp., K

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

300 400 500 600 800 1000 1500

4.86 0.864 −1.85 −4.61 −7.49 −8.53 −7.37

9.04 12.6 15.5 17.5 20.1 21.6 23.9

5.69 7.37 8.89 10.5 13.1 15.2 17.9

11.4 13.9 15.7 17.5 19.4 20.4 20.6

11.9 14.0 16.0 17.3 19.4 20.4 21.1

15.3 17.0 19.4 20.3 22.3 22.9 22.5

12.7 16.2 17.9 20.1 21.5 22.4 22.1

16.8 18.9 20.2 21.4 22.4 22.8 22.6

18.7 20.5 22.1 23.3 25.0 25.4 24.6

17.8 19.9 21.2 22.4 23.4 23.8 23.0

14.6 17.5 19.6 20.9 23.2 23.9 24.1

15.8 18.3 20.0 21.1 22.3 22.8 23.2

11.5 14.7 17.0 18.3 20.8 22.2 24.2

18.0 20.9 21.6 22.8 23.0 23.4 24.2

19.5 20.8 21.7 22.1 23.0 23.3 23.3

Parameter a

n = number of different atomic groups contained in the molecule Ni = number of atomic groups i contained in the molecule ∆Gi = numeric value of atomic group i obtained from Table 2-388

Results and parameters may be interpolated between temperatures. Average errors are between 2 and 6 percent, with the higher errors at the lower temperatures. Example 9 Using Eq. 2-48 to estimate the ideal gas heat capacity of acetone (C3H6O) at 600 K: Cp° = −4.61 + (17.5)(3) + (10.5)(6) + (17.5)(1) = 128.39 J/mol K

Average errors of 8 to 9 kJ/mol may be expected.

Daubert et al.24 report a value of 121.8 J/mol K.

Enthalpy of Formation The ideal gas standard enthalpy (heat) of formation (∆H°f 298) of a chemical compound is the increment of enthalpy associated with the reaction of forming that compound in the ideal gas state from the constituent elements in their standard states, defined as the existing phase at a temperature of 298.15 K and one atmosphere (101.3 kPa). Sources for data are Refs. 15, 23, 24, 104, 115, and 116. The most accurate, but again complicated, estimation method is that of Benson et al.6,7 A compromise between complexity and accuracy is based on the additive atomic group-contribution scheme of Joback44; his original units of kcal/mol have been converted to kJ/mol by the conversion 1 kcal/mol = 4.1868 kJ/mol:

Example 11 The ∆G°f 298 of phenol is estimated using Table 2-388. The molecular groups are

(2-49)

i=1

where ∆H°f 298 = enthalpy of formation at 298.15 K, kJ/mol n = number of different atomic groups contained in the molecule Ni = number of atomic groups i contained in the molecule ∆Hi = numeric value of atomic group i obtained from Table 2-388. Average expected errors are about 9 kJ/mol. For other temperatures: ∆H°f T = ∆H°f 298 +



T

C°p dT

(2-50)

298

C

1

OH (phenol)

(both ring)

∆G°f 298 = 53.88 + 5(11.30) + (54.05) + (−197.37) = −32.94 kJ/mol

∆G°f T = ∆H°f T − T ∆S°f T

where

(2-52)

∆G°f T = Gibbs energy of formation at T, kJ/mol ∆H°f T = enthalpy of formation at T, kJ/mol (see above) ∆S°f T = entropy of formation at T, kJ/mol K (see below)

Entropy of Formation The ideal gas standard entropy of formation (∆S°f 298) of a chemical compound is the increment of entropy associated with the reaction of forming that compound in the ideal gas state from the constituent elements in their standard state defined as the existing phase at a temperature of 298.15 K and one atmosphere (101.325 kPa). Thus: n

r ∆S°f 298 = S°compound − Ni Selement i

Example 10 The ∆H°f 298 of 2-butanol is estimated using Table 2-388. The molecular groups are 2CH3, 1CH2, 1CH (all nonring), and 1OH (alcohol). Therefore: ∆H°f 298 = 68.29 + 2(−76.45) + (−20.64) + (29.89) + (−208.04) = −283.40 kJ/mol The value from Daubert et al.24 is −292.9 kJ/mol.

Gibbs Energy of Formation The ideal gas standard Gibbs energy of formation (∆G°f 298) of a chemical compound is the increment of Gibbs energy associated with the reaction of forming that compound in the ideal gas state from the constituent elements in their standard state defined as the existing phase at a temperature of 298.15 K and one atmosphere (101.325 kPa). Refs. 15, 23, 24, 104, 115, and 116 are good sources of data. The additive atomic groupcontribution scheme of Joback44 may be used to estimate ∆G°f 298; his original units of kcal/mol have been converted to kJ/mol by the conversion 1 kcal/mol = 4.1868 kJ/mol: n

(2-51)

i=1

where

1

The value from Daubert et al.24 is −32.64 kJ/mol. For other temperatures, the exact Eq. (2-52) may be used at temperature T (K):

See above for discussion of the ideal gas heat capacity (C°p ).

∆G°f 298 = 53.88 + Ni ∆Gi

CH

Therefore,

n

∆H°f 298 = 68.29 + Ni ∆Hi

5

∆G°f 298 = Gibbs energy of formation at 298.15 K, kJ/mol

(2-53)

i=1

where

∆S°f 298 = entropy of formation at 298.15 K and 1 atm, J/mol K S°compound = ideal gas absolute entropy of the compound at 298.15 K and 1 atm, J/mol K n = number of different elements contained in the compound Ni = moles of element i contained in one mole of compound r Selement i = absolute entropy of element i in its standard state at 298.15 K and 1 atm, J/mol K.

Ideal gas absolute entropies of many compounds may be found in Daubert et al.,24 Daubert and Danner,23 JANAF Thermochemical Tables,15 TRC Thermodynamic Tables,115,116 and Stull et al.104 Otherwise, the estimation method of Benson et al.6,7 is reasonably accurate, with average errors of 1–2 J/mol K. Elemental standard-state absolute entropies may be found in Cox et al.21 Values from this source for some common elements are listed in Table 2-389. ∆S°f 298 may also be calculated from Eq. (2-52) if values for ∆H°f 298 and ∆G°f 298 are known.

PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES TABLE 2-388

Atomic Group Contributions to Estimate ⌬H f°298 and ⌬G f°298 ∆H

Nonring Increments CH3

∆G

−76.45

−43.96

CH2

−20.64

8.42

CH

29.89

58.36

∆H

∆G

Oxygen Increments (Cont.) CHO (aldehyde)

−162.03

−143.48

COOH (acid)

−426.72

−387.87

COO

−337.92

−301.95

−247.61

−250.83

−22.02

14.07

NH (nonring)

53.47

89.39

NH (ring)

31.65

75.61

(ester)

O (except for above) C

82.23

116.02

CH2

−9.63

3.77

CH

37.97

48.53

C

83.99

92.36

C CH C Ring Increments CH2 CH

C

123.34

163.16

(nonring)

23.61

—

79.30

77.71

N

(ring)

55.52

79.93

115.51

109.82

NH

93.70

119.66

−26.80

−3.68

CN

88.43

89.22

NO2

−66.57

−16.83

8.67

40.99

Sulfur Increments SH S (nonring) S (ring)

−17.33 41.87 39.10

−22.99 33.12 27.76

−251.92 −71.55 −29.48 21.06

−247.19 −64.31 −38.06 5.74

87.88

46.43

54.05

−208.04

−189.20

OH (phenol)

−221.65

−197.37

O

(nonring)

−132.22

−105.00

O

(ring)

−138.16

−98.22

C

O (nonring)

−133.22

−120.50

C

O (ring)

−164.50

−126.27

Halogen Increments F Cl Br I

where

ENTHALPY OF VAPORIZATION AND FUSION Enthalpy of Vaporization The enthalpy (heat) of vaporization ∆HV is defined as the difference of the enthalpies of a unit mole or mass of a saturated vapor and saturated liquid of a pure component; i.e., at a temperature (below the critical temperature) and corresponding vapor pressure. ∆HV is related to vapor pressure by the thermodynamically exact Clausius-Clapeyron equation: d ln Psat ∆Hv = −R ∆ZV  d (1/T)

(nonring)

N

11.30

(aromatic or cyclic olefin)

N 136.70

2.09

Oxygen Increments OH (alcohol)

Nitrogen Increments NH2

142.14

79.72

CH (aromatic or cyclic olefin) C

2-349

(2-54)

R = gas constant in energy units ∆ZV = ZG − ZL ZG = compressibility factor of the saturated vapor ZL = compressibility factor of the saturated liquid Psat = vapor pressure T = absolute temperature

(2-55)

If accurate ZG and ZL data are available, excellent ∆HV values can be obtained by differentiating a vapor pressure correlation and using Eq. (2-54). If not, ∆ZV may be esimated by Haggenmacher’s equation34:

2-350

PHYSICAL AND CHEMICAL DATA where Tr 1,2 = reduced temperature = T1 /Tc or T2 /Tc T1,2 = temperature, K Tc = critical temperature, K

TABLE 2-389 Standard-State Entropy of Elements at 298.15 K and 1 Atmosphere Element

State

Absolute entropy J/mol K

C H2 O2 N2 S F2 Cl2 Br2 I2

crystal (graphite) gas gas gas crystal (rhombic) gas gas liquid crystal

5.74 130.571 205.043 191.500 32.054 202.682 222.972 152.21 116.14





P 1/2 ∆Zv = 1 − r Tr where Pr = reduced pressure = P/Pc Tr = reduced temperature = T/Tc 3

Equation (2-59) works best between the normal boiling and critical temperatures, producing values of engineering accuracy.

Example 14 Estimate ∆Hv of ethyl acetate at 450 K, using the normal boiling point values as a basis (see Example 13). ∆Hv1 = 32.23 kJ/mol, Tr1 = 0.6692, and Tr2 = 450.0/523.3 = 0.8599. Substituting in Eq. (2-59): 1 − 0.8599 0.38 ∆Hv(450) = 32.23  = 23.25 kJ/mol 1 − 0.6692 A value of 23.16 kJ/mol is obtained from Daubert et al.24



(2-56)

However, Eq. (2-56) should be used only near or below the normal boiling point; even then, the accuracy of the resulting ∆HV is significantly reduced. The corresponding states approach suggested by Pitzer et al.82 requires only the critical temperature and acentric factor of the compound. For a close approximation, an analytical representation of this method proposed by Reid et al.86 for 0.6 < Tr < 1.0 is: ∆HV /RTc = 7.08(1 − Tr)0.354 + 10.95ω(1 − Tr)0.456

(2-57)

where ∆HV = enthalpy of vaporization, kJ/mol R = gas constant = 0.008314 kJ/mol K Tc = critical temperature, K Tr = reduced temperature, T/Tc T = temperature, K ω = acentric factor

Enthalpy of Fusion The enthalpy (heat) of fusion ∆Hfus is defined as the difference of the enthalpies of a unit mole or mass of a solid and liquid at its melting temperature and one atmosphere pressure of a pure component. There are no generally applicable estimation techniques that are very accurate. However, if the melting temperature is known, the atomic group contribution method of Chickos et al.16 yields approximate results: ∆Hfus = Tmlt∆Sfus where

a = 35.19 NR + 4.289 (NCH2 − 3 NR) where

Example 12 Estimate ∆HV of Propionaldehyde at 350 K. The required properties from Daubert et al.24 are Tc = 504.4 K and ω = 0.2559. Tr = 350.0/504.4 = 0.6939. Substituting in Eq. (2-57): ∆HV /RTc = (7.08)(1 − 0.6939)0.354 + (10.95)(0.2559)(1 − 0.6939)0.456 = 6.289 ∆HV = (6.289)(0.008314)(504.4) = 26.37 kJ/mol The reported value is 26.85 kJ/mol.24 The enthalpy of vaporization at the normal boiling temperature ∆Hvb (kJ/mol) can be estimated by an equation suggested by Riedel90: (ln [Pc /101.325] − 1) (2-58) ∆Hvb = 1.093 RTc Tbr  0.930 − Tbr where R = gas constant = 0.008314 kJ/mol K. Tc = critical temperature, K Tbr = reduced normal boiling temperature = Tb /Tr Tb = normal boiling temperature, K Pc = critical pressure, kPa



Average errors are about 2 percent.

Example 13 Estimate ∆HVb of Ethyl Acetate. The required properties for ethyl acetate are from Daubert et al.24: Tc = 523.3 K, Tb = 350.2 K, and Pc = 3880.0 kPa. Tbr = 350.2/523.3 = 0.6692. Substituting in Eq. (2-58): (ln [3880.0/101.325] − 1) ∆HVb = (1.093)(0.008314)(523.3) (0.6692)  0.930 − 0.6692







= 32.28 kJ/mol The value from Daubert et al.24 is 32.23 kJ/mol. The enthalpy of vaporization decreases with temperature and is zero at the critical point. If the value of an enthalpy of vaporization ∆Hv1 is known at temperature T1, this temperature dependency can be represented by the Watson relation127 to calculate another enthalpy of vaporization ∆Hv2 at any other temperature T2: 1 − Tr 0.38 (2-59) ∆Hv2 = ∆Hv1 2 1 − Tr1





(2-60)

∆Hfus = enthalpy of fusion at the melting temperature, J/mol Tmlt = melting temperature, K ∆Sfus = a + b = entropy of fusion at the melting temperature, J/mol K.

It should be noted that the methodology for a and b results in a ∆Sfus associated with the phase change from a solid at 0 K to the liquid at Tmlt. No entropy changes resulting from solid transitions are taken into account, and ∆Sfus for a substance that undergoes such a transition will be overestimated by this technique.

Maximum errors are in the order of 8 percent.





(2-61)

NR = number of nonaromatic rings NCH2 = number of CH2 atomic groups in nonaromatic ring(s) required to form a cyclic paraffin of the same ring size(s) as contained in the molecule of interest.

Example: For

O

, NCH 2 = 5; a = 0 if there are no nonaromatic rings

in the molecule of interest. If a nonaromatic ring in fact contains a CH2 atomic group, then no consideration of that group in the b term in Eq. (2-62) is required. ng

ns

nf

i=1

j=1

k=1

b = (Ng)i(∆s)i + (Ns)j(Cs)j(∆s)j + (Nf )k(Ct)k(∆s)k where

(2-62)

ng = number of different nonring or aromatic C-H atomic groups bonded to other carbon atoms in the molecule of interest (Ng)i = number of C-H atomic groups i bonded to other carbon atoms in the molecule of interest ns = number of different nonring or aromatic C-H atomic groups bonded to at least one functional group or atom in the molecule of interest (Ns)j = number of C-H atomic groups j bonded to at least one functional group or atom in the molecule of interest nf = number of different functional groups or atoms in the molecule of interest (Nf)k = number of functional groups or atoms k in the molecule of interest (Cs)j = coefficient for C-H atomic group j bonded to at least one functional group or atom in the molecule of interest; numeric values for C-H atomic groups are found in Table 2-390 Ct = coefficient for the functional group or atom k in the molecule of interest, where t = the total number of functional groups or atoms in the molecule of interest. Exception: Molecules containing any number of fluorine atoms are treated as having only one functional fluorine atom. Numeric values of C1–C4 are in Table 2-391 for functional groups or atoms (∆s)i, j, k = contribution of the atomic group or atom i, j, or k to the entropy of fusion, J/mol K. Numeric values for C-H atomic groups are in Table 2-390; values for functional groups or atoms are in Table 2-391

PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES TABLE 2-390 Cs and ⌬s Values for C-H Atomic Groups to Estimate ⌬Hfus Cs

∆s

CH3

1.0

18.33

CH2

1.0

9.41

CH

0.69

−16.19

C

0.67

−38.70

CH2

1.0

14.56

CH

3.23

4.85

C

1.0

−11.38

CH

1.0

10.88

C

1.0

2.18

1.0

6.44

Nonring

2-351

There are no reliable prediction methods for solid heat capacity as a function of temperature. However, the atomic element contribution method of Hurst and Harrison,41 which is a modification of Kopp’s Rule,50 provides estimations at 298.15 K and is easy to use: n

Aromatic CH C

(bonded to paraffinic C)

1.0

−10.33

C

(bonded to olefinic C or non-C)

1.0

−4.27

CpS = Ni ∆Ei

(2-63)

i=1

where CpS = solid heat capacity at 298.15 K, J/mol K n = number of different atomic elements in the compound Ni = number of atomic elements i in the compound ∆Ei = numeric value of the contribution of atomic element i found in Table 2-393 Average errors are in the 9–10 percent range. Example 15 Estimate solid heat capacity of dibenzothiophene, C12H8S. The required atomic element contributions from Table 2-393 are: C = 10.89, H = 7.56, and S = 12.36. Substituting in Eq. (2-63): CpS = (12)(10.89) + (8)(7.56) + (1)(12.36) = 203.52 J/mol K Daubert et al.24 report a value of 198.5 J/mol K.

Liquid Heat Capacity The two commonly used liquid heat capacities are either at constant pressure or at saturated conditions. There is negligible difference between them for most compounds up to a reduced temperature (temperature/critical temperature) of 0.7. Liquid heat capacity increases with increasing temperature, although a minimum occurs near the triple point for many compounds. There are a number of reliable estimating techniques for obtaining pure-component liquid heat capacity as a function of temperature, including Ruzicka and Dolmalski,93,94 Tarakad and Danner,112 and Lee and Kesler.55 These methods are somewhat complicated. The relatively simple atomic group contribution approach of Chueh and Swanson17 for liquid heat capacity at 293.15 K is presented here: n

CpL = Ni ∆cpi + 18.83m

(2-64)

i=1

1.0

−2.51

CH

0.76

−15.98

C

1.0

−32.97

CH

0.62

−4.35

C

0.86

−11.72

1.0

−5.36

C

(bonded to acetylenic C)

Ring

C or

C

where CpL = liquid heat capacity at 293.15 K, J/mol K. n = number of different atomic groups in the compound Ni = number of atomic groups i in the compound ∆cpi = numeric value of the contribution of atomic element i found in Table 2-394. The original units of cal/mol K have been converted to J/mol K by the conversion 1 cal/mol K = 4.184 J/mol K m = number of carbon groups requiring an additional contribution, which are those that are joined by a single bond to a carbon group, which in turn is connected to a third carbon group by a double or triple bond. If a carbon group meets this criterion in more than one way, m should be increased by one for each of the ways. Exceptions: CH3 groups or carbon groups in a ring never require an additional contribution; and the first additional contribution for a CH2 group is 10.46 J/mol K rather than 18.83 J/mol K. However, if the CH2 group meets the criterion in a second way, the second additional contribution reverts to the 18.83 J/mol K value (see Example 17, below).

Chickos et al.16 report an average error of 2050 J/mol for monofunctional molecules and 3180 J/mol for multifunctional molecules when using their method to estimate ∆Hfus. Four example estimations are shown in Table 2-392.

Errors should be less than 6 percent for all compounds except for acids, amines, and halides.

SOLID AND LIQUID HEAT CAPACITY

Example 16 Estimate the liquid heat capacity at 293.15 K of 2-butanol. The atomic groups are:

The heat capacity is defined as the amount of energy required to change the temperature of a unit mass or mole one degree; typical units are J/kg·K or J/kmol·K. Solid Heat Capacity Solid heat capacity increases with increasing temperature, with steep rises near the triple point for many compounds. When experimental data are available, a simple polynomial equation in temperature is often used to correlate the data. It should be noted that step changes in heat capacity occur if the compound undergoes crystalline state changes at different temperatures.

2

CH3

1

CH2

1 CHOH Substituting in Eq. (2-64) the atomic group contributions from Table 2-394 with m = 0: CpL = (2)(36.82) + (1)(30.38) + (1)(76.15) = 180.17 J/mol K

The value from Daubert et al.24 is 190.3 J/mol K.

TABLE 2-391 Ct and Ds Values for Functional Groups and Atoms to Estimate DHfus ∆s

C1

C2

C3

C4

OH (alcohol)

1.0

12.6

18.9

26.4

1.13

OH (phenol)

1.0

1.0

1.0

1.0

16.57

O

(ether, nonring)

1.0

1.0

1.0

1.0

1.09

O

(ether, ring)

1.0

1.0

1.0

1.0

1.34

C

O (ketone, nonring)

1.0

1.0

3.14

C

O (ketone, ring)

1.0

1.0

−1.88

CHO (aldehyde)

1.0

COOH (acid)

1.0

1.83

1.88

1.72

14.90

COO

(ester)

1.0

1.0

1.0

1.0

3.68

NH2 (aliphatic)

1.0

1.82

16.23

NH2 (aromatic)

1.0

1.0

15.48

NH (nonring)

1.0

1.0

−2.18

NH (ring)

1.0

1.84

N

(nonring)

1.0

−15.90

N

(ring)

1.0

1.0

−17.07

N

(ring)

1.0

1.0

1.67

N

(aromatic)

1.0

1.0

CN (nitrile)

1.0

1.4

NO2

1.0

1.0

1.0

1.0

26.19

1.0

1.0

−0.42

1.0

1.0

17.99

19.66

1.0

7.32 9.62

1.0

17.36

O C NH2 O C NH SH S S

(nonring)

1.0

(ring)

1.0

–

2-352

0.36 1.0

7.20 2.18

SO2 (nonring)

1.0

F (on

C)

1.0

1.0

1.0

1.0

14.73

F (on

C)

1.0

1.0

1.0

1.0

13.01

F (on ring C)

1.0

1.0

1.0

1.0

15.90

Cl

1.0

2.0

2.0

1.93

8.37

Br

1.0

1.0

1.0

0.82

17.95

I

1.0

1.0

3.26

16.95

PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES TABLE 2-392

2-353

Examples of Estimations of DHfus, J/mol Melting temp., K

Molecule

Atomic group cyclopentane

(35.19)(1) + (4.289) [5 − (3)(1)]

=

43.77

(1)(0.76)(−15.98)

=

−12.14

(1)(1.0)(17.99) ∆Sfus

= =

17.99 49.62

∆Hfus = (49.62)(155.4) ∆Hfus = (experimental)16

= =

(35.19) (1) + (4.289)[5 − (3)(1)]

=

43.77

CH (aromatic)

(4)(6.44)

=

25.76

C

(ring)

(1)(−11.72)

=

−11.72

C

(ring)

(1)(0.86)(−11.72)

=

−10.08

CH (ring)

(1)(−4.35)

=

−4.35

CH (ring)

(1)(0.62)(−4.35)

=

−2.69

(1)(1.0)(2.18) ∆Sfus

= =

2.18 42.87

∆Hfus = (42.87)(304.5) ∆Hfus = (experimental)16

= =

CH (aromatic)

(4)(6.44)

=

25.76

C

(aromatic)

(1)(1.0)(−4.27)

=

−4.27

C)

(1)(1.0)(13.01)

=

13.01

(1)(−10.33)

=

−10.33

CH

(1)(0.69)(−16.19)

=

−11.17

COOH

(1)(1.88)(14.90)

=

28.01

OH (alcohol)

(1)(18.9)(1.13) ∆Sfus

= =

21.36 62.37

∆Hfus = (62.37)(363.0) ∆Hfus = (experimental)16

= =

CH2

(2)(1.0) 9.41

=

18.82

CH

(2)(0.69)(−16.19)

=

−22.34

OH (alcohol)

(2)(26.4)(1.13)

=

59.66

Br

(2)(0.82)(17.95) ∆Sfus

= =

29.44 85.58

∆Hfus = (85.58)(338.2) ∆Hfus = (experimental)16

= =

CH SH

155.4

Contribution

(ring)

SH

(t = 1) cyclopentane

7710.9 7802

S 304.5 (t = 1)

S

(ring)

13054 11823

F CH

363.0

COOH

F (on

OH C

(aromatic)

(t = 3)

HO

CH2

H

Br

C

C

Br H (t = 4)

338.2 CH2

OH

22640 20959

28943 29291

2-354

PHYSICAL AND CHEMICAL DATA

TABLE 2-393 Atomic Element Contributions to Estimate Solid Heat Capacity at 298.15 K Atomic element

∆E

Atomic element

∆E

Atomic element

∆E

C H O N S F Cl Br I Al B

10.89 7.56 13.42 18.74 12.36 26.16 24.69 25.36 25.29 18.07 10.10

Ba Be Ca Co Cu Fe Hg K Li Mg Mn

32.37 12.47 28.25 25.71 26.92 29.08 27.87 28.78 23.25 22.69 28.06

Mo Na Ni Pb Si Sr Ti V W Zr All other

29.44 26.19 25.46 31.60 17.00 28.41 27.24 29.36 30.87 26.82 26.63

TABLE 2-394

Example 17 Estimate liquid heat capacity at 293.15 K of 1,4-pentadiene, CH2苷CHCH2CH苷CH2. The atomic groups are: 2

CH2

2

CH

1

CH2

The CH2 group is twice joined by a single bond to a carbon group, which in turn is connected to a third carbon group by a double bond, and m = 2. However, by the second exception, the first additional contribution is 10.46 J/mol K rather than 18.83 J/mol K. Substituting in Eq. (2-64) the atomic group contributions from Table 2-394: CpL = (2)(21.76) + (2)(21.34) + (1)(30.38) + 10.46 + 18.83 = 145.87 J/mol K

Atomic Group Contributions to Estimate Liquid Heat Capacity at 293.15 K ∆cp

Nonring Increments CH3

36.82

CH2

30.38

CH

20.92

∆cp Oxygen Increments (Cont.) CH2OH CHOH

7.36

CH2

21.76

CH

21.34

C

15.90

CH C Ring Increments CH2 CH

OH (except for above) ONO2 Nitrogen Increments NH2

C

CH

43.93

24.69

N

31.38

24.69

N

25.94

CN

(ring)

Sulfur Increments SH

12.13

Halogen Increments F

18.83 58.16

44.77 33.47

16.74

Cl (first or second on a carbon)

35.98

Cl (third or fourth on a carbon)

25.10

Br

37.66

I

35.98

22.18

Oxygen Increments O C

58.58

NH

18.41

or

44.77 119.24

S C

76.15

111.29

COH C

73.22

O

35.15 52.97

CHO (aldehyde)

52.97

COOH (acid)

79.91

COO

60.67

(ester)

Hydrogen Increment H (for formic acid, formates, hydrogen cyanide, etc.)

14.64

PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES Daubert et al.24 report a value of 145.6 J/mol K. For liquid mixtures, the values of the pure components can be molefraction-averaged. This procedure neglects any heat of mixing effects.

DENSITY Density is defined as the mass of a substance contained in a unit volume. In the SI system of units, the ratio of the density of a substance to the density of water at 15°C is known as its relative density, while the older term specific gravity is the ratio relative to water at 60°F. Various units of density, such as kg/m3, lb-mass/ft3, and g/cm3, are commonly used. In addition, molar densities, or the density divided by the molecular weight, is often specified. This section briefly discusses methods of correlation of density as a function of temperature and presents the most common accurate methods for prediction of vapor, liquid, and solid density. Correlation Methods Vapor densities are not correlated as functions of temperature alone, as pressure and temperature are both important. At high temperatures and very low pressures, the ideal gas law can be applied; while at moderate temperature and low pressure, vapor density is usually correlated by the virial equation. Both methods will be discussed later. Molar liquid density (ρ) is best correlated by an equation adopted from the Rackett predictor. The equation has the form of Eq. (2-65): A ρ = D (2-65) T B 1 −  C The regression constants A, B, and D are determined from the nonlinear regression of available data, while C is usually taken as the critical temperature. The liquid density decreases approximately linearly from the triple point to the normal boiling point and then nonlinearly to the critical density (the reciprocal of the critical volume). A few compounds such as water cannot be fit with this equation over the entire range of temperature. Liquid density data to be regressed should be at atmospheric pressure up to the normal boiling point, above which saturated liquid data should be used. Constants for 1500 compounds are given in the DIPPR compilation.24 Solid density data are sparse and usually only available over a narrow temperature range, for which the general decrease in density with temperature is approximately linear. Vapor Density Prediction A myriad of methods exist for prediction of vapor density as a function of temperature and pressure. This section will only present the most accurate and generally used methods. For simple molecules at temperatures above the critical and at pressures no more than a few atmospheres, the ideal gas law, Eq. (2-66), may be used to estimate vapor density.





1 P ρ== V RT

(2-66)

At slightly higher pressures up to a reduced pressure of about 0.4, the truncated virial equation, Eq. (2-67), is commonly used for all types of organic fluids. PV B Z==1+ (2-67a) RT V P Z =  = 1 + Bρ ρRT

(2-67b)

Second virial coefficients, B, are a function of temperature and are available for about 1500 compounds in the DIPPR compilation.24 The second virial coefficient can be regressed from experimental PVT data or can be reasonably and accurately predicted. Tsonopoulos117 proposed a prediction method for nonpolar compounds that requires the critical temperature, critical pressure, and acentric factor. Equations (2-68) through (2-70) describe the method. BPc (2-68)  = B0 + ωB1 RTc



 

 



 

 

 

2-355

0.330 0.1385 0.0121 0.000607 B0 = 0.1445 −  −  −  −  Tr T r2 T 3r T r5

 (2-69)



0.0331 0.423 0.008 −  −  (2-70) B1 = 0.0637 +  T r2 T 3r T 8r For non-hydrogen-bonding polar compounds such as carbonyls and ethers, Tsonopoulos117 recommends that Eq. (2-68) be expanded to a third term that is a function of the reduced dipole moment (µr) as described by Eqs. (2-71) through (2-73): BPc (2-71)  = B0 + ωB1 + B2 RTc B2 = −0.0002410λr − 4.308 × 10−21λ8r (2-72) 105λ2p Pc λr =  (2-73) T c2 The dipole moment λp in Eq. (2-73) is in debyes, while Pc is in atm and Tc is in K. Units must be watched carefully. For hydrogen-bonding molecules, Eq. (2-71) can be used with a value of B2 calculated by Eq. (2-74). a b B2 = 6 − 8 (2-74) Tr Tr Variables a and b are specific constants reported by Tsonopoulos117 for some alcohols and water (e.g., methanol: a = 0.0878, b = 0.0560; and water: a = 0.0279, b = 0.0229). Tsonopoulos also gives specific prediction methods for haloalkanes118 and water pollutants.119 Example 18 Estimate the molar volume of isobutane at 155°C and 1.0 MPa pressure. Properties of isobutane are Tc = 135.0°C, Pc = 3.647 MPa, and ω = 0.1170. 1.0 155 + 273.1 Tr =  = 1.05 Pr =  = 0.274 135 + 273.1 3.647 Since reduced pressure is below 0.4, use virial equation (2-67a). Calculate B by the Tsonopoulos method, Eq. (2-68). BPc  = B0 + ωB1 RTc Using Eq. (2-69):



 

 

 

0.330 0.1385 0.0121 0.000607 −  −  B0 = 0.1445 −  −  1.05 1.052 1.053 1.055



B0 = 0.1445 − 0.3143 − 0.1256 − 0.0105 − 0.0005 = −0.3064 Using Eq. (2-70):



 

 

0.0331 0.423 0.008 −  −  B1 = 0.0637 +  1.052 1.053 1.058



B1 = 0.0637 + 0.0300 − 0.3654 − 0.0054 = −0.2771 [−0.3064 + (0.1770)(−0.2771)](8314)(408.1) B =  = −0.3307 (3.647 × 106) B PV =1+ RT V 106 V 0.3307  = 1 −  (8314)(428.1) V Trial and error or the quadratic formula can be used for the solution. If you opt for trial and error, start with the ideal gas value (B = 0) where V = 3.559 m3/ kmole [Eq. (2-66)]. Solving, V = 3.190 m3/kmole. For prediction of vapor density of pure hydrocarbon and nonpolar gases, the corresponding states method of Pitzer et al.82 is the most accurate method, with errors of less than 1 percent except in the critical region where errors of up to 30 percent can occur. The method correlates the compressibility factor by Eq. (2-75), after which the density can be calculated by Eq. (2-75): Z = Z(0) + ωZ(1) (2-75) P 1 ρ== (2-76) ZRT V Z(0) is the compressibility factor for the simple fluid, while Z(1) is the correction term for molecular acentricity, both of which are functions of Tr and Pr. Both

2-356

PHYSICAL AND CHEMICAL DATA

plots and detailed tabulations of the functions are available in the Technical Data Book.23 Critical temperature and pressure and the acentric factor from tabulations or as predicted are required. For hydrogen, Tc and Pc should be taken as 41.7 K and 2100 kPa, respectively. For approximate calculations, Figs. 2-35 and 2-36 should be used for calculating Z(0) and Z(1), respectively, for superheated vapors with 0.2 ≤ Pr ≤ 10. Z(0) will approach 1, and Z(1) will approach 0 for Pr < 0.2. More accurately, Eq. (2-77) can be used for Pr between 0 and 0.2.

FIG. 2-35

P Z = 1 + r [(0.1445 + 0.073ω) − (0.330 − 0.46ω)T r−1 Tr − (0.1385 + 0.50ω)T r−2 − (0.0121 + 0.097ω)T r−3 − 0.0073ωT r−8]

(2-77)

Extension of the pressure range to Pr = 14 is available in the Technical Data Book. For saturated vapor densities, the values of Z(0) and Z(1) are tabulated as a

Generalized compressibility factors—Pitzer Method, simple fluid term.

PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES function of reduced pressure in Table 2-395. If the saturation temperature rather than the saturation pressure is known, the vapor pressure of the compound can be determined either from data or the vapor pressure prediction methods discussed earlier.

As high pressure, use Eq. (2-75) to calculate Z and then Eq. (2-76) to estimate the molar volume. The properties of isobutane necessary are Tc = 135.0°C, Pc = 3.647 MPa, and ω = 0.1770.

Example 19 Estimate the molar volume of isobutane at 155°C and 8.6 MPa pressure.

FIG. 2-36

2-357

Generalized compressibility factors—Pitzer Method, correction term.

155 + 273.1 Tr =  = 1.05 135 + 273.1

8.6 Pr =  = 2.36 3.647

2-358

PHYSICAL AND CHEMICAL DATA

TABLE 2-395

Saturated Vapor Density Parameters

Pr

Z (0)

Z (1)

Pr

Z (0)

Z (1)

1.00 0.99 0.98 0.97 0.96 0.95 0.94 0.92 0.90 0.85 0.80 0.75 0.70

0.291 0.35 0.38 0.40 0.41 0.42 0.43 0.45 0.47 0.50 0.53 0.56 0.59

−0.080 −0.083 −0.085 −0.087 −0.088 −0.089 −0.089 −0.090 −0.091 −0.090 −0.087 −0.081 −0.075

0.65 0.60 0.55 0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05

0.615 0.64 0.665 0.688 0.711 0.734 0.758 0.783 0.809 0.835 0.864 0.896 0.935

−0.069 −0.063 −0.056 −0.049 −0.041 −0.033 −0.025 −0.018 −0.012 −0.008 −0.005 −0.002 0.000

Using Fig. 2-35, Z(0) = 0.385. Using Fig. 2-36, Z(1) = −0.063. Z = Z(0) + ω Z(1) Z = 0.385 + 0.1770(−0.063) = 0.374 ZRT (0.374)(8314)(428.1) V =  =  P 8.6 × 106 V = 0.155 m3/kmol An experimental value for Z is 0.377. Note that use of the Lee-Kesler fit [Eq. (2-78)] would give a slightly more accurate answer than the graphical method, and this fit is used for any computer applications. Lee and Kesler55 fit the entire Pitzer method to equations, rewriting the basic Eq. (2-75) with respect to a heavy reference fluid (n-octane) as shown by Eq. (2-78). ω (h) Z = Z(0) +  (Z − Z(0)) ω(h)

(2-78)

where h specifies the heavy reference fluid with an acentric factor of 0.3978. The parameters in the equation are calculated for the simple fluid and the heavy reference fluid with an acentric factor of 0.3978. The parameters in the equation are calculated for the simple fluid and the heavy reference fluid from Eq. (2-79) PrVr B C D c4 γ −γ Z(i) =  = 1 +  + 2 + 5 +  β + 2 exp 2 Tr Vr V r Vr T r3 V 2r Vr Vr



  

(2-79)

     

b2 b3 b4 B = b1 −  − 2 − 3 Tr Tr Tr

   

c2 c3 C = c1 −  − 3 Tr Tr

 

d2 D = d1 +  Tr where

Z(i) = Z(0) when the constants in the equation correspond to the simple fluid and Z(h) when the constants in the equation correspond to the heavy reference fluid P = pressure, kPa Pc = critical pressure of the compound whose density is sought, kPa V = molar volume of the simple fluid or the heavy reference fluid, as the case may be, in m3/kmole. R = gas constant = 8.3140 m3 kPa/k mole K Tc = critical temperature of the compound whose density is sought, K T = temperature, K

Constant

Simple fluid

Heavy reference fluid

b1 b2 b3 b4 c1 c2 c3 c4 d1 × 104 d2 × 104 β γ

0.1181193 0.265728 0.154790 0.030323 0.0236744 0.0186984 0.0 0.042724 0.155488 0.623689 0.65392 0.060167

0.2026579 0.331511 0.027655 0.203488 0.0313385 0.0503618 0.016901 0.041577 0.48736 0.0740336 1.226 0.03754

For hydrocarbon and nonpolar gas mixtures, the Pitzer pure component method can be used to predict vapor density by replacing the true critical properties with pseudocritical properties defined in Eqs. (2-80) and (2-81) by Kay.47 n

Tpc = xiTci

(2-80)

i=1 n

Ppc = xiPci

(2-81)

i=1

The mixture acentric factor, Eq. (2-82), can also be used. n

ω = xiωi

(2-82)

i=1

Errors in compressibility factors tabulated for over 6500 data points rarely exceed 2 percent except in the critical region, where 15 percent errors may be expected and 50 percent errors can occur. For mixtures near the critical point, special techniques are available as discussed in the sixth chapter of the Technical Data Book. For pure organic vapors, the Lydersen et al.63 corresponding states method is the most accurate technique for predicting compressibility factors and, hence, vapor densities. Critical temperature, critical pressure, and critical compressibility factor defined by Eq. (2-21) are used as input parameters. Figure 2-37 is used to predict the compressibility factor at Zc = 0.27, and the result is corrected to the Zc of the desired fluid using Eq. (2-83). Z = Z@ Zc = 0.27 + Di(Zc − 0.27)

(2-83)

Di is equal to Da read from Fig. 2-38 if Zc > 0.27; and Di is equal to Db read from Fig. 2-39 if Zc < 0.27. At reduced temperatures less than 0.9, Di can be taken as 0. The density is then calculated from Eq. (2-76). All families of organic compounds except mercaptans and carboxylic acids are predicted within an average deviation of 5 percent. No specific mixing rules have been tested for predicting compressibility factors for defined organic mixtures. However, the Lydersen method using pseudocritical properties as defined in Eqs. (2-80), (2-81), and (2-82) in place of true critical properties will give a reasonable estimate of the compressibility factor and hence the vapor density. Vapor densities for pure compounds can also be predicted by cubic equations of state. For hydrocarbons, relatively accurate RedlichKwong-type equations such as the Soave98 and Peng-Robinson81 equations are often used. Both require only Tc, Pc, and ω as inputs. For organic compounds, the Lee-Erbar-Edmister54 equation (which requires the same input parameters) has been used with errors essentially equivalent to those determined for the Lydersen method. While analytical equations of state are not often used when only densities are required, values from equations of state are used as inputs to equation of state formulations for thermal and equilibrium properties. Liquid Density Prediction Methods for the prediction of pure saturated hydrocarbons and nonhydrocarbon organics, compressed hydrocarbon liquids, and defined and undefined hydrocarbon mixtures were evaluated. Only the most accurate and convenient methods are included here. The most convenient method for predicting the saturated liquid density of both pure hydrocarbons and pure organic liquids is the method of Rackett84 as modified by Spencer and Danner.99 Equation (2-84) is used to calculate the saturated liquid molar density at any temperature using input parameters of Tc, Pc, and ZRA. ZRA is a parameter regressed from experimental data. Values for some common substances are given in Table 2-396. Extensive tabulations are given in the Technical Data Book23 for hydrocarbons and nonhydrocarbons as well as organic and inorganic gases. Additional values are given in the Data Prediction Manual22 for nonhydrocarbons. RTc 1 (2-84)  = Vsat =  Z nRA ρsat Pc

 

n = 1.0 + (1.0 − Tr)2/7 Errors for hydrocarbons between the triple and critical points average about 0.7 percent, with organics averaging about 1.2 percent. The cor-

PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES

FIG. 2-37

2-359

Generalized compressibility factor—Lydersen Method, Zc = 0.27.

relation is especially sensitive to the value of ZRA near the critical point. If no value of ZRA is available or derivable, the critical compressibility factor can be used in Eq. (2-84) as originally proposed by Rackett. Use of Zc increases the average error to about 3.0 percent.

Example 20 Estimate the density of saturated liquid propane at 0°C. Use Eq. (2-84).

1   Z 

Pc ρsat =  RTc

n RA

n = 1.0 + (1.0 − Tr)2/7

2-360

PHYSICAL AND CHEMICAL DATA Pure component properties of propane are M = 44.1, Tc = 96.7°C, and Pc = 4.246 MPa. From Table 2-388, ZRA = 0.2763. 273.1 Tr =  = 0.739, n = 1.0 + (1.0 − 0.739)2/7 = 1.6813 369.8 n = 1.6813 (4.246 × 106) 1 ρsat =   (8314)(369.8) 0.27631.6813





ρsat = 1.201 kmole/m = 529.6 kg/m3 3

An accepted experimental value is 531 kg/m3. An alternate method with approximately the same accuracy as the Rackett method is the COSTALD method of Hankinson and Thomson.35 The critical temperature, a characteristic volume near the critical volume, and an acentric factor optimized for vapor pressure prediction by the Soave98 equation of state are required input parameters. The method is detailed in the Technical Data Book.23 Prediction of the density of compressed pure liquid hydrocarbons and their defined mixtures are readily and accurately predicted by the method of Lu.62 One value of low pressure liquid density and the critical temperature and pressure are required to predict the density at higher pressures from Eq. (2-85).

 

C ρ2 = 2 ρ1 C1

(2-85)

The constants C1 and C2 are both obtained from Fig. 2-40: C1, usually from the saturated liquid line; and C2, at the higher pressure. Errors should be less than 1 percent for pure hydrocarbons except at reduced temperatures above 0.95 where errors of up to 10 percent may occur. The method can be used for defined mixtures substituting pseudocritical properties for critical properties. For mixtures, the Technical Data Book—Petroleum Refining gives a more complex and accurate mixing rule than merely using the pseudocritical properties. The saturated low pressure value should be obtained from experiment or from prediction procedures discussed in this section for both pure and mixed liquids. FIG. 2-38

Compressibility factor correction for Lydersen Method, Zc > 0.27.

Example 21 Estimate the liquid density of n-nonane at 104.5°C and 6.893 MPa pressure. A tabulated value of liquid density at 60°F (15.5°C) and 1 atm is 719.8 kg/m3. Pure component properties are Tc = 321.5°C and Pc = 2.288 MPa. Use Eq. (2-85) to correct a low pressure density. C ρ2 = 2 ρ1 C1 where ρ1 is a reference density. C2 and C1 are functions of Tr and Pr. 288.6 377.6 Tr2 =  = 0.635 Tr1 =  = 0.485 594.6 594.6

 

0.1013 6.893 Pr1 =  = 0.0443 Pr2 =  = 3.013 2.288 2.288 From Fig. 2-40, C1 = 1.08 and C2 = 0.998.





0.998 ρ2 =  (719.8) = 665.1 kg/m3 1.08 An accepted experimental value at this temperature is 658.5 kg/m3, a 1 percent difference. An analytical method for the prediction of compressed liquid densities was proposed by Thomson et al.114 The method requires the saturated liquid density at the temperature of interest, the critical temperature, the critical pressure, an acentric factor (preferably the one optimized for vapor pressure data), and the vapor pressure at the temperature of interest. All properties not known experimentally may be estimated. Errors range from about 1 percent for hydrocarbons to 2 percent for nonhydrocarbons. For prediction of the densities of a defined liquid mixture at its bubble point (ρbp), the method of Spencer and Danner100 is the simplest. The density is calculated from Eq. (2-86) using inputs from Eqs. (2-87) and (2-88). For hydrocarbons, Tmc is calculated by Eqs. (2-89) through (2-92) if high accuracy is desired or by Eq. (2-93) for a less accurate answer. 1 =R ρbp

 x P Z n

Tc

i

i

i=1

ci

[1 + (1 − Tr)2/7] RAm

(2-86)

n

ZRAm = xi ZRAi

(2-87)

i=1

T Tr =  Tmc n

FIG. 2-39

Compressibility factor correction for Lydersen Method, Zc < 0.27.

(2-88) n

Tmc = φiφjTcij i=1 j=i

(2-89)

PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES TABLE 2-396

2-361

The Modified Rackett Equation Input Parameters for Calculating Pure Saturated Liquid Densities

Liquid

ZRA

Hydrocarbons Methane Ethane Propane n-Butane 2-Methylpropane (isobutane) n-Pentane n-Hexane n-Heptane n-Octane n-Nonane n-Decane n-Dodecane n-Tetradecane n-Hexadecane n-Octadecane n-Eicosane Cyclopentane Methylcyclopentane Cyclohexane Methylcyclohexane Ethene (ethylene) Propene (propylene) 1-Butene cis-2-Butene trans-2-Butene 2-Methylpropene (isobutylene) 1-Hexene 1,3-Butadiene 2-Methyl-1,3-butadiene Ethyne (acetylene) Propyne (methylacetylene) Benzene Methylbenzene (toluene) Ethylbenzene 1,2-Dimethylbenzene (o-xylene) 1,3-Dimethylbenzene (m-xylene) 1,4-Dimethylbenzene (p-xylene) Isopropylbenzene (cumene) Biphenyl Naphthalene

Liquid

0.2880 0.2819 0.2763 0.2730 0.2760 0.2685 0.2637 0.2610 0.2569 0.2555 0.2527 0.2471 0.2270 0.2386 0.2292 0.2281 0.2709 0.2712 0.2729 0.2702 0.2813 0.2783 0.2735 0.2705 0.2722 0.2727 0.2654 0.2713 0.2680 0.2707 0.2703 0.2696 0.2645 0.2619 0.2626 0.2594 0.2590 0.2616 0.2746 0.2611

xiVci φi =  n

xjVc j

(2-90)

j=1

 Tcij = T c Tc (1 − kij)

(2-91)

j

i



1/3 1/3  V V ci  cj 

kij = 1.0 −  (Vc1/3 + Vc1/3 )/2 i j



3

(2-92)

n

Tmc = xi Tci

(2-93)

i=1

Errors for binary hydrocarbon systems average about 2.5 percent except near the critical, where errors can approach 20 percent. If inorganic gases are included, errors average 4 percent except at high concentrations of carbon dioxide or hydrogen, where higher errors would be expected. If the simplified method for predicting Tmc is used, average errors of 5 to 7 percent should be expected for both binary hydrocarbon and nonhydrocarbon systems. No data are available to test the method for systems with more than two components. A similarly accurate but slightly more complex method for prediction of densities of defined liquid hydrocarbon mixtures at their bubble points was published by Hankinson and Thomson35 and was previously cited for prediction of pure liquid hydrocarbons. For undefined hydrocarbon mixtures, the liquid density may be predicted at any temperature (T) from the mean average boiling point (MeABP) and the specific gravity (sp gr) by Eq. (2-94), adopted from Ritter et al.92



P = 62.3636 (sp gr)2 (1.2655)(sp gr) − 0.5098 + 8.011 × 10−5MeABP(T − 519.67) 1/2 −  (2-94) MeABP 3 The density is calculated in lbm /ft if the temperatures are both in °R. Errors



ZRA

Organics Acetic acid Methanol Ethanol 2-Propanol Acetaldehyde Acetone Methyl ethyl ketone Methyl isobutyl ketone Ethylamine Aniline Methyl formate Methyl acetate Ethyl acetate Ethyl acrylate Methyl-n-butyl ether Diethyl ether Diisopropyl ether

0.2242 0.2340 0.2523 0.2508 0.2387 0.2448 0.2524 0.2589 0.2640 0.2607 0.2581 0.2553 0.2538 0.2583 0.2655 0.2643 0.2699

Halogen Compounds Methyl chloride Dichloromethane Chloroform Tetrachloromethane Chlorobenzene Propionitrile

0.2679 0.2619 0.2751 0.2721 0.2650 0.2156

Inorganics Ammonia Argon Carbon dioxide Carbon disulfide Carbon monoxide Chlorine Hydrogen Hydrogen chloride Hydrogen sulfide Nitrogen Oxygen Sulfur dioxide Sulfur trioxide

0.2466 0.2933 0.2729 0.2850 0.2898 0.2781 0.3218 0.2673 0.2818 0.2893 0.2890 0.2667 0.2513

average about 0.3 percent at atmospheric pressure. At high pressures, the liquid density of undefined hydrocarbon mixtures can be predicted from the low pressure value by the method of Wright131 fully outlined in the Technical Data Book.23

Solid Density Prediction The prediction of solid density is an inexact science and sometimes is taken as the liquid density at the triple point, although the solid density normally is higher than this value with a discontinuity at the triple point. Based on solid density data reviewed for the DIPPR compilation,24 the solid density at the triple point can be estimated for organic compounds as 1.17 times the liquid density at the triple point. As liquid density at low temperatures varies little with temperature, the density of the liquid at the lowest estimable point above the triple point can be used with little degradation of the result. As solid density only decreases very slightly with increasing temperature and very little data on solid density as a function of temperature exist, no methods have been developed for predicting the solid density vs. temperature. VISCOSITY Viscosity is defined as the shear stress per unit area at any point in a confined fluid divided by the velocity gradient in the direction perpendicular to the direction of flow. If this ratio is constant with time at a given temperature and pressure for any species, the fluid is called a Newtonian fluid. This section is limited to Newtonian fluids, which include all gases and most nonpolymeric liquids and their mixtures. Most polymers, pastes, slurries, waxy oils, and some silicate esters are examples of non-Newtonian fluids. The absolute viscosity (µ) is defined as the sheer stress at a point

2-362

PHYSICAL AND CHEMICAL DATA

FIG. 2-40

Densities of compressed pure liquid hydrocarbons and their defined mixtures.

divided by the velocity gradient at that point. The most common unit is the poise (1 g/cm sec). The SI unit is the Pa·sec (1 kg/m sec). As many common fluids have viscosities in the hundredths of a poise the centipoise (cp) is often used. One centipoise is then equal to one mPa sec. The kinematic viscosity (ν) is defined as the ratio of the absolute viscosity to density at the same temperature and pressure. The most common unit corresponding to the poise is the stoke (1 cm2/sec). The SI unit would be m2/sec. Correlation Methods This section briefly discusses methods for correlating viscosities as a function of temperature and presents the most common accurate methods for prediction of vapor and liquid viscosity. Vapor viscosity is accurately correlated as a function of temperature by Eq. (2-95). AT B µv =  (2-95) C D 1 +  + 2 T T

If data are available over a wide range, all four regression constants (A, B, C, and D) are usually used. Over narrow temperature ranges, only constants A and B are necessary. Liquid viscosity is accurately correlated as a function of temperature by the modified Riedel equation previously discussed for correlation of vapor pressure and shown by Eq. (2-96). B µᐉ = exp A +  + C ln (T) + DT E (2-96) T For most systems, only the first three terms are used. Only the first two terms are used for narrow ranges. If data are available in a wide range extending far above the normal boiling point, all four terms are used, with values of E varying in integers from −10 to 10 (excluding 0 and −1). Constants for about 1500 compounds for both viscosities are available in the DIPPR compilation.24 Vapor Viscosity Methods for prediction of vapor viscosity abound such that only the most accurate and generally used methods are included.





PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES For prediction of the vapor viscosity of pure hydrocarbons at low pressure (below Tr of 0.6), the method of Stiel and Thodos101 is the most accurate. Only the molecular weight, the critical temperature, and the critical pressure are required. Equation (2-97) with values of N from Eqs. (2-98) and (2-99) is used. NM1/2P c2/3 µv = 4.60 × 10−4  T c1/6 0.94 N = 0.0003400Tr for Tr ≤ 1.5 N = 0.0001778(4.58Tr − 1.67)0.625

(2-97) (2-98) for Tr > 1.5

(2-99)

The resultant viscosity is in centipoise (mPa·sec) if Tc and Pc are given in K and Pa, respectively. This method can also be used for light nonhydrocarbon gases except for hydrogen where, special N’s are required. For hydrocarbons below ten carbon atoms, average errors of about 3 percent can be expected, with errors increasing to 5–10 percent for heavier hydrocarbons. Example 22 Estimate the vapor viscosity of propane at 101.3 kPa and 80°C. Use Eq. (2-97).

Tc = 96.7°C,

Z = Z(0) + ωZ(1) 398.1 Tr =  = 1.62 245.5

Pc = 4.246 MPa,

and M = 44.1.

To determine whether Eq. (2-98) or (2-99) should be used to calculate N, calculate Tr. 80 + 273.1 Tr =  = 0.955 96.7 + 273.1 Use Eq. (2-98):

Example 23 Estimate the vapor viscosity of a mixture of propane and methane. Assume 60 mole percent methane and 40 mole percent propane at 125°C and 10.34 MPa total pressure. The low pressure viscosity is 0.0123 cp. Use Eq. (2-102): M1/2P 2/3 c µ − µo = 5.0 × 10−8  [exp (1.439ρr) − exp (−1.11ρ1.858 )] r T1/6 c Properties of the pure components are: Methane. M = 16.04, Tc = −110.4°C, Pc = 4.593 MPa Propane. M = 44.10, Tc = 96.7°C, Pc = 4.246 MPa For the mixture: Tpc = (0.60)(−110.4 + 273.1) + (0.40)(96.7 + 273.1) = 245.5 K Ppc = (0.60)(4.593) + (0.40)(4.246) = 4.454 MPa Mm = (0.60)(16.04) + (0.40)(44.10) = 27.26 To calculate ρr, use Eq. (2-76) to calculate ρ using Pitzer corresponding states first to calculate Z. Then calculate ρr = ρ/ρc = ρVpc Additional properties required are: Methane ω = 0.0115, Vc = 0.0986 m3/kmole Propane ω = 0.1523, Vc = 0.2002 m3/kmole For the mixture: ωm = (0.60)(0.0115) + (0.40)(0.1523) = 0.0678 Vpc = (0.60)(0.0986) + (0.40)(0.2002) = 0.1392 m3/kmole Using Eq. (2-58):

M1/2Pc2/3 µv = 4.60 × 10−4N  T c1/6

 

x 1 + Qij j xi j=1 j≠1

      

µi 1/2 Mj 1/4 2   Mi µj Qij =  Mi 1/2 8 1 +  Mj 1+

(2-100)

P (10.34) ρ =  =  ZRT (0.88)(8.314 × 10−3)(398.1)

µ − µo = 0.0037 µ = 0.0037 + 0.0123 = 0.0160 cp An experimental value of 0.0167 cp compares favorably. For pure nonhydrocarbon polar gases at low pressures, the viscosity can be estimated by the method of Reichenberg85 given by Eq. (2-103). ATr µ =  (2-103) [1 + 0.36Tr(Tr − 1)]1/6 For organic compounds: M1/2Tc A=

niCi

(2-101)

Errors, when tested against binary and multicomponent mixtures of both hydrocarbons and nonhydrocarbon gas mixtures, average about 3 percent. For prediction of the vapor viscosity of gaseous hydrocarbons and mixtures of hydrocarbons at high pressures (not applicable to nonhydrocarbon gases) above a Tr of 0.6, low pressure values are calculated from Eq. (2-97) and/or (2-100) and then corrected for pressure by the method of Dean and Stiel25 given by Eq. (2-102). M1/2P c2/3 [exp (1.439ρr) − exp (−1.11ρ1.858 )] µ − µo = 5.0 × 10−8  r T c1/6

Z = 0.87 + (0.0678)(.20) = 0.88

(5.08 × 10−8)(5.22)(27,085) µ − µo =  [2.0358 − 0.7413] (2.503)

An experimental value of 0.0095 cp compares favorably. For prediction of the vapor viscosity of gaseous mixtures of hydrocarbons and nonhydrocarbon gases at low pressures below a Tr of 0.6, the method of Bromley and Wilke13 is recommended. The mixing rule is given by Eq. (2-100) with the interaction parameter Q for each pair of components defined by Eq. (2-101). µi

Z(0)  0.20

(5.0 × 10−8)(27.26)1/2(4.454 × 106)2/3 1.439(.494) −1.11(.494)1.858 µ − µo =  [e −e ] (245.5)1/6

µv = 0.0097 cp

n

From Fig. 2-36:

ρ = 3.55 kmole/m3

(4.60 × 10−4)(3.255 × 10−4)(44.1)1/2(4.246 × 106)2/3 µv =  (369.8)1/6

n

Z(0) = 0.87

10.34 Pr =  = 2.32 4.454

ρr = ρVpc = (3.55)(.1392) = 0.494

Thus, N = 3.255 × 10−4.

i=1

From Fig. 2-35:

Hence:

N = 0.0003400Tr0.94

µm =

2-363

(2-102)

If critical pressure and critical temperature are given in Pa and K, respectively, viscosities in centipoise result. The variable µo is either the low pressure pure component or mixture viscosity according to whether a pure component or mixture is being considered. For mixtures, simple molar average pseudocritical temperature (Kay’s rule), pressure, and density, and molar average molecular weight are used. The vapor density can be predicted by the methods previously discussed. Errors of above 5 percent are common for hydrocarbons and their mixtures. Experimental densities will reduce the errors slightly.

For inorganic gases:

(2-104)





M1/2P c2/3 (2-105) A = 1.6104 × 10−10  T c1/6 Viscosities are calculated in Pa sec (103 cp = 1 Pa sec) with Tc in K and Pc in Pa. Group contributions based on atomic structure for organic compounds necessary for Eq. (2-104) are tabulated in Table 2-397. Errors average about 5 percent for most organics, with slightly higher errors for inorganic gases. For pure nonhydrocarbon nonpolar gases, an alternate method is the method of Yoon and Thodos,132 which requires the same input parameters as the Reichenberg method. The method, when evaluated for the Technical Data Manual, showed errors of about 3 percent for compounds with low dipole moments and requires special correlations for hydrogen and helium.

Example 24 Estimate the vapor viscosity of isopropyl alcohol at 251°C and atmospheric pressure. Use Eq. (2-103) with A determined from Eq. (2-104). ATr µ =  [1 + 0.36Tr(Tr − 1)]1/6 M1/2Tc A=

niCi

2-364

PHYSICAL AND CHEMICAL DATA

TABLE 2-397

Group Contribution Values for Eq. (2-104) Contribution Ci × 10−8 m⋅s⋅K/kg

Group CH3

0.904

CH2 (nonring)

0.647

CH

0.267

Pure component properties are Tc = 508.3 K and M = 60.1. Isopropyl alcohol group contributions for niCi are calculated using Table 2-397.

2 1 1

C

(nonring)

CH2

0.768

CH

0.553

(nonring)

C

(nonring)

0.178

CH

0.741 (nonring)

C

0.524

CH2 (ring)

0.691

group = (1)(.267 × 108) +

OH group = (1)(.796 × 108) = 2.871 × 108

60.11/2(508.3) A =  × 108 = 1.373 × 10−5 Pa sec 2.871 (1.373 × 10−5)(1.031) µ =  = 1.413 × 10−5 Pa sec [1 + 0.36(1.031)(.031)]1/6 (An experimental value of 1.380 × 10−5 Pa sec compares favorably.) For pure nonhydrocarbon polar and nonpolar gases at high pressure, a method of prediction attributed to Stiel and Thodos102 depending on the reduced density as a corrector to the low pressure gas viscosity (µo) takes various forms for polar gases according to the reduced density (ρr) as shown in Eqs. (2-106) through (2-109). ρr ≤ 0.1

(µ − µo)B = 1.656 × 10−7ρ1.111 r

(2-106)

0.1 ≤ ρr ≤ 0.9

(µ − µo)B = 6.07 × 10−9(9.045ρr + 0.63)1.739

(2-107)

0.9 ≤ ρr < 2.2

log {4 − log10 [(µ − µo)107B]} = 0.6439 − 0.1005ρr

(2-108)

2.2 ≤ ρr < 2.6

log {4 − log10 [(µ − µo)107B]} = 0.6439 − 0.1005ρr B = 2173.424 T

C

(ring)

CH C

0.023

(ring)

0.590

(ring)

0.359

F

0.446

Cl

1.006

Br

1.283

OH (alcohols)

0.796

O (nonring)

0.359

C

O (nonring)

1.202

CHO (aldehydes)

1.402

COOH (acids)

1.865

COO

(esters) or HCOO

(formates)

1.341 0.971

NH2 NH (nonring)

0.368

N

0.497

(ring)

− 4.75 × 10−4(ρ3r − 10.65)2

(2-109)

M−1/2 P c−2/3

(2-110)

0.116

(ring)

CH

CH

524.1 Tr =  = 1.031 508.3

−0.153

(nonring)

CH3 groups = (2)(.904 × 108) +

CN

1.813

S (ring)

0.886

1/6 c

For nonpolar gases, Jossi et al.45 extended the method as shown in Eq. (2-111) for 0.1 < ρr < 3.0. [(µ − µo)B107 + 1]1/4 = 1.0230 + 0.23364ρr + 0.58533ρr2 − 0.40758ρr3 + 0.093324ρr4 (2-111) In all cases viscosities are in Pa sec with Tc in K and Pc in Pa. For nonpolars, errors are very small; while for polars, average errors reach 11 percent.

Example 25 Estimate the vapor viscosity of carbon dioxide at 350 K and a total pressure of 20 MPa. An experimental low pressure viscosity at 350 K is 1.7386 × 10−5 Pa sec. Use Eq. (2-111) with B calculated by Eq. (2-110). The pure component properties necessary are Tc = 304.19 K, Pc = 7.3815 MPa, Vc = 0.094 m3/kmole, and M = 44.01. Using the Lee-Kesler form of the Pitzer method [Eq. (2-78)], Z = 0.4983. (0.094)(2 × 107) V Vc P ρr = c =  =  = 1.2965 V ZRT (0.4983)(8314)(350) (2173.424)(304.19)1/6 2173.424T1/6 c B =  =  M1/2P 2/3 (44.01)1/2(7.3815 × 106)2/3 c B = 2.2411 × 10−2 [(µ − µo)B107 + 1]1/4 = 1.0230 + 0.23364(1.2965) + 0.58533(1.2965)2 − 0.40758(1.2965)3 + 0.093324(1.2965)4 = 1.6852 (µ − 1.7386 × 10−5)(2.2411 × 10−2)(107) + 1 = 8.0659 µ − 1.7386 × 10−5 = 3.1529 × 10−5 µ = 4.89 × 10−5 Pa sec An experimental value of 4.73 × 10−5 Pa sec compares favorably. For both polar and nonpolar nonhydrocarbon gaseous mixtures at low pressures, the most accurate viscosity prediction method is the method of Brokaw.10,11 The method is quite accurate but requires the dipole moment and the Stockmayer energy parameter (ε /k) for polar components as well as pure component viscosities, molecular weights, the normal boiling point, and the liquid molar volume at the normal boiling point. The Technical Data Manual should be consulted for the full method. For nonpolar, nonhydrocarbon vapor mixtures at high pressures, the method of Dean and Stiel25 [Eq. (2-102)] discussed earlier can be used. The accuracy of the method is excellent and dependent on the pure component viscosity values used as input parameters.

PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES Liquid Viscosity The viscosity of both pure hydrocarbon and pure nonhydrocarbon liquids are most accurately predicted by the method of van Velzen et al.122 The basic equation (2-112) depends on group contributions which are dependent on structure for the calculation of compound-specific constants B and To. 1 1 log µ = B  −  − 3.0 (2-112) T To Resultant viscosities are in Pa sec. If the −3.0 on the right is deleted,



TABLE 2-398



2-365

answers are in cp. To is calculated by Eq. (2-114) or (2-115) according to the value of an adjusted carbon number N* calculated by Eq. (2-113) using the actual carbon number N and group contributions from Table 2-398. N* = N + ∆Ni (2-113) i

N* ≤ 20

To = 28.86 + 37.439N* − 1.3547N*2 + 0.02076N*3

(2-114)

N* > 20

To = 8.164N* + 238.59

(2-115)

Group Contribution Values for Liquid Viscosity Prediction

Structures or functional group n-Alkanes Isoalkanes Saturated hydrocarbons with two methyl groups in isoposition n-Alkanes n-Alkadienes Isoalkanes Isoalkadienes Hydrocarbon with one double bond and two methyl groups in isoposition Hydrocarbon with two double bonds and two methyl groups in isoposition Cyclopentanes Cyclohexanes Alkyl benzenes Polyphenols Alcohols Primary Secondary Tertiary Diols (correction) Phenols (correction) OH on side chain to aromatic ring (correction) Acids Isoacids Acids with aromatic nucleus in structure (correction) Esters Esters with aromatic nucleus in structure (correction) Ketones Ketones with aromatic nucleus in structure (correction) Ethers Aromatic ethers Amines Primary Primary amine in side chain of aromatic compound (correction) Secondary Tertiary Primary amines with NH2 group on aromatic nucleus Secondary or tertiary amine with at least one aromatic group attached to amino nitrogen Nitro compounds 1-Nitro 2-Nitro 3-Nitro 4-Nitro; 5-Nitro Aromatic nitrocompounds Nitrile

∆ Ni

∆Bi

0 1.389 − 0.238N 2.319 − 0.238N

0 15.51 15.51

−0.152 − 0.042N −0.304 − 0.084N 1.237 − 0.280N 1.085 − 0.322N 2.626 − 0.518N

−44.94 + 5.410N* −44.94 + 5.410N* −36.01 + 5.410N* −36.01 + 5.410N* −36.01 + 5.410N*

2.474 − 0.560N

−36.01 + 5.410N*

0.205 + 0.069N 3.971 − 0.172N 1.48 6.517 − 0.311N 0.60 3.055 − 0.161N −5.340 + 0.815N

−45.96 + 2.224N* −339.67 + 23.135N* −272.85 + 25.041N* −272.85 + 25.041N* −140.04 + 13.869N* −140.04 + 13.869N* −188.40 + 9.558N*

10.606 − 0.276N 11.200 − 0.605N 11.200 − 0.605N See remarks 16.17 − N −0.16

−589.44 + 70.519N* 497.58 928.83 557.77 213.68 213.68

b

6.795 + 0.365N 10.71 See remarks

−249.12 + 22.449N* −249.12 + 22.449N* −249.12 + 22.449N*

4.81

−188.40 + 9.558N*

N < 11, not recommended for N = 1,2 N ≥ 11 Calculate ∆B as for straight-chain acid; calculate ∆N for straight-chain acid but reduce ∆N by 0.24 for each methyl group in isoposition

Remarks

For any additional CH3 groups in isoposition, increase ∆N by 1.389 − 0.238N For any additional CH3 groups in isoposition, increase ∆N by 1.389 − 0.238N N ≤ 16; not recommended for N = 5,6 N ≥ 16 N < 17; not recommended for N = 6,7 N ≥ 17 N < 16; not recommended for N = 6,7a,e,f N ≥ 16a,e, f

a

b b

For ∆N, use alcohol contributions and add N − 2.50

a,c,d

4.337 − 0.230N −1.174 + 0.376N

−149.13 + 18.695N* −140.04 + 13.869N*

If hydrocarbon groups have isoconfiguration, see e Add to values of ∆N, ∆B calculated for ester

3.265 − 0.122N 2.70

−117.21 + 15.781N* −760.65 + 50.478N*

If hydrocarbon groups have isoconfiguration, see e Add to values of ∆N, ∆B calculated for ketone

0.298 + 0.209N 11.5 − N

−9.39 + 2.848N* −140.04 + 13.869N*

If hydrocarbon groups have isoconfiguration, see e The ∆N value is not a correction to regular ether value, but the ∆B value is a correction to regular etherc

3.581 + 0.325N −0.16

25.39 + 8.744N* 0

If hydrocarbon groups have isoconfiguration, see e Corrections to be added to amine calculatione

1.390 − 0.461N 3.27 15.04 − N

25.39 + 8.744N* 25.39 + 8.744N* 0

If hydrocarbon groups have isoconfiguration, see e If hydrocarbon groups have isoconfiguration, see e The ∆N value is not a correction to regular amine value; to find ∆B, use primary amine valuea,c

f

7.812 − 0.236N 5.84 5.56 5.36 7.182 − 0.236N 4.039 − 0.0103N

f

−213.14 + 18.330N* −213.4 + 18.330N* −338.01 + 25.086N* −338.01 + 25.086N* −213.14 + 18.330N* −241.66 + 27.937N*

Note alkene contribution is necessary For aromatic correction, see f

2-366

PHYSICAL AND CHEMICAL DATA

TABLE 2-398

Group Contribution Values for Liquid Viscosity Prediction (Concluded ) ∆ Ni

∆Bi

−0.7228 + 0.1755N 2.321 − 0.2357N 10.452 − 1.1276N

286.26 − 31.009N* −26.063 − 11.516N* 3599.9 − 199.96N*

Structures or functional group Amines (Cont.) Isomethyl on nitrile Aromatic nitrile Dinitrile Halogenated compounds Fluoride Chloride Bromide Iodide Special configurations (corrections) C(Cl)x CClCCl C(Br)x CBrCBr CF3 In alcohols In other compounds Aldehydes Aldehydes with an aromatic nucleus in structure (correction) Anhydrides Anhydrides with an aromatic nucleus in structure (correction) Amides Amides with an aromatic nucleus in structure (correction) Sulfide Isomethyl on sulfide a

1.43 3.21 4.39 5.76

Remarks

5.75 −17.03 −101.97 + 5.954N* −85.32

1.91 − 1.459x 0.96 0.50 1.60

e, f e, f e, f

−26.38 0 81.34 − 86.850x −57.73

−3.93 −3.93 3.38 2.70

341.68 25.55 146.45 − 25.11N* −760.65 + 50.478N*

7.97 − 0.50N 2.70

−33.50 −760.65 + 50.478N*

13.12 + 1.49N 2.70

524.63 − 20.72N* −760.65 + 50.478N*

3.9965 − 0.1861N 0.1601

−76.676 + 8.1403N* −25.026

For substitutions on an aromatic nucleus in more than one position, additional corrections are required: Ortho

∆N = 0.51

Meta Para

∆N = 0.11 ∆N = −0.04




∆B = −571.94 (with OH) 54.84 (without OH) ∆B = 27.25 ∆B = 17.57

For alcohols, if there is a methyl group in the isoposition, increase ∆ N by 0.24 and ∆ B by 94.23. If the compound has an aromatic OH or NH2, or if there is an aromatic ether, use ∆N contribution in table but neglect other substituents on the ring such as halogen, CH3, NO2, and the like. For the calculation of ∆ B, however, such substituents must be taken into account. d For aromatic alcohols and compounds with an OH on a side chain, the alcohol contribution (primary, etc.) must be included. For example, o-chlorophenol: ∆ B = ∆B (primary alcohol) + ∆ B (chlorine) + ∆ B (phenol) + ∆ B (ortho correction—see footnote a) With N* = 16.17 (see footnote c): ∆B = (−589.44 + 70.519 × 16.17) + (−17.03) + (213.68) + (−571.94) = 175.56 Ba = 745.94 B = Ba + ∆B = 921.50 2-Phenylethanol: N = 8; ∆N = ∆N (primary alcohol) + ∆N (correction) = [10.606 − (0.276)(8)] + (−0.16) = 8.24 N* = N + ∆N = 8 + 8.24 = 16.24 ∆ B = ∆B (primary alcohol) + ∆ B (correction) = [−589.44 + (70.519)(16.24)] + 213.68 = 769.47 Ba = 747.43 B = Ba + ∆B = 1516.9 e For esters, alkylbenzenes, halogenated hydrocarbons, and ketones: If the hydrocarbon chain has a methyl group in an isoposition, decrease ∆N by 0.24 and increase ∆ B by 8.93 for each such grouping. For ethers and amines, decrease ∆ N by 0.50 and increase ∆ B by 8.93 for each isogroup. f For alkylbenzenes, nitrobenzenes, halogenated benzenes and for secondary or tertiary amines where at least one aromatic group is connected to an amino nitrogen, add the following corrections for each aromatic nucleus. If N < 16, increase ∆N by 0.60; if N ≥ 16, increase ∆ N by 3.055 − 0.161N for each aromatic group. For any N, increase ∆B by (−140.04 + 13.869N*). From van Velzen et al. (122). b c

B is calculated by Eq. (2-116) using values of Ba calculated from Eq. (2-117) or (2-118) according to the value of N* and group contributions from Table 2-398. B = Ba + ∆Bi

(2-116)

N* ≤ 20

Ba = 24.79 + 66.885N* − 1.3173N*2 − 0.00377N*3

(2-117)

N* > 20

Ba = 530.59 + 13.740N*

(2-118)

i

The method should not be used for the first member of a homologous series or for temperatures much above the normal boiling point (Tr ≈ 0.75). Errors for both hydrocarbons and nonhydrocarbons average 15 percent for a wide variety of compounds. Higher errors are noted for amines, diols, ethers, and fluorides. Table 2-398 gives ∆N and ∆B contributions for most common groups. Space prohibits examples for

each type of compound or inclusion of specialized cases that are more fully discussed in the Technical Data Manual. Example 26 Estimate the liquid viscosity of cis-1,4-dimethylcyclohexane at 0°C. Use Eq. (2-112) with N* calculated from Eq. (2-113) and B calculated from Eq. (2-116). Determine group contributions from Table 2-398.

Cyclohexanes n-Alkanes

(1) (2)

∆Ni

∆Bi

1.48 0

−272.85 + 25.041 N* 0

N* = N + ni∆Ni i

N* = 8 + (1)(1.48) + (2)(0) = 9.48

PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES Use Eq. (2-114) to calculate To: To = 28.86 + 37.439(9.48) − 1.3547(9.48)2 + (.02076)(9.48)3 To = 279.72 B = Ba + ni∆Bi i

Use Eq. (2-117) to calculate Ba: Ba = 24.79 + 66.885(9.48) − 1.3173(9.48)2 − 0.00377(9.48)3 Ba = 537.26





1 1 log µ = B  −  − 3.0 T To





1 1 log µ = 501.80  −  − 3.0 273.15 279.72 log µ = −2.9569 µ = 0.001104 Pa sec = 1.104 cp An experimental value at 0°C is 1.224 cp. A mixing rule developed by Kendall and Monroe48 is useful for determining the liquid viscosity of defined hydrocarbon mixtures. Equation (2-119) depends only on the pure component viscosities at the given temperature and pressure and the mixture composition. µm =

 xµ  1/3 i

i

(2-119)

For mixtures of the same chemical family, errors average less than 3 percent, while errors overall average 5–6 percent, with errors of mixed families averaging from 10–15 percent. For estimating the liquid viscosity of defined nonhydrocarbon mixtures, a mixing rule shown by Eq. (2-120) was recommended by the Technical Data Manual. ln µm = xi ln µi

(2-120)

i

Errors average near 15 percent. For interpolating viscosities of hydrocarbon mixtures within a limited range knowing viscosities at two temperatures, ASTM Procedure D341-89, including both charts and equations, is recommended. Several recommended methods for predicting the viscosity of undefined hydrocarbon mixtures, such as petroleum fractions and coal liquids, are presented and evaluated in the Technical Data Book—Petroleum Refining. In addition, several methods for determining the liquid viscosity of blends of hydrocarbon mixtures and a method for liquid viscosity of pure hydrocarbons blended with undefined mixtures are given. The most accurate method for predicting the liquid viscosity of hydrocarbons containing fewer than 20 carbon atoms at high pressure is a corresponding states method developed by Graboski and included in the Technical Data Book—Petroleum Refining. Critical temperature, critical pressure, the acentric factor, and knowing or being able to calculate at least one viscosity at a reference temperature and pressure are required. Errors average about 5 percent. For high molecular weight hydrocarbons at high pressure, low pressure viscosities can be converted instead of using the method of Kouzel,51 which requires a low pressure liquid viscosity as input. Compounds of more than 20 carbon atoms and their mixtures are treated in this way. For predicting the liquid viscosity of pure hydrocarbon mixtures at high temperatures, the method of Letsou and Stiel58 is available. Error analyses with only a small amount of data shows errors averaging 34 percent in the reduced temperature range of 0.76 to 0.98. Equation (2-121) defines the method with inputs of Eqs. (2-122) and (2-123). T1/6 c µ 2173.424  = µ(0) + ωµ(1) (2-121) M1/2P 2/3 c





Cp may be assumed to be the ideal gas heat capacity, Cpo. Average errors can be expected to be less than 5 percent.

3

n

i=1



µ = 1.5174 × 10 − 2.135 × 10−5Tr + 7.5 × 10−6T r2

(2-122)

µ(1) = 4.2552 × 10−5 − 7.674 × 10−5Tr + 3.4 × 10−5T r2

(2-123)

(0)

Gases For pure component, low pressure (<350 kPa) hydrocarbon gases, Misic and Thodos70,71 recommend the following equations. For methane and cyclic compounds below reduced temperatures of 1.0: C kG = 4.45 × 10−7 Tr p (2-124) λ For these hydrocarbons above reduced temperatures of 1.0 and for other hydrocarbons at any temperature: C kG = 10−7 (14.52 Tr − 5.14)2/3 p (2-125) λ In these equations, 2/3 1/2 101.325 λ = T 1/6 (2-126)  c M Pc where kG = vapor thermal conductivity, W/m K Tr = reduced temperature, T/Tc T = temperature, K Tc = critical temperature, K Cp = heat capacity at constant pressure, J/kmol K M = molecular weight Pc = critical pressure, kPa

 

B = 537.26 + (1)[−272.85 + 25.041 (9.48)] B = 501.80



2-367

−5

Results are in Pa sec with inputs of Tc in K and Pc in Pa.

VAPOR AND LIQUID THERMAL CONDUCTIVITY Thermal conductivity describes the ease with which conductive heat can flow through a vapor, liquid, or solid layer of a substance. It is defined as the proportionality constant in Fourier’s law of heat conduction in units of energy ⋅ length/time ⋅ area ⋅ temperature; e.g., W/m K.

Example 27 Estimate thermal conductivity for n-hexane. For n-hexane at 373.15 K and low pressure, the required properties from Daubert et al.24 are: Tc = 507.6 K, M = 86.18, Pc = 3025.0 kPa, and Cpo = 1.721 × 105. Tr = 373.15/ 507.6 = 0.7351. Using Eq. (2-126): 101.325 2/3 λ = (507.6)1/6 (86.18)1/2  3025.0





λ = 2.724 Substituting into Eq. (2-125): 1.721 × 105 kG = 10−7 [(14.52)(0.7351) − 5.14]2/3  2.724





kG = 0.01977 W/m K The reported value is 0.02025 W/m K.24 For pure nonhydrocarbon gases at low pressures (up through 1 atm), the following equations may be used at temperature T (K): ηGCv Monatomic gases12: kG = 2.5  (2-127) M Linear molecules12:

ηG 2928.8 kG =  1.30 Cv + 14644.0 −  M Tr





ηG kG =  (1.15 Cv + 16903.36) M kG = vapor thermal conductivity, W/m K ηG = vapor viscosity, Pa·s Cv = heat capacity at constant volume, J/kmol K M = molecular weight Tr = reduced temperature, T/Tc Tc = critical temperature, K

Nonlinear molecules102: where

(2-128) (2-129)

Cv may be calculated as Cpo − R, where Cpo is the ideal gas heat capacity in J/kmol K and R is the gas constant, 8314 J/kmol K. Average errors are in the 8–10 percent range but may be higher for polar compounds. This method should not be used for molecules that associate; e.g., organic acids.

Example 28 Estimate thermal conductivity of carbon dioxide at 370 K and low pressure. For carbon dioxide (a linear molecule) at 370.0 K and low pressure, the required properties from Daubert et al.24 are: Tc = 304.2 K, ηG = 1.828 × 10−5 Pa·s, M = 44.01, and Cpo = 40520 J/kmol K. Tr = 370.0/304.2 = 1.2163, and Cv = 40520 − 8314 = 32206 J/kmol K. Substituting in Eq. (2-128): 1.828 × 10−5 2928.8 kG =  (1.30) (32206) + 14644.0 −  44.01 1.2163



kG = 0.02247 W/m K A value of 0.0220 W/m K is obtained from Vargaftik et al.124



2-368

PHYSICAL AND CHEMICAL DATA

For pure gases above atmospheric pressure, the method of Stiel and Thodos102 may be used: A × 10−4(eBρr + C) kG = k′G +  (2-130) Tc1/ 6 M1/2 5 Zc  2/3 Pc



ρr < 0.5 0.5 < ρr > 2.0 2.0 < ρr > 2.8

A = 2.702 A = 2.528 A = 0.574



B = 0.535 B = 0.670 B = 1.155

373.15 K and low pressure. The required pure component properties from Daubert et al.24 are: k1 = 0.02504 W/m K, k2 = 0.01587 W/m K, µ1 = 1.161 × 10−5 Pa·s, µ2 = 1.361 × 10−5 Pa·s, M1 = 46.07, M2 = 50.49, Tb1 = 248.3 K, and Tb2 = 248.9 K. By Eq. (2-134), S1 = (1.5)(248.3) = 372.45, and S2 = (1.5)(248.9) = 373.35. By Eq. (2-133), S12 = S21 = (1.0)[(372.45)(373.35)]1/2 = 372.90. Substituting in Eq. (2-132): A11 = A22 = 1.0

C = −1.000 C = −1.069 C = 2.016

where kG = vapor thermal conductivity at the temperature T (K) and pressure P of interest, W/m K k′G = vapor thermal conductivity at T and atmospheric pressure, W/m K ρr = reduced density = Vc /V Vc = critical molar volume, m3/kmol V = molar volume at T and P, m3/kmol Tc = critical temperature, K M = molecular weight Pc = critical pressure, MPa Zc = critical compressibility factor = PcVc /RTc R = gas constant = 0.008314 MPa m3/kmol K Errors in the range of 5–6 percent are typical with this method but may be higher for branched compounds.

Example 29 Estimate the thermal conductivity of carbon dioxide at

1.161 × 10−5

373.15 + 372.45  
 1.361 × 10  46.07   373.15 + 373.35  

1 A12 =  1 + 4

50.49

3/4

1/2 2

−5

373.15 + 372.90   = 0.956 373.15 + 372.45





373.15 + 373.35 1.361 × 10  
 1.161 × 10  50.49   373.15 + 372.45  

1 A21 =  1 + 4

−5

46.07

3/4

1/2 2

−5

373.15 + 372.90   = 1.047 373.15 + 373.35





Using Eq. (2-131): (0.23)(0.02504) (0.77)(0.01587) km =  +  = 0.01805 W/m K (0.23)(1.0) + (0.77)(0.956) (0.23)(1.047) + (0.77)(1.0) The experimental value is 0.01778 W/m K.66

24

370 K and 10 MPa pressure. The required properties from Daubert et al. are: Tc = 304.2 K, M = 44.01, Pc = 7.383 MPa, and Vc = 0.0940 m3/kmol; Zc = (7.383) × (0.0940)/(0.008314)(304.2) = 0.274. k′G = 0.0220 W/m K123, and V = 0.22809 m3/ kmol3; ρr = 0.0940/0.22809 = 0.4121. Substituting in Eq. (2-130) using the equation constants A, B, and C for ρr < 0.5: (2.702 × 10−4)(e(0.535)(0.4121) − 1.000) kG = 0.0220 +  = 0.0315 W/m K (304.2)1/6(44.01)1/2 (0.274)5  (7.383)2/3



124

Vargaftik et al.



report a value of 0.0308 W/m K.

The thermal conductivity of low pressure (1 atm or less) gas mixtures can be determined from the relation of Wassiljewa126: n yi ki km =  (2-131) n i=1

yj Aij j−1

where km = mixture thermal conductivity, W/m K n = number of components yi, j = mole fraction of component i or j in the vapor mixture ki = thermal conductivity of pure component i at the temperature of interest The binary interaction parameter Aij is obtained by the method of Lindsay and Bromley61: 1 µi Mj 3/4 T + Si 1/2 2 T + Sij Aij =  1 +   (2-132)   4 µj Mi T + Sj T + Si where µi, j = vapor viscosity of pure component i or j at the temperature T of interest and low pressure, Pa·s Mi, j = molecular weight of pure component i or j T = temperature, K Sij = Sji; see Eq. (2-133) C = 1.0 except when either or both components i and j are very polar; then C = 0.73 Si, j = 79 K for helium, hydrogen, and neon; for all others, see Eq. (2-134) Tbi, j = normal boiling temperature of pure component i or j, K


  

Sji = C(SiSj)1/2 Si,j = 1.5Tbi, j

  



(2-133) (2-134)

Expected errors for this method are 4–5 percent. At higher pressures, a pressure correction using Eq. (2-130) may be used. The mixture is treated as a hypothetical pure component with mixture critical properties obtained via Eqs. (2-5), (2-8), and (2-17) and with the molecular weight being mole-averaged. Example 30 Estimate thermal conductivity of a mixture of 0.23 mole fraction dimethylether (1) and 0.77 mole fraction methyl chloride (2) at

Liquids For pure component hydrocarbon liquids at reduced temperatures between 0.25 and 0.8 and at pressures below 3.4 MPa, an equation based on the methods of Pachaiyappan et al.79 and Riedel88 may be used: 3 + 20(1 − Tr)2/3 kL = CρM n  (2-135) 293.15 2/3 3 + 20 1 −  Tc









where kL = liquid thermal conductivity, W/m K M = molecular weight ρ = molar density at 293.15 K, kmol/m3 Tr = reduced temperature, T/Tc T = temperature, K Tc = critical temperature, K For unbranched, straight chain hydrocarbons, n = 1.001 and C = 1.811 × 10−4. For branched and cyclic hydrocarbons, n = 0.7717 and C = 4.407 × 10−4. Average errors are 5 percent when this equation is used. For pressures greater than 3.4 MPa, the thermal conductivity from Eq. (2-135) may be corrected by the technique suggested by Lenoir.57 The correction factor is the ratio of conductivity factors F/F′, where F is at the desired temperature and higher pressure, and F′ is at the same temperature and lower pressure (usually atmospheric). The conductivity factors are calculated from: 2.054T 2r F = 17.77 + 0.065Pr − 7.764Tr −  (2-136) e0.2 Pr where Tr = reduced temperature as in Eq. (2-135) Pr = reduced pressure, P/Pc P = pressure, MPa Pc = critical pressure, MPa The average error in the pressure correction alone is typically 3 percent. Example 31 Estimate thermal conductivity of n-octane at 373.15 K and pressures of 0.1 MPa and 20.0 MPa. The required properties from Daubert et 24 al. are: ρ at 293.15 K = 6.155 kmol/m3, M = 114.2, Tc = 568.7 K, and Pc = 2.490 MPa. Tr = 373.15/568.7 = 0.6561, n = 1.001, and C = 1.811 × 10−4. Substituting in Eq. (2-135) for the thermal conductivity at 0.1 MPa:



3 + (20)(1 − 0.6561)2/3 kL = (1.811 × 10−4)(6.155)(114.2)1.001  293.15 2/3 3 + (20) 1 −  568.7 = 0.107 W/m K







PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES The reported value is 0.111 W/m K.123 To correct to 20 MPa, Pr = 20.0/2.490 = 8.032, and P′r = 0.1/2.490 = 0.04016. Substituting in Eq. (2-136) to calculate F and F′: (2.054)(0.6561)2 F = 17.77 + (0.065)(8.032) − (7.764)(0.6561) −  e(0.2)(8.032) F = 13.02 (2.054)(0.6561)2 F′ = 17.77 + (0.065)(0.04016) − (7.764)(0.6561) −  e(0.2)(0.04016) F′ = 11.80 Thus, the thermal conductivity of n-octane at 373.15 K and 20.0 MPa is estimated to be (0.111)(13.02/11.80) = 0.122 W/m K as compared with an experimental value of 0.121 W/m K.123 For pure component hydrocarbon liquids above the normal boiling point and all pressures, the method of Kanitkar and Thodos46 is recommended: α eβρr P < 10,000 kPa (2-137) kL =  λ βρ 2.596 × 10−4 P 1.6 r + α e r kL =  P > 10,000 kPa (2-138) λ where kL = liquid thermal conductivity at the temperature T (K) and pressure P (kPa) of interest, W/m K α = 0.0112 β−3.322 (2-139) β = 0.40 + 0.986e−0.64λ (2-140)





101.325 2/3 1/2  λ = T 1/6 c M Pc ρr = reduced density = Vc /V Vc = critical molar volume, m3/kmol V = molar volume at T and P, m3/kmol Tc = critical temperature, K M = molecular weight Pc = critical pressure, kPa Pr = reduced pressure, P/Pc

(2-141)



β = 0.40 + 0.986e−(0.64)(3.639)

= 0.496

α = (0.0112)(0.496)

= 0.115

−3.322

Using Eq. (2-138): (2.596 × 10−4)(6.024)1.6 + 0.115e(0.496)(2.478) kL =  = 0.109 W/m K 3.639 The reported value is 0.106 W/m K.123 The thermal conductivity of pure component nonhydrocarbon liquids may be estimated by the method of Baroncini et al.,5 with a modification by Myers75 for silicon compounds, at reduced temperatures between 0.3 and 0.8 and at pressures below 3.5 MPa: abc (1 − Tr)0.38 kL =   (2-142) T 1/6 m r where kL = liquid thermal conductivity, W/m K Tr = reduced temperature, T/Tc T = temperature, K Tc = critical temperature, K a = constant parameter b = constant parameter = function of normal boiling temperature Tb, K c = constant paramerer = function of critical temperature Tc, K m = constant parameter = function of molecular weight M

 

Compound class

a

b

c

m

Acids* Alcohols, phenols Esters† Ethers Halides‡ Refrigerants R20–R23 Ketones Alkoxysilanes Alkyl-(aryl)-chlorosilanes

0.00319 0.00339 0.0415 0.0385 0.494 0.562 0.00383 0.00482 0.6510

T b6/5 T b6/5 T b6/5 T b6/5 1.0 1.0 T b6/5 T b6/5 1.0

T c−1/6 T c−1/6 T c−1/6 T c−1/6 T 1/6 c T 1/6 c T c−1/6 T c−1/6 T 1/6 c

M 1/2 M 1/2 M M M 1/2 M 1/2 M 1/2 M 1/2 M 1/2

*Do not use for formic, myristic, or oleic acids. †Do not use for butyl stearate. ‡Do not use for refrigerants R20–R23 (CHCl3, CHFCl2, CHClF2, or CHF3). 390.8 K. Tr = 360.0/563.0 = 0.6394. From Table 2-399, a = 0.00339, b = (Tb)6/5 = (390.8)6/5 = 1289.3, c = (Tc)−1/6 = (563.0)−1/6 = 0.3480, and m = M1/2 = (74.12)1/2 = 8.609. Substituting in Eq. (2-142): (0.00339)(1289.3)(0.3480) (1 − 0.6394)0.38 kL =   = 0.1292 W/m K 8.609 (0.6394)1/6 123 The reported value is 0.1429 W/m K. For pure component nonhydrocarbon liquids for which Eq. (2-142) is not applicable, the method of Missenard72,73 may be used at temperature T (K) and below pressures of 3.5 MPa: 3 + 20 (1 − Tr)2/3 kL = kLr  (2-143) 273.15 2/3 3 + 20 1 −  Tc













(Tbρr)1/2 Cpr kLr = 2.656 × 10−7  M1/2 N1/4

Example 32 Estimate thermal conductivity of n-octane at 473.15 K and 15,000 kPa. The required properties from Daubert et al.24 are: Tc = 568.7 K, M = 114.2, Pc = 2490.0 kPa, and Vc = 0.4860 m3/kmol. The specific volume at 473.15 K and 15,000 kPa is 0.001717 m3/kg123; V = (0.001717)(114.2) = 0.1961 m3/kmol. Thus, Pr = 15000/2490 = 6.024, and ρr = 0.4860/0.1961 = 2.478. Substituting in Eqs. (2-141), (2-140), and (2-139): 101.325 2/3 λ = (568.7)1/6 (114.2)1/2  = 3.639 2490



TABLE 2-399 Values of Constant Parameters in Eq. (2-142) for Various Compound Classes



Average errors can be expected to be in the order of 10 percent.



Values of a, b, c, and m for various compound classes are found in Table 2-399. Average errors are about 8 percent.

Example 33 Estimate thermal conductivity of n-butanol. The properties required to estimate the liquid thermal conductivity of n-butanol at 360.0 K and 0.1 MPa from Daubert et al.24 are: Tc = 563.0 K, M = 74.12, and Tb =

2-369

where

(2-144)

kL = liquid thermal conductivity, W/m K kLr = liquid thermal conductivity at 273.15 K, W/m K Tr = reduced temperature, T/Tc Tc = critical temperature, K Tb = normal boiling temperature, K ρr = molar density at 273.15 K, kmol/m3 Cpr = molar heat capacity at 273.15 K, J/kmol K M = molecular weight N = number of atoms in the molecule

Errors in the order of 8 percent can be expected.

Example 34 Estimate the thermal conductivity of n-propionaldehyde (CH3CH2CHO) at 318.15 K and low pressure (0.1 MPa); the necessary properties from Daubert et al.24 are: Tc = 504.4 K, Tb = 321.1 K, ρr = 14.11 kmol/m3, C pr = 1.309 × 105 J/kmol K, and M = 58.08. N = 10, and Tr = 318.15/504.4 = 0.6307. Substituting in Eq. (2-144): [(321.1) (14.11)]1/2 (1.309 × 105) kLr = (2.656 × 10−7)  = 0.1727 W/m K (58.08)1/2 (10)1/4 Using Eq. (2-143): 3 + 20 (1 − 0.6307)2/3 = 0.1542 W/m K kL = 0.1727  3 + 20 (1 − 273.15/504.4)2/3





A value from the literature is 0.1541 W/m K.24 For pressures greater than 3.5 MPa, the correction factor suggested by Missenard74 may be used to obtain the thermal conductivity of pure component nonhydrocarbon liquids. Thus: Pr kL = k′L 0.98 + 0.0079 PrT r1.4 + 0.63 T 1.2 (2-145)  r 30 + Pr where kL = liquid thermal conductivity at the desired temperature T (K) and pressure P (MPa), W/m K k′L = liquid thermal conductivity at T and pressure of 0.1 MPa, W/m K Pr = reduced pressure, P/Pc Pc = critical pressure, MPa Tr = reduced temperature, T/Tc Tc = critical temperature, K







Average errors are in the range of 5–20 percent.

Example 35 Estimate thermal conductivity of n-butanol. The required properties at 360 K and 15 MPa from Daubert et al.24 are: Tc = 563.0 K and

2-370

PHYSICAL AND CHEMICAL DATA

Pc = 4.423 MPa. Tr = 360.0/563.0 = 0.6394 and Pr = 15.0/4.423 = 3.391. k′L = 0.1429 W/m K.123 Substituting in Eq. (2-145): 3.391 kL = 0.1429 0.98 + (0.0079) (3.391) (0.6394)1.4 + (0.63) (0.6394)1.2  30 + 3.391







= 0.1474 W/m K Vargaftik123 reports a value of 0.1494 W/m K. For both aqueous and nonaqueous liquid mixtures, the method of Li59 is suggested for pressures below 3.5 MPa: n

km =

n

φ φ k i

j

ij

(2-146)

i=1 j=1

xi Vi φi =  n

xj Vj

(2-147)

Gas Diffusivity For prediction of the gas diffusivity of binary hydrocarbon-hydrocarbon gas systems at low pressures (below about 500 psia [3.5 MPa]) the method of Gilliland32 given by Eq. (2-150) is recommended. 1 1 0.5 0.1014T 1.5  +  M1 M2 (2-150) D12 =  1/3 2 P(V 1/3 1 + V2 )



Component 1 is the solute, while component 2 is the solvent. Units of T, P, and V are °R, psia, and cm3/gmole, respectively. Diffusivity is then in ft2/hr. The molar volumes V1 and V2 at the normal boiling point are estimated by Tyn and Calus,120 Eq. (2-151). Vi = 0.285Vc1.048 i

j=1

2 kij =  (2-148) (1/ki) + (1/kj) where km = mixture liquid thermal conductivity at temperature T (K), W/m K n = number of components xi, j = mole fraction of component i or j in the liquid mixture Vi, j = liquid molar volume of pure component i or j at temperature T, m3/kmol ki, j = liquid thermal conductivity of pure component i or j at temperature T, W/m K Expected errors are in the 4–6 percent range. At pressures greater than 3.5 MPa, a pressure correction using Eq. (2-145) may be used. The mixture is treated as a hypothetical pure component with mixture critical properties obtained via Eqs. (2-5) and (2-8).

Example 36 Estimate thermal conductivity of a mixture of 0.302 mole fraction diethyl ether (1) and 0.698 mole fraction methanol (2) at 273.15 K and 0.1 MPa. The required properties from Daubert et al.24 are: V1 = 0.1007 m3/kmol, V2 = 0.03942 m3/kmol, k1 = 0.1383 W/m K, and k2 = 0.2069 W/m K. Substituting in Eq. (2-148): 2 k12 =  = 0.1658 1/0.1383 + 1/0.2069 Then, using Eq. (2-147) to obtain φ1 and φ2: (0.302)(0.1007) 0.03041 φ1 =  =  = 0.525 (0.302)(0.1007) + (0.698)(0.03942) 0.05793

(2-151)

The method gives average errors of less than 4 percent. For prediction of the gas diffusivity of binary air-hydrocarbon or nonhydrocarbon gas mixtures at low pressures, the method of Fuller et al.30 given by Eq. (2-152) is recommended. 1 1 0.5 0.1013T 1.75  +  M1 M2 D12 =  (2-152) P[( v1)1/3 + ( v2)1/3]2





Units of T and P are K and Pa, respectively, with the resulting diffusivity in m2/sec. All vi are group contribution values for the subscript component summed over atoms, groups, and structural features given in Table 2-400. For air-hydrocarbon systems, average deviations do not exceed 9 percent. For general nonhydrocarbon gas systems, the average deviation is about 6 percent. Example 37 Estimate the diffusivity of benzene vapor diffusing into air at 30°C and 96.5 kPa total pressure. Use Eq. (2-152). 1 1 0.5 0.1013T1.75  +  M M 1 2 D12 =  P[( v1)1/3 + ( v2)1/3]2





T = 303.1 K, M1 = 78.1, M2 = 28.86, and P = 96,500 Pa.

(0.698)(0.03942) φ2 =  = 0.475 0.05793 Substituting in Eq. (2-146):

Now consult Table 2-400. For benzene (C6H6):

km = (0.525)2(0.1383) + 2(0.525)(0.475)(0.1658) + (0.475)2(0.2069) = 0.167 W/m K

For air, v2 = 20.1:

v1 = 6(16.5) + 6(1.98) − 20.2

v1 = 90.68





1 1 0.5 (.01013)303.11.75  +  78.1 28.86 D12 =  (96,500)(90.681/3 + 20.11/3)2

Jamieson and Hastings42 report a value of 0.173 W/m K for this mixture.

DIFFUSIVITY Diffusion is the molecular transport of mass without flow. The diffusivity (D) or diffusion coefficient is the proportionality constant between the diffusion and the concentration gradient causing diffusion. It is usually defined by Fick’s first law for one-dimensional, binary component diffusion for molecular transport without turbulence shown by Eq. (2-149) dC1 N1 (2-149)  = −D12  A dL The molar flow of species 1(N1) per unit area (A) is directly proportional to the change in concentration of species 1 (C1) per distance diffused (L). The usual units of diffusivity are m2/sec. In chemical engineering, the primary application of the diffusivity is to calculate the Schmidt number (µ/ρD) used to correlate mass transfer properties. This number is also used in reaction rate calculations involving transport to and away from catalyst surfaces. Experimental diffusion coefficients are scarce and not highly accurate, especially in the liquid phase, leading to prediction methods with marginal accuracy. However, use of the values predicted are generally suitable for engineering calculations. At concentrations above about 10 mole percent, predicted values should be used with caution. Diffusivities in liquids are 104–105 times lower than those in gases.



D12 = 9.68 × 10−6 m2/sec This value is within 1 percent of an available experimental value. TABLE 2-400

Atomic Diffusion Volumes for Use in Eq. (2-153)

Atomic and structural diffusion-volume increments ν C H O N

16.5 1.98 5.481 (5.69)

Cl S Aromatic ring Heterocyclic ring

(19.5) (17.0) −20.2 −20.2

Diffusion volumes for simple molecules H2 D2 He N2 O2 Air Ar Kr Xe

7.07 6.70 2.88 17.9 16.6 20.1 16.1 22.8 (37.9)

CO CO2 N2O NH3 H2O CCl2F2 SF6 CI2 Br2 SO2

18.9 26.9 35.9 14.9 12.7 (114.8) (69.7) (37.7) (67.2) (41.1)

Parentheses indicate that the value listed is based only on a few data points.

PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES TABLE 2-401

Parameters for Eq. (2-153)

Pr

(D 12 P)R

A

B

C

0.1 0.2 0.3 0.4 0.5 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.5 3.0 4.0 5.0

1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.02 1.02 1.02 1.02 1.03 1.03 1.04 1.05 1.06 1.07

0.38042 0.067433 0.098371 0.137610 0.175081 0.216376 0.314051 0.385736 0.514553 0.599184 0.557725 0.593007 0.696001 0.790770 0.502100 0.837452 0.890390

1.52267 2.16794 2.42910 2.77605 2.98256 3.11384 0.50264 3.07773 3.54744 3.61216 3.41882 3.18415 3.3760 3.27984 2.39031 3.23513 3.13001

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.141211 0.278407 0.372683 0.504894 0.678469 0.0665702 0.0 0.602907 0.0 0.0

E

13.45454 14.00000 10.00900 8.57519 10.37483 11.21674

D′12 D12 = 1.013 × 105  (D12 P)R(1 − AT r−B)(1 − CT −E r ) P

6.19043

(2-153)

Example 38 Estimate the diffusivity of hydrogen (1) in nitrogen (2) at 60°C and 17.23 MPa. A value of the low pressure diffusivity obtained using Eq. (2-152) is D′12 = 9.2 × 10−5 m2/sec. Use Eq. (2-153): D′12 D12 = 1.013 × 105  (D12P)R(1 − AT r−B)(1 − CT −E r ) P For nitrogen, Tc = 126.2 K and Pc = 3.394 MPa. 17.23 333.2 Pr =  = 5.077 Tr =  = 2.640 126.2 3.394 From Table 2-401: (D12P)R = 1.07, A = 0.89039, B = 3.1300, C = 0 9.2 × 10−5 D12 = 1.013 × 105 7 (1.07)[1 − .89039(2.64)−3.13] 1.723 × 10



D12 = 5.54 × 10 m /sec

y D j

(1.1728 × 10−16)(298.15)(1.0 × 112.56)1/2 D12 =  (7.548 × 10−4)(0.0745)0.6 D12 = 2.33 × 10−9 m2/sec

Liquid Diffusivity Liquid diffusivities are in general not as accurately predicted as vapor diffusivities, and specialized methods have been developed. References to each method determined to be accurate are given, but only the most common methods will be presented. For predicting liquid diffusivities of binary nonpolar liquid systems at high solute dilution, Umesi120 developed a method that only depends on the viscosity of the solvent (2) and the radius of gyration of the solvent (2) and the solute (1). The Technical Data Book— Petroleum Refining gives the method and values of the radii of gyration for common hydrocarbons. Errors average 16 percent but reach 30 percent at times. For predicting diffusivities in binary polar or associating liquid systems at high solute dilution, the method of Wilke and Chang129 defined in Eq. (2-156) can be utilized. The Tyn and Calus equation (2-152) can be used to determine the molar volume of the solute at the normal boiling point. Errors average 20 percent, with occasional errors of 35 percent. The method is not considered to be accurate above a solute concentration of 5 mole percent. For concentrated binary nonpolar liquid systems (more than 5 mole percent solute), the diffusivity can be estimated by a molar average mixing rule developed by Caldwell and Babb,14 Eq. (2-156). D1m = x1D21 + (1 − x1)D12

(2-156)

D21 and D12 are dilute solution binary diffusivities. Errors depend on the procedure used to determine the dilute solution diffusivities. For multicomponent nonpolar liquid systems, Leffler and Cullinan56 developed a mixing rule, Eq. (2-157). 2

An available experimental value is 4.89 × 10−7 m2/sec. For prediction of gas phase diffusion coefficients in multicomponent hydrocarbon/nonhydrocarbon gas systems, the method of Wilke128 shown in Eq. (2-154) is used. 1 − yi

For chlorobenzene, M2 = 112.56, χ2 = 1.0, µ2 @ 25°C = 7.548 × 10−4 Pa sec, and V1 = 0.0745 m3/kmole.

D1mµm = (D12µ2)x (D13µ3)x . . .

2

Dim =

T(χ2M2)1/2 D12 = 1.1728 × 10−16  µ2V 10.6

An experimental value of 2.77 × 10−9 m2/sec is available.

D′12 is the low pressure diffusivity at the temperature of interest. (D12P)R is a reduced diffusivity pressure product at infinite reduced temperature; and A, B, C, and E are constants. All are a function of Pr tabulated in Table 2-401. Component 1 is the diffusing species, while component 2 is the concentrated species. Critical properties are for the solvent. The pressure is given in Pa. The diffusivity is in m2/sec. Errors from evaluation average near 15 percent.

−7

molecular weight are required input parameters. For the common solvents, χ decreases to 1 as polarity decreases, with values of 2.6 for water, 1.9 for methanol, 1.5 for ethanol, and 1.0 for less polar solvents. When tested with both hydrocarbon and nonhydrocarbon systems, average errors are about 25 percent—not excessive, considering the magnitude of the diffusivity.

Example 39 Estimate the infinite dilution diffusivity of propane (1) in chlorobenzene (2) at 25°C. Use Eq. (2-113):

An alternate method for gas diffusivity of binary gas mixtures at low pressures is the method of Hirschfelder et al.40 The method requires several molecular parameters and, when evaluated, gives an average absolute error of about 10 percent. The method is discussed in detail in the Data Prediction Manual. For predicting the diffusivity of binary gas mixtures at high pressures, the method of Takahashi,109 Eq. (2-153), applies.



2-371

(2-154)

ij

j j≠1

This mixing rule is used to determine the diffusivity of any component in a j + 1 component mixture and requires binary diffusivities of component i with all other components. It has been estimated that errors are about 5 percent greater than the greatest error in the binary diffusivities. Fairbanks and Wilke,28 using the same Eq. (2-154), made the same recommendation with essentially the same errors. For prediction of the diffusivity of a dilute dissolved gas (hydrocarbon or nonhydrocarbon) in a liquid, the standard method is that of Wilke and Chang129 shown by Eq. (2-155). T(χ2M2)1/2 (2-155) D12 = 1.1728 × 10−16  µ2V 0.6 1 Component 1 is the diffusing gas, while component 2 is the solvent. The solvent viscosity µ2 in Pa sec, the solute molar volume at the normal boiling point V1 in m3/kmole, and the solvent association parameter χ2 multiplied by the solvent

(2-157)

3

The diffusivity of solute 1 in the mixture is related to the binary infinite dilution diffusivities for each of the other components calculated from Eq. (2-155) or the Umesi method. The viscosities are calculated by the methods in the previous section. Errors are not quantifiable, as little experimental data exist, although these errors would be related to those assumed for the binary pairs. For concentrated binary liquid nonhydrocarbon systems, the method of Caldwell and Babb,14 Eq. (2-156) has been modified by introduction of a thermodynamic correction term as shown in Eq. (2-158). d ln γ1 D1mµm = [xiD21 + (1 − x1)D12] 1 +  d ln x1





(2-158)

The activity coefficient (γ) based corrector is calculated using any applicable activity correlating equation such as the van Laar (slightly polar) or Wilson (more polar) equations. The average absolute error is 20 percent. An alternate method for binary concentrated liquid systems where activity coefficients are not available or estimable is the method of Leffler and Cullinan56 previously given in Eq. (2-156). Absolute errors average 25 percent. For estimating the diffusivity of the dilute solute (10 mole percent) in water, the method of Hayduk and Laudie,37 Eq. (2-159), applies.

2-372

PHYSICAL AND CHEMICAL DATA 8.621 × 10−14 D12 =  µ1.14 V 0.589 2 1

Component 1 is the solute, while component 2 is water. The molar volume of the solute in m3/kmole is at the solute normal boiling point, while the viscosity of water in Pa sec is at the temperature of the system resulting in a diffusivity in m2/sec. The average error is about 9 percent when tested on 36 experimental systems. For estimating the diffusivity of a dilute solute (<10 mole percent) in any solvent except water, the method of King et al.,49 Eq. (2-160), applies. D12 = 4.4 × 10

−15

λ2  λ1

  

T V2   µ2 V1

1/6

1/2 1/2 ln (D1m µ1/2 m ) = x2 ln (D* 12 µ 2 ) + x3 ln (D* 13 µ 3 )



σ = a − bT

(2-162)

where a and b are listed for most of the substances. To obtain values at a higher temperature than the upper temperature limit indicated by Jasper, the following expression may be used: σ = d(1 − Tr)e

(2-163)

where d and e are determined such that σ and dσ/dT at the upper temperature limit THI have the same values when calculated from both Eqs. (2-162) and (2-163): b(Tc − THI) e =  (2-164) a − b THI d = (a − b THI)(1 − THI)−e



σ = 4.601 × 10−4(5.49 × 106)2/3(499.15)1/3(0.6170)(1 − 0.6073)11/9 σ = 22.36 mN/m The reported experimental value is 22.68 mN/m.43 For hydrocarbon and polar chemicals, the approach originally suggested by Macleod65 as further developed by Sugden106,107 can be used:




(2-165)

where Tc = critical temperature, °C Tr = reduced temperature = (T + 273.15)/(Tc + 273.15) Eq. (2-163) correctly predicts that the surface tension becomes zero at the critical point.87 For nonpolar, nonhydrocarbon chemicals not found in Jasper,43 use can be made of the corresponding states approach of Brock and Bird9 as modified by Miller69 at temperature T (K): σ = 4.601 × 10−4 P c2/3 T c1/3 Q (1 − Tr)11/9

(2-166)

Tbr(ln Pc − 11.5261) Q = 0.1207 1 +  − 0.281 1 − Tbr

(2-167)



[P] σ =  (ρL − ρG) 1000

(2-161)

The molecules in a gas-liquid interface are in tension and tend to contract to a minimum surface area. This tension may be quantified by the surface tension, which is defined as the force in the plane of the surface per unit length. Jasper43 has made a critical evaluation of experimental surface tension data for approximately 2200 pure chemicals. He correlates surface tension σ (mN/m = dyn/cm) with temperature T (°C) over a specified temperature range as

σ = surface tension, mN/m Pc = critical pressure, Pa

0.6173 (ln 5.49 × 106 − 11.5261) Q = 0.1207 1 +  − 0.281 1 − 0.6173 Q = 0.6170

SURFACE TENSION

where

Example 40 Estimate surface tension for ethyl mercaptan. The required properties from Daubert et al.24 are Pc = 5.49 × 106 Pa, Tc = 499.15 K, and Tb = 308.15 K. For T = 303.15 K, Tr = 303.15/499.15 = 0.6073, and Tbr = 308.15/ 499.15 = 0.6173. Substituting into Eqs. (2-167) and (2-166):

(2-160)

The solute 1 is dissolved in a solvent pair of 2 and 3. D* are infinite dilution binary diffusivities estimated by the proper method discussed previously. The mixture viscosity can be predicted by methods of the previous section. The average absolute error when tested on 40 systems is 25 percent. The method gives higher errors if the solute is gaseous.



Errors are usually less than 5 percent.

1/2

Component 1 is the solute, while component 2 is the solvent. The latent heats, λ, are at the normal boiling point, as are the molar volumes. Using T in K and µ in Pa sec yields a diffusivity in m2/sec. The average error is 21 percent when tested on 237 experimental systems. For prediction of the liquid diffusivity of a solute in a pair of mixed solvents, the method of Tang and Himmelblau,111 Eq. (2-161), is recommended.



Tc = critical temperature, K Tr = reduced temperature, T/Tc Tb = normal boiling temperature, K Tbr = reduced normal boiling temperature, Tb /Tc

(2-159)

where

4

(2-168)

σ = surface tension, mN/m ρL = liquid density, kmol/m3 ρG = vapor density, kmol/m3

The temperature-independent parachor [P] may be calculated by the additive scheme proposed by Quale.83 The atomic group contributions for this method, with contributions for silicon, boron, and aluminum from Myers,75 are shown in Table 2-402. At low pressures, where ρL >> ρG, the vapor density term may be neglected. Errors using Eq. (2-168) are normally less than 5 to 10 percent.

Example 41 Estimate surface tension for isobutyric acid. For isobutyric acid, the liquid density from Daubert et al.24 is 10.77 kmol/m3 at 293.15 K. [P] is determined from Table 2-402: O

CH3 CH

C

CH3

OH

is made up of the two groups CH3CH(CH3) and COOH. Therefore, [P] = 133.3 + 73.8 = 207.1. With Eq. (2-168), neglecting the vapor density,



 = 24.75 mN/m

207.1 σ =  (10.77) 1000

4

Jasper43 quotes a value of 25.04 mN/m at 293.15 K. In general, the surface tension of a liquid mixture is not a simple function of the pure component surface tensions because the composition of the mixture surface is not the same as the bulk. For nonaqueous solutions of n components, the method of Winterfeld, Scriven, and Davis130 is applicable: n n xi xi (2-169) σm = ρ2   (σiσj)1/2 ρLi ρLi i=1 j=1

  

n

1 xi (2-170) =  ρ i = 1 ρLi where σm = mixture surface tension, mN/m xi, j = mole fraction of component i or j in the liquid mixture ρLi, j = pure component liquid density of component i or j, kmol/m3 σi, j = pure component surface tension of component i or j, mN/m Accuracies of 3–4 percent average deviation are typical when using this method.

Example 42 Estimate surface tension of a mixture. At 298.15 K, Daubert et al.24 report the liquid density of n-pentane to be 8.617 kmol/m3 and its surface tension to be 15.47 mN/m. From the same source, the corresponding values for dichloromethane are 15.52 kmol/m3 and 27.22 mN/m. Using Eqs. (2-170) and (2-169) for a mixture of 0.1606 mole fraction n-pentane and 0.8394 mole fraction dichloromethane: 1 0.1606 0.8394 =+ ρ 8.617 15.52 ρ = 13.75 kmol/m3

PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES TABLE 2-402

2-373

Atomic Group Contributions for Calculation of the Parachor [P ]

Atomic group

[P]

Carbon-hydrogen C H CH3 (CH2)n n = 1 − 12 n > 12 CH3CH(CH3) CH3CH2  CH(CH3) CH3CH2CH2CH(CH3) CH3CH(CH3)CH2 CH3CH2CH(C2H5) CH3C(CH3)2 CH3CH2C(CH3)2 CH3CH(CH3)CH(CH3) CH3CH(CH3)C(CH3)2 C6H5

Atomic group

9.0 15.5 55.5 40.0 40.3 133.3 171.9 211.7 173.3 209.5 170.4 207.5 207.9 243.5 189.6

Special Groups H in OH H in HN O OH O2 in acids, esters COO COOH N NH2 S P Si Si (silanes)

10.0 12.5 19.8 29.8 54.8 63.8 73.8 17.5 42.5 49.1 40.5 30.3 43.3

[P]

Special Groups (Cont.) B Al F Cl Br I

13.2 34.9 26.1 55.2 68.0 90.3

Ethylenic Bond Terminal* 2,3-position 3,4-position†

19.1 17.7 16.3

Triple Bond

40.6

Ring Closure 3-membered 4-membered 5-membered 6-membered 7-membered

12.5 6.0 3.0 0.8 4.0

ⴝ0 (ketone) 3 carbon atoms 4 carbon atoms 5 carbon atoms 6 carbon atoms 7 carbon atoms 8 carbon atoms 9 carbon atoms 10 carbon atoms 11 carbon atoms

22.3 20.0 18.5 17.3 17.3 15.1 14.1 13.0 12.6

* Use the value for double bonds in cyclic compound. Assume 3 double bonds for the aromatic ring. † Use 16.3 for double bonds in the 3, 4 or higher positions.



 (15.47) 0.1606 0.8394 + 2(13.75)     [(15.47)(27.22)] 8.617 15.52 0.8394 + (13.75)    (27.22) 15.52

0.1606 σm = (13.75)2  8.617

2

2

1/2

σm = 23.89 mN/m De Soria et al.26 give an experimental value of 24.24 mN/m for this mixture. Surface tensions for aqueous solutions are more difficult to predict than those for nonaqueous mixtures because of the nonlinear dependence on mole fraction. Small concentrations of the organic material may significantly affect the mixture surface tension value. For many binary organic-water mixtures, the method of Tamura, Kurata, and Odani110 may be used: where

(2-172)

ψw is defined by the relation:



Chloroacetic acid

V(chloroacetic acid) q = 2  V(acetic acid) Expected errors are less than 10 percent when q is less than 5 and within 20 percent when q is greater than 5.

Example 43 Estimate surface tension of a water-methanol mixture. Equation (2-171) can be used with a water-methanol mixture at 303.15 K when the methanol mole fraction is 0.122. From Jasper,43 σw = 71.40 mN/m, and σo = 21.73 mN/m. The density of water (per Ref. 24) is 55.16 kmol/m3; Vw = 0.01813 m3/kmol. The density of methanol is 24.49 kmol/m3 (Ref. 24); Vo = 0.04083 m3/kmol. For methanol, q = 1. Using Eq. (2-173) to obtain ψw: ψw (0.878)(0.01813) log10  = log10  1 − ψw (0.122)(0.04083)



(ψw)q (xwVw)q log10  = log10  (xwVw + xoVo)1 − q (1 − ψw) xoVo





q σoV 2/3 o + 44.1   − σwV w2/3 T q where

Example Acetic acid: q = 2 Acetone: q = 2

(2-171)

σm = mixture surface tension, mN/m σw = surface tension of pure water, mN/m σo = surface tension of pure organic component, mN/m ψo = 1 − ψw



q Number of carbon atoms One less than the number of carbon atoms Number of carbon atoms times the ratio of the molar volume of the halogen derivative to the parent fatty acid

Halogen derivatives of fatty acids

2

2

1/4 σ m1/4 = ψwσ 1/4 w + ψoσ o

Organic component Fatty acids, alcohols Ketones

(2-173)

xw = bulk mole fraction of pure water xo = bulk mole fraction of pure organic component Vw = molar volume of pure water, m3/kmol Vo = molar volume of pure organic component, m3/kmol T = temperature, K q = constant depending upon the size and type of the organic component; see table:



44.1 +  [(21.73)(0.04083)2/3 − (71.40)(0.01813)2/3] 303.15 = 0.505 − 0.342 = 0.163 ψw  = 100.163 = 1.455 1 − ψw 1.455 ψw =  = 0.593 2.455

2-374

PHYSICAL AND CHEMICAL DATA

Using Eq. (2-172): ψo = 1 − 0.593 = 0.407. Substituting into Eq. (2-171): σ m1/4 = (0.593)(71.40)1/4 + (0.407)(21.73)1/4 σ m1/4 = 2.603 σm = 45.91 mN/m The reported experimental value is 46.1 mN/m.110

FLAMMABILITY PROPERTIES Flash points, lower and upper flammability limits, and autoignition temperatures are the three properties used to indicate safe operating limits of temperature when processing organic materials. Prediction methods are somewhat erratic, but, together with comparisons with reliable experimental values for families or similar compounds, they are valuable in setting a conservative value for each of the properties. The DIPPR compilation includes evaluated values for over 1000 common organics. Detailed examples of most of the methods discussed are available in Danner and Daubert.22 The flash point is the lowest temperature at which a liquid gives off sufficient vapor to form an ignitable mixture with air near the surface of the liquid or within the vessel used. ASTM test methods include procedures using a closed cup (ASTM D56, ASTM D93, and ASTM D3828), which is preferred, and an open cup (ASTM D92 and ASTM D1310). When several values are available, the lowest temperature is usually taken in order to assure safe operation of the process. The method of Shebeko et al.96 is the preferred flash point prediction method. The formula of the compound, the system pressure, and vapor pressure data for the compound must be available or estimable. Equation (2-174) is the basic equation. P Psat =  = 0 1 + 4.76(2β − 1)

(2-174)

(NH − NX) NO β = NC + NS +  −  4 2 N’s are the numbers of atoms of carbon (C), sulfur (S), hydrogen (H), halogens (X), and oxygen (O) in the molecule. P is the total system pressure. Psat is the vapor pressure of the compound at the flash point temperature. If Psat is available as a function of temperature, Eq. (2-174) can be solved directly for the flash point temperature. Otherwise, trial and error with a table of Psat vs. T is required. Errors average about 5°C but may be as much as 15°C. An alternate method for flash point prediction is the method of Gmehling and Rasmussen33 and depends on the lower flammability limit (discussed later). Vapor pressure as a function of temperature is also required. The method is generally not as accurate as the preceding method as flammability limit errors are propagated. The authors have also extended the method to defined mixtures of organics. The upper and lower flammability limits are the boundary-line mixtures of vapor or gas with air, which, if ignited, will just propagate flame and are given in terms of percent by volume of gas or vapor in the air. Each of these limits also has a temperature at which the flammability limits are reached. The temperature corresponding to the lower-limit partial vapor pressure should equal the flash point. The

temperature corresponding to the upper-limit partial vapor pressure is somewhat above the lower limit and is usually considerably below the autoignition temperature. Flammability limits are calculated at one atmosphere total pressure and normally are considered synonymous with explosive limits. Limits in oxygen rather than air are sometimes measured and available. Limits are generally reported at 298 K and 1 atm. If temperature or pressure are increased, the lower limit will decrease while the upper limit will increase, giving a wider range of compositions over which flame will propagate. The most generally applicable method for prediction of the property is the method of Seaton,95 which depends only on the molecular structure of the molecule and utilizes second order (Benson-type) groups to construct the molecule. Equation (2-175) sums the groups’ number of each type group (ni) to get both the upper and lower limits. zu or

(ni fi ) zl =  nf

ii gi

 

(2-175)

Two sets of fi and gi are given in the article for each second-order group to cover both upper (u) and lower (l) limits (z) in volume percent units. A study of this method for about 80 organic compounds in 14 families shows absolute errors of 0.15 percent and 2.3 percent for the lower and upper limits, respectively. The upper limit prediction should not be used for ethers. Alternate group contribution methods dependent only on molecular structure are the method of Shebeko et al.97 modified by High and Siegel39 for lower flammability limit and the method of High and Danner38 for upper flammability limit. Both methods are detailed by Danner and Daubert.22 A study comparing these methods with the Seaton method shows slightly higher absolute errors of 0.23 percent and 2.9 percent for the lower and upper limits, respectively. The upper limit prediction should not be used for ethers. Both methods are recommended to be used only for qualitative guidance. Lower flammability limits can also be back-calculated from a known flash point by the method of Gmehling and Rasmussen33 discussed earlier. The autoignition temperature is the minimum temperature for a substance to initiate self-combustion in air in the absence of a spark or flame. The temperature is no lower than and is generally considerably higher than the temperature corresponding to the upper flammability limit. Large differences can occur in reported values determined by different procedures. The lowest reasonable value should be accepted in order to assure safety. Values are also sometimes given in oxygen rather than in air. Values for hydrocarbons other than alkynes and alkadienes can be predicted by the method of Suzuki et al.108 The best model includes the descriptors Tc, Pc, the parachor, the molecular surface area (which can be approximated by the van der Waals area), and the zero-order connectivity index. Excluding alkynes and alkadienes, a study for 58 alkanes, aromatics, and cycloalkanes showed an average deviation from experimental values of about 30 K. Another method of estimating autoignition temperatures is to compare values for a compound with other members of its homologous series on a plot vs. carbon number as the temperature decreases and carbon number increases. Affens1 gives a formal procedure for such estimation.

Section 3

Mathematics*

Bruce A. Finlayson, Ph.D., Rehnberg Professor and Chair, Department of Chemical Engineering, University of Washington: Member, National Academy of Engineering (Numerical methods and all general material; section editor) James F. Davis, Ph.D., Professor of Chemical Engineering, Ohio State University (Intelligent Systems) Arthur W. Westerberg, Ph.D., Swearingen University Professor of Chemical Engineering, Carnegie Mellon University: Member, National Academy of Engineering (Optimization) Yoshiyuki Yamashita, Ph.D., Associate Professor of Chemical Engineering, Tohoku University, Sendai, Japan (Intelligent Systems)

MATHEMATICS General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Miscellaneous Mathematical Constants. . . . . . . . . . . . . . . . . . . . . . . . . . The Real-Number System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Progressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Algebraic Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3-7 3-8 3-8 3-9 3-9

MENSURATION FORMULAS Plane Geometric Figures with Straight Boundaries . . . . . . . . . . . . . . . . Plane Geometric Figures with Curved Boundaries . . . . . . . . . . . . . . . . Solid Geometric Figures with Plane Boundaries . . . . . . . . . . . . . . . . . . Solids Bounded by Curved Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . Miscellaneous Formulas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Irregular Areas and Volumes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3-10 3-10 3-11 3-11 3-12 3-12

ELEMENTARY ALGEBRA Operations on Algebraic Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . The Binomial Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Progressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Permutations, Combinations, and Probability. . . . . . . . . . . . . . . . . . . . . Theory of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3-12 3-13 3-13 3-14 3-14

ANALYTIC GEOMETRY Plane Analytic Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solid Analytic Geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3-16 3-18

PLANE TRIGONOMETRY Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Functions of Circular Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inverse Trigonometric Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relations between Angles and Sides of Triangles . . . . . . . . . . . . . . . . . . Hyperbolic Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Approximations for Trigonometric Functions . . . . . . . . . . . . . . . . . . . . .

3-20 3-20 3-21 3-22 3-22 3-23

DIFFERENTIAL AND INTEGRAL CALCULUS Differential Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multivariable Calculus Applied to Thermodynamics . . . . . . . . . . . . . . . Integral Calculus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3-23 3-26 3-27

INFINITE SERIES Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Operations with Infinite Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tests for Convergence and Divergence. . . . . . . . . . . . . . . . . . . . . . . . . . Series Summation and Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3-30 3-31 3-31 3-32

COMPLEX VARIABLES Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Special Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Trigonometric Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Powers and Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Elementary Complex Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Complex Functions (Analytic) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3-33 3-33 3-33 3-33 3-33 3-34

* The contributions of William F. Ames (retired), Georgia Institute of Technology; Arthur E. Hoerl (deceased), University of Delaware; and M. Zuhair Nashed, University of Delaware, to material that was used from the sixth edition is gratefully acknowledged. 3-1

3-2

MATHEMATICS

DIFFERENTIAL EQUATIONS Ordinary Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ordinary Differential Equations of the First Order . . . . . . . . . . . . . . . . Ordinary Differential Equations of Higher Order . . . . . . . . . . . . . . . . . Special Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3-35 3-36 3-36 3-37 3-38

DIFFERENCE EQUATIONS Elements of the Calculus of Finite Differences . . . . . . . . . . . . . . . . . . . Difference Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3-41 3-41

INTEGRAL EQUATIONS Classification of Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relation to Differential Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Methods of Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3-42 3-43 3-43

INTEGRAL TRANSFORMS (OPERATIONAL METHODS) Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Convolution Integral. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . z-Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fourier Cosine Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3-44 3-45 3-45 3-46 3-46

MATRIX ALGEBRA AND MATRIX COMPUTATIONS Matrix Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Matrix Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3-47 3-48

NUMERICAL APPROXIMATIONS TO SOME EXPRESSIONS Approximation Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

NUMERICAL ANALYSIS AND APPROXIMATE METHODS Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical Solution of Linear Equations. . . . . . . . . . . . . . . . . . . . . . . . . Numerical Solution of Nonlinear Equations in One Variable . . . . . . . . Interpolation and Finite Differences. . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical Integration (Quadrature) . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical Solution of Ordinary Differential Equations as Initial Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ordinary Differential Equations-Boundary Value Problems . . . . . . . . . Numerical Solution of Integral Equations. . . . . . . . . . . . . . . . . . . . . . . . Monte Carlo Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical Solution of Partial Differential Equations. . . . . . . . . . . . . . . Spline Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fast Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3-54 3-57 3-60 3-60 3-61 3-64 3-64

OPTIMIZATION Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conditions for Optimality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Strategies of Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3-65 3-66 3-67

STATISTICS Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Enumeration Data and Probability Distributions . . . . . . . . . . . . . . . . . . Measurement Data and Sampling Densities. . . . . . . . . . . . . . . . . . . . . . Tests of Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Least Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Error Analysis of Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Factorial Design of Experiments and Analysis of Variance . . . . . . . . . .

3-69 3-71 3-72 3-76 3-83 3-87 3-87

DIMENSIONAL ANALYSIS PROCESS SIMULATION

3-49

3-49 3-50 3-50 3-51 3-53 3-53

INTELLIGENT SYSTEMS IN PROCESS ENGINEERING

GENERAL REFERENCES: The list of references for this section is selected to provide a broad perspective on classical and modern mathematical methods that are useful in chemical engineering. The references supplement and extend the treatment given in this section. Also included are selected references to important areas of mathematics that are not covered in the Handbook but that may be useful for certain areas of chemical engineering, e.g., additional topics in numerical analysis and software, optimal control and system theory, linear operators, and functional-analysis methods. Readers interested in brief summaries of theory, together with many detailed examples and solved problems on various topics of college mathematics and mathematical methods for engineers, are referred to the Schaum’s Outline Series in Mathematics, published by the McGraw-Hill Book Comapny.

32. 33. 34. 35. 36. 37. 38. 39.

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31.

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237. Rall, L. B. Computational Solution of Nonlinear Operator Equations, Wiley, New York (1969) and Dover, New York (1981). 238. Ralston, A., and A. Rabinowitz. A First Course in Numerical Analysis, 2d ed., McGraw-Hill, New York (1978). 239. Ramirez, W. F. Computational Methods for Process Simulations, Butterworths, Boston (1989). 240. Rauch, J. Partial Differential Equations, Springer-Verlag (1991). 241. Reddy, J. N., and D. K. Gartling. The Finite Element Method in Heat Transfer and Fluid Dynamics, CRC Press (1994). 242. Reklaitis, G. V. Introduction to Material and Energy Balances, Wiley (1983). 243. Rekalitis, G. V., and H. D. Spriggs. Proceedings of the First International Conference on Foundations of Computer-Aided Operations, Elsevier Science Publishers, Inc., New York (1987). 244. Reklaitis, G. V., A. Ravindran, and K. M. Ragsdell. Engineering Optimization Methods and Applications, Wiley, New York (1983). 245. Rhee, H.-K., R. Aris, and N. R. Amundson. First-Order Partial Differential Equations, vol. I., Prentice Hall, Englewood Cliffs, NJ (1986). 246. ———. Matrix Computations and Mathematical Software, McGrawHill, New York (1981). 247. ———. Numerical Methods, Software, and Analysis, 2d ed., Academic, New York (1993). 248. Rich, E., and K. Kevin. Artificial Intelligence, 2d ed., McGraw-Hill, New York (1991). 249. Riggs, J. B. An Introduction to Numerical Methods for Chemical Engineers, Texas Tech Univ. Press, Lubbock, TX (1994). 250. Rippin, D. W. T., J. C. Hale, and J. F. Davis (ed.). Proceedings of the Second International Conference on Foundations of Computer-Aided Operations, CACHE Corporation, Austin, TX (1994). 251. Ritchmyer, R., and K. Morton. Difference Methods for Initial-Value Problems, 2d ed., Interscience, New York (1967). 252. Saaty, T. L., and J. Bram. Nonlinear Mathematics, McGraw-Hill, New York (1964) and Dover, New York (1981). 253. Schiesser, W. E. The Numerical Method of Lines, Academic Press (1991). 254. Schittkowski, K. Num. Math. 38 (1982): 83. 255. Seader, J. D. Computer Modeling of Chemical Processes, AIChE Monog. Ser. No. 15 (1985). 256. Seborg, D. E., T. F. Edgar, and D. A. Mellichamp. Process Dynamics and Control, Wiley, New York (1989). 257. Shampine, L. Numerical Solution of Ordinary Differential Equations, Chapman & Hall (1994). 258. Shapiro, S. C., D. Eckroth et. al (ed.). Encyclopedia of Artificial Intelligence, Wiley, New York (1987). 259. Shanno, D. F. Math. Comp. 24 (1970): 647. 260. Shenk, A. Calculus and Analytic Geometry, Goodyear Publishing Company, Santa Monica, CA (1977). 261. Shockley, J. E. Calculus and Analytic Geometry, Saunders, Philadelphia (1982). 262. Siirola, J. J., I. E. Grossmann, and G. Stephanopoulos. Proceedings of the Second International Conference on Foundations of Computer-Aided Design, Elsevier Science Publishers, Inc., New York (1990). 263. Simmons, G. F. Differential Equations, McGraw-Hill, New York (1972). 264. Simmonds, J. G. A Brief on Tensor Analysis, Springer-Verlag (1994). 265. Sincich, T., and Mendenhall, W. Statistics for Engineering and the Sciences, 4th ed., Prentice Hall, Englewood Cliffs, NJ (1995). 266. Sincovec, R. F. Math. Comp. 26 (1972): 893–895. 267. Smith, I. M., J. L. Siemienivich, and I. Gladwell. “A Comparison of Old and New Methods for Large Systems of Ordinary Differential Equations Arising from Parabolic Partial Differential Equations,” Num. Anal. Rep. Department of Engineering, no. 13, University of Manchester, England (1975). 268. Smith, W. K. Analytic Geometry, Macmillan (1972). 269. Sobel, M. A., and N. Lerner. College Algebra, Prentice Hall, Englewood Cliffs, NJ (1983). 270. Sod, G. A. Numerical Methods in Fluid Dynamics, Cambridge Univ. Press (1985). 271. Sokolnikoff, I. S., and Sokolnikoff, E. S. Higher Mathematics for Engineers and Physicists, McGraw-Hill, New York (1941). 272. Spiegel, M. R. Applied Differential Equations, 3d ed., Prentice Hall, Englewood Cliffs, NJ (1981). 273. Stakgold, I. Green’s Functions and Boundary Value Problems, Interscience, New York (1979). 274. Stein, S. K. Calculus and Analytic Geometry, 3d ed., McGraw-Hill, New York (1982). 275. Stephanopoulos, G., and J. F. Davis (eds.). Artificial Intelligence in Process Engineering, CACHE Monograph Series, CACHE, Austin (1990–1992).

276. Stephanopoulos, G., and H. Chonghun. “Intelligent Systems in Process Engineering: A Review,” Proceedings of PSE ’94, Korea (1994). 277. Stillwell, J. C. Elements of Algebra, CRC Press, New York (1994). 278. Stoer, J., and R. Bulirsch. Introduction to Numerical Analysis, Springer, New York (1993). 279. Strang, G. Linear Algebra and Its Applications, 2d ed., Academic, New York (1980). 280. Strang, G. Introduction to Linear Algebra, Wellesley-Cambridge, Cambridge, MA (1993). 281. ——— and G. Fix. An Analysis of the Finite Element Method, Prentice Hall, Englewood Cliffs, NJ (1973). 282. Swokowski, E. W. Calculus with Analytic Geometry, 2d ed., Prindle, Weber & Schmidt, Boston (1981). 283. Taylor, A. E., and D. C. Lay. Introduction to Functional Analysis, 2d ed., Wiley, New York (1980). 284. Umeda, T., and A. Ichikawa. I&EC Proc. Design Develop. 10 (1971): 229. 285. Vasantharajan, S., and L. T. Biegler. Computers and Chemical Engineering 12 (1988): 1087. 286. Vemuri, V., and W. Karplus. Digital Computer Treatment of Partial Differential Equations, Prentice Hall, Englewood Cliffs, NJ (1981). 287. Vichnevetsky, R. Computer Methods for Partial Differential Equations, vols. 1 and 2, Prentice Hall, Englewood Cliffs, NJ (1981, 1982). 288. Villadsen, J. V., and M. L. Michelsen. Solution of Differential Equation Models by Polynomial Approximation, Prentice Hall, Englewood Cliffs, NJ (1978). 289. Villadsen, J., and W. E. Stewart. Chem. Eng. Sci. 22 (1967): 1483–1501. 290. Walas, S. M. Modeling with Differential Equations in Chemical Engineering, Butterworth-Heinemann, Stoneham, MA (1991). 291. Weisberg, S. Applied Linear Regression, 2d ed., Wiley, New York (1985). 292. Weld, D. S., and J. de Kleer (ed.). Readings in Qualitative Reasoning About Physical Systems, Morgan Kaufman, San Mateo, CA (1990). 293. Westerberg, A. W., H. P. Hutchison, R. L. Motard, and P. Winter. Process Flowsheeting, Cambridge University Press, London (1979). 294. Westerberg, A. W., and H. H. Chien (ed.). Proceedings of the Second International Conference on Foundations of Computer-Aided Design, CACHE Corporation, Austin, TX (1984). 295. Westerberg, A. W. “Optimization” in A. K. Sunol, D. W. T. Rippin, G. V. Reklaitis, O. Hortacsu (eds.), Batch Processing Systems Engineering: Current Status and Future Directions, vol. 143, NATO ASI Series F, Springer, Berlin (1995). 296. Whipkey, K. L., and M. N. Whipkey. The Power of Calculus, 3d ed., Wiley, New York (1979). 297. Wilkinson, J. H. The Algebraic Eigenvalue Problem, Clarendon Press, Oxford (1988). 298. Williams, G. Computational Linear Algebra with Models, 2d ed., Allyn and Bacon, Boston (1981). 299. Wolfram, S. Mathematica, Addison-Wesley, New York (1988). 300. Wouk, A. A Course of Applied Functional Analysis, Interscience, New York (1979). 301. Wylie, C. R. Advanced Engineering Mathematics, 5th ed., McGraw-Hill, New York (1982). 302. Young, D. M. Iterative Solution for Large Linear Systems, Academic, New York (1971). 303. Zienkiewicz, O. C., and R. L. Taylor. The Finite Element Method, McGraw-Hill, London (1989). 304. ——— and K. Morgan. Finite Elements and Approximations, Wiley, New York (1983). REFERENCES FOR GENERAL AND SPECIFIC TOPICS Advanced engineering mathematics: Upper undergraduate level, 19, 80, 127, 144, 156, 173, 194, 203, 209, 301. Graduate level, 79, 127, 212, 273. Mathematical tables, mathematical dictionaries, and handbooks of mathematical functions and formulas, 1, 28, 48, 57, 118, 126, 188, 208, 217, 222, 234. Mathematical modeling of physical phenomena, 17, 19, 31, 44, 98, 132, 194, 273. Mathematical theory of reaction, diffusion, and transport processes, 10, 16, 19, 88, 223. Mathematical methods in chemical engineering, 13, 15, 61, 85, 104, 106, 138, 239, 249, 288. Inequalities, 28, 126, 139, 290. Vector and tensor analysis, 18, 163, 173, 264. Special functions in physics and engineering, 190, 197, 235. Green’s functions and applications, 75, 127, 273. Perturbation and asymptotic methods in applied mathematics, 170, 215, 216. Approximation theory and interpolation, 86, 87. Functional analysis; linear operators, 129, 147, 162, 174, 196, 216, 226, 283, 300. Generalized inverses and least-squares problems, 33, 128, 189.

MATHEMATICS

3-7

MATHEMATICS GENERAL The basic problems of the sciences and engineering fall broadly into three categories: 1. Steady state problems. In such problems the configuration of the system is to be determined. This solution does not change with time but continues indefinitely in the same pattern, hence the name “steady state.” Typical chemical engineering examples include steady temperature distributions in heat conduction, equilibrium in chemical reactions, and steady diffusion problems. 2. Eigenvalue problems. These are extensions of equilibrium problems in which critical values of certain parameters are to be determined in addition to the corresponding steady-state configurations. The determination of eigenvalues may also arise in propagation problems. Typical chemical engineering problems include those in heat transfer and resonance in which certain boundary conditions are prescribed. 3. Propagation problems. These problems are concerned with predicting the subsequent behavior of a system from a knowledge of the initial state. For this reason they are often called the transient (time-varying) or unsteady-state phenomena. Chemical engineering examples include the transient state of chemical reactions (kinetics), the propagation of pressure waves in a fluid, transient behavior of an adsorption column, and the rate of approach to equilibrium of a packed distillation column. The mathematical treatment of engineering problems involves four basic steps: 1. Formulation. The expression of the problem in mathematical language. That translation is based on the appropriate physical laws governing the process. 2. Solution. Appropriate mathematical operations are accomplished so that logical deductions may be drawn from the mathematical model. 3. Interpretation. Development of relations between the mathematical results and their meaning in the physical world. 4. Refinement. The recycling of the procedure to obtain better predictions as indicated by experimental checks. Steps 1 and 2 are of primary interest here. The actual details are left to the various subsections, and only general approaches will be discussed. The formulation step may result in algebraic equations, difference equations, differential equations, integral equations, or combinations of these. In any event these mathematical models usually arise from statements of physical laws such as the laws of mass and energy conservation in the form. Input of conserved quantity − output of conserved quantity + conserved quantity produced = accumulation of conserved quantity Rate of input of conserved quantity − rate of output of conserved quantity + rate of conserved quantity produced = rate of accumulation of conserved quantity These statements may be abbreviated by the statement Input − output + production = accumulation

FIG. 3-1

Boundary conditions.

is called a boundary-value problem (Fig. 3-1). Schematically, the problem is characterized by a differential equation plus an open region in which the equation holds and, on the boundaries of the region, by certain conditions (boundary conditions) that are dictated by the physical problem. The solution of the equation must satisfy the differential equation inside the region and the prescribed conditions on the boundary. In mathematical language, the propagation problem is known as an initial-value problem (Fig. 3-2). Schematically, the problem is characterized by a differential equation plus an open region in which the equation holds. The solution of the differential equation must satisfy the initial conditions plus any “side” boundary conditions. The description of phenomena in a “continuous” medium such as a gas or a fluid often leads to partial differential equations. In particular, phenomena of “wave” propagation are described by a class of partial differential equations called “hyperbolic,” and these are essentially different in their properties from other classes such as those that describe equilibrium (“elliptic”) or diffusion and heat transfer (“parabolic”). Prototypes are: 1. Elliptic. Laplace’s equation ∂2u ∂2u
2 +
2 = 0 ∂x ∂y Poisson’s equation

∂2u ∂2u
2 +
2 = g(x,y) ∂x ∂y These do not contain the variable t (time) explicitly; accordingly, their solutions represent equilibrium configurations. Laplace’s equation corresponds to a “natural” equilibrium, while Poisson’s equation corresponds to an equilibrium under the influence of an external force of density proportional to g(x, y). 2. Parabolic. The heat equation ∂u ∂2u ∂2u
=
2 +
2 ∂t ∂x ∂y describes nonequilibrium or propagation states of diffusion as well as heat transfer. 3. Hyperbolic. The wave equation ∂2u ∂2u ∂2u =
2 +
2
∂t2 ∂x ∂y describes wave propagation of all types when the assumption is made that the wave amplitude is small and that interactions are linear. The solution phase has been characterized in the past by a concentration on methods to obtain analytic solutions to the mathematical

When the basic physical laws are expressed in this form, the formulation is greatly facilitated. These expressions are quite often given the names, “material balance,” “energy balance,” and so forth. To be a little more specific, one could write the law of conservation of energy in the steady state as Rate of energy in − rate of energy out + rate of energy produced = 0 Many general laws of the physical universe are expressible by differential equations. Specific phenomena are then singled out from the infinity of solutions of these equations by assigning the individual initial or boundary conditions which characterize the given problem. In mathematical language one such problem, the equilibrium problem,

FIG. 3-2

Propagation problem.

3-8

MATHEMATICS

equations. These efforts have been most fruitful in the area of the linear equations such as those just given. However, many natural phenomena are nonlinear. While there are a few nonlinear problems that can be solved analytically, most cannot. In those cases, numerical methods are used. Due to the widespread availability of software for computers, the engineer has quite good tools available. Numerical methods almost never fail to provide an answer to any particular situation, but they can never furnish a general solution of any problem. The mathematical details outlined here include both analytic and numerical techniques useful in obtaining solutions to problems. Our discussion to this point has been confined to those areas in which the governing laws are well known. However, in many areas, information on the governing laws is lacking. Interest in the application of statistical methods to all types of problems has grown rapidly since World War II. Broadly speaking, statistical methods may be of use whenever conclusions are to be drawn or decisions made on the basis of experimental evidence. Since statistics could be defined as the technology of the scientific method, it is primarily concerned with the first two aspects of the method, namely, the performance of experiments and the drawing of conclusions from experiments. Traditionally the field is divided into two areas: 1. Design of experiments. When conclusions are to be drawn or decisions made on the basis of experimental evidence, statistical techniques are most useful when experimental data are subject to errors. The design of experiments may then often be carried out in such a fashion as to avoid some of the sources of experimental error and make the necessary allowances for that portion which is unavoidable. Second, the results can be presented in terms of probability statements which express the reliability of the results. Third, a statistical approach frequently forces a more thorough evaluation of the experimental aims and leads to a more definitive experiment than would otherwise have been performed. 2. Statistical inference. The broad problem of statistical inference is to provide measures of the uncertainty of conclusions drawn from experimental data. This area uses the theory of probability, enabling scientists to assess the reliability of their conclusions in terms of probability statements. Both of these areas, the mathematical and the statistical, are intimately intertwined when applied to any given situation. The methods of one are often combined with the other. And both in order to be successfully used must result in the numerical answer to a problem—that is, they constitute the means to an end. Increasingly the numerical answer is being obtained from the mathematics with the aid of computers. MISCELLANEOUS MATHEMATICAL CONSTANTS Numerical values of the constants that follow are approximate to the number of significant digits given. π = 3.1415926536 e = 2.7182818285 γ = 0.5772156649 ln π = 1.1447298858 log π = 0.4971498727 Radian = 57.2957795131° Degree = 0.0174532925 rad Minute = 0.0002908882 rad Second = 0.0000048481 rad

Pi Napierian (natural) logarithm base Euler’s constant Napierian (natural) logarithm of pi, base e Briggsian (common logarithm of pi, base 10

THE REAL-NUMBER SYSTEM The natural numbers, or counting numbers, are the positive integers: 1, 2, 3, 4, 5, . . . . The negative integers are −1, −2, −3, . . . . A number in the form a/b, where a and b are integers, b ≠ 0, is a rational number. A real number that cannot be written as the quotient of two integers is called an irrational number, e.g.,
2,
3,
5, π, 3 e,
2. There is a one-to-one correspondence between the set of real numbers and the set of points on an infinite line (coordinate line).

Order among Real Numbers; Inequalities a > b means that a − b is a positive real number. If a < b and b < c, then a < c. If a < b, then a  c < b  c for any real number c. If a < b and c > 0, then ac < bc. If a < b and c < 0, then ac > bc. If a < b and c < d, then a + c < b + d. If 0 < a < b and 0 < c < d, then ac < bd. If a < b and ab > 0, then 1/a > 1/b. If a < b and ab < 0, then 1/a < 1/b. Absolute Value For any real number x, |x| = x −x



if x ≥ 0 if x < 0

Properties If |x| = a, where a > 0, then x = a or x = −a. |x| = |−x|; −|x| ≤ x ≤ |x|; |xy| = |x| |y|. If |x| < c, then −c < x < c, where c > 0. ||x| − |y|| ≤ |x + y| ≤ |x| + |y|.
x2 = |x|. a = c , then a + b = c + d , a − b = c − d , Proportions If





b d d b d b a−b c−d = .

a+b c+d Indeterminants Form

Example

(∞)(0) 00 ∞0 1∞

xe−x xx (tan x)cos x (1 + x)1/x

x→∞ x → 0+ − x→aπ x → 0+

∞−∞ 0
0 ∞
∞

1 −
x− 1
x+ sin x
x ex
x

x→∞ x→0 x→∞

Limits of the type 0/∞, ∞/0, 0∞, ∞ ⋅ ∞, (+∞) + (+∞), and (−∞) + (−∞) are not indeterminate forms. Integral Exponents (Powers and Roots) If m and n are positive integers and a, b are numbers or functions, then the following properties hold: a−n = 1/an a≠0 (ab)n = anbn (an)m = anm, n


a  = a1/n

anam = an + m if

a>0

mn

 =
a, a > 0
a 
 m

n

n

am a > 0 am/n = (am)1/n =
, a0 = 1 (a ≠ 0) 0a = 0 (a ≠ 0) Infinity (∞) is not a real number. It is possible to extend the realnumber system by adjointing to it “∞” and “−∞,” and within the extended system, certain operations involving +∞ or −∞ are possible. For example, if 0 < a < 1, then a∞ = lim x→∞ ax = 0, whereas if a > 1, then a∞ = ∞, ∞a = ∞ (a > 0), ∞a = 0 (a < 0). Care should be taken in the case of roots and fractional powers of a product; e.g.,
 xy ≠
x
y if x and y are negative. This rule applies if one is careful about the domain of the functions involved; so
 xy =
 x
y if x > 0 and y > 0. Given any number b > 0, there is a unique function f(x) defined for all real numbers x such that (1) f(x) = bx for all rational x; (2) f is increasing if b > 1, constant if b = 1, and decreasing if 0 < b < 1. This function is called the exponential function bx. For any b > 0, f(x) = bx is

MATHEMATICS a continuous function. Also with a,b > 0 and x,y any real numbers, we have (ab)x = axbx bx by = bx + y (bx)y = bxy The exponential function with base b can also be defined as the inverse of the logarithmic function. The most common exponential function in applications corresponds to choosing b the transcendental number e. Logarithms log ab = log a + log b, a > 0, b > 0 log an = n log a log (a/b) = log a − log b n log
a = (1/n) log a The common logarithm (base 10) is denoted log a or log10a. The natural logarithm (base e) is denoted ln a (or in some texts loge a). Roots If a is a real number, n is a positive integer, then x is called the nth root of a if xn = a. The number of nth roots is n, but not all of them are necessarily real. The principal nth root means the following: (1) if a > 0 the principal nth root is the unique positive root, (2) if a < 0, and n odd, it is the unique negative root, and (3) if a < 0 and n even, it is any of the complex roots. In cases (1) and (2), the root can be found on a calculator by taking y = ln a/n and then x = e y. In case (3), see the section on complex variables. PROGRESSIONS Arithmetic Progression n−1

1

n

 (a + kd) = na +
2 n(n − 1)d =
2 (a + ᐉ)

k=0

where ᐉ is the last term, ᐉ = a + (n − 1)d. Geometric Progression n a(rn − 1) ar k − 1 =
(r ≠ 1)  r−1 k=1 Arithmetic-Geometric Progression n−1 a − [a + (n − 1)d]r n dr(1 − r n − 1) (a + kd)r k =

+

 1−r (1 − r)2 k=0 n 1 2 n (n + 1)2(2n2 + 2n − 1)  k5 =
12 k=1 2

k=1 n

 (2k − 1)

3



2

n

n

n

k k

k

k=1

2

k

k=1

2

k=1

The equality holds if, and only if, the vectors a, b are linearly dependent (i.e., one vector is scalar times the other vector). Minkowski’s Inequality Let a1, a2, . . . , an and b1, b2, . . . , bn be any two sets of complex numbers. Then for any real number p > 1,

  |a + b |  ≤   |a |  +   |b |  1/p

n

k

k

1/p

n

p

k

k=1

1/p

n

p

k

k=1

p

k=1

Hölder’s Inequality Let a1, a2, . . . , an and b1, b2, . . . , bn be any two sets of complex numbers, and let p and q be positive numbers with 1/p + 1/q = 1. Then

  a b  ≤   |a |    |b |  n

1/p

n

k k

k

k=1

1/q

n

p

q

k

k=1

k=1

The equality holds if, and only if, the sequences |a1|p, |a2|p, . . . , |an|p and |b1|q, |b2|q, . . . , |bn|q are proportional and the argument (angle) of the complex numbers akb k is independent of k. This last condition is of course automatically satisfied if a1, . . . , an and b1, . . . , bn are positive numbers. Lagrange’s Inequality Let a1, a2, . . . , an and b1, b2, . . . , bn be real numbers. Then 2

n

n

2 k

k=1



2 k

k=1

(akbj − aj bk)2

1≤k≤j≤n

Example Two chemical engineers, John and Mary, purchase stock in the same company at times t1, t2, . . . , tn, when the price per share is respectively p1, p2, . . . , pn. Their methods of investment are different, however: John purchases x shares each time, whereas Mary invests P dollars each time (fractional shares can be purchased). Who is doing better? While one can argue intuitively that the average cost per share for Mary does not exceed that for John, we illustrate a mathematical proof using inequalities. The average cost per share for John is equal to n

x  pi 1 n Total money invested i=1



=
=
 pi Number of shares purchased n i=1 nx

nP n
=
n n P 1



i i = 1 pi = 1 pi

= n2(2n2 − 1)



1 γ = lim 
− ln n = 0.577215 n→∞ m m=1 ALGEBRAIC INEQUALITIES Arithmetic-Geometric Inequality Let An and Gn denote respectively the arithmetic and the geometric means of a set of positive numbers a1, a2, . . . , an. The An ≥ Gn, i.e., a1 + a2 + ⋅⋅⋅ + an

≥ (a1a2 ⋅⋅⋅ an)1/n n The equality holds only if all of the numbers ai are equal. Carleman’s Inequality The arithmetic and geometric means just defined satisfy the inequality n

G

r

r=1

  a b  ≤   |a |   |b | 

The average cost per share for Mary is

1 =
n(4n2 − 1) 3

k=1

n

⋅⋅⋅ ar)1/r ≤ neAn

where e is the best possible constant in this inequality. Cauchy-Schwarz Inequality Let a = (a1, a2, . . . , an), b = (b1, b2, . . . , bn), where the ai’s and bi’s are real or complex numbers. Then

k=1

k=1

2

1 2

r=1

k k

n

n

n

 (a a

n

 (2k − 1) = n  (2k − 1)

or, equivalently,

  a b  =   a   b  −

(r ≠ 1)

3-9

≤ neAn

Thus the average cost per share for John is the arithmetic mean of p1, p2, . . . , pn, whereas that for Mary is the harmonic mean of these n numbers. Since the harmonic mean is less than or equal to the arithmetic mean for any set of positive numbers and the two means are equal only if p1 = p2 = ⋅⋅⋅ = pn, we conclude that the average cost per share for Mary is less than that for John if two of the prices pi are distinct. One can also give a proof based on the Cauchy-Schwarz inequality. To this end, define the vectors a = (p1−1/2, p2−1/2, . . . , pn−1/2)

b = (p11/2, p21/2, . . . , pn1/2)

Then a ⋅ b = 1 + ⋅⋅⋅ + 1 = n, and so by the Cauchy-Schwarz inequality n 1 (a ⋅ b)2 = n2 ≤ 
i = 1 pi

n

p

i

i=1

with the equality holding only if p1 = p2 = ⋅⋅⋅ = pn. Therefore n

 pi n i=1

n ≤ 1 

n i = 1 pi

3-10

MATHEMATICS

MENSURATION FORMULAS Let A denote areas and V, volumes, in the following.

Radius R of Circumscribed Circle abc R =


4
 s( s −a )( s − b )( s −c)

PLANE GEOMETRIC FIGURES WITH STRAIGHT BOUNDARIES Triangles (see also “Plane Trigonometry”) A = a bh where b = base, h = altitude. Rectangle A = ab where a and b are the lengths of the sides. Parallelogram (opposite sides parallel) A = ah = ab sin α where a, b are the lengths of the sides, h the height, and α the angle between the sides. See Fig. 3-3. Rhombus (equilateral parallelogram) A = aab where a, b are the lengths of the diagonals. Trapezoid (four sides, two parallel) A = a(a + b)h where the lengths of the parallel sides are a and b, and h = height. Quadrilateral (four-sided) A = aab sin θ where a, b are the lengths of the diagonals and the acute angle between them is θ. Regular Polygon of n Sides See Fig. 3-4. 1 180° A =
nl 2 cot
where l = length of each side 4 n l 180° R =
csc
where R is the radius of the circumscribed circle 2 n l 180° r =
cot
where r is the radius of the inscribed circle 2 n 360° β=
n (n − 2)180° θ =

n β β l = 2r tan
= 2R sin
2 2 Inscribed and Circumscribed Circles with Regular Polygon of n Sides Let l = length of one side.

Figure

n

Area

Radius of circumscribed circle

Equilateral triangle Square Pentagon Hexagon Octagon Decagon Dodecagon

3 4 5 6 8 10 12

0.4330 l2 1.0000 l2 1.7205 l2 2.5981 l2 4.8284 l2 7.6942 l2 11.1962 l2

0.5774 l 0.7071 l 0.8507 l 1.0000 l 1.3065 l 1.6180 l 1.8660 l

Radius of inscribed circle 0.2887 l 0.5000 l 0.6882 l 0.8660 l 1.2071 l 1.5388 l 1.9318 l

Radius r of Circle Inscribed in Triangle with Sides a, b, c r=

FIG. 3-3

 (s − a)(s − b)(s − c)

s

Parallelogram.

where s = a(a + b + c)

FIG. 3-4

Regular polygon.

Area of Regular Polygon of n Sides Inscribed in a Circle of Radius r A = (nr 2/2) sin (360°/n) Perimeter of Inscribed Regular Polygon P = 2nr sin (180°/n) Area of Regular Polygon Circumscribed about a Circle of Radius r A = nr 2 tan (180°/n) Perimeter of Circumscribed Regular Polygon 180° P = 2nr tan
n PLANE GEOMETRIC FIGURES WITH CURVED BOUNDARIES Circle (Fig. 3-5) Let C = circumference r = radius D = diameter A = area S = arc length subtended by θ l = chord length subtended by θ H = maximum rise of arc above chord, r − H = d θ = central angle (rad) subtended by arc S C = 2πr = πD (π = 3.14159 . . .) S = rθ = aDθ l = 2
 r2 − d2 = 2r sin (θ/2) = 2d tan (θ/2) 1 1 θ d =

4 r2 −l2 =
l cot
2 2 2 S −1 d −1 l θ =
= 2 cos
= 2 sin
r r D A (circle) = πr2 = dπD2 A (sector) = arS = ar 2θ A (segment) = A (sector) − A (triangle) = ar 2(θ − sin θ) r−H  = r2 cos−1
− (r − H)
2rH − H2 r Ring (area between two circles of radii r1 and r2 ) The circles need not be concentric, but one of the circles must enclose the other. A = π(r1 + r2)(r1 − r2)

FIG. 3-5

Circle.

r1 > r2

MENSURATION FORMULAS

3-11

Volume and Surface Area of Regular Polyhedra with Edge l

FIG. 3-6

Ellipse.

Ellipse (Fig. 3-6)

FIG. 3-7

Parabola.

Let the semiaxes of the ellipse be a and b A = πab C = 4aE(k)

where e2 = 1 − b2/a2 and E(e) is the complete elliptic integral of the second kind, π 1 2 E(e) =
1 −
e2 + ⋅⋅⋅ 2 2





2 2+ b ]. [an approximation for the circumference C = 2π
(a )/2 Parabola (Fig. 3-7) y2 x2 +y2 2x +
4 Length of arc EFG =
4 x2 +y2 +
ln

2x y

4 Area of section EFG =
xy 3 Catenary (the curve formed by a cord of uniform weight suspended freely between two points A, B; Fig. 3-8) y = a cosh (x/a) Length of arc between points A and B is equal to 2a sinh (L/a). Sag of the cord is D = a cosh (L/a) − 1. SOLID GEOMETRIC FIGURES WITH PLANE BOUNDARIES Cube Volume = a3; total surface area = 6a2; diagonal = a
3, where a = length of one side of the cube. Rectangular Parallelepiped Volume = abc; surface area = 2  2(ab + ac + bc); diagonal =
 a2 + b +c2, where a, b, c are the lengths of the sides. Prism Volume = (area of base) × (altitude); lateral surface area = (perimeter of right section) × (lateral edge). Pyramid Volume = s (area of base) × (altitude); lateral area of regular pyramid = a (perimeter of base) × (slant height) = a (number of sides) (length of one side) (slant height). Frustum of Pyramid (formed from the pyramid by cutting off the top with a plane

1⋅A 2)h V = s (A1 + A2 +
A where h = altitude and A1, A2 are the areas of the base; lateral area of a regular figure = a (sum of the perimeters of base) × (slant height).

FIG. 3-8

Catenary.

Type of surface

Name

Volume

Surface area

4 equilateral triangles 6 squares 8 equilateral triangles 12 pentagons 20 equilateral triangles

Tetrahedron Hexahedron (cube) Octahedron Dodecahedron Icosahedron

0.1179 l3 1.0000 l3 0.4714 l3 7.6631 l3 2.1817 l3

1.7321 l2 6.0000 l2 3.4641 l2 20.6458 l2 8.6603 l2

SOLIDS BOUNDED BY CURVED SURFACES Cylinders (Fig. 3-9) V = (area of base) × (altitude); lateral surface area = (perimeter of right section) × (lateral edge). Right Circular Cylinder V = π (radius)2 × (altitude); lateral surface area = 2π (radius) × (altitude). Truncated Right Circular Cylinder V = πr 2h; lateral area = 2πrh h = a (h1 + h2) Hollow Cylinders Volume = πh(R2 − r 2), where r and R are the internal and external radii and h is the height of the cylinder. Sphere (Fig. 3-10) V (sphere) = 4⁄ 3πR3, jπD3 V (spherical sector) = wπR2h = jπh1(3r 22 + h12) V (spherical segment of one base) = jπh1(3r 22 + h12) V (spherical segment of two bases) = jπh 2(3r 12 + 3r 22 + h 22 ) A (sphere) = 4πR2 = πD2 A (zone) = 2πRh = πDh A (lune on the surface included between two great circles, the inclination of which is θ radians) = 2R2θ. Cone V = s (area of base) × (altitude). Right Circular Cone V = (π/3) r 2h, where h is the altitude and r is the radius of the base; curved surface area = πr
 r2 + h2, curved sur2 face of the frustum of a right cone = π(r1 + r2)
 h2 +( r1 −r 2), where r1, r2 are the radii of the base and top, respectively, and h is the altitude; volume of the frustum of a right cone = π(h/3)(r 21 + r1r2 + r 22) = h/3(A1 + A2 +
 A 1A2), where A1 = area of base and A2 = area of top. Ellipsoid V = (4 ⁄3) πabc, where a, b, c are the lengths of the semiaxes. Torus (obtained by rotating a circle of radius r about a line whose distance is R > r from the center of the circle) V = 2π2Rr 2

Surface area = 4π2Rr

Prolate Spheroid (formed by rotating an ellipse about its major axis [2a]) Surface area = 2πb2 + 2π(ab/e) sin−1 e

V = 4 ⁄3πab2

where a, b are the major and minor axes and e = eccentricity (e < 1). Oblate Spheroid (formed by the rotation of an ellipse about its minor axis [2b]) Data as given previously.

FIG. 3-9

Cylinder.

FIG. 3-10

Sphere.

3-12

MATHEMATICS b2 1+e Surface area = 2πa2 + π
ln
e 1−e

V = 4 ⁄3πa2b

MISCELLANEOUS FORMULAS See also “Differential and Integral Calculus.” Volume of a Solid Revolution (the solid generated by rotating a plane area about the x axis) V=π

[ f(x)] dx b

2

a

where y = f(x) is the equation of the plane curve and a ≤ x ≤ b. Area of a Surface of Revolution

y ds

IRREGULAR AREAS AND VOLUMES

b

S = 2π

a

y/d x) 2 dx and y = f(x) is the equation of the plane where ds =
1 +(d curve rotated about the x axis to generate the surface. Area Bounded by f(x), the x Axis, and the Lines x = a, x = b A=

1. If a plane area is revolved about a line which lies in its plane but does not intersect the area, then the volume generated is equal to the product of the area and the distance traveled by the area’s center of gravity. 2. If an arc of a plane curve is revolved about a line that lies in its plane but does not intersect the arc, then the surface area generated by the arc is equal to the product of the length of the arc and the distance traveled by its center of gravity. These theorems are useful for determining volumes V and surface areas S of solids of revolution if the centers of gravity are known. If S and V are known, the centers of gravity may be determined.

f(x) dx b

[ f(x) ≥ 0]

Irregular Areas Let y0, y1, . . . , yn be the lengths of a series of equally spaced parallel chords and h be their distance apart. The area of the figure is given approximately by any of the following: AT = (h/2)[(y0 + yn) + 2(y1 + y2 + ⋅⋅⋅ + yn − 1)] As = (h/3)[(y0 + yn) + 4(y1 + y3 + y5 + ⋅⋅⋅ + yn − 1) + 2(y2 + y4 + ⋅⋅⋅ + yn − 2)]

a

Length of Arc of a Plane Curve If y = f(x), Length of arc s =

(n even, Simpson’s rule)

The greater the value of n, the greater the accuracy of approximation. Irregular Volumes To find the volume, replace the y’s by crosssectional areas Aj and use the results in the preceding equations.

dy 1 + 
 dx dx b

(trapezoidal rule)

2

a

If x = g(y), Length of arc s =

dx 1 + 
 dy dy 2

d

c

If x = f(t), y = g(t), Length of arc s =

dy dx  + 
dt
dt  dt  t1

2

2

t0

In general, (ds)2 = (dx)2 + (dy)2. Theorems of Pappus (for volumes and areas of surfaces of revolution)

FIG. 3-11

Irregular area.

ELEMENTARY ALGEBRA REFERENCES:

20, 102, 108, 191, 269, 277.

Dividend Divisor ex + 1 | 3e2x + ex + 1 3ex − 2 quotient 3e2x + 3ex

OPERATIONS ON ALGEBRAIC EXPRESSIONS

−2ex + 1 −2ex − 2

An algebraic expression will here be denoted as a combination of letters and numbers such as 3ax − 3xy + 7x2 + 7x 3/ 2 − 2.8xy Addition and Subtraction Only like terms can be added or subtracted in two algebraic expressions. Example (3x + 4xy − x ) + (3x + 2x − 8xy) = 5x − 4xy + 2x . 2

2

2

Example (2x + 3xy − 4x1/2) + (3x + 6x − 8xy) = 2x + 3x + 6x − 5xy − 4x1/2. Multiplication Multiplication of algebraic expressions is term by term, and corresponding terms are combined. Example (2x + 3y − 2xy)(3 + 3y) = 6x + 9y + 9y − 6xy . 2

2

Division This operation is analogous to that in arithmetic. Example Divide 3e2x + ex + 1 by ex + 1.

+ 3 (remainder) Therefore, 3e + e + 1 = (e + 1)(3e − 2) + 3. 2x

x

x

x

Operations with Zero All numerical computations (except division) can be done with zero: a + 0 = 0 + a = a; a − 0 = a; 0 − a = −a; (a)(0) = 0; a0 = 1 if a ≠ 0; 0/a = 0, a ≠ 0. a/0 and 0/0 have no meaning. Fractional Operations x −x x −x x −x x ax −
= −
=
=
;
=
;
=
, if a ≠ 0. y −y −y y y −y y ay

 

x z xz

=
; y y y


y 
t  =
yt ; x

z

xz

  

xt x/y x t
=

=
z/t y z yz

Factoring That process of analysis consisting of reducing a given expression into the product of two or more simpler expressions called factors. Some of the more common expressions are factored here: (1) (x2 − y2) = (x − y)(x + y)

ELEMENTARY ALGEBRA s (n − 1)d 2s a = l − (n − 1)d =
−
=
− l n 2 n

(2) x2 + 2xy + y2 = (x + y)2 (3) x + ax + b = (x + c)(x + d) where c + d = a, cd = b 2

l−a 2(s − an) 2(nl − s) d=
=
=
n−1 n(n − 1) n(n − 1)

(4) by2 + cy + d = (ey + f)(gy + h) where eg = b, fg + eh = c, fh = d (5) x2 + y2 + z2 + 2yz + 2xz + 2xy = (x + y + z)2

2l + d +
 (2 l + d )2 −8 ds 2s l−a n =
+ 1 =
=


2d d l+a

(6) x2 − y2 − z2 − 2yz = (x − y − z)(x + y + z) (7) x2 + y2 + z2 − 2xy − 2xz + 2yz = (x − y − z)2 (8) x3 − y3 = (x − y)(x2 + xy + y2) (9) (x + y ) = (x + y)(x − xy + y ) 3

3

2

2

(10) (x4 − y4) = (x − y)(x + y)(x2 + y2) (11) x5 + y5 = (x + y)(x4 − x3y + x2y2 − xy3 + y4) (12) xn − yn = (x − y)(xn − 1 + xn − 2y + xn − 3y2 + ⋅⋅⋅ + yn − 1) Laws of Exponents (an)m = anm; an + m = an ⋅ am; an/m = (an)1/m; an − m = an/am; a1/m = m
a; a1/2 =
a;
 x2 = |x| (absolute value of x). For x > 0, y > 0,
xy  =
x n
 y; for x > 0
 xm = xm/n;
1 /x = 1/
x

The arithmetic mean or average of two numbers a, b is (a + b)/2; of n numbers a1, . . . , an is (a1 + a2 + ⋅⋅⋅ + an)/n. A geometric progression is a succession of terms such that each term, except the first, is derivable from the preceding by the multiplication of a quantity r called the common ratio. All such progressions have the form a, ar, ar 2, . . . , ar n − 1. With a = first term, l = last term, r = ratio, n = number of terms, s = sum of the terms, the following relations hold: [a + (r − 1)s] (r − 1)sr n − 1 l = ar n − 1 =

=

r rn − 1 a(r n − 1) a(1 − r n) rl − a lr n − l s=
=
=
=
r−1 1−r r − 1 rn − rn − 1

THE BINOMIAL THEOREM

l (r − 1)s s−a log l − log a a=
=
r =
log r =

rn − l rn − 1 s−l n−1

If n is a positive integer, n(n − 1) (a + b)n = an + nan − 1b +
an − 2 b2 2! n n(n − 1)(n − 2) n n−j j +

an − 3b3 + ⋅⋅⋅ + bn =  a b j 3! j=0



where

n! = number of combinations of n things taken j at  nj =
j!(n − j)!

log l − log a log[a + (r − 1)s] − log a n =

+ 1 =


log r log r ; of n The geometric mean of two nonnegative numbers a, b is
ab numbers is (a1a2 . . . an)1/n. Example Find the sum of 1 + a + d + ⋅⋅⋅ + 1⁄64. Here a = 1, r = a, n = 7. Thus a(1⁄64) − 1 s =

= 127/64 a−1

a time. n! = 1 ⋅ 2 ⋅ 3 ⋅ 4 ⋅⋅⋅ n, 0! = 1. Example Find the sixth term of (x + 2y) . The sixth term is obtained by setting j = 5. It is 12 12 − 5 x (2y)5 = 792x7(2y)5 5

a ar n s = a + ar + ar 2 + ⋅⋅⋅ + ar n − 1 =
−
1−r 1−r

12

 

  j  = (1 + 1) 14

Example

n

14

3-13

a lim s =
1−r which is called the sum of the infinite geometric progression. If |r| < 1,

then

n→∞

Example The present worth (PW) of a series of cash flows Ck at the end

= 214.

of year k is

j=0

PROGRESSIONS

n Ck PW = 
k k = 1 (1 + i) where i is an assumed interest rate. (Thus the present worth always requires specification of an interest rate.) If all the payments are the same, Ck = R, the present worth is n 1 PW = R 
k k = 1 (1 + i) This can be rewritten as n R 1 R n−1 1 =

PW =

1 + i k = 1 (1 + i)k − 1 1 + i j = 0 (1 + i) j

An arithmetic progression is a succession of terms such that each term, except the first, is derivable from the preceding by the addition of a quantity d called the common difference. All arithmetic progressions have the form a, a + d, a + 2d, a + 3d, . . . . With a = first term, l = last term, d = common difference, n = number of terms, and s = sum of the terms, the following relations hold:

This is a geometric series with r = 1/(1 + i) and a = R/(1 + i). The formulas above give R (1 + i)n − 1 PW (=s) =


(1 + i)n i The same formula applies to the value of an annuity (PW) now, to provide for equal payments R at the end of each of n years, with interest rate i.

If n is not a positive integer, the sum formula no longer applies and an infinite series results for (a + b)n. The coefficients are obtained from the first formulas in this case. Example (1 + x)1/2 = 1 + ax − a ⋅ dx2 + a ⋅ d ⋅ 3⁄ 6 x3 ⋅⋅⋅ (convergent for

x2 < 1).

Additional discussion is under “Infinite Series.”

d l = a + (n − 1)d = −
+ 2

2 d s + a −
d2  2

s (n − 1) =
+
d n 2 n n n s =
[2a + (n − 1)d] =
(a + l) =
[2l − (n − 1)d] 2 2 2

A progression of the form a, (a + d )r, (a + 2d )r 2, (a + 3d )r 3, etc., is a combined arithmetic and geometric progression. The sum of n such terms is a − [a + (n − 1)d]r n rd(1 − r n − 1) s =

+

I−r (1 − r)2 a If |r| < 1, lim s =
+ rd/(1 − r)2. n→∞ 1−r

3-14

MATHEMATICS

The non-zero numbers a, b, c, etc., form a harmonic progression if their reciprocals 1/a, 1/b, 1/c, etc., form an arithmetic progression. Example The progression 1, s, 1⁄5, 1⁄7, . . . , 1⁄31 is harmonic since 1, 3, 5, 7, . . . , 31 form an arithmetic progression. The harmonic mean of two numbers a, b is 2ab/(a + b). PERMUTATIONS, COMBINATIONS, AND PROBABILITY Each separate arrangement of all or a part of a set of things is called a permutation. The number of permutations of n things taken r at a time, written n! P(n, r) =
= n(n − 1)(n − 2) ⋅⋅⋅ (n − r + 1) (n − r)! Example The permutations of a, b, c two at a time are ab, ac, ba, ca, cb, and bc. The formula is P(3,2) = 3!/1! = 6. The permutations of a, b, c three at a time are abc, bac, cab, acb, bca, and cba. Each separate selection of objects that is possible irrespective of the order in which they are arranged is called a combination. The number of combinations of n things taken r at a time, written C(n, r) = n!/ [r!(n − r)!].

has three complex roots. If the coefficients are real numbers, then at least one of the roots must be real. The cubic equation x3 + bx2 + cx + d = 0 may be reduced by the substitution x = y − (b/3) to the form y3 + py + q = 0, where p = s(3c − b2), q = 1⁄27(27d − 9bc + 2b3). This equation has the solutions y1 = A + B, y2 = −a(A + B) + (i
3/2)(A − B), 3 y3 = −a(A + B) − (i
3/2)(A − B), where i2 = −1, A =
 − q /2 + R,
 3 3 2 B =
 − q /2 − R, and R = (p/3) + (q/2) . If b, c, d are all real and if
 R > 0, there are one real root and two conjugate complex roots; if R = 0, there are three real roots, of which at least two are equal; if R < 0, there are three real unequal roots. If R < 0, these formulas are impractical. In this case, the roots are given by yk =  2
 − p /3 cos [(φ/3) + 120k], k = 0, 1, 2 where φ = cos−1

Example Two dice may be thrown in 36 separate ways. What is the probability of throwing such that their sum is 7? Seven may arise in 6 ways: 1 and 6, 2 and 5, 3 and 4, 4 and 3, 5 and 2, 6 and 1. The probability of shooting 7 is j.

2

3

and the upper sign applies if q > 0, the lower if q < 0. Example x3 + 3x2 + 9x + 9 = 0 reduces to y3 + 6y + 2 = 0 under x = y − 1. 3 3 4. The desired roots in y are Here p = 6, q = 2, R = 9. Hence A =
2, B =
− 3 3 3 3 3 3 4 and −a(
2 −
4)  (i
3/2)(
2 +
4). The roots in x are x =
2 −
− y − 1. Example y3 − 7y + 7 = 0. p = −7, q = 7, R < 0. Hence 28 φ
cos 
+ 120k  3 3 27 φ φ = 
,
= 3°37′52″. 28 3

xk = −

Example The combinations of a, b, c taken 2 at a time are ab, ac, bc; taken 3 at a time is abc. An important relation is r! C(n, r) = P(n, r). If an event can occur in p ways and fail to occur in q ways, all ways being equally likely, the probability of its occurrence is p/(p + q), and that of its failure q/(p + q).

q /4
 −p /27

where

The roots are approximately −3.048916, 1.692020, and 1.356897.

Example Many equations of state involve solving cubic equations for the compressibility factor Z. For example, the Redlich-Kwong-Soave equation of state requires solving Z 3 − Z 2 + cZ + d = 0,

d<0

THEORY OF EQUATIONS

where c and d depend on critical constants of the chemical species. In this case, only positive solutions, Z > 0, are desired.

Linear Equations A linear equation is one of the first degree (i.e., only the first powers of the variables are involved), and the process of obtaining definite values for the unknown is called solving the equation. Every linear equation in one variable is written Ax + B = 0 or x = −B/A. Linear equations in n variables have the form

Quartic Equations See Ref. 118. General Polynomials of the nth Degree Denote the general polynomial equation of degree n by

a11 x1 + a12 x2 + ⋅⋅⋅ + a1n xn = b1 a21 x1 + a22 x2 + ⋅⋅⋅ + a2n xn = b2 ⯗ am1 x1 + am2 x2 + ⋅⋅⋅ + amn xn = bm The solution of the system may then be found by elimination or matrix methods if a solution exists (see “Matrix Algebra and Matrix Computations”). Quadratic Equations Every quadratic equation in one variable is expressible in the form ax 2 + bx + c = 0. a ≠ 0. This equation has two solutions, say, x1, x2, given by

4ac  −b 
 x1 b2− =

x2 2a

If a, b, c are real, the discriminant b − 4ac gives the character of the roots. If b2 − 4ac > 0, the roots are real and unequal. If b2 − 4ac < 0, the roots are complex conjugates. If b2 − 4ac = 0 the roots are real and equal. Two quadratic equations in two variables can in general be solved only by numerical methods (see “Numerical Analysis and Approximate Methods”). If one equation is of the first degree, the other of the second degree, a solution may be obtained by solving the first for one unknown. This result is substituted in the second equation and the resulting quadratic equation solved. Cubic Equations A cubic equation, in one variable, has the form x3 + bx2 + cx + d = 0. Every cubic equation having complex coefficients 2

P(x) = a0 x n + a1 x n − 1 + ⋅⋅⋅ + an − 1 x + an = 0 If n > 4, there is no formula which gives the roots of the general equation. For fourth and higher order (even third order), the roots can be found numerically (see “Numerical Analysis and Approximate Methods”). However, there are some general theorems that may prove useful. Remainder Theorems When P(x) is a polynomial and P(x) is divided by x − a until a remainder independent of x is obtained, this remainder is equal to P(a). Example P(x) = 2x4 − 3x2 + 7x − 2 when divided by x + 1 (here a = −1) results in P(x) = (x + 1)(2x3 − 2x2 − x + 8) − 10 where −10 is the remainder. It is easy to see that P(−1) = −10. Factor Theorem If P(a) is zero, the polynomial P(x) has the factor x − a. In other words, if a is a root of P(x) = 0, then x − a is a factor of P(x). If a number a is found to be a root of P(x) = 0, the division of P(x) by (x − a) leaves a polynomial of degree one less than that of the original equation, i.e., P(x) = Q(x)(x − a). Roots of Q(x) = 0 are clearly roots of P(x) = 0. Example P(x) = x3 − 6x2 + 11x − 6 = 0 has the root + 3. Then P(x) = (x − 3)(x2 − 3x + 2). The roots of x2 − 3x + 2 = 0 are 1 and 2. The roots of P(x) are therefore 1, 2, 3. Fundamental Theorem of Algebra Every polynomial of degree n has exactly n real or complex roots, counting multiplicities.

ELEMENTARY ALGEBRA Every polynomial equation a0 x n + a1 x n − 1 + ⋅⋅⋅ + an = 0 with rational coefficients may be rewritten as a polynomial, of the same degree, with integral coefficients by multiplying each coefficient by the least common multiple of the denominators of the coefficients. Example The coefficients of 3⁄2 x4 + 7⁄3 x3 − 5⁄6 x2 + 2x − j = 0 are rational

numbers. The least common multiple of the denominators is 2 × 3 = 6. Therefore, the equation is equivalent to 9x4 + 14x3 − 5x2 + 12x − 1 = 0.

Upper Bound for the Real Roots Any number that exceeds all the roots is called an upper bound to the real roots. If the coefficients of a polynomial equation are all of like sign, there is no positive root. Such equations are excluded here since zero is the upper bound to the real roots. If the coefficient of the highest power of P(x) = 0 is negative, replace the equation by −P(x) = 0. If in a polynomial P(x) = c0 x n + c1 x n − 1 + ⋅⋅⋅ + cn − 1 x + cn = 0, with c0 > 0, the first negative coefficient is preceded by k coefficients which are positive or zero, and if G denotes the greatest of the numerical values of the negative coefficients, then each real root is less than k 1 +
 G /c0. A lower bound to the negative roots of P(x) = 0 may be found by applying the rule to P(−x) = 0. Example P(x) = x + 2x + 4x − 8x − 32 = 0. Here3 k = 5 (since 2 coefficients are zero), G = 32, c0 = 1. The upper bound is 1 +
32 = 3. P(−x) = −x7 − 5 2x + 4x4 − 8x2 − 32 = 0. −P(−x) = x73+ 2x5 − 4x4 + 8x2 + 32 = 0. Here k = 3, G = 4, c0 = 1. The lower bound is −(1 +
4) ≈ −2.587. Thus all real roots r lie in the range −2.587 < r < 3. 7

5

4

2

Descartes Rule of Signs The number of positive real roots of a polynomial equation with real coefficients either is equal to the number v of its variations in sign or is less than v by a positive even integer. The number of negative roots of P(x) = 0 either is equal to the number of variations of sign of P(−x) or is less than that number by a positive even integer. Example P(x) = x4 + 3x3 + x − 1 = 0. v = 1; so P(x) has one positive root. P(−x) = x4 − 3x3 − x − 1. Here v = 1; so P(x) has one negative root. The other two roots are complex conjugates. Example P(x) = x4 − x2 + 10x − 4 = 0. v = 3; so P(x) has three or one positive roots. P(−x) = x4 − x2 − 10x − 4. v = 1; so P(x) has exactly one negative root. Numerical methods are often used to find the roots of polynomials. A detailed discussion of these techniques is given under “Numerical Analysis and Approximate Methods.” Determinants Consider the system of two linear equations a11x1 + a12x2 = b1 a21x1 + a22x2 = b2 If the first equation is multiplied by a22 and the second by −a12 and the results added, we obtain (a11a22 − a21a12)x1 = b1a22 − b2a12 The expression a11a22 − a21a12 may be represented by the symbol a11





a11 a12 a a22 |A| = 21 ⯗ an1 an2

a13 ⋅⋅⋅ a1n ⋅⋅⋅⋅⋅ a2n



an3 ⋅⋅⋅ ann

is called a determinant. The n2 quantities aij are called the elements of the determinant. In the determinant |A| let the ith row and jth column be deleted and a new determinant be formed having n − 1 rows and columns. This new determinant is called the minor of aij denoted Mij.



Example





a11 a12 a13 a21 a22 a23 = a31A31 + a32A32 + a33A33 a31 a32 a33 = a31

aa









a13 a a a a − a32 11 13 + a33 11 12 a23 a21 a23 a21 a22

12 22



In general, Aij will be determinants of order n − 1, but they may in turn be expanded by the rule. Also, n

a

j=1

n

ji



A jk =  a ij A jk = |A| 0 j=1

i=k i≠k

Fundamental Properties of Determinants 1. The value of a determinant |A| is not changed if the rows and columns are interchanged. 2. If the elements of one row (or one column) of a determinant are all zero, the value of |A| is zero. 3. If the elements of one row (or column) of a determinant are multiplied by the same constant factor, the value of the determinant is multiplied by this factor. 4. If one determinant is obtained from another by interchanging any two rows (or columns), the value of either is the negative of the value of the other. 5. If two rows (or columns) of a determinant are identical, the value of the determinant is zero. 6. If two determinants are identical except for one row (or column), the sum of their values is given by a single determinant obtained by adding corresponding elements of dissimilar rows (or columns) and leaving unchanged the remaining elements. Example

 31 25  +  47 25  = 13 + 6 = 19

Directly

 78 25  = 35 − 16 = 19

By rule 6

7. The value of a determinant is not changed if to the elements of any row (or column) are added a constant multiple of the corresponding elements of any other row (or column). 8. If all elements but one in a row (or column) are zero, the value of the determinant is the product of that element times its cofactor. The evaluation of determinants using the definition is quite laborious. The labor can be reduced by applying the fundamental properties just outlined. The solution of n linear equations (not all bi zero) a11 x1 + a12x2 + ⋅⋅⋅ + a1n xn = b1



21



The cofactor Aij of the element aij is the signed minor of aij determined by the rule Aij = (−1) i + jMij. The value of |A| is obtained by forming any of the n n equivalent expressions  j = 1 aij Aij ,  t = 1 aij Aij, where the elements aij must be taken from a single row or a single column of A.

a12 = a11a22 − a21a12 a22 This symbol is called a determinant of second order. The value of the square array of n2 quantities aij, where i = 1, . . . , n is the row index, j = 1, . . . , n the column index, written in the form

a



a11 a12 a13 a a a21 a22 a23 The minor of a23 is M23 = 11 12 a31 a32 a31 a32 a33

Example

3-15

a21 x1 + a22x2 + ⋅⋅⋅ + a2n xn = b2 ⯗⯗⯗ ⯗ an1x1 + an2x2 + ⋅⋅⋅ + annxn = bn a11 ⋅⋅⋅ a1n a21 ⋅⋅⋅ a2n where |A| = ⯗ ≠0 an1 ⋅⋅⋅ ann





has a unique solution given by x1 = |B1|/ |A|, x2 = |B2|/ |A|, . . . , xn = |Bn|/ |A|, where Bk is the determinant obtained from A by replacing its kth column by b1, b2, . . . , bn. This technique is called Cramer’s rule. It requires more labor than the method of elimination and should not be used for computations.

3-16

MATHEMATICS

ANALYTIC GEOMETRY REFERENCES: 108, 188, 193, 260, 261, 268, 274, 282. Analytic geometry uses algebraic equations and methods to study geometric problems. It also permits one to visualize algebraic equations in terms of geometric curves, which frequently clarifies abstract concepts.

PLANE ANALYTIC GEOMETRY Coordinate Systems The basic concept of analytic geometry is the establishment of a one-to-one correspondence between the points of the plane and number pairs (x, y). This correspondence may be done in a number of ways. The rectangular or cartesian coordinate system consists of two straight lines intersecting at right angles (Fig. 3-12). A point is designated by (x, y), where x (the abscissa) is the distance of the point from the y axis measured parallel to the x axis, positive if to the right, negative to the left. y (ordinate) is the distance of the point from the x axis, measured parallel to the y axis, positive if above, negative if below the x axis. The quadrants are labeled 1, 2, 3, 4 in the drawing, the coordinates of points in the various quadrants having the depicted signs. Another common coordinate system is the polar coordinate system (Fig. 3-13). In this system the position of a point is designated by the pair (r, θ), r =
 x2 +y2 being the distance to the origin 0(0,0) and θ being the angle the line r makes with the positive x axis (polar axis). To change from polar to rectangular coordinates, use x = r cos θ and y = r sin θ. To change from rectangular to polar coordinates, use r =
 x2 +y2 and θ = tan−1 (y/x) if x ≠ 0; θ = π/2 if x = 0. The distance between two points (x1, y1), (x2, y2) is defined 2 2 by d =
 (x1 −x +(y1 −y 2) 2) in rectangular coordinates or by d = 2 2 r 1 +r −2 r1 r2cos (θ1 − θ
 2 2) in polar coordinates. Other coordinate systems are sometimes used. For example, on the surface of a sphere latitude and longitude prove useful. The Straight Line (Fig. 3-14) The slope m of a straight line is the tangent of the inclination angle θ made with the positive x axis. If (x1, y1) and (x2, y2) are any two points on the line, slope = m = (y2 − y1)/ (x2 − x1). The slope of a line parallel to the x axis is zero; parallel to the y axis, it is undefined. Two lines are parallel if and only if they have the same slope. Two lines are perpendicular if and only if the product of their slopes is −1 (the exception being that case when the lines are parallel to the coordinate axes). Every equation of the type Ax + By + C = 0 represents a straight line, and every straight line has an equation of this form. A straight line is determined by a variety of conditions:

origin are given (see Fig. 3-15). Let p = length of the perpendicular and α the angle that the perpendicular makes with the positive x axis. The equation of the line is x cos  + y sin  = p. The equation of a line perpendicular to a given line of slope m and passing through a point (x1, y1) is y − y1 = −(1/m) (x − x1). The distance from a point (x1, y1) to a line with equation Ax + by + C = 0 is |Ax1 + By1 + C| d =

A2 + B2
 Example If it is known that centigrade C and Fahrenheit F are linearly related and when C = 0°, F = 32°; C = 100°, F = 212°, find the equation relating C and F and that point where C = F. By using the two-point form, the equation is 212 − 32 F − 32 =
(C − 0) 100 − 0 or F = 9⁄ 5C + 32. Equivalently 100 − 0 C − 0 =
(F − 32) 212 − 32 or C = 5⁄ 9(F − 32). Letting C = F, we have from either equation F = C = −40.

Occasionally some nonlinear algebraic equations can be reduced to linear equations under suitable substitutions or changes of variables. In other words, certain curves become the graphs of lines if the scales or coordinate axes are appropriately transformed. Example Consider y = bxn. B = log b. Taking logarithms log y = n log x + log b. Let Y = log y, X = log x, B = log b. The equation then has the form Y = nX + B, which is a linear equation. Consider k = k0 exp (−E/RT), taking logarithms loge k = loge k0 − E/(RT). Let Y = loge k, B = loge k0, and m = −E/R, X = 1/T, and the result is Y = mX + B. Next consider y = a + bxn. If the substitution t = x n is made, then the graph of y is a straight line versus t. Asymptotes The limiting position of the tangent to a curve as the point of contact tends to an infinite distance from the origin is called an asymptote. If the equation of a given curve can be expanded in a Laurent power series such that n n b f(x) =  ak x k + 
kk k=0 k=0 x n

lim f(x) =  akxk

and Given conditions (1) (2) (3) (4) (5)

Parallel to x axis Parallel y axis Point (x1, y1) and slope m Intercept on y axis (0, b), m Intercept on x axis (a, 0), m

(6)

Two points (x1, y1), (x2, y2)

(7)

Two intercepts (a, 0), (0, b)

Equation of line y = constant x = constant y − y1 = m(x − x1) y = mx + b y = m(x − a) y2 − y1 y − y1 =
(x − x1) x2 − x1 x/a + y/b = 1

The angle β a line with slope m1 makes with a line having slope m2 is given by tan β = (m2 − m1)/(m1m2 + 1). A line is determined if the length and direction of the perpendicular to it (the normal) from the

FIG. 3-12

Rectangular coordinates.

x→∞

FIG. 3-13

Polar coordinates.

k=0

then the equation of the asymptote is y =  k = 0 ak x k. If n = 1, then the asymptote is (in general oblique) a line. In this case, the equation of the asymptote may be written as n

y = mx + b

m = lim f ′(x) x→∞

b = lim [ f(x) − xf ′(x)] x→∞

Geometric Properties of a Curve When the Equation Is Given The analysis of the properties of an equation is facilitated by the investigation of the equation by using the following techniques: 1. Points of maximum, minimum, and inflection. These may be investigated by means of the calculus.

FIG. 3-14

Straight line.

FIG. 3-15

Determination of line.

ANALYTIC GEOMETRY 2.

Symmetry. Let F(x, y) = 0 be the equation of the curve.

The table characterizes the curve represented by the equation.

Condition on F(x, y)

Symmetry

B2 − 4AC < 0

F(x, y) = F(−x, y) F(x, y) = F(x, −y) F(x, y) = F(−x, −y) F(x, y) = F(y, x)

With respect to y axis With respect to x axis With respect to origin With respect to the line y = x

A < 0 A ≠ C, an ellipse A < 0 A = C, a circle A > 0, no locus

3. Extent. Only real values of x and y are considered in obtaining the points (x, y) whose coordinates satisfy the equation. The extent of them may be limited by the condition that negative numbers do not have real square roots. 4. Intercepts. Find those points where the curves of the function cross the coordinate axes. 5. Asymptotes. See preceding discussion. 6. Direction at a point. This may be found from the derivative of the function at a point. This concept is useful for distinguishing among a family of similar curves. Example y2 = (x2 + 1)/(x2 − 1) is symmetric with respect to the x and y axis, the origin, and the line y = x. It has the vertical asymptotes x = 1. When x = 0, y2 = −1; so there are no y intercepts. If y = 0, (x2 + 1)/(x2 − 1) = 0; so there are no x intercepts. If |x| < 1, y2 is negative; so |x| > 1. From x2 = ( y2 + 1)/(y2 − 1), y = 1 are horizontal asymptotes and |y| > 1. As x → 1+, y → + ∞; as x → + ∞, y → + 1. The graph is given in Fig. 3-16. Conic Sections The curves included in this group are obtained from plane sections of the cone. They include the circle, ellipse, parabola, hyperbola, and degeneratively the point and straight line. A conic is the locus of a point whose distance from a fixed point called the focus is in a constant ratio to its distance from a fixed line, called the directrix. This ratio is the eccentricity e. If e = 0, the conic is a circle; if 0 < e < 1, the conic is an ellipse; if e = 1, the conic is a parabola; if e > 1, the conic is a hyperbola. Every conic section is representable by an equation of second degree. Conversely, every equation of second degree in two variables represents a conic. The general equation of the second degree is Ax2 + Bxy + Cy2 + Dx + Ey + F = 0. Let  be defined as the determinant



2A = B D

FIG. 3-16

B 2C E

D E 2F



Graph of y2 = (x2 + 1)/(x2 − 1)

≠0

=0

B2 − 4AC = 0

B2 − 4AC > 0

Parabola

Hyperbola

2 parallel lines if Q = D2 + E2 − 4(A + C)F > 0 1 straight line if Q = 0, no locus if Q < 0

Point

2 intersecting straight lines

Example 3x2 + 4xy − 2y2 + 3x − 2y + 7 = 0.



6 = 4 3

4 −4 −2



3 −2 = −596 ≠ 0, B2 − 4AC = 40 > 0 14

The curve is therefore a hyperbola.

To translate the axes to a new origin at (h, k), substitute for x and y in the original equation x + h and y + k. Translation of the axes can always be accomplished to eliminate the linear terms in the seconddegree equation in two variables having no xy term. Example x2 + y2 + 2x − 4y + 2 = 0. Rewrite this as x2 + 2x + 1 + y2 − 4y + 4 − 5 + 2 = 0 or (x + 1)2 + (y − 2)2 = 3. Let u = x + 1, v = y − 2. Then u2 + v2 = 3. The axis has been translated to the new origin (−1, 2). The type of curve determined by a specific equation of the second degree can also be easily determined by reducing it to a standard form by translation and/or rotation. In the case in which the equation has no xy term, the procedure is merely to complete the squares of the terms in x and y separately. To rotate the axes through an angle α, substitute for x the quantity x cos α − y sin α and for y the quantity x sin α + y cos α. A rotation of the axes through α = a cot−1 (A − C)/B will eliminate the crossproduct term in the general second-degree equation. Example Consider 3x2 + 2xy + y2 − 2x + 3y = 7. A rotation of axes through α = a cot−1 1 = 22a° eliminates the cross-product term. The following tabulation gives the form of the more common equations. Polar equation

Type of curve

(1) r = a (2) r = 2a cos θ (3) r = 2a sin θ (4) r2 − 2br cos (θ − β) + b2 − a2 = 0

Circle Circle Circle Circle at (b, β), radius a

ke (5) r =

1 − e cos θ

e = 1 parabola 0 < e < 1 ellipse e > 1 hyperbola

Some common equations in parametric form are given below. (1) (x − h)2 + (y − k)2 = a2 (x − h)2 (y − k)2 (2)
+
=1 a2 b2

(3) z2 + y2 = a2 x (4) y = a cosh
a (5) Cycloid

3-17

x = h + a cos θ y = k + a sin θ x = h + a cos φ y = k + a sin φ −at x=
t2 +1
 a y=
t2 + 1


Circle (Fig. 3-23) Parameter is angle θ.

s x = a sinh−1
a y2 = a2 + s2 x = a(φ − sin φ) y = a(1 − cos φ)

Catenary (Fig. 3-24; such as hanging cable under gravity) Parameter s = arc length from (0, a) to (x, y). See Fig. 3-24.

Ellipse (Fig. 3-20) Parameter is angle φ. dy Circle Parameter is t =
= slope of tangent at (x, y). dx

3-18

MATHEMATICS

FIG. 3-17

Circle center (0,0) r = a.

FIG. 3-18

Circle center (a,0) r = 2a cos θ.

FIG. 3-19

Circle center (0,a) r = 2a sin θ.

FIG. 3-20

Ellipse, 0 < e < 1.

FIG. 3-21

Hyperbola, e > 1, r = ke/(1 − e cos θ).

FIG. 3-22

Parabola, e = 1.

Circle at (b, β), radius a: r2 − 2br cos (θ − β)  b2 − a2 = 0.

Graphs of Polar Equations The equation r = 0 corresponds to x = 0, y = 0 regardless of θ. The same point may be represented in several different ways; thus the point (2, π/3) or (2, 60°) has the following representations: (2, 60°), (2, −300°). These are summarized in (2, 60° + n 360°), n = 0, 1, 2, or in radian measure [2, (π/3) + 2nπ], n = 0, 1, 2. Plotting of polar equations can be facilitated by the following steps: 1. Find those points where r is a maximum or minimum. 2. Find those values of θ where r = 0, if any. 3. Symmetry: The curve is symmetric about the origin if the equation is unchanged when θ is replaced by θ  π, symmetric about the x axis if the equation is unchanged when θ is replaced by −θ, and symmetric about the y axis if the equation is unchanged when θ is replaced by π − θ. Parametric Equations It is frequently useful to write the equations of a curve in terms of an auxiliary variable called a parameter. For example, a circle of radius a, center at (0, 0), can be written in the equivalent form x = a cos θ, y = a sin φ where θ is the parameter. Similarly, x = a cos φ, y = b sin φ are the parametric equations of the ellipse x2/a2 + y2/b2 = 1 with parameter φ.

FIG. 3-23

Circle.

FIG. 3-24

Cycloid.

SOLID ANALYTIC GEOMETRY Coordinate Systems The commonly used coordinate systems are three in number. Others may be used in specific problems (see Ref. 212). The rectangular (cartesian) system (Fig. 3-25) consists of mutually orthogonal axes x, y, z. A triple of numbers (x, y, z) is used to represent each point. The cylindrical coordinate system (r, θ, z; Fig. 3-26) is frequently used to locate a point in space. These are essentially the polar coordinates (r, θ) coupled with the z coordinate. As

FIG. 3-25

Cartesian coordinates.

FIG. 3-26

Cylindrical coordinates.

ANALYTIC GEOMETRY before, x = r cos θ, y = r sin θ, z = z and r2 = x2 + y2, y/x = tan θ. If r is held constant and θ and z are allowed to vary, the locus of (r, θ, z) is a right circular cylinder of radius r along the z axis. The locus of r = C is a circle, and θ = constant is a plane containing the z axis and making an angle θ with the xz plane. Cylindrical coordinates are convenient to use when the problem has an axis of symmetry. The spherical coordinate system is convenient if there is a point of symmetry in the system. This point is taken as the origin and the coordinates (ρ, φ, θ) illustrated in Fig. 3-27. The relations are x = ρ sin φ cos θ, y = ρ sin φ sin θ, z = ρ cos φ, and r = ρ sin φ. θ = constant is a plane containing the z axis and making an angle θ with the xz plane. φ = constant is a cone with vertex at 0. ρ = constant is the surface of a sphere of radius ρ, center at the origin 0. Every point in the space may be given spherical coordinates restricted to the ranges 0 ≤ φ ≤ π, ρ ≥ 0, 0 ≤ θ < 2π. Lines and Planes The distance between two points (x1, y1, z1), 2 2 2 (x2, y2, z2) is d =
 (x −x +(y −y +( z1 −z 1 2) 1 2) 2). There is nothing in the geometry of three dimensions quite analogous to the slope of a line in the plane case. Instead of specifying the direction of a line by a trigonometric function evaluated for one angle, a trigonometric function evaluated for three angles is used. The angles α, β, γ that a line segment makes with the positive x, y, and z axes, respectively, are called the direction angles of the line, and cos α, cos β, cos γ are called the direction cosines. Let (x1, y1, z1), (x2, y2, z2) be on the line. Then cos α = (x2 − x1)/d, cos β = (y2 − y1)/d, cos γ = (z2 − z1)/d, where d = the distance between the two points. Clearly cos2 α + cos2 β + cos2 γ = 1. If two lines are specified by the direction cosines (cos α1, cos β1, cos γ1), (cos α2, cos β2, cos γ2), then the angle θ between the lines is cos θ = cos α1 cos α2 + cos β1 cos β2 + cos γ1 cos γ2. Thus the lines are perpendicular if and only if θ = 90° or cos α1 cos α2 + cos β1 cos β2 + cos γ1 cos γ2 = 0. The equation of a line with direction cosines (cos α, cos β, cos γ) passing through (x1, y1, z1) is (x − x1)/cos α = (y − y1)/cos β = (z − z1)/cos γ. The equation of every plane is of the form Ax + By + Cz + D = 0. The numbers A B C

,

,

2 2 2 2 2 2       A + B + C2
 A + B + C2
 A + B + C2


FIG. 3-27

Spherical coordinates.

FIG. 3-29

y2 z2 x2 Ellipsoid.
2 +
2 +
2 = 1 (sphere if a = b = c) b c a

FIG. 3-30

y2 z2 x2 Hyperboloid of one sheet.
2 +
2 −
2 = 1 b c a

FIG. 3-31

y2 z2 x2 Hyperboloid of two sheets.
2 +
2 −
2 = −1 b c a

FIG. 3-28

Parabolic cylinder.

are direction cosines of the normal lines to the plane. The plane through the point (x1, y1, z1) whose normals have these as direction cosines is A(x − x1) + B(y − y1) + C(z − z1) = 0. Example Find the equation of the plane through (1, 5, −2) perpendicular to the line (x + 9)/7 = (y − 3)/−1 = z/8. The numbers (7, −1, 8) are called direction numbers. They are a constant multiple of the direction cosines. cos α = 7/114, cos β = −1/114, cos γ = 8/114. The plane has the equation 7(x − 1) − 1(y − 5) + 8(z + 2) = 0 or 7x − y + 8z + 14 = 0. The distance from the point (x1, y1, z1) to the plane Ax + By + Cz + D = 0 is |Ax1 + By1 + Cz1 + D| d =


2+   B2+ C2
A

Space Curves Space curves are usually specified as the set of points whose coordinates are given parametrically by a system of equations x = f(t), y = g(t), z = h(t) in the parameter t. Example The equation of a straight line in space is (x − x1)/a = (y − y1)/b = (z − z1)/c. Since all these quantities must be equal (say, to t), we may write x = x1 + at, y = y1 + bt, z = z1 + ct, which represent the parametric equations of the line. Example The equations z = a cos βt, y = a sin βt, z = bt, a, β, b positive constants, represent a circular helix. Surfaces The locus of points (x, y, z) satisfying f(x, y, z) = 0, broadly speaking, may be interpreted as a surface. The simplest surface is the plane. The next simplest is a cylinder, which is a surface generated by a straight line moving parallel to a given line and passing through a given curve. Example The parabolic cylinder y = x2 (Fig. 3-28) is generated by a straight line parallel to the z axis passing through y = x2 in the plane z = 0.

3-19

3-20

MATHEMATICS

A surface whose equation is a quadratic in the variables x, y, and z is called a quadric surface. Some of the more common such surfaces are tabulated and pictured in Figs. 3-29 to 3-37.

Elliptic paraboloid.

FIG. 3-33

y2 z2 x2 FIG. 3-32 Cone.
2 +
2 +
2 = 0 b c a

y2 x2
2 +
2 + 2z = 0 b a

FIG. 3-35

y2 x2 Elliptic cylinder.
2 +
2 = 1 b a

FIG. 3-36

Hyperbolic cylinder. y2 x2
2 −
2 = 1 b a

y2 x2 FIG. 3-34 Hyperbolic paraboloid.
2 −
2 + 2z = 0 b a

FIG. 3-37

Parabolic cylinder. y2 + 2ax = 0

PLANE TRIGONOMETRY REFERENCES:

20, 108, 131, 158, 166, 202.

ANGLES An angle is generated by the rotation of a line about a fixed center from some initial position to some terminal position. If the rotation is clockwise, the angle is negative; if it is counterclockwise, the angle is positive. Angle size is unlimited. If α, β are two angles such that α + β = 90°, they are complementary; they are supplementary if α + β = 180°. Angles are most commonly measured in the sexagesimal system or by radian measure. In the first system there are 360 degrees in one complete revolution; one degree = 1⁄90 of a right angle. The degree is subdivided into 60 minutes; the minute is subdivided into 60 seconds. In the radian system one radian is the angle at the center of a circle subtended by an arc whose length is equal to the radius of the circle. Thus 2 rad = 360°; 1 rad = 57.29578°; 1° = 0.01745 rad; 1 min = 0.00029089 rad. The advantage of radian measure is that it is dimensionless. The quadrants are conventionally labeled as Fig. 3-38 shows.

FIG. 3-38

Quadrants.

FUNCTIONS OF CIRCULAR TRIGONOMETRY The trigonometric functions of angles are the ratios between the various sides of the reference triangles shown in Fig. 3-39 for the various quadrants. Clearly r =
 x2 +y2 ≥ 0. The fundamental functions (see Figs. 3-40, 3-41, 3-42) are

FIG. 3-39

Triangles.

FIG. 3-40

Graph of y = sin x.

PLANE TRIGONOMETRY

3-21

Relations between Functions of a Single Angle sec θ = 1/ cos θ; csc θ = 1/sin θ, tan θ = sin θ/cos θ = sec θ/csc θ = 1/cot θ; sin2 θ + cos2 θ = 1; 1 + tan2 θ = sec2 θ; 1 + cot2 θ = csc2 θ. For 0 ≤ θ ≤ 90° the following results hold: sin θ = cos θ/cot θ =
1 −co s2 θ = cos θ tan θ

FIG. 3-41

tan θ 1 θ θ =

=

= 2 sin
cos
+t an2 θ
1 +co t2 θ
1 2 2

 

Graph of y = cos x.

and

1 2θ =

cos θ =
1 −sin 2 θ +t an
1 cot θ sin θ θ θ =

=
= cos2
− sin2
2 2 +co t2 θ tan θ
1





The cofunction property is very important. cos θ = sin (90° − θ), sin θ = cos (90° − θ), tan θ = cot (90° − θ), cot θ = tan (90° − θ), etc. Functions of Negative Angles sin (−θ) = −sin θ, cos (−θ) = cos θ, tan (−θ) = −tan θ, sec (−θ) = sec θ, csc (−θ) = −csc θ, cot (−θ) = −cot θ. FIG. 3-42

Graph of y = tan x.

Plane Trigonometry Sine of θ = sin θ = y/r Cosine of θ = cos θ = x/r Tangent of θ = tan θ = y/x

Secant of θ = sec θ = r/x Cosecant of θ = csc θ = r/y Cotangent of θ = cot θ = x/y

Magnitude and Sign of Trigonometric Functions 0 ≤ θ ≤ 360° Function

0° to 90°

90° to 180°

180° to 270°

270° to 360°

sin θ csc θ cos θ sec θ tan θ cot θ

+0 to +1 +∞ to +1 +1 to 0 +1 to +∞ +0 to +∞ +∞ to +0

+1 to +0 +1 to +∞ −0 to −1 −∞ to −1 −∞ to −0 −0 to −∞

−0 to −1 −∞ to −1 −1 to −0 −1 to −∞ +0 to +∞ +∞ to +0

−1 to −0 −1 to −∞ +0 to +1 +∞ to +1 −∞ to −0 −0 to −∞

Identities Sum and Difference Formulas Let x, y be two angles. sin (x  y) = sin x cos y  cos x sin y; cos (x  y) = cos x cos y  sin x sin y; tan (x  y) = (tan x  tan y)/(1  tan x tan y); sin x  sin y = 2 sin a(x  y) cos a(x  y); cos x + cos y = 2 cos a(x + y) cos a(x − y); cos x − cos y = −2 sin a(x + y) sin a(x − y); tan x  tan y = [sin (x  y)]/(cos x cos y); sin2 x − sin2 y = cos2 y − cos2 x = sin (x + y) sin (x − y); cos2 x − sin2 y = cos2 y − sin2 x = cos (x + y) cos (x − y); sin (45° + x) = cos (45° − x); sin (45° − x) = cos (45° + x); tan (45°  x) = cot (45°  x). A cos x + 2 2   B sin x =
 A + B2 sin (α + x) =
 A + B2 cos (β − x) where tan α = A/B, tan β = B/A; both α and β are positive acute angles. Multiple and Half Angle Identities Let x = angle, sin 2x = 2 sin x cos x; sin x = 2 sin ax cos ax; cos 2x = cos2 x − sin2x = 1 − 2 sin2x = 2 cos2x − 1. tan 2x = (2 tan x)/(1 − tan2 x); sin 3x = 3 sin x − 4 sin3x; cos 3x = 4 cos3 x − 3 cos x. tan 3x = (3 tan x − tan3 x)/(1 − 3 tan2 x); sin 4x = 4 sin x cos x − 8 sin3 x cos x; cos 4x = 8 cos4 x − 8 cos2 x + 1. x sin
=
 a (1 −co s x) 2

 

x (1   cos
=
a +o csx) 2

Values of the Trigonometric Functions for Common Angles θ°

θ, rad

sin θ

cos θ

tan θ

0 30 45 60 90

0 π/6 π/4 π/3 π/2

0 1/2
2/2
3/2 1

1
3/2
2/2 1/2 0

0
3/3 1
3 +∞

If 90° ≤ θ ≤ 180°, sin θ = sin (180° − θ); cos θ = −cos (180° − θ); tan θ = −tan (180° − θ). If 180° ≤ θ ≤ 270°, sin θ = −sin (270° − θ); cos θ = −cos (270° − θ); tan θ = tan (270° − θ). If 270° ≤ θ ≤ 360°, sin θ = −sin (360° − θ); cos θ = cos (360° − θ); tan θ = −tan (360° − θ). The reciprocal properties may be used to find the values of the other functions. If it is desired to find the angle when a function of it is given, the procedure is as follows: There will in general be two angles between 0° and 360° corresponding to the given value of the function.

1 − cos x sin x 1 − cos x
=
=
   1 + cos x 1 + cos x sin x

x tan
= 2

Relations between Three Angles Whose Sum Is 180° Let x, y, z be the angles. x y z six x + sin y + sin z = 4 cos
cos
cos
2 2 2

  

  

x y z cos x + cos y + cos z = 4 sin
sin
sin
+ 1 2 2 2

  

x y z sin x + sin y − sin z = 4 sin
sin
cos
2 2 2

sin x + sin y + sin z = 2 cos x cos y cos z + 2; tan x + tan y + tan z = tan x tan y tan z; sin 2x + sin 2y + sin 2z = 4 sin x sin y sin z. 2

2

2

INVERSE TRIGONOMETRIC FUNCTIONS Given (a > 0)

Find an acute angle θ0 such that

Required angles are

sin θ = +a cos θ = +a tan θ = +a sin θ = −a cos θ = −a tan θ = −a

sin θ0 = a cos θ0 = a tan θ0 = a sin θ0 = a cos θ0 = a tan θ0 = a

θ0 and (180° − θ0) θ0 and (360° − θ0) θ0 and (180° + θ0) 180° + θ0 and 360° − θ0 180° − θ0 and 180° + θ0 180° − θ0 and 360° − θ0

y = sin −1 x = arcsin x is the angle y whose sine is x. Example y = sin−1 a, y is 30°. The complete solution of the equation x = sin y is y = (−1)n sin−1 x + n(180°), −π/2 ≤ sin−1 x ≤ π/2 where sin−1 x is the principal value of the angle whose sine is x. The range of principal values of the cos−1 x is 0 ≤ cos−1 x ≤ π and −π/2 ≤ tan−1 x ≤ π/2. If these restrictions are allowed to hold, the following formulas result:

3-22

MATHEMATICS

x −x2
1 sin−1 x = cos−1
1 −x2 = tan−1
2 = cot−1
x −x
1 1 1 π = sec−1
2 = csc−1
=
− cos−1 x x 2 −x
1 −x2
1 −x2 = tan−1
cos−1 x = sin−1
1 x FIG. 3-44

x 1 = cot−1
2 = sec−1
x −x
1

Right triangle.

a =
(c  + b)( c−  b) = c sin α = b tan α

x 1 tan−1 x = sin−1
2 = cos−1
2 −x +x
1
1 +x2 1
1 = cot−1
= sec−1
1 +x2 = csc−1
x x RELATIONS BETWEEN ANGLES AND SIDES OF TRIANGLES Solutions of Triangles (Fig. 3-43) Let a, b, c denote the sides and α, β, γ the angles opposite the sides in the triangle. Let 2s = a + b + c, A = area, r = radius of the inscribed circle, R = radius of the circumscribed circle, and h = altitude. In any triangle α + β + γ = 180°.

 +a)( c−  a) = c cos α = a cot α b =
(c a b a 2 2 a  b, sin α =
, cos α =
, tan α =
, β = 90° − α c =
 c c b

1 a2 b2 tan α c2 sin 2α A =
ab =
=
=
2 2 tan α 2 4 Oblique Triangles (Fig. 3-45) There are four possible cases. 1. Given b, c and the included angles α, b−c 1 1 1 1 tan
(β + γ)
(β + γ) = 90° −
α; tan
(β − γ) =
b +c 2 2 2 2 1 1 1 1 b sin α β =
(β + γ) +
(β − γ); γ =
(β + γ) −
(β − γ); a =
2 2 2 2 sin β 2. Given the three sides a, b, c, s = a (a + b + c); r=

Triangle.

Law of Sines sin α/a = sin β/ b = sin γ/ c. Law of Tangents a + b tan a(α + β) b + c tan a(β + γ) a + c tan a(α + γ)
=

;
=

;
=

a − b tan a(α − β) b − c tan a(β − γ) a − c tan a(α − γ) Law of Cosines a2 = b2 + c2 − 2bc cos α; b2 = a2 + c2 − 2ac cos β; c = a2 + b2 − 2ab cos γ. Other Relations In this subsection, where appropriate, two more formulas can be generated by replacing a by b, b by c, c by a, α by β, β by γ, and γ by α. cos α = (b2 + c2 − a2)/2bc; a = b cos γ + c = cos β; sin α = (2/bc)
 s( s −a )( s − b )( s −c) ; 2

     

α sin
= 2

(s − b)(s − c) α

; cos
= bc 2

s(s − a) 1
; A =
bh bc 2

1 a2 sin β sin γ =
ab sin γ =

=
s( s−  a)( s−  b)( s− c) = rs 2 2 sin α where r =

Oblique triangle.

Right Triangle (Fig. 3-44) Given one side and any acute angle α or any two sides, the remaining parts can be obtained from the following formulas:

1 π = csc−1
2 =
− sin−1 x  −x 2
1

FIG. 3-43

FIG. 3-45

(s − a)(s − b)(s − c)

 s

R = a/(2 sin α) = abc/4A; h = c sin a = a sin γ = 2rs/b. Example a = 5, b = 4, α = 30°. Use the law of sines. 0.5/5 = sin β/4, sin β = 2⁄ 5, β = 23°35′, γ = 126°25′. So c = sin 126°25′/ 1⁄10 = 10(.8047) = 8.05. The relations given here suffice to solve any triangle. One method for each triangle is given.

(s − a)(s − b)(s − c)

 s

1 r 1 r 1 r tan
α =
; tan
β =
; tan
γ =
2 s−a 2 s−b 2 s−c 3. Given any two sides a, c and an angle opposite one of them α, sin γ = (c sin α)/a; β = 180° − a − γ; b = (a sin β)/(sin α). There may be two solutions here. γ may have two values γ1, γ2; γ1 < 90°, γ2 = 180° − γ1 > 90°. If α + γ2 > 180°, use only γ1. This case may be impossible if sin γ > 1. 4. Given any side c and two angles α and β, γ = 180° − α − β; a = (c sin α)/(sin γ); b = (c sin β)/(sin γ). HYPERBOLIC TRIGONOMETRY The hyperbolic functions are certain combinations of exponentials ex and e−x. ex + e−x ex − e−x sinh x ex − e−x cosh x =
; sinh x =
; tanh x =
=
2 2 cosh x ex + e−x ex + e−x 1 cosh x 1 2 coth x =
=
=
; sech x =
=
; ex − e−x tanh x sinh x cosh x ex + e−x 2 1 csch x =
=
sinh x ex − e−x Fundamental Relationships sinh x + cosh x = ex; cosh x − sinh x = e−x; cosh2 x − sinh2 x = 1; sech2 x + tanh2 x = 1; coth2 x − csch2 x = 1; sinh 2x = 2 sinh x cosh x; cosh 2x = cosh2 x + sinh2 x = 1 + 2 sinh2 x = 2 cosh2 x − 1. tanh 2x = (2 tanh x)/(1 + tanh2 x); sinh (x  y) = sinh x cosh y  cosh x sinh y; cosh (x  y) = cosh x cosh y  sinh x sinh y; 2 sinh2 x/2 = cosh x − 1; 2 cosh2 x/2 = cosh x + 1; sinh (−x) = −sinh x; cosh (−x) = cosh x; tanh (−x) = −tanh x. When u = a cosh x, v = a sinh x, then u2 − v2 = a2; which is the equation for a hyperbola. In other words, the hyperbolic functions in the parametric equations u = a cosh x, v = a sinh x have the same relation to the hyperbola u2 − v2 = a2 that the equations u = a cos θ, v = a sin θ have to the circle u2 + v2 = a2.

DIFFERENTIAL AND INTEGRAL CALCULUS Inverse Hyperbolic Functions If x = sinh y, then y is the inverse hyperbolic sine of x written y = sinh−1 x or arcsinh x. sinh−1 x = loge (x +
 x2 +1) 1 1+x −1 x2 +1 ); tanh −1 x =
loge
; cosh x = loge (x +
 2 1−x 1 x+1 −x2 1 +
1 coth−1 x =
loge
; sech−1 x = loge

; 2 x−1 x





1 +
1 +x2 csch−1 = loge

x Magnitude of the Hyperbolic Functions cosh x ≥ 1 with equality only for x = 0; −∞ < sinh x < ∞; −1 < tanh x < 1. cosh x ∼ ex/2 as x → ∞; sinh x → ex/2 as x → ∞.





3-23

APPROXIMATIONS FOR TRIGONOMETRIC FUNCTIONS For small values of θ (θ measured in radians) sin θ ≈ θ, tan θ ≈ θ; cos θ ≈ 1 − (θ2/2). The following relations actually hold: sin θ < θ < tan θ; cos θ < sin θ/θ < 1; θ
1 −θ2 < sin θ < θ; cos θ < θ/tan θ < 1; θ θ2 θ 1 −
< sin θ < θ and θ < tan θ <
2 − θ
1 2





The behavior ratio of the functions as θ → 0 is given by the following: lim sin θ/θ = 1; sin θ/tan θ = 1. θ→0

DIFFERENTIAL AND INTEGRAL CALCULUS REFERENCES: 114, 158, 260, 261, 274, 282, 296. See also “General References: References for General and Specific Topics—Advanced Calculus.” For computer evaluations of the calculus described here, see Refs. 68, 299.

Limits The limit of function f(x) as x approaches a (a is finite or else x is said to increase without bound) is the number N. lim f(x) = N x→a

DIFFERENTIAL CALCULUS An Example of Functional Notation Suppose that a storage warehouse of 16,000 ft3 is required. The construction costs per square foot are $10, $3, and $2 for walls, roof, and floor respectively. What are the minimum cost dimensions? Thus, with h = height, x = width, and y = length, the respective costs are Walls = 2 × 10hy + 2 × 10hx = 20h(y + x) Roof = 3xy Floor = 2xy Total cost = 2xy + 3xy + 20h(x + y) = 5xy + 20h(x + y)

This states that f(x) can be calculated as close to N as desirable by making x sufficiently close to a. This does not put any restriction on f(x) when x = a. Alternatively, for any given positive number ε, a number δ can be found such that 0 < |a − x| < δ implies that |N − f(x)| < ε. The following operations with limits (when they exist) are valid: lim bf(x) = b lim f(x) x→a

x→a

(3-1)

x→a

x→a

lim f(x) x→a f(x) lim
=
lim g(x) x→a g(x)

(3-2)

x→a

Solving for h from Eq. (3-2), h = volume/xy = 16,000/xy 320,000 Cost = 5xy +
(y + x) = 5xy + 320,000 xy In this form it can be shown that the minimum x = y; therefore Cost = 5x2 + 640,000 (1/x)

x→a

(3-3)


x +
y  1

1

(3-4)

cost will occur for

By evaluation, the smallest cost will occur when x = 40. Cost = 5(1600) + 640,000/40 = $24,000 The dimensions are then x = 40 ft, y = 40 ft, h = 16,000/(40 × 40) = 10 ft. Symbolically, the original cost relationship is written Cost = f(x, y, h) = 5xy + 20h(y + x) and the volume relation Volume = g(x, y, h) = xyh = 16,000 In terms of the derived general relationships (3-1) and (3-2), x, y, and h are independent variables—cost and volume, dependent variables. That is, the cost and volume become fixed with the specification of dimensions. However, corresponding to the given restriction of the problem, relative to volume, the function g(x, y, z) = xyh becomes a constraint function. In place of three independent and two dependent variables the problem reduces to two independent (volume has been constrained) and two dependent as in functions (3-3) and (3-4). Further, the requirement of minimum cost reduces the problem to three dependent variables (x, y, h) and no degrees of freedom, that is, freedom of independent selection.

x→a

lim [ f(x)g(x)] = lim f(x) ⋅ lim g(x)

and the restriction Total volume = xyh

x→a

lim [ f(x) + g(x)] = lim f(x) + lim g(x) x→a

if

lim g(x) ≠ 0 x→a

Continuity A function f(x) is continuous at the point x = a if lim [ f(a + h) − f(a)] = 0 h→0

Rigorously, it is stated f(x) is continuous at x = a if for any positive ε there exists a δ > 0 such that |f(a + h) − f(a)| < ε for all x with |x − a| < δ. For example, the function (sin x)/x is not continuous at x = 0 and therefore is said to be discontinuous. Discontinuities are classified into three types: 1. Removable 2. Infinite 3. Jump

y = sin x/x at x = 0 y = 1/x at x = 0 y = 10/(1 + e1/x) at x = 0+ y = 0+ x=0 y=0 x = 0− y = 10

Derivative The function f(x) has a derivative at x = a, which can be denoted as f ′(a), if f(a + h) − f(a) lim

h→0 h exists. This implies continuity at x = a. Conversely, a function may be continuous but not have a derivative. The derivative function is df f(x + h) − f(x) f ′(x) =
= lim

dx h→0 h Differentiation Define ∆y = f(x + ∆x) − f(x). Then dividing by ∆x ∆y f(x + ∆x) − f(x)
=

∆x ∆x

3-24

MATHEMATICS ∆y dy lim
=
∆x dx

Call

∆x→0

Example Find the derivative of y = sin x. dy sin (x + ∆x) − sin(x)
= lim

dx ∆x→0 ∆x sin x cos ∆x + sin ∆x cos x − sin x = lim



∆x→0 ∆x sin ∆x cos x sin x(cos ∆x − 1) = lim

+ lim

∆x→0 ∆x→0 ∆x ∆x sin ∆x = cos x since lim
= 1 ∆x→0 ∆x

Differential Operations The following differential operations are valid: f, g, . . . are differentiable functions of x, c is a constant; e is the base of the natural logarithms. dc (3-5)
=0 dx dx
=1 dx

(3-6)

d df dg
(f + g) =
+
dx dx dx

(3-7)

d dg df
(f × g) = f
+ g
dx dx dx dy 1
=
dx dx/dy

if

(3-8) dx
≠0 dy

d df
f n = nf n − 1
dx dx d f g(df/dx) − f(dg/dx)

=

dx g g2



df df dv
=
×
dx dv dx

d tan x = sec x dx

(3-20)

d cot x = −csc2 x dx

(3-21)

d sec x = tan x sec x dx

(3-22)

d csc x = −cot x csc x dx

(3-23)

d sin−1 x = (1 − x2)−1/2 dx

(3-24)

d cos−1x = −(1 − x2)−1/2 dx

(3-25)

d tan−1 x = (1 + x2)−1 dx

(3-26)

d cot−1 x = −(1 + x2)−1 dx

(3-27)

d sec−1 x = x−1(x2 − 1)−1/2 dx

(3-28)

d csc−1 x = −x−1(x2 − 1)−1/2 dx

(3-29)

d sinh x = cosh x dx

(3-30)

d cosh x = sinh x dx

(3-31)

d tanh x = sech2 x dx

(3-32)

d coth x = −csch2 x dx

(3-33)

d sech x = −sech x tanh x dx

(3-34)

d csch x = −csch x coth x dx

(3-35)

d sinh x = (x + 1)

(3-36)

−1

(3-9)

−1/2

2

dx

d cosh−1 = (x2 − 1)−1/2 dx

(3-37)

d tanh−1 x = (1 − x2)−1 dx

(3-38)

d coth−1 x = −(x2 − 1)−1 dx

(3-39)

d sech−1 x = −(1/x)(1 − x2)−1/2 dx

(3-40)

d csch x = −x (x + 1)

(3-41)

−1

−1

−1/2

2

dx

Example Find dy/dx for y =
x cos (1 − x ). 2

(3-10)

dy d d
=
x
cos (1 − x2) + cos (1 − x2)

x dx dx dx

(3-11)

d d
cos (1 − x2) = −sin (1 − x2)
(1 − x2) dx dx

Using (3-8) (3-19)

= −sin (1 − x )(0 − 2x) 2

(chain rule)

(3-12)

g

df df dg
= gf g − 1
+ f g ln f
dx dx dx

(3-13)

dax
= (ln a) ax dx

(3-14)

(3-5), (3-10)

d
x 1
=
x−1/2 dx 2

(3-10)

1 dy
= 2x3/2 sin (1 − x2) +
x−1/2 cos (1 − x2) dx 2

Example Find the derivative of tan x with respect to sin x. v = sin x

Example Derive dy/dx for x2 + y3 = x + xy + A. Here

(3-19)

2

f(x + ∆x) − f(x) dy lim


= ∆x→0 dx ∆x

then

d cos x = −sin x dx

y = tan x

d d d d d
x2 +
y3 =
x +
xy +
A dx dx dx dx dx

d tan x dy dy dx
=
=

d sin x dv dx dv 1 d tan x =

d sin x dx

dx = sec2 x/cos x

dy dy 2x + 3y2
= 1 + y + x
+ 0 dx dx by rules (3-10), (3-10), (3-6), (3-8), and (3-5) respectively. dy 2x − 1 − y Thus
=

dx x − 3y2

Differentials dex = ex dx

(3-15a)

d(ax) = ax log a dx

(3-15b)

d ln x = (1/x) dx

(3-16)

d log x = (log e/x)dx

(3-17)

d sin x = cos x dx

(3-18)

Using (3-12) (3-9) (3-18), (3-20)

Very often in experimental sciences and engineering functions and their derivatives are available only through their numerical values. In particular, through measurements we may know the values of a function and its derivative only at certain points. In such cases the preceding operational rules for derivatives, including the chain rule, can be applied numerically. Example Given the following table of values for differentiable functions f and g; evaluate the following quantities:

DIFFERENTIAL AND INTEGRAL CALCULUS x

f(x)

f′(x)

g(x)

g′(x)

1 3 4

3 0 −2

1 2 10

4 4 3

−4 7 6

d
[ f(x) + g(x)]| x = 4 = f ′(4) + g′(4) = 10 + 6 = 16 dx f ′

f ′(1)g(1) − f(1)g′(1)

1 ⋅ 4 − 3(−4)

=

=
= 1 
g  (1) =


[g(1)] (−4) 16 2

16

Example Given f(x) = 3x3 + 2x + 1, calculate all derivative values at x = 3. df(x)
= 9x2 + 2 dx

x = 3, f′(3) = 9(9) + 2 = 83

d2f(x) = 18x
dx2

x = 3, f″(3) = 18(3) = 54

d3f(x) = 18
dx3

x = 3, f ″(3) = 18

dnf(x) =0
dxn

for n ≥ 4

If f ′(x) > 0 on (a, b), then f is increasing on (a, b). If f ′(x) < 0 on (a, b), then f is decreasing on (a, b). The graph of a function y = f(x) is concave up if f ′ is increasing on (a, b); it is concave down if f ′ is decreasing on (a, b). If f ″(x) exists on (a, b) and if f ″(x) > 0, then f is concave up on (a, b). If f ″(x) < 0, then f is concave down on (a, b). An inflection point is a point at which a function changes the direction of its concavity. Indeterminate Forms: L’Hospital’s Theorem Forms of the type 0/0, ∞/∞, 0 × ∞, etc., are called indeterminates. To find the limiting values that the corresponding functions approach, L’Hospital’s theorem is useful: If two functions f(x) and g(x) both become zero at x = a, then the limit of their quotient is equal to the limit of the quotient of their separate derivatives, if the limit exists or is + ∞ or − ∞. sin x x sin x d sin x cos x lim
= lim
= lim
= 1 x→0 x→0 x→0 x dx 1

Example Find lim
. n→0 Here

(1.1) x

Example Find lim .
1000 x→∞ (1.1)x d(1.1)x (ln 1.1)(1.1)x = lim
= lim

lim
x→∞ x1000 x→∞ dx1000 x→∞ 1000x999 1.1x Obviously lim
= ∞ since repeated application of the rule will reduce the x→∞ x1000 denominator to a finite number 1000! while the numerator remains infinitely large.

x3 6 lim x3 e−x = lim
x = lim
x = 0 x→∞ x→∞ e x→∞ e

∂e ∂z ∂x
= y
+ ey
∂x ∂x ∂x

∂ey ∂z 2 ∂y
= ex
+ x
∂y ∂y ∂y

= 2xyex + e y

= ex + xey

2

2

Order of Differentiation It is generally true that the order of differentiation is immaterial for any number of differentiations or variables provided the function and the appropriate derivatives are continuous. For z = f(x, y) it follows: ∂3f ∂3f ∂3f =
=

∂y2 ∂x ∂y ∂x ∂y ∂x ∂y2 General Form for Partial Differentiation 1. Given f(x, y) = 0 and x = g(t), y = h(t). df ∂f dx ∂f dy Then
=

+

dt ∂x dt ∂y dt d 2f ∂2f dx
2 =
2
dt ∂x dt

∂2

∂2

f dy f ∂f d x

+

+

  +2
∂x ∂y dt dt ∂y  dt  ∂x dt 2

dx

dy

y = (1 − x)

1/x

ln y = (1/x) ln (1 − x) ln(1 − x) lim (ln y) = lim
= −1 x→0 x→0 x lim y = e−1

2

2

2

2

∂f d 2y +

∂y dt2 Example Find df/dt for f = xy, x = ρ sin t, y = ρ cos t. df ∂(xy) d ρ sin t ∂(xy) d ρ cos t
=

+

dt ∂x dt ∂y dt = y(ρ cos t) + x(−ρ sin t) = ρ2 cos2 t − ρ2 sin2 t









Given f(x, y) = 0 and x = g(t, s), y = h(t, s). ∂f ∂f ∂x ∂f ∂y Then
=

+

∂t ∂x ∂t ∂y ∂t 2.

∂f ∂f ∂x ∂f ∂y
=

+

∂s ∂x ∂s ∂y ∂x Differentiation of Composite Function dy ∂f/∂x ∂f Rule 1. Given f(x, y) = 0, then
= −

≠ 0 . dx ∂f/∂y ∂y Rule 2. Given f(u) = 0 where u = g(x), then df du
= f ′(u)
dx dx du   + f ′(u)
dx

d 2f du
2 = f ″(u)
dx dx

2

2

2

df d sin2 u d
1 −x2
=


dx du dx



1 = 2 sin u cos u
(−2x)(1 − x2)−1/2 2

Example Find lim (1 − x)1/x. x→0

x→0

2

x2

Example Find df/dx for f = sin2 u and u =
1−x2

Example Find lim x3 e−x. x→∞

Therefore,

Example Find ∂z/∂x and ∂z/∂y for z = ye x + xey.



x

Let

Partial Derivative The abbreviation z = f(x, y) means that z is a function of the two variables x and y. The derivative of z with respect to x, treating y as a constant, is called the partial derivative with respect to x and is usually denoted as ∂z/∂x or ∂f(x, y)/∂x or simply fx. Partial differentiation, like full differentiation, is quite simple to apply. Conversely, the solution of partial differential equations is appreciably more difficult than that of differential equations.

2

Higher Differentials The first derivative of f(x) with respect to x is denoted by f′ or df/dx. The derivative of the first derivative is called the second derivative of f(x) with respect to x and is denoted by f ″, f (2), or d 2 f/dx 2; and similarly for the higher-order derivatives.

3-25

− u
1 = −2
sin u cos u u 2

Rule 3. Given f(u) = 0 where u = g(x,y), then ∂f ∂u ∂f ∂u
= f ′(u)
+
= f ′(u)
∂x ∂x ∂y ∂y



3-26

MATHEMATICS ∂2f ∂u
2 = f ″
∂x ∂x

∂   + f′
∂x 2

2

u

V is specified once T and p are specified; it is therefore a state function. All applications are for closed systems with constant mass. If a process is reversible and only p-V work is done, the first law and differentials can be expressed as follows.

2

∂2f ∂u ∂u ∂2u
= f ″

+ f′
∂x ∂y ∂x ∂y ∂x ∂y ∂2f ∂u
2 = f ″
∂y ∂y

dU = T dS − p dV dH = T dS + V dp dA = −S dT − p dV dG = −S dT + V dp

∂2u

  + f′
∂y 2

2

MULTIVARIABLE CALCULUS APPLIED TO THERMODYNAMICS Many of the functional relationships needed in thermodynamics are direct applications of the rules of multivariable calculus. This section reviews those rules in the context of the needs of themodynamics. These ideas were expounded in one of the classic books on chemical engineering thermodynamics.151 State Functions State functions depend only on the state of the system, not on past history or how one got there. If z is a function of two variables, x and y, then z(x,y) is a state function, since z is known once x and y are specified. The differential of z is dz = M dx + N dy The line integral



(M dx + N dy)

is independent of the path in x-y space if and only if ∂M ∂N
=
∂y ∂x The total differential can be written as ∂z ∂z dz =
dx +
dy ∂x y ∂y x and the following condition guarantees path independence. ∂ ∂z ∂ ∂z

=

∂y ∂x y ∂x ∂y x

 

 

 

∂2z ∂2z
=
∂y ∂x ∂x ∂y

or

 

(3-42)

(3-43)

x

z

∂z ∂z ∂y (∂y/∂x)z (3-45)
=−

=−
∂x y ∂x z ∂y x (∂y/∂z)x Alternatively, divide Eq. (3-43) by dy when holding some other variable w constant to obtain ∂z ∂z ∂x ∂z (3-46)
=

+
∂y w ∂x v ∂y w ∂y x Also divide both numerator and denominator of a partial derivative by dw while holding a variable y constant to get

    

      

(∂z/∂w) ∂z ∂w = 
 
 
∂z∂x  =
(∂x/∂w) ∂w ∂x y y

y

 

∂T ∂p = −
 
∂V  ∂S

(3-47)

y

Themodynamic State Functions In thermodynamics, the state functions include the internal energy, U; enthalpy, H; and Helmholtz and Gibbs free energies, A and G, respectively, defined as follows: H=U+pV A = U − TS G = H − TS = U + pV − TS = A + pV S is the entropy, T the absolute temperature, p the pressure, and V the volume. These are also state functions, in that the entropy is specified once two variables (like T and p) are specified, for example. Likewise,

V

This is one of the Maxwell relations, and the other Maxwell relations can be derived in a similar fashion by applying Eq. (3-44). ∂T ∂V
=
∂p S ∂S p ∂S ∂p
=
∂V T ∂T V

        ∂S ∂V = −
 
∂p  ∂T

p

In process simulation it is necessary to calculate enthalpy as a function of state variables. This is done using the following formulas, derived from the above relations by considering S and H as functions of T and p. ∂V dH = Cp dT + V − T
dp ∂T p Enthalpy differences are then given by the following formula. T2 p 2 ∂V H(T2, p2) − H(T1, p1) = Cp(T, p1) dT + V−T
dp T1 p1 ∂T p T2,p The same manipulations can be done for internal energy as a function of T and V. (∂V/∂T) dU = CV dT − p + T
p dV (∂V/∂p)T



Rearrangement gives

y

 

T

(3-44)

  dy

 

S

Example Suppose z is constant and apply Eq. (3-43). ∂z ∂z 0 =
dx +
∂x y ∂y

 

This is

C

 

Alternatively, if the internal energy is considered a function of S and V, then the differential is: ∂U ∂U dU =
dS +
dV ∂S V ∂V S This is the equivalent of Eq. (3-43) and gives the following definitions. ∂U ∂U T=
, p=−
∂S V ∂V S Since the internal energy is a state function, then Eq. (3-44) must be satisfied. ∂2U ∂2U
=
∂V ∂S ∂S ∂V



 



  



Partial Derivatives of All Thermodynamic Functions The various partial derivatives of the thermodynamic functions can be classified into six groups. In the general formulas below, the variables U, H, A, G or S are denoted by Greek letters, while the variables V, T, or p are denoted by Latin letters. Type I (3 possibilities plus reciprocals) ∂a ∂p General:
; Specific:
∂b c ∂T V Eq. (3-45) gives ∂p ∂V ∂p (∂V/∂T)p
=−

=−
∂T V ∂T p ∂V T (∂V/∂p)T

 

 

    

Type II

(30 possibilities) ∂α ∂G General:
; Specific:
∂b c ∂T

 

 

V

DIFFERENTIAL AND INTEGRAL CALCULUS The differential for G gives ∂G ∂p
= −S + V
∂T V ∂T V Using the other equations for U, H, A, or S gives the other possibilities. Type III (15 possibilities plus reciprocals) ∂a ∂V General:
; Specific:
∂b α ∂T S First expand the derivative using Eq. (3-45). (∂S/∂T)V ∂V ∂S ∂V
=−

=−
∂T S ∂T V ∂S T (∂S/∂V)T Then evaluate the numerator and denominator as type II derivatives. CV ∂V



T ∂V CV ∂p T



= − ∂V ∂p =
∂T S T ∂V −





∂T p ∂V T ∂T p

 

 

 

f ′(x) dx = f(x) + c where c is an arbitrary constant to be determined by the problem. By virtue of the known formulas for differentiation the following relationships hold (a is a constant):

(du + dv + dw) = du + dv + dw a dv = a dv v v dv =
+ c (n ≠ −1) n+1

 

    

 

n

dv
= ln |v| + c v



∂G

∂G

v

 

∂T

=

=

(∂A/∂T) ∂A   ∂T   ∂A  p

p

(∂G/∂T)p

p

p

This operation has created two type II derivatives; by substitution we obtain ∂G S
=

∂A p S + p (∂V/∂T)p Type V (60 possibilities) ∂α ∂G General:
; Specific:
∂b β ∂p A Start from the differential for dG. Then we get ∂G ∂T
= −S
+ V ∂p A ∂p A The derivative is type III and can be evaluated by using Eq. (3-45). ∂G (∂A/∂p)T
= S
+ V ∂p A (∂A/∂T)p The two type II derivatives are then evaluated. ∂G Sp (∂V/∂p)T
=

+ V ∂p A S + p (∂V/∂T)p These derivatives are also of interest for free expansions or isentropic changes. Type VI (30 possibilities plus reciprocals) ∂α ∂G General:
; Specific:
∂β γ ∂A H We use Eq. (3-47) to obtain two type V derivatives. (∂G/∂T)H ∂G
=
∂A H (∂A/∂T)H These can then be evaluated using the procedures for Type V derivatives.

 

 

(3-54) (3-55) (3-56)

2

(3-57) (3-58) (3-59) (3-60)

2

−1

2

2

v
+c a

(3-61)

dv 1 v−a
=
ln 
 + c v −a 2a v+a 2

(3-62)

2

dv

= ln |v +
 v a| + c v a
 sec v dv = ln (sec v + tan v) + c csc v dv = ln (csc v − cot v) + c

 

 

(3-53)

2

dv
= sin a −v


2

2

 

 

v

2

 

 

(3-52)

−1

 

 

(3-50)

v

v

 

 

(3-49)

(3-51)

a a dv =
+c ln a e dv = e + c sin v dv = −cos v + c cos v dv = sin v + c sec v dv = tan v + c csc v dv = −cot v + c sec v tan v dv = sec v + c csc v cot v dv = −csc v + c dv 1 v
=
tan
+ c v +a a a

These derivatives are of importance for reversible, adiabatic processes (such as in an ideal turbine or compressor), since then the entropy is constant. An example is the Joule-Thomson coefficient. ∂T 1 ∂V
=
−V + T
∂p H Cp ∂T p Type IV (30 possibilities plus reciprocals) ∂α ∂G General:
; Specific:
∂β c ∂A p Use Eq. (3-47) to introduce a new variable.

 

(3-48)

n+1

   

  

3-27

2

2

(3-63) (3-64) (3-65)

Example Derive ∫ av dv = (av/ln a) + c. By reference to the differentiation formula dav/dv = av ln a, or in the more usable form d(av/ln a) = av dv, let f ′ = av dv; then f = av/ln a and hence ∫ av dv = (av/ln a) + c. Example Find ∫ (3x2 + ex − 10) dx using Eq. (3-48). ∫ (3x2 + ex − 10) dx = 3 ∫ x2 dx + ∫ ex dx − 10 ∫ dx = x3 + ex − 10x + c (by Eqs. 3-50, 3-53). 7x dx
. Let v = 2 − 3x ; dv = −6x dx 2 − 3x 7x dx x dx 7 −6x dx
=7
=−

2 − 3x 2 − 3x 6 2 − 3x 7 dv =−

6 v

Example Find Thus

2

2

2

2

2

INTEGRAL CALCULUS

7 = −
ln |v| + c 6

Indefinite Integral If f ′(x) is the derivative of f(x), an antiderivative of f ′(x) is f(x). Symbolically, the indefinite integral of f ′(x) is

7 = −
ln |2 − 3x2| + c 6

3-28

MATHEMATICS

Example—Constant of Integration By definition the derivative of x3 is 3x2, and x3 is therefore the integral of 3x2. However, if f = x3 + 10, it follows that f′ = 3x2, and x3 + 10 is therefore also the integral of 3x2. For this reason the constant c in ∫ 3x2 dx = x3 + c must be determined by the problem conditions, i.e., the value of f for a specified x. Methods of Integration In practice it is rare when generally encountered functions can be directly integrated. For example, the integrand in ∫
 si nx dx which appears quite simple has no elementary function whose derivative is
 si nx. In general, there is no explicit way of determining whether a particular function can be integrated into an elementary form. As a whole, integration is a trial-and-error proposition which depends on the effort and ingenuity of the practitioner. The following are general procedures which can be used to find the elementary forms of the integral when they exist. When they do not exist or cannot be found either from tabled integration formulas or directly, the only recourse is series expansion as illustrated later. Indefinite integrals cannot be solved numerically unless they are redefined as definite integrals (see “Definite Integral”), i.e., F(x) = ∫ f(x) dx, indefinite, whereas F(x) = ∫ ax f(t) dt, definite. Direct Formula Many integrals can be solved by transformation in the integrand to one of the forms given previously. Example Find ∫ x
3 x + 10 dx. Let v = 3x + 10 for which dv = 9x dx. 2

Thus

3

3

x
3x +10 dx = (3x + 10) 2

3

3

1/2

(x2 dx)

(3x + 10)

1 =
9

v

1/2

x dx

= (3 + 4x) 1/4

1 =
4

1/2

(9x2 dx)

dv

3/2

1 v =

+c 9 3⁄ 2

General The number of possible transformations one might use are unlimited. No specific overall rules can be given. Success in handling integration problems depends primarily upon experience and ingenuity. The following example illustrates the extent to which alternative approaches are possible.

e dx− 1

Example Find
. Let ex = y; then ex dx = dy or dx = 1/y dy. x dx (1/y) dy e −2 dy y−1
=
=
= ln
= ln
e −1 y−1 y −y e y x

x

A B C =
+
+
x+2 x−2 x−1 A(x − 2)(x − 1) + B(x + 2)(x − 1) + C(x + 2)(x − 2) =





(x + 2)(x − 2)(x − 1) x2(A + B + C) + x(−3A + B) + (2A − 2B − 4C) =




(x + 2)(x − 2)(x − 1) Equate coefficients and solve for A, B, and C. A+B+C=0 −3A + B = 0 2A − 2B − 4C = 1 A = 1⁄12, B = d, C = −s 1 1 1 1 =
+
−


x3 − x2 − 4x + 4 12(x + 2) 4(x − 2) 3(x − 1)

x +a Let x = a tan θ
 2

a2 −x2 Let x = a sin θ
  − 9 x
4 2 2

dx. Let x =
sin θ; then dx =
cos θ dθ. x 3 3 2

2

 (2 /3 ) −x θ 2 2/3
1 −sin dx = 3


cos θ dθ


x (2/3) sin θ  3 2

2

2

2

2

=3

cos θ
dθ sin θ

=3

cot θ dθ

Hence

dx dx dx dx

=
+
−
x − x − 4x + 4 12(x + 2) 4(x − 2) 3(x − 1)

2

3

2

d(uv) = u dv + v du

2

= −3 cot θ − 3θ + c by trigonometric transform − 9 x2 3
4 = −

− 3 sin−1
x + c in terms of x x 2

Algebraic Substitution Functions containing elements of the type (a + bx)1/n are best handled by the algebraic transformation yn = a + bx. Example Find

. Let 3 + 4x = y4; then 4dx = 4y3 dy and 1/4

2

Parts An extremely useful formula for integration is the relation

2

(3 +x dx4x)

x

2

Partial Fractions Rational functions are of the type f(x)/g(x) where f(x) and g(x) are polynomial expressions of degrees m and n respectively. If the degree of f is higher than g, perform the algebraic division—the remainder will then be at least one degree less than the denominator. Consider the following types: Type 1 Reducible denominator to linear unequal factors. For example, 1 1 =




x3 − x2 − 4x + 4 (x + 2)(x − 2)(x − 1)

x2 −a2 Let x = a sec θ


3

4

1 1 =
(3 + 4x)7/4 −
(3 + 4x)3/4 + c 28 4

[by Eq. (3-50)]

Trigonometric Substitution This technique is particularly well adapted to integrands in the form of radicals. For these the function is transformed into a trigonometric form. In the latter form they may be more easily recognizable relative to the identity formulas. These functions and their transformations are

Example Find

2

1 y7 3 y3 =

−

+c 4 7 4 3

2 =
(3x3 + 10)3/2 + c 27

2

y (y − 3) dy

2

1 =
9

3

y4 − 3

y3 dy 4

y

and or

u dv + v du u dv = uv − v du uv =

No general rule for breaking an integrand can be given. Experience alone limits the use of this technique. It is particularly useful for trigonometric and exponential functions. Example Find xex dx. Let u=x du = dx

and

dv = ex dx v = ex

DIFFERENTIAL AND INTEGRAL CALCULUS

xe dx = xe − e dx x

Therefore

x

Techniques for determining when integration is valid under these conditions are available in the references. However, the following simplified rules will, in general, serve as a guide for most practical applications. Rule 1 For the integral ∞ φ(x) dx
0 xn

x

= xex − ex + c

Example Find ex sin x dx. Let u = ex



dv = sin x dx v = −cos x

du = ex dx

e sin x dx = −e cos x + e cos x dx x

x

x

u = ex

Again

dv = cos x dx v = sin x

du = ex dx

e sin x dx = −e cos x + e sin x − e sin x dx + c x

x

x

x

c = (ex/2)(sin x − cos x) +
2

Series Expansion When an explicit function cannot be found, the integration can sometimes be carried out by a series expansion. Example Find e− x dx. Since 2

4

6

x x 2 e− x = 1 − x2 +
−
+ ⋅⋅⋅ 2! 3!

e

−x2

dx =

dx − x dx +
2!x dx −
3!x dx + ⋅⋅⋅ 4



if φ(x) is bounded, the integral will converge for n < 1 and diverge for n ≥ 1. Thus 1 1
dx x 0




will converge (exist) since a = n < 1. Properties The fundamental theorem of calculus states

f(x) dx = F(b) − F(a)

6

for all x

b

a

dF(x)/dx = f(x)

where

Definite Integral The concept and derivation of the definite integral are completely different from those for the indefinite integral. These are by definition different types of operations. However, the formal operation ∫ as it turns out treats the integrand in the same way for both. Consider the function f(x) = 10 − 10e−2x. Define x1 = a and xn = b, and suppose it is desirable to compute the area between the curve and the coordinate axis y = 0 and bounded by x1 = a, xn = b. Obviously, by a sufficiently large number of rectangles this area could be approximated as closely as desired by the formula



if φ(x) is bounded, the integral will converge for n > 1 and not converge for n ≤ 1. ∞ It is easily seen that ∫ 0 e−x dx converges by noting 1/x2 > 1/ex > 0 for large x. Rule 2 For the integral b φ(x)
n dx, a (a − x)

2

x3 x5 x7 = x −
+
−
+ ⋅⋅⋅ 3 5.2! 7.3!

n−1

Other properties of the definite integral are

c[ f(x) dx] = c f(x) dx b

[ f (x) + f (x)] dx = f (x) dx + f (x) dx

a

i

2

a

a

f(x) dx = − f(x) dx b

a

a

b

f(x) dx = b

c

a

f(x) dx +

a

f(x) dx b

c

f(x) dx = (b − a) f(ξ) for some ξ in (a, b)

xi − 1 ≤ ξi − 1 ≤ xi

b

i+1

− xi)

i=1

f(x) dx indefinite integral where dF/dx = f(x) F(a, b) = f(x) dx definite integral F(α) = f(x, α) dx

∂
∂b

f(x) dx = f(b)

∂
∂a

f(x) dx = −f(a)

dF(α)
= dα

F(x) =

a b

a

b

a

∂f(x, α)
dx if a and b are constant ∂α b

a

b

d

a

when F(x) =



d

c

c

b

a

b(x)

f(x, y) dy

a(x)

a

There are certain restrictions of the integration definition, “The function f(x) must be continuous in the finite interval (a, b) with at most a finite number of finite discontinuities,” which must be observed before integration formulas can be generally applied. Two of these restrictions give rise to so-called improper integrals and require special handling. These occur when ∞ 1. The limits of integration are not both finite, i.e., ∫ 0 e−x dx. 2. The function becomes infinite within the interval of integration, i.e., 1 1
dx x 0


b

dx f(x, α) dα = dα f(x, α) dx

b



b

1

a

where the points x1, x2, . . . , xn are equally spaced. For a rigorous definition of the definite integral the references should be consulted. Thus, the value of a definite integral depends on the limits a, b, and any selected variable coefficients in the function but not on the dummy variable of integration x. Symbolically

or

2

a

n

n→∞

b

1

The definite integral of f(x) is defined as

f(x) dx = lim  f(ξ )(x

a

b

i=1

+ ⋅⋅⋅ + f(ξn − 1)(b − xn − 1)

b

a

f(ξi)(xi + 1 − xi) = f(ξ1)(x2 − a) + f(ξ2)(x3 − x2)

b

3-29

the Leibniz rule gives dF db da
=
f [x, b(x)] −
f [x, a(x)] + dx dx dx





b(x)

a(x)

∂f
dy ∂x

π/2

Example Find

sin x dx.

0



π/2

0

since

π sin x dx = [−cos x]0π/2 = − cos
− cos 0 = 1 2



−d cos x/dx = sin x



(3-66)

3-30

MATHEMATICS dx
. Direct application of the formula would yield (x − 1) 2

Example Find the incorrect value

1 dx
= −
 (x − 1) x−1 2

2

0

c = π/2 φ(α) = −tan−1 α + π/2

2

= −2

0

It should be noted that f(x) = 1/(x − 1) becomes unbounded as x → 1 and by Rule 2 the integral diverges and hence is said not to exist. Methods of Integration All the methods of integration available for the indefinite integral can be used for definite integrals. In addition, several others are available for the latter integrals and are indicated below. Change of Variable This substitution is basically the same as previously indicated for indefinite integrals. However, for definite integrals, the limits of integration must also be changed: i.e., for x = φ(t), 2

f (x) dx = b

Integration It is sometimes useful to generate a double integral to solve a problem. By this approach, the fundamental theorem indicated by Eq. (3-66) can be used.

0

x 1

α

Consider

0

dα x b

α

dα x a

2

π/2

0

cos2 θ dθ = 16[aθ + d sin 2θ]0π/2 = 4π



∞

e−αx sin x
dx (α > 0) x Since this is a continuous function of α, it may be differentiated under the integral sign ∞ dφ
= − e−αx sin x dx 0 dα φ(α) =

b

dα b+1
= ln
α+1 a+1





= −1/(1 + α ) φ(α) = −tan−1 α + c 2

1

b

α



1

dα =

a

0

b+1 xb − xα
dx = ln
ln x a+1



xb − xα
dx ln x



Complex Variable Certain definite integrals can be evaluated by the technique of complex variable integration. This is described in the references for “Complex Variables.” Numerical Because of the property of definite integrals another method for obtaining their solution is available which cannot be applied to indefinite integrals. This involves a numerical approximation based on the previously outlined summation definition:

0



dx x 0



Therefore

Differentiation Here the application of the general rules for differentiating under the integral sign may be useful. Example Find



a

dx =

0

1

x = 4 sin θ (x = 0, θ = 0) dx = 4 cos θ dθ (x = 4, θ = π/2) 0

1

α


16−x dx. Let 2

0

dx =

0

But also b


16−x dx = 16

1

a

t0

4

Then

1 dx =
(α > −1) α+1

Then multiplying both sides by dα and integrating between a and b,

4

0

xb − xα
dx ln x

1

f [φ(t)]φ′(t) dt

t = t0 when x = a t = t1 when x = b

Example Find



Example Find

t1

a

where

and since φ(α) → 0 as α → ∞,

2

0

n−1

lim  f(ξi)(xi + 1 − xi) =

n→∞

where x1 = a

and

1

f(x) dx b

a

xn = b

Examples of this procedure are given in the subsection “Numerical Analysis and Approximate Methods.”

INFINITE SERIES REFERENCES: 53, 126, 127, 163. For asymptotic series and asymptotic methods, see Refs. 51, 127.

DEFINITIONS

while valid for any finite value of r and n now takes on a different interpretation. In this sense it is necessary to consider the limit of Sn as n increases indefinitely: S = lim Sn

A succession of numbers or terms that are formed according to some definite rule is called a sequence. The indicated sum of the terms of a sequence is called a series. A series of the form a0 + a1(x − c) + a2(x − c)2 + ⋅⋅⋅ + an(x − c)n + ⋅⋅⋅ is called a power series. Consider the sum of a finite number of terms in the geometric series (a special case of a power series). Sn = a + ar + ar 2 + ar 3 + ⋅⋅⋅ + ar n − 1

(3-67)

For any number of terms n, the sum equals 1 − rn Sn = a
1−r In this form, the geometric series is assumed finite. In the form of Eq. (3-67), it can further be defined that the terms in the series be nonending and therefore an infinite series. S = a + ar + ar 2 + ⋅⋅⋅ + ar n + ⋅⋅⋅ However, the defined sum of the terms [Eq. (3-67)] 1 − rn Sn = a
r≠1 1−r

(3-68)

n→∞

1 − rn = a lim
n→∞ 1 − r For this, it is stated the infinite series converges if the limit of Sn approaches a fixed finite value as n approaches infinity. Otherwise, the series is divergent. On this basis an analysis of 1 − rn S = a lim
n→∞ 1 − r shows that if r is less than 1 but greater than −1, the infinite series is convergent. For values outside of the range −1 < r < 1, the series is divergent because the sum is not defined. The range −1 < r < 1 is called the region of convergence. (We assume a ≠ 0.) Consider the divergence of Eq. (3-68) when r = −1 and +1. For the former case r = −1, S = a + a(−1) + a(−1)2 + a(−1)3 + ⋅⋅⋅ + a(−1)n + ⋅⋅⋅ = a − a + a − a + a − ⋅⋅⋅

INFINITE SERIES and for which 1 − rn S = a lim
n→∞ 1 − r 1 − (−1)n = a lim
undefined limit (if a ≠ 0) n→∞ 1+1 Since the limit sum does not exist, the series is divergent. This is defined as a bounded or oscillating divergent series. Similarly for the value r = +1, S = a + a(1) + a(1)2 + a(1)3 + ⋅⋅⋅ + a(1)n + ⋅⋅⋅ S = a + a + a + a + ⋅⋅⋅ + a + ⋅⋅⋅

The series is also divergent but defined as an unbounded divergent series. There are also two types of convergent series. Consider the new series 1 1 1 1 S = 1 −
+
−
+ ⋅⋅⋅ + (−1)n + 1
+ ⋅⋅⋅ (3-69) 2 3 4 n It can be shown that the series (3-69) does converge to the value S = log 2. However, if each term is replaced by its absolute value, the series becomes unbounded and therefore divergent (unbounded divergent): 1 1 1 1 S = 1 +
+
+
+
+ ⋅⋅⋅ (3-70) 2 3 4 5 In this case the series (3-69) is defined as a conditionally convergent series. If the replacement series of absolute values also converges, the series is defined to converge absolutely. Series (3-69) is further defined as an alternating series, while series (3-70) is referred to as a positive series. OPERATIONS WITH INFINITE SERIES 1. The convergence or divergence of an infinite series is unaffected by the removal of a finite number of finite terms. This is a trivial theorem but useful to remember, especially when using the comparison test to be described in the subsection “Tests for Convergence and Divergence.” 2. If a series is conditionally convergent, its sums can be made to have any arbitrary value by a suitable rearrangement of the series; it can in fact be made divergent or oscillatory (Riemann’s theorem). This seemingly paradoxical theorem can be illustrated by the following example. 1 1 1 1 1 2 3 4 5 6 The series is rearranged so that each positive term is followed by two negative terms: 1 1 1 1 1 1 1 1 t = 1 −
−
+
−
−
+
−
−
+ ⋅⋅⋅ 2 4 3 6 8 5 10 12 Define t3n for the first 3n terms in the series

Example S = 1 −
+
−
+
−
+ ⋅⋅⋅













1 1 1 1 1 1 1 1 t3n = 1 −
−
+
−
−
+ ⋅⋅⋅ +
−
−
2 4 3 6 8 2n − 1 4n − 2 4n 1 1 1 1 1 1 =
−
+
−
+ ⋅⋅⋅ +
−
2 4 6 8 4n − 2 4n



1 1 1 1 1 1 =
1 −
+
−
+ ⋅⋅⋅ +
−
2 2 3 4 2n − 1 2n

3. A series of positive terms, if convergent, has a sum independent of the order of its terms; but if divergent, it remains divergent however its terms are rearranged. 4. An oscillatory series can always be made to converge by grouping the terms in brackets. Example Consider the series 1 2 3 4 5 1 −
+
−
+
−
+ ⋅⋅⋅ 2 3 4 5 6 which oscillates between the values 0.306 and 1.306. However, the series 1 2 3 4 5 1 1 1 1 1 −
+
−
+
−
+ ⋅⋅⋅ =
−
−
−
− ⋅⋅⋅  0.306 ⋅⋅⋅ 2 3 4 5 6 2 12 30 56



(a ≠ 0)

3-31

 

 



and



 

 



1 1 2 3 4 5 6 1 1 1 −
−
−
−
−
−
+ ⋅⋅⋅ = 1 +
+
+
+ ⋅⋅⋅ = 1.306 ⋅⋅⋅ 2 3 4 5 6 7 6 20 42

5. A power series can be inverted, provided the first-degree term is not zero. Given y = b1 x + b2 x2 + b3 x3 + b4 x4 + b5 x5 + b6 x6 + b7 x7 + ⋅⋅⋅ then

x = B1 y + B2 y2 + B3 y3 + B4 y4 + B5 y5 + B6 y6 + B7 y7 + ⋅⋅⋅

where B1 = 1/b1 B2 = −b2 /b13 B3 = (1/b15 ) (2b22 − b1b3 ) B4 = (1/b17 )(5b1b2 b3 − b12 b4 − 5b23 ) Additional coefficients are available in the references. 6. Two series may be added or subtracted term by term provided each is a convergent series. The joint sum is equal to the sum (or difference) of the individuals. 7. The sum of two divergent series can be convergent. Similarly, the sum of a convergent series and a divergent series must be divergent. Example Given 1+n

∞

=
+
+
+
+ ⋅⋅⋅  
n  1 4 9 16

n=1

2

3

1−n

∞

5

4

(a divergent series)

2

= −
−
−
+ ⋅⋅⋅  
n  4 9 16

n=1

1

2

3

(a divergent series)

2

1+n

1−n

1+n+1−n

+  
 =  

  
n  n n

However,

2

2

2

1 = 2 
2 n

(convergent)

8. A power series may be integrated term by term to represent the integral of the function within an interval of the region of convergence. If f(x) = a0 + a1x + a2x2 + ⋅⋅⋅, then



x2

x1

f(x) dx =



x2

x1

a0 dx +



x2

x1

a1x dx +



x2

a2 x2 dx + ⋅⋅⋅

x1

9. A power series may be differentiated term by term and represents the function df(x)/dx within the same region of convergence as f(x). TESTS FOR CONVERGENCE AND DIVERGENCE



1 =
S2n 2 where S2n is the sum of the first 2n terms of the original series. Thus 1 lim t3n = lim
S2n n→∞ n→∞ 2 1 t=
S 2 and since lim t3n + 2 = lim t3n + 1 = lim t3n, it follows the sum of the series t is (a) S. Hence a rearrangement of the terms of an alternating series alters the sum of the series.

In general, the problem of determining whether a given series will converge or not can require a great deal of ingenuity and resourcefulness. There is no all-inclusive test which can be applied to all series. As the only alternative, it is necessary to apply one or more of the developed theorems in an attempt to ascertain the convergence or divergence of the series under study. The following defined tests are given in relative order of effectiveness. For examples, see references on advanced calculus. 1. Comparison Test. A series will converge if the absolute value of each term (with or without a finite number of terms) is less than the corresponding term of a known convergent series. Similarly, a positive series is divergent if it is termwise larger than a known divergent series of positive terms.

3-32

MATHEMATICS

2. nth-Term Test. A series is divergent if the nth term of the series does not approach zero as n becomes increasingly large. 3. Ratio Test. If the absolute ratio of the (n + 1) term divided by the nth term as n becomes unbounded approaches a. A number less than 1, the series is absolutely convergent b. A number greater than 1, the series is divergent c. A number equal to 1, the test is inconclusive 4. Alternating-Series Leibniz Test. If the terms of a series are alternately positive and negative and never increase in value, the absolute series will converge, provided that the terms tend to zero as a limit. 5. Cauchy’s Root Test. If the nth root of the absolute value of the nth term, as n becomes unbounded, approaches a. A number less than 1, the series is absolutely convergent b. A number greater than 1, the series is divergent c. A number equal to 1, the test is inconclusive 6. Maclaurin’s Integral Test. Suppose  an is a series of positive terms and f is a continuous decreasing function such that f(x) ≥ 0 for 1∞ ≤ x < ∞ and f(n) = an. Then the series and the improper integral ∫ 1 f(x) dx either both converge or both diverge. SERIES SUMMATION AND IDENTITIES Sums for the First n Numbers to Integer Powers n



j=1 n



j=1 n



j=1 n



j=1

n(n + 1) j =
= 1 + 2 + 3 + 4 + ⋅⋅⋅ + n 2 n(n + 1)(2n + 1) j 2 =

= 12 + 22 + 32 + 42 + ⋅⋅⋅ + n2 6

Taylor’s Series x2 x3 f(x + h) = f(h) + xf ′(h) +
f ″(h) +
f ′′′(h) + ⋅⋅⋅ 2! 3! f ″(x0) f ′′′(x0) or f(x) = f(x0) + f ′(x0) (x − x0) +
(x − x0)2 +
(x − x0)3 + ⋅⋅⋅ 2! 3! Example Find a series expansion for f(x) = ln (1 + x) about x0 = 0. f′(x) = (1 + x)−1, thus

f(0) = 0,

f″(x) = −(1 + x)−2,

f′(0) = 1,

f ″(0) = −1,

which converges for −1 < x ≤ 1.

Maclaurin’s Series x2 x3 f(x) = f(0) + xf ′(0) +
f ″(0) +
f ′′′(0) + ⋅⋅⋅ 2! 3! This is simply a special case of Taylor’s series when h is set to zero. Exponential Series x2 x3 xn ex = 1 + x +
+
+ ⋅⋅⋅ +
+ ⋅⋅⋅ − ∞ < x < ∞ 2! 3! n! Logarithmic Series

n2(n + 1)2 j =

= 13 + 23 + 33 + ⋅⋅⋅ + n3 4

x−1 1 x−1 ln x =
+

x 2 x

n(n + 1)(2n + 1)(3n2 + 3n − 1) j 4 =



= 14 + 24 + 34 + ⋅⋅⋅ + n4 30

ln x = 2

n

 [a + (k − 1)d] = a + (a + d ) + (a + 2d )

+ (a + 3d ) + ⋅⋅⋅ + [a + (n − 1)]d

k=1



 ar

j−1

3

1

x−1 +

+ ⋅⋅⋅


x + 1 3 x + 1 3

1

(x > a)

(x > 0)

−∞ < x < ∞ −∞ < x < ∞

x3 1 3 x5 1 3 5 x7 sin−1 x = x +
+
⋅
⋅
+
⋅
⋅
⋅
+ ⋅⋅⋅ 6 2 4 5 2 4 6 7

= a + ar + ar 2 + ar 3 + ⋅⋅⋅ + ar n − 1 1 − rn =a
1−r

2

x2 x4 x 6 cos x = 1 −
+
−
+ ⋅⋅⋅ 2! 4! 6!

Geometric Progression j=1

x−1

x−1 + ⋅⋅⋅  +
3 
x 

Trigonometric Series* x3 x5 x7 sin x = x −
+
−
+ ⋅⋅⋅ 3! 5! 7!

1 = na +
n(n − 1)d 2

n

f ′′′(1) = 2, etc.

x2 x3 x4 xn ln (x + 1) = x −
+
−
+ ⋅⋅⋅ + (−1)n + 1
+ ⋅⋅⋅ 2 3 4 n

3

Arithmetic Progression

f′′′(x) = 2(1 + x)−3, etc.

r≠1

1 1 1 tan−1 x = x −
x3 +
x5 −
x7 + ⋅⋅⋅ 3 5 7

∂f f(x, y) = f(x0, y0) +
∂x

n

1 1 1 1 1 1 1 +
+
+
+ ⋅⋅⋅ +

=
a +
a + d a + 2d a + 3d a + 4d a + nd k = 0 a + kd

1 ∂2f +

2 2! ∂x



The reciprocals of the terms of the arithmetic-progression series are called harmonic progression. No general summation formulas are available for this series. n(n − 1) (x + y)n = xn + nx n − 1y +
x n − 2 y2 2! n(n − 1)(n − 2) n − 3 3 n! +

x y + ⋅⋅⋅ +
x n − ryr + ⋅⋅⋅ + yn 3! (n − r)!r! n(n − 1) n(n − 1)(n − 2) (1  x)n = 1  nx +
x2 

x3 + ⋅⋅⋅ (x2 < 1) 2! 3!

(x2 < 1)

Taylor Series The Taylor series for a function of two variables, expanded about the point (x0, y0), is

Harmonic Progression

Binomial Series

(x2 < 1)



x 0, y0

∂f (x − x0) +
∂y

∂2f (x − x0)2 + 2
x 0, y0 ∂x∂y





(y − y0) x 0, y0

(x − x0)(y − y0) x 0 , y0

∂2f +
2 ∂y





(y − y0)2 + ⋅⋅⋅ x 0, y0

Partial Sums of Infinite Series, and How They Grow Calculus textbooks devote much space to tests for convergence and divergence of series that are of little practical value, since a convergent

* tan x series has awkward coefficients and should be computed as sin x (sign)

. 2  −sin x
1





COMPLEX VARIABLES series either converges rapidly, in which case almost any test (among those presented in the preceding subsections) will do; or it converges slowly, in which case it is not going to be of much use unless there is

3-33

some way to get at its sum without adding up an unreasonable number of terms. To find out, as accurately as possible, how fast a convergent series converges and how fast a divergent series diverges, see Ref. 34.

COMPLEX VARIABLES REFERENCES: General. 73, 163, 172, 179. Applied and computational complex analysis. 141, 146, 179.

Numbers of the form z = x + iy, where x and y are real, i = −1, are called complex numbers. The numbers z = x + iy are representable in the plane as shown in Fig. 3-46. The following definitions and terminology are used: 1. Distance OP = r = modulus of z written |z|. |z| =
 x2 + y2. 2. x is the real part of z. 3. y is the imaginary part of z. 4. The angle θ, 0 ≤ θ < 2π, measured counterclockwise from the positive x axis to OP is the argument of z. θ = arctan y/x = arcsin y/r = arccos x/r if x ≠ 0, θ = π/2 if x = 0 and y > 0. 5. The numbers r, θ are the polar coordinates of z. 6. z = x − iy is the complex conjugate of z. 2

ALGEBRA Let z1 = x1 + iy1, z2 = x2 + iy2. Equality z1 = z2 if and only if x1 = x2 and y1 = y2. Addition z1 + z2 = (x1 + x2) + i(y1 + y2). Subtraction z1 − z2 = (x1 − x2) + i(y1 − y2). Multiplication z1 ⋅ z2 = (x1x2 − y1y2) + i(x1y2 + x2y1). x2y1 − x1y2 x1x2 + y1y2 Division z1 /z2 =

+ i

, z2 ≠ 0. x 22 + y22 x 22 + y22 SPECIAL OPERATIONS zz = x2 + y2 = |z|2; z1z2 = z1  z2; z1 = z1; z1z2 = z1z2; |z1 ⋅ z2| = |z1| ⋅ |z2|; arg (z1 ⋅ z2) = arg z1 + arg z2; arg (z1 /z2) = arg z1 − arg z2; i4n = 1 for n any integer; i2n = −1 where n is any odd integer; z + z = 2x; z − z = 2iy. Every complex quantity can be expressed in the form x + iy. TRIGONOMETRIC REPRESENTATION By referring to Fig. 3-46, there results x = r cos θ, y = r sin θ so that z = x + iy = r (cos θ + i sin θ), which is called the polar form of the complex number. cos θ + i sin θ = e iθ. Hence z = x + iy = re iθ. z = x − iy = re−iθ. Two important results from this are cos θ = (eiθ + e−iθ)/2 and sin θ = (eiθ − e−iθ)/2i. Let z1 = r1e iθ1, z2 = r2e iθ2. This form is convenient for multiplication for z1z2 = r1 r2 e i(θ1 + θ2) and for division for z1 /z2 = (r1 /r2)ei(θ1 − θ2), z2 ≠ 0. POWERS AND ROOTS If n is a positive integer, z = (re ) = r e If n is a positive integer, n

FIG. 3-46

Complex plane.

iθ n

n inθ

= r (cos nθ + i sin nθ). n

θ + 2kπ θ + 2kπ z1/ n = r 1/ nei[(θ + 2kπ)/n] = r 1/n cos
+ i sin
n n









and selecting values of k = 0, 1, 2, 3, . . . , n − 1 give the n distinct values of z1/n. The n roots of a complex quantity are uniformly spaced around a circle, with radius r 1/n, in the complex plane in a symmetric fashion. Example Find the three cube roots of −8. Here r = 8, θ = π. The roots are z0 = 2(cos π/3 + i sin π/3) = 1 + i
3, z1 = 2(cos π + i sin π) = −2, z2 = 2(cos 5π/3 + i sin 5π/3) = 1 − i
3. ELEMENTARY COMPLEX FUNCTIONS Polynomials A polynomial in z, anzn + an − 1zn − 1 + ⋅⋅⋅ + a0, where n is a positive integer, is simply a sum of complex numbers times integral powers of z which have already been defined. Every polynomial of degree n has precisely n complex roots provided each multiple root of multiplicity m is counted m times. Exponential Functions The exponential function ez is defined by the equation ez = ex + iy = ex ⋅ eiy = ez(cos y + i sin y). Properties: e0 = 1; ez1 ⋅ ez2 = ez1 + z2; ez1/ez2 = ez1 − z2; ez + 2kπi = ez. Trigonometric Functions sin z = (eiz − e−iz)/2i; cos z = (eiz + e−iz)/2; tan z = sin z/cos z; cot z = cos z/sin z; sec z = 1/cos z; csc z = 1/sin z. Fundamental identities for these functions are the same as their real counterparts. Thus cos2 z + sin2 z = 1, cos (z1  z2) = cos z1 cos z2  sin z1 sin z2, sin (z1  z2) = sin z1 cos z2  cos z1 sin z2. The sine and cosine of z are periodic functions of period 2π; thus sin (z + 2π) = sin z. For computation purposes sin z = sin (x + iy) = sin x cosh y + i cos x sinh y, where sin x, cosh y, etc., are the real trigonometric and hyperbolic functions. Similarly, cos z = cos x cosh y − i sin x sinh y. If x = 0 in the results given, cos iy = cosh y, sin iy = i sinh y. Example Find all solutions of sin z = 3. From previous data sin z = sin x cosh y + i cos x sinh y = 3. Equating real and imaginary parts sin x cosh y = 3, cos x sinh y = 0. The second equation can hold for y = 0 or for x = π/2, 3π/2, . . . . If y = 0, cosh 0 = 1 and sin x = 3 is impossible for real x. Therefore, x = π/2, 3π/2, . . . (2n + 1)π/2, n = 0, 1, 2, . . . . However, sin 3π/2 = −1 and cosh y ≥ 1. Hence x = π/2, 5π/2, . . . . The solution is z = [(4n + 1)π]/2 + i cosh−13, n = 0, 1, 2, 3, . . . . Example Find all solutions of ez = −i. ez = ex(cos y + i sin y) = −i. Equating real and imaginary parts gives ex cos y = 0, ex sin y = −1. From the first y = π/2, 3π/2, . . . . But ex > 0. Therefore, y = 3π/2, 7π/2, −π/2, . . . . Then x = 0. The solution is z = i[(4n + 3)π]/2. Two important facets of these functions should be recognized. First, the sin z is unbounded; and, second, ez takes all complex values except 0. Hyperbolic Functions sinh z = (ez − e−z)/2; cosh z = (ez + e−z)/2; tanh z = sinh z/cosh z; coth z = cosh z/sinh z; csch z = 1/sinh z; sech z = 1/cosh z. Identities are: cosh2 z − sinh2 z = 1; sinh (z1 + z2) = sinh z1 cosh z2 + cosh z1 sinh z2; cosh (z1 + z2) = cosh z1 cosh z2 + sinh z1 sinh z2; cosh z + sinh z = ez; cosh z − sinh z = e−z. The hyperbolic sine and hyperbolic cosine are periodic functions with the imaginary period 2πi. That is, sinh (z + 2πi) = sinh z. Logarithms The logarithm of z, log z = log |z| + i(θ + 2nπ), where log |z| is taken to the base e and θ is the principal argument of z, that is, the particular argument lying in the interval 0 ≤ θ < 2π. The logarithm of z is infinitely many valued. If n = 0, the resulting logarithm is called the principal value. The familiar laws log z1 z2 = log z1 + log z2, log z1 /z2 = log z1 − log z2, log zn = n log z hold for the principal value.

3-34

MATHEMATICS

 π4



Example log (1 + i) = log
2 + i
+ 2nπ . General powers of z are defined by zα = eα log z. Since log z is infinitely many valued, so too is zα unless α is a rational number. DeMoivre’s formula can be derived from properties of ez. zn = rn (cos θ + i sin θ)n = rn (cos nθ + i sin nθ) (cos θ + i sin θ)n = cos nθ + i sin nθ

Thus

Example ii = eilog i = ei[log|i| + i(π/2 + 2nπ)] = e−(π/2 + 2nπ). Thus i i is real with principal value (n = 0) = e−π/2. Example (
2)1 + i = e(1 + i) log
2 = elog
2. ei log
2 =
2. (cos log
2 + i sin log
2) =
2[cos (0.3466) + i sin (0.3466)]. Inverse Trigonmetric Functions cos−1 z = −i log (z 
 z2 −1); i + z i −1 2 −1 sin z = −i log (iz 
1 −z); tan z =
log
. These functions 2 i−z are infinitely many valued. Inverse Hyperbolic Functions cosh−1 z = log (z 
 z2−1); 1+z 1 z2 +1); tanh−1 z =
log
. sinh−1 z = log (z 
 2 1−z

  



COMPLEX FUNCTIONS (ANALYTIC) In the real-number system a greater than b(a > b) and b less than c(b < c) define an order relation. These relations have no meaning for complex numbers. The absolute value is used for ordering. Some important relations follow: |z| ≥ x; |z| ≥ y; |z1  z2| ≤ |z1| + |z2|; |z1 − z2| ≥ ||z1| − |z2||; |z| ≥ (|x| + |y|)/
2. Parts of the complex plane, commonly called regions or domains, are described by using inequalities.  Example |z − 3| ≤ 5. This is equivalent to
(x −3 )2 +y2 ≤ 5, which is the set of all points within and on the circle, centered at x = 3, y = 0 of radius 5.

Example |z − 1| ≤ x represents the set of all points inside and on the parabola 2x = y2 + 1 or, equivalently, 2x ≥ y2 + 1. Functions of a Complex Variable If z = x + iy, w = u + iv and if for each value of z in some region of the complex plane one or more values of w are defined, then w is said to be a function of z, w = f(z). Some of these functions have already been discussed, e.g., sin z, log z. All functions are reducible to the form w = u(x, y) + iv(x, y), where u, v are real functions of the real variables x and y.

∂v/∂x = −∂u/∂y. The last two equations are called the CauchyRiemann equations. The derivative dw ∂u ∂v ∂v ∂u
=
+i
=
−i
dz ∂x ∂x ∂y ∂y If f(z) possesses a derivative at zo and at every point in some neighborhood of z0, then f(z) is said to be analytic at z0. If the Cauchy-Riemann equations are satisfied and ∂u ∂u ∂v ∂v u, v,
,
,
,
∂x ∂y ∂x ∂y are continuous in a region of the complex plane, then f(z) is analytic in that region. Example w = zz = x2 + y2. Here u = x2 + y2, v = 0. ∂u/∂x = 2x, ∂u/∂y = 2y, ∂v/∂x = ∂v/∂y = 0. These are continuous everywhere, but the Cauchy-Riemann equations hold only at the origin. Therefore, w is nowhere analytic, but it is differentiable at z = 0 only. Example w = ez = ex cos y + iex sin y. u = ex cos y, v = ex sin y. ∂u/∂x = ex cos y, ∂u/∂y = −ex sin y, ∂v/∂x = ex sin y, ∂v/∂y = ex cos y. The continuity and CauchyRiemann requirements are satisfied for all finite z. Hence ez is analytic (except at ∞) and dw/∂z = ∂u/∂x + i(∂v/∂x) = ez. 1 z

x − iy x +y

x x +y

y x +y

Example w =
=
=
−i
2 2 2 2 2 2 It is easy to see that dw/dz exists except at z = 0. Thus 1/z is analytic except at z = 0.

Singular Points If f(z) is analytic in a region except at certain points, those points are called singular points. Example 1/z has a singular point at zero. Example tan z has singular points at z = (2n + 1)(π/2), n = 0, 1, 2, . . . . The derivatives of the common functions, given earlier, are the same as their real counterparts. Example (d/dz)(log z) = 1/z, (d/dz)(sin z) = cos z. Harmonic Functions Both the real and the imaginary parts of any analytic function f = u + iv satisfy Laplace’s equation ∂2φ/∂x2 + ∂2φ/∂y2 = 0. A function which possesses continuous second partial derivatives and satisfies Laplace’s equation is called a harmonic function.

Example z3 = (x + iy)3 = x3 + 3x2(iy) + 3x(iy)2 + (iy)3 = (x3 − 3xy2) +

Example ez = ex cos y + iex sin y. u = ex cos y, ∂u/∂x = ex cos y, ∂2u/∂x2 = ex cos y, ∂u/∂y = −ex sin y, ∂2u/∂y2 = −ex cos y. Clearly ∂2u/∂x2 + ∂2u/∂y2 = 0. Similarly, v = ex sin y is also harmonic.

Example cos z = cos x cosh y − i sin x sinh y.

If w = u + iv is analytic, the curves u(x, y) = c and v(x, y) = k intersect at right angles, if wi(z) ≠ 0.

i(3x2y − y3).

Differentiation The derivative of w = f(z) is dw f(z + ∆z) − f(z) lim


= ∆z→0 dz ∆z and for the derivative to exist the limit must be the same no matter how ∆z approaches zero. If w1, w2 are differentiable functions of z, the following rules apply: dw2 d(w1w2) dw1 dw2 d(w1  w2) dw1
= w2
+ w1


=

dz dz dz dz dz dz d(w1/w2) w2(dw1/dz) − w1(dw2/dz)
=


dz w22 dw1n dw1
= nw1n − 1
dz dz For w = f(z) to be differentiable, it is necessary that ∂u/∂x = ∂v/∂y and and

Example z3 = (x3 − 3xy2) + i(3x2y − y3). Set u = x3 − 3xy2 = c, v = 3x2y − y3 = k. By implicit differentiation there results, respectively, dy/dx = (x2 − y2)/2xy, dy/dx = 2xy/(y2 − x2), which are clearly negative reciprocals, the condition for perpendicularity. Integration In much of the work with complex variables a simple extension of integration called line or curvilinear integration is of fundamental importance. Since any complex line integral can be expressed in terms of real line integrals, we define only real line integrals. Let F(x,y) be a real, continuous function of x and y and c be any continuous curve of finite length joining the points A and B (Fig. 3-47). F(x,y) is not related to the curve c. Divide c up into n segments, ∆si, whose projection on the x axis is ∆xi and on the y axis is ∆yi. Let (εi, ηi) be the coordinates of an arbitrary point on ∆si. The limits of the sums

 F(ε , η ) ∆s = F(x, y) ds n

lim

∆si→0

i

i=1

i

i

c

DIFFERENTIAL EQUATIONS

Conformal transformation.

FIG. 3-48

Line integral.

FIG. 3-47

grals are the same as those for ordinary integrals. That is, ∫c [ f(z)  g(z)] dz = ∫c f(z) dz  ∫c g(z) dz; ∫c kf(z) dz = k ∫c f(z) dz for any constant k, etc.

 F(ε , η ) ∆x = F(x, y) dx n

lim

∆si→0

i

i

i

c

i=1

 F(ε , η ) ∆ y = F(x, y) dy

lim

∆si→0

i

i

c

2

are known as line integrals. Much of the initial strangeness of these integrals will vanish if it be observed that the ordinary definite integral b ∫a f(x) dx is just a line integral in which the curve c is a line segment on the x axis and F(x, y) is a function of x alone. The evaluation of line integrals can be reduced to evaluation of ordinary integrals. Example ∫c y(1 + x) dy, where c: y = 1 − x21 from (−1, 0) to (1, 0). Clearly y = 1 − x2, dy = −2x dx. Thus ∫c y(1 + x) dy = −2 ∫ −1 (1 − x2)(1 + x)x dx = −8⁄15. Example ∫c x2y ds, c is the square whose vertices are (0, 0), (1, 0), (1, 1),  x2 + dy2. When dx = 0, ds = dy. From (0, 0) to (1, 0), y = 0, dy = (0, 1). ds =
d 0. Similar arguments for the other sides give

x y ds = 0.x dx + y dy + x dx + 0.y dy = a − s = ⁄ 1

2

1

0

2

0

0

2

0

16

1

2

c

i

i=1

(x + iy) dz along c: y = x, 0 to 1 + i. This becomes (x + iy) dz = (x dx − y dy) + i (y dx + x dy) = x dx − x dx + i x dx + i x dx = − ⁄

Example

n

c

3-35

1

Let f(z) be any function of z, analytic or not, and c any curve as above. The complex integral is calculated as ∫c f(z) dz = ∫c (u dx − v dy) + i ∫c (v dx + u dy), where f(z) = u(x, y) + iv(x, y). Properties of line inte-

2

c

c

1

2

c

1

1

1

2

0

2

0

0

16

+ 5i/6

0

Conformal Mapping Every function of a complex variable w = f(z) = u(x, y) + iv(x, y) transforms the x, y plane into the u, v plane in some manner. A conformal transformation is one in which angles between curves are preserved in magnitude and sense. Every analytic function, except at those points where f ′(z) = 0, is a conformal transformation. See Fig. 3-48. Example w = z2. u + iv = (x2 − y2) + 2ixy or u = x2 − y2, v = 2xy. These are the transformation equations between the (x, y) and (u, v) planes. Lines parallel to the x axis, y = c1 map into curves in the u, v plane with parametric equations u = x2 − c12, v = 2c1x. Eliminating x, u = (v2/4c12) − c12, which represents a family of parabolas with the origin of the w plane as focus, the line v = 0 as axis and opening to the right. Similar arguments apply to x = c2. The principles of complex variables are useful in the solution of a variety of applied problems. See the references for additional information.

DIFFERENTIAL EQUATIONS REFERENCES: Ordinary Differential Equations: Elementary level, 41, 44, 62, 81, 204, 236, 263. Intermediate level, 30, 43, 144. Theory and Advanced topics, 252. Applications, 9, 263. Partial Differential Equations: Elementary level and solution methods, 9, 41, 61, 72, 144, 156, 229. Theory and advanced level, 79, 220, 240. See also “Numerical Analysis and Approximate Methods” and “General References: References for General and Specific Topics—Advanced Engineering Mathematics” for additional references on topics in ordinary and partial differential equations.

The natural laws in any scientific or technological field are not regarded as precise and definitive until they have been expressed in mathematical form. Such a form, often an equation, is a relation between the quantity of interest, say, product yield, and independent variables such as time and temperature upon which yield depends. When it happens that this equation involves, besides the function itself, one or more of its derivatives it is called a differential equation. Example The homogeneous bimolecular reaction A + B → C is characterized by the differential equation dx/dt = k(a − x)(b − x), where a = initial concentration of A, b = initial concentration of B, and x = x(t) = concentration of C as a function of time t. k

Example The differential equation of heat conduction in a moving fluid with velocity components vx, vy is ∂u ∂u ∂u K ∂2u ∂2u
+ vx
+ vy
=

2 +
2 ∂t ∂x ∂y ρcp ∂x ∂y





where u = u(x, y, t) = temperature, K = thermal conductivity, ρ = density, and cp = specific heat at constant pressure.

ORDINARY DIFFERENTIAL EQUATIONS When the function involved in the equation depends upon only one variable, its derivatives are ordinary derivatives and the differential equation is called an ordinary differential equation. When the function depends upon several independent variables, then the equation is called a partial differential equation. The theories of ordinary and partial differential equations are quite different. In almost every respect the latter is more difficult. Whichever the type, a differential equation is said to be of nth order if it involves derivatives of order n but no higher. The equation in the first example is of first order and that in the second example of second order. The degree of a differential equation is the power to which the derivative of the highest order is raised after the equation has been cleared of fractions and radicals in the dependent variable and its derivatives. A relation between the variables, involving no derivatives, is called a solution of the differential equation if this relation, when substituted in the equation, satisfies the equation. A solution of an ordinary differential equation which includes the maximum possible number of “arbitrary” constants is called the general solution. The maximum number of “arbitrary” constants is exactly equal to the order of the dif-

3-36

MATHEMATICS

ferential equation. If any set of specific values of the constants is chosen, the result is called a particular solution.

are distinct real roots, r1 and r2, say, the solution is y = Aer1x + Ber2x, where A and B are arbitrary constants.

Example The general solution of (d2x/dt2) + k2x = 0 is x = A cos kt + B sin kt, where A, B are arbitrary constants. A particular solution is x = a cos kt + 3 sin kt.

Example y″ + 4y′ + 3 = 0. The characteristic equation is m2 + 4m + 3 = 0. The roots are −3 and −1, and the general solution is y = Ae−3x + Be−x.

In the case of some equations still other solutions exist called singular solutions. A singular solution is any solution of the differential equation which is not included in the general solution. Example y = x(dy/dx) − d(dy/dx) has the general solution y = cx − dc , where c is an arbitrary constant; y = x2 is a singular solution, as is easily verified. 2

2

ORDINARY DIFFERENTIAL EQUATIONS OF THE FIRST ORDER Equations with Separable Variables Every differential equation of the first order and of the first degree can be written in the form M(x, y) dx + N(x, y) dy = 0. If the equation can be transformed so that M does not involve y and N does not involve x, then the variables are said to be separated. The solution can then be obtained by quadrature, which means that y = ∫ f(x) dx + c, which may or may not be expressible in simpler form. Example Two liquids A and B are boiling together in a vessel. Experimentally it is found that the ratio of the rates at which A and B are evaporating at any time is proportional to the ratio of the amount of A (say, x) to the amount of B (say, y) still in the liquid state. This physical law is expressible as (dy/dt)/(dx/dt) = ky/x or dy/dx = ky/x, where k is a proportionality constant. This equation may be written dy/y = k(dx/x), in which the variables are separated. The solution is ln y = k ln x + ln c or y = cxk. Exact Equations The equation M(x, y) dx + N(x, y) dy = 0 is exact if and only if ∂M/∂y = ∂N/∂x. In this case there exists a function w = f(x, y) such that ∂f /∂x = M, ∂f /∂y = N, and f(x, y) = C is the required solution. f(x, y) is found as follows: treat y as though it were constant and evaluate ∫ M(x, y) dx. Then treat x as though it were constant and evaluate ∫ N(x, y) dy. The sum of all unlike terms in these two integrals (including no repetitions) is f(x, y). Example (2xy − cos x) dx + (x2 − 1) dy = 0 is exact for ∂M/∂y = 2x, ∂N/∂x = 2x. ∫ M dx = ∫ (2xy − cos x) dx = x2y − sin x, ∫ N dy = ∫ (x2 − 1) dy = x2y − y. The solution is x2y − sin x − y = C, as may easily be verified. Linear Equations A differential equation is said to be linear when it is of first degree in the dependent variable and its derivatives. The general linear first-order differential equation has the form dy/dx + P(x)y = Q(x). Its general solution is y = e−∫ P dx

Qe

∫ P dx

dx + C



Example A tank initially holds 200 gal of a salt solution in which 100 lb is dissolved. Six gallons of brine containing 4 lb of salt run into the tank per minute. If mixing is perfect and the output rate is 4 gal/min, what is the amount A of salt in the tank at time t? The differential equation of A is dA/dt + [1/(100 + t)]A = 4. Its general solution is A = 2(100 + t) + C/(100 + t). At t = 0, ′A = 100; so the particular solution is A = 2(100 + t) − 104/(100 + t). ORDINARY DIFFERENTIAL EQUATIONS OF HIGHER ORDER The higher-order differential equations, especially those of order 2, are of great importance because of physical situations describable by them. Equation y(n) = f(x) Such a differential equation can be solved by n integrations. The solution will contain n arbitrary constants. Linear Differential Equations with Constant Coefficients and Right-Hand Member Zero (Homogeneous) The solution of y″ + ay′ + by = 0 depends upon the nature of the roots of the characteristic equation m2 + am + b = 0 obtained by substituting the trial solution y = emx in the equation. Distinct Real Roots If the roots of the characteristic equation

Multiple Real Roots If r1 = r2, the solution of the differential equation is y = e r1x(A + Bx). Example y″ + 4y + 4 = 0. The characteristic equation is m2 + 4m + 4 = 0 with roots −2 and −2. The solution is y = e−2x(A + Bx). Complex Roots If the characteristic roots are p  iq, then the solution is y = epx(A cos qx + B sin qx). Example The differential equation My″ + Ay′ + ky = 0 represents the vibration of a linear system of mass M, spring constant k, and damping constant , the roots of the characteristic equation A. If A < 2
kM A Mm2 + Am + k = 0 are complex −
 i 2M and the solution is y = e−(At/2M)

k A − 

 M 2M 

2

A − 
t + c sin  − 
t

c cos  M 2M   M 2M   k

A

1

2

k

2

2

This solution is oscillatory, representing undercritical damping.

All these results generalize to homogeneous linear differential equations with constant coefficients of order higher than 2. These equations (especially of order 2) have been much used because of the ease of solution. Oscillations, electric circuits, diffusion processes, and heat-flow problems are a few examples for which such equations are useful. Second-Order Equations: Dependent Variable Missing Such an equation is of the form dy d 2y F x,
,
2 = 0 dx dx It can be reduced to a first-order equation by substituting p = dy/dx and dp/dx = d 2y/dx2. Second-Order Equations: Independent Variable Missing Such an equation is of the form dy d 2y F y,
,
2 = 0 dx dx









dy d 2y dp
= p,
2 = p
dx dx dy The result is a first-order equation in p, dp F y, p, p
= 0 dy Set





Example The capillary curve for one vertical plate is given by


dx  

d 2y 4y 1+
2 =
dx c2 Its solution by this technique is c 2 c2 −y2 −
 c2 − h x +
 0 =
2 where c, h0 are physical constants.

dy

2 3/2

cosh

−1

c c
− cosh−1
y h0



Example The equation governing chemical reaction in a porous catalyst in plane geometry of thickness L is dc d 2c D
2 = k f(c),
(0) = 0, c(L) = c0 dx dx where D is a diffusion coefficient, k is a reaction rate parameter, c is the concentration, k f(c) is the rate of reaction, and c0 is the concentration at the boundary. Making the substitution gives k dp p
=
f(c) dc D

DIFFERENTIAL EQUATIONS



c p2 k f(c) dc
=
2 D c(0) If the reaction is very fast, c(0) ≈ 0 and the average reaction rate is related to p(L). See Ref. 106. This variable is given by 1/2 2k c0 p(L) =
f(c) dc D 0 Thus, the average reaction rate can be calculated without solving the complete problem.

Integrating gives





Linear Nonhomogeneous Differential Equations Linear Differential Equations Right-Hand Member f(x) ≠ 0 Again the specific remarks for y″ + ay′ + by = f(x) apply to differential equations of similar type but higher order. We shall discuss two general methods. Method of Undetermined Coefficients Use of this method is limited to equations exhibiting both constant coefficients and particular forms of the function f(x). In most cases f(x) will be a sum or product of functions of the type constant, xn (n a positive integer), emx, cos kx, sin kx. When this is the case, the solution of the equation is y = H(x) + P(x), where H(x) is a solution of the homogeneous equations found by the method of the preceding subsection and P(x) is a particular integral found by using the following table subject to these conditions: (1) When f(x) consists of the sum of several terms, the appropriate form of P(x) is the sum of the particular integrals corresponding to these terms individually. (2) When a term in any of the trial integrals listed is already a part of the homogeneous solution, the indicated form of the particular integral is multiplied by x. Form of Particular Integral

If f(x) is

Then P(x) is

a (constant) axn aerx c cos kx d sin kx

A (constant) An xn + An − 1 xn − 1 + ⋅⋅⋅A1 x + A0 Berx

gxnerx cos kx hxnerx sin kx

A cos kx + B sin kx

(Anxn + ⋅⋅⋅ + A0)erx cos kx + (Bn xn + ⋅⋅⋅ + B0)erx sin kx

d 2y

1 dy

dx

x dx

Example (1 − x2)
2 −

= x. The homogeneous equation d 2y 1 dy (1 − x2)
2 −

= 0 x dx dx dp dx
=
p x(1 − x2) when we set dy/dx = p. Upon integrating twice, y = c1
 x2 − 1 + c2 is the homogeneous solution. Now assume that the particular solution has the form y = u
 x2 −1 + v. The equations for u and v become reduces to

u′ = du/dx =
 x2 −1 dv 2 v′ =
= 1 − x dx so that 1 x2 −1 − ln (x +
 x2 −1)] and v = x − x3/3. u =
[x
 2 The complete solution is x x3 1 x2 −1 + c2 +
−
−

 x2 −1 ln (x +
 x2 −1). y = c1
 2 6 2

Perturbation Methods If the ordinary differential equation has a parameter that is small and is not multiplying the highest derivative, perturbation methods can give solutions for small values of the parameter. Example Consider the differential equation for reaction and diffusion in a catalyst; the reaction is second order: c″ = ac2, c′(0) = 0, c(1) = 1. The solution is expanded in the following Taylor series in a. c(x, a) = c0(x) + ac1(x) + a2c2(x) + … The goal is to find equations governing the functions {ci(x)} and solve them. Substitution into the equations gives the following equations: c0″(x) + a c″1(x) + a2c″2(x) + … = a[c0(x) + ac1(x) + a2c2(x) + …]2 c′0(0) + ac′1(0) + a2c′2(0) + … = 0 c0(1) + ac1(1) + a2c2(1) + … = 1 Like terms in powers of a are collected to form the individual problems. c″0 = 0,

Since the form of the particular integral is known, the constants may be evaluated by substitution in the differential equation. Example y″ + 2y′ + y = 3e − cos x + x . The characteristic equation is (m + 1)2 = 0 so that the homogeneous solution is y = (c1 + c2x)e−x. To find a particular solution we use the trial solution from the table, y = a1e2x + a2 cos x + a3 sin x + a4x3 + a5x2 + a6x + a7. Substituting this in the differential equation collecting and equating like terms, there results a1 = s, a2 = 0, a3 = −a, a4 = 1, a5 = −6, a6 = 18, and a7 = −24. The solution is y = (c1 + c2x)e−x + se2x − a sin x + x3 − 6x2 + 18x − 24. 2x

3

Method of Variation of Parameters This method is applicable to any linear equation. The technique is developed for a second-order equation but immediately extends to higher order. Let the equation be y″ + a(x)y′ + b(x)y = R(x) and let the solution of the homogeneous equation, found by some method, be y = c1 f1(x) + c2 f2(x). It is now assumed that a particular integral of the differential equation is of the form P(x) = uf1 + vf2 where u, v are functions of x to be determined by two equations. One equation results from the requirement that uf1 + vf2 satisfy the differential equation, and the other is a degree of freedom open to the analyst. The best choice proves to be u′f1 + v′f2 = 0

and

u′f 1′ + v′f′2 = R(x)

f2 du Then u′ =
= −

R(x) dx f1 f 2′ − f2 f 1′ f1 dv v′ =
=

R(x) dx f1 f 2′ − f2 f 1′ and since f1, f2, and R are known u, v may be found by direct integration.

3-37

c″1 = c 02,

c′0(0) = 0, c′1(0) = 0,

c″2 = 2c0 c1,

c0(1) = 1 c1(1) = 0

c′2(0) = 0,

c2(1) = 0

The solution proceeds in turn. c0(x) = 1,

(x2 − 1) c1(x) =
, 2

5 − 6x2 + x4 c2(x) =

12

SPECIAL DIFFERENTIAL EQUATIONS (SEE REF. 1) Euler’s Equation The linear equation x ny(n) + a1x n − 1y(n − 1) + ⋅⋅⋅ + an − 1xy′ + any = R(x) can be reduced to a linear equation with constant coefficients by the change of variable x = et. To solve the homogeneous equation substitute y = x r into it, cancel the powers of x, which are the same for all terms, and solve the resulting polynomial for r. In case of multiple or complex roots there results the form y = x r(log x)r and y = x α[cos (β log x) + i sin (β log x)]. Example Solve x2y″ − 2y = 0. By setting y = xr, xr[r(r − 1) − 2] = 0. The roots of r 2 − r − 2 = 0 are r = 2, −1. The general solution is y = Ax2 + B/x. The equation (ax + b)ny(n) + a1(ax + b)n − 1y(n − 1) + ⋅⋅⋅ + an y = R(x) can be reduced to the Euler form by the substitution ax + b = z. It may be treated without change of variable, the homogeneous equation having solutions of the form y = (ax + b)r. Bessel’s Equation The linear equation x2(d 2y/dx2) + (1 − 2α) x(dy/dx) + [β 2 γ 2 x 2γ + (α2 − p2γ 2)]y = 0 is the general Bessel equation. By series methods, not to be discussed here, this equation can be shown to have the solution y = AxαJp(βxγ) + BxαJ−p(βxγ)

p not an integer or zero

y = AxαJp(βxγ) + BxαYp(βxγ)

p an integer

3-38 where

MATHEMATICS ∞

PARTIAL DIFFERENTIAL EQUATIONS

  

k!Γ(p + k + 1)

x Jp(x) =
2

p

The analysis of situations involving two or more independent variables frequently results in a partial differential equation.

k=0

 Γ(n) = x

x J−p(x) =
2

(−1)k(x/2)2k

–p

∞



k=0

(−1)k(x/2)2k

k!Γ(k + 1 − p)

∞

n − 1 −x

e dx

p not an integer

n>0

0

is the gamma function. For p an integer x p ∞ (−1)k(x/2)2k Jp(x) =


2 k = 0 k!(p + k)!



(Bessel function of the first kind of order p) [ Jp(x) cos (pπ) − J−p(x)] Yp(x) =


sin (pπ) (replace right-hand side by limiting value if P is an integer or zero). The series converge for all x. Much of the importance of Bessel’s equation and Bessel functions lies in the fact that the solutions of numerous linear differential equations can be expressed in terms of them. Example d 2y/dx2 + [9x − (63/4x2)]y = 0. In general form this is x2(d 2y/dx2) + (9x3 − 63⁄ 4)y = 0. Thus α = a, γ = 3⁄ 2; β = 2, p = 8⁄ 3. The solution is (since p integer) y = Ax1/2J8/3(2x3/2) + Bx1/2J–8/3(2x3/2). Tables are available for the evaluation of many of these functions. Example The heat flow through a wedge-shaped fin is characterized by the equation x2(d 2y/dx2) + x(dy/dx) − a2xy = 0, where y = T − Tair, α is a combination of physical constants, and x = distance from fin end. By comparing this with the standard equation, there results α = 0, p = 0, γ = a, β 2 = −4a2 or β = 2ai. The solution is y = AJ0(2ai
x) + BY0(2ai
x). Legendre’s Equation The Legendre equation (1 − x2)y″ − 2xy′ + n(n + 1)y = 0, n ≥ 0, has the solution y = Aun(x) + Bvn(x) for n not an integer where n(n + 1) n(n − 2)(n + 1)(n + 3) un(x) = 1 −
x2 +


x4 2! 4! n(n − 2)(n − 4)(n + 1)(n + 3)(n + 5) 6 −



x + ⋅⋅⋅ 6! (n − 1)(n + 2) 3 (n − 1)(n − 3)(n + 2)(n + 4) 5 vn(x) = x −

x +


x − ⋅⋅⋅⋅ 3! 5! If n is an even integer or zero, un is a polynomial in x. If n is an odd integer, then vn is a polynomial. The interval of convergence for the series is −1 < x < 1. If n is an integer, set un(x) vn(x) Pn(x) =
(n even or zero), Pn =
(n odd) un(1) vn(1) The polynomials Pn are the so-called Legendre polynomials, P0(x) = 1, P1(x) = x, P2(x) = a(3x2 − 1), P3(x) = a(5x3 − 3x), . . . . Laguerre’s Equation The Laguerre equation x(d 2y/dx2) + (c − x) (dy/dx) − ay = 0 is satisfied by the confluent hypergeometric function. See Refs. 1 and 173. Hermite’s Equation The Hermite equation y″ − 2xy′ + 2ny = 0 is satisfied by the Hermite polynomial of degree n, y = AHn(x) if n is a positive integer or zero. H0(x) = 1, H1(x) = 2x, H2(x) = 4x2 − 2, H3(x) = 8x3 − 12x, H4(x) = 16x4 − 48x2 + 12, Hr + 1(x) = 2xHr(x) − 2rHr − 1(x). Example y″ − 2xy′ + 6y = 0. Here n = 3; so y = AH3 = A(8x3 − 12x) is a solution.

Chebyshev’s Equation The equation (1 − x2)y″ − xy′ + n2y = 0 for n a positive integer or zero is satisfied by the nth Chebyshev polynomial y = ATn(x). T0(x) = 1, T1(x) = x, T2(x) = 2x2 − 1, T3(x) = 4x3 − 3x, T4(x) = 8x4 − 8x2 + 1; Tr + 1(x) = 2xTr(x) − Tr − 1(x). Example (1 − x2)y″ − xy′ + 36y = 0. Here n = 6. A solution is y = T6(x) = 2xT5(x) − T4(x) = 2x(2xT4 − T3) − T4 = 32x6 − 48x4 + 18x2 − 1. Further details on these special equations and others can be found in the literature.

Example The equation ∂T/∂t = K(∂2T/∂x2) represents the unsteady onedimensional conduction of heat. Example The equation for the unsteady transverse motion of a uniform beam clamped at the ends is ∂4y ρ ∂2y =0
4 +

∂x EI ∂t2 Example The expansion of a gas behind a piston is characterized by the simultaneous equations ∂u ∂u c2 ∂ρ ∂ρ ∂ρ ∂u
+ u
+

= 0 and
+ u
+ ρ
= 0 ∂t ∂x ρ ∂x ∂t ∂x ∂x Example The heating of a diathermanous solid is characterized by the equation α(∂2θ/∂x2) + βe−γz = ∂θ/∂t. The partial differential equation ∂2f/∂x ∂y = 0 can be solved by two integrations yielding the solution f = g(x) + h(y), where g(x) and h(y) are arbitrary differentiable functions. This result is an example of the fact that the general solution of partial differential equations involves arbitrary functions in contrast to the solution of ordinary differential equations, which involve only arbitrary constants. A number of methods are available for finding the general solution of a partial differential equation. In most applications of partial differential equations the general solution is of limited use. In such applications the solution of a partial differential equation must satisfy both the equation and certain auxiliary conditions called initial and/or boundary conditions, which are dictated by the problem. Examples of these include those in which the wall temperature is a fixed constant T(x0) = T0, there is no diffusion across a nonpermeable wall, and the like. In ordinary differential equations these auxiliary conditions allow definite numbers to be assigned to the constants of integration. In partial differential equations the boundary conditions demand that the arbitrary functions resulting from integration assume specific forms. Except for a few cases (some first-order equations, D’Alembert’s solution of the wave equation, and others) a procedure which first determines the arbitrary functions and then specializes them to fit the boundary conditions is usually not feasible. A more fruitful attack is to determine directly a set of particular solutions and then combine them so that the boundary conditions are satisfied. The only area in which much analysis has been accomplished is for linear homogeneous partial differential equations. Such equations have the property that if f1, f2, . . . , ∞ fn, . . . are individually solutions, then the function f =  i=1 fi is also a solution, provided the series converges and is differentiable up to the order (termwise) of the equation. Partial Differential Equations of Second and Higher Order Many of the applications to scientific problems fall naturally into partial differential equations of second order, although there are important exceptions in elasticity, vibration theory, and elsewhere. A second-order differential equation can be written as ∂2u ∂2u ∂2u a
2 + b
+ c
2 = f ∂x ∂x∂y ∂y where a, b, c, and f depend upon x, y, u, ∂u/∂x, and ∂u/∂y. This equation is hyperbolic, parabolic, or elliptic, depending on whether the discriminant b2 − 4ac is >0, =0, or <0, respectively. Since a, b, c, and f depend on the solution, the type of equation can be different at different x and y locations. If the equation is hyperbolic, discontinuities can be propagated. See Refs. 11, 79, 105, 159, and 192. Phenomena of propagation such as vibrations are characterized by equations of “hyperbolic” type which are essentially different in their properties from other classes such as those which describe equilibrium (elliptic) or unsteady diffusion and heat transfer (parabolic). Prototypes are as follows: Elliptic Laplace’s equation ∂2u/∂x2 + ∂2u/∂y2 = 0 and Poisson’s equation ∂2u/∂x2 + ∂2u/∂y2 = g(x, y) do not contain the variable time

DIFFERENTIAL EQUATIONS explicitly and consequently represent equilibrium configurations. Laplace’s equation is satisfied by static electric or magnetic potential at points free from electric charges or magnetic poles. Other important functions satisfying Laplace’s equation are the velocity potential of the irrotational motion of an incompressible fluid, used in hydrodynamics; the steady temperature at points in a homogeneous solid, and the steady state of diffusion through a homogeneous body. The gravitational potential V at points occupied by mass of density d satisfies Poisson’s equation ∂2 V/∂x2 + ∂2 V/∂y2 + ∂2 V/∂z2 = −4πd. Parabolic The heat equation ∂T/∂t = ∂ 2T/∂x2 + ∂2T/∂y2 represents nonequilibrium or unsteady states of heat conduction and diffusion. Hyperbolic The wave equation ∂2u/∂t2 = c2(∂2u/∂x2 + ∂2u/∂y2) represents wave propagation of many varied types. Quasilinear first-order differential equations are like ∂u ∂u a
+b
=f ∂x ∂y where a, b, and f depend on x, y, and u, with a2 + b2 ≠ 0. This equation can be solved using the method of characteristics, which writes the solution in terms of a parameter s, which defines a path for the characteristic. dx dy du
= a,
= b,
= f ds ds ds These equations are integrated from some initial conditions. For a specified value of s, the value of x and y shows the location where the solution is u. The equation is semilinear if a and b depend just on x and y (and not u), and the equation is linear if a, b, and f all depend on x and y, but not u. Such equations give rise to shock propagation, and conditions have been derived to deduce the presence of shocks, Ref. 245. For further information, see Refs. 79, 159, 192, and 245. An example of a linear hyperbolic equation is the advection equation for flow of contaminants when the x and y velocity components are u and v, respectively. ∂c ∂c ∂c
+u
+v
=0 ∂t ∂x ∂y The equations for flow and adsorption in a packed bed or chromatography column give a quasilinear equation. ∂c ∂c df ∂c φ
+ φ u
+ (1 − φ)

= 0 ∂t ∂x dc ∂t Here n = f(c) is the relation between concentration on the adsorbent and fluid concentration. The solution of problems involving partial differential equations often revolves about an attempt to reduce the partial differential equation to one or more ordinary differential equations. The solutions of the ordinary differential equations are then combined (if possible) so that the boundary conditions as well as the original partial differential equation are simultaneously satisfied. Three of these techniques are illustrated. Similarity Variables The physical meaning of the term “similarity” relates to internal similitude, or self-similitude. Thus, similar solutions in boundary-layer flow over a horizontal flat plate are those for which the horizontal component of velocity u has the property that two velocity profiles located at different coordinates x differ only by a scale factor. The mathematical interpretation of the term similarity is a transformation of variables carried out so that a reduction in the number of independent variables is achieved. There are essentially two methods for finding similarity variables, “separation of variables” (not the classical concept) and the use of “continuous transformation groups.” The basic theory is available in Ames (see the references). Example The equation ∂θ/∂x = (A/y)(∂2θ/∂y2) with the boundary conditions θ = 0 at x = 0, y > 0; θ = 0 at y = ∞, x > 0; θ = 1 at y = 0, x > 0 represents the nondimensional temperature θ of a fluid moving past an infinitely wide flat plate immersed in the fluid. Turbulent transfer is neglected, as is molecular transport except in the y direction. It is now assumed that the equation and the boundary conditions can be satisfied by a solution of the form θ = f(y/xn) = f(u), where θ =

3-39

0 at u = ∞ and θ = 1 at u = 0. The purpose here is to replace the independent variables x and y by the single variable u when it is hoped that a value of n exists which will allow x and y to be completely eliminated in the equation. In this case since u = y/xn, there results after some calculation ∂θ/∂x = −(nu/x)(dθ/du), ∂2θ/∂y2 = (1/x2n)(d2θ/du2), and when these are substituted in the equation, −(1/x)nu(dθ/du) = (1/x3n)(A/u)(d2θ/du2). For this to be a function of u only, choose n = s. There results (d2θ/du2) + (u2/3A)(dθ/du) = 0. Two integrations and use of the boundary conditions for this ordinary differential equation give the solution θ=



∞

u



exp (−u3/9A) du 

∞

exp (−u3/9A) du

0

Group Method The type of transformation can be deduced using group theory. For a complete exposition, see Refs. 9, 12, and 145; a shortened version is in Ref. 106. Basically, a similarity transformation should be considered when one of the independent variables has no physical scale (perhaps it goes to infinity). The boundary conditions must also simplify (and combine) since each transformation leads to a differential equation with one fewer independent variable. Example A similarity variable is found for the problem ∂c ∂ ∂c
=
D(c)
, c(0,t) = 1, c(∞,t) = 0, c(x,0) = 0 ∂t ∂x ∂x Note that the length dimension goes to infinity, so that there is no length scale in the problem statement; this is a clue to try a similarity transformation. The transformation examined here is





t = a α t,

x = a β x, c = a γ c With this substitution, the equation becomes ∂c ∂ ∂c aα − γ
= a2β − γ
D(a−γ c)
∂t ∂x ∂x Group theory says a system is conformally invariant if it has the same form in the new variables; here, that is





γ = 0, α − γ = 2β − γ,

or α = 2β

The invariants are β δ=
α

x η =
δ , t and the solution is

c(x, t) = f(η)tγ/α We can take γ = 0 and δ = β/α = a. Note that the boundary conditions combine because the point x = ∞ and t = 0 give the same value of η and the conditions on c at x = ∞ and t = 0 are the same. We thus make the transformation x η =
, c(x, t) = f(η) 4 D
 0t The use of the 4 and D0 makes the analysis below simpler. The result is





df d df
D(c)
+ 2η
= 0, dη dη dη

f(0) = 1,

f(∞) = 0

Thus, we solve a two-point boundary value problem instead of a partial differential equation. When the diffusivity is constant, the solution is the error function, a tabulated function. c(x,t) = 1 − erf η = erfc η erf η =



η

0



e−ξ dξ  2

∞

2

e−ξ dξ

0

Separation of Variables This is a powerful, well-utilized method which is applicable in certain circumstances. It consists of assuming that the solution for a partial differential equation has the form U = f(x)g(y). If it is then possible to obtain an ordinary differential equation on one side of the equation depending only on x and on the other side only on y, the partial differential equation is said to be separable in the variables x, y. If this is the case, one side of the equation is a function of x alone and the other of y alone. The two can be equal only if each is a constant, say λ. Thus the problem has again been reduced to the solution of ordinary differential equations. Example Laplace’s equation ∂2V/∂x2 + ∂2V/∂y2 = 0 plus the boundary conditions V(0, y) = 0, V(l, y) = 0, V(x, ∞) = 0, V(x, 0) = f(x) represents the steadystate potential in a thin plate (in z direction) of infinite extent in the y direction

3-40

MATHEMATICS

and of width l in the x direction. A potential f(x) is impressed (at y = 0) from x = 0 to x = 1, and the sides are grounded. To obtain a solution of this boundaryvalue problem assume V(x, y) = f(x)g(y). Substitution in the differential equation yields f ″(x)g(y) + f(x)g″(y) = 0, or g″(y)/g(y) = −f″(x)/f(x) = λ2 (say). This system becomes g″(y) − λ2g(y) = 0 and f ″(x) + λ2f(x) = 0. The solutions of these ordinary differential equations are respectively g(y) = Aeλy + Be−λy, f(x) = C sin λx + D cos λx. Then f(x)g(y) = (Aeλy + Be−λy) (C sin λx + D cos λx). Now V(0, y) = 0 so that f(0)g(y) = (Aeλy + Be−λy) D  0 for all y. Hence D = 0. The solution then has the form sin λx (Aeλy + Be−λy) where the multiplicative constant C has been eliminated. Since V(l, y) = 0, sin λl(Aeλy + Be−λy)  0. Clearly the bracketed function of y is not zero, for the solution would then be the identically zero solution. Hence sin λl = 0 or λn = nπ/l, n = 1, 2, . . . where λn = nth eigenvalue. The solution now has the form sin (nπx/l)(Aenπy/l + Be−nπy/l). Since V(x, ∞) = 0, A must be taken to be zero because ey becomes arbitrarily large as y → ∞. The solution then reads Bn sin (nπx/l)e−nπy/l, where Bn is the multiplicative constant. The differential equation is linear and homogeneous so that n∞= 1 Bne−nπy/l sin (nπx/l) is also a solution. Satisfaction of the last boundary condition is ensured by taking 2 Bn =
l

f(x) sin (nπx/l) dx = Fourier sine coefficients of f(x) l

0

Further, convergence and differentiability of this series are established quite easily. Thus the solution is ∞ nπx V(x, y) =  Bne−nπy/l sin
l n=1

Example The diffusion problem ∂c ∂ ∂c
=
D(c)
, c(0, t) = 1, c(∞, t) = 0, c(x, 0) = 0 ∂t ∂x ∂x can be solved by separation of variables. First transform the problem so that the boundary conditions are homogeneous (having zeroes on the right-hand side). Let c(x, t) = 1 − x + u(x, t) Then u(x, t) satisfies





∂u ∂ 2u
= D
2 , ∂t ∂x

u(x, 0) = x − 1,

u(0, t) = 0,

u(1, t) = 0

Assume a solution of the form u(x, t) = X(x) T(t), which gives dT d 2X X
=DT
dt dx2 Since both sides are constant, this gives the following ordinary differential equations to solve. 1 d 2X 1 dT = −λ

= −λ,

D T dt X dx2 The solution of these is T=Ae

−λDt

,

 x + E sin
λ x X = B cos
λ

The combined solution for u(x,t) is

 x + E sin
λ  x) e−λDt u = A (B cos
λ Apply the boundary condition that u(0,t) = 0 to give B = 0. Then the solution is

 x)e−λDt u = A (sin
λ where the multiplicative constant E has been eliminated. Apply the boundary condition at x = L.

 L)e−λDt 0 = A (sin
λ This can be satisfied by choosing A = 0, which gives no solution. However, it can also be satisfied by choosing λ such that

 L = 0,
λ L=nπ sin
λ n2π2 Thus λ=
L2 The combined solution can now be written as





sin nπx 2 2 2 u = A
e−n π Dt/L L

Since the initial condition must be satisfied, we use an infinite series of these functions. ∞ sin nπx 2 2 2 u =  An
e−n π Dt/L L n=1 At t = 0, we satisfy the initial condition. ∞ sin nπx x − 1 =  An
L n=1 This is done by multiplying the equation by sin mπx
L and integrating over x: 0 → L. (This is the same as minimizing the mean-square error of the initial condition.) This gives L AmL
= (x − 1) sin mπx dx 0 2 which completes the solution.











Integral-Transform Method A number of integral transforms are used in the solution of differential equations. Only one, the Laplace transform, will be discussed here [for others, see “Integral Transforms (Operational Methods)”]. The one-sided Laplace transform indicated ∞ by L[ f(t)] is defined by the equation L[ f(t)] = ∫0 f(t)e−st dt. It has numerous important properties. The ones of interest here are L[ f′(t)] = sL[f(t)] − f(0); L[ f″(t)] = s2L[f(t)] − sf(0) − f′(0); L[ f (n)(t)] = snL[ f(t)] − sn − 1f(0) − sn − 2f′(0) − ⋅⋅⋅ − f (n − 1)(0) for ordinary derivatives. For partial derivatives an indication of which variable is being transformed avoids confusion. Thus, if ∂y y = y(x, t), Lt
= sL[y(x, t)] − y(x, 0) ∂t



∂y dLt[y(x, t)] Lt
=

∂x dx since L[y(x, t)] is “really” only a function of x. Otherwise the results are similar. These facts coupled with the linearity of the transform, i.e., L[af(t) + bg(t)] = aL[ f(t)] + bL[g(t)], make it a useful device in solving some linear differential equations. Its use reduces the solution of ordinary differential equations to the solution of algebraic equations for L[y]. The solution of partial differential equations is reduced to the solution of ordinary differential equations. In both situations the inverse transform must be obtained either from tables, of which there are several, or by use of complex inversion methods.



whereas

Example The equation ∂c/∂t = D(∂2c/∂x2) represents the diffusion in a semi-infinite medium, x ≥ 0. Under the boundary conditions c(0, t) = c0, c(x, 0) = 0 find a solution of the diffusion equation. By taking the Laplace transform of both sides with respect to t, ∞ ∂2c 1 ∞ −st ∂c e−st
2 dt =
e
dt 0 ∂x D 0 ∂t





sF d2F = (1/D)sF − c(x, 0) =

dx2 D where F(x, s) = Lt[c(x, t)]. Hence d 2F s −
F=0
dx2 D The other boundary condition transforms into F(0, s) = c0 /s. Finally the solution of the ordinary differential equation for F subject to F(0, s) = c0 /s and F remains x. Reference to a table shows that the funcfinite as x → ∞ is F(x, s) = (c0 /s)e−
s/D tion having this as its Laplace transform is or

 



2 c(x, t) = c0 1 −

π



t x/2
D

0



e−u du 2

Matched-Asymptotic Expansions Sometimes the coefficient in front of the highest derivative is a small number. Special perturbation techniques can then be used, provided the proper scaling laws are found. See Refs. 32, 170, and 180.

DIFFERENCE EQUATIONS

3-41

DIFFERENCE EQUATIONS REFERENCES:

30, 43.

Certain situations are such that the independent variable does not vary continuously but has meaning only for discrete values. Typical illustrations occur in the stagewise processes found in chemical engineering such as distillation, staged extraction systems, and absorption columns. In each of these the operation is characterized by a finite between-stage change of the dependent variable in which the independent variable is the integral number of the stage. The importance of difference equations is twofold: (1) to analyze problems of the type described and (2) to obtain approximate solutions of problems which lead, in their formulation, to differential equations. In this subsection only problems of analysis are considered; the application to approximate solutions is considered under “Numerical Analysis and Approximate Methods.” ELEMENTS OF THE CALCULUS OF FINITE DIFFERENCES Let y = f(x) be defined for discrete equidistant values of x, which will be denoted by xn. The corresponding value of y will be written yn = f(xn). The first forward difference of f(x) denoted by ∆f(x) = f(x + h) − f(x) where h = xn − xn − 1 = interval length. Example Let f(x) = x2. Then ∆f(x) = (x + h)2 − x2 = 2hx + h2. The second forward difference is obtained by taking the difference of the first; thus ∆∆f(x) = ∆2f(x) = ∆f(x + h) − ∆f(x) = f(x + 2h) − 2f(x + h) + f(x). Example f(x) = x2, ∆2f(x) = ∆[∆f(x)] = ∆2hx + ∆h2 = 2h(x + h) − 2hx + h2 −

h2 = 2h2.

Similarly the nth forward difference is defined by the relation ∆nf(x) = ∆[∆n − 1f(x)]. Other difference relations are also quite useful. Some of these are ∇f(x) = f(x) − f(x − h), which is called the backward difference, and δf(x) = f [x + (h/2)] − f [x − (h/2)], called the central difference. Some properties of the operator ∆ are quite important. If C is any constant, ∆C = 0; if f(x) is any function of period h, ∆f(x) = 0 (in fact, periodic functions of period h play the same role here as constants do in the differential calculus); ∆[ f(x) + g(x)] = ∆f(x) + ∆g(x); ∆m[∆nf(x)] = ∆m + nf(x); ∆[f(x)g(x)] = f(x) ∆g(x) + g(x + h) ∆f(x) g(x) ∆f(x) − f(x) ∆g(x) f(x) ∆
=


g(x) g(x)g(x + h)



Example ∆(x sin x) = x∆ sin x + sin (x + h) ∆x = 2x sin (h/2) cos [x + (h/2)] +

h sin (x + h).

DIFFERENCE EQUATIONS A difference equation is a relation between the differences and the independent variable, φ(∆ny, ∆n − 1y, . . . , ∆y, y, x) = 0, where φ is some given function. The general case in which the interval between the successive points is any real number h, instead of 1, can be reduced to that with interval size 1 by the substitution x = hx′. Hence all further difference-equation work will assume the interval size between successive points is 1. Example f(x + 1) − (α + 1)f(x) + αf(x − 1) = 0. Common notation usually is yx = f(x). This equation is then written yx + 1 − (α + 1)yx + αyx − 1 = 0. Example yx + 2 + 2yx yx + 1 + yx = x2. Example yx + 1 − yx = 2x. The order of the difference equation is the difference between the largest and smallest arguments when written in the form of the second example. The first and second examples are both of order 2, while the third example is of order 1. A linear difference equation involves no

products or other nonlinear functions of the dependent variable and its differences. The first and third examples are linear, while the second example is nonlinear. A solution of a difference equation is a relation between the variables which satisfies the equation. If the difference equation is of order n, the general solution involves n arbitrary constants. The techniques for solving difference equations resemble techniques used for differential equations. Equation Dny = a The solution of ∆ny = a, where a is a constant, is a polynomial of degree n plus an arbitrary periodic function of period 1. That is, y = (axn/n!) + c1xn − 1 + c2xn − 2 + ⋅⋅⋅ + cn + f(x), where f(x + 1) = f(x). Example ∆3y = 6. The solution is y = x3 + c1x2 + c2x + c3 + f(x); c1, c2, c3 are arbitrary constants, and f(x) is an arbitrary periodic function of period 1. Equation yx + 1 - yx = φ(x) This equation states that the first difference of the unknown function is equal to the given function φ(x). The solution by analogy with solving the differential equation dy/dx = φ(x) by integration is obtained by “finite integration” or summation. When there are only a finite number of data points, this is easily accomplished by writing yx = y0 +  tx = 1 φ(t − 1), where the data points are numbered from 1 to x. This is the only situation considered here. Examples If φ(x) = 1, yx = x. If φ(x) = x, yx = [x(x − 1)]/2. If φ(x) = ax, a ≠ 0, yx = ax/(a − 1). In all cases y0 = 0. Other examples may be evaluated by using summation, that is, y2 = y1 + φ(1), y3 = y2 + φ(2) = y1 + φ(1) + φ(2), y4 = y3 + φ(3) = y1 + φ(1) + φ(2) + φ(3), . . . , yx = y1 +  tx =− 11 φ(t). Example yx + 1 − ryx = 1, r constant, x > 0 and y0 = 1. y1 = 1 + r, y2 = 1 + r + r2, . . . , yx = 1 + r + ⋅⋅⋅ + rx = (1 − rx + 1)/(1 − r) for r ≠ 1 and yx = 1 + x for r = 1. Linear Difference Equations The linear difference equation of order n has the form Pnyx + n + Pn − 1yx + n − 1 + ⋅⋅⋅ + P1yx + 1 + P0yx = Q(x) with Pn ≠ 0 and P0 ≠ 0 and Pj ; j = 0, . . . , n are functions of x. Constant Coefficient and Q(x) = 0 (Homogeneous) The solution is obtained by trying a solution of the form yx = cβ x. When this trial solution is substituted in the difference equation, a polynomial of degree n results for β. If the solutions of this polynomial are denoted by β1, β2, . . . , βn then the following cases result: (1) if all the βj’s are n real and unequal, the solution is yx =  j = 1 cj β jx, where the c1, . . . , cn are arbitrary constants; (2) if the roots are real and repeated, say, βj has multiplicity m, then the partial solution corresponding to βj is β jx(c1 + c2 x + ⋅⋅⋅ + cm xm − 1); (3) if the roots are complex conjugates, say, a + ib = peiθ and a − ib = pe−iθ, the partial solution corresponding to this pair is px(c1 cos θx + c2 sin θx); and (4) if the roots are multiple complex conjugates, say, a + ib = peiθ and a − ib = pe−iθ are m-fold, then the partial solution corresponding to these is px[(c1 + c2 x + ⋅⋅⋅ + cm x m − 1) cos θx + (d1 + d2 x + ⋅⋅⋅ + dm x m − 1) sin θx]. Example The equation yx + 1 − (α + 1)yx + αyx − 1 = 0, y0 = c0 and ym + 1 = xm + 1/k represents the steady-state composition of transferable material in the raffinate stream of a staged countercurrent liquid-liquid extraction system. Clearly y is a function of the stage number x. α is a combination of system constants. By using the trial solution yx = cβx, there results β2 − (α + 1)β + α = 0, so that β1 = 1, β2 = α. The general solution is yx = c1 + c2αx. By using the side conditions, c1 = c0 − c2, c2 = (ym + 1 − c0)/(αm + 1 − 1). The desired solution is (yx − c0)/(ym + 1 − c0) = (αx − 1)/(αm + 1 − 1). Example yx + 3 − 3yx + 2 + 4yx = 0. By setting yx = cβ x, there results β3 − 3β2 + 4 = 0 or β1 = −1, β2 = 2, β3 = 2. The general solution is yx = c1(−1)x + 2x(c2 + c3x). Example yx + 1 − 2yx + 2yx − 1 = 0. β1 = 1 + i, β2 = 1 − i. p =
1 + 1 =
2, θ = π/4. The solution is yx = 2 x/2 [c1 cos (xπ/4) + c2 sin (xπ/4)]. Constant Coefficients and Q(x) ≠ 0 (Nonhomogeneous) In this case the general solution is found by first obtaining the homoge-

3-42

MATHEMATICS

neous solution, say, yHx and adding to it any particular solution with Q(x) ≠ 0, say, y xP. There are several means of obtaining the particular solution. Method of Undetermined Coefficients If Q(x) is a product or linear combination of products of the functions ebx, a x, x p (p a positive integer or zero) cos cx and sin cx, this method may be used. The “families” [ax], [ebx], [sin cx, cos cx] and [x p, x p − 1, . . . , x, 1] are defined for each of the above functions in the following way: The family of a term fx is the set of all functions of which fx and all operations of the form a x + y, cos c(x + y), sin c(x + y), (x + y)p on fx and their linear combinations result in. The technique involves the following steps: (1) Solve the homogeneous system. (2) Construct the family of each term. (3) If the family has no representative in the homogeneous solution, assume y Px is a linear combination of the families of each term and determine the constants so that the equation is satisfied. (4) If a family has a representative in the homogeneous solution, multiply each member of the family by the smallest integral power of x for which all such representatives are removed and revert to step 3. Example yx + 1 − 3yx + 2yx − 1 = 1 + ax. a ≠ 0. The homogeneous solution is yxH = c1 + c2 2x. The family of 1 is 1 and of ax is ax. However, 1 is a solution of the homogeneous system. Therefore, try yxP = Ax + Bax. Substituting in the equation there results a yx = c1 + c2 2x − x +

axa ≠ 1, a ≠ 2 (a − 1)(a − 2) If a = 1, yx = c1 + c22x − 2x. If a = 2, yx = c1 + c22x − x + x2x.

Example The family of x23x is [x23x, x3x, 3x]. Method of Variation of Parameters This technique is applicable to general linear difference equations. It is illustrated for the second-order system yx + 2 + Ayx + 1 + Byx = φ(x). Assume that the homogeneous solution has been found by some technique and write yxH = c1ux + c2vx. Assume that a particular solution yxP = Dxux + Exvx. Ex and Dx can be found by solving the equations: ux + 1φ(x) Ex + 1 − Ex =


ux + 1vx + 2 − ux + 2vx + 1 vx + 1φ(x) Dx + 1 − Dx =


vx + 1ux + 2 − vx + 2ux + 1 by summation. The general solution is then yx = yxP + yxH. Variable Coefficients The method of variation of parameters applies equally well to the linear difference equation with variable coefficients. Techniques are therefore needed to solve the homogeneous system with variable coefficients. Equation yx + 1 - axyx = 0 By assuming that this equation is valid for x ≥ 0 and y0 = c, the solution is yx = c  xn = 1 an − 1. x+2 x+1

Example yx + 1 +
yx = 0. The solution is x x+1 n+1 2 3 yx = c  −
= c(−1)x
⋅
⋅⋅⋅
= (−1)xc(x + 1) n 1 2 x n=1





Example yx + 1 − xyx = 0. The solution is yx = c(x − 1)!

Reduction of Order If one homogeneous solution, say, ux, can be found by inspection or otherwise, an equation of lower order can be obtained by the substitution vx = yx /ux. The resultant equation must be satisfied by vx = constant or ∆vx = 0. Thus the equation will be of reduced order if the new variable Ux = ∆(yx /ux) is introduced. Example (x + 2)yx + 2 − (x + 3)yx + 1 + yx = 0. By observation ux = 1 is a solution. Set Ux = ∆yx = yx + 1 − yx. There results (x + 2)Ux + 1 − Ux = 0, which is of degree one lower than the original equation. The complete solution for yx is finally x 1 yx = c0 
+ c1 n = 0 n! Factorization If the difference equation can be factored, then the general solution can be obtained by solving two or more successive equations of lower order. Consider yx + 2 + Axyx + 1 + Bxyx = φ(x). If there exists ax, bx such that ax + bx = −Ax and axbx = Bx, then the difference equation may be written yx + 2 − (ax + bx) yx + 1 + axbxyx = φ(x). First solve Ux + 1 − bxUx = φ(x) and then yx + 1 − axyx = Ux. Example yx + 2 − (2x + 1)yx + 1 + (x2 + x)yx = 0. Set ax = x, bx = x + 1. Solve ux + 1 − (x + 1)ux = 0 and then yx + 1 − xyx = ux. Substitution If it is possible to rearrange a difference equation so that it takes the form afx + 2yx + 2 + bfx + 1yx + 1 + cfxyx = φ(x) with a, b, c constants, then the substitution ux = fxyx reduces the equation to one with constant coefficients. Example (x + 2)2yx + 2 − 3(x + 1)2yx + 1 + 2x2yx = 0. Set ux = x2yx. The equation becomes ux + 2 − 3ux + 1 + 2ux = 0, which is linear and easily solved by previous methods. The substitution ux = yx / fx reduces afx fx + 1yx + z + bfx fx + 2yx + 1 + cfx + 1 fx + 2yx = φ(x) to an equation with constant coefficients. Example x(x + 1)yx + 2 + 3x(x + 2)yx + 1 − 4(x + 1)(x + 2)yx = x. Set ux = yx /fx = yx /x. Then yx = xux, yx + 1 = (x + 1)ux + 1 and yx + 2 = (x + 2)ux + 2. Substitution in the equation yields x(x + 1)(x + 2)ux + 2 + 3x(x + 2)(x + 1)uu + 1 − 4x(x + 1)(x + 2) ux = x or ux + 2 + 3ux + 1 − 4ux = 1/(x + 1)(x + 2), which is a linear equation with constant coefficients. Nonlinear Difference Equations: Riccati Difference Equation The Riccati equation yx + 1yx + ayx + 1 + byx + c = 0 is a nonlinear difference equation which can be solved by reduction to linear form. Set y = z + h. The equation becomes zx + 1zx + (h + a)zx + 1 + (h + b)zx + h2 + (a + b)h + c = 0. If h is selected as a root of h2 + (a + b)h + c = 0 and the equation is divided by zx + 1zx there results [(h + b)/zx + 1] + [(h + a)/zx] + 1 = 0. This is a linear equation with constant coefficients. The solution is 1 yx = h +



a+h x 1 c −

−

b+h (a + h) + (b + h)





Example This equation is obtained in distillation problems, among others, in which the number of theoretical plates is required. If the relative volatility is assumed to be constant, the plates are theoretically perfect, and the molal liquid and vapor rates are constant, then a material balance around the nth plate of the enriching section yields a Riccati difference equation.

INTEGRAL EQUATIONS REFERENCES: 75, 79, 105, 195, 273. See also “Numerical Analysis and Approximate Methods.”

An integral equation is any equation in which the unknown function appears under the sign of integration and possibly outside the sign of integration. If derivatives of the dependent variable appear elsewhere in the equation, the equation is said to be integrodifferential.

CLASSIFICATION OF INTEGRAL EQUATIONS Volterra integral equations have an integral with a variable limit. The Volterra equation of the second kind is u(x) = f(x) + λ

K(x, t)u(t) dt x

a

INTEGRAL EQUATIONS whereas a Volterra equation of the first kind is u(x) = λ

METHODS OF SOLUTION

K(x, t)u(t) dt x

a

Equations of the first kind are very sensitive to solution errors so that they present severe numerical problems. Volterra equations are similar to initial value problems. A Fredholm equation of the second kind is u(x) = f(x) + λ

K(x, t)u(t) dt b

a

whereas a Fredholm equation of the first kind is u(x) =

b

a

The limits of integration are fixed, and these problems are analogous to boundary value problems. An eigenvalue problem is a homogeneous equation of the second kind, and solutions exist only for certain λ.

K(x, t)u(t) dt b

a

See Refs. 105 and 195 for further information and existence proofs. If the unknown function u appears in the equation in any way except to the first power, the integral equation is said to be nonlinear. b The equation u(x) = f(x) + ∫a K(x, t)[u(t)]3/2 dt is nonlinear. The differential equation du/dx = g(x, u) is equivalent to the nonlinear integral x equation u(x) = c + ∫a g[t, u(t)] dt. An integral equation is said to be singular when either one or both of the limits of integration become infinite or if K(x, t) becomes infinite for one or more points of the interval under discussion.



∞

cos (xt)u(t) dt and f(x) =

The Leibniz rule (see “Integral Calculus”) can be used to show the equivalence of the initial-value problem consisting of the secondorder differential equation d 2y/dx2 + A(x)(dy/dx) + B(x)y = f(x) together with the prescribed initial conditions y(a) = y0, y′(a) = y′0 to the integral equation. y(x) =

K(x, t)y(t) dt + F(x) x

and

(x − t)f(t) dt + [A(a)y + y′](x − a) + y x

0





0

0

a

This integral equation is a Volterra equation of the second kind. Thus the initial-value problem is equivalent to a Volterra integral equation of the second kind. Example d 2y/dx2 + x2(dy/dx) + xy = x, y(0) = 1, y′(0) x= 0. Here A(x) = x2, B(x) = x, f(x) = x. The equivalent integral equation is y(x) = ∫ 0 K(x, t)y(t) dt + F(x) x where K(x, t) = t(x − t) − t2 and F(x) = ∫ 0 (x − t)t dt + 1 = x3/6 + 1. Combining these x y(x) = ∫ 0 t[x − 2t]y(t) dt + x3/6 + 1. Eigenvalue problems can also be related. For example, the problem (d y/dx2) + λy = 0 with y(0) = 0, y(a) = 0 is equivalent to the integral a equation y(x) = λ ∫ 0 K(x, t)y(t) dt, where K(x, t) = (t/a)(a − x) when t < x and K(x, t) = (x/a)(a − t) when t > x. The differential equation may be recovered from the integral equation by differentiating the integral equation by using the Leibniz rule.







Method of Successive Approximations Consider the equation b y(x) = f(x) + λ ∫ a K(x, t)y(t) dt. In this method a unique solution is obtained in sequence form as follows: Substitute in the right-hand member of the equation y0(t) for y(t). Upon integration there results b y1(t) = f(x) + λ ∫ a K(x, t)y0(t) dt. Continue in like manner by replacing y0 by y1, y1 by y2, etc. A series of functions y0(x), y1(x), y2(x), . . . are obtained which satisfy the equations yn(x) = f(x) + λ

K(x, t)y b

(t) dt

n−1

a

Then yn(x) = f(x) + λ K(x, t)f(t) dt + λ K(x, t) ∫ a K(t, t1)f(t1) dt1 dt + b λ3 ∫ ab K(x, t) ∫ a K(t, t ) K(t1, t2)f(t2) dt2 dt1 dt + ⋅⋅⋅ + Rn, where Rn is the remainder, and max. y0 |Rn| ≤ |λn| Mn(b − a)n a≤x≤b 2 b ∫a

∫ ab b 1 ∫a



b



where M = maximum value of |K| in the rectangle a ≤ t ≤ b, a ≤ x ≤ b. If |λ|M(b − a) < 1, lim Rn = 0. Then yn(x) → y(x), which is the unique n→∞ solution. 1

Example Consider the equation y(x) = 1 + λ ∫ 0 (1 − 3xt)y(t) dt.

(1 − 3xt) dt + λ (1 − 3xt) (1 − 3tt ) dt dt + ⋅⋅⋅ 1

1

1

2

1

0

K(x, t) = (t − x)[B(t) − A′(t)] − A(t) F(x) =



L[1] 1/p p E(t) = L−1

= L−1
2 = L−1
= cosh
 λt 1 − λL[K(t)] 1 − λ/p p2 − λ

y(x) = 1 + λ

a

where

Example In a certain linear system, the effect E(t) due to a cause C = λE at time τ is a function only of the elapsed time t − τ. If the system has the activity level 1 at time t < 0, thet cause λE and effect (E) relation is given by the integralt equation E(t) = 1 + λ ∫ 0 K(t − τ)E(τ) dτ. Let K(t − τ) = t − τ. Then E(t) = 1 + λ ∫ 0 (t − τ)E(τ) dτ. By using the transform method

x

RELATION TO DIFFERENTIAL EQUATIONS



L[ f(x)] L[ f(x)] L[u(x)] =

, u(x) = L−1

1 − λL[K(x)] 1 − λL[K(x)] Equations of the type considered here occur quite frequently in practice in what can be called “cause-and-effect” systems.



u(t)
dt are both 0 0 x − t singular. The kernel of the first equation is cos (xt), and that of the second is (x − t)−1.

Example u(x) = x +

In general, the solution of integral equations is not easy, and a few exact and approximate methods are given here. Often numerical methods must be employed, as discussed in “Numerical Solution of Integral Equations.” Equations of Convolution Type The equation u(x) = f(x) + x λ ∫ 0 K(x − t)u(t) dt is a special case of the linear integral equation of the second kind of Volterra type. The integral part is the convolution integral discussed under “Integral Transforms (Operational Methods)”; so the solution can be accomplished by Laplace transforms; L[u(x)] = L[ f(x)] + λL[u(x)]L[K(x)] or



K(x, t)u(t) dt

u(x) = λ

3-43

0

1

0

3 1 1 3 λ4 1 3 = 1 + λ 1 −
x + λ2
+
λ3 1 −
x +
+
λ5 1 −
x + ⋅⋅⋅ 2 4 4 2 16 16 2







λ λ 3 = 1 +
+
+ ⋅⋅⋅ 1 + λ 1 −
x 4 16 2



2

 

4

1 + λ(1 − ⁄ x) =

, 1 − dλ2 32









|λ|<2

Example dy/dx = x2 x+ y. x0 = 0, y0 = 1. This problem is equivalent to the integral equation y = 1 + ∫ 0 (x2 + y) dx. Let the initial approximation for y be 1. Then x x3 y(1) = 1 + (x2 + 1) dx = 1 + x +
0 3



2

y(2) = 1 +

[x + y x

2

0

] dx = 1 +

(1)

x + 1 + x +
x3  dx x

3

2

0

x4 x2 x3 = 1 + x +
+
+
, etc. 2 3 12

3-44

MATHEMATICS

INTEGRAL TRANSFORMS (OPERATIONAL METHODS) REFERENCES:

63, 64, 71, 72, 97, 137, 217.

The term “operational method” implies a procedure of solving differential and difference equations by which the boundary or initial conditions are automatically satisfied in the course of the solution. The technique offers a very powerful tool in the applications of mathematics, but it is limited to linear problems. Most integral transforms are special cases of the equation g(s) = b ∫ a f(t)K(s, t) dt in which g(s) is said to be the transform of f(t) and K(s, t) is called the kernel of the transform. A tabulation of the more important kernels and the interval (a, b) of applicability follows. The first three transforms are considered here.

Properties of the Laplace Transform 1. The Laplace transform is a linear operator: L{af(t) + bg(t)} = aL{ f(t)} + bL{g(t)} for any constants a, b and any two functions f and g whose Laplace transforms exist. 2. The Laplace transform of a real-valued function is real for real s. If f(t) is a complex-valued function, f(t) = u(t) + iv(t), where u and v are real, then L{ f(t)} = L{u(t)} + iL{v(t)}. Thus L{u(t)} is the real part of L{ f(t)}, and L{v(t)} is the imaginary part of L{ f(t)}. 3. The Laplace transform of a function in the class Λ has derivatives of all orders, and L{tkf(t)} = (−1)kd kF(s)/ds k, k = 1, 2, 3, . . . .

Name of transform

(a, b)

Laplace

(0, ∞)

Fourier

(−∞, ∞)

Fourier cosine

(0, ∞)

Fourier sine

(0, ∞)

Mellin

(0, ∞)

Hankel

(0, ∞)

K(s, t) e−st 1
e−ist 
2π


π2 cos st 
π2 sin st



2as a e−st sin at dt =
, s > 0. By property 3,
= (s2 + a2)2 s2 + a2 e−st t sin at dt = L{t sin at}.

Example

0

0

Example By applying property 3 with f(t) = 1 and using the preceding results, we obtain k! dk 1 L{tk} = (−1)k
k
=
ds s sk + 1



provided Re s > 0; k = 1, 2, . . . . Similarly, we obtain



t

tJν (st), ν ≥ −a

The Laplace transform of a function f(t) is defined by F(s) = L{f(t)} = ∞ ∫ 0 e−stf(t) dt, where s is a complex variable. Note that the transform is an improper integral and therefore may not exist for all continuous functions and all values of s. We restrict consideration to those values of s and those functions f for which this improper integral converges. The function L[f(t)] = g(s) is called the direct transform, and L−1[g(s)] = f(t) is called the inverse transform. Both the direct and the inverse transforms are tabulated for many often-occurring functions. In general, 1  + i∞ st L−1[g(s)] =
e g(s) ds 2πi  − i∞ and to evaluate this integral requires a knowledge of complex variables, the theory of residues, and contour integration. A function is said to be piecewise continuous on an interval if it has only a finite number of finite (or jump) discontinuities. A function f on 0 < t < ∞ is said to be of exponential growth at infinity if there exist constants M and α such that |f(t)| ≤ Meαt for sufficiently large t. Sufficient Conditions for the Existence of Laplace Transform Suppose f is a function which is (1) piecewise continuous on every finite interval 0 < t < T, (2) of exponential growth at infinity, and (3) ∫ 0δ |f(t)| dt exist (finite) for every finite δ > 0. Then the Laplace transform of f exists for all complex numbers s with sufficiently large real part. Note that condition 3 is automatically satisfied if f is assumed to be piecewise continuous on every finite interval 0 ≤ t < T. The function f(t) = t−1/2 is not piecewise continuous on 0 ≤ t ≤ T but satisfies conditions 1 to 3. Let Λ denote the class of all functions on 0 < t < ∞ which satisfy conditions 1 to 3.



Example Let f(t) be the Heaviside step function at t = t0; i.e., f(t) = 0 for t ≤ t0, and f(t) = 1 for t > t0. Then ∞ T 1 e−st0 e−st dt = lim e−st dt = lim
(e−st0 − e−sT) =
L{ f(t)} = T→∞ t0 T→∞ s t0 s provided s > 0.



Example Let f(t) = eat, t ≥ 0, where a is a real number. Then L{eat} = ∞ ∫ 0 e−(s − a) dt = 1/(s − a), provided Re s > a.



dk 1 k! L{tkeat} = (−1)k
k
=

ds s − a (s − a)k + 1

s−1

LAPLACE TRANSFORM





∞

∞

4. Frequency-shift property (or, equivalently, the transform of an exponentially modulated function). If F(s) is the Laplace transform of a function f(t) in the class Λ, then for any constant a, L{eat f(t)} = F(s − a). 1 (s + a)

Example L{te−at} =
2 , s > 0. 5. Time-shift property. Let u(t − a) be the unit step function at t = a. Then L{f (t − a)u(t − a)} = e−asF(s). 6. Transform of a derivative. Let f be a differentiable function such that both f and f ′ belong to the class Λ. Then L{ f ′(t)} = sF(s) − f(0). 7. Transform of a higher-order derivative. Let f be a function which has continuous derivatives up to order n on (0, ∞), and suppose that f and its derivatives up to order n belong to the class Λ. Then L{ f ( j)(t)} = s jF(s) − s j − 1f(0) − s j − 2 f ′(0) − ⋅ ⋅ ⋅ − sf ( j − 2)(0) − f ( j − 1)(0) for j = 1, 2, . . . , k. Example L{ f″(t)} = s2L{ f(t)} − sf(0) − f′(0) L{ f″′(t)} = s3L{ f(t)} − s2f(0) − sf ′(0) − f″(0)

Example Solve y″ + y = 2et, y(0) = y′(0) = 2. L[y″] = −y′(0) − sy(0) + s2L[y] = −2 − 2s + s2L[y]. Thus 2 −2 − 2s + s2L[y] + L[y] = 2L[et] =
s−1 1 1 s 2s2 L[y] =

=
+
+
(s − 1)(s2 + 1) s − 1 s2 + 1 s2 + 1 Hence y = et + cos t + sin t.

A short table (Table 3-1) of very common Laplace transforms and inverse transforms follows. The references include more detailed ∞ tables. NOTE: Γ(n + 1) = ∫ 0 xne−x dx (gamma function); Jn(t) = Bessel function of the first kind of order n. t 1 1 0 8. L f(t) dt =
L[ f(t)] +
f(t) dt a s s a







1 s

s −1 a 

Example Find f(t) if L[ f(t)] =
2
2 2 Therefore f(t) =

sinh at
1a sinh at dt dt =
a1
− t a t

t

0

0

2





1 1 L
sinh at =
a s2 − a2

INTEGRAL TRANSFORMS (OPERATIONAL METHODS) TABLE 3-1

Laplace Transforms

f(t)

g(s)

f(t)

g(s)

1

1/s

e−at(1 − at)

s
2 (s + a)

tn, (n a + integer)

n!
sn + 1

t sin at
2a

s
(s2 + a2)2

tn, n ≠ + integer

Γ(n + 1)
sn + 1

1
2 sin at sinh at 2a

s
s4 + 4a4

cos at

s
s2 + a2

cos at cosh at

s2
s4 + 4a4

a
s2 + a2

sin at

2

s
s4 − a4

1
(sinh at + sin at) 2a

cosh at

s
s2 − a2

a (cosh at + cos at)

s2
s4 − a4

sinh at

a
s2 − a2

sin at
t

a tan−1
s

e−at

1
s+a

J0(at)

1
s2 +a2


e−bt cos at

s+b

(s + b)2 + a2

Jn(at) nan
t

1

s2 +a2 − sn


e−bt sin a

a

(s + b)2 + a2

) J0 (2
at

1
e−a/s s

3-45

I0 = zero-order Bessel function of an imaginary argument. For large u, In(u) ∼ eu/
2π u . Hence for large n, exp [−(
θ  −
n )2] y(n, θ) ∼


2π1/2(nθ)1/4 or for sufficiently large n, the peak concentration occurs near θ = n.

Other applications of Laplace transforms are given under “Differential Equations.” CONVOLUTION INTEGRAL The convolution integral (faltung) of two functions f(t), r(t) is x(t) = f(t)°r(t) = ∫ 0t f(τ)r(t − τ) dτ. Example t° sin t =

τ sin (t − τ) dτ = t − sin t. t

0

L[ f(t)]L[h(t)] = L[ f(t)°h(t)] Z-TRANSFORM

See Refs. 198, 218, and 256. The z-transform is useful when data is available at only discrete points. Let f *(t) = f(tk) be the value of f at the sample points tk = k ∆t,

k = 0, 1, 2, . . .

Then the function f *(t) is ∞

9.



f(t) L
= t

 ⋅⋅⋅

∞

f(t) L
= tk

g(s) ds

s

∞

s

f *(t) =  f(tk) δ(t − tk) k=0

∞

k

g(s)(ds)

Take the Laplace transform of this.

s



sin at Example L
= t



10.



∞

L[sin at] ds =

s

s

a ds s
= cot−1
s2 + a2 a

01

ta

∞0

k=0

∞

k=0

L[u(t − a)] = e−as/s

at t = a elsewhere

L[u′(t − a)] = e−as

12. L−1[e−asg(s)] = f(t − a)u(t − a) (second shift theorem). 13. If f(t) is periodic of period b, i.e., f(t + b) = f(t), then b 1 L[f(t)] =
e−stf(t) dt 1 − e−bs 0



k=0

g*(z) =  f(tk) z−k

The unit impulse function is δ(a) = u′(t − a) =

∞

For convenience, replace e−s∆t by z and call g*(z) the z-transform of f *(t).

The unit step function u(t − a) =

11.

∞

∞

g*(s) = L[ f *(t)] =  f(tk) e−stk =  f(tk) e−s∆tk

k integrals



Example The partial differential equations relating gas composition to position and time in a gas chromatograph are ∂y/∂n + ∂x/∂θ = 0, ∂y/∂n = x − y, where x = mx′, n = (kGaP/Gm)h, θ = (mkGaP/ρB)t and GM = molar velocity, y = mole fraction of the component in the gas phase, ρB = bulk density, h = distance from the entrance, P = pressure, kG = mass-transfer coefficient, and m = slope of the equilibrium line. These equations are equivalent to ∂2y/∂n ∂θ + ∂y/∂n + ∂y/∂θ = 0, where the boundary conditions considered here are y(0, θ) = 0 and x(n, 0) = y(n, 0) + (∂y/∂n) (n, 0) = δ(0) (see property 11). The problem is conveniently solved by using the Laplace transform of y with respect to n; write ∞ g(s, θ) = ∫ 0 e−nsy(n, θ) dn. Operating on the partial differential equation gives s(dg/dθ) − (∂y/∂θ) (0, θ) + sg − y(0, θ) + dg/dθ = 0 or (s + 1) (dg/dθ) + sg = (∂y/∂θ) (0, θ) + y(0, θ) = 0. The second boundary condition gives g(s, 0) + sg(s, 0) − y(0, 0) = 1 or g(s, 0) + sg(s, 0) = 1 (L[δ(0)] = 1). A solution of the ordinary differential equation for g consistent with this second condition is 1 g(s, θ) =
e−sθ /(s + 1) s+1

θ) where Inversion of this transform gives the solution y(n, θ) = e−(n + θ) I0(2
n

The z-transform is used in process control when the signals are at intervals of ∆t. A brief table (Table 3-2) is provided here. The z-transform can also be used to solve difference equations, just like the Laplace transform can be used to solve differential equations. TABLE 3-2

z-Transforms

f(k)

g*(z)

1(k)

1
1 − z−1

k ∆t

∆t z−1
(1 − z−1)2

(k ∆t)n − 1

∂n − 1 1 lim (−1)n − 1


a→0 ∂an − 1 1 − e−a∆tz−1

sin a k ∆t

z−1 sin a ∆t


(1 − 2 z−1 cos a ∆t + z−2)

cos a k ∆t

1 − z−1 cos a ∆t


(1 − 2 z−1 cos a ∆t + z−2)

e−ak∆t

1

1 − e−a∆tz−1

e−bk∆t cos a k ∆t

1 − z−1 e−b∆t cos a ∆t



1 − 2 z−1 e−b∆t cos a ∆t + z−2 e−2b∆t

1
e−bk∆t sin a k ∆t b

1 z−1 e−b∆t sin a ∆t




b 1 − 2 z−1 e−b∆t cos a ∆t + z−2 e−2b∆t





3-46

MATHEMATICS

Example The difference equation for y(k) is

and its inverse by

y(k) + a1 y(k − 1) + a2 y(k − 2) = b1u(k)

F c−1[g(s)] = f(t) =

Take the z-transform −2

The inverse transform must be found, usually from a table of inverse transforms.



The Fourier transform is given by ∞ 1 F[ f(t)] =
f(t)e−ist dt = g(s)   2 π
−∞ and its inverse by ∞ 1 g(s)eist dt = f(t) F−1[g(s)] =
π −∞
2





1−a≤t≤a has F[ f(t)] = 0 elsewhere a a a 2 sin sa eist dt + e−ist dt = 2 cos st dt =
0 0 0 s





a

−a



Example Find F[e−a|t|] by property 7. e−a|t| is even. So L[e−at] = 1/(s + a). Therefore, F[e−a|t|] = 1/(is + a) + 1/(−is + a) = 2a/(s2 + a2). Tables of this transform may be found in Higher Transcendental Functions, vols. I, II, and III, A. Erdelyi, et al., McGraw-Hill, New York, 1953–1955. FOURIER COSINE TRANSFORM The Fourier cosine transform is given by


π2

a

cos st dt =

0



 



4. Fc[t−2nf(t)] = (−1)n (d 2ng/ds2n). 5. Fc[t2n + 1f(t)] = (−1)n (d 2n + 1/ds2n + 1) Fs[ f(t)]. A short table (Table 3-3) of Fourier cosine transforms follows.

e−ist dt =

Properties of the Fourier Transform Let F [ f (t)] = g(s); F−1[g(s)] = f(t). 1. F[ f (n)(t)] = (is)nF[ f(t)]. 2. F[af(t) + bh(t)] = aF[ f(t)] + bF [h(t)]. 3. F[ f(−t)] = g(−s). 1 s 4. F[ f(at)] =
g
, a > 0. a a 5. F[e−iwt f(t)] = g(s + w). 6. F[ f(t + t1)] = eist1g(s). 7. F [ f(t)] = G(is) + G(−is) if f(t) = f(−t) ( f even) F[ f(t)] = G(is) − G(−is) if f(t) = −f(−t) ( f odd) where G(s) = L[f(t)]. This result allows the use of the Laplacetransform tables to obtain the Fourier transforms.

Fc[f(t)] = g(s) =


π2

1 s+b s−b Fc[ f(at) cos bt] =
g
+ g
, a, b > 0 2a a a

3.

In brief, the condition for the Fourier transform to exist is that ∞ ∫ -∞ |f(t)| dt < ∞, although certain functions may have a Fourier transform even if this is violated.



0
10

Properties of the Fourier Cosine Transform Fc[ f(t)] = g(s). 1. Fc[af(t) + bh(t)] = aFc[ f(t)] + bFc[h(t)]. 2. Fc[ f(at)] = (1/a)g(s/a).





Example Fc[ f(t)], f(t) = 2 sin as

π s

FOURIER TRANSFORM

Example The function f(t) =

g(s) cos st ds

The Fourier sine transform Fs is obtainable by replacing the cosine by the sine in these integrals.

u*(z) y*(z) =

1 + a1z−1 + a2z−2

Then

∞

0

(1 + a1z + a2z ) y*(z) = u*(z) −1


π2

∞

f(t) cos st dt

0

TABLE 3-3

Fourier Transforms g(s)

2/π

f(t) t 2−t 0 t−1/2 0 (t − a)−1/2 (t2 + a2)−1 e−at

0
1
2 [2 cos s − 1 − cos 2s] s π1/2(2s)−1/2

0
π1/2(2s)−1/2 [cos as − sin as]

a>0

a πa−1e−as

e−at

2

a>0

a
s2 + a2 2 aπ1/2a−1/2e−s /4a

sin at
t

a>0



π/2 π/4 0

sa

Example The temperature in the semi-infinite rod 0 ≤ x < ∞ is determined by the differential equation ∂ ∂t = k(∂2 ∂x2) and the condition = 0 when t = 0, x ≥ 0; ∂ ∂x = −µ = constant when x = 0, t > 0. By using the Fourier cosine transform a solution may be found as 2µ

(x, t) =
π



∞

0

cos px
(1 − e−kp2t ) dp. p

MATRIX ALGEBRA AND MATRIX COMPUTATIONS REFERENCES: General (textbooks that cover at an introductory level a variety of topics that constitute a core of numerical methods for practicing engineers), 2, 3, 4, 22, 56, 59, 70, 77, 133, 135, 143, 150, 155, 219. Numerical solution of nonlinear equations, 153, 171, 237, 302. Numerical solution of ordinary differential equations, 76, 117, 127, 185, 257. Numerical solution of integral equa-

tions, 23, 26, 129, 162. Numerical solution of partial differential equations, 11, 76, 127, 133, 155, 210, 251, 286, 287, 213, 233, 253. Spline functions and applications, 38, 56, 70, 230. Finite elements and applications, 5, 29, 83, 130, 164, 210, 241, 281, 287, 303, 304. Fast Fourier transforms, 47, 56, 135, 238. Software, 187, 231.

MATRIX ALGEBRA AND MATRIX COMPUTATIONS MATRIX ALGEBRA Matrices A rectangular array of rows and n columns a11 a A = (aij) = 21 ⯗ am1



mn quantities, arranged in m



… …

a1n a2n

…

amn

is called a matrix. The elements aij may be real or complex. The notation aij means the element in the ith row and jth column, i is called the row index, j the column index. If m = n the matrix is said to be square and of order n. A matrix, even if it is square, does not have a numerical value, as a determinant does. However, if the matrix A is square, a determinant can be formed which has the same elements as the matrix A. This is called the determinant of the matrix and is written det (A) or |A|. If A is square and det (A) ≠ 0, A is said to be nonsingular; if det (A) = 0, A is said to be singular. A matrix A has rank r if and only if it has a nonvanishing determinant of order r and no nonvanishing determinant of order > r. Equality of Matrices Let A = (aij), B = (bij). Two matrices A and B are equal (=) if and only if they are identical; that is, they have the same number of rows and the same number of columns and equal corresponding elements (aij = bij for all i and j). Addition and Subtraction The operations of addition (+) and subtraction (−) of two or more matrices are possible if and only if they have the same number of rows and columns. Thus A  B = (aij  bij); i.e., addition and subtraction are of corresponding elements. Transposition The matrix obtained from A by interchanging the rows and columns of A is called the transpose of A, written A′ or AT.



1 2 Example A = 12 31 46 , AT = 3 1 4 6





Note that (AT )T = A.

Multiplication Let A = (aij), i = 1, . . . , m1; j = 1, . . . , m2. B = (bij), i = 1, . . . , n1, j = 1, . . . , n2. The product AB is defined if and only if the number of columns of A (m2) equals the number of rows of B(n1), i.e., n1 = m2. For two such matrices the product P = AB is defined by summing the element by element products of a row of A by a column of B. This is the row by column rule. Thus n1

pij =  aikbkj k=1

The resulting matrix has m1 rows and n2 columns. Example



3 1 5

2 1 4

0 −2

1 0

5 1



6 = 3



−4 −2 −8

3 1 5

17 6 29

24 9 42



It is helpful to remember that the element pij is formed from the ith row of the first matrix and the jth column of the second matrix. The matrix product is not commutative. That is, AB ≠ BA in general. Inverse of a Matrix A square matrix A is said to have an inverse if there exists a matrix B such that AB = BA = I, where I is the identity matrix of order n. 1 0.... 0 0 1.. ⯗ 1 0 0 ....0 1





−1

The inverse B is a square matrix of the order of A, designated by A . Thus AA−1 = A−1A = I. A square matrix A has an inverse if and only if A is nonsingular. Certain relations are important: (1)

(AB)−1 = B−1A−1

(2)

(AB)T = BTAT

(3)

(A−1)T = (AT )−1

(4)

(ABC)−1 = C−1B−1A−1

3-47

Scalar Multiplication Let c be any real or complex number. Then cA = (caij). Adjugate Matrix of a Matrix Let Aij denote the cofactor of the element aij in the determinant of the matrix A. The matrix BT where B = (Aij) is called the adjugate matrix of A written adj A = BT. The elements bij are calculated by taking the matrix A, deleting the ith row and jth column, and calculating the determinant of the remaining matrix times (−1)i + j. Then A−1 = adj A/ |A|. This definition may be used to calculate A−1. However, it is very laborious and the inversion is usually accomplished by numerical techniques shown under “Numerical Analysis and Approximate Methods.”



3

0

3

6

Example Let A = −1 2

adj A = BT =



0 6 −12



−1 1 3

Form B = (A i j), B =



0 −6 2

6 12 −2

−12 −18 6





−6 2 12 −2 ; |A| = 12 −18 6





−a 1 −3 ⁄ 2

0 adj A A−1 =
= a |A| −1

1 ⁄6 −1⁄6 a

Linear Equations in Matrix Form Every set of n nonhomogeneous linear equations in n unknowns a11 x1 + a12 x2 + ⋅⋅⋅ + a1n xn = b1 a21 x1 + a22 x2 + ⋅⋅⋅ + a2n xn = b2   an1 x1 + an2 x2 + ⋅⋅⋅ + ann xn = bn can be written in matrix form as AX = B, where A = (aij), X T = [x1 ⋅⋅⋅ xn], and BT = [b1 ⋅⋅⋅ bn]. The solution for the unknowns is X = A−1B. Special Square Matrices 1. A triangular matrix is a matrix all of whose elements above or below the main diagonal (set of elements a11, . . . , ann) are zero. If A is triangular, det (A) = a11. a22 . . . ann. 2. A diagonal matrix is one such that all elements both above and below the main diagonal are zero (i.e., aij = 0 for all i ≠ j). If all diagonal elements are equal, the matrix is called scalar. If A is diagonal, A = (aij), A−1 = (1/aij). 3. If aij = aji for all i and j (i.e., A = AT ), the matrix is symmetric. 4. If aij = −aji for i ≠ j but the aij are not all zero, the matrix is skew. 5. If aij = −aji for all i and j (i.e., aii = 0), the matrix is skew symmetric. 6. If AT = A−1, the matrix A is orthogonal. 7. If the matrix A* = (aij)T, aij = complex conjugate of aij, A* is the hermitian conjugate of A. 8. If A = A−1, A is involutory. 9. If A = A*, A is hermitian. 10. If A = −A*, A is skew hermitian. 11. If A−1 = A*, A is unitary. If A is any matrix, then AAT and ATA are square symmetric matrices, usually of different order.



5

Example Let A = 3



35 AAT = 22 8

2

1 3 4 1 −2 0



22 51 3

8 3 , 9



3 4 1 5

2 −2 0 1

38 13 18 17

13 21 7 18

18 7 10 5

5 0 1 5 , AT = 3 1 0



ATA =



 

17 18 5 26

Using a program such as MATLAB, these are easily calculated.

Matrix Calculus Differentiation Let the elements of A = [aij(t)] be differentiable dA daij(t) functions of t. Then
=
. dt dt





3-48

MATHEMATICS







interchange two rows of A to produce a (nonzero) pivot, such as if we start with 0 2 A= 1 6



sin t cos t dA cos t −sin t Example A = −cos , = . t sin t
sin t cos t dt



Integration The integral ∫ A dt = [∫ aij(t) dt].









2 t 2t Example A = t2 e2t , ∫ A dt = tt3/2 . /3 et

The matrix B = A − λI is called the characteristic (eigen) matrix of A. Here A is square of order n, λ is a scalar parameter, and I is the n × n identity. det B = det (A − λI) = 0 is the characteristic (eigen) equation for A. The characteristic equation is always of the same degree as the order of A. The roots of the characteristic equation are called the eigenvalues of A.



 

Example A = 13 28 , B = 13 28 − λ0 λ0 = 1−λ3 8−λ2



is the characteristic matrix and f(λ) = det (B) = det (A − λI) = (1 − λ)(8 − λ) − 6 = 2 − 9λ + λ2 = 0 is the characteristic equation. The eigenvalues of A are the roots of λ2 − 9λ + 2 = 0, which are (9 
73)/2.

A nonzero matrix Xi, which has one column and n rows, called a column vector satisfying the equation (A − λI)Xi = 0 and associated with the ith characteristic root λi is called an eigenvector. Vector and Matrix Norms To carry out error analysis for approximate and iterative methods for the solutions of linear systems, one needs notions for vectors in Rn and for matrices that are analogous to the notion of length of a geometric vector. Let Rn denote the set of all vectors with n components, x = (x1, . . . , xn). In dealing with matrices it is convenient to treat vectors in Rn as columns, and so x = (x1, . . . , xn)T; however, we shall here write them simply as row vectors. A norm on Rn is a real-valued function f defined on Rn with the following properties: 1. f(x) ≥ 0 for all x
Rn. 2. f(x) = 0 if and only if x = (0, 0, . . . , 0). 3. f(ax) = |a| f(x) for all real numbers a and x
Rn. 4. f(x + y) f(x) + f(y) for all x, y
Rn. The usual notation for a norm is f(x) =  x. The norm of a matrix is κ(A)  A  A−1 where

If A is a square matrix and if principal submatrices of A are all nonsingular, then we may choose P as the identity in the preceding factorization and obtain A = LU. This factorization is unique if L is normalized (as assumed previously), so that it has unit elements on the main diagonal. Solution of Ax = b by Using LU Factorization Suppose that the indicated system is compatible and that A = LU (the case PA = LU is similarly handled and amounts to rearranging the equations). Let z = Ux. Then Ax = LUx = b implies that Lz = b. Thus to solve Ax = b we first solve Lz = b for z and then solve Ux = z for x. This procedure does not require that A be invertible and can be used to determine all solutions of a compatible system Ax = b. Note that the systems Lz = b and Ux = z are both in triangular forms and thus can be easily solved. The LU decomposition is essentially a Gaussian elimination, arranged for maximum efficiency (Ref. 112). The chief reason for doing an LU decomposition is that it takes fewer multiplications than would be needed to find an inverse. Also, once the LU decomposition has been found, it is possible to solve for multiple right-hand sides with little increase in work. The multiplication count for an n × n matrix and m right-hand sides is 1 1 operation count =
n3 −
n + mn2 3 3 If an inverse is desired, it can be calculated by solving for the LU decomposition and then solving n problems with right-hand sides consisting of all zeroes except one entry. Thus 4n2/3 − n/3 multiplications are required for the inverse. The determinant is given by n

Det A =  aii(i) i=1

where aii(i) are the diagonal elements obtained in the LU decomposition. A tridiagonal matrix is one in which the only nonzero entries lie on the main diagonal and the diagonal just above and just below the main diagonal. The set of equations can be written as ai xi − 1 + bi xi + ci xi + 1 = di The LU decomposition is b1 = b1 for k=2,n do ak ak a′k =
, b′k = bk −
ck − 1 b′k − 1 b′k − 1 enddo d′1 = d1 for k=2,n do d′k = dk − a′k d′k − 1 enddo xn = d n′ / b′n for k=n−1,1 do d k′ − ck xk + 1 xk =

b′k enddo

n  A x A = supx ≠ 0
= maxk  |ajk| j=1  x

The norm is useful when doing numerical calculations. If the computer’s floating-point precision is 10−6, then κ = 106 indicates an illconditioned matrix. If the floating-point precision is 10−12 (double precision), then a matrix with κ = 1012 may be ill-conditioned. Two other measures are useful and are more easily calculated: (k) |det A| maxk |akk | Ratio =

V =

, αi = (a 2i1 + a 2i2 + . . . a 2in)1/2 (k) , mink |akk | α1 α2 . . . αn where akk(k) are the diagonal elements of the LU decomposition. MATRIX COMPUTATIONS The principal topics in linear algebra involve systems of linear equations, matrices, vector spaces, linear transformations, eigenvalues and eigenvectors, and least-squares problems. The calculations are routinely done on a computer. LU Factorization of a Matrix To every m × n matrix A there exists a permutation matrix P, a lower triangular matrix L with unit diagonal elements, and an m × n (upper triangular) echelon matrix U such that PA = LU. The Gauss elimination is in essence an algorithm to determine U, P, and L. The permutation matrix P may be needed since it may be necessary in carrying out the Gauss elimination to

The operation count for an n × n matrix with m right-hand sides is 2(n − 1) + m(3n − 2) If

|bi| > |ai| + |ci |

no pivoting is necessary, and this is true for many boundary-value problems and partial-differential equations. Sparse matrices are ones in which the majority of the elements are

NUMERICAL ANALYSIS AND APPROXIMATE METHODS zero. If the structure of the matrix is exploited, the solution time on a computer is greatly reduced. See Refs. 27, 55, 95, 96, 101, and 246. The conjugate gradient method is one method for solving sparse matrix problems, since it only involves multiplication of a matrix times a vector. Thus the sparseness of the matrix is easy to exploit. The conjugate gradient method is an iterative method that converges for sure in n iterations where the matrix is an n × n matrix. See Refs. 142 and 206. The singular value decomposition is useful when the matrix is singular or nearly singular (see Ref. 231). Matrix methods, in particular finding the rank of the matrix, can be used to find the number of independent reactions in a reaction set. If the stoichiometric numbers for the reactions and molecules are put in the form of a matrix, the rank of the matrix gives the number of independent reactions (see Ref. 13). Pivoting in Gauss Elimination It might seem that the Gauss elimination completely disposes of the problem of finding solutions of linear systems, and theoretically it does. In practice, however, things are not so simple. Example Assume three-decimal floating arithmetic (i.e., only the three most significant digits of any number are retained), and solve the following system by Gauss elimination: 0.000100x1 + 1.00x2 = 1.00 1.00x1 + 1.00x2 = 2.00 We obtain 0.100 × 10−3x1 + 0.100 × 101x2 = 0.100 × 101 −0.100 × 105x2 = −0.100 × 105 so that x2 = 1.00 and x1 = 0.00.

3-49

We check our solution by computing the residual vector r = b − Ax: r1 = 0.100 × 101 − 0.100 × 10−3x1 − 0.100 × 101x2 = 0.00 r2 = 0.200 × 101 − 0.100 × 101x1 − 0.100 × 101x2 = 0.100 × 101 The fact that r2 = 1 indicates that our “solution” is not very good. Indeed the exact solution of the system is x1 = 1.00010 and x2 = 0.99990, so the result computed by Gauss elimination is pretty bad. Now reverse the order of the equations (that is, pivot) and solve 0.100 × 101x1 + 0.100 × 101x2 = 0.200 × 101 0.100 × 101x2 = 0.100 × 101 so that x2 = 1.00 and x1 = 1.00. In this case the residual vector is r1 = 0.00 and r2 = 0.100 × 10−3, a considerable improvement over the previous result. In fact, the solution is as good as one could hope for by using three-digit arithmetic.

The moral of the preceding example is that the order of equations can make a large difference in how good an answer is obtained. It should be clear that the poor results in the first case are caused by having the large multiplier (0.100 × 101)/(0.100 × 10−3), which resulted from dividing by a relatively small a11. It is not enough just to avoid zero “pivots”; one must also avoid using pivots that are relatively small. This magnification of errors can be reduced if we arrange that the pivot at any stage is larger in magnitude than any remaining element in the column. If this is done, the multipliers will then be less than or equal to 1 in magnitude. Gauss elimination modified in this manner is called pivotal condensation or partial pivoting. This is routinely done by computer programs.

NUMERICAL APPROXIMATIONS TO SOME EXPRESSIONS APPROXIMATION IDENTITIES For the following relationships the sign  means approximately equal to, when X is small: Approximation 1
1X 1X 1+Y
1+YX 1X

Approximation X  X1

1 2 −n

(1  X)  1  nX

Approximation

Approximation

(1  X)n  1  nX

X (1  X)−1/2  1 
2

(a  X)2  a2  2aX

ez  1 + X

sin X  X(X rad)

tan X  X

2Y + X (Y  + X) 

Y 2

X2 X 2 +  X2  Y +

small
Y 2Y Y





NUMERICAL ANALYSIS AND APPROXIMATE METHODS REFERENCES: General (textbooks that cover at an introductory level a variety of topics that constitute a core of numerical methods for practicing engineers), 2, 3, 4, 22, 56, 59, 70, 77, 133, 135, 143, 150, 155, 219. Numerical solution of nonlinear equations, 153, 171, 237, 302. Numerical solution of ordinary differential equations, 76, 117, 127, 185, 257. Numerical solution of integral equations, 23, 26, 129, 162. Numerical solution of partial differential equations, 11, 76, 127, 133, 155, 210, 251, 286, 287, 213, 233, 253. Spline functions and applications, 38, 56, 70, 230. Finite elements and applications, 5, 29, 83, 130, 164, 210, 241, 281, 287, 303, 304. Fast Fourier transforms, 47, 56, 135, 238. Software, 187, 231.

INTRODUCTION The goal of approximate and numerical methods is to provide convenient techniques for obtaining useful information from mathematical formulations of physical problems. Often this mathematical statement is not solvable by analytical means. Or perhaps analytic solutions are available but in a form that is inconvenient for direct interpretation

numerically. In the first case it is necessary either to attempt to approximate the problem satisfactorily by one which will be amenable to analysis, to obtain an approximate solution to the original problem by numerical means, or to use the two techniques in combination. Numerical techniques therefore do not yield exact results in the sense of the mathematician. Since most numerical calculations are inexact, the concept of error is an important feature. The error associated with an approximate value is defined as True value = approximate value + error The four sources of error are as follows: 1. Gross errors. These result from unpredictable human, mechanical, or electrical mistakes. 2. Round-off errors. These are the consequence of using a number specified by m correct digits to approximate a number which requires more than m digits for its exact specification. For example, approximate the irrational number
2 by 1.414. Such errors are often

3-50

MATHEMATICS

present in experimental data, in which case they may be called inherent errors, due either to empiricism or to the fact that the computer dictates the number of digits. Such errors may be especially damaging in areas such as matrix inversion or the numerical solution of partial differential equations when the number of algebraic operations is extremely large. 3. Truncation errors. These errors arise from the substitution of a finite number of steps for an infinite sequence of steps which would yield the exact result. To illustrate this error consider the infinite series for e−x ⋅ e−x = 1 − x + x2/2 − x3/6 + ET(x), where ET is the truncation error, ET = (1/24)e−εx4, 0 < ε < x. If x is positive, ε is also positive. Hence e−ε < 1. The approximation e−x ≈ 1 − x + x2/2 − x3/6 is in error by a positive amount smaller than (1/24)x4. 4. Inherited errors. These arise as a result of errors occurring in the previous steps of the computational algorithm. The study of errors in a computation is related to the theory of probability. In what follows a relation for the error will be given in certain instances. NUMERICAL SOLUTION OF LINEAR EQUATIONS See the section entitled “Matrix Algebra and Matrix Computation.” NUMERICAL SOLUTION OF NONLINEAR EQUATIONS IN ONE VARIABLE Special Methods for Polynomials Consider a polynomial equation of degree n: P(x) = a0 xn + a1xn − 1 + a2 xn − 2 + ⋅⋅⋅ + an − 1x + an = 0

(3-71)

with real coefficients. P(x) has exactly n roots, which may be real or complex. If all the coefficients of P(x) are integers, then any rational root, say, r/s (r, s integers, having no common divisors) of P(x), must be such that r is an integral divisor of an and s is an integral divisor of a0. Further, any polynomial with rational coefficients may be converted into one with integral coefficients by multiplying by the lowest common multiple of the denominators of the coefficients. Example 3x4 − 5⁄ 3 x2 + 1⁄5 x − 2 = 0. The lowest common multiple of the denominator is 15. Thus multiplying by 15 (which does not change the roots) gives 45x4 − 25x2 + 3x − 30 = 0. The only possible rational roots r/s are such that r may have the values 30, 15, 10, 6, 5, 3, 2, 1. s may have the values 45, 15, 9, 5, 3, 1. The possible rational roots may then be formed from all possible quotients, having no common factor. In addition to these results, one can obtain an upper and lower bound for the real roots by the following device: If a0 > 0 in Eq. (3-71) and if in Eq. (3-71) the first negative coefficient is preceded by k coefficients which are positive or zero, and if G is the greatest of the absolute values of the negative coefficients, then each real root is less k than 1 +
 G/a 0. Example P(x) = x5 + 3x4 − 7x2 − 40x + 2 = 0. Here a03 = 1, G = 40, and k = 3

Example f(x) = x4 − 13x2 + 4x − 2 = 0 has three changes in sign; therefore, there are either three or one positive roots. f(−x) = x4 − 13x2 − 4x − 2 has one change in sign. Therefore, there is one negative root. General Methods for Nonlinear Equations in One Variable Successive Substitutions Let f(x) = 0 be the nonlinear equation to be solved. If this is rewritten as x = F(x), then an iterative scheme can be set up in the form xk + 1 = F(xk). To start the iteration an initial guess must be obtained graphically or otherwise. The convergence or divergence of the procedure depends upon the method of writing x = F(x), of which there will usually be several forms. However, if a is a root of f(x) = 0, and if |F′(a)| < 1, then for any initial approximation sufficiently close to a, the method converges to a. This process is called first order because the error in xk + 1 is proportional to the first power of the error in xk for large k. Example f(x) = x3 − x − 1 = 0. A rough plot shows a real root of approximately 1.3. The equation can be written in the form x = F(x) in several ways such as x = x3 − 1, x = 1/(x2 − 1), and x = (1 + x)1/3. In the first case F′(x) = 3x2 = 5.07 at x = 1.3, in the second F(1.3) = −5.46, and only in the third case is F′(1.3) < 1. Hence only the third iterative process has a chance to converge. This is illustrated in the following table. Step k

x = x3 − 1

x = 1/(x2 − 1)

x = (1 + x)1/3

0 1 2 3 4

1.3 1.197 0.7150 −0.6345

1.3 1.4493 0.9088 −5.742

1.3 1.32 1.3238 1.3245 1.3247

Another way of writing the equation is xk + 1 = xk + β f(xk). The choice of β is made such that |1 + β df/dx(a)| < 1. Convergence is guaranteed by the theorem given for simultaneous equations.

Methods of Perturbation Let f(x) = 0 be the equation. In general, the iterative relation is xk + 1 = xk − [ f(xk)/ak] where the iteration begins with x0 as an initial approximation and αk as some functional. Newton-Raphson Procedure This variant chooses αk = f ′(xk) where f ′ = df/dx and geometrically consists of replacing the graph of f(x) by the tangent line at x = xk in each successive step. If f′(x) and f ″(x) have the same sign throughout an interval a ≤ x ≤ b containing the solution, with f(a), f(b) of opposite signs, then the process converges starting from any x0 in the interval a ≤ x ≤ b. The process is second order. (0.5)x − 0.5 0.3

Example f(x) = x − 1 +

f′ (x) = 1 − 2.3105[0.5]x An approximate root (obtained graphically) is 2.

since we must supply 0 as the coefficient for x . Thus 1 +
40 ≈ 4.42 is an upper bound for the real roots.

Step k

xk

f(xk)

f′(xk)

A lower bound to the real roots may be found by applying the criterion to the equation P(−x).

0 1 2

2 1.6054 1.4632

0.1667 0.0342 0.0055

0.4224 0.2407 0.1620

3

Example P(−x) = −x5 + 3x4 − 7x2 + 40x + 2 = 0, which is equivalent to x5 − 3x4 + 7x2 − 40x − 2 = 0 since a0 must be +. Then a0 = 1, G = 40, and k = 1. Hence −(1 + 40) = −41 is a lower bound. Thus all real roots −41 < r < 4.42. One last result is helpful in getting an estimate of how many positive and negative real roots there are. Descartes Rule The number of positive real roots of a polynomial with real coefficients is either equal to the number of changes in sign v or is less than v by a positive even integer. The number of negative roots of f(x) is either equal to the number of variations of sign of f(−x) or is less than this by a positive even integer.

Method of False Position This variant is commenced by finding x0 and x1 such that f(x0), f(x1) are of opposite signs. Then α1 = slope of secant line joining [x0, f(x0)] and [x1, f(x1)] so that x1 − x0 x2 = x1 −

f(x1) f(x1) − f(x0) In each following step αk is the slope of the line joining [xk, f(xk)] to the most recently determined point where f(xj) has the opposite sign from that of f(xk). This method is of first order. If one uses the most recently determined point (regardless of sign), the method is a secant method.

NUMERICAL ANALYSIS AND APPROXIMATE METHODS Method of Wegstein This is a variant of the method of successive substitutions which forces and/or accelerates convergence. The iterative procedure xk + 1 = F(xk) is revised by setting ˆxk + 1 = F(xk) and then taking xk + 1 = qxk + (1 − q)ˆxk + 1, where q is a suitably chosen number which may be taken as constant throughout or may be adjusted at each step. Wegstein found that suitable q’s are: Behavior of successive substitution process

Range of optimum q

Oscillatory convergence Oscillatory divergence Monotonic convergence Monotonic divergence

0
At each step q may be calculated to give a locally optimum value by setting xˆ k + 1 − xˆ k q =

xˆ k + 1 − 2 xˆ k + 1 + xˆ k− 1 The Wegstein method is a secant method applied to g(x)  x − F(x). Numerical Solution of Simultaneous Nonlinear Equations The techniques illustrated here will be demonstrated for two simultaneous equations f(x, y) = 0, g(x, y) = 0. They immediately generalize to more than two simultaneous equations. Method of Successive Substitutions Write a system of equations as αi = fi(α),

or α = f(α)

The following theorem guarantees convergence. Let α be the solution to αi = fi(α). Assume that given h > 0, there exists a number 0 < µ < 1 such that n ∂fi
≤ µ for |xi − αi| < h, i = 1, . . . , n  ∂x j=1 j

 

x Then

k+1 i

= fi(x ik)

x ik → αi

Step k

xk

yk

f(xk, yk)

g(xk, yk)

0 1 2 3

0.4 0.43673 0.42672 0.42925

0.3 0.24184 0.25573 0.24943

−0.26 0.078 −0.0170 0.0077

0.07 0.0175 −0.007 0.0010

Method of Continuity (Homotopy) In the case of n equations in n unknowns, when n is large, determining the approximate solution may involve considerable effort. In such a case the method of continuity is admirably suited for use on digital computers. It consists basically of the introduction of an extra variable into the n equations fi(x1, x2, . . . , xn) = 0

i = 1, . . . , n

(3-72)

and replacing them by fi(x1, x2, . . . , xn, λ) = 0

i = 1, . . . , n

(3-73)

where λ is introduced in such a way that the functions (3-73) depend in a simple way upon λ and reduce to an easily solvable system for λ = 0 and to the original equations (3-72) for λ = 1. A system of ordinary differential equations, with independent variable λ, is then constructed by differentiating Eqs. (3-73) with respect to λ. There results n ∂fi dxf ∂fi (3-74)

+
=0  ∂λ j = 1 ∂xj dλ where x1, . . . , xn are considered as functions of λ. Equations (3-74) are integrated, with initial conditions obtained from Eqs. (3-73) with λ = 0, from λ = 0 to λ = 1. If the solution can be continued to λ = 1, the values of x1, . . . , xn for λ = 1 will be a solution of the original equations. If the integration becomes infinite, the parameter λ must be introduced in a different fashion. Integration of the differential equations (which are usually nonlinear in λ) may be accomplished by using techniques described under “Numerical Solution of Ordinary Differential Equations.” Other Methods Other methods can be found in the literature. See Ref. 66. INTERPOLATION AND FINITE DIFFERENCES

as k increases (see Ref. 106). Newton-Raphson Method To solve the set of equations Fi(x1, x2, . . . , xn) = 0,

or Fi({xj}) = 0,

or Fi(x) = 0

one uses a truncated Taylor series to give n ∂F 0 = Fi({x k}) + 
i j = 1 ∂xj



xk

(x jk + 1 − x jk)

Linear Interpolation If a function f(x) is approximately linear in a certain range, then the ratio f(x1) − f(x0)

= f [x0, x1] x1 − x0 is approximately independent of x0, x1 in the range. The linear approximation to the function f(x), x0 < x < x1 then leads to the interpolation formula f(x) ≈ f(x0) + (x − x0)f [x0, x1] x − x0 ≈ f(x0) +
[ f(x1) − f(x0)] x1 − x0 1 ≈
[(x1 − x) f(x0) − (x0 − x)f(x1)] x1 − x0

Thus one solves iteratively from one point to another. n

A

k ij

(x

k+1 j

− x ) = −Fi({x }) k

k j

j=1

where

3-51

∂F A kij =
i ∂xj



xk

This method requires solution of sets of linear equations until the functions are zero to some tolerance or the changes of the solution between iterations is small enough. Convergence is guaranteed provided the norm of the matrix A is bounded, F(x) is bounded for the initial guess, and the second derivative of F(x) with respect to all variables is bounded. See Refs. 106 and 155. Example f(x, y) = 4x2 + 6x − 4xy + 2y2 − 3 g(x, y) = 2x2 − 4xy + y2

Example Find cosh 0.83 by linear interpolation given cosh 0.8 and cosh 0.9. xi

f(xi)

xi − 0.83

0.8 0.9

1.33743 1.43309

−0.03 +0.07

f(0.83) ≈ 1/0.10[(0.07)(1.33743) − (−0.03)(1.43309)] f(0.83) ≈ 1.36613

By plotting one of the approximate roots is found to be x0 = 0.4, y0 = 0.3. At this point there results ∂f/∂x = 8, ∂f/∂y = −0.4, ∂g/∂x = 0.4, and ∂g/∂y = −1.

Since the true five-place value is 1.36468, it is seen that here linear interpolation gives three significant figures.

8(xk + 1 − xk) − 0.4( yk + 1 − yk) = +0.26

Divided Differences of Higher Order and Higher-Order Interpolation The first-order divided difference f [x0, x1] was defined previously. Divided differences of second and higher order are defined iteratively by

0.4(xk + 1 − xk) − 1( yk + 1 − yk) = −0.07 The first few iteration steps are as follows:

3-52

MATHEMATICS f [x1, x2] − f[x0, x1] f [x0, x1, x2] =

x2 − x0 ⯗ f [x1, . . . , xk] − f[x0, x1, . . . , xk − 1] f [x0, x1, . . . , xk] =



xk − x0

and a convenient form for computational purposes is k′ f(xj) f [x0, x1, . . . , xk] = 


j = 0 (xj − x0)(xj − x1) ⋅⋅⋅ (xj − xk) for any k ≥ 0, where the ′ means that the term (xj − xj) is omitted in the denominator. For example, f(x0) f(x1) f(x2) f [x0, x1, x2] =

+

+

(x0 − x1)(x0 − x2) (x1 − x0)(x1 − x2) (x2 − x0)(x2 − x1) If the accuracy afforded by a linear approximation is inadequate, a generally more accurate result may be based upon the assumption that f(x) may be approximated by a polynomial of degree 2 or higher over certain ranges. This assumption leads to Newton’s fundamental interpolation formula with divided differences f(x) ≈ f(x0) + (x − x0) f[x0, x1] + (x − x0)(x − x1) f [x0, x1, x2] + ⋅⋅⋅ + (x − x0)(x − x1) ⋅⋅⋅ (x − xn − 1) f [x0, x1, . . . , xn] + En(x) 1 En(x) = error =
f n + 1(ε)π(x) (n + 1)! where minimum (x0, . . . , x) < ε < maximum (x0, x1, . . . , xn, x) and π(x) = (x − x0)(x − x1) ⋅⋅⋅ (x − xn). In order to use the previous equation most effectively one may first form a divided-difference table. For example, for third-order interpolation the difference table is

where

and the finite interpolation formula through the points y0, y1, . . . , yn is written as follows: α (α − 1) α (α − 1) ⋅⋅⋅ (α − n + 1) yα = y0 + α ∆y0 +
∆2y0 + ⋅⋅⋅ +


∆ny0 2! n! Keeping only the first two terms gives a straight line through (x0, y0)–(x1, y1); keeping the first three terms gives a quadratic function of position going through those points plus (x2, y2). The value α = 0 gives x = x0; α = 1 gives x = x1, and so on. Equally Spaced Backward Differences Backward differences are defined by ∇yn = yn − yn − 1 ∇ 2yn = ∇yn − ∇yn − 1 = yn − 2 yn − 1 + yn − 2 The interpolation polynomial of order n through the points y0, y−1, . . . , y−n is α (α + 1) α (α + 1) ⋅⋅⋅ (α + n − 1) yα = y0 + α ∇y0 +
∇2y0 + ⋅⋅⋅ +


∇ny0 2! n! The value of α = 0 gives x = x0; α = −1 gives x = x−1, and so on. Alternatively, the interpolation polynomial of order n through the points y1, y0, y−1, . . . , y−n is α (α − 1) yα = y1 + (α − 1) ∇y1 +
∇ 2y1 2! (α − 1) α (α + 1) ⋅⋅⋅ (α + n − 2) + ⋅⋅⋅ +



∇ny1 n! Now α = 1 gives x = x1; α = 0 gives x = x0. Central Differences The central difference denoted by h h δf(x) = f x +
− f x −
2 2



 











h h δ2f(x) = δn − 1 f x +
− δn − 1 f x −
2 2 is useful for calculating at the interior points of tabulated data. Lagrange Interpolation Formulas A global polynomial is defined over the entire region of space m

where each entry is given by taking the difference between diagonally adjacent entries to the left, divided by the abscissas corresponding to the ordinates intercepted by the diagonals passing through the calculated entry. Equally Spaced Forward Differences If the ordinates are equally spaced, i.e., xj − xj − 1 = ∆x for all j, then the first differences are denoted by ∆f(x0) = f(x1) − f(x0) or ∆y0 = y1 − y0, where y = f(x). The differences of these first differences, called second differences, are denoted by ∆2y0, ∆2y1, . . . , ∆2yn. Thus ∆2y0 = ∆y1 − ∆y0 = y2 − y1 − y1 + y0 = y2 − 2y1 + y0 And in general j

∆ y0 =  (−1) j

n

n=0



j y n j−n



j! j where =
= binomial coefficients. n n!( j − n)! If the ordinates are equally spaced, xn + 1 − xn = ∆x yn = y(xn) then the first and second differences are denoted by ∆yn = yn + 1 − yn ∆2yn = ∆yn + 1 − ∆yn = yn + 2 − 2yn + 1 + yn A new variable is defined xα − x0 α=
∆x

Pm(x) =  cj x j j=0

This polynomial is of degree m (highest power is x m) and order m + 1 (m + 1 parameters {cj}). If we are given a set of m + 1 points y1 = f(x1), y2 = f(x2), . . . , ym + 1 = f(xm + 1) then Lagrange’s formula gives a polynomial of degree m that goes through the m + 1 points: (x − x2) (x − x3) ⋅⋅⋅ (x − xm + 1) Pm(x) =



y1 (x1 − x2) (x1 − x3) ⋅⋅⋅ (x1 − xm + 1) (x − x1) (x − x3) ⋅⋅⋅ (x − xm + 1) +



y2 + ⋅⋅⋅ (x2 − x1) (x2 − x3) ⋅⋅⋅ (x2 − xm + 1) (x − x1) (x − x2) ⋅⋅⋅ (x − xm + 1) +



ym + 1 (xm + 1 − x1) (xm + 1 − x2) ⋅⋅⋅ (xm + 1 − xm) Note that each coefficient of yj is a polynomial of degree m that vanishes at the points {xj} (except for one value of j) and takes the value of 1.0 at that point: Pm(xj) = yj,

j = 1, 2, . . . , m + 1

If the function f(x) is known, the error in the approximation is, per Ref. 14, |xm + 1 − x1|m + 1 |error(x)| ≤

maxx1 ≤ x ≤ xm + 1 | f (m + 2)(x)| (m + 2)! The evaluation of Pm(x) at a point other than at the defining points can be made with Neville’s algorithm (Ref. 231). Let P1 be the value at x of the unique function passing through the point (x1, y1); or P1 = y1. Let

NUMERICAL ANALYSIS AND APPROXIMATE METHODS P12 be the value at x of the unique polynomial passing through the points x1 and x2. Likewise, Pijk . . . r is the unique polynomial passing through the points xi, xj, xk, . . . xr. Then use the table x1 y1 = P1 x2 y2 = P2 x3 y3 = P3 x4 y4 = P4

3-53

f 0′ = 1/10h [−2f−2 − f−1 + f1 + 2f2] If derivatives are required at end points, with all points and ordinates to one side, the derivatives are f0′ = 1/20h [−21 f0 + 13 f1 + 17 f2 − 9 f3]

P12

P123

P23

P234

P34

f0′ = 1/20h [21 f0 − 13 f−1 − 17 f−2 + 9 f−1] f−1 ′ = 1/20h [ 11 f0 − 3 f−1 − 7 f−2 − f−1]

These entries are defined using (x − xi + m) Pi(i + 1) . . . (i + m − 1) + (xi − x) P(i + 1)(i + 2) . . . (i + m) Pi(i + 1) . . . (i + m) =





xi − xi + m For example, consider P1234. The terms on the right-hand side involve P123 and P234. The “parents,” P123 and P234, already agree at points 2 and 3. Here i = 1, m = 3; thus, the parents agree at xi + 1, . . . , xi + m − 1 already. The formula makes Pi(i + 1) . . . (i + m) agree with the function at the additional points xi + m and xi. Thus, Pi(i + 1) . . . (i + m) agrees with the function at all the points {xi, xi + 1, . . . xi + m}. NUMERICAL DIFFERENTIATION Numerical differentiation should be avoided whenever possible, particularly when data are empirical and subject to appreciable observation errors. Errors in data can affect numerical derivatives quite strongly; i.e., differentiation is a roughening process. When such a calculation must be made, it is usually desirable first to smooth the data to a certain extent. Use of Interpolation Formula If the data are given over equidistant values of the independent variable x, an interpolation formula such as the Newton formula (see Refs. 143 and 185) may be used and the resulting formula differentiated analytically. If the independent variable is not at equidistant values, then Lagrange’s formulas must be used. By differentiating three- and five-point Lagrange interpolation formulas the following differentiation formulas result for equally spaced tabular points: Three-Point Formulas Let x0, x1, x2 be the three points. 1 h2 f ′(x0) =
[−3f(x0) + 4f(x1) − f(x2)] +
f ′″(ε) 2h 3 1 h2 f ′(x1) =
[−f(x0) + f(x2)] −
f ′″(ε) 2h 6 h2 1 f ′(x2) =
[ f(x0) − 4f(x1) + 3f(x2)] +
f ′″(ε) 2h 3 where the last term is an error term min. xj < ε < max. xj. j

f1′ = 1/20h [−11 f0 + 3 f1 + 7 f2 + f3]

P1234

j

Smoothing Techniques These techniques involve the approximation of the tabular data by a least-squares fit of the data by using some known functional form, usually a polynomial (for the concept of least squares see “Statistics”). In place of approximating f(x) by a single least-squares polynomial of degree n over the entire range of the tabulation, it is often desirable to replace each tabulated value by the value taken on by a least-squares polynomial of degree n relevant to a subrange of 2M + 1 points centered, when possible, at the point for which the entry is to be modified. Thus each smoothed value replaces a tabulated value. Let fj = f(xj) be the tabular points and yj = smoothed values. First-Degree Least Squares with Three Points y0 = j[5f0 + 2 f1 − f2] y1 = s[ f0 + f1 + f2] y2 = j[−f0 + 2 f1 + 5f2] Second-Degree Least Squares with Five Points For five evenly spaced points x−2, x−1, x0, x1, and x2 (separated by distance h) and their ordinates f−2, f−1, f0, f1, and f2, assume a parabola is fit by least squares. Then the derivative at the center point is

Numerical Derivatives The results given above can be used to obtain numerical derivatives when solving problems on the computer, in particular for the Newton-Raphson method and homotopy methods. Suppose one has a program, subroutine, or other function evaluation device that will calculate f given x. One can estimate the value of the first derivative at x0 using f [x0(1 + ε)] − f [x0] df
≈

dx x0 ε (a first-order formula) or f [x0(1 + ε)] − f [x0(1 − ε)] df
≈


dx x0 2ε





(a second-order formula). The value of ε is important; a value of 10−6 is typical, but smaller or larger values may be necessary depending on the computer precision and the application. One must also be sure that the value of x0 is not zero and use a different increment in that case. NUMERICAL INTEGRATION (QUADRATURE) A multitude of formulas have been developed to accomplish numerical integration, which consists of computing the value of a definite integral from a set of numerical values of the integrand. Newton-Cotes Integration Formulas (Equally Spaced Ordinates) for Functions of One Variable The definite integral b ∫ a f(x) dx is to be evaluated. Trapezoidal Rule This formula consists of subdividing the interval a ≤ x ≤ b into n subintervals a to a + h, a + h to a + 2h, . . . and replacing the graph of f(x) by the result of joining the ends of adjacent ordinates by line segments. If fj = f(xj) = f(a + jh), f0 = f(a), fn = f(b), the integration formula is b h f(x) dx =
[ f0 + 2f1 + 2f2 + ⋅⋅⋅ + 2fn − 1 + fn] + En a 2



nh3 (b − a)3 |En| =
|f″ (ε)| =
| f ″(ε)| a<ε


nh5 (b − a)5 (IV) |En| =
| f (IV)(ε)| =
| f (ε)| a<ε
x = a + (b − a)u,

dx = (b − a)du

f(x) dx = (b − a) f(u) du b

a

1

0

3-54

MATHEMATICS

f(u) du =  W f(u ) 1

m

i

0

i

i=1

The quadrature is exact when f is a polynomial of degree 2m − 1 in x. Because there are m weights and m Gauss points, we have 2m parameters that are chosen to exactly represent a polynomial of degree 2m − 1, which has 2m parameters. The Gauss points and weights are given in the table. Gaussian Quadrature Points and Weights m

ui

Wi

2

0.21132 48654 0.78867 51346 0.11270 16654 0.50000 00000 0.88729 83346 0.06943 18442 0.33000 94783 0.66999 05218 0.93056 81558 0.04691 00771 0.23076 53450 0.50000 00000 0.76923 46551 0.95308 99230

0.50000 00000 0.50000 00000 0.27777 77778 0.44444 44445 0.27777 77778 0.17392 74226 0.32607 25774 0.32607 25774 0.17392 74226 0.11846 34425 0.23931 43353 0.28444 44444 0.23931 43353 0.11846 34425

3

4

5

Example Evaluate the same integral (3-75) using the trapezoid rule and then apply the Romberg method. To achieve four-digit accuracy, any result from J2 through L1 are suitable, even though the base results (I1–I4) are not that close. I1 = 0.967058363

I2 = 0.923704741 J1 = 0.909253534

f(x) dx = b

1/a

a

1/b

b

g2(x)

sin x dx

f(x, y) dx dy =

g1(x)

(3-75)

Using the Gaussian quadrature formulas gives the following values for various values of m. Clearly, three internal points, requiring evaluation of the integrand at only three points, gives excellent results.

1 2 3 4 5

0.908185 0.910089 0.909336367 0.909330666 0.909330674

Romberg’s Method Romberg’s method uses extrapolation techniques to improve the answer (Ref. 231). If we let I1 be the value of the integral obtained using interval size h = ∆x, and I2 be the value of I obtained when using interval size h/2, and I0 the true value of I, then the error in a method is approximately hm, or I1 ≈ I0 + chm h I2 ≈ I0 + c
2

 

m

Replacing the ≈ by an equality (an approximation) and solving for c and I0 gives 2mI2 − I1 I0 =
2m − 1 This process can also be used to obtain I1, I2, . . . , by halving h each time, and then calculating new estimates from each pair, calling them J1, J2, . . . ; that is, in the formula above, replace I0 with J1. The formulas are reapplied for each pair of J to obtain K1, K2, . . . The process continues until the required tolerance is obtained. I1

I2 I3 I4 J1 J2 J3 K1 K2 L1

Romberg’s method is most useful for a low-order method (small m) because significant improvement is then possible.

b



g2(x)

f(x, y) dy

Gaussian quadrature can also be used in two dimensions, provided the integration is on a square or can be transformed to one. (Domain transformations might be used to convert the domain to a square.)

f(x, y) dx dy =  W 1

I

G(x) dx g1(x)

0

m

ab > 0

a

G(x) =

1

−x

 

1 1
2 f
du u u

Two-Dimensional Formula Two-dimensional integrals can be calculated by breaking down the integral into one-dimensional integrals. a

e

I4 = 0.910227902 J3 = 0.909330366 K2 = 0.909331898 L1 = 0.909325916

Singularities When the integrand has singularities, a variety of techniques can be tried. The integral may be divided into one part that can be integrated analytically near the singularity and another part that is integrated numerically. Sometimes a change of argument allows analytical integration. Series expansion might be helpful, too. When the domain is infinite, it is possible to use Gauss-Legendre or Gauss-Hermite quadrature. Also a transformation can be made (Ref. 26). For example, let u = 1/x and then

Example Calculate the value of the following integral. I=

I3 = 0.912920511 J2 = 0.909325768 K1 = 0.909349846

1

mx

xi

0

0

i=1

my

W

yj

f(xi, yj)

j=1

NUMERICAL SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS AS INITIAL VALUE PROBLEMS A differential equation for a function that depends on only one variable, often time, is called an ordinary differential equation. The general solution to the differential equation includes many possibilities; the boundary or initial conditions are needed to specify which of those are desired. If all conditions are at one point, then the problem is an initial value problem and can be integrated from that point on. If some of the conditions are available at one point and others at another point, then the ordinary differential equations become two-point boundary value problems, which are treated in the next section. Initial value problems as ordinary differential equations arise in control of lumped parameter models, transient models of stirred tank reactors, and in all models where there are no spatial gradients in the unknowns. A higher-order differential equation y(n) + F(y(n − 1), y(n − 2), . . . , y′, y) = 0 with initial conditions Gi(y(n − 1)(0), y(n − 2)(0), . . . , y(0), y(0)) = 0,

i = 1, . . . , n

can be converted into a set of first-order equations using d(i − 1)y d dyi − 1 =
y(i − 2) =
yi  y(i − 1) =
dt(i − 1) dt dt The higher-order equation can be written as a set of first-order equations. dy1
= y2 dt dy2
= y3 dt

NUMERICAL ANALYSIS AND APPROXIMATE METHODS dy3
= y4 dt ... dyn
= −F(yn − 1, yn − 2, . . . , y2, y1) dt The initial conditions would have to be specified for variables y1(0), . . . , yn(0), or equivalently y(0), . . . , y(n − 1)(0). The set of equations is then written as dy
= f(y, t) dt All the methods in this section are described for a single equation; the methods apply to multiple equations. See Refs. 106 and 185 for more details. Euler’s method is first-order. yn + 1 = yn + ∆t f(yn) and errors are proportional to ∆t. The second-order Adams-Bashforth method is ∆t yn + 1 = yn +
[3 f(yn) − f(yn − 1)] 2 Errors are proportional to ∆ t2, and high-order methods are available. Notice that the higher-order explicit methods require knowing the solution (or the right-hand side) evaluated at times in the past. Since these were calculated to get to the current time, this presents no problem except for starting the problem. Then it may be necessary to use Euler’s method with a very small step size for several steps in order to generate starting values at a succession of time points. The error terms, order of the method, function evaluations per step, and stability limitations are listed in Ref. 106. The advantage of the high-order Adams-Bashforth method is that it uses only one function evaluation per step yet achieves high-order accuracy. The disadvantage is the necessity of using another method to start. Runge-Kutta methods are explicit methods that use several function evaluations for each time step. Runge-Kutta methods are traditionally written for f(t, y). The first-order Runge-Kutta method is Euler’s method. A second-order Runge-Kutta method is ∆t yn + 1 = yn +
[ f n + f(tn + ∆t, yn + ∆t f n)] 2 while the midpoint scheme is also a second-order Runge-Kutta method. ∆t ∆t yn + 1 = yn + ∆t f tn +
, yn +
f n 2 2 A popular fourth-order Runge-Kutta method is the Runge-KuttaFeldberg formulas (Ref. 111), which have the property that the method is fourth-order but achieves fifth-order accuracy. The popular integration package RKF45 is based on this method.



k1 = ∆t f(tn, yn) ∆t k1 k2 = ∆t f tn +
, yn +
4 4

 



 









12 1932 7200 7296 k4 = ∆t f tn +
∆t, yn +
k1 −
k2 +
k3 13 2197 2197 2197

439 3680 845 k5 = ∆t f tn + ∆t, yn +
k1 − 8k2 +
k3 −
k4 216 513 4104

∆t 8 3544 1859 11 k6 = ∆t f tn +
, yn −
k1 + 2k2 −
k3 +
k4 −
k5 2 27 2565 4104 40



1 25 1408 2197 yn + 1 = yn +
k1 +
k3 +
k4 −
k5 216 2565 4104 5

2 16 6656 28561 9 zn + 1 = yn +
k1 +
k3 +
k4 −
k5 +
k6 135 12825 56430 50 55 The value of yn + 1 − zn + 1 is an estimate of the error in yn + 1 and can be used in step-size control schemes. Usually one would use a high-order method to achieve high accuracy. The Runge-Kutta-Feldberg method is popular because it is high order and does not require a starting method (as does an AdamsBashforth method). However, it does require four function evaluations per time step, or four times as many as a fourth-order Adams-Bashforth method. For problems in which the function evaluations are a significant portion of the calculation time, this might be important. Given the speed of present-day computers and the widespread availability of microcomputers (which can be run while you are doing something else, if need be), the efficiency of the methods is most important only for very large problems that are going to be solved many times. For other problems, the most important criterion for choosing a method is probably the time the user spends setting up the problem. The stability limits for the explicit methods are based on the largest eigenvalue of the linearized system of equations (see Ref. 106). For linear problems, the eigenvalues do not change, so that the stability and oscillation limits must be satisfied for every eigenvalue of the matrix A. When solving nonlinear problems, the equations are linearized about the solution at the local time, and the analysis applies for small changes in time, after which a new analysis about the new solution must be made. Thus, for nonlinear problems, the eigenvalues keep changing, and the largest stable time step changes, too. The stability limits are: Euler method, λ ∆t ≤ 2 Runge-Kutta, 2nd order, λ ∆t < 2 Runge-Kutta-Feldberg, λ ∆t < 3.0 Richardson extrapolation can be used to improve the accuracy of a method. Suppose we step forward one step ∆t with a pth-order method. Then redo the problem, this time stepping forward from the same initial point, but in two steps of length ∆t/2, thus ending at the same point. Call the solution of the one-step calculation y1 and the solution of the two-step calculation y2. Then an improved solution at the new time is given by 2 py2 − y1 y=
2p − 1 This gives a good estimate provided ∆t is small enough that the method is truly convergent with order p. This process can also be repeated in the same way Romberg’s method was used for quadrature. The error term in the various methods can be used to deduce a step size that will give a user-specified accuracy. Most packages today are based on a user-specified tolerance; the step-size is changed during the calculation to achieve that accuracy. The accuracy itself is not guaranteed, but it improves as the tolerance is decreased. Implicit Methods By using different interpolation formulas involving yn + 1, it is possible to derive implicit integration methods. Implicit methods result in a nonlinear equation to be solved for yn + 1 so that iterative methods must be used. The backward Euler method is a first-order method. yn + 1 = yn + ∆t f(yn + 1)

3 3 9 k3 = ∆t f t +
∆t, yn +
k1 +
k2 8 32 32 n

3-55



Errors are proportional to ∆t for small ∆t. The trapezoid rule is a second-order method. ∆t yn + 1 = yn +
[ f(yn) + f(yn + 1)] 2 Errors are proportional to ∆ t2 for small ∆ t. When the trapezoid rule is used with the finite difference method for solving partial differential equations, it is called the Crank-Nicolson method. The implicit methods are stable for any step size but do require the solution of a set of nonlinear equations, which must be solved iteratively. The set of equations can be solved using the successive substitution method or Newton-Raphson method. See Ref. 36 for an application to dynamic distillation problems. The best packages for stiff equations (see below) use Gear’s back-

3-56

MATHEMATICS

ward difference formulas. The formulas of various orders are, per Refs. 59 and 117, (1) yn + 1 = yn + ∆t f(yn + 1) 4 1 2 (2) yn + 1 =
yn −
yn − 1 +
∆t f(yn + 1) 3 3 3 18 9 2 6 (3) yn + 1 =
yn −
yn − 1 +
yn − 2 +
∆t f(yn + 1) 11 11 11 11 48 36 16 3 12 (4) yn + 1 =
yn −
yn − 1 +
yn − 2 −
yn − 3 +
∆t f(yn + 1) 25 25 25 25 25 300 300 200 75 12 (5) yn + 1 =
yn −
yn − 1 +
yn − 2 −
yn − 3 +
yn − 4 137 137 137 137 137 60 +
∆t f(yn + 1) 137 Stiffness The concept of stiffness is described for a system of linear equations. dy
=Ay dt Let λi be the eigenvalues of the matrix A (Ref. 267). Then, per Ref. 181, the stiffness ratio is defined as maxi |Re (λi)| SR =

mini |Re (λi)| SR = 20 is not stiff, SR = 103 is stiff, and SR = 106 is very stiff. If the problem is nonlinear, then the solution is expanded about the current state. n dyi ∂fi
= fi [y(tn)] + 
[yj − yj(tn)] dt j = 1 ∂yj The question of stiffness then depends on the solution at the current time. Consequently nonlinear problems can be stiff during one time period and not stiff during another. While the chemical engineer may not actually calculate the eigenvalues, it is useful to know that they determine the stability and accuracy of the numerical scheme and the step size used. Problems are stiff when the time constants for different phenomena have very different magnitudes. Consider flow through a packed bed reactor. The time constants for different phenomena are: 1. Time for device flow-through L φAL tflow =
=
u Q where Q is the volumetric flow rate, A is the cross sectional area, L is the length of the packed bed, and φ is the void fraction; 2. Time for reaction 1 tr × n =
k where k is a rate constant (time−1); 3. Time for diffusion inside the catalyst εR2 tinternal diffusion =
De where ε is the porosity of the catalyst, R is the catalyst radius, and De is the effective diffusion coefficient inside the catalyst; 4. Time for heat transfer is R2 ρsCsR2 tinternal heat transfer =
=
α ke where ρs is the catalyst density, Cs is the catalyst heat capacity per unit mass, ke is the effective thermal conductivity of the catalyst, and α is the thermal diffusivity. For example, in the model of a catalytic converter for an automobile (Ref. 103), the time constants for internal diffusion was 0.3 seconds; internal heat transfer, 21 seconds; and device flow-through, 0.003 seconds. The device flow-through is so fast that it might as well be instantaneous. The stiffness is approximately 7000.

Implicit methods must be used to integrate the equations. Alternatively, a quasistate model can be developed (Ref. 239). Differential-Algebraic Systems Sometimes models involve ordinary differential equations subject to some algebraic constraints. For example, the equations governing one equilibrium stage (as in a distillation column) are dxn M
= Vn + 1yn + 1 − Lnxn − V nyn + Ln − 1xn − 1 dt xn − 1 − xn = En(xn − 1 − x*,n) N

x =1 i

i=1

where x and y are the mole fraction in the liquid and vapor, respectively; L and V are liquid and vapor flow rates, respectively; M is the holdup; and the superscript is the stage number. The efficiency is E, and the concentration in equilibrium with the vapor is x*. The first equation is an ordinary differential equation for the mass of one component on the stage, while the third equation represents a constraint that the mass fractions add to one. This is a differential-algebraic system of equations. Differential-algebraic equations can be written in the general notation dy F t, y,
= 0 dt





To solve the general problem using the backward Euler method, replace the nonlinear differential equation with the nonlinear algebraic equation for one step. yn + 1 − yn F t, yn + 1,
= 0 ∆t This equation must be solved for yn + 1. The Newton-Raphson method can be used, and if convergence is not achieved within a few iterations, the time step can be reduced and the step repeated. In actuality, the higher-order backward-difference Gear methods are used in DASSL (Ref. 224). Differential-algebraic systems are more complicated than differential systems because the solution may not always be defined. Pontelides et al. (Ref. 226) introduced the term index to identify the possible problems. The index is defined as the minimum number of times the equations need to be differentiated with respect to time to convert the system to a set of ordinary differential equations. These higher derivatives may not exist, and the process places limits on which variables can be given initial values. Sometimes the initial values must be constrained by the algebraic equations (Ref. 226). For a differential-algebraic system modeling a distillation tower, Ref. 226 shows that the index depends on the specification of pressure for the column. Byrne and Ponzi (Ref. 58) also list several chemical engineering examples of differential-algebraic systems and solve one involving two-phase flow. Computer Software Efficient computer packages are available for solving ordinary differential equations as initial value problems. The packages are widely available and good enough that most chemical engineers use them and do not write their own. Here we discuss three of them: RKF45, LSODE, and EPISODE. In each of the packages, the user specifies the differential equation to be solved and a desired error criterion. The package then integrates in time and adjusts the step size to achieve the error criterion within the limitations imposed by stability. A popular explicit, Runge-Kutta package is RKF45, developed by Forsythe et al. (Ref. 111). The method is based on the Runge-KuttaFeldberg formulas. Notice there that an estimate of the truncation error at each step is available. Then the step size can be reduced until this estimate is below the user-specified tolerance. The method is thus automatic, and the user is assured of the results. Note, however, that the tolerance is set on the local truncation error, namely from one step to another, whereas the user is usually interested in the global truncation error, or the error after several steps. The global error is generally made smaller by making the tolerance smaller, but the absolute accuracy is not the same as the tolerance. If the problem is stiff, then very





NUMERICAL ANALYSIS AND APPROXIMATE METHODS small step sizes are used; the computation becomes very lengthy. The RKF45 code discovers this and returns control to the user with a message indicating the problem is too hard to solve with RKF45. A popular implicit package is LSODE, a version of Gear’s method (Ref. 117) written by Alan Hindmarsh at Lawrence Livermore Laboratory (Ref. 148). In this package, the user specifies the differential equation to be solved and the tolerance desired. Now the method is implicit and therefore stable for any step size. The accuracy may not be acceptable, however, and sets of nonlinear equations must be solved. Thus, in practice the step size is limited but not nearly so much as in the Runge-Kutta methods. In these packages, both the step size and order of the method are adjusted by the package. Suppose we are calculating with a kth order method. The truncation error is determined by the (k + 1)th order derivative. This is estimated using difference formulas and the values of the right-hand sides at previous times. An estimate is also made for the kth and (k + 2)th derivative. Then it is possible to estimate the error in a (k − 1)th order method, a kth order method, and a (k + 1)th order method. Furthermore, the step size needed to satisfy the tolerance with each of these methods can be determined. Then we can choose the method and step size for the next step that achieves the biggest step, with appropriate adjustments due to the different work required for each order. The package generally starts with a very small step size and a first-order method, the backward Euler method. Then it integrates along, adjusting the order up (and later down) depending on the error estimates. The user is thus assured that the local truncation error meets the tolerance. There is a further difficulty, since the set of nonlinear equations must be solved. Usually a good guess of the solution is available, since the solution is evolving in time and past history can be extrapolated. Thus, the Newton-Raphson method will usually converge. The package protects itself, though, by only doing three iterations. If convergence is not reached within this many iterations, then the step size is reduced and the calculation is redone for that time step. The convergence theorem for the Newton-Raphson method (p. 3-50) indicates that the method will converge if the step size is small enough. Thus the method is guaranteed to work. Further economies are possible. The Jacobian needed in the Newton-Raphson method can be fixed over several time steps. Then, if the iteration does not converge, the Jacobian can be reevaluated at the current time-step. If the iteration still does not converge, then the step-size is reduced and a new Jacobian is evaluated. Also the successive substitution method can be used, which is even faster, except that it may not converge. However, it, too, will converge if the time step is small enough. Comparisons of the methods and additional details are provided for chemical engineering problems by Refs. 59 and 106. Generally, the Runge-Kutta methods give extremely good accuracy, especially when the step size is kept small for stability reasons. When the computation time is comparable for LSODE and RKF45, the RKF45 package generally gives much more accurate results. The RKF45 package is unsuitable, however, for many chemical reactor problems because they are so stiff. Generally, though, standard packages must have a highorder explicit method (usually a version of Runge-Kutta) and a multistep, implicit method (usually a version of GEAR, EPISODE, or LSODE). The package DASSL (Ref. 224) uses similar principles to solve the differential-algebraic systems. The software described here is available by electronic mail over the Internet. Sending the message send index to [email protected]

will retrieve an index and descriptions of how to obtain the software. Stability, Bifurcations, Limit Cycles Some aspects of this subject involve the solution of nonlinear equations; other aspects involve the integration of ordinary differential equations; applications include chaos and fractals as well as unusual operation of some chemical engineering equipment. Ref. 176 gives an excellent introduction to the subject and the details needed to apply the methods. Ref. 66 gives more details of the algorithms. A concise survey with some chemical engineering examples is given in Ref. 91. Bifurcation results are closely connected with stability of the steady states, which is essentially a transient phenomenon.

3-57

Sensitivity Analysis When solving differential equations, it is frequently necessary to know the solution as well as the sensitivity of the solution to the value of a parameter. Such information is useful when doing parameter estimation (to find the best set of parameters for a model) and for deciding if a parameter needs to be measured accurately. See Ref. 105. ORDINARY DIFFERENTIAL EQUATIONS-BOUNDARY VALUE PROBLEMS Diffusion problems in one dimension lead to boundary value problems. The boundary conditions are applied at two different spatial locations: at one side the concentration may be fixed and at the other side the flux may be fixed. Because the conditions are specified at two different locations, the problems are not initial value in character. It is not possible to begin at one position and integrate directly because at least one of the conditions is specified somewhere else and there are not enough conditions to begin the calculation. Thus, methods have been developed especially for boundary value problems. Shooting Methods The first method is one that utilizes the techniques for initial value problems but allows for an iterative calculation to satisfy all the boundary conditions. Consider the nonlinear boundary value problem d 2y dy
2 = f x, y,
, y(0) = α, y(1) = β dx dx Convert this second-order equation into two first-order equations along with the boundary conditions written to include a parameter s to represent the unknown value of v(0) = dy/dx(0). dy dv
= v,
= f(x, y, v), y(0) = α, v(0) = s dx dx The parameter s is chosen so that the last boundary condition is satisfied: y(1) = β. Define the function





χ(s) = y(1, s) − β and iterate on s to make χ(s) = 0. Note that the condition at x = 0 is satisfied for any s, the differential equation is satisfied by the integration routine, and only the last boundary condition is yet to be satisfied. Both successive substitution and the Newton-Raphson methods can be used. The technique can be used when the boundary conditions are more general and convergence can be proved (see Refs. 106 and 167). Computer software exists: the IMSL program DTPTB uses DVERK, which employs Runge-Kutta integration to integrate the ordinary differential equations (Ref. 55). Finite Difference Method To apply the finite difference method, we first spread grid points through the domain. Figure 3-49 shows a uniform mesh of n points (nonuniform meshes are possible, too). The unknown, here c(x), at a grid point xi is assigned the symbol ci = c(xi). The finite difference method can be derived easily by using a Taylor expansion of the solution about this point. Expressions for the derivatives are: dc dc ci + 1 − ci d 2c ∆x ci − ci − 1 d 2c ∆x
=
−
2
+ ⋅⋅⋅ ,
=
+
2
+ ⋅⋅⋅ dx i ∆x dx i 2 dx i ∆x dx i 2









dc ci + 1 − ci − 1 d c ∆x
=

−
3
dx i 2∆x dx i 3! The truncation error in the first two expressions is proportional to ∆x, and the methods are said to be first-order. The truncation error in the third expression is proportional to ∆x2, and the method is said to be second-order. Usually the last equation is used to insure the best accuracy. The finite difference representation of the second derivative is:



FIG. 3-49

3

Finite difference mesh; ∆x uniform.



2

3-58

MATHEMATICS

d 2c ci + 1 − 2ci + ci − 1 d4c 2∆x2 −
4
+ ⋅⋅⋅
2 =

dx i ∆x2 dx i 4! The truncation error is proportional to ∆x2. To solve a differential equation, it is evaluated at a point i and then these expressions are inserted for the derivatives.





FIG. 3-50

Example Consider the equation for convection, diffusion, and reaction in

Finite difference grid with variable spacing.

a tubular reactor. 1 d 2c dc

2 −
= Da R(c) Pe dx dx The finite difference representation is 1 ci + 1 − 2ci + ci − 1 ci + 1 − ci − 1 −

= Da R(ci)


Pe ∆x2 2∆x This equation is written for i = 2 to n − 1, or the internal points. The equations would then be coupled but would also involve the values of c1 and cn, as well. These are determined from the boundary conditions. If the boundary condition involves a derivative, it is important that the derivatives be evaluated using points that exist. Three possibilities exist: dc c2 − c1
=
dx 1 ∆x



dc −3c1 + 4c2 − c3
=

dx 1 2∆x The third alternative is to add a false point, outside the domain, as c0 = c(x = −∆x). c2 − c0 dc
=
dx 1 2∆x Since this equation introduces a new variable, c0, another equation is needed and is obtained by writing the finite difference equation for i = 1, too. The sets of equations can be solved using the Newton-Raphson method. The first form of the derivative gives a tridiagonal system of equations, and the standard routines for solving tridiagonal equations suffice. For the other two options, some manipulation is necessary to put them into a tridiagonal form (see Ref. 105). Frequently, the transport coefficients, such as diffusion coefficient or thermal conductivity, depend on the dependent variable, concentration, or temperature, respectively. Then the differential equation might look like dc d
D(c)
= 0 dx dx This could be written as two equations. dJ dc −
=0 J = −D(c)
dx dx Because the coefficient depends on c, the equations are more complicated. A finite difference method can be written in terms of the fluxes at the midpoints, i + 1/2. ci + 1 − ci Ji + 1/2 − Ji − 1/2 −

=0 Ji +1/2 = −D(ci + 1/2)
∆x ∆x These are combined to give the complete equation. D(ci + 1/2) (ci + 1 − ci) − D(ci − 1/2) (ci − ci − 1) =0




∆x2 This represents a set of nonlinear algebraic equations that can be solved with the Newton-Raphson method. However, in this case, a viable iterative strategy is to evaluate the transport coefficients at the last value and then solve D(cki + 1/2) (cik++11 − c ik + 1) − D(cik− 1/2) (cik + 1 − c ik−+11) =0





∆x2 The advantage of this approach is that it is easier to program than a full NewtonRaphson method. If the transport coefficients do not vary radically, then the method converges. If the method does not converge, then it may be necessary to use the full Newton-Raphson method. There are three common ways to evaluate the transport coefficient at the midpoint. The first one uses the transport coefficient evaluated at the average value of the solutions on either side. 1 D (ci + 1/2) ≈ D
(ci + 1 + ci) 2 The truncation error of this approach is ∆x2 (Ref. 106). The second approach uses the average of the transport coefficients on either side. 1 D (ci + 1/2) ≈
[D(ci + 1) + D(ci)] 2













The truncation error of this approach is also ∆x2 (Ref. 106). The third approach uses an “upstream” transport coefficient. D (ci + 1/2) ≈ D(ci + 1),

when D (ci + 1) > D(ci)

D (ci + 1/2) ≈ D(ci),

when D (ci + 1) < D(ci)

This approach is used when the transport coefficients vary over several orders of magnitude, and the “upstream” direction is defined as the one in which the transport coefficient is larger. The truncation error of this approach is only ∆x (Refs. 106 and 107), but this approach is useful if the numerical solutions show unrealistic oscillations. If the grid spacing is not uniform, the formulas must be revised. The notation is shown in Fig. 3-50. The finite-difference form of the equations is then Ji + 1/2 − Ji − 1/2 ci + 1 − ci ci − ci − 1 −

= 0 Ji + 1/2 = −Di + 1/2
, Ji − 1/2 = −Di − 1/2
a(∆xi + ∆xi + 1) ∆xi + 1 ∆xi If average diffusion coefficients are used, then the finite difference equation is as follows.





1 1 1


(Di + 1 + Di) (ci + 1 − ci) −
(Di + Di − 1) (ci − ci − 1) = 0 ∆xi + 1 + ∆xi ∆xi + 1 ∆xi Rigorous error bounds are discussed for linear ordinary differential equations solved with the finite difference method by Isaacson and Keller (Ref. 107). Computer software exists to solve two-point boundary value problems. The IMSL routine DVCPR uses the finite difference method with a variable step size (Ref. 247). Finlayson (Ref. 106) gives FDRXN for reaction problems.

Example A reaction diffusion problem is solved with the finite difference method. d 2c
2 = φ2c, dx

dc
(0) = 0, dx

c(1) = 1

The solution is derived for φ = 2. It is solved several times, first with two intervals and three points (at x = 0, 0.5, 1), then with four intervals, then with eight intervals. The reason is that when an exact solution is not known, one must use several ∆x and see that the solution converges as ∆x approaches zero. With two intervals, the equations are as follows. The points are x1 = 0, x2 = 0.5, and x3 = 1.0; and the solution at those points are c1, c2, and c3, respectively. A false boundary is used at x0 = −0.5. c0 − c2
= 0, 2∆x

c0 − 2c1 + c2 − φ2c1 = 0,

∆x2

c1 − 2c2 + c3 − φ2c2 = 0,

∆x2

c3 = 1

The solution is c1 = 0.2857, c2 = 0.4286, and c3 = 1.0. Since the solution is only an approximation and approaches the exact solution only as ∆x approaches zero, it is necessary to find out if ∆x is small enough to be considered zero. This is done by solving the problem again with more grid points. The value of concentration at x = 0 takes the following values for different ∆x. These values are extrapolated using the Richardson extrapolation technique to give c(0) = 0.265826. Using this value as the best estimate of the exact solution, the errors in the solution are tabulated versus ∆x. Clearly the errors go as ∆x2 (decreasing by a factor of 4 when ∆x decreases by a factor of 2), thus validating the solution. The exact solution (given below) is 0.265802. n−1

∆x

c(0)

2 4 8

0.5 0.25 0.125

0.285714 0.271043 0.267131

n−1

∆x

Error in c(0)

2 4 8

0.5 0.25 0.125

0.01989 0.00521 0.00130

Finite Difference Methods Solved with Spreadsheets A convenient way to solve the finite difference equations for simple

NUMERICAL ANALYSIS AND APPROXIMATE METHODS

3-59

The residual is then made orthogonal to the set of basis functions. db b (x)  a
− φ R  a b (x) dx = 0 dx 1

2

NT

j

i

0

FIG. 3-51

Finite difference method using spreadsheets.

problems is to use a computer spreadsheet. The equations for the problem solved in the example can be cast into the following form 2c2 c1 =

2 + φ2∆x2

i=1

NT

i

2

i i

2

j = 1, . . . , NT

i=1

This is the process that makes the method a Galerkin method. The basis for the orthogonality condition is that a function that is made orthogonal to each member of a complete set is then zero. The residual is being made orthogonal, and if the basis functions are complete and you use infinitely many of them, then the residual is zero. Once the residual is zero, the problem is solved. This equation is integrated by parts to give the following equation NT

− i=1



1

0

b (x)R  a b (x) dx

dbj dbi

dxai = φ2 dx dx

1

NT

j

i i

0

i=1

j = 1, . . . , NT − 1

ci + 1 + ci − 1 ci =

2 + φ2∆x2 cn + 1 = 1 Let us solve the problem using 6 nodes, or 5 intervals. Then the connection between the cell in the spreadsheet and the nodal value is shown in Fig. 3-51. The following equations are placed into the various cells. A1: = 2*B1/(2.+(phi*dx)**2) B1: = (A1 + C1)/(2.+(phi*dx)**2) F1: = 1.

(3-76)

This equation defines the Galerkin method and a solution that satisfies this equation (for all j = 1, . . . , ∞) is called a weak solution. For an approximate solution, the equation is written once for each member of the trial function, j = 1, . . . , NT − 1, and the boundary condition is applied. NT

 a b (1) = c i

i

B

i=1

The equation in cell B1 is copied into cells C1 though E1. Then turn on the iteration scheme in the spreadsheet and watch the solution converge. Whether or not convergence is achieved can depend on how you write the equations, so some experimentation may be necessary. Theorems for convergence of the successive substitution method are useful in this regard. Orthogonal Collocation The orthogonal collocation method has found widespread application in chemical engineering, particularly for chemical reaction engineering. In the collocation method, the dependent variable is expanded in a series of orthogonal polynomials, and the differential equation is evaluated at certain collocation points. The collocation points are the roots to an orthogonal polynomial, as first used by Lanczos (Refs. 182 and 183). A major improvement was proposed by Villadsen and Stewart (Refs. 288 and 289), who proposed that the entire solution process be done in terms of the solution at the collocation points rather than the coefficients in the expansion. This method is especially useful for reaction-diffusion problems that frequently arise when modeling chemical reactors. It is highly efficient when the solution is smooth, but the finite difference method is preferred when the solution changes steeply in some region of space. See Ref. 105 for comparisons. Galerkin Finite Element Method In the finite element method, the domain is divided into elements and an expansion is made for the solution on each finite element. In the Galerkin finite element method an additional idea is introduced: the Galerkin method is used to solve the equation. The Galerkin method is explained before the finite element basis set is introduced, using the equations for reaction and diffusion in a porous catalyst pellet. d 2c
2 = φ2R(c) dx dc
(0) = 0, c(1) = 1 dx The unknown solution is expanded in a series of known functions {bi(x)} with unknown coefficients {ai}. NT

c(x) =  ai bi(x) i=1

The trial solution is substituted into the differential equation to obtain the residual. NT NT d 2bi Residual =  ai
− φ2R  aibi(x) 2 dx i=1 i=1





The Galerkin finite element method results when the Galerkin method is combined with a finite element trial function. The domain is divided into elements separated by nodes, as in the finite difference method. The solution is approximated by a linear (or sometimes quadratic) function of position within the element. These approximations are substituted into Eq. (3-76) to provide the Galerkin finite element equations. The element integrals are defined as 1 e BJI = −
∆xe



1

0

dNJ dNI

du, du du

F Je = φ2∆xe

N (u)R  c N (u) du 1

NP

e I

J

0

I

I=1

and the entire method can be written in the following compact notation:

B

c =  FJe

e e JI I

e

e

The matrices for various terms are given in the table. This equation can also be written in the form AAc = f where the matrix AA is sparse; if linear elements are used, the matrix is tridiagonal. Once the solution is found, the solution at any point can be recovered from ce(u) = c eI = 1(1 − u) + c eI = 2u for linear elements. Element Matrices for Galerkin Method with Linear Shape Functions

dN dN N1 = 1 − u, N2 = u,
1 = −1,
2 = 1 du du



−a a du =   N
−a a du N N du = sj js, N du = aa, N u du = js 1

0



dNJ dNI 1 −1 ,

du = −1 1 du du

1

dNI

J

0

1

J

0

1

1

I

J

0

j

0

Example Solve the specified problem when φ = 2, the rate expression is linear, R(c) = c, and the boundary condition is 1.0. The Galerkin finite element method is used with ∆x = 0.33333. The element nodes are at x = 0, 0.3333, 0.6667, and 1.0. The solution at x = 1.0 is c4 = 1.0. The Galerkin equations for one element are obtained from the table.

−11 −11 = φ ∆x  s⁄ s⁄ 

−

2

16

2

16

3-60

MATHEMATICS

When these are summed over all elements the result is



−1 1 0 0

1 −1 − 1 1 0

0 1 −1 − 1 0

 

0 0 1 1

1

c1 ⁄6 s 0 1 4 1⁄6 s + s c2 ⁄6 =
1 c3 0 ⁄ 6 s + s 9 0 0 c4 0

0 0 ⁄6 0

1

   c1 0 0 c2 + 0 c3 1 c4

u(x) = f(x) + λ(b − a)

After rearrangement this is



31⁄27

−25⁄27 0

−25⁄27 62⁄27 −25⁄27

0

−25⁄27 62⁄27

   c1 c2 = c3

25

⁄27

Cubic B-Splines Cubic B-splines can also be used to solve differential equations (Refs. 105 and 266). Adaptive Meshes In many two-point boundary value problems, the difficulty in the problem is the formation of a boundary layer region, or a region in which the solution changes very dramatically. In such cases, it is prudent to use small mesh spacing there, either with the finite difference method or the finite element method. If the region is known a priori, small mesh spacings can be assumed at the boundary layer. If the region is not known, though, other techniques must be used. These techniques are known as adaptive mesh techniques. The mesh size is made small where some property of the solution is large. For example, if the truncation error of the method is nth order, then the nth-order derivative of the solution is evaluated and a small mesh is used where it is large. Alternatively, the residual (the differential equation with the numerical solution substituted into it) can be used as a criterion. See Refs. 21 and 107. It is also possible to define the error that is expected from a method one order higher and one order lower. Then a decision about whether to increase or decrease the order of the method can be made, taking into account the relative work of the different orders. This provides a method of adjusting both the mesh spacing (∆x, or sometimes called h) and the degree of polynomial (p). Such methods are called h-p methods. Singular Problems and Infinite Domains If the solution being sought has a singularity, it may be difficult to find a good numerical solution. Sometimes even the location of the singularity may not be known (Ref. 11). One method of solving such problems is to refine the mesh near the singularity, relying on the better approximation due to a smaller ∆x. Another approach is to incorporate the singular trial function into the approximation. Thus, if the solution approaches f(x) as x goes to zero and f(x) becomes infinite, one may define a new variable u(x) = y(x) − f(x) and derive an equation for u. The differential equation is more complicated, but the solution is better near the singularity. See Refs. 39 and 231. Sometimes the domain is semi-infinite, as in boundary layer flow. The domain can be transformed from the x domain (0–∞) to the η domain (1–0) using the transformation η = exp (−x). Another approach is to use a variable mesh, perhaps with the same transformation. For example, use η = exp (−βx) and a constant mesh size in η; the value of β is found experimentally. Still another approach is to solve on a finite mesh in which the last point is far enough away that its location does not influence the solution (Ref. 59). A location that is far enough away must be found by trial and error. NUMERICAL SOLUTION OF INTEGRAL EQUATIONS

b

a

for u(x)

n

i

i

(3-78)

i

where t1, . . . , tn are points of subdivision of the t axis, a ≤ t ≤ b, and the c’s are coefficients whose values depend upon the type of numerical integration formula used. Now Eq. (3-78) must hold for all values of x, a ≤ x ≤ b; so it must hold for x = t1, x = t2, . . . , x = tn. Substituting for x successively t1, t2, . . . , tn and setting u(ti) = ui, f(ti) = fi, we get n linear algebraic equations for the n unknowns u1, . . . , un. That is, ui = fi + (b − a)[c1k(ti, t1)u1 + c2 k(ti, t2)u2 + ⋅⋅⋅ + cnk(ti, tn)un]

i = 1, 2, . . . , n

These uj may be solved for by the methods under “Numerical Solution of Linear Equations and Associated Problems” and substituted into Eq. (3-78) to yield an approximate solution for Eq. (3-77). Example Solve numerically u(x) = x + s ∫ 01 (t + x)u(t) dt. In this example a = 0, b = 1. Take n = 3, t1 = 0, t2 = a, t3 = 1. Then Eq. (3-78) takes the form (for which we have used the parabolic rule) a u(x) = x + (s)
[(t1 + x)u(t1) + 4(t2 + x)u(t2) + (t3 + x)u(t3)] 3 = x + (1/18)[(t1 + x)u(t1) + 4(t2 + x)u(t2) + (t3 + x)u(t3)] This must hold for all x, 0 ≤ x ≤ 1. Here t1 = 0, t2 = a, and t3 = 1. Evaluate at x = ti. u(t1) = t1 + 1⁄18[2t1u(t1) + 4(t2 + t1)u(t2) + (t3 + t1)u(t3)] u(t2) = t2 + 1⁄18[(t1 + t2)u(t1) + 4(2t2)u(t2) + (t3 + t2)u(t3)] u(t3) = t3 + 1⁄18[(t1 + t3)u(t1) + 4(t2 + t3)u(t2) + 2t3u(t3)] By setting in the values of t1, t2, t3 and u(ti) = ui, 18u1 − 2u2 − u3 = 0 −u1 + 28u2 − 3u3 = 18 −u1 − 6u2 + 16u3 = 18 with the solution u1 = 12⁄ 71, u2 = 57⁄71, u3 = 102⁄71. Thus u(x) = x + 1⁄18[x12⁄71 + 4(a + x)57⁄71 + (1 + x)102⁄71] = 90⁄ 71x + 12⁄71

Because of the work involved in solving large systems of simultaneous linear equations it is desirable that only a small number of u’s be computed. Thus the gaussian integration formulas are useful because of the economy they offer. See references on numerical solutions of integral equations. Solutions for Volterra equations are done in a similar fashion, except that the solution can proceed point by point, or in small groups of points depending on the quadrature scheme. See Refs. 105 and 195. There are methods that are analogous to the usual methods for integrating differential equations (Runge-Kutta, predictor-corrector, Adams methods, etc.). Explicit methods are fast and efficient until the time step is very small to meet the stability requirements. Then implicit methods are used, even though sets of simultaneous algebraic equations must be solved. The major part of the calculation is the evaluation of integrals, however, so that the added time to solve the algebraic equations is not excessive. Thus, implicit methods tend to be preferred (Ref. 195). Volterra equations of the first kind are not well posed, and small errors in the solution can have disastrous consequences. The boundary element method uses Green’s functions and integral equations to solve differential equations (Refs. 45 and 200). MONTE CARLO SIMULATIONS

In this subsection is considered a method of solving numerically the Fredholm integral equation of the second kind:

k(x, t)u(t) dt

 c k(x, t )u(t ) i=1

0 0

The solution is c1 = 0.2560, c2 = 0.3174, c3 = 0.5312, and c4 = 1. The exact solution is derived using the section entitled “Ordinary Differential Equations: Linear Differential Equations with Constant Coefficients.” e2x + e−2x cosh (2x) c=
=
e2 + e−2 cosh (2) The values of the exact solution at the same finite element nodes are c1 = 0.2658, c2 = 0.3271, c3 = 0.5392, and c4 = 1, indicating that the three-element finite element solution is accurate within 3 percent. When the exact solution is not known, the problem must be solved several times, each with a different number of elements, so that convergence is seen as the number of elements increases.

u(x) = f(x) + λ

The method discussed arises because a definite integral can be closely approximated by any of several numerical integration formulas (each of which arises by approximating the function by some polynomial over an interval). Thus the definite integral in Eq. (3-77) can be replaced by an integration formula, and Eq. (3-77) may be written

(3-77)

Some physical problems, such as those involving interaction of molecules, are usually formulated as integral equations. Monte Carlo methods are especially well-suited to their solution. This section cannot give a comprehensive treatment of such methods, but their use in

NUMERICAL ANALYSIS AND APPROXIMATE METHODS

3-61

calculating the value of an integral will be illustrated. Suppose we wish to calculate the integral G=



Ω0

g(x)f(x) dx

where the distribution function f(x) satisfies: f(x) ≥ 0,



Ω0

f(x) dx = 1

The distribution function f(x) can be taken as constant; for example, 1/Ω0. We choose variables x1, x2, . . . , xN randomly from f(x) and form the arithmetic mean 1 GN =
 g(xi) N i The quantity GN is an estimation of G, and the fundamental theorem of Monte Carlo guarantees that the expected value of GN is G, if G exists (Ref. 161). The error in the calculation is given by σ1 ε=
N1/2 where σ 21 is calculated from σ 21 =



Ω0

g2(x) f(x) dx − G2

Thus the number of terms needed to achieve a specified accuracy can be calculated once an estimate of σ 21 is known. σ12 N=
ε Various methods, such as influence sampling, can be used to reduce the number of calculations needed (Ref. 161). NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS Parabolic Equations in One Dimension By combining the techniques applied to initial value problems and boundary value problems it is possible to easily solve parabolic equations in one dimension. The method is often called the method of lines. It is illustrated here using the finite difference method, but the Galerkin finite element method and the orthogonal collocation method can also be combined with initial value methods in similar ways. The analysis is done by example. Example Consider the diffusion equation, with boundary and initial conditions.

∂c ∂2c
= D
2 ∂t ∂x c(x, 0) = 0 c(0, t) = 1, c(1, t) = 0

We denote by ci the value of c(xi, t) at any time. Thus, ci is a function of time, and differential equations in ci are ordinary differential equations. By evaluating the diffusion equation at the ith node and replacing the derivative with a finite difference equation, the following working equation is derived for each node i, i = 2, . . . , n (see Fig. 3-52). dci ci + 1 − 2ci + ci − 1
= D

dt ∆x2 This can be written in the general form of a set of ordinary differential equations by defining the matrix AA. dc
= AAc dt This set of ordinary differential equations can be solved using any of the standard methods, and the stability of the integration of these equations is governed by the largest eigenvalue of AA. If Euler’s method is used for integration, the time step is limited by 2 ∆t ≤
|λ|max

FIG. 3-52

Computational molecules. h = ∆x = ∆y.

whereas, if the Runge-Kutta-Feldberg method is used, the 2 in the numerator is replaced by 3.0. The largest eigenvalue of AA is bounded by Gerschgorin’s Theorem (Ref. 155, p. 135). n 4D |λ|max ≤ max 2 < j < n  |AA ji | =
2 ∆x i=2 This gives the well-known stability limit D 1 ∆t
2 ≤
∆x 2

3-62

MATHEMATICS

The smallest eigenvalue is independent of ∆x (it is Dπ 2/L2) so that the ratio of largest to smallest eigenvalue is proportional to 1/∆x2. Thus, the problem becomes stiff as ∆x approaches zero (Ref. 106). Another way to study the stability of explicit equations is to use the positivity theorem. For Euler’s method, the equations can be written in the form c in + 1 − cin cni+ 1 − 2c ni + cni− 1

= D

∆t ∆x2 n n where c i = c(xi, t ). Then the new value is given by





D∆t n D∆t n D∆t n c in + 1 =
ci + 1 + 1 − 2
ci +
ci − 1 ∆x2 ∆x2 ∆x2 Theorem. If c ni + 1 = Acni+ 1 + Bc in + Ccni− 1 and A, B, and C are positive and A + B + C ≤ 1, then the scheme is stable and the errors die out. Here the theorem requires D∆t 1−2
>0 ∆x2 which gives the same stability condition (Ref. 106). Implicit methods can also be used. Write a finite difference form for the time derivative and average the right-hand sides, evaluated at the old and new time. c in + 1 − cin cni+ 1 − 2cni + cni− 1 cin++11 − 2c in + 1 + cni −+11 + Dθ




= D(1 − θ)

2 ∆t ∆x ∆x2 Now the equations are of the form









D∆tθ n + 1 D∆tθ n + 1 D∆tθ n + 1 −
ci + 1 + 1 + 2
ci −
ci − 1 ∆x2 ∆x2 ∆x2 D∆t(1 − θ) n = c in +

(ci + 1 − 2c in + cni− 1) ∆x2 and require solving a set of simultaneous equations, which have a tridiagonal structure. Using θ = 0 gives the Euler method (as above), θ = 0.5 gives the Crank-Nicolson method, and θ = 1 gives the backward Euler method. The Crank-Nicolson method is also the same as applying the trapezoid rule to do the integration. The stability limit is given by D∆t 0.5 ≤

∆x2 1 − 2θ If the ∆t satisfies the following equation, then the solution will not oscillate from node to node (a numerical artifact). See Ref. 106. 0.25 D∆t ≤

∆x2 1−θ Other methods can be used in space, such as the finite element method, the orthogonal collocation method, or the method of orthogonal collocation on finite elements (see Ref. 106). Spectral methods employ Chebyshev polynomials and the Fast Fourier Transform and are quite useful for hyperbolic or parabolic problems on rectangular domains (Ref. 125). Packages exist that use various discretizations in the spatial direction and an integration routine in the time variable. PDECOL uses B-splines for the spatial direction and various GEAR methods in time (Ref. 247). PDEPACK and DSS (Ref. 247) use finite differences in the spatial direction and GEARB in time (Ref. 66). REACOL (Ref. 106) uses orthogonal collocation in the radial direction and LSODE in the axial direction, while REACFD uses finite difference in the radial direction; both codes are restricted to modeling chemical reactors. Elliptic Equations Elliptic equations can be solved with both finite difference and finite element methods. One-dimensional elliptic problems are two-point boundary value problems. Two- and threedimensional elliptic problems are often solved with iterative methods when the finite difference method is used and direct methods when the finite element method is used. So there are two aspects to consider: how the equations are discretized to form sets of algebraic equations and how the algebraic equations are then solved.

The prototype elliptic problem is steady-state heat conduction or diffusion, ∂2T ∂2T +
=Q k
∂x2 ∂y2





possibly with a heat generation term per unit volume, Q. The boundary conditions taken here are T = f(x, y) on the boundary (S) with f a known function. Illustrations are given for constant thermal conductivity k while Q is a known function of position. The finite difference formulation is given using the following nomenclature: Ti, j = T(i∆x, j∆y) The finite difference formulation is then (see Fig. 3-52) Ti + 1, j − 2Ti, j + Ti − 1, j Ti, j + 1 − 2Ti, j + Ti, j − 1 +


= Qi, j


∆x2 ∆y2

(3-79)

Ti, j = f(xi, yj) on S If the boundary is parallel to a coordinate axis any derivative is evaluated as in the section on boundary value problems, using either a onesided, centered difference or a false boundary. If the boundary is more irregular and not parallel to a coordinate line then more complicated expressions are needed and the finite element method may be the better method. Equation (3-79) is rewritten in the form ∆ x2 ∆x2 Qi, j 2 1 +
2 Ti, j = Ti + 1, j + Ti − 1, j +
2 (Ti, j + 1 + Ti, j − 1) − ∆x2
∆y ∆y k





The relaxation method solves this equation iteratively. Qi, j ∆ x2 ∆ x2 2 1 +
2 T*i, j = Tis+ 1, j + T si −+ 1,1 j +
2 (T i,s j + 1 + T i,s +j −1 1) − ∆ x2
∆y ∆y k





T si,+j 1 = Ti,s j + β(T*i, j − Ti,s j) If β = 1, this is the Gauss-Seidel method. If β > 1, it is overrelaxation; if β < 1 it is underrelaxation. The value of β may be chosen empirically, 0 < β < 2, but it can be selected theoretically for simple problems like this (Refs. 106 and 221). In particular, these equations can be programmed in a spreadsheet and solved using the iteration feature, provided the boundaries are all rectangular. The alternating direction method can be used for elliptic problems by using sequences of iteration parameters (Refs. 106 and 221). The method is well suited to transient problems as well. These are the classical iterative techniques. Recently preconditioned conjugate gradient methods have been developed (see Ref. 100). In these methods, a series of matrix multiplications are done iteration by iteration; and the steps lend themselves to the efficiency available in parallel computers. In the multigrid method, the problem is solved on several grids, each more refined than the previous one. As one iterates between the solutions on the different grids, one converges to the solution of the algebraic equations. See Juncu and Mihail (Ref. 68) for a chemical engineering application. The Galerkin finite element method (FEM) is useful for solving elliptic problems and is particularly effective when the domain or geometry is irregular. As an example, cover the domain with triangles and define a trial function on each triangle. The trial function takes the value 1.0 at one corner and 0.0 at the other corners and is linear in between. See Fig. 3-53. These trial functions on each triangle are pieced together to give a trial function on the whole domain. General treatments of the finite element method are available (see references). The steps in the solution method are similar to those described for boundary value problems, except now the problems are much bigger so that the numerical analysis must be done very carefully to be efficient. Most engineers, though, just use a finite element program without generating it. There are three major caveats that must be addressed. The first one is that the solution is dependent on the mesh laid down, and the only way to assess the accuracy of the solution is to solve the problem with a more refined mesh. The second concern is that the solution obeys the shape of the trial function inside

NUMERICAL ANALYSIS AND APPROXIMATE METHODS the element. Thus, if linear functions are used on triangles, a threedimensional view of the solution, plotting the solution versus x and y, consists of a series of triangular planes joined together at the edges, as in a geodesic dome. The third caveat is that the Galerkin finite element method is applied to both the differential equations and the boundary conditions. Computer programs are usually quite general and may allow the user to specify boundary conditions that are not realistic. Also, natural boundary conditions are satisfied if no other boundary condition (ones involving derivatives) is set at a node. Thus, the user of finite element codes must be very clear what boundary conditions and differential equations are built into the computer code. When the problem is nonlinear, the Newton-Raphson method is used to iterate from an initial guess. Nonlinear problems lead to complicated integrals to evaluate, and they are usually evaluated using Gaussian quadrature. One nice feature of the finite element method is the use of natural boundary conditions. It may be possible to solve the problem on a domain that is shorter than needed to reach some limiting condition (such as at an outflow boundary). The externally applied flux is still applied at the shorter domain, and the solution inside the truncated domain is still valid. Examples are given in Refs. 67 and 107. The effect of this is to allow solutions in domains that are smaller, thus saving computation time and permitting the solution in semi-infinite domains. A general purpose package for general two-dimensional domains and rectangular three-dimensional rectangular domains is ELLPACK (Ref. 247). This package allows choice of a variety of methods: finite difference, Hermite collocation, spline Galerkin, collocation, as well as others. Comparisons of the various methods are available (Ref. 154). The program FISHPAK solves the Helmholtz equation in multiple dimensions when the domain is separable (since fast methods like FFT are used). See Ref. 247. Hyperbolic Equations The most common situation yielding hyperbolic equations involves unsteady phenomena with convection. Two typical equations are the convective diffusive equation ∂c ∂2c ∂c
+ u
= D
2 ∂t ∂x ∂x and the chromatography equation (Ref. 245) ∂c ∂c df ∂c φ
+ φu
+ (1 − φ)

= 0 ∂t ∂x dc ∂t where φ is the void fraction and f(c) gives the equilibrium relation between the concentration in the fluid phase and the concentration in the solid phase. If the diffusion coefficient is zero, the convective diffusion equation is hyperbolic. If D is small, the phenomenon may be essentially hyperbolic, even though the equations are parabolic. Thus the numerical methods for hyperbolic equations may be useful even for parabolic equations. Equations for several methods are given here, as taken from the book by Finlayson (Ref. 107). If the convective term is treated with a centered difference expression, the solution exhibits oscillations from node to node, and these only go away if a very fine grid is used. The

FIG. 3-53 Trial functions for Galerkin finite element method: linear polynomial on triangle.

3-63

simplest way to avoid the oscillations with a hyperbolic equation is to use upstream derivatives. If the flow is from left to right, this would give dci ci − ci − 1 ci + 1 − 2ci + ci − 1
+ u
= D

dt ∆x ∆x2 d ci − ci − 1
[ φci + (1 − φ)f(ci)] + φui
= 0 dt ∆x (See Ref. 227 for the reason the equation is written in this form.) The effect of using upstream derivatives is to add artificial or numerical diffusion to the model. This can be ascertained by rearranging the finite difference form of the convective diffusion equation dci ci + 1 − ci − 1 u∆x ci + 1 − 2ci + ci − 1
+ u

= D +


dt 2∆x 2 ∆x2 Thus the diffusion coefficient has been changed from u∆x D to D +
2 Another method often used for hyperbolic equations is the MacCormack method. This method has two steps, and it is written here for the convective diffusion equation. u∆t D∆t n c*i n + 1 = cin −
(cni+ 1 − c ni ) +
(c i + 1 − 2c ni + cni − 1) ∆x ∆x2





1 u∆t c in + 1 =
(cin + c*i n + 1) −
(c*i n + 1 − c*i −n 1+ 1) 2 2∆x D∆t +
2 (c*i +n 1+ 1 − 2c*i n + 1 + c*i −n 1+ 1) 2∆x The concentration profile is steeper for the MacCormack method than for the upstream derivatives, but oscillations can still be present. The flux-corrected transport method can be added to the MacCormack method. A solution is obtained both with the upstream algorithm and the MacCormack method and then they are combined to add just enough diffusion to eliminate the oscillations without smoothing the solution too much. The algorithm is complicated and lengthy but well worth the effort (Refs. 37, 107, and 270). Stability conditions can be constructed in terms of Co = u∆t/∆x and r = D∆t/∆x2 by using Fourier analysis (Ref. 107). All the methods require u∆t Co =
≤ 1 ∆x where Co is the Courant number. How much Co should be less than one depends on the method and on r = D∆t/∆x2. For example, the upstream method requires Co ≤ 1 − 2r. The MacCormack method depends less on r and is stable for most Co as long as r ≤ 0.5. Each of these methods is trying to avoid oscillations that would disappear if the mesh were fine enough. For the steady convective diffusion equation, these oscillations do not occur provided u∆x
≤1 2D For large velocity u, the ∆x must be small to meet this condition. An alternative is to use a small ∆x in regions where the solution changes drastically. Since these regions change in time, it is necessary that the elements or grid points move. The criteria to move the grid points can be quite complicated, and typical methods are reviewed in Ref. 107. Similar considerations apply to the nonlinear chromatography problem (Ref. 227). See especially Ref. 192. Parabolic Equations in Two or Three Dimensions Computations become much more lengthy when there are two or more spatial dimensions. For example, we may have the unsteady heat conduction equation ∂T ∂2T ∂2T ρCp
= k
+
−Q ∂t ∂x2 ∂y2





3-64

MATHEMATICS

In the finite difference method an explicit technique would evaluate the right-hand side at the nth time level. T i,n j+ 1 − T i,nj k n ρCp

=
2 (T in+ 1, j − 2T i, j + T in− 1, j) ∆t ∆x k +
2 (T ni, j + 1 − 2T i,n j + Ti,nj − 1) − Q ∆y When Q = 0 and ∆x = ∆y, the time step limit can be found using the positivity rule. ∆x2ρC ∆x2 ∆t ≤
p or
4k 4D These time steps are smaller than for one-dimensional problems. For three dimensions, the limit is ∆x2 ∆t ≤
6D To avoid such small time steps, which become smaller as ∆x decreases, an implicit method could be used. This leads to large, sparse matrices rather than convenient tridiagonal matrices. These can be solved, but the alternating direction method is also useful (Ref. 221). This reduces a problem on an n n grid to a series of 2n one-dimensional problems on an n grid. SPLINE FUNCTIONS Splines are functions that match given values at the points x1, . . . , xNT and have continuous derivatives up to some order at the knots, or the points x2, . . . , xNT − 1. Cubic splines are most common; see Ref. 38. The function is represented by a cubic polynomial within each interval (xi, xi + 1) and has continuous first and second derivatives at the knots. Two more conditions can be specified arbitrarily. These are usually the second derivatives at the two end points, which are commonly taken as zero; this gives the natural cubic splines. Take yi = y(xi) at each of the points xi, and let ∆xi = xi + 1 − xi. Then, in the interval (xi, xi + 1), the function is represented as a cubic polynomial.

FAST FOURIER TRANSFORM (REF. 231) Suppose a signal y(t) is sampled at equal intervals

n = . . . , −2, −1, 0, 1, 2, . . .

∆ = sampling rate (e.g., number of samples per second) The Fourier transform and inverse transform are Y(ω) =



∞

−∞

y(t)eiωt dt



1 ∞ Y(ω)e−iωt dt y(t) =
2π −∞ The Nyquist critical frequency or critical angular frequency is 1 π fc =
, ωc =
2∆ ∆ If a function y(t) is bandwidth-limited to frequencies smaller than fc, such as Y(ω) = 0

for ω > ωc

then the function is completely determined by its samples yn. Thus, the entire information content of a signal can be recorded by sampling at a rate ∆−1 = 2fc. If the function is not bandwidth-limited, then aliasing occurs. Once a sample rate ∆ is chosen, information corresponding to frequencies greater than fc is simply aliased into that range. The way to detect this in a Fourier transform is to see if the transform approaches zero at fc; if not, aliasing has occurred, and a higher sampling rate is needed. Next, suppose we have N samples, where N is even yk = y(tk)

tk = k∆

k = 0,1,2, . . . , N − 1

and the sample rate is ∆. With only N values {yk}, it is not possible to determine the complete Fourier transform Y(ω). We calculate the value Y(ωn) at the discrete points 2πn N N ωn =
, n = −
, . . . , 0, . . . ,
N∆ 2 2 Yn =

N−1

 ye k

2πikn/N

k=0

Y(ωn) = ∆Yn

Ci(x) = a0i + a1i x + a2i x2 + a3i x3 The interpolating function takes on specified values at the knots and has continuous first and second derivatives at the knots. Within the ith interval, the function is (x − xi) Ci(x) = Ci(xi) + C′i (xi)(x − xi) + C″i (xi)
2 (x − xi)3 + [C″i (xi + 1) − C″i (xi)]
6∆xi where Ci(xi) = yi. The second derivative C″i (xi) = y″i is found by solving the following tridiagonal system of equations: yi − yi − 1 yi + 1 − yi y″i − 1∆xi − 1 + y″i 2(∆xi − 1 + ∆xi) + y″i + 1∆xi = 6
−
∆xi − 1 ∆xi Since the continuity conditions apply only for i = 2, . . . , NT − 1, we have only NT − 2 conditions for the NT values of y″i . Two additional conditions are needed, and these are usually taken as the value of the second derivative at each end of the domain, y″1 , y″NT. If these values are zero, we get the natural cubic splines; they can also be set to achieve some other purpose, such as making the first derivative match some desired condition at the two ends. With these values taken as zero in the natural cubic spline, we have a NT − 2 system of tridiagonal equations, which is easily solved. Once the second derivatives are known at each of the knots, the first derivatives are given by ∆x yi + 1 − yi ∆x y′i =
− y″i
i − y″i + 1
i ∆xi 3 6 The function itself is then known within each element.



yn = y(n∆),



The discrete inverse Fourier transform is 1 N−1 yk =
 Yne−2πikn/N N n=0 The fast Fourier transform (FFT) is used to calculate the Fourier transform as well as the inverse Fourier transform. A discrete Fourier transform of length N can be written as the sum of two discrete Fourier transforms, each of length N/2. Yk = Yke + W k Yko Here Yk is the kth component of the Fourier transform of y, and Yke is the kth component of the Fourier transform of the even components of {yj} and is of length N/2. Similarly, Yko is the kth component of the Fourier transform of the odd components of {yj} and is of length N/2. W is a constant, which is taken to the kth power. W = e2πi/N Since Yk has N components, while Y ek and Yko have N/2 components, Y ek and Yko are repeated once to give N components in the calculation of Yk. This decomposition can be used recursively. Thus, Y ek is split into even and odd terms of length N/4. Yke = Ykee + W k Ykeo Yko = Ykoe + W k Ykoo This process is continued until there is only one component. For this reason, the number N is taken as a power of 2. The vector {yj } is filled with zeroes, if need be, to make N = 2 p for some p. For the computer program, see Ref. 26. The standard Fourier transform takes N 2 operations to calculation, whereas the fast Fourier transform takes only N log2 N. For large N, the difference is significant; at N = 100 it is a factor of 15, but for N = 1000 it is a factor of 100.

OPTIMIZATION The discrete Fourier transform can also be used for differentiating a function, and this is used in the spectral method for solving differential equations. Suppose we have a grid of equidistant points L xn = n∆x, n = 0, 1, 2, . . . , 2N − 1, ∆x =
2N The solution is known at each of these grid points {Y(xn)}. First the Fourier transform is taken. 1 2N − 1 yk =
 Y(xn)e−2ikπx n /L 2N n = 0 The inverse transformation is 1 N Y(x) =
 yke2ikπx/L L k = −N Differentiate this to get dY 1 N 2πik
=
 yk
e2ikπx/L dx L k = −N L

3-65

Thus at the grid points



dY 1 N 2πik
=
 yk
e2ikπx n /L dx n L k = −N L The process works as follows. From the solution at all grid points the Fourier transform is obtained using FFT, {yk}. Then this is multiplied by 2πik/L to obtain the Fourier transform of the derivative. 2πik yk = yk
L Then the inverse Fourier transform is taken using FFT, giving the value of the derivative at each of the grid points.



dY 1 N
=
 yke2ikπx n /L dx n L k = −N

OPTIMIZATION INTRODUCTION Optimization should be viewed as a tool to aid in decision making. Its purpose is to aid in the selection of better values for the decisions that can be made by a person in solving a problem. To formulate an optimization problem, one must resolve three issues. First, one must have a representation of the artifact that can be used to determine how the artifact performs in response to the decisions one makes. This representation may be a mathematical model or the artifact itself. Second, one must have a way to evaluate the performance—an objective function—which is used to compare alternative solutions. Third, one must have a method to search for the improvement. This section concentrates on the third issue, the methods one might use. The first two items are difficult ones, but discussing them at length is outside the scope of this section. Example optimization problems are: (1) determining the optimal thickness of pipe insulation; (2) finding the best equipment sizes and operating schedules for the design of a new batch process to make a given slate of products; (3) choosing the best set of operating conditions for a set of experiments to determine the constants in a kinetic model for a given reaction; (4) finding the amounts of a given set of ingredients one should use for making a carbon rod to be used as an electrode in an arc welder. For the first problem, one will usually write a mathematical model of how insulation of varying thicknesses restricts the loss of heat from a pipe. Evaluation requires that one develop a cost model for the insulation (a capital cost in dollars) and the heat that is lost (an operating cost in dollars/year). Some method is required to permit these two costs to be compared, such as a present worth analysis. Finally, if the model is simple enough, the method one can use is to set the derivative of the evaluation function to zero with respect to wall thickness to find candidate points for its optimal thickness. For the second problem, selecting a best operating schedule involves discrete decisions, which will generally require models that have integer variables. It may not be possible to develop a mathematical model for the fourth problem if not enough is known to characterize the performance of a rod versus the amounts of the various ingredients used in its manufacture. The rods may have to be manufactured and judged by ranking the rods relative to each other, perhaps based partially or totally on opinions. Pattern search methods have been devised to attack problems in this class. In this section assume a mathematical model is possible for the problem to be solved. The model may be encoded in a subroutine and be known only implicitly, or the equations may be known explicitly. A general form for such an optimization problem is min F = F(z), such that h(z) = 0 and g(z) ≤ 0

where F represents a specified objective function that is to be minimized. Functions h and g represent equality and inequality constraints that must be satisfied at the final problem solution. Variables z are used to model such things as flows, mole fractions, physical properties, temperatures, and sizes. The objective function F is generally assumed to be a scalar function, one which represents such things as cost, net present value, safety, or flexibility. Sometimes several objective functions are specified (e.g., minimizing cost while maximizing reliability); these are commonly combined into one function, or else one is selected for the optimization while the others are specified as constraints. Equations h(z) = 0 are typically algebraic equations, linear or nonlinear, when modeling steady-state processes, or algebraic coupled with ordinary and/or partial differential equations when optimizing time-varying processes. Inequalities g(z) ≤ 0 put limits on the values variables can take, such as a minimum and maximum temperature, or they restrict one pressure to be greater than another. An important issue is how to solve large problems that occur in distributed systems. The optimization of distributed systems is discussed in Refs. 52, 120, 244, and 285. For further reading on optimization, readers are directed to Refs. 120 and 244 as well as introductory texts on optimization applied to chemical engineering (Refs. 99 and 225). The material in this section is part of a more advanced treatment (Ref. 295). Packages There are a number of packages available for optimization, some of which are listed here. 1. Frameworks • GAMS. This framework is commercially available. It provides a uniform language to access several different optimization packages, many of them listed below. It will convert the model as expressed in “GAMS” into the form needed to run the package chosen. • AMPL. This framework is by Fourier and coworkers (Ref. 113) at Northwestern University. It is well suited for constructing complex models. • ASCEND. This framework is by Westerberg and coworkers (Ref. 295) at Carnegie-Mellon University. It features an objectoriented modeling language and is well suited for constructing complex models. 2. Algebraic optimization with equality and inequality constraints • SQP. A package by Biegler at Carnegie-Mellon University. • MINOS5.4. A package available from Stanford Research Institute (affiliated with Stanford University). This package is the state of the art for mildly nonlinear programming problems. • GRG. A package from Lasdon at the University of Texas, Dept. of Management Science. 3. Linear programming. Most current commercial codes for lin-

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MATHEMATICS

ear programming extend the Simplex algorithm, and they can typically handle problems with up to 15,000 constraints. • MPSX. From IBM • SCICONIC. From the company of that name • MINOS5.4 • Cplex. A package by R. Bixby at Rice University and Cplx, Inc. CONDITIONS FOR OPTIMALITY


du dF

i

∆h = 0

= 0,

i = 1, 2, . . . , r

Equality Constrained Problems—Lagrange Multipliers Form a scalar function, called the Lagrange function, by adding each of the equality constraints multiplied by an arbitrary multiplier to the objective function. m

Local Minimum Point for Unconstrained Problems Consider the following unconstrained optimization problem: Min {F(u) | u
Rr} u

If F is continuous and has continuous first and second derivatives, it is necessary that F is stationary with respect to all variations in the independent variables u at a point û, which is proposed as a minimum to F; that is, ∂F or ∇u F = 0 at u = û (3-80)
= 0, i = 1, 2, . . . , r ∂ui These are only necessary conditions, as point û may be a minimum, maximum, or saddle point. Sufficient conditions are that any local move away from the optimal point û gives rise to an increase in the objective function. Expand F in a Taylor series locally around the candidate point û up to second-order terms: 1 2 F(u) = F(û) + ∇u F T  û (u − û) +
(u − û)T∇uu F  û (u − û) + … 2 If û satisfies necessary conditions [Eq. (3-80)], the second term disappears in this last line. Sufficient conditions for the point to be a local 2 minimum are that the matrix of second partial derivatives ∇uu F is positive definite. This matrix is symmetric, so all of its eigenvalues are real; to be positive definite, they must all be greater than zero. Constrained Derivatives—Equality Constrained Problems Consider minimizing the objective function F written in terms of n variables z and subject to m equality constraints h(z) = 0, or Mzin {F(z) | h(z) = 0, z
Rn, h:Rn → Rm}

(3-81)

The point zˆ is tested to see if it could be a minimum point. It is necessary that F be stationary for all infinitesimal moves for z that satisfy the equality constraints. Linearize the m equality constraints around zˆ, getting h(ˆz + ∆z) = h(ˆz) + ∇z hT  zˆ ∆z

(3-82)

where ∆z = z − zˆ . There are m constraints here, so m of the variables are dependent, leaving r = n − m independent variables. Partition the variables ∆z into a set of m dependent variables ∆x and r = n − m independent variables ∆u. Equation (3-82), rearranged and then rewritten in terms of these variables, becomes

L(x, u, λ) = F(x, u) +  λi hi(x, u) = F(x, u) + λT h(x, u) i=1

At any point where the functions h(z) are zero, the Lagrange function equals the objective function. Next differentiate L with respect to variables x, u, and λ. ∇x LT  zˆ = ∇x F T  zˆ + λT∇h xT  zˆ

= 0T

(3-83)

∇u L  zˆ = ∇u F  zˆ + λ ∇h 

=0

(3-84)

∇λ LT  zˆ = hT(x, u)

=0

T

T

T

T u zˆ

T T

Solve Eq. (3-83) for the Lagrange multipliers λT = −∇x F T[∇h Tx ]−1

(3-85)

and then eliminate these multipliers from Eq. (3-84). ∇u LT = ∇u F T − ∇x F T[∇hTx ]∇h Tu = 0 T ∇uL is equal to the constrained derivatives for the problem, which should be zero at the solution to the problem. Also, these stationarity conditions very neatly provide the necessary conditions for optimality of an equality-constrained problem. Lagrange multipliers are often referred to as shadow prices, adjoint variables, or dual variables, depending on the context. Suppose the variables are at an optimum point for the problem. Perturb the variables such that only constraint hi changes. We can write ∆L = ∆F + λi ∆hi = 0 which is zero because, as just shown, the Lagrange function is at a stationary point at the optimum. Solving for the change in the objective function: ∆F = −λi∆hi The multiplier tells how the optimal value of the objective function changes for this small change in the value of a constraint while holding all the other constraints at zero. It is for this reason that they are often called shadow prices. Equality- and Inequality-Constrained Problems—KuhnTucker Multipliers Next a point is tested to see if it is an optimum one when there are inequality constraints. The problem is Mzin {F(z) | h(z) = 0, g(z) ≤ 0, z
Rn, F:Rn → R1, h:Rn → Rm, g:Rn → Rp}

The Lagrange function here is similar to that used above.

∆h = ∇x hT  zˆ ∆x + ∇uhT  zˆ ∆u = 0

L(z, λ, µ)  F(z) + λTh(z) + µTg(z)

This enables the solution for ∆x. Linearize the objective function F(z) in terms of the partitioned variables

Each of the inequality constraints gi(z) multiplied by what is called a Kuhn-Tucker multiplier µi is added to form the Lagrange function. The necessary conditions for optimality, called the Karush-KuhnTucker conditions for inequality-constrained optimization problems, are

∆F = ∇x F T  zˆ ∆x + ∇u F T  zˆ ∆u and substitute for ∆x. ∆F = {∇x F T − ∇u F T[∇x hT]−1 ∇u hT}zˆ ∆u



dF =
du

T ∆h = 0

r



dF ∆u = 
dui i=1

∆h = 0

∆ui

There is one term for each ∆ui in the row vector which is in the curly braces {}. These terms are called constrained derivatives, which tells how the object function changes when the independent variables ui are changed while keeping the constraints satisfied (by varying the dependent variables xi). Necessary conditions for optimality are that these constrained derivatives are zero; that is,

∇z L zˆ = ∇z F zˆ + ∇z h zˆ λ + ∇z g zˆ µ = 0 ∇λL = h(z) = 0 g(z) ≤ 0 µi gi(z) = 0 ,

i = 1, 2, . . . , p

µi ≥ 0 ,

i = 1, 2, . . . , p

(3-86)

Conditions in Eq. (3-86), called complementary slackness conditions, state that either the constraint gi(z) = 0 and/or its corresponding multiplier µi is zero. If constraint gi(z) is zero, it is behaving like an equality constraint, and its multiplier µi is exactly the same as a Lagrange multiplier for an equality constraint. If the constraint is

OPTIMIZATION away from zero, it is not a part of the problem and should not affect it. Setting its multiplier to zero removes it from the problem. As the goal is to minimize the objective function, releasing the constraint into the feasible region must not decrease the objective function. Using the shadow price argument above, it is evident that the multiplier must be nonnegative (Ref. 177). Sufficiency conditions to assure that a Kuhn-Tucker point is a local minimum point require one to prove that the objective function will increase for any feasible move away from such a point. To carry out such a test, one has to generate the matrix of second derivatives of the Lagrange function with respect to all the variables z evaluated at zˆ . The test is seldom done, as it requires too much work. STRATEGIES OF OPTIMIZATION The theory just covered tells if a candidate point is or is not the optimum point, but how is the candidate point found? The simplest strategy is to place a grid of points throughout the feasible space, evaluating the objective function at every grid point. If the grid is fine enough, then the point yielding the highest value for the objective function can be selected as the optimum. Twenty variables gridded over only ten points would take place over 1020 points in our grid, and, at one nanosecond per evaluation, it would take in excess of four thousand years to carry out these evaluations. Most strategies limit themselves to finding a local minimum point in the vicinity of the starting point for the search. Such a strategy will find the global optimum only if the problem has a single minimum point or a set of “connected” minimum points. A “convex” problem has only a global optimum. Pattern Search Suppose the optimization problem is to find the right mix of a given set of ingredients and the proper baking temperature and time to make the best cake possible. A panel of judges can be formed to judge the cakes; assume they are only asked to rank the cakes and that they can do that task in a consistent manner. Our approach will be to bake several cakes and ask the judges to rank them. For this type of problem, pattern-search methods can be used to find the better conditions for manufacturing the product. We shall only describe the ideas behind this approach. Details on implementing it can be found in Ref. 284. The complex method is one such pattern search method (see Fig. 3-54). First, form a “complex” of at least r + 1 (r = 2 and 4 points are used in Fig. 3-54) different points at which to bake the cakes by picking a range of suitable values for the r independent variables for the baking process. Bake the cakes and then ask the judges to identify the worst cake. For each independent variable, form the average value at which it was run in the complex. Draw a line from the coordinates of the worst cake through the average point—called the centroid—and continue on that line a distance that is twice that between these two points. This point will be the next test point. First decide if it is feasible. If so, bake the cake and discover if it leads to a cake that is better than the worst cake from the last set of cakes. If it is not feasible or it is not better, then return half the distance toward the average values from the last

FIG. 3-54

Complex method, a pattern search optimization method.

3-67

test and try again. If it is better, toss out the worst point of the last test and replace it with this new one. Again, ask the judges to find the worst cake. Continue as above until the cakes are all the same quality in the most recent test. It might pay to restart at this point, stopping finally if the restart leads to no improvement. The method takes large steps if the steps are being successful in improving the recipe. It collapses onto a set of points quite close to each other otherwise. The method works reasonably well, but it requires one to bake lots of cakes. The following strategies are all examples of Generalized Reduced Gradient (GRG) methods. Optimization of Unconstrained Objective Assume the objective function F is a function of independent variables ui, i = 1 ⋅⋅⋅ r. A computer program, given the values for the independent variables, can calculate F and its derivatives with respect to each ui. Assume that F is well approximated as an as-yet-unknown quadratic function in u. 1 F ≈ a + bTu +
uTQu 2 where a is a scalar; b, a vector; and Q, an r × r symmetric positive definite matrix. The gradient of the approximate function is ∇u F = b + Qu Setting the gradient to zero allows an estimate for its minimum. u = −Q−1b

(3-87)

Initially, Q and b are not known and the calculation proceeds as follows: b contains r unknown coefficients and Q another r(r + 1)/2. To estimate b and Q, the computer code is used repeatedly, getting r equations each time—namely (∇u F)(1) = b + Qu(1) (∇u F)(2) = b + Qu(2) ... (∇u F )(t) = b + Qu(t)

(3-88)

As soon as there are as many independent equations as there are unknown coefficients, these linear equations are solved for b and Q. A proper choice of the points u(i) guarantees getting independent equations to solve here. Given b and Q, Eq. (3-87) provides a new estimate for u as a candidate minimum point. The subroutine is used again to obtain the gradient of F at this point. If the gradient is essentially zero, the calculations stop, since a point has been found that satisfies the necessary conditions for optimality. If not, the equations are written in the form of Eq. (3-88) for this new point, adding them to the set while removing the oldest set of equations. The new set of equations for b and Q are solved, and the calculations continue until a minimum point is found. If removal of the oldest equations from the set in Eq. (3-88) leads to a singular set of equations, then different equations have to be selected for removal. Alternatively, all the older equations can be kept, with the new ones added to the top of the list. Pivoting can be done by proceeding down the list until a nonsingular set of equations is found. Then the older equations are used only if necessary. Also, since only one set of equations is being replaced, clever methods are available to find the solution to the equations with much less work than is required to solve the set of equations the first time (Refs. 89 and 259). Quadratic Fit for the Equality Constrained Case Next consider solving a problem of the form of Eq. (3-82). For each iteration k: 1. Enter with values provided for variables u(k). 2. Given values for u(k), solve equations h(x, u) = 0 for x(k). These will be m equations in m unknowns. If the equations are nonlinear, solving can be done using a variant of the Newton-Raphson method. 3. Use Eq. (3-85) to solve for the Lagrange multipliers λ(k). If the Newton-Raphson method (or any or several variants to it) is used to solve the equations, the jacobian matrix ∇ xT h| z (k) and its LU factors are already known so solving Eq. (3-85) requires very little effort. 4. Substitute λ(k) into Eq. (3-84), which in general will not be zero. The gradient ∇u L(k) computed will be the constrained derivatives of F with respect to the independent variables u(k). 5. Return.

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MATHEMATICS

The calculations begin with given values for the independent variables u and exit with the (constrained) derivatives of the objective function with respect to them. Use the routine described above for the unconstrained problem where a succession of quadratic fits is used to move toward the optimal point for an unconstrained problem. This approach is a form of the generalized reduced gradient (GRG) approach to optimizing, one of the better ways to carry out optimization numerically. Inequality Constrained Problems To solve inequality constrained problems, a strategy is needed that can decide which of the inequality constraints should be treated as equalities. Once that question is decided, a GRG type of approach can be used to solve the resulting equality constrained problem. Solving can be split into two phases: phase 1, where the goal is to find a point that is feasible with respect to the inequality constraints; and phase 2, where one seeks the optimum while maintaining feasibility. Phase 1 is often accomplished by ignoring the objective function and using instead p

F=

i=1



g 2i (z) if gi(z) > 0) 0 otherwise

until all the inequality constraints are satisfied. Then at each point, check which of the inequality constraints are active, or exactly equal to zero. These can be placed into the active set and treated as equalities. The remaining can be put aside to be used only for testing. A step can then be proposed using the GRG algorithm. If it does not cause one to violate any of the inactive inequality constraints, the step is taken. Otherwise one can add the closest inactive inequality constraint to the active set. Finding the closet inactive equality will almost certainly require a line search in the direction proposed by the GRG algorithm. When one comes to a stationary point, one has to test the active inequality constraints at that point to see if they should remain active. This test is done by examining the sign (they should be nonnegative if they are to remain active) of their respective Kuhn-Tucker multipliers. If any should be released, it has to be done carefully as the release of a constraint changes the multipliers for all the constraints. One can find oneself cycling through the testing to decide whether to release the constraints. A correct approach is to add slack variables s to the problem to convert the inequality constraints to equalities and then require the slack variables to remain positive. The multipliers associated with the inequalities s ≥ 0 all behave independently, and their sign tells one directly to keep or release the constraints. In other words, simultaneously release all the slack variables that have multipliers strictly less than zero. If released, the slack variables must be treated as a part of the set of independent variables until one is well away from the associated constraints for this approach to work. Successive Quadratic Programming (SQP) The above approach to finding the optimum is called a feasible path method, as it attempts at all times to remain feasible with respect to the equality and inequality constraints as it moves to the optimum. A quite different method exists called the Successive Quadratic Programming (SQP) method, which only requires one be feasible at the final solution. Tests that compare the GRG and SQP methods generally favor the SQP method so it has the reputation of being one of the best methods known for nonlinear optimization for the type of problems considered here. Assume certain inequality constraints will be active at the final solution. The necessary conditions for optimality are ∇z L(z, µ, λ) = ∇F + ∇g A µ + ∇hλ = 0,

gA(z) = 0,

h(z) = 0

Then one can apply Newton’s method to the necessary conditions for optimality, which are a set of simultaneous (non)linear equations. The Newton equations one would write are



∇zzL[z(i), u(i), λ(i)]

∇gA[z(i)]

∇h[z(i)]

∇gA[z(i)]T

0

0

∇h[z(i)]T

0

0

 

∆z(i)

∇zL[z(i), µ(i), λ(i)]

∆µ(i) = −

gA[z(i)]

∆λ(i)

h[z(i)]



A sufficient condition for a unique Newton direction is that the matrix of constraint derivatives is of full rank (linear independence

of constraints) and the Hessian matrix of the Lagrange function [∇zz L(z, µ, λ)] projected into the space of the linearized constraints is positive definite. The linearized system actually represents the solution of the following quadratic programming problem: 1 Min ∇F [z(i)]T∆z +
∆ zT ∇zz L[z(i), µ(i), λ(i)]∆z ∆z 2 subject to gA[z(i)] + ∇g A[z(i)]T∆z = 0

and

h[z(i)] + ∇h[z(i)]T∆z = 0

Reformulating the necessary conditions as a linear quadratic program has an interesting side effect. We can simply add linearizations of the inactive inequalities to the problem and let the active set be selected by the algorithm used to solve the linear quadratic program. Problems with calculating second derivatives as well as maintaining positive definiteness of the Hessian matrix can be avoided by approximating this matrix by B(i) using a quasi-Newton formula such as BFGS (Refs. 50, 84, 109, 110, 122, and 259). One maintains positive definiteness by skipping the update if it causes the matrix to lose this property. Here gradients of the Lagrange function are used to calculate the update formula (Refs. 136 and 228). The resulting quadratic program, which generates the search direction at each iteration i, becomes: 1 Min ∇F[z(i)]T∆z +
∆zTB(i)∆z ∆z 2 subject to

g[z(i)] + ∇g[z(i)]T∆z ≤ 0 h[z(i)] + ∇h[z(i)]T∆z = 0

This linear quadratic program will have a unique solution if B(i) is kept positive definite. Efficient solution methods exist for solving it (Refs. 119 and 123). Finally, to ensure convergence of this algorithm from poor starting points, a step size α is chosen along the search direction so that the point at the next iteration (zi + 1 = zi + αd) is closer to the solution of the NLP (Refs. 65, 136, and 254). These problems get very large as the Lagrange function involves all the variables in the problem. If one has a problem with 5000 variables z and the problem has only 10 degrees of freedom (i.e., the partitioning will select 4990 variables x and only 10 variables u), one is still faced with maintaining a matrix B that is 5000 × 5000. See Westerberg (Ref. 40) for references to this case. Interior Point Algorithms for Linear Programming Problems There has been considerable excitement in the popular press about so-called interior point algorithms (Ref. 23) for solving extremely large linear programming problems. Computational demands for these algorithms grow less rapidly than for the Simplex algorithm, with a breakeven point being a few thousand constraints. A key idea for an interior method is that one heads across the feasible region to locate the solution rather than around its edges as one does for the Simplex algorithm. This move is found by computing the direction of steepest descent for the objective function with respect to changing the slack variables. Variables u are computed in terms of the slack variables by using the inequality constraints. The direction of steepest descent is a function of the scaling of the variables used for the problem. See Refs. 6, 124, 199, and 295. Linear Programming The combined term linear programming is given to any method for finding where a given linear function of several variables takes on an extreme value, and what that value is, when the variables are nonnegative and are constrained by linear equalities or inequalities. A very general problem consists of maximizing f = nf = 1 cj zj subject to the constraints zj ≥ 0 ( j = 1, 2, . . . , n) n and j = 1 aij zj ≤ bi (i = 1, 2, . . . , m). With S the set of all points whose coordinates zj satisfy all the constraints, we must ask three questions: (1) Are the constraints consistent? If not, S is empty and there is no solution. (2) If S is not empty, does the function f become unbounded on S? If so, the problem has no solution. If not, then there is a point P of S that is optimal in the sense that if Q is any point of S then f(Q) ≤ f(P). (3) How can we find P? The simplex algorithm, in a sense, prepares the problem before cal-

STATISTICS culation in such a way that favorable answers to these questions are tentatively assumed for the given problem and can be guaranteed for the prepared problem. The calculations then reveal whether or not those assumptions are justified for the given problem. The simplex algorithm terminates automatically, yielding full information on the given problem and so-called dual problem. The dual of the general problem of linear programming is to minimizem d(µ1, . . . , µm) = m  i = 1 µi bi subject to µi ≥ 0 (i = 1, 2, . . . , m) and  i = 1 µ i aif ≥ cj( j = 1, 2, . . . , n). Let A be the matrix [aij], c = [cj], U = [µi] be row vectors, and B = [bi]T, Z = [zj]T be column vectors. In matrix form the original (primal) problem is to maximize f(Z) = CZ subject to Z ≥ 0, AZ ≤ B. The dual is to minimize d(U) = UB subject to U ≥ 0, UA ≥ C. Example Maximize 3z1 + 4z2 subject to the constraints z1 ≥ 0, z2 ≥ 0, 2z1 + 4z2 ≤ 8, and 4z1 + 3z2 ≤ 10. The dual problem is to minimize 8µ1 + 10µ2 subject to the constraints µ1 ≥ 0, µ2 ≥ 0, 2µ1 + 4µ2 ≥ 3, and 5µ1 + 3µ2 ≥ 4. Simplex Method 1. Original problem. Let the column vector [zj]T = z ( j = 1, 2, n . . . , n) and the row vector [cj] = c. To maximize f(z) =  j = 1 cj zj = cTz subject to then n constraints zj ≥ 0 ( j = 1, . . . , n) and m further constraints hi:  j = 1 aij zj° ibi (i = 1, 2, . . . , m) where °i can be ≥ = or ≤. If any bi ≤ 0, multiply hi by − 1; thus we may assume bi ≥ 0. We suppose the m constraints have been arranged so that °i is ≥ for i = 1, . . . , g; °i is = for i = g + 1, . . . , g + e; °i is ≤ for i = g + e + 1, . . . , g + e + l = m. 2. Adjusted original problem. Introduce m + g further variables

3-69

with associated constraints and coefficients for use in f. Thus, replacing j by j + m, f becomes f(x) =  mj =+mn + 1 cj zj and constraints zj ≥ 0 and hi:  mj =+mn + 1 aij zj° i bi, i = 1, . . . , m. 3. Prepared problem. For i = 1, . . . , g replace hi by Hi: zi +  mj =+mn + 1 aij zj zm + n + i = bi, define ci = −M(M > 0 and “large”) and Cm + n + i = 0, and add the constraints zi ≥ 0, zm + n + i ≥ 0. For i = g + e + 1, . . . , m replace hi by Hi. xi +  mj =+mn + 1 aij zj = bi, define ci = 0, and adjoin zi ≥ 0. Let J run from 1 to N = n + m + g; put Z = [zj]T and j = m + 1, . . . , m + n. The new function to be maximized is f(Z) = N g=e vm + n  J = 1 cj z j. Actually this is f(Z) = −M  i = 1 zi +  j = m + 1 cjzj, for all other coefficients are zero. Thus for J = g + e + 1, . . . , m and m + n + 1, . . . , N the variables z j make no contribution to f. They are called slack variables, since they take up the slack permitted by the inequalities (≤ and ≥) in hi. Any variable zj, i = 1, . . . , g + e whose value is not zero gives rise to a large negative term −Mzi. Such a term will keep f(Z) less than it would be with that zi = 0. The effect of ci = −M(i = 1, . . . , g + e) is to make it likely that the optimal solution will have the artificial variables zi = 0(i = 1, . . . , g + e). N The prepared problem now has the form—maximize f(Z) =  j = 1 cjzJ N subject to zJ ≥ 0 and Hi:  j = 1 aijzj = (i = 1, . . . , m), where bi ≥ 0, aiβ = δiβ =

i≠β (β = 1, . . . , m), ai,m + n + β = −δiβ (β = 1, . . . , g) i≠β

l

0

and aij came from hi. The set of feasible points SP (points satisfying all constraints) for the prepared problem is not empty, and f(Z) is bounded above on SP.

STATISTICS GENERAL REFERENCES:

69, 93, 134, 169, 186, 211, 265, 291.

INTRODUCTION Statistics represents a body of knowledge which enables one to deal with quantitative data reflecting any degree of uncertainty. There are six basic aspects of applied statistics. These are: 1. Type of data 2. Random variables 3. Models 4. Parameters 5. Sample statistics 6. Characterization of chance occurrences From these can be developed strategies and procedures for dealing with (1) estimation and (2) inferential statistics. The following has been directed more toward inferential statistics because of its broader utility. Detailed illustrations and examples are used throughout to develop basic statistical methodology for dealing with a broad area of applications. However, in addition to this material, there are many specialized topics as well as some very subtle areas which have not been discussed. The references should be used for more detailed information. Type of Data In general, statistics deals with two types of data: counts and measurements. Counts represent the number of discrete outcomes, such as the number of defective parts in a shipment, the number of lost-time accidents, and so forth. Measurement data are treated as a continuum. For example, the tensile strength of a synthetic yarn theoretically could be measured to any degree of precision. A subtle aspect associated with count and measurement data is that some types of count data can be dealt with through the application of techniques which have been developed for measurement data alone. This ability is due to the fact that some simplified measurement statistics serve as an excellent approximation for the more tedious count statistics. Random Variables Applied statistics deals with quantitative data. In tossing a fair coin the successive outcomes would tend to be

different, with heads and tails occurring randomly over a period of time. Given a long strand of synthetic fiber, the tensile strength of successive samples would tend to vary significantly from sample to sample. Counts and measurements are characterized as random variables, that is, observations which are susceptible to chance. Virtually all quantitative data are susceptible to chance in one way or another. Models Part of the foundation of statistics consists of the mathematical models which characterize an experiment. The models themselves are mathematical ways of describing the probability, or relative likelihood, of observing specified values of random variables. For example, in tossing a coin once, a random variable x could be defined by assigning to x the value 1 for a head and 0 for a tail. Given a fair coin, the probability of observing a head on a toss would be a .5, and similarly for a tail. Therefore, the mathematical model governing this experiment can be written as x

P(x)

0 1

.5 .5

where P(x) stands for what is called a probability function. This term is reserved for count data, in that probabilities can be defined for particular outcomes. The probability function that has been displayed is a very special case of the more general case, which is called the binomial probability distribution. For measurement data which are considered continuous, the term probability density is used. For example, consider a spinner wheel which conceptually can be thought of as being marked off on the circumference infinitely precisely from 0 up to, but not including, 1. In spinning the wheel, the probability of the wheel’s stopping at a specified marking point at any particular x value, where 0 ≤ x < 1, is zero, for example, stopping at the value x =
 .5. For the spinning wheel, the probability density function would be defined by f(x) = 1 for 0 ≤ x < 1. Graphically, this is shown in Fig. 3-55. The relative-probability

3-70

MATHEMATICS 1 n x =
 xi with n = 6 n i=1 = (1/6)(.1009 + .3754 + ⋅⋅⋅ + .6606) = .3899 The sample standard deviation s is defined by the computation s= Density function.

FIG. 3-55

=

concept refers to the fact that density reflects the relative likelihood of occurrence; in this case, each number between 0 and 1 is equally likely. For measurement data, probability is defined by the area under the curve between specified limits. A density function always must have a total area of 1. P [0 ≤ x ≤ .4] = .4 P [.2 ≤ x ≤ .9] = .7 P[.6 ≤ x < 1] = .4 and so forth. Since the probability associated with any particular point value is zero, it makes no difference whether the limit point is defined by a closed interval (≤ or ≥) or an open interval (< or >).

Many different types of models are used as the foundation for statistical analysis. These models are also referred to as populations. Parameters As a way of characterizing probability functions and densities, certain types of quantities called parameters can be defined. For example, the center of gravity of the distribution is defined to be the population mean, which is designated as µ. For the coin toss µ = .5, which corresponds to the average value of x; i.e., for half of the time x will take on a value 0 and for the other half a value 1. The average would be .5. For the spinning wheel, the average value would also be .5. Another parameter is called the standard deviation, which is designated as σ. The square of the standard deviation is used frequently and is called the popular variance, σ2. Basically, the standard deviation is a quantity which measures the spread or dispersion of the distribution from its mean µ. If the spread is broad, then the standard deviation will be larger than if it were more constrained. For specified probability and density functions, the respective means and variances are defined by the following: Probability functions

Probability density functions E(x) = µ =

Var(x) = σ =  (x − µ) P(x) 2

x f(x) dx x

x

Var(x) = σ =

2

2

(x − µ) f(x) dx 2

x

x

(xi − x)2

n−1

  

n x 2i − ( xi)2

n(n − 1)

In effect, this represents the root of a statistical average of the squares. The divisor quantity (n − 1) will be referred to as the degrees of freedom. The value of n − 1 is used in the denominator because the deviations from the sample average must total zero, or

 (x − x) = 0 i

Thus knowing n − 1 values of xi − x permits calculation of the nth value of xi − x. The sample value of the standard deviation for the data given is .3686. The following is a tabulation of the deviations (xi − xj) for the data:

Example For the density of Fig. 3-55 the

E(x) = µ =  x P(x)

 

where E(x) is defined to be the expected or average value of x. Sample Statistics Many types of sample statistics will be defined. Two very special types are the sample mean, designated as x, and the sample standard deviation, designated as s. These are, by definition, random variables. Parameters like µ and σ are not random variables; they are fixed constants. Example In an experiment, six random numbers (rounded to four decimal places) were observed from the uniform distribution f(x) = 1 for 0 ≤ x < 1: .1009 .3754 .0842 .9901 .1280 .6606 The sample mean corresponds to the arithmetic average of the observations, which will be designated as x1 through x6, where

x .1009 .3754 .0842 .9901 .1280 .6606

x − x −.2890 −.0145 −.3057 .6002 −.2619 .2707

x = .3899

s = .3686

In effect, the standard deviation quantifies the relative magnitude of the deviation numbers, i.e., a special type of “average” of the distance of points from their center. In statistical theory, it turns out that the corresponding variance quantities s2 have remarkable properties which make possible broad generalities for sample statistics and therefore also their counterparts, the standard deviations. For the corresponding population, the parameter values are µ = .50 and σ = .2887. If, instead of using individual observations only, averages of six were reported, then the corresponding population parameter values would be µ = .50 and σx = σ/
6 = .1179. The corresponding variance for an average will be written occasionally as Var (x) = var (x)/n. In effect, the variance of an average is inversely proportional to the sample size n, which reflects the fact that sample averages will tend to cluster about µ much more closely than individual observations. This is illustrated in greater detail under “Measurement Data and Sampling Densities.” Characterization of Chance Occurrences To deal with a broad area of statistical applications, it is necessary to characterize the way in which random variables will vary by chance alone. The basic foundation for this characteristic is laid through a density called the gaussian, or normal, distribution. Determining the area under the normal curve is a very tedious procedure. However, by standardizing a random variable that is normally distributed, it is possible to relate all normally distributed random variables to one table. The standardization is defined by the identity z = (x − µ)/σ, where z is called the unit normal. Further, it is possible to standardize the sampling distribution of averages x by the identity z = (x − µ)/(σ/
 n). A remarkable property of the normal distribution is that, almost regardless of the distribution of x, sample averages x will approach the gaussian distribution as n gets large. Even for relatively small values of n, of about 10, the approximation in most cases is quite close. For example, sample averages of size 10 from the uniform distribution will have essentially a gaussian distribution. Also, in many applications involving count data, the normal distribution can be used as a close approximation. In particular, the approximation is quite close for the binomial distribution within certain guidelines.

STATISTICS ENUMERATION DATA AND PROBABILITY DISTRIBUTIONS Introduction Many types of statistical applications are characterized by enumeration data in the form of counts. Examples are the number of lost-time accidents in a plant, the number of defective items in a sample, and the number of items in a sample that fall within several specified categories. The sampling distribution of count data can be characterized through probability distributions. In many cases, count data are appropriately interpreted through their corresponding distributions. However, in other situations analysis is greatly facilitated through distributions which have been developed for measurement data. Examples of each will be illustrated in the following subsections. Binomial Probability Distribution Nature Consider an experiment in which each outcome is classified into one of two categories, one of which will be defined as a success and the other as a failure. Given that the probability of success p is constant from trial to trial, then the probability of observing a specified number of successes x in n trials is defined by the binomial distribution. The sequence of outcomes is called a Bernoulli process, Nomenclature n = total number of trials x = number of successes in n trials p = probability of observing a success on any one trial pˆ = x/n, the proportion of successes in n trials Probability Law x n x P(x) = P
= p (1 − p)n − x x = 0, 1, 2, . . . , n x n

  

where

 x =
x!(n − x)! n

n!

Properties E(x) = np E( p ˆ)=p

5. The number of spinning machines that break down during 1 day at a large plant. Nomenclature x = total number of arrivals in a total length L or total period T a = average rate of arrivals for a unit length or unit time λ = aL = expected or average number of arrivals for the total length L λ = aT = expected or average number of arrivals for the total time T Probability Law Given that a is constant for the total length L or period T, the probability of observing x arrivals in some period L or T is given by λx P(x) =
e−λ x = 0, 1, 2, . . . x! Properties E(x) = λ

Var (x) = λ

Example The number of broken filaments in a threadline has been averaging .015 per yard. What is the probability of observing exactly two broken filaments in the next 100 yd? In this example, a = .015/yd and L = 100 yd; therefore λ = (.015)(100) = 1.5: (1.5)2 P(x = 2) =
e−1.5 = .2510 2! Example A commercial item is sold in a retail outlet as a unit product. In the past, sales have averaged 10 units per month with no seasonal variation. The retail outlet must order replacement items 2 months in advance. If the outlet starts the next 2-month period with 25 items on hand, what is the probability that it will stock out before the end of the second month? Given a = 10/month, then λ = 10 × 2 = 20 for the total period of 2 months: ∞

25

26

0

P(x ≥ 26) =  P(x) = 1 −  P(x)



2025 20x −20 −20 20 202 e =e 1 +
+
+ ⋅⋅⋅ +
0
x! 1 2! 25! 25



= .887815

Var(x) = np(1 − p) Var( p ˆ ) = p(1 − p)/n

Geometric Probability Distribution Nature Consider an experiment in which each outcome is classified into one of two categories, one of which will be defined as a success and the other as a failure. Given that the probability of success p is constant from trial to trial, then the probability of observing the first success on the xth trial is defined by the geometric distribution. Nomenclature p = probability of observing a success on any one trial x = the number of trials to obtain the first success Probability Law P(x) = p(1 − p)x − 1

x = 1, 2, 3, . . .

Properties E(x) = 1/p

3-71

Var (x) = (1 − p)/p2

Poisson Probability Distribution Nature In monitoring a moving threadline, one criterion of quality would be the frequency of broken filaments. These can be identified as they occur through the threadline by a broken-filament detector mounted adjacent to the threadline. In this context, the random occurrences of broken filaments can be modeled by the Poisson distribution. This is called a Poisson process and corresponds to a probabilistic description of the frequency of defects or, in general, what are called arrivals at points on a continuous line or in time. Other examples include: 1. The number of cars (arrivals) that pass a point on a high-speed highway between 10:00 and 11:00 A.M. on Wednesdays 2. The number of customers arriving at a bank between 10:00 and 10:10 A.M. 3. The number of telephone calls received through a switchboard between 9:00 and 10:00 A.M. 4. The number of insurance claims that are filed each week

Therefore P(x ≥ 26) = .112185 or roughly an 11 percent chance of a stockout.

Hypergeometric Probability Distribution Nature In an experiment in which one samples from a relatively small group of items, each of which is classified in one of two categories, A or B, the hypergeometric distribution can be defined. One example is the probability of drawing two red and two black cards from a deck of cards. The hypergeometric distribution is the analog of the binomial distribution when successive trials are not independent, i.e., when the total group of items is not infinite. This happens when the drawn items are not replaced. Nomenclature N = total group size n = sample group size X = number of items in the total group with a specified attribute A N − X = number of items in the total group with the other attribute B x = number of items in the sample with a specified attribute A n − x = number of items in the sample with the other attribute B

Category A Category B Total

Probability Law P(x) =

Population

Sample

X N−X N

x n−x n

Nn −− Xx   XxNn

nX E(x) =
N N−n var (x) = nP(1 − P)
N−1

3-72

MATHEMATICS

Example What is the probability that an appointed special committee of 4 has no female members when the members are randomly selected from a candidate group of 10 males and 7 females? 10 7 4 0 P(x = 0) = = .0882 17 4

    

Example A bin contains 300 items, of which 240 are good and 60 are defective. In a sample of 6 what is the probability of selecting 4 good and 2 defective items by chance?

 4  2  = .2478 300 6 240

P(x) =

60

Multinomial Distribution Nature For an experiment in which successive outcomes can be classified into two or more categories and the probabilities associated with the respective outcomes remain constant, then the experiment can be characterized through the multinomial distribution. Nomenclature n = total number of trials k = total number of distinct categories pj = probability of observing category j on any one trial, j = 1, 2, . . . , k xj = total number of occurrences in category j in n trials Probability Law n! P(x1, x2, . . . , xk) =

p1x1 p2x2 ⋅⋅⋅ pkx k x1! x2! . . . xk! Example In tossing a die 12 times, what is the probability that each face value will occur exactly twice? 12! 1 2 1 2 1 2 1 2 1 2 1 2 p(2, 2, 2, 2, 2, 2) =







= .003438 2!2!2!2!2!2! 6 6 6 6 6 6

Monday 16

Tuesday 17

Wednesday 18

Thursday 19

Friday 20

Groups of 25 pooled

37.68 36.38 38.43 39.07 33.06 x = 36.92 s = 2.38

35.97 35.92 36.51 33.89 36.01 35.66 1.02

33.71 32.34 33.29 32.81 37.13 33.86 1.90

35.61 37.13 31.37 35.89 36.33 35.27 2.25

36.65 37.91 42.18 39.25 33.32 37.86 3.27

35.91 2.52

Pooled sample of 100: x = 35.16

Even if the process were at steady state, tensile strength, a key property would still reflect some variation. Steady state, or stable operation of any process, has associated with it a characteristic variation. Superimposed on this is the testing method, which is itself a process with its own characteristic variation. The observed variation is a composite of these two variations. Assume that the table represents “typical” production-line performance. The numbers themselves have been generated on a computer and represent random observations from a population with µ = 35 and a population standard deviation σ = 2.45. The sample values reflect the way in which tensile strength can vary by chance alone. In practice, a production supervisor unschooled in statistics but interested in high tensile performance would be despondent on the eighth day and exuberant on the twentieth day. If the supervisor were more concerned with uniformity, the lowest and highest points would have been on the eleventh and seventeenth days. An objective of statistical analysis is to serve as a guide in decision making in the context of normal variation. In the case of the production supervisor, it is to make a decision, with a high probability of being correct, that something has in fact changed the operation. Suppose that an engineering change has been made in the process and five new tensile samples have been tested with the results: 36.81 38.34 34.87 39.58 36.12

      

MEASUREMENT DATA AND SAMPLING DENSITIES Introduction The following example data are used throughout this subsection to illustrate concepts. Consider, for the purpose of illustration, that five synthetic-yarn samples have been selected randomly from a production line and tested for tensile strength on each of 20 production days. For this, assume that each group of five corresponds to a day, Monday through Friday, for a period of 4 weeks: Monday 1

Tuesday 2

Wednesday 3

Thursday 4

Friday 5

36.48 35.33 35.92 32.28 31.61 x = 34.32 s = 2.22

38.06 31.86 33.81 30.30 35.27 33.86 3.01

35.28 36.58 38.81 33.31 33.88 35.57 2.22

36.34 36.25 30.46 37.37 37.52 35.59 2.92

36.73 37.17 33.07 34.27 36.94 35.64 1.85

6

7

8

9

10

38.67 32.08 33.79 32.85 35.22 x = 34.52 s = 2.60

36.62 33.05 35.43 36.63 31.46 34.64 2.30

35.03 36.22 32.71 32.52 27.23 32.74 3.46

35.80 33.16 35.19 32.91 35.44 34.50 1.36

36.82 36.49 32.83 32.43 34.16 34.54 2.03

11

12

13

14

15

39.63 34.38 36.51 30.00 39.64 x = 36.03 s = 4.04

34.52 37.39 34.16 35.76 37.63 35.89 1.59

36.05 35.36 35.00 33.61 36.98 35.40 1.25

36.64 31.18 36.13 37.51 39.05 36.10 2.96

31.57 36.21 33.84 35.01 34.95 34.32 1.75

Groups of 25 pooled

34.19 2.35

x = 37.14 s = 1.85

In this situation, management would inquire whether the product has been improved by increased tensile strength. To answer this question, in addition to a variety of analogous questions, it is necessary to have some type of scientific basis upon which to draw a conclusion. A scientific basis for the evaluation and interpretation of data is contained in the accompanying table descriptions. These tables characterize the way in which sample values will vary by chance alone in the context of individual observations, averages, and variances.

Table number

35.00 2.40

s = 2.47

Designated symbol

Variable

Sampling distribution of

3-4

z

x−µ
σ

3-4

z

x − µ
σ/
 n

Averages

3-5

t

x − µ
s/
 n

Averages when σ is unknown*

3-6

χ2

(s2/σ2)(df)

3-7

F

s12 /s 22

Observations*

Variances* Ratio of two independent sample variances*

*When sampling from a gaussian distribution.

35.55 2.42

Normal Distribution of Observations Many types of data follow what is called the gaussian, or bell-shaped, curve; this is especially true of averages. Basically, the gaussian curve is a purely mathematical function which has very special properties. However, owing to some mathematically intractable aspects primary use of the function is restricted to tabulated values.

STATISTICS

3-73

0 and 2 is .4772; therefore, the area between 1 and 2 is .4772 − .3413 = .1359. Also, since the normal curve is symmetric, areas to the left can be determined in exactly the same way. For example, the area between −2 and +1 would include the area between −2 and 0, .4772 (the same as 0 to 2), plus the area between 0 and 1, .3413, or a total area of .8185. Any types of observation which are applicable to the normal curve can be transformed to Z values by the relationship z = (x − µ)/σ and, conversely, Z values to x values by x = µ + σz, as shown in Fig. 3-56. For example, for tensile strength, with µ = 35 and σ = 2.45, this would dictate z = (x − 35)/2.45 and x = 35 + 2.45z. Example What proportion of tensile values will fall between 34 and 36? FIG. 3-56

z1 = (34 − 35)/2.45 = −.41

Transformation of z values.

z2 = (36 − 35)/2.45 = .41

P[−.41 ≤ z ≤ .41] = .3182, or roughly 32 percent

Basically, the tabled values represent area (proportions or probability) associated with a scaling variable designated by Z in Fig. 3-56. The normal curve is centered at 0, and for particular values of Z, designated as z, the tabulated numbers represent the corresponding area under the curve between 0 and z. For example, between 0 and 1 the area is .3413. (Get this number from Table 3-4. The value of A includes the area on both sides of zero. Thus we want A/2. For z = 1, A = 0.6827, A/2 = 0.3413. For z = 2, A/2 = 0.4772.) The area between

TABLE 3-4

The value 0.3182 is interpolated from Table 3-4 using z = 0.40, A = 0.3108, and z = 0.45, A = 0.3473.

Example What midrange of tensile values will include 95 percent of the sample values? Since P[−1.96 ≤ z ≤ 1.96] = .95, the corresponding values of x are x1 = 35 − 1.96(2.45) = 30.2 x2 = 35 + 1.96(2.45) = 39.8 P[30.2 ≤ x ≤ 39.8] = .95

or

Ordinates and Areas between Abscissa Values -z and +z of the Normal Distribution Curve

z

X

Y

A

1−A

z

X

0 0.05  .10  .15  .20

µ µ  0.05σ µ  .10σ µ  .15σ µ  .20σ

0.399 .398 .397 .394 .391

0.0000 .0399 .0797 .1192 .1585

1.0000 0.9601 .9203 .8808 .8415

1.50 1.55 1.60 1.65 1.70

µ  1.50σ µ  1.55σ µ  1.60σ µ  1.65σ µ  1.70σ

0.1295 .1200 .1109 .1023 .0940

0.8664 .8789 .8904 .9011 .9109

0.1336 .1211 .1096 .0989 .0891

    

.25 .30 .35 .40 .45

µ µ µ µ µ

.25σ .30σ .35σ .40σ .45σ

.387 .381 .375 .368 .361

.1974 .2358 .2737 .3108 .3473

.8026 .7642 .7263 .6892 .6527

1.75 1.80 1.85 1.90 1.95

µ  1.75σ µ  1.80σ µ  1.85σ µ  1.90σ µ  1.95σ

.0863 .0790 .0721 .0656 .0596

.9199 .9281 .9357 .9446 .9488

.0801 .0719 .0643 .0574 .0512

    

.50 .55 .60 .65 .70

µ µ+ µ µ µ

.50σ .55σ .60σ .65σ .70σ

.352 .343 .333 .323 .312

.3829 .4177 .4515 .4843 .5161

.6171 .5823 .5485 .5157 .4839

2.00 2.05 2.10 2.15 2.20

µ  2.00σ µ  2.05σ µ  2.10σ µ  2.15σ µ  2.20σ

.0540 .0488 .0440 .0396 .0335

.9545 .9596 .9643 .9684 .9722

.0455 .0404 .0357 .0316 .0278

    

.75 .80 .85 .90 .95

µ µ µ µ µ

.75σ .80σ .85σ .90σ .95σ

.301 .290 .278 .266 .254

.5467 .5763 .6047 .6319 .6579

.4533 .4237 .3953 .3681 .3421

2.25 2.30 2.35 2.40 2.45

µ  2.25σ µ  2.30σ µ  2.35σ µ  2.40σ µ  2.45σ

.0317 .0283 .0252 .0224 .0198

.9756 .9786 .9812 .9836 .9857

.0244 .0214 .0188 .0164 .0143

1.00 1.05 1.10 1.15 1.20

µ  1.00σ µ  1.05σ µ  1.10σ µ  1.15σ µ  1.20σ

.242 .230 .218 .206 .194

.6827 .7063 .7287 .7499 .7699

.3173 .2937 .2713 .2501 .2301

2.50 2.55 2.60 2.65 2.70

µ  2.50σ µ  2.55σ µ  2.60σ µ  2.65σ µ  2.70σ

.0175 .0154 .0136 .0119 .0104

.9876 .9892 .9907 .9920 .9931

.0124 .0108 .0093 .0080 .0069

1.25 1.30 1.35 1.40 1.45

µ  1.25σ µ  1.30σ µ  1.35σ µ  1.40σ µ  1.45σ

.183 .171 .160 .150 .139

.7887 .8064 .8230 .8385 .8529

.2113 .1936 .1770 .1615 .1471

2.75 2.80 2.85 2.90 2.95

µ  2.75σ µ  2.80σ µ  2.85σ µ  2.90σ µ  2.95σ

.0091 .0079 .0069 .0060 .0051

.9940 .9949 .9956 .9963 .9968

.0060 .0051 .0044 .0037 .0032

1.50

µ  1.50σ

.130

.8664

.1336

3.00 4.00 5.00

µ  3.00σ µ  4.00σ µ  5.00s

.0044 .0001 .000001

.9973 .99994 .9999994

.0027 .00006 .0000006

0.000  .126  .253  .385  .524  .674  .842

µ µ  0.126σ µ  .253σ µ  .385σ µ  .524σ µ  .674σ µ  .842σ

0.3989 .3958 .3863 .3704 .3477 .3178 .2800

.0000 .1000 .2000 .3000 .4000 .5000 .6000

1.0000 0.9000 .8000 .7000 .6000 .5000 .4000

1.036 1.282 1.645 1.960 2.576 3.291 3.891

µ  1.036σ µ  1.282σ µ  1.645σ µ  1.960σ µ  2.576σ µ  3.291σ µ  3.891σ

Y

0.2331 .1755 .1031 .0584 .0145 .0018 .0002

A

0.7000 .8000 .9000 .9500 .9900 .9990 .9999

1−A

0.3000 .2000 .1000 .0500 .0100 .0010 .0001

3-74

MATHEMATICS

Normal Distribution of Averages An examination of the tensile-strength data previously tabulated would show that the range (largest minus the smallest) of tensile strength within days averages 5.72. The average range in x values within each week is 2.37, while the range in the four weekly averages is 1.72. This reflects the fact that averages tend to be less variable in a predictable way. Given that the variance of x is var (x) = σ2, then the variance of x based on n observations is var (x) = σ2/n. For averages of n observations, the corresponding relationship for the Z-scale relationship is σ z = (x − µ/σ/
 n) or z x = µ +

 n Example What proportion of daily tensile averages will fall between 34 and 36? z1 = (34 − 35)/(2.45/
5) = −.91

z2 = (36 − 35)/(2.45/
5) = .91

P[−.91 ≤ z ≤ .91] = .6372, or roughly 64 percent

Example What midrange of daily tensile averages will include 95 percent of the sample values? x1 = 35 − 1.96(2.45/
5) = 32.85 x2 = 35 + 1.96(2.45/
5) = 37.15 P[32.85 ≤ x ≤ 37.15] = .95

or

Example What proportion of weekly tensile averages will fall between 34 and 36? z1 = (34 − 35)/(2.45/
25) = −2.04 z2 = (36 − 35)/(2.45/
25) = 2.04 P[−2.04 ≤ z ≤ 2.04] = .9586, or roughly 96 percent

Distribution of Averages The normal curve relies on a knowledge of σ, or in special cases, when it is unknown, s can be used with the normal curve as an approximation when n > 30. For example, with n > 30 the intervals x  s and x  2s will include roughly 68 and 95 percent of the sample values respectively when the distribution is normal. In applications sample sizes are usually small and σ unknown. In these cases, the t distribution can be used where t = (x − µ)/(s/
n ) or  x = µ + ts/
n The t distribution is also symmetric and centered at zero. It is said to be robust in the sense that even when the individual observations x are not normally distributed, sample averages of x have distributions which tend toward normality as n gets large. Even for small n of 5 through 10, the approximation is usually relatively accurate. In reference to the tensile-strength table, consider the summary statistics x and s by days. For each day, the t statistic could be computed. If this were repeated over an extensive simulation and the resultant t quantities plotted in a frequency distribution, they would match the corresponding distribution of t values summarized in Table 3-5. Since the t distribution relies on the sample standard deviation s, the resultant distribution will differ according to the sample size n. To designate this difference, the respective distributions are classified according to what are called the degrees of freedom and abbreviated as df. In simple problems, the df are just the sample size minus 1. In more complicated applications the df can be different. In general, degrees of freedom are the number of quantities minus the number of constraints. For example, four numbers in a square which must have row and column sums equal to zero have only one df, i.e., four numbers minus three constraints (the fourth constraint is redundant).

TABLE 3-5 df

t.30

t.20

t.10

t.05

t.025

t.01

t.005

1 2 3 4 5

0.325 .289 .277 .271 .267

0.727 .617 .584 .569 .559

1.376 1.061 0.978 .941 .920

3.078 1.886 1.638 1.533 1.476

6.314 2.920 2.353 2.132 2.015

12.706 4.303 3.182 2.776 2.571

31.821 6.965 4.541 3.747 3.365

63.657 9.925 5.841 4.604 4.032

6 7 8 9 10

.265 .263 .262 .261 .260

.553 .549 .546 .543 .542

.906 .896 .889 .883 .879

1.440 1.415 1.397 1.383 1.372

1.943 1.895 1.860 1.833 1.812

2.447 2.365 2.306 2.262 2.228

3.143 2.998 2.896 2.821 2.764

3.707 3.499 3.355 3.250 3.169

11 12 13 14 15

.260 .259 .259 .258 .258

.540 .539 .538 .537 .536

.876 .873 .870 .868 .866

1.363 1.356 1.350 1.345 1.341

1.796 1.782 1.771 1.761 1.753

2.201 2.179 2.160 2.145 2.131

2.718 2.681 2.650 2.624 2.602

3.106 3.055 3.012 2.977 2.947

16 17 18 19 20

.258 .257 .257 .257 .257

.535 .534 .534 .533 .533

.865 .863 .862 .861 .860

1.337 1.333 1.330 1.328 1.325

1.746 1.740 1.734 1.729 1.725

2.120 2.110 2.101 2.093 2.086

2.583 2.567 2.552 2.539 2.528

2.921 2.898 2.878 2.861 2.845

21 22 23 24 25

.257 .256 .256 .256 .256

.532 .532 .532 .531 .531

.859 .858 .858 .857 .856

1.323 1.321 1.319 1.318 1.316

1.721 1.717 1.714 1.711 1.708

2.080 2.074 2.069 2.064 2.060

2.518 2.508 2.500 2.492 2.485

2.831 2.819 2.807 2.797 2.787

26 27 28 29 30

.256 .256 .256 .256 .256

.531 .531 .530 .530 .530

.856 .855 .855 .854 .854

1.315 1.314 1.313 1.311 1.310

1.706 1.703 1.701 1.699 1.697

2.056 2.052 2.048 2.045 2.042

2.479 2.473 2.467 2.462 2.457

2.779 2.771 2.763 2.756 2.750

40 60 120 ∞

.255 .254 .254 .253

.529 .527 .526 .524

.851 .848 .845 .842

1.303 1.296 1.289 1.282

1.684 1.671 1.658 1.645

2.021 2.000 1.980 1.960

2.423 2.390 2.358 2.326

2.704 2.660 2.617 2.576

Above values refer to a single tail outside the indicated limit of t. For example, for 95 percent of the area to be between −t and +t in a two-tailed t distribution, use the values for t0.025 or 2.5 percent for each tail.

Example What is the sample value of t for the first day of tensile data? Sample t = (34.32 − 35)/(2.22/
5) = −.68 Note that on the average 90 percent of all such sample values would be expected to fall within the interval 2.132.

t Distribution for the Difference in Two Sample Means Population Variances Are Equal The t distribution can be readily extended to the difference in two sample means when the respective populations have the same variance σ2: (x1 − x2) − (µ1 − µ2) t =

sp
1 /n +1 /n2 1  where s 2p is a pooled variance defined by (n1 − 1)s12 + (n2 − 1)s22 s 2p =


(n1 − 1) + (n2 − 1) In this application, the t distribution has (n1 + n2 − 2) df. Population Variances Are Unequal When population variances are unequal, an approximate t quantity can be used:

Example For a sample size n = 5, what values of t define a midarea of 90 percent. For 4 df the tabled value of t corresponding to a midarea of 90 percent is 2.132; i.e., P[−2.132 ≤ t ≤ 2.132] = .90. Example For a sample size n = 25, what values of t define a midarea of 95 percent? For 24 df the tabled value of t corresponding to a midarea of 95 percent is 2.064; i.e., P[−2.064 ≤ t ≤ 2.064] = .95.

Values of t

t.40

with and

(x1 − x2) − (µ1 − µ2) t =

a + b
 a = s12 / n1 b = s22 / n2 (a + b)2 df =


a2/(n1 − 1) + b2/(n2 − 1)

STATISTICS Chi-Square Distribution For some industrial applications, product uniformity is of primary importance. The sample standard deviation s is most often used to characterize uniformity. In dealing with this problem, the chi-square distribution can be used where χ 2 = (s2/σ2) (df). The chi-square distribution is a family of distributions which are defined by the degrees of freedom associated with the sample variance. For most applications, df is equal to the sample size minus 1. The probability distribution function is −(df )2 p(y) = y0ydf − 2 exp
2 where y0 is chosen such that the integral of p(y) over all y is one. In terms of the tensile-strength table previously given, the respective chi-square sample values for the daily, weekly, and monthly figures could be computed. The corresponding df would be 4, 24, and 99 respectively. These numbers would represent sample values from the respective distributions which are summarized in Table 3-6. In a manner similar to the use of the t distribution, chi square can be interpreted in a direct probabilistic sense corresponding to a midarea of (1 − α):





P[χ 21 ≤ (s2/σ2)(df) ≤ χ 22 ] = 1 − α where χ 21 corresponds to a lower-tail area of α/2 and χ 22 an upper-tail area of α/2. The basic underlying assumption for the mathematical derivation of chi square is that a random sample was selected from a normal distribution with variance σ2. When the population is not normal but skewed, square probabilities could be substantially in error. Example On the basis of a sample size n = 5, what midrange of values will include the sample ratio s/σ with a probability of 90 percent? Use Table 3-6 for 4 df and read χ 21 = 0.484 for a lower tail area of 0.05/2, 2.5 percent, and read χ 22 = 11.1 for an upper tail area of 97.5 percent.

TABLE 3-6

3-75

P[.484 ≤ (s2/σ2)(4) ≤ 11.1] = .90 P[.35 ≤ s/σ ≤ 1.66] = .90

or

Example On the basis of a sample size n = 25, what midrange of values will include the sample ratio s/σ with a probability of 90 percent? P[12.4 ≤ (s2/σ2)(24) ≤ 39.4] = .90 P[.72 ≤ s/σ ≤ 1.28] = .90

or

This states that the sample standard deviation will be at least 72 percent and not more than 128 percent of the population variance 90 percent of the time. Conversely, 10 percent of the time the standard deviation will underestimate or overestimate the population standard deviation by the corresponding amount. Even for samples as large as 25, the relative reliability of a sample standard deviation is poor.

The chi-square distribution can be applied to other types of application which are of an entirely different nature. These include applications which are discussed under “Goodness-of-Fit Test” and “Two-Way Test for Independence of Count Data.” In these applications, the mathematical formulation and context are entirely different, but they do result in the same table of values. F Distribution In reference to the tensile-strength table, the successive pairs of daily standard deviations could be ratioed and squared. These ratios of variance would represent a sample from a distribution called the F distribution or F ratio. In general, the F ratio is defined by the identity F(γ1, γ 2) = s12 / s22 where γ1 and γ2 correspond to the respective df’s for the sample variances. In statistical applications, it turns out that the primary area of interest is found when the ratios are greater than 1. For this reason, most tabled values are defined for an upper-tail area. However, defining F2 to be that value corresponding to an upper-tail area of α/2, then F1 for a lower-tail area of α/2 can be determined through the identity

Percentiles of the c2 Distribution Percent

df

10

90

95

97.5

0.000039 .0100 .0717 .207 .412

0.00016 .0201 .115 .297 .554

0.00098 .0506 .216 .484 .831

0.0039 .1026 .352 .711 1.15

0.0158 .2107 .584 1.064 1.61

2.71 4.61 6.25 7.78 9.24

3.84 5.99 7.81 9.49 11.07

5.02 7.38 9.35 11.14 12.83

6.63 9.21 11.34 13.28 15.09

7.88 10.60 12.84 14.86 16.75

6 7 8 9 10

.676 .989 1.34 1.73 2.16

.872 1.24 1.65 2.09 2.56

1.24 1.69 2.18 2.70 3.25

1.64 2.17 2.73 3.33 3.94

2.20 2.83 3.49 4.17 4.87

10.64 12.02 13.36 14.68 15.99

12.59 14.07 15.51 16.92 18.31

14.45 16.01 17.53 19.02 20.48

16.81 18.48 20.09 21.67 23.21

18.55 20.28 21.96 23.59 25.19

11 12 13 14 15

2.60 3.07 3.57 4.07 4.60

3.05 3.57 4.11 4.66 5.23

3.82 4.40 5.01 5.63 6.26

4.57 5.23 5.89 6.57 7.26

5.58 6.30 7.04 7.79 8.55

17.28 18.55 19.81 21.06 22.31

19.68 21.03 22.36 23.68 25.00

21.92 23.34 24.74 26.12 27.49

24.73 26.22 27.69 29.14 30.58

26.76 28.30 29.82 31.32 32.80

16 18 20 24 30

5.14 6.26 7.43 9.89 13.79

5.81 7.01 8.26 10.86 14.95

6.91 8.23 9.59 12.40 16.79

7.96 9.39 10.85 13.85 18.49

9.31 10.86 12.44 15.66 20.60

23.54 25.99 28.41 33.20 40.26

26.30 28.87 31.41 36.42 43.77

28.85 31.53 34.17 39.36 46.98

32.00 34.81 37.57 42.98 50.89

34.27 37.16 40.00 45.56 53.67

40 60 120

20.71 35.53 83.85

22.16 37.48 86.92

24.43 40.48 91.58

26.51 43.19 95.70

29.05 46.46 100.62

51.81 74.40 140.23

55.76 79.08 146.57

59.34 83.30 152.21

63.69 88.38 158.95

66.77 91.95 163.64

1 2 3 4 5

0.5

1

2.5

5

99

99.5

For large values of degrees of freedom the approximate formula



2 χ 2a = n 1 −
+ za 9n

2
  9n

3

where za is the normal deviate and n is the number of degrees of freedom, may be used. For example, χ.299 = 60[1 − 0.00370 + 2.326(0.06086)]3 = 60(1.1379)3 = 88.4 for the 99th percentile for 60 degrees of freedom.

3-76

MATHEMATICS F1(γ1, γ 2) = 1/F2(γ 2, γ1)

With 103 df and α = .05, t = 1.96 using t.025 in Table 3-5. Therefore

The F distribution, similar to the chi square, is sensitive to the basic assumption that sample values were selected randomly from a normal distribution. Example For two sample variances with 4 df each, what limits will bracket their ratio with a midarea probability of 90 percent? Use Table 3-7 with 4 df in the numerator and denominator and upper 5 percent points (to get both sides totaling 10 percent). The entry is 6.39. Thus: P[1/6.39 ≤ s 12 /s 22 ≤ 6.39] = .90 P[.40 ≤ s1 /s2 ≤ 2.53] = .90

or

Confidence Interval for a Mean For the daily sample tensilestrength data with 4 df it is known that P[−2.132 ≤ t ≤ 2.132] = .90. This states that 90 percent of all samples will have sample t values which fall within the specified limits. In fact, for the 20 daily samples exactly 16 do fall within the specified limits (note that the binomial with n = 20 and p = .90 would describe the likelihood of exactly none through 20 falling within the prescribed limits—the sample of 20 is only a sample). Consider the new daily sample (with n = 5, x = 37.14, and s = 1.85) which was observed after a process change. In this case, the same probability holds. However, in this instance the sample value of t cannot be computed, since the new µ, under the process change, is not known. Therefore P[−2.132 ≤ (37.14 − µ)/(1.85/
5) ≤ 2.132] = .90. In effect, this identity limits the magnitude of possible values for µ. The magnitude of µ can be only large enough to retain the t quantity above −2.132 and small enough to retain the t quantity below +2.132. This can be found by rearranging the quantities within the bracket; i.e., P[35.78 ≤ µ ≤ 38.90] = .90. This states that we are 90 percent sure that the interval from 35.78 to 38.90 includes the unknown parameter µ. In general, s s P x − t
≤ µ ≤ x + t
= 1 − α
 n
 n





where t is defined for an upper-tail area of α/2 with (n − 1) df. In this application, the interval limits (x + t s/
n ) are random variables which will cover the unknown parameter µ with probability (1 − α). The converse, that we are 100 (1 − α) percent sure that the parameter value is within the interval, is not correct. This statement defines a probability for the parameter rather than the probability for the interval. Example What values of t define the midarea of 95 percent for weekly samples of size 25, and what is the sample value of t for the second week? P[−2.064 ≤ t ≤ 2.064] = .95 (34.19 − 35)/(2.35/
25) = 1.72.

and

Example For the composite sample of 100 tensile strengths, what is the 90 percent confidence interval for µ? Use Table 3-5 for t.05 with df ≈ ∞.





2.47 2.47 P 35.16 − 1.645
< µ < 35.16 + 1.645
= .90
 100
 100 or

P[34.75 ≤ µ ≤ 35.57] = .90

Confidence Interval for the Difference in Two Population Means The confidence interval for a mean can be extended to include the difference between two population means. This interval is based on the assumption that the respective populations have the same variance σ2:

 1/n 2 ≤ µ1 − µ2 ≤ (x1 − x2) + tsp
1/n   2 (x1 − x2) − tsp
1/n 1/n 1 + 1 + Example Compute the 95 percent confidence interval based on the original 100-point sample and the subsequent 5-point sample: 99(2.47)2 + 4(1.85)2 s p2 =


= 5.997 103 or

sp = 2.45

00  = −1.98  2.20 +1/5 (35.16 − 37.14)  1.96(2.45)
1/1 or

−4.18 ≤ (µ1 − µ2) ≤ .22

Note that if the respective samples had been based on 52 observations each rather than 100 and 5, the uncertainty factor would have been .94 rather than the observed 2.20. The interval width tends to be minimum when n1 = n2.

Confidence Interval for a Variance The chi-square distribution can be used to derive a confidence interval for a population variance σ2 when the parent population is normally distributed. For a 100(1 − α) percent confidence interval (df )s2 (df )s2 ≤ σ2 ≤

2 χ2 χ12 where χ 12 corresponds to a lower-tail area of α/2 and χ 22 to an upper-tail area of α/2. Example For the first week of tensile-strength samples compute the 90 percent confidence interval for σ2 (df = 24, corresponding to n = 25, using 5 percent and 95 percent in Table 3-6): 24(2.40)2 24(2.40)2
≤ σ2 ≤
36.4 13.8 3.80 ≤ σ2 ≤ 10.02 or

1.95 ≤ σ ≤ 3.17

TESTS OF HYPOTHESIS General Nature of Tests The general nature of tests can be illustrated with a simple example. In a court of law, when a defendant is charged with a crime, the judge instructs the jury initially to presume that the defendant is innocent of the crime. The jurors are then presented with evidence and counterargument as to the defendant’s guilt or innocence. If the evidence suggests beyond a reasonable doubt that the defendant did, in fact, commit the crime, they have been instructed to find the defendant guilty; otherwise, not guilty. The burden of proof is on the prosecution. Jury trials represent a form of decision making. In statistics, an analogous procedure for making decisions falls into an area of statistical inference called hypothesis testing. Suppose that a company has been using a certain supplier of raw materials in one of its chemical processes. A new supplier approaches the company and states that its material, at the same cost, will increase the process yield. If the new supplier has a good reputation, the company might be willing to run a limited test. On the basis of the test results it would then make a decision to change suppliers or not. Good management would dictate that an improvement must be demonstrated (beyond a reasonable doubt) for the new material. That is, the burden of proof is tied to the new material. In setting up a test of hypothesis for this application, the initial assumption would be defined as a null hypothesis and symbolized as H0. The null hypothesis would state that yield for the new material is no greater than for the conventional material. The symbol µ0 would be used to designate the known current level of yield for the standard material and µ for the unknown population yield for the new material. Thus, the null hypothesis can be symbolized as H0: µ ≤ µ0. The alternative to H0 is called the alternative hypothesis and is symbolized as H1: µ > µ0. Given a series of tests with the new material, the average yield x would be compared with µ0. If x < µ0, the new supplier would be dismissed. If x > µ0, the question would be: Is it sufficiently greater in the light of its corresponding reliability, i.e., beyond a reasonable doubt? If the confidence interval for µ included µ0, the answer would be no, but if it did not include µ0, the answer would be yes. In this simple application, the formal test of hypothesis would result in the same conclusion as that derived from the confidence interval. However, the utility of tests of hypothesis lies in their generality, whereas confidence intervals are restricted to a few special cases.

TABLE 3-7 F Distribution Upper 5% Points (F.95)

Degrees of freedom for denominator

Degrees of freedom for numerator 1

2

3

4

5

6

7

8

9

10

12

15

20

24

30

40

60

120

∞

1 2 3 4 5

161 18.5 10.1 7.71 6.61

200 19.0 9.55 6.94 5.79

216 19.2 9.28 6.59 5.41

225 19.2 9.12 6.39 5.19

230 19.3 9.01 6.26 5.05

234 19.3 8.94 6.16 4.95

237 19.4 8.89 6.09 4.88

239 19.4 8.85 6.04 4.82

241 19.4 8.81 6.00 4.77

242 19.4 8.79 5.96 4.74

244 19.4 8.74 5.91 4.68

246 19.4 8.70 5.86 4.62

248 19.4 8.66 5.80 4.56

249 19.5 8.64 5.77 4.53

250 19.5 8.62 5.75 4.50

251 19.5 8.59 5.72 4.46

252 19.5 8.57 5.69 4.43

253 19.5 8.55 5.66 4.40

254 19.5 8.53 5.63 4.37

6 7 8 9 10

5.99 5.59 5.32 5.12 4.96

5.14 4.74 4.46 4.26 4.10

4.76 4.35 4.07 3.86 3.71

4.53 4.12 3.84 3.63 3.48

4.39 3.97 3.69 3.48 3.33

4.28 3.87 3.58 3.37 3.22

4.21 3.79 3.50 3.29 3.14

4.15 3.73 3.44 3.23 3.07

4.10 3.68 3.39 3.18 3.02

4.06 3.64 3.35 3.14 2.98

4.00 3.57 3.28 3.07 2.91

3.94 3.51 3.22 3.01 2.85

3.87 3.44 3.15 2.94 2.77

3.84 3.41 3.12 2.90 2.74

3.81 3.38 3.08 2.86 2.70

3.77 3.34 3.04 2.83 2.66

3.74 3.30 3.01 2.79 2.62

3.70 3.27 2.97 2.75 2.58

3.67 3.23 2.93 2.71 2.54

11 12 13 14 15

4.84 4.75 4.67 4.60 4.54

3.98 3.89 3.81 3.74 3.68

3.59 3.49 3.41 3.34 3.29

3.36 3.26 3.18 3.11 3.06

3.20 3.11 3.03 2.96 2.90

3.09 3.00 2.92 2.85 2.79

3.01 2.91 2.83 2.76 2.71

2.95 2.85 2.77 2.70 2.64

2.90 2.80 2.71 2.65 2.59

2.85 2.75 2.67 2.60 2.54

2.79 2.69 2.60 2.53 2.48

2.72 2.62 2.53 2.46 2.40

2.65 2.54 2.46 2.39 2.33

2.61 2.51 2.42 2.35 2.29

2.57 2.47 2.38 2.31 2.25

2.53 2.43 2.34 2.27 2.20

2.49 2.38 2.30 2.22 2.16

2.45 2.34 2.25 2.18 2.11

2.40 2.30 2.21 2.13 2.07

16 17 18 19 20

4.49 4.45 4.41 4.38 4.35

3.63 3.59 3.55 3.52 3.49

3.24 3.20 3.16 3.13 3.10

3.01 2.96 2.93 2.90 2.87

2.85 2.81 2.77 2.74 2.71

2.74 2.70 2.66 2.63 2.60

2.66 2.61 2.58 2.54 2.51

2.59 2.55 2.51 2.48 2.45

2.54 2.49 2.46 2.42 2.39

2.49 2.45 2.41 2.38 2.35

2.42 2.38 2.34 2.31 2.28

2.35 2.31 2.27 2.23 2.20

2.28 2.23 2.19 2.16 2.12

2.24 2.19 2.15 2.11 2.08

2.19 2.15 2.11 2.07 2.04

2.15 2.10 2.06 2.03 1.99

2.11 2.06 2.02 1.98 1.95

2.06 2.01 1.97 1.93 1.90

2.01 1.96 1.92 1.88 1.84

21 22 23 24 25

4.32 4.30 4.28 4.26 4.24

3.47 3.44 3.42 3.40 3.39

3.07 3.05 3.03 3.01 2.99

2.84 2.82 2.80 2.78 2.76

2.68 2.66 2.64 2.62 2.60

2.57 2.55 2.53 2.51 2.49

2.49 2.46 2.44 2.42 2.40

2.42 2.40 2.37 2.36 2.34

2.37 2.34 2.32 2.30 2.28

2.32 2.30 2.27 2.25 2.24

2.25 2.23 2.20 2.18 2.16

2.18 2.15 2.13 2.11 2.09

2.10 2.07 2.05 2.03 2.01

2.05 2.03 2.01 1.98 1.96

2.01 1.98 1.96 1.94 1.92

1.96 1.94 1.91 1.89 1.87

1.92 1.89 1.86 1.84 1.82

1.87 1.84 1.81 1.79 1.77

1.81 1.78 1.76 1.73 1.71

30 40 60 120 ∞

4.17 4.08 4.00 3.92 3.84

3.32 3.23 3.15 3.07 3.00

2.92 2.84 2.76 2.68 2.60

2.69 2.61 2.53 2.45 2.37

2.53 2.45 2.37 2.29 2.21

2.42 2.34 2.25 2.18 2.10

2.33 2.25 2.17 2.09 2.01

2.27 2.18 2.10 2.02 1.94

2.21 2.12 2.04 1.96 1.88

2.16 2.08 1.99 1.91 1.83

2.09 2.00 1.92 1.83 1.75

2.01 1.92 1.84 1.75 1.67

1.93 1.84 1.75 1.66 1.57

1.89 1.79 1.70 1.61 1.52

1.84 1.74 1.65 1.55 1.46

1.79 1.69 1.59 1.50 1.39

1.74 1.64 1.53 1.43 1.32

1.68 1.58 1.47 1.35 1.22

1.62 1.51 1.39 1.25 1.00

Upper 1% Points (F.99)

Degrees of freedom for denominator

Degrees of freedom for numerator 1

2

3

4

5

6

7

8

9

10

12

15

20

24

30

40

60

120

∞

1 2 3 4 5

4052 98.5 34.1 21.2 16.3

5000 99.0 30.8 18.0 13.3

5403 99.2 29.5 16.7 12.1

5625 99.2 28.7 16.0 11.4

5764 99.3 28.2 15.5 11.0

5859 99.3 27.9 15.2 10.7

5928 99.4 27.7 15.0 10.5

5982 99.4 27.5 14.8 10.3

6023 99.4 27.3 14.7 10.2

6056 99.4 27.2 14.5 10.1

6106 99.4 27.1 14.4 9.89

6157 99.4 26.9 14.2 9.72

6209 99.4 26.7 14.0 9.55

6235 99.5 26.6 13.9 9.47

6261 99.5 26.5 13.8 9.38

6287 99.5 26.4 13.7 9.29

6313 99.5 26.3 13.7 9.20

6339 99.5 26.2 13.6 9.11

6366 99.5 26.1 13.5 9.02

6 7 8 9 10

13.7 12.2 11.3 10.6 10.0

10.9 9.55 8.65 8.02 7.56

9.78 8.45 7.59 6.99 6.55

9.15 7.85 7.01 6.42 5.99

8.75 7.46 6.63 6.06 5.64

8.47 7.19 6.37 5.80 5.39

8.26 6.99 6.18 5.61 5.20

8.10 6.84 6.03 5.47 5.06

7.98 6.72 5.91 5.35 4.94

7.87 6.62 5.81 5.26 4.85

7.72 6.47 5.67 5.11 4.71

7.56 6.31 5.52 4.96 4.56

7.40 6.16 5.36 4.81 4.41

7.31 6.07 5.28 4.73 4.33

7.23 5.99 5.20 4.65 4.25

7.14 5.91 5.12 4.57 4.17

7.06 5.82 5.03 4.48 4.08

6.97 5.74 4.95 4.40 4.00

6.88 5.65 4.86 4.31 3.91

11 12 13 14 15

9.65 9.33 9.07 8.86 8.68

7.21 6.93 6.70 6.51 6.36

6.22 5.95 5.74 5.56 5.42

5.67 5.41 5.21 5.04 4.89

5.32 5.06 4.86 4.70 4.56

5.07 4.82 4.62 4.46 4.32

4.89 4.64 4.44 4.28 4.14

4.74 4.50 4.30 4.14 4.00

4.63 4.39 4.19 4.03 3.89

4.54 4.30 4.10 3.94 3.80

4.40 4.16 3.96 3.80 3.67

4.25 4.01 3.82 3.66 3.52

4.10 3.86 3.66 3.51 3.37

4.02 3.78 3.59 3.43 3.29

3.94 3.70 3.51 3.35 3.21

3.86 3.62 3.43 3.27 3.13

3.78 3.54 3.34 3.18 3.05

3.69 3.45 3.25 3.09 2.96

3.60 3.36 3.17 3.00 2.87

16 17 18 19 20

8.53 8.40 8.29 8.19 8.10

6.23 6.11 6.01 5.93 5.85

5.29 5.19 5.09 5.01 4.94

4.77 4.67 4.58 4.50 4.43

4.44 4.34 4.25 4.17 4.10

4.20 4.10 4.01 3.94 3.87

4.03 3.93 3.84 3.77 3.70

3.89 3.79 3.71 3.63 3.56

3.78 3.68 3.60 3.52 3.46

3.69 3.59 3.51 3.43 3.37

3.55 3.46 3.37 3.30 3.23

3.41 3.31 3.23 3.15 3.09

3.26 3.16 3.08 3.00 2.94

3.18 3.08 3.00 2.92 2.86

3.10 3.00 2.92 2.84 2.78

3.02 2.92 2.84 2.76 2.69

2.93 2.83 2.75 2.67 2.61

2.84 2.75 2.66 2.58 2.52

2.75 2.65 2.57 2.49 2.42

21 22 23 24 25

8.02 7.95 7.88 7.82 7.77

5.78 5.72 5.66 5.61 5.57

4.87 4.82 4.76 4.72 4.68

4.37 4.31 4.26 4.22 4.18

4.04 3.99 3.94 3.90 3.86

3.81 3.76 3.71 3.67 3.63

3.64 3.59 3.54 3.50 3.46

3.51 3.45 3.41 3.36 3.32

3.40 3.35 3.30 3.26 3.22

3.31 3.26 3.21 3.17 3.13

3.17 3.12 3.07 3.03 2.99

3.03 2.98 2.93 2.89 2.85

2.88 2.83 2.78 2.74 2.70

2.80 2.75 2.70 2.66 2.62

2.72 2.67 2.62 2.58 2.53

2.64 2.58 2.54 2.49 2.45

2.55 2.50 2.45 2.40 2.36

2.46 2.40 2.35 2.31 2.27

2.36 2.31 2.26 2.21 2.17

30 40 60 120 ∞

7.56 7.31 7.08 6.85 6.63

5.39 5.18 4.98 4.79 4.61

4.51 4.31 4.13 3.95 3.78

4.02 3.83 3.65 3.48 3.32

3.70 3.51 3.34 3.17 3.02

3.47 3.29 3.12 2.96 2.80

3.30 3.12 2.95 2.79 2.64

3.17 2.99 2.82 2.66 2.51

3.07 2.89 2.72 2.56 2.41

2.98 2.80 2.63 2.47 2.32

2.84 2.66 2.50 2.34 2.18

2.70 2.52 2.35 2.19 2.04

2.55 2.37 2.20 2.03 1.88

2.47 2.29 2.12 1.95 1.79

2.39 2.20 2.03 1.86 1.70

2.30 2.11 1.94 1.76 1.59

2.21 2.02 1.84 1.66 1.47

2.11 1.92 1.73 1.53 1.32

2.01 1.80 1.60 1.38 1.00

Interpolation should be performed using reciprocals of the degrees of freedom. 3-77

3-78

MATHEMATICS

Test of Hypothesis for a Mean Procedure Nomenclature µ = mean of the population from which the sample has been drawn σ = standard deviation of the population from which the sample has been drawn µ0 = base or reference level H0 = null hypothesis H1 = alternative hypothesis α = significance level, usually set at .10, .05, or .01 t = tabled t value corresponding to the significance level α. For a two-tailed test, each corresponding tail would have an area of α/2, and for a one-tailed test, one tail area would be equal to α. If σ2 is known, then z would be used rather than the t. t = (x − µ0)/(s/
 n) = sample value of the test statistic. Assumptions 1. The n observations x1, x2, . . . , xn have been selected randomly. 2. The population from which the observations were obtained is normally distributed with an unknown mean µ and standard deviation σ. In actual practice, this is a robust test, in the sense that in most types of problems it is not sensitive to the normality assumption when the sample size is 10 or greater. Test of Hypothesis 1. Under the null hypothesis, it is assumed that the sample came from a population whose mean µ is equivalent to some base or reference designated by µ0. This can take one of three forms: Form 1

Form 2

Form 3

H0: µ = µ0 H1: µ ≠ µ0 Two-tailed test

H0: µ ≤ µ0 H1: µ > µ0 Upper-tailed test

H0: µ ≥ µ0 H1: µ < µ0 Lower-tailed test

2. If the null hypothesis is assumed to be true, say, in the case of a two-sided test, form 1, then the distribution of the test statistic t is known. Given a random sample, one can predict how far its sample value of t might be expected to deviate from zero (the midvalue of t) by chance alone. If the sample value of t does, in fact, deviate too far from zero, then this is defined to be sufficient evidence to refute the assumption of the null hypothesis. It is consequently rejected, and the converse or alternative hypothesis is accepted. 3. The rule for accepting H0 is specified by selection of the α level as indicated in Fig. 3-57. For forms 2 and 3 the α area is defined to be in the upper or the lower tail respectively. 4. The decision rules for each of the three forms are defined as follows: If the sample t falls within the acceptance region, accept H0 for lack of contrary evidence. If the sample t falls in the critical region, reject H0 at a significance level of 100α percent. Example Application. In the past, the yield for a chemical process has been established at 89.6 percent with a standard deviation of 3.4 percent. A new supplier of raw materials will be used and tested for 7 days. Procedure 1. The standard of reference is µ0 = 89.6 with a known σ = 3.4. 2. It is of interest to demonstrate whether an increase in yield is achieved with the new material; H0 says it has not; therefore, H0: µ ≤ 89.6

H1: µ > 89.6

3. Select α = .05, and since σ is known (the new material would not affect the day-to-day variability in yield), the test statistic would be z with a corresponding critical value cv(z) = 1.645 (Table 3-5, df = ∞). 4. The decision rule: Accept H0 if sample z < 1.645 Reject H0 if sample z > 1.645 5. A 7-day test was carried out, and daily yields averaged 91.6 percent with a sample standard deviation s = 3.6 (this is not needed for the test of hypothesis). 6. For the data sample z = (91.6 − 89.6)/(3.4/
7) = 1.56. 7. Since the sample z < cv(z), accept the null hypothesis for lack of contrary evidence; i.e., an improvement has not been demonstrated beyond a reasonable doubt.

Example Application. In the past, the break strength of a synthetic yarn has averaged 34.6 lb. The first-stage draw ratio of the spinning machines has been increased. Production management wants to determine whether the break strength has changed under the new condition. Procedure 1. The standard of reference is µ0 = 34.6. 2. It is of interest to demonstrate whether a change has occurred; therefore, H0: µ = 34.6

Accept H0 if −2.086 < sample t < 2.086 Reject H0 if sample t < −2.086 or > 2.086 6. For the data sample t = (35.55 − 34.6)/(2.041/
21) = 2.133. 7. Since 2.133 > 2.086, reject H0 and accept H1. It has been demonstrated that an improvement in break strength has been achieved.

Two-Population Test of Hypothesis for Means Nature Two samples were selected from different locations in a plastic-film sheet and measured for thickness. The thickness of the respective samples was measured at 10 close but equally spaced points in each of the samples. It was of interest to compare the average thickness of the respective samples to detect whether they were significantly different. That is, was there a significant variation in thickness between locations? From a modeling standpoint statisticians would define this problem as a two-population test of hypothesis. They would define the respective sample sheets as two populations from which 10 sample thickness determinations were measured for each. In order to compare populations based on their respective samples, it is necessary to have some basis of comparison. This basis is predicated on the distribution of the t statistic. In effect, the t statistic characterizes the way in which two sample means from two separate populations will tend to vary by chance alone when the population means and variances are equal. Consider the following: Population 1

Acceptance region.

Population 2

Normal

Sample 1

Normal

Sample 2

µ1

n1 x1 s12

µ2

n2 x2 s22

σ

FIG. 3-57

H1: µ ≠ 34.6

3. Select α = .05, and since with the change in draw ratio the uniformity might change, the sample standard deviation would be used, and therefore t would be the appropriate test statistic. 4. A sample of 21 ends was selected randomly and tested on an Instron with the results x = 35.55 and s = 2.041. 5. For 20 df and a two-tailed α level of 5 percent, the critical values of t are given by 2.086 with a decision rule (Table 3-5, t.025, df = 20):

2 1

σ

2 2

Consider the hypothesis µ1 = µ2. If, in fact, the hypothesis is correct, i.e., µ1 = µ2 (under the condition σ 12 = σ 22), then the sampling distribution of (x1 − x2) is predictable through the t distribution. The observed sample values then can be compared with the corresponding t distribution. If the sample values are reasonably close (as reflected through the α level), that is, x1 and x2 are not “too different” from each other on the basis of the t distribution, the null hypothesis would be accepted. Conversely, if they deviate from each other “too much” and the deviation is therefore not ascribable to chance, the conjecture would be questioned and the null hypothesis rejected.

STATISTICS Example Application. Two samples were selected from different locations in a plastic-film sheet. The thickness of the respective samples was measured at 10 close but equally spaced points. Procedure 1. Demonstrate whether the thicknesses of the respective sample locations are significantly different from each other; therefore, H0: µ1 = µ2 2. 3.

H1: µ1 ≠ µ2

Select α = .05. Summarize the statistics for the respective samples: Sample 1

Sample 2

1.473 1.484 1.484 1.425 1.448

1.367 1.276 1.485 1.462 1.439

1.474 1.501 1.485 1.435 1.348

1.417 1.448 1.469 1.474 1.452

x1 = 1.434

s1 = .0664

x2 = 1.450

s2 = .0435

4. As a first step, the assumption for the standard t test, that σ 12 = σ 22, can be tested through the F distribution. For this hypothesis, H0: σ 12 = σ 22 would be tested against H1: σ 12 ≠ σ 22. Since this is a two-tailed test and conventionally only the upper tail for F is published, the procedure is to use the largest ratio and the corresponding ordered degrees of freedom. This achieves the same end result through one table. However, since the largest ratio is arbitrary, it is necessary to define the true α level as twice the value of the tabled value. Therefore, by using Table 3-7 with α = .05 the corresponding critical value for F(9,9) = 3.18 would be for a true α = .10. For the sample,

Test of Hypothesis for Matched Pairs: Procedure Nomenclature di = sample difference between the ith pair of observations s = sample standard deviation of differences µ = population mean of differences σ = population standard deviation of differences µ0 = base or reference level of comparison H0 = null hypothesis H1 = alternative hypothesis α = significance level t = tabled value with (n − 1) df t = (d n), the sample value of t  − µ0)/(s/
 Assumptions 1. The n pairs of samples have been selected and assigned for testing in a random way. 2. The population of differences is normally distributed with a mean µ and variance σ2. As in the previous application of the t distribution, this is a robust procedure, i.e., not sensitive to the normality assumption if the sample size is 10 or greater in most situations. Test of Hypothesis 1. Under the null hypothesis, it is assumed that the sample came from a population whose mean µ is equivalent to some base or reference level designated by µ0. For most applications of this type, the value of µ0 is defined to be zero; that is, it is of interest generally to demonstrate a difference not equal to zero. The hypothesis can take one of three forms:

Sample F = (.0664/.0435)2 = 2.33

Form 1

Form 2

Therefore, the ratio of sample variances is no larger than one might expect to observe when in fact σ 12 = σ 22. There is not sufficient evidence to reject the null hypothesis that σ 12 = σ 22. 5. For 18 df and a two-tailed α level of 5 percent the critical values of t are given by 2.101 (Table 3-5, t0.025, df = 18). 6. The decision rule:

H0: µ = µ0 H1: µ ≠ µ0 Two-tailed test

H0: µ ≤ µ0 H1: µ > µ0 Upper-tailed test

Accept H0 if −2.101 ≤ sample t ≤ 2.101 Reject H0 otherwise 7.

For the sample the pooled variance estimate is given by 9(.0664)2 + 9(.0435)2 (.0664)2 + (.0435)2 s p2 =


=

= .00315 9+9 2 sp = .056

or 8.

The sample statistic value of t is 1.434 − 1.450 Sample t =

= −.64 0 0 .056
1/1 + 1/1

9. Since the sample value of t falls within the acceptance region, accept H0 for lack of contrary evidence; i.e., there is insufficient evidence to demonstrate that thickness differs between the two selected locations.

Test of Hypothesis for Paired Observations Nature In some types of applications, associated pairs of observations are defined. For example, (1) pairs of samples from two populations are treated in the same way, or (2) two types of measurements are made on the same unit. For applications of this type, it is not only more effective but necessary to define the random variable as the difference between the pairs of observations. The difference numbers can then be tested by the standard t distribution. Examples of the two types of applications are as follows: 1. Sample treatment a. Two types of metal specimens buried in the ground together in a variety of soil types to determine corrosion resistance b. Wear-rate test with two different types of tractor tires mounted in pairs on n tractors for a defined period of time 2. Same unit a. Blood-pressure measurements made on the same individual before and after the administration of a stimulus b. Smoothness determinations on the same film samples at two different testing laboratories

3-79

Form 3 H0: µ ≥ µ0 H1: µ < µ0 Lower-tailed test

2. If the null hypothesis is assumed to be true, say, in the case of a lower-tailed test, form 3, then the distribution of the test statistic t is known under the null hypothesis that limits µ = µ0. Given a random sample, one can predict how far its sample value of t might be expected to deviate from zero by chance alone when µ = µ0. If the sample value of t is too small, as in the case of a negative value, then this would be defined as sufficient evidence to reject the null hypothesis. 3. Select α. 4. The critical values or value of t would be defined by the tabled value of t with (n − 1) df corresponding to a tail area of α. For a twotailed test, each tail area would be α/2, and for a one-tailed test there would be an upper-tail or a lower-tail area of α corresponding to forms 2 and 3 respectively. 5. The decision rule for each of the three forms would be to reject the null hypothesis if the sample value of t fell in that area of the t distribution defined by α, which is called the critical region. Otherwise, the alternative hypothesis would be accepted for lack of contrary evidence. Example Application. Pairs of pipes have been buried in 11 different locations to determine corrosion on nonbituminous pipe coatings for underground use. One type includes a lead-coated steel pipe and the other a bare steel pipe. Procedure 1. The standard of reference is taken as µ0 = 0, corresponding to no difference in the two types. 2. It is of interest to demonstrate whether either type of pipe has a greater corrosion resistance than the other. Therefore, H0: µ = 0

H1: µ ≠ 0

3. Select α = .05. Therefore, with n = 11 the critical values of t with 10 df are defined by t = 2.228 (Table 3.5, t.025). 4. The decision rule: Accept H0 if −2.228 ≤ sample t ≤ 2.228 Reject H0 otherwise 5. The sample of 11 pairs of corrosion determinations and their differences are as follows:

3-80

MATHEMATICS

Soil type

Lead-coated steel pipe

Bare steel pipe

d = difference

A B C D E F

27.3 18.4 11.9 11.3 14.8 20.8

41.4 18.9 21.7 16.8 9.0 19.3

−14.1 −0.5 −9.8 −5.5 5.8 1.5

G H I J K

17.9 7.8 14.7 19.0 65.3

32.1 7.4 20.7 34.4 76.2

−14.2 0.4 −6.0 −15.4 −10.9

6.

The sample statistics: 11  d 2 − ( d)2 s2 =

= 52.56 11 × 10

d = −6.245 

s = 7.25

or

Sample t = (−6.245 − 0)/(7.25/
11) = −2.86 7. Since the sample t of −2.86 < tabled t of −2.228, reject H0 and accept H1; that is, it has been demonstrated that, on the basis of the evidence, lead-coated steel pipe has a greater corrosion resistance than bare steel pipe.

Example Application. A stimulus was tested for its effect on blood pressure. Ten men were selected randomly, and their blood pressure was measured before and after the stimulus was administered. It was of interest to determine whether the stimulus had caused a significant increase in the blood pressure. Procedure 1. The standard of reference was taken as µ0 ≤ 0, corresponding to no increase. 2. It was of interest to demonstrate an increase in blood pressure if in fact an increase did occur. Therefore, H0: µ0 ≤ 0

H1: µ0 > 0

3. Select α = .05. Therefore, with n = 10 the critical value of t with 9 df is defined by t = 1.833 (Table 3-5, t.05, one-sided). 4. The decision rule: Accept H0 if sample t < 1.833 Reject H0 if sample t > 1.833 5. The sample of 10 pairs of blood pressure and their differences were as follows:

proportion of workers in a plant who are out sick, (2) lost-time worker accidents per month, (3) defective items in a shipment lot, and (4) preference in consumer surveys. The procedure for testing the significance of a sample proportion follows that for a sample mean. In this case, however, owing to the nature of the problem the appropriate test statistic is Z. This follows from the fact that the null hypothesis requires the specification of the goal or reference quantity p0, and since the distribution is a binomial proportion, the associated variance is [p0(1 − p0)]n under the null hypothesis. The primary requirement is that the sample size n satisfy normal approximation criteria for a binomial proportion, roughly np > 5 and n(1 − p) > 5. Test of Hypothesis for a Proportion: Procedure Nomenclature p = mean proportion of the population from which the sample has been drawn p0 = base or reference proportion [p0(1 − p0)]/n = base or reference variance p ˆ = x/n = sample proportion, where x refers to the number of observations out of n which have the specified attribute H0 = assumption or null hypothesis regarding the population proportion H1 = alternative hypothesis α = significance level, usually set at .10, .05, or .01 z = Tabled Z value corresponding to the significance level α. The sample sizes required for the z approximation according to the magnitude of p0 are given in Table 3-5. z = (p ˆ − p0)/
 p 1 − p /n, the sample value of the test 0( 0) statistic Assumptions 1. The n observations have been selected randomly. 2. The sample size n is sufficiently large to meet the requirement for the Z approximation. Test of Hypothesis 1. Under the null hypothesis, it is assumed that the sample came from a population with a proportion p0 of items having the specified attribute. For example, in tossing a coin the population could be thought of as having an unbounded number of potential tosses. If it is assumed that the coin is fair, this would dictate p0 = 1/2 for the proportional number of heads in the population. The null hypothesis can take one of three forms:

Individual

Before

After

d = difference

Form 1

Form 2

Form 3

1 2 3 4 5

138 116 124 128 155

146 118 120 136 174

8 2 −4 8 19

H0: p = p0 H1: p ≠ p0 Two-tailed test

H0: p ≤ p0 H1: p > p0 Upper-tailed test

H0: p ≥ p0 H1: p < p0 Lower-tailed test

6 7 8 9 10

129 130 148 143 159

133 129 155 148 155

4 −1 7 5 −4

2. If the null hypothesis is assumed to be true, then the sampling distribution of the test statistic Z is known. Given a random sample, it is possible to predict how far the sample proportion x/n might deviate from its assumed population proportion p0 through the Z distribution. When the sample proportion deviates too far, as defined by the significance level α, this serves as the justification for rejecting the assumption, that is, rejecting the null hypothesis. 3. The decision rule is given by Form 1: Accept H0 if lower critical z < sample z < upper critical z Reject H0 otherwise Form 2: Accept H0 if sample z < upper critical z Reject H0 otherwise Form 3: Accept H0 if lower critical z < sample z Reject H0 otherwise

6.

The sample statistics: d  = 4.4

s = 6.85

Sample t = (4.4 − 0)/(6.85/
10) = 2.03 7. Since the sample t = 2.03 > critical t = 1.833, reject the null hypothesis. It has been demonstrated that the population of men from which the sample was drawn tend, as a whole, to have an increase in blood pressure after the stimulus has been given. The distribution of differences d seems to indicate that the degree of response varies by individuals.

Example Test of Hypothesis for a Proportion Nature Some types of statistical applications deal with counts and proportions rather than measurements. Examples are (1) the

Application. A company has received a very large shipment of rivets. One product specification required that no more than 2 percent of the rivets have diameters greater than 14.28 mm. Any rivet with a diameter greater than this would be classified as defective. A random sample of 600 was selected and

STATISTICS tested with a go–no-go gauge. Of these, 16 rivets were found to be defective. Is this sufficient evidence to conclude that the shipment contains more than 2 percent defective rivets? Procedure 1. The quality goal is p ≤ .02. It would be assumed initially that the shipment meets this standard; i.e., H0: p ≤ .02. 2. The assumption in step 1 would first be tested by obtaining a random sample. Under the assumption that p ≤ .02, the distribution for a sample proportion would be defined by the z distribution. This distribution would define an upper bound corresponding to the upper critical value for the sample proportion. It would be unlikely that the sample proportion would rise above that value if, in fact, p ≤ .02. If the observed sample proportion exceeds that limit, corresponding to what would be a very unlikely chance outcome, this would lead one to question the assumption that p ≤ .02. That is, one would conclude that the null hypothesis is false. To test, set H0: p ≤ .02

H1: p > .02

3. Select α = .05. 4. With α = .05, the upper critical value of Z = 1.645 (Table 3-5, t.05, df = ∞, one-sided). 5. The decision rule:

Test of Hypothesis 1. Under the null hypothesis, it is assumed that the respective two samples have come from populations with equal proportions p1 = p2. Under this hypothesis, the sampling distribution of the corresponding Z statistic is known. On the basis of the observed data, if the resultant sample value of Z represents an unusual outcome, that is, if it falls within the critical region, this would cast doubt on the assumption of equal proportions. Therefore, it will have been demonstrated statistically that the population proportions are in fact not equal. The various hypotheses can be stated: Form 1

Form 2

Form 3

H0: p1 = p2 H1: p1 ≠ p2 Two-tailed test

H0: p1 ≤ p2 H1: p1 > p2 Upper-tailed test

H0: p1 ≥ p2 H1: p1 < p2 Lower-tailed test

2.

Accept H0 if sample z < 1.645 Reject H0 if sample z > 1.645 6.

The sample z is given by (16/600) − .02 Sample z =

02)( .9 8)/ 600
(. = 1.17

7. Since the sample z < 1.645, accept H0 for lack of contrary evidence; there is not sufficient evidence to demonstrate that the defect proportion in the shipment is greater than 2 percent.

Test of Hypothesis for Two Proportions Nature In some types of engineering and management-science problems, we may be concerned with a random variable which represents a proportion, for example, the proportional number of defective items per day. The method described previously relates to a single proportion. In this subsection two proportions will be considered. A certain change in a manufacturing procedure for producing component parts is being considered. Samples are taken by using both the existing and the new procedures in order to determine whether the new procedure results in an improvement. In this application, it is of interest to demonstrate statistically whether the population proportion p2 for the new procedure is less than the population proportion p1 for the old procedure on the basis of a sample of data. Test of Hypothesis for Two Proportions: Procedure Nomenclature p1 = population 1 proportion p2 = population 2 proportion n1 = sample size from population 1 n2 = sample size from population 2 x1 = number of observations out of n1 that have the designated attribute x2 = number of observations out of n2 that have the designated attribute p ˆ 1 = x1/n1, the sample proportion from population 1 p ˆ 2 = x2/n2, the sample proportion from population 2 α = significance level H0 = null hypothesis H1 = alternative hypothesis z = tabled Z value corresponding to the stated significance level α ˆ2 p ˆ1−p z =



, the sample value of Z
pˆ 1(1 − pˆ 1)/n1 + pˆ 2(1 − pˆ 2)/n2 Assumptions 1. The respective two samples of n1 and n2 observations have been selected randomly. 2. The sample sizes n1 and n2 are sufficiently large to meet the requirement for the Z approximation; i.e., x1 > 5, x2 > 5.

3-81

The decision rule for form 1 is given by Accept H0 if lower critical z < sample z < upper critical z Reject H0 otherwise

Example Application. A change was made in a manufacturing procedure for component parts. Samples were taken during the last week of operations with the old procedure and during the first week of operations with the new procedure. Determine whether the proportional numbers of defects for the respective populations differ on the basis of the sample information. Procedure 1. The hypotheses are H0: p1 = p2

H1: p1 ≠ p2

2. Select α = .05. Therefore, the critical values of z are 1.96 (Table 3-4, A = 0.9500). 3. For the samples, 75 out of 1720 parts from the previous procedure and 80 out of 2780 parts under the new procedure were found to be defective; therefore, p ˆ 1 = 75/1720 = .0436 4.

p ˆ 2 = 80/2780 = .0288

The decision rule: Accept H0 if −1.96 ≤ sample Z ≤ 1.96 Reject H0 otherwise

5.

The sample statistic: .0436 − .0288 Sample z =




0436)( .9 564)/ 1720 288)( .9 712)/ 2780 +(.0
(. = 2.53

6. Since the sample z of 2.53 > tabled z of 1.96, reject H0 and conclude that the new procedure has resulted in a reduced defect rate.

Goodness-of-Fit Test Nature A standard die has six sides numbered from 1 to 6. If one were really interested in determining whether a particular die was well balanced, one would have to carry out an experiment. To do this, it might be decided to count the frequencies of outcomes, 1 through 6, in tossing the die N times. On the assumption that the die is perfectly balanced, one would expect to observe N/6 occurrences each for 1, 2, 3, 4, 5, and 6. However, chance dictates that exactly N/6 occurrences each will not be observed. For example, given a perfectly balanced die, the probability is only 1 chance in 65 that one will observe 1 outcome each, for 1 through 6, in tossing the die 6 times. Therefore, an outcome different from 1 occurrence each can be expected. Conversely, an outcome of six 3s would seem to be too unusual to have occurred by chance alone. Some industrial applications involve the concept outlined here. The basic idea is to test whether or not a group of observations follows a preconceived distribution. In the case cited, the distribution is uniform; i.e., each face value should tend to occur with the same frequency. Goodness-of-Fit Test: Procedure Nomenclature Each experimental observation can be classified into one of r possible categories or cells.

3-82

MATHEMATICS

r = total number of cells Oj = number of observations occurring in cell j Ej = expected number of observations for cell j based on the preconceived distribution N = total number of observations f = degrees of freedom for the test. In general, this will be equal to (r − 1) minus the number of statistical quantities on which the Ej’s are based (see the examples which follow for details). Assumptions 1. The observations represent a sample selected randomly from a population which has been specified. 2. The number of expectation counts Ej within each category should be roughly 5 or more. If an Ej count is significantly less than 5, that cell should be pooled with an adjacent cell. Computation for Ej On the basis of the specified population, the probability of observing a count in cell j is defined by pj. For a sample of size N, corresponding to N total counts, the expected frequency is given by Ej = Npj. Test Statistics: Chi Square r (Oj − Ej)2 χ2 = 
with f df Ej j=1 Test of Hypothesis 1. H0: The sample came from the specified theoretical distribution H1: The sample did not come from the specified theoretical distribution 2. For a stated level of α, Reject H0 if sample χ2 > tabled χ2 Accept H0 if sample χ2 < tabled χ2 Example Application A production-line product is rejected if one of its characteristics does not fall within specified limits. The standard goal is that no more than 2 percent of the production should be rejected. Computation 1. Of 950 units produced during the day, 28 units were rejected. 2. The hypotheses: H0: the process is in control H1: the process is not in control 3. Assume that α = .05; therefore, the critical value of χ2(1) = 3.84 (Table 3-6, 95 percent, df = 1). One degree of freedom is defined since (r − 1) = 1, and no statistical quantities have been computed for the data. 4. The decision rule: Reject H0 if sample χ2 > 3.84 Accept H0 otherwise 5. Since it is assumed that p = .02, this would dictate that in a sample of 950 there would be on the average (.02)(950) = 19 defective items and 931 acceptable items:

Category

Observed Oj

Expectation Ej = 950pj

Acceptable Not acceptable Total

922 28 950

931 19 950

(922 − 931)2 (28 − 19)2 Sample χ =

+

931 19 = 4.35 with critical χ2 = 3.84 2

6. Conclusion. Since the sample value exceeds the critical value, it would be concluded that the process is not in control.

Example Application A frequency count of workers was tabulated according to the number of defective items that they produced. An unresolved question is whether the observed distribution is a Poisson distribution. That is, do observed and expected frequencies agree within chance variation? Computation 1. The hypotheses: H0: there are no significant differences, in number of defective units, between workers H1: there are significant differences

2. 3.

Assume that α = .05. Test statistic: No. of defective units 0 1 2 3 4 5 6 7 8 9 ≥10 Sum

Oj 8 7 9 12 9 6 3 2 0 1 0 52

Ej

10

2.06 6.64 10.73 11.55 9.33 6.03

8.70 pool

3.24 1.50 .60 .22 .10 52

6

5.66 pool

The expectation numbers Ej were computed as follows: For the Poisson distribution, λ = E(x); therefore, an estimate of λ is the average number of defective units per worker, i.e., λ = (1/52)(0 × 3 + 1 × 7 + ⋅⋅⋅ + 9 × 1) = 3.23. Given this approximation, the probability of no defective units for a worker would be (3.23)0/0!)e−3.23 = .0396. For the 52 workers, the number of workers producing no defective units would have an expectation E = 52(0.0396) = 2.06, and so forth. The sample chi-square value is computed from (10 − 8.70)2 (9 − 10.73)2 (6 − 5.66)2 χ2 =

+

+ ⋅⋅⋅ +

8.70 10.73 5.66 = .53 4. The critical value of χ2 would be based on four degrees of freedom. This corresponds to (r − 1) − 1, since one statistical quantity λ was computed from the sample and used to derive the expectation numbers. 5. The critical value of χ2(4) = 9.49 (Table 3-6) with α = .05; therefore, accept H0.

Two-Way Test for Independence for Count Data Nature When individuals or items are observed and classified according to two different criteria, the resultant counts can be statistically analyzed. For example, a market survey may examine whether a new product is preferred and if it is preferred due to a particular characteristic. Count data, based on a random selection of individuals or items which are classified according to two different criteria, can be statistically analyzed through the χ2 distribution. The purpose of this analysis is to determine whether the respective criteria are dependent. That is, is the product preferred because of a particular characteristic? Two-Way Test for Independence for Count Data: Procedure Nomenclature 1. Each observation is classified into each of two categories: a. The first one into 2, 3, . . . , or r categories b. The second one into 2, 3, . . . , or c categories 2. Oij = number of observations (observed counts) in cell (i, j) with i = 1, 2, . . . , r j = 1, 2, . . . , c 3. N = total number of observations 4. Eij = computed number for cell (i,j) which is an expectation based on the assumption that the two characteristics are independent 5. Ri = subtotal of counts in row i 6. Cj = subtotal of counts in column j 7. α = significance level 8. H0 = null hypothesis 9. H1 = alternative hypothesis 10. χ2 = critical value of χ2 corresponding to the significance level α and (r − 1)(c − 1) df c,r (Oij − Eij)2 11. Sample χ2 = 

Eij i, j Assumptions 1. The observations represent a sample selected randomly from a large total population.

STATISTICS 2. The number of expectation counts Eij within each cell should be approximately 2 or more for arrays 3 × 3 or larger. If any cell contains a number smaller than 2, appropriate rows or columns should be combined to increase the magnitude of the expectation count. For arrays 2 × 2, approximately 4 or more are required. If the number is less than 4, the exact Fisher test should be used. Test of Hypothesis Under the null hypothesis, the classification criteria are assumed to be independent, i.e., H0: the criteria are independent H1: the criteria are not independent For the stated level of α, Reject H0 if sample χ2 > tabled χ2 Accept H0 otherwise Computation for Eij Compute Eij across rows or down columns by using either of the following identities: R Eij = Cj
i across rows N

 

 

Cj Eij = Ri
down columns N Sample c Value 2

(Oij − Eij)2 χ2 = 

Eij i, j

In the special case of r = 2 and c = 2, a more accurate and simplified formula which does not require the direct computation of Eij can be used: [|O11O22 − O12O21| − aN]2N χ2 =


R1R2C1C2 Example Application A market research study was carried out to relate the subjective “feel” of a consumer product to consumer preference. In other words, is the consumer’s preference for the product associated with the feel of the product, or is the preference independent of the product feel? Procedure 1. It was of interest to demonstrate whether an association exists between feel and preference; therefore, assume H0: feel and preference are independent H1: they are not independent 2. A sample of 200 people was asked to classify the product according to two criteria: a. Liking for this product b. Liking for the feel of the product

127/200 = 63.5 percent suggest further that there are other attributes of the product which tend to nullify the beneficial feel of the product.

LEAST SQUARES When experimental data is to be fit with a mathematical model, it is necessary to allow for the fact that the data has errors. The engineer is interested in finding the parameters in the model as well as the uncertainty in their determination. In the simplest case, the model is a linear equation with only two parameters, and they are found by a least-squares minimization of the errors in fitting the data. Multiple regression is just linear least squares applied with more terms. Nonlinear regression allows the parameters of the model to enter in a nonlinear fashion. The following description of maximum likelihood applies to both linear and nonlinear least squares (Ref. 231). If each measurement point yi has a measurement error ∆yi that is independently random and distributed with a normal distribution about the true model y(x) with standard deviation σi, then the probability of a data set is N 1 yi − y(xi) 2 P =  exp −

∆y 2 σi i=1

 











Yes

No

Ri

114 55 169

13 18 31

= 127 = 73 200

3. Select α = .05; therefore, with (r − 1)(c − 1) = 1 df, the critical value of χ2 is 3.84 (Table 3-6, 95 percent). 4. The decision rule: Accept H0 if sample χ2 < 3.84 Reject H0 otherwise 5.







∞

ta − 1e−t dt

0

Like feel Yes No Cj



Here, yi is the measured value, σi is the standard deviation of the ith measurement, and ∆y is needed to say a measured value ∆y has a certain probability. Given a set of parameters (maximizing this function), the probability that this data set plus or minus ∆y could have occurred is P. This probability is maximized (giving the maximum likelihood) if the negative of the logarithm is minimized. N yi − y(xi) 2
− N log ∆y 
2 σi i=1 Since N, σi, and ∆y are constants, this is the same as minimizing χ2. N yi − y(xi; a1, . . . , aM) 2 χ2 = 


σi i=1 with respect to the parameters {aj}. Note that the standard deviations {σi} of the measurements are expected to be known. The goodness of fit is related to the number of degrees of freedom, ν = N − M. The probability that χ2 would exceed a particular value (χ0)2 is ν 1 P = 1 − Q
,
χ 20 2 2 where Q(a, x) is the incomplete gamma function x 1 Q(a, x) =
e−t ta − 1 dt (a > 0) Γ(a) 0 and Γ(a) is the gamma function Γ(a) =

Like product

3-83

The sample value of χ2 by using the special formula is [|114 × 18 − 13 × 55| − 100]2200 Sample χ2 =



(169)(31)(127)(73) = 6.30

6. Since the sample χ2 of 6.30 > tabled χ2 of 3.84, reject H0 and accept H1. The relative proportionality of E11 = 169(127/200) = 107.3 to the observed 114 compared with E22 = 31(73/200) = 11.3 to the observed 18 suggests that when the consumer likes the feel, the consumer tends to like the product, and conversely for not liking the feel. The proportions 169/200 = 84.5 percent and

Both functions are tabulated in mathematical handbooks (Ref. 1). The function P gives the goodness of fit. Call χ02 the value of χ2 at the minimum. Then P > 0.1 represents a believable fit; if Q > 0.001, it might be an acceptable fit; smaller values of Q indicate the model may be in error (or the σi are really larger.) A “typical” value of χ2 for a moderately good fit is χ2 ∼ ν. Asymptotically for large ν, the statistic χ2 becomes normally distributed with a mean ν and a standard deviation (2 ν) (Ref. 231).
 If values σi are not known in advance, assume σi = σ (so that its value does not affect the minimization of χ2). Find the parameters by minimizing χ2 and compute: N [yi − y(xi)]2 σ2 = 

N i=1 This gives some information about the errors (i.e., the variance and standard deviation of each data point), although the goodness of fit, P, cannot be calculated. The minimization of χ2 requires yi − y(xi) ∂y(xi; a1, . . . , aM)

= 0, 


σ ∂a N

i=1

2 i

k

k = 1, . . . , M

3-84

MATHEMATICS

Linear Least Squares When the model is a straight line yi − a − bxi χ2(a, b) = 

σi i=1 N

Define

N 1 S = 
2 , i = 1 σi N



N xi Sx = 
, 2 i = 1 σi

2 i



y = a + bx + ε where x = strip-method determination y = magnetic-method determination

N y Sy = 
i2 i = 1 σi



N

x Sxx = 
2 , i = 1 σi

Nomenclature. The calibration between the magnetic and the stripping methods can be determined through the model

2

xiyi Sxy = 
, 2 i = 1 σi

Sample data



Thickness, 10−5 ln

N

1 S ti =
xi −
x , σi S

Stt =  t 2i

Stripping method, x

Magnetic method, y

1 σ b2 =
Stt

104 114 116 129 132 139

85 115 105 127 120 121

We thus get the values of a and b with maximum likelihood as well as the variances of a and b. Using the value of χ2 for this a and b, we can also calculate the goodness of fit, P. In addition, the linear correlation coefficient r is related by

174 312 338 465 720

155 250 310 443 630

Then

1 N tiyi b=

, Stt i = 1 σi

Sy − Sxb a=
, S

Sx Cov (a, b) = −
, SStt



i=1



1 S 2x σ a2 =
1 +
, S SStt Cov (a, b) rab =
σaσb

N

χ2 = (1 − r2)  (yi − y)2 i=1

(xi − x)(yi − y)

σ2i i=1 r =


N N (xi − x )2 (yi − y)2



  2 σi σ 2i i=1 i=1

Computations. The normal equations are defined by na + ( x)b =  y

N

Here

 

Values of r near 1 indicate a positive correlation; r near −1 means a negative correlation and r near zero means no correlation. The form of the equations here is given to provide good accuracy when many terms are used and to provide the variances of the parameters. Another form of the equations for a and b is simpler, but is sometimes inaccurate unless many significant digits are kept in the calculations. The minimization of χ2 when σi is the same for all i gives the following equations for a and b. N

N

i=1

i=1

aN + b  xi =  yi N

N

N

a  xi + b  x =  yi xi 2 i

i=1

i=1

i=1

N

The solution is

N

N

N  yi xi −  xi  yi i=1 i=1 i=1 b =


N N 2 2 N  x i −  xi



i=1

N



N

y =  yi N, i=1

i=1





( x)a + ( x2)b =  xy

For the sample

11a + 2743b = 2461 2743a + 1,067,143b = 952,517 with  y2 = 852,419. The solution to the normal equations is given by a = 3.19960

χ2 =  y2 − a  y − b  xy if a sufficient number of significant digits is retained (usually six or seven digits are sufficient). Here χ2 = 2175.14 If the normalized method is used in addition, the value of Stt is 3.8314 × 105/σ2, where σ2 is the variance of the measurement of y. The values of a and b are, of course, the same. The variances of a and b are σ a2 = 0.2532σ2, σ b2 = 2.610 × 10−6σ2. The correlation coefficient is 0.996390, which indicates that there is a positive correlation between x and y. The small value of the variance for b indicates that this parameter is determined very well by the data. The residuals show no particular pattern, and the predictions are plotted along with the data in Fig. 3-58. If the variance of the measurements of y is known through repeated measurements, then the variance of the parameters can be made absolute.

Multiple Regression A general linear model is one expressed as M

x =  xi N i=1

a = y − bx

The value of χ2 can be calculated from the formula N

N

N

i=1

i=1

i=1

χ2 =  y 2i − a  yi − b  yi xi It is usually advisable to plot the observed pairs of yi versus xi to support the linearity assumption and to detect potential outliers. Suspected outliers can be omitted from the least-squares “fit” and then subsequently tested on the basis of the least-squares fit.

y(x) =  akXk(x) k=1

where the parameters are {ak}, and the expression is linear with respect to them, and Xk(x) can be any (nonlinear) functions of x, not depending on the parameters {ak}. Then: N M 1
2 yi −  aj Xj(xi) Xk(xi) = 0, k = 1, . . . , M  i = 1 σi j=1 This is rewritten as M N N 1 yi
2 Xj(xi)Xk(xi) aj = 
2 Xk(xi)   j = 1 i = 1 σi i = 1 σi









M

Example Application. Brenner (Magnetic Method for Measuring the Thickness of Non-magnetic Coatings on Iron and Steel, National Bureau of Standards, RP1081, March 1938) suggests an alternative way of measuring the thickness of nonmagnetic coatings of galvanized zinc on iron and steel. This procedure is based on a nondestructive magnetic method as a substitute for the standard destructive stripping method. A random sample of 11 pieces was selected and measured by both methods.

b = .884362

The error sum of squares can be computed from the formula

or as

α

a = βk

kj j

j=1

Solving this set of equations gives the parameters {aj}, which maximize the likelihood. The variance of aj is σ2(aj) = Cj j where Cj k = α−1 j k, or C is the inverse of α. The covariance of aj and ak

STATISTICS

FIG. 3-58

3-85

Plot of data and correlating line. 4.5βˆ 0 + 2.85βˆ 1 + 2.025βˆ 2 = 14.08957

is given by Cj k. If rounding errors affect the result, then we try to make the functions orthogonal. For example, using Xk(x) = xk − 1

2.85βˆ 0 + 2.025βˆ 1 + 1.5333βˆ 2 = 8.828813 The algebraic solution to the simultaneous equations is βˆ = 3.19513 βˆ = .4425 βˆ = −.7653

will cause rounding errors for a smaller M than

0

Xk(x) = Pk − 1(x)

1

2

The inverse of the product matrix

where Pk − 1 are orthogonal polynomials. If necessary, a singular value decomposition can be used. Various global and piecewise polynomials can be used to fit the data. Most approximations are to be used with M < N. One can sometimes use more and more terms, and calculating the value of χ2 for each solution. Then stop increasing M when the value of χ2 no longer increases with increasing M. Example Application. Merriman (“The Method of Least Squares Applied to a Hydraulic Problem,” J. Franklin Inst., 233–241, October 1877) reported on a study of stream velocity as a function of relative depth of the stream. Sample data

α=



α−1 =

is

10 4.5 2.85



4.5 2.85 2.025

.6182 −2.5909 2.2727

2.85 2.025 1.5333



−2.5909 16.5530 −17.0455



2.2727 −17.0455 18.9394

The variances are then the diagonal elements of the inverse of matrix α (0.6182, 16.5530, 18.9394) times the variance of the measurement of y, σ y2. The value of χ2 is 5.751 × 10−5, the correlation coefficient r = 0.99964, and σ = 0.002398. t values. A sample t value can be computed for each regression coefficient j through the identity tj = βˆ j /(ˆσ
cjj), where cjj is the ( j,j) element in the inverse. For the two variables x1 and x2,

Velocity, y, ft/s

Coefficient

cjj

Sample t value

.1 .2 .3 .4

3.1950 3.2299 3.2532 3.2611 3.2516

.4425 −.7653

16.55 18.94

45.3 −73.3

.5 .6 .7 .8 .9

3.2282 3.1807 3.1266 3.0594 2.9759

Depth* 0

*As a fraction of total depth. Model. Owing to the curvature of velocity with depth, a quadratic model was specified:

Computational note. From a computational standpoint, it is usually advisable to define the variables in deviation units. For example, in the problem presented, let x1 = depth −  dep th  = depth − .45 For expansion terms such as a square, define x2 = x21 − x¯21 ( x¯21 = .0825) For the previous sample data, Deviation units

Velocity = β0 + β1x1 + β2x2 where x2 = x 12 . Normal equations. The three normal equations are defined by (n)βˆ + ( x )βˆ + ( x )βˆ =  y 0



1

1



2

2

( x1)βˆ 0 + ( x12)βˆ 1 + ( x1x2)βˆ 2 =  x1 y ( x2)βˆ 0 + ( x1x2)βˆ 1 + ( x22)βˆ 2 =  x2 y For the sample data, the normal equations are 10βˆ + 4.5βˆ + 2.85 βˆ = 31.7616 0

1

2

x1

x2

x1

x2

−.45 −.35 −.25 −.25 −.15 −.05

.12 .04 −.02 −.02 −.06 −.08

.05 .15 .25 .35 .45

−.08 −.06 −.02 .04 .12

The resultant analysis-of-variance tables will remain exactly the same. However,

3-86

MATHEMATICS

the corresponding coefficient t value for the linear coefficient will usually be improved. This is an idiosyncrasy of regression modeling. With the coded data presented, the least-squares solution is given by Yˆ = 3.17616 − .2462x1 − .7653x2 with a corresponding t value for βˆ 1 = −.2462 of t = −63.63. When expansion terms are used but not expanded about the mean, the corresponding t values for the generating terms should not be used. For example, if x3 = x1x2 is used rather than the correct expansion (x1 − x1)(x2 − x2), then the corresponding t values for x1 and x2 should not be used.

Nonlinear Least Squares There are no analytic methods for determining the most appropriate model for a particular set of data. In many cases, however, the engineer has some basis for a model. If the parameters occur in a nonlinear fashion, then the analysis becomes more difficult. For example, in relating the temperature to the elapsed time of a fluid cooling in the atmosphere, a model that has an asymptotic property would be the appropriate model (temp = a + b exp(−c time), where a represents the asymptotic temperature corresponding to t → ∞. In this case, the parameter c appears nonlinearly. The usual practice is to concentrate on model development and computation rather than on statistical aspects. In general, nonlinear regression should be applied only to problems in which there is a welldefined, clear association between the two variables; therefore, a test of hypothesis on the significance of the fit would be somewhat ludicrous. In addition, the generalization of the theory for the associate confidence intervals for nonlinear coefficients is not well developed. The Levenberg-Marquardt method is used when the parameters of the model appear nonlinearly (Ref. 231). We still define N yi − y(xi; a) χ2(a) = 

σ 2i i=1





2

1. Choose a and calculate χ2(a). 2. Choose λ, say λ = 0.001. 3. Solve Eq. (3-91) for ak + 1 and evaluate χ2(ak + 1). 4. If χ2(ak + 1) ≥ χ2(ak) then increase λ by a factor of, say, 10 and go back to step 3. This makes the step more like a steepest descent. 5. If χ2(ak + 1) < χ2(ak) then update a, i.e., use a = ak + 1, decrease λ by a factor of 10, and go back to step 3. 6. Stop the iteration when the decrease in χ2 from one step to another is not statistically meaningful, i.e., less than 0.1 or 0.01 or 0.001. 7. Set λ = 0 and compute the estimated covariance matrix: C = α−1. This gives the standard errors in the fitted parameters a. For normally distributed errors the parameter region in which χ2 = constant can give boundaries of the confidence limits. The value of a obtained in the Marquardt method gives the minimum χ2min. If we set χ2 = χ2min + ∆χ for some ∆χ and then look at contours in parameter space where χ12 = constant then we have confidence boundaries at the probability associated with χ 12. For example, in a chemical reactor with radial dispersion the heat transfer coefficient and radial effective heat conductivity are closely connected: decreasing one and increasing the other can still give a good fit. Thus, the confidence boundaries may look something like Fig. 3-59. The ellipse defined by ∆χ2 = 2.3 contains 68.3 percent of the normally distributed data. The curve defined by ∆χ2 = 6.17 contains 95.4 percent of the data. Example Application. Data were collected on the cooling of water in the atmosphere as a function of time. Sample data

and near the optimum represent χ2 by 1 χ2(a) = χ02 − dT ⋅ a +
aT ⋅ D ⋅ a 2 where d is an M × 1 vector and D is an M × M matrix. We then calculate iteratively D ⋅ (ak + 1 − ak) = −∇χ2(ak)

(3-89)

The notation alk means the lth component of a evaluated on the kth iteration. If ak is a poor approximation to the optimum, we might use steepest descent instead. ak + 1 − ak = −constant × ∇χ2(ak)

(3-90)

and choose the constant somehow to decrease χ2 as much as possible. The gradient of χ2 is N ∂χ2 yi − y(xi; a) ∂y(xi; a)
= −2 


∂ak σ 2i ∂ak i=1

k = 1, 2, . . . , M

The second derivative (in D) is N ∂2χ2 1 ∂y(xi; a) ∂y(xi; a) ∂2y(xi; a)
= 2 
2

− [yi − y(xi; a)]
∂ak∂al ∂ak ∂al ∂ak∂al i=1 σi



Time x

Temperature y

0 1 2 3 5

92.0 85.5 79.5 74.5 67.0

7 10 15 20

60.5 53.5 45.0 39.5

Model form. On the basis of the nature of the data, an exponential model was selected initially to represent the trend y = a + becx. In this example, the resultant temperature would approach as an asymptotic (a with c negative) the wet-bulb temperature of the surrounding atmosphere. Unfortunately, this temperature was not reported. Using a computer package in MATLAB gives the following results: a = 33.54, b = 57.89, c = 0.11. The value of χ2 is 1.83. An alternative form of model is y = a + b/(c + x). For this model the results were a = 9.872, b = 925.7, c = 11.27, and the value of χ2 is 0.19. Since this model had a smaller value of χ2, it might be the chosen one, but it is only a fit of the specified data and may not be generalized beyond that. Both curve fits give an equivalent plot. The second form is shown in Fig. 3-60.

Both Eq. (3-89) and Eq. (3-90) are included if we write M

 α′

kl

(a kl + 1 − a lk ) = βk

(3-91)

l=1

where

N 1 ∂y(xi; a) ∂y(xi; a) α′kl = 
2

∂ak ∂al i=1 σi N 1 ∂y(xi; a) α′kk = 
2
∂ak i=1 σi



k≠1

 (1 + λ) 2

N yi − y(xi; a) ∂y(xi; a) β k = 


σ 2i ∂ak i=1

The second term in the second derivative is dropped because it is usually small [remember that yi will be close to y(xi, a)]. The LevenbergMarquardt method then iterates as follows

FIG. 3-59

Parameter estimation for heat transfer.

STATISTICS

FIG. 3-60

Data in nonlinear regression example.

ERROR ANALYSIS OF EXPERIMENTS Consider the problem of assessing the accuracy of a series of measurements. If measurements are for independent, identically distributed observations, then the errors are independent and uncorrelated. Then y, the experimentally determined mean, varies about E(y), the true mean, with variance σ2/n, where n is the number of observations in y. Thus, if one measures something several times today, and each day, and the measurements have the same distribution, then the variance of the means decreases with the number of samples in each day’s measurement, n. Of course, other factors (weather, weekends) may make the observations on different days not distributed identically. Consider next the problem of estimating the error in a variable that cannot be measured directly but must be calculated based on results of other measurements. Suppose the computed value Y is a linear combination of the measured variables {yi}, Y = α1y1 + α2y2 + … . Let the random variables y1, y2, . . . have means E(y1), E(y2), . . . and variances σ2(y1), σ2(y2), . . . . The variable Y has mean E(Y) = α1E(y1) + α2 E(y2) + … and variance (Ref. 82) n

n

σ2(Y) =  α2i σ2(yi) + 2  i=1

n



αi αj Cov (yi, yj)

i=1 j=i+1

If the variables are uncorrelated and have the same variance, then σ2(Y) =

3-87

  α σ n

2 i

2

i=1

Next suppose the model relating Y to {yi} is nonlinear, but the errors are small and independent of one another. Then a change in Y is related to changes in yi by ∂Y ∂Y dY =
dy1 +
dy2 + … ∂y1 ∂y2 If the changes are indeed small, then the partial derivatives are constant among all the samples. Then the expected value of the change, E(dY), is zero. The variances are given by the following equation (Refs. 25 and 40): N ∂Y 2 σ2(dY) = 
σ i2 ∂yi i=1 Thus, the variance of the desired quantity Y can be found. This gives an independent estimate of the errors in measuring the quantity Y from the errors in measuring each variable it depends upon.

 

Example Suppose one wants to measure the thermal conductivity of a solid (k). To do this, one needs to measure the heat flux (q), the thickness of the

sample (d), and the temperature difference across the sample (∆T). Each measurement has some error. The heat flux (q) may be the rate of electrical heat ˙ divided by the area (A), and both quantities are measured to some tolinput (Q) erance. The thickness of the sample is measured with some accuracy, and the temperatures are probably measured with a thermocouple to some accuracy. These measurements are combined, however, to obtain the thermal conductivity, and it is desired to know the error in the thermal conductivity. The formula is d ˙ k=
Q A∆T The variance in the thermal conductivity is then k 2 k 2 k 2 2 k 2 σ k2 =
σ d2 +
σ Q˙ +
σ A2 +
σ 2∆T ˙ Q d A ∆T

 

 

 

 

FACTORIAL DESIGN OF EXPERIMENTS AND ANALYSIS OF VARIANCE Statistically designed experiments consider, of course, the effect of primary variables, but they also consider the effect of extraneous variables and the interactions between variables, and they include a measure of the random error. Primary variables are those whose effect you wish to determine. These variables can be quantitative or qualitative. The quantitative variables are ones you may fit to a model in order to determine the model parameters (see the section “Least Squares”). Qualitative variables are ones you wish to know the effect of, but you do not try to quantify that effect other than to assign possible errors or magnitudes. Qualitative variables can be further subdivided into Type I variables, whose effect you wish to determine directly, and Type II variables, which contribute to the performance variability and whose effect you wish to average out. For example, if you are studying the effect of several catalysts on yield in a chemical reactor, each different type of catalyst would be a Type I variable because you would like to know the effect of each. However, each time the catalyst is prepared, the results are slightly different due to random variations; thus, you may have several batches of what purports to be the same catalyst. The variability between batches is a Type II variable. Since the ultimate use will require using different batches, you would like to know the overall effect including that variation, since knowing precisely the results from one batch of one catalyst might not be representative of the results obtained from all batches of the same catalyst. A randomized block design, incomplete block design, or Latin square design (Ref. 40), for example, all keep the effect of experimental error in the blocked variables from influencing the effect of the primary variables. Other uncontrolled variables are accounted for by introducing randomization in parts of the experimental design. To study all variables and their interaction requires a factorial design, involving all possible

3-88

MATHEMATICS

combinations of each variable, or a fractional factorial design, involving only a selected set. Statistical techniques are then used to determine which are the important variables, what are the important interactions, and what the error is in estimating these effects. The discussion here is only a brief overview of the excellent Ref. 40. Suppose we have two methods of preparing some product and we wish to see which treatment is best. When there are only two treatments, then the sampling analysis discussed in the section “TwoPopulation Test of Hypothesis for Means” can be used to deduce if the means of the two treatments differ significantly. When there are more treatments, the analysis is more detailed. Suppose the experimental results are arranged as shown in the table: several measurements for each treatment. The goal is to see if the treatments differ significantly from each other; that is, whether their means are different when the samples have the same variance. The hypothesis is that the treatments are all the same, and the null hypothesis is that they are different. The statistical validity of the hypothesis is determined by an analysis of variance. Estimating the Effect of Four Treatments Treatment

Treatment average Grand average

1

2

3

4

— — —

— — — —

— — — — —

— — — — — — —

—

— —

—

The data for k = 4 treatments is arranged in the table. For each treatment, there are nt experiments and the outcome of the ith experiment with treatment t is called yti. Compute the treatment average nt

y

Basically the test for whether the hypothesis is true or not hinges on a comparison of the within-treatment estimate sR2 (with νR = N − k degrees of freedom) with the between-treatment estimate s2T (with νT = k − 1 degrees of freedom). The test is made based on the F distribution for νR and νT degrees of freedom (Table 3-7). Next consider the case that uses randomized blocking to eliminate the effect of some variable whose effect is of no interest, such as the batch-to-batch variation of the catalysts in the chemical reactor example. Suppose there are k treatments and n experiments in each treatment. The results from nk experiments can be arranged as shown in the block design table; within each block, the various treatments are applied in a random order. Compute the block average, the treatment average, as well as the grand average as before. Block Design with Four Treatments and Five Blocks Treatment

1

2

3

4

Block average

Block 1 Block 2 Block 3 Block 4 Block 5

— — — — —

— — — — —

— — — — —

— — — — —

— — — — —

The following quantities are needed for the analysis of variance table. Name

Formula

dof

average

2 SA = nky 

blocks

2 SB = k  i = 1 (y i − y)

n−1

treatments

ST = n  t = 1 (y t − y)

k−1

residuals

SR =  t = 1  i = 1 (yti − yi − yt + y)2

(n − 1)(k − 1)

total

S = t = 1  i = 1 y

N = nk

1

n

k

2

k

k

n

n

2 ti

ti

i=1

The key test is again a statistical one, based on the value of

yt =
nt

sT2 ,
sR2

Also compute the grand average k

 n y

t t

k

N =  nt

t=1

y =

, N

t=1

Next compute the sum of squares of deviations from the average within the tth treatment nt

St =  (yti − yt)2 i=1

Since each treatment has nt experiments, the number of degrees of freedom is nt − 1. Then the sample variances are St s t2 =
nt − 1 The within-treatment sum of squares is k

SR =  St t=1

and the within-treatment sample variance is SR sR2 =
N−k Now, if there is no difference between treatments, a second estimate of σ2 could be obtained by calculating the variation of the treatment averages about the grand average. Thus compute the betweentreatment mean square k ST 2 sT2 =
, ST =  nt(y t − y) k−1 t=1

ST sT2 =
, k−1

SR s R2 =

(n − 1)(k − 1)

and the F distribution for νR and νT degrees of freedom (Table 3-7). The assumption behind the analysis is that the variations are linear (Ref. 40). There are ways to test this assumption as well as transformations to make if it is not true. Reference 40 also gives an excellent example of how the observations are broken down into a grand average, a block deviation, a treatment deviation, and a residual. For twoway factorial design in which the second variable is a real one rather than one you would like to block out, see Ref. 40. To measure the effects of variables on a single outcome a factorial design is appropriate. In a two-level factorial design, each variable is considered at two levels only, a high and low value, often designated as a + and −. The two-level factorial design is useful for indicating trends, showing interactions, and it is also the basis for a fractional factorial design. As an example, consider a 23 factorial design with 3 variables and 2 levels for each. The experiments are indicated in the factorial design table. Two-Level Factorial Design with Three Variables Variable Run

1

2

3

1 2 3 4 5 6 7 8

− + − + − + − +

− − + + − − + +

− − − − + + + +

DIMENSIONAL ANALYSIS The main effects are calculated by calculating the difference between results from all high values of a variable and all low values of a variable; the result is divided by the number of experiments at each level. For example, for the first variable: [(y2 + y4 + y6 + y8) − (y1 + y3 + y5 + y7)] Effect of variable 1 =



4 Note that all observations are being used to supply information on each of the main effects and each effect is determined with the precision of a fourfold replicated difference. The advantage of a one-at-atime experiment is the gain in precision if the variables are additive and the measure of nonadditivity if it occurs (Ref. 40). Interaction effects between variables 1 and 2 are obtained by calculating the difference between the results obtained with the high and low value of 1 at the low value of 2 compared with the results obtained with the high and low value 1 at the high value of 2. The 12-interaction is [(y4 − y3 + y8 − y7) − (y2 − y1 + y6 − y5)] 12-interaction =



2

3-89

The key step is to determine the errors associated with the effect of each variable and each interaction so that the significance can be determined. Thus, standard errors need to be assigned. This can be done by repeating the experiments, but it can also be done by using higher-order interactions (such as 123 interactions in a 24 factorial design). These are assumed negligible in their effect on the mean but can be used to estimate the standard error (see Ref. 40). Then, calculated effects that are large compared with the standard error are considered important, while those that are small compared with the standard error are considered to be due to random variations and are unimportant. In a fractional factorial design one does only part of the possible experiments. When there are k variables, a factorial design requires 2k experiments. When k is large, the number of experiments can be large; for k = 5, 25 = 32. For a k this large, Box et al. (Ref. 82, p. 376) do a fractional factorial design. In the fractional factorial design with k = 5, only 16 experiments are done. Cropley (Ref. 82) gives an example of how to combine heuristics and statistical arguments in application to kinetics mechanisms in chemical engineering.

DIMENSIONAL ANALYSIS Dimensional analysis allows the engineer to reduce the number of variables that must be considered to model experiments or correlate data. Consider a simple example in which two variables F1 and F2 have the units of force and two additional variables L1 and L2 have the units of length. Rather than having to deduce the relation of one variable on the other three, F1 = fn (F2, L1, L2), dimensional analysis can be used to show that the relation must be of the form F1 /F2 = fn (L1 /L2). Thus considerable experimentation is saved. Historically, dimensional analysis can be done using the Rayleigh method or the Buckingham pi method. This brief discussion is equivalent to the Buckingham pi method but uses concepts from linear algebra; see Ref. 13 for further information. The general problem is posed as finding the minimum number of variables necessary to define the relationship between n variables. Let {Qi} represent a set of fundamental units, like length, time, force, and so on. Let [Pi] represent the dimensions of a physical quantity Pi; there are n physical quantities. Then form the matrix αij

Q1 Q2 … Qm

[P1]

[P2]

…

[Pn]

α11 α21

α12 α22

… …

α1n α2n

αm1

αm2

…

αmn

in which the entries are the number of times each fundamental unit appears in the dimensions [Pi]. The dimensions can then be expressed as follows.

Fluid thermal conductivity = k = (F/θT) Fluid specific heat = cp = (FL/MT) Dimensional constant = gc = (ML/Fθ2) The matrix α in this case is as follows. [Pi]

Qj

F M L θ T

Example: Buckingham Pi Method—Heat-Transfer Film Coefficient It is desired to determine a complete set of dimensionless groups with which to correlate experimental data on the film coefficient of heat transfer between the walls of a straight conduit with circular cross section and a fluid flowing in that conduit. The variables and the dimensional constant believed to be involved and their dimensions in the engineering system are given below: Film coefficient = h = (F/LθT) Conduit internal diameter = D = (L) Fluid linear velocity = V = (L/θ) Fluid density = ρ = (M/L3) Fluid absolute viscosity = µ = (M/Lθ)

D

V

ρ

µ

k

Cp

gc

1 0 −1 −1 −1

0 0 1 0 0

0 0 1 −1 0

0 1 −3 0 0

0 1 −1 −1 0

1 0 0 −1 −1

1 −1 1 0 −1

−1 1 1 −2 0

Here m ≤ 5, n = 8, p ≥ 3. Choose D, V, µ, k, and gc as the primary variables. By examining the 5 × 5 matrix associated with those variables, we can see that its determinant is not zero, so the rank of the matrix is m = 5; thus, p = 3. These variables are thus a possible basis set. The dimensions of the other three variables h, ρ, and Cp must be defined in terms of the primary variables. This can be done by inspection, although linear algebra can be used, too. hD h [h] = D−1k+1; thus
=
is a dimensionless group D−1 k k ρVD ρ [ρ] = µ1V −1D −1; thus

=
is a dimensionless group µ1V−1D−1 µ Cpµ Cp [Cp] = k+1µ−1; thus
=
is a dimensionless group k+1 µ−1 k Thus, the dimensionless groups are [Pi] hD ρVD Cp µ :
,
,


Q1α1i Q2α2i⋅⋅⋅Qmαmi k µ k

[Pi] = Q1α1i Q2α2i⋅⋅⋅Qmαmi Let m be the rank of the α matrix. Then p = n − m is the number of dimensionless groups that can be formed. One can choose m variables {Pi} to be the basis and express the other p variables in terms of them, giving p dimensionless quantities.

h

The dimensionless group hD/k is called the Nusselt number, NNu, and the group Cp µ /k is the Prandtl number, NPr. The group DVρ/µ is the familiar Reynolds number, NRe , encountered in fluid-friction problems. These three dimensionless groups are frequently used in heat-transfer-film-coefficient correlations. Functionally, their relation may be expressed as φ(NNu, NPr, NRe) = 0 or as

(3-91)

NNu = φ1(NPr, NRe)

It has been found that these dimensionless groups may be correlated well by an equation of the type hD/k = K(cpµ/k)a(DVρ/µ)b in which K, a, and b are experimentally determined dimensionless constants. However, any other type of algebraic expression or perhaps simply a graphical relation among these three groups that accurately fits the experimental data would be an equally valid manner of expressing Eq. (3-91).

3-90

MATHEMATICS

Naturally, other dimensionless groups might have been obtained in the example by employing a different set of five repeating quantities that would not form a dimensionless group among themselves. Some of these groups may be found among those presented in Table 3-8. Such a complete set of three dimensionless groups might consist of Stanton, Reynolds, and Prandtl numbers or of Stanton, Peclet, and Prandtl numbers. Also, such a complete set different from that obtained in the preceding example will result from a multiplication of appropriate powers of the Nusselt, Prandtl, and Reynolds numbers. For such a set to be complete, however, it must satisfy the condition that each of the three dimensionless groups be independent of the other two.

TABLE 3-8 Dimensionless Groups in the Engineering System of Dimensions Biot number Condensation number Number used in condensation of vapors Euler number Fourier number Froude number Graetz number Grashof number Mach number Nusselt number Peclet number Prandtl number Reynolds number Schmidt number Stanton number Weber number

NBi NCo NCv NEu NFo NFr NGz NGr NMa NNu NPe NPr NRe NSc NSt NWe

hL/k (h/k)(µ2/ρ2g)1/3 L3ρ2gλ/kµ∆t gc(−dp)/ρV2 kθ/ρcL2 V2/Lg wc/kL L3ρ2βg∆t/µ2 V/Va hD/k DVρc/k cµ/k DVρ/µ µ/ρDυ h/cVρ LV2ρ/σgc

PROCESS SIMULATION Classification Process simulation refers to the activity in which mathematical models of chemical processes and refineries are modeled with equations, usually on the computer. The usual distinction must be made between steady-state models and transient models, following the ideas presented in the introduction to this section. In a chemical process, of course, the process is nearly always in a transient mode, at some level of precision, but when the time-dependent fluctuations are below some value, a steady-state model can be formulated. This subsection presents briefly the ideas behind steady-state process simulation (also called flowsheeting), which are embodied in commercial codes. The transient simulations are important for designing startup of plants and are especially useful for the operating of chemical plants. Process Modules The usual first step in process simulation is to perform a mass and energy balance for a chosen process. The most important aspect of the simulation is that the thermodynamic data of the chemicals be modeled correctly. The computer results of vaporliquid equilibria, for example, must be checked against experimental data to insure their validity before using the data in more complicated computer calculations. At this first level of detail, it is not necessary to know the internal parameters for all the units, since what is desired is just the overall performance. For example, in a heat exchanger design, it suffices to know the heat duty, the total area, and the temperatures of the output streams; the details like the percentage baffle cut, tube layout, or baffle spacing can be specified later when the details of the proposed plant are better defined. Each unit operation is modeled by a subroutine, which is governed by equations (presented throughout this book). Some of the inputs to the units are known, some are specified by the user as design variables, and some are to be found using the simulation. It is important to know the number of degrees of freedom for each option of the unit operation, because at least that many parameters must be specified in order for the simulation to be able to calculate unit outputs. Sometimes the quantities the user would like to specify are targets, and parameters in the unit operation are to be changed to meet that target. This is not always possible, and the designer will have to adjust the parameters of the unit operation to achieve the desired target, possibly using the convergence tools discussed below. For example, in a reaction/separation system, if there is an impurity that must be purged, a common objective is to set the purge fraction so that the impurity concentration into the reactor is kept at some moderate value. Yet the solution techniques do not readily lend themselves to this connection, so convergence strategies must be employed.

Solution Strategies Consider a chemical process consisting of a series of units, such as distillation towers, reactors, and so forth. If the feed to the process is known and the operating parameters of the unit operations are specified by the user, then one can begin with the first unit, take the process input, calculate the unit output, carry that output to the input of the next unit, and continue the process. In this way, one can simulate the entire process. However, if the process involves a recycle stream, as nearly all chemical processes do, then when the calculation is begun, it is discovered that the recycle stream is unknown. Thus the calculation cannot begin. This situation leads to the need for an iterative process: the flow rates, temperature, and pressure of the unknown recycle stream are guessed and the calculations proceed as before. When one reaches the end of the process, where the recycle stream is formed to return to the inlet, it is necessary to check to see if the recycle stream is the same as assumed. If not, an iterative procedure must be used to cause convergence. The techniques like Wegstein (see “Numerical Solution of Nonlinear Equations in One Variable”) can be used to accelerate the convergence. When doing these iterations, it is useful to analyze the process using precedence ordering and tearing to minimize the number of recycle loops (Refs. 201, 242, 255, and 293). When the recycle loops interact with one another the iterations may not lead to a convergent solution. The designer usually wants to specify stream flow rates or parameters in the process, but these may not be directly accessible. For example, the desired separation may be known for a distillation tower, but the simulation program requires the specification of the number of trays. It is left up to the designer to choose the number of trays that lead to the desired separation. In the example of the purge stream/ reactor impurity, a controller module may be used to adjust the purge rate to achieve the desired reactor impurity. This further complicates the iteration process. An alternative method of solving the equations is to solve them as simultaneous equations. In that case, one can specify the design variables and the desired specifications and let the computer figure out the process parameters that will achieve those objectives. It is possible to overspecify the system or give impossible conditions. However, the biggest drawback to this method of simulation is that large sets (10,000s) of algebraic equations must be solved simultaneously. As computers become faster, this is less of an impediment. For further information, see Refs. 90, 175, 255, and 293. For information on computer software, see the Annual CEP Software Directory (Ref. 8) and other articles (Refs. 7 and 175).

INTELLIGENT SYSTEMS IN PROCESS ENGINEERING

3-91

INTELLIGENT SYSTEMS IN PROCESS ENGINEERING REFERENCES: General, 232, 248, 258, 275, 276. Knowledge-Based Systems, 49, 232, 275. Neural Networks, 54, 140. Qualitative Simulation, 178, 292. Fuzzy Logic, 94. Genetic Algorithms, 121. Applications, 15, 24, 205, 232, 250, 262, 294.

Intelligent system is a term that refers to computer-based systems that include knowledge-based systems, neural networks, fuzzy logic and fuzzy control, qualitative simulation, genetic algorithms, natural language understanding, and others. The term is often associated with a variety of computer programming languages and/or features that are used as implementation media, although this is an imprecise use. Examples include object-oriented languages, rule-based languages, prolog, and lisp. The term intelligent system is preferred over the term artificial intelligence. The three intelligent-system technologies currently seeing the greatest amount of industrial application are knowledge-based systems, fuzzy logic, and artificial neural networks. These technologies are components of distributed systems. Mathematical models, conventional numeric and statistical approaches, neural networks, knowledge-based systems, and the like, all have their place in practical implementation and allow automation of tasks not welltreated by numerical algorithms. Fundamentally, intelligent-system techniques are modeling techniques. They allow the encoding of qualitative models that draw upon experience and expertise, thereby extending modeling capacity beyond mathematical description. An important capability of intelligent system techniques is that they can be used not only to model physical behaviors but also decision-making processes. Decision processes reflect the selection, application, and interpretation of highly relevant pieces of information to draw conclusions about complex situations. Activity-specific decision processes can be expressed at a functional level, such as diagnosis, design, planning, and scheduling, or as their generic components, such as classification, abduction, and simulation. Decision process models address how information is organized and structured and then assimilated into active decisions. Knowledge-Based Systems Knowledge-based system (KBS) approaches capture the structural and information processing features of qualitative problem solving associated with sequential consideration, selection, and search. These technologies not only provide the means of capturing decision-making knowledge but also offer a medium for exploiting efficient strategies used by experts. KBSs, then, are computer programs that model specific ways of organizing problem-specific fragments of knowledge and then searching through them by establishing appropriate relationships to reach correct conclusions. Deliberation is a general label for the algorithmic process for sorting through the knowledge fragments. The basic components of KBSs are knowledge representation (structure) and search. They are the programming mechanisms that facilitate the use and application of the problem-specific knowledge appropriate to solving the problem. Together they are used to form conclusions, decisions, or interpretations in a symbolic form. See Refs. 49, 232, and 275. Qualitative simulation is a specific KBS model of physical processes that are not understood well enough to develop a physics-based numeric model. Corrosion, fouling, mechanical wear, equipment failure, and fatigue are not easily modeled, but decisions about them can be based on qualitative reasoning. See Refs. 178 and 292. Qualitative description of physical behaviors require that each continuous variable space be quantized. Quantization is typically based on landmark values that are boundary points separating qualitatively distinct regions of continuous values. By using these qualitative quantity descriptions, dynamic relations between variables can be modeled as qualitative equations that represent the structure of the system. The

solution to the equations represents the possible sequences of qualitative states as well as the explanations for changes in behaviors. Building and explaining a complex model requires a unified view called an ontology. Methods of qualitative reasoning can be based on different viewpoints; the dominant viewpoints are device, process, and constraints. Behavior generation is handled with two approaches: (1) simulating successive states from one or more initial states, and (2) determining all possible state-to-state transitions once all possible states are determined. Fuzzy Logic Fuzzy logic is a formalism for mapping between numerical values and qualitative or linguistic interpretations. This is useful when it is difficult to define precisely such terms as “high” and “low,” since there may be no fixed threshold. Fuzzy sets use the concept of degree of membership to overcome this problem. Degree of membership allows a descriptor to be associated with a range of numeric values but in varying degrees. A fuzzy set is explicitly defined by a degree of membership for each linguistic variable that is applicable, mA(x) where mA is the degree of membership for linguistic variable A. For fuzzy sets, logical operators, such as complement (NOT), intersection (AND), and union (OR) are defined. The following are typical definitions. NOT: mNOT A(x) = 1 − mA(x) AND: mA AND B(x) = min [mA(x), mB(x)] OR: mA OR B(x) = max [mA(x), mB(x)] Using these operators, fuzzy inference mechanisms are then developed to manipulate rules that include fuzzy values. The largest difference between fuzzy inference and ordinary inference is that fuzzy inference allows “partial match” of input and produces an “interpolated” output. This technology is useful in control also. See Ref. 94. Artificial Neural Networks An artificial neural network (ANN) is a collection of computational units that are interconnected in a network. Knowledge is captured in the form of weights, and input-output mappings are produced by the interactions of the weights and the computational units. Each computational unit combines weighted inputs and generates an output base on an activation function. Typical activation functions are (1) specified limit, (2) sigmoid, and (3) gaussian. ANNs can be feedforward, with multiple layers of intermediate units, or feedback (sometimes called recurrent networks). The ability to generalize on given data is one of the most important performance characteristics. With appropriate selection of training examples, an optimal network architecture, and appropriate training, the network can map a relationship between input and output that is complete but bounded by the coverage of the training data. Applications of neural networks can be broadly classified into three categories: 1. Numeric-to-numeric transformations are used as empirical mathematical models where the adaptive characteristics of neural networks learn to map between numeric sets of input-output data. In these modeling applications, neural networks are used as an alternative to traditional data regression schemes based on regression of plant data. Backpropagation networks have been widely used for this purpose. 2. Numeric-to-symbolic transformations are used in patternrecognition problems where the network is used to classify input data vectors into specific labeled classes. Pattern recognition problems include data interpretation, feature identification, and diagnosis. 3. Symbolic-to-symbolic transformations are used in various symbolic manipulations, including natural language processing and rulebased system implementation. See Refs. 54 and 140.

Section 4

Thermodynamics

Hendrick C. Van Ness, D.Eng., Howard P. Isermann Department of Chemical Engineering, Rensselaer Polytechnic Institute; Fellow, American Institute of Chemical Engineers; Member, American Chemical Society Michael M. Abbott, Ph.D., Howard P. Isermann Department of Chemical Engineering, Rensselaer Polytechnic Institute; Member, American Institute of Chemical Engineers

INTRODUCTION Postulate 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Postulate 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Postulate 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Postulate 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Postulate 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4-3 4-3 4-4 4-4 4-4

VARIABLES, DEFINITIONS, AND RELATIONSHIPS Constant-Composition Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Enthalpy and Entropy as Functions of T and P. . . . . . . . . . . . . . . . . . Internal Energy and Entropy as Functions of T and V. . . . . . . . . . . . Heat-Capacity Relations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Ideal Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Systems of Variable Composition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Partial Molar Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gibbs/Duhem Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Partial Molar Gibbs Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Ideal Gas State and the Compressibility Factor . . . . . . . . . . . . . Residual Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4-5 4-5 4-6 4-6 4-6 4-7 4-7 4-7 4-8 4-8 4-8

SOLUTION THERMODYNAMICS Ideal Gas Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fugacity and Fugacity Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fundamental Residual-Property Relation. . . . . . . . . . . . . . . . . . . . . . . . The Ideal Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fundamental Excess-Property Relation . . . . . . . . . . . . . . . . . . . . . . . . . Summary of Fundamental Property Relations . . . . . . . . . . . . . . . . . . . . Property Changes of Mixing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Behavior of Binary Liquid Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4-8 4-9 4-9 4-10 4-10 4-11 4-11 4-12

EVALUATION OF PROPERTIES Residual-Property Formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Liquid/Vapor Phase Transition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4-14 4-15

Liquid-Phase Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Properties from PVT Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pitzer’s Corresponding-States Correlation . . . . . . . . . . . . . . . . . . . . . Alternative Property Formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . Virial Equations of State. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Generalized Correlation for the Second Virial Coefficient. . . . . . . . . Cubic Equations of State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Benedict/Webb/Rubin Equation of State . . . . . . . . . . . . . . . . . . . . . . Expressions for the Excess Gibbs Energy . . . . . . . . . . . . . . . . . . . . . . . .

4-15 4-15 4-16 4-16 4-19 4-20 4-20 4-21 4-22

EQUILIBRIUM Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Phase Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 1: Application of the Phase Rule . . . . . . . . . . . . . . . . . . . . . Vapor/Liquid Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gamma/Phi Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Data Reduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solute/Solvent Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K-Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equation-of-State Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Liquid/Liquid and Vapor/Liquid/Liquid Equilibria . . . . . . . . . . . . . . . . Chemical-Reaction Stoichiometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chemical-Reaction Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Standard Property Changes of Reaction . . . . . . . . . . . . . . . . . . . . . . . Equilibrium Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 2: Single-Reaction Equilibrium . . . . . . . . . . . . . . . . . . . . . . Complex Chemical-Reaction Equilibria . . . . . . . . . . . . . . . . . . . . . . . Example 3: Minimization of Gibbs Energy . . . . . . . . . . . . . . . . . . . . .

4-24 4-24 4-25 4-25 4-25 4-26 4-27 4-28 4-28 4-30 4-31 4-31 4-31 4-32 4-32 4-33 4-34

THERMODYNAMIC ANALYSIS OF PROCESSES Calculation of Ideal Work. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lost Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Analysis of Steady-State, Steady-Flow Processes . . . . . . . . . . . . . . . . . . Example 4: Lost-Work Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4-34 4-35 4-35 4-36

4-1

4-2

THERMODYNAMICS

Nomenclature and Units Symbols are omitted that are correlation- or application-specific. Symbol A âi B C D B′ C′ D′ Bij Cijk CP CV EK EP fi fˆi G g g H Ki Kj ki M

Mi Mi  ∆M ∆M°j m m ˙ n ni P Pc Pisat pi Q Q˙ R

U.S. customary units

Definition

SI units

Helmholtz energy Activity of species i in solution 2d virial coefficient, density expansion 3d virial coefficient, density expansion 4th virial coefficient, density expansion 2d virial coefficient, pressure expansion 3d virial coefficient, pressure expansion 4th virial coefficient, pressure expansion Interaction 2d virial coefficient Interaction 3d virial coefficient Heat capacity at constant pressure Heat capacity at constant volume Kinetic energy Gravitational potential energy Fugacity of pure species i Fugacity of species i in solution Molar or unit-mass Gibbs energy Acceleration of gravity
GE/RT Molar or unit-mass enthalpy

J Dimensionless

Equilibrium K-value, yi/xi Equilibrium constant for chemical reaction j Henry’s constant Molar or unit-mass value of any extensive thermodynamic property of a solution Molar or unit-mass value of any extensive property of pure species i Partial molar property of species i in solution Property change of mixing Standard property change of reaction j Mass Mass flow rate Number of moles Number of moles of species i Absolute pressure Critical pressure Saturation or vapor pressure of species i Partial pressure of species i in gas mixture (
yiP) Heat Rate of heat transfer Universal gas constant

Dimensionless Dimensionless

Btu/lb mol or Btu/lbm Dimensionless Dimensionless

kPa

psi

cm3/mol 6

Btu Dimensionless cm3/mol

2

cm /mol

cm6/mol2

cm9/mol3

cm9/mol3

kPa−1

kPa−1

kPa−2

kPa−2

−3

kPa

kPa−3

cm3/mol

cm3/mol

cm6/mol2

cm6/mol2

J/(mol⭈K)

Btu/(lb mol⭈R)

J/(mol⭈K)

Btu/(lb mol⭈R)

J J kPa kPa

Btu Btu psi psi

J/mol or J/kg

Btu/lb mol or Btu/lbm ft/s2

m/s2 J/mol or J/kg

Symbol

Definition

SI units

S

Molar or unit-mass entropy

T Tc U

Absolute temperature Critical temperature, Molar or unit-mass internal energy Velocity Molar or unit-mass volume

u V W Ws ˙s W xi

yi Z z

Work Shaft work for flow process Shaft power for flow process Mole fraction in general or liquid-phase mole fraction of species i in solution Vapor-phase mole fraction of species i in solution Compressibility factor Elevation above a datum level

J/(mol⭈K) or J/(kg⭈K) K K J/mol or J/kg

U.S. customary units

m/s m3/mol or m3/kg J J J/s Dimensionless

Btu/(lb mol⭈R) or Btu/(lb⭈R) R R Btu/lb mol or Btu/lbm ft/s ft3/lb mol or ft3/lbm Btu Btu Btu/s Dimensionless

Dimensionless

Dimensionless

Dimensionless m

Dimensionless ft

Superscripts E id ig l lv R t v ∞

Denotes excess thermodynamic property Denotes value for an ideal solution Denotes value for an ideal gas Denotes liquid phase Denotes phase transition from liquid to vapor Denotes residual thermodynamic property Denotes a total value of a thermodynamic property Denotes vapor phase Denotes a value at infinite dilution

C c H r rev

Denotes a value for a colder heat reservoir Denotes a value for the critical state Denotes a value for a hotter heat reservoir Denotes a reduced value Denotes a reversible process

α,β

As superscripts, identify phases Volume expansivity, species i Reaction coordinate for reaction j Defined by Eq. (4-72) Heat-capacity ratio, CP/CV Activity coefficient of species i in solution Chemical potential of species i Stoichiometric number of species i in reaction j Molar density As a subscript, denotes a heat reservoir Defined by Eq. (4-283) Fugacity coefficient of pure species i Fugacity coefficient of species i in solution Acentric factor

Subscripts

Greek letters

βi εj Γi(T) γ γi

kg kg/s

lbm lbm/s

kPa kPa kPa

psi psi psi

ρ σ

kPa

psi

Φi φi

J J/s J/(mol⭈K)

Btu Btu/s Btu/(lb mol⭈R)

µi νi, j

φˆ i ω

K−1 mol

R−1 lb mol

J/mol Dimensionless Dimensionless

Btu/lb mol Dimensionless Dimensionless

J/mol Dimensionless

Btu/lb mol Dimensionless

mols/m3

lb moles/ft3

Dimensionless Dimensionless

Dimensionless Dimensionless

Dimensionless

Dimensionless

Dimensionless

Dimensionless

GENERAL REFERENCES: Abbott, M.M., and H.C. Van Ness, Schaum’s Outline of Theory and Problems of Thermodynamics, 2d ed., McGraw-Hill, New York, 1989. Tester, J.W. and M. Modell, Thermodynamics and its Applications, 3d ed., Prentice-Hall, Englewood Cliffs, N.J., 1996. Prausnitz, J.M., R.N. Lichtenthaler, and E.G. de Azevedo, Molecular Thermodynamics of Fluid-Phase Equilibria, 2d ed., Prentice-Hall, Englewood Cliffs, N.J., 1986. Reid, R.C., J.M. Prausnitz, and B.E. Poling, The Properties of Gases and Liquids, 4th ed.,

McGraw-Hill, New York, 1987. Sandler, S.I., Chemical and Engineering Thermodynamics, 2d ed., Wiley, New York, 1989. Smith, J.M., H.C. Van Ness, and M.M. Abbott, Introduction to Chemical Engineering Thermodynamics, 5th ed., McGraw-Hill, New York, 1996. Van Ness, H.C., and M.M. Abbott, Classical Thermodynamics of Nonelectrolyte Solutions: With Applications to Phase Equilibria, McGraw-Hill, New York, 1982.

INTRODUCTION Thermodynamics is the branch of science that embodies the principles of energy transformation in macroscopic systems. The general restrictions which experience has shown to apply to all such transformations are known as the laws of thermodynamics. These laws are primitive; they cannot be derived from anything more basic. The first law of thermodynamics states that energy is conserved; that, although it can be altered in form and transferred from one place to another, the total quantity remains constant. Thus, the first law of thermodynamics depends on the concept of energy; but, conversely, energy is an essential thermodynamic function because it allows the first law to be formulated. This coupling is characteristic of the primitive concepts of thermodynamics. The words system and surroundings are similarly coupled. A system is taken to be any object, any quantity of matter, any region, and so on, selected for study and set apart (mentally) from everything else, which is called the surroundings. The imaginary envelope which encloses the system and separates it from its surroundings is called the boundary of the system. Attributed to this boundary are special properties which may serve either (1) to isolate the system from its surroundings, or (2) to provide for interaction in specific ways between system and surroundings. An isolated system exchanges neither matter nor energy with its surroundings. If a system is not isolated, its boundaries may permit exchange of matter or energy or both with its surroundings. If the exchange of matter is allowed, the system is said to be open; if only energy and not matter may be exchanged, the system is closed (but not isolated), and its mass is constant. When a system is isolated, it cannot be affected by its surroundings. Nevertheless, changes may occur within the system that are detectable with such measuring instruments as thermometers, pressure gauges, and so on. However, such changes cannot continue indefinitely, and the system must eventually reach a final static condition of internal equilibrium. For a closed system which interacts with its surroundings, a final static condition may likewise be reached such that the system is not only internally at equilibrium but also in external equilibrium with its surroundings. The concept of equilibrium is central in thermodynamics, for associated with the condition of internal equilibrium is the concept of state. A system has an identifiable, reproducible state when all its properties, such as temperature T, pressure P, and molar volume V, are fixed. The concepts of state and property are again coupled. One can equally well say that the properties of a system are fixed by its state. Although the properties T, P, and V may be detected with measuring instruments, the existence of the primitive thermodynamic properties (see Postulates 1 and 3 following) is recognized much more indirectly. The number of properties for which values must be specified in order to fix the state of a system depends on the nature of the system and is ultimately determined from experience. When a system is displaced from an equilibrium state, it undergoes a process, a change of state, which continues until its properties attain new equilibrium values. During such a process the system may be

caused to interact with its surroundings so as to interchange energy in the forms of heat and work and so to produce in the system changes considered desirable for one reason or another. A process that proceeds so that the system is never displaced more than differentially from an equilibrium state is said to be reversible, because such a process can be reversed at any point by an infinitesimal change in external conditions, causing it to retrace the initial path in the opposite direction. Thermodynamics finds its origin in experience and experiment, from which are formulated a few postulates that form the foundation of the subject. The first two deal with energy: POSTULATE 1 There exists a form of energy, known as internal energy, which for systems at internal equilibrium is an intrinsic property of the system, functionally related to its characteristic coordinates. POSTULATE 2 (FIRST LAW OF THERMODYNAMICS) The total energy of any system and its surroundings is conserved. Internal energy is quite distinct from such external forms as the kinetic and potential energies of macroscopic bodies. Although a macroscopic property characterized by the macroscopic coordinates T and P, internal energy finds its origin in the kinetic and potential energies of molecules and submolecular particles. In applications of the first law of thermodynamics, all forms of energy must be considered, including the internal energy. It is therefore clear that Postulate 2 depends on Postulate 1. For an isolated system, the first law requires that its energy be constant. For a closed (but not isolated) system, the first law requires that energy changes of the system be exactly compensated by energy changes in the surroundings. Energy is exchanged between such a system and its surroundings in two forms: heat and work. Heat is energy crossing the system boundary under the influence of a temperature difference or gradient. A quantity of heat Q represents an amount of energy in transit between a system and its surroundings, and is not a property of the system. The convention with respect to sign makes numerical values of Q positive when heat is added to the system and negative when heat leaves the system. Work is again energy in transit between a system and its surroundings, but resulting from the displacement of an external force acting on the system. Like heat, a quantity of work W represents an amount of energy, and is not a property of the system. The sign convention, analogous to that for heat, makes numerical values of W positive when work is done on the system by the surroundings and negative when work is done on the surroundings by the system. When applied to closed (constant-mass) systems for which the only form of energy that changes is the internal energy, the first law of thermodynamics is expressed mathematically as dU t = dQ + dW (4-1) 4-3

4-4

THERMODYNAMICS

where U t is the total internal energy of the system. Note that dQ and dW, differential quantities representing energy exchanges between the system and its surroundings, serve to account for the energy change of the surroundings. On the other hand, dU t is directly the differential change in internal energy of the system. Integration of Eq. (4-1) gives for a finite process ∆Ut = Q + W

(4-2)

where ∆U is the finite change given by the difference between the final and initial values of U t. The heat Q and work W are finite quantities of heat and work; they are not properties of the system nor functions of the thermodynamic coordinates that characterize the system.

encountered in chemical technology is one for which the primary characteristic variables are temperature T, pressure P, molar volume V, and composition, not all of which are necessarily independent. Such systems are usually made up of fluids (liquid or gas) and are called PVT systems. For closed systems of this kind, the work of a reversible process may always be calculated from dWrev = −P dV t

t

POSTULATE 3 There exists a property called entropy, which for systems at internal equilibrium is an intrinsic property of the system, functionally related to the measurable coordinates which characterize the system. For reversible processes, changes in this property may be calculated by the equation: dS t = dQrev /T

(4-3)

where St is the total entropy of the system and T is the absolute temperature of the system. POSTULATE 4 (SECOND LAW OF THERMODYNAMICS) The entropy change of any system and its surroundings, considered together, resulting from any real process is positive, approaching zero when the process approaches reversibility. In the same way that the first law of thermodynamics cannot be formulated without the prior recognition of internal energy as a property, so also the second law can have no complete and quantitative expression without a prior assertion of the existence of entropy as a property. The second law requires that the entropy of an isolated system either increase or, in the limit, where the system has reached an equilibrium state, remain constant. For a closed (but not isolated) system it requires that any entropy decrease in either the system or its surroundings be more than compensated by an entropy increase in the other part or that in the limit, where the process is reversible, the total entropy of the system plus its surroundings be constant. The fundamental thermodynamic properties that arise in connection with the first and second laws of thermodynamics are internal energy and entropy. These properties, together with the two laws for which they are essential, apply to all types of systems. However, different types of systems are characterized by different sets of measurable coordinates or variables. The type of system most commonly

(4-4)

t

where P is the absolute pressure and V is the total volume of the system. This equation follows directly from the definition of mechanical work. POSTULATE 5 The macroscopic properties of homogeneous PVT systems at internal equilibrium can be expressed as functions of temperature, pressure, and composition only. This postulate imposes an idealization, and is the basis for all subsequent property relations for PVT systems. The PVT system serves as a satisfactory model in an enormous number of practical applications. In accepting this model one assumes that the effects of fields (e.g., electric, magnetic, or gravitational) are negligible and that surface and viscous-shear effects are unimportant. Temperature, pressure, and composition are thermodynamic coordinates representing conditions imposed upon or exhibited by the system, and the functional dependence of the thermodynamic properties on these conditions is determined by experiment. This is quite direct for molar or specific volume V, which can be measured, and leads immediately to the conclusion that there exists an equation of state relating molar volume to temperature, pressure, and composition for any particular homogeneous PVT system. The equation of state is a primary tool in applications of thermodynamics. Postulate 5 affirms that the other molar or specific thermodynamic properties of PVT systems, such as internal energy U and entropy S, are also functions of temperature, pressure, and composition. These molar or unit-mass properties, represented by the plain symbols V, U, and S, are independent of system size and are called intensive. Temperature, pressure, and the composition variables, such as mole fraction, are also intensive. Total-system properties (V t, U t, St ) do depend on system size, and are extensive. For a system containing n moles of fluid, M t = nM, where M is a molar property. Applications of the thermodynamic postulates necessarily involve the abstract quantities internal energy and entropy. The solution of any problem in applied thermodynamics is therefore found through these quantities.

VARIABLES, DEFINITIONS, AND RELATIONSHIPS Consider a single-phase closed system in which there are no chemical reactions. Under these restrictions the composition is fixed. If such a system undergoes a differential, reversible process, then by Eq. (4-1) dU t = dQrev + dWrev

where i is an index identifying the chemical species present. When U, S, and V represent specific (unit-mass) properties, n is replaced by m. Equation (4-5) shows that for the single-phase, nonreacting, closed system specified, nU = u(nS, nV)

Substitution for dQrev and dWrev by Eqs. (4-3) and (4-4) gives dU t = T dS t − P dV t

Then

Although derived for a reversible process, this equation relates properties only and is valid for any change between equilibrium states in a closed system. It may equally well be written d(nU) = T d(nS) − P d(nV)

(4-5)

where n is the number of moles of fluid in the system and is constant for the special case of a closed, nonreacting system. Note that n
n1 + n2 + n3 + ⋅⋅⋅ =  ni i

∂(nU) d(nU) = ᎏ ∂(nS)





nV,n

∂(nU) d(nS) + ᎏ ∂(nV)





d(nV) nS,n

where the subscript n indicates that all mole numbers ni (and hence n) are held constant. Comparison with Eq. (4-5) shows that ∂(nU) =T (4-6) ᎏ ∂(nS) nV,n





∂(nU) ᎏ ∂(nV) 

= −P nS,n

(4-7)

VARIABLES, DEFINITIONS, AND RELATIONSHIPS Consider now an open system consisting of a single phase and assume that nU =
(nS, nV, n1, n2, n3, . . .) Then ∂(nU) ∂(nU) ∂(nU) d(nS) + ᎏ d(nV) +  ᎏ dni d(nU) = ᎏ ∂(nS) nV,n ∂(nV) nS,n ∂ni nS,nV,n i where the summation is over all species present in the system and subscript nj indicates that all mole numbers are held constant except the ith. Let ∂(nU) µi
ᎏ ∂ni nS,nV,nj















j



Together with Eqs. (4-6) and (4-7), this definition allows elimination of all the partial differential coefficients from the preceding equation: d(nU) = T d(nS) − P d(nV) +  µi dni

(4-8)

i

Equation (4-8) is the fundamental property relation for singlephase PVT systems, from which all other equations connecting properties of such systems are derived. The quantity µi is called the chemical potential of species i, and it plays a vital role in the thermodynamics of phase and chemical equilibria. Additional property relations follow directly from Eq. (4-8). Since ni = xin, where xi is the mole fraction of species i, this equation may be rewritten: d(nU) − T d(nS) + P d(nV) −  µi d(xin) = 0 i

Upon expansion of the differentials and collection of like terms, this becomes



 



dU − T dS + P dV −  µi dxi n + U − TS + PV −  xiµi dn = 0 i

i

Since n and dn are independent and arbitrary, the terms in brackets must separately be zero. Then dU = T dS − P dV +  µi dxi

(4-9)

i

U = TS − PV +  xiµi

(4-10)

i

Equations (4-8) and (4-9) are similar, but there is an important difference. Equation (4-8) applies to a system of n moles where n may vary; whereas Eq. (4-9) applies to a system in which n is unity and invariant. Thus Eq. (4-9) is subject to the constraint that i xi = 1 or that i dxi = 0. In this equation the xi are not independent variables, whereas the ni in Eq. (4-8) are. Equation (4-10) dictates the possible combinations of terms that may be defined as additional primary functions. Those in common use are: Enthalpy Helmholtz energy Gibbs energy

H
U + PV A
U − TS G
U + PV − TS = H − TS

(4-11) (4-12) (4-13)

Additional thermodynamic properties are related to these and arise by arbitrary definition. Multiplication of Eq. (4-11) by n and differentiation yields the general expression: d(nH) = d(nU) + P d(nV) + nV dP d(nH) = T d(nS) + nV dP +  µi dni

and so on, as a function of a particular set of independent variables; these are the canonical variables for the property. The choice of which equation to use in a particular application is dictated by convenience. However, the Gibbs energy G is special, because of its unique functional relation to T, P, and the ni, which are the variables of primary interest in chemical processing. A similar set of equations is developed from Eq. (4-9). This set also follows from the preceding set when n = 1 and ni = xi. The two sets are related exactly as Eq. (4-8) is related to Eq. (4-9). The equations written for n = 1 are, of course, less general. Furthermore, the interdependence of the xi precludes those mathematical operations which depend on independence of these variables. CONSTANT-COMPOSITION SYSTEMS For 1 mole of a homogeneous fluid of constant composition Eqs. (4-8) and (4-14) through (4-16) simplify to: dU = T dS − P dV dH = T dS + V dP dA = −S dT − P dV dG = −S dT + V dP Implicit in these are the following: ∂U ∂H T= ᎏ = ᎏ ∂S V ∂S

    ∂U ∂A −P =  ᎏ  =  ᎏ  ∂V ∂V ∂H ∂G V = ᎏ = ᎏ ∂P ∂P

(4-14)

i

The total differentials of nA and nG are obtained similarly: d(nA) = −nS dT − P d(nV) +  µi dni

(4-15)

d(nG) = −nS dT + nV dP +  µi dni

(4-16)

i

i

Equations (4-8) and (4-14) through (4-16) are equivalent forms of the fundamental property relation. Each expresses a property nU, nH,

(4-17) (4-18) (4-19) (4-20)

(4-21)

P

S

T

S

T

(4-22) (4-23)

∂A ∂G (4-24) −S = ᎏ = ᎏ ∂T V ∂T P In addition, the common Maxwell equations result from application of the reciprocity relation for exact differentials: ∂T ∂P (4-25) ᎏ =− ᎏ ∂V S ∂S V

   

    ∂T ∂V = ᎏ ᎏ ∂P   ∂S  S

∂P ∂S = ᎏ ᎏ ∂T   ∂V  V

(4-26) P

(4-27) T

∂V ∂S = − ᎏ  ᎏ ∂T  ∂P P

(4-28)

T

In all these equations the partial derivatives are taken with composition held constant. Enthalpy and Entropy as Functions of T and P At constant composition the molar thermodynamic properties are functions of temperature and pressure (Postulate 5). Thus ∂H ∂H dH = ᎏ dT + ᎏ dP (4-29) ∂T P ∂P T

 

 

∂S ∂S (4-30) dS = ᎏ dT + ᎏ dP ∂T P ∂P T The obvious next step is to eliminate the partial-differential coefficients in favor of measurable quantities. The heat capacity at constant pressure is defined for this purpose: ∂H CP
ᎏ (4-31) ∂T P It is a property of the material and a function of temperature, pressure, and composition. Equation (4-18) may first be divided by dT and restricted to constant pressure, and then be divided by dP and restricted to constant temperature, yielding the two equations:

 

Substitution for d(nU) by Eq. (4-8) reduces this result to:

4-5

 

 

4-6

THERMODYNAMICS ∂H

∂S = T ᎏ ᎏ ∂T  ∂T P

P

∂H ∂S ᎏ =T ᎏ +V ∂P T ∂P T In view of Eq. (4-31), the first of these becomes ∂S CP (4-32) ᎏ =ᎏ ∂T P T and in view of Eq. (4-28), the second becomes ∂H ∂V (4-33) ᎏ =V−T ᎏ ∂P T ∂T P Combination of Eqs. (4-29), (4-31), and (4-33) gives ∂V dH = CP dT + V − T ᎏ dP (4-34) ∂T P and in combination Eqs. (4-30), (4-32), and (4-28) yield CP ∂V dS = ᎏ dT − ᎏ dP (4-35) T ∂T P Equations (4-34) and (4-35) are general expressions for the enthalpy and entropy of homogeneous fluids at constant composition as functions of T and P. The coefficients of dT and dP are expressed in terms of measurable quantities. Internal Energy and Entropy as Functions of T and V Because V is related to T and P through an equation of state, V rather than P can serve as an independent variable. In this case the internal energy and entropy are the properties of choice; whence ∂U ∂U dU = ᎏ dT + ᎏ dV (4-36) ∂T V ∂V T

 

 

 

 

 



 

 

 

 

∂S ∂S (4-37) dS = ᎏ dT + ᎏ dV ∂T V ∂V T The procedure now is analogous to that of the preceding section. Define the heat capacity at constant volume by ∂U CV
ᎏ (4-38) ∂T V It is a property of the material and a function of temperature, pressure, and composition. Two relations follow immediately from Eq. (4-17): ∂U ∂S ᎏ =T ᎏ ∂T V ∂T V

 

 

 

    ∂U ∂S = T ᎏ − P ᎏ ∂V  ∂V T

T

As a result of Eq. (4-38) the first of these becomes CV ∂S (4-39) ᎏ =ᎏ ∂T V T and as a result of Eq. (4-27), the second becomes ∂U ∂P (4-40) ᎏ =T ᎏ −P ∂V T ∂T V Combination of Eqs. (4-36), (4-38), and (4-40) gives ∂P dU = CV dT + T ᎏ − P dV (4-41) ∂T V and Eqs. (4-37), (4-39), and (4-27) together yield ∂P CV dS = ᎏ dT + ᎏ dV (4-42) T ∂T V Equations (4-41) and (4-42) are general expressions for the internal energy and entropy of homogeneous fluids at constant composition as functions of temperature and molar volume. The coefficients of dT and dV are expressed in terms of measurable quantities.

 

 

 

     

Heat-Capacity Relations In Eqs. (4-34) and (4-41) both dH and dU are exact differentials, and application of the reciprocity relation leads to ∂CP ∂2V (4-43) ᎏ = −T ᎏ2 ∂P T ∂T P

 

 

∂CV

∂2P

= T ᎏ ᎏ ∂V  ∂T 2

T

(4-44)

V

Thus, the pressure or volume dependence of the heat capacities may be determined from PVT data. The temperature dependence of the heat capacities is, however, determined empirically and is often given by equations such as CP = α + βT + γ T 2 Equations (4-35) and (4-42) both provide expressions for dS, which must be equal for the same change of state. Equating them and solving for dT gives T ∂V T ∂P dT = ᎏ ᎏ dP + ᎏ ᎏ dV CP − CV ∂T P CP − CV ∂T V However, at constant composition T = T(P,V), and ∂T ∂T dT = ᎏ dP + ᎏ dV ∂P V ∂V P Equating coefficients of either dP or dV in these two expressions for dT gives ∂V ∂P CP − CV = T ᎏ ᎏ (4-45) ∂T P ∂T V Thus the difference between the two heat capacities may be determined from PVT data. Division of Eq. (4-32) by Eq. (4-39) yields the ratio of these heat capacities: CP (∂S/∂T)P (∂S/∂V)P(∂V/∂T)P ᎏ = ᎏ = ᎏᎏ CV (∂S/∂T)V (∂S/∂P)V(∂P/∂T)V

 

 

 

 

  

Replacement of each of the four partial derivatives through the appropriate Maxwell relation gives finally C ∂V ∂P γ
ᎏP = ᎏ (4-46) ᎏ CV ∂P T ∂V S where γ is the symbol conventionally used to represent the heatcapacity ratio. The Ideal Gas The simplest equation of state is the ideal gas equation: PV = RT

  

where R is a universal constant, values of which are given in Table 1-9. The following partial derivatives are obtained from the ideal gas equation: ∂P R P ∂2P ᎏ =ᎏ=ᎏ ᎏ2 = 0 ∂T V V T ∂T V

 

 

∂V R V ∂V =ᎏ=ᎏ =0 ᎏ ᎏ ∂T  P T ∂T  2

2

P

P

∂P P ᎏ =−ᎏ ∂V T V The general equations for constant-composition fluids derived in the preceding subsections reduce to very simple forms when the relations for an ideal gas are substituted into them:

 

∂U ∂H = ᎏ =0 ᎏ ∂V   ∂P  T

T

∂S R ∂S R =−ᎏ =ᎏ ᎏ ᎏ ∂P  P ∂V  V T

dU = CV dT

T

VARIABLES, DEFINITIONS, AND RELATIONSHIPS dH = CP dT CV R dS = ᎏ dT + ᎏ dV T V

 

 

 

 

Combining these expressions with Eq. (4-48) and collecting like terms gives ∂M ∂M dM − ᎏ dT − ᎏ dP −   Mi dxi n + M −   Mi xi dn = 0 ∂T P,x ∂P T,x i i



CP R dS = ᎏ dT − ᎏ dP T P ∂CV

∂CP

∂ ln P C γ
ᎏP = − ᎏ CV ∂ ln V





and

SYSTEMS OF VARIABLE COMPOSITION The composition of a system may vary because the system is open or because of chemical reactions even in a closed system. The equations developed here apply regardless of the cause of composition changes. Partial Molar Properties Consider a homogeneous fluid solution comprised of any number of chemical species. For such a PVT system let the symbol M represent the molar (or unit-mass) value of any extensive thermodynamic property of the solution, where M may stand in turn for U, H, S, and so on. A total-system property is then nM, where n =  ini and i is the index identifying chemical species. One might expect the solution property M to be related solely to the properties Mi of the pure chemical species which comprise the solution. However, no such generally valid relation is known, and the connection must be established experimentally for every specific system. Although the chemical species which make up a solution do not in fact have separate properties of their own, a solution property may be arbitrarily apportioned among the individual species. Once an apportioning recipe is adopted, then the assigned property values are quite logically treated as though they were indeed properties of the species in solution, and reasoning on this basis leads to valid conclusions. For a homogeneous PVT system, Postulate 5 requires that nM = (T, P, n1, n2, n3 , . . .) The total differential of nM is therefore





P,n

∂(nM) dT + ᎏ ∂P





T,n

∂(nM) dP +  ᎏ ∂ni i





dni T,P,nj

where subscript n indicates that all mole numbers ni are held constant, and subscript nj signifies that all mole numbers are held constant except the ith. This equation may also be written ∂M d(nM) = n ᎏ ∂T

 

P,x

∂M dT + n ᎏ ∂P

 

T,x

∂(nM) dP +  ᎏ ∂ni i





dni T,P,nj

where subscript x indicates that all mole fractions are held constant. The derivatives in the summation are called partial molar properties M i; by definition, ∂(nM) M (4-47) i
ᎏ ∂ni T,P,nj





The basis for calculation of partial properties from solution properties is provided by this equation. Moreover, the preceding equation becomes ∂M ∂M d(nM) = n ᎏ dT + n ᎏ dP +  M (4-48) i dni ∂T P,x ∂P T,x i

 

 

 

Important equations follow from this result through the relations: d(nM) = n dM + M dn dni = d(xin) = xi dn + n dxi



and arbitrary, the terms in brackets ∂M

ᎏ ∂P 

T,x

dP +  M i dxi

(4-49)

i

M =  xiM i

(4-50)

i

S

These equations clearly show that for an ideal gas U, H, CP, and CV are functions of temperature only and are independent of P and V. The entropy of an ideal gas, however, is a function of both T and P or of both T and V.

∂(nM) d(nM) = ᎏ ∂T

 

 

T

CP − CV = R

 

Since n and dn are independent must separately be zero; whence ∂M dM = ᎏ dT + ∂T P,x

= ᎏ =0 ᎏ ∂V   ∂P  T

4-7

Equation (4-49) is merely a special case of Eq. (4-48); however, Eq. (4-50) is a vital new relation. Known as the summability equation, it provides for the calculation of solution properties from partial properties. Thus, a solution property apportioned according to the recipe of Eq. (4-47) may be recovered simply by adding the properties attributed to the individual species, each weighted by its mole fraction in solution. The equations for partial molar properties are also valid for partial specific properties, in which case m replaces n and the xi are mass fractions. Equation (4-47) applied to the definitions of Eqs. (4-11) through (4-13) yields the partial-property relations: Hi =  Ui + PV  i A Ui − TS i i =  G Hi − TS i i =  Pertinent examples on partial molar properties are presented in Smith, Van Ness, and Abbott (Introduction to Chemical Engineering Thermodynamics, 5th ed., Sec. 10.3, McGraw-Hill, New York, 1996). Gibbs/Duhem Equation Differentiation of Eq. (4-50) yields dM =  xi dM i +  M i dxi i

i

Since this equation and Eq. (4-49) are both valid in general, their right-hand sides can be equated, yielding ∂M ∂M (4-51) ᎏ dT + ᎏ dP −  xi dM i = 0 ∂T P,x ∂P T,x i

 

 

This general result, the Gibbs/Duhem equation, imposes a constraint on how the partial molar properties of any phase may vary with temperature, pressure, and composition. For the special case where T and P are constant:  =0  x dM i

(constant T, P)

i

(4-52)

i

Symbol M may represent the molar value of any extensive thermodynamic property; for example, V, U, H, S, or G. When M
H, the derivatives (∂H/∂T)P and (∂H/∂P)T are given by Eqs. (4-31) and (4-33). Equations (4-49), (4-50), and (4-51) then become ∂V dH = CP dT + V − T ᎏ dP +  H (4-53) i dxi ∂T P,x i



 

H =  xiH i

(4-54)

i

∂V CP dT + V − T ᎏ dP −  xi dH (4-55) i = 0 ∂T P,x i Similar equations are readily derived when M takes on other identities. Equation (4-47), which defines a partial molar property, provides a general means by which partial property values may be determined. However, for a binary solution an alternative method is useful. Equation (4-50) for a binary solution is



 

M = x1 M 1 + x2 M 2

(4-56)

Moreover, the Gibbs/Duhem equation for a solution at given T and P, Eq. (4-52), becomes x1 dM 1 + x2 dM 2 = 0

(4-57)

4-8

THERMODYNAMICS

These two equations can be combined to give dM M 1 = M + x2 ᎏ dx1

The following equation is a mathematical identity:

dM M2 = M − x1 ᎏ  dx1

nG 1 nG d ᎏ
ᎏ d(nG) − ᎏ2 dT RT RT RT

(4-58b)

Substitution for d(nG) by Eq. (4-16) and for G by H − TS (Eq. [4-13]) gives, after algebraic reduction,

Thus for a binary solution, the partial properties are given directly as functions of composition for given T and P. For multicomponent solutions such calculations are complex, and direct use of Eq. (4-47) is appropriate. Partial Molar Gibbs Energy Implicit in Eq. (4-16) is the relation ∂(nG) µi = ᎏ ∂ni T,P,nj





In view of Eq. (4-47), the chemical potential and the partial molar Gibbs energy are therefore identical: µi =  Gi

(4-59)

The reciprocity relation for an exact differential applied to Eq. (416) produces not only the Maxwell relation, Eq. (4-28), but also two other useful equations: ∂µi ∂(nV) = Vi (4-60) ᎏ = ᎏ ∂P T,n ∂ni T,P,nj

    ∂µ ∂(nS) = − ᎏ  ᎏ ∂T  ∂n i

P,n

i

T,P,nj

= −S i

(4-61)

In a solution of constant composition, µi = µ(T,P); whence ∂µi ∂µi dµi
dG i = ᎏ dT + ᎏ dP ∂T P,n ∂P T,n

 

or

 

dG i = −S i dT + V i dP

(4-62)

Comparison with Eq. (4-20) provides an example of the parallelism that exists between the equations for a constant-composition solution and those for the corresponding partial properties. This parallelism exists whenever the solution properties in the parent equation are related linearly (in the algebraic sense). Thus, in view of Eqs. (4-17), (4-18), and (4-19): dU i = T dS i − P dV i Vi dP dH i = T dS i +  dA i = −S i dT − P dV i

 

(4-58a)

(4-63) (4-64) (4-65)

nG nV nH µi d ᎏ = ᎏ dP − ᎏ2 dT +  ᎏ dni RT RT RT RT i

 

(4-66)

Equation (4-66) is a useful alternative to the fundamental property relation given by Eq. (4-16). All terms in this equation have the units of moles; moreover, the enthalpy rather than the entropy appears on the right-hand side. The Ideal Gas State and the Compressibility Factor The simplest equation of state for a PVT system is the ideal gas equation: PV ig = RT ig

where V is the ideal-gas–state molar volume. Similarly, H ig, S ig, and G ig are ideal gas–state values; that is, the molar enthalpy, entropy, and Gibbs energy values that a PVT system would have were the ideal gas equation the correct equation of state. These quantities provide reference values to which actual values may be compared. For example, the compressibility factor Z compares the true molar volume to the ideal gas molar volume as a ratio: V V PV Z=ᎏ =ᎏ=ᎏ V ig RT/P RT Generalized correlations for the compressibility factor are treated in Sec. 2. Residual Properties These quantities compare true and ideal gas properties through differences: MR
M − M ig

(4-67)

where M is the molar value of an extensive thermodynamic property of a fluid in its actual state and Mig is the corresponding value for the ideal gas state of the fluid at the same T, P, and composition. Residual properties depend on interactions between molecules and not on characteristics of individual molecules. Since the ideal gas state presumes the absence of molecular interactions, residual properties reflect deviations from ideality. Most commonly used of the residual properties are: V R
V − V ig HR
H − H ig SR
S − S ig GR
G − G ig

Residual volume Residual enthalpy Residual entropy Residual Gibbs energy

Note that these equations hold only for species in a constantcomposition solution.

SOLUTION THERMODYNAMICS M iig(T, P) = Miig(T, pi)

IDEAL GAS MIXTURES An ideal gas is a model gas comprising imaginary molecules of zero volume that do not interact. Each chemical species in an ideal gas mixture therefore has its own private properties, uninfluenced by the presence of other species. The partial pressure of species i in a gas mixture is defined as pi = xiP

The partial molar property, other than the volume, of a constituent species in an ideal gas mixture is equal to the corresponding molar property of the species as a pure ideal gas at the mixture temperature but at a pressure equal to its partial pressure in the mixture.

This is expressed mathematically for generic partial property M iig by the equation

(4-68)

Miig = Miig  ig

where Mi is evaluated at the mixture T and P. Thus, for the enthalpy,

(i = 1, 2, . . . , N)

where xi is the mole fraction of species i. The sum of the partial pressures clearly equals the total pressure. Gibbs’ theorem for a mixture of ideal gases may be stated as follows:

(M ≠ V)

For those properties of an ideal gas that are independent of P, for example, U, H, and CP, this becomes simply

Hiig = Hiig  The entropy of an ideal gas does depend on pressure: dSiig = −R d ln P

(constant T )

Integration from pi to P gives P P Siig(T, P) − Siig(T, pi) = −R ln ᎏ = −R ln ᎏ = R ln xi pi xi P Whence

Siig(T, pi) = Siig(T, P) − R ln xi

(4-69)

SOLUTION THERMODYNAMICS Substituting this result into Eq. (4-68) written for the entropy gives Sl = S − R ln xi  ig

ig i

(4-70)

where Siig is evaluated at the mixture T and P. For the Gibbs energy of an ideal gas mixture, Gig = Hig − TSig; the parallel relation for partial properties is Giig =  Hiig − TSiig In combination with Eqs. (4-69) and (4-70), this becomes G Hiig − TSiig + RT ln xi iig =  or

µiig
 Giig = Giig + RT ln xi

(4-72)

where Γi(T), the integration constant for a given temperature, is a function of temperature only. Equation (4-71) now becomes µigi = Γi(T) + RT ln xi P

(4-73)

FUGACITY AND FUGACITY COEFFICIENT The chemical potential µi plays a vital role in both phase and chemicalreaction equilibria. However, the chemical potential exhibits certain unfortunate characteristics which discourage its use in the solution of practical problems. The Gibbs energy, and hence µi, is defined in relation to the internal energy and entropy, both primitive quantities for which absolute values are unknown. Moreover, µi approaches negative infinity when either P or xi approaches zero. While these characteristics do not preclude the use of chemical potentials, the application of equilibrium criteria is facilitated by introduction of the fugacity, a quantity that takes the place of µi but which does not exhibit its less desirable characteristics. The origin of the fugacity concept resides in Eq. (4-72), an equation valid only for pure species i in the ideal gas state. For a real fluid, an analogous equation is written: Gi
Γi(T) + RT ln f i

(4-74)

in which a new property f i replaces the pressure P. This equation serves as a partial definition of the fugacity f i. Subtraction of Eq. (4-72) from Eq. (4-74), both written for the same temperature and pressure, gives f Gi − Giig = RT ln ᎏi P According to the definition of Eq. (4-67), Gi − Giig is the residual Gibbs energy, GiR. The dimensionless ratio fi /P is another new property called the fugacity coefficient φi. Thus, (4-75) G = RT ln φi fi where φi
ᎏ (4-76) P The definition of fugacity is completed by setting the ideal-gas–state fugacity of pure species i equal to its pressure: R i

f iig = P Thus, for the special case of an ideal gas, GiR = 0, φi = 1, and Eq. (4-72) is recovered from Eq. (4-74). The definition of the fugacity of a species in solution is parallel to the definition of the pure-species fugacity. An equation analogous to the ideal gas expression, Eq. (4-73), is written for species i in a fluid mixture: µ
Γ (T) + RT ln fˆ (4-77) i

i

i in solution. Since it is not a partial molar property, it is identified by a circumflex rather than an overbar. Subtracting Eq. (4-73) from Eq. (4-77), both written for the same temperature, pressure, and composition, yields fˆi µi − µiig = RT ln ᎏ xiP Analogous to the defining equation for the residual Gibbs energy of a mixture, GR
G − Gig, is the definition of a partial molar residual Gibbs energy: G Gi −  Gigi = µi − µigi iR


(4-71)

Elimination of Giig from this equation is accomplished by Eq. (4-20), written for pure species i as: RT dGiig = Viig dP = ᎏ dP = RT d ln P (constant T) P Integration gives Giig = Γi(T) + RT ln P

4-9

i

where the partial pressure xi P is replaced by fˆi , the fugacity of species

GiR = RT ln φˆ i (4-78)  ˆ f i where by definition φˆ i
ᎏ (4-79) xiP The dimensionless ratio φˆ i is called the fugacity coefficient of species i in solution. Eq. (4-78) is the analog of Eq. (4-75), which relates φi to GiR. For an ideal gas,  GiR is necessarily 0; therefore φˆ iig = 1, and fˆ ig = x P Therefore

i

i

Thus, the fugacity of species i in an ideal gas mixture is equal to its partial pressure. Pertinent examples are given in Smith, Van Ness, and Abbott (Introduction to Chemical Engineering Thermodynamics, 5th ed., Secs. 10.5–10.7, McGraw-Hill, New York, 1996). FUNDAMENTAL RESIDUAL-PROPERTY RELATION In view of Eq. (4-59), the fundamental property relation given by Eq. (4-66) may be written Gi  nH nG nV d ᎏ = ᎏ dP − ᎏ2 dT +  ᎏ dni (4-80) RT RT RT RT i

 

This equation is general, and may be written for the special case of an ideal gas: G nGig nVig nHig iig d ᎏ = ᎏ dP − ᎏ2 dT +  ᎏ dni RT RT RT RT i

 

Subtraction of this equation from Eq. (4-80) gives GiR nGR nVR nHR  d ᎏ = ᎏ dP − ᎏ2 dT +  ᎏ dni RT RT RT RT i

 

(4-81)

where the definitions GR
G − Gig and G iR
G i − G iig have been imposed. Equation (4-81) is the fundamental residual-property relation. An alternative form follows by introduction of the fugacity coefficient as given by Eq. (4-78): nGR nVR nHR d ᎏ = ᎏ dP − ᎏ2 dT +  ln φˆ i dni (4-82) RT RT RT i

 

These equations are of such generality that for practical application they are used only in restricted forms. Division of Eq. (4-82) by dP and restriction to constant T and composition leads to: VR ∂(GR/RT) ᎏ = ᎏᎏ RT ∂P





(4-83)

T,x

Similarly, division by dT and restriction to constant P and composition gives HR ∂(GR/RT) ᎏ = −T ᎏᎏ RT ∂T





(4-84)

P,x

Also implicit in Eq. (4-82) is the relation ∂(nGR/RT) ln φˆ i = ᎏᎏ (4-85) T,P,nj ∂ni This equation demonstrates that ln φˆ i is a partial property with respect to GR/RT. The partial-property analogs of Eqs. (4-83) and (4-84) are therefore:





4-10

THERMODYNAMICS ∂ ln φ ᎏ ∂P  ∂ ln φˆ ᎏ ∂T  ˆi

ViR  =ᎏ T,x RT

(4-86)

H iR = − ᎏ2 (4-87) P,x RT The partial-property relationship of ln φˆ i to GR/RT also means that the summability relation applies; thus GR (4-88) ᎏ =  xi ln φˆ i RT i i

THE IDEAL SOLUTION The ideal gas is a useful model of the behavior of gases and serves as a standard to which real gas behavior can be compared. This is formalized by the introduction of residual properties. Another useful model is the ideal solution, which serves as a standard to which real solution behavior can be compared. This is formalized by introduction of excess properties. The partial molar Gibbs energy of species i in an ideal gas mixture is given by Eq. (4-71). This equation takes on new meaning when Giig, the Gibbs energy of pure species i in the ideal gas state, is replaced by Gi, the Gibbs energy of pure species i as it actually exists at the mixture T and P and in the same physical state (real gas, liquid, or solid) as the mixture. It then becomes applicable to species in real solutions; indeed, to liquids and solids as well as to gases. The ideal solution is therefore defined as one for which G iid
Gi + RT ln xi

(4-89)

where superscript id denotes an ideal-solution property. This equation is the basis for development of expressions for all other thermodynamic properties of an ideal solution. Equations (4-60) and (4-61), applied to an ideal solution with µi replaced by  Gi, can be written id id ∂G ∂G i i V and Siid = − ᎏ iid = ᎏ  ∂P T,x ∂T P,x Appropriate differentiation of Eq. (4-89) in combination with these relations and Eqs. (4-23) and (4-24) yields









id i

Thus, the fugacity coefficient of species i in an ideal solution is equal to the fugacity coefficient of pure species i in the same physical state as the solution and at the same T and P. Ideal solution behavior is often approximated by solutions comprised of molecules not too different in size and of the same chemical nature. Thus, a mixture of isomers conforms very closely to ideal solution behavior. So do mixtures of adjacent members of a homologous series. FUNDAMENTAL EXCESS-PROPERTY RELATION The residual Gibbs energy and the fugacity coefficient are useful where experimental PVT data can be adequately correlated by equations of state. Indeed, if convenient treatment of all fluids by means of equations of state were possible, the thermodynamic-property relations already presented would suffice. However, liquid solutions are often more easily dealt with through properties that measure their deviations from ideal solution behavior, not from ideal gas behavior. Thus, the mathematical formalism of excess properties is analogous to that of the residual properties. If M represents the molar (or unit-mass) value of any extensive thermodynamic property (e.g., V, U, H, S, G, and so on), then an excess property M E is defined as the difference between the actual property value of a solution and the value it would have as an ideal solution at the same temperature, pressure, and composition. Thus, M E
M − Mid

(4-99)

This definition is analogous to the definition of a residual property as given by Eq. (4-67). However, excess properties have no meaning for pure species, whereas residual properties exist for pure species as well as for mixtures. In addition, analogous to Eq. (4-99) is the partialproperty relation, MiE =  Mi −  Miid (4-100) 

Gid =  xiGi + RT  xi ln xi

(4-93)

Vid =  xiVi

(4-94)

where  M iE is a partial excess property. The fundamental excessproperty relation is derived in exactly the same way as the fundamental residual-property relation and leads to analogous results. Equation (4-80), written for the special case of an ideal solution, is subtracted from Eq. (4-80) itself, yielding: GiE nGE nV E nH E  d ᎏ = ᎏ dP − ᎏ2 dT +  ᎏ dni (4-101) RT RT RT RT i This is the fundamental excess-property relation, analogous to Eq. (4-81), the fundamental residual-property relation. The excess Gibbs energy is of particular interest. Equation (4-77) may be written: G  = Γ (T ) + RT ln fˆ

S =  xiSi − R  xi ln xi

(4-95)

In accord with Eq. (4-97) for an ideal solution, this becomes

Hid =  xiHi

(4-96)

Viid = Vi 

(4-90)

S iid = Si − R ln xi

(4-91)

G + TS Since H i =   , substitutions by Eqs. (4-89) and (4-91) yield id

same T and P. Division of both sides of Eq. (4-97) by xiP and substitution of φˆ iid for fˆiid/xiP (Eq. [4-79]) and of φi for fi /P (Eq. [4-76]) gives an alternative form: φˆ iid = φi (4-98)

id i

Hiid = Hi 

(4-92)

The summability relation, Eq. (4-50), written for the special case of an ideal solution, may be applied to Eqs. (4-89) through (4-92): i

i

i

id

i

i

i

i

i

i

When this equation and Eq. (4-74) are combined with Eq. (4-89), Γi(T) is eliminated, and the resulting expression reduces to fˆiid = xi fi (4-97) This equation, known as the Lewis/Randall rule, applies to each species in an ideal solution at all conditions of T, P, and composition. It shows that the fugacity of each species in an ideal solution is proportional to its mole fraction; the proportionality constant is the fugacity of pure species i in the same physical state as the solution and at the

i

i

G iid = Γi (T ) + RT ln xi fi

i

A simple equation for the fugacity of a species in an ideal solution follows from Eq. (4-89). Written for the special case of species i in an ideal solution, Eq. (4-77) becomes µ id
 G id = Γ (T) + RT ln fˆid i

 

By difference

fˆi G Giid = RT ln ᎏ i −  xi fi The left-hand side is the partial excess Gibbs energy  GiE; the dimensionless ratio fˆi/xifi appearing on the right is called the activity coefficient of species i in solution, and is given the symbol γi. Thus, by definition, fˆi γi
ᎏ (4-102) xi fi and

G iE = RT ln γi

(4-103)

Comparison with Eq. (4-78) shows that Eq. (4-103) relates γi to G iE exactly as Eq. (4-78) relates φˆ i to  GiR. For an ideal solution,G iE =0, and therefore γi = 1.

SOLUTION THERMODYNAMICS VE ∂(GE/RT) ᎏ = ᎏᎏ RT ∂P

An alternative form of Eq. (4-101) follows by introduction of the activity coefficient through Eq. (4-103):

  E

E



E

nH nG nV d ᎏ = ᎏ dP − ᎏ2 dT +  ln γi dni RT RT RT i



(4-117)

T,x

HE ∂(GE/RT) ᎏ = −T ᎏᎏ RT ∂T



(4-104)

4-11



(4-118)

P,x

and from Eq. (4-104) SUMMARY OF FUNDAMENTAL PROPERTY RELATIONS For convenience, the three other fundamental property relations, Eqs. (4-16), (4-80), and (4-82), expressing the Gibbs energy and related properties as functions of T, P, and the ni, are collected here: d(nG) = nV dP − nS dT +  µi dni

(4-16)

 

(4-80)

i

G nG nV nH i d ᎏ = ᎏ dP − ᎏ2 dT +  ᎏ dni RT RT RT RT i

  R

R

R

nG nV nH d ᎏ = ᎏ dP − ᎏ2 dT +  ln φˆ i dni RT RT RT i

(4-82)

These equations and Eq. (4-104) may also be written for the special case of 1 mole of solution by setting n = 1 and ni = xi. The xi are then subject to the constraint that i xi = 1. If written for 1 mole of a constant-composition solution, they become: dG = V dP − S dT

(4-105)

 

(4-106)

 

(4-107)

 

(4-108)

G V H d ᎏ = ᎏ dP − ᎏ2 dT RT RT RT GR VR HR d ᎏ = ᎏ dP − ᎏ2 dT RT RT RT GE VE HE d ᎏ = ᎏ dP − ᎏ2 dT RT RT RT

These equations are, of course, valid as a special case for a pure species; in this event they are written with subscript i affixed to the appropriate symbols. The partial-property analogs of these equations are: dG Vi dP −  Si dT i = dµi =  µi G V H i i i d ᎏ = d ᎏ = ᎏ dP − ᎏ2 dT RT RT RT RT

   

 

G Vi H    d ᎏ = d ln φˆ i = ᎏ dP − ᎏ2 dT RT RT RT R i

R

R i

 

H G ViE iE  iE d ᎏ = d ln γi = ᎏ dP − ᎏ2 dT RT RT RT

(4-110)







∂ ln γi

ᎏ ∂T 

H iE = − ᎏ2 (4-121) P,x RT Finally, an especially useful form of the Gibbs/Duhem equation follows from Eq. (4-116):

 x d ln γ = 0 i

(constant T,P)

i

Since ln γi is a partial property with respect to GE/RT, the following form of the summability equation is valid: GE (4-123) ᎏ =  xi ln γi RT i The analogy between equations derived from the fundamental residual- and excess-property relations is apparent. Whereas the fundamental residual-property relation derives its usefulness from its direct relation to equations of state, the excess-property formulation is useful because VE, HE, and γi are all experimentally accessible. Activity coefficients are found from vapor/liquid equilibrium data, and VE and HE values come from mixing experiments. PROPERTY CHANGES OF MIXING If M represents a molar thermodynamic property of a homogeneous fluid solution, then by definition, ∆M
M −  xi Mi

(4-124)

i

(4-111) (4-112)

where by definition

(4-113)

G V H i ᎏ dP − ᎏ2 dT =  xi d ᎏ RT RT RT i

(4-114)

VR HR ᎏ dP − ᎏ2 dT =  xi d ln φˆ i RT RT i

(4-115)

∆M =  xi ∆ M i

(4-125)

Mi − Mi ∆ M i


(4-126)

i

All three quantities are for the same T, P, and physical state. Eq. (4-126) defines a partial molar property change of mixing, and Eq. (4-125) is the summability relation for these properties. Each of Eqs. (4-93) through (4-96) is an expression for an ideal solution property, and each may be combined with the defining equation for an excess property (Eq. [4-99]), yielding GE = G −  xiGi − RT  xi ln xi

(4-127)

V = V −  xiVi

(4-128)

i

i

E

HE VE ᎏ dP − ᎏ2 dT =  xi d ln γi RT RT i

(4-122)

i

where ∆M is the property change of mixing, and Mi is the molar property of pure species i at the T and P of the solution and in the same physical state (gas or liquid). The summability relation, Eq. (4-50), may be combined with Eq. (4-124) to give

i

 



(4-109)

Finally, a Gibbs/Duhem equation is associated with each fundamental property relation: V dP − S dT =  xi dµi

∂(nGE/RT) ln γi = ᎏᎏ (4-119) T,P,nj ∂ni The last relation demonstrates that ln γi is a partial property with respect to GE/RT. The partial-property analogs of Eqs. (4-117) and (4-118) follow from Eq. (4-112): V ∂ ln γi iE (4-120) ᎏ =ᎏ ∂P T,x RT

(4-116)

This depository of equations stores an enormous amount of information. The equations themselves are so general that their direct application is seldom appropriate. However, by inspection one can write a vast array of relations valid for particular applications. For example, Eqs. (4-83) and (4-84) come directly from Eq. (4-107); Eqs. (4-86) and (4-87), from (4-111). Similarly, from Eq. (4-108),

i

SE = S −  xiSi + R  xi ln xi

(4-129)

HE = H −  xiHi

(4-130)

i

i

i

In view of Eq. (4-124), these may be written G E = ∆G − RT  xi ln xi i

(4-131)

4-12

THERMODYNAMICS V E = ∆V

(4-132)

S = ∆S + R  xi ln xi E

H E = ∆H

i

(4-133) (4-134)

where ∆G, ∆V, ∆S, and ∆H are the Gibbs energy change of mixing, the volume change of mixing, the entropy change of mixing, and the enthalpy change of mixing. For an ideal solution, each excess property is zero, and for this special case ∆Gid = RT  xi ln xi ∆Vid = 0

i

(4-135) (4-136)

∆S = −R  xi ln xi id

(4-137)

i

(4-138) ∆Hid = 0 Property changes of mixing and excess properties are easily calculated one from the other. The most commonly encountered property changes of mixing are the volume change of mixing ∆V and the enthalpy change of mixing ∆H, commonly called the heat of mixing. These properties are directly measurable and are identical to the corresponding excess properties. Pertinent examples are given in Smith, Van Ness, and Abbott (Introduction to Chemical Engineering Thermodynamics, 5th ed., Sec. 11.4, McGraw-Hill, New York, 1996). BEHAVIOR OF BINARY LIQUID SOLUTIONS Property changes of mixing and excess properties find greatest application in the description of liquid mixtures at low reduced tempera-

tures, that is, at temperatures well below the critical temperature of each constituent species. The properties of interest to the chemical engineer are V E (
∆V), H E (
∆H), S E, ∆S, G E, and ∆G. The activity coefficient is also of special importance because of its application in phase-equilibrium calculations. The behavior of binary liquid solutions is clearly displayed by plots of M E, ∆M, and ln γi vs. x1 at constant T and P. The volume change of mixing (or excess volume) is the most easily measured of these quantities and is normally small. However, as illustrated by Fig. 4-1, it is subject to individualistic behavior, being sensitive to the effects of molecular size and shape and to differences in the nature and magnitude of intermolecular forces. The heat of mixing (excess enthalpy) and the excess Gibbs energy are also experimentally accessible, the heat of mixing by direct measurement and GE (or ln γi) indirectly as a product of the reduction of vapor/liquid equilibrium data. Knowledge of HE and GE allows calculation of SE by Eq. (4-13) written for excess properties, HE − GE SE = ᎏ T

(4-139)

with ∆S then given by Eq. (4-133). Figure 4-2 displays plots of ∆H, ∆S, and ∆G as functions of composition for 6 binary solutions at 50°C. The corresponding excess properties are shown in Fig. 4-3; the activity coefficients, derived from Eq. (4-119), appear in Fig. 4-4. The properties shown here are insensitive to pressure, and for practical purposes represent solution properties at 50°C (122°F) and low pressure (P ≈ 1 bar [14.5 psi]).

FIG. 4-1 Excess volumes at 25°C for liquid mixtures of cyclohexane(1) with some other C6 hydrocarbons.

SOLUTION THERMODYNAMICS

(a)

(b)

(c)

(d)

(e)

(f)

Property changes of mixing at 50°C for 6 binary liquid systems: (a) chloroform(1)/n-heptane(2); (b) acetone(1)/ methanol(2); (c) acetone(1)/chloroform(2); (d) ethanol(1)/n-heptane(2); (e) ethanol(1)/chloroform(2); ( f) ethanol(1)/water(2). FIG. 4-2

(a)

(b)

(c)

(d)

(e)

(f)

Excess properties at 50°C for 6 binary liquid systems: (a) chloroform(1)/n-heptane(2); (b) acetone(1)/methanol(2); (c) acetone(1)/chloroform(2); (d) ethanol(1)/n-heptane(2); (e) ethanol(1)/chloroform(2); ( f) ethanol(1)/water(2). FIG. 4-3

4-13

4-14

THERMODYNAMICS

(a)

(b)

(c)

(d)

(e)

(f)

Activity coefficients at 50°C for 6 binary liquid systems: (a) chloroform(1)/n-heptane(2); (b) acetone(1)/ methanol(2); (c) acetone(1)/chloroform(2); (d) ethanol(1)/n-heptane(2); (e) ethanol(1)/chloroform(2); ( f) ethanol(1)/ water(2).

FIG. 4-4

EVALUATION OF PROPERTIES RESIDUAL-PROPERTY FORMULATIONS The most satisfactory calculational procedure for thermodynamic properties of gases and vapors requires PVT data and ideal gas heat capacities. The primary equations are based on the concept of the ideal gas state and the definitions of residual enthalpy and residual entropy: H = Hig + HR

and

S = Sig + SR

The enthalpy and entropy are simple sums of the ideal gas and residual properties, which are evaluated separately. For the ideal gas state at constant composition, dH = C dT ig P

ig

dT dP dSig = CPig ᎏ − R ᎏ T P Integration from an initial ideal gas reference state at conditions T0 and P0 to the ideal gas state at T and P gives: Hig = H0ig +

C

ig P

dT

C

ig P

dT P ᎏ − R ln ᎏ T P0

T

T0

Sig = S0ig +

T

T0

H = H0ig +

C T

T0

ig P

dT + HR

(4-140)

T

T0

ig P

CPig = A + BT + CT 2 + DT −2

(4-142)

where A, B, C, and D are constants characteristic of the particular gas, ig and either C or D is 0. Evaluation of the integrals ∫ CP dT and ig ig ∫ (CP /T)dT is accomplished by substitution for CP , followed by formal integration. For temperature limits of T0 and T the results are conveniently expressed as follows: T B C D τ−1 ig CP dT = AT0(τ − 1) + ᎏ T02(τ2 − 1) + ᎏ T03 (τ3 − 1) + ᎏ ᎏ T0 2 3 T0 τ (4-143) ig T CP D τ+1 2 and ᎏ dT = A ln τ + BT0 + CT0 + ᎏ ᎏ (τ − 1) T0 T τ 2T02 2 (4-144) T where τ
ᎏ T0







Substitution into the equations for H and S yields

C

dT P (4-141) ᎏ − R ln ᎏ + SR T P0 The reference state at T0 and P0 is arbitrarily selected, and the values assigned to H0ig and S0ig are also arbitrary. In practice, only changes in H and S are of interest, and the reference-state values ultimately cancel in their calculation. ig The ideal-gas–state heat capacity CP is a function of T but not of P. ig For a mixture, the heat capacity is simply the molar average i xi CP . ig Empirical equations giving the temperature dependence of CP are available for many pure gases, often taking the form S = S0ig +











EVALUATION OF PROPERTIES Equations (4-140) and (4-141) may sometimes be advantageously expressed in alternative form through use of mean heat capacities: H = H0ig + 〈CPig〉H(T − T0) + HR (4-145) T P ig R ig S = S0 + 〈CP 〉S ln ᎏ − R ln ᎏ + S (4-146) T0 P0 where 〈CPig〉H and 〈CPig〉S are mean heat capacities specific respectively to enthalpy and entropy calculations. They are given by the following equations: B C D 〈CPig〉H = A + ᎏ T0 (τ + 1) + ᎏ T02(τ 2 + τ + 1) + ᎏ2 (4-147) 2 3 τ T0





D 〈CPig〉S = A + BT0 + CT02 + ᎏ τ 2 T02

τ+1 τ−1 ᎏ  ᎏ 2   ln τ 

(4-148)

LIQUID/VAPOR PHASE TRANSITION When a differential amount of a pure liquid in equilibrium with its vapor in a piston-and-cylinder arrangement evaporates at constant temperature T and vapor pressure Pisat, Eq. (4-16) applied to the process reduces to d(niGi) = 0, whence ni dGi + Gi dni = 0

B ln P sat = A − ᎏ (4-152) T+C A principal advantage of this equation is that values of the constants A, B, and C are readily available for a large number of species. The accurate representation of vapor-pressure data over a wide temperature range requires an equation of greater complexity. The Wagner equation, one of the best, expresses the reduced vapor pressure as a function of reduced temperature: Aτ + Bτ 1.5 + Cτ 3 + Dτ 6 ln Prsat = ᎏᎏᎏ (4-153) 1−τ where here τ
1 − Tr and A, B, C, and D are constants. Values of the constants either for this equation or the Antoine equation are given for many species by Reid, Prausnitz, and Poling (The Properties of Gases and Liquids, 4th ed., App. A, McGraw-Hill, New York, 1987). LIQUID-PHASE PROPERTIES Given saturated-liquid enthalpies and entropies, the calculation of these properties for pure compressed liquids is accomplished by integration at constant temperature of Eqs. (4-34) and (4-35):

(4-149)

where Gil and Giv are the molar Gibbs energies of the individual phases. If the temperature of a two-phase system is changed and if the two phases continue to coexist in equilibrium, then the vapor pressure must also change in accord with its temperature dependence. Since Eq. (4-149) holds throughout this change, dGil = dGiv Substituting the expressions for dGil and dGiv given by Eq. (4-16) yields Vil dPisat − Sil dT = Viv dPisat − Siv dT which upon rearrangement becomes dPisat Siv − Sil ∆Silv =ᎏ ᎏ=ᎏ v l dT Vi − Vi ∆Vilv The entropy change ∆Silv and the volume change ∆Vilv are the changes which occur when a unit amount of a pure chemical species is transferred from phase l to phase v at constant temperature and pressure. Integration of Eq. (4-18) for this change yields the latent heat of phase transition: ∆ Hilv = T∆Silv Thus, ∆Si = ∆Hi /T, and substitution in the preceding equation gives lv

lv

dPisat ∆ Hilv (4-150) ᎏ=ᎏ dT T∆Vilv Known as the Clapeyron equation, this is an exact thermodynamic relation, providing a vital connection between the properties of the liquid and vapor phases. Its use presupposes knowledge of a suitable vapor pressure vs. temperature relation. Empirical in nature, such relations are approximated by the equation B ln P sat = A − ᎏ (4-151) T where A and B are constants for a given species. This equation gives a rough approximation of the vapor-pressure relation for its entire temperature range. Moreover, it is an excellent basis for interpolation between values that are reasonably spaced. The Antoine equation, which is more satisfactory for general use, has the form

 −

P

Hi = Hisat +

Since the system is closed, dni = 0 and, therefore, dGi = 0; this requires the molar (or specific) Gibbs energy of the vapor to be identical with that of the liquid: Gil = Giv

4-15

sat

Vi(1 − βiT)dP

(4-154)

βiVi dP

(4-155)

Pi

Si = Sisat

P sat

Pi

where the volume expansivity of species i at temperature T is 1 ∂Vi βi
ᎏ ᎏ (4-156) Vi ∂T P Since βi and Vi are weak functions of pressure for liquids, they are usually assumed constant at the values for the saturated liquid at temperature T.

 

PROPERTIES FROM PVT CORRELATIONS The empirical representation of the PVT surface for pure materials is treated later in this section. We first present general equations for evaluation of reduced properties from such representations. Equation (4-83), applied to a pure material, may be written GR VR d ᎏ = ᎏ dP (constant T) RT RT Integration from zero pressure to arbitrary pressure P gives P GR VR (constant T) ᎏ = ᎏ dP 0 RT RT where at the lower limit GR/RT is set equal to zero on the basis that the zero-pressure state is an ideal gas state. The residual volume is related directly to the compressibility factor: ZRT RT RT V R
V − V ig = ᎏ − ᎏ = (Z − 1) ᎏ P P P

 



VR Z − 1 ᎏ=ᎏ RT P

whence



(4-157)

P GR dP (constant T ) (4-158) ᎏ = (Z − 1) ᎏ 0 RT P Differentiation of Eq. (4-158) with respect to temperature in accord with Eq. (4-84), gives P HR ∂Z dP (constant T ) (4-159) ᎏ = −T ᎏ ᎏ 0 RT ∂T P P Equation (4-13) written for residual properties becomes SR HR GR (4-160) ᎏ=ᎏ−ᎏ R RT RT

Therefore

 

4-16

THERMODYNAMICS

In view of Eq. (4-75), Eqs. (4-158) and (4-160) may be expressed alternatively as P dP ln φ = (Z − 1) ᎏ (constant T ) (4-161) 0 P

Again, in alternative form, HR (HR)0 (HR)1 ᎏ=ᎏ+ωᎏ RTc RTc RTc

SR HR and (4-162) ᎏ = ᎏ − ln φ R RT Values of Z and of (∂Z/∂T)P come from experimental PVT data, and the integrals in Eqs. (4-158), (4-159), and (4-161) may be evaluated by numerical or graphical methods. Alternatively, the integrals are expressed analytically when Z is given by an equation of state. Residual properties are therefore evaluated from PVT data or from an appropriate equation of state. Pitzer’s Corresponding-States Correlation A three-parameter corresponding-states correlation of the type developed by Pitzer, K.S. (Thermodynamics, 3d ed., App. 3, McGraw-Hill, New York, 1995) is described in Sec. 2. It has as its basis an equation for the compressibility factor:

where



Z = Z0 + ωZ1 0

(4-163)

1

where Z and Z are each functions of reduced temperature Tr and reduced pressure Pr. The acentric factor ω is defined by Eq. (2-23). The Tr and Pr dependencies of functions Z0 and Z1 are shown by Figs. 2-1 and 2-2. Generalized correlations are developed here for the residual enthalpy, residual entropy, and the fugacity coefficient. Equations (4-161) and (4-159) are put into generalized form by substitution of the relationships P = Pc Pr

T = Tc Tr

dP = Pc dPr

dT = Tc dTr

The resulting equations are: ln φ =



Pr

0

dP (Z − 1) ᎏr Pr

HR ᎏ = −T r2 RTc

and

∂Z  ᎏ ∂T 

(4-164)

Pr

0

r

Pr

dPr ᎏ Pr

(4-165)

(HR)0 ᎏ = −T r2 RTc

∂Z  ᎏ ∂T  ∂Z  ᎏ ∂T  Pr

0

0 r

Pr

(4-168)

dPr ᎏ Pr

Pr 1 (HR)1 dPr ᎏ = −T r2 ᎏ 0 RTc r Pr Pr The residual entropy is given by Eq. (4-162), here written SR 1 HR (4-169) ᎏ = ᎏ ᎏ − ln φ R Tr RTc Pitzer’s original correlations for Z and the derived quantities were determined graphically and presented in tabular form. Since then, analytical refinements to the tables have been developed, with extended range and accuracy. The most popular Pitzer-type correlation is that of Lee and Kesler (AIChE J., 21, pp. 510–527 [1975]). These tables cover both the liquid and gas phases, and span the ranges 0.3 ≤ Tr ≤ 4.0 and 0.01 ≤ Pr ≤ 10.0. Shown by Figs. 4-5 and 4-6 are isobars of −(HR)0/RTc and −(HR)1/RTc with Tr as independent variable drawn from these tables. Figures 4-7 and 4-8 are the corresponding plots for −ln φ0 and −ln φ1. Figures 4-9 and 4-10 are isotherms of φ0 and φ1 with Pr as independent variable. Although the Pitzer correlations are based on data for pure materials, they may also be used for the calculation of mixture properties. A set of recipes is required relating the parameters Tc, Pc, and ω for a mixture to the pure-species values and to composition. One such set is given by Eqs. (2-80) through (2-82) in Sec. 2, which define pseudoparameters, so called because the defined values of Tc, Pc, and ω have no physical significance for the mixture. Alternative Property Formulations Direct application of Eqs. (4-159) and (4-161) can be made only to equations of state that are solvable for volume, that is, that are volume explicit. Most equations of state are in fact pressure explicit, and alternative equations are required.

 

The terms on the right-hand sides of these equations depend only on the upper limit Pr of the integrals and on the reduced temperature at which they are evaluated. Thus, values of ln φ and HR/RTc may be determined once and for all at any reduced temperature and pressure from generalized compressibility factor data. Substitution for Z in Eq. (4-164) by Eq. (4-163) yields ln φ =



Pr

0

dP (Z0 − 1) ᎏr + ω Pr



Pr

0

dP Z1 ᎏr Pr

This equation may be written in alternative form as ln φ = ln φ0 + ω ln φ1 where

ln φ0




ln φ1




Pr

0 Pr

0

(4-166)

dP (Z0 − 1) ᎏr Pr dP Z1 ᎏr Pr

Since Eq. (4-166) may also be written φ = (φ0)(φ1)ω

(4-167)

correlations may be presented for φ0 and φ1 as well as for their logarithms. Differentiation of Eq. (4-163) yields ∂Z ∂Z = ᎏ ᎏ ∂T  ∂T 0

r

Pr

r

Pr

∂Z1 +ω ᎏ ∂Tr

 

Pr

Substitution for (∂Z/∂Tr)Pr in Eq. (4-165) gives: HR ᎏ = −T r2 RTc

∂Z  ᎏ ∂T  Pr

0

0 r

dPr ᎏ − ωT r2 Pr Pr

∂Z  ᎏ ∂T  Pr

0

1 r

Pr

dPr ᎏ Pr

Correlation of −(HR)0/RTc, drawn from the tables of Lee and Kesler (AIChE J., 21, pp. 510–527 [1975]).

FIG. 4-5

EVALUATION OF PROPERTIES

4-17

Correlation of −(HR)1/RTc, drawn from the tables of Lee and Kesler (AIChE J., 21, pp. 510–527 [1975]).

FIG. 4-6

Equation (4-158) is converted through application of the general relation PV = ZRT. Differentiation at constant T gives P dV + V dP = RT dZ which is readily transformed to dP dZ dV ᎏ=ᎏ−ᎏ P Z V

(constant T )

(constant T )

FIG. 4-8 Correlation of [−ln φ1] vs. Tr, drawn from the tables of Lee and Kesler (AIChE J., 21, pp. 510–527 [1975]).

Substitution into Eq. (4-158) leads to V GR dV (4-170) ᎏ = Z − 1 − ln Z − (Z − 1) ᎏ ∞ RT V The molar volume may be eliminated in favor of the molar density, ρ = V−1, to give ρ GR dρ (4-171) ᎏ = Z − 1 − ln Z + (Z − 1) ᎏ 0 RT ρ For a pure material, Eq. (4-75) shows that GR/RT = ln φ, in which case Eqs. (4-170) and (4-171) directly yield values of ln φ: V dV ln φ = Z − 1 − ln Z − (Z − 1) ᎏ (4-172) ∞ V









ρ

dρ (Z − 1) ᎏ (4-173) ρ where subscript i is omitted for simplicity. The corresponding equations for HR are most readily found from Eq. (4-107) applied to a pure material. In view of Eqs. (4-75) and (4-157), this equation may be written HR dP d ln φ = (Z − 1) ᎏ − ᎏ2 dT RT P Division by dT and restriction to constant V gives, upon rearrangement, HR Z − 1 ∂P ∂ ln φ ᎏ2 = ᎏ ᎏ − ᎏ RT P ∂T V ∂T V Differentiation of P = ZRT/V provides the first derivative on the right and differentiation of Eq. (4-172) provides the second. Substitution then leads to V HR ∂Z dV (4-174) ᎏ=Z−1+T ᎏ ᎏ ∞ RT ∂T V V ln φ = Z − 1 − ln Z +

0

  

FIG. 4-7 Correlation of [−ln φ0] vs. Tr, drawn from the tables of Lee and Kesler (AIChE J., 21, pp. 510–527 [1975]).

 



4-18

THERMODYNAMICS

FIG. 4-10 Correlation of φ1 vs. Pr, drawn from the tables of Lee and Kesler (AIChE J., 21, pp. 510–527 [1975]).

Direct application of these results is possible only to equations of state explicit in volume. For pressure-explicit equations of state, alternative recipes are required. The basis is Eq. (4-82), which in view of Eq. (4-157) may be written nGR n(Z − 1) nHR d ᎏ = ᎏ dP − ᎏ2 dT +  ln φˆ i dni RT P RT i Division by dni and restriction to constant T, nV, and nj ( j ≠ i) leads to ∂(nGR/RT) n(Z − 1) ∂P ln φˆ i = ᎏᎏ −ᎏ ᎏ T,nV,nj ∂ni P ∂ni T,nV,nj But P = (nZ)RT/nV, and therefore ∂P P ∂(nZ) =ᎏ ᎏ ᎏ ∂ni T,nV,nj nZ ∂ni T,nV,nj Combination of the last two equations gives ∂(nGR/RT) Z − 1 ∂(nZ) ln φˆ i = ᎏᎏ − ᎏ ᎏ (4-178) T,nV,nj ∂ni Z ∂ni T,nV,nj Alternatively, ∂(nGR/RT) Z − 1 ∂(nZ) ln φˆ i = ᎏᎏ − ᎏ ᎏ (4-179) T,ρ/n,nj ∂ni Z ∂ni T,ρ/n,nj These equations may either be applied to the results of integrations of Eqs. (4-170) and (4-171) or directly to Eqs. (4-170) and (4-171) as written for a mixture. In the latter case the following analogs of Eq. (4-176) are obtained: V ∂(nZ) dV ln φˆ i = − − 1 ᎏ − ln Z (4-180) ᎏ ∞ ∂ni T,nV,nj V

 

FIG. 4-9 Correlation of φ0 vs. Pr, drawn from the tables of Lee and Kesler (AIChE J., 21, pp. 510–527 [1975]).

  ρ

H ∂Z dρ (4-175) ᎏ=Z−1−T ᎏ ᎏ 0 RT ∂T ρ ρ As before, the residual entropy is found by Eq. (4-162). In applications to equilibrium calculations, the fugacity coefficients of species in a mixture φˆ i are required. Given an expression for GR/RT as determined from Eq. (4-158) for a constant-composition mixture, the corresponding recipe for ln φˆ i is found through the partialproperty relation ∂(nGR/RT) ln φˆ i = ᎏᎏ (4-85) T,P,nj ∂ni There are two ways to proceed: operate on the result of the integration of Eq. (4-158) in accord with Eq. (4-85) or apply Eq. (4-85) directly to Eq. (4-158), obtaining P dP ln φˆ i = (Z (4-176) i − 1) ᎏ 0 P where  Zi is the partial compressibility factor, defined as ∂(nZ) Z (4-177) i
ᎏ ∂ni T,P,nj R

Alternatively,















 

 

























   ∂(nZ) ln φˆ = −   ᎏ  ∂n ρ

i

0

i

T,ρ/n,nj





dρ − 1 ᎏ − ln Z ρ

(4-181)

EVALUATION OF PROPERTIES Virial Equations of State The virial equation in density is an infinite-series representation of the compressibility factor Z in powers of molar density ρ (or reciprocal molar volume V−1) about the real-gas state at zero density (zero pressure): Z = 1 + Bρ + Cρ2 + Dρ3 + ⋅ ⋅ ⋅

(4-182)

The density-series virial coefficients B, C, D, . . . , depend on temperature and composition only. The composition dependencies are given by the exact recipes B =  yiyjBij

(4-183)

C =    yiyjykCijk

(4-184)

i

i

j

j

k

where yi, yj, and yk are mole fractions for a gas mixture, with indices i, j, and k identifying species. The coefficient Bij characterizes a bimolecular interaction between molecules i and j, and therefore Bij = Bji. Two kinds of second virial coefficient arise: Bii and Bjj, wherein the subscripts are the same (i = j); and Bij, wherein they are different (i ≠ j). The first is a virial coefficient for a pure species; the second is a mixture property, called a cross coefficient. Similarly for the third virial coefficients: Ciii, Cjjj, and Ckkk are for the pure species; and Ciij = Ciji = Cjii, and so on, are cross coefficients. Although the virial equation itself is easily rationalized on empirical grounds, the “mixing rules” of Eqs. (4-183) and (4-184) follow rigorously from the methods of statistical mechanics. The temperature derivatives of B and C are given exactly by dB dBij (4-185) ᎏ =   yiyj ᎏ dT dT i j dC dCijk (4-186) ᎏ =    yiyjyk ᎏ dT dT i j k An alternative form of the virial equation expresses Z as an expansion in powers of pressure about the real-gas state at zero pressure (zero density): Z = 1 + B′P + C′P 2 + D′P 3 + ⋅ ⋅ ⋅ (4-187) Equation (4-187) is the virial equation in pressure, and B′, C′, D′, . . . , are the pressure-series virial coefficients. Like the density-series coefficients, they depend on temperature and composition only. Moreover, the two sets of coefficients are related: B B′ = ᎏ (4-188) RT

  





















P (4-196) ln φˆ i = 2  ykBki − B ᎏ RT k Equation (4-191) is explicit in pressure, and Eqs. (4-173), (4-175), and (4-181) are therefore applicable. Direct substitution of Eq. (4-191) into Eq. (4-173) yields 3 ln φ = 2Bρ + ᎏ Cρ2 − ln Z (4-197) 2

Whence

(4-189)

∂Z

dC =ᎏρ+ᎏρ ᎏ ∂T  dT dT

Moreover,

dB

2

ρ

and so on Application of an infinite series to practical calculations is, of course, impossible, and truncations of the virial equations are in fact employed. The degree of truncation is conditioned not only by the temperature and pressure but also by the availability of correlations or data for the virial coefficients. Values can usually be found for B (see Sec. 2), and often for C (see, e.g., De Santis and Grande, AIChE J., 25, pp. 931–938 [1979]), but rarely for higher-order coefficients. Application of the virial equations is therefore usually restricted to two- or three-term truncations. For pressures up to several bars, the two-term expansion in pressure, with B′ given by Eq. (4-188), is usually preferred: BP Z=1+ᎏ (4-190) RT For supercritical temperatures, it is satisfactory to ever-higher pressures as the temperature increases. For pressures above the range where Eq. (4-190) is useful, but below the critical pressure, the virial expansion in density truncated to three terms is usually suitable:

(4-191)

Equations for derived properties may be developed from each of these expressions. Consider first Eq. (4-190), which is explicit in volume. Equations (4-159), (4-161), and (4-176) are therefore applicable. Direct substitution for Z in Eq. (4-161) gives BP ln φ = ᎏ (4-192) RT Differentiation of Eq. (4-190) yields ∂Z dB B P ᎏ = ᎏ−ᎏ ᎏ ∂T P dT T RT Whence, by Eq. (4-159) HR P B dB (4-193) ᎏ=ᎏ ᎏ−ᎏ RT R T dT and by Eq. (4-162), SR P dB (4-194) ᎏ=−ᎏᎏ R R dT Multiplication of Eq. (4-190) by n gives P nZ = n + (nB) ᎏ RT Differentiation in accord with Eq. (4-177) yields ∂(nB) P Z i = 1 + ᎏ ᎏ ∂ni T,nj RT Whence, by Eq. (4-176), ∂(nB) P ln φˆ i = ᎏ ᎏ ∂ni T,nj RT Equation (4-183) can be written 1 nB = ᎏ   nknlBkl n k l from which, by differentiation, ∂(nB) = 2  ykBki − B (4-195) ᎏ ∂ni T,nj k



and so on

C − B2 C′ = ᎏ (RT)2

Z = 1 + Bρ + Cρ2

4-19



 



HR dB T dC (4-198) ᎏ = B − T ᎏ ρ + C − ᎏ ᎏ ρ2 RT dT 2 dT The residual entropy is given by Eq. (4-162). Application of Eq. (4-181) provides an expression for ln φˆ i. First, from Eq. (4-191), whence

∂(nZ) ᎏ ∂n 

∂(nB) ∂(nC) =1+ B+ ᎏ ρ + 2C + ᎏ ∂ni T,nj ∂ni Substitution into Eq. (4-181) gives, on integration, i



T,ρ/n,nj





 ρ

2

T,nj

∂(nB) 1 ∂(nC) ln φˆ i = B + ᎏ ρ + ᎏ 2C + ᎏ ρ2 − ln Z ∂ni T,nj 2 ∂ni T,nj The mole-number derivative of nB is given by Eq. (4-195); the corresponding derivative of nC, similarly found from Eq. (4-184), is ∂(nC) = 3   ykylCkli − 2C (4-199) ᎏ ∂ni T,nj k l















4-20

THERMODYNAMICS

Finally, 3 ln φˆ i = 2ρ  ykBki + ᎏ ρ2   ykylCkli − ln Z (4-200) 2 k k l In a process calculation, T and P, rather than T and ρ (or T and V), are usually the favored independent variables. Application of Eqs. (4-197), (4-198), and (4-200) therefore requires prior solution of Eq. (4-191) for Z or ρ. Since Z = P/ρRT, Eq. (4-191) may be written in two equivalent forms: BP CP 2 Z3 − Z2 − ᎏ Z − ᎏ2 = 0 (4-201) RT (RT)

 

 

 

B 1 P (4-202) ρ3 + ᎏ ρ2 + ᎏ ρ − ᎏ = 0 C C CRT In the event that three real roots obtain for these equations, only the largest Z (smallest ρ) appropriate for the vapor phase has physical significance, because the virial equations are suitable only for vapors and gases. Generalized Correlation for the Second Virial Coefficient Perhaps the most useful of all Pitzer-type correlations is the one for the second virial coeffieient. The basic equation (see Eq. [2-68]) is BPc (4-203) ᎏ = B0 + ωB1 RTc where for a pure material B0 and B1 are functions of reduced temperture only. Substitution for B by this expression in Eq. (4-190) yields P Z = 1 + (B0 + ωB1) ᎏr (4-204) Tr By differentiation, ∂Z dB0/dTr B0 dB1/dT B1 + ωPr ᎏr − ᎏ ᎏ = Pr ᎏ − ᎏ 2 ∂Tr Pr Tr Tr Tr Tr2 Substitution of these equations into Eqs. (4-164) and (4-165) and integration gives P ln φ = (B0 + ωB1) ᎏr (4-205) Tr or

 















HR dB0 dB1 (4-206) ᎏ = Pr B0 − Tr ᎏ + ω B1 − Tr ᎏ RTc dTr dTr The residual entropy follows from Eq. (4-162): SR dB0 dB1 (4-207) ᎏ = −Pr ᎏ + ω ᎏ R dTr dTr In these equations, B0 and B1 and their derivatives are well represented by 0.422 B0 = 0.083 − ᎏ (4-208) Tr1.6 and





0.172 B1 = 0.139 − ᎏ Tr4.2 dB0 0.675 ᎏ=ᎏ dTr Tr2.6

(4-209) (4-210)

dB1 0.722 (4-211) ᎏ=ᎏ dTr Tr5.2 Though limited to pressures where the two-term virial equation in pressure has approximate validity, this correlation is applicable to most chemical-processing conditions. As with all generalized correlations, it is least accurate for polar and associating molecules. Although developed for pure materials, this correlation can be extended to gas or vapor mixtures. Basic to this extension is the mixing rule for second virial coefficients and its temperature derivative: B =   yiyjBij

(4-183)

dB dBij ᎏ =   yiyj ᎏ dT dT i j

(4-185)

i

j

Values for the cross coefficients and their derivatives in these equations are provided by writing Eq. (4-203) in extended form: RTcij 0 Bij = ᎏ (B + ωij B1) (4-212) Pcij where B0, B1, dB0/dTr, and dB1/dTr are the same functions of Tr as given by Eqs. (4-208) through (4-211). Differentiation produces dBij RTcij dB0 dB1 ᎏ = ᎏ ᎏ + ωij ᎏ dT Pcij dT dT







0

1



dBij R dB dB (4-213) ᎏ = ᎏ ᎏ + ωij ᎏ dT Pcij dTrij dTrij where Trij = T/Tcij. The following are combining rules for calculation of ωij, Tcij, and Pcij as given by Prausnitz, Lichtenthaler, and de Azevedo (Molecular Thermodynamics of Fluid-Phase Equilibria, 2d ed., pp. 132 and 162, Prentice-Hall, Englewood Cliffs, N.J., 1986): ωi + ωj ωij = ᎏ (4-214) 2 Tcij = (Tci Tcj)1/2(1 − kij) (4-215) Zcij RTcij Pcij = ᎏ (4-216) Vcij Zci + Zcj (4-217) with Zcij = ᎏ 2 1/3 1/3 3 Vci + Vc j (4-218) Vcij = ᎏᎏ 2 In Eq. (4-215), kij is an empirical interaction parameter specific to an i-j molecular pair. When i = j and for chemically similar species, kij = 0. Otherwise, it is a small (usually) positive number evaluated from minimal PVT data or in the absence of data set equal to zero. When i = j, all equations reduce to the appropriate values for a pure species. When i ≠ j, these equations define a set of interaction parameters having no physical significance. For a mixture, values of Bij and dBij /dT from Eqs. (4-212) and (4-213) are substituted into Eqs. (4183) and (4-185) to provide values of the mixture second virial coefficient B and its temperature derivative. Values of HR and SR for the mixture are then given by Eqs. (4-193) and (4-194), and values of ln φˆ i for the component fugacity coefficients are given by Eq. (4-196). Cubic Equations of State The simplest expressions that can (in principle) represent both the vapor- and liquid-phase volumetric behavior of pure fluids are equations cubic in molar volume. All such expressions are encompassed by the generic equation or





RT a(V − η) P = ᎏ − ᎏᎏᎏ (4-219) V − b (V − b)(V 2 + δV + ε) where parameters b, θ, δ, ε, and η can each depend on temperature and composition. Special cases are obtained by specification of values or expressions for the various parameters. The modern development of cubic equations of state started in 1949 with publication of the Redlich/Kwong equation (Redlich and Kwong, Chem. Rev., 44, pp. 233–244 [1949]): RT a(T) P=ᎏ−ᎏ (4-220) V − b V(V + b) a a(T) = ᎏ T1/2 and a and b are functions of composition only. This equation, like other cubic equations of state, has three volume roots, of which two may be complex. Physically meaningful values of V are always real, positive, and greater than the constant b. When T > Tc, solution for V at any positive value of P yields only one real positive root. When T = Tc, this is also true, except at the critical pressure, where there are three roots, all equal to Vc. For T < Tc, only one real positive root exists at high pressures, but for a range of lower pressures there are three real positive roots. Here, the middle root is of no significance; the smallest root is a liquid or liquidlike volume, and the largest root is a vapor or vaporlike volume. The volumes of saturated liquid and satuwhere

EVALUATION OF PROPERTIES rated vapor are given by the smallest and largest roots when P is the saturation vapor pressure Psat. The application of cubic equations of state to mixtures requires expression of the equation-of-state parameters as functions of composition. No exact theory like that for the virial coefficients prescribes this composition dependence, and empirical mixing rules provide approximate relationships. The mixing rules that have found general favor for the Redlich/Kwong equation are: with aij = aji, and

a =   yi yj aij

(4-221)

b =  yi bi

(4-222)

i

j

i

The aij are of two types: pure-species parameters (like subscripts) and interaction parameters (unlike subscripts). The bi are parameters for the pure species. Parameter evaluation may be accomplished with the equations 0.42748R2 Tcij2.5 aij = ᎏᎏ (4-223) Pcij 0.08664RTci (4-224) bi = ᎏᎏ Pci where Eqs. (4-215) through (4-218) provide for the calculation of the Tcij and Pcij. Multiplication of the Redlich/Kwong equation (Eq. [4-220]) by V/RT leads to its expression in alternative form: a 1 h Z=ᎏ−ᎏ (4-225) ᎏ 1 − h bRT1.5 1 + h









a h h Z−1=ᎏ−ᎏ ᎏ 1 − h bRT1.5 1 + h

Whence

(4-226)

bP h=ᎏ (4-227) ZRT Equations (4-170) and (4-174) in combination with Eq. (4-226) give GR a ln (1 + h) (4-228) ᎏ = Z − 1 − ln (1 − h)Z − ᎏ RT bRT1.5 where



R







H 3a ln (1 + h) (4-229) ᎏ=Z−1− ᎏ RT 2 bRT 1.5 Once a and b are determined by Eqs. (4-221) through (4-224), then for given T and P values of Z, GR/RT, and HR/RT are found by Eqs. (4-225), (4-228), and (4-229) and SR/R by Eq. (4-160). The procedure requires initial solution of Eqs. (4-225) and (4-227) for Z and h. The original Redlich/Kwong equation is rarely satisfactory for vapor/liquid equilibrium calculations, and equations have been developed specific to this purpose. The two most popular are the Soave/ Redlich/Kwong (SRK) equation, a modification of the Redlich/Kwong equation (Soave, Chem. Eng. Sci., 27, pp. 1197–1203 [1972]), and the Peng/Robinson (PR) equation (Peng and Robinson, Ind. Eng. Chem. Fundam., 15, pp. 59–64 [1976]). Both equations are designed specifically to yield reasonable vapor pressures for pure fluids. Thus, there is no assurance that molar volumes calculated by these equations are more accurate than values given by the original Redlich/Kwong equation. Written for pure species i the SRK and PR equations are special cases of the following: RT ai(T ) P = ᎏ − ᎏᎏ (4-230) Vi − bi (Vi + εbi)(Vi + σbi) and

where

Ωaα(Tri; ωi)R2 Tci2 ai(T) = ᎏᎏ Pci

Ωb RTci bi = ᎏ Pci and ε σ, Ωa, and Ωb are equation-specific constants. For the Soave/ Redlich/Kwong equation: α(Tri; ωi) = [1 + (0.480 + 1.574ωi − 0.176ω2i ) (1 − Tri1/2)]2

4-21

For the Peng/Robinson equation: α(Tri; ωi) = [1 + (0.37464 + 1.54226ωi − 0.26992ω i2) (1 − Tri1/2)]2 Written for a mixture, Eq. (4-230) becomes RT a(T) P = ᎏ − ᎏᎏ (4-231) V − b (V + εb)(V + σb) where a and b are mixture values, related to the ai and bi by mixing rules. Equation (4-170) applied to Eq. (4-231) leads to ai  bi bi (V − b)Z a/bRT V + σb  ln φˆ i = ᎏ (Z − 1) − ln ᎏ + ᎏ 1 + ᎏ − ᎏ ln ᎏ b a b V ε−σ V + εb (4-232)





where ai and bi are partial parameters for species i, defined by ∂(na) (4-233) ai = ᎏ ∂ni T,nj

  ∂(nb) b = ᎏ  ∂n

and

(4-234)

i

i

T,nj

These are general equations that do not depend on the particular mixing rules adopted for the composition dependence of a and b. The mixing rules given by Eqs. (4-221) and (4-222) can certainly be employed with these equations. However, for purposes of vapor/liquid equilibrium calculations, a special pair of mixing rules is far more appropriate, and will be introduced when these calculations are treated. Solution of Eq. (4-232) for fugacity coefficient φˆ i at given T and P requires prior solution of Eq. (4-231) for V, from which is found Z = PV/RT. Benedict/Webb/Rubin Equation of State The BWR equation of state with Z as the dependent variable is written A0 C0 a Z = 1 + B0 − ᎏ −ᎏ ρ + b − ᎏ ρ2 RT RT 3 RT



 



c aα + ᎏ ρ5 + ᎏ3 ρ2 (1 + γρ2) exp (−γρ2) (4-235) RT RT All eight parameters depend on composition; moreover, parameters C0, b, and γ are for some applications treated as functions of T. By Eq. (4-171), the residual Gibbs energy is GR C0 3 a A0 6aα ᎏ = 2 B0 − ᎏ − ᎏ3 ρ + ᎏ b − ᎏ ρ2 + ᎏ ρ5 RT RT RT 2 RT 5RT













c + ᎏ3 (2γ 2ρ4 + γρ2 − 2) exp (−γ ρ2) + 2 − ln Z (4-236) 2γRT With allowance for T dependence of C0, b, and γ, Eq. (4-175) yields HR 1 dC0 2A0 4C0 ᎏ = B0 − ᎏ − ᎏ3 + ᎏ2 ᎏ ρ RT RT RT RT dT









1 db 3a 6aα − ᎏ T ᎏ − 2b + ᎏ ρ2 + ᎏ ρ5 2 dT RT 5RT





c + ᎏ3 (2γ 2ρ4 − γρ2 − 6) exp (−γρ2) + 6 2γRT





c dγ −ᎏ (4-237) ᎏ (γ 2ρ4 + 2γρ2 + 2) exp (−γρ2) − 2 2γ 2RT 2 dT The residual entropy is given by Eq. (4-160). Computation of ln φˆ i is done via Eq. (4-181). The result is A0 + A 0 C0 + C  0i ln φˆ i = B0 +  B0i − ᎏi − ᎏ ρ RT RT 3





4aα + aα 2a + ai 2 1 i + αai ρ5 bi − ᎏ ρ + ᎏᎏ + ᎏ 2b +  5RT RT 2



c + ᎏ3 2γRT



2 γi



1 + ᎏγ  γ ρ + ᎏγ − ᎏc  ρ

ci γi −2 1+ᎏ −ᎏ c γ



γi



2 4

ci

2

γi

 exp (−γρ ) + 2 1 + ᎏc − ᎏγ  − ln Z 2

ci

(4-238)

4-22

THERMODYNAMICS

Here the quantities with overbars are partial parameters for species i, defined for arbitrary parameter π by ∂(nπ) πi
ᎏ (4-239)  ∂ni T,nj Application of these equations requires specific mixing rules. For example, if



π=



 y π 

r

1/r k k

(4-240)

The symmetrical nature of these relations is evident. The infinitedilution values of the activity coefficients are ln γ 1∞ = ln γ 2∞ = B. If D = ⋅ ⋅ ⋅ = 0, then GE ᎏ = B + C(x1 − x2) = B + C(2x1 − 1) x1 x2 RT and in this case GE/x1 x2 RT is linear in x1. The substitutions, B + C = A21 and B − C = A12 transform this expression into the Margules equation: GE/x1 x2 RT = A21 x1 + A12 x2

k

where r is a small integer, the recipe for  πi is πi 1/r π (4-241) i = π r ᎏ − (r − 1) π Specifically, if r = 3 for π
c; then ci 1/3 −2 ci = c 3 ᎏ c where ci is the parameter for pure i and c is the parameter for the mixture, given by

 



 

c=



 y c 

3

1/3 k k

k

Equation-of-state examples are given in Smith, Van Ness, and Abbott (Introduction to Chemical Engineering Thermodynamics, 5th ed., Secs. 3.4–3.7 and 6.2–6.6, McGraw-Hill, New York, 1996). EXPRESSIONS FOR THE EXCESS GIBBS ENERGY In principle, equation-of-state procedures can be used for the calculation of liquid-phase as well as gas-phase properties, and much has been accomplished in the development of PVT equations of state suitable for both phases. However, a widely used alternative for the liquid phase is application of excess properties. The excess property of primary importance for engineering calculations is the excess Gibbs energy GE, because its canonical variables are T, P, and composition, the variables usually specified or sought in a design calculation. Knowing GE as a function of T, P, and composition, one can in principle compute from it all other excess properties (see, for example, Eqs. [4-117] through [4-119]). As noted with respect to Fig. 4-1, the excess volume for liquid mixtures is usually small; the pressure dependence of GE may then be safely ignored. Thus, the engineering efforts at describing GE center on representing its composition and temperature dependence. For binary systems at constant T, GE is a function of just x1, and the quantity most conveniently represented by an equation is GE/x1x2RT. The simplest procedure is to express this quantity as a power series in x1: GE ᎏ = a + bx1 + cx 12 + ⋅ ⋅ ⋅ (constant T) x1 x2 RT An equivalent power series with certain advantages is known as the Redlich/Kister expansion (Redlich, Kister, and Turnquist, Chem. Eng. Progr. Symp. Ser. No. 2, 48, pp. 49–61 [1952]): GE ᎏ = B + C(x1 − x2) + D(x1 − x2)2 + ⋅ ⋅ ⋅ x1x2 RT In application, different truncations of this series are appropriate. For each particular expression representing GE/x1x2 RT, specific expressions for ln γ1 and ln γ2 result from application of Eq. (4-119). When all parameters are zero, GE/RT = 0, and the solution is ideal. If C = D = ⋅ ⋅ ⋅ = 0, then GE ᎏ=B x1 x2 RT where B is a constant for a given temperature. The corresponding equations for ln γ1 and ln γ2 are ln γ1 = Bx22

(4-242)

ln γ2 = Bx 21

(4-243)

(4-244)

Application of Eq. (4-119) yields ln γ1 = x 22 [A12 + 2(A21 − A12)x1]

(4-245)

ln γ2 = x 21 [A21 + 2(A12 − A21)x2]

(4-246)

An alternative equation is obtained when the reciprocal quantity x1x2RT/GE is expressed as a linear function of x1: x1x2 ᎏ = B′ + C′ (x1 − x2) = B′ + C′ (2x1 − 1) GE/RT This may also be written: x1x2 = B′ (x1 + x2) + C′ (x1 − x2) = (B′ + C′)x1 + (B′ − C′)x2 ᎏ GE/RT The substitutions B′ + C′ = 1/A′21 and B′ − C′ = 1/A′12 produce x1x2 x1 x2 A′12 x1 + A′21 x2 = ᎏ + ᎏ = ᎏᎏ ᎏ GE/RT A′21 A′12 A′12 A′21 or

GE A′12 A′21 ᎏ = ᎏᎏ x1x2 RT A′12 x1 + A′21 x2

(4-247)

The activity coefficients implied by this equation are given by A′12 x1 −2 (4-248) ln γ1 = A′12 1 + ᎏ A′21 x2









A′21 x2 ln γ2 = A′21 1 + ᎏ A′12 x1

−2

(4-249)

These are known as the van Laar equations. When x1 = 0, ln γ 1∞ = A′12; when x2 = 0, ln γ 2∞ = A′21. The Redlich/Kister expansion, the Margules equations, and the van Laar equations are all special cases of a very general treatment based on rational functions, that is, on equations for GE given by ratios of polynomials (Van Ness and Abbott, Classical Thermodynamics of Nonelectrolyte Solutions: With Applications to Phase Equilibria, Sec. 5-7, McGraw-Hill, New York, 1982). Although providing great flexibility in the fitting of VLE data for binary systems, they are without theoretical foundation, with no rational basis for their extension to multicomponent systems. Nor do they incorporate an explicit temperature dependence for the parameters. Modern theoretical developments in the molecular thermodynamics of liquid-solution behavior are often based on the concept of local compositon, presumed to account for the short-range order and nonrandom molecular orientations that result from differences in molecular size and intermolecular forces. Introduced with the publication of a model of GE behavior known as the Wilson equation ( J. Am. Chem. Soc., 86, pp. 127–130 [1964]), it prompted the development of alternative local-composition models, most notably the NRTL (Non-Random-Two-Liquid) equation of Renon and Prausnitz (AIChE J., 14, pp. 135–144 [1968]) and the UNIQUAC (UNIversal QUAsi-Chemical) equation of Abrams and Prausnitz (AIChE J., 21, pp. 116–128 [1975]). A further significant development, based on the UNIQUAC equation, is the UNIFAC method (UNIQUAC Functional-group Activity Coefficients). Proposed by Fredenslund, Jones, and Prausnitz (AIChE J., 21, pp. 1086–1099 [1975]) and given detailed treatment by Fredenslund, Gmehling, and Rasmussen (Vapor-Liquid Equilibrium Using UNIFAC, Elsevier, Amsterdam, 1977), it provides for the calculation of activity coefficients from contributions of the various groups making up the molecules of a solution.

EVALUATION OF PROPERTIES The Wilson equation, like the Margules and van Laar equations, contains just two parameters for a binary system (Λ12 and Λ21), and is written: GE (4-250) ᎏ = −x1 ln (x1 + x2Λ12) − x2 ln (x2 + x1Λ21) RT Λ12 Λ21 ln γ1 = −ln (x1 + x2Λ12) + x2 ᎏᎏ − ᎏᎏ x1 + x2Λ12 x2 + x1Λ21



Λ21 Λ12 ln γ2 = −ln (x2 + x1Λ21) − x1 ᎏᎏ − ᎏᎏ x1 + x2Λ12 x2 + x1Λ21



whence





(4-252)

Both Λ12 and Λ21 must be positive numbers. The NRTL equation contains three parameters for a binary system and is written: GE G21τ21 G12τ12 (4-253) ᎏ = ᎏᎏ + ᎏᎏ x1x2 RT x1 + x2 G21 x2 + x1 G12 G21 ln γ1 = x 22 τ21 ᎏᎏ x1 + x2 G21

 

G τ  + ᎏᎏ (x + x G ) 

(4-254)

 

G τ  + ᎏᎏ (x + x G ) 

(4-255)

G12 ln γ2 = x 21 τ12 ᎏᎏ x2 + x1 G12 Here

G12 = exp (−ατ12)

12 12

2

1

12

2

21 21

1

2

21

2

b12 b21 τ21 = ᎏ τ12 = ᎏ RT RT where α, b12, and b21, parameters specific to a particular pair of species, are independent of composition and temperature. The infinite-dilution values of the activity coefficients are given by the equations:

and

ln γ 1∞ = τ21 + τ12 exp (−ατ12) ln γ 2∞ = τ12 + τ 21 exp (−ατ 21)

j

Function g contains pure-species parameters only, whereas function g R incorporates two binary parameters for each pair of molecules. For a multicomponent system, Φ θ gC =  xi ln ᎏi + 5  qi xi ln ᎏi (4-260) xi Φi i i gR = − qi xi ln i

where

 θ τ 

(4-261)

j ji

j

xi ri Φi
ᎏ  xj rj

(4-262)

xi qi θi
ᎏ  xj qj

(4-263)

j

and

Subscript i identifies species, and j is a dummy index; all summations are over all species. Note that τji ≠ τij; however, when i = j, then τii = τjj = 1. In these equations ri (a relative molecular volume) and qi (a relative molecular surface area) are pure-species parameters. The influence of temperature on g enters through the interaction parameters τji of Eq. (4-261), which are temperature dependent: −(uji − uii) τji = exp ᎏᎏ (4-264) RT Parameters for the UNIQUAC equation are therefore values of (uji − uii). An expression for ln γi is found by application of Eq. (4-119) to the UNIQUAC equation for g (Eqs. [4-259] through [4-261]). The result is given by the following equations: ln γ i = ln γ iC + ln γ iR

The local-composition models have limited flexibility in the fitting of data, but they are adequate for most engineering purposes. Moreover, they are implicitly generalizable to multicomponent systems without the introduction of any parameters beyond those required to describe the constituent binary systems. For example, the Wilson equation for multicomponent systems is written: GE (4-256) ᎏ = − xi ln  xj Λij RT i j and

(4-259)

j

G21 = exp (−ατ21)

xkΛki ln γi = 1 − ln  xj Λij −  ᎏ  xj Λkj j k

g = gC + g R C

ln γ 2∞ = −ln Λ21 + 1 − Λ12

2

binary (in contrast to multicomponent) systems. This makes parameter determination for the local-composition models a task of manageable proportions. The UNIQUAC equation treats g
GE/RT as comprised of two additive parts, a combinatorial term g C, accounting for molecular size and shape differences, and a residual term g R (not a residual property), accounting for molecular interactions:

(4-251)

ln γ 1∞ = −ln Λ12 + 1 − Λ21

2

4-23

(4-257)

where Λij = 1 for i = j, and so on. All indices in these equations refer to the same species, and all summations are over all species. For each ij pair there are two parameters, because Λij ≠ Λji. For example, in a ternary system the three possible ij pairs are associated with the parameters Λ12, Λ21; Λ13, Λ31; and Λ23, Λ32. The temperature dependence of the parameters is given by: −a V Λij = ᎏj exp ᎏij (i ≠ j) (4-258) Vi RT where Vj and Vi are the molar volumes at temperature T of pure liquids j and i, and aij is a constant independent of composition and temperature. Thus the Wilson equation, like all other local-composition models, has built into it an approximate temperature dependence for the parameters. Moreover, all parameters are found from data for

(4-265)



J J ln γ i = 1 − Ji + ln Ji − 5qi 1 − ᎏi + ln ᎏi Li Li C

τij ln γ iR = qi 1 − ln si −  θj ᎏ sj j





where in addition to Eqs. (4-263) and (4-264) ri Ji = ᎏ  rj xj



(4-266) (4-267)

(4-268)

j

qi Li = ᎏ  qj xj

(4-269)

si =  θl τ li

(4-270)

j

l

Again subscript i identifies species, and j and l are dummy indicies. Values for the parameters ri, qi, and (uij − ujj) are given by Gmehling, Onken, and Arlt (Vapor-Liquid Equilibrium Data Collection, Chemistry Data Series, vol. I, parts 1–8, DECHEMA, Frankfurt/Main, 1974–1990). The Wilson parameters Λij, NRTL parameters Gij, and UNIQUAC parameters τij all inherit a Boltzmann-type T dependence from the origins of the expressions for GE, but it is only approximate. Computations of properties sensitive to this dependence (e.g., heats of mixing and liquid/liquid solubility) are in general only qualitatively correct.

4-24

THERMODYNAMICS

EQUILIBRIUM CRITERIA The equations developed in preceding sections are for PVT systems in states of internal equilibrium. The criteria for internal thermal and mechanical equilibrium are well known, and need not be discussed in detail. They simply require uniformity of temperature and pressure throughout the system. The criteria for phase and chemical-reaction equilibria are less obvious. Consider a closed PVT system, either homogeneous or heterogeneous, of uniform T and P, which is in thermal and mechanical equilibrium with its surroundings, but which is not initially at internal equilibrium with respect to mass transfer or with respect to chemical reaction. Changes occurring in the system are then irreversible, and must necessarily bring the system closer to an equilibrium state. The first and second laws written for the entire system are dU t = dQ + dW dQ dSt ≥ ᎏ T

dW = −P dV t dU t + P dV t − T dSt ≤ 0

The inequality applies to all incremental changes toward the equilibrium state, whereas the equality holds at the equilibrium state where any change is reversible. Various constraints may be put on this expression to produce alternative criteria for the directions of irreversible processes and for the condition of equilibrium. For example, it follows immediately that ≤0

Alternatively, other pairs of properties may be held constant. The most useful result comes from fixing T and P, in which case d(Ut + PVt − TSt)T,P ≤ 0 t ≤0 dGT,P

or

i

dn″i = −dn′i

 (µ′ − µ″) dn′ = 0

Therefore

i

i

i

i

Since the dn′i are independent and arbitrary, it follows that µ′i = µ″i This is the criterion of two-phase equilibrium. It is readily generalized to multiple phases by successive application to pairs of phases. The general result is

i

Since mechanical equilibrium is assumed,

dU

i

Since the system is closed and without chemical reaction, material balances require that

(4-272)

Substitution for each µi by Eq. (4-77) produces the equivalent result (4-273) fˆ ′ = fˆ ″ = fˆ ′′′ = ⋅ ⋅ ⋅

dU t − dW − T dSt ≤ 0

t St,V t

t dG T,P
d(nG)T,P =  µ′i dn′i +  µ″i dn″i = 0

µ′i = µ″i = µ′′′ i = ⋅ ⋅ ⋅

Combination gives

Whence

If the two-phase system is at equilibrium, then application of Eq. (4-271) yields

This expression shows that all irreversible processes occurring at constant T and P proceed in a direction such that the total Gibbs energy of the system decreases. Thus the equilibrium state of a closed system is the state with the minimum total Gibbs energy attainable at the given T and P. At the equilibrium state, differential variations may occur in the system at constant T and P without producing a change in Gt. This is the meaning of the equilibrium criterion t =0 dGT,P

(4-271)

This equation may be applied to a closed, nonreactive, two-phase system. Each phase taken separately is an open system, capable of exchanging mass with the other, and Eq. (4-16) may be written for each phase: d(nG)′ = −(nS)′ dT + (nV)′ dP +  µ′i dn′i i

d(nG)″ = −(nS)″ dT + (nV)″ dP +  µ″i dn″i i

where the primes and double primes denote the two phases and the presumption is that T and P are uniform throughout the two phases. The change in the Gibbs energy of the two-phase system is the sum of these equations. When each total-system property is expressed by an equation of the form nM = (nM)′ + (nM)″

i

i

These are the criteria of phase equilibrium applied in the solution of practical problems. For the case of equilibrium with respect to chemical reaction within a single-phase closed system, combination of Eqs. (4-16) and (4-271) leads immediately to

 µ dn = 0 i

i

(4-274)

i

For a system in which both phase and chemical-reaction equilibrium prevail, the criteria of Eqs. (4-272) and (4-274) are superimposed. THE PHASE RULE The intensive state of a PVT system is established when its temperature and pressure and the compositions of all phases are fixed. However, for equilibrium states these variables are not all independent, and fixing a limited number of them automatically establishes the others. This number of independent variables is given by the phase rule, and is called the number of degrees of freedom of the system. It is the number of variables which may be arbitrarily specified and which must be so specified in order to fix the intensive state of a system at equilibrium. This number is the difference between the number of variables needed to characterize the system and the number of equations that may be written connecting these variables. For a system containing N chemical species distributed at equilibrium among π phases, the phase-rule variables are temperature and pressure, presumed uniform throughout the system, and N − 1 mole fractions in each phase. The number of these variables is 2 + (N − 1)π. The masses of the phases are not phase-rule variables, because they have nothing to do with the intensive state of the system. The equations that may be written connecting the phase-rule variables are: 1. Equation (4-272) for each species, giving (π − 1)N phaseequilibrium equations. 2. Equation (4-274) for each independent chemical reaction, giving r equations. The total number of independent equations is therefore (π − 1)N + r. In their fundamental forms these equations relate chemical potentials, which are functions of temperature, pressure, and composition, the phase-rule variables. Since the degrees of freedom of the system F is the difference between the number of variables and the number of equations,

this sum is given by

F = 2 + (N − 1)π − (π − 1)N − r

d(nG) = (nV) dP − (nS) dT +  µ′i dn′i +  µ″i dn″i i

i

or

F=2−π+N−r

(4-275)

EQUILIBRIUM The number of independent chemical reactions r can be determined as follows: 1. Write formation reactions from the elements for each chemical compound present in the system. 2. Combine these reaction equations so as to eliminate from the set all elements not present as elements in the system. A systematic procedure is to select one equation and combine it with each of the other equations of the set so as to eliminate a particular element. This usually reduces the set by one equation for each element eliminated, though two or more elements may be simultaneously eliminated. The resulting set of r equations is a complete set of independent reactions. More than one such set is often possible, but all sets number r and are equivalent. Example 1: Application of the Phase Rule a. For a system of two miscible nonreacting species in vapor/liquid equilibrium, F=2−π+N−r=2−2+2−0=2 The two degrees of freedom for this system may be satisfied by setting T and P, or T and y1, or P and x1, or x1 and y1, and so on, at fixed values. Thus, for equilibrium at a particular T and P, this state (if possible at all) exists only at one liquid and one vapor composition. Once the two degrees of freedom are used up, no further specification is possible that would restrict the phase-rule variables. For example, one cannot in addition require that the system form an azeotrope (assuming this possible), for this requires x1 = y1, an equation not taken into account in the derivation of the phase rule. Thus, the requirement that the system form an azeotrope imposes a special constraint and reduces the number of degrees of freedom to one. b. For a gaseous system consisting of CO, CO2, H2, H2O, and CH4 in chemical-reaction equilibrium, F=2−π+N−r=2−1+5−2=4 The value of r = 2 is found from the formation reactions: C + a O2 → CO C + O2 → CO2 H2 + a O2 → H2O C + 2H2 → CH4 Systematic elimination of C and O2 from this set of chemical equations reduces the set to two. Three possible pairs of equations may result, depending on how the combination of equations is effected. Any pair of the following three equations represents a complete set of independent reactions, and all pairs are equivalent. CH4 + H2O → CO + 3H2 CO + H2O → CO2 + H2 CH4 + 2H2O → CO2 + 4H2 The result, F = 4, means that one is free to specify, for example, T, P, and two mole fractions in an equilibrium mixture of these five chemical species, provided nothing else is arbitrarily set. Thus, it cannot simultaneously be required that the system be prepared from specified amounts of particular constituent species.

Since the phase rule treats only the intensive state of a system, it applies to both closed and open systems. Duhem’s theorem, on the other hand, is a rule relating to closed systems only: For any closed system formed initially from given masses of prescribed chemical species, the equilibrium state is completely determined by any two properties of the system, provided only that the two properties are independently variable at the equilibrium state. The meaning of completely determined is that both the intensive and extensive states of the system are fixed; not only are T, P, and the phase compositions established, but so also are the masses of the phases. VAPOR/LIQUID EQUILIBRIUM Vapor/liquid equilibrium (VLE) relationships (as well as other interphase equilibrium relationships) are needed in the solution of many engineering problems. The required data can be found by experiment, but such measurements are seldom easy, even for binary systems, and they become rapidly more difficult as the number of constituent species increases. This is the incentive for application of thermodynamics to the calculation of phase-equilibrium relationships.

4-25

The general VLE problem involves a multicomponent system of N constituent species for which the independent variables are T, P, N − 1 liquid-phase mole fractions, and N − 1 vapor-phase mole fractions. (Note that i xi = 1 and i yi = 1, where xi and yi represent liquid and vapor mole fractions respectively.) Thus there are 2N independent variables, and application of the phase rule shows that exactly N of these variables must be fixed to establish the intensive state of the system. This means that once N variables have been specified, the remaining N variables can be determined by simultaneous solution of the N equilibrium relations: fˆ l = fˆ v (i = 1, 2, . . . , N) (4-276) i

i

where superscripts l and v denote the liquid and vapor phases, respectively. In practice, either T or P and either the liquid-phase or vapor-phase composition are specified, thus fixing 1 + (N − 1) = N independent variables. The remaining N variables are then subject to calculation, provided that sufficient information is available to allow determination of all necessary thermodynamic properties. Gamma/Phi Approach For many VLE systems of interest the pressure is low enough that a relatively simple equation of state, such as the two-term virial equation, is satisfactory for the vapor phase. Liquid-phase behavior, on the other hand, may be conveniently described by an equation for the excess Gibbs energy, from which activity coefficients are derived. The fugacity of species i in the liquid phase is then given by Eq. (4-102), written fˆ l = γ x f i

i i i

while the vapor-phase fugacity is given by Eq. (4-79), written fˆiv = φˆ iv yi P Equation (4-276) is now expressed as (i = 1, 2, . . . , N ) γi xi fi = φˆ i yiP

(4-277)

The identifying superscripts l and v are omitted here with the understanding that γi and fi are liquid-phase properties, whereas φˆ i is a vapor-phase property. Applications of Eq. (4-277) represent what is known as the gamma/phi approach to VLE calculations. Evaluation of φˆ i is usually by Eq. (4-196), based on the two-term virial equation of state, but other equations, such as Eq. (4-200), are also applicable. The activity coefficient γi is evaluated by Eq. (4-119), which relates ln γi to GE/RT as a partial property. Thus, what is required for the liquid phase is a relation between GE/RT and composition. Equations in common use for this purpose have already been described. The fugacity fi of pure compressed liquid i must be evaluated at the T and P of the equilibrium mixture. This is done in two steps. First, one calculates the fugacity coefficient of saturated vapor φiv = φisat by an integrated form of Eq. (4-161), written for pure species i and evaluated at temperature T and the corresponding vapor pressure P = Pisat. Equation (4-276) written for pure species i becomes f iv = f il = f isat

(4-278)

sat

where fi indicates the value both for saturated liquid and for saturated vapor. The corresponding fugacity coefficient is fisat φisat = ᎏ (4-279) Pisat This fugacity coefficient applies equally to saturated vapor and to saturated liquid at given temperature T. Equation (4-278) can therefore equally well be written φ iv = φ il

(4-280)

The second step is the evaluation of the change in fugacity of the liquid with a change in pressure to a value above or below Pisat. For this isothermal change of state from saturated liquid at Pisat to liquid at pressure P, Eq. (4-105) is integrated to give Gi − Gisat =



P

sat Pi

Vi dP

Equation (4-74) is then written twice: for Gi and for Gisat. Subtraction provides another expression for Gi − Gisat:

4-26

THERMODYNAMICS

fi Gi − Gisat = RT ln ᎏ fisat Equating the two expressions for Gi − Gisat yields P 1 fi ln ᎏ = ᎏ sat Vi dP sat RT Pi fi Since Vi, the liquid-phase molar volume, is a very weak function of P at temperatures well below Tc, an excellent approximation is often obtained when evaluation of the integral is based on the assumption that Vi is constant at the value for saturated liquid, Vil: fi Vil(P − Pisat) ln ᎏ = ᎏᎏ sat fi RT Substituting fisat = φ isatPisat (Eq. [4-279]), and solving for fi gives Vil(P − Pisat) fi = φ isat Pisat exp ᎏᎏ (4-281) RT The exponential is known as the Poynting factor. Equation (4-277) may now be written



yiPΦi = xiγi Pisat

(i = 1, 2, . . . , N)

(4-282)

φˆ i −Vi (P − Pi ) Φi = ᎏ exp ᎏᎏ (4-283) φisat RT If evaluation of φisat and φˆ i is by Eqs. (4-192) and (4-196), this reduces to PB i − PisatBii − Vil(P − Pisat) Φi = exp ᎏᎏᎏ (4-284) RT where  Bi is given by Eq. (4-195): ∂(nB) Bi
ᎏ = 2  ykBki − B (4-285)  ∂ni T,nj k with B evaluated by Eq. (4-183). The N equations represented by Eq. (4-282) in conjunction with Eq. (4-284) may be used to solve for N unspecified phase-equilibrium variables. For a multicomponent system the calculation is formidable, but well suited to computer solution. The types of problems encountered for nonelectrolyte systems at low to moderate pressures (well below the critical pressure) are discussed by Smith, Van Ness, and Abbott (Introduction to Chemical Engineering Thermodynamics, 5th ed., McGraw-Hill, New York, 1996). When Eq. (4-282) is applied to VLE for which the vapor phase is an ideal gas and the liquid phase is an ideal solution, it reduces to a very simple expression. For ideal gases, fugacity coefficients φˆ i and φisat are unity, and the right-hand side of Eq. (4-283) reduces to the Poynting factor. For the systems of interest here this factor is always very close to unity, and for practical purposes Φi = 1. For ideal solutions, the activity coefficients γi are also unity. Equation (4-282) therefore reduces to

 

where

l

sat





yiP = xiPisat





(i = 1, 2, . . . , N)

an equation which expresses Raoult’s law. It is the simplest possible equation for VLE, and as such fails to provide a realistic representation of real behavior for most systems. Nevertheless, it is useful as a standard of comparison. When an appropriate correlating equation for G E is not available, reliable estimates of activity coefficients may often be obtained from a group-contribution correlation. The Analytical Solution of Groups (ASOG) method (Kojima and Tochigi, Prediction of Vapor-Liquid Equilibrium by the ASOG Method, Elsevier, Amsterdam, 1979) and the UNIFAC method are both well developed. Additional references of interest include Hansen et al. (Ind. Eng. Chem. Res., 30, pp. 2352– 2355 [1991]), Gmehling and Schiller (Ibid., 32, pp. 178–193 [1993]); Larsen et al. (Ibid., 26, pp. 2274–2286 [1987]); and Tochigi et al. ( J. Chem. Eng. Japan, 23, pp. 453–463 [1990]). Data Reduction Correlations for GE and the activity coefficients are based on VLE data taken at low to moderate pressures. The ASOG and UNIFAC group-contribution methods depend for validity on parameters evaluated from a large base of such data. The process

of finding a suitable analytic relation for g (
GE/RT) as a function of its independent variables T and x1, thus producing a correlation of VLE data, is known as data reduction. Although g is in principle also a function of P, the dependence is so weak as to be universally and properly neglected. Given here is a brief description of the treatment of data taken for binary systems under isothermal conditions. A more comprehensive development is given by Van Ness ( J. Chem. Thermodyn., 27, pp. 113–134 [1995]; Pure & Appl. Chem., 67, pp. 859–872 [1995]). Presumed in all that follows is the existence of an equation inherently capable of representing correct values of GE for the liquid phase as a function of x1: g
GE/RT =  (x1; α, β, . . .)

(4-286)

where α, β, and so on, represent adjustable parameters. The measured variables of binary VLE are x1, y1, T, and P. Experimental values of the activity coefficient of species i in the liquid are related to these variables by Eq. (4-282), written: y*i P* γ*i = ᎏ Φi (i = 1, 2) (4-287) xiPisat where Φi is given by Eq. (4-283), and the asterisks denote experimental values. A simple summability relation analogous to Eq. (4-123) defines an experimental value of g*: g*
x1 ln γ 1* + x2 ln γ 2*

(4-288)

Moreover, Eq. (4-122), the Gibbs/Duhem equation, may be written for experimental values in a binary system as d ln γ1* d ln γ 2* x1 ᎏ + x2 ᎏ = 0 (4-289) dx1 dx1 Because experimental measurements are subject to systematic error, sets of values of ln γ 1* and ln γ 2* determined by experiment may not satisfy, that is, may not be consistent with, the Gibbs/Duhem equation. Thus, Eq. (4-289) applied to sets of experimental values becomes a test of the thermodynamic consistency of the data, rather than a valid general relationship. Values of g given by the correlating equation, Eq. (4-286), are called derived values, and associated derived values of the activity coefficients are given by specialization of Eqs. (4-58): dg γ1 = exp g + x2 ᎏ (4-290) dx1









dg (4-291) γ2 = exp g − x1 ᎏ dx1 These two equations may be combined to yield dg γ1 (4-292) ᎏ = ln ᎏ dx1 γ2 This equation applies to derived property values. The corresponding experimental values are given by differentiation of Eq. (4-288): dg* d ln γ 1* d ln γ 2* ᎏ = x1 ᎏ + ln γ 1* + x2 ᎏ − ln γ 2* dx1 dx1 dx1 dg* γ 1* d ln γ 1* d ln γ 2* ᎏ = ln ᎏ + x1 ᎏ + x2 ᎏ dx1 γ 2* dx1 dx1 Subtraction of Eq. (4-293) from Eq. (4-292) gives

or

(4-293)

γ1 dg dg* γ 1* d ln γ 1* d ln γ 2* ᎏ − ᎏ = ln ᎏ − ln ᎏ − x1 ᎏ + x2 ᎏ dx1 dx1 γ2 γ 2* dx1 dx1 The differences between like terms represent residuals between derived and experimental values. Defining these residuals as γ γ γ 1* δ g
g − g* and δ ln ᎏ1
ln ᎏ1 − ln ᎏ γ2 γ2 γ 2* puts this equation into the form dδg γ1 d ln γ 1* d ln γ 2* ᎏ = δ ln ᎏ − x1 ᎏ + x2 ᎏ dx1 γ2 dx1 dx1









EQUILIBRIUM If a data set is reduced so as to make the δg residuals scatter about zero, then the derivative on the left is effectively zero, and the preceding equation becomes γ d ln γ1* d ln γ 2* δ ln ᎏ1 = x1 ᎏ + x2 ᎏ (4-294) γ2 dx1 dx1 The right-hand side of this equation is exactly the quantity that Eq. (4-289), the Gibbs/Duhem equation, requires to be zero for consistent data. The residual on the left is therefore a direct measure of deviations from the Gibbs/Duhem equation. The extent to which values of this residual fail to scatter about zero measures the departure of the data from consistency with respect to this equation. The data-reduction procedure just described provides parameters in the correlating equation for g that make the δg residuals scatter about zero. This is usually accomplished by finding the parameters that minimize the sum of squares of the residuals. Once these parameters are found, they can be used for the calculation of derived values of both the pressure P and the vapor composition y1. Equation (4-282) is solved for yiP and written for species 1 and for species 2. Adding the two equations gives x1γ1P1sat x2γ2P2sat P=ᎏ +ᎏ (4-295) Φ1 Φ2 whence by Eq. (4-282), x1γ1P1sat y1 = ᎏ (4-296) Φ1P These equations allow calculation of the primary residuals: δP
P − P*

and

δy1
y1 − y1*

If the experimental values P* and y1* are closely reproduced by the correlating equation for g, then these residuals, evaluated at the experimental values of x1, scatter about zero. This is the result obtained when the data are thermodynamically consistent. When they are not, these residuals do not scatter about zero, and the correlation for g does not properly reproduce the experimental values P* and y1*. Such a correlation is, in fact, unnecessarily divergent. An alternative is to process just the P-x1 data; this is possible because the P-x1-y1 data set includes more information than necessary. Assuming that the correlating equation is appropriate to the data, one merely searches for values of the parameters α, β, and so on, that yield pressures by Eq. (4-295) that are as close as possible to the measured values. The usual procedure is to minimize the sum of squares of the residuals δP. Known as Barker’s method (Austral. J. Chem., 6, pp. 207–210 [1953]), it provides the best possible fit of the experimental pressures. When the experimental data do not satisfy the Gibbs/Duhem equation, it cannot precisely represent the experimental y1 values; however, it provides a better fit than does the procedure that minimizes the sum of the squares of the δg residuals. Worth noting is the fact that Barker’s method does not require experimental y1* values. Thus the correlating parameters α, β, and so on, can be evaluated from a P-x1 data subset. Common practice now is, in fact, to measure just such data. They are, of course, not subject to a test for consistency by the Gibbs/Duhem equation. The world’s store of VLE data has been compiled by Gmehling et al. (VaporLiquid Equilibrium Data Collection, Chemistry Data Series, vol. I, parts 1–8, DECHEMA, Frankfurt am Main, 1979–1990). Solute/Solvent Systems The gamma/phi approach to VLE calculations presumes knowledge of the vapor pressure of each species at the temperature of interest. For certain binary systems species 1, designated the solute, is either unstable at the system temperature or is supercritical (T > Tc ). Its vapor pressure cannot be measured, and its fugacity as a pure liquid at the system temperature f1 cannot be calculated by Eq. (4-281). Equations (4-282) and (4-283) are applicable to species 2, designated the solvent, but not to the solute, for which an alternative approach is required. Figure 4-11 shows a typical plot of the liquidphase fugacity of the solute fˆ1 vs. its mole fraction x1 at constant temperature. Since the curve representing fˆ1 does not extend all the way to x1 = 1, the location of f1, the liquid-phase fugacity of pure species 1, is not established. The tangent line at the origin, representing Henry’s

FIG. 4-11

4-27

Plot of solute fugacity fˆ1 vs. solute mole fraction.

law, provides alternative information. The slope of the tangent line is Henry’s constant, defined as fˆ1 k1
lim ᎏ (4-297) x1→0 x 1 This is the definition of k1 for temperature T and for a pressure equal to the vapor pressure of the pure solvent P2sat. The activity coefficient of the solute at infinite dilution is fˆ1 1 fˆ1 lim γ1 = lim ᎏ = ᎏ lim ᎏ x1→0 x1→0 x f x1→0 x f 1 1 1 1 In view of Eq. (4-297), this becomes γ1∞ = k1/f1, or k1 f1 = ᎏ (4-298) γ ∞1 where γ 1∞ represents the infinite-dilution value of the activity coefficient of the solute. Since both k1 and γ 1∞ are evaluated at P2sat, this pressure also applies to f1. However, the effect of P on a liquid-phase fugacity, given by a Poynting factor, is very small, and for practical purposes may usually be neglected. The activity coefficient of the solute, given by fˆ1 y1P φˆ 1 γ1
ᎏ =ᎏ x1 f1 x1 f1 then becomes y1P φˆ 1γ1∞ γ1 = ᎏ x1k1 For the solute, this equation takes the place of Eqs. (4-282) and (4-283). Solution for y1 gives x1(γ1/γ1∞)k1 y1 = ᎏᎏ (4-299) φˆ 1P For the solvent, species 2, the analog of Eq. (4-296) is x2γ 2 P2sat y2 = ᎏ (4-300) Φ2P Since y1 + y2 = 1, x1(γ1/γ1∞)k1 x2γ2 P2sat P = ᎏᎏ +ᎏ (4-301) φˆ 1 Φ2 Note that the same correlation that provides for the evaluation of γ1 also allows evaluation of γ ∞1 . There remains the problem of finding Henry’s constant from the available VLE data. For equilibrium fˆ1
fˆ1l = fˆ1v = y1Pφˆ 1 Division by x1 gives fˆ1 y1 ᎏ = Pφˆ 1 ᎏ x1 x1

4-28

THERMODYNAMICS

Henry’s constant is defined as the limit as x1 → 0 of the ratio on the left; therefore y k1 = P2sat φˆ 1∞ lim ᎏ1 x1→0 x 1 The limiting value of y1/x1 can be found by plotting y1/x1 vs. x1 and extrapolating to zero. K-Values A measure of how a given chemical species distributes itself between liquid and vapor phases is the equilibrium ratio: y Ki
ᎏi (4-302) xi

with the theoretically based Wong/Sandler mixing rules for parameters a and b these equations provide the means for accurate correlation and prediction of VLE data. The first of the Wong/Sandler mixing rules relates the difference in mixture quantities b and a/RT to the corresponding differences (identified by subscripts) for the pure species: a b − ᎏ =   xp xq E pq (4-305) RT p q where





1 ap aq E pq
ᎏ bp − ᎏ + bq − ᎏ (1 − k pq) 2 RT RT

(4-306)

Usually called simply a K-value, it adds nothing to thermodynamic knowledge of VLE. However, its use may make for computational convenience, allowing formal elimination of one set of mole fractions {yi} or {xi} in favor of the other. Moreover, it characterizes lightness of a constituent species. For a light species, tending to concentrate in the vapor phase, K > 1; for a heavy species, tending to concentrate in the liquid phase, K < 1. Empirical correlations for K-values found in the older literature have little relation to thermodynamics. Their proper evaluation comes directly from Eq. (4-277): γifi y Ki
ᎏi = ᎏ (4-303) xi φˆ i P

Binary interaction parameters kpq are determined for each pq pair (p ≠ q) from experimental data. Note that kpq = kqp and kpp = kqq = 0. Since the quantity on the left-hand side of Eq. (4-305) represents the second virial coefficient as predicted by Eq. (4-231), the basis for Eq. (4-305) lies in Eq. (4-183), which expresses the quadratic dependence of the mixture second virial coefficient on mole fraction. The second Wong/Sandler mixing rule relates ratios of a/RT to b: a (4-307) ᎏ=1−D bRT

When Raoult’s law applies, this becomes Ki = Pisat/P. In general, Kvalues are functions of T, P, liquid composition, and vapor composition, making their direct and accurate correlation impossible. Those correlations that do exist are approximate and severely limited in application. The DePriester correlation, for example, gives K-values for light hydrocarbons (Chem. Eng. Prog. Symp. Ser. No. 7, 49, pp. 1–43 [1953]). Equation-of-State Approach Although the gamma/phi approach to VLE is in principle generally applicable to systems comprised of subcritical species, in practice it has found use primarily where pressures are no more than a few bars. Moreover, it is most satisfactory for correlation of constant-temperature data. A temperature dependence for the parameters in expressions for GE is included only for the localcomposition equations, and it is at best only approximate. A generally applicable alternative to the gamma/phi approach results when both the liquid and vapor phases are described by the same equation of state. The defining equation for the fugacity coefficient, Eq. (4-79), may be applied to each phase:

The quantity GE/RT is given by an appropriate correlation for the excess Gibbs energy of the liquid phase, and is evaluated at the mixture composition, regardless of whether the mixture is liquid or vapor. The constant c is specific to the equation of state. The theoretical basis for these equations can be found in the literature (Wong and Sandler, op. cit.; Ind. Eng. Chem. Res., 31, pp. 2033–2039 [1992]; Eubank, et al., Ind. Eng. Chem. Res., 34, pp. 314–323 [1995]). Elimination of a from Eq. (4-305) by Eq. (4-307) provides an expression for b: 1 b = ᎏ   xp xq Epq (4-309) D p q

Liquid: Vapor:

fˆil = φˆ il xiP fˆiv = φˆ iv yiP (i = 1, 2, . . . , N)

(4-304)

This introduces the compositions xi and yi into the equilibrium equations, but neither is explicit, because the φˆ i are functions, not only of T and P, but of composition. Thus Eq. (4-304) represents N complex relationships connecting T, P, the xi, and the yi, suitable for computer solution. Given an appropriate equation of state, one or another of Eqs. (4-178) through (4-181) provides for expression of the φˆ i as functions of T, P, and composition. Because of inadequacies in empirical mixing rules, such as those given by Eqs. (4-221) and (4-222), the equation-of-state approach was long limited to systems exhibiting modest and well-behaved deviations from ideal solution behavior in the liquid phase; for example, to systems containing hydrocarbons and cryogenic fluids. However, the introduction by Wong and Sandler (AIChE J., 38, pp. 671–680 [1992]) of a new class of mixing rules for cubic equations of state has greatly expanded their useful application to VLE. The Soave/Redlich/Kwong (SRK) and the Peng/Robinson (PR) equations of state, both expressed by Eqs. (4-230) and (4-231), were developed specifically for VLE calculations. The fugacity coefficients implicit in these equations are given by Eq. (4-232). When combined

GE ap D
1 + ᎏ −  xp ᎏ cRT bp RT p

(4-308)

Mixture parameter a then follows from Eq. (4-307): a = bRT(1 − D)

(4-310)

Equations (4-233) and (4-234) may now be applied for the evaluation of partial parameters ai and  bi: 1 ln γ ai bi = ᎏ 2 xj Eij − b 1 + ᎏi − ᎏ  D c biRT j



and

Equation (4-276) now becomes xi φˆ il = yi φˆ iv

where





ai ln γi bi  ai = bRT ᎏ − ᎏ + a ᎏ − 1 bi RT c b



 



(4-311) (4-312)

For pure species i, Eq. (4-232) reduces to (Vi − bi)Zi ai /biRT Vi + σbi ln φi = Zi − 1 − ln ᎏᎏ + ᎏ ln ᎏ Vi ε−σ Vi + εbi

(4-313)

This equation may be applied separately to the liquid phase and to the vapor phase to yield the pure-species values φil and φiv. For vapor/ liquid equilibrium (Eq. [4-280]), these two quantities are equal. Given parameters ai and bi, the pressure P in Eq. (4-230) that makes these two values equal is Pisat, the equilibrium vapor pressure of pure species i as predicted by the equation of state. The correlations for α(Tri; ωi) that follow Eq. (4-230) are designed to provide values of ai that yield pure-species vapor pressures which, on average, are in reasonable agreement with experiment. However, reliable correlations for Pisat as a function of temperature are available for many pure species. Thus when Pisat is known for a particular temperature, ai should be evaluated so that the equation of state correctly predicts this known value. The procedure is to write Eq. (4-313) for each of the phases, combining the two equations in accord with Eq. (4-280), written ln φil = ln φiv

EQUILIBRIUM The resulting expression may be solved for ai: Vi − bi bi RT(ε − σ) ln ᎏ ᎏ + Ziv − Zil Viv − bi ai = (Vil + σbi)(Viv + εbi) ln ᎏᎏᎏ (Vil + εbi)(Viv + σbi)





l

(4-314)

where ln γi∞ comes from the GE correlation and ln φi is given by Eq. (4-313) written for the liquid phase. Equation (4-315) supplies a value for ln φˆ i∞ which, when used with Eq. (4-232), ultimately (see following) leads to values for kpq. For a binary system comprised of species p and q, Eqs. (4-232), (4-312), and (4-315) may be written for species p at infinite dilution. The three resulting equations are then combined to yield b p∞ ln γ p∞ + ln φp − Mp ᎏ = ᎏᎏ bq Zq − 1

(4-316)

where (Vq − bq)Zq 1 ap ln γ p∞ Vq + σbq Mp
−ln ᎏᎏ +ᎏ ᎏ −ᎏ ln ᎏ (4-317) Vq ε − σ bp RT c Vq + εbq By Eq. (4-311) written for species p at infinite dilution in a pq binary, 2Epq ln γ p∞ ap ᎏᎏ − 1 − ᎏᎏ + ᎏᎏ ∞ b b c RT q p bp  ᎏ = ᎏᎏᎏ (4-318) aq 1 − ᎏᎏ bq bqRT





Equations (4-316) and (4-318) are set equal, Epq is eliminated by Eq. (4-306), and kpq is replaced by kp, its infinite-dilution value at xp → 0. Solution for kp then yields ln γ + ln φ − M ln γ a ᎏ + b 1 + ᎏᎏ − ᎏᎏ b − ᎏRaᎏT  ᎏ Z −1 c b RT q

q

∞ p

p

p

∞ p

q

p

kp = 1 − ᎏᎏᎏᎏᎏᎏ ap aq bp − ᎏᎏ + bq − ᎏᎏ RT RT (4-319) q

p

where ln φp comes from Eq. (4-313). All values in Eq. (4-319) are for the liquid phase at P = Pqsat. The analogous equation for kq, the infinitedilution value of k pq at xq → 0 is written ln γ + ln φ − M ln γ a ᎏ + b 1 + ᎏᎏ − ᎏᎏ b − ᎏRaᎏT  ᎏ c Z −1 b RT p

p

∞

q

q

q

∞

p

q

q

kq = 1 − ᎏᎏᎏᎏᎏᎏ ap aq bp − ᎏᎏ + bq − ᎏᎏ RT RT (4-320) p

A second advantage is that the procedure, applied for infinite dilution of each species, yields two values of kpq from which a composition-dependent function can be generated, a simple linear relation proving fully satisfactory: kpq = kp xq + kq xp

where Ziv = Pisat Viv/RT and Zil = Pisat Vil/RT. Values of Viv and Vil come from solution of Eq. (4-230) for each phase with P = Pisat at temperature T. Since a value of ai is required for these calculations, an iterative procedure is implemented with an initial value for ai from the appropriate correlation for α(Tri; ωi). The binary interaction parameters kpq are evaluated from liquidphase GE correlations for binary systems. The most satisfactory procedure is to apply at infinite dilution the relation between a liquid-phase activity coefficient and its underlying fugacity coefficients, γ i∞ = φˆ i∞/φi. Rearrangement of the logarithmic form yields ln φˆ i∞ = ln γ i∞ + ln φi (4-315)

q

where Mq is given by an equation analogous to Eq. (4-317) but with subscripts reversed. All values in Eq. (4-320) are for the liquid phase at P = Ppsat. One advantage of this procedure is that kp and kq are found directly from the pure-species parameters ap, aq, bp, and bq. In addition, the required values of ln γp∞ and ln γq∞ can be found from experimental data for the pq binary system, independent of the correlating expression used for GE.

4-29

(4-321)

The two values kp and kq are usually not very different, and kpq is not strongly composition dependent. Nevertheless, the quadratic dependence of b − (a/RT) on composition indicated by Eq. (4-305) is not exactly preserved. Since this quantity is not a true second virial coefficient, only a value predicted by a cubic equation of state, a strict quadratic dependence is not required. Moreover, the compositiondependent kpq leads to better results than does use of a constant value. The equation-specific constants for the SRK and PR equations are given by the following table:

ε σ Ωa Ωb c

SRK equation

PR equation

0 1 0.42748 0.08664 0.69315

−0.414214 2.414214 0.457235 0.077796 0.62323

Outlined below are the steps required for of a VLE calculation of vapor-phase composition and pressure, given the liquid-phase composition and temperature. A choice must be made of an equation of state. Only the Soave/Redlich/Kwong and Peng/Robinson equations, as represented by Eqs. (4-230) and (4-231), are considered here. These two equations usually give comparable results. A choice must also be made of a two-parameter correlating expression to represent the liquid-phase composition dependence of GE for each pq binary. The Wilson, NRTL (with α fixed), and UNIQUAC equations are of general applicability; for binary systems, the Margules and van Laar equations may also be used. The equation selected depends on evidence of its suitability to the particular system treated. Reasonable estimates of the parameters in the equation must also be known at the temperature of interest. These parameters are directly related to infinite-dilution values of the activity coefficients for each pq binary. Input information includes the known values of T and {xi}, as well as the equation-of-state and GE-expression parameters. Estimates are also needed of P and {yi}, the quantities to be evaluated, and these require some preliminary calculations: 1. For the chosen equation of state (with appropriate values of ε, σ, and c), find values of bi and preliminary values of ai for each species from the information following Eq. (4-230). 2. If the vapor pressure Pisat for species i at temperature T is known, determine a new value for ai by Eqs. (4-314) and (4-230). 3. Evaluate kp and kq by Eqs. (4-319) and (4-320) for each pq binary. 4. Although pressure P is to be determined, an estimate is required to permit any VLE calculations at all. A reasonable initial value is the sum of the pure-species vapor pressures, each weighted by its known liquid-phase mole fraction. 5. The vapor-phase composition is also to be determined, and it, too, is required to initiate calculations. Assuming both the liquid and vapor phases to be ideal solutions, Eqs. (4-98) and (4-304) combine to give φil yi = xi ᎏ φiv Evaluation of the pure-species values φil and φiv by Eq. (4-313) then provides values for yi. Since these are not constrained to sum to unity, they should be normalized to yield an initial vapor-phase composition. Given estimates for P and {yi} an iterative procedure can be initiated: 1. At the known liquid-phase composition, evaluate D by Eq. (4-308), b and a by Eqs. (4-309) and (4-310), and {b i} and {ai} by Eqs. (4-311) and (4-312).

4-30

THERMODYNAMICS

2. Evaluate {φˆ il }. The mixture volume V is determined from the equation of state, Eq. (4-231), applied to the liquid phase at the given composition, T, and P. 3. Repeat the two preceding items for the vapor-phase composition, thus evaluating {φˆ vi }. 4. Eq. (4-304) is now written φˆ l yi = xi ᎏvi φˆ i The values of yi so calculated are normalized by division by  i yi. 5. Recalculate the φˆ iv, and continue this iterative procedure until it converges to a fixed value for  i yi. This sum is appropriate to the pressure P for which the calculations have been made. Unless the sum is unity, the pressure is adjusted and the iteration process is repeated. Systematic adjustment of pressure P continues until  i yi = 1. The pressure and vapor compositions so found are the equilibrium values for the given temperature and liquid-phase composition as predicted by the equation of state. A vast store of liquid-phase excess-property data for binary systems at temperatures near 30°C and somewhat higher is available in the literature. Effective use of these data to extend G E correlations to higher temperatures is critical to the procedure considered here. The key relations are Eq. (4-118),

 

GE HE d ᎏ = − ᎏ2 dT RT RT

(constant P,x)

and the excess-property analog of Eq. (4-31), dH E = CPE dT

Integration of the first of these equations from T0 to T gives

  

T

GE GE HE (4-322) ᎏ= ᎏ − ᎏ dT RT RT T0 T0 RT 2 Similarly, the second equation may be integrated from T1 to T:



T

E

CP dT

(4-323)

T1

In addition, we may write ∂CPE dC PE = ᎏ ∂T Integration from T2 to T yields



CPE = CPE2 +



dT

P,x

∂C  ᎏ ∂T  T

E P

dT

P,x

T2

Combining this equation with Eqs. (4-322) and (4-323) leads to H T T ᎏ − 1 ᎏ   − ᎏ RT   T T

GE GE ᎏ= ᎏ RT RT

E

T0

1

T1

0

  ∂C 1 I
 ᎏ   ᎏ RT ∂T

 

CPE2 T T T −ᎏ ln ᎏ − ᎏ − 1 ᎏ1 − I R T0 T0 T

T

where

T0

2

T

T

T1

T2

E P



 

(4-324)

dT dT dT

P,x

This general equation makes use of excess Gibbs-energy data at temperature T0, excess enthalpy (heat-of-mixing) data at T1, and excess heat-capacity data at T2. Evaluation of the integral I requires informaE tion with respect to the temperature dependence of CP . Because of the relative paucity of excess heat-capacity data, the most reasonable assumption is that this quantity is constant, independent of T. In this event, the integral is zero, and the closer T0 and T1 are to T, the less the influence of this assumption. When no information is available with E respect to CP , and excess enthalpy data are available at only a single temperature, the excess heat capacity must be assumed zero. In this case only the first two terms on the right-hand side of Eq. (4-324) are retained, and it more rapidly becomes imprecise as T increases. Our primary interest in Eq. (4-324) is its application to binary systems at infinite dilution of one of the constituent species. For this pur-





E P







 

 

C T T T − ᎏ ln ᎏ − ᎏ − 1 ᎏ1 x1x2R T0 T0 T As shown by Smith, Van Ness and Abbott (Introduction to Chemical Engineering Thermodynamics, 5th ed., Chap. 11, McGraw-Hill, New York, 1996), GE
ln γi∞ ᎏ x1x2 RT xi = 0 The preceding equation may therefore be written HE T T1 ln γi∞ = (ln γi∞)T0 − ᎏ ᎏ−1 ᎏ x1x2 RT T1,xi = 0 T0 T











E P





 

C T T T − ᎏ ln ᎏ − ᎏ − 1 ᎏ1 (4-325) x1x2 R xi = 0 T0 T0 T The methanol(1)/acetone(2) system serves as a specific example in conjunction with the Peng/Robinson equation of state. At a base temperature T0 of 323.15 K (50°C), both VLE data (Van Ness and Abbott, Int. DATA Ser., Ser. A, Sel. Data Mixtures, 1978, p. 67 [1978]) and excess enthalpy data (Morris, et al., J. Chem. Eng. Data, 20, pp. 403– 405 [1975]) are available. From the former, (ln γ 1∞)T0 = 0.6281

(constant P,x)

HE = H1E +

E

pose, we divide Eq. (4-324) by the product x1x2. For CP independent of T (and thus with I = 0), it then becomes GE GE HE T T1 ᎏ= ᎏ − ᎏ ᎏ−1 ᎏ x1x2RT x1x2RT T0 x1x2RT T1 T0 T

and

(ln γ 2∞)T0 = 0.6557

and from the latter HE HE = 1.3636 and = 1.0362 ᎏ ᎏ x1x2 RT T0,x 1 = 0 x1x2 RT T0 ,x 2 = 0 The Margules equations (Eqs. [4-244], [4-245], and [4-246]) are well suited to this system, and the parameters for this equation are given as







A12 = ln γ 1∞

and



A21 = ln γ 2∞

This information allows prediction of VLE at 323.15 K and at the higher temperatures, 372.8, 397.7, and 422.6 K, for which measured VLE values are given by Wilsak, et al. (Fluid Phase Equilibria, 28, pp. 13–37 [1986]). Values of ln γi∞ and hence of the Margules parameters at the higher temperatures are given by Eq. (4-325) with CPE = 0. The pure-species vapor pressures in all cases are the measured values reported with the data sets. Results of these calculations are displayed in Table 4-1, where the parentheses enclose values from the gamma/ phi approach as reported in the papers cited. The results at 323.15 K (581.67 R) show both the suitability of the Margules equation for correlation of data for this system and the capability of the equation-of-state method to reproduce the data. Results for the three higher temperatures indicate the quality of predictions based only on vapor-pressure data for the pure species and on mixture data at 323.15 K (581.67 R). Extrapolations based on the same data to still higher temperatures can be expected to become progressively less accurate. When Eq. (4-325) can no longer be expected to produce reasonable values, better results are obtained for higher temperatures by assuming that the parameters, A12, A21, k1, and k2, do not change further at still-higher temperatures. This is also the course to be followed for extrapolation to supercritical temperatures. Only the Wilson, NRTL, and UNIQUAC equations are suited to the treatment of multicomponent systems. For such systems, the parameters are determined for pairs of species exactly as for binary systems. Examples treating the calculation of VLE are given in Smith, Van Ness, and Abbott (Introduction to Chemical Engineering Thermodynamics, 5th ed., Chap. 12, McGraw-Hill, New York, 1996). LIQUID/LIQUID AND VAPOR/LIQUID/LIQUID EQUILIBRIA Equation (4-273) is the basis for both liquid/liquid equilibria (LLE) and vapor/liquid/liquid equilibria (VLLE). Thus, for LLE with superscripts α and β denoting the two phases, Eq. (4-273) is written

EQUILIBRIUM TABLE 4-1

VLE Results for Methanol(1)/Acetone(2)

T, K

ln γ1∞

ln γ 2∞

k1

k2

RMS δP, kPa

RMS % δP

323.15

0.6281 (0.6281) 0.4465 (0.4607) 0.3725 (0.3764) 0.3072 (0.3079)

0.6557 (0.6557) 0.5177 (0.5271) 0.4615 (0.4640) 0.4119 (0.3966)

0.1395

0.0955

0.12

0.1432

0.1056

0.1454

0.1118

0.1480

0.1192

0.08 (0.06) 0.85 (0.83) 2.46 (1.39) 7.51 (2.38)

372.8 397.7 422.6

fˆiα = fˆiβ

(i = 1, 2, . . . , N)

(4-326)

Eliminating fugacities in favor of activity coefficients gives x αi γ iα = xiβ γ iβ

(i = 1, 2, . . . , N)

(4-327)

For most LLE applications, the effect of pressure on the γ i can be ignored, and thus Eq. (4-327) constitutes a set of N equations relating equilibrium compositions to each other and to temperature. For a given temperature, solution of these equations requires a single expression for the composition dependence of GE suitable for both liquid phases. Not all expressions for GE suffice, even in principle, because some cannot represent liquid/liquid phase splitting. The UNIQUAC equation is suitable, and therefore prediction is possible by the UNIFAC method. A special table of parameters for LLE calculations is given by Magnussen, et al. (Ind. Eng. Chem. Process Des. Dev., 20, pp. 331–339 [1981]). A comprehensive treatment of LLE is given by Sorensen, et al. (Fluid Phase Equilibria, 2, pp. 297–309 [1979]; 3, pp. 47–82 [1979]; 4, pp. 151–163 [1980]). Data for LLE are collected in a three-part set compiled by Sorensen and Arlt (Liquid-Liquid Equilibrium Data Collection, Chemistry Data Series, vol. V, parts 1–3, DECHEMA, Frankfurt am Main, 1979–1980). For vapor/liquid/liquid equilibria, Eq. (4-273) gives fˆiα = fˆiβ = fˆiv (i = 1, 2, . . . , N) (4-328) where α and β designate the two liquid phases. With activity coefficients applied to the liquid phases and fugacity coefficients to the vapor phase, the 2N equilibrium equations for subcritical VLLE are xiα γiα fiα = yi φˆ iP (all i) (4-329) xiβ γiβ fiβ = yi φˆ iP



E

As for LLE, an expression for G capable of representing liquid/liquid phase splitting is required; as for VLE, a vapor-phase equation of state for computing the φˆ i is also needed.

0.32 0.55

∆nN, j ∆n1, j ∆n2, j (4-331) ᎏ=ᎏ=⋅⋅⋅=ᎏ ν1, j ν2, j νN, j Since all of these terms are equal, they can be equated to the change in a single quantity εj, called the reaction coordinate for reaction j, thereby giving i = 1, 2, . . . , N ∆ni, j = νi, j ∆εj (4-332) j = I, II, . . . , r



Since the total change in mole number ∆ni is just the sum of the changes ∆ni,j resulting from the various reactions, ∆ni =  ∆ni, j =  νi, j ∆εj j

sign (νi,j ) =

(4-333)

If the initial number of moles of species i is ni0 and if the convention is adopted that εj = 0 for each reaction in this initial state, then ni = ni0 +  νi, j εj

(i = 1, 2, . . . , N)

(4-334)

j

Equation (4-334) is the basic expression of material balance for a closed system in which r chemical reactions occur. It shows for a reacting system that at most r mole number–related quantities εj are capable of independent variation. Note the absence of implied restrictions with respect to chemical-reaction equilibria; the reactioncoordinate formalism is merely an accounting scheme, valid for tracking the progress of each reaction to any arbitrary level of conversion. The reaction coordinate has units of moles. A change in εj of 1 mole signifies a mole of reaction, meaning that reaction j has proceeded to such an extent that the change in mole number of each reactant and product is equal to its stoichiometric number. CHEMICAL-REACTION EQUILIBRIA The general criterion of chemical-reaction equilibria is given by Eq. (4-274). For a system in which just a single reaction occurs, Eq. (4-334) becomes dni = νi dε

whence

Substitution for dni in Eq. (4-274) leads to

 νµ = 0 i i

where the |νi| are stoichiometric coefficients and the Ai stand for chemical formulas. The νi themselves are called stoichiometric numbers, and associated with them is a sign convention such that the value is positive for a product and negative for a reactant. More generally, for a system containing N chemical species, any or all of which can participate in r chemical reactions, the reactions can be represented by the equations:

where

(i = 1, 2, . . . , N)

j

|ν1|A1 + |ν2|A2 + ⋅ ⋅ ⋅ → |ν3|A3 + |ν4|A4 + ⋅ ⋅ ⋅

( j = I, II, . . . , r)

0.004 (0.006) 0.014 (0.013) 0.009 (0.006)

ni = ni0 + νiε

Consider a phase in which a chemical reaction occurs according to the equation

0 = νi, j Ai

RMS δy1

0.22

CHEMICAL-REACTION STOICHIOMETRY

i

4-31

(4-330)

− for a reactant species

+ for a product species

If species i does not participate in reaction j, then νi,j = 0. The stoichiometric numbers provide relations among the changes in mole numbers of chemical species which occur as the result of chemical reaction. Thus, for reaction j:

(4-335)

i

Generalization of this result to multiple reactions produces

ν

i, j

µi = 0

( j = I, II, . . . , r)

(4-336)

i

Standard Property Changes of Reaction A standard property change for the reaction aA + bB → lL + mM is defined as the property change that occurs when a moles of A and b moles of B in their standard states at temperature T react to form l moles of L and m moles of M in their standard states also at temperature T. A standard state of species i is its real or hypothetical state as a pure species at temperature T and at a standard-state pressure P°. The standard property change of reaction j is given the symbol ∆Mj°, and its general mathematical definition is ∆Mj°
 νi, j Mi° i

(4-337)

4-32

THERMODYNAMICS

For species present as gases in the actual reactive system, the standard state is the pure ideal gas at pressure P°. For liquids and solids, it is usually the state of pure real liquid or solid at P°. The standard-state pressure P° is fixed at 100 kPa. Note that the standard states may represent different physical states for different species; any or all of the species may be gases, liquids, or solids. The most commonly used standard property changes of reaction are ∆Gj°
 νi, jGi° =  νi,j µi°

(4-338)

∆Hj°
 νi, j Hi°

(4-339)

∆C°Pi
 νi, j C°Pi

(4-340)

i

i

i

i

The standard Gibbs-energy change of reaction ∆Gj° is used in the calculation of equilibrium compositions. The standard heat of reaction ∆Hj° is used in the calculation of the heat effects of chemical reaction, and the standard heat-capacity change of reaction is used for extrapolating ∆Hj° and ∆Gj° with T. Numerical values for ∆Hj° and ∆Gj° are computed from tabulated formation data, and ∆C°Pi is determined from empirical expressions for the T dependence of the C°Pi (see, e.g., Eq. [4-142]). Equilibrium Constants For practical application, Eq. (4-336) must be reformulated. The initial step is elimination of the µi in favor of fugacities. Equation (4-74) for species i in its standard state is subtracted from Eq. (4-77) for species i in the equilibrium mixture, giving µi = Gi° + RT ln âi

(4-341)

where, by definition, âi
fˆi /fi° and is called an activity. Substitution of this equation into Eq. (4-341) yields, upon rearrangement,

 [ν

(Gi° + RT ln âi)] = 0

i, j

i

or

 (ν

Gi°) + RT  ln âiνi,j = 0

i, j

i

i

− (νi,jGi°)

ln  âiνi, j = ᎏᎏ RT i The right-hand side of this equation is a function of temperature only for given reactions and given standard states. Convenience suggests setting it equal to ln Kj; whence i

or

 (y φˆ ) i i

νi, j

i

νj

P ᎏ P° 

= Kj

(all j)

(4-344)

where νj
i νi,j and P° is the standard-state pressure of 100 kPa, expressed in the same units used for P. The yi may be eliminated in favor of equilibrium values of the reaction coordinates εj. Then, for fixed temperature Eqs. (4-344) relate the εj to P. In principle, specification of the pressure allows solution for the εj. However, the problem may be complicated by the dependence of the φˆ i on composition, that is, on the εj. If the equilibrium mixture is assumed an ideal solution, then each φˆ i becomes φi, the fugacity coefficient of pure species i at the mixture T and P. This quantity does not depend on composition and may be determined from experimental data, from a generalized correlation, or from an equation of state. An important special case of Eq. (4-344) is obtained for gas-phase reactions when the phase can be assumed an ideal gas. In this event φˆ i = 1, and P νj = Kj (all j) (4-345) i (yi)νi, j ᎏ P° In the general case the evaluation of the φˆ i requires an iterative process. An initial step is to set the φˆ i equal to unity and to solve the problem by Eq. (4-345). This provides a set of yi values, allowing evaluation of the φˆ i by, for example, Eq. (4-196), (4-200), or (4-231). Equation (4-344) can then be solved for a new set of yi values, and the process continues to convergence. For liquid-phase reactions, Eq. (4-342) is modified by introduction of the activity coefficient, γi = fˆi /xi fi, where xi is the liquid-phase mole fraction. The activity is then f fˆi âi
ᎏ = γi xi ᎏi fi° fi° Both fi and fi° represent fugacity of pure liquid i at temperature T, but at pressures P and P°, respectively. Except in the critical region, pressure has little effect on the properties of liquids, and the ratio fi/fi° is often taken as unity. When this is not acceptable, this ratio is evaluated by the equation f 1 P Vi(P − P°) ln ᎏi = ᎏ Vi dP ᎏᎏ fi° RT P° RT When the ratio fi /fi° is taken as unity, âi = γi xi, and Eq. (4-342) becomes

 



 (γ x ) i i

νi,j

= Kj

(all j)

(4-346)

i

â

i

νi, j

= Kj

(all j)

(4-342)

i

−∆Gj° (4-343) Kj
exp ᎏ RT Quantity Kj is the chemical-reaction equilibrium constant for reaction j, and ∆Gj° is the corresponding standard Gibbs-energy change of reaction (see Eq. [4-338]). Although called a “constant,” Kj is a function of T, but only of T. The activities in Eq. (4-342) provide the connection between the equilibrium states of interest and the standard states of the constituent species, for which data are presumed available. The standard states are always at the equilibrium temperature. Although the standard state need not be the same for all species, for a particular species it must be the state represented by both G°i and the fi° upon which the activity âi is based. The application of Eq. (4-342) requires explicit introduction of composition variables. For gas-phase reactions this is accomplished through the fugacity coefficient: âi
fˆi/fi° = yi φˆ iP/fi° where

and Eq. (4-342) becomes





However, the standard state for gases is the ideal gas state at the standard-state pressure, for which fi° = P°. Therefore yi φˆ iP âi = ᎏ P°

Here the difficulty is to determine the γi , which depend on the xi . This problem has not been solved for the general case. Two courses are open: the first is experiment; the second, assumption of solution ideality. In the latter case, γi = 1, and Eq. (4-346) reduces to

 (x ) i

νi,j

= Kj

(all j)

(4-347)

i

the law of mass action. The significant feature of Eqs. (4-345) and (4-347), the simplest expressions for gas- and liquid-phase reaction equilibrium, is that the temperature-, pressure-, and compositiondependent terms are distinct and separate. Example 2: Single-Reaction Equilibrium Consider the equilibrium state at 1,000 K and atmospheric pressure for the reaction CO + H2O → CO2 + H2 Let the feed stream contain 3 mol CO, 1 mol H2O, and 2 mol CO2 for every mole of H2 present. This initial constitution forms the basis for calculation, and for this single reaction, Eq. (4-334) becomes ni = ni0 + νiε. Whence nCO = 3 − ε nH2O = 1 − ε nCO2 = 2 + ε nH2 = 1 + ε

n =7 i

i

Each mole fraction is therefore given by yi = ni/7.

EQUILIBRIUM At 1,000 K, ∆G° = −2680 J per mole of reaction; whence by Eq. (4-343) 2680 K = exp ᎏᎏ = 1.38 (8.314)(1000) For the given conditions, the assumption of ideal gases is appropriate; Eq. (4-345) written for a single reaction (subscript j omitted) with ν = 0 becomes

ᎏ2 7+ᎏε  ᎏ1 7+ᎏε  = K = 1.38  y = ᎏᎏ ᎏ3 7−ᎏε  ᎏ1 7−ᎏε  i

or

νi i

limited to gas-phase reactions for which the problem is to find the equilibrium composition for given T and P and for a given initial feed. 1. Formulate the constraining material-balance equations, based on conservation of the total number of atoms of each element in a system comprised of w elements. Let subscript k identify a particular atom, and define Ak as the total number of atomic masses of the kth element in the feed. Further, let aik be the number of atoms of the kth element present in each molecule of chemical species i. The material balance for element k is then

 na

i ik

na

or

ε = 0.258

i ik

2.

nCO = 2.74 mol nH2O = 0.74 mol nCO2 = 2.26 mol nH2 = 1.26 mol

yCO = 0.391 yH2O = 0.106 yCO2 = 0.323 yH2 = 0.180

 i ni = 7.00 mol

 i yi = 1.000

(4-353)

− Ak = 0

(k = 1, 2, . . . , w)



− Ak = 0

(k = 1, 2, . . . , w)

 λ  n a k

i ik

k

∆CP° ᎏ dT R

(4-351)

where for simplicity subscript j has been suppressed. A convenient integrated form of Eq. (4-349) is −∆G° ∆H0° − ∆G0° ∆H° 1 ln K = ᎏ = ᎏᎏ −ᎏ+ᎏ RT RT RT T

i ik

Summed over k, these equations give

Integration of Eq. (4-350) from reference temperature T0 (usually 298.15 K) to temperature T gives T0

 n a i

For an endothermic reaction ∆Hj° is positive; for an exothermic reaction it is negative. The temperature dependence of ∆ Hj° is given by d∆Hj° (4-350) ᎏ = ∆C°Pj dT

T

Multiply each element balance by λk, a Lagrange multiplier: λk

The effect of temperature on the equilibrium constant follows from Eq. (4-106): d(∆Gj°/RT) −∆Hj° (4-348) ᎏᎏ = ᎏ dT RT 2 The total derivative is appropriate here because property changes of reaction are functions of temperature only. In combination with Eq. (4-343) this gives d ln Kj ∆ Hj° (4-349) ᎏ=ᎏ dT RT 2



(k = 1, 2, . . . , w)

i

Thus, for the equilibrium mixture,

∆H° = ∆H0° + R

= Ak

i

(2 + ε)(1 + ε) ᎏᎏ = 1.38 (3 − ε)(1 − ε)

whence

4-33



T

T0

CP° ᎏ dT R

(4-352)

where ∆H°/RT is given by Eq. (4-351). In the more extensive compilations of data, values of ∆G° and ∆H° for formation reactions are given for a wide range of temperatures, rather than just at the reference temperature of 298.15 K. (See in particular TRC Thermodynamic Tables—Hydrocarbons and TRC Thermodynamic Tables—Non-hydrocarbons, serial publications of the Thermodynamics Research Center, Texas A & M University System, College Station, Tex.; “The NBS Tables of Chemical Thermodynamic Properties,” J. Physical and Chemical Reference Data, 11, supp. 2 [1982]. Where data are lacking, methods of estimation are available; these are reviewed by Reid, Prausnitz, and Poling, The Properties of Gases and Liquids, 4th ed., Chap. 6, McGraw-Hill, New York, 1987. For an estimation procedure based on molecular structure, see Constantinou and Gani, Fluid Phase Equilibria, 103, pp. 11–22 [1995]. (See also Sec. 2.) Complex Chemical-Reaction Equilibria When the composition of an equilibrium mixture is determined by a number of simultaneous reactions, calculations based on equilibrium constants become complex and tedious. A more direct procedure (and one suitable for general computer solution) is based on minimization of the total Gibbs energy Gt in accord with Eq. (4-271). The treatment here is

3.

i



− Ak = 0

Form a function F by addition of this sum to Gt: F = Gt +  λk k

 n a

i ik

i

− Ak



t

Function F is identical with G , because the summation term is zero. However, the partial derivatives of F and Gt with respect to ni are different, because function F incorporates the constraints of the material balances. 4. The minimum value of both F and Gt is found when the partial derivatives of F with respect to ni are set equal to zero: ∂F ∂Gt = ᎏ +  λkaik = 0 ᎏ ∂ni T,P,nj ∂ni T,P,nj k The first term on the right is the definition of the chemical potential; whence

 

 

µi +  λk aik = 0

(i = 1, 2, . . . , N)

(4-354)

k

However, the chemical potential is given by Eq. (4-341); for gas-phase reactions and standard states as the pure ideal gases at P°, this equation becomes fˆi µi = G°i + RT ln ᎏ P° If Gi° is arbitrarily set equal to zero for all elements in their standard states, then for compounds G°i = ∆Gf°, the standard Gibbs-energy i change of formation for species i. In addition, the fugacity is eliminated in favor of the fugacity coefficient by Eq. (4-79), fˆi = yiφˆ iP. With these substitutions, the equation for µi becomes yi φˆ iP µi = ∆Gf°i + RT ln ᎏ P° Combination with Eq. (4-354) gives yi φˆ iP ∆Gf°i + RT ln ᎏ +  λkaik = 0 P° k

(i = 1, 2, . . . , N)

(4-355)

If species i is an element, ∆Gf°i is zero. There are N equilibrium equations (Eqs. [4-355]), one for each chemical species, and there are w material-balance equations (Eqs. [4-353]), one for each element—a total of N + w equations. The unknowns in these equations are the ni (note that yi = ni / i ni), of which there are N, and the λk, of which there are w—a total of N + w unknowns. Thus, the number of equations is sufficient for the determination of all unknowns. Equation (4-355) is derived on the presumption that the φˆ i are known. If the phase is an ideal gas, then each φˆ i is unity. If the phase is an ideal solution, each φˆ i becomes φi, and can at least be estimated. For real gases, each φˆ i is a function of the yi , the quantities being calculated. Thus an iterative procedure is indicated, initiated with each φˆ i

4-34

THERMODYNAMICS

set equal to unity. Solution of the equations then provides a preliminary set of yi. For low pressures or high temperatures this result is usually adequate. Where it is not satisfactory, an equation of state with the preliminary yi gives a new and more nearly correct set of φˆ i for use in Eq. (4-355). Then a new set of yi is determined. The process is repeated to convergence. All calculations are well suited to computer solution. In this procedure, the question of what chemical reactions are involved never enters directly into any of the equations. However, the choice of a set of species is entirely equivalent to the choice of a set of independent reactions among the species. In any event, a set of species or an equivalent set of independent reactions must always be assumed, and different assumptions produce different results. Example 3: Minimization of Gibbs Energy Calculate the equilibrium compositions at 1,000 K and 1 bar of a gas-phase system containing the species CH4, H2O, CO, CO2, and H2. In the initial unreacted state there are present 2 mol of CH4 and 3 mol of H2O. Values of ∆Gf° at 1,000 K are ∆G°f CH = 19,720 J/mol

The five equations for the five species then become: CH4:

19,720 nCH λC 4λH ᎏ + ln ᎏ4 + ᎏ + ᎏ = 0 i ni RT RT RT

H2O:

−192,420 nH2O 2λH λO +ᎏ+ᎏ=0 ᎏᎏ + ln ᎏ i ni RT RT RT

CO:

−200,240 nCO λC λO ᎏᎏ + ln ᎏ + ᎏ + ᎏ = 0 i ni RT RT RT

CO2:

−395,790 nCO λC 2λO ᎏᎏ + ln ᎏ2 + ᎏ + ᎏ = 0 i ni RT RT RT

H2:

nH2 2λH ln ᎏ +ᎏ=0 i ni RT

The three material-balance equations (Eq. [4-353]) are:

∆G°f H O = −192,420 J/mol

C:

nCH4 + nCO + nCO2 = 2

∆G°f CO = −200,240 J/mol

H:

4nCH4 + 2nH2O + 2nH2 = 14

∆G°f CO = −395,790 J/mol

O:

nH2O + nCO + 2nCO2 = 3

4

2

2

The required values of Ak are determined from the initial numbers of moles, and the values of aik come directly from the chemical formulas of the species. These are shown in the accompanying table.

Simultaneous computer solution of these eight equations, with RT = 8,314 J/mol and

n = n i

Carbon

Oxygen

Hydrogen

Ak = no. of atomic masses of k in the system AC = 2 Species i CH4 H2O CO CO2 H2

AO = 3

AH = 14

aik = no. of atoms of k per molecule of i aCH4,C = 1 aH2O,C = 0 aCO,C = 1 aCO2,C = 1 aH2,C = 0

aCH4,O = 0 aH2O,O = 1 aCO,O = 1 aCO2,O = 2 aH2,O = 0

CH4

+ nH2O + nCO + nCO2 + nH2

i

Element k

aCH4,H = 4 aH2O,H = 2 aCO,H = 0 aCO2,H = 0 aH2,H = 2

At 1 bar and 1,000 K the assumption of ideal gases is justified, and the φˆ i are all unity. Since P = 1 bar, Eq. (4-355) is written: ∆Gf° ni λk ᎏi + ln ᎏ +  ᎏ aik = 0 RT Σi ni RT k

produces the following results (yi = ni/i ni): yCH4 = 0.0196

λC ᎏ = 0.7635 RT

yH2O = 0.0980 yCO = 0.1743

λO ᎏ = 25.068 RT

yCO2 = 0.0371 yH2 = 0.6711

λH ᎏ = 0.1994 RT

 y = 1.000 i

i

The values of λk/RT are of no significance, but are included to make the results complete.

THERMODYNAMIC ANALYSIS OF PROCESSES Real irreversible processes can be subjected to thermodynamic analysis. The goal is to calculate the efficiency of energy use or production and to show how energy loss is apportioned among the steps of a process. The treatment here is limited to steady-state, steady-flow processes, because of their predominance in chemical technology. CALCULATION OF IDEAL WORK In any steady-state, steady-flow process requiring work, a minimum amount must be expended to bring about a specific change of state in the flowing fluid. In a process producing work, a maximum amount is attainable for a specific change of state in the flowing fluid. In either case, the limiting value obtains when the specific change of state is accomplished completely reversibly. The implications of this requirement are: 1. The process is internally reversible within the control volume. 2. Heat transfer external to the control volume is reversible. The second item means that heat exchange between system and surroundings must occur at the temperature of the surroundings, presumed to constitute a heat reservoir at a constant and uniform temperature

Tσ. This may require Carnot engines or heat pumps internal to the system that provide for the reversible transfer of heat from the temperature of the flowing fluid to that of the surroundings. Since Carnot engines and heat pumps are cyclic, they undergo no net change of state. The entropy change of the surroundings, found by integration of Eq. (4-3), is ∆Sσ = Qσ/Tσ; whence Qσ = Tσ ∆Sσ (4-356) Since heat transfer with respect to the surroundings and with respect to the system are equal but of opposite sign, Qσ = −Q. Moreover, the second law requires for a reversible process that the entropy changes of system and surroundings be equal but of opposite sign: ∆Sσ = −∆St. Equation (4-356) can therefore be written Q = Tσ∆St. In terms of rates this becomes ˙ = Tσ ∆(Sm) Q ˙ fs (4-357) ˙ where Q = rate of heat transfer with respect to the system m ˙ = mass rate of flow of fluid In addition, ∆ denotes the difference between exit and entrance streams, and fs indicates that the term applies to all flowing streams.

THERMODYNAMIC ANALYSIS OF PROCESSES The energy balance for a steady-state steady-flow process resulting from the first law of thermodynamics is 1 ˙ +W ˙s ∆ H + ᎏ u2 + zg m ˙ =Q (4-358) fs 2 where H = specific enthalpy of flowing fluid u = velocity of flowing fluid z = elevation of flowing fluid above datum level g = local acceleration of gravity Ws = shaft work ˙ in Eq. (4-358) by Eq. (4-357) gives Eliminating Q



 



 

1 ˙ s(rev) ∆ H + ᎏ u2 + zg m ˙ = Tσ ∆(Sm) ˙ fs + W fs 2 ˙ s(rev) indicates that the shaft work is for a completely where W ˙ ideal. Thus reversible process. This work is called the ideal work W 1 2 ˙ ideal = ∆ H + ᎏ W u + zg m ˙ − Tσ ∆(Sm) ˙ fs (4-359) fs 2 In most applications to chemical processes, the kinetic- and potential-energy terms are negligible compared with the others; in this event Eq. (4-359) is written ˙ ideal = ∆(Hm) W ˙ fs − Tσ ∆(Sm) ˙ fs (4-360)



 

For the special case of a single stream flowing through the system, Eq. (4-360) becomes ˙ ideal = m(∆H W ˙ − Tσ ∆S) (4-361) Division by m ˙ puts this equation on a unit-mass basis Wideal = ∆H − Tσ ∆S (4-362) A completely reversible processes is hypothetical, devised solely to find the ideal work associated with a given change of state. Its only connection with an actual process is that it brings about the same change of state as the actual process, allowing comparison of the actual work of a process with the work of the hypothetical reversible process. Equations (4-359) through (4-362) give the work of a completely reversible process associated with given property changes in the flowing streams. When the same property changes occur in an actual ˙ s (or Ws) is given by an energy balance, and process, the actual work W comparison can be made of the actual work with the ideal work. When ˙ ideal (or Wideal) is positive, it is the minimum work required to bring W about a given change in the properties of the flowing streams, and is ˙ s. In this case a thermodynamic efficiency ηt is defined smaller than W as the ratio of the ideal work to the actual work: ˙ ideal W ηt (work required) = ᎏ (4-363) ˙s W ˙ ideal| is the maximum work ˙ ideal (or Wideal) is negative, | W When W obtainable from a given change in the properties of the flowing ˙ s|. In this case, the thermodynamic effistreams, and is larger than |W ciency is defined as the ratio of the actual work to the ideal work: ˙s W ηt(work produced) = ᎏ (4-364) ˙ ideal W LOST WORK Work that is wasted as the result of irreversibilities in a process is ˙ lost, and is defined as the difference between the called lost work W actual work of a process and the ideal work for the process. Thus, by definition, Wlost
Ws − Wideal (4-365) In terms of rates this is written ˙ lost
W ˙ s −W ˙ ideal W (4-366) The actual work rate comes from Eq. (4-358) 1 2 ˙ s=∆ H+ᎏ ˙ W u + zg m ˙ −Q fs 2 Subtracting the ideal work rate as given by Eq. (4-359) yields ˙ ˙ lost = Tσ∆(Sm) ˙ fs − Q (4-367) W



 

4-35

For the special case of a single stream flowing through the control volume, ˙ lost = mT ˙ W ˙ σ∆S − Q (4-368) Division of this equation by m ˙ gives Wlost = Tσ∆S − Q

(4-369)

where the basis is now a unit amount of fluid flowing through the control volume. The total rate of entropy increase (in both system and surroundings) as a result of a process is ˙ Q S˙ total = ∆(Sm) ˙ fs − ᎏ (4-370) Tσ For a single stream, division by m ˙ provides an equation based on a unit amount of fluid flowing through the control volume: Q Stotal = ∆S − ᎏ (4-371) Tσ Multiplication of Eq. (4-370) by Tσ gives ˙ ˙ fs − Q TσS˙ total = Tσ∆(Sm) Since the right-hand sides of this equation and of Eq. (4-367) are identical, it follows that ˙ lost = TσS˙ total W (4-372) For flow of a single stream on the basis of a unit amount of fluid, this becomes Wlost = TσStotal

(4-373)

Since the second law of thermodynamics requires that S˙ total ≥ 0 and Stotal ≥ 0 it follows that ˙ lost ≥ 0 W

and

Wlost ≥ 0

When a process is completely reversible, the equality holds, and the lost work is zero. For irreversible processes the inequality holds, and the lost work, that is, the energy that becomes unavailable for work, is positive. The engineering significance of this result is clear: The greater the irreversibility of a process, the greater the rate of entropy production and the greater the amount of energy that becomes unavailable for work. Thus, every irreversibility carries with it a price. ANALYSIS OF STEADY-STATE, STEADY-FLOW PROCESSES Many processes consist of a number of steps, and lost-work calculations are then made for each step separately. Writing Eq. (4-372) for each step of the process and summing gives

 W˙ lost = Tσ  S˙ total Dividing Eq. (4-372) by this result yields ˙ lost S˙ total W =ᎏ ᎏ ˙ lost  S˙ total W Thus, an analysis of the lost work, made by calculation of the fraction that each individual lost-work term represents of the total lost work, is the same as an analysis of the rate of entropy generation, made by expressing each individual entropy-generation term as a fraction of the sum of all entropy-generation terms. An alternative to the lost-work or entropy-generation analysis is a work analysis. This is based on Eq. (4-366), written ˙ lost = W ˙ s −W ˙ ideal W (4-374) For a work-requiring process, all of these work quantities are positive ˙ s >W ˙ ideal. The preceding equation is then expresed as and W ˙ ideal +  W ˙ lost ˙ s =W (4-375) W A work analysis gives each of the individual work terms on the right as ˙ s. a fraction of W

4-36

THERMODYNAMICS TABLE 4-2

States and Values of Properties for the Process of Fig. 4-12*

Point

P, bar

T, K

Composition

State

H, J/mol

S, J/(mol ⋅ K)

1 2 3 4 5 6 7

55.22 1.01 1.01 55.22 1.01 1.01 1.01

300 295 295 147.2 79.4 90 300

Air Pure O2 91.48% N2 Air 91.48% N2 pure O2 Air

Superheated Superheated Superheated Superheated Saturated vapor Saturated vapor Superheated

12,046 13,460 12,074 5,850 5,773 7,485 12,407

82.98 118.48 114.34 52.08 75.82 83.69 117.35

*Properties on the basis of Miller and Sullivan, U.S. Bur. Mines Tech. Pap. 424 (1928).

˙ s and W ˙ ideal are negative, and For a work-producing process, W ˙ ideal | > |W ˙ s|. Equation (4-374) in this case is best written: |W ˙ s| + ˙ lost ˙ ideal| = |W W (4-376) |W



A work analysis here expresses each of the individual work terms on ˙ ideal|. A work analysis cannot be carried out the right as a fraction of |W ˙ ideal is negative, indiin the case where a process is so inefficient that W ˙ s is positive, indicating that the process should produce work, but W cating that the process in fact requires work. A lost-work or entropy-generation analysis is always possible. Example 4: Lost-Work Analysis Make a work analysis of a simple Linde system for the separation of air into gaseous oxygen and nitrogen, as depicted in Fig. 4-12. Table 4-2 lists a set of operating conditions for the numbered points of the diagram. Heat leaks into the column of 147 J/mol of entering air and into the exchanger of 70 J/mol of entering air have been assumed. Take Tσ = 300 K. The basis for analysis is 1 mol of entering air, assumed to contain 79 mol % N2 and 21 mol % O2. By a material balance on the nitrogen, 0.79 = 0.9148 x; whence x = 0.8636 mol of nitrogen product 1 − x = 0.1364 mol of oxygen product Calculation of Ideal Work If changes in kinetic and potential energies are neglected, Eq. (4-360) is applicable. From the tabulated data, ∆(Hm) ˙ fs = (13,460)(0.1364) + (12,074)(0.8636) − (12,407)(1) = −144 J ∆(Sm) ˙ fs = (118.48)(0.1364) + (114.34)(0.8636) − (117.35)(1) = −2.4453 J/K

Thus, by Eq. (4-360), ˙ ideal = −144 − (300)(−2.4453) = 589.6 J W Calculation of Actual Work of Compression For simplicity, the work of compression is calculated by the equation for an ideal gas in a three-stage reciprocating machine with complete intercooling and with isentropic compression in each stage. The work so calculated is assumed to represent 80 percent of the actual work. The following equation may be found in any number of textbooks on thermodynamics: nγRT1 P2 (γ − 1)/nγ ˙ s = ᎏᎏ W −1 ᎏ (0.8)(γ − 1) P1

 

where



n = number of stages, here taken as 3 γ = ratio of heat capacities, here taken as 1.4 T1 = initial absolute temperature, 300 K P2 /P1 = overall pressure ratio, 54.5 R = universal gas constant, 8.314 J/(mol⋅K)

The efficiency factor of 0.8 is already included in the equation. Substitution of the remaining values gives (3)(1.4)(8.314)(300) ˙ s = ᎏᎏᎏ W (54.5)0.4/(3)(1.4) − 1 = 15,171 J (0.8)(0.4) The heat transferred to the surroundings during compression as a result of intercooling and aftercooling to 300 K is found from the first law:





Diagram of simple Linde system for air separation.

FIG. 4-12

˙ s = (12,046 − 12,407) − 15,171 = −15,532 J Q˙ = m(∆H) ˙ −W Calculation of Lost Work Equation (4-367) may be applied to each of the major units of the process. For the compressor/cooler, ˙ lost = (300)[(82.98)(1) − (117.35)(1)] − (−15,532) W = 5,221.0 J For the exchanger, ˙ lost = (300)[(118.48)(0.1364) + (114.34)(0.8636) + (52.08)(1) W − (75.82)(0.8636) − (83.69)(0.1364) − (82.98)(1)] − 70 = 2,063.4 J Finally, for the rectifier, ˙ lost = (300)[(75.82)(0.8636) + (83.69)(0.1364) − (52.08)(1)] − 147 W = 7,297.0 J Work Analysis Since the process requires work, Eq. (4-375) is appropriate for a work analysis. The various terms of this equation appear as entries in the following table, and are on the basis of 1 mol of entering air. ˙s % of W ˙ ideal W ˙ lost: W ˙ lost: W ˙ lost: W ˙s W

Compressor/cooler Exchanger Rectifier

589.6 J 5,221.0 J 2,063.4 J 7,297.0 J

3.9 34.4 13.6 48.1

15,171.0 J

100.0

The thermodynamic efficiency of this process as given by Eq. (4-363) is only 3.9 percent. Significant inefficiencies reside with each of the primary units of the process.

Section 5

Heat and Mass Transfer*

James G. Knudsen, Ph.D., Professor Emeritus of Chemical Engineering, Oregon State University; Member, American Institute of Chemical Engineers, American Chemical Society; Registered Professional Engineer (Oregon). (Conduction and Convection; Condensation, Boiling; Section Coeditor) Hoyt C. Hottel, S.M., Professor Emeritus of Chemical Engineering, Massachusetts Institute of Technology; Member, National Academy of Sciences, American Academy of Arts and Sciences, American Institute of Chemical Engineers, American Chemical Society, Combustion Institute. (Radiation) Adel F. Sarofim, Sc.D., Lammot du Pont Professor of Chemical Engineering and Assistant Director, Fuels Research Laboratory, Massachusetts Institute of Technology; Member, American Institute of Chemical Engineers, American Chemical Society, Combustion Institute. (Radiation) Phillip C. Wankat, Ph.D., Professor of Chemical Engineering, Purdue University; Member, American Institute of Chemical Engineers, American Chemical Society, International Adsorption Society. (Mass Transfer Section Coeditor) Kent S. Knaebel, Ph.D., President, Adsorption Research, Inc.; Member, American Institute of Chemical Engineers, American Chemical Society, International Adsorption Society. Professional Engineer (Ohio). (Mass Transfer Section Coeditor)

HEAT TRANSFER Modes of Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5-8

HEAT TRANSFER BY CONVECTION

HEAT TRANSFER BY CONDUCTION Fourier’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Three-Dimensional Conduction Equation . . . . . . . . . . . . . . . . . . . . . . . Thermal Conductivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Steady-State Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . One-Dimensional Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conduction through Several Bodies in Series . . . . . . . . . . . . . . . . . . . Conduction through Several Bodies in Parallel. . . . . . . . . . . . . . . . . . Several Bodies in Series with Heat Generation . . . . . . . . . . . . . . . . . Example 1. Steady-State Conduction with Heat Generation . . . . . . . Two-Dimensional Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Unsteady-State Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . One-Dimensional Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Two-Dimensional Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conduction with Change of Phase. . . . . . . . . . . . . . . . . . . . . . . . . . . .

5-8 5-8 5-9 5-9 5-9 5-9 5-10 5-10 5-10 5-10 5-10 5-10 5-11 5-11

Coefficient of Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Energy Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Individual Coefficient of Heat Transfer. . . . . . . . . . . . . . . . . . . . . . . . Overall Coefficient of Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . Representation of Heat-Transfer Film Coefficients . . . . . . . . . . . . . . Natural Convection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nusselt Equation for Various Geometries . . . . . . . . . . . . . . . . . . . . . . Simplified Dimensional Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . Simultaneous Loss by Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Enclosed Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Forced Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Analogy between Momentum and Heat Transfer . . . . . . . . . . . . . . . . Laminar Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Transition Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Turbulent Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 2. Calculation of j Factors in an Annulus . . . . . . . . . . . . . . . Jackets and Coils of Agitated Vessels . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5-12 5-12 5-12 5-12 5-12 5-12 5-12 5-12 5-12 5-12 5-14 5-14 5-15 5-16 5-16 5-17 5-19

* The contribution to the section on Interphase Mass Transfer of Mr. William M. Edwards (editor of Sec. 14), who was an author for the sixth edition, is acknowledged. 5-1

5-2

HEAT AND MASS TRANSFER

Nonnewtonian Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Liquid Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5-19 5-19

HEAT TRANSFER WITH CHANGE OF PHASE Condensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Condensation Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Condensation Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Boiling (Vaporization) of Liquids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Boiling Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Boiling Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5-20 5-20 5-20 5-22 5-22 5-22

HEAT TRANSFER BY RADIATION General References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nomenclature for Radiative Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . Nature of Thermal Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Blackbody Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Radiative Exchange between Surfaces and Solids . . . . . . . . . . . . . . . . . Emittance and Absorptance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Black-Surface Enclosures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 3: Calculation of View Factor . . . . . . . . . . . . . . . . . . . . . . . . Example 4: Calculation of Exchange Area. . . . . . . . . . . . . . . . . . . . . . Non-Black Surface Enclosures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 5: Radiation in a Furnace Chamber . . . . . . . . . . . . . . . . . . . Emissivities of Combustion Products . . . . . . . . . . . . . . . . . . . . . . . . . . . Gaseous Combustion Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 6: Calculation of Gas Emissivity and Absorptivity . . . . . . . . Flames and Particle Clouds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Radiative Exchange between Gases or Suspended Matter and a Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 7: Radiation in Gases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Single-Gas-Zone/Two-Surface-Zone Systems . . . . . . . . . . . . . . . . . . . The Effect of Nongrayness of Gas on Total-Exchange Area . . . . . . . Example 8: Effective Gas Emissivity . . . . . . . . . . . . . . . . . . . . . . . . . . Treatment of Refractory Walls Partially Enclosing a Radiating Gas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Combustion Chamber Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 9: Radiation in a Furnace . . . . . . . . . . . . . . . . . . . . . . . . . . . MASS TRANSFER General References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fick’s First Law. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mutual Diffusivity, Mass Diffusivity, Interdiffusion Coefficient . . . .

5-23 5-23 5-24 5-24 5-25 5-25 5-27 5-29 5-29 5-29 5-31 5-32 5-33 5-34 5-35 5-36 5-36 5-37 5-37 5-38 5-39 5-40 5-41

5-42 5-42 5-42 5-46

Self Diffusivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tracer Diffusivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mass-Transfer Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problem Solving Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Continuity and Flux Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Material Balances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flux Expressions: Simple Integrated Forms of Fick’s First Law . . . . Stefan-Maxwell Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Diffusivity Estimation—Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Binary Mixtures—Low Pressure—Nonpolar Components . . . . . . . . Binary Mixtures—Low Pressure—Polar Components . . . . . . . . . . . . Self-Diffusivity—High Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Supercritical Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Low-Pressure/Multicomponent Mixtures . . . . . . . . . . . . . . . . . . . . . . Diffusivity Estimation—Liquids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stokes-Einstein and Free-Volume Theories . . . . . . . . . . . . . . . . . . . . Dilute Binary Nonelectrolytes: General Mixtures . . . . . . . . . . . . . . . Binary Mixtures of Gases in Low-Viscosity, Nonelectrolyte Liquids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dilute Binary Mixtures of a Nonelectrolyte in Water . . . . . . . . . . . . . Dilute Binary Hydrocarbon Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . Dilute Binary Mixtures of Nonelectrolytes with Water as the Solute. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dilute Dispersions of Macromolecules in Nonelectrolytes . . . . . . . . Concentrated, Binary Mixtures of Nonelectrolytes . . . . . . . . . . . . . . Binary Electrolyte Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multicomponent Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Diffusion of Fluids in Porous Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interphase Mass Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mass-Transfer Principles: Dilute Systems . . . . . . . . . . . . . . . . . . . . . . Mass-Transfer Principles: Concentrated Systems . . . . . . . . . . . . . . . . HTU (Height Equivalent to One Transfer Unit) . . . . . . . . . . . . . . . . NTU (Number of Transfer Units) . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definitions of Mass-Transfer Coefficients ˆkG and ˆkL . . . . . . . . . . . . . Simplified Mass-Transfer Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . Mass-Transfer Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effects of Total Pressure on ˆkG and ˆkL . . . . . . . . . . . . . . . . . . . . . . . . . Effects of Temperature on ˆkG and ˆkL . . . . . . . . . . . . . . . . . . . . . . . . . . Effects of System Physical Properties on ˆkG and ˆkL . . . . . . . . . . . . . . Effects of High Solute Concentrations on ˆkG and ˆkL . . . . . . . . . . . . . Influence of Chemical Reactions on ˆkG and ˆkL . . . . . . . . . . . . . . . . . . Effective Interfacial Mass-Transfer Area a . . . . . . . . . . . . . . . . . . . . . Volumetric Mass-Transfer Coefficients Kˆ Ga and Kˆ La . . . . . . . . . . . . . Chilton-Colburn Analogy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5-46 5-46 5-46 5-46 5-46 5-46 5-47 5-47 5-48 5-48 5-49 5-49 5-49 5-50 5-50 5-50 5-50 5-51 5-52 5-52 5-52 5-52 5-52 5-53 5-54 5-54 5-54 5-54 5-56 5-57 5-57 5-57 5-57 5-58 5-61 5-64 5-66 5-69 5-69 5-74 5-78 5-79

HEAT AND MASS TRANSFER

5-3

Nomenclature and Units Specialized heat transfer nomenclature used for radiative heat transfer is defined in the subsection “Heat Transmission by Radiation.” Nomenclature for mass transfer is defined in the subsection “Mass Transfer.” Symbol a ax a′ A

b b′ bf B c1, c2, etc. c, cp C Cr dm D Dc De Dj Dotl Dp Ds Dt D1, D2 EH EM f F Fa Fg Fc Ft Fs Fw FD FT g, gL gc G Gmax Gmf h ham, hlm hb hcg hc + hr

Definition Proportionality coefficient Cross-sectional area of a fin Proportionality factor Area of heat transfer surface; Ai for inside; Ao for outside; Am for mean; Aavg for average; A1, A2, and A3 for points 1, 2, and 3 respectively; AB for bare surface of finned tube; Af for finned portion of tube; Auf for external area of unfinned portion of finned tube; Aof for external area of finned tube before fins are attached, equals Ao; Aoe for effective area of finned surface; AT for total external area of finned tube; Ad for surface area of dirt (scale) deposit Proportionality coefficient Proportionality factor Height of fin Material constant = 5D−0.5 Constants of integration Specific heat at constant pressure; cs for specific heat of solid; cg for specific heat of gas Thermal conductance, equals kA/x, hA, or UA; C1, C2, C3, Cn, thermal conductance of sections 1, 2, 3, and n respectively of a composite body Correlating constant; proportionality coefficient Depth of divided solids bed Diameter; Do for outside; Di for inside; Dr for root diameter of finned tube Diameter of a coil or helix Equivalent diameter of a cross section, usually 4 times free area divided by wetted perimeter; Dw for equivalent diameter of window Diameter of a jacketed cylindrical vessel Outside diameter of tube bundle Diameter of packing in a packed tube Inside diameter of heat-exchanger shell Solids-processing vessel diameter Diameter at points 1 and 2 respectively; inner and outer diameter of annulus respectively Eddy conductivity of heat Eddy viscosity Fanning friction factor; f1 for inner wall and f2 for outer wall of annulus; fk for ideal tube bank; skin friction drag coefficient Entrance factors Dry solids feed rate Gas volumetric flow rate Fraction of total tubes in cross-flow; Fbp for fraction of cross-flow area available for bypass flow Factor, ratio of temperature difference across tube-side film to overall mean temperature difference Factor, ratio of temperature difference across shell-side film to overall mean temperature difference Factor, ratio of temperature difference across retaining wall to overall mean temperature difference between bulk fluids Factor, ratio of temperature difference across combined dirt or scale films to overall mean temperature difference between bulk fluids Temperature-difference correction factor Acceleration due to gravity Conversion factor Mass velocity, equals V␳ or W/S; Gv for vapor mass velocity Mass velocity through minimum free area between rows of tubes normal to the fluid stream Minimum fluidizing mass velocity Local individual coefficient of heat transfer, equals dq/(dA)(∆T) Film coefficient based on arithmetic-mean temperature difference and logarithmic-mean temperature difference respectively Film coefficient delivered at base of fin Effective combined coefficient for simultaneous gas-vapor cooling and vapor condensation Combined coefficient for conduction, convection, and radiation between surface and surroundings

SI units

U.S. customary units

Dimensionless m2

Dimensionless ft2

m2

ft2

m

ft

J/(kg⋅K)

Btu/(lb⋅°F)

J/(s⋅K)

Btu/(h⋅°F)

Dimensionless m m

Dimensionless ft ft

m m

ft ft

m m m m m m

ft ft ft ft ft ft

J/(s⋅m⋅K) Pa⋅s Dimensionless

Btu/(h⋅ft⋅°F) lb/(ft⋅h) Dimensionless

kg/(s⋅m2) m3 (s⋅m2 of bed area)

lb/(h⋅ft2) ft3/(h⋅ft2 of bed area)

Dimensionless

Dimensionless

Dimensionless

Dimensionless

Dimensionless

Dimensionless

Dimensionless

Dimensionless

981 m/s2 1.0 (kg⋅m)/(N⋅s2) kg/(m2⋅s) kg/(m2⋅s)

(4.18)(108) ft/h2 (4.17)(108)(lb⋅ft)/(lbf⋅h2) lb/(h⋅ft2) lb/(h⋅ft2)

kg/(m2⋅s) J/(m2⋅s⋅K) J/(m2⋅s⋅K)

lb/(h⋅ft2) Btu/(h⋅ft2⋅°F) Btu/(h⋅ft2⋅°F)

J/(m2⋅s⋅K) J/(m2⋅s⋅K)

Btu/(h⋅ft2⋅°F) Btu/(h⋅ft2⋅°F)

J/(m2⋅s⋅K)

Btu/(h⋅ft2⋅°F)

5-4

HEAT AND MASS TRANSFER

Nomenclature and Units (Continued ) Symbol hdo, hdi hf hfi hfo hF, hs hi, ho hk h1 h′ hlm hr hT hw j J Jb, Jc, Jl, Jr k kv kavg, km kf kw K′ lc L Lo Lu LF LH Lp m M M n nt nr n′ nb Nr N

NB Nd NGr NNu NPe NPr NRe NSt Nss p pf p, p′

Definition Film coefficient for dirt or scale on outside or inside respectively of a surface Film coefficient for finned-tube exchangers based on total external surface Effective outside film coefficient of a finned tube based on inside area Film coefficient for air film of an air-cooled finned-tube exchanger based on external bare surface Effective film coefficient for dirt or scale on heat-transfer surface Film coefficient for heat transfer for inside and outside surface respectively Film coefficient for ideal tube bank; hs for shell side of baffled exchanger; hsv for coefficient at liquid-vapor interface Condensing coefficient on top tube; hN coefficient for N tubes in a vertical row Film coefficient for enclosed spaces Film coefficient based on log-mean temperature difference Heat-transfer coefficient for radiation Coefficient of total heat transfer by conduction, convection, and radiation between the surroundings and the surface of a body subject to unsteady-state heat transfer Equivalent coefficient of retaining wall, equals k/x Ordinate, Colburn j factor, equals f/2; jH for heat transfer; jH1 for inner wall of annulus; jH2 for outer wall of annulus; jk for heat transfer for ideal tube bank Mechanical equivalent of heat Correction factors for baffle bypassing, baffle configuration, baffle leakage, and adverse temperature gradient respectively Thermal conductivity; k1, k2, k3, thermal conductivities of bodies 1, 2, and 3 Thermal conductivity of vapor; k1 for liquid thermal conductivity; ks for thermal conductivity of solid Mean thermal conductivity Thermal conductivity of fluid at film temperature Thermal conductivity of retaining-wall material Property of non-Newtonian fluid Baffle cut; ls for baffle spacing Length of heat-transfer surface Flow rate Undisturbed length of path of fluid flow Thickness of dirt or scale deposit Depth of fluidized bed Diameter of agitator blade Ratio, term, or exponent as defined where used Molecular weight Weight of fluid Position ratio or number Number of tubes in parallel in a heat exchanger Number of rows in a vertical plane Flow-behavior index for nonnewtonian fluids Number of baffle-type coils Speed of agitator Number of tubes in a vertical row; or number of tubes in a bundle; Nb for number of baffles; NT for total number of tubes in exchanger; Nc for number of tubes in one cross-flow section; Ncw for number of cross-flow rows in each window Biot number, hT ∆x/k Proportionality coefficient, dimensionless group Grashof number, L3ρ2gβ ∆t/µ2 Nusselt number, hD/k or hL/k Peclet number, DGc/k Prandtl number, cµ/k Reynolds number, DG/µ Stanton number, NNu/NRe NPr Number of sealing strips Pressure Perimeter of a fin Center-to-center spacing of tubes in tube bundle (tube pitch); pn for tube pitch normal to flow; pp for tube pitch parallel to flow

SI units

U.S. customary units

J/(m2⋅s⋅K)

Btu/(h⋅ft2⋅°F)

J/(m2⋅s⋅K)

Btu/(h⋅ft2⋅°F)

J/(m2⋅s⋅K) J/(m2⋅s⋅K)

Btu/(h⋅ft2⋅°F) Btu/(h⋅ft2⋅°F)

J/(m2⋅s⋅K) J/(m2⋅s⋅K)

Btu/(h⋅ft2⋅°F) Btu/(h⋅ft2⋅°F)

J/(m2⋅s⋅K)

Btu/(h⋅ft2⋅°F)

J/(m2⋅s⋅K)

Btu/(h⋅ft2⋅°F)

J/(m2⋅s⋅K) J/(m2⋅s⋅K) J/(m2⋅s⋅K) J/(m2⋅s⋅K)

Btu/(h⋅ft2⋅°F) Btu/(h⋅ft2⋅°F) Btu/(h⋅ft2⋅°F) Btu/(h⋅ft2⋅°F)

J/(m2⋅s⋅K) Dimensionless

Btu/(h⋅ft2⋅°F) Dimensionless

1.0(N⋅m)/J

778(ft⋅lbf)/Btu

J/(m⋅s⋅K)

(Btu⋅ft)/(h⋅ft2⋅°F)

J/(m⋅s⋅K)

(Btu⋅ft)/(h⋅ft2⋅°F)

J/(m⋅s⋅K) J/(m⋅s⋅K) J/(m⋅s⋅K)

(Btu⋅ft)/(h⋅ft2⋅°F) (Btu⋅ft)/(h⋅ft2⋅°F) (Btu⋅ft)/(h⋅ft2⋅°F)

m m kg/s m m m m

ft ft lb/h ft ft ft ft

kg/mol kg Dimensionless

lb/mol lb Dimensionless

rad/s

r/h

Dimensionless

Dimensionless

kPa m m

lbf/ft2 abs ft ft

HEAT AND MASS TRANSFER

5-5

Nomenclature and Units (Continued ) Symbol ∆p P P′ ∆Pbk, ∆Pwk q q′ (q/A)max Q Q r

rj R

Rj S

Sr s t t1, t2, tn t′ t′1, t′2 t″1, t″2 tb tH, tL ts tsv tw t∞ TH, TL T ∆T, ∆t

∆tam, ∆tlm ∆tom ∆TH, ∆tH ∆TL, ∆tL ∆Tm, ∆tm u u° U U1, U2 Uco, Ucv, Uct, Ura Um v Vr

Definition Pressure of the vapor in a bubble minus saturation pressure of a flat liquid surface Absolute pressure; Pc for critical pressure Spacing between adjacent baffles on shell side of a heat exchanger (baffle pitch) Pressure drop for ideal-tube-bank cross-flow and ideal window respectively; ∆Ps for shell side of baffled exchanger Rate of heat flow, equals Q/θ Rate of heat generation Maximum heat flux in nucleate boiling Quantity of heat; rate of heat transfer Quantity of heat; QT for total quantity Radius; cylindrical and spherical coordinate; distance from midplane to a point in a body; r1 for inner wall of annulus; r2 for outer wall of annulus; ri for inside radius of tube; rm for distance from midplane or center of a body to the exterior surface of the body Inside radius Thermal resistance, equals x/kA, 1/UA, 1/hA; R1, R2, R3, Rn for thermal resistance of sections 1, 2, 3, and n of a composite body; RT for sum of individual resistances of several resistances in series or parallel; Rdi and Rdo for dirt or scale resistance on inner and outer surface respectively Ratio of total outside surface of finned tube to area of tube having same root diameter Cross-sectional area; Sm for minimum cross-sectional area between rows of tubes, flow normal to tubes; Stb for tube-to-baffle leakage area for one baffle; Ssb for shell-to-baffle area for one baffle; Sw for area for flow through window; Swg for gross window area; Swt for window area occupied by tubes Slope of rotary shell Specific gravity of fluid referred to liquid water Bulk temperature; temperature at a given point in a body at time θ Temperature at points 1, 2, and n in a system through which heat is being transferred Temperature of surroundings Inlet and outlet temperature respectively of hotter fluid Inlet and outlet temperature respectively of colder fluid Initial uniform bulk temperature of a body; bulk temperature of a flowing fluid High and low temperature respectively on tube side of a heat exchanger Surface temperature Saturated-vapor temperature Wall temperature Temperature of undisturbed flowing stream High and low temperature respectively on shell side of a heat exchanger Absolute temperature; Tb for bulk temperature; Tw for wall temperature; Tv for vapor temperature; Tc for coolant temperature; Te for temperature of emitter; Tr for temperature of receiver Temperature difference; ∆t1, ∆t2, and ∆t3 temperature difference across bodies 1, 2, and 3 or at points 1, 2, and 3; ∆To, ∆to for overall temperature difference; ∆tb for temperature difference between surface and boiling liquid Arithmetic- and logarithmic-mean temperature difference respectively Mean effective overall temperature difference Greater terminal temperature difference Lesser terminal temperature difference Mean temperature difference Velocity in x direction Friction velocity Overall coefficient of heat transfer; Uo for outside surface basis; U′ for overall coefficient between liquid-vapor interface and coolant Overall coefficient of heat transfer at points 1 and 2 respectively Overall coefficients for divided solids processing by conduction, convection, contact, and radiation mechanism respectively Mean overall coefficient of heat transfer Velocity in y direction Volume of rotating shell

SI units

U.S. customary units

kPa

lbf/ft2 abs

kPa m

lbf/ft2 ft

kPa

lbf/ft2

W, J/s J/(s⋅m3) J/(s⋅m2) J/s J m

Btu/h Btu/(h⋅ft3) Btu/(h⋅ft2) Btu/h Btu ft

Dimensionless (s⋅K)/J

Dimensionless (h⋅°F)/Btu

m2

ft2

K K

°F °F

K K K K

°F °F °F °F

K

°F

K K K K K

°F °F °F °F °F

K

°R

K

°F, °R

K K K K K m/s m/s J/(s⋅m2⋅K)

°F °F °F °F °F ft/h ft/h Btu/(h⋅ft2⋅°F)

J/(s⋅m2⋅K) J/(s⋅m2⋅K)

Btu/(h⋅ft2⋅°F) Btu/(h⋅ft2⋅°F)

J/(s⋅m2⋅K) m/s m3

Btu/(h⋅ft2⋅°F) ft/h ft3

5-6

HEAT AND MASS TRANSFER

Nomenclature and Units (Concluded ) Symbol V V′, Vs VF Vg, Vl V′max w w W Wr W1, Wo xq x X y y+ Y z zp ZH

Definition

SI units

Velocity Velocity Face velocity of a fluid approaching a bank of finned tubes Specific volume of gas, liquid Maximum velocity through minimum free area between rows of tubes normal to the fluid stream Velocity in z direction Flow rate Total mass rate of flow; mass rate of vapor generated; WF for total rate of vapor condensation in one tube Weight rate of flow Total mass rate of flow on tube side and shell side respectively of a heat exchanger Vapor quality, xi for inlet quality, xo for outlet quality Coordinate direction; length of conduction path; xs for thickness of scale; x1, x2, and x3 at positions 1, 2, and 3 in a body through which heat is being transferred Factor Coordinate direction Wall distance Factor Coordinate direction Distance (perimeter) traveled by fluid across fin Ratio of sensible heat removed from vapor to total heat transferred

U.S. customary units

m/s m/s m/s m3/kg m/s

ft/h ft/s ft/h ft3/lb ft/h

m/s kg/s kg/s

ft/h lb/h lb/h

kg/(s⋅tube) kg/s

lb/(h⋅tube) lb/h

kg/s m

lb/h ft

Dimensionless m Dimensionless Dimensionless m m Dimensionless

Dimensionless ft Dimensionless Dimensionless ft ft Dimensionless

m2/s

ft2/h

K−1 ° kg/(s2 − n′⋅m) kg/(s⋅m)

°F−1 ° lb/(ft⋅s2 − n′) lb/(h⋅ft)

m m m2/s

ft ft ft2/h

s

h

J/kg m Pa⋅s

Btu/lb ft lb/(h⋅ft)

m2/s kg/m3

ft2/h lb/ft3

N/m

lbf/ft

N/m2

lbf/ft2

rad Dimensionless

rad Dimensionless

Greek symbols α β β′ γ Γ δs δ δsb ε εv η θ θb λ λm µ

ν ρ σ
τ φ φp Φ ω Ω

Thermal diffusivity, equals k/ρc; αe for effective thermal diffusivity of powdered solids Volumetric coefficient of thermal expansion Contact angle of a bubble Fluid consistency Mass rate of flow of a falling film from a tube or surface per unit perimeter, equals w/πD for vertical tube, w/2L for horizontal tube Correction factor, ratio of nonnewtonian to newtonian shear rates Cell width Diametral shell-to-baffle clearance Eddy diffusivity; εM for eddy diffusivity of momentum; εH for eddy diffusivity of heat Fraction of voids in porous bed Fluidization efficiency Time Baffle cut Latent heat (enthalpy) of vaporization (condensation) Radius of maximum velocity Viscosity; µw for viscosity at wall temperature; µb for viscosity at bulk temperature; µf for viscosity at film temperature; µG, µg, and µv for viscosity of gas or vapor; µL, µl for viscosity of liquid; µw for viscosity at wall; µl for viscosity of fluid at inner wall of annulus Kinematic viscosity Density; ρL, ρl for density of liquid; ρG, ρv for density of gas or vapor; ρs for density of solid Surface tension between a liquid and its vapor Term indicating summation of variables Shear stress τw for shear stress at the wall Velocity-potential function Particle sphericity Viscous-dissipation function Angle of repose of powdered solid Fin efficiency

GENERAL REFERENCES: Becker, Heat Transfer, Plenum, New York, 1986. Bejan, Convection Heat Transfer, Wiley, New York, 1984. Bird, Stewart, and Lightfoot, Transport Phenomena, Wiley, New York, 1960. Carslaw and Jaeger, Conduction of Heat in Solids, Clarendon Press, Oxford, 1959. Chapman, Heat Transfer, 2d ed., Macmillan, New York, 1967. Drew and Hoopes, Advances in Chemical Engineering, Academic, New York, vol. 1, 1956; vol. 2, 1958; vol. 5, 1964; vol. 6, 1966; vol. 7, 1968. Dusinberre, Heat Transfer Calculations by Finite Differences, International Textbook, Scranton, Pa., 1961. Eckert and Drake, Heat and Mass Transfer, 2d ed., McGraw-Hill, New York, 1959. Gebhart, Heat Transfer, McGraw-Hill, New York, 1961. Irvine and Hartnett, Advances in Heat Transfer, Academic, New York, vol. 1, 1964; vol. 2, 1965; vol. 3, 1966. Grigull and Sandner, Heat Conduction, Hemisphere Publishing, 1984. Jakob, Heat Transfer, Wiley, New York, vol. 1, 1949; vol. 2, 1957. Jakob and Hawkins, Elements of Heat Transfer, 3d ed., Wiley, New York, 1957. Kakac, Bergles, and Mayinger, Heat Exchangers: Thermal Hydraulic Fundamentals and Design, Hemisphere Publishing, Washington, 1981. Kakac and Yener, Convective Heat Transfer, Hemisphere Publishing, Washington, 1980. Kay, An Introduction to Fluid Mechanics and Heat Transfer, 2d ed., Cambridge University Press, Cambridge, England, 1963. Kays, Convective Heat and Mass Transfer, McGraw-Hill, New York, 1966. Kays and London, Compact Heat Exchangers, 3d ed., McGraw-Hill, New York, 1984. Kern, Process Heat Transfer, McGrawHill, New York, 1950. Knudsen and Katz, Fluid Dynamics and Heat Transfer, McGraw-Hill, New York, 1958. Kraus, Analysis and Evaluation of Extended Surface Thermal Systems, Hemisphere Publishing, Washington, 1982. Kutatladze, A Concise Encyclopedia of Heat Transfer, 1st English ed., Pergamon, New York, 1966. Lykov, Heat and Mass Transfer in Capillary Porous Bodies, translated from Russian, Pergamon, New York, 1966. McAdams, Heat Transmission, 3d ed., McGraw-Hill, New York, 1954. Mickley, Sherwood, and Reed, Applied Mathematics in Chemical Engineering, 2d ed., McGraw-Hill, New York, 1957. Rohsenow and Choi, Heat, Mass, and Momentum Transfer, Prentice-Hall, Englewood Cliffs, N.J., 1961. Schlünder (ed.), Heat Exchanger Design Handbook, Hemisphere Publishing, Washington, 1983 (Book 2; Chapter 2.4 [Conduction], Chapter 2.5 [Convection], Chapter 2.6 [Condensation], Chapter 2.7 [Boiling]). Skelland, Non-Newtonian Flow and Heat Transfer, Wiley, New York, 1967. Taborek and Bell, Process Heat Exchanger Design, Hemisphere Publishing, Washington, 1984. Taborek, Hewitt, and Afghan, Heat Exchangers: Theory and Practice, Hemisphere Publishing, Washington, 1983. TSederberg, Thermal Conductivity of Liquids and Gases, M.I.T., Cambridge, Mass., 1965. Welty, Wicks, and Wilson, Fundamentals of Momentum, Heat and Mass Transfer, 3d ed., Wiley, New York, 1984. Zenz and Othmer, Fluidization and Fluid Particle Systems, Reinhold, New York, 1960.

31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65.

Lee and Thodos, Ind. Eng. Chem. Fundam., 22, 17–26 (1983). Lee and Thodos, Ind. Eng. Chem. Res., 27, 992–997 (1988). Lees and Sarram, J. Chem. Eng. Data, 16, 1, 41 (1971). Leffler and Cullinan, Ind. Eng. Chem. Fundam., 9, 84, 88 (1970). Lugg, Anal. Chem., 40, 1072 (1968). Marrero and Mason, AIChE J., 19, 498 (1973). Mathur and Thodos, AIChE J., 11, 613 (1965). Matthews and Akgerman, AIChE J., 33, 881 (1987). Matthews, Rodden and Akgerman, J. Chem. Eng. Data, 32, 317 (1987). Olander, AIChE J., 7, 175 (1961). Passut and Danner, Chem. Eng. Prog. Symp. Ser., 140, 30 (1974). Perkins and Geankoplis, Chem. Eng. Sci., 24, 1035–1042 (1969). Pinto and Graham, AIChE J., 32, 291 (1986). Pinto and Graham, AIChE J., 33, 436 (1987). Quale, Chem. Rev., 53, 439 (1953). Rathbun and Babb, Ind. Eng. Chem. Proc. Des. Dev., 5, 273 (1966). Reddy and Doraiswamy, Ind. Eng. Chem. Fundam., 6, 77 (1967). Riazi and Whitson, Ind. Eng. Chem. Res., 32, 3081 (1993). Robinson, Edmister, and Dullien, Ind. Eng. Chem. Fundam., 5, 75 (1966). Rollins and Knaebel, AIChE J., 37, 470 (1991). Siddiqi, Krahn, and Lucas, J. Chem. Eng. Data, 32, 48 (1987). Siddiqi and Lucas, Can. J. Chem. Eng., 64, 839 (1986). Smith and Taylor, Ind. Eng. Chem. Fundam., 22, 97 (1983). Sridhar and Potter, AIChE J., 23, 4, 590 (1977). Stiel and Thodos, AIChE J., 7, 234 (1961). Sun and Chen, Ind. Eng. Chem. Res., 26, 815 (1987). Tanford, Phys. Chem. of Macromolecule, Wiley, New York, NY (1961). Taylor and Webb, Comput. Chem. Eng., 5, 61 (1981). Tyn and Calus, J. Chem. Eng. Data, 20, 310 (1975). Umesi and Danner, Ind. Eng. Chem. Process Des. Dev., 20, 662 (1981). Van Geet and Adamson, J. Phys. Chem., 68, 2, 238 (1964). Vignes, Ind. Eng. Chem. Fundam., 5, 184 (1966). Wilke, Chem. Eng. Prog., 46, 2, 95 (1950). Wilke and Chang, AIChE J., 1, 164 (1955). Wilke and Lee, Ind. Eng. Chem., 47, 1253 (1955).

REFERENCES FOR DIFFUSIVITIES IN POROUS SOLIDS, TABLE 5-20 66. Ruthven, Principles of Adsorption & Adsorption Processes, Wiley, 1984. 67. Satterfield, Mass Transfer in Heterogeneous Catalysis, MIT Press, 1970. 68. Suzuki, Adsorption Engineering, Kodansha—Elsevier, 1990. 69. Yang, Gas Separation by Adsorption Processes, Butterworths, 1987.

REFERENCES FOR DIFFUSIVITIES 1. Akita, Ind. Eng. Chem. Fundam., 10, 89 (1981). 2. Asfour and Dullien, Chem. Eng. Sci., 41, 1891 (1986). 3. Blanc, J. Phys., 7, 825 (1908). 4. Brokaw, Ind. Eng. Chem. Process Des. and Dev., 8, 2, 240 (1969). 5. Caldwell and Babb, J. Phys. Chem., 60, 51 (1956). 6. Catchpole and King, Ind. Eng. Chem. Res., 33, 1828 (1994). 7. Chen and Chen, Chem. Eng. Sci., 40, 1735 (1985). 8. Chung, Ajlan, Lee and Starling, Ind. Eng. Chem. Res., 27, 671 (1988). 9. Condon and Craven, Aust. J. Chem., 25, 695 (1972). 10. Cullinan, AIChE J., 31, 1740–1741 (1985). 11. Cullinan, Can. J. Chem. Eng., 45, 377–381 (1967). 12. Cussler, AIChE J., 26, 1 (1980). 13. Darken, Trans. Am. Inst. Mining Met. Eng., 175, 184 (1948). 14. Debenedetti and Reid, AIChE J., 32, 2034 (1986); see errata: AIChE J., 33, 496 (1987). 15. Elliott, R. W. and H. Watts, Can. J. Chem., 50, 31 (1972). 16. Erkey and Akgerman, AIChE J., 35, 443 (1989). 17. Ertl, Ghai, and Dullien, AIChE J., 20, 1, 1 (1974). 18. Fairbanks and Wilke, Ind. Eng. Chem., 42, 471 (1950). 19. Fuller, Schettler and Giddings, Ind. Eng. Chem., 58, 18 (1966). 20. Ghai, Ertl, and Dullien, AIChE J., 19, 5, 881 (1973). 21. Gordon, J. Chem. Phys., 5, 522 (1937). 22. Graham and Dranoff, Ind. Eng. Chem. Fundam., 21, 360–365 (1982). 23. Graham and Dranoff, Ind. Eng. Chem. Fundam., 21, 365–369 (1982). 24. Gurkan, AIChE J., 33, 175–176 (1987). 25. Hayduk and Laudie, AIChE J., 20, 3, 611 (1974). 26. Hayduk and Minhas, Can. J. Chem. Eng., 60, 195 (1982). 27. Hildebrand, Science, 174, 490 (1971). 28. Hiss and Cussler, AIChE J., 19, 4, 698 (1973). 29. Jossi, Stiel, and Thodos, AIChE J., 8, 59 (1962). 30. Krishnamurthy and Taylor, Chem. Eng. J., 25, 47 (1982).

REFERENCES FOR TABLES 5-21 TO 5-28 70. Bahmanyar, Chang-Kakoti, Garro, Liang, and Slater, Chem. Engr. Rsch. Des., 68, 74 (1990). 71. Beenackers and van Swaaij, Chem. Engr. Sci., 48, 3109 (1993). 72. Bird, Stewart and Lightfoot, Transport Phenomena, Wiley, 1960. 73. Blatt, Dravid, Michaels, and Nelson in Flinn (ed.), Membrane Science and Technology, 47, Plenum, 1970. 74. Bolles and Fair, Institution Chem. Eng. Symp. Ser., 56, 3/35 (1979). 75. Bolles and Fair, Chem. Eng., 89(14), 109 (July 12, 1982). 76. Bravo and Fair, Ind. Eng. Chem. Process Des. Dev., 21, 162 (1982). 77. Bravo, Rocha and Fair, Hydrocarbon Processing, 91 (Jan. 1985). 78. Brian and Hales, AIChE J., 15, 419 (1969). 79. Calderbank and Moo-Young, Chem. Eng. Sci., 16, 39 (1961). 80. Chilton and Colburn, Ind. Eng. Chem., 26, 1183 (1934). 81. Cornell, Knapp, and Fair, Chem. Engr. Prog., 56(7), 68 (1960). 82. Cornet and Kaloo, Proc. 3rd Int’l. Congr. Metallic Corrosion—Moscow, 3, 83 (1966). 83. Cussler, Diffusion: Mass Transfer in Fluid Systems, Cambridge, 1984. 84. Dwivedi and Upadhyay, Ind. Eng. Chem. Process Des. Develop, 16, 1657 (1977). 85. Eisenberg, Tobias, and Wilke, Chem. Engr. Prog. Symp. Sec., 51(16), 1 (1955). 86. Elzinga and Banchero, Chem. Engr. Progr. Symp. Ser., 55(29), 149 (1959). 87. Fair, “Distillation” in Rousseau (ed.), Handbook of Separation Process Technology, Wiley, 1987. 88. Faust, Wenzel, Clump, Maus, and Andersen, Principles of Unit Operations, 2d ed., Wiley, 1980. 89. Frossling, Gerlands Beitr. Geophys., 52, 170 (1938). 90. Garner and Suckling, AIChE J., 4, 114 (1958). 91. Geankoplis, Transport Processes and Unit Operations, 3d ed., Prentice Hall, 1993. 5-7

5-8 92. 93. 94. 95. 96. 97. 98. 99. 100. 101. 102. 103. 104. 105. 106. 107. 108. 109. 110. 111. 112. 113. 114. 115. 116. 117. 118. 119. 120. 121. 122. 123. 124. 125. 126. 127. 128.

HEAT AND MASS TRANSFER Gibilaro, Davies, Cooke, Lynch, and Middleton, Chem. Engr. Sci., 40, 1811 (1985). Gilliland and Sherwood, Ind. Engr. Chem., 26, 516 (1934). Griffith, Chem. Engr. Sci., 12, 198 (1960). Gupta and Thodos, AIChE J., 9, 751 (1963). Gupta and Thodos, Ind. Eng. Chem. Fundam., 3, 218 (1964). Harriott, AIChE J., 8, 93 (1962). Hausen, Verfahrenstech. Beih. Z. Ver. Dtsch. Ing., 4, 91 (1943). Heertjes, Holve, and Talsma, Chem. Engr. Sci., 3, 122 (1954). Hines and Maddox, Mass Transfer: Fundamentals and Applications, Prentice Hall, 1985. Hsiung and Thodos, Int. J. Heat Mass Transfer, 20, 331 (1977). Hsu, Sato, and Sage, Ind. Engr. Chem., 46, 870 (1954). Hughmark, Ind. Eng. Chem. Fundam., 6, 408 (1967). Johnson, Besic, and Hamielec, Can. J. Chem. Engr., 47, 559 (1969). Johnstone and Pigford, Trans. AIChE, 38, 25 (1942). Kafesjian, Plank, and Gerhard, AIChE J., 7, 463 (1961). Kelly and Swenson, Chem. Eng. Prog., 52, 263 (1956). King, Separation Processes, 2d ed., McGraw-Hill (1980). Kirwan, “Mass Transfer Principles” in Rousseau, Handbook of Separation Process Technology, Wiley, 1987. Klein, Ward, and Lacey, “Membrane Processes—Dialysis and ElectroDialysis” in Rousseau, Handbook of Separation Process Technology, Wiley, 1987. Kohl, “Absorption and Stripping” in Rousseau, Handbook of Separation Process Technology, Wiley, 1987. Kojima, Uchida, Ohsawa, and Iguchi, J. Chem. Engng. Japan, 20, 104 (1987). Koloini, Sopcic, and Zumer, Chem. Engr. Sci., 32, 637 (1977). Lee, Biochemical Engineering, Prentice Hall, 1992. Lee and Foster, Appl. Catal., 63, 1 (1990). Lee and Holder, Ind. Engr. Chem. Res., 34, 906 (1995). Levich, Physicochemical Hydrodynamics, Prentice Hall, 1962. Levins and Gastonbury, Trans. Inst. Chem. Engr., 50, 32, 132 (1972). Lim, Holder, and Shah, J. Supercrit. Fluids, 3, 186 (1990). Linton and Sherwood, Chem. Engr. Prog., 46, 258 (1950). Ludwig, Applied Process Design for Chemical and Petrochemical Plants, 2d ed., vol. 2, Gulf Pub. Co., 1977. McCabe, Smith, and Harriott, Unit Operations of Chemical Engineering, 5th ed., McGraw-Hill, 1993. Nelson and Galloway, Chem. Engr. Sci., 30, 7 (1975). Notter and Sleicher, Chem. Eng. Sci., 26, 161 (1971). Ohashi, Sugawara, Kikuchi, and Konno, J. Chem. Engr. Japan, 14, 433 (1981). Onda, Takeuchi, and Okumoto, J. Chem. Engr. Japan, 1, 56 (1968). Pasternak and Gauvin, AIChE J., 7, 254 (1961). Pasternak and Gauvin, Can. J. Chem. Engr., 38, 35 (April 1960).

129. 130. 131. 132. 133. 134. 135. 136. 137. 138. 139. 140. 141. 142. 143. 144. 145. 146. 147. 148. 149. 150. 151. 152. 153. 154. 155. 156. 157. 158. 159. 160. 161. 162. 163. 164. 165. 166. 167. 168. 169.

Perez and Sandall, AIChE J., 20, 770 (1974). Petrovic and Thodos, Ind. Eng. Chem. Fundam., 7, 274 (1968). Pinczewski and Sideman, Chem. Engr. Sci., 29, 1969 (1974). Prandtl, Phys. Zeit., 29, 487 (1928). Prasad and Sirkar, AIChE J., 34, 177 (1988). Rahman and Streat, Chem. Engr. Sci., 36, 293 (1981). Ranz and Marshall, Chem. Engr. Prog., 48, 141, 173 (1952). Reiss, Ind. Eng. Chem. Process Des. Develop, 6, 486 (1967). Riet, Ind. Eng. Chem. Process Des. Dev., 18, 357 (1979). Rowe, Chem. Engr. Sci., 30, 7 (1975). Rowe, Claxton, and Lewis, Trans. Inst. Chem. Engr. London, 43, 14 (1965). Ruckenstein and Rajagopolan, Chem. Engr. Commun., 4, 15 (1980). Ruthven, Principles of Adsorption & Adsorption Processes, Wiley, 1984. Satterfield, AIChE J., 21, 209 (1975). Schluter and Deckwer, Chem. Engr. Sci., 47, 2357 (1992). Schmitz, Steiff, and Weinspach, Chem. Engng. Technol., 10, 204 (1987). Sherwood, Brian, Fisher, and Dresner, Ind. Eng. Chem. Fundam., 4, 113 (1965). Sherwood, Pigford, and Wilke, Mass Transfer, McGraw-Hill, 1975. Shulman, Ullrich, Proulx, and Zimmerman, AIChE J., 1, 253 (1955). Shulman and Margolis, AIChE J., 3, 157 (1957). Siegel, Sparrow, and Hallman, Appl. Sci. Res. Sec. A., 7, 386 (1958). Sissom and Pitts, Elements of Transport Phenomena, McGraw-Hill, 1972. Skelland, Diffusional Mass Transfer, Wiley (1974). Skelland and Cornish, AIChE J., 9, 73 (1963). Skelland and Moeti, Ind. Eng. Chem. Res., 29, 2258 (1990). Skelland and Tedder, “Extraction—Organic Chemicals Processing” in Rousseau, Handbook of Separation Process Technology, Wiley, 1987, pp. 405–466. Skelland and Wellek, AIChE J., 10, 491, 789 (1964). Slater, “Rate Coefficients in Liquid-Liquid Extraction Systems” in Godfrey and Slater, Liquid-Liquid Extraction Equipment, Wiley, 1994, pp. 45–94. Steinberger and Treybal, AIChE J., 6, 227 (1960). Steiner, L., Chem. Eng. Sci., 41, 1979 (1986). Taylor and Krishna, Multicomponent Mass Transfer, Wiley, 1993. Tournie, Laguerie, and Couderc, Chem. Engr. Sci., 34, 1247 (1979). Treybal, Mass Transfer Operations, 3d ed., McGraw-Hill, 1980. Von Karman, Trans. ASME, 61, 705 (1939). Wakao and Funazkri, Chem. Engr. Sci., 33, 1375 (1978). Wankat, Equilibrium-Staged Separations, Prentice Hall, 1988. Wankat, Rate-Controlled Separations, Chapman-Hall, 1990. Wilson and Geankoplis, Ind. Eng. Chem. Fundam., 5, 9 (1966). Yagi and Yoshida, Ind. Eng. Chem. Process Des. Dev., 14, 488 (1975). Yang, Gas Separation by Adsorption Processes, Butterworths, 1987. Yoshida, Ramaswami, and Hougen, AIChE J., 8, 5 (1962).

HEAT TRANSFER MODES OF HEAT TRANSFER There are three fundamental types of heat transfer: conduction, convection, and radiation. All three types may occur at the same time, and it is advisable to consider the heat transfer by each type in any particular case. Conduction is the transfer of heat from one part of a body to another part of the same body, or from one body to another in physical contact with it, without appreciable displacement of the particles of the body.

Convection is the transfer of heat from one point to another within a fluid, gas, or liquid by the mixing of one portion of the fluid with another. In natural convection, the motion of the fluid is entirely the result of differences in density resulting from temperature differences; in forced convection, the motion is produced by mechanical means. When the forced velocity is relatively low, it should be realized that “free-convection” factors, such as density and temperature difference, may have an important influence. Radiation is the transfer of heat from one body to another, not in contact with it, by means of wave motion through space.

HEAT TRANSFER BY CONDUCTION FOURIER’S LAW Fourier’s law is the fundamental differential equation for heat transfer by conduction: dQ/dθ = −kA(dt/dx)

direction of the flow of heat, i.e., the temperature gradient. The factor k is called the thermal conductivity; it is a characteristic property of the material through which the heat is flowing and varies with temperature.

(5-1)

where dQ/dθ (quantity per unit time) is the rate of flow of heat, A is the area at right angles to the direction in which the heat flows, and −dt/dx is the rate of change of temperature with the distance in the

THREE-DIMENSIONAL CONDUCTION EQUATION Equation (5-1) is used as a basis for derivation of the unsteady-state three-dimensional energy equation for solids or static fluids:

HEAT TRANSFER BY CONDUCTION ∂t ∂t ∂ ∂t ∂ ∂t ∂ cρ ᎏ = ᎏ k ᎏ + ᎏ k ᎏ + ᎏ k ᎏ + q′ ∂θ ∂x ∂x ∂y ∂y ∂z ∂z

 

 

 

(5-2)

where x, y, z are distances in the rectangular coordinate system and q′ is the rate of heat generation (by chemical reaction, nuclear reaction, or electric current) in the solid per unit of volume. Solution of Eq. (5-2) with appropriate boundary and initial conditions will give the temperature as a function of time and location in the material. Equation (5-2) may be transformed into spherical or cylindrical coordinates to conform more closely to the physical shape of the system. THERMAL CONDUCTIVITY Thermal conductivity varies with temperature but not always in the same direction. The thermal conductivities for many materials, as a function of temperature, are given in Sec. 2. Additional and more comprehensive information may often be obtained from suppliers of the materials. Impurities, especially in metals, can give rise to variations in thermal conductivity of from 50 to 75 percent. In using thermal conductivities, engineers should remember that conduction is not the sole method of transferring heat and that, particularly with liquids and gases, radiation and convection may be much more important. The thermal conductivity at a given temperature is a function of the apparent, or bulk, density. Thus, at 0°C (32°F), k for asbestos wool is 0.09 J/(m⭈s⭈K) [0.052 Btu/(hr⭈ft⭈°F)] when the bulk density is 400 kg/m3 (24.9 lb/ft3) and is 0.19 (0.111) for a density of 700 (43.6). In determining the apparent thermal conductivities of granular solids, such as granulated cork or charcoal grains, Griffiths (Spec. Rep. 5, Food Investigation Board, H. M. Stationery Office, 1921) found that air circulates within the mass of granular solid. Under a certain set of conditions, the apparent thermal conductivity of a charcoal was 9 percent greater when the test section was vertical than when it was horizontal. When the apparent conductivity of a mixture of cellular or porous nonhomogeneous solid is determined, the observed temperature coefficient may be much larger than for the homogeneous solid alone, because heat is transferred not only by the mechanism of conduction but also by convection in the gas pockets and by radiation from surface to surface of the individual particles. If internal radiation is an important factor, a plot of the apparent conductivity as ordinate versus temperature should show a curve concave upward, since radiation increases with the fourth power of the absolute temperature. Griffiths noted that cork, slag, wool, charcoal, and wood fibers, when of good quality and dry, have thermal conductivities about 2.2 times that of still air, whereas a highly cellular form of rubber, 112 kg/m3 (7 lb/ft3), had a thermal conductivity only 1.6 times that of still air. In measuring the apparent thermal conductivity of diathermanous substances such as quartz (especially when exposed to radiation emitted at high temperatures), it should be remembered that a part of the heat is transmitted by radiation. Bridgman [Proc. Am. Acad. Arts Sci., 59, 141 (1923)] showed that the thermal conductivity of liquids is increased by only a few percent under a pressure of 100,330 kPa (1000 atm). The thermal conductivity of some liquids varies with temperature through a maximum. It is often necessary for the engineer to estimate thermal conductivities; methods are indicated in Sec. 2. Equation (5-2) considers the thermal conductivity to be variable. If k is expressed as a function of temperature, Eq. (5-2) is nonlinear and difficult to solve analytically except for certain special cases. Usually in complicated systems numerical solution by means of computer is possible. A complete review of heat conduction has been given by Davis and Akers [Chem. Eng., 67(4), 187, (5), 151 (1960)] and by Davis [Chem. Eng., 67(6), 213, (7), 135 (8), 137 (1960)]. STEADY-STATE CONDUCTION For steady flow of heat, the term dQ/dθ in Eq. (5-1) is constant and may be replaced by Q/θ or q. Likewise, in Eq. (5-2) the term ∂ t/∂θ is zero. Hence, for constant thermal conductivity, Eq. (5-2) may be expressed as ∇ 2 t = (q′/k)

(5-3)

5-9

One-Dimensional Conduction Many heat-conduction problems may be formulated into a one-dimensional or pseudo-onedimensional form in which only one space variable is involved. Forms of the conduction equation for rectangular, cylindrical, and spherical coordinates are, respectively, ∂2 t q′ (5-4a) ᎏ2 = − ᎏ ∂x k

 

1 d dt q′ ᎏᎏ rᎏ =−ᎏ r dr dr k



(5-4b)



1 d dt q′ (5-4c) ᎏ2 ᎏ r 2 ᎏ = − ᎏ r dr dr k These are second-order differential equations which upon integration become, respectively, t = −(q′x2/2k) + c1x + c2

(5-5a)

t = −(q′r /4k) + c1 ln r + c2

(5-5b)

t = −(q′r 2/6k) − (c1/r) + c2

(5-5c)

2

Constants of integration c1 and c2 are determined by the boundary conditions, i.e., temperatures and temperature gradients at known locations in the system. For the case of a solid surface exposed to surroundings at a different temperature and for a finite surface coefficient, the boundary condition is expressed as hT (ts − t′ ) = −k(dt/dx)surf

(5-6)

Inspection of Eqs. (5-5a), (5-5b), and (5-5c) indicates the form of temperature profile for various conditions and geometries and also reveals the effect of the heat-generation term q′ upon the temperature distributions. In the absence of heat generation, one-dimensional steady-state conduction may be expressed by integrating Eq. (5-1): x2 t2 dx q (5-7) ᎏ = − k dt x1 t1 A Area A must be known as a function of x. If k is constant, Eq. (5-7) is expressed in the integrated form





q = kAavg (t1 − t2)/(x2 − x1)

(5-8)



x2

1 dx (5-9) Aavg = ᎏ ᎏ x2 − x1 x1 A Examples of values of Aavg for various functions of x are shown in the following table. where

Area proportional to

Aavg

Constant x

A1 = A2 A2 − A1 ln (A2/A1) 2 A1 A

x2

Usually, thermal conductivity k is not constant but is a function of temperature. In most cases, over the ranges of values used the relation is linear. Integration of Eq. (5-7), with k linear in t, gives x2 dx q (5-10) ᎏ = kavg(t1 − t2) x1 A where kavg is the arithmetic-average thermal conductivity between temperatures t1 and t2. This average probably gives results which are correct within the precision of the data in the majority of cases, though a special integration can be made whenever k is known to be greatly different from linear in temperature. Conduction through Several Bodies in Series Figure 5-1 illustrates diagrammatically the temperature gradients accompanying the steady conduction of heat in series through three solids.



5-10

HEAT AND MASS TRANSFER For fuel, at x = 0, t = 570°C (1050°F), dt/dx = 0 (this follows if the temperature is finite at the midplane). For fuel and cladding, at x = x1, tf = tc, kf (dt/dx) = kc (dt/dx) For cladding, at x = x2, tc − 400 = −(kc /42,600)(dt/dx) For the fuel, the first integration of Eq. (10-4a) gives dtf /dx = −(q′/kf )x + c1 which gives c1 = 0 when the boundary condition is applied. Thus the second integration gives tf = −(q′/2kf )x2 + c2 from which c2 is determined to be 570 (1050) upon application of the boundary condition. Thus the temperature profile in the fuel is tf = −(q′/2kf )x2 + 570

FIG. 5-1

Temperature gradients for steady heat conduction in series through three solids.

The temperature profile in the cladding is obtained by integrating Eq. (10-4a) twice with q′ = 0. Hence

Since the heat flow through each of the three walls must be the same,

There are now three unknowns, c1, c2, and q′, and three boundary conditions by which they can be determined. At x = x1,

(dtc /dx) = c1

q = (k1 A1 ∆t1/x1) = (k2 A2 ∆ t2 /x2) = (k3 A3 ∆ t3 /x3)

(5-11)

then

∆t1 = qR1

∆ t2 = qR2

(5-12) ∆ t3 = qR3

(5-13)

Adding the individual temperature drops, noting that q is uniform, q(R1 + R2 + R3) = ∆t1 + ∆ t2 + ∆ t3 =
∆ t

(5-14)

q =
∆ t/RT = (t1 − t4)/RT

(5-15)

or

where RT is the overall resistance and is the sum of the individual resistances in series, then RT = R1 + R2 + ⋅ ⋅ ⋅ + Rn

(5-16)

When a wall is constructed of several layers of solids, the joints at adjacent layers may not perfectly exclude air spaces, and these additional resistances should not be overlooked. Conduction through Several Bodies in Parallel For n resistances in parallel, the rates of heat flow are additive: q = ∆t/R1 + ∆ t/R2 + ⋅ ⋅ ⋅ + ∆ t/Rn



1 1 1 q= ᎏ+ᎏ+⋅⋅⋅+ᎏ R1 R2 Rn

 ∆t

q = (C1 + C2 + ⋅ ⋅ ⋅ + Cn)∆ t =
C ∆ t

tc = c1 x + c2

q′x12 /2kf + 570 = c1x1 + c2 − kf q′x1/kf = kc c1 At x = x2,

Since, by definition, individual thermal resistance R = x/ kA

and

(5-17a) (5-17b)

c1 x 2 + c2 − 200 = −(kc /42,600)c1 From which q′ = (2.53)(109) J/(m3⋅s)[(2.38)(108)Btu/(h⋅ft3)] c1 = −(1.92)(105) c2 = 724

Two-Dimensional Conduction If the temperature of a material is a function of two space variables, the two-dimensional conduction equation is (assuming constant k) ∂ 2t/∂x2 + ∂ 2 t/∂y2 = −q′/k

(5-18)

When q′ is zero, Eq. (5-18) reduces to the familiar Laplace equation. The analytical solution of Eq. (10-18) as well as of Laplace’s equation is possible for only a few boundary conditions and geometric shapes. Carslaw and Jaeger (Conduction of Heat in Solids, Clarendon Press, Oxford, 1959) have presented a large number of analytical solutions of differential equations applicable to heat-conduction problems. Generally, graphical or numerical finite-difference methods are most frequently used. Other numerical and relaxation methods may be found in the general references in the “Introduction.” The methods may also be extended to three-dimensional problems. UNSTEADY-STATE CONDUCTION

(5-17c)

where R1 to Rn are the individual resistances and C1 to Cn are the individual conductances; C = kA/x. Several Bodies in Series with Heat Generation The simple Fourier type of equation indicated by Eq. (5-15) may not be used when heat generation occurs in one of the bodies in the series. In this case, Eq. (5-5a), (5-5b), or (5-5c) must be solved with appropriate boundary conditions. Example 1: Steady-State Conduction with Heat Generation A plate-type nuclear fuel element, consisting of a uranium-zirconium alloy (3.2)(10−3) m (0.125 in) thick clad on each side with a (6.4)(10−4)-m- (0.025-in-) thick layer of zirconium, is cooled by water under pressure at 200°C (400°F), the heat-transfer coefficient being 42,600 J/(m2⋅s⋅K) [7500 Btu/(h⋅ft2⋅°F)]. If the temperature at the center of the fuel must not exceed 570°C (1050°F), determine the maximum rate of heat generation in the fuel. The zirconium and zirconium alloy have a thermal conductivity of 21 J/(m⋅s⋅K) [12 Btu/(h⋅ft2)(°F/ft)]. Solution. Equation (5-4a) may be integrated for each material. The heat generation is zero in the cladding, and its value for the fuel may be determined from the integrated equations. Let x = 0 at the midplane of the fuel. Then x1 = (1.6)(10−3) m (0.0625 in) at the cladding-fuel interface and x2 = (2.2)(10−3) m (0.0875 in) at the cladding-water interface. Let the subscripts c, f refer to cladding and fuel respectively. The boundary conditions are:

When temperatures of materials are a function of both time and space variables, more complicated equations result. Equation (5-2) is the three-dimensional unsteady-state conduction equation. It involves the rate of change of temperature with respect to time ∂ t/ ∂θ. Solutions to most practical problems must be obtained through the use of digital computers. Numerous articles have been published on a wide variety of transient conduction problems involving various geometrical shapes and boundary conditions. One-Dimensional Conduction The one-dimensional transient conduction equations are (for constant physical properties) ∂ t/∂θ = α(∂2t/∂x2) + q′/cρ ∂t α ∂ ∂t q′ ᎏ=ᎏᎏ rᎏ +ᎏ ∂θ r dr ∂r cρ

 

(rectangular coordinates)

(5-19a)

(cylindrical coordinates)

(5-19b)

q′ ∂t α ∂ ∂t (spherical coordinates) (5-19c) ᎏ = ᎏ2 ᎏ r2 ᎏ + ᎏ ∂θ r ∂r ∂r cρ These equations have been solved analytically for solid slabs, cylinders, and spheres. The solutions are in the form of infinite series, and usually the results are plotted as curves involving four ratios [Gurney and Lurie, Ind. Eng. Chem., 15, 1170 (1923)] defined as follows with q′ = 0:

 

HEAT TRANSFER BY CONDUCTION Y = (t′ − t)/(t′ − tb)

X = kθ/ρcrm2

5-11

Since each ratio is dimensionless, any consistent units may be employed in any ratio. The significance of the symbols is as follows: t′ = temperature of the surroundings; tb = initial uniform temperature of the body; t = temperature at a given point in the body at the time θ measured from the start of the heating or cooling operations; k = uniform thermal conductivity of the body; ρ = uniform density of the body; c = specific heat of the body; hT = coefficient of total heat transfer between the surroundings and the surface of the body expressed as heat transferred per unit time per unit area of the surface per unit difference in temperature between surroundings and surface; r = distance, in the direction of heat conduction, from the midpoint or midplane of the body to the point under consideration; rm = radius of a sphere or cylinder, one-half of the thickness of a slab heated from both faces, the total thickness of a slab heated from one face and insulated perfectly at the other; and x = distance, in the direction of heat conduction, from the surface of a semi-infinite body (such as the surface of the earth) to the point under consideration. In making the integrations which lead to the curves shown, the following factors were assumed constant: c, hT, k, r, rm, t′, x, and ρ. The working curves are shown in Figs. 5-2 to 5-5 for cylinders of infinite length, spheres, slabs of infinite faces, and semi-infinite solids respectively, with Y plotted as ordinates on a logarithmic scale versus X as abscissas to an arithmetic scale, for various values of the ratios m and n. To facilitate calculations involving instantaneous rates of cooling or heating of the semi-infinite body, Fig. 5-5 shows also a curve of dY/dX versus X. Similar plots to a larger scale are given in McAdams,

Brown and Marco, Schack, and Stoever (see “Introduction: General References”). For a solid of infinite thickness (Fig. 5-5) and with m = 0, 2 z (5-21) Y = ᎏ exp (−z2) dz π 0  where z = 1/ 2X and the “error integral” may be evaluated from standard mathematical tables. Various numerical and graphical methods are used for unsteadystate conduction problems, in particular the Schmidt graphical method (Foppls Festschrift, Springer-Verlag, Berlin, 1924). These methods are very useful because any form of initial temperature distribution may be used. Two-Dimensional Conduction The governing differential equation for two-dimensional transient conduction is q′ ∂t ∂2t ∂2t (5-22) ᎏ = α ᎏ2 + ᎏ2 + ᎏ ∂θ ∂x ∂y cρ McAdams (Heat Transmission, 3d ed., McGraw-Hill, New York, 1954) gives various forms of transient difference equations and methods of solving transient conduction problems. The availability of computers and a wide variety of computer programs permits virtually routine solution of complicated conduction problems. Conduction with Change of Phase A special type of transient problem (the Stefan problem) involves conduction of heat in a material when freezing or melting occurs. The liquid-solid interface moves with time, and in addition to conduction, latent heat is either generated or absorbed at the interface. Various problems of this type are discussed by Bankoff [in Drew et al. (eds.), Advances in Chemical Engineering, vol. 5, Academic, New York, 1964].

Heating and cooling of a solid cylinder having an infinite ratio of length to diameter.

FIG. 5-4 Heating and cooling of a solid slab having a large face area relative to the area of the edges.

m = k/hTrm

n = r/rm

FIG. 5-2

(5-20a,b) (5-20c,d)





Heating and cooling of a solid of infinite thickness, neglecting edge effects. (This may be used as an approximation in the zone near the surface of a body of finite thickness.)

FIG. 5-5 FIG. 5-3

Heating and cooling of a solid sphere.



5-12

HEAT AND MASS TRANSFER

HEAT TRANSFER BY CONVECTION COEFFICIENT OF HEAT TRANSFER In many cases of heat transfer involving either a liquid or a gas, convection is an important factor. In the majority of heat-transfer cases met in industrial practice, heat is being transferred from one fluid through a solid wall to another fluid. Assume a hot fluid at a temperature t1 flowing past one side of a metal wall and a cold fluid at t7 flowing past the other side to which a scale of thickness xs adheres. In such a case, the conditions obtaining at a given section are illustrated diagrammatically in Fig. 5-6. For turbulent flow of a fluid past a solid, it has long been known that, in the immediate neighborhood of the surface, there exists a relatively quiet zone of fluid, commonly called the film. As one approaches the wall from the body of the flowing fluid, the flow tends to become less turbulent and develops into laminar flow immediately adjacent to the wall. The film consists of that portion of the flow which is essentially in laminar motion (the laminar sublayer) and through which heat is transferred by molecular conduction. The resistance of the laminar layer to heat flow will vary according to its thickness and can range from 95 percent of the total resistance for some fluids to about 1 percent for other fluids (liquid metals). The turbulent core and the buffer layer between the laminar sublayer and turbulent core each offer a resistance to heat transfer which is a function of the turbulence and the thermal properties of the flowing fluid. The relative temperature difference across each of the layers is dependent upon their resistance to heat flow. The Energy Equation A complete energy balance on a flowing fluid through which heat is being transferred results in the energy equation (assuming constant physical properties): ∂t ∂t ∂t ∂t cρ ᎏ + u ᎏ + v ᎏ + w ᎏ ∂θ ∂x ∂y ∂z





∂2 t ∂ 2 t ∂ 2 t = k ᎏ2 + ᎏ2 + ᎏ2 + q′ + Φ (5-23) ∂x ∂y ∂z where Φ is the term accounting for energy dissipation due to fluid viscosity and is significant in high-speed gas flow and in the flow of highly viscous liquids. Except for the time term, the left-hand terms of Eq. (523) are the so-called convective terms involving the energy carried by the fluid by virtue of its velocity. Therefore, the solution of the equation is dependent upon the solution of the momentum equations of flow. Solutions of Eq. (5-23) exist only for several simple flow cases and geometries and mainly for laminar flow. For turbulent flow the difficulties of expressing the fluid velocity as a function of space and time coordinates and of obtaining reliable values of the effective thermal conductivity of the flowing fluid have prevented solution of the equation unless simplifying assumptions and approximations are made.





Individual Coefficient of Heat Transfer Because of the complicated structure of a turbulent flowing stream and the impracticability of measuring thicknesses of the several layers and their temperatures, the local rate of heat transfer between fluid and solid is defined by the equations dq = hi d Ai (t1 − t3) = ho d Ao (t5 − t7)

(5-24)

where hi and ho are the local heat-transfer coefficients inside and outside the wall, respectively, and temperatures are defined by Fig. 5-6. The definition of the heat-transfer coefficient is arbitrary, depending on whether bulk-fluid temperature, centerline temperature, or some other reference temperature is used for t1 or t7. Equation (5-24) is an expression of Newton’s law of cooling and incorporates all the complexities involved in the solution of Eq. (5-23). The temperature gradients in both the fluid and the adjacent solid at the fluid-solid interface may also be related to the heat-transfer coefficient: dt dt dq = h i dAi (t1 − t3) = −k ᎏ = −k ᎏ (5-25) dx fluid dx solid Equation (5-25) holds for the liquid only if laminar flow exists immediately adjacent to the solid surface. The integration of Eq. (5-24) will give out out dq dq Ai = or Ao = (5-26) ᎏ ᎏ in in hi ∆ ti ho ∆ to which may be evaluated only if the quantities under the integral can be expressed in terms of a single variable. If q is a linear function of ∆t and h is constant, then Eq. (5-26) gives hA(∆tin − ∆ tout) q = ᎏᎏ (5-27) ln (∆ tin/∆ tout) where the ∆t factor is the logarithmic-mean temperature difference between the wall and the fluid. Frequently experimental data report average heat-transfer coefficients based upon an arbitrarily defined temperature difference, the two most common being hlm A(∆tin − ∆ tout) q = ᎏᎏ (5-28a) ln(∆ tin/∆ tout) ham A(∆tin + ∆ tout) q = ᎏᎏ (5-28b) 2 where hlm and ham are average heat-transfer coefficients based upon the logarithmic-mean temperature difference and the arithmeticaverage temperature difference, respectively. Overall Coefficient of Heat Transfer In testing commercial heat-transfer equipment, it is not convenient to measure tube temperatures (t3 or t4 in Fig. 5-6), and hence the overall performance is expressed as an overall coefficient of heat transfer U based on a convenient area dA, which may be dAi, dAo, or an average of dAi and dAo; whence, by definition,













dq = U dA (t1 − t7)

(5-29)

U is called the “overall coefficient of heat transfer,” or merely the “overall coefficient.” The rate of conduction through the tube wall and scale deposit is given by kdAavg(t3 − t4) dq = ᎏᎏ = hd dAd(t4 − t5) (5-30) x Upon eliminating t3, t4, t5 from Eqs. (5-24), (5-29), and (5-30), the complete expression for the steady rate of heat flow from one fluid through the wall and scale to a second fluid, as illustrated in Fig. 5-6, is t1 − t7 dq = ᎏᎏᎏᎏ = U dA (t1 − t7) (5-31)* 1 x 1 1 ᎏᎏ + ᎏᎏ + ᎏᎏ + ᎏᎏ hi dAi k dAavg hd dAd ho dAo Temperature gradients for a steady flow of heat by conduction and convection from a warmer to a colder fluid separated by a solid wall.

FIG. 5-6

* Normally, dirt and scale resistance must be considered on both sides of the tube wall. The area dA is any convenient reference area.

HEAT TRANSFER BY CONVECTION

tally and are given in Table 5-1. Fluid properties are evaluated at tf = (ts + t′)/2. For vertical plates and cylinders and 1 < NPr < 40, Kato, Nishiwaki, and Hirata [Int. J. Heat Mass Transfer, 11, 1117 (1968)] recommend the relations 0.36 0.175 NNu = 0.138NGr (NPr − 0.55) (5-33a)

Representation of Heat-Transfer Film Coefficients There are two general methods of expressing film coefficients: (1) dimensionless relations and (2) dimensional equations. The dimensionless relations are usually indicated in either of two forms, each yielding identical results. The preferred form is that suggested by Colburn [Trans. Am. Inst. Chem. Eng., 29, 174–210 (1933)]. It relates, primarily, three dimensionless groups: the Stanton number h/cG, the Prandtl number cµ/k, and the Reynolds number DG/µ. For more accurate correlation of data (at Reynolds number <10,000), two additional dimensionless groups are used: ratio of length to diameter L/D and ratio of viscosity at wall (or surface) temperature to viscosity at bulk temperature. Colburn showed that the product of the Stanton number and the two-thirds power of the Prandtl number (and, in addition, power functions of L/D and µw /µ for Reynolds number <10,000) is approximately equal to half of the Fanning friction factor f/2. This product is called the Colburn j factor. Since the Colburn type of equation relates heat transfer and fluid friction, it has greater utility than other expressions for the heat-transfer coefficient. The classical (and perhaps more familiar) form of dimensionless expressions relates, primarily, the Nusselt number hD/k, the Prandtl number cµ/k, and the Reynolds number DG/µ. The L/D and viscosityratio modifications (for Reynolds number <10,000) also apply. The dimensional equations are usually expansions of the dimensionless expressions in which the terms are in more convenient units and in which all numerical factors are grouped together into a single numerical constant. In some instances, the combined physical properties are represented as a linear function of temperature, and the dimensional equation resolves into an equation containing only one or two variables.

for NGr > 109, and 0.25 0.25 NNu = 0.683NGr NPr [NPr /(0.861 + NPr)]0.25

h = b(∆t)mL3m − 1 (5-34) Values of b in SI and U.S. customary units are given in Table 5-1 for air, water, and organic liquids. Simultaneous Loss by Radiation The heat transferred by radiation is often of significant magnitude in the loss of heat from surfaces to the surroundings because of the diathermanous nature of atmospheric gases (air). It is convenient to represent radiant-heat transfer, for this case, as a radiation film coefficient which is added to the film coefficient for convection, giving the combined coefficient for convection and radiation (hc + hr). In Fig. 5-7 values of the film coefficient for radiation hr are plotted against the two surface temperatures for emissivity = 1.0. Table 5-2 shows values of (hc + hr) from single horizontal oxidized pipe surfaces. Enclosed Spaces The rate of heat transfer across an enclosed space is calculated from a special coefficient h′ based upon the temperature difference between the two surfaces, where h′ = (q/A)/ (ts1 − ts2). The value of h′L/k may be predicted from Eq. (5-32) by using the values of a and m given in Table 5-3. For vertical enclosed cells 10 in high and up to 2-in gap width, Landis and Yanowitz (Proc. Third Int. Heat Transfer Conf., Chicago, 1966, vol. II, p. 139) give

Natural convection occurs when a solid surface is in contact with a fluid of different temperature from the surface. Density differences provide the body force required to move the fluid. Theoretical analyses of natural convection require the simultaneous solution of the coupled equations of motion and energy. Details of theoretical studies are available in several general references (Brown and Marco, Introduction to Heat Transfer, 3d ed., McGraw-Hill, New York, 1958; and Jakob, Heat Transfer, Wiley, New York, vol. 1, 1949; vol. 2, 1957) but have generally been applied successfully to the simple case of a vertical plate. Solution of the motion and energy equations gives temperature and velocity fields from which heat-transfer coefficients may be derived. The general type of equation obtained is the so-called Nusselt equation: L3ρ2gβ ∆t cµ m hL (5-32a) ᎏ = a ᎏᎏ ᎏ k µ2 k

q

0.84

(NGr NPr)0.28



3

7

hδ δ ᎏ = 0.2 + 0.145 ᎏ NGr k D1





0.25

δ exp −0.02 ᎏ D1



b, air at Configuration

Y = NGrNPr

a

m

Vertical surfaces L = vertical dimension < 3 ft

<104 104 < Y < 109 >109 <10−5 −5 10 < Y < 10−3 10−3 < Y < 1 1 < Y < 104 104 < Y < 109 >109 105 < Y < 2 × 107(FU) 2 × 107 < Y < 3 × 1010(FU) 3 × 105 < Y < 3 × 1010(FD)

1.36 0.59 0.13 0.49 0.71 1.09 1.09 0.53 0.13 0.54 0.14 0.27

1⁄ 5

Horizontal flat surface



(5-36)

for 0.55 < δ/D1 < 2.65, where NGr is based upon gap width δ and D1 is the core diameter of the annulus.

Values of a, m, and b for Eqs. (5-32) and (5-34)

Horizontal cylinder L = diameter < 8 in

(5-35)

for 2 × 10 < NGr NPr (δ/L) < 10 , where q/A is the uniform heat flux and ∆ t is the temperature difference at L/2. Equation (5-35) is applicable for air, water, and silicone oils. For horizontal annuli Grugal and Hauf (Proc. Third Int. Heat Transfer Conf., Chicago, 1966, vol. II, p 182) report 3

(5-32b) NNu = a(NGr NPr) Nusselt Equation for Various Geometries Natural-convection coefficients for various bodies may be predicted from Eq. (5-32). The various numerical values of a and m have been determined experimen-

NOTE:

δ

= 0.123(δ/L) ᎏA  ᎏ k ∆t

m

TABLE 5-1

(5-33b)

for NGr < 109. Simplified Dimensional Equations Equation (5-32) is a dimensionless equation, and any consistent set of units may be used. Simplified dimensional equations have been derived for air, water, and organic liquids by rearranging Eq. (5-32) into the following form by collecting the fluid properties into a single factor:

NATURAL CONVECTION



5-13

d s 0 1⁄25 1⁄ 10 1⁄ 5 d s d s d

b, water at

b, organic liquid at

21°C

70°F

21°C

70°F

21°C

70°F

1.37 1.24

0.28 0.18

127

26

59

12

1.32 1.24 1.86

0.27 0.18 0.38

0.88

0.18

FU = facing upward; FD = facing downward. b in SI units is given in °C column; b in U.S. customary units, in °F column.

5-14

HEAT AND MASS TRANSFER siderable effort has been directed toward deriving some simple relationship between momentum and heat transfer. The methodology has been to use easily observed velocity profiles to obtain a measure of the diffusivity of momentum in the flowing stream. The analogy between heat and momentum is invoked by assuming that diffusion of heat and diffusion of momentum occur by essentially the same mechanism so that a relatively simple relationship exists between the diffusion coefficients. Thus, the diffusivity of momentum is used to predict temperature profiles and thence by Eq. (10-25) to predict the heat-transfer coefficient. The analogy has been reasonably successful for simple geometries and for fluids of very low Prandtl number (liquid metals). For highPrandtl-number fluids the empirical analogy of Colburn [Trans. Am. Inst. Chem. Eng., 29, 174 (1933)] has been very successful. A j factor for momentum transfer is defined as j = f/2, where f is the friction factor for the flow. The j factor for heat transfer is assumed to be equal to the j factor for momentum transfer j = h/cG(cµ/k)2/3

(5-37)

More involved analyses for circular tubes reduce the equations of motion and energy to the form (ν + εM)du τgc (5-38a) ᎏ = − ᎏᎏ dy ρ (α + εH)dt q/A (5-38b) ᎏ = − ᎏᎏ dy cρ where εH is the eddy diffusivity of heat and εM is the eddy diffusivity of momentum. The units of diffusivity are L2/θ. The eddy viscosity is EM = ρεM, and the eddy conductivity of heat is EH = εHcρ. Values of εM are determined via Eq. (5-38a) from experimental velocitydistribution data. By assuming εH/εM = constant (usually unity), Eq. (10-38b) is solved to give the temperature distribution from which the heat-transfer coefficient may be determined. The major difficulties in solving Eq. (5-38b) are in accurately defining the thickness of the various flow layers (laminar sublayer and buffer layer) and in obtaining a suitable relationship for prediction of the eddy diffusivities. For assistance in predicting eddy diffusivities, see Reichardt (NACA Tech. Memo 1408, 1957) and Strunk and Chao [Am. Inst. Chem. Eng. J., 10, 269 (1964)]. Internal and External Flow Two main types of flow are considered in this subsection: internal or conduit flow, in which the fluid completely fills a closed stationary duct, and external or immersed flow, in which the fluid flows past a stationary immersed solid. With internal flow, the heat-transfer coefficient is theoretically infinite at the location where heat transfer begins. The local heat-transfer coefficient rapidly decreases and becomes constant, so that after a certain length the average coefficient in the conduit is independent of the length. The local coefficient may follow an irregular pattern, however, if obstructions or turbulence promoters are present in the duct. For immersed flow, the local coefficient is again infinite at the point where heating begins, after which it decreases and may show various irregularities depending upon the configuration of the body. Usually in this instance the local coefficient never becomes constant as flow proceeds downstream over the body. When heat transfer occurs during immersed flow, the rate is dependent upon the configuration of the body, the position of the body, the proximity of other bodies, and the flow rate and turbulence of the

FIG. 5-7 Radiation coefficients of heat transfer hr. To convert British thermal units per hour-square foot-degrees Fahrenheit to joules per square meter– second–kelvins, multiply by 5.6783; °C = (°F − 32)/1.8.

FORCED CONVECTION Forced-convection heat transfer is the most frequently employed mode of heat transfer in the process industries. Hot and cold fluids, separated by a solid boundary, are pumped through the heat-transfer equipment, the rate of heat transfer being a function of the physical properties of the fluids, the flow rates, and the geometry of the system. Flow is generally turbulent, and the flow duct varies in complexity from circular tubes to baffled and extended-surface heat exchangers. Theoretical analyses of forced-convection heat transfer have been limited to relatively simple geometries and laminar flow. Analyses of turbulent-flow heat transfer have been based upon some mechanistic model and have not generally yielded relationships which were suitable for design purposes. Usually for complicated geometries only empirical relationships are available, and frequently these are based upon limited data and special operating conditions. Heattransfer coefficients are strongly influenced by the mechanics of flow occurring during forced-convection heat transfer. Intensity of turbulence, entrance conditions, and wall conditions are some of the factors which must be considered in detail as greater accuracy in prediction of coefficients is required. Analogy between Momentum and Heat Transfer The interrelationship of momentum transfer and heat transfer is obvious from examining the equations of motion and energy. For constant fluid properties, the equations of motion must be solved before the energy equation is solved. If fluid properties are not constant, the equations are coupled, and their solutions must proceed simultaneously. ConTABLE 5-2

Nominal pipe diameter, in 1 3 5 10

Values of (hc + hr)*

Btu/(h⋅ft2⋅°F from pipe to room) For horizontal bare standard steel pipe of various sizes in a room at 80°F Temperature difference, °F

30

50

100

150

200

250

300

350

400

450

500

550

600

650

700

2.16 1.97

2.26 2.05 1.95 1.87

2.50 2.25 2.15 2.07

2.73 2.47 2.36 2.29

3.00 2.73 2.61 2.54

3.29 3.00 2.90 2.82

3.60 3.31 3.20 3.12

3.95 3.69 3.54 3.47

4.34 4.03 3.90 3.84

4.73 4.43

5.16 4.85

5.60 5.26

6.05 5.71

6.51 6.19

6.98 6.66

1.80

*Bailey and Lyell [Engineering, 147, 60 (1939)] give values for (hc + hr) up to ∆ts of 1000°F. °C = (°F − 32)/1.8; 5.6783 Btu/(h⋅ft2⋅°F) = J/(m2⋅s⋅/K).

HEAT TRANSFER BY CONVECTION TABLE 5-3

Values of a and m for Eq. (5-32)

Configuration Vertical spaces Horizontal spaces

NGr NPr(δ/L)3

a

m

2 × 104 to 2 × 105 2 × 105 to 107 104 to 3 × 105 3 × 105 to 107

0.20 (δ/L)−5/36 0.071 (δ/L)1/9 0.21 (δ/L)−1/4 0.075

d s d s

stream. The heat-transfer coefficient varies over the immersed body, since both the thermal and the momentum boundary layers vary in thickness. Relatively simple relationships are available for simple configurations immersed in an infinite flowing fluid. For complicated configurations and assemblages of bodies such as are found on the shell side of a heat exchanger, little is known about the local heattransfer coefficient; empirical relationships giving average coefficients are all that are usually available. Research that has been conducted on local coefficients in complicated geometries has not been extensive enough to extrapolate into useful design relationships. Laminar Flow Normally, laminar flow occurs in closed ducts when NRe < 2100 (based on equivalent diameter De = 4 × free area ÷ perimeter). Laminar-flow heat transfer has been subjected to extensive theoretical study. The energy equation has been solved for a variety of boundary conditions and geometrical configurations. However, true laminar-flow heat transfer very rarely occurs. Natural-convection effects are almost always present, so that the assumption that molecular conduction alone occurs is not valid. Therefore, empirically derived equations are most reliable. Data are most frequently correlated by the Nusselt number (NNu)lm or (NNU)am, the Graetz number NGz = (NRe NPr D/L), and the Grashof (natural-convection effects) number NGr. Some correlations consider only the variation of viscosity with temperature, while others also consider density variation. Theoretical analyses indicate that for very long tubes (NNu)lm approaches a limiting value. Limiting Nusselt numbers for various closed ducts are shown in Table 5-4. Circular Tubes For horizontal tubes and constant wall temperature, several relationships are available, depending on the Graetz number. For 0.1 < NGz < 104, Hausen’s [Allg. Waermetech., 9, 75 (1959)], the following equation is recommended. 0.8 µb 0.14 0.19NGz (NNu)lm = 3.66 + ᎏᎏ (5-39) ᎏ 0.467 1 + 0.117NGz µw

 

For NGz > 100, the Sieder-Tate relationship [Ind. Eng. Chem., 28, 1429 (1936)] is satisfactory for small diameters and ∆t’s: 1/3 (NNu)am = 1.86NGz (µb /µw)0.14

(5-40)

A more general expression covering all diameters and ∆ t’s is 1/3 obtained by including an additional factor 0.87(1 + 0.015NGr ) on the right side of Eq. (5-40). The diameter should be used in evaluating NGr. An equation published by Oliver [Chem. Eng. Sci., 17, 335 (1962)] is also recommended.

TABLE 5-4 Values of Limiting Nusselt Number in Laminar Flow in Closed Ducts Limiting Nusselt number NGr < 4.0 Configuration Circular tube Concentric annulus Equilateral triangle Rectangles Aspect ratio: 1.0 (square) 0.713 0.500 0.333 0.25 0 (parallel planes)

For laminar flow in vertical tubes a series of charts developed by Pigford [Chem. Eng. Prog. Symp. Ser. 17, 51, 79 (1955)] may be used to predict values of ham. Annuli Approximate heat-transfer coefficients for laminar flow in annuli may be predicted by the equation of Chen, Hawkins, and Solberg [Trans. Am. Soc. Mech. Eng., 68, 99 (1946)]:

Constant wall temperature

Constant heat flux

3.66

4.36 Eq. (10-42) 3.00

2.89

3.63 3.78 4.11 4.77 5.35 8.24

3.39 7.60

µb

ᎏ   ᎏ D  µ 

0.45 0.5 De (NNu)am = 1.02N Re N Pr ᎏ L

δ = cell width, L = cell length.

5-15

0.4

D2

0.8

1

0.14 0.05 NGr

(5-41)

1

Limiting Nusselt numbers for slug-flow annuli may be predicted (for constant heat flux) from Trefethen (General Discussions on Heat Transfer, London, ASME, New York, 1951, p. 436): 8(m − 1)(m2 − 1)2 (NNu)lm = ᎏᎏᎏ 4m4 ln m − 3m4 + 4m2 − 1

(5-42)

where m = D2/D1. The Nusselt and Reynolds numbers are based on the equivalent diameter, D2 − D1. Limiting Nusselt numbers for laminar flow in annuli have been calculated by Dwyer [Nucl. Sci. Eng., 17, 336 (1963)]. In addition, theoretical analyses of laminar-flow heat transfer in concentric and eccentric annuli have been published by Reynolds, Lundberg, and McCuen [Int. J. Heat Mass Transfer, 6, 483, 495 (1963)]. Lee [Int. J. Heat Mass Transfer, 11, 509 (1968)] presented an analysis of turbulent heat transfer in entrance regions of concentric annuli. Fully developed local Nusselt numbers were generally attained within a region of 30 equivalent diameters for 0.1 < NPr < 30, 104 < NRe < 2 × 105, 1.01 < D2 /D1 < 5.0. Parallel Plates and Rectangular Ducts The limiting Nusselt number for parallel plates and flat rectangular ducts is given in Table 5-4. Norris and Streid [Trans. Am. Soc. Mech. Eng., 62, 525 (1940)] report for constant wall temperature 1/3 (NNu)lm = 1.85NGz

(5-43)

for NGz > 70. Both Nusselt number and Graetz numbers are based on equivalent diameter. For large temperature differences it is advisable to apply the correction factor (µb/µw)0.14 to the right side of Eq. (5-43). For rectangular ducts Kays and Clark (Stanford Univ., Dept. Mech. Eng. Tech. Rep. 14, Aug. 6, 1953) published relationships for heating and cooling of air in rectangular ducts of various aspect ratios. For most noncircular ducts Eqs. (5-39) and (5-40) may be used if the equivalent diameter (= 4 × free area/wetted perimeter) is used as the characteristic length. See also Kays and London, Compact Heat Exchangers, 3d ed., McGraw-Hill, New York, 1984. Immersed Bodies When flow occurs over immersed bodies such that the boundary layer is completely laminar over the whole body, laminar flow is said to exist even though the flow in the mainstream is turbulent. The following relationships are applicable to single bodies immersed in an infinite fluid and are not valid for assemblages of bodies. In general, the average heat-transfer coefficient on immersed bodies is predicted by NNu = Cr(NRe)m(NPr)1/3

(5-44)

Values of Cr and m for various configurations are listed in Table 5-5. The characteristic length is used in both the Nusselt and the Reynolds numbers, and the properties are evaluated at the film temperature = (tw + t∞)/2. The velocity in the Reynolds number is the undisturbed free-stream velocity. Heat transfer from immersed bodies is discussed in detail by Eckert and Drake, Jakob, and Knudsen and Katz (see “Introduction: General References”), where equations for local coefficients and the effects of unheated starting length are presented. Equation (5-44) may also be expressed as m−1 NSt NPr2/3 = Cr NRe = f/2

(5-45)

where f is the skin-friction drag coefficient (not the form drag coefficient). Falling Films When a liquid is distributed uniformly around the periphery at the top of a vertical tube (either inside or outside) and allowed to fall down the tube wall by the influence of gravity, the fluid

5-16

HEAT AND MASS TRANSFER

TABLE 5-5

Laminar-Flow Heat Transfer over Immersed Bodies [Eq. (5-44)] Configuration

Characteristic length

Flat plate parallel to flow Circular cylinder axes perpendicular to flow

Plate length Cylinder diameter

Non-circular cylinder, axis Perpendicular to flow, characteristic Length perpendicular to flow

Square, short diameter Square, long diameter Hexagon, short diameter Hexagon, long diameter

Sphere*

Diameter

NRe

NPr

Cr

m

103 to 3 × 105 1–4 4 – 40 40 – 4000 4 × 103 – 4 × 104 4 × 104 – 2.5 × 105 5 × 103 – 105 5 × 103 – 105 5 × 103 – 105 5 × 103 – 2 × 104 2 × 104 – 105 1 – 7 × 104

>0.6

0.648 0.989 0.911 0.683 0.193 0.0266 0.104 0.250 0.155 0.162 0.0391 0.6

0.50 0.330 0.385 0.466 0.618 0.805 0.675 0.588 0.638 0.638 0.782 0.50

>0.6

>0.6 0.6 – 400

*Replace NNu by NNu − 2.0 in Eq. (5-44).

does not fill the tube but rather flows as a thin layer. Similarly, when a liquid is applied uniformly to the outside and top of a horizontal tube, it flows in layer form around the periphery and falls off the bottom. In both these cases the mechanism is called gravity flow of liquid layers or falling films. For the turbulent flow of water in layer form down the walls of vertical tubes the dimensional equation of McAdams, Drew, and Bays [Trans. Am. Soc. Mech. Eng., 62, 627 (1940)] is recommended: hlm = bΓ1/3

(5-46)

where b = 9150 (SI) or 120 (U.S. customary) and is based on values of Γ = WF/πD ranging from 0.25 to 6.2 kg/(m⭈s) [600 to 15,000 lb/(h⭈ft)] of wetted perimeter. This type of water flow is used in vertical vaporin-shell ammonia condensers, acid coolers, cycle water coolers, and other process-fluid coolers. The following dimensional equations may be used for any liquid flowing in layer form down vertical surfaces: For

k3ρ2g 4Γ ᎏ > 2100 hlm = 0.01 ᎏ µ µ2

For

4Γ k2ρ4/3cg2/3 ᎏ < 2100 ham = 0.50 ᎏᎏ µ Lµ1/3





4Γ  ᎏk  ᎏ µ  1/3

cµ

1/3

µ

1/3

(5-47a)

ᎏ  ᎏ µ   µ  1/3

1/4

4Γ

1/9

(5-47b)

w

Equation (5-47b) is based on the work of Bays and McAdams [Ind. Eng. Chem., 29, 1240 (1937)]. The significance of the term L is not clear. When L = 0, the coefficient is definitely not infinite. When L is large and the fluid temperature has not yet closely approached the wall temperature, it does not appear that the coefficient should necessarily decrease. Within the finite limits of 0.12 to 1.8 m (0.4 to 6 ft), this equation should give results of the proper order of magnitude. For falling films applied to the outside of horizontal tubes, the Reynolds number rarely exceeds 2100. Equations may be used for falling films on the outside of the tubes by substituting πD/2 for L. For water flowing over a horizontal tube, data for several sizes of pipe are roughly correlated by the dimensional equation of McAdams, Drew, and Bays [Trans. Am. Soc. Mech. Eng., 62, 627 (1940)]. ham = b (Γ/D0)1/3

4, 91 (1934)] fits both the laminar extreme and the fully turbulent extreme quite well. D 2/3 µb 0.14 2/3 (NNu)am = 0.116(NRe − 125)NPr1/3 1 + ᎏ (5-49) ᎏ L µw between 2100 and 10,000. It is customary to represent the probable magnitude of coefficients in this region by hand-drawn curves (Fig. 5-8). Equation (5-40) is plotted as a series of curves ( j factor versus Reynolds number with L/D as parameters) terminating at Reynolds number = 2100. Continuous curves for various values of L/D are then hand-drawn from these terminal points to coincide tangentially with the curve for forced-convection, fully turbulent flow [Eq. (5-50c)].

    

Turbulent Flow Circular Tubes Numerous relationships have been proposed for predicting turbulent flow in tubes. For high-Prandtl-number fluids, relationships derived from the equations of motion and energy through the momentum-heat-transfer analogy are more complicated and no more accurate than many of the empirical relationships that have been developed. For NRe > 10,000, 0.7 < NPr < 170, for properties based on the bulk temperature and for heating, the Dittus-Boelter equation [Boelter, Cherry, Johnson and Martinelli, Heat Transfer Notes, McGrawHill, New York (1965)] may be used: 0.8 0.4 NNu = 0.0243 NRe NPr (µ b /µw)0.14

(5-50a)

For cooling, the relationship is 0.14 0.3 NNu = 0.0265 N 0.8 Re N Pr (µb /µw)

(5.50b)

The Colburn correlation is 2/3 −0.2 (µw /µb)0.14 = 0.023NRe jH = NSt NPr

(5-50c)

In Eq. (5-50c), the viscosity-ratio factor may be neglected if properties are evaluated at the film temperature (tb + tw)/2.

(5-48)

where b = 3360 (SI) or 65.6 (U.S. customary) and Γ ranges from 0.94 to 4 kg/m⋅s) [100 to 1000 lb/(h⋅ft)]. Falling films are also used for evaporation in which the film is both entirely or partially evaporated (juice concentration). This principle is also used in crystallization (freezing). The advantage of high coefficient in falling-film exchangers is partially offset by the difficulties involved in distribution of the film, maintaining complete wettability of the tube, and pumping costs required to lift the liquid to the top of the exchanger. Transition Region Turbulent-flow equations for predicting heat transfer coefficients are usually valid only at Reynolds numbers greater than 10,000. The transition region lies in the range 2000 < NRe < 10,000. No simple equation exists for accomplishing a smooth mathematical transition from laminar flow to turbulent flow. Of the relationships proposed, Hausen’s equation [Z. Ver. Dtsch. Ing. Beih. Verfahrenstech., No.

Graphical representation of the Colburn j factor for the heating and cooling of fluids inside tubes. The curves for NRe below 2100 are based on Eq. (5-40). L is the length of each pass in feet. The curve for NRe above 10,000 is represented by Eq. (5-50c).

FIG. 5-8

HEAT TRANSFER BY CONVECTION For the transition and turbulent regions, including diameter to length effects, Gnielinski [Int. Chem. Eng., 16, 359 (1976)] recommends a modification of an equation suggested by Petukhov and Popov [High Temp., 1, 69 (1963)]. This equation applies in the ranges 0 < D/L < 1, 0.6 < NPr < 2000, 2300 < NRe < 106. ( f/2)(NRe − 1000) NPr D 1+ ᎏ NNu = ᎏᎏᎏ 2/3 1 + 12.7( f/2)0.5(NPr − 1) L

µb

   ᎏ µ  2/3

0.14

(5-51a)

w

The Fanning friction f is determined by an equation recommended by Filonenko [Teploenergetika, 1, 40 (1954)] f = 0.25 (1.82 log10 NRe − 1.64)−2

(5-51b)

Any other appropriate friction factor equation for smooth tubes may be used. Approximate predictions for rough pipes may be obtained from Eq. (5-50c) if the right-hand term is replaced by f / 2 for the rough pipe. For air, Nunner (Z. Ver. Dtsch. Ing. Forsch., 1956, p. 455) obtains frough (NNu)rough (5-52) ᎏᎏ = ᎏ (NNu)smooth fsmooth Dippery and Sabersky [Int. J. Heat Mass Transfer, 6, 329 (1963)] present a complete discussion of the influence of roughness on heat transfer in tubes. Dimensional Equations for Various Conditions For gases at ordinary pressures and temperatures based on cµ/k = 0.78 and µ = (1.76)(10−5) Pa⋅s [0.0426 lb/(ft⋅h)] h = bcρ0.8(V 0.8/D0.2)

(5-53)

where b = (3.04)(10−3) (SI) or (1.44)(10−2) (U.S. customary). For air at atmospheric pressure h = b(V 0.8/D0.2)

(5-54)

where b = 3.52 (SI) or (4.35)(10−4) (U.S. customary). For water [based on a temperature range of 5 to 104°C (40 to 220°F)] h = 1057 (1.352 + 0.02t) (V 0.8/D0.2)

(5-55a)

in SI units with t = °C, or h = 0.13(1 + 0.011t)(V 0.8/D0.2)

(5-55b)

in U.S. customary units with t = °F. For organic liquids, based on c = 2.092 J/kg⋅K)[0.5 Btu/(lb⋅°F)], k = 0.14 J/(m⋅s⋅K) [0.08 Btu/(h⋅ft⋅°F)], µb = (1)(10−3) Pa⋅s (1.0 cP), and ρ = 810 kg/m3 (50 lb/ft3), h = b(V 0.8/D0.2)

(5-56)

where b = 423 (SI) or (5.22)(10 ) (U.S. customary). Within reasonable limits, coefficients for organic liquids are about one-third of the values obtained for water. Entrance effects are usually not significant industrially if L/D > 60. Below this limit Nusselt recommended the conservative equation for 10 < L/D < 400 and properties evaluated at bulk temperature −2

0.8 NNu = 0.036NRe NPr1/3(L/D)−0.054

(5-57)

It is common to correlate entrance effects by the equation hm /h = 1 + F(D/L)

(5-58)

where h is predicted by Eq. (5-50a) or (5-50b), and hm is the mean coefficient for the pipe in question. Values of F are reported by Boelter, Young, and Iverson [NACA Tech. Note 1451, 1948] and tabulated by Kays and Knudsen and Katz (see “Introduction: General References”). Selected values of F are as follows: Fully developed velocity profile 1.4 Abrupt contraction entrance 6 90° right-angle bend 7 180° round bend 6

5-17

For large temperature differences different equations are necessary and usually are specifically applicable to either gases or liquids. Gambill (Chem. Eng., Aug. 28, 1967, p. 147) provides a detailed review of high-flux heat transfers to gases. He recommends 0.8 0.4 0.021NRe NPr NNu = ᎏᎏ (5-59) (Tw /Tb)0.29 + 0.0019 (L/D) for 10 < L/D < 240, 110 < Tb < 1560 K (200 < Tb < 2800°R), 1.1 < (Tw /Tb) < 8.0, and properties evaluated at Tb. For liquids, Eq. (5-50c) is generally satisfactory. Annuli For diameter ratios D1/D2 > 0.2, Monrad and Pelton’s equation [Trans. Am. Inst. Chem. Eng., 38, 593 (1942)] is recommended for either or both the inner and outer tube: 0.8 1/3 NNu = 0.020NRe NPr (D2/D1)0.53 (5-60a) Equation (5-51a) may also be used for smooth annuli as follows: (NNu)ann D1 (5-60b) ᎏ=φ ᎏ (NNu)tube D2 The hydraulic diameter D2 − D1 is used in NNu, NRe, and D/L is used for the annulus. The function on the right of Eq. (5-60b) is given by Petukhov and Roizen [High Temp., 2, 65 (1964)] as follows: Inner tube heated 0.86 (D1 /D2)−0.16 Outer tube heated 1 − 0.14 (D1 /D2)0.6 If both tubes are heated, the function is the sum of the above two functions divided by 1 + D1/D2 [Stephan, Chem. Ing. Tech., 34, 207 (1962)]. The Colburn form of relationship may be employed for the individual walls of the annulus by using the individual friction factor for each wall [see Knudsen, Am. Inst. Chem. Eng. J., 8, 566 (1962)]: jH1 = (NSt)1NPr2/3 = f1 /2 (5-61a)

 

jH2 = (NSt)2 NPr2/3 = f2 /2 (5-61b) Rothfus, Monrad, Sikchi, and Heideger [Ind. Eng. Chem., 47, 913 (1955)] report that the friction factor f2 for the outer wall bears the same relation to the Reynolds number for the outer portion of the annular stream 2(r22 − λm)Vρ /r2 µ as the friction factor for circular tubes does to the Reynolds number for circular tubes, where r2 is the radius of the outer tube and λm is the position of maximum velocity in the annulus, estimated from r22 − r12 λm = ᎏ (5-62)* ln (r2/r1)2 To calculate the friction factor f1 for the inner tube use the relation f2 r2(λm − r12) f1 = ᎏᎏ (5-63) r1(r22 − λm) There have been several analyses of turbulent heat transfer in annuli: for example, Deissler and Taylor (NACA Tech. Note 3451, 1955), Kays and Leung [Int. J. Heat Mass Transfer, 6, 537 (1963)], Lee [Int. J. Heat Transfer, 11, 509 (1968)], Sparrow, Hallman and Siegel [Appl. Sci. Res., 7A, 37 (1958)], and Johnson and Sparrow [Am. Soc. Mech. Eng. J. Heat Transfer, 88, 502 (1966)]. The reader is referred to these for details of the analyses. For annuli containing externally finned tubes the heat-transfer coefficients are a function of the fin configurations. Knudsen and Katz (Fluid Dynamics and Heat Transfer, McGraw-Hill, New York, 1958) present relationships for transverse finned tubes, spined tubes, and longitudinal finned tubes in annuli. Noncircular Ducts Equations (5-50a) and (5-50b) may be employed for noncircular ducts by using the equivalent diameter De = 4 × free area per wetted perimeter. Kays and London (Compact Heat Exchangers, 3rd ed., McGraw-Hill, New York, 1984) give charts for various noncircular ducts encountered in compact heat exchangers. Vibrations and pulsations generally tend to increase heat-transfer coefficients. Example 2: Calculation of j Factors in an Annulus Calculate the heat-transfer j factors for both walls of an annulus for the following condi-

* Equation (5-62) predicts the point of maximum velocity for laminar flow in annuli and is only an approximate equation for turbulent flow. Brighton and Jones [Am. Soc. Mech. Eng. Basic Eng., 86, 835 (1964)] and Macagno and McDougall [Am. Inst. Chem. Eng. J., 12, 437 (1966)] give more accurate equations for predicting the point of maximum velocity for turbulent flow.

5-18

HEAT AND MASS TRANSFER

tions: D1 = 0.0254 m (1.0 in); D2 = 0.0635 m (2.5 in); water at 15.6°C (60°F); µ/ρ = (1.124)(10−6) m2/s [(1.21)(10−5) ft2/s]; velocity = 1.22 m/s (4 ft/s). 0.06352 − 0.02542 λm = ᎏᎏᎏ2 = (4.621)(10−4) m2 (0.716 in2) 4 ln (0.0635/0.0254) 2(r 22 − λm)Vρ 2[0.03182 − (4.621)(10−4)(1.22)] Re2 = ᎏᎏ = ᎏᎏᎏᎏ = (3.74)(104) r2µ (0.0318)(1.124)(10−6) From Eq. (5-51b), f2 = 0.0055. Hence 2/3 jH2 = (NSt)2 N Pr = 0.00275 From Eq. (5-63), (0.0055)(0.0318)[(4.621)(10−4) − 0.01272] = 0.00754 f1 = ᎏᎏᎏᎏᎏ (0.0127)[0.03182 − (4.621)(10−4)] 2/3 from which jH1 = (NSt)1N Pr = 0.00377. These results indicate that for this system the heat-transfer coefficient on the inner tube is about 40 percent greater than on the outer tube.

Coils For flow inside helical coils, Reynolds number above 10,000, multiply the value of the film coefficient obtained from the applicable equation for straight tubes by the term (1 + 3.5 Di /Dc). For flow inside helical coils, Reynolds number less than 10,000, substitute the term (Dc /Di)1/2 for (L/Di) where the latter appears in the applicable equation for straight tubes (frequently as part of the Graetz number). For flat spiral (pancake) coils, in which the ratio Dc /Di varies for each turn, a different value of coefficient will be obtained for each turn; a weighted average based on length per turn is used. For flow outside helical coils use the equation for flow normal to a bank of tubes, in-line flow. Finned Tubes (Extended Surface) When the film coefficient on the outside of a metal tube is much lower than that on the inside, as when steam condensing in a pipe is being used to heat air, externally finned (or extended) heating surfaces are of value in increasing substantially the rate of heat transfer per unit length of tube. The data on extended heating surfaces, for the case of air flowing outside and at right angles to the axes of a bank of finned pipes, can be represented approximately by the dimensional equation derived from 0.6 VF0.6 p′ hf = b ᎏ (5-64) ᎏ 0.4 D0 p′ − D0 −3 where b = 5.29 (SI) or (5.39)(10 ) (U.S. customary); hf is the film coefficient of heat transfer on the air side; VF is the face velocity of the air; p′ is the center-to-center spacing, m, of the tubes in a row; and D0 is the outside diameter, m, of the bare tube (diameter at the root of the fins). In atmospheric air-cooled finned tube exchangers, the air-film coefficient from Eq. (5-64) is sometimes converted to a value based on outside bare surface as follows: Af + Auf A hfo = hf ᎏ = hf ᎏT (5-65) Aof Ao in which hfo is the air-film coefficient based on external bare surface; hf is the air-film coefficient based on total external surface; AT is total external surface, and Ao is external bare surface of the unfinned tube; Af is the area of the fins; Auf is the external area of the unfinned portion of the tube; and Aof is area of tube before fins are attached. Fin efficiency is defined as the ratio of the mean temperature difference from surface to fluid divided by the temperature difference from fin to fluid at the base or root of the fin. Graphs of fin efficiency for extended surfaces of various types are given by Gardner [Trans. Am. Soc. Mech. Eng., 67, 621 (1945)]. Heat-transfer coefficients for finned tubes of various types are given in a series of papers [Trans. Am. Soc. Mech. Eng., 67, 601 (1945)]. For flow of air normal to fins in the form of short strips or pins, Norris and Spofford [Trans. Am. Soc. Mech. Eng., 64, 489 (1942)] correlate their results for air by the dimensionless equation of Pohlhausen: hm cpµ 2/3 zpGmax −0.5 = 1.0 ᎏ (5-66) ᎏ ᎏ cpGmax k µ for values of zpGmax/µ ranging from 2700 to 10,000.



 







For the general case, the treatment suggested by Kern (Process Heat Transfer, McGraw-Hill, New York, 1950, p. 512) is recommended. Because of the wide variations in fin-tube construction, it is convenient to convert all film coefficients to values based on the inside bare surface of the tube. Thus to convert the film coefficient based on outside area (finned side) to a value based on inside area Kern gives the following relationship: hfi = (ΩAf + Ao)(hf /Ai)

(5-67)

in which hfi is the effective outside film coefficient based on the inside area, hf is the outside film coefficient calculated from the applicable equation for bare tubes, Af is the surface area of the fins, Ao is the surface area on the outside of the tube which is not finned, Ai is the inside area of the tube, and Ω is the fin efficiency defined as Ω = (tanh mbf)/mbf

(5-68)

m = (hf pf / kax)1/2 m−1 (ft−1)

(5-69)

in which and bf = height of fin. The other symbols are defined as follows: pf is the perimeter of the fin, ax is the cross-sectional area of the fin, and k is the thermal conductivity of the material from which the fin is made. Fin efficiencies and fin dimensions are available from manufacturers. Ratios of finned to inside surface are usually available so that the terms A f, Ao, and Ai may be obtained from these ratios rather than from the total surface areas of the heat exchangers. Banks of Tubes For heating and cooling of fluids flowing normal to a bank of circular tubes at least 10 rows deep the following equations are applicable: Colburn type:

 

h cµ ᎏ ᎏ cGmax k

2/3

a = ᎏᎏ =j (DoGmax/µ)0.4

(5-70)

cµ  ᎏ k 

(5-71)

Nusselt type:



hD DoGmax ᎏ=a ᎏ k µ

0.6

1/3

The dimensionless constant a in these equations varies depending upon conditions. Conditions, Reynolds number > 3000

Value of a

Flow normal to apex of diamond, staggered arrangement No leakage Normal leakage in baffled exchanger Flow normal to flat side of diamond, not staggered (in-line) arrangement No leakage Normal leakage in baffled exchanger

0.330 0.198 0.260 0.156

For Reynolds number less than 3000, Eq. (5-70) would give conservative results, but greater accuracy (if desired) may be obtained by using the following equation.

 

h cµ ᎏ ᎏ cGmax k

2/3

a = ᎏᎏ =j (DoGmax /µ)m

(5-72)

in which the constant a and exponent m are as follows: Reynolds number

m

Tube pitch

100–300

0.492

Staggered In-line

1–100

0.590

Staggered In-line

Leakage

a

None Normal None Normal None Normal None Normal

0.695 0.416 0.548 0.329 1.086 0.650 0.855 0.513

HEAT TRANSFER BY CONVECTION The following dimensional equations (5-73 to 5-77) are based on flow normal to a bank of staggered tubes without leakage. Multiply the values obtained for h by 0.6 for normal leakage and, in addition, by 0.79 for in-line (not staggered) tube arrangement. c1/3k2/3ρ0.6 V 0.6 max h = b ᎏᎏ (5-73) µ0.267 D00.4 where b = 0.33 (SI) or 0.261 (U.S. customary). For gases at ordinary pressures and temperatures, based on cµ/k = 0.78; µ = (1.76)(10−5) Pa⭈s [0.0426 lb/(ft⭈h)], 0.6 Gmax h = bc ᎏ (5-74) 0.4 D0 where b = (4.82)(10−3) (SI) or 0.109 (U.S. customary). For air at atmospheric pressure 0.6 Vmax h=bᎏ (5-75) D00.4 where b = 5.33 (SI) or (5.44)(10−3) (U.S. customary). For water based on a temperature range 7 to 104°C (40 to 220°F) 0.6 Vmax h = 986(1.21 + 0.0121t) ᎏ (5-76a) D00.4 in SI units and t in °C. 0.6 Vmax h = 1.01(1 + 0.0067t) ᎏ (5-76b) D00.4 in U.S. customary units and t in °F. For organic liquids, based on c = 2.22 J/(kg⭈K) [0.53 Btu/(lb⭈°F)], k = 0.14 J/(m⭈s⭈K) [0.08 Btu/ (h⭈ft⭈°F)], µb = (1)(10−3) Pa⭈s (1.0 cP), ρ = 810 kg/m3 (50 lb/ft3), V 0.6 max h=bᎏ (5-77) D00.4 where b = 400 (SI) or 0.408 (U.S. customary).

5-19

LIQUID METALS Liquid metals constitute a class of heat-transfer media having Prandtl numbers generally below 0.01. Heat-transfer coefficients for liquid metals cannot be predicted by the usual design equations applicable to gases, water, and more viscous fluids with Prandtl numbers greater than 0.6. Relationships for predicting heat-transfer coefficients for liquid metals have been derived from solution of Eqs. (5-38a) and (5-38b). By the momentum-transfer-heat-transfer analogy, the eddy conductivity of heat is kNPr(EM /µ) ≈ k for small NPr. Thus in the solution of Eqs. (5-38a) and (5-38b) the knowledge of the thickness of various layers of flow is not critical. In fact, assumption of slug flow and constant conductivity (=k) across the duct gives reasonable values of heat-transfer coefficients for liquid metals. For constant heat flux: NNu = 5 + 0.025(NRe NPr)0.8

(5-81)

For constant wall temperature: NNu = 7 + 0.025(NRe NPr)0.8

(5-82)

For 0.003 < NPr < 0.05 and constant heat flux, Sleicher and Rouse [Int. J. Heat Mass Transfer, 18, 677 (1975)] obtained the correlation 0.85 NNu = 6.3 + 0.0167 NRe NPr0.93

(5-83)

For parallel plates and annuli with D2 /D1 < 1.4 and uniform heat flux, Seban [Trans. Am. Soc. Mech. Eng., 72, 789 (1950)] obtained the equation NNu = 5.8 + 0.020(NRe NPr)0.8

(5-84)

JACKETS AND COILS OF AGITATED VESSELS

For annuli only, application of a factor of 0.70(D2 /D1)0.53 is recommended for Eqs. (5-81) and (5-82). For more accurate semiempirical relationships for tubes, annuli, and rod bundles, refer to Dwyer [Am. Inst. Chem. Eng. J., 9, 261 (1963)]. Hsu [Int. J. Heat Mass Transfer, 7, 431 (1964)] and Kalish and Dwyer [Int. J. Heat Mass Transfer, 10, 1533 (1967)] discuss heat transfer to liquid metals flowing across banks of tubes. Hsu recommends the equations

See Sec. 18.

NNu = 0.81NRe NPr (φ/D)1/2

(for uniform heat flux)

NONNEWTONIAN FLUIDS

NNu = 0.096NRe NPr (φ/D)

(for cosine surface temperature)

1/2

A wide variety of nonnewtonian fluids are encountered industrially. They may exhibit Bingham-plastic, pseudoplastic, or dilatant behavior and may or may not be thixotropic. For design of equipment to handle or process nonnewtonian fluids, the properties must usually be measured experimentally, since no generalized relationships exist to predict the properties or behavior of the fluids. Details of handling nonnewtonian fluids are described completely by Skelland (NonNewtonian Flow and Heat Transfer, Wiley, New York, 1967). The generalized shear-stress rate-of-strain relationship for nonnewtonian fluids is given as d ln (D ∆P/4L) n′ = ᎏᎏ (5-78) d ln (8V/D) as determined from a plot of shear stress versus velocity gradient. For circular tubes, NGz > 100, n′ > 0.1, and laminar flow 1/3 (NNu)lm = 1.75 δ1/3 s NGz

(5-79)

where δs = (3n′ + 1)/4n′. When natural-convection effects are considered, Metzer and Gluck [Chem. Eng. Sci., 12, 185 (1960)] obtained the following for horizontal tubes: NPr NGr D 0.4 1/3 γb 0.14 (NNu)lm = 1.75 δ 1/3 NGz + 12.6 ᎏ (5-80) ᎏ s L γw where properties are evaluated at the wall temperature, i.e., γ = gc K′8n′ − 1 and τw = K′(8V/D)n′. Metzner and Friend [Ind. Eng. Chem., 51, 879 (1959)] present relationships for turbulent heat transfer with nonnewtonian fluids. Relationships for heat transfer by natural convection and through laminar boundary layers are available in Skelland’s book (op. cit.).





  

(5-85) (5-86)

where the heat-transfer coefficient is based on the average circumferential temperature around the tubes, the Reynolds number is based on the superficial velocity through the tube bank, D is the tube outside diameter, and φ is a velocity potential function having the following values:

D/p′

φ/D square pitch

φ/D equilateral triangular pitch

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

2.00 2.02 2.07 2.16 2.30 2.52 2.84 3.34 4.23

2.00 2.02 2.06 2.15 2.27 2.45 2.71 3.11 3.80

Equations (5-85) and (5-86) are useful in calculating tube-surface temperatures. Further information on liquid-metal heat transfer in tube banks is given by Hsu for spheres and elliptical rod bundles [Int. J. Heat Mass Transfer, 8, 303 (1965)] and by Kalish and Dwyer for oblique flow across tube banks [Int. J. Heat Mass Transfer, 10, 1533 (1967)]. For additional details of heat transfer with liquid metals for various systems see Dwyer (1968 ed., Na and Nak supplement to Liquid Metals Handbook) and Stein (“Liquid Metal Heat Transfer,” in Advances in Heat Transfer, vol. 3, Academic, New York, 1966).

5-20

HEAT AND MASS TRANSFER

HEAT TRANSFER WITH CHANGE OF PHASE In any operation in which a material undergoes a change of phase, provision must be made for the addition or removal of heat to provide for the latent heat of the change of phase plus any other sensible heating or cooling that occurs in the process. Heat may be transferred by any one or a combination of the three modes—conduction, convection, and radiation. The process involving change of phase involves mass transfer simultaneous with heat transfer. CONDENSATION Condensation Mechanisms Condensation occurs when a saturated vapor comes in contact with a surface whose temperature is below the saturation temperature. Normally a film of condensate is formed on the surface, and the thickness of this film, per unit of breadth, increases with increase in extent of the surface. This is called film-type condensation. Another type of condensation, called dropwise, occurs when the wall is not uniformly wetted by the condensate, with the result that the condensate appears in many small droplets at various points on the surface. There is a growth of individual droplets, a coalescence of adjacent droplets, and finally a formation of a rivulet. Adhesional force is overcome by gravitational force, and the rivulet flows quickly to the bottom of the surface, capturing and absorbing all droplets in its path and leaving dry surface in its wake. Film-type condensation is more common and more dependable. Dropwise condensation normally needs to be promoted by introducing an impurity into the vapor stream. Substantially higher (6 to 18 times) coefficients are obtained for dropwise condensation of steam, but design methods are not available. Therefore, the development of equations for condensation will be for the film type only. The physical properties of the liquid, rather than those of the vapor, are used for determining the film coefficient for condensation. Nusselt [Z. Ver. Dtsch. Ing., 60, 541, 569 (1916)] derived theoretical relationships for predicting the film coefficient of heat transfer for condensation of a pure saturated vapor. A number of simplifying assumptions were used in the derivation. The Reynolds number of the condensate film (falling film) is 4Γ/µ, where Γ is the weight rate of flow (loading rate) of condensate per unit perimeter kg/(s⭈m) [lb/(h⭈ft)]. The thickness of the condensate film for Reynolds number less than 2100 is (3µΓ/ρ2g)1/3. Condensation Coefficients Vertical Tubes For the following cases Reynolds number < 2100 and is calculated by using Γ = WF /πD. The Nusselt equation for the heat-transfer coefficient for condensate films may be written in the following ways (using liquid physical properties and where L is the cooled length and ∆t is tsv − ts): Colburn type: h cµ 5.35 ᎏ ᎏ=ᎏ cG k 4Γ/µ Γ WF2 ρ2g where G = ᎏᎏ = ᎏ (3µΓ/ρ2g)1/3 29.6D2µ





(5-87)

1/3

kg/(s⭈m2) [lb/(h⭈ft2)]





1/4

L3ρ2g = 0.925 ᎏ µΓ





1/3

(5-88)

Dimensional: h = b(k3ρ2D/µbWF)1/3

(5-89)

where b = 127 (SI) or 756 (U.S. customary). For steam at atmospheric pressure, k = 0.682 J/(m⭈s⭈K) [0.394 Btu/(h⭈ft⭈°F)], ρ = 960 kg/m3 (60 lb/ft3), µb = (0.28)(10−3) Pa⭈s (0.28 cP), h = b(D/WF)1/3

h = b(D/WF)1/3

(5-91)

where b = 457 (SI) or 1080 (U.S. customary). Horizontal Tubes For the following cases Reynolds number < 2100 and is calculated by using Γ = WF /2L. Colburn type: h cµ 4.4 (5-92) ᎏᎏ=ᎏ cG k 4Γ/µ Γ WF2 ρ2g 1/3 kg/(s⭈m2) [lb/(h⭈ft2)] G = ᎏᎏ = ᎏ (3µΓ/ρ2g)1/3 12L2µ Nusselt type: D3ρ2g 1/3 hD D3ρ2gλ 1/4 = 0.76 ᎏ (5-93)* ᎏ = 0.73 ᎏ k kµ ∆t µΓ Dimensional:













h = b(k3ρ2L/µbWF)1/3

(5-90)

(5-94)

where b = 205.4 (SI) or 534 (U.S. customary). For steam at atmospheric pressure h = b(L/WF)1/3

(5-95)

where b = 2080 (SI) or 4920 (U.S. customary). For organic vapors at normal boiling point h = b(L/WF)1/3

(5-96)

where b = 324 (SI) or 766 (U.S. customary). Figure 5-9 is a nomograph for determining coefficients of heat transfer for condensation of pure vapors. Banks of Horizontal Tubes (NRe < 2100) In the idealized case of N tubes in a vertical row where the total condensate flows smoothly from one tube to the one beneath it, without splashing, and still in laminar flow on the tube, the mean condensing coefficient hN for the entire row of N tubes is related to the condensing coefficient for the top tube h1 by hN = h1N−1/4

(5-97)

Dukler Theory The preceding expressions for condensation are based on the classical Nusselt theory. It is generally known and conceded that the film coefficients for steam and organic vapors calculated by the Nusselt theory are conservatively low. Dukler [Chem. Eng. Prog., 55, 62 (1959)] developed equations for velocity and temperature distribution in thin films on vertical walls based on expressions of Deissler (NACA Tech. Notes 2129, 1950; 2138, 1952; 3145, 1959) for the eddy viscosity and thermal conductivity near the solid boundary. According to the Dukler theory, three fixed factors must be known to establish the value of the average film coefficient: the terminal Reynolds number, the Prandtl number of the condensed phase, and a dimensionless group Nd defined as follows: 2/3 2 0.553 0.78 0.16 Nd = (0.250µ1.173 ρG ) L µG /(g D ρL

Nusselt type: hL L3ρ2gλ ᎏ = 0.943 ᎏ k kµ ∆t

where b = 2954 (SI) or 6978 (U.S. customary). For organic vapors at normal boiling point, k = 0.138 J/(m⭈s⭈K) [0.08 Btu/(h⭈ft⭈°F)], ρ = 720 kg/m3 (45 lb/ft3), µb = (0.35)(10−3) Pa⭈s (0.35 cP),

(5-98)

Graphical relationships of these variables are available in Document 6058, ADI Auxiliary Publications Project, Library of Congress, Washington. If rigorous values for condensing-film coefficients are desired, especially if the value of Nd in Eq. (5-98) exceeds (1)(10−5), it is suggested that these graphs be used. For the case in which interfacial shear is zero, Fig. 5-10 may be used. It is interesting to note that, according to the Dukler development, there is no definite transition Reynolds number; deviation from Nusselt theory is less at low Reynolds numbers; and when the Prandtl number of a fluid is less * If the vapor density is significant, replace ρ2 with ρl(ρl − ρv).

HEAT TRANSFER WITH CHANGE OF PHASE

5-21

Chart for determining film coefficient hm for film-type condensation of pure vapor, based on Eqs. 5-88 and 5-93. For 4 ρ 2k3/µ  is in U.S. customary units; to convert feet to vertical tubes multiply hm by 1.2. If 4Γ/µf exceeds 2100, use Fig. 5-10. λ meters, multiply by 0.3048; to convert inches to centimeters, multiply by 2.54; and to convert British thermal units per hour– square foot–degrees Fahrenheit to watts per square meter–kelvins, multiply by 5.6780.

FIG. 5-9

than 0.4 (at Reynolds number above 1000), the predicted values for film coefficient are lower than those predicted by the Nusselt theory. The Dukler theory is applicable for condensate films on horizontal tubes and also for falling films, in general, i.e., those not associated with condensation or vaporization processes.

Vapor Shear Controlling For vertical in-tube condensation with vapor and liquid flowing cocurrently downward, if gravity controls, Figs. 5-9 and 5-10 may be used. If vapor shear controls, the Carpenter-Colburn correlation (General Discussion on Heat Transfer, London, 1951, ASME, New York, p. 20) is applicable:

5-22

HEAT AND MASS TRANSFER

Dukler plot showing average condensing-film coefficient as a function of physical properties of the condensate film and the terminal Reynolds number. (Dotted line indicates Nusselt theory for Reynolds number < 2100.) [Reproduced by permission from Chem. Eng. Prog., 55, 64 (1959).] FIG. 5-10

1/2 hµl / kl ρl1/2 = 0.065(NPr)1/2 l Fvc

(5-99a)

Fvc = fG2vm /2ρv 2 1/2 Gvi2 + GviGvo + Gvo Gvm = ᎏᎏᎏ 3 and f is the Fanning friction factor evaluated at

where



(5-99b)



(5-99c)

(NRe)vm = DiGvm /µv

(5-99d)

and the subscripts vi and vo refer to the vapor inlet and outlet, respectively. An alternative formulation, directly in terms of the friction factor, is h = 0.065 (cρkf/2µρv)1/2Gvm

(5-99e)

expressed in consistent units. Another correlation for vapor-shear-controlled condensation is the Boyko-Kruzhilin correlation [Int. J. Heat Mass Transfer, 10, 361 (1967)], which gives the mean condensing coefficient for a stream between inlet quality xi and outlet quality xo: hDi DiGT 0.8 (ρ /ρ (ρ /ρ  m)i +  m)o (5-100a) ᎏ = 0.024 ᎏ (NPr)l0.43 ᎏᎏᎏ kl µl 2 where GT = total mass velocity in consistent units ρ ρl − ρv (5-100b) ᎏ = 1 + ᎏ xi ρm i ρv





 

ρ ρ −ρ =1+ᎏx ᎏ ρ  ρ l

and

v

(5-100c)

o

m

o

v

For horizontal in-tube condensation at low flow rates Kern’s modification (Process Heat Transfer, McGraw-Hill, New York, 1950) of the Nusselt equation is valid: k 3l ρl (ρl − ρv)gλ 1/4 Lk 3l ρl(ρl − ρv)g 1/3 hm = 0.761 ᎏᎏ = 0.815 ᎏᎏ (5-101) WF µl π µl Di ∆t where WF is the total vapor condensed in one tube and ∆ t is tsv − ts . A more rigorous correlation has been proposed by Chaddock [Refrig. Eng., 65(4), 36 (1957)]. Use consistent units. At high condensing loads, with vapor shear dominating, tube orientation has no effect, and Eq. (5-100a) may also be used for horizontal tubes. Condensation of pure vapors under laminar conditions in the presence of noncondensable gases, interfacial resistance, superheating, variable properties, and diffusion has been analyzed by Minkowycz and Sparrow [Int. J. Heat Mass Transfer, 9, 1125 (1966)].









BOILING (VAPORIZATION) OF LIQUIDS Boiling Mechanisms Vaporization of liquids may result from various mechanisms of heat transfer, singly or combinations thereof.

For example, vaporization may occur as a result of heat absorbed, by radiation and convection, at the surface of a pool of liquid; or as a result of heat absorbed by natural convection from a hot wall beneath the disengaging surface, in which case the vaporization takes place when the superheated liquid reaches the pool surface. Vaporization also occurs from falling films (the reverse of condensation) or from the flashing of liquids superheated by forced convection under pressure. Pool boiling refers to the type of boiling experienced when the heating surface is surrounded by a relatively large body of fluid which is not flowing at any appreciable velocity and is agitated only by the motion of the bubbles and by natural-convection currents. Two types of pool boiling are possible: subcooled pool boiling, in which the bulk fluid temperature is below the saturation temperature, resulting in collapse of the bubbles before they reach the surface, and saturated pool boiling, with bulk temperature equal to saturation temperature, resulting in net vapor generation. The general shape of the curve relating the heat-transfer coefficient to ∆tb, the temperature driving force (difference between the wall temperature and the bulk fluid temperature) is one of the few parametric relations that are reasonably well understood. The familiar boiling curve was originally demonstrated experimentally by Nukiyama [ J. Soc. Mech. Eng. ( Japan), 37, 367 (1934)]. This curve points out one of the great dilemmas for boiling-equipment designers. They are faced with at least six heat-transfer regimes in pool boiling: natural convection (+), incipient nucleate boiling (+), nucleate boiling (+), transition to film boiling (−), stable film boiling (+), and film boiling with increasing radiation (+). The signs indicate the sign of the derivative d(q/A)/d ∆ tb. In the transition to film boiling, heat-transfer rate decreases with driving force. The regimes of greatest commercial interest are the nucleate-boiling and stable-film-boiling regimes. Heat transfer by nucleate boiling is an important mechanism in the vaporization of liquids. It occurs in the vaporization of liquids in kettle-type and natural-circulation reboilers commonly used in the process industries. High rates of heat transfer per unit of area (heat flux) are obtained as a result of bubble formation at the liquid-solid interface rather than from mechanical devices external to the heat exchanger. There are available several expressions from which reasonable values of the film coefficients may be obtained. The boiling curve, particularly in the nucleate-boiling region, is significantly affected by the temperature driving force, the total system pressure, the nature of the boiling surface, the geometry of the system, and the properties of the boiling material. In the nucleate-boiling regime, heat flux is approximately proportional to the cube of the temperature driving force. Designers in addition must know the minimum ∆t (the point at which nucleate boiling begins), the critical ∆t (the ∆t above which transition boiling begins), and the maximum heat flux (the heat flux corresponding to the critical ∆t). For designers who do not have experimental data available, the following equations may be used. Boiling Coefficients For the nucleate-boiling coefficient the Mostinski equation [Teplenergetika, 4, 66 (1963)] may be used: q 0.7 P 0.17 P 1.2 P 10 h = bPc0.69 ᎏ 1.8 ᎏ + 4 ᎏ + 10 ᎏ (5-102) A Pc Pc Pc where b = (3.75)(10−5)(SI) or (2.13)(10−4) (U.S. customary), Pc is the critical pressure and P the system pressure, q/A is the heat flux, and h is the nucleate-boiling coefficient. The McNelly equation [J. Imp. Coll. Chem. Eng. Soc., 7(18), (1953)] may also be used: 0.33 qc 0.69 Pkl 0.31 ρl h = 0.225 ᎏl (5-103) ᎏ ᎏ−1 Aλ σ ρv where cl is the liquid heat capacity, λ is the latent heat, P is the system pressure, kl is the thermal conductivity of the liquid, and σ is the surface tension. An equation of the Nusselt type has been suggested by Rohsenow [Trans. Am. Soc. Mech. Eng., 74, 969 (1952)].

    

 

 

    



hD/k = Cr(DG/µ)2/3(cµ/k)−0.7 in which the variables assume the following form: 1/2 1/2 hβ′ gcσ β′ gcσ W 2/3 cµ = Cr ᎏ ᎏᎏ ᎏ ᎏ ᎏ ᎏ k g(ρL − ρv) µ g(ρL − ρv) A k





 



 

(5-104a) −0.7

(5-104b)

The coefficient Cr is not truly constant but varies from 0.006 to 0.015.* It is possible that the nature of the surface is partly responsible for the variation in the constant. The only factor in Eq. (5-104b) not readily available is the value of the contact angle β′. Another Nusselt-type equation has been proposed by Forster and Zuber:† 0.62 1/3 NNu = 0.0015NRe NPr (5-105) which takes the following form:

α  W 2σ cρLπ ᎏᎏ ᎏ kρv A ∆p

ρL

  ᎏ ∆pg  α  ρ cρ ∆T π cµ = 0.0015  ᎏ  ᎏᎏ  ᎏ  µ λρ k 1/2

1/4

c

L

2 0.62

L

1/ 2

(5-106)

v

where α = k/ρc (all liquid properties) ∆p = pressure of the vapor in a bubble minus saturation pressure of a flat liquid surface Equations (5-104b) and (5-106) have been arranged in dimensional form by Westwater. The numerical constant may be adjusted to suit any particular set of data if one desires to use a certain criterion. However, surface conditions vary so greatly that deviations may be as large as ⫾25 percent from results obtained. The maximum heat flux may be predicted by the KutateladseZuber [Trans. Am. Soc. Mech. Eng., 80, 711 (1958)] relationship, using consistent units: (ρl − ρv)σg 1/4 q = 0.18gc1/4ρv λ ᎏᎏ (5-107) ᎏ A max ρ2v Alternatively, Mostinski presented an equation which approximately represents the Cichelli-Bonilla [Trans. Am. Inst. Chem. Eng., 41, 755 (1945)] correlation:

 





HEAT TRANSFER BY RADIATION

5-23

  1 − ᎏPP 

(5-108)

(q/A)max P ᎏ=b ᎏ Pc Pc

0.35

0.9

c

where b = 0.368(SI) or 5.58 (U.S. customary); Pc is the critical pressure, Pa absolute; P is the system pressure; and (q/A)max is the maximum heat flux. The lower limit of applicability of the nucleate-boiling equations is from 0.1 to 0.2 of the maximum limit and depends upon the magnitude of natural-convection heat transfer for the liquid. The best method of determining the lower limit is to plot two curves: one of h versus ∆ t for natural convection, the other of h versus ∆ t for nucleate boiling. The intersection of these two curves may be considered the lower limit of applicability of the equations. These equations apply to single tubes or to flat surfaces in a large pool. In tube bundles the equations are only approximate, and designers must rely upon experiment. Palen and Small [Hydrocarbon Process., 43(11), 199 (1964)] have shown the effect of tube-bundle size on maximum heat flux. p q gσ(ρl − ρv) 1/4 = b ᎏ ρv λ ᎏᎏ (5-109) ᎏ Do NT A max ρ2v

 





where b = 0.43 (SI) or 61.6 (U.S. customary), p is the tube pitch, Do is the tube outside diameter, and NT is the number of tubes (twice the number of complete tubes for U-tube bundles). For film boiling, Bromley’s [Chem. Eng. Prog., 46, 221 (1950)] correlation may be used: kv3(ρl − ρv)ρv g h = b ᎏᎏ µv Do ∆ tb





1/4

(5-110)

where b = 4.306 (SI) or 0.620 (U.S. customary). Katz, Myers, and Balekjian [Pet. Refiner, 34(2), 113 (1955)] report boiling heat-transfer coefficients on finned tubes.

HEAT TRANSFER BY RADIATION GENERAL REFERENCES: Much of the pertinent literature on radiative heat transfer has been surveyed in the following texts: Goody, Atmospheric Radiation, Clarendon Press, Oxford, 1964. Sparrow and Cess, Radiation Heat Transfer, Brooks/Cole Publishing Company, Belmont, Calif., 1966. Hottel and Sarofim, Radiative Transfer, McGraw-Hill, New York, 1967. Love, Radiative Heat Transfer, Merrill, Columbus, 1968. Siegel and Howell, Thermal Radiation Heat Transfer, NASA SP-164, GPO, Washington, 1968. Edwards, Radiation Heat Transfer Notes, Hemisphere Publishing Corp., 1981; Howell and Siegel, Thermal Radiative Heat Transfer, McGraw-Hill, 3d ed., 1992; Brewster, Thermal Radiative Transfer and Properties, Wiley, 1992; Modest, Radiative Heat Transfer, McGraw-Hill, 1993. Additional sources are the Journal of Applied Optics and the Journal of the Optical Society of America, particularly for surface properties; the Journal of Quantitative Spectroscopy and Radiative Transfer for gas properties; the Journal of Heat Transfer and the International Journal of Heat and Mass Transfer for broad coverage; and the Journal of the Institute of Energy for applications to industrial furnaces.

Thermal radiation—electromagnetic energy in transport—is emitted within matter excited by temperature; it is absorbed in other matter at distances from the source which depend on the mean free path of the photons emitted. The ratio of the mean free path involved in an energy-transport process to a characteristic dimension of the system of interest determines the mathematical structure of the formulation. In molecular conduction this ratio is minute (unless the system or the density of matter is minute, which is the case of free molecular flow), and a differential equation of energy diffusion is involved. In gas radiation the ratio is generally large enough to give rise to an integral equation, with an unknown function inside the integral. Solids gener-

ally have small enough photon mean free paths (high enough absorption coefficients) for the radiation escaping through the surface to have originated close to the surface; radiative loss is then identifiable with its surface temperature, but an integral equation is still involved if all the surfaces of an enclosure filled with a diathermanous medium like air are not specified as to temperature or are not black. Radiation differs from conduction and convection not only in mathematical structure but in its much higher sensitivity to temperature. It is of dominating importance in furnaces because of their temperature, and in cryogenic insulation because of the vacuum existing between particles. The temperature at which it accounts for roughly half of the total heat loss from a surface in air depends on such factors as surface emissivity and the convection coefficient. For pipes in free convection, this is room temperature; for fine wires of low emissivity it is above red heat. Gases at combustion-chamber temperatures lose more than 90 percent of their energy by radiation from the carbon dioxide, water vapor, and particulate matter. NOMENCLATURE FOR RADIATIVE TRANSFER Terms that are defined at specific places in the text are excluded. a = effective energy fraction of blackbody spectrum in which a nongray gas absorbs. A = area. c = number concentration of particles in a cloud. c1, c2 = first and second Planck-law constants.

* Reported by Westwater in Drew and Hoopes, Advances in Chemical Engineering, vol. I, Academic, New York, 1956, p. 15. † Forster, J. Appl. Phys., 25, 1067 (1954); Forster and Zuber, J. Appl. Phys., 25, 474 (1954); Forster and Zuber, Conference on Nuclear Engineering, University of California, Los Angeles, 1955; excellent treatise on boiling of liquids by Westwater in Drew and Hoopes, Advances in Chemical Engineering, vol. I, Academic, New York, 1956.

5-24

HEAT AND MASS TRANSFER

C = axis-to-axis distance of separation of tubes. Cb = mean specific heat of combustion products from base temperature To to leaving-gas temperature TE. C = cold-surface fraction of a furnace enclosure. CW = correction factor for pressure broadening of radiation from water vapor. d = particle diameter. D = tube diameter; characteristic dimension; dimensionless firing density. D′ = reduced firing density. E = hemispherical emissive power of a blackbody. f = fraction of blackbody radiation lying below λ. fv = volume fraction of space occupied by particles. F = direct view factor; Fij, fraction of isotropic radiation from Ai intercepted directly by Aj. F = total view factor from black source to black sink, with  allowance for refractory surfaces (subscripts identify source and sink). Ᏺij = total view factor, radiation from i to j both directly and indirectly, expressed as fraction of blackbody radiation from Ai. gs = direct-exchange area between gas volume and surface. G S  = total-exchange area between gas and surface; subscript R indicates allowance for radiatively adiabatic surfaces. h = coefficient of convective heat transfer. H˙ = enthalpy of fuel plus air entering combustion chamber. I = intensity, radiant-energy-flux density per unit solid angle of divergence. ij = shorthand for sisj. k = absorption or emission coefficient; or thermal conductivity. K = constant defined in connection with Eq. (5-147). L = mean beam length; L0, at vanishingly small optical thickness; Lm, average value. L = wall-loss group. Lc = dimensionless convective loss. Lo = dimensionless wall-opening loss. Lr = dimensionless refractory-wall loss. ˙ = mass flow rate. m n = refractive index. p = partial pressure, atm.; subscript c, CO2; subscript w, water vapor. P = total pressure atm. q = heat-flux density, energy per time-area. ˙ = heat flux, energy per time. Q r = separating distance; or electrical resistivity; or refractory (radiatively adiabatic) surface. ss AF, direct-exchange area (subscripts identify surface zones). SS   AᏲ, total-exchange area. T = absolute temperature. Subscript 1 (or G), radiating surface (or gas) temperature; subscript E, exit-gas; subscript o, base temperature; subscript F, pseudoadiabatic flame temperature based on C p averaged from To to TE. U = overall coefficient of heat transfer, gas convection to refractory wall to ambient air. W = total leaving-flux density (also radiosity). α = absorptivity or absorptance; α12, absorptance of surface 1 for radiation from surface 2. ∆ = difference between radiating temperature and leaving-gas temperature divided by the pseudoadiabatic flame temperature TF. ε = emissivity or emittance. η = thermal efficiency. Subscript G, gas-side; subscript 1, sinkside. θ = polar angle. λ = wavelength. µm = micrometer (m−6). ρ = reflectance; ρs, specular reflectance. σ = Stefan-Boltzmann constant. τ = ratio of temperature to TF. Subscript G, gas; subscript 1 sink; subscript o, base. τ = transmittance. Ω = solid angle. ω = albedo of a surface.

NATURE OF THERMAL RADIATION Consider a pencil of radiation, defined as all the rays passing through each of two small widely separated areas dA1 and dA2. The rays at dA1 will have a solid angle of divergence dΩ1 equal to the apparent area of dA2 viewed from dA1, divided by the square of the separating distance. Let the normal to dA1 make the angle θ1 with the pencil. The flux density q (energy per time-area) normal to the beam and per unit solid ˙1 angle of its divergence is called the intensity I, and the flux dQ (energy per time) through the area dA1 (of apparent area dA1 cos θ1 normal to the beam) is therefore given by ˙ 1 = dA1(cos θ1)q = I dA1(cos θ1) dΩ1 dQ (5-111) The intensity I along a pencil, in the absence of absorption or scatter, is constant (unless the beam passes into a medium of different refractive index n; then I1/n21 = I2 /n22). The emissive power* of a surface is the flux density (energy per time-surface area) due to emission from it throughout a hemisphere. If the intensity I of emission from a surface is independent of the angle of emission, Eq. (5-111) may be integrated to show that the surface emissive power is πI, though the emission is throughout 2π sr. Blackbody Radiation Engineering calculations of thermal radiation from surfaces are best keyed to the radiation characteristics of the blackbody, or ideal radiator. The characteristic properties of a blackbody are that it absorbs all the radiation incident on its surface and that the quality and intensity of the radiation it emits are completely determined by its temperature. The total radiative flux throughout a hemisphere from a black surface of area A and absolute temperature T is given by the Stefan-Boltzmann law: ˙ = AσT 4 Q or q = σT 4 (5-112) The Stefan-Boltzmann constant σ has the value (0.1713)(10−8) Btu/ (ft2 ⭈h⭈°R4); (1.00)(10−8) CHU/(ft2 ⭈h⭈K4); (4.88)(10−8) kcal/(m2 ⭈h⭈K4); (1.356)(10−12) cal (cm2 ⭈s⭈K4); (5.67)(10−12) W/(cm2 ⭈K4); (5.67)(10−8) W/(m2 ⭈K4); or in terms of Planck constants, c1(π/c2)4/15. From the definition of emissive power, σT 4 is the total emissive power of a blackbody, called E; the intensity IB of blackbody emission is E/π or σT 4/π. The spectral distribution of energy flux from a black body is expressed by Planck’s law: Eλ dλ = (2πhc2 n2 λ−5)/(ehc/kλT − 1) dλ

(n2c1λ−5)/(ec / λT − 1) dλ 2

(5-113) (5-114)

where Eλ dλ is the hemispherical flux density lying in the wavelength range λ to λ + dλ; h is Planck’s constant, (6.6256)(10−27) erg⭈s; c is the velocity of light in vacuo, (2.9979)(1010) cm/s; k is the Boltzmann constant, (1.3805)(10−16) erg/K; λ is the wavelength measured in vacuo; and n is the refractive index of the emitter (λ = nλm, where λm is the wavelength measured in the medium; Eλ d λ = Eλm d λm, where Eλ and Eλm are both measured in the medium; engineers commonly use Eλ). Equation (10-191) may be written Eλ c1(λT)−5 =ᎏ (5-115) ᎏ 2 5 nT ec /λT − 1 The first and second Planck-law constants c1 and c2 are respectively (3.740)(10−16) (J⭈m2)/s and (1.4388)(10−2) m⭈K. The term Eλ/n2T 5, clearly a function only of the product λT, is given in Fig. 5-11 which may be visualized as the monochromatic emissive power versus wavelength measured in vacuo of a black surface at 1 K discharging in vacuo. The wavelength of maximum intensity is seen to be inversely proportional to the absolute temperature. The relation is known as Wien’s displacement law: λmaxT = (2.898)(10−3) m⭈K. This can be misleading, however, since the wavelength of maximum intensity depends on whether intensity is defined in terms of wavelength interval or frequency interval. More useful displacement laws refer to the value of λT corresponding to maximum energy per unit fractional change in wavelength or frequency [(3.67)(10−3) m⭈K] or to the value of λT corresponding to half of the energy [(4.11)(10−3) m⭈K]. Figure 2

* Variously called, in the literature, emittance, total hemispherical intensity, or radiant flux density.

HEAT TRANSFER BY RADIATION

5-25

Distribution of energy in the spectrum of a blackbody. To convert microns to micrometers, multiply by unity. To convert ergs per square centimeter–second–micron–K5 to watts per square meter, per meter, per K5, multiply by 10−3. FIG. 5-11

5-11 carries, at the top, a scale giving the fraction f of the total energy in the spectrum that lies below λT. A generalization useful for identifying the spectral range of greatest interest in evaluations of radiative transfer is that roughly half of the energy from a black surface lies within the twofold range of λT geometrically centered on 3.67 × 10−3, i.e., from λT = (3.67/2)(10−3) to (3.67 × 2)(10−3) m⭈K. One limiting form of the Planck equation, approached as λT → 0, is the Wien equation [Eqs. (5-113) and (5-114)] with the 1 missing in the denominator. The error is less than 1 percent when λT < (3)(10−3) m⭈K or when T < 4800 K if an optical pyrometer with red screen (λ = 0.65µm) is used. RADIATIVE EXCHANGE BETWEEN SURFACES OF SOLIDS Emittance and Absorptance The ratio of the total radiating power of a real surface to that of a black surface at the same temperature is called the emittance of the surface (for a perfectly plane surface, the emissivity), designated by ε. Subscripts λ, θ, and n may be assigned to differentiate monochromatic, directional, and surfacenormal values respectively from the total hemispherical value. If radi-

ation is incident on a surface, the fraction absorbed is called the absorptance (absorptivity), a term to which two subscripts may be appended, the first to identify the temperature of the surface and the second to identify the spectral energy distribution of the surface. According to Kirchhoff’s law, the emissivity and absorptivity of a surface in surroundings at its own temperature are the same for both monochromatic and total radiation. When the temperatures of the surface and its surroundings differ, the total emissivity and absorptivity of the surface often are found to be different, but, because absorptivity is substantially independent of irradiation density, the monochromatic emissivity and absorptivity of surfaces are for all practical purposes the same. The difference between total emissivity and absorptivity depends on the variation, with wavelength, of ελ and on the difference between the emitter temperature and the effective source temperature. Consider radiative exchange between a body of area A1 and temperature T1 and black surroundings at T2. The net interchange is given by ˙ 1 = 2 = A1 Q



∞

[ελEλ(T1) − αλEλ(T2)] dλ

0

= A1(ε1σT 41 − α12σT 42 )

(5-116)

5-26 where

HEAT AND MASS TRANSFER

 ε df =  ε df

ε1 =

1

λ

λT 1

(5-117)

λ

λT 2

(5-118)

0

and

α12

1

0

The value of ε1 (or α12, the absorptivity of surface A1 for blackbody radiation at T2) is the area under a curve of ελ versus f, the latter read as a function of λT1 (or λT2) from the top ordinate of Fig. 5-11. For a gray surface, ε1 = α12 = ελ. A selective surface is one whose ελ changes dramatically with wavelength. If this change is unidirectional, ε1 and α12 are, according to Eqs. (5-116) to (5-118), markedly different when the absolute-temperature ratio is far from 1; e.g., when T1 = 294 K (530°R; ambient temperature), and T2 = 6000 K (10,800°R; effective solar temperature), ε1 = 0.9 and α12 = 0.1 to 0.2 for a white paint, but ε1 can be as low as 0.12 and α12 above 0.9 for a thin layer of copper oxide on bright aluminum. The effect of radiation-source temperature on the low-temperature absorptivity of a number of additional materials is presented in Fig. 5-12. It will be noted that polished aluminum (curve 15) and anodized (surface-oxidized) aluminum (curve 13), representative of metals and nonmetals respectively, respond oppositely to a change in the temperature of the radiation source. The absorptance of surfaces for solar

FIG. 5-12 Variation of absorptivity with temperature of radiation source. (1) Slate composition roofing. (2) Linoleum, red brown. (3) Asbestos slate. (4) Soft rubber, gray. (5) Concrete. (6) Porcelain. (7) Vitreous enamel, white. (8) Red brick. (9) Cork. (10) White dutch tile. (11) White chamotte. (12) MgO, evaporated. (13) Anodized aluminum. (14) Aluminum paint. (15) Polished aluminum. (16) Graphite. The two dashed lines bound the limits of data on gray paving brick, asbestos paper, wood, various cloths, plaster of paris, lithopone, and paper. To convert degrees Rankine to kelvins, multiply by (5.556)(10−1).

radiation may be read from the right of Fig. 10-45, if solar radiation is assumed to consist of blackbody radiation from a source at 5800 K (10,440°R). Although values of emittance and absorptance depend in very complex ways on the real and imaginary components of the refractive index and on the geometrical structure of the surface layer, the generalizations that follow are possible. Polished Metals 1. ε λ in the infrared is governed by free-electron contributions, is quite low, and is a function of the resistivity-wavelength quotient r/ λ (Fig. 5-13). For λ > 8µm, ε λ,n is approximately 0.0365 r/ λ , where r is in ohm-meters and λ in micrometers (the Drude or HagenRubens relation). At shorter wavelengths, bound-electron contributions become significant and ελ increases, sometimes exhibiting maxima; values of 0.4 to 0.8 are common in the visible spectrum (0.4 to 0.7 µm). ελ is approximately proportional to the square root of the absolute temperature (ελ ∝ r, and r ∝ T) in the far infrared (λ > 8µm), is temperature-insensitive in the near infrared (0.7 to 1.5 µm), and decreases slightly as temperature increases in the visible. 2. Total emittance is substantially proportional to absolute temperature; at moderate temperature, εn = 0.058T rT, where T is in kelvin. 3. The total absorptance of a metal at T1 for radiation from a black or gray source at T2 is equal to the emissivity evaluated at the geometric mean of T1 and T2. Figure 5-13 gives values of ελ, ελ,n, and their ratio as a function of r/λ (dashed lines); and total emissivities ε, εn and their ratio as a function of rT (solid lines). Although the figure is based on free-electron contributions to emissivity in the far infrared, the relations for total emissivity are remarkably good even at high temperatures. Unless extraordinary pains are taken to prevent oxidation, however, a metallic surface may exhibit several times the emittance or absorptance of a polished specimen. The emittance of iron and steel, for example, varies widely with degree of oxidation and roughness; clean metallic surfaces have an emittance of from 0.05 to 0.45 at ambient temperatures to 0.4 to 0.7 at high temperatures; oxidized and/or rough surfaces range from 0.6 to 0.95 at low temperatures to 0.9 to 0.95 at high temperatures. Refractory Materials Grain size and concentration of trace impurities are important. 1. Most refractory materials have an ελ of 0.8 to 1.0 at wavelengths beyond 2 to 4 µm; ελ decreases rapidly toward shorter wavelengths for materials that are white in the visible but retains its high value for black materials such as FeO and Cr2O3. Small concentrations of FeO

FIG. 5-13 Hemispherical and normal emissivities of metals and their ratio. Dashed lines: monochromatic (spectral) values versus r/λ. Solid lines: total values versus rT. To convert ohm-centimeter-kelvins to ohm-meter-kelvins, multiply by 10−2.

HEAT TRANSFER BY RADIATION and Cr2O3 or other colored oxides can cause marked increases in the emittance of materials that normally are white. The sensitivity of the emittance of refractory oxides to small additions of absorbing materials is demonstrated by the results of calculations, shown in Fig. 5-14, of the emittance of a semi-infinite absorbing-scattering medium as a function of its albedo: the ratio of the scatter coefficient to the sum of scatter and absorption coefficients. The results, pertinent to the radiative properties of fibrous materials, paints, oxide coatings, and refractories, show that when absorption accounts for only 0.5 percent (10 percent) of the total attenuation within the medium, the emittance is greater than 0.15 (0.5). ελ for refractory materials varies little with temperature, with the exception of some white oxides which at high temperatures become good emitters in the visible spectrum as a consequence of the induced electronic transitions. 2. Refractory materials generally have a total emittance which is high (0.7 to 1.0) at ambient temperatures and decreases with increase in temperature; a change from 1000 to 1570°C (1850 to 2850°F) may cause a decrease in ε of one-fourth to one-third. 3. The emittance and absorptance increase with increase in grain size over a grain-size range of 1 to 200 µm. 4. The ratio ε/εn of hemispherical to normal emissivity of polished surfaces varies with refractive index n from 1 at n = 1.0 to 0.93 at n = 1.5 (common glass) and back to 0.96 at n = 3. 5. The ratio ε/εn for a surface composed of particulate matter which scatters isotropically varies with ε from 1 when ε = 1 to 0.8 when ε = 0.07 (see Fig. 5-14). 6. The total absorptance shows a decrease with increase in temperature of the radiation source similar to the decrease in emittance with increase in the specimen temperature. Figure 5-12 shows a regular variation of α12 with T2. When T2 is not very different from T1, α12 may be expressed as ε1(T2 /T1)m. It may be shown that Eq. (5-116) is then approximated by m ˙ 1,net = σA1εav 1 + ᎏ Q (T 14 − T 42) (5-119) 4 where εav is evaluated at the arithmetic mean of T1 and T2. For metals m is about 0.5; for nonmetals it is small and negative. Table 5-6, based on a critical evaluation of early data, is illustrative of the emittance of materials encountered in engineering practice; it shows the wide variation possible in the emissivity of a particular material due to variations in surface roughness and thermal pretreatment. (With few exceptions the values refer to emission normal to the surface; see above for conversion to hemispherical values.) More recent data support the range of emittance values given in Table 5-6 and their dependence on surface conditions. Extensive compilations of data are provided by Schmidt and Furthmann (Mitt. KaiserWilhelm-Inst. Eisenforsch., 109, 225), covering data to 1928; by Gubareff, Jansen, and Torborg [Thermal Radiation Properties Sur-





5-27

vey, Honeywell Research Center, Minneapolis, 1960), covering data to 1940; and by Goldsmith, Waterman, and Hirschhorn [Thermophysical Properties of Matter, Purdue University (Touloukian, Ed.), Plenum, 1970]. For opaque materials, the reflectance ρ is the complement of the absorptance. The directional distribution of the reflected radiation depends on the material, its degree of roughness or grain size, and, if a metal, its state of oxidation. Polished surfaces of homogeneous materials reflect specularly. In contrast, the intensity of the radiation reflected from a perfectly diffuse, or Lambert, surface is independent of direction. The directional distribution of reflectance of many oxidized metals, refractory materials, and natural products approximates that of a perfectly diffuse reflector. A better model, adequate for many calculational purposes, is achieved by assuming that the total reflectance ρ is the sum of diffuse and specular components ρD and ρ S . Black-Surface Enclosures View Factor and Direct-Exchange Area When several surfaces are present, the need arises for evaluating a geometrical factor F, called the direct view factor. In the following discussion, restriction is to black surfaces, the intensity from which is independent of angle of emission. Define F12 as the fraction of the radiation leaving surface A1 in all directions which is intercepted by surface A2. Since the net interchange between A1 and A2 must be zero when their temperatures are alike, it follows that A1F12 = A2F21. This product, having the dimensions of area, is called the direct-exchange area and is designated for brevity by 12( 2 1). It is sometimes designated s1s2. Clearly, 11 + 12 + 13 + ⭈⭈⭈ = A1; and when A1 cannot “see” itself, 11 = 0. From Eq. (5-111) and the definition of F: Q1 − 2 dA1(cos θ1) dΩ1 A1F12 s1s2 ᎏ = ᎏᎏ A1 A2 E1 π





dA1(cos θ1) dA2 (cos θ2) (5-120) ᎏᎏᎏ πr 2 where A(cos θ) is the projection of A normal to r, the line connecting dA1 and dA2. Values of s1s2 (or of F12) may be obtained by integrating either Eq. (5-120) or an equivalent contour integral (see Hottel and Sarofim, Radiative Transfer, McGraw-Hill, New York, 1967, chap. 2). Such values are given for opposed parallel disks or rectangles in Fig. 5-15. For rectangles of dimensions L1 and L2, F   F 1F2, where F1 and F2 are for squares of sides L1 and L2. The view factor for rectangles in perpendicular planes and having a common edge length x and ratios Y and Z of widths to common length is given by =

A1

1 FYZ = ᎏ πY







1 Y2 ᎏ ln (1 + Y 2) ᎏ2 4 1+Y

A2

Z 1+Z ᎏᎏ  ᎏ Y + Z  1 + Y + Z  Y2

Z2

2

2

2

1 − Y 2 − Z2

2

2

2

1 1 1 2+  + Y tan−1 ᎏ + Z tan−1 ᎏ − Y Z2 tan−1 ᎏᎏ Y2 + Z2  Y Z



(5-121) The direct-exchange area is given by x2 sysz = ᎏ (5-122) π When the maximum dimensions of each of two plane surfaces is small relative to their center-to-center separating distance r, Eq. (5-120) gives A1(cos θ1)A2(cos θ2) 12 = ᎏᎏᎏ (5-123) πr2 and when, in addition, the normals to A1 and A2 are in a common plane, 12 = A1A2n1n2/πr2

Hemispherical emittance εh and the ratio of hemispherical to normal emittance εh/εn for a semi-infinite absorbing-scattering medium.

FIG. 5-14

(5-124)

where n1 is the normal-to-A1 component of the distance to A2. Equation (5-124) is, for example, in error only by +7 percent for the case of opposed squares separated by 3 times their side dimension. The view factors are given for finite coaxial coextensive cylinders in Fig. 5-16,

5-28

HEAT AND MASS TRANSFER

TABLE 5-6

Normal Total Emissivity of Various Surfaces A. Metals and Their Oxides Surface

Aluminum Highly polished plate, 98.3% pure Polished plate Rough plate Oxidized at 1110°F Aluminum-surfaced roofing Calorized surfaces, heated at 1110°F. Copper Steel Brass Highly polished: 73.2% Cu, 26.7% Zn 62.4% Cu, 36.8% Zn, 0.4% Pb, 0.3% Al 82.9% Cu, 17.0% Zn Hard rolled, polished: But direction of polishing visible But somewhat attacked But traces of stearin from polish left on Polished Rolled plate, natural surface Rubbed with coarse emery Dull plate Oxidized by heating at 1110°F Chromium; see Nickel Alloys for Ni-Cr steels Copper Carefully polished electrolytic copper Commercial, emeried, polished, but pits remaining Commercial, scraped shiny but not mirrorlike Polished Plate, heated long time, covered with thick oxide layer Plate heated at 1110°F Cuprous oxide Molten copper Gold Pure, highly polished Iron and steel Metallic surfaces (or very thin oxide layer): Electrolytic iron, highly polished Polished iron Iron freshly emeried Cast iron, polished Wrought iron, highly polished Cast iron, newly turned Polished steel casting Ground sheet steel Smooth sheet iron Cast iron, turned on lathe Oxidized surfaces: Iron plate, pickled, then rusted red Completely rusted Rolled sheet steel Oxidized iron Cast iron, oxidized at 1100°F Steel, oxidized at 1100°F Smooth oxidized electrolytic iron Iron oxide Rough ingot iron

t, °F*

Emissivity*

440–1070 73 78 390–1110 100

0.039–0.057 0.040 0.055 0.11–0.19 0.216

390–1110 390–1110

0.18–0.19 0.52–0.57

476–674 494–710 530

0.028–0.031 0.033–0.037 0.030

70 73 75 100–600 72 72 120–660 390–1110 100–1000

0.038 0.043 0.053 0.096 0.06 0.20 0.22 0.61–0.59 0.08–0.26

176

0.018

66

0.030

72 242

0.072 0.023

77 390–1110 1470–2010 1970–2330

0.78 0.57 0.66–0.54 0.16–0.13

440–1160

0.018–0.035

350–440 800–1880 68 392 100–480 72 1420–1900 1720–2010 1650–1900 1620–1810

0.052–0.064 0.144–0.377 0.242 0.21 0.28 0.435 0.52–0.56 0.55–0.61 0.55–0.60 0.60–0.70

68 67 70 212 390–1110 390–1110 260–980 930–2190 1700–2040

0.612 0.685 0.657 0.736 0.64–0.78 0.79 0.78–0.82 0.85–0.89 0.87–0.95

Surface Sheet steel, strong rough oxide layer Dense shiny oxide layer Cast plate: Smooth Rough Cast iron, rough, strongly oxidized Wrought iron, dull oxidized Steel plate, rough High temperature alloy steels (see Nickel Alloys). Molten metal Cast iron Mild steel Lead Pure (99.96%), unoxidized Gray oxidized Oxidized at 390°F. Mercury Molybdenum filament Monel metal, oxidized at 1110°F Nickel Electroplated on polished iron, then polished Technically pure (98.9% Ni, + Mn), polished Electroplated on pickled iron, not polished Wire Plate, oxidized by heating at 1110°F Nickel oxide Nickel alloys Chromnickel Nickelin (18–32 Ni; 55–68 Cu; 20 Zn), gray oxidized KA-2S alloy steel (8% Ni; 18% Cr), light silvery, rough, brown, after heating After 42 hr. heating at 980°F. NCT-3 alloy (20% Ni; 25% Cr.), brown, splotched, oxidized from service NCT-6 alloy (60% Ni; 12% Cr), smooth, black, firm adhesive oxide coat from service Platinum Pure, polished plate Strip Filament Wire Silver Polished, pure Polished Steel, see Iron. Tantalum filament Tin—bright tinned iron sheet Tungsten Filament, aged Filament Zinc Commercial, 99.1% pure, polished Oxidized by heating at 750°F. Galvanized sheet iron, fairly bright Galvanized sheet iron, gray oxidized

t, °F* 75 75 73 73 100–480 70–680 100–700

2370–2550 2910–3270 260–440 75 390 32–212 1340–4700 390–1110 74

Emissivity* 0.80 0.82 0.80 0.82 0.95 0.94 0.94–0.97

0.29 0.28 0.057–0.075 0.281 0.63 0.09–0.12 0.096–0.292 0.41–0.46 0.045

440–710

0.07–0.087

68 368–1844 390–1110 1200–2290

0.11 0.096–0.186 0.37–0.48 0.59–0.86

125–1894

0.64–0.76

70

0.262

420–914 420–980

0.44–0.36 0.62–0.73

420–980

0.90–0.97

520–1045

0.89–0.82

440–1160 1700–2960 80–2240 440–2510

0.054–0.104 0.12–0.17 0.036–0.192 0.073–0.182

440–1160 100–700

0.0198–0.0324 0.0221–0.0312

2420–5430 76

0.194–0.31 0.043 and 0.064

80–6000 6000

0.032–0.35 0.39

440–620 750 82 75

0.045–0.053 0.11 0.228 0.276

260–1160

0.81–0.79

1900–2560 206–520 209–362

0.526 0.952 0.959–0.947

B. Refractories, Building Materials, Paints, and Miscellaneous Asbestos Board Paper Brick Red, rough, but no gross irregularities Silica, unglazed, rough Silica, glazed, rough Grog brick, glazed See Refractory Materials below.

74 100–700 70 1832 2012 2012

0.96 0.93–0.945 0.93 0.80 0.85 0.75

Carbon T-carbon (Gebr. Siemens) 0.9% ash (this started with emissivity at 260°F. of 0.72, but on heating changed to values given) Carbon filament Candle soot Lampblack-waterglass coating

HEAT TRANSFER BY RADIATION TABLE 5-6

5-29

Normal Total Emissivity of Various Surfaces (Concluded ) B. Refractories, Building Materials, Paints, and Miscellaneous Surface

Same Thin layer on iron plate Thick coat Lampblack, 0.003 in. or thicker Enamel, white fused, on iron Glass, smooth Gypsum, 0.02 in. thick on smooth or blackened plate Marble, light gray, polished Oak, planed Oil layers on polished nickel (lube oil) Polished surface, alone +0.001-in. oil +0.002-in. oil +0.005-in. oil Infinitely thick oil layer Oil layers on aluminum foil (linseed oil) Al foil +1 coat oil +2 coats oil Paints, lacquers, varnishes Snowhite enamel varnish or rough iron plate Black shiny lacquer, sprayed on iron Black shiny shellac on tinned iron sheet Black matte shellac Black lacquer Flat black lacquer White lacquer

t, °F*

Emissivity*

260–440 69 68 100–700 66 72

0.957–0.952 0.927 0.967 0.945 0.897 0.937

70 72 70 68

0.903 0.931 0.895 0.045 0.27 0.46 0.72 0.82

212 212 212

0.087† 0.561 0.574

Surface Oil paints, sixteen different, all colors Aluminum paints and lacquers 10% Al, 22% lacquer body, on rough or smooth surface 26% Al, 27% lacquer body, on rough or smooth surface Other Al paints, varying age and Al content Al lacquer, varnish binder, on rough plate Al paint, after heating to 620°F. Paper, thin Pasted on tinned iron plate On rough iron plate On black lacquered plate Plaster, rough lime Porcelain, glazed Quartz, rough, fused Refractory materials, 40 different poor radiators

t, °F*

Emissivity*

212

0.92–0.96

212

0.52

212

0.3

212 70 300–600

0.27–0.67 0.39 0.35

66 66 66 50–190 72 70 1110–1830

0.924 0.929 0.944 0.91 0.924 0.932

good radiators 73 76 70 170–295 100–200 100–200 100–200

0.906 0.875 0.821 0.91 0.80–0.95 0.96–0.98 0.80–0.95

Roofing paper Rubber Hard, glossy plate Soft, gray, rough (reclaimed) Serpentine, polished Water

69 74 76 74 32–212



0.65 – 0.75 0.70 0.80 – 0.85 0.85 – 0.90 0.91

} }{



0.945 0.859 0.900 0.95–0.963

*When two temperatures and two emissivities are given, they correspond, first to first and second to second, and linear interpolation is permissible. °C = (°F − 32)/1.8. †Although this value is probably high, it is given for comparison with the data by the same investigator to show the effect of oil layers. See Aluminum, Part A of this table.

and for an infinite plane parallel to a system of rows of parallel tubes as curves 1 and 3 of Fig. 5-17. The exchange area between any two area elements of a sphere is independent of their relative shape and position and is simply the product of the areas divided by the area of the whole sphere; i.e., any spot on a sphere has equal views of all other spots. For surfaces in two-dimensional systems (with third dimension infinite), A1F12 per unit length in the third dimension may be obtained simply by evaluating, in a cross-sectional view, the sum of lengths of crossed strings from the ends of A1 to the ends of A2 less the sum of uncrossed strings from and to the same points, all divided by 2. The strings must be so drawn that all the flux from one surface to the other must cross each of a pair of crossed strings and neither of a pair of uncrossed ones. If one surface can see the other around both sides of an obstruction, two more pairs of strings are involved.

Example 3: Calculation of View Factor Evaluate the view factor between two parallel circular tubes long enough compared with their diameter D or their axis-to-axis separating distance C to make the problem twodimensional. With reference to Fig. 5-18, the crossed-strings method yields, per unit of axial length, 1/2 2(EFGH − HJ) D C 2 C A1F12 = ᎏᎏ = D sin−1 ᎏ + ᎏ − 1 − ᎏ 2 C D D



 

Results for a large number of other cases are given by Hottel and Sarofim (op. cit., chap. 2) and Hamilton and Morgan (NACA-TN2836, December 1952). A comprehensive bibliography is provided by Siegel and Howell (Thermal Radiation Heat Transfer, McGraw-Hill, 1992). The view factor F may often be evaluated from that for simpler configurations by the application of three principles: that of reciprocity, AiFij = AjFji; that of conservation,
Fij = 1; and that due to Yamauti [Res. Electrotech. Lab. (Tokyo), 148, 1924; 194, 1927; 250, 1929], showing that the exchange areas AF between two pairs of surfaces are equal when there is a one-to-one correspondence for all sets of symmetrically placed pairs of elements in the two surface combinations. Example 4: Calculation of Exchange Area The exchange area between the two squares 1 and 4 of Fig. 5-19 is to be evaluated. The following exchange areas may be obtained from Eq. 5-121. F for common-side rectangles: 13 = 0.24, 24 = 2 × 0.29 = 0.5S, (1 + 2)(3 + 4) = 3 × 0.32 = 0.96. Expression of (1 + 2)(3 + 4) in terms of its components yields (1 + 2)(3 3+ 4) = 13 + 14 + 23  + 24. And by the Yamauti principle 14 = 23, since for every pair of elements in 1 and 4 there is a corresponding pair in 2 and 3. Therefore, (1 1+ 2)(3 3+ 4) − 13 − 24 14 = ᎏᎏᎏ = 0.07 2 Figure 5-16 may be used in the same way.

FIG. 5-15

Radiation between parallel planes, directly opposed.

Non-Black-Surface Enclosures In the following discussion we are concerned with enclosures containing gray sources and sinks, radiatively adiabatic surfaces, and no absorbing gas. The calculation of interchange between a source and a sink under conditions involving successive multiple reflections from other source-sink surfaces in the

5-30

HEAT AND MASS TRANSFER

(a) FIG. 5-16

(b)

View factors for a system of two concentric coaxial c to inner cylinder. (b) Inner surface of outer cylinder to itself.

FIG. 5-18

Direct exchange between parallel circular tubes.

FIG. 5-19

Illustration of the Yamauti principle.

Distribution of radiation to rows of tubes irradiated from one side. Dashed lines: direct view factor F from plane to tubes. Solid lines: total view factor  F for black tubes backed by a refractory surface. FIG. 5-17

enclosure, as well as reradiation from refractory surfaces which are in radiative equilibrium, can become complicated. Zone Method Let a zone of a furnace enclosure be an area small enough to make all elements of itself have substantially equivalent “views” of the rest of the enclosure. (In a furnace containing a symmetry plane, parts of a single zone would lie on either side of the plane.) Zones are of two classes: source-sink surfaces, designated by numerical subscripts and having areas A1, A2, . . . , and emissivities ε1, ε2, . . . ; and surfaces at which the net radiant-heat flux is zero (fulfilled by the average refractory wall in which difference between internal convection and external loss is minute compared with incident radiation), designated by letter subscripts starting with r, and having areas Ar, As,. . . . It may be shown (see, for example, Hottel and Sarofim, op. cit., chap. 3) that the net radiation interchange between source-sink zones i and j is given by ˙ i = j = AiᏲijσ T i4 − AjᏲjiσ T 4j Q (5-125)

Ᏺij is called the total view factor from i to j, and the term AiᏲij, sometimes designated S iS j, is called the total interchange area shared by areas Ai and Aj and depends on the shape of the enclosure and the emissivity and absorptivity of the source and sink zones. Restriction here is to gray source-sink zones, for which AiᏲij = AjᏲji; the more general case is treated elsewhere (Hottel and Sarofim, op. cit., chap. 5). Evaluation of the AᏲ’s that characterize an enclosure involves solution of a system of radiation balances on the surfaces. If the assumption is made that all the zones of the enclosure are gray and emit and reflect diffusely,* then the direct-exchange area ij, as evaluated for the black-surface pair Ai and Aj, applies to emission and reflections between them. If at a surface the total leaving-flux density, emitted plus reflected, is denoted by W (and called by some the radiosity and by others the exitance), radiation balances take the form: * So-called Lambert surfaces, which emit or reflect with an intensity independent of angle; approximately satisfied by most nonmetallic, tarnished, oxidized, or rough surfaces.

HEAT TRANSFER BY RADIATION For source-sink j, AjεjEj + ρj
(ij)Wi = AjWj

(5-126)

i

For adiabatic surface r,


(ir)W = A W i

r

(5-127)

r

i

where ρ is reflectance and the summation is over all surfaces in the enclosure. In matrix notation, Eq. (10-196) becomes, with source or sink zones represented by 1, 2, 3 . . . and adiabatic zones by r, s, t . . . ,



A 11  − ᎏᎏ1 ρ1

12

1r

1s

12

A 22 − ᎏᎏ2 ρ2

2r

2s

1r 1s

2r 2s

rr − Ar rs

rs ss − As

W2

=

Refractory temperature is obtained from Wr = Er = σTr4. The more general use of Eq. (5-128) is to obtain the set of total interchange areas AᏲ⋅ which constitute a complete description of the effect of shape, size, and emissivity on radiative flux, independent of the presence or absence of other transfer mechanisms. It may be shown that Aiεi Ajεj D′ij AiᏲij AjᏲji  SiS j = ᎏ ᎏ − ᎏ − δijεj ρi ρj D





(5-130)

where D is the determinant of the square coefficient matrix in Eq. 5-128) and D′ij is the cofactor of its ith row and jth column, or −1)i + j times the minor of D formed by crossing out the ith row and ith column, and δij is the Kronecker delta, 1 when i = j, otherwise 0. As an example, consider radiation between two surfaces A1 and A2, which together form a complete enclosure. Equation (5-130) takes the form



12 A 11 − ᎏᎏ1 ρ1 12

12 A 22 − ᎏᎏ2 ρ2



(5-131)

Only one direct-view factor F12 or direct-exchange area 12 is needed because F11 equals 1 − F12 and F22 equals 1 − F21 equals 1 − F12 A1 /A2. Then 11 equals A1 − 12 and 22 equals A2 − 21. With these substitutions, Eq. (5-131) becomes A1 A1Ᏺ12 = ᎏᎏᎏ 1 1 A1 1 ᎏᎏ + ᎏᎏ − 1 + ᎏᎏ ᎏᎏ − 1 F12 ε1 A2 ε2





Ᏺ12 = ε1

(5-135)

F12( A2 F21) represents the total interchange where the expression A1 area for the limiting case of a black source and black sink (the refractory emissivity is of no moment). The factor F  is known exactly for a few geometrically simple cases and may be approximated for others. If A1 and A2 are equal parallel disks, squares, or rectangles, connected by nonconducting but reradiating refractory walls, then  F is given by Fig. 5-15, curves 5 to 8. If A1 represents an infinite plane and A2 is one or two rows of infinite parallel tubes in a parallel plane and if the only other surface is a refractory surface behind the tubes,  F12 is given by curve 5 or 6 of Fig. 5-17. If an enclosure may be divided into several radiant-heat sources or sinks A1, A2, etc., and the rest of the enclosure (reradiating refractory surface) may be lumped together as Ar at a uniform temperature Tr, then the total interchange area for zone pairs in the black system is given by (1r)(r2) A1F (5-137) 12( A2F 21) = 12 + ᎏ Ar − rr For the two-source-sink-zone system to which Eq. (5-136) applies, Eq. (5-137) simplifies to

(5-128)

  



0 0

This represents a system of simultaneous equations equal in number to the number of rows of the square matrix. Each equation consists, on the left, of the sum of the products of the members of a row of the square matrix and the corresponding members of the W-column matrix and, on the right, of the member of that row in the third matrix. With this set of equations solved for Wi, the net flux at any surface Ai is given by Q˙ i,net = (Aiεi /ρi)(Ei − Wi) (5-129)

A1ε1 A2ε2 A1Ᏺ12 = ᎏ ᎏ ρ1 ρ2

 

Aε − ᎏ1ᎏ1 E1 ρ1 A2ε2 − ᎏᎏE2 ρ2

Wr Ws

 

2. Sphere of area A1 concentric with surrounding sphere of area A2. F12 = 1. Then A1 A1Ᏺ12 = ᎏᎏ (5-134) 1 A1 1 ᎏᎏ + ᎏᎏ ᎏᎏ ε1 A2 ε2 − 1 3. Body of surface A1 having no negative curvature, surrounded by very much larger surface A2. F12 = 1 and A1/A2 → 0. Then Many furnace problems are adequately handled by dividing the enclosure into but two source-sink zones A1 and A2 and any number of no-flux zones Ar, As,. . . . For this case Eq. (5-130) yields 1 1 1 1 1 1 1 ᎏ ᎏ = ᎏ ᎏ − 1 + ᎏ ᎏ − 1 + ᎏ (5-136) A1Ᏺ12 A2Ᏺ21 A1 ε1 A2 ε2 A1F 12

  W1

5-31

(5-132)

Special cases include 1. Parallel plates, large compared to clearance. Substitution of F12 = 1 and A1 = A2 gives A1 (5-133) A1Ᏺ12 = ᎏᎏ 1 1 ᎏᎏ + ᎏᎏ − 1 ε1 ε2













A 1F 12 = 12 + 1/(1/1 r + 1/2 r)

(5-138)

and if A1 and A2 each can see none of itself, there is further simplification to 1 A1F 12 = 12 + ᎏᎏᎏ 1/(A1 − 12) + 1/(A2 − 12) A1A2 − (1 2)2 = ᎏᎏ (5-139) A1 + A2 − 2(1 2) which necessitates the evaluation of but one geometrical factor F. Equation (5-136) covers many of the problems of radiant-heat interchange between source and sink in a furnace enclosure. The error due to single zoning of source and sink is small even if the views of the enclosure from different parts of each zone are quite different, provided the emissivity is fairly high; the error in F  is zero if it is obtainable from Fig. 5-15 or 5-17, small if Eq. (5-137) is used and the variation in temperature over the refractory is small. An approach to any desired accuracy can be made by use of Eqs. (5-126) and (5-130) with division of the surfaces into more zones. From the definitions of F,  F, and Ᏺ it is to be noted that F11 + F12 + F13 + ⋅⋅⋅ + F1r + F1s + ⋅⋅⋅ = 1 F F12 +  F13 + ⋅⋅⋅ = 1 11 +  Ᏺ11 + Ᏺ12 + Ᏺ13 + ⋅⋅⋅ = ε1 Example 5: Radiation in a Furnace Chamber A furnace chamber of rectangular parallelepipedal form is heated by the combustion of gas inside vertical radiant tubes lining the sidewalls. The tubes are of 0.127-m (5-in) outside diameter on 0.305-m (12-in) centers. The stock forms a continuous plane on the hearth. Roof and end walls are refractory. Dimensions are shown in Fig. 5-20. The radiant tubes and stock are gray bodies having emissivities of 0.8 and 0.9 respectively. What is the net rate of heat transmission to the stock by radiation when the mean temperature of the tube surface is 816°C (1500°F) and that of the stock is 649°C (1200°F)?

5-32

FIG. 5-20

HEAT AND MASS TRANSFER

Furnace-chamber cross section. To convert feet to meters, multiply

by 0.3048.

This problem must be broken up into two parts, first considering the walls with their refractory-backed tubes. To imaginary planes A2 of area 1.83 by 3.05 m (6 by 10 ft) and located parallel to and inside the rows of radiant tubes, the tubes emit radiation σ T 14A1Ᏺ12, which equals σT 14A2 Ᏺ21. To find Ᏺ21, use Fig. 5-17, curve 5, from which F 21 = 0.81. Then from Eq. (10-200) 1 ᎏᎏᎏᎏ Ᏺ21 = 1 1 = 0.702 12 1 ᎏᎏ − 1 + ᎏᎏ ᎏᎏ − 1 + ᎏᎏ 1 5π 0.8 0.81 This amounts to saying that the system of refractory-backed tubes is equal in radiating power to a continuous plane A2 replacing the tubes and refractory back of them, having a temperature equal to that of the tubes and an equivalent or effective emissivity of 0.702. The new simplified furnace now consists of an enclosure formed by two 1.83by 3.05-m (6- by 10-ft) radiating sidewalls (area A2, emissivity 0.702), a 1.52- by 3.05-m (5- by 10-ft) receiving plane on the floor A3, and refractory surfaces Ar to complete the enclosure (ends, roof, and floor side strips). The desired heat transfer is Q˙ 2 ⭵ 3 = σ (T 14 − T 34)A2 Ᏺ23



  



To evaluate Ᏺ23, start with the direct interchange factor F23. F23 = F from A2 to (A3 + a strip of Ar alongside A3, which has a common edge with A2) minus F from A2 to the strip only. These two F’s may be evaluated from Fig. 5-20 and Eq. (5-121). For the first F, Y = 1.83/3.05, Z = 1.98/3.05, and F = 0.239; for the second F, Y = 1.83/3.05, Z = 0.46/3.05, and F = 0.100. Then F23 = 0.239 − 0.10 = 0.139. Now F  may be evaluated. From Eq. 5-137 et seq., 1 1 A2F23 = 23 ÷ ᎏᎏ F 23 = F23 + ᎏᎏᎏ 1/2r + 1/3r (1/F2r) + (A2/A3)(1/F3r) Since A2 “sees” Ar, A3, and some of itself (the plane opposite), F2r = 1 − F22 − F23. F22, the direct interchange factor between parallel 1.83- by 3.05-m (6- by 10-ft) rectangles separated by 2.44 m (8 ft), may be taken as the geometric mean of the factors for 1.83-m (6-ft) squares separated by 2.44 m (8 ft) and for 3.05-m (10-ft) squares separated by 2.44 m (8 ft). These come from Fig. 5-15, curve 2, according to which F22 = 0 .13 ×0 .255 = 0.182. Then F2r = 1 − 0.182 − 0.139 = 0.679. The other required direct factor is F3r = 1 − F32 = 1 − F23 A2/A3 = 1 − (0.139)(11.14)/4.65 = 0.666. Then 1 F 23 = 0.139 + ᎏᎏᎏᎏ = 0.336 (1/0.679) + (11.14/4.65)(1/0.666) Having  F23, we may now evaluate the factor Ᏺ23: 1 Ᏺ23 = ᎏᎏᎏᎏᎏᎏ = 0.273 (1/0.336) + [(1/0.702) − 1] + (11.14/4.65)[(1/0.9) − 1] ˙ net = σ(T 14 − T 43)A2 Ᏺ23 = 5.67(10.894 − 9.224)(11.15)(0.273) Q = 118,000 J/s (402,000 Btu/h) A result of interest is obtained by dividing the term A2 Ᏺ23(11.15 × 0.273, or 3.04) by the actual area A1 of the radiating tubes (0.127π)(18.3)(2) = 14.6 m2 (157 ft2). Thus 3.04/14.6 = 0.208, which means that the net radiation from a tube to the stock is 20.8 percent as much as if the tube were black and completely surrounded by black stock.

Integral Formulation The zone method has the purpose of dodging the solution of an integral equation. If in Eq. (5-126) the zone on which the radiation balance is formulated is decreased to a differential element, that equation becomes dAi dAf (cos θi)(cos θj)Wi dA jεj Ej + ρj ᎏᎏᎏ = dA jWj (5-140) r2



which is an integral equation with the unknown function W inside the integral. Integration is over the entire surface area. Exact solutions have been carried out for only a few simple cases. One of these is the evaluation of emittance of an isothermal spherical cavity, for which dAi dAf (cos θi)(cos θj)/r2 in the integral of Eq. (5-140) becomes dAi dAj /4πR2, where R is the sphere radius. For this special case W is, from Eq. (10-202), constant over the inner surface of the cavity and given by εE W = ᎏᎏ (5-141) 1 − ρ(1 − A1/4πR2) where A1 is the curved area of a hole in the sphere’s surface. The ratio W/E is the effective emittance of the hole as sensed by a narrow-angle receiver viewing the cavity interior. If the material of construction of the cavity is a diffuse emitter and reflector and has an emissivity of 0.5 and the cavity is to appear at least 98 percent black, the curved area A1 of the hole must be smaller than 2 percent of the total surface area of the sphere. Enclosures of Surfaces That Are Not Diffuse Reflectors If no restriction that the surfaces be diffuse emitters and reflectors is imposed, Eq. (5-140) becomes much more complex. The W’s are replaced by πI’s and εj, Ii, and Ij all become functions of the angle of the leaving beam, and ρj goes inside the integral and becomes a function of angles of incidence and reflection. Seldom are such details of reflectance known. When they are and a solution is needed, the Monte Carlo method of tracing the history of a large number of beams emitted from random positions and in random initial directions is probably the best method of obtaining a solution. Another approach is possible, however, because of the tendency of most surfaces to fit a simpler reflection model. The total reflectance ρ( 1 − ε) can be represented by the sum of a diffuse component ρD and a specular component ρS. For applications see Hottel and Sarofim (op. cit., chap. 5). The method yields the following relation for exchange between concentric spheres or infinite cylinders: A1Ᏺ12  S1S 2 1 = ᎏᎏᎏᎏᎏᎏ (5-142) 1/A1ε1 + (1/A2)(1/ε2 − 1) + [ρs2 /(1 − ρs2)](1/A1 − 1/A2) When there is no specular reflectance, the third term in the denominator drops out, in agreement with Eqs. (5-134) and (5-135). When the reflectance is exclusively specular, the denominator becomes 1/A1ε1 + ρs2 /A1(1 − ρs2), easily derivable from first principles. EMISSIVITIES OF COMBUSTION PRODUCTS The radiation from a flame is due to radiation from burning soot particles of microscopic and submicroscopic dimensions, from suspended larger particles of coal, coke, or ash, and from the water vapor and carbon dioxide in the hot gaseous combustion products. The contribution of radiation emitted by the combustion process itself, so-called chemiluminescence, is relatively negligible. Common to these problems is the effect of the shape of the emitting volume on the radiative flux; this is considered first. Mean Beam Lengths Evaluation of radiation from a nonisothermal volume is beyond the scope of this section (see Hottel and Sarofim, Radiative Transfer, McGraw-Hill, New York, 1967, chap. 11). Consider an isothermal gas confined within the volume bounded by the solid angle dΩ with vertex at dA and making the angle θ with the normal to dA. The ratio of the emission to dA from the gas to that from a blackbody at the gas temperature and filling the field of view dΩ is called the gas emissivity ε. Clearly, ε depends on the path length L through the volume to dA. A hemispherical volume radiating to a spot on the center of its base represents the only case in which L is independent of direction. Flux at that spot relative to hemispherical blackbody flux is thus an alternative way to visualize emissivity. The flux density to a small area of interest on the envelope of an emitter volume of any shape can be matched by that at the base of a hemispherical volume of some radius L, which is called the mean beam length. It is found that although the ratio of L to a characteristic dimension D of the shape varies with opacity, the variation is small enough for most engineering purposes to permit use of a constant ratio LM/D,

HEAT TRANSFER BY RADIATION where LM is the average mean beam length. LM can be defined to apply either to a spot on the envelope or to any finite portion of its area. An important limiting case is that of opacity approaching zero (pD → 0, where p = partial pressure of the emitter constituent). For this case, L (called L0) equals 4V/A (V = gas volume; A = bounding area) when interest is in radiation to the entire envelope. For the range of pD encountered in practice, the optimum value of L (now LM) lies between 0.8 to 0.95 times L0. For shapes not reported in Table 5-7, a factor of 0.88 (or LM = 0.88L0 = 3.5V/A) is recommended. Instead of using the average-mean-beam-length concept to approximate A1[ε(Lm)] (the flux per unit black emissive power from a gas volume partially bounded by a surface of area A1), one may calculate the flux rigorously by integration, over the gas volume and over A1, of the expression 4k dv τ(r) dA cos θ/πr 2. Here k is the emission coefficient of the gas, and τ(r) is the transmittance through the distance r between dv and dA. The result has the dimensions of area and, by analogy to ss, is called gs1, the direct-exchange area between the gas zone and the surface zone (Hottel and Sarofim, op. cit., chap. 7). The use of A1[ε(Lm)] instead of gs1 is adequate when the problem is such that all the gas can be treated as a single zone in contact with A1, and having a mean radiating temperature, but the gs concept is clearly useful if allowance is to be made for temperature variations within the gas. Gaseous Combustion Products Radiation from water vapor and carbon dioxide occurs in spectral bands in the infrared. In magnitude it overshadows convection at furnace temperatures. Carbon Dioxide The contribution εc to the emissivity of a gas containing CO2 depends on gas temperature TG, on the CO2 partial pressure-beam length product pc L and, to a much lesser extent, on the total pressure P. Constants for use in evaluating εc at a total pressure of 101.3 kPa (1 atm) are given in Table 5-8 (more on this later). The gas absorptivity αc equals the emissivity when the absorbing gas and the emitter are at the same temperature. When the emitter surface temperature is T1, αc is (TG /T1)0.65 times εc, evaluated using Table 5-8 at T1 instead of TG and at pc LT1 /TG instead of pc L. Line broadening, due to TABLE 5-7

increases either in total pressure or in partial pressure of CO2, makes a correction necessary. However, at a total pressure of 101.3 kPa (1 atm) the correction factor may be ignored, since it decreases with increase in temperature and is never more than 4 percent at temperatures above 1111 K(2000°R). Estimations of the correction in systems up to 1013.3 kPa (10 atm) are given by Hottel and Sarofim (op. cit., p. 228), and by Edwards [J. Opt. Soc. Am., 50, 617 (1960)] who in addition presents data on CO2-band emission for use in calculations involving spectrally selective surfaces. The principal emission bands of CO2 are at about 2.64 to 2.84, 4.13 to 4.5, and 13 to 17 µm. Water Vapor The contribution εw to the emissivity of a gas containing H2O depends on TG and pw L and on total pressure P and partial pressure pw. Table 5-8 gives constants for use in evaluating εw. Allowance for departure from the special pressure conditions is made by multiplying εw by a correction factor Cw read from Fig. 5-21 as a function of (pw + P) and pw L. The absorptivity αw of water vapor for blackbody radiation is εw evaluated from Table 5-8 but at T1 instead of TG and at pw LT1 /TG instead of pw L. Multiply by (TG /T1)0.45. The correction factor Cw still applies. Spectral data for water vapor, tabulated for 371 wavelength intervals from 1 to 40 µm, are also available [Ferriso, Ludwig, and Thompson, J. Quant. Spectros. Radiat. Transfer, 6, 241–273 (1966)]. The principal emission is in bands at about 2.55 to 2.84, 5.6 to 7.6, and 12 to 25 µm. Carbon Dioxide–Water-Vapor Mixtures When these gases are present together, the total radiation due to both is somewhat less than the sum of the separately calculated effects, because each gas is somewhat opaque to radiation from the other in the wavelength regions 2.7 and 15 µm. Allowance for spectral overlap, the effect of pressure, and the effect of soot luminosity would make computation tedious. Table 5-8 gives constants for use in direct calculation, for H2O/CO2 mixtures, of the product εGT . The product term is used because it varies much less with T than does εG alone. Constants are given for mixtures, in nonradiating gases, of water vapor alone, CO2 alone, and four pw /pc mixtures.

Mean Beam Lengths for Volume Radiation Shape

Sphere Infinite cylinder Semi-infinite cylinder, radiating to: Center of base Entire base Right-circle cylinder, ht. = diam. radiating to: Center of base Whole surface Right-circle cylinder, ht. = 0.5 diam. radiating to: End Side Total surface Right-circle cylinder, ht. = 2 × diam. radiating to: End Side Total surface Infinite cylinder, half-circle cross section radiating to middle of flats Rectangular parallelepipeds: 1:1:1 (cube) 1:1:4, radiating to: 1 × 4 face 1 × 1 face Whole surface 1:2:6, radiating to: 2 × 6 face 1 × 6 face 1 × 2 face Whole surface Infinite parallel planes Space outside bank of parallel tubes on equilateral triangular centers Tube diam. = clearance Tube diam. = a clearance Tube centers on squares, diam. = clearance

5-33

Characteristic dimension, D

L 0/D

L M /D

Diameter Diameter

0.67 1.0

0.63 0.94

Diameter Diameter

1.0 0.81

0.90 0.65

Diameter Diameter

0.76 0.67

0.71 0.60

Diameter Diameter Diameter

0.47 0.52 0.50

0.43 0.46 0.45

Diameter Diameter Diameter

0.73 0.82 0.80

0.60 0.76 0.73

Radius

1.26

Edge

0.67

0.60

Shortest edge Shortest edge Shortest edge

0.90 0.86 0.89

0.82 0.71 0.81

Shortest edge Shortest edge Shortest edge Shortest edge Clearance

1.18 1.24 1.18 1.2 2.00

1.76

Clearance 3.4 Clearance 4.45 Clearance 4.1

2.8 3.8 3.5

0.82 0.85 0.85

5-34

HEAT AND MASS TRANSFER

TABLE 5-8

Emissivity eG of H2O:CO2 Mixtures Limited range for furnaces, valid over 25-fold range of pw + cL, 0.046–1.15 m atm (0.15–3.75 ft. atm)

pw /pc pw ᎏ pw + pc

0

a

1

2

3

∞

0

s(0.3–0.42)

a(0.42–0.5)

w(0.6–0.7)

e(0.7–0.8)

1

corresponding to (CH6)x, covering future high H2 fuels

H2O only

CO2 only

corresponding to (CH)x, covering coal, heavy oils, pitch

corresponding to (CH2)x, covering distillate oils, paraffins, olefines

corresponding to CH4, covering natural gas and refinery gas

Constants b and n of Eq., εGT = b(pL − 0.015)n, pL = m atm, T = K T, K

b

n

b

n

b

n

b

n

b

n

b

n

1000 1500 2000

188 252 267

0.209 0.256 0.316

384 448 451

0.33 0.38 0.45

416 495 509

0.34 0.40 0.48

444 540 572

0.34 0.42 0.51

455 548 594

0.35 0.42 0.52

416 548 632

0.400 0.523 0.640

T, °R

b

n

b

n

b

n

b

n

b

n

b

n

1800 2700 3600

264 335 330

0.209 0.256 0.316

467 514 476

0.33 0.38 0.45

501 555 519

0.34 0.40 0.48

534 591 563

0.34 0.42 0.51

541 600 577

0.35 0.42 0.52

466 530 532

0.400 0.523 0.640

Constants b and n of Eq., εGT = b(pL − 0.05)n, pL = ft. atm, T = °R

Full range, valid over 2000-fold range of pw + cL, 0.005–10.0 m atm (0.016–32.0 ft. atm) Constants of Eq., log10 εGTG = a0 + a1 log pL + a2 log2 pL + a3 log3 pL pL = m atm, T = K pw ᎏ pc

pw ᎏ pw + pc

0

0

a

s

1

a

2

w

3

e

∞

1

pL = ft. atm, T = °R

T, K

a0

a1

a2

a3

T, °R

a0

a1

a2

a3

1000 1500 2000 1000 1500 2000 1000 1500 2000 1000 1500 2000 1000 1500 2000 1000 1500 2000

2.2661 2.3954 2.4104 2.5754 2.6451 2.6504 2.6090 2.6862 2.7029 2.6367 2.7178 2.7482 2.6432 2.7257 2.7592 2.5995 2.7083 2.7709

0.1742 0.2203 0.2602 0.2792 0.3418 0.4279 0.2799 0.3450 0.4440 0.2723 0.3386 0.4464 0.2715 0.3355 0.4372 0.3015 0.3969 0.5099

−0.0390 −0.0433 −0.0651 −0.0648 −0.0685 −0.0674 −0.0745 −0.0816 −0.0859 −0.0804 −0.0990 −0.1086 −0.0816 −0.0981 −0.1122 −0.0961 −0.1309 −0.1646

0.0040 0.00562 −0.00155 0.0017 −0.0043 −0.0120 −0.0006 −0.0039 −0.0135 0.0030 −0.0030 −0.0139 0.0052 0.0045 −0.0065 0.0119 0.00123 −0.0165

1800 2700 3600 1800 2700 3600 1800 2700 3600 1800 2700 3600 1800 2700 3600 1800 2700 3600

2.4206 2.5248 2.5143 2.6691 2.7074 2.6686 2.7001 2.7423 2.7081 2.7296 2.7724 2.7461 2.7359 2.7811 2.7599 2.6720 2.7238 2.7215

0.2176 0.2695 0.3621 0.3474 0.4091 0.4879 0.3563 0.4561 0.5210 0.3577 0.4384 0.5474 0.3599 0.4403 0.5478 0.4102 0.5330 0.6666

−0.0452 −0.0521 −0.0627 −0.0674 −0.0618 −0.0489 −0.0736 −0.0756 −0.0650 −0.0850 −0.0944 −0.0871 −0.0896 −0.1051 −0.1021 −0.1145 −0.1328 −0.1391

0.0040 0.00562 −0.00155 0.0017 −0.0043 −0.0120 −0.0006 −0.0039 −0.0135 0.0030 −0.0030 −0.0139 0.0052 0.0045 −0.0065 0.0119 0.00123 −0.0165

NOTE: pw /(pw + pc) of s, a, w, and e may be used to cover the ranges 0.2–0.4, 0.4–0.6, 0.6–0.7, and 0.7–0.8, respectively, with a maximum error in εG of 5 percent at pL = 6.5 m atm, less at lower pLs. Linear interpolation reduces the error generally to less than 1 percent. Linear interpolation or extrapolation on T introduces an error generally below 2 percent, less than the accuracy of the original data.

Four suffice, since a change halfway from one mixture ratio to the adjacent one changes the emissivity by a maximum of but 5 percent; linear interpolation may be used if considered necessary. The constants are given for three temperatures, adequate for linear interpolation since εGT  changes a maximum of one-sixth due to a change from one temperature base halfway to the adjacent one. The interpolation relation, with TH and TL representing the higher and lower base temperatures bracketing T, and with the brackets in the term [A(x)] indicating that the parentheses refer not to a multiplier but to an argument, is [εGT H(pL)](TG − TL) + [εGT L(pL)](TH − TG) εGT G = ᎏᎏᎏᎏᎏ (5-143) 500 Extrapolation to a temperature that is above the highest or below the lowest of the three base temperatures in Table 5-8 uses the same formulation, but one of its terms becomes negative. The gas absorptivity may also be obtained from the constants for emissivities. The product αG1T1 (gas absorptivity for black surface radiation), x (surface temperature), is εGT 1 evaluated at T1 instead of TG and at pLT1/TG instead of pL, then multiplied by (TG /T1)0.5, or pLT1 TG 0.5 αG1T1 = εGT (5-144) 1 ᎏ ᎏ TG T1

 

  

The exponent 0.5 is an adequate average of the exponents for the pure components. The interpolation relation for absorptivity is TG 0.5 T1 − TL pLTH αG1T1 = εGT ᎏ H ᎏ ᎏ TG TH 500

 

     

pLTL + εGT L ᎏ TG



T T −T ᎏ  ᎏ T   500  G

0.5

H

1

(5-145)

L

The base temperature pair TH and TL can be different for evaluating εG and αG1 if TG and T1 are far enough apart. Example 6: Calculation of Gas Emissivity and Absorptivity This example will use only SI units, except that pressure will be in atm, not kPa. Flue gas containing 6 percent CO2 and 11 percent H2O vapor, wet basis, flows through a bank of tubes of 0.1016 in (4-in) outside diameter on equilateral 0.2032 m (8-in) triangular centers. In a section in which the gas and tube surface temperatures are 691°C (964 K) and 413°C (686 K), what are the emissivity and absorptivity of the gas? From Table 5-8, Lm = (2.8)(0.01016) = 0.2845 m (only SI units will be used in this example). p = pw + pc = 0.17 atm; pL = 0.0484 m atm, barely large enough to justify the short method, the top part of Table 5-8. pw /pc = 11/6, near enough to 2 to use col. 5. Since both TG and T1 are below the lowest T in the top part of the table, use the nearest pair, TH = 1500 K and TL = 1000 K. At TH, b = 540, n = 0.42. εGT H = 540(0.0484 − 0.015)0.42 = 129.5. At TL,

HEAT TRANSFER BY RADIATION

5-35

cles are small relative to the wavelength of the radiation of interest [diameters (2)(10−8) to (1.4)(10−7) m (200 to 1400 Å)], the monochromatic emissivity ελ depends on the total particle volume per unit volume of space fv regardless of particle size. It is given by ελ = 1 − e−Kf v L/λ

(5-146)

where L is the path length. Use of the perfect gas law and a material balance allows the restatement of the above to ελ = 1 − e−KPSL/λT

Correction factor for converting emissivity of water vapor to values of Pw and PT other than 0 to 1 atm respectively. To convert atmosphere-feet to kilopascal-meters, multiply by 30.89; to convert atmospheres to kilo-pascals, multiply by (1.0133)(102).

FIG. 5-21

(5-147)

where P is the total pressure (atm) and S is the mole fraction of soot in the gas. S depends on the fractional conversion fc of the fuel carbon to soot and is the mole fraction, wet basis, of carbon in gaseous form (CO2, CO, CH4, etc.) times fc /(1 − fc) or, with negligible error, times fc, which is a very small number. Evaluation of K is complex, and its numerical value depends somewhat on the age of the soot, the temperature at which it is formed, and its hydrogen content. It is recommended that K = 0.526 [K/atm]. The total emissivity of soot εs is obtained by integration over the wavelength spectrum, giving KPSL 15 εs = 1 − ᎏ ψ(3) 1 + ᎏ , (5-148) 4 c2 where ψ(3)(x) is the pentagamma function of x. It may be shown that an excellent approximation to Eq. (5-148) is

 



εs = 1 − [1 + 34.9SPL]−4 b = 444, n = 0.34, εGT L = 444(0.0484 − 0.015)0.34 = 139.8. From interpolation Eq. (5-143) (here extrapolation), εGT G = [129.5 (964 − 1000) + 139.5 (1500 − 964)]/500 = 140.2. For αG1T1 with T1 = TH = 1500, pLTH /TG = 0.0753. From Eq. (5-144),  αGHT αGLT H = 540 (0.0753 − 0.015)0.42(964/1500)0.5 = 133.1. Similarly,  L = 444 (0.0502 − 0.015)0.34 (964/1000)0.5 = 139.7. From Eq. (5-145) 133.1(686 − 1000) + 139.7(1500 − 686) αG1T1 = ᎏᎏᎏᎏᎏ = 143.8 500 Then εG = 140.2/964 = 0.145 and αG1 = 143.8/686 = 0.210. If the longer method (the bottom part of Table 5-8) were used, εG = 0.141 and αG1 = 0.206.

Other Gases Because of their practical importance, the emissivities of CO2 and H2O have been studied much more extensively than those of other gases, and the values summarized in the preceding paragraphs are based on extensive measurement of both total and integrated spectral values. Correction for pressure has reduced the disagreement among experimenters. A summary of the less adequate information on other gases appears in Table 5-9. Flames and Particle Clouds Luminous Flames Luminosity conventionally refers to soot radiation; it is important when combustion occurs under such conditions that the hydrocarbons in the flame are subject to heat in the absence of sufficient air well mixed on a molecular scale. Because soot parti-

TABLE 5-9

(5-149)

where PL is in atm m. The error is less the lower εs, and is only 0.5 percent at εs = 0.5 and 0.8 percent at 0.67. There is at present no method of predicting soot concentration of a luminous flame analytically; reliance must be placed on experimental measurement on flames similar to that of interest. Visual observation is misleading; a flame so bright as to hide the wall behind it may be far from a “black” radiator. The chemical kinetics and fluid mechanics of soot burnout have not progressed far enough to evaluate the soot fraction fc for relatively complex systems. Additionally, the soot in a combustion chamber is highly localized, and a mean value is needed for calculation of the radiative heat transfer performance of the chamber. On the basis of limited experience with fitting data to a model, the following procedure is recommended when total combustion chamber performance is being estimated: (1) When pitch, or a highly aromatic fuel, is burned, one percent of the fuel carbon appears as soot. This produces values of εs of 0.4–0.5 and εG + s of 0.6–0.7. These values are lower than some measurements on pitch flames, but the measurements are usually taken through the flame at points of high luminosity. (2) When No. 2 fuel oil is burned, 33 percent of the fuel carbon appears as soot (but that number varies greatly with burner design). (3) When natural gas is burned, any soot contribution to emissivity may be ignored. Admittedly, the numbers given should be functions of burner design and excess air, and they should be considered tentative, subject to change when good data show they are off target. The Inter-

Total Emissivities of Some Gases 1000°R

1600°R

2200°R

2800°R

Temperature Px L, (atm)(ft)

0.01

0.1

1.0

0.01

0.1

1.0

0.01

0.1

1.0

0.01

0.1

1.0

NH3a SO2b CH4c COd NOe HCl f

0.047 0.020 0.020 0.011 0.0046 0.00022

0.20 0.13 0.060 0.031 0.018 0.00079

0.61 0.28 0.15 0.061 0.060 0.0020

0.020 0.013 0.023 0.022 0.0046 0.00036

0.120 0.090 0.072 0.057 0.021 0.0013

0.44 0.32 0.194 0.10 0.070 0.0033

0.0057 0.0085 0.022 0.022 0.0019 0.00037

0.051 0.051 0.070 0.050 0.010 0.0014

0.25 0.27 0.185 0.080 0.040 0.0036

(0.001) 0.0058 0.019 (0.012) 0.00078 0.00029

(0.015) 0.043 0.059 (0.035) 0.004 0.0010

(0.14) 0.20 0.17 (0.050) 0.025 0.0027

NOTE: Figures in this table are taken from plots in Hottel and Sarofim, Radiative Transfer, McGraw-Hill, New York, 1967, chap. 6. Values in parentheses are extrapolated. To convert degrees Rankine to kelvins, multiply by (5.556)(10−1). To convert atmosphere-feet to kilopascal-meters, multiply by 30.89. a Total-radiation measurements of Port (Sc.D. thesis in chemical engineering, MIT, 1940) at 1-atm total pressure, L = 1.68 ft, T to 2000°R. b Calculations of Guerrieri (S.M. thesis in chemical engineering, MIT, 1932) from room-temperature absorption measurements of Coblentz (Investigations of Infrared Spectra, Carnegie Institution, Washington, 1905) with poor allowance for temperature. c Band measurements of Lee and Happel [Ind. Eng. Chem. Fundam., 3, 167 (1964)] at T up to 2050°R plus calculations to extrapolate temperature to 3800°R. d Total-radiation measurements of Ullrich (Sc.D. thesis in chemical engineering, MIT, 1953) at 1-atm total pressure, L = 1.68 ft, T to 2200°R. e Calculations of Malkmus and Thompson [J. Quant. Spectros. Radiat. Transfer, 2, 16 (1962)], to T = 5400°R and PL = 30 atm ⋅ ft. f Calculations of Malkmus and Thompson [J. Quant. Spectros. Radiat. Transfer, 2, 16 (1962)], to T = 5400°R and PL = 300 atm ⋅ ft.

5-36

HEAT AND MASS TRANSFER

where a/v is the projected area of the particles per unit volume of space. If the particles have no negative curvature (a particle can see none of itself) and are randomly oriented, a is a′/4, where a′ is the actual surface area; and if the particles are uniform, a/v = cA = cA′/4 where A and A′ are the projected and total areas of each particle and c is the number concentration of particles. For spherical particles, this gives εM = 1 − e−(π/4)cd L = 1 − e−1.5fvL/d (5-153)

Evaluation of αG1 is unnecessary when T1 is less than one-half TG; αG1 may then be assumed equal to εG. If the bounding surface is gray rather than black, multiplication of Eq. (5-154) by surface emissivity ε1 allows properly for reduction of the primary beams, gas-to-surface or surface-to-gas, but secondary reflections are ignored. The correction then lies between ε1 and 1, and for most industrially important surfaces with ε1 > 0.8 a value of (1 + ε1)/ 2 is adequate. Rigorous allowance for this and other factors is presented later, e.g., Eq. (5-163). If the bounding walls are mostly sink-type surfaces of area A1 and temperature T1, but in small part refractory surfaces of area Ar in radiative equilibrium at unknown temperature Tr, an energy balance on Ar is in principle necessary to determine Tr and the effect on energy flux. However, the total heat transfer to the sink may be visualized as corresponding to its having an effective area equal to its own plus a fraction x of that of the refractory, with the only temperatures involved being those of the gas and the heat sink. The fraction x varies from zero when the ratio of refractory to heat-sink surface is very high to unity when the ratio is very low and the value of εG is low. If Ar is small compared with A1, a value for x of 0.7 may be used in the approximate method. Long Exchanger This case, in which axial radiative flux is ignored, includes most radiatively modified heat exchangers of interest to chemical engineers. When the gas temperature transverse to the flow direction is reasonably uniform and the chamber is long compared with its mean hydraulic radius, the opposed upstream and downstream fluxes through the flow cross section will substantially cancel (hot combustion products through tubes or across tube banks, tunnel kilns, billet-reheating furnaces, Example 7). Under these conditions, the radiative contribution to local flux density q may be formulated in terms of local temperatures and beam lengths or exchange areas evaluated for a two-dimensional system infinite in the flow direction. The local flux density at the sink A1 is then

As an example, consider heavy fuel oil (CH1.5, specific gravity, 0.95) atomized to a surface mean particle diameter of d, burned with 20 percent excess air to produce coke-residue particles having the original drop diameter and suspended in combustion products at 1204°C (2200°F). The flame emissivity due to the particles along a path of L m will be, with d in micrometers, εM = 1 − e−24.3L/d (5-154)

where h is the local convective heat-transfer coefficient and qr(TG, T1) the radiation contribution calculated from TG, T1, εG, and ε1 by using the approximate treatment in the preceding subsection or the more rigorous treatment in the following subsection. If m ˙ Cp is the hourly heat capacity of the gas stream, the temperature of which changes by dTG over the sink-area increment dA1, then

national Flame Foundation has recorded data on many luminous flames from gas, oil, and coal (see J. Inst. Energy, formerly J. Inst. Fuel, 1956 to present). Combined Soot, H2O, and CO2 Radiation The spectral overlap of H2O and CO2 radiation has been taken into account by the constants for obtaining εG. Additional overlap occurs when soot emissivity εs is added. If the emission bands of water vapor and CO2 were randomly placed in the spectrum and soot radiation were gray, the combined emissivity would be εG plus εs minus an overlap correction εGεs. But monochromatic soot emissivity is higher the shorter the wavelength, and in a highly sooted flame at 1500 K half the soot emission lies below 2.5 µm where H2O and CO2 emission is negligible. Then the correction εGεs must be reduced, and the following is recommended: εG + s = εG + εs − MεGεs (5-150) where M depends mostly on TG and to a much less extent on optical density SPL. Values that have been calculated from this simple model can be represented with acceptable error by T M = 1.07 + 18 SPL − 0.27 ᎏ (5-151) 1000 Clouds of Large Black Particles The emissivity εM of a cloud of particles with a perimeter large compared with wavelength λ is εM = 1 − e−(a/v)L (5-152)





2

With 200-µm particles and an L of 3.05 m (10 ft), the particle contribution to emissivity will be 0.31. Soot luminosity will increase this; particle burnout will decrease it. Clouds of Nonblack Particles The correction for nonblackness of the particles is complicated by multiple scatter of the radiation reflected by each particle. The emissivity εM of a cloud of gray particles of individual surface emissivity ε1 can be estimated by the use of Eq. (5-151), with its exponent multiplied by ε1, if the optical thickness (a/v)L does not exceed about 2. Modified Eq. (5-151) would predict an approach of εM to 1 as L → ∞, an impossibility in a scattering system; the asymptotic value of εM can be read from Fig. 5-14 as εh, with albedo ω given by particle-surface reflectance 1 − ε1. Particles with a perimeter lying between 0.5 and 5 times the wavelength of interest can be handled with difficulty by use of the Mie equations (see Hottel and Sarofim, op. cit., chaps. 12 and 13). Summation of Separate Contributions to Gas or Flame Emissivity Flame emissivity εG + s due to joint emission from gas and soot has already been treated. If massive-particle emissivity εM, such as from fly ash, coal char, or carbonaceous cenospheres from heavy fuel oil, are present, it is recommended that the total emissivity be approximated by εG + s + εM − (εG + s)(εM) RADIATIVE EXCHANGE BETWEEN GASES OR SUSPENDED MATTER AND A BOUNDARY ˙ between an Local Radiative Exchange The interchange rate Q isothermal gas mass at TG and its isothermal black bounding surface of area A1 is given by ˙ = A1σ(TG4 εG − T 41 αG1) Q (5-155)

q(TG, T1) = qr(TG, T1) + h(TG − T1)

˙ Cp dTG [q(TG, T1)]dA1 = −m from which

(5-156)

(5-157)



TG,inlet

Cp dTG (5-158) ᎏ q(TG, T1) The area under a curve of Cp /q versus TG or 1/q versus the specific enthalpy i may be used to solve for the area A1 required to obtain a given outlet temperature or to obtain the outlet temperature given A1. Three points generally suffice to determine the area under the curve within 10 percent. Instead of using graphical integration, which can handle any complexity of variation of flux density q with TG and T1 along an interchanger flow path, one may evaluate a mean flux density based on mean gas and sink temperatures, based in turn on terminal temperatures. It has been found empirically that fair results are obtained by the use of a mean surface temperature equal to the arithmetic mean of the terminal surface temperatures and by the use of a mean gas temperature equal to the mean surface temperature plus the logarithmic mean of the temperature difference, gas to surface, at the two ends of the exchanger. When radiation dominates the transfer process, however, graphical integration is safer. ˙ A1 = m

TG,outlet

Example 7: Radiation in Gases Flue gas containing 6 percent carbon dioxide and 11 percent water vapor by volume (wet basis) flows through the convection bank of an oil tube still consisting of rows of 0.102-m (4-in) tubes on 0.203-m (8-in) centers, nine 7.62-m (25-ft) tubes in a row, the rows staggered to put the tubes on equilateral triangular centers. The flue gas enters at 871°C (1144 K, 1600°F) and leaves at 538°C (811 K, 1000°F). The oil flows in a countercurrent direction to the gas and rises from 316 to 427°C (600 to 800°F). Tube surface emissivity is 0.8. What is the average heat-input rate, due to gas radiation alone, per square meter of external tube area?

HEAT TRANSFER BY RADIATION With each row of tubes there is associated (0.203)(3/2) = 0.176 m (0.577 ft) of wall height, of area [(0.203)(9)(2) + (7.62)(2)]0.176 − (9)(2)(π)(0.0508)2 = 3.18 m2 (34.2 ft2). One row of tubes has an area of (π)(0.102)(7.62)(9) = 22.0 m2 (236 ft2). If the recommended factor of 0.7 on the refractory area is used, the effective area of the tubes is [22.0 + (0.7)(3.18)]/22.0 = 1.10 m2/m2 of actual area. The exact evaluation of the outside tube temperature from the known oil temperature would involve a knowledge of the oil-film coefficient, tube-wall resistance, and rate of heat flow into the tube, the evaluation usually involving trial and error. However, for the present purpose the temperature drop through the tube wall and oil film will be assumed to be 41.7°C (75°F), making the tube surface temperatures 357°C (675°F) and 468°C (875°F); the average is 412°C (775°F). The radiating gas temperature is (871 − 468) − (538 − 357) tg = 412 + ᎏᎏᎏᎏ 2.3 log [(871 − 468)/(538 − 357)] = 412 + 278 = 690°C (1274°F) These temperatures, partial pressures, and dimensions were used in Example 6 to determine gas emissivity and absorptivity, εG = 0.145; αG1 = 0.210. The approximate effective emissivity of the boundary is (0.8 + 1)/2 = 0.9. Then from Eq. (5-155), modified to allow for sink emissivity and for the presence of a small amount of refractory boundary, ˙ 1 = q = (0.9)(1.10)σ(TG4 εG − T41αG) Q/A = (0.9)(1.10)(5.67)[(9.63)4(0.145) − (6.85)4(0.210)] = 4405 J/(m2 tube area⋅s) [1396, Btu/(ft2 tube area⋅h)] This is equivalent to a convection coefficient of 4405/278, or 15.85 W/(m2)(K) which is of the order of magnitude expected of the convection coefficient itself. Radiation rapidly becomes dominant as the system temperature rises.

Total-Exchange Areas  SS GS  and   The arguments leading to the development of the interchange factor AiᏲij ( S iS j) between surfaces apply to the case of absorption within the gas volume if in the evaluation of the direct-exchange areas allowance is made for attenuation of the radiant beam through the gas. This necessitates nothing more than redefinition, in Eqs. (5-126) to (5-130), of every term ij( Si S j AiFij) to represent, per unit black emissive power, flux from Ai through an absorbing gas to Aj. This may be visualized as multiplication of AiFij by the mean gas transmittance Tij(= 1 − εG for a gray gas). In a system containing an isothermal gas and source-sink boundaries of areas A1 . . . An, the total emission from A1 per unit of its black emissive power is ε1A1, of which  S1S 1 + S 1S 2 + ⋅⋅⋅ + S 1S n is absorbed in the various source-sink surfaces by multiple reflections. The difference has been absorbed in the gas and is called the gas-surface totalexchange area  GS 1 G S1 = A1ε1 −
S  1S i

(5-159)

i

Note that though S 1S i is never used in calculating radiative exchange, its value is necessary for use of Eq. (5-159) to calculate  GS . If the gas volume is not isothermal and is zoned, an additional magnitude, the gas-to-gas total-exchange area  GiG j, arises (see Hottel and Sarofim. Radiative Transfer, McGraw-Hill, New York, 1967, chap. 11). Space does not permit derivations of special cases; only the single-gas-zone system is treated here. Single-Gas-Zone/Two-Surface-Zone Systems An enclosure consisting of but one isothermal gas zone and two gray surface zones can, properly specified, model so many industrially important radiation problems as to merit detailed presentation. One can evaluate the total radiation flux between any two of the three zones, including multiple reflection at all surfaces. ˙ G⇔1 =  Q GS 1σ(TG4 − T 41) ˙ 1⇔2 =  Q S1S 2σ(T14 − T 42 ) The total-exchange area takes a relatively simple closed form, even when important allowance is made for gas radiation not being gray and when a reduction of the number of system parameters is introduced by assuming that one of the surface zones, if refractory, is radiatively adiabatic. Before allowance is made for these factors, the case of a gray gas enclosed by two source-sink surface zones will be presented. Modification of Eq. 5-130, as discussed in the first paragraph of this subsection, combined with the assumption that a single mean beam length applies to all transfers; that is, that there is but one gas transmittance τ(= 1 − εG), gives

5-37

A1ε1ε2F12 S (5-160) 1S 2 = ᎏᎏᎏᎏᎏ 1/τ + τρ1ρ2(1 − F12 /C2) − ρ1(1 − F12) − ρ2(1 − F21) A1ε12(F11 + ρ2τ(F12 /C2 − 1) (5-161) S 1S 1 = ᎏᎏᎏᎏᎏ 1/τ + τρ1ρ2(1 − F12 /C2) − ρ1(1 − F12) − ρ2(1 − F21) A1ε1εG(1/τ + ρ2(F12 /C2 − 1)) (5-162) G S 1 = ᎏᎏᎏᎏᎏ 1/τ + τρ1ρ2(1 − F12 /C2) − ρ1(1 − F12) − ρ2(1 − F21) These three expressions suffice to formulate total-exchange areas for gas-enclosing arrangements which include, for example, the four cases illustrated in Table 5-10. An additional surface arrangement of importance is a single-zone surface enclosing gas. With the gas assumed gray, the simplest derivation of G S 1 is to note that the emission from surface A1 per unit of its blackbody emissive power is A1ε1, of which the fractions εG and (1 − εG)ε1 are absorbed by the gas and the surface, respectively, and the surface-reflected residue always repeats this distribution. Therefore, εG A1 G GS (5-163) S single surface  1 = A1ε1 ᎏᎏ = ᎏᎏ zone surround1 1 εG + (1 − εG)ε1 ing gray gas ᎏᎏ + ᎏᎏ − 1 εG ε1 Alternatively, G S1 could be obtained from Case 1 of Table 5-10 by  letting plane area A1 approach 0, leaving A2 as the sole surface zone. Departure of gas from grayness has so marked an effect on radiative transfer that the subject will be presented prior to discussion of the systems covered by Table 5-10. The Effect of Nongrayness of Gas on Total-Exchange Area A radiating gas departs from grayness in two ways: (1) Its transmittance τ through successive path lengths Lm due to surface reflection, instead of being constant, keeps increasing because at the wavelengths of high absorption the incremental absorption decreases with increasing pathlength; (2) Gas emissivity εG and absorptivity αG1 are not the same unless T1 equals TG. The total emissivity of a real gas, the spectral emissivity, and absorptivity ελ that varies in any way with λ can be expressed as the a-weighted mean of a suitable number of gray-gas emissivity or absorptivity terms εG,i or αG,i, representing the gray-gas emissivity or absorptivity in the energy fractions ai of the blackbody spectrum. Then n

n

0

0

εG =
aiεG,i =
ai(1 − e−k i pL)

(5-164)

For simplicity, n should be as low as is consistent with small error. The retention of but two terms is feasible when one considers that if αG1 is so fitted that the first absorption and the second following surface reflection are correct, then further attenuation of the beam by successive surface reflections makes the errors in those absorptions decrease in importance. Let the gas be modeled as the sum of one gray gas plus a clear gas, with the gray gas occupying the energy fraction a of the blackbody spectrum and the clear gas the fraction (1 − a). Then [εG(pL)] = a(1 − e−kpL) + (1 − a)0 [εG(2pL)] = a(1 − e−2kpL) + (1 − a)0

(5-165)

Solution of these gives εG(pL) a = ᎏᎏ εG(2pL) 2−ᎏ ᎏ εG(pL) εG(pL) kpL = −ln 1 − ᎏ a





(5-166)

Note that these values are specific to the subject problem in which the mean beam length is Lm, with εGs evaluated from basic data, such as Table 5-8. (1 − e−kpL) in Eq. (5-165) represents the emissivity of a gray gas, which will be called εG,i. For later use, note that, εG(pL) εG,i = ᎏ (5-167) a To allow for the difference between emissivity and absorptivity and combine them into a single emissivity-absorptivity term called effec-

5-38

HEAT AND MASS TRANSFER

TABLE 5-10

Total-Exchange Areas for Four Arrangements of Two-Zone-Surface Enclosures of a Gray Gas

A plane surface A1 and a surface A2 completing the enclosure

Infinite parallel planes

Concentric spherical or infinite cylindrical surface zones, A1 inside

Two-surface-zone spherical surface, each zone one or more parts; or speckled enclosure, any shape

F12 = 1

F12 = F21 = 1

F12 = 1; F21 = A1/A2

F12 = F22 = C2; F21 = F11 = C1

S 1S 2 ε1ε2 ᎏ=ᎏ A1 D1

S 1S 2 ε1ε2 ᎏ=ᎏ A1 D2

S 1S 2 ε1ε2 ᎏ=ᎏ A1 D3

S 1S 2 ε1ε2C2 ᎏ=ᎏ A1 D4

G S 1 ε1εG(1/τ + ρ2A1/A2) ᎏ = ᎏᎏ A1 D1

G S 1 ε1εG(1/τ + ρ2) ᎏ = ᎏᎏ A1 D2

G S 1 ε1εG(1/τ + ρ2A1/A2) ᎏ = ᎏᎏ A1 D3

G S 1 ε1εG/τ ᎏ=ᎏ A1 D4

G S 2 ε2εG(1/τ + ρ1A1/A2) ᎏ = ᎏᎏ A2 D1

S 1S 1 ε 12 ρ2τ ᎏ=ᎏ A1 D2

G S 2 ε2εG(1/τ + ρ1A1/A2) ᎏ = ᎏᎏ A2 D3

S 1S 1 ε 21 C1 ᎏ=ᎏ A1 D4

S 1S 1 ε 12 τρ2A1/A2 ᎏ = ᎏᎏ A1 D1

D2 1/τ − τρ1ρ2

S 1S 1 ε 12 ρ2τA1/A2 ᎏ = ᎏᎏ A1 D3

D4 1/τ − ρ1C1 − ρ2C2









1 A1 D1 ᎏ − ρ2 1 − ᎏ (1 − τρ1) τ A2

1 A1 D3 ᎏ − ρ2 1 − ᎏ (1 − τρ1) τ A2

tive emissivity εG,e, one must first evaluate absorptivity αG1 using Eq. (5-161). Formulation of the net direct exchange can then be used to define εG,e: σ(εGTG4 − αG1T 41) σεG,e(T G4 − T 41) εG,e − αG1(T41/ TG4 ) (5-168) or εG,e = ᎏᎏ 1 − (T1 / TG)4 The emissivity and absorptivity of use in converting gray-gas totalexchange areas to real-gas values are εG,e and ae, the latter obtained by using Eq. (5-166), except that εG,e (pL) replaces εG(pL); the same for εG,e (2pL). This means that, for the conversion, four terms will have to be formulated: εG(pL), εG(2pL), αG1(pL), and αG1(2pL). The gray emissivity term εG,i of Eq. (5-167) now becomes εG,e /ae. Conversion of gray-gas total exchange areas  GS SS  and   to their nongray form depends on the fact that the relation between radiative transfer and blackbody emissive power σT 4 is linear and proportional. The gray-gas-equivalent emissivity εG,e /ae is applicable only to the energy fraction ae of σT 4. In consequence, to convert  GS SS  or   to its nongray form, wherever εG or τ appears in G S  it must be replaced by εG,e /ae or (1 − εG,e /ae), respectively; the overall result is then multiplied by ae. The converted S S  is in two parts: The gray-gas contribution involves, as above, replacement of εG by εG,e /ae and τ by (1 − εG,e /ae), and the result multiplied by ae; for the clear-gas contribution, εG is replaced by 0 and τ by 1, and the result is multiplied by (1 − ae) and added to the gray-gas contribution. The simplest application of this simple gray-plus-clear model of gas radiation is the case of a single gas zone surrounded by a single surface zone, Eq. (5-163) for a gray gas. The gray-plus-clear model gives ae GS  1 (5-169) ᎏ = ᎏᎏ ae 1 A1 ᎏᎏ + ᎏᎏ − 1 εG,e ε1

at 1000 K. The surrounding gas is at 1500 K and is well stirred. Find the effective gas emissivity εG,e, the weighting factor ae, and the surface radiative flux density. Solution: Combustion is 1CH4 + 2 × 1.2 O2 + 2 × 1.2 × 79/21 N2 + 2 × 1.2 × 100/21 × 0.0088 H2O, going to 1 CO2 + [2 + 2 × 1.2 × (100/21) × 0.0088] H2O + 0.4 O2 + 9.03 N2 = 12.53 moles per mole of CH4; pc + pw = (1 + 2.1)/12.53 = 0.2474 atm. The mean beam length, Lm = 0.88 × 4V/AT = 0.88 × 4(10 × 3 × 5)/ [2 × (10 × 3 + 10 × 5 + 3 × 5)] = 2.779 m. pLm = 0.2474 × 2.779 = 0. 6875 m atm. From emissivity Table 5-8, b(1500) = 540; n(1500) = 0.42; b(1000) = 444; n(1000) = 0.34. εG(pL) = 540(0.6875 − 0.015)0.42/1500 = 0.3047; αG1(pL) = 444(0.6875 × 1000/1500 − 0.015)0.34 (1500/1000)0.5/1000 = 0.4124. Then εG,e(pL) = [0.3047 − 0.4124(1000/1500)4]/[1 − (1000/1500)4] = 0.2782. Repeat all 3 computations for pL = 2 × 0.6875 to give εG(2pL) = 0.4096, αG1 (2pL) = 0.5250, εG,e(2pL) = 0.3812. Then ae = 0.2782/(2 − 0.3812/0.2782) = 0.4418 and the emissivity substitute = 0.2782/0.4418 = 0.6297. For a single enveloping surface zone, the total-exchange area comes from Eq. (5-169); G S 1/A1 = ae/(ae/εG,e + 1/ε1 − 1) = ˙ 0.4418/(0.4418/0.2782 + 1/0.8 − 1) = 0.2404. The flux density is Q/A = q = 4 4 (G S 1/A)σ(TG − T 1) = 0.2404 × 56.7 × [(1500/1000)4 − (1000/1000)4] = 55.37 2 kW/m (17,550 Btu/sq ft hr). (Note that allowing for average humidity in air adds 5 percent to H2O and approximately 2 percent to gas emissivity.)

Example 8: Effective Gas Emissivity Methane is burned to completion with 20 percent excess air (air half-saturated with water vapor at 298 K (60°F), 0.0088 mols H2O/mol dry air) in a furnace chamber of floor dimensions 3 × 10 m and height 5 m. The whole surface is a gray-energy sink of emissivity 0.8

Total-exchange areas for the basic one-gas two-surface model [Eqs. (5-160) to (5-162)], used to evaluate the cases in Table 5-10, take the following form when converted by the above described procedure to their nongray form: ae A1F12ε1ε2 S 1S 2 = ᎏᎏᎏᎏᎏ 1/(1 − εG,e /ae) + (1 − εG,e /ae)ρ1ρ2(1 − F12C2) − ρ1(1 − F12) − ρ2(1 − F21) (1 − ae)A1F12ε1ε2 + ᎏᎏᎏᎏᎏ 1 + ρ1ρ2(1 − F12C2) − ρ1(1 − F12) − ρ2(1 − F21) A1ε1εG,e(1/(1 − εG,e /ae) + ρ2(F12 /C2 − 1) G S 1 = ᎏᎏᎏᎏᎏ 1/(1 − εG,e /ae) + (1 − εG,e /ae)ρ1ρ2(1 − F12C2) − ρ1(1 − F12) − ρ2(1 − F21)

(5-170) (5-171)

Modification of Table 5-10 to make the total-exchange areas conform to the gray-plus-clear gas model is straightforward, following the instructions presented above. The results are given in Table 5-11.

HEAT TRANSFER BY RADIATION TABLE 5-11

5-39

Conversion of Some Total-Exchange Areas to Their Gray-Plus-Clear Values

1. Plane slab A1 and surface A2, completing an enclosure of gas (F12 = 1) 1S S 2 aε1ε2 (1 − a)ε1ε2 ᎏ=ᎏ + ᎏᎏ A1 D1 1 − ρ2(1 − ε1A1/A2) ρ1εG

− ρ 1 −  ᎏ ε + ᎏ  D = 1ᎏ − ε /a A a 1

A1

1

2

1

G

2

S G 1 ε1εG(1/(1 − εG /a) + ρ2A1 /A2) ᎏ = ᎏᎏᎏ A1 D1 S G 2 ε2εG(1/(1 − εG /a) + ρ1A1/A2) ᎏ = ᎏᎏᎏ A2 D1 2. Infinite parallel planes, gas between (F12 = F21 = 1) S S aε1ε2 (1 − a)ε1ε2 1 2 ᎏ=ᎏ + ᎏᎏ A1 D2 1 − ρ1ρ2 εG

− 1 − ᎏ ρ ρ D = ᎏ 1 − ε /a  a 1

2

1 2

G

S G 1 ε1εG(1/(1 − εG /a) + ρ2) ᎏ = ᎏᎏᎏ A1 D2 3. Concentric spherical or infinite cylindrical surface zones, A1 inside (F12 = 1; F21 = A1/A2) 1S S 2 aε1ε2 (1 − a)ε1ε2 ᎏ = ᎏ + ᎏᎏ A1 D3 1 − ρ2(1 − ε1A1/A2) 1 ρ1εG A D3 = ᎏᎏ − ρ2 1 − ᎏ1 ε1 + ᎏ (1 − εG /a) A2 a

  



S G 1 ε1εG(1/(1 − εG /a) + ρ2A1/A2) ᎏ = ᎏᎏᎏ A1 D3 S G 2 ε2εG(1/(1 − εG /a) + ρ1A1/A2) ᎏ = ᎏᎏᎏ A2 D3 4. Spherical enclosure of two surface zones or “speckled” A1:A2 enclosure (F12 = F22 = C2; F21 = F11 = C1) S S aε1ε2C2 (1 − a)ε1ε2C2 1 2 ᎏ = ᎏ + ᎏᎏ A1 D4 1 − ρ1C1 − ρ2C2 −ρC −ρC D = ᎏᎏ (1 − ε /a) 1

4

1

1

2

2

G

C = Aᎏ +A A1

1

1

2

GS  1 ε1εG /(1 − εG /a) ᎏ = ᎏᎏ A1 D4

Treatment of Refractory Walls Partially Enclosing a Radiating Gas Another modification of the results in Table 5-10 becomes important when one of the surface zones is radiatively adiabatic; the need to find its temperature can be eliminated. If surface A2, now called Ar, is radiatively adiabatic, its net radiative exchange with A1 must equal its net exchange with the gas.

or

4 4 G SS S r(TG4 − T 4r ) =  r(T 1 r − T 1) TG4 − T 4r T 4r − T 41 T G4 − T 41 ᎏ = ᎏ = ᎏᎏ 1/G 1/S 1/G S r rS 1 S r + 1/S rS 1

(5-172)

GS The net flux from gas G is  GS 1σ(TG4 − T14) +  rσ(TG4 − Tr4) which, with replacement of the last term using Eq. (5-172), gives the single term



1 σ(TG4 − T14) G S1 + ᎏᎏ  1 1 ᎏᎏ + ᎏᎏ G Sr S  rS 1



The bracketed term is called (G S 1)R, the total exchange area from G to A1 with assistance from a refractory surface. In summary,





1 ˙ G ⇔ 1 = (G S 1 + ᎏᎏ σ(TG4 − T14) Q S 1)Rσ(TG4 − T 14) = G 1 1 ᎏᎏ + ᎏᎏ G Sr S  rS 1

(5-173)

Table 5-10 supplies the forms for the three terms needed to formulate (G S 1)R, with Ar substituted for A2. If, in addition, allowance is to be made for the gas not being gray, εG,e and ae are evaluated using values of the emissivity and absorptivity calculated using Table 5-8, and the procedure described in the previous subsection is followed with εG,e /ae replacing εG together with the addition of a clear-gas contribution, when S S  is at issue. It is tempting to say that a surface A2 (or Ar) could be made radiatively adiabatic simply by assigning its reflectance ρ a value of 1, making the terms in the brackets of Eq. (5-173) much easier to evaluate and the result much simpler. This is valid only if the gas is gray. If it is not, Ar is a net absorber of radiation

5-40

HEAT AND MASS TRANSFER

in the spectral energy fraction a (or ae) and a net emitter in the cleargas fraction (1 − a). Conversion of  GS  to (G S 1)R will be carried out for two of the four cases of Table 5-10. Case 1 is an idealization of a metal-heating slab furnace or glass furnace, with its plane sink A1 combining with refractory surface Ar to complete the enclosure. With insertion into Eq. (5173) of G SrS S 1, G S r, and  1 after converting each to its gray plus clear form, one obtains



(G C1 1 S 1)R εG ᎏ = ᎏ ε1 ρ2 ᎏ + ᎏ A1 D1 Cr 1 − ᎏεGᎏ a

+



εr εr + ρrε1(C1/Cr) 1 ᎏᎏᎏ + ρ1 + (Cr /C1)[1 − (εG/a)] ε1 C1 ᎏ a εr + ρrε1 ᎏ + (1 − a)D1 εG Cr







(5-174) where D1 = 1/(1 − εG/a) − ρr{1 − (C1/Cr)[1 − ρ1(1 − εG/a)]}. Conversion of (G S ) 1 R to applicability to a gray gas comes by making a equal 1, producing the enormous simplification to  ᎏ A (GS1)R 1

gray gas

1 = ᎏᎏᎏ 1 + C1(1/εG − 2) ρ1 ᎏᎏ + ᎏᎏ 1 − C1εG ε1

(5-175)

Note that the emissivity and reflectance of the refractory are without effect on (G S 1)R if the gas is gray. The second conversion of G S ) S  to (G 1 R will be Case 4 of Table 5-10, the two-surface-zone enclosure with computation simplified by assuming that the direct-view factor from any spot to a surface equals the fraction of the whole enclosure that the surface occupies (the speckled-furnace model). This case can be considered an idealization of many processing furnaces such as distilling and cracking coil furnaces, with parts of the enclosure tube-covered and part left refractory. (But the refractory under the tubes is not to be classified as part of the refractory zone.) Again, one starts with substitution into Eq. (5-173) of the terms G SrS S 1, G S r, and  1 from Table 5-10, Case 4, with all terms first converted to their gray-plus-clear form. To indicate the procedure, one of the components,  SrS 1, wil be formulated. Sr S1  Crε1εr Crε1εr ᎏ = a ᎏ + (1 − a) ᎏᎏ A1 D4 1 − ρ1C1 − ρrCr Crε1εr εG(1 − a)(a − εG) = ᎏ 1 + ᎏᎏ D4 1 − ρ1C1 − ρrCr With D4 = 1/(1 − εG/a) − ρ1C1 − ρrCr, the result of the full substitution simplifies to





1 (G S 1)R ᎏ = ᎏᎏᎏᎏ A1 1 1 1 1/a − 1 C1 ᎏᎏ − ᎏᎏ + ᎏᎏ + ᎏ ᎏ εG a ε1 ε1 + εr(Cr /C1)





For a gray gas (a = 1), the above becomes 1 (G S 1)R ᎏ = ᎏᎏ 1 1 A1 C1 ᎏᎏ − 1 + ᎏᎏ εG ε1





(5-176)

(5-177)

Eq. (5-176) has wide applicability. COMBUSTION CHAMBER HEAT TRANSFER Treatment of radiative transfer in combustion chambers is available at varying levels of complexity, including allowance for temperature variation in both gas and refractory walls (Hottel and Sarofim,

Radiative Transfer, McGraw-Hill, New York, 1967, chap. 14). A less rigorous treatment suffices, however, for handling many problems. There are two limiting cases: the long chamber with gas temperature varying only in the direction of gas flow (already treated) and the compact chamber containing a gas or a flame to which can be assigned an effective or average radiating temperature. The latter will be considered. Stirred-Chamber Model; Refractory Wall Loss Negligible What furnace engineers most need is a closed-form solution of the problem, theoretically sound in structure and therefore containing a minimum number of parameters and no empirical constants and, preferably, physically visualizable. They can then (1) correlate data on existing furnaces, (2) develop a performance equation for standard design, or (3) estimate performance of a new furnace type on which no data are available. An equation representing an energy balance on a combustion chamber of two surface zones, a heat sink A1 at temperature T1, and a refractory surface Ar assumed radiatively adiabatic at Tr, is most simply solved if the total enthalpy input H is expressed as mC ˙ p(TF − To); m ˙ is the mass rate of fuel plus air; and TF is a pseudoadiabatic flame temperature based on a mean specific heat from base temperature To up to the gas exit temperature TE rather than up to TF. The heat transfer rate Q˙ out of the gas is then H˙ − mC ˙ p(TE − To) or mC ˙ p(TF − TE). The energy balance, with ambient temperature taken as conventional base To, is ˙ − mC (Q˙ =)H ˙ p(TE − To) = (G S 1)R σ(TG4 − T 14 ) + h1A1(TG − T1) + AoFoσ(TG4 − To4) + UAr(TG − To)

(5-178)

S 1)RσTF4 To make the relation dimensionless, divide through by (G and let all temperatures, expressed as ratios to TF, be called τs. For clarity, the terms are tabulated: mC ˙ p/(G S 1)RσTF3 = D, dimensionless firing density l.h.s. term = D(1 − τE) 1st r.h.s. term = τ G4 − τ 14 (h1A1/(G GS  1)R σTF3) = Lc, convection number (dimensionless) AoFo /(G S 1)R = Lo, wall-openings loss number, (dimensionless) UAr /(G S 1)RσTF3 = Lr, refractory-wall loss number (dimensionless) The equation then becomes D(1 − τE) = τG4 − τ14 + Lc(τG − τ1) + Lo(τG4 − τo4) + Lr(τG − τo)

(5-179)

This equation has two unknowns (τG and τE), and an empirical relation between them is needed. Many have been tried, and one of the best is to assume that the excess of TG over TE expressed as a ratio to TF (zero for a perfectly stirred chamber) is a constant ∆ [ (TG − TE)/ TF]. Although ∆ should vary with burner type, the effects of firing rate and percent excess air are small. In the absence of performance data on the kind of furnace under study, assume ∆ = 300/TF, °R or 170/TF, K. The left side of Eq. (5-178) then becomes D(1 − τG + ∆), and with coefficients of τG and τ G4 collected, the equation becomes D + Lc + Lr τ14 + Lcτ1 + Loτ 4o + Lrτo + D(1 + ∆) τG4 + ᎏᎏ τG − ᎏᎏᎏᎏ = 0 1 + Lo 1 + Lo (5-180)



 



Though this is a quartic equation, it is capable of explicit solution because of the absence of second and third degree terms. Trial-anderror enters, however, because (G S 1)R and C p are mild functions of TG and related TE, respectively, and a preliminary guess of TG is necessary. An ambiguity can exist in interpretation of terms. If part of the enclosure surface consists of screen tubes over the chamber-gas exit to a convection section, radiative transfer to those tubes is included in the chamber energy balance, but convection is not, because it has no effect on chamber gas temperature. With Eq. (5-180) solved, the gas-side efficiency ηG is (1 − τG + ∆)/ (1 − τo). The sink-side efficiency η1 is less by the amount (Lo(τG4 − τL4) + Lr(τG − τo))/D(1 − τo) and is also given by [(G GS  1)Rσ(TG4 − T 41) + h1A1(TG − T1)]/H˙. It must be remembered that the efficiency η1

HEAT TRANSFER BY RADIATION includes the losses through the wall from the backside of any wall–mounted heat sinks. Though the results must be considered approximations, depending as they do on the empirical ∆, the equation may be used to find the effect of firing rate, excess air, and air preheat on efficiency. With some performance data available, the small effect of various factors on ∆ may be found. The first term on the right side of Eq. (5-179) is so nearly dominant for most furnaces that consideration of the main features of chamber performance is clarified by ignoring the loss terms Lo and Lr or by assuming that they and Lc have a constant mean value. The relation of a modified chamber efficiency ηG(1 − τo) to a modified firing density D/(1 − τo) and to the normalized sink temperature τ = T1 /TF is shown in Fig. 5-23, which is based on Eq. (5-178), with the radiative and convective transfer terms (G S 1)Rσ(TG4 − T 41) + h1A1(TG − T1) replaced by a combined radiation/conduction term (G GS  1)R,cσ(TG4 − T 41), where 3 (G GS GS ; TG1 is adequately approximated by the  1)R,c = (G  1)R + h1A1/4σTG1 arithmetic mean of TG and T1. Example 9: Radiation in a Furnace Consider a furnace 3 m × 10 m × 5 m fired with methane and 20 percent excess air, at a methane firing rate of 2500 kg/hr. Two rows of 5-inch (0.127 m) tubes (outer diameter) are mounted on equilateral triangular centers, with center-to-center distance twice the tube diameter, on 60 percent of the interior surface of the chamber. The radiative properties of the gases for an enclosure of these dimensions, containing the same combustion products, have been estimated in Example 8 for a gas temperature of 1500 K and a sink temperature of 1000 K: 12.53 moles of combustion products are generated per mole of fuel, with a mean molar heat capacity between a base temperature of 298 K and the exit gas exit temperature TE, adequately represented for this example by M C p = 7.01 + 0.875 (T/1000) over a TE range of 800 to 1600 K. The lower heating value of CH4 is 191,760 cal/g mole. The air is preheated to 600°C, and has a mean  MC p of 7.31 cal/g mole. The alloy tube emissivity ε1 is 0.7 and may be assumed gray; the mean tube surface temperature is 700°C. The convection coefficient, gas-to-tube plane and to refractory surface is 0.0170 kW/m2 °C; hc + r on the outside surface is 0.0114 kW/m2 °C. The 0.343-m-thick refractory walls and roof have a k of 0.00050 kW/m °C and an assumed εr of 0.6; the walls are pierced by four 0.10-m × 0.23-m peepholes. The gas exit area, 1 m × 10 m, is tube-screen-covered. What is the sink-side efficiency η1, the gas exit temperature TE, and the mean flux density through the tube surface? Solution: Temporary basis—1 mole entering CH4. Since no molal change occurs when CH4 burns completely with half-saturated entering air 20 percent in excess of stoichiometric, the total number of moles produced equal 11.53 moles. Entering enthalpy = 191,760 + 11.53 × 7.31(600 − 25) = 240,220 Kcal/kg mol CH4. H˙ [of Eq. (5-178)] = 240,220 × (2500/16.04) = 37.44E6 Kcal/hr × 4.186/3600 = 43.54E3 Kw. m ˙ = [16.04 + 2 × 1.2(100/21)(29 + 0.0088 × 18.016)](2500/16.04) = 54,440 kg/hr. Trial and error solution necessitates several sets of computations of TG to check assumed TE; only the last of these will be given. The first, to save time by using results attained elsewhere, assumes TG = 1500 K; the resulting TG is 369 K higher. The second set assumes TG = 2017 K; the resulting TG is 80 K lower. Linear interpolation indicates the third set should assume TG = 1934 K. That set is presented: TE = 1934 − 170 = 1764.  M Cp = 7.01 + 0.875 × (1764/1000) = 8.554 cal/(gmol)(K). TF = 240,220/(12.53 × 8.554) +

5-41

298 = 2539 K.  Cp = (8.554/16.04)(4.186/3600) = 0.6201E-3 kw-hr/(kg)(K). For εG, pLm = 0.247 × 2.779 = 0.6875 m atm. Use of Table 5-8, with TH = 2000 K (b = 572, n = 0.51) and TL = 1500 (b = 540, n = 0.42) gives εG(pL) = [572(0.6875 − 0.015)0.51(1934 − 1500) + 540(0.6725)0.42(2000 − 1934)]/(500 × 1934) = 0.2409. For αG1, with T1 = 1000, Table 5-8 gives b = 444, n = 0.34. αG1(pL) = 444(0.6875 × 1000/1934 − 0.015)0.34(1934/1000)0.5/1000 = 0.4281. εG,e(pL) = [0.2409 − 0.4281(1000/1934)4]/(1 − 1/1.9344) = 0.2265. 2pLm = 2 × 0.6875 = 1.375 m atm. εG(2pL) = [572(1.375 − 0.015)0.51 × 434 + 540(1.36)0.42 × 66]/(500 × 1934) = 0.3422. αG1(2pL) = 444 × (1.375 × 1000/1934 − 0.015)0.34(1.934)0.5/1000 = 0.5459. εG,e(2pL) = [0.3422 − 0.5459(1/1.934)4/(1 − 1/1.9344) = 0.3265. ae = 0.2265/(2 − 0.3265/0.2265) = 0.4056. Of all these, only εG,e(pL) and ae will be used from here on. From Eq. (5-176), (G S1)R / A1 = 1/[0.6(1/0.2265 − 1/0.4056) +  1/0.87 + (1/0.4056 − 1)/(0.87 + 0.6 (0.4/0.6))] = 0.2879. A1 = (3 × 10 + 3 × 5 + 2 10 × 5) × 2 = 190 × 0.6 = 114 m . Ar (with floor area omitted for loss) = 190 × 0.4 − 10 × 3 = 46 m2. (G S 1)R = 0.2879 × 114 = 32.82 m2. (G S 1)RσTF3 = 32.82 × (56.7E − 12) × 25393 = 30.46 kw/K. D = 54,440 × (0.6201E − 3)/30.46 = 1.1083. ∆ = 170/2539 = 0.06696. Lc = 0.017 × 114/30.46 = 0.0636. Lo = 4 × 0.1 × 0.23 × 0.335/32.82 = 9.4E − 4. Lr = 0.0012 × 46/30.46 = 0.001812. In Eq. (5-180), the coefficient of τG equals (1.1083 + 0.0636 + 0.0018)/1.00094 = 1.1726. The constant in the equation equals [(100/2539)4 + 0.0636(1/2.539) + 0.00094 × 0.11744 + 0.00181(298/2539) + 1.1083(1 + 170/2539)]/1.00094 = 1.2307. The equation to solve is: τG4 + 1.1726τG − 1.2307 = 0. Solution gives τG = 0.7620; TG = 0.762 × 2539 = 1935 K. Te = 1765 K, only 1 K above value assumed for obtaining C p and εG. ηG = (1 − τG)/(1 − τo) = (1 − 1765/2539)/(1 − 298/2539) = 0.3454. Sink-side efficiency η1 0.3454 − [0.00094(0.7624 − 0.117444)] + 0.001812(0.762 − 0.1174)/1.1083(1 − 0.1174) = 0.344, not including convection to screen tubes ˙ η1)/A1 = 43,540 × 0.3439/114 = 131.3 kw/m2. covering gas exit. qplane of sink = (H qtube surf = 131.3 (2D/2πD) = 41.8 kw/m2 × 3412 × 0.30482 = 13,300 Btu/(ft2)(hr). TE = 1765 K = 1492°C = 2717°F.

In Fig. 5-22, the shaded areas indicate the operating regimes of a wide range of furnace types. Note the significant properties of the function presented. (1) As firing rate D′ goes down, the efficiency rises and approaches 1 − τ1 in the limit. (This conclusion is modified if wall losses are significant.) (2) Changes in sink temperature have little effect if τ1 < 0.3. (3) As the furnace walls approach complete coverage by a black sink [Cε1 → 1 in Eqs. (5-176) and (5-177)] and as convection becomes unimportant, the effect of flame emissivity on D becomes one of inverse proportionality; thus at very high firing rates at which efficiency approaches inverse proportionality to D, the efficiency of heat transfer varies directly as εG (gas-turbine chambers), but at low firing rates εG has relatively little effect. (4) When Cε1 << 1 because of a nonblack sink or much refractory surface, the effect of changing flame emissivity is to produce a much less than proportional effect on heat flux. The factor ∆, the allowance for imperfect stirring, must be estimated. Values in the range of 93 to 149°C (200 to 300°F) have been found to produce data correlation for a series of tests on marine boilers. Equation (5-179) and Fig. 5-22 serve as a framework for correlating the performance of furnaces with flow patterns—plug flow, parabolic

FIG. 5-22 Thermal performance of well-stirred furnace chambers; reduced efficiency as a function of reduced firing density D and reduced sink temperature τ1. (a) Radiant section, oil tube stills, cracking coils. (b) Domestic boiler combustion chambers. (c) Open-hearth furnaces. (d) Soaking pits.

5-42

HEAT AND MASS TRANSFER

FIG. 5-23

Effect of wall-loss factor L on combustion-chamber performance; L′ = 0.02, and τ = 0.5 and 0.7.

profile, and recirculatory flow—which differ from the well-stirred model [Hottel and Sarofim, Int. J. Mass Heat Transfer, 8, 1153 (1965)]. As expected, plug-flow furnaces show somewhat higher efficiency, mild-recirculation types somewhat lower efficiency, and strong-recirculation furnaces a performance closely similar to that of the well-stirred model.

If data on several furnaces of a single class are available, a similar treatment can lead to a partially empirical equation based on simplified rules for obtaining (G S1)R,c or an effective ∆. Because Eq. (5-178) has a  structure which covers a wide range of furnace types and has a sound theoretical basis, it provides safer structures of empirical design equations than many such equations available in the engineering literature.

MASS TRANSFER GENERAL REFERENCES: Bird, Stewart, and Lightfoot, Transport Phenomena, Wiley, New York, 1960. Cussler, Diffusion: Mass Transfer in Fluid Systems, Cambridge University Press, Cambridge, 1984. Danner and Daubert, Manual for Predicting Chemical Process Design Data, AIChE, New York, 1983. Daubert and Danner, Physical and Thermodynamic Properties of Pure Chemicals, Taylor and Francis, Bristol, PA, 1989–1995. Fahien, Fundamentals of Transport Phenomena, McGraw-Hill, New York, 1983. Foust, Wenzel, Clump, Maus, and Andersen, Principles of Unit Operations, 2d ed., Wiley, New York, 1980. Gammon, Marsh, and Dewan, Transport Properties and Related Thermodynamic Data of Binary Mixtures, AIChE, New York, Part 1, 1993; Part 2, 1994. Geankoplis, Transport Processes and Unit Operations, 3d ed., Prentice Hall, Englewood Cliffs, NJ, 1993. Hines and Maddox, Mass Transfer: Fundamentals and Applications, Prentice Hall, Englewood Cliffs, NJ, 1985. Kirwan, “Mass Transfer Principles,” Chap. 2 in Rousseau, R. W. (ed.), Handbook of Separation Process Technology, Wiley, New York, 1987. McCabe, Smith, and Harriott, Unit Operations of Chemical Engineering, 5th ed., McGraw-Hill, New York, 1993. Reid, Prausnitz, and Poling, The Properties of Gases and Liquids, 4th ed., McGrawHill, New York, 1987. Schwartzberg and Chao, Food Technol., 36(2), 73 (1982). Sherwood, Pigford, and Wilke, Mass Transfer, McGraw-Hill, New York, 1975. Skelland, Diffusional Mass Transfer, Wiley, New York, 1974. Taylor and Krishna, Multicomponent Mass Transfer, Wiley, New York, 1993. Treybal, Mass-Transfer Operations, 3d ed., McGraw-Hill, New York, 1980.

INTRODUCTION This part of Sec. 5 provides a concise guide to solving problems in situations commonly encountered by chemical engineers. It deals with

diffusivity and mass-transfer coefficient estimation and common flux equations, although material balances are also presented in typical coordinate systems to permit a wide range of problems to be formulated and solved. Mass-transfer calculations involve transport properties, such as diffusivities, and other empirical factors that have been found to relate mass-transfer rates to measured “driving forces” in myriad geometries and conditions. The context of the problem dictates whether the fundamental or more applied coefficient should be used. One key distinction is that, whenever there is flow parallel to an interface through which mass transfer occurs, the relevant coefficient is an empirical combination of properties and conditions. Conversely, when diffusion occurs in stagnant media or in creeping flow without transverse velocity gradients, ordinary diffusivities may be suitable for solving the problem. In either case, it is strongly suggested to employ data, whenever available, instead of relying on correlations. Units employed in diffusivity correlations commonly followed the cgs system. Similarly, correlations for mass transfer correlations used the cgs or English system. In both cases, only the most recent correlations employ SI units. Since most correlations involve other properties and physical parameters, often with mixed units, they are repeated here as originally stated. Common conversion factors are listed in Table 1-4. Fick’s First Law This law relates flux of a component to its composition gradient, employing a constant of proportionality called a

MASS TRANSFER

5-43

Nomenclature and Units—Mass Transfer Symbols a

Definition

Acs A′ ap c ci cP d db dc d drop dimp dpore DA′A DAB D°AB Deff DK DL DS E ES f F g gc G Ga GM h′ hT H H

Effective interfacial mass transfer area per unit volume Cross-sectional area of vessel Constant (see Table 5-28-I) See a Concentration = P/RT for an ideal gas Concentration of component i = xi c at gas-liquid interface Specific heat Characteristic length Bubble diameter Column diameter Sauter mean diameter Impeller diameter Pore diameter Self-diffusivity (= DA at xA = 1) Mutual diffusivity Mutual diffusivity at infinite dilution of A in B Effective diffusivity within a porous solid = εpD/τ Knudson diffusivity for gases in small pores Liquid phase diffusion coefficient Surface diffusivity Energy dissipation rate/mass Activation energy for surface diffusion Friction factor for fluid flow Faraday’s constant Acceleration due to gravity Conversion factor Gas-phase mass flux Dry air flux Molar gas-phase mass flux Heat transfer coefficient Total height of tower packing Compartment height Henry’s law constant

H′

Henry’s law constant

HG HOG HL HOL HTU jD jH jM m JA

J JSi k k k k′

Height of one transfer unit based on gas-phase resistance Height of one overall gas-phase mass-transfer unit Height of one transfer unit based on liquid-phase resistance Height of one overall liquid-phase mass-transfer unit Height of one transfer unit (general) Chilton-Colburn factor for mass transfer, Eq. (5-291) Chilton-Colburn factor for heat transfer See jD Mass flux of A by diffusion with respect to the mean mass velocity Molar flux of A by diffusion with respect to mean molar velocity Molar flux of A with respect to mean volume velocity Molar flux by surface diffusion Boltzmann’s constant Film mass transfer coefficient Thermal conductivity Mass-transfer coefficient for dilute systems

kG

Gas-phase mass-transfer coefficient for dilute systems

k′G

Gas-phase mass-transfer coefficient for dilute systems

kGa kˆ Ga

Volumetric gas-phase mass-transfer Overall volumetric gas-phase mass-transfer coefficient for concentrated systems Liquid phase mass transfer coefficient for pure absorption (no reaction) Liquid-phase mass-transfer coefficient for dilute systems

J

M A

V A

ˆL k° kL k′L kˆ L kLa K K KG

Liquid-phase mass-transfer coefficient for dilute systems Liquid-phase mass-transfer coefficient for concentrated systems Volumetric liquid-phase mass-transfer coefficient for dilute systems Overall mass transfer coefficient α/R = specific conductance Overall gas-phase mass-transfer coefficient for dilute systems

SI units

U.S. customary units

m2/m3

ft2/ft3

m2 or cm2

ft2

mol/m3 or mol/l or gequiv/l mol/m3 or mol/l or gequiv/l kJ/(kg⋅K) m or cm m m or cm m m m or cm m2/s or cm2/s m2/s or cm2/s m2/s or cm2/s m2/s m2/s or cm2/s m2/s m2/s or cm2/s

lbmol/ft3 lbmol/ft3 Btu/(lb⋅°F) ft ft ft ft ft ft ft2/h ft2/h ft2/h ft2/h ft2/h ft2/h ft2/h

J/mol or cal/mol Dimensionless 96,487 Coulomb/gequiv m/s2 1.0 kg/(s⋅m2) kg/(s⋅m2) kmol/(s⋅m2) W/(m2⋅K) = J/(s⋅m2⋅K) m m kPa/(mole-fraction solute in liquid phase) kPa/[kmol/(m3 solute in liquid phase)]

Dimensionless

m m m m m Dimensionless Dimensionless

ft/h2 4.17 × 108 lb ft/[lbf⋅h2] lb/(h⋅ft2) lb/(h⋅ft2) (lbmol)/(h⋅ft2) Btu/(h⋅ft2⋅°F) ft ft (lbf/in2)/(mole-fraction solute in liquid phase) (lbf/in2)/[(lbmol)/(ft3 solute in liquid phase)] or atm/[(lbmole)/ (ft3 solute in liquid phase)] ft ft ft ft ft Dimensionless Dimensionless

kmol/(m2⋅s) or mol/(cm2⋅s)

lbmol/(ft2⋅h)

kmol/(m2⋅s) or mol/(cm2⋅s)

lbmol/(ft2⋅h)

kmol/(m2⋅s) kmol/(m2⋅s) or gmol/(cm2⋅s) 8.9308 × 10−10 gequiv ohm/s m/s or cm/s (J⋅m)/(s⋅m2⋅K) kmol/[(s⋅m2)(kmol/m3)] or m/s kmol/[(s⋅m2)(kPa solute partial pressure)] kmol/[(s⋅m2)(mole fraction in gas)] kmol/[(s⋅m3)(mole fraction)] kmol/(s⋅m3)

lbmol/(ft2 ⋅h) lbmol/(ft2⋅h) ft/hr Btu/(h⋅ft⋅°F) lbmol/[(h⋅ft2)(lbmol/ft3)] or ft/hr lbmol/[(h⋅ft2)lbf/in2 solute partial pressure)] lbmol/[(h⋅ft2)(mole fraction in gas)] (lbmol)/[(h⋅ft3)(mole fraction)] lbmol/(h⋅ft3)

kmol/(s⋅m2)

lbmol/(h⋅ft2)

kmol/[(s⋅m2)(mole-fraction solution in liquid)] kmol/[(s⋅m2)(kmol/m3)] or m/s kmol/(s⋅m2)

(lbmol)/[(h⋅ft2)(mole-fraction solute in liquid)] (lbmol)/[(h⋅ft2)(lbmol/ft3)] or ft/h lbmol/(h⋅ft2)

kmol/[(s⋅m3)(mole fraction)]

(lbmol)/[(h⋅ft3)(mole fraction)]

m/s or cm/s ohm/cm kmol/[(s⋅m2)(mole fraction)]

ft/h (lbmol)/[(h⋅ft2)(mole fraction)]

5-44

HEAT AND MASS TRANSFER

Nomenclature and Units—Mass Transfer (Continued ) Symbols Kˆ G

Definition

KGa

Overall gas-phase mass-transfer coefficient for concentrated systems Overall volumetric gas-phase mass-transfer dilute systems

K′Ga

Overall volumetric gas-phase mass-transfer dilute systems

(Ka)H KL Kˆ L

Overall enthalpy mass-transfer coefficient Overall liquid-phase mass-transfer coefficient Liquid-phase mass-transfer coefficient for concentrated systems Overall volumetric liquid-phase mass-transfer coefficient for dilute systems Overall volumetric liquid-phase mass-transfer coefficient for concentrated systems Liquid-phase mass flux Molar liquid-phase mass flux Slope of equilibrium curve = dy/dx (mole-fraction solute in gas)/(mole-fraction solute in liquid) Molality of solute Molecular weight of species i Mass in a control volume V Valences of cationic and anionic species See Table 5-28-I Mass flux of A with respect to fixed coordinates Impeller speed Number deck levels Interphase mass-transfer rate of solute A per interfacial area with respect to fixed coordinates Number of components Froude Number (dimp N2/g)

KLa Kˆ La L LM m m Mi M |n+ n −| n′ nA N N′ NA Nc NFr

NOG NOL NTU NKn NPr NRe NSc NSh NSt NW p pB,M pi pT P P Pc Per Q rA R

ρ∞ gx3 Grashof number ᎏ2 ᎏ −1 (µ/ρ) ρo Number of overall gas-phase mass-transfer units Number of overall liquid-phase mass-transfer units Number of transfer units (general) Knudson number = l/dpore Prandtl number (cpµ/k) Reynolds number (Gd/µG) Schmidt number (µG/ρGDAB) or (µL/ρLDL) Sherwood number (ˆkGRTd/DABpT) Stanton number (ˆkG/GM) or (ˆkL/LM) Weber number (ρcN2d3imp/σ) Solute partial pressure in bulk gas Log mean partial pressure difference of stagnant gas B Solute partial pressure at gas-liquid interface Total system pressure Pressure Power Critical pressure Perimeter/area Volumetric flow rate Radius of dilute spherical solute Gas constant

R Ri s S t tf T Tb Tc Tr u, v uo u∞ uL vs vT vTS V V Vb Vi vi

Solution electrical resistance Radius of gyration of the component i molecule Fractional surface-renewal rate Tower cross-sectional area = πd2/4 Contact time Formation time of drop Temperature Normal boiling point Critical temperature Reduced temperature = T/Tc Fluid velocity Blowing or suction velocity Velocity away from object Superficial liquid velocity in vertical direction Slip velocity Terminal velocity Stokes law terminal velocity Packed volume in tower Control volume Volume at normal boiling point Molar volume of i at its normal boiling point Partial molar volume of i

NGr







SI units

U.S. customary units

kmol/(s⋅m2)

lbmol/(h⋅ft2)

kmol/[(s⋅m3)(mole-fraction solute in gas)] kmol/[(s⋅m3)(kPa solute partial pressure)] kmol/[(s⋅m2)(mole fraction)] kmol/[(s⋅m2)(mole fraction)] kmol/(s⋅m2)

(lbmol)/[(h⋅ft3(mole-fraction solute in gas)] (lbmol)/[(h⋅ft3)(lbf/in2 solute partial pressure)] lb/[(h⋅ft3)(lb water/lb dry air)] (lbmol)/[(h⋅ft2)(mole fraction)] (lbmol)/(h⋅ft2)

kmol/[(s⋅m3)(mole-fraction solute in liquid)] kmol/(s⋅m3)

(lbmol)/[(h⋅ft3)(mole-fraction solute in liquid)] (lbmol)/(h⋅ft3)

kg/(s⋅m2) kmol/(s⋅m2) Dimensionless

lb/(h⋅ft2) (lbmol)/(h⋅ft2) Dimensionless

mol/1000 g solvent kg/kmol or g/mol kg or g Dimensionless Dimensionless kg/(s⋅m2) Revolution/s Dimensionless kmol/(s⋅m2)

lb/lbmol lb Dimensionless Dimensionless lb/(h⋅ft2) Revolution/min Dimensionless (lbmol)/(h⋅ft2)

Dimensionless Dimensionless

Dimensionless Dimensionless

Dimensionless

Dimensionless

Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless kPa Dimensionless kPa kPa Pa Watts Pa m−1 m3/s Å 8.314 J/mol K = 8.314 Pa m3/(mol K) = 82.057 atm cm3/mol K ohm Å s−1 m2 s s K K K Dimensionless m/s or cm/s m/s m/s m/s m/s m/s m/s m3 m3 or cm3 m3/kmol or cm3/mol m3/kmol or cm3/mol m3/kmol or cm3/mol

Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless lbf/in2 Dimensionless lbf/in2 lbf/in2 lbf/in2 or atm lbf/in2 or atm ft−1 ft3/h 10.73 ft3 psia/lbmol⋅h

h−1 ft2 h h °R °R °R Dimensionless ft/h ft/h ft/h ft/h ft/h ft/h ft/h ft3 ft3 ft3/lbmol ft3/lbmol ft3/lbmol

MASS TRANSFER

5-45

Nomenclature and Units—Mass Transfer (Concluded ) Symbols Vmli Vtower w x x xA xo xBM o BM

x

xi y yBM o yBM yi yio z

Definition

SI units

Molar volume of the liquid-phase component i at the melting point Tower volume per area Width of film Length along plate Mole-fraction solute in bulk-liquid phase Mole fraction of component A Mole-fraction solute in bulk liquid in equilibrium with bulk-gas solute concentration y Logarithmic-mean solvent concentration between bulk liquid and interface values Logarithmic-mean inert-solvent concentration between bulk-liquid value and value in equilibrium with bulk gas Mole-fraction solute in liquid at gas-liquid interface Mole-fraction solute in bulk-gas phase Logarithmic-mean inert-gas concentration (5-262) Logarithmic-mean inert-gas concentration Mole fraction solute in gas at interface Mole-fraction solute in gas at interface in equilibrium with the liquid-phase interfacial solute concentration xi Direction of unidimensional diffusion

U.S. customary units

m3/kmol or cm3/mol

ft3/lbmol

m3/m2 m m (kmol solute)/(kmol liquid) kmole A/kmole fluid (kmol solute)/(kmol liquid)

ft3/ft2 ft ft (lbmol solute)/(lb mol liquid) lbmole A/lb mole fluid (lbmol solute)/(lbmol liquid)

(kmol solvent)/(kmol liquid)

(lbmol solvent)/(lbmol liquid)

(kmol solvent)/(kmol liquid)

(lbmol solvent)/(lbmol liquid)

(kmol solute)/(kmol liquid) (kmol solute)/(kmol gas) (kmol inert gas)/(kmol gas) (kmol inert gas)/(kmol gas) (kmole solute)/(kmol gas) (kmol solute)/(kmol gas)

(lbmol solute)/(lbmol liquid) (lbmol solute)/(lbmol gas) (lbmol inert gas)/(lbmol gas) (lbmol inert gas)/(lbmol gas) (lbmol solute)/(lbmol gas) (lbmol solute)/(lbmol gas)

m

ft

Dimensionless cm−1 Dimensionless m Dimensionless

Dimensionless

m3/m3 Dimensionless Dimensionless Dimensionless Dimensionless cm2/(gequiv⋅ohm) cm2/ohm gequiv cm2/gequiv ohm Debeyes cP or Pa s kg/(s⋅m) kg/(s⋅m) m2/s kg/m3 or g/cm3 kg/m3 or g/cm3 kg/m3 kg/m3 kmol/m3 kg/m3 or g/cm3 Dimensionless

ft3/ft3 Dimensionless Dimensionless Dimensionless Dimensionless

Dimensionless Dimensionless dyn/cm Å dyn/cm Å Dimensionless Dimensionless Radians/s Dimensionless

Dimensionless Dimensionless lbf/ft

Greek symbols α α β δ ε ε εA εAB γi γ⫾ λ+ λ− Λ Λo µi µi µG µL ν ρ ρc ρc ρG ρ L ρp ρr ψi ψ Ψ σ σi σi σAB τ ω ω Ω

1 + NB/NA Conductance cell constant (measured) 5/6 MA1/2 P1/3 c /T c Effective thickness of stagnant-film layer Fraction of discontinuous phase in continuous phase for two-phase flow Void fraction available for gas flow or fractional gas holdup Characteristic Lennard-Jones energy (εAεB)1/2 Activity coefficient of solute i Mean ionic activity coefficient of solute Infinite dilution conductance of cation and anion 1000 K/C = λ+ + λ− = Λo + f(C) Infinite dilution conductance Dipole moment of i Viscosity of pure i Gas-phase viscosity Liquid-phase viscosity Kinematic viscosity = ρ/µ Density of A Critical density of A Density continuous phase Gas-phase density Average molar density of liquid phase Particle density Reduced density = ρ/ρc Parachor of component i = Viσ1/4 Parameter, Table 5-28-G Shape factor, Table 5-27-A Interfacial tension Characteristic length Surface tension of component i Binary pair characteristic length = (σA + σB)/2 Intraparticle tortuosity Pitzer’s acentric factor = −[1.0 + log10(P*/Pc)] Rotational velocity Diffusion collision integral = f(kT/εAB) Subscript

A B G m L super

Solute component in liquid or gas phase Inert-gas or inert-solvent component Gas phase Mean value Liquid phase Superficial velocity

*

At equilibrium

Superscript

Dimensionless ft Dimensionless

lb/(h⋅ft) lb/(h⋅ft) lb/(h⋅ft) ft2/h lb/ft3 lb/ft3 lb/ft3 lb/ft3 (lbmol)/ft3 lb/ft3 Dimensionless

Dimensionless Dimensionless

5-46

HEAT AND MASS TRANSFER

diffusivity. It can be written in several forms, depending on the units and frame of reference. Three that are related but not identical are dcA dxA dwA (5-181) V JA = −DAB ᎏ ≈ M JA = −cDAB ᎏ ∝ m JA = −ρDAB ᎏ dz dz dz The first equality (on the left-hand side) corresponds to the molar flux with respect to the volume average velocity, while the equality in the center represents the molar flux with respect to the molar average velocity and the one on the right is the mass flux with respect to the mass average velocity. These must be used with consistent flux expressions for fixed coordinates and for NC components, such as: NC

NC

NC

J + wA
ni

m A

i=1

NA = V JA + cA
NiV i = M JA + xA
Ni = ᎏᎏ MA i=1 i=1

(5-182)

In each case, the term containing the summation accounts for conveyance, which is the amount of component A carried by the net flow in the direction of diffusion. Its impact on the total flux can be as much as 10 percent. In most cases it is much less, and it is frequently ignored. Some people refer to this as the “convective” term, but that conflicts with the other sense of convection which is promoted by flow perpendicular to the direction of flux. Mutual Diffusivity, Mass Diffusivity, Interdiffusion Coefficient Diffusivity is denoted by DAB and is defined by Fick’s first law as the ratio of the flux to the concentration gradient, as in Eq. (5-181). It is analogous to the thermal diffusivity in Fourier’s law and to the kinematic viscosity in Newton’s law. These analogies are flawed because both heat and momentum are conveniently defined with respect to fixed coordinates, irrespective of the direction of transfer or its magnitude, while mass diffusivity most commonly requires information about bulk motion of the medium in which diffusion occurs. For liquids, it is common to refer to the limit of infinite dilution of A in B using the symbol, D°AB. When the flux expressions are consistent, as in Eq. (5-182), the diffusivities in Eq. (5-181) are identical. As a result, experimental diffusivities are often measured under constant volume conditions but may be used for applications involving open systems. It turns out that the two versions are very nearly equivalent for gas-phase systems because there is negligible volume change on mixing. That is not usually true for liquids, however. Self Diffusivity Self-diffusivity is denoted by DA′A and is the measure of mobility of a species in itself; for instance, using a small concentration of molecules tagged with a radioactive isotope so they can be detected. Tagged and untagged molecules presumably do not have significantly different properties. Hence, the solution is ideal, and there are practically no gradients to “force” or “drive” diffusion. This kind of diffusion is presumed to be purely statistical in nature. In the special case that A and B are similar in molecular weight, polarity, and so on, the self-diffusion coefficients of pure A and B will be approximately equal to the mutual diffusivity, DAB. Second, when A and B are the less mobile and more mobile components, respectively, their self-diffusion coefficients can be used as rough lower and upper bounds of the mutual diffusion coefficient. That is, DA′A ≤ DAB ≤ DB′B. Third, it is a common means for evaluating diffusion for gases at high pressure. Self-diffusion in liquids has been studied by many [Easteal AIChE J. 30, 641 (1984), Ertl and Dullien, AIChE J. 19, 1215 (1973), and Vadovic and Colver, AIChE J. 18, 1264 (1972)]. Tracer Diffusivity Tracer diffusivity, denoted by DA′B is related to both mutual and self-diffusivity. It is evaluated in the presence of a second component B, again using a tagged isotope of the first component. In the dilute range, tagging A merely provides a convenient method for indirect composition analysis. As concentration varies, tracer diffusivities approach mutual diffusivities at the dilute limit, and they approach self-diffusivities at the pure component limit. That is, at the limit of dilute A in B, DA′ B → D°AB and DB ′A → DB ′ B; likewise at the limit of dilute B in A, DB ′A → D°BA and DA ′B → DA′A. Neither the tracer diffusivity nor the self-diffusivity has much practical value except as a means to understand ordinary diffusion and as

order-of-magnitude estimates of mutual diffusivities. Darken’s equation [Eq. (5-220)] was derived for tracer diffusivities but is often used to relate mutual diffusivities at moderate concentrations as opposed to infinite dilution. Mass-Transfer Coefficient Denoted by kc, kx, Kx, and so on, the mass-transfer coefficient is the ratio of the flux to a concentration (or composition) difference. These coefficients generally represent rates of transfer that are much greater than those that occur by diffusion alone, as a result of convection or turbulence at the interface where mass transfer occurs. There exist several principles that relate that coefficient to the diffusivity and other fluid properties and to the intensity of motion and geometry. Examples that are outlined later are the film theory, the surface renewal theory, and the penetration theory, all of which pertain to idealized cases. For many situations of practical interest like investigating the flow inside tubes and over flat surfaces as well as measuring external flow through banks of tubes, in fixed beds of particles, and the like, correlations have been developed that follow the same forms as the above theories. Examples of these are provided in the subsequent section on mass-transfer coefficient correlations. Problem Solving Methods Most, if not all, problems or applications that involve mass transfer can be approached by a systematic course of action. In the simplest cases, the unknown quantities are obvious. In more complex (e.g., multicomponent, multiphase, multidimensional, nonisothermal, and/or transient) systems, it is more subtle to resolve the known and unknown quantities. For example, in multicomponent systems, one must know the fluxes of the components before predicting their effective diffusivities and vice versa. More will be said about that dilemma later. Once the known and unknown quantities are resolved, however, a combination of conservation equations, definitions, empirical relations, and properties are applied to arrive at an answer. Figure 5-24 is a flowchart that illustrates the primary types of information and their relationships, and it applies to many mass-transfer problems. CONTINUITY AND FLUX EXPRESSIONS Material Balances Whenever mass-transfer applications involve equipment of specific dimensions, flux equations alone are inadequate to assess results. A material balance or continuity equation must also be used. When the geometry is simple, macroscopic balances suffice. The following equation is an overall mass balance for such a unit having Nm bulk-flow ports and Nn ports or interfaces through which diffusive flux can occur: Nm Nn dM (5-183) ᎏ =
mi +
ni Acsi dt i = 1 i=1 where M represents the mass in the unit volume V at any time t; mi is the mass flow rate through the ith port; and ni is the mass flux through the ith port, which has a cross-sectional area of Acsi. The corresponding balance equation for individual components includes a reaction term: Nm Nn dMj (5-184) ᎏ =
mij +
nij Acsi + rj V dt i=1 i=1 For the jth component, mij = miwij is the component mass flow rate in stream i; wij is the mass fraction of component j in stream i; and rj is the net reaction rate (mass generation minus consumption) per unit volume V that contains mass M. If it is inconvenient to measure mass flow rates, the product of density and volumetric flow rate is used instead. In addition, most situations that involve mass transfer require material balances, but the pertinent area is ambiguous. Examples are packed columns for absorption, distillation, or extraction. In such cases, flow rates through the discrete ports (nozzles) must be related to the mass-transfer rate in the packing. As a result, the mass-transfer rate is determined via flux equations, and the overall material balance incorporates the stream flow rates mi and integrated fluxes. In such instances, it is common to begin with the most general, differential material balance equations. Then, by eliminating terms that are negligible, the simplest applicable set of equations remains to be solved.

MASS TRANSFER

5-47

Flowchart illustrating problem solving approach using mass-transfer rate expressions in the context of mass conservation.

FIG. 5-24

Table 5-12 provides material balances for Cartesian, cylindrical, and spherical coordinates. The generic form applies over a unit crosssectional area and constant volume: ∂ρ j (5-185a) ᎏ = −∇ ⋅ nj + rj ∂t where nj = ρvj. Applying Fick’s law and expressing composition as concentration gives ∂cj (5-185b) ᎏ = −v ⋅ ∇cj + Dj∇2cj + rj ∂t Flux Expressions: Simple Integrated Forms of Fick’s First Law Simplified flux equations that arise from Eqs. (5-181) and (5-182) can be used for unidimensional, steady-state problems with binary mixtures. The boundary conditions represent the compositions xAL and xAR at the left-hand and right-hand sides of a hypothetical layer having thickness ∆z. The principal restriction of the following equations is that the concentration and diffusivity are assumed to be constant. As written, the flux is positive from left to right, as depicted in Fig. 5-25. 1. Equimolar counterdiffusion (NA = −NB) dxA DAB NA = M JA = −DAB c ᎏ = ᎏ c (xAL − xAR) (5-189) dz ∆z 2. Unimolar diffusion (NA ≠ 0, NB = 0) 1 − xA DAB NA = M JA + xANA = ᎏ c ln ᎏR (5-190) ∆z 1 − xAL 3. Steady state diffusion (NA ≠ −NB ≠ 0) NA ᎏᎏ − xAR NA + NB NA DAB ᎏᎏ (5-191) NA = M JA + xA(NA + NB) = ᎏ ᎏ c ln NA NA + NB ∆z ᎏᎏ − xAL NA + NB TABLE 5-12

Nc

Ni = −cDim ∇xi + xi
Nj

In the late 1800s, the development of the kinetic theory of gases led to a method for calculating multicomponent gas diffusion (e.g., the flux of each species in a mixture). The methods were developed simultaneously by Stefan and Maxwell. The problem is to determine the diffusion coefficient Dim. The Stefan-Maxwell equations are simpler in principle since they employ binary diffusivities: Nc

1 ∇xi =
ᎏ (xiNj − xj Ni) j = 1 cDij





(5-186)

∂nzj 1 ∂rnrj 1 ∂nθj ∂ρj ᎏ = − ᎏ ᎏ + ᎏ ᎏ + ᎏ + rj ∂t ∂z r ∂r r ∂θ



Cylindrical



(5-187)

2 ∂nθj sin θ ∂nφj ∂ρj 1 ∂r nrj 1 1 ᎏ = − ᎏ2 ᎏ + ᎏ ᎏᎏ + ᎏ ᎏ + rj ∂t ∂r r sin θ ∂θ r sin θ ∂φ r



(5-193)

If Eqs. (5-192) and (5-193) are combined, the multicomponent diffusion coefficient may be assessed in terms of binary diffusion coefficients [see Eq. (5-204)]. For gases, the values Dij of this equation are approximately equal to the binary diffusivities for the ij pairs. The Stefan-Maxwell diffusion coefficients may be negative, and the method may be applied to liquids, even for electrolyte diffusion [Kraaijeveld, Wesselingh, and Kuiken, Ind. Eng. Chem. Res., 33, 750 (1994)]. Approximate solutions have been developed by linearization [Toor, H.L., AIChE J., 10, 448 and 460 (1964); Stewart and Prober, Ind. Eng. Chem. Fundam., 3, 224 (1964)]. Those differ in details but yield about the same accuracy. More recently, efficient algorithms for solving the equations exactly have been developed (see Taylor and Krishna, Krishnamurthy and Taylor, and Taylor and Webb).

Equation ∂nyj ∂nzj ∂nxj ∂ρj ᎏ = − ᎏ + ᎏ + ᎏ + rj ∂t ∂x ∂y ∂z

Cartesian

(5-192)

j=1

Continuity Equation in Various Coordinate Systems

Coordinate System

Spherical

The unfortunate aspect of the last relationship is that one must know a priori the ratio of the fluxes to determine the magnitudes. It is not possible to solve simultaneously the pair of equations that apply for components A and B because the equations are not independent. Stefan-Maxwell Equations Following Eq. (5-182), a simple and intuitively appealing flux equation for applications involving Nc components is



(5-188)

5-48

FIG. 5-25

HEAT AND MASS TRANSFER Before using diffusivities from either data or correlations, it is a good idea to check their reasonableness with respect to values that have been commonly observed in similar situations. Table 5-13 is a compilation of several rules of thumb. These values are not authoritative; they simply represent guidelines based on experience. Diffusivity correlations for gases are outlined in Table 5-14. Specific parameters for individual equations are defined in the specific text regarding each equation. References are given after Table 5-19. The errors reported for Eq. (5-194) through (5-197) were compiled by Reid et al., who compared the predictions with 68 experimental values of DAB. Errors cited for Eqs. (5-198) to (5-202) were reported by the authors. Binary Mixtures—Low Pressure—Nonpolar Components Many evaluations of correlations are available (Elliott and Watts; Lugg; Marrero and Mason). The differences in accuracy of the correlations are minor, and thus the major concern is ease of calculation. The Fuller-Schettler-Giddings equation is usually the simplest correlation to use and is recommended by Reid et al. Chapman-Enskog (Bird et al.) and Wilke-Lee The inherent assumptions of these equations are quite restrictive (i.e., low density,

Hypothetical film and boundary conditions.

DIFFUSIVITY ESTIMATION—GASES Whenever measured values of diffusivities are available, they should be used. Typically, measurement errors are less than those associated with predictions by empirical or even semitheoretical equations. A few general sources of data are Sec. 2 of this handbook, Schwartzberg and Chao; Reid et al.; Gammon et al.; and Daubert and Danner. Many other more restricted sources are listed under specific topics later in this subsection. TABLE 5-13

Rules of Thumb for Diffusivities (See Cussler, Reid et al., Schwartzberg and Chao) Di magnitude

Di range

m2/s

cm2/s

m2/s

Gas at atmospheric pressure

10−5

0.1

10−4–10−6

Liquid Liquid occluded in solid matrix Polymers and glasses

10−9 10−10 10−12

10−5 10−6 10−8

10−8–10−10 10−8–10−12 10−10–10−14

10−4–10−6 10−4–10−8 10−6–10−10

Solid

10−14

10−10

10−10–10−34

10−6–10−30

Continuous phase

TABLE 5-14

cm2/s 1–10−2

Comments Accurate theories exist, generally within ⫾10%; DiP  constant; Di ∝ T1.66 to 2.0 Approximate correlations exist, generally within ⫾25% Hard cell walls: Deff /Di = 0.1 to 0.2. Soft cell walls: Deff /Di = 0.3 to 0.9 Approximate theories exist for dilute and concentrated limits; strong composition dependence Approximate theories exist; strong temperature dependence

Correlations of Diffusivities for Gases

Authors*

Equation

Error

1. Binary Mixtures—Low Pressure—Nonpolar Chapman-Enskog

1/2 0.001858T 3/2 MAB DAB = ᎏᎏ Pσ 2AB ΩD

(5-194)

7.3%

Wilke-Lee [65]

1/2 1/2 (0.0027 − 0.0005 MAB ) T 3/2MAB DAB = ᎏᎏᎏᎏ 2 PσAB ΩD

(5-195)

7.0%

Fuller-Schettler-Giddings [19]

1/2 0.001T1.75MAB DAB = ᎏᎏᎏ 2 P [(
v)A1/3 + (
v)1/3 B ]

(5-196)

5.4%

(5-197)

9.0%

2. Binary Mixtures—Low Pressure—Polar Brokaw [4]

1/2 0.001858T 3/2MAB DAB = ᎏᎏ 2 Pσ ABΩD

3. Self-Diffusivity—High Pressure Mathur-Thodos [37]

10.7 × 10−5T DAA = ᎏᎏr {ρr ≤ 1.5} βρr

(5-198)

5%

Lee-Thodos [31]

0.77 × 10−5Tr DAA = ᎏᎏ {ρr ≤ 1} ρrδ

(5-199)

0.5%

Lee-Thodos [32]

(0.007094G + 0.001916)2.5Tr DAA = ᎏᎏᎏ , [ρr > 1, G < 1] δ

(5-200)

17%

4. Supercritical Mixtures Sun and Chen [56]

1.23 × 10−10T DAB = ᎏᎏ µ0.799VC0.49 A

(5-201)

5%

Catchpole and King [6]

(ρ−0.667 − 0.4510) (1 + MA/MB) R DAB = 5.152 DcTr ᎏᎏᎏᎏ (1 + (VcB /VcA)0.333)2

(5-202)

10%

*References are listed on pages 5-7 and 5-8.

MASS TRANSFER TABLE 5-15

5-49

Estimates for εi and σi (K, Å, atm, cm3, mol)

Critical point

ε/k = 0.75 Tc

1/3 σ = 0.841 V1/3 c or 2.44 (Tc/Pc)

Critical point

ε/k = 65.3 Tc zc3.6

1.866 V1/3 c σ = ᎏᎏ z1.2 c

Normal boiling point Melting point

ε/k = 1.15 Tb ε/k = 1.92 Tm

σ = 1.18 V1/3 b σ = 1.222 V1/3 m

Acentric factor

ε/k = (0.7915 + 0.1693 ω) Tc

T σ = (2.3551 − 0.087 ω) ᎏc Pc

 

1/3

NOTE: These values may not agree closely, so usage of a consistent basis is suggested (e.g., data at the normal boiling point).

spherical atoms), and the intrinsic potential function is empirical. Despite that, they provide good estimates of DAB for many polyatomic gases and gas mixtures, up to about 1000 K and a maximum of 70 atm. The latter constraint is because observations for many gases indicate that DABP is constant up to 70 atm. The characteristic length is σAB = (σA + σB)/2 in A˚. In order to estimate ΩD for Eqs. (5-194) or (5-195), two empirical equations are available. The first is: ΩD = (44.54T *−4.909 + 1.911T *−1.575)0.10 (5-203a)

the gas pairs [e.g., (CH3)2O & CH3Cl]. Despite that, Reid, op. cit., found the average error was 9.0 percent for combinations of mixtures (including several polar-nonpolar gas pairs), temperatures and pressures. In this equation, ΩD is calculated as described previously, and other terms are:

where T* = kT/εAB and εAB = (εA εB)1/2. Estimates for σi and εi are given in Table 5-15. This expression shows that ΩD is proportional to temperature roughly to the −0.49 power at low temperatures and to the −0.16 power at high temperature. Thus, gas diffusivities are proportional to temperatures to the 2.0 power and 1.66 power, respectively, at low and high temperatures. The second is: G A C E ΩD = ᎏ + ᎏᎏ + ᎏᎏ + ᎏᎏ (5-203b) T*B exp (DT*) exp (FT*) exp (HT*)

Self-Diffusivity—High Pressure The criterion of high pressure is vague at best. For most “permanent” gases, such as the major constituents of air, it would mean P > 70 atm. For less volatile components, the criterion would be lower. At present, accurate prediction of mutual diffusion coefficients for dense gas mixtures is not possible. One major reason for this is the scarcity of data. Most high-pressure diffusion experiments have measured the self-diffusion coefficient. The general observation is that the product DP is near constant at low pressure, is not constant at high pressure, but rather decreases as pressure increases. In addition, although there are usually negligible composition effects on diffusivity of gases at low pressures, the effects are not negligible at high pressures. Mathur-Thodos showed that for reduced densities less than unity, the product DAAρ is approximately constant at a given temperature. Thus, by knowing the value of the product at low pressure, it is possible to estimate its value at a higher pressure. They found at higher pressures the density increases, but the product DAAρ decreases rapidly. In their correlation, β = MA1/2PC1/3/T C5/6. Lee-Thodos presented a generalized treatment of self-diffusivity for gases (and liquids). These correlations have been tested for more than 500 data points each. The average deviation of the first is 0.51 percent, and that of the second is 17.2 percent. δ = MA1/2/ 2 5/6 0.1 P1/2 c V c , s/cm , and where G = (X* − X)/(X* − 1), X = ρr /T r , and X* = ρr /T r0.1 evaluated at the solid melting point. Lee and Thodos expanded their earlier treatment of self-diffusivity to cover 58 substances and 975 data points, with an average absolute deviation of 5.26 percent. Their correlation is too involved to repeat here, but those interested should refer to the original paper. Supercritical Mixtures Debenedetti-Reid showed that conventional correlations based on the Stokes-Einstein relation (for liquid phase) tend to overpredict diffusivities in the supercritical state. Nevertheless, they observed that the Stokes-Einstein group DABµ/T was constant. Thus, although no general correlation applies, only one data point is necessary to examine variations of fluid viscosity and/or temperature effects. They explored certain combinations of aromatic solids in SF6 and CO2. Sun-Chen examined tracer diffusion data of aromatic solutes in alcohols up to the supercritical range and found their data correlated with average deviations of 5 percent and a maximum deviation of 17 percent for their rather limited set of data. Catchpole-King examined binary diffusion data of near-critical fluids in the reduced density range of 1 to 2.5 and found that their data correlated with average deviations of 10 percent and a maximum deviation of 60 percent. They observed two classes of behavior. For the first, no correction factor was required (R = 1). That class was comprised of alcohols as solvents with aromatic or aliphatic solutes, or carbon dioxide as a solvent with aliphatics except ketones as solutes, or

where A = 1.06036, B = 0.15610, C = 0.1930, D = 0.47635, E = 1.03587, F = 1.52996, G = 1.76474, and H = 3.89411. Fuller-Schettler-Giddings The parameters and constants for this correlation were determined by regression analysis of 340 experimental diffusion coefficient values of 153 binary systems. Values of
vi used in this equation are in Table 5-16. Binary Mixtures—Low Pressure—Polar Components The Brokaw correlation was based on the Chapman-Enskog equation, but σAB* and ΩD* were evaluated with a modified Stockmayer potential for polar molecules. Hence, slightly different symbols are used. That potential model reduces to the Lennard-Jones 6-12 potential for interactions between nonpolar molecules. As a result, the method should yield accurate predictions for polar as well as nonpolar gas mixtures. Brokaw presented data for 9 relatively polar pairs along with the prediction. The agreement was good: an average absolute error of 6.4 percent, considering the complexity of some of TABLE 5-16 Atomic Diffusion Volumes for Use in Estimating DAB by the Method of Fuller, Schettler, and Giddings Atomic and Structural Diffusion–Volume Increments, vi (cm3/mol) C H O (N)

16.5 1.98 5.48 5.69

(Cl) (S) Aromatic ring Heterocyclic ring

19.5 17.0 −20.2 −20.2

Diffusion Volumes for Simple Molecules, Σvi (cm3/mol) H2 D2 He N2 O2 Air Ar Kr (Xe) Ne

7.07 6.70 2.88 17.9 16.6 20.1 16.1 22.8 37.9 5.59

CO CO2 N2O NH3 H2O (CCl2F2) (SF5) (Cl2) (Br2) (SO2)

18.9 26.9 35.9 14.9 12.7 114.8 69.7 37.7 67.2 41.1

Parentheses indicate that the value listed is based on only a few data points.

ΩD* = ΩD + 0.19 δ 2AB/T* σAB* = (σA* σB*)1/2 δAB = (δA δB)1/2 εAB* = (εA*εB*)1/2

T* = kT/εAB* σi* = [1.585 Vbi /(1 + 1.3 δ2i )]1/3 δi = 1.94 × 103 µ2i /VbiTbi εi* /k = 1.18 (1 + 1.3 δ 2i )Tbi

5-50

HEAT AND MASS TRANSFER

ethylene as a solvent with aliphatics except ketones and naphthalene as solutes. For the second class, the correction factor was R = X 0.17. The class was comprised of carbon dioxide with aromatics; ketones and carbon tetrachloride as solutes; and aliphatics (propane, hexane, dimethyl butane), sulfur hexafluoride, and chlorotrifluoromethane as solvents with aromatics as solutes. In addition, sulfur hexafluoride combined with carbon tetrachloride, and chlorotrifluoromethane combined with 2-propanone were included in that class. In all cases, X = (1 + (VCB /VCA)1/3)2/(1 + MA /MB) was in the range of 1 to 10. Low-Pressure/Multicomponent Mixtures These methods are outlined in Table 5-17. Stefan-Maxwell equations were discussed earlier. Smith-Taylor compared various methods for predicting multicomponent diffusion rates and found that Eq. (5-204) was superior among the effective diffusivity approaches, though none is very good. They also found that linearized and exact solutions are roughly equivalent and accurate. Blanc provided a simple limiting case for dilute component i diffusing in a stagnant medium (i.e., N ≈ 0), and the result, Eq. (5-205), is known as Blanc’s law. The restriction basically means that the compositions of all the components, besides component i, are relatively large and uniform. Wilke obtained solutions to the Stefan-Maxwell equations. The first, Eq. (5-206), is simple and reliable under the same conditions as Blanc’s law. This equation applies when component i diffuses through a stagnant mixture. It has been tested and verified for diffusion of toluene in hydrogen + air + argon mixtures and for diffusion of ethyl propionate in hydrogen + air mixtures (Fairbanks and Wilke). When the compositions vary from one boundary to the other, Wilke recommends that the arithmetic average mole fractions be used. Wilke also suggested using the Stefan-Maxwell equation, which applies when the fluxes of two or more components are significant. In this situation, the mole fractions are arithmetic averages of the boundary conditions, and the solution requires iteration because the ratio of fluxes is not known a priori. DIFFUSIVITY ESTIMATION—LIQUIDS Many more correlations are available for diffusion coefficients in the liquid phase than for the gas phase. Most, however, are restricted to binary diffusion at infinite dilution D°AB or to self-diffusivity DA′A. This reflects the much greater complexity of liquids on a molecular level. For example, gas-phase diffusion exhibits negligible composition effects and deviations from thermodynamic ideality. Conversely, liquid-phase diffusion almost always involves volumetric and thermodynamic effects due to composition variations. For concentrations greater than a few mole percent of A and B, corrections are needed to obtain the true diffusivity. Furthermore, there are many conditions that do not fit any of the correlations presented here. Thus, careful consideration is needed to produce a reasonable estimate. Again, if diffusivity data are available at the conditions of interest, then they are strongly preferred over the predictions of any correlations. Stokes-Einstein and Free-Volume Theories The starting point for many correlations is the Stokes-Einstein equation. This equation is derived from continuum fluid mechanics and classical thermodynamics for the motion of large spherical particles in a liquid.

For this case, the need for a molecular theory is cleverly avoided. The Stokes-Einstein equation is (Bird et al.) kT (5-207) DAB = ᎏ 6πrAµB where A refers to the solute and B refers to the solvent. This equation is applicable to very large unhydrated molecules (M > 1000) in lowmolecular-weight solvents or where the molar volume of the solute is greater than 500 cm3/mol (Reddy and Doraiswamy; Wilke and Chang). Despite its intellectual appeal, this equation is seldom used “as is.” Rather, the following principles have been identified: (1) The diffusion coefficient is inversely proportional to the size rA  VA1/3 of the solute molecules. Experimental observations, however, generally indicate that the exponent of the solute molar volume is larger than one-third. (2) The term DABµB /T is approximately constant only over a 10-to-15 K interval. Thus, the dependence of liquid diffusivity on properties and conditions does not generally obey the interactions implied by that grouping. For example, Robinson et al. found that: ln DAB ∝ −1/T. (3) Finally, pressure does not affect liquid-phase diffusivity much, since µB and VA are only weakly pressure-dependent. Pressure does have an impact at very high levels. Another advance in the concepts of liquid-phase diffusion was provided by Hildebrand, who adapted a theory of viscosity to selfdiffusivity. He postulated that DA′A = B(V − Vms)/Vms, where DA′A is the self-diffusion coefficient, V is the molar volume, and Vms is the molar volume at which fluidity is zero (i.e., the molar volume of the solid phase at the melting temperature). The difference (V − Vms) can be thought of as the free volume, which increases with temperature; and B is a proportionality constant. Ertl and Dullien [ibid.] found that Hildebrand’s equation could not fit their data with B as a constant. They modified it by applying an empirical exponent n (a constant greater than unity) to the volumetric ratio. The new equation is not generally useful, however, since there is no means for predicting n. The theory does identify the free volume as an important physical variable, since n > 1 for most liquids implies that diffusion is more strongly dependent on free volume than is viscosity. Dilute Binary Nonelectrolytes: General Mixtures These correlations are outlined in Table 5-18. Wilke-Chang This correlation for D°AB is one of the most widely used, and it is an empirical modification of the Stokes-Einstein equation. It is not very accurate, however, for water as the solute. Otherwise, it applies to diffusion of very dilute A in B. The average absolute error for 251 different systems is about 10 percent. φB is an association factor of solvent B that accounts for hydrogen bonding. Component B

φB

Water Methanol Ethanol Propanol Others

2.26 1.9 1.5 1.2 1.0

The value of φB for water was originally stated as 2.6, although when the original data were reanalyzed, the empirical best fit was 2.26.

TABLE 5-17 Relationships for Diffusivities of Multicomponent Gas Mixtures at Low Pressure Authors* Stefan-Maxwell, Smith and Taylor [53]

Blanc [13]

Wilke [63]

Equation



Dim = 1 − xi Dim = Dim =

NC

NC

j=1

xj

NC

*References are listed at the beginning of this subsection.

i

j=1

xiNi

j

ij

(5-204)

i

(5-205)

ij


 j=1 j≠i

j

−1


ᎏ D  j=1

NC

D 
N /N /
x − ᎏ N /

xj ᎏ Dij

−1

(5-206)

MASS TRANSFER TABLE 5-18

5-51

Correlations for Diffusivities of Dilute, Binary Mixtures of Nonelectrolytes in Liquids

Authors*

Equation

Error

1. General Mixtures Wilke-Chang [64]

7.4 × 10−8 (φBMB)1/2 T D°AB = ᎏᎏᎏ µB VA0.6

(5-208)

20%

Tyn-Calus [59]

8.93 × 10−8 (VA/VB2 )1/6 (ψB/ψA)0.6 T D°AB = ᎏᎏᎏᎏ µB

(5-209)

10%

Umesi-Danner [60]

2.75 × 10−8 (RB/RA2/3) T D°AB = ᎏᎏᎏ µB

(5-210)

16%

Siddiqi-Lucas [52]

9.89 × 10−8 VB0.265 T D°AB = ᎏᎏᎏ V A0.45 µB0.907

(5-211)

13%

  ᎏ V 

(5-212)

18%

(5-213)

6%

2. Gases in Low Viscosity Liquids Sridhar-Potter [54]

Chen-Chen [7]

Vc D°AB = DBB ᎏB VcA

2/3

VB

mlB

 

(βVcB)2/3(RTcB)1/2 T (Vr − 1) ᎏ D°AB = 2.018 × 10−9 ᎏᎏ MA1/6 (MBVcA)1/3 TcB

1/2

3. Aqueous Solutions Hayduk-Laudie [25]

13.16 × 10−5 D°AW = ᎏᎏ µ w1.14 VA0.589

(5-214)

18%

Siddiqi-Lucas [52]

D°AW = 2.98 × 10−7 VA−0.5473 µ w−1.026 T

(5-215)

13%

Hayduk-Minhas [26]

A − 0.791) D°AB = 13.3 × 10−8 T1.47 µ(10.2/V VA−0.71 B

(5-216)

5%

Matthews-Akgerman [38]

D°AB = 32.88 MA−0.61VD−1.04 T 0.5 (VB − VD)

(5-217)

5%

Riazi-Whitson [48]

(ρDAB)° µ DAB = 1.07 ᎏ ᎏ ρ µ°

(5-218)

15%

4. Hydrocarbon Mixtures

−0.27 − 0.38 ω + (−0.05 + 0.1 ω)P r

 

*References are listed on pages 5-7 and 5-8.

Random comparisons of predictions with 2.26 versus 2.6 show no consistent advantage for either value, however. It has been suggested to replace the exponent of 0.6 with 0.7 and to use an association factor of 0.7 for systems containing aromatic hydrocarbons. These modifications, however, are not recommended by Umesi and Danner. Lees and Sarram present a comparison of the association parameters. The average absolute error for 87 different solutes in water is 5.9 percent. Tyn-Calus This correlation requires data in the form of molar volumes and parachors ψi = Viσ1/4 i (a property which, over moderate temperature ranges, is nearly constant), measured at the same temperature (not necessarily the temperature of interest). The parachors for the components may also be evaluated at different temperatures from each other. Quale has compiled values of ψi for many chemicals. Group contribution methods are available for estimation purposes (Reid et al.). The following suggestions were made by Reid et al.: The correlation is constrained to cases in which µB < 30 cP. If the solute is water or if the solute is an organic acid and the solvent is not water or a short-chain alcohol, dimerization of the solute A should be assumed for purposes of estimating its volume and parachor. For example, the appropriate values for water as solute at 25°C are VW = 37.4 cm3/mol and ψW = 105.2 cm3g1/4/s1/2mol. Finally, if the solute is nonpolar, the solvent volume and parachor should be multiplied by 8µB. Umesi-Danner They developed an equation for nonaqueous solvents with nonpolar and polar solutes. In all, 258 points were involved ˚ of the component in the regression. Ri is the radius of gyration in A molecule, which has been tabulated by Passut and Danner for 250 compounds. The average absolute deviation was 16 percent, compared with 26 percent for the Wilke-Chang equation.

Siddiqi-Lucas In an impressive empirical study, these authors examined 1275 organic liquid mixtures. Their equation yielded an average absolute deviation of 13.1 percent, which was less than that for the Wilke-Chang equation (17.8 percent). Note that this correlation does not encompass aqueous solutions; those were examined and a separate correlation was proposed, which is discussed later. Binary Mixtures of Gases in Low-Viscosity, Nonelectrolyte Liquids Sridhar-Potter derived an equation for predicting gas diffusion through liquid by combining existing correlations. Hildebrand had postulated the following dependence of the diffusivity for a gas in a liquid: D°AB = DB′B(VcB /VcA)2/3, where DB′B is the solvent self-diffusion coefficient and Vci is the critical volume of component i, respectively. To correct for minor changes in volumetric expansion, Sridhar and Potter multiplied the resulting equation by VB / VmlB, where VmlB is the molar volume of the liquid B at its melting point and DB′B can be estimated by the equation of Ertl and Dullien (see p. 5-50). Sridhar and Potter compared experimentally measured diffusion coefficients for twenty-seven data points of eleven binary mixtures. Their average absolute error was 13.5 percent, but Chen and Chen analyzed about 50 combinations of conditions and 3 to 4 replicates each and found an average error of 18 percent. This correlation does not apply to hydrogen and helium as solutes. However, it demonstrates the usefulness of self-diffusion as a means to assess mutual diffusivities and the value of observable physical property changes, such as molar expansion, to account for changes in conditions. Chen-Chen Their correlation was based on diffusion measurements of 50 combinations of conditions with 3 to 4 replicates each and exhibited an average error of 6 percent. In this correlation, Vr = VB / [0.9724 (VmlB + 0.04765)] and VmlB = the liquid molar volume at the

5-52

HEAT AND MASS TRANSFER

melting point, as discussed previously. Their association parameter β [which is different from the definition of that symbol in Eq. (5-219)] accounts for hydrogen bonding of the solvent. Values for acetonitrile and methanol are: β = 1.58 and 2.31, respectively. Dilute Binary Mixtures of a Nonelectrolyte in Water The correlations that were suggested previously for general mixtures, unless specified otherwise, may also be applied to diffusion of miscellaneous solutes in water. The following correlations are restricted to the present case, however. Hayduk-Laudie They presented a simple correlation for the infinite dilution diffusion coefficients of nonelectrolytes in water. It has about the same accuracy as the Wilke-Chang equation (about 5.9 percent). There is no explicit temperature dependence, but the 1.14 exponent on µw compensates for the absence of T in the numerator. That exponent was misprinted (as 1.4) in the original article and has been reproduced elsewhere erroneously. Siddiqi-Lucas These authors examined 658 aqueous liquid mixtures in an empirical study. They found an average absolute deviation of 19.7 percent. In contrast, the Wilke-Chang equation gave 35.0 percent and the Hayduk-Laudie correlation gave 30.4 percent. Dilute Binary Hydrocarbon Mixtures Hayduk-Minhas presented an accurate correlation for normal paraffin mixtures that was developed from 58 data points consisting of solutes from C5 to C32 and solvents from C5 to C16. The average error was 3.4 percent for the 58 mixtures. Matthews-Akgerman The free-volume approach of Hildebrand was shown to be valid for binary, dilute liquid paraffin mixtures (as well as self-diffusion), consisting of solutes from C8 to C16 and solvents of C6 and C12. The term they referred to as the “diffusion volume” was simply correlated with the critical volume, as VD = 0.308 Vc. We can infer from Table 5-15 that this is approximately related to the volume at the melting point as VD = 0.945 Vm. Their correlation was valid for diffusion of linear alkanes at temperatures up to 300°C and pressures up to 3.45 MPa. Matthews et al. and Erkey and Akgerman completed similar studies of diffusion of alkanes, restricted to n-hexadecane and n-octane, respectively, as the solvents. Riazi-Whitson They presented a generalized correlation in terms of viscosity and molar density that was applicable to both gases and liquids. The average absolute deviation for gases was only about 8 percent, while for liquids it was 15 percent. Their expression relies on the Chapman-Enskog correlation [Eq. (5-194)] for the low-pressure diffusivity and the Stiel-Thodos correlation for low-pressure viscosity: xAµ°A MA1/2 + xBµ°B MB1/2 µ° = ᎏᎏᎏ xAMA1/2 + xBMB1/2 i where µ°i ξi = 3.4 × 10−4 Tr0.94 for Tr i < 1.5 or µ°i ξi = 1.778 × 10−4 (4.58 2/3 i /Pc i Tr i − 1.67)5/8 for Tr i > 1.5. In these equations, ξi = Tc1/6 M1/2 i , and units are in cP, atm, K, and mol. For dense gases or liquids, the Chung et al. or Jossi-Stiel-Thodos correlation may be used to estimate viscosity. The latter is: (µ − µ°) ξ + 10−4 = (0.1023 + 0.023364 ρr + 0.058533 ρ 2r − 0.040758 ρ3r + 0.093324 ρ4r)4 where

(xA TcA + xBTcB)1/6 ξ = ᎏᎏᎏᎏ (xAMA + xBMB)1/2 (xA PcA + xB PcB)

and

ρr = (xA VcA + xB VcB)ρ.

Dilute Binary Mixtures of Nonelectrolytes with Water as the Solute Olander modified the Wilke-Chang equation to adapt it to the infinite dilution diffusivity of water as the solute. The modification he recommended is simply the division of the right-hand side of the Wilke-Chang equation by 2.3. Unfortunately, neither the WilkeChang equation nor that equation divided by 2.3 fit the data very well. A reasonably valid generalization is that the Wilke-Chang equation is accurate if water is very insoluble in the solvent, such as pure hydrocarbons, halogenated hydrocarbons, and nitro-hydrocarbons. On the other hand, the Wilke-Chang equation divided by 2.3 is accurate for solvents in which water is very soluble, as well as those that have low viscosities. Such solvents include alcohols, ketones, carboxylic acids,

and aldehydes. Neither equation is accurate for higher-viscosity liquids, especially diols. Dilute Dispersions of Macromolecules in Nonelectrolytes The Stokes-Einstein equation has already been presented. It was noted that its validity was restricted to large solutes, such as spherical macromolecules and particles in a continuum solvent. The equation has also been found to predict accurately the diffusion coefficient of spherical latex particles and globular proteins. Corrections to StokesEinstein for molecules approximating spheroids is given by Tanford. Since solute-solute interactions are ignored in this theory, it applies in the dilute range only. Hiss-Cussler Their basis is the diffusion of a small solute in a fairly viscous solvent of relatively large molecules, which is the opposite of the Stokes-Einstein assumptions. The large solvent molecules investigated were not polymers or gels but were of moderate molecular weight so that the macroscopic and microscopic viscosities were the same. The major conclusion is that D°AB µ2/3 = constant at a given temperature and for a solvent viscosity from 5 × 10−3 to 5 Pa s or greater (5 to 5 × 103 cP). This observation is useful if D°AB is known in a given high-viscosity liquid (oils, tars, etc.). Use of the usual relation of D°AB ∝ 1/µ for such an estimate could lead to large errors. Concentrated, Binary Mixtures of Nonelectrolytes Several correlations that predict the composition dependence of DAB are summarized in Table 5-19. Most are based on known values of D°AB and D°BA. In fact, a rule of thumb states that, for many binary systems, D°AB and D°BA bound the DAB vs. xA curve. Cullinan’s equation predicts diffusivities even in lieu of values at infinite dilution, but requires accurate density, viscosity, and activity coefficient data. Since the infinite dilution values D°AB and D°BA are generally unequal, even a thermodynamically ideal solution like γA = γB = 1 will exhibit concentration dependence of the diffusivity. In addition, nonideal solutions require a thermodynamic correction factor to retain the true “driving force” for molecular diffusion, or the gradient of the chemical potential rather than the composition gradient. That correction factor is: ∂ ln γA βA = 1 + ᎏ (5-219) ∂ ln xA Caldwell-Babb Darken observed that solid-state diffusion in metallurgical applications followed a simple relation. His equation related the tracer diffusivities and mole fractions to the mutual diffusivity: DAB = (xA DB + xB DA) βA

(5-220)

Caldwell and Babb used virtually the same equation to evaluate the mutual diffusivity for concentrated mixtures of common liquids. Van Geet and Adamson tested that equation for the n-dodecane (A) and n-octane (B) system and found the average deviation of DAB from experimental values to be −0.68 percent. In addition, that equation was tested for benzene + bromobenzene, n-hexane + n-dodecane, benzene + CCl 4, octane + decane, heptane + cetane, benzene + diphenyl, and benzene + nitromethane with success. For systems that depart significantly from thermodynamic ideality, it breaks down, sometimes by a factor of eight. For example, in the binary systems acetone + CCl 4, acetone + chloroform, and ethanol + CCl 4, it is not accurate. Thus, it can be expected to be fairly accurate for nonpolar hydrocarbons of similar molecular weight but not for polar-polar mixtures. Siddiqi et al. found that this relation was superior to those of Vignes and Leffler and Cullinan for a variety of mixtures. Umesi and Danner found an average absolute deviation of 13.9 percent for 198 data points. Rathbun-Babb suggested that Darken’s equation could be improved by raising the thermodynamic correction factor βA to a power, n, less than unity. They looked at systems exhibiting negative deviations from Raoult’s law and found n = 0.3. Furthermore, for polarnonpolar mixtures, they found n = 0.6. In a separate study, Siddiqi and Lucas followed those suggestions and found an average absolute error of 3.3 percent for nonpolar-nonpolar mixtures, 11.0 percent for polarnonpolar mixtures, and 14.6 percent for polar-polar mixtures. Siddiqi et al. examined a few other mixtures and found that n = 1 was probably best. Thus, this approach is, at best, highly dependent on the type of components being considered.

MASS TRANSFER

5-53

TABLE 5-19 Correlations of Diffusivities for Concentrated, Binary Mixtures of Nonelectrolyte Liquids Authors*

Equation

Caldwell-Babb [5] Rathbun-Babb [46] Vignes [62] Leffler-Cullinan [34]

DAB = (xA DBA ° + xB DAB ° )βA DAB = (xA DBA ° + xB DAB ° )βAn DAB = DAB ° xB DBA ° xA βA DAB µ mix = (DAB ° µ B)xB (DBA ° µA) xA βA

Cussler [12]

K ∂ ln xA DAB = D0 1 + ᎏ ᎏ −1 xA xB ∂ ln aA

Cullinan [10]

kT 2πxA xB βA DAB = ᎏᎏ ᎏᎏ 2πµ mix (V/A)1/3 1 + βA (2πxA xB − 1)

Asfour-Dullien [2]

DAB ° DAB = ᎏ µB





−1/2







(5-221) (5-222) (5-223) (5-224) (5-225)



1/2

ζµ β  ᎏ µ  xB

DBA °

xA

(5-227)

A

A

DAB = (CB V B D AB ° + CA V A D BA ° )βA

Siddiqi-Lucas [52]

(5-226)

(5-228)

Relative errors for the correlations in this table are very dependent on the components of interest and are cited in the text. *See pages 5-7 and 5-8 for references.

Vignes empirically correlated mixture diffusivity data for 12 binary mixtures. Later Ertl et al. evaluated 122 binary systems, which showed an average absolute deviation of only 7 percent. None of the latter systems, however, was very nonideal. Leffler-Cullinan modified Vignes’ equation using some theoretical arguments to arrive at Eq. (5-224), which the authors compared to Eq. (5-223) for the 12 systems mentioned above. The average absolute maximum deviation was only 6 percent. Umesi and Danner, however, found an average absolute deviation of 11.4 percent for 198 data points. For normal paraffins, it is not very accurate. In general, the accuracies of Eqs. (5-223) and (5-224) are not much different, and, since Vignes’ is simpler to use, it is suggested. The application of either should be limited to nonassociating systems that do not deviate much from ideality (0.95 < βA < 1.05). Cussler studied diffusion in concentrated associating systems and has shown that, in associating systems, it is the size of diffusing clusters rather than diffusing solutes that controls diffusion. Do is a reference diffusion coefficient discussed hereafter; aA is the activity of component A; and K is a constant. By assuming that Do could be predicted by Eq. (5-223) with β = 1, K was found to be equal to 0.5 based on five binary systems and validated with a sixth binary mixture. The limitations of Eq. (5-225) using Do and K defined previously have not been explored, so caution is warranted. Gurkan showed that K should actually be closer to 0.3 (rather than 0.5) and discussed the overall results. Cullinan presented an extension of Cussler’s cluster diffusion theory. His method accurately accounts for composition and temperature dependence of diffusivity. It is novel in that it contains no adjustable constants, and it relates transport properties and solution thermodynamics. This equation has been tested for six very different mixtures by Rollins and Knaebel, and it was found to agree remarkably well with data for most conditions, considering the absence of adjustable parameters. In the dilute region (of either A or B), there are systematic errors probably caused by the breakdown of certain implicit assumptions (that nevertheless appear to be generally valid at higher concentrations). Asfour-Dullien developed a relation for predicting alkane diffusivities at moderate concentrations that employs: Vfm 2/3 MxAMxB ᎏ ζ= ᎏ (5-229) Vf xAVf xB Mm





where Vfxi = V ; the fluid free volume is Vf i = Vi − Vml i for i = A, B, and m, in which Vml i is the molar volume of the liquid at the melting point and xA2 2 xA xB xB2 −1 Vmlm = ᎏ +ᎏ +ᎏ VmlA VmlAB VmlB xi fi



and

1/3 V1/3 mlA + V mlB VmlAB = ᎏᎏ 2







3

and µ is the mixture viscosity; Mm is the mixture mean molecular weight; and βA is defined by Eq. (5-219). The average absolute error of this equation is 1.4 percent, while the Vignes equation and the Leffler-Cullinan equation give 3.3 percent and 6.2 percent, respectively. Siddiqi-Lucas suggested that component volume fractions might be used to correlate the effects of concentration dependence. They found an average absolute deviation of 4.5 percent for nonpolarnonpolar mixtures, 16.5 percent for polar-nonpolar mixtures, and 10.8 percent for polar-polar mixtures. Binary Electrolyte Mixtures When electrolytes are added to a solvent, they dissociate to a certain degree. It would appear that the solution contains at least three components: solvent, anions, and cations. If the solution is to remain neutral in charge at each point (assuming the absence of any applied electric potential field), the anions and cations diffuse effectively as a single component, as for molecular diffusion. The diffusion of the anionic and cationic species in the solvent can thus be treated as a binary mixture. Nernst-Haskell The theory of dilute diffusion of salts is well developed and has been experimentally verified. For dilute solutions of a single salt, the well-known Nernst-Haskell equation (Reid et al.) is applicable: 1 1 1 1 ᎏᎏ + ᎏᎏ ᎏᎏ + ᎏᎏ n− n n RT n+ + − ᎏᎏ = 8.9304 × 10−10 T ᎏᎏ (5-230) D°AB = ᎏ 1 1 F 2 ᎏ1ᎏ + ᎏ1ᎏ ᎏᎏ0 + ᎏᎏ0 λ0+ λ0− λ+ λ−

   

   

where D°AB = diffusivity based on molarity rather than normality of dilute salt A in solvent B, cm2/s. The previous definitions can be interpreted in terms of ionicspecies diffusivities and conductivities. The latter are easily measured and depend on temperature and composition. For example, the equivalent conductance Λ is commonly tabulated in chemistry handbooks as the limiting (infinite dilution) conductance Λo and at standard concentrations, typically at 25°C. Λ = 1000 K/C = λ+ + λ− = Λo + f(C), (cm2/ohm gequiv); K = α/R = specific conductance, (ohm cm)−1; C = solution concentration, (gequiv/ᐉ); α = conductance cell constant (measured), (cm−1); R = solution electrical resistance, which is measured (ohm); and f(C) = a complicated function of concentration. The resulting equation of the electrolyte diffusivity is |z+| + |z−| DAB = ᎏᎏᎏ (5-231) (|z−| / D+) + (|z+| / D−) where |z⫾| represents the magnitude of the ionic charge and where the cationic or anionic diffusivities are D⫾ = 8.9304 × 10−10 T λ⫾ / |z⫾| cm2/s. The coefficient is kN0 /F 2 = R/F 2. In practice, the equivalent conductance of the ion pair of interest would be obtained and supplemented with conductances of permutations of those ions and one independent cation and anion. This would allow determination of all the ionic conductances and hence the diffusivity of the electrolyte solution.

5-54

HEAT AND MASS TRANSFER

Gordon Typically, as the concentration of a salt increases from infinite dilution, the diffusion coefficient decreases rapidly from D°AB. As concentration is increased further, however, DAB rises steadily, often becoming greater than D°AB. Gordon proposed the following empirical equation, which is applicable up to concentrations of 2N: ln γ⫾ 1 µB 1+ᎏ (5-232) DAB = D°AB ᎏ ᎏ CBVB µ ln m where D°AB is given by the Nernst-Haskell equation. References that tabulate γ⫾ as a function of m, as well as other equations for DAB, are given by Reid et al. Multicomponent Mixtures No simple, practical estimation methods have been developed for predicting multicomponent liquiddiffusion coefficients. Several theories have been developed, but the necessity for extensive activity data, pure component and mixture volumes, mixture viscosity data, and tracer and binary diffusion coefficients have significantly limited the utility of the theories (see Reid et al.). The generalized Stefan-Maxwell equations using binary diffusion coefficients are not easily applicable to liquids since the coefficients are so dependent on conditions. That is, in liquids, each Dij can be strongly composition dependent in binary mixtures and, moreover, the binary Dij is strongly affected in a multicomponent mixture. Thus, the convenience of writing multicomponent flux equations in terms of binary coefficients is lost. Conversely, they apply to gas mixtures because each Dij is practically independent of composition by itself and in a multicomponent mixture (see Taylor and Krishna for details). One particular case of multicomponent diffusion that has been examined is the dilute diffusion of a solute in a homogeneous mixture (e.g., of A in B + C). Umesi and Danner compared the three equations given below for 49 ternary systems. All three equations were equivalent, giving average absolute deviations of 25 percent. Perkins-Geankoplis





n

Dam µ m0.8 =
xj D°A j µ0.8 j

(5-233)

j=1 j≠A

Cullinan This is an extension of Vignes’ equation to multicomponent systems:

applications of interest are outlined in Table 5-20. Applications of these equations are found in Secs. 16, 22, and 23. Diffusion within the largest cavities of a porous medium is assumed to be similar to ordinary or bulk diffusion except that it is hindered by the pore walls (see Eq. 5-236). The tortuosity τ that expresses this hindrance has been estimated from geometric arguments. Unfortunately, measured values are often an order of magnitude greater than those estimates. Thus, the effective diffusivity Deff (and hence τ) is normally determined by comparing a diffusion model to experimental measurements. The normal range of tortuosities for silica gel, alumina, and other porous solids is 2 ≤ τ ≤ 6, but for activated carbon, 5 ≤ τ ≤ 65. In small pores and at low pressures, the mean free path ᐉ of the gas molecule (or atom) is significantly greater than the pore diameter dpore. Its magnitude may be estimated from 3.2 µ RT 1/2 ᐉ = ᎏ ᎏ , in m P 2πM As a result, collisions with the wall occur more frequently than with other molecules. This is referred to as the Knudsen mode of diffusion and is contrasted with ordinary or bulk diffusion, which occurs by intermolecular collisions. At intermediate pressures, both ordinary diffusion and Knudsen diffusion may be important [see Eqs. (5-239) and (5-240)]. For gases and vapors that adsorb on the porous solid, surface diffusion may be important, particularly at high surface coverage [see Eqs. (5-241) and (5-244)]. The mechanism of surface diffusion may be viewed as molecules hopping from one surface site to another. Thus, if adsorption is too strong, surface diffusion is impeded, while if adsorption is too weak, surface diffusion contributes insignificantly to the overall rate. Surface diffusion and bulk diffusion usually occur in parallel [see Eqs. (5-245) and (5-246)]. Although Ds is expected to be less than Deff, the solute flux due to surface diffusion may be larger than that due to bulk diffusion if ∂qi / ∂z >> ∂Ci / ∂z. This can occur when a component is strongly adsorbed and the surface coverage is high. For all that, surface diffusion is not well understood. The references in Table 5-20 should be consulted for further details.





INTERPHASE MASS TRANSFER

DIFFUSION OF FLUIDS IN POROUS SOLIDS

Transfer of material between phases is important in most separation processes in which two phases are involved. When one phase is pure, mass transfer in the pure phase is not involved. For example, when a pure liquid is being evaporated into a gas, only the gas-phase mass transfer need be calculated. Occasionally, mass transfer in one of the two phases may be neglected even though pure components are not involved. This will be the case when the resistance to mass transfer is much larger in one phase than in the other. Understanding the nature and magnitudes of these resistances is one of the keys to performing reliable mass transfer. In this section, mass transfer between gas and liquid phases will be discussed. The principles are easily applied to the other phases. Mass-Transfer Principles: Dilute Systems When material is transferred from one phase to another across an interface that separates the two, the resistance to mass transfer in each phase causes a concentration gradient in each, as shown in Fig. 5-26 for a gas-liquid interface. The concentrations of the diffusing material in the two phases immediately adjacent to the interface generally are unequal, even if expressed in the same units, but usually are assumed to be related to each other by the laws of thermodynamic equilibrium. Thus, it is assumed that the thermodynamic equilibrium is reached at the gas-liquid interface almost immediately when a gas and a liquid are brought into contact. For systems in which the solute concentrations in the gas and liquid phases are dilute, the rate of transfer may be expressed by equations which predict that the rate of mass transfer is proportional to the difference between the bulk concentration and the concentration at the gas-liquid interface. Thus

Diffusion in porous solids is usually the most important factor controlling mass transfer in adsorption, ion exchange, drying, heterogeneous catalysis, leaching, and many other applications. Some of the

where NA = mass-transfer rate, k′G = gas-phase mass-transfer coefficient, k′L = liquid-phase mass-transfer coefficient, p = solute partial pressure in

Dam =

n

 (D° ) Aj

xj

(5-234)

j=1 j≠A

Leffler-Cullinan They extended their binary relation to an arbitrary multicomponent mixture, as follows: Dam µm =

n

 (D° µ ) Aj

j

xj

(5-235)

j=1 j≠A

where DAj is the dilute binary diffusion coefficient of A in j; DAm is the dilute diffusion of A through m; xj is the mole fraction; µj is the viscosity of component j; and µm is the mixture viscosity. Akita Another case of multicomponent dilute diffusion of significant practical interest is that of gases in aqueous electrolyte solutions. Many gas-absorption processes use electrolyte solutions. Akita presents experimentally tested equations for this case. Graham-Dranoff They studied multicomponent diffusion of electrolytes in ion exchangers. They found that the Stefan-Maxwell interaction coefficients reduce to limiting ion tracer diffusivities of each ion. Pinto-Graham Pinto and Graham studied multicomponent diffusion in electrolyte solutions. They focused on the Stefan-Maxwell equations and corrected for solvation effects. They achieved excellent results for 1-1 electrolytes in water at 25°C up to concentrations of 4M.

NA = k′G(p − pi) = k′L(ci − c)

(5-248)

MASS TRANSFER TABLE 5-20

5-55

Relations for Diffusion in Porous Solids

Mechanism

Equation

Bulk diffusion in pores

ε pD Deff = ᎏ τ

Knudsen diffusion

T DK = 48.5 dpore ᎏ M

 

Applies to

1/2

in m2/s

References*

(5-236)

Gases or liquids in large pores. NK n = ᐉ/d pore < 0.01

[67]

(5-237)

Dilute (low pressure) gases in small pores. NK n = ᐉ/d pore > 10

Geankoplis, [68, 69]

(5-238)

"

"

"

"

(5-239)

"

"

"

"

ε pDK DKeff = ᎏ τ dC Ni = −DK ᎏi dz Combined bulk and Knudsen diffusion

1 − α xA 1 Deff = ᎏ + ᎏ Deff DKeff



−1



NB α=1+ᎏ NA



1 1 Deff = ᎏ + ᎏ Deff DKeff Surface diffusion

−1



(5-240)

NA = NB

dq JSi = −DSeff ρp ᎏi dz

(5-241)

Adsorbed gases or vapors

ε p DS DSeff = ᎏ τ

(5-242)

"

DSθ = 0 DSθ = ᎏ (1 − θ)

(5-243)

θ = fractional surface coverage ≤ 0.6

(5-244)

"

"

"

"

(5-245)

"

"

"

"

(5-246)

"

"

"

"

(5-247)

"

"

"

"

 

−E DS = DS′ (q) exp ᎏS RT

 

Parallel bulk and surface diffusion

Geankoplis, [66, 69]

NA ≠ NB

  

 

dp dq J = − Deff ᎏi + DSeff ρp ᎏi dz dz

 

dp J = −Dapp ᎏi dz

 

dqi Dapp = Deff + DSeff ρp ᎏ dpi

"

"

[66, 68, 69]

"

[68]

*See pages 5-7 and 5-8 for references.

bulk gas, pi = solute partial pressure at interface, c = solute concentration in bulk liquid, and ci = solute concentration in liquid at interface. The mass-transfer coefficients k′G and k′L by definition are equal to the ratios of the molal mass flux NA to the concentration driving forces (p − pi) and (ci − c) respectively. An alternative expression for the rate of transfer in dilute systems is given by NA = kG(y − yi) = kL(xi − x) (5-249)

where NA = mass-transfer rate, kG = gas-phase mass-transfer coefficient, kL = liquid-phase mass-transfer coefficient, y = mole-fraction solute in bulk-gas phase, yi = mole-fraction solute in gas at interface, x = mole-fraction solute in bulk-liquid phase, and xi = mole-fraction solute in liquid at interface. The mass-transfer coefficients defined by Eqs. (5-248) and (5-249) are related to each other as follows: kG = k′G pT

(5-250)

kL = k′L ρ (5-251) L where pT = total system pressure employed during the experimental determinations of k′G values and ρ L = average molar density of the liquid phase. The coefficient kG is relatively independent of the total system pressure and therefore is more convenient to use than k′G, which is inversely proportional to the total system pressure. The above equations may be used for finding the interfacial concentrations corresponding to any set of values of x and y provided the ratio of the individual coefficients is known. Thus (y − yi)/(xi − x) = kL /kG = k′Lρ L/k′G pT = LMHG /GMHL

FIG. 5-26

Concentration gradients near a gas-liquid interface.

(5-252)

where LM = molar liquid mass velocity, GM = molar gas mass velocity, HL = height of one transfer unit based on liquid-phase resistance, and HG = height of one transfer unit based on gas-phase resistance. The last term in Eq. (5-252) is derived from Eqs. (5-271) and (5-273). Equation (5-252) may be solved graphically if a plot is made of the equilibrium vapor and liquid compositions and a point representing

5-56

HEAT AND MASS TRANSFER

the bulk concentrations x and y is located on this diagram. A construction of this type is shown in Fig. 5-27, which represents a gasabsorption situation. The interfacial mole fractions yi and xi can be determined by solving Eq. (5-252) simultaneously with the equilibrium relation y°i = F(xi) to obtain yi and xi. The rate of transfer may then be calculated from Eq. (5-249). If the equilibrium relation y°i = F(xi) is sufficiently simple, e.g., if a plot of y°i versus xi is a straight line, not necessarily through the origin, the rate of transfer is proportional to the difference between the bulk concentration in one phase and the concentration (in that same phase) which would be in equilibrium with the bulk concentration in the second phase. One such difference is y − y°, and another is x° − x. In this case, there is no need to solve for the interfacial compositions, as may be seen from the following derivation. The rate of mass transfer may be defined by the equation NA = KG(y − y°) = kG(y − yi) = kL(xi − x) = KL(x° − x) (5-253) where KG = overall gas-phase mass-transfer coefficient, KL = overall liquid-phase mass-transfer coefficient, y° = vapor composition in equilibrium with x, and x° = liquid composition in equilibrium with vapor of composition y. This equation can be rearranged to the formula 1 1 y − y° 1 1 yi − y° 1 1 yi − y° ᎏ=ᎏ ᎏ =ᎏ+ᎏ ᎏ =ᎏ+ᎏ ᎏ KG kG y − yi kG kG y − yi kG kL xi − x













(5-254) in view of Eq. (5-252). Comparison of the last term in parentheses with the diagram of Fig. 5-27 shows that it is equal to the slope of the chord connecting the points (x,y°) and (xi,yi). If the equilibrium curve is a straight line, then this term is the slope m. Thus 1/KG = (1/ kG + m/ kL) (5-255) When Henry’s law is valid (pA = HxA or pA = H′CA), the slope m can be computed according to the relationship m = H/ pT = H′ρ (5-256) L/ pT where m is defined in terms of mole-fraction driving forces compatible with Eqs. (5-249) through (5-255), i.e., with the definitions of kL, kG, and KG. If it is desired to calculate the rate of transfer from the overall concentration difference based on bulk-liquid compositions (x° − x), the appropriate overall coefficient KL is related to the individual coefficients by the equation 1/KL = (1/kL + 1/mkG) (5-257) Conversion of these equations to a k′G, k′L basis can be accomplished readily by direct substitution of Eqs. (5-250) and (5-251). Occasionally one will find k′L or K′L values reported in units (SI) of meters per second. The correct units for these values are kmol/

[(s⋅m2)(kmol/m3)], and Eq. (5-251) is the correct equation for converting them to a mole-fraction basis. When k′G and K′G values are reported in units (SI) of kmol/[(s⋅m2) (kPa)], one must be careful in converting them to a mole-fraction basis to multiply by the total pressure actually employed in the original experiments and not by the total pressure of the system to be designed. This conversion is valid for systems in which Dalton’s law of partial pressures (p = ypT) is valid. Comparison of Eqs. (5-255) and (5-257) shows that for systems in which the equilibrium line is straight, the overall mass transfer coefficients are related to each other by the equation KL = mKG

(5-258)

When the equilibrium curve is not straight, there is no strictly logical basis for the use of an overall transfer coefficient, since the value of m will be a function of position in the apparatus, as can be seen from Fig. 5-27. In such cases the rate of transfer must be calculated by solving for the interfacial compositions as described above. Experimentally observed rates of mass transfer often are expressed in terms of overall transfer coefficients even when the equilibrium lines are curved. This procedure is empirical, since the theory indicates that in such cases the rates of transfer may not vary in direct proportion to the overall bulk concentration differences (y − y°) and (x° − x) at all concentration levels even though the rates may be proportional to the concentration difference in each phase taken separately, i.e., (xi − x) and (y − yi). In most types of separation equipment such as packed or spray towers, the interfacial area that is effective for mass transfer cannot be accurately determined. For this reason it is customary to report experimentally observed rates of transfer in terms of transfer coefficients based on a unit volume of the apparatus rather than on a unit of interfacial area. Such volumetric coefficients are designated as KGa, kLa, etc., where a represents the interfacial area per unit volume of the apparatus. Experimentally observed variations in the values of these volumetric coefficients with variations in flow rates, type of packing, etc., may be due as much to changes in the effective value of a as to changes in k. Calculation of the overall coefficients from the individual volumetric coefficients is made by means of the equations 1/KGa = (1/kGa + m/kLa)

(5-259)

1/KLa = (1/kLa + 1/mkGa)

(5-260)

Because of the wide variation in equilibrium, the variation in the values of m from one system to another can have an important effect on the overall coefficient and on the selection of the type of equipment to use. For example, if m is large, the liquid-phase part of the overall resistance might be extremely large where kL might be relatively small. This kind of reasoning must be applied with caution, however, since species with different equilibrium characteristics are separated under different operating conditions. Thus, the effect of changes in m on the overall resistance to mass transfer may partly be counterbalanced by changes in the individual specific resistances as the flow rates are changed. Mass-Transfer Principles: Concentrated Systems When solute concentrations in the gas and/or liquid phases are large, the equations derived above for dilute systems no longer are applicable. The correct equations to use for concentrated systems are as follows: NA = kˆ G(y − yi)/yBM = kˆ L(xi − x)/xBM = Kˆ G(y − y°)/y°BM = Kˆ L(x° − x)/x°BM

(5-261)

where (NB = 0)

Identification of concentrations at a point in a countercurrent absorption tower.

FIG. 5-27

(1 − y) − (1 − yi) yBM = ᎏᎏ ln [(1 − y)/(1 − yi)] (1 − y) − (1 − y°) y°BM = ᎏᎏᎏ ln [(1 − y)/(1 − y°)] (1 − x) − (1 − xi) xBM = ᎏᎏ ln [(1 − x)/(1 − xi)] (1 − x) − (1 − x°) x°BM = ᎏᎏ ln [(1 − x)/(1 − x°)]

(5-262) (5-263) (5-264) (5-265)

MASS TRANSFER and where kˆ G and kˆ L are the gas-phase and liquid-phase mass-transfer coefficients for concentrated systems and Kˆ G and Kˆ L are the overall gas-phase and liquid-phase mass-transfer coefficients for concentrated systems. These coefficients are defined later in Eqs. (5-268) to (5-270). The factors yBM and xBM arise from the fact that, in the diffusion of a solute through a second stationary layer of insoluble fluid, the resistance to diffusion varies in proportion to the concentration of the insoluble stationary fluid, approaching zero as the concentration of the insoluble fluid approaches zero. See Eq. (5-190). The factors y°BM and x°BM cannot be justified on the basis of masstransfer theory since they are based on overall resistances. These factors therefore are included in the equations by analogy with the corresponding film equations. In dilute systems the logarithmic-mean insoluble-gas and nonvolatileliquid concentrations approach unity, and Eq. (5-261) reduces to the dilute-system formula. For equimolar counter diffusion (e.g., binary distillation), these log-mean factors should be omitted. See Eq. (5-189). Substitution of Eqs. (5-262) through (5-265) into Eq. (5-261) results in the following simplified formula: NA = kˆ G ln [(1 − yi)/(1 − y)] = Kˆ G ln [(1 − y°)/(1 − y)] = kˆ L ln [(1 − x)/(1 − xi)] (5-266) = Kˆ L ln [(1 − x)/(1 − x°)] ˆ ˆ ˆ ˆ Note that the units of kG, KG, kL, and KL are all identical to each other, i.e., kmol/(s⋅m2) in SI units. The equation for computing the interfacial gas and liquid compositions in concentrated systems is (y − yi)/(xi − x) = kˆ LyBM / kˆ GxBM = LMHGyBM /GMHL x BM = kL /kG

(5-267) This equation is identical to the one for dilute systems since kˆ G = kGyBM and kˆ L = kLxBM. Note, however, that when kˆ G and kˆ L are given, the equation must be solved by trial and error, since xBM contains xi and yBM contains yi. The overall gas-phase and liquid-phase mass-transfer coefficients for concentrated systems are computed according to the following equations: 1 yBM 1 xBM 1 yi − y° =ᎏ (5-268) ᎏ ᎏ+ᎏᎏ ᎏ Kˆ G y°BM kˆ G y°BM kˆ L xi − x





1 xBM 1 yBM 1 x° − xi (5-269) =ᎏᎏ+ᎏᎏ ᎏ ᎏ Kˆ L x°BM kˆ L x°BM kˆ G y − yi When the equilibrium curve is a straight line, the terms in parentheses can be replaced by the slope m as before. In this case the overall mass-transfer coefficients for concentrated systems are related to each other by the equation Kˆ L = m Kˆ G(x°BM /y°BM) (5-270)





All these equations reduce to their dilute-system equivalents as the inert concentrations approach unity in terms of mole fractions of inert concentrations in the fluids. HTU (Height Equivalent to One Transfer Unit) Frequently the values of the individual coefficients of mass transfer are so strongly dependent on flow rates that the quantity obtained by dividing each coefficient by the flow rate of the phase to which it applies is more nearly constant than the coefficient itself. The quantity obtained by this procedure is called the height equivalent to one transfer unit, since it expresses in terms of a single length dimension the height of apparatus required to accomplish a separation of standard difficulty. The following relations between the transfer coefficients and the values of HTU apply: H = G /k ay = G /kˆ a (5-271) G

M

G

BM

M

G

HOG = GM /KGay°BM = GM/Kˆ Ga H = L /k ax = L /kˆ a

(5-272)

HOL = LM /KLax°BM = LM/Kˆ La

(5-274)

L

M

L

BM

M

L

(5-273)

5-57

The equations that express the addition of individual resistances in terms of HTUs, applicable to either dilute or concentrated systems, are yBM mGM xBM HOG = ᎏ HG + ᎏ (5-275) ᎏ HL y°BM LM y°BM xBM LM yBM (5-276) HOL = ᎏ HL + ᎏ ᎏ HG x°BM mGM x°BM These equations are strictly valid only when m, the slope of the equilibrium curve, is constant, as noted previously. NTU (Number of Transfer Units) The NTU required for a given separation is closely related to the number of theoretical stages or plates required to carry out the same separation in a stagewise or plate-type apparatus. For equimolal counterdiffusion, such as in a binary distillation, the number of overall gas-phase transfer units NOG required for changing the composition of the vapor stream from y1 to y2 is y1 dy (5-277) NOG = ᎏ y2 y − y° When diffusion is in one direction only, as in the absorption of a soluble component from an insoluble gas, y1 y°BM dy (5-278) NOG = ᎏᎏ y2 (1 − y)(y − y°) The total height of packing required is then





hT = HOGNOG

(5-279)

When it is known that HOG varies appreciably within the tower, this term must be placed inside the integral in Eqs. (5-277) and (5-278) for accurate calculations of hT. For example, the packed-tower design equation in terms of the overall gas-phase mass-transfer coefficient for absorption would be expressed as follows: y1 y°BM dy GM hT = (5-280) ᎏ ᎏᎏ y2 KGay°BM (1 − y)(y − y°) where the first term under the integral can be recognized as the HTU term. Convenient solutions of these equations for special cases are discussed later. Definitions of Mass-Transfer Coefficients kˆ G and kˆ L The mass-transfer coefficient is defined as the ratio of the molal mass flux NA to the concentration driving force. This leads to many different ways of defining these coefficients. For example, gas-phase masstransfer rates may be defined as N = k (y − y ) = k′ (p − p ) = kˆ (y − y )/y (5-281)

 

A

G

i



G

i

G

i

BM

where the units (SI) of kG are kmol/[(s⋅m2)(mole fraction)], the units of k′G are kmol/[(s⋅m2)(kPa)], and the units of kˆ G are kmol/(s⋅m2). These coefficients are related to each other as follows: kG = kGyBM = k′G pT yBM

(5-282)

where pT is the total system pressure (it is assumed here that Dalton’s law of partial pressures is valid). In a similar way, liquid-phase mass-transfer rates may be defined by the relations N = k (x − x) = k′ (c − c) = kˆ (x − x)/x (5-283) A

L

i

L

i

L

i

BM

where the units (SI) of kL are kmol/[(s⋅m2)(mole fraction)], the units of k′L are kmol/[(s⋅m2)(kmol/m3)] or meters per second, and the units of kˆ L are kmol/(s⋅m2). These coefficients are related as follows: (5-284) kˆ L = kLxBM = k′Lρ LxBM is the molar density of the liquid phase in units (SI) of kilowhere ρ L moles per cubic meter. Note that, for dilute solutions where xBM ⬟ 1, kL and kˆ L will have identical numerical values. Similarly, for dilute gases kˆ G ⬟ kG. Simplified Mass-Transfer Theories In certain simple situations, the mass-transfer coefficients can be calculated from first principles. The film, penetration, and surface-renewal theories are attempts to extend these theoretical calculations to more complex sit-

5-58

HEAT AND MASS TRANSFER

uations. Although these theories are often not accurate, they are useful to provide a physical picture for variations in the mass-transfer coefficient. For the special case of steady-state unidirectional diffusion of a component through an inert-gas film in an ideal-gas system, the rate of mass transfer is derived as 1 − yi DABpT (y − yi) DABpT NA = ᎏ (5-285) ᎏ = ᎏ ln ᎏ RT δG yBM RT δG 1−y where DAB = the diffusion coefficient or “diffusivity,” δG = the “effective” thickness of a stagnant-gas layer which would offer a resistance to molecular diffusion equal to the experimentally observed resistance, and R = the gas constant. [Nernst, Z. Phys. Chem., 47, 52 (1904); Whitman, Chem. Mat. Eng., 29, 149 (1923), and Lewis and Whitman, Ind. Eng. Chem., 16, 1215 (1924)]. The film thickness δG depends primarily on the hydrodynamics of the system and hence on the Reynolds number and the Schmidt number. Thus, various correlations have been developed for different geometries in terms of the following dimensionless variables: NSh = kˆ GRTd/DABpT = f(NRe,NSc) (5-286) where NSh is the Sherwood number, NRe (= Gd/µG) is the Reynolds number based on the characteristic length d appropriate to the geometry of the particular system; and NSc (= µG /ρGDAB) is the Schmidt number. According to this analysis one can see that for gas-absorption problems, which often exhibit unidirectional diffusion, the most appropriate driving-force expression is of the form (y − yi)/yBM, and the most appropriate mass-transfer coefficient is therefore kˆ G. This concept is to be found in all the key equations for the design of mass-transfer equipment. The Sherwood-number relation for gas-phase mass-transfer coefficients as represented by the film diffusion model in Eq. (5-286) can be rearranged as follows: NSh = (kˆ G /GM)NReNSc = NStNReNSc = f (NRe,NSc) (5-287) where NSt = kˆ G /GM = k′G pBM /GM is known as the Stanton number. This equation can now be stated in the alternative functional forms NSt = kˆ G /GM = g(NRe,NSc) (5-288) 2 /3 (5-289) jD = NSt ⋅ NSc where j is the Chilton-Colburn “j factor” for mass transfer (discussed later). The important point to note here is that the gas-phase masstransfer coefficient kˆ G depends principally upon the transport properties of the fluid (NSc) and the hydrodynamics of the particular system involved (NRe). It also is important to recognize that specific masstransfer correlations can be derived only in conjunction with the investigator’s particular assumptions concerning the numerical values of the effective interfacial area a of the packing. The stagnant-film model discussed previously assumes a steady state in which the local flux across each element of area is constant; i.e., there is no accumulation of the diffusing species within the film. Higbie [Trans. Am. Inst. Chem. Eng., 31, 365 (1935)] pointed out that industrial contactors often operate with repeated brief contacts between phases in which the contact times are too short for the steady state to be achieved. For example, Higbie advanced the theory that in a packed tower the liquid flows across each packing piece in laminar flow and is remixed at the points of discontinuity between the packing elements. Thus, a fresh liquid surface is formed at the top of each piece, and as it moves downward, it absorbs gas at a decreasing rate until it is mixed at the next discontinuity. This is the basis of penetration theory. If the velocity of the flowing stream is uniform over a very deep region of liquid (total thickness, δT >>  Dt), the time-averaged masstransfer coefficient according to penetration theory is given by k′L = 2 D πt (5-290) L/ where k′L = liquid-phase mass-transfer coefficient, DL = liquid-phase diffusion coefficient, and t = contact time. In practice, the contact time t is not known except in special cases in which the hydrodynamics are clearly defined. This is somewhat

similar to the case of the stagnant-film theory in which the unknown quantity is the thickness of the stagnant layer δ (in film theory, the liquid-phase mass-transfer coefficient is given by k′L = DL /δ). The penetration theory predicts that k′L should vary by the square root of the molecular diffusivity, as compared with film theory, which predicts a first-power dependency on D. Various investigators have reported experimental powers of D ranging from 0.5 to 0.75, and the Chilton-Colburn analogy suggests a w power. Penetration theory often is used in analyzing absorption with chemical reaction because it makes no assumption about the depths of penetration of the various reacting species, and it gives a more accurate result when the diffusion coefficients of the reacting species are not equal. When the reaction process is very complex, however, penetration theory is more difficult to use than film theory, and the latter method normally is preferred. Danckwerts [Ind. Eng. Chem., 42, 1460 (1951)] proposed an extension of the penetration theory, called the surface renewal theory, which allows for the eddy motion in the liquid to bring masses of fresh liquid continually from the interior to the surface, where they are exposed to the gas for finite lengths of time before being replaced. In his development, Danckwerts assumed that every element of fluid has an equal chance of being replaced regardless of its age. The Danckwerts model gives

s k′L = D

(5-291)

where s = fractional rate of surface renewal. Note that both the penetration and the surface-renewal theories predict a square-root dependency on D. Also, it should be recognized that values of the surface-renewal rate s generally are not available, which presents the same problems as do δ and t in the film and penetration models. The predictions of correlations based on the film model often are nearly identical to predictions based on the penetration and surfacerenewal models. Thus, in view of its relative simplicity, the film model normally is preferred for purposes of discussion or calculation. It should be noted that none of these theoretical models has proved adequate for making a priori predictions of mass-transfer rates in packed towers, and therefore empirical correlations such as those outlined later in Table 5-28. must be employed. Mass-Transfer Correlations Because of the tremendous importance of mass transfer in chemical engineering, a very large number of studies have determined mass-transfer coefficients both empirically and theoretically. Some of these studies are summarized in Tables 5-21 to 5-28. Each table is for a specific geometry or type of contactor, starting with flat plates, which have the simplest geometry (Table 5-21); then wetted wall columns (Table 5-22); flow in pipes and ducts (Table 5-23); submerged objects (Table 5-24); drops and bubbles (Table 5-25); agitated systems (Table 5-26); packed beds of particles for adsorption, ion exchange, and chemical reaction (Table 5-27); and finishing with packed bed two-phase contactors for distillation, absorption and other unit operations (Table 5-28). Graphical correlations for the Bolles and Fair correlation (Table 5-28-G) are in Figs. 5-28 to 5-30. Although extensive, these tables are not meant to be encyclopedic. For simple geometries, one may be able to determine a theoretical (T) form of the mass-transfer correlation. For very complex geometries, only an empirical (E) form can be found. In systems of intermediate complexity, semiempirical (S) correlations where the form is determined from theory and the coefficients from experiment are often useful. Although the major limitations and constraints in use are usually included in the tables, obviously many details cannot be included in this summary form. Readers are strongly encouraged to check the references before using the correlations in important situations. Note that even authoritative sources occasionally have typographical errors in the fairly complex correlation equations. Thus, it is a good idea to check several sources, including the original paper. The references will often include figures comparing the correlations with data. These figures are very useful since they provide a visual picture of the scatter in the data. Since there are often several correlations that are applicable, how does one choose the correlation to use? First, the engineer must determine which correlations are closest to the current situation. This

MASS TRANSFER TABLE 5-21

5-59

Mass Transfer Correlations for a Single Flat Plate or Disk—Transfer to or from Plate to Fluid Situation

A. Laminar, local, flat plate, forced flow

Laminar, average, flat plate, forced flow

Comments E = Empirical, S = Semiempirical, T = Theoretical

Correlation k′x NSh,x = ᎏ = 0.323(NRe,x)1/2(NSc)1/3 D Coefficient 0.332 is a better fit.

B. Laminar, local, flat plate, blowing or suction and forced flow

C. Laminar, local, flat plate, natural convection vertical plate

[T] Low M.T. rates. Low mass-flux, constant property systems. NSh,x is local k. Use with arithmetic difference in concentration. Coefficient 0.323 is Blasius’ approximate solution.

[100] p. 183 [108] p. 526 [146] p. 79 [150] p. 518

xu ∞ ρ NRe,x = ᎏ , x = length along plate µ

[151] p. 110

Lu ∞ ρ NRe,L = ᎏ , 0.664 (Polhausen) µ

k′m L NSh,avg = ᎏ = 0.646(NRe,L)1/2(NSc)1/3 D k′m is mean mass-transfer coefficient for dilute systems.

j-factors

References*

f jD = jH = ᎏ = 0.664(NRe,L)−1/2 2 k′x NSh,x = ᎏ = (Slope)y = 0 (NRe,x)1/2(NSc)1/3 D

k′x −1/4 1/4 NSh,x = ᎏ = 0.508 N1/2 N Gr Sc (0.952 + NSc) D

is a better fit for NSc > 0.6, NRe,x < 3 × 105. [S] Analogy. Nsc = 1.0, f = drag coefficient. jD is defined in terms of k′m.

[151] p. 271

[T] Blowing is positive. Other conditions as above. uo Re ,x ᎏᎏ N u∞ 0.6 0.5 0.25 0.0 −2.5 ᎏᎏ ᎏᎏᎏ (Slope)y = 0 0.01 0.06 0.17 0.332 1.64

[100] p. 185

[T] Low MT rates. Dilute systems, ∆ρ/ρ << 1. NGr NSc < 108. Use with arithmetic concentration difference. x = length from plate bottom.

[151] p. 120

[150] p. 271

gx 3 ρ∞ NGr = ᎏ2 ᎏ −1 (µ/ρ) ρ0



D. Laminar, stationary disk

Laminar, spinning disk

E. Laminar, inclined, plate

k′d disk 8 NSh = ᎏ = ᎏ D π k′d disk 1/3 NSh = ᎏ = 0.879N 1/2 Re N Sc D

3 2 1/3 x ρ g sinα NSh,avg = 0.783 N1/9 ᎏᎏ Re,film N Sc µ2







[T] Stagnant fluid. Use arithmetic concentration difference.

[146] p. 240

[T] Asymptotic solution for large NSc. NRe < ∼ 104 u = ωddisk/2, ω = rotational speed, rad/s. Rotating disks are often used in electrochemical research.

[117] p. 60

[T] Constant-property liquid film with low masstransfer rates. Use arithmetic concentration difference.

[151] p. 130

[146] p. 240

2/9

4Qρ NRe,film = ᎏ < 2000 µ2 k′m x NSh,avg = ᎏ D



3µQ δ film = ᎏᎏ wρg sinα

F. Turbulent, local flat plate, forced flow

Turbulent, average, flat plate, forced flow

w = width of plate, δ f = film thickness, α = angle of inclination, x = distance from start soluble surface.



1/3

= film thickness

k′x 0.8 NSh,x = ᎏ = 0.0292N Re,x , D k′L 0.8 , average coefficient NSh,avg = ᎏ = 0.0365N Re,L D

Newtonian fluid. Solute does not penetrate past region of linear velocity profile. Differences between theory and experiment.

[146] p. 209

[S] Low mass-flux with constant property system. Use with arithmetic concentration difference. NSc = 1.0, NRe,x > 105

[100] p. 191 [146] p. 201 [151] p. 221

Based on Prandtl’s 1/7-power velocity law,



y u ᎏ= ᎏ u∞ δ G. Laminar and turbulent, flat plate, forced flow

f −0.2 jD = jH = ᎏ = 0.037 N Re,L 2

1/7

Chilton-Colburn analogies, NSc = 1.0, (gases), f = drag coefficient. Corresponds to item 5-21-F and refers to same conditions. 8000 < NRe < 300,000. Can apply analogy, jD = f/2, to entire plate (including laminar portion) if average values are used.

[100] p. 193 [109] p. 112 [146] p. 201 [151] p. 271

5-60

HEAT AND MASS TRANSFER

TABLE 5-21

Mass Transfer Correlations for a Single Flat Plate or Disk—Transfer to or from Plate to Fluid (Concluded ) Situation

H. Laminar and turbulent, flat plate, forced flow

Comments E = Empirical, S = Semiempirical, T = Theoretical

Correlation 0.8 NSh,avg = 0.037N 1/3 Sc (N Re,L − 15,500) to NRe,L = 320,000

k′m L NSh,avg = ᎏ , NSc > 0.5 D Entrance effects are ignored. NRe,Cr is transition laminar to turbulent.

NSh,avg = 0.037N 1/3 Sc



References*

[E] Use arithmetic concentration difference.



0.664 1/2 0.8 0.8 × N Re,L − N Re,Cr + ᎏ N Re,Cr 0.037

[109] p. 112 [146] p. 201

in range 3 × 105 to 3 × 106. I. Turbulent, local flat plate, natural convection, vertical plate Turbulent, average, flat plate, natural convection, vertical plate

k′x 2/5 7/15 NSh,x = ᎏ = 0.0299N Gr N Sc D 2/3 −2/5 × (1 + 0.494N Sc ) 2/5 7/15 2/3 −2/5 NSh,avg = 0.0249N Gr N Sc × (1 + 0.494N Sc )

[S] Low solute concentration and low transfer rates. Use arithmetic concentration difference. gx3 ρ∞ NGr = ᎏ2 ᎏ −1 (µ/ρ) ρ0



[151] p. 225



NGr > 1010 Assumes laminar boundary layer is small fraction of total. k′m L NSh,avg = ᎏ D

J. Turbulent, vertical plate

k′m x x 3ρ 2g 2/9 NSh,avg = ᎏ = 0.327N Re,film N 1/3 Sc ᎏ µ2 D



 

Q2 δ film = 0.172 ᎏ w 2g K. Turbulent, spinning disk

L. Mass transfer to a flat plate membrane in a stirred vessel



2/9

[E] See 5-21-E for terms.

[151] p. 229

4Qρ NRe,film = ᎏ > 2360 wµ2 Solute remains in laminar sublayer.

1/3

k ′ddisk 1.1 1/3 NSh = ᎏ = 5.6N Re N Sc D

k′dtank b c NSh = ᎏ = aN Re N Sc D a depends on system. a = 0.0443 [73, 165]; b is often 0.65–0.70 [110]. If

[E] Use arithmetic concentration difference. 6 × 105 < NRe < 2 × 106 120 < NSc < 1200 u = ωddisk /2 where ω = rotational speed, radians/s. NRe = ρωd 2/2µ.

[82] [146] p. 241

[E] Use arithmetic concentration difference. ω = stirrer speed, radians/s. Useful for laboratory dialysis, R.O., U.F., and microfiltration systems.

[73] [110] p. 965 [165] p. 738

2 ωd tank ρ NRe = ᎏ µ b = 0.785 [73]. c is often 0.33 but other values have been reported [110].

*See pages 5-7 and 5-8 for references.

involves recognizing the similarity of geometries, which is often challenging, and checking that the range of parameters in the correlation is appropriate. For example, the Bravo, Rocha, and Fair correlation for distillation with structured packings with triangular cross-sectional channels (Table 5-28-H) uses the Johnstone and Pigford correlation for rectification in vertical wetted wall columns (Table 5-22-D). Recognizing that this latter correlation pertains to a rather different application and geometry was a nontrivial step in the process of developing a correlation. If several correlations appear to be applicable, check to see if the correlations have been compared to each other and to the data. When a detailed comparison of correlations is not available, the following heuristics may be useful: 1. Mass-transfer coefficients are derived from models. They must be employed in a similar model. For example, if an arithmetic concentration difference was used to determine k, that k should only be used in a mass-transfer expression with an arithmetic concentration difference. 2. Semiempirical correlations are often preferred to purely empirical or purely theoretical correlations. Purely empirical correlations are dangerous to use for extrapolation. Purely theoretical correlations may predict trends accurately, but they can be several orders of magnitude off in the value of k. 3. Correlations with broader data bases are often preferred.

4. The analogy between heat and mass transfer holds over wider ranges than the analogy between mass and momentum transfer. Good heat transfer data (without radiation) can often be used to predict mass-transfer coefficients. 5. More recent data is often preferred to older data, since end effects are better understood, the new correlation often builds on earlier data and analysis, and better measurement techniques are often available. 6. With complicated geometries, the product of the interfacial area per volume and the mass-transfer coefficient is required. Correlations of kap or of HTU are more accurate than individual correlations of k and ap since the measurements are simpler to determine the product kap or HTU. 7. Finally, if a mass-transfer coefficient looks too good to be true, it probably is incorrect. To determine the mass-transfer rate, one needs the interfacial area in addition to the mass-transfer coefficient. For the simpler geometries, determining the interfacial area is straightforward. For packed beds of particles a, the interfacial area per volume can be estimated as shown in Table 5-27-A. For packed beds in distillation, absorption, and so on in Table 5-28, the interfacial area per volume is included with the mass-transfer coefficient in the correlations for HTU. For agitated liquid-liquid systems, the interfacial area can be estimated

MASS TRANSFER

5-61

TABLE 5-22 Mass Transfer Correlations for Falling Films with a Free Surface in Wetted Wall Columns—Transfer between Gas and Liquid Situation A. Laminar, vertical wetted wall column

Comments E = Empirical, S = Semiempirical, T = Theoretical

Correlation x k′m x NSh,avg = ᎏ ≈ 3.41 ᎏ D δfilm (first term of infinite series)



3µQ δfilm = ᎏ wρg



1/3

[T] Low rates M.T. Use with log mean concentration difference. Parabolic velocity distribution in films. 4Qρ NRe,film = ᎏ < 20 wµ Derived for flat plates, used for tubes if

= film thickness

C. Turbulent, vertical wetted wall column with ripples





[E] Use with log mean concentration difference for correlations in B and C. NRe is for gas. NSc for vapor in gas. 2000 < NRe ≤ 35,000, 0.6 ≤ NSc ≤ 2.5. Use for gases, dt = tube diameter.

0.15

[E] For gas systems with rippling.





4Qρ Fits B for ᎏ = 1000 wµ





4Qρ 30 ≤ ᎏ < 1200 wµ k′m d t 0.8 1/3 NSh,avg = ᎏ = 0.023N Re N Sc D

D. Rectification in vertical wetted wall column with turbulent vapor flow, Johnstone and Pigford correlation

k′G dcol pBM 0.33 NSh,avg = ᎏᎏ = 0.0328(N R′ e) 0.77 N Sc Dv p 3000 < N R′ e < 40,000, 0.5 < NSc < 3 d col vrel ρv N′Re = ᎏ , v rel = gas velocity relative to µv 3 liquid film = ᎏ uavg in film 2

[151] p. 137 [161] p. 50

 

k′m dt 0.83 0.44 N Sc NSh,avg = ᎏ = 0.023N Re D A coefficient 0.0163 has also been reported using NRe′, where v = v of gas relative to liquid film. k′m d t 0.83 0.44 4Qρ NSh,avg = ᎏ = 0.00814N Re N Sc ᎏ D wµ

[146] p. 78

ρg 1/2 rtube ᎏ > 3.0. σ = surface tension 2σ If NRe,film > 20, surface waves and rates increase. An approximate solution Dapparent can be used. Ripples are suppressed with a wetting agent good to NRe = 1200.

w = film width (circumference in column)

B. Turbulent, vertical wetted wall column

References*

[88] p. 266 [93] [100] p.181 [146] p. 211 [151] p. 265 [159] p. 212 [161] p. 71 [88] p. 266 [106] [146] p. 213

[E] “Rounded” approximation to include ripples. Includes solid-liquid mass-transfer data to find s 0.83 coefficient on NSc. May use N Re . Use for liquids. See also Table 5-23. [E] Use logarithmic mean driving force at two ends of column. Based on four systems with gas-side resistance only. pBM = logarithmic mean partial pressure of nondiffusing species B in binary mixture. p = total pressure Modified form is used for structured packings (See Table 5-28-H).

[105] [146] p. 214

*See pages 5-7 and 5-8 for references.

from the dispersed phase holdup and mean drop size correlations. Godfrey, Obi, and Reeve [Chem. Engr. Prog. 85, 61 (Dec. 1989)] summarize these correlations. For many systems,  ddrop/dimp = −0.6 (const)NWe where NWe = ρc N 2d 3imp /σ. Effects of Total Pressure on kˆ G and kˆ L The influence of total system pressure on the rate of mass transfer from a gas to a liquid or to a solid has been shown to be the same as would be predicted from stagnant-film theory as defined in Eq. (5-285), where kˆ G = DABpT /RT δG (5-292) Since the quantity DAB pT is known to be relatively independent of the pressure, it follows that the rate coefficients kˆ G, kGyBM, and k′G pTyBM (= k′G pBM) do not depend on the total pressure of the system, subject to the limitations discussed later. Investigators of tower packings normally report k′Ga values measured at very low inlet-gas concentrations, so that yBM = 1, and at total pressures close to 100 kPa (1 atm). Thus, the correct rate coefficient for use in packed-tower designs involving the use of the driving force (y − yi)/yBM is obtained by multiplying the reported k′Ga values by the value of pT employed in the actual test unit (e.g., 100 kPa) and not the total pressure of the system to be designed.

From another point of view one can correct the reported values of k′Ga in kmol/[(s⋅m3)(kPa)], valid for a pressure of 101.3 kPa (1 atm), to some other pressure by dividing the quoted values of k′Ga by the design pressure and multiplying by 101.3 kPa, i.e., (k′Ga at design pressure pT) = (k′Ga at 1 atm) × 101.3/pT. One way to avoid a lot of confusion on this point is to convert the experimentally measured k′Ga values to values of kˆ Ga straightaway, before beginning the design calculations. A design based on the rate coefficient kˆ Ga and the driving force (y − yi)/yBM will be independent of the total system pressure with the following limitations: caution should be employed in assuming that kˆ Ga is independent of total pressure for systems having significant vapor-phase nonidealities, for systems that operate in the vicinity of the critical point, or for total pressures higher than about 3040 to 4050 kPa (30 to 40 atm). Experimental confirmations of the relative independence of kˆ G with respect to total pressure have been widely reported. Deviations do occur at extreme conditions. For example, Bretsznajder (Prediction of Transport and Other Physical Properties of Fluids, Pergamon Press, Oxford, 1971, p. 343) discusses the effects of pressure on the DABpT product and presents experimental data on the self-diffusion of CO2 which show that the D-p product begins to decrease at a pressure of

5-62

HEAT AND MASS TRANSFER

TABLE 5-23

Mass-Transfer Correlations for Flow in Pipes and Ducts—Transfer is from Wall to Fluid

Situation A. Tubes, laminar, fully developed parabolic velocity profile, developing concentration profile, constant wall concentration

Comments E = Empirical, S = Semiempirical, T = Theoretical

Correlation 0.0668(d t /x)NRe NSc k′d t NSh = ᎏ = 3.66 + ᎏᎏᎏ 1 + 0.04[(d t /x)NRe NSc]2/3 D

[T] Use log mean concentration difference. For x/d t ᎏ < 0.10, NRe < 2100. NRe NSc x = distance from tube entrance. Good agreement with experiment at values

References* [98] [100] p. 176 [108] p. 525 [151] p. 159

π d 104 > ᎏ ᎏt NReNSc > 10 4 x B. Tubes, fully developed concentration profile

k′d NSh = ᎏt = 3.66 D

[T] Subset of 5-23-A for fully developed concentration profile. x/d t ᎏ > 0.1 NRe NSc

C. Tubes, approximate solution

 

k′d d NSh,x = ᎏt = 1.077 ᎏt D x

 

E. Laminar, fully developed parabolic velocity profile, constant mass flux at wall

F. Laminar, alternate

(NRe NSc)1/3

[T] For arithmetic concentration difference.

(NRe NSc)1/3



∞

 

 

Graetz solution for heat transfer written for M.T.

11 1 ∞ exp [−λ 2j (x /rt)/(NRe NSc)] NSh, x = ᎏ − ᎏ
ᎏᎏᎏ 48 2 j = 1 Cj λ 4j



λ 2j 25.68 83.86 174.2 296.5 450.9



[T] Use arithmetic concentration difference. Fits W gas data well, for ᎏ < 50 (fit is fortuitous). Dρx NSh,avg = (k′m d t)/D. a1 = 2.405, a 2 = 5.520, a 3 = 8.654, a4 = 11.792, a 5 = 14.931. Graphical solutions are in references.

H. Vertical tubes, laminar flow, forced and natural convection

cj 7.630 × 10−3 2.058 × 10−3 0.901 × 10−3 0.487 × 10−3 0.297 × 10−3

0.023(dt /L)NRe NSc NSh = 4.36 + ᎏᎏᎏ 1 + 0.0012(dt/L)NRe NSc

[T] Use log mean concentration difference. NRe < 2100





[149] [151] p. 167

k′d NSh,x = ᎏt D vd t ρ NRe = ᎏ µ [T]

k′dt 48 NSh = ᎏ = ᎏ = 4.3636 D 11 (NGr NSc d/L)3/4 1/3 1 ⫾ 0.0742 ᎏᎏ NSh,avg = 1.62N Gz NGz

[91] p. 443 [120] [151] p. 150

−1



k′d Nsh = ᎏt D Use log mean concentration difference. NRe < 2100 G. Laminar, fully developed concentration and velocity profile

[151] p. 166

W ᎏ > 400 ρDx Leveque’s approximation: Concentration BL is thin. Assume velocity profile is linear. High mass velocity. Fits liquid data well.

1/3

−2 a 2j (x/rt) 1 − 4
a −2 j exp ᎏᎏ 1 dt NRe NSc j=1 NSh,avg = ᎏ ᎏ NRe NSc ᎏᎏᎏᎏ ∞ 2 L −2 a 2j (x/rt) −2 1 + 4
a j exp ᎏᎏ NRe NSc j=1

j 1 2 3 4 5

[151] p. 165

1/3

k′d d NSh,avg = ᎏt = 1.615 ᎏt D L

D. Tubes, laminar, uniform plug velocity, developing concentration profile, constant wall concentration

[98]

[98]

[100] p. 176

[T] Use log mean concentration difference. NRe < 2100

[98] [151] p. 167

[T] Approximate solution. Use minus sign if forced and natural convection oppose each other. NRe NSc d NGz = ᎏ L

[140]

1/3

g∆ρd 3 NGr = ᎏ ρν 2 Good agreement with experiment. I. Tubes, laminar, RO systems

J. Tubes and parallel plates, laminar RO

 

k′m d t ud 2 NSh,avg = ᎏ = 1.632 ᎏt D DL

1/3

Graphical solutions for concentration polarization. Uniform velocity through walls.

Use arithmetic concentration difference. Thin concentration polarization layer, not fully developed. NRe < 2000, L = length tube.

[73] [165] p. 738

[T]

[145] [165] p. 762

MASS TRANSFER TABLE 5-23

5-63

Mass-Transfer Correlations for Flow in Pipes and Ducts—Transfer is from Wall to Fluid (Continued ) Comments E = Empirical, S = Semiempirical, T = Theoretical

References*

Graphical solution

[T] Low transfer rates.

[151] p. 176

k′(2h) NSh = ᎏ = 7.6 D

[T] h = distance between plates. Use log mean concentration difference.

[151] p. 177

Situation K. Parallel plates, laminar, parabolic velocity, developing concentration profile, constant wall concentration L. 5-23-K, fully developed

Correlation

NRe NSc ᎏ < 20 x/(2h) M. Parallel plates, laminar, parabolic velocity, developing concentration profile, constant mass flux at wall N. 5-23-M, fully developed

O. Laminar flow, vertical parallel plates, forced and natural convection

Graphical solution

[T] Low transfer rates.

[151] p. 176

k′(2h) NSh = ᎏ = 8.23 D

[T] Use log mean concentration difference.

[151] p. 177

NRe NSc ᎏ < 20 x/(2h)



(NGr NSc h/L)3/4 NSh,avg = 1.47N 1/3 Gz 1 ⫾ 0.0989 ᎏᎏ NGz

1/3

[T] Approximate solution. Use minus sign if forced and natural convection oppose each other.

 

uH 2p k′(2Hp) NSh,avg = ᎏ = 2.354 ᎏ D DL

Q. Tubes, turbulent

k′m d t 1/3 NSh,avg = ᎏ = 0.023N 0.83 Re N Sc D

R. Tubes, turbulent

k′m d t 0.44 NSh,avg = ᎏ = 0.023N 0.83 Re N Sc D

S. Tubes, turbulent

k′d t 0.346 NSh = ᎏ = 0.0096 N 0.913 Re N Sc D

T. Tubes, turbulent, smooth tubes, Reynolds analogy

f k′d t NSh = ᎏ = ᎏ NRe NSc D 2 f = Fanning friction faction

1/3



f jD = jH = ᎏ 2 f NSh −0.2 = 0.023N Re If ᎏ = 0.023N −0.2 Re , jD = ᎏ 2 NRe N 1/3 Sc

Thin concentration polarization layer. Short tubes, concentration profile not fully developed. Use arithmetic concentration difference.

[73] [165] p. 738

[E] Use with log mean concentration difference at two ends of tube. 2100 < NRe < 35,000 0.6 < NSc < 3000 From wetted wall column and dissolution data— see Table 5-22-B. Good fit for liquids.

[88] p. 266

[E] Evaporation of liquids. Use with log mean concentration difference. See item above. Better fit for gases. 2000 < NRe < 35,000 0.6 < NSc < 2.5.

[93][100] p. 181 [109] p. 112 [146] p. 211

[E] 430 < NSc < 100,000. Dissolution data. Use for high NSc.

[122] p. 668

[T] Use arithmetic concentration difference. NSc near 1.0 Turbulent core extends to wall. Of limited utility.

[91] p. 438 [100] p. 171 [151] p. 239 [159] p. 250

[T] Use log-mean concentration difference. Relating jD to f/2 approximate. NPr and NSc near 1.0. Low concentration. Results about 20% lower than experiment. 3 × 104 < NRe < 106

[72] p. 400, 647 [80][88] p. 269 [151] p. 264 [159] p. 251

k′d t NSh = ᎏ D jD = jH = f (NRe, geometry and B.C.)

V. Tubes, turbulent, smooth tubes, constant surface concentration, Prandtl analogy *See pages 5-7 and 5-8 for references.

[140]

NRe NSc h NGz = ᎏ L g∆ρh3 NGr = ᎏ ρν 2 Good agreement with experiment.

P. Parallel plates, laminar, RO systems

U. Tubes, turbulent, smooth tubes, Chilton-Colburn analogy



k′d ( f /2)NRe NSc NSh = ᎏt = ᎏᎏ D 1 + 5f2 /(NSc − 1) f −0.25 ᎏ = 0.04 NRe 2

[100] p. 181 [120] [161] p. 72

[E] Good over wide ranges. [72] p. 647 [80] [T] Use arithmetic concentration difference. Improvement over Reynolds analogy. Best for NSc near 1.0.

[100] p. 173 [132] [151] p. 241

5-64

HEAT AND MASS TRANSFER

TABLE 5-23

Mass-Transfer Correlations for Flow in Pipes and Ducts—Transfer is from Wall to Fluid (Concluded )

Situation

Comments E = Empirical, S = Semiempirical, T = Theoretical

Correlation

References*

W. Tubes, turbulent, smooth tubes, Constant surface concentration, Von Karman analogy

( f/2)NReNSc ᎏᎏᎏᎏᎏ NSh = 5 2 (NSc − 1) + ln 1 + ᎏᎏ (NSc − 1) 1 + 5f/ 6 f −0.25 ᎏ = 0.04 N Re 2

[T] Use arithmetic concentration difference. NSh = k′dt /D. Improvement over Prandtl, NSc < 25.

[100] p. 173 [151] p. 243 [159] p. 250 [162]

X. Tubes, turbulent, smooth tubes, constant surface concentration

For 0.5 < NSc < 10:

[S] Use arithmetic concentration difference. Based on partial fluid renewal and an infrequently replenished thin fluid layer for high Nsc. Good fit to available data.

[100] p. 179 [131]







9/10 1/2 NSh,avg = 0.0097 N Re N Sc −1/3 −1/6 × (1.10 + 0.44N Sc − 0.70N Sc )

For 10 < NSc < 1000:

u bulk d t NRe = ᎏ ν

NSh,avg

k′avg d t NSh,avg = ᎏ D

9/10 1/2 −1/3 −1/6 0.0097N Re N Sc (1.10 + 0.44N Sc − 0.70N Sc ) = ᎏᎏᎏᎏᎏ 1/2 −1/3 −1/6 1 + 0.064 N Sc (1.10 + 0.44N Sc − 0.70N Sc )

For NSc > 1000: 9/10 1/3 N Sh,avg = 0.0102 N Re N Sc

Y. Turbulent flow, tubes

NSh NSh −0.12 −2/3 NSt = ᎏ =ᎏ = 0.0149N Re N Sc NPe NRe NSc

Z. Turbulent flow, noncircular ducts

Use correlations with

[E] Smooth pipe data. Data fits within 4% except at NSc > 20,000, where experimental data is underpredicted.

[124]

NSc > 100, 105 > NRe > 2100 Can be suspect for systems with sharp corners.

4 cross-sectional area d eq = ᎏᎏᎏ wetted perimeter Parallel plates:

[151] p. 289 [165] p. 738

2hw d eq = 4 ᎏ 2w + 2h *See pages 5-7 and 5-8 for references.

approximately 8100 kPa (80 atm). For reduced temperatures higher than about 1.5, the deviations are relatively modest for pressures up to the critical pressure. However, deviations are large near the critical point (see also p. 5-49). The effect of pressure on the gas-phase viscosity also is negligible for pressures below about 5060 kPa (50 atm). For the liquid-phase mass-transfer coefficient kˆ L, the effects of total system pressure can be ignored for all practical purposes. Thus, when using kˆ G and kˆ L for the design of gas absorbers or strippers, the primary pressure effects to consider will be those which affect the equilibrium curves and the values of m. If the pressure changes affect the hydrodynamics, then kˆ G, kˆ L, and a can all change significantly. Effects of Temperature on kˆ G and kˆ L The Stanton-number relationship for gas-phase mass transfer in packed beds, NSt = kˆ G /GM = g(NRe,NSc) (5-293)

With regard to the liquid-phase mass-transfer coefficient, Whitney and Vivian found that the effect of temperature upon kLa could be explained entirely by variations in the liquid-phase viscosity and diffusion coefficient with temperature. Similarly, the oxygen-desorption data of Sherwood and Holloway [Trans. Am. Inst. Chem. Eng., 36, 39 (1940)] show that the influence of temperature upon HL can be explained by the effects of temperature upon the liquid-phase viscosity and diffusion coefficients. It is important to recognize that the effects of temperature on the liquid-phase diffusion coefficients and viscosities can be very large and therefore must be carefully accounted for when using kˆ L or HL data. For liquids the mass-transfer coefficient kˆ L is correlated in terms of design variables by relations of the form NSt = kˆ L/LM = f(NRe,NSc) (5-294)

indicates that for a given system geometry the rate coefficient kˆ G depends only on the Reynolds number and the Schmidt number. Since the Schmidt number for a gas is independent of temperature, the principal effect of temperature upon kˆ G arises from changes in the gas viscosity with changes in temperature. For normally encountered temperature ranges, these effects will be small owing to the fractional powers involved in Reynolds-number terms (see Tables 5-21 to 5-28). It thus can be concluded that for all practical purposes kˆ G is independent of temperature and pressure in the normal ranges of these variables. For modest changes in temperature the influence of temperature upon the interfacial area a may be neglected. For example, in experiments on the absorption of SO2 in water, Whitney and Vivian [Chem. Eng. Prog., 45, 323 (1949)] found no appreciable effect of temperature upon k′Ga over the range from 10 to 50°C.

A general relation for HL which may be used as the basis for applying temperature corrections is as follows: a HL = bNRe N1/2 Sc

(5-295)

where b is a proportionality constant and the exponent a may range from about 0.2 to 0.5 for different packings and systems. The liquidphase diffusion coefficients may be corrected from a base temperature T1 to another temperature T2 by using the Einstein relation as recommended by Wilke [Chem. Eng. Prog., 45, 218 (1949)]: D2 = D1(T2 /T1)(µ1/µ2)

(5-296)

The Einstein relation can be rearranged to the following equation for relating Schmidt numbers at two temperatures: NSc2 = NSc1(T1 /T2)(ρ1 /ρ2)(µ2 /µ1)2

(5-297)

MASS TRANSFER TABLE 5-24

5-65

Mass Transfer Correlations for Flow Past Submerged Objects

Situation A. Single sphere

B. Single sphere, creeping flow with forced convection

Comments E = Empirical, S = Semiempirical, T = Theoretical

Correlation k′G pBLM RTd s 2r NSh = ᎏᎏ =ᎏ PD r − rs r/rs 2 5 10 50 ∞ (asymptotic limit) NSh 4.0 2.5 2.22 2.04 2.0 k′d NSh = ᎏ = [4.0 + 1.21(NRe NSc)2/3]1/2 D

k′d NSh = ᎏ = a(NRe NSc)1/3 D a = 1.01, 1.0, or 0.991 C. Single spheres, molecular diffusion, and forced convection, low flow rates

1/2 1/3 N Sh = 2.0 + AN Re N Sc A = 0.5 to 0.62

A = 0.60.

[T] Use with log mean concentration difference. r = distance from sphere, rs, ds = radius and diameter of sphere. No convection.

[151] p. 18

[T] Use with log mean concentration difference. Average over sphere. Numerical calculations. (NRe NSc) < 10,000 NRe < 1.0. Constant sphere diameter. Low mass-transfer rates.

[78][109] p. 114 [122] p. 671 [146] p. 214

[T] Fit to above ignoring molecular diffusion.

[117] p. 80

1000 < (NReNSc) < 10,000.

146 p. 215

[E] Use with log mean concentration difference. Average over sphere. Frössling Eq. (A = 0.552), 2 ≤ NRe ≤ 800, 0.6 ≤ Nsc ≤ 2.7. NSh lower than experimental at high NRe. [E] Ranz and Marshall 2 ≤ NRe ≤ 200, 0.6 ≤ Nsc ≤ 2.5. See also Table 5-27-L.

[72] [89] p. 409, 647 [100], p. 194 109 p. 114 151 p. 276 [72] p. 409, 647 [135] [146] p. 217 [151] p. 276 [90][91] p. 446 [146] p. 217 [139][151] p. 276 [102][151] p. 276

[E] Liquids 2 ≤ NRe ≤ 2,000. Graph in Ref. 146, p. 217–218. [E] 100 ≤ NRe ≤ 700; 1,200 ≤ NSc ≤ 1525.

A = 0.95. A = 0.95. A = 0.544.

References*

[E] Use with arithmetic concentration difference. NSc = 1; 50 ≤ NRe ≤ 350.

D. 5-24-C

k′d 0.35 NSh = ᎏs = 2.0 + 0.575N 1/2 Re N Sc D

[E] Use with log mean concentration difference. NSc ≤ 1, NRe < 1.

[94][151] p. 276

E. 5-24-C

k′d 0.53 1/3 NSh = ᎏs = 2.0 + 0.552N Re N Sc D

[E] Use with log mean concentration difference. 1.0 < NRe ≤ 48,000 Gases: 0.6 ≤ NSc ≤ 2.7.

[91] p. 446

[S] Correlates large amount of data and compares to published data. vr = relative velocity between fluid and sphere, m/s. CDr = drag coefficient for single particle fixed in fluid at velocity vr. See 5-27-G for calculation details and other applications.

[125]

[E] Use with arithmetic concentration difference. Liquids, 2000 < NRe < 17,000. High NSc, graph in Ref. 146, p. 217–218.

[91] p. 446 [157] [146] p. 217

k′d 0.6 1/3 NSh = ᎏs = 0.33N Re N Sc D

[E] 1500 ≤ NRe ≤ 12,000.

[151] p. 276

k′d 0.56 1/3 NSh = ᎏs = 0.43N Re N Sc D

[E] 200 ≤ NRe ≤ 4 × 104, “air” ≤ NSc ≤ “water.”

[151] p. 276

k′d 0.514 1/3 NSh = ᎏs = 0.692N Re N Sc D

[E] 500 ≤ NRe ≤ 5000.

[128] [151] p. 276

k′d 1/3 NSh = ᎏs = AN 1/2 Re N Sc , A = 0.82 D

[E] 100 < NRe ≤ 3500, NSc = 1560.

[151] p. 276

A = 0.74

[E] 120 ≤ NRe ≤ 6000, NSc = 2.44.

A = 0.582

[E] 300 ≤ NRe ≤ 7600, NSc = 1200.

[151] p. 276 [152]

jD = 0.600(N Re)−0.487

[E] Use with arithmetic concentration difference.

[151] p. 276

k′d cyl NSh = ᎏ D

50 ≤ NRe ≤ 50,000; gases, 0.6 ≤ NSc ≤ 2.6; liquids; 1000 ≤ NSc ≤ 3000. Data scatter ⫾ 30%.

[91] p. 450

F. Single spheres, forced concentration, any flow rate

E1/3d p4/3ρ k′L d s NSh = ᎏ = 2.0 + 0.59 ᎏ µ D





0.57 1/3 N Sc

Energy dissipation rate per unit mass of fluid (ranges 570 < NSc < 1420): ᎏ   ᎏ d  s m2

v 3r

CDr E= ᎏ 2

3

p

E d ρ 2 < ᎏᎏ < 63,000 µ



G. Single spheres, forced convection, high flow rates, ignoring molecular diffusion

H. Single cylinders, perpendicular flow

1/3

4/3 p



k′d 0.62 1/3 NSh = ᎏs = 0.347N Re N Sc D

5-66

HEAT AND MASS TRANSFER

TABLE 5-24

Mass Transfer Correlations for Flow Past Submerged Objects (Concluded )

Situation

Comments E = Empirical, S = Semiempirical, T = Theoretical

Correlation Can use jD = jH. Graphical correlation.

I. 5-24-H

J. Rotating cylinder in an infinite liquid, no forced flow

[E] Used with linear concentration difference.

[72] p. 408, 647

k′d s NSh = ᎏ D

[146] p. 236 [151] p. 273

k′ 0.644 −0.30 j′D = ᎏ N Sc = 0.0791N Re v

[E] Used with arithmetic concentration difference.

[85]

Results presented graphically to NRe = 241,000.

112 < NRe ≤ 100,000. 835 < NSc < 11490

[146] p. 238

vdcyl µ ω dcyl NRe = ᎏ where v = ᎏ = peripheral velocity ρ 2

K. Oblate spheroid, forced convection

References*

NSh −0.5 = 0.74 N Re jD = ᎏ NRe N 1/3 Sc

k′ = mass-transfer coefficient, cm/s; ω = rotational speed, radian/s. Useful geometry in electrochemical studies. [E] Used with arithmetic concentration difference. 120 ≤ NRe ≤ 6000; standard deviation 2.1%. Eccentricities between 1:1 (spheres) and 3:1.

total surface area dch vρ NRe = ᎏ , dch = ᎏᎏᎏ µ perimeter normal to flow

[151] p. 284 [152]

e.g., for cube with side length a, dch = 1.27a. k′d ch NSh = ᎏ D L. Other objects, including prisms, cubes, hemispheres, spheres, and cylinders; forced convection

M. Other objects, molecular diffusion limits

jD = 0.692N

Shape is often approximated by drops. [109] p. 115

v d ch ρ , N Re,p = ᎏ µ

−0.486 Re,p

[E] Used with arithmetic concentration difference.

Terms same as in 5-24-J.

k′d ch NSh = ᎏ =A D

500 ≤ N Re, p ≤ 5000. Turbulent. Agrees with cylinder and oblate spheroid results, ⫾15%. Assumes molecular diffusion and natural convection are negligible.

[127, 128] [151] p. 285

[T] Use with arithmetic concentration difference. Hard to reach limits in experiments.

[109] p. 114

[E] Use with logarithmic mean concentration difference.

[133]

 Spheres and cubes A = 2, tetrahedrons A = 26 . octahedrons 22 N. Shell side of microporous hollow fiber module for solvent extraction

0.33 NSh = β[d h(1 − ϕ)/L]N 0.6 Re N Sc

Kdh  NSh = ᎏ D β = 5.8 for hydrophobic membrane.

d h vρ N Re = ᎏ ,K  = overall mass-transfer coefficient µ

β = 6.1 for hydrophilic membrane.

dh = hydraulic diameter 4 × cross-sectional area of flow = ᎏᎏᎏᎏ wetted perimeter ϕ = packing fraction of shell side. L = module length. Based on area of contact according to inside or outside diameter of tubes depending on location of interface between aqueous and organic phases. Can also be applied to gas-liquid systems with liquid on shell side.

See Table 5-27 for flow in packed beds. *See pp. 5-7 and 5-8 for references.

Substitution of this relation into Eq. (5-295) shows that for a given geometry the effect of temperature on HL can be estimated as HL2 = HL1(T1 /T2)1/2(ρ1 /ρ2)1/2(µ2 /µ1)1 − a

(5-298)

In using these relations it should be noted that for equal liquid flow rates HL2 /HL1 = (kˆ La)1/(kˆ La)2 (5-299) ˆ L When ˆ G and k Effects of System Physical Properties on k designing packed towers for nonreacting gas-absorption systems for

which no experimental data are available, it is necessary to make corrections for differences in composition between the existing test data and the system in question. For example, the test data of Fellinger for ammonia-water absorption on various packings are frequently used as a base (see Table 5-28-B). In these tests it is estimated that HG = 0.9HOG, so that one may wish to use these data as the basis for estimating HG or kˆ Ga values for other systems. This may be done by tak0.5 −0.5 ing HG proportional to NSc and kˆ Ga proportional to NSc , based on a value of NSc for NH3-air of 0.66 at 25°C. The coefficient kG varies as the diffusivity DAB to the 0.5 power. It should be noted, however, that

MASS TRANSFER TABLE 5-25

5-67

Mass-Transfer Correlations for Drops and Bubbles

Conditions A. Single liquid drop in immiscible liquid, drop formation, discontinuous (drop) phase coefficient

Correlations ρd kˆ d,f = A ᎏ Md

   Dd ᎏ π tf

av



24 A = ᎏ (0.8624) (extension by fresh surface 7 elements) kˆ df = 0.0432 d ρd × ᎏp ᎏ tf Md

−0.334

µd ᎏᎏ d gc d pσ ρ

     

C. Single liquid drop in immiscible liquid, drop formation, continuous phase coefficient D. 5-25-C

av

0.089

uo ᎏ dp g

G. 5-25-E, continuous phase coefficient, stagnant drops, spherical

H. 5-25-E, oblate spheroid



Dc

av

f

   

F. 5-25-E

−0.601

kL,c = 0.386 ρc × ᎏ Mc

E. Single liquid drop in immiscible liquid, free rise or fall, discontinuous phase coefficient, stagnant drops

d p2 ᎏ t f Dd

ᎏ    πt

ρc kˆ cf = 4.6 ᎏ Mc

Dc ᎏ tf

av

−dp ρd kL,d,m = ᎏ ᎏ 6t M d

0.5

ρcσgc ᎏ ∆ ρgt fµ c

   0.407

gt f2 ᎏ dp

0.148

−Dd j 2π 2 t

∞

  ln ᎏπ6
ᎏj1 exp ᎏ (d /2) 

−d p ρd kˆ L,d,m = ᎏ ᎏ 6t Md

[T,S] Use arithmetic mole fraction difference.

[151] p. 399

Fits some, but not all, data. Low mass transfer rate. Md = mean molecular weight of dispersed phase; tf = formation time of drop. kL,d = mean dispersed liquid phase M.T. coefficient kmole/[s ⋅ m2 (mole fraction)].

A = 1.31 (semiempirical value)

B. 5-25-A

References*

1/2

24 A = ᎏ (penetration theory) 7



Comments E = Empirical, S = Semiempirical, T = Theoretical

2

av

2

j=1

p

2

πD t   ln 1 − ᎏ d /2 1/2 1/2 d

av

p

kL,c,m dc ρc NSh = ᎏ = 0.74 ᎏ Dc Mc

 N

(NSc)1/3

1/2 Re

av

ρc kL,c,m d3 NSh = ᎏ = 0.74 ᎏ Dc Mc

  (N

)1/2(NSc,c)1/3

Re,3

av

[E] Use arithmetic mole fraction difference. Based on 23 data points for 3 systems. Average absolute deviation 26%. Use with surface area of drop after detachment occurs. uo = velocity through nozzle; σ = interfacial tension.

[151] p. 401

[T] Use arithmetic mole fraction difference. Based on rate of bubble growth away from fixed orifice. Approximately three times too high compared to experiments.

[151] p. 402

[E] Average absolute deviation 11% for 20 data points for 3 systems.

[151] p. 402 [154] p. 434

[T] Use with log mean mole fraction differences based on ends of column. t = rise time. No continuous phase resistance. Stagnant drops are likely if drop is very viscous, quite small, or is coated with surface active agent. kL,d,m = mean dispersed liquid M.T. coefficient.

[151] p. 404 [154] p. 435

[S] See 5-25-E. Approximation for fractional extractions less than 50%.

[151] p. 404 [154] p. 435

vs d p ρc [E] NRe = ᎏ , special case Eq. (5-254). µc

[151] p. 407

vs = slip velocity between drop and continuous phase.

[152][154] p. 436

[E] Used with log mean mole fraction. Differences based on ends of extraction column; 100 measured values ⫾2% deviation. Based on area oblate spheroid.

[151] p. 285, 406, 407

[154] p. 434

vsd3ρc NRe,3 = ᎏ µc total drop surface area vs = slip velocity, d3 = ᎏᎏᎏ perimeter normal to flow I. Single liquid drop in immiscible liquid, Free rise or fall, discontinuous phase coefficient, circulating drops

J. 5-25-I



dp 3 kdr,circ = − ᎏ ln ᎏ 6θ 8

∞


B

j=1

2 j

λ j64Ddθ exp − ᎏ d p2





Eigenvalues for Circulating Drop k d d p /Dd

λ1

λ2

3.20 10.7 26.7 107 320 ∞

0.262 0.680 1.082 1.484 1.60 1.656

0.424 4.92 5.90 7.88 8.62 9.08

d p ρd kˆ L,d,circ = − ᎏ ᎏ 6θ Md

λ3

B1

B2

B3

15.7 19.5 21.3 22.2

1.49 1.49 1.49 1.39 1.31 1.29

0.107 0.300 0.495 0.603 0.583 0.596

0.205 0.384 0.391 0.386

R πD θ   ln 1 − ᎏᎏ d /2 1/2

av

1/2 1/2 d

p

[T] Use with arithmetic concentration difference.

[86][99][151] p. 405

θ = drop residence time. A more complete listing of eigenvalues is given by Refs. 86 and 99.

[161] p. 523

k′L,d,circ is m/s.

[E] Used with mole fractions for extraction less than 50%, R ≈ 2.25.

[151] p. 405

5-68

HEAT AND MASS TRANSFER

TABLE 5-25

Mass-Transfer Correlations for Drops and Bubbles (Continued )

Conditions K. 5-25-I

ˆk L,d,circ d p NSh = ᎏ Dd ρd = 31.4 ᎏ Mf

2 −0.125 d p v s ρc N Sc,d ᎏ σg c

−0.34

  ᎏ d 

L. Liquid drop in immiscible liquid, free rise or fall, continuous phase coefficient, circulating single drops

Comments E = Empirical, S = Semiempirical, T = Theoretical

Correlations

4Dd t 2 p

av



−0.37



k′L,c d p NSh,c = ᎏ Dd





d p g1/3 0.484 = 2 + 0.463N Re,drop N 0.339 ᎏ Sc,c D c2/3

1/8 K = N Re,drop

M. 5-25-L, circulating, single drop

µcvs ᎏ σg c

[E] Used with log mean mole fraction difference. dp = diameter of sphere with same volume as drop. 856 ≤ NSc ≤ 79,800, 2.34 ≤ σ ≤ 4.8 dynes/cm.

[154] p. 435 [155]

[E] Used as an arithmetic concentration difference.

[103]

d pv sρc NRe,drop = ᎏ µc

 F 0.072

F = 0.281 + 1.615K + 3.73K 2 − 1.874K µc ᎏ µd

References*

Solid sphere form with correction factor F.

   1/4

k L,c d p ρc NSh = ᎏ = 0.6 ᎏ Dc Mc

1/6

 N

1/2 Re,drop

1/2 N Sc,c

[E] Used as an arithmetic concentration difference. Low σ.

av

[151] p. 407

d pv sρc NRe,drop = ᎏ µc N. 5-25-L, circulating swarm of drops

O. Liquid drops in immiscible liquid, free rise or fall, discontinuous phase coefficient, oscillating drops

P. 5-25-O

Q. Single liquid drop in immiscible liquid, range rigid to fully circulating

ρc k L,c = 0.725 ᎏ Mc

 N

−0.43 Re,drop

−0.58 N Sc,c v s (1 − φ d)

av

k L,d,osc d p NSh = ᎏ Dd ρd = 0.32 ᎏ Md

−0.14

t   ᎏ d  4Dd 2 p

av

σ 3g c3 ρ2c 0.68 N Re,drop ᎏ gµ 4c∆ρ





0.10

0.00375v k L,d,osc = ᎏᎏs 1 + µ d /µ c

Rigid drops: 104 < NPe,c < 106

[E] Used as an arithmetic concentration difference. Low σ, disperse-phase holdup of drop swarm. φ d = volume fraction dispersed phase.

[151] p. 407 [154] p. 436

[E] Used with a log mean mole fraction difference. Based on ends of extraction column.

[151] p. 406

d pvsρc NRe,drop = ᎏ µc d p = diameter of sphere with volume of drop. Average absolute deviation from data, 10.5%. 411 ≤ NRe ≤ 3114 Low interfacial tension (3.5–5.8 dynes), µc < 1.35 centipoise.

[154] p. 435 [155]

[T] Use with log mean concentration difference. Based on end of extraction column. No continuous phase resistance. kL,d,osc in cm/s, vs = drop velocity relative to continuous phase.

[146] p. 228 [151] p. 405

[E] Allows for slight effect of wake.

[156] p. 58 [158]

[E] Used with log mean mole fraction difference. 23 data points. Average absolute deviation 25%. t f = formation time.

[151] p. 408

[E] Used with log mean mole fraction difference. 20 data points. Average absolute deviation 22%.

[151] p. 409

kcdp 0.5 0.33 NSh,c,rigid = ᎏ = 2.43 + 0.774N Re N Sc Dc 0.33 + 0.0103NReN Sc

Circulating drops: 10 < NRe < 1200, 190 < NSc < 241,000, 103 < NPe,c < 106



2 NSh,c,fully circular = ᎏ N 0.5 Pe,c π 0.5 Drops in intermediate range: NSh,c − NSh,c,rigid ᎏᎏᎏ = 1 − exp [−(4.18 × 10−3)N 0.42 Pe,c] NSh,c,fully circular − NSh,c,rigid R. Coalescing drops in immiscible liquid, discontinuous phase coefficient

S. 5-25-R, continuous phase coefficient

d p ρd kˆ d,coal = 0.173 ᎏ ᎏ t f Md

µ   ᎏ ρ D 

∆ ρgd p2 × ᎏ σg c



−1.115

d

av

d

d

v  ᎏ D  1.302

2 s f

t

0.146

d

ρ kˆ c,coal = 5.959 × 10−4 ᎏ M

 

ρdu3s

av

d p2 ρc ρd v3s

ᎏᎏ   ᎏ g µ   µ σg 

D × ᎏc tf

0.5

c

0.332

d

c

0.525

MASS TRANSFER TABLE 5-25

5-69

Mass-Transfer Correlations for Drops and Bubbles (Concluded )

Conditions T. Single liquid drops in gas, gas side coefficient

Comments E = Empirical, S = Semiempirical, T = Theoretical

Correlations kˆg Mg d p P 1/2 N 1/3 ᎏ = 2 + AN Re,g Sc,g Dgas ρg A = 0.552 or 0.60.

References*

[E] Used for spray drying (arithmetic partial pressure difference).

[88] p. 489

d pρgvs NRe,g = ᎏ , vs = slip velocity between drop µg

[111] p. 388 [135]

and gas stream. Sometimes written with: MgP ᎏ = RT ρg U. Single water drop in air, liquid side coefficient

 

DL kL = 2 ᎏ πt

1/2

, short contact times

DL k L = 10 ᎏ , long contact times dp V. Single bubbles of gas in liquid, continuous phase coefficient, very small bubbles

k′c d b NSh = ᎏ = 1.0(NReNSc)1/3 Dc

W. 5-25-V, medium to large bubbles

k′c d b NSh = ᎏ = 1.13(NReNSc)1/2 Dc

X. 5-25-W

k′c d b db NSh = ᎏ = 1.13(NReNSc)1/2 ᎏᎏ Dc 0.45 + 0.2d b

Y. Rising small bubbles of gas in liquid, continuous phase





k′c d b 1/3 NSh = ᎏ = 2 + 0.31(NGr)1/3N Sc , d b < 0.25 cm Dc

[T] Use arithmetic concentration difference. Penetration theory. t = contact time of drop. Gives plot for k G a also. Air-water system.

[111] p. 389

[T] Solid-sphere Eq. (see Table 5-24-B). d b < 0.1 cm, k′c is average over entire surface of bubble.

[122] p. 673 [146] p. 214

[T] Use arithmetic concentration difference. Droplet equation: d b > 0.5 cm.

[146] p. 231

[S] Use arithmetic concentration difference. Modification of above (W), db > 0.5 cm. 500 ≤ NRe ≤ 8000. No effect SAA for dp > 0.6 cm.

[104][146] p. 231

[E] Use with arithmetic concentration difference.

[79][91] p. 451

d |ρG − ρL|g NRa = ᎏᎏ = Raleigh number µ L DL Note that NRa = NGr NSc. Valid for single bubbles or swarms. Independent of agitation as long as bubble size is constant.

[109] p. 119 [161] p. 156

[E] Use with arithmetic concentration difference. Hg = fractional gas holdup, volume gas/total volume. For large bubbles, k′c is independent of bubble size and independent of agitation or liquid velocity. Resistance is entirely in liquid phase for most gas-liquid mass transfer.

[79][91] p. 452 [109] p. 119

3 b

Z. 5-25-Y, large bubbles

k′c d b NSh = ᎏ = 0.42 (NGr)1/3N 1/2 Sc , d b > 0.25 cm Dc 6 Hg Interfacial area ᎏᎏ = a = ᎏ volume db

[114] p. 249

See Table 5-26 for agitated systems. *See pages 5-7 and 5-8 for references.

there is conflicting evidence concerning this exponent (w versus a) as discussed by Yadav and Sharma [Chem. Eng. Sci., 34, 1423 (1979)]. The existing data indicate that kˆ La is proportional to the square root of the solute-diffusion coefficient, and since the interfacial area a does not depend on DL, it follows that kˆ L is proportional to DL0.5. An analysis of the design variables involved indicates that kˆ L should be −0.5 proportional to NSc when the Reynolds number is held constant. It should be noted that the influence of substituting solvents of widely differing viscosities upon the interfacial area a can be very large. One therefore should be cautious about extrapolating kˆ La data to account for viscosity effects between different solvent systems. ˆ G and k ˆ L As disEffects of High Solute Concentrations on k cussed previously, the stagnant-film model indicates that kˆ G should be independent of yBM and kG should be inversely proportional to yBM. The data of Vivian and Behrman [Am. Inst. Chem. Eng. J., 11, 656 (1965)] for the absorption of ammonia from an inert gas strongly suggest that the film model’s predicted trend is correct. This is another indication that the most appropriate rate coefficient to use is kˆ G and the proper driving-force term is of the form (y − yi)/yBM.

The use of the rate coefficient kˆ L and the driving force (xi − x)/xBM is believed to be appropriate. For many practical situations the liquidphase solute concentrations are low, thus making this assumption unimportant. ˆ G and k ˆ L When a chemInfluence of Chemical Reactions on k ical reaction occurs, the transfer rate may be influenced by the chemical reaction as well as by the purely physical processes of diffusion and convection within the two phases. Since this situation is common in gas absorption, gas absorption will be the focus of this discussion. One must consider the impacts of chemical equilibrium and reaction kinetics on the absorption rate in addition to accounting for the effects of gas solubility, diffusivity, and system hydrodynamics. There is no sharp dividing line between pure physical absorption and absorption controlled by the rate of a chemical reaction. Most cases fall in an intermediate range in which the rate of absorption is limited both by the resistance to diffusion and by the finite velocity of the reaction. Even in these intermediate cases the equilibria between the various diffusing species involved in the reaction may affect the rate of absorption.

5-70

HEAT AND MASS TRANSFER

TABLE 5-26

Mass-Transfer Correlations for Particles, Drops, and Bubbles in Agitated Systems Comments E = Empirical, S = Semiempirical, T = Theoretical

References*

k′LT d p 1/3 ᎏ = 2 + 0.6N 1/2 Re,T N Sc D

[S] Use log mean concentration difference. Modified Frossling equation:

[97][146] p. 220

Replace vslip with v T = terminal velocity. Calculate Stokes’ law terminal velocity

v Ts d p ρc NRe,Ts = ᎏ µc

Situation A. Solid particles suspended in agitated vessel containing vertical baffles, continuous phase coefficient

Correlation

d p2|ρ p − ρc|g v Ts = ᎏᎏ 18µ c

(Reynolds number based on Stokes’ law.) v T d p ρc NRe,T = ᎏ µc

and correct: NRe,Ts 1 v T /v Ts 0.9

10 0.65

100 0.37

1,000 0.17

10,000 0.07

100,000 0.023

Approximate: k′L = 2k′LT B. 5-26-A C. Solid, neutrally buoyant particles, continuous phase coefficient

Graphical comparisons experiments and correlations.

 

k′Ld p d imp 0.36 NSh = ᎏ = 2 + 0.47N 0.62 ᎏ Re,p N Sc D d tank

(terminal velocity Reynolds number.) k′L almost independent of d p. Harriott suggests different correction procedures. Range k′L /k′LT is 1.5 to 8.0.

[87]

[E,S] For spheres. Includes transpiration effects and changing diameters.

[78][146] p. 222

[E] Use log mean concentration difference. Density unimportant if particles are close to neutrally buoyant. [E] E = energy dissipation rate per unit mass fluid

[109] p. 115 [118] p. 132

0.17

Graphical comparisons are in Ref. 109, p. 116.

[161] p. 523

Pgc =ᎏ , P = power Vtank ρc E1/3d p4/3 NRe,p = ᎏ ν Also used for drops. Geometric effect (d imp/d tank) is usually unimportant. Ref. 118 gives a variety of references on correlations. D. 5-26-C, small particles

0.52 NSh = 2 + 0.52N Re,p N 1/3 Sc , NRe,p < 1.0





E. Solid particles with significant density difference

F. Small solid particles, gas bubbles or liquid drops, dp < 2.5 mm

k′L d p d 3p |ρp − ρ c| NSh = ᎏ = 2 + 0.31 ᎏᎏ µ cD D

G. Highly agitated systems; solid particles, drops, and bubbles; continuous phase coefficient

H. Liquid drops in baffled tank with flat six-blade turbine







 

[E] Use log mean concentration difference. NSh standard deviation 11.1%. vslip calculated by methods given in reference.

[118]

[E] Use log mean concentration difference. g = 9.80665 m/s 2. Second term RHS is free-fall or rise term. For large bubbles, see Table 5-25-Z.

[79][91] p. 451 [114] p. 249

[E] Use arithmetic concentration difference. Use when gravitational forces overcome by agitation. Up to 60% deviation. Correlation prediction is low (Ref. 118). (P/Vtank) = power dissipated by agitator per unit volume liquid.

[79][83] p. 231 [91] p. 452

[E] Use arithmetic concentration difference. Studied for five systems.

[154] p. 437

1/4

1.582

2 NRe = d imp Nρc /µ c , NOh = µ c /(ρc d impσ)1/2

1.025 N 1.929 Re N Oh

φ = volume fraction dispersed phase. N = impeller speed (revolutions/time). For dtank = htank, average absolute deviation 23.8%.

k′c d p 1/3 NSh = ᎏ = 1.237 × 10 −5 N Sc N 2/3 D ρd d p2

ᎏ  ᎏ D   σ 

5/12 d imp × N Fr ᎏ dp

[109] p. 116

1/3

(ND)1/2 k′c a = 2.621 × 10−3 ᎏ d imp d imp × φ 0.304 ᎏ d tank

I. Liquid drops in baffled tank, low volume fraction dispersed phase

0.38 N Sc



(P/Vtank)µ cg c 2/3 k′LN Sc = 0.13 ᎏᎏ ρc2

[E] Terms same as above.

1/2

d pv slip k′L d p NSh = ᎏ = 2 + 0.44 ᎏ D ν

dp

1/2

5/4

φ−1/2

tank

Stainless steel flat six-blade turbine. Tank had four baffles. Correlation recommended for φ ≤ 0.06 [Ref. 156] a = 6φ /dˆ 32, where dˆ 32 is Sauter mean diameter when 33% mass transfer has occurred.

[E] 180 runs, 9 systems, φ = 0.01. kc is timeaveraged. Use arithmetic concentration difference.







2 d impN 2 d imp NSc NRe = ᎏ , NFr = ᎏ µc g



d p = particle or drop diameter; σ = interfacial tension, N/m; φ = volume fraction dispersed phase; a = interfacial volume, 1/m; and kcαD c2/3 implies rigid drops. Negligible drop coalescence. Average absolute deviation—19.71%. Graphical comparison given by Ref. 153.

[153, 156] p. 78

MASS TRANSFER TABLE 5-26

5-71

Mass-Transfer Correlations for Particles, Drops, and Bubbles in Agitated Systems (Concluded )

Situation J. Gas bubble swarms in sparged tank reactors

Comments E = Empirical, S = Semiempirical, T = Theoretical

Correlation ν 1/3 P/VL a qG ν 1/3 b k′L a ᎏ2 =C ᎏ ᎏ ᎏ2 g ρ(νg 4)1/3 VL g Rushton turbines: C = 7.94 × 10−4, a = 0.62, b = 0.23. Intermig impellers: C = 5.89 × 10−4, a = 0.62, b = 0.19.

 



  

 

K. 5-26-J

P k′L a = 2.6 × 10−2 ᎏ VL

L. 5-26-J

P k′L a = 2.0 × 10−3 ᎏ VL

M. 5-26-J, baffled tank with standard blade Rushton impeller

P k′L a = 93.37 ᎏ VL

N. 5-26-M

d 2imp µ eff k′L a ᎏ = 7.57 ᎏ D ρD

 

 

References*

[E] Use arithmetic concentration difference. Done for biological system, O2 transfer. htank /Dtank = 2.1; P = power, kW. VL = liquid volume, m3. qG = gassing rate, m3/s. k′L a = s −1. Since a = m2/m3, ν = kinematic viscosity, m2/s. Low viscosity system. Better fit claimed with qG /VL than with uG (see 5-26-K to O).

[143]

u 0.5 G

[E] Use arithmetic concentration difference. Ion free water VL < 2.6, uG = superficial gas velocity in m/s. 500 < P/VL < 10,000. P/VL = watts/m3, VL = liquid volume, m3.

[115, 137]

u 0.2 G

[E] Use arithmetic concentration difference. Water with ions. 0.002 < VL < 4.4, 500 < P/VL < 10,000. Same definitions as 5-26-J.

[115, 117]

[E] Air-water. Same definitions as 5-26-J. 0.005 < uG < 0.025, 3.83 < N < 8.33, 400 < P/VL < 7000 h = Dtank = 0.305 or 0.610 m. VG = gas volume, m3, N = stirrer speed, rpm. Method assumes perfect liquid mixing.

[92, 115]

[E] Use arithmetic concentration difference. CO2 into aqueous carboxyl polymethylene. Same definitions as 5-26-M. µeff = effective viscosity from power law model, Pa⋅s. σ = surface tension liquid, N/m.

[115, 129]

[E] Use arithmetic concentration difference. O2 into aqueous glycerol solutions. O2 into aqueous millet jelly solutions. Same definitions as 5-26-M.

[115, 167]

[E] Use arithmetic concentration difference. Solids are glass beads, d p = 320 µm. ε s = solids holdup m3/m3 liquid, (k′L a)o = mass transfer in absence of solids. Ionic salt solution— noncoalescing.

[71, 144]

[E] Use arithmetic concentration difference. Variety of solids, d p > 150 µm (glass, amberlite, polypropylene). Tap water. Slope very different than item P. Coalescence may have occurred.

[71, 112]

0.4

0.7

0.76

u G0.45

µ  ᎏ µ



0.5

d 2impNρL × ᎏ µ eff

G

0.694

eff

u d ᎏ σ  1.11

G

0.447

d imp = impeller diameter, m; D = diffusivity, m2/s O. 5-26-M, bubbles

P. Gas bubble swarm in sparged stirred tank reactor with solids present

Q. 5-26-P



k′Lad 2imp d 2impNρ ᎏ = 0.060 ᎏ µ eff D

d N µ u ᎏ ᎏ g   σ  2 imp

2

0.19

eff

k′La ᎏ = 1 − 3.54(εs − 0.03) (k′La)o 300 ≤ P/Vrx < 10,000 W/m3, 0.03 ≤ εs ≤ 0.12 0.34 ≤ uG ≤ 4.2 cm/s, 5 < µ L < 75 Pa⋅s k′La ᎏ = 1 − εs (k′La)o

G

0.6

See also Table 5-25. *See pages 5-7 and 5-8 for references.

The gas-phase rate coefficient kˆ G is not affected by the fact that a chemical reaction is taking place in the liquid phase. If the liquidphase chemical reaction is extremely fast and irreversible, the rate of absorption may be governed completely by the resistance to diffusion in the gas phase. In this case the absorption rate may be estimated by knowing only the gas-phase rate coefficient kˆ G or else the height of one gas-phase transfer unit HG = GM /(kˆ Ga). It should be noted that the highest possible absorption rates will occur under conditions in which the liquid-phase resistance is negligible and the equilibrium back pressure of the gas over the solvent is zero. Such situations would exist, for instance, for NH3 absorption into an acid solution, for SO2 absorption into an alkali solution, for vaporization of water into air, and for H2S absorption from a dilute-gas stream into a strong alkali solution, provided there is a large excess of reagent in solution to consume all the dissolved gas. This is known as the gas-phase mass-transfer limited condition, when both the liquidphase resistance and the back pressure of the gas equal zero. Even when the reaction is sufficiently reversible to allow a small back pres-

sure, the absorption may be gas-phase-controlled, and the values of kˆ G and HG that would apply to a physical-absorption process will govern the rate. The liquid-phase rate coefficient kˆ L is strongly affected by fast chemical reactions and generally increases with increasing reaction rate. Indeed, the condition for zero liquid-phase resistance (m/kˆ L) implies that either the equilibrium back pressure is negligible, or that kˆ L is very large, or both. Frequently, even though reaction consumes the solute as it is dissolving, thereby enhancing both the mass-transfer coefficient and the driving force for absorption, the reaction rate is slow enough that the liquid-phase resistance must be taken into account. This may be due either to an insufficient supply of a second reagent or to an inherently slow chemical reaction. In any event the value of kˆ L in the presence of a chemical reaction normally is larger than the value found when only physical absorption occurs, kˆ L0 . This has led to the presentation of data on the effects of chemical reaction in terms of the “reaction factor” or “enhancement factor” defined as

5-72

HEAT AND MASS TRANSFER

TABLE 5-27

Mass Transfer Correlations for Fixed and Fluidized Beds Transfer is to or from particles.

Situation A. Heat or mass transfer in packed bed for gases and liquids

(shape factor, Ψ)

Comments E = Empirical, S = Semiempirical, T = Theoretical

Correlation −0.51 jD = jH = 0.91ΨN Re , 0.01 < NRe < 50 1/3 Equivalent NSh = 0.91ΨN 0.49 Re N Sc . −0.41 jD = jH = 0.61ΨN Re , 50 < NRe < 1000 1/3 Equivalent NSh = 0.61ΨN 0.59 Re N Sc .

particle 1.00

sphere 0.91

[E] Different constants and shape factors reported in other references. Evaluate terms at film temperature or composition. cylinder 0.81

References* [100] p. 194 [169]

k′d NSh NSh = ᎏs , jD = ᎏ NRe N 1/3 D Sc vsuper ρ NRe = ᎏ , v super = superficial velocity µΨa surface area a = ᎏᎏ = 6(1 − ε)/d p volume For spheres, dp = diameter. ar t. u .A re a For nonspherical: d p = 0.567 P Srf Results are from too-short beds—use with caution.

B. For gases, fixed and fluidized beds, Gupta and Thodos correlation

2.06 jH = jD = ᎏ , 90 ≤ NRe ≤ A εN 0.575 Re

v superd p ρ [E] For spheres. NRe = ᎏ µ

[95, 96]

Equivalent:

A = 2453 [Ref. 151], A = 4000 [Ref. 100]. For NRe > 1900, j H = 1.05j D. Heat transfer result is in absence of radiation.

[100] p. 195 [151]

2.06 0.425 1/3 N Sc NSh = ᎏ N Re ε For other shapes:

k′d NSh = ᎏs D

ε jD ᎏ = 0.79 (cylinder) or 0.71 (cube) (ε j D)sphere Graphical results are available for NRe from 1900 to 10,300. C. For gases, for fixed beds, Petrovic and Thodos correlation

0.357 0.641 1/3 NSh = ᎏ N Re N Sc ε

D. For gases and liquids, fixed and fluidized beds

0.4548 jD = ᎏ , 10 ≤ NRe ≤ 2000 εN 0.4069 Re NSh k′d , NSh = ᎏs jD = ᎏ NReN 1/3 D Sc

E. For gases, fixed beds

F. For liquids, fixed bed, Wilson and Geankoplis correlation

0.499 jD = ᎏ 0.382 εN Re 1.09 jD = ᎏ , 0.0016 < NRe < 55 2/3 εN Re 165 ≤ NSc ≤ 70,600, 0.35 < ε < 0.75 Equivalent: 1.09 1/3 NSh = ᎏ N 1/3 Re N Sc ε 0.25 jD = ᎏ , 55 < NRe < 1500, 165 ≤ NSc ≤ 10,690 εN 0.31 Re Equivalent: 0.25 0.69 1/3 N Sc NSh = ᎏ N Re ε

[E] Packed spheres, deep beds, 3 < NRe < 900 can be extrapolated to NRe < 2000. Corrected for axial dispersion with axial Peclet number = 2.0. Prediction is low at low NRe. NRe defined as in 5-27-A and B.

[130][141] p. 214 [163]

[E] Packed spheres, deep bed. Average deviation ⫾20%, NRe = dpvsuperρ/µ. Can use for fluidized beds. 10 ≤ NRe ≤ 4000.

[85][91] p. 447

[E] Data on sublimination of naphthalene spheres dispersed in inert beads. 0.1 < NRe < 100, NSc = 2.57. Correlation coefficient = 0.978.

[101]

[E] Beds of spheres,

[91] p. 448

d pVsuperρ NRe = ᎏ µ Deep beds. k′d s NSh = ᎏ D

[100] p. 195 [151] p. 287 [166]

MASS TRANSFER TABLE 5-27

5-73

Mass Transfer Correlations for Fixed and Fluidized Beds (Continued )

Situation G. For liquids, fixed beds, Ohashi et al. correlation

Comments E = Empirical, S = Semiempirical, T = Theoretical

Correlation E 1/3d 4/3 k′d p ρ NSh = ᎏs = 2 + 0.51 ᎏ µ D





References*

0.60

N 1/3 Sc

E = Energy dissipation rate per unit mass of fluid

[S] Correlates large amount of published data. Compares number of correlations, v r = relative velocity, m/s. In packed bed, v r = v super /ε.

[125]

 

v 3r = 50(1 − ε)ε 2 CDo ᎏ , m2/s 3 dp 50(1 − ε)CD = ᎏᎏ ε



ᎏ d  v 3super p

General form: α E 1/3 D 4/3 p ρ β NSh = 2 + K ᎏᎏ N Sc µ





applies to single particles, packed beds, two-phase tube flow, suspended bubble columns, and stirred tanks with different definitions of E.

CDo = single particle drag coefficient at v super cal−m culated from CDo = AN Re . i NRe 0 to 5.8 5.8 to 500 >500

A 24 10 0.44

m 1.0 0.5 0

Ranges for packed bed: 0.001 < NRe < 1000 505 < NSc < 70600 E 1/3d p4/3ρ 0.2 < ᎏᎏ < 4600 µ Compares different situations versus general correlation. See also 5-24-F.

H. For liquids, fixed and fluidized beds

1.1068 ε jD = ᎏ , 1.0 < N Re ≤ 10 N 0.72 Re NSh k′d ε jD = ᎏ , NSh = ᎏs NRe N 1/3 D Sc

I. For gases and liquids, fixed and fluidized beds, Dwivedi and Upadhyay correlation

0.765 0.365 +ᎏ ε jD = ᎏ N 0.82 N 0.386 Re Re Gases: 10 ≤ N Re ≤ 15,000. Liquids: 0.01 ≤ N Re ≤ 15,000.

[E] Spheres: d pv superρ NRe = ᎏ µ [E] Deep beds of spheres, NSh jD = ᎏ N Re N 1/3 Sc

[84][91] p. 448

[84][100] p. 196

d pv superρ k′d s NRe = ᎏ , NSh = ᎏ µ D Based on 20 gas studies and 17 liquid studies. Recommended instead of 5-27-D or F. J. For gases and liquids, fixed bed

K. For liquids, fixed and fluidized beds, Rahman and Streat correlation L. For liquids and gases, Ranz and Marshall correlation

−0.415 , 10 ≤ NRe ≤ 2500 jD = 1.17N Re

[E] Spheres:

k′ pBM 2/3 jD = ᎏ ᎏ NSc vav P

d pv superρ NRe = ᎏ µ

Comparison with other results are shown.

Variation in packing that changes ε not allowed for. Extensive data referenced. 0.5 < NSc < 15,000.

0.86 NSh = ᎏ NReN1/3 Sc , 2 ≤ NRe ≤ 25 ε

[E] Can be extrapolated to NRe = 2000. NRe = dpvsuperρ/µ. Done for neutralization of ion exchange resin.

[134]

[E] Based on freely falling, evaporating spheres (see 5-24-C). Has been applied to packed beds. Prediction is low compared to experimental data for packed beds. Limit of 2.0 at low NRe is too high. Not corrected for axial dispersion.

[135][141] p. 214 [163][168] p. 106

k′d 1/2 NSh = ᎏ = 2.0 + 0.6N 1/3 Sc N Re D d pv superρ NRe = ᎏ µ

M. For liquids and gases, Wakao and Funazkri correlation

0.6 NSh = 2.0 + 1.1N 1/3 Sc N Re , 3 < NRe < 10,000

k′film d p NSh = ᎏ t D Graphical comparison with data shown by Refs. 141, p. 215, and 163.

[146] p. 241

ρf vsuperρ [E] NRe = ᎏ µ

[141] p. 214 [163]

Correlate 20 gas studies and 16 liquid studies. Corrected for axial dispersion with:

[165] p. 376 [168] p. 106

εDaxial ᎏ = 10 + 0.5NScNRe D Daxial is axial dispersion coefficient.

5-74

HEAT AND MASS TRANSFER

TABLE 5-27

Mass Transfer Correlations for Fixed and Fluidized Beds (Concluded )

Situation

N. Liquid fluidized beds

Comments E = Empirical, S = Semiempirical, T = Theoretical

Correlation (2ξ/εm)(1 − ε)1/2 2ξ/ε m + ᎏᎏ − 2 tan h (ξ/ε m) [1 − (1 − ε)1/3]2 NSh = ᎏᎏᎏᎏᎏ ξ/ε m ᎏᎏ − tan h (ξ/ε m) 1 − (1 − ε1/2)





where α 1 1/2 − 1 ᎏ N 1/3 ξ= ᎏ Sc N Re (1 − ε)1/3 2





This simplifies to: α2

ε 1 − 2m 1 NSh = ᎏ ᎏ−1 (1 − ε)1/3 (1 − ε)1/3



O. Liquid fluidized beds

ᎏ2 N

0.306 ρ s − ρ NSh = 0.250N 0.023 ᎏ Re N Ga ρ











m = 1 for NRe > 2; m = 0.5 for NRe < 1.0; ε = voidage; α = const. Best fit data is α = 0.7. Comparison of theory and experimental ion exchange results in Ref. 113.

0.410 N Sc

(ε < 0.85)

[E] Correlate amount of data from literature. Compare large number of published correlations.

(ε > 0.85)

k′L d p d p ρvsuper NSh = ᎏ , NRe = ᎏ D µ



0.297 0.404 N Sc

0.300 0.400 N Sc

[113, 123, 138]

Vsuper d pξ NRe = ᎏ µ

2/3 N Sc (NRe < 0.1)

This can be simplified (with slight loss in accuracy at high ε) to 0.323 ρ s − ρ NSh = 0.245N Ga ᎏ ρ

k′L d p NSh = ᎏ D

Re

0.282

−0.057 0.332 ρs − ρ NSh = 0.304N Re N Ga ᎏ ρ

[S] Modification of theory to fit experimental data. For spheres, m = 1, NRe > 2.

References*

[160]

µ d p3 ρ 2g , NSc = ᎏ NGa = ᎏ µ2 ρD 1.6 < NRe < 1320, 2470 < NGa < 4.42 × 106 ρs − ρ 0.27 < ᎏ < 1.114, 305 < NSc < 1595 ρ Predicts very little dependence of NSh on velocity.

P. Liquid film flowing over solid particles with air present, trickle bed reactors, fixed bed

Q. Supercritical fluids in packed bed

kL 1/3 NSh = ᎏ = 1.8N 1/2 Re N Sc , 0.013 < NRe < 12.6 aD

L [E] NRe = ᎏ aµ

two-phases, liquid trickle, no forced flow of gas. 1/2 1/3 NSh = 0.8N Re N Sc , one-phase, liquid only.

L = superficial liquid flow rate, kg/m2s. a = surface area/col. volume, m2/m3. Irregular granules of benzoic acid, 0.29 ≤ dp ≤ 1.45 cm.



2 NSh N Re N 1/3 Sc = 0.1813 ᎏ ᎏᎏ NGr (NSc NGr)1/4



1/4



1/3 3/4 (N 1/2 Re N Sc )

2 1/3 N Re N Sc + 1.2149 ᎏ NGr

R. Supercritical fluids in packed bed





3/4

1/3 (N 1/2 NSh Re N Sc ) = 0.5265 ᎏᎏ ᎏᎏ (NSc NGr)1/4 (NSc NGr)1/4



2 N Re N 1/3 Sc + 2.48 ᎏ NGr





0.6439



− 0.01649



[E] Natural and forced convection, 4 < NRe < 135.

[119]

[E] Natural and forced convection. 0.3 < NRe < 135. Improvement of correlation in Q.

[116]

1/3

1.6808

− 0.8768

[142]

1.553

NOTE: For NRe < 3 convective contributions which are not included may become important. Use with logarithmic concentration difference (integrated form) or with arithmetic concentration difference (differential form). *See pages 5-7 and 5-8 for references.

φ = kˆ L / kˆ L0 ≥ 1 (5-300) 0 ˆ ˆ where kL = mass-transfer coefficient with reaction and kL = masstransfer coefficient for pure physical absorption. It is important to understand that when chemical reactions are involved, this definition of kˆ L is based on the driving force defined as the difference between the concentration of unreacted solute gas at the interface and in the bulk of the liquid. A coefficient based on the total of both unreacted and reacted gas could have values smaller than the physical-absorption mass-transfer coefficient kˆ L0 . When liquid-phase resistance is important, particular care should be taken in employing any given set of experimental data to ensure that the equilibrium data used conform with those employed by the original author in calculating values of kˆ L or HL. Extrapolation to widely different concentration ranges or operating conditions should

be made with caution, since the mass-transfer coefficient kˆ L may vary in an unexpected fashion, owing to changes in the apparent chemicalreaction mechanism. Generalized prediction methods for kˆ L and HL do not apply when chemical reaction occurs in the liquid phase, and therefore one must use actual operating data for the particular system in question. A discussion of the various factors to consider in designing gas absorbers and strippers when chemical reactions are involved is presented by Astarita, Savage, and Bisio, Gas Treating with Chemical Solvents, Wiley (1983) and by Kohl and Ricsenfeld, Gas Purification, 4th ed., Gulf (1985). Effective Interfacial Mass-Transfer Area a In a packed tower of constant cross-sectional area S the differential change in solute flow per unit time is given by −d(GMSy) = NAa dV = NAaS dh

(5-301)

MASS TRANSFER

5-75

TABLE 5-28 Mass Transfer Correlations for Packed Two-Phase Contactors—Absorption, Distillation, Cooling Towers, and Extractors (Packing Is Inert) Situation A. Absorption, counter-current, liquid-phase coefficient HL, Sherwood and Holloway correlation for random packings

Comments E = Empirical, S = Semiempirical, T = Theoretical

Correlations

 

L n 0.5 HL = a L ᎏ N Sc,L , L = lb/hr ft 2 µL LM HL = ᎏ kˆ L a L M = lbmoles/hr ft 2, kˆ L = lbmoles/hr ft 2, a = ft 2/ft 3, µ L in lb/(hr ft). Ranges for 5-28-B (G and L) Packing

aG

b

c

G

L

aL

References*

[E] From experiments on desorption of sparingly soluble gases from water. Graphs [Ref. 146], p. 606. Equation is dimensional. A typical value of n is 0.3 [Ref. 91] p. 633 has constants in kg, m, and s units for use in 5-28-A and B with kˆ G in kgmole/s m2 and kˆ L in kgmole/s m2 (kgmole/m3). Constants for other packings are given by Refs. 121 p. 187 and 161, p. 239.

[121] p. 187 [122] p. 714 [146] p. 606 [164] p. 660

[E] Based on ammonia-water-air data in Fellinger’s 1941 MIT thesis. Curves: Refs. 121, p. 186 and 146 p. 607. Constants given in 5-28-A. The equation is dimensional.

[91] p. 633 [121] p. 189 [146] p. 607 [164] p. 660

[E] Compared napthalene sublimination to aqueous absorption to obtain kˆ G , a, and kˆ L separately. Raschig rings and Berl saddles. d p = diameter of sphere with same surface area as packing piece. ε Lo = operating void space = ε − φ Li , where ε = void fraction w/o liquid, and φ Li = liquid holdup. Same definition as 5-28-A and B. Onda et al. correlation (5-28-D) is preferred. G = ρGv super,gas

[76][123] p. 174, 186 [151, 158] [161] p. 203

n

Raschig rings 3/8 inch 1 1 2

2.32 7.00 6.41 3.82

0.45 0.39 0.32 0.41

0.47 0.58 0.51 0.45

32.4 0.30 0.811 0.30 1.97 0.36 5.05 0.32

0.74 0.24 0.40 0.45

200–500 200–800 200–600 200–800

500–1500 0.00182 0.46 400–500 0.010 0.22 500–4500 — — 500–4500 0.0125 0.22

Berl saddles 1/2 inch 1/2 1 1.5

200–700 200–800 200–800 200–1000

500–1500 0.0067 0.28 400–4500 — — 400–4500 0.0059 0.28 400–4500 0.0062 0.28

Range for 5-28-A is 400 < L < 15,000 lb/hr ft2 B. Absorption counter-current, gasphase coefficient HG, for random packing

0.5 a G(G) bN Sc,v HG = ᎏᎏ , G = lb/hr ft 2 (L) c

GM HG = ᎏ kˆ G a G M = lbmoles/hr ft 2, kˆ G = lbmoles/hr ft 2.

C. Absorption, counter-current, gasliquid individual coefficients and interfacial area, Shulman data for random packings



2/3 dpG k GN Sc,v ᎏ = 1.195 ᎏᎏ GM µ G(1 − ε Lo)

dp L kˆ L d p ᎏ = 25.1 ᎏ DL µL

 



−0.36

0.45

N 0.5 Sc,L

Interfacial area a per volume given for Racshig rings and Berl saddles in graphical form by Refs. 78 and 121 p. 178, and in equation form by Ref. 161, p. 205. Liquid holdups are given by Refs. 161 (p. 206), 148, or 121, p. 174. D. Absorption and and distillation, counter-current, gas and liquid individual coefficients and wetted surface area, Onda et al. correlation for random packings



G k′G RT ᎏ=A ᎏ a pDG a pµ G



0.7

−2.0 N 1/3 Sc,G (a pd′p)

A = 5.23 for packing ≥ 1/2 inch (0.012 m) A = 2.0 for packing < 1/2 inch (0.012 m) k′G = lbmoles/hr ft 2 atm [kg mol/s m2 (N/m2)] ρL k L′ ᎏ µ Lg

 

1/3



L = 0.0051 ᎏ aw µL



2/3

−1/2 N Sc,L (a p d′p)0.4

[E] Gas absorption and desorption from water and organics plus vaporization of pure liquids for Raschig rings, saddles, spheres, and rods. d′p = nominal packing size, a p = dry packing surface area/volume, a w = wetted packing surface area/volume. Equations are dimensionally consistent, so any set of consistent units can be used. σ = surface tension, dynes/cm.

k′L = lbmoles/hr ft 2 (lbmoles/ft 3) [kgmoles/s m2 (kgmoles/m3)] aw ᎏ = 1 − exp ap



σ −1.45 ᎏc σ

  ᎏ aµ 

L2a p × ᎏ ρ L2 g

0.75

0.1

L

p

L

 ᎏ ρ σa  −0.05

L

L

p

0.2

Critical surface tensions, σ C = 61 (ceramic), 75 (steel), 33 (polyethylene), 40 (PVC), 56 (carbon) dynes/cm. Graphical comparison with data in Ref. 126.

L 4 < ᎏ < 400 aw µ L G 5 < ᎏ < 1000 ap µG Most data ± 20% of correlation, some ± 50%.

[76][88] p. 399 [111] p. 380 [126][159] p. 355

5-76

HEAT AND MASS TRANSFER

TABLE 5-28 Mass Transfer Correlations for Packed Two-Phase Contactors—Absorption, Distillation, Cooling Towers, and Extractors (Packing Is Inert) (Continued ) Situation E. Distillation and absorption, counter-current, random packings, modification of Onda correlation, Bravo and Fair correlation

Comments E = Empirical, S = Semiempirical, T = Theoretical

Correlations Use Onda’s correlations (5-28-D) for k′G and k′L. Calculate: G L HG = ᎏ , HL = ᎏ k G′ aePMG k′LaeρL HOG = HG + λHL where m λ=ᎏ LM/GM Using

References*

[E] Use’s Bolles & Fair (Ref. 75) data base to determine new effective area ae to use with Onda et al. (Ref. 126) correlation. Same definitions as 5-28-D. P = total pressure, atm; MG = gas, molecular weight; m = local slope of equilibrium curve; LM/GM = slope operating line; Z = height of packing in feet. Equation for ae is dimensional. Fit to data for effective area quite good for distillation. Good for absorption at low values of (Nca,L × NRe,G), but correlation is too high at higher values of (NCa,L × NRe,G).

[76]

[E] Based on oxygen transfer from water to air 77°F. Liquid film resistance controls. (Dwater @ 77°F = 2.4 × 10−5). Equation is dimensional. Data was for thin-walled polyethylene Raschig rings. Correlation also fit data for spheres. Fit ⫾25%. See Reiss for graph.

[136] [142] p. 217

σ0.5 ae = 0.498ap ᎏ (NCa,LNRe,G)0.392 Z0.4 where

 

6G NRe,G = ᎏ , apµG LµL NCa,L = ᎏ (dimensionless) ρLσgc F. Absorption, co-current downward flow, random packings

Air-oxygen-water results correlated by k′La = 0.12EL0.5. Extended to other systems.



DL k′La = 0.12EL0.5 ᎏ 2.4 × 105



0.5

∆p EL = ᎏ vL ∆L 2-phase −1 k′La = s DL = cm/s EL = ft, lbf/s ft3 vL = superficial liquid velocity, ft/s

 

∆p ᎏ = pressure loss in two-phase flow = lbf/ft2 ft ∆L k′G a = 2.0 + 0.91EG2/3 for NH3 ∆p Eg = ᎏ vg ∆L 2-phase vg = superficial gas velocity, ft/s.

 

G. Absorption, stripping, distillation, counter-current, HL, and HG, random packings, Cornell et al. correlation, and Bolles and Fair correlation

For Raschig rings, Berl saddles, and spiral tile: Z 0.15 φCflood 0.5 HL = ᎏ N Sc,L ᎏ 3.28 3.05 Cflood = 1.0 if below 40% flood—otherwise, use Fig. 5-28. φ shown in Fig. 5-29 for different packings and sizes. Range 0.02 < φ < 0.300. Aψ(d′col)mZ0.33N0.5 Sc,G ᎏᎏᎏᎏ HG = µL 0.16 ρwater 1.25 σwater 0.8 n L ᎏᎏ ᎏᎏ ᎏᎏ µwater ρL σL

 



 

 

[E] Ammonia absorption into water from air at 70°F. Gas-film resistance controls. Thin-walled polyethylene Raschig rings and 1-inch Intalox saddles. Fit ⫾25%. See Reiss for fit. Terms defined as above. [E] Z = packed height, m of each section with its own liquid distribution. The original work is reported in English units. Cornell et al. (Ref. 81) review early literature. Improved fit of Cornell’s φ values given by Bolles and Fair (Refs. 74 and 75) and in Fig. 5-29.



A = 0.017 (rings) or 0.029 (saddles) d′col = column diameter in m (if diameter > 0.6 m, use d′col = 0.6) m = 1.24 (rings) or 1.11 (saddles) n = 0.6 (rings) or 0.5 (saddles) ψ is given in Fig. 5-30. Range: 25 < ψ < 190 m.

L = liquid rate, kg/(sm2), µwater = 1.0 Pa ⋅ s, ρwater = 1000 kg/m3, σwater = 72.8 mN/m (72.8 dynes/cm). HG and HL will vary from location to location. Design each section of packing separately.

[136]

[74, 75, 81] [100] p. 428 [111] p. 381 [151] p. 353 [164] p. 651

MASS TRANSFER

5-77

TABLE 5-28 Mass Transfer Correlations for Packed Two-Phase Contactors—Absorption, Distillation, Cooling Towers, and Extractors (Packing Is Inert) (Concluded ) Situation H. Distillation and absorption. Counter-current flow. Structured packings. Gauze-type with triangular flow channels, Bravo, Rocha, and Fair correlation

Correlations

[T] Check of 132 data points showed average deviation 14.6% from theory. Johnstone and Pigford [Ref. 105] correlation (5-22-D) has exponent on NRe rounded to 0.8. Assume gauze packing is completely wet. Thus, aeff = ap to calculate HG and HL. Same approach may be used generally applicable to sheet-metal packings, but they will not be completely wet and need to estimate transfer area. L = liquid flux, kg/s m2. G = vapor flux, kg/s m2. Fit to data shown in Ref. 77.

Equivalent channel:



1 1 deq = Bh ᎏ + ᎏ B + 2S 2S

Comments E = Empirical, S = Semiempirical, T = Theoretical



Use modified correlation for wetted wall column (See 5-22-D) k′vdeq 0.333 NSh,v = ᎏ = 0.0338N0.8 Re,vNSc,v Dv

References* [77] [87] p. 310, 326 [159] p. 356, 362

G L HG = ᎏ , HL = ᎏ k′vapρv k′LapρL

deqρv(Uv,eff + UL,eff) NRe,v = ᎏᎏ µv where effective velocities Uv,super Uv,eff = ᎏ ε sin θ 3Γ ρL2 g UL,eff = ᎏ ᎏ 2ρL 3µLΓ





0.333

L ,Γ=ᎏ Per

Perimeter 4S + 2B Per = ᎏᎏ = ᎏ Area Bh Calculate k′L from penetration model (use time for liquid to flow distance s). k′L = 2(DLUL,eff /πS)1/2. I. High-voidage packings, cooling towers, splash-grid packings

J. Liquid-liquid extraction, packed towers

K. Liquid-liquid extraction in Rotating-disc contactor (RDC)

(Ka)HVtower L −n′ ᎏᎏ = 0.07 + A′N′ ᎏ L Ga A′ and n′ depend on deck type (Ref. 107), 0.060 ≤ A′ ≤ 0.135, 0.46 ≤ n′ ≤ 0.62. General form fits the graphical comparisons (Refs. 146 and 164).

 



lb water lb/(h)(ft3) ᎏ lb dry air Vtower = tower volume, ft3/ft2. If normal packings are used, use absorption masstransfer correlations or Ref. 88, p. 452. [156] p. 79

[70][156] p. 79

kd,RDC N H ᎏ = 1.0 + 1.825 ᎏ ᎏ kd NCr Dtank

kc, kd are for drops (Table 5-25) N = impeller speed Breakage occurs when N > NCr. Maximum enhancement before breakage was factor of 2.0. H = compartment height, Dtank = tank diameter, σ = interfacial tension, N/m. Done in 0.152 and 0.600 m RDC.

See Table 5-26-F, G, H, and I.

[E]

 

kc,RDC N ᎏ = 1.0 + 2.44 ᎏ kc NCr



2.5

ᎏ D  H

tank

 

*See pages 5-7 and 5-8 for references.



[107][121] p. 220 [146] p. 286 [164] p. 681

[E]

Use k values for drops (Table 5-25). Enhancement due to packing is at most 20%. Packing decreases drop size and increases interfacial area.

σ NCr = 7.6 × 10−4 ᎏ ddrop µc

L. Liquid-liquid extraction, stirred tanks

[E] General form. Ga = lb dry air/hr ft2. L = lb/h ft2, N′ = number of deck levels. (Ka)H = overall enthalpy transfer coefficient =

5-78

HEAT AND MASS TRANSFER

Liquid-film correction factor (Table 5-28-G) for operation at high percent of flood. [Cornell et al., Chem. Eng. Prog., 56(8), 68 (1960).]

FIG. 5-28

where a = interfacial area effective for mass transfer per unit of packed volume and V = packed volume. Owing to incomplete wetting of the packing surfaces and to the formation of areas of stagnation in the liquid film, the effective area normally is significantly less than the total external area of the packing pieces. The effective interfacial area depends on a number of factors, as discussed in a review by Charpentier [Chem. Eng. J., 11, 161 (1976)]. Among these factors are (1) the shape and size of packing, (2) the packing material (for example, plastic generally gives smaller interfacial areas than either metal or ceramic), (3) the liquid mass velocity, and (4), for small-diameter towers, the column diameter. Whereas the interfacial area generally increases with increasing liquid rate, it apparently is relatively independent of the superficial gas mass velocity below the flooding point. According to Charpentier’s review, it appears valid to assume that the interfacial area is independent of the column height when specified in terms of unit packed volume (i.e., as a). Also, the existing data for chemically reacting gas-liquid systems (mostly aqueous electrolyte solutions) indicate that the interfacial area is independent of the chemical system. However, this situation may not hold true for systems involving large heats of reaction. Rizzuti et al. [Chem. Eng. Sci., 36, 973 (1981)] examined the influence of solvent viscosity upon the effective interfacial area in packed columns and concluded that for the systems studied the effective interfacial area a was proportional to the kinematic viscosity raised to the 0.7 power. Thus, the hydrodynamic behavior of a packed absorber is strongly affected by viscosity effects. Surface-tension effects also are important, as expressed in the work of Onda et al. (see Table 5-28-D). In developing correlations for the mass-transfer coefficients kˆ G and kˆ L, the various authors have assumed different but internally compatible correlations for the effective interfacial area a. It therefore would be inappropriate to mix the correlations of different authors unless it has been demonstrated that there is a valid area of overlap between them. Volumetric Mass-Transfer Coefficients Kˆ Ga and Kˆ La Experimental determinations of the individual mass-transfer coefficients kˆ G and kˆ L and of the effective interfacial area a involve the use of extremely difficult techniques, and therefore such data are not plentiful. More often, column experimental data are reported in terms of overall volumetric coefficients, which normally are defined as follows: and

K′Ga = nA /(hTSpT ∆y°1m)

(5-302)

KLa = nA /(hTS ∆x°1m)

(5-303)

where K′Ga = overall volumetric gas-phase mass-transfer coefficient, KLa = overall volumetric liquid-phase mass-transfer coefficient, nA =

Hl correlation for various packings (Table 5-28-G). To convert meters to feet, multiply by 3.281; to convert pounds per hour-square foot to kilograms per second-square meter, multiply by 0.001356; and to convert millimeters to inches, multiply by 0.0394. [Bolles and Fair, Inst. Chem. Eng. Symp. Ser., no. 56, 3.3/35 (1969).]

FIG. 5-29

overall rate of transfer of solute A, hT = total packed depth in tower, S = tower cross-sectional area, pT = total system pressure employed during the experiment, and ∆x°1m and ∆y°1m are defined as (y − y°)1 − (y − y°)2 ∆y°1m = ᎏᎏᎏ ln [(y − y°)1/(y − y°)2]

(5-304)

MASS TRANSFER

5-79

vary with the total system pressure except when the liquid-phase resistance is negligible (i.e., when either m = 0, or kˆ La is very large, or both). Extrapolation of KGa data for absorption and stripping to conditions other than those for which the original measurements were made can be extremely risky, especially in systems involving chemical reactions in the liquid phase. One therefore would be wise to restrict the use of overall volumetric mass-transfer-coefficient data to conditions not too far removed from those employed in the actual tests. The most reliable data for this purpose would be those obtained from an operating commercial unit of similar design. Experimental values of HOG and HOL for a number of distillation systems of commercial interest are also readily available. Extrapolation of the data or the correlations to conditions that differ significantly from those used for the original experiments is risky. For example, pressure has a major effect on vapor density and thus can affect the hydrodynamics significantly. Changes in flow patterns affect both masstransfer coefficients and interfacial area. Chilton-Colburn Analogy When a fluid moves over either a liquid or a solid surface, the eddy motion that causes mass transfer also causes heat transfer and fluid friction owing to the transfer of thermal energy and momentum respectively. This close similarity among the mechanisms for the transfer of mass, heat, and momentum was brought out in the Reynolds analogy (see Table 5-23-T), which stated that the following dimensionless ratios are equal: kˆ G/GM = h′/cpG = f/2 (5-306) where h′ = heat-transfer coefficient, cp = specific heat, G = mass flux, and f = friction factor. Experimental data for mass transfer into gas streams agree approximately with Eq. (5-306) when the Schmidt number is close to unity and in smooth, straight tubes or along flat plates when the pressure drop is due entirely to skin friction against the surface. It does not, however, agree for cases involving “form” drag as well as skin friction. Also, it does not account for the mass-transfer resistance of the region of fluid near the liquid or solid boundary in which mass transfer occurs principally by molecular (as opposed to turbulent) motion. Colburn [Trans. Am. Inst. Chem. Eng., 29, 174 (1933)] and Chilton and Colburn [Ind. Eng. Chem., 26, 1183 (1934)] showed empirically that the resistance of the laminar sublayer can be expressed by the following modification of the Reynolds analogy: ( kˆ G/GM)NSc2/3 = jM = (h′/cpG)NPr2/3 = jH = f/2 (5-307) Hg correlation for various packings (Table 5-28-G). To convert meters to feet, multiply by 3.281; to convert millimeters to inches, multiply by 0.03937. [Bolles and Fair, Inst. Chem. Eng. Symp. Ser., no. 56, 3.3/35 (1979).] FIG. 5-30

(x° − x)2 − (x° − x)1 ∆x°1m = ᎏᎏᎏ (5-305) ln [(x° − x)2/(x° − x)1] where subscripts 1 and 2 refer to the bottom and top of the tower respectively. Experimental K′Ga and KLa data are available for most absorption and stripping operations of commercial interest (see Sec. 15). The solute concentrations employed in these experiments normally are very low, so that KLa ⬟ Kˆ La and K′GapT ⬟ Kˆ Ga, where pT is the total pressure employed in the actual experimental-test system. Unlike the individual gas-film coefficient kˆ Ga, the overall coefficient Kˆ Ga will and

for turbulent flow through straight tubes (see Table 5-23-U) and across plane surfaces (see Table 5-21-G), and jM = jH ≤ f/2

(5-308)

for turbulent flow around cylinders (see Table 5-24-I), where jM = mass-transfer factor, jH = heat-transfer factor, NPr = cp µ/k = Prandtl number, and k = thermal conductivity; other symbols are as defined earlier. On occasion one will find that heat-transfer-rate data are available for a system in which mass-transfer-rate data are not readily available. The Chilton-Colburn analogy provides a procedure for developing estimates of the mass-transfer rates based on heat-transfer data. Extrapolation of experimental jM or jH data obtained with gases to predict liquid systems (and vice versa) should be approached with caution, however. When pressure-drop or friction-factor data are available, one may be able to place an upper bound on the rates of heat and mass transfer, according to Eq. (5-308).

Section 6

Fluid and Particle Dynamics*

James N. Tilton, Ph.D., P.E., Senior Consultant, Process Engineering, E. I. du Pont de Nemours & Co.; Member, American Institute of Chemical Engineers; Registered Professional Engineer (Delaware)

FLUID DYNAMICS Nature of Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Deformation and Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Viscosity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rheology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kinematics of Fluid Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Compressible and Incompressible Flow . . . . . . . . . . . . . . . . . . . . . . . Streamlines, Pathlines, and Streaklines . . . . . . . . . . . . . . . . . . . . . . . . One-dimensional Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rate of Deformation Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vorticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Laminar and Turbulent Flow, Reynolds Number . . . . . . . . . . . . . . . . Conservation Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Macroscopic and Microscopic Balances . . . . . . . . . . . . . . . . . . . . . . . Macroscopic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mass Balance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Momentum Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Total Energy Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mechanical Energy Balance, Bernoulli Equation . . . . . . . . . . . . . . . . Microscopic Balance Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mass Balance, Continuity Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . Stress Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cauchy Momentum and Navier-Stokes Equations . . . . . . . . . . . . . . . Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 1: Force Exerted on a Reducing Bend. . . . . . . . . . . . . . . . . Example 2: Simplified Ejector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 3: Venturi Flowmeter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 4: Plane Poiseuille Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . Incompressible Flow in Pipes and Channels. . . . . . . . . . . . . . . . . . . . . . Mechanical Energy Balance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Friction Factor and Reynolds Number . . . . . . . . . . . . . . . . . . . . . . . . Laminar and Turbulent Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Velocity Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Entrance and Exit Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Residence Time Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Noncircular Channels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nonisothermal Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Open Channel Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Non-Newtonian Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Economic Pipe Diameter, Turbulent Flow . . . . . . . . . . . . . . . . . . . . .

6-4 6-4 6-4 6-4 6-5 6-5 6-5 6-5 6-5 6-5 6-5 6-6 6-6 6-6 6-6 6-6 6-6 6-7 6-7 6-7 6-7 6-7 6-8 6-8 6-8 6-8 6-9 6-9 6-9 6-9 6-9 6-10 6-11 6-11 6-11 6-12 6-12 6-12 6-13 6-14

Economic Pipe Diameter, Laminar Flow . . . . . . . . . . . . . . . . . . . . . . Vacuum Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Molecular Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Slip Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Frictional Losses in Pipeline Elements . . . . . . . . . . . . . . . . . . . . . . . . . . Equivalent Length and Velocity Head Methods . . . . . . . . . . . . . . . . . Contraction and Entrance Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 5: Entrance Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Expansion and Exit Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fittings and Valves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 6: Losses with Fittings and Valves . . . . . . . . . . . . . . . . . . . . Curved Pipes and Coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Screens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jet Behavior. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flow through Orifices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Compressible Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mach Number and Speed of Sound . . . . . . . . . . . . . . . . . . . . . . . . . . Isothermal Gas Flow in Pipes and Channels. . . . . . . . . . . . . . . . . . . . Adiabatic Frictionless Nozzle Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 7: Flow through Frictionless Nozzle . . . . . . . . . . . . . . . . . . Adiabatic Flow with Friction in a Duct of Constant Cross Section. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 8: Compressible Flow with Friction Losses . . . . . . . . . . . . . Convergent/Divergent Nozzles (De Laval Nozzles) . . . . . . . . . . . . . . Multiphase Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Liquids and Gases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gases and Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fluid Distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Perforated-Pipe Distributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 9: Pipe Distributor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Slot Distributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Turning Vanes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Perforated Plates and Screens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Beds of Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Other Flow Straightening Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . Fluid Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stirred Tank Agitation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pipeline Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tube Banks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Turbulent Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Transition Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6-14 6-14 6-15 6-15 6-16 6-16 6-16 6-16 6-17 6-17 6-17 6-18 6-19 6-20 6-21 6-22 6-22 6-22 6-22 6-23 6-23 6-25 6-25 6-26 6-26 6-29 6-32 6-32 6-33 6-33 6-33 6-33 6-34 6-34 6-34 6-34 6-35 6-36 6-36 6-38

* The author acknowledges the contribution of B. C. Sakiadis, editor of this section in the sixth edition of the Handbook. 6-1

6-2

FLUID AND PARTICLE DYNAMICS

Laminar Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Beds of Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fixed Beds of Granular Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Porous Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tower Packings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fluidized Beds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Boundary Layer Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flat Plate, Zero Angle of Incidence . . . . . . . . . . . . . . . . . . . . . . . . . . . Cylindrical Boundary Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Continuous Flat Surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Continuous Cylindrical Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vortex Shedding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Coating Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Falling Films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Minimum Wetting Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Laminar Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Turbulent Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effect of Surface Traction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flooding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hydraulic Transients. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Water Hammer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6-38 6-38 6-38 6-39 6-40 6-40 6-40 6-40 6-40 6-40 6-41 6-41 6-42 6-42 6-42 6-42 6-43 6-43 6-43 6-44 6-44

Example 10: Response to Instantaneous Valve Closing . . . . . . . . . . . Pulsating Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cavitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Time Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Closure Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Eddy Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Computational Fluid Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dimensionless Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6-44 6-44 6-44 6-45 6-45 6-46 6-46 6-47 6-48

PARTICLE DYNAMICS Drag Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Terminal Settling Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spherical Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nonspherical Rigid Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hindered Settling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Time-dependent Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gas Bubbles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Liquid Drops in Liquids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Liquid Drops in Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wall Effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6-50 6-50 6-50 6-51 6-52 6-52 6-53 6-53 6-54 6-54

Nomenclature and Units* In this listing, symbols used in this section are defined in a general way and appropriate SI units are given. Specific definitions, as denoted by subscripts, are stated at the place of application in the section. Some specialized symbols used in the section are defined only at the place of application. Some symbols have more than one definition; the appropriate one is identified at the place of application. Symbol

Definition

SI units

U.S. customary units

a A b b c cf C Ca C0 CD d D De Dij

Pressure wave velocity Area Wall thickness Channel width Acoustic velocity Friction coefficient Conductance Capillary number Discharge coefficient Drag coefficient Diameter Diameter Dean number Deformation rate tensor components Elastic modulus Energy dissipation rate Eotvos number Fanning friction factor Vortex shedding frequency Force Cumulative residence time distribution Froude number Acceleration of gravity Mass flux Enthalpy per unit mass Liquid depth Ratio of specific heats Kinetic energy of turbulence Power law coefficient Viscous losses per unit mass Length Mass flow rate Mass Mach number Morton number Molecular weight Power law exponent Blend time number Best number Power number Pumping number Pressure Entrained flow rate Volumetric flow rate Throughput (vacuum flow) Heat input per unit mass Radial coordinate Radius Ideal gas universal constant Volume fraction of phase i Reynolds number Density ratio

m/s m2 m m m/s Dimensionless m3/s Dimensionless Dimensionless Dimensionless m m Dimensionless 1/s

ft/s ft2 in ft ft/s Dimensionless ft3/s Dimensionless Dimensionless Dimensionless ft ft Dimensionless 1/s

Pa J/s Dimensionless Dimensionless 1/s N Dimensionless

lbf/in2 ft ⋅ lbf/s Dimensionless Dimensionless 1/s lbf Dimensionless

Dimensionless m/s2 kg/(m2 ⋅ s) J/kg m Dimensionless J/kg kg/(m ⋅ s2 − n) J/kg m kg/s kg Dimensionless Dimensionless kg/kgmole Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Pa m3/s m3/s Pa ⋅ m3/s J/kg m m J/(kgmole ⋅ K) Dimensionless Dimensionless Dimensionless

Dimensionless ft/s2 lbm/(ft2 ⋅ s) Btu/lbm ft Dimensionless ft ⋅ lbf/lbm lbm/(ft ⋅ s2 − n) ft ⋅ lbf/lbm ft lbm/s lbm Dimensionless Dimensionless lbm/lbmole Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless lbf/in2 ft3/s ft3/s lbf ⋅ ft3/s Btu/lbm ft ft Btu/(lbmole ⋅ R) Dimensionless Dimensionless Dimensionless

E E˙ v Eo f f F F Fr g G h h k k K lv L m ˙ M M M Mw n Nb ND NP NQ p q Q Q δQ r R R Ri Re s

Symbol

Definition

SI units

U.S. customary units

s S S S St t t T u u U v V V We ˙s W δWs x y z z

Entropy per unit mass Slope Pumping speed Surface area per unit volume Strouhal number Time Force per unit area Absolute temperature Internal energy per unit mass Velocity Velocity Velocity Velocity Volume Weber number Rate of shaft work Shaft work per unit mass Cartesian coordinate Cartesian coordinate Cartesian coordinate Elevation

J/(kg ⋅ K) Dimensionless m3/s l/m Dimensionless s Pa K J/kg m/s m/s m/s m/s m3 Dimensionless J/s J/kg m m m m

Btu/(lbm ⋅ R) Dimensionless ft3/s l/ft Dimensionless s lbf/in2 R Btu/lbm ft/s ft/s ft/s ft/s ft3 Dimensionless Btu/s Btu/lbm ft ft ft ft

α α β β γ˙ Γ

Velocity profile factor Included angle Velocity profile factor Bulk modulus of elasticity Shear rate Mass flow rate per unit width Boundary layer or film thickness Kronecker delta Pipe roughness Void fraction Turbulent dissipation rate Residence time Angle Mean free path Viscosity Kinematic viscosity Density Surface tension Cavitation number Components of total stress tensor Shear stress Time period Components of deviatoric stress tensor Energy dissipation rate per unit volume Angle of inclination Vorticity

Greek symbols

δ δij


θ θ λ µ ν ρ σ σ σij τ τ τij Φ φ ω

Dimensionless Radians Dimensionless Pa l/s kg/(m ⋅ s)

Dimensionless Radians Dimensionless lbf/in2 l/s lbm/(ft ⋅ s)

m

ft

Dimensionless m Dimensionless J/(kg ⋅ s) s Radians m Pa ⋅ s m2/s kg/m3 N/m Dimensionless Pa

Dimensionless ft Dimensionless ft ⋅ lbf/(lbm ⋅ s) s Radians ft lbm/(ft ⋅ s) ft2/s lbm/ft3 lbf/ft Dimensionless lbf/in2

Pa s Pa

lbf/in2 s lbf/in2

J/(m3 ⋅ s)

ft ⋅ lbf/(ft3 ⋅ s)

Radians 1/s

Radians 1/s

* Note that with U.S. Customary units, the conversion factor gc may be required to make equations in this section dimensionally consistent; gc = 32.17 (lbm⋅ft)/lbf⋅s2). 6-3

6-4

FLUID AND PARTICLE DYNAMICS

FLUID DYNAMICS GENERAL REFERENCES: Batchelor, An Introduction to Fluid Dynamics, Cambridge University, Cambridge, 1967; Bird, Stewart, and Lightfoot, Transport Phenomena, Wiley, New York, 1960; Brodkey, The Phenomena of Fluid Motions, Addison-Wesley, Reading, Mass., 1967; Denn, Process Fluid Mechanics, Prentice-Hall, Englewood Cliffs, N.J., 1980; Landau and Lifshitz, Fluid Mechanics, 2d ed., Pergamon, 1987; Govier and Aziz, The Flow of Complex Mixtures in Pipes, Van Nostrand Reinhold, New York, 1972, Krieger, Huntington, N.Y., 1977; Panton, Incompressible Flow, Wiley, New York, 1984; Schlichting, Boundary Layer Theory, 8th ed., McGraw-Hill, New York, 1987; Shames, Mechanics of Fluids, 3d ed., McGraw-Hill, New York, 1992; Streeter, Handbook of Fluid Dynamics, McGraw-Hill, New York, 1971; Streeter and Wylie, Fluid Mechanics, 8th ed., McGraw-Hill, New York, 1985; Vennard and Street, Elementary Fluid Mechanics, 5th ed., Wiley, New York, 1975; Whitaker, Introduction to Fluid Mechanics, Prentice-Hall, Englewood Cliffs, N.J., 1968, Krieger, Huntington, N.Y., 1981.

NATURE OF FLUIDS Deformation and Stress A fluid is a substance which undergoes continuous deformation when subjected to a shear stress. Figure 6-1 illustrates this concept. A fluid is bounded by two large parallel plates, of area A, separated by a small distance H. The bottom plate is held fixed. Application of a force F to the upper plate causes it to move at a velocity U. The fluid continues to deform as long as the force is applied, unlike a solid, which would undergo only a finite deformation.

A

F V

y

H

Fluids without any solidlike elastic behavior do not undergo any reverse deformation when shear stress is removed, and are called purely viscous fluids. The shear stress depends only on the rate of deformation, and not on the extent of deformation (strain). Those which exhibit both viscous and elastic properties are called viscoelastic fluids. Purely viscous fluids are further classified into time-independent and time-dependent fluids. For time-independent fluids, the shear stress depends only on the instantaneous shear rate. The shear stress for time-dependent fluids depends on the past history of the rate of deformation, as a result of structure or orientation buildup or breakdown during deformation. A rheogram is a plot of shear stress versus shear rate for a fluid in simple shear flow, such as that in Fig. 6-1. Rheograms for several types of time-independent fluids are shown in Fig. 6-2. The Newtonian fluid rheogram is a straight line passing through the origin. The slope of the line is the viscosity. For a Newtonian fluid, the viscosity is independent of shear rate, and may depend only on temperature and perhaps pressure. By far, the Newtonian fluid is the largest class of fluid of engineering importance. Gases and low molecular weight liquids are generally Newtonian. Newton’s law of viscosity is a rearrangement of Eq. (6-1) in which the viscosity is a constant: du τ = µγ˙ = µ  (6-2) dy All fluids for which the viscosity varies with shear rate are nonNewtonian fluids. For non-Newtonian fluids the viscosity, defined as the ratio of shear stress to shear rate, is often called the apparent viscosity to emphasize the distinction from Newtonian behavior. Purely viscous, time-independent fluids, for which the apparent viscosity may be expressed as a function of shear rate, are called generalized Newtonian fluids.

x Deformation of a fluid subjected to a shear stress.

The SI units of viscosity are kg/(m ⋅ s) or Pa ⋅ s (pascal second). The cgs unit for viscosity is the poise; 1 Pa ⋅ s equals 10 poise or 1000 centipoise (cP) or 0.672 lbm/(ft ⋅ s). The terms absolute viscosity and shear viscosity are synonymous with the viscosity as used in Eq. (6-1). Kinematic viscosity ν ⬅ µ/ρ is the ratio of viscosity to density. The SI units of kinematic viscosity are m2/s. The cgs stoke is 1 cm2/s. Rheology In general, fluid flow patterns are more complex than the one shown in Fig. 6-1, as is the relationship between fluid deformation and stress. Rheology is the discipline of fluid mechanics which studies this relationship. One goal of rheology is to obtain constitutive equations by which stresses may be computed from deformation rates. For simplicity, fluids may be classified into rheological types in reference to the simple shear flow of Fig. 6-1. Complete definitions require extension to multidimensional flow. For more information, several good references are available, including Bird, Armstrong, and Hassager, (Dynamics of Polymeric Liquids, vol. 1: Fluid Mechanics, Wiley, New York, 1977); Metzner, (“Flow of Non-Newtonian Fluids” in Streeter, Handbook of Fluid Dynamics, McGraw-Hill, New York, 1971); and Skelland (Non-Newtonian Flow and Heat Transfer, Wiley, New York, 1967).

ti c as m tic ha las g p n o Bi pl

τy

Ps eu d

The force is directly proportional to the area of the plate; the shear stress is τ = F/A. Within the fluid, a linear velocity profile u = Uy/H is established; due to the no-slip condition, the fluid bounding the lower plate has zero velocity and the fluid bounding the upper plate moves at the plate velocity U. The velocity gradient γ˙ = du/dy is called the shear rate for this flow. Shear rates are usually reported in units of reciprocal seconds. The flow in Fig. 6-1 is a simple shear flow. Viscosity The ratio of shear stress to shear rate is the viscosity, µ. τ µ= (6-1) γ˙

Shear stress τ

FIG. 6-1

lat Di

t an

an toni Ne w Shear rate |du/dy| FIG. 6-2

Shear diagrams.

Non-Newtonian fluids include those for which a finite stress τy is required before continuous deformation occurs; these are called yield-stress materials. The Bingham plastic fluid is the simplest yield-stress material; its rheogram has a constant slope µ∞, called the infinite shear viscosity. τ = τy + µ∞γ˙ (6-3) Highly concentrated suspensions of fine solid particles frequently exhibit Bingham plastic behavior. Shear-thinning fluids are those for which the slope of the rheogram decreases with increasing shear rate. These fluids have also been called pseudoplastic, but this terminology is outdated and discouraged. Many polymer melts and solutions, as well as some solids suspensions, are shear-thinning. Shear-thinning fluids without yield stresses typically obey a power law model over a range of shear rates. τ = Kγ˙ n (6-4) The apparent viscosity is µ = Kγ˙ n − 1

(6-5)

FLUID DYNAMICS The factor K is the consistency index or power law coefficient, and n is the power law exponent. The exponent n is dimensionless, while K is in units of kg/(m ⋅ s2 − n). For shear-thinning fluids, n < 1. The power law model typically provides a good fit to data over a range of one to two orders of magnitude in shear rate; behavior at very low and very high shear rates is often Newtonian. Shear-thinning power law fluids with yield stresses are sometimes called Herschel-Bulkley fluids. Numerous other rheological model equations for shear-thinning fluids are in common use. Dilatant, or shear-thickening, fluids show increasing viscosity with increasing shear rate. Over a limited range of shear rate, they may be described by the power law model with n > 1. Dilatancy is rare, observed only in certain concentration ranges in some particle suspensions (Govier and Aziz, pp. 33–34). Extensive discussions of dilatant suspensions, together with a listing of dilatant systems, are given by Green and Griskey (Trans. Soc. Rheol, 12[1], 13–25 [1968]); Griskey and Green (AIChE J., 17, 725–728 [1971]); and Bauer and Collins (“Thixotropy and Dilatancy,” in Eirich, Rheology, vol. 4, Academic, New York, 1967). Time-dependent fluids are those for which structural rearrangements occur during deformation at a rate too slow to maintain equilibrium configurations. As a result, shear stress changes with duration of shear. Thixotropic fluids, such as mayonnaise, clay suspensions used as drilling muds, and some paints and inks, show decreasing shear stress with time at constant shear rate. A detailed description of thixotropic behavior and a list of thixotropic systems is found in Bauer and Collins (ibid.). Rheopectic behavior is the opposite of thixotropy. Shear stress increases with time at constant shear rate. Rheopectic behavior has been observed in bentonite sols, vanadium pentoxide sols, and gypsum suspensions in water (Bauer and Collins, ibid.) as well as in some polyester solutions (Steg and Katz, J. Appl. Polym. Sci., 9, 3, 177 [1965]). Viscoelastic fluids exhibit elastic recovery from deformation when stress is removed. Polymeric liquids comprise the largest group of fluids in this class. A property of viscoelastic fluids is the relaxation time, which is a measure of the time required for elastic effects to decay. Viscoelastic effects may be important with sudden changes in rates of deformation, as in flow startup and stop, rapidly oscillating flows, or as a fluid passes through sudden expansions or contractions where accelerations occur. In many fully developed flows where such effects are absent, viscoelastic fluids behave as if they were purely viscous. In viscoelastic flows, normal stresses perpendicular to the direction of shear are different from those in the parallel direction. These give rise to such behaviors as the Weissenberg effect, in which fluid climbs up a shaft rotating in the fluid, and die swell, where a stream of fluid issuing from a tube may expand to two or more times the tube diameter. A parameter indicating whether viscoelastic effects are important is the Deborah number, which is the ratio of the characteristic relaxation time of the fluid to the characteristic time scale of the flow. For small Deborah numbers, the relaxation is fast compared to the characteristic time of the flow, and the fluid behavior is purely viscous. For very large Deborah numbers, the behavior closely resembles that of an elastic solid. Analysis of viscoelastic flows is very difficult. Simple constitutive equations are unable to describe all the material behavior exhibited by viscoelastic fluids even in geometrically simple flows. More complex constitutive equations may be more accurate, but become exceedingly difficult to apply, especially for complex geometries, even with advanced numerical methods. For good discussions of viscoelastic fluid behavior, including various types of constitutive equations, see Bird, Armstrong, and Hassager (Dynamics of Polymeric Liquids, vol. 1: Fluid Mechanics, vol. 2: Kinetic Theory, Wiley, New York, 1977); Middleman (The Flow of High Polymers, Interscience (Wiley) New York, 1968); or Astarita and Marrucci (Principles of Non-Newtonian Fluid Mechanics, McGraw-Hill, New York, 1974). Polymer processing is the field which depends most on the flow of non-Newtonian fluids. Several excellent texts are available, including Middleman (Fundamentals of Polymer Processing, McGraw-Hill, New York, 1977) and Tadmor and Gogos (Principles of Polymer Processing, Wiley, New York, 1979).

6-5

There is a wide variety of instruments for measurement of Newtonian viscosity, as well as rheological properties of non-Newtonian fluids. They are described in Van Wazer, Lyons, Kim, and Colwell, (Viscosity and Flow Measurement, Interscience, New York, 1963); Coleman, Markowitz, and Noll (Viscometric Flows of Non-Newtonian Fluids, Springer-Verlag, Berlin, 1966); Dealy and Wissbrun (Melt Rheology and its Role in Plastics Processing, Van Nostrand Reinhold, 1990). Measurement of rheological behavior requires wellcharacterized flows. Such rheometric flows are thoroughly discussed by Astarita and Marrucci (Principles of Non-Newtonian Fluid Mechanics, McGraw-Hill, New York, 1974). KINEMATICS OF FLUID FLOW Velocity The term kinematics refers to the quantitative description of fluid motion or deformation. The rate of deformation depends on the distribution of velocity within the fluid. Fluid velocity v is a vector quantity, with three cartesian components vx, vy, and vz. The velocity vector is a function of spatial position and time. A steady flow is one in which the velocity is independent of time, while in unsteady flow v varies with time. Compressible and Incompressible Flow An incompressible flow is one in which the density of the fluid is constant or nearly constant. Liquid flows are normally treated as incompressible, except in the context of hydraulic transients (see following). Compressible fluids, such as gases, may undergo incompressible flow if pressure and/or temperature changes are small enough to render density changes insignificant. Frequently, compressible flows are regarded as flows in which the density varies by more than 5 to 10 percent. Streamlines, Pathlines, and Streaklines These are curves in a flow field which provide insight into the flow pattern. Streamlines are tangent at every point to the local instantaneous velocity vector. A pathline is the path followed by a material element of fluid; it coincides with a streamline if the flow is steady. In unsteady flow the pathlines generally do not coincide with streamlines. Streaklines are curves on which are found all the material particles which passed through a particular point in space at some earlier time. For example, a streakline is revealed by releasing smoke or dye at a point in a flow field. For steady flows, streamlines, pathlines, and streaklines are indistinguishable. In two-dimensional incompressible flows, streamlines are contours of the stream function. One-dimensional Flow Many flows of great practical importance, such as those in pipes and channels, are treated as onedimensional flows. There is a single direction called the flow direction; velocity components perpendicular to this direction are either zero or considered unimportant. Variations of quantities such as velocity, pressure, density, and temperature are considered only in the flow direction. The fundamental conservation equations of fluid mechanics are greatly simplified for one-dimensional flows. A broader category of one-dimensional flow is one where there is only one nonzero velocity component, which depends on only one coordinate direction, and this coordinate direction may or may not be the same as the flow direction. Rate of Deformation Tensor For general three-dimensional flows, where all three velocity components may be important and may vary in all three coordinate directions, the concept of deformation previously introduced must be generalized. The rate of deformation tensor Dij has nine components. In Cartesian coordinates, ∂v ∂v Dij = i + j ∂xj ∂xi

冢

冣

(6-6)

where the subscripts i and j refer to the three coordinate directions. Some authors define the deformation rate tensor as one-half of that given by Eq. (6-6). Vorticity The relative motion between two points in a fluid can be decomposed into three components: rotation, dilatation, and deformation. The rate of deformation tensor has been defined. Dilatation refers to the volumetric expansion or compression of the fluid, and vanishes for incompressible flow. Rotation is described by a tensor ωij = ∂vi / ∂xj − ∂vj / ∂xi. The vector of vorticity given by one-half the

6-6

FLUID AND PARTICLE DYNAMICS

curl of the velocity vector is another measure of rotation. In twodimensional flow in the x-y plane, the vorticity ω is given by ∂v 1 ∂v (6-7) ω =  y − x 2 ∂x ∂y Here ω is the magnitude of the vorticity vector, which is directed along the z axis. An irrotational flow is one with zero vorticity. Irrotational flows have been widely studied because of their useful mathematical properties and applicability to flow regions where viscous effects may be neglected. Such flows without viscous effects are called inviscid flows. Laminar and Turbulent Flow, Reynolds Number These terms refer to two distinct types of flow. In laminar flow, there are smooth streamlines and the fluid velocity components vary smoothly with position, and with time if the flow is unsteady. The flow described in reference to Fig. 6-1 is laminar. In turbulent flow, there are no smooth streamlines, and the velocity shows chaotic fluctuations in time and space. Velocities in turbulent flow may be reported as the sum of a time-averaged velocity and a velocity fluctuation from the average. For any given flow geometry, a dimensionless Reynolds number may be defined for a Newtonian fluid as Re = LU ρ /µ where L is a characteristic length. Below a critical value of Re the flow is laminar, while above the critical value a transition to turbulent flow occurs. The geometry-dependent critical Reynolds number is determined experimentally.

冢

冣

CONSERVATION EQUATIONS Macroscopic and Microscopic Balances Three postulates, regarded as laws of physics, are fundamental in fluid mechanics. These are conservation of mass, conservation of momentum, and conservation of energy. In addition, two other postulates, conservation of moment of momentum (angular momentum) and the entropy inequality (second law of thermodynamics) have occasional use. The conservation principles may be applied either to material systems or to control volumes in space. Most often, control volumes are used. The control volumes may be either of finite or differential size, resulting in either algebraic or differential conservation equations, respectively. These are often called macroscopic and microscopic balance equations. Macroscopic Equations An arbitrary control volume of finite size Va is bounded by a surface of area Aa with an outwardly directed unit normal vector n. The control volume is not necessarily fixed in space. Its boundary moves with velocity w. The fluid velocity is v. Figure 6-3 shows the arbitrary control volume. Mass balance Applied to the control volume, the principle of conservation of mass may be written as (Whitaker, Introduction to Fluid Mechanics, Prentice-Hall, Englewood Cliffs, N.J., 1968, Krieger, Huntington, N.Y., 1981) d ρ dV + ρ(v − w) ⋅ n dA = 0 (6-8)  Aa dt Va This equation is also known as the continuity equation.

冕

冕

Area Aa

Volume Va

Simplified forms of Eq. (6-8) apply to special cases frequently found in practice. For a control volume fixed in space with one inlet of area A1 through which an incompressible fluid enters the control volume at an average velocity V1, and one outlet of area A2 through which fluid leaves at an average velocity V2, as shown in Fig. 6-4, the continuity equation becomes V1 A1 = V2 A2 (6-9) The average velocity across a surface is given by V = (1/A)

冕 v dA A

where v is the local velocity component perpendicular to the inlet surface. The volumetric flow rate Q is the product of average velocity and the cross-sectional area, Q = VA. The average mass velocity is G = ρV. For steady flows through fixed control volumes with multiple inlets and/or outlets, conservation of mass requires that the sum of inlet mass flow rates equals the sum of outlet mass flow rates. For incompressible flows through fixed control volumes, the sum of inlet flow rates (mass or volumetric) equals the sum of exit flow rates, whether the flow is steady or unsteady. Momentum Balance Since momentum is a vector quantity, the momentum balance is a vector equation. Where gravity is the only body force acting on the fluid, the linear momentum principle, applied to the arbitrary control volume of Fig. 6-3, results in the following expression (Whitaker, ibid.). d ρv dV + ρv(v − w) ⋅ n dA = ρg dV + tn dA (6-10)  Aa Va Aa dt Va Here g is the gravity vector and tn is the force per unit area exerted by the surroundings on the fluid in the control volume. The integrand of the area integral on the left-hand side of Eq. (6-10) is nonzero only on the entrance and exit portions of the control volume boundary. For the special case of steady flow at a mass flow rate m ˙ through a control volume fixed in space with one inlet and one outlet, (Fig. 6-4) with the inlet and outlet velocity vectors perpendicular to planar inlet and outlet surfaces, giving average velocity vectors V1 and V2, the momentum equation becomes

冕

冕

冕

冕

m(β ˙ 2V2 − β1V1) = −p1A1 − p2A2 + F + Mg

n outwardly directed unit normal vector

V2 2

w boundary velocity

V1

v fluid velocity FIG. 6-3

Arbitrary control volume for application of conservation equations.

(6-11)

where M is the total mass of fluid in the control volume. The factor β arises from the averaging of the velocity across the area of the inlet or outlet surface. It is the ratio of the area average of the square of velocity magnitude to the square of the area average velocity magnitude. For a uniform velocity, β = 1. For turbulent flow, β is nearly unity, while for laminar pipe flow with a parabolic velocity profile, β = 4/3. The vectors A1 and A 2 have magnitude equal to the areas of the inlet and outlet surfaces, respectively, and are outwardly directed normal to the surfaces. The vector F is the force exerted on the fluid by the nonflow boundaries of the control volume. It is also assumed that the stress vector tn is normal to the inlet and outlet surfaces, and that its magnitude may be approximated by the pressure p. Equation (6-11) may be generalized to multiple inlets and/or outlets. In such cases, the mass flow rates for all the inlets and outlets are not equal. A distinct flow rate m ˙ i applies to each inlet or outlet i. To generalize the equation, ⴚpA terms for each inlet and outlet, − mβV ˙ terms for each inlet, and mβV ˙ terms for each outlet are included. Balance equations for angular momentum, or moment of momentum, may also be written. They are used less frequently than the lin-

FIG. 6-4

Fixed control volume with one inlet and one outlet.

FLUID DYNAMICS ear momentum equations. See Whitaker (Introduction to Fluid Mechanics, Prentice-Hall, Englewood Cliffs, N.J., 1968, Krieger, Huntington, N.Y., 1981; or Shames (Mechanics of Fluids, 3d ed., McGraw-Hill, New York, 1992). Total Energy Balance The total energy balance derives from the first law of thermodynamics. Applied to the arbitrary control volume of Fig. 6-3, it leads to an equation for the rate of change of the sum of internal, kinetic, and gravitational potential energy. In this equation, u is the internal energy per unit mass, v is the magnitude of the velocity vector v, z is elevation, g is the gravitational acceleration, and q is the heat flux vector: d  dt

冕 ρ冢u + v2 + gz冣 dV + 冕 ρ冢u + v2 + gz冣(v − w) ⋅ n dA = 冕 (v ⋅ t ) dA − 冕 (q ⋅ n) dA (6-12) 2

2

Va

Aa

n

Aa

Aa

The first integral on the right-hand side is the rate of work done on the fluid in the control volume by forces at the boundary. It includes both work done by moving solid boundaries and work done at flow entrances and exits. The work done by moving solid boundaries also includes that by such surfaces as pump impellers; this work is called ˙ S. shaft work; its rate is W A useful simplification of the total energy equation applies to a particular set of assumptions. These are a control volume with fixed solid boundaries, except for those producing shaft work, steady state conditions, and mass flow at a rate m ˙ through a single planar entrance and a single planar exit (Fig. 6-4), to which the velocity vectors are perpendicular. As with Eq. (6-11), it is assumed that the stress vector tn is normal to the entrance and exit surfaces and may be approximated by the pressure p. The equivalent pressure, p + ρgz, is assumed to be uniform across the entrance and exit. The average velocity at the entrance and exit surfaces is denoted by V. Subscripts 1 and 2 denote the entrance and exit, respectively. V 12 V 22 h1 + α1  + gz1 = h2 + α2  + gz2 − δQ − δWS 2 2

(6-13)

Here, h is the enthalpy per unit mass, h = u + p/ρ. The shaft work per ˙ s /m. unit of mass flowing through the control volume is δWS = W ˙ Similarly, δQ is the heat input rate per unit of mass. The factor α is the ratio of the cross-sectional area average of the cube of the velocity to the cube of the average velocity. For a uniform velocity profile, α = 1. In turbulent flow, α is usually assumed to equal unity; in turbulent pipe flow, it is typically about 1.07. For laminar flow in a circular pipe with a parabolic velocity profile, α = 2. Mechanical Energy Balance, Bernoulli Equation A balance equation for the sum of kinetic and potential energy may be obtained from the momentum balance by forming the scalar product with the velocity vector. The resulting equation, called the mechanical energy balance, contains a term accounting for the dissipation of mechanical energy into thermal energy by viscous forces. The mechanical energy equation is also derivable from the total energy equation in a way that reveals the relationship between the dissipation and entropy generation. The macroscopic mechanical energy balance for the arbitrary control volume of Fig. 6-3 may be written, with p = thermodynamic pressure, as d  dt

冕 ρ冢v2 + gz冣 dV + 冕 ρ冢v2 + gz冣(v − w) ⋅ n dA = 冕 p ⵱ ⋅ v dV + 冕 (v ⋅ t ) dA − 冕 Φ dV 2

2

Va

Aa

n

Va

Aa

(6-14)

Va

The last term is the rate of viscous energy dissipation to internal energy, E˙ v = 冕Va Φ dV, also called the rate of viscous losses. These losses are the origin of frictional pressure drop in fluid flow. Whitaker and Bird, Stewart, and Lightfoot provide expressions for the dissipation function Φ for Newtonian fluids in terms of the local velocity gradients. However, when using macroscopic balance equations the local velocity field within the control volume is usually unknown. For such cases additional information, which may come from empirical correlations, is needed.

6-7

For the same special conditions as for Eq. (6-13), the mechanical energy equation is reduced to p 2 dp V12 V 22 (6-15) α1  + gz1 + δWS = α2  + gz2 +  + lv p1 2 2 ρ ˙ is the energy dissipation per unit mass. This equation Here lv = E˙ v /m has been called the engineering Bernoulli equation. For an incompressible flow, Eq. (6-15) becomes p2 p1 V12 V22 (6-16)  + α1  + gz1 + δWS =  + α2  + gz2 + lv ρ ρ 2 2 The Bernoulli equation can be written for incompressible, inviscid flow along a streamline, where no shaft work is done. p1 V12 p2 V22 (6-17)  +  + gz1 =  +  + gz2 ρ 2 ρ 2 Unlike the momentum equation (Eq. [6-11]), the Bernoulli equation is not easily generalized to multiple inlets or outlets. Microscopic Balance Equations Partial differential balance equations express the conservation principles at a point in space. Equations for mass, momentum, total energy, and mechanical energy may be found in Whitaker (ibid.), Bird, Stewart, and Lightfoot (Transport Phenomena, Wiley, New York, 1960), and Slattery (Momentum, Heat and Mass Transfer in Continua, 2d ed., Krieger, Huntington, N.Y., 1981), for example. These references also present the equations in other useful coordinate systems besides the cartesian system. The coordinate systems are fixed in inertial reference frames. The two most used equations, for mass and momentum, are presented here. Mass Balance, Continuity Equation The continuity equation, expressing conservation of mass, is written in cartesian coordinates as ∂ρ ∂ρvx ∂ρvy ∂ρvz (6-18) +++=0 ∂t ∂x ∂y ∂z In terms of the substantial derivative, D/Dt, Dρ ∂ρ ∂ρ ∂ρ ∂ρ ∂vx ∂vy ∂vz  ⬅  + vx  + vy  + vz  = −ρ  +  +  (6-19) Dt ∂t ∂x ∂y ∂z ∂x ∂y ∂z The substantial derivative, also called the material derivative, is the rate of change in a Lagrangian reference frame, that is, following a material particle. In vector notation the continuity equation may be expressed as Dρ (6-20)  = −ρ∇ ⋅ v Dt For incompressible flow, ∂v ∂v ∂v ∇ ⋅ v = x + y + z = 0 (6-21) ∂x ∂y ∂z Stress Tensor The stress tensor is needed to completely describe the stress state for microscopic momentum balances in multidimensional flows. The components of the stress tensor σij give the force in the j direction on a plane perpendicular to the i direction, using a sign convention defining a positive stress as one where the fluid with the greater i coordinate value exerts a force in the positive i direction on the fluid with the lesser i coordinate. Several references in fluid mechanics and continuum mechanics provide discussions, to various levels of detail, of stress in a fluid (Denn; Bird, Stewart, and Lightfoot; Schlichting; Fung [A First Course in Continuum Mechanics, 2d. ed., Prentice-Hall, Englewood Cliffs, N.J., 1977]; Truesdell and Toupin [in Flügge, Handbuch der Physik, vol. 3/1, Springer-Verlag, Berlin, 1960]; Slattery [Momentum, Energy and Mass Transfer in Continua, 2d ed., Krieger, Huntington, N.Y., 1981]). The stress has an isotropic contribution due to fluid pressure and dilatation, and a deviatoric contribution due to viscous deformation effects. The deviatoric contribution for a Newtonian fluid is the threedimensional generalization of Eq. (6-2):

冕

冢

冣

τij = µDij

(6-22)

σij = (−p + λ∇ ⋅ v)δij + τij

(6-23)

The total stress is

6-8

FLUID AND PARTICLE DYNAMICS

The identity tensor δij is zero for i ≠ j and unity for i = j. The coefficient λ is a material property related to the bulk viscosity, κ = λ + 2µ/3. There is considerable uncertainty about the value of κ. Traditionally, Stokes’ hypothesis, κ = 0, has been invoked, but the validity of this hypothesis is doubtful (Slattery, ibid.). For incompressible flow, the value of bulk viscosity is immaterial as Eq. (6-23) reduces to σij = −pδij + τij

(6-24)

Similar generalizations to multidimensional flow are necessary for non-Newtonian constitutive equations. Cauchy Momentum and Navier-Stokes Equations The differential equations for conservation of momentum are called the Cauchy momentum equations. These may be found in general form in most fluid mechanics texts (e.g., Slattery [ibid.]; Denn; Whitaker; and Schlichting). For the important special case of an incompressible Newtonian fluid with constant viscosity, substitution of Eqs. (6-22) and (6-24) lead to the Navier-Stokes equations, whose three Cartesian components are ∂v ∂v ∂v ∂v ρ x + vx x + vy x + vz x ∂t ∂x ∂y ∂z

冢

冣

∂p ∂2vx ∂2vx ∂2vx =−+µ  + + + ρgx (6-25) ∂x ∂x2 ∂y2 ∂z2

冢

∂vy ∂vy ∂vy ∂vy ρ  + vx  + vy  + vz  ∂t ∂x ∂y ∂z

冢

冣

冣

∂p ∂2vy ∂2vy ∂2vy =−+µ  + + + ρgy ∂y ∂x2 ∂y2 ∂z2

冢

∂vz ∂vz ∂vz ∂vz ρ  + vx  + vy  + vz  ∂t ∂x ∂y ∂z

冢

冣

(6-26)

冣

∂p ∂2vz ∂2vz ∂2vz =−+µ  + + + ρgz (6-27) ∂z ∂x2 ∂y2 ∂z2

冢

冣

Stokes equations, Dirichlet and Neumann, or essential and natural, boundary conditions may be satisfied by different means. Fluid statics, discussed in Sec. 10 of the Handbook in reference to pressure measurement, is the branch of fluid mechanics in which the fluid velocity is either zero or is uniform and constant relative to an inertial reference frame. With velocity gradients equal to zero, the momentum equation reduces to a simple expression for the pressure field, ∇p = ρg. Letting z be directed vertically upward, so that gz = −g where g is the gravitational acceleration (9.806 m2/s), the pressure field is given by dp/dz = −ρg

Example 1: Force Exerted on a Reducing Bend An incompressible fluid flows through a reducing elbow (Fig. 6-5) situated in a horizontal plane. The inlet velocity V1 is given and the pressures p1 and p2 are measured. Selecting the inlet and outlet surfaces 1 and 2 as shown, the continuity equation Eq. (6-9) can be used to find the exit velocity V2 = V1A1/A2. The mass flow rate is obtained by m ˙ = ρV1A1. Assume that the velocity profile is nearly uniform so that β is approximately unity. The force exerted on the fluid by the bend has x and y components; these can be found from Eq. (6-11). The x component gives Fx = m(V ˙ 2x − V1x) + p1A1x + p2 A2x while the y component gives Fy = m(V ˙ 2y − V1y) + p1 A1y + p2 A2y

In vector notation, Dv ∂v (6-28) ρ  =  + (v ⋅ ∇)v = −∇p + µ∇2v + ρg Dt ∂t The pressure and gravity terms may be combined by replacing the pressure p by the equivalent pressure P = p + ρgz. The left-hand side terms of the Navier-Stokes equations are the inertial terms, while the terms including viscosity µ are the viscous terms. Limiting cases under which the Navier-Stokes equations may be simplified include creeping flows in which the inertial terms are neglected, potential flows (inviscid or irrotational flows) in which the viscous terms are neglected, and boundary layer and lubrication flows in which certain terms are neglected based on scaling arguments. Creeping flows are described by Happel and Brenner (Low Reynolds Number Hydrodynamics, Prentice-Hall, Englewood Cliffs, N.J., 1965); potential flows by Lamb (Hydrodynamics, 6th ed., Dover, New York, 1945) and Milne-Thompson (Theoretical Hydrodynamics, 5th ed., Macmillan, New York, 1968); boundary layer theory by Schlichting (Boundary Layer Theory, 8th ed., McGraw-Hill, New York, 1987), and lubrication theory by Batchelor (An Introduction to Fluid Dynamics, Cambridge University, Cambridge, 1967) and Denn (Process Fluid Mechanics, Prentice-Hall, Englewood Cliffs, N.J., 1980). Because the Navier-Stokes equations are first-order in pressure and second-order in velocity, their solution requires one pressure boundary condition and two velocity boundary conditions (for each velocity component) to completely specify the solution. The no slip condition, which requires that the fluid velocity equal the velocity of any bounding solid surface, occurs in most problems. Specification of velocity is a type of boundary condition sometimes called a Dirichlet condition. Often boundary conditions involve stresses, and thus velocity gradients, rather than the velocities themselves. Specification of velocity derivatives is a Neumann boundary condition. For example, at the boundary between a viscous liquid and a gas, it is often assumed that the liquid shear stresses are zero. In numerical solution of the Navier-

(6-29)

This equation applies to any incompressible or compressible static fluid. For an incompressible liquid, pressure varies linearly with depth. For compressible gases, p is obtained by integration accounting for the variation of ρ with z. The force exerted on a submerged planar surface of area A is given by F = pc A where pc is the pressure at the geometrical centroid of the surface. The center of pressure, the point of application of the net force, is always lower than the centroid. For details see, for example, Shames, where may also be found discussion of forces on curved surfaces, buoyancy, and stability of floating bodies. Examples Four examples follow, illustrating the application of the conservation equations to obtain useful information about fluid flows.

The velocity components are V1x = V1, V1y = 0, V2x = V2 cos θ, and V2y = V2 sin θ. The area vector components are A1x = −A1, A1y = 0, A 2x = A 2 cos θ, and A 2y = A 2 sin θ. Therefore, the force components may be calculated from Fx = m(V ˙ 2 cos θ − V1) − p1A1 + p2A2 cos θ Fy = mV ˙ 2 sin θ + p2A2 sin θ The force acting on the fluid is F; the equal and opposite force exerted by the fluid on the bend is ⴚF.

Example 2: Simplified Ejector Figure 6-6 shows a very simplified sketch of an ejector, a device that uses a high velocity primary fluid to pump another (secondary) fluid. The continuity and momentum equations may be

V2 θ

V1 F

y x Force at a reducing bend. F is the force exerted by the bend on the fluid. The force exerted by the fluid on the bend is ⴚF. FIG. 6-5

FLUID DYNAMICS

6-9

y H

x

FIG. 6-8

FIG. 6-6

Draft-tube ejector.

applied on the control volume with inlet and outlet surfaces 1 and 2 as indicated in the figure. The cross-sectional area is uniform, A1 = A2 = A. Let the mass flow rates and velocities of the primary and secondary fluids be m ˙ p, m ˙ s, Vp and Vs. Assume for simplicity that the density is uniform. Conservation of mass gives m˙2 = m ˙p +m ˙ s. The exit velocity is V2 = m ˙ 2 /(ρA). The principle momentum exchange in the ejector occurs between the two fluids. Relative to this exchange, the force exerted by the walls of the device are found to be small. Therefore, the force term F is neglected from the momentum equation. Written in the flow direction, assuming uniform velocity profiles, and using the extension of Eq. (6-11) for multiple inlets, it gives the pressure rise developed by the device: (p2 − p1)A = (m˙p + m ˙ s)V2 − m ˙ pVp − m ˙ sVs Application of the momentum equation to ejectors of other types is discussed in Lapple (Fluid and Particle Dynamics, University of Delaware, Newark, 1951) and in Sec. 10 of the Handbook.

Example 3: Venturi Flowmeter An incompressible fluid flows

Plane Poiseuille flow.

ish in the x direction. Since the flow field is infinite in the z direction, all velocity derivatives should be zero in the z direction. Therefore, velocity components are a function of y alone. It is also assumed that there is no flow in the z direction, so vz = 0. The continuity equation Eq. (6-21), with vz = 0 and ∂vx / ∂x = 0, reduces to dvy  = 0. dy Since vy = 0 at y = H/2, the continuity equation integrates to vy = 0. This is a direct result of the assumption of fully developed flow. The Navier-Stokes equations are greatly simplified when it is noted that vy = vz = 0 and ∂vx /∂x = ∂vx /∂z = ∂vx /∂t = 0. The three components are written in terms of the equivalent pressure P: ∂P ∂2vx 0=−+µ ∂x ∂y2 ∂P 0=− ∂y ∂P 0=− ∂z

through the venturi flowmeter in Fig. 6-7. An equation is needed to relate the flow rate Q to the pressure drop measured by the manometer. This problem can be solved using the mechanical energy balance. In a well-made venturi, viscous losses are negligible, the pressure drop is entirely the result of acceleration into the throat, and the flow rate predicted neglecting losses is quite accurate. The inlet area is A and the throat area is a. With control surfaces at 1 and 2 as shown in the figure, Eq. (6-17) in the absence of losses and shaft work gives

The latter two equations require that P is a function only of x, and therefore ∂P/∂x = dP/dx. Inspection of the first equation shows one term which is a function only of x and one which is only a function of y. This requires that both terms are constant. The pressure gradient −dP/dx is constant. The x-component equation becomes d 2vx 1 dP =  dy2 µ dx

p1 V 12 p2 V 22 +=+ ρ 2 ρ 2 The continuity equation gives V2 = V1A/a, and V1 = Q/A. The pressure drop measured by the manometer is p1 − p2 = (ρm − ρ)g∆z. Substituting these relations into the energy balance and rearranging, the desired expression for the flow rate is found. 2(ρm − ρ)g∆z 1 Q =   A ρ[(A/a)2 − 1]

Two integrations of the x-component equation give

冪莦

Example 4: Plane Poiseuille Flow An incompressible Newtonian fluid flows at a steady rate in the x direction between two very large flat plates, as shown in Fig. 6-8. The flow is laminar. The velocity profile is to be found. This example is found in most fluid mechanics textbooks; the solution presented here closely follows Denn. This problem requires use of the microscopic balance equations because the velocity is to be determined as a function of position. The boundary conditions for this flow result from the no-slip condition. All three velocity components must be zero at the plate surfaces, y = H/2 and y = −H/2. Assume that the flow is fully developed, that is, all velocity derivatives van-

1

2

1 dP vx =   y2 + C1y + C2 2µ dx where the constants of integration C1 and C2 are evaluated from the boundary conditions vx = 0 at y = H/2. The result is

冢

冣冤 冢 冣 冥

H2 dP 2y 2 vx =  −  1 −  8µ dx H This is a parabolic velocity distribution. The average velocity V = H/2 (1/H) 冕 −H/2 vx dy is dP H2 V= − 12µ dx

冢

This flow is one-dimensional, as there is only one nonzero velocity component, vx, which, along with the pressure, varies in only one coordinate direction.

INCOMPRESSIBLE FLOW IN PIPES AND CHANNELS Mechanical Energy Balance The mechanical energy balance, Eq. (6-16), for fully developed incompressible flow in a straight circular pipe of constant diameter D reduces to p2 p1 (6-30)  + gz1 =  + gz 2 + lv ρ ρ In terms of the equivalent pressure, P ⬅ p + ρgz, P1 − P2 = ρ lv

∆z

FIG. 6-7

Venturi flowmeter.

冣

(6-31)

The pressure drop due to frictional losses lv is proportional to pipe length L for fully developed flow and may be denoted as the (positive) quantity ∆P ⬅ P1 − P2. Friction Factor and Reynolds Number For a Newtonian fluid in a smooth pipe, dimensional analysis relates the frictional pressure drop per unit length ∆P/L to the pipe diameter D, density ρ, and average velocity V through two dimensionless groups, the Fanning friction factor f and the Reynolds number Re.

6-10

FLUID AND PARTICLE DYNAMICS D∆P f⬅ 2ρV 2L

(6-32)

DVρ Re ⬅  (6-33) µ For smooth pipe, the friction factor is a function only of the Reynolds number. In rough pipe, the relative roughness
/D also affects the friction factor. Figure 6-9 plots f as a function of Re and
/D. Values of
for various materials are given in Table 6-1. The Fanning friction factor should not be confused with the Darcy friction factor used by Moody (Trans. ASME, 66, 671 [1944]), which is four times greater. Using the momentum equation, the stress at the wall of the pipe may be expressed in terms of the friction factor: ρV 2 τw = f  2

(6-34)

Laminar and Turbulent Flow Below a critical Reynolds number of about 2,100, the flow is laminar; over the range 2,100 < Re < 5,000 there is a transition to turbulent flow. For laminar flow, the Hagen-Poiseuille equation 16 f = , Re

Re ≤ 2,100

(6-35)

may be derived from the Navier-Stokes equation and is in excellent agreement with experimental data. It may be rewritten in terms of volumetric flow rate, Q = VπD2/4, as π∆PD4 Q = , Re ≤ 2,100 (6-36) 128µL

TABLE 6-1 Values of Surface Roughness for Various Materials* Surface roughness ε, mm

Material Drawn tubing (brass, lead, glass, and the like) Commercial steel or wrought iron Asphalted cast iron Galvanized iron Cast iron Wood stove Concrete Riveted steel

0.00152 0.0457 0.122 0.152 0.259 0.183–0.914 0.305–3.05 0.914–9.14

* From Moody, Trans. Am. Soc. Mech. Eng., 66, 671–684 (1944); Mech. Eng., 69, 1005–1006 (1947). Additional values of ε for various types or conditions of concrete wrought-iron, welded steel, riveted steel, and corrugated-metal pipes are given in Brater and King, Handbook of Hydraulics, 6th ed., McGraw-Hill, New York, 1976, pp. 6-12–6-13. To convert millimeters to feet, multiply by 3.281 × 10−3.

For turbulent flow in smooth tubes, the Blasius equation gives the friction factor accurately for a wide range of Reynolds numbers. 0.079 f= , 4,000 < Re < 105 (6-37) Re0.25 The Colebrook formula (Colebrook, J. Inst. Civ. Eng. [London], 11, 133–156 [1938–39]) gives a good approximation for the f-Re-(
/D) data for rough pipes over the entire turbulent flow range: 1.256 1
 = −4 log  +  3.7D Re兹f苶 兹苶f

冤

冥

Re > 4,000

Fanning Friction Factors. Reynolds number Re = DVρ/µ, where D = pipe diameter, V = velocity, ρ = fluid density, and µ = fluid viscosity. (Based on Moody, Trans. ASME, 66, 671 [1944].)

FIG. 6-9

(6-38)

FLUID DYNAMICS An equation by Churchill (Chem. Eng., 84[24], 91–92 [Nov. 7, 1977]) for both smooth and rough tubes offers the advantage of being explicit in f: 1 0.27
 = −4 log  + (7/Re)0.9 Re > 4,000 (6-39) D 兹苶f In laminar flow, f is independent of
/D. In turbulent flow, the friction factor for rough pipe follows the smooth tube curve for a range of Reynolds numbers (hydraulically smooth flow). For greater Reynolds numbers, f deviates from the smooth pipe curve, eventually becoming independent of Re. This region, often called complete turbulence, is frequently encountered in commercial pipe flows. The Reynolds number above which f becomes essentially independent of Re is (Davies, Turbulence Phenomena, Academic, New York, 1972, p. 37) 20[3.2 − 2.46 ln (
/D)] Re =  (6-40) (
/D) Roughness may also affect the transition from laminar to turbulent flow (Schlichting). Common pipe flow problems include calculation of pressure drop given the flow rate (or velocity) and calculation of flow rate (or velocity) given pressure drop. When flow rate is given, the Reynolds number is first calculated to determine the flow regime, so that the appropriate relations between f and Re (or pressure drop and velocity or flow rate) are used. When pressure drop is given and the velocity is unknown, the Reynolds number and flow regime cannot be immediately determined. It is necessary to assume the flow regime and then verify by checking Re afterward. With experience, the initial guess for the flow regime will usually prove correct. When solving Eq. (6-38) for velocity when pressure drop is given, it is useful to note that the right-hand side is independent of velocity since Re兹苶f = (D3/2/µ)兹苶 ρ苶 ∆苶 P苶 /(苶2苶 L苶). As Fig. 6-9 suggests, the friction factor is uncertain in the transition range 2,100 < Re < 4,000 and a conservative choice should be made for design purposes. Velocity Profiles In laminar flow, the solution of the NavierStokes equation, corresponding to the Hagen-Poiseuille equation, gives the velocity v as a function of radial position r in a circular pipe of radius R in terms of the average velocity V = Q/A. The parabolic profile, with centerline velocity twice the average velocity, is shown in Fig. 6-10. r2 v = 2V 1 − 2 (6-41) R In turbulent flow, the velocity profile is much more blunt, with most of the velocity gradient being in a region near the wall, described by a universal velocity profile. It is characterized by a viscous sublayer, a turbulent core, and a buffer zone in between.

冤

冥

冢

冣

Viscous sublayer u+ = y+

for

y+ < 5

(6-42)

u+ = 5.00 ln y+ − 3.05

for

5 < y+ < 30

(6-43)

for

y+ > 30

(6-44)

Buffer zone Turbulent core u+ = 2.5 ln y+ + 5.5

Here, u+ = v/u* is the dimensionless, time-averaged axial velocity, u* =

r z

R

(

2 v = 2V 1 – r 2 R

v max = 2V FIG. 6-10

velocity V.

Parabolic velocity profile for laminar flow in a pipe, with average

6-11

兹苶 τw/ρ 苶 is the friction velocity and τw = fρV 2/2 is the wall stress. The

friction velocity is of the order of the root mean square velocity fluctuation perpendicular to the wall in the turbulent core. The dimensionless distance from the wall is y+ = yu ρ/µ. The universal velocity * profile is valid in the wall region for any cross-sectional channel shape. For incompressible flow in constant diameter circular pipes, τw = ∆P/4L where ∆P is the pressure drop in length L. In circular pipes, Eq. (6-44) gives a surprisingly good fit to experimental results over the entire cross section of the pipe, even though it is based on assumptions which are valid only near the pipe wall. For rough pipes, the velocity profile in the turbulent core is given by u+ = 2.5 ln y/
+ 8.5

for

y+ > 30

(6-45)

when the dimensionless roughness
+ =
u*ρ/µ is greater than 5 to 10; for smaller
+, the velocity profile in the turbulent core is unaffected by roughness. For velocity profiles in the transition region, see Patel and Head ( J. Fluid Mech., 38, part 1, 181–201 [1969]) where profiles over the range 1,500 < Re < 10,000 are reported. Entrance and Exit Effects In the entrance region of a pipe, some distance is required for the flow to adjust from upstream conditions to the fully developed flow pattern. This distance depends on the Reynolds number and on the flow conditions upstream. For a uniform velocity profile at the pipe entrance, the computed length in laminar flow required for the centerline velocity to reach 99 percent of its fully developed value is (Dombrowski, Foumeny, Ookawara and Riza, Can. J. Chem. Engr., 71, 472–476 [1993]) Lent /D = 0.370 exp (−0.148Re) + 0.0550Re + 0.260

(6-46)

In turbulent flow, the entrance length is about Lent /D = 40

(6-47)

The frictional losses in the entrance region are larger than those for the same length of fully developed flow. (See the subsection, “Frictional Losses in Pipeline Elements,” following.) At the pipe exit, the velocity profile also undergoes rearrangement, but the exit length is much shorter than the entrance length. At low Re, it is about one pipe radius. At Re > 100, the exit length is essentially 0. Residence Time Distribution For laminar Newtonian pipe flow, the cumulative residence time distribution F(θ) is given by θavg F(θ) = 0 for θ< 2 1 θavg 2 θavg F(θ) = 1 −   for θ≥ (6-48) 4 θ 2 where F(θ) is the fraction of material which resides in the pipe for less than time θ and θavg is the average residence time, θ = V/L. The residence time distribution in long transfer lines may be made narrower (more uniform) with the use of flow inverters or static mixing elements. These devices exchange fluid between the wall and central regions. Variations on the concept may be used to provide effective mixing of the fluid. See Godfrey (“Static Mixers,” in Harnby, Edwards, and Nienow, Mixing in the Process Industries, 2d ed., Butterworth Heinemann, Oxford, 1992); Gretta and Smith (Trans. ASME J. Fluids Eng., 115, 255–263 [1993]); Kemblowski and Pustelnik (Chem. Eng. Sci., 43, 473–478 [1988]). A theoretically derived equation for flow in helical pipe coils by Ruthven (Chem. Eng. Sci., 26, 1113–1121 [1971]; 33, 628–629 [1978]) is given by θavg 1 θavg 2.81 F(θ) = 1 −   for 0.5 <  < 1.63 (6-49) 4 θ θ and was substantially confirmed by Trivedi and Vasudeva (Chem. Eng. Sci., 29, 2291–2295 [1974]) for 0.6 < De < 6 and 0.0036 < D/Dc < 0.097 where De = Re兹苶 D苶 /Dc苶 is the Dean number and Dc is the diameter of curvature of the coil. Measurements by Saxena and Nigam (Chem. Eng. Sci., 34, 425–426 [1979]) indicate that such a distribution will hold for De > 1. The residence time distribution for helical coils is narrower than for straight circular pipes, due to the secondary flow which exchanges fluid between the wall and center regions.

冢 冣

冢 冣冤 冥

6-12

FLUID AND PARTICLE DYNAMICS

In turbulent flow, axial mixing is usually described in terms of turbulent diffusion or dispersion coefficients, from which cumulative residence time distribution functions can be computed. Davies (Turbulence Phenomena, Academic, New York, 1972, p. 93), gives DL = 1.01νRe0.875 for the longitudinal dispersion coefficient. Levenspiel (Chemical Reaction Engineering, 2d ed., Wiley, New York, 1972, pp. 253–278) discusses the relations among various residence time distribution functions, and the relation between dispersion coefficient and residence time distribution. Noncircular Channels Calculation of frictional pressure drop in noncircular channels depends on whether the flow is laminar or turbulent, and on whether the channel is full or open. For turbulent flow in ducts running full, the hydraulic diameter DH should be substituted for D in the friction factor and Reynolds number definitions, Eqs. (6-32) and (6-33). The hydraulic diameter is defined as four times the channel cross-sectional area divided by the wetted perimeter. For example, the hydraulic diameter for a circular pipe is DH = D, for an annulus of inner diameter d and outer diameter D, DH = D − d, for a rectangular duct of sides a, b, DH = ab/[2(a + b)]. The hydraulic radius RH is defined as one-fourth of the hydraulic diameter. With the hydraulic diameter subsititued for D in f and Re, Eqs. (6-37) through (6-40) are good approximations. Note that V appearing in f and Re is the actual average velocity V = Q/A; for noncircular pipes; it is not Q/(πDH2 /4). The pressure drop should be calculated from the friction factor for noncircular pipes. Equations relating Q to ∆P and D for circular pipes may not be used for noncircular pipes with D replaced by DH because V ≠ Q/(πDH2 /4). Turbulent flow in noncircular channels is generally accompanied by secondary flows perpendicular to the axial flow direction (Schlichting). These flows may cause the pressure drop to be slightly greater than that computed using the hydraulic diameter method. For data on pressure drop in annuli, see Brighton and Jones (J. Basic Eng., 86, 835–842 [1964]); Okiishi and Serovy (J. Basic Eng., 89, 823–836 [1967]); and Lawn and Elliot (J. Mech. Eng. Sci., 14, 195–204 [1972]). For rectangular ducts of large aspect ratio, Dean (J. Fluids Eng., 100, 215–233 [1978]) found that the numerator of the exponent in the Blasius equation (6-37) should be increased to 0.0868. Jones (J. Fluids Eng., 98, 173–181 [1976]) presents a method to improve the estimation of friction factors for rectangular ducts using a modification of the hydraulic diameter–based Reynolds number. The hydraulic diameter method does not work well for laminar flow because the shape affects the flow resistance in a way that cannot be expressed as a function only of the ratio of cross-sectional area to wetted perimeter. For some shapes, the Navier-Stokes equations have been integrated to yield relations between flow rate and pressure drop. These relations may be expressed in terms of equivalent diameters DE defined to make the relations reduce to the second form of the Hagen-Poiseulle equation, Eq. (6-36); that is, DE ⬅ (128QµL/π∆P)1/4. Equivalent diameters are not the same as hydraulic diameters. Equivalent diameters yield the correct relation between flow rate and pressure drop when substituted into Eq. (6-36), but not Eq. (6-35) because V ≠ Q/(πDE/4). Equivalent diameter DE is not to be used in the friction factor and Reynolds number; f ≠ 16/Re using the equivalent diameters defined in the following. This situation is, by arbitrary definition, opposite to that for the hydraulic diameter DH used for turbulent flow. Ellipse, semiaxes a and b (Lamb, Hydrodynamics, 6th ed., Dover, New York, 1945, p. 587): 32a3b3 1/4 (6-50) DE =  a2 + b2 Rectangle, width a, height b (Owen, Trans. Am. Soc. Civ. Eng., 119, 1157–1175 [1954]): 128ab3 1/4 DE =  (6-51) πK a/b = 1 1.5 2 3 4 5 10 ∞ K = 28.45 20.43 17.49 15.19 14.24 13.73 12.81 12

冢

冣

冢

冣

Annulus, inner diameter D1 outer diameter D2 (Lamb, op. cit., p. 587):

D22 − D12 DE = (D22 − D12) D22 + D12 −  ln (D2 /D1)

冦

冤

冥冧

1/4

(6-52)

For isosceles triangles and regular polygons, see Sparrow (AIChE J., 8, 599–605 [1962]), Carlson and Irvine (J. Heat Transfer, 83, 441–444 [1961]), Cheng (Proc. Third Int. Heat Transfer Conf., New York, 1, 64–76 [1966]), and Shih (Can. J. Chem. Eng., 45, 285–294 [1967]). The critical Reynolds number for transition from laminar to turbulent flow in noncircular channels varies with channel shape. In rectangular ducts, 1,900 < Rec < 2,800 (Hanks and Ruo, Ind. Eng. Chem. Fundam., 5, 558–561 [1966]). In triangular ducts, 1,600 < Rec < 1,800 (Cope and Hanks, Ind. Eng. Chem. Fundam., 11, 106–117 [1972]; Bandopadhayay and Hinwood, J. Fluid Mech., 59, 775–783 [1973]). Nonisothermal Flow For nonisothermal flow of liquids, the friction factor may be increased if the liquid is being cooled or decreased if the liquid is being heated, because of the effect of temperature on viscosity near the wall. In shell and tube heat-exchanger design, the recommended practice is to first estimate f using the bulk mean liquid temperature over the tube length. Then, in laminar flow, the result is divided by (µa /µw)0.23 in the case of cooling or (µa /µw)0.38 in the case of heating. For turbulent flow, f is divided by (µa /µw)0.11 in the case of cooling or (µa /µw)0.17 in case of heating. Here, µa is the viscosity at the average bulk temperature and µw is the viscosity at the average wall temperature (Seider and Tate, Ind. Eng. Chem., 28, 1429–1435 [1936]). In the case of rough commercial pipes, rather than heat-exchanger tubing, it is common for flow to be in the “complete” turbulence regime where f is independent of Re. In such cases, the friction factor should not be corrected for wall temperature. If the liquid density varies with temperature, the average bulk density should be used to calculate the pressure drop from the friction factor. In addition, a (usually small) correction may be applied for acceleration effects by adding the term G2[(1/ρ2) − (1/ρ1)] from the mechanical energy balance to the pressure drop ∆P = P1 − P2, where G is the mass velocity. This acceleration results from small compressibility effects associated with temperature-dependent density. Christiansen and Gordon (AIChE J., 15, 504–507 [1969]) present equations and charts for frictional loss in laminar nonisothermal flow of Newtonian and non-Newtonian liquids heated or cooled with constant wall temperature. Frictional dissipation of mechanical energy can result in significant heating of fluids, particularly for very viscous liquids in small channels. Under adiabatic conditions, the bulk liquid temperature rise is given by ∆T = ∆P/Cv ρ for incompressible flow through a channel of constant cross-sectional area. For flow of polymers, this amounts to about 4°C per 10 MPa pressure drop, while for hydrocarbon liquids it is about 6°C per 10 MPa. The temperature rise in laminar flow is highly nonuniform, being concentrated near the pipe wall where most of the dissipation occurs. This may result in significant viscosity reduction near the wall, and greatly increased flow or reduced pressure drop, and a flattened velocity profile. Compensation should generally be made for the heat effect when ∆P exceeds 1.4 MPa (203 psi) for adiabatic walls or 3.5 MPa (508 psi) for isothermal walls (Gerard, Steidler, and Appeldoorn, Ind. Eng. Chem. Fundam., 4, 332–339 [1969]). Open Channel Flow For flow in open channels, the data are largely based on experiments with water in turbulent flow, in channels of sufficient roughness that there is no Reynolds number effect. The hydraulic radius approach may be used to estimate a friction factor with which to compute friction losses. Under conditions of uniform flow where liquid depth and cross-sectional area do not vary significantly with position in the flow direction, there is a balance between gravitational forces and wall stress, or equivalently between frictional losses and potential energy change. The mechanical energy balance reduces to lv = g(z1 − z2). In terms of the friction factor and hydraulic diameter or hydraulic radius, 2 f V 2L f V 2L lv =  =  = g(z1 − z2) (6-53) DH 2RH The hydraulic radius is the cross-sectional area divided by the wetted perimeter, where the wetted perimeter does not include the free sur-

FLUID DYNAMICS face. Letting S = sin θ = channel slope (elevation loss per unit length of channel, θ = angle between channel and horizontal), Eq. (6-53) reduces to 2gSRH V=  (6-54) f The most often used friction correlation for open channel flows is due to Manning (Trans. Inst. Civ. Engrs. Ireland, 20, 161 [1891]) and is equivalent to 29n2 f= (6-55) RH1/3 where n is the channel roughness, with dimensions of (length)1/6. Table 6-2 gives roughness values for several channel types. For gradual changes in channel cross section and liquid depth, and for slopes less than 10°, the momentum equation for a rectangular channel of width b and liquid depth h may be written as a differential equation in the flow direction x.

冪莦

dh h db f V 2(b + 2h)  (1 − Fr) − Fr   = S −  dx b dx 2gbh

冢 冣

(6-56)

For a given fixed flow rate Q = Vbh, and channel width profile b(x), Eq. (6-56) may be integrated to determine the liquid depth profile h(x). The dimensionless Froude number is Fr = V 2/gh. When Fr = 1, the flow is critical, when Fr < 1, the flow is subcritical, and when Fr > 1, the flow is supercritical. Surface disturbances move at a wave velocity c = 兹苶 gh; they cannot propagate upstream in supercritical flows. The specific energy Esp is nearly constant. 2

V Esp = h +  2g

(6-57)

This equation is cubic in liquid depth. Below a minimum value of Esp there are no real positive roots; above the minimum value there are two positive real roots. At this minimum value of Esp the flow is critical; that is, Fr = 1, V = 兹苶 gh, and Esp = (3/2)h. Near critical flow conditions, wave motion and sudden depth changes called hydraulic jumps are likely. Chow (Open Channel Hydraulics, McGraw-Hill, New York, 1959), discusses the numerous surface profile shapes which may exist in nonuniform open channel flows. For flow over a sharp-crested weir of width b and height L, from a liquid depth H, the flow rate is given approximately by 2 Q =  Cd b兹2苶g苶(H − L)3/2 3

(6-58)

where Cd ≈ 0.6 is a discharge coefficient. Flow through notched weirs is described under flow meters in Sec. 10 of the Handbook.

TABLE 6-2 Eq. (6-55)

Average Values of n for Manning Formula, Surface

n, m1/6

n, ft1/6

Cast-iron pipe, fair condition Riveted steel pipe Vitrified sewer pipe Concrete pipe Wood-stave pipe Planed-plank flume Semicircular metal flumes, smooth Semicircular metal flumes, corrugated Canals and ditches Earth, straight and uniform Winding sluggish canals Dredged earth channels Natural-stream channels Clean, straight bank, full stage Winding, some pools and shoals Same, but with stony sections Sluggish reaches, very deep pools, rather weedy

0.014 0.017 0.013 0.015 0.012 0.012 0.013 0.028

0.011 0.014 0.011 0.012 0.010 0.010 0.011 0.023

0.023 0.025 0.028

0.019 0.021 0.023

0.030 0.040 0.055 0.070

0.025 0.033 0.045 0.057

SOURCE: Brater and King, Handbook of Hydraulics, 6th ed., McGraw-Hill, New York, 1976, p. 7–22. For detailed information, see Chow, Open-Channel Hydraulics, McGraw-Hill, New York, 1959, pp. 110–123.

6-13

Non-Newtonian Flow For isothermal laminar flow of timeindependent non-Newtonian liquids, integration of the Cauchy momentum equations yields the fully developed velocity profile and flow rate–pressure drop relations. For the Bingham plastic fluid described by Eq. (6-3), in a pipe of diameter D and a pressure drop per unit length ∆P/L, the flow rate is given by πD3τw 4τy τ y4 Q= 1−+ 32µ∞ 3τw 3τ w4

冤

冥

(6-59)

where the wall stress is τw = D∆P/(4L). The velocity profile consists of a central nondeforming plug of radius rP = 2τy /(∆P/L) and an annular deforming region. The velocity profile in the annular region is given by 1 ∆P vz =   (R2 − r2) − τy(R − r) , rP ≤ r ≤ R (6-60) µ∞ 4L where r is the radial coordinate and R is the pipe radius. The velocity of the central, nondeforming plug is obtained by setting r = rP in Eq. (6-60). When Q is given and Eq. (6-59) is to be solved for τw and the pressure drop, multiple positive roots for the pressure drop may be found. The root corresponding to τw < τy is physically unrealizable, as it corresponds to rp > R and the pressure drop is insufficient to overcome the yield stress. For a power law fluid, Eq. (6-4), with constant properties K and n, the flow rate is given by ∆P 1/n n Q=π  (6-61)  R(1 + 3n)/n 2KL 1 + 3n and the velocity profile by ∆P 1/n n (6-62) vz =   [R(1 + n)/n − r (1 + n)/n] 2KL 1+n Similar relations for other non-Newtonian fluids may be found in Govier and Aziz and in Bird, Armstrong, and Hassager (Dynamics of Polymeric Liquids, vol. 1: Fluid Mechanics, Wiley, New York, 1977). For steady-state laminar flow of any time-independent viscous fluid, at average velocity V in a pipe of diameter D, the RabinowitschMooney relations give a general relationship for the shear rate at the pipe wall. 8V 1 + 3n′ γ˙ w =   (6-63) D 4n′ where n′ is the slope of a plot of D∆P/(4L) versus 8V/D on logarithmic coordinates, d ln [D∆P/(4L)] n′ =  (6-64) d ln (8V/D) By plotting capillary viscometry data this way, they can be used directly for pressure drop design calculations, or to construct the rheogram for the fluid. For pressure drop calculation, the flow rate and diameter determine the velocity, from which 8V/D is calculated and D∆P/(4L) read from the plot. For a Newtonian fluid, n′ = 1 and the shear rate at the wall is γ˙ = 8V/D. For a power law fluid, n′ = n. To construct a rheogram, n′ is obtained from the slope of the experimental plot at a given value of 8V/D. The shear rate at the wall is given by Eq. (6-63) and the corresponding shear stress at the wall is τw = D∆P/(4L) read from the plot. By varying the value of 8V/D, the shear rate versus shear stress plot can be constructed. The generalized approach of Metzner and Reed (AIChE J., 1, 434 [1955]) for time-independent non-Newtonian fluids defines a modified Reynolds number as Dn′V 2 − n′ρ ReMR ⬅  (6-65) K′8n′ − 1 where K′ satisfies 8V n′ D∆P (6-66)  = K′  4L D With this definition, f = 16/ReMR is automatically satisfied at the value of 8V/D where K′ and n′ are evaluated. Equation (6-66) may be obtained by integration of Eq. (6-64) only when n′ is a constant, as, for

冤

冥

冢

冢

冣 冢

冣 冢

冣

冣

冢

冣

冢 冣

6-14

FLUID AND PARTICLE DYNAMICS

example, the cases of Newtonian and power law fluids. For Newtonian fluids, K′ = µ and n′ = 1; for power law fluids, K′ = K[(1 + 3n)/ (4n)]n and n′ = n. For Bingham plastics, K′ and n′ are variable, given as a function of τw (Metzner, Ind. Eng. Chem., 49, 1429–1432 [1957]). K = τ 1w− n′

冤

µ∞  1 − 4τy /3τw + (τy /τw)4/3

冥

3n + 1 µeff = K  4n

冢

n′

(6-67)

1 − 4τy /(3τw) + (τy /τw)4/3 n′ =  1 − (τy /τw)4

(6-68)

For laminar flow of power law fluids in channels of noncircular cross section, see Schecter (AIChE J., 7, 445–448 [1961]), Wheeler and Wissler (AIChE J., 11, 207–212 [1965]), Bird, Armstrong, and Hassager (Dynamics of Polymeric Liquids, vol. 1: Fluid Mechanics, Wiley, New York, 1977), and Skelland (Non-Newtonian Flow and Heat Transfer, Wiley, New York, 1967). Steady state, fully developed laminar flows of viscoelastic fluids in straight, constant-diameter pipes show no effects of viscoelasticity. The viscous component of the constitutive equation may be used to develop the flow rate–pressure drop relations, which apply downstream of the entrance region after viscoelastic effects have disappeared. A similar situation exists for time-dependent fluids. The transition to turbulent flow begins at ReMR in the range of 2,000 to 2,500 (Metzner and Reed, AIChE J., 1, 434 [1955]). For Bingham plastic materials, K′ and n′ must be evaluated for the τw condition in question in order to determine ReMR and establish whether the flow is laminar. An alternative method for Bingham plastics is by Hanks (Hanks, AIChE J., 9, 306 [1963]; 14, 691 [1968]; Hanks and Pratt, Soc. Petrol. Engrs. J., 7, 342 [1967]; and Govier and Aziz, pp. 213–215). The transition from laminar to turbulent flow is influenced by viscoelastic properties (Metzner and Park, J. Fluid Mech., 20, 291 [1964]) with the critical value of ReMR increased to beyond 10,000 for some materials. For turbulent flow of non-Newtonian fluids, the design chart of Dodge and Metzner (AIChE J., 5, 189 [1959]), Fig. 6-11, is most widely used. For Bingham plastic materials in turbulent flow, it is generally assumed that stresses greatly exceed the yield stress, so that the friction factor–Reynolds number relationship for Newtonian fluids applies, with µ∞ substituted for µ. This is equivalent to setting n′ = 1 and τy /τw = 0 in the Dodge-Metzner method, so that ReMR = DVρ/µ∞. Wilson and Thomas (Can. J. Chem. Eng., 63, 539–546 [1985]) give friction factor equations for turbulent flow of power law fluids and Bingham plastic fluids. Power law fluids: 1−n 1+n 1 1  =  + 8.2  + 1.77 ln  1 + n 2 fN 兹苶f 兹苶

冢

where fN is the friction factor for Newtonian fluid evaluated at Re = DVρ/µeff where the effective viscosity is

冣

(6-69)

FIG. 6-11 Fanning friction factor for non-Newtonian flow. (From Dodge and Metzner, Am. Inst. Chem. Eng. J., 5, 189 [1959]).

n−1

冣 冢 D冣 8V

n−1

(6-70)

Bingham fluids: (1 − ξ)2 1 1  =  + 1.77 ln  + ξ(10 + 0.884ξ) 1+ξ fN 兹苶f 兹苶

冢

冣

(6-71)

where fN is evaluated at Re = DVρ/µ∞ and ξ = τy /τw. Iteration is required to use this equation since τw = fρV 2/2. Drag reduction in turbulent flow can be achieved by adding soluble high molecular weight polymers in extremely low concentration to Newtonian liquids. The reduction in friction is generally believed to be associated with the viscoelastic nature of the solutions effective in the wall region. For a given polymer, there is a minimum molecular weight necessary to initiate drag reduction at a given flow rate, and a critical concentration above which drag reduction will not occur (Kim, Little and Ting, J. Colloid Interface Sci., 47, 530–535 [1974]). Drag reduction is reviewed by Hoyt (J. Basic Eng., 94, 258–285 [1972]); Little, et al. (Ind. Eng. Chem. Fundam., 14, 283–296 [1975]) and Virk (AIChE J., 21, 625–656 [1975]). At maximum possible drag reduction in smooth pipes, 1 50.73  = −19 log  (6-72) Re兹苶f 兹苶f

冢

or, approximately,

冣

0.58 f= Re0.58

(6-73)

for 4,000 < Re < 40,000. The actual drag reduction depends on the polymer system. For further details, see Virk (ibid.). Economic Pipe Diameter, Turbulent Flow The economic optimum pipe diameter may be computed so that the last increment of investment reduces the operating cost enough to produce the required minimum return on investment. For long cross-country pipelines, alloy pipes of appreciable length and complexity, or pipelines with control valves, detailed analyses of investment and operating costs should be made. Peters and Timmerhaus (Plant Design and Economics for Chemical Engineers, 4th ed., McGraw-Hill, New York, 1991) provide a detailed method for determining the economic optimum size. For pipelines of the lengths usually encountered in chemical plants and petroleum refineries, simplified selection charts are often adequate. In many cases there is an economic optimum velocity that is nearly independent of diameter, which may be used to estimate the economic diameter from the flow rate. For low-viscosity liquids in schedule 40 steel pipe, economic optimum velocity is typically in the range of 1.8 to 2.4 m/s (5.9 to 7.9 ft/s). For gases with density ranging from 0.2 to 20 kg/m3 (0.013 to 1.25 lbm/ft3), the economic optimum velocity is about 40 m/s to 9 m/s (131 to 30 ft/s). Charts and rough guidelines for economic optimum size do not apply to multiphase flows. Economic Pipe Diameter, Laminar Flow Pipelines for the transport of high-viscosity liquids are seldom designed purely on the basis of economics. More often, the size is dictated by operability considerations such as available pressure drop, shear rate, or residence time distribution. Peters and Timmerhaus (ibid., Chap. 10) provide an economic pipe diameter chart for laminar flow. For non-Newtonian fluids, see Skelland (Non-Newtonian Flow and Heat Transfer, Chap. 7, Wiley, New York, 1967). Vacuum Flow When gas flows under high vacuum conditions or through very small openings, the continuum hypothesis is no longer appropriate if the channel dimension is not very large compared to the mean free path of the gas. When the mean free path is comparable to the channel dimension, flow is dominated by collisions of molecules with the wall, rather than by collisions between molecules. An approximate expression based on Brown, et al. (J. Appl. Phys., 17, 802–813 [1946]) for the mean free path is 8RT  冢 冣 冪莦 πM

2µ λ=  p

w

(6-74)

FLUID DYNAMICS The Knudsen number Kn is the ratio of the mean free path to the channel dimension. For pipe flow, Kn = λ/D. Molecular flow is characterized by Kn > 1.0; continuum viscous (laminar or turbulent) flow is characterized by Kn < 0.01. Transition or slip flow applies over the range 0.01 < Kn < 1.0. Vacuum flow is usually described with flow variables different from those used for normal pressures, which often leads to confusion. Pumping speed S is the actual volumetric flow rate of gas through a flow cross section. Throughput Q is the product of pumping speed and absolute pressure. In the SI system, Q has units of Pa ⋅ m3/s. Q = Sp

(6-75)

The mass flow rate w is related to the throughput using the ideal gas law. Mw w=Q (6-76) RT Throughput is therefore proportional to mass flow rate. For a given mass flow rate, throughput is independent of pressure. The relation between throughput and pressure drop ∆ p = p1 − p2 across a flow element is written in terms of the conductance C. Resistance is the reciprocal of conductance. Conductance has dimensions of volume per time. Q = C∆p (6-77) The conductance of a series of flow elements is given by 1 1 1 1  =  +  +  + ⋅⋅⋅ C C1 C2 C3 while for elements in parallel,

(6-78)

C = C1 + C2 + C3 + ⋅⋅⋅

(6-79)

For a vacuum pump of speed Sp withdrawing from a vacuum vessel through a connecting line of conductance C, the pumping speed at the vessel is SpC S= (6-80) Sp + C Molecular Flow Under molecular flow conditions, conductance is independent of pressure. It is proportional to 兹苶 T苶 /Mw苶, with the proportionality constant a function of geometry. For fully developed pipe flow, πD3 RT C=  (6-81) 8L Mw For an orifice of area A, RT (6-82) C = 0.40A  Mw Conductance equations for several other geometries are given by Ryans and Roper (Process Vacuum System Design and Operation, Chap. 2, McGraw-Hill, New York, 1986). For a circular annulus of outer and inner diameters D1 and D2 and length L, the method of Guthrie and Wakerling (Vacuum Equipment and Techniques, McGraw-Hill, New York, 1949) may be written

冪莦

1 K =  1 + (L/D)

for

0 ≤ L/D ≤ 0.75

6-15 (6-84)

1 + 0.8(L/D) K = 2 for L/D > 0.75 (6-85) 1 + 1.90(L/D) + 0.6(L/D) For L/D > 100, the error in neglecting the end correction by using the fully developed pipe flow equation (6-81) is less than 2 percent. For rectangular channels, see Normand (Ind. Eng. Chem., 40, 783–787 [1948]). Yu and Sparrow ( J. Basic Eng., 70, 405–410 [1970]) give a theoretically derived chart for slot seals with or without a sheet located in or passing through the seal, giving mass flow rate as a function of the ratio of seal plate thickness to gap opening. Slip Flow In the transition region between molecular flow and continuum viscous flow, the conductance for fully developed pipe flow is most easily obtained by the method of Brown, et al. (J. Appl. Phys., 17, 802–813 [1946]), which uses the parameter X=

2µ RT  冪莦π8 冢Dλ 冣 = 冢 M p D 冣 冪莦

(6-86)

m

where pm is the arithmetic mean absolute pressure. A correction factor F, read from Fig. 6-12 as a function of X, is applied to the conductance for viscous flow. πD4pm C=F (6-87) 128µL For slip flow through square channels, see Milligan and Wilkerson (J. Eng. Ind., 95, 370–372 [1973]). For slip flow through annuli, see Maegley and Berman (Phys. Fluids, 15, 780–785 [1972]). The pump-down time θ for evacuating a vessel in the absence of air in-leakage is given approximately by p1 − p0 V θ = t ln  (6-88) S0 p2 − p0 where Vt = volume of vessel plus volume of piping between vessel and pump; S0 = system speed as given by Eq. (6-80), assumed independent of pressure; p1 = initial vessel pressure; p2 = final vessel pressure; and p0 = lowest pump intake pressure attainable with the pump in question. See Dushman and Lafferty (Scientific Foundations of Vacuum Technique, 2d ed., Wiley, New York, 1962). The amount of inerts which has to be removed by a pumping system after the pump-down stage depends on the in-leakage of air at the various fittings, connections, and so on. Air leakage is often correlated with system volume and pressure, but this approach introduces uncer-

冢 冣 冢

冣

冪莦

(D1 − D2)2(D1 + D2) C = 0.42K  L

RT  冪莦 M

(6-83)

w

where K is a dimensionless constant with values given in Table 6-3. For a short pipe of circular cross section, the conductance as calculated for an orifice from Eq. (6-82) is multiplied by a correction factor K which may be approximated as (Kennard, Kinetic Theory of Gases, McGraw-Hill, New York, 1938, pp. 306–308) TABLE 6-3

Constants for Circular Annuli

D2 /D1

K

D2 /D1

K

0 0.259 0.500

1.00 1.072 1.154

0.707 0.866 0.966

1.254 1.430 1.675

FIG. 6-12 Correction factor for Poiseuille’s equation at low pressures. Curve A: experimental curve for glass capillaries and smooth metal tubes. (From Brown, et al., J. Appl. Phys., 17, 802 [1946].) Curve B: experimental curve for iron pipe (From Riggle, Courtesy of E. I. du Pont de Nemours & Co.)

6-16

FLUID AND PARTICLE DYNAMICS

tainty because the number and size of leaks does not necessily correlate with system volume, and leakage is sensitive to maintenance quality. Ryans and Roper (Process Vacuum System Design and Operation, McGraw-Hill, New York, 1986) present a thorough discussion of air leakage.

Contraction and Entrance Losses For a sudden contraction at a sharp-edged entrance to a pipe or sudden reduction in crosssectional area of a channel, as shown in Fig. 6-13a, the loss coefficient based on the downstream velocity V2 is given for turbulent flow in Crane Co. Tech Paper 410 (1980) approximately by

冢

A K = 0.5 1 − 2 A1

FRICTIONAL LOSSES IN PIPELINE ELEMENTS The viscous or frictional loss term in the mechanical energy balance for most cases is obtained experimentally. For many common fittings found in piping systems, such as expansions, contractions, elbows and valves, data are available to estimate the losses. Substitution into the energy balance then allows calculation of pressure drop. A common error is to assume that pressure drop and frictional losses are equivalent. Equation (6-16) shows that in addition to frictional losses, other factors such as shaft work and velocity or elevation change influence pressure drop. Losses lv for incompressible flow in sections of straight pipe of constant diameter may be calculated as previously described using the Fanning friction factor: ∆P 2 f V 2L lv =  =  (6-89) ρ D where ∆P = drop in equivalent pressure, P = p + ρgz, with p = pressure, ρ = fluid density, g = acceleration of gravity, and z = elevation. Losses in the fittings of a piping network are frequently termed minor losses or miscellaneous losses. These descriptions are misleading because in process piping fitting losses are often much greater than the losses in straight piping sections. Equivalent Length and Velocity Head Methods Two methods are in common use for estimating fitting loss. One, the equivalent length method, reports the losses in a piping element as the length of straight pipe which would have the same loss. For turbulent flows, the equivalent length is usually reported as a number of diameters of pipe of the same size as the fitting connection; Le /D is given as a fixed quantity, independent of D. This approach tends to be most accurate for a single fitting size and loses accuracy with deviation from this size. For laminar flows, Le /D correlations normally have a size dependence through a Reynolds number term. The other method is the velocity head method. The term V 2/2g has dimensions of length and is commonly called a velocity head. Application of the Bernoulli equation to the problem of frictionless discharge at velocity V through a nozzle at the bottom of a column of liquid of height H shows that H = V 2/2g. Thus H is the liquid head corresponding to the velocity V. Use of the velocity head to scale pressure drops has wide application in fluid mechanics. Examination of the Navier-Stokes equations suggests that when the inertial terms dominate the viscous terms, pressure gradients are expected to be proportional to ρV 2 where V is a characteristic velocity of the flow. In the velocity head method, the losses are reported as a number of velocity heads K. Then, the engineering Bernoulli equation for an incompressible fluid can be written ρV22 ρV12 ρV 2 p1 − p2 = α2  − α1  + ρg(z2 − z1) + K  (6-90) 2 2 2 where V is the reference velocity upon which the velocity head loss coefficient K is based. For a section of straight pipe, K = 4 f L/D.

(a) FIG. 6-13

(b)

冣

(6-91)

Example 5: Entrance Loss Water, ρ = 1000 kg/m3, flows from a large vessel through a sharp-edged entrance into a pipe at a velocity in the pipe of 2 m/s. The flow is turbulent. Estimate the pressure drop from the vessel into the pipe. With A2 /A1 ∼ 0, the viscous loss coefficient is K = 0.5 from Eq. (6-91). The mechanical energy balance, Eq. (6-16) with V1 = 0 and z2 − z1 = 0 and assuming uniform flow (α2 = 1) becomes ρV22 ρV22 p1 − p2 =  + 0.5  = 4,000 + 2,000 = 6,000 Pa 2 2 Note that the total pressure drop consists of 0.5 velocity heads of frictional loss contribution, and 1 velocity head of velocity change contribution. The frictional contribution is a permanent loss of mechanical energy by viscous dissipation. The acceleration contribution is reversible; if the fluid were subsequently decelerated in a frictionless diffuser, a 4,000 Pa pressure rise would occur.

For a trumpet-shaped rounded entrance, with a radius of rounding greater than about 15 percent of the pipe diameter (Fig. 6-13b), the turbulent flow loss coefficient K is only about 0.1 (Vennard and Street, Elementary Fluid Mechanics, 5th ed., Wiley, New York, 1975, pp. 420–421). Rounding of the inlet prevents formation of the vena contracta, thereby reducing the resistance to flow. For laminar flow the losses in sudden contraction may be estimated for area ratios A2 /A1 < 0.2 by an equivalent additional pipe length Le given by Le /D = 0.3 + 0.04Re

(6-92)

where D is the diameter of the smaller pipe and Re is the Reynolds number in the smaller pipe. For laminar flow in the entrance to rectangular ducts, see Shah (J. Fluids Eng., 100, 177–179 [1978]) and Roscoe (Philos. Mag., 40, 338–351 [1949]). For creeping flow, Re < 1, of power law fluids, the entrance loss is approximately Le/D = 0.3/n (Boger, Gupta, and Tanner, J. Non-Newtonian Fluid Mech., 4, 239–248 [1978]). For viscoelastic fluid flow in circular channels with sudden contraction, a toroidal vortex forms upstream of the contraction plane. Such flows are reviewed by Boger (Ann. Review Fluid Mech., 19, 157–182 [1987]). For creeping flow through conical converging channels, inertial acceleration terms are negligible and the viscous pressure drop ∆p = ρlv may be computed by integration of the differential form of the Hagen-Poiseuille equation Eq. (6-36), provided the angle of convergence is small. The result for a power law fluid is 3n + 1 ∆p = K  4n

冢

8V 1  1 − 冢  冣 冥冧 冣 冢 D 冣 冦 6n tan (α/2) 冤 D n

2

n

D2

2

3n

1

where D1 = inlet diameter D2 = exit diameter V2 = velocity at the exit α = total included angle

(c)

(d)

Contractions and enlargements: (a) sudden contraction, (b) rounded contraction, (c) sudden enlargement, and (d) uniformly diverging duct.

(6-93)

FLUID DYNAMICS Equation (6-93) agrees with experimental data (Kemblowski and Kiljanski, Chem. Eng. J. (Lausanne), 9, 141–151 [1975]) for α < 11°. For Newtonian liquids, Eq. (6-93) simplifies to 8V 1 D 3 ∆p = µ 2  1 − 2 (6-94) D2 6 tan (α/2) D1 For creeping flow through rectangular or two-dimensional converging channels, the differential form of the Hagen-Poiseulle equation with equivalent diameter given by Eq. (6-49) may be used, provided the convergence is gradual. Expansion and Exit Losses For ducts of any cross section, the frictional loss for a sudden enlargement (Fig. 6-13c) with turbulent flow is given by the Borda-Carnot equation: A 2 V12 − V22 V12 lv =  = 1 − 1 (6-95) 2 2 A2 where V1 = velocity in the smaller duct V2 = velocity in the larger duct A1 = cross-sectional area of the smaller duct A2 = cross-sectional area of the larger duct

冢 冣冦

冤 冢 冣 冥冧

冢

冣

Equation (6-95) is valid for incompressible flow. For compressible flows, see Benedict, Wyler, Dudek, and Gleed ( J. Eng. Power, 98, 327–334 [1976]). For an infinite expansion, A1/A2 = 0, Eq. (6-95) shows that the exit loss from a pipe is 1 velocity head. This result is easily deduced from the mechanical energy balance Eq. (6-90), noting that p1 = p2. This exit loss is due to the dissipation of the discharged jet; there is no pressure drop at the exit. For creeping Newtonian flow (Re < 1), the frictional loss due to a sudden enlargement should be obtained from the same equation for a sudden contraction (Eq. [6-92]). Note, however, that Boger, Gupta, and Tanner (ibid.) give an exit friction equivalent length of 0.12 diameter, increasing for power law fluids as the exponent decreases. For laminar flows at higher Reynolds numbers, the pressure drop is twice that given by Eq. (6-95). This results from the velocity profile factor α in the mechanical energy balance being 2.0 for the parabolic laminar velocity profile. If the transition from a small to a large duct of any cross-sectional shape is accomplished by a uniformly diverging duct (see Fig. 6-13d) with a straight axis, the total frictional pressure drop can be computed by integrating the differential form of Eq. (6-89), dlv /dx = 2 f V 2/D over the length of the expansion, provided the total angle α between the diverging walls is less than 7°. For angles between 7 and 45°, the loss coefficient may be estimated as 2.6 sin (α/2) times the loss coefficient for a sudden expansion; see Hooper (Chem. Eng., Nov. 7, 1988). Gibson (Hydraulics and Its Applications, 5th ed., Constable, London 1952, p. 93) recommends multiplying the sudden enlargement loss by 0.13 for 5° < α < 7.5° and by 0.0110α1.22 for 7.5° < α < 35°. For angles greater than 35 to 45°, the losses are normally considered equal to those for a sudden expansion, although in some cases the losses may be greater. Expanding flow through standard pipe reducers should be treated as sudden expansions. Trumpet-shaped enlargements for turbulent flow designed for constant decrease in velocity head per unit length were found by Gibson (ibid., p. 95) to give 20 to 60 percent less frictional loss than straight taper pipes of the same length. A special feature of expansion flows occurs when viscoelastic liquids are extruded through a die at a low Reynolds number. The extrudate may expand to a diameter several times greater than the die diameter, whereas for a Newtonian fluid the diameter expands only 10 percent. This phenomenon, called die swell, is most pronounced with short dies (Graessley, Glasscock, and Crawley, Trans. Soc. Rheol., 14, 519–544 [1970]). For velocity distribution measurements near the die exit, see Goulden and MacSporran (J. Non-Newtonian Fluid Mech., 1, 183–198 [1976]) and Whipple and Hill (AIChE J., 24, 664–671 [1978]). At high flow rates, the extrudate becomes distorted, suffering melt fracture at wall shear stresses greater than 105 N/m2. This phenomenon is reviewed by Denn (Ann. Review Fluid Mech., 22, 13–34 [1990]). Ramamurthy (J. Rheol., 30, 337–357 [1986]) has found a dependence of apparent stick-slip behavior in melt fracture to be dependent on the material of construction of the die.

6-17

Fittings and Valves For turbulent flow, the frictional loss for fittings and valves can be expressed by the equivalent length or velocity head methods. As fitting size is varied, K values are relatively more constant than Le /D values, but since fittings generally do not achieve geometric similarity between sizes, K values tend to decrease with increasing fitting size. Table 6-4 gives K values for many types of fittings and valves. Manufacturers of valves, especially control valves, express valve capacity in terms of a flow coefficient Cv, which gives the flow rate through the valve in gal/min of water at 60°F under a pressure drop of 1 lbf/in2. It is related to K by C1d 2 Cv =  (6-96) K 兹苶 where C1 is a dimensional constant equal to 29.9 and d is the diameter of the valve connections in inches. For laminar flow, data for the frictional loss of valves and fittings are meager. (Beck and Miller, J. Am. Soc. Nav. Eng., 56, 62–83 [1944]; Beck, ibid., 56, 235–271, 366–388, 389–395 [1944]; De Craene, Heat. Piping Air Cond., 27[10], 90–95 [1955]; Karr and Schutz, J. Am. Soc. Nav. Eng., 52, 239–256 [1940]; and Kittredge and Rowley, Trans. ASME, 79, 1759–1766 [1957]). The data of Kittredge and Rowley indicate that K is constant for Reynolds numbers above 500 to 2,000, but increases rapidly as Re decreases below 500. Typical values for K for laminar flow Reynolds numbers are shown in Table 6-5. Methods to calculate losses for tee and wye junctions for dividing and combining flow are given by Miller (Internal Flow Systems, 2d ed., Chap. 13, BHRA, Cranfield, 1990), including effects of Reynolds number, angle between legs, area ratio, and radius. Junctions with more than three legs are also discussed. The sources of data for the loss coefficient charts are Blaisdell and Manson (U.S. Dept. Agric. Res. Serv. Tech. Bull. 1283 [August 1963]) for combining flow and Gardel (Bull. Tech. Suisses Romande, 85[9], 123–130 [1957]; 85[10], 143–148 [1957]) together with additional unpublished data for dividing flow. Miller (Internal Flow Systems, 2d ed., Chap. 13, BHRA, Cranfield, 1990) gives the most complete information on losses in bends and curved pipes. For turbulent flow in circular cross-section bends of constant area, as shown in Fig. 6-14a, a more accurate estimate of the loss coefficient K than that given in Table 6-4 is K = K*CReCoCf

(6-97)

where K*, given in Fig. 6-14b, is the loss coefficient for a smoothwalled bend at a Reynolds number of 106. The Reynolds number correction factor CRe is given in Fig. 6-14c. For 0.7 < r/D < 1 or for K* < 0.4, use the CRe value for r/D = 1. Otherwise, if r/D < 1, obtain CRe from K* CRe =  K* + 0.2(1 − CRe, r/D = 1)

(6-98)

The correction Co (Fig. 6-14d) accounts for the extra losses due to developing flow in the outlet tangent of the pipe, of length Lo. The total loss for the bend plus outlet pipe includes the bend loss K plus the straight pipe frictional loss in the outlet pipe 4f Lo /D. Note that Co = 1 for Lo /D greater than the termination of the curves on Fig. 6-14d, which indicate the distance at which fully developed flow in the outlet pipe is reached. Finally, the roughness correction is frough Cf =  fsmooth

(6-99)

where frough is the friction factor for a pipe of diameter D with the roughness of the bend, at the bend inlet Reynolds number. Similarly, fsmooth is the friction factor for smooth pipe. For Re > 106 and r/D ≥ 1, use the value of Cf for Re = 106. Example 6: Losses with Fittings and Valves It is desired to calculate the liquid level in the vessel shown in Fig. 6-15 required to produce a discharge velocity of 2 m/s. The fluid is water at 20°C with ρ = 1,000 kg/m3 and µ = 0.001 Pa ⋅ s, and the butterfly valve is at θ = 10°. The pipe is 2-in Schedule 40, with an inner diameter of 0.0525 m. The pipe roughness is 0.046 mm. Assuming the flow is turbulent and taking the velocity profile factor α = 1, the engineering Bernoulli equation Eq. (6-16), written between surfaces 1 and 2, where the

6-18

FLUID AND PARTICLE DYNAMICS

TABLE 6-4 Additional Frictional Loss for Turbulent Flow through Fittings and Valvesa

Type of fitting or valve 45° ell, standardb,c,d,e,f 45° ell, long radiusc 90° ell, standardb,c,e,f,g,h Long radius b,c,d,e Square or miter h 180° bend, close returnb,c,e Tee, standard, along run, branch blanked off e Used as ell, entering rung,i Used as ell, entering branchc,g,i Branching flowi,j,k Coupling c,e Unione Gate valve,b,e,m open e openn a openn d openn Diaphragm valve,o open e openn a openn d openn Globe valve,e,m Bevel seat, open a openn Composition seat, open a openn Plug disk, open e open n a open n d open n Angle valve,b,e open Y or blowoff valve,b,m open Plug cockp θ = 5° θ = 10° θ = 20° θ = 40° θ = 60° Butterfly valve p θ = 5° θ = 10° θ = 20° θ = 40° θ = 60° Check valve,b,e,m swing Disk Ball Foot valvee Water meter,h disk Piston Rotary (star-shaped disk) Turbine-wheel a

Additional friction loss, equivalent no. of velocity heads, K 0.35 0.2 0.75 0.45 1.3 1.5 0.4 1.0 1.0 1l 0.04 0.04 0.17 0.9 4.5 24.0 2.3 2.6 4.3 21.0 6.0 9.5 6.0 8.5 9.0 13.0 36.0 112.0 2.0 3.0 0.05 0.29 1.56 17.3 206.0 0.24 0.52 1.54 10.8 118.0 2.0q 10.0q 70.0q 15.0 7.0r 15.0r 10.0r 6.0r

Lapple, Chem. Eng., 56(5), 96–104 (1949), general survey reference. “Flow of Fluids through Valves, Fittings, and Pipe,” Tech. Pap. 410, Crane Co., 1969. c Freeman, Experiments upon the Flow of Water in Pipes and Pipe Fittings, American Society of Mechanical Engineers, New York, 1941. d Giesecke, J. Am. Soc. Heat. Vent. Eng., 32, 461 (1926). e Pipe Friction Manual, 3d ed., Hydraulic Institute, New York, 1961. f Ito, J. Basic Eng., 82, 131–143 (1960). g Giesecke and Badgett, Heat. Piping Air Cond., 4(6), 443–447 (1932). h Schoder and Dawson, Hydraulics, 2d ed., McGraw-Hill, New York, 1934, p. 213. i Hoopes, Isakoff, Clarke, and Drew, Chem. Eng. Prog., 44, 691–696 (1948). j Gilman, Heat. Piping Air Cond., 27(4), 141–147 (1955). k McNown, Proc. Am. Soc. Civ. Eng., 79, Separate 258, 1–22 (1953); discussion, ibid., 80, Separate 396, 19–45 (1954). For the effect of branch spacing on junction losses in dividing flow, see Hecker, Nystrom, and Qureshi, Proc. Am. Soc. Civ. Eng., J. Hydraul. Div., 103(HY3), 265–279 (1977). l This is pressure drop (including friction loss) between run and branch, based on velocity in the mainstream before branching. Actual value depends on the flow split, ranging from 0.5 to 1.3 if mainstream enters run and from 0.7 to 1.5 if mainstream enters branch. m Lansford, Loss of Head in Flow of Fluids through Various Types of 1a-in. Valves, Univ. Eng. Exp. Sta. Bull. Ser. 340, 1943. b

pressures are both atmospheric and the fluid velocities are 0 and V = 2 m/s, respectively, and there is no shaft work, simplifies to V2 gZ =  + lv 2 Contributing to lv are losses for the entrance to the pipe, the three sections of straight pipe, the butterfly valve, and the 90° bend. Note that no exit loss is used because the discharged jet is outside the control volume. Instead, the V 2/2 term accounts for the kinetic energy of the discharging stream. The Reynolds number in the pipe is DVρ 0.0525 × 2 × 1000 Re =  =  = 1.05 × 105 µ 0.001 From Fig. 6-9 or Eq. (6-38), at
/D = 0.046 × 10−3/0.0525 = 0.00088, the friction factor is about 0.0054. The straight pipe losses are then 4fL V 2 lv(sp) =   D 2

冢 冣

4 × 0.0054 × (1 + 1 + 1) V 2 =   0.0525 2

冢

冣

V2 = 1.23  2 The losses from Table 6-4 in terms of velocity heads K are K = 0.5 for the sudden contraction and K = 0.52 for the butterfly valve. For the 90° standard radius (r/D = 1), the table gives K = 0.75. The method of Eq. (6-94), using Fig. 6-14, gives K = K*CReCoCf

冢

0.0054 = 0.24 × 1.24 × 1.0 ×  0.0044 = 0.37

冣

This value is more accurate than the value in Table 6-4. The value fsmooth = 0.0044 is obtainable either from Eq. (6-37) or Fig. 6-9. The total losses are then V2 V2 lv = (1.23 + 0.5 + 0.52 + 0.37)  = 2.62  2 2 and the liquid level Z is 1 V2 V2 V2 Z =   + 2.62  = 3.62  g 2 2 2g

冢

冣

3.62 × 2 =  = 0.73 m 2 × 9.81 2

Curved Pipes and Coils For flow through curved pipe or coil, a secondary circulation perpendicular to the main flow called the Dean effect occurs. This circulation increases the friction relative to straight pipe flow and stabilizes laminar flow, delaying the transition Reynolds number to about D Recrit = 2,100 1 + 12  (6-100) Dc where Dc is the coil diameter. Equation (6-100) is valid for 10 < Dc / D < 250. The Dean number is defined as Re De =  (6-101) (Dc /D)1/2 In laminar flow, the friction factor for curved pipe fc may be expressed in terms of the straight pipe friction factor f = 16/Re as (Hart, Chem. Eng. Sci., 43, 775–783 [1988])

冢

冪莦冣

TABLE 6-5 Additional Frictional Loss for Laminar Flow through Fittings and Valves Additional frictional loss expressed as K Type of fitting or valve

Re = 1,000

500

100

50

90° ell, short radius Gate valve Globe valve, composition disk Plug Angle valve Check valve, swing

0.9 1.2 11 12 8 4

1.0 1.7 12 14 8.5 4.5

7.5 9.9 20 19 11 17

16 24 30 27 19 55

SOURCE: From curves by Kittredge and Rowley, Trans. Am. Soc. Mech. Eng., 79, 1759–1766 (1957).

FLUID DYNAMICS

V2 = 2 m/s 2

1

m

1

Z

90° horizontal bend 1m FIG. 6-15

1m

6-19

10, 328–332 [1971]). For friction loss in laminar flow through semicircular ducts, see Masliyah and Nandakumar (AIChE J., 25, 478– 487 [1979]); for curved channels of square cross section, see Cheng, Lin, and Ou (J. Fluids Eng., 98, 41–48 [1976]). For non-Newtonian (power law) fluids in coiled tubes, Mashelkar and Devarajan (Trans. Inst. Chem. Eng. (London), 54, 108–114 [1976]) propose the correlation fc = (9.07 − 9.44n + 4.37n2)(D/Dc)0.5(De′)−0.768 + 0.122n (6-104) where De′ is a modified Dean number given by 1 6n + 2 De′ =   8 n

冢

Tank discharge example.

D  冣 Re 冪莦 D n

MR

(6-105)

c

De1.5 fc /f = 1 + 0.090  (6-102) 70 + De For turbulent flow, equations by Ito (J. Basic Eng, 81, 123 [1959]) and Srinivasan, Nandapurkar, and Holland (Chem. Eng. [London] no. 218, CE113-CE119 [May 1968]) may be used, with probable accuracy of 15 percent. Their equations are similar to 0.0073 0.079 fc =  + (6-103) Re0.25 兹(D 苶c苶/D 苶)苶 The pressure drop for flow in spirals is discussed by Srinivasan, et al. (loc. cit.) and Ali and Seshadri (Ind. Eng. Chem. Process Des. Dev.,

where ReMR is the Metzner-Reed Reynolds number, Eq. (6-65). This correlation was tested for the range De′ = 70 to 400, D/Dc = 0.01 to 0.135, and n = 0.35 to 1. See also Oliver and Asghar (Trans. Inst. Chem. Eng. [London], 53, 181–186 [1975]). Screens The pressure drop for incompressible flow across a screen of fractional free area α may be computed from

(a)

(b)

(c)

(d)

冢

冣

Loss coefficients for flow in bends and curved pipes: (a) flow geometry, (b) loss coefficient for a smooth-walled bend at Re = 106, (c) Re correction factor, (d) outlet pipe correction factor (From D. S. Miller, Internal Flow Systems, 2d. ed., BHRA, Cranfield, U.K., 1990.)

FIG. 6-14

6-20

FLUID AND PARTICLE DYNAMICS

FIG. 6-16

Screen discharge coefficients, plain square-mesh screens. (Courtesy of E. I. du Pont de Nemours

& Co.)

ρV 2 ∆p = K  2

(6-106)

where ρ = fluid density V = superficial velocity based upon the gross area of the screen K = velocity head loss 1 − α2

冢 冣 冢 α 冣

1 K = 2 C

2

(6-107)

The discharge coefficient for the screen C with aperture Ds is given as a function of screen Reynolds number Re = Ds(V/α)ρ/µ in Fig. 6-16 for plain square-mesh screens, α = 0.14 to 0.79. This curve fits most of the data within 20 percent. In the laminar flow region, Re < 20, the discharge coefficient can be computed from

苶e苶 C = 0.1兹R

(6-108)

Coefficients greater than 1.0 in Fig. 6-16 probably indicate partial pressure recovery downstream of the minimum aperture, due to rounding of the wires. Grootenhuis (Proc. Inst. Mech. Eng. [London], A168, 837–846 [1954]) presents data which indicate that for a series of screens, the total pressure drop equals the number of screens times the pressure drop for one screen, and is not affected by the spacing between screens or their orientation with respect to one another, and presents a correlation for frictional losses across plain square-mesh screens and sintered gauzes. Armour and Cannon (AIChE J., 14, 415–420 [1968]) give a correlation based on a packed bed model for plain, twill, and “dutch” weaves. For losses through monofilament fabrics see Pedersen (Filtr. Sep., 11, 586–589 [1975]). For screens inclined at an angle θ, use the normal velocity component V ′ V′ = V cos θ

(6-109)

(Carothers and Baines, J. Fluids Eng., 97, 116–117 [1975]) in place of V in Eq. (6-106). This applies for Re > 500, C = 1.26, α ≤ 0.97 and 0 < θ < 45°, for square-mesh screens and diamond-mesh netting. Screens inclined at an angle to the flow direction also experience a tangential stress. For non-Newtonian fluids in slow flow, friction loss across a square-woven or full-twill-woven screen can be estimated by considering the screen as a set of parallel tubes, each of diameter equal to the average minimal opening between adjacent wires, and length twice the diameter, without entrance effects (Carley and Smith, Polym. Eng. Sci., 18, 408–415 [1978]). For screen stacks, the losses of individual screens should be summed.

JET BEHAVIOR A free jet, upon leaving an outlet, will entrain the surrounding fluid, expand, and decelerate. To a first approximation, total momentum is conserved as jet momentum is transferred to the entrained fluid. For practical purposes, a jet is considered free when its cross-sectional area is less than one-fifth of the total cross-sectional flow area of the region through which the jet is flowing (Elrod, Heat. Piping Air Cond., 26(3), 149–155 [1954]), and the surrounding fluid is the same as the jet fluid. A turbulent jet in this discussion is considered to be a free jet with Reynolds number greater than 2,000. Additional discussion on the relation between Reynolds number and turbulence in jets is given by Elrod (ibid.). Abramowicz (The Theory of Turbulent Jets, MIT Press, Cambridge, 1963) provides a thorough discourse on the theory of turbulent jets. Hussein, et al. (J. Fluid Mech., 258, 31–75 [1994]) give extensive velocity data for a free jet, as well as an extensive discussion of free jet experimentation and comparison of data with momentum conservation equations. A turbulent free jet is normally considered to consist of four flow regions (Tuve, Heat. Piping Air Cond., 25(1), 181–191 [1953]; Davies, Turbulence Phenomena, Academic, New York, 1972) as shown in Fig. 6-17: 1. Region of flow establishment—a short region whose length is about 6.4 nozzle diameters. The fluid in the conical core of the same length has a velocity about the same as the initial discharge velocity. The termination of this potential core occurs when the growing mixing or boundary layer between the jet and the surroundings reaches the centerline of the jet. 2. A transition region that extends to about 8 nozzle diameters. 3. Region of established flow—the principal region of the jet. In this region, the velocity profile transverse to the jet is self-preserving when normalized by the centerline velocity.

FIG. 6-17

Configuration of a turbulent free jet.

FLUID DYNAMICS TABLE 6-6

Turbulent Free-Jet Characteristics Where Both Jet Fluid and Entrained Fluid Are Air Rounded-inlet circular jet

Longitudinal distribution of velocity along jet center line*† x Vc D0 for 7 <  < 100  = K V0 D0 x K=5 for V0 = 2.5 to 5.0 m/s K = 6.2 for V0 = 10 to 50 m/s Radial distribution of longitudinal velocity†

冢 冣

冢冣

V r log c = 40  Vr x

2

x for 7 <  < 100 D0

Jet angle°† x for  < 100 D0

α ⯝ 20° Entrainment of surrounding fluid‡

x for 7 <  < 100 D0

q x  = 0.32  q0 D0

Rounded-inlet, infinitely wide slot jet Longitudinal distribution of velocity along jet centerline‡

冢 冣

x Vc B0 0.5 for 5 <  < 2,000 and V0 = 12 to 55 m/s  = 2.28  V0 B0 x Transverse distribution of longitudinal velocity‡

冢 冣

冢冣

y V log c = 18.4  Vx x

2

x for 5 <  < 2,000 B0

Jet angle‡ α is slightly larger than that for a circular jet Entrainment of surrounding fluid‡

冢 冣

q x  = 0.62  q0 B0

0.5

x for 5 <  < 2,000 B0

*Nottage, Slaby, and Gojsza, Heat, Piping Air Cond., 24(1), 165–176 (1952). †Tuve, Heat, Piping Air Cond., 25(1), 181–191 (1953). ‡Albertson, Dai, Jensen, and Rouse, Trans. Am. Soc. Civ. Eng., 115, 639–664 (1950), and Discussion, ibid., 115, 665–697 (1950).

4. A terminal region where the residual centerline velocity reduces rapidly within a short distance. For air jets, the residual velocity will reduce to less than 0.3 m/s, (1.0 ft/s) usually considered still air. Several references quote a length of 100 nozzle diameters for the length of the established flow region. However, this length is dependent on initial velocity and Reynolds number. Table 6-6 gives characteristics of rounded-inlet circular jets and rounded-inlet infinitely wide slot jets (aspect ratio > 15). The information in the table is for a homogeneous, incompressible air system under isothermal conditions. The table uses the following nomenclature: B0 = slot height D0 = circular nozzle opening q = total jet flow at distance x q0 = initial jet flow rate r = radius from circular jet centerline y = transverse distance from slot jet centerline Vc = centerline velocity Vr = circular jet velocity at r Vy = velocity at y Witze (Am. Inst. Aeronaut. Astronaut. J., 12, 417–418 [1974]) gives equations for the centerline velocity decay of different types of subsonic and supersonic circular free jets. Entrainment of surrounding fluid in the region of flow establishment is lower than in the region of established flow (see Hill, J. Fluid Mech., 51, 773–779 [1972]). Data of Donald and Singer (Trans. Inst. Chem. Eng. [London], 37, 255–267

6-21

[1959]) indicate that jet angle and the coefficients given in Table 6-6 depend upon the fluids; for a water system, the jet angle for a circular jet is 14° and the entrainment ratio is about 70 percent of that for an air system. Most likely these variations are due to Reynolds number effects which are not taken into account in Table 6-6. Rushton (AIChE J., 26, 1038–1041 [1980]) examined available published results for circular jets and found that the centerline velocity decay is given by D0 Vc (6-110)  = 1.41Re0.135  V0 x where Re = D0V0ρ/µ is the initial jet Reynolds number. This result corresponds to a jet angle tan α/2 proportional to Re−0.135. Characteristics of rectangular jets of various aspect ratios are given by Elrod (Heat., Piping, Air Cond., 26[3], 149–155 [1954]). For slot jets discharging into a moving fluid, see Weinstein, Osterle, and Forstall (J. Appl. Mech., 23, 437–443 [1967]). Coaxial jets are discussed by Forstall and Shapiro (J. Appl. Mech., 17, 399–408 [1950]), and double concentric jets by Chigier and Beer (J. Basic Eng., 86, 797–804 [1964]). Axisymmetric confined jets are described by Barchilon and Curtet (J. Basic Eng., 777–787 [1964]). Restrained turbulent jets of liquid discharging into air are described by Davies (Turbulence Phenomena, Academic, New York, 1972). These jets are inherently unstable and break up into drops after some distance. Lienhard and Day (J. Basic Eng. Trans. AIME, p. 515 [September 1970]) discuss the breakup of superheated liquid jets which flash upon discharge. Density gradients affect the spread of a single-phase jet. A jet of lower density than the surroundings spreads more rapidly than a jet of the same density as the surroundings, and, conversely, a denser jet spreads less rapidly. Additional details are given by Keagy and Weller (Proc. Heat Transfer Fluid Mech. Inst., ASME, pp. 89–98, June 22–24 [1949]) and Cleeves and Boelter (Chem. Eng. Prog., 43, 123–134 [1947]). Few experimental data exist on laminar jets (see Gutfinger and Shinnar, AIChE J., 10, 631–639 [1964]). Theoretical analysis for velocity distributions and entrainment ratios are available in Schlichting and in Morton (Phys. Fluids, 10, 2120–2127 [1967]). Theoretical analyses of jet flows for power law non-Newtonian fluids are given by Vlachopoulos and Stournaras (AIChE J., 21, 385–388 [1975]), Mitwally (J. Fluids Eng., 100, 363 [1978]), and Sridhar and Rankin (J. Fluids Eng., 100, 500 [1978]).

冢 冣

FLOW THROUGH ORIFICES Section 10 of this Handbook describes the use of orifice meters for flow measurement. In addition, orifices are commonly found within pipelines as flow-restricting devices, in perforated pipe distributing and return manifolds, and in perforated plates. Incompressible flow through an orifice in a pipeline as shown in Fig. 6-18, is commonly described by the following equation for flow rate Q in terms of pressure drop across the orifice ∆p, the orifice area Ao, the pipe crosssectional area A, and the density ρ. Q = Co Ao

2∆p/ρ  冪莦 [1 − (A /A) ] o

2

Vena contracta Pipe area A Orifice area A o FIG. 6-18

Flow through an orifice.

(6-111)

6-22

FLUID AND PARTICLE DYNAMICS Mach Number and Speed of Sound The Mach number M = V/c is the ratio of fluid velocity, V, to the speed of sound or acoustic velocity, c. The speed of sound is the propagation velocity of infinitesimal pressure disturbances and is derived from a momentum balance. The compression caused by the pressure wave is adiabatic and frictionless, and therefore isentropic.

.90

Co, orifice number

.85 .80

Data scatter ±2%

c=

∂p 莦 冪冢莦莦 ∂ρ 冣

(6-112)

s

.75

The derivative of pressure p with respect to density ρ is taken at constant entropy s. For an ideal gas,

.70

∂p

= 冢 ∂ρ 冣 M kRT

s

.65 0

50

100

150

200

ρg∆p , Froude number Do Orifice coefficient vs. Froude number. (Courtesy E. I. duPont de Nemours & Co.)

FIG. 6-19

where

w

k = ratio of specific heats, Cp /Cv R = universal gas constant (8,314 J/kgmol K) T = absolute temperature Mw = molecular weight

Hence for an ideal gas, c=

kRT  冪莦 M

(6-113)

w

The velocity of approach term [1 − (Ao /A)2] accounts for the kinetic energy approaching the orifice, while the orifice coefficient or discharge coefficient Co accounts for the vena contracta effect which causes the fluid to accelerate to velocity greater than Q/Ao. The downstream pressure measurement corresponding to ∆p in Eq. (6-111) is taken at the vena contracta. Downstream of the vena contracta, the velocity decelerates and some pressure recovery may be expected. Any pressure recovery is completed about 4 to 8 pipe diameters downstream of the orifice. As an approximation, the pressure recovery, expressed as a fraction of the orifice pressure drop, is approximately equal to the area ratio Ao /A. When the orifice discharges into a large chamber, instead of being installed within a pipe, there is negligible pressure recovery. Equation (6-111) may also be used for flow across a perforated plate with open area Ao and total area A. The orifice coefficient has a value of about 0.62 at large Reynolds numbers (Re = DoVoρ/µ > 20,000), although values ranging from 0.60 to 0.70 are frequently used. At lower Reynolds numbers, the orifice coefficient varies with both Re and with the area or diameter ratio. See Sec. 10 for more details. When liquids discharge vertically downward from orifices into gas, gravity increases the discharge coefficient. Figure 6-19 shows this effect, giving the discharge coefficient in terms of a modified Froude number, Fr = ρg∆p/Do. The orifice coefficient deviates from its value for sharp-edged orifices when the orifice wall thickness exceeds about 75 percent of the orifice diameter. Some pressure recovery occurs within the orifice and the orifice coefficient increases. Pressure drop across segmental orifices is roughly 10 percent greater than that for concentric circular orifices of the same open area. COMPRESSIBLE FLOW Flows are typically considered compressible when the density varies by more than 5 to 10 percent. In practice compressible flows are normally limited to gases, supercritical fluids, and multiphase flows containing gases. Liquid flows are normally considered incompressible, except for certain calculations involved in hydraulic transient analysis (see following) where compressibility effects are important even for nearly incompressible liquids with extremely small density variations. Textbooks on compressible gas flow include Shapiro (Dynamics and Thermodynamics of Compressible Fluid Flow, vol. I and II, Ronald Press, New York [1953]) and Zucrow and Hofmann (Gas Dynamics, vol. I and II, Wiley, New York [1976]). In chemical process applications, one-dimensional gas flows through nozzles or orifices and in pipelines are the most important applications of compressible flow. Multidimensional external flows are of interest mainly in aerodynamic applications.

Most often, the Mach number is calculated using the speed of sound evaluated at the local pressure and temperature. When M = 1, the flow is critical or sonic and the velocity equals the local speed of sound. For subsonic flow M < 1 while supersonic flows have M > 1. Compressibility effects are important when the Mach number exceeds 0.1 to 0.2. A common error is to assume that compressibility effects are always negligible when the Mach number is small. The proper assessment of whether compressibility is important should be based on relative density changes, not on Mach number. Isothermal Gas Flow in Pipes and Channels Isothermal compressible flow is often encountered in long transport lines, where there is sufficient heat transfer to maintain constant temperature. Velocities and Mach numbers are usually small, yet compressibility effects are important when the total pressure drop is a large fraction of the absolute pressure. For an ideal gas with ρ = pMw /RT, integration of the differential form of the momentum or mechanical energy balance equations, assuming a constant friction factor f over a length L of a channel of constant cross section and hydraulic diameter DH, yields,

冤

冢 冣冥

p1 RT 4 f L p21 − p22 = G2   + 2 ln  Mw DH p2

(6-114)

where the mass velocity G = w/A = ρV is the mass flow rate per unit cross-sectional area of the channel. The logarithmic term on the righthand side accounts for the pressure change caused by acceleration of gas as its density decreases, while the first term is equivalent to the calculation of frictional losses using the density evaluated at the average pressure (p1 + p2)/2. Solution of Eq. (6-114) for G and differentiation with respect to p2 reveals a maximum mass flux Gmax = p2兹苶 Mw苶/苶 (R苶 T苶) and a corresponding exit velocity V2,max = 兹苶 R苶 T苶 /M苶w and exit Mach number M2 = 1/兹苶k. This apparent choking condition, though often cited, is not physically meaningful for isothermal flow because at such high velocities, and high rates of expansion, isothermal conditions are not maintained. Adiabatic Frictionless Nozzle Flow In process plant pipelines, compressible flows are usually more nearly adiabatic than isothermal. Solutions for adiabatic flows through frictionless nozzles and in channels with constant cross section and constant friction factor are readily available. Figure 6-20 illustrates adiabatic discharge of a perfect gas through a frictionless nozzle from a large chamber where velocity is effectively zero. A perfect gas obeys the ideal gas law ρ = pMw /RT and also has constant specific heat. The subscript 0 refers to the stagnation conditions in the chamber. More generally, stagnation conditions refer to the conditions which would obtained by isentropically decelerating a gas flow to zero velocity. The minimum area section, or throat, of the nozzle is at the nozzle exit. The flow through the nozzle is isentropic

FLUID DYNAMICS

p0

FIG. 6-20

p1

from upstream pressure p0 to external pressure p2, Equations (6-115) through (6-122) are best used as follows. The critical pressure is first determined from Eq. (6-119). If p2 > p*, then the flow is subsonic (subcritical, unchoked). Then p1 = p2 and M1 may be obtained from Eq. (6-115). Substitution of M1 into Eq. (6-118) then gives the desired mass velocity G. Eqs. (6-116) and (6-117) may be used to find the exit temperature and density. On the other hand, if p2 ≤ p*, then the flow is choked and M1 = 1. Then p1 = p*, and the mass velocity is G* obtained from Eq. (6-122). The exit temperature and density may be obtained from Eqs. (6-120) and (6-121). When the flow is choked, G = G* is independent of external downstream pressure. Reducing the downstream pressure will not increase the flow. The mass flow rate under choking conditions is directly proportional to the upstream pressure.

p2

Isentropic flow through a nozzle.

because it is frictionless (reversible) and adiabatic. In terms of the exit Mach number M1 and the upstream stagnation conditions, the flow conditions at the nozzle exit are given by p0 k − 1 2 k / (k − 1) (6-115)  = 1 +  M1 p1 2

冢

冣

T0 k−1 2  = 1 +  M1 T1 2

(6-116)

ρ0 k−1 2 (6-117)  = 1 +  M1 ρ1 2 The mass velocity G = w/A, where w is the mass flow rate and A is the nozzle exit area, at the nozzle exit is given by 1 / (k − 1)

冢

G = p0

冣

kM M   冪莦 RT k−1 w 0

1

(k + 1) / 2(k − 1)

冢1 + 2 M 冣

(6-118)

2 1

冣

T* 2 = T0 k + 1 ρ* 2 =  ρ0 k+1

冢

G* = p0

(6-120) 1/(k − 1)

冣

(6-121)

kM 2 莦  冣莦 冪冢莦莦 k + 1 冣 莦莦莦莦莦冢莦莦 RT (k + 1)/(k − 1)

Example 7: Flow through Frictionless Nozzle Air at p0 and temperature T0 = 293 K discharges through a frictionless nozzle to atmospheric pressure. Compute the discharge mass flux G, the pressure, temperature, Mach number, and velocity at the exit. Consider two cases: (1) p0 = 7 × 105 Pa absolute, and (2) p0 = 1.5 × 105 Pa absolute. 1. p0 = 7.0 × 105 Pa. For air with k = 1.4, the critical pressure ratio from Eq. (6-119) is p*/p0 = 0.5285 and p* = 0.5285 × 7.0 × 105 = 3.70 × 105 Pa. Since this is greater than the external atmospheric pressure p2 = 1.01 × 105 Pa, the flow is choked and the exit pressure is p1 = 3.70 × 105 Pa. The exit Mach number is 1.0, and the mass flux is equal to G* given by Eq. (6-118). G* = 7.0 × 105 ×

2 1.4 × 29 = 1,650 kg/m ⋅ s 莦 冪冢莦莦 冢 1.4 + 1 冣 莦莦莦莦莦莦莦莦 8314 × 293 冣莦 (1.4 + 1)/(1.4 − 1)

2

The exit temperature, since the flow is choked, is 2 T* T* =  T0 =  × 293 = 244 K T0 1.4 + 1

冢 冣 冢

冣

苶T 苶*/ 苶M 苶w苶 = 313 m/s. The exit velocity is V = Mc = c* = 兹kR 2. p0 = 1.5 × 105 Pa. In this case p* = 0.79 × 105 Pa, which is less than p2. 5 Hence, p1 = p2 = 1.01 × 10 Pa. The flow is unchoked (subsonic). Equation (6-115) is solved for the Mach number. 1.4/(1.4 − 1) 1.5 × 105 1.4 − 1 5 = 1 +  M 21 1.01 × 10 2

冢

冣

M1 = 0.773

These equations are consistent with the isentropic relations for a perfect gas p/p0 = (ρ/ρ0)k, T/T0 = (p/p0)(k − 1)/k. Equation (6-116) is valid for adiabatic flows with or without friction; it does not require isentropic flow. However, Eqs. (6-115) and (6-117) do require isentropic flow. The exit Mach number M1 may not exceed unity. At M1 = 1, the flow is said to be choked, sonic, or critical. When the flow is choked, the pressure at the exit is greater than the pressure of the surroundings into which the gas flow discharges. The pressure drops from the exit pressure to the pressure of the surroundings in a series of shocks which are highly nonisentropic. Sonic flow conditions are denoted by *; sonic exit conditions are found by substituting M1 = M*1 = 1 into Eqs. (6-115) to (6-118). p* 2 k/(k − 1) (6-119) =  p0 k+1

冢

6-23

w

(6-122)

0

Note that under choked conditions, the exit velocity is V = V* = c* = 兹苶 kR苶 T苶 */苶 M苶w, not 兹kR 苶T 苶苶 苶苶w. Sonic velocity must be evaluated at the 0/M exit temperature. For air, with k = 1.4, the critical pressure ratio p*/p0 is 0.5285 and the critical temperature ratio T*/T0 = 0.8333. Thus, for air discharging from 300 K, the temperature drops by 50 K (90 R). This large temperature decrease results from the conversion of internal energy into kinetic energy and is reversible. As the discharged jet decelerates in the external stagant gas, it recovers its initial enthalpy. When it is desired to determine the discharge rate through a nozzle

Substitution into Eq. (6-118) gives G. G = 1.5 × 105 ×

1.4 × 29

 冪莦 8,314 × 293

0.773 ×  = 337 kg/m2 ⋅ s (1.4 + 1)/2(1.4 − 1) 1.4 − 1 1 +  × 0.7732 2 The exit temperature is found from Eq. (6-116) to be 261.6 K or −11.5°C. The exit velocity is

冢 冢

V = Mc = 0.773 ×

冣

冣

1.4 × 8314 × 261.6  = 250 m/s 冪莦莦 29

Adiabatic Flow with Friction in a Duct of Constant Cross Section Integration of the differential forms of the continuity, momentum, and total energy equations for a perfect gas, assuming a constant friction factor, leads to a tedious set of simultaneous algebraic equations. These may be found in Shapiro (Dynamics and Thermodynamics of Compressible Fluid Flow, vol. I, Ronald Press, New York, 1953) or Zucrow and Hofmann (Gas Dynamics, vol. I, Wiley, New York, 1976). Lapple’s (Trans. AIChE., 39, 395–432 [1943]) widely cited graphical presentation of the solution of these equations contained a subtle error, which was corrected by Levenspiel (AIChE J., 23, 402–403 [1977]). Levenspiel’s graphical solutions are presented in Fig. 6-21. These charts refer to the physical situation illustrated in Fig. 6-22, where a perfect gas discharges from stagnation conditions in a large chamber through an isentropic nozzle followed by a duct of length L. The resistance parameter is N = 4fL/DH, where f = Fanning friction factor and DH = hydraulic diameter. The exit Mach number M2 may not exceed unity. M2 = 1 corresponds to choked flow; sonic conditions may exist only at the pipe exit. The mass velocity G* in the charts is the choked mass flux for an isentropic nozzle given by Eq. (6-118). For a pipe of finite length,

6-24

FLUID AND PARTICLE DYNAMICS

(a)

(b) Design charts for adiabatic flow of gases; (a) useful for finding the allowable pipe length for given flow rate; (b) useful for finding the discharge rate in a given piping system. (From Levenspiel, Am. Inst. Chem. Eng. J., 23, 402 [1977].) FIG. 6-21

the mass flux is less than G* under choking conditions. The curves in Fig. 6-21 become vertical at the choking point, where flow becomes independent of downstream pressure. The equations for nozzle flow, Eqs. (6-114) through (6-118), remain valid for the nozzle section even in the presence of the discharge pipe. Equations (6-116) and (6-120), for the temperature variation, may also be used for the pipe, with M2, p2 replacing M1, p1 since they are valid for adiabatic flow, with or without friction. The graphs in Fig. 6-21 are based on accurate calculations, but are

L p0

p1

p2

p3

D FIG. 6-22

entrance.

Adiabatic compressible flow in a pipe with a well-rounded

FLUID DYNAMICS difficult to interpolate precisely. While they are quite useful for rough estimates, precise calculations are best done using the equations for one-dimensional adiabatic flow with friction, which are suitable for computer programming. Let subscripts 1 and 2 denote two points along a pipe of diameter D, point 2 being downstream of point 1. From a given point in the pipe, where the Mach number is M, the additional length of pipe required to accelerate the flow to sonic velocity (M = 1) is denoted Lmax and may be computed from k+1  M 2 2  4f Lmax 1 − M 2 k + 1 k−1 + ln (6-123) =  1 +  M 2 D kM 2 2k 2

冢

冣

With L = length of pipe between points 1 and 2, the change in Mach number may be computed from

冢

4fL 4f Lmax =  D D

冣 − 冢 D 冣 4f Lmax

1

(6-124) 2

Eqs. (6-116) and (6-113), which are valid for adiabatic flow with friction, may be used to determine the temperature and speed of sound at points 1 and 2. Since the mass flux G = ρv = ρcM is constant, and ρ = PMw /RT, the pressure at point 2 (or 1) can be found from G and the pressure at point 1 (or 2). The additional frictional losses due to pipeline fittings such as elbows may be added to the velocity head loss N = 4 f L/DH using the same velocity head loss values as for incompressible flow. This works well for fittings which do not significantly reduce the channel crosssectional area, but may cause large errors when the flow area is greatly reduced, as, for example, by restricting orifices. Compressible flow across restricting orifices is discussed in Sec. 10 of this Handbook. Similarly, elbows near the exit of a pipeline may choke the flow even though the Mach number is less than unity due to the nonuniform velocity profile in the elbow. For an abrupt contraction rather than rounded nozzle inlet, an additional 0.5 velocity head should be added to N. This is a reasonable approximation for G, but note that it allocates the additional losses to the pipeline, even though they are actually incurred in the entrance. It is an error to include one velocity head exit loss in N. The kinetic energy at the exit is already accounted for in the integration of the balance equations. Example 8: Compressible Flow with Friction Losses Calculate the discharge rate of air to the atmosphere from a reservoir at 106 Pa gauge and 20°C through 10 m of straight 2-in Schedule 40 steel pipe (inside diameter = 0.0525 m), and 3 standard radius, flanged 90° elbows. Assume 0.5 velocity heads lost for the elbows. For commercial steel pipe, with a roughness of 0.046 mm, the friction factor for fully rough flow is about 0.0047, from Eq. (6-38) or Fig. 6-9. It remains to be verified that the Reynolds number is sufficiently large to assume fully rough flow. Assuming an abrupt entrance with 0.5 velocity heads lost, 10 N = 4 × 0.0047 ×  + 0.5 + 3 × 0.5 = 5.6 0.0525 The pressure ratio p3 /p0 is 1.01 × 105 = 0.092  (1 × 106 + 1.01 × 105) From Fig. 6-21b at N = 5.6, p3 /p0 = 0.092 and k = 1.4 for air, the flow is seen to be choked. At the choke point with N = 5.6 the critical pressure ratio p2 /p0 is about 0.25 and G/G* is about 0.48. Equation (6-122) gives G* = 1.101 × 106 ×

2 1.4 × 29 = 2,600 kg/m ⋅ s 莦 冪冢莦莦 冢 1.4 + 1 冣 莦莦莦莦莦莦莦莦 8,314 × 293.15 冣莦 (1.4 + 1)/(1.4 − 1)

and a Reynolds number of 3.6 × 106. Over the entire pipe length the Reynolds number is very large and the fully rough flow friction factor choice was indeed valid.

Once the mass flux G has been determined, Fig. 6-21a or 6-21b can be used to determine the pressure at any point along the pipe, simply by reducing 4fL/DH and computing p2 from the figures, given G, instead of the reverse. Charts for calculation between two points in a pipe with known flow and known pressure at either upstream or downstream locations have been presented by Loeb (Chem. Eng., 76[5], 179–184 [1969]) and for known downstream conditions by Powley (Can. J. Chem. Eng., 36, 241–245 [1958]). Convergent/Divergent Nozzles (De Laval Nozzles) During frictionless adiabatic one-dimensional flow with changing crosssectional area A the following relations are obeyed: dA dp 1 − M 2 dρ dV (6-125)  = 2 (1 − M 2) =   = −(1 − M 2)  M2 A ρV ρ V Equation (6-125) implies that in converging channels, subsonic flows are accelerated and the pressure and density decrease. In diverging channels, subsonic flows are decelerated as the pressure and density increase. In subsonic flow, the converging channels act as nozzles and diverging channels as diffusers. In supersonic flows, the opposite is true. Diverging channels act as nozzles accelerating the flow, while converging channels act as diffusers decelerating the flow. Figure 6-23 shows a converging/diverging nozzle. When p2 /p0 is less than the critical pressure ratio (p*/p0), the flow will be subsonic in the converging portion of the nozzle, sonic at the throat, and supersonic in the diverging portion. At the throat, where the flow is critical and the velocity is sonic, the area is denoted A*. The cross-sectional area and pressure vary with Mach number along the converging/ diverging flow path according to the following equations for isentropic flow of a perfect gas: (k + 1) / 2(k − 1) A 1 2 k−1 (6-126)  =   1 +  M2 A* M k + 1 2

冤

冢

冣冥

k / (k − 1) p0 k−1 (6-127)  = 1 +  M2 p 2 The temperature obeys the adiabatic flow equation for a perfect gas, k−1 T0 (6-128)  = 1 +  M2 T 2 Equation (6-128) does not require frictionless (isentropic) flow. The sonic mass flux through the throat is given by Eq. (6-122). With A set equal to the nozzle exit area, the exit Mach number, pressure, and temperature may be calculated. Only if the exit pressure equals the ambient discharge pressure is the ultimate expansion velocity reached in the nozzle. Expansion will be incomplete if the exit pressure exceeds the ambient discharge pressure; shocks will occur outside the nozzle. If the calculated exit pressure is less than the ambient discharge pressure, the nozzle is overexpanded and compression shocks within the expanding portion will result. The shape of the converging section is a smooth trumpet shape similar to the simple converging nozzle. However, special shapes of the diverging section are required to produce the maximum supersonic exit velocity. Shocks result if the divergence is too rapid and excessive boundary layer friction occurs if the divergence is too shallow. See Liepmann and Roshko (Elements of Gas Dynamics, Wiley, New York, 1957, p. 284). If the nozzle is to be used as a thrust device, the diverg-

冢

冣

2

Multiplying by G/G* = 0.48 yields G = 1,250 kg/m2 ⋅ s. The discharge rate is w = GA = 1,250 × π × 0.05252/4 = 2.7 kg/s. Before accepting this solution, the Reynolds number should be checked. At the pipe exit, the temperature is given by Eq. (6-120) since the flow is choked. Thus, T2 = T* = 244.6 K. The viscosity of air at this temperature is about 1.6 × 10−5 Pa ⋅ s. Then DVρ DG 0.0525 × 1,250 Re =  =  =  = 4.1 × 106 µ µ 1.6 × 10−5 At the beginning of the pipe, the temperature is greater, giving greater viscosity

6-25

FIG. 6-23

Converging/diverging nozzle.

6-26

FLUID AND PARTICLE DYNAMICS

ing section can be conical with a total included angle of 30° (Sutton, Rocket Propulsion Elements, 2d ed., Wiley, New York, 1956). To obtain large exit Mach numbers, slot-shaped rather than axisymmetric nozzles are used. MULTIPHASE FLOW Multiphase flows, even when restricted to simple pipeline geometry, are in general quite complex, and several features may be identified which make them more complicated than single-phase flow. Flow pattern description is not merely an identification of laminar or turbulent flow. The relative quantities of the phases and the topology of the interfaces must be described. Because of phase density differences, vertical flow patterns are different from horizontal flow patterns, and horizontal flows are not generally axisymmetric. Even when phase equilibrium is achieved by good mixing in two-phase flow, the changing equilibrium state as pressure drops with distance, or as heat is added or lost, may require that interphase mass transfer, and changes in the relative amounts of the phases, be considered. Wallis (One-dimensional Two-phase Flow, McGraw-Hill, New York, 1969) and Govier and Aziz present mass, momentum, mechanical energy, and total energy balance equations for two-phase flows. These equations are based on one-dimensional behavior for each phase. Such equations, for the most part, are used as a framework in which to interpret experimental data. Reliable prediction of multiphase flow behavior generally requires use of data or correlations. Two-fluid modeling, in which the full three-dimensional microscopic (partial differential) equations of motion are written for each phase, treating each as a continuum, occupying a volume fraction which is a continuous function of position, is a rapidly developing technique made possible by improved computational methods. For some relatively simple examples not requiring numerical computation, see Pearson (Chem. Engr. Sci., 49, 727–732 [1994]). Constitutive equations for two-fluid models are not yet sufficiently robust for accurate general-purpose two-phase flow computation, but may be quite good for particular classes of flows. Liquids and Gases For cocurrent flow of liquids and gases in vertical (upflow), horizontal, and inclined pipes, a very large literature of experimental and theoretical work has been published, with less work on countercurrent and cocurrent vertical downflow. Much of the effort has been devoted to predicting flow patterns, pressure drop, and volume fractions of the phases, with emphasis on fully developed flow. In practice, many two-phase flows in process plants are not fully developed. The most reliable methods for fully developed gas/liquid flows use mechanistic models to predict flow pattern, and use different pressure drop and void fraction estimation procedures for each flow pattern. Such methods are too lengthy to include here, and are well suited to incorporation into computer programs; commercial codes for gas/liquid pipeline flows are available. Some key references for mechanistic methods for flow pattern transitions and flow regime– specific pressure drop and void fraction methods include Taitel and Dukler (AIChE J., 22, 47–55 [1976]), Barnea, et al. (Int. J. Multiphase Flow, 6, 217–225 [1980]), Barnea (Int. J. Multiphase Flow, 12, 733–744 [1986]), Taitel, Barnea, and Dukler (AIChE J., 26, 345–354 [1980]), Wallis (One-dimensional Two-phase Flow, McGraw-Hill, New York, 1969), and Dukler and Hubbard (Ind. Eng. Chem. Fundam., 14, 337–347 [1975]). For preliminary or approximate calculations, flow pattern maps and flow regime–independent empirical correlations, are simpler and faster to use. Such methods for horizontal and vertical flows are provided in the following. In horizontal pipe, flow patterns for fully developed flow have been reported in numerous studies. Transitions between flow patterns are gradual, and subjective owing to the visual interpretation of individual investigators. In some cases, statistical analysis of pressure fluctuations has been used to distinguish flow patterns. Figure 6-24 (Alves, Chem. Eng. Progr., 50, 449–456 [1954]) shows seven flow patterns for horizontal gas/liquid flow. Bubble flow is prevalent at high ratios of liquid to gas flow rates. The gas is dispersed as bubbles which move at velocity similar to the liquid and tend to concentrate near the top of the pipe at lower liquid velocities. Plug flow describes a pattern in which alternate plugs of gas and liquid move along the upper

Gas/liquid flow patterns in horizontal pipes. (From Alves, Chem. Eng. Progr., 50, 449–456 [1954].)

FIG. 6-24

part of the pipe. In stratified flow, the liquid flows along the bottom of the pipe and the gas flows over a smooth liquid/gas interface. Similar to stratified flow, wavy flow occurs at greater gas velocities and has waves moving in the flow direction. When wave crests are sufficiently high to bridge the pipe, they form frothy slugs which move at much greater than the average liquid velocity. Slug flow can cause severe and sometimes dangerous vibrations in equipment because of impact of the high-velocity slugs against bends or other fittings. Slugs may also flood gas/liquid separation equipment. In annular flow, liquid flows as a thin film along the pipe wall and gas flows in the core. Some liquid is entrained as droplets in the gas core. At very high gas velocities, nearly all the liquid is entrained as small droplets. This pattern is called spray, dispersed, or mist flow. Approximate prediction of flow pattern may be quickly done using flow pattern maps, an example of which is shown in Fig. 6-25 (Baker, Oil Gas J., 53[12], 185–190, 192–195 [1954]). The Baker chart remains widely used; however, for critical calculations the mechanistic model methods referenced previously are generally preferred for their greater accuracy, especially for large pipe diameters and fluids with physical properties different from air/water at atmospheric pressure. In the chart, λ = (ρ′G ρ′L)1/2 1 µ′L ψ =  2 σ′ (ρ′L )

冤

(6-129)

冥

1/3

(6-130)

Flow-pattern regions in cocurrent liquid/gas flow through horizontal pipes. To convert lbm/(ft2 ⋅ s) to kg/(m2 ⋅ s), multiply by 4.8824. (From Baker, Oil Gas J., 53[12], 185–190, 192, 195 [1954].)

FIG. 6-25

FLUID DYNAMICS GL and GG are the liquid and gas mass velocities, µ′L is the ratio of liquid viscosity to water viscosity, ρ′G is the ratio of gas density to air density, ρ′L is the ratio of liquid density to water density, and σ′ is the ratio of liquid surface tension to water surface tension. The reference properties are at 20°C (68°F) and atmospheric pressure, water density 1,000 kg/m3 (62.4 lbm/ft3), air density 1.20 kg/m3 (0.075 lbm/ft3), water viscosity 0.001 Pa ⋅ s, (1.0 cp) and surface tension 0.073 N/m (0.0050 lbf/ft). The empirical parameters λ and ψ provide a crude accounting for physical properties. The Baker chart is dimensionally inconsistent since the dimensional quantity GG / λ is plotted against a dimensionless one, GLλψ/GG, and so must be used with GG in lbm/(ft2 ⋅ s) units on the ordinate. To convert to kg/(m2 ⋅ s), multiply by 4.8824. Rapid approximate predictions of pressure drop for fully developed, incompressible horizontal gas/liquid flow may be made using the method of Lockhart and Martinelli (Chem. Eng. Prog., 45, 39–48 [1949]). First, the pressure drops that would be expected for each of the two phases as if flowing alone in single-phase flow are calculated. The Lockhart-Martinelli parameter X is defined in terms of the ratio of these pressure drops: (∆ p/L) 1/2 X = L (6-131) (∆ p/L)G The two-phase pressure drop may be then be estimated from either of the single-phase pressure drops, using ∆p ∆p (6-132)  = YL  L TP L L

冤

冥

冢 冣

冢 冣

∆p 冢 L 冣

冢 冣

∆p = YG  (6-133) L G where YL and YG are read from Fig. 6-26 as functions of X. The curve labels refer to the flow regime (laminar or turbulent) found for each of the phases flowing alone. The common turbulent-turbulent case is approximated well by 1 20 YL = 1 +  + 2 (6-134) X X Lockhart and Martinelli (ibid.) correlated pressure drop data from pipes 25 mm (1 in) in diameter or less within about 50 percent. In general, the predictions are high for stratified, wavy, and slug flows and low for annular flow. The correlation can be applied to pipe diameters up to about 0.1 m (4 in) with about the same accuracy. or

TP

Parameters for pressure drop in liquid/gas flow through horizontal pipes. (Based on Lockhart and Martinelli, Chem. Engr. Prog., 45, 39 [1949].)

FIG. 6-26

6-27

The volume fraction, sometimes called holdup, of each phase in two-phase flow is generally not equal to its volumetric flow rate fraction, because of velocity differences, or slip, between the phases. For each phase, denoted by subscript i, the relations among superficial velocity Vi, in situ velocity vi, volume fraction Ri, total volumetric flow rate Qi, and pipe area A are (6-135) Qi = Vi A = vi Ri A Vi (6-136) vi =  Ri The slip velocity between gas and liquid is vs = vG − vL . For two-phase gas/liquid flow, RL + RG = 1. A very common mistake in practice is to assume that in situ phase volume fractions are equal to input volume fractions. For fully developed incompressible horizontal gas/liquid flow, a quick estimate for RL may be obtained from Fig. 6-27, as a function of the Lockhart-Martinelli parameter X defined by Eq. (6-131). Indications are that liquid volume fractions may be overpredicted for liquids more viscous than water (Alves, Chem. Eng. Prog., 50, 449–456 [1954]), and underpredicted for pipes larger than 25 mm diameter (Baker, Oil Gas J., 53[12], 185–190, 192–195 [1954]). A method for predicting pressure drop and volume fraction for non-Newtonian fluids in annular flow has been proposed by Eisenberg and Weinberger (AIChE J., 25, 240–245 [1979]). Das, Biswas, and Matra (Can. J. Chem. Eng., 70, 431–437 [1993]) studied holdup in both horizontal and vertical gas/liquid flow with non-Newtonian liquids. Farooqi and Richardson (Trans Inst. Chem. Engrs., 60, 292–305, 323–333 [1982]) developed correlations for holdup and pressure drop for gas/non-Newtonian liquid horizontal flow. They used a modified Lockhart-Martinelli parameter for non-Newtonian liquid holdup. They found that two-phase pressure drop may actually be less than the single-phase liquid pressure drop with shear thinning liquids in laminar flow. Pressure drop data for a 1-in feed tee with the liquid entering the run and gas entering the branch are given by Alves (Chem. Eng. Progr., 50, 449–456 [1954]). Pressure drop and division of two-phase annular flow in a tee are discussed by Fouda and Rhodes (Trans. Inst. Chem. Eng. [London], 52, 354–360 [1974]). Flow through tees can result in unexpected flow splitting. Further reading on gas/liquid flow through tees may be found in Mudde, Groen, and van den Akker (Int. J. Multiphase Flow, 19, 563–573 [1993]); Issa and Oliveira (Computers and Fluids, 23, 347–372 [1993]) and Azzopardi and Smith (Int. J. Multiphase Flow, 18, 861–875 [1992]). Results by Chenoweth and Martin (Pet. Refiner, 34[10], 151–155 [1955]) indicate that single-phase data for fittings and valves can be used in their correlation for two-phase pressure drop. Smith, Murdock, and Applebaum (J. Eng. Power, 99, 343–347 [1977]) evaluated existing correlations for two-phase flow of steam/water and other gas/liquid mixtures through sharp-edged orifices meeting ASTM standards for flow measurement. The correlation of Murdock (J. Basic Eng., 84, 419–433 [1962]) may be used for these orifices. See also Collins and Gacesa (J. Basic Eng., 93, 11–21 [1971]), for measurements with steam and water beyond the limits of this correlation.

Liquid volume fraction in liquid/gas flow through horizontal pipes. (From Lockhart and Martinelli, Eng. Prog., 45, 39 [1949].) FIG. 6-27

6-28

FLUID AND PARTICLE DYNAMICS

For pressure drop and holdup in inclined pipe with upward or downward flow, see Beggs and Brill (J. Pet. Technol., 25, 607–617 [1973]); the mechanistic model methods referenced above may also be applied to inclined pipes. Up to 10° from horizontal, upward pipe inclination has little effect on holdup (Gregory, Can. J. Chem. Eng., 53, 384–388 [1975]). For fully developed incompressible cocurrent upflow of gases and liquids in vertical pipes, a variety of flow pattern terminologies and descriptions have appeared in the literature; some of these have been summarized and compared by Govier, Radford, and Dunn (Can. J. Chem. Eng., 35, 58–70 [1957]). One reasonable classification of patterns is illustrated in Fig. 6-28. In bubble flow, gas is dispersed as bubbles throughout the liquid, but with some tendency to concentrate toward the center of the pipe. In slug flow, the gas forms large Taylor bubbles of diameter nearly equal to the pipe diameter. A thin film of liquid surrounds the Taylor bubble. Between the Taylor bubbles are liquid slugs containing some bubbles. Froth or churn flow is characterized by strong intermittency and intense mixing, with neither phase easily described as continuous or dispersed. There remains disagreement in the literature as to whether churn flow is a real fully developed flow pattern or is an indication of large entry length for developing slug flow (Zao and Dukler, Int. J. Multiphase Flow, 19, 377–383 [1993]; Hewitt and Jayanti, Int. J. Multiphase Flow, 19, 527–529 [1993]). Ripple flow has an upward-moving wavy layer of liquid on the pipe wall; it may be thought of as a transition region to annular, annular mist, or film flow, in which gas flows in the core of the pipe while an annulus of liquid flows up the pipe wall. Some of the liquid is entrained as droplets in the gas core. Mist flow occurs when all the liquid is carried as fine drops in the gas phase; this pattern occurs at high gas velocities, typically 20 to 30 m/s (66 to 98 ft/s). The correlation by Govier, et al. (Can. J. Chem. Eng., 35, 58–70 [1957]), Fig. 6-29, may be used for quick estimate of flow pattern. Slip, or relative velocity between phases, occurs for vertical flow as well as for horizontal. No completely satisfactory, flow regime–independent correlation for volume fraction or holdup exists for vertical flow. Two frequently used flow regime–independent methods are those by Hughmark and Pressburg (AIChE J., 7, 677 [1961]) and Hughmark (Chem. Eng. Prog., 58[4], 62 [April 1962]). Pressure drop in upflow may be calculated by the procedure described in Hughmark (Ind. Eng. Chem. Fundam., 2, 315–321 [1963]). The mechanistic, flow regime–based methods are advisable for critical applications. For upflow in helically coiled tubes, the flow pattern, pressure drop, and holdup can be predicted by the correlations of Banerjee,

FIG. 6-29 Flow-pattern regions in cocurrent liquid/gas flow in upflow through vertical pipes. To convert ft/s to m/s, multiply by 0.3048. (From Govier, Radford, and Dunn, Can. J. Chem. Eng., 35, 58–70 [1957].)

Rhodes, and Scott (Can. J. Chem. Eng., 47, 445–453 [1969]) and Akagawa, Sakaguchi, and Ueda (Bull JSME, 14, 564–571 [1971]). Correlations for flow patterns in downflow in vertical pipe are given by Oshinowo and Charles (Can. J. Chem. Eng., 52, 25–35 [1974]) and Barnea, Shoham, and Taitel (Chem. Eng. Sci., 37, 741–744 [1982]). Use of drift flux theory for void fraction modeling in downflow is presented by Clark and Flemmer (Chem. Eng. Sci., 39, 170–173 [1984]). Downward inclined two-phase flow data and modeling are given by Barnea, Shoham, and Taitel (Chem. Eng. Sci., 37, 735–740 [1982]). Data for downflow in helically coiled tubes are presented by Casper (Chem. Ing. Tech., 42, 349–354 [1970]). The entrance to a drain is flush with a horizontal surface, while the entrance to an overflow pipe is above the horizontal surface. When such pipes do not run full, considerable amounts of gas can be drawn down by the liquid. The amount of gas entrained is a function of pipe diameter, pipe length, and liquid flow rate, as well as the drainpipe outlet boundary condition. Extensive data on air entrainment and liquid head above the entrance as a function of water flow rate for pipe diameters from 43.9 to 148.3 mm (1.7 to 5.8 in) and lengths from about 1.22 to 5.18 m (4.0 to 17.0 ft) are reported by Kalinske (Univ. Iowa Stud. Eng., Bull. 26, pp. 26–40 [1939–1940]). For heads greater than the critical, the pipes will run full with no entrainment. The critical head h for flow of water in drains and overflow pipes is given in Fig. 6-30. Kalinske’s results show little effect of the height of protrusion of overflow pipes when the protrusion height is greater than about one pipe diameter. For conservative design, McDuffie (AIChE J., 23, 37–40 [1977]) recommends the following relation for minimum liquid height to prevent entrainment. h 2 Fr ≤ 1.6  (6-137) D

冢 冣

FIG. 6-28 Flow patterns in cocurrent upward vertical gas/liquid flow. (From Taitel, Barnea, and Dukler, AIChE J., 26, 345–354 [1980]. Reproduced by permission of the American Institute of Chemical Engineers © 1980 AIChE. All rights reserved.)

FIG. 6-30 Critical head for drain and overflow pipes. (From Kalinske, Univ. Iowa Stud. Eng., Bull. 26 [1939–1940].)

FLUID DYNAMICS where the Froude number is defined by VL Fr ⬅  g(苶 ρL苶苶 −苶 ρ苶 D苶 /ρL苶 兹苶 G)苶

(6-138)

where g = acceleration due to gravity VL = liquid velocity in the drain pipe ρL = liquid density ρG = gas density D = pipe inside diameter h = liquid height For additional information, see Simpson (Chem. Eng., 75(6), 192–214 [1968]). A critical Froude number of 0.31 to ensure vented flow is widely cited. Recent results (Thorpe, 3d Int. Conf. Multi-phase Flow, The Hague, Netherlands, 18–20 May 1987, paper K2, and 4th Int. Conf. Multi-phase Flow, Nice, France, 19–21 June 1989, paper K4) show hysteresis, with different critical Froude numbers for flooding and unflooding of drain pipes, and the influence of end effects. Wallis, Crowley, and Hagi (Trans. ASME J. Fluids Eng., 405–413 [ June 1977]) examine the conditions for horizontal discharge pipes to run full. Flashing flow and condensing flow are two examples of multiphase flow with phase change. Flashing flow occurs when pressure drops below the bubble point pressure of a flowing liquid. A frequently used one-dimensional model for flashing flow through nozzles and pipes is the homogeneous equilibrium model which assumes that both phases move at the same in situ velocity, and maintain vapor/ liquid equilibrium. It may be shown that a critical flow condition, analogous to sonic or critical flow during compressible gas flow, is given by the following expression for the mass flux G in terms of the derivative of pressure p with respect to mixture density ρm at constant entropy: ∂p Gcrit = ρm (6-139)  ∂ρm s

冪冢莦莦莦冣

苶p 苶/∂ 苶ρ 苶苶 The corresponding acoustic velocity 兹(∂ m)s苶 is normally much less than the acoustic velocity for gas flow. The mixture density is given in terms of the individual phase densities and the quality (mass flow fraction vapor) x by 1 x 1−x (6-140) =+ ρm ρG ρL Choked and unchoked flow situations arise in pipes and nozzles in the same fashion for homogeneous equilibrium flashing flow as for gas flow. For nozzle flow from stagnation pressure p0 to exit pressure p1, the mass flux is given by p1 dp G2 = −2ρ2m1 (6-141)  p0 ρm The integration is carried out over an isentropic flash path: flashes at constant entropy must be carried out to evaluate ρm as a function of p. Experience shows that isenthalpic flashes provide good approximations unless the liquid mass fraction is very small. Choking occurs when G obtained by Eq. (6-141) goes through a maximum at a value of p1 greater than the external discharge pressure. Equation (6-139) will also be satisfied at that point. In such a case the pressure at the nozzle exit equals the choking pressure and flashing shocks occur outside the nozzle exit. For homogeneous flow in a pipe of diameter D, the differential form of the Bernoulli equation (6-15) rearranges to dp G2 1 dx′ G2 =0 (6-142)  + g dz +  d  + 2f   ρm ρm ρm D ρm2 where x′ is distance along the pipe. Integration over a length L of pipe assuming constant friction factor f yields

冕

冕

−

p

2

p1

ρm dp − g

冕

z2

z1

ρm2 dz

G =  ln (ρm1 /ρm2) + 2f L/D 2

(6-143)

Frictional pipe flow is not isentropic. Strictly speaking, the flashes must be carried out at constant h + V 2/2 + gz, where h is the enthalpy

6-29

per unit mass of the two-phase flashing mixture. The flash calculations are fully coupled with the integration of the Bernoulli equation; the velocity V must be known at every pressure p to evaluate ρm. Computational routines, employing the thermodynamic and material balance features of flowsheet simulators, are the most practical way to carry out such flashing flow calculations, particularly when multicompent systems are involved. Significant simplification arises when the mass fraction liquid is large, for then the effect of the V 2/2 term on the flash splits may be neglected. If elevation effects are also negligible, the flash computations are decoupled from the Bernoulli equation integration. For many horizontal flashing flow calculations, this is satisfactory and the flash computatations may be carried out first, to find ρm as a function of p from p1 to p2, which may then be substituted into Eq. (6-143). With flashes carried out along the appropriate thermodynamic paths, the formalism of Eqs. (6-139) through (6-143) applies to all homogeneous equilibrium compressible flows, including, for example, flashing flow, ideal gas flow, and nonideal gas flow. Equation (6-118), for example, is a special case of Eq. (6-141) where the quality x = 1 and the vapor phase is a perfect gas. Various nonequilibrium and slip flow models have been proposed as improvements on the homogeneous equilibrium flow model. See, for example, Henry and Fauske (Trans. ASME J. Heat Transfer, 179–187 [May 1971]). Nonequilibrium and slip effects both increase computed mass flux for fixed pressure drop, compared to homogeneous equilibrium flow. For flow paths greater than about 100 mm, homogeneous equilibrium behavior appears to be the best assumption (Fischer, et al., Emergency Relief System Design Using DIERS Technology, AIChE, New York [1992]). For shorter flow paths, the best estimate may sometimes be given by linearly interpolating (as a function of length) between frozen flow (constant quality, no flashing) at 0 length and equilibrium flow at 100 mm. In a series of papers by Leung and coworkers (AIChE J., 32, 1743–1746 [1986]; 33, 524–527 [1987]; 34, 688–691 [1988]; J. Loss Prevention Proc. Ind., 2[2], 78–86 [April 1989]; 3(1), 27–32 [January 1990]; Trans. ASME J. Heat Transfer, 112, 524–528, 528–530 [1990]; 113, 269–272 [1991]) approximate techniques have been developed for homogeneous equilibrium calculations based on pseudo–equation of state methods for flashing mixtures. Relatively less work has been done on condensing flows. Slip effects are more important for condensing than for flashing flows. Soliman, Schuster, and Berenson (J. Heat Transfer, 90, 267–276 [1968]) give a model for condensing vapor in horizontal pipe. They assume the condensate flows as an annular ring. The LockhartMartinelli correlation is used for the frictional pressure drop. To this pressure drop is added an acceleration term based on homogeneous flow, equivalent to the G2d(1/ρm) term in Eq. (6-142). Pressure drop is computed by integration of the incremental pressure changes along the length of pipe. For condensing vapor in vertical downflow, in which the liquid flows as a thin annular film, the frictional contribution to the pressure drop may be estimated based on the gas flow alone, using the friction factor plotted in Fig. 6-31, where ReG is the Reynolds number for the gas flowing alone (Bergelin, et al., Proc. Heat Transfer Fluid Mech. Inst., ASME, June 22–24, 1949, pp. 19–28). 2fG′ ρGVG2 dp −  =  (6-144) dz D To this should be added the GG2 d(1/ρG)/dx term to account for velocity change effects. Gases and Solids The flow of gases and solids in horizontal pipe is usually classified as either dilute phase or dense phase flow. Unfortunately, there is no clear dilineation between the two types of flow, and the dense phase description may take on more than one meaning, creating some confusion (Knowlton, et al., Chem. Eng. Progr., 90(4), 44–54 [April 1994]). For dilute phase flow, achieved at low solids-to-gas weight ratios (loadings), and high gas velocities, the solids may be fully suspended and fairly uniformly dispersed over the pipe cross section (homogeneous flow), particularly for low-density or small particle size solids. At lower gas velocities, the solids may bounce along the bottom of the pipe. With higher loadings and lower gas velocities, the particles may settle to the bottom of the pipe, form-

6-30

FLUID AND PARTICLE DYNAMICS

Friction factors for condensing liquid/gas flow downward in vertical pipe. In this correlation Γ/ρL is in ft2/h. To convert ft2/h to m2/s, multiply by 0.00155. (From Bergelin, et al., Proc. Heat Transfer Fluid Mech. Inst., ASME, 1949, p. 19.) FIG. 6-31

ing dunes, with the particles moving from dune to dune. In dense phase conveying, solids tend to concentrate in the lower portion of the pipe at high gas velocity. As gas velocity decreases, the solids may first form dense moving strands, followed by slugs. Discrete plugs of solids may be created intentionally by timed injection of solids, or the plugs may form spontaneously. Eventually the pipe may become blocked. For more information on flow patterns, see Coulson and Richardson (Chemical Engineering, vol. 2, 2d ed., Pergamon, New York, 1968, p. 583); Korn (Chem. Eng., 57[3], 108–111 [1950]); Patterson ( J. Eng. Power, 81, 43–54 [1959]); Wen and Simons (AIChE J., 5, 263–267 [1959]); and Knowlton, et al. (Chem. Eng. Progr., 90[4], 44–54 [April 1994]). For the minimum velocity required to prevent formation of dunes or settled beds in horizontal flow, some data are given by Zenz (Ind. Eng. Chem. Fundam., 3, 65–75 [1964]), who presented a correlation for the minimum velocity required to keep particles from depositing on the bottom of the pipe. This rather tedious estimation procedure may also be found in Govier and Aziz, who provide additional references and discussion on transition velocities. In practice, the actual conveying velocities used in systems with loadings less than 10 are generally over 15 m/s, (49 ft/s) while for high loadings (>20) they are generally less than 7.5 m/s (24.6 ft/s) and are roughly twice the actual solids velocity (Wen and Simons, AIChE J., 5, 263–267 [1959]). Total pressure drop for horizontal gas/solid flow includes acceleration effects at the entrance to the pipe and frictional effects beyond the entrance region. A great number of correlations for pressure gradient are available, none of which is applicable to all flow regimes. Govier and Aziz review many of these and provide recommendations on when to use them. For upflow of gases and solids in vertical pipes, the minimum conveying velocity for low loadings may be estimated as twice the terminal settling velocity of the largest particles. Equations for terminal settling velocity are found in the “Particle Dynamics” subsection, following. Choking occurs as the velocity is dropped below the minimum conveying velocity and the solids are no longer transported, collapsing into solid plugs (Knowlton, et al., Chem. Eng. Progr., 90[4], 44–54 [April 1994]). See Smith (Chem. Eng. Sci., 33, 745–749 [1978]) for an equation to predict the onset of choking. Total pressure drop for vertical upflow of gases and solids includes acceleration and frictional affects also found in horizontal flow, plus potential energy or hydrostatic effects. Govier and Aziz review many of the pressure drop calculation methods and provide recommendations for their use. See also Yang (AIChE J., 24, 548–552 [1978]). Drag reduction has been reported for low loadings of small diameter particles (<60 µm diameter), ascribed to damping of turbulence near the wall (Rossettia and Pfeffer, AIChE J., 18, 31–39 [1972]). For dense phase transport in vertical pipes of small diameter, see

Sandy, Daubert, and Jones (Chem. Eng. Prog., 66, Symp. Ser., 105, 133–142 [1970]). The flow of bulk solids through restrictions and bins is discussed in symposium articles (J. Eng. Ind., 91[2] [1969]) and by Stepanoff (Gravity Flow of Bulk Solids and Transportation of Solids in Suspension, Wiley, New York, 1969). Some problems encountered in discharge from bins include (Knowlton, et al., Chem. Eng. Progr., 90[4], 44–54 [April 1994]) flow stoppage due to ratholing or arching, segregation of fine and coarse particles, flooding upon collapse of ratholes, and poor residence time distribution when funnel flow occurs. Solid/liquid or slurry flow may be divided roughly into two categories based on settling behavior (see Etchells in Shamlou, Processing of Solid-Liquid Suspensions, Chap. 12, Butterworth-Heinemann, Oxford, 1993). Nonsettling slurries are made up of very fine, highly concentrated, or neutrally buoyant particles. These slurries are normally treated as pseudohomogeneous fluids. They may be quite viscous and are frequently non-Newtonian. Slurries of particles that tend to settle out rapidly are called settling slurries or fast-settling slurries. While in some cases positively buoyant solids are encountered, the present discussion will focus on solids which are more dense than the liquid. For horizontal flow of fast-settling slurries, the following rough description may be made (Govier and Aziz). Ultrafine particles, 10 µm or smaller, are generally fully syspended and the particle distributions are not influenced by gravity. Fine particles 10 to 100 µm (3.3 × 10−5 to 33 × 10−5 ft) are usually fully suspended, but gravity causes concentration gradients. Medium-size particles, 100 to 1000 µm, may be fully suspended at high velocity, but often form a moving deposit on the bottom of the pipe. Coarse particles, 1,000 to 10,000 µm, (0.0033 to .033 ft), are seldom fully suspended and are usually conveyed as a moving deposit. Ultracoarse particles larger than 10,000 µm (0.033 ft) are not suspended at normal velocities unless they are unusually light. Figure 6-32, taken from Govier and Aziz, schematically indicates four flow pattern regions superimposed on a plot of pressure gradient vs. mixture velocity VM = VL + VS = (QL + QS)/A where VL and VS are the superficial liquid and solid velocities, QL and QS are liquid and solid volumetric flow rates, and A is the pipe cross-sectional area. VM4 is the transition velocity above which a bed exists in the bottom of the pipe, part of which is stationary and part of which moves by saltation, with the upper particles tumbling and bouncing over one another, often with formation of dunes. With a broad particle-size distribution, the finer particles may be fully suspended. Near VM4, the pressure gra-

FIG. 6-32 Flow pattern regimes and pressure gradients in horizontal slurry flow. (From Govier and Aziz, The Flow of Complex Mixtures in Pipes, Van Nostrand Reinhold, New York, 1972.)

FLUID DYNAMICS dient rapidly increases as VM decreases. Above VM3, the entire bed moves. Above VM2, the solids are fully suspended; that is, there is no deposit, moving or stationary, on the bottom of the pipe. However, the concentration distribution of solids is asymmetric. This flow pattern is the most frequently used for fast-settling slurry transport. Typical mixture velocities are in the range of 1 to 3 m/s (3.3 to 9.8 ft/s). The minimum in the pressure gradient is found to be near VM2. Above VM1, the particles are symmetrically distributed, and the pressure gradient curve is nearly parallel to that for the liquid by itself. The most important transition velocity, often regarded as the minimum transport or conveying velocity for settling slurries, is VM2. The Durand equation (Durand, Minnesota Int. Hydraulics Conf., Proc., 89, Int. Assoc. for Hydraulic Research [1953]; Durand and Condolios, Proc. Colloq. On the Hyd. Transport of Solids in Pipes, Nat. Coal Board [UK], Paper IV, 39–35 [1952]) gives the minimum transport velocity as VM2 = FL[2gD(s − 1)]0.5 where

(6-145)

g = acceleration of gravity D = pipe diameter s = ρS /ρL = ratio of solid to liquid density FL = a factor influenced by particle size and concentration

Probably FL is a function of particle Reynolds number and concentration, but Fig. 6-33 gives Durand’s empirical correlation for FL as a function of particle diameter and the input, feed volume fraction solids, CS = QS /(QS + QL). The form of Eq. (6-145) may be derived from turbulence theory, as shown by Davies (Chem. Eng. Sci., 42, 1667–1670 [1987]). No single correlation for pressure drop in horizontal solid/liquid flow has been found satisfactory for all particle sizes, densities, concentrations, and pipe sizes. However, with reference to Fig. 6-32, the following simplifications may be considered. The minimum pressure gradient occurs near VM2 and for conservative purposes it is generally desirable to exceed VM2. When VM2 is exceeded, a rough guide for pressure drop is 25 percent greater than that calculated assuming that the slurry behaves as a psuedohomogeneous fluid with the density of the mixture and the viscosity of the liquid. Above the transition velocity to symmetric suspension, VM1, the pressure drop closely approaches the pseuodohomogeneous pressure drop. The following

Durand factor for minimum suspension velocity. (From Govier and Aziz, The Flow of Complex Mixtures in Pipes, Van Nostrand Reinhold, New York, 1972.)

FIG. 6-33

6-31

correlation by Spells (Trans. Inst. Chem. Eng. [London], 33, 79–84 [1955]) may be used for VM1. DVM1ρM 0.775 2 = 0.075  gDS(s − 1) (6-146) VM1 µ where D = pipe diameter DS = particle diameter (such that 85 percent by weight of particles are smaller than DS ρM = the slurry mixture density µ = liquid viscosity s = ρS /ρL = ratio of solid to liquid density

冢

冣

Between VM2 and VM1 the concentration of solids gradually becomes more uniform in the vertical direction. This transition has been modeled by several authors as a concentration gradient where turbulent diffusion balances gravitational settling. See, for example, Karabelas (AIChE J., 23, 426–434 [1977]). Published correlations for pressure drop are frequently very complicated and tedious to use, may not offer significant accuracy advantages over the simple guide given here, and many of them are applicable only for velocities above VM2. One which does include the effect of sliding beds is due to Gaessler (Doctoral Dissertation, Technische Hochshule, Karlsruhe, Germany [1967]; reproduced by Govier and Aziz, pp. 668–669). Turian and Yuan (AIChE J., 23, 232–243 [1977]; see also Turian and Oroskar, AIChE J., 24, 1144 [1978]) segregated a large body of data into four flow regime groups and developed empirical correlations for predicting pressure drop in each flow regime. Pressure drop data for the flow of paper stock in pipes are given in the data section of Standards of the Hydraulic Institute (Hydraulic Institute, 1965). The flow behavior of fiber suspensions is discussed by Bobkowicz and Gauvin (Chem. Eng. Sci., 22, 229–241 [1967]), Bugliarello and Daily (TAPPI, 44, 881–893 [1961]), and Daily and Bugliarello (TAPPI, 44, 497–512 [1961]). In vertical flow of fast-settling slurries, the in situ concentration of solids with density greater than the liquid will exceed the feed concentration C = QS /(QS + QL) for upflow and will be smaller than C for downflow. This results from slip between the phases. The slip velocity, the difference between the in situ average velocities of the two phases, is roughly equal to the terminal settling velocity of the solids in the liquid. Specification of the slip velocity for a pipe of a given diameter, along with the phase flow rates, allows calculation of in situ volume fractions, average velocities, and holdup ratios by simple material balances. Slip velocity may be affected by particle concentration and by turbulence conditions in the liquid. Drift-flux theory, a framework incorporating certain functional forms for empirical expressions for slip velocity, is described by Wallis (One-Dimensional Two-Phase Flow, McGraw-Hill, New York, 1969). Minimum transport velocity for upflow for design purposes is usually taken as twice the particle settling velocity. Pressure drop in vertical pipe flow includes the effects of kinetic and potential energy (elevation) changes and friction. Rose and Duckworth (The Engineer, 227[5,903], 392 [1969]; 227[5,904], 430 [1969]; 227[5,905], 478 [1969]; see also Govier and Aziz, pp. 487–493) have developed a calculation procedure including all these effects, which may be applied not only to vertical solid/liquid flow, but also to gas/solid flow and to horizontal flow. For fast-settling slurries, ensuring conveyance is usually the key design issue while pressure drop is somewhat less important. For nonsettling slurries conveyance is not an issue, because the particles do not separate from the liquid. Here, viscous and rheological behavior, which control pressure drop, take on critical importance. Fine particles, often at high concentration, form nonsettling slurries for which useful design equations can be developed by treating them as homogeneous fluids. These fluids are usually very viscous and often non-Newtonian. Shear-thinning and Bingham plastic behavior are common; dilatancy is sometimes observed. Rheology of such fluids must in general be empirically determined, although theoretical results are available for some very limited circumstances. Further discussion of both fast-settling and nonsettling slurries may be found in Shook (in Shamlou, Processing of Solid-Liquid Suspensions, Chap. 11, Butterworth-Heinemann, Oxford, 1993).

6-32

FLUID AND PARTICLE DYNAMICS

FLUID DISTRIBUTION Uniform fluid distribution is essential for efficient operation of chemical-processing equipment such as contactors, reactors, mixers, burners, heat exchangers, extrusion dies, and textile-spinning chimneys. To obtain optimum distribution, proper consideration must be given to flow behavior in the distributor, flow conditions upstream and downstream of the distributor, and the distribution requirements of the equipment. Even though the principles of fluid distribution have been well developed for more than three decades, they are frequently overlooked by equipment designers, and a significant fraction of process equipment needlessly suffers from maldistribution. In this subsection, guides for the design of various types of fluid distributors, taking into account only the flow behavior within the distributor, are given. Perforated-Pipe Distributors The simple perforated pipe or sparger (Fig. 6-34) is a common type of distributor. As shown, the flow distribution is uniform; this is the case in which pressure recovery due to kinetic energy or momentum changes, frictional pressure drop along the length of the pipe, and pressure drop across the outlet holes have been properly considered. In typical turbulent flow applications, inertial effects associated with velocity changes may dominate frictional losses in determining the pressure distribution along the pipe, unless the length between orifices is large. Application of the momentum or mechanical energy equations in such a case shows that the pressure inside the pipe increases with distance from the entrance of the pipe. If the outlet holes are uniform in size and spacing, the discharge flow will be biased toward the closed end. Disturbances upstream of the distributor, such as pipe bends, may increase or decrease the flow to the holes at the beginning of the distributor. When frictional pressure drop dominates the inertial pressure recovery, the distribution is biased toward the feed end of the distributor.

FIG. 6-34

Perforated-pipe distributor.

For turbulent flow, with roughly uniform distribution, assuming a constant friction factor, the combined effect of friction and inertial (momentum) pressure recovery is given by 4fL ρVi2 ∆p =  − 2K  (discharge manifolds) (6-147) 3D 2 where ∆p = net pressure drop over the length of the distributor L = pipe length D = pipe diameter f = Fanning friction factor Vi = distributor inlet velocity

冢

冣

The factor K would be 1 in the case of full momentum recovery, or 0.5 in the case of negligible viscous losses in the portion of flow which remains in the pipe after the flow divides at a takeoff point (Denn, pp. 126–127). Experimental data (Van der Hegge Zijnen, Appl. Sci. Res., A3, 144–162 [1951–1953]; and Bailey, J. Mech. Eng. Sci., 17, 338–347 [1975]), while scattered, show that K is probably close to 0.5 for discharge manifolds. For inertially dominated flows, ∆p will be negative. For return manifolds the recovery factor K is close to 1.0, and the pressure drop between the first hole and the exit is given by ρVe2 4fL ∆p =  + 2K  (return manifolds) (6-148) 3D 2 where Ve is the pipe exit velocity. One means to obtain a desired uniform distribution is to make the average pressure drop across the holes ∆ po large compared to the pressure variation over the length of pipe ∆ p. Then, the relative variation in pressure drop across the various holes will be small, and so will be the variation in flow. When the area of an individual hole is

冢

冣

small compared to the cross-sectional area of the pipe, hole pressure drop may be expressed in terms of the discharge coefficient Co and the velocity across the hole Vo as 1 ρV o2 (6-149) ∆po =   C o2 2 Provided Co is the same for all the holes, the percent maldistribution, defined as the percentage variation in flow between the first and last holes, may be estimated reasonably well for small maldistribution by (Senecal, Ind. Eng. Chem., 49, 993–997 [1957]) ∆p − |∆p|  冣 冢 冪莦 ∆p

Percent maldistribution = 100 1 −

o

(6-150)

o

This equation shows that for 5 percent maldistribution, the pressure drop across the holes should be about 10 times the pressure drop over the length of the pipe. For discharge manifolds with K = 0.5 in Eq. (6-147), and with 4f L/3D << 1, the pressure drop across the holes should be 10 times the inlet velocity head, ρVi2/2 for 5 percent maldistribution. This leads to a simple design equation. Discharge manifolds, 4f L/3D << 1, 5% maldistribution: Vo Ap  =  = 兹1苶0苶Co Vi Ao

(6-151)

Here Ap = pipe cross-sectional area and Ao is the total hole area of the distributor. Use of large hole velocity to pipe velocity ratios promotes perpendicular discharge streams. In practice, there are many cases where the 4f L/3D term will be less than unity but not close to zero. In such cases, Eq. (6-151) will be conservative, while Eqs. (6-147), (6-149), and (6-150) will give more accurate design calculations. In cases where 4f L/(3D) > 2, friction effects are large enough to render Eq. (6-151) nonconservative. When significant variations in f along the length of the distributor occur, calculations should be made by dividing the distributor into small enough sections that constant f may be assumed over each section. For return manifolds with K = 1.0 and 4f L/(3D) << 1, 5 percent maldistribution is achieved when hole pressure drop is 20 times the pipe exit velocity head. Return manifolds, 4f L/3D << 1, 5% maldistribution: Vo Ap (6-152)  =  = 兹苶2苶0Co Ve Ao When 4f L/3D is not negligible, Eq. (6-152) is not conservative and Eqs. (6-148), (6-149), and (6-150) should be used. One common misconception is that good distribution is always provided by high pressure drop, so that increasing flow rate improves distribution by increasing pressure drop. Conversely, it is mistakenly believed that turndown of flow through a perforated pipe designed using Eqs. (6-151) and (6-152) will cause maldistribution. However, when the distribution is nearly uniform, decreasing the flow rate decreases ∆p and ∆po in the same proportion, and Eqs. (6-151) and (6-152) are still satisfied, preserving good distribution independent of flow rate, as long as friction losses remain small compared to inertial (velocity head change) effects. Conversely, increasing the flow rate through a distributor with severe maldistribution will not generally produce good distribution. Often, the pressure drop required for design flow rate is unacceptably large for a distributor pipe designed for uniform velocity through uniformly sized and spaced orifices. Several measures may be taken in such situations. These include the following: 1. Taper the diameter of the distributor pipe so that the pipe velocity and velocity head remain constant along the pipe, thus substantially reducing pressure variation in the pipe. 2. Vary the hole size and/or the spacing between holes to compensate for the pressure variation along the pipe. This method may be sensitive to flow rate and a distributor optimized for one flow rate may suffer increased maldistribution as flow rate deviates from design rate. 3. Feed or withdraw from both ends, reducing the pipe flow velocity head and required hole pressure drop by a factor of 4.

FLUID DYNAMICS

6-33

The orifice discharge coefficient Co is usually taken to be about 0.62. However, Co is dependent on the ratio of hole diameter to pipe diameter, pipe wall thickness to hole diameter ratio, and pipe velocity to hole velocity ratio. As long as all these are small, the coefficient 0.62 is generally adequate. Example 9: Pipe Distributor A 3-in schedule 40 (inside diameter 7.793 cm) pipe is to be used as a distributor for a flow of 0.010 m3/s of water (ρ = 1,000 kg/m3, µ = 0.001 Pa ⋅ s). The pipe is 0.7 m long and is to have 10 holes of uniform diameter and spacing along the length of the pipe. The distributor pipe is submerged. Calculate the required hole size to limit maldistribution to 5 percent, and estimate the pressure drop across the distributor. The inlet velocity computed from Vi = Q/A = 4Q/(πD2) is 2.10 m/s, and the inlet Reynolds number is DVi ρ 0.07793 × 2.10 × 1000 Re =  =  = 1.64 × 105 µ 0.001 For commercial pipe with roughness
= 0.046 mm, the friction factor is about 0.0043. Approaching the last hole, the flow rate, velocity and Reynolds number are about one-tenth their inlet values. At Re = 16,400 the friction factor f is about 0.0070. Using an average value of f = 0.0057 over the length of the pipe, 4 fL/3D is 0.068 and may reasonably be neglected so that Eq. (6-151) may be used. With Co = 0.62, Vo A p  =  = 兹1苶0苶Co = 兹1苶0苶 × 0.62 = 1.96 Vi Ao With pipe cross-sectional area Ap = 0.00477 m2, the total hole area is 0.00477/1.96 = 0.00243 m2. The area and diameter of each hole are then 0.00243/10 = 0.000243 m2 and 1.76 cm. With Vo /Vi = 1.96, the hole velocity is 1.96 × 2.10 = 4.12 m/s and the pressure drop across the holes is obtained from Eq. (6-149). 1000(4.12)2 1 ρVo2 1 ∆ po =   = 2 ×  = 22,100 Pa Co2 2 0.62 2 Since the hole pressure drop is 10 times the pressure variation in the pipe, the total pressure drop from the inlet of the distributor may be taken as approximately 22,100 Pa. Further detailed information on pipe distributors may be found in Senecal (Ind. Eng. Chem., 49, 993–997 [1957]). Much of the information on tapered manifold design has appeared in the pulp and paper literature (Spengos and Kaiser, TAPPI, 46[3], 195–200 [1963]; Madeley, Paper Technology, 9[1], 35–39 [1968]; Mardon, et al., TAPPI, 46[3], 172–187 [1963]; Mardon, et al., Pulp and Paper Magazine of Canada, 72[11], 76–81 [November 1971]; Trufitt, TAPPI, 58[11], 144–145 [1975]). Slot Distributors These are generally used in sheeting dies for extrusion of films and coatings and in air knives for control of thickness of a material applied to a moving sheet. A simple slotted pipe for turbulent flow conditions may give severe maldistribution because of nonuniform discharge velocity, but also because this type of design does not readily give perpendicular discharge (Koestel and Tuve, Heat. Piping Air Cond., 20[1], 153–157 [1948]; Senecal, Ind. Eng. Chem., 49, 993–997 [1957]; Koestel and Young, Heat. Piping Air Cond., 23[7], 111–115 [1951]). For slots in tapered ducts where the duct cross-sectional area decreases linearly to zero at the far end, the discharge angle will be constant along the length of the duct (Koestel and Young, ibid.). One way to ensure an almost perpendicular discharge is to have the ratio of the area of the slot to the cross-sectional area of the pipe equal to or less than 0.1. As in the case of perforatedpipe distributors, pressure variation within the slot manifold and pressure drop across the slot must be carefully considered. In practice, the following methods may be used to keep the diameter of the pipe to a minimum consistent with good performance (Senecal, Ind. Eng. Chem., 49, 993–997 [1957]): 1. Feed from both ends. 2. Modify the cross-sectional design (Fig. 6-35); the slot is thus farther away from the influence of feed-stream velocity. 3. Increase pressure drop across the slot; this can be accomplished by lengthening the lips (Fig. 6-35). 4. Use screens (Fig. 6-35) to increase overall pressure drop across the slot. Design considerations for air knives are discussed by Senecal (ibid.). Design procedures for extrusion dies when the flow is laminar,

FIG. 6-35

Modified slot distributor.

as with highly viscous fluids, are presented by Bernhardt (Processing of Thermoplastic Materials, Rheinhold, New York, 1959, pp. 248– 281). Turning Vanes In applications such as ventilation, the discharge profile from slots can be improved by turning vanes. The tapered duct is the most amenable for turning vanes because the discharge angle remains constant. One way of installing the vanes is shown in Fig. 6-36. The vanes should have a depth twice the spacing (Heating, Ventilating, Air Conditioning Guide, vol. 38, American Society of Heating, Refrigerating and Air-Conditioning Engineers, 1960, pp. 282–283) and a curvature at the upstream end of the vanes of a circular arc which is tangent to the discharge angle θ of a slot without vanes and perpendicular at the downstream or discharge end of the vanes (Koestel and Young, Heat. Piping Air Cond., 23[7], 111–115 [1951]). Angle θ can be estimated from Cd As cot θ =  (6-153) Ad where A s = slot area A d = duct cross-sectional area at upstream end Cd = discharge coefficient of slot Vanes may be used to improve velocity distribution and reduce frictional loss in bends, when the ratio of bend turning radius to pipe diameter is less than 1.0. For a miter bend with low-velocity flows, simple circular arcs (Fig. 6-37) can be used, and with high-velocity flows, vanes of special airfoil shapes are required. For additional details and references, see Ower and Pankhurst (The Measurement of Air Flow, Pergamon, New York, 1977, p. 102); Pankhurst and Holder (Wind-Tunnel Technique, Pitman, London, 1952, pp. 92–93); Rouse (Engineering Hydraulics, Wiley, New York, 1950, pp. 399–401); and Jorgensen (Fan Engineering, 7th ed., Buffalo Forge Co., Buffalo, 1970, pp. 111, 117, 118). Perforated Plates and Screens A nonuniform velocity profile in turbulent flow through channels or process equipment can be smoothed out to any desired degree by adding sufficient uniform resistance, such as perforated plates or screens across the flow channel, as shown in Fig. 6-38. Stoker (Ind. Eng. Chem., 38, 622–624 [1946]) provides the following equation for the effect of a uniform resistance on velocity profile:

冪莦莦

(V1,max /V)2 + α2 − α1 + α2K V2,max (6-154)  =  V 1+K Here, V is the area average velocity, K is the number of velocity heads of pressure drop provided by the uniform resistance, ∆ p = KρV 2/2, and α is the velocity profile factor used in the mechanical energy bal-

FIG. 6-36 Turning vanes in a slot distributor.

6-34

FLUID AND PARTICLE DYNAMICS entrance to a vessel, it is sometimes necessary to diffuse the momentum of the inlet jet discharging from the feed pipe in order to produce a more uniform velocity profile within the vessel. Methods for this application exist, but remain largely in the domain of proprietary, commercial design. FLUID MIXING

FIG. 6-37

Miter bend with vanes.

ance, Eq. (6-13). It is ratio of the area average of the cube of the velocity, to the cube of the area average velocity V. The shape of the exit velocity profile appears twice in Eq. (6-154), in V2,max / V and α2. Typically, K is on the order of 10, and the desired exit velocity profile is fairly uniform so that α2 ∼ 1.0 may be appropriate. Downstream of the resistance, the velocity profile will gradually reestablish the fully developed profile characteristic of the Reynolds number and channel shape. The screen or perforated plate open area required to produce the resistance K may be computed from Eqs. (6-107) or (6-111). Screens and other flow restrictions may also be used to suppress stream swirl and turbulence (Loehrke and Nagib, J. Fluids Eng., 98, 342–353 [1976]). Contraction of the channel, as in a venturi, provides further reduction in turbulence level and flow nonuniformity. Beds of Solids A suitable depth of solids can be used as a fluid distributor. As for other types of distribution devices, a pressure drop of 10 velocity heads is typically used, here based on the superficial velocity through the bed. There are several substantial disadvantages to use of particle beds for flow distribution. Heterogeneity of the bed may actually worsen rather than improve distribution. In general, uniform flow may be found only downstream of the point in the bed where sufficient pressure drop has occurred to produce uniform flow. Therefore, inefficiency results when the bed also serves reaction or mass transfer functions, as in catalysts, adsorbents, or tower packings for gas/liquid contacting, since portions of the bed are bypassed. In the case of trickle flow of liquid downward through column packings, inlet distribution is critical since the bed itself is relatively ineffective in distributing the liquid. Maldistribution of flow through packed beds also arises when the ratio of bed diameter to particle size is less than 10 to 30. Other Flow Straightening Devices Other devices designed to produce uniform velocity or reduce swirl, sometimes with reduced pressure drop, are available. These include both commercial devices of proprietary design and devices discussed in the literature. For pipeline flows, see the references under flow inverters and static mixing elements previously discussed in the “Incompressible Flow in Pipes and Channels” subsection. For large area changes, as at the

FIG. 6-38

Smoothing out a nonuniform profile in a channel.

Mixing of fluids is a discipline of fluid mechanics. Fluid motion is used to accelerate the otherwise slow processes of diffusion and conduction to bring about uniformity of concentration and temperature, blend materials, facilitate chemical reactions, bring about intimate contact of multiple phases, and so on. As the subject is too broad to cover fully, only a brief introduction and some references for further information are given here. Several texts are available. These include Harnby, Edwards, and Nienow (Mixing in the Process Industries, 2d ed., Butterworths, London, 1992), Oldshue (Fluid Mixing Technology, McGraw-Hill, New York, 1983), Tatterson (Fluid Mixing and Gas Dispersion in Agitated Tanks, McGraw-Hill, New York, 1991), Uhl and Gray (Mixing, vols. I–III, Academic, New York, 1966, 1967, 1986), and Nagata (Mixing: Principles and Applications, Wiley, New York, 1975). A good overview of stirred tank agitation is given in the series of articles from Chemical Engineering (110–114, Dec. 8, 1975; 139–145, Jan. 5, 1976; 93–100, Feb. 2, 1976; 102–110, Apr. 26, 1976; 144–150, May 24, 1976; 141– 148, July 19, 1976; 89–94, Aug. 2, 1976; 101–108, Aug. 30, 1976; 109– 112, Sept. 27, 1976; 119–126, Oct. 25, 1976; 127–133, Nov. 8, 1976). Process mixing is commonly carried out in pipeline and vessel geometries. The terms radial mixing and axial mixing are commonly used. Axial mixing refers to mixing of materials which pass a given point at different times, and thus leads to backmixing. For example, backmixing or axial mixing occurs in stirred tanks where fluid elements entering the tank at different times are intermingled. Mixing of elements initially at different axial positions in a pipeline is axial mixing. Radial mixing occurs between fluid elements passing a given point at the same time, as, for example, between fluids mixing in a pipeline tee. Turbulent flow, by means of the chaotic eddy motion associated with velocity fluctuation, is conducive to rapid mixing and, therefore, is the preferred flow regime for mixing. Laminar mixing is carried out when high viscosity makes turbulent flow impractical. Stirred Tank Agitation Turbine impeller agitators, of a variety of shapes, are used for stirred tanks, predominantly in turbulent flow. Figure 6-39 shows typical stirred tank configurations and timeaveraged flow patterns for axial flow and radial flow impellers. In order to prevent formation of a vortex, four vertical baffles are normally installed. These cause top-to-bottom mixing and prevent mixing-ineffective swirling motion. For a given impeller and tank geometry, the impeller Reynolds number determines the flow pattern in the tank: D2Nρ ReI =  (6-155) µ where D = impeller diameter, N = rotational speed, and ρ and µ are the liquid density and viscosity. Rotational speed N is typically reported in revolutions per minute, or revolutions per second in SI units. Radians per second are almost never used. Typically, ReI > 104 is required for fully turbulent conditions throughout the tank. A wide transition region between laminar and turbulent flow occurs over the range 10 < ReI < 104. The power P drawn by the impeller is made dimensionless in a group called the power number: P NP =  (6-156) ρN 3D5 Figure 6-40 shows power number vs. impeller Reynolds number for a typical configuration. The similarity to the friction factor vs. Reynolds number behavior for pipe flow is significant. In laminar flow, the power number is inversely proportional to Reynolds number, reflecting the dominance of viscous forces over inertial forces. In turbulent flow, where inertial forces dominate, the power number is nearly constant.

FLUID DYNAMICS Nb = tb N

FIG. 6-39 Typical stirred tank configurations, showing time-averaged flow patterns for axial flow and radial flow impellers. (From Oldshue, Fluid Mixing Technology, McGraw-Hill, New York, 1983.)

Impellers are sometimes viewed as pumping devices; the total volumetric flow rate Q discharged by an impeller is made dimensionless in a pumping number: Q NQ = 3 (6-157) ND Blend time tb, the time required to achieve a specified maximum standard deviation of concentration after injection of a tracer into a stirred tank, is made dimensionless by multiplying by the impeller rotational speed:

6-35 (6-158)

Dimensionless pumping number and blend time are independent of Reynolds number under fully turbulent conditions. The magnitude of concentration fluctuations from the final well-mixed value in batch mixing decays exponentially with time. The design of mixing equipment depends on the desired process result. There is often a tradeoff between operating cost, which depends mainly on power, and capital cost, which depends on agitator size and torque. For some applications bulk flow throughout the vessel is desired, while for others high local turbulence intensity is required. Multiphase systems introduce such design criteria as solids suspension and gas dispersion. In very viscous systems, helical ribbons, extruders, and other specialized equipment types are favored over turbine agitators. Pipeline Mixing Mixing may be carried out with mixing tees, inline or motionless mixing elements, or in empty pipe. In the latter case, large pipe lengths may be required to obtain adequate mixing. Coaxially injected streams require lengths on the order of 100 pipe diameters. Coaxial mixing in turbulent single-phase flow is characterized by the turbulent diffusivity (eddy diffusivity) DE which determines the rate of radial mixing. Davies (Turbulence Phenomena, Academic, New York, 1972) provides an equation for DE which may be rewritten as DE ∼ 0.015DVRe−0.125 where

(6-159)

D = pipe diameter V = average velocity Re = pipe Reynolds number, DVρ/µ ρ = density µ = viscosity

Properly designed tee mixers, with due consideration given to main stream and injected stream momentum, are capable of producing high degrees of uniformity in just a few diameters. Forney (Jet Injection for Optimum Pipeline Mixing, in “Encyclopedia of Fluid Mechanics,” vol. 2., Chap. 25, Gulf Publishing, 1986) provides a thorough discussion of tee mixing. Inline or motionless mixers are generally of

FIG. 6-40 Dimensionless power number in stirred tanks. (Reprinted with permission from Bates, Fondy, and Corpstein, Ind. Eng. Chem. Process Design Develop., 2, 310 [1963].)

6-36

FLUID AND PARTICLE DYNAMICS TUBE BANKS Pressure drop across tube banks may not be correlated by means of a single, simple friction factor—Reynolds number curve, owing to the variety of tube configurations and spacings encountered, two of which are shown in Fig. 6-41. Several investigators have allowed for configuration and spacing by incorporating spacing factors in their friction factor expressions or by using multiple friction factor plots. Commercial computer codes for heat-exchanger design are available which include features for estimating pressure drop across tube banks. Turbulent Flow The correlation by Grimison (Trans. ASME, 59, 583–594 [1937]) is recommended for predicting pressure drop for turbulent flow (Re ≥ 2,000) across staggered or in-line tube banks for tube spacings [(a/Dt), (b/Dt)] ranging from 1.25 to 3.0. The pressure drop is given by

FIG. 6-41

2 4 f NrρVmax ∆p =  2

Tube-bank configurations.

where proprietary commercial design, and may be selected for viscous or turbulent, single or multiphase mixing applications. They substantially reduce required pipe length for mixing.

(6-160)

f = friction factor Nr = number of rows of tubes in the direction of flow ρ = fluid density Vmax = fluid velocity through the minimum area available for flow.

Upper chart: Friction factors for staggered tube banks with minimum fluid flow area in transverse openings. Lower chart: Friction factors for staggered tube banks with minimum fluid flow area in diagonal openings. (From Grimison, Trans. ASME, 59, 583 [1937].) FIG. 6-42

FLUID DYNAMICS

FIG. 6-43

6-37

Friction factors for in-line tube banks. (From Grimison, Trans. ASME, 59, 583 [1937].)

For banks of staggered tubes, the friction factor for isothermal flow is obtained from Fig. (6-42). Each “fence” (group of parametric curves) represents a particular Reynolds number defined as DtVmaxρ Re =  (6-161) µ where Dt = tube outside diameter and µ = fluid viscosity. The numbers along each fence represent the transverse and inflow-direction spacings. The upper chart is for the case in which the minimum area for flow is in the transverse openings, while the lower chart is for the case in which the minimum area is in the diagonal openings. In the latter case, Vmax is based on the area of the diagonal openings and Nr is the number of rows in the direction of flow minus 1. A critical comparison of this method with all the data available at the time showed an average deviation of the order of 15 percent. (Boucher and Lapple, Chem. Eng. Prog., 44, 117–134 [1948]). For tube spacings greater than 3 tube diameters, the correlation by Gunter and Shaw (Trans. ASME, 67, 643–660 [1945]) can be used as an approximation. As an approximation, the pressure drop can be taken as 0.72 velocity head (based on Vmax per row of tubes for tube spacings commonly encountered in practice (Lapple, et al., Fluid and Particle Mechanics, University of Delaware, Newark, 1954). For banks of in-line tubes, f for isothermal flow is obtained from Fig. 6-43. Average deviation from available data is on the order of 15 percent. For tube spacings greater than 3Dt, the charts of Gram, Mackey, and Monroe (Trans. ASME, 80, 25–35 [1958]) can be used. As an approximation, the pressure drop can be taken as 0.32 velocity head (based on Vmax) per row of tubes (Lapple, et al., Fluid and Particle Mechanics, University of Delaware, Newark, 1954). For turbulent flow through shallow tube banks, the average friction factor per row will be somewhat greater than indicated by Figs. 6-42 and 6-43, which are based on 10 or more rows depth. A 30 percent increase per row for 2 rows, 15 percent per row for 3 rows and 7 percent per row for 4 rows can be taken as the maximum likely to be encountered (Boucher and Lapple, Chem. Eng. Prog., 44, 117–134 [1948]). For a single row of tubes, the friction factor is given by Curve B in Fig. 6-44 as a function of tube spacing. This curve is based on the

the data of several experimenters, all adjusted to a Reynolds number of 10,000. The values should be substantially independent of Re for 1,000 < Re < 100,000. For extended surfaces, which include fins mounted perpendicularly to the tubes or spiral-wound fins, pin fins, plate fins, and so on, friction data for the specific surface involved should be used. For details, see Kays and London (Compact Heat Exchangers, 2d ed., McGraw-Hill, New York, 1964). If specific data are unavailable, the correlation by Gunter and Shaw (Trans. ASME, 67, 643–660 [1945]) may be used as an approximation. When a large temperature change occurs in a gas flowing across a

FIG. 6-44 Friction factors vs. transverse spacing for single row of tubes. (From Boucher and Lapple, Chem. Eng. Prog., 44, 117 [1948].)

6-38

FLUID AND PARTICLE DYNAMICS

tube bundle, gas properties should be evaluated at the mean temperature Tm = Tt + K ∆Tlm

(6-162)

where Tt = average tube-wall temperature K = constant ∆Tlm = log-mean temperature difference between the gas and the tubes. Values of K averaged from the recommendations of Chilton and Genereaux (Trans. AIChE, 29, 151–173 [1933]) and Grimison (Trans. ASME, 59, 583–594 [1937]) are as follows: for in-line tubes, 0.9 for cooling and −0.9 for heating; for staggered tubes, 0.75 for cooling and −0.8 for heating. For nonisothermal flow of liquids across tube bundles, the friction factor is increased if the liquid is being cooled and decreased if the liquid is being heated. The factors previously given for nonisothermal flow of liquids in pipes (“Incompressible Flow in Pipes and Channels”) should be used. For two-phase gas/liquid horizontal cross flow through tube banks, the method of Diehl and Unruh (Pet. Refiner, 37[10], 124–128 [1958]) is available. Transition Region This region extends roughly over the range 200 < Re < 2,000. Figure 6-45 taken from Bergelin, Brown, and Doberstein (Trans. ASME, 74, 953–960 [1952]) gives curves for friction factor fT for five different configurations. Pressure drop for liquid flow is given by 2 µs 0.14 4 fT Nr ρVmax ∆p =  (6-163)  2 µb where Nr = number of major restrictions encountered in flow through the bank (equal to number of rows when minimum flow area occurs in transverse openings, and to number of rows minus 1 when it occurs in the diagonal openings); ρ = fluid density; Vmax = velocity through minimum flow area; µs = fluid viscosity at tube-surface temperature and µb = fluid viscosity at average bulk temperature. This method gives the friction factor within about 25 percent. Laminar Region Bergelin, Colburn, and Hull (Univ. Delaware

冢 冣

Eng. Exp. Sta. Bull., 2 [1950]) recommend the following equations for pressure drop with laminar flow (Rev < 100) of liquids across banks of plain tubes with pitch ratios P/Dt of 1.25 and 1.50: 2 280Nr Dt 1.6 µs m ρV max (6-164) ∆p =     Rev P µb 2

冢 冣 冢 冣冢

冣

0.57 m= (6-165) (Rev)0.25 where Rev = DvVmaxρ/µb; Dv = volumetric hydraulic diameter [(4 × free-bundle volume)/(exposed surface area of tubes)]; P = pitch (= a for in-line arrangements, = a or c [whichever is smaller] for staggered arrangements), and other quantities are as defined following Eq. (6-163). Bergelin, et al. (ibid.) show that pressure drop per row is independent of the number of rows in the bank with laminar flow. The pressure drop is predicted within about 25 percent. The validity of extrapolating Eq. (6-164) to pitch ratios larger than 1.50 is unknown. The correlation of Gunter and Shaw (Trans. ASME, 67, 643–660 [1945]) may be used as an approximation in such cases. For laminar flow of non-Newtonian fluids across tube banks, see Adams and Bell (Chem. Eng. Prog., 64, Symp. Ser., 82, 133–145 [1968]). Flow-induced tube vibration occurs at critical fluid velocities through tube banks, and is to be avoided because of the severe damage that can result. Methods to predict and correct vibration problems may be found in Eisinger (Trans. ASME J. Pressure Vessel Tech., 102, 138–145 [May 1980]) and Chen (J. Sound Vibration, 93, 439–455 [1984]). BEDS OF SOLIDS Fixed Beds of Granular Solids Pressure-drop prediction is complicated by the variety of granular materials and of their packing arrangement. For flow of a single incompressible fluid through an incompressible bed of granular solids, the pressure drop may be estimated by the correlation given in Fig. 6-46 (Leva, Chem. Eng., 56[5], 115–117 [1949]), or Fluidization, McGraw-Hill, New York, 1959). The modified friction factor and Reynolds number are defined by Dpρφ3s − n
3|∆p| fm ⬅  2G2L(1 −
)3 − n

(6-166)

DpG Re′ ⬅  (6-167) µ where −∆ p = pressure drop L = depth of bed Dp = average particle diameter, defined as the diameter of a sphere of the same volume as the particle
= void fraction n = exponent given in Fig. 6-46 as a function of Re′ φs = shape factor defined as the area of sphere of diameter Dp divided by the actual surface area of the particle G = fluid superficial mass velocity based on the empty chamber cross section ρ = fluid density µ = fluid viscosity As for any incompressible single-phase flow, the equivalent pressure P = p + ρgz where g = acceleration of gravity z = elevation, may be used in place of p to account for gravitational effects in flows with vertical components. In creeping flow (Re′ < 10), 100 fm =  (6-168) Re′ At high Reynolds numbers the friction factor becomes nearly constant, approaching a value of the order of unity for most packed beds. In terms of S, particle surface area per unit volume of bed, Friction factors for transition region flow across tube banks. (Pitch is the minimum center-to-center tube spacing.) (From Bergelin, Brown, and Doberstein, Trans. ASME, 74, 953 [1952].)

FIG. 6-45

6(1 −
) Dp =  φsS

(6-169)

FLUID DYNAMICS

FIG. 6-46

6-39

Friction factor for beds of solids. (From Leva, Fluidization, McGraw-Hill, New York, 1959, p. 49.)

Porous Media Packed beds of granular solids are one type of the general class referred to as porous media, which include geological formations such as petroleum reservoirs and aquifers, manufactured materials such as sintered metals and porous catalysts, burning coal or char particles, and textile fabrics, to name a few. Pressure drop for incompressible flow across a porous medium has the same qualitative behavior as that given by Leva’s correlation in the preceding. At low Reynolds numbers, viscous forces dominate and pressure drop is proportional to fluid viscosity and superficial velocity, and at high Reynolds numbers, pressure drop is proportional to fluid density and to the square of superficial velocity. Creeping flow (Re′ <∼ 1) through porous media is often described in terms of the permeability k and Darcy’s Law: −∆P µ (6-170) =V L k where V = superficial velocity. The SI units for permeability are m2. Creeping flow conditions generally prevail in geological porous media. For multidimensional flows through isotropic porous media, the superficial velocity V and pressure gradient ∇P vectors replace the corresponding one-dimensional variables in Eq. (6-170). µ ∇P = −  V (6-171) k For isotropic homogeneous porous media (uniform permeability and porosity), the pressure for creeping incompressible single phase-flow may be shown to satisfy the LaPlace equation: ∇ 2P = 0

(6-172)

For anisotropic or oriented porous media, as are frequently found in geological media, permeability varies with direction and a permeability tensor K, with nine components Kij giving the velocity compenent in the i direction due to a pressure gradient in the j direction, may be introduced. For further information, see Slattery (Momentum, Energy and Mass Transfer in Continua, Krieger, Huntington, New York, 1981, p. 193–218). See also Dullien (Chem. Eng. J. [Laussanne], 10, 1,034 [1975]) for a review of pressure-drop methods in singlephase flow. Solutions for Darcy’s law for several geometries of interest in petroleum reservoirs and aquifers, for both incompressible and compressible flows, are given in Craft and Hawkins (Applied Petro-

leum Reservoir Engineering, Prentice-Hall, Englewood Cliffs, N.J., 1959). See also Todd (Groundwater Hydrology, 2nd ed., Wiley, New York, 1980). For granular solids of mixed sizes the average particle diameter may be calculated as 1 xi (6-173) =冱 Dp Dp,i i where xi = weight fraction of particles of size Dp,i. For isothermal compressible flow of a gas with constant compressibility factor Z through a packed bed of granular solids, an equation similar to Eq. (6-114) for pipe flow may be derived: 2ZRG2T v 2fmL(1 −
)3 − n p21 − p22 =  ln 2 +  Mw v1 φ 3s − n
3Dp

冤

冥

(6-174)

where p1 = upstream absolute pressure p2 = downstream absolute pressure R = gas constant T = absolute temperature Mw = molecular weight v1 = upstream specific volume of gas v2 = downstream specific volume of gas For creeping flow of power law non-Newtonian fluids, the method of Christopher and Middleton (Ind. Eng. Chem. Fundam., 4, 422–426 [1965]) may be used: 150HLV n(1 −
)2 −∆p =  (6-175) Dp2 φ 2s
3 K 3 n Dp2 φ2s
4 (1 − n)/2 H= 9+ (6-176)  12 n (1 −
)2 where V = G/ρ = superficial velocity, K, n = power law material constants, and all other variables are as defined in Eq. (6-166). This correlation is supported by data from Christopher and Middleton (ibid.), Gregory and Griskey (AIChE J., 13, 122–125 [1967]), Yu, Wen, and Bailie (Can. J. Chem. Eng., 46, 149–154 [1968]), Siskovic, Gregory, and Griskey (AIChE J., 17, 176–187 [1978]), Kemblowski and Mertl (Chem. Eng. Sci., 29, 213–223 [1974]), and Kemblowski and Dziu-

冢

冣冤

冥

6-40

FLUID AND PARTICLE DYNAMICS

minski (Rheol. Acta, 17, 176–187 [1978]). The measurements cover the range n = 0.50 to 1.60, and modified Reynolds number Re′ = 10−8 to 10, where DpV 2 − nρ Re′ =  (6-177) H For the case n = 1 (Newtonian fluid), Eqs. (6-175) and (6-176) give a pressure drop 25 percent less than that given by Eqs. (6-166) through (6-168). For viscoelastic fluids see Marshall and Metzner (Ind. Eng. Chem. Fundam., 6, 393–400 [1967]), Siskovic, Gregory, and Griskey (AIChE J., 13, 122–125 [1967]) and Kemblowski and Dziubinski (Rheol. Acta, 17, 176–187 [1978]). For gas flow through porous media with small pore diameters, the slip flow and molecular flow equations previously given (see the “Vacuum Flow” subsection) may be applied when the pore is of the same or smaller order as the mean free path, as described by Monet and Vermeulen (Chem. Eng. Prog., 55, Symp. Ser., 25 [1959]). Tower Packings For the flow of a single fluid through a bed of tower packing, pressure drop may be estimated using the preceding methods. See also Sec. 14 of this Handbook. For countercurrent gas/liquid flow in commercial tower packings, both structured and unstructured, several sources of data and correlations for pressure drop and flooding are available. See, for example, Strigle (Random Packings and Packed Towers, Design and Applications, Gulf Publishing, Houston, 1989; Chem. Eng. Prog., 89[8], 79–83 [August 1993]), Hughmark (Ind. Eng. Chem. Fundam., 25, 405–409 [1986]), Chen (Chem. Eng. Sci., 40, 2139–2140 [1985]), Billet and Mackowiak (Chem. Eng. Technol., 11, 213–217 [1988]), Krehenwinkel and Knapp (Chem. Eng. Technol., 10, 231–242 [1987]), Mersmann and Deixler (Ger. Chem. Eng., 9, 265–276 [1986]) and Robbins (Chem. Eng. Progr., 87[5], 87–91 [May 1991]). Data and correlations for flooding and pressure drop for structured packings are given by Fair and Bravo (Chem. Eng. Progr., 86[1], 19–29 [January 1990]). Fluidized Beds When gas or liquid flows upward through a vertically unconstrained bed of particles, there is a minimum fluid velocity at which the particles will begin to move. Above this minimum velocity, the bed is said to be fluidized. Fluidized beds are widely used, in part because of their excellent mixing and heat and mass transfer characteristics. See Sec. 17 of this Handbook for detailed information. BOUNDARY LAYER FLOWS Boundary layer flows are a special class of flows in which the flow far from the surface of an object is inviscid, and the effects of viscosity are manifest only in a thin region near the surface where steep velocity gradients occur to satisfy the no-slip condition at the solid surface. The thin layer where the velocity decreases from the inviscid, potential flow velocity to zero (relative velocity) at the solid surface is called the boundary layer. The thickness of the boundary layer is indefinite because the velocity asymptotically approaches the free-stream velocity at the outer edge. The boundary layer thickness is conventionally taken to be the distance for which the velocity equals 0.99 times the free-stream velocity. The boundary layer may be either laminar or turbulent. Particularly in the former case, the equations of motion may be simplified by scaling arguments. Schlichting (Boundary Layer Theory, 8th ed., McGraw-Hill, New York, 1987) is the most comprehensive source for information on boundary layer flows. Flat Plate, Zero Angle of Incidence For flow over a wide, thin flat plate at zero angle of incidence with a uniform free-stream velocity, as shown in Fig. 6-47, the critical Reynolds number at which the boundary layer becomes turbulent is normally taken to be

where

xVρ Rex =  = 500,000 µ V = free-stream velocity ρ = fluid density µ = fluid viscosity x = distance from leading edge of the plate

(6-178)

Uniform free-stream velocity V

y

δ(x )

x L FIG. 6-47

Boundary layer on a flat plate at zero angle of incidence.

However, the transition Reynolds number depends on free-stream turbulence and may range from 3 × 105 to 3 × 106. The laminar boundary layer thickness δ is a function of distance from the leading edge: δ ≈ 5.0xRe −0.5 x

(6-179)

The total drag on the plate of length L and width b for a laminar boundary layer, including the drag on both surfaces, is: FD = 1.328bLρV 2ReL−0.5

(6-180)

For non-Newtonian power law fluids (Acrivos, Shah, and Peterson, AIChE J., 6, 312–317 [1960]; Hsu, AIChE J., 15, 367–370 [1969]), FD = CbLρV 2ReL′− 1/(1 + n)

(6-181)

n = 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 C = 2.075 1.958 1.838 1.727 1.627 1.538 1.460 1.390 1.328 where Re′L = ρV 2 − nLn/K and K and n are the power law material constants (see Eq. [6-4]). For a turbulent boundary layer, the thickness may be estimated as δ ≈ 0.37xRe −0.2 x

(6-182)

and the total drag force on both sides of the plate of length L is 0.455 1,700 −  ρbLV 2 5 × 105 < ReL < 109 (6-183) FD =  (log ReL)2.58 ReL Here the second term accounts for the laminar leading edge of the boundary layer and assumes that the critical Reynolds number is 500,000. Cylindrical Boundary Layer Laminar boundary layers on cylindrical surfaces, with flow parallel to the cylinder axis, are described by Glauert and Lighthill (Proc. R. Soc. [London], 230A, 188–203 [1955]), Jaffe and Okamura (Z. Angew. Math. Phys., 19, 564–574 [1968]) and Stewartson (Q. Appl. Math., 13, 113–122 [1955]). For a turbulent boundary layer, the total drag may be estimated as

冤

冥

FD = 苶cj苶πrLρV (6-184) where r = cylinder radius, L = cylinder length, and the average friction coefficient is given by (White, J. Basic Eng., 94, 200–206 [1972]) L 0.4 ReL−1/3 (6-185) 苶cj苶 = 0.0015 + 0.30 + 0.015  r 6 9 6 for ReL = 10 to 10 and L/r < 10 . Continuous Flat Surface Boundary layers on continuous surfaces drawn through a stagnant fluid are shown in Fig. 6-48. Figure 6-48a shows the continuous flat surface (Sakiadis, AIChE J., 7, 26–28, 221–225, 467–472 [1961]). The critical Reynolds number for transition to turbulent flow may be greater than the 500,000 value for the finite flat-plate case discussed previously (Tsou, Sparrow, and Kurtz, J. Fluid Mech., 26, 145–161 [1966]). For a laminar boundary layer, the thickness is given by 2

冤

冢 冣冥

δ = 6.37xRe x−0.5

(6-186)

FLUID DYNAMICS

6-41

(a)

FIG. 6-49 Boundary layer parameters for continuous cylindrical surfaces. (From Sakiadis, Am. Inst. Chem. J., 7, 467 [1961].)

(b) Continuous surface: (a) continuous flat surface, (b) continuous cylindrical surface. (From Sakiadis, Am. Inst. Chem. Eng. J., 7, 221, 467 [1961].)

FIG. 6-48

(Chem. Eng. Sci., 32, 1517–1525 [1977] and Crane (Z. Angew. Math. Phys., 26, 619–622 [1975]). VORTEX SHEDDING

and the total drag exerted on the two surfaces is FD = 1.776bLρV 2ReL−0.5

(6-187)

The total flow rate of fluid entrained by the surface is q = 3.232bLVReL−0.5

(6-188)

The theoretical velocity field was experimentally verified by Tsou, Sparrow, and Goldstein (Int. J. Heat Mass Transfer, 10, 219–235 [1967]) and Szeri, Yates, and Hai ( J. Lubr. Technol., 98, 145–156 [1976]). For non-Newtonian power law fluids see Fox, Erickson, and Fan (AIChE J., 15, 327–333 [1969]). For a turbulent boundary layer, the thickness is given by δ = 1.01xRe −0.2 x

(6-189)

and the total drag on both sides by FD = 0.056bLρV 2ReL−0.2

(6-190)

and the total entrainment by q = 0.252bLVReL−0.2

(6-191)

When the laminar boundary layer is a significant part of the total length of the object, the total drag should be corrected by subtracting a calculated turbulent drag for the length of the laminar section and then adding the laminar drag for the laminar section. Tsou, Sparrow, and Goldstein (Int. J. Heat Mass Transfer, 10, 219–235 [1967]) give an improved analysis of the turbulent boundary layer; their data indicate that Eq. (6-190) underestimates the drag by about 15 percent. Continuous Cylindrical Surface The continuous surface shown in Fig. 6-48b is applicable, for example, for a wire drawn through a stagnant fluid (Sakiadis, AIChE J., 7, 26–28, 221–225, 467–472 [1961]). The critical-length Reynolds number for transition is Rex = 200,000. The laminar boundary layer thickness, total drag, and entrainment flow rate may be obtained from Fig. 6-49; the drag and entrainment rate are obtained from the momentum area Θ and displacement area ∆ evaluated at x = L. FD = ρV 2Θ

(6-192)

q = V∆

(6-193)

Further laminar boundary layer analysis is given by Crane (Z. Angew. Math. Phys., 23, 201–212 [1972]). For a turbulent boundary layer, the total drag may be roughly estimated using Eqs. (6-184) and (6-185) for finite cylinders. Measured forces by Kwon and Prevorsek (J. Eng. Ind., 101, 73–79 [1979]) are greater than predicted this way. The laminar boundary layer on deforming continuous surfaces with velocity varying with axial position is discussed by Vleggaar

When fluid flows past objects or through orifices or similar restrictions, vortices may periodically be shed downstream. Objects such as smokestacks, chemical-processing columns, suspended pipelines, and electrical transmission lines can be subjected to damaging vibrations and forces due to the vortices, especially if the shedding frequency is close to a natural vibration frequency of the object. The shedding can also produce sound. See Krzywoblocki (Appl. Mech. Rev., 6, 393–397 [1953]) and Marris (J. Basic Eng., 86, 185–196 [1964]). Development of a vortex street, or von Kármán vortex street is shown in Fig. 6-50. Discussions of the vortex street may be found in Panton (pp. 387–393). The Reynolds number is DVρ Re =  (6-194) µ where D = diameter of cylinder or effective width of object V = free-stream velocity ρ = fluid density µ = fluid viscosity For flow past a cylinder, the vortex street forms at Reynolds numbers above about 40. The vortices initially form in the wake, the point of formation moving closer to the cylinder as Re is increased. At a Reynolds number of 60 to 100, the vortices are formed from eddies attached to the cylinder surface. The vortices move at a velocity slightly less than V. The frequency of vortex shedding f is given in terms of the Strouhal number, which is approximately constant over a wide range of Reynolds numbers. fD St ⬅  (6-195) V For 40 < Re < 200 the vortices are laminar and the Strouhal number has a nearly constant value of 0.2 for flow past a cylinder. Between Re = 200 and 400 the Strouhal number is no longer constant and the wake becomes irregular. Above about Re = 400 the vortices become turbulent, the wake is once again stable, and the Strouhal number remains constant at about 0.2 up to a Reynolds number of about 10 5.

FIG. 6-50

Vortex street behind a cylinder.

6-42

FLUID AND PARTICLE DYNAMICS

Above Re = 105 the vortex shedding is difficult to see in flow visualization experiments, but velocity measurements still show a strong spectral component at St = 0.2 (Panton, p. 392). Experimental data suggest that the vortex street disappears over the range 5 × 105 < Re < 3.5 × 106, but is reestablished at above 3.5 × 106 (Schlichting). Vortex shedding exerts alternating lateral forces on a cylinder, perpendicular to the flow direction. Such forces may lead to severe vibration or mechanical failure of cylindrical elements such as heatexchanger tubes, transmission lines, stacks, and columns when the vortex shedding frequency is close to resonant bending frequency. According to Den Hartog (Proc. Nat. Acad. Sci., 40, 155–157 [1954]), the vortex shedding and cylinder vibration frequency will shift to the resonant frequency when the calculated shedding frequency is within 20 percent of the resonant frequency. The well-known Tacoma Narrows bridge collapse resulted from resonance between a torsional oscillation and vortex shedding (Panton, p. 392). Spiral strakes are sometimes installed on tall stacks so that vortices at different axial positions are not shed simultaneously. The alternating lateral force FK, sometimes called the von Kármán force, is given by (Den Hartog, Mechanical Vibrations, 4th ed., McGraw-Hill, New York, 1956, pp. 305–309): ρV 2 FK = CK A  (6-196) 2 where CK = von Kármán coefficient A = projected area perpendicular to the flow ρ = fluid density V = free-stream fluid velocity For a cylinder, CK = 1.7. For a vibrating cylinder, the effective projected area exceeds, but is always less than twice, the actual cylinder projected area (Rouse, Engineering Hydraulics, Wiley, New York, 1950). The following references pertain to discussions of vortex shedding in specific structures: steel stacks (Osker and Smith, Trans. ASME, 78, 1381–1391 [1956]; Smith and McCarthy, Mech. Eng., 87, 38–41 [1965]); chemical-processing columns (Freese, J. Eng. Ind., 81, 77–91, [1959]); heat exchangers (Eisinger, Trans. ASME J. Pressure Vessel Tech., 102, 138–145 [May 1980]; Chen, J. Sound Vibration, 93, 439–455 [1984]; Gainsboro, Chem. Eng. Prog., 64[3], 85–88 [1968]; “Flow-Induced Vibration in Heat Exchangers,” Symp. Proc., ASME, New York, 1970); suspended pipe lines (Baird, Trans. ASME, 77, 797–804 [1955]); and suspended cable (Steidel, J. Appl. Mech., 23, 649–650 [1956]). COATING FLOWS In coating flows, liquid films are entrained on moving solid surfaces. For general discussions, see Ruschak (Ann. Rev. Fluid Mech., 17, 65–89 [1985]), Cohen and Gutoff (Modern Coating and Drying Technology, VCH Publishers, New York, 1992), and Middleman (Fundamentals of Polymer Processing, McGraw-Hill, New York, 1977). It is generally important to control the thickness and uniformity of the coatings. In dip coating, or free withdrawal coating, a solid surface is withdrawn from a liquid pool, as shown in Fig. 6-51. It illustrates many of the features found in other coating flows, as well. Tallmadge and Gutfinger (Ind. Eng. Chem., 59[11], 19–34 [1967]) provide an early review of the theory of dip coating. The coating flow rate and film thickness are controlled by the withdrawal rate and the flow behavior in the meniscus region. For a withdrawal velocity V and an angle of inclination from the horizontal φ, the film thickness h may be estimated for low withdrawal velocities by ρg h  σ

冢 冣

where

1/2

0.944 =  Ca2/3 (1 − cos φ)1/2

g = acceleration of gravity Ca = µV/σ = capillary number µ = viscosity σ = surface tension

(6-197)

FIG. 6-51

Dip coating.

Equation (6-197) is asymptotically valid as Ca → 0 and agrees with experimental data up to capillary numbers in the range of 0.01 to 0.03. In practice, where high production rates require high withdrawal speeds, capillary numbers are usually too large for Eq. (6-197) to apply. Approximate analytical methods for larger capillary numbers have been obtained by numerous investigators, but none appears wholly satisfactory, and some are based on questionable assumptions (Ruschak, Ann. Rev. Fluid Mech., 17, 65–89 [1985]). With the availability of high-speed computers and the development of the field of computational fluid dynamics, numerical solutions accounting for two-dimensional flow aspects, along with gravitational, viscous, inertial, and surface tension forces are now the most effective means to analyze coating flow problems. Other common coating flows include premetered flows, such as slide and curtain coating, where the film thickness is an independent parameter that may be controlled within limits, and the curvature of the mensiscus adjusts accordingly; the closely related blade coating; and roll coating and extrusion coating. See Ruschak (ibid.), Cohen and Gutoff (Modern Coating and Drying Technology, VCH Publishers, New York, 1992) and Middleman (Fundamentals of Polymer Processing, McGraw-Hill, New York, 1977). For dip coating of wires, see Taughy (Int. J. Numerical Meth. Fluids, 4, 441–475 [1984]). Many coating flows are subject to instabilities that lead to unacceptable coating defects. Three-dimensional flow instabilities lead to such problems as ribbing. Air entrainment is another common defect. FALLING FILMS Minimum Wetting Rate The minimum liquid rate required for complete wetting of a vertical surface is about 0.03 to 0.3 kg/m ⋅ s for water at room temperature. The minimum rate depends on the geometry and nature of the vertical surface, liquid surface tension, and mass transfer between surrounding gas and the liquid. See Ponter, et al. (Int. J. Heat Mass Transfer, 10, 349–359 [1967]; Trans. Inst. Chem. Eng. [London], 45, 345–352 [1967]), Stainthorp and Allen (Trans. Inst. Chem. Eng. [London], 43, 85–91 [1967]) and Watanabe, et al. (J. Chem. Eng. [Japan], 8[1], 75 [1975]). Laminar Flow For films falling down vertical flat surfaces, as shown in Fig. 6-52, or vertical tubes with small film thickness compared to tube radius, laminar flow conditions prevail for Reynolds numbers less than about 2,000, where the Reynolds number is given by 4Γ Re =  (6-198) µ where Γ = liquid mass flow rate per unit width of surface and µ = liq-



,,  V

Γ = mass flow rate per unit width of surface

δ

FIG. 6-52

Falling film.

uid viscosity. For a flat film surface, the following equations may be derived. The film thickness δ is 3Γµ 1/3 δ=  (6-199) ρ2g The average film velocity is Γ gρδ2 V== (6-200) ρδ 3µ The downward velocity profile u(x) where x = 0 at the solid surface and x = δ at the liquid/gas interface is given by 2x x 2 (6-201) u = 1.5V  −  δ δ These equations assume that there is no drag force at the gas/liquid interface, such as would be produced by gas flow. For a flat surface inclined at an angle θ with the horizontal, the preceding equations may be modified by replacing g by g sin θ. For films falling inside vertical tubes with film thickness up to and including the full pipe radius, see Jackson (AIChE J., 1, 231–240 [1955]). These equations have generally given good agreement with experimental results for low-viscosity liquids (<0.005 Pa ⋅ s) (< 5 cp) whereas Jackson (ibid.) found film thicknesses for higher-viscosity liquids (0.01 to 0.02 Pa⋅s (10 to 20 cp) were significantly less than predicted by Eq. (6-197). At Reynolds numbers of 25 or greater, surface waves will be present on the liquid film. West and Cole (Chem. Eng. Sci., 22, 1388– 1389 [1967]) found that the surface velocity u(x = δ) is still within 7 percent of that given by Eq. (6-201) even in wavy flow. For laminar non-Newtonian film flow, see Bird, Armstrong, and Hassager (Dynamics of Polymeric Liquids, vol. 1: Fluid Mechanics, Wiley, New York, 1977, p. 215, 217), Astarita, Marrucci, and Palumbo (Ind. Eng. Chem. Fundam., 3, 333–339 [1964]) and Cheng (Ind. Eng. Chem. Fundam., 13, 394–395 [1974]). Turbulent Flow In turbulent flow, Re > 2,000, for vertical surfaces, the film thickness may be estimated to within 25 percent using Γ1.75µ0.25 1/3 δ = 0.304  (6-202) ρ2g Replace g by g sin θ for a surface inclined at angle θ to the horizontal. The average film velocity is V = Γ/ρδ. Tallmadge and Gutfinger (Ind. Eng. Chem., 59[11], 19–34 [1967]) discuss prediction of drainage rates from liquid films on flat and cylindrical surfaces. Effect of Surface Traction If a drag is exerted on the surface of the film because of motion of the surrounding fluid, the film thickness will be reduced or increased, depending upon whether the drag acts with or against gravity. Thomas and Portalski (Ind. Eng. Chem., 50, 1081–1088 [1958]), Dukler (Chem. Eng. Prog., 55[10], 62–67 [1959]) and Kosky (Int. J. Heat Mass Transfer, 14, 1220–1224 [1971]) have presented calculations of film thickness and film velocity. Film thickness data for falling water films with cocurrent and countercurrent air flow in pipes are given by Zhivaikin (Int. Chem. Eng., 2, 337–341 [1962]). Zabaras, Dukler, and Moalem-Maron (AIChE J., 32, 829–843

冢 冣

冤

冢

冢 冣冥

FLUID DYNAMICS

6-43

[1986]) and Zabaras and Dukler (AIChE J., 34, 389–396 [1988]) present studies of film flow in vertical tubes with both cocurrent and countercurrent gas flow, including measurements of film thickness, wall shear stress, wave velocity, wave amplitude, pressure drop, and flooding point for countercurrent flow. Flooding With countercurrent gas flow, a condition is reached with increasing gas rate for which flow reversal occurs and liquid is carried upward. The mechanism for this flooding condition has been most often attributed to waves either bridging the pipe or reversing direction to flow upward at flooding. However, the results of Zabaras and Dukler (ibid.) suggest that flooding may be controlled by flow conditions at the liquid inlet and that wave bridging or upward wave motion does not occur, at least for the 50.8-mm diameter pipe used for their study. Flooding mechanisms are still incompletely understood. Under some circumstances, as when the gas is allowed to develop its normal velocity profile in a “calming length” of pipe beneath the liquid draw-off, the gas superficial velocity at flooding will be increased, and increases with decreasing length of wetted pipe (Hewitt, Lacy, and Nicholls, Proc. Two-Phase Flow Symp., University of Exeter, paper 4H, AERE-4 4614 [1965]). A bevel cut at the bottom of the pipe with an angle 30° from the vertical will increase the flooding velocity in small-diameter tubes at moderate liquid flow rates. If the gas approaches the tube from the side, the taper should be oriented with the point facing the gas entrance. Figures 6-53 and 6-54 give correlations for flooding in tubes with square and slant bottoms (courtesy Holmes, DuPont Co.) The superficial mass velocities of gas and liquid GG and GL, and the physical property parameters λ and ψ are the same as those defined for the Baker chart (“Multiphase Flow” subsection, Fig. 6-25). For tubes larger than 50 mm (2 in), flooding velocity appears to be relatively insensitive to diameter and the flooding curves for 1.98-in diameter may be used.

Flooding in vertical tubes with square top and square bottom. To convert lbm/(ft2 ⋅s) to kg/(m2 ⋅s), multiply by 4.8824; to convert in to mm, multiply by 25.4. (Courtesy of E. I. du Pont de Nemours & Co.) FIG. 6-53

冣

Flooding in vertical tubes with square top and slant bottom. To convert lbm/(ft2 ⋅s) to kg/(m2 ⋅s), multiply by 4.8824; to convert in to mm, multiply by 25.4. (Courtesy of E. I. du Pont de Nemours & Co.) FIG. 6-54

6-44

FLUID AND PARTICLE DYNAMICS

HYDRAULIC TRANSIENTS Many transient flows of liquids may be analyzed by using the full timedependent equations of motion for incompressible flow. However, there are some phenomena that are controlled by the small compressibility of liquids. These phenomena are generally called hydraulic transients. Water Hammer When liquid flowing in a pipe is suddenly decelerated to zero velocity by a fast-closing valve, a pressure wave propagates upstream to the pipe inlet, where it is reflected; a pounding of the line commonly known as water hammer is often produced. For an instantaneous flow stoppage of a truly incompressible fluid in an inelastic pipe, the pressure rise would be infinite. Finite compressibility of the fluid and elasticity of the pipe limit the pressure rise to a finite value. The Joukowski formula gives the maximum pressure rise as ∆ p = ρa∆V where

(6-203)

ρ = liquid density ∆V = change in liquid velocity a = pressure wave velocity

The wave velocity is given by β苶 /ρ 兹苶 a =  +苶(苶 β苶 /E苶 )(苶 D苶 /b苶) 兹苶1苶 where

(6-204)

β = liquid bulk modulus of elasticity E = elastic modulus of pipe wall D = pipe inside diameter b = pipe wall thickness

The numerator gives the wave velocity for perfectly rigid pipe, and the denominator corrects for wall elasticity. This formula is for thin-walled pipes; for thick-walled pipes, the factor D/ b is replaced by (Do2 + Di2 ) 2  (Do2 − Di2 ) where Do = pipe outside diameter Di = pipe inside diameter Example 10: Response to Instantaneous Valve Closing Compute the wave speed and maximum pressure rise for instantaneous valve closing, with an initial velocity of 2.0 m/s, in a 4-in Schedule 40 steel pipe with elastic modulus 207 × 109 Pa. Repeat for a plastic pipe of the same dimensions, with E = 1.4 × 109 Pa. The liquid is water with β = 2.2 × 109 Pa and ρ = 1,000 kg/m3. For the steel pipe, D = 102.3 mm, b = 6.02 mm, and the wave speed is 苶/ρ 苶 兹β a =  苶)( 苶D 苶/b 苶)苶 +苶β (苶/E 兹1苶苶 苶苶 ×苶10苶9苶/1苶0苶0苶0苶 兹2苶.2 =  9 9 苶苶 苶0苶2苶.3 苶/6 苶.0 苶2苶)苶 +苶(2苶.2 ×苶10苶苶 /2苶0苶7苶苶 ×苶10苶苶 )(1 兹1苶苶 = 1365 m/s The maximum pressure rise is ∆ p = ρa∆V = 1,000 × 1,365 × 2.0 = 2.73 × 106 Pa For the plastic pipe,

苶苶 ×苶10苶9苶/1苶0苶0苶0苶 兹2苶.2 a =  9 9 苶苶 苶苶 苶0苶2苶.3 苶/6 苶.0 苶2苶)苶 +苶(2苶.2 ×苶10苶苶 /1苶.4 ×苶 10苶苶 )(1 兹1苶苶 = 282 m/s ∆p = ρa∆V = 1,000 × 282 × 2.0 = 5.64 × 105 Pa

The maximum pressure surge is obtained when the valve closes in less time than the period τ required for the pressure wave to travel from the valve to the pipe inlet and back, a total distance of 2L. 2L τ= (6-205) a The pressure surge will be reduced when the time of flow stoppage exceeds the pipe period τ, due to cancellation between direct and

reflected waves. Wood and Jones (Proc. Am. Soc. Civ. Eng., J. Hydraul. Div., 99, (HY1), 167–178 [1973]) present charts for reliable estimates of water-hammer pressure for different valve closure modes. Wylie and Streeter (Hydraulic Transients, McGraw-Hill, New York, 1978) describe several solution methods for hydraulic transients, including the method of characteristics, which is well suited to computer methods for accurate solutions. A rough approximation for the peak pressure for cases where the valve closure time tc exceeds the pipe period τ is (Daugherty and Franzini, Fluid Mechanics with Engineering Applications, McGraw-Hill, New York, 1985): τ ∆p ≈  ρa∆V (6-206) tc Successive reflections of the pressure wave between the pipe inlet and the closed valve result in alternating pressure increases and decreases, which are gradually attenuated by fluid friction and imperfect elasticity of the pipe. Periods of reduced pressure occur while the reflected pressure wave is traveling from inlet to valve. Degassing of the liquid may occur, as may vaporization if the pressure drops below the vapor pressure of the liquid. Gas and vapor bubbles decrease the wave velocity. Vaporization may lead to what is often called liquid column separation; subsequent collapse of the vapor pocket can result in pipe rupture. In addition to water hammer induced by changes in valve setting, including closure, numerous other hydraulic transient flows are of interest, as, for example (Wylie and Streeter, Hydraulic Transients, McGraw-Hill, New York, 1978), those arising from starting or stopping of pumps; changes in power demand from turbines; reciprocating pumps; changing elevation of a reservoir; waves on a reservoir; turbine governor hunting; vibration of impellers or guide vanes in pumps, fans, or turbines; vibration of deformable parts such as valves; draft-tube instabilities due to vortexing; and unstable pump or fan characteristics. Tube failure in heat exhangers may be added to this list. Pulsating Flow Reciprocating machinery (pumps and compressors) produces flow pulsations, which adversely affect flow meters and process control elements and can cause vibration and equipment failure, in addition to undesirable process results. Vibration and damage can result not only from the fundamental frequency of the pulse producer but also from higher harmonics. Multipiston double-acting units reduce vibrations. Pulsation dampeners are often added. Damping methods are described by M. W. Kellogg Co. (Design of Piping Systems, rev. 2d ed., Wiley, New York, 1965). For liquid phase pulsation damping, gas-filled surge chambers, also known as accumulators, are commonly used; see Wylie and Streeter (Hydraulic Transients, McGraw-Hill, New York, 1978). Software packages are commercially available for simulation of hydraulic transients. These may be used to analyze piping systems to reveal unsatisfactory behavior, and they allow the assessment of design changes such as increases in pipe-wall thickness, changes in valve actuation, and addition of check valves, surge tanks, and pulsation dampeners. Cavitation Loosely regarded as related to water hammer and hydraulic transients because it may cause similar vibration and equipment damage, cavitation is the phenomenon of collapse of vapor bubbles in flowing liquid. These bubbles may be formed anywhere the local liquid pressure drops below the vapor pressure, or they may be injected into the liquid, as when steam is sparged into water. Local low-pressure zones may be produced by local velocity increases (in accordance with the Bernoulli equation; see the preceding “Conservation Equations” subsection) as in eddies or vortices, or near boundary contours; by rapid vibration of a boundary; by separation of liquid during water hammer; or by an overall reduction in static pressure, as due to pressure drop in the suction line of a pump. Collapse of vapor bubbles once they reach zones where the pressure exceeds the vapor pressure can cause objectionable noise and vibration and extensive erosion or pitting of the boundary materials. The critical cavitation number at inception of cavitation, denoted σi, is useful in correlating equipment performance data: (p − pv) σi =  (6-207) ρV 2/2

冢 冣

FLUID DYNAMICS where

p = static pressure in undisturbed flow pv = vapor pressure ρ = liquid density V = free-stream velocity of the liquid

The value of the cavitation number for incipient cavitation for a specific piece of equipment is a characteristic of that equipment. Cavitation numbers for various head forms of cylinders, for disks, and for various hydrofoils are given by Holl and Wislicenus (J. Basic Eng., 83, 385–398 [1961]) and for various surface irregularities by Arndt and Ippen (J. Basic Eng., 90, 249–261 [1968]), Ball (Proc. ASCE J. Constr. Div., 89(C02), 91–110 [1963]), and Holl (J. Basic Eng., 82, 169–183 [1960]). As a guide only, for blunt forms the cavitation number is generally in the range of 1 to 2.5, and for somewhat streamlined forms the cavitation number is in the range of 0.2 to 0.5. Critical cavitation numbers generally depend on a characteristic length dimension of the equipment in a way that has not been explained. This renders scale-up of cavitation data questionable. For cavitation in flow through orifices, Fig. 6-55 (Thorpe, Int. J. Multiphase Flow, 16, 1023–1045 [1990]) gives the critical cavitation number for inception of cavitation. To use this cavitation number in Eq. (6-207), the pressure p is the orifice backpressure downstream of the vena contracta after full pressure recovery, and V is the average velocity through the orifice. Fig. 6-55 includes data from Tullis and Govindarajan (ASCE J. Hydraul. Div., HY13, 417–430 [1973]) modified to use the same cavitation number definition; their data also include critical cavitation numbers for 30.50- and 59.70-cm pipes (12.00- to 23.50-in). Very roughly, compared with the 15.40-cm pipe, the cavitation number is about 20 percent greater for the 30.50-cm (12.01-in) pipe and about 40 percent greater for the 59.70-cm (23.50in) diameter pipe. Inception of cavitation appears to be related to release of dissolved gas and not merely vaporization of the liquid. For further discussion of cavitation, see Eisenberg and Tulin (Streeter, Handbook of Fluid Dynamics, Sec. 12, McGraw-Hill, New York, 1961). TURBULENCE Turbulent flow occurs when the Reynolds number exceeds a critical value above which laminar flow is unstable; the critical Reynolds number depends on the flow geometry. There is generally a transition regime between the critical Reynolds number and the Reynolds number at which the flow may be considered fully turbulent. The transition regime is very wide for some geometries. In turbulent flow, variables such as velocity and pressure fluctuate chaotically; statistical methods are used to quantify turbulence. Time Averaging In turbulent flows it is useful to define timeaveraged and fluctuation values of flow variables such as velocity com-

6-45

ponents. For example, the x-component velocity fluctuation v′x is the difference between the actual instantaneous velocity vx and the timeaveraged velocity v苶苶x:

(6-208) v′x(x, y, z, t) = vx(x, y, z, t) − v苶苶x(x, y, z) The actual and fluctuating velocity components are, in general, functions of the three spatial coordinates x, y, and z and of time t. The time-averaged velocity v苶苶x is independent of time for a stationary flow. Nonstationary processes may be considered where averages are defined over time scales long compared to the time scale of the turbulent fluctuations, but short compared to longer time scales over which the time-averaged flow variables change due, for example, to time-varying boundary conditions. The time average over a time interval 2T centered at time t of a turbulently fluctuating variable ζ(t) is defined as 1 t+T ζ(τ) dτ (6-209) 苶ζ(苶t苶)苶 =  2T t − T where τ = dummy integration variable. For stationary turbulence, ζ苶 does not vary with time. 1 t+T ζ苶 = lim  ζ(τ) dτ (6-210) T→∞ 2T t − T

冕

冕

The time average of a fluctuation ζ苶′苶 = ζ  苶ζ = 0. Fluctuation magnitudes are quantified by root mean squares. 苶x′苶 v˜ ′x = 冪(v ) 2苶

(6-211)

In isotropic turbulence, statistical measures of fluctuations are equal in all directions. v˜ ′x = v˜ ′y = v˜ ′z

(6-212)

In homogeneous turbulence, turbulence properties are independent of spatial position. The kinetic energy of turbulence k is given by 1 (6-213) k =  (˜v′x2 + v˜ ′y2 + v˜ ′z2) 2 Turbulent velocity fluctuations ultimately dissipate their kinetic energy through viscous effects. Macroscopically, this energy dissipation requires pressure drop, or velocity decrease. The energy dissipation rate per unit mass is usually denoted
. For steady flow in a pipe, the average energy dissipation rate per unit mass is given by 2f V3
= (6-214) D where ρ = fluid density f = Fanning friction factor D = pipe inside diameter When the continuity equation and the Navier-Stokes equations for incompressible flow are time averaged, equations for the timeaveraged velocities and pressures are obtained which appear identical to the original equations (6-18 through 6-28), except for the appearance of additional terms in the Navier-Stokes equations. Called Reynolds stress terms, they result from the nonlinear effects of momentum transport by the velocity fluctuations. In each i-component (i = x, y, z) Navier-Stokes equation, the following additional terms appear on the right-hand side: 3 ∂τ(t) ji  冱 ∂xj j=1 with j components also being x, y, z. The Reynolds stresses are given by τ(t) vi′苶 v′j (6-215) ij = −ρ苶

FIG. 6-55 Critical cavitation number vs. diameter ratio β. (Reprinted from Thorpe, “Flow regime transitions due to cavitation in the flow through an orifice,” Int. J. Multiphase Flow, 16, 1023–1045. Copyright © 1990, with kind permission from Elsevier Science, Ltd., The Boulevard, Langford Lane, Kidlington OX5 1GB, United Kingdom.)

The Reynolds stresses are nonzero because the velocity fluctuations vi′苶 v′j in general in different coordinate directions are correlated so that 苶 is nonzero. Although direct numerical simulations under limited circumstances have been carried out to determine (unaveraged) fluctuating velocity fields, in general the solution of the equations of motion for turbulent flow is based on the time-averaged equations. This requires semi-

6-46

FLUID AND PARTICLE DYNAMICS

empirical models to express the Reynolds stresses in terms of timeaveraged velocities. This is the closure problem of turbulence. In all but the simplest geometries, numerical methods are required. Closure Models Many closure models have been proposed. A few of the more important ones are introduced here. Many employ the Boussinesq approximation, simplified here for incompressible flow, which treats the Reynolds stresses as analogous to viscous stresses, introducing a scalar quantity called the turbulent or eddy viscosity µt . ∂v苶苶i ∂v苶j苶 vi′苶 v′j = µt  + (6-216) −ρ 苶 ∂xj ∂xi An additional turbulence pressure term equal to −wkδij, where k = turbulent kinetic energy and δij = 1 if i = j and δij = 0 if i ≠ j, is sometimes included in the right-hand side. To solve the equations of motion using the Boussinesq approximation, it is necessary to provide equations for the single scalar unknown µt (and k, if used) rather than the nine unknown tensor components τ (t) ij . With this approximation, and using the effective viscosity µeff = µ + µt, the time-averaged momentum equation is similar to the original Navier-Stokes equation, with time-averaged variables and µeff replacing the instantaneous variables and molecular viscosity. However, solutions to the time-averaged equations for turbulent flow are not identical to those for laminar flow because µeff is not a constant. The universal turbulent velocity profile near the pipe wall presented in the preceding subsection “Incompressible Flow in Pipes and Channels” may be developed using the Prandtl mixing length approximation for the eddy viscosity, d 苶vx苶 µt = ρ lP2  (6-217) dy where lP is the Prandtl mixing length. The turbulent core of the universal velocity profile is obtained by assuming that the mixing length is proportional to the distance from the wall. The proportionality constant is one of two constants adjusted to fit experimental data. The Prandtl mixing length concept is useful for shear flows parallel to walls, but is inadequate for more general three-dimensional flows. A more complicated semiempirical model commonly used in numerical computations, and found in most commercial software for computational fluid dynamics (CFD; see the following subsection), is the k–
model described by Launder and Spaulding (Lectures in Mathematical Models of Turbulence, Academic, London, 1972). In this model the eddy viscosity is assumed proportional to the ratio k2/
. k2 (6-218) µt = ρCµ 
where the value Cµ = 0.09 is normally used. Semiempirical partial differential conservation equations for k and
derived from the NavierStokes equations with simplifying closure assumptions are coupled with the equations of continuity and momentum: ∂ ∂  (ρk) +  (ρv苶苶ik) ∂t ∂xi

冢

冣

冨 冨

∂ µt ∂k ∂v苶苶i ∂v苶j苶 ∂v苶苶i +   − ρ
=    + µt  ∂xi σk ∂xi ∂xj ∂xi ∂xj

冢

冣

冢

冣

(6-219)

∂ ∂  (ρ
) +  (ρv苶苶i
) ∂t ∂xi ρ
2 ∂ µt ∂

µt ∂v苶苶i ∂v苶j苶 ∂v苶苶i +   − C2
 (6-220) =    + C1
  ∂xi σ
∂xi k ∂xj ∂xi ∂xj k In these equations summations over repeated indices are implied. The values for the empirical constants C1
= 1.44, C2
= 1.92, σk = 1.0, and σ
= 1.3 are widely accepted (Launder and Spaulding, The Numerical Computation of Turbulent Flows, Imperial Coll. Sci. Tech. London, NTIS N74-12066 [1973]). The k–
model has proved reasonably accurate for many flows without highly curved streamlines or significant swirl. It usually underestimates flow separation and overestimates turbulence production by normal straining. The k–
model is suitable for high Reynolds number flows. See Virendra, Patel, Rodi,

冢

冣

冢

冣

and Scheuerer (AIAA J., 23, 1308–1319 [1984]) for a review of low Reynolds number k–
models. More advanced models, more complex and computationally intensive, are being developed. For example, the renormalization group theory (Yakhot and Orszag, J. Scientific Computing, 1, 1–51 [1986]; Yakhot, Orszag, Thangam, Gatski, and Speziale, Phys. Fluids A, 4, 1510–1520 [1992]) modification of the k–
model provides theoretical values of the model constants and provides substantial improvement in predictions of flows with stagnation, separation, normal straining, transient behavior such as vortex shedding, and relaminarization. Stress transport models provide equations for all nine Reynolds stress components, rather than introducing eddy viscosity. Algebraic closure equations for the Reynolds stresses are available, but are no longer in common use. Differential Reynolds stress models (e.g., Launder, Reece, and Rodi, J. Fluid Mech., 68, 537–566 [1975]) use differential conservation equations for all nine Reynolds stress components. In direct numerical simulation of turbulent flows, the solution of the unaveraged equations of motion is sought. Due to the extreme computational intensity, solutions to date have been limited to relatively low Reynolds numbers (Re < about 10,000 to 20,000) in simple geometries such as channel flow. See, for example, Kim, Moin, and Moser (J. Fluid Mech., 177, 133 [1987]). Since computational grids must be sufficiently fine to resolve even the smallest eddies, the computational difficulty rapidly becomes prohibitive as Reynolds number increases. Large eddy simulations use models for subgrid turbulence while solving for larger-scale fluctuations. Eddy Spectrum The energy that produces and sustains turbulence is extracted from velocity gradients in the mean flow, principally through vortex stretching. At Reynolds numbers well above the critical value there is a wide spectrum of eddy sizes, often described as a cascade of energy from the largest down to the smallest eddies. The largest eddies are of the order of the equipment size. The smallest are those for which viscous forces associated with the eddy velocity fluctuations are of the same order as inertial forces, so that turbulent fluctuations are rapidly damped out by viscous effects at smaller length scales. Most of the turbulent kinetic energy is contained in the larger eddies, while most of the dissipation occurs in the smaller eddies. Large eddies, which extract energy from the mean flow velocity gradients, are generally anisotropic. At smaller length scales, the directionality of the mean flow exerts less influence, and local isotropy is approached. The range of eddy scales for which local isotropy holds is called the equilibrium range. Davies (Turbulence Phenomena, Academic, New York, 1972) presents a good discussion of the spectrum of eddy lengths for welldeveloped isotropic turbulence. The smallest eddies, usually called Kolmogorov eddies (Kolmogorov, Compt. Rend. Acad. Sci. URSS, 30, 301; 32, 16 [1941]), have a characteristic velocity fluctuation v˜ K′ given by v˜ K′ = (ν
)1/4

(6-221)

where ν = kinematic viscosity and
= energy dissipation per unit mass. The size of the Kolmogorov eddy scale is lK = (ν3/
)1/4

(6-222)

The Reynolds number for the Kolmogorov eddy, ReK = lK v˜ ′k /ν, is equal to unity by definition. In the equilibrium range, which exists for welldeveloped turbulence and extends from the medium eddy sizes down to the smallest, the energy dissipation at the smaller length scales is supplied by turbulent energy drawn from the bulk flow and passed down the spectrum of eddy lengths according to the scaling rule (˜v′)3
= (6-223) l which is consistent with Eqs. (6-221) and (6-222). For the medium, or energy-containing, eddy size, (˜ve′)3
= (6-224) le For turbulent pipe flow, the friction velocity u* = 兹苶 τw 苶 /ρ used earlier in describing the universal turbulent velocity profile may be used as an estimate for v˜ e′. Together with the Blasius equation for the friction fac-

FLUID DYNAMICS tor from which
may be obtained (Eq. 6-214), this provides an estimate for the energy-containing eddy size in turbulent pipe flow: le = 0.05DRe−1/8

(6-225)

where D = pipe diameter and Re = pipe Reynolds number. Similarly, the Kolmogorov eddy size is lK = 4DRe−0.78

(6-226)

Most of the energy dissipation occurs on a length scale about 5 times the Kolmogorov eddy size. The characteristic fluctuating velocity for these energy-dissipating eddies is about 1.7 times the Kolmogorov velocity. The eddy spectrum is normally described using Fourier transform methods; see, for example, Hinze (Turbulence, McGraw-Hill, New York, 1975), and Tennekes and Lumley (A First Course in Turbulence, MIT Press, Cambridge, 1972). The spectrum E(κ) gives the fraction of turbulent kinetic energy contained in eddies of wavenumber ∞ between κ and κ + dκ, so that k = 冕0 E(κ) dκ. The portion of the equilibrium range excluding the smallest eddies, those which are affected by dissipation, is the inertial subrange. The Kolmogorov law gives E(κ) ∝ κ−5/3 in the inertial subrange. Several texts are available for further reading on turbulent flow, including Tennekus and Lumley (ibid.), Hinze (Turbulence, McGrawHill, New York, 1975), Landau and Lifshitz (Fluid Mechanics, 2d ed., Chap. 3, Pergamon, Oxford, 1987) and Panton (Incompressible Flow, Wiley, New York, 1984). COMPUTATIONAL FLUID DYNAMICS Computational fluid dynamics (CFD) emerged in the 1980s as a significant tool for fluid dynamics both in research and in practice, enabled by rapid development in computer hardware and software. Commercial CFD software is widely available. Computational fluid dynamics is the numerical solution of the equations of continuity and momentum (Navier-Stokes equations for incompressible Newtonian fluids) along with additional conservation equations for energy and material species in order to solve problems of nonisothermal flow, mixing, and chemical reaction. Textbooks include Fletcher (Computational Techniques for Fluid Dynamics, vol. 1: Fundamental and General Techniques, and vol. 2: Specific Techniques for Different Flow Categories, Springer-Verlag, Berlin, 1988), Hirsch (Numerical Computation of Internal and External Flows, vol. 1: Fundamentals of Numerical Discretization, and vol. 2: Computational Methods for Inviscid and Viscous Flows, Wiley, New York, 1988), Peyret and Taylor (Computational Methods for Fluid Flow, Springer-Verlag, Berlin, 1990), Canuto, Hussaini, Quarteroni, and Zang (Spectral Methods in Fluid Dynamics, Springer-Verlag, Berlin, 1988), Anderson, Tannehill, and Pletcher (Computational Fluid Mechanics and Heat Transfer, Hemisphere, New York, 1984), and Patankar (Numerical Heat Transfer and Fluid Flow, Hemisphere, Washington, D.C., 1980). A wide variety of numerical methods has been employed, but three basic steps are common to all CFD methods: 1. Subdivision or discretization of the flow domain into cells or elements. There are methods, called boundary element methods, in which the surface of the flow domain, rather than the volume, is discretized, but the vast majority of CFD work uses volume discretization. Discretization produces a set of grid lines or curves which define a mesh and a set of nodes at which the flow variables are to be calculated. The equations of motion are solved approximately on a domain defined by the grid. Curvilinear or body-fitted coordinate system grids may be used to ensure that the discretized domain accurately represents the true problem domain. 2. Discretization of the governing equations. In this step, the exact partial differential equations to be solved are replaced by approximate algebraic equations written in terms of the nodal values of the dependent variables. Among the numerous discretization methods, finite difference, finite volume, and finite element methods are the most common. The finite difference method estimates spatial derivatives in terms of the nodal values and spacing between nodes. The governing equations are then written in terms of

6-47

the nodal unknowns at each interior node. Finite volume methods, related to finite difference methods, may be derived by a volume integration of the equations of motion, with application of the divergence theorem, reducing by one the order of the differential equations. Equivalently, macroscopic balance equations are written on each cell. Finite element methods are weighted residual techniques in which the unknown dependent variables are expressed in terms of basis functions interpolating among the nodal values. The basis functions are substituted into the equations of motion, resulting in error residuals which are multiplied by the weighting functions, integrated over the control volume, and set to zero to produce algebraic equations in terms of the nodal unknowns. Selection of the weighting functions defines the various finite element methods. For example, Galerkin’s method uses the nodal interpolation basis functions as weighting functions. Each method also has its own method for implementing boundary conditions. The end result after discretization of the equations and application of the boundary conditions is a set of algebraic equations for the nodal unknown variables. Discretization in time is also required for the ∂/∂t time derivative terms in unsteady flow; finite differencing in time is often used. The descretized equations represent an approximation of the exact equations, and their solution gives an approximation for the flow variables. The accuracy of the solution improves as the grid is refined; that is, as the number of nodal points is increased. 3. Solution of the algebraic equations. For creeping flows, the algebraic equations are linear and a linear matrix equation is to be solved. Both direct and iterative solvers have been used. For most flows, the nonlinear inertial terms in the momentum equation are important and the algebraic discretized equations are therefore nonlinear. Solution yields the nodal values of the unknowns. CFD solutions, especially for complex three-dimensional flows, generate very large quantities of solution data. Computer graphics have greatly improved the ability to examine CFD solutions and visualize flow. CFD methods are used for incompressible and compressible, creeping, laminar and turbulent, Newtonian and non-Newtonian, and isothermal and nonisothermal flows. Chemically reacting flows, particularly in the field of combustion, have been simulated. Solution accuracy must be considered from several perspectives. These include convergence of the algorithms for solving the nonlinear discretized equations and convergence with respect to refinement of the mesh so that the discretized equations better approximate the exact equations and, in some cases, so that the mesh more accurately fits the true geometry. The possibility that steady-state solutions are unstable must always be considered. In addition to numerical sources of error, modeling errors are introduced in turbulent flow, where semiempirical closure models are used to solve time-averaged equations of motion, as discussed previously. Most commercial CFD codes include the k–
turbulence model, which has been by far the most widely used. More accurate models, such as differential Reynolds stress and renormalization group theory models, are also becoming available. Significant solution error is known to result in some problems from inadequacy of the turbulence model. Closure models for nonlinear chemical reaction source terms may also contribute to inaccuracy. Direct numerical simulation and large eddy simulation, which involve solutions for velocity fluctuations, under limited conditions or with certain modeling assumptions, remain primarily research areas. In its general sense, multiphase flow is not currently solvable by computational fluid dynamics. However, in certain cases reasonable solutions are possible. These include well-separated flows where the phases are confined to relatively well-defined regions separated by one or a few interfaces and flows in which a second phase appears as discrete particles of known size and shape whose motion may be approximately computed with drag coefficient formulations, or rigorously computed with refined meshes applying boundary conditions at the particle surface. Two-fluid modeling, in which the phases are treated as overlapping continua, with each phase occupying a volume fraction that is a continuous function of position (and time) is a useful approximation which is becoming available in commercial software. See Elghobashi and Abou-Arab ( J. Physics Fluids, 26, 931–938 [1983]) for a k–
model for two-fluid systems.

6-48

FLUID AND PARTICLE DYNAMICS

Figure 6-56 gives an example CFD calculation for time-dependent flow past a square cylinder at a Reynolds number of 22,000 (Choudhury, et al., Trans. ASME Fluids Div., Lake Tahoe, Nev. [1994]). The computation was done with an implementation of the renormalization group theory k–
model. The series of contour plots of stream function shows a sequence in time over about 1 vortex-shedding period. The calculated Strouhal number (Eq. [6-195]) is 0.146, in excellent agreement with experiment, as is the time-averaged drag coefficient, CD = 2.24. Similar computations for a circular cylinder at Re = 14,500 have given excellent agreement with experimental measurements for St and CD (Introduction to the Renormalization Group Method and Turbulence Modeling, Fluent, Inc., 1993).

DIMENSIONLESS GROUPS For purposes of data correlation, model studies, and scale-up, it is useful to arrange variables into dimensionless groups. Table 6-7 lists many of the dimensionless groups commonly found in fluid mechanics problems, along with their physical interpretations and areas of application. More extensive tabulations may be found in Catchpole and Fulford (Ind. Eng. Chem., 58[3], 46–60 [1966]) and Fulford and Catchpole (Ind. Eng. Chem., 60[3], 71–78 [1968]).

FIG. 6-56 Computational fluid dynamic simulation of flow over a square cylinder, showing one vortex shedding period. (From Choudhury, et al., Trans. ASME Fluids Div., TN-076 [1994].)

FLUID DYNAMICS TABLE 6-7

6-49

Dimensionless Groups and Their Significance

Name

Symbol

Formula

Archimedes number

Ar

gL (ρp − ρ)ρ  µ2

inertial forces × buoyancy forces  (viscous forces)2

Particle settling

Bingham number

Bm

τ yL  µ ∞V

yield stress  viscous stress

Flow of Bingham plastics = yield number, Y

Bingham Reynolds number

ReB

LVρ  µ∞

inertial force  viscous force

Flow of Bingham plastics

Blake number

B

Vρ  µ(1 −
)s

inertial force  viscous force

Beds of solids

Bond number

Bo

(ρL − ρG)L2g  σ

gravitational force  surface-tension force

Atomization = Eotvos number, Eo

Capillary number

Ca

µV  σ

viscous force  surface-tension force

Two-phase flows, free surface flows

Cauchy number

C

ρV 2  β

inertial force  compressibility force

Compressible flow, hydraulic transients

Cavitation number

σ

p − pv  ρV 2/2

excess pressure above vapor pressure  velocity head

Cavitation

Dean number

De

Re  (Dc/D)1/2

inertial force Reynolds number ×  centrifugal force

Flow in curved channels

Deborah number

De

λ

fluid relaxation time  flow characteristic time

Viscoelastic flow

Drag coefficient

CD

FD  AρV 2/2

drag force  projected area × velocity head

Flow around objects, particle settling

Elasticity number

El

λµ 2 ρL

elastic force  inertial force

Viscoelastic flow

Euler number

Eu

∆p 2 ρV

frictional pressure loss  2 × velocity head

Fluid friction in conduits

Fanning friction factor

f

D∆ p 2τ w =  2ρV 2L ρV 2

wall shear stress  velocity head

Fluid friction in conduits Darcy friction factor = 4f

Froude number

Fr

V2  gL

inertial force  gravity force

苶 Often defined as Fr = V/兹gL

Densometric Froude number

Fr′

ρV 2  (ρd − ρ)gL

inertial force  gravity force

or

Hedstrom number

He

L2 τYρ  µ ∞2

Bingham Reynolds number × Bingham number

Flow of Bingham plastics

Hodgson number

H

V′ω∆p  q 苶p 苶

time constant of system  period of pulsation

Pulsating gas flow

Mach number

M

V  c

fluid velocity  sonic velocity

Compressible flow

Ohnesorge number

Z

µ  (ρLσ)1/2

viscous force  (inertial force × surface tension force)1/2

Weber number Atomization =  Reynolds number

Peclet number

Pe

LV  D

convective transport  diffusive transport

Heat, mass transfer, mixing

Pipeline parameter

Pn

aVo  2gH

maximum water-hammer pressure rise  2 × static pressure

Water hammer

3

Physical interpretation

Comments

V Fr′ =  苶d苶− 苶苶 苶L 苶/ρ 苶 ρ)g 兹(ρ

Newton number

6-50

FLUID AND PARTICLE DYNAMICS

TABLE 6-7

Dimensionless Groups and Their Significance (Concluded )

Name

Symbol

Formula

Physical interpretation

Comments

Power number

Po

P  ρN 3L5

impeller drag force  inertial force

Prandtl velocity ratio

v+

v  (τ w /ρ)1/2

velocity normalized by friction velocity

Reynolds number

Re

LVρ  µ

inertial force  viscous force

Strouhal number

St

f′ L  V

vortex shedding frequency × characteristic flow time scale

Weber number

We

ρV 2L  σ

Nomenclature a A c D Dc D′ f′ FD g H L N p pv p 苶 ∆p

Wave speed Projected area Sonic velocity Diameter of pipe Diameter of curvature Diffusivity Vortex shedding frequency Drag force Acceleration of gravity Static head Characteristic length Rotational speed Pressure Vapor pressure Average static pressure Frictional pressure drop

Agitation Turbulent flow near a wall, friction velocity = 兹苶 τw苶 /ρ

inertial force  surface tension force SI Units Nomenclature m/s m m/s m m m2/s 1/s N m/s m m 1/s Pa Pa Pa Pa

P q 苶 s v V V′ 
λ µ µ∞ ρ ρG, ρL ρd σ ω

Vortex shedding, von Karman vortex streets Bubble, drop formation SI Units

Power Average volumetric flow rate Particle area/particle volume Local fluid velocity Characteristic or average fluid velocity System volume Bulk modulus Void fraction Fluid relaxation time Fluid viscosity Infinite shear viscosity (Bingham plastics) Fluid density Gas, liquid densities Dispersed phase density Surface tension Characteristic frequency or reciprocal time scale of flow

Watts m3/s 1/m m/s m/s m3 Pa m3 s Pa ⋅ s Pa ⋅ s kg/m3 kg/m3 kg/m3 N/m 1/s

PARTICLE DYNAMICS GENERAL REFERENCES: Brodkey, The Phenomena of Fluid Motions, AddisonWesley, Reading, Mass., 1967; Clift, Grace, and Weber, Bubbles, Drops and Particles, Academic, New York, 1978; Govier and Aziz, The Flow of Complex Mixtures in Pipes, Van Nostrand Reinhold, New York, 1972, Krieger, Huntington, N.Y., 1977; Lapple, et al., Fluid and Particle Mechanics, University of Delaware, Newark, 1951; Levich, Physicochemical Hydrodynamics, PrenticeHall, Englewood Cliffs, N.J., 1962; Orr, Particulate Technology, Macmillan, New York, 1966; Shook and Roco, Slurry Flow, Butterworth-Heinemann, Boston, 1991; Wallis, One-dimensional Two-phase Flow, McGraw-Hill, New York, 1969.

DRAG COEFFICIENT Whenever relative motion exists between a particle and a surrounding fluid, the fluid will exert a drag upon the particle. In steady flow, the drag force on the particle is CD APρu2 FD =  (6-227) 2 where FD = drag force CD = drag coefficient AP = projected particle area in direction of motion ρ = density of surrounding fluid u = relative velocity between particle and fluid The drag force is exerted in a direction parallel to the fluid velocity. Equation (6-227) defines the drag coefficient. For some solid bodies, such as aerofoils, a lift force component perpendicular to the liquid velocity is also exerted. For free-falling particles, lift forces are generally not important. However, even spherical particles experience lift forces in shear flows near solid surfaces.

TERMINAL SETTLING VELOCITY A particle falling under the action of gravity will accelerate until the drag force balances gravitational force, after which it falls at a constant terminal or free-settling velocity ut, given by ut = where

2gm (ρ − ρ)  冪莦 ρρ A C p

p

p

P

(6-228)

D

g = acceleration of gravity m p = particle mass ρp = particle density

and the remaining symbols are as previously defined. Settling particles may undergo fluctuating motions owing to vortex shedding, among other factors. Oscillation is enhanced with increasing separation between the mass and geometric centers of the particle. Variations in mean velocity are usually less than 10 percent. The drag force on a particle fixed in space with fluid moving is somewhat lower than the drag force on a particle freely settling in a stationary fluid at the same relative velocity. Spherical Particles For spherical particles of diameter dp, Eq. (6-228) becomes 4gdp(ρp − ρ) ut =  (6-229) 3ρCD The drag coefficient for rigid spherical particles is a function of particle Reynolds number, Rep = dpρu/µ where µ = fluid viscosity, as shown in Fig. 6-57. At low Reynolds number, Stokes’ Law gives 24 CD =  Rep < 0.1 (6-230) Rep

冪莦

PARTICLE DYNAMICS

6-51

FIG. 6-57 Drag coefficients for spheres, disks, and cylinders: Ap = area of particle projected on a plane normal to direction of motion; C = overall drag coefficient, dimensionless; Dp = diameter of particle; Fd = drag or resistance to motion of body in fluid; Re = Reynolds number, dimensionless; u = relative velocity between particle and main body of fluid; µ = fluid viscosity; and ρ = fluid density. (From Lapple and Shepherd, Ind. Eng. Chem., 32, 605 [1940].)

which may also be written FD = 3π µudp

Rep < 0.1

(6-231)

and gives for the terminal settling velocity gdp2 (ρp − ρ) Rep < 0.1 (6-232) ut =  18µ In the intermediate regime (0.1 < Rep < 1,000), the drag coefficient may be estimated within 6 percent by 24 CD =  1 + 0.14Re 0.70 0.1 < Rep < 1,000 (6-233) p Rep In the Newton’s Law regime, which covers the range 1,000 < Rep < 350,000, CD = 0.445, within 13 percent. In this region, Eq. (6-227) becomes gdp(ρp − ρ) ut = 1.73  1,000 < Rep < 350,000 (6-234) ρ 6 Between about Rep = 350,000 and 1 × 10 , the drag coefficient drops dramatically in a drag crisis owing to the transition to turbulent flow in the boundary layer around the particle, which delays aft separation, resulting in a smaller wake and less drag. Beyond Re = 1 × 106, the drag coefficient may be estimated from (Clift, Grace, and Weber): 8 × 104 CD = 0.19 −  Rep > 1 × 106 (6-235) Rep Drag coefficients may be affected by turbulence in the free-stream flow; the drag crisis occurs at lower Reynolds numbers when the free stream is turbulent. Torobin and Guvin (AIChE J., 7, 615–619 [1961]) found that the drag crisis Reynolds number decreases with increasing free-stream turbulence, reaching a value of 400 when the relative turbulence intensity, defined as 兹苶 u苶苶′/U u苶′ is the rms 苶R is 0.4. Here 兹苶 fluctuating velocity and 苶 UR is the relative velocity between the particle and the fluid. For particles settling in non-Newtonian fluids, correlations are

冢 冣冢

冪莦

冣

given by Dallon and Christiansen (Preprint 24C, Symposium on Selected Papers, part III, 61st Ann. Mtg. AIChE, Los Angeles, Dec. 1–5, 1968) for spheres settling in shear-thinning liquids, and by Ito and Kajiuchi (J. Chem. Eng. Japan, 2[1], 19–24 [1969]) and Pazwash and Robertson (J. Hydraul. Res., 13, 35–55 [1975]) for spheres settling in Bingham plastics. Beris, Tsamopoulos, Armstrong, and Brown ( J. Fluid Mech., 158 [1985]) present a finite element calculation for creeping motion of a sphere through a Bingham plastic. Nonspherical Rigid Particles The drag on a nonspherical particle depends upon its shape and orientation with respect to the direction of motion. The orientation in free fall as a function of Reynolds number is given in Table 6-8. The drag coefficients for disks (flat side perpendicular to the direction of motion) and for cylinders (infinite length with axis perpendicular to the direction of motion) are given in Fig. 6-57 as a function of Reynolds number. The effect of length-to-diameter ratio for cylinders in the Newton’s law region is reported by Knudsen and Katz (Fluid Mechanics and Heat Transfer, McGraw-Hill, New York, 1958). Pettyjohn and Christiansen (Chem. Eng. Prog., 44, 157–172 [1948]) present correlations for the effect of particle shape on freesettling velocities of isometric particles. For Re < 0.05, the terminal or free-settling velocity is given by TABLE 6-8

Free-Fall Orientation of Particles

Reynolds number*

Orientation

0.1–5.5

All orientations are stable when there are three or more perpendicular axes of symmetry. Stable in position of maximum drag. Unpredictable. Disks and plates tend to wobble, while fuller bluff bodies tend to rotate. Rotation about axis of least inertia, frequently coupled with spiral translation.

5.5–200 200–500 500–200,000 SOURCE:

From Becker, Can. J. Chem. Eng., 37, 85–91 (1959). *Based on diameter of a sphere having the same surface area as the particle.

6-52

FLUID AND PARTICLE DYNAMICS gd s2(ρp − ρ) ut = K1  18µ

(6-236)

ψ (6-237) K1 = 0.843 log  0.065 where ψ = sphericity, the surface area of a sphere having the same volume as the particle, divided by the actual surface area of the particle; ds = equivalent diameter, equal to the diameter of the equivalent sphere having the same volume as the particle; and other variables are as previously defined. In the Newton’s law region, the terminal velocity is given by

冢

ut =

冣

4d (ρ − ρ)g  冪莦 3K ρ s

p

(6-238)

3

K3 = 5.31 − 4.88ψ

(6-239)

Equations (6-236) to (6-239) are based on experiments on cubeoctahedrons, octahedrons, cubes, and tetrahedrons for which the sphericity ψ ranges from 0.906 to 0.670, respectively. See also Clift, Grace, and Weber. A graph of drag coefficient vs. Reynolds number with ψ as a parameter may be found in Brown, et al. (Unit Operations, Wiley, New York, 1950) and in Govier and Aziz. For particles with ψ < 0.67, the correlations of Becker (Can. J. Chem. Eng., 37, 85–91 [1959]) should be used. Reference to this paper is also recommended for intermediate region flow. Settling characteristics of nonspherical particles are discussed by Clift, Grace, and Weber, Chaps. 4 and 6. The terminal velocity of axisymmetric particles in axial motion can be computed from Bowen and Masliyah (Can. J. Chem. Eng., 51, 8–15 [1973]) for low–Reynolds number motion: V′ gDs2(ρp − ρ) ut =   (6-240) 18µ K2 K2 = 0.244 + 1.035 − 0.7122 + 0.4413

(6-241)

where Ds = diameter of sphere with perimeter equal to maximum particle projected perimeter V′ = ratio of particle volume to volume of sphere with diameter Ds  = ratio of surface area of particle to surface area of a sphere with diameter Ds and other variables are as defined previously. Hindered Settling When particle concentration increases, particle settling velocities decrease because of hydrodynamic interaction between particles and the upward motion of displaced liquid. The suspension viscosity increases. Hindered settling is normally encountered in sedimentation and transport of concentrated slurries. Below 0.1 percent volumetric particle concentration, there is less than a 1 percent reduction in settling velocity. Several expressions have been given to estimate the effect of particle volume fraction on settling velocity. Maude and Whitmore (Br. J. Appl. Phys., 9, 477–482 [1958]) give, for uniformly sized spheres, ut = ut0 (1 − c)n

(6-242)

where ut = terminal settling velocity ut0 = terminal velocity of a single sphere (infinite dilution) c = volume fraction solid in the suspension n = function of Reynolds number Rep = dput0 ρ/µ as given Fig. 6-58 In the Stokes’ law region (Rep < 0.3), n = 4.65 and in the Newton’s law region (Rep > 1,000), n = 2.33. Equation (6-242) may be applied to particles of any size in a polydisperse system, provided the volume fraction corresponding to all the particles is used in computing terminal velocity (Richardson and Shabi, Trans. Inst. Chem. Eng. [London], 38, 33–42 [1960]). The concentration effect is greater for nonspherical and angular particles than for spherical particles (Steinour, Ind. Eng. Chem., 36, 840–847 [1944]). Theoretical developments for low–Reynolds number flow assemblages of spheres are given by Happel and Brenner (Low Reynolds Number Hydrodynamics, Prentice-

Values of exponent n for use in Eq. (6-240). (From Maude and Whitmore, Br. J. Appl. Phys., 9, 481 [1958]. Courtesy of the Institute of Physics and the Physical Society.)

FIG. 6-58

Hall, Englewood Cliffs, N.J., 1965) and Famularo and Happel (AIChE J., 11, 981 [1965]) leading to an equation of the form ut0 ut =  1 + γc1/3

(6-243)

where γ is about 1.3. As particle concentration increases, resulting in interparticle contact, hindered settling velocities are difficult to predict. Thomas (AIChE J., 9, 310 [1963]) provides an empirical expression reported to be valid over the range 0.08 < ut /ut0 < 1: ut (6-244) ln  = −5.9c ut0

冢 冣

Time-Dependent Motion The time-dependent motion of particles is computed by application of Newton’s second law, equating the rate of change of particle motion to the net force acting on the particle. Rotation of particles may also be computed from the net torque. For large particles moving through low-density gases, it is usually sufficient to compute the force due to fluid drag from the relative velocity and the drag coefficient computed for steady flow conditions. For two- and three-dimensional problems, the velocity appearing in the particle Reynolds number and the drag coefficient is the amplitude of the relative velocity. The drag force, not the relative velocity, is to be resolved into vector components to compute the particle acceleration components. Clift, Grace, and Weber (Bubbles, Drops and Particles, Academic, London, 1978) discuss the complexities that arise in the computation of transient drag forces on particles when the transient nature of the flow is important. Analytical solutions for the case of a single particle in creeping flow (Rep = 0) are available. For example, the creeping motion of a sphericial particle released from rest in a stagnant fluid is described by dU ρ dU ρpV  = g(ρp − ρ)V − 3πµdpU −  V  dt 2 dt

冢冣

3 苶ρ 苶µ 苶 −  d 2p 兹π 2

(dU/dt) ds 冕  苶苶s 兹t苶− t

t=s

(6-245)

0

Here, U = particle velocity, positive in the direction of gravity, and V = particle volume. The first term on the right-hand side is the net gravitational force on the particle, accounting for buoyancy. The second is the steady-state Stokes drag (Eq. 6-231). The third is the added mass or virtual mass term, which may be interpreted as the inertial effect of the fluid which is accelerated along with the particle. The volume of the added mass of fluid is half the particle volume. The last term, the Basset force, depends on the entire history of the transient motion, with past motions weighted inversely with the square root of elapsed time. Clift, et al. provide integrated solutions. In turbulent flows, particle velocity will closely follow fluid eddy velocities when (Clift et al.)

PARTICLE DYNAMICS

6-53

Terminal velocity of air bubbles in water at 20°C. (From Clift, Grace, and Weber, Bubbles, Drops and Particles, Academic, New York, 1978). FIG. 6-59

d p2 [(2ρp /ρ) + 1] τ0 >>  (6-246) 36ν where τ0 = oscillation period or eddy time scale, the right-hand side expression is the particle relaxation time, and ν = kinematic viscosity. Gas Bubbles Fluid particles, unlike rigid solid particles, may undergo deformation and internal circulation. Figure 6-59 shows rise velocity data for air bubbles in stagnant water. In the figure, Eo = Eotvos number, g(ρL − ρG)d b2 /σ, where ρL = liquid density, ρG = gas density, db = bubble diameter, and σ = surface tension. Small bubbles (<1-mm [0.04-in] diameter) remain spherical and rise in straight lines. The presence of surface active materials generally renders small bubbles rigid, and they rise roughly according to the drag coefficient and terminal velocity equations for spherical solid particles. Bubbles roughly in the range 2- to 8-mm (0.079- to 0.32-in) diameter assume flattened, ellipsoidal shape, and rise in a zig-zag or spiral pattern. This motion increases dissipation and drag, and the rise velocity may actually decrease with increasing bubble diameter in this region, characterized by rise velocities in the range of 20 to 30 cm/s (0.7 to 1.0 ft/s). Large bubbles, >8-mm (0.32-in) diameter, are greatly deformed, assuming a mushroomlike, spherical cap shape. These bubbles are unstable and may break into smaller bubbles. Carefully purified water, free of surface active materials, allows bubbles to freely circulate even when they are quite small. Under creeping flow conditions Reb = dburρL /µL < 1, where ur = bubble rise velocity and µL = liquid viscosity, the bubble rise velocity may be computed analytically from the Hadamard-Rybczynski formula (Levich, Physicochemical Hydrodynamics, Prentice-Hall, Englewood Cliffs, N.J., 1962, p. 402). When µG /µL << 1, which is normally the case, the rise velocity is 1.5 times the rigid sphere Stokes law velocity. However, in practice, most liquids, including ordinary distilled water, contain sufficient surface active materials to render small bubbles rigid. Larger bubbles undergo deformation in both purified and ordinary liquids; however, the variation in rise velocity for large bubbles with degree of purity is quite evident in Fig. 6-59. For additional discussion, see Clift, et al., Chap. 7. Figure 6-60 gives the drag coefficient as a function of bubble or drop Reynolds number for air bubbles in water and water drops in air, compared with the standard drag curve for rigid spheres. Information on bubble motion in non-Newtonian liquids may be found in Astarita and Apuzzo (AIChE J., 11, 815–820 [1965]); Calderbank, Johnson, and Loudon (Chem. Eng. Sci., 25, 235–256 [1970]); and Acharya, Mashelkar, and Ulbrecht (Chem. Eng. Sci., 32, 863–872 [1977]).

Liquid Drops in Liquids Very small liquid drops in immisicibile liquids behave like rigid spheres, and the terminal velocity can be approximated by use of the drag coefficient for solid spheres up to a Reynolds number of about 10 (Warshay, Bogusz, Johnson, and Kintner, Can. J. Chem. Eng., 37, 29–36 [1959]). Between Reynolds numbers of 10 and 500, the terminal velocity exceeds that for rigid spheres owing to internal circulation. In normal practice, the effect of drop phase viscosity is neglected. Grace, Wairegi, and Nguyen (Trans. Inst. Chem. Eng., 54, 167–173 [1976]; Clift, et al., op. cit., pp. 175–177) present a correlation for terminal velocity valid in the range M < 10−3 Eo < 40 Re > 0.1 (6-247) where M = Morton number = gµ4∆ρ/ρ2σ3 Eo = Eotvos number = g∆ρd 2/σ Re = Reynolds number = duρ/µ ∆ρ = density difference between the phases ρ = density of continuous liquid phase d = drop diameter µ = continuous liquid viscosity σ = surface tension u = relative velocity

FIG. 6-60 Drag coefficient for water drops in air and air bubbles in water. Standard drag curve is for rigid spheres. (From Clift, Grace, and Weber, Bubbles, Drops and Particles, Academic, New York, 1978.)

6-54

FLUID AND PARTICLE DYNAMICS

The correlation is represented by J = 0.94H0.757

(2 < H ≤ 59.3)

(6-248)

J = 3.42H0.441

(H > 59.3)

(6-249)

µ 4 H =  EoM−0.149  3 µw

冢 冣

where

−0.14

(6-250)

J = ReM 0.149 + 0.857

(6-251)

Note that the terminal velocity may be evaluated explicitly from µ u =  M −0.149( J − 0.857) ρd

(6-252)

In Eq. (6-250), µ = viscosity of continuous liquid and µw = viscosity of water, taken as 0.9 cP (0.0009 Pa ⋅ s). For drop velocities in non-Newtonian liquids, see Mhatre and Kinter (Ind. Eng. Chem., 51, 865–867 [1959]); Marrucci, Apuzzo, and Astarita (AIChE J., 16, 538–541 [1970]); and Mohan, et al. (Can. J. Chem. Eng., 50, 37–40 [1972]). Liquid Drops in Gases Liquid drops falling in stagnant gases appear to remain spherical and follow the rigid sphere drag relationships up to a Reynolds number of about 100. Large drops will deform, with a resulting increase in drag, and in some cases will shatter. The largest water drop which will fall in air at its terminal velocity is about 8 mm (0.32 in) in diameter, with a corresponding velocity of about 9 m/s (30 ft/s). Drops shatter when the Weber number defined as ρGu2d We =  σ

(6-253)

exceeds a critical value. Here, ρG = gas density, u = drop velocity, d = drop diameter, and σ = surface tension. A value of Wec = 13 is often cited for the critical Weber number. Terminal velocities for water drops in air have been correlated by Berry and Prnager (J. Appl. Meteorol., 13, 108–113 [1974]) as Re = exp [−3.126 + 1.013 ln ND − 0.01912(ln ND)2]

(6-254)

for 2.4 < ND < 107 and 0.1 < Re < 3550. The dimensionless group ND (often called the Best number [Clift, et al.]) is given by 4ρ∆ρgd 3 ND =  3µ2

(6-255)

and is proportional to the similar Archimedes and Galileo numbers. Figure 6-61 gives calculated settling velocities for solid spherical particles settling in air or water using the standard drag coefficient curve for spherical particles. For fine particles settling in air, the Stokes-Cunningham correction has been applied to account for particle size comparable to the mean free path of the gas. The correction is less than 1 percent for particles larger than 16 µm settling in air. Smaller particles are also subject to Brownian motion. Motion of particles smaller than 0.1 µm is dominated by Brownian forces and gravitational effects are small. Wall Effects When the diameter of a settling particle is significant compared to the diameter of the container, the settling velocity is reduced. For rigid spherical particles settling with Re < 1, the correction given in Table 6-9 may be used. The factor kw is multiplied by the settling velocity obtained from Stokes’ law to obtain the corrected setTABLE 6-9 Wall Correction Factor for Rigid Spheres in Stokes’ Law Region β*

kw

β

kw

0.0 0.05 0.1 0.2 0.3

1.000 0.885 0.792 0.596 0.422

0.4 0.5 0.6 0.7 0.8

0.279 0.170 0.0945 0.0468 0.0205

SOURCE:

From Haberman and Sayre, David W. Taylor Model Basin Report

1143, 1958. *β = particle diameter divided by vessel diameter.

FIG. 6-61 Terminal velocities of spherical particles of different densities settling in air and water at 70°F under the action of gravity. To convert ft/s to m/s, multiply by 0.3048. (From Lapple, et al., Fluid and Particle Mechanics, University of Delaware, Newark, 1951, p. 292.)

tling rate. For values of diameter ratio β = particle diameter/vessel diameter less than 0.05, kw = 1/(1 + 2.1β) (Zenz and Othmer, Fluidization and Fluid-Particle Systems, Reinhold, New York, 1960, pp. 208–209). In the range 100 < Re < 10,000, the computed terminal velocity for rigid spheres may be multiplied by k′w to account for wall effects, where k′w is given by (Harmathy, AIChE J., 6, 281 [1960]) 1 − β2 k′w = 4 (6-256) +苶 β苶 兹苶1苶 For gas bubbles in liquids, there is little wall effect for β < 0.1. For β > 0.1, see Uto and Kintner (AIChE J., 2, 420–424 [1956]), Maneri and Mendelson (Chem. Eng. Prog., 64, Symp. Ser., 82, 72–80 [1968]), and Collins (J. Fluid Mech., 28, part 1, 97–112 [1967]).

Section 7

Reaction Kinetics

Stanley M. Walas, Ph.D., Professor Emeritus, Department of Chemical and Petroleum Engineering, University of Kansas; Fellow, American Institute of Chemical Engineers

REACTION KINETICS Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Primary Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7-3 7-3 7-3

IDEAL REACTORS

RATE EQUATIONS Rate of Reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Law of Mass Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effect of Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concentration, Moles, Partial Pressure, and Mole Fraction . . . . . . . . . Typical Units of Specific Rates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 1: Rates of Change at Constant V or Constant P . . . . . . . . . Reaction Time in Flow Reactors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Constants of the Rate Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . From the Differential Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . From the Integrated Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . From Half-Times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Complex Rate Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multiple Reactions and Stoichiometric Balances . . . . . . . . . . . . . . . . . . Single Reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multiple Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stoichiometric Balances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 2: Analysis of Three Simultaneous Reactions . . . . . . . . . . . Mechanisms of Some Complex Reactions. . . . . . . . . . . . . . . . . . . . . . . . Phosgene Synthesis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ozone and Chlorine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hydrogen Bromide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Enzyme Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chain Polymerization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solid Catalyzed Reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . With Diffusion between Phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Catalysis by Solids: Langmuir-Hinshelwood Mechanism . . . . . . . . . . . . Adsorptive Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dissociation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Different Sites. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dual Sites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reactant in the Gas Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chemical Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 3: Phosgene Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chemical Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 4: Reaction between Methane and Steam . . . . . . . . . . . . . . Approach to Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 5: Percent Approach to Equilibrium . . . . . . . . . . . . . . . . . . Integration of Rate Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 6: Laplace Transform Application . . . . . . . . . . . . . . . . . . . .

7-5 7-5 7-5 7-5 7-7 7-7 7-7 7-8 7-8 7-8 7-8 7-8 7-8 7-8 7-8 7-10 7-10 7-10 7-10 7-10 7-10 7-10 7-11 7-11 7-11 7-11 7-11 7-11 7-11 7-12 7-12 7-12 7-12 7-13 7-13 7-14 7-14 7-14 7-15

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Material and Energy Balances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Batch Reactors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Daily Yield. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Filling and Emptying Periods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimum Operation of Reversible Reactions . . . . . . . . . . . . . . . . . . . Continuous Stirred Tank Reactors (CSTR) . . . . . . . . . . . . . . . . . . . . . . . Example 7: A Four-Stage Unit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 8: Consecutive Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 9: Comparison of Batch and CSTR Volumes . . . . . . . . . . . . Different Sizes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Selectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tubular and Packed Bed Flow Reactors . . . . . . . . . . . . . . . . . . . . . . . . . Frictional Pressure Drop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Recycle and Separation Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 10: Reactor Size with Recycle. . . . . . . . . . . . . . . . . . . . . . . . Heat Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Batch Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CSTR Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plug Flow Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Packed Bed Reactors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Unsteady Conditions with Accumulation Terms . . . . . . . . . . . . . . . . . . . Example 11: Balances of a Semibatch Process . . . . . . . . . . . . . . . . . .

7-15 7-15 7-15 7-16 7-16 7-16 7-17 7-17 7-17 7-17 7-19 7-19 7-19 7-19 7-20 7-20 7-21 7-21 7-22 7-22 7-22 7-22 7-23

LARGE SCALE OPERATIONS Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multiple Steady States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nonideal Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Laminar Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Residence Time Distribution (RTD) . . . . . . . . . . . . . . . . . . . . . . . . . . Segregated Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 12: Segregated Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Maximum Mixedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimum Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Objective Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Heterogeneous Reactions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reaction and Separation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7-23 7-23 7-23 7-23 7-24 7-25 7-25 7-25 7-25 7-25 7-25 7-25 7-25 7-26 7-26 7-27 7-1

7-2

REACTION KINETICS

ACQUISITION OF DATA Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Batch Reactors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flow Reactors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multiple Phases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solid Catalysts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References for Laboratory Reactors . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7-27 7-27 7-27 7-27 7-28 7-28 7-28 7-28

SOLVED PROBLEMS Equilibrium of Formation of Ethylbenzene . . . . . . . . . . . . . . . . . Optimum Cycle Period with Downtime . . . . . . . . . . . . . . . . . . . .

7-28 7-29

P1. P2.

P3. P4. P5. P6. P7. P8. P9. P10. P11. P12. P13. P14. P15. P16.

Parallel Reactions of Butadiene . . . . . . . . . . . . . . . . . . . . . . . . . . . Batch Reaction with Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . A Semibatch Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimum Reaction Temperature with Downtime. . . . . . . . . . . . . Rate Equations from CSTR Data . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of Batch and CSTR Operations . . . . . . . . . . . . . . . . . Instantaneous and Gradual Feed Rates . . . . . . . . . . . . . . . . . . . . . Filling and Unsteady Operating Period of a CSTR . . . . . . . . . . . Second-Order Reaction in Two Stages . . . . . . . . . . . . . . . . . . . . . Butadiene Dimerization in a TFR . . . . . . . . . . . . . . . . . . . . . . . . Autocatalytic Reaction with Recycle . . . . . . . . . . . . . . . . . . . . . . Minimum Residence Time in a PFR . . . . . . . . . . . . . . . . . . . . . . Heat Transfer in a Cylindrical Reactor. . . . . . . . . . . . . . . . . . . . . Pressure Drop and Conversion in a PFR . . . . . . . . . . . . . . . . . . .

7-29 7-29 7-30 7-30 7-30 7-31 7-31 7-31 7-32 7-32 7-32 7-32 7-33 7-33

Nomenclature and Units Following is a listing of typical nomenclature expressed in SI and U.S. customary units. Specific definitions and units are stated at the place of application in this section.

Symbol A, B, C, . . . •

A Ca C0 Cp CSTR D, De, Dx Deff DK E(t) E(tr) fa F(t) ∆G Ha ∆Hr K, Ke, Ky, Kφ k, kc, kp L n na n′a nt pa Pe PFR Q r ra R Re Sc

Definition Names of substances, or their concentrations Free radical, as CH •3 Concentration of substance A Initial mean concentration in vessel Heat capacity Continuous stirred tank reactor Dispersion coefficient Effective diffusivity Knudsen diffusivity Residence time distribution Normalized residence time distribution Ca /Ca 0 or na /na 0, fraction of A remaining unconverted Age function of tracer Gibbs energy change Hatta number Heat of reaction Chemical equilibrium constant Specific rate of reaction Length of path in reactor Parameter of Erlang or Gamma distribution, or number of stages in a CSTR battery Number of mols of A present Number of mols flowing per unit time; the prime (′) may be omitted when context is clear Total number of mols Partial pressure of substance A Peclet number for dispersion Plug flow reactor Heat transfer Radial position Rate of reaction of A per unit volume Radius of cylindrical vessel Reynolds number Schmidt number

SI units

U.S. customary units

kg mol/m3 lb mol/ft3 kg mol/m3 lb mol/ft3 kJ/(kg⋅K)

Btu/(lbm⋅°F)

m2/s m2/s m2/s

ft2/s ft2/s ft2/s

Symbol t t
tr TFR u u(t) V V′ Vr x xa z

Definition Time Mean residence time t/t
, reduced time Tubular flow reactor Linear velocity Unit step input Volume of reactor contents Volumetric flow rate Volume of reactor Axial position in a reactor 1 − fa = 1 − Ca /Ca0 or 1 − na /na0, fraction of A converted x/L, normalized axial position

U.S. customary units

SI units s s

s s

m/s

ft/s

m3 m3/s m3 m

ft3 ft3/s ft3 ft

Pa⋅s m2/s Pa kg/m3

lbm/(ft⋅s) ft2/s psi lbm/ft3

Greek letters

kJ

Btu

β γ3(t) δ(t)

kJ/kg mol

Btu/lb mol

ε

Variable m

Variable ft

␪ η Λ(t) µ ν π ρ ρ

kPa

psi

kJ m Variable

Btu ft Variable

m

ft

σ2(t) σ2(tr) τ τ φ φm

r/R, normalized radial position Skewness of distribution Unit impulse input, Dirac function Fraction void space in a packed bed t/t
, reduced time, fraction of surface covered by adsorbed species Effectiveness of porous catalyst Intensity function Viscosity υ/ρ, kinematic viscosity Total pressure Density r/R, normalized radial position in a pore Variance Normalized variance t/t
, reduced time Tortuosity Thiele modulus Modified Thiele modulus Subscripts

0

Subscript designating initial or inlet conditions, as in Ca0, na0, V′0 , . . .

REACTION KINETICS GENERAL REFERENCES 1. Aris, Elementary Chemical Reactor Analysis, Prentice-Hall, 1969. 2. Bamford and Tipper (eds.), Comprehensive Chemical Kinetics, Elsevier, 1969–date. 3. Boudart, Kinetics of Chemical Processes, Prentice-Hall, 1968. 4. Brotz, Fundamentals of Chemical Reaction Engineering, Addison-Wesley, 1965. 5. Butt, Reaction Kinetics and Reactor Design, Prentice-Hall, 1980. 6. Capello and Bielski, Kinetic Systems: Mathematical Description of Kinetics in Solution, Wiley, 1972. 7. Carberry, Chemical and Catalytic Reaction Engineering, McGraw-Hill, 1976. 8. Carberry and Varma (eds.), Chemical Reaction and Reactor Engineering, Dekker, 1987. 9. Chen, Process Reactor Design, Allyn & Bacon, 1983. 10. Cooper and Jeffreys, Chemical Kinetics and Reactor Design, PrenticeHall, 1971. 11. Cremer and Watkins (eds.), Chemical Engineering Practice, vol. 8: Chemical Kinetics, Butterworths, 1965. 12. Denbigh and Turner, Chemical Reactor Theory, Cambridge, 1971. 13. Fogler, Elements of Chemical Reaction Engineering, Prentice-Hall, 1992. 14. Froment and Bischoff, Chemical Reactor Analysis and Design, Wiley, 1990. 15. Hill, An Introduction to Chemical Engineering Kinetics and Reactor Design, Wiley, 1977. 16. Holland and Anthony, Fundamentals of Chemical Reaction Engineering, Prentice-Hall, 1989.

17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32.

7-3

Horak and Pasek, Design of Industrial Chemical Reactors from Laboratory Data, Heyden, 1978. Kafarov, Cybernetic Methods in Chemistry and Chemical Engineering, Mir Publishers, 1976. Laidler, Chemical Kinetics, Harper & Row, 1987. Levenspiel, Chemical Reaction Engineering, Wiley, 1972. Lewis (ed.), Techniques of Chemistry, vol. 4: Investigation of Rates and Mechanisms of Reactions, Wiley, 1974. Naumann, Chemical Reactor Design, Wiley, 1987. Panchenkov and Lebedev, Chemical Kinetics and Catalysis, Mir Publishers, 1976. Petersen, Chemical Reaction Analysis, Prentice-Hall, 1965. Rase, Chemical Reactor Design for Process Plants: Principles and Case Studies, Wiley, 1977. Rose, Chemical Reactor Design in Practice, Elsevier, 1981. Smith, Chemical Engineering Kinetics, McGraw-Hill, 1981. Steinfeld, Francisco, and Hasse, Chemical Kinetics and Dynamics, Prentice-Hall, 1989. Ulrich, Guide to Chemical Engineering Reactor Design and Kinetics, Ulrich, 1993. Walas, Reaction Kinetics for Chemical Engineers, McGraw-Hill, 1959; reprint, Butterworths, 1989. Walas, Chemical Reaction Engineering Handbook of Solved Problems, Gordon & Breach Publishers, 1995. Westerterp, van Swaaij, and Beenackers, Chemical Reactor Design and Operation, Wiley, 1984.

REACTION KINETICS INTRODUCTION From an engineering viewpoint, reaction kinetics has these principal functions: Establishing the chemical mechanism of a reaction Obtaining experimental rate data Correlating rate data by equations or other means Designing suitable reactors Specifying operating conditions, control methods, and auxiliary equipment to meet the technological and economic needs of the reaction process Reactions can be classified in several ways. On the basis of mechanism they may be: 1. Irreversible 2. Reversible 3. Simultaneous 4. Consecutive A further classification from the point of view of mechanism is with respect to the number of molecules participating in the reaction, the molecularity: 1. Unimolecular 2. Bimolecular and higher Related to the preceding is the classification with respect to order. In the power law rate equation r = kCap Cbq, the exponent to which any particular reactant concentration is raised is called the order p or q with respect to that substance, and the sum of the exponents p + q is the order of the reaction. At times the order is identical with the molecularity, but there are many reactions with experimental orders of zero or fractions or negative numbers. Complex reactions may not conform to any power law. Thus, there are reactions of: 1. Integral order 2. Nonintegral order 3. Non–power law; for instance, hyperbolic With respect to thermal conditions, the principal types are: 1. Isothermal at constant volume 2. Isothermal at constant pressure 3. Adiabatic 4. Temperature regulated by heat transfer According to the phases involved, reactions are: 1. Homogeneous, gaseous, liquid or solid 2. Heterogeneous:

Controlled by diffusive mass transfer Controlled by chemical factors A major distinction is between reactions that are: 1. Uncatalyzed 2. Catalyzed with homogeneous or solid catalysts Equipment is also a basis for differentiation, namely: 1. Stirred tanks, single or in series 2. Tubular reactors, single or in parallel 3. Reactors filled with solid particles, inert or catalytic: Fixed bed Moving bed Fluidized bed, stable or entrained Finally, there are the operating modes: 1. Batch 2. Continuous flow 3. Semibatch or semiflow Clearly, these groupings are not mutually exclusive. The chief distinctions are between homogeneous and heterogeneous reactions and between batch and flow reactions. These distinctions most influence the choice of equipment, operating conditions, and methods of design. PRIMARY NOMENCLATURE The participant A is identified by the subscript a. Thus, the concentration is Ca; the number of mols is na; the fractional conversion is xa; the partial pressure is pa; and the rate of decomposition is ra. Capital letters are also used to represent concentration on occasion; thus, A instead of Ca. The flow rate in mol is n′a but the prime (′) is left off when the meaning is clear from the context. The volumetric flow rate is V′; reactor volume is Vr or simply V of batch reactors; the total pressure is π; and the temperature is T. The concentration is Ca = na /V or n′a / V′. Throughout this section, equations are presented without specification of units. Use of any consistent unit set is appropriate. SUMMARY Basic kinetic relations of this section are summarized in Table 7-1.

TABLE 7-1

Basic Rate Equations The relation between ra and Ca must be established (numerically if need be) from the second line before the integration can be completed.

1. The reference reaction is νa A + νbB + ⋅ ⋅ ⋅ → νrR + νsS + ⋅ ⋅ ⋅ ∆ν = νr + νs + ⋅ ⋅ ⋅ − (νa + νb + ⋅ ⋅ ⋅) 2. Stoichiometric balance for any component i: ν ni = ni0 ⫾ ᎏi (na0 − na) νa + for product (right-hand side, RHS) − for reactant (left-hand side, LHS) ν Ci = Ci0 ⫾ ᎏi (Ca0 − Ca), at constant T and V only νa

 

{

 

∆ν nt = nt0 + ᎏ (na0 − na) νa

 

 

ν ra = kCαa [Cb0 − ᎏb (Ca0 − Ca)]β⋅ ⋅ ⋅ νa where it is not necessarily true that α = νa′, β = νb′, ⋅ ⋅ ⋅

 

4. At constant volume, Ca = na/Vr Ca0

Ca



na0

na

5. Ideal gases at constant pressure:



  

α − 1 na0 na



[nt0 + (∆ν/νa)(na0 − na)]α − 1 dna ᎏᎏᎏ nαa

b′ −E k = k∞ exp ᎏ = exp a′ − ᎏ RT T E = energy of activation





k1

A+B→C+D (1) k2 (2) C+D→A+B k3 (3) A+C→E The rates are related by: ra = ra1 + ra2 + ra3 = k1CaCb − k2CcCd + k3CaCc rb = −rd = k1CaCb − k2CcCd rc = k1CaCb + k2CcCd + k3CaCc re = −k3CaCc The number of independent rate equations is the same as the number of independent stoichiometric relations. In the present example, Reactions (1) and (2) are reversible reactions and are not independent. Accordingly, Cc and Cd, for example, can be eliminated from the equations for ra and rb which then become an integrable system. Usually only systems of linear differential equations with constant coefficients are solvable analytically. 8. Mass transfer resistance: Cai = interfacial concentration of reactant A





Ca0

Ca

7-4

1 dCa ᎏᎏ (Ca − ra/kd)α

ra = kPaPbθ3v 1 θv = ᎏᎏᎏ (1 + K
a
Pa
+ KbPb + ⋅ ⋅ ⋅)

(3)

n Pa = ᎏa P nt ni ni0 ⫾ (νi/νa)(na0 − na) Pi = ᎏ P = ᎏᎏᎏ P nt nt0 + (∆ν/νa)(na0 − na)

ntRT P

kt =



na0

na

dna ᎏ2 , VPaPbθv

⯗

dC r ra = − ᎏa = kd(Ca − Cai) = kCαai = k Ca − ᎏa kd dt

(2)

• Reaction A2 + B → R + S, with A2 dissociated upon adsorption and with surface reaction rate controlling:

for a Case (2) batch reaction

10. A continuously stirred tank reactor (CSTR) battery Material balances: n′a0 = n′a + ra1Vr1

7. Simultaneous reactions. The overall rate is the algebraic sum of the rates of the individual reactions. For example, take the three reactions:

kt =

summation over all substances absorbed

V=ᎏ

6. Temperature effect on the specific rate:

 

Ke = PrPs /PaPb (equilibrium constant)

products, RHS +− for for reactants, LHS

∆ν ntRT RT Vr = ᎏ = ᎏ nt0 + ᎏ (na0 − na) P P νa ra = kCαa RT kt = ᎏ P

(1)

• At constant P and T the Pi are eliminated in favor of ni and the total pressure by:

1 dCa ᎏᎏᎏᎏ Caα[Cb0 − (νb /νa)(Ca0 − Ca)]β ⋅ ⋅ ⋅

Vr−1 + α + β dna ᎏᎏᎏ nαa [nb0 + (νb /νa)(na0 − na)]β⋅ ⋅ ⋅ Completed integrals for some values of α and β are in Table 7-4. kt =



1 θv = ᎏᎏ , 1 + KjPj

ν = kC [Cb0 − ᎏb (Ca0 − Ca)]νb⋅ ⋅ ⋅ νa



/

Ka PrPs θv = 1 1 + ᎏ ᎏ + KbPb + KrPr + KsPs + KlPl Ke Pb

r = kPaPbθv2

1 dna ra = − ᎏ ᎏ = kC νaaCbνb⋅ ⋅ ⋅ Vr dt

kt =

1 dn ra = − ᎏ ᎏa = kPaθv V dt

l is an adsorbed substance that is chemically inert. • Surface reaction rate controlling:

3. Law of mass action:

νa a

9. Solid-catalyzed reactions. Some Langmuir-Hinshelwood mechanisms for the reference reaction A + B → R + S (see also Tables 7.2, 7.3): • Adsorption rate of A controlling:

n′a,j − 1 = n′aj + rajVrj,

for the jth stage

For a first-order reaction, with ra = kCa: Caj 1 ᎏ = ᎏᎏᎏᎏ Ca0 (1 + k1t
1)(1 + k2 t
2)⋅ ⋅ ⋅(1 + kj t
j) 1 = ᎏj (1 + kt
i) for j tanks in series with the same temperatures and residence times t
i = Vri /V′i, where V′ is the volumetric flow rate.

11. Plug flow reactor (PFR): α



dn′a rs = − ᎏ = kCαa Cbβ⋅ ⋅ ⋅ dVr α

β

n′ ⋅⋅⋅   ᎏ V′ 

n′a =k ᎏ V′

b

(4)

RATE EQUATIONS TABLE 7-1

Basic Rate Equations (Concluded )

12. Material and energy balances for batch, CSTR, and PFR in Tables 7-5, 7-6, and 7-7. SOURCE:

7-5

13. Notation. A, B, R, S are participants in the reaction; the letters also are used to represent concentrations. Ci = ni/Vr or n′i /V′, concentration ni = mol of component i in the reactor n′i = molal flow rate of component i Vr = volume of reactor V′ = volumetric flow rate νi = stoichiometric coefficient ri = rate of reaction of substance i [mol/(unit time)(unit volume)] α,β = empirical exponents in a rate equation

Adapted from Walas, Chemical Process Equipment Selection and Design, Butterworth-Heinemann, 1990.

RATE EQUATIONS RATE OF REACTION The term rate of reaction means the rate of decomposition per unit volume, 1 dn ra = − ᎏ ᎏa , mol/(unit time) (unit volume) (7-1) V dt na0 dxa (7-2) = ᎏ ᎏ, n0 = na0(1 − xa) V dt where xa is the fractional conversion of substance A. A rate of formation will have the opposite sign. The negative sign is required for the rate of decomposition to be a positive number. When the volume is constant, dC ra = − ᎏa only at constant volume (7-3) dt Law of Mass Action The effect of concentration on the rate is isolated as ra = kf(Ca, Cb, . . .)

the rate equation is (7-6)

dCa ⇒−ᎏ at constant volume (7-7) dt The exponents (p, q, r, . . .) are empirical, but they are identical with the stoichiometric coefficients (νa, νb, νc, . . .) when the stoichiometric equation truly represents the mechanism of reaction. The first group of exponents identifies the order of the reaction, the stoichiometric coefficients the molecularity. Effect of Temperature The Arrhenius equation relates the specific rate to the absolute temperature, −E k = k0 exp ᎏ (7-8) RT

  B = exp A − ᎏ  T

CONCENTRATION, MOLES, PARTIAL PRESSURE, AND MOLE FRACTION Any property of a reacting system that changes regularly as the reaction proceeds can be formulated as a rate equation which should be convertible to the fundamental form in terms of concentration, Eq. (7-4). Examples are the rates of change of electrical conductivity, of pH, or of optical rotation. The most common other variables are partial pressure pi and mole fraction Ni. The relations between these units are nt pi ni = VCi = nt Ni = ᎏ (7-11) π where the subscript t denotes the total mol and π the total pressure. For ideal gases, nt RT V=ᎏ π

(7-4)

where the specific rate k is independent of concentration but does depend on temperature, catalysts, and other factors. The law of mass action states that the rate is proportional to the concentrations of the reactants. For the reaction νa A + νb B + νcC + . . . ⇒ νr R + νsS + . . . (7-5) 1 dn ra = − ᎏ ᎏa = kCap Cbq Ccr . . . V dt

last equation, the reaction is believed to have a complex mechanism (Fig. 7-1g).

(7-9)

B ln k = A − ᎏ (7-10) T E is called the activation energy and k0 the preexponential factor. When presumably accurate data deviate from linearity as stated by the

nt RT ni V πV (7-12) ni = ᎏ Ci = ᎏ pi = ᎏ pi = ᎏ Ni π π RT RT Other volume-explicit equations of state are sometimes required, such as the compressibility equation V = zRT/P or the truncated virial equation V = (1 + B′P)RT/P. The quantities z and B′ are not constants, so some kind of averaging will be required. More accurate equations of state are even more difficult to use but are not often justified for kinetic work. Designate δa as the increase in the total mol per mol decrease of substance A according to the stoichiometric equation Eq. (7-5): (νr + νs + . . .) − (νa + νb + νc + . . .) δa = ᎏᎏᎏᎏ (7-13) νa The total number of mols present is nt = nt0 + δa(na0 − na) = nt0 + δa xa = nt0 + δb xb = . . . (7-14) Accordingly, dn dn dn dn − ᎏt = δa ᎏa = δ b ᎏb = δc ᎏc = . . . dt dt dt dt The various differentials are 1 nt dni = d(VCi) = ᎏ d(Vpi) = d(nt Ni) = ᎏ dNi RT 1 + δi Ni The rate equation 1 dCa ra = − ᎏ ᎏ = kcCaα V dt

(7-15)

(7-16)

(7-17)

ln C

l/C q – 1

REACTION KINETICS

t r = kCq, slope = k(q – 1) (a)

t r = kC, slope = –k (b)

( b)

ln r

ln t1/2

(a)

ln C

ln C0 r = kCq, slope = 1 – q (d)

r = kCq, slope = q (c)

(c)

( d)

108

ln k

ln k4

107 106 105 104

l/T k = exp(A – E/RT), slope = –E/R (e) (e)

Range 800–3000 K 103 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1000/T Variable activation energy (f) ( f)

106 k

7-6

220 Oxidation of acetylene on manganese dioxide 200 catalyst 180 External diffusion 160 140 Pore diffusion 120 100 80 60 Chemical rate 40 20 10 12 14 16 18 20 22 24 26 28 30 10,000/T (g)

Constants of the power law and Arrhenius equations by linearization: (a) integrated equation, (b) integrated first order, (c) differential equation, (d) half-time method, (e) Arrhenius equation, ( f) variable activation energy, and (g) change of mechanism with temperature (T in K).

FIG. 7-1

RATE EQUATIONS can be expressed in terms of pressure and mole fraction, α

 p

1 d(Vpa ) 1 − ᎏ ᎏ = kc ᎏ RTV dt RT or at constant volume,

α a

 

(7-19)

1 α−1 (7-20) kp = kc ᎏ RT Typical Units of Specific Rates For order α, typical units are:

 

kc kp

(L/g mol)α − 1 ⋅s−1, and s−1 when first order (g mol)/L⋅ s⋅atmα

α−1

1 N   ᎏ 1+δ N 

α a

π α−1 1 = kc ᎏ ᎏ N aα RT 1 + δa Na Various derivatives are evaluated in numerical Example 1.



(7-21)

Consider the ideal gas reaction 2A ⇒ B + 2C occurring at 800°R, starting with 5 lb mol of pure A at 10 atm. The rate equation is

1 dn ra = − ᎏ ᎏa = 700 Ca2 lb mol/(ft3 ⋅h) V dt Evaluate the various rates of change at the time when the rate of reaction is ra = 0.1 lb mol/(ft3⋅h) and the reaction proceeds at (1) constant volume, and (2) constant pressure. 1 dn ra = − ᎏ ᎏa = 700 Ca2 = 0.1 lb mol/(ft3⋅h) V dt 0.1 ᎏ = 0.01195 lb mol/ft

 700

3

na 0 RT 5(0.729)(800) V0 = ᎏ = ᎏᎏ = 291.6 ft3 π0 10 5 na 0 Ca0 = ᎏ = ᎏ = 0.01715 lb mol/ft3 V0 291.6 nt = 0.5(3na 0 − na) lb mol (3na 0 − na) RT V = ᎏᎏ = 29.16(15 − na) ft3 2π0 n π = ᎏt π0 = 3na 0 − na atm nt 0 At constant volume, na = V0Ca = 291.6(0.01195) = 3.4853 lb mol dna dCa ᎏ = V0 ᎏ = −291.6(0.1) = −29.16 lb mol/h dt dt n 2na Na = ᎏa = ᎏ nt 3na 0 − na

 

dNa 6 dna 6 ᎏ = ᎏᎏ2 ᎏ = ᎏᎏ2 (−29.16) dt na 0(3 − na /na 0) dt 5(3 − 3.4853/5) = −6.598 h−1 na RT pa = ᎏ atm V0

 

V = 29.16 (15 − 3.8768) = 324.4 ft3 n na Ca = ᎏa = ᎏᎏ lb mol/ft3 V 29.16(15 − na )







2π0na pa = Naπ0 = ᎏ atm 3na 0 − na

Example 1: Rates of Change at Constant V or Constant P

Ca =

since

dNa 6na 0 dna 30 ᎏ = ᎏᎏ2 ᎏ = ᎏᎏ2 (−32.44) = −7.8658 h−1 dt (3na 0 − na) dt (15 − 3.8768)

a

  

dna ᎏ = −Vra = −324.4(0.1) = −32.44 lb mol/h dt

= 0.1349 lb mol/(ft3 ⋅h)

α a

or a



dπ 5 dna ᎏ = − ᎏ ᎏ = 29.16 atm/h dt na 0 dt At constant pressure, na = VCa = 29.16(15 − na) (0.01195) = 3.8768 lb mol



α

 N

dN n − ᎏa = kc ᎏt dt V



dCa 1 15 dna 1 15 ᎏ = ᎏ ᎏᎏ2 ᎏ = ᎏ ᎏᎏ2 (−32.44) dt 29.16 (15 − na) dt 29.16 (15 − 3.8768)

Furthermore, nt dNa nt − ᎏᎏ ᎏ = kc ᎏ V(1 + δaNa) dt V

n 3na 0 − na na π = ᎏt π0 = ᎏᎏ π0 = 5 3 − ᎏ atm nt 0 2na0 na 0

(7-18)

dpa 1 α−1 α pa = k p paα − ᎏ = kc ᎏ dt RT where the specific rate in terms of partial pressure is

7-7

dpa RT dna 0.729(800) ᎏ = ᎏ ᎏ = ᎏᎏ (−29.16) = −58.32 atm/h dt V0 dt 291.6

dpa 6π0 dna (6)(10)(5) ᎏ = ᎏᎏ2 ᎏ = ᎏᎏ2 (−32.44) = −78.66 atm/h dt (3na 0 − na) dt (15 − 3.8768) (3na 0 − na)RT 3 V = ᎏᎏ ft 2π0





dV RT dna 0.729(800) ᎏ = ᎏ − ᎏ = ᎏᎏ (32.44) = 945.95 ft3/h dt 2π0 dt 20 d na 0 − na 1 dxa 1 dna ᎏ = ᎏ ᎏ = − ᎏ ᎏ = − ᎏ (−32.44) = 6.488 h−1 dt dt na0 na0 dt 5





SUMMARY Rate

At constant V

At constant P

dna dt, lb mol/h dNa /dt, h−1 dpa /dt, atm/h dπ/dt, atm/h dV/dt, ft3/h dxa /dt, h−1

−29.16 −6.598 −58.32 29.16 0 5.832

−32.44 −7.866 −78.66 0 946.0 6.488

REACTION TIME IN FLOW REACTORS Flow reactors usually operate at nearly constant pressure, and thus at variable density when there is a change of moles of gas or of temperature. An apparent residence time is the ratio of reactor volume and the inlet volumetric flow rate, Vr t
app = ᎏ (7-22) V′0 The true residence time is obtained by integration of the rate equation, t
=

dn dV dn ᎏ =  ᎏ =  ᎏᎏ V′ V′r kV′(n/V′) r

q

(7-23)

The apparent time is readily evaluated and is popularly used to indicate the loading of a flow reactor. A related concept is that of space velocity, which is a ratio of a flow rate at STP (usually 60°F, 1 atm) to the size of the reactor. The most common versions in typical units are: GHSV (gas hourly space velocity) = (volumes of feed as gas at STP/h)/(volume of reactor or its content of catalyst) = SCFH gas feed/ft3. LHSV (liquid hourly space velocity) = (volume of liquid feed at 60°F/h)/(ft3 of reactor) = SCFH liquid feed/ft3. WHSV (weight hourly space velocity) = (lb feed/h)/(lb catalyst). It is usually advisable to spell out the units when the acronym is used, since the units are arbitrary.

7-8

REACTION KINETICS

CONSTANTS OF THE RATE EQUATION The problem is to apply experimental data to find the constants of assumed rate equations, of which some of the simpler examples are: dC r = − ᎏ = kCq dt

(7-24)





(7-25)

dCa q (7-26) ra = − ᎏ = kCap Cb dt Experimental data that are most easily obtained are of (C, t), (p, t), (r, t), or (C, T, t). Values of the rate are obtainable directly from measurements on a continuous stirred tank reactor (CSTR), or they may be obtained from (C, t) data by numerical means, usually by first curve fitting and then differentiating. When other properties are measured to follow the course of reaction—say, conductivity—those measurements are best converted to concentrations before kinetic analysis is started. The most common ways of evaluating the constants are from linear rearrangements of the rate equations or their integrals. Figure 7-1 examines power law and Arrhenius equations, and Fig. 7-2 has some more complex cases. From the Differential Equation Linear regression can be applied with the differential equation to obtain constants. Taking logarithms of Eq. (7-25), or

b ln r = a − ᎏ + q ln C (7-27) T The variables that are combined linearly are ln r, 1/T, and ln C. Multilinear regression software can be used to find the constants, or only three sets of the data suitably spaced can be used and the constants found by simultaneous solution of three linear equations. For a linearized Eq. (7-26) the variables are logarithms of r, Ca, and Cb. The logarithmic form of Eq. (7-24) has only two constants, so the data can be plotted and the constants read off the slope and intercept of the best straight line. From the Integrated Equation The integral of Eq. (7-24) is 1 C k = ᎏ ln ᎏ0 , t − t0 C

when q = 1

(7-28)

C0 − 1 C0 q − 1 − 1 , when q ≠ 1 (7-29) ᎏᎏ ᎏ (t − t0) (q − 1) C A value of q is assumed and values of k are calculated for each data point. The correct value of q has been chosen when the values of k are nearly constant or show no drift. This procedure is applicable for a rate equation of any complexity if it can be integrated. Eqs. (7-28) and (7-29) can also be put into linear form: q

 

 

C ln ᎏ0 = k(t − t0), C q−1



when q = 1

(7-30)

1 q−1 = ᎏ + k(q − 1) (t − t0), when q ≠ 1 (7-31) C0 When the plots are collinear, the correct value of k is found from the slope of the best straight line. From Half-Times The time by which one-half of the reactant has been converted is called the half-time. From Eq. (7-24),

ᎏC  1

 

kt1/2 = ln 2,

[r − f (C , a, b, . . .)] i

dC b r = − ᎏ = exp a − ᎏ Cq dt t

or

Complex Rate Equations Complex rate equations may require individual treatment, although the examples in Fig. 7-2 are all linearizable. A perfectly general procedure is nonlinear regression. For instance, when r = f(C, a, b, . . .) where (a, b, . . .) are the constants to be found, the condition is

q=1

2q − 1 − 1 q≠1 (7-32) ᎏᎏ q (q − 1)C0 − 1 When several sets of (C0, t1/2) are known, values of q are tried until one is found that makes all k values substantially the same. Alternatively, the constants may be found from a linearized plot, 2q − 1 − 1 ln t1/2 = ln ᎏ + (1 − q) ln C0 (q − 1)k

(7-33)

2

i

⇒ Minimum

(7-34)

∂Σ ∂Σ (7-35) ᎏ=ᎏ=...=0 ∂a ∂b Much professional software is devoted to this problem. A diskette for sets of differential and algebraic equations with parameters to be found by this method is by Constantinides (Applied Numerical Methods with Personal Computers, McGraw-Hill, 1987). The acquisition of kinetic data and parameter estimation can be at quite a sophisticated level, particularly for solid catalytic reactions: statistical design of experiments, refined equipment, computer monitoring of data acquisition, and statistical evaluation of the data. Two papers are devoted to this topic by Hofmann (in Chemical Reaction Engineering, ACS Advances in Chemistry, 109, 519–534 [1972]; in de Lasa, ed., Chemical Reactor Design and Technology, Martinus Nijhoff, 1985, pp. 69–105).

and

MULTIPLE REACTIONS AND STOICHIOMETRIC BALANCES Single Reaction For the stoichiometric equation, Eq. (7-5), the relations between the conversions of the several participants are x na0 − na nb0 − nb nr0 − nr ns0 − ns ᎏ=ᎏ=ᎏ=...=−ᎏ=−ᎏ=... νa νa νb νr νs n na0 − x Ca = ᎏa = ᎏ , V V

nb0 − νb x/νa Cb = ᎏᎏ , V

(7-36)

Cc0 − νc x/νa Cc = ᎏᎏ , and so on V (7-37)

ν Cb = Cb0 − ᎏb (Ca0 − Ca) νa

Also,

νc (7-38) Cc = Cc0 − ᎏ (Ca0 − Ca), and so on νa Accordingly, the rate equation can be written in terms of the single dependent variable x; thus, 1 dn 1 dx ra = − ᎏ ᎏa = ᎏ ᎏ V dt V dt na0 − x p nb0 − νb x/νa =k ᎏ ᎏᎏ V V and in terms of concentrations,





n − ν x /ν . . .  ᎏᎏ  V q

c0

c

a

r

(7-39)

dC q ra = − ᎏa = kCap Cb Ccr . . . dt q r νb ν Cc0 − ᎏc (Ca0 − Ca) . . . (7-40) = kCap Cb0 − ᎏ (Ca0 − Ca) νa νa Eq. (7-39) becomes integrable when V is properly expressed in terms of the composition of the system, and Eq. (7-40) can be integrated as it stands. Multiple Reactions When a substance participates in several reactions at the same time, its net rate of decomposition is the algebraic sum of its rates in the individual reactions. Identify the rates of the individual steps with subscripts, (dC/dt)1, (dC/dt)2, . . . . Take this case of three reactions,





1

A+B⇒C 2

A+C⇒D+E 3

D+E⇒A+C



RATE EQUATIONS 12

6

11

5 4

9

C/r

t/(C0 – C)

10

3

8 2

7 6

1

y = 5.03 + 0.99×

C/r = 0.72 + 5.73C 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 C (b)

5 1

2

3 4 5 In (C0/C)/(C0 – C)

6

7

(a) (a )

(b)

20 19

110

18

100

17 y2

90

y2

15

80

y1

y1

16 14

70

13 12 11 10

60 50 0

1

2

3

4

5

x1, x2

(c)) (c 9.00

–3.25

Regression data Calculated value

7.80

–3.75 –4.25

5.40

–4.75

4.20

–5.25

y

6.60

3.00 0

2

4

6 8 Point no. (d)

10

–5.75 0

12

2

4

6 8 Point no. (e)

10

12

FIG. 7-2 Linear analysis of catalytic rate equations. (a), (b) Sucrose hydrolysis with an enzyme, r = kM/(M + C). Data are (C, t) curve-fitted with a fourth-degree polynomial and differentiated for r − (−dC/dt). Integrated equation, t 1 M ln (C0 /C) k = 0.199, M = 4.98 ᎏ = ᎏ + ᎏ ᎏ, C0 − C k k C0 − C Linearized rate equation, C M C M = 4.13 ᎏ = ᎏ + ᎏ, r k k poor agreement. (c) For a solid catalyzed reaction, two possible equations in linear form are y1 = Pa /r = a + bpa and y2 = Pa /r
= a + bpa, of which the second appears to fit. (d), (e) Hydrogenation of octenes, Hougen and Watson (Chemical Process Principles, Wiley, 1947, p. 943). The hyperbolic and power law fits are of about equal quality. Pressure in atm; r in lb mol/(ft3 ⋅h).

y=

PP ᎏ = a + bp + cp + dp

 r u h

u

s

h

= 2.7655 + 1.5247pu + 1.0092ps + 1.1291ph ln r = ln k + a ln Pu + b ln ps + c ln ph = −4.059 + 0.469 ln pu − 0.2356 ln ps + 0.5997 ln ph r = 0.0173 P u0.469 p −0.2356 P h0.5997 s

7-9

7-10

REACTION KINETICS

The overall rates of the several participants are

dC ᎏ = 3k1AB2 − k2AC − k3CD dt

ra = ra1 + ra2 + ra3 = −k1CaCb − k2CaCc + k3CdCe

= 3k1AB2 − C{k2A + k3[D0 + 3(A0 − A) − 3(B0 − B)]}

rb = −k1CaCb rc = k1CaCb − k2CaCc + k3CdCe rd = re = k2CaCe − k3CdCe

(7-41)

The number of independent rate equations is the same as the number of independent stoichiometric relations. In this example, reactions 2 and 3 are reversible and are not independent, so there are only two independent rate equations. Some reactions apparently represented by single stoichiometric equations are in reality the result of several reactions, often involving short-lived intermediates. After a set of such elementary reactions is postulated by experience, intuition, and exercise of judgment, a rate equation is deduced and checked against experimental rate data. Several examples are given under “Mechanisms of Some Complex Reactions,” following. Stoichiometric Balances The amounts of all participants in a group of reactions can be expressed in terms of a number of key components equal to the number of independent stoichiometric relations. The independent rate equations will then involve only those key components and will be, in principle, integrable. For a single equation, Eqs. (7-36) and (7-37) relate the amounts of the several participants. For multiple reactions, the procedure for finding the concentrations of all participants starts by assuming that the reactions proceed consecutively. Key components are identified. Intermediate concentrations are identified by subscripts. The resulting concentration from a particular reaction is the starting concentration for the next reaction in the series. The final value carries no subscript. After the intermediate concentrations are eliminated algebraically, the compositions of the excess components will be expressible in terms of the key components.

MECHANISMS OF SOME COMPLEX REACTIONS The rates of many reactions are not represented by application of the law of mass action on the basis of their overall stoichiometric relations. They appear, rather, to proceed by a sequence of first- and secondorder processes involving short-lived intermediates which may be new species or even unstable combinations of the reactants; for 2A + B ⇒ C, the sequence could be A + B ⇒ AB followed by A + AB ⇒ C. Free radicals are molecular fragments having one or more unpaired electrons, usually short-lived (milliseconds) and highly reactive. They are detectable spectroscopically and some have been isolated. They occur as initiators and intermediates in such basic phenomena as oxidation, combustion, photolysis, and polymerization. The rate equation of a process in which they are involved is developed on the postulate that each free radical is at equilibrium or its net rate of formation is zero. Several examples of free radical and catalytic mechanisms will be cited, all possessing nonintegral power law or hyperbolic rate equations. Phosgene Synthesis CO + Cl2 ⇒ COCl2, but with the sequence: Cl2 ⇔ 2Cl• Cl• + CO ⇔ COCl• COCl• + Cl2 ⇒ COCl2 + Cl•

1

Assuming the first two reactions to be in equilibrium, an expression is found for the concentration of COCl• and when this is substituted into the third equation the rate becomes

2

rCOCl 2 = k(CO)(Cl2)3/2

Example 2: Analysis of Three Simultaneous Reactions Consider the three reactions A + 2B ⇒ 3C A + C ⇒ 2D C + D ⇒ 2E

Cl2 + O3 ⇒ ClO• + ClO•2

with A, B, and C the key components. Apply Eq. (7-37), B0 − B C1 − C0 A0 − A1 = ᎏ =ᎏ 2 3

ClO•2 + O3 ⇒ ClO•3 + O2 ClO•3 + O3 ⇒ ClO•2 + 2O2

D2 − D0 A1 − A = C1 − C2 = ᎏ 2

ClO•3 + ClO•3 ⇒ Cl2 + 3O2

E − E0 (7-42) C2 − C = D2 − D = ᎏ 2 Elimination of the concentrations with subscripts 1 and 2 will find D and E in terms of A, B, and C, with the same results that are achieved by the following method. This alternative procedure is called the xyz method. The amount of change by the first reaction is x, by the second y, and by the third z. For the same example, A = A0 − x − y

(7-47)

Hydrogen Bromide H2 + Br2 ⇒ 2HBr (Bodenstein, 1906). The chain of reactions is: 1

Br2 ⇒ 2Br• 2

Br + H2 ⇒ HBr •

H• + Br2 ⇒ HBR + Br•

C = C0 + 3x − y − z

Br• + Br• ⇒ Br2 (7-43)

Elimination of x, y, and z gives for the excess components: D0 − D = −3(A0 − A) + 3(B0 − B) E0 − E = 2(A0 − A) − 4(B0 − B) − 2(C0 − C)

rO3 = k(Cl 2)1/ 2 (O3)3/ 2

H• + HBr ⇒ H2 + Br•

D = D0 + 2y − z

The differential equations for the three key components become: dA ᎏ = −k1AB2 − k2AC dt

The chain carriers ClO•2 and ClO•3 are assumed to attain steady state. Then,

3

B = B0 − 2x

E = E0 + 2z

(7-46)

Ozone and Chlorine The assumed sequence is:

3

dB ᎏ = −2k1AB2 dt

(7-45)

These equations will have to be solved numerically for A, B, and C as functions of time; then D and E can be found by algebra. Alternatively, five differential equations can be written and solved directly for the five participants as functions of time, thus avoiding the use of stoichiometric balances, although these are really involved in the formulation of the differential equations.

(7-44)

Assuming equilibrium for the concentrations of the free radicals, the rate equation becomes d(HBr) ᎏ = k1(Br•)(H2) + k2(H•)(Br2) − k3(H•)(HBr) dt k1(H2)(Br2)3/ 2 = ᎏᎏ (7-48) k2(Br2) + k3(HBr) Enzyme Kinetics The enzyme E and the reactant S are assumed to form a complex ES that then dissociates into product P and uncombined enzyme.

RATE EQUATIONS 1

S+E⇔ ES 2 3

ES ⇒ E + P If equilibrium holds, (S)(E) (S)[(E0) − (ES)] ᎏ = ᎏᎏ = K m (ES) (ES) where (E0) is the total of the free and combined enzyme and K m is a dissociation constant. Solve for (ES) and substitute into the rate equation, d(P) k(E0)(S) rp = ᎏ = k(ES) = ᎏ (7-49) dt K m + (S) This hyperbolic equation is named after Michaelis and Menten (Biochem. Zeit., 49, 333 [1913]). Chain Polymerization The growth process of a polymer postulates a three-step mechanism: 1. An initiator I generates a free radical R• 2. The free radical reacts repeatedly with monomer by a process called propagation. 3. The free radical eventually disappears by some reaction, called termination. The stoichiometric equations are 1

I ⇒ 2R• 2

R + M ⇒ RM•, •

initiation

kp

RM• + M ⇒ RM2• or

kp

RMn• + M ⇒ RM•n + 1,

propagation

kt

RM + RM ⇒ R2Mn + m • n

or

• m

RMn + RMm,

termination

The rates of formation of the free radicals R• and M• reach steady states, dR•n ᎏ = 2k1(I) − k2(R•)(M) = 0 dt dM• ᎏ = k2(R•)(M) − 2kt(M•)2 = 0 dt These equations are solved for (R•) and (M•) and substituted into the propagation equation. The rate of polymerization becomes dM k 1/2 rp = − ᎏ = kp(M•)(M) = kp ᎏ1 (M)(I)1/2 (7-50) dt kt Thus, the process of chain polymerization is first-order with respect to monomer and half-order with respect to initiator. Solid Catalyzed Reaction The pioneers were Langmuir ( J. Am. Chem. Soc., 40, 1361 [1918]) and Hinshelwood (Kinetics of Chemical Change, Oxford, 1940). For a gas phase reaction A + B ⇒ Products, catalyzed by a solid, the postulated mechanism consists of the following: 1. The reactants are first adsorbed on the surface, where they subsequently react and the product is desorbed. 2. The rate of adsorption is proportional to the partial pressure and to the fraction of uncovered surface ϑv. 3. The rate of desorption of A is proportional to the fraction ϑa of the surface covered by A. 4. Adsorptive equilibrium is maintained. 5. The rate of reaction between adsorbed species is proportional to their amounts on the surface. The net rates of adsorption are:

 

ra = ka paϑv − k−aϑa ⇒ 0 Substitute ϑv = 1 − ϑa − ϑb and solve for the coverages: ka ϑa = ᎏ paϑv = Ka paϑv k−a

  k ϑ = ᎏ p ϑ = K p ϑ k b

b

−b

v

b

b

y=

(7-51)

pp 1+KP +K p ᎏ = ᎏᎏ

 r

K
kK a

b

a a

b

a

b

b

More about this topic is presented later. WITH DIFFUSION BETWEEN PHASES When reactants are distributed between several phases, migration between phases ordinarily will occur: with gas/liquid, from the gas to the liquid; with fluid/solid, from the fluid to the solid; between liquids, possibly both ways because reactions can occur in either or both phases. The case of interest is at steady state, where the rate of mass transfer equals the rate of reaction in the destined phase. Take a hyperbolic rate equation for the reaction on a surface. Then, r = rd = rs k2(C − r/k1) k2Cs = k1(C − Cs) = ᎏ = ᎏᎏ (7-52) 1 + k3Cs 1 + k3(C − r/k1) The unknown intermediate concentration Cs has been mathematically eliminated from the last term. In this case, r can be solved for explicitly, but that is not always possible with surface rate equations of greater complexity. The mass transfer coefficient k1 is usually obtainable from correlations. When the experimental data are of (C, r) the other constants can be found by linear plotting. CATALYSIS BY SOLIDS: LANGMUIR-HINSHELWOOD MECHANISM A plausible mechanism of solid catalytic reactions is that the participants chemisorb on the surface and react while in the adsorbed state. The process of adsorption of A on an active site of the surface σ is represented by A + σ ⇒ Aσ and the reaction between adsorbed molecules, for instance, by Aσ + Bσ ⇒ Cσ + Dσ Adsorptive Equilibrium The fraction of the surface covered by A at equilibrium is ϑa = Ka paϑv

(7-53)

1 ϑv = ᎏᎏᎏᎏ 1 + Ka pa + Kb pb + KcPc + Kd pd + . . .

(7-54)

where terms may be added for adsorbed inerts that may be present, and analogous expressions for the other participants. The rate of reaction between species in adsorptive equilibrium is then r = kpa pbϑv2

(7-55)

Dissociation A diatomic molecule A2 may adsorb as atoms, A2 + 2σ ⇒ 2Aσ with the result,
a
pa
K ϑa = ᎏᎏᎏ = K

apa
ϑv
a
1 + K pa
+ Kbpb + . . .

rb = kb pbϑv − k−bϑb ⇒ 0

b

1 ϑv = ᎏᎏ 1 + Ka pa + K b pb The rate of surface reaction is: kKaKb papb r = kϑaϑb = ᎏᎏ2 (1 + Ka pa + K b pb) The linearized form can be used to find the constants,

7-11

v

and the rate of the reaction is 2Aσ + Bσ ⇒ Products r = k′ϑa2 ϑb = kpa pbϑv3

(7-56)

Different Sites When A and B adsorb on chemically different sites σ1 and σ2, the rate of the reaction A + B ⇒ Unadsorbed products

7-12

REACTION KINETICS

is

Walas (Reaction Kinetics for Chemical Engineers, McGraw-Hill, 1959; Butterworths, 1989, pp. 153–164), and Rase (Chemical Reactor Design for Process Plants, vol. 1, Wiley, 1977, pp. 178–191). All of the relations developed here assume that only one step is controlling. A more general case is that of the reaction A ⇒ B with five steps controlling, namely

kpa pb r = ᎏᎏᎏ (7-57) (1 + Kapa)(1 + Kbpb) Dual Sites When the numbers of moles of reactants and products are unequal, A ⇔ M + N, the mechanism is assumed to be

r = k1(pag − pai)

Aσ + σ ⇔ Mσ + Nσ

θa r = k2 paiθv − ᎏ k3



and the rate ϑmϑn pmpn r = k ϑaϑv − ᎏ = k pa − ᎏ ϑv2 K K



 





Adsorption of A

r = k4θa

k(pa − pmpn /K) = ᎏᎏᎏ (7-58) (1 + Kapa + Kmpm + Knpn)2 Reactant in the Gas Phase When A in the gas phase reacts with adsorbed B,

Surface reaction

θb r = k5 pbiθv − ᎏ k6





Desorption of B

r = k7(pbi − pbg)

Diffusion of B from the surface

(7-62)

where θv = 1 − θa − θb . At steady state these rates are all the same. Upon elimination of the unmeasurable quantities pai, pbi, ϑa, ϑb, and ϑv, the relation becomes k5(k7 pbg + r) r r = ᎏᎏ 1−ᎏ k7 k4

A + Bσ ⇒ Products kpa pb r = kpaϑb = kpapbϑv = ᎏᎏ (7-59) (1 + ΣKi pi) Chemical Equilibrium When A is not in adsorptive equilibrium, it is assumed to be in chemical equilibrium, with p*a = pmpn /Ke pb. This expression is substituted for pa wherever it appears in the rate equation. Then













r 1 r k1r 1 1 − k5 pbg + ᎏ + ᎏ 1 − ᎏ − ᎏ (7-63) ᎏ+ᎏ k7 k6 k4 k1 pag − r k2 k3 k4 Combinations of several adsorption and surface reaction steps are usually not felt to be necessary, since so many alternatives are available individually. Single steps in combination with diffusion to the surface are usually adequate, as in the case leading to Eq. (7-52). Over the usual limited range of conditions, a power law rate equation often appears to be as satisfactory a fit of the data as a more complex Langmuir-Hinshelwood equation. The example of the hydrogenation of octenes is shown in Fig. 7-2d and 7-2e, and another case follows.

kpm pn/Ke r = kp*a pb ϑv2 = ᎏᎏᎏᎏᎏ (7-60) (1 + K a pmpn /K e pb + K b pb + K m pm + K n pn)2 All of these relations are brought together in the fundamental form (kinetic term)(driving force) r = ᎏᎏᎏ (7-61) adsorption term Table 7-2 summarizes the cases when all substances are in adsorptive equilibrium and the surface reaction controls. In Table 7-3, substance A is not in adsorptive equilibrium, so its adsorption rate is controlling. Details of the derivations of these and some other equations are presented by Yang and Hougen (Chem. Eng. Prog., 46, 146 [1950]),

TABLE 7-2

Diffusion of A to the surface

Example 3: Phosgene Synthesis Rate data were obtained by Potter and Baron (Chem. Eng. Prog., 47, 478 [1951]) for the reaction CO (A) + Cl2 (B2) ⇒ COCl2 (C) at 30.6. Three correlations of approximately equal statis-

Surface-reaction Controlling (Adsorptive Equilibrium Maintained of All Participants)

Reaction 1. A → M + N A→M+N A→M+N

Special condition General case Sparsely covered surface Fully covered surface

2. A A M

Basic rate equation

Driving force

Adsorption term

r = kθa r = kθa r = kθa

pa pa 1

1 + Kapa + Kmpm + Knpn 1 1

r = k1θa − k−1θm

pm pa − ᎏ K

1 + Kapa + Kmpm (1 + Kapa + Kmpm + Knpn)2

3. A A M + N

Adsorbed A reacts with vacant site

r = k1θaθv − k−1θmθn

pmpn pa − ᎏ K

4. A2 A M

Dissociation of A2 upon adsorption

r = k1θa2 − k−1θmθv

5. A + B → M + N A+B→M+N

Adsorbed B reacts with A in gas but not with adsorbed A

r = kθaθb r = kpaθb

pm pa − ᎏ K papb papb

2

(1 + K ap
a + Kmpm)

(1 + Kapa + Kbpb + Kmpm + Knpn)2 1 + Kapa + Kbpb + Kmpm + Knpn

6. A + B A M

r = k1θaθb − k−1θmθv

pm papb − ᎏ K

(1 + Kapa + Kbpb + Kmpm)2

7. A + B A M + N

r = k1θaθb − k−1θmθn

pmpn papb − ᎏ K

(1 + Kapa + Kbpb + Kmpm + Knpn)2

r = k1θa2θb − k−1θmθnθv

pmpn papb − ᎏ K

3

(1 + K apa
+ Kbpb + Kmpm + Knpn)

8. A2 + B A M + N NOTE:

Dissociation of A2 upon adsorption

The rate equation is: k (driving force) r = ᎏᎏ adsorption term When an inert substance I is adsorbed, the term Kipi is to be added to the adsorption term. SOURCE: From Walas, Reaction Kinetics for Chemical Engineers, McGraw-Hill, 1959; Butterworths, 1989.

RATE EQUATIONS TABLE 7-3

7-13

Adsorption-rate Controlling (Rapid Surface Reaction)

Reaction

Special condition

Basic rate equation

1. A → M + N

r = kpaθv

2. A A M

θa r = k paθv − ᎏ Ka

3. A A M + N

θa r = k paθv − ᎏ Ka

 

Driving force

Adsorption term

pa

Kapmpn 1+ᎏ + Kmpm + Knpn K



pm pa − ᎏ K

Kapm 1+ᎏ + Kmpm K



pmpn pa − ᎏ K

Kapmpn 1+ᎏ + Kmpm + Knpn K

pm pa − ᎏ K

ᎏ+K p  1 +  K

pa

Kapmpn 1 + ᎏ + Kbpb + Kmpm + Knpn Kpb

4. A2 A M

Dissociation of A2 upon adsorption

θa2 r = k paθv2 − ᎏ Ka

5. A + B → M + N

Unadsorbed A reacts with adsorbed B

r = kpaθv





2

Kapm

m m

6. A + B A M

θa r = k paθv − ᎏ Ka



pm pa − ᎏ Kpb

Kn pm 1 + ᎏ + Kbpb + Kmpm Kpb

7. A + B A M + N

θa r = k paθv − ᎏ Ka



pmpn pa − ᎏ Kpb

Kapmpn 1 + ᎏ + Kbpb + Kmpm + Knpn Kpb

θa2 r = k paθv2 − ᎏ Ka

pmpn pa − ᎏ Kpb

ᎏ+Kp +K p 1 +  Kp

8. A2 + B A M + N

 

Dissociation of A2 upon adsorption





Kapmpn

b b

m m



+ Knpn

b

2

NOTES:

The rate equation is: k (driving force) r = ᎏᎏ adsorption term Adsorption rate of substance A is controlling in each case. When an inert substance I is adsorbed, the term Kipi is to be added to the adsorption term. SOURCE: From Walas, Reaction Kinetics for Chemical Engineers, McGraw Hill, 1959; Butterworths, 1989.

tical validity are:

Aσ + 2Bσ ⇒ C + 3σ papb 1/3
b
− 0.00046pc) y= ᎏ = 0.34(1 − 0.061pa + 0.0032p r 2. Aσ + B2σ ⇒ C + 2σ papb 1/2 y= ᎏ = 2.38(1 + 1.98pb + 0.59pc) r 0.58 −0.68 3. r = 0.02p1.33 a pb pc The data are partial pressures, atm and the rate r, g mol phosgene made/(h⋅g catalyst). The first is ruled out because the constants physically cannot be negative. Although the other correlations are equally valid statistically, the Langmuir-Hinshelwood may be preferred to the power law form because it is more likely to be amenable to extrapolation. 1.

   

CHEMICAL EQUILIBRIUM

d ln K ∆Hr ᎏ = ᎏ2 dT RT This is integrable to 1 ln K = ln K298 + ᎏ R



T

∆Hr298 + 298 ∆Cp dT dT ᎏᎏᎏ 298 T2



T

(7-65)

where ∆Hr is the enthalpy change of reaction. Over a moderate temperature range, an adequate form of relation is





b K = exp a + ᎏ (7-66) t Gaseous equilibria are expressed in terms of fugacities or fugacity coefficients. In terms of partial pressures, pi = yiπ,

The rate of a reversible reaction k1

aA + bB ⇔ cC + dD k2

may be written



found by simultaneous solution of the several equations. The equilibrium composition of a mixture of known chemical species also can be found by a process of Gibbs energy minimization without the formulation of stoichiometric equations. Examples of the calculation of equilibria are in books on thermodynamics and in Walas (Phase Equilibria in Chemical Engineering, Butterworths, 1985). The equilibrium constant depends on temperature according to



Ccc Cdd r = k1 Caa Cbb − ᎏ (7-64) Ke In terms of the compositions at equilibrium, the equilibrium constant is d Ccec Cde Ke = ᎏ a b Cae Cbe With the aid of the stoichiometric “degree of advancement,” Ca0 − Ca Cb0 − Cb CC0 − Cc Cd0 − Cd ε=ᎏ=ᎏ=−ᎏ=−ᎏ a b c d the equilibrium constant can be written in terms of a single variable. When several reactions occur simultaneously, each reaction is characterized by its own εi. When the Kes are known, the composition can be

pcc pdd Kp = ᎏ = K yπ c + d − a − b (7-67) paa pbb Pressure affects the composition of an equilibrium mixture, but not the equilibrium constant itself. Although the equilibrium constant can be evaluated in terms of kinetic data, it is usually found independently so as to simplify finding the other constants of the rate equation. With K e known, the correct exponents of Eq. (7-64) can be found by choosing trial sets until k1 comes out approximately constant. When the exponents are small integers or simple fractions, this process is not overly laborious. Example 4: Reaction between Methane and Steam At 600°C the principal reactions between methane and steam are CH4 + H2O ⇔ CO + 3H2

CO +

1−x 5−x

x−y 5−x−y

x

3x

H2O ⇔ CO2 + H2 y

3x + y

Σ = 6 + 2x

7-14

REACTION KINETICS

where K1 = 0.574, K2 = 2.21. Starting with 1 mol methane and 5 mol steam, (x − y)(3x + y)3 y(3x + y) ᎏᎏᎏ2 = 0.574, ᎏᎏ = 2.21 (1 − x)(5 − x − y)(6 + 2x) (x − y)(5 − x − y) Simultaneous solution by the Newton-Raphson method yields x = 0.9121, y = 0.6328. Accordingly, the fractional compositions are: (1 − x) CH4 = ᎏ = 0.0112 (6 + 2x) (x − y) CO = ᎏ = 0.0357 (6 + 2x) y CO2 = ᎏ = 0.0809 (6 + 2x) (5 − x − y) H2O = ᎏᎏ = 0.4416 (6 + 2x)

Example 5: Percent Approach to Equilibrium For a reversible reaction with rate equation r = k[A2 − (1 − A)2/16], the size function kVr /V′ of a plug flow reactor will be found in terms of percent approach to equilibrium: 1 dA kVr , Aequilib = 0.2000 ᎏ = ᎏᎏ 2 2 A A − (1 − A) /16 V′



Percent approach

70

90

95

98

99

99.5

100

A kVr /V′

0.440 1.319

0.280 3.053

0.240 4.309

0.216 6.090

0.208 7.600

0.204 9.315

0.2 ∞

The volume escalates rapidly at high percent approaches.

(3x + y) H2 = ᎏ = 0.4306 (6 + 2x)

INTEGRATION OF RATE EQUATIONS

Approach to Equilibrium As equilibrium is approached the rate of reaction falls off, and the reactor size required to achieve a specified conversion goes up. At some point, the cost of increased reactor size will outweigh the cost of discarded or recycled unconverted material. No simple rule for an economic appraisal is really possible, but sometimes a basis of 95 percent of equilibrium converTABLE 7-4

sion is taken. For adiabatic operation, a certain approach to equilibrium temperature is common practice, say within 10 to 20°C (18 to 36°F), a number possibly based on experience with a particular process.

In either batch or flow systems, many single-rate equations lead to integrands that are ratios of low-degree polynomials that can be integrated by inspection or with the briefest of integral tables. Some of the cases of frequent occurrence are summarized in Table 7-4. When the problem is to relate C and t, the constants are known, and the polynomials are of second degree or higher, numerical integration may save

Some Isothermal Rate Equations and Their Integrals

1. A → Products: dA − ᎏ kAq dt A q=1 ᎏ = exp [−k(t − t0)], A0 1 ᎏᎏᎏ q−1 1 + (q − 1)kA0 (t − t0)



RT RT ∆ν V = nt ᎏ = nt0 + ᎏ (na0 − na) ᎏ P νa P



1/(q − 1)

,

k(t − t0) =

q≠1

2. A + B → Products: dA − ᎏ = kAB = kA(A + B0 − A0) dt 1 A0(A + B0 − A0) k(t − t0) = ᎏ ln ᎏᎏ B0 − A0 AB0

k1

k1

4. Reversible reaction, second order, A + B A R + S: k2

SOURCE:

1 ᎏn1 − ᎏ n   , when α = 2 a

a0

k2

These linear equations are solved for the transforms as D = s2 + (k1 + k2 + k3)s + k1k2

q=0

2αA + β + q ᎏᎏ , ᎏᎏ 2αA + β + q  2αA + β − q 

dA − ᎏ = k1A − k2B dt dB − ᎏ = −k1A + (k2 + k3)B dt dC − ᎏ = −k2B dt (s + k1)A
+ k3B
= A0 −k1A
+ (s + k2 + k3)B
= B0 −k2B
+ sC
= C0

q≠0

0

5. The reaction νaA → νrR + νsS between ideal gases at constant T and P: knαa dn − ᎏa = ᎏ dt Vα − 1



Laplace transformations are made and rearranged to

dA − ᎏ = k1AB − k2RS = k1A(A + B0 − A0) dt − k2(A0 + R0 − A)(A0 + S0 − A) = αA2 + βA − γ α = k1 − k2 β = k1(B0 − A0) + k2(2A0 + R0 + S0) γ = k2(A0 + R0)(A0 + S0)
2
+

α q = β 4
γ


2αA0 + β − q

RT ∆ν ᎏ nb0 + ᎏ P νa na0 ∆ν − ᎏ ln ᎏ νa na

in general

Rate equations are

k2

1 ᎏ ln q

V ᎏ dna, nαa

k3

dA − ᎏ = k1A − k2(A0 + B0 − A) = (k1 + k2)A − k2(A0 + B0) dt k1A0 − k2B0 (k1 + k2)(t − t0) = ln ᎏᎏᎏ (k1 + k2)A − k2(A0 + B0)

{

na

AA B →C

k1

k(t − t0) =





α−1

6. Equations readily solvable by Laplace transforms. For example:

3. Reversible reaction A AB:

2αA0 + β ᎏ, 2αA + β





na0

A0s + (k2 + k3)A0 + K3B0 A
= ᎏᎏᎏ D B0s + k1(A0 + B0) B
= ᎏᎏ D k2
B + C0 C
=ᎏ s Inversion of the transforms can be made to find the concentrations A, B, and C as functions of the time t.

Adapted from Walas, Chemical Process Equipment Selection and Design, Butterworth-Heinemann, 1990.

IDEAL REACTORS time and preserve reliability. Some 40 cases of integrations at constant volume are developed by Capellos and Bielski (Kinetic Systems Mathematical Descriptions of Chemical Kinetics, Wiley, 1972). Sets of first-order rate equations are solvable by Laplace transform (Rodiguin and Rodiguina, Consecutive Chemical Reactions, Van Nostrand, 1964). The methods of linear algebra are applied to large sets of coupled first-order reactions by Wei and Prater (Adv. Catal., 13, 203 [1962]). Reactions of petroleum fractions are examples of this type. Example 6: Laplace Transform Application For the reaction A ⇒B ⇔C 1

2

3

B0 = C0 = 0 dA ᎏ = k1A dt

with the rate equations are

sA
− A0 = −k1A


The transforms are

k3A0 sB A − (k2 + k3)
B
= ᎏ + (k1 − k3)
s Explicitly,

A0 A
=ᎏ s + k1

The reactions are 2A ⇒ B ⇒ C. The partial solutions are A0 A = ᎏᎏ (1 + 2k1A0 t) dB k1A20 ᎏ + k2B = ᎏᎏ2 dt (1 + 2k1 A0t) Although the differential equation is first-order linear, its integration requires evaluation of an infinite series of integrals of increasing difficulty. 1 2 2. The reactions are A ⇒ B and A + B ⇒ C. After A is expressed in terms of B by elimination of t, dB B−k k k = ᎏ1 ᎏ = (k1 − k2B) A0 − B0 + B − 2k ln ᎏ , dt B+k k2 but this cannot be integrated analytically. 1 2 1 2 1 3. For the reactions A ⇒ B, 2B ⇒ C; 2A ⇒ B ⇒ C; 2A ⇒ B, 2 2B ⇒ C; the rate equations are solved in terms of higher transcendental functions by Chien ( J. Am. Chem. Soc., 76, 2256 [1948]). For the first case, with B0 = 0: A = exp (−k1t) 1

1.

2



dB ᎏ = k1A − k2B + k3(A0 − A − B) dt

k3 k1 − k3 A0 B
= ᎏᎏ ᎏ + ᎏ s + k2 + k3 s s + k1









where

A and B as functions of t are found by inversion with a table of L-T pairs.

When even second-order reactions are included in a group to be analyzed, individual integration methods may be needed. Three cases of coupled first- and second-order reactions will be touched on. All of them are amenable only with difficulty to the evaluation of specific rates from kinetic data. Numerical integrations are often necessary.

7-15

τ iJ1(γ) − βH1 (γ) B = A0 ᎏ ᎏᎏ K J0(γ) + β i H0(i) τ = exp (−k1t) K = k1k2 A0
τ
γ = 2iK β = iJ1(γ)/H1(i)(γ) (i)

The notation of the Bessel functions is that of Jahnke and Emde (Tables of Functions with Formulas and Curves, Dover, 1945; Teubner, 1960).

IDEAL REACTORS INTRODUCTION A useful classification of kinds of reactors is in terms of their concentration distributions. The concentration profiles of certain limiting cases are illustrated in Fig. 7-3; namely, of batch reactors, continuously stirred tanks, and tubular flow reactors. Basic types of flow reactors are illustrated in Fig. 7-4. Many others, employing granular catalysts and for multiphase reactions, are illustrated throughout Sec. 23. The present material deals with the sizes, performances and heat effects of these ideal types. They afford standards of comparison. In a batch reactor, all the reactants are loaded at once; the concentration then varies with time, but at any one time it is uniform throughout. Agitation serves to mix separate feeds initially and to enhance heat transfer. In a semibatch operation, some of the reactants are charged at once and the others are then charged gradually. In an ideal continuously stirred tank reactor (CSTR), the conditions are uniform throughout and the condition of the effluent is the same as the condition in the tank. When a battery of such vessels is employed in series, the concentration profile is step-shaped if the abscissa is the total residence time or the stage number. The residence time of individual molecules varies exponentially from zero to infinity, as illustrated in Fig. 7-3e. In another kind of ideal flow reactor, all portions of the feed stream have the same residence time; that is, there is no mixing in the axial direction but complete mixing radially. It is called a plug flow reactor (PFR), or a tubular flow reactor (TFR), because this flow pattern is characteristic of tubes and pipes. As the reaction proceeds, the concentration falls off with distance. Often, complete mixing cannot be approached for economic reasons. Inactive or dead zones, bypassing, and limitations of energy input are common causes. Packed beds are usually predominantly used in plug flow reactors, but they may also have small mixing zones

superimposed in series or in parallel. In tubular reactors for viscous fluids, laminar or non-Newtonian behavior gives rise to variations of residence time. Deviations from ideal behavior are analyzed at length in Sec. 23. MATERIAL AND ENERGY BALANCES These balances are based on the general conservation law, Input + Sources = Outputs + Sinks + Accumulation

(7-68)

The terms may be quantities or rates of flow of material or enthalpy. Inputs and outputs are streams that cross the vessel boundaries. A heat of reaction within the vessel is a source. A depletion of reactant in the vessel is a sink. Accumulation is the time derivative of the content of the reference quantity in the vessel; of the volume times the concentration, ∂VrCa /∂t; or of the total enthalpy of the vessel contents, ∂[WCp(T − Tref )] /∂t. BATCH REACTORS Batch reactors are tanks, usually provided with agitation and some mode of heat transfer to maintain temperature within a desirable range. They are primarily employed for relatively slow reactions of several hours duration, since the downtime for filling and emptying large equipment may be an hour or so. Agitation maintains uniformity and improves heat transfer. Modes of heat transfer are illustrated in Figs. 23-1 and 23-2. Except in the laboratory, batch reactors are mostly liquid phase. In semibatch operation, a gas of limited solubility may be fed in gradually as it is used up. Batch reactors are popular in practice because of their flexibility with respect to reaction time and to the kinds and quantities of reactions that they can process.

Concentration

REACTION KINETICS

Concentration

7-16

(b) (b)

(a)

Concentration

Time

Concentration

Time (a) (a)

1

2 3 4 5 Stage number

(b)

(c)

Distance along reactor

(c)

(d)

1.0 0.9 0.8

(d)

n=1

0.7

n=5 PFR

C/C0

0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.0

0.5

1.0 1.5 Reduced time, t / t

2.0

(e)

2.5

(f)

Types of flow reactors: (a) stirred tank battery, (b) vertically staged, (c) compartmented, (d) single-jacketed tube, (e) shell and tube, ( f) semiflow stirred tank.

FIG. 7-4

(e) Concentration profiles in batch and continuous flow: (a) batch time profile, (b) semibatch time profile, (c) five-stage distance profile, (d) tubular flow distance profile, (e) residence time distributions in single, five-stage, and PFR; the shaded area represents the fraction of the feed that has a residence time between the indicated abscissas.

FIG. 7-3

Material and energy balances of a nonflow reactor are summarized in Table 7-5. Several batch operations are summarized in Fig. 7-5. Daily Yield Say the downtime for filling and emptying a reactor is td and no reaction occurs during these periods. The reaction time tr of a first-order reaction, for instance, is given by ktr = −ln (1 − x). The daily yield with n batches per day will be 24Vr kC0 x 24VrC0 x y = nVr(C0 − C) = ᎏ = ᎏᎏ (7-69) tr + td −ln (1 − x) + ktd Some conditions at which the daily yield is a maximum are ktd x

0.01 0.13

0.1 0.45

0.5 0.68

5.0 0.88

Thus, the required conversion goes up as the downtime increases. Details are in Problem P2 of the “Solved Problems” subsection. Filling and Emptying Periods Say the pumping rate is V′, the full tank volume is Vr1, and the rate of reaction is r = kCaq. For t ≤ Vr1/V ′, the material balances with Eq. (7-68) are as follows.

Filling: Vr = V′t,

Ca = Ca0 , when t = 0, d(VrCa) dC V′Ca0 = 0 + rVr + ᎏ = kVrCaq + V′Ca + V′t ᎏa dt dt

 

Vr dCa (7-70) Ca0 = Ca + k ᎏ Caq + t ᎏ V′ dt where the variables are separable. Emptying: Vr = Vr1 − V′t, Ca = Ca1, when t = 0, d(VrCa) dC q 0 = V′Ca + kVrCa + ᎏ = V′Ca + kVrCaq − V′Ca + Vr ᎏa dt dt or

dCa (7-71) ᎏ = −kCaq dt This is the same equation as for the full tank, but applies only for t ≤ Vr1/V′. Figure 7-5e shows a complete batch cycle. Optimum Operation of Reversible Reactions Often, equilibrium composition becomes less favorable and the rate of reaction becomes more favorable as the temperature increases, so a best condition may exist. If the temperature is adjusted at each composition to

IDEAL REACTORS TABLE 7-5 Material and Energy Balances of a Nonflow Reactor

Changes in density because of reaction or temperature changes are often small enough to be ignored. Then the volumetric flow rate is uniform and the balance becomes

Rate equations: α

 

1 dn n ra = − ᎏ ᎏa = kC αa = k ᎏa Vr dθ Vr



b′ k = exp a′ − ᎏ T′

(1)





T

∆Cp dT

Rate of heat transfer: Q′ = UA(Ts − T)

(5)

d(VrCa) V′0 Ca0 = V′Ca + Vr ra + ᎏ dt where the reactor volume Vr is a known function of time. For a power law rate equation at steady state,

Enthalpy balance: dT 1 UA(Ts − T) ᎏ = ᎏ ∆Hr + ᎏᎏα dna ρVrC Vrk(na /Vr)
p



dT 1 UA(Ts − T) ᎏ = ᎏ ∆Hr + ᎏᎏ dCa ρC VrkCaα
p



when





Ca = Ca0

(7) (8)

Solve Eq. (6) to find T = f(Ca); combine Eqs. (1) and (2) and integrate as 1 ᎏᎏᎏ dCa Caαexp [a′ − b′/f(Ca)]

(9)

SOURCE: Adapted from Walas, Chemical Process Equipment Selection and Design, Butterworth-Heinemann, 1990.

Ca, n − 1 = Can +
tnran

Example 7: A Four-Stage Unit When the material balances are Cn − 1 = Cn + 1.5[Cn /(0.2 + Cn)]2 and C0 = 2, the successive outlet concentrations are found by RootSolver to be 0.985, 0.580, 0.389, and 0.281. The simplest problem is when all of the stages have the same kt
; then one of the three variables (kt
, n, or Can) can be found when the others are specified. For first-order reactions, 1 Can ᎏ = ᎏᎏᎏᎏ Ca0 (1 + k1t
1)(1 + k2t
2) . . . (1 + knt
n)

(7-80)

1 ⇒ ᎏᎏn (1 + kt
total /n)

(7-81)

for identical stages. For multiple reactions, material balances are required for each stoichiometry.

k1 = A1 exp (−B1 /T) and k2 = A2 exp (−B2 /T). The condition (∂r/∂T)x = 0

Example 8: Consecutive Reactions Take the reaction A ⇒ B ⇒ C, with B0 = C0 = 0. Define ϑ = k1t
A0, α = 1/(1 + k1 t
), β = 1/(1 + k 2 t
). Then by setting up successive material balances, equations for the effluent from the nth stage are derived as 1

(7-72)

which tells what the temperature must be at each fractional conversion for the minimum reactor size. Practically, it may be difficult to vary the temperature of a batch reactor in this way, but the operation may be more nearly feasible with a CSTR battery or a PFR. Figure 7-5f shows an example of such a temperature profile for a batch reactor. CONTINUOUS STIRRED TANK REACTORS (CSTR) Flow reactors are used for greater production rates when the reaction time is comparatively short, when uniform temperature is desired, when labor costs are high. CSTRs are used singly or in multiple units in series, in either separate vessels or single, compartmented shells. Material and energy balances are based on the conservation law, Eq. (7-69). In the operation of liquid phase reactions at steady state, the input and output flow rates are constant so the holdup is fixed. The usual control of the discharge is on the liquid level in the tank. When the mixing is adequate, concentration and temperature are uniform, and the effluent has these same properties. The steady state material balance on a reactant A is V 0′ Ca0 = V′Ca + Vr ra

(7-79)

The set of equations for all stages can be solved in succession, starting with the inlet to the first stage as Ca0.

r = k1(1 − x) − k2 x

leads to

(7-78)

A summary of material and energy balances is in Table 7-6. For each vessel of a series,

make the rate a maximum, then a minimum reactor size or maximum conversion will result. Take the first-order reversible process,

B1 − B2 T = ᎏᎏ A1B1(1 − x) ln ᎏᎏ A2B2x

(7-77)

Ca0 = Ca + kt
Caq (6)

1 C
p = ᎏ niCpi ρVr

Ca

(7-75)

(4)

(the simplest case is when UA and Ts are constant)

Ca0

Vr
t = ᎏ V′

Ca0 − Ca (7-76) ra = ᎏ t
emphasizes how CSTR measurements can provide data for the development of rate equations without integrating them. During startup or discharge the material balance becomes

(3)

298



(7-74)

A useful rearrangement,

∆Hr = ∆Hr298 +

θ=

Ca0 = Ca +
tra where the residence time is

(2)

Heat of reaction:

T = T0

7-17

(7-73)

αβϑ Bn = ᎏ (α n − β n) α−β When n ⇒ ∞, this equation reduces to An = A0αn

Cn = A0 − An − Bn

2

(7-82)

B k1 (7-83) ᎏ = ᎏ [exp (−k1t
) − exp (−k2t
)] A0 k2 − k1 This is the equation for a plug flow reactor. It can be derived directly from the rate equations with the aid of Laplace transforms. The sequences of secondorder reactions of Figs. 7-5a and 7-5c required numerical integrations.

When CSTRs are operated in series, the sum of the reactor volumes drops off sharply with the number of stages. An economical number often is only 3 to 6, since the benefit of reduced volume may be outweighed by the increased cost of multiple agitators, pumps, and controls. When all stages are in a single shell, the economics are more favorable to large numbers of stages, but the single-shell arrangements lose some of the flexibility of the multiple-tank designs. Example 9: Comparison of Batch and CSTR Volumes For a first-order reaction, the ratio of n-stage CSTR and batch volumes is n[(C0 /C)1/n − 1] Ratio = ᎏᎏ ln(C0 /C)

REACTION KINETICS 1

1.0 0.9 0.8 0.7 0.6

c0 c3

.5 c2

c1 0 0

A/A0

ci /c00

c4

C/A0

A/A0

Batch

0.5 0.4

One-stage B/A0

0.3 0.2

.8

1.6

2.4

0.1 0.0 0

3.2

Two-stage

2

1

3

b/c00

Time, t

(a) (a)

(b) (b)

2

4

5

Concentration

2.0 1.8 1.6

1

C B A

0 0

k1 = 2, k2 = 1

1

2

3

4

5

1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0

C B A 2

4

t

6 8 10 12 14 CSTR residence time

16

18

20

(d) (d)

(c) (c)

0.50

2000 1800 (0.7494, 20)

.6 .5 .4 .3

(0.1375, 174)

Temperature

Concentration, A

2200 1 .9 .8 .7

(0.1498, 154) 0

40

80 120 Time, min (e)

160

200

0.40

Temperature

1600

0.35

1400

0.30

1200

0.25

Equilibrium at 600

1000

0.20 0.15

800 600

.2 .1 0

0.45

Time below T = 600

400 200 0.0

Time

7-18

0.10

Time at 600 0.1

0.2

0.3 0.4 0.5 0.6 0.7 Fraction converted, f

0.8

0.9

0.05 0.00 1.0

(f)

FIG. 7-5 Some batch operations, the P-code refers to detailed solutions in Walas (Chemical Reaction Engineering Handbook of Solved Problems, Gordon & Breach, 1995). (a) Methane chlorination in batch reactor or PFR; abscissa is ratio of chlorine to methane; P4.03.20. (b) Product yields of A ⇒ B ⇒ C with k1 = 2 and k2 = 1 in batch reactor and CSTR; P4.04.60. (c) The reactions 2A ⇒ B and 2B + 2C ⇒ D with k1 = 1.0 in batch reactor; P4.04.46. (d) Same as c but in CSTR. (e) Fill for 20 min, react, and discharge for 20 min with ra = 0.03[Ca(0.2 + Ca) − 0.04(1 − Ca)2]; P4.09.18. ( f ) Best temperature profile for a reversible reaction, r = k1(1 − x) − k2x; when t = 0.1, x = 0.37 at 600 R, and x = 0.51 with optimum temperature profile; x = 0.81 when t = 0.5 and final temperature is 250 R; the full range is impractical; P4.11.02.

IDEAL REACTORS TABLE 7-6

Material and Energy Balance of a CSTR

when

Mean residence time: Vr t
= ᎏ V′

(1)

Temperature dependence: b′ k = exp a′ − ᎏ T





(2)

Rate equation: (Ca0 − Ca) x = ᎏᎏ Ca0

ra = kCαa = kCαa0(1 − x)α,

C1q + 1 + C2q [(q − 1)C1 − qC0] = 0

and

The sketch identifies the nomenclature.

(3)

Material balance: Ca0 = Ca + kt
Ca

(4) (5)

x = ktCαa0− 1(1 − x)α

7-19

q = 1,



C1 = C 0C
, 2

and
t1 =
t2

For higher orders
t1 ≠ t
2 and the sum is less than twice the sum of equal stages, although usually not much different from that sum. As an example, when the second-order reaction between benzoquinone and cyclopentadiene is done in a three-stage unit, the reactor sizes are 3.25, 4.68, and 6.27, totaling 14.20, as compared to 14.56 with three equal stages (where consistent units are used). Details are in Walas (Chemical Reaction Engineering Handbook of Solved Problems, p. 4.11.15, Gordon & Breach, 1995). Selectivity A significant respect in which CSTRs may differ from batch (or PFR) reactors is in the product distribution of complex reactions. However, each particular set of reactions must be treated individually to find the superiority. For the consecutive reactions A ⇒ B ⇒ C, Fig. 7-5b shows that a higher peak value of B is reached in batch reactors than in CSTRs; as the number of stages increases the batch performance is approached. TUBULAR AND PACKED BED FLOW REACTORS

Enthalpy balance:

n′H − n′ H i

i

i0

i0

= Q′ − ∆Hr(n′a0 − n′a)

Hi =



(6)

T

Cpi dT

(7)

298

∆Hr = ∆Hr298 +



T

∆Cp dT

(8)

298

For the reaction aA + bB → rR + sS: ∆Cp = rCpr + sCps − aCpa − bCpb

(9)

When the heat capacities are equal and constant, the heat balance is: C
pρV′(T − T0) = Q′ − ∆Hr298V′(Ca0 − Ca)

(10)

SOURCE: Adapted from Walas, Chemical Process Equipment Selection and Design, Butterworth-Heinemann, 1990.

Some values are C0 /C n

2

10

20

1 5

2.89 1.07

3.91 1.27

6.34 1.37

The ratio goes up sharply as the conversion increases and down sharply as the number of stages increases. For higher-order reactions the numbers are of comparable magnitudes.

Different Sizes Ordinarily, it is most economical to make all stages of a CSTR battery the same size. For a first-order reaction the resulting total volume is a minimum for a specified performance, but not so for other orders. Take a two-stage battery: C0 − C1 C1 − C2 +ᎏ
t1 +
t2 = ᎏ kC1q kC2q With C0 and C2 specified, the condition for a minimum is ∂(t
1 +
t2) ᎏ=0 ∂C1

Tubular reactors are made up of one or more tubes in parallel, each of less than approximately 100-mm (3.94-in) diameter. With fluids of normal viscosity, plug flow exists in tubes of this size, with all molecules having essentially the same residence time. In packed beds of larger diameters, large-scale convection may be inhibited to such an extent that plug flow is also approached. Continuous gas phase reactions are predominantly done in such units, as are many liquid phase processes. Immiscible liquids are best handled in stirred tanks, although in-line mixers can facilitate such reactions in pipes. Reaction times are mostly short, made feasible by elevated temperatures. In such large-scale operations as oil cracking, the tubes may be several hundred meters long in a trombonelike arrangement. Temperature control is by heat transfer through the walls or by cold-shot injection. Shell-and-tube arrangements can provide large amounts of heat transfer. Product distribution of complex reactions is like that of batch reactors, but different from that of CSTRs. Material and energy balances of a plug flow reactor are summarized in Table 7-7. For convenience, the loading on a flow reactor is expressed as a size of reactor per unit of flow rate, say Vr /V′, and is labeled the space velocity. Some of the units in practical use are stated in the Introduction. How the actual residence time is calculated when the density of flow varies is illustrated in Table 7-8. Tubular flow reactors operate at nearly constant pressure. How the differential material balance is integrated for a number of secondorder reactions will be explained. When na is the molal flow rate of reactant A the flow reactor equation is −dna = na0 dx = −V′dCa = ra dVr (7-84) na0 dna (7-85) or Vr = ᎏ na ra The equation is rendered integrable by application of the stoichiometry of the reaction, the ideal gas law, and, for instance, the power law for rate of reaction. Some details are shown in Table 7-9. Frictional Pressure Drop Usually this does not have a significant effect on the reactor size, except perhaps when the flow is twophase. Some approximate relations will be cited that are adequate for pressure-drop calculations of homogeneous flow reactions in pipelines. The pressure drop is given by f ρu2 −dP = ᎏ dL (7-86) 2gD A good approximation to the friction factor in the turbulent flow range is µD 0.2 f = 0.046(Re)−0.2 = 0.044 ᎏ (7-87) W The mass flow rate is



 

W = 0.7854D2ρu

7-20

REACTION KINETICS

TABLE 7-7

Material and Energy Balances of a Plug Flow Reactor (PFR)

The balances are made over a differential volume dVr of the reactor. Rate equation:

At constant Ts, Eq. (7) may be integrated numerically to yield the temperature as a function of the number of moles T = φ(n′a)

−dn′a dVr = ᎏ ra

(8)

(1) Then the reactor volume is found by integration

1 V′ α = − ᎏ ᎏ dn′a k n′a

  

b′ = −exp −a′ + ᎏ T

(2) α

dn′ ᎏ Pn′  n′tRT

a

(3)

a



n′a0

Vr =

n′a

1 dn′a ᎏᎏᎏᎏ exp [a′ − b′/φ(n′a)][Pn′a /n′tRφ(n′a)]α

(9)

Adiabatic process: dQ = 0

(10)

Enthalpy balance: ∆Hr = ∆Hr298 +



T

∆Cp dT

(4)

298

4U dQ = U(Ts − T) dAp = ᎏ (Ts − T) dVr D 4U(Ts − T) = − ᎏᎏ dn′a Dra dQ + ∆Hr dn′a = ni dHi = niCpi dT dT ᎏ = ∆Hr − 4U(Ts − T)/Dra = f(T, Ts, n′a) dn′a ᎏᎏᎏ nicpi

(5) The balance around one end of the reactor is (6) (7)

n

Hi0 − Hr0(n′a0 − n′a) = niHi = ni

i0

C

pi

dT

(11)

With reference temperature at T0, enthalpies Hi0 = 0 ∆Hr0 = ∆Hr298 +



T0

∆Cp dT

(12)

298

Substituting Eq. (12) into Eq. (10)

−∆H

r298

+



T0

298



∆Cp dT (n′a0 − n′a) = ni



T

Cpi dT

(13)

T0

Adiabatic process with ∆Cp = 0 and with constant heat capacities ∆Hr298(n′a0 − n′a) T = T0 − ᎏᎏ niCpi

(14)

This expression is substituted instead of Eq. (8) to find the volume with Eq. (9).

The density in terms of the molecular weight M is M PM0nt0 ρ=ᎏ=ᎏ V RTnt Also, in terms of the tube length dL, dVr = 0.7854D2dL Combining,

0.046W1.8µ0.2RT[nt0 + δa(na0 − na)] −dP = ᎏᎏᎏᎏ dVr gD6.8M0nt0P

(7-88)

This is to be solved simultaneously with the flow reactor equation, Eq. (7-84). Alternatively, dVr can be eliminated from Eq. (7-88) for a direct relation between P and na. More accurate relations than Eqs. (7-86) and (7-87) are described in Sec. 11 of this Handbook. RECYCLE AND SEPARATION MODES All reactor modes can sometimes be advantageously operated with recycling of part of the product or intermediate streams. Heated or cooled recycle streams serve to moderate undesirable temperature travels, and they can be processed for changes in composition before being returned. Say the recycle flow rate in a PFR is V r′ and the fresh feed rate is V0′, with the ratio R = Vr′ /V′0. With a fresh feed concentration of C0 and a product of C2 the composite feed concentration is C0 + RC2 C1 = ᎏ 1+R

(7-89)

The change in concentration across the reactor becomes C2 − C0 ∆C = C1 − C2 = ᎏ (7-90) 1+R Accordingly, the change in concentration (or in temperature) across the reactor can be made as small as desired by upping the recycle ratio. Eventually, the reactor can become a differential unit with substantially constant temperature, while substantial differences will concurrently arise between the fresh feed inlet and the product withdrawal outlet. Such an operation is useful for obtaining experimental data for analysis of rate equations. In the simplest case, where the product is recycled without change, the flow reactor equation at constant density with a power law is −V′0 (1 + R)dC = kCqdVr C1 dC and Vr = V′0 (1 + R) (7-91) ᎏq C2 kC Recycling increases the size of the reactor and degrades the plug flow characteristics, so there must be practical compensation by adjustment of the temperature or composition.



Example 10: Reactor Size with Recycle For first-order reaction with C0 /C2 = 10 and R = 5, C1 /C2 = 2.5. The relative reactor sizes with recycle and without are (1 + 5) ln (C1 /C2) 5.497 Ratio = ᎏᎏ = ᎏ = 1.193 ln (C0 /C2) 4.605 With reversible reactions, recycling is warranted when improvement in conversion can be realized by removing some of the product in a separator and returning only unconverted material. In some CSTR operations, the product is removed continuously by extraction or azeotropic distillation. The gasoline addi-

IDEAL REACTORS TABLE 7-8

True Contact Time in A PFR

The ratio Vr /V′0 is of the volume of the reactor to the incoming volumetric rate and has the dimensions of time. It will be compared with the true residence time when the number of mols changes as reaction goes on, or P and T also change.

     π =k ᎏ RT

q

n rα = k ᎏα V′

q

nα ᎏ nt

q

t

q

(1)

α

The definition of the rate of reaction and the law of mass action is:

 

1 dn n rα = − ᎏ ᎏα = k ᎏα V′ dt V′

q

Rearrange to

 

n dn   ᎏ n 

1 dnα 1 V′ q 1 RT dt = − ᎏ ᎏ = − ᎏ ᎏ dnα = − ᎏ ᎏ V′ rα kV′ nα k π

q−1

q−1

t

q α

Example: Take q = 1, nt0 = nα0. 1 (δα + 1)nα0 − δαnα 1 ᎏᎏ dnα = ᎏ (δα + 1) ln ᎏ − δαx nα 1−x k

nα0

nα

α0

t

t0

α

The rate equations will be stated in these terms for a number of reactions. In all these cases, the integrands are ratios of second-degree equations. The moderately complex integrations are accomplished with the aid of a table of integrals, or by MATHEMATICA, or numerically when the constants are known. 2A⇒M (1) A+B⇒M (2) 2A ⇔ M (3) A+B⇔M (4) Part (1):

nt = nt0 + δα(nα0 − nα)



α

(2)

α

Eqs. (1) and (2) are the desired comparison. Before integrating, substitute



reactor volume ᎏᎏ molal input rate

nα π Cα = ᎏ = ᎏ V′ RT

α

t0

1 k

dx ᎏ, rα

π n −x ᎏ  ᎏnn  = ᎏ RT  n + δ x 

nt q ᎏ dnα nα q−1

r

t=ᎏ

x

x0

nt = nt0 + δα(nα0 − nα) = nt0 + δαx

   dV 1 RT n ᎏ = − ᎏ  ᎏ   ᎏ  dn V′ kn π n

nα



V′ = ntRT/π

1 RT dVr = − ᎏ ᎏ k π



−dnα = nα0dx = −V′dCα = V′Cα0dx = rαdVr

Vr ᎏ= nα0

−dnα = rαdVr

Vr ᎏ= V′0

Tubular flow reactors usually operate at nearly constant pressure. For a reactant A, the differential material balance is:

One form of the integration is:

The differential balance on the reactant is:

nα0

TABLE 7-9 Integration of Rate Equations of a PFR at Constant Pressure

q

ntRT V′ = ᎏ π

0

7-21



1 1 dnα ᎏ = ᎏ ln ᎏ nα 1−x k

(1 − 2) δα = ᎏ = −0.5 2

 

RT V′ = ᎏ (nt0 − 0.5x) π Vr ᎏ= nα0

dx 1 n − 0.5x = ᎏ (R
T
)   ᎏᎏ  dx ᎏ r n −x k 2

x

2

t0

x0

α

α0

Part (2): (1 − 2) δα = ᎏ = −1 1

The ratio t δαx y = ᎏ = δα + 1 − ᎏᎏ ln [1/(1 − x)] Vr /V′0 >1 when δα < 0 <1 when δα > 0

  (n − x)(n − x) V 1 π ᎏ = ᎏ  ᎏ   ᎏᎏ dx n (n − x) k RT RT V′ = ᎏ (nt0 − x) π 2 x

r

α0

x0

α0

b0

2

t0

Part (3): tive methyl-tert-butyl ether is made in a distillation column where reaction and simultaneous separation occur.

HEAT EFFECTS The heat balance of a reactor is made up of three terms: Heat of reaction + Heat transfer = Gain of sensible and latent heats by the mixture. This establishes the temperature as a function of the composition T = f(na)

δa = − 0.5

  RT n −x RT n + 0.5n x r = k  ᎏ   ᎏᎏ  − k  ᎏ  ᎏᎏ  π n − 0.5x π n − 0.5x

RT V′ = ᎏ (nt0 − 0.5x) π 2

α

α0

1

2

α0

m0

2

t0

t0

Part (4): δα = −0.5

which may be substituted into the equations of the specific rate and the equilibrium constant B k = exp A + ᎏ f(na)









D K e = exp C + ᎏ f(na) With these substitutions the rate equation remains a function of the composition alone. Heat balances of several kinds of reactors are summarized in Tables 7-5, 7-6, 7-7 and 7-10. Enthalpy changes of processes depend only on the end states. Normally the enthalpy change of reaction is known at some standard tem-

RT 2 (nα0 − x)(nb0 − x) RT rα = k1 ᎏ ᎏᎏ − k2 ᎏ π (nt0 − 0.5x)2 π

 

nm0 + 0.5nα0x

  ᎏᎏ n − 0.5x  t0

perature, Tb = 298 K (536 R), for instance. The simplest formulation of the heat balance, accordingly, is to consider the reaction to occur at this temperature, transfer whatever heat is required, and raise the enthalpy of the reaction products to their final values. Batch Reactions For a batch reaction, the heat balance is −(∆Hr)Tb(na0 − na) + Q = ni(HiT − HiTb)

(7-92)

7-22

REACTION KINETICS −(∆Hr)Tb(na0 − na) + UA(Tm − T) = VrρCp(T − Tb) (7-95) UA or −(∆Hr)Tb(Ca0 − Ca) + ᎏ (Tm − T) = ρCp(T − Tb) (7-96) Vr CSTR Reactions For a CSTR reaction, the quantities ni are molal flow rates. Per unit of time,

TABLE 7-10 Material and Energy Balances of a Packed Bed Reactor

 

Diffusivity and thermal conductivity are taken appreciable only in the radial direction. Material balance equation: ρ ∂x D ∂2x 1 ∂x ᎏ − ᎏ ᎏ2 + ᎏ ᎏ − ᎏ rc = 0 ∂z u ∂r u0C0 r ∂r Energy balance equation:





⇒ ni

∆Hrρ ∂T k ∂2T 1 ∂T + ᎏ ᎏ + ᎏ rc = 0 ᎏ−ᎏ ᎏ ∂z GCp ∂r2 GCp r ∂r



−(∆Hr)TbVrra + Q = ni(HiT − HiTb)

(1)



(2)

At the inlet: x(0, r) = x0

(3) (4)

T(0, r) = T0

(5)

∂x ᎏ=0 ∂r

(6)

∂T U ᎏ = ᎏ (T′ − T) ∂r k

(7)

At the wall:

When the temperature T′ of the heat-transfer medium is not constant, another enthalpy balance must be formulated to relate T′ with the process temperature T. A numerical solution of these equations may be obtained in terms of finite difference equivalents, taking m radial increments and n axial ones. With the following equivalents for the derivatives, the solution may be carried out by direct iteration: r = m(∆r)

(7-98)

 

⇒ ntCpt dT

∂x ∂T ᎏ=ᎏ=0 ∂r ∂r

r = R,

Cpi dT

Tb

The last equation applies in the absence of phase change. Plug Flow Reactions The differential relations in a cylindrical vessel are 4 4UA dA = ᎏ dVr −∆HrT dna + ᎏ dVr = niCpi dT (7-99) D D

At the center: r = 0,



(7-97)

T

(7-100)

Note that the enthalpy change of reaction is a function of temperature, but a mean value often is adequate. The various heat balances are to be solved simultaneously with the appropriate material balances, but when the temperatures can be solved for explicitly their equivalents are simply substituted into the equations for k and Ke and the material balance is solved alone. Packed Bed Reactors The commonest vessels are cylindrical. They will have gradients of composition and temperature in the radial and axial directions. The partial differential equations of the material and energy balances are summarized in Table 7-10. Example 4 of “Modeling of Chemical Reactions” in Sec. 23 is an application of such equations. A variety of provisions for heat transfer are illustrated in Figs. 23-1 to 23-3 and elsewhere in Sec. 23. UNSTEADY CONDITIONS WITH ACCUMULATION TERMS

z = n(∆z)

(8)

∂T Tm,n + 1 − Tm,n ᎏ = ᎏᎏ ∂z ∆z

(9)

∂T Tm + 1,n − Tm,n ᎏ = ᎏᎏ ∂r ∆r

(10)

∂2T Tm + 1,n − 2Tm,n + Tm − 1,n = ᎏᎏᎏ ᎏ ∂r2 (∆r)2

(11)

Expressions for the x derivatives are of the same form: rc = rate of reaction, a function of s and T G = mass flow rate, mass/(time)(superficial cross section) u = linear velocity D = diffusivity k = thermal conductivity SOURCE: Adapted from Walas, Chemical Process Equipment Selection αnd Design, Butterworth-Heinemann, 1990.

The solvent, as well as any other inerts, and the mass of the vessel are included in this summation. The heat exchange through a jacket or coils at temperature Tm is Q = UA(Tm − T)

(7-93)

When phase changes are absent, −(∆Hr)Tb(na0 − na) + UA(Tm − T) = ni



T

Cpi dT

(7-94)

Tb

When the mixture can be characterized by an overall heat capacity,

Unsteady material and energy balances are formulated with the conservation law, Eq. (7-68). The sink term of a material balance is Vr ca and the accumulation term is the time derivative of the content of reactant in the vessel, or ∂(VrCa)/∂t, where both Vr and Ca depend on the time. An unsteady condition in the sense used in this section always has an accumulation term. This sense of unsteadiness excludes the batch reactor where conditions do change with time but are taken account of in the sink term. Startup and shutdown periods of batch reactors, however, are classified as unsteady; their equations are developed in the “Batch Reactors” subsection. For a semibatch operation in which some of the reactants are preloaded and the others are fed in gradually, equations are developed in Example 11, following. For a CSTR the unsteady material balance is d(VrCa) V′Ca0 = V′Ca + Vrra + ᎏ (7-101) dt Enthalpy balances also will have accumulation terms. Conditions that give rise to unsteadiness are changes in feed rate, composition, or temperature. In the case of Fig. 7-6, a sinusoidal input of feed rate is introduced. The output concentration also appears to vary sinusoidally. The amplitude of the response is lower as the specific rate is increased. If a sinusoidal variation of the temperature of the heat transfer medium in the jacket or coil occurs, say Tm = Tm0(1 + α sin β t)

(7-102)

the balances will be dT −∆HrVrra + UA(Tm − T) = V′ρCp(T − T0) + ρVrCp ᎏ dt

(7-103)

dCa (7-104) V′Ca0 = V′Ca + Vrra + Vr ᎏ dt Since each input of mass to a perfect plug flow unit is independent of what has been input previously, its condition as it moves along the

LARGE SCALE OPERATIONS

Example 11: Balances of a Semibatch Process The reaction A + B ⇒ Products is carried out by first charging B into the vessel to a concentration Cb0 and a volume Vr0, then feeding a solution of concentration Ca0 at volumetric rate V′ for a time t. Volume of solution in the tank:

1.50 Feed rate 1.20

Vr = Vr0 + V′t

1.00 0.90

(7-105)

Stoichiometric balance:

Output C, k = 1

0.618

0.60

V′tCa 0 − VrCa = Vr 0Cb 0 − Vr Cb Vr 0Cb0 − V′tCa0 Cb = Ca + ᎏᎏ Vr0 + V′t

Output C, k = 5 0.358

0.30 0.00 0

7-23

(7-106)

Material balance on A: Input = Output + Sink + Accumulation

3

6

9

d(CaVr) V′Ca0 = 0 + kVrCaCb + ᎏ dt

12

t

dCa = kVrCaCb + Vr ᎏ + CaV′ dt

FIG. 7-6 Sinusoidal input of feed rate to a CSTR. Input: F = 1 + 0.2 sin (t); output: dC/dt = (1 − C)[1 + 0.2 sin (t)] − kc2. Straight lines are for constant feed rate.

reactor will be determined solely by its initial condition and its residence time, independently of what comes before and after. Practically, of course, some interaction will occur at the boundaries of successive inputs of different compositions or temperatures. This is governed by diffusional behaviors that are beyond the scope of the present work.

dCa V′Ca V′Ca0 ᎏ + kCaCb + ᎏ = ᎏ dt Vr Vr

(7-107)

Eqs. (7-105), (7-106), and (7-107) are combined into dCa V′ Vr0Cb0 − V′tCa0 ᎏ = ᎏ (Ca0 − Ca) − kCa Ca + ᎏᎏ dt Vr0 + V′t Vr0 + V′t





(7-108)

A numerical integration is required.

LARGE SCALE OPERATIONS INTRODUCTION In this category are included a number of topics that become especially significant on the industrial scale. Some of this material is covered at length in Sec. 23, so only an outline is provided here. MULTIPLE STEADY STATES Phenomena of multiple steady states and instabilities occur particularly with nonisothermal CSTRs. Some isothermal processes with hyperbolic rate equations and processes with porous catalysts also can have such behavior. Mathematically, multiplicities become evident when heat and material balances are combined. Both are functions of temperature, the latter through the rate equation which depends on temperature by way of the Arrhenius law. The curves representing these balances may intersect in several points. For first order in a CSTR, the material balance in terms of the fraction converted can be written kt
b x = ᎏ, k = exp a − ᎏ (7-109) 1 + kt
T and the energy balance Heat generation = Sensible heat gain −∆HrVrCf (1 − x) ᎏᎏ + UA(Tm − T) = ρCpVr (T − Tf ) (7-110) t
These balances can be plotted two ways, as shown in Fig. 7-7: 1. x from both equations can be plotted against T, with the intersections at the steady state values of T and corresponding values of x. 2. The LHS (heat generation) and RHS (heat removal) of Eq. (7-110) are plotted against T after x has been eliminated between the two balances; the intersections identify the same steady state temperatures as the plot in Fig. 7-7a. Conditions at which the slope of the heat generation line is greater than that of the heat removal line are unstable, and where it is less the condition is stable (see Fig. 7-7b). At an unstable point, any fluctua-





tion in conditions will move the temperature to a neighboring point. Control systems always produce small fluctuations of the process variables, as in the sinusoidal case of Fig. 7-6. If the fluctuations occur while the system is at an unstable point, the steadiness will disappear. In the case of Fig. 7-7c, as the unstable position is approached (T = 280, C = 2.4) the profiles of T and C become erratic and eventually degenerate to the condition at the stable point on the right (Figs. 7-7d and 7-7e). Either of the two stable operating conditions can be selected by adjusting the positions of the curves so that only one intersection is obtained. In a plant, long-time unstable operation is unlikely because of imprecise temperature control. Plug flow reactors with recycle exhibit some of the characteristics of CSTRs, including the possibility of multiple steady states. This topic is explored by Perlmutter (Stability of Chemical Reactors, PrenticeHall, 1972). Endothermic reactions possess only one steady state. For complex reactions and with multistage CSTRs, more than three steady states can exist (as in Fig. 23-17c). Most of the work on multiplicities and instabilities has been done only on paper. No plant studies and a very few laboratory studies are mentioned in the comprehensive reviews of Razon and Schmitz (Chem. Eng. Sci., 42, 1,005–1,047 [1987]) and Morbidelli et al. (in Carberry and Varma, Chemical Reaction and Reactor Engineering, Dekker, 1987, pp. 973–1,054). NONIDEAL BEHAVIOR Reactors that are nominally CSTRs or PFRs may in practice deviate substantially from ideal mixing or nonmixing. This topic is developed at length in Sec. 23, so only a few summary statements are made here. More information about this topic also may be found in Nauman and Buffham (Mixing in Continuous Flow Systems, Wiley, 1983). Laminar Flow With highly viscous fluids the linear velocity along a streamline varies with the radial position. Laminar flow is characteristic of some polymeric systems. Figure 23-21 shows how the conversion is poorer in laminar flow than with uniform flow over the

7-24

REACTION KINETICS

1.0

100

0.9

90 Mass balance

Heat generation

80

0.7

70 Heat balance

0.6 0.5 0.4

Qg and Qr

Fraction converted

0.8

Heat removal

60 50 40

0.3

30

0.2

20

0.1 0.0 350 360 370 380 390 400 410 420 430 440 450 Temperature

10 0 350 360 370 380 390 400 410 420 430 440 450 Temperature

(a) (a)

(b) (b)

3.0 2.7

Concentration, C

2.4 2.1

Tf = 250

Heat balance Tf = 300

1.8 1.5 1.2 0.9 0.6

Material balance

0.3 0.0 250 270 290 310 330 350 370 390 410 430 450 Temperature, T (c) (c)

450

Tf = 250

1.5

0 0

T

C

3

200

400

600

800

1000

Tf = 250

350

250 0

200

400

600

t

t

(d)

(e)

800

1000

Multiplicity and instability of first-order reactions in CSTRs. For (a) and (b), k = exp (25 − 10,000/T), x = k/(1 + k), 200x + (350 − T) = T − 350. For curves (c) to (e), k = exp (25 − 7,550/T), C = 3/(1 + 300k), C = 3 − 0.2(T − Tf). Unsteady state, 300dC/dt = 3 − (1 + 300k)C, C0 = 0; 300dT/dt = Tf − T + 15,000kC, T0 = 300. Temperature in °C. FIG. 7-7

cross section for first- and second-order reactions. Another adverse effect with viscous solutions is poor heat transfer. Accordingly, stirred tanks are often preferred to tubular units for such applications. The equations for radial and axial distributions of composition and temperature in laminar flow are studied by Nauman (Chemical Reactor Design, Wiley, 1987, pp. 165–203). Residence Time Distribution (RTD) This is established by injecting a known amount of tracer into the feed stream and monitor-

ing its concentration in the effluent. At present there are no correlations of this kind of behavior that could be used for design of a new process, but such information about existing units is of value for diagnostic purposes. The RTD is a distinctive characteristic of mixing behavior. In Fig. 7-3e, the CSTR has an RTD that varies as the negative exponential of the time and the PFR is represented by a vertical line at tr = 1. Multistage units and many packed beds have bell-shaped RTDs, like that of

LARGE SCALE OPERATIONS the five-stage unit of Fig. 7-3e and the large-scale units of Figs. 23-10 and 23-11. An equation that represents such shapes is called the Erlang, nn RTD = ᎏ trn − 1 exp (−ntr) (7-111) Γ(n) where n is interpreted as a number of CSTRs in series. When n is integral, Γ(n) is replaced by (n − 1)!. Plots of these curves are shown in Fig. 23-9. When the RTD of a vessel is known, its performance as a reactor for a first-order reaction, and the range within which its performance will fall for other orders, can be predicted. Segregated Flow A real example is bead polymerization of styrene and some other materials. The reactant is in the form of individual small beads suspended in a fluid and retarded from agglomeration by colloids on their surfaces. Accordingly, they go through the reactor as independent bodies and attain conversions under batch conditions with their individual residence times. This is called segregated flow. With a particular RTD, conversion is a maximum with this flow pattern. The mean conversion of all the segregated elements then is given by ∞ C C
(RTD) ᎏ dtr (7-112) ᎏ= 0 C0 C0 batch



For first order

 

C ᎏ C 

batch

ᎏ C 

batch

0

For second order

C

0

= exp (−kt
tr)

(7-113)

1 = ᎏᎏ (1 + kC0t
tr)

(7-114)

Example 12: Segregated Flow The pilot unit of Fig. 23-11 with

n = 9.3 has

RTD = 13188tr8.3 exp (−9.3tr) Some values of mean concentration ratio
C/C0 of first- and second-order reactions obtained with Eq. (7-112) are: kt
or kC0t


1

2

5

10

First order C
/C0 Second order
C/C0 Second order PFR

0.386 0.513 0.500

0.163 0.349 0.333

0.018 0.180 0.167

0.100 0.091

Maximum Mixedness With a particular RTD, this pattern provides a lower limit to the attainable conversion. It is explained in Sec. 23. Some comparisons of conversions with different flow patterns are made in Fig. 23-14. Segregated conversion is easier to calculate and is often regarded as a somewhat plausible mechanism, so it is often the only one taken into account. Dispersion In tubes, and particularly in packed beds, the flow pattern is disturbed by eddies whose effect is taken into account by a dispersion coefficient in Fick’s diffusion law. A PFR has a dispersion coefficient of 0 and a CSTR of ∞. Some rough correlations of the Peclet number uL/D in terms of Reynolds and Schmidt numbers are Eqs. (23-47) to (23-49). There is also a relation between the Peclet number and the value of n of the RTD equation, Eq. (7-111). The dispersion model is sometimes said to be an adequate representation of a reactor with a “small” deviation from plug flow, without specifying the magnitude of small. As a point of superiority to the RTD model, the dispersion model does have the empirical correlations that have been cited and can therefore be used for design purposes within the limits of those correlations. OPTIMUM CONDITIONS Optimization of a process is an activity whereby the best conditions are found for attainment of a maximum or minimum of some desired objective. In the broadest sense, an industrial process has maximum profit as its goal, but there are also problems with less-ambitious goals that do not involve money or the whole plant.

7-25

The best quality to be found may be a temperature, a temperature program, a concentration, a conversion, a yield of preferred product, a cycle period for a batch reaction, a daily production level, a kind of reactor, a size for a reactor, an arrangement of reactor elements, provisions for heat transfer, profit or cost, and so on—a maximum or minimum of some of these factors. Among the constraints that may be imposed on the process are temperature range, pressure range, corrosiveness, waste disposal, and others. Once the objective and the constraints have been set, a mathematical model of the process can be subjected to a search strategy to find the optimum. Simple calculus is adequate for some problems, or Lagrange multipliers can be used for constrained extrema. When a full plant simulation can be made, various alternatives can be put through the computer. Such an operation is called flowsheeting. A chapter is devoted to this topic by Edgar and Himmelblau (Optimization of Chemical Processes, McGraw-Hill, 1988) where they list a number of commercially available software packages for this purpose, one of the first of which was Flowtran. With many variables and constraints, linear and nonlinear programming may be applicable, as well as various numerical gradient search methods. Maximum principle and dynamic programming are laborious and have had only limited applications in this area. The various mathematical techniques are explained and illustrated, for instance, by Edgar and Himmelblau (Optimization of Chemical Processes, McGraw-Hill, 1988). A few specific conclusions about optimum performance can be stated: 1. The minimum total volume of a CSTR battery for first-order reaction, and near-minimum for second-order, is obtained when all vessels are the same size. 2. An economical optimum number of CSTRs and their auxiliaries in series is 4 to 5. 3. In a sequence of PFR and CSTR, better performance is obtained with the PFR last. Performance of reversible reactions is improved with the CSTR at a higher temperature. 4. For the consecutive reactions A ⇒ B ⇒ C, a higher yield of intermediate B is obtained in batch reactors or PFRs than in CSTRs. 5. When the desirable product of a complex reaction is favored by a high concentration of some reactant, batch or semibatch reactors can be made superior to CSTRs. 6. Conversion by a reversible reaction is enhanced by starting out at high temperature and ending at low temperature if equilibrium conversion drops off at high temperature. 7. For a reversible reaction, the minimum size or maximum conversion is obtained when the rate of reaction is kept at a maximum at each conversion by adjustment of the temperature. Variables It is possible to identify a large number of variables that influence the design and performance of a chemical reactor with heat transfer, from the vessel size and type; catalyst distribution among the beds; catalyst type, size, and porosity; to the geometry of the heattransfer surface, such as tube diameter, length, pitch, and so on. Experience has shown, however, that the reactor temperature, and often also the pressure, are the primary variables; feed compositions and velocities are of secondary importance; and the geometric characteristics of the catalyst and heat-exchange provisions are tertiary factors. Tertiary factors are usually set by standard plant practice. Many of the major optimization studies cited by Westerterp et al. (1984), for instance, are devoted to reactor temperature as a means of optimization. The complexity of temperature regulation of three major commercial reversible processes are represented in Figs. 23-3a, 23-3e, and 23-3f. Presumably, these profiles have been established by fine-tuning the operations over a period of time. Objective Function This is the quantity for which a minimax is sought. For a complete manufacturing plant, it is related closely to the economy of the plant. Subsidiary problems may be to optimize conversion, production, selectivity, energy consumption, and so on in terms of temperature, pressure, catalyst, or other pertinent variables. Case Studies Several collections of more or less detailed solutions of optimization problems are cited, as follows. 1. Of the 23 studies listed under “Modeling of Chemical Reactors” in Sec. 23, a number are optimization oriented. Added to them

7-26

REACTION KINETICS

may be a detailed study of an existing sulfuric acid plant by Crowe et al. (Chemical Plant Simulation, Prentice-Hall, 1971). 2. Chen (Process Reactor Design, Allyn & Bacon, 1983) does the following examples mostly with simple calculus: Batch reactors—optimum residence time for series and complex reactions, minimum cost, optimal operating temperature, and maximum rate of reaction CSTRs—minimum volume of battery, maximum yield, optimal temperature for reversible reaction, minimum total cost, reactor volume with recycle, maximum profit for reversible reaction with recycle, and heat loss Tubular flow reactors—minimum volume for second-order reversible reactions, maximum yield of consecutive reactions, minimum cost with and without recycle, and maximum profit with recycle Packed bed reactor optimization Size comparison for first- and second-order and reversible reactions Selectivity of parallel and consecutive reactions and of reactions in a porous catalyst 3. Edgar and Himmelblau (Optimization of Chemical Processes, McGraw-Hill, 524–550, 1988) supply many references to other problems in the literature: Optimal residence time for the reactions A ⇔ B followed by B ⇒ P or X Optimal time for a biochemical CSTR Selection of feedstock for thermal cracking to ethylene by linear programming Maximum yield from a four-stage CSTR by nonlinear programming Optimal design of ammonia synthesis by differential equation solution and a numerical gradient search A C4 alkylation process by sequential quadratic programming 4. Westerterp, van Swaaij, and Beenackers (Chemical Reactor Design and Operation, Wiley, 1984, pp. 674–746) also supply many references to other problems in the literature: Optimized costs for several gas phase reactions: (1) A + B ⇒ P; (2) A + B ⇔ P; and (3) A + B ⇒ P, A + 2B ⇒ X, P + B ⇒ X Ammonia cold-shot converter Maximum yield of first-order consecutive reactions in CSTR by application of Lagrange multipliers Autothermal reactor for methanol synthesis using a numerical search technique Minimum reactor volumes of isothermal and nonisothermal cascades by dynamic programming Optimum temperature profiles of 2A ⇒ B ⇒ P by the maximum principle Optimizing the temperature for A ⇒ P and A ⇒ X by the maximum principle Westerterp et al. (1984; see Case Study 4, preceding) conclude, “Thanks to mathematical techniques and computing aids now available, any optimization problem can be solved, provided it is realistic and properly stated. The difficulties of optimization lie mainly in providing the pertinent data and in an adequate construction of the objective function.” HETEROGENEOUS REACTIONS Heterogeneous reactions of industrial significance occur between all combinations of gas, liquid, and solid phases. The solids may be inert or reactive or catalysts in granular form. Some noncatalytic examples are listed in Table 7-11, and processes with solid catalysts are listed under “Catalysis” in Sec. 23. Equipment and operating conditions of heterogeneous processes are covered at some length in Sec. 23; only some highlights will be pointed out here. Reactants migrate between phases in order to react: from gas phase to liquid, from fluid to solid, and between liquids when the reaction occurs in both phases. One of the liquids usually is aqueous. Resistance to mass transfer may have a strong effect on the overall rate of reaction. A principal factor is the interfacial area. Its magnitude is enhanced by agitation, spraying, sparging, use of trays or packing, and by size reduction or increase of the porosity of solids. These are the same operations that are used to effect physical mass transfer between

TABLE 7-11

Industrial Noncatalytic Heterogeneous Reaction

Gas/solid Action of chlorine on uranium oxide to recover volatile uranium chloride Removal of iron oxide impurity from titanium oxide by volatilization by action of chlorine Combustion and gasification of coal Manufacture of hydrogen by action of steam on iron Manufacture of blue gas by action of steam on carbon Calcium cyanamide by action of atmospheric nitrogen on calcium carbide Burning of iron sulfide ores with air Nitriding of steel Liquid/solid Ion exchange Acetylene by action of water on calcium carbide Cyaniding of steel Hydration of lime Action of liquid sulfuric acid on solid sodium chloride or on phosphate rock or on sodium nitrate Leaching of uranium ores with sulfuric acid Gas/liquid Sodium thiosulfate by action of sulfur dioxide on aqueous sodium carbonate and sodium sulfide Sodium nitrite by action of nitric oxide and oxygen on aqueous sodium carbonate Sodium hypochlorite by action of chlorine on aqueous sodium hydroxide Ammonium nitrate by action of ammonia on aqueous nitric acid Nitric acid by absorption of nitric oxide in water Recovery of iodine by action of sulfur dioxide on aqueous sodium iodate Hydrogenation of vegetable oils with gaseous hydrogen Desulfurization of gases by scrubbing with aqueous ethanolamines Liquid/liquid Caustic soda by reaction of sodium amalgam and water Nitration of organic compounds with aqueous nitric acid Formation of soaps by action of aqueous alkalies on fats or fatty acids Sulfur removal from petroleum fractions by aqueous ethanolamines Treating of petroleum products with sulfuric acid Solid/solid Manufacture of cement Boron carbide from boron oxide and carbon Calcium silicate from lime and silica Calcium carbide by reaction of lime and carbon Leblanc soda ash Gas/liquid/solid Hydrogenation or liquefaction of coal in oil slurry SOURCE: Adapted from Walas, Reaction Kinetics for Chemical Engineers, McGraw-Hill, 1959; Butterworths, 1989.

phases and the equipment can be similar, except that more heat transfer may be needed because of substantial heats of reaction. Chemical reaction always enhances the rate of mass transfer between phases. The possible magnitudes of such enhancements are indicated in Tables 23-6 and 23-7. They are no more predictable than are specific rates of chemical reactions and must be found experimentally for each case, or in the relatively sparse literature on the subject. Mechanisms The most widely investigated heterogeneous reactions have been gas/liquid and fluid/solid catalyst. The Hatta theory or Langmuir-Hinshelwood mechanisms can suggest the forms of rate equations but they always involve parameters to be found empirically. Because liquid diffusivities are low, most liquid/liquid reactions are believed to be mass-transfer controlled. In some cases the phase in which reaction occurs has been identified, but there are cases where both phases are active. Phase-transfer catalysts enhance the transfer of reactant from an aqueous to an organic phase and thus speed up reactions. Mass transfer responds less strongly to change of temperature than does chemical rate, so this feature can be used to discriminate between possible controlling mechanisms. A sensitivity to stirring rate or to a change in linear velocity also will indicate the presence of major resistance to mass transfer. No single pattern appears to hold for reactions of solids, but much is known about the behavior of important operations like cement manufacturing, ore roasting, and lime burning.

ACQUISITION OF DATA Reaction and Separation Some multiphase operations combine simultaneous reaction and separation. A few examples follow. 1. The yield of furfural from xylose is improved by countercurrent extraction with tetralin (Schoenemann, Proc. 2d Europ. Symp. Chem. React. Eng., Pergamon, 1961, p. 30). 2. The reaction of vinyl acetate and stearic acid makes vinyl stearate and acetic acid but also some unwanted ethylidene acetate. A high selectivity is obtained by reaction in a distillation column with acetic acid overhead and vinyl stearate to the bottom (Geelen and Wiffels, Proc. 3d Europ. Symp. Chem. React. Eng., Pergamon, 1964, p. 125).

7-27

3. The hydrolysis of fats is improved by running in a countercurrent extraction column (Donders et al., Proc. 4th Europ. Symp. Chem. React. Eng., Pergamon, 1968, pp. 159–168). 4. In the production of KNO3 from KCl and HNO3, the product HCl is removed continuously from the aqueous phase by contact with amyl alcohol, thus forcing the reaction to completion (Baniel and Blumberg, Chim. Ind., 4, 27 [1957]). 5. Methyl-tert-butyl ether, a gasoline additive, is made from isobutene and methanol with distillation in a bed of acidic ionexchange resin catalyst. The MTBE goes to the bottom with purity above 99 percent and unreacted materials overhead.

ACQUISITION OF DATA INTRODUCTION Kinetic data are acquired in the laboratory as a basis for design of large-scale equipment or for an understanding of its performance, or for the interpretation of possible reaction mechanisms. All levels of sophistication of equipment, statistical design of experiments, execution, and statistical analysis of the data are reported in the literature. Before serious work is undertaken, the appropriate literature should be consulted. The bibliography of Shah (Gas-Liquid-Solid Reactor Design, McGraw-Hill, 1979), for instance, has 145 items classified into 22 categories of reactor types. The criteria for selection of laboratory reactors include equipment cost, ease of operation, ease of data analysis, accuracy, versatility, temperature uniformity, and controllability, suitability for mixed phases, and scale-up feasibility. A number of factors limit the accuracy with which parameters for the design of commercial equipment can be determined. The parameters may depend on transport properties for heat and mass transfer that have been determined under nonreacting conditions. Inevitably, subtle differences exist between large and small scale. Experimental uncertainty is also a factor, so that under good conditions with modern equipment kinetic parameters can never be determined more precisely than ⫾5 to 10 percent (Hofmann, in de Lasa, Chemical Reactor Design and Technology, Martinus Nijhoff, 1986, p. 72). Composition The law of mass action is expressed as a rate in terms of chemical compositions of the participants, so ultimately the variation of composition with time must be found. The composition is determined in terms of a property that is measured by some instrument and calibrated in terms of composition. Among the measures that have been used are titration, pressure, refractive index, density, chromatography, spectrometry, polarimetry, conductimetry, absorbance, and magnetic resonance. In some cases the composition may vary linearly with the observed property, but in every case a calibration is needed. Before kinetic analysis is undertaken, the data are converted to composition as a function of time (C, t), or to composition and temperature as functions of time (C, T, t). In a steady CSTR the rate is observed as a function of residence time. When a reaction has many participants, which may be the case even of apparently simple processes like pyrolysis of ethane or synthesis of methanol, a factorial or other experimental design can be made and the data subjected to a response surface analysis (Davies, Design and Analysis of Industrial Experiments, Oliver & Boyd, 1954). A quadratic of this type for the variables x1, x2, and x3 is r = k1 x1 + k2 x2 + k3 x3 + k11 x12 + k22 x22 + k33 x32 + k12x1x2 + k13x1x3 + k23 x2 x3 (7-115) Analysis of such a correlation may reveal the significant variables and interactions, and may suggest some model, say of the L-H type, that could be analyzed in more detail by a regression process. The variables xi could be various parameters of heterogeneous processes as well as concentrations. An application of this method to isomerization of n-pentane is given by Kittrel and Erjavec (Ind. Eng. Chem. Proc. Des. Dev., 7, 321 [1968]).

The constants of rate equations of single reactions often can be found by one of the linearization schemes of Fig. 7-1. Nonlinear regression methods can treat any kind of rate equation, even models made up of differential and algebraic equations together, for instance dA ᎏ = −k1A dt dB ᎏ = k1A − k2B2 + k3C dt C = A0 + B0 + C0 − A − B Software for these procedures is supplied, for example, by Constantinides (Applied Numerical Methods with Personal Computers, McGraw-Hill, 1987, pp. 577–614, with diskette) and by the commercial product SimuSolv (Mitchell and Gauthier Associates, 200 Baker Street, Concord, MA 01742). These do the integration, find the constants and their statistical criteria, and make the plots. SimuSolv is claimed “to provide maximum efficiency in problem solving with minimum involvement in computational procedures.” Since the computer does the work, many possibilities may be considered. For the reaction cyclohexanol to cyclohexanone, 36 experiments at 6 temperature levels were made and more than 50 rate equations were tested (Hofmann, in de Lasa, Chemical Reactor Design and Technology, Martinus Nijhoff, 1986, p. 72). A rate equation for methanol from CO2 and H2 was selected from 44 possibilities by Beenackers and Graaf (in Cheremisinoff, Handbook of Heat and Mass Transfer, vol. 3, Gulf Publishing, 1989, pp. 671–699). They used a spinning basket reactor like the item shown in Fig. 23-29c. EQUIPMENT Many configurations of laboratory reactors have been employed. Rase (Chemical Reactor Design for Process Plants, Wiley, 1977) and Shah (Gas-Liquid-Solid Reactor Design, McGraw-Hill, 1979) each have about 25 sketches, and Shah’s bibliography has 145 items classified into 22 categories of reactor types. Jankowski et al. (Chemische Technik, 30, 441–446 [1978]) illustrate 25 different kinds of gradientless laboratory reactors for use with solid catalysts. Laboratory reactors are of two main types: 1. Designed to obtain such fundamental data as chemical rates free of mass transfer resistances or other complications. Some of the heterogeneous reactors of Fig. 23-29, for instance, employ known interfacial areas, thus avoiding one uncertainty. 2. Simulations of the kinds of reactor intended for the pilot or plant scale. How to do the scale-up to the plant size, however, is a sizable problem in itself. Batch Reactors In the simplest kind of investigation, reactants can be loaded into a number of ampules, kept in a thermostatic bath for various periods, and analyzed. In terms of cost and versatility, the stirred batch reactor is the unit of choice for homogeneous or slurry reactions and even gas/liquid reactions when provision is made for recirculation of the gas. They are especially suited to reactions with half-lives in excess of 10 min. Sam-

7-28

REACTION KINETICS

ples are taken at intervals and the reaction is stopped by cooling, usually by at least 50°C (122°F), by dilution, or by destroying a residual reactant such as an acid or base; analysis can then be made at leisure. Analytic methods that do not necessitate termination of reaction include measurements of (1) the amount of gas produced, (2) the gas pressure in a constant volume vessel, (3) absorption of light, (4) electrical or thermal conductivity, (5) polarography, (6) viscosity of polymerization, and so on. The readings of any instrument should be calibrated to chemical composition or concentration. Operation may be isothermal, with the important effect of temperature determined from several isothermal runs, or the composition and temperature may be recorded simultaneously and the data regressed simultaneously. Finding the parameters of the nonisothermal equation r = exp (a + b/T) Cq is only a little more difficult than for r = kCq. Rates, dC/dt, are found by numerical differentiation of (C, t) data. On the laboratory scale, it is usually safe to assume that a batch reactor is stirred to uniform composition, but for critical cases such as high viscosities this could be checked with tracer tests. CSTRs and other devices that require flow control are more expensive and difficult to operate. Particularly in steady operation, however, the great merit of CSTRs is their isothermicity and the fact that their mathematical representation is algebraic, involving no differential equations, thus making data analysis simpler. For laboratory research purposes, CSTRs are considered feasible for holding times of 1 to 4,000 s, reactor volumes of 2 to 400 cm3 (0.122 to 24.4 in3) and flow rates of 0.1 to 2.0 cm3/s. Flow Reactors Fast reactions and those in the gas phase are generally done in tubular flow reactors, just as they are often done on the commercial scale. Some heterogeneous reactors are shown in Fig. 23-29; the item in Fig. 23-29g is suited to liquid/liquid as well as gas/liquid. Stirred tanks, bubble and packed towers, and other commercial types are also used. The operation of such units can sometimes be predicted from independent data of chemical and mass transfer rates, correlations of interfacial areas, droplet sizes, and other data. Usually it is not possible to measure compositions along a TFR, although temperatures can sometimes be measured. Mostly TFRs are kept at nearly constant temperatures. Small-diameter tubes immersed in a fluidized sand bed or molten lead or salt can hold quite constant temperatures of a few hundred degrees. A recycle unit like that shown in Fig. 23-29a can be operated as a differential reactor with arbitrarily small conversion and temperature change. This and the CSTR are the preferred laboratory devices nowadays, unless the budget allows for only a batch stirred flask. Test work in a tubular flow unit may be desirable if the commercial unit is to be of that type, although rate data from any kind of laboratory equipment are adaptable to the design of most kinds of large-scale equipment. Larger TFRs may be used in pilot plants to test predictions by data from gradientless reactors. Multiple Phases Reactions between gas/liquid, liquid/liquid, and fluid/solid phases are often tested in CSTRs. Other laboratory types are suggested by the commercial units depicted in appropriate sketches in Sec. 23. Liquids can be reacted with gases of low solubili-

ties in stirred vessels, with the liquid charged first and the gas fed continuously at the rate of reaction or dissolution, sometimes with recirculation in larger units. The reactors of Fig. 23-29 are designed to have known interfacial areas. Most equipment for gas absorption without reaction is adaptable to absorption with reaction. The many types of equipment for liquid/liquid extraction also are adaptable to reactions of immiscible phases. Solid Catalysts Processes with solid catalysts are affected by diffusion of heat and mass (1) within the pores of the pellet, (2) between the fluid and the particle, and (3) axially and radially within the packed bed. Criteria in terms of various dimensionless groups have been developed to tell when these effects are appreciable. They are discussed by Mears (Ind. Eng. Chem. Proc. Des. Devel., 10, 541–547 [1971]; Ind. Eng. Chem. Fund., 15, 20–23 [1976]) and Satterfield (Heterogeneous Catalysis in Practice, McGraw-Hill, 1991, p. 491). For catalytic investigations, the rotating basket or fixed basket with internal recirculation are the standard devices nowadays, usually more convenient and less expensive than equipment with external recirculation. In the fixed basket type, an internal recirculation rate of 10 to 15 or so times the feed rate effectively eliminates external diffusional resistance, and temperature gradients. A unit holding 50 cm3 (3.05 in3) of catalyst can operate up to 800 K (1440 R) and 50 bar (725 psi). When deactivation occurs rapidly (in a few seconds during catalytic cracking, for instance), the fresh activity can be found with a transport reactor through which both reactants and fresh catalyst flow without slip and with short contact time. Since catalysts often are sensitive to traces of impurities, the time-deactivation of the catalyst usually can be evaluated only with commercial feedstock, preferably in a pilot plant. Physical properties of catalysts also may need to be checked periodically, including pellet size, specific surface, porosity, pore size and size distribution, and effective diffusivity. The effectiveness of a porous catalyst is found by measuring conversions with successively smaller pellets until no further change occurs. These topics are touched on by Satterfield (Heterogeneous Catalysis in Industrial Practice, McGraw-Hill, 1991). REFERENCES FOR LABORATORY REACTORS Berty, “Laboratory reactors for catalytic studies”, in Leach, ed., Applied Industrial Catalysis, vol. 1, Academic, 1983, pp. 41–57. Danckwerts, Gas-Liquid Reactions, McGraw-Hill, 1970. Hofmann, “Industrial process kinetics and parameter estimation”, in ACS Advances in Chemistry, 109, 519–534 (1972); “Kinetic data analysis and parameter estimation”, in de Lasa, ed., Chemical Reactor Design and Technology, Martinus Nijhoff, 1986, pp. 69–105. Horak and Pasek, Design of Industrial Chemical Reactors from Laboratory Data, Heyden, 1978. Rase, Chemical Reactor Design for Process Plants, Wiley, 1977, pp. 195–259. 124 references. Shah, Gas-Liquid-Solid Reactor Design, McGraw-Hill, 1979, pp. 149–179. 145 references.

SOLVED PROBLEMS These numerical problems deal with ideal types of batch, continuously stirred, and plug flow reactors, for which the formulas are summarized in Tables 7-5 to 7-7. They find parameters of rate equations, conversions, vessel sizes, or operating conditions. Numerical methods are adopted for most integrations and differential equations. Several ODE softwares are readily available, including POLYMATH, which is obtainable through the AIChE. A larger and broader collection of solutions is provided by Walas (Chemical Reaction Engineering Handbook of Solved Problems, Gordon & Breach, 1995). P1.

EQUILIBRIUM OF FORMATION OF ETHYLBENZENE

Ethylbenzene is made from benzene and ethylene in the gas phase at 260°C and 40 atm.

C6H6 + C2H4 ⇔ C6H5C2H5 Equimolal proportions of the reactants are used. Thermodynamic data at 298 K are tabulated. The specific heats are averages. Find: (1) the enthalpy change of reaction at 298 and 573 K; (2) equilibrium constant at 298 and 573 K; (3) fractional conversion at 573 K.

C6H6 C2H4 C6H5C2H5 ∆

Cp

∆Hf

∆Gf

28 5 38 −5

19,820 12,496 7,120 −25,196

30,989 16,282 31,208 −16,063

SOLVED PROBLEMS ∆HT = ∆H298 +



T

∆Cp dT = −25,196 −5(T − 298)

298

= −26,576 at 573 K

(1)

−∆G298 16,063 ln K298 = ᎏ = ᎏᎏ = 26.90 298R (1.987)(298)

dA − ᎏ = k1AB = 5.9AB dt

(1)

dB − ᎏ = k1AB + k2B2 = 5.9AB + 1.443B2 dt

(2)

Dividing these equations,

d ln K ∆H 23,706 5 −ᎏ ᎏ = ᎏ2 = − ᎏ dT RT RT2 RT

dB k2B ᎏ−ᎏ=1 dA k1A This is a linear equation whose solution is

11,854 2.50  ᎏ + ᎏ  dT = 6.17 T T 573

(2)

2

298

K = 485

k1 B = ᎏ A + IAk 2 /k 1 = 1.32A − 0.010A0.245 k1 − k2

(4)

The integration constant was evaluated with A0 = B0 = 0.010. Substituting (4) into (1),

x(2 − x) = ᎏᎏ2 40(1 − x) x = 0.9929, fraction converted P2.

(3)

dA − ᎏ = 5.9A(1.32A − 0.010A0.245) dt

(3)

OPTIMUM CYCLE PERIOD WITH DOWNTIME

Find the optimum cycle period for a first-order batch reaction with a downtime of ϑd h per batch.

The variables are separable, but an integration in closed form is not possible because of the odd exponent. Numerical integration followed by substitution into (4) will provide both A and B as functions of t. The plots, however, are of solutions of the original differential equations with ODE.

dC − ᎏ = kC dϑ

.01

.01

 

Butadiene

1 C0 ϑ = ᎏ ln ᎏ k C

(5)

gmol/liter

ln K298 = 26.9 −

7-29

A, Acrolein

Number of daily batches: B, Butadiene

24 n=ᎏ ϑ + ϑd

0 0

0 0

.01

10

20

30

40

50

t

Acrolein

Daily yield: 24Vr(C0 − C) 24kVrC0(1 − C/C0) y = Vr(C0 − C)n = ᎏᎏ = ᎏᎏ 1 k ln (C0 /C) + ϑd ln (C0 /C) + kϑd The ordinate of the plot is y/24kVrC0 which is proportional to the daily yield. The peaks in this curve are at these values of the parameters: kϑd C/C0

0.01 0.87

0.10 0.65

1 0.32

5 0.12

BATCH REACTION WITH HEAT TRANSFER

A second-order reaction proceeds in a batch reactor provided with heat transfer. Initial conditions are T0 = 350 and C0 = 1. Other data are:



5,000 k = exp 16 − ᎏ T



∆Hr = −(5000 + 5T )

ft3/(lb mol⋅h)

(1)

Btu/lb mol

(2)

Btu/h

(3)

ρCp = 50

1.0 0.9

l – C/C0 ln(C0 /C) + kθd

P4.

The rate of heat transfer is

kθd = 0.01

0.8 0.7

Q = UA(300 − T )

0.1

0.6

The temperature T and the time t will be found in terms of fractional conversion x when UA/Vr = 0 or 150. The rate equation may be written:

0.5 0.4 0.3 0.2

1.0

0.1 0.0 0.0

5.0 0.1

0.2 0.3

0.4

0.5 0.6 C/C0

0.7

0.8 0.9

1.0

dt 1 ᎏ = ᎏᎏ2 dx kC0(1 − x)

(4)

The differential heat balance is ρCpVr dT = Qdt − ∆HrVr C0dx

P3.

PARALLEL REACTIONS OF BUTADIENE

Butadiene (B) reacts with acrolein (A) and also forms a dimer according to the reactions C4H6 + C3H4O ⇒ C7H10O, 2 C4H6 ⇒ C8H12 1

2

The reaction is carried out in a closed vessel at 330°C, starting at 1 atm with equal concentrations of A and B, 0.010 g mol /L each. Specific rates are k1 = 5.900 and k2 = 1.443 L /(g mol⋅min). Find (1) B as a function of A; (2) A and B as functions of t.

Substituting for dt from Eq. (4) and rearranging,



dT 1 Q ᎏ = ᎏ ᎏᎏ2 − ∆HrC0 dx ρCp Vr kC0(1 − x)



UA(300 − T) = 0.02 ᎏᎏ + 5000 + 5T Vr k(1 − x)2





(5)

Equations (1), (4), and (5) are solved simultaneously with UA/Vr = 0 or 150. In the adiabatic case, the temperature tends to run away.

7-30

REACTION KINETICS 0.2

500 UA/Vr = 0



0.1

150

t

T

150 400

0 350

300 0

.2

.4

.6

.8

0 1

X

P5.

The downtime is 1 h per batch. Find the temperature at which the daily production is a maximum. The reaction time of one batch is 1 f df tb = ᎏ ᎏᎏ (1) k 0 (1 − f )2 − f 2/Ke 24 Batches/day = ᎏ tb + 1 24 Daily production = ᎏ VrCa0 f tb + 1 Maximize P = f/(tb + 1) as a function of temperature. Eq. (1) is integrated with POLYMATH for several temperatures and the results are plotted. The tabulation gives the integration at 550 K. The peak value of P = f/(tb + 1) = 0.1941 at 550 K, tb = 0.6, and f = 0.3105.

A SEMIBATCH PROCESS

A tank is charged initially with vr0 = 100 L of a solution of concentration Cb0 = 2 g mol/L. Another solution is then pumped in at V′ = 5 L/min with concentration Ca0 = 0.8 until a stoichiometric amount has been added. The rate equation is

Maximum daily production = 0.1941(24)VrCa0 = 4.66Vr kg mol/d 2.2

r = 0.015CaCb g mol/(L⋅min)

2.0

Find the concentration during the filling period and for 50 min afterward.

1.8 1.6 tb, 10P

Vr = 100 + 5t 100(2) − 5(0.8)t 40 − 0.8t Cb = Ca + ᎏᎏ = Ca + ᎏ 100 + 5t 20 + t

(1)

1.4 1.2

(2)

1.0

dCa 0.8 − Ca 40 − 0.8t (3) ᎏ = ᎏ − 0.015Ca Ca + ᎏ dt 20 + t 20 + t The input is continued until 200 lb mol of A have been added, which is for 50 min. Eq. (3) is integrated for this time interval. After input is discontinued the rate equation is dC − ᎏa = kCa2 (4) dt At t = 50, Ca = Ca1 = 0.4467. 1 1 (5) Ca = ᎏᎏ = ᎏᎏᎏ 1/Ca1 + k(t − 50) 2.2386 + 0.015(t − 50) Plots are shown for several specific rates, including k = 0 when no reaction takes place.

0.6





0.4 0.2 400 425 450 475 500 525 550 575 600 625 650 Temperature, T

The equations:







p = f /(t + 1)

k = 0.015 Ca



d( f ) 2 ᎏ = k∗ (1 − f )∗∗2 − f ∗∗ ᎏ d(t) ke x = 550 2,500 k = exp 4.5 − ᎏ x 5,178 ke = exp 28.8 − ᎏᎏ x − .037 ∗ x

 

k=0

Initial values:

t0 = 0.0

Final value:

tf = 2.0000

0.3 k = 0.050

100

50 t

tb

0.8

0.6

0 0

10P

P6. OPTIMUM REACTION TEMPERATURE WITH DOWNTIME A liquid phase reaction 2A ⇔ B + C has the rate equation 2 C C f2 b c 2 (1 − f )2 − ᎏ , kg mol/(m3⋅h) ra = k Ca2 − ᎏ = kCa0 Ke Ke where f = fractional conversion Ca0 = 1 2,500 k = exp 4.5 − ᎏ T 5,178 Ke = exp 28.8 − 0.037T − ᎏ T

f0 = 0.0

t

f

p

0.0 0.2000 0.4000 0.6000 0.8000 1.0000 1.2000 1.4000 1.6000 1.8000 2.0000

0.0 0.1562 0.2535 0.3105 0.3427 0.3605 0.3702 0.3755 0.3784 0.3799 0.3807

0.0 0.1302 0.1811 0.1941 0.1904 0.1802 0.1683 0.1565 0.1455 0.1357 0.1269

1





 









P7.

RATE EQUATIONS FROM CSTR DATA

For the consecutive reactions 2A ⇒ B and 2B ⇒ C, concentrations were measured as functions of residence time in a CSTR. In all experiments, Ca0 = 1 lb mol/ft3. Volumetric flow rate was constant. The data are tabulated in the first three columns. Check the proposed rate equations, ra = k1Caα β

rb = −0.5k1Caα + k2Cb

SOLVED PROBLEMS Write and rearrange the material balances on the CSTR.

7-31

The solution is,

Ca0 = Ca +
tra Ca0 − Ca ra = ᎏ = k1Caα t


(1)

Cb0 − Cb β β (2) rb = ᎏ = −0.5k1Caα + k2Cb = −0.5ra + k2Cb t
Numerical values of ra, rb, and rb + 0.5ra are tabulated. The constants of the rate equations are evaluated from the plots of the linearized equations, ln ra = ln k1 + α ln Ca = −2.30 + 2.001 ln Ca ln (rb + 0.5ra ) = ln k2 + β ln Cb = −4.606 + 0.9979 ln Cb which make the rate equations ra = 0.1003Ca2.00

(3)

rb = −0.0502Ca2 + 0.01Cb0.998

(4)


t

Ca

Cb

ra

−rb

rb + 0.5ra

10 20 40 100 450

1.000 0.780 0.592 0.400 0.200

0.4545 0.5083 0.5028 0.400 0.1636

0.100 0.061 0.0352 0.0160 0.0040

0.04545 0.02542 0.01257 0.0040 0.000364

0.00455 0.00508 0.00503 0.0040 0.00164

2

P9.

C1 = 0.1994,

= 60.1% conversion

C2 = 0.1025,

= 79.5% conversion

INSTANTANEOUS AND GRADUAL FEED RATES

Initially a reactor contains 2 m3 of a solvent. A solution containing 2 kg mol/m3 of reactant A is pumped in at the rate of 0.06 m3/min until the volume becomes 4 m3. The rate equation is ra = 0.25Ca, 1/min. Compare the time-composition profile of this operation with charging all of the feed instantaneously. During the filling period, Vr = 2 + 0.06t d(VrCa) dC dV V′Ca0 = kVrCa + ᎏ = kVrCa + Vr ᎏa + Ca ᎏr dt dt dt dCa 0.06(2) = 0.25(2 + 0.06t)Ca + (2 + 0.06t) ᎏ + 0.06Ca dt dCa 0.12 −(0.56 + 0.015t)Ca Ca0 = 0 (1) ᎏ = ᎏᎏᎏ , dt 2 + 0.06t When all of A is charged at the beginning, dCa Ca0 = 0.5 (2) ᎏ = −0.25Ca, dt The integrals of these two equations are plotted. A peak value, Ca = 0.1695, is reached in the first operation at t = 10. .5

10–1

.4 3 2

(Ca, ra) .3 Ca

10–2

Charged at once

.2 Charged gradually

3 2 10–3 10–1

(Cb, 0.5ra + rb) 2

3 4 Ca, Cb

5

6

.1 7 8 9 100

0 0

8

16

24

32

40

t

P8.

COMPARISON OF BATCH AND CSTR OPERATIONS

A solution containing 0.5 lb mol/ft3 of reactive component is to be treated at 25 ft3/h. The rate equation is −dC r = ᎏ = 2.33C1.7 lb mol/(ft3⋅h) dt 1. If the downtime is 45 min per batch, what size reactor is needed for 90% conversion? 2. What percentage conversion is attained with a two-stage CSTR, each vessel being 50 ft3? Part 1: The integral of the rate equation is solved for the time, 1 1 1 1 t = ᎏ (C−0.7 −C−0.7 −ᎏ 0 ) = ᎏ ᎏ 0.7k 0.7(2.33) 0.050.7 0.50.7



= 4.00 h 24 Number of batches = ᎏᎏ = 5.053/d (4 + 0.75) 24(25) Reactor volume Vr = ᎏ = 118.7 ft3 5.053 Part 2: 50 τ=ᎏ=2 25 0.5 = C1 + τr = C1 + 2(2.33)C1.7 1 C1 = C2 + 2(2.33)C1.7 2



P10. FILLING AND UNSTEADY OPERATING PERIOD OF A CSTR A stirred reactor is being charged at 5 ft3/min with a concentration of 2 mol/ft3. The reactor has a capacity of 150 ft3 but is initially empty. The rate of reaction is r = 0.02C2 lb mol/(ft3⋅min). After the tank is filled, pumping is continued and overflow is permitted at the same flow rate. Find the concentration in the tank when it first becomes full, and find how long it takes for the effluent concentration to get within 95% of the steady state value. Filling period: Vr = V′t





dVrC dC V′C0 = kVrC2 + ᎏ = kVrC2 + C + t ᎏ V′ dt dt dC C0 − ktC2 − C 2 − 0.02tC2 − C C = 2 when t = 0 ᎏ = ᎏᎏ = ᎏᎏ , dt t t The numerical solution is C = 1.3269 when t = 30. Unsteady period: dC V′C0 = V′C + kVrC2 + Vr ᎏ dt dC C0 − C − kτC2 2 − C − 0.02(30)C2 ᎏ = ᎏᎏ = ᎏᎏ , dt τ 30 C = 1.3269 when t = 30.

REACTION KINETICS nt = ns + na + nb = 0.5na0 + na + 0.5(na0 − na) = na0 + 0.5na na Pa = ᎏᎏ (1) na0 + 0.5na

The variables are separable, but the plot is of a numerical solution. The steady state concentration is 1.1736. At 95% approach to steady state from the condition at t = 30, C = 0.05(1.3269) + 0.95(1.1736) = 1.1813

0.5(na0 − na) Pb = ᎏᎏ na0 + 0.5na

From a printout of the solution, t = 67.4 min at this value. 2

Pb 118 na2 na0 − na (1) ra = k Pa2 − ᎏ = ᎏᎏ ᎏᎏ − ᎏ Ke na0 + 0.5na na0 + 0.5na 2(1.27) Put na0 = 6, substitute Eq. (1) into the flow reactor equation, and integrate numerically. 6 dna Vr = ᎏ = 0.0905 m3 3.6 ra

C



1.5

(67.4, 1.1813)

P11.

50 t

100

P13.

A second order reaction is conducted in two equal CSTR stages. The residence time per stage is τ = 1 and the specific rate is kC0 = 0.5. Feed concentration is C0. Two cases are to be examined: (1) with pure solvent initially in the tanks; and (2) with concentrations C0 initially in both tanks, that is, with C10 = C20 = C0. The unsteady balances on the two reactors are C fi = ᎏi C0 dC1 FC0 = FC1 + VrkC12 + Vr ᎏ dt df 1 = f1 + 0.5f 12 + ᎏ1 dt

AUTOCATALYTIC REACTION WITH RECYCLE

Part of the effluent from a PFR is returned to the inlet. The recycle ratio is R, fresh feed rate is F0 F R = ᎏr F0 Ft = Fr + F0 = F0(R + 1) The concentration of the mixed feed is Ca0 + RCaf Cat = ᎏᎏ 1+R where Caf is the outlet concentration. For the autocatalytic reaction A ⇒ B, the rate equation is ra = kCaCb = kCa(Ca0 − Ca)

(1)

The flow reactor equation is

dC2 FC1 = FC2 + Vr kC22 + Vr ᎏ dt

−Ft dCa = −F0(R + 1)dCa = ra dVr = kCa(Ca0 − Ca)dVr

df f1 = f2 + 0.5f 22 + ᎏ2 (2) dt The steady state values are the same for both starting conditions, obtained by zeroing the derivatives in Eqs. (1) and (2). Then

kVr ᎏ = (R + 1) F0

kVr /Fo

5

P14. .8

1.6

2.4

3.2

4

t

P12.

BUTADIENE DIMERIZATION IN A TFR

A mixture of 0.5 mol of steam per mol of butadiene is dimerized in a tubular reactor at 640°C and 1 atm. The forward specific rate is k = 118 g mol/(L⋅h⋅atm2) and the equilibrium constant is 1.27. Find the length of 10-cm ID tube for 40% conversion when the total feed rate is 9 kg mol/h. 2A ⇔ B na0 = 6 kg mol/h

1.6 1.4 1.2

Left

1

4

0.8

3

0.6

2

0.4

1 0 10–3 2 3

.2 .1 0 0

1.8 Right

6

Stage 2

f20 = 0

.3

2.0

7

f10 = 0

.5 .4

dCa ᎏᎏ Ca(Ca0 − Ca)

8

Stage 1

f20 = 1

caf

9

1 .9

.6

cat

10

f2 = 0.7321

f10 = 1



The plot is for Ca0 = 2 and Caf = 0.04. The minimum reactor size is at a recycle ratio R = 0.23 and mixed feed Cat = 1.57.

The plots are of numerical solutions.

.8 .7



0.0905(106) L = ᎏᎏ = 1,153 cm 78.5

SECOND-ORDER REACTION IN TWO STAGES

f1 = 0.5702,





(30, 1.3269)

1 0



Cat

7-32

0.2 0.0 10–2

2 3 10–1 2 3 Recycle ratio, R

100

2 3

101

MINIMUM RESIDENCE TIME IN A PFR

A reversible reaction A ⇔ B is conducted in a plug flow reactor. The rate equation is x r = kCa0 1 − x − ᎏ Ke where Ca0 = 4 5,800 k = exp 17.2 − ᎏ T



  9,000 K = exp −24.7 + ᎏ  T e



SOLVED PROBLEMS = na0[20(1 − x) + 15(2x)dT + 24U(T − Tw)dVr

Find the conditions for minimum Vr /V′ when conversion is 80%. The flow reactor equation is

dT −∆Hrra − 24U(T − Tw) 8,000ra −120(T − 630) ᎏ = ᎏᎏᎏ = ᎏᎏᎏ d(Vr /na0) 20 − 10x 20 − 10x

1−x−x −dna = V′Ca0dx = kCa0 ᎏ dVr Ke







Vr 1 0.8 dx 1 1 (1) ᎏ=ᎏ ᎏᎏ = ᎏᎏ ln ᎏᎏ V′ k 0 1 − x − x/Ke k(1 + 1/Ke) 0.2 − 0.8/Ke The plot of this equation shows the minimum to be Vr /V′ = 2.04 at T = 340 K.

(4)

Differential Eqs. (3) and (4) are solved simultaneously with auxiliary Eqs. (1) and (2) by ODE. The solutions with U = 5 and U = 0 are shown. 1000

1

4 2

4.00

1. 2. 3. 4.

3.50 T

3.25

800

3.00

T, U = 5 T, U = 0 x, U = 5 x, U = 0

0.5

x

3

3.75

Vr /V′

7-33

1

2.75 2.50 2.25 600

2.00

Minimum 2.04 at 340 K 1.75 1.50 330 332 334 336 338 340 342 344 346 348 350

0

T

P15.

P16.

HEAT TRANSFER IN A CYLINDRICAL REACTOR

A reaction A ⇒ 2B runs in a tube provided with a cooling jacket that keeps the wall at 630 R. Inlet is pure A at 650 R and 50 atm. Other data are stated in the following. Find the profiles of temperature and conversion along the reactor, both with heat transfer and adiabatically. 1 Tube diameter D = ᎏ ft 6 Cpa = 20, Cpb = 15 Btu/(lb mol R)

(1)

Heat transfer area dA = (4/D)dVr = 24dVr Rate equation: na na π 50k na ra = k ᎏ = k ᎏ ᎏ = ᎏ ᎏ V RT nt 0.729T 2na0 − na





 









dx P(1 − x) (1) ᎏ = 0.02 ᎏ dL 1 + 2x where several factors have been combined into the numerical coefficient. The pressure gradient due to friction is proportional to the flowing mole rate, 1 + 2x, and inversely to the density or the pressure. Here again, several factors are incorporated into a numerical coefficient, making



 (2)

dP 1 + 2x − ᎏ = 0.6 ᎏ (2) dL P The numbered equations are integrated and plotted. They show the typical fall in pressure as conversion with an increase in the number of moles proceeds at constant pressure, x = 0.48 when L = 10.

na x=1−ᎏ na0 Flow reactor:

.5 P

−dna = na0dx = radVr

∆Hr dna = −∆Hrra dVr = niCpi dT + U(T − Tw)dA

(3)

x and P/10

.4

dx ᎏ = ra d(Vr /na0) Heat balance over a differential volume dVr:



or



68.6k 1 − x =ᎏ ᎏ T 1+x

n na kP 1 − x kP ra = k ᎏa = ᎏ ᎏᎏ =ᎏ ᎏ V′ RT 3na0 − na RT 1 + 2x When put into the plug flow equation,



3,000 k = exp 7.82 − ᎏ T Heat transfer coefficient U = 5 Btu/(ft3⋅h R)

    

PRESSURE DROP AND CONVERSION IN A PFR

A reaction A ⇒ 3B takes place in a tubular flow reactor at constant temperature and an inlet pressure of 5 atm. The rate equation is

kP 1 − x na0 dx = ᎏ ᎏ AdL RT 1 + 2x

∆Hr = −8,000 Btu/(lb mol A)



0 1

0.5 Vr /na0

.3

x

.2 .1 0 0

2

4

6 L

8

10

Section 8

Process Control

Thomas F. Edgar, Ph.D., Professor of Chemical Engineering, University of Texas, Austin, TX. (Advanced Control Systems, Process Measurements, Section Editor) Cecil L. Smith, Ph.D., Principal, Cecil L. Smith Inc., Baton Rouge, LA. (Batch Process Control, Telemetering and Transmission, Digital Technology for Process Control, Process Control and Plant Safety) F. Greg Shinskey, B.S.Ch.E., Consultant (retired from Foxboro Co.), North Sandwich, NH. (Fundamentals of Process Dynamics and Control, Unit Operations Control) George W. Gassman, B.S.M.E., Senior Research Specialist, Final Control Systems, Fisher Controls International, Inc., Marshalltown, IA. (Controllers, Final Control Elements, and Regulators) Paul J. Schafbuch, Ph.D., Senior Research Specialist, Final Control Systems, Fisher Controls International, Inc., Marshalltown, IA. (Controllers, Final Control Elements, and Regulators) Thomas J. McAvoy, Ph.D., Professor of Chemical Engineering, University of Maryland, College Park, MD. (Fundamentals of Process Dynamics and Control) Dale E. Seborg, Ph.D., Professor of Chemical Engineering, University of California, Santa Barbara, CA. (Advanced Control Systems)

FUNDAMENTALS OF PROCESS DYNAMICS AND CONTROL The General Control System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-4 Feedback Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-4 Feedforward Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-4 Computer Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-4 Process Dynamics and Mathematical Models . . . . . . . . . . . . . . . . . . . . . 8-4 Open-Loop versus Closed-Loop Dynamics . . . . . . . . . . . . . . . . . . . . 8-4 Physical Models versus Empirical Models . . . . . . . . . . . . . . . . . . . . . 8-5 Nonlinear versus Linear Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-6 Simulation of Dynamic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-6 Laplace Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-6 Transfer Functions and Block Diagrams . . . . . . . . . . . . . . . . . . . . . . . 8-6 Continuous versus Discrete Models . . . . . . . . . . . . . . . . . . . . . . . . . . 8-7 Process Characteristics in Transfer Functions. . . . . . . . . . . . . . . . . . . 8-7 Fitting Dynamic Models to Experimental Data . . . . . . . . . . . . . . . . . 8-10 Feedback Control System Characteristics. . . . . . . . . . . . . . . . . . . . . . . . 8-11 Closing the Loop. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-11 On/Off Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-12 Proportional Control. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-12 Proportional-plus-Integral (PI) Control . . . . . . . . . . . . . . . . . . . . . . . 8-12 Proportional-plus-Integral-plus-Derivative (PID) Control . . . . . . . . 8-12 Controller Tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-13 Controller Performance Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-13 Tuning Methods Based on Known Process Models . . . . . . . . . . . . . . 8-14 Tuning Methods When Process Model Is Unknown . . . . . . . . . . . . . 8-15

ADVANCED CONTROL SYSTEMS Benefits of Advanced Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Advanced Control Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Feedforward Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cascade Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Time-Delay Compensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Selective and Override Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Adaptive Control. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fuzzy Logic Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Statistical Process Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multivariable Control Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Control Strategies for Multivariable Control Problems . . . . . . . . . . . Decoupling Control Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pairing of Controlled and Manipulated Variables . . . . . . . . . . . . . . . . RGA Method for 2 × 2 Control Problems . . . . . . . . . . . . . . . . . . . . . . RGA Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Model Predictive Control. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Advantages and Disadvantages of MPC . . . . . . . . . . . . . . . . . . . . . . . Economic Incentives for Automation Projects . . . . . . . . . . . . . . . . . . Basic Features of MPC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Implementation Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Integration of MPC and On-Line Optimization . . . . . . . . . . . . . . . . . Real-Time Process Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Essential Features of Optimization Problems. . . . . . . . . . . . . . . . . . .

8-16 8-16 8-16 8-18 8-19 8-19 8-20 8-21 8-21 8-22 8-22 8-23 8-23 8-24 8-24 8-25 8-25 8-25 8-25 8-25 8-27 8-27 8-27 8-28 8-1

8-2

PROCESS CONTROL

Development of Process (Mathematical) Models . . . . . . . . . . . . . . . . Formulation of the Objective Function. . . . . . . . . . . . . . . . . . . . . . . . Unconstrained Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Single Variable Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multivariable Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Constrained Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nonlinear Programming. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Expert Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8-28 8-29 8-30 8-30 8-30 8-30 8-31 8-31

UNIT OPERATIONS CONTROL Process and Instrumentation Diagrams. . . . . . . . . . . . . . . . . . . . . . . . . . Control of Heat Exchangers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Steam-Heated Exchangers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exchange of Sensible Heat. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Distillation Column Control. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Controlling Quality of a Single Product. . . . . . . . . . . . . . . . . . . . . . . . Controlling Quality of Two Products . . . . . . . . . . . . . . . . . . . . . . . . . . Chemical Reactors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Composition Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Temperature Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Controlling Evaporators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Drying Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8-31 8-32 8-32 8-32 8-33 8-33 8-33 8-33 8-33 8-35 8-35 8-36

BATCH PROCESS CONTROL Batch versus Continuous Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Batches and Recipes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Routing and Production Monitoring . . . . . . . . . . . . . . . . . . . . . . . . . . Production Scheduling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Batch Automation Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interlocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Discrete Device States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Process States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Regulatory Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sequence Logic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Batch Production Facilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equipment Suite. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Process Unit or Batch Unit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Item of Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Device . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Structured Batch Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Product Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Process Technology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8-37 8-38 8-39 8-39 8-39 8-39 8-39 8-40 8-40 8-41 8-41 8-42 8-42 8-42 8-42 8-42 8-42 8-42 8-42

PROCESS MEASUREMENTS General Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Continuous Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Accuracy and Repeatability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dynamics of Process Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . Selection Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Calibration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Temperature Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermocouples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Resistance Thermometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermistors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Filled-System Thermometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bimetal Thermometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pyrometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Accuracy of Pyrometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pressure Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Liquid-Column Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Elastic-Element Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electrical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flow Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Orifice Meter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Venturi Meter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rotameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Turbine Meter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vortex-Shedding Flowmeters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ultrasonic Flowmeters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Magnetic Flowmeters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Coriolis Mass Flowmeters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Level Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Float-Actuated Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Head Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electrical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermal Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sonic Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Physical Property Measurements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Density and Specific Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Viscosity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8-43 8-43 8-43 8-44 8-44 8-45 8-45 8-45 8-45 8-45 8-46 8-46 8-46 8-47 8-47 8-47 8-47 8-48 8-48 8-48 8-48 8-48 8-48 8-48 8-49 8-49 8-49 8-49 8-49 8-49 8-49 8-50 8-50 8-50 8-50 8-50

Refractive-Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dielectric Constant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermal Conductivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chemical Composition Analyzers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chromatographic Analyzers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Infrared Analyzers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ultraviolet and Visible-Radiation Analyzers . . . . . . . . . . . . . . . . . . . . Paramagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electroanalytical Instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conductometric Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Measurement of pH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Specific-Ion Electrodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Moisture Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dew-Point Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Piezoelectric Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Capacitance Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Oxide Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Photometric Moisture Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Other Transducers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gear Train . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Differential Transformer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hall-Effect Sensors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sampling Systems for Process Analyzers . . . . . . . . . . . . . . . . . . . . . . . . . Selecting the Sampling Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sample Withdrawal from Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sample Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sample Conditioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8-50 8-50 8-50 8-50 8-51 8-51 8-51 8-51 8-51 8-51 8-51 8-51 8-51 8-51 8-52 8-52 8-52 8-52 8-52 8-52 8-52 8-52 8-52 8-52 8-52 8-53 8-53

TELEMETERING AND TRANSMISSION Analog Signal Transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Digital Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Analog Input and Outputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pulse Inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Serial Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Microprocessor-Based Transmitters . . . . . . . . . . . . . . . . . . . . . . . . . . Transmitter/Actuator Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Filtering and Smoothing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alarms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8-53 8-53 8-53 8-54 8-54 8-54 8-54 8-55 8-55

DIGITAL TECHNOLOGY FOR PROCESS CONTROL Hierarchy of Information Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Measurement Devices and Actuators . . . . . . . . . . . . . . . . . . . . . . . . . Regulatory Controls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Supervisory Controls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Production Controls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Corporate Information Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Distributed Control Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Distributed Database and the Database Manager . . . . . . . . . . . . . . . . . Historical Database Subsystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Process Control Language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Single-Loop Controls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Programmable Logic Controllers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Intercomputer Communications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8-56 8-56 8-57 8-57 8-57 8-57 8-57 8-58 8-59 8-59 8-59 8-60 8-60

CONTROLLERS, FINAL CONTROL ELEMENTS, AND REGULATORS Electronic and Pneumatic Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . Electronic Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pneumatic Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Control Valves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Valves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Special Application Valves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Actuators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Valve-Control Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Valve Application Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Other Process Valves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Valves for On/Off Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pressure Relief Valves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Check Valves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Adjustable Speed Pumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Regulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Self-Operated Regulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pilot-Operated Regulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Over-Pressure Protection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8-61 8-61 8-62 8-64 8-64 8-65 8-66 8-67 8-73 8-77 8-77 8-78 8-78 8-79 8-79 8-80 8-81 8-81

PROCESS CONTROL AND PLANT SAFETY Role of Automation in Plant Safety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Integrity of Process Control Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . Considerations in Implementation of Safety Interlock Systems. . . . . . . Interlocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8-81 8-82 8-82 8-83 8-84

Nomenclature Symbol

Definition

A Aa Ac Am Av A1 b B B i* c,C cA Cd Ci C i* CL Co Cr Cv D D i* D(s) e E f F,f FL gc gi G Gc Gf GL Gm Gp Gt Gv hi h1 H i Ii j J k kf kr K Kc KL Km Kp Ku L Lp m,M mc Mo Mr Mw n N p1 pa pv P Pu q qb Qa rc R

Area Actuator area Amplitude of controlled variable Output amplitude limits Cross sectional area of valve Cross sectional area of tank Controller output bias Bottoms flow rate Limit on control Controlled variable Concentration of A Discharge coefficient Inlet concentration Limit on control move Specific heat of liquid Integration constant Heat capacity of reactants Valve flow coefficient Distillate flow rate Limit on output Decoupler transfer function Error Economy of evaporator Function of time Feed flow rate Pressure recovery factor Unit conversion constant Algebraic inequality constraint Transfer function Controller transfer function Feedforward controller transfer function Load transfer function Sensor transfer function Process transfer function Transmitter transfer function Valve transfer function Algebraic equality constraints Liquid head in tank Latent heat of vaporization Summation index Impulse response coefficient Time index Objective function or performance index Time index Flow coefficient Kinetic rate constant Gain Controller gain Load transfer function gain Measurement gain Process gain Ultimate controller gain (stability) Disturbance or load variable Sound pressure level Manipulated variable Number of constraints Mass flow Mass of reactants Molecular weight Number of data points, number of stages or effects Number of inputs/outputs, model horizon Pressure Actuator pressure Vapor pressure Proportional band (%) Proportional band (ultimate) Radiated energy flux Energy flux to a black body Flow rate Number of constraints Equal percentage valve characteristic

Symbol

Definition

R,r RT R1 s s Si t T Tb Tf TR u U V Vs w wi W x xi xT X y Y z zi Z

Set point Resistance in temperature sensor Valve resistance Laplace transform variable Search direction Step response coefficient Time Temperature Base temperature Exhaust temperature Reset time Controller output Heat transfer coefficient Volume Product value Mass flow rate Weighting factor Steam flow rate Mass fraction Optimization variable Pressure drop ratio factor Transform of deviation variable Process output, controlled variable, valve travel Controller tuning law, expansion factor z-transform variable Feed mole fraction (distillation) Compressibility factor

α αT β γ δ ∆q ∆t ∆T ∆u ε ζ θ λ Λ ξ ρ σ Στ τ τD τF τI τL τn τp τo φPI

Digital filter coefficient Temperature coefficient of resistance Resistance thermometer parameter Ratio of specific heats Move suppression factor Load step change Time step Temperature change Control move Spectral emissivity, step size Damping factor (second order system) Time delay Relative gain array parameter, wavelength Relative gain array Deviation variable Density Stefan-Boltzmann constant Total response time Time constant Derivative time (PID controller) Filter time constant Integral time (PID controller) Load time constant Natural period of closed loop Process time constant Period of oscillation Phase lag

A b c eff F i L m p s set t u v

Species A Best Controller Effective Feedforward Initial, inlet Load, disturbance Measurement or sensor Process Steady state Set point value Transmitter Ultimate Valve

Greek symbols

Subscripts

8-3

FUNDAMENTALS OF PROCESS DYNAMICS AND CONTROL THE GENERAL CONTROL SYSTEM A process is shown in Fig. 8-1 with a manipulated input M, a load input L, and a controlled output C, which could be flow, pressure, liquid level, temperature, composition, or any other inventory, environmental, or quality variable that is to be held at a desired value identified as the set point R. The load may be a single variable or aggregate of variables acting either independently or manipulated for other purposes, affecting the controlled variable much as the manipulated variable does. Changes in load may occur randomly as caused by changes in weather, diurnally with ambient temperature, manually when operators change production rate, stepwise when equipment is switched in or out of service, or cyclically as the result of oscillations in other control loops. Variations in load will drive the controlled variable away from set point, requiring a corresponding change in the manipulated variable to bring it back. The manipulated variable must also change to move the controlled variable from one set point to another. An open-loop system positions the manipulated variable either manually or on a programmed basis, without using any process measurements. This operation is acceptable for well-defined processes without disturbances. An automanual transfer switch is provided to allow manual adjustment of the manipulated variable in case the process or the control system is not performing satisfactorily. A closed-loop system uses the measurement of one or more process variables to move the manipulated variable to achieve control. Closedloop systems may include feedforward, feedback, or both. Feedback Control In a feedback control loop, the controlled variable is compared to the set point R, with the difference, deviation, or error e acted upon by the controller to move m in such a way as to minimize the error. This action is specifically negative feedback, in that an increase in deviation moves m so as to decrease the deviation. (Positive feedback would cause the deviation to expand rather than diminish and therefore does not regulate.) The action of the controller is selectable to allow use on process gains of both signs. The controller has tuning parameters related to proportional, integral, derivative, lag, deadtime, and sampling functions. A negative feedback loop will oscillate if the controller gain is too high, but if it is too low, control will be ineffective. The controller parameters must be properly related to the process parameters to ensure closed-loop stability while still providing effective control. This is accomplished first by the proper selection of control modes to satisfy the requirements of the process, and second by the appropriate tuning of those modes. Feedforward Control A feedforward system uses measurements of disturbance variables to position the manipulated variable in such a way as to minimize any resulting deviation. The disturbance

FIG. 8-1

8-4

variables could be either measured loads or the set point, the former being more common. The feedforward gain must be set precisely to offset the deviation of the controlled variable from the set point. Feedforward control is usually combined with feedback control to eliminate any offset resulting from inaccurate measurements and calculations and unmeasured load components. The feedback controller can either bias or multiply the feedforward calculation. Computer Control Computers have been used to replace analog PID controllers, either by setting set points of lower level controllers in supervisory control, or by driving valves directly in direct digital control. Single-station digital controllers perform PID control in one or two loops, including computing functions such as mathematical operations, characterization, lags, and deadtime, with digital logic and alarms. Distributed control systems provide all these functions, with the digital processor shared among many control loops; separate processors may be used for displays, communications, file servers, and the like. A host computer may be added to perform highlevel operations such as scheduling, optimization, and multivariable control. More details on computer control are provided later in this section. PROCESS DYNAMICS AND MATHEMATICAL MODELS GENERAL REFERENCES: Seborg, Edgar, and Mellichamp, Process Dynamics and Control, Wiley, New York, 1989; Marlin, Process Control, McGraw-Hill, New York, 1995; Ogunnaike and Ray, Process Dynamics Modeling and Control, Oxford University Press, New York, 1994; Smith and Corripio, Principles and Practices of Automatic Process Control, Wiley, New York, 1985

Open-Loop versus Closed-Loop Dynamics It is common in industry to manipulate coolant in a jacketed reactor in order to control conditions in the reactor itself. A simplified schematic diagram of such a reactor control system is shown in Fig. 8-2. Assume that the reactor temperature is adjusted by a controller that increases the coolant flow in proportion to the difference between the desired reactor temperature and the temperature that is measured. The proportionality constant is Kc. If a small change in the temperature of the inlet stream occurs, then depending on the value of Kc, one might observe the reactor temperature responses shown in Fig. 8-3. The top plot shows the case for no control (Kc = 0), which is called the open loop, or the normal dynamic response of the process by itself. As Kc increases, several effects can be noted. First, the reactor temperature responds faster and faster. Second, for the initial increases in Kc, the maximum deviation in the reactor temperature becomes smaller. Both of these effects are desirable so that disturbances from normal operation have

Block diagram for feedforward and feedback control.

FUNDAMENTALS OF PROCESS DYNAMICS AND CONTROL

FIG. 8-2

Reactor control system.

as small an effect as possible on the process under study. As the gain is increased further, eventually a point is reached where the reactor temperature oscillates indefinitely, which is undesirable. This point is called the stability limit, where Kc = Ku, the ultimate controller gain. Increasing Kc further causes the magnitude of the oscillations to increase, with the result that the control valve will cycle between full open and closed. The responses shown in Fig. 8-3 are typical of the vast majority of regulatory loops encountered in the process industries. Figure 8-3 shows that there is an optimal choice for Kc, somewhere between 0 (no control) and Ku (stability limit). If one has a dynamic model of a process, then this model can be used to calculate controller settings. In Fig. 8-3, no time scale is given, but rather the figure shows relative responses. A well-designed controller might be able to speed up the response of a process by a factor of roughly two to four. Exactly how fast the control system responds is determined by the dynamics of the process itself. Physical Models versus Empirical Models In developing a dynamic process model, there are two distinct approaches that can be

FIG. 8-3

Typical control system responses.

8-5

taken. The first involves models based on first principles, called physical models, and the second involves empirical models. The conservation laws of mass, energy, and momentum form the basis for developing physical models. The resulting models typically involve sets of differential and algebraic equations that must be solved simultaneously. Empirical models, by contrast, involve postulating the form of a dynamic model, usually as a transfer function, which is discussed below. This transfer function contains a number of parameters that need to be estimated. For the development of both physical and empirical models, the most expensive step normally involves verification of their accuracy in predicting plant behavior. To illustrate the development of a physical model, a simplified treatment of the reactor, shown in Fig. 8-2 is used. It is assumed that the reactor is operating isothermally and that the inlet and exit volumetric flows and densities are the same. There are two components, A and B, in the reactor, and a single first order reaction of A → B takes place. The inlet concentration of A, which we shall call ci, varies with time. A dynamic mass balance for the concentration of A (cA) can be written as follows: dcA Vᎏ = Fci − FcA − krVc (8-1) dt In Eq. (8-1), the flow in of A is Fci, the flow out is FcA, and the loss via reaction is krVcA, where V = reactor volume and kr = kinetic rate constant. In this example, ci is the input, or forcing variable, and cA is the output variable. If V, F, and kr are constant, Eq. (8-1) can be rearranged by dividing by (F + krV) so that it only contains two groups of parameters. The result is: dc τ ᎏ = Kci − cA (8-2) dt where τ = V/(F + krV) and K = F/(F + krV). For this example, the resulting model is a first-order differential equation in which τ is called the time constant and K the process gain.

8-6

PROCESS CONTROL

As an alternative to deriving Eq. (8-2) from a dynamic mass balance, one could simply postulate a first-order differential equation to be valid (empirical modeling). Then it would be necessary to estimate values for τ and K so that the postulated model described the reactor’s dynamic response. The advantage of the physical model over the empirical model is that the physical model gives insight into how reactor parameters affect the values of τ, and K, which in turn affects the dynamic response of the reactor. Nonlinear versus Linear Models If V, F, and k are constant, then Eq. (8-1) is an example of a linear differential equation model. In a linear equation, the output and input variables and their derivatives only appear to the first power. If the rate of reaction were second order, then the resulting dynamic mass balance would be: dc V ᎏ = Fci − FcA − kr VcA2 (8-3) dt Since cA appears in this equation to the second power, the equation is nonlinear. The difference between linear systems and nonlinear systems can be seen by considering the steady state behavior of Eq. (8-1) compared to Eq. (8-3) (the left-hand side is zero; i.e., dcA /dt = 0). For a given change in ci, ∆ci, the change in cA calculated from Eq. (8-1), or ∆c, is always proportional to ∆ci, and the proportionality constant is K [see Eq. (8-2)]. The change in the output of a system divided by a change in the input to the system is called the process gain. Linear systems have constant process gains for all changes in the input. By contrast, Eq. (8-3) gives a ∆c that varies in proportion to ∆ ci but with the proportionality factor being a function of the concentration levels in the reactor. Thus, depending on where the reactor operates, a change in ci produces different changes in cA. In this case, the process has a nonlinear gain. Systems with nonlinear gains are more difficult to control than linear systems that have constant gains. Simulation of Dynamic Models Linear dynamic models are particularly useful for analyzing control-system behavior. The insight gained through linear analysis is invaluable. However, accurate dynamic process models can involve large sets of nonlinear equations. Analytical solution of these models is not possible. Thus, in these cases, one must turn to simulation approaches to study process dynamics and the effect of process control. Equation (8-3) will be used to illustrate the simulation of nonlinear processes. If dcA /dt on the left-hand side of Eq. (8-3) is replaced with its finite difference approximation, one gets: cA(t) + ∆ t ⋅ [Fci(t) − FcA(t) − krVcA(t)2] cA(t + ∆ t) = ᎏᎏᎏᎏ V

(8-4)

Starting with an initial value of cA and knowing ci(t), Eq. (8-4) can be solved for cA(t + ∆ t). Once cA(t + ∆ t) is known, the solution process can be repeated to calculate cA(t + 2∆ t), and so on. This approach is called the Euler integration method; while it is simple, it is not necessarily the best approach to numerically integrating nonlinear differential equations. To achieve accurate solutions with an Euler approach, one often needs to take small steps in time, ∆t. A number of more sophisticated approaches are available that allow much larger step sizes to be taken but require additional calculations. One widely used approach is the fourth-order Runge Kutta method, which involves the following calculations: Fci(t) − FcA − krVcA2 define f(cA,t) = ᎏᎏᎏ (8-5) V then with

cA(t + ∆t) = cA(t) + ∆t(m1 + 2m2 + 2m3 + m4) m1 = f [cA(t), t]

(8-6) (8-7)

m1∆t ∆t m2 = f cA(t) + ᎏ , t + ᎏ 2 2



(8-8)

∆t m2∆t m3 = f cA(t) + ᎏ , t + ᎏ 2 2



(8-9)





m4 = f [cA(t) + m3∆t, t + ∆t]

(8-10)

In this method, the mi’s are calculated sequentially in order to take a step in time. Even though this method requires calculation of the four additional mi values, for equivalent accuracy the fourth-order Runge Kutta method can result in a faster numerical solution, since a larger step, ∆t, can be taken with it. Increasingly sophisticated simulation packages are being used to calculate the dynamic behavior of processes and test control system behavior. These packages have good user interfaces, and they can handle stiff systems where some variables respond on a time scale that is much much faster or slower than other variables. A simple Euler approach cannot effectively handle stiff systems, which frequently occur in chemical-process models. Laplace Transforms When mathematical models are used to describe process dynamics in conjunction with control-system analysis, the models generally involve linear differential equations. Laplace transforms are very effective for solving linear differential equations. The key advantage of using Laplace transforms is that they convert differential equations into algebraic equations. The resulting algebraic equations are easier to solve than the original differential equations. When the Laplace transform is applied to a linear differential equation in time, the result is an algebraic equation in a new variable, s, called the Laplace variable. To get the solution to the original differential equation, one needs to invert the Laplace transform. Table 8-1 gives a number of useful Laplace transform pairs, and more extensive tables are available (Seborg, Edgar, and Mellichamp, Process Dynamics and Control, Wiley, New York, 1989). To illustrate how Laplace transforms work, consider the problem of solving Eq. (8-2), subject to the initial condition that cA = 0 at t = 0, and ci is constant. If cA were not initially zero, one would define a deviation variable between cA and its initial value (cA − c0). Then the transfer function would be developed using this deviation variable. Taking the Laplace transform of both sides of Eq. (8-2) gives: τdcA £ ᎏ = £(Kci) − £(cA) (8-11) dt Denoting the £(c) as CA(s) and using the relationships in Table 8-1 gives: Kc τsCA(s) = ᎏi − CA(s) (8-12) s Equation (8-12) can be solved for CA to give: Kci /s CA(s) = ᎏ (8-13) τs + 1 Using the entries in Table 8-1, Eq. (8-13) can be inverted to give the transient response of cA as:





cA(t) = (Kci)(1 − e−t/τ)

(8-14)

Equation (8-14) shows that cA starts from 0 and builds up exponentially to a final concentration of Kci. Note that to get Eq. (8-14), it was only necessary to solve the algebraic Eq. (8-12) and then find the inverse of CA(s) in Table 8-1. The original differential equation was not solved directly. In general, techniques such as partial fraction expansion must be used to solve higher order differential equations with Laplace transforms. Transfer Functions and Block Diagrams A very convenient and compact method of representing the process dynamics of linear systems involves the use of transfer functions and block diagrams. A transfer function can be obtained by starting with a physical model as TABLE 8-1

Frequently Used Laplace Transforms

Time function, f(t)

Transform, F(s)

A At Ae−at A(1 − e−t/τ) A sin (ωt) f(t − θ) df/dt ∫ f(t) dt

A/s2 A/s A/(s + a) A/[s(τs + 1)] Aω/(s2 + ω2) e−θsF(s) sF(s) − f(0) F(s)/s

FUNDAMENTALS OF PROCESS DYNAMICS AND CONTROL

8-7

discussed previously. If the physical model is nonlinear, then it first needs to be linearized around an operating point. The resulting linearized model is then approximately valid in a region around this operating point. To illustrate how transfer functions are developed, Eq. (8-2) will again be used. First, one defines deviation variables, which are the process variables minus their steady state values at the operating point. For Eq. (8-2), there would be deviation variables for both cA and ci, and these are defined as: ξ = cA − cs

(8-15)

ξi = ci − cis

(8-16)

where the subscript s stands for steady state. Substitution of Eq. (8-15) and (8-16) into Eq. (8-2) gives: dξ τ ᎏ = K ξi − ξ + (Kcis − cs) (8-17) dt The term in parentheses in Eq. (8-17) is zero at steady state and thus it can be dropped. Next the Laplace transform is taken, and the resulting algebraic equation solved. Denoting X(s) as the Laplace transform of ξ and X i(s) as the transform of ξ i, the final transfer function can be written as: K X (8-18) ᎏ=ᎏ Xi τs + 1 Equation (8-18) is an example of a first-order transfer function. As mentioned above, an alternative to formally deriving Eq. (8-18) involves simply postulating its form and then identifying its two parameters, the process gain K and time constant τ, to fit the process under study. In fitting the parameters, data can be generated by forcing the process. If step forcing is used, then the resulting response is called the process reaction curve. Often transfer functions are placed in block diagrams, as shown in Fig. 8-4. Block diagrams show how changes in an input variable affect an output variable. Block diagrams are a means of concisely representing the dynamics of a process under study. Since linearity is assumed in developing a block diagram, if more than one variable affects an output, the contributions from each can be added together. Continuous versus Discrete Models The preceding discussion has focused on systems where variables change continuously with time. Most real processes have variables that are continuous in nature, such as temperature, pressure, and flow. However, some processes involve discrete events, such as the starting or stopping of a pump. In addition, modern plants are controlled by digital computers, which are discrete by nature. In controlling a process, a digital system samples variables at a fixed rate, and the resulting system is a sampled data system. From one sampling instant until the next, variables are assumed to remain fixed at their sampled values. Similarly, in controlling a process, a digital computer sends out signals to control elements, usually valves, at discrete instants of time. These signals remain fixed until the next sampling instant. Figure 8-5 illustrates the concept of sampling a continuous function. At integer values of the sampling rate, ∆ t, the value of the variable to be sampled is measured and held until the next sampling instant. To deal with sampled data systems, the z transform has been developed. The z transform of the function given in Fig. 8-5 is defined as

FIG. 8-5

Sampled data example.

In an analogous manner to Laplace transforms, one can develop transfer functions in the z domain as well as block diagrams. Tables of z transform pairs have been published (Seborg, Edgar, and Mellichamp, Process Dynamics and Control, Wiley, New York, 1989) so that the discrete transfer functions can be inverted back to the time domain. The inverse gives the value of the function at the discrete sampling instants. Sampling a continuous variable results in a loss of information. However, in practical applications, sampling is fast enough that the loss is typically insignificant and the difference between continuous and discrete modeling is small in terms of its effect on control. Increasingly, model predictive controllers that make use of discrete dynamic models are being used in the process industries. The purpose of these controllers is to guide a process to optimum operating points. These model predictive control algorithms are typically run at much slower sampling rates than are used for basic control loops such as flow control or pressure control. The discrete dynamic models used are normally developed from data generated from plant testing as discussed hereafter. For a detailed discussion of modeling sampled data systems, the interested reader is referred to textbooks on digital control (Astrom and Wittenmark, Computer Controlled Systems, Prentice Hall, Englewood Cliffs, NJ, 1984). Process Characteristics in Transfer Functions In many cases, process characteristics are expressed in the form of transfer functions. In the previous discussion, a reactor example was used to illustrate how a transfer function could be derived. Here, another system involving flow out of a tank, shown in Fig. 8-6, is considered. Proportional Element First, consider the outflow through the exit valve on the tank. If the flow through the line is turbulent, then Bernoulli’s equation can be used to relate the flow rate through the valve to the pressure drop across the valve as: f1 = kf Av2(h gc− h 1  0)

where f1 = flow rate, kf = flow coefficient, Av = cross sectional area of the restriction, gc = constant, h1 = liquid head in tank, and h0 = atmo-

∞

Z( f ) =  f(n ∆t)z−n

(8-19)

n=0

FIG. 8-4

First-order transfer function.

(8-20)

FIG. 8-6

Single tank with exit valve.

8-8

PROCESS CONTROL

spheric pressure. This relationship between flow and head is nonlinear, and it can be linearized around a particular operating point to give: 1 f1 − f1s = ᎏ (h1 − h1s) (8-21) R1 2 2 where R1 = f1s /(gc kf A ) is called the resistance of the valve in analogy with an electrical resistance. The transfer function relating changes in flow to changes in head is shown in Fig. 8-7, and it is an example of a pure gain system with no dynamics. In this case, the process gain is K = 1/R1. Such a system has an instantaneous dynamic response, and for a step change in head, there is an immediate step change in flow, as shown in Fig. 8-8. The exact magnitude of the step in flow depends on the operating flow, f1s, as the definition of R1 shows. First-Order Lag (Time Constant Element) Next consider the system to be the tank itself. A dynamic mass balance on the tank gives: dh A1 ᎏ1 = fi − f1 (8-22) dt where A1 is the cross sectional area of the tank and fi is the inlet flow. By substituting Eq. (8-21) into Eq. (8-22), and following the approach discussed above for deriving transfer functions, one can develop the transfer function relating changes in h1 to changes in fi. The resulting transfer function is another example of a first-order system, shown in Fig. 8-4, and it has a gain, K = R1, and a time constant, τ1 = R1A1. For a step change in fi, h1 follows a decaying exponential response from its initial value, h1s, to a final value of h1s + R1∆fi (Fig. 8-9). At a time equal to τ1, the transient in h1 is 63 percent finished; and at 3τ1, the response is 95 percent finished. These percentages are the same for all firstorder processes. Thus, knowledge of the time constant of a first-order process gives insight into how fast the process responds to sudden input changes. Capacity Element Now consider the case where the valve in Fig. 8-7 is replaced with a pump. In this case, it is reasonable to assume that the exit flow from the tank is independent of the level in the tank. For such a case, Eq. (8-22) still holds, except that f1 no longer depends on h1. For changes in fi, the transfer function relating changes in h1 to changes in fi is shown in Fig. 8-10. This is an example of a pure capacity process, also called an integrating system. The cross sectional area of the tank is the chemical process equivalent of an electrical capacitor. If the inlet flow is step forced while the outlet is held

 

FIG. 8-7

FIG. 8-8

FIG. 8-9

Response of first-order system.

constant, then the level builds up linearly as shown in Fig. 8-11. Eventually the liquid would overflow the tank. Second-Order Element Because of their linear nature, transfer functions can be combined in a straightforward manner. Consider the two tank system shown in Fig. 8-12. For tank 1, the transfer function relating changes in f1 to changes in fi can be obtained by combining two first order transfer functions to give: F1(s) 1 (8-23) ᎏ = ᎏᎏ Fi(s) R1A1s + 1 Since f1 is the inlet flow to tank 2, the transfer function relating changes in h2 to changes in f1 has the same form as that given in Fig. 8-4: R2 H2(s) (8-24) ᎏ = ᎏᎏ F1(s) A2 R2 s + 1 Equations (8-23) and (8-24) can be multiplied together to give the final transfer function relating changes in h2 to changes in fi as shown in Fig. 8-13. This is an example of a second-order transfer function. This transfer function has a gain R1R2 and two time constants, R1A1 and R2 A2. For two equal tanks, a step change in fi produces the S-shaped response in level in the second tank shown in Fig. 8-14. General Second-Order Element Figure 8-3 illustrates the fact that closed loop systems often exhibit oscillatory behavior. A general

Proportional element transfer function.

Response of proportional element.

FIG. 8-10

Pure capacity transfer function.

FIG. 8-11

Response of pure capacity system.

FUNDAMENTALS OF PROCESS DYNAMICS AND CONTROL

FIG. 8-12

Two tanks in series.

FIG. 8-13

Second-order transfer function.

FIG. 8-15

General second-order transfer function.

FIG. 8-16

Response of general second-order system.

8-9

Higher-Order Lags If a process is described by a series of n firstorder lags, the overall system response becomes proportionally slower with each lag added. The special case of a series of n first-order lags with equal time constants has a transfer function given by: K G(s) = ᎏn (8-25) (τs + 1) The step response of this transfer function is shown in Fig. 8-19. Note that all curves reach about 60 percent of their final value at t = nτ.

FIG. 8-14

Response of second-order system.

second-order transfer function that can exhibit oscillatory behavior is important for the study of automatic control systems. Such a transfer function is given in Fig. 8-15. For a step input, the transient responses shown in Fig. 8-16 result. As can be seen when ζ < 1, the response oscillates and when ζ > 1, the response is S-shaped. Few open-loop chemical processes exhibit an oscillating response; most exhibit an S-shaped step response. Distance-Velocity Lag (Dead-Time Element) The dead-time element, commonly called a distance-velocity lag, is often encountered in process systems. For example, if a temperature-measuring element is located downstream from a heat exchanger, a time delay occurs before the heated fluid leaving the exchanger arrives at the temperature measurement point. If some element of a system produces a dead-time of θ time units, then an input to that unit, f(t), will be reproduced at the output as f(t − θ). The transfer function for a pure dead-time element is shown in Fig. 8-17, and the transient response of the element is shown in Fig. 8-18.

FIG. 8-17

Dead-time transfer function.

FIG. 8-18

Response of dead-time system.

8-10

FIG. 8-19

PROCESS CONTROL

FIG. 8-20

Example of 2 × 2 transfer function.

FIG. 8-21

Plot of experimental data.

Response of nth order lags.

Higher-order systems can be approximated by a first or second-order plus dead-time system for control system design. Multiinput, Multioutput Systems The dynamic systems considered up to this point have been examples of single-input, singleoutput (SISO) systems. In chemical processes, one often encounters systems where one input can affect more than one output. For example, assume that one is studying a distillation tower in which both reflux and boilup are manipulated for control purposes. If the output variables are the top and bottom product compositions, then each input affects both outputs. For this distillation example, the process is referred to as a 2 × 2 system to indicate the number of inputs and outputs. In general, multiinput, multioutput (MIMO) systems can have n inputs and m outputs with n ≠ m, and they can be nonlinear. Such a system would be called an n × m system. An example of a transfer function for a 2 × 2 linear system is given in Fig. 8-20. Note that since linear systems are involved, the effects of the two inputs on each output are additive. In many process-control systems, one input is selected to control one output in a MIMO system. For m output there would be m such selections. For this type of control strategy, one needs to consider which inputs and outputs to couple together, and this problem is referred to as loop pairing. Another important issue that arises involves interaction between control loops. When one loop makes a change in its manipulated variable, the change affects the other loops in the system. These changes are the direct result of the multivariable nature of the process. In some cases, the interaction can be so severe that overall control-system performance is drastically reduced. Finally, some of the modern approaches to process control tackle the MIMO problem directly, and they simultaneously use all manipulated variables to control all output variables rather than pairing one input to one output (see later section on multivariable control). Fitting Dynamic Models to Experimental Data In developing empirical transfer functions, it is necessary to identify model parameters from experimental data. There are a number of approaches to process identification that have been published. The simplest approach involves introducing a step test into the process and recording the response of the process, as illustrated in Fig. 8-21. The x’s in the figure represent the recorded data. For purposes of illustration, the process under study will be assumed to be first order with deadtime and have the transfer function:

C(s) G(s) = ᎏ = K exp(−θs)/(τs + 1) (8-26) M(s) The response produced by Eq. (8-26), c(t), can be found by inverting the transfer function, and it is also shown in Fig. 8-21 for a set of model parameters, K, τ, and θ, fitted to the data. These parameters are calculated using optimization to minimize the squared difference between the model predictions and the data, i.e., a least squares approach. Let each measured data point be represented by cj (measured response), tj (time of measured response), j = 1 to n. Then the least squares problem can be formulated as: min τ,θ,K

n

ˆ )]  [c − c(t j

j

2

(8-27)

j=0

which can be solved to calculate the optimal values of K, τ, and θ. A number of software packages are available for minimizing Eq. (8-27). One operational problem that step forcing causes is the fact that the process under study is moved away from its steady state operating point. Plant managers may be reluctant to allow large steady state changes, since normal production will be disturbed by the changes. As a result, alternative methods of forcing actual processes have been developed, and these included pulse testing and pseudo random binary signal (PRBS) forcing, both of which are illustrated in Fig. 8-22. With pulse forcing, one introduces a step, and then after a period of time the input is returned to its original value. The result is that the process dynamics are excited, but after the forcing, the process returns to its original steady state. PRBS forcing involves a series of pulses of fixed height and random duration, as shown in Fig. 8-22. The advantage of PRBS is that forcing can be concentrated on particular frequency ranges that are important for control-system design.

FUNDAMENTALS OF PROCESS DYNAMICS AND CONTROL

8-11

FIG. 8-23 Both load regulation and setpoint response require high gains for the feedback controller.

FIG. 8-22

Pulse and PRBS testing.

Transfer function models are linear in nature, but chemical processes are known to exhibit nonlinear behavior. One could use the same type of optimization objective as given in Eq. (8-26) to determine parameters in nonlinear first-principle models, such as Eq. (8-3) presented earlier. Also, nonlinear empirical models, such as neural network models, have recently been proposed for process applications. The key to the use of these nonlinear empirical models is having high-quality process data, which allows the important nonlinearities to be identified. FEEDBACK CONTROL SYSTEM CHARACTERISTICS GENERAL REFERENCES: Shinskey, Feedback Controllers for the Process Industries, McGraw-Hill, New York, 1994; Seborg, Edgar, and Mellichamp, Process Dynamics and Control, Wiley, New York, 1989.

FIG. 8-24

There are two objectives in applying feedback control: regulating the controlled variable at set point following changes in load, and responding to set-point changes; the latter called servo operation. In fluid processes, almost all control loops must contend with variations in load; therefore, regulation is of primary importance. While most loops will operate continuously at fixed set points, frequent changes in set points can occur in flow loops and in batch production. The most common mechanism for achieving both objectives is feedback control, because it is the simplest and most universally applicable approach to the problem. Closing the Loop The simplest representation of the closed feedback loop is shown in Fig. 8-23. The load is shown entering the process at the same point as the manipulated variable because that is the most common point of entry, and also because, lacking better information, the transfer function gains in the path of the manipulated variable are the best estimates of those in the load path. In general, the load never impacts directly on the controlled variable without passing through the dominant lag in the process. Where the load is unmeasured, its current value can be observed to be the controller output required to keep the controlled variable C at set point R. If the loop is opened, either by placing the controller in manual operation or by setting its gains to zero, the load will have complete influence over the controlled variable, and the set point will have none. Only by closing the loop with controller gains as high as possible will the influence of the load be minimized and that of the set point be maximized. There is a practical limit to the controller gains, however, at the point where the controlled variable develops a uniform oscillation (see Fig. 8-24). This is defined as the limit of stability, and it is reached when the product of gains in the loop GcGvGp for that frequency of oscillation is equal to 1.0. If a change in a parameter in the loop causes an increase from this condition, oscillations will expand, creating a dangerous situation where safe limits of operation could be exceeded. Consequently, control loops should be left in a condition

Transition to instability as controller gain increases.

8-12

PROCESS CONTROL

where the loop gain is less than 1.0 by a safe margin that allows for possible variations in process parameters. In controller design, a choice must be made between performance and robustness. Performance is a measure of how well a given controller with certain parameter settings regulates a variable relative to the best loop performance with optimal controller settings. Robustness is a measure of how small a change in a process parameter is required to bring the loop from its current state to the limit of stability. Increasing controller performance by raising its gains can be expected to decrease robustness. Both performance and robustness are functions of the process being controlled, the selection of the controller, and the tuning of the controller parameters. On/Off Control An on/off controller is used for manipulated variables having only two states. They commonly control temperatures in homes, electric water-heaters and refrigerators, and pressure and liquid level in pumped storage systems. On/off control is satisfactory where slow cycling is acceptable because it always leads to cycling when the load lies between the two states of the manipulated variable. The cycle will be positioned symmetrically about the set point only if the normal value of the load is equidistant between the two states of the manipulated variable. The period of the symmetrical cycle will be approximately 4θ, where θ is the deadtime in the loop. If the load is not centered between the states of the manipulated variable, the period will tend to increase, and the cycle follows a sawtooth pattern. Every on/off controller has some degree of deadband, also known as lockup, or differential gap. Its function is to prevent erratic switching between states, thereby extending the life of contacts and motors. Instead of changing states precisely when the controlled variable crosses set point, the controller will change states at two different points for increasing and decreasing signals. The difference between these two switching points is the deadband (see Fig. 8-25); it increases the amplitude and period of the cycle, similar to the effect of dead time. A three-state controller is used to drive either a pair of independent on/off actuators such as heating and cooling valves, or a bidirectional motorized actuator. The controller is actually two on/off controllers, each with deadband, separated by a dead zone. When the controlled variable lies within the dead zone, neither output is energized. This controller can drive a motorized valve to the point where the manipulated variable matches the load, thereby avoiding cycling. Proportional Control A proportional controller moves its output proportional to the deviation in the controlled variable from set point: 100 u = Kce + b = ᎏ e + b (8-28) P where e = ⫾(r − c), the sign selected to produce negative feedback. In

FIG. 8-25

On/off controller characteristics.

some controllers, proportional gain Kc is expressed as a pure number; in others, it is set as 100/P, where P is the proportional band in percent. The output bias b of the controller is also known as manual reset. The proportional controller is not a good regulator, because any change in output to a change in load results in a corresponding change in the controlled variable. To minimize the resulting offset, the bias should be set at the best estimate of the load and the proportional band set as low as possible. Processes requiring a proportional band of more than a few percent will control with unacceptable values of offset. Proportional control is most often used for liquid level where variations in the controlled variable carry no economic penalty, and where other control modes can easily destabilize the loop. It is actually recommended for controlling the level in a surge tank when manipulating the flow of feed to a critical downstream process. By setting the proportional band just under 100 percent, the level is allowed to vary over the full range of the tank capacity as inflow fluctuates, thereby minimizing the resulting rate of change of manipulated outflow. This technique is called averaging level control. Proportional-plus-Integral (PI) Control Integral action eliminates the offset described above by moving the controller output at a rate proportional to the deviation from set point. Although available alone in an integral controller, it is most often combined with proportional action in a PI controller: 100 1 u = ᎏ e + ᎏ e dt + C0 (8-29) P τI where τI is the integral time constant in minutes; in some controllers, it is introduced as integral gain or reset rate 1/τI in repeats per minute. The last term in the equation is the constant of integration, the value the controller output has when integration begins. The PI controller is by far the most commonly used controller in the process industries. The summation of the deviation with its integral in the above equation can be interpreted in terms of frequency response of the controller (Seborg, Edgar, and Mellichamp, Process Dynamics and Control, Wiley, New York, 1989). The PI controller produces a phase lag between zero and 90 degrees: τ0 φPI = −tan−1 ᎏ (8-30) 2πτI where τ0 is the period of oscillation of the loop. The phase angle should be kept between 15 degrees for lag-dominant processes and 45 degrees for dead-time-dominant processes for optimum results. Proportional-plus-Integral-plus-Derivative (PID) Control The derivative mode moves the controller output as a function of the rate-of-change of the controlled variable, which adds phase lead to the controller, increasing its speed of response. It is normally combined





FUNDAMENTALS OF PROCESS DYNAMICS AND CONTROL with proportional and integral modes. The noninteracting form of the PID controller appears functionally as:



100 1 u=ᎏ e+ᎏ P τI

dc e dt + τ ᎏ +C dt  D

0

(8-31)

where τD is the derivative time constant. Note that derivative action is applied to the controlled variable rather than to the deviation, as it should not be applied to the set point; the selection of the sign for the derivative term must be consistent with the action of the controller. Figure 8-26 compares typical loop responses for P, PI, and PID controllers, along with the uncontrolled case. In some analog PID controllers, the integral and derivative terms are combined serially rather than in parallel as done in the last equation. This results in interaction between these modes, such that the effective values of the controller parameters differ from their set values as follows: τIeff = τI + τD 1 τDeff = ᎏᎏ 1/τD + 1/τI τD 100 Kc = ᎏ 1 + ᎏ P τI





(8-32)

The performance of the interacting controller is almost as good as the noninteracting controller on most processes, but the tuning rules differ because of the above relationships. With digital PID controllers, the noninteracting version is commonly used. There is always a gain limit placed upon the derivative term—a value of 10 is typical. However, interaction decreases the derivative gain below this value by the factor 1 + τD /τI, which is the reason for the decreased performance of the interacting PID controller. Sampling in a digital controller has a similar effect, limiting derivative gain to the ratio of derivative time to the sample interval of the controller. Noise on the controlled variable is amplified by derivative action, preventing its use in controlling flow and liquid level. Derivative action is recommended for control of temperature and composition, reducing the integrated error (IE) by a factor of two over PI control with no loss in robustness (Shinskey, Feedback Controllers for the Process Industries, McGraw-Hill, New York, 1994). CONTROLLER TUNING The performance of a controller depends as much on its tuning as its design. Tuning must be applied by the end user to fit the controller to the controlled process. There are many different approaches to controller tuning based on the particular performance criteria selected,

whether load or set-point changes are most important, whether the process is lag- or deadtime-dominant, and the availability of information about the process dynamics. The earliest definitive work in this field was done at the Taylor Instrument Company by Ziegler and Nichols (Trans. ASME, 759, 1942), tuning PI and interacting PID controllers for optimum response to step load changes applied to lag-dominant processes. While these tuning rules are still in use, they are approximate and do not apply to set-point changes, dead-timedominant processes, or noninteracting PID controllers (Seborg, Edgar, and Mellichamp, Process Dynamics and Control, Wiley, New York, 1989). Controller Performance Criteria The most useful measures of controller performance in an industrial setting are the maximum deviation in the controlled variable resulting from a disturbance and its integral. The disturbance could be to the set point or to the load, depending on the variable being controlled and its context in the process. The size of the deviation and its integral are proportional to the size of the disturbance (if the loop is linear at the operating point). While actual disturbances in a plant setting may appear to be random, the controller needs a reliable test to determine how well it is tuned. The disturbance of choice for test purposes is the step, because it can be applied manually, and by containing all frequencies including zero, it exercises all modes of the controller. When tuned optimally for step disturbances, the controller should be well-tuned for most other disturbances as well. Figure 8-27 shows the optimum response of a controlled variable to a step change in load. A step change in load may be simulated by stepping the controller output while it is in the manual mode followed immediately by transfer to automatic. The maximum deviation is the most important criterion for variables that could exceed safe operating levels such as steam pressure, drum level, and steam temperature in a boiler. The same rule applies to product quality, which could violate specifications and therefore be rejected. If the product can be accumulated in a downstream storage tank, however, its average quality is more important, and this is a function of the deviation integrated over the residence time in the tank. Deviation in the other direction, where the product is better than specification, is safe, but it increases production costs in proportion to the integrated deviation because quality is given away. For a PI or PID controller, the integrated deviation—better known as the integrated error IE—is related to the controller settings: ∆ uPτ IE = ᎏI (8-33) 100 where ∆u is the difference in controller output between two steady states, as required by a change in load or set point. The proportional band P and integral time τI are the indicated settings of the controller

Response for a step change in disturbance with tuned P, PI, and PID controllers and with no control. FIG. 8-26

8-13

8-14

PROCESS CONTROL

FIG. 8-27

The minimum-IAE response to a step load change has little overshoot and is well-damped.

for both interacting and noninteracting PID controllers. Although the derivative term does not appear in the relationship, its use typically allows a 50 percent reduction in integral time and therefore in IE. The integral time in the IE expression should be augmented by the sample interval if the controller is digital, the time constant of any filter used, and the value of any deadtime compensator. It would appear from the above that minimizing IE is simply a matter of minimizing the P and τI settings of the controller. However, settings will be reached that produce uniform oscillations—an unacceptable situation. It is preferable, instead, to find a combination of controller settings that minimize integrated absolute error IAE, which for both load and set-point changes is a well-damped response with minimal overshoot. Figure 8-27 is an example of a minimum-IAE response to a step change in load for a lag-dominant process. Because of the very small overshoot, the IAE will be only slightly larger than the IE. Loops that are tuned to minimize IAE tend to be close to minimum IE and also minimum peak deviation. The performance of a controller (and its tuning) must be based on what is achievable for a given process. The concept of best practical IE (IEb) for a step change in load ∆q can be estimated (Shinskey, Feedback Controllers for the Process Industries, McGraw-Hill, New York, 1994): IEb = ∆qKLτL(1 − e−θ/τ L )

(8-34)

where KL is the gain and τL the primary time constant in the load path, and θ the dead time in the manipulated path to the controlled variable. If the load or its gain is unknown, ∆ u and K(= Kv Kp) may be substituted. If the process is non-self-regulating (i.e., it is an integrator), the relationship is ∆qθ2 IEb = ᎏ (8-35) τ1 where τ1 is the time constant of the process integrator. The peak deviation with the best practical response curve is: IEb eb = ᎏ (8-36) θ + τ2 where τ2 is the time constant of a common secondary lag (e.g., in the measuring device). The performance for any controller can be measured against this standard by comparing the IE it achieves in responding to a step load change with the best practical IE. Potential performance improvements by tuning PI controllers on lag-dominant processes lie in the 20–30 percent range, while for PID controllers they fall between 40–60 percent, varying with secondary lags.

Tuning Methods Based on Known Process Models The most accurate tuning rules for controllers have been based on simulation, where the process parameters can be specified and IAE and IE can be integrated during the simulation as an indication of performance. Controller settings are then iterated until a minimum IAE is reached for a given disturbance. These optimum settings are then related to the parameters of the simulated process in tables, graphs, or equations, as a guide to tuning controllers for processes whose parameters are known (Seborg, Edgar, and Mellichamp, Process Dynamics and Control, Wiley, New York, 1989). This is a multidimensional problem, however, in that the relationships change as a function of process type, controller type, and source of disturbance. Table 8-2 summarizes these rules for minimum-IAE load response for the most common controllers. The process gain K and time constant τm are obtained from the product of Gv and Gp in Fig. 8-23. Derivative action is not effective for dead-time-dominant processes. For non-self-regulating processes, τ is the time constant of the integrator. The last category of distributed lag includes all heat-transfer processes, backmixed vessels, and processes having multiple interacting lags such as distillation columns;  τ represents the total response time of these processes (i.e., the time required for 63 percent complete response to a step input). Any secondary lag, sampling interval, or filter time constant should be added to deadtime θ. The principal limitation to using these rules is that the true process parameters are often unknown. Steady-state gain K can be calculated from a process model or determined from the steady-state results of a step test as ∆c/∆u, as shown in Fig. 8-28. The test will not be viable, however, if the time constant of the process τm is longer than a few TABLE 8-2

Tuning Rules Using Known Process Parameters

Process

Controller

P

τI

Dead-time-dominant Lag-dominant

PI PI PIDn PIDi PI PIDn PIDi PI PIDn PIDi

250K 106K θ/τm 77K θ/τm 106K θ/τm 106 θ/τ1 78 θ/τ1 108 θ/τ1 20K 10K 15K

0.5 θ 4.0 θ 1.8 θ 1.5 θ 4.0 θ 1.9 θ 1.6 θ 0.50  τ 0.30  τ 0.25  τ

Non-self-regulating Distributed lags

NOTE:

n = noninteracting; i = interacting controller modes

τD

0.45 θ 0.55 θ 0.48 θ 0.58 θ 0.09  τ 0.10  τ

FUNDAMENTALS OF PROCESS DYNAMICS AND CONTROL

8-15

If a steady state can be reached, gain K and time constant τ can be estimated from a step response; if not, use τ1 instead.

FIG. 8-28

minutes, since five time constants must elapse to approach a steady state within one percent, and unrequested disturbances may intervene. Estimated dead-time θ is the time from the step to the intercept of a straight line tangent to the steepest part of the response curve. The estimated time constant τ is the time from that point to 63 percent of the complete response. In the presence of a secondary lag, these results will not be completely accurate, however. The time for 63 percent response may be more accurately calculated as the residence time of the process: its volume divided by current volumetric flow rate. Tuning Methods When Process Model Is Unknown Ziegler and Nichols developed two tuning methods for processes with unknown parameters. The open-loop method uses a step test without waiting for a steady state to be reached and is therefore applicable to very slow processes. Deadtime is estimated from the intercept of the tangent in Fig. 8-28, whose slope is also used. If the process is non-self-regulating, the controlled variable will continue to follow this slope, changing an amount equal to ∆u in a time equal to its time constant. This time estimate τ1 is used along with θ to tune controllers according to Table 8-3, applicable to lag-dominant processes. A more recent tuning approach uses integral criteria such as the integral of the squared error (ISE), integral of the absolute error (IAE), and the time-weighted IAE (ITAE) of Seborg, Edgar, and Mellichamp (Process Dynamics and Control, Wiley, New York, 1989). The controller parameters are selected to minimize various integrals. Power-law correlations for PID controller settings have been tabulated for a range of first-order model parameters. The best tuning parameters have been fitted using a general equation, Y = A(θ/τ)B, where Y depends on the particular controller mode to be evaluated (KC, τI, τD). There are several features of the correlations that should be noted:

TABLE 8-3

Tuning Rules Using Slope and Intercept

Controller

P

τI

τD

PI PIDn PIDi

150 θ/τ 75 θ/τ 113 θ/τ

3.5 θ 2.1 θ 1.8 θ

— 0.63 θ 0.70 θ

NOTE:

1. The controller gain is inversely proportional to the process gain for constant dead time and time constant. 2. The allowable controller gain is higher when the ratio of dead time to time constant becomes smaller. This is because dead time has a destabilizing effect on the control system, limiting the controller gain, while a larger time constant generally demands a higher controller gain. A recent addition to the model-based tuning correlations is Internal Model Control (Rivera, Morari, and Skogestad, “Internal Model Control 4: PID Controller Design,” IEC Proc. Des. Dev., 25, 252, 1986), which offers some advantages over the other methods described here. However, the correlations are similar to the ones discussed above. Other plant testing and controller design approaches such as frequency response can be used for more complicated models. The Ziegler and Nichols closed-loop method requires forcing the loop to cycle uniformly under proportional control. The natural period τn of the cycle—the proportional controller contributes no phase shift to alter it—is used to set the optimum integral and derivative time constants. The optimum proportional band is set relative to the undamped proportional band Pu, which produced the uniform oscillation. Table 8-4 lists the tuning rules for a lag-dominant process. A uniform cycle can also be forced using on/off control to cycle the manipulated variable between two limits. The period of the cycle will be close to τn if the cycle is symmetrical; the peak-to-peak amplitude of the controlled variable Ac divided by the difference between the output limits Am is a measure of process gain at that period and is therefore related to Pu for the proportional cycle: π A Pu = 100 ᎏ ᎏc (8-37) 4 Am The factor π/4 compensates for the square wave in the output. Tuning rules are given in Table 8-4.

n = noninteracting, i = interacting controller modes

TABLE 8-4

Tuning Rules Using Proportional Cycle

Controller

P

τI

τD

PI PIDn PIDi

1.70 Pu 1.30 Pu 1.80 Pu

0.81 τn 0.48 τn 0.39 τn

— 0.11 τn 0.14 τn

8-16

PROCESS CONTROL

ADVANCED CONTROL SYSTEMS BENEFITS OF ADVANCED CONTROL The economics of most processes are determined by the steady-state operating conditions. Excursions from these steady-state conditions generally average out and have an insignificant effect on the economics of the process, except when the excursions lead to off-specification products. In order to enhance the economic performance of a process, the steady-state operating conditions must be altered in a manner that leads to more efficient process operation. The following hierarchy is used for process control: Level 0: Measurement devices and actuators Level 1: Regulatory control Level 2: Supervisory control Level 3: Production control Level 4: Information technology Levels 2, 3, and 4 clearly affect the process economics, as all three levels are directed to optimizing the process in some manner. However, level 0 (measurement devices and actuators) and level 1 (regulatory control) would appear to have no effect on process economics. Their direct effect is indeed minimal, but indirectly, they have a major effect. Basically, these levels provide the foundation for all higher levels. A process cannot be optimized until it can be operated consistently at the prescribed targets. Thus, a high degree of regulatory control must be the first goal of any automation effort. In turn, the measurements and actuators provide the process interface for regulatory control. For most processes, the optimum operating point is determined by a constraint. The constraint might be a product specification (a product stream can contain no more than 2 percent ethane); violation of this constraint causes off-specification product. The constraint might be an equipment limit (vessel pressure rating is 300 psig); violation of this constraint causes the equipment protection mechanism (pressure relief device) to activate. As the penalties are serious, violation of such constraints must be very infrequent. If the regulatory control system were perfect, the target could be set exactly equal to the constraint (that is, the target for the pressure controller could be set at the vessel relief pressure). However, no regulatory control system is perfect. Therefore, the value specified for the target must be on the safe side of the constraint, thus giving the control system some “elbow room.” How much depends on the following: 1. The performance of the control system (i.e., how effectively it responds to disturbances). The faster the control system reacts to a disturbance, the closer the process can be operated to the constraint. 2. The magnitude of the disturbances to which the control system must respond. If the magnitude of the major disturbances can be reduced, the process can be operated closer to the constraint. One measure of the performance of a control system is the variance of the controlled variable from the target. Both improving the control system and reducing the disturbances will lead to a lower variance in the controlled variable. In a few applications, improving the control system leads to a reduction in off-specification product and thus improved process economics. However, in most situations, the process is operated sufficiently far from the constraint that very little, if any, off-specification product results from control system deficiencies. Management often places considerable emphasis on avoiding off-spec production, so consequently the target is actually set far more conservatively than it should be. In most applications, simply improving the control system does not directly lead to improved process economics. Instead, the control system improvement must be accompanied by shifting the target closer to the constraint. There is always a cost of operating a process in a conservative manner. The cost may be a lower production rate, a lower process efficiency, a product giveaway, or otherwise. When management places extreme emphasis on avoiding off-spec production, the natural reaction is to operate very conservatively, thus incurring other costs.

The immediate objective of an advanced control effort is to reduce the variance in an important controlled variable. However, this effort must be coupled with a commitment to adjust the target for this controlled variable so that the process is operated closer to the constraint. In large throughput (commodity) processes, very small shifts in operating targets can lead to large economic returns. ADVANCED CONTROL TECHNIQUES GENERAL REFERENCES: Seborg, Edgar, and Mellichamp, Process Dynamics and Control, John Wiley and Sons, New York, 1989. Stephanopoulos, Chemical Process Control: An Introduction to Theory and Practice, Prentice Hall, Englewood Cliffs, New Jersey, 1984. Shinskey, Process Control Systems, 3d ed., McGraw-Hill, New York, 1988. Ogunnaike and Ray, Process Dynamics, Modeling, and Control, Oxford University Press, New York, 1994.

While the single-loop PID controller is satisfactory in many process applications, it does not perform well for processes with slow dynamics, time delays, frequent disturbances, or multivariable interactions. We discuss several advanced control methods hereafter that can be implemented via computer control, namely feedforward control, cascade control, time-delay compensation, selective and override control, adaptive control, fuzzy logic control, and statistical process control. Feedforward Control If the process exhibits slow dynamic response and disturbances are frequent, then the application of feedforward control may be advantageous. Feedforward (FF) control differs from feedback (FB) control in that the primary disturbance or load (L) is measured via a sensor and the manipulated variable (m) is adjusted so that deviations in the controlled variable from the set point are minimized or eliminated (see Fig. 8-29). By taking control action based on measured disturbances rather than controlled variable error, the controller can reject disturbances before they affect the controlled variable c. In order to determine the appropriate settings for the manipulated variable, one must develop mathematical models that relate: 1. The effect of the manipulated variable on the controlled variable 2. The effect of the disturbance on the controlled variable These models can be based on steady-state or dynamic analysis. The performance of the feedforward controller depends on the accuracy of both models. If the models are exact, then feedforward control offers the potential of perfect control (i.e., holding the controlled variable precisely at the set point at all times because of the ability to predict the appropriate control action). However, since most mathematical models are only approximate and since not all disturbances are measurable, it is standard practice to utilize feedforward control in conjunction with feedback control. Table 8-5 lists the relative advantages and disadvantages of feedforward and feedback control. By combining the two control methods, the strengths of both schemes can be utilized. FF control therefore attempts to eliminate the effects of measurable disturbances, while FB control would correct for unmeasurable

FIG. 8-29

Block diagram for feedforward control configuration.

ADVANCED CONTROL SYSTEMS

8-17

TABLE 8-5 Relative Advantages and Disadvantages of Feedback and Feedforward Advantages

Disadvantages Feedforward

• Acts before the effect of a disturbance has been felt by the system

• Requires measurement of all possible disturbances and their direct measurement

• Good for systems with large time constant or deadtime

• Cannot cope with unmeasured disturbances

• Does not introduce instability in the closed-loop response

• Sensitive to process/model error

Feedback • Does not require identification and measurement of any disturbance for corrective action

• Control action not taken until the effect of the disturbance has been felt by the system

• Does not require an explicit process model

• Unsatisfactory for processes with large time constants and frequent disturbances

• Controller can be robust to process/ model errors

• May cause instability in the closed-loop response

disturbances and modeling errors. This is often referred to as feedback trim. These controllers have become widely accepted in the chemical process industries since the 1960s. Design Based on Material and Energy Balances Consider a heat exchanger example (see Fig. 8-30) to illustrate the use of FF and FB control. The control objective is to maintain T2, the exit liquid temperature, at the desired value (or set point) Tset despite variations in inlet liquid flow rate F and inlet liquid temperature T1. This is done by manipulating W, the steam flow rate. A feedback control scheme would entail measuring T2, comparing T2 to Tset, and then adjusting W. A feedforward control scheme requires measuring F and T1, and adjusting W (knowing Tset), in order to control exit temperature, T2. Figure 8-31 shows the control system diagrams for FB and FF control. A feedforward control algorithm can be designed for the heat exchanger in the following manner. Using a steady-state energy balance and assuming no heat loss from the heat exchanger, WH = FC(T2 − T1)

(8-38)

(a)

(b) (a) Feedback control of a heat exchanger. (b) Feedforward control of a heat exchanger.

FIG. 8-31

where H = latent heat of vaporization CL = specific heat of liquid. Rearranging Eq. (8-38), CL W=ᎏ F(T2 − T1) H

(8-39)

or

W = K1F(T2 − T1)

(8-40)

with

CL K1 = ᎏ H

(8-41)

W = K1F(Tset − T1)

(8-42)

Replace T2 by Tset:

FIG. 8-30

A heat exchanger diagram.

Equation (8-42) can be used in the FF calculation, assuming one knows the physical properties CL and H. Of course, it is probable that the model will contain errors (e.g., unmeasured heat losses, incorrect CL or H). Therefore, K1 can be designated as an adjustable parameter that can be tuned. The use of a physical model for FF control is desirable since it provides a physical basis for the control law and gives an a priori estimate of what the tuning parameters are. Note that such a model could be nonlinear [e.g., in Eq. (8-42), F and Tset are multiplied]. Block Diagram Analysis One shortcoming of this feedforward design procedure is that it is based on the steady-state characteristics of the process and as such, neglects process dynamics (i.e., how fast the controlled variable responds to changes in the load and manipulated variables). Thus, it is often necessary to include “dynamic compensation” in the feedforward controller. The most direct method of designing the FF dynamic compensator is to use a block diagram of a general process, as shown in Fig. 8-32. Gt represents the disturbance transmitter, Gf is the feedforward controller, GL relates the load to the controlled variable, Gv is the valve, and Gp is the process. Gm is the output transmitter and Gc is the feedback controller. All blocks correspond to transfer functions (via Laplace transforms). Using block diagram algebra and Laplace transform variables, the controlled variable C(s) is given by GtGf L(s) + GLL(s) C(s) = ᎏᎏ 1 + GmGcGvGp

(8-43)

8-18

PROCESS CONTROL

FIG. 8-32

Block diagram for feedback-feedforward control.

For disturbance rejection [L(s) ≠ 0] we require that C(s) = 0, or zero error. Solving Eq. (8-43) for Gf, −GL Gf = ᎏ (8-44) GtGvGp Suppose the dynamics of GL and Gp are first order; in addition, assume that Gv = Kv and Gt = Kt (constant gains for simplicity). KL C(s) GL(s) = ᎏ =ᎏ (8-45) τLs + 1 L(s)

GcGvGpKm C(s) ᎏ = ᎏᎏ R(s) 1 + GcGvGpGm

(8-48)

−K(τps + 1) KL τps + 1 (8-47) Gf(s) = − ᎏ ⋅ ᎏ = ᎏᎏ τLs + 1 KpKvKt τLs + 1 The above FF controller can be implemented using analog elements or more commonly by a digital computer. Figure 8-33 compares typical responses for PID FB control, steady-state FF control (s = 0), dynamic FF control, and combined FF/FB control. In practice, the engineer can tune K, τp and τL in the field to improve the performance of the FF controller. The feedforward controller can also be simplified to provide steady-state feedforward control. This is done by setting s = 0 in Gf(s). This might be appropriate if there is uncertainty in the dynamic models for GL and Gp. Other Considerations in Feedforward Control The tuning of feedforward and feedback control systems can be performed independently. In analyzing the block diagram in Fig. 8-32, note that Gf is chosen to cancel out the effects of the disturbance L(s) as long as there are no model errors. For the feedback loop, therefore, the effects of L(s) can also be ignored, which for the servo case is:

Note that the characteristic equation will be unchanged for the FF + FB system, hence system stability will be unaffected by the presence of the FF controller. In general, the tuning of the FB controller can be less conservative than for the case of FB alone, since smaller excursions from the set point will result. This in turn would make the dynamic model Gp(s) more accurate. The tuning of the controller in the feedback loop can be theoretically performed independent of the feedforward loop (i.e., the feedforward loop does not introduce instability in the closed-loop response). For more information on feedforward/feedback control applications and design of such controllers, refer to the general references. Cascade Control One of the disadvantages of using conventional feedback control for processes with large time lags or delays is that disturbances are not recognized until after the controlled variable deviates from its set point. In these processes, correction by feedback control is generally slow and results in long-term deviation from set point. One way to improve the dynamic response to load changes is by using a secondary measurement point and a secondary controller; the secondary measurement point is located so that it recognizes the upset condition before the primary controlled variable is affected. One such approach is called cascade control, which is routinely used in most modern computer control systems. Consider a chemical reactor, where reactor temperature is to be controlled by coolant flow to the jacket of the reactor (Fig. 8-34). The reactor temperature can be influenced by changes in disturbance variables such as feed rate or feed temperature; a feedback controller could be employed to compensate for such disturbances by adjusting a valve on the coolant flow to the reactor jacket. However, suppose an increase occurs in the

(a)

(b)

Kp C(s) Gp(s) = ᎏ = ᎏ τps + 1 U(s) Using Eq. (8-44),

FIG. 8-33

control.

(8-46)

(a) Comparison of FF (steady state model) and PID FB control for load change; (b) comparison of FF (dynamic model) and combined FF/FB

ADVANCED CONTROL SYSTEMS coolant temperature as a result of changes in the plant coolant system. This will cause a change in the reactor temperature measurement, although such a change will not occur quickly, and the corrective action taken by the controller will be delayed. Cascade control is one solution to this problem (see Fig. 8-35). Here the jacket temperature is measured, and an error signal is sent from this point to the coolant control valve; this reduces coolant flow, maintaining the heat transfer rate to the reactor at a constant level and rejecting the disturbance. The cascade control configuration will also adjust the setting of the coolant control valve when an error occurs in reactor temperature. The cascade control scheme shown in Fig. 8-35 contains two controllers. The primary controller is the reactor temperature coolant temperature controller. It measures the reactor temperature, compares it to the set point, and computes an output, which is the set point for the coolant flow rate controller. This secondary controller compares the set point to the coolant temperature measurement and adjusts the valve. The principal advantage of cascade control is that the secondary measurement (jacket temperature) is located closer to a potential disturbance in order to improve the closed-loop response. Figure 8-36 shows the block diagram for a general cascade control system. In tuning of a cascade control system, the secondary controller (in the inner loop) is tuned first with the primary controller in manual. Often only a proportional controller is needed for the secondary loop, since offset in the secondary loop can be treated by using proportional plus integral action in the primary loop. When the primary controller is transferred to automatic, it can be tuned using the techniques described earlier in this section. For more information on theoretical analysis of cascade control systems, see the general references for a discussion of applications of cascade control. Time-Delay Compensation Time delays are a common occurrence in the process industries because of the presence of recycle loops, fluid-flow distance lags, and “dead time” in composition measurements resulting from use of chromatographic analysis. The presence of a time delay in a process severely limits the performance of a conventional PID control system, reducing the stability margin of the closed-loop control system. Consequently, the controller gain must be reduced below that which could be used for a process without delay. Thus, the response of the closed-loop system will be sluggish compared to that of the system with no time delay. In order to improve the performance of time-delay systems, special control algorithms have been developed to provide time-delay com-

FIG. 8-34

Conventional control of an exothermic chemical reactor.

FIG. 8-35

8-19

Cascade control of an exothermic chemical reactor.

pensation. The Smith predictor technique is the best known algorithm; a related method is called the analytical predictor. Various investigators have found that based on integral squared error, the performance of the Smith predictor can be as much as 30 percent better than for a conventional controller. The Smith predictor is a model-based control strategy that involves a more complicated block diagram than that for a conventional feedback controller, although a PID controller is still central to the control strategy (see Fig. 8-37). The key concept is based on better coordination of the timing of manipulated variable action. The loop configuration takes into account the fact that the current controlled variable measurement is not a result of the current manipulated variable action, but the value taken 0 time units earlier. Time-delay compensation can yield excellent performance; however, if the process model parameters change (especially the time delay), the Smith predictor performance will deteriorate and is not recommended unless other precautions are taken. Selective and Override Control When there are more controlled variables than manipulated variables, a common solution to this problem is to use a selector to choose the appropriate process variable from among a number of available measurements. Selectors can be based on either multiple measurement points, multiple final control elements, or multiple controllers, as discussed below. Selectors are used to improve the control system performance as well as to protect equipment from unsafe operating conditions. One type of selector device chooses as its output signal the highest (or lowest) of two or more input signals. This approach is often referred to as auctioneering. On instrumentation diagrams, the symbol HS denotes high selector and LS a low selector. For example, a high selector can be used to determine the hot-spot temperature in a fixed-bed chemical reactor. In this case, the output from the high selector is the input to the temperature controller. In an exothermic catalytic reaction, the process may run away due to disturbances or changes in the reactor. Immediate action should be taken to prevent a dangerous rise in temperature. Because a hot spot may potentially develop at one of several possible locations in the reactor, multiple (redundant) measurement points should be employed. This approach minimizes the time required to identify when a temperature has risen too high at some point in the bed. The use of high or low limits for process variables is another type of selective control, called an override. The feature of anti-reset windup in feedback controllers is a type of override. Another example is a distillation column with lower and upper limits on the heat input to the column reboiler. The minimum level ensures that liquid will remain

8-20

PROCESS CONTROL

FIG. 8-36 Block diagram of the cascade control system. For a chemical reactor, L1 would correspond to a feed temperature or composition disturbance, while L2 would be a change in the cooling water temperature.

on the trays, while the upper limit is determined by the onset of flooding. Overrides are also used in forced-draft combustion-control systems to prevent an imbalance between air flow and fuel flow, which could result in unsafe operating conditions. Other types of selective systems employ multiple final control elements or multiple controllers. In some applications, several manipulated variables are used to control a single process variable (also called split-range control). Typical examples include the adjustment of both inflow and outflow from a chemical reactor in order to control reactor pressure or the use of both acid and base to control pH in waste-water treatment. In this approach, the selector chooses from several controller outputs which final control element should be adjusted (Marlin, Process Control, McGraw-Hill, New York, 1995).

Adaptive Control Process control problems inevitably require on-line tuning of the controller constants to achieve a satisfactory degree of control. If the process operating conditions or the environment changes significantly, the controller may have to be retuned. If these changes occur quite frequently, then adaptive control techniques should be considered. An adaptive control system is one in which the controller parameters are adjusted automatically to compensate for changing process conditions. During the 1980s, several adaptive controllers were field-tested and commercialized in the U.S. and abroad, including products by ASEA (Sweden), Leeds and Northrup, Foxboro, and Sattcontrol. At the present time, some form of adaptive tuning is available on almost all PID controllers. The ASEA adaptive controller, Novatune, was

˜

˜ = G*e ˜ − θs (G* ˜ = model withFIG. 8-37 Block diagram of the Smith predictor. The process model used in the controller is G ˜ out delay; e−θs = time delay element).

ADVANCED CONTROL SYSTEMS announced in 1983 and is generally based on minimum-variancecontrol algorithms. Both feedforward and feedback control capabilities reside in the hardware. The unit has been tested successfully in reactor and paper machine control applications in Europe and in pH control of wastewater in the United States. Foxboro developed a self-tuning PID controller that is based on a so-called “expert system” approach for adjustment of the controller parameters. The on-line tuning of Kc, τI, and τD is based on the closedloop transient response to a step change in set point. By evaluating the salient characteristics of the response (e.g., the decay ratio, overshoot, and closed-loop period), the controller parameters can be updated without actually finding a new process model. The details of the algorithm, however, are proprietary. The Sattcontroller (also marketed by Fisher-Rosemount) has an autotuning function that is based on placing the process in a controlled oscillation at very low amplitude, comparable with that of the noise level of the process. This is done via a relay-type step function with hysteresis. The autotuner identifies the dynamic parameters of the process (the ultimate gain and period) and automatically calculates Kc, τI, and τD using empirical tuning rules. Gain scheduling can also be implemented with this controller, using up to three sets of PID controller parameters. The subject of adaptive control is one of current interest. New algorithms are presently under development, but these need to be fieldtested before industrial acceptance can be expected. It is clear, however, that digital computers will be required for implementation of self-adaptive controllers due to their complexity. An adaptive controller is inherently nonlinear and therefore more complicated than the conventional PID controller. Fuzzy Logic Control The application of fuzzy logic to process control requires the concepts of fuzzy rules and fuzzy inference. A fuzzy rule, also known as a fuzzy IF-THEN statement, has the form: If x then y where x specifies a vector of input variables and corresponding membership values and y specifies an output variable and its corresponding membership value. For example, if input1 = high and input2 = low, then output = medium. Three functions are required to perform logical inferencing with the fuzzy rules. The fuzzy AND is the product of a rule’s input membership values, generating a weight for the rule’s output. The fuzzy OR is a normalized sum of the weights assigned to each rule that contributes to a particular decision. The third function used is defuzzification, which generates a crisp final output. In one approach, the crisp output is the weighted average of the peak element values:  [w(i) p(i)]/ [w(i)]. With a single feedback control architecture, information that is readily available to the algorithm includes the error signal, the difference between the process variable and the set point variable, change in error from previous cycles to the current cycle, changes to the set point variable, change of the manipulated variable from cycle to cycle, and the change in the process variable from past to present. In addition, multiple combinations of the system response data are available. As long as the irregularity lies in that dimension wherein fuzzy decisions are being based or associated, the result should be enhanced performance. This enhanced performance should be demonstrated in both the transient and steady-state response. If the system tends to have changing dynamic characteristics or exhibits nonlinearities, fuzzy logic control should offer a better alternative to using constant PID settings. Most fuzzy logic software begins building its information base during the autotune function. In fact, the majority of the information used in the early stages of system startup comes from the autotune solutions. In addition to single-loop process controllers, products that have benefited from the implementation of fuzzy logic are: • Camcorders with automatic compensation for operator-injected noise such as shaking and moving • Elevators with decreased wait time, making intelligent floor decisions and minimizing travel and power consumption

8-21

• Antilock braking systems with quickly reacting independent wheel decisions based on current and acquired knowledge • Television with automatic color, brightness, and acoustic control based on signal and environmental conditions Sometimes fuzzy logic controllers are combined with pattern recognition software such as artificial neural networks (Kosko, Neural Networks and Fuzzy Systems, Prentice Hall, Englewood Cliffs, New Jersey, 1992). Statistical Process Control Statistical process control (SPC), also called statistical quality control (SQC), involves the application of statistical concepts to determine whether a process is operating satisfactorily. The ideas involved in statistical quality control are over fifty years old, but only recently with the growing worldwide focus on increased productivity have applications of SPC become widespread. If a process is operating satisfactorily (or “in control”), then the variation of product quality falls within acceptable bounds, usually the minimum and maximum values of a specified composition or property (product specification). Figure 8-38 illustrates the typical spread of values of the controlled variable that might be expected to occur under steady-state operating conditions. The mean and root mean square (RMS) deviation are identified in Fig. 8-38 and can be computed from a series of n observations c1, c2, . . . cn as follows: 1 n mean: c = ᎏ  ci (8-49) n i=1 RMS deviation: 1/ 2 1 n (8-50) σ = ᎏ  (ci − c)2 n i=1 The RMS deviation is a measure of the spread of values for c around the mean. A large value of σ indicates that wide variations in c occur. The probability that the controlled variable lies between the values of c1 and c2 is given by the area under the distribution between c1 and c2 (histogram). If the histogram follows a normal probability distribution, then 99.7 percent of all observations should lie with ⫾3σ of the mean (between the lower and upper control limits). These limits are used to determine the quality of control. If all data from a process lie within the ⫾3σ limits, then we conclude that nothing unusual has happened during the recorded time period. The process environment is relatively unchanged, and the product quality lies within specification. On the other hand, if repeated violations of the ⫾3σ limits occur, then the process environment has changed and the process is out of control. One way to codify abnormal behavior is the so-called Western Electric rules, which identify cases where a process is out of control: 1. One point that occurs outside the upper or lower control limits 2. Any seven consecutive points lying on the same side of the center line (mean) 3. Any seven consecutive points that increase or decrease 4. Any nonrandom pattern In the above list, one assumes that sample values are independent (i.e., not correlated). There are important economic consequences of a process being out of control; for example, product waste and customer dissatisfaction. Hence, statistical process control does provide a way to continuously monitor process performance and improve product quality. A typical process may go out of control due to several reasons, including • Persistent disturbances from the weather • An undetected grade change in raw materials • A malfunctioning instrument or control system Statistical quality control is a diagnostic tool—that is, an indicator of quality problems—but it does not identify the source of the problem or the corrective action to be taken. The Shewhart chart provides a way to analyze variability of a single measurement, as discussed in the following example. The data in Fig. 8-39 were obtained from the monitoring of pH in a yarn-soaking kettle used in textile manufacturing (Seborg, Edgar, and Mellichamp, Process Dynamics and Control, Wiley, New York, 1989). Because pH has a crucial influence on color and durability of the yarn, it is important to maintain pH within a range that gives the best results for both characteristics. The pH is considered to be in control between values of 4.25 and 4.64. At the






8-22

PROCESS CONTROL

Histogram plotting frequency of occurrence. c = mean, σ = rms deviation. Also shown is fit by normal probability distribution.

FIG. 8-38

25th day, the data show that pH is out of control; this might imply that a property change in the raw material has occurred and must be corrected with the supplier. However, a real-time correction would be preferable. In Fig. 8-39, the pH was adjusted by slowly adding more acid to the vats until it came back into control (on day 29). In continuous processes where automatic feedback control has been implemented, the feedback mechanism theoretically ensures that product quality is at or near the set point regardless of process disturbances. This, of course, requires that an appropriate manipulated variable has been identified for adjusting the product quality. However, even under feedback control, there may be daily variations of product quality because of disturbances or equipment or instrument malfunctions. These occurrences can be analyzed using the concepts of statistical quality control. More details on statistical process control are available in several textbooks (Grant and Leavenworth, Statistical Quality Control, McGraw-Hill, New York, 1980; Montgomery, Introduction to Statistical Quality Control, Wiley, New York, 1985). MULTIVARIABLE CONTROL PROBLEMS GENERAL REFERENCES: Shinskey, F. G., Process Control Systems, 3d ed., McGraw-Hill, New York, 1988. Seborg, D. E., T. F. Edgar, and D. A. Mellichamp, Process Dynamics and Control, Wiley, New York, 1989. McAvoy, T. J., Interaction Analysis, ISA, Research Triangle Park, North Carolina, 1983.

Process control books and journal articles tend to emphasize problems with a single controlled variable. In contrast, most practical

FIG. 8-39

problems are multivariable control problems because many process variables must be controlled. In fact, for virtually any important industrial process, at least two variables must be controlled: product quality and throughput. In this section, strategies for multivariable control problems are considered. Three examples of simple multivariable control problems are shown in Fig. 8-40. The in-line blending system blends pure components A and B to produce a product stream with flow rate w and mass fraction of A, x. Adjusting either inlet flow rate wA or wB affects both of the controlled variables w and x. For the pH neutralization process in Figure 8-40(b), liquid level h and the pH of the exit stream are to be controlled by adjusting the acid and base flow rates wa and wb. Each of the manipulated variables affects both of the controlled variables. Thus, both the blending system and the pH neutralization process are said to exhibit strong process interactions. In contrast, the process interactions for the gas-liquid separator in Fig. 8-40(c) are not as strong because one manipulated variable, liquid flow rate L, has only a small and indirect effect on one controlled variable, pressure P. Strong process interactions can cause serious problems if a conventional multiloop feedback control scheme (e.g., PI or PID controllers) is employed. The process interactions can produce undesirable control loop interactions where the controllers fight each other. Also, it may be difficult to determine the best pairing of controlled and manipulated variables. For example, in the in-line blending process in Fig. 8-40(a), should w be controlled with wA and x with wB, or vice versa? Control Strategies for Multivariable Control Problems If a conventional multiloop control strategy performs poorly due to control loop interactions, a number of solutions are available:

Process control chart for the average daily pH readings.

ADVANCED CONTROL SYSTEMS

(a)

(b)

(c) FIG. 8-40

Physical examples of multivariable control problems.

a. Detune one or more of the control loops b. Choose different controlled or manipulated variables (or pairings) c. Use a decoupling control system d. Use a multivariable control scheme (e.g., model predictive control) Detuning a controller (e.g., using a smaller controller gain or a larger reset time) tends to reduce control loop interactions by sacrificing the performance for the detuned loops. This approach may be acceptable if some of the controlled variables are faster or less important than others. The selection of controlled and manipulated variables is of crucial importance in designing a control system. In particular, a judicious choice may significantly reduce control loop interactions. For the blending process in Fig. 8-40(a), a straightforward control strategy would be to control x by adjusting wA, and w by adjusting wB. But

8-23

physical intuition suggests that it would be better to control x by adjusting the ratio wA /(wA + wB) and to control product flow rate w by the sum wA + wB. Thus, the new manipulated variables would be: M1 = wA /(wA + wB) and M2 = wA + wB. In this control scheme, M1 only affects x and M2 only affects w. Thus, the control loop interactions have been eliminated. Similarly, for the pH neutralization process in Fig. 8-40(b), the control loop interactions would be greatly reduced if pH is controlled by M1 = wa /(wa + wb) and liquid level h is controlled by M2 = wa + wb. Decoupling Control Systems Decoupling control systems provide an alternative approach for reducing control loop interactions. The basic idea is to use additional controllers called “decouplers” to compensate for undesirable process interactions. As an illustrative example, consider the simplified block diagram for a representative decoupling control system shown in Fig. 8-41. The two controlled variables C1 and C2 and two manipulated variables M1 and M2 are related by four process transfer functions, Gp11, Gp12, and so on. For example, Gp11 denotes the transfer function between M1 and C1: C1(s) (8-51) ᎏ = Gp11(s) M1(s) Figure 8-41 includes two conventional feedback controllers: Gc1 controls C1 by manipulating M1, and Gc2 controls C2 by manipulating M2. The output signals from the feedback controllers serve as input signals to the two decouplers D12 and D21. The block diagram is in a simplified form because the load variables and transfer functions for the final control elements and sensors have been omitted. The function of the decouplers is to compensate for the undesirable process interactions represented by Gp12 and Gp21. Suppose that the process transfer functions are all known. Then the ideal design equations are: Gp12(s) D12(s) = − ᎏ (8-52) Gp11(s) Gp21(s) (8-53) D21(s) = − ᎏ Gp22(s) These decoupler design equations are very similar to the ones for feedforward control in an earlier section. In fact, decoupling can be interpreted as a type of feedforward control where the input signal is the output of a feedback controller rather than a measured load variable. In principle, ideal decoupling eliminates control loop interactions and allows the closed-loop system to behave as a set of independent control loops. But in practice, this ideal behavior is not attained for a variety of reasons, including imperfect process models and the presence of saturation constraints on controller outputs and manipulated variables. Furthermore, the ideal decoupler design equations in (8-52) and (8-53) may not be physically realizable and thus would have to be approximated. In practice, other types of decouplers and decoupling control configurations have been employed. For example, in partial decoupling, only a single decoupler is employed (i.e., either D12 or D21 in Fig. 8-41 is set equal to zero). This approach tends to be more robust than complete decoupling and is preferred when one of the controlled variables is more important than the other. Static decouplers can be used to reduce the steady-state interactions between control loops. They can be designed by replacing the transfer functions in Eqs. (8-52) and (8-53) with the corresponding steady-state gains, Kp12 D12(s) = − ᎏ (8-54) Kp11 Kp21 (8-55) D21(s) = − ᎏ Kp22 The advantage of static decoupling is that less process information is required: namely, only steady-state gains. Nonlinear decouplers can be used when the process behavior is nonlinear. Pairing of Controlled and Manipulated Variables A key decision in multiloop-control-system design is the pairing of manipu-

8-24

PROCESS CONTROL

FIG. 8-41

Decoupling control system.

lated and controlled variables. This is referred to as the controllerpairing problem. Suppose there are N controlled variables and N manipulated variables. Then N! distinct control configurations exist. For example, if N = 5, then there are 120 different multiloop control schemes. In practice, many of them would be rejected based on physical insight or previous experience. But a smaller number (e.g., 5–15) may appear to be feasible and further analysis would be warranted. Thus, it is very useful to have a simple method for choosing the most promising control configuration. The most popular and widely used technique for determining the best controller pairing is the relative gain array (RGA) method (Bristol, “On a New Measure of Process Interaction,” IEEE Trans. Auto. Control, AC-11, 133, 1966). The RGA method provides two important items of information: 1. A measure of the degree of process interactions between the manipulated and controlled variables 2. A recommended controller pairing An important advantage of the RGA method is that it requires minimal process information: namely, steady-state gains. Another advantage is that the results are independent of both the physical units used and the scaling of the process variables. The chief disadvantage of the RGA method is that it neglects process dynamics, which can be an important factor in the pairing decision. Thus, the RGA analysis should be supplemented with an evaluation of process dynamics. Although extensions of the RGA method that incorporate process dynamics have been reported, these extensions have not been widely applied. RGA Method for 2 ⴛ 2 Control Problems To illustrate the use of the RGA method, consider a control problem with two inputs and two outputs. The more general case of N × N control problems is considered elsewhere (McAvoy, Interaction Analysis, ISA, Research Triangle Park, North Carolina, 1983). As a starting point, it is assumed that a linear, steady-state process model is available, C1 = K11 M1 + K12 M2

(8-56)

C2 = K21 M1 + K22 M2

(8-57)

where M1 and M2 are steady-state values of the manipulated inputs; C1 and C2 are steady-state values of the controlled outputs; and the values K are steady-state gains. The C and M variables are deviation variables from nominal steady-state values. This process model could be obtained in a variety of ways, such as by linearizing a theoretical model or by calculating steady-state gains from experimental data or a steady-state simulation.

By definition, the relative gain λij between the ith manipulated variable and the jth controlled variable is defined as: open-loop gain between Ci and Mj λij = ᎏᎏᎏᎏ (8-58) closed-loop gain between Ci and Mj where the open-loop gain is simply Kij from Eqs. (8-56) and (8-57). The closed-loop gain is defined to be the steady-state gain between Mj and Ci when the other control loop is closed and no offset occurs due to the presence of integral control action. The RGA for the 2 × 2 process is denoted by Λ=

λλ

11 21



λ12 λ22

(8-59)

The RGA has the important normalization property that the sum of the elements in each row and each column is exactly one. Consequently, the RGA in Eq. (8-59) can be written as Λ=

1 −λ λ



1−λ λ

(8-60)

where λ can be calculated from the following formula: 1 λ = ᎏᎏ K12 K21 1 − ᎏᎏ K11 K22

(8-61)

Ideally, the relative gains that correspond to the proposed controller pairing should have a value of one since Eq. (8-58) implies that the open and closed-loop gains are then identical. If a relative gain equals one, the steady-state operation of this loop will not be affected when the other control loop is changed from manual to automatic, or vice versa. Consequently, the recommendation for the best controller pairing is to pair the controlled and manipulated variables so that the corresponding relative gains are positive and close to one. RGA Example In order to illustrate use of the RGA method, consider the following steady-state version of a transfer function model for a pilot-scale, methanol-water distillation column (Wood and Berry, “Terminal Composition Control of a Binary Distillation Column,” Chem. Eng. Sci., 28, 1707, 1973): K11 = 12.8, K12 = −18.9, K21 = 6.6, and K22 = −19.4. It follows that λ = 2 and Λ=

−12 −12

(8-62)

Thus it is concluded that the column is fairly interacting and the recommended controller pairing is to pair C1 with M1 and C2 with M2.

ADVANCED CONTROL SYSTEMS MODEL PREDICTIVE CONTROL Introduction The model-based control strategy that has been most widely applied in the process industries is model predictive control (MPC). It is a general method that is especially well-suited for difficult multiinput, multioutput (MIMO) control problems where there are significant interactions between the manipulated inputs and the controlled outputs. Unlike other model-based control strategies, MPC can easily accommodate inequality constraints on input and output variables such as upper and lower limits or rate-of-change limits. A key feature of MPC is that future process behavior is predicted using a dynamic model and available measurements. The controller outputs are calculated so as to minimize the difference between the predicted process response and the desired response. At each sampling instant, the control calculations are repeated and the predictions updated based on current measurements. In typical industrial applications, the set point and target values for the MPC calculations are updated using on-line optimization based on a steady-state model of the process. Constraints on the controlled and manipulated variables can be routinely included in both the MPC and optimization calculations. The extensive MPC literature includes survey articles (Garcia, Prett, and Morari, Automatica, 25, 335, 1989; Richalet, Automatica, 29, 1251, 1993) and books (Prett and Garcia, Fundamental Process Control, Butterworths, Stoneham, Massachusetts, 1988; Soeterboek, Predictive Control—A Unified Approach, Prentice Hall, Englewood Cliffs, New Jersey, 1991). The current widespread interest in MPC techniques was initiated by pioneering research performed by two industrial groups in the 1970s. Shell Oil (Houston, TX) reported their Dynamic Matrix Control (DMC) approach in 1979, while a similar technique, marketed as IDCOM, was published by a small French company, ADERSA, in 1978. Since then, there have been over one thousand applications of these and related MPC techniques in oil refineries and petrochemical plants around the world. Thus, MPC has had a substantial impact and is currently the method of choice for difficult multivariable control problems in these industries. However, relatively few applications have been reported in other process industries, even though MPC is a very general approach that is not limited to a particular industry. Advantages and Disadvantages of MPC Model Predictive Control offers a number of important advantages: 1. It is a general control strategy for MIMO processes with inequality constraints on input and output variables. 2. It can easily accommodate difficult or unusual dynamic behavior such as large time delays and inverse responses. 3. Since the control calculations are based on optimizing control system performance, MPC can be readily integrated with on-line optimization strategies to optimize plant performance. 4. The control strategy can be easily updated on-line to compensate for changes in process conditions, constraints, or performance criteria. But current versions of MPC have significant disadvantages: 1. The MPC strategy is very different from conventional multiloop control strategies and thus initially unfamiliar to plant personnel. 2. The MPC calculations can be relatively complicated (e.g., solving an LP or QP problem at each sampling instant) and thus require a significant amount of computer resources and effort. 3. The development of a dynamic model from plant data is time consuming, typically requiring one to three weeks of around-the-clock plant tests. 4. Since empirical models are generally used, they are only valid over the range of conditions that were considered during the plant tests. 5. Theoretical studies have demonstrated that MPC can perform poorly for some types of process disturbances, especially when output constraints are employed (Lundstrom, Lee, Morari, and Skogestad, Computers Chem. Eng., 19, 409, 1995). Since MPC has been widely used and has had considerable impact, there is a broad consensus that its advantages far outweigh its disadvantages. Economic Incentives for Automation Projects Industrial applications of advanced process control strategies such as MPC are

8-25

motivated by the need for improvements regarding safety, product quality, environmental standards, and economic operation of the process. One view of the economics incentives for advanced automation techniques is illustrated in Fig. 8-42. Distributed control systems (DCS) are widely used for data acquisition and conventional singleloop (PID) control. Usually, they are the most expensive part of the entire control system. The addition of advanced regulatory control systems such as decouplers, selective controls, and time-delay compensation can provide additional benefits for a modest incremental cost. But experience has indicated that the major benefits can be obtained for relatively small incremental costs through a combination of MPC and on-line optimization. The results in Fig. 8-42 are shown qualitatively, rather than quantitatively, because the actual costs and benefits are application-dependent. A key reason why MPC has become a major commercial and technical success is that there are numerous vendors who are licensed to market MPC products and install them on a turnkey basis. Consequently, even medium-sized companies are able to take advantage of this new technology. Payout times of 3–12 months have been reported. Basic Features of MPC Model predictive control strategies have a number of distinguishing features: 1. A dynamic model of the process is used to predict the future outputs over a prediction horizon consisting of the next p sampling periods. 2. A reference trajectory is used to represent the desired output response over the prediction horizon. 3. Inequality constraints on the input and output variables can be included as an option. 4. At each sampling instant, a control policy consisting of the next m control moves is calculated. The control calculations are based on minimizing a quadratic or linear performance index over the prediction horizon while satisfying the constraints. 5. The performance index is expressed in terms of future control moves and the predicted deviations from the reference trajectory. 6. A receding horizon approach is employed. At each sampling instant, only the first control move (of the m moves that were calculated) is actually implemented. Then the predictions and control calculations are repeated at the next sampling instant. These distinguishing features of MPC will now be described in more detail.

FIG. 8-42

industries.

Economic incentives for automation projects in the process

8-26

PROCESS CONTROL

A key feature of MPC is that a dynamic model of the process is used to predict future values of the controlled outputs. There is considerable flexibility concerning the choice of the dynamic model. For example, a physical model based on first principles (e.g., mass and energy balances) or an empirical model could be selected. Also, the empirical model could be a linear model (e.g., transfer function, step response model, or state space model) or a nonlinear model (e.g., neural net model). However, most industrial applications of MPC have relied on linear empirical models, which may include simple nonlinear transformations of process variables. The original formulations of MPC (i.e., DMC and IDCOM) were based on empirical linear models expressed in either step-response or impulse-response form. For simplicity, we will consider only a singleinput, single-output (SISO) model. However, the SISO model can be easily generalized to the MIMO models that are used in industrial applications. The step response model relating a single controlled variable y and a single manipulated variable u can be expressed as

sion of MPC is based on step-response models, while IDCOM utilizes impulse response models. The receding horizon feature of MPC is shown in Fig. 8-44 with the current sampling instant denoted by k. Past input signals [u(i) for i < k] are used to predict the output at the next p sampling instants [y(k ˆ + i) for i = 1, 2, . . . , p]. The control calculations are performed to generate an m-step control policy [u(k), u(k + 1), . . . , u(k + m)], which optimizes the performance index. The first control action, u(k), is implemented. Then at the next sampling instant (k + 1), the prediction and control calculations are repeated in order to determine u(k + 1). In Fig. 8-44, the reference trajectory (or target) is considered to be constant. Other possibilities include a gradual or step set point change that can be generated by on-line optimization. The performance index for MPC applications is usually a linear or quadratic function of the predicted errors and calculated future control moves. For example, the following quadratic performance index has been widely used:

N

y(k) ˆ =  Si ∆u(k − i) + y(0)

∆u(k)

i=1

where y(k) ˆ is the predicted value of y at the k-sampling instant; u(k) is the value of the manipulated input at time k; and the model parameters Si are referred to as the step-response coefficients. The initial value y(0) is assumed to be known. The change in the manipulated input from one sampling instant to the next is denoted by ∆u(k) = u(k) − u(k − 1) 䉭

(8-64)

The step-response model is also referred to as a finite impulse response (FIR) model or a discrete convolution model. In principle, the step-response coefficients can be determined from the output response to a step change in the input. A typical response to a unit step change in input u is shown in Fig. 8-43. The step response coefficients Si are simply the values of the output variable at the sampling instants, after the initial value y(0) has been subtracted. Theoretically, they can be determined from a single-step response, but, in practice, a number of “bump tests” are required to compensate for unanticipated disturbances, process nonlinearities, and noisy measurements. The step-response model in Eq. (8-63) is equivalent to the following impulse response model:

where the impulse response coefficients Ii are related to the stepresponse coefficients by Ii = Si − Si − 1. Step- and impulse-response models typically contain a large number of parameters because the model horizon N is usually quite large (30 < N < 70). In fact, these models are often referred to as nonparametric models. The DMC ver-

FIG. 8-43

Step response for u, a unit step change in the input.

i=1

i=1

ˆ + i) e(k + i) = r(k + i) − y(k 䉭

(8-66)

(8-67)

where r(k + i) is the reference value at time k + i, and ∆u(k) denotes the vector of current and future control moves over the next m sampling instants: ∆u(k) = [∆u(k), ∆u(k + 1), . . . , ∆u(k + m − 1)]T

(8-68)

Equation (8-66) contains two types of design parameters that can also be used for tuning purposes. The move suppression factor δ penalizes large control moves, while the weighting factors wi allow the predicted errors to be weighed differently at each time step, if desired. Inequality constraints on the future inputs or their rates of change are widely used in the MPC calculations. For example, if both upper and lower limits are required, the constraints could be expressed as: Bi∗ ≤ u(k + i) ≤ B*i

for i = 1, 2, . . . , m

(8-69)

Ci∗ ≤ ∆u(k + i) ≤ C*i

for i = 1, 2, . . . , m

(8-70)

where Bi and Ci are constants. Constraints on the predicted outputs are sometimes included as well: Di∗ ≤ y(k ˆ + i) ≤ D *i

(8-65)

i=1

m

The value e(k + i) denotes the predicted error at time (k + i),

N

y(k) ˆ =  Ii u(k − i) + y(0)

p

min J =  wie2(k + i) + δ  ∆u2(k + i − 1)

(8-63)

for i = 1, 2, . . . , p

(8-71)

The minimization of the quadratic performance index in Eq. (8-66), subject to the constraints in Eq. (8-69) to (8-71) and the step-response model in Eq. (8-63), can be formulated as a standard quadratic pro-

FIG. 8-44

The “moving horizon” approach of model predictive control.

ADVANCED CONTROL SYSTEMS gramming (QP) problem. Consequently, efficient QP solution techniques can be employed. When the inequality constraints in Eqs. (8-69) to (8-71) are omitted, the optimization problem has an analytical solution (Prett and Garcia, Fundamental Process Control, Butterworths, Stoneham, Massachusetts, 1988; Soeterboek, Predictive Control—A Unified Approach, Prentice Hall, Englewood Cliffs, New Jersey, 1991). If the quadratic terms in Eq. (8-66) are replaced by linear terms, a linear programming program (LP) problem results that can also be solved using standard methods. This MPC formulation for SISO control problems can easily be extended to MIMO problems. Implementation Issues A critical factor in the successful application of any model-based technique is the availability of a suitable dynamic model. In typical MPC applications, an empirical model is identified from data acquired during extensive plant tests. The experiments generally consist of a series of bump tests in the manipulated variables. Typically, the manipulated variables are adjusted one at a time and the plant tests require a period of one to three weeks. The step or impulse response coefficients are then calculated using linearregression techniques such as least-squares methods. However, details concerning the procedures utilized in the plant tests and subsequent model identification are considered to be proprietary information. The scaling and conditioning of plant data for use in model identification and control calculations can be key factors in the success of the application. The MPC control problem illustrated in Eqs. (8-66) to (8-71) contains a variety of design parameters: model horizon N, prediction horizon p, control horizon m, weighting factors wi, move suppression factor δ, the constraint limits Bi, Ci, and Di, and the sampling period ∆t. Some of these parameters can be used to tune the MPC strategy, notably the move suppression factor δ, but details remain largely proprietary. One commercial controller, Honeywell’s RMPCT ® (Robust Multivariable Predictive Control Technology), provides default tuning parameters based on the dynamic process model and the model uncertainty. Integration of MPC and On-Line Optimization As indicated in Fig. 8-42, significant potential benefits can be realized by using a combination of MPC and on-line optimization. At the present time, most commercial MPC packages integrate the two methodologies in a hierarchical configuration such as the one shown in Fig. 8-45. The MPC calculations are performed quite often (e.g., every 1–10 min) and implemented as set points for PID control loops at the DCS level. The targets and constraints for the MPC calculations are generated by solving a steady-state optimization problem (LP or QP) based on a linear process model. These calculations may be performed as often as the MPC calculations. As an option, the targets and constraints for the LP or QP optimization can be generated from a nonlinear process model using a nonlinear optimization technique. These calculations tend to be performed less frequently (e.g., every 1–24 hours) due to the complexity of the calculations and the process models. The combination of MPC and frequent on-line optimization has been successfully applied in oil refineries and petrochemical plants around the world. REAL-TIME PROCESS OPTIMIZATION GENERAL REFERENCES: Biles and Swain, Optimization and Industrial Experimentation, Wiley—Interscience, New York, 1980. Dantzig, Linear Programming and Extensions, Princeton University Press, Princeton, New Jersey, 1963. Edgar and Himmelblau, Optimization of Chemical Processes, McGraw-Hill, New York, 1987. Fletcher, Practical Methods of Optimization. Wiley, New York, 1980. Gill, Murray, and Wright, Practical Optimization, Academic Press, New York, 1981. Murtagh, Advanced Linear Programming, McGraw-Hill, New York, 1983. Murty, Linear Programming, Wiley, New York, 1983. Reklaitis, Ravindran, and Ragsdell, Engineering Optimization, Wiley—Interscience, New York, 1984.

The chemical industry has undergone significant changes during the past 20 years due to the increased cost of energy and raw materials, more stringent environmental regulations, and intense worldwide competition. Modifications of both plant-design procedures and plant operating conditions have been implemented in order to reduce costs

FIG. 8-45

8-27

Hierarchical control configuration for MPC and on-line optimiza-

tion.

and meet constraints. One of the most important engineering tools that can be employed in such activities is optimization. As plant computers have become more powerful, the size and complexity of problems that can be solved by optimization techniques have correspondingly expanded. A wide variety of problems in the operation and analysis of chemical plants (as well as many other industrial processes) can be solved by optimization. Real-time optimization means that the process-operating conditions (set points) are evaluated on a regular basis and optimized. Sometimes this is called steady-state optimization or supervisory control. This section examines the basic characteristics of optimization problems and their solution techniques and describes some representative benefits and applications in the chemical and petroleum industries. Typical problems in chemical engineering process design or plant operation have many possible solutions. Optimization is concerned with selecting the best among the entire set by efficient quantitative methods. Computers and associated software make the computations involved in the selection feasible and cost-effective. Engineers work to improve the initial design of equipment and strive for enhancements in the operation of the equipment once it is installed in order to realize the most production, the greatest profit, the maximum cost, the least energy usage, and so on. In plant operations, benefits arise from improved plant performance, such as improved yields of valuable products (or reduced yields of contaminants), reduced energy consumption, higher processing rates, and longer times between shutdowns. Optimization can also lead to reduced maintenance costs, less equipment wear, and better staff utilization. It is helpful to systematically identify the objective, constraints, and degrees of freedom in a process or a plant if such benefits as improved quality of designs, faster and more reliable troubleshooting, and faster decision making are to be achieved. Optimization can take place at many levels in a company, ranging

8-28

PROCESS CONTROL

from a complex combination of plants and distribution facilities down through individual plants, combinations of units, individual pieces of equipment, subsystems in a piece of equipment, or even smaller entities. Problems that can be solved by optimization can be found at all these levels. While process design and equipment specification are usually performed prior to the implementation of the process, optimization of operating conditions is carried out monthly, weekly, daily, hourly, or even every minute. Optimization of plant operations determines the set points for each unit at the temperatures, pressures, and flow rates that are the best in some sense. For example, the selection of the percentage of excess air in a process heater is quite critical and involves a balance on the fuel-air ratio to assure complete combustion and at the same time make the maximum use of the heating potential of the fuel. Typical day-to-day optimization in a plant minimizes steam consumption or cooling water consumption, optimizes the reflux ratio in a distillation column, or allocates raw materials on an economic basis [Latour, Hydro Proc., 58(6), 73, 1979, and Hydro. Proc., 58(7), 219, 1979]. A real-time optimization (RTO) system determines set point changes and implements them via the computer control system without intervention from unit operators. The RTO system completes all data transfer, optimization calculations, and set point implementation before unit conditions change and invalidate the computed optimum. In addition, the RTO system should perform all tasks without upsetting plant operations. Several steps are necessary for implementation of RTO, including determination of the plant steady state, data gathering and validation, updating of model parameters (if necessary) to match current operations, calculation of the new (optimized) set points, and the implementation of these set points. To determine if a process unit is at steady state, a program monitors key plant measurements (e.g., compositions, product rates, feed rates, and so on) and determines if the plant is steady enough to start the sequence. Only when all of the key measurements are within the allowable tolerances is the plant considered steady and the optimization sequence started. Tolerances for each measurement can be tuned separately. Measured data are then collected by the optimization computer. The optimization system runs a program to screen the measurements for unreasonable data (gross error detection). This validity checking automatically modifies the model updating calculation to reflect any bad data or when equipment is taken out of service. Data validation and reconciliation (on-line or off-line) is an extremely critical part of any optimization system. The optimization system then may run a parameter-fitting case that updates model parameters to match current plant operation. The integrated process model calculates such items as exchanger heat transfer coefficients, reactor performance parameters, furnace efficiencies, and heat and material balances for the entire plant. Parameter fitting allows for continual updating of the model to account for plant deviations and degradation of process equipment. After completion of the parameter fitting, the information regarding the current plant constraints, the control status data, and the economic values for feed products, utilities, and other operating costs are collected. The economic values are updated by the planning and scheduling department on a regular basis. The optimization system then calculates the optimized set points. The steady-state condition of the plant is rechecked after the optimization case is successfully completed. If the plant is still steady, then the values of the optimization targets are transferred to the process-control system for implementation. After a line-out period, the process-control computer resumes the steadystate detection calculations, restarting the cycle. Essential Features of Optimization Problems The solution of optimization problems involves the use of various tools of mathematics. Consequently, the formulation of an optimization problem requires the use of mathematical expressions. From a practical viewpoint, it is important to mesh properly the problem statement with the anticipated solution technique. Every optimization problem contains three essential categories: 1. An objective function to be optimized (revenue function, cost function, etc.) 2. Equality constraints (equations)

3. Inequality constraints (inequalities) Categories 2 and 3 comprise the model of the process or equipment; category 1 is sometimes called the economic model. No single method or algorithm of optimization exists that can be applied efficiently to all problems. The method chosen for any particular case will depend primarily on (1) the character of the objective function, (2) the nature of the constraints, and (3) the number of independent and dependent variables. Table 8-6 summarizes the six general steps for the analysis and solution of optimization problems (Edgar and Himmelblau, Optimization of Chemical Processes, McGraw-Hill, New York, 1988). You do not have to follow the cited order exactly, but you should cover all of the steps eventually. Shortcuts in the procedure are allowable, and the easy steps can be performed first. Steps 1, 2, and 3 deal with the mathematical definition of the problem: identification of variables and specification of the objective function and statement of the constraints. If the process to be optimized is very complex, it may be necessary to reformulate the problem so that it can be solved with reasonable effort. Later in this section, we discuss the development of mathematical models for the process and the objective function (the economic model). Step 5 in Table 8-6 involves the computation of the optimum point. Quite a few techniques exist to obtain the optimal solution for a problem. We describe several classes of methods below; Fig. 8-46 is a diagram for selection of individual optimization techniques. In general, the solution of most optimization problems involves the use of a digital computer to obtain numerical answers. Over the past 15 years, substantial progress has been made in developing efficient and robust digital methods for optimization calculations. Much is known about which methods are most successful. Virtually all numerical optimization methods involve iteration, and the effectiveness of a given technique often depends on a good first guess for the values of the variables at the optimal solution. After the optimum is computed, a sensitivity analysis for the objective function value should be performed to determine the effects of errors or uncertainty in the objective function, mathematical model, or other constraints. Development of Process (Mathematical) Models Constraints in optimization problems arise from physical bounds on the variables, empirical relations, physical laws, and so on. The mathematical relations describing the process also comprise constraints. Two general categories of models exist: 1. Those based on physical theory 2. Those based on strictly empirical descriptions Mathematical models based on physical and chemical laws (e.g., mass and energy balances, thermodynamics, chemical reaction kinetics) are frequently employed in optimization applications. These models are conceptually attractive because a general model for any system size can be developed before the system is constructed. On the other hand, an empirical model can be devised that simply correlates inputoutput data without any physiochemical analysis of the process. For TABLE 8-6 Problems

The Six Steps Used to Solve Optimization

1. Analyze the process itself so that the process variables and specific characteristics of interest are defined (i.e., make a list of all of the variables). 2. Determine the criterion for optimization and specify the objective function in terms of the above variables together with coefficients. This step provides the performance model (sometimes called the economic model when appropriate). 3. Develop via mathematical expressions a valid process or equipment model that relates the input-output variables of the process and associated coefficients. Include both equality and inequality constraints. Use well-known physical principles (mass balances, energy balances), empirical relations, implicit concepts, and external restrictions. Identify the independent and dependent variables (number of degrees of freedom). 4. If the problem formulation is too large in scope, (a) break it up into manageable parts and/or (b) simplify the objective function and model. 5. Apply a suitable optimization technique to the mathematical statement of the problem. 6. Check the answers and examine the sensitivity of the result to changes in the coefficients in the problem and the assumptions.

ADVANCED CONTROL SYSTEMS

FIG. 8-46

8-29

Diagram for selection of optimization techniques with algebraic constraints and objective function.

these models, optimization is often used to fit a model to process data, using a procedure called parameter estimation. The well-known least squares curve-fitting procedure is based on optimization theory, assuming that the model parameters are contained linearly in the model. One example is the yield matrix, where the percentage yield of each product in a unit operation is estimated for each feed component using process data rather than employing a mechanistic set of chemical reactions.

Formulation of the Objective Function The formulation of objective functions is one of the crucial steps in the application of optimization to a practical problem. You must be able to translate the desired objective into mathematical terms. In the chemical process industries, the objective function often is expressed in units of currency (e.g., U.S. dollars) because the normal industrial goal is to minimize costs or maximize profits subject to a variety of constraints. A typical economic model involves the costs of raw materials, values

8-30

PROCESS CONTROL

of products, costs of production, as functions of operating conditions, projected sales figures, and the like. An objective function can be expressed in terms of these quantities; for example, annual operating profit ($/yr) might be expressed as: J =  FsVs −  FrCr − OC s

where

(8-72)

r

J = profit/time

s FsVs = sum of product flow rates times respective product values (income) F C r

r

r

= sum of feed flows times respective unit costs

OC = operating costs/time

Unconstrained Optimization Unconstrained optimization refers to the case where no inequality constraints are present and all equality constraints can be eliminated by solving for selected dependent variables followed by substitution for them in the objective function. Very few realistic problems in process optimization are unconstrained. However, it is desirable to have efficient unconstrained optimization techniques available since these techniques must be applied in real time and iterative calculations cost computer time. The two classes of unconstrained techniques are single-variable optimization and multivariable optimization. Single Variable Optimization Many process optimization problems can be reduced to the variation of a single variable so as to maximize profit or some other overall process objective function. Some examples of single variable optimization include optimizing the reflux ratio in a distillation column or the air/fuel ratio in a furnace (Martin, Latour, and Richard, Chem. Engr. Prog., 77, September, 1981). While most processes actually are multivariable processes with several operating degrees of freedom, often we choose to optimize only the most important variable in order to keep the strategy uncomplicated. One characteristic implicitly required in a single variable optimization problem is that the objective function J be unimodal in the variable x. The selection of a method for one-dimensional search is based on the trade-off between number of function evaluations versus computer time. We can find the optimum by evaluating the objective function at many values of x using a small grid spacing (∆x) over the allowable range of x values, but this method is generally inefficient. There are three classes of techniques that can be used efficiently for one-dimensional search: 1. Indirect 2. Region elimination 3. Interpolation Indirect methods seek to solve the necessary condition dJ/dx = 0 by iteration, but these methods are not as popular as the second two classes. Region elimination methods include equal interval search, dichotomous search (or bisecting), Fibonacci search, and golden section. These methods do not use information on the shape of the function (other than being unimodal) and thus tend to be rather conservative. The third class of techniques uses repeated polynomial fitting to predict the optimum. These interpolation methods tend to converge rapidly to the optimum without being very complicated. Two interpolation methods, quadratic and cubic interpolation, have been used in many optimization packages. Multivariable Optimization In multivariable optimization problems, there is no guarantee that the optimum can be reached in a reasonable amount of computer time. The numerical optimization of general nonlinear multivariable objective functions requires that efficient and robust techniques be employed. Efficiency is important since iteration is employed. For example, in multivariable “grid” search for a problem with four independent variables, an equally spaced grid for each variable is prescribed. For ten values of each of the four variables, there would be 104 total function evaluations required to find the best answer for the grid intersections. However, this computational effort still may not yield a result close enough to the true optimum, requiring further search. Therefore, grid search is a very inefficient method for most problems involving many variables. In multivariable optimization, the difficulty of dealing with multi-

variable functions is usually resolved by treating the problem as a series of one-dimensional searches. For a given starting point, a search direction s is specified, and the optimum is found by searching along that direction. The step size ε is the distance moved along s. Then a new search direction is determined, followed by another onedimensional search. The algorithm used to specify the search direction depends on the optimization method. There are two basic types of unconstrained optimization algorithms: (1) those requiring function derivatives and (2) those that do not. The nonderivative methods are of interest in optimization applications because these methods can be readily adapted to the case in which experiments are carried out directly on the process. In such cases, an actual process measurement (such as yield) can be the objective function, and no mathematical model for the process is required. Methods that do not require derivatives are called direct methods and include sequential simplex (Nelder-Meade) and Powell’s method. The sequential simplex method is quite satisfactory for optimization with two or three independent variables, is simple to understand, and is fairly easy to execute. Powell’s method is more efficient than the simplex method and is based on the concept of conjugate search directions. The second class of multivariable optimization techniques in principle requires the use of partial derivatives, although finite difference formulas can be substituted for derivatives; such techniques are called indirect methods and include the following classes: 1. Steepest descent (gradient) method 2. Conjugate gradient (Fletcher-Reeves) method 3. Newton’s method 4. Quasi-Newton methods The steepest descent method is quite old and utilizes the intuitive concept of moving in the direction where the objective function changes the most. However, it is clearly not as efficient as the other three. Conjugate gradient utilizes only first-derivative information, as does steepest descent, but generates improved search directions. Newton’s method requires second derivative information but is very efficient, while quasi-Newton retains most of the benefits of Newton’s method but utilizes only first derivative information. All of these techniques are also used with constrained optimization. Constrained Optimization When constraints exist and cannot be eliminated in an optimization problem, more general methods must be employed than those described above, since the unconstrained optimum may correspond to unrealistic values of the operating variables. The general form of a nonlinear programming problem allows for a nonlinear objective function and nonlinear constraints, or Minimize

J(x1, x2, . . . , xn)

Subject to

hi(x1, x2, . . . , xn) = 0

(i = 1, rc)

gi(x1, x2, . . . , xn) ≥ 0

(i = 1, mc)

(8-73)

In this case, there are n process variables with rc equality constraints and mc inequality constraints. Such problems pose a serious challenge to performing optimization calculations in a reasonable amount of time. Typical constraints in chemical process optimization include operating conditions (temperatures, pressures, and flows have limits), storage capacities, and product purity specifications. An important class of constrained optimization problems is one in which both the objective function and constraints are linear. The solution of these problems is highly structured and can be obtained rapidly. The accepted procedure, linear programming (LP), has become quite popular in the past twenty years, solving a wide range of industrial problems. It is increasingly being used for on-line optimization. For processing plants, there are several different kinds of linear constraints that may arise, making the LP method of great utility. 1. Production limitation due to equipment throughput restrictions, storage limits, or market constraints. 2. Raw material (feedstock) limitation. 3. Safety restrictions on allowable operating temperatures and pressures. 4. Physical property specifications placed on the composition of the final product. For blends of various products, we usually assume that a composite property can be calculated through the averaging of pure component physical properties.

UNIT OPERATIONS CONTROL 5. Material and energy balances of the steady-state model. The optimum in linear programming lies at the constraint intersections, which was generalized to any number of variables and constraints by George Dantzig. The Simplex algorithm is a matrix-based numerical procedure for which many digital computer codes exist, both for mainframe and microcomputers (Edgar and Himmelblau, Optimization of Chemical Processes, McGraw-Hill, New York, 1987; Schrage, Linear, Integer, and Quadratic Programming with LINDO, Scientific Press, Palo Alto, California, 1983). The algorithm can handle virtually any number of inequality constraints and any number of variables in the objective function and utilizes the observation that only the constraint boundaries need to be examined to find the optimum. In some instances, nonlinear optimization problems even with nonlinear constraints can be linearized so that the LP algorithm can be employed to solve them (called successive linear programming or SLP). In the process industries, the Simplex algorithm has been applied to a wide range of problems, including refinery scheduling, olefins production, the optimal allocation of boiler fuel, and the optimization of a total plant. Nonlinear Programming The most general case for optimization occurs when both the objective function and constraints are nonlinear, a case referred to as nonlinear programming. While the idea behind the search methods used for unconstrained multivariable problems are applicable, the presence of constraints complicates the solution procedure. In practice, one of the best current general algorithms (best on the basis of many tests) using iterative linearization is the Generalized Reduced Gradient algorithm (GRG). The GRG algorithm employs linear or linearized constraints, defines new variables that are normal to the constraints, and expresses the gradient (or other search direction) in terms of this normal basis (Liebman, Lasdon, Schrage, and Waren, GINO, Scientific Press, Palo Alto, California, 1986). Other established types of constrained optimization methods include the following types of algorithms: 1. Penalty functions with augmented Lagrangian method (an enhancement of the classical Lagrange multiplier method) 2. Successive quadratic programming All of these methods have been utilized to solve nonlinear programming problems in the field of chemical engineering design and operations (Lasdon and Waren, Oper. Res., 5, 34, 1980). Nonlinear programming is receiving increased usage in the area of real-time optimization. One important class of nonlinear programming techniques is called quadratic programming (QP), where the objective function is quadratic and the constraints are linear. While the solution is iterative, it can be obtained quickly as in linear programming. This is the basis for the newest type of constrained multivariable control algorithms called model predictive control. The dominant method used in the refining industry utilizes the solution of a QP and is called dynamic matrix con-

8-31

trol or DMC. See the earlier subsection on model predictive control for more details. EXPERT SYSTEMS An expert system is a computer program that uses an expert’s knowledge in a particular domain to solve a narrowly focused, complex problem. An off-line system uses information entered manually and produces results in visual form to guide the user in solving the problem at hand. An on-line system uses information taken directly from process measurements to perform tasks automatically or instruct or alert operating personnel to the status of the plant. Each expert system has a rule base created by the expert to respond the way the expert would to sets of input information. Expert systems used for plant diagnostics and management usually have an open rule base, which can be changed and augmented as more experience accumulates and more tasks are to be automated. The system begins as an empty shell with an assortment of functions such as equation-solving, logic, and simulation, as well as input and display tools to allow an expert to construct a proprietary rule base. The “expert” in this case would be the person or persons having the deepest knowledge about the process, its problems, its symptoms, and remedies. Converting these inputs into meaningful outputs is the principal task in constructing a rule base. Skill at computer programming is especially helpful, although most shells allow rules to be entered in the vernacular. Firstprinciples models (deep knowledge) produce the most accurate results, although heuristics are always required to establish limits. A closed expert system is one designed by an expert to be sold in quantity for use by others (where open systems tend to be unique). It is closed to keep users from altering the rule base and thereby changing the product. Common examples in process control are autotuning and self-tuning controllers whose rule base is designed by one or more experts in that field. Once packaged and sold, its rule base cannot be changed in the field no matter how poorly it performs the task; revisions must be made by the manufacturer in later releases as for any software product. The development vehicle used to create and test the rule base must be as flexible as possible, allowing easy alterations and expansion of the rule base with whatever displays can convey the most information. The delivery vehicle, however, should be virtually transparent to the user, conveying only as much information as needed to solve the problem at hand. Self-tuning controllers can perform their task without explicitly informing users, but their output and status is available on demand, and their operation may be easily limited or interrupted. To be successful, the scope of an expert system must be limited to a narrow group of common problems that are readily solved by conventional means, and where the return on investment is greatest. Widening the scope usually requires more complex methods and treats less common problems having lower return.

UNIT OPERATIONS CONTROL PROCESS AND INSTRUMENTATION DIAGRAMS GENERAL REFERENCES: Shinskey, Process Control Systems, 3d ed., McGrawHill, New York, 1988. Luyben, Practical Distillation Control, Van Nostrand Reinhold, New York, 1992.

The process and instrumentation (P&I) diagram provides a graphical representation of the control configuration for the process. The P&I diagrams illustrate the measurement devices that provide inputs to the control strategy, the actuators that will implement the results of the control calculations, and the function blocks that provide the control logic. The symbology for drawing P&I diagrams generally follows standards developed by one of the following organizations: 1. International Society for Measurement and Control (ISA). The chemicals, refining, and foods industries generally follow this standard.

2. Scientific Apparatus Manufacturers Association (SAMA). The fossil-fuel electric utility industry generally follows this standard. Both organizations update their standards from time to time, primarily because the continuing evolutions in control-system hardware provide additional possibilities for implementing control schemes. Although arguments can be made for the advantages of each symbology, the practices within an industry seem to be mainly the result of historical practice with no indication of any significant shift. Most companies adopt one of the standards but then tailor or extend the symbology to best suit their internal practices. Such companies maintain an internal document and/or drawing that specifies the symbology used on their P&I diagrams. Their internal personnel and all contractors are instructed to adhere to this symbology when developing P&I diagrams. Figure 8-47 presents a P&I diagram for a simple temperature control loop that adheres to the ISA symbology. The measurement

8-32

PROCESS CONTROL

Temperature leaving a heat exchanger responds as a distributed lag, the gain and time constant of which vary inversely with flow.

FIG. 8-48

FIG. 8-47

Example of a process and instrument diagram.

devices and most elements of the control logic are represented by circles. In Figure 8-47, circles are used to designate the following: 1. TT102 is the temperature measurement device. 2. TC102 is the temperature controller. 3. TY102 is the current-to-pneumatic (I/P) transducer. The symbol for the control valve in Fig. 8-47 is for a pneumatic positioning valve without a valve positioner. Electronic signals (that is, 4–20 milliamp current loops) are represented by dashed lines. In Fig. 8-47, these include the following: 1. The signal from the measurement device to the controller. 2. The signal from the controller to the I/P transducer. Pneumatic signals are represented by solid lines that are crosshatched intermittently. The signal from the I/P transducer to the pneumatic positioning valve is pneumatic. The ISA symbology provides different symbols for different types of actuators. Furthermore, variations for the controller symbol distinguish control algorithms implemented in DCS technology from panel-mounted single-loop controllers. CONTROL OF HEAT EXCHANGERS Steam-Heated Exchangers Steam, the most common heating medium, transfers its latent heat in condensing, causing heat flow to be proportional to steam flow. Thus, a measurement of steam flow is essentially a measure of heat transfer. Consider raising a liquid from temperature T1 to T2 by condensing steam: Q = WH = FCL(T2 − T1)

(8-74)

where W and H are the mass flow of steam and its latent heat, F and CL are the mass flow and specific heat of the liquid, and Q is the rate of heat transfer. The response of controlled temperature to steam flow is linear: H dT2 (8-75) ᎏ=ᎏ dW FCL However, the steady-state process gain described by this derivative varies inversely with liquid flow: Adding a given increment of heat flow to a smaller flow of liquid produces a greater temperature rise. Dynamically, the response of liquid temperature to a step in steam flow is that of a distributed lag, shown in Fig. 8-48. The time required to reach 63 percent complete response,  τ, is essentially the residence time of the fluid in the exchanger, which is its volume divided

by its flow. The residence time then varies inversely with flow. Table 8-2 gives optimum settings for PI and PID controllers for distributed lags, the proportional band varying directly with steady-state gain, and integral and derivative settings directly with  τ. Since both these parameters vary inversely with liquid flow, fixed settings for the temperature controller are optimal at only one flow rate. Undamped oscillations will be produced when the flow decreases by one-third from the value at which the controller was optimally tuned, whereas increasing flow rates produces an overdamped response. The stable operating range can be broadened to one-half the original flow by using an equal-percentage steam valve whose gain varies directly with flow. The best solution is to adapt the PID settings to change inversely with measured flow, thereby keeping the controller optimally tuned for all flow rates. Feedforward control can also be applied by multiplying the liquid flow measurement—after dynamic compensation—by the output of the temperature controller, the result used to set steam flow in cascade. Feedforward is capable of a reduction in integrated error as much as a hundredfold but requires the use of a steam-flow loop and dynamic compensator to approach this. Steam flow is sometimes controlled by manipulating a valve in the condensate line rather than the steam line, because it is smaller and hence less costly. Heat transfer, then, is changed by raising or lowering the level of condensate flooding the heat-transfer surface, an operation that is slower than manipulating a steam valve. Protection also needs to be provided against an open condensate valve blowing steam into the condensate system. Exchange of Sensible Heat When there is no change in phase, heat transfer is no longer linear with flow of the manipulated stream, as illustrated by Fig. 8-49. Here again, an equal-percentage valve should be used on that stream to linearize the loop. The variable dynamics of the distributed lag apply, limiting the stable operating range in the same way as for the steam-heated exchanger. These heat exchangers are also sensitive to variations in the temperature of the manipulated stream, an increasingly common problem where heat is being recovered at variable temperatures for reuse in heat transfer. Figure 8-50 shows a temperature controller (TC) setting a heatflow controller (QC) in cascade. A measurement of the manipulated flow is multiplied by its temperature difference across the heat exchanger to calculate the current heat-transfer rate, using the right side of Eq. (8-74). Variations in supply temperature, then, appear as variations in calculated heat transfer, which the QC can quickly correct by adjusting the manipulated flow. An equal-percentage valve is still required to linearize the secondary loop, but the primary loop of temperature-setting heat flow is linear. Feedforward can be added by multiplying the dynamically compensated flow measurement of the other fluid by the output of the temperature controller.

UNIT OPERATIONS CONTROL F=D+B

8-33 (8-76)

as well as a balance on each component: Fzi = Dyi + Bxi

(8-77)

where z, y, and x are mol fractions of component i in the respective streams. Combining these equations gives a relationship between the composition of the products and their relative portion of the feed: D B zi − xi (8-78) ᎏ=1−ᎏ=ᎏ F F yi − xi

Heat-transfer rate in sensible-heat exchange varies nonlinearly with flow of the manipulated fluid. FIG. 8-49

Manipulating heat flow linearizes the loop and protects against variations in supply temperature.

FIG. 8-50

When manipulating a stream whose flow is independently determined, such as flow of a product or a heat-transfer fluid from a fired heater, a three-way valve is used to divert the required flow to the heat exchanger. This does not alter the linearity of the process or its sensitivity to supply variations and even adds the possibility of independent flow variations. The three-way valve should have equal-percentage characteristics, and heat-flow control may be even more beneficial. DISTILLATION COLUMN CONTROL Distillation columns have four or more closed loops—increasing with the number of product streams and their specifications—all of which interact with each other to some extent. Because of this interaction, there are many possible ways to pair manipulated and controlled variables through controllers and other mathematical functions with widely differing degrees of effectiveness. Columns also differ from each other, so that no single rule of configuring control loops can be applied successfully to all. The following rules apply to the most common separations. Controlling Quality of a Single Product If one of the products of a column is far more valuable than the others, its quality should be controlled to satisfy given specifications, and its recovery should be maximized by minimizing losses of its principal component in other streams. This is achieved by maximizing reflux ratio consistent with flooding limits on trays, which means maximizing the flow of reflux or vapor, whichever is limiting. The same rule should be followed when heating and cooling have little value. A typical example is the separation of high-purity propylene from much lower-valued propane, usually achieved with waste heat from quench water from the cracking reactors. The most important factor affecting product quality is the material balance. In separating a feed stream F into distillate D and bottom B products, an overall mole-flow balance must be maintained:

From the above, it can be seen that control of either xi or yi requires both product flow rates to change with feed rate and feed composition. Figure 8-51 shows a propylene-propane fractionator controlled at maximum boilup by the differential pressure controller (DPC) across the trays. This loop is fast enough to reject upsets in the temperature of the quench water quite easily. Pressure is controlled by manipulating the heat-transfer surface in the condenser through flooding. If the condenser should become overloaded, pressure will rise above set point, but this has no significant effect on the other control loops. Temperature measurements on this column are not helpful, as the difference between the component boiling points is too small. Propane content in the propylene distillate is measured by a chromatographic analyzer sampling the overhead vapor for fast response and is controlled by the analyzer controller (AC) manipulating the ratio of distillate to feed rates. The feedforward signal from feed rate is dynamically compensated by f(t) and nonlinearly characterized by f(x) to account for variations in propylene recovery as feed rate changes. Distillate flow can be measured and controlled more accurately than reflux flow by a factor equal to the reflux ratio—in this column, typically between 10 and 20. Therefore, reflux flow is placed under accumulator level control (LC). Yet composition responds to the difference between boilup and reflux. To eliminate the lag inherent in the response of the level controller, reflux flow is driven by the subtractor in the direction opposite to distillate flow—this is essential to fast response of the composition loop. Controlling Quality of Two Products Where the two products have similar value, or where heating and cooling costs are comparable to product losses, the compositions of both products should be controlled. This introduces the possibility of strong interaction between the two composition loops, as they tend to have the same speed of response. To minimize interaction, most columns should have distillate composition controlled by reflux ratio and bottom composition by boilup or preferably boilup-to-bottom ratio. These loops are insensitive to variations in feed rate, eliminating the need for feedforward control, and they also reject heat-balance upsets quite effectively. Figure 8-52 shows a depropanizer controlled by reflux and boilup ratios. The actual mechanism through which these ratios are manipulated is as D/(L + D) and B/(V + B), where L is reflux flow and V is vapor boilup, which decouples the temperature loops from the liquid-level loops. Column pressure here is controlled by flooding both condenser and accumulator; however, there is no LC on the accumulator, so this arrangement will not function with an overloaded condenser. Temperatures are used as indications of composition in this column because of the substantial difference in boiling points between propane and butanes. However, off-key components such as ethane do effect the accuracy of the relationship so that an analyzer controller is used to set the top temperature controller (TC) in cascade. If the products from a column are especially pure, even this configuration may produce excessive interaction between the composition loops. Then the composition of the less pure product should be controlled by manipulating its own flow; the composition of the remaining product should be controlled by manipulating reflux ratio if it is the distillate or boilup ratio if it is the bottom product. CHEMICAL REACTORS Composition Control The first requirement for successful control of a chemical reactor is to establish the proper stoichiometry, that is, to control the flow rates of the reactants in the proportions needed

8-34

PROCESS CONTROL

FIG. 8-51

FIG. 8-52

The quality of high-purity propylene should be controlled by manipulating the material balance.

Depropanizers require control of both products, here using reflux-ratio and boilup-ratio manipulation.

UNIT OPERATIONS CONTROL to satisfy the reaction chemistry. In a continuous reactor, this begins by setting ingredient flow rates in ratio to one another. However, because of variations in the purity of the feed streams and inaccuracy in flow metering, some indication of excess reactant such as pH or a composition measurement should be used to trim the ratios. Many reactions are incomplete, leaving one or more reactants unconverted. They are separated from the products of the reaction and recycled to the reactor, usually contaminated with inert components. While reactants can be recycled to complete conversion (extinction), inerts can accumulate to the point of impeding the reaction and must be purged from the system. Inerts include noncondensible gases that must be vented and nonvolatiles from which volatile products must be stripped. If one of the reactants differs in phase from the others and the products, it may be manipulated to close the material balance on that phase. For example, a gas reacting with liquids to produce a liquid product may be added, as it is consumed to control reactor pressure; a gaseous purge would be necessary. Similarly, a liquid reacting with a gas to produce a gaseous product could be added, as it is consumed to control liquid level in the reactor; a liquid purge would be required. Where a large excess of one reactant A is used to minimize side reactions, the unreacted excess is typically sent to a storage tank for recycling. Its flow from the recycle storage tank is set in the desired ratio to the flow of reactant B, with the flow of fresh A manipulated to control recycle tank level if the feed is a liquid or tank pressure if it is a gas. Some catalysts travel with the reactants and must be recycled in the same way. With batch reactors, it may be possible to add all reactants in their proper quantities initially if the reaction rate can be controlled by injection of initiator or adjustment of temperature. In semibatch operation, one key ingredient is flow-controlled into the batch at a rate that sets the production. This ingredient should not be manipulated for temperature control of an exothermic reactor, as the loop includes two dominant lags—concentration of the reactant and heat capacity of the reaction mass—and can easily go unstable. Temperature Control Reactor temperature should always be controlled by heat transfer. Endothermic reactions require heat and therefore are eminently self-regulating. Exothermic reactions produce heat, which tends to raise reaction temperature, thereby increasing reaction rate and producing more heat. This positive feedback is countered by negative feedback in the cooling system, which removes more heat as reactor temperature rises. Most continuous reactors have enough heat-transfer surface relative to reaction mass so that negative feedback dominates and they are self-regulating. But most batch reactors do not and are therefore steady-state unstable. Unstable reactors are controllable, but the temperature controller requires a high gain, and the cooling system must have enough margin to accommodate the largest expected disturbance in heat load. Figure 8-53 shows the recommended system for controlling the temperature of an exothermic reactor, either continuous or batch. The circulating pump on the coolant loop is absolutely essential to effective temperature control in keeping dead time minimum and constant—without it, dead time varies inversely with cooling load, causing limit cycling at low loads. Heating is usually required to raise the temperature to reaction conditions, although it is often locked out in a batch reactor once initiator is introduced. The valves are operated in split range, the heating valve opening from 50–100 percent of controller output, and the cooling valve opening from 0–50 percent. The cascade system linearizes the reactor temperature loop, speeds its response, and protects it from disturbances in the cooling system. The flow of heat removed per unit of coolant flow is directly proportional to the temperature rise of the coolant, which varies with both the temperature of the reactor and the rate of heat transfer from it. Using an equal-percentage cooling valve helps compensate for this nonlinearity, although it is incomplete—a preferred arrangement would be to manipulate coolant flow using a heat-flow controller as described in Fig. 8-50. The flow of heat across the heat-transfer surface is linear with both temperatures, leaving the primary loop with a constant gain. Using the coolant exit rather than inlet temperature as the secondary controlled variable moves the jacket dynamics from the primary to the secondary

8-35

The reactor temperature controller sets coolant outlet temperature in cascade, with primary integral feedback taken from the secondary temperature measurement.

FIG. 8-53

loop, reducing the period of the primary loop. Performance and robustness are both improved by using the secondary temperature measurement as the feedback signal to the integral mode of the primary controller. (This feature may only be available with controllers that integrate by positive feedback.) This places the entire secondary loop in the integral path of the primary controller, effectively pacing its integral time to the rate at which the secondary temperature is able to respond. The primary controller may also be left in the automatic mode at all times without integral windup. The primary time constant of the reactor is MrCr τ1 = ᎏ (8-79) UA where Mr and Cr are the mass and heat capacity of the reactants, and U and A are the overall heat-transfer coefficient and area respectively. This system was tested on a pilot reactor where the heat-transfer area and mass could both be changed by a factor of two, changing τ1 by a factor of four as confirmed by observations of rates of temperature rise. Yet the controllers configured as described in Fig. 8-53 did not require retuning as τ1 varied. The primary controller should be PID, and the secondary controller at least PI in this system; if the secondary controller has no integral mode, the primary controller will control with offset. Set point overshoot in batch reactor control can be avoided by setting derivative time of the primary controller higher than its integral time, but this is only effective with interacting PID controllers. CONTROLLING EVAPORATORS The most important consideration in controlling the quality of concentrate from an evaporator is forcing the vapor rate to match the flow of excess solvent entering in the feed. The mass flow of solid material entering and leaving are equal in the steady state: M0 x0 = Mn xn

(8-80)

where M0 and x0 are the mass flow and solid fraction of the feed, and Mn and xn are their values in the product after n effects of evaporation. The total solvent evaporated from all the effects must then be

W=M

0



x − Mn = M0 1 − ᎏ0 xn



(8-81)

8-36

PROCESS CONTROL

For a steam-heated evaporator, each unit of steam W0 applied produces a known amount of evaporation based on the number of effects and their fractional economy E:

 W = nEW0

(8-82)

(A comparable statement can be made with regard to the power applied to a mechanical recompression evaporator.) In summary, the steam flow required to increase the solid content of the feed from x0 to xn is M0(1 − x0 / xn) W0 = ᎏᎏ (8-83) nE The usual measuring device for feed flow is a magnetic flowmeter, which is a volumetric device whose output F must be multiplied by density ρ to produce mass flow M0. For most aqueous solutions which are fed to evaporators, the product of density and the function of solid content appearing above is linear with density: x Fρ 1 − ᎏ0 ≈ F [1 − m(ρ − 1)] (8-84) xn where slope m is determined by the desired product concentration, and density is in g/ml. The required steam flow in lb/h for feed measured in gal/min is then 500F [1 − m(ρ − 1)] W0 = ᎏᎏᎏ (8-85) nE where the factor of 500 converts gal/min of water to lb/h. The factor nE is about 1.74 for a double-effect evaporator and 2.74 for a tripleeffect. Using a thermocompressor (ejector) driven with 150-lb/in2 steam on a single-effect evaporator gives an nE of 2.05; it essentially adds the equivalent of one effect to the evaporator train. A cocurrent evaporator train with its controls is illustrated in Fig. 8-54. The control system applies equally well to countercurrent or mixed-feed evaporators, the principal difference being the tuning of the dynamic compensator f(t), which must be done in the field to minimize the short-term effects of changes in feed flow on product quality. Solid concentration in the product is usually measured as density; feedback trim is applied by the AC adjusting slope m of the density function, which is the only term related to xn. This recalibrates the system whenever xn must move to a new set point. The accuracy of the system depends on controlling heat flow; therefore, if steam pressure varies, compensation must be applied to cor-



FIG. 8-54



rect for both steam density and enthalpy as a function of pressure. Some evaporators must use unreliable sources of low-pressure steam. In this case, the measurement of pressure-compensated steam flow can be used to set feed flow by solving the last equation for F using W0 as a variable. The steam-flow controller would be set for a given production rate, but the dynamically compensated steam-flow measurement would be the input signal to calculate the feed-flow set point. Both of these configurations are widely used in controlling corn-syrup concentrators. DRYING OPERATIONS Controlling dryers is much different than controlling evaporators because on-line measurements of feed rate and composition and product composition are rarely available. Most dryers transfer moisture from wet feed into hot dry air in a single pass. The process is generally very self-regulating, in that moisture becomes progressively harder to remove from the product as it dries: This is known as fallingrate drying. Controlling the temperature of the air leaving a cocurrent dryer tends to regulate the moisture in the product, as long as feed rate and the moisture in the feed and air are reasonably constant. At constant outlet air temperature, product moisture tends to rise with all three of these variables. In the absence of moisture analyzers, regulation of product quality can be improved by raising the temperature of the exhaust air in proportion to the evaporative load. The evaporative load can be estimated by the loss in temperature of the air passing through the dryer in the steady state. Changes in load are first observed in upsets in exhaust temperature at a given inlet temperature; the controller then responds by returning the exhaust air to its original temperature by changing that of the inlet air. Figure 8-55 illustrates the simplest application of this principal as the linear relationship T0 = Tb + K∆T

(8-86)

where T0 is the set point for exhaust temperature elevated above a base temperature Tb corresponding to zero-load operation, and ∆T is the drop in air temperature from inlet to outlet. Coefficient K must be set to regulate product moisture over the expected range of evaporative load. If set too low, product moisture will increase with increasing load; if set too high, it will decrease with increasing load. While K can be estimated from the model of a dryer, it does depend on the rate-of-

Controlling evaporators requires matching steam flow and evaporative load, here using feedforward control.

BATCH PROCESS CONTROL

FIG. 8-55

8-37

Product moisture from a cocurrent dryer can be regulated through temperature control indexed to heat load.

drying curve for the product, its particle size, and whether the load variations are due primarily to changes in feed rate or feed moisture. It is important to have the most accurate measurement of exhaust temperature attainable. Note that Fig. 8-55 shows the sensor inserted into the dryer upstream of the rotating seal, because leakage there could cause the temperature in the exhaust duct to read low—even lower than the wet-bulb temperature, an impossibility without leakage of either heat or outside air. The calculation of exhaust-temperature set point forms a positivefeedback loop capable of destabilizing the dryer. For example, an increase in load causes the controller to raise inlet temperature, which will in turn raise the calculated set point calling for a further increase in inlet temperature. The gain in the set point loop, K, typically is well below the gain of the exhaust temperature measurement responding to the same change in inlet temperature. Negative feedback then dominates in the steady state, but the response of the exhaust temperature measurement is delayed by the dryer. A similar lag f(t) is shown inserted in the set point loop to prevent positive feedback from dominating in the short term, which could cause cycling. If product moisture is measured off-line, analytical results can be used to adjust K and Tb manually. If an on-line analyzer is used, the analyzer controller would be most effective in adjusting the bias Tb, as shown in the figure. While the rotary dryer shown is commonly used for grains and minerals, this system has been successfully applied to fluid-bed drying of plastic pellets, air-lift drying of wood fibers, and spray drying of milk solids. The air may be steam-heated as shown or heated by direct combustion of fuel, provided that a representative measurement of inlet air temperature can be made. If it cannot, then evaporative load can be inferred from a measurement of fuel flow, replacing ∆T in the set point calculation.

If the feed flows countercurrent to the air, as is the case when drying granulated sugar, exhaust temperature does not respond to variations in product moisture. For these dryers, product moisture can better be regulated by controlling its temperature at the point of discharge. Conveyor-type dryers are usually divided into a number of zones, each separately heated with recirculation of air which raises its wet-bulb temperature. Only the last two zones may require indexing of exhaust-air temperature as a function of ∆T. Batch drying, used on small lots like pharmaceuticals, begins operation by blowing air at constant inlet temperature through saturated product in constant-rate drying, where ∆T is constant at its maximum value ∆Tc. When product moisture reaches the point where fallingrate drying begins, the exhaust temperature begins to rise. The desired product moisture will be reached at a corresponding exhaust temperature Tf, which is related to the temperature Tc observed during constant-rate drying, as well as ∆Tc: Tf = Tc + K∆Tc

(8-87)

The control system requires the values of Tc and ∆Tc observed during the first minutes of operation to be stored as the basis for the above calculation of end point. When the exhaust temperature then reaches the value calculated, drying is terminated. Coefficient K can be estimated from models but requires adjustment on-line to reach product specifications repeatedly. Products having different moisture specifications or particle size will require different settings of K, but the system does compensate for variations in feed moisture, batch size, air moisture, and inlet temperature. Some exhaust air may be recirculated to control the dewpoint of the inlet air, thereby conserving energy toward the end of the batch and when the ambient air is especially dry.

BATCH PROCESS CONTROL BATCH VERSUS CONTINUOUS PROCESSES GENERAL REFERENCES: Fisher, Batch Control Systems: Design, Application, and Implementation, ISA, Research Triangle Park, North Carolina, 1990; Rosenof and Ghosh, Batch Process Automation, Van Nostrand Reinhold, New York, 1987.

When categorizing process plants, the following two extremes can be identified:

1. Commodity plants. These plants are custom-designed to produce large amounts of a single product (or a primary product plus one or more secondary products). An example is a chlorine plant, where the primary product is chlorine and the secondary products are hydrogen and sodium hydroxide. Usually the margins (product value less manufacturing costs) for the products from commodity plants are small, so the plants must be designed and operated for best possible efficiencies. Although a few are batch, most commodity plants are

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PROCESS CONTROL

continuous. Factors such as energy costs are life-and-death issues for such plants. 2. Specialty plants. These plants are capable of producing small amounts of a variety of products. Such plants are common in fine chemicals, pharmaceuticals, foods, and so on. In specialty plants, the margins are usually high, so factors such as energy costs are important but not life-and-death issues. As the production amounts are relatively small, it is not economically feasible to dedicate processing equipment to the manufacture of only one product. Instead, batch processing is utilized so that several products (perhaps hundreds) can be manufactured with the same process equipment. The key issue in such plants is to manufacture consistently each product in accordance with its specifications. The above two categories represent the extremes in process configurations. The term semibatch designates plants in which some processing is continuous but other processing is batch. Even processes that are considered to be continuous can have a modest amount of batch processing. For example, the reformer unit within a refinery is thought of as a continuous process, but the catalyst regeneration is normally a batch process. In a continuous process, the conditions within the process are largely the same from one day to the next. Variations in feed composition, plant utilities (e.g., cooling water temperature), catalyst activities, and other variables occur, but normally these changes are either about an average (e.g., feed compositions) or exhibit a gradual change over an extended period of time (e.g., catalyst activities). Summary data such as hourly averages, daily averages, and the like are meaningful in a continuous process. In a batch process, the conditions within the process are continually changing. The technology for making a given product is contained in the product recipe that is specific to that product. Such recipes normally state the following: 1. Raw material amounts. This is the stuff needed to make the product. 2. Processing instructions. This is what must be done with the stuff in order to make the desired product. This concept of a recipe is quite consistent with the recipes found in cookbooks.

FIG. 8-56

Batch control overview.

Sometimes the term recipe is used to designate only the raw material amounts and other parameters to be used in manufacturing a batch. Although appropriate for some batch processes, this concept is far too restrictive for others. For some products, the differences from one product to the next are largely physical as opposed to chemical. For such products, the processing instructions are especially important. The term formula is more appropriate for the raw material amounts and other parameters, with recipe designating the formula and the processing instructions. The above concept of a recipe permits the following three different categories of batch processes to be identified: 1. Cyclical batch. Both the formula and the processing instructions are the same from batch to batch. Batch operations within processes that are primarily continuous often fall into this category. The catalyst regenerator within a reformer unit is a cyclical batch process. 2. Multigrade. The processing instructions are the same from batch to batch, but the formula can be changed to produce modest variations in the product. In a batch PVC plant, the different grades of PVC are manufactured by changing the formula. In a batch pulp digester, the processing of each batch or cook is the same, but at the start of each cook, the process operator is permitted to change the formula values for chemical-to-wood ratios, cook time, cook temperature, and so on. 3. Flexible batch. Both the formula and the processing instructions can change from batch to batch. Emulsion polymerization reactors are a good example of a flexible batch facility. The recipe for each product must detail both the raw materials required and how conditions within the reactor must be sequenced in order to make the desired product. Of these, the flexible batch is by far the most difficult to automate and requires a far more sophisticated control system than either the cyclical batch or the multigrade batch facility. Batches and Recipes Each batch of product is manufactured in accordance with a product recipe, which contains all information (formula and processing instructions) required to make a batch of the product (see Fig. 8-56). For each batch of product, there will be one and only one product recipe. However, a given product recipe is nor-

BATCH PROCESS CONTROL mally used to make several batches of product. To uniquely identify a batch of product, each batch is assigned a unique identifier called the batch ID. Most companies adopt a convention for generating the batch ID, but this convention varies from one company to the next. In most batch facilities, more than one batch of product will be in some stage of production at any given time. The batches in progress may or may not be using the same recipe. The maximum number of batches that can be in progress at any given time is a function of the equipment configuration for the plant. The existence of multiple batches in progress at a given time presents numerous opportunities for the process operator to make errors, such as charging a material to the wrong batch. Charging a material to the wrong batch is almost always detrimental to the batch to which the material is incorrectly charged. Unless this error is recognized quickly so that the proper charge can be made, the error is also detrimental to the batch to which the charge was supposed to be made. Such errors usually lead to an off-specification batch, but the consequences could be more serious and result in a hazardous condition. Recipe management refers to the assumption of such duties by the control system. Each batch of product is tracked throughout its production, which may involve multiple processing operations on various pieces of processing equipment. Recipe management assures that all actions specified in the product recipe are performed on each batch of product made in accordance with that recipe. As the batch proceeds from one piece of processing equipment to the next, recipe management is also responsible for assuring that the proper type of process equipment is used and that this processing equipment is not currently in use by another batch. By assuming such responsibilities, the control system greatly reduces the incidences where operator error results in off-specification batches. Such a reduction in error is essential to implement just-intime production practices, where each batch of product is manufactured at the last possible moment. When a batch (or batches) are made today for shipment by overnight truck, there is insufficient time for producing another batch to make up for an off-specification batch. Routing and Production Monitoring In some facilities, batches are individually scheduled. However, in most facilities, production is scheduled by product runs, where a run is the production of a stated quantity of a given product. From the stated quantity and the standard yield of each batch, the number of batches can be determined. As this is normally more than one batch of product, a production run is normally a sequence of some number of batches of the same product. In executing a production run, the following issues must be addressed (see Fig. 8-56): 1. Processing equipment must be dedicated to making the run. More than one run is normally in progress at a given time. The maximum number of runs simultaneously in progress depends on the equipment configuration of the plant. Routing involves determining which processing equipment will be used for each production run. 2. Raw material must be utilized. When a production run is scheduled, the necessary raw materials must be allocated to the production run. As the individual batches proceed, the consumption of raw materials must be monitored for consistency with the allocation of raw materials to the production run. 3. The production quantity for the run must be achieved by executing the appropriate number of batches. The number of batches is determined from a standard yield for each batch. However, some batches may achieve yields higher than the standard yield, but other batches may achieve yields lower than the standard yield. The actual yields from each batch must be monitored and significant deviations from the expected yields must be communicated to those responsible for scheduling production. The last two activities are key components of production monitoring, although production monitoring may also involve other activities such as tracking equipment utilization. Production Scheduling In this regard, it is important to distinguish between scheduling runs (sometimes called long-term scheduling) and assigning equipment to runs (sometimes called routing or short-term scheduling). As used herein, production scheduling refers to scheduling runs and is usually a corporate-level as opposed to a

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plant-level function. Short-term scheduling or routing was previously discussed and is implemented at the plant level. The long-term scheduling is basically a material resources planning (MRP) activity involving the following: 1. Forecasting. Orders for long-delivery raw materials are issued at the corporate level based on the forecast for the demand for products. The current inventory of such raw materials is also maintained at the corporate level. This constitutes the resources from which products can be manufactured. 2. The orders for products. Orders are normally received at the corporate level and then assigned to individual plants for production and shipment. Although the scheduling of some products is based on required product inventory levels, scheduling based on orders and shipping directly to the customer (usually referred to as just-in-time) avoids the costs associated with maintaining product inventories. 3. Plant locations and capacities. While producing a product at the nearest plant usually lowers transportation costs, plant capacity limitations sometimes dictate otherwise. Any company competing in the world economy needs the flexibility to accept orders on a worldwide basis and then assign them to individual plants to be filled. Such a function is logically implemented within the corporate-level information technology framework. BATCH AUTOMATION FUNCTIONS Automating a batch facility requires a spectrum of functions. Interlocks Some of these are provided for safety and are properly called safety interlocks. However, others are provided to avoid mistakes in processing the batch and are properly called process interlocks. Discrete Device States Discrete devices such as two-position valves can be driven to either of two possible states. Such devices can be optionally outfitted with limit switches that indicate the state of the device. For two-position valves, the following combinations are possible: 1. No limit switches 2. One limit switch on the closed position 3. One limit switch on the open position 4. Two limit switches In process-control terminology, the discrete device driver is the software routine that generates the output to a discrete device such as a valve and also monitors the state feedback information to ascertain that the discrete device actually attains the desired state. Given the variety of discrete devices used in batch facilities, this logic must include a variety of capabilities. For example, valves do not instantly change states, but instead each valve exhibits a travel time for the change from one state to another. To accommodate this characteristic of the field device, the processing logic within the discrete device driver must provide for a user-specified transition time for each field device. When equipped with limit switches, the potential states for a valve are as follows: 1. Open. The valve has been commanded to open, and the limit switch inputs are consistent with the open state. 2. Closed. The valve has been commanded to close, and the limit switch inputs are consistent with the closed state. 3. Transition. This is a temporary state that is only possible after the valve has been commanded to change state. The limit switch inputs are not consistent with the commanded state, but the transition time has not expired. 4. Invalid. The transition time has expired, and the limit switch inputs are not consistent with the commanded state for the valve. The invalid state is an abnormal condition that is generally handled in a manner similar to process alarms. The transition state is not considered to be an abnormal state but may be implemented in either of the following ways: 1. Drive and wait. Further actions are delayed until the device attains its commanded state. 2. Drive and proceed. Further actions are initiated while the device is in the transition state. The latter is generally necessary for devices with long travel times, such as flush-fitting reactor discharge valves that are motor-driven.

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Closing such valves is normally via drive and wait; however, drive and proceed is usually appropriate when opening the valve. Although two-state devices are most common, the need occasionally arises for devices with three or more states. For example, an agitator may be on high speed, on slow speed, or off. Process States Batch processing usually involves imposing the proper sequence of states on the process. For example, a simple blending sequence might be as follows: 1. Transfer specified amount of material from tank A to tank R. The process state is “Transfer from A.” 2. Transfer specified amount of material from tank B to tank R. The process state is “Transfer from B.” 3. Agitate for specified period of time. The process state is “Agitate without cooling.” 4. Cool (with agitation) to specified target temperature. The process state is “Agitate with cooling.” For each process state, the various discrete devices are expected to be in a specified device state. For process state “Transfer from A,” the device states might be as follows: 1. Tank A discharge valve: open. 2. Tank R inlet valve: open. 3. Tank A transfer pump: running. 4. Tank R agitator: off. 5. Tank R cooling valve: closed. For many batch processes, process state representations are a very convenient mechanism for representing the batch logic. A grid or table can be constructed, with the process states as rows and the discrete device states as columns (or vice versa). For each process state, the state of every discrete device is specified to be one of the following: 1. Device state 0, which may be valve closed, agitator off, and so on 2. Device state 1, which may be valve open, agitator on, and so on 3. No change or don’t care This representation is easily understandable by those knowledgeable about the process technology and is a convenient mechanism for conveying the process requirements to the control engineers responsible for implementing the batch logic. Many batch software packages also recognize process states. A configuration tool is provided to define a process state. With such a mechanism, the batch logic does not need to drive individual devices but can simply command that the desired process state be achieved. The system software then drives the discrete devices to the device states required for the target process state. This normally includes the following: 1. Generating the necessary commands to drive each device to its proper state 2. Monitoring the transition status of each device to determine when all devices have attained their proper states 3. Continuing to monitor the state of each device to assure that the devices remain in their proper states Should any discrete device not remain in its target state, failure logic must be initiated. We will use the control of a simple mixing process (Fig. 8-57) to demonstrate various batch control strategies found in commercial systems. To start the operation sequence, a solenoid valve (VN7) is opened to introduce liquid A. When the liquid level in the tank reaches an intermediate level (LH2), flow B is started to turn on the mixer. When the liquid level is high (LXH2), flow B is stopped and the discharge valve is opened (VN9). The discharge valve is closed and the motor stopped when the tank level reaches the low limit (LL2). The operator may start another mixing cycle by depressing the start button again. It should be noted that this simplified control strategy does not deal with emergency process conditions. Timing of equipment sequencing, such as making sure valve 8 is closed before opening the discharge valve, is not considered. However, this example fully demonstrates the device interlocking and signal latching often encountered in sequential process control. This process is event triggered and can be easily programmed using sequential logic [Figure 8-58a]. Many PLC implementations start the programming phase with sequential logic design. Gate 1 ensures that

FIG. 8-57

Process schematics of a mixing tank.

the process will not start, when requested, if the tank level is not low. Gate 3 opens valve 7 for flow A only if valve 8 is not opened. Gate 2 latches the operator request once valve 7 is opened such that the operator may release the push button. Gate 4 starts flow B and the mixer motor when the intermediate level is reached. The start signal is fed into gate 3 to terminate flow A. At the high tank level, gate 6 opens the discharge valve. This signal is fed into gate 4 to stop flow B and the mixer motor. Gate 5 latches in the discharge signal until the tank is drained. Note that for a DCS, this sequential logic can be entered entirely as Boolean functional blocks. Figure 8-58b is the ladder logic diagram for the same mixing process. It involves rungs of parallel circuits containing relays (the circles) and contacts. Parallel bars on the rungs represent contacts. A slashed pair of bars depict a normally closed contact. A normally open momentary contact is shown on rung 1 in Fig. 8-58b. The ladder logic and diagram builder in PLCs can be programmed easily because there are only a limited number of symbols required in ladder logic diagrams. The translation from sequential logic to ladder logic is straightforward. In general, two or more contacts on the same rung forms an AND gate. Contacts on branches of a rung form an OR gate. For example, contact C1 on rung 1 is normally open, unless the tank level is low. Contact CR8 is normally closed unless relay CR8 on rung 2 is energized. An operator-actuated push button, HS4, and the contact C1 forms an AND gate equivalent to gate 1 in Fig. 8-58a. Therefore, when the operator depresses the push button when the tank level is low, relay CR7 is energized, which closes contact CR7 on branch rung 1A. Once contact CR7 is latched in, the operator may release the button. The junction connecting rungs 1 and 1A is equivalent to the output of the OR gate 2 in Fig. 8-58a. Regulatory Control For most batch processes, the discrete logic requirements overshadow the continuous control requirements. For many batch processes, the continuous control can be provided by simple loops for flow, pressure, level, and temperature. However, very sophisticated advanced control techniques are occasionally applied. As temperature control is especially critical in reactors, the simple feedback approach is replaced by model-based strategies that rival if not exceed the sophistication of advanced control loops in continuous plants. In some installations, alternative approaches for regulatory control

BATCH PROCESS CONTROL

(a)

(b) FIG. 8-58

Logic diagrams for the control of the mixing tank.

may be required. Where a variety of products are manufactured, the reactor may be equipped with alternative heat-removal capabilities, including the following: 1. Jacket filled with cooling water. Most such jackets are oncethrough, but some are recirculating. 2. Heat exchanger in a pump-around loop. 3. Reflux condenser. The heat removal capability to be used usually depends on the product being manufactured. Therefore, regulatory loops must be configured for each possible option, and sometimes for certain combinations of the possible options. These loops are enabled and disabled depending on the product being manufactured. The interface between continuous controls and discrete controls is also important. For example, a feed might be metered into a reactor at a variable rate, depending on another feed or possibly on reactor temperature. However, the product recipe calls for a specified quantity of this feed. The flow must be totalized (i.e., integrated), and when the flow total attains a specified value, the feed must be terminated. The discrete logic must have access to operational parameters such as controller modes. That is, the discrete logic must be able to switch a controller to manual, auto, or cascade. Furthermore, the discrete logic must be able to force the controller output to a specified value. Sequence Logic Sequence logic must not be confused with discrete logic. Discrete logic is especially suitable for interlocks or per-

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missives, e.g., the reactor discharge valve must be closed in order for the feed valve to be opened. Sequence logic is used to force the process to attain the proper sequence of states. For example, a feed preparation might be to first charge A, then charge B, then mix, and finally cool. Although discrete logic can be used to implement sequence logic, other alternatives are often more attractive. Sequence logic is often, but not necessarily, coupled with the concept of a process state. Basically, the sequence logic determines when the process should proceed from the current state to the next, and sometimes what the next state should be. Sequence logic must encompass both normal and abnormal process operations. Thus, sequence logic is often viewed as consisting of two distinct but related parts: 1. Normal logic. This sequence logic provides for the normal or expected progression from one process state to another. 2. Failure logic. This logic provides for responding to abnormal conditions, such as equipment failures. Of these, the failure logic can easily be the most demanding. The simplest approach is to stop or hold on any abnormal condition, and let the process operator sort things out. However, this is not always acceptable. Some failures lead to hazardous conditions that require immediate action; waiting for the operator to decide what to do is not acceptable. The appropriate response to such situations is best determined in conjunction with the process hazards analysis. No single approach has evolved as the preferred way to implement sequence logic. The approaches utilized include the following: 1. Discrete logic. Sequence logic can be implemented via ladder logic, and this approach is common when sequence logic is implemented in programmable logic controllers (PLCs). 2. Programming languages. Traditional procedural languages do not provide the necessary constructs for implementing sequence logic. This necessitates one of the following: a. Special languages. The necessary extensions for sequence logic are provided by extending the syntax of the programming language. This is the most common approach within distributed control systems (DCSs). The early implementations used BASIC as the starting point for the extensions; the later implementations used C as the starting point. A major problem with this approach is portability, especially from one manufacturer to the next but sometimes from one product version to the next within the same manufacturer’s product line. b. Subroutine or function libraries. The facilities for sequence logic are provided via subroutines or functions that can be referenced from programs written in FORTRAN or C. This requires a generalpurpose program development environment and excellent facilities to trap the inevitable errors in such programs. Operating systems with such capabilities have long been available on the larger computers, but not for the microprocessors utilized within DCS systems. However, such operating systems are becoming more common within DCS systems. 3. State machines. This technology is commonly applied within the discrete manufacturing industries. However, its migration to process batch applications has been limited. 4. Graphical implementations. For sequence logic, the flowchart traditionally used to represent the logic of computer programs must be extended to provide parallel execution paths. Such extensions have been implemented in a graphical representation called Grafcet. As process engineers have demonstrated a strong dislike for ladder logic, PLC manufacturers are considering providing Grafcet either in addition to or as an alternative to ladder logic. As none of the above have been able to dominate the industry, it is quite possible that future developments will provide a superior approach for implementing sequence logic. BATCH PRODUCTION FACILITIES Especially for flexible batch applications, the batch logic must be properly structured in order to be implemented and maintained in a reasonable manner. An underlying requirement is that the batch process equipment be properly structured. The following structure is appropriate for most batch production facilities.

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Plant A plant is the collection of production facilities at a geographical site. The production facilities at a site normally share warehousing, utilities, and the like. Equipment Suite An equipment suite is the collection of equipment available for producing a group of products. Normally, this group of products is similar in certain respects. For example, they might all be manufactured from the same major raw materials. Within the equipment suite, material transfer and metering capabilities are available for these raw materials. The equipment suite contains all of the necessary types of processing equipment (reactors, separators, and so on) required to convert the raw materials into salable products. A plant may consist of only one suite of equipment, but large plants usually contain multiple equipment suites. Process Unit or Batch Unit A process unit is a collection of processing equipment that can, at least at certain times, be operated in a manner completely independent from the remainder of the plant. A process unit normally provides a specific function in the production of a batch of product. For example, a process unit might be a reactor complete with all associated equipment (jacket, recirculation pump, reflux condenser, and so on). However, each feed preparation tank is usually a separate process unit. With this separation, preparation of the feed for the next batch can be started as soon as the feed tank is emptied for the current batch. All but the very simplest equipment suites contain multiple process units. The minimum number of process units is one for each type of processing equipment required to make a batch of product. However, many equipment suites contain multiple process units of each type. In such equipment suites, multiple batches and multiple production runs can be in progress at a given time. Item of Equipment An item of equipment is a hardware item that performs a specific purpose. Examples are pumps, heat exchangers, agitators, and the like. A process unit could consist of a single item of equipment, but most process units consist of several items of equipment that must be operated in harmony in order to achieve the function expected of the process unit. Device A device is the smallest element of interest to batch logic. Examples of devices include measurement devices and actuators. STRUCTURED BATCH LOGIC

product must be heated to a specified temperature. Whether heating is undertaken with steam or hot oil is irrelevant to the product technology. By restricting the product recipe to a given product technology, the same product recipe can be used to make products at different sites. Timing diagrams (such as Fig. 8-59) are one way to represent a recipe. At a given site, the specific approach to be used to heat a vessel is important. The traditional approach is for an engineer at each site to expand the product recipe into a document that explains in detail how the product is to be made at the site. This document goes by various names, although standard operating procedure or SOP is a common one. Depending on the level of detail to which it is written, the SOP could specify exactly which valves must be opened in order to heat the contents of a vessel. Thus, the SOP is site-dependent, and contains both product technology and process technology. In structuring the logic for a flexible batch application, the following organization permits product technology to be cleanly separated from process technology: • A recipe consists of a formula and one or more processing operations. Ideally, only product technology is contained in a recipe. • A processing operation consists of one or more phases. Ideally, only product technology is contained in a processing operation. • A phase consists of one or more actions. Ideally, only process technology is contained in a phase. In this structure, the recipe and processing operations would be the same at each site that manufactures the product. However, the logic that comprises each phase would be specific to a given site. Using the heating example from above, each site would require a phase to heat the contents of the vessel. However, the logic within the phase at one site would accomplish the heating by opening the appropriate steam valves, while the logic at the other site would accomplish the heating by opening the appropriate hot oil valves. Usually the critical part of structuring batch logic is the definition of the phases. There are two ways to approach this: 1. Examine the recipes for the current products for commonality, and structure the phases to reflect this commonality. 2. Examine the processing equipment to determine what processing capabilities are possible, and write phases to accomplish each possible processing capability.

Flexible batch applications must be pursued using a structured approach to batch logic. In such applications, the same processing equipment is used to make a variety of products. In most facilities, little or no proprietary technology is associated with the equipment itself; the proprietary technology is how this equipment is used to produce each of the products. The primary objective of the structured approach is to separate cleanly the following two aspects of the batch logic: Product Technology Basically, this encompasses the product technology, such as how to mix certain molecules to make other molecules. This technology ultimately determines the chemical and physical properties of the final product. The product recipe is the principal source for the product technology. Process Technology The process equipment permits certain processing operations (e.g., heat to a specified temperature) to be undertaken. Each processing operation will involve certain actions (e.g., opening appropriate valves). The need to keep these two aspects separated is best illustrated by a situation where the same product is to be made at different plants. While it is possible that the processing equipment at the two plants is identical, this is rarely the case. Suppose one plant uses steam for heating its vessels, but the other uses a hot oil system as the source of heat. When a product recipe requires that material is to be heated to a specified temperature, each plant can accomplish this objective, but will go about it in quite different ways. The ideal case for a product recipe is as follows: 1. Contains all of the product technology required to make a product 2. Contains no equipment-dependent information, that is, no process technology In the previous example, such a recipe would simply state that the

FIG. 8-59

Sample process timing diagram.

PROCESS MEASUREMENTS There is the additional philosophical issue of whether to have a large number of simple phases with few options each, or a small number of complex phases with numerous options. The issues are a little different from structuring a complex computer program into subprograms. Each possible alternative will have advantages and disadvantages. As the phase contains no product technology, the implementation of a phase must be undertaken by those familiar with the process equipment. Furthermore, they should undertake this on the basis that the result will be used to make a variety of products, not just those that are initially contemplated. The development of the phase logic must also encompass all equipment-related safety issues. The phase should accomplish a clearly defined objective, so the implementers should be able to thoroughly consider all relevant issues in accomplishing this objective. The phase logic is defined in detail, implemented in the control system, and then thoroughly tested. Except when the processing equipment is modified, future modifications to the phase should be infrequent. The result should be a very dependable module that can serve as a building block for batch logic.

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Even for flexible batch applications, a comprehensive menu of phases should permit most new products to be implemented using currently existing phases. By reusing exising phases, numerous advantages accrue: 1. The engineering effort to introduce a new recipe at a site is reduced. 2. The product is more likely to be on-spec the first time, thus avoiding the need to dispose of off-spec product. 3. The new product can be supplied to customers sooner, hopefully before competitors can supply the product. There is also a distinct advantage in maintenance. When a problem with a phase is discovered and the phase logic is corrected, the correction is effectively implemented in all recipes that use the phase. If a change is implemented in the processing equipment, the affected phases must be modified accordingly and then thoroughly tested. These modifications are also effectively implemented in all recipes that use these phases.

PROCESS MEASUREMENTS GENERAL REFERENCES: Benedict, Fundamentals of Temperature, Pressure, and Flow Measurements, Wiley, New York, 1969. Considine, Process Instruments and Control Handbook, McGraw-Hill, New York, 1993. Considine and Ross, Handbook of Applied Instrumentation, McGraw-Hill, New York, 1964. Doebelin, Measurement Systems: Application and Design, 4th ed. McGrawHill, New York, 1990. Ginesi and Annarummo, “User Tips for Mass, Volume Flowmeters,” Tech, 41, April, 1994. ISA Transducer Compendium, 2d ed., Plenum, New York, 1969. Liptak, Instrument Engineers Handbook, Chilton, Philadelphia, 1995. Michalski, Eckersdorf, and McGhee, Temperature Measurement, Wiley, Chichester, 1991. Nichols, G. D., On-Line Process Analyzers, Wiley, New York, 1988.

GENERAL CONSIDERATIONS Process measurements encompass the application of the principles of metrology to the process in question. The objective is to obtain values for the current conditions within the process and make this information available in a form usable by either the control system, process operators, or any other entity that needs to know. The term “measured variable” or “process variable” designates the process condition that is being determined. Process measurements fall into two categories: 1. Continuous measurements. An example of a continuous measurement is a level measurement device that determines the liquid level in a tank (in meters). 2. Discrete measurements. An example of a discrete measurement is a level switch that indicates the presence or absence of liquid at the location at which the level switch is installed. In continuous processes, most process control applications rely on continuous measurements. In batch processes, many of the process control applications will utilize discrete as well as continuous measurements. In both types of processes, the safety interlocks and process interlocks rely largely on discrete measurements. Continuous Measurements In most applications, continuous measurements are considerably more ambitious than discrete measurements. Basically, discrete measurements involve a yes/no decision, whereas continuous measurements may entail considerable signal processing. The components of a typical continuous measurement device are as follows: • Sensor. This component produces a signal that is related in a known manner to the process variable of interest. The sensors in use today are primarily of the electrical analog variety, and the signal is in the form of a voltage, a resistance, a capacitance, or some other directly measurable electrical quantity. Prior to the mid 1970s, instruments tended to use sensors whose signal was mechanical in nature, and thus compatible with pneumatic technology. Since that time the

fraction of sensors that are digital in nature has grown considerably, often eliminating the need for analog-to-digital conversion. • Signal processing. The signal from most sensors is related in a nonlinear fashion to the process variable of interest. In order for the output of the measurement device to be linear with respect to the process variable of interest, linearization is required. Furthermore, the signal from the sensor might be affected by variables other than the process variable. In this case, additional variables must be sensed and the signal from the sensor compensated to account for the other variables. For example, reference junction compensation is required for thermocouples (except when used for differential temperature measurements). • Transmitter. The measurement device output must be a signal that can be transmitted over some distance. Where electronic analog transmission is used, the low range on the transmitter output is 4 milliamps, and the upper range is 20 milliamps. Microprocessor-based transmitters (often referred to as smart transmitters) are usually capable of transmitting the measured variable digitally in engineering units. Accuracy and Repeatability Definitions of terminology pertaining to process measurements can be obtained from standard S51.1 from the International Society of Measurment and Control (ISA) and standard RC20-11 from the Scientific Apparatus Manufacturers Association (SAMA), both of which are updated periodically. An appreciation of accuracy and repeatability is especially important. Some applications depend on the accuracy of the instrument, but other applications depend on repeatability. Excellent accuracy implies excellent repeatability; however, an instrument can have poor accuracy but excellent repeatability. In some applications, this is acceptable, as discussed below. Range and Span A continuous measurement device is expected to provide credible values of the measured value between a lower range and an upper range. The difference between the upper range and the lower range is the span of the measurement device. The maximum value for the upper range and the minimum value for the lower range depend on the principles on which the measurement device is based and on the design chosen by the manufacturer of the measurement device. If the measured variable is greater than the upper range or less than the lower range, the measured variable is said to be out-of-range or the measurement device is said to be overranged. Accuracy Accuracy refers to the difference between the measured value and the true value of the measured variable. Unfortunately, the true value is never known, so in practice accuracy refers to the difference between the measured value and an accepted standard value for the measured variable.

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Accuracy can be expressed in a number of ways: 1. As an absolute difference in the units of the measured variable 2. As a percent of the current reading 3. As a percent of the span of the measured variable 4. As a percent of the upper range of the span For process measurements, accuracy as a percent of span is the most common. Manufacturers of measurement devices always state the accuracy of the instrument. However, these statements always specify specific or reference conditions at which the measurement device will perform with the stated accuracy, with temperature and pressure most often appearing in the reference conditions. When the measurement device is applied at other conditions, the accuracy is affected. Manufacturers usually also provide some statements on how accuracy is affected when the conditions of use deviate from the referenced conditions in the statement of accuracy. Although appropriate calibration procedures can minimize some of these effects, rarely can they be totally eliminated. It is easily possible for such effects to cause a measurement device with a stated accuracy of 0.25 percent of span at reference conditions to ultimately provide measured values with accuracies of 1 percent or less. Microprocessor-based measurement devices usually provide better accuracy than the traditional electronic measurement devices. In practice, most attention is given to accuracy when the measured variable is the basis for billing, such as in custody transfer applications. However, whenever a measurement device provides data to any type of optimization strategy, accuracy is very important. Repeatability Repeatability refers to the difference between the measurements when the process conditions are the same. This can also be viewed from the opposite perspective. If the measured values are the same, repeatability refers to the difference between the process conditions. For regulatory control, repeatability is of major interest. The basic objective of regulatory control is to maintain uniform process operation. Suppose that on two different occasions, it is desired that the temperature in a vessel be 80°C. The regulatory control system takes appropriate actions to bring the measured variable to 80°C. The difference between the process conditions at these two times is determined by the repeatability of the measurement device. In the use of temperature measurement for control of the separation in a distillation column, repeatability is crucial but accuracy is not. Composition control for the overhead product would be based on a measurement of the temperature on one of the trays in the rectifying section. A target would be provided for this temperature. However, at periodic intervals, a sample of the overhead product is analyzed in the laboratory and the information provided to the process operator. Should this analysis be outside acceptable limits, the operator would adjust the set point for the temperature. This procedure effectively compensates for an inaccurate temperature measurement; however, the success of this approach requires good repeatability from the temperature measurement. Dynamics of Process Measurements Especially where the measurement device is incorporated into a closed loop control configuration, dynamics are important. The dynamic characteristics depend on the nature of the measurement device, and also on the nature of components associated with the measurement device (for example, thermowells and sample conditioning equipment). The term measurement system designates the measurement device and its associated components. The following dynamics are commonly exhibited by measurement systems: • Time constants. Where there is a capacity and a throughput, the measurement device will exhibit a time constant. For example, any temperature measurement device has a thermal capacity (mass times heat capacity) and a heat flow term (heat transfer coefficient and area). Both the temperature measurement device and its associated thermowell will exhibit behavior typical of time constants. • Dead time. Probably the best example of a measurement device that exhibits pure dead time is the chromatograph, because the analysis is not available for some time after a sample is injected. Additional dead time results from the transportation lag within the sample

system. Even continuous analyzer installations are plagued by dead time from the sample system. • Underdamped behavior. Measurement devices with mechanical components often have a natural harmonic and can exhibit underdamped behavior. The displacer type of level measurement device is capable of such behavior. While the manufacturers of measurement devices can supply some information on the dynamic characteristics of their devices, interpretation is often difficult. Measurement device dynamics are quoted on varying bases, such as rise time, time to 63 percent response, settling time, and so on. Even where the time to 63 percent response is quoted, it might not be safe to assume that the measurement device exhibits first-order behavior. Where the manufacturer of the measurement device does not supply the associated equipment (thermowells, sample conditioning equipment, and the like), the user must incorporate the characteristics of these components to obtain the dynamics of the measurement system. An additional complication is that most dynamic data are stated for configurations involving reference materials such as water, air, and so on. The nature of the process material will affect the dynamic characteristics. For example, a thermowell will exhibit different characteristics when immersed in a viscous organic emulsion than when immersed in water. It is often difficult to extrapolate the available data to process conditions of interest. Similarly, it is often impossible, or at least very difficult, to experimentally determine the characteristics of a measurement system under the conditions where it is used. It is certainly possible to fill an emulsion polymerization reactor with water and determine the dynamic characteristics of the temperature measurement system. However, it is not possible to determine these characteristics when the reactor is filled with the emulsion under polymerization conditions. The primary impact of unfavorable measurement dynamics is on the performance of closed loop control systems. This explains why most control engineers are very concerned with measurement dynamics. The goal to improve the dynamic characteristics of measurement devices is made difficult because the discussion regarding measurement dynamics is often subjective. Selection Criteria The selection of a measurement device entails a number of considerations given below, some of which are almost entirely subjective. 1. Measurement span. The measurement span required for the measured variable must lie entirely within the instrument’s envelope of performance. 2. Performance. Depending on the application, accuracy, repeatability, or perhaps some other measure of performance is appropriate. Where closed loop control is contemplated, speed of response must be included. 3. Reliability. Data available from the manufacturers can be expressed in various ways and at various reference conditions. Often, previous experience with the measurement device within the purchaser’s organization is weighted most heavily. 4. Materials of construction. The instrument must withstand the process conditions to which it is exposed. This encompasses considerations such as operating temperatures, operating pressures, corrosion, and abrasion. For some applications, seals or purges may be necessary. 5. Prior use. For the first installation of a specific measurement device at a site, training of maintenance personnel and purchases of spare parts might be necessary. 6. Potential for releasing process materials to the environment. Fugitive emissions are receiving ever increasing attention. Exposure considerations, both immediate and long term, for maintenance personnel are especially important when the process fluid is either corrosive or toxic. 7. Electrical classification. Article 500 of the National Electric Code provides for the classification of the hazardous nature of the process area in which the measurement device will be installed. If the measurement device is not inherently compatible with this classification, suitable enclosures must be purchased and included in the installation costs.

PROCESS MEASUREMENTS 8. Physical access. Subsequent to installation, maintenance personnel must have physical access to the measurement device for maintenance and calibration. If additional structural facilities are required, they must be included in the installation costs. 9. Cost. There are two aspects of the cost: a. Initial purchase and installation (capital costs). b. Recurring costs (operational expense). This encompasses instrument maintenance, instrument calibration, consumables (for example, titrating solutions must be purchased for automatic titrators), and any other costs entailed in keeping the measurement device in service. Calibration Calibration entails the adjustment of a measurement device so that the value from the measurement device agrees with the value from a standard. The International Standards Organization (ISO) has developed a number of standards specifically directed to calibration of measurement devices. Furthermore, compliance with the ISO 9000 standards requires that the working standard used to calibrate a measurement device must be traceable to an internationally recognized standard such as those maintained by the National Institute of Standards and Technology (NIST). Within most companies, the responsibility for calibrating measurement devices is delegated to a specific department. Often, this department may also be responsible for maintaining the measurement device. The specific calibration procedures depend on the type of measurement device. The frequency of calibration is normally predetermined, but earlier action may be dictated if the values from the measurement device become suspect. Calibration of some measurement devices involves comparing the measured value with the value from the working standard. Pressure and differential pressure transmitters are calibrated in this manner. Calibration of analyzers normally involves using the measurement device to analyze a specially prepared sample whose composition is known. These and similar approaches can be applied to most measurement devices. Flow is an important measurement whose calibration presents some challenges. When a flow measurement device is used in applications such as custody transfer, provision is made to pass a known flow through the meter. However, such a provision is costly and is not available for most in-process flowmeters. Without such a provision, a true calibration of the flow element itself is not possible. For orifice meters, calibration of the flowmeter normally involves calibration of the differential pressure transmitter, and the orifice plate is usually only inspected for deformation, abrasion, and so on. Similarly, calibration of a magnetic flowmeter normally involves calibration of the voltage measurement circuitry, which is analogous to calibration of the differential pressure transmitter for an orifice meter. TEMPERATURE MEASUREMENTS Measurement of the hotness or coldness of a body or fluid is commonplace in the process industries. Temperature-measuring devices utilize systems with properties that vary with temperature in a simple, reproducible manner and thus can be calibrated against known references (sometimes called secondary thermometers). The three dominant measurement devices used in automatic control are thermocouples, resistance thermometers, and pyrometers and are applicable over different temperature regimes. Thermocouples Temperature measurements using thermocouples are based on the discovery by Seebeck in 1821 that an electric current flows in a continuous circuit of two different metallic wires if the two junctions are at different temperatures. The thermocouple may be represented diagrammatically as shown in Fig. 8-60. A and B are the two metals, and T1 and T2 are the temperatures of the junctions. Let T1 and T2 be the reference junction (cold junction) and the measuring junction, respectively. If the thermoelectric current i flows in the direction indicated in Fig. 8-60, metal A is customarily referred to as thermoelectrically positive to metal B. Metal pairs used for thermocouples include platinum-rhodium (the most popular and accurate), chromel-alumel, copper-constantan, and iron-constantan. The thermal emf is a measure of the difference in temperature between T2 and T1. In control systems the reference junction is usually located at

FIG. 8-60

8-45

Basic circuit of Seebeck effect.

the emf-measuring device. The reference junction may be held at constant temperature such as in an ice bath or a thermostated oven, or it may be at ambient temperature but electrically compensated (coldjunction-compensated circuit) so that it appears to be held at a constant temperature. Resistance Thermometers The resistance thermometer depends upon the inherent characteristics of materials to change in electrical resistance when they undergo a change in temperature. Industrial resistance thermometers are usually constructed of platinum, copper, or nickel, and more recently semiconducting materials such as thermistors are being used. Basically, a resistance thermometer is an instrument for measuring electrical resistance that is calibrated in units of temperature instead of in units of resistance (typically ohms). Several common forms of bridge circuits are employed in industrial resistance thermometry, the most common being the Wheatstone bridge. A resistance thermometer detector (RTD) consists of a resistance conductor (metal), which generally shows an increase in resistance with temperature. The following equation represents the variation of resistance with temperature (°C): RT = R0(1 + a1T + a2T 2 + . . . + anT n) R0 = resistance at 0°C

(8-88)

The temperature coefficient of resistance αT is expressed as: 1 dR αT = ᎏ ᎏT (8-89) RT dT For most metals, αT is positive. For many pure metals, the coefficient is essentially constant and stable over large portions of their useful range. Typical resistance versus temperature curves for platinum, copper, and nickel are given in Fig. 8-61, with platinum usually the metal of choice. Platinum has a useful range of −200°C to 800°C, while Nickel (−80°C to 320°C) and copper (−100°C to 100°C) are more limited. Detailed resistance versus temperature tables are available from the National Bureau of Standards and suppliers of resistance thermometers. Table 8-7 gives recommended temperature measurement ranges for thermocouples and RTDs. Resistance thermometers are receiving increased usage because they are about ten times more accurate than thermocouples. Thermistors Thermistors are nonlinear temperature-dependent resistors, and normally only the materials with negative temperature TABLE 8-7 Recommended Temperature Measurement Ranges for RTDs and Thermocouples Resistance thermometer detectors (RTDs) −200°C–+850°C −80°C–+320°C

100V Pt 120V Ni Thermocouples Type B Type E Type J Type K Type N Type R Type S Type T

700°C–+1820°C −175°C–+1000°C −185°C–+1200°C −175°C–+1372°C 0°C–+1300°C 125°C–+1768°C 150°C–+1768°C −170°C–+400°C

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Typical resistance-thermometer curves for platinum, copper, and nickel wire, where RT = resistance at temperature T and R0 = resistance at 0°C.

FIG. 8-61

coefficient of resistance (NTC type) are used. The resistance is related to temperature as: 1 1 RT = RTr exp β ᎏ − ᎏ (8-90) T Tr






where Tr is a reference temperature, which is generally 298 K. Thus 1 dR αT = ᎏ ᎏT (8-91) RT dT The value of β is of the order of 4000, so at room temperature (298 K), αT = −0.045 for thermistor and 0.0035 for 100 Ω Platinum RTD. Compared with RTDs, NTC type thermistors are advantageous in that the detector dimension can be made small, the resistance value is higher (less affected by the resistances of the connecting leads), the temperature sensitivity is higher, and the thermal inertia of the sensor is low. Disadvantages of thermistors to RTDs include nonlinear characteristics and low measuring temperature range. Filled-System Thermometers The filled-system thermometer is designed to provide an indication of temperature some distance removed from the point of measurement. The measuring element (bulb) contains a gas or liquid that changes in volume, pressure, or vapor pressure with temperature. This change is communicated through a capillary tube to a Bourdon tube or other pressure- or volume-sensitive device. The Bourdon tube responds so as to provide a motion related to the bulb temperature. Those systems that respond to volume changes are completely filled with a liquid. Systems that respond to pressure changes either are filled with a gas or are partially filled with a volatile liquid. Changes in gas or vapor pressure with changes in bulb temperatures are carried through the capillary to the Bourdon. The latter bulbs are sometimes constructed so that the capillary is filled with a nonvolatile liquid. Fluid-filled bulbs deliver enough power to drive controller mechanisms and even directly actuate control valves. These devices are characterized by large thermal capacity, which sometimes leads to slow response, particularly when they are enclosed in a thermal well for process measurements. Filled-system thermometers are used extensively in industrial processes for a number of reasons. The simplicity

of these devices allows rugged construction, minimizing the possibility of failure with a low level of maintenance, and inexpensive overall design of control equipment. In case of system failure, the entire unit must be replaced or repaired. As normally used in the process industries, the sensitivity and percentage of span accuracy of these thermometers are generally the equal of those of other temperature-measuring instruments. Sensitivity and absolute accuracy are not the equal of those of short-span electrical instruments used in connection with resistance-thermometer bulbs. Also, the maximum temperature is somewhat limited. Bimetal Thermometers Thermostatic bimetal can be defined as a composite material made up of strips of two or more metals fastened together. This composite, because of the different expansion rates of its components, tends to change curvature when subjected to a change in temperature. With one end of a straight strip fixed, the other end deflects in proportion to the temperature change, the square of the length, and inversely as the thickness, throughout the linear portion of the deflection characteristic curve. If a bimetallic strip is wound into a helix or a spiral and one end is fixed, the other end will rotate when heat is applied. For a thermometer with uniform scale divisions, a bimetal must be designed to have linear deflection over the desired temperature range. Bimetal thermometers are used at temperatures ranging from 580°C down to −180°C and lower. However, at the low temperatures the rate of deflection drops off quite rapidly. Bimetal thermometers do not have long-time stability at temperatures above 430°C. Pyrometers Planck’s distribution law gives the radiated energy flux qb(λ, T)d λ in the wavelength range λ to λ + dλ from a black surface: 1 C1 qb(λ, T) = ᎏ (8-92) ᎏ λ5 ec2 /λT − 1 where C1 = 3.7418 × 1010 µW µm4 cm−2, and C2 = 14,388 µm K. If the target object is a black body and if the pyrometer has a detector that measures the specific wavelength signal from the object, the temperature of the object can be exactly estimated from Eq. (8-92). While it is possible to construct a physical body that closely approxi-

PROCESS MEASUREMENTS

8-47

mates black body behavior, most real-world objects are not black bodies. The deviation from a black body can be described by the spectral emissivity q(T) εT = ᎏ (8-93) qb(T)

the ratio of the radiant intensities of the nonblack body, whose temperature is to be measured at the same wavelength, then Wien’s law gives ελ1 exp−C2/λ1T exp−C2/λ1Tc =ᎏ (8-99) ᎏᎏ ελ2 exp−C2/λ2T exp−C2/λ2Tc

where q(λ, T) is the radiated energy flux from a real body in the wavelength range λ to λ + dλ and 0 < ελ,T < 1. Integrating Eq. (8-92) over all wavelengths gives the Stefan-Boltzmann equation

where T is the true temperature of the body. Rearranging Eq. (8-99) gives

qb(T) =



∞

qb(λ, T) dλ

0

= σT 4

(8-94)

where σ is the Stefan-Boltzmann constant. Similar to Eq. (8-93), the emissivity εT for the total radiation is q(T) εT = ᎏ (8-95) qb(T) where q(T) is the radiated energy flux from a real body with emissivity εT. Total Radiation Pyrometers In total radiation pyrometers, the thermal radiation is detected over a large range of wavelengths from the object at high temperature. The detector is normally a thermopile, which is built by connecting several thermocouples in series to increase the temperature measurement range. The pyrometer is calibrated for black bodies, so the indicated temperature Tp should be converted for non-black body temperature. Photoelectric Pyrometers Photoelectric pyrometers belong to the class of band radiation pyrometers. The thermal inertia of thermal radiation detectors does not permit the measurement of rapidly changing temperatures. For example, the smallest time constant of a thermal detector is about 1 msec, while the smallest time constant of a photoelectric detector can be about 1 or 2 sec. Photoelectric pyrometers may use photoconductors, photodiodes, photovoltaic cells, or vacuum photocells. Photoconductors are built from glass plates with thin film coatings of 1 µm thickness, using PbS, CdS, PbSe or PbTe. When the incident radiation has the same wavelength as the materials are able to absorb, the captured incident photons free photoelectrons, which form an electric current. Photodiodes in germanium or silicon are operated with a reverse bias voltage applied. Under the influence of the incident radiation their conductivity as well as their reverse saturation current is proportional to the intensity of the radiation within the spectral response band from 0.4 to 1.7 µm for Ge and 0.6 to 1.1 µm for Si. Because of the above characteristics, the operating range of a photoelectric pyrometer can be either spectral or in a specific band. Photoelectric pyrometers can be applied for a specific choice of the wavelength. Disappearing Filament Pyrometers Disappearing filament pyrometers can be classified as spectral pyrometers. The brightness of a lamp filament is changed by adjusting the lamp current until the filament disappears against the background of the target, at which point the temperature is measured. Since the detector is the human eye, it is difficult to calibrate for on-line measurements. Ratio Pyrometers The ratio pyrometer is also called the twocolor pyrometer. Two different wavelengths are utilized for detecting the radiated signal. If one uses Wien’s law for small values of λT, the detected signals from spectral radiant energy flux emitted at the wavelengths λ1 and λ2 with emissivities ελ1 and ελ2 are −C2/λ1T Sλ1 = KC1ελ1λ−5 1 exp

(8-96)

−C2/λ2T Sλ2 = KC1ελ2λ−5 2 exp

(8-97)

The ratio of the signals Sλ1 and Sλ2 is ελ1 λ2 Sλ ᎏ1 = ᎏ ᎏ Sλ2 ελ2 λ1

C 1 ᎏ − ᎏ    exp
ᎏ T λ λ 5

1

2

2

(8-98)

1

Nonblack or nongrey bodies are characterized by wavelength dependence of their spectral emissivity. Let Tc be defined as the temperature of the body corresponding to the temperature of a black body. If the ratio of its radiant intensities at the wavelengths λ1, and λ2 equals




ln ελ1/ελ2 1 T = ᎏᎏ +ᎏ 1 1 Tc C2 ᎏᎏ − ᎏᎏ λ1 λ2







−1

(8-100)

For black or grey bodies, Eq. (8-98) reduces to 1 Sλ λ2 5 C2 1 (8-101) ᎏ1 = ᎏ exp ᎏ ᎏ − ᎏ Sλ2 λ1 T λ2 λ1 Thus, by measuring Sλ1 and Sλ2, the temperature T can be estimated. Accuracy of Pyrometers Most of the temperature estimation methods for pyrometers assume that the object is either a grey body or has known emissivity values. The emissivity of the nonblack body depends on the internal state or the surface geometry of the objects. Also, the medium through which the thermal radiation passes is not always transparent. These inherent uncertainties of the emissivity values make the accurate estimation of the temperature of the target objects difficult. Proper selection of the pyrometer and accurate emissivity values can provide a high level of accuracy.

 






PRESSURE MEASUREMENTS Pressure defined as force per unit area is usually expressed in terms of familiar units of weight-force and area or the height of a column of liquid that produces a like pressure at its base. Process pressuremeasuring devices may be divided into three groups: (1) those that are based on the measurement of the height of a liquid column, (2) those that are based on the measurement of the distortion of an elastic pressure chamber, and (3) electrical sensing devices. Liquid-Column Methods Liquid-column pressure-measuring devices are those in which the pressure being measured is balanced against the pressure exerted by a column of liquid. If the density of the liquid is known, the height of the liquid column is a measure of the pressure. Most forms of liquid-column pressure-measuring devices are commonly called manometers. When the height of the liquid is observed visually, the liquid columns are contained in glass or other transparent tubes. The height of the liquid column may be measured in length units or be calibrated in pressure units. Depending on the pressure range, water and mercury are the liquids most frequently used. Since the density of the liquid used varies with temperature, the temperature must be taken into account for accurate pressure measurements. Elastic-Element Methods Elastic-element pressure-measuring devices are those in which the measured pressure deforms some elastic material (usually metallic) within its elastic limit, the magnitude of the deformation being approximately proportional to the applied pressure. These devices may be loosely classified into three types: Bourdon tube, bellows, and diaphragm. Bourdon-Tube Elements Probably the most frequently used process pressure-indicating device is the C-spring Bourdon-tube pressure gauge. Gauges of this general type are available in a wide variety of pressure ranges and materials of construction. Materials are selected on the basis of pressure range, resistance to corrosion by the process materials, and effect of temperature on calibration. Gauges calibrated with pressure, vacuum, compound (combination pressure and vacuum), and suppressed-zero ranges are available. Bellows Element The bellows element is an axially elastic cylinder with deep folds or convolutions. The bellows may be used unopposed, or it may be restrained by an opposing spring. The pressure to be measured may be applied either to the inside or to the space outside the bellows, with the other side exposed to atmospheric pressure. For measurement of absolute pressure either the inside or the space outside of the bellows can be evacuated and sealed. Differential pres-

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sures may be measured by applying the pressures to opposite sides of a single bellows or to two opposing bellows. Diaphragm Elements Diaphragm elements may be classified into two principal types: those that utilize the elastic characteristics of the diaphragm and those that are opposed by a spring or other separate elastic element. The first type usually consists of one or more capsules, each composed of two diaphragms bonded together by soldering, brazing, or welding. The diaphragms are flat or corrugated circular metallic disks. Metals commonly used in diaphragm elements include brass, phosphor bronze, beryllium copper, and stainless steel. Ranges are available from fractions of an inch of water to about 206.8 kPa gauge. The second type of diaphragm is used for containing the pressure and exerting a force on the opposing elastic element. The diaphragm is a flexible or slack diaphragm of rubber, leather, impregnated fabric, or plastic. Movement of the diaphragm is opposed by a spring that determines the deflection for a given pressure. This type of diaphragm is used for the measurement of extremely low pressure, vacuum, or differential pressure. Electrical Methods Strain Gauges When a wire or other electrical conductor is stretched elastically, its length is increased and its diameter is decreased. Both of these dimensional changes result in an increase in the electrical resistance of the conductor. Devices utilizing resistancewire grids for measuring small distortions in elastically stressed materials are commonly called strain gauges. Pressure-measuring elements utilizing strain gauges are available in a wide variety of forms. They usually consist of one of the elastic elements described earlier to which one or more strain gauges have been attached to measure the deformation. There are two basic strain-gauge forms: bonded and unbonded. Bonded strain gauges are those which are bonded directly to the surface of the elastic element whose strain is to be measured. The unbonded-strain-gauge transducer consists of a fixed frame and an armature which moves with respect to the frame in response to the measured pressure. The strain-gauge wire filaments are stretched between the armature and frame. The strain gauges are usually connected electrically in a Wheatstone-bridge configuration. Strain-gauge pressure transducers are manufactured in many forms for measuring gauge, absolute, and differential pressures and vacuum. Full-scale ranges from 25.4 mm of water to 10,134 MPa are available. Strain gauges bonded directly to a diaphragm pressure-sensitive element usually have an extremely fast response time and are suitable for high-frequency dynamic-pressure measurements. Piezoresistive Transducers A variation of the conventional strain-gauge pressure transducer uses bonded single-crystal semiconductor wafers, usually silicon, whose resistance varies with strain or distortion. Transducer construction and electrical configurations are similar to those using conventional strain gauges. A permanent magnetic field is applied perpendicular to the resonating sensor. An AC current causes the resonator to vibrate, and the resonant frequency is a function of the pressure (tension) of the resonator. The principal advantages of piezoresistive transducers are a much higher bridge voltage output and smaller size. Full-scale output voltages of 50 to 100 mV/V of excitation are typical. Some newer devices provide digital rather than analog output. Piezoelectric Transducers Certain crystals produce a potential difference between their surfaces when stressed in appropriate directions. Piezoelectric pressure transducers generate a potential difference proportional to a pressure-generated stress. Because of the extremely high electrical impedance of piezoelectric crystals at low frequency, these transducers are usually not suitable for measurement of static process pressures. FLOW MEASUREMENTS Flow, defined as volume per unit of time at specified temperature and pressure conditions, is generally measured by positive-displacement or rate meters. The term “positive-displacement meter” applies to a device in which the flow is divided into isolated measured volumes when the number of fillings of these volumes is counted in some man-

ner. The term “rate meter” applies to all types of flowmeters through which the material passes without being divided into isolated quantities. Movement of the material is usually sensed by a primary measuring element that activates a secondary device. The flow rate is then inferred from the response of the secondary device by means of known physical laws or from empirical relationships. The principal classes of flow-measuring instruments used in the process industries are variable-head, variable-area, positive-displacement, and turbine instruments, mass flowmeters, vortex-shedding and ultrasonic flowmeters, magnetic flowmeters, and more recently, Coriolis mass flowmeters. Head meters are covered in more detail in Sec. 5. Orifice Meter The most widely used flowmeter involves placing a fixed-area flow restriction (an orifice) in the pipe carrying the fluid. This flow restriction causes a pressure drop that can be related to flow rate. The sharp-edge orifice is popular because of its simplicity, low cost, and the large amount of research data on its behavior. For the orifice meter, the flow rate Qa for a liquid is given by CdA2 Qa = ᎏᎏ ⋅ −A (2A /1)2 1

2(p − p ) ᎏᎏ  ρ 1

2

(8-102)

where p1 − p2 is the pressure drop, ρ is the density, A1 is the pipe crosssectional area, A2 is the orifice cross-sectional area, and Cd is the discharge coefficient. The discharge coefficient Cd varies with the Reynolds number at the orifice and can be calibrated with a single fluid, such as water (typically Cd ≈ 0.6). If the orifice and pressure taps are constructed according to certain standard dimensions, quite accurate (about 0.4 to 0.8 percent error) values of Cd may be obtained. It should also be noted that the standard calibration data assume no significant flow disturbances such as elbows, valves, and so on, for a certain minimum distance upstream of the orifice. The presence of such disturbances close to the orifice can cause errors of as much as 15 percent. Accuracy in measurements limits the meter to a range of 3:1. The orifice has a relatively large permanent pressure loss that must be made up by the pumping machinery. Venturi Meter The venturi tube operates on exactly the same principle as the orifice [see Eq. (8-102)]. Discharge coefficients of venturis are larger than those for orifices and vary from about 0.94 to 0.99. A venturi gives a definite improvement in power losses over an orifice and is often indicated for measuring very large flow rates, where power losses can become economically significant. The initial higher cost of a venturi over an orifice may thus be offset by reduced operating costs. Rotameter A rotameter consists of a vertical tube with a tapered bore in which a float changes position with the flow rate through the tube. For a given flow rate the float remains stationary since the vertical forces of differential pressure, gravity, viscosity, and buoyancy are balanced. The float position is the output of the meter and can be made essentially linear with flow rate by making the tube area vary linearly with the vertical distance. Turbine Meter If a turbine wheel is placed in a pipe containing a flowing fluid, its rotary speed depends on the flow rate of the fluid. A turbine can be designed whose speed varies linearly with flow rate. The speed can be measured accurately by counting the rate at which turbine blades pass a given point, using magnetic pickup to produce voltage pulses. By feeding these pulses to an electronic pulse-rate meter, one can measure flow rate by summing the pulses during a timed interval. Turbine meters are available with full-scale flow rates ranging from about 0.1 to 30,000 gpm for liquids and 0.1 to 15,000 ft3/min for air. Nonlinearity can be less than 0.05 percent in the larger sizes. Pressure drop across the meter varies with the square of flow rate and is about 3 to 10 psi at full flow. Turbine meters can follow flow transients quite accurately since their fluid/mechanical time constant is of the order of 2 to 10 msec. Vortex-Shedding Flowmeters These flowmeters take advantage of vortex shedding, which occurs when a fluid flows past a nonstreamlined object (a blunt body). The flow cannot follow the shape of the object and separates from it, forming turbulent vortices or eddies at the object’s side surfaces. As the vortices move downstream, they grow in size and are eventually shed or detached from the object.

PROCESS MEASUREMENTS Shedding takes place alternately at either side of the object, and the rate of vortex formation and shedding is directly proportional to the volumetric flow rate. The vortices are counted and used to develop a signal linearly proportional to the flow rate. The digital signals can easily be totaled over an interval of time to yield the flow rate. Accuracy can be maintained regardless of density, viscosity, temperature, or pressure when the Reynolds number is greater than 10,000. There is usually a low flow cutoff point below which the meter output is clamped at zero. This flowmeter is recommended for use with relatively clean, low viscosity liquids, gases, and vapors, and rangeability of 10:1 to 20:1 is typical. A sufficient length of straight-run pipe is necessary to prevent distortion in the fluid velocity profile. Ultrasonic Flowmeters All ultrasonic flowmeters are based upon the variable time delays of received sound waves that arise when a flowing liquid’s rate of flow is varied. Two fundamental measurement techniques, depending upon liquid cleanliness, are generally used. In the first technique, two opposing transducers are inserted in a pipe so that one transducer is downstream from the other. These transducers are then used to measure the difference between the velocity at which the sound travels with the direction of flow and the velocity at which it travels against the direction of flow. The differential velocity is measured either by (1) direct time delays using sound wave burst or (2) frequency shifts derived from beat-together, continuous signals. The frequency-measurement technique is usually preferred because of its simplicity and independence of the liquid static velocity. A relatively clean liquid is required to preserve the uniqueness of the measurement path. In the second technique, the flowing liquid must contain scatters in the form of particles or bubbles that will reflect the sound waves. These scatters should be traveling at the velocity of the liquid. A Doppler method is applied by transmitting sound waves along the flow path and measuring the frequency shift in the returned signal from the scatters in the process fluid. This frequency shift is proportional to liquid velocity. Magnetic Flowmeters The principle behind these flowmeters is Faraday’s law of electromagnetic inductance. The magnitude of the voltage induced in a conductive medium moving at right angles through a magnetic field is directly proportional to the product of the magnetic flux density, the velocity of the medium, and the path length between the probes. A minimum value of fluid conductivity is required to make this approach viable. The pressure of multiple phases or undissolved solids can affect the accuracy of the measurement if the velocities of the phases are different than that for straightrun pipe. Magmeters are very accurate over wide flow ranges and are especially accurate at low flow rates. Typical applications include metering viscous fluids, slurries, or highly corrosive chemicals. Because magmeters should be filled with fluid, the preferred installation is in vertical lines with flow going upwards. However, magmeters can be used in tight piping schemes where it is impractical to have long pipe runs, typically requiring lengths equivalent to five or more pipe diameters. Coriolis Mass Flowmeters Coriolis mass flowmeters utilize a vibrating tube in which Coriolis acceleration of a fluid in a flow loop can be created and measured. They can be used with virtually any liquid and are extremely insensitive to operating conditions, with high pressure over ranges of 100:1. These meters are more expensive than volumetric meters and range in size from g to 6 inches. Due to the circuitous path of flow through the meter, Coriolis flowmeters exhibit higher than average pressure changes. The meter should be installed so that it will remain full of fluid, with the best installation in a vertical pipe with flow going upward. There is no Reynolds number limitation with this meter, and it is quite insensitive to velocity profile distortions and swirl, hence there is no requirement for straight piping upstream. LEVEL MEASUREMENTS The measurement of level can be defined as the determination of the location of the interface between two fluids, separable by gravity, with respect to a fixed datum plane. The most common level measurement is that of the interface between a liquid and a gas. Other level mea-

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surements frequently encountered are the interface between two liquids, between a granular or fluidized solid and a gas, and between a liquid and its vapor. A commonly used basis for classification of level devices is as follows: float-actuated, displacer, and head devices, and a miscellaneous group that depends mainly on fluid characteristics. Float-Actuated Devices Float-actuated devices are characterized by a buoyant member that floats at the interface between two fluids. Since a significant force is usually required to move the indicating mechanism, float-actuated devices are generally limited to liquid-gas interfaces. By properly weighting the float, they can be used to measure liquid-liquid interfaces. Float-actuated devices may be classified on the basis of the method used to couple the float motion to the indicating system as discussed below. Chain or Tape Float Gauge In these types of gauges, the float is connected to the indicating mechanism by means of a flexible chain or tape. These gauges are commonly used in large atmospheric storage tanks. The gauge-board type is provided with a counterweight to keep the tape or chain taut. The tape is stored in the gauge head on a spring-loaded reel. The float is usually a pancake-shaped hollow metal float with guide wires from top to bottom of the tank to constrain it. Lever and Shaft Mechanisms In pressurized vessels, floatactuated lever and shaft mechanisms are frequently used for level measurement. This type of mechanism consists of a hollow metal float and lever attached to a rotary shaft, which transmits the float motion to the outside of the vessel through a rotary seal. Magnetically Coupled Devices A variety of float-actuated level devices that transmit the float motion by means of magnetic coupling have been developed. Typical of this class of devices are magnetically operated level switches and magnetic-bond float gauges. A typical magnetic-bond float gauge consists of a hollow magnet-carrying float that rides along a vertical nonmagnetic guide tube. The follower magnet is connected and drives an indicating dial similar to that on a conventional tape float gauge. The float and guide tube are in contact with the measured fluid and come in a variety of materials for resistance to corrosion and to withstand high pressures or vacuum. Weighted floats for liquid-liquid interfaces are available. Head Devices A variety of devices utilize hydrostatic head as a measure of level. As in the case of displacer devices, accurate level measurement by hydrostatic head requires an accurate knowledge of the densities of both heavier-phase and lighter-phase fluids. The majority of this class of systems utilize standard-pressure and differential-pressure measuring devices. Bubble-Tube Systems The commonly used bubble-tube system sharply reduces restrictions on the location of the measuring element. In order to eliminate or reduce variations in pressure drop due to the gas flow rate, a constant differential regulator is commonly employed to maintain a constant gas flow rate. Since the flow of gas through the bubble tube prevents entry of the process liquid into the measuring system, this technique is particularly useful with corrosive or viscous liquids, liquids subject to freezing, and liquids containing entrained solids. Electrical Methods Two electrical characteristics of fluids— conductivity and dielectric constant—are frequently used to distinguish between two phases for level-measurement purposes. An application of electrical conductivity is the fixed-point level detection of a conductive liquid such as high and low water levels. A voltage is applied between two electrodes inserted into the vessel at different levels. When both electrodes are immersed in the liquid, a current flows. Capacitance-type level measurements are based on the fact that the electrical capacitance between two electrodes varies with the dielectric constant of the material between them. A typical continuous level-measurement system consists of a rod electrode positioned vertically in a vessel, the other electrode usually being the metallic vessel wall. The electrical capacitance between the electrodes is a measure of the height of the interface along the rod electrode. The rod is usually conductively insulated from process fluids by a coating of plastic. The dielectric constant of most liquids and solids is markedly higher than that of gases and vapors. The dielectric constant of water and other polar liquids is also higher than that of hydrocarbons and other nonpolar liquids.

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Thermal Methods Level-measuring systems may be based on the difference in thermal characteristics between the fluids, such as temperature or thermal conductivity. A fixed-point level sensor based on the difference in thermal conductivity between two fluids consists of an electrically heated thermistor inserted into the vessel. The temperature of the thermistor and consequently its electrical resistance increase as the thermal conductivity of the fluid in which it is immersed decreases. Since the thermal conductivity of liquids is markedly higher than that of vapors, such a device can be used as a point level detector for liquid-vapor interface. Sonic Methods A fixed-point level detector based on sonicpropagation characteristics is available for detection of a liquid-vapor interface. This device uses a piezoelectric transmitter and receiver, separated by a short gap. When the gap is filled with liquid, ultrasonic energy is transmitted across the gap, and the receiver actuates a relay. With a vapor filling the gap, the transmission of ultrasonic energy is insufficient to actuate the receiver. PHYSICAL PROPERTY MEASUREMENTS Physical-property measurements are sometimes equivalent to composition analyzers, because the composition can frequently be inferred from the measurement of a selected physical property. Density and Specific Gravity For binary or pseudobinary mixtures of liquids or gases or a solution of a solid or gas in a solvent, the density is a function of the composition at a given temperature and pressure. Specific gravity is the ratio of the density of a noncompressible substance to the density of water at the same physical conditions. For nonideal solutions, empirical calibration will give the relationship between density and composition. Several types of measuring devices are described below. Liquid Column Density may be determined by measuring the gauge pressure at the base of a fixed-height liquid column open to the atmosphere. If the process system is closed, then a differential pressure measurement is made between the bottom of the fixed height liquid column and the vapor over the column. If vapor space is not always present, the differential-pressure measurement is made between the bottom and top of a fixed-height column with the top measurement being made at a point below the liquid surface. Displacement There are a variety of density-measurement devices based on displacement techniques. A hydrometer is a constant-weight, variable-immersion device. The degree of immersion, when the weight of the hydrometer equals the weight of the displaced liquid, is a measure of the density. The hydrometer is adaptable to manual or automatic usage. Another modification includes a magnetic float suspended below a solenoid, the varying magnetic field maintaining the float at a constant distance from the solenoid. Change in position of the float, resulting from a density change, excites an electrical system which increases or decreases the current through the solenoid. Direct Mass Measurement One type of densitometer measures the natural vibration frequency and relates the amplitude to changes in density. The density sensor is a U-shaped tube held stationary at its node points and allowed to vibrate at its natural frequency. At the curved end of the U is an electrochemical device that periodically strikes the tube. At the other end of the U, the fluid is continuously passed through the tube. Between strikes, the tube vibrates at its natural frequency. The frequency changes directly in proportion to changes in density. A pickup device at the curved end of the U measures the frequency and electronically determines the fluid density. This technique is useful because it is not affected by the optical properties of the fluid. However, particulate matter in the process fluid can affect the accuracy. Radiation-Density Gauges Gamma radiation may be used to measure the density of material inside a pipe or process vessel. The equipment is basically the same as for level measurement, except that here the pipe or vessel must be filled over the effective, irradiated sample volume. The source is mounted on one side of the pipe or vessel and the detector on the other side with appropriate safety radiation shielding surrounding the installation. Cesium 137 is used as the radi-

ation source for path lengths under 610 mm (24 in) and cobalt 60 above 610 mm. The detector is usually an ionization gauge. The absorption of the gamma radiation is a function of density. Since the absorption path includes the pipe or vessel walls, an empirical calibration is used. Appropriate corrections must be made for the source intensity decay with time. Viscosity Continuous viscometers generally measure either the resistance to flow or the drag or torque produced by movement of an element (moving surface) through the fluid. Each installation is normally applied over a narrow range of viscosities. Empirical calibration over this range allows use on both newtonian and nonnewtonian fluids. One such device uses a piston inside a cylinder. The hydrodynamic pressure of the process fluid raises the piston to a preset height. Then the inlet valve closes and the piston is allowed to free-fall, and the time of travel (typically a few seconds) is a measure of viscosity. Other geometries include the rotation of a spindle inside a sample chamber and a vibrating probe immersed in the fluid. Because viscosity depends on temperature, the viscosity measurement must be thermostated with a heater or cooler. Refractive-Index When light travels from one medium (e.g., air or glass) into another (e.g., a liquid), it undergoes a change of velocity and, if the angle of incidence is not 90°, a change of direction. For a given interface, angle, temperature, and wavelength of light the amount of deviation or refraction will depend on the composition of the liquid. If the sample is transparent, the normal method is to measure the refraction of light transmitted through the glass-sample interface. If the sample is opaque, the reflectance near the critical angle at a glass-sample interface is measured. In an on-line refractometer, the process fluid is separated from the optics by a prism material. A beam of light is focused on a point in the fluid which creates a conic section of light at the prism, striking the fluid at different angles (greater than or less than the critical angle). The critical angle depends on the species concentrations; as the critical angle changes, the proportions of reflected and refracted light change. A photodetector produces a voltage signal proportional to the light refracted, when compared to a reference signal. Refractometers can be used with opaque fluids and in streams that contain particulates. Dielectric Constant The dielectric constant of material represents its ability to reduce the electric force between two charges separated in space. This property is useful in process control for polymers, ceramic materials, and semiconductors. Dielectric constants are measured with respect to vacuum (1.0); typical values range from 2 (benzene) to 33 (methanol) to 80 (water). The value for water is higher than for most plastics. A measuring cell is made of glass or some other insulating material and is usually doughnut-shaped, with the cylinders coated with metal, which constitute the plates of the capacitor. Thermal Conductivity All gases and vapor have the ability to conduct heat from a heat source. At a given temperature and physical environment, radiation, and convection heat losses will be stabilized and the temperature of the heat source will be mainly dependent on the thermal conductivity and thus the composition of the surrounding gases. Thermal-conductivity analyzers normally consist of a sample cell and a reference cell, each containing a combined heat source and detector. These cells are normally contained in a metal block with two small cavities in which the detectors are mounted. The sample flows through the sample-cell cavity past the detector. The reference cell is an identical cavity with a detector through which a known gas flows. The combined heat source and detectors are normally either wire filaments or thermistors heated by a constant current. Since their resistance is a function of temperature, the sample-detector resistance will vary with sample composition while the reference-detector resistance will remain constant. The output from the detector bridge will be a function of sample composition. CHEMICAL COMPOSITION ANALYZERS A number of composition analyzers used for process monitoring and control require chemical conversion of one or more sample components preceding quantitative measurement. These reactions include

PROCESS MEASUREMENTS formation of suspended solids for turbidimetric measurement, formation of colored materials for colorimetric detection, selective oxidation or reduction for electrochemical measurement, and formation of electrolytes for measurement by electrical conductance. Some nonvolatile materials may be separated and measured by gas chromatography after conversion into volatile derivatives. Chromatographic Analyzers Chromatographic analyzers are widely used for the separation and measurement of volatile compounds and of compounds that can be quantitatively converted into volatile derivatives. These materials are separated by placing a portion of the sample in a chromatographic column and carrying the compounds through the column with a gas stream. As a result of the different affinities of the sample components for the column packing, the compounds emerge successively as binary mixtures with the carrier gas. A detector at the column outlet measures some physical property which can be related to the concentrations of the compounds in the carrier gas. Both the concentration peak height and the peak height-time integral, (i.e., peak area) can be related to the concentration of the compound in the original sample. The two detectors most commonly used for process chromatographs are the thermalconductivity detector and the hydrogen-flame ionization detector. Thermal-conductivity detectors, discussed earlier, require calibration for the thermal response of each compound. Hydrogen-flame ionization detectors are more complicated than thermal-conductivity detectors but are capable of 100 to 10,000 times greater sensitivity for hydrocarbons and organic compounds. For ultrasensitive detection of trace impurities, carrier gases must be specially purified. Infrared Analyzers Many gaseous and liquid compounds absorb infrared radiation to some degree. The degree of absorption at specific wavelengths depends on molecular structure and concentration. There are two common detector types for nondispersive infrared analyzers. These analyzers normally have two beams of radiation, an analyzing and a reference beam. One type of detector consists of two gas-filled cells separated by a diaphragm. As the amount of infrared energy absorbed by the detector gas in one cell changes, the cell pressure changes. This causes movement in the diaphragm, which in turn causes a change in capacitance between the diaphragm and a reference electrode. This change in electrical capacitance is measured as the output. The second type of detector consists of two thermopiles or two bolometers, one in each of the two radiation beams. The infrared radiation absorbed by the detector is measured by a differential thermocouple output or a resistance-thermometer (bolometer) bridge circuit. With gas-filled detectors, a chopped light system is normally used in which one side of the detector sees the source through the analyzing beam and the other side the reference beam, alternating at a frequency of a few hertz. Ultraviolet and Visible-Radiation Analyzers Many gas and liquid compounds absorb radiation in the near-ultraviolet or visible region. For example, organic compounds containing aromatic and carbonyl structural groups are good absorbers in the ultraviolet region. Also many inorganic salts and gases absorb in the ultraviolet or visible region. In contrast, straight-chain and saturated hydrocarbons, inert gases, air, and water vapor are essentially transparent. Process analyzers are designed to measure the absorbance in a particular wavelength band. The desired band is normally isolated by means of optical filters. When the absorbance is in the visible region, the term “colorimetry” is used. A phototube is the normal detector. Appropriate optical filters are used to limit the energy reaching the detector to the desired level and the desired wavelength region. Since absorption by the sample is logarithmic if a sufficiently narrow wavelength region is used, an exponential amplifier is sometimes used to compensate and produce a linear output. Paramagnetism A few gases including O2, NO, and NO2 exhibit paramagnetic properties as a result of unpaired electrons. In a nonuniform magnetic field, paramagnetic gases, because of their magnetic susceptibility, tend to move toward the strongest part of the field, thus displacing diamagnetic gases. Paramagnetic susceptibility of these gases decreases with temperature. These effects permit measurement of the concentration of the strongest paramagnetic gas, oxy-

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gen. This analyzer used a dumbbell suspended in the magnetic field which is repelled or attracted toward the magnetic field depending on the magnetic susceptibility of the gas. ELECTROANALYTICAL INSTRUMENTS Conductometric Analysis Solutions of electrolytes in ionizing solvents (e.g., water) conduct current when an electrical potential is applied across electrodes immersed in the solution. Conductance is a function of ion concentration, ionic charge, and ion mobility. Conductance measurements are ideally suited for measurement of the concentration of a single strong electrolyte in dilute solutions. At higher concentrations, conductance becomes a complex, nonlinear function of concentration requiring suitable calibration for quantitative measurements. Measurement of pH The primary detecting element in pH measurement is the glass electrode. A potential is developed at the pH-sensitive glass membrane as a result of differences in hydrogen ion activity in the sample and a standard solution contained within the electrode. This potential measured relative to the potential of the reference electrode gives a voltage that is expressed as pH. Instrumentation for pH measurement is among the most widely used process-measurement devices. Rugged electrode systems and highly reliable electronic circuits have been developed for this use. After installation, the majority of pH measurement problems are sensor-related, mostly on the reference side, including junction plugging, poisoning, and depletion of electrolyte. For the glass (measuring electrode), common difficulties are broken or cracked glass, coating, and etching or abrasion. Symptoms such as drift, sluggish response, unstable readings, and inability to calibrate are indications of measurement problems. On-line diagnostics such as impedance measurements, wiring checks, and electrode temperature are now available in most instruments. Other characteristics that can be measured off-line include efficiency or slope and asymmetry potential (offset), which indicate whether the unit should be cleaned or changed [Nichols, Chem. Engr. Prog., 90(12), 64, 1994; McMillan, Chem. Engr. Prog., 87(12), 30, 1991]. Specific-Ion Electrodes In addition to the pH glass electrode specific for hydrogen ions, a number of electrodes that are selective for the measurement of other ions have been developed. This selectivity is obtained through the composition of the electrode membrane (glass, polymer, or liquid-liquid) and the composition of the electrode. These electrodes are subject to interference from other ions, and the response is a function of the total ionic strength of the solution. However, electrodes have been designed to be highly selective for specific ions, and when properly used, these provide valuable process measurements. MOISTURE MEASUREMENT Moisture measurements are important in the process industries because moisture can foul products, poison reactions, damage equipment, or cause explosions. Moisture measurements include both absolute-moisture methods and relative-humidity methods. The absolute methods are those that provide a primary output that can be directly calibrated in terms of dew-point temperature, molar concentration, or weight concentration. Loss of weight on heating is the most familiar of these methods. The relative-humidity methods are those that provide a primary output that can be more directly calibrated in terms of percentage of saturation of moisture. Dew-Point Method For many applications, the dew point is the desired moisture measurement. When concentration is desired, the relation between water content and dew point is well-known and available. The dew-point method requires an inert surface whose temperature can be adjusted and measured, a sample gas stream flowing past the surface, a manipulated variable for adjusting the surface temperature to the dew point, and a means of detecting the onset of condensation. Although the presence of condensate can be detected electrically, the original and most often used method is the optical detection of

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change in light reflection from an inert metallic-surface mirror. Some instruments measure the attenuation of reflected light at the onset of condensation. Others measure the increase of light dispersed and scattered by the condensate instead of, or in addition to, the reflectedlight measurement. Surface cooling is obtained with an expendable refrigerant liquid, conventional mechanical refrigeration, or thermoelectric cooling. Surface-temperature measurement is usually made with a thermocouple or a thermistor. Piezoelectric Method A piezoelectric crystal in a suitable oscillator circuit will oscillate at a frequency dependent on its mass. If the crystal has a stable hygroscopic film on its surface, the equivalent mass of the crystal varies with the mass of water sorbed in the film. Thus, the frequency of oscillation depends on the water in the film. The analyzer contains two such crystals in matched oscillator circuits. Typically, valves alternately direct the sample to one crystal and a dry gas to the other on a 30-s cycle. The oscillator frequencies of the two circuits are compared electronically, and the output is the difference between the two frequencies. This output is then representative of the moisture content of the sample. The output frequency is usually converted to a variable DC voltage for meter readout and recording. Multiple ranges are provided for measurement from about 1 ppm to near saturation. The dry reference gas is preferably the same as the sample except for the moisture content of the sample. Other reference gases which are adsorbed in a manner similar to the dried sample gas may be used. The dry gas is usually supplied by an automatic dryer. The method requires a vapor sample to the detector. Mist striking the detector destroys the accuracy of measurement until it vaporizes or is washed off the crystals. Water droplets or mist may destroy the hygroscopic film, thus requiring crystal replacement. Vaporization or gasliquid strippers may sometimes be used for the analysis of moisture in liquids. Capacitance Method Several analyzers utilize the high dielectric constant of water for its detection in solutions. The alternating electric current through a capacitor containing all or part of the sample between the capacitor plates is measured. Selectivity and sensitivity are enhanced by increasing the concentration of moisture in the cell by filling the capacitor sample cell with a moisture-specific sorbent as part of the dielectric. This both increases the moisture content and reduces the amount of other interfering sample components. Granulated alumina is the most frequently used sorbent. These detectors may be cleaned and recharged easily and with satisfactory reproducibility if the sorbent itself is uniform. Oxide Sensors Aluminum oxide can be used as a sensor for moisture analysis. A conductivity cell has one electrode node of aluminum, which is anodized to form a thin film of aluminum oxide, followed by coating with a thin layer of gold (the opposite electrode). Moisture is selectively adsorbed through the gold layer and into the hygroscopic aluminum oxide layer, which in turn determines the electrical conductivity between gold and aluminum oxide. This value can be related to ppm water in the sample. This sensor can operate between near vacuum to several hundred atmospheres, and it is independent of flow rate (including static conditions). Temperature, however, must be carefully monitored. A similar device is based on phosphorous pentoxide. Moisture content influences the electrical current between two inert metal electrodes, which are fabricated as a helix on the inner wall of a tubular nonconductive sample cell. For a constant DC voltage applied to the electrodes, a current flows which is proportional to moisture. The moisture is absorbed into the hygroscopic phosphorous pentoxide, where the current electrolyzes the water molecules into hydrogen and oxygen. This sensor will handle moisture up to 1000 ppm and 6 atm pressure. Similar to the aluminum oxide ion, temperature control is very important. Photometric Moisture Analysis This analyzer requires a light source, a filter wheel rotated by a synchronous motor, a sample cell, a detector to measure the light transmitted, and associated electronics. Water has two absorption bands in the near infrared region at 1400 and 1900 nm. This analyzer can measure moisture in liquid or gaseous samples at levels from 5 ppm up to 100 percent, depending on other chemical species in the sample. Response time is less than 1 s, and samples can be run up to 300°C and 400 psig.

OTHER TRANSDUCERS Gear Train Rotary motion and angular position are easily transduced by various types of gear arrangements. A gear train in conjunction with a mechanical counter is a direct and effective way to obtain a digital readout of shaft rotations. The numbers on the counter can mean anything desired, depending on the gear ratio and the actuating device used to turn the shaft. A pointer attached to a gear train can be used to indicate a number of revolutions or a small fraction of a revolution for any specified pointer rotation. Differential Transformer These devices produce an AC electrical output from linear movement of an armature. They are very versatile in that they can be designed for a full range of output with any range of armature travel up to several inches. The transformers have one or two primaries and two secondaries connected to oppose each other. With an AC voltage applied to the primary, the output voltage depends on the position of the armature and the coupling. Such devices produce accuracies of 0.5 to 1.0 percent of full scale and are used to transmit forces, pressures, differential pressures, or weights up to 1500 m. They can also be designed to transmit rotary motion. Hall-Effect Sensors Some semiconductor materials exhibit a phenomenon in the presence of a magnetic field which is adaptable to sensing devices. When a current is passed through one pair of wires attached to a semiconductor, such as germanium, another pair of wires properly attached and oriented with respect to the semiconductor will develop a voltage proportional to the magnetic field present and the current in the other pair of wires. Holding the exciting current constant and moving a permanent magnet near the semiconductor produce a voltage output proportional to the movement of the magnet. The magnet may be attached to a process-variable measurement device which moves the magnet as the variable changes. Hall-effect devices provide high speed of response, excellent temperature stability, and no physical contact. SAMPLING SYSTEMS FOR PROCESS ANALYZERS The sampling system consists of all the equipment required to present a process analyzer with a clean representative sample of a process stream and to dispose of that sample. When the analyzer is part of an automatic control loop, the reliability of the sampling system is as important as the reliability of the analyzer or the control equipment. Sampling systems have several functions. The sample must be withdrawn from the process, transported, conditioned, introduced into the analyzer, and disposed. Probably the most common problem in sample-system design is the lack of realistic information concerning the properties of the process material at the sampling point. Another common problem is the lack of information regarding the conditioning required so that the analyzer may utilize the sample without malfunction for long periods of time. Some samples require enough conditioning and treating that the sampling systems become equivalent to miniature online processing plants. These systems possess many of the same fabrication, reliability, and operating problems as small-scale pilot plants except that the sampling system must generally operate reliably for much longer periods of time. Selecting the Sampling Point The selection of the sampling point is based primarily on supplying the analyzer with a sample whose composition or physical properties are pertinent to the control function to be performed. Other considerations include selecting locations that provide representative homogeneous samples with minimum transport delay, locations that collect a minimum of contaminating material, and locations that are accessible for test and maintenance procedures. Sample Withdrawal from Process A number of considerations are involved in the design of sample-withdrawal devices that will provide representative samples. For example, in a horizontal pipe that conveys process fluid, a sample point on the bottom of the pipe will collect a maximum amount of rust, scale, or other solid materials being carried along by the process fluid. In a gas stream, such a location will also collect a maximum amount of liquid contaminants. A sample point on the top side of a pipe will, for liquid streams, collect a

TELEMETERING AND TRANSMISSION maximum amount of vapor contaminants being carried along. Bends in the piping that produce swirls or cause centrifugal concentration of the denser phase may cause maximum contamination to be at unexpected locations. Two-phase process materials are difficult to sample for a total-composition representative sample. A typical method for obtaining a sample of process fluid well away from vessel or pipe walls is an eduction tube inserted through a packing gland. This sampling method withdraws liquid sample and vaporizes it for transporting to the analyzer location. The transport lag time from the end of the probe to the vaporizer is minimized by using tubing having a small internal volume compared with pipe and valve volumes. This sample probe may be removed for maintenance and reinstalled without shutting down the process. The eduction tube is made of material that will not corrode so that it will slide through the packing gland even after long periods of service. There may be a small amount of process-fluid leakage until the tubing is withdrawn sufficiently to close the gate valve. A swaged ferrule on the end of the tube prevents accidental ejection of the eduction tube prior to removal of the packing gland. The section of pipe surrounding the eduction tube and extending into the process vessel provides mechanical protection for the eduction tube. Sample Transport Transport time, the time elapsed between sample withdrawal from the process and its introduction into the analyzer, should be minimized, particularly if the analyzer is an automatic analyzer-controller. Any sample-transport time in the analyzercontroller loop must be treated as equivalent to process dead time in determining conventional feedback controller settings or in evaluating controller performance. Reduction in transport time usually means transporting the sample in the vapor state.

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Design considerations for sample-lines are as follows: 1. The structural strength or protection must be compatible with the area through which the sample line runs. 2. Line size and length must be small enough to meet transporttime requirements without excessive pressure drop or excessive bypass of sample at the analyzer input. 3. Line size and internal-surface quality must be adequate to prevent clogging by the contaminants in the sample. 4. The prevention of a change of state of the sample may require insulation, refrigeration, or heating of the sample line. 5. Sample-line material must be such as to minimize corrosion due to sample or environment. Sample Conditioning Sample conditioning usually involves the removal of contaminants or some deleterious component from the sample mixture and/or the adjustment of temperature, pressure, and flow rate of the sample to values acceptable to the analyzer. Some of the more common contaminants that must be removed are rust, scale, corrosion products, deposits due to chemical reactions, and tar. In sampling some process streams, the material to be removed may include the primary-process product such as polymer or the main constituent of the stream such as oil. In other cases, the material to be removed is present in trace quantities. For example, water in an online chromatograph sample can damage the chromatographic column packing. When contaminants or other materials that will hinder analysis represent a large percentage of the stream composition, their removal may significantly alter the integrity of the sample. In some cases, removal must be done as part of the analysis function so that removed material can be accounted for. In other cases, proper calibration of the analyzer output will suffice.

TELEMETERING AND TRANSMISSION ANALOG SIGNAL TRANSMISSION Modern control systems permit the measurement device, the control unit, and the final actuator to be physically separated by several hundred meters, if necessary. This requires the transmission of the measured variable from the measurement device to the control unit, and the transmission of the controller output from the control unit to the final actuator. In each case, transmission of a single value in only one direction is required. Such requirements can be met by analog signal transmission. A span is defined for the value to be transmitted, and the value is basically transmitted as a percent of this span. For the measured variable, the logical span is the measurement span. For the controller output, the logical span is the range of the final actuator (e.g., valve fully closed to valve fully open). For pneumatic transmission systems, the signal range used for the transmission is 3 to 15 psig. In each pneumatic transmission system, there can be only one transmitter, but there can be any number of receivers. When most measurement devices were pneumatic, pneumatic transmission was the logical choice. However, with the displacement of pneumatic measurement devices by electronic devices, pneumatic transmission is becoming less common but is unlikely to totally disappear. In order for electronic transmission systems to be less susceptible to interference from magnetic fields, current is used for the transmission signal instead of voltage. The signal range is 4 to 20 milliamps. In each circuit or “current loop,” there can be only one transmitter. There can be more than one receiver, but not an unlimited number. For each receiver, a 250 ohm “range resistor” is inserted into the current loop, which provides a 1- to 5-volt input to the receiving device. The number of receivers is limited by the power available from the transmitter. Both pneumatic and electronic transmission use a “live zero.” This

enables the receiver to distinguish a transmitted value of zero percent of span from a transmitter or transmission system failure. Transmission of zero percent of span provides a signal of 4 milliamps in electronic transmission. Should the transmitter or the transmission system fail (i.e., an open circuit in a current loop), the signal level would be zero milliamps. For most measurement variable transmissions, the lower range of the measurement span corresponds to 4 milliamps and the upper range of the measurement span corresponds to 20 milliamps. On an open circuit, the measured variable would fail to its lower range. In some applications, this is undesirable. For example, in a fired heater that is heating material to a target temperature, failure of the temperature measurement to its lower span value would drive the output of the combustion control logic to the maximum possible firing rate. In such applications, the analog transmission signal is normally inverted, with the upper range of the measurement span corresponding to 4 milliamps and the lower range of the measurement span corresponding to 20 milliamps. On an open circuit, the measured variable would fail to its upper range. For the fired heater, failure of the measured variable to its upper span would drive the output of the combustion control logic to the minimum firing rate. DIGITAL SYSTEMS With the advent of the microprocessor, digital technology began to be used for data collection, feedback control, and all other information processing requirements in production facilities. Such systems must acquire data from a variety of measurement devices, and control systems must drive final actuators. Analog Input and Outputs Analog inputs are generally divided into two categories:

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1. High level. Where the source is a process transmitter, the range resistor in the current loop converts the 4–20 milliamp signal into a 1–5 volt signal. The conversion equipment can be unipolar (i.e., capable of processing only positive voltages). 2. Low level. The most common low level signals are inputs from thermocouples. These inputs rarely exceed 30 millivolts, and could be zero or even negative. The conversion equipment must be bipolar (i.e., capable of processing positive and negative voltages). Ultimately, such signals are converted to digital values via an analog-to-digital (A/D) converter. However, the A/D converter is normally preceded by two other components: 1. Multiplexer. This permits one A/D converter to service multiple analog inputs. The number of inputs to a multiplexer is usually between 8 and 256. 2. Amplifier. As A/D converters require high level signals, a high gain amplifier is required to convert low-level signals into high-level signals. One of the important parameters for the A/D converter is its resolution. The resolution is stated in terms of the number of significant binary digits (bits) in the digital value. As the repeatability of most process transmitters is around 0.1 percent, the minimum acceptable resolution for a bipolar A/D converter is 12 bits, which translates to 11 data bits plus one bit for the sign. With this resolution, the analog input values can be represented to 1 part in 211, or one part in 2048. Normally, a 5-volt input is converted to a digital value of 2000, which effectively gives a resolution of 1 part in 2000 or 0.05 percent. Very few process control systems utilize resolutions higher than 14 bits, which translates to a resolution of 1 part in 8000 or 0.0125 percent. For 4–20 milliamp inputs, the resolution is not quite as good as stated above. For a 12-bit bipolar A/D converter, 1-volt converts to a digital value of 400. Thus, the range for the digital value is 400 to 2000, making the effective input resolution 1 part in 1600, or 0.0625 percent. On the output side, dedicated digital-to-analog converters are provided for each analog output. Outputs are normally unipolar, and require a lower resolution than inputs. A 10-bit resolution is normally sufficient, giving a resolution of 1 part in 1000 or 0.1%. Pulse Inputs Where the sensor within the measurement device is digital in nature, analog-to-digital conversion can be avoided. For rotational devices, the rotational element can be outfitted with a shaft encoder that generates a known number of pulses per revolution. The digital system can process such inputs in either of the following ways: 1. Count the number of pulses over a fixed interval of time. 2. Determine the time for a specified number of pulses. 3. Determine the duration of time between the leading (or trailing) edges of successive pulses. Of these, the first option is the most commonly used in process applications. Turbine flowmeters are probably the most common example where pulse inputs are used. Another example is a watt-hour meter. Basically any measurement device that involves a rotational element can be interfaced via pulses. Occasionally, a nonrotational measurement device can generate pulse outputs. One example is the vortex shedding meter, where a pulse can be generated when each vortex passes over the detector. Serial Interfaces Some very important measurement devices cannot be reasonably interfaced via either analog or pulse inputs. Two examples are the following: 1. Chromatographs can perform a total composition analysis for a sample. It is possible but inconvenient to provide an analog input for each component. Furthermore, it is often desirable to capture other information, such as the time that the analysis was made (normally the time the sample was injected). 2. Load cells are capable of resolutions of 1 part in 100,000. A/D converters for analog inputs cannot even approach such resolutions. One approach to interfacing with such devices is serial interfaces. This involves two aspects: 1. Hardware interface. The RS-232 interface standard is the basis for most serial interfaces. 2. Protocol. This is interpreting the sequence of characters

transmitted by the measurement device. There are no standards for protocols, which means that custom software is required. One advantage of serial interfaces is that two-way communication is possible. For example, a “tare” command can be issued to a load cell. Microprocessor-Based Transmitters The cost of microprocessor technology has declined to the point where it is economically feasible to incorporate a microprocessor into each transmitter. Such microprocessor-based transmitters are often referred to as “smart” transmitters. As opposed to conventional or “dumb” transmitters, the smart transmitters offer the following capabilities: 1. Checks on the internal electronics, such as verifying that the voltage levels of internal power supplies are within specifications. 2. Checks on environmental conditions within the instruments, such as verifying that the case temperature is within specifications. 3. Compensation of the measured value for conditions within the instrument, such as compensating the output of a pressure transmitter for the temperature within the transmitter. Smart transmitters are much less affected by temperature and pressure variations than conventional transmitters. 4. Compensation of the measured value for other process conditions, such as compensating the output of a capacitance level transmitter for variations in process temperature. 5. Linearizing the output of the transmitter. Functions such as square root extraction of the differential pressure for a head-type flowmeter can be done within the instrument instead of within the control system. 6. Configuring the transmitter from a remote location, such as changing the span of the transmitter output. 7. Automatic recalibration of the transmitter. Although this is highly desired by users, the capabilities, if any, in this respect depend on the type of measurement. Due to these capabilities, smart transmitters offer improved performance over conventional transmitters. Transmitter/Actuator Networks With the advent of smart transmitters and smart actuators, the limitations of the 4–20 milliamp analog signal transmission retard the full utilization of the capabilities of the smart devices. For smart transmitters, the following capabilities are required: 1. Transmission of more than one value from a transmitter. Information beyond the measured variable is available from the smart transmitter. For example, a smart pressure transmitter can also report the temperature within its housing. Knowing that this temperature is above normal values permits corrective action to be taken before the device fails. Such information is especially important during the initial commissioning of a plant. 2. Bidirectional transmission. Configuration parameters such as span, engineering units, resolution, and so on, must be communicated to the smart transmitter. Similar capabilities are required for smart actuators. In order to meet their initial requirements, several manufacturers have developed digital communications capabilities for communicating with smart transmitters. These can be used either in addition to or in lieu of the 4–20 milliamp signal. Although most manufacturers release enough information on their communications features to permit another manufacturer to provide compatible instruments (and in some cases provide an open communication standard), the communications capability provided by a manufacturer may be proprietary. Users purchase their transmitters from a variety of manufacturers, so this situation limits the full utilization of the capabilities of smart transmitters and valves. Efforts to develop a standard for a communications network have not proceeded smoothly. The International Society for Measurement and Control (ISA) has attempted to develop a standard generally referred to as fieldbus. The standards effort attempted to develop a world standard, encompassing European, Japanese, and American products. This effort focused on developing a single standard with which all manufacturers would comply. Currently, efforts are mostly being directed to providing the capability for interoperability between the products of the manufacturers with competing communications networks. Meanwhile, users are reluctant to make major commitments, and are continuing to rely primarily on the traditional 4–20 milliamp transmission.

TELEMETERING AND TRANSMISSION FILTERING AND SMOOTHING A signal received from a process transmitter generally contains the following features: 1. Low-frequency process disturbances. The control system is expected to react to these disturbances. 2. High-frequency process disturbances. The frequency of these disturbances is beyond the capability of the control system to effectively react. 3. Measurement noise. 4. Stray electrical pickup, primarily 50- or 60-cycle AC. Frequencies are measured in Hertz (Hz), with 60-cycle AC being a 60-Hz frequency. The objective of filtering and smoothing is to remove the last three components, leaving only the low frequency process disturbances. Normally this has to be accomplished using the proper combination of analog and digital filters. Sampling a continuous signal results in a phenomenon often referred to as aliasing or foldover. In order to represent a sinusoidal signal, a minimum of four samples are required during each cycle. That is, the sampling interval must be at least 1/4th the period of the sinusoid. Consequently, when a signal is sampled at a frequency ωs, all frequencies higher than (π/2)ωs cannot be represented at their original frequency. Instead, they are present in the sampled signal with their original amplitude but at a lower frequency harmonic. Because of the aliasing or foldover issues, a combination of analog and digital filtering is usually required. The sampler (i.e., the A/D converter) must be preceded by an analog filter that rejects those highfrequency components such as stray electrical pickup that would result in foldover when sampled. In commercial products, analog filters are normally incorporated into the input processing hardware by the manufacturer. The software then permits the user to specify digital filtering to remove any undesirable low-frequency components. On the analog side, the filter is often the conventional resistorcapacitor or RC filter. However, other possibilities exist. For example, one type of A/D converter is called an “integrating A/D” because the converter basically integrates the input signal over a fixed interval of time. By making the interval 1/60th second, this approach provides excellent rejection of any 60-Hz electrical noise. On the digital side, the input processing software generally provides for smoothing via the exponentially weighted moving average, which is the digital counterpart to the RC network analog filter. The smoothing equation is as follows: yi = αxi + (1 − α)yi − 1 where

(8-103)

xi = current value of input yi = current output from filter yi − 1 = previous output from filter α = filter coefficient

The degree of smoothing is determined by the filter coefficient α, with α = 1 being no smoothing and α = 0 being infinite smoothing (no effect of new measurements). The filter coefficient α is related to the filter time constant τF and the sampling interval ∆t by the following equation: −∆t α = 1 − exp ᎏ (8-104) τF or by the approximation ∆t α=ᎏ (8-105) ∆t + τF Another approach to smoothing is to use the arithmetic moving average, which is represented by the following equation:

 


 x  n

i+1−j

j=1

yi = ᎏᎏ n

(8-106)

The term “moving” is applied because the filter software maintains a storage array with the previous n values of the input. When a new value is received, the oldest value in the storage array is replaced with

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the new value, and the arithmetic average recomputed. This permits the filtered value to be updated each time a new input value is received. In process applications, determining τF (or α) for the exponential filter and n for the moving average filter is often done merely by observing the behavior of the filtered value. If the filtered value is “bouncing,” the degree of smoothing (that is, τF or n) is increased. This can easily lead to an excessive degree of filtering, which will limit the performance of any control system that uses the filtered value. The degree of filtering is best determined from the frequency spectrum of the measured input, but such information is rarely available for process measurements. ALARMS The purpose of an alarm is to alert the process operator to a process condition that requires immediate attention. An alarm is said to occur whenever the abnormal condition is detected and the alert is issued. An alarm is said to return to normal when the abnormal condition no longer exists. Analog alarms can be defined on measured variables, calculated variables, controller outputs, and the like. For analog alarms, the following possibilities exist: 1. High/low alarms. A high alarm is generated when the value is greater than or equal to the value specified for the high-alarm limit. A low alarm is generated when the value is less than or equal to the value specified for the low-alarm limit. 2. Deviation alarms. An alarm limit and a target are specified. A high deviation alarm is generated when the value is greater than or equal to the target plus the deviation alarm limit. A low deviation alarm is generated when the value is less than or equal to the target minus the deviation alarm limit. 3. Trend or rate-of-change alarms. A limit is specified for the maximum rate of change, usually specified as a change in the measured value per minute. A high trend alarm is generated when the rate of change of the variable is greater than or equal to the value specified for the trend alarm limit. A low trend alarm is generated when the rate of change of the variable is less than or equal to the negative of the value specified for the trend alarm limit. Most systems permit multiple alarms of a given type to be configured for a given value. For example, configuring three high alarms provides a high alarm, a high-high alarm, and a high-high-high alarm. One operational problem with analog alarms is that noise in the variable can cause multiple alarms whenever its value approaches a limit. This can be avoided by defining a deadband on the alarm. For example, a high alarm would be processed as follows: 1. Occurrence. The high alarm is generated when the value is greater than or equal to the value specified for the high-alarm limit. 2. Return to normal. The high-alarm return to normal is generated when the value is less than or equal to the high alarm limit less the deadband. As the degree of noise varies from one input to the next, the deadband must be individually configurable for each alarm. Discrete alarms can be defined on discrete inputs, limit switch inputs from on/off actuators and so on. For discrete alarms, the following possibilities exist: 1. Status alarms. An expected or normal state is specified for the discrete value. A status alarm is generated when the discrete value is other than its expected or normal state. 2. Change-of-state alarm. A change-of-state alarm is generated on any change of the discrete value. The expected sequence of events on an alarm is basically as follows: 1. The alarm occurs. This usually activates an audible annunciator. 2. The alarm occurrence is acknowledged by the process operator. When all alarms have been acknowledged, the audible annunciator is silenced. 3. Corrective action is initiated by the process operator. 4. The alarm condition returns to normal. However, additional requirements are imposed at some plants. Sometimes the process operator must acknowledge the alarm’s return to normal. Some plants require that the alarm occurrence be reissued

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if the alarm remains in the occurred state longer than a specified period of time. Consequently, some “personalization” of the alarming facilities is done. When alarms were largely hardware-based (i.e., the panel alarm systems), the purchase and installation of the alarm hardware imposed a certain discipline on the configuration of alarms. With digital systems, the suppliers have made it extremely easy to configure alarms. In fact, it is sometimes easier to configure alarms on a measured value than not to configure the alarms. Furthermore, the engineer assigned the responsibility for defining alarms should ensure that an abnormal process condition will not go undetected because an alarm has not been configured. When alarms are defined on every measured and calculated variable, the result is an excessive number of alarms, most of which are duplicative and unnecessary. The accident at the Three Mile Island nuclear plant clearly demonstrated that an alarm system can be counterproductive. An excessive number of alarms can distract the operator’s attention from the real problem that needs to be addressed. Alarms that merely tell the operator something that is already known do the same. In fact, a very good definition of a nuisance alarm is one that informs the operator of a situation of which the operator is already aware. The only problem with applying this definition is determining what the operator already knows. Unless some discipline is imposed, engineering personnel, especially where contractors are involved, will define far more alarms than plant operations require. This situation may be addressed by simply setting the alarm limits to values such that the alarms never occur. However, changes in alarms and alarm limits are changes from the perspective of the Process Safety Management regulations. It is prudent to impose the necessary discipline to avoid an excessive number of alarms. Potential guidelines are as follows: 1. For each alarm, a specific action is expected from the process operator. Operator actions such as “call maintenance” are inappropriate with modern systems. If maintenance needs to know, modern systems can inform maintenance directly. 2. Alarms should be restricted to abnormal situations for which the process operator is responsible. A high alarm on the temperature in one of the control system cabinets should not be issued to the process operator. Correcting this situation is the responsibility of maintenance, not the process operator. 3. Process operators are expected to be exercising normal surveillance of the process. Therefore, alarms are not appropriate for situations known to the operator either through previous alarms or through normal process surveillance. The “sleeping operator” problem can be addressed by far more effective means than the alarm system.

4. When the process is operating normally, no alarms should be triggered. Within the electric utility industry, this design objective is known as “darkboard.” Application of darkboard is especially important in batch plants, where much of the process equipment is operated intermittently. Ultimately, guidelines such as those above will be taken seriously only if production management carefully configures the alarms. The consequences of excessive and redundant alarms will be felt primarily by those responsible for production operations. Therefore, production management must make adequate resources available for reviewing and analyzing the proposed alarm configurations. Another serious distraction to a process operator is the multiple alarm event, where a single event within the process results in multiple alarms. When the operator must individually acknowledge each alarm, considerable time can be lost in silencing the obnoxious annunciator before the real problem is addressed. Air-handling systems are especially vulnerable to this, where any fluctuation in pressure (for example, resulting from a blower trip) can cause a number of pressure alarms to occur. Point alarms (high alarms, low alarms, status alarms, etc.) are especially vulnerable to the multiple alarm event. This can be addressed in one of two ways: 1. Ganging alarms. Instead of individually issuing the point alarms, all alarms associated with a certain aspect of the process are simply wired to give a single trouble alarm. The responsibility rests entirely with the operator to determine the nature of the problem. 2. Intelligent alarms. Logic is incorporated into the alarm system to determine the nature of the problem and then issue a single alarm to the process operator. Sometimes this is called an expert system. While the intelligent alarm approach is clearly preferable, substantial process analysis is required to support intelligent alarming. Meeting the following two objectives is quite challenging: 1. The alarm logic must consistently detect abnormal conditions within the process. 2. The alarm logic must not issue an alert to an abnormal condition when in fact none exists. Often the latter case is more challenging than the former. Logically, the intelligent alarm effort must be linked to the process hazards analysis. Developing an effective intelligent alarming system requires substantial commitments of effort, involving both process engineers, control systems engineers, and production personnel. Methodologies such as expert systems can facilitate the implementation of an intelligent alarming system, but they must still be based on a sound analysis of the potential process hazards.

DIGITAL TECHNOLOGY FOR PROCESS CONTROL GENERAL REFERENCES: Fortier, Design and Analysis of Distributed RealTime Systems, McGraw-Hill, New York, 1985; Hawryszkiewycs, Database Analysis and Design, Science Research Associates Inc., Chicago, 1984; Khambata, Microprocessors/Microcomputers: Architecture, Software, and Systems, 2d ed., Wiley, New York, 1987; Liptak, Instrument Engineers Handbook, Chilton Book Company, Philadelphia, 1995; Mellichamp (ed.), Real-Time Computing with Applications to Data Acquisition and Control, Van Nostrand Reinhold, New York, 1983.

Since the 1970s, process controls have evolved from pneumatic analog technology to electronic analog technology to microprocessorbased controls. Electronic analog technology has virtually disappeared from process controls. Pneumatic controls continue to be manufactured, but they are relegated to special situations where pneumatics can offer a unique advantage. Process controls are dominated by programmable electronic systems (PES), most of which are based on microprocessor technology.

HIERARCHY OF INFORMATION SYSTEMS Coupling digital controls with networking technology permits information to be passed from level-to-level within a corporation at high rates of speed. This technology is capable of presenting the measured variable from a flow transmitter installed in a plant in a remote location anywhere in the world to the company headquarters in less than a second. A hierarchical representation of the information flow within a company leads to a better understanding of how information is passed from one layer to the next. Such representations can be developed in varying degrees of detail, and most companies have developed one that describes their specific practices. The following hierarchy consists of five levels. Measurement Devices and Actuators Often referred to as level 0, this layer couples the control and information systems to the process. The measurement devices provide information on the cur-

DIGITAL TECHNOLOGY FOR PROCESS CONTROL rent conditions within the process. The actuators permit control decisions to be imposed on the process. Although traditionally analog, smart transmitters and smart valves based on microprocessor technology will eventually dominate this layer. Regulatory Controls The objective of this layer is to operate the process at or near the targets supplied by others, be it the process operator or a higher layer in the hierarchy. In order to achieve consistent process operations, a high degree of automatic control is required from the regulatory layer. The direct result is a reduction in variance in the key process variables. More uniform product quality is an obvious benefit. However, consistent process operation is a prerequisite for optimizing the process operations. To ensure success for the upper level functions, the first objective of any automation effort must be to achieve a high degree of regulatory control. Supervisory Controls The regulatory layer blindly attempts to operate the process at the specified targets, regardless of the appropriateness of these targets. Determining the most appropriate targets is the responsibility of the supervisory layer. Given the current production targets for a unit, supervisory control determines how the process can be best operated to meet the production targets. Usually this optimization has a limited scope, being confined to a single production unit or possibly even a single unit operation within a production unit. Supervisory control translates changes in factors such as current process efficiencies, current energy costs, cooling medium temperatures, and so on, to changes in process operating targets so as to optimize process operations. Production Controls The nature of the production control logic differs greatly between continuous and batch plants. A good example of production control in a continuous process is refinery optimization. From the assay of the incoming crude oil, the values of the various possible refined products, the contractual commitments to deliver certain products, the performance measures of the various units within a refinery, and the like, it is possible to determine the mix of products that optimizes the economic return from processing this crude. The solution of this problem involves many relationships and constraints and is solved with techniques such as linear programming. In a batch plant, production control often takes the form of routing or short-term scheduling. For a multiproduct batch plant, determining the long term schedule is basically a manufacturing resource planning (MRP) problem, where the specific products to be manufactured and the amounts to be manufactured are determined from the outstanding orders, the raw materials available for production, the production capacities of the process equipment, and other factors. The goal of the MRP effort is the long-term schedule, which is a list of the products to be manufactured over a specified period of time (often one week). For each product on the list, a target amount is also specified. To manufacture this amount usually involves several batches. The term “production run” often refers to the sequence of batches required to make the target amount of product, so in effect the long term schedule is a list of production runs. Most multiproduct batch plants have more than one piece of equipment of each type. Routing refers to determining the specific pieces of equipment that will be used to manufacture each run on the long term production schedule. For example, the plant might have five reactors, eight neutralization tanks, three grinders, and four packing machines. For a given run, a rather large number of possible routes are possible. Furthermore, rarely is only one run in progress at a given time. The objective of routing is to determine the specific pieces of production equipment to be used for each run on the long-term production schedule. Given the dynamic nature of the production process (equipment failures, insertion/deletion of runs into the longterm schedule, etc.), the solution of the routing problem continues to be quite challenging. Corporate Information Systems Terms such as management information systems (MIS) and information technology (IT) are frequently used to designate the upper levels of computer systems within a corporation. From a control perspective, the functions performed at this level are normally long-term and/or strategic. For example, in a processing plant, long-term contracts are required with the providers of the feedstocks. A forecast must be developed for the demand for possible products from the plant. This demand must be translated into

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needed raw materials, and then contracts executed with the suppliers to deliver these materials on a relatively uniform schedule. While most companies within the process industries recognize the importance of information technology in managing their businesses, this technology has been a source of considerable frustration and disappointment. Schedule delays, cost overruns, and failure of the final product to perform as expected have often eroded the credibility of information technology. However, immense potential remains for the technology, and process companies have no choice but to seek continuous improvement. DISTRIBUTED CONTROL SYSTEMS Although digital control technology was first applied to process control in 1959, the total dependence of the early centralized architectures on a single computer for all control and operator interface functions resulted in complex systems with dubious reliability. Adding a second processor increased both the complexity and the cost. Consequently, many installations provided analog backup systems to protect against a computer malfunction. Microprocessor technology permitted these technical issues to be addressed in a cost-effective manner. In the mid-1970s, a process control architecture referred to as a distributed control system (DCS) was introduced and almost instantly became a commercial success. A DCS consists of some number of microprocessor-based nodes that are interconnected by a digital communications network, often called a data highway. The key features of this architecture are as follows: 1. The process control functions and the operator interface, also referred to as man-machine interface (MMI) or human-machine interface (HMI), is provided by separate nodes. This approach is referred to as split-architecture, and it permits considerable flexibility in choosing a configuration that most appropriately meets the needs of the application. 2. The process control functions can be distributed functionally and/or geographically. Functional distribution permits related control functions to be grouped and implemented in a single node. Geographical distribution permits the process control nodes to be physically located near the equipment being controlled. As the digital communications network is based on local area network (LAN) technology, the nodes within the DCS can be physically separated by thousands of meters. 3. Redundancy can be provided where appropriate, the following being typical: a. Multiple operator interface nodes can be provided to reduce the impact of an operator interface node failure. b. The digital communications network is normally redundant to the extent that at least two independent paths are available between any two nodes of the DCS. c. Consisting basically of processor and memory, the process control nodes are highly reliable, with mean-times-between-failures approaching 100 years. Redundant configurations are available for especially critical applications. 4. As the data within the DCS are digital in nature, interfaces to upper level computers are technically easier to implement. Unfortunately, the proprietary nature of the communications networks within commercial DCS products complicate the implementation of such interfaces. Truly open DCS architectures, at least as the term “open” is used in the mainstream of computing, are not yet available. Figure 8-62 depicts a hypothetical distributed control system. A number of different unit configurations are illustrated. This system consists of many commonly used DCS components, including multiplexers (MUXs), single/multiple-loop controllers, programmable logic controllers (PLCs), and smart devices. A typical system includes the following elements as well: • Host computers. These are the most powerful computers in the system, capable of performing functions not normally available in other units. They act as the arbitrator unit to route internodal communications. An operator interface is supported and various peripheral devices are coordinated. Computationally intensive tasks, such as optimization or advanced control strategies, are processed here. • Data highway. This is the communication link between com-

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FIG. 8-62

A typical DDC system.

ponents of a network. Coaxial cable is often used. A redundant pair is normally supplied to reduce possibility of link failure. • Real-time clocks (RTCs). Real-time systems are required to respond to events, as they occur, in a timely manner. This is especially crucial in process control systems where control actions applied at the wrong time may amplify process deviations or destabilize the processes. The nodes in the systems are interrupted periodically by the real-time clocks to maintain the actual elapsed times. • Operator control stations. These typically consist of color graphics monitors with special keyboards, in addition to a conventional alphanumeric keyboard, containing keys to perform dedicated functions. Operators may supervise and control processes from these stations. A control station may contain a number of printers for alarm logging, report printing, or hard-copying process graphics. • Remote control units. These units are used to control unit processes. Basic control functions such as the PID algorithm are implemented here. Depending on other hardware components used, data acquisition capability may be required to perform digital control. They may be configured to supply process set points to single-loop controllers. Radio telemetry may be installed to communicate with MUX units located at great distances. • Programmer consoles. These are programming terminals. Developing system software on the host machines is a common prac-

tice by many system suppliers. This eliminates compatibility problems between development and target environments. Programming capability is normally retained when the system is delivered such that system users may develop their own application programs. • Mass storage device. Typically, fixed-head hard disk drives are used to store active data, including on-line and historical databases and non-memory-resident programs. Memory-resident programs are stored to allow loading at system startups. The tape drives are used for archives and backups. DISTRIBUTED DATABASE AND THE DATABASE MANAGER A database is a centralized location for data storage. The use of databases enhances system performance by maintaining complex relations between data elements while reducing data redundancy. A database may be built based on the relational model, the entity relationship model, or some other model. The database manager is a system utility program or programs acting as the gatekeeper to the databases. All functions retrieving or modifying data must submit a request to the manager. Information required to access the database include the tag name of the database entity, often referred to as a point, the attributes to be accessed, and the values if modifying. The database manager maintains the integrity of the databases by executing a request only

DIGITAL TECHNOLOGY FOR PROCESS CONTROL when not processing other conflicting requests. Although a number of functions may read the same data item at the same time, writing by a number of functions or simultaneous read and write of the same data item is not permitted. To allow flexibility, the database manager must also perform point addition or deletion. However, the ability to create a point type or to add or delete attributes of a point type is not normally required because, unlike other data processing systems, a process control system normally involves a fixed number of point types and related attributes. For example, analog and binary input and output types are required for process I/O points. Related attributes for these point types include tag names, values, and hardware addresses. Different system manufacturers may define different point types using different data structures. We will discuss other commonly used point types and attributes as they appear. Historical Database Subsystem We have discussed the use of on-line databases. An historical database is built similar to an on-line database. Unlike their on-line counterparts, the information stored in a historical database is not normally accessed directly by other subsystems for process control and monitoring. Periodic reports and longterm trends are generated based on the archived data. The reports are often used for long-term planning and system performance evaluations such as statistical process (quality) control. The trends may be used to detect process drifts or to compare process variations at different times. The historical data is sampled at user-specified intervals. A typical process plant contains a large number of data points, but it is not feasible to store data for all points at all times. The user determines if a data point should be included in the list of archive points. Most systems provide archive-point menu displays. The operators are able to add or delete data points to the archive point lists. The sampling periods are normally some multiples of their base scan frequencies. However, some systems allow historical data sampling of arbitrary intervals. This is necessary when intermediate virtual data points that do not have the scan frequency attribute are involved. The archive point lists are continuously scanned by the historical database software. On-line databases are polled for data. The times of data retrieval are recorded with the data obtained. To conserve storage space, different data compression techniques are employed by various manufacturers. The historical data may be used for long-term trending. The live trends data are displayed but not stored. Therefore, these trends cannot be recalled once cleared off the screens. The historical trend of any archive point may be displayed at any time because the values used are extracted from the archived data. Zooming, that is, axis scaling, is allowed by most systems. As a result, the displayed data point intervals may not be multiples of stored data intervals. Many systems provide data interpolation and smoothing functions to process the stored data when they are displayed. The live and historical trend displays are superior to strip charts in many ways. In addition to conventional trend recording, the zoom-in capability allows close examination of recorded data, whereas zoom-out compresses long-term data within a screen. Exact data sampled at any time point can be extracted by cursor positioning. Strip-chart recorders have disappeared from many modern plants. Periodic reports, including shift, daily, weekly, monthly, and quarterly reports, are printed based on archived data. Some reports may contain simply the stored data in certain specific arrangements. More often, quantities such as mean values, standard deviations, or other calculated values are included. Instead of hard-coding reports to user specifications, many system suppliers provide report generation packages in the form of metalanguages. These packages allow users to configure report formats suitable for their particular requirements. The report generator interprets the configuration files prepared by the users to create reports. Due to the infrequent execution, the report generator is normally operated in the batch mode. DCS manufacturers have devoted considerable efforts to make it easy to implement and enhance process control configurations within their products. Although programming in the traditional sense is possible within most products, the majority of the functions required for a process control application can be implemented by configuring as opposed to programming.

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PROCESS CONTROL LANGUAGE A digital control system involves software development. The introduction of high-level programming languages such as FORTRAN and BASIC in the 1960s was considered a major breakthrough. For process control applications, some companies have incorporated libraries of software routines for these languages, but others have developed speciality pseudolanguages. These implementations are characterized by their statement-oriented language structure. Although substantial savings in time and efforts can be realized, software development costs continue to be significant. The most successful and user-friendly approach, which is now adopted by virtually all commercial systems, is the fill-in-the-forms or table-driven process control languages (PCLs). The core of these languages is a number of basic functional blocks or software modules. All modules are defined as database points. Using a module is analogous to calling a subroutine in conventional programs. In general, each module contains some inputs and an output. The programming involves soft-wiring outputs of blocks to inputs of other blocks. Some modules may require additional parameters to direct module execution. The users are required to fill in the sources of input values, the destinations of output values, and the parameters in blanks of forms or tables prepared for the modules. The source and destination blanks may be filled with process I/Os when appropriate. To connect modules, some systems require filling the tag names of modules originating or receiving data. Additional programs are often required to resolve ambiguities when connecting multiple input-output modules. Another method involves the use of intermediate data points. The blanks in a pair of interconnecting modules are filled with the tag name of the same data point. Batch jobs and/or interactive data entry may be performed to fill the databases. A completed control strategy resembles a data flow diagram. The soft-wiring of modules is similar to hard-wiring analog-electronic circuits in analog computers. Additional database space must be allocated when intermediate data points are used. A system can be designed to use process I/O points as intermediates. However, the data acquisition software must be programmed to bypass these points when scanned. All system builders provide virtual data point types if the intermediate data storage scheme is adopted. These points are not scanned by the data acquisition software. Memory space requirements are reduced by eliminating unnecessary attributes such as hardware addresses and scan frequencies. It should be noted that the fill-in-the-forms technique is applicable to all data point types. All process control languages contain PID controller blocks. The digital PID controller is normally programmed to execute in velocity form. A pulse duration output may be used to receive the velocity output directly. Where positional signal is expected, an operating mode bit is used to enable an internal integrator. This flexibility is not normally available in analog controllers. Unlike an analog controller, the three modes in a digital PID controller do not interact. This simplifies the tuning effort. In addition to the tuning constants, a typical digital PID controller contains some entries not normally found in an analog controller: • When a process error is below certain tolerable deadband, the controller ceases modifying output. This is referred to as gap action. • The magnitude of change in a velocity output is limited by a change clamp. • A pair of output clamps is used to restrict a positional output value from exceeding specified limits. • The controller action can be disabled by triggering a binary deactivate input signal, during process startup, shutdown, or when some abnormal conditions exist. Although modules are supplied and their internal configurations are different from system to system, their basic functionalities are the same. SINGLE-LOOP CONTROLS With the exception of pneumatic controllers for special applications, commercial single-loop controllers are almost entirely microprocessorbased. The most basic products provide only the PID control algo-

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rithm, but the more powerful versions provide a set of general-purpose algorithms comparable to those in a DCS. For applications such as cascade control and multivariable control, the manufacturers of singleloop controllers provide multiloop versions of their products. These multiloop controllers have much in common with the process control node in a DCS. Single-loop controllers provide both the process control functions and the operator interface function. This makes them ideally suited to very small applications, where only two or three loops are required. However, it is possible to couple single-loop controllers to a personal computer (PC) to provide the operator interface function. Such installations are extremely cost effective, and with the keen competition in PC-based products, the capabilities are comparable and sometimes even better than that provided by a DCS. However, this approach makes sense only up to about 25 loops. Initially, the microprocessor-based single-loop controllers made the power of digital control affordable to those with small processes. To compete with these products in small applications, the DCS suppliers have introduced micro-DCS versions of their products. As a PC-based operator interface is usually a component of the micro-DCS, there is sometimes little distinction between a micro-DCS and a system consisting of single-loop controllers coupled to a PC-based operator interface. PROGRAMMABLE LOGIC CONTROLLERS The programmable logic controller (PLC) was the first digital technology to successfully compete with conventional technology in industrial control applications. Initially developed in the early 1970s for applications within the manufacturing industries (principally automotive), the PLC proved to be superb for implementing discrete logic. The earliest PLCs were limited to discrete I/O, basic Boolean logic functions (AND, OR, NOT), timers, and counters. However, versions soon appeared with analog I/O, math functions, PID control algorithms, and other functions required for process control applications. Developed to replace hard-wired relay logic, the early PLCs were “programmed” using the same ladder logic diagrams used to represent logic implemented with hard-wired relays. As the initial target market was electrical, programming in ladder logic was a definite

advantage, and some union contracts specifically required that such discrete logic be presented as ladder diagrams. However, ladder logic is not the programming medium preferred by instrument engineers, which hampers the acceptance of PLCs for process control. Alternatives to ladder logic are available for programming PLCs, but established perceptions are slow to change. Developed specifically for implementing discrete logic, PLCs continue to provide the best route to implementing such logic. The manufacturers of PLCs provide robust, cost-effective discrete I/O modules. Regardless of its acceptability, ladder logic is the most efficient means for implementing discrete logic. Because PLCs scan the discrete logic very rapidly, a 100-millisecond scan rate is considered very slow for a PLC. The process control modules of a DCS often implement discrete logic using function blocks, which is less efficient than ladder logic and normally results in a slower scan rate. A few DCS process control modules have used ladder logic to implement discrete logic, but their discrete I/O capabilities and slow scan rates rarely match that of a PLC. Consequently, for applications heavy with discrete logic, most DCS suppliers will incorporate one or more PLCs into their system. Being excellent at discrete logic, PLCs are a potential candidate for implementing interlocks. Process interlocks are clearly acceptable for implementation within a PLC. Implementation of safety interlocks in programmable electronic systems (such as a PLC) is not universally accepted. Many organizations continue to require that all safety interlocks be hard-wired, but implementing safety interlocks in a PLC that is dedicated to safety functions is accepted by some as being equivalent to the hard-wired approach. INTERCOMPUTER COMMUNICATIONS A group of computers becomes a network when intercomputer communication is established. Prior to the 1980s, all system suppliers used proprietary protocols to network their systems. Ad hoc approaches were sometimes used to connect third-party equipment, which was not cost-effective in system maintenance, upgrade, and expansion. The recent introduction of standardized protocols has led to a decrease in initial capital cost. Most current DCS network protocol designs are based on the ISO-OSI* seven-layer model. The most notable effort in standardizing plant automation protocols

* Abbreviated from International Standards Organization-Open System Interconnection. They are the physical, data link, network, transports session, presentation, and application layers. Only the physical, data link, and application layers are present in the mini-MAP.

FIG. 8-63

A DCS using broadband data highway and fieldbus.

CONTROLLERS, FINAL CONTROL ELEMENTS, AND REGULATORS was initiated by General Motors in the early 1980s. This culminated in the Manufacturing Automation Protocol (MAP), which adopted the ISO-OSI standards as its basis. MAP specifies a broadband backbone local area network (LAN), which incorporates a selection of existing standard protocols suitable for manufacturing automation while defining additional protocols where no standard previously existed. Although intended for discrete component systems, MAP has evolved to address the integration of DCSs used in process control as well. Due to various technical reasons, MAP has gained limited acceptance by the process industries as of 1990. Engineering societies, including ISA and IEEE, and many operating companies are collaborating to refine MAP for wider support. More microprocessor-based process equipment, such as smart instruments and single-loop controllers, with digital communications capability are now becoming available and are used extensively in process plants. A fieldbus, which is a low-cost protocol, is necessary to perform efficient communication between the DCS and these devices. So-called mini-MAP architecture was developed to satisfy process control and instrumentation requirements while incorporating existing ISA standards. It is intended to improve access time while

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allowing communications to a large number of microprocessor-based devices. The mini-MAP contains only three of the seven layers specified by the ISO-OSI model; therefore, a mini-MAP device cannot communicate with devices on the MAP bus directly. The development of MAP/EPA (Enhanced Performance Architecture) is parallel to that of the mini-MAP. This scheme adopts the full seven-layer model with a reduced set of MAP protocols. The MAP/EPA is compatible to both the complete MAP and the mini-MAP. Another benefit of standardizing the fieldbus is that it encourages third-party traditional equipment manufacturers to enter the smart equipment market, resulting in increased competition and improved equipment quality. Figure 8-63 illustrates a LAN-based DCS. Irrespective of the protocol used, communication programs act as servers to the database manager. When some functions request data from a remote node, the database manager will transfer the request to the remote node database manager via the communication programs. The remote node communication programs will relay the request to the resident database manager and return the obtained data. The remote database access and the existence of communications equipment and software are transparent to plant operators.

CONTROLLERS, FINAL CONTROL ELEMENTS, AND REGULATORS GENERAL REFERENCES: Baumann, Control Valve Primer, 2d ed., ISA, 1994; Considine, Process/Industrial Instruments & Controls Handbook, 4th ed., McGraw-Hill, 1993; Driskell, Control-Valve Selection and Sizing, ISA, 1983; Hammitt, Cavitation and Multiphase Flow Phenomena, McGraw-Hill, 1980; Norton, Fundamentals of Noise and Vibration Analysis for Engineers, Cambridge University Press, 1989; Ulanski, Valve and Actuator Technology, McGraw-Hill, 1991.

External control of the process is achieved by devices that are specially designed, selected and configured for the intended processcontrol application. The text below covers three very common function classifications of process-control devices: controllers, final control elements, and regulators. The process controller is the “master” of the process-control system. It accepts a set point and other inputs and generates an output or outputs that it computes from a rule or set of rules that are part of its internal configuration. The controller output serves as an input to another controller or, more often, as an input to a final control element. The final control element is the device that affects the flow in the piping system of the process. The final control element serves as an interface between the process controller and the process. Control valves and adjustable speed pumps are the principal types discussed. Regulators, though not controllers or final control elements, perform the combined function of these two devices (controller and final control element) along with the measurement function commonly associated with the process variable transmitter. The uniqueness, control performance, and widespread usage of the regulator make it deserving of a functional grouping of its own. ELECTRONIC AND PNEUMATIC CONTROLLERS Electronic Controllers Almost all of the electronic process controllers used today are microprocessor-based devices. These processor-based controllers contain, or have access to, input/output (I/O) interface electronics that allow various types of signals to enter and leave the controller’s processor. The controller, depending on its type, uses sufficient read-only-memory (ROM) and read/write-accessiblememory (RAM) to perform the controller function. The resolution of the analog I/O channels of the controller vary somewhat, with 12-bit and 14-bit conversions quite common. Sample rates for the majority of the constant sample rate controllers range from 1 to 10 samples/second. Hard-wired single-pole, low-pass filters are installed on the analog inputs to the controller to protect the sampler from aliasing errors.

Distributed Control Systems Some knowledge of the distributed control system (DCS) is useful in understanding electronic controllers. A DCS is a process control system with sufficient performance to support large-scale real-time process applications. The DCS has (1) an operations workstation with a cathode ray tube (CRT) for display; (2) a controller subsystem that supports various types of controllers and controller functions; (3) an I/O subsystem for transducing data; (4) a higher-level computing platform for performing process supervision, information processing, and analysis; and (5) communication networks to tie the DCS subsystems, plant areas, and other plant systems together. The component controllers used in the controller subsystem portion of the DCS can be of various types and include multiloop controllers, programmable logic controllers, personal computer controllers, singleloop controllers, and fieldbus controllers. The type of electronic controller utilized depends on the size and functional characteristic of the process application being controlled. See the earlier section on distributed control systems. Multiloop Controllers The multiloop controller is a DCS network device that uses a single 32-bit microprocessor to provide control functions to many process loops. The controller operates independent of the other devices on the DCS network and can support from 20 to 500 loops. Data acquisition capability for up to 1000 analog and discrete I/O channels or more can also be provided by this controller. The multiloop controller contains a variety of function blocks (for example, PID, totalizer, lead/lag compensator, ratio control, alarm, sequencer, and Boolean) that can be “soft-wired” together to form complex control strategies. The multiloop controller, as part of a DCS, communicates with other controllers and man/machine interface (MMI) devices also on the DCS network. Programmable Logic Controllers The programmable logic controller (PLC) originated as a solid-state replacement for the hardwired relay control panel and was first used in the automotive industry for discrete manufacturing control. Today, PLCs are used to implement Boolean logic functions, timers, counters, and some math functions and PID control. PLCs are often used with on/off process control valves. PLCs are classified by the number of the I/O functions supported. There are several sizes available, with the smallest PLCs supporting less than 128 I/O channels and the largest supporting over 1023 I/O channels. I/O modules are available that support high-current motor loads, general-purpose voltage and current loads, discrete inputs, ana-

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log I/O and special-purpose I/O for servomotors, stepping motors, high-speed pulse counting, resolvers, decoders, multiplexed displays, and keyboards. PLCs often come with lights or other discrete indicators to determine the status of key I/O channels. When used as an alternative to a DCS, the PLC is programmed with a handheld or computer-based loader. The PLC is typically programmed with basic ladder logic or a high-level computer language such as BASIC, FORTRAN, or C. Programmable logic controllers use 16- or 32-bit microprocessors and offer some form of point-to-point communications such as RS-232C, RS-422, or RS-485. Personal Computer Controller Because of its high performance at low cost and its unexcelled ease of use, application of the personal computer (PC) as a platform for process controllers is growing. When configured to perform scan, control, alarm, and data acquisition (SCADA) functions and combined with a spreadsheet or database management application, the PC controller can be a lowcost, basic alternative to the DCS or PLC. Using the PC for control requires installation of a board into the expansion slot in the computer, or the PC can be connected to an external I/O module using a standard communication port on the PC (RS-232, RS-422, or IEEE-488). The controller card/module supports 16- or 32-bit microprocessors. Standardization and high volume in the PC market has produced a large selection of hardware and software tools for PC controllers. Single-Loop Controller The single-loop controller (SLC) is a process controller that produces a single output. SLCs can be pneumatic, analog electronic, or microprocessor-based. Pneumatic SLCs are discussed in the pneumatic controller section, and analog electronic SLC is not discussed because it has been virtually replaced by the microprocessor-based design. The microprocessor-based SLC uses an 8- or 16-bit microprocessor with a small number of digital and analog process input channels with control logic for the I/O incorporated within the controller. Analog inputs and outputs are available in the standard ranges (1–5 volts DC and 4–20 mA DC). Direct process inputs for temperature sensors (thermistor RTD and thermocouple types) are available. Binary outputs are also available. The face of the SLC has some form of visible display and pushbuttons that are used to view or adjust control values and configuration. SLCs are available for mounting in panel openings as small as 48 mm by 48 mm (1.9 by 1.9 inches). The processor based SLC allows the operator to select a control strategy from a predefined set of control functions. Control functions include PID, on/off, lead/lag, adder/subtractor, multiply/divider, filter functions, signal selector, peak detector, and analog track. SLCs feature auto/manual transfer switching, multi-setpoint, self diagnostics, gain scheduling, and perhaps also time sequencing. Most processorbased SLCs have self-tuning PID control algorithms. Sample times for the microprocessor-based SLCs vary from 0.1 to 0.4 seconds. Low-pass analog electronic filters are installed on the process inputs to stop aliasing errors caused by fast changes in the process signal. Input filter time constants are typically in the range from 0.1 to 1 s. Microprocessor-based SLCs may be made part of a DCS by using the communication port (RS-488 is common) on the controller or may be operated in a standalone mode independent of the DCS. Fieldbus Controller The benefits of eliminating all analog communication links to and from the devices in the process loop (including final control elements and measurement transmitters) have stimulated considerable interest in standardizing a suitable digital fieldbus communication network. Although a universal network standard is not currently complete (see “Digital Field Communications” in this section), several manufacturers have made available field devices that feature basic process-controller functionality. These controllers, known as fieldbus controllers, reside in the final control element or measurement transmitter and are considered to be an option available with these control devices. A suitable communications modem is present in the device to interface with a proprietary, PCbased, or hybrid analog/digital bus network. Presently, fieldbus controllers are single-loop controllers with 8- and 16-bit microprocessors and are options to digital field-control devices. These controllers support the basic PID control algorithm

and are projected to increase in functionality as the controller market develops. Parameters relating to intrinsic safety, communication type and bit rate, level of DCS support, and ultimate controller performance differentiate currently available fieldbus controllers. Controller Reliability and Application Trends Critical process-control applications demand a high level of reliability from the electronic controller. Some methods that improve the reliability of electronic controllers include: (1) focusing on robust circuit design using quality components; (2) using redundant circuits, modules, or subsystems where necessary; (3) using small-sized backup systems when needed; (4) reducing repair time and using more powerful diagnostics; and (5) distributing functionality to more independent modules to limit the impact of a failed module. Currently, the trend in process control is away from centralized process control and toward an increased number of small distributedcontrol or PLC systems. This trend will put emphasis on the evolution of the fieldbus controller and continued growth of the PC-based controller. Also, as hardware and software improves, the functionality of the controller will increase, and the supporting hardware will be physically smaller. Hence, the traditional lines between the DCS and the PLC will become less distinct as systems will be capable of supporting either function set. Pneumatic Controllers The pneumatic controller is an automatic controller that uses pneumatic pressure as a power source and generates a single pneumatic output pressure. The pneumatic controller is used in single-loop control applications and is often installed on the control valve or on an adjacent pipestand or wall in close proximity to the control valve and/or measurement transmitter. Pneumatic controllers are used in areas where it would be hazardous to use electronic equipment, in locations without power, in situations where maintenance personnel are more familiar with pneumatic controllers, or in applications where replacement with modern electronic controls has not been justified. Process-variable feedback for the controller is achieved by one of two methods. The process variable can (1) be measured and transmitted to the controller by using a separate measurement transmitter with a 0.2–1.0-bar (3–15-psig) pneumatic output, or (2) be sensed directly by the controller, which contains the measurement sensor within its enclosure. Controllers with integral sensing elements are available that sense pressure, differential pressure, temperature, and level. Some controller designs have the set point adjustment knob in the controller, making set point adjustment a local and manual operation. Other types receive a set point from a remotely located pneumatic source, such as a manual air set regulator or another controller, to achieve set point adjustment. There are versions of the pneumatic controller that support the useful one-, two-, and three-mode combinations of proportional, integral, and derivative actions. Other options include auto/manual transfer stations, antireset windup circuitry, on/off control, and process-variable and set point indicators. Pneumatic controllers are made of Bourdon tubes, bellows, diaphragms, springs, levers, cams, and other fundamental transducers to accomplish the control function. If operated on clean, dry plant air, they offer good performance and are extremely reliable. Pneumatic controllers are available with one or two stages of pneumatic amplification, with the two-stage designs having faster dynamic response characteristics. An example of a pneumatic PI controller is shown in Fig. 8-64a. This controller has two stages of pneumatic amplification and a Bourdon tube input element that measures process pressure. The Bourdon tube element is a flattened tube that has been formed into a curve so that changes in pressure inside the tube cause vertical motions to occur at the ungrounded end. This motion is transferred to the left end of the beam, as shown. The resulting motion of the beam is detected by the pneumatic nozzle amplifier, which, by proper sizing of the nozzle and fixed orifice diameters, causes the pressure internal to the nozzle to rise and fall with vertical beam motion. The internal nozzle pressure is routed to the pneumatic relay. The relay, which is constructed like the booster relay described in the “Valve Control Devices” subsection, has a direct linear input-to-output pressure characteristic. The output of the relay is the controller’s output and is piped away to the final control element.

CONTROLLERS, FINAL CONTROL ELEMENTS, AND REGULATORS

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(a)

(b) FIG. 8-64

Pneumatic controller: (a) example; (b) frequency response characteristic.

To generate the P and I control modes, a feedback circuit consisting of two bellows and two small metering valves has been added to the pneumatic amplifier system described above. The first valve is the proportional gain valve and is adjusted to provide an output pressure that is ratioed to the output pressure from the relay. The ratio used is set by manual adjustment of this valve. The output from the proportional gain valve is connected to a proportional feedback bellows, which provides negative feedback around the pneumatic amplifiers. The output pressure from the proportional gain valve is also connected to another valve, called the reset valve. This valve meters flow to and from a second bellows, known as the reset bellows. The resistance provided by the valve and the capacitance relating to the volume of the bellows forms a time constant that becomes the reset time constant for the controller. The frequency response characteristics of a pneumatic PI controller and an ideal PI controller are shown in Fig. 8-64b. Notice that the gain of the pneumatic controller reaches a limit at the lowest frequencies. This limit is due to a less-than-infinite amount of forward amplifier gain. The manufacturer of the controller designs the forward gain term to be as high as possible to better approximate ideal reset action in the controller but never to reach the ideal. Reset gain for available pneumatic controllers runs between 20 and 100 times the gain implied by a proportional gain of unity. Unity proportional gain implies that a 100 percent change in process input (i.e., a full range change) will generate a 100 percent change in controller output.

The reset time, which is user-adjustable, can range from 0.05 seconds to 80 minutes or more, depending on controller design. The reset time constant, when converted to frequency 1/2(TR) Hz (where TR is the reset time in seconds), determines the frequency where the reset and proportional response characteristics of the controller merge (see Fig. 8-64b). Tuning the reset adjustment on the controller moves the reset frequency to the left or right along the frequency axis and thereby affects the reset action of the controller. The response limit for the controller is a function of the design of the relay, the size of the load volume to which the controller is attached, the setting of the proportional band valve, and the forward gain designed into the pneumatic amplifiers. The frequency response limit (sometimes called the controller bandwidth) for a pneumatic controller into a small instrument load volume is in the 5- to 8-Hz range for two-stage pneumatic designs like the one shown. The dynamic response of the controller to set point changes is essentially the same as that indicated for process-variable changes. The set point adjustment mechanism affects the vertical motion of the nozzle over the beam and results in actions in the controller similar to those produced by changes in the process pressure. The main shortcomings of the pneumatic controller is its lack of flexibility when compared to modern electronic controller designs. Increased range of adjustability, choice of alternate control algorithms, the communication link to the control system, and other features and services provided by the electronic controller make it a superior choice in most of today’s applications.

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CONTROL VALVES A control valve consists of a valve, an actuator, and possibly one or more valve-control devices. The valves discussed in this section are applicable to throttling control (i.e., where flow through the valve is regulated to any desired amount between maximum and minimum limits). Other valves such as check, isolation, and relief valves are addressed in the next subsection. As defined, control valves are automatic control devices that modify the fluid flow rate as specified by the controller. Valves Valves are categorized according to their design style. These styles can be grouped into type of stem motion—linear or rotary. The valve stem is the rod, shaft, or spindle that connects the actuator with the closure member (i.e., a movable part of the valve that is positioned in the flow path to modify the rate of flow). Motion of either type is known as travel. The major categories are described briefly below. Globe and Angle The most common linear stem-motion control valve is the globe valve. The name comes from the globular shaped cavities around the port. In general, a port is any fluid passageway, but often the reference is to the passage that is blocked off by the closure member when the valve is closed. In globe valves, the closure member is called a plug. The plug in the valve shown in Fig. 8-65 is guided by a large-diameter port and moves within the port to provide the flow control orifice of the valve. A very popular alternate construction is a cage-guided plug as illustrated in Fig. 8-66. In many such designs, openings in the cage provide the flow control orifices. The valve seat is the zone of contact between the moving closure member and the stationary valve body, which shuts off the flow when the valve is closed. Often the seat in the body is on a replaceable part known as a seat ring. This stationary seat can also be designed as an integral part of the cage. Plugs may also be port-guided by wings or a skirt that fits snugly into the seat-ring bore. One distinct advantage of cage guiding is the use of balanced plugs in single-port designs. The unbalanced plug depicted in Fig. 8-65 is subjected to a static pressure force equal to the port area times the valve pressure differential (plus the stem area times the downstream pressure) when the valve is closed. In the balanced design (Fig. 8-66),

Actuator yoke mounting boss

note that both the top and bottom of the plug are subjected to the same downstream pressure when the valve is closed. Leakage via the plug-to-cage clearance is prevented by a plug seal. Both plug types are subjected to hydrostatic force due to internal pressure acting on the stem area and to dynamic flow forces when the valve is flowing. The plug, cage, seat ring, and associated seals are known as the trim. A key feature of globe valves is that they allow maintenance of the trim via a removable bonnet without removing the valve body from the line. Bonnets are typically bolted on but may be threaded in smaller sizes. Angle valves are an alternate form of the globe valve. They often share the same trim options and have the top-entry bonnet style. Angle valves can eliminate the need for an elbow but are especially useful when direct impingement of the process fluid on the body wall is to be avoided. Sometimes it is not practical to package a long trim within a globe body, so an angle body is used. Some angle bodies are self draining, which is an important feature for dangerous fluids. Butterfly The classic design of butterfly valves is shown in Fig. 8-67. Its chief advantage is high capacity in a small package and a very low initial cost. Much of the size and cost advantage is due to the wafer body design, which is clamped between two pipeline flanges. In the simplest design, there is no seal as such, merely a small clearance gap between the disc OD and the body ID. Often a true seal is provided by a resilient material in the body that is engaged via an interference fit with the disc. In a lined butterfly valve, this material covers the entire body ID and extends around the body ends to eliminate the need for pipeline joint gaskets. In a fully lined valve, the disc is also coated to minimize corrosion or erosion. A high-performance butterfly valve has a disc that is offset from the shaft center line. This eccentricity causes the seating surface to move away from the seal once the disc is out of the closed position, reducing friction and seal wear. Also known as an eccentric disc valve, its advantage is improved shutoff while maintaining high ultimate capacity at a

Stem

Graphite packing

Spring-loaded PTFE V-ring packing

Bonnet

Bonnet gasket

Backup ring

Guide bushing Plug

Cage

Body

Seat ring

Gasket

Post-guided contour-plug globe valve with metal seat and raisedface flange end connections. (Courtesy Fisher-Rosemount.)

FIG. 8-65

Seal ring Valve plug

Seat

PTFE disk

Metal disk retainer

Cage-guided balanced-plug globe valve with polymer seat and plug seal. (Courtesy Fisher-Rosemount.)

FIG. 8-66

CONTROLLERS, FINAL CONTROL ELEMENTS, AND REGULATORS

8-65

Partial cutaway of wafer-style lined butterfly valve. (Courtesy Fisher-Rosemount.)

Bodies are usually in two or three pieces or have a removable retainer to facilitate installing seals. End connections are usually flanged or threaded in small sizes, although segmented-ball valves are offered in wafer style also. Plug There are two substantially different rotary-valve design categories referred to as plug valves. The first consists of a cylindrical or slightly conical plug with a port through it. The plug rotates to vary the flow much like a ball valve. The body is top-entry but is geometrically simpler than a globe valve and thus can be lined with fluorocarbon polymer to protect against corrosion. These plug valves have excellent shutoff but are generally not for modulating service due to high friction. A variation of the basic design (similar to the eccentric butterfly disc) only makes sealing contact in the closed position and is used for control. The other rotary plug design is portrayed in Fig. 8-69. The seating surface is substantially offset from the shaft, producing a ball-valve-like motion with the additional cam action of the plug into the seat when closing. In reverse flow, high-velocity fluid motion is directed inward— impinging on itself and only contacting the plug and seat ring. Multi-Port This term refers to any valve or manifold of valves with more than one inlet or outlet. For throttling control, the threeway body is used for blending (two inlets, one outlet) or as a divertor (one inlet, two outlets). A three-way valve is most commonly a special globelike body with special trim that allows flow both over and under the plug. Two rotary valves and a pipe tee can also be used. Special three-, four-, and five-way ball-valve designs are used for switching applications.

reasonable cost. This cost advantage relative to other design styles is particularly true in sizes above 6-inch nominal pipe size (NPS). Improved shutoff is due to advances in seal technologies, including polymer, flexing metal, combination metal with polymer inserts, and so on, many utilizing pressure assist. Ball Ball valves get their name from the shape of the closure member. One version uses a full spherical member with a cylindrical bore through it. The ball is rotated d turn from the full-closed to the full-open position. If the bore is the same diameter as the mating-pipe fitting ID, the valve is referred to as full-bore. If the hole is undersized, the ball valve is considered to be a venturi style. A segmented ball is a portion of a hollow sphere—large enough to block the port when closed. Segmented balls often have a V-shaped contour along one edge, which provides a desirable flow characteristic (see Fig. 8-68). Both full-ball and segmented-ball valves are known for their low resistance to flow when full open. Shutoff leakage is minimized through the use of flexing or spring-loaded elastomeric or metal seals.

Special Application Valves Digital Valves True digital valves consist of discrete solenoidoperated flow ports that are sized according to binary weighing. The valve can be designed with sharp-edged orifices or with streamlined nozzles that can be used for flow metering. Precise control of the throttling-control orifice is the strength of the digital valve. Digital valves are mechanically complicated and expensive, and they have considerably reduced maximum flow capacities over the globe and rotary valve styles. Cryogenic Service Valves designed to minimize heat absorption for throttling liquids and gases below 80 K are called cryogenic service valves. These valves are designed with small valve bodies to minimize heat absorption and long bonnets between the valve and actuator to allow for extra layers of insulation around the valve. For extreme cases, vacuum jacketing can be constructed around the entire valve to minimize heat influx. High Pressure Valves used for pressures nominally above 760

Stem PTFE molded to inside diameter and sides of backup ring Bearing Shaft

Body

Special disc design to reduce dynamic torque

FIG. 8-67

V-notch ball Ball seal

Seal protector ring Gasket

Actuator mounting Follower shaft

Groove pin Body

Drive shaft Bearing Taper key

FIG. 8-68

Packing follower

Segmented ball valve. Partial view of actuator mounting shown 90° out of position. (Courtesy Fisher-Rosemount.)

8-66

PROCESS CONTROL

Eccentric plug valve shown in erosion-resistant reverse flow direction. Shaded components can be made of hard metal or ceramic materials. (Courtesy Fisher-Rosemount.)

FIG. 8-69

bar (11,000 psi, pressures above ANSI Class 4500) are often customdesigned for specific applications. Normally, these valves are of the plug type and use specially hardened plug and seat assemblies. Internal surfaces are polished, and internal corners and intersecting bores are smoothed to reduce high localized stresses in the valve body. Steam loops in the valve body are available to raise the body temperature to increase ductility and impact strength of the body material. High-Viscous Process Used most extensively by the polymer industry, the valve for high-viscous fluids is designed with smooth finished internal passages to prevent stagnation and polymer degradation. These valves are available with integral body passages through which a heat-transfer fluid is pumped to keep the valve and process fluid heated. Pinch The industrial equivalent of controlling flow by pinching a soda straw is the pinch valve. Valves of this type use fabric-reinforced elastomer sleeves that completely isolate the process fluid from the metal parts in the valve. The valve is actuated by applying air pressure directly to the outside of the sleeve, causing it to contract or pinch. Another method is to pinch the sleeve with a linear actuator with a specially attached foot. Pinch valves are used extensively for corrosive material service and erosive slurry service. This type of valve is used in applications with pressure drops up to 10 bar (145 psi). Fire-Safe Valves that handle flammable fluids may have additional safety-related requirements for minimal external leakage, minimal internal (downstream) leakage, and operability during and after a fire. Being fire-safe does not mean being totally impervious to fire, but a sample valve must meet particular specifications such as American Petroleum Institute (API) 607, Factory Mutual Research Corp. (FM) 7440, or the British Standard 5146 under a simulated fire test. Due to very high flame temperature, metal seating (either primary or as a backup to a burned-out elastomer) is mandatory. Solids Metering The control valves described earlier are primarily used for the control of fluid (liquid or gas) flow. Sometimes these valves, particularly the ball, butterfly, or sliding gate valves, are used to throttle dry or slurry solids. More often, special throttling mechanisms like venturi ejectors, conveyers, knife-type gate valves, or rotating vane valves are used. The particular solids-metering valve hardware

depends on the volume, density, particle shape, and coarseness of the solids to be handled. Actuators An actuator is a device that applies the force (torque) necessary to cause a valve’s closure member to move. Actuators must overcome pressure and flow forces; friction from packing, bearings or guide surfaces, and seals; and provide the seating force. In rotary valves, maximum friction occurs in the closed position and the moment necessary to overcome it is referred to as breakout torque. The rotary valve shaft torque generated by steady-state flow and pressure forces is called dynamic torque. It may tend to open or close the valve depending on valve design and travel. Dynamic torque per unit pressure differential is largest in butterfly valves at roughly 70° open. In linear stem-motion valves, the flow forces should not exceed available actuator force, but this is usually accounted for by default when the seating force is provided. Actuators often provide a failsafe function. In the event of an interruption in the power source, the actuator will place the valve in a predetermined safe position, usually either full open or full closed. Safety systems are often designed to trigger local failsafe action at specific valves to cause a needed action to occur, which may not be a complete process or plant shutdown. Actuators are classified according to their power source. The nature of these sources leads naturally to design features that make their performance characteristics distinct. Pneumatic Despite the availability of more sophisticated alternatives, the pneumatically driven actuator is still by far the most popular type. Historically the most common has been the spring and diaphragm design (Fig. 8-70). The compressed air input signal fills a chamber sealed by an elastomeric diaphragm. The pressure force on the diaphragm plate causes a spring to be compressed and the actuator stem to move. This spring provides the failsafe function and contributes to the dynamic stiffness of the actuator. If the accompanying valve is “push-down-to-close,” the actuator depicted in Fig. 8-70 would be described as “air-to-close” or synonymously as fail-open. A Air connection Diaphragm casing Diaphragm and stem shown in up position Diaphragm plate Lower diaphragm casing Actuator spring Actuator stem Spring seat

Spring adjustor Stem connector

Yoke Travel indicator disk Indicator scale FIG. 8-70 Spring and diaphragm actuator with an “up” fail-safe mode. Spring adjuster allows slight alteration of bench set. (Courtesy Fisher-Rosemount.)

CONTROLLERS, FINAL CONTROL ELEMENTS, AND REGULATORS slightly different design yields “air-to-open” or fail-closed action. The spring is typically precompressed to provide a significant available force in the failed position (e.g., to provide seating load). The spring also provides a proportional relationship between the force generated by air pressure and stem position. The pressure range over which a spring and diaphragm actuator strokes in the absence of valve forces is known as the bench set. The chief advantages of spring and diaphragm actuators are their high reliability, low cost, adequate dynamic response, and failsafe action—all of which are inherent in their simple design. Alternately, the pressurized chamber can be formed by a circular piston with a seal on its outer edge sliding within a cylindrical bore. Higher operating pressure (6 bar [∼90 psig] is typical) and longer strokes are possible. Piston actuators can be spring-opposed but many times are in a dual-acting configuration (i.e., compressed air is applied to both sides of the piston with the net force determined from the pressure difference—see Fig. 8-71). Dynamic stiffness is usually higher with piston designs than with spring and diaphragm actuators; see “Positioner/Actuator Stiffness.” Failsafe action, if necessary, is achieved without a spring through the use of additional solenoid valves, trip valves, or relays. See “Valve Control Devices.” Motion Conversion Actuator power units with translational output can be adapted to rotary valves that generally need 90° or less rotation. A lever is attached to the rotating shaft and a link with pivoting means on the end connects to the linear output of the power unit, an arrangement similar to an internal combustion engine crankshaft, connecting rod, and piston. When the actuator piston, or more commonly the diaphragm plate, is designed to tilt, one pivot can be eliminated (see Fig. 8-71). Scotch yoke and rack and pinion arrangements are also commonly used, especially with piston power units. Friction and changing mechanical advantage of these motion-conversion mechanisms means the available torque may vary greatly with travel. One notable exception is vane-style rotary actuators whose offset “piston” pivots, giving direct rotary output. Hydraulic The design of typical hydraulic actuators is similar to double-acting piston pneumatic types. One key advantage is the high pressure (typically 35 to 70 bar [500 to 1000 psi]), which leads to high thrust in a smaller package. The incompressible nature of the

O-ring seal Piston

Piston rod

Cylinder Sliding seal

Lever

Double-acting piston rotary actuator with lever and tilting piston for motion conversion. (Courtesy Fisher-Rosemount.)

FIG. 8-71

8-67

hydraulic oil means these actuators have very high dynamic stiffness. The incompressibility and small chamber size connote fast stroking speed and good frequency response. The disadvantages include high initial cost, especially when considering the hydraulic supply. Maintenance is much more difficult than with pneumatics, especially on the hydraulic positioner. Electrohydraulic actuators have similar performance characteristics and cost/maintenance ramifications. The main difference is that they contain their own electric-powered hydraulic pump. The pump may run continuously or be switched on when a change in position is required. Their main application is remote sites without an air supply when a failsafe spring return is needed. Electric The most common electric actuators use a typical motor—three-phase AC induction, capacitor-start split-phase induction, or DC. Normally the motor output passes through a large gear reduction and, if linear motion output is required, a ball screw or thread. These devices can provide large thrust, especially given their size. Lost motion in the gearing system does create backlash, but if not operating across a thrust reversal, this type of actuator has very high stiffness. Usually the gearing system is self-locking, which means that forces on the closure member cannot move it by spinning a nonenergized motor. This behavior is called a lock-in-last-position failsafe mode. Some gear systems (e.g., low-reduction spur gears) can be backdriven. A solenoid-activated mechanical brake or locking current to motor field coils is added to provide lock-in-last-position fail mode. A battery backup system for a DC motor can guard against power failures. Otherwise, an electric actuator is not acceptable if fail-open/ closed action is mandatory. Using electrical power requires environmental enclosures and explosion protection, especially in hydrocarbon-processing facilities; see the full discussion in “Valve Control Devices.” Unless sophisticated speed-control power electronics is used, position modulation is achieved via bang-bang control. Mechanical inertia causes overshoot, which is (1) minimized by braking and/or (2) hidden by adding dead band to the position control. Without these provisions, high starting currents would cause motors to overheat from constant “hunting” within the position loop. Travel is limited with power interruption switches or with force (torque) electromechanical cutouts when the closed position is against a mechanical stop (e.g., a globe valve). Electric actuators are often used for on/off service. Stepper motors can be used instead, and they, as their name implies, move in fixed incremental steps. Through gear reduction, the typical number of increments for 90° rotation range from 5000 to 10,000; hence positioning resolution at the actuator is excellent. Position overshoot is not an issue, and added dead band need only be a few steps away. An electromagnetic solenoid can be used to directly actuate the plug on very small linear stem-motion valves. A solenoid is usually designed as a two-position device, so this valve control is on/off. Special solenoids with position feedback can provide proportional action for modulating control. Force requirements of medium-sized valves can be met with piloted plug designs, which use process pressure to assist the solenoid force. Piloted plugs are also used to minimize the size of common pneumatic actuators, especially when there is need for high seating load. Manual A manually positioned valve is by definition not an automatic control valve, but it may be involved with process control. For rotary valves, the manual operator can be as simple as a lever, but a wheel driving a gear reduction is necessary in larger-size valves. Linear motion is normally created with a wheel turning a screw-type device. A manual override is usually available as an option for the powered actuators listed above. For spring-opposed designs, an adjustable travel stop will work as a one-way manual input. In more complex designs the handwheel can provide loop control override via an engagement means. Some gear-reduction systems of electric actuators allow the manual positioning to be independent from the automatic positioning without declutching. Valve-Control Devices Devices mounted on the control valve that interface various forms of input signals, monitor and transmit valve position, or modify valve response are valve-control devices. In some applications, several auxiliary devices are used together on the

8-68

PROCESS CONTROL

same control valve. For example, mounted on the control valve, one may find a current-to-pressure transducer, a valve positioner, a volume booster relay, a solenoid valve, a trip valve, a limit switch, a process controller, and/or a stem-position transmitter. Figure 8-72 shows a valve positioner mounted on the yoke leg of a spring and diaphragm actuator. As most throttling control valves are still operated by pneumatic actuators, the control-valve device descriptions that follow relate primarily to devices that are used with pneumatic actuators. The function of hydraulic and electrical counterparts are very similar. Specific details on a particular valve-control device are available from the vendor of the device. Transducers The current-to-pressure transducer (I/P transducer) is a conversion interface that accepts a standard 4–20 mA input current from the process controller and converts it to a pneumatic output in a standard pneumatic pressure range (normally 0.2–1.0 bar [3–15 psig] or, less frequently, 0.4–2.0 bar [6–30 psig]). The output pressure generated by the transducer is connected directly to the pressure connection on a spring-opposed diaphragm actuator or to the input of a pneumatic valve positioner. Figure 8-73a is the schematic of a basic I/P transducer. The transducer shown is characterized by (1) an input conversion that generates an angular displacement of the beam proportional to the input current, (2) a pneumatic amplifier stage that converts the resulting angu-

(a)

(b)

(c) Current to pressure transducer components parts: (a) direct current to pressure conversion; (b) pneumatic booster amplifier (relay); (c) block diagram of a modern I/P transducer. FIG. 8-73

FIG. 8-72 Valve and actuator with valve positioner attached. (Courtesy FisherRosemount.)

lar displacement to pneumatic pressure, and (3) a pressure area that serves as a means to return the beam back to very near its original position when the new output pressure is achieved. The result is a device that generates a pressure output that tracks the input current signal. The transducer shown in Fig. 8-73a is used to provide pressure to small load volumes (normally 4.0 in3 or less), such as a positioner or booster input. With only one stage of pneumatic amplification, the flow capacity of this transducer is limited and not sufficient to provide responsive load pressure directly to a pneumatic actuator. The flow capacity of the transducer can be increased by adding a booster relay like the one shown in Fig. 8-73b. The flow capacity of the booster relay is nominally fifty to one hundred times that of the nozzle amplifier shown in Fig. 8-73a and makes the combined transducer/booster suitably responsive to operate pneumatic actuators. This type of transducer is stable into all sizes of load volumes and produces measured accuracy (see Instrument Society of America [ISA]S51.1-1979, “Process Instrumentation Terminology” for the definition of measured accuracy) of 0.5 percent to 1.0 percent of span. Better measured accuracy results from the transducer design shown in Fig. 8-73c. In this design, pressure feedback is taken at the output of the booster-relay stage and fed back to the main summer. This allows the transducer to correct for errors generated in the pneumatic booster as well as errors in the I/P-conversion stage. Also, particularly with the new analog electric and digital versions of this design, PID control is used in the transducer-control network to give extremely good static accuracy, fast dynamic response, and reasonable

CONTROLLERS, FINAL CONTROL ELEMENTS, AND REGULATORS stability into a wide range of load volumes (small instrument bellows to large actuators). Also, environmental factors such as temperature change, vibration, and supply pressure fluctuation affect this type of transducer the least. Even a perfectly accurate I/P transducer cannot compensate for stem-position errors generated by friction, backlash, and varying force loads coming from the actuator and valve. To do this compensation, a different control-valve device, known as a valve positioner, is required. Valve Positioners The valve positioner, when combined with an appropriate actuator, forms a complete closed-loop valve-position control system. This system makes the valve stem conform to the input signal coming from the process controller in spite of force loads that the actuator may encounter while moving the control valve. Usually, the valve positioner is contained in its own enclosure and is mounted on the control valve. The key parts of the positioner/actuator system, shown in Fig. 8-74a, are (1) an input-conversion network, (2) a stem-position feedback network, (3) a summing junction, (4) an amplifier network, and (5) an actuator. The input-conversion network shown is the interface between the input signal and the summer. This block converts the input current or pressure (from an I/P transducer or a pneumatic process controller) to a voltage, an electric current, a force, torque, displacement or other particular variable that can be directly used by the summer. The input conversion usually contains a means to adjust the slope and offset of the block to provide for a means of spanning and zeroing the positioner during calibration. In addition, means for changing the sense (known as “action”) of the input/output characteristic are oftentimes addressed in this block. Also, exponential, logarithmic or other predetermined characterization can be put in this block to provide a characteristic that is useful in offsetting or reinforcing a nonlinear valve or process characteristic. The stem-position feedback network converts stem travel to a useful form for the summer. This block includes the feedback linkage,

(a)

(b) Positioner/actuators: (a) generic block diagram; (b) an example of a pneumatic positioner/actuator.

FIG. 8-74

8-69

which varies with actuator type. Depending on positioner design, the stem-position feedback network can provide span and zero and characterization functions similar to that described for the inputconversion block. The amplifier network provides signal conversion and suitable static and dynamic compensation for good positioner performance. Control from this block usually reduces down to a form of proportional or proportional plus derivative control. The output from this block in the case of a pneumatic positioner is a single connection to the spring and diaphragm actuator or two connections for push-pull operation of a springless piston actuator. The action of the amplifier network and the action of the stem-position feedback can be reversed together to provide for reversed positioner action. By design, the gain of the amplifier network shown in Fig. 8-74a is made very large. Large gain in the amplifier network means that only a small proportional deviation will be required to position the actuator through its active range of travels. This means that the signals into the summer track very closely and that the gain of the input-conversion block and the stem-position feedback block determine the closedloop relationship between the input signal and the stem travel. Large amplifier gain also means that only a small amount of additional stem-travel deviation will result when large external force loads are applied to the actuator stem. For example, if the positioner’s amplifier network has a gain of 50 and assuming that high packing-box friction loads require 25 percent of the actuator’s range of thrust to move the actuator, then only 25 percent/50 or 0.5 percent deviation between input signal and output travel will result due to valve friction. Figure 8-74b is an example of a pneumatic positioner/actuator. The input signal is a pneumatic pressure that (1) moves the summing beam, which (2) operates the spool valve amplifier, which (3) provides flow to and from the piston actuator, which (4) causes the actuator to move and continue moving until (5) the feedback force returns the beam to its original position and stops valve travel at a new position. Typical positioner operation is thereby achieved. Static performance measurements related to positioner/actuator operation are: conformity, measured accuracy, hysteresis, dead band, repeatability, and locked stem-pressure gain. Definitions and standardized test procedures for determining these measurements can be found in ISA-S75.13-1989, “Method of Evaluating the Performance of Positioners with Analog Input Signals and Pneumatic Output”. Dynamics of Pneumatic Positioners Dynamically, the pneumatic positioner is characterized by the combined effects of gain and capacitance and nonlinear effects such as valve friction and flowcapacity saturation. Generally, there is a threshold level of input signal below which the positioner output will not respond at all. This band is on the order of 0.1 percent of input span for pneumatic positioners but can be larger if significant valve friction is present. Above this threshold level, but below the level that causes velocity saturation, the positioner is approximately linear and likened to a second-order lowpass filter (see Fig. 8-75a). Natural frequencies range from 0.3 to 3.0 Hz and damping ratios of 0.6 to 2.0 are common and dependent on positioner design and the physical size of the actuator volume. At higher drive levels, the flow capacity of the positioner is reached and attenuation of the resulting travel begins. Positioner/Actuator Stiffness Minimizing the effect of dynamic loads on valve-stem travel is an important characteristic of the positioner/actuator. Stem position must be maintained in spite of changing reaction forces caused by valve throttling. These forces can be random in nature (buffeting force) or result from a negative sloped force/stem travel characteristic (negative gradient); either could result in valvestem instability and loss of control. To reduce and eliminate the effect of these forces, the effective stiffness of the positioner/actuator must be made sufficiently high to maintain adequate stability of the valve stem. The stiffness characteristic of the positioner/actuator varies with frequency. Figure 8-75b indicates the stiffness of the positioner/actuator is high at low frequencies and is directly related to the lockedstem pressure gain provided by the positioner. As frequency increases, a dip in the stiffness curve results from dynamic gain attenuation in the pneumatic amplifiers in the positioner. The value at the bottom of the dip is the sum of the mechanical stiffness of the spring in the actu-

8-70

PROCESS CONTROL

(a)

(b) Frequency response curves for a pneumatic positioner/actuator: (a) input signal to stem travel for a 69-inch2 spring and diaphragm actuator with a 1.5-inch total travel and 3–15 psig input pressure; (b) dynamic stiffness for the same positioner/actuator. FIG. 8-75

CONTROLLERS, FINAL CONTROL ELEMENTS, AND REGULATORS ator and the air spring effect produced by air enclosed in the actuator casing. The air spring effect results from adiabatic expansion and compression of the air in the actuator casing. Numerically, the small perturbation value for air spring stiffness in Newtons/meter is given by Eq. (8-107). γPaA2a Air spring rate = ᎏ (8-107) V where γ is ratio of specific heats (1.4 for air), Pa is the actuator pressure in Pascal absolute, Aa is the actuator pressure area in m2, and V is the internal actuator volume in m3. Notice in the figure that the minimum stiffness value (mechanical spring stiffness + air spring stiffness) is several times larger than the stiffness produced by the spring in the actuator (shown as a dotted line) by itself. This indicates that the air spring stiffness is quite significant and worth considering in actuator design and actuator sizing. To the right of the dip, the inertia effects of the mass of the moving parts of the valve and actuator cause the overall system stiffness to rise with increasing frequency. Positioner Application Positioners are widely used on pneumatic valve actuators. More often than not, they provide improved process-loop control because they reduce valve-related nonlinearity. Dynamically, positioners maintain their ability to improve controlvalve performance for sinusoidal input frequencies up to about one half of the positioner bandwidth. At input frequencies greater than this, the attenuation in the positioner amplifier network gets large, and valve nonlinearity begins to affect final control-element performance more significantly. Because of this, the most successful use of the positioner occurs when the positioner-response bandwidth is greater than twice that of the most dominant time lag in the process loop. Some typical examples of where the dynamics of the positioner are sufficiently fast to improve process control are the following: 1. In a distributed control system (DCS) process loop with an electronic transmitter. The DCS controller and the electronic transmitter have time constants that are dominant over the positioner response. Positioner operation is therefore beneficial in reducing valve-related nonlinearity. 2. In a process loop with a pneumatic controller and a large process time constant. Here the process time constant is dominant, and the positioner will improve the linearity of the final control element. Some common processes with large time constants that benefit from positioner application are liquid level, temperature, large volume gas pressure, and mixing. 3. Additional situations where valve positioners are used are as follows: • On springless actuators where the actuator is not usable for throttling control without position feedback. • When split ranging is required to control two or more valves sequentially. In the case of two valves, the smaller control valve is calibrated to open in the lower half of the input signal range and a larger valve is calibrated to open in the upper half of the input signal range. Calibrating the input command signal range in this way is known as split-range operation and increases the practical range of throttling process flows over that of a single valve. • In open-loop control applications where best static accuracy is needed. On occasion, positioner use can degrade process control. Such is the case when the process controller, the process, and the process transmitter have time constants that are similar or smaller than that of the positioner/actuator. This situation is characterized by low processcontroller P gain (P gain < 0.5), and hunting or limit cycling of the process variable is observed. Improvements here can be made by doing one of the following: • Install a dominant first-order low-pass filter in the loop ahead of the positioner and retune the process loop. This should allow increased proportional gain in the process loop and reduce hunting. Possible means for adding the filter include adding it to the firmware of the DCS controller, by adding an external RC network on the output of the process controller or by enabling the filter function in the

8-71

input of the positioner if it is available. Also, some transducers, when connected directly to the actuator, form a dominant first-order lag that can be used to stabilize the process loop. • Select a positioner with a faster response characteristic. Booster Relays The booster relay is a single-stage power amplifier having a fixed gain relationship between the input and output pressures. The device is packaged as a complete standalone unit with pipe-thread connections for input, output, and supply pressure. The booster amplifier shown in Fig. 8-73b shows the basic construction of the booster relay. Enhanced versions are available that provide specific features such as: (1) variable gain to split the output range of a pneumatic controller to operate more than one valve or to provide additional actuator force; (2) low hysteresis for relaying measurement and control signals; (3) high flow capacity for increased actuatorstroking speed; and (4) arithmetic, logic, or other compensation functions for control system design. A particular type of booster relay, called a dead-band booster, is shown in Fig. 8-76. This booster is designed to be used exclusively between the output of a valve positioner and the input to a pneumatic actuator. It is designed to provide extra flow capacity to stroke the actuator faster than with the positioner alone. The dead-band booster is designed intentionally with a large dead band (approximately 5 percent of the input span), elastomer seats for tight shutoff, and an adjustable bypass valve connected between the input and the output of the booster. The bypass valve is tuned to provide the best compromise between increased actuator stroking speed and positioner/actuator stability. With the exception of the dead-band booster, the application of booster relays has diminished somewhat by the increased use of current-to-pressure transducers, electropneumatic positioners, and electronic control systems. Transducers and valve positioners serve much the same functionality as the booster relay in addition to interfacing with the electronic process controller. Solenoid Valves The electric solenoid valve has two output states. When sufficient electric current is supplied to the coil, an internal armature moves against a spring to an extreme position. This motion causes an attached pneumatic or hydraulic valve to operate. When current is removed, the spring returns the armature and the attached solenoid valve to the deenergized position. An intermediate pilot stage is sometimes used when additional force is required to operate the main solenoid valve. Generally, solenoid valves are used to pressurize or vent the actuator casing for on/off control-valve application and safety shutdown applications. Input signal Diaphragms

Exhaust port

Bypass valve adjusting screw Adjustable restriction

Exhaust

Supply port

Supply

FIG. 8-76

Output to actuator

Dead-band booster relay. (Courtesy Fisher-Rosemount.)

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PROCESS CONTROL

Trip Valves The trip valve is part of a system that is used where a specific valve action (i.e., fail up, fail down, or lock in last position) is required when pneumatic supply pressure to the control valve falls below a preset level. Trip systems are used primarily on springless piston actuators requiring fail-open or fail-closed action. An air storage or “volume” tank and a check valve are used with the trip valve to provide power to stroke the valve when supply pressure is lost. Trip valves are designed with hysteresis around the trip point to avoid instability when the trip pressure and the reset pressure settings are too close to the same value. Limit Switches and Stem-Position Transmitters Travel-limit switches, position switches, and valve-position transmitters are devices that, when mounted on the valve, actuator, damper, louver, or other throttling element, detect the component’s relative position. The switches are used to operate alarms, signal lights, relays, solenoid valves, or discrete inputs into the control system. The valve-position transmitter generates a 4–20-mA output that is proportional to the position of the valve. Fire and Explosion Protection Electrical equipment can be a source of ignition in environments with combustible concentrations of gas, liquid, dust, fibers, or flyings. Most of the time it is possible to locate the electronic equipment away from these hazardous areas. However, where electric or electronic valve-mounted instruments must be used in areas where there is a hazard of fire or explosion, the equipment must be designed to meet requirements for safety. Articles 500 through 504 of the National Electrical Code address definitions of hazardous locations and the requirements for electrical devices in locations where fire or explosion hazard exists. NFPA (National Fire Protection Agency) 497M addresses the properties and group classification of gas, vapor, and dust for electrical devices in hazardous locations. With valve-mounted accessories, the approved protection concepts most often used for safety protection are explosion-proof, intrinsically safe, nonincendive, and dust-ignition-proof. The explosion-proof enclosure is designed such that an explosion in the interior of the enclosure containing the electronic circuits will be contained. The enclosure will not allow sufficient flame to escape to the exterior to cause an ignition. Also, a surface temperature rating is given to the device. This rating must indicate a lower surface temperature than the ignition temperature of the gas in the hazardous area. Explosion-proof enclosures are characterized by strong metal enclosures with special close-fitting access covers and breathers that contain an ignition to the inside of the enclosure. Field wiring in the hazardous environment is enclosed in a metal conduit of the mineralinsulated-cable type. All conduit and cable connections or cable terminations are threaded and explosion-proof. Conduit seals are put into the conduit or cable system at locations defined by the National Electric Code (Article 501) to prevent gas and vapor leakage and to prevent flames from passing from one part of the conduit system to the other. The intrinsically safe (I.S.) control-valve device contains circuits that are incapable of releasing sufficient electrical or thermal energy to cause ignition of a specified hazardous mixture under normal or fault operating conditions of the circuit. I.S. circuits are designed with voltage- and current-limiting networks added where necessary to achieve approved levels of safety. I.S. field wiring need not be enclosed in metal conduit but must be kept separate from wiring for nonintrinsically safe circuits. Intrinsically safe field wiring must be energy limited, usually by a Zener diode barrier circuit located in the control room. The manufacture of the intrinsically safe control-valve devices must list the identification number of the control drawing on the nameplate attached to the approved device. The control drawing contains information showing approved combinations of accessories and other connected apparatus such as Zener diode energy barriers. ANSI/ISA S12.12, “Nonincendive Electrical Equipment for Use in Class I and Class II, Division 2 and Class III, Division 1 and 2 Hazardous (Classified) Locations,” addresses requirements for nonincendive electrical equipment and wiring. Nonincendive apparatus and/or field wiring refers to approved equipment or wiring that is incapable of imparting sufficient energy to ignite the specified hazardous atmosphere under normal circuit operating conditions. Nonincendive protection is considered for applications where hazardous concentrations

of flammable gas and vapors or combustible dusts are only present under abnormal operating conditions or in those applications where easily ignited fibers or flyings are present in sufficient quantities to cause ignitable mixtures. For applications where hazardous concentrations of flammable gas and vapors or combustible dust are present continuously, intermittently, or periodically, more stringent protection (see Division 1, National Electrical Code, Article 500) offered by explosion-proof or intrinsically safe concepts is required. The dust-ignition-proof protection concept excludes dust from entering the device enclosure and will not permit arcs, sparks, or heat generated by the device to cause ignition of external suspensions or accumulations of the dust. Enclosure requirements can be found in ANSI/UL 1203-1994, “Explosion-Proof and Dust-Ignition-Proof Electrical Equipment for Use in Hazardous Locations.” Certified testing and approval for control-valve devices used in hazardous locations is normally procured by the manufacturer of the device. The manufacturer often goes to a third party laboratory for testing and certification. Applicable approval standards are available from CSA, CENELEC, FM, SAA, and UL. Environmental Enclosures Enclosures for valve accessories are sometimes required to provide protection from specific environmental conditions. The National Electrical Manufacturers Association (NEMA) provides descriptions and test methods for equipment used in specific environmental conditions in NEMA 250. Protection against rain, windblown dust, hose-directed water, and external ice formation are examples of environmental conditions that are covered by NEMA standards. Also, the electronic control-valve device’s level of immunity to, and emission of, electromagnetic interference (EMI) can be an issue in the chemical-valve environment. EMI requirements for the controlvalve devices are presently mandatory in the European Community but voluntary in the United States, Japan, and the rest of the world. International Electrotechnical Commission (IEC) 801, Parts 1 through 4, “Electromagnetic Compatibility for Industrial Process Measurement and Control Equipment,” defines tests and requirements for control-device immunity. Immunity and emission standards are addressed in CENELEC (European Committee for Electrotechnical Standardization) EN 50 081-1:1992, EN 50 081-2:1993, EN 50 082-1:1992, and prEN 50 082-2:1994. Digital Field Communications An increasing number of valvemounted devices are available that support digital communications in addition to, or in place of, the traditional 4–20 mA current signal. These control-valve devices have increased functionality, resulting in reduced setup time, improved control, combined functionality of traditionally separate devices, and control-valve diagnostic capability. Digital communications also allow the control system to become completely distributed where, for example, the process PID controller could reside in the valve positioner or in the process transmitter. The high-performance, all-digital, multidrop communication protocol for use in the process-control industry is known as fieldbus. Presently there are several regional and industry-based fieldbus standards including the French standard, FIP (NFC 4660x approved by UTE), the German standard, Profibus (DIN 19245 approved by DKE), and proprietary standards by DCS vendors. As of 1997, none of these fieldbus standards have been adopted by international standards organizations. The International Electrotechnical Commission (IEC) Standards Committee 65C Working Group 6 (IEC SC65C WG6) and the ISA Standards and Practices Committee 50 (ISA SP50) are presently working on a fieldbus standard, but at the time of this writing, the standard is unfinished. One interim solution supported by some valvedevice products is the hybrid communication method, where both analog and digital communication capabilities are present in the same device. This scheme has the advantage of allowing the communicating valve-control device to be retrofit into a traditional 4–20 mA current loop and still support digital communications between the final control element and the control room. Here the current signal is used to communicate the primary signal value, and the digital communication channel carries secondary variable information, configuration information, calibration information, and alert and diagnostic information. An example of a hybrid protocol that is open (not proprietary) and

CONTROLLERS, FINAL CONTROL ELEMENTS, AND REGULATORS

FIG. 8-77 Hybrid point-to-point communications between the control room and the control valve device.

in use by several manufacturers of control-valve devices is known as HART ®* (Highway Addressable Remote Transducer) protocol (see Fig. 8-77). With this protocol, the digital communications occur over the same two wires that provide the 4–20 mA process control signal without disrupting the process signal. The protocol uses the frequency-shift keying (FSK) technique where two individual frequencies, one representing the mark and the other representing the space, are superimposed on the 4–20 mA current signal. As the average value of the signals used is zero, there is no DC offset value added to the 4–20 mA signal. Valve Application Technology Functional requirements and the properties of the controlled fluid determine which valve and actuator types are best for a specific application. If demands are modest and no unique valve features are required, the valve-design style selection may be determined solely by cost. If so, general-purpose globe or angle valves provide exceptional value, especially in sizes less than 3-inch NPS and hence are very popular. Beyond type selection, there are many other valve specifications that must be determined properly in order to ultimately yield-improved process control. Materials and Pressure Ratings Valves must be constructed from materials that are sufficiently immune to corrosive or erosive action by the process fluid. Common body materials are cast iron, steel, stainless steel, high-nickel alloys, and copper alloys such as bronze. Trim materials usually need a greater immunity due to the higher fluid velocity in the throttling region. High hardness is desirable in erosive and cavitating applications. Heat-treated and precipitation-hardened stainless steels are common. High hardness is also good for guiding, bearing, and seating surfaces; cobalt-chromium alloys are utilized in cast or wrought form and frequently as welded overlays called hard facing. In less stringent situations, chrome plating, heat-treated nickel coatings, and ion nitriding are used. Tungsten carbide and ceramic trim are warranted in extremely erosive services. See Sec. 28, “Materials of Construction,” for specific material properties. Since the valve body is a pressurized vessel, it is usually designed to comply with a standardized system of pressure ratings. Two common systems are described in the standards ANSI B16.34 and DIN 2401. * HART is a trademark owned by Rosemount, Inc.

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Internal pressure limits under these standards are divided into broad classes, with specific limits being a function of material and temperature. Manufacturers also assign their own pressure ratings based on internal design rules. A common insignia is “250 WOG,” which means a pressure rating of 250 psig (∼17 bar) in water, oil, or gas at ambient temperature. “Storage and Process Vessels” in Sec. 10 provides introductory information on compliance of pressure-vessel design to industry codes (e.g., ASME Boiler and Pressure Vessel Code—Section VIII, ASME B31.3 Chemical Plant and Petroleum Refinery Piping). Valve bodies are also standardized to mate with common piping connections: flanged, butt-weld end, socket-weld end, and screwed end. Dimensional information for some of these joints and class pressure-temperature ratings are included in Sec. 10, “Process Plant Piping.” Control valves have their own standardized face-to-face dimensions that are governed by ISA Standards S75.03, 04, 12, 14, 15, 16, 20, and 22. Butterfly valves are also governed by API 609 and Manufacturers Standardization Society (MSS) SP-67 and 68. Sizing Throttling control valves must be selected to pass the required flow rate given expected pressure conditions. Sizing is not merely matching the end connection size with surrounding piping; it is a key step in ensuring that the process can be properly controlled. Sizing methods range from simple models based on elementary fluid mechanics to very complex models when unusual thermodynamics or nonideal behaviors occur. Basic sizing practices have been standardized upon (e.g., ISA S75.01) and are implemented as PC-based programs by manufacturers. The following is a discussion of very basic sizing equations and the associated physics. Regardless of the particular process variable being controlled (e.g., temperature, level, pH), the output of a control valve is flow rate. The throttling valve performs its function of manipulating flow rate by virtue of being an adjustable resistance to flow. Flow rate and pressure conditions are normally known when a process is designed and the valve resistance range must be matched accordingly. In the tradition of orifice and nozzle discharge coefficients, this resistance is embodied in the valve flow coefficient Cv. The mass flow rate (w) in kg/h is given for a liquid by (p 1−  w = 27.3Cvρ p 2)

(8-108)

where p1 and p2 are upstream and downstream static pressure in bar, respectively. The density of the fluid ρ is expressed in kg/m3. This equation is valid for nonvaporizing, turbulent-flow conditions for a valve with no attached fittings. The relationship can be derived from the principles of conservation of mass and energy. A more complete presentation of sizing relationships is given in ISA S75.01, including provisions for pipe reducers, vaporizing liquids, and Reynolds number effects. While the above equation gives the relationship between pressure and flow from a macroscopic point of view, it does not explain what is going on inside the valve. Valves create a resistance to flow by restricting the cross sectional area of the flow passage and also by forcing the fluid to change direction as it passes through the body and trim. The conservation of mass principle dictates that, for steady flow, density × average velocity × cross sectional area equals a constant. The average velocity of the fluid stream at the minimum restriction in the valve is therefore much higher than at the inlet. Note that due to the abrupt nature of the flow contraction that forms the minimum passage, the main fluid stream may separate from the passage walls and form a jet that has an even smaller cross section, the so-called vena contracta. The ratio of minimum stream area to the corresponding passage area is called the contraction coefficient. As the fluid expands from the minimum cross sectional area to the full passage area in the downstream piping, large amounts of turbulence are generated. Direction changes can also induce significant amounts of turbulence. Some of the potential energy that was stored in the fluid by pressurizing it (e.g., the work done by a pump) is first converted into the kinetic energy of the fast-moving fluid at the vena contracta. Some of that kinetic energy turns into the kinetic energy of turbulence. As the turbulent eddies break down into smaller and smaller structures, viscous effects ultimately convert all of the turbulent energy into heat. Therefore, a valve converts fluid energy from one form to another. For many valve constructions, it is reasonable to approximate the

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PROCESS CONTROL

fluid transition from the valve inlet to the minimum cross section of the flow stream as an isentropic or lossless process. This minimum pressure, pvc, can be estimated from the Bernoulli relationship. See Sec. 6 (“Fluid and Particle Mechanics”) for more background information. Downstream of the vena contracta, the flow is definitely not lossless due to all the turbulence that is generated. As the flow passage area increases and the fluid slows down, some of the kinetic energy of the fluid is converted back to potential energy as the pressure recovers. The remaining energy that is permanently lost via turbulence accounts for the permanent pressure or head loss of the valve. The relative amount of pressure that is recouped determines whether the valve is considered to be high or low recovery. See Fig. 8-78 for an illustration of how the mean pressure changes as fluid moves through a valve. The flow-passage geometry at and downstream of the vena contracta primarily determines the amount of recovery. The amount of recovery is quantified by the liquid pressure recovery factor FL where FL =

p −p ᎏ  p −p 1

2

1

vc

(8-109)

A key limitation of sizing Eq. (8-109) is the limitation to incompressible fluids. For gases and vapors, density is dependent on pressure. For convenience, compressible fluids are often assumed to follow the ideal-gas-law model. Deviations from ideal behavior are corrected for, to first order, with nonunity values of compressibility factor Z. (See Sec. 2, “Physical and Chemical Data,” for definitions and data for common fluids.) For compressible fluids

w = 94.8Cv P1Y

xM ᎏ  TZ w

where P1 is in bar absolute, T1 is inlet temperature in K, Mw is the molecular weight, and x is the dimensionless pressure-drop ratio (p1 − p2)/p1. The expansion factor Y accounts for changes in the fluid density as the fluid passes through the valve and for variation in the contraction coefficient with pressure drop. For convenience the experimental data is approximated by a simple relationship: 1.4x xTγ Y=1−ᎏ for x≤ᎏ (8-111) 3xTγ 1.4 where γ is the ratio of specific heats and xT is the pressure drop ratio factor. Even though a fluid may be compressible, if the value of x is small, the flow will behave as though it is incompressible. In the limit as x goes to zero, Eq. (8-110) reduces to the incompressible form Eq. (8-108) with ρ expressed via the ideal-gas equation of state. Compressible fluids exhibit a phenomenon known as choking. Given a nozzle geometry with fixed inlet conditions, the mass flow rate will increase as P2 is decreased up to a maximum amount at the critical pressure drop. The velocity at the vena contracta has reached sonic and a standing shock has formed. This shock causes a step change in pressure as flow passes through it, and further reduction in P2 does not increase mass flow. xT is a parameter of the flow model that relates to the critical pressure-drop ratio but also accounts for valve geometry effects. The value of xT varies with flow-path geometry; a rough estimate for conventional valves is one-half. In the choked case, xTγ x>ᎏ and Y = 0.67 (8-112) 1.4 Noise Control Sound is a fluctuation of air pressure that can be detected by the human ear. Sound travels through any fluid (e.g., the air) as a compression/expansion wave. This wave travels radially outward in all directions from the sound source. The pressure wave induces an oscillating motion in the transmitting medium that is superimposed on any other net motion it may have. These waves are reflected, refracted, scattered, and absorbed as they encounter solid objects. Sound is transmitted through solids in a complex array of types of elastic waves. Sound is characterized by its amplitude, frequency, phase, and direction of propagation. Sound strength is therefore location-dependent and is often quantified as a sound pressure level (Lp) in dB based on the root-meansquare (rms) sound-pressure (pS) value, where



pS Lp = 10 log10 ᎏ preference

FIG. 8-78 Generic depictions of average pressure at subsequent cross sections throughout a control valve. FLs selected for illustration are 0.9 and 0.63 for low and high recovery, respectively. Internal pressure in the high-recovery valve is shown as a dashed line for flashing conditions (p2 < pv) with pv = B.

(8-110)

1



2

(8-113)

For airborne sound, the reference pressure is 2 × 10−5 Pa (29 × 10−10 psi), which is nominally the human threshold of hearing at 1000 Hz. The corresponding sound pressure level is 0 dB. Conversation is about 50 dB, and a jackhammer operator is subject to 100 dB. Extreme levels such as a jet engine at takeoff might produce 140 dB at a distance of 3 m, which is a pressure amplitude of 200 Pa (29 × 10−3 psi). These examples demonstrate both the sensitivity and wide dynamic range of the human ear. Traveling sound waves carry energy. Sound intensity I is a measure of the power passing through a unit area in a specified direction and is related to pS. Measuring sound intensity in a process plant gives clues as to the location of the source. As one moves away from the source, the fact that the energy is spread over a larger area requires that sound pressure level decrease. For example, doubling one’s distance from a point source reduces the Lp by 6 dB. Viscous action from the induced fluid motion absorbs additional acoustic energy. However, in free air, this viscous damping is negligible over short distances (on the order of a meter). Noise is a group of sounds with many nonharmonic frequency components of varying amplitudes and random phase. The turbulence generated by a throttling valve creates noise. As a valve converts potential energy to heat, some of it becomes acoustic energy as an intermediate step. Valves handling large amounts of compressible fluid through a large pressure change create the most noise because more total power is being transformed. Liquid flows are noisy only

CONTROLLERS, FINAL CONTROL ELEMENTS, AND REGULATORS under special circumstances as will be seen in the next subsection. Due to the random nature of turbulence and the broad distribution of length and velocity scales of turbulent eddies, valve-generated sound is usually random, broad-spectrum noise. Total sound pressure level from two such statistically uncorrelated sources is (in dB): (pS1)2 + (pS2)2 Lp = 10 log10 ᎏᎏ (8-114) (preference)2 For example, two sources of equal strength combine to create an Lp that is 3 dB higher. While noise is annoying to listen to, the real reasons for being concerned about noise are its impact on people and equipment. Hearing loss can occur due to long-term exposure to moderately high or even short exposure to very high noise levels. The U.S. Occupational Safety and Health Act (OSHA) has specific guidelines for permissible levels and exposure times. The human ear has a frequency-dependent sensitivity to sound. When the effect on humans is the criteria, Lp measurements are weighted to account for the ear’s response. This socalled A-weighted scale is defined in ANSI S1.4 and is commonly reported as LpA. Figure 8-79 illustrates the difference between actual and perceived airborne sound pressure level. At sufficiently high levels, noise and the associated vibration can damage equipment. There are two approaches to fluid-generated noise control—source or path treatment. Path treatment means absorbing or blocking the transmission of noise after it has been created. The pipe itself is a barrier. The sound pressure level inside a standard schedule pipe is roughly 40–60 dB higher than on the outside. Thicker walled pipe reduces levels somewhat more, and adding acoustical insulation on the outside of the pipe reduces ambient levels up to 10 dB per inch of thickness. Since noise propagates relatively unimpeded inside the




FIG. 8-79



Valve-generated sound pressure level spectrums.

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pipe, barrier approaches require the entire downstream piping system to be treated in order to be totally effective. In-line silencers place absorbent material inside the flow stream, thus reducing the level of the internally propagating noise. Noise reductions up to 25 dB can be achieved economically with silencers. The other approach to valve noise problems is the use of quiet trim. Two basic strategies are used to reduce the initial production of noise—dividing the flow stream into multiple paths and using several flow resistances in series. Lp is proportional to mass flow and is dependent on vena contracta velocity. If each path is an independent source, it is easy to show from Eq. (8-114) that p2s is inversely proportional to the number of passages; additionally, smaller passage size shifts the predominate spectral content to higher frequencies, where structural resonance may be less of a problem. Series resistances or multiple stages can reduce maximum velocity and/or produce back pressure to keep jets issuing from multiple passages acting independently. While some of the basic principles are understood, predicting noise for a particle-flow passage requires some empirical data as a basis. Valve manufacturers have developed noise-prediction methods for the valves they build. ISA S75.17 is a public-domain methodology for standard (nonlow noise) valve types, although treatment of some multistage, multipath types is underway. Low-noise hardware consists of special cages in linear stem valves, perforated domes or plates and multichannel inserts in rotary valves, and separate devices that use multiple fixed restrictions. Cavitation and Flashing From the discussion on pressure recovery it was seen that the pressure at the vena contracta can be much lower than the downstream pressure. If the pressure on a liquid falls below its vapor pressure (pv ), the liquid will vaporize. Due to the effect of surface tension, this vapor phase will first appear as bubbles. These bubbles are carried downstream with the flow, where they collapse if the pressure recovers to a value above pv. This pressure-driven process of vapor-bubble formation and collapse is known as cavitation. Cavitation has three negative side effects in valves—noise and vibration, material removal, and reduced flow. The bubble-collapse process is a violent asymmetrical implosion that forms a high-speed microjet and induces pressure waves in the fluid. This hydrodynamic noise and the mechanical vibration that it can produce are far stronger than other noise-generation sources in liquid flows. If implosions occur adjacent to a solid component, minute pieces of material can be removed, which, over time, will leave a rough, cinderlike surface. The presence of vapor in the vena contracta region puts an upper limit on the amount of liquid that will pass through a valve. A mixture of vapor and liquid has a lower density than the liquid alone. While Eq. (8-108) is not applicable to two-phase flows because pressure changes are redistributed due to varying density and the two phases do not necessarily have the same average velocity, it does suggest that lower density reduces total mass flow rate. Figure 8-80 illustrates a typical flow-rate-to-pressure-drop relationship. As with compressible gas flow at a given p1, flow increases as p2 is decreased until the flow chokes (i.e., no additional fluid will pass). The transition between incompressible and choked flow is gradual because, within the convoluted flow passages of valves, the pressure is actually an uneven distribution at each cross section, and, consequently, vapor-formation zones increase gradually. In fact, isolated zones of bubble formation or incipient cavitation often occur at pressure drops well below that at which a reduction in flow is noticeable. The similarity between liquid and gas choking is not serendipitous; it is surmised that the two-phase fluid is traveling at the mixture’s sonic velocity in the throat when choked. Complex fluids with components having varying vapor pressures and/or entrained noncondensable gases (e.g., crude oil) will exhibit soft vaporization/implosion transitions. There are several methods to reduce cavitation or at least its negative side effects. Material damage is slowed by using harder materials and by directing the cavitating stream away from passage walls (e.g., with an angle body flowing down). Sometimes the system can be designed to place the valve in a higher p2 location or add downstream resistance, which creates back pressure. A low recovery valve has a higher minimum pressure for a given p2 and so is a means to eliminate the cavitation itself, not just its side effects. In Fig. 8-78, if pv < “B” neither valve will cavitate substantially. For pv > “B” but < “A,” the

8-76

PROCESS CONTROL and seating surfaces become nicked or worn. Leak passages across the seat-contact line, known as wire drawing, may form and become worse over time—even in hard metal seats under sufficiently high pressure differentials. Polymers used for seat and plug seals and internal static seals include: PTFE (polytetrafluoroethylene) and other fluorocarbons, polyethylene, nylon, polyether-ether-ketone, and acetal. Fluorocarbons are often carbon or glass-filled to improve mechanical properties and heat resistance. Temperature and chemical compatibility with the process fluid are the key selection criteria. Polymer-lined bearings and guides are used to decrease friction, which lessens dead band and reduces actuator force requirements. See Sec. 28, “Materials of Construction,” for properties. Packing forms the pressure-tight seal, where the stem protrudes through the pressure boundary. Packing is typically made from PTFE or, for high temperature, a bonded graphite. If the process fluid is toxic, more sophisticated systems such as dual packing, live-loaded, or a flexible metal bellows may be warranted. Packing friction can significantly degrade control performance. Pipe, bonnet, and internal-trim joint gaskets are typically a flat sheet composite. Gaskets intended to absorb dimensional mismatch are typically made from filled spiralwound flat stainless steel wire with PTFE or graphite filler. The use of asbestos in packing and gaskets has largely been eliminated. Flow Characteristics The relationship between valve flow and valve travel is called the valve-flow characteristic. The purpose of flow characterization is to make loop dynamics independent of load, so that a single controller tuning remains optimal for all loads. Valve gain is one factor affecting loop dynamics. In general, gain is the ratio of change in output to change in input. The input of a valve is travel (y) and the output is flow (w). Since pressure conditions at the valve can depend on flow (hence travel), valve gain is dw ∂w dCv ∂w dp1 ∂w dp2 ᎏ=ᎏᎏ+ᎏᎏ+ᎏᎏ dy ∂Cv dy ∂p1 dy ∂p2 dy

FIG. 8-80

Liquid flow rate versus pressure drop (assuming constant P1 and Pv).

high recovery valve will cavitate substantially, but the low recovery valve will not. Special anticavitation trims are available for globe/angle valves and more recently for some rotary valves. These trims use multiple contraction/expansion stages or other distributed resistances to boost FL to values sometimes near unity. If p2 is below pv, the two-phase mixture will continue to vaporize in the body outlet and/or downstream pipe until all liquid phase is gone, a condition known as flashing. The resulting huge increase in specific volume leads to high velocities, and any remaining liquid droplets acquire much of the higher vapor-phase velocity. Impingement of these droplets can produce material damage, but it differs from cavitation damage because it exhibits a smooth surface. Hard materials and directing the two-phase jets away from solid surfaces are means to avoid this damage. Seals, Bearings, and Packing Systems In addition to their control function, valves often need to provide shutoff. ANSI B16.104, FCI 70-2 (1991), and IEC 534-4 all recognize six standard classifications and define their as-shipped qualification tests. Class I is an amount agreed to by user and supplier with no test needed. Classes II, III, and IV are based on an air test with maximum leakage of 0.5 percent, 0.1 percent, and 0.01 percent of rated capacity, respectively. Class V restricts leakage to 5 × 10−6 ml of water per second per mm of port diameter per bar differential. Class VI allows 0.15 to 6.75 ml per minute of air to escape depending on port size; this class implies the need for interference-fit elastomeric seals. With the exception of Class V, all classes are based on standardized pressure conditions that may not represent actual conditions. Therefore, it is difficult to estimate leakage in service. Leakage normally increases over time as seals

(8-115)

An inherent valve flow characteristic is defined as the relationship between flow rate and travel, under constant pressure conditions. Since the last two terms in Eq. (8-115) are zero in this case, the inherent characteristic is necessarily also the relationship between flow coefficient and travel. Figure 8-81 shows three common inherent characteristics. A linear characteristic has a constant slope, meaning the inherent valve gain is a constant. The most popular characteristic is equal-percentage, which gets its name from the fact that equal changes in travel produce equal-percentage changes in the existing flow coefficient. In other words, the slope of the curve is proportional to Cv or equivalently that inherent valve gain is proportional to flow. The equal-percentage characteristic can be expressed mathematically by




 

y Cv(y) = (rated Cv) exp ᎏ − 1 ln R rated y

(8-116)

This expression represents a set of curves parameterized by R. Note that Cv (y = 0) equals (rated Cv)/R rather than zero; real equalpercentage characteristics deviate from theory at some small travel to meet shutoff requirements. An equal-percentage characteristic provides perfect compensation for a process that has gain inversely proportional to flow (e.g., liquid pressure). Quick opening does not have a standardized mathematical definition. Its shape arises naturally from high-capacity plug designs used in on/off service globe valves. Frequently, pressure conditions at the valve will change with flow rate. This so-called process influence [the last two terms on the right hand side of Eq. (8-115)] combine with inherent gain to express the installed valve gain. The flow-versus-travel relationship for a specific set of conditions is called the installed flow characteristic. Typically, valve ∆p decreases with load, since pressure losses in the piping system increase with flow. Figure 8-82 illustrates how allocation of total system head to the valve influences the installed flow characteristics. For a linear or quick-opening characteristic, this transition toward a concave down shape would be more extreme. This effect of typical process pressure variation, which causes equal-percentage character-

CONTROLLERS, FINAL CONTROL ELEMENTS, AND REGULATORS

FIG. 8-81

Typical inherent flow characteristics.

istics to have fairly constant installed gain, is one reason the equalpercentage characteristic is the most popular. Due to clearance flow, flow force gradients, seal friction, and the like, flow cannot be throttled to an arbitrarily small value. Installed rangeability is the ratio of maximum to minimum controllable flow. The actuator and positioner, as well as the valve, influence the installed rangeability. Inherent rangeability is defined as the ratio of the largest to the smallest Cv within which the characteristic meets specified criteria (see ISA S75.11). The R value in the equalpercentage definition is a theoretical rangeability only. While high installed rangeability is desirable, it is also important not to oversize a valve; otherwise, turndown (ratio of maximum normal to minimum controllable flow) will be limited. Sliding stem valves are characterized by altering the contour of the plug when the port and plug determine the minimum (controlling) flow area. Passage area versus travel is also easily manipulated in characterized cage designs. Inherent rangeability varies widely, but typical values are 30 for contoured plugs and 20–50 for characterized cages. While these types of valves can be characterized, the degree to which manufacturers conform to the mathematical ideal is revealed by plotting measured Cv versus travel. Note that ideal equal percentage will plot as a straight line on a semilog graph. Custom characteristics that compensate for a specific process are possible. Rotary stem-valve designs are normally offered only in their naturally occurring characteristic, since it is difficult to appreciably alter this. If additional characterization is required, the positioner or controller may be characterized. However, these approaches are less direct, since it is possible for device nonlinearity and dynamics to distort the compensation.

8-77

Installed flow characteristic as a function of percent of total system head allocated to the control valve (assuming constant head pump, no elevation head loss, and an R equal 30 equal-percentage inherent characteristic).

FIG. 8-82

OTHER PROCESS VALVES In addition to the throttling control valve, other types of process valves are used to manipulate the process. Valves for On/Off Applications Valves are often required for service that is primarily nonthrottling in nature. Valves in this category, depending on the service requirements, may be of the same design as the types used for throttling control or, as in the case of gate valves, different in design. Valves in this category usually have tight shutoff when they are closed and low pressure drops when they are wide open. The on/off valve can be operated manually, such as by handwheel or lever; or automatically, with pneumatic or electric actuators. Batch Batch process operation is an application requiring on/off valve service. Here the valve is opened and closed to provide reactant, catalyst, or product to and from the batch reactor. Like the throttling control valve, the valve used in this service must be designed to open and close thousands of times. For this reason, valves used in this application are often the same valves used in continuous throttling applications. Ball valves are especially useful in batch operations. The ball valve has a straight-through flow passage that reduces pressure drop in the wide-open state and provides tight shutoff capability when closed. In addition, the segmented ball valve provides for shearing action between the ball and the ball seat that promotes closure in slurry service. Isolation A means for pressure-isolating control valves, pumps, and other piping hardware for installation and maintenance is another

8-78

PROCESS CONTROL

common application for an on/off valve. In this application, the valve is required to have tight shutoff so that leakage is stopped when the piping system is under repair. As the need to cycle the valve in this application is far less than that of a throttling control valve, the wear characteristics of the valve are less important. Also, because there are many required in a plant, the isolation valve needs to be reliable, simple in design and simple in operation. The gate valve, shown in Figure 8-83, is the most widely used valve in this application. The gate valve is composed of a gate-like disc that moves perpendicular to the flow stream. The disc is moved up and down by a threaded screw that is rotated to effect disc movement. Because the disc is large and at right angles to the process pressure, large seat loading for tight shutoff is possible. Wear produced by high seat loading during the movement of the disk prohibits the use of the gate valve for throttling applications. Pressure Relief Valves Definitions for pressure relief valves, relief valves, pilot-operated pressure relief valves and safety valves, are found in the ASME Boiler and Pressure Vessel Code, Section VIII, Division 1, “Rules for Construction of Pressure Vessels,” Paragraphs UG-125 and UG-126. The pressure-relief valve is an automatic pressure relieving device designed to open when normal conditions are exceeded and to close again when normal conditions are restored. Within this class there are relief valves, pilot operated pressure relief valves, and safety valves. Relief valves (see Fig. 8-84) have spring-loaded disks that close a main orifice against a pressure source. As pressure rises, the disk begins to rise off the orifice and a small amount of fluid passes through the valve. Continued rise in pressure above the opening pressure causes the disk to open the orifice in a proportional fashion. The main orifice reduces and closes when the pressure returns to the set pres-

FIG. 8-84

FIG. 8-83

Gate valve. (Courtesy Crane Valves.)

Relief valve. (Courtesy Teledyne Fluid Systems, Farris Engineering.)

sure. Additional sensitivity to over-pressure conditions can be improved by adding an auxiliary pressure relief valve (pilot) to the basic pressure relief valve. This combination is known as a pilot-operated pressure relief valve. The safety valve is similar to the relief valve except it is designed to open fully, or pop, with only a small amount of pressure over the rated limit. Conventional safety valves are sensitive to downstream pressure and may have unsatisfactory operating characteristics in variable back pressure applications. The balanced safety relief valve is available and minimizes the effect of downstream pressure on performance. Check Valves The purpose of a check valve is to allow relatively unimpeded flow in the desired direction but to prevent flow in the reverse direction. Two common designs are swing-type and lift-type check valves—the names of which denote the motion of the closure member. In the forward direction, flow forces overcome the weight of

CONTROLLERS, FINAL CONTROL ELEMENTS, AND REGULATORS the member or a spring to open the flow passage. With reverse pressure conditions, flow forces drive the closure member into the valve seat, thus providing shutoff. ADJUSTABLE SPEED PUMPS An alternative to throttling a process with a process-control valve and a fixed speed pump is by adjusting the speed of the process pump and not using a throttling control valve at all. Pump speed can be varied by using variable-speed prime movers such as turbines, motors with magnetic or hydraulic couplings, and electric motors. Each of these methods of modulating pump speed has its own strengths and weaknesses but all offer energy savings and dynamic performance advantages over throttling with a control valve. The centrifugal pump directly driven by a variable-speed electric motor is the most commonly used hardware combination for adjustable speed pumping. The motor is operated by an electronicmotor speed controller whose function is to generate the voltage or current waveform required by the motor to make the speed of the motor track the input command input signal from the process controller. The most popular form of motor speed control for adjustable-speed pumping is the voltage-controlled pulse-width-modulated (PWM) frequency synthesizer and AC squirrel-cage induction motor combination. The flexibility of application of the PWM motor drive and its 90 percent+ electrical efficiency along with the proven ruggedness of the traditional AC induction motor makes this combination popular. From an energy-consumption standpoint, the power required to maintain steady process flow with an adjustable-speed pump system (three-phase PWM drive and a squirrel-cage induction motor driving a centrifugal pump on water) is less than that required with a conventional control valve and a fixed speed pump. Figure 8-85 shows this to be the case for a system where 100 percent of the pressure loss is due to flow velocity losses. At 75 percent flow, Fig. 8-85 shows the constant speed-pump/control-valve use at a 10.1-kW rate where throttling with the adjustable speed pump and no control valve used at a 4.1-kW rate. This trend of reduced energy consumption is true for the entire range of flows, although amounts vary. From a dynamic-response standpoint, the adjustable speed pump has a dynamic characteristic that is more suitable in process-control applications than those characteristics of control valves. The small amplitude response of an adjustable speed pump does not contain the dead band or the dead time commonly found in the small amplitude response of the control valve. Nonlinearities associated with frictions in the valve and discontinuities in the pneumatic portion of the control-valve instrumentation are not present with electronic

8-79

variable-speed drive technology. As a result, process control with the adjustable speed pump does not exhibit limit cycles, problems related to low controller gain and generally degraded process loop performance caused by control valve nonlinearities. Unlike the control valve, the centrifugal pump has poor or nonexistent shutoff capability. A flow check valve or an automated on/off valve may be required to achieve shutoff requirements. This requirement may be met by automating an existing isolation valve in retrofit applications. REGULATORS A regulator is a compact device that maintains the process variable at a specific value in spite of disturbances in load flow. It combines the functions of the measurement sensor, controller, and final control element into one self-contained device. Regulators are available to control pressure, differential pressure, temperature, flow, liquid level, and other basic process variables. They are used to control the differential across a filter press, heat exchanger, or orifice plate. Regulators are used for monitoring pressure variables for redundancy, flow check, and liquid surge relief. Regulators may be used in gas blanketing systems to maintain a protective environment above any liquid stored in a tank or vessel as the liquid is pumped out. When the temperature of the vessel is suddenly cooled, the regulator maintains the tank pressure and protects the walls of the tank from possible collapse. Regulators are known for their fast dynamic response. The absence of time delay that often comes with more sophisticated control systems makes the regulator useful in applications requiring fast corrective action. Regulators are designed to operate on the process pressures in the pipeline without any other sources of energy. Upstream and downstream pressures are used to supply and exhaust the regulator. Exhausting is back to the downstream piping so that no contamination or leakage to the external environment occurs. This makes regulators useful in remote locations where power is not available or where external venting is not allowed. The regulator is limited to operating on processes with clean, nonslurry process fluids. The small orifice and valve assemblies contained in the regulator can plug and malfunction if the process fluid that operates the regulator is not sufficiently clean. Regulators are normally not suited to systems that require constant set point adjustment. Although regulators are available with capability to respond to remote set point adjustment, this feature adds complexity to the regulator and may be better addressed by a control-valvebased system. In the simplest of regulators, tuning of the regulator for best control is accomplished by changing a spring, an orifice, or a nozzle.

FIG. 8-85 Pressure, flow, and power for a throttling process using (1) a control valve and a constant speed pump and (2) an adjustable speed pump.

8-80

PROCESS CONTROL

Spring Main throttling valve Diaphragm

(b)

(a) FIG. 8-86

Regulators: (a) self-operated; (b) pilot-operated. (Courtesy Fisher-Rosemount.)

Self-Operated Regulators Self-operated regulators are the simplest form of regulator. This regulator (see Fig. 8-86a) is composed of a main throttling valve, a diaphragm or piston to sense pressure, and a spring. The self-contained regulator is completely operated by the process fluid, and no outside control lines or pilot stage is used. In general, self-operated regulators are simple in construction, easy to operate and maintain, and are usually stable devices. Except for some of the pitot tube types, self-operated regulators have very good dynamic response characteristics. This is because any change in the

FIG. 8-87

controlled variable registers directly and immediately upon the main diaphragm to produce a quick response to the disturbance. The disadvantage of the self-operated regulator is that it is not generally capable of maintaining a set point as load flow is increased. Because of the proportional nature of the spring and diaphragmthrottling effect, offset from set point occurs in the controlled variable as flow increases. Figure 8-87 shows a typical regulation curve for the self-contained regulator. Reduced set point offset with increasing load flow can be achieved

Pressure regulation curves for three regulator types.

PROCESS CONTROL AND PLANT SAFETY by adding a pitot tube to the self-operated regulator. The tube is positioned somewhere near the vena contracta of the main regulator valve. As flow though the valve increases, the measured feedback pressure from the pitot tube drops below the control pressure. This causes the main valve to open or boost more than it would if the static value of control pressure were acting on the diaphragm. The resultant effect keeps the control pressure closer to set point and thus prevents a large drop in process pressure during high-load-flow conditions. Figure 8-87 shows the improvement that the pitot-tube regulator provides over the regulator without the tube. A side effect of adding a pitot-tube method is that the response of the regulator can be slowed due to the restriction provided by the pitot tube. Pilot-Operated Regulators Another category of regulators uses a pilot stage to provide the load pressure on the main diaphragm. This pilot is a regulator itself that has the ability to multiply a small change in downstream pressure into a large change in pressure applied to the regulator diaphragm. Due to this high-gain feature, pilot-operated regulators can achieve a dramatic improvement in steady-state accuracy over that achieved with a self-operated regulator. Figure 8-87 shows for regulation at high flows the pilot-operated regulator is best of the three regulators shown.

8-81

The main limitation of the pilot-operated regulator is stability. When the gain in the pilot amplifier is raised too much, the loop can become unstable and oscillate or hunt. The two-path pilot regulator (see b) is also available. This regulator combines the effects of selfoperated and the pilot-operated styles and mathematically produces the equivalent of proportional plus reset control of the process pressure. Over-Pressure Protection Figure 8-87 shows a characteristic rise in control pressure that occurs at low or zero flow. This lockup tail is due to the effects of imperfect plug and seat alignment and the elastomeric effects of the main throttle valve. If for some reason the main throttle valve fails to completely shut off, or if the valve shuts off but the control pressure continues to rise for other reasons, the lockup tail could get very large, and the control pressure could rise to extremely high valves. Damage to the regulator or the downstream pressure volume could occur. To avoid this situation, some regulators are designed with a built-in over-pressure relief mechanism. Over-pressure relief circuits usually are composed of a spring-opposed diaphragm and valve assembly that vents the downstream piping when the control pressure rises above the set point pressure.

PROCESS CONTROL AND PLANT SAFETY GENERAL REFERENCES: Guidelines for Safe Automation of Chemical Processes, AIChE Center for Chemical Process Safety, New York, 1993.

Accidents in chemical plants make headline news, especially when there is loss of life or the general public is affected in even the slightest way. This increases the public’s concern and may lead to government action. The terms hazard and risk are defined as follows: • Hazard. A potential source of harm to people, property, or the environment • Risk. Possibility of injury, loss, or an environmental accident created by a hazard Safety is the freedom from hazards and thus the absence of any associated risks. Unfortunately, absolute safety cannot be realized. The design and implementation of safety systems must be undertaken with a view of two issues: • Regulatory. The safety system must be consistent with all applicable codes and standards as well as “generally accepted good engineering practices.” • Technical. Just meeting all applicable regulations and “following the crowd” does not relieve a company of its responsibilities. The safety system must work. The regulatory environment will continue to change. As of this writing, the key regulatory instrument is OSHA 29 CFR 1910.119 that pertains to process safety management within plants in which certain chemicals are present. In addition to government regulation, industry groups and professional societies are producing documents ranging from standards to guidelines. Instrument Society of America Standard S84.01, “Application of Safety Instrumented Systems for the Process Industries,” is in draft form at the date of this writing. The Guidelines for Safe Automation of Chemical Processes from the American Institute of Chemical Engineers’ Center for Chemical Process Safety (1993) provides a comprehensive coverage of the various aspects of safety, and, although short on specifics, it is very useful to operating companies developing their own specific safety practices (that is, it does not tell you what to do, but it helps you decide what is proper for your plant). The ultimate responsibility for safety rests with the operating company; OSHA 1910.119 is clear on this. Each company is expected to develop (and enforce) its own practices in the design, installation, testing, and maintenance of safety systems. Fortunately, some companies make these documents public. Monsanto’s Safety System Design Practices was published in its entirety in the proceedings of the International Symposium and Workshop on Safe Chemical Process Automation, Houston, Texas, September 27–29, 1994 (available from

the American Institute of Chemical Engineers’ Center for Chemical Process Safety). ROLE OF AUTOMATION IN PLANT SAFETY As microprocessor-based controls displaced hardwired electronic and pneumatic controls, the impact on plant safety has definitely been positive. When automated procedures replace manual procedures for routine operations, the probability of human errors leading to hazardous situations is lowered. The enhanced capability for presenting information to the process operators in a timely manner and in the most meaningful form increases the operator’s awareness of the current conditions in the process. Process operators are expected to exercise due diligence in the supervision of the process, and timely recognition of an abnormal situation reduces the likelihood that the situation will progress to the hazardous state. Figure 8-88 depicts the layers of safety protection in a typical chemical plant. Although microprocessor-based process controls enhance plant safety, their primary objective is efficient process operation. Manual

FIG. 8-88

Layers of safety protection in chemical plants.

8-82

PROCESS CONTROL

operations are automated to reduce variability, to minimize the time required, to increase productivity, and so on. Remaining competitive in the world market demands that the plant be operated in the best manner possible, and microprocessor-based process controls provide numerous functions that make this possible. Safety is never compromised in the effort to increase competitiveness, but enhanced safety is a by-product of the process-control function and is not a primary objective. By attempting to maintain process conditions at or near their design values, the process controls also attempt to prevent abnormal conditions from developing within the process. Although process controls can be viewed as a protective layer, this is really a by-product and not the primary function. Where the objective of a function is specifically to reduce risk, the implementation is normally not within the process controls. Instead, the implementation is within a separate system specifically provided to reduce risk. This system is generally referred to as the safety interlock system. As safety begins with the process design, an inherently safe process is the objective of modern plant designs. When this cannot be achieved, process hazards of varying severity will exist. Where these hazards put plant workers and/or the general public at risk, some form of protective system is required. Process safety management addresses the various issues, ranging from assessment of the process hazard to assuring the integrity of the protective equipment installed to cope with the hazard. When the protective system is an automatic action, it is incorporated into the safety interlock system, not within the process controls. INTEGRITY OF PROCESS CONTROL SYSTEMS Ensuring the integrity of process controls involves both hardware issues, software issues, and human issues. Of these, the hardware issues are usually the easiest to assess and the software issues the most difficult. The hardware issues are addressed by providing various degrees of redundancy, by providing multiple sources of power and/or an uninterruptible power supply, and the like. The manufacturers of process controls provide a variety of configuration options. Where the process is inherently safe and infrequent shutdowns can be tolerated, nonredundant configurations are acceptable. For more demanding situations, an appropriate requirement might be that no single component failure can render the process-control system inoperable. For the very critical situations, triple-redundant controls with voting logic might be appropriate. The difficulty is assessing what is required for a given process. Another difficulty is assessing the potential for human errors. If redundancy is accompanied with increased complexity, the resulting increased potential for human errors must be taken into consideration. Redundant systems require maintenance procedures that can correct problems in one part of the system while the remainder of the system is in full operation. When conducting maintenance in such situations, the consequences of human errors can be rather unpleasant. The use of programmable systems for process control present some possibilities for failures that do not exist in hard-wired electromechanical implementations. Probably the one of most concern is latent defects or “bugs” in the software, either the software provided by the supplier or the software developed by the user. The source of this problem is very simple. There is no methodology available that can be applied to obtain absolute assurance that a given set of software is completely free of defects. Increased confidence in a set of software is achieved via extensive testing, but no amount of testing results in absolute assurance that there are no defects. This is especially true of real-time systems, where the software can easily be exposed to a sequence of events that was not anticipated. Just because the software performs correctly for each event individually does not mean that it will perform correctly when two (or more) events occur at nearly the same time. This is further complicated by the fact that the defect may not be in the programming; it may be in how the software was designed to respond to the events.

The testing of any collection of software is made more difficult as the complexity of the software increases. Software for process control has progressively become more complex, mainly because the requirements have progressively become more demanding. To remain competitive in the world market, processes must be operated at higher production rates, within narrower operating ranges, closer to equipment limits, and so on. Demanding applications require sophisticated control strategies, which translate into more complex software. Even with the best efforts of both supplier and user, complex software systems are unlikely to be completely free of defects. CONSIDERATIONS IN IMPLEMENTATION OF SAFETY INTERLOCK SYSTEMS Where hazardous conditions can develop within a process, a protective system of some type must be provided. Sometimes these are in the form of process hardware such as pressure relief devices. However, sometimes logic must be provided for the specific purpose of taking the process to a state where the hazardous condition cannot exist. The term safety interlock system is normally used to designate such logic. The purpose of the logic within the safety interlock system is very different from the logic within the process controls. Fortunately, the logic within the safety interlock system is normally much simpler than the logic within the process controls. This simplicity means that a hardwired implementation of the safety interlock system is usually an option. Should a programmable implementation be chosen, this simplicity means that latent defects in the software are less likely to be present. Most safety systems only have to do simple things, but they must do them very, very well. The difference in the nature of process controls and safety interlock systems leads to the conclusion that these two should be physically separated (see Fig. 8-89). That is, safety interlocks should not be piggy-backed onto a process-control system. Instead, the safety interlocks should be provided by equipment, either hard-wired or programmable, that is dedicated to the safety functions. As the process controls become more complex, faults are more likely. Separation means that faults within the process controls have no consequences in the safety interlock system. Modifications to the process controls are more frequent than modifications to the safety interlock system. Therefore, physically separating the safety interlock system from the process controls provides the following benefits: 1. The possibility of a change to the process controls leading to an unintentional change to the safety interlock system is eliminated. 2. The possibility of a human error in the maintenance of the process controls having consequences for the safety interlock system is eliminated. 3. Management of change is simplified. 4. Administrative procedures for software-version control are more manageable. Separation also applies to the measurement devices and actuators. Although the traditional point of reference for safety interlock systems is a hard-wired implementation, a programmed implementation is an alternative. The potential for latent defects in software implementation is a definite concern. Another concern is that solid-state components are not guaranteed to fail to the safe state. The former is addressed by extensive testing; the latter is addressed by manufacturer-supplied and/or user-supplied diagnostics that are routinely executed by the processor within the safety interlock system. Although issues must be addressed in programmable implementations, the hard-wired implementations are not perfect either. Where a programmed implementation is deemed to be acceptable, the choice is usually a programmable logic controller (PLC) that is dedicated to the safety function. PLCs are programmed using the traditional relay ladder diagrams used for hard-wired implementations. The facilities for developing, testing, and troubleshooting PLCs are excellent. However, for PLCs used in safety interlock systems, administrative procedures must be developed and implemented to address the following issues:

PROCESS CONTROL AND PLANT SAFETY

FIG. 8-89

8-83

Total control system with parallel tasks.

1. Version controls for the PLC program must be implemented and rigidly enforced. Revisions to the program must be reviewed in detail and thoroughly tested before implementing in the PLC. The various versions must be clearly identified so that there can be no doubt as to what logic is provided by each version of the program. 2. The version of the program that is currently being executed by the PLC must be known with absolute certainty. It must simply not be possible for a revised version of the program undergoing testing to be downloaded to the PLC. Constant vigilance is required to prevent lapses in such administrative procedures. INTERLOCKS An interlock is a protective response initiated on the detection of a process hazard. The interlock system consists of the measurement devices, logic solvers, and final control elements that recognize the hazard and initiate an appropriate response. Most interlocks consist of one or more logic conditions that detect out-of-limit process conditions and respond by driving the final control elements to the safe states. For example, one must specify that a valve fails open or fails closed. The potential that the logic within the interlock could contain a defect or bug is a strong incentive to keep it simple. Within process plants, most interlocks are implemented with discrete logic, which means either hard-wired electromechanical devices or programmable logic controllers. Interlocks within process plants can be broadly classified as follows: 1. Safety interlocks. These are designed to protect the public, the plant personnel, and possibly the plant equipment from process hazards. 2. Process interlocks. These are designed to prevent process conditions that would unduly stress equipment (perhaps leading to minor damage), lead to off-specification product, and so on. Basically, the process interlocks address hazards whose consequences essentially lead to a monetary loss, possibly even a short plant shut-

down. The more serious hazards are addressed by the safety interlocks. Implementation of process interlocks within process control systems is perfectly acceptable. Furthermore, it is also permissible (and probably advisable) that responsible operations personnel be authorized to bypass or ignore a process. Safety interlocks must be implemented within the separate safety interlock system. Bypassing or ignoring safety interlocks by operations personnel is simply not permitted. When this is necessary for actions such as verifying that the interlock continues to be functional, such situations must be infrequent and incorporated into the design of the interlock. Safety interlocks are assigned to categories that reflect the severity of the consequences should the interlock fail to perform as intended. The specific categories used within a company is completely at the discretion of the company. However, most companies use categories that distinguish among the following: 1. Hazards that pose a risk to the public. Complete redundancy is normally required. 2. Hazards that could lead to injury of company personnel. Partial redundancy is often required (for example, redundant measurements but not redundant logic). 3. Hazards that could result in major equipment damage and consequently lengthy plant downtime. No redundancy is normally required for these, although redundancy is always an option. Situations that result in minor equipment damage that can be quickly repaired do not generally require a safety interlock; however, a process interlock might be appropriate. A process hazards analysis is intended to identify the safety interlocks required for a process and to provide the following for each: 1. The hazard that is to be addressed by the safety interlock. 2. The classification of the safety interlock. 3. The logic for the safety interlock, including inputs from measurement devices and outputs to actuators. The process hazards analysis is conducted by an experienced, multidisciplinary team that examines the process design, the plant equipment, operating procedures, and so on, using techniques such as

8-84

PROCESS CONTROL

hazard and operability studies (HAZOP), failure mode and effect analysis (FEMA), and others. The process hazards analysis recommends appropriate measures to reduce the risk, including (but not limited to) the safety interlocks to be implemented in the safety interlock system. Diversity is recognized as a useful approach to reduce the number of defects. The team that conducts the process hazards analysis does not implement the safety interlocks but provides the specifications for the safety interlocks to another organization for implementation. This organization reviews the specifications for each safety interlock, seeking clarifications as necessary from the process hazards analysis team and bringing any perceived deficiencies to the attention of the process hazards analysis team. Diversity can be used to further advantage in redundant configurations. Where redundant measurement devices are required, different technology can be used for each. Where redundant logic is required, one can be programmed and one hard-wired. Reliability of the interlock systems has two aspects: 1. It must react should the hazard arise. 2. It must not react when there is no hazard. Emergency shutdowns often pose risks in themselves, and therefore they should be undertaken only when truly appropriate. The need to avoid extraneous shutdowns is not just to avoid disruption in production operations. Although safety interlocks can inappropriately initiate shutdowns, the process interlocks are usually the major source of problems. It is possible to configure so many process interlocks that it is not possible to operate the plant.

TESTING As part of the detailed design of each safety interlock, written test procedures must be developed for the following purposes: 1. Assure that the initial implementation complies with the requirements defined by the process hazards analysis team. 2. Assure that the interlock (hardware, software, and I/O) continues to function as designed. The design must also determine the time interval on which this must be done. Often these tests must be done with the plant in full operation. The former is the responsibility of the implementation team and is required for the initial implementation and following any modification to the interlock. The latter is the responsibility of plant maintenance, with plant management responsible for seeing that it is done on the specified interval of time. Execution of each test must be documented, showing when it was done, by whom, and the results. Failures must be analyzed for possible changes in the design or implementation of the interlock. These tests must encompass the complete interlock system, from the measurement devices through the final control elements. Merely simulating inputs and checking the outputs is not sufficient. The tests must duplicate the process conditions and operating environments as closely as possible. The measurement devices and final control elements are exposed to process and ambient conditions and thus are usually the most likely to fail. Valves that remain in the same position for extended periods of time may stick in that position and not operate when needed. The easiest component to test is the logic; however, this is the least likely to fail.

Section 9

Process Economics*

F. A. Holland, D.Sc., Ph.D., Consultant in Heat Energy Recycling; Research Professor, University of Salford, England; Fellow, Institution of Chemical Engineers, London. (Section Editor) J. K. Wilkinson, M.Sc., Consultant Chemical Engineer; Fellow, Institution of Chemical Engineers, London.

INVESTMENT AND PROFITABILITY Annual Costs, Profits, and Cash Flows . . . . . . . . . . . . . . . . . . . . . . . . . . Contribution and Breakeven Charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . Capital Costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Depreciation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Traditional Measures of Profitability . . . . . . . . . . . . . . . . . . . . . . . . . . . . Time Value of Money . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 1: Capitalized Cost of Equipment . . . . . . . . . . . . . . . . . . . . Modern Measures of Profitability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 2: Net Present Value for Different Depreciation Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 3: Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Learning Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 4: Estimation of Average Cost of Incremental Units. . . . . . Risk and Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 5: Probability Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 6: Calculation of Probability of Meeting a Sales Demand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 7: Calculation of Probability of Sales . . . . . . . . . . . . . . . . . . Example 8: Calculation of Probability of Equipment Breakdowns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 9: Calculation of Probability of Machine Failures . . . . . . . . Example 10: Logistics Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 11: Parameter Method of Risk Analysis . . . . . . . . . . . . . . . . Example 12: Expected Value of Net Profit . . . . . . . . . . . . . . . . . . . . . Example 13: Evaluation of Investment Priorities Using Probability Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 14: Estimation of Probability of a Research and Development Program Breaking Even . . . . . . . . . . . . . Example 15: Utility Function Curve . . . . . . . . . . . . . . . . . . . . . . . . . . Inflation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 16: Effect of Inflation on Net Present Value . . . . . . . . . . . . Example 17: Effect of Fuel Cost on Project Economics . . . . . . . . . .

9-5 9-7 9-7 9-7 9-8 9-10 9-13 9-13 9-16 9-19 9-20 9-20 9-22 9-23 9-24 9-24 9-25 9-25 9-25 9-27 9-28 9-30 9-30 9-32 9-33 9-34 9-34 9-38

ACCOUNTING AND COST CONTROL Principles of Accounting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Financing Assets by Equity and Debt . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparative Company Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cost of Capital . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 18: Risk-Free Cost of Capital . . . . . . . . . . . . . . . . . . . . . . . . Management and Cost Accounting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Allocation of Overheads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 19: Overhead in Two Different Products . . . . . . . . . . . . . . . Inventory Evaluation and Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 20: Inventory Computations . . . . . . . . . . . . . . . . . . . . . . . . . Working Capital . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Budgets and Cost Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . MANUFACTURING-COST ESTIMATION General Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . One Main Product Plus By-Products. . . . . . . . . . . . . . . . . . . . . . . . . . . . Two Main Products. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 21: Calculation of Contributions to Income for Multiple Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Direct Manufacturing Costs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Indirect Manufacturing Costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rapid Manufacturing-Cost Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . Manufacturing Cost as a Basis for Product Pricing. . . . . . . . . . . . . . . . . Standard Costs for Budgetary Control. . . . . . . . . . . . . . . . . . . . . . . . . . . Example 22: Direct-Material-Mixture Variance . . . . . . . . . . . . . . . . . Contribution Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Valuation of Recycled Heat Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . FIXED-CAPITAL-COST ESTIMATION Total Capital Cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cost Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9-39 9-42 9-44 9-47 9-47 9-48 9-48 9-49 9-49 9-50 9-52 9-54

9-55 9-55 9-56 9-56 9-57 9-57 9-57 9-58 9-59 9-60 9-61 9-62

9-63 9-63

* The contribution of the late Mr. F. A. Watson, who was an author for the Sixth edition, is acknowledged. 9-1

9-2

PROCESS ECONOMICS

Types and Accuracy of Estimates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rapid Estimations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 23: Estimation of Total Installed Cost of a Plant . . . . . . . . . Equipment Costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Piping Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electrical and Instrumentation Estimation . . . . . . . . . . . . . . . . . . . . . . .

9-63 9-64 9-68 9-72 9-73 9-73

Auxiliaries Estimation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Use of Computers in Cost Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . Startup Costs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Construction Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Project Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Overseas Construction Costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9-74 9-75 9-76 9-76 9-77 9-78

Nomenclature and Units Symbol a A AA AD (ATR) b bc B c c cB cD cI cL c° C CCT CDS (CEQ)DEL CK CL CRS CRW CWS (CI) (COP)A (CR) (CRR) (CSR) d d (DR) (DCFRR) e e exp (a) (EMIP) fAF fAP

Definition Empirical constant in general equations Annual income or expenditure particularized by the subscript Annual allowances against tax other than for depreciation of fixed assets Annual writing down (depreciation) of fixed assets, allowable against tax Asset-turnover ratio defined by Eq. (9-131) Empirical constant in general equations Deviation from budgeted capacity Parametric constant in Eq. (9-204) Empirical constant in general equations Cost (or income) per unit of sales or production particularized by the subscript Cost of base heat supply Cost of heat energy delivered by a heat pump defined by Eq. (9-240) Cost of high-grade energy supplied to the compressor of a vapor compression heat pump Cost of labor per unit of production Standard cost particularized by the subscript Cost particularized by the subscript Installed cost of a cooling tower Installed cost of a demineralized-water system Delivered-equipment cost Capitalized cost of a fixed asset defined by Eq. (9-47) Cost of land and other nondepreciable assets Installed cost of a refrigeration system Installed cost of a river-water supply system Installed cost of a water-softening system Cost index as used in Eq. (9-246) Actual coefficient of performance of a heat pump Capital ratio defined by Eq. (9-134) Capital-rate-of-return ratio defined by Eq. (9-56) Contribution-sales ratio defined by Eq.(9-236) Empirical constant in general equations Symbol indicating differentiation Debt ratio defined by Eq. (9-139) Discounted-cash-flow rate of return Empirical constant in general equations Base of natural logarithms, 2.71828 Exponential function of a, ea Equivalent maximum investment period defined by Eq. (9-55) Annuity future-worth factor, i[(1 + i)n − 1]−1 Annuity present-worth factor, fAF(1 + i)n

Units Various $/year $/year $/year Dimensionless Various

Symbol fd fi fk fp f(x) F Fn i

Dimensionless Dimensionless Various

ie

$/unit

ir i′ I

$/unit $/GJ $/GJ $/hour $/hour

im

kn K ln (a) log (a) m

$ $ $

m

$ $

n N

$

N (NPV) p(x)

$ $ $ Dimensionless Dimensionless

(MSF)

P Pa Pb Pe

Year Year

P1

Dimensionless

Ps

Various Dimensionless Dimensionless Year−1 Various Dimensionless Dimensionless Year Dimensionless Dimensionless

Ps′ Pw (PBP) (PM) (PSR) q QD r R R° RB

Definition

Units

Discount factor, (1 + i)−n Compound-interest factor, (1 + i)n Capitalized-cost factor, fAP/i Piping-cost factor defined by Eq. (9-249) Distribution function of x variously defined Future value of a sum of money Sum of fd for Years 1 to n Interest rate per period, usually annual, often the cost of capital Effective interest rate defined by Eq. (9-111) Minimum acceptable interest rate defined by Eq. (9-107) Entrepreneurial-risk interest rate Nominal annual interest rate Value of inventory particularized by the subscript Constants in Eq. (9-81) Effective value of the first unit of production Logarithm to the base e of a Logarithm to the base 10 of a Number of interest periods due per year Number of units removed from inventory Measured-survival function defined by Eq. (9-106) Number of years, units, etc. Slope of the learning curve defined by Eq. (9-64) Number of inventory orders per year Net present value Probability of the variable having the value x Present value of a sum of money Production time worked Budgeted production

Dimensionless Dimensionless Dimensionless Dimensionless

Production efficiency defined by Eq. (9-216) Level of productive activity defined by Eq. (9-217) Actual production rate Book value of asset at the end of year s′ Budgeted working time Payback period defined by Eq. (9-30) Profit margin defined by Eq. (9-127) Profit-sales ratio defined by Eq. (9-235) Quantity defining the scale of operation Process-heat-rate requirement Fraction of range of the independent variable Production rate Standard production rate Breakeven production rate

Dimensionless $ Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless $ Various $/unit, time/ unit, etc. Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless $ Dimensionless $ Hour Standard hour Dimensionless Dimensionless Standard hour $ Hour Year Dimensionless Dimensionless Various GJ/hour Dimensionless Units/year Units/year Units/year

9-3

9-4

PROCESS ECONOMICS

Nomenclature and Units (Concluded ) Symbol R0 RS (ROA) (ROE) (ROI) s s′ s° S t tC tSU T U V W x x
X y y Y Y Y
z

Definition

Units

Scheduled production rate Sales rate Return on assets defined by Eq. (9-129) Return on equity defined by Eq. Eq. (9-130) Return on investment defined by Eq. (9-128) Scheduled number of productive years Number of productive years to date Sample standard deviation Scrap value of a depreciable asset Fractional tax rate payable on adjusted income Time taken to construct plant Time taken to start up plant Auxiliary variable defined by Eq. (9-92) Size of inventory order Variable cost of inventory order Power supplied at shaft of a heat pump General variable Mean value of x Cumulative production from startup Cumulative probability Operating time of a heat pump Cumulative average cost, production time, etc. Operating-labor rate in Eq. (9-204) Cumulative-average batch cost, etc. Standard score defined by Eq. (9-73)

Units/year Units/year Dimensionless Dimensionless Dimensionless

Various $ Dimensionless Years Years Various Units $/unit GJ/hour Various Units Dimensionless Hours/year $/unit, hour/ unit, etc. labor-hour/ton $/unit, etc. Dimensionless

Greek symbols α β β δ ∆ η η θ σ  φ φP ψ χ χ

Proportionality factor in Eq. (9-168) Proportionality factor in Eq. (9-171) Exponent in Eqs. (9-106) and (9-117) Symbol indicating partial differentiation Symbol indicating a difference of like quantities Contribution efficiency defined by Eq. (9-119) Margin of safety defined by Eq. (9-229) Time taken to produce a given amount of product Population standard deviation Symbol indicating a sum of like quantities Fractional increase in production rate Parameter defined with Eq. (9-254) Parameter defined with Eq. (9-241) Plant capacity in Eq. (9-204) Weight of product per unit of raw material Subscripts

A BD BL

Allowance against tax other than for capital depreciation Depreciation allowance shown in company balance sheet Within project boundary limits

Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Hour Various Dimensionless Dimensionless Dimensionless Dimensionless Tons/day Dimensionless

Symbol BOH CF CI DCF DME FC FE FGE FIFO FIN FME FOH GE GP IME INV IO IT IW L L LIFO max M ME N NCI NOH NNP NP OH P RM s′ S SAV ST SVOH TC TE TFE TVE U U VE VGE VME VOH W WC WAV 1, 2, j, n

Definition Budgeted overhead Cash flow after payment of tax and expenses Cash income after payment of expenses Discounted cash flow Direct manufacturing expense Fixed capital Fixed expense Fixed general expense On a first-in–first-out basis Financial-resources inventory Fixed manufacturing expense Fixed overhead General expense Gross profit Indirect manufacturing expense Inventory Inventory-orders cost Income tax payable Inventory working cost Labor-earnings index Lower-quartile value of the variable Last-in–first-out basis Maximum value Median value of the variable Manufacturing expense At agreed normal production rate Net cash income after payment of tax Overhead cost at agreed normal production rate Net profit after payment of tax Net profit before payment of tax Overhead cost Profit Raw material In the s′th productive year From sales and other income On a simple-average basis Steel-price index Semivariable overhead Total capital Total expense Total fixed expense Total variable expense Utilities Upper-quartile value of the variable Variable expense Variable general expense Variable manufacturing expense Variable overhead expense Weighted value Working capital On a weighted-average basis 1st, 2d, jth, nth item, year, etc.

GENERAL REFERENCES: Allen, D. H., Economic Evaluation of Projects, 3d ed., Institution of Chemical Engineers, Rugby, England, 1991. Aries, R. S. and R. D. Newton, Chemical Engineering Cost Estimation, McGraw-Hill, New York, 1955. Baasel, W. D., Preliminary Chemical Engineering Plant Design, 2d ed., Van Nostrand Reinold, New York, 1989. Barish, N. N. and S. Kaplan, Economic Analysis for Engineering and Management Decision Making, 2d ed., McGraw-Hill, New York, 1978. Bierman, H., Jr. and S. Smidt, The Capital Budgeting Decision, Economic Analysis and Financing of Investment Projects, 7th ed., Macmillan, London, 1988. Canada, J. R. and J. A. White, Capital Investment Decision: Analysis for Management and Engineering, 2d ed., Prentice Hall, Englewood Cliffs, NJ, 1980. Carsberg, B. and A. Hope, Business Investment Decisions under Inflation, Macdonald & Evans, London, 1976. Chemical Engineering (ed.), Modern Cost Engineering, McGraw-Hill, New York, 1979. Garvin, W. W., Introduction to Linear Programming, McGraw-Hill, New York, 1960. Gass, S. I., Linear Programming, McGraw-Hill, New York, 1985. Granger, C. W. J., Forecasting in Business and Economics, Academic Press, New York, 1980. Hackney, J. W. and K. K. Humphreys (ed.), Control and Management of Capital Projects, 2d ed., McGraw-Hill, New York, 1991. Happel, J., W. H. Kapfer, B. J. Blewitt, P. T. Shannon, and D. G. Jordan, Process Economics, American Institute of Chemical Engineers, New York, 1974. Hill, D. A. and L. E. Rockley, Secrets of Successful Financial Management, Heinemann, London, 1990. Holland, F. A., F. A. Watson, and J. K. Wilkinson, Introduction to Process Economics, 2d ed., Wiley, London, 1983. Humphreys, K. K. (ed.), Jelen’s Cost and Optimization Engineering, 3d ed., McGraw-Hill, New York, 1991. Institution of Chemical Engineers (ed.), A Guide to Capital Cost Estimation, Institution of Chemical Engineers, Rugby, England, 1988. Jordan, R. B., How to Use the Learning Curve, Materials Management Institute, Boston, 1965. Kharbanda, O. P. and E. A. Stallworthy, Capital Cost Estimating in the Process Indus-

tries, 2d ed., Butterworth-Heinemann, London, 1988. Kirkman, P. R., Accounting under Inflationary Conditions, 2d ed., Routledge, Chapman & Hall, London, 1978. Liddle, C. J. and A. M. Gerrard, The Application of Computers to Capital Cost Estimation, Institution of Chemical Engineers, Rugby, England, 1975. Loomba, N. P., Linear Programming, McGraw-Hill, New York, 1964. Merrett, A. J. and A. Sykes, The Finance and Analysis of Capital Projects, Longman, London, 1963. Merrett, A. J. and A. Sykes, Capital Budgeting and Company Finance, Longman, London, 1966. Ostwald, P. F., Engineering Cost Estimating, 3d ed., Prentice Hall, Englewood Cliffs, NJ, 1991. Park, W. R. and D. E. Jackson, Cost Engineering Analysis, 2d ed., Wiley, New York, 1984. Peters, M. S. and K. D. Timmerhaus, Plant Design and Economics for Chemical Engineers, 4th ed., McGraw-Hill, New York, 1991. Pilcher, R., Principles of Construction Management, 3d ed., McGraw-Hill, New York, 1992. Popper, H. (ed.), Modern Cost Estimating Techniques, McGraw-Hill, New York, 1970. Raiffa, H. and R. Schlaifer, Applied Statistical Decision Theory, Harper & Row (Harvard Business), New York, 1984. Ridge, W. J., Value Analysis for Better Management, American Management Association, New York, 1969. Rose, L. M., Engineering Investment Decisions: Planning under Uncertainty, Elsevier, Amsterdam, 1976. Rudd, D. F. and C. C. Watson, The Strategy of Process Engineering, Wiley, New York, 1968. Thorne, H. C. and J. B. Weaver (ed.), Investment Appraisal for Chemical Engineers, American Institute of Chemical Engineers, New York, 1991. Weaver, J. B., ‘Project Selection in the 1980’s’, Chem. Eng. News 37–46 (Nov. 2, 1981). Wells, G. L., Process Engineering with Economic Objectives, Wiley, New York, 1973. Wilkes, F. M., Capital Budgeting Techniques, 2d ed., Wiley, London, 1983. Wood, E. G., Costing Matters for Managers, Beekman Publications, London, 1977. Woods, D. R., Process Design and Engineering, Prentice Hall, Englewood Cliffs, NJ, 1993. Wright, M. G., Financial Management, McGraw-Hill, London, 1970.

NOMENCLATURE An attempt has been made to bring together most of the methods currently available for project evaluation and to present them in such a way as to make the methods amenable to modern computational techniques. To this end the practices of accountants and others have been reduced, where possible, to mathematical equations which are usually solvable with an electronic hand calculator equipped with scientific function keys. To make the equations suitable for use on high-speed computers an attempt has been made to devise a nomenclature which is suitable for machines using ALGOL, COBOL, or FORTRAN compilers. The number of letters and numbers used to define a variable has usually been limited to five. The letters are mnemonic in English wherever possible and are derived in two ways. First, when a standard accountancy phrase exists for a term, this has been abbreviated in capital letters and enclosed in parentheses, e.g., (ATR), for assets-toturnover ratio; (DCFRR), for discounted-cash-flow rate of return. Clearly, the parentheses are omitted when the letter group is used to define the variable name for the computer. Second, a general symbol is defined for a type of variable and is modified by a mnemonic subscript, e.g., an annual cash quantity ATC, annual total capital outlay, $/year. Clearly, the symbols are written on one line when the letter group is used to define a variable name for the computer. In other cases, when well-known standard symbols exist, they have been

adopted, e.g., z for the standard score as used in the normal distribution. Also, a, b, c, d, and e have been used to denote empirical constants and x and y to denote general variables where their use does not clash with other meanings of the same symbols. The coverage in this section is so wide that nomenclature has sometimes proved a problem which has required the use of primes, asterisks, and other symbols not universally acceptable in the naming of computer variables. However, it is realized that each individual will program only his or her preferred methods, which will release some symbols for other uses. Also, it is not difficult to replace a forbidden symbol by an acceptable one; e.g., cRM might be rendered CARM and PS′ as PSP by using A for asterisk and P for prime. For compilers which recognize only one alphabetical case, an extra prefix can be used to distinguish between uppercase and lowercase letters, for which purpose the letters U and L have been used only in a restricted way in the nomenclature. It is, of course, impossible to allow for all possible variations of equation requirements and machine capability, but it is hoped that the nomenclature in the table presented at the beginning of the section will prove adequate for most purposes and will be capable of logical extension to other more specialized requirements.

INVESTMENT AND PROFITABILITY In order to assess the profitability of projects and processes it is necessary to define precisely the various parameters. Annual Costs, Profits, and Cash Flows To a large extent, accountancy is concerned with annual costs. To avoid confusion with other costs, annual costs will be referred to by the letter A.

The revenue from the annual sales of product AS, minus the total annual cost or expense required to produce and sell the product ATE, excluding any annual provision for plant depreciation, is the annual cash income ACI: A CI = AS − A TE (9-1) 9-5

9-6

PROCESS ECONOMICS

Net annual cash income ANCI is the annual cash income ACI, minus the annual amount of tax AIT: A NCI = ACI − AIT

(9-2)

Taxable income is (ACI − AD − AA), where AD is the annual writingdown allowance and AA is the annual amount of any other allowances. A distinction is made between the writing-down allowance permissible for the computation of tax due, the actual depreciation in value of an asset, and the book depreciation in value of that asset as shown in the company position statement. There is no necessary connection between these values unless specified by law, although the first two or all three are often assigned the same value in practice. Some governments give cash incentives to encourage companies to build plants in otherwise unattractive areas. Neither AD nor AA involves any expenditure of cash, since they are merely book transactions. The annual amount of tax AIT is given by AIT = (A CI − AD − AA) t

(9-3)

where t is the fractional tax rate. The value of t is determined by the appropriate tax authority and is subject to change. For most developed countries the value of t is about 0.35 or 35 percent. The annual amount of tax AIT included in Eq. (9-2) does not necessarily correspond to the annual cash income ACI in the same year. The tax payments in Eq. (9-2) should be those actually paid in that year. In the United States, companies pay about 80 percent of the tax on estimated current-year earnings in the same year. In the United Kingdom, companies do not pay tax until at least 9 months after the end of the accounting period, which, for the most part, amounts to paying tax on the previous year’s earnings. When assessing projects for different countries, engineers should acquaint themselves with the tax situation in those countries. In modern methods of profitability assessment, cash flows are more meaningful than profits, which tend to be rather loosely defined. The net annual cash flow after tax is given by A CF = A NCI − ATC

(9-4)

where ATC is the annual expenditure of capital, which is not necessarily zero after the plant has been built. For example, working capital, plant additions, or modifications may be required in future years. The total annual expense ATE required to produce and sell a product can be written as the sum of the annual general expense AGE and the annual manufacturing cost or expense A ME: A TE = A GE + A ME

(9-5)

Annual general expense A GE arises from the following items: adminis-

tration, sales, shipping of product, advertising and marketing, technical service, research and development, and finance. The terms gross annual profit A GP and net annual profit ANP are commonly used by accountants and misused by others. Normally, both A GP and A NP are calculated before tax is deducted. Gross annual profit A GP is given by A GP = AS − A ME − A BD

A NP = AGP − AGE

(9-7)

Equation (9-7) can also be written as A NP = ACI − A BD

(9-8)

Net annual profit after tax ANNP can be written as A NNP = A NCI − ABD

(9-9)

The relationships among the various annual costs given by Eqs. (9-1) through (9-9) are illustrated diagrammatically in Fig. 9-1. The top half of the diagram shows the tools of the accountant; the bottom half, those of the engineer. The net annual cash flow ACF, which excludes any provision for balance-sheet depreciation ABD, is used in two of the more modern methods of profitability assessment: the net-presentvalue (NPV) method and the discounted-cash-flow-rate-of-return (DCFRR) method. In both methods, depreciation is inherently taken care of by calculations which include capital recovery. Annual general expense AGE can be written as the sum of the fixed and variable general expenses: AGE = AFGE + AVGE

(9-10)

Similarly, annual manufacturing expense AME can be written as the sum of the fixed and variable manufacturing expenses: AME = AFME + A VME

(9-11)

A variable expense is considered to be one which is directly proportional to the rate of production RP or of sales RS as is most appropriate to the case under consideration. Unless the variation in finishedproduct inventory is large when compared with the total production over the period in question, it is usually sufficiently accurate to consider RP and RS to be represented by the same-numerical-value R units of sale or production per year. A fixed expense is then considered to be one which is not directly proportional to R, such as overhead charges. Fixed expenses are not necessarily constant but may be sub-

Relationship between annual costs, annual profits, and cash flows for a project. ABD = annual depreciation allowance; ACF = annual net cash flow after tax; ACI = annual cash income; AGE = annual general expense; AGP = annual gross profit; AIT = annual tax; AME = annual manufacturing cost; ANCI = annual net cash income; ANNP = annual net profit after taxes; ANP = annual net profit; AS = annual sales; ATC = annual total cost; (DCFRR) = discounted-cash-flow rate of return; (NPV) = net present value. FIG. 9-1

(9-6)

where ABD is the balance-sheet annual depreciation charge, which is not necessarily the same as AD used in Eq. (9-3) for tax purposes. Net annual profit ANP is simply

INVESTMENT AND PROFITABILITY

9-7

Breakeven chart showing relationship between contribution and fixed expense. FIG. 9-4

FIG. 9-2

Conventional breakeven chart.

ject to stepwise variation at different levels of production. Some authors consider such steps as included in a semivariable expense, which is less amenable to mathematical analysis than the above division of expenses. Contribution and Breakeven Charts These can be used to give valuable preliminary information prior to the use of the more sophisticated and time-consuming methods based on discounted cash flow. If the sales price per unit of sales is cS and the variable expense is cVE per unit of production, Eq. (9-7) can be rewritten as ANP = R(cS − cVE) − AFE

(9-12)

where R(cS − cVE) is known as the annual contribution. The net annual profit is zero at an annual production rate RB = AFE/(cS − cVE)

(9-13)

where RB is the breakeven production rate. Breakeven charts can be plotted in any of the three forms shown in Figs. 9-2, 9-3, and 9-4. The abscissa shown as annual sales volume R is also frequently plotted as a percentage of the designed production or sales capacity R0. In the case of ships, aircraft, etc., it is then called the percentage utilization. The percentage margin of safety is defined as 100(R0 − RB)/R0. A decrease in selling price cS will decrease the slope of the lines in Figs. 9-2, 9-3, and 9-4 and increase the required breakeven value RB for a given level of fixed expense AFE. Capital Costs The total capital cost CTC of a project consists of the fixed-capital cost CFC plus the working-capital cost CWC, plus the cost of land and other nondepreciable costs CL:

CTC = CFC + CWC + CL

TABLE 9-1

FIG. 9-3

Breakeven chart showing fixed expense as a burden cost.

(9-14)

The project may be a complete plant, an addition to an existing plant, or a plant modification. The working-capital cost of a process or a business normally includes the items shown in Table 9-1. Since working capital is completely recoverable at any time, in theory if not in practice, no tax allowance is made for its depreciation. Changes in working capital arising from varying trade credits or payroll or inventory levels are usually treated as a necessary business expense except when they exceed the tax debt due. If the annual income is negative, additional working capital must be provided and included in the ATC for that year. The value of land and other nondepreciables often increases over the working life of the project. These are therefore not treated in the same way as other capital investments but are shown to have made a (taxable) profit or loss only when the capital is finally recovered. Working capital may vary from a very small fraction of the total capital cost to almost the whole of the invested capital, depending on the process and the industry. For example, in jewelry-store operations, the fixed capital is very small in comparison with the working capital. On the other hand, in the chemical-process industries, the working capital is likely to be in the region of 10 to 20 percent of the value of the fixed-capital investment. Depreciation The term “depreciation” is used in a number of different contexts. The most common are: 1. A tax allowance 2. A cost of operation 3. A means of building up a fund to finance plant replacement 4. A measure of falling value In the first case, the annual taxable income is reduced by an annual depreciation charge or allowance which has the effect of reducing the annual amount of tax payable. The annual depreciation charge is merely a book transaction and does not involve any expenditure of cash. The method of determining the annual depreciation charge must be agreed to by the appropriate tax authority. In the second case, depreciation is considered to be a manufacturing cost in the same way as labor cost or raw-materials cost. However, Working-Capital Costs

Raw materials for plant startup Raw-materials, intermediate, and finished-product inventories Cost of handling and transportation of materials to and from stores Cost of inventory control, warehouse, associated insurance, security arrangements, etc. Money to carry accounts receivable (i.e., credit extended to customers) less accounts payable (i.e., credit extended by suppliers) Money to meet payrolls when starting up Readily available cash for emergencies Any additional cash required to operate the process or business

9-8

PROCESS ECONOMICS

it is more difficult to estimate a depreciation cost per unit of product than it is to do so for labor or raw-materials costs. In the net-presentvalue (NPV) and discounted-cash-flow-rate-of-return (DCFRR) methods of measuring profitability, depreciation, as a cost of operation, is implicitly accounted for. (NPV) and (DCFRR) give measures of return after a project has generated sufficient income to repay, among other things, the original investment and any interest charges that the invested money would otherwise have brought into the company. In the third case, depreciation is considered as a means of providing for plant replacement. In the rapidly changing modern chemicalprocess industries, many plants will never be replaced because the processes or products have become obsolete during their working life. Management should be free to invest in the most profitable projects available, and the creation of special-purpose funds may hinder this. However, it is desirable to designate a proportion of the retained income as a fund from which to finance new capital projects. These are likely to differ substantially from the projects that originally generated the income. In the fourth case, a plant or a piece of equipment has a limited useful life. The primary reason for the decrease in value is the decrease in future life and the consequent decrease in the number of years for which income will be earned. At the end of its life, the equipment may be worth nothing, or it may have a salvage or scrap value S. Thus a fixed-capital cost CFC depreciates in value during its useful life of s years by an amount that is equal to (CFC − S). The useful life is taken from the startup of the plant. On the basis of straight-line depreciation, the average annual amount of depreciation AD over a service life of s years is given by AD = (CFC − S)/s

(9-15)

The book value after the first year P1 is given by P1 = CFC − AD

(9-16)

The book value at the end of a specified number of years s′ is given by Ps′ = CFC − s′AD

(9-17)

The principal use of a particular depreciation rate is for tax purposes. The permitted annual depreciation is subtracted from the annual income before the latter is taxed. The basis for depreciation in a particular case is a matter of agreement between the taxation authority and the company, in conformity with tax laws. Other commonly used methods of computing depreciation are the declining-balance method (also known as the fixed-percentage method) and the sum-of-years-digits method. On the basis of declining-balance (fixed-percentage) depreciation, the book value at the end of the first year is given by P1 = CFC(1 − r)

(9-18)

where r is a fraction to be agreed with the taxation authority. The book value at the end of specified number of years s′ is given by Ps′ = CFC(1 − r)s′

(9-19)

When the fraction r is chosen to be 2/s, i.e., twice the reciprocal of the service life s, the method is called the double-declining-balance method. The declining-balance method of depreciation allows equipment or plant to be depreciated by a greater amount during the earlier years than during the later years. This method does not allow equipment or plant to be depreciated to a zero value at the end of the service life. On the basis of sum-of-years-digits depreciation, the annual amount of depreciation for a specified number of years s′ for a plant of fixed-capital cost CFC, scrap value S, and service life s is given by s − s′ + 1 ADs′ = ᎏᎏ (CFC − S) 1+2+3+⋅⋅⋅+s





(9-20)













AD CFC − S = ᎏ fAF where fAF is the annuity future-worth factor given by

or

(9-24)

fAF = i / [(1 + i)s − 1] In the sinking-fund method of depreciation, the effect of interest is to make the annual decrease of the book value of the equipment or plant less in the early than in the later years with consequent higher tax due in the earlier years when recovery of the capital is most important. It is preferable not to think of annual depreciation as a contribution to a fund to replace equipment at the end of its life but as part of the difference between the revenue and the expenditure, which difference is tax-free. Some of the preceding methods of computing depreciation are not allowed by taxation authorities in certain countries. When calculating depreciation, it is necessary to obtain details of the methods and rates permitted by the appropriate authority and to use the information provided. Figure 9-5 shows the fall in book value with time for a piece of equipment having a fixed-capital cost of $120,000, a useful life of 10 years, and a scrap value of $20,000. This fall in value is calculated by using (1) straight-line depreciation, (2) double-declining depreciation, and (3) sum-of-years-digits depreciation. Traditional Measures of Profitability Rate-of-Return Methods Although traditional rate-of-return methods have the advantage of simplicity, they can yield very misleading results. They are based on the relation Percent rate of return = [(annual profit)/(invested capital)]100

(9-25)

Since different meanings are ascribed to both annual profit and invested capital in Eq. (9-25), it is important to define the terms precisely. The invested capital may refer to the original total capital investment, the depreciated investment, the average investment, the current value of the investment, or something else. The annual profit may refer to the net annual profit before tax A NP, the net annual profit after tax A NNP, the annual cash income before tax A CI, or the annual cash income after tax A NCI. The fractional interest rate of return based on the net annual profit after tax and the original investment is i = ANNP /CTC

(9-26)

which can be written in terms of Eq. (9-9) as

Equation (9-20) can also be rewritten in the form 2(s − s′ + 1) ADs′ = ᎏᎏ (CFC − S) s(s + 1)

It can be shown that the book value at the end of a particular year s′ is 1 + 2 + ⋅ ⋅ ⋅ + (s − s′) Ps′ = 2 ᎏᎏᎏ (CFC − S) + S (9-22) s(s + 1) The sum-of-years-digits depreciation allows equipment or plant to be depreciated by a greater amount during the early years than during the later years. A fourth method of computing depreciation (now seldom used) is the sinking-fund method. In this method, the annual depreciation AD is the same for each year of the life of the equipment or plant. The series of equal amounts of depreciation AD, invested at a fractional interest rate i and made at the end of each year over the life of the equipment or plant of s years, is used to build up a future sum of money equal to (CFC − S). This last is the fixed-capital cost of the equipment or plant minus its salvage or scrap value and is the total amount of depreciation during its useful life. The equation relating (CFC − S) and AD is simply the annual cost or payment equation, written either as (1 + i)s − 1 CFC − S = AD ᎏᎏ (9-23) i

i = (ANCI /CTC) − (ABD / CTC) (9-21)

(9-27)

where ABD is the balance-sheet annual depreciation. The main disadvantage of using Eq. (9-27) is that the fractional depreciation rate

INVESTMENT AND PROFITABILITY

FIG. 9-7 FIG. 9-5

Book value against time for various depreciation methods.

9-9

Effect of double-declining depreciation on rate of return for a proj-

ect.

Estimated salvage value of plant items S = $20,000 ABD /CTC is arbitrarily assessed. Its value will affect the fractional rate of return considerably and may lead to erroneous conclusions when making comparisons between different companies. This is particularly true when making international comparisons. Figures 9-6, 9-7, and 9-8 show the effect of the depreciation method on profit for a project described by the following data: Net annual cash income after tax ANCI = $25,500 in each of 10 years Fixed-capital cost CFC = $120,000

Working capital CWC = $10,000 Cost of land CL = $20,000 In Eq. (9-27), i can be taken either on the basis of the net annual cash income for a particular year or on the basis of an average net annual cash income over the length of the life of the project. The equations corresponding to Eq. (9-26) based on depreciated and average investment are given respectively as follows: and

Effect of straight-line depreciation on rate of return for a project. ABD = annual depreciation allowance; ANCI = annual net cash income after tax; ANNP = annual net profit after payment of tax; CTC = total capital cost.

i = A NNP /(Ps′ + CWC + CL)

(9-28)

i = 2A NNP /(CFC + S + 2CWC + 2CL)

(9-29)

FIG. 9-6

FIG. 9-8

project.

Effect of sum-of-years-digits depreciation on rate of return for a

9-10

PROCESS ECONOMICS

where Ps′ is the book value of the fixed-capital investment at the end of a particular year s′. If i is taken on the basis of average values for ANNP over the length of the project, an average value for the working capital CWC must be used. In Eqs. (9-28) and (9-29), the computations are based on unchanging values of the cost of land and other nondepreciable costs CL. This is unrealistic, since the value of land has a tendency to rise. In such circumstances, the accountancy principle of conservatism requires that the lowest valuation be adopted. Payback Period Another traditional method of measuring profitability is the payback period or fixed-capital-return period. Actually, this is really a measure not of profitability but of the time it takes for cash flows to recoup the original fixed-capital expenditure. The net annual cash flow after tax is given by A CF = A NCI − A TC (9-4) where ATC is the annual expenditure of capital, which is not necessarily zero after the plant has been built. The payback period (PBP) is the time required for the cumulative net cash flow taken from the startup of the plant to equal the depreciable fixed-capital investment (CFC − S). It is the value of s′ that satisfies

Annual Compound Interest It is more common to use compound interest, in which F and P are related by F = P(1 + i)n F = Pfi

or

where the compound-interest factor fi = (1 + i)n. Values for compound-interest factors are readily available in tables. The present value P of a future sum of money F is P = F/(1 + i)n F = P/fd

or



A CF = CFC − S

(9-30)

The payback-period method takes no account of cash flows or profits received after the breakeven point has been reached. The method is based on the premise that the earlier the fixed capital is recovered, the better the project. However, this approach can be misleading. Let us consider projects A and B, having net annual cash flows as listed in Table 9-2. Both projects have initial fixed-capital expenditures of $100,000. On the basis of payback period, project A is the more desirable since the fixed-capital expenditure is recovered in 3 years, compared with 5 years for project B. However, project B runs for 7 years with a cumulative net cash flow of $110,000. This is obviously more profitable than project A, which runs for only 4 years with a cumulative net cash flow of only $10,000. Time Value of Money A large part of business activity is based on money that can be loaned or borrowed. When money is loaned, there is always a risk that it may not be returned. A sum of money called interest is the inducement offered to make the risk acceptable. When money is borrowed, interest is paid for the use of the money over a period of time. Conversely, when money is loaned, interest is received. The amount of a loan is known as the principal. The longer the period of time for which the principal is loaned, the greater the total amount of interest paid. Thus, the future worth of the money F is greater than its present worth P. The relationship between F and P depends on the type of interest used. Table 9-3 gives examples of compound-interest factors and example compound-interest calculations. Simple Interest When simple interest is used, F and P are related by F = P(1 + ni) (9-31) where i is the fractional interest rate per period and n is the number of interest periods. Normally, the interest period is 1 year, in which case i is known as the effective interest rate. TABLE 9-2

Cash Flows for Two Projects

(9-34) (9-35)

where the discount factor fd is fd = 1 / fi = 1/[(1 + i)n] Values for the discount factors are readily available in tables which show that it will take 7.3 years for the principal to double in amount if compounded annually at 10 percent per year and 14.2 years if compounded annually at 5 percent per year. For the case of different annual fractional interest rates (i1,i2, . . . ,in in successive years), Eq. (9-32) should be written in the form F = P(1 + i1)(1 + i2)(1 + i3) ⋅ ⋅ ⋅ (1 + in)

s′ = (PBP)

s′ = 0

(9-32) (9-33)

(9-36)

Short-Interval Compound Interest If interest payments become due m times per year at compound interest, mn payments are required in n years. The nominal annual interest rate i′ is divided by m to give the effective interest rate per period. Hence, F = P[1 + (i′/m)]mn

(9-37)

It follows that the effective annual interest i is given by i = [1 + (i′/m)]m − 1

(9-38)

The annual interest rate equivalent to a compound-interest rate of 5 percent per month (i.e., i′/m = 0.05) is calculated from Eq. (9-38) to be i = (1 + 0.05)12 − 1 = 0.796, or 79.6 percent/year Continuous Compound Interest As m approaches infinity, the time interval between payments becomes infinitesimally small, and in the limit Eq. (9-37) reduces to F = P exp (i′n)

(9-39)

A comparison of Eqs. (9-32) and (9-39) shows that the nominal interest rate i′ on a continuous basis is related to the effective interest rate i on an annual basis by exp (i′n) = (1 + i)n

(9-40)

Numerically, the difference between continuous and annual compounding is small. In practice, it is probably far smaller than the errors in the estimated cash-flow data. Annual compound interest conforms more closely to current acceptable accounting practice. However, the small difference between continuous and annual compounding may be significant when applied to very large sums of money. Let us suppose that $100 is invested at a nominal interest rate of 5 percent. We then compute the future worth of the investment after 2 years and also compute the effective annual interest rate for the following kinds of interest: (1) simple, (2) annual compound, (3) monthly compound, (4) daily compound, and (5) continuous compound. The following tabulation shows the results of the calculations, along with the appropriate equation to be used:

Cash flows ACF Year

Project A

Project B

0 1 2 3 4 5 6 7  A CF

$100,000 50,000 30,000 20,000 10,000 0 0 0 $ 10,000

$100,000 0 10,000 20,000 30,000 40,000 50,000 60,000 $110,000

3 years

5 years

Payback period (PBP)

Interest type

Equation

Future worth F

Effective rate i, %

Equation

1 2 3 4 5

(9-31) (9-32) (9-37) (9-37) (9-39)

$110.000 $110.250 $110.495 $110.516 $110.517

5 5 5.117 5.1267 5.1271

(9-31) (9-38) (9-38) (9-38) (9-38)

When computing the effective annual rate for continuous compounding, the first term of Eq. (9-38), [1 + (i′/m)]m, approaches ei′ as m approaches infinity.

TABLE 9-3

Compound Interest Factors* (For examples demonstrating use see end of table.) Single payment

Uniform annual series

Single payment

Uniform annual series

Compoundamount factor

Presentworth factor

Sinkingfund factor

Capitalrecovery factor

Compoundamount factor

Presentworth factor

Compoundamount factor

Presentworth factor

Sinkingfund factor

Capitalrecovery factor

Compoundamount factor

Presentworth factor

n

Given P, to find F (1 + i)n

Given F, to find P 1 ᎏn (1 + i)

Given F, to find A i ᎏᎏ (1 + i)n − 1

Given P, to find A i(1 + i)n ᎏᎏ (1 + i)n − 1

Given A, to find F (1 + i)n − 1 ᎏᎏ i

Given A, to find P (1 + i)n − 1 ᎏᎏ i(1 + i)n

Given P, to find F (1 + i)n

Given F, to find P 1 ᎏn (1 + i)

Given F, to find A i ᎏᎏ (1 + i)n − 1

Given P, to find A i(1 + i)n ᎏᎏ (1 + i)n − 1

Given A, to find F (1 + i)n − 1 ᎏᎏ i

Given A, to find P (1 + i)n − 1 ᎏᎏ i(1 + i)n

1 2 3 4 5

1.050 1.103 1.158 1.216 1.276

0.9524 .9070 .8638 .8227 .7835

1.00000 0.48780 .31721 .23201 .18097

1.05000 0.53780 .36721 .28201 .23097

1.000 2.050 3.153 4.310 5.526

0.952 1.859 2.723 3.546 4.329

1.060 1.124 1.191 1.262 1.338

0.9434 .8900 .8396 .7921 .7473

1.00000 0.48544 .31411 .22859 .17740

1.06000 0.54544 .37411 .28859 .23740

1.000 2.060 3.184 4.375 5.637

0.943 1.833 2.673 3.465 4.212

1 2 3 4 5

6 7 8 9 10

1.340 1.407 1.477 1.551 1.629

.7462 .7107 .6768 .6446 .6139

.14702 .12282 .10472 .09069 .07940

.19702 .17282 .15472 .14069 .12950

6.802 8.142 9.549 11.027 12.578

5.076 5.786 6.463 7.108 7.722

1.419 1.504 1.594 1.689 1.791

.7050 .6651 .6274 .5919 .5584

.14336 .11914 .10104 .08702 .07587

.20336 .17914 .16104 .14702 .13587

6.975 8.394 9.897 11.491 13.181

4.917 5.582 6.210 6.802 7.360

6 7 8 9 10

11 12 13 14 15

1.710 1.796 1.886 1.980 2.079

.5847 .5568 .5303 .5051 .4810

.07039 .06283 .05646 .05102 .04634

.12039 .11283 .10646 .10102 .09634

14.207 15.917 17.713 19.599 21.579

8.306 8.863 9.394 9.899 10.380

1.898 2.012 2.133 2.261 2.397

.5268 .4970 .4688 .4423 .4173

.06679 .05928 .05296 .04758 .04296

.12679 .11928 .11296 .10758 .10296

14.972 16.870 18.882 21.015 23.276

7.887 8.384 8.853 9.295 9.712

11 12 13 14 15

16 17 18 19 20

2.183 2.292 2.407 2.527 2.653

.4581 .4363 .4155 .3957 .3769

.04227 .03870 .03555 .03275 .03024

.09227 .08870 .08555 .08275 .08024

23.657 25.840 28.132 30.539 33.066

10.838 11.274 11.690 12.085 12.462

2.540 2.693 2.854 3.026 3.207

.3936 .3714 .3503 .3305 .3118

.03895 .03544 .03236 .02962 .02718

.09895 .09544 .09236 .08962 .08718

25.673 28.213 30.906 33.760 36.786

10.106 10.477 10.828 11.158 11.470

16 17 18 19 20

21 22 23 24 25

2.786 2.925 3.072 3.225 3.386

.3589 .3418 .3256 .3101 .2953

.02800 .02597 .02414 .02247 .02095

.07800 .07597 .07414 .07247 .07095

35.719 38.505 41.430 44.502 47.727

12.821 13.163 13.489 13.799 14.094

3.400 3.604 3.820 4.049 4.292

.2942 .2775 .2618 .2470 .2330

.02500 .02305 .02128 .01968 .01823

.08500 .08305 .08128 .07968 .07823

39.993 43.392 46.996 50.816 54.865

11.764 12.042 12.303 12.550 12.783

21 22 23 24 25

26 27 28 29 30

3.556 3.733 3.920 4.116 4.322

.2812 .2678 .2551 .2429 .2314

.01956 .01829 .01712 .01605 .01505

.06956 .06829 .06712 .06605 .06505

51.113 54.669 58.403 62.323 66.489

14.375 14.643 14.898 15.141 15.372

4.549 4.822 5.112 5.418 5.743

.2198 .2074 .1956 .1846 .1741

.01690 .01570 .01459 .01358 .01265

.07690 .07570 .07459 .07358 .07265

59.156 63.706 68.528 73.640 79.058

13.003 13.211 13.406 13.591 13.765

26 27 28 29 30

31 32 33 34 35

4.538 4.765 5.003 5.253 5.516

.2204 .2099 .1999 .1904 .1813

.01413 .01328 .01249 .01176 .01107

.06413 .06328 .06249 .06176 .06107

70.761 75.299 80.064 85.067 90.320

15.593 15.803 16.003 16.193 16.374

6.088 6.453 6.841 7.251 7.686

.1643 .1550 .1462 .1379 .1301

.01179 .01100 .01027 .00960 .00897

.07179 .07100 .07027 .06960 .06897

84.802 90.890 97.343 104.184 111.435

13.929 14.084 14.230 14.368 14.498

31 32 33 34 35

40 45 50

7.040 8.985 11.467

.1420 .1113 .0872

.00828 .00626 .00478

.05828 .05626 .05478

120.800 159.700 209.348

17.159 17.774 18.256

10.286 13.765 18.420

.0972 .0727 .0543

.00646 .00470 .00344

.06646 .06470 .06344

154.762 212.744 290.336

15.046 15.456 15.762

40 45 50

55 60 65 70 75

14.636 18.679 23.840 30.426 38.833

.0683 .0535 .0419 .0329 .0258

.00367 .00283 .00219 .00170 .00132

.05367 .05283 .05219 .05170 .05132

272.713 353.584 456.798 588.529 756.654

18.633 18.929 19.161 19.343 19.485

24.650 32.988 44.145 59.076 79.057

.0406 .0303 .0227 .0169 .0126

.00254 .00188 .00139 .00103 .00077

.06254 .06188 .06139 .06103 .06077

394.172 533.128 719.083 967.932 1,300.949

15.991 16.161 16.289 16.385 16.456

55 60 65 70 75

80 85 90 95 100

49.561 63.254 80.730 103.035 131.501

.0202 .0158 .0124 .0097 .0076

.00103 .00080 .00063 .00049 .00038

.05103 .05080 .05063

971.229 1,245.087 1,594.607 2,040.694 2,610.025

19.596 19.684 19.752 19.806 19.848

105.796 141.579 189.465 253.546 339.302

.0095 .0071 .0053 .0039 .0029

.00057 .00043 .00032 .00024 .00018

.06057 .06043 .06032 .06024 .06018

1,746.600 2,342.982 3,141.075 4,209.104 5,638.368

16.509 16.549 16.579 16.601 16.618

80 85 90 95 100

5% Compound Interest Factors

.05038

n

6% Compound Interest Factors

9-11

9-12

PROCESS ECONOMICS TABLE 9-3

Compound Interest Factors (Concluded )

Examples of Use of Table and Factors Given: $2500 is invested now at 5 percent. Required: Accumulated value in 10 years (i.e., the amount of a given principal). Solution:

F = P(1 + i)n = $2500 × 1.0510 Compound-amount factor = (1 + i)n = 1.0510 = 1.629 F = $2500 × 1.629 = $4062.50

Given: $19,500 will be required in 5 years to replace equipment now in use. Required: With interest available at 3 percent, what sum must be deposited in the bank at present to provide the required capital (i.e., the principal which will amount to a given sum)? 1 1 Solution: P = F ᎏn = $19,500 ᎏ5 (1 + i) 1.03 Present-worth factor = 1/(1 + i)n = 1/1.035 = 0.8626 P = $19,500 × 0.8626 = $16,821 Given: $50,000 will be required in 10 years to purchase equipment. Required: With interest available at 4 percent, what sum must be deposited each year to provide the required capital (i.e., the annuity which will amount to a given fund)? i 0.04 Solution: A = F ᎏᎏ = $50,000 ᎏᎏ (1 + i)n − 1 1.0410 − 1 i 0.04 Sinking-fund factor = ᎏᎏ = ᎏᎏ = 0.08329 (1 + i)n − 1 1.0410 − 1 A = $50,000 × 0.08329 = $4,164 Given: $20,000 is invested at 10 percent interest. Required: Annual sum that can be withdrawn over a 20-year period (i.e., the annuity provided by a given capital). i(1 + i)n 0.10 × 1.1020 Solution: A = P ᎏᎏ = $20,000 ᎏᎏ (1 + i)n − 1 1.1020 − 1 i(1 + i)n 0.10 × 1.1020 Capital-recovery factor = ᎏᎏ = ᎏᎏ = 0.11746 (1 + i)n − 1 1.1020 − 1 A = $20,000 × 0.11746 = $2349.20 Given: $500 is invested each year at 8 percent interest. Required: Accumulated value in 15 years (i.e., amount of an annuity). (1 + i)n − 1 1.0815 − 1 Solution: F = A ᎏᎏ = $500 ᎏᎏ i 0.08 (1 − i)n − 1 1.0815 − 1 Compound-amount factor = ᎏᎏ = ᎏᎏ = 27.152 i 0.08 F = $500 × 27.152 = $13,576 Given: $8000 is required annually for 25 years. Required: Sum that must be deposited now at 6 percent interest. (1 + i)n − 1 1.0625 − 1 Solution: P = A ᎏᎏ = $8000 ᎏᎏ i(1 + i)n 0.06 × 1.0625 (1 + i)n − 1 1.0625 − 1 Present-worth factor = ᎏᎏ = ᎏᎏ = 12.783 i(1 + i)n 0.06 × 1.0625 P = $8000 × 12.783 = $102,264 *Factors presented for two interest rates only. By using the appropriate formulas, values for other interest rates may be calculated.

INVESTMENT AND PROFITABILITY Annual Cost or Payment A series of equal annual payments A invested at a fractional interest rate i at the end of each year over a period of n years may be used to build up a future sum of money F. These relations are given by (1 + i)n − 1 F = A ᎏᎏ (9-41) i





F = A/fAF

or

(9-42)

where the annuity future-worth factor is fAF = i / [(1 + i)n − 1] Values for fAF are readily available in tables. Equation (9-41) can be combined with Eq. (9-34) to yield (1 + i)n − 1 P = A ᎏᎏ i(1 + i)n





P = A/ fAP

(9-43)

fAP = [i(1 + i)n]/[(1 + i)n − 1] Values for fAP are also available in tables. Alternatively, the annual payment A required to build up a future sum of money F with a present value of P is given by A = FfAF

(9-45) (9-46)

A = PfAP

Equation (9-41) represents the future sum of a series of uniform annual payments that are invested at a stated interest rate over a period of years. This procedure defines an ordinary annuity. Other forms of annuities include the annuity due, in which payments are made at the beginning of the year instead of at the end; and the deferred annuity, in which the first payment is deferred for a definite number of years. Capitalized Cost A piece of equipment of fixed-capital cost CFC will have a finite life of n years. The capitalized cost of the equipment CK is defined by (CK − CFC)(1 + i)n = CK − S (9-47) CK is in excess of CFC by an amount which, when compounded at an annual interest rate i for n years, will have a future worth of CK less the salvage or scrap value S. If the renewal cost of the equipment remains constant at (CFC − S) and the interest rate remains constant at i, then CK is the amount of capital required to replace the equipment in perpetuity. Equation (9-47) may be rewritten as S (1 + i)n CK = CFC − ᎏn ᎏᎏ (9-48) (1 + i) (1 + i)n − 1



or





CK = (CFC − Sfd) f k

(9-49)

where fd is the discount factor and fk, the capitalized-cost factor, is fk = [(1 + i)n]/[(1 + i)n − 1] Values for each factor are available in tables. Example 1: Capitalized Cost of Equipment A piece of equipment has been installed at a cost of $100,000 and is expected to have a working life of 10 years with a scrap value of $20,000. Let us calculate the capitalized cost of the equipment based on an annual compound-interest rate of 5 percent. Therefore, we substitute values into Eq. (9-48) to give $20,000 (1 + 0.05)10 CK = $100,000 − ᎏᎏ ᎏᎏ (1 + 0.05)10 (1 + 0.05)10 − 1



the venture and the greater the justification for putting the capital at risk. A profitability estimate is an attempt to quantify the desirability of taking this risk. The ways of assessing profitability to be considered in this section are (1) discounted-cash-flow rate of return (DCFRR), (2) net present value (NPV) based on a particular discount rate, (3) equivalent maximum investment period (EMIP), (4) interest-recovery period (IRP), and (5) discounted breakeven point (DBEP). Cash Flow Let us consider a project in which CFC = $1,000,000, CWC = $90,000, and CL = $10,000. Hence, CTC = $1,100,000 from Eq. (9-14). If all this capital expenditure occurs in Year 0 of the project, then ATC = $1,100,000 in Year 0 and −ATC = −$1,100,000. From Eq. (9-4), it is seen that any capital expenditure makes a negative contribution to the net annual cash flow ACF. Let us consider another project in which the fixed-capital expenditure is spread over 2 years, according to the following pattern: CFC = CFC0 + CFC1

(9-44)

where P is the present worth of the series of future equal annual payments A and the annuity present-worth factor is





CK = [$100,000 − ($20,000/1.62889)](2.59009) CK = $227,207

Modern Measures of Profitability An investment in a manufacturing process must earn more than the cost of capital for it to be worthwhile. The larger the additional earnings, the more profitable

9-13

Year 0

Year 1

CFC0 = $400,000 CL = 10,000 ATC = 410,000

CFC1 = $600,000 CWC = 90,000 ATC = 690,000

In the final year of the project, the working capital and the land are recovered, which in this case cost a total of $100,000. Thus, in the final year of the project, ATC = −$100,000 and −ATC = +$100,000. From Eq. (9-4), it is seen that any capital recovery makes a positive contribution to the net annual cash flow. During the development and construction stages of a project, ACI and AIT are both zero in Eqs. (9-2) and (9-4). For this period, the cash flow for the project is negative and is given by ACF = −ATC

(9-50)

Figure 9-9 shows the cash-flow stages in a project. The expenditure during the research and development stage is normally relatively small. It will usually include some preliminary process design and a market survey. Once the decision to go ahead with the project has been taken, detailed process-engineering design will commence, and the rate of expenditure starts to increase. The rate is increased still further when equipment is purchased and construction gets under way. There is no return on the investment until the plant is started up. Even during startup, there is some additional expenditure. Once the plant is operating smoothly, an inflow of cash is established. During the early stages of a project, there may be a tax credit because of the existence of expenses without corresponding income. Discounted Cash Flow The present value P of a future sum of money F is given by P = Ffd (9-51) where fd = 1/(1 + i)n, the discount factor. Values for this factor are readily available in tables. For example, $90,909 invested at an annual interest rate of 10 percent becomes $100,000 after 1 year. Similarly, $38,554 invested at 10 percent becomes $100,000 after 10 years. Thus, cash flow in the early years of a project has a greater value than the same amount in the later years of a project. Therefore, it pays to receive money as soon as possible and to delay paying out money for as long as possible. Time is taken into account by using the annual discounted cash flow ADCF, which is related to the annual cash flow ACF and the discount factor fd by (9-52) ADCF = A CF fd Thus, at the end of any year n, (ADCF)n = (ACF)n / (1 + i)n The sum of the annual discounted cash flows over n years,  ADCF, is known as the net present value (NPV) of the project: n

(NPV) =  (ADCF)n 0

(9-53)

9-14

PROCESS ECONOMICS

FIG. 9-9

Effect of discount rate on cash flows.

The value of (NPV) is directly dependent on the choice of the fractional interest rate i. An interest rate can be selected to make (NPV) = 0 after a chosen number of years. This value of i is found from n (ACF)n (ACF)0 (ACF)1 +ᎏ +⋅⋅⋅+ᎏ =0 (9-54) 0 (ADCF)n = ᎏ (1 + i)0 (1 + i)1 (1 + i)n Equation (9-54) may be solved for i either graphically or by an iterative trial-and-error procedure. The value of i given by Eq. (9-54) is known as the discounted-cash-flow rate of return (DCFRR). It is also known as the profitability index, true rate of return, investor’s rate of return, and interest rate of return. Cash-Flow Curves Figure 9-9 shows the cash-flow stages in a project together with their discounted-cash-flow values for the data given in Table 9-4. In addition to cash-flow and discounted-cash-flow curves, it is also instructive to plot cumulative-cash-flow and cumulative-discounted-cash-flow curves. These are shown in Fig. 9-10 for the data in Table 9-4. The cost of capital may also be considered as the interest rate at which money can be invested instead of putting it at risk in a manufacturing process. Let us consider the process data listed in Table 9-4 and plotted in Fig. 9-10. If the cost of capital is 10 percent, then the appropriate discounted-cash-flow curve in Fig. 9-10 is abcdef. Up to point e, or 8.49 years, the capital is at risk. Point e is the discounted breakeven point (DBEP). At this point, the manufacturing process TABLE 9-4

has paid back its capital and produced the same return as an equivalent amount of capital invested at a compound-interest rate of 10 percent. Beyond the breakeven point, the capital is no longer at risk and any cash flow above the horizontal baseline, ADCF = 0, is in excess of the return on an equivalent amount of capital invested at a compoundinterest rate of 10 percent. Thus, the greater the area above the baseline, the more profitable the process. When (NPV) and (DCFRR) are computed, depreciation is not considered as a separate expense. It is simply used as a permitted writingdown allowance to reduce the annual amount of tax in accordance with the rules applying in the country of earning. The tax payable is deducted in accordance with Eq. (9-2) in the year in which it is paid, which may differ from the year in which the corresponding income was earned. A (DCFRR) of, say, 15 percent implies that 15 percent per year will be earned on the investment, in addition to which the project generates sufficient money to repay the original investment plus any interest payable on borrowed capital plus all taxes and expenses. It is not normally possible to make a comprehensive assessment of profitability with a single number. The shape of the cumulative-cashflow and cumulative-discounted-cash-flow curves both before and after the breakeven point is an important factor. D. H. Allen [Chem. Eng., 74, 75–78 (July 3, 1967)] accounted for the shape of the cumulative-undiscounted-cash-flow curve up to the

Annual Cash Flows and Discounted Cash Flows for a Project Discounted at 10%

Discounted at 20%

Discounted at 25%

Year

ACF, $

 ACF, $

fd

ADCF, $

 ADCF, $

fd

A DCF, $

 ADCF, $

fd

A DCF, $

 ADCF, $

0 1 2 3 4

−10,000 −30,000 −60,000 −750,000 −150,000

−10,000 −40,000 −100,000 −850,000 −1,000,000

1.00000 0.90909 0.82645 0.75131 0.68301

−10,000 −27,273 −49,587 −563,483 −102,452

−10,000 −37,273 −86,860 −650,343 −752,795

1.00000 0.83333 0.69444 0.57870 0.48225

−10,000 −25,000 −41,666 −434,025 −72,338

−10,000 −35,000 −76,666 −510,691 −583,029

1.00000 0.80000 0.64000 0.51200 0.40960

−10,000 −24,000 −38,400 −384,000 −61,440

−10,000 −34,000 −72,400 −456,400 −517,840

5 6 7 8 9

+200,000 +300,000 +400,000 +400,000 +360,000

−800,000 −500,000 −100,000 +300,000 +660,000

0.62092 0.56447 0.51316 0.46651 0.42410

+124,184 +169,341 +205,264 +186,604 +152,676

−628,611 −459,270 −254,006 −67,402 +85,274

0.40188 0.33490 0.27908 0.23257 0.19381

+80,376 +100,470 +111,632 +93,028 +69,772

−502,653 −402,183 −290,551 −197,523 −127,751

0.32768 0.26214 0.20972 0.16777 0.13422

+65,536 +78,642 +83,888 +67,108 +48,319

−452,304 −373,662 −289,774 −222,666 −174,347

10 11 12 13 14

+320,000 +280,000 +240,000 +240,000 +400,000

+980,000 +1,260,000 +1,500,000 +1,740,000 +2,140,000

0.38554 0.35049 0.31863 0.28966 0.26333

+123,373 +98,137 +76,471 +69,518 +105,332

+208,647 +306,784 +383,255 +452,773 +558,105

0.16151 0.13459 0.11216 0.09346 0.07789

+51,683 +37,685 +26,918 +22,430 +31,156

−76,068 −38,383 −11,465 +10,965 +42,121

0.10737 0.08590 0.06872 0.05498 0.04398

+34,358 +24,052 +16,493 +13,195 +17,592

−139,989 −115,937 −99,444 −86,249 −68,657

NOTE: ACF is net annual cash flow, ADCF is net annual discounted cash flow, fd is discount factor at stated interest,  ACF is cumulative cash flow, and  ADCF is cumulative discounted cash flow.

INVESTMENT AND PROFITABILITY

FIG. 9-10

Effect of discount rate on cumulative cash flows.

breakeven point e0 in Fig. 9-10 by using a parameter known as the equivalent maximum investment period (EMIP), which is defined as area (a0 to e0) (EMIP) = ᎏᎏ for ACF ≤ 0 (9-55) ( ACF)max where the area (a0 to e0) refers to the area below the horizontal baseline ( ACF = 0) on the cumulative-cash-flow curve in Fig. 9-10. The sum ( ACF)max is the maximum cumulative expenditure on the project, which is given by point d0 in Fig. 9-10. (EMIP) is a time in years. It is the equivalent period during which the total project debt would be outstanding if it were all incurred at one instant and all repaid at one instant. Clearly, the shorter the (EMIP), the more attractive the project. Allen accounted for the shape of the cumulative-cash-flow curve

FIG. 9-11

9-15

beyond the breakeven point by using a parameter known as the interest-recovery period (IRP). This is the time period (illustrated in Fig. 9-11) that makes the area (e0 to f0) above the horizontal baseline equal to the area (a0 to e0) below the horizontal baseline on the cumulativecash-flow curve. C. G. Sinclair [Chem. Process. Eng., 47, 147 (1966)] has considered similar parameters to the (EMIP) and (IRP) based on a cumulativediscounted-cash-flow curve. Consideration of the cash-flow stages in Fig. 9-10 shows the factors that can affect the (EMIP) and (IRP). If the required capital investment is increased, it is necessary to increase the rate of income after startup for the (EMIP) to remain the same. In order to have the (EMIP) small, it is necessary to keep the research and development, design, and construction stages short.

Cumulative cash flow against time, showing interest recovery period.

9-16

PROCESS ECONOMICS ACF = $110,000/year − (−$100,000/year)

Example 2: Net Present Value for Different Depreciation Methods The following data describe a project. Revenue from annual sales and the total annual expense over a 10-year period are given in the first three columns of Table 9-5. The fixed-capital investment CFC is $1,000,000. Plant items have a zero salvage value. Working capital CWC is $90,000, and cost of land CL is $10,000. There are no tax allowances other than depreciation; i.e., AA is zero. The fractional tax rate t is 0.50. We shall calculate for these data the net present value (NPV) for the following depreciation methods and discount factors: a. Straight-line, 10 percent b. Straight-line, 20 percent c. Double-declining, 10 percent d. Sum-of-years-digits, 10 percent e. Straight-line, 10 percent; income tax delayed for 1 year In addition, we shall calculate the discounted-cash-flow rate of return (DCFRR) with straight-line depreciation. a. We begin the calculations for this example by finding the total capital cost CTC for the project from Eq. (9-14). Here, CTC = $1,100,000. In Year 0, this amount is the same as the net annual capital expenditure ATC and is listed in Table 9-5. The annual rate of straight-line depreciation of the fixed-capital investment CFC, from $1,000,000 at startup to a salvage value S, of zero at the end of a productive life s of 10 years, is given by AD = (CFC − S)/s AD = ($1,000,000 − $0)/10 years = $100,000/year The annual cash income ACI for Year 1, when AS = $400,000 per year and ATE = $100,000 per year, is, from Eq. (9-1), $300,000 per year. Values for subsequent years are calculated in the same way and listed in Table 9-4. Annual amount of tax AIT for Year 1, when ACI = $300,000 per year, AD = $100,000 per year, AA = $0 per year, and t = 0.5, is found from Eq. (9-3) to be AIT = [($300,000 − $100,000 − $0)/year](0.5) = $100,000/year Values for subsequent years are calculated in the same way and listed in Table 9-4. Net annual cash flow (after tax) ACF for Year 0, when ACI = $0 per year, AIT = $0 per year, and ATC = $1,100,000 per year, is found from Eq. (9-4) to be ACF = $0/year − $1,100,000/year = −$1,100,000/year Net annual cash flow (after tax) ACF for Year 1, when ACI = $300,000 per year, AIT = $100,000 per year, and ATC = $0 per year, is found from Eqs. (9-2) and (9-4) to be ACF = $200,000/year − $0/year = $200,000/year Values for the years up to and including Year 9 are calculated in the same way and listed in Table 9-5. At the end of Year 10, the working capital (CWC = $90,000) and the cost of land (CL = $10,000) are recovered, so that the annual expenditure of capital ATC in Year 10 is −$100,000 per year. Hence, the net annual cash flow (after tax) for Year 10 must reflect this recovery. By using Eq. (9-4),

TABLE 9-5

= $210,000/year The net annual discounted cash flow ADCF for Year 1, when ACF = $200,000 per year and fd = 0.90909 (for i = 10 percent), is found from Eq. (9-52) to be ADCF = ($200,000/year)(0.90909) = $181,820/year Values for subsequent years are calculated in the same way and listed in Table 9-5. The net present value (NPV) is found by summing the values of ADCF for each year, as in Eq. (9-53). The net present value is found to be $276,210, as given by the final entry in Table 9-5. b. The same procedure is used for i = 20 percent. The discount factors to be used in a table similar to Table 9-5 must be those for 20 percent. The (NPV) is found to be −$151,020. c. The calculations are similar to those for subexample a except that depreciation is computed by using the double-declining method of Eq. (9-19). The net present value is found to be $288,530. d. Again, the calculations are similar to those for subexample a except that depreciation is computed by using the sum-of-years-digits method of Eq. (9-20). The net present value is found to be $316,610. e. The calculations follow the same procedure as for subexample a, but the annual amount of tax AIT is calculated for a particular year and then deducted from the annual cash income ACI for the following year. The net present value for Year 11 is found to be $341,980. The discounted-cash-flow rate of return (DCFRR) can readily be obtained approximately by interpolation of the (NPV) for i = 10 percent and i = 20 percent: (DCFRR) = 0.100 + [($276,210)(0.20 − 0.10)]/[$276,210 − (−$151,020)] (DCFRR) = 0.164, or 16.4 percent The calculation of (DCFRR) usually requires a trial-and-error solution of Eq. (9-57), but rapidly convergent methods are available [N. H. Wild, Chem. Eng., 83, 153–154 (Apr. 12, 1976)]. For simplicity linear interpolation is often used. A comparison of the (NPV) values for a 10 percent discount factor shows clearly that double-declining depreciation is more advantageous than straightline depreciation and that sum-of-years-digits depreciation is more advantageous than the double-declining method. However, a significant advantage is obtained by delaying the payment of tax for 1 year even with straight-line depreciation.

This example is a simplified one. The cost of the working capital is assumed to be paid for in Year 0 and returned in Year 10. In practice, working capital increases with the production rate. Thus there may be an annual expenditure on working capital in a number of years subsequent to Year 0. Except in loss-making years, this is usually treated as an expense of the process. In loss-making years the cash injection for working capital is included in the ATC for that year. Analysis of Techniques Both the (NPV) and the (DCFRR) methods are based on discounted cash flows and in that sense are vari-

Annual Cash Flows, Straight-Line Depreciation, and 10 Percent Discount Factor Before tax

After tax

Year

AS, $

ATE, $

ACI, $

AD + AA, $

AIT, $

ATC, $

ACF, $

fd

ADCF, $

(NPV), $

0 1 2 3

0 400,000 500,000 500,000

0 100,000 100,000 110,000

0 300,000 400,000 390,000

0 100,000 100,000 100,000

0 200,000 300,000 290,000

0 100,000 150,000 145,000

+1,100,000 0 0 0

−1,100,000 200,000 250,000 245,000

1.0000 0.90909 0.82645 0.75131

−1,100,000 181,820 206,610 184,070

−1,100,000 −918,180 −711,570 −527,500

4 5 6 7

500,000 520,000 520,000 520,000

120,000 130,000 130,000 140,000

380,000 390,000 390,000 380,000

100,000 100,000 100,000 100,000

280,000 290,000 290,000 280,000

140,000 145,000 145,000 140,000

0 0 0 0

240,000 245,000 245,000 240,000

0.68301 0.62092 0.56447 0.51316

163,920 152,120 138,300 123,160

−363,580 −211,460 −73,160 +50,000

8 9 10

390,000 350,000 280,000

140,000 150,000 160,000

250,000 200,000 120,000

100,000 100,000 100,000

150,000 100,000 20,000

75,000 50,000 10,000

0 0 −100,000

175,000 150,000 210,000

0.46651 0.42410 0.38554

81,640 63,610 80,960

+131,640 +195,250 +276,210

AS = revenue from annual sales. ATE = total annual expense. ACI = annual cash income. AD + AA = annual depreciation and other tax allowances. ACI − AD − AA = taxable income. AIT = (ACI − AD − AA)t = amount of tax at t = 0.5.

ACI − AD − AA, $

ATC = total annual capital expenditure. ACF = ACI − AIT − ATC = net annual cash flow. fd = discount factor at 10%. ADCF = net annual discounted cash flow. (NPV) =  ADCF = net present value.

INVESTMENT AND PROFITABILITY ations of the same basic method. However, when ranking different projects on the basis of profitability, they can produce different results. Discounted-cash-flow rate of return (DCFRR) has the advantage of being unique and readily understood. However, when used alone, it gives no indication of the scale of the operation. The (NPV) indicates the monetary return, but unlike that of the (DCFRR) its value depends on the base year chosen for the calculation. Additional information is needed before its significance can be appreciated. However, when a company is considering investment in a portfolio of projects, individual (NPV)s have the advantage of being additive. This is not true of (DCFRR)s. Increasing use is being made of the capital-rate-of-return ratio (CRR), which is the net present value (NPV) divided by the maximum cumulative expenditure or maximum net outlay, −( ACF)max (CRR) = (NPV)/(ACF)max

for

ACF ≤ 0

(9-56)

The maximum net outlay is very important, since no matter how profitable a project is, the matter is academic if the company is unable to raise the money to undertake the project. An (NPV) or (DCFRR) estimation will be no better than the accuracy of the projected cash flows over the life of the project. Clearly, one is likely to predict cash flows more accurately for 2 or 3 years ahead than, say, for 9 or 10 years ahead. However, since the cash flows for the later years are discounted to a greater extent than the cash flows for the earlier years, the latter have less effect on the overall estimation. Nevertheless, the difficulty of predicting cash flows in later years and the inherent lack of confidence in these predictions are serious disadvantages of the (DCFRR) method. In this respect (NPV)s are more useful since they are calculated for each year of a project. Thus, a project with a favorable (NPV) in the early years is a promising one. One way of overcoming these disadvantages of the (DCFRR) method is to make estimates of the times required to reach certain values of (DCFRR). For example, how many years will it take to reach (DCFRR)s of 10 percent, 15 percent, 20 percent per year, etc.? Although (DCFRR) trial-and-error calculations and (NPV) calculations are tedious if done manually, computer programs which are suitable for programmable pocket calculators can readily be written to make calculations easier. It is possible for some projects to reach a stage at which repairs, replacements, etc., can exceed net earnings in a particular year. In this case the cumulative-discounted-cash-flow or net-present-value curve plotted against time has a genuine maximum. It is important when appraising by (NPV) and (DCFRR) not to consider the past in profitability estimations. Good money should never follow bad. It is unwise to continue to put money into a project if a more profitable project exists, even though this course may involve scrapping an expensive plant. Other considerations may, however, outweigh purely financial criteria in a particular case. No single value for a profitability estimate should be accepted without further consideration. An intelligent consideration of the cumulative-cash-flow and cumulative-discounted-cash-flow curves such as those shown in Fig. 9-10, together with experience and good judgment, is the best way of assessing the financial merit of a project. When considering future projects, top management will most likely require the discounted-cash-flow rate of return and the payback period. However, the estimators should also supply management with the following: Cumulative discounted-cash-flow or (NPV) curve for a discount rate of 10 percent per year or other agreed aftertax cost of capital Maximum net outlay, ( ACF)max, for ACF ≤ 0 Discounted breakeven point (DBEP) Plot of capital-return ratio (CRR) against time over the life of the project for a discount rate at the cost of capital Number of years to reach discounted-cash-flow rates of return of, say, 15 and 25 percent per year respectively Comparisons on the basis of time can be summarized by the following: Duration of the project Breakeven point (BEP)

9-17

Discounted breakeven point (DBEP) Equivalent maximum investment period (EMIP) Interest-recovery period (IRP) Payback period (PBP) Comparisons on the basis of cash can be summarized by the following: Maximum cumulative expenditure on the project, ( ACF)max, for ACF ≤ 0 Maximum discounted cumulative expenditure on the project Cumulative net annual cash flow  ACF Cumulative net annual discounted cash flow  ADCF or net present value (NPV) Capitalized cost CK Comparisons on the basis of interest can be summarized as (1) the net present value (NPV) and (2) the discounted-cash-flow rate of return (DCFRR), which from Eqs. (9-53) and (9-54) is given formally as the fractional interest rate i which satisfies the relationship n

(NPV) =  (ADCF)n = 0

(9-57)

0

When comparing project profitability, the ranking on the basis of net present value (NPV) may differ from that on the basis of discounted-cash-flow rate of return (DCFRR). Let us consider the data for two projects: Cost of capital

Project C

Project D

i, % 4 8 12 16

(NPV), $ +100,000 +41,000 −2,000 −32,000

(NPV), $ +62,000 +28,000 +10,000 −4,000

These (NPV) data are plotted against the cost of capital, as shown in Fig. 9-12. The discounted-cash-flow rate of return is the value of i that satisfies Eq. (9-5). From Fig. 9-12, (NPV) = 0 at a (DCFRR) of 11.8 percent for project C and 14.7 percent for project D. Thus, on the basis of (DCFRR), project D is more profitable than project C. The (NPV) of project C is equal to that of project D at a cost of capital i = 9.8 percent. If the cost of capital is greater than 9.8 percent, project D has the higher (NPV) and is, therefore, the more profitable. If the cost of capital is less than 9.8 percent, project C has the higher (NPV) and is the more profitable. Benefit of Early Cash Flows It pays to receive cash inflows as early as possible and to delay cash outflows as long as possible. Let us consider the net annual cash flows (after tax) ACF for projects E, F, and G, listed in Table 9-6. The cumulative annual cash flows  ACF and cumulative discounted annual cash flows  ADCF, using a discount of 10 percent for these projects, are also listed in Table 9-6. We notice that the cumulative annual cash flow for each project is +$1000.

FIG. 9-12

Effect of cost of capital on net present value.

9-18

PROCESS ECONOMICS

TABLE 9-6

Cash-Flow Data for Projects E, F, and G

TABLE 9-7

Cash-Flow Data for Projects H and I

Discounted at 10% Year

ACF, $

 ACF, $

0 1 2 3

−5000 +3000 +2000 +1000

−5000 −2000 0 +1000

0 1 2 3

−5000 +1000 +2000 +3000

−5000 −4000 −2000 +1000

0 1 2 3

−5000 +2000 +2000 +2000

−5000 −3000 −1000 +1000

Discounted at 10%

fd

ADCF, $

 ADCF = (NPV), $

1.0000 0.90909 0.82645 0.75131

−5000 +2727 +1653 +751

ADCF, $

 ADCF = (NPV), $

1.0000 0.90909 0.82645 0.75131 0.68301

−50,000 9,091 8,265 7,513 6,830

−50,000 −40,909 −32,644 −25,131 −18,301

+10,000 +10,000 +10,000 +10,000 +10,000

0.62092 0.56447 0.51316 0.46651 0.42410

6,209 5,645 5,132 4,665 4,241

−12,092 −6,447 −1,315 +3,350 +7,591

−40,000 +20,000 +20,000 +20,000 +20,000 +20,000 +20,000

0.75131 0.68301 0.62092 0.56447 0.51316 0.46651 0.42410

−30,052 +13,660 +12,418 +11,289 +10,263 +9,330 +8,482

−30,052 −16,392 −3,974 +7,315 +17,578 +26,908 +35,390

Year

ACF, $

fd

−5000 −2273 −620 +131

0 1 2 3 4

−50,000 +10,000 +10,000 +10,000 +10,000

−5000 +909 +1653 +2254

−5000 −4091 −2438 −184

5 6 7 8 9

−5000 +1818 +1653 +1503

−5000 −3182 −1529 −26

3 4 5 6 7 8 9

Project E

Project H

Project F 1.0000 0.90909 0.82645 0.75131

Project G 1.0000 0.90909 0.82645 0.75131

The (DCFRR) is the discount rate that satisfies Eq. (9-57) in the final year of the project. We can approximate the (DCFRR) for each project as follows: For project E,  ACF = +$1000 in Year 3 for i = 0 percent  ADCF = +$131 in Year 3 for i = 10 percent  ADCF = $0 in Year 3 for i = (DCFRR) Therefore, 1000/(1000 − 131)  (DCFRR)/10 (DCFRR)  11.5 percent Similarly for project F,

Project I

land have been neglected since the latter is the same for each project and the former would also favor project I. Incremental Comparisons A company may have the choice of, say, investing $10,000 in project J, which will give a (DCFRR) of 16 percent, or $7000 in project K, which will give a (DCFRR) of 18 percent. Should it spend $10,000 on project J or spend only $7000 on project K and invest the difference of $3000 elsewhere? Both projects have lives of 10 years and constant positive net annual cash flows ACF of $2069 and $1558 for projects J and K respectively. The corresponding (NPV)s at a discount factor of 10 percent are +$2710 and +$2560 respectively. These data are summarized as follows:

1000/(1000 + 184)  (DCFRR)/10 (DCFRR)  8.4 percent Similarly for project G, 1000/(1000 + 26)  (DCFRR)/10 (DCFRR)  9.7 percent In terms of net present value (NPV), the projects in order of merit are E, G, and F, with (NPV)s of +$131, −$26, and −$184 respectively. In terms of (DCFRR), the projects in order of merit are also E, G, and F, with (DCFRR) values of 11.5 percent, 9.7 percent, and 8.4 percent respectively. When to Scrap an Existing Process Let us suppose that a company invests $50,000 in a manufacturing process that has positive net annual flows (after tax) ACF of $10,000 in each year. During the third year of operation, an alternative process becomes available. The new process would require an investment of $40,000 but would have positive net annual cash flows (after tax) of $20,000 in each year. The cost of capital is 10 percent, and it is estimated that a market will exist for the product for at least 6 more years. Should the company continue with the existing process (project H), or should it scrap project H and adopt the new process (project I)? The net annual cash flows ACF and cumulative discounted annual cash flow  ADCF for a discount factor of 10 percent are listed in Table 9-7 for the two projects. At the end of Year 9, the net present values are (NPV) = +$35,390 for project I (NPV) = +$7591 for project H The difference is +$27,779, which is numerically greater than the money lost by the end of Year 3 for project H. Thus project H should be scrapped, and the new project I adopted if only economic reasons need to be considered. Recovery of working capital and the cost of

ACF, $, in Year 0 ACF, $, in each of Years 1–10 (NPV), i = 10 percent, $ (DCFRR), percent

Project J

Project K

−10,000 +2,069 +2,710 16

−7,000 +1,558 +2,560 18

Project ( J − K) −3,000 +511 +150 12.4

From the difference in cash flows between the projects, the discounted-cash-flow rate of return (DCFRR) for project (J-K) can be shown as 12.4 percent. This is significantly lower than for either project J or project K. Thus, if the $3000 can be invested to give a return greater than 12.4 percent, project K should be chosen in preference to project J. Comparisons on the Basis of Capitalized Cost A machine in a process generates a positive net cash flow of $1000. Two alternatives are available: machine L, costing $2000, requires replacement every 4 years, and machine M, costing $3000, requires replacement every 6 years. Neither machine has any scrap value. The cost of capital is 10 percent. Which machine is the more profitable to operate? In this case, the lives of the machines are unequal, and the comparison is conveniently made on the basis of capitalized cost. This puts lives on the same basis, which is an infinite number of years. The net annual cash flows generated by each machine are equal. The capitalized cost CK of a piece of fixed-capital cost CFC is the amount of capital required to ensure that the equipment may be renewed in perpetuity. For a piece of equipment with no scrap value, CK is given by (1 + i)n CK = CFC ᎏᎏ (9-58) (1 + i)n − 1 For machine L, CK = ($2000)(3.15471) = $6309.42





INVESTMENT AND PROFITABILITY For machine M, CK = ($3000)(2.29607) = $6888.21 Thus, machine L with the lower capitalized cost is the more profitable to operate. Relationship between (PBP) and (DCFRR) For the case of a single lump-sum capital expenditure CFC which generates a constant annual cash flow ACF in each subsequent year, the payback period is given by the equation (PBP) = CFC / ACF

(9-59)

if the scrap value of the capital outlay may be taken as zero. For this simplified case the net present value (NPV) after n years with money invested at a required aftertax compound annual fractional interest rate i is given by the equation (NPV) = CFC − ACF Fn

(9-60)

n

where

1 Fn =  ᎏn (1 + i) 1

When (NPV) = 0, the value of i given by Eq. (9-60) is the discountedcash-flow rate of return (DCFRR), and in this case Eqs. (9-59) and (9-60) can be combined to give: (PBP) = Fn

(9-61)

Figure 9-13 is a plot of Eq. (9-61) in the form of the number of years n required to reach a certain discounted-cash-flow rate of return (DCFRR) for a given payback period (PBP). The figure is a modification of plots previously published by A. G. Bates [Hydrocarbon Process., 45, 181–186 (March 1966)], C. Estrup [Br. Chem. Eng., 16, 171 (February–March 1971)], and F. A. Holland and F. A. Watson [Process Eng. Econ., 1, 293–299 (December 1976)]. In the limiting case when n approaches infinity, Eq. (9-61) can be written as (DCFRR)max = 1/(PBP)

(9-62)

which means, for example, that if the payback period is 4 years, the maximum possible discounted-cash-flow rate of return which can be reached is 25 percent. The corresponding (DCFRR) for (PBP) = 10 years is 10 percent.

FIG. 9-13

9-19

Equations (9-59), (9-60), (9-61), and (9-62) may be used as they stand to assess expenditure on energy-conservation measures since a constant amount of energy is saved in each year subsequent to the capital outlay. However, the annual cash flows ACF corresponding to the energy savings remain constant only if there is no inflation or if the money values are corrected to their purchasing power at the time of the capital expenditure. Sensitivity Analysis An economic study should pinpoint the areas most susceptible to change. It is easier to predict expenses than either sales or profits. Fairly accurate estimates of capital costs and processing costs can be made. However, for the most part, errors in these estimates have a correspondingly smaller effect than changes in sales price, sales volume, and the costs of raw materials and distribution. Sales and raw-materials prices may be affected by any of the following: discounts and allowances, availability of substitutes, contract pricing, government regulations, quality and form of the materials, and competition. Sales volume may be affected by any of the following: new uses for the product, new markets, advertising, quality, overcapacity, replacement by another product, competition, and timing of entry into the market. Distribution costs depend on plant location, physical state of the material (whether liquid, gas, or solid), nature of the material (whether corrosive, explosive, flammable, perishable, or toxic), freight rates, and labor costs. Distribution costs may be affected by any of the following: new methods of materials handling, safety regulations, productivity agreements, wage rates, transportation systems, storage systems, quality, losses, and seasonal effects. It is worthwhile to make tables or plot curves that show the effect of variations in costs and prices on profitability. This procedure is called sensitivity analysis. Its purpose is to determine to which factors the profitability of a project is most sensitive. Sensitivity analysis should always be carried out to observe the effect of departures from expected values. For many years, companies and countries have lived with the problem of inflation, or the falling value of money. Costs—in particular, labor costs—tend to rise each year. Failure to account for this trend in predicting future cash flows can lead to serious errors and misleading profitability estimates. Another important factor is the tendency of product prices to fall as the total national or international volume of production increases. Sales prices may fall by 20 percent for a doubling in volume or production.

Relationship between payback period and discounted-cash-flow rate of return.

9-20

PROCESS ECONOMICS

FIG. 9-14

Net present value against time, showing effect of adverse changes in cash flows.

No profitability estimate is better than the inherent accuracy of the data. Example 3: Sensitivity Analysis The following data describe a project. Revenue from annual sales and total annual expense over a 10-year period are given in the first three columns of Table 9-5. The fixed-capital investment CFC is $1 million. Plant items have a zero salvage value. Working capital CWC is $90,000, and the cost of land CL is $10,000. There are no tax allowances other than depreciation; i.e., AA is zero. The fractional tax rate t is 0.50. For this project, the net present value for a 10 percent discount factor and straight-line depreciation was shown to be $276,210 and the discounted-cash-flow rate of return to be 16.4 percent per year. We shall use these data and the accompanying information of Table 9-5 as the base case and calculate for straight-line depreciation the net present value (NPV) with a 10 percent discount factor and the discounted-cash-flow rate of return (DCFRR) for the project with the following situations.

Case a b c d e

Modification Revenue AS reduced by 10 percent per year Revenue AS reduced by 20 percent per year Total expense ATE increased by 10 percent per year Fixed-capital investment increased by 10 percent AS reduced by 10 percent per year, ATE increased by 10 percent per year, and CFC increased by 10 percent

The results are shown in Figs. 9-14 and 9-15 and Tables 9-8 and 9-9.

Learning Curves It is usual to learn from experience. Consequently, the time taken to produce an article, the number of spoiled batches, the cost per unit of production, etc., tend to decrease with the number of units produced. The relationships are expressed for the ideal case by

TABLE 9-8 Annual Cash Flows, Straight-Line Depreciation, and 10 Percent Discount Factor When Revenue Is Reduced by 10 Percent per Year Base case Year

fd

∆ADCF, $

∆(NPV), $

Reduced (NPV), $

0 40,000 50,000 50,000

1.0000 0.90909 0.82645 0.75131

0 18,180 20,660 18,780

0 18,180 38,840 57,620

−1,100,000 −936,360 −750,410 −585,120

−363,580 −211,460 −73,160 +50,000

50,000 52,000 52,000 52,000

0.68301 0.62092 0.56447 0.51316

17,070 16,140 14,680 13,340

74,690 90,830 105,510 118,850

−438,270 −302,290 −178,670 −68,850

+131,640 +195,250 +276,210

39,000 35,000 28,000

0.46651 0.42410 0.38554

9,100 7,420 5,390

127,950 135,370 140,760

+3,690 +59,880 +135,450

AS, $

ATE, $

(NPV), $

0 1 2 3

0 400,000 500,000 500,000

0 100,000 100,000 110,000

−1,100,000 −918,180 −711,570 −527,500

4 5 6 7

500,000 520,000 520,000 520,000

120,000 130,000 130,000 140,000

8 9 10

390,000 350,000 280,000

140,000 150,000 160,000

AS = base revenue from annual sales before tax. ATE = base total annual expense before tax. (NPV) = base net present value after tax. ∆AS = decrease in annual revenue.

∆AS, $

fd = discount factor at 10%. ∆ADCF = decrease in net discounted cash flow at income tax rate = 0.5. ∆(NPV) =  ∆ADCF = decrease in net present value. Reduced (NPV) =  ADCF = reduced net present value after tax.

INVESTMENT AND PROFITABILITY

FIG. 9-15

where Y = X= K= N=

Decrease in net present value against time resulting from adverse changes in cash flows.

Y = KX N (9-63) cumulative-average cost, production time, etc., per unit cumulative production, units effective value of first unit produced slope of straight-line plot of Y versus X on log-log paper

The particular learning curve is usually characterized by the percentage reduction in the cumulative average value Y when the number of units X is doubled. From this definition it follows that N = log (characteristic/100)/log 2

(9-64)

The cost cME of the last unit of a block bringing the cumulative production to X units is, from Eq. (9-63), cME = K[X N + 1 − (X − 1) N + 1]

(9-65)

These unit costs, or the time taken to produce the last unit, etc., may be plotted on cartesian coordinates against the number of units produced to provide a standard against which the performance of a new employee, a new machine, etc., can be judged. Figure 9-16 shows such a plot for the subsequent example. In general, cost data will be available for multiple units. Typically, the cost of production for 1 week or of a specific order is computed and an average cost per unit obtained. This average value Y
for the batch should be plotted against the corresponding learning-curve value
X calculated by Eq. (9-66): XN = (X2N + 1 − X1N + 1)/(X2 − X1)


(9-66)

where X1 and X2 are the cumulative production before and after the batch. This form of the equation is useful when only the previous production history of the process is known, from the serial numbers or otherwise. TABLE 9-9

Summary of Results of Sensitivity Analysis

Case Base case A S reduced 10% per year A S reduced 20% per year A TE increased 10% per year CFC increased 10% Combined: A S reduced 10% per year A TE increased 10% per year CFC increased 10%

}

9-21

(NPV), $ i = 10%

(DCFRR), %

276,210 135,450 −5,330 238,430 206,890

16.4 12.9 9.8 15.0 14.0

28,420

10.6

A straight line may be fitted to the (X,Y) or (X
,Y
) pairs of data when plotted on log-log graph paper from which the slope N and the intercept log K with X = 1 may be read. Alternatively, the method of least squares may be used to estimate the values of K and N, giving the best fit to the available data. It will be noted that a value of N = 0, corresponding to a characteristic of 100 percent for the learning curve, implies that the value of Y is independent of X. This would imply that learning by experience was not possible and thus corresponds to an optimally designed process or one for which the costs are determined by external factors. Similarly, a value of N = −1, corresponding to the 50 percent learning curve, implies that the cost of production is inversely proportional to the number produced, which is absurd. Projects having characteristics less than 70 percent are impractical. Low characteristics are typical of hasty entry into a market in an attempt to preempt it. Characteristics tend to increase with experience, so that established and mature projects are likely to have characteristics around 95 percent. Characteristics close to 100 percent are unlikely to be achieved because of random factors such as changes in personnel, accidents, supply delays, etc. Figure 9-17 represents a typical practical case, from which it can be seen that the curve has a point of inflexion but eventually settles down to an approximately straight line of lower slope than that of the conventionally defined learning curve. At some point it is useful to change to the equation of this mature project line. Significant changes in working, such as the introduction of new equipment, the influx of a large number of inexperienced workers, or a temporary reduction in skills after a long shutdown, may produce a sudden increase in all the cumulative-average curves. The simplest way to handle this, when the next accurate costing is available, is to deduct the value of X obtained from the curve from that actually achieved and to use this value as a constant correction to X until the next break in the curve is reached. If the causes of such steps recur, the size of the step can often be related to a particular cause. In such cases the estimated step change can be used for predictions until the next accurately determined values are obtained. Applications for the learning curve are already extensive, and new uses can often be found. Care is needed in applying the techniques to ensure that it is possible for learning to take place. In projecting prices, etc., unusual items, such as the cost of the special setting up of tools or factory rearrangements, should be excluded from the production costs used to establish the learning curve. In times of inflation, costs should be corrected for the effects of inflation in the manner to be shown subsequently. Production times or spoilage rates are not affected by cost allocations or inflation and may prove to be better

9-22

PROCESS ECONOMICS

FIG. 9-16

FIG. 9-17

Cartesian plot of learning curve.

Logarithmic plot of learning curve.

standards of performance where appropriate. However, the learning curve is often required when preparing quotations for batch production runs, particularly when competition is likely to be keen. In such cases the average cost of the production run Y
between cumulative production totals of X1 and X2 may be estimated by Eq. (9-67) when the previous cumulative-average cost Y1 is known:

extensively and provides many tables of factors. The uses considered include estimating starting costs, determining labor requirements, establishing factory cost targets, checking employee-training progress, the make-or-buy decision, aid in purchasing negotiations, and aid in establishing a selling price.

Y
= Y1[(X2 / X1)N + 1 − 1] / [(X2 / X1) − 1]

The cost of an initial batch of 21 units, exclusive of special tools and setting-up costs, averaged $120 per unit. The average cost of the next batch of 80 units was $75.81. Let us establish the learning curve implied by these data and hence estimate the probable average cost of the next 50 units. We shall establish also the unit-cost curve to be used as a control during follow-up orders. If the batch units are capable of continuous subdivision, we proceed as follows. We substitute the given values of the cumulative-average cost Y and cumulative production X for the first batch into Eq. (9-63) to give, by taking logarithms of each side,

(9-67)

In process engineering, fractional units can often be produced so that the learning curve can be treated as being continuous. When only discrete numbers of units can be produced, the learning curve is strictly a histogram. In order to allow for this it is sufficient to increase the value of X by half a unit before applying the above equations. The difference is significant at small values of X, such as may be used for the initial estimates of K and N. As the project matures, it is better to use the equations as presented, as the cost of the first unit K is an entirely notional one. Major technological changes should, of course, be treated as the start of a new project. R. B. Jordan (How to Use the Learning Curve, Materials Management Institute, Boston, 1965) discusses the uses of the learning curve

Example 4: Estimation of Average Cost of Incremental Units

log 120 = log K + N log 21 The cost of the first batch is 120 × 21 = $2520, and that of the second batch is 75.81 × 80 = $6065. The total cost of the first 101 units is therefore $8585, with a cumulative-average unit cost of $85. We substitute as before to give log 85 = log K + N log 101

INVESTMENT AND PROFITABILITY

FIG. 9-18

9-23

Effect of learning on the average cost of a product.

From these equations it follows that K = $234.15 and N = −0.2196. This line is plotted in Fig. 9-18. From Eq. (9-64) it follows that the value of the characteristic of this learning curve = 100 antilog (−0.2196 log 2) = 85.0 percent. From Eq. (9-65) the production cost of the third unit is

tion. In many cases, not all the x values will be different. In such circumstances, Eq. (9-68) can be written as

cME = ($234.15)(30.7804 − 20.7804) = $149.70

where f(xi) is the frequency with which a particular value xi occurs. It is often convenient to divide the frequency of occurrence by the total number of items. In this case, f(xi) becomes the relative frequency of occurrence of the value xi, and  f(xi) = 1. The values of x may be either discrete or continuous. The number of sales of, say, automobiles in any one day must be an integer. If a business sells 4 automobiles, this represents all possible values of x in the range of 3.5 to 4.5. When x represents a continuous variable quantity, it is sometimes convenient to take the total or relative frequency of occurrences within a given range of x values. These frequencies can then be plotted against the midvalues of x to form a histogram. In this case, the ordinate should be the frequency per unit of width x. This makes the area under any bar proportional to the probability that the value of x will lie in the given range. If the relative frequency is plotted as ordinate, the sum of the areas under the bars is unity. If x is a continuous variable and the interval ranges are made smaller and smaller, a smooth curve will eventually result. The area under such a curve between x1 and x2 represents the probability that a randomly selected item will have a value of x lying in the range x1 to x2. This is the information that is desired. Data available from past experience can be used to generate frequency distribution curves. It is essential for a company to have an efficient commercial-intelligence system to assess market conditions. Accuracy of sales forecasting can also be increased by a careful study of past sales records, price trends, etc. However, the uncertainty of an estimate increases the farther into the future that the estimate is projected. Estimates of sales income and other types of forecasts are usually based on the opinions of experts. Experts should be able to estimate maximum, minimum, and most likely, or modal, values for a quantity. The modal value is not necessarily midway between the minimum and maximum values, since many distributions are skewed. An expert may be asked to estimate the probability of the occurrence of certain values on each side of the mode. When experts are questioned separately, the procedure is known as the Delphic method. Strictly speaking, this method requires that the opinion of each expert be assessed by a coordinator, who then feeds the results back to see if the opinions of one expert are modified by those of others. The process is repeated until agreement is reached. In practice, the procedure is too tedious to be repeated more than once. It is useful to compare the past predictions of each expert with the results obtained in practice. This information enables the opinions to be weighted by the coordinator. When the experts work in close collaboration, it is not possible to avoid some collusion. In this case, it is

Values calculated in this way are plotted in Fig. 9-16 and also in Fig. 9-18. It will be noted that after about 10 units this latter curve becomes parallel to the cumulative-average-cost curve and that the Y values are (N + 1) times those obtained from the latter curve. Since the cumulative-average cost Y2 of the first 101 units was $85, it follows from Eq. (9-67) that the average cost of the third batch of 50 units, bringing the cumulative total to 151, is given by Y
3 = ($85)[(151/101)0.7804 − 1]/[(151/101) − 1] = $63.30 per unit This may be used as a cost guide when quoting the order. If the units of production may not be subdivided, the procedure is similar except that all X values are increased by 0.5 unit in establishing the curves. The results are not sufficiently different to be significant for estimation purposes. To the above costs must be added back any unit costs omitted from those to which learning might bring improvement. These will normally include overheads and specific charges on the project such as the unit cost of special tools, jigs, etc.

Risk and Uncertainty Discounted-cash-flow rates of return (DCFRR) and net present values (NPV) for future projects can never be predicted absolutely because the cash-flow data for such projects are subject to uncertainty. Therefore, when stating predicted values of (DCFRR) and (NPV) for projects, it is also desirable to give a measure of confidence in the predictions. For example, for a particular project it may be estimated that there is a 90 percent chance of the (DCFRR) being greater than 10 percent, a 50 percent chance of its being greater than 16 percent, and only a 10 percent chance of its being greater than 20 percent. Management retains the power of decision to proceed with the project or not, but the probability data provide desirable information for the decision. The estimation of probabilities requires the use of statistics. Thus statistical methods play an increasing role in decision making. Predictions from Limited Data Predictions of future sales price, sales volume, etc., are normally based on a very limited amount of data about past events. Furthermore, it would not be convenient to use the entire population of past events even if it were available. A statistic is a measure, based on limited information from a sample, that allows the corresponding parameter of the population to be estimated. The mean value x
of a property x is a statistic based on a sample of n items defined by x
= (x1 + x2 + x3 + ⋅ ⋅ ⋅ + xn)/n (9-68) The mean
x is the statistic corresponding to the population parameters µ, which is the arithmetic average of all the items in the popula-

x
= [ xi f(xi)] / [ f(xi)]

(9-69)

9-24

PROCESS ECONOMICS

better to arrive at a single consensus opinion by a free and open discussion. This is the think-tank method. Its main disadvantage is that rank or aggressiveness might unduly weight one or more opinions. The opinions of the experts, however obtained, provide a basis for plotting a frequency or probability distribution curve. If the relative frequency is plotted as ordinate, the total area under the curve is unity. The area under the curve between two values of the quantity is the probability that a randomly selected value will fall in the range between the two values of the quantity. These probabilities are mere estimates, and their reliability depends on the skill of the forecasters. The estimated (DCFRR) and the estimated (NPV) are both functions of the estimated cumulative revenue from annual sales  AS, the estimated cumulative total annual cost or expense  ATE, and the estimated fixed capital cost CFC of the plant. The revenue from annual sales for each year is in turn the product of the sales price and sales volume. Initially it is desirable to select those values from the distribution curves of  AS,  ATE, and CFC which enable the maximum and minimum (DCFRR) and (NPV) to be calculated. If the maximum values of (DCFRR) and (NPV) are not acceptable to the company, the project should promptly be rejected. If the minimum values of (DCFRR) and (NPV) are acceptable, a detailed assessment should be made. If the maximum values of (DCFRR) and (NPV) are acceptable but the minimum values are not, the feasibility study should be continued. Mathematical Models for Distribution Curves Mathematical models have been developed to fit the various distribution curves. It is most unlikely that any frequency distribution curve obtained in practice will exactly fit a curve plotted from any of these mathematical models. Nevertheless, the approximations are extremely useful, particularly in view of the inherent inaccuracies of practical data. The most common are the binomial, Poisson, and normal, or gaussian, distributions. A normal distribution curve is bell-shaped (see Sec. 3). The curve obeys the relationship exp − [(x − µ)2/2σ2] f(x) = ᎏᎏᎏ (9-70) σ(2π)0.5 where σ is known as the true standard deviation. The standard deviation s° from a sample is given by (xi −
x)2f(xi) s° = ᎏᎏ  f(xi) − 1





0.5

(9-71)

The standard deviation s° for the sample corresponds to the true standard deviation σ for the whole population in the same way that the mean
x of the sample corresponds to the arithmetic average µ for the whole population. Equation (9-70) can be written more compactly as f(z) = [exp (−z2 / 2) ]/ [(2π)0.5]

(9-72)

where the standard score z is z = (x − µ)/σ

(9-73)

The area under the curve of f(z) is unity if the abscissa extends from minus infinity to plus infinity. The area under the curve between z1 and z2 is the probability that a randomly selected value of x will lie in the range z1 and z2, since this is the relative frequency with which that range of values would be represented in an infinite number of trials. An event that will definitely occur has a probability of unity. An event that will definitely not occur has a probability of zero. Equation (9-72) can be integrated between limits to determine the probability that a random value lies between the selected limits. Extensive tables of f(z) and the associated integral are available (see Sec. 3). A frequency distribution curve can be used to plot a cumulativefrequency curve. This is the curve of most importance in business decisions and can be plotted from a normal frequency distribution curve (see Sec. 3). The cumulative curve represents the probability of a random value z having a value of, say, z1 or less. If a property or variable c is a function of several other variables x1, x2, etc., it can be written in the form c = φ(x1, x2, . . . xn)

(9-74)

If each x is a normally distributed independent variable, then ∂c 2 ∂c 2 ∂c 2 σc2 = ᎏ σ12 + ᎏ σ22 + ⋅ ⋅ ⋅ + ᎏ σn2 (9-75) ∂x1 ∂x2 ∂xn where σc is the standard deviation of the variable c and σ1, σ2, etc., are the standard deviations of the variables x1, x2, etc. Many distributions occurring in business situations are not symmetrical but skewed, and the normal distribution curve is not a good fit. However, when data are based on estimates of future trends, the accuracy of the normal approximation is usually acceptable. This is particularly the case as the number of component variables x1, x2, etc., in Eq. (9-74) increases. Although distributions of the individual variables (x1, x2, etc.) may be skewed, the distribution of the property or variable c tends to approach the normal distribution. Let us consider an event that must have one of two outcomes. It must either occur with probability p1 or fail to occur with probability p2. Since these are exclusive events and the probability that something will happen is unity, it follows that

 

 

 

p1 + p2 = 1

(9-76)

Provided that no learning process is involved (so that the value of p1 is not influenced by previous results), the probability of x successes in n trials is given by the term containing p1x in the expansion of the binomial: n! (p1 + p2)n = p1n + ⋅ ⋅ ⋅ ᎏ p1x p2(n − x) ⋅ ⋅ ⋅ + p2n (9-77) x!(n − x)! where x and n are integers and x! (read as x factorial) is the product of all integers from unity to x. Example 5: Probability Calculation If a six-sided die marked with the numbers 1, 2, 3, 4, 5, and 6 is thrown, the probability that any given number will be uppermost is 1/6. If the die is thrown twice in succession, then the probability of a given sequence of numbers occurring, say, 5 followed by 6, is (1/6)(1/6) = 1/36. The chance of any particular number occurring 0, 1, 2, 3, or 4 times in four throws of the die (or in a simultaneous throw of four dice) is given by the successive terms of Eq. (9-77), expanded as (5⁄6 + j)4 = (1)(5⁄6)4(j)0 + (4)(5⁄6)3(j)1 + (6)(5⁄6)2(j)2 + (4)(5⁄6)1(j)3 + (1)(5⁄6)0(j)4 = 0.4823 + 0.3858 + 0.1157 + 0.0154 + 0.0008 = 1 The distribution of the number of successes is skewed toward the low numbers. In particular, there is only a slightly better than even-money chance that any given number will occur even once in four throws. Such highly unsymmetrical distributions cannot be approximated by the normal distribution curve. However, an increasing number of throws will result in totals that are close to the normal distribution. This fact can be used to approximate such a distribution without the enormous labor of the calculations required by the use of Eq. (9-77). Possible values of the total of four throws of a die are integers from 4 to 24 and hence represent values in the range from 3.5 to 24.5. The mean value
x of this range is given by Eq. (9-68) as
x = (3.5 + 24.5)/2 = 14.0. The cumulative probability of a normally distributed variable lying within 4 standard deviations of the mean is 0.49997. Therefore, it is more than 99.99 percent (0.49997/0.50000) certain that a random value will be within ⫾ 4σ from the mean. For practical purposes, σ may be taken as one-eighth of the range of certainty, and the standard deviation can be obtained: s °  (24.5 − 3.5)/8 = 2.625 From Eq. (9-73) the standard score becomes z = (x − µ)/σ  (x −
x)/s° For a total score of 4 (i.e., x = 4), the standard score is approximately z = (4 − 14)/2.625 = −3.81. Since the normal curve is symmetrical about z = 0, the height of the ordinate at z = −3.81 is the same as that at z = +3.81. From tables of values of cumulative probabilities of the normal distribution, the height of the ordinate is 0.0003 in units of 1/σ. The relative frequency of 4 occurring is thus approximately 0.0003/2.625 = 0.0001. This concept can be used to translate Delphic or other opinions into probability distributions and hence into useful decision-making tools.

Example 6: Calculation of Probability of Meeting a Sales Demand A store that is open 5 days a week is to promote a new product. The manager believes that not more than 5 units will be sold in any one day, but he cannot be more precise about the probable sales pattern. Stocks are delivered once per week. What size should the first order be to give a 95 percent certainty of meeting demand?

INVESTMENT AND PROFITABILITY Since the product is sold in units, the possible range of weekly sales is from −0.5 to +25.5 units. Therefore, the mean of the sales distribution will be
x  [25.5 − (−0.5)]/2 = 13 The standard deviation for this example will be

9-25

Since something must happen, the probability of 4 or more breakdowns is 1 − 0.0498 − 0.1496 − 0.2240 − 0.2240 = 0.3526 A simple trial will show how much more easily the preceding calculation is carried out than direct use of Eq. (9-77).

s°  [25.5 − (−0.5)]/8 = 3.25 From this, the approximate frequency distribution of daily sales can be derived by using Eqs. (9-70), (9-72), or (9-73). The desired area to the right of z = 0 for the normal probability distribution curve is 0.95 − 0.50 = 0.45. For this value the standard score z = 1.645. x  13 + (1.645)(3.25) = 18.35 Hence, to be 95 percent certain of meeting demand, 19 units should be purchased.

If the value of n in Eq. (9-77) is large and neither p1 nor p2 is too close to zero, the binomial distribution can be approximated by x − np1 z=ᎏ (9-78)
n
p

1p2 The approximation of Eq. (9-78) is good enough for most purposes if np1 and np2 are each greater than 5.

The necessary value of λ may often be established as in the following example. Example 9: Calculation of Probability of Machine Failures In a production period of 100 days, 0, 1, 2, 3, and 4 machine failures occurred in a single day on 41, 37, 15, 6, and 1 occasions respectively. Let us fit a Poisson distribution to the data and estimate the maximum number of machine failures likely to occur in 1 day of a 300-day year. The mean number of failures is found from Eq. (9-69) to be 0(41) + 1(37) + 2(15) + 3(6) + 4(1)
x = ᎏᎏᎏᎏ = 0.89 41 + 37 + 15 + 6 + 1 The standard deviation is found from Eq. (9-71):



Example 7: Calculation of Probability of Sales The records of a business show that never more than 1 item is sold in a day and that 2 sales per week can be expected. What is the probability of selling between 90 and 120 items in a 300-day year? In a year consisting of 50 weeks of 6 days, the mean or expected value of the distribution is 100 items. The probability of a sale of an item on a given day is p1 = 100/300 = 1/3, and of no sale is p2 = 2/3. From Eq. (9-78),

(xi −
x) (0 − 0.89)2 (1 − 0.89)2 (2 − 0.89)2 (3 − 0.89)2 (4 − 0.89)2

There are times when the frequency measurement is an integral number of events in a given segment of a continuum, for example, the number of automobiles passing a given point in 1 h or the number of leaks in a given length of hosepipe. In such cases, the correct frequency distribution is the Poisson distribution, in which the probability of x events per unit of a continuum occurring is given by f(x) = λxe−λ/x!

(9-79)

where x is an integer, e is the base of natural logarithms, and λ is a parameter of the system λ = µ = σ2. As λ increases, the Poisson distribution approaches the normal distribution, with the relationship


z = (x − λ)/ λ

(9-80)

When the value of p1 is very close to zero in Eq. (9-77), so that the occurrence of the event is rare, the binomial distribution can be approximated by the Poisson distribution with λ = np1 when n > 50 while np1 < 5. Example 8: Calculation of Probability of Equipment Breakdown The daily chance of a breakdown in a production line operated continuously for 300 days per year is estimated at 1 percent from past performance. Let us estimate the probability of 4 or more breakdowns in the coming year. For n = 300 and p1 = 0.01, λ  np1 = 3 < 5, the probability of no breakdown is found from Eq. (9-79) to be f(0) = (3)0e−3/0! = 1/e3 = 0.0498 Similarly, Breakdowns

Probability

1 2 3

f(1) = (3)1e−3/1! = 0.1496 f(2) = (3)2e−3/2! = 0.2240 f(3) = (3)3e−3/3! = 0.2240

f(xi)

(xi −
x)2f(xi)

41 37 15 6 1  (xi −
x)−2 f(xi)  f(xi) − 1

32.48 0.45 18.48 26.71 9.67 = 87.79 = 99

2

x − (300)(1/3) x − 100 z = ᎏᎏ = ᎏ 8.165
0
0
)(
1
/3
)(
2
/3
)
(3 The integral range of 90 to 120 items contains all possible values of x from 89.5 to 120.5. For x = 89.5, z = −1.286; and for x = 120.5, z = 2.511. The cumulative probability of a standard score of 1.286 is 0.11, while that of a standard score of 2.511 is 0.99. Therefore, the probability of annual sales in the range of 90 to 120 items is (0.99 − 0.11) = 0.88, or 88 percent.



 (xi −
x)2f(xi) 0.5 s° = ᎏᎏ  f(xi) − 1 The steps for calculating the numerator and denominator for this equation are tabulated as follows:

Therefore, s° = (87.79/99)0.5 = 0.9417. The Poisson distribution is a good fit since λ = µ  x
= 0.8900 λ = σ2  (s°)2  0.8868

and

The Poisson distribution is found from Eq. (9-79) to be f(x) = [(0.89)xe−0.89]/x! By substituting the appropriate values of x for this example into the preceding equation, we find f(0) = 0.4107, f(1) = 0.3655, f(2) = 0.1627, f(3) = 0.0483, f(4) = 0.0107, f(5) = 0.0019, and f(6) = 0.0003. Hence, in 300 days the expected maximum number of breakdowns in 1 day is 5 since (300)f(6) = 0.09 occurrence.

In many business applications, Eq. (9-74) can be reduced to the linear relationship c = k1x1 + k2x2 + ⋅ ⋅ ⋅ + knxn

(9-81)

where the k’s are constants. Equation (9-75) then becomes σc2 = k12σ12 + k22σ22 + ⋅ ⋅ ⋅ + kn2σn2

(9-82)

On the other hand, for a product function such as c = x1x2

(9-83)

Eq. (9-75) can be written in the form σc2/c2 = σ12/x12 + σ22 /x12

(9-84)

The discounted-cash-flow rate of return (DCFRR) and net present value (NPV) are functions of the cumulative revenue from annual sales  ATE and the fixed-capital cost of the plant CFC, among other factors. Equation (9-75) can be written for (DCFRR) and for (NPV) as



∂(DCFRR) 2 σ (DCFRR) = ᎏᎏ ∂ AS

σ 2

2

 AS



∂(DCFRR) + ᎏᎏ ∂ ATE

σ 2

2

 ATE

∂(DCFRR) + ᎏᎏ ∂CFC



σ 2

2 CFC

(9-85)

9-26

PROCESS ECONOMICS

∂(NPV)2 2 ∂(NPV)2 2 ∂(NPV)2 2 σ2(NPV) = ᎏ σ  AS + ᎏ σ  ATE + ᎏ σ CFC ∂ AS ∂ ATE ∂CFC

(9-86)

The revenue from annual sales AS of a product at an annual production rate R and sales price of cs per unit of production is AS = RcS

(9-87)

Equation (9-84) can be written as: σ AS = (AS/R)2σR + (AS/cs)2σ c2s 2

2

(9-88)

An extensive example illustrating the use of Eqs. (9-81) through (9-86) in establishing the probability of attaining a given value of the net present value or less in a particular year of a project was presented by Holland et al. [F. A. Holland, F. A. Watson, and J. K. Wilkinson, Chem. Eng., 81, 105–110 (Jan. 7, 1974)]. The result is shown in Fig. 9-19. Decision makers often prefer to have graphs showing the probability of attaining a value greater than a given value. Such curves are easily obtained by subtracting the probability of achieving a given value or less from 100 percent. Figure 9-20 was obtained in this way and shows the probability of attaining a (DCFRR) greater than a given value. Monte Carlo Method The Monte Carlo method makes use of random numbers. A digital computer can be used to generate pseudorandom numbers in the range from 0 to 1. To describe the use of random numbers, let us consider the frequency distribution curve of a particular factor, e.g., sales volume. Each value of the sales volume has a certain probability of occurrence. The cumulative probability of that value (or less) being realized is a number in the range from 0 to 1. Thus, a random number in the same range can be used to select a random value of the sales volume. In the same way, random values of the other factors can be obtained. These can then be combined to give random values of (DCFRR) and (NPV) and, in turn, used to plot cumulative-probability curves for (DCFRR) and (NPV). The computer may be required to perform some 10,000 to 50,000 calculations. The use of the Monte Carlo method in project appraisal was illustrated by Holland et al. [F. A. Holland, F. A. Watson, and J. K. Wilkinson, Chem. Eng., 81, 76–79 (Feb. 4, 1974)]. The cumulativeprobability curves of (DCFRR) and (NPV) can never be more accurate than the opinions on which they are based, and comparable accuracy can be obtained by the use of S-shaped curves with relatively small computational effort. S-Shaped Curves K. D. Tocher (The Art of Simulation, rev. ed., English Universities Press, London, 1967) presented a comprehen-

FIG. 9-20 Probability of a given discounted-cash-flow rate of return or more for a project.

sive treatment of the generation of random and pseudorandom numbers and their use in a wide range of simulated processes. He also considered sampling techniques from the various statistical distributions and the design of simulated processes. It will be noted that the cumulative distribution curves are S-shaped, and Tocher (op. cit., p. 16) recommended as a general equation for such curves x = a + by + cy2 + d(1 − y)2 ln y + ey2 ln (1 − y)

(9-89)

in which x varies from −∞ to +∞ as y varies from 0 to 1. The underlying frequency curve corresponding to Eq. (9-89) is 1 dx 1−y ᎏ = ᎏ = b + 2cy + d(1 − y) ᎏ − 2 ln y p(x) dy y







y + ey 2 ln (1 − y) − ᎏ 1−y



(9-90)

If necessary, the fit can be improved by increasing the order of the polynomial part of Eq. (9-89), so that this approach provides a very flexible method of simulation of a cumulative-frequency distribution. The method can even be extended to J-shaped curves, which are characterized by a maximum frequency at x = 0 and decreasing frequency for increasing values of x, by considering the reflexion of the curve in the y axis to exist. The resulting single maximum curve can then be sampled correctly by Monte Carlo methods if the vertical scale is halved and only absolute values of x are considered. When the data do not warrant the accuracy of Eq. (9-89) or Eq. (9-90), simpler curves will usually suffice if the frequency distribution may be assumed to have a single maximum value. Let us consider a product which is sold entirely on the basis of personal recommendation. The rate of sale will depend on the number of people who have already bought the product. Thus initially sales will increase exponentially. Eventually the market will be saturated, and only replacement purchases will be made. If the frequency curve may be assumed to be symmetrical about a single maximum value, the cumulative distribution curve is known as the logistics curve and is defined by Eq. (9-91): y = c / [1 + a exp (−bx)] FIG. 9-19

Probability of a given net present value or less for a project.

(9-91)

where y varies between zero and c as x ranges from −∞ to +∞.

INVESTMENT AND PROFITABILITY

9-27

Although only three constants appear explicitly in Eq. (9-91), two further constants are implied by the choice of zero as the lower bound of y and the point of inflexion at y = c/2. The usual use of Eq. (9-91) is in sales forecasting, in which case y is sales demand and x is time. If such a curve already exists, the value of c can be read as the upper asymptote and a and b obtained by the use of an auxiliary variable T where

When a cumulative-frequency curve can be satisfactorily represented by a logistics curve, the underlying frequency curve can be obtained by differentiation of Eq. (9-91) as

The probability-density function for the normal distribution curve calculated from Eq. (9-95) by using the values of a, b, and c obtained in Example 10 is also compared with precise values in Table 9-10. In such symmetrical cases the best fit is to be expected when the median or 50 percentile xM is used in conjunction with the lower quartile or 25 percentile xL or with the upper quartile or 75 percentile xU. These statistics are frequently quoted, and determination of values of a, b, and c by using xM with xL and with xU is an indication of the symmetry of the curve. When the agreement is reasonable, the mean values of b so determined should be used to calculate the corresponding value of a. In practice most distribution curves are not symmetrical about the median but are inherently skewed. The effect of an advertising campaign is usually to increase the rate of sales in the early years. It may also increase the level of mature demand for the product, but this mature demand must be asymptotic to a finite upper limit of sales c. Such a curve is positively skewed since (xM − xL) < (xU − xM). This situation can often be approximated by the Gompertz curve defined by Eq. (9-96):

T = x2 (at y = r2c) − x1 (at y = r1c)

(9-92)

b = [ln (1/r1 − 1) − ln (1/r2 − 1)]/T

(9-93)

a = (1/r1 − 1) exp (bx1)

(9-94)

a = (1/r2 − 1) exp (bx2)

or

If the values of a obtained from Eq. (9-94) differ significantly, the logistics curve is not a suitable representation of the data. Example 10: Logistics Curve We shall derive the logistics curve representing the cumulative-frequency distributions of the normal distribution curve defined by Eqs. (9-72) and (9-73). In this case, y varies between a cumulative probability of zero and unity as z varies from −∞ to +∞. Since the upper bound is unity, c = 1. From Table 9-10 the area under the right-hand side of the curve between z = 0 and z = z may be read. Since the frequency curve is symmetrical about the mean, this is also the area between z = 0 and z = z. Hence, the area under the frequency curve, which represents the cumulative probability, is 0.50000 at z = 0 and the 80 percentile, for which the area is 0.80000, corresponds to the value z = 0.842. We substitute these values into Eqs. (9-92) through (9-94) to give T = 0.842 − 0.000 = 0.842

abc exp (−bx) dy p(x) = ᎏ = ᎏᎏ2 dx [1 + a exp (−bx)]

ln y = ln c − a exp (−bx)

a = 1.0000 or 1.00000

b = 0.8794/(xU − xM)

From Eq. (9-91) the corresponding logistic curve is y = [1 + exp (−1.6464z)]

−1

The cumulative-frequency function calculated from this simple expression is compared with the precise value in Table 9-10.

Data for Normal Distribution Curve Area under normal curve, cumulative probability, Ordinate of normal distribution curve, p(z)

p(z) dz z

y=

(9-96)

which has its point of inflexion at 0.3679 c. In terms of the upper and lower quartiles and the median,

b = [ln (1/0.50 − 1) − ln (1/0.80 − 1)]/0.842 = 1.6464

TABLE 9-10

(9-95)

Precise

Estimated

Precise

Estimated

0.000 0.100 0.200 0.253 0.300

0.3989 0.3970 0.3910 0.3864 0.3814

0.4116 0.4088 0.4006 0.3943 0.3875

0.0000 0.03983 0.07926 0.10000 0.11791

0.00000 0.04017 0.08158 0.10265 0.12103

0.400 0.500 0.524 0.600 0.700

0.3683 0.3521 0.3478 0.3332 0.3123

0.3700 0.3491 0.3436 0.3255 0.3003

0.15542 0.19146 0.20000 0.22575 0.25804

0.15894 0.19492 0.20323 0.22866 0.25996

0.800 0.842 0.900 1.000 1.200

0.2897 0.2798 0.2661 0.2420 0.1942

0.2744 0.2634 0.2484 0.2231 0.1761

0.28814 0.30000 0.31594 0.34134 0.38493

0.28870 0.30000 0.31484 0.33840 0.37822

1.282 1.400 1.600 1.645 1.800

0.1953 0.1497 0.1109 0.1032 0.0790

0.1587 0.1358 0.1029 0.0964 0.0769

0.40000 0.41234 0.44520 0.45000 0.46407

0.39194 0.40929 0.43303 0.43752 0.45092

2.000 2.500 3.000 4.000 ∞

0.0540 0.0175 0.0044 0.0001 0.0000

0.0569 0.0260 0.0116 0.0023 0.0000

0.47725 0.49379 0.49865 0.49997 0.50000

0.46418 0.48395 0.49289 0.49862 0.50000

(9-98)

a = 0.6931 exp (bxM)

(9-99)

The suitability of the Gompertz fit to the curve can be assessed by comparing the values of b calculated from Eqs. (9-97) and (9-98), and, if suitable, the average value of b may be used in Eq. (9-99) to calculate the corresponding value of a to ensure a fit at the median and reasonable accuracy over the more important practical range within a couple of standard deviations on either side of the median. The underlying frequency distribution curve of the Gompertz curve may be obtained by differentiation of Eq. (9-96) to give p(x) = dy/dx = yab exp (−bx)

0

Standard score, z

(9-97)

b = 0.6931/(xM − xL)

(9-100)

The logistic and Gompertz curves are of the general shape illustrated by Fig. 9-19. They may be adapted to fit curves of the general shape illustrated by Fig. 9-20 by a little mathematical manipulation. As an example, let us consider the current ratio, the ratio of current assets to current debts, as is quoted in Dun & Bradstreet statistics. A typical value for United States industrial chemical companies might be listed as xL = 1.82, xM = 2.59, and xU = 3.25. First, we notice that (xM − xL) > (xU − xM). This curve is, therefore, negatively skewed, or reversed S-shaped, and the logistics curve is not suitable. Nor can the Gompertz equation be used directly. However, it is clear that if the curve is drawn upside down and backward, the transformed curve will be positively skewed. Mathematically, this is equivalent to interchanging the upper and lower bounds and considering the dependent variable to be (c − y). In the present case the quoted values represent the cumulative probabilities that the current ratio will be less than the quoted value and hence the value of y ranges between zero and unity. Hence, c = 1. In the transformed curve xL = 3.25, xM = 2.59, and xU = 1.82. Hence, from Eq. (9-97) b = 0.8794/(1.82 − 2.59) = −1.1421 and from Eq. (9-98) b = 0.6931/(2.59 − 3.25) = −1.0502 The variation is within 5 percent of the mean value of b = −1.0961, and the transformed curve should be sufficiently accurate for many purposes. From Eq. (9-99) a = 0.6931 exp [(−1.0961)(2.59)] = 0.04054

9-28

PROCESS ECONOMICS In all such S-shaped curves the range of x is from −∞ to +∞, so that there is always a finite possibility of negative values of x occurring. In the present case the definition of the current ratio makes values of x below zero meaningless. The error of some 4 percent in the cumulative-probability curve implied by this factor may be tolerable in a given case. It can be shown [J. J. Molder and E. G. Rogers, Manage. Sci., 15, B-76 (1968)] that for continuous events it is possible to estimate the mean and standard deviation of a skewed distribution from estimates of a low value, a most likely or modal value, and a high value. It is suggested, since it is difficult to make very fine subjective judgments as to probabilities, that the range most likely to be accurate is that for which there is a 10 percent chance of a value less than the low value and a 10 percent chance of a value greater than the high value. These values will usually imply a skewed distribution. For the suggested 80 percent confidence level the best available estimates are x
= (low value) + [(2)(modal value)] + (high value) s° = [(high value) − (low value)] / 2.65

(9-101) (9-102)

On this basis an alternative approach to risk analysis is the parameter method [D. O. Cooper and L. B. Davidson, Chem. Eng. Prog., 72, 73–78 (November 1976)].

FIG. 9-21

Cumulative probability of a given current ratio.

Hence, from Eq. (9-96) the equation of the transformed curve is ln (1 − y) = −0.04054 exp (1.0961x) Since d(1 − y) = −dy, the corresponding underlying frequency distribution curve is from Eq. (9-100): p(x) = +0.0444(1 − y) exp (1.0961x) Values of y and p(x) calculated from last two equations are plotted in Figs. 9-21 and 9-22 respectively.

Example 11: Parameter Method of Risk Analysis Let us consider the project outlined in Table 9-5. It is estimated that the basic data represent the most likely values and that there is a 10 percent chance that AS will be reduced by more than 20 percent or will be increased by more than 5 percent. In the same way the low and high levels at 10 percent probability for ATE are considered to be 5 percent below and 25 percent above the base figures respectively. The low and high values for CFC are considered to be 5 percent below and 30 percent above the base figure, while changes in other parameters are considered to be immaterial. With a cost of capital i of 10 percent the various cash flows can be discounted and summed. Thus for the base cases  As fd = $2,815,600,  ATE fd = $754,716,  AD fd = $614,457, and  CWC fd = $61,446. With corporate taxes payable at 50 percent the aftertax cash flows of the first three items are (1 − 0.50) of the sums calculated above. The discounted working capital and the fixed-capital outlay are not subject to tax. These most probable values are listed and summed in Table 9-11 and, after adjustment for tax, give the modal value of the (NPV) as $276,224. A reduction of 20 percent in As for each year will result in a 20 percent reduction in  As fd below the modal value, i.e., a reduction of (0.20)($2,815,600) = $563,120. The aftertax effect of this reduction on the contribution to (NPV) is (1 − 0.50)($563,120) = $281,560, making the low value $1,407,800 − $281,560 = $1,126,240 or, more directly, (0.8)($1,407,800). Other values in Table 9-11 are calculated in a similar manner. The mean value of each of the distributions is obtained from these high, modal, and low values by the use of Eq. (9-101). If the distribution is skewed, the mean and the mode will not coincide. However, the mean values may be summed to give the mean value of the (NPV) as $161,266. The standard deviation of each of the distributions is calculated by the use of Eq. (9-75). The fact that the (NPV) of the mean or the mode is the sum of the individual mean or modal values implies that Eq. (9-81) is appropriate with all the k’s equal to unity. Hence, by Eq. (9-81) the standard deviation of the (NPV) is the root mean square of the individual standard deviations. In the present case s° = $166,840 for the (NPV). If the resulting distribution is assumed to be normal, then the cumulative distribution curve can immediately be generated. From Table 9-10, a standard score of 4 corresponds to a probability of 0.5 + 0.49997 = 0.99997 and one of −4 to a probability of 0.5 − 0.49997 = 0.00003, virtually unity and zero respectively. From Eq. (9-73) a standard score of 4 corresponds to an (NPV) of $161,266 + (4)($166,840) = $828,626 and one of −4 to an (NPV) of $506,094. Values of (NPV) corresponding to other confidence limits may be calculated in the same way and plotted to give the curve of Fig. 9-23. TABLE 9-11

Data for Risk Analysis (NPV), $/year

FIG. 9-22

Probability of a given current ratio.

Parameter

Low

Modal

High

Mean

Standard deviation

AS ATE CFC CWC

1,126,240 −471,698 −900,603 −61,446

1,407,800 −377,358 −692,772 −61,446

1,478,190 −358,490 −658,133 −61,446

1,355,008 −396,226 −736,070 −61,446

132,811 −42,720 −91,498 0

INVESTMENT AND PROFITABILITY

9-29

used in place of Eq. (9-86) to estimate the overall variance of the main variable. When the estimates are well founded, the skewness may be preserved by using a distribution such as the Gompertz. The median of that curve occurs as y = 0.5 c, while the point of inflexion corresponds to the mode at y = c/exp (1) = 0.3679 c. The statistician Karl Pearson suggested as a simple measure of skewness Skewness = 3 (mean − median)/σ

(9-103)

with an empirical approximation in terms of the mode given by (Mean − mode) = 3 (mean − median)

(9-104)

Applying these equations to the present problem, (Mean − mode) = $161,266 − $276,224 = −$114,958 Skewness = −$114,958/$166,840 = −0.6890 For symmetrical distributions, such as the logistic or normal, the skewness should be zero. The Gompertz distribution requires the distribution to be positively skewed, which can be achieved by treating −(NPV) as the independent variable and (c − y) as the dependent variable. From Eq. (9-104) the median of the distribution is given approximately as Median = [$161,266 − (−$114,958)]/3 = $199,585 Substituting values into Eq. (9-96) with −(NPV) as the independent variable to give, since the range of y is zero to unity, ln (1 − 0.5) = ln (1) − a exp [(−b)(−$199,585)] ln (1 − 1/e) = ln (1) − a exp [(−b)(−$276,224)]

Cumulative probability of a given net present value or less for a project showing normal and Gompertz approximations.

FIG. 9-23

As has been stated, with the uncertainties attached to many business assessments of the range of various factors, the central-limit theorem implies that the assumption of a normal distribution of the main variable is sufficiently accurate provided that there are several factors contributing to that main variable. The results are as informative as most Monte Carlo estimates and have the advantage that they can be rapidly obtained without recourse to a digital computer, although a good desk calculator speeds the work. Strictly, the variables should be independent and additive. Thus it is better, for example, to treat (AS − ATVE) as a single variable since both sales income and total variable expense are related to the annual rate of sales R. In such cases the standard deviation of ATVE would be added to or subtracted from that of AS before squaring to obtain the variance according as the uncertainty of the group was greater or less than that of the individual factors. Also, when a product such as AS = RcS is involved, Eq. (9-84) should be used to estimate the variance rather than Eq. (9-82). When the predominant uncertainties are multiplied together, a log-normal distribution may provide a better final distribution. A similar technique may be applied to the (DCFRR) provided that Eq. (9-85) is

FIG. 9-24

−ln [ln (0.5)/ln (1 − 1/e)] whence b = ᎏᎏᎏ = −5.388 × 10−6/$ [(−$199,585) − (−$276,224)] −ln 0.5 a = ᎏᎏᎏᎏ = 2.0315 exp [(5.388 × 10−6/$)(−$199,585)] The Gompertz curve of the distribution is then, in terms of (NPV), ln y = −2.0315 exp [−5.388 × 10−6(NPV)] For the same degree of certainty as before, the minimum value of the (NPV) is likely to be ln (0.00003) ln ᎏᎏ − 5.388 × 10−6 = −$303,365 −2.0315 and the maximum of $2,064,569 calculated in the same way for y = 0.99997. Other values are calculated in the same way and are plotted as in Fig. 9-23. Decision Trees In a typical decision tree, illustrated in a very simplified form by Fig. 9-24, each node represents a decision point (DP) at which one or more alternatives are available. Some quantifiable result of each alternative is chosen as a basis for comparison: for example, the net present value (NPV). A value is assigned to the probability of attaining each result, either cumulative or not as required. These may be obtained by the methods just described or otherwise. The estimates are subject to the restriction that the sum of the proba-

Effect of decision-tree options on net present value.





9-30

PROCESS ECONOMICS

bilities for all branches leaving each node shall be unity since some decision must be taken there. In considering two investments, we shall let option B be a safe investment having a base net present value (NPV)B that is independent of any competition. We shall let option A yield a net present value (NPV)A1 if no competition exists and (NPV)A2 if competition exists. We shall then let the probabilities of no competition and competition be p1 and p2 respectively. Then p2 must equal (1 − p1). The expected (NPV) for option A can be written, from Eq. (9-105), which follows, as (NPV)WA = p1(NPV)A1 + (1 − p1)(NPV)A2 where (NPV)WA is the weighted net present value for option A based on the probabilities of encountering no competition p1 and of encountering competition (1 − p1). In the same way the expected (NPV) for option B is given by (NPV)WB = p1(NPV)B + (1 − p1)(NPV)B = (NPV)B The gain in the expected value of option A over option B is thus ∆(NPV)W = (NPV)WA − (NPV)WB Let us suppose that the options represented in Fig. 9-24 were such that (NPV)B = (0.5)(NPV)A1 = 2(NPV)A2. Then substitution leads to ∆(NPV)W = [2p1 + 0.5(1 − p1) − 1](NPV)B = (1.5p1 − 0.5)(NPV)B The choice is immaterial when ∆(NPV)W = 0, i.e., when p1 = 1/3. If the probability of no competition is greater than 1/3, option A should be chosen; otherwise option B should be chosen. The technique is based on the methods of linear algebra and the theory of games. When the problem contains many multibranched decision points, a computer may be needed to follow all possible paths and list them in order of desirability in terms of the quantitative criterion chosen. The decision maker may then concentrate on the routes at the top of the list and choose from among them by using other, possibly subjective criteria. The technique has many uses which are well covered in an extensive literature and will not be further considered here. Numerical Measures of Risk Without risk and the reward for successfully accepting risk, there would be no business activity. In estimating the probabilities of attaining various levels of net present value (NPV) and discounted-cash-flow rate of return (DCFRR), there was a spread in the possible values of (NPV) and (DCFRR). A number of methods have been suggested for assessing risks and rewards to be expected from projects. Let us consider a proposed project in which there is a probability p1 that a net present value (NPV)1 will result, a probability p2 that (NPV)2 will result, etc. A weighted average (NPV)w, known as the expected value, can then be calculated from (NPV)w = p1(NPV)1 + p2(NPV)2 + ⋅ ⋅ ⋅

(9-105)

where p1 + p2 + ⋅ ⋅ ⋅ = 1.0. Analogous equations may be written for other additive measures of profitability such as net profit. Example 12: Expected Value of Net Profit Let us consider a contractor who stands to make a net profit of $100,000 on a contract. The cost of preparing the bid on the contract is $10,000. There are four competing contractors, each with a probability p1 = 0.25 of obtaining the contract. Thus, each contractor has a probability p2 = 0.75 of not obtaining the contract. Therefore, the expected value of the project is 0.25($100,000) + 0.75(−$10,000) = $17,500 In this case, the potential gain is 10 times greater than the potential loss.

If the potential loss can bankrupt the company, then decisions are not necessarily made on the basis of expected value even though the potential gain may be very high. Also, decisions are not necessarily made on the basis of expected value if the potential loss represents a relatively small amount of money to the company. Between these two extremes, expected value can be a very useful criterion, particularly for a company with a large number of projects.

A company may be considering a project with a very high potential rate of return and a low risk, but it may prove impossible to raise the money to start the project. Conversely, the company may be prepared to undertake an extremely risky project if the investment is trivial. Thus, the attitude of a company to risk depends on the circumstances. Money does not hold the same value for each company or each individual. A dollar may keep a pauper from starvation while being a trivial amount to the person who gave it. Attempts have been made to quantify a company’s attitude to money, risk, and uncertainty by asking business executives a number of questions such as the following: “Your company has signed a business contract with potential aftertax proceeds of $P. The probability of achieving the net gain of $P is, say, p1 = 0.75, and the probability of a net loss of $P is p2 = 0.25. If you would rather keep the contract, how much cash would you accept for your interest in it? If you would rather be released from the contract, how much cash would you pay to be released from it?” The same questions may then be asked for different values of the probabilities p1 and p2. The answers to these questions can give an indication of the importance to the company of $P at various levels of risk and are used to plot the utility curve in Fig. 9-25. Positive values are the amounts of money that the company would accept in order to forgo participation. Negative values are the amounts the company would pay in order to avoid participation. Only when the utility value and the expected value (i.e., the straight line in Fig. 9-25) are the same can net present value (NPV) and discounted-cash-flow rate of return (DCFRR) be justified as investment criteria. Since the utility curve has such a subjective basis, most companies prefer the objectivity of (NPV) and (DCFRR) over the range of the normal income and expenditure budget. Subjective methods tend to be reserved for exceptionally high risk projects. A utility curve such as that in Fig. 9-25 is specific to a certain sum of money. The curve is likely to be different for, say, P = $10,000. Figure 9-25 can only be used to consider projects that fall within the range of −$100,000 to +$100,000. Other utility curves must be used to cover projects that lie outside this range. R. O. Swalm [“Utility Theory—Insight into Risk Taking,” Harv. Bus. Rev., 44, 123–136 (November–December 1966)] found that many business executives had difficulty in appreciating fine shades of odds and confined his considerations to even-money bets. He asked various executives to state what guaranteed sum of money they considered equivalent to a gamble related to the toss of a coin. If the coin fell on one side, they would win a given sum of money; if the coin fell on the other side, they would get nothing. Swalm started by considering a sum of money equivalent to twice the maximum expenditure that the executive could authorize in 1 year. This was used to obtain a further utility. In this way, a utility curve could be sketched. Swalm chose an arbitrary utility scale based on a range of −120 utiles to +120 utiles. (NOTE: It is as incorrect to compare utiles by ratio as it is to imply that an object at 30°C is twice as hot as an object at 15°C.) Swalm found that most executives are conservative in their expenditure and that the patterns of utility curves are very similar if plotted with an ordinate range of ⫾1 unit. The unit, in this case, is the maximum authorized annual expenditure of the executive. Such curves may appear to differ quite widely when plotted in terms of absolute money values. The curves also show that executives tend to be more conservative when considering a loss than they do when considering a reduced gain. Example 13: Evaluation of Investment Priorities Using Prob-

ability Calculations A company is considering investment in one or more of three projects, A, B, and C. We wish to evaluate the investment priorities if the probabilities of attaining various net present values (NPV) are as listed in the third column of Table 9-11. Equation (9-105) gives the expected value for (NPV)w. Hence for project A, (NPV)w is computed from the data in Table 9-12 and found to be (NPV) w = 0.1($95,000) + 0.8($45,000) + 0.1(−$75,000) (NPV) w = $9,500 + $36,000 − $7,500 (NPV) w = $38,000 Corresponding values for projects B and C are calculated in the same way and are listed in Table 9-12.

INVESTMENT AND PROFITABILITY TABLE 9-12 Comparison of Projects in Terms of Expected Value and Expected Utility

Project

(NPV), Probability, $ p

Equivalent probability of winning Probable Expected Expected $100,000 utility value, $ utility, $

A

95,000 45,000 −75,000

0.1 0.8 0.1 1.0

0.80 0.34 0.03

0.080 0.272 0.003 0.355

9,500 36,000 −7,500 38,000

8,000 27,200 300 35,500

B

50,000 20,000 −60,000

0.2 0.6 0.2 1.0

0.37 0.23 0.04

0.074 0.138 0.008 0.220

10,000 12,000 −12,000 10,000

7,400 13,800 800 22,000

C

45,000 10,000 −60,000

0.1 0.6 0.3 1.0

0.35 0.20 0.04

0.035 0.120 0.012 0.167

4,500 6,000 −18,000 −7,500

3,500 12,000 1,200 16,700

In project A, the probability p = 0.1 for (NPV) = $95,000. Figure 9-25 shows that $95,000 is the amount of money that this company would pay for a 0.8 probability of gaining $100,000. There is, therefore, a 0.2 probability of losing $100,000. In this case, a probability of p = 0.1 of attaining $95,000 is equivalent to a probable utility of (0.1)(0.8) = 0.08 of gaining $100,000. Equation (9-105) can be used to calculate the expected utility if the probabilities p1, p2, etc., are replaced by the probable utilities and if the net present values (NPV)1, (NPV)2, etc., are each replaced by $100,000. For project A, the expected utility Uw is U w = [0.1(0.8) + 0.8(0.34) + 0.1(0.03)]$100,000 U w = $8,000 + $27,200 + $300 U w = $35,500 Corresponding values for projects B and C are calculated in the same way and are listed in Table 9-12. The straight line in Fig. 9-25 represents the situation in which the expected value and the expected utility are equal over the range of −$100,000 to

FIG. 9-25

Utility-function plot for $100,000.

9-31

+$100,000. In this case, decisions can be taken on the basis of the highest expected value as a routine matter. In other cases, decisions should be made on the basis of the highest expected utility. The utility curve in Fig. 9-25 represents the present attitude of management to $100,000. This curve should be updated as the company’s business position changes. In this example, the utility curve is above the straight line. This represents a tendency on the part of the company’s decision makers to gamble. When it is below the straight line, the utility curve implies conservatism. The investment priorities should be to implement project A and then, if finance is available, project B. It might appear that project C should also be considered in view of the expected utility of $16,700. However, it is better to do nothing than to implement project C. The utility of doing nothing, which is equivalent to paying $0, is read from Fig. 9-25 to be 0.17. This gives a corresponding probable utility of (1.0)(0.17)($100,000), or $17,000. This is a better result than investing in project C. In this example, the order of priorities based on expected utilities is the same as that based on expected values. However, the order of priorities is clear-cut on the basis of expected value but much less so on the basis of expected utility.

Capital is at risk until the breakeven point has been reached. It is common practice to give consideration to the discounted breakeven point (DBEP), the time at which the (NPV) is zero when discounting at the cost of capital. At any time after the (DBEP), the project will have recovered its cost and provided a greater return on the capital than the cost of capital. It is customary for management to spread risk by diversifying the activities of a company among a portfolio of projects. R. L. Reul [Chem. Eng. (London), 238, CE 120–125 (May 1970)] has defined a parameter, which he calls the measured-survival function (MSF), given by (MSF) = 1 − (1 − p)β

(9-106)

where (MSF) is the probability that a portfolio of bets with a similar strategy will at least break even, and β is the amount of one win divided by the amount of each bet. Reul has applied Eq. (9-106) to the research and development activities of a company. Equation (9-106) is based on the simplified assumption that a project either succeeds with probability p and achieves the expected reward or fails completely with probability (1 − p). Therefore, (MSF) is the probability of at least

9-32

PROCESS ECONOMICS

one success when β similar projects are undertaken and represents a conservative measure of risk. It follows that β > 1 and hence that (MSF) > p. Many projects may result in greater returns or have an increased probability of attaining a given return if more money is spent. Each alternative derivable result from a given project is treated as a separate risk in the portfolio. Research and development activities do not, in themselves, produce a salable product. Thus, they cannot directly generate a return on capital outlay. A successful research and development project is one that results in an activity that earns revenue for the company. The life cycle of the revenue from an individual product may be as shown in Fig. 9-26. This revenue has to pay not only for the successful project but for all the unsuccessful research and development activities. It is common practice to consider all R&D as a portfolio. Disbursements for R&D are relatively flexible and can be switched from less favorable to more favorable projects at short notice. When considering individual projects, β should be taken as the lesser of expected proceeds if project is successful β = ᎏᎏᎏᎏᎏ disbursement on project total expenditure on all projects over budget period β = ᎏᎏᎏᎏᎏᎏ expenditure on project over budget period Because the projects in a portfolio will usually have different probabilities of success and different rewards for success, β and p in Eq. (9-106) are conservatively estimated as follows: total annual proceeds if all projects are successful β = ᎏᎏᎏᎏᎏᎏ total annual disbursements on all projects or

total expected value of all projects p = ᎏᎏᎏᎏᎏ total proceeds if all projects are successful The expected value can be calculated from Eq. (9-105). The relationship between (MSF), p, and 1/β in Eq. (9-106) is shown graphically in Fig. 9-27. It is the responsibility of management to decide on an acceptable value of the (MSF) for its company. The value chosen will depend on the company’s attitude to risk that can be quantified in the form of a utility curve such as the one shown in Fig. 9-25, from which a value of equivalent (MSF) can be obtained. It is also the responsibility of management to estimate the probabilities for the success of individual projects after due consideration of all the data provided by the various departments. The rate of return on investment that is acceptable to management is a function of these responsibilities. Each industry has a reasonably well defined return on investment that reflects the degree of risk inherent in that industry. If management decisions are faulty, the company either will overspend or will miss opportunities. With a disbursement of $1000 in Year 0, the discounted breakeven point (DBEP) will be reached in 3 years at a compound-interest rate of 30 percent if the annual net profit ANP = $550.63 per year. Thus, a

FIG. 9-27

Measured-survival-function plot.

discounted-cash-flow rate of return (DCFRR) of 30 percent corresponds to 1 $1000 ᎏ = ᎏᎏᎏ = 0.61 β (3 years)($550.63/year) For 1/β = 0.61 and an (MSF) = 0.999, the probability of individual success is read from Fig. 9-27 to be p = 0.985. Similarly, it can be deduced that if (MSF) = 0.999 and p = 0.95, a (DCFRR) of 45 percent is required; if breakeven in 20 years is acceptable, then a (DCFRR) of only 10 percent is needed. Example 14: Estimation of Probability of a Research and Development Program Breaking Even Details of the estimates for the current research and development program of a company are given in Table 9-13. We shall estimate the probability that this portfolio will at least break even. The total annual proposed disbursement for R&D is $500,000. The effective total annual income if all projects reached their anticipated income would be $1,300,000. Therefore, β = $1,300,000/$500,000 = 2.600 Project A has an expected value of (0.95)($500,000/year) = $475,000/year; project B has an expected value of (0.90)($400,000/year) = $360,000/year; and so on. We sum these values to obtain the total expected value of the portfolio as $1,109,500 per year. Hence, p = ($1,109,500/year)/($1,300,000/year) = 0.8535 TABLE 9-13 Example of a Portfolio of Projects for a Research and Development Program

FIG. 9-26

Life cycle of products.

Project

Proposed disbursement for coming year

Annual aftertax income if successful

A B C D E F G H Totals

$125,000 100,000 100,000 80,000 50,000 20,000 20,000 5,000 $500,000

$ 500,000 400,000 125,000 100,000 60,000 30,000 70,000 15,000 $1,400,000

Probability of success 0.95 0.90 0.80 0.75 0.70 0.65 0.50 0.20

INVESTMENT AND PROFITABILITY From Fig. 9-27, for a probability of success p = 0.8535 and a value 1/β = 1/2.600 = 0.3846, the (MSF) is 99.3 percent. This is the probability that this portfolio will at least break even. Alternatively, we can substitute the values for p and β into Eq. (9-106) to get (MSF) = 1 − (1 − 0.8535)2.6 = 0.9932, or 99.32 percent The (MSF) and utility curves can be related.

Example 15: Utility-Function Curve Let us sketch a utilityfunction curve that is equivalent to the following pattern of measured-survival functions (MSF), which expresses the observed strategy of a particular manager when spending an authorized annual budget of $1,000,000: Case

Potential proceeds annually, $

(MSF), %

a b c d

Above 600,000 300,000–600,000 0–300,000 Losses

99.9 95.0 65.0 75.0

We shall plot the resultant curve on a utility scale of ⫾120 utiles against a potential gain of ⫾$1,000,000. The required axes range from −$1,000,000 per year to +$1,000,000 per year, and from −120 utiles to +120 utiles. Utiles can be compared by ratio on an absolute scale only. Hence, for purposes of calculation the axes are moved to provide a working range of $0 per year to $2,000,000 per year and 0 to 240 utiles as in Fig. 9-28. On these axes, a potential gain of $600,000 per year corresponds to an absolute amount of (600,000 + 1,000,000) = $1,600,000 per year, and a potential loss of $200,000 per year to an absolute amount of (−200,000 + 1,000,000) = $800,000 per year. a. For annual proceeds above $600,000 per year, (MSF) is 99.9 percent. If the certainty of an annual gain of $600,000 has to be abandoned in an effort to obtain an annual gain of $1,000,000, then on an absolute scale β = ($2,000,000/year)/($1,600,000) = 1.2500 With 1/β = 0.8000 and (MSF) = 99.9 percent, we find the required probability of success by solving Eq. (9-106) for p: p = 1 − (1 − 0.999)0.800 = 0.996 The utility of an amount of money is its utility when it is certain to be obtained, multiplied by its probability of being attained. On a scale in which an absolute annual income of $2,000,000 per year has a utility of 240 utiles, the utility of $1,600,000 is (0.996)(240), or 239 utiles. b. For annual proceeds between $300,000 and $600,000, (MSF) = 95 percent. If the certainty of an annual gain of $300,000 has to be abandoned to obtain an annual gain of $600,000, then, as before, 1/β = $1,300,000/year/$1,600,000/year = 0.8125 p = 1 − (1 − 0.95)0.8125 = 0.912 Since, to this manager, the utility of an absolute income of $1,600,000 is 239 utiles, the value of $1,300,000 is (0.912)(239) = 218 utiles. On the original scales, potential annual proceeds of $300,000 have a utility of (218 − 120), or 98 utiles.

9-33

c and d. Values of utility at other potential annual gains are calculated in the same way and shown graphically in Fig. 9-28.

This strategy is extremely conservative when high gains are possible but becomes less so for smaller potential gains. If potential losses are involved, the strategy is a fair one for which (NPV) would be an accurate guide for choosing alternatives. Insurance and Risk In the venture-premium method of assessment, risky investments are required to yield a rate of return that adds a premium to the cost of finance. D. F. Rudd and C. C. Watson (The Strategy of Process Engineering, Wiley, New York, 1968, p. 91) consider this relationship: im = i + ir (9-107) where i is the cost of capital, i m is the minimum acceptable interest rate of return on the investment, and i r is known as the risk rate. They suggested that each project should pay an insurance premium ir to guarantee the expected profits. The magnitude of ir is proportional to the amount of capital to be risked. It is also a function of the degree of risk involved. Working capital and capital for auxiliary facilities are assumed to be risk-free. Thus, the risk rate is applied only to the fraction of the capital investment likely to be lost if the project is unexpectedly terminated. The main objection to the venture-premium method is that the assessment of the riskiness of a project may be too subjective. This could lead to the rejection of potentially attractive proposals and the acceptance of projects that merely appear to be risk-free. Insurance is protection against risk. Commercial insurance companies minimize their own risks by covering a large number of individuals against a given risk and also by offering coverage on a wide variety of different types of risk. It is frequently quite difficult to assess the probability of success of a particular research and development project. It is much easier for an insurance company to assess its probabilities from its casualty tables. Businesses tend to provide their own insurance cover when individual claims are likely to be a small fraction of the available capital. The cost of commercial insurance is about 30 percent higher than would be necessary to cover the same risk in one’s own company. However, for low-probability, high-cost risks, most businesses prefer to insure with a commercial insurance company. Such risks include loss of plant or buildings due to fire and losses of revenue due to delays in startup or strikes. It is also becoming necessary to insure against factors not normally considered until recently. These include possible lawsuits for polluting the environment. The cost of insurance increases the annual total expense ATE. Thus, overinsurance can lead to an unnecessary decrease in profitability. The management of a company must ultimately judge its own risks. As an example, let us calculate the required risk rate for a project that is described by the following: (1) risk strategy is equivalent to an (MSF) of 99 percent, (2) payback of risk capital is 3 years, (3) cost of capital i is 10 percent, and (4) probability of complete success of the project is estimated as 95 percent. First, we calculate the value of β in Eq. (9-106). For this project, with (MSF) = 0.99 and p = 0.95, log [1 − (MSF)] log (0.01) β = ᎏᎏ = ᎏ = 1.537 log (1 − p) log (0.05) To recover this amount of capital and interest in 3 years, the average net annual cash flow ACF required is ACF = 1.537/3 = 0.5124 ($/year)/($ invested) In effect, in computing the average net annual cash flow per dollar invested, the value of fAP of Eq. (9-46) has been obtained for this example. From tables of the annuity present-worth factor fAP the value of the interest rate is found to be im = 0.25 when fAP = 0.5124 with n = 3 years. Hence, by substituting appropriate values into Eq. (9-107) and solving for the required risk rate,

FIG. 9-28

Utility-function plot illustrating managerial strategy.

ir = im − i = 25 − 10 = 15 percent

9-34

PROCESS ECONOMICS

based on the payback period of the risk capital. All capital CWC is completely recoverable without risk and requires interest only at 10 percent. The unrecovered part of the risk capital CFC attracts the additional risk interest rate of 15 percent, which should be reduced as the risk capital is written down. A different view of risk is expressed in Eq. (9-108): [1 + (DCFRR)] = (1 + i)(1 + i′r)

(9-108)

The (DCFRR) represents the return on all capital invested after such capital has been paid back, together with any interest incurred by borrowing it, and after payment of all expenses, including taxes, associated with the project. It thus represents the entrepreneurial return to the company for managing the total capital employed. If the cost of capital i is set at the best risk-free use of that capital, such as the interest rate on a bank deposit or on government bonds, etc., i′r represents the increased entrepreneurial return on the capital for taking the risks involved. This is a useful concept since the probability of achieving a given (DCFRR), and hence of a particular value of i′r, may be estimated by the methods detailed previously. We notice that i, as so defined, implies that all taxes and interest have been paid. Thus, $100 deposited in a bank at a rate of 10 percent with half of the money borrowed at 15 percent and corporation tax at 40 percent would result in a risk-free income after tax of [($100)(0.10/year) − (0.5)($100)(0.15/year)](1 − 0.40) = $1.5/year The same money invested in a project with a (DCFRR) of 10 percent would, by Eq. (9-108), obtain an entrepreneurial return i′r = 8.37 percent on the whole investment, i.e., $8.37/$100. Investment of the entrepreneur’s own money would only achieve an aftertax return of (0.1)(1 − 0.40) = 6 percent on $50, or $3/$100 of total investment. The incentive to the entrepreneur to manage the project thus corresponds to a tax-free income of $5.37/$100 of total investment. In practice, money is borrowed from more than one source at different interest rates and at different tax liabilities. The effective cost of capital in such cases can be obtained by an extension of the above reasoning and is treated in detail by A. J. Merrett and A. Sykes (Capital Budgeting and Company Finance, Longmans, London, 1966, pp. 30–48). Inflation It is currently necessary to evaluate the profitability of proposed investments whose future earnings are virtually certain to be eroded by inflation. It has been common practice to ignore the effects of inflation. This is done on the reasonable grounds that predicting the market rate of interest, and thus the appropriate discount rate for future cash flows, is difficult enough without having to worry about inflation as well. But failure at least to try to predict inflation rates and take them into account can greatly distort a project’s economics, especially at the double-digit rates that have been found throughout the world. It is the common experience that a given amount of money buys less and less of goods and services as time goes by. The problem is to express this experience quantitatively. Published figures for inflation rates are based on some particular mixture of goods and services that is chosen to represent the material wants of the average citizen. If a given quantity of this specific mixture cost $100 last year and now costs $120, then the mix has suffered a 20 percent rate of inflation. The purchasing power of the currency (i.e., of the $120) in respect of these goods and services has consequently fallen by a factor of ($120 − 100)/$120, or 16.7 percent. Two kinds of inflation can be considered: general, or open, inflation and repressed, or differential, inflation. In the first case, all costs and TABLE 9-14

prices increase at a uniform rate. Thus, the same rate of inflation will be calculated regardless of the particular mixture of goods and services chosen. In the second case, the rate of inflation will depend on the spending spectrum of the individual or company. For instance, a given company’s labor costs and material costs may inflate at different rates. To quite a large extent, inflation becomes repressed, or differential, in such fields as taxation, import control, and price restriction. The effect of inflation on the real value of future earnings from a project should not be confused with the effect of the market rates of interest on those earnings. Strictly speaking, the market interest rate and the inflation rate are not fully independent, at least according to some economic theorists. However, they are here treated as being separate. Because of each effect, a dollar of project income next year has a smaller true value than does a dollar in hand today. The interestrate effect could be offset because a dollar could be financially invested at the prevailing interest rate and the dollar plus interest earnings recouped in a year. By contrast, the inflation effect comes about simply because a dollar can buy more now than a year hence because of an irreversible rise in prices. The distinction is clarified in the following subsections. Effect of Inflation on (NPV) When computing the (NPV) for a proposed project, error arises if the actual cash flows are simply added together instead of adjusting all the values to their purchasing power in a particular year. The reason lies in the basis of (NPV) calculations. We shall rewrite Eq. (9-57) to give n ACFn (NPV) = ACF0 +  ᎏ (9-109) (1 + i)n 1 Equation (9-109) is valid for the case of no inflation. In the case of general inflation at a fractional rate ii, this equation can be written in the modified form n A CF0 (NPV) = ACF0 +  ᎏᎏ (9-110) n (1 + i) (1 + ii)n 1 Equation (9-110) enables all the net annual cash flows to be corrected to their purchasing power in Year 0. If the inflation rate is zero, Eq. (9-110) becomes identical with Eq. (9-109). The following example illustrates the effect of inflation on (NPV) as well as on the taxes the company pays. Example 16: Effect of Inflation on Net Present Value Let us consider a simplified project in which $1,100,000 of capital is spent in Year 0, $1,000,000 for fixed-capital items and $100,000 for working capital. The fixed capital is depreciated on a straight-line basis to a book value of zero at the end of Year 5. The annual sales revenue in Years 1 through 5 is $500,000. There is no inflation. The $100,000 of working capital is recovered at the end of Year 5. The taxation rate is 50 percent, and the market interest rate is 10 percent. Table 9-14 lists the cash-flow data for this project, showing that the (NPV) at the end of Year 5 is $99,326 by using Eq. (9-109). Let us modify this example by assuming that there is a general inflation rate of 20 percent per year and that the project analyst ignores the inflation and (inappropriately) applies Eq. (9-109). The revenue and expense data for this case are shown in Table 9-15, yielding an (NPV) of $431,269. When Eq. (9-109) is (inappropriately) used for the same example with various other rates of inflation, the resulting (NPV)s can be plotted as the upper line in Fig. 9-29. If the inflation is correctly taken into account by applying Eq. (9-110), the results are strikingly different. By further discounting the discounted cash flows ADCF of Table 9-15 by the fd factors corresponding to an inflation rate of 20 percent before summing, it can be seen that the project actually incurs a negative (NPV) of $208,733 in uninflated-money terms. The lower line in Fig. 9-29

(NPV) Calculations with No Inflation Revenue from sales, AS

Total Cash income, Depreciation Taxable Amount of tax at charge, income, t = 0.5, AIT expenses, ACI (= AS − ATE ATE) AD (ACI − AD) [= (ACI − AD)t]

Net cash flow Discount factor at i = 10%, fd after tax, ACF 1 (= ACI − = ᎏᎏn AIT − ATC) (1 + 0.1)

Discounted net cash flow, ADCF [= ACF( fd)]

Net present value (NPV),

Year, n

Net capital expenditure, ATC

0 1 2 3 4 5

$1,100,000 0 0 0 0 −100,000

0 $500,000 500,000 500,000 500,000 500,000

0 $100,000 100,000 100,000 100,000 100,000

−$1,100,000 300,000 300,000 300,000 300,000 400,000

−$1,100,000 272,727 247,935 225,393 204,903 248,368

−$1,100,000 −827,273 −579,338 −353,945 −149,042 +99,326

0 $400,000 400,000 400,000 400,000 400,000

0 $200,000 200,000 200,000 200,000 200,000

0 $200,000 200,000 200,000 200,000 200,000

0 $100,000 100,000 100,000 100,000 100,000





1.00000 0.90909 0.82645 0.75131 0.68301 0.62092

n

=  A  DCF

0

INVESTMENT AND PROFITABILITY TABLE 9-15

(NPV) Calculations with Inflation Present But Not Allowed For

Year, n

Net capital expenditure, ATC

Revenue from sales, AS

Total expenses, ATE

Cash income, ACI

0 1 2 3 4 5

$1,100,000 0 0 0 0 −100,000

0 $500,000 600,000 720,000 864,000 1,036,800

0 $100,000 120,000 144,000 172,800 207,360

0 $400,000 480,000 576,000 691,200 829,440

Depreciation charge, AD

Taxable income, (ACI − AD)

Amount of tax at t = 0.5, AIT

0 $200,000 200,000 200,000 200,000 200,000

0 $200,000 280,000 376,000 491,200 629,440

0 $100,000 140,000 188,000 245,600 314,720

extends the example by assuming other rates of inflation. Figure 9-29 shows that the effect of inflation, if not taken into account, is to make a project seem more profitable than it actually is. Table 9-15 shows that the total amount of tax actually paid over the 5-year period was $988,320. This becomes $534,272 in uninflated-money terms when the tax for each year is corrected to its purchasing power in Year 0, using fd factors for the 20 percent inflation rate employed for the example. Calculations for other rates of inflation can also be made, and the results plotted as in Fig. 9-30.

This confirms that although the tax paid will increase with inflation, the gain to the government is more apparent than real. It is interesting to note that although the tax paid corrected to its purchasing power in Year 0 is almost constant irrespective of the inflation rate, it does go through a maximum at an inflation rate of about 17 percent in this example. Effect of Inflation on (DCFRR) A net annual cash flow ACF will have a cash value of ACF(1 + i) 1 year later if invested at a fractional interest rate i. If there is inflation at an annual rate ii, then an effective rate of return or interest rate ie can be defined by the equation ACF(1 + ie) = [ACF(1 + i)]/(1 + ii)

FIG. 9-29

9-35

Effect of inflation rate on net present value for a project.

(9-111)

Net cash flow, ACF

Discount factor at i = 10%, fd

Discounted net cash flow, ADCF

Net present value (NPV)

−$1,100,000 300,000 340,000 388,000 445,600 614,720

1.00000 0.90909 0.82645 0.75131 0.68301 0.62092

−$1,100,000 272,727 280,993 291,508 304,349 381,692

−$1,100,000 −827,273 −546,280 −254,772 +49,577 +431,269

which can be simplified and rewritten to give ie = i − ii − ie ii

(9-112)

In the context of the discounted-cash-flow rate of return, Eq. (9-112) becomes ie = (DCFRR) − ii − ie ii

(9-113)

In this equation, (DCFRR) can be viewed as the nominal discountedcash-flow rate of return uncorrected for inflation and ie can be thought of as the true or real discounted-cash-flow rate of return. Instead of using Eq. (9-113), it is unfortunately common practice to try to obtain the true or effective rate of return by calculating the nominal (DCFRR), based on actual net annual cash flows uncorrected for inflation, and then subtracting the inflation rate from it as if ie = (DCFRR) − ii

(9-114)

Equation (9-113) shows that Eq. (9-114) is only approximately true and should be used, if at all, solely for low interest rates. Let us consider the case of a nominal (DCFRR) of 5 percent and an inflation rate of 3 percent. Equation (9-14) yields an approximate effective return rate of 2 percent, compared with the real effective rate of 1.94 percent given by Eq. (9-113); i.e., there is an error of 3.1 percent. Now let us consider the case of a nominal (DCFRR) of 25 percent and an inflation rate of 23 percent. Equation (9-114) yields an approximate effective return rate of 2 percent, compared with 1.63 percent from Eq. (9-113); in this case, the error that results is 22.7 percent. Inflation, (DCFRR), and Payback Period More insight into the effect of inflation on (DCFRR) calculations can be gained by considering the payback period (PBP), which is defined as the elapsed time necessary for the positive aftertax cash flows from the project to

FIG. 9-30

Effect of inflation rate on taxes paid for a project.

9-36

PROCESS ECONOMICS

recoup the original fixed-capital expenditure. In this definition, the cash flows are not discounted to allow for the market rate of interest or for the inflation rate, so that a project with a given (PBP) could show various values for its (DCFRR) and a given (DCFRR) could pertain to projects with various payback periods. We shall consider the simple case of (1) a single capital expenditure made immediately before the start of production and (2) equal positive net annual cash flows ACF in all the productive years of the project. For this case, Eq. (9-109) can be rewritten in terms of the payback period and the (DCFRR) as follows: n 1 (PBP) =  ᎏᎏn (9-115) [(1 + (DCFRR)] 1

ignores it and mistakenly takes a (DCFRR) of the project at its nominal value instead of converting it to an ie. Equation (9-115) rearranged into the form ie = [1 + (DCFRR)]/(1 + ii) − 1

(9-116)

shows that in order to achieve an ie of 20 percent when the general inflation rate is likewise 20 percent, a project must generate a nominal (DCFRR) of 44 percent. This is the basis for the uppermost line in Fig. 9-31. Other lines pertaining to other rates of inflation could be plotted in the same way. Let us assume that 20 percent inflation prevails but that the analyst

shows that with a nominal (DCFRR) of 20 percent and a general inflation rate of 20 percent, the true or effective rate of interest is zero. This is the basis for the lowest line in Fig. 9-31. Points for lines corresponding to other rates of inflation could be plotted onto that figure. Plots similar to Fig. 9-31 can be drawn for other (DCFRR) values. Figure 9-31 shows that the elapsed time necessary to reach a nominal (DCFRR) for a given project decreases sharply with inflation. This figure, like Fig. 9-29, shows that the effect of inflation is to make a project seem more profitable than it actually is. The magnitude of the effect comes through even more clearly in Fig. 9-32, a plot of the time needed to reach a nominal (DCFRR) of 20 percent against the inflation rate for various values of (PBP). This plot also shows that the longer the payback period, the greater the increase in apparent profitability of the project. The true rates of return ie can be calculated from Eq. (9-116) to be 20, 9.09, 0, and −7.69 percent respectively for general inflation rates of 0, 10, 20, and 30 percent. Thus, although the time required for a project with a payback period of 4 years to reach a nominal (DCFRR) of 20 percent is reduced from almost 9 years under conditions of no inflation to less than 3a years for 30 percent inflation, the true rate of return that prevails for the latter condition is −7.69 percent, implying that the project loses money in real terms. It is interesting to note that, in order to reach a real (DCFRR) or ie of 20 percent within a reasonable project lifetime when the general inflation rate is 20 percent, it follows from Fig. 9-31 that the payback period for the project must not be much in excess of 2 years. Although it is difficult to carry out economic-feasibility studies on projects in a time of high inflation, it is important to try to predict inflation rates and allow for them in such studies. When different people talk about inflation, they often adopt different concepts without realizing it. The area of conceptual uncertainty

FIG. 9-31 Effect of inflation rate on the relationship between the payback period and the discounted-cash-flow rate of return.

FIG. 9-32

The relationship set out in Eq. (9-115) can also be viewed via a different chain of causality with (DCFRR) as a given parameter, (PBP) as the independent variable, and n as the variable whose value is being sought. Such an approach is the basis for the lines in Fig. 9-31, each of which shows the number of years of project life required to achieve an effective interest rate or a (DCFRR) of 20 percent by projects having various payback periods. The three lines differ from each other with respect to the matter of inflation. If there is no inflation, then the middle line pertains. Because there is no inflation, the nominal (DCFRR) is equal to or identical with ie, the real discounted-cash-flow rate of return, as can be seen from the relationship expressed in Eq. (9-113). When inflation does exist, the relevant parameter is ie, which is different from the nominal (DCFRR). Equation (9-113), manipulated into equivalent form, (DCFRR) = (1 + ie)(1 + ii) − 1

Adverse effect of inflation for higher payback periods.

INVESTMENT AND PROFITABILITY

9-37

Relationship between measured-survival function, number of payback periods, and contribution efficiency.

FIG. 9-33

can be said to lie somewhere between the upper and lower lines shown on Fig. 9-31 in most cases. Inflation and the (MSF) By applying the measured-survivalfunction concept to manufacturing projects rather than to research and development, we can define a modified (MSF) for a given project as (MSF) = 1 − (1 − η)β

(9-117)

Here, β is the number of payback periods that have elapsed since the project started to generate positive net annual cash flows ACF up to any given year n since project startup. It is given by n=n

 (A

)

CF n

n=0

β = ᎏᎏ CFC − S

(9-118)

If all the net annual cash flows in Eq. (9-118) are based on their purchasing power in Year 0, then β is independent of inflation. As for the contribution efficiency η, it is the ratio of (1) the annual profit that can actually be achieved in a given year for a given sales volume to (2) the profit that could be obtained if no repayment of capital or interest were required and all fixed-expense items were credited free to the project. It is defined by η = [R(cs − cVE) − AFE]/[R(cs − cVE)]

(9-119)

where R is the annual production rate or sales volume in physical units, cS is the sales price per unit, cVE is the variable production and selling cost per unit, and AFE is the annual fixed cost. If the project gets a “free ride,” i.e., if AFE is zero, then η takes on its maximum possible value of unity. Conversely, if the project and its production rate are only at the breakeven point, then η becomes zero. Therefore, contribution efficiency can be regarded as a measure of the probability of success for the project. The relationship between the number of payback periods, the contribution efficiency, and the measured-survival function as set out in Eq. (9-117) is plotted in Fig. 9-33. The contribution efficiency defined by Eq. (9-119) may vary from year to year. In that case, Eq. (9-117) can be written in the modified form (MSF) = 1 − [(1 − η1)(1 − η2) ⋅ ⋅ ⋅ (1 − ηn)]β/n

manufacturing projects. That decision reflects and helps quantify the company’s attitude toward risk. Thus, (MSF) should in practice be regarded as a given or predetermined variable, and Eq. (9-117) accordingly becomes more useful if it is rearranged. For instance, the values of contribution efficiency for a given value of (MSF) are related to the number of elapsed payback periods by η = 1 − [1 − (MSF)]1/β

(9-121)

If the acceptable (MSF) is 0.9, this can be satisfied by a project having η = 0.9 and β = 1, or a project having η = 0.684 and β = 2, and so on. Once Eq. (9-121) has been used to calculate a required contribution efficiency [given the (MSF) and the expected number of payback periods of project life], Eq. (9-119) can be applied to determine the necessary selling price if R, cVE, and AFE are known. Similarly, Eq. (9-119) can be used to find the required production rate if cS is known. It is also possible to combine (MSF) considerations with evaluation of the true discounted-cash-flow rate of return (DCFRR) by using Eq. (9-62). The relationship of Eq. (9-59) is independent of inflation if all money values are based on those prevailing in the startup year. For this case, Fig. 9-34 shows the true (DCFRR) reached in a given time, expressed as the number of elapsed payback periods β for various values of the payback period. Let us consider a project having a contribution efficiency of 0.684 and a payback period of 3 years. Figure 9-33 shows that when two payback periods have elapsed, a measured-survival function of 0.9 has

(9-120)

where η1, η2, . . . ηn are the contribution efficiencies in Years 1, 2, . . . n respectively. As in the case of the (MSF) defined by Reul for research and development projects, it is the responsibility of management in a particular manufacturing company to decide on an acceptable level of (MSF) for

FIG. 9-34 Real discounted-cash-flow rate of return against number of payback periods for various payback periods.

9-38

PROCESS ECONOMICS

been attained. In addition, Fig. 9-34 shows that the discounted-cashflow rate of return reached at that time is 24 percent. Effects of Differential Inflation Inflation can be general or differential. In the first case, all costs and prices increase at a uniform rate. In the second, government controls and other factors cause the various costs and prices to inflate at different rates. The onset of general inflation does not change the value of the contribution efficiency η, as can be seen from Eq. (9-119), and it does not affect the value of β if the cash flows in Eq. (9-118) are converted to their purchasing power in Year 0. Thus, general inflation does not cause the measured-survival function to change. Differential inflation, on the other hand, can affect the measuredsurvival function. We shall assume, for instance, that the sales price per unit product cS in Eq. (9-119) is frozen at a constant level while some or all of the production costs are allowed to rise. This causes the value of η to decrease; therefore, (MSF) likewise decreases, as can be seen from Eq. (9-117). Let us consider the effect of differential inflation on the overall profitability of the project of the last example. The effect of general inflation on this project showed that the apparent profitability rises sharply, to an (NPV) of $431,269 at a general inflation rate of 20 percent. However, when the cash flows of the (NPV) are properly corrected to their purchasing power in Year 0, the (NPV) instead becomes $208,733. The effect of differential inflation on this project emerges in Fig. 9-35, with all (NPV)s corrected to their purchasing power in Year 0. The top line shows (NPV) for various rates of general inflation. The bottom line shows (NPV) for the differential-inflation case in which only the costs are allowed to increase while product selling price and thus cash income remain constant from year to year. The middle line shows the effect of general inflation when the price rises are delayed by 1 year. The figure confirms that both of these situations take away from the attractiveness of the project.

The effect upon total taxes paid, when they are corrected to their purchasing power in Year 0, is shown in Fig. 9-36. Differential inflation not only decreases the profitability of the project to its owner but also decreases the revenue received by the taxing authority. The method of calculation is identical to that of the earlier example. Another instance of differential inflation occurs when the prices of goods and services rise uniformly but the cost of borrowing money, the interest rate charged on a loan, does not rise. If the fractional inflation rate is ii, a fractional interest rate iL on a loan can be corrected to an effective rate of interest by Eq. (9-116) with iL substituted for (DCFRR). The effect of various amounts of loan, borrowed at various interest rates iL, on the net present value of a particular, fairly simple project is shown in Fig. 9-37. Thus, if $25,000 were borrowed at an interest rate of 15 percent for the project, the (NPV) would be about $43,000 at a zero inflation rate. But if the inflation for goods and services ii is 10 percent, the effective interest rate for that loan can be calculated from Eq. (9-116) to be only 4.55 percent. It is seen from Fig. 9-37 that this increases the (NPV) of the project to $48,000. This confirms the economic advantage of borrowing at a fixed interest rate in a time of general inflation. A topical aspect of differential inflation is the question of energy costs. Will the cost of a particular fuel rise or fall in relation to prices in general, and if so, what effect will this have on the economics of a project? Example 17: Effect of Fuel Cost on Project Economics A process unit is heated by gas. We assume that $100 spent on energyconservation measures for this particular unit at the end of 1980 would save 200 therms (21.1 GJ) of gas energy in each subsequent year. If the cost of gas in 1980 is $x per therm, the annual dollar savings at 1980 prices is $200x. The (NPV) at the end of year n for this project is n (200)x (NPV) = −100 +  ᎏn (1 + i) 1

if the appropriate discount factor is i. This is independent of inflation provided that the cost of gas rises in line with any general rate of inflation. However, if the real cost of gas rises at a fractional annual rate r over and above the general inflation rate, it should be modified into the form n (200x)(1 + r)n (NPV) = −100 +  ᎏᎏ (1 + i)n 1 This equation confirms that as the gas price rises because of inflation, the attractiveness of the conservation project also rises.

FIG. 9-35

Effect of differential inflation on inflation-corrected net present

value.

FIG. 9-36

Effect of differential inflation on inflation-corrected tax revenue.

FIG. 9-37

Effect of loan interest rate on the net present value of a project.

ACCOUNTING AND COST CONTROL

9-39

ACCOUNTING AND COST CONTROL Principles of Accounting Accounting is the art of recording business transactions in a systematic manner. Financial statements are both the basis for and the result of management decisions. Such statements can tell managers or engineers a great deal about their company, provided that they can interpret the information correctly. Since a fair allocation of costs requires considerable technical knowledge of operations in the chemical-process industries, a close liaison between the senior process engineers and the accountants in a company is desirable. Indeed, the success of a company depends on a combination of financial, technical, and managerial skills. Accounting is also the language of business, and the different departments of management use it to communicate within a broad context of financial and cost terms. Engineers involved in feasibility studies and detailed process evaluations are dependent for financial information on the company accountants, especially for information on the way in which the company intends to allocate its overhead costs. It is vital that engineers correctly interpret such information and that they can, if necessary, make the accountants understand the effect of the chosen method of allocation. The method of allocating overheads can seriously affect the assigned costs of a project and hence the apparent cash flows for that project. Since these cash flows are used to assess profitability by the net-present-value (NPV) and discounted-cash-flow-rate-of-return (DCFRR) methods, unfair allocation of overhead costs can result in a wrong choice between alternative projects. In addition to understanding the principles of accountancy and obtaining a working knowledge of its practical techniques, engineers should be aware of possible inaccuracies of accounting information in the same way that they allow for errors in any technical data. At first acquaintance, the language of accountancy appears illogical to most engineers. Although accountants normally express themselves in tabular form, the basis of all their practice can be simply expressed by Capital = assets − liabilities (9-122) Equation (9-122) can alternatively be written as Assets = capital + liabilities (9-123) Capital, often referred to as net worth, is the money value of the business, since assets are the money values of things the business owns while liabilities are the money values of the things the business owes. Most engineers have great difficulty in thinking of capital (also known as ownership) as a liability. This is easily overcome once it is realized that a business is a legal entity in its own right, owing money to the individuals who own it. This realization is absolutely essential when considering large companies with stockholders and is used for consistency even for sole ownerships and partnerships. If an individual puts up $10,000 capital to start a business, then that business has a liability to repay $10,000 to the individual. It is even more difficult to think of profit as being a liability. Profit is the increase in money value available for distribution to the owners and effectively represents the interest obtained on the capital. If the profit is not distributed, it represents an increase in capital by the normal concept of compound interest. Thus, if the individual’s business makes a profit of $5000, the liability to the individual is increased to $15,000. With this concept in mind, Eq. (9-123) can be expanded to Assets = capital + liabilities + profit (9-124) where the capital is considered as the cash investment in the business and is distinguished from the resultant profit in the same way that principal and interest are separated. Profit (as referred to above) is the difference between the total cash revenue from sales and the total of all costs and other expenses incurred in making those sales. With this definition, Eq. (9-124) can be further expanded to Assets + expenses = capital + liabilities + revenue from sales (9-125) Engineers usually have the greatest difficulty in regarding an expense as being equivalent to an asset, as is implied by Eq. (9-125). Let us consider a one-person business. We assume for a given period

a profit of $5000 and total expenses excluding the individual’s earnings of $8000. Also we assume that the individual’s labor to the business in this period is worth $12,000. The revenue required from sales would be $25,000. Effectively, the individual has made a personal income of $17,000 in the period but has apportioned it to the business as $12,000 expense for the individual’s labor and $5000 return on capital. In larger businesses, there will also be those who receive salaries but do not hold stock and, therefore, receive no profits and stockholders who receive profits but no salaries. Thus, the difference between expenses and profits is very practical. The period covered by the published accounts of a company is usually 1 year, but the details from which these accounts are compiled are entered daily in a journal. The journal is a chronological listing of every transaction of the business, with details of the corresponding income or expenditure. For the smallest businesses, this may provide sufficient documentation, but in most cases the unsystematic nature of the journal can lead to computational errors. Therefore, the usual practice is to keep accounts that are listings of transactions related to a specific topic such as “purchase-of-oil account.” This account would list the cost of each purchase of oil, together with the date of purchase, as extracted from the journal. Principles of Double-Entry Accounting Many of the accounts involve both income and expenditure. The general practice is to keep accounts by the double-entry system, which may be summarized by Debits = credits

(9-126)

The principle of double entry dates from the fifteenth century and is based on the premise that every transaction involves a giver and a receiver of value. Double entry requires that each transaction be entered into two accounts, the convention being that the account of the giver is credited and the account of the receiver is debited with the same amount of money, as noted in the journal. For convenience, each account is divided centrally, and the debit items are entered on the right-hand side. It is also usual to provide a cross-reference to the journal entry so that errors and omissions can be checked. Let us consider the purchase of $50,000 worth of plant equipment by company A, paid for by check. The accounting entries are: debit the plant-equipment account $50,000, and credit the bank account $50,000. The plant-equipment account is then said to have a debit balance of $50,000, and the bank account a credit balance of $50,000, if these happen to be the only entries. If company A then sells $100,000 worth of product that is paid for by check, the accounting entries are: credit the sales account $100,000, and debit the bank account $100,000. The bank account will now have a debit balance of ($100,000 − $50,000) = $50,000, and the sales account a credit balance of $100,000, if this happens to be the only sale to date in the accounting period. In principle, the debiting and crediting of accounts are relatively straightforward. However, a great deal of practice is essential in order to achieve proficiency. Although it is not at all necessary for engineers to compete with professional accountants in this field, engineers should appreciate what accountants do and why they do it. Of the accounts considered in the preceding illustrations, the plantequipment and bank accounts are asset accounts, and the sales account is a liability account. To increase an asset, debit the asset account; to increase a liability, credit the liability account. Conversely, to decrease an asset, credit the asset account; to decrease a liability, debit the liability account. Closing the Books At the end of the accounting period, the individual accounts are closed by balancing each in accordance with Eq. (9-126). The balances are transferred either to the balance sheet in the case of capital expenditure or to the income statement in the case of revenue expenditure. An alternative name for the balance sheet is the position statement; the income statement is also called the trading and profit-and-loss account. The purpose of capital expenditure, such as the purchase of a piece of plant equipment for $50,000, is to earn future revenue. In contrast, the purpose of revenue expenditure is to maintain existing business.

9-40

PROCESS ECONOMICS

TABLE 9-16

Income Statement for ABC Company

Revenue Sales revenue Other revenue

$1,900,000 100,000 $2,000,000

Expenses Raw materials Wages Utilities Depreciation Other expenses Income taxes

953,000 185,000 44,000 68,000 376,000 194,000 1,820,000

Net profit (after tax)

$ 180,000

Revenue expenditure includes the direct material costs and direct labor costs incurred in the manufacture of a product, together with the associated overheads that include maintenance of the plant. Since these expenses are debits, the debit balance for a given accounting period is obtained by adding up the debit balances from each individual expenditure account. Similarly, since revenues from sales and other income are credits, the credit balance for a given accounting period is obtained by adding up the credit balances from each individual income or revenue account. To ascertain profit or loss (calculated as income minus expenditure for a given accounting period), income and expenditure must be matched. For example, any rent paid in advance beyond the current accounting period should not be included in the profit or loss calculation. Similarly, goods sold but not yet paid for in a given accounting period should not be included in the revenue total for that period. An income statement such as the one shown in Table 9-16 is used to obtain the profit or loss for a given period. The debit and credit balances of all the accounts that do not represent expenditure or income for a given accounting period are entered as assets and liabilities in a balance sheet such as that shown in Table 9-17. There is no rigid format for either the income statement or the balance sheet. Tables 9-16 and 9-17 show common layouts for the income statement and balance sheet respectively, but these are not the only forms. For example, vertical balance sheets, with the assets listed above the liabilities and equity, are also popular. Some expenditures are partly capital and partly revenue. For example, repair and improvement work may be done on a plant simultaneously. In this case, the repair work should be classified as revenue expenditure and the plant-improvement work as capital expenditure. TABLE 9-17

Accounting Concepts and Conventions Accounting is based on the following concepts: (1) money measurement, (2) business entity, (3) going concern, (4) cost, and (5) matching. Concept 1. “Money measurement” means that only those facts that can be represented in monetary terms are recorded. The balance sheet and income statement for a company give no indication as to what might happen in the future. The company may be about to be successfully sued for a large sum of money, or a competitor may be launching a new product that will seriously reduce future sales of the company’s products. Concept 2. “Business entity” means that accounts are kept for the company quite independently of the people who may own the company. For example, if an individual puts an additional $10,000 into a one-person business, the accounts show that the business is $10,000 richer. They do not show that the individual’s personal wealth has been depleted by $10,000. Concept 3. “Going concern” means that the accounting is based on the premise that the business will continue indefinitely. It is most unlikely that the values of the assets shown in the balance sheet are what the assets would realize if sold. No attempt is made in normal accounting to measure the value of the business to a potential buyer. Concept 4. “Cost” means that the assets are normally shown in the balance sheet at cost price together with their subsequent depreciation. Some assets such as land may be considerably more valuable than when originally purchased, but no indication of this is given in the balance sheet. However, some governments now require a note giving the current estimated value of the land. Concept 5. “Matching” means that the revenue in a given accounting period should correspond to the expenses for that accounting period. Accounting is also based on the following conventions: (1) materiality, (2) conservatism, or prudence, and (3) consistency. Materiality deals with determining whether certain expenditures will have a significant effect on a company’s accounting procedures. This is a matter of judgment that is to be made by each company. Obviously, the purchase of a vehicle is a material item, but writing paper or tools for maintenance are less obvious. Although such items may last well beyond the current accounting period, it may not be worth the accounting effort to treat them as material items. Some companies will treat a particular item as capital; other companies, as expenditure. Clearly, the purchase of a piece of equipment costing, say, $1000, will be regarded as less material by a giant company than by a small one. Conservatism, or prudence, means monetary values that tend to understate rather than overstate the profit are taken. Consistency means that accounting items are normally treated in

Balance Sheet for XYZ Company Liabilities and stockholders’ equity (thousands of dollars)

Assets (thousands of dollars) Current assets Cash Notes and accounts receivable Inventories: Finished products Work in process Raw materials and supplies (at cost) Total inventories Total current assets Investments and long-term receivables (at cost) Property, plant, and equipment (at cost) Land Buildings Machinery and equipment Less accumulated depreciation Net property, plant and equipment Prepaid and deferred, charges Total assets

$ 38,893 110,740 17,396 56,690 35,790 109,876 259,509 94,009 6,110 63,848 106,185 176,143 75,163 100,980 6,094 $460,592

Current liabilities Notes payable Accounts payable and accrued liabilities Accrued taxes Total current liabilities Long-term liabilities Deferred income taxes Other deferred credits Stockholders’ equity Common stock, $20 par value Shares authorized, 7,750,000 Shares issued, 4,794,450 Capital in excess of par value of common stock Retained earnings Total stockholders’ equity

Total liabilities and stockholders’ equity

$ 34,507 106,433 7,264 148,204 67,677 13,225 2,307

95,889 31,798 101,492 229,179

$460,592

ACCOUNTING AND COST CONTROL the same way over an indefinite number of years. For example, an individual item would not be treated as an expenditure during one year and as a capital item during the next year without good reason being given. Balance Sheet The balance sheet, also called the position statement, presents an accounting view of the financial status of a company at a particular point in time. A typical balance sheet is shown in Table 9-17. Although a balance sheet has two sides that balance, it is not part of the double-entry system. In fact, it is not an account but rather a statement listing all the assets of a company and the various claims against these assets on the last day of the accounting period. The assets must be equal to the claims against them at all times. Those who have claims against the assets are the owners (stockholders in a business corporation) and the people to whom the company owes money. In the case of the latter, the company is said to have liabilities to its creditors. The total claim against the assets is often labeled “liabilities and owners’ equity.” Assets are classified as current or fixed, and liabilities as current or long-term. Fixed assets are material items that have a relatively long life and normally include land, buildings, plant, vehicles, etc. They are held for the specific purpose of earning revenue and are not for sale in the normal course of business. Current assets include cash and those items that can be fairly easily converted into cash, such as rawmaterials inventories, etc. In contrast to fixed assets, current assets are acquired for the specific purpose of conversion into cash in the normal course of business. However, what is regarded as a fixed asset by one type of company might be regarded as a current asset by another. For example, a chemical company would normally classify its vehicles as a fixed asset. However, a company whose primary business was to sell vehicles would classify them as a current asset. Similarly, the distinction between current and long-term liabilities is also not clear-cut. Current liabilities include accounts payable (money owed to creditors), taxes payable, dividends payable, etc., if due within a year. Long-term liabilities include deferred income taxes, bonds, notes, etc., that do not have to be paid within a year. The owners’ equity includes the par, or face, value of the capital received from stockholders and any retained earnings. The balance sheet shows only the nominal value and not the current or real value of this capital. A balance sheet includes items that are not regarded as assets or liabilities in normal language, such as expenditures carried forward and accumulated profits. Accountants regard assets as resources that have not yet been used up. Assets are normally shown on the balance sheet at cost minus accumulated depreciation. In this sense, the depreciation charge for an accounting period is the means of converting a part of an asset into a current expenditure that is then listed as an expense in the income statement. Let us consider plant equipment costing $1 million and purchased on Jan. 1, 1988. Table 9-18 shows the provision for the depreciation account for 1988, 1989, and 1990 for straight-line depreciation, assuming a service life of 10 years and zero scrap value. The credit entries of $100,000 for the depreciation in each year are balanced by the depreciation charge of $100,000 debited to the income statement (or trading and profit-and-loss account) in each year. Table 9-19 shows the correTABLE 9-18 Account

Provision for Depreciation of Plant-Equipment

1988 Dec. 31 Balance carried down

$100,000

Jan. 1 Balance brought down Dec. 31 Debited to income statement

0 $100,000

1989 Dec. 31 Balance carried down

$200,000 $200,000

Jan. 1 Balance brought down Dec. 31 Debited to income statement

$100,000 100,000 $200,000

1990 Dec. 31 Balance carried down

$300,000 $300,000

Jan. 1 Balance brought down Dec. 31 Debited to income statement

$200,000 100,000 $300,000

TABLE 9-19

9-41

Balance-Sheet Entries

As of Dec. 31, 1988 Plant equipment at cost Less depreciation to date

$1,000,000 100,000 $900,000

As of Dec. 31, 1989 Plant equipment at cost Less depreciation to date

$1,000,000 200,000 $800,000

As of Dec. 31, 1990 Plant equipment at cost Less depreciation to date

$1,000,000 300,000 $700,000

sponding entries in the balance sheets for the years 1988, 1989, and 1990. Entries for subsequent years are made in the same way. A balance sheet is true only for one particular point in time; it tells nothing about the trends in a company. However, by comparing balance sheets for successive years, management can follow changes in the various items. If the observed trend is undesirable, management can take corrective action. Since the accounting period of 1 year is long for most businesses, it is usual to draw up balance sheets at more frequent intervals for control purposes. These may be less formal than those issued annually to the stockholders. In general, balance sheets are less useful to management than are income statements. Income Statement Income statements range from the very simple presentation shown in Table 9-16 to the more informative and more complex presentation shown in Table 9-20. The income statement shows the revenue and the corresponding expenses that were incurred to earn that revenue over a period of time. It is the most obvious measure of the efficiency of a business. Although published income statements are normally for 1-year periods, many companies use monthly income statements for internal purposes. Income statements are very useful tools to assist management in controlling a business and planning for the future. Since management needs to follow the trends of the normal expenses, extraordinary expenses such as those incurred as a result of a major fire or flood should be shown separately. If revenue and expenses are not properly matched, an understatement or an overstatement of profit may occur. If raw materials were previously purchased at a lower cost than their current cost, profit will be overstated. Any overstatement of profit will mean that more tax will be paid. One of the most important items in an income statement is depreciation expense. Although depreciation should not be thought of as a means to build up a fund to replace plant, it nevertheless does enable money to be retained in the business by reducing the profit available for distribution to stockholders. It is of course a duty of both accountants and management to see that sufficient money is retained in the business to replace assets and to invest such money in other processes or outside investment. A further duty of accountants and management is to ensure that the company always has sufficient working capital to enable it to carry on its business. Types of Accountancy The traditional work of accountants has been to prepare balance sheets and income statements. Nowadays, accountants are becoming increasingly concerned with forward planning. Modern accountancy can roughly be divided into two branches, financial accountancy and management or cost accountancy. Financial accountancy is concerned with stewardship. This involves the preparation of balance sheets and income statements that represent the interest of stockholders and are consistent with existing legal requirements. Taxation is an important element of financial accounting. Management accounting is concerned with decision making and control. This is the branch of accountancy closest to the interest of most process engineers. Management accounting is concerned with standard costing, budgetary control, and investment decisions. Accounting statements present only facts that can be expressed in financial terms. They do not indicate whether a company is developing new products that will ensure a sound business future. A company

9-42

PROCESS ECONOMICS

TABLE 9-20 Income Statement for a Mature Year for a New Chemical Product, Produced at 10 Million lb/Year

Revenue from annual sales AS Direct manufacturing expense ADME Raw materials Catalysts and solvents Operating labor Operating supervision Utilities Operating maintenance Operating supplies Royalties and patents Total ADME Indirect manufacturing expense AIME Payroll overhead Central laboratory General plant overhead Packaging and storage Property taxes Insurance Total AIME

Unit values, cents/lb

% sales revenue, %

20.00

100.0

$ 884,000 69,000 102,000 20,000 22,000

8.84 0.69 1.02 0.20 0.22

44.2 3.4 5.1 1.0 1.1

21,000 4,000 10,000 $1,132,000

$1,132,000

0.21 0.04 0.10 11.32

1.1 0.2 0.5 56.6

$132,000

0.28 0.10 0.52 0.22 0.14 0.06 1.32

1.4 0.5 2.6 1.1 0.7 0.3 6.6

$1,264,000

12.64

63.2

68,000

0.68

3.4

74,000 124,000

0.74 1.24

3.7 6.2

40,000 10,000

0.40 0.10

2.0 0.5

308,000

0.60 3.08

3.0 15.4

$1,640,000 $360,000

16.40 3.60

82.0 18.0

$2,000,000

28,000 10,000 52,000 22,000 14,000 6,000 $132,000

Total manufacturing expense (excluding depreciation) AME Depreciation ABD Other expenses Administration Sales and shipping Advertising and marketing Technical service Research and development Total other expenses Total expense ATE Net annual profit ANP

60,000 308,000

net annual profit (ROA) = ᎏᎏ 100 (9-129) total assets where (ROA) is called the return on assets. In the second case, Eq. (9-128) can be written as net annual profit (ROE) = ᎏᎏᎏ 100 (9-130) stockholders’ equity where (ROE) is the return on equity. Asset-turnover ratio (ATR) is a commonly used measure of company performance, defined as revenue from annual sales (ATR) = ᎏᎏᎏ 100 (9-131) total assets A comparison between Eqs. (9-127), (9-129), and (9-131) shows that (ROA) = (ATR)(PM) (9-132) Thus (ROA) can be improved by increasing either (ATR) or (PM). A variation of Eq. (9-131) is the fixed-asset turnover ratio (FATR), defined as revenue from annual sales (FATR) = ᎏᎏᎏ 100 (9-133) fixed assets Clearly, (FATR) is of less value than (ATR) when applied to companies that use relatively large amounts of working capital. The (FATR) is the inverse of the capital ratio (CR) for single projects. (CR) is defined as (CR) = CFC / AS (9-134) where CFC is the fixed-capital cost for a green-fields (grass-roots) site and AS is the revenue from annual sales. The fixed assets in Eq. (9-133) and those included in the total assets in Eqs. (9-129) and (9-131) are usually taken at their written-down, or book, value, which may differ significantly from their market value. This is one disadvantage in using Eqs. (9-129), (9-131), and (9-133). The revenue from annual sales referred to in Eqs. (9-127), (9-131), and (9-132) is normally taken to be the gross turnover, which includes intergroup sales. However, intergroup sales are eliminated in consolidated or group accounts. Again, revenue from annual sales must be clearly defined before comparisons are made with other companies. Let us consider the simplified balance-sheet or position statement shown in Table 9-21. Essentially, total assets are related to liabilities and stockholders’ equity by Total assets = stockholders’ equity + total debt

(9-135)

Equation (9-135) can also be written as may have impressive current financial statements and yet be heading for bankruptcy in a few years’ time if provision is not being made for the introduction of sufficient new products or services. Financing Assets by Equity and Debt Financial Ratios Probably the most commonly mentioned ratio is the profit margin (PM), defined as net annual profit (PM) = ᎏᎏᎏ 100 (9-127) revenue from annual sales Another common ratio is the return on investment (ROI), defined as net annual profit (ROI) = ᎏᎏ 100 (9-128) investment In both Eq. (9-127) and Eq. (9-128), the net annual profit can be either before or after tax. It can also include interest and dividends receivable, etc. Obviously, the net annual profit must be clearly defined before comparisons are made with other companies. Similarly the term “investment” in Eq. (9-128) can have a variety of meanings. The two most common ones (used when assessing the profitability of companies as opposed to projects) are total assets and owners’ equity or capital employed. In the first case, Eq. (9-128) can be written as

Stockholders’ equity = total assets − total debt

(9-136)

Equations (9-130) and (9-136) can be combined to give net annual profit (ROE) = ᎏᎏᎏ 100 (9-137) total assets − total debt Equation (9-137) can also be written to include a quantity called the debt ratio (DR), which gives net annual profit 100 (ROE) = ᎏᎏ ᎏ (9-138) total assets 1 − (DR)





TABLE 9-21 Simplified Balance Sheets for Companies X and Y X Company balance sheet

Total assets $100,000

Total debt Stockholders’ equity

0 $100,000

Total liabilities and stockholders’ equity

$100,000

Y Company balance sheet

Total assets $100,000

Total debt Stockholders’ equity

$ 50,000 $ 50,000

Total liabilities and stockholders’ equity

$100,000

ACCOUNTING AND COST CONTROL

9-43

where (DR) is the debt ratio as given by total debt (DR) = ᎏᎏ (9-139) total assets Return on assets (ROA) can be related to the return on equity (ROE) by combining Eqs. (9-129) and (9-138): (ROA) = (ROE)/[1 − (DR)]

(9-140)

(ROE) can also be related to the asset-turnover ratio (ATR) and the profit margin (PM) by combining Eqs. (9-132) and (9-140): (ROE) = [(ATR)(PM)]/[1 − (DR)]

(9-141)

Financing by Debt, or Leverage The debt ratio (DR) is also known as the leverage, or gearing, ratio. Highly levered companies have a high proportion of debt to total assets. At first glance, it may appear that the use of leverage is a simple way of increasing the return on equity (ROE). However, interest charges have to be paid on the debt. Whether leverage is a good thing or not will depend on exactly what the interest charges are in relation to the return on assets and the return on equity. Let us consider the simplified balance sheets of two companies, X and Y, shown in Table 9-21. Companies X and Y have a debt, or leverage, ratio of zero and 0.5 respectively. Let us assume that the debt is of the debenture type for tax purposes and that the interest rate is 10 percent per annum. The return on equity (ROE) after tax is given in Table 9-22 for companies X and Y for various values of net annual profit ANP before tax. ANNP is the net annual profit after tax. The data for Table 9-21 are plotted in Fig. 9-38. This figure shows that leverage has no effect on the (ROE) when the interest rate charged for the borrowed money is equal to the return on assets (ROA) before tax. Leverage provides increased (ROE) values when the (ROA) is greater than the interest rate charged for the borrowed money and decreased (ROE) values when it is less. The greater the debt, or leverage, ratio (DR), the more sensitive the (ROE) is to a change in (ROA) and the steeper the slope of the line in Fig. 9-38. Dividends to stockholders are paid out of the net annual profit after tax ANNP, from which the (ROE) after tax in Fig. 9-38 is calculated. Thus, the higher the leverage, the greater the financial risk to the stockholder. This risk is not the same as the business risk of the company, which is a function of its overall prospects in its particular industry. Leverage increases the return to the stockholders when the (ROA) is higher than the interest rate on debt and decreases the return when the (ROA) is lower than the interest rate. Whether the assets of a company are financed largely by stockholders’ equity (also called net worth), or largely by debt, or by some combination of the two depends on a number of factors. If sales do not fluctuate, a company is in a good position to pay the fixed interest charges on debt. This is also the case if the revenue from sales is steadily increasing. In this case, any new common stock issued by the company is likely to command a good price, and it also increases the attractiveness of equity financing. The attitude of management is also an important factor in deterTABLE 9-22 Return on Equity after Tax for Companies X and Y (ROA) before tax

5%

10%

15%

20%

X company ANP Less tax at 50%

$5,000 ($2,500)

$10,000 ($ 5,000)

$15,000 ($ 7,500)

$20,000 ($10,000)

ANNP (ROE) after tax

$2,500 2a %

$ 5,000 5%

$ 7,500 7a %

$10,000 10%

Y company ANP before interest Less interest

$5,000 ($5,000)

$10,000 ($ 5,000)

$15,000 ($ 5,000)

$20,000 ($ 5,000)

ANP after interest Less tax at 50%

0 0

$ 5,000 ($ 2,500)

$10,000 ($ 5,000)

$15,000 ($ 7,500)

ANNP (ROE) after tax

0 0

$ 2,500 5%

$ 5,000 10%

$ 7,500 15%

FIG. 9-38

Effect of leverage on the return on equity.

mining how much debt financing is used. In a small firm in which management owns most of the equity, management may be very reluctant to issue further amounts of common stock that would lead to a dilution of its control. Furthermore, if management has great confidence in future prospects, it will wish to ensure the maximum return for itself. In contrast, the equity in a large company is widely distributed, and the issue of further amounts of common stock has little effect on the control of the company. The difference between equity financing and debt financing is not always clear-cut. For example, preferred stock can be classified as stockholders’ equity or debt, depending on who is doing the financial analysis. Equity Financing Typically, the company balance sheet will show the stockholders’ equity and list the preferred stock, common stock, and retained earnings as in Table 9-23. The issue of common stock is the basic method of financing a company. Common stockholders take the ultimate risk in a business because they have no right to a return on their investment. However, they have the right to elect the directors of the company, who in turn are responsible for the management of the business. Stockholders are likely to vote the board of directors out if adequate dividends are not paid. Usually the liability of stockholders is limited to the nominal, or par, value of their stock, and hence they can lose only what they have already paid for the stock. If the liability is not limited by law, the personal assets of the stockholders are at risk in the event of company bankruptcy, in proportion to the amount of stock held. Preferred stock is often used as an alternative to debt when companies do not wish to issue additional common stock or to incur the fixed interest charges required to finance debt. Preferred stockholders are not normally allowed to vote for the board of directors. They have the right to receive fixed amounts of dividends before common stockholders are paid any dividends. However, a company does not have to pay dividends. The board of directors may decide to pay small or no dividends in a particular year. Holders of cumulative preferred stock TABLE 9-23 Stockholders’ Equity as Shown in Section of a Company’s Balance Sheet Preferred stock, par value $100 per share Authorized 2000 issued and outstanding Less discount on preferred stock Preferred-stock equity Common stock, par value $10 per share Authorized and issued 100,000 shares Amount paid in excess of par Retained earnings Common-stock equity Total stockholders’ equity

1,500

$ 150,000 (10,000) $ 140,000

$1,000,000 100,000 $1,100,000 100,000 $1,200,000 $1,340,000

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PROCESS ECONOMICS

are entitled to receive compensation for the previous underpayment of dividends when the company again pays dividends. Common stockholders have a right to the residual assets of a company in the event of dissolution or liquidation but only after all the creditors and then any liabilities to the preferred stockholders have been paid. The larger the proportion of debt financing in a company, the smaller the amount the common stockholders are likely to receive if the company is liquidated. Common stockholders normally have a preemptive right to the first option to purchase any additional issues of common stock. This prevents management from using an additional issue of common stock to override the control exercised by existing stockholders. Preemptive rights also protect existing stockholders from having the value of their shares decreased by such dilution, since the same net earnings would be spread over more units of stock. Let us consider the very simplified case of a company with 100,000 shares of common stock, each with a market value of $10, giving a total market value of $1,000,000. If a further 50,000 shares are sold at $4 each, the total market value of the 150,000 shares is $1,200,000, or $8 each. This means that the new stockholders have gained at the expense of the original ones. The preemptive right is designed to prevent this. In practice, the situation is rather more complex than is indicated here. Both common and preferred stocks normally have a par, or nominal, value. In the case of common stock, the market value at the time of issue usually differs from the par value. Stock can be issued either at a premium or at a discount, depending on prevailing economic conditions and the strength of the company. The difference between the actual amount paid and the par value is listed in the stockholders’equity section of the balance sheet, as shown in Table 9-23. The issuance of stock at a premium or a discount is done to protect existing stockholders. In the case of preferred stock, the par value has more meaning than with common stock, since it is the amount due preferred stockholders if the company goes into liquidation, provided that this is a condition of issue. The advantage of using common stock to finance assets is that it does not incur fixed interest charges. Furthermore, there is no maturity date, as there is with all loans and most preference issues. Common stock can often be issued more easily than debt can be financed. However, the flotation costs of common stock can be quite high, especially when stock values are depressed, so that large discounts for the stock are needed to induce purchase. Stockholders’ equity in a company is made up of the capital contributed by the stockholders and the capital generated from retained earnings. The presence of retained earnings on a balance sheet, as shown in Table 9-23, does not necessarily mean that they are matched by an equal amount of cash. In fact, there may be little or no cash available. The retained earnings shown on a balance sheet may be largely fictitious. For example, the assets on a balance sheet may be worth less than shown by at least the value of the retained earnings. Purchase and Sale of Equities Stockholders usually require an adequate return on their investment, and the quoted price of the stock reflects the consensus opinion of investors as to the current health of the company. Purchases or sales are normally made through stockbrokers. Most stock transactions are completed through organized security exchanges on which the stock is listed. Such exchanges have physical existence in the form of buildings located in different regions of the country. Each exchange has members who are often the nominated representatives of large brokerage firms having offices in various cities. These offices are in constant telephone and telegraph communication with the members at the exchange, passing on requests to buy or sell specified stocks. Since brokers live by commissions and charges on transactions, they attempt to match such requests either directly or by dealings with other brokers. In the United Kingdom, brokers must deal through an independent “jobber,” similar in function to a specialist broker, who quotes a low price for sales and a higher one for purchases before the jobber knows whether the broker is buying or selling. The difference represents the jobber’s margin, or “turn.” If requests to buy exceed offers for sale, the price of the stock rises until someone is tempted to sell. Conversely, if an excess of stock is offered for sale, the price is likely to fall.

It is an advantage to a company to be listed on a stock exchange since its investors can more easily sell their stock if they decide to do so. This increased liquidity makes investors more willing to accept a lower rate of return, which effectively lowers the cost of capital to the company. Because dealings in the stock of a listed company are published, a healthy company engenders confidence that makes it easier to obtain other forms of finance. In the absence of a regular market, stock transactions are necessarily infrequent, and prices are liable to wide fluctuation, which may make creditors wary and possibly lead to bankruptcy proceedings. Such dealings are usually referred to as “over-thecounter” and are confined to the relatively few specialist brokers who hold inventories of such stock and are prepared to “make a market” in them or are limited to private transactions. Retained Earnings Much confusion is caused by the practice of dividing retained earnings under various headings such as reserve for replacement of plant, reserve for contingencies, etc. This procedure also restricts the flexibility of management in expenditure decisions. The amount of retained earnings shown on a balance sheet should not be taken as a measure of the amount of future dividends that the company is likely to pay. A contract may exist that specifies a minimum balance of retained earnings, which is then not available for dividends until bonds issued by the company have been retired. Dividends can be paid either as cash or in the form of an additional issue of stock. A stock split is really a stock dividend, and both are used to reduce the price of stock when management considers that it is too high. A stock dividend is essentially a transfer of retained earnings to the common-stock account and makes the amount transferred unavailable for future dividends. A stock dividend may be used in place of a cash dividend when a company is short of cash. Debt Financing In practice, debt financing covers a variety of fixed-income securities, both long-term and short-term. The most common forms of long-term debt are bonds, mortgages, and debentures. A bond is simply a long-term promissory note. It is a contract established between borrower and lender in a document called an indenture. A bond indenture includes a detailed description of assets that are pledged, together with any protective clauses and provisions for redemption. A trustee is appointed to look after the interest of the bondholders. The trustee is normally a commercial bank. Bonds may be issued with a call provision that enables a company to redeem its bonds at any date earlier than scheduled. Obviously, this would be an advantage to a company in times of falling interest rates. However, a company has to pay more than the par value of the bond for this privilege. The additional amount is called the bond premium. Sometimes a company uses a sinking fund to retire a bond. A series of equal annual payments A, invested at a fractional interest rate i and made at the end of each year over a period of n years, is equivalent to a sum of money of present value P, given by A = PfAP

(9-46)

where fAP is the annuity present-worth factor, which is fAP = [i(1 + i)n]/[(1 + i)n − 1] A company may use a sinking fund in a variety of ways, but the simplest is to pay a fixed amount A at the end of each year to buy and retire bonds until after n years all the bonds have been retired. This annual payment may prove a significant strain on the resources of a company. Failure to make the payment could result in bankruptcy. In the case of income bonds, a company is required only to pay interest when it earns it. A mortgage is a bond in which specific real assets are pledged as security. A senior mortgage has a prior claim on assets. A junior mortgage is normally a second mortgage on the residual value of the assets. A blanket mortgage is a pledge on all real property owned by a company. A debenture is an unsecured bond. Strong companies are in a better position to issue debentures than weak companies since they have less need to pledge specific assets. Debenture holders are really general creditors. Subordinated debenture holders have claims on assets only after the claims of certain other claimants have been met. The issue of subordinated debentures provides a tax advantage for a com-

ACCOUNTING AND COST CONTROL TABLE 9-24

9-45

Comparative Ratios for Selected United States Industry Groups for 1992*

Industry Number of companies ᎏᎏᎏ Ratio

Agricultural chemicals 2391

Paints and allied products 238

Petroleum refining 98

Plastic materials and resins

Soap and other detergents

234

66

Current assets ᎏᎏ Current debt

2.0

2.8

1.3

1.8

2.2

Net profit ⫻ 100, % ᎏᎏᎏ net sales

1.9

1.8

1.5

1.5

3.8

Net profit ⫻ 100, % ᎏᎏᎏ net worth

9.3

6.8

6.7

14.3

16.3

Net sales ᎏᎏᎏ Net working capital

9.3

5.2

15.8

9.9

5.9

Collection period, days

27.8

39.6

34.8

39.3

44.7

Net sales ᎏᎏ Inventory

11.4

8.1

14.6

11.9

8.9

Fixed assets ⫻ 100, % ᎏᎏᎏ Net worth

43.0

30.8

131.5

20.0

42.3

Current debt ⫻ 100, % ᎏᎏᎏ Net worth

52.1

42.0

68.4

96.8

46.9

Total debt ⫻ 100, % ᎏᎏᎏ Net worth

71.9

66.2

147.7

111.0

61.4

116.1

89.0

193.7

143.6

119.4

Current debt ⫻ 100, % ᎏᎏᎏ Inventory

*Reprinted with the special permission of Dun & Bradstreet International. Numbers above are median values.

pany compared with the issue of preferred stock because the interest payable is a tax-deductible expense. A financial analyst looking at a company from a potential common stockholder’s point of view is likely to classify preferred stock as debt. In contrast, bondholders and general creditors are likely to regard preferred stock as additional equity. Since preferred stock is a hybrid type of security, it may be issued by a company whose management is divided over the question of whether to use equity or debt to finance additional assets. However, preferred stock does have the disadvantage that the dividends are not allowed as a tax-deductible expense. Comparative Company Data Table 9-24 gives comparative company data that have been compiled by Dun & Bradstreet for various types of processing industries. The median value for each ratio is given. Row 1 in Table 9-24 is the current assets Current ratio = ᎏᎏ (9-142) current liabilities liquid assets Compare Quick ratio = ᎏᎏ (9-143) current liabilities Row 2 in Table 9-24 is the profit margin (PM) of Eq. (9-127). In this case, the net profit referred to is the net annual profit after tax and depreciation ANNP. The net sales is the revenue from annual sales AS after deductions for returns, allowances, and discounts for gross sales. Row 3 in Table 9-24 is the return on equity (ROE) of Eq. (9-130). In this case, the net worth is the tangible net worth representing the sum of the preferred and common stocks and the surplus and undistributed profits or retained earnings, less any intangible items such as goodwill, etc. Row 7 in Table (9-24) is the Average collection period average value of accounts receivable = ᎏᎏᎏᎏ (9-144) revenue from sales per day The funded debt (referred to in row 14) consists of mortgages,

bonds, debentures, serial notes, or other obligations with maturity of more than 1 year from the statement date. Robert Morris Associates also compiles extensive comparative company data for various industries. In addition to ratios similar to the Dun & Bradstreet ratios shown in Table 9-24, Robert Morris Associates gives very useful breakdowns of assets and liabilities for various industries. Table 9-25 shows a breakdown of assets and liabilities for United States manufacturers of industrial inorganic chemicals. Application of Overall Company Ratios The various ratios for a hypothetical company are listed in Table 9-26. The balance sheet shown in Table 9-27 has been built up from the ratios in Table 9-26 in terms of the revenue from net annual sales AS. Let us calculate the following values for the right-hand side of the balance sheet as follows: From ratio 5 Net worth = AS/2.50 = 0.4 AS From ratio 11 Total debt = (0.4 AS)(0.65) = 0.26 AS From ratio 10 Current debt = (0.4 AS)(0.35) = 0.14 AS Long-term debt = total debt − current debt Long-term debt = 0.26 AS − 0.14 AS = 0.12 AS We calculate the following values for the left-hand side of the balance sheet: From ratio 99-45 Fixed assets = (0.4 AS)(0.74) = 0.29 AS From ratio 1 Current assets = (0.14 AS)(2.60) = 0.36 AS From ratio 8 Inventory = AS /7.14 = 0.14 AS From ratio 7 Accounts receivable = (AS /365)(61) = 0.167 AS

9-46

PROCESS ECONOMICS TABLE 9-25 Typical Balance Sheet for a United States Manufacturer of Industrial Inorganic Chemicals Assets

%

Cash Marketable securities Net receivables Net inventory All other current assets Total current assets Fixed assets All other noncurrent assets

4.3 2.2 26.2 24.6 0.9 58.3 36.3 5.4

Total assets

100.0

Liabilities Short-term due to banks Due to trade Income taxes Current maturities long-term debt All other current liabilities Total current debt Noncurrent debt unsubordinated Total unsubordinated debt Subordinated debt Tangible net worth Total liabilities and stockholders’ equity

% 5.6 14.5 3.1 2.1 5.0 30.3 18.3 48.6 3.1 48.3 100.0

Abridged from Annual Statement Studies, 1973 ed., copyright 1973 by Robert Morris Associates, Philadelphia.

Cash and short-term investments = total current assets − (inventory + accounts receivable) Cash and short-term investments = 0.364 AS − 0.307 AS = 0.057 AS In addition to the data for the balance sheet, we calculate the net annual profit (after tax), i.e., ratio 2, to be ANNP = 0.04 AS. TABLE 9-26

Ratios for a Typical Industrial Chemical Company

No.

Ratio

1

Current assets ᎏᎏ = 2.60 Current debt

2

100 = 4.00 ᎏᎏ Net sales 

3

100 = 10.0 ᎏᎏ Net worth 

4

100 = 18.18 ᎏᎏᎏ Net working capital 

5

Net sales ᎏᎏ = 2.50 Net worth

6

Net sales ᎏᎏᎏ = 4.50 Net working capital

7

accounts receivable Collection period = ᎏᎏᎏ = 61 days sales per day

8

Net sales ᎏᎏ = 7.14 Inventory

9



Net profit

Net profit

Net profit



Fixed assets ᎏᎏ 100 = 74.00 Net worth

10

100 = 35.00 ᎏᎏ Net worth 

11

100 = 65.00 ᎏᎏ Net worth 

12

100 = 63.00 ᎏᎏᎏ Net working capital 

13



Current debt ᎏᎏ 100 = 100.00 Inventory

14



Funded debt ᎏᎏᎏ 100 = 76.50 Net working capital

Current debt

Total debt

Inventory





In practice, the ratios are obtained from the information published in the balance sheet. The advantage of the above presentation is that it relates everything to the revenue from net annual sales and hence underlines the importance of sales. Careful study of the ratios can produce many inferences as to the health of the company. For example, the leverage, or debt, ratio (DR) for this example is total debt 0.260 A (DR) = ᎏᎏ = ᎏS = 0.40 total assets 0.660 AS This value is quite low and does not present any problems of control by debtors, such as can arise when (DR) is greater than 1. From Table 9-27 we calculate the ratio for 0.140 AS Current debt ᎏᎏᎏᎏ = ᎏ = 2.45 Cash + short-term investments 0.057 AS Therefore, requests for early repayment by more than 40 percent of the debtors could be met. Hence, no liquidity problems are likely to arise, and advantage can be taken of discounts for early payment. Also, the current debt could be met by sale of the inventory, which takes (0.140 AS /AS)(365), or 51 days. The quick ratio is 1/2.45 = 0.407. If it is assumed that current debtors are due for payment within 61 days, the same time as that allowed to creditors, no bankruptcy petitions are likely. The profit of 10 percent, indicated by ratio 3 in Table 9-26, will be reduced by any dividend due to preferred stockholders, because such payments are not part of fixed-debt expenses; the residue is shared among the ordinary stockholders. If all the long-term debts were in redeemable 6 percent preferred shares, then (from ratio 3) the net annual profit (after tax) is ANNP = 0.10(0.40 AS), or 0.04 AS. Interest due on preferred shares is 0.06(0.12 AS), or 0.0072 AS. Therefore, the earnings for the ordinary shares are (0.04 AS − 0.0072 AS)/(0.4 AS − 0.12 AS) = 0.1171 This value corresponds to 11.71 cents per dollar of common stockholders’ equity. If it is assumed that available interest rates offered by banks, government, etc., for no-risk investment of capital are 10 percent, then the maximum economic market price of $100 stock units in this hypothetical company is about $117. If all the debt is in bonds, etc., earnings on ordinary stock would be 10 cents per dollar of net worth, and the maximum economic price of the stock would be about $100 unless stock prices were expected to rise. Other ratios can easily be deduced from those listed. For example, the return on assets (ROA) and the asset-turnover ratio (ATR) are net annual profit 0.04 A (ROA) = ᎏᎏ 100 = ᎏS = 6.06 total assets 0.66 AS revenue from annual sales (ATR) = ᎏᎏᎏ 100 total assets (ATR) = (AS /0.66 AS)100 = 151.5

ACCOUNTING AND COST CONTROL

9-47

TABLE 9-27 Balance Sheet for a Typical Industrial Chemical Company, Dec. 31, 1991 Assets Current assets Cash Accounts receivable Inventory Total current assets Fixed assets Total assets

Liabilities and stockholders’ equity 0.057 AS 0.167 AS 0.140 AS 0.364 AS 0.296 AS 0.660 AS

Liabilities Current debt Long-term debt

0.140 AS 0.120 AS

Total debt Net worth Total liabilities and stockholders’ equity

0.260 AS 0.400 AS 0.660 AS

Cost of Capital The value of the interest rate of return used in calculating the net present value (NPV) of a project is usually referred to as the cost of capital. It is not a constant value since it depends on the financial structure of the company, the policy of the company toward a particular project, the local method of assessing taxation, and, in some cases, the measure of risk associated with the particular project. The last-named factor is best dealt with by calculating the entrepreneur’s risk allowance inherent in the project i′r from Eq. (9-108), written in the form i′r = [1 + (DCFRR)]/(1 + i) − 1 where i is the cost of capital exclusive of the risk allowance. The value of i′r should be compared with the probability of exceeding or of failing to achieve an (NPV) of zero when using that value of i. The decision to proceed can then be made with a full knowledge of the odds against success. The decision can be related to the company attitude to budgets of the relevant size by the use of probable utilities, as has already been discussed. Cash flows used in calculating (NPV) and (DCFRR) should, of course, be corrected for the anticipated rates of inflation, preferably to the time when the utility curve was obtained. This is important since inflation is likely to have a distorting effect on utility curves obtained at different times. This may be due to an unconscious wish to protect against inflation by achieving higher rewards while assigning less importance to any losses incurred, thus tending toward a gambling outlook. In the absence of a risk allowance the cost of capital becomes a technical financial computation based on sources of funds and company policy. As such it will usually be presented as a figure specified for use in a particular appraisal and is therefore of little concern to the project assessor. However, the following résumé indicates the kinds of factors to be considered. In most companies the objective of company policy is to maximize the financial return to the equity stockholders. This is not invariably the case, since a young company will often plow back an unusually large proportion of its profits to encourage growth. Also, it is increasingly the case that projects are undertaken to restore or preserve an environmental amenity or to bring work into a particular locality. In such circumstances a low value of the cost of capital might be assigned to the project. In many government projects a limited loss is acceptable, in which case the value of i would be negative. When the objective is to maximize the aftertax return to the stockholders, a balance must be struck between the proportion of aftertax company profits which are retained to permit growth of the company assets and the proportion which are distributed to provide an income for the stockholders in the form of dividends. The latter will usually be subject to personal income taxes, sometimes at higher-than-normal rates. The growth potential should be reflected in an increased value of the stock as quoted on the stock exchange. Such growth may result in the imposition of capital gains or inheritance taxes. The selection of the right proportion of earnings to be retained is crucial since this affects the appeal of the company to investors and hence its credit worthiness in the eyes of creditors. The optimum split is influenced by the type of investor since institutional tax rates and exemptions often differ from those applied to private investors. It is for this reason that the optimum split is sensitive to local taxation policy. Most companies can maintain a given level of business only by continuous reinvestment in plant and equipment. If company growth is

required, additional investment is essential. In general, a company has only three sources of new money, namely, cash received from the sale of newly issued shares, retained earnings, and debt capital of all kinds including deferred taxes. In certain circumstances cash grants may be forthcoming from government sources. Each of these sources has its own effective rate of interest, and it is the weighted average of these rates which constitutes the cost of capital exclusive of risk allowance. There is no interest payable directly on equity stocks, but there is a concealed rate expected by investors. Without the expectation of a certain return on their investment they would not invest in a new issue, nor would they retain existing holdings of stock. The sale of stocks on the stock exchange does not affect the cash holding of the company, but new issues must be at prices lower than existing values quoted on the exchange unless great confidence exists that the new money will produce an increased income greatly in excess of the reduction in earnings per share caused by the new issue. Stock carrying a fixed interest rate normally has the interest treated as an allowable expense before tax in the same way as a bank overdraft, which is a relatively short-term source of debt. Deferred taxes carry an interest rate which, like an overdraft, is normally compounded daily at a nominal annual rate but naturally is not an allowable expense for tax purposes. Cash owing on outstanding bills carries a notional rate of interest since in many cases prompt settlement of bills would attract a cash discount. Example 18: Risk-Free Cost of Capital A company requires an investment of $100,000 in new plant to maintain its present sales. Let us determine the current cost of capital to the company and the risk-free cost of capital that it should assign to the plant-replacement project, given the following data. Company assets: from stock sales $ 300,000 from retained earnings 200,000 as bills due 100,000 as deferred taxes 200,000 as bank overdraft 200,000 Total assets Current annual income

$1,000,000 $ 200,000

Bills are due on monthly account with a 2 percent discount for cash. Overdraft and deferred-tax interest are compounded daily at nominal annual interest rates of 15 and 9 percent respectively. Corporation tax, capital gains tax, and personal income tax rates are 50, 40, and 30 percent respectively. The current rate of inflation is at 8 percent per year. The traditional return expected by investors is 7 percent per year net of all taxes in real terms. The interest-rate equivalent of the cash discounts is 2 percent per month, since this discount could be obtained every month if payment were to be made at the beginning of the month rather than, as at present, at its end. Since the bills are settled monthly, the notional interest is paid monthly and should not be compounded. The discount is equivalent to 12 monthly simple-interest payments per year. Hence, from Eq. (9-31) the effective annual interest rate on discounts = (12)(0.02) = 0.24 = 24 percent. It would, therefore, be a good use of surplus cash to reduce this debt as quickly as possible. This would require cash equivalent to one-sixth of the annual bills due, or $16,700, to be available. It can, therefore, be assumed that this level of liquidity is not available for capital projects, either as working capital to reduce the debt or for fixed-capital projects. Further, since the new project will not increase sales, it cannot generate further debt of this kind. Hence, this source is not available to capitalize the new project. Since the overdraft is payable daily at a nominal annual interest rate of 15 percent, it follows from Eq. (9-38) that the effective annual interest rate on overdraft = (1 + 0.15/365)365 − 1 = 16.18 percent. Similarly, the effective annual interest rate on deferred tax = (1 + 0.09/365)365 − 1 = 9.42 percent.

9-48

PROCESS ECONOMICS

The new plant will not increase sales and will therefore not increase the tax debt, so that this source is not available to capitalize the project. An increase in overdraft may be available, subject to a maximum imposed by the acceptable gearing of the company. Since the liquidity of the company is so low, it is possible that it is already extended to its maximum debt, in which case the gearing $300,000 + $200,000 Total equity ᎏᎏ = ᎏᎏᎏᎏ = 1.00 Total debt $100,000 + $200,000 + $200,000 Since neither increased bills due nor increased tax debt is available to finance the new project, this implies that the required $100,000 of new capital will be available as $50,000 from increased overdraft and $50,000 from increased equity. The effective interest rate on the equity involved must therefore be calculated. Equity is available from two sources. First, the company can sell new stock which, if in the form of ordinary shares, carries no interest payment. Although this course appears cheap, its use for projects which do not increase earnings, at least to a compensatory level, is usually inadvisable. This leaves retained earnings as the most likely source of equity for the present project. Equity holders require a real return on their outlay, which they assume to be at the stock-market price if this differs from the face value of the stock, of 7 percent net of all taxes. Retained earnings attract a 40 percent capital gains tax; hence the actual interest rate required on distribution forgone is 7/(1 − 0.40) = 11.67 percent. This is in real terms and at a time of 8 percent inflation rate must be increased in cash terms to (1 + 0.1167)(1.08) − 1 = 20.60 percent. In the same way the effective interest rate on distributed earnings, on which an income tax of 30 percent is payable, may be calculated to be [0.07/(1 − 0.30) + 1](1.08) − 1 = 18.80 percent. If the shares are currently valued at par by the stock exchange, this would require distribution, on the $300,000 of issued equity, of ($300,000)(0.188) = $56,400. This amount is required after corporation tax has been paid on the earnings of $200,000. Thus the earnings which can be retained are ($200,000)(1 − 0.50) − $56,400 = $43,600. This is close to the $50,000 required. If the company were well regarded on the stock exchange, a slight reduction in distributed dividends might not reduce share values since the purpose is to maintain future earnings. However, it would be possible for share values to be reduced by up to ($56,400 − $50,000)/$56,400 = 0.1135 = 11.35 percent. If this happened, the effective interest rate on the retained earnings should be increased to 20.60/(1 − 0.1135) = 23.24 percent. These rates are net of corporation tax, which is the correct form for use in (NPV) calculations. The interest rate due on deferred tax is also net of corporation tax at 9.42 percent. The interest payable on overdrafts is an expense fully allowable against tax, so that the effective aftertax rate is reduced to (16.18)(1 − 0.50) = 8.09 percent. Similarly, as the advantage forgone on the discounts would tend to increase company profits and hence tax due, the effective aftertax gain is reduced to (24)(1 − 0.50) = 12.00 percent. The present cost of capital to the company is the weighted-average interest payable on the various sources of funds on an after-corporation-tax basis. This is readily calculable to be at least [($300,000)(0.00) + ($200,000)(20.60) + ($100,000)(12.00) + ($200,000)(9.42) + ($200,000)(8.09)]/($1,000,000) = 8.82 percent. The cost of capital to the new project, with only two sources, should be [($50,000)(20.60) + ($50,000)(8.09)]/($100,000) = 14.35 percent. Since this project is essential if current production is to be maintained, many companies would assess the cost of capital at somewhere near the lower value. Values of cost of capital in the region of 10 percent are to be expected in developed countries at the present time. We notice in particular that inflation does not affect quoted interest rates when assessing present values of cost of capital. It must, however, be taken into account in assessing the interest rate on the dividend which will be expected by investors. As has been stated, it is alternatively possible to assign to the cost of capital the best risk-free return available on the money. The assessment then proceeds as discussed in connection with Eq. (9-108).

Management and Cost Accounting In any given time period, cost may be divided into expired and unexpired cost. An expired cost is an expense; an unexpired cost is an asset. This division is the basis for income statements and balance sheets. Cost accounting is the name traditionally given to accounting for manufacturing costs. The manufacturing cost of a product is traditionally taken as the sum of the costs for (1) direct materials, (2) direct labor, (3) manufacturing overheads, and (4) administration, selling, and finance. Two methods are in general use in accounting for manufacturing costs, absorption costing and marginal costing. In absorption costing, which is the traditional method, all manufacturing overhead costs are included in the cost of sales. In marginal costing only variable manufacturing overhead costs are included in the cost of sales. Marginal costing is more valuable than absorption costing in decision making.

However, it is sometimes quite difficult to separate costs and particularly manufacturing overhead costs into fixed and variable components. In the long term virtually all costs are variable. The difference between the two methods assumes great importance in inventory evaluation. In cost accounting, costs are identified with cost centers. These are accounting devices which may or may not have a physical existence. In the simplest case of a plant manufacturing a single product, the entire plant may be the cost center. In practice there are two major classifications of cost accounting systems, job costing and process costing. In the former, costs are collected for each job or batch irrespective of the accounting period. This system is normally used in construction work. Process costing is normally used in continuous and semicontinuous processes. Costs are collected for a specific accounting period. Allocation of Overheads How overheads are allocated can affect the total cost of a product and, hence, the estimated future cash flows for a project. Since these cash flows are used in the net-presentvalue (NPV) and discounted-cash-flow-rate-of-return (DCFRR) methods for estimating profitability, erroneous allocations could result in the wrong choice of project. The modern trend is for overhead costs to become an increasing proportion of total product costs. This results from the ever-greater sophistication of process plants. Therefore, it is highly desirable that chemical engineers should have some say in the allocation of overheads and that this should not be left entirely to accountants. Direct costs are those that can be directly charged to a single product. The most obvious direct cost is for raw materials, of which the quantity consumed is directly proportional to the amount of product manufactured. Direct process labor is also considered to be a direct cost. However, many costs cannot be directly charged to an individual product. These so-called indirect, burden, or overhead costs range from the lighting and heating required for the plant and offices to the cafeteria and medical facilities provided. When several products are made in a plant, it becomes increasingly difficult to allocate overheads correctly among the various products. A number of different methods are commonly used to estimate the amount of overhead to be allocated to an individual product. These methods are necessary because accountancy costs become prohibitive for charging all costs directly to an individual product. Unfortunately, there is always an arbitrary element inherent in the process of allocation. Overheads in the chemical-process industries are commonly calculated as a percentage of (1) direct materials cost, (2) direct labor cost, or (3) prime or direct costs. Other methods of allocating overheads are on the basis of (1) plant area, (2) number of employees, (3) capital value, and (4) electric power. These listings do not include all the methods in use. The validity of a particular method depends on the process and the industry. An inappropriate method can lead to misleading and even absurd results. Let us consider the manufacture of metal ornaments. The processing cost, exclusive of material, may vary very little for a wide range of materials. However, the direct materials cost will be much greater for precious than for base metals. In this case, an overhead allocation on the basis of direct material costs could be very misleading, while one based on direct labor cost could be quite accurate. Problems can also arise when allocating overheads on the basis of direct labor cost. Let us consider a company that evaluates overheads at 125 percent of direct labor cost. A process plant employs seven operators, each with a direct cost of $10,000 per budget period. As a result of a works-study exercise, it is found that the plant can operate satisfactorily with six operators. The actual cost saving is likely to be far nearer to the direct labor savings of $10,000 per period than to the calculated saving of $10,000 + $10,000(125/100) = $22,500 per period. The $22,500 calculated saving is the direct labor cost plus overheads taken as 125 percent of the direct labor cost. A thorough analysis should be made before production is stopped on a product that is losing money. Although direct costs of the discontinued product will be saved, overheads are not eliminated, as might be inferred from taking overheads as a percentage of direct material, direct labor, or prime costs. The plant is still there, together with its

ACCOUNTING AND COST CONTROL associated costs for interest charges, insurance, painting, some maintenance, etc. Continued production can still make a useful contribution to such overheads. This contribution is lost if production is stopped and must then be borne by other products. Problems can arise with each of the methods used for allocating overheads. Two process plants may occupy similar areas yet have vastly different material or labor costs. Problems can also arise with an individual plant that can be used to make different products, as Example 19 will show. Example 19: Overhead in Two Different Projects Let us consider a plant that can make either product A or product B. At normal capacity, the overhead cost is known to be $2.50 per unit. Product A has a direct materials cost of $8 per unit and a direct labor cost of $2 per unit. For simplicity, the prime cost is here taken as the sum of these two costs, i.e., $10 per unit. In Table 9-28, a correct overhead cost of $2.50 per unit at normal capacity is calculated by taking either 31.25 percent of the direct materials cost, 125 percent of the direct labor cost, or 25 percent of the prime cost. All these methods give a total cost of $12.50 per unit and a profit of $1.50 per unit for a selling price of $14 per unit. The alternative, product B, has a direct materials cost of $6 per unit and a direct labor cost of $4 per unit. In Table 9-28 overhead costs of $1.875 per unit, $5 per unit, and $2.50 per unit are calculated by taking 31.25 percent of direct materials cost, 125 percent of direct labor cost, and 25 percent of prime cost respectively. Total costs are $11.875 per unit, $15 per unit, and $12.50 per unit respectively and profits of $2.125 per unit, −$1 per unit, and $1.50 per unit respectively, for a selling price of $14 per unit. An alternative to allocating overheads by using a single method is to classify the various overheads into groups and to use the most appropriate allocation for each group. For example, depreciation would be allocated on the basis of capital cost, while indirect labor might be allocated either on the basis of direct labor cost or on the number of employees. Clearly, this alternative method is more complex, increases the associated accountancy costs, and is prone to misinterpretation and possibly abuse. Inventory Evaluation and Control Inventory Effect on Cash Income and Profit When the annual production rate is equal to the annual sales volume, the revenue from annual sales AS is AS = Rcs

(9-87)

where R is the production rate, units per year, and cs is the sales price per unit. In this case, the annual cash income ACI and the net annual profit ANP before tax are given respectively from Eqs. (9-1), (9-8), and (9-12) by ACI = R(cs − cTVE) − ATFE

(9-145)

ANP = R(cs − cTVE) − ATFE − ABD

(9-146)

where cTVE is the total variable expense per unit of production, ATFE is the total annual fixed expense required to produce and sell a product but excluding any annual provision for plant depreciation, and ABD is the balance-sheet annual depreciation charge. However, in a given accounting period the sales volume may differ from the volume of production. In this case, the inventory of finished product I1 at the beginning of the accounting period will differ from

TABLE 9-28

that at the end, I2, and Eqs. (9-145) and (9-146) need to be written in modified form as ACI = R(cS − cTVE) − ATFE + (I1 − I2)

(9-147)

ANP = R(cS − cTVE) − (ATFE + ABD) + (I1 − I2)

(9-148)

If the annual sales volume exceeds the annual production rate R by an amount ∆R, then Eqs. (9-147) and (9-148) can be written as ACI = R(cS − cTVE) − ATFE + ∆R(cS − cINV)

(9-149)

ANP = R(cS − cTVE) − (ATFE + ABD) + ∆R(cS − cINV)

(9-150)

where cINV is the value per unit of inventory of the finished product. Clearly, the value of cINV affects both the annual cash income and the net annual profit. Since annual cash incomes are the basic data for (NPV) and (DCFRR) methods of estimating profitability, the actual value per unit of inventory is of direct importance for chemical engineers engaged in economic assessments. Let us divide Eq. (9-150) by Eq. (9-87) to give ANP cTVE (ATFE + ABD) ∆R cINV ᎏ = 1 − ᎏ − ᎏᎏ + ᎏ 1 − ᎏ AS cS AS R cS





   c A (A + A ) ∆R ᎏ = (CSR) − ᎏᎏ +  ᎏ 1 − ᎏ  A A R c NP

TFE

S

BD

INV

S

(9-151) (9-152)

S

where (CSR) = 1 − (cTVE /cS) is the contribution to the sales-price ratio. When the ratio of net annual profit to revenue from annual sales ANP /AS is expressed as a percentage it is known as the profit margin. The terms (CSR) and (ATFE + ABD)/AS have a similar order of magnitude in chemical processing. For example, (CSR) values of 0.1 to 0.4 are typical, and these are quite close to a value of 0.2 that is common in general chemical processing for (ATFE + ABD)/AS. Thus, the profit margin is very sensitive to the value cINV per unit of inventory. For example, let us consider that both (CSR) and (ATFE + ABD)/AS are equal to 0.3. In this case, Eq. (9-152) can be written as ANP /AS = (∆R/R)(1 − cINV/cS)

(9-153)

If the sales volume exceeds the annual production rate by 10 percent and the inventory is valued at the sales price, then Eq. (9-153) shows that the profit margin is (ANP /AS)100 = 0 percent. If the inventory is valued at the total variable cost, then the profit margin (ANP /AS)100 = (0.1)(1 − 0.7) (100) = 3 percent. Hence, the value of the inventory is of vital importance. Unfortunately, there is no universally accepted method for valuing inventory. The value of cINV per unit can be taken on any of the following bases: 1. Direct material plus direct labor cost. 2. Direct material plus direct labor plus other direct production expenses. (This is the total variable production cost.) 3. Total variable production cost plus fixed production overhead cost. 4. Total variable cost. (This includes both production and general expenses.) 5. Total cost. (This includes variable and fixed production and general expenses.) Methods 1, 2, and 4 are termed direct costing, variable costing, and marginal costing respectively. Although direct costing is being increas-

Profits of Products with Different Methods of Overhead Allocation Product A—basis of overhead allocation

Direct materials cost Direct labor cost Prime or direct cost Overhead cost Total cost Selling price Profit/(loss)

9-49

Product B—basis of overhead allocation

31.25% of direct materials cost, $/unit

125% of direct labor cost, $/unit

25% of prime cost, $/unit

31.25% of direct materials cost, $/unit

125% of direct labor cost, $/unit

25% of prime cost, $/unit

8.00 2.00 10.00 2.50 12.50 14.00 1.50

8.00 2.00 10.00 2.50 12.50 14.00 1.50

8.00 2.00 10.00 2.50 12.50 14.00 1.50

6.000 4.000 10.000 1.875 11.875 14.000 2.125

6.000 4.000 10.000 5.000 15.000 14.000 (1.000)

6.000 4.000 10.000 2.500 12.500 14.000 1.500

9-50

PROCESS ECONOMICS

ingly used for internal accounting and control purposes, it is not acceptable to tax authorities as a basis for calculating profit. Tax authorities and most accountants favor method 3, which is known as absorption costing. We have already seen that this method has the disadvantage that the fixed-overhead cost per unit is determined for a particular normal production rate. If the production rate exceeds the normal, there is overabsorption of fixed overheads. Conversely, if the production rate falls below the normal, there is underabsorption of fixed overheads. Method 5 is known as full-absorption costing. Most people agree that general expenses incurred in administration, selling, distribution, etc., should not be included in the cost of inventory. In fact, many feel that no costs should be absorbed before they have been incurred. In general, method 2 is favored by engineers and method 3 by most accountants. However, the accountancy convention is to value at either cost or market value, whichever is the lower. In the methods considered, either actual or standard costs can be used. Note that method 3 shows a higher profit than method 2 when sales volume exceeds the production rate and a lower profit when the production rate exceeds sales volume. The total profits (before tax) over the life of a project are independent of the method used to value inventory. Over a project life of n years, Eq. (9-148) can be written as n

A

n

NP

0

n

p

χ <m 1

Hence, the value of the inventory, IFIFO at any given time is IFIFO = CI − CFIFO

0

(9-154)

0

where the last term in Eq. (9-154) becomes  n0 (Im − Im + 1) = I0 − In + 1. Since there will be no material in inventory in the year before the project starts or in the year after it terminates, I0 = In + 1 = 0. Hence, total profits do not depend on individual values for I. However, the annual profit ANP (before tax) does depend on the value of the inventory. Since the tax payable in any individual year is based on ANP, the net annual profit ANNP (after tax) is also dependent on the method chosen for valuing inventory. Frequently, a particular method for valuing inventory is chosen to delay payment of tax as long as it is legally possible to do so. So far, only the inventory of finished product has been considered. There are also inventories of raw materials and work in process, i.e., partially processed materials or intermediate products, to be considered. It is necessary to modify Eqs. (9-147) through (9-154) accordingly to take these inventories into account. Effect of Raw-Materials Prices Raw materials for the chemicalprocess industries are subject to relatively wide variations in price. These effects on profits will now be considered. When the price of raw materials varies from week to week, not all the units in storage will have been purchased at the same price. Let us consider χ units in storage at the start of the inventory period, purchased at a price c1 per unit. Additional quantities χ2, χ3, etc., are purchased at prices c2, c3, etc., per unit respectively until finally χn units are purchased at the latest price of cn per unit at the end of the inventory period. The total value of the inventory CINV at the end of the inventory period (in the absence of any withdrawal) is given by n

CINV =  χ j cj

(9-155)

p



p



CFIFO =  χ j cj + m −  χ j cp + 1 1

n

p

1

1

providing that the value of m satisfies n

m <  χj 0

1

(9-156)

p





IFIFO =  χ j cj −  χ jcj − m −  χ j cp + 1 1

(9-158)

In the LIFO method, the m units taken out of storage are valued at their purchase price, beginning with the latest item purchased. In a similar manner, the value of the material CLIFO taken out of inventory is given by



n



n

CLIFO =  χ j cj + m −  χ j cp − 1 p

p

(9-159)

where p is the smallest integer, such that n

χ

j

≤m

p

Hence, the value of the inventory ILIFO at any given time is ILIFO = CI − CLIFO

(9-160)

In terms of Eqs. (9-155) and (9-159), Eq. (9-160) can be written n

n

1

p





n

ILIFO =  χ j cj −  χ j cj − m −  χ j cp − 1 p

(9-161)

Example 20: Inventory Computation Let us consider 10 successive batches of raw materials, of 1000 units, purchased in a time of rising prices in which c1 = $0.10 per unit, c2 = $0.11 per unit, etc., as listed in Table 9-29. The total cost of the purchases in the inventory is found from Eq. (9-155) to be $1450. Let us calculate the value of the raw-materials inventory after, say, 5500 units have been withdrawn from inventory, first by using FIFO and then by using LIFO. We substitute the appropriate quantities into Eq. (9-158) for FIFO, keeping in mind that p = 5, n = 10, and m = 5500, to get IFIFO = $1450 − $600 − (5500 − 5000)($0.15) IFIFO = $775 In a similar manner, we substitute values into Eq. (9-161) for LIFO (except that p is now 6) to get ILIFO = $1450 − $850 − (5500 − 5000)($0.14) ILIFO = $530 Thus, in a time of rising raw-materials prices, the FIFO method gives a higher value for the remaining inventory than will LIFO. In a time of falling prices, the FIFO method will give a lower value for the remaining inventory than will LIFO.

Average-Cost Basis for Inventory Either a simple average or a weighted average can be used to value inventory cost. Using the simple-average method, the value CSAV of the material taken out of inventory is given by CSAV = m(c1 + cn)/2

1

The value of the inventory I at any given time depends on the values ascribed to the units withdrawn from inventory. There are five methods for valuing inventory: (1) FIFO (first-in–first-out), (2) LIFO (last-in–first-out), (3) average cost, (4) standard cost, and (5) market value. In the FIFO method, the units taken out of storage are valued at their purchase price beginning with the earliest item purchased. If a number of units m are removed from inventory during the period, the total cost of these items on a FIFO basis is given by:

(9-157)

In terms of Eqs. (9-155) and (9-156), Eq. (9-157) can be written as

n

=  R(cS − cTVE) −  (ATFE + ABD) +  (Im − Im + 1) 0

and where p is the largest integer, such that

TABLE 9-29

(9-162)

Costs of Inventory with Rising Prices p

p

n

n

x,

xc,

x,

xc,

units

$

units

$

j

j j

j

j j

p

xj, units

cj , $/unit

xj cj , $

1 2 3 4 5

1,000 1,000 1,000 1,000 1,000

0.10 0.11 0.12 0.13 0.14

100 110 120 130 140

1,000 2,000 3,000 4,000 5,000

100 210 330 460 600

10,000 9,000 8,000 7,000 6,000

1,450 1,350 1,240 1,120 990

6 7 8 9 10

1,000 1,000 1,000 1,000 1,000

0.15 0.16 0.17 0.18 0.19

150 160 170 180 190

6,000 7,000 8,000 9,000 10,000

750 910 1,080 1,260 1,450

5,000 4,000 3,000 2,000 1,000

850 700 540 370 190

1

1

p

p

ACCOUNTING AND COST CONTROL The value of the inventory ISAV at any given time is ISAV = CI − CSAV

(9-163)

In terms of Eqs. (9-155) and (9-162), Eq. (9-163) can be written as n

ISAV =  χ j cj − [m(c1 + cn)/2]

(9-164)

1

We shall consider the value of the inventory after 5500 units have been withdrawn, using the data listed in Table 9-29. On the basis of a simple average, the materials withdrawn are priced at ($0.10 + $0.19)/2 = $0.145 per unit. Since the total cost of the purchases in the raw-materials inventory is found from Eq. (9-155) to be $1450, the value of the inventory after 5500 units have been withdrawn is calculated from Eq. (9-164) to be ISAV = $1450 − (5500)($0.145) = $652.50 For the weighted-average method, the value of the material CWAV taken out of inventory is given by n

n

1

1

CWAV = m  χ j cj  χ j

(9-165)

IWAV = CI − CWAV

(9-166)

In terms of Eqs. (9-153) and (9-164), Eq. (9-166) can be written n

 

n

IWAV =  χ j cj 1 − m χ j 1

1



(9-167)

Let us use Eq. (9-167) to value the inventory, after 5500 units have been withdrawn, by employing the data of Table 9-29. IWAV = $1450 [1 − (5500/10,000)] = $652.50 For this example, the values of ISAV and IWAV are the same because the batches purchased are of equal size and the prices are linearly progressive. This is a combination rarely found in practice. A more realistic example, in which the buyer seeks to purchase at the lowest price, is provided by the data of Table 9-30. The quantities bought vary according to price, but some may have been made at high prices to maintain production. On the basis of Table 9-30, when 5500 items have been removed from inventory, the value of the inventory by using the FIFO, LIFO, simple-average, and weighted average methods respectively is

p

n

xc,

x,

xc,

units

$

units

$

100 165 65 200 30

1,000 2,500 3,000 5,000 5,200

100 265 330 530 560

10,000 9,000 7,500 7,000 5,000

1,175 1,075 910 845 645

140 275 45 20 135

6,200 8,700 9,000 9,100 10,000

700 975 1,020 1,040 1,175

4,800 3,800 1,300 1,000 900

615 475 200 155 135

xj , units

cj , $/unit

xj cj , $

1 2 3 4 5

1,000 1,500 500 2,000 200

0.10 0.11 0.13 0.10 0.15

6 7 8 9 10

1,000 2,500 300 100 900

0.14 0.11 0.15 0.20 0.15

j

1

j j

1

(9-169)

j

p

j j

p

(9-170)

For a given number of orders per year N, the value of the inventory is proportional to the magnitude of the average individual order U. Hence, Eq. (9-170) can be written as AI = βU + (FR/U) + VR

n

x,

p

(9-168)

where α is the proportionality factor which is of the order of 0.25 for many industries. In contrast, some costs can be reduced as a result of larger inventories. For example, larger discounts can be obtained on bulk purchases and deliveries. In addition, larger inventories reduce the risk of losing sales and goodwill through interruptions to production and consequently running out of stocks of finished goods. The cost of placing an order for materials is partly fixed and partly variable. The annual cost of ordering AIO is given by

AI = αI + FN + VR

Costs of Inventory with Fluctuating Prices p

AIW = αI

where F is the fixed cost per order, N is the number of orders per year, V is the variable cost of ordering per unit of production, and R is the annual production rate. Although the administrative cost of placing an order will be more or less fixed, the shipping costs are proportional to the size of the order and, hence, for the year are proportional to the annual production rate. The total annual cost of inventory AI is the sum of Eqs. (9-168) and (9-169):

Pros and Cons of Inventory Valuation In the standard-price method of inventory valuation, all materials are taken out at the same TABLE 9-30

price. In addition to simplicity, the method has the advantage that the efficiency of raw-materials purchase is constantly checked. In both the average-cost and the standard-cost methods of valuing inventory, materials are not charged out at actual cost. Thus, the amount of profit or loss for the period may be varied by the method chosen to value the inventory. For this reason, accountants usually insist that the method of inventory valuation be consistent from period to period. This causes inertia but does not prevent a change of method when it can be justified. In such cases, it is usual to inform stockholders of the change because the influence on declared profits can be large. Unfortunately, there is no right or wrong way to value inventory, although certain methods are not allowed in certain countries for taxassessment purposes. For example, LIFO is not allowed in the United Kingdom. As a general rule, the method used should be the one that gives the lowest tax liability. However, it is generally accepted that consistency is also a virtue in inventory valuation. It is important to realize that the method used to value inventory for cost accounting purposes is not necessarily the one used to draw up the balance sheet and financial accounts. In this case, inventory is valued either at the cost given by another method or at the market value, whichever is lower. Inventory Control The optimum size of inventory depends on the type of industry and on the skills available to the individual company. Inventories are high in the tobacco industry and low in perishable-foods businesses. The larger the inventory, the larger the warehousing and associated costs. These costs include insurance, taxes, depreciation, handling and security charges, etc., and can be taken as roughly proportional to the value of the inventory I. The annual cost AIW of maintaining an inventory of value I is given by

AIO = FN + VR

IFIFO = $1175 − $560 − (5500 − 5200)($0.10) IFIFO = $573.00 ILIFO = $1175 − $645 − (5500 − 5000)($0.10) ILIFO = $480.00 ISAV = $1175 − 5500 [($0.10 + $0.15)/2] ISAV = $487.50 IWAV = $1175[1 − (5000/10,000)] IWAV = $528.75

9-51

(9-171)

where β is a proportionality factor and the number of orders per year N has been written as R/U. By differentiating Eq. (9-171) with respect to U, the optimum size of order can be estimated. The differential is: dAI /dU = β − (FR/U 2)

(9-172)

Setting the right-hand side of Eq. (9-172) equal to zero yields


R
/β
U = F

(9-173)

where U is now the optimum size of order. A number of models have been developed to enable managers to handle inventories in the most profitable manner. These models can be applied to other elements of working capital, such as cash.

9-52

PROCESS ECONOMICS

Working Capital The amount and disposition of working capital and the efficiency of its use determine the immediate prospects for future growth in a company. The bulk of managerial effort in a company is directly or indirectly concerned with the manipulation of working capital. Insufficient or misused working capital is the commonest cause of business failure. Engineers concerned with cost estimations tend to make estimates of the fixed-capital cost of a project, leaving considerations of working capital to the accountants. Although the estimation of fixed-capital cost is more straightforward from an engineering point of view, the estimation of working capital is of vital importance both for an individual project and for the company as a whole. Working capital can range from about 10 percent to almost 100 percent of the invested capital, depending on the industry, and is an important factor in the profitability index of a business. For this reason, it is best to compare the performance of an individual company with that of others that are as similar as possible. Gross working capital is normally defined as total current assets, while net working capital is current assets minus current liabilities. Current assets normally amount to more than half of the total assets of a company. When accountants wish to emphasize working capital in a company, they present the balance sheet in the vertical form, as shown in Table 9-31. In this table, the stockholders’ equity of $91,650 funds the sum of $38,650 (net working capital) and the sum of $53,000 (fixed assets less the long-term loan). The flow of working capital is diagrammatically illustrated in Fig. 9-39. The necessary working capital varies with sales volume or production rate. For example, sales and hence accounts receivable will double for a doubling in sales volume. In addition, an increase in sales volume normally requires increased inventories of raw materials, work in progress, and finished goods, all of which tie up capital. An increase in sales volume may lead to a relative shortage of working capital. In turn, this may mean that accounts payable cannot be paid in time and that valuable cash discounts may be lost or interest and penalty charges incurred. Creditors may take legal action to obtain payments and thereby put an additional strain on the current assets of the company to provide legal fees. Often, such action leads other creditors to take similar steps, which may lead a fundamentally sound company into bankruptcy. A shortage of cash may prevent a company from taking advantage of large discounts available for bulk purchase of raw materials. The importance of the availability of adequate cash or near cash can be seen by considering an account payable within 28 days, with a 2 percent discount allowed if paid within 7 days. If cash is not available to pay the account within 7 days, this is then equivalent to paying 2 percent interest on the money for the remaining 21-day period, or an annual compound-interest rate of more than 41 percent. Adequate cash and a history of prompt payment of accounts strengthen the credit standing of a company and make it easier to obtain bank loans, etc. A company needs additional cash as a contingency against fires, floods, strikes, etc., as well as for additional adver-

FIG. 9-39

TABLE 9-31

Balance Sheet for BCD Company, Dec. 31, 1991

Current assets Cash Accounts receivable Inventories Raw materials Work in process Finished product Gross working capital Current liabilities Accounts payable Notes payable Bank loans Accruals payable

$14,575 35,575 6,000 3,750 10,000 $69,900 25,000 3,000 3,000 250 31,250 $38,650

Net working capital Fixed assets Long-term loan

75,000 22,000 $53,000

Stockholders’ equity

$91,650

tising required to counteract the activities of competitors. This additional money is normally held as interest-bearing investments that can be turned into cash on short notice. Working-Capital Ratios Financial analysts make extensive use of ratios in assessing the economic health of a company. For evaluating the ability of a company to successfully maintain and develop its immediate business activities, analysts apply a current ratio and a quick (or acid-test) ratio, as given by current assets Current ratio = ᎏᎏ (9-142) current liabilities liquid assets Quick ratio = ᎏᎏ (9-143) current liabilities Liquid assets are those that can be realized almost immediately, such as cash, accounts receivable, and marketable securities. Although inventories are current assets, they must not be regarded as liquid assets because they cannot usually be converted into cash without winding up the business. Although a high current ratio is desirable, this may be achieved by having unnecessarily high inventories that bring no profit except when commodity prices are rising rapidly. The quick ratio is less misleading in this respect. Good management practice will hold inventories at the lowest possible levels consistent with customer satisfaction and efficient plant operation. Excessive inventories are unproductive and are an investment having little or no rate of return. Excessive inventories should be maintained only when supplies are erratic or rising in price. Management should normally aim for a high inventory-turnover ratio, as given by:

Flow of working capital showing relationship between current liabilities and current assets.

ACCOUNTING AND COST CONTROL revenue from annual sales Inventory-turnover ratio = ᎏᎏᎏ average value of inventory

(9-174)

Similarly, good management practice is to hold accounts receivable at a low level and to have a high accounts-receivable-turnover ratio, as given by Accounts-receivable-turnover ratio revenue from annual sales = ᎏᎏᎏᎏ (9-175) average value of accounts receivable Alternatively, it is good practice to have a low average collection period, as given by Average collection period average value of accounts receivable = ᎏᎏᎏᎏ (9-144) revenue from sales per day Equation (9-144) gives the number of days during which sales are tied up in receivables. An accounts-receivable-turnover ratio of 12 is considered fairly good for a manufacturing company. This implies an average collection period of about 1 month. The price obtained for the goods should include an allowance for interest (otherwise obtainable) on the money tied up for such a period. Since ratios, like balance sheets, refer to a particular point in time, they have a limited use unless they are compared with previous values. A study of ratio trends indicates whether or not a company is approaching a working-capital or a liquidity crisis and may enable management to compare the performance of the company with that of competitors. Funds Statement A typical funds statement is shown in Table 9-32. It displays the change in net working capital and can be obtained from a statement of changes in working capital, such as the one shown in Table 9-33. A funds statement shows where the cash came from and how it was used. Changes in working capital ∆CWC in an annual accounting period can be represented by ∆CWC = AS − ATE −  ∆CFC −  ∆CL +  ∆CFIN

(9-176)

where AS is the revenue from annual sales of a product, ATE is the total cost or expense required to produce and sell the product but excluding any annual provision for plant depreciation,  ∆CFC is the sum of the changes in depreciable fixed assets,  ∆CL is the sum of the changes in nondepreciable fixed assets, and  ∆CFIN is the sum of the

TABLE 9-32

Funds Statement for Year Ending Dec. 31, 1991

Sources of funds Cash income from operations New finance Sale of plant equipment Total sources of funds Applications of funds Purchase of land Cash dividend on stock Total applications of funds Increase in net working capital

TABLE 9-33

$15,000 10,000 2,000 $27,000 $3,000 9,000 $12,000 $15,000

Statement of Changes in Working Capital

Cash Accounts receivable Inventories Total current assets Current liabilities Net working capital Current ratio Quick ratio

Jan. 1, 1991

Jan. 1, 1992

Change

$ 80,000 20,000 30,000 $130,000 ($ 60,000) $ 70,000 2.17 1.67

$ 94,000 35,000 25,000 $154,000 ($ 69,000) $ 85,000 2.23 1.87

+$14,000 +15,000 −5,000 +$24,000 (+$ 9,000) +$15,000

9-53

changes in financial resources such as loans, bonds, preferred stock, common stock, etc. Equation (9-176) can also be written as ∆CWC = ACI −  ∆CFC −  ∆CL +  ∆CFIN

(9-177)

where ACI is the annual cash income, which is the main source of funds for most companies. In this case, the annual cash income excludes all noncash expenses such as the balance-sheet annual depreciation charge ABD, which is purely a book transaction. A positive value of any term in Eq. (9-177) implies an increase in working capital, and a negative value a decrease. For example, the sale of fixed assets such as plant, buildings, land, etc., is a source of cash, and the purchase of fixed assets uses up cash. Similarly, an increase in financial resources in the form of loans and stock and bond issues is a source of cash, and a decrease in financial resources in the form of repayment of loans, retirement of stocks and bonds, and the payment of cash dividends uses up cash. (Note that a stock dividend as opposed to a cash dividend does not use up cash.) The relation between (1) net annual cash flow ACF after tax for individual projects, (2) the annual amount of tax AIT, and (3) the annual expenditure of capital A⌼C is ACF = ACI − AIT − A⌼C

(9-178)

Equation (9-178) for a single project is really analogous to Eq. (9-177) for a company. An income statement or profit-and-loss account gives the net annual profit A NP before tax. In order to assess the annual cash income A CI as a source of funds from the value of the net annual profit A NP given in the income statement, it is necessary to add back all noncash expenses such as the balance-sheet annual depreciation charge A BD. This practice sometimes erroneously suggests that depreciation is a source of funds, whereas cash income is the only source of funds. Although ABD does not affect working capital in any way, the annual depreciation charge AD does affect the annual amount of tax AIT given by A IT = (A CI − AD − AA)t

(9-3)

where t is the fractional tax rate and AA is the annual amount of any other allowances. Thus, the net annual cash income is affected by depreciation allowances, as follows: A NCI = A CI − A IT

(9-2)

In this sense, depreciation makes working capital available by reducing the cash outflow for taxes. When a fixed asset is sold at a price that differs from its book value, an accounting gain or loss is recorded. This gain or loss does not affect working capital, which has simply been increased by the amount of cash received from the sale. For example, if an item of plant is sold for $40,000 (whatever its book value), the increase in working capital resulting from the sale is $40,000. If the book value of the plant had been $50,000, the accounting loss of $10,000 on the sale would be included as an expense when calculating the cash income. This $10,000 must be added back to the cash income in order to get the cash income excluding noncash expenses as required for ACI. Conversely, any accounting gain on the sale of a fixed asset must be subtracted if it has been included in the cash income. Book values of fixed assets are determined by the balance-sheet annual depreciation charges ABD, which do not affect working capital. Although the accounting gain or loss on the sale of a fixed asset is based on its book value, working capital is not affected by depreciation assessments. Transactions that change the character of the net working capital but do not affect its value occur in a company. For example, a cash payment of $10,000 for accounts payable reduces both the current asset of cash by $10,000 and the current liability of accounts payable by $10,000, leaving the net working capital unchanged. However, this transaction affects both the current and the quick ratios. After the balance sheet and the income statement or profit-and-loss account, the funds statement is generally regarded as the most important financial document. However, many financial managers regard a statement showing changes in cash as being of equal importance. As with balance sheets and income statements, there is no rigid format for funds statements. These vary in the amount of detail given and

9-54

PROCESS ECONOMICS

TABLE 9-34 Statement of Changes in Working Capital and Sources and Applications of Funds for BCD Company Balance sheets Cash Accounts receivable Inventories Total current assets Accounts payable Notes payable Bank loans Accruals payable Total current liabilities Net working capital Fixed assets Net assets Long-term loan Stockholders’ equity

Change in funds

Dec. 31, 1991

Dec. 31, 1992

$ 14,575 35,575 19,750 $ 69,900 $ 25,000 3,000 3,000 250 $ 31,250 $ 38,650 75,000 $113,650 $ 22,000 $ 91,650

$ 13,000 36,000 20,750 $ 69,750 $ 27,000 2,500 2,000 300 $ 31,800 $ 37,950 85,000 $122,950 $ 22,000 $100,950

Applications Sources $ 1,575 $

425 1,000 2,000 500 1,000 50

10,000

$12,925

9,300 $12,925

also in their layout. Funds statements help management in planning for the future, for example, in timing and making financial provision for future expenditures. A typical management accountant’s statement for changes in working capital and sources and applications of funds is shown in Table 9-34. This is based on the following relation: an increase in application of funds equals an increase in sources of funds. The relation can also be expressed as follows: an increase in assets plus a decrease in liabilities equals an increase in liabilities plus a decrease in assets. In Table 9-34, the cash income for the year has been absorbed in the increase in stockholders’ equity, which consists of issued stock plus retained profits or income. General Considerations Many people regard the management of working capital as essentially a cash-flow problem. Certainly, expansion involves increased investment in fixed assets and in the various items that comprise the current assets, all of which require cash. If this leads to a shortage of cash so that a company cannot pay its liabilities, then a situation called overtrading results, and the company may ultimately be forced into liquidation. Therefore, it is essential that a company have access to readily available cash or sources of short-term financing, preferably without having to specify a particular asset as collateral, since such collateral can only be pledged to an agreed realizable value that is much less than its true value. Accounts payable, also called trade credit, are the major source of short-term financing. Accounts payable normally amount to about 40 percent of the current liabilities of a manufacturing company. Such short-term financing is relatively expensive when available discounts are lost. The second most important source of short-term financing is notes payable from commercial banks. Banks normally require a borrower to maintain a compensating balance. For example, if a company requires a loan of $100,000, it must borrow more than this, say, $120,000 (on which it pays interest), in order to maintain a minimum checking-account balance of $20,000. Commercial banks also provide a wide variety of other services that can be of great help to companies in temporary financial difficulties. The third most important source of short-term financing is the commercial paper, or promissory notes, of large companies. This is the

cheapest form of finance. However, the amount of money available is a function of the excess liquidity of the large companies at a given time, and the money may have to be repaid at relatively short notice. Sometimes, fixed assets are purchased via short-term loans, which can lead to liquidity problems. For the most part, fixed assets should be financed from long-term or permanent capital such as stocks or bonds. The proven ability of management to handle working capital efficiently will put a company in a better position to obtain such longterm capital when required, because the confidence of bankers and stockholders will have been obtained. Budgets and Cost Control R. J. Bull (Accounting in Business, 2d ed., Butterworth, London, 1972, p. 163) defined a budget as a comprehensive and coordinated plan, expressed in monetary terms, directing and controlling the resources and trading activities of an enterprise for some specified period in the future. A budget is not a forecast. A forecast is an estimate of the future which may or may not be attained. A budget is an overall objective based on a forecast. In addition, a budget defines the detailed objectives to be achieved by various levels of management in an organization. Since the achievement of these objectives requires the complete cooperation of management, the targets set must be realistic in terms of resources and past performance. A comparison of actual with budgeted results can be used as the basis for control at the company, departmental, plant, or project level. In addition, a continuing record of performance should be maintained to provide the data for the preparation of further budgets. Since company accounts are normally published annually, 1 year is commonly taken as a budget period. However, budget periods can vary widely depending on the nature of the operation. For example, a sales budget may be for a period of, say, 3 or 6 months, while the budget period for the installation of, say, a nuclear power station would extend over many years. The basic objectives of budgets are planning and control. The first step is to determine the limiting factor. For example, budgeted sales cannot exceed the maximum productive capacity of the available plant. Since all the activities in the plan are interrelated, the extent of the plan is determined by the limiting factor. After the plan is put into operation, actual progress is monitored against established standards. These data may subsequently lead to the plan’s being modified in order to achieve the objectives more effectively. R. Pilcher (Appraisal and Control of Project Costs, McGraw-Hill, Maidenhead, England, 1973, p. 233) stated the main purposes of a cost control system to be: 1. To provide immediate warning of uneconomic operations in both the long and the short terms 2. To provide the relevant feedback, carefully qualified in detail by all the conditions under which the work has been carried out, to the estimator who is responsible for establishing the standards in the past and in the future 3. To provide data to assist in the valuation of those variations that will arise during the course of the work 4. To promote cost consciousness 5. To summarize progress Budgeted income statements are identical in form to actual income statements. However, the budgeted numbers are objectives rather than achievements. Budgetary models based on mathematical equations are increasingly being used. These may be used to determine rapidly the effect of changes in variables. Variance analysis is discussed in the treatment of manufacturing-cost estimation.

MANUFACTURING-COST ESTIMATION The annual manufacturing cost or expense AME can be written as the sum of the direct manufacturing or prime cost ADME and the indirect manufacturing or overhead cost AIME: AME = ADME + AIME

(9-179)

The determination of direct or prime costs is more straightforward than the determination of indirect or overhead costs. When more than one product is involved, the question arises as to the correct distribution of overhead costs between the various products. In addition to fixed and variable costs there are mixed or semivari-

MANUFACTURING-COST ESTIMATION able costs. These have a fixed and a variable element. The variable element can vary with production linearly, stepwise, or in a curvilinear manner. However, it is a convenient simplification to divide all manufacturing costs into fixed and linearly variable costs. General Considerations Manufacturing costs are best considered in the context of the manufacturing, trading, and profit-and-loss accounts. Typical examples of these are shown in Tables 9-35, 9-36, and 9-37, respectively. These are based on the conventional accountancy period of 1 year. The gross annual profit A GP in Table 9-36 is dependent on the balance-sheet annual depreciation charge A BD, which is not necessarily the same as the depreciation allowance used for tax purposes. Since A BD is arbitrarily chosen, it can be used to make the gross annual profit A GP high or low according to the company policy. The gross annual profit A GP is also dependent on the method of valuing the inventory. For example, raw materials may have been purchased at the beginning of the accounting year at, say, 9 cents per kilogram. The purchase price may have risen to, say, 12 cents per kilogram at the end of the accounting year. If valuation of the inventory is made at the higher purchase price, the production cost is lower and the gross profit higher than if the valuation is made at the lower purchase price. Although in this case the profit looks better, a higher tax is payable. TABLE 9-35

$

Less inventory of raw materials Dec. 31, 1991 Cost of raw materials consumed Direct wages Direct utilities Other direct expenses Prime cost or direct manufacturing expense, ADME Payroll overhead General plant overhead Other indirect expenses Indirect manufacturing expense, AIME Depreciation, ABD

35,000 870,000 25,000 $ 930,000 51,000 $ 879,000 122,000 22,000 104,000 $1,127,000 28,000 52,000 52,000 $ 132,000 68,000 $ 200,000

Add work in progress, Jan. 1, 1991 Less work in progress, Dec. 31, 1991 Production cost of products

TABLE 9-36

Less inventory of finished products, Dec. 31, 1991 Gross profit Sales

TABLE 9-37

$1,127,000

Net profit before taxes Gross profit

$ 74,000 124,000 40,000 10,000 60,000 $308,000

AS = RcS

(9-180)

(9-181)

AVGE = RcVGE

(9-182)

AVME = RcVME

(9-183)

In practice, annual direct variable costs such as raw materials, utilities, etc., are not always proportional to the production rate. Substituting Eqs. (9-182) and (9-183) into the first term on the right of Eq. (9-180) yields AS − (AVGE + AVME) = R[cS − (cVGE + cVME)]

200,000 $1,327,000 30,000 $1,357,000 35,000 $1,322,000

$ 200,000 1,322,000 $1,522,000 190,000 $1,332,000 668,000 $2,000,000

Profit-and-Loss Account

Administration Sales and shipping Advertising and marketing Technical service Research and development

ACI = [AS − (AVGE + AVME)] − (AFGE + AFME)

where AVGE and AVME are the annual variable general and variable manufacturing expenses respectively and AFGE and AFME are the annual fixed general and fixed manufacturing expenses respectively. Revenue from annual sales is

Likewise, for AVME,

Trading Account

Inventory of finished products, Jan. 1, 1991 Add production cost of products

The profit is also dependent on the method of valuing the work in progress. We shall consider the manufacture of 100,000 kg of product with a prime or direct manufacturing cost of 10 cents per kilogram and an additional indirect manufacturing expense of 5 cents per kilogram. We assume that 90,000 kg of the product is sold and 10,000 kg is stored in inventory. The value of the inventory is $1000 on the basis of prime or direct cost and $1500 when the indirect manufacturing expense is included. It is still a controversial question as to whether manufacturing overheads should be absorbed in the cost. The latter is known as absorption costing and is the traditional accounting method. Direct costing is being increasingly used and is particularly favored by engineers. The annual cash income ACI before tax, in terms of the revenue from annual sales AS of a product and the components of the total annual cost or expense required to produce and sell the product (excluding any allowance for plant depreciation), is expressed by

where R is the production rate and cS the sales price per unit of production. Similarly, AVGE can be taken as proportional to the annual production rate R and the variable general expense per unit of production cVGE:

Manufacturing Account

Inventory of raw materials, Jan. 1, 1991 Add purchases Add carriage inward

9-55

$308,000 360,000 $668,000

(9-184)

In Eq. (9-184), the term [cS − (cVGE + cVME)] is the contribution to cash income. For simultaneous production of more than one product, Eq. (9-184) can be written as AS − (AVGE + AVME) = R1 {(cS)1 − [(cVGE)1 + (cVME)1]} + R2 {(cS)2 − [(cVGE)2 + (cVME)2]} + . . .

(9-185)

where the subscripts 1, 2, etc., refer to the various coproducts. The contribution to cash income made by a particular product depends on the method of accounting. Widely different values for the contribution can be calculated by using different methods of assigning manufacturing expenses. The cost of each item in a cost estimate should be presented in such a way that the estimate can be modified and updated at any time in the future when revised data become available. Published data and shortcut estimating methods can be used to calculate the approximate manufacturing cost of a new product. However, most companies have extensive data on various items of cost such as overheads, property taxes, etc. These data should be used whenever possible to give the estimate that is most valid for a particular company. For a new product, the ratio of manufacturing expense to sales price cME/cS should be compared with the ratio of total manufacturing cost or expense to sales revenue for the company as a whole. If the ratio cME/cS is less than or equal to the ratio for the company, then the proposed sales price appears to be reasonable and the product is probably commercially viable. This comparison is, of course, used only as an approximate guide in preliminary assessments. One Main Product Plus By-Products We shall let one unit of raw material yield χ1, χ2, etc., weights of products 1, 2, etc., respectively. The variable general expense per unit of raw material will be

9-56

PROCESS ECONOMICS

cVGE and the variable manufacturing expense per unit of raw material will be cVME. The unit of raw material may be either a single material or a mixture of several components. I. Leibson and G. A. Trischman [Chem. Eng., 78, 69–74 (May 31, 1971)] showed the effects on manufacturing expenses of alternate feedstocks having different compositions and costs for producing the same products. If product 1 is the main product, then this carries all the variable expenses, so that (cVGE)1 = cVGE / χ1

(9-186)

(cVME)1 = cVME / χ1

(9-187)

By Value We shall let one unit of raw material yield χ1 and χ2 weights of products 1 and 2 respectively, with values of χ1(cS)1 and χ2(cS)2 respectively; (cS)1 and (cS)2 are the sales prices of products 1 and 2 per unit of production. We shall let the variable general expense per unit of raw material be cVGE and the variable manufacturing expense per unit of raw material be cVME. If the variable expenses are shared by value then χ1(cS)1cVGE (9-198) (cVGE)1 = ᎏᎏ [χ1(cS)1 + χ2(cS)2]χ1

where (cVGE)1 and (cVME)1 are the variable general and variable manufacturing expenses respectively per unit of product 1. For by-product 2, etc., (cVGE)2 = 0 and (cVME)2 = 0, etc. For this case, Eq. (9-184) becomes AS − (AVGE + AVME) = R1 [(cS)1 − (cVGE + cVME)/χ1] + R2(cS)2 + . . .

(9-188)

If R is the annual rate at which the raw material is consumed, then R1 = χ1 R

(9-189)

R2 = χ2 R

(9-190)

In terms of one unit of raw material, Eq. (9-185) can be combined with Eq. (9-189) and (9-190) and written as [AS − (AVGE + AVME)]/R = χ1(cS)1 − (cVGE + cVME) + χ2(cS)2 + . . .

(9-191)

Equation (9-191) gives the contribution of products 1, 2, etc., per unit of raw material. In this particular case, the contribution of the main product per unit weight of raw material is χ1(cS)1 − (cVGE + cVME). The contribution of product 2 is χ2(cS)2, i.e., its selling price per unit weight of raw material.

χ1(cS)1cVME (cVME)1 = ᎏᎏ [χ1(cS)1 + χ2(cS)2]χ1

(9-199)

χ2(cS)2cVGE (cVGE)2 = ᎏᎏ [χ1(cS)1 + χ2(cS)2]χ2

(9-200)

χ2(cS)2cVME (9-201) (cVME)2 = ᎏᎏ [χ1(cS)1 + χ2(cS)2]χ2 where (cVGE)1 and (cVME)1 are the variable general and variable manufacturing expenses per unit of product 1 respectively and (cVGE)2 and (cVME)2 are the comparable expenses for product 2. For this case, Eq. (9-185) becomes (cS)1(cVGE + cVME) AS − (AVGE + AVME) = R1 (cS)1 − ᎏᎏ [χ1(cS)1 + χ2(cS)2]



(cS)2(cVGE + cVME) + R2 (cS)2 − ᎏᎏ (9-202) [χ1(cS)1 + χ2(cS)2] This analysis can be extended for any number of coproducts. In terms of one unit of raw material, Eq. (9-202) can be combined with Eqs. (9-189) and (9-190) and written as AS − (AVGE + AVME) (cS)1(cVGE + cVME) ᎏᎏ = χ1 (cS)1 − ᎏᎏ R [χ1(cS)1 + χ2(cS)2]





Two Main Products By Weight We shall let one unit of raw material yield χ1 and χ2 weights of products 1 and 2 respectively. The variable general expense per unit of raw material will be cVGE and the variable manufacturing expense per unit of raw material cVME. In practice, it is rare for χ1 + χ2 to be exactly unity. If the variable expenses are shared by weight, then

(cS)2(cVGE + cVME) + χ2 (cS)2 − ᎏᎏ (9-203) [χ1(cS)1 + χ2(cS)2] Equation (9-203) gives the contribution of products 1 and 2 per unit weight of raw material.

(cVGE)1 = χ1 cVGE / [(χ1 + χ2)χ1]

0.32 kg of product 1 and χ2 = 0.64 kg of product 2. The balance of the raw material goes to waste. Product 1 sells at (cS)1 = 40 cents per kilogram, and product 2 sells at (cS)2 = 12 cents per kilogram. The variable general and variable manufacturing expenses, including raw materials, cVGE + cVME, total 10 cents per kilogram. Let us calculate (a) the total contribution to cash income per kilogram of raw material and (b) the individual contribution of each product to cash income per kilogram of raw material for the following:

(9-192)

(cVME)1 = χ1 cVME / [(χ1 + χ2)χ1]

(9-193)

(cVGE)2 = χ2 cVGE / [(χ1 + χ2)χ2]

(9-194)

(cVME)2 = χ2 cVME / [(χ1 + χ2)χ2]

(9-195)

where (cVGE)1 and (cVME)1 are the variable general and variable manufacturing expenses per unit of product 1 respectively and (cVGE)2 and (cVME)2 are the comparable expenses for product 2. For this example, Eq. (9-185) becomes (cVGE + cVME) AS − (AVGE + AVME) = R1 (cS)1 − ᎏᎏ (χ1 + χ2)





(cVGE + cVME) + R2 (cS)2 − ᎏᎏ (9-196) (χ1 + χ2) In Eqs. (9-192) through (9-196) the sum of χ1 and χ2 may be equal to or less than 1. The above analysis can be extended for any number of coproducts. In terms of one unit of raw material, Eq. (9-196) can be combined with Eqs. (9-189) and (9-190) and written as AS − (AVGE + AVME) (cVGE + cVME) ᎏᎏ = χ1 (cS)1 − ᎏᎏ R (χ1 + χ2)









(cVGE + cVME) + χ2 (cS)2 − ᎏᎏ (9-197) (χ1 + χ2) Equation (9-197) gives the contribution of products 1 and 2 per unit weight of raw material.







Example 21: Calculation of Contributions to Income for Multiple Products One kilogram of raw material is used to manufacture χ1 =

Case

Condition

1 2 3 4

Product 1 as the main product charged for all the variable expenses Product 2 as the main product charged for all the variable expenses Products 1 and 2 sharing the variable expenses on the basis of weight Products 1 and 2 sharing the variable expenses on the basis of value

Case 1. The total contribution per kilogram of raw material is obtained by substituting the appropriate values into Eq. (9-191). For these conditions, χ1(cS)1 + χ2(cS)2 − (cVGE + cVME) = 0.32(40 cents/kg) + 0.64 (12 cents/kg) − 10 cents/kg = 12.8 cents/kg + 7.68 cents/kg − 10 cents/kg = 10.48 cents/kg The individual contribution of product 1 as the main product is 12.8 cents − 10 cents = 2.8 cents per kilogram of raw material. The individual contribution of product 2 as the by-product is 7.68 cents per kilogram of raw material. Case 2. The total contribution when product 2 is the main product is also obtained from Eq. (9-191) and as in Case 1 is found to be 10.48 cents per kilogram.

MANUFACTURING-COST ESTIMATION The individual contribution of product 2 as the main product is 7.68 cents − 10 cents = −2.32 cents per kilogram of raw material. The individual contribution of product 1 as the by-product is 12.8 cents per kilogram of raw material. Case 3. When products 1 and 2 share the variable expenses on the basis of weight, the total contribution per kilogram of raw material is found by substituting the unit costs into Eq. (9-197). Values for each term are χ1(cS)1 = 12.8 cents/kg

(same as Case 1)

χ2(cS)2 = 7.68 cents/kg

(same as Case 1)

[χ1 /(χ1 + χ2)](cVGE + cVME) = (0.32/0.96)(10 cents/kg) = 3.34 cents/kg [χ2 /(χ1 + χ2)](cVGE + cVME) = (0.64/0.96)(10 cents/kg) = 6.66 cents/kg Therefore, the total contribution becomes 12.8 + 7.68 − 3.34 − 6.66 = 10.48 cents per kilogram. The individual contribution of product 1 is 12.8 cents − 3.34 cents = 9.46 cents per kilogram of raw material. The individual contribution of product 2 is 7.68 cents − 6.66 cents = 1.02 cents per kilogram of raw material. Case 4. When products 1 and 2 share the variable expenses on the basis of value, the total contribution per kilogram of raw material is found by substituting the unit costs into Eq. (9-203). Values of each term are χ1(cs)1 = 12.8 cents/kg

(same as Case 1)

χ2(cs)2 = 7.68 cents/kg

(same as Case 1)

χ1(cs)1(cVGE + cVME) (12.8)(10) ᎏᎏ = ᎏᎏ = 6.24 cents/kg χ1(cs)1 + χ2(cs)2 12.8 + 7.68 χ2(cs)2(cVGE + cVME) (7.68)(10) ᎏᎏ = ᎏᎏ = 3.76 cents/kg χ1(cs)1 + χ2(cs)2 12.8 + 7.68 Therefore, the total contribution becomes 12.8 + 7.68 − 6.24 − 3.76 = 10.48 cents per kilogram. The individual contribution of product 1 is found as 12.8 cents − 6.24 cents = 6.56 cents per kilogram of raw material. The individual contribution of product 2 then becomes 7.68 cents − 3.76 cents = 3.92 cents per kilogram of raw material.

Direct Manufacturing Costs Direct manufacturing costs include raw materials, operating labor, utilities, and some miscellaneous items. A summary of the characteristics of each follows. Raw Materials The cost of raw materials is normally the largest item of expense in the manufacturing cost of a product. The quantities of raw materials consumed can be calculated from material balances. Material costs are conveniently presented in tables that give the following: name of material, form and grade, method of delivery, unit of measure, cost per unit, source of cost, annual consumption, annual cost, fractional consumption per unit of production, and cost per unit of production. Net consumption of materials should be used for catalysts, solvents, filter aids, etc., that may have a recovery value. Current prices of chemicals are published in various trade journals. However, quotations from suppliers should be used whenever possible. It may be possible for a company to negotiate the purchase of a material at a cost per unit that is significantly lower than the current published price. This is particularly true if large quantities are involved. Thus, estimates should be presented for both minimum and maximum costs. Price trends, availability, and quality are other factors that should be considered. A knowledge of price trends is particularly important for a product that a company may not manufacture for several years. The yield in a chemical reaction determines the quantities of materials in the material balance. Assumed yields are used to obtain approximate exploratory estimates. In this case, possible ranges should be given. Firmer estimates require yields based on laboratory or, preferably, pilot-plant work. Operating Labor The cost of operating labor is the second largest item of expense in the manufacturing cost. Labor requirements for a process can be estimated from an intelligent study of the equipment flow sheet, paying careful attention to the various primary process steps such as fractionation, filtration, etc. The hourly wage rate should be that currently paid in the company. Once the number of persons required per shift has been estimated for a particular production rate, the annual labor cost and the labor cost per unit of production can be estimated. H. E. Wessel [Chem. Eng., 59, 209–210 (July 1952)] made a study of the operating-labor requirements in the United States chemical

9-57

industry and presented the data as a plot of labor-hours per ton per processing step versus plant capacity in tons per day. These data can be represented by: log10 Y = 0.783 log10 χ + 1.252 + B

(9-204)

where Y is the operating-labor-hours per ton per processing step; χ is plant capacity, tons per day; and B is a constant having values of 0.132 when multiple units are used to increase capacity or when the process is completely batch, of 0 for the average chemical-processing plant, and of −0.167 for large, highly automated plants or plants concerned with fluid processing. Wessel’s data for the United States chemical industry refer to the short ton equal to 2000 lb, or 907.2 kg. Labor requirements are higher in countries with lower productivities. The approximate cost of supervision for operating labor is equivalent to 10 percent of the labor cost for simple operations and 25 percent for complex operations. Utilities These include steam, cooling water, process water, electricity, fuel, compressed air, and refrigeration. The consumption of utilities can be estimated from the material and energy balances for the process, together with the equipment flow sheet. Let us consider a cooler in the equipment flow sheet. The required rate of heat removal is known from the balances, and the rate of cooling water can be calculated once the inlet and outlet temperatures of the water have been specified. The calculation of the consumption of other utilities is also straightforward. Allowances should be made for wastage. The current cost per unit for each utility is usually well known in a company. Thus, the annual cost for utilities and the utilities cost per unit of production can be estimated. The latter is normally much smaller than the raw-materials and labor costs. However, a great deal more work is involved in calculating the utilities cost than for any other item in the manufacturing cost. Unfortunately, there are no satisfactory shortcut methods for doing this. When the utilities cost is relatively small, it may be possible to make an intelligent guess on the basis of known costs for similar processes in the company. Alternatively, published data for the consumption of utilities per unit of production for various processes may be used. Miscellaneous Direct Costs Estimates for the cost of maintenance and repairs, operating supplies, royalties, and patents are best based on company records for similar processes. A rough average value for the annual cost of maintenance is 6 percent of the capital cost of the plant. This percentage can vary from 2 to 10 percent, depending on the severity of plant operation. Approximately half of the maintenance costs are for materials and half for labor. Royalty and patents costs are in the order of 1 to 5 percent of the sales price of the product. Indirect Manufacturing Costs Estimates for the cost of payroll overhead, control laboratory, general plant overhead, packaging, and storage facilities are best based on company records for similar processes. Payroll overhead includes the cost of pensions, holidays, sick pay, etc., and is normally between 15 and 20 percent of the operating-labor cost. Laboratory work is required for product quality control, and its cost is approximately 10 to 20 percent of the operating-labor cost. Plant overhead includes the cost of medical, safety, recreational, effluent-disposal, and warehousing facilities, etc. In general, the larger the plant, the lower the overhead per unit of production. Plantoverhead costs can vary between 15 and 150 percent of the operatinglabor cost. Packaging costs depend on the physical and chemical nature of the product as well as on its use and value. The cost of packaging is as high as one-third of the selling price for soaps and pharmaceuticals. Rapid Manufacturing-Cost Estimates Fixed manufacturing costs are a function of the fixed-capital investment and are independent of the production rate of the plant. Property taxes or rates depend on location. They may be taken as 2 percent of the fixedcapital cost of the plant in the absence of specific data. The cost of insurance depends on both location and the hazardous nature of the materials handled. This cost is normally of the order of 1 percent of the fixed-capital cost of the plant.

9-58

PROCESS ECONOMICS

The manufacturing cost of a product is the sum of the processing or conversion cost and the cost of raw materials. The processing cost can be roughly broken down into three parts: investment-related cost, labor-related cost, and utility cost. Companies usually include in the charge for overhead the following items: operating supplies, supervision, indirect payroll expenses, plant protection, plant office, general plant overhead, and control laboratory. This overhead charge is frequently taken as an equivalent percentage of the direct labor cost. The percentage is best obtained from company records. Although it can vary over a wide range, a reasonable value is 125 percent. In this case, the labor-related cost would be 2.25 times the direct labor cost. For a 6000-h year and N persons per shift earning cL$ per hour, the annual labor-related cost would be 13,500 (cL N). Let us consider a plant of fixed-capital cost CFC. If the annual property taxes are taken as 0.02 CFC, insurance as 0.01 CFC, and maintenance as 0.06 CFC, the annual investment-related cost would be 0.09 CFC. Annual utilities cost is AU. The annual processing cost Ap can be represented by A p = η1CFC + η2cL N + AU

(9-205)

where the factors η1 and η2 can be obtained from the data available in a particular company and have the dimensions of year −1 and hours per year respectively. Substituting the information previously given into Eq. (9-205) yields the relationship A p = 0.09CFC + 13,500cL N + AU

(9-206)

Equation (9-206) represents very closely the manufacturing costs of a particular company and is typical of the coefficients to be expected. The annual processing cost AP2 for a similar plant of a different size designed for an annual production rate R2 can be approximately calculated from an equation of the form AP2 = η1CFC1(R2/R1)0.7 + η2cLN1(R2/R1)0.25 + AU1(R2/R1)

(9-207)

(F. A. Holland, F. A. Watson, and J. K. Wilkinson, Introduction to Process Economics, 2d ed. Wiley, London and New York, 1983, p. 158). The processing cost per unit of production for a plant with an annual production R2 can be approximately calculated from AP2 100 R R2 0.7 R2 0.25 + AU1 ᎏ2 (9-208) ᎏ = ᎏ η1 CFC ᎏ + η2cL N1 ᎏ R2 R2 R1 R1 R1 where AP2 /R2 is in cents per kilogram. Equation (9-208) can be used to compute data for plots such as Fig. 9-40, which shows the decrease in processing cost per unit of production AP2 /R2 with increasing plant size. Manufacturing Cost as a Basis for Product Pricing Pricing on the basis of cost plus a fair profit has the disadvantage of ignoring demand. The modern approach is to price on the basis of market research. However, the classic cost-plus-fair-profit approach can still give useful complementary information. This can be done by any of the following three methods:



 

 

 

1. Absorption pricing 2. Rate-of-return pricing 3. Marginal pricing The gross annual profit A GP for a product is given by AGP = AS − AME − ABD

(9-6)

where AS is the revenue from annual sales, AME the annual manufacturing cost, and ABD the balance-sheet annual depreciation charge. Equation (9-6) can also be rewritten in the form AS = A GP + AVME + AFME + ABD

(9-209)

where AVME and AFME are the annual variable and fixed manufacturing costs or expenses respectively. Equation (9-209) can also be rewritten as cS = cVME + (AGP /R) + (AFME + ABD)/R

(9-210)

where R is the annual sales volume taken as equal to the annual production rate, cS is the sales price per unit of production, and cVME is the variable manufacturing expense per unit of production. Absorption pricing is based on a normal annual production rate R. The gross profit per unit AGP/R is taken as a fixed percentage χ of the fixed plus variable manufacturing costs given by the equation AGP χ AFME + ABD (9-211) ᎏ = ᎏ cVME + ᎏᎏ R 100 R We combine Eqs. (9-210) and (9-211) to give 100 + χ AFME + ABD (9-212) cS = ᎏ cVME + ᎏᎏ 100 R Equations (9-210), (9-211), and (9-212) are based on a fixed normal annual production rate R. Let us consider a change in annual production rate to R + ∆R. In order to maintain the gross profit per unit as AGP /R, the sales price per unit of production would need to be cS − ∆cS. For this case Eq. (9-210) can be written in the modified form



 











cS − ∆ cS = cVME + (AGP/R) + [(AFME + ABD)/(R + ∆R)]

(9-213)

We subtract Eq. (9-213) from Eq. (9-212) to give ∆R AFME + ABD (9-214) ∆cS = ᎏ ᎏᎏ R R + ∆R Equation (9-214) gives the overpricing ∆ cS per unit of production for an increase in annual production rate ∆ R. Equation (9-214) also gives the underpricing ∆ cS per unit of production for a decrease in annual production rate ∆ R. In the first case the fixed costs or overheads are said to be overabsorbed and in the second case underabsorbed. Absorption pricing is rigid and arbitrary and may result in business being turned away if the fixed sales price cS cannot be obtained even though the business may give a useful contribution to fixed costs. Rate-of-return pricing is a modified form of absorption pricing. It is based on the equation AGP AGP CTC (9-215) ᎏ= ᎏ ᎏ AME AME CTC where CTC is the total capital employed. Equation (9-215) can also be written as





  

Percentage markup on cost = (capital-turnover ratio)(projected rate of return on capital)

FIG. 9-40

Decrease in processing cost against increase in plant size.

The percentage markup on cost is calculated for a known capitalturnover ratio and a desired rate of return on capital. As with absorption pricing, the percentage markup on manufacturing cost per unit of production is calculated for a normal annual production rate. If this production rate is exceeded, the rate of return on capital will be higher than projected because of the decrease in unit cost. Conversely, if the production rate is lower than normal, the rate of return on capital will be lower than projected because of the increase in unit cost. For production rates both higher and lower than the normal production rate, the percentage markup is based on the normal unit cost. Thus the method is strictly valid only for the normal production rate.

MANUFACTURING-COST ESTIMATION With marginal pricing a company chooses its selling prices so as to maximize the total contribution  R(cS − cVME) from its various products. The method is particularly useful for large multiproduct companies with extensive existing facilities since it is the marginal cost that must be considered as the base case when entering into competition with another company. The lowest acceptable price for a product is that which gives the lowest worthwhile contribution to fixed costs. Marginal pricing enables a company to develop a more aggressive pricing policy than when using absorption or rate-of-return pricing. Absorption, rate-of-return, and marginal pricing have been considered here on the basis of manufacturing cost. Total cost, which is the sum of manufacturing and general costs, can also be considered as the basis. In this case the appropriate profit to consider is the net annual profit rather than the gross annual profit. Standard Costs for Budgetary Control For convenience and simplicity, we shall consider the total cost of a manufactured product to be the sum of the material, labor, and overhead costs. Standard costs are those that have been predetermined and budgeted for the manufacture of a given amount of product in a given time. The deviation of the actual cost from the standard cost is called the variance. It is far easier to make comparisons between periods by using variances than by using actual production data. The different variances for material, labor, and overhead costs are listed in Table 9-38. Standard costing is extensively used in budgetary-control systems. Criteria for the establishment of standards range from the maximum possible under ideal conditions to those expected under normal conditions. Past or historical costs are not always the best basis for setting up standards because past performance may have been unnecessarily inefficient. Static and Flexible Budgets Overhead cost can significantly affect the profitability of a project and is the only cost outside the control of the project manager. The project is expected to contribute a definite amount toward the expenses of the company and will be charged this amount even if the production rate is zero. This is the fixed component of the overhead cost and will include directly allocable costs such as depreciation and a proportion of general costs such as office salaries and heating. Other nonproduction costs such as indirect labor may vary linearly with the production rate and represent the variable component of the overhead. Costs that are neither fixed nor variable but occur in disTABLE 9-38

Variances

Revenue from sales RcS − R*c*S =

Sales price R(cS − c*S) +

Sales volume c*S(R − R*)

Profit RcNP − R*c*NP

=

Unit profit R(cNP − c*NP) +

Quantity c*NP(R − R*)

Total cost RcTE − R*c*TE

=

Unit cost R(cTE − c*TE) +

Production rate c*TE(R − R*)

Direct material cost Material price RcRM − R*c*RM = R(cRM − c*RM) +

Material usage c*RM(R − R*)

Direct labor cost θcL = θ*c*L

=

Overhead cost COH − θ*c*NOH

=

Wage rate Labor efficiency θ(cL − c*L) + c*L(θ − θ*) Budgeted cost (COH − θcBOH) +

Volume Efficiency θ(cBOH − c*NOH) + c*NOH(θ − θ*)

cBOH = Flexible budgeted overhead cost, $/h. cL = Actual labor cost, $/h. c*NOH = Standard overhead cost based on normal production rate, $/h. cNP = Actual net profit before tax, $/unit. COH = Actual overhead cost, $/period. cRM = Actual raw-materials cost, $/unit. cS = Actual selling price, $/unit. cTE = Actual total cost, $/unit. R = Actual quantity, units/period. R* = Standard quantity, units/period. θ = Actual time to produce a given quantity, h. θ* = Standard time to produce a given quantity, h. cNP = cS − cTE. NOTE: The asterisk on all items not otherwise defined indicates the standard cost for that item.

TABLE 9-39 Overhead cost, $/month Fixed Variable Semivariable Total

9-59

Flexible Budget for Overhead Costs Production, 1 million lb/month 8

9

10

11

12

40,000 40,000 40,000 120,000

40,000 45,000 40,000 125,000

40,000 50,000 60,000 150,000

40,000 55,000 60,000 155,000

40,000 60,000 90,000 190,000

crete steps at various production levels (such as supervisory labor) are the semivariable component of the overhead cost. It is an easy matter to determine these various components for various production rates and list them as shown in Table 9-39. Two types of overhead budget are currently in use. The static (often referred to as the fixed) budgeted overhead cost is related to the standard budgeted production rate. The flexible budgeted overhead cost is that shown as the total cost in Table 9-39. Values for intermediate production rates are often obtained by interpolation. This is justifiable only when semivariable costs are a negligible part of overhead costs. Flexible budgeting is more widely used than static budgeting despite certain logical difficulties. This is so because production in many cases is seasonal and the use of a static production norm might distort evaluation of performance. Variances are the difference between the actual costs expended and the budgeted costs expected. Variances are unfavorable if positive and favorable if negative. Any variance should be explained and, if necessary, controlled; the largest variance should be considered first. Let us consider the overhead-cost data for Table 9-39 with 10 million kg per month as the standard production rate. The static budgeted overhead is then $150,000 per month, or 1.5 cents per kilogram. We assume that the actual overhead is $186,000 for a month in which 12 million kg was produced. Then, the static budgeted overhead cost would be 12 million(1.5), or $180,000 per month. Therefore, the variance is $186,000 − $180,000 = +$6000, which is unfavorable because $6000 more was spent than was anticipated. From Table 9-39 we find that the flexible budgeted overhead cost for a production rate of 12 million kg per month is $190,000. The corresponding variance is $186,000 minus $190,000, or −$4,000, which is favorable because $4,000 less was spent than was anticipated. Thus, the use of flexible budgeting makes this particular performance look better without changing either the production rate or a single cost of the planned budget. The Standard Hour The standard hour can be defined as the number of units of output expected to be produced in 1 h. It is often used as a measure of output rate by cost accountants. Let us consider a batch processing unit that can produce either 1000 kg of product A in a cycle time of 5 h or 900 kg of product B in a cycle time of 3 h. Thus, for this processing unit a standard hour is 200 kg of product A or 300 kg of product B. In a budget period of, say, 1000 h, it is possible to produce 200,000 kg of product A, or 300,000 kg of product B, or any appropriate combination of the two products. For example, let us assume that production requirements are twice as great for product A as for product B, i.e., a ratio of 600 kg, or 3 standard hours, of product A to 300 kg, or 1 standard hour, of product B. On this basis, for a budget period of 1000 h, 750 standard hours [(e)1000] would be used to produce 750(200) = 150,000 kg of product A, and 250 standard hours [(1/4)1,000] would be used to produce 250(300) = 75,000 kg of product B. Production efficiency Pe can be calculated from Pe = (Ps /Pa )100 (9-216) where Ps is the actual production rate in standard hours and Pa is the actual hours worked. The level of production activity Pᐉ can be calculated from Pᐉ = (Ps /Pb )100 (9-217) where Pb is the budgeted production in standard hours. The deviation from budgeted capacity bc can be calculated from bc = (Pa /Pw)100 (9-218) where Pw is the budgeted number of working hours.

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PROCESS ECONOMICS

TABLE 9-40

Sales, Profits, and Manufacturing Costs

Component

Actual, $/period

Standard, $/period

Variance, $/period

Direct materials cost Direct labor cost Overhead cost Manufacturing cost Revenue from sales Gross profit

325,080 55,001 75,000 455,081 588,240 133,159

350,000 51,600 68,370 469,970 570,000 100,030

−24,920 +3,401 +6,630 −14,889 +18,240 +33,129

Actual Costs versus Standard Costs Let us consider the sales, profits, and manufacturing-cost data in Table 9-40. The gross profit is $33,129 per period better than expected. Clearly, there is less incentive to investigate overall costs when the profit variance is favorable than if the profit were less than expected. However, standard costing enables an objective analysis of the data, whether good or bad, to be made. The individual variances in Table 9-40 show that the increased profit is due to reduced material costs, which affect manufacturing costs to a greater extent than increased labor and overhead costs. The individual variances also show that to an even greater extent the increased profit is due to the increase in sales revenue. Clearly, management will wish to investigate both labor and overhead costs for any inefficiencies and to ascertain the reasons for the improved sales revenue. If necessary, the standard values can be revised. We notice that profit is obtained as the difference between two large cash sums and that variances of some 3 percent in manufacturing costs and sales revenue have resulted in a variance of some 33 percent in gross profit. Table 9-40 is a very simplified presentation. In a full standard costing system, the direct material, direct labor, and overhead variances are broken down into component parts to enable an even closer look at the operation. Standard costing is an invaluable aid to management for controlling a business. Variances: Direct Material Cost Since the variance in direct material cost ∆ (RcRM) is the difference between actual cost and standard cost, ∆ (RcRM) = RcRM − R*c*RM (9-219) where R is the actual quantity and R* is the standard quantity, units per period, cRM is the actual price, and c*RM is the standard price, $ per unit. Equation (9-219) can be written in an expanded form: ∆ (RcRM) = R(cRM − c*RM) + c*RM(R − R*)

(9-220)

where R(cRM − c*RM), known as the direct-material-price variance, is the actual quantity multiplied by the deviation in unit price, and c*RM(R − R*), known as the direct-material-usage variance, is the standard unit price multiplied by the deviation in quantity. By using the data of Table 9-41, let us calculate the direct materials cost and the standard direct materials cost as RcRM = 1,806,000(0.18) = $325,080/period R*c*RM = 1,750,000(0.20) = $350,000/period

of (1,806,000)(0.18 − 0.20) = −$36,120 per period. This variance is favorable. Likewise, we calculate a direct-material-usage variance of (0.20)(1,806,000 − 1,750,000) = $11,200 per period. This variance is unfavorable. In this case, the favorable direct-material-cost variance was achieved because of a lower unit price despite an inefficient material usage, which needs to be investigated. (There is no room for complacency since the lower unit price may well be temporary.) In the case of mixtures of raw materials, the direct-material-usage variance can be further subdivided into (1) a direct-material-mixture variance and (2) a direct-material-yield variance. The former is due to the difference between the actual and standard mixture compositions, and the latter to the difference between the actual and standard yields. Here, the standard yield is the output expected from the standard input of material. The yield variance denotes the extent of loss of material. The direct-material-mixture variance can be illustrated by Example 22. Example 22: Direct-Material-Mixture Variance A standard mixture of 100 units of material contains 70 percent of material A at $0.08 per unit and 30 percent of material B at $0.12 per unit. The standard mixture cost is 70(0.08) + 30(0.12) = $9.20. Now let us consider a mixture of 100 units containing 75 percent of material A and 25 percent of material B. The cost of this mixture at standard prices is 75(0.08) + 25(0.12) = $9.00. The direct-material-mixture variance is $9.00 − $9.20 = −$0.20 and is favorable. The favorable variance has been brought about by using more of the lower-priced material A and less of the higher-priced material B. The direct-material-yield variance is illustrated as follows. Let us assume that the standard mixture (cost $9.20 for 100 units) has a standard loss of 20 percent, making the cost $9.20 for 80 units, or $0.115 per unit of output. Now let us consider the actual loss to be 30 percent, leaving 70 units of output for each 100 units of input. The directmaterial-yield variance is 0.115(80 − 70) = $1.15 and is unfavorable. Variances: Direct Labor Cost Since the variance in direct labor cost ∆ (θcL) is the difference between actual cost and standard cost, ∆ (θcL ) = θcL − θ*c*L

(9-221)

where cL is the actual pay or wage rate, $ per hour; c*L is the standard pay or wage rate, $ per hour; θ is the actual time taken to produce a given quantity of product in a given period, hours; and θ* is the standard time taken to produce a given quantity of product in a given period, hours. Equation (9-221) can also be written in expanded form: ∆ (θcL ) = θ(cL − c*L ) + c*L(θ − θ*)

(9-222)

where θ(cL − c*L), known as the direct pay, or wage-rate, variance, is the actual time taken to produce a given output multiplied by the deviation in wage rate, and c*L(θ − θ*), known as the direct-laborefficiency variance, is the standard wage rate multiplied by the deviation in time taken to produce a given output. By using the data of Table 9-41, we calculate that 1 standard hour corresponds to (1,750,000/6250) = 280 units/standard hour Standard time to actual production is

From Eq. (9-219) we calculate the direct-material-cost variance as −$24,920 per period. This variance is favorable. However, by using the relations of Eq. (9-220) we calculate a direct-material-price variance

θ* = (1,806,000/280) = 6450 standard hours/period Direct labor cost is θcL = 6509(8.45) = $55,001/period

TABLE 9-41

Cost Data for Problems

Standard direct labor cost is

Factor

Actual value

Standard value

Raw-materials cost, $/unit Direct labor cost, $/unit Production rate, units/period Production time h/period Fixed and semivariable overhead cost, $/period Variable overhead cost, $/h Overhead cost, $/period

cRM = 0.18 cL = 8.45 R = 1,806,000 θ = 6509

c *RM = 0.20 c *L = 8.00 R* = 1,750,000 θ *N = 6250

COH = 75,000

C *FOH = 60,000 C *VOH = 1.00

θ*c*L = 6450(8.00) = $51,600/period From Eq. (9-219) we calculate the direct-labor-cost variance as $3401 per period. This variance is unfavorable. However, by using the relations of Eq. (9-220), we calculate a direct pay, or wage-rate, variance of θ(cL − c*L) = 6509(8.45 − 8.00) = $2929 per period. This direct pay variance is unfavorable. Likewise, we calculate the direct-laborefficiency variance as c*L(θ − θ*) = 8.00(6509 − 6450) = $472 per period. This variance is also unfavorable.

MANUFACTURING-COST ESTIMATION For this example, the adverse direct-labor cost variance of $3401 is due to both a higher wage rate per hour and a higher number of laborhours. The direct-labor-cost variance can, if necessary, be broken down into a direct-labor-idle-time variance in addition to the direct-wagerate and direct-labor-efficiency variances. The direct-labor-idle-time variance is simply the number of idle labor-hours in the period multiplied by the standard wage rate. This is rarely relevant to the conditions existing in process plants except when maintenance is involved. Variances: Overhead Cost The variance in overhead cost ∆COH is the difference between actual overhead cost and static standard overhead cost, ∆COH = COH − θ*c*NOH

(9-223)

where COH is the actual overhead cost incurred in a given period, $ per period; c*NOH is the static standard overhead cost based on the normal production rate, $ per hour; θ is the actual time taken to produce a given quantity of product in a given period; and θ* is the standard time taken to produce a given quantity of product in a given period, hours. Equation (9-223) can also be written in expanded form ∆COH = (COH − θcBOH) + θ(cBOH − c*NOH) + c*NOH (θ − θ*)

(9-224)

where cBOH is the flexible budgeted overhead cost at the actual production rate or operating capacity. In Eq. (9-223), (COH − θcBOH) is known as the budgeted overheadcost variance, θ(cBOH − c*NOH) as the overhead-volume variance, and c*NOH(θ − θ*) as the overhead-efficiency variance. The last is analogous to the labor-efficiency variance and is the standard overhead rate multiplied by the deviation in time taken to produce a given output. Also in Eq. (9-224), cBOH is simply the flexible budgeted overhead cost in dollars per hour for the actual production rate, and the overhead-volume variance θ(cBOH − c*NOH) is the actual time taken to produce a given output multiplied by the difference between the flexible budgeted overhead cost and the standard overhead cost in dollars per hour. The budgeted overhead-cost variance (COH − θcBOH) is the difference between the actual overhead cost and the actual time (in hours) required to produce the given output multiplied by the flexible budgeted overhead cost (in dollars per hour). We shall write the fixed overhead cost for the budget period as C* FOH, the semivariable overhead cost as C* SVOH, and the standard hours to produce the agreed normal production as θ*N. The standard overhead cost at the agreed normal production rate can then be calculated from c*NOH = [(C*FOH + C*SVOH)/θ*N] + c*VOH

(9-225)

where c*VOH is the standard variable overhead cost, $ per hour. For production rates that differ from the agreed normal rate, the flexible budgeted overhead cost is given by c BOH = [(C*FOH + C*SVOH)/θ] + c*VOH

(9-226)

where θ is the actual hours taken to produce a given amount of product. For production rates lower than normal, the fixed overheads are underused, and the flexible budgeted overhead cost cBOH is greater than the standard overhead cost c*NOH. For production rates higher than normal, cBOH is less than c*NOH. It is common practice in cost accountancy to treat the standard semivariable cost C*SVOH at the normal production rate as part of the standard fixed cost. In this case, Eqs. (9-225) and (9-226) can be written respectively as c*NOH = (C*FOH/θ*N) + c*VOH

(9-227)

cBOH = (C*FOH /θ) + c*VOH

(9-228)

By using the data of Table 9-40 in Eq. (9-227), we calculate the standard overhead cost to be c*NOH = (60,000/6250) + 1.00 = $10.60/h From Eq. (9-228) and the data in Table 9-40 we calculate the flexible budgeted overhead cost to be cBOH = (60,000/6509) + 1.00 = $10.22/h

9-61

By substituting into Eq. (9-223), we calculate the overhead cost variance: ∆cOH = 75,000 − 6450(10.60) = $6630 This variance is unfavorable. (Note that standard time for actual production was previously calculated to be 6450 h per period.) The overhead cost variance comprises (1) budgeted overhead-cost variance, (2) overhead-volume variance, and (3) overhead-efficiency variance. The calculations for each follow: cOH − θcBOH = 75,000 − 6509(10.22) = $8478 Budgeted overhead-cost variance is positive and, therefore, unfavorable. θ(cBOH − c*NOH) = 6509(10.22 − 10.60) = −$2473 Overhead volume variance is negative and favorable. c*NOH(θ − θ*) = 10.60(6509 − 6450) = $625 Overhead efficiency variance is positive and unfavorable. The total variances in each category are listed in Table 9-40. Chemical engineers usually make detailed evaluations of costs rather than evaluations for profits or sales. However, the latter can be analyzed in a similar manner to costs by using the equations shown in Table 9-38. For this purpose, the sign convention will be reversed because an increase in sales or profits would be considered favorable, whereas an increase in cost would be considered unfavorable. The equations can be applied to both batch and continuous processes. Budgets can be used for both forward planning and control. Variances show managers what their costs should have been and how near they came to meeting budgeted values. Managers will be able to assess, over a number of budget periods, the rate of improvement in performance in their areas of responsibility. A good budgetary system not only should provide detailed information and an appraisal of performance but also should motivate people to improve performance. Contribution Analysis Contribution analysis can be used to make rapid assessments of the effect of changes in manufacturing costs on profitability. A dimensionless contribution efficiency η can be defined by rewriting Eq. (9-12) in the form R(cS − cVE) − AFE η = ᎏᎏ (9-229) R(cS − cVE) [F. A. Holland and F. A. Watson, Eng. Process Econ., 1, 135–143 (1976)]. This represents the ratio of the net annual profit ANP actually achieved divided by the profit which could be obtained if no repayment of capital or interest were required and all fixed-expense items were credited free to the project. The contribution efficiency η is also the profit per unit of contribution. A value for η of unity would be obtained for a very high production rate R whether cS is greater or less than cVE. For the unusual case of cS being equal to c VE, the value of η would become negatively infinite for a finite annual fixed expense A FE or positively infinite if A FE became negative because of excessive subsidy of expenses. However, for most projects which are intended to pay their own expenses and taxes, ANP must be positive, and hence cS is usually greater than cVE, so that η will normally have values in the range of zero to unity. For projects which are not intended to make a profit but are provided for their social or amenity value, the aim should be to bring the value of η as near to zero as possible. The breakeven production rate RB is defined by Eq. (9-13) as the production rate at which the project makes neither a profit nor a loss. Equation (9-13) and (9-229) can be combined to give η = (R − R B)/R

(9-230)

which shows that the contribution efficiency η is a function of the production rate R and that η has the value zero when the production rate is the breakeven production rate RB. For all real projects R B will be positively finite while R cannot be less than zero, and hence the practical range of η is from negative infinity to unity. At first glance it might appear that it is desirable to have a value of η as near to unity as possible. However, this is not necessarily so. Reference to Eq. (9-229) will show that if the unit contribution (cS − cVE)

9-62

PROCESS ECONOMICS

is positive, as it must be if the project is ever to make a profit, a value of unity for the contribution efficiency η implies a negligible value of the annual fixed expense AFE when compared with the annual contribution R(cS − cVE). In such cases either R(cS − cVE) is very large, thus attracting competition in spite of high capital charges, or the fixed expenses are very low, so that it is easy for many small competitors to enter the market. The result of such competition often leads to a rapid reduction in sales price cS. Ball-point pens and electronic calculators were both drastically reduced in price as a result of competitors entering the market. Since the variable expense per unit of production cVE is by definition independent of production rate, the unit contribution (cS − cVE) and hence the value of η will be reduced. On the other hand, a very large company may be in a stronger and more stable position with a modest contribution efficiency and relatively high fixed costs which will deter competitors from entering the market and thereby depressing the sales price. In this argument it is implicit that cS is also independent of output from the project under consideration. In many cases this will be the case since if many small buyers can choose from many alternative producers, the individual producer cannot adjust the price to suit its output, while at the opposite extreme a group of producers that are in a position to make such adjustments are also likely to attract the attention of antitrust legislation. It is of interest to be able to examine the effect of changes in productivity on the profitability of projects. Historically, labor costs have been regarded as variable costs, implying that if workers doubled their output their net wages also doubled. This may have been the case for some piecework rates, but it is generally not true today. It is not normally possible to reduce the work force in step with falling demand or to recruit and train labor in step with increasing demand. In general it is better to consider labor as a fixed cost, with any part of a production bonus which is truly proportional to output included in the variable expense cVE. If the annual fixed expense AFE varies significantly with production rate R owing to this factor, then the breakeven chart will consist of curves, the simplicity of the method is lost, and it will be assumed that a particular change in productivity agreements implies a step change in AFE and/or in cVE. Let us consider an increase in productivity for the same fixed labor cost, with other fixed costs remaining the same. We shall let the original production rate R be increased by an increase in productivity by a fraction φ. Therefore, ∆R = Rφ

(9-231)

where ∆R is the increase in sales volume or production rate. By substitution into Eq. (9-229), the resulting increase in profit ∆A NP is given by A NP + ∆ANP = AS + ∆ AS − (AVE + ∆ AVE) − (AFE + 0)

(9-232)

We shall subtract ANP from Eq. (9-232) to give ∆ A NP = ∆ AS − ∆ AVE = ∆ R(cS − cVE)

(9-233)

It follows from Eqs. (9-12), (9-229), and (9-233) that ∆ ANP /ANP = φ/η

(9-234)

Thus a change of productivity φ of 10 percent will result in a 10 percent increase in profit when η = 1 and a very large increase in profit when η is close to zero. If η is negative, increased productivity reduces the profit (or increases the loss). Equation (9-234) illustrates the enormous influence that go-slow tactics can have on the profitability of companies and processes which have low contribution efficiencies, since a slowdown has little effect on AFE. It is sometimes the case that in different countries productivity per worker varies considerably in similar industries. When poor productivity is not the result of technical or capital inadequacy, it should be possible to increase profitability without a proportionate increase in AFE. Breakeven charts present a snapshot of the present situation by means of graphs which are generally drawn in the manner shown in Figs. 9-2, 9-3, and 9-4. Since the lines are straight, this implies that cS, cVE, and AFE will remain constant over the range of variation of R, which is of interest. The values would be based on the production rate currently achieved (or scheduled), since all the data are available from

the financial analysis of current production, so that cS would be (AS)0 /R0, and so on. Two well-known ratios used by financial analysts are the profit-tosales ratio (PSR) and the contribution-to-sales ratio (CSR). The profitto-sales ratio is simply Eq. (9-127) for profit margin (PM) rewritten as the ratio (PSR) = ANP /RcS

(9-235)

The contribution-to-sales ratio (CSR) = (AS − AVE)/AS = (cS − cVE)/cS

(9-236)

We substitute these values into Eq. (9-229) to give η = (PSR)/(CSR)

(9-237)

The contribution efficiency at the scheduled output η0 is given by substituting the value of the scheduled output into Eq. (9-229) to give R0 (cS − cVE) − AFE η0 = ᎏᎏ R0 (cS − cVE) The characteristic shape of a given breakeven chart of the type represented by Figs. (9-2), (9-3), and (9-4) can be defined by the two ratios (CSR) and η0, while the scale of the project can be defined by a single annual cost such as AFE. This information may be used for the rapid investigation of the likely effect on current profits obtainable by changes in various factors such as prices, expenses, and throughput. It should be noted that this technique is not intended to replace discounted methods of investment appraisal but to provide a rapid assessment of the probable effect of changes in current conditions. If the current profitability is always maximized, then the discountedcash-flow present value will always be made as great as conditions in a changing world will permit. Valuation of Recycled Heat Energy The rising cost of energy is having an inflationary effect on manufacturing costs. One obvious way to reduce energy costs is to recycle heat energy whenever possible [S. A. K. El-Meniawy, F. A. Watson, and F. A. Holland, Indian Chem. Eng., 22 (July–September 1980)]. Heat pumps are particularly suitable for recycling heat energy in the chemical-process industries. For the outlay of an additional fixedcapital expenditure CFC on a heat-pump system, a considerable reduction in the annual heating cost can be effected. Let us consider a process unit requiring heat at the rate of QD GJ/h operating for y h in a year. We shall let the unit cost of this base heating requirement be cB $ per gigajoule. Therefore the annual heating cost for this unit is QDycB $ per year. We then consider the use of a heat pump to supply this heat so that QD = W(COP)A

(9-238)

where W is the rate of energy input to the compressor in gigajoules per hour and (COP)A is the actual coefficient of performance of the heat pump. When interest charges are involved, the fixed-capital expenditure on a heat-pump system CFC can be related to an annual cost AFC for the estimated life of the heat pump in years by the equation AFC = CFC fAP

(9-239)

where fAP = [i(1 + i)n]/[(1 + i)n − 1], the annuity present-worth factor, and i is the fractional interest rate per year payable on the borrowed money. A given value for AFC enables a cost in, say, dollars per gigajoule, to be assigned to the heat energy made available by a heat pump. We shall consider a heat-pump system which operates for y h/year and consumes W GJ/h of high-grade energy to drive the compressor. We shall let the unit cost of the input energy to the compressor be cI $ per gigajoule. The annual amount of heat delivered by the heat pump is QDy, which in terms of Eq. (9-238) can be written as W(COP)Ay in gigajoules per year. The annual cost of this delivered heat, neglecting any maintenance cost, is (WcI y + AFC) in dollars per year. Therefore the unit cost cD of the heat energy delivered by a compressor-driven pump is cD = (WcI y + AFC)/[W(COP)Ay] $/GJ

(9-240)

FIXED-CAPITAL-COST ESTIMATION The ratio of the unit costs of the delivered heat energy and the input energy can be obtained by combining Eqs. (9-239) and (9-240) to give (cD /cI) = [(1 + ψ)/(COP)A] (9-241) where ψ = (CFC /W)( fAP /ycI). ψ is a dimensionless parameter which contains cost and usage data for a particular heat pump; it should have as low a value as possible in order to minimize the unit cost of the delivered heat cD. The cost CFC/W is the fixed-capital cost per unit of input energy in $ (GJh−1)−1, and this should also be as low as possible consistent with a long life and good reliability for the heat pump. Clearly y, the number of operating hours per year, should approach as closely as possible to the maximum value of 8760 h for a 365-day year. Since all costs refer to a given year, Eqs. (9-239), (9-240), and (9-241) are independent of inflation. The annual cost of heat delivered by the heat pump is QDycD, where the unit cost cD is given by Eq. (9-240). Therefore, the annual saving on heating costs in dollars per year is QDy(cB − cD), which can also be written in terms of Eq. (9-238) as W(COP)Ay(cB − cD). The payback period in years for a heat-pump system is the additional fixed-capital cost CFC divided by the annual saving on heating costs. This can be written as CFC 1 (PBP) = ᎏ (9-242) ᎏᎏ W (COP)Ay(cB − cD) which can also be written in terms of Eq. (9-241) as CFC 1 (9-243) (PBP) = ᎏ ᎏᎏᎏ W y[(COP)AcB − cI(1 + ψ)] For the special case of the unit cost of the input energy to the compressor being the same as the unit cost of the base heat supply, i.e., cB = cI, Eq. (9-243) simplifies to CFC 1 (PBP) = ᎏ (9-244) ᎏᎏᎏ W ycI[(COP)A − (1 + ψ)]

 

 



 

9-63

Equation (9-244) can be used to calculate the payback period when electricity, oil, or gas, etc., is used to drive the compressor and also to provide the base heating. Equation (9-244) shows that, to have a low payback period (PBP), CFC/W and ψ should be small and y, (COP)A, and cI large. Clearly as the unit cost of input energy cI increases, the economics of heat pumps becomes more favorable. The value of ψ will for most cases be less than 0.2 and with the right application may well be less than 0.1. Values for the annuity presentworth factor will in most cases be less than 0.15. Since 1 bbl (0.159 m3) of oil is normally quoted as having a thermalenergy value of 6.12 GJ, a world oil price of, say, US$40 per barrel is equivalent to US$6.54 per gigajoule. For simplicity, we substitute cI = US$6 w per gigajoule and a conservative value of [(COP)A − (1 + ψ)] = 3 into Eq. (9-244) to give (PBP) = (CFC / W)(1/20y)

(9-245)

where (PBP) is the payback period in years, y is the operating hours per year, and (CFC/W ) is the fixed-capital cost in US$ (GJh−1)−1 of primary-energy input. Equation (9-245) shows that in this particular case the fixed-capital cost per unit of input energy (CFC/W) must not exceed $160,000 (GJh−1)−1, or $576 per kilowatt, to have a 1-year payback period if the heat pump is operational for 8000 h/year. For this case the corresponding value of ψ is about 0.12 for a heat pump with an operating life of 10 years purchased with money borrowed at a 10 percent rate of interest. Equation (9-245) also shows that the fixed-capital cost per unit of energy input (CFC/W) must not exceed $40,000 (GJh−1)−1, or $144 per kilowatt, to have a 1-year payback period if the heat pump is operational for only 2000 h/year. For this case the corresponding value of ψ is also about 0.12 for a heat pump with an operating life of 10 years purchased with money borrowed at a 10 percent rate of interest.

FIXED-CAPITAL-COST ESTIMATION Total Capital Cost The installed cost of the fixed-capital investment CFC is obviously an essential item which must be forecast before an investment decision can be made. It forms part of the total capital investment CTC, defined by Eq. (9-14). The fixed-capital investment is usually regarded as the capital needed to provide all the depreciable facilities. It is sometimes divided into two classes by defining battery limits and auxiliary facilities for the project. The boundary for battery limits includes all manufacturing equipment but excludes administrative offices, storage areas, utilities, and other essential and nonessential auxiliary facilities. Cost Indices The value of money will change because of inflation and deflation. Hence cost data can be accurate only at the time when they are obtained and soon go out of date. Data from cost records of equipment and projects purchased in the past may be converted to present-day values by means of a cost index. The present cost of the item is found by multiplying the historical cost by the ratio of the present cost index divided by the index applicable at the previous date. Ideally each cost item affected by inflation should be forecast separately. Labor costs, construction costs, raw-materials and energy prices, and product prices all change at different rates. Composite indices are derived by adding weighted fractions of the component indices. Most cost indices represent national averages, and local values may differ considerably. Table 9-42 presents information on some cost indices for the United States. Engineering News-Record updates its constructioncost index in March, June, September, and December. The Oil and Gas Journal gives the Nelson-Farrar refinery indices in the first issue of each quarter. The Chemical Engineering plant-cost index and Marshall and Swift equipment-cost index are given in each issue of the publication Chemical Engineering. Derivation of the base values is referred to in the respective publications.

Table 9-43 is based on the method suggested by J. Cran [Eng. Process Econ., 2, 89–90 (1977)]. He showed that reasonably accurate plant-cost indices for various countries could be derived by using two component indices in the equation (CI)P = 0.327(CI)ST + 0.673(CI)L

(9-246)

where (CI)ST is the steel-price index and (CI)L the earnings index for labor in the particular country. Most of the data required can be obtained from the United Nations Monthly Bulletin of Statistics or the Organization for Economic Cooperation and Development (OECD) annual review of the iron and steel industry. In Table 9-43, the plantcost indices have been brought to a common base of 1980 = 100. The values given do not relate costs in one country to those in another country, as this involves many complex and difficult problems. However, the table indicates the inflationary trends in plant costs since 1980 for each of the countries listed. Types and Accuracy of Estimates Capital-cost estimates may be required for a variety of reasons, among others to enable feasibility studies to be carried out, to enable a manufacturing company to select from alternative investments, to assist in selection from alternative designs, to provide information for planning the appropriation of capital, and to enable a contractor to bid on a new project. It is therefore essential to achieve the greatest accuracy of estimation with a minimum expenditure of time and money. Two simple rules are invaluable in aiding the production of consistently accurate estimates: 1. Check the completeness of the project scope. 2. Reduce the effect of bias by using statistically proven methods of estimation based on experience. Estimates which are lower than actual project costs are often the result of sizable omissions of equipment, services, or auxiliary facilities

9-64

PROCESS ECONOMICS

TABLE 9-42

Cost Indices for the United States Price indices*

Construction cost indices

Year

Consumer

Producer

Industrial chemicals

1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993

100 110 117 121 126 131 133 138 144 151 159 165 170 175

100 108 113 114 115 115 114 117 122 127 132 132 133 136

100 113 106 103 102 100 97 102 106 108 111 111 110 111

Chemical equipment†

Process plants‡

Petroleum refinery§

General construction¶

660 721 746 761 780 790 798 814 852 895 915 931 943 967

261 297 314 317 323 325 318 324 343 355 358 361 358 359

286 315 340 357 370 374 380 391 406 417 427 436 445 453

303 330 357 380 387 392 401 412 422 429 441 452 470 489

*Bureau of Labor Statistics, 1980 = 100. †Marshall & Swift, Chem. Eng., 1926 = 100. ‡Chem. Eng., plant index, 1957–1959 = 100. §Nelson-Farrar refinery index, Oil Gas J., 1967 = 100. ¶Eng. News Rec., 1967 = 100.

rather than of errors in pricing or estimation methods. To avoid this, the use of a checklist of items involved in a new project as given in Table 9-44 can be invaluable. The first stage toward producing an accurate estimate is to use a standard cost code for all construction projects. Table 9-45 shows a suitable numerical cost code, and Table 9-46 shows a typical alphabetical-numerical code. The cost-code system can be used throughout the estimating and construction stages for the collection of cost data by manual or computer methods. There are numerous types of fixedcapital-cost estimates, but in 1958 the American Association of Cost Engineers defined five types as follows: 1. Order-of-magnitude estimate (ratio estimate). Rule-of-thumb method based on cost data for previous similar types of plant; probable error within 10 to 50 percent. 2. Study estimate ( factored estimate). Better than order-ofmagnitude; requires knowledge of major items of equipment; used for feasibility surveys; probable error up to 30 percent. 3. Preliminary estimate (budget-authorization estimate). Requires more detailed information than study estimate; probable error up to 20 percent. 4. Definitive estimate (project-control estimate). Based on considerable data prior to preparation of completed drawings and specifications; probable error within 10 percent. 5. Detailed estimate ( firm or contractor’s estimate). Requires completed drawings, specifications, and site surveys; probable error within 5 percent. Greater accuracy of estimation may be achieved, within limits, by the expenditure of more time and money. The greater the accuracy required, the greater the time and effort needed to obtain the design and cost data prior to making the estimate. W. R. Park investigated the cost and accuracy of estimates for a project with a total cost of $1 million as shown in Fig. 9-41 (Cost EngiTABLE 9-43

neering Analysis, Wiley, New York, 1973, p. 133). Table 9-47 shows typical average costs for producing estimates [adapted from A. Pikulik and H. E. Diaz, Chem. Eng., 84, 106–122 (Oct. 10, 1977)]. Rapid Estimations Ratio Methods J. E. Haselbarth [Chem. Eng., 74, 214–215 (Dec. 4, 1967)] published data giving the total capital investment per unit of annual production capacity CTC/R. Table 9-48 lists data for many processes involving production units constructed on a previously developed site. Plants built on a green-field site would cost about 30 to 40 percent more, but enlargements of an existing plant would cost about 20 to 30 percent less than the values given in Table 9-48. Total fixed-capital investments for installations within the battery limits are given in Table 9-48. These refer to North American values corresponding to a Marshall and Swift index of 1000 and are adapted from the data of D. R. Woods (Financial Decision Making in the Process Industry, Prentice Hall, Englewood Cliffs, NJ, 1975, pp. 288–290). L. Lynn and R. F. Howland [Chem. Eng., 67, 131–136 (Feb. 8, 1960)] studied the capital ratios for 17 process industries and summarized data for more than 1000 processes. The capital ratio (CR) for a plant erected on a green-field site is defined as the ratio of the fixed-capital investment CFC to the annual sales revenue AS: (CR) = CFC/AS

(9-247)

However, Lynn and Howland included in the fixed-capital cost not only money invested in production and storage facilities but also that invested in land, research and development costs, and any auxiliary facilities necessary to support the process. Typical values of capital ratios for the year 1958 are listed in Table 9-49. Both of the preceding methods are relatively inaccurate and can be used only for rough screening. They have the advantage that an esti-

International Plant-Cost Indices*

Year

Australia

Belgium

Canada

Denmark

France

Germany

Italy

Japan

Netherlands

Spain

Sweden

United Kingdom

United States

1985 1986 1987 1988 1989 1990 1991 1992 1993

151 160 170 180 193 203 208 215 218

134 135 138 141 147 152 161 167 171

137 142 146 153 160 166 173 174 180

143 148 159 163 177 184 193 199 198

171 178 186 191 200 204 210 217 223

121 124 128 133 137 142 150 160 162

201 206 217 231 253 260 269 286 292

112 111 110 115 122 126 127 127 129

119 121 123 124 126 130 133 140 142

190 207 222 233 249 267 283 303 315

151 162 171 183 199 215 227 232 244

148 156 166 182 200 216 232 241 240

124 122 123 131 137 136 138 136 136

*From Process Eng., London, U.K. (monthly). Annual averages for each year (1980 = 100).

FIXED-CAPITAL-COST ESTIMATION TABLE 9-44

9-65

Checklist of Items for Fixed-Capital-Cost Estimates

Land: Surveys Fees Property cost Site development: Site clearing Grading Roads, access and on-site Walkways Railroads Fence Parking areas Other paved areas Wharves and piers Recreational facilities Landscaping Process buildings: (List as required) Include in each as required substructure, superstructures, platforms, supports, stairways, ladders, access ways, cranes, monorails, hoists, elevators Auxiliary buildings: Administration and office Medical or dispensary Cafeteria Garage Product warehouse(s) Parts or stores warehouse Maintenance shops—electric, piping, sheet metal, machine, welding, carpenters, instrument Guard and safety Hose houses Change houses Smoking stations (in hazardous plants) Personnel building Shipping office and platforms Research laboratory Control laboratories Building services: Plumbing Heating Ventilation Dust collection Air conditioning Sprinkler systems Elevators, escalators Building lighting Telephones Fire alarm Paging Intercommunication systems Painting Process equipment: (List carefully from checked flow sheets) Nonprocess equipment: Office furniture and equipment Cafeteria equipment Safety and medical equipment Shop equipment Automotive heavy maintenance and yard material-handling equipment Laboratory equipment Lockers and locker-room benches Garage equipment Shelves, bins, pallets, hand trucks Housekeeping equipment Fire extinguishers, hoses, fire engines Process appurtenances: Piping—carbon steel, alloy, cast iron, lead-lined, aluminum, copper, asbestos-cement, ceramic, plastic, rubber, reinforced concrete Pipe hangers, fittings, valves Insulation—piping, equipment Instruments Instrument panels Electrical—panels, switches, motors, conduit, wire, fittings, feeders, grounding, instrument and control wiring

Utilities: Boiler plant Incinerator Ash disposal Boiler feed-water treatment Electric generation Electrical substations Refrigeration plant Air plant Wells River intake Primary water treatment—filtration, coagulation, aeration Secondary water treatment—deionization, demineralization, pH and hardness control Cooling towers Water storage Effluent outfall Process-waste sewers Process-waste pumping stations Sanitary-waste sewers Sanitary-waste pumping stations Impounders, collection basins Waste treatment, including gases Storm sewers Yard distribution and facilities (outside battery limits): Process pipe lines—steam, condensate, water, gas, fuel oil, air, fire, instrument, and electric lines Raw-material and finished-product handling equipment—elevators, hoists, conveyors, airveyors, cranes Raw-material and finished-product storage—tanks, spheres, drums, bins, silos Fuel receiving, blending, and storage Product loading stations Track and truck scales Miscellaneous: Demolition and alteration work Catalysts Chemicals (initial charge only) Spare parts and noninstalled equipment spares Surplus equipment, supplies and equipment allowance Equipment rentals (for construction) Premium time (for construction) Inflation cost allowance Freight charges Taxes and insurance Duties Allowance for modifications and extra construction work during startup Engineering costs: Administrative Process, project, and general engineering Drafting Cost engineering Procurement, expediting, and inspection Travel and living expense Reproductions Communications Scale model Outside architect and engineering fees Construction expense: Construction, operation, and maintenance of temporary sheds, offices, roads, parking lots, railroads, electrical, piping, communication, and fencing Construction tools and equipment Warehouse personnel and expense Construction supervision Accounting and timekeeping Purchasing, expediting, and traffic Safety and medical Guards and watchmen Travel and transportation allowance for craft labor Fringe benefits Housekeeping Weather protection Permits, special licenses, field tests Rental of off-site space Contractor’s home office expense and fees Taxes and insurance, interest

9-66

PROCESS ECONOMICS

TABLE 9-45 Code Group Direct costs: 0–349 350–399 400–449 450–489 490–499 500–599 600–699

700–729 730–749 750 760–769 770–799 800–819 820–830 Indirect costs: 850–874 875–889 900–969 970–994 995 996

Standard Cost Code (Numerical) Summary Subdivision Process section (process equipment listed in flow-sheet sequence 1, 2, 3, 4, 5, . . . , N, preceded by process identification number) Site development Process buildings Auxiliary buildings Nonprocess equipment Process appurtenances (piping, insulation, instrumentation, electrical, etc.) Utilities and yard services (boiler plant, refrigeration, compressed air, water supply and treatment, effluents, fire protection, yard piping, yard electrical, yard materials handling, raw and finished-product storage) Substructures Superstructures Painting Building services Demolition and alteration to existing structure Surplus equipment, supplies and materials, royalty payments Design modifications, construction modifications, and extra work during startup Home office engineering Architect’s and engineer’s charges and fees Construction expenses Taxes and insurance Contractor’s home office expenses Contractor’s fees

mate can be made in a few minutes, and they do not require design work or process flow sheets. Step Count Methods These methods, used for order-ofmagnitude estimates, are based on the definition of the functional units required to carry out the process. A functional unit is a signifiTABLE 9-46 A B C D E F G H J K L M N O P Q R S T U V W X 100 200 300 320 340 360 380 400 500 600 700 800 900 950

Standard Cost Code (Alphabetical-Numerical)

Site work and foundations Buildings (less foundations) Steel structures and platforms (other than buildings) Heat exchangers Fractionating towers Tanks and drums Pumps and pump drivers Compressors and blowers Reactors and converters Grinding, crushing, and classifying equipment Materials-handling equipment Fired heaters Catalysts and chemicals Laboratory equipment Piping Instruments and controls Electrical Insulation and painting Utility equipment (boilers, generators, refrigeration) Plant and building accessories (railroads, fence, etc.) Laboratory equipment Safety equipment Warehouse spares Equipment and materials delivered and installed (A through X) Sales and use taxes Temporary facilities Construction equipment, tools, and supplies Construction supervision Field office expense Warehousing expense Payroll taxes and insurance Home office engineering costs Procurement costs Resident engineering Royalty payments Engineering administrative overhead and profit Constructor’s administrative overhead and profit

FIG. 9-41

Relationship between cost and accuracy of cost estimations.

cant process step, including all the equipment and ancillaries to operate. The sum of the costs of the functional units gives an estimate of the total capital cost of the plant. Usually a functional unit is a unit operation, a unit process, or a separation involving energy transfer, moving parts, and possibly a high level of internals. Pumping and heat exchange form part of a functional unit. “Inprocess” storage is ignored, but large storage requirements for raw materials, intermediates, or products are usually estimated separately. Sometimes there are difficulties in the identification and definition of the functional units involved in a process. For a particular process, the capital cost per functional unit is given by CF = f (q, T, p, Mc, CI) (9-248) where q is capacity, T is operating temperature, p is pressure, Mc is a materials of construction factor, and CI is a relevant cost index. Typical methods are those of F. C. Zevnik and R. L. Buchanan [Chem. Eng. Progr., 59, 70–77 (Feb. 1963)] and J. H. Taylor (Eng. & Proc. Econ., 2, 259–267, 1977). The former is mainly a graphical method of estimating the cost per functional unit (CF) based on the capacity, the maximum pressure, the maximum temperature, and the materials of construction. The Taylor method requires the determination of the costliness index, which is dependent on the complexity of the process. A simpler method was suggested by S. R. Timms (M.Phil. thesis, Aston University, England, 1980) to give the battery limits cost for gas phase processes only in U.S. dollars with a Marshall and Swift index of 1000. The simple equation is CBL = 11,800Nq0.615

(9-249a)

Taking into account materials of construction, temperature, and pressure effects, this becomes CBL = 5500Nq0.639Mc(Tmax)0.066(pmax)−0.016

(9-249b)

where N is the number of functional units; q is the capacity (tons/y); Mc is the materials of construction factor: 1.0 for carbon steel, 1.15 for TABLE 9-47 (1990)*

Typical Average Costs for Making Estimates Cost of project

Type of estimate Order of magnitude estimate Study estimate Preliminary estimate Definitive estimate Detailed estimate

Less than $2 million

$2 to $10 million

$10 to $100 million

$

$

$

3,000 20,000 50,000 80,000 200,000

6,000 40,000 80,000 160,000 520,000

13,000 80,000 130,000 320,000 1,000,000

*Adapted from A. Pikulik and H. E. Diaz, “Cost Estimating for Major Process Equipment.” Chem. Eng., 84, 106–122 (Oct. 10, 1977).

FIXED-CAPITAL-COST ESTIMATION TABLE 9-48

9-67

Capital-Cost Data for Processing Plants*

Product

Process route

Size, 1000 metric tons/year

Acetaldehyde Acetic acid Acetone Ammonia Ammonium nitrate Butanol Chlorine Ethylene Ethylene oxide Formaldehyde (37%) Glycol Hydrofluoric acid Methanol Nitric acid (conc.) Phosphoric acid Polyethylene (high density) Propylene Sulfuric acid Urea

Ethylene Methanol/CO Propylene Steam reforming Ammonia/nitric acid Propylene/CO/H2O Electrolysis of NaCl Refinery gases Ethylene/O2 Methanol Ethylene/Cl2 Hydrogen fluoride/H2O CO2/natural gas/steam Ammonia oxidation Calcium phosphate/H2SO4 Ethylene

50 10 100 100 100 50 50 50 50 10 5 10 60 100 5 5

Refinery gases Sulfur Ammonia/CO2

10 100 60

Alkylation (H2SO4) Coking (delayed) Coking (fluid) Cracking (fluid) Cracking Distillation (atm) Distillation (vac.) Hydrotreating Reforming Polymerization

Catalytic Thermal Thermal Catalytic Thermal 65% vaporized 65% vaporized Catalytic desulfurization Catalytic Catalytic

10 10 10 10 10 100 100 10 10 10

Total capital cost ($) ᎏᎏᎏ metric ton/yr of product

Size range 1000 metric tons/year

Exponent n‡

13.7 6.6 35.0 26.0 6.0 44.0 31.0 14.4 55.0 17.7 16.6 8.8 14.4 6.6 3.3 17.7

274 660 350 260 60 880 620 288 1100 1770 3320 880 240 66 660 3540

20–150 3–30 30–300 30–300 30–300 20–150 20–150 20–150 20–150 3–30 2–20 3–30 20–200 30–300 2–20 2–20

0.70 0.68 0.45 0.70 0.65 0.40 0.45 0.83 0.78 0.55 0.75 0.68 0.60 0.60 0.60 0.65

3.3 3.3 8.8

330 33 147

3–30 30–300 20–200

0.70 0.65 0.70

2100 2880 1770 1770 550 354 210 330 3200 550

3–30 3–30 3–30 3–30 3–30 30–300 30–300 3–30 3–30 3–30

0.60 0.38 0.42 0.70 0.70 0.90 0.70 0.65 0.60 0.58

Approximate cost† $ × 106

Chemical plants

Refinery units 21.0 28.8 17.7 17.7 5.5 35.4 21.0 3.3 32.0 5.5

*Adapted from M. S. Peters and K. D. Timmerhaus, “Plant Design and Economics for Chemical Engineers”, 4th ed., McGraw-Hill, New York, 1991. †All costs are approximate U.S.A. values with M & S = 1000, assuming 330 operating days per year. ‡Exponents apply roughly for threefold capacity ratio extending either way from the plant size given.

low-grade stainless steel, 1.2 for medium-grade stainless steel, 1.3 for high-grade stainless steel; Tmax is the maximum process temperature (K); and pmax is the maximum process pressure (bars). For liquid and/or solid handling, A. V. Bridgwater and F. D. G. Bossom (Proceedings of the 6th International Cost Engineering Conference, Mexico, October, 1980) suggested the following equations: CBL = 2160N(q/s)0.675

(9-250)

for plant capacities (q) above 60,000 tons/y, where CBL is the capital cost for battery limits in U.S. dollars (M & S = 1,000); N is the number of functional units; and s is the reactor conversion (weight of desired TABLE 9-49

Capital Ratios for Process Industries Industry

Capital ratio,* 1958

Chemicals, general Carbon black Explosives Glass Fibers, synthetic Foodstuffs, processed Inorganics, heavy Nonferrous metals Petroleum Pharmaceuticals Pigments, paints, and inks Pulp and paper Resins and plastics Rubber Soap and detergents Steel Sulfur Average

2.02 3.98 1.64 1.46 3.44 0.66 2.24 3.31 3.08 0.92 1.04 2.01 1.90 1.04 0.69 2.78 1.97 2.01

*Capital ratio = (fixed-capital investment)/(annual sales revenue).

reactor product/weight of reactor input). Thus, q/s represents the feed throughput in tons per year. For plant capacities below 60,000 tons per year, the equation becomes: CBL = 189,300N(q/s)0.30

(9-251)

In general, the step count method of estimation can be applied to any special situation to derive a model equation for that particular industry or group of processes. Exponential Methods Rapid capital-cost estimates can be made by using capacity-ratio exponents based on existing cost data of a company or drawn from published correlations. If the cost of a piece of equipment or plant of size or capacity q1 is C1, then the cost of a similar piece of equipment or plant of size or capacity q2 can be calculated from C2 = C1(q2 /q1)n

(9-252)

where the value of the exponent n depends on the type of equipment or plant. Cost indices should be used to bring the cost data to a common year. Table 9-48 gives typical values of n for various processes along with the cost of a plant of given capacity at a particular time and the capacity range of applicability. For process plants, capacity is expressed in terms of annual production capacity in metric tons per year. Exponential cost correlations have been developed for individual items of equipment. Care must be taken in determining whether the cost of the equipment has been expressed as free on board (FOB), delivered (DEL), or installed (INST), as this is not always clearly stated. In many cases the cost must be correlated in terms of parameters related to capacity such as surface area for heat exchangers or power for grinding equipment. There are four main sources of error in such cost correlations: 1. Oversimplification by correlating the cost of equipment in terms of a single variable

9-68

PROCESS ECONOMICS

2. Representation of data by using a simple exponential relationship 3. Failure to include the effects of technological improvements 4. Errors incurred because of special circumstances Table 9-50 gives typical values of the exponent n for many types of equipment. Prices are North American with a Marshall and Swift index of 1000, mainly for carbon steel equipment. Factor Estimations Most factor methods for estimating the total installed cost of a process plant are based on a combination of materials, labor, and overhead cost components. These can be conveniently grouped as 1. Cost of major items of equipment 2. Cost of complete installation of equipment 3. Auxiliary equipment to make the process work 4. Engineering and field expenses 5. Contractor’s fees and contingencies A great variety of factors are in use, depending on the time available and the accuracy expected. Normally the input information required is the base cost. Determination of this cost usually requires a knowledge of equipment sizes, probably using mass and energy balances for the proposed process. Equipment or Base Cost The total cost of the main-plant items is generally used as the base cost. Again, care must be taken with equipment costs which may be quoted as installed (INST), delivered to site (DEL), or free on board the delivery vehicle at the place of manufacture or other specified location (FOB). Base equipment includes all equipment within the battery limits whose cost is as significant as the cost of a pump. For example, storage tanks, knockout drums, accumulators, heat exchangers, and pumps are classed as main-plant items (MPI). Early in the development of the process-flow diagram, it is advisable to increase the estimated (MPI) cost by 10 to 20 percent to allow for later additions. When the scope of the process has been well defined, (MPI) costs should be increased by 1 to 10 percent. For order-of-magnitude estimates the cost of equipment delivered (CEQ)DEL varies approximately from 1.1 to 1.25 times the FOB cost (CEQ)FOB. The factor would be at the lower end of the range for domestic purchases and at the higher end for imports. Installation costs include unpacking, mounting, and connecting up to existing auxiliaries or utilities. The cost of equipment installed (CEQ)INST varies with type and size but generally ranges from 1.4 to 2.2 times the deliveredequipment cost (CEQ)DEL. Single-factor methods collect the various items of expenditure into one factor, which is usually used to multiply the total cost of delivered equipment  (CEQ)DEL to give the fixed-capital cost for plant within the battery limits: (CFC)BL = f

 (CEQ)DEL

(9-253)

Typical values for single factors f for battery-limit-plant costs (for a carbon steel plant including auxiliaries but not land) are as follows: Solids processing (S) 3.8 Solids-fluid processing (S-F) 4.1 Fluid processing (F) 4.8 Thus the factors vary with the type of processing, although the boundaries between the classifications are not clear-cut and considerable judgment is required in selection of the correct factor. Multiple-factor methods include the cost contributions for each given activity, which can be added together to give an overall factor. This factor can be used to multiply the total cost of delivered equipment  (CEQ)DEL to produce an estimate of the total fixed-capital investment either for grass-roots or for battery-limit plants. The costs may be divided into four groups: 1. Cost of plant within battery limits 2. Cost of auxiliaries 3. Cost of engineering and field expenses 4. Cost of contractor’s fees plus contingency allowance Table 9-51 gives typical values of such factors for carbon steel installations taken from the data of D. R. Woods (Financial Decision Making in the Process Industry, Prentice Hall, Englewood Cliffs, NJ, 1975, p. 184). Auxiliaries and site preparation are given as factors of the delivered-equipment cost in Table 9-51, whereas C. A. Miller [Chem.

Eng., 72, 226–236 (Sept. 13. 1965)] expresses auxiliary costs as factors of the battery-limit (BL) cost. Table 9-52 gives the factors from the breakdown of Miller, which is more detailed than that of Woods. Example 23: Estimation of Total Installed Cost of a Plant Let us estimate the total installed cost for a grass-roots plant producing an organic chemical (S-F process) on a continuous basis. We assume that the total cost of delivered equipment  (CEQ)DEL is $1 million and use suitable factors from Table 9-51. The estimated values for the various contributions are given in Table 9-53, resulting in an estimate of $4,280,000 for the total fixed-capital investment, including a contingency factor. A multiple-factor method for predesign cost estimating has been put forward by D. H. Allen and R. C. Page [Chem. Eng., 82, 142–150 (Mar. 3, 1975)] for fluid-type plants (F) that include some vapor processing. The method requires the following input information: 1. Plant flow sheet giving main-plant items and process streams 2. Total process-stream input per year 3. Extreme temperature and pressure conditions, if any 4. Materials of construction for main-plant items 5. Operating phases for each main-plant item 6. Expectation of any unusually high or low, direct or indirect initial costs By means of 12 procedural steps involving the input information, several equations, graphs, and tables, the total cost of delivered equipment  (CEQ)DEL is estimated. This is then converted into a grass-roots investment estimate by dividing  (CEQ)DEL by a single factor ranging from 15 to 30 percent (average value, 21 percent). The method is rapid and is claimed to be accurate within −20 to +25 percent, but it has only been tested by using data published in the literature for eight plants. Multiple-Factor Methods That Separate Materials and Labor These methods have become increasingly popular. While they are similar to the preceding methods, labor and materials costs are considered separately. Hence it is possible to allow for variations in efficiency and labor costs in different localities or countries. H. C. Bauman (Fundamentals of Cost Engineering in the Chemical Industry, Van Nostrand Reinhold, New York, 1964, p. 295) divides most of the components of Table 9-51 into material and labor components, quoting the data as ranges and medians of the percentage of the total fixed-capital investment. In Table 9-54, Bauman’s data have been converted to factors of the delivered-equipment cost for a grass-roots installation. A study has been made by A. V. Bridgwater [Chem. Eng., 86, 119–121 (Nov. 5, 1979)] of the geographical variations in capital costs. He concluded that because of trade and competition basic equipment costs do not vary significantly in the industrialized countries of the western world. The main differences in construction costs at various international locations are due to variations in labor costs and productivity, the use of specialized equipment, and sundry local factors. Table 9-55 gives location factors for the construction of chemical plants of similar function in various countries (1993 values). The factors have been corrected by Bridgwater for location variations in labor costs and efficiency and converted at the average value of the exchange rate. Factor Methods Using the Modular Approach These are methods used for estimating the cost of major-equipment units and have been proposed by several authors. Perhaps the most comprehensive is the method suggested by K. M. Guthrie [Chem. Eng., 76, 114–142 (Mar. 24, 1969)]. Table 9-56 gives average factors for majorequipment items based on a (CEQ)FOB cost for carbon steel units. To the FOB cost of the item is added, by means of factors, the total materials cost to complete the module M. Erection and setting costs L are added as a factor or calculated from the L/M cost ratio to give M + L = X, the direct module cost. Indirect costs, such as freight, taxes, insurance, engineering, and field expense, are added to (M + L) to give the total module cost. This excludes contingency allowances, contractor’s fees, auxiliaries, site development, land, and industrial buildings, which may have to be added when applicable. The factors in Table 9-56 were based on mid-1968 prices for a United States Gulf Coast location.

FIXED-CAPITAL-COST ESTIMATION TABLE 9-50

Typical Exponents for Equipment Cost versus Capacity Equipment

Agitator, turbine, top entry, open, FOB Agitator, turbine, top entry, closed, FOB Blower, centrifugal, 4 lbf/in2 (27.6 kN/m2), DEL, excluding motor Cone crusher, FOB, crusher only Jaw crusher, FOB, excluding motor Jaw crusher, FOB, excluding motor Centrifugal pump, C/S, FOB, excluding motor Centrifugal pump, C/S, FOB, excluding motor Conveyor, belt, C/S, FOB, excluding motor Conveyor, screw, C/S, DEL, excluding motor Centrifuge, automatic batch, horizontal, C/S, FOB Compressor, reciprocating, <1000 lbf/in2, FOB, including motor Crystallizer, forced circulation, C/S, FOB Dryer, drum, C/S, FOB, excluding motor Dryer, vacuum, shelf, C/S, FOB, excluding trays, vacuum equipment Dust collector, cloth, shaker type, FOB, including motors Dust collector, multicyclones, FOB Electrostatic precipitator, FOB Ejector, single-stage, 100 psig, steam, FOB Ejector, two-stage, FOB, including condenser, piping Ejector, multistage, FOB, including condenser, piping Filter, vertical-pressure leaf, C/S, DEL Filter, plate and frame, C/S, DEL Filter, vacuum rotary drum, C/S, FOB, including motor Heat exchanger, shell-tube, floating head, C/S, DEL; fixed tube × 0.85; U tube × 0.87; kettle × 1.35 Heat exchanger, thermal screw, C/S, FOB, excluding motor Kettle, jacketed, glass-lined, FOB Motors, ac induction, wound rotor, T