Fundamentals of Turbulent and Multi-Phase Combustion

Author: Kenneth Kuan-yun Kuo | Ragini Acharya

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FUNDAMENTALS OF TURBULENT AND MULTIPHASE COMBUSTION

Fundamentals of Turbulent and Multiphase Combustion Copyright © 2012 John Wiley & Sons, Inc.

Kenneth K. Kuo and Ragini Acharya

FUNDAMENTALS OF TURBULENT AND MULTIPHASE COMBUSTION

KENNETH K. KUO RAGINI ACHARYA

JOHN WILEY & SONS, INC.

This book is printed on acid-free paper. Copyright © 2012 by John Wiley & Sons, Inc. All rights reserved Published by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 646-8600, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at www.wiley.com/go/permissions. Limit of Liability/Disclaimer of Warranty: While the publisher and the author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and speciﬁcally disclaim any implied warranties of merchantability or ﬁtness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor the author shall be liable for damages arising herefrom. For general information about our other products and services, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002. Wiley publishes in a variety of print and electronic formats and by print-on-demand. Some material included with standard print versions of this book may not be included in e-books or in print-on-demand. If this book refers to media such as a CD or DVD that is not included in the version you purchased, you may download this material at http://booksupport.wiley.com. For more information about Wiley products, visit www.wiley.com. Library of Congress Cataloging-in-Publication Data: Kuo, Kenneth K. KuFundamentals of turbulent and multiphase combustion / Kenneth K. Kuo, Ragini Acharya. — 1st ed. Ragini p. cm. RaIncludes bibliographical references and index. RagiISBN 978-0-470-22622-3 (hardback); 978-111-8-09929-2 (ebk.); 978-111-8-09931-5 (ebk.); 978-111-8-09932-2 (ebk.); 978-111-8-10767-6 (ebk.); 978-111-8-10768-3 (ebk.); 978-111-8-10770-6 (ebk.) R1. Combustion engineering. 2. Turbulence. 3. Combustion—Mathematical models. I. Acharya, Ragini. II. Acharya, Ragini. III. Title. RaQD516.K858 2012 Ra541 .361—dc23 2011024787 ISBN: 978-0-470-22622-3 Printed in the United States of America 10 9 8 7 6 5 4 3 2 1

Ken Kuo would like to dedicate this book to his wife, Olivia (Jeon-lin), and their daughters, Phyllis and Angela, for their love, understanding, patience, and support, and to his mother, Mrs. Wen-Chen Kuo, for her love and encouragement. Ragini Acharya would like to dedicate this book to her parents, Meenakshi and Krishnama Acharya, for their love, patience, and support and for having endless faith in her.

CONTENTS

Preface

xix

1 Introduction and Conservation Equations

1

1.1 Why Is Turbulent and Multiphase Combustion Important?, 3 1.2 Different Applications for Turbulent and Multiphase Combustion, 3 1.2.1 Applications in High Rates of Combustion of Materials for Propulsion Systems, 5 1.2.2 Applications in Power Generation, 7 1.2.3 Applications in Process Industry, 7 1.2.4 Applications in Household and Industrial Heating, 7 1.2.5 Applications in Safety Protections for Unwanted Combustion, 7 1.2.6 Applications in Ignition of Various Combustible Materials, 8 1.2.7 Applications in Emission Control of Combustion Products, 8 1.2.8 Applications in Active Control of Combustion Processes, 8 1.3 Objectives of Combustion Modeling, 8 1.4 Combustion-Related Constituent Disciplines, 9 1.5 General Approach for Solving Combustion Problems, 9 vii

viii

CONTENTS

1.6 Governing Equations for Combustion Models, 11 1.6.1 Conservation Equations, 11 1.6.2 Transport Equations, 11 1.6.3 Common Assumptions Made in Combustion Models, 11 1.6.4 Equation of State, 12 1.6.4.1 High-Pressure Correction, 13 1.7 Deﬁnitions of Concentrations, 14 1.8 Deﬁnitions of Energy and Enthalpy Forms, 16 1.9 Velocities of Chemical Species, 19 1.9.1 Deﬁnitions of Absolute and Relative Mass and Molar Fluxes, 20 1.10 Dimensionless Numbers, 23 1.11 Derivation of Species Mass Conservation Equation and Continuity Equation for Multicomponent Mixtures, 23 1.12 Momentum Conservation Equation for Mixture, 29 1.13 Energy Conservation Equation for Multicomponent Mixture, 33 1.14 Total Unknowns versus Governing Equations, 40 Homework Problems, 41 2 Laminar Premixed Flames

43

2.1 Basic Structure of One-Dimensional Premixed Laminar Flames, 46 2.2 Conservation Equations for One-Dimensional Premixed Laminar Flames, 47 2.2.1 Various Models for Diffusion Velocities, 49 2.2.1.1 Multicomponent Diffusion Velocities (First-Order Approximation), 49 2.2.1.2 Various Models for Describing Source Terms due to Chemical Reactions, 54 2.2.2 Sensitivity Analysis, 66 2.3 Analytical Relationships for Premixed Laminar Flames with a Global Reaction, 68 2.3.1 Three Analysis Procedures for Premixed Laminar Flames, 77 2.3.2 Generalized Expression for Laminar Flame Speeds, 80 2.3.2.1 Reduced Reaction Mechanism for HC-Air Flame, 81

CONTENTS

ix

2.3.3 Dependency of Laminar Flame Speed on Temperature and Pressure, 82 2.3.4 Premixed Laminar Flame Thickness, 84 2.4 Effect of Flame Stretch on Laminar Flame Speed, 86 2.4.1 Deﬁnitions of Stretch Factor and Karlovitz Number, 86 2.4.2 Governing Equation for Premixed Laminar Flame Surface Area, 94 2.4.3 Determination of Unstretched Premixed Laminar Flame Speeds and Markstein Lengths, 95 2.5 Modeling of Soot Formation in Laminar Premixed Flames, 103 2.5.1 Reaction Mechanisms for Soot Formation and Oxidation, 104 2.5.1.1 Empirical Models for Soot Formation, 106 2.5.1.2 Detailed Models for Soot Formation and Oxidation, 108 2.5.1.3 Formation of Aromatics, 109 2.5.1.4 Growth of Aromatics, 110 2.5.1.5 Migration Reactions, 112 2.5.1.6 Oxidation of Aromatics, 113 2.5.2 Mathematical Formulation of Soot Formation Model, 114 Homework Problems, 124 3 Laminar Non-Premixed Flames

125

3.1 Basic Structure of Non-Premixed Laminar Flames, 128 3.2 Flame Sheet Model, 129 3.3 Mixture Fraction Deﬁnition and Examples, 130 3.3.1 Balance Equations for Element Mass Fractions, 134 3.3.2 Temperature-Mixture Fraction Relationship, 138 3.4 Flamelet Structure of a Diffusion Flame, 142 3.4.1 Physical Signiﬁcance of the Instantaneous Scalar Dissipation Rate, 145 3.4.2 Steady-State Combustion and Critical Scalar Dissipation Rate, 147 3.5 Time and Length Scales in Diffusion Flames, 151 3.6 Examples of Laminar Diffusion Flames, 153 3.6.1 Unsteady Mixing Layer, 153

x

CONTENTS

3.6.2 Counterﬂow Diffusion Flames, 155 3.6.3 Coﬂow Diffusion Flame or Jet Flames, 165 3.7 Soot Formation in Laminar Diffusion Flames, 172 3.7.1 Soot Formation Model, 173 3.7.1.1 Particle Inception, 174 3.7.1.2 Surface Growth and Oxidation, 174 3.7.2 Appearance of Soot, 175 3.7.3 Experimental Studies by Using Coﬂow Burners, 176 3.7.3.1 Sooting Zone, 178 3.7.3.2 Effect of Fuel Structure, 182 3.7.3.3 Inﬂuence of Additives, 183 3.7.3.4 Coﬂow Ethylene/Air Laminar Diffusion Flames, 186 3.7.3.5 Modeling of Soot Formation, 191 Homework Problems, 204 4 Background in Turbulent Flows

206

4.1 Characteristics of Turbulent Flows, 210 4.1.1 Some Pictures, 212 4.2 Statistical Understanding of Turbulence, 213 4.2.1 Ensemble Averaging, 214 4.2.2 Time Averaging, 215 4.2.3 Spatial Averaging, 215 4.2.4 Statistical Moments, 215 4.2.5 Homogeneous Turbulence, 216 4.2.6 Isotropic Turbulence, 217 4.3 Conventional Averaging Methods, 217 4.3.1 Reynolds Averaging, 218 4.3.1.1 Correlation Functions, 222 4.3.2 Favre Averaging, 225 4.3.3 Relation between Time Averaged-Quantities and Mass-Weighted Averaged Quantities, 227 4.3.4 Mass-Weighted Conservation and Transport Equations, 228 4.3.4.1 Continuity and Momentum Equations, 228 4.3.4.2 Energy Equation, 230 4.3.4.3 Mean Kinetic Energy Equation, 231

CONTENTS

xi

4.3.4.4 4.3.4.5 4.3.4.6 4.3.4.7

4.4 4.5

4.6 4.7

4.8

Reynolds-Stress Transport Equations, 232 Turbulence-Kinetic-Energy Equation, 234 Turbulent Dissipation Rate Equation, 236 Species Mass Conservation Equation, 242 4.3.5 Vorticity Equation, 243 4.3.6 Relationship between Enstrophy and the Turbulent Dissipation Rate, 246 Turbulence Models, 247 Probability Density Function, 249 4.5.1 Distribution Function, 250 4.5.2 Joint Probability Density Function, 252 4.5.3 Bayes’ Theorem, 254 Turbulent Scales, 256 4.6.1 Comment on Kolmogorov Hypotheses, 260 Large Eddy Simulation, 266 4.7.1 Filtering, 268 4.7.2 Filtered Momentum Equations and Subgrid Scale Stresses, 270 4.7.3 Modeling of Subgrid-Scale Stress Tensors, 274 Direct Numerical Simulation, 279 Homework Problems, 280

5 Turbulent Premixed Flames

283

5.1 Physical Interpretation, 289 5.2 Some Early Studies in Correlation Development, 291 5.2.1 Damk¨ohler’s Analysis (1940), 292 5.2.2 Schelkin’s Analysis (1943), 295 5.2.3 Karlovitz, Denniston, and Wells’s Analysis (1951), 296 5.2.4 Summerﬁeld’s Analysis (1955), 297 5.2.5 Kovasznay’s Characteristic Time Approach (1956), 298 5.2.6 Limitations of the Preceding Approaches, 299 5.3 Characteristic Scale of Wrinkles in Turbulent Premixed Flames, 304 5.3.1 Schlieren Photographs, 305 5.3.2 Observations on the Structure of Wrinkled Laminar Flames, 305

xii

CONTENTS

5.4

5.5 5.6

5.7 5.8 5.9

5.10

5.3.3 Measurements of Scales of Unburned and Burned Gas Lumps, 307 5.3.4 Length Scale of Wrinkles, 310 Development of Borghi Diagram for Premixed Turbulent Flames, 310 5.4.1 Physical Interpretation of Various Regimes in Borghi’s Diagram, 311 5.4.1.1 Wrinkled Flame Regime, 311 5.4.1.2 Wrinkled Flame with Pockets Regime (also Called Corrugated Flame Regime), 311 5.4.1.3 Thickened Wrinkled Flames, 313 5.4.1.4 Thickened Flames with Possible Extinctions/Thick Flames, 314 5.4.2 Klimov-Williams Criterion, 314 5.4.3 Construction of Borghi Diagram, 316 5.4.3.1 Thick Flames (or Distributed Reaction Zone or Well-Stirred Reaction Zone), 318 5.4.4 Wrinkled Flames, 318 5.4.4.1 Wrinkled Flamelets (Weak Turbulence), 320 5.4.4.2 Corrugated Flamelets (Strong Turbulence), 322 Measurements in Premixed Turbulent Flames, 324 Eddy-Break-up Model, 324 5.6.1 Spalding’s EBU Model, 335 5.6.2 Magnussen and Hjertager’s EBU Model, 336 Intermittency, 337 Flame-Turbulence Interaction, 339 5.8.1 Effects of Flame on Turbulence, 341 Bray-Moss-Libby Model, 342 5.9.1 Governing Equations, 349 5.9.2 Gradient Transport, 353 5.9.3 Countergradient Transport, 354 5.9.4 Closure of Transport Terms, 357 5.9.4.1 Gradient Closure, 357 5.9.4.2 BML Closure, 358 5.9.5 Effect of Pressure Fluctuations Gradients, 361 5.9.6 Summary of DNS Results, 364 Turbulent Combustion Modeling Approaches, 368

CONTENTS

xiii

5.11 Geometrical Description of Turbulent Premixed Flames and G-Equation, 368 5.11.1 Level Set Approach for the Corrugated Flamelets Regime, 371 5.11.2 Level Set Approach for the Thin Reaction Zone Regime, 374 5.12 Scales in Turbulent Combustion, 376 5.13 Closure of Chemical Reaction Source Term, 380 5.14 Probability Density Function Approach to Turbulent Combustion, 381 5.14.1 Derivation of the Transport Equation for Probability Density Function, 386 5.14.2 Moment Equations and PDF Equations, 391 5.14.3 Lagrangian Equations for Fluid Particles, 392 5.14.4 Gradient Transport Model in Composition PDF Method, 395 5.14.5 Determination of Overall Reaction Rate, 397 5.14.6 Lagrangian Monte Carlo Particle Methods, 398 5.14.7 Filtered Density Function Approach, 398 5.14.8 Prospect of PDF Methods, 399 Homework Problems, 400 Project No. 1, 400 Project No. 2, 401 6 Non-premixed Turbulent Flames 6.1 6.2 6.3 6.4

402

Major Issues in Non-premixed Turbulent Flames, 404 Turbulent Damk¨ohler number, 406 Turbulent Reynolds Number, 407 Scales in Non-premixed Turbulent Flames, 407 6.4.1 Direct Numerical Simulation and Scales, 411 6.5 Turbulent Non-premixed Combustion Regime Diagram, 414 6.6 Turbulent Non-premixed Target Flames, 418 6.6.1 Simple Jet Flames, 419 6.6.1.1 CH4 /H2 /N2 Jet Flame, 420 6.6.1.2 Effect of Jet Velocity, 430 6.6.2 Piloted Jet Flames, 432 6.6.2.1 Comparison of Simple Jet Flame and Sandia Flames D and F, 448

xiv

CONTENTS

6.7

6.8

6.9

6.10

6.11

6.6.3 Bluff Body Flames, 452 6.6.4 Swirl Stabilized Flames, 455 Turbulence-Chemistry Interaction, 456 6.7.1 Inﬁnite Chemistry Assumption, 456 6.7.1.1 Unity Lewis Number, 457 6.7.1.2 Nonunity Lewis Number, 458 6.7.2 Finite-Rate Chemistry, 458 Probability Density Approach for Turbulent Non-premixed Combustion, 462 6.8.1 Physical Models, 465 6.8.2 Turbulent Transport in Velocity-Composition Pdf Methods, 466 6.8.2.1 Stochastic Mixing Model, 467 6.8.2.2 Stochastic Reorientation Model, 468 6.8.3 Molecular Transport and Scalar Mixing Models, 469 6.8.3.1 Interaction by Exchange with the Mean Model, 471 6.8.3.2 Modiﬁed Curl Mixing Model, 471 6.8.3.3 Euclidean Minimum Spanning Tree Model, 472 Flamelet Models, 476 6.9.1 Laminar Flamelet Assumption, 477 6.9.2 Unsteady Flamelet Modeling, 478 6.9.3 Flamelet Models and PDF, 479 Interactions of Flame and Vortices, 480 6.10.1 Flame Rolled Up in a Single Vortex, 482 6.10.2 Flame in a Shear Layer, 483 6.10.3 Jet Flames, 483 6.10.4 K´arm´an Vortex Street/V-Shaped Flame Interaction, 484 6.10.5 Burning Vortex Ring, 484 6.10.6 Head-on Flame/Vortex Interaction, 485 6.10.7 Experimental Setups for Flame/Vortex Interaction Studies, 486 6.10.7.1 Reaction Front/Vortex Interaction in Liquids, 486 6.10.7.2 Jet Flames, 487 6.10.7.3 Counterﬂow Diffusion Flames, 488 Generation and Dissipation of Vorticity Effects, 492

CONTENTS

xv

6.12 Non-premixed Flame–Vortex Interaction Combustion Diagram, 493 6.13 Flame Instability in Non-premixed Turbulent Flames, 496 6.14 Partially Premixed Flames or Edge Flames, 500 6.14.1 Formation of Edge Flames, 501 6.14.2 Triple Flame Stabilization of Lifted Diffusion Flame, 502 6.14.3 Analysis of Edge Flames, 503 Homework Problems, 506 Project No. 6.1, 506 Project No. 6.2, 507 Project No. 6.3, 507

7 Background in Multiphase ﬂows with Reactions

509

7.1 Classiﬁcation of Multiphase Flow Systems, 512 7.2 Practical Problems Involving Multiphase Systems, 514 7.3 Homogeneous versus Multi-component/Multiphase Mixtures, 515 7.4 CFD and Multiphase Simulation, 516 7.5 Averaging Methods, 520 7.5.1 Eulerian Average—Eulerian Mean Values, 522 7.5.2 Lagrangian Average—Lagrangian Mean Values, 523 7.5.3 Boltzmann Statistical Average, 524 7.5.4 Anderson and Jackson’s Averaging for Dense Fluidized Beds, 525 7.6 Local Instant Formulation, 533 7.7 Eulerian-Eulerian Modeling, 536 7.7.1 Fluid-Fluid Modeling, 536 7.7.1.1 Closure Models, 538 7.7.2 Fluid-Solid Modeling, 540 7.7.2.1 Closure Models, 541 7.7.2.2 Dense Particle Flows, 547 7.7.2.3 Dilute Particle Flows, 549 7.8 Eulerian-Lagrangian Modeling, 550 7.8.1 Fluid-Solid Modeling, 551 7.8.1.1 Fluid Phase, 551 7.8.1.2 Solid Phase, 552 7.9 Interfacial Transport (Jump Conditions), 555

xvi

CONTENTS

7.10 Interface-Tracking/Capturing, 561 7.10.1 Interface Tracking, 563 7.10.1.1 Markers on Interface (Surface Marker Techniques), 564 7.10.1.2 Surface-Fitted Method, 567 7.10.2 Interface Capturing, 568 7.10.2.1 Markers in Fluid (MAC Formulation), 568 7.10.2.2 Volume of Fluid Method, 569 7.11 Discrete Particle Methods, 573 Homework Problems, 575 8 Spray Atomization and Combustion

576

8.1 Introduction to Spray Combustion, 578 8.2 Spray-Combustion Systems, 580 8.3 Fuel Atomization, 582 8.3.1 Injector Types, 582 8.3.2 Atomization Characteristics, 584 8.4 Spray Statistics, 584 8.4.1 Particle Characterization, 584 8.4.2 Distribution Function, 585 8.4.2.1 Logarithmic Probability Distribution Function, 588 8.4.2.2 Rosin-Rammler Distribution Function, 588 8.4.2.3 Nukiyama-Tanasawa Distribution Function, 589 8.4.2.4 Upper-Limit Distribution Function of Mugele and Evans, 589 8.4.3 Transport Equation of the Distribution Function, 590 8.4.4 Simpliﬁed Spray Combustion Model for Liquid-Fuel Rocket Engines, 591 8.5 Spray Combustion Characteristics, 594 8.6 Classiﬁcation of Models Developed for Spray Combustion Processes, 602 8.6.1 Simple Correlations, 602 8.6.2 Droplet Ballistic Models, 603 8.6.3 One-Dimensional Models, 603 8.6.4 Stirred-Reactor Models, 604

CONTENTS

8.7

8.8

8.9

8.10

xvii

8.6.5 Locally Homogeneous-Flow Models, 605 8.6.6 Two-Phase-Flow (Dispersed-Flow) Models, 605 Locally Homogeneous Flow Models, 605 8.7.1 Classiﬁcation of LHF Models, 606 8.7.2 Mathematical Formulation of LHF Models, 609 8.7.2.1 Basic Assumptions, 609 8.7.2.2 Equation of State, 609 8.7.2.3 Conservation Equations, 615 8.7.2.4 Turbulent Transport Equations, 619 8.7.2.5 Boundary Conditions, 620 8.7.2.6 Solution Procedures, 620 8.7.2.7 Comparison of LHF-Model Predictions with Experimental Data, 626 Two-Phase-Flow (Dispersed-Flow) Models, 634 8.8.1 Particle-Source-in-Cell Model (Discrete-Droplet Model), 637 8.8.1.1 Models for Single Drop Behavior, 639 8.8.2 Drop Breakup Process and Mechanism, 654 8.8.2.1 Drop Breakup Process, 654 8.8.2.2 Multi-component Droplet Breakup by Microexplosion, 659 8.8.3 Deterministic Discrete Droplet Models, 662 8.8.3.1 Gas-Phase Treatment in DDDMs, 664 8.8.3.2 Liquid-Phase Treatment in DDDMs, 666 8.8.3.3 Results of DDDMs, 667 8.8.4 Stochastic Discrete Droplet Models, 669 8.8.5 Comparison of Results between DDDMs and SDDMs, 671 8.8.6 Dense Sprays, 682 8.8.6.1 Introduction, 682 8.8.6.2 Background, 684 8.8.6.3 Jet Breakup Models, 690 8.8.6.4 Impinging Jet Atomization, 699 Group-Combustion Models of Chiu, 700 8.9.1 Group-Combustion Numbers, 701 8.9.2 Modes of Group Burning in Spray Flames, 703 Droplet Collison, 706 8.10.1 Droplet-Droplet Collisions, 707 8.10.2 Droplet-Wall Collision, 708 8.10.3 Interacting Droplet in a Many-Droplet System, 710

xviii

CONTENTS

8.11 Optical Techniques for Particle Size Measurements, 710 8.11.1 Types of Optical Particle Sizing Methods, 711 8.11.2 Single Particle Counting Methods, 711 8.11.2.1 Scattering Ratio Technique, 712 8.11.2.2 Intensity Deconvolution Method, 713 8.11.2.3 Interferometric Method (Phase-Shift Method), 713 8.11.2.4 Visibility Method Using a Laser Doppler Velocimeter LDV, 713 8.11.2.5 Phase Doppler Sizing Anemometer, 713 8.11.3 Ensemble Particle Sizing Techniques, 714 8.11.3.1 Extinction Measurement Techniques, 714 8.11.3.2 Multiple Angle Scattering Technique, 714 8.11.3.3 Fraunhofer Diffraction Particle Analyzer, 715 8.11.3.4 Integral Transform Solutions for Near-Forward Scattering, 716 8.12 Effect of Droplet Spacing on Spray Combustion, 717 8.12.1 Evaporation and Combustion of Droplet Arrays, 717 Homework Problems, 720 Appendix A: Useful Vector and Tensor Operations

723

Appendix B: Constants and Conversion Factors Often Used in Combustion 751 Appendix C: Naming of Hydrocarbons 755 Appendix D: Detailed Gas-Phase Reaction Mechanism for Aromatics Formation 759 Appendix E: Particle Size–U.S. Sieve Size and Tyler Screen Mesh Equivalents 795 Bibliography Index 869

799

PREFACE

There is an ever-increasing need to understand turbulent and multiphase combustion due to their broad application in energy, environment, propulsion, transportation, industrial safety, and nanotechnology. More engineers and scientists with skills in these areas are needed to solve many multifaceted problems. Turbulence itself is one of the most complex problems the scientiﬁc community faces. Its complexity increases with chemical reactions and even more in the presence of multiphase ﬂows. A number of useful books have been published recently in the areas of theory of turbulence, multiphase ﬂuid dynamics, turbulent combustion, and combustion of propellants. These include Theoretical and Numerical Combustion by Poinsot and Veynante; Turbulent Flows by Pope; Introduction to Turbulent Flow by Mathieu and Scott; Turbulent Combustion by Peters; Multiphase Flow Dynamics by Kolev; Combustion Physics by Law; Fluid Dynamics and Transport of Droplet and Sprays by Sirignano; Compressible, Turbulence, and High-Speed Flow by Gatski and Bonnet; Combustion by Glassman and Yetter, among others. Kenneth Kuo, the ﬁrst author of this book, previously published Principles of Combustion. The second edition, published in 2005, contains comprehensive material on laminar ﬂames, chemical thermodynamics, reaction kinetics, and transport properties for multicomponent mixtures. As the research in laminar ﬂames was overwhelming, he decided to develop two separate books dedicated entirely to turbulent and multiphase combustion. Turbulence, turbulent combustion, and multiphase reacting ﬂows have been major research topics for many decades, and research in these areas is expected to continue at even a greater pace. Usually the research has focused on experimental studies with phenomenological approaches, resulting in the development of empirical correlations. Theoretical approaches have achieved some degree of success. However, in the past 20 years, advances in computational capability xix

xx

PREFACE

have enabled signiﬁcant progress to be made toward comprehensive theoretical modeling and numerical simulation. Experimental diagnostics, especially nonintrusive laser-based measurement techniques, have been developed and used to obtain accurate data, which have been used for model validation. There is a greater synergy between the experimental and theoretical/numerical approaches. Due to these ongoing developments and advancements, theoretical modeling and numerical simulation hold great potential for future solutions of problems. In these two new books, we have attempted to integrate the fundamental theories of turbulence, combustion, and multiphase phenomena as well as experimental techniques, so that readers can acquire a ﬁrm background in both contemporary and classical approaches. The ﬁrst book volume is called Fundamentals of Turbulent and Multiphase Combustion; the second is called Applications of Turbulent and Multiphase Combustion. The ﬁrst volume can serve as a graduate-level textbook that covers the area of turbulent combustion and multiphase reacting ﬂows as well as material that builds on these fundamentals. This volume also can be useful for research purpose. It is oriented toward the theories of combustion, turbulence, multiphase ﬂows, and turbulent jets. Whenever appropriate, experimental setups and results are provided. The ﬁrst volume addresses eight basic topical areas in combustion and multiphase ﬂows, including laminar premixed and nonpremixed ﬂames; theory of turbulence; turbulent premixed and nonpremixed ﬂames; background of multiphase ﬂows; and spray atomization and combustion. A deep understanding of these topics is necessary for researchers in the ﬁeld of combustion. The six chapters in the second volume build on the ground covered in the ﬁrst volume. Its chapters include: solid propellant combustion, thermal decomposition and combustion of nitramines burning behavior of homogeneous solid propellants, chemically reacting boundary-layer ﬂows, ignition and combustion of combustion of single energetic solid particles, and combustion of solid particles in multiphase ﬂows. The major reason for including solid-propellant combustion here is to provide concepts for condensed-phase combustion modeling as an example. Nitramines are explosive or propellant ingredients; their decomposition and reaction mechanisms are also good examples for combustion behavior of condensed-phase materials. Chapters in Volume 2 focus on the application aspect of fundamental concepts and can form the framework for an advanced graduate-level course in combustion of condensed-phase materials. However, the selection of materials for instruction depends extirely on the interests of instructors and students. Although several chapters address solid propellant combustion, this volume is not a textbook for solid propellant combustion; many topics in this area are not included due to space limitations. VOLUME 1, FUNDAMENTALS OF TURBULENT AND MULTIPHASE COMBUSTION

Chapter 1 introduces and stresses the importance of combustion and multiphase ﬂows in research. It also provides a succinct review of major conservation

PREFACE

xxi

equations. Appendix A provides the vector and tensor operations frequently used in the formulation and manipulation of these equations. Chapter 2 covers the basic structure of laminar premixed ﬂames, conservation equations, various models for diffusion velocities in a multicomponent gas system with increasing complexities, laminar ﬂame thickness, asymptotic analyses, and ﬂame speeds. Effect of ﬂame stretch on laminar ﬂame speed, Karlovitz number, and Markstein lengths are also discussed in detail along with soot formation in laminar premixed ﬂames. Chapter 3 discusses the basic structure of laminar nonpremixed ﬂames and provides detailed descriptions of mixture fraction deﬁnition, balance equations for mixture fraction, temperature-mixture fraction relationship, and examples, since mixture fraction is a very important parameter in the study of nonpremixed ﬂames. The chapter also discusses laminar ﬂamelet structure and equations, critical scalar dissipation rate, steady-state combustion, and examples of laminar diffusion ﬂames with equations and solutions. Since pollution, speciﬁcally soot formation, has become a major topic of interest, it is also covered in this chapter with respect to laminar diffusion ﬂames. Appendix D provides a detailed soot formation mechanism and rate constants that was proposed by Wang and Frenklach. Chapter 4 is devoted entirely to turbulent ﬂows. It covers the fundamental understanding of turbulence from a statistical point of view; homogeneous and/or isotropic turbulence, averaging procedures, statistical moments, and correlation functions; Kolmogorov hypotheses; turbulent scales; ﬁltering and large-eddy simulation (LES) concepts along with various subgrid scale models; and basic deﬁnitions to prepare readers for the probability density function (pdf) approach in later chapters. This chapter also includes the governing equations for compressible ﬂows. A short introduction of the direct numerical simulation (DNS) approach is also provided at the end of the chapter. Chapters 5 and 6 focus on the turbulent premixed and nonpremixed ﬂames, respectively. Chapter 5 consists of physical interpretation; studies for turbulent ﬂame-speed correlation development; Borghi diagram and physical interpretation of various regimes; eddy breakup models; measurements in premixed turbulent ﬂames; ﬂame-turbulence interaction (effects of turbulence on ﬂame as well as effect of ﬂame on turbulence); turbulence combustion modeling approaches; Bray-Moss-Libby model (gradient and counter-gradient transport); level set approach and G-equation for ﬂame surfaces; and the pdf approach and closure of chemical reaction source term. In Chapter 6, the discussion focuses on major problems in nonpremixed turbulent combustion; turbulent Damk¨ohler number and Reynolds number; scales in nonpremixed turbulent ﬂames; regime diagrams; target ﬂames; turbulence-chemistry interaction; pdf approach; ﬂamelet models; ﬂame-vortex interaction; ﬂame instability; partially premixed ﬂames; and edge ﬂames. The fundamentals of multiphase ﬂows are covered in Chapter 7, which has sections on classiﬁcation of multiphase ﬂows; homogeneous versus multiphase mixtures; averaging methods; local instant formulation; Eulerian-Eulerian modeling; Eulerian-Lagrangian modeling; interface transport (tracking and capturing)

xxii

PREFACE

methods (volume of ﬂuid, surface ﬁtted method, markers on interface); and discrete particle methods. This chapter also provides many contemporary approaches for modeling two-phase ﬂows. Spray combustion is an extremely important topic for combustion, and Chapter 8 provides a comprehensive account of various modeling approaches to spray combustion associated with single drop behavior, drop breakup mechanisms, jet breakup models, group combustion models, droplet-droplet collisions, and dense sprays. Experimental approaches and results are also presented in this chapter. VOLUME 2, APPLICATIONS OF TURBULENT AND MULTIPHASE COMBUSTION

Chapter 1 provides a background in solid propellants and their combustion behavior, including desirable characteristics; oxygen balance; homogeneous and heterogeneous propellants; fuel binders, oxidizer ingredients, curing and crosslinking agents, and aging; hazard classiﬁcations; material characterization of solid propellants; and gun performance parameters including thrust, speciﬁc impulse, and stable/unstable burning behavior. Chapter 2 focuses on nitramine decomposition and combustion; phase transformation; and three different approaches for thermal decomposition of royal demolition explosive (RDX) as well as gas-phase reactions. This chapter also describes a modeling approach for RDX combustion. Chapter 3 covers the burning behavior of homogeneous (e.g., double-base) propellants, describing both the experimental and modeling approaches to study and predict the burning rate and temperature sensitivities of common solid propellants. The transient burning characteristics of a typical homogeneous propellant is also presented in detail, including the Zel’dovich map technique and the Novozhilov stability parameters. Chapter 4 covers reacting turbulent boundary-layer ﬂows, a topic of research for the last six decades. The chapter discusses the modeling approaches from 1940s to the current date. Graphite nozzle erosion process by high-temperature combustion product gases through heterogeneous chemical reactions is covered in detail. Turbulent wall ﬁres are also covered. Chapter 5 contains the ignition and combustion studies of single energetic particles (such as micron-size boron and aluminum particles) including multistage combustion models for cases with and without the presence of oxide layers, kinetic mechanisms, criterion for diffusion-controlled combustion versus, kinetic controlled combustion, effect of oxidizers (such as oxygen- and ﬂuorinecontaining species), combustion of nano-size energetic particles, and their strong dependency on kinetic rates. Chapter 6 addresses the two-phase reacting ﬂow simulation and focuses on granular bed combustion with different solution techniques for the governing equations. It also includes experimental validation of the calculated results. We would like to acknowledge the contributions of many of our combustion and turbulence colleagues for reviewing and providing a critical assessment

PREFACE

xxiii

of multiple chapters of these volumes includes Professor Forman A. Williams of the University of California-San Diego; Professor Stephen B. Pope, Cornell University; Dr. Richard Behrens, Jr. of Sandia National Laboratory; Dr. William R. Anderson of the U.S. Army Research Laboratory; Professor Luigi T. DeLuca of Politecnico di Milano, Italy; and Professors James G. Brasseur, Daniel C. Haworth, and Michael M. Micci of Pennsylvania State University. They spent their valuable time reading chapters and helped us to improve the material covered in Volume 1 and Volume 2. We also want to thank Professor Michael Frenklach of University of California-Berkeley for providing us the detailed information on soot formation kinetics used in Appendix D of Volume 1. We also like to thank Professor William A. Sirignano of University of California-Irvine for his valuable input on evaporation and combustion of droplet arrays. Professor Norbert Peters of the Institut f¨ur Technische Mechanik of Aachen, Germany, was very geneous to provide his book draft to Kenneth Kuo while he was visiting the Pennsylvania State University. His notes were very helpful in explaining turbulent combustion topics. During the sabbatical leave of the ﬁrst author at the U.S. Army Research Lab (ARL), Dr. Brad E. Forch of ARL and Dr. Ralph A. Anthenien Jr. of the Army Research Ofﬁce (ARO) hosted and supported a series of his lectures. The lecture materials, which we prepared jointly, were used in the development of several chapters of Volume 2. We greatly appreciate the encouragement and support of Dr. Forch and Dr. Anthenien. Kenneth Kuo would like to take this opportunity to thank his many research project sponsors, since his in-depth understanding of many topics in turbulent and multiphase combustion has been acquired through multi-year research. These sponsors include: Drs. Richard S. Miller, Judah Goldwasser, and Clifford D. Bedford of ONR of the U.S. Navy; Drs. David M. Mann, Robert W. Shaw, Ralph A. Anthenien, Jr. of ARO; Dr. Martin S. Miller of ARL; Mr. Carl Gotzmer of NSWC-Indian Head; Dr. Rich Bowen of NAVSEA of the US Navy, Drs. William H. Wilson and Suhithi Peiris of the Defense Threat Reduction Agency (DTRA); and Drs. Jeff Rybak, Claudia Meyer, and Matthew Cross of NASA. The authors would like to thank Mr. Henry T. Rand of ARDEC and Mr. Jack Sacco of Savit Corporation for sponsoring our project on granular propellant combustion. Ragini Acharya would like to thank several professors at The Pennsylvania State University for developing the framework and knowledge base to aid her in writing the book manuscript, including Professors Andr´e L. Boehman, James G. Brasseur, John H. Mahaffy, Daniel C. Haworth, and Richard A. Yetter. We both would like to acknowledge the generosity of Professor Peyman Givi of the University of Pittsburgh for granting us full permission to use some of his numerical simulation results of RANS, LES, and DNS of a turbulent jet ﬂame on the jacket of Volume 1. For the cover of Volume 2, we would like to thank Dr. Larry P. Goss of Innovative Scientiﬁc Solutions, Inc and Dr. J. Eric Boyer of the High Pressure Combustion Lab of PSU for the photograph of metalized propellant combustion. Also, Professor Luigi De Luca and his colleagues Dr. Filippo Maggi at the Polytechnic Institute of Milan for granting the permission to

xxiv

PREFACE

use their close-up photographs of the burning surface region of metallized solid propellants, showing the dynamic motion of the burning of aluminum/Al2 O3 particles. We would also like to thank Ms. Petek Jinkins and Ms. Aqsa Ahmed for typing references, preliminary proofreading, and miscellaneous help with the preparation of the manuscript. We also want to thank John Wiley & Sons for their patience and cooperation. Last but not least, we also would like to thank our family members for their sacriﬁce during the long and difﬁcult process of manuscript preparation. Kenneth K. Kuo and Ragini Acharya University Park, Pennsylvania

1 INTRODUCTION AND CONSERVATION EQUATIONS

SYMBOLS

Symbol Bi C Ci d DAb eij Eak fi F FS h ht I Ji J∗i K

Description

Dimension

Body force per unit volume in i-direction (vector) Molar concentration Molar concentration of the i th species Molecular diameter Binary mass diffusivity for A-B system Strain rate tensor Activation energy for the kth reaction External force per unit mass on species i (vector) Force (vector) Surface force (vector) Enthalpy per unit mass Total enthalpy per unit mass Identity matrix or vector form of Kronecker delta δij Mass ﬂux of species i relative to mass-average velocity (vector) Molar ﬂux of species i relative to molar-average velocity (vector) Boltzmann constant

F/L3 N/L3 N/L3 L L2 /t t−1 Q/N F/M F F Q/M Q/M –

Fundamentals of Turbulent and Multiphase Combustion Copyright © 2012 John Wiley & Sons, Inc.

Kenneth K. Kuo and Ragini Acharya

M/L2 t N/L2 t (Q/T)/molecule 1

2

INTRODUCTION AND CONSERVATION EQUATIONS

Symbol l ˙ m mi mt Mwi n˙ ni Ni NA q T◦ u ui v V vi ν∗ Vi Vi∗ Xi y Yi z Z

Description

Dimension

Mean free path Mass ﬂux (vector) Mass of the i th species in the mixture Total mass of a multi component gaseous mixture Molecular weight of i th species Molar ﬂux (vector) Number of moles of i th species in the gaseous mixture Number of moles of species i Avogadro’s number, 6.02252 × 1023 molecules/mole Heat-ﬂux vector (vector) Fixed standard reference temperature, at 298.15 K Arithmetic-mean molecular speed Velocity component in i th -direction Mass-average velocity (vector) Control volume Velocity of i th species with respect to stationary coordinate axes (vector) Molar-average velocity (vector) Mass diffusion velocity of i th species (vector) Molar diffusion velocity of i th species (vector) Mole fraction of the i th species Space coordinate in y-direction Mass fraction of the i th species Space coordination in z -direction Frequency of molecular collisions of gaseous species per unit surface area

L M/L2 t M M M/N NL2 t N

Greek Symbols α Thermal diffusivity αi Thermal diffusion coefﬁcient for species i l Thermal conductivity or second viscosity μ Dynamic viscosity or ﬁrst viscosity μ Bulk viscosity μij Reduced mass of molecules of species i and j σij , σ˜ Total stress tensor τij Viscous stress tensor ˙i Molar rate or production of species i ω˙ i Mass rate of production of species i

— – Q/L2 t T L/t L/t L/t L3 L/t L/t L/t L/t — L — L L−2 t−1

L2 /t L2 /t Q/tLT or Ft/L2 Ft/L2 Ft/L2 M F/L2 F/L2 N/(tL3 ) M/(tL3 )

DIFFERENT APPLICATIONS FOR TURBULENT AND MULTIPHASE COMBUSTION

3

This chapter ﬁrst discusses turbulent and multiphase combustion as a major area of research for understanding and importance of solution of multiple challenging and interesting problems related to energy, environment, transportation, and chemical propulsion, among other ﬁelds. The second topic provides a summary of the major conservation equations used by researchers in the combustion community. 1.1 WHY IS TURBULENT AND MULTIPHASE COMBUSTION IMPORTANT?

Currently, a very high percentage (∼80%) of energy is generated by combustion of liquids (such as gasoline and hydrocarbon fuels), solids (such as coal and wood), and gases (such as natural gas composed of largely methane and other hydrocarbons like ethane, propane, butanes and pentanes). For example, during the ﬁrst decades of the twenty-ﬁrst century more than 50% of the electricity in the United States was generated by coal-ﬁred furnaces. This trend is expected to continue for several decades. Thus, energy generation will continue to rely heavily on combustion technology. Most practical devices involve turbulent combustion, which requires understanding of both turbulence and combustion, as well as their effects on each other. Industrial furnaces, diesel engines, liquid rocket engines, and devices using solid propellants involve multiphase and turbulent combustion. Single-phase turbulent reacting ﬂows are complicated enough for modeling and numerical solutions, some of these ﬂows are still unresolved problems of our time. The complexity of the problem increases even further with the presence of multiple phases. In recent years, there has been a greater move to increase combustion efﬁciency while keeping the emissions level as low as possible. We live in times in which energy has become a very critical commodity. Therefore, it is important that the unresolved problems of combustion should be understood and solved. Well-trained combustion engineers and scientists are needed to engage in numerous challenging combustion problems. This chapter provides some general background about the applications of turbulent and multiphase combustion, the general concept of modeling, and basic conservation equations for gas-phase mixtures containing multiple species. 1.2 DIFFERENT APPLICATIONS FOR TURBULENT AND MULTIPHASE COMBUSTION

There are various applications of turbulent and multiphase combustion associated closely with our daily life. Some of these are: • Power generation from combustion (one example of two-phase turbulent combustion used for energy generation from coal-ﬁred burners can be seen in Figure 1.1)

4

INTRODUCTION AND CONSERVATION EQUATIONS PFB Gasifier

Gas Cleaning

Two-Phase, Turbulent Combustion

Air to Gasifier Air

Combustor Exhaust to Char Combustor Electricity

Coal and Sorbent Char Circulating FB Combustor

Compressor Gas Turbine and Generator Waste Heat Recovery

Air from Compressor

Gas Cleaning External Heat Exchanger Steam

FB: Fluidized Bed PFB: Pressurized

Gas to Stack

Condenser Electricity

Pressure Let Down Two-Phase, Turbulent Combustion

Solids Return

Gas Turbine Exhaust

Steam Turbine and Generator

Figure 1.1 Schematic of a hybrid power generation system using coal-air combustion (modiﬁed from http://fossil.energy.gov).

• High rates of combustion of energetic materials for various propulsion systems • Process industry for production of engineering materials (e.g., ceramics, H2 , nanosized particles) • Household and industrial heating; • Active control of combustion processes; • Safety protections for unwanted combustion; • Ignition of various condensed-phase combustible materials (like solid propellants airbags in automobiles) for safety enhancement under emergency situation • Pollutant emission control of combustion products (about one-third of carbon emissions in the United States comes from coal-ﬁred power plants, onethird from transportation, and the rest from the industrial, commercial, and residential sources) Figure 1.2 shows the distribution of total emissions estimates in the United States by source category for speciﬁc pollutants in 2008. The major air pollutants are particulate matter, CO, CO2 , SOx , NOx , VOCs (volatile organic compounds), NH3 , mercury, and lead. Electric utilities contribute about 70%

5

DIFFERENT APPLICATIONS FOR TURBULENT AND MULTIPHASE COMBUSTION Direct PM2.5

Air Pollutants

Direct PM10 NH3 SO2 NOX VOC CO Lead 0

20

40 60 Percentage of Emissions

Source Category Stationary Fuel Combustion

Industrial and Other Processes

Highway Vehicles

Non-Road Mobile

80

100

PM2.5: Particulate matter of size = 2.5 μm PM10: Particulate matter of size = 10 μm VOC: Volatile organic compounds NOx: Oxides of nitrogen NO, NO2, N2O, etc.

Figure 1.2 Distribution of national total emissions estimates by source category for speciﬁc pollutants in year 2008 (modiﬁed from EPA report).

of national SO2 emissions. Agricultural operations (other processes) contribute over 80% of national NH3 emissions. Almost 50% of the national VOC emissions originate from solvent use (other processes) and highway vehicles. Highway vehicles and nonroad mobile sources (e.g., aircrafts, agricultural vehicles, ships, etc.) together contribute approximately 80% of national CO emissions. Fossil fuel combustion is the primary source contributing to CO2 emissions. In 2007, fossil fuel combustion contributed almost 94% of the total CO2 emissions. Major sources of fossil fuel combustion include electricity generation, transportation (including personal and heavy-duty vehicles), industrial processes, residential, and commercial. Electricity generation contributed approximately 42% of CO2 emissions from fossil fuel combustion while transportation contributed approximately 33%. Advance in combustion technology can lead to higher burning efﬁciency and less production of harmful compounds.

1.2.1 Applications in High Rates of Combustion of Materials for Propulsion Systems

Many propulsion systems employ combustion of condensed phase materials to generate thermal energy. Some of these are: • Gas turbine engines for aircrafts; • Liquid fuels and oxidizers for liquid rocket engines (see Figure 1.3); • Spray of liquid fuels for diesel engines, bipropellant rockets, and ramjets, and the like • Prevaporized hydrocarbons for reciprocating engines • Solid propellants in rocket motors for space and missile propulsion

6

INTRODUCTION AND CONSERVATION EQUATIONS

Oxidizer Spray Combustion

Fuel

•

M

Combustion Chamber

Subsonic Region

Transonic Region

Converging Section

Throat

Diverging Section Nozzle

Pa Supersonic Region Viscous Boundary Layer Pe

•

Ve

Thrust = (MVe + PeAe) – PaAe •

M = Engine mass flow rate Ve = Gas velocity at nozzle exit Pe = Static pressure at nozzle exit Ae = Area of nozzle exit Pa = Ambient pressure

Throat Area

Ae

Figure 1.3 Combustion and energy conversion in a nozzle of bipropellant liquid rocket (Modiﬁed from O’ Leary and Beck, 1992).

• Solid fuels for hybrid rocket motors, ramjets, scramjets • Monopropellants for space thrusters • Solid propellants for gun and artillery propulsion systems As shown in Figure 1.3, chemical energy is converted into thermal energy by combustion. The thrust of a propulsion system is proportional to the momentum of the exhaust jet. The speciﬁc impulse (Isp ), deﬁned as the thrust per propellant weight ﬂow rate, is known to be proportional to the square root of the ﬂame temperature divided by the average molecular weight of the combustion products, as shown in Equation 1.1. Isp ∝ Tf /Mw (1.1)

DIFFERENT APPLICATIONS FOR TURBULENT AND MULTIPHASE COMBUSTION

7

More detailed description of this relationship is given in Chapter 1 of Kuo Acharya, Applications of Turbulent and Multiphase Combustion (2012). 1.2.2

Applications in Power Generation

Condensed phase and gas-phase material are turned in various power generation systems. For example: • Coal particles: Burned in furnaces of power stations to produce steam for driving turbines in order to generate electricity (see Figure 1.1) • Liquid fuels: Used as the source of energy for transportation purposes with automobiles, aircrafts, and ships • Natural gases: Used for gas turbines and reciprocating engines • Incineration of waste materials 1.2.3

Applications in Process Industry

In the material processing industry, combustion of different types of fuels has been used for obtained elevated temperature conditions in the manufacturing process. For example: • Production of iron, steel, glass, ceramics, cement, carbon black, and reﬁned fuels through thermal heating processes • Direct fabrication of ceramic materials by self-propagating high-temperature synthesis (SHS) processes • Combustion synthesis of nanosize powders 1.2.4

Applications in Household and Industrial Heating

For various heating systems, chemical energies of fuels and oxidizers are converted to thermal energy by turbulent and multiphase combustion processes. • Thermal energy generated by combustion: Used for heating of residences, factories, ofﬁces, hospitals, schools, and various types of buildings; and heating of International Space Station (ISS) and many special facilities 1.2.5

Applications in Safety Protections for Unwanted Combustion

Knowledge of turbulent and multiphase combustion is also very useful for various ﬁre and hazard prevention systems, such as: • • • •

Fire prevention for forest ﬁres Fire prevention for building ﬁres Reduction of industrial explosions Reduction of susceptibility for deﬂagration-to-detonation transitions (DDT) and shock-to-detonation transition (SDT) leading to catastrophic hazards

8

INTRODUCTION AND CONSERVATION EQUATIONS

1.2.6

Applications in Ignition of Various Combustible Materials

Many safety protection systems depend upon the reliable ignition of various combustion materials, for example • • • • 1.2.7

For safety enhancement under emergency situations Inﬂation of airbags during, collisions automobile Actuation of ejection pilot seats and other emergency escape systems Fire extinguishment by strong-ﬂow gas generators Applications in Emission Control of Combustion Products

The success of emission control of combustion products depend strongly upon the knowledge of the turbulent and multiphase combustion with application in different aspects, such as: • • • •

1.2.8

For reduction of pollutants generated from combustion Reduction of formation of NOx , SOx , and CO2 Reduction of formation of particulates such as soot and coke Control of the temperature and chemical compositions of combustion products Applications in Active Control of Combustion Processes

To achieve better combustion performance and to reduce combustion instabilities in various propulsion systems, certain active control systems can be employed: • To enhance combustion efﬁciencies of reactors by external energy sources, such as acoustic energy emission • To enhance combustion efﬁciencies of certain systems with injection of nanosize energetic particles 1.3

OBJECTIVES OF COMBUSTION MODELING

With signiﬁcant advancements in computational power and numerical schemes in recent years, simulation of complicated combustion problems could be tractable. Several major objectives for combustion modeling are listed below • To simulate certain turbulent combustion processes involving single and/or multi-phase combustible materials • To develop predictive capability for combustion systems under various operating conditions • To help in interpreting and understanding observed combustion phenomena • To substitute for difﬁcult or expensive experiments • To guide the design of combustion experiments

GENERAL APPROACH FOR SOLVING COMBUSTION PROBLEMS

9

• To determine the effect of individual parameters in combustion processes by parametric studies 1.4

COMBUSTION-RELATED CONSTITUENT DISCIPLINES

The science of turbulent and multiphase combustion often involves inticate intercoupling and interactions between many constituent disciplines. Background in the following areas would be very helpful for scientists and engineers to acquire and to apply to various unresolved combustion problems • • • • • • • • • • • • • • •

1.5

Thermodynamics Chemical kinetics Fluid mechanics Heat and mass transfer Turbulence Transport phenomena Statistical mechanics Instrumentation and diagnostic techniques Quantum chemistry and physics Materials structure and behavior Mathematical and statistical theories Numerical methods Design of combustion test apparatus Data analysis and correlation methods Safety and hazard analysis

GENERAL APPROACH FOR SOLVING COMBUSTION PROBLEMS

For solving combustion problems, one can consider the following methods: • Theoretical and numerical methods • Experimental methods • Any combination of the above methods A theoretical model for a combustion problem consists of a set of governing equations that must be solved with multiple input parameters and initial and boundary conditions, as shown in Figure 1.4. As one can observe, there is a signiﬁcant level of coupling between the intermediate solution from governing equations and the input parameters, such as reaction mechanism, turbulence closure conditions, and diffusion/transport mechanisms. The major output of the

10

INTRODUCTION AND CONSERVATION EQUATIONS

Initial conditions and physical model of a given problem

Reaction mechanism and kinetic data Empirical input data and/or correlations with a set of assumptions

Boundary conditions Thermodynamic and transport properties

Governing Equations: 1. Conservation equations 2. Equation of state 3. Transport equations

Material properties and structural characteristics

Turbulence closure considerations

Intermediate solution for major variables, e.g., T, Yi U, p, etc.

Convergence criteria and numerical method

Diffusion and transport mechanism Final Output: Flame structure, flame speed, flame surface area, mass consumption rate, etc.

Figure 1.4

General structure of a theoretical model.

model consists of ﬂame structure, speed, surface area, burning rate, ﬂow ﬁeld structure, and the like. A combustion problem can be solved by using different numerical approaches. Currently there are three major categories of such approaches: Reynolds average Navies-Stokes (RANS) simulation, large-eddy simulation (LES), and direct numerical simulation (DNS). A discussion of these methods is provided in Chapter 4. The effect of these different numerical approaches on the ﬁnal solution can be seen in Figure 1.5, which shows the predicted results for a diffusion ﬂame. Currently RANS is most commonly employed in industry, but its range of validity is limited. DNS is the most detailed, but it is too

(a)

(b)

(c)

Figure 1.5 Predicted results for a diffusion ﬂame by using (a) DNS, (b) LES, and (c) RANS (from Givi, 2009; http://cfd.engr.pitt.edu/).

GOVERNING EQUATIONS FOR COMBUSTION MODELS

11

computationally demanding for most realistic engineering problems. LES is a compromise between the two and provides excellent reliability and applicability. 1.6 1.6.1

GOVERNING EQUATIONS FOR COMBUSTION MODELS Conservation Equations

The ﬁve groups of conservation equations consist of: 1. 2. 3. 4. 5.

Conservation Conservation Conservation Conservation Conservation

of of of of of

mass (continuity equation) molecular species (or conservation of atomic species) momentum (for each independent spatial direction) energy angular momentum

These equations are used together with the transport equations and the equation of state to solve for ﬂow property distributions, including temperature, density, pressure, velocity, and concentrations of chemical species. Note that the conservation equation of angular momentum is not often used unless the problems involve external torque with signiﬁcant amounts of swirling or with polar ﬂuids ﬂowing in magnetic ﬁelds. 1.6.2

Transport Equations

Transport equations are usually required for turbulent combustion problems. They include: 1. Transport of 2. Transport of pation rate) 3. Transport of 4. Transport of 5. Transport of

turbulent kinetic energy turbulence dissipation rate (or turbulent kinetic energy dissiturbulent Reynolds stresses probability density function moments such as 2 Y , Y 2 , u Y , Y 2 , T 2 , etc. u i i ,T i i

1.6.3

Common Assumptions Made in Combustion Models

Certain commonly used assumptions are listed below. Renders must recognize that some of these assumptions can be relaxed nowadays due to the advancements in numerical predictive schemes and/or the availability of thermal and transport property data. • Reacting ﬂuid can be treated as a continuum. • Inﬁnitely fast chemistry (chemical equilibrium) can be applied for hightemperature combustion problems.

12

INTRODUCTION AND CONSERVATION EQUATIONS

• Simple, one-step, forward irreversible global reaction can sometimes be applied for less comprehensive models. • Ideal gas law can be used for low pressure with moderately high temperature reacting ﬂow problems • Lewis, Schmidt, and Prandtl numbers may be assumed equal to 1, under certain combustion conditions. • Equal mass diffusivities of all species was used by many researchers when there were no diffusivity data available. • Fick’s law of species mass diffusion can be assumed to be valid in many circumstances. • Constant speciﬁc heats of the gas-phase species had been assumed when no thermal data were available. • Reacting solid surfaces are sometimes assumed to be energetically homogeneous. • Uniform pressure can be assumed for the region having low-speed combustion situations. • Dufour and Soret effects are often assumed to be negligible • Bulk viscosity is often assumed to be negligibly small. • Under certain conditions, negligible combustion-generated turbulence can be assumed. These assumptions must be examined for validity before they are adopted in modeling work. 1.6.4

Equation of State

The simplest equation of state is that for an ideal gas. The ideal gas law, which applies to both pure components and mixtures, has been established from empirical observation and is accurate for gases at low density or up to tens of atmospheric pressure for most compounds. For nondissociating molecules, this relationship holds for low to moderate pressures. pV = nRu T = m

Ru T = mRT Mw

(1.2)

where Ru is the universal gas constant [= 8.3144 J/(mol K)] Other Forms of Ideal Gas Law ρ=

p pMw m = = = V RT Ru T

p Ru T

N i=1

c=

n p p = = V Ru T RT Mw

Yi Mwi

(1.3)

(1.4)

GOVERNING EQUATIONS FOR COMBUSTION MODELS

13

In terms of speciﬁc volume v, the ideal gas law can be written as: pv = RT

where v =

V m

(1.5)

1.6.4.1 High-Pressure Correction Van der Waals Equation of State The van der Waals equation of state is one of the best-known generalized equations of state. It is essentially a modiﬁed version of the ideal gas law, expressed by Equation 1.5, except that it accounts for the intermolecular forces that exist between molecules (represented by the term a/υ 2 ) and also corrects for the covolume, b, occupied by the molecules themselves. The van der Waals equation of state is: a p + 2 (v − b) = RT (1.6) v

where a and b are evaluated from the general behavior of gases. These constants are related to the critical temperatures and pressures of pure substances by a=

27 R 2 Tc2 64 pc

and

b=

RTc 8pc

(1.7)

If a is equal to 0, then the van der Waals equation of state is called the Noble-Abel equation of state. p=

RT (v − b)

(1.8)

Redlich-Kwong Equation of State The Redlich-Kwong equation of state (and many of its variants) is representative of the commonly used empirical cubic equations of state. It is considerably more accurate than the van der Waals equation and has been shown to be very successful not only for pure substances but also for mixture calculations and phase equilibrium correlations. The original Redlich-Kwong equation is given as

p= where a=

a Ru T − v − b v (v + b) T 1/2

0.42748Ru2 Tc2.5 pc

and

b=

0.08664Ru Tc pc

(1.9)

(1.10)

The values of critical pressure (pc ) and critical temperature (Tc ) for various hydrocarbon fuels are listed in Kuo (2005), Appendix C. Soave-Redlich-Kwong and Peng-Robinson Equations of State The Soave’s modiﬁed RK equation or (SRK) and the Peng-Robinson equations of state are both “cubic” equations of state developed to improve the Redlich-Kwong form. Both approaches have used the same method to set the parameters a and b. That is,

14

INTRODUCTION AND CONSERVATION EQUATIONS

TABLE 1.1. Summary of Four Common “Cubic” Equations of State and their Constants Equation

u

w

b

a

Van der Waals

0

0

R u Tc 8Pc

27 Ru2 Tc2 64 Pc

Redlich-Kwong

1

0

0.08664Ru Tc Pc

Soave or SoaveRedlich-Kwong (SRK)

1

0

0.08664Ru Tc Pc

0.42748Ru2 Tc2.5 Pc T 0.5 0.42748Ru2 Tc2

2 1 + f (ω) 1 − Tr0.5 Pc where f (ω) = 0.48 + 1.574ω − 0.176ω2

Peng-Robinson

2

–1

0.07780Ru Tc Pc

2 0.42748Ru2 Tc2 1 + f (ω) 1 − Tr0.5 Pc f (ω) = 0.37464 + 1.5423ω −0.26992ω2

Note: Values of ω for various substances can be found in Appendix A of R.C. Reid, J. M. Prausnitz and B.E. Poling, The Properties of Gases and Liquids, 4th ed., McGraw Hill, 1987.

both the ﬁrst and second partial derivatives of pressure with respect to speciﬁc volume are set to zero, as was done previously for the Redlich-Kwong equation of state. For brevity, the cubic form of the equations and their coefﬁcients are provided in Table 1.1 for common cubic equations of state. The last four equations of state discussed above can be classiﬁed as cubic equations of state; that is, if expanded, the equations would contain volume terms raised to the ﬁrst, second, or third power. These equations (containing two parameters a and b) can be expressed by the following equation: p=

a Ru T − 2 v − b v + ubv + wb2

(1.11)

More detailed discussion of the equation of state and the mixing rules for multi-component mixtures are given in Appendix A of Kuo (2005). 1.7

DEFINITIONS OF CONCENTRATIONS

There are four ways to express concentration of various species in a multicomponent gas mixture: 1. Mass concentration ρi is the mass of the i th species per unit volume of mixture or solution; 2. Molar concentration Ci ≡ ρi /Mwi is the number of moles of the i th species per unit volume. 3. Mass fraction Yi ≡ ρi /ρ = mi /mt is the mass of the i th species divided by the total mass of the mixture.

DEFINITIONS OF CONCENTRATIONS

15

4. Mole fraction Xi ≡ Ci /C is the molar concentration of the i th species divided by the total molar concentration of the gaseous mixture or liquid solution. Mole Numbers: Gaseous molecules and atoms are conveniently counted in terms of amount of substances or mole numbers. One mole (1 mol) of compound corresponds to 6.02252 × 1023 molecules (or atoms). Avogadro’s number (NA ) is therefore 6.02252 × 1023 molecules/mol. Mole Fractions: ni ni = Xi = (1.12) N n ni i=1

Mass Fractions:

Yi =

mi N

=

mi

mi m

(1.13)

i=1

Average Molecular Weight: The mole fraction Xi and mass fraction Yi are related by:Xi = Yi Mw/Mwi , where Mw is the average molecular weight of the multicomponent gas mixture in the control volume. It can be evaluated by: N N Mw = Xi Mwi = 1 (1.14) (Yi /Mwi ) i=1

i=1

The relationship’s between Yi and Xi are given below. Yi =

Mwi ni N j =1

Mwj nj

=

Mwi Xi N

=

Mwi Xi

Mwj Xj

Mw

j =1

Yi Yi Xi = Mw = Mwi Mwi

N Yj Mwj

(F /O) φ= (F /O)st

(1.16)

j =1

Fuel-Oxidant Ratio, F/O: F mass of fuel F O ≡ O = mass of oxidant Equivalence Ratio:

(1.15)

⎧ ⎨ 0 < φ < 1 fuel-lean φ=1 stoichiometric condition ⎩ 1 < φ < ∞ fuel-rich

(1.17)

(1.18)

16

INTRODUCTION AND CONSERVATION EQUATIONS

TABLE 1.2. Deﬁnitions of Mass Fractions, Mole Fractions, Molar Concentrations, and Useful Relations Quantity

Physical Deﬁnition

Mass fraction, Yi

Mass of i

th

species/Total Mass

Mole fraction, Xi

Moles of i species/Total number of moles

Molar concentration, Ci

Moles of i th species/Total volume

th

Mathematical Expression Yi = mi /mt

Xi ≡ ni /ni = Yi Mw/Mwi Ci ≡ ni /Vt = ρ (Yi /MWi ) = ρ Xi /Mw

The next sections provide readers with basic deﬁnitions of many important parameters utilized in the conservation equations as well as various forms of these equations in different coordinate systems. The detailed derivation of these conservation equations is given in Kuo (2005), Chap. 3. The physical meaning of various terms in the conservation equations are also described in these sections. 1.8

DEFINITIONS OF ENERGY AND ENTHALPY FORMS

Several deﬁnitions of energy are useful in the conservation equations. It is very important to have a clear understanding of the physical meaning and mathematical expression of each of these energy forms as well as their relationships with each other. Sensible internal energy of i th species (es,i ) can be determined with temperature measurements; therefore, it is called sensible. When the heat of formation of the i th species is added to the sensible internal energy, their sum is represented by ei as shown in Table 1.3. The total internal energy of the i th species (et,i ) includes sensible, kinetic, and chemical energies. The total nonchemical energy (etnc,i ) includes sensible and kinetic energies only, as shown in Table 1.3. The same deﬁnitions are used for enthalpy terms. TABLE 1.3. Deﬁnitions of Internal Energy and Enthalpy Forms of the i th Species Quantity

Internal Energy

Sensible es,i

T = Cv,i dT + es,i Tref Tref

Sensible + chemical

Enthalpy T hs,i = Tref

=−Ru Tref /Mwi

T ei = es,i + h0f,i =

o Cv,i dT + ef,i

Cp,i dT + hs,i Tref

hi = hs,i + h0f,i

Tref

Total Total nonchemical

uj uj et,i = ei + 2 uj uj etnc,i = es,i + 2

uj uj 2 uj uj = hs,i + 2

ht,i = hi + htnc,i

=0

DEFINITIONS OF ENERGY AND ENTHALPY FORMS

17

The enthalpy and internal energies are related by: es,i = hs,i − pi /ρi

(1.19)

ei = hi − pi /ρi

(1.20)

T hi = hs,i +

h0f,i

= Tref

Cp,i dT + hs,i Tref +h0f,i

(1.21)

=0

The sensible internal energy is deﬁned to satisfy hs,i = es,i + pi /ρi . The sensible internal energy for the i th species is deﬁned as: T (1.22) es,i = Cv,i dT + es,i Tref Tref

Since at reference temperature of 298.15 K, the sensible enthalpy is deﬁned to be zero, that is, h T = 0, we can conclude from Equation 1.19 that s,i ref es,i Tref = −pi /ρi = −Ru Tref /Mwi . Thus, pi pi = hs,i + hof,i − = es,i + hof,i ρi ρi

o ei = es,i + ef,i = hi −

Therefore,

o = hof,i ef,i

(1.23)

(1.24)

The mass-based enthalpy of formation of the k th species (hof,i ) is related to the molar enthalpy of formation (ho,m f,i ) by Equation 1.25. hof,i = ho,m f,i /Mwi

(1.25)

The negative value of the enthalpy of formation indicates that when 1 mole of i th species is formed from its elements at the standard state of Tref = 298.15 K and p = 1 bar, there is exothermic heat release. The standard state of an element is the stable form of that element at room temperature and 1 bar pressure. For example, H2(g) , O2(g) , N2(g) , Hg(l) , C(s, graphite) are called elements in thermochemical terms. Heats of formation of various compounds are tabulated in various sources. For example, see Kuo (2005), Chap. 1. The mass-based constant-pressure heat capacities (Cp,i ) of the i th species is m ) by: related to the molar heat capacities (Cp,i m Cp,i = Cp,i /Mwi

(1.26)

For a perfect diatomic gas: m = 3.5Ru Cp,i

and

Cp,i = 3.5Ru /Mwi

(1.27)

In many combustion problems, the change of Cp,i with T is quite signiﬁcant in chemically reacting ﬂows. Cp,i values usually are tabulated as polynomial

18

INTRODUCTION AND CONSERVATION EQUATIONS

functions of temperature (see JANAF tables compiled by Stull and Prophet, 1971). Usually the Cp increases with temperature due to an increase in the stored internal energies of different modes, including vibrational, rotational, and translational modes at higher temperatures. Near room temperature, the molar heat capacity of diatomic gases such as N2 and H2 are very close to 3.5Ru ; however, their heat capacities increase rapidly at high temperatures. The mass-based and molar-based constant-volume speciﬁc heats are related to the constant-pressure speciﬁc heats by: Cv,i = Cp,i − Ru /Mwi

m m Cv,i = Cv,i − Ru

or

(1.28)

The constant-pressure heat capacity of the mixture Cp is deﬁned by: Cp =

N

Cp,i Yi =

N

i=1

m Cp,i

i=1

Yi Mwi

(1.29)

The constant-volume heat capacity of the mixture Cv is deﬁned by: Cv =

N

Yi Cv,i =

N

i=1

Yi

i=1

m Cv,i

The speciﬁc enthalpy of the mixture is deﬁned by: N N T hi Yi = Yi Cp,i dT + hof,i = h= i=1

Tref

i=1

(1.30)

Mwi

T

Tref

Cp dT +

N

Yi hof,i

i=1

(1.31) The speciﬁc internal energy of the mixture e = h − p/ρ can be written as: ⎛ ⎞ ⎜ T ⎟ ⎜ ⎟ o ⎟ ⎜ e= Yi ⎜ Cp,i dT − Ru T /MWi +hf,i ⎟ ⎝ Tref ⎠ i=1 pi /ρi N

hs,i

⎛

⎞

⎜ T ⎟ ⎜ ⎟ o = Yi e i = Yi ⎜ Cv,i dT − Ru Tref /MWi +hf,i ⎟ ⎜ ⎟ ⎝ Tref ⎠ i=1 i=1 N

N

(1.32)

es,i

=

T Tref

Cv dT − Ru Tref /Mw + es

N

Yi hof,i

i=1

Table 1.4 summarizes the deﬁnitions of different from of energy and enthalpy of the mixture containing multi-component chemical species.

19

VELOCITIES OF CHEMICAL SPECIES

TABLE 1.4. Deﬁnitions of Different Forms of Energy and Enthalpy Quantity

Energy T

Sensible es =

Tref

Sensible + chemical

Enthalpy T

Cv dT + es Tref

e = es +

hs = Tref

=−Ru Tref /Mw N

Yi hof,i

h = hs +

i=1

Total Total nonchemical

1.9

Cp dT + hs Tref =0

N

Yi hof,i

i=1

uj uj et = e + ; j = 1, 2, 3 2 uj uj etnc = es + ; j = 1, 2, 3 2

uj uj ht = h + ; j = 1, 2, 3 2 uj uj htnc = hs + ; j = 1, 2, 3 2

VELOCITIES OF CHEMICAL SPECIES

In a multicomponent system, various chemical species move at different average velocities. For a mixture of N species with respect to the stationary coordinate axis, the local mass-average velocity v can be deﬁned as: N

v=

N

ρi vi

i=1 N

= ρi

ρi vi

i=1

ρ

=

N

Yi vi

(1.33)

i=1

i=1

The local molar-average velocity v∗ can be deﬁned as N

v∗ =

N

Ci vi

i=1 N

= Ci

Ci vi

i=1

C

=

N

Xi vi

(1.34)

i=1

i=1

The molar-averaged velocity v* differs from the mass-averaged velocity v in both magnitude and direction. Often we are interested in velocity of a given species with respect to the bulk mass-averaged or molar-averaged velocity rather than with respect to stationary coordinates. Therefore, two diffusion velocities are introduced. • Mass diffusion velocity of the i th species is deﬁned as: Vi ≡ vi − v

(1.35)

20

INTRODUCTION AND CONSERVATION EQUATIONS

Vi *

Vi

v

vi v*

Figure 1.6

Vector description of various local velocities in a multispecies system.

TABLE 1.5. Deﬁnitions of diffusion velocities Quantity

Physical Deﬁnition

Mass diffusion velocity of i th species

Vi ≡ vi − v, where v is local mass-average velocity

Mathematical Expression N

v=

N

ρi vi

i=1 N

=

ρi vi

i=1

ρi

=

ρ

N

Yi vi

i=1

i=1

Molar diffusion velocity of i th species

V∗i ≡ vi − v∗ , where v∗ is local molar-average velocity

N ∗

v =

N

C i vi

i=1 N

= Ci

C i vi

i=1

C

=

N

Xi vi

i=1

i=1

• Molar diffusion velocity of the i th species is deﬁned as: V∗i ≡ vi − v∗

(1.36)

These diffusion velocities indicate average motion of component i relative to the local motion of the mixture in the control volume. These velocity components are shower in Fig. 1.6 and also summarized in Table 1.5. 1.9.1

Deﬁnitions of Absolute and Relative Mass and Molar Fluxes

Absolute mass or molar ﬂux of species i is a vector quantity denoting the mass or number of moles of species i that passes through a unit area per unit time. They are deﬁned as: ˙ i ≡ ρi vi (mass ﬂux) m (1.37) n˙ i ≡ Ci vi (molar ﬂux) (1.38)

VELOCITIES OF CHEMICAL SPECIES

21

Relative mass and molar ﬂuxes are deﬁned as: Ji ≡ ρi (vi − v) = ρi Vi J∗i ≡ Ci vi − v∗ = Ci V∗i

(1.39) (1.40)

In a multicomponent system, the relative molar ﬂux J∗i and absolute molar ﬂux n˙ i are related to each other. From the deﬁnitions of v∗ and J∗i N Ci J∗i ≡ Ci vi − v∗ = Ci vi − Cj vj C

(1.41)

j =1

From the deﬁnitions of n˙ i and Xi J∗i = n˙ i − Xi

N

n˙ j

(1.42)

j =1

Summation of Equation 1.42 from i = 1 to i = N gives N

J∗i = 0

(1.43)

i=1

Fick’s Law of Diffusion In a binary system with two chemical species, species A always diffuses in the direction from high concentration of A to low concentration of A, and species B always diffuses from high concentration of B to low concentration of B. The binary mass diffusivity can be expressed by DBA or DAB with dimensions of (L2 /t), usually given in (m2 /s). Fick’s ﬁrst law of diffusion in terms of molar diffusion ﬂux J∗A for the binary system is:

J∗A = −C DAB ∇XA

(1.44)

Equation 1.44 states that species A diffuses in the direction of decreasing mole fraction of A. This is similar to heat transfer by conduction in the direction of decreasing temperature. Molar ﬂux relative to stationary coordinates can now be given as the sum of two molar ﬂuxes n˙ A = CA v∗ − C DAB ∇XA

(1.45)

The ﬁrst term represents the molar ﬂux of A from the bulk motion of the ﬂuid, while the second term with the minus sign represents the relative molar ﬂux of A resulting from the diffusion of species A. In terms of mass ﬂux relative to stationary coordinates, Fick’s law also can be written as the sum of two mass ﬂuxes: ˙ A = ρA v − ρDAB ∇YA m where JA = −ρDAB ∇YA

(1.46)

22

INTRODUCTION AND CONSERVATION EQUATIONS

Note that the mathematical form of Fick’s law of mass transport for a constant density situation in the transverse direction (y-direction) of a binary system is similar to Newton’s law of momentum transport and Fourier’s law of energy transport in the transverse direction. ∂ ρCp T ∂y ∂ τyx = −v (ρvx ) ∂y ∂ JAy = −DAB ρ ∂y A qy = −α

Fourier’s law for constant ρCp

(1.47)

(Newton’s law for constant ρ)

(1.48)

(Fick’s law for constant ρ)

(1.49)

Mass diffusivity D AB for binary mixtures of nonpolar gases (without any dipole moments) is predictable within about 5% by kinetic theory. For a nonpolar gas containing two molecular species A and A* with the same mass mA and the same size and shape, with constant temperature T and molar concentration C, the random motion molecular velocity relative to ﬂuid velocity v has an average magnitude: 8 kB T u= (1.50) πmA where kB = Boltzmann constant = Ru /NA with the Avogadro’s number, NA = 6.02252 × 1023 molecules/mol, and universal gas constant, Ru = 8.3144 J/(mol · K). A schematic representation of the bulk and random velocities is shown in Fig. 1.7. The frequency of molecular collisions per unit area (Z) on a stationary surface exposed to the gas is 1 Z = nu (1.51) ˜ 4 where n˜ represents molecules per unit volume, which is constant since the molar concentration C is constant and n˜ = C × NA . The mean free path l from kinetic theory is 1 l=√ (1.52) 2πdA2 n˜ u

v

Figure 1.7

Schematic representation of bulk and random velocities.

DERIVATION OF SPECIES MASS CONSERVATION EQUATION

23

where dA is the diameter of the molecule A. The new molar ﬂux equation corresponds to Fick’s law of diffusion in the y-direction, with DAA∗ approximately given by 1 DAA∗ = ul (1.53) 3 Substituting for u and l into Equation 1.53, we have kB 3 T 3/2 1 1 8 kB T 1 2 DAA∗ = = √ 3 πmA 2πdA2 n˜ 3 π 3 mA dA2 nk ˜ BT

(1.54)

˜ B T allows calFurther substitution using the perfect gas law p = CRu T = nk culation of an approximate value for DAA∗ from kB 3 T 3/2 2 T 3/2 DAA∗ = ∝ (1.55) 3 π 3 mA pd2A p DAA∗ represents the mass diffusivity of a mixture of two species of rigid spheres of identical mass and diameter. Calculation of DAB for rigid spheres of unequal mass and diameter results in DAB

1.10

2 = 3

kB 3 π3

1/2

1 1 + 2mA 2mB

1/2

T

3/2

dA + dB p 2

2

(1.56)

DIMENSIONLESS NUMBERS

Mass diffusivity (D), momentum diffusivity (ν), and thermal diffusivity (α) all have the same dimensions. Schmidt number, Prandtl number, and Lewis number can then be deﬁned as the ratios between these quantities (see Table 1.6). Sc ≡ ν/D

(1.57)

Pr ≡ ν/α

(1.58)

Le ≡ α/D

(1.59)

1.11 DERIVATION OF SPECIES MASS CONSERVATION EQUATION AND CONTINUITY EQUATION FOR MULTICOMPONENT MIXTURES

We start with a mass balance over an inﬁnitesimal differential ﬂuid element in a binary mixture to derive the mass conservation equation of each species in a multicomponent mixture. We then apply the law of conservation of mass of species A to a volume element xyz ﬁxed in space through which a binary mixture of A and B is ﬂowing (see Fig. 1.8).

24

INTRODUCTION AND CONSERVATION EQUATIONS

TABLE 1.6. Deﬁnitions of Three Important Dimensional Numbers Quantity

Physical Meaning

Mathematical Deﬁnition

Schmidt number

Ratio of momentum transport to mass transport Ratio of momentum transport to thermal transport Ratio of thermal transport to mass transport

Sc ≡ ν/D

Prandtl number Lewis number

Pr ≡ ν/α Le ≡ α/D

y

•

•

mΑx

Δy

x

mΑx

x+Δx

Δz Δx

x z

Figure 1.8 Fixed inﬁnitesimal control volume xyz through which a ﬂuid is ﬂowing.

The rate of accumulation of mass of species A is: ∂ρA xyz ∂t The rate of mass of species A ﬂowing into the control volume due to the x -direction mass ﬂux at the x station is: m ˙ Ax |x yz The rate of mass of species A ﬂowing out of the control volume due to the x -direction mass ﬂux at the x + x station is: ˙ Ax |x yz + m ˙ Ax |x+x yz = m

∂m ˙ Ax xyz x

Within this inﬁnitesimal control volume, species A can be produced by chemical reactions at a net rate of ω˙ A (kg m−3 s−1 ). The net rate of production of species A by chemical reactions is: ω˙ A xyz

DERIVATION OF SPECIES MASS CONSERVATION EQUATION

25

Adding the input and output terms in the y and z directions and dividing the entire mass balance by xyz, it yields: ∂m ˙ Ay ∂m ˙ Ax ∂ρA ∂m ˙ Az (1.60) + + + = ω˙ A ∂t ∂x ∂y ∂z which is the mass conservation equation of species A in a binary mixture. Equation 1.60 can be rewritten in a vector form as: ∂ρA ˙ A = ω˙ A (1.61) + ∇ ·m ∂t ˙A = m ˙ Ax , m ˙ Ay , m ˙ Az is the mass ﬂux vector with m ˙ Ay , m ˙ Az compowhere m ˙ Ax , m nents in rectangular coordinates. Similarly, the mass conservation equation of species B is ∂ρB ˙ B = ω˙ B + ∇ ·m ∂t

(1.62)

When the equations of continuity for components A and B are added together, the result is ∂ρ + ∇ · (ρv) = 0 (1.63) ∂t which is the equation of continuity for the mixture. This equation makes use of ˙ B = ρv and the law of conservation of mass in the form ˙A+m the relation m ω˙ A + ω˙ B = 0, since the combustion process does not produce or destroy mass. The combustion process converts one group of species (reactants) into another group of species (products). ˙ A ) in Equation 1.61 by using Fick’s law Substituting the mass ﬂux term (m shown in Eqaution 1.46, we have: ∂ρA + ∇ · ρA v = ∇ · ρDAB ∇YA + ω˙ A ∂t

(1.64)

Using the relationships that ρi = Yi ρ and vi = v + Vi for a multicomponent system, Equation 1.61 can be generalized into this form: ∂ (ρYi ) + ∇ · [ρYi (v + Vi )] = ω˙ i ∂t

(1.65)

The divergence form in Equation 1.65 can be reduced to the Euler form by ﬁrst expending parts of the terms on the left-hand side ρ

∂Yi ∂ρ + Yi + Yi ∇ · (ρv) + ρv · ∇Yi + ∇ · (ρYi Vi ) = ω˙ i ∂t ∂t

(1.66)

Then, using the overall continuity equation, the Euler form is obtained ρ

∂Yi + ρv · ∇Yi + ∇ · (ρYi Vi ) = ω˙ i ∂t

i = 1, 2, . . . , N

(1.67)

26

INTRODUCTION AND CONSERVATION EQUATIONS

In a general multicomponent system, there are N equations of the Euler form. All values of Yi are considered as unknown in the numerical solution. It is not necessary to solve all N partial differential equations for Yi , since N i=1

Yi = 1

(1.68)

This allows one of the N species conservation equations to be replaced by the above algebraic Equation 1.68. Usually N − 1 independent equations for Yi are solved with other conservation equations for the chemically reacting mixture. ˙ A for the molar rate of production Using CA for molar concentration and per unit volume, the continuity (or mass conservation) equation for species A can be written as: ∂CA ˙A (1.69) + ∇ · n˙ A = ∂t Substituting the molar ﬂux equation yields ∂CA ˙A + ∇ · CA v∗ = ∇ · CDAB ∇XA + ∂t

(1.70)

In a generalized form, the species conservation equation for the i th species in terms of molar concentration can be written as: ∂Ci ˙i + ∇ · n˙ i = ∂t

(1.71)

Substituting the molar ﬂux equation yields ∂Ci ˙i + ∇ · Ci v∗ = ∇ · CDim ∇Xi + ∂t

(1.72)

In the Equation 1.72, Dim is the mass diffusivity of the i th species with respect to the rest of the mixture. A detailed treatment of diffusion velocity representations and mass diffusivities is given in Chapter 2, where the equations for Dim are also shown. It is also shown that a correction velocity Vc is required in order to satisfy the overall mass conservation. By summing over all the species from 1 to N , the summed species conservation equation is: N ∂ (ρYi ) i=1

∂t

+

N i=1

∇ · [ρYi (v + Vi )] =

N

ω˙ i

i=1

Taking the summation inside, we get: ⎛ ⎞ ⎡ ⎛ ⎞⎤ =1 N N =1 N N =0 ∂ ⎝ ⎠ ⎣ ⎝ ⎠ ⎦ ρ Yi + ∇ · ρ v Yi + = Yi Vi ω˙ i ∂t i−1 i−1 i=1 i−1

(1.73)

DERIVATION OF SPECIES MASS CONSERVATION EQUATION

or

27

% $ N ∂ρ =0 Yi Vi + ∇ · (ρv) +∇ · ρ ∂t i=1 =0

This equation implies that the next relationship must be satisﬁed by the diffusion velocity deﬁnition to achieve overall mass conservation: N

Yi Vi = 0

(1.74)

i=1

Readers should refer to Section 2.1 of Chapter 2 to understand the requirement for a correction velocity and the derivation of the expression for a correction velocity. For convenience, the correction velocity expression is given next: Vc = −

N

Yi Vi

(1.75)

i=1

With this correction velocity, the species conservation equation then becomes: ∂ (ρYi ) + ∇ · [ρYi (v + Vi + Vc )] = ω˙ i ∂t

(1.76)

Different models for Vi by using multicomponent species diffusion, the Hirschfelder-Curtiss approximation, Fick’s law, the constant Lewis number for the i th species, or unity Lewis number approaches are shown in Table 2.1 of Chapter 2. A summary of overall mass conservation equation (or continuity equation) in different coordinate systems is given in Table 1.7. TABLE 1.7. Equation of Continuity in Several Coordinate Systems Rectangular coordinates (x, y, z): ∂ ∂ ∂ρ ∂ + ρuy + (ρux ) + (ρuz ) = 0 ∂t ∂x ∂y ∂z

(1.77)

Cylindrical coordinates (r, θ, z):a ∂ρ 1 ∂ ∂ 1 ∂ + (ρrur ) + (ρuθ ) + (ρuz ) = 0 ∂t r ∂r r ∂θ ∂z

(1.78)

Spherical coordinates (r, θ, φ):b ∂ρ 1 ∂ 2 ∂ ∂ 1 1 + 2 ρr ur + ρuφ = 0 (ρuθ sin θ) + ∂t r ∂r r sin θ ∂θ r sin θ ∂φ ar br

≥ 0, 2π ≥ θ ≥ 0. ≥ 0, 2π ≥ φ ≥ 0, π ≥ θ ≥ 0.

(1.79)

28

INTRODUCTION AND CONSERVATION EQUATIONS

A summary of species mass conservation equation in different coordinate systems is given in Table 1.8. In the model of Hirschfelder, Curtiss, and Bird (1954), an approximate diffusion coefﬁcient for i th species against the rest of the mixture is calculated by the following equation: ⎛

N

∗ Dim = (1 − Yi ) ⎝Mw

j =1, j =i

⎞−1

Yj ⎠ Mwj Dij

(1 − Yi )

=

N

(1.80)

Xj /Dij

j =1, j =i

TABLE 1.8. Mass Conservation Equation for i th Species in Several Coordinate Systems Rectangular coordinates (x, y, z): ρ

∂Yi ∂Yi ∂Yi ∂Yi + ux + uy + uz ∂t ∂x ∂y ∂z

+

∂ ∂ ∂ ρYi Viy + (ρYi Vix ) + (ρYi Viz ) ∂x ∂y ∂z

+

∂ ∂ ∂ ρYi Vc,x + ρYi Vc,y + ρYi Vc,z = ω˙ i ∂x ∂y ∂z

(1.81)

Mass diffusion velocities by Hirschfelder-Curtiss approximation: Vix = −

D∗im ∂Yi , Yi ∂x

Viy = −

D∗im ∂Yi , Yi ∂y

Viz = −

D∗im ∂Yi Yi ∂z

Cylindrical coordinates (r, θ, z): ρ

∂Yi ∂Yi uθ ∂Yi ∂Yi + ur + + uz ∂t ∂r r ∂θ ∂z

1 ∂ ∂ 1 ∂ (rρYi Vir ) + (rρYi Viθ ) + (ρYi Viz ) r ∂r r ∂θ ∂z 1 ∂ 1 ∂ ∂ + rρYi Vc,r + rρYi Vc,θ + ρYi Vc,z = ω˙ i r ∂r r ∂θ ∂z +

Mass diffusion velocities by Hirschfelder-Curtiss approximation: Vir = −

D∗im ∂Yi , Yi ∂r

Viθ = −

D∗im ∂Yi , Yi r∂θ

Viz = −

D∗im ∂Yi Yi ∂z

(1.82)

MOMENTUM CONSERVATION EQUATION FOR MIXTURE

29

TABLE 1.8. (continued ) Spherical coordinates (r, θ, φ): ρ

∂Yi ∂Yi uθ ∂Yi uθ ∂Yi + ur + + ∂t ∂r r ∂θ r sin θ ∂φ

+

1 1 ∂ 2 ∂ ∂ 1 r ρYi Vir + ρYi Viφ (sin θρYi Viθ ) + 2 r ∂r r sin θ ∂θ r sin θ ∂φ

+

∂ ∂ 1 ∂ 2 1 1 r ρYi Vc,r + sin θρYi Vc,θ + ρYi Vc,φ = ω˙ i 2 r ∂r r sin θ ∂θ r sin θ ∂φ

(1.83)

Mass diffusion velocities by Hirschfelder-Curtiss approximation: Vir = −

1.12

D∗im ∂Yi , Yi ∂r

Viθ = −

D∗im ∂Yi , Yi r ∂θ

Viφ = −

D∗im ∂Yi Yi r sin θ ∂φ

MOMENTUM CONSERVATION EQUATION FOR MIXTURE

In this section we present the momentum equations in the form of partial differential equations. The basic assumption is that we are dealing with continuous, isotropic, homogeneous, and Newtonian ﬂuids. For Newtonian ﬂuids, there is a linear relationship between shear stress and rate of deformation. Readers interested in the derivation of the momentum equation by various approaches are referred to Kuo (2005), Chap. 3. For a Newtonian ﬂuid, the stress tensor can be written as: & ' ∂uj 2 ∂ui ∂uk δij + μ + (1.84) σij = −pδij + τij = −pδij + μ − μ 3 ∂xk ∂xj ∂xi In this constitutive relationship between stress and strain rate, the coefﬁcient μ is usually called the dynamic viscosity or the ﬁrst viscosity and μ is called the bulk viscosity. For monatomic gas mixtures, kinetic theory shows that μ = 0. For most practical purposes, μ can be treated as zero. In Equation 1.84, the Kronecker delta function, δij , is deﬁned in such way that ( 1, i = j (1.85) δij = 0, i = j In Equation 1.84, the total stress tensor is expressed as a sum of the hydrostatic pressure component and the viscous stress component, which is further expressed in terms of the volume dilatation contribution due to (∂uk /∂xk ) and strain-rate tensor eij contribution, where eij ≡ ∂ui /∂xj + ∂uj /∂xi /2 (1.86)

30

INTRODUCTION AND CONSERVATION EQUATIONS

The i th direction momentum equation can be written in the Euler form as: ' N ∂σji ∂τji ∂p ∂ui ∂ui = + Bi = − + +ρ ρ + uj (Yk fk )i ∂t ∂xj ∂xj ∂xi ∂xj k=1 Pressure &

Inertial force

gradient force

Viscous stress force

(1.87)

Body forces

This equation represents the balance of four different forces: inertial force, pressure gradient force, viscous stress force, and body forces. The body forces act on the control volume due to gravity or the Lorenz force acting in distance. If the ﬂuid mixture in the control volume consists of N species, the body forces acting on different chemical species may differ. For example, some species could be ionized. If the reacting mixture ﬂows through a magnetic ﬁeld, these ionized species will experience Lorenz forces depending the degree of ionization and the mass of each species. Thus, for a multicomponent system, we have Bi = ρ

N

(Yk fk )i

(1.88)

k=1

where fk,i is the force per unit mass of k th species in i th direction, A summary of momentum conservation equation in rectangular, cylindrical, and spherical coordinate systems are given in Table 1.9, Table 1.10, and TABLE 1.9. Momentum Conservation Equation in Rectangular Coordinate Systems (Modiﬁed* from Bird, Stewart, and Lightfoot, 1960) In terms of viscous stress, τ : ∂ux ∂ux ∂ux ∂p ∂τxx ∂τyx ∂τzx ∂ux +ux +uy +uz =− + + + +Bx x: ρ ∂t ∂x ∂y ∂z ∂x ∂x ∂y ∂z ∂uy ∂uy ∂uy ∂τxy ∂τyy ∂τzy ∂uy ∂p +ux +uy +uz =− + + + +By y: ρ ∂t ∂x ∂y ∂z ∂y ∂x ∂y ∂z ∂uz ∂uz ∂uz ∂p ∂τxz ∂τyz ∂τzz ∂uz +ux +uy +uz =− + + + +Bz z: ρ ∂t ∂x ∂y ∂z ∂z ∂x ∂y ∂z

(1.89) (1.90) (1.91)

In terms of velocity gradients for Newtonian ﬂuid with constant ρ and μ: 2 ∂ux ∂ux ∂ux ∂p ∂ ux ∂ 2 ux ∂ 2 ux ∂ux +Bx (1.92) +ux +uy +uz = − +μ x: ρ + + ∂t ∂x ∂y ∂z ∂x ∂x 2 ∂y 2 ∂z2 2 ∂uy ∂uy ∂uy ∂uy ∂ uy ∂ 2 uy ∂ 2 uy ∂p +By (1.93) y: ρ +ux +uy +uz = − +μ + + ∂t ∂x ∂y ∂z ∂y ∂x 2 ∂y 2 ∂z2 2 ∂uz ∂uz ∂uz ∂uz ∂p ∂ uz ∂ 2 uz ∂ 2 uz +Bz z: ρ +ux +uy +uz = − +μ + + (1.94) ∂t ∂x ∂y ∂z ∂z ∂x 2 ∂y 2 ∂z2 *These equation numbers are continuous with these in the main text

MOMENTUM CONSERVATION EQUATION FOR MIXTURE

31

TABLE 1.10. Momentum Conservation Equation in Cylindrical Coordinate Systems (Modiﬁed from Bird, Stewart, and Lightfoot, 1960) In terms of viscous stress, τ : u2 ∂ur ∂ur uθ ∂ur ∂ur + ur + − θ + uz ρ ∂t ∂r r ∂θ r ∂z r: 1 ∂τrθ 1 ∂ τθ θ ∂τrz ∂p + − + + Br =− (rτrr ) + r ∂r r ∂θ r ∂z ∂r ∂uθ ∂uθ uθ ∂uθ ur uθ ∂uθ + ur + − + uz ρ ∂t ∂r r ∂θ r ∂z θ: 1 ∂ 2 1 ∂τθ θ ∂τθ z 1 ∂p + 2 r τrθ + + + Bθ =− r ∂r r ∂θ ∂z r ∂θ ∂uz ∂uz ∂uz uθ ∂uz + ur + + uz ρ ∂t ∂r r ∂θ ∂z z: 1 ∂τθ z 1 ∂ ∂τzz ∂p + + + Bz =− (rτrz ) + ∂z r ∂r r ∂θ ∂z

(1.95)

(1.96)

(1.97)

In terms of velocity gradients of Newtonian ﬂuids with constant ρ & μ: u2 ∂ur ∂ur ∂ur uθ ∂ur ρ + ur + − θ + uz ∂t ∂r r ∂θ r ∂z r: (1.98) ' & 1 ∂ 2 ur ∂ 1 ∂ ∂ 2 ur 2 ∂uθ ∂p + Br +μ + − =− (rur ) + 2 ∂r r ∂r r ∂θ 2 r 2 ∂θ ∂z2 ∂r ∂uθ ∂uθ ∂uθ uθ ∂uθ ur uθ ρ + ur + + + uz ∂t ∂r r ∂θ r ∂z θ: (1.99) ' & 1 ∂ 2 uθ ∂ 1 ∂ ∂ 2 uθ 2 ∂ur 1 ∂p + Bθ +μ + + 2 =− (ruθ ) + 2 2 ∂r r ∂r r ∂θ ∂z2 r ∂θ r ∂θ ∂uz ∂uz uθ ∂uz ∂uz + ur + + uz ρ ∂t ∂r r ∂θ ∂z (1.100) z: ' & 1 ∂ ∂uz 1 ∂ 2 uz ∂ 2 uz ∂p + Bz +μ r + 2 + =− ∂r r ∂r ∂r r ∂θ 2 ∂z2

Table 1.11, respectively. In each of these tables, there are two sets of momentum equations; the ﬁrst set is written in terms of the viscous stress components, and the second set is written in terms of velocity components with the constant density and constant viscosity assumptions. Readers interested in compressible ﬂuids and/or variable viscosity cases can substitute the constitutive relationship given by Equation 1.84. The stress tensor components in different coordinate systems are given in Bird, Stewart, and Lightfoot (1960), Chap. 3 and Kuo (2005), Chap. 3. The term ρu2θ /r in the r-direction momentum equation [Equation’s 1.95 and 1.98] is the centrifugal force. It gives the effective force in the r-direction resulting from ﬂuid motion in the θ-direction. This term arises automatically on transformation from rectangular to cylindrical coordinates. The term ρur uθ /r

32

INTRODUCTION AND CONSERVATION EQUATIONS

TABLE 1.11. Momentum Conservation Equation in Spherical Coordinate Systems (Modiﬁed from Bird, Stewart, and Lightfoot, 1960) In terms of viscous stress, τ : u2θ + u2φ ∂ur ∂ur uθ ∂ur uφ ∂ur ρ + ur + + − ∂t ∂r r ∂θ r sin θ ∂φ r 1 ∂ 2 ∂ 1 ∂τrφ ∂p 1 R: + 2 r τrr + =− (τrθ sin θ) + ∂r r ∂r r sin θ ∂θ r sin θ ∂φ τθ θ + τφφ + Br − r u2φ cot θ ∂uθ uθ ∂uθ uφ ∂uθ ur uθ ∂uθ + ur + + + − ρ ∂t ∂r r ∂θ r sin θ ∂φ r r 1 ∂ 2 ∂ τrθ 1 ∂τθ φ 1 ∂p 1 θ: + 2 r τrθ + + =− (τθ θ sin θ) + r ∂θ r ∂r r sin θ ∂θ r sin θ ∂φ r cot θ − τφφ + Bθ r ∂uφ uθ ∂uφ uφ ∂uφ uφ ur uθ uφ ∂uφ + ur + + + + cot θ ρ ∂t ∂r r ∂θ r sin θ ∂φ r r 1 ∂ 2 1 ∂τφθ 1 ∂τφφ τrφ 1 ∂p + 2 r τrφ + + + =− φ: r sin θ ∂φ r ∂r r ∂θ r sin θ ∂φ r 2 cot θ τθ φ + Bφ − r

(1.101)

(1.102)

(1.103)

In terms of velocity gradients of Newtonian ﬂuids with constant ρ and μ: u2θ + u2φ ∂ur uθ ∂ur uφ ∂ur ∂ur + ur + + − ρ ∂t ∂r r ∂θ r sin θ ∂φ r R: (1.104) 2 2 2 ∂uθ 2 ∂uφ ∂p 2 + μ ∇ ur − 2 ur − 2 − 2 uθ cot θ − 2 + Br =− ∂r r r ∂θ r r sin θ ∂φ u2φ cot θ ∂uθ uθ ∂uθ uφ ∂uθ ur uθ ∂uθ + ur + + + − ρ ∂t ∂r r ∂θ r sin θ ∂φ r r (1.105) θ: 2 cos θ ∂uφ uθ 2 ∂ur 1 ∂p 2 − + μ ∇ uθ + 2 − + Bθ =− r ∂θ r ∂θ r 2 sin2 θ r 2 sin2 θ ∂φ ∂uφ ∂uφ uθ ∂uφ uφ ∂uφ uφ ur uθ uφ ρ + ur + + + + cot θ ∂t ∂r r ∂θ r sin θ ∂φ r r 2 ∂ur u 1 ∂p φ φ: + 2 + μ ∇ 2 uφ − (1.106) =− 2 2 r sin θ ∂φ r sin θ ∂φ r sin θ 2 cos θ ∂uθ + Bφ + r 2 sin2 θ ∂φ

ENERGY CONSERVATION EQUATION FOR MULTICOMPONENT MIXTURE

33

in the θ-direction momentum equation Equations.1.96 and 1.99 is the Coriolis force. It is an effective force in the θ-direction when there is ﬂow in both the r and θ directions. This term also arises automatically in the coordinate transformation. The Coriolis force arises in the problem of ﬂow near a rotating disk. (See, e.q., Schlichting, 1968), Chap. 5. In Table 1.10, the Laplacian operator (∇ 2 ) is given as: 2 ∂ 1 ∂ ∂ 1 ∂ ∂ 1 (1.107) r2 + 2 sin θ + ∇2 = 2 2 2 2 r ∂r ∂r r sin θ ∂θ ∂θ ∂φ r sin θ 1.13 ENERGY CONSERVATION EQUATION FOR MULTICOMPONENT MIXTURE

The energy conservation equation requires the greatest attention because multiple forms exist. Note ﬁrst that because of continuity, the relation shown in Equation 1.108 (which may be used in all left-hand sides of enthalpy, energy, or temperature equations) holds for any quantity f : ∂f ∂f ∂ρui f ∂ρf Df (1.108) = =ρ + ui + ρ Dt ∂t ∂xi ∂t ∂xi In this equation, D/Dt is called the material derivative or substantial derivative. In the Lagrangian point of view, this time derivative is taken while following the motion of the ﬂuid particle with a ﬁxed mass. In the Eulerian frame of reference, the D/Dt operator can be expressed by the sum of four terms on the right-hand side of Equation 1.109, since there are four independent variables in the Eulerian coordinates. Thus, ∂ ∂ ∂ D ∂ d + u2 + u3 ≡ ≡ + u1 dt Dt ∂t ∂x1 ∂x2 ∂x3

(1.109)

As shown in Table 1.4, there are eight different forms of energy for the gaseous mixture. The energy conservation equation can be written in terms of any of these eight forms. In addition, the energy conservation equation also can be given in terms of temperature. Although there are many different choices for writing the energy conservation equation, only one energy equation for the gaseous mixture can be used since all forms of energy are interrelated. Readers can ﬁnd the detailed derivation of energy equation in Kuo (2005), Chap. 3. Next we present different forms of energy equations. In terms of total energy (internal with chemical and kinetic) et , the energy conservation equation can be written as: N ∂σji ui ∂et ∂qi ∂et ˙ + + ρui = − + Q +ρ Yk fk,i ui + Vk,i ρ ∂t ∂x ∂x ∂xj k=1 i i Net rate of Rate of accumulation of internal and kinetic energy per unit volume stored in control volume

Net rate of energy transported out of control volume by advection

Net rate of heat addition to control volume by conduction, interdiffusion, & Dufour ﬂux

external energy input per unit volume to control volume

Work done by surface stress induced forces on control volume

Body force work

(1.110)

34

INTRODUCTION AND CONSERVATION EQUATIONS

where qi is the i th component of the ﬂux vector q, which contains conduction heat ﬂux, interdiffusion heat ﬂux, and the Dufour heat ﬂux; that is q = q conduction + q interdiffusion + q Dufour N N N Xj DTk = −l∇T + ρ hk Yk Vk + Ru T Vk − Vj (1.111) Mwk Dkj k=1

k=1 j =1

By neglecting the Dufour effect, qi can be written as: N

qi = −l

∂T +ρ hk Yk Vk,i ∂xi

(1.112)

k=1

The kinetic energy equation can be written as shown in Equation 1.113, by using the product of ui with the momentum conservation equation: N ∂ 12 ui ui ∂ 12 ui ui ∂σji ρ = ui +ρ Yk fk,i ui (1.113) + ρuj ∂t ∂xj ∂xj k=1

Substituting Equation 1.113 in Equation 1.110, we obtain a conservation equation for sensible and chemical energy, e: N

ρ

De ∂ui ∂qi ˙ +ρ + σji +Q Yk fk,i Vk,i =− Dt ∂xi ∂xj

(1.114)

k=1

The equation for sensible internal energy es is:

ρ

⎛

N N Des ∂qi ∂ui ∂ ˙− ω˙ k hof,k − + σji +Q hof,k =− Dt ∂xi ∂xj ∂xi k=1 k=1 ω˙ T

+ρ

N

∂ = ω˙ T + ∂xi +ρ

⎜ ∂Yk ⎟ ⎜ ⎟ ⎜ρ Dk ⎟ ⎝ ∂xi ⎠ =−Yk Vk,i

Yk fk,i Vk,i

k=1

N

⎞

∂T l ∂xi

(1.115)

∂ui ˙− ∂ +Q + σji ∂xj ∂xi

ρ

N

hs,k Yk Vk,i

k=1

Yk fk,i Vk,i

k=1

In Equation 1.115, the source term due to heat released by chemical reactions is ω˙ T , and it is deﬁned as: ω˙ T ≡ −

N k=1

ω˙ k hof,k

(1.116)

35

ENERGY CONSERVATION EQUATION FOR MULTICOMPONENT MIXTURE

The equation for total nonchemical energy (sensible + kinetic) energy etnc is: N ∂σij ui ∂ ∂ ∂T Detnc ˙− l +Q hs,k Yk Vk,i + = ω˙ T + ρ ρ Dt ∂xi ∂xi ∂xj ∂xi +ρ

N

k=1

Yk fk,i ui + Vk,i

(1.117)

k=1

The conservation equation for (sensible + chemical) enthalpy is: ρ

∂ui +τji + ∂xj

Dp ∂qi Dh = − Dt Dt ∂xi

˙ +ρ Q

N

Yk fk,i Vk,i

k=1

=Viscous dissipation

Dp ∂ = + Dt ∂xi +ρ

N

∂T l ∂xi

∂ − ∂xi

ρ

N

hk Yk Vk,i

+ τji

k=1

∂ui ˙ +Q ∂xj

(1.118)

Yk fk,i Vk,i

k=1

The conservation equation kinetic energy) is: ∂p ∂ τji ui Dht ˙− +Q = + ρ Dt ∂t ∂xj ∂p ∂ τji ui ˙+ = +Q + ∂t ∂xj +ρ

N

for total enthalpy (sensible + chemical + N ∂qi +ρ Yk fk,i ui + Vk,i ∂xi k=1 N ∂ ∂ ∂T l hk Yk Vk,i − ρ ∂xi ∂xi ∂xi

(1.119)

k=1

Yk fk,i ui + Vk,i

k=1

The conservation equation for sensible enthalpy is: ρ

Dp ∂qi Dhs + = ω˙ T + − Dt Dt ∂xi

∂ui τji ∂xj

⎛

N ∂ ˙− + Q hof,k ∂xi k=1

⎜ ∂Yk ⎟ ⎜ ⎟ ⎜ρ Dk ⎟ ⎝ ∂xi ⎠

=Viscous dissipation

+ρ

N

⎞

=−Yk Vk,i

Yk fk,i Vk,i

k=1

Dp ∂ = ω˙ T + + Dt ∂xi ˙ +ρ +Q

N k=1

∂T l ∂xi

Yk fk,i Vk,i

∂ − ∂xi

ρ

N k=1

hs,k Yk Vk,i

+ τji

∂ui (1.120) ∂xj

36

INTRODUCTION AND CONSERVATION EQUATIONS

The conservation equation for total nonchemical (sensible + kinetic energy) enthalpy is: N ∂τij ui ∂T ∂p ∂ Dhtnc ∂ ˙− ρ l +Q hs,k Yk Vk,i + ρ = ω˙ T + + Dt ∂t ∂xi ∂xi ∂xj ∂xi k=1 N (1.121) Yk fk,i ui + Vk,i +ρ k=1

The energy conservation equation in terms of temperature can be very useful. The enthalpy (sensible + chemical) can be written as: T h=

Cp dT +

N

Yi h0f,i

=

i=1

Tref

T N Tref

Cp,i Yi dT +

i=1

N

Yi h0f,i

=

i=1

N

hi Yi

i=1

(1.122) Since the mass fraction of the i th species is an independent variable, the fractional change in enthalpy (sensible + chemical) can be written as: N (1.123) Cp,i Yi dT dh = Cp dT = i=1

dhk = Cp,k dT

or

∂hk ∂T = Cp,k ∂xi ∂xi

(1.124)

The constant-pressure speciﬁc heat of the i th species is a function of temperature; therefore, Cp = Cp (Yi , T )

and

h = h (Yi , T )

(1.125)

By applying the chain rule, the time derivative and spatial gradients of the enthalpy (sensible + chemical) can be written as: ∂h ∂T ∂h ∂Yi ∂h = + ∂t ∂T ∂t ∂Yi ∂t

(1.126)

∂h ∂h ∂T ∂h ∂Yk = + ∂xi ∂T ∂xi ∂Yk ∂xi

(1.127)

∂h = Cp (Yk , T ) ∂T

(1.128)

From Equation 1.123,

∂h ∂ = ∂Yk ∂Yk

N k=1

hk (T ) Yk

=

N k=1

∂hk (T ) Yk ∂Y k

=0

∂Yk + hk (T ) ∂Yk

=

N k=1

hk (T )

(1.129)

ENERGY CONSERVATION EQUATION FOR MULTICOMPONENT MIXTURE

Therefore,

37

N

∂T ∂Yk ∂h hk (T ) = Cp (Yk , T ) + ∂t ∂t ∂t

(1.130)

k=1 N

∂T ∂Yk ∂h = Cp (Yk , T ) + hk (T ) ∂xi ∂xi ∂xi

(1.131)

k=1

The material derivative of enthalpy (sensible + chemical) can be written as: Dh ∂h ∂h (1.132) = + ui Dt ∂t ∂xi Substituting Equations 1.130 and 1.131 into Equation 1.132, we have: N N ∂T ∂T ∂Yk ∂Yk Dh hk (T ) + hk (T ) = Cp (Yk , T ) + + ui Cp (Yk , T ) Dt ∂t ∂t ∂xi ∂xi k=1 k=1 N (1.133) DT DYk = Cp (Yk , T ) hk (T ) + Dt Dt k=1

Therefore, N

ρCp (Yk , T )

DT DYk Dh hk (T ) =ρ −ρ Dt Dt Dt

(1.134)

k=1

By substituting the species conservation equation, we have: N

ρCp (Yk , T )

DT Dh hk (T ) [ω˙ k − ∇ · (ρYk Vk )] =ρ − Dt Dt

(1.135)

k=1

Next, substituting the energy conservation equation Equation 1.118 into Equation 1.135, we get: N DT ∂ ∂T ∂ui Dp ∂ l hk Yk Vk,i + τji − = + ρ ρCp (Yk , T ) Dt Dt ∂xi ∂xi ∂xi ∂xj k=1 & ' N N ∂ ˙ + Q+ρ Yk fk,i Vk,i − hk (T ) ω˙ k − ρYk Vk,i ∂xi k=1

k=1

(1.136) Equation 1.136 can be simpliﬁed by the following step: N DT ∂ ∂T Dp ∂ l hk Yk Vk,i − = + ρ ρCp (Yk , T ) Dt Dt ∂xi ∂xi ∂xi + τji

+

∂ui ˙ +ρ +Q ∂xj

N k=1

hk (T )

N

k=1

Yk fk,i Vk,i −

k=1

∂ ρYk Vk,i ∂xi

N k=1

hk (T )ω˙ k

=ω˙ T

(1.137)

38

INTRODUCTION AND CONSERVATION EQUATIONS

The last term on the RHS of Equation 1.137 can be written as: N

hk (T )

k=1

N N ∂ ∂ ∂hk (T ) ρYk Vk,i ρYk Vk,i = ρYk Vk,i hk (T ) − ∂xi ∂xi ∂xi k=1

k=1

N

N ∂hk (T ) ρYk Vk,i ρYk Vk,i hk (T ) − ∂xi k=1 k=1 (1.138) Substituting Equation 1.138 into Equation 1.137, we get: N Dp DT ∂T ∂ui ∂ ˙ l +Q+ρ Yk fk,i Vk,i + τji ρCp = ω˙ T + + Dt Dt ∂xi ∂xi ∂xj k=1 (1.139) N ∂hk ρYk Vk,i − ∂xi

=

∂ ∂xi

k=1

where ω˙ T ≡ −

N

hk (T ) ω˙ k = −

k=1

N

hs,k (T ) ω˙ k −

k=1

N

h0f,k (T ) ω˙ k

(1.140)

k=1

By using Equation 1.124 and substituting it in Equation 1.139, we have: N Dp DT ∂T ∂ui ∂ ˙ l +Q+ρ Yk fk,i Vk,i + τji = ω˙ T + + ρCp Dt Dt ∂xi ∂xi ∂xj k=1 N (1.141) ∂T − ρ Yk Vk,i Cp,k ∂xi k=1

Similarly, we can show that Equation 1.141 can be written by using constantvolume speciﬁc heat: N ∂ DT ∂T ∂ui l + Q˙ + ρ Yk fk,i Vk,i + σji = ω˙ T + ρCv Dt ∂xi ∂xi ∂xj k=1 N (1.142) N Yk Vk,i ∂ ∂T − ρ Yk Vk,i Cp,k − Ru T ρ ∂xi ∂xi Mwk k=1

where ω˙ T

≡−

N k=1

ek (T ) ω˙ k = −

k=1

N k=1

es,k (T ) ω˙ k −

N

h0f,k (T ) ω˙ k

(1.143)

k=1

All 10 forms of the energy conservation equation are summarized in Table 1.12. These are most general forms of the energy conservation equations, for which the speciﬁc heats are considered temperature dependent quantities. Also, the ﬂuid is considered compressible.

ENERGY CONSERVATION EQUATION FOR MULTICOMPONENT MIXTURE

39

TABLE 1.12. Energy Conservation Equation in Various Forms In terms of energy: et

ρ

N ∂ei ∂et ∂qi ˙ + ∂σj i ui + ρ + ρui =− +Q Yk fk,i ui + Vk,i ∂t ∂xi ∂xi ∂xj

(1.144)

k=1

N

e

ρ

∂ui ∂qi De ˙ +ρ =− + σj i +Q Yk fk,i Vk,i Dt ∂xi ∂xj

(1.145)

k=1

ρ

N ∂T ∂ui Des ∂ ˙− ∂ ρ + σj i = ω˙ T + l +Q hs,k Yk Vk,i Dt ∂xi ∂xi ∂xj ∂xi k=1

es +ρ

N

(1.146)

Yk fk,i Vk,i

k=1

N ∂σij ui ∂T Detnc ∂ ∂ ˙− + ρ = ω˙ T + ρ l +Q hs,k Yk Vk,i Dt ∂xi ∂xi ∂xj ∂xi k=1

etnc

N

+ρ

(1.147)

Yk fk,i ui + Vk,i

k=1

In terms of enthalpy: N ∂ τj i ui ∂ ∂T ∂p Dhi ∂ ˙ − ρ = + ρ +Q+ l hk Yk Vk,i Dt ∂t ∂xj ∂xi ∂xi ∂xi k=1

ht +ρ

N

(1.148)

Yk fk,i ui + Vk,i

k=1

N ∂ ∂T ∂ui Dp ∂ Dh ˙ − ρ = + l hk Yk Vk,i + τj i +Q ρ Dt Dt ∂xj ∂xi ∂xi ∂xj k=1

h +ρ

N

(1.149)

Yk fk,i Vk,i

k=1

N ∂ ∂T ∂ui ∂ Dhs Dp ˙ − ρ = ω˙ T + + ρ l hs,k Yk Vk,i + τj i +Q Dt Dt ∂xj ∂xi ∂xi ∂xj k=1

hs +ρ

N

(1.150)

Yk fk,i Vk,i

k=1

N ∂τij ui ∂T ∂ Dhtnc ∂p ∂ ˙ + ρ = ω˙ T + + ρ l +Q− hs,k Yk Vk,i Dt ∂t ∂xi ∂xi ∂xi ∂xi k=1

htnc +ρ

N

Yk fk,i ui + Vk,i

(1.151)

k=1

(continued overleaf )

40

INTRODUCTION AND CONSERVATION EQUATIONS

TABLE 1.12. (continued ) Temperature ρCp T, Cp

N DT ∂T ∂ui ∂ Dp ˙ +ρ + τj i = ω˙ T + + l +Q Yk fk,i Vk,i Dt Dt ∂xi ∂xi ∂xj k=1 N ∂T − ρ Yk Vk,i Cp,k ∂xi

(1.152)

k=1

ρCv T, Cv

N DT ∂T ∂ui ∂ ˙ +ρ + σj i = ω˙ T + l +Q Yk fk,i Vk,i Dt ∂xi ∂xi ∂xj k=1 N N Yk Vk,i ∂T ∂ ρ − ρ Yk Vk,i Cp,k − Ru T ∂xi ∂xi Mwk k=1

(1.153)

k=1

In addition to the conservation equations shown in Table 1.12, there is a set of independent equations for conservation of angular momentum. In the absence of external torques, the angular momentum is automatically conserved since it can be obtained by taking the moment of the linear momentum conservation equation. If an external torque is present, the angular momentum conservation equation cannot be obtained directly just by taking the moment of the linear momentum conservation equation. Major applications of angular momentum conservation equations include polar ﬂuids in magnetic ﬁelds and combustion systems with externally applied torque. Readers interested in the derivation of the angular momentum conservation equation are referred to Yamaguchi (2008), Chap. 2.

1.14

TOTAL UNKNOWNS VERSUS GOVERNING EQUATIONS

Depending on the treatment of the diffusion velocity, the total number of unknowns and required governing equations for combustion problems in laminar ﬂows are listed in Table 1.13 and Table 1.14. TABLE 1.13. Unknowns versus Available Equations when the Fick’s Law Is Used for Diffusion Velocity Unknowns

Equations

ρ,p, T, ui = (u1 , u2 , u3 ), Yk = (Y1 , Y2 . . . . . . , YN )

1 continuity, 1 energy, 1 equation of state, 3 linear momentum, N − 1 species conservation equation, and

N k=1

Number of unknowns = N+6

Number of equations = N +6

Yk = 1

HOMEWORK PROBLEMS

41

TABLE 1.14. Unknowns versus Available Equations when the Hirschfelder-Curtiss Approximation Is Used for Diffusion Velocity Unknowns

Equations

ρ,p, T, ui = (u1 , u2 , u3 ), Yk = (Y1 , Y2 . . . . . . , YN ) ⎞ ⎛ V1,1 , V2,1 . . . . . . . . . . . . . . . .., VN,1 Vk,i = ⎝ V1,2 , V2,2 . . . . . . . . . . . . . . . .., VN,2 ⎠ V1,3 , V2,3 . . . . . . . . . . . . . . . .., VN,3 Xk = (X1 , X2 . . . . . . , XN )

1 continuity, 1 energy, 1 equation of state 3 linear momentum, N − 1 species conservation equation, N Yk = 1

Number of unknowns= 5N +6

Number of equations = 5N +6

k=1

3N diffusion equations for all chemical species, and N relationships between Xk and Yk

For turbulent reacting ﬂows, the turbulent transport equations and closure problems must be considerd. HOMEWORK PROBLEMS

1.

Show that the expression for the j th component of the correction velocity ∂ (ρYk ) Vc in the species conservation equation + ∇ · [ρYk (v + Vk + Vc )] ∂t = ω˙ k can be written as: Vc,j =

N

Dk

k=1

MWk ∂Xk Mw ∂xj

where Dk can be written in the following form, based upon the Hirschfelder and Curtiss approximation for the diffusion velocity. Dk =

1 − Yk N

Xl /Dlk

l =k

Start the problem by adopting the above equation for Dk and then substitute the diffusion velocity into the following species conservation equation: ∂ ∂ρYk [ρ (ui + Vki ) Yk ] = ω˙ k + ∂t ∂xi 2.

for

k = 1, 2, . . . , N

Make sure that you understand the equivalence of these two forms of the continuity equation: ∂ρ + ∇ · (ρv) = 0 ∂t

and

Dρ + ρ∇ · v = 0. Dt

42

INTRODUCTION AND CONSERVATION EQUATIONS

Also, express ∇ · v in terms of the density variations with respect to time in order to understand the meaning of volume dilatation. 3.

Show that the momentum equation given in vector form can be written as: ' & ' & v · v ∂v ∂v Dv − v × (∇ × v) =ρ + (v · ∇) v = ρ +∇ ρ Dt ∂t ∂t 2 = f + ∇ · σ = f − ∇p + ∇ · τ where σ is the total stress tensor, τ is the viscous stress tensor, and f is the body force. In the Cartesian coordinate, ∂τyx ∂τxy ∂τyy ∂τyz ∂τxx ∂τxz ∇ ·τ = + + ex + + + ey ∂x ∂y ∂z ∂x ∂y ∂z ∂τzy ∂τzx ∂τzz + + + ez ∂x ∂y ∂z

4.

Familiarize yourself with the following vector algebra and a set of vector identities involving del operators (∇). At the end of this list given in section A.14 of Appendix A, there are several equations associated with the Gauss divergence theorem. Make sure that you can to utilize them.

2 LAMINAR PREMIXED FLAMES

SYMBOLS

Symbol A af DkT fv Ka Ka* Kc k Le LM Ma N Ni P or p Pr PSDF QF q˙r Qm F

Description

Dimension

Arrhenius factor (for a reaction of order m) Flame area per unit mass of gas mixture Thermal diffusion coefﬁcient of the k th species Soot volume fraction Karlovitz number, deﬁned in Equation 2.142 Modiﬁed Karlovitz number, deﬁned in Equation 2.162 Equilibrium constant, see Equations 2.36 and 2.37 Speciﬁc reaction-rate constant (for a reaction of order of m) Lewis number, deﬁned in Equations 1.59 and 2.27 Markstein length, see Equations 2.158 and 2.159 Markstein number, deﬁned by Equation 2.160 Total number of chemical species Number density of soot particles formed from ith monomer units Pressure Prandtl number, deﬁned in Equations 1.58 and 2.26 Soot particle size distribution function Heating value of fuel, positive for exothermic reaction, see Equation 2.48 Radiative heat ﬂux Molar heating value of the fuel, normally positive for fuels

(N/L3 )1-m /t L2 /M M/(Lt) — — — — (N/L3 )1-m /t

Fundamentals of Turbulent and Multiphase Combustion Copyright © 2012 John Wiley & Sons, Inc.

Kenneth K. Kuo and Ragini Acharya

— L — — 1/(L3 L) F/L2 — — Q/M Q/(L2 t) Q/N

43

44

LAMINAR PREMIXED FLAMES

Symbol RRi S Sa Sc Scons Sd Sj SL Ta TSI Vc Y

Description

Dimension

Reaction rate of the ith elementary reaction, see Equation 2.34 Surface area of soot particle Absolute speed of the ﬂame surface, see Equation 2.109 Schmidt number, deﬁned in Equations 1.57 and 2.25 Consumption speed of laminar premixed ﬂame, see Equation 2.95 Displacement speed of the ﬂame surface, see Equation 2.109 Percent sensitivity of SL on speciﬁc reaction rate constant kj , see Equation 2.41 Laminar ﬂame speed against unburned mixture Activation temperature, ≡ Ea /Ru Threshold soot index, deﬁned by Equation 2.170, indicating tendency to soot Correction velocity, deﬁned in Equations 1.75 and 2.20 Reduced fuel mass fraction, deﬁned in Equation 2.59

N/(L3 t)

Greek Symbols α Dimensionless temperature ratio deﬁned by Equation 2.70 Fraction of the soot particle surface area available for αs chemical reaction β Dimensionless temperature parameter deﬁned by Equations 2.71 or 2.167 Size-dependent frequency factor βi,j δ Premixed laminar ﬂame thickness δL, B Premixed laminar ﬂame thickness, proposed by Blint δL, slope Premixed laminar ﬂame thickness based upon the temperature slope Total thickness of premixed laminar ﬂame δL, total δr Reaction zone thickness of premixed laminar ﬂame, see Equation 2.105 Rate of progress of a single-step global reaction per ε˙ v volume Reduced temperature parameter, deﬁned in Equation 2.60 κ Flame stretch factor deﬁned by Equation 2.106 Hydrodynamic length characterizing the combustor or ﬂame Kinematic viscosity of the kth species νk

L2 L/t — L/t L/t — L/t T — L/t —

— — — L3 /t L L L L L 1/(L3 t) — 1/t L L2 /t

45

SYMBOLS

Symbol νk νk νik φc χk∗ F ω˙ T

Description

Dimension

Stoichiometric coefﬁcient of the kth reactant Stoichiometric coefﬁcient of the kth product νik ≡ νik − νik for the kth species in the ith reaction Flame surface density, deﬁned by Equation 2.146 Critical equivalence ratio, above which onset of soot formation begins Dimensionless thermal diffusion ratio, deﬁned by Equation 2.22 Total fuel consumption rate per unit area of the ﬂame, see Equation 2.53 Rate of heat release due to chemical reaction, see Equation 2.45

— or N — or N — or N 1/L — — M/(L2 t) Q/(L3 t)

Subscripts a Activation b Backward reaction or Burned e Equilibrium f Forward reaction p Pressure t Total u Unburned

This chapter gives readers a quick summary of the premixed laminar ﬂame theories so that an adequate background is provided for usage in premixed turbulent ﬂame (Chapter 5). Readers are referred to Principles of Combustion (Kuo, 2005, Chap. 5), which is devoted to premixed laminar ﬂames. Another recent source of information on this topic is Theoretical and Numerical Combustion (Poinsot and Veynante, 2005). This chapter covers recent work on premixed laminar ﬂames and provides a comprehensive description of premixed ﬂamelet theory. In order to understand turbulent ﬂames, readers must have a thorough background in laminar ﬂames. Traditionally, the closure for the chemical kinetics in the turbulent ﬂames is achieved by using the probability density function approach, initially proposed by O’Brien (1980) and later advanced signiﬁcantly by Pope (1985) (see probability density function (pdf) approach in Chapter 4). To reduce the computational cost for the solution of turbulent ﬂames, another approach called ﬂamelet theory was initiated by F. A. Williams (1975) and later continued by N. Peters (1984), among others. In this theory, the chemical reactions take place in thin ﬂamelets where the turbulent effects are negligible. In these regions, laminar ﬂame concepts can be used; therein lays the signiﬁcance of studying laminar ﬂame theory. The laminar ﬂamelet approach also has become very useful in reducing the computational cost for the numerical simulation of turbulent ﬂames. This approach is discussed in detail later in this chapter.

46

LAMINAR PREMIXED FLAMES

2.1 BASIC STRUCTURE OF ONE-DIMENSIONAL PREMIXED LAMINAR FLAMES

The laminar ﬂame speed (SL ) is deﬁned as the normal velocity of the premixed reactant mixture ﬂowing into the ﬂame zone (see Figure 2.1a). Note that this is the simplest case, where the ﬂow is perpendicular to the ﬂame front. In general, the ﬂow velocity of the unburned mixture could be at any angle to the laminar ﬂame front. Figure 2.1b shows a stationary ﬂame at an oblique angle, α to the incoming stream. The thickness of the laminar ﬂame (δL ) can be taken as the sum of the preheat zone and chemical reaction zone. There is usually a small pressure difference between the unburned and burned zones. However, the mixture temperature increases substantially across the laminar ﬂame, thereby decreasing the gas density signiﬁcantly in the burned zone. In order to satisfy the continuity condition across the ﬂame, it is necessary that the gas velocity in the burned zone be much higher than SL . The general structure of a premixed laminar ﬂame is shown in Figure 2.2. The temperature rise in the reaction zone is due to the heat release process from the chemical reaction. The reactant concentration decreases through the ﬂame zone while the product concentrations increase. Some intermediate product species (certain radicals and smaller molecules) could reach their peak values in the reaction zone and reduce to nearly zero in either preheat or reaction zones. Usually the study of premixed laminar ﬂame focuses on the solution of the detailed distributions of chemical species and temperature as well as the laminar ﬂame speed. The study also involves the dependency of these parameters on the operating conditions, such as initial temperature, chamber pressure, and reactant concentration. A typical solution of the hydrogen-oxygen premixed ﬂame at 1 atm and ambient temperature condition was solved by Warnatz (1981) and is shown in Figure 2.3. In recent years, since gas-phase kinetics has been studied extensively, the solution for laminar ﬂames is generally obtained by using CHEMKIN software. The information about the chemical kinetics and the solution technique are discussed in Kuo (2005, Chaps. 2 and 5).

Unburned premixed gas (u)

Burned product gases (b) ub

Burned product gases (b) u b > SL

uu = SL

Premixed laminar flame with thickness dL x = 0 x = dL

ut,u

un,b

uu

Stationary oblique laminar flame α

uu

(a) Flow normal to the flame front

Figure 2.1

Unburned premixed gas (u) SL,u = un,u

ut,b = ut,u

(b) Flow at angle α to the flame front

One-dimensional premixed laminar ﬂame.

CONSERVATION EQUATIONS FOR 1-D PREMIXED LAMINAR FLAMES

47

T, Yi , or ω T

Tf

Yreactant

Energy release rate or mean reaction rate ω T or ω Yproduct, i

Ti

x=0

Figure 2.2

General structure of the premixed laminar ﬂame.

Xi T

1.0

x = dL

x

T, K 2000

1.0

1000

0.5

2 × H2O

2 × H2

pinitial = 1 atm, Ti = 298 K

Xi 5 × O2

1000 × HO2

0.5 20 × O 0 0.0

0.5

1.0

10 × H 20 × OH

1.5

0 0.0

0.5

x, mm

1.0

1.5

x, mm

Figure 2.3 Calculated solution of a H2 -O2 premixed laminar ﬂame (modiﬁed from Warnatz, 1981).

2.2 CONSERVATION EQUATIONS FOR ONE-DIMENSIONAL PREMIXED LAMINAR FLAMES

Solving for laminar premixed ﬂames is of interest because: • Detailed comparison between experiments and theoretical/numerical results can be performed. • Close comparison can be used for validation of proposed chemical reaction mechanism(s). • It forms a basis for studying the instabilities of ﬂame fronts. • It is one of the elementary building blocks of turbulent ﬂames; premixed laminar ﬂame is a ﬁrst step toward more complex conﬁgurations. Depending on the complexity of the chemistry and transport property considerations, there are many ways to compute the structure and speed (SL ) of

48

LAMINAR PREMIXED FLAMES

a premixed laminar ﬂame. With complex chemical reaction kinetics, numerical techniques are needed. For detailed coverage of kinetics see Principles of Combustion (Kuo, 2005, Chap. 2) or other chemical kinetics books. Conservation equations and numerical solutions of 1-D premixed laminar ﬂames are given below. Readers may want to refer to Chapter 1 for information about the conservation equations in multiple coordinate systems. • Mass conservation equation: ∂ρ ∂ + (ρu) = 0 ∂t ∂x

(2.1)

• Species conservation equation: ∂ ∂ρYk + (ρ (u + Vk ) Yk ) = ω˙ k ∂t ∂x

(2.2)

• Energy conservation equation: ρCp

∂T ∂T +u ∂t ∂x

=

ω˙ T

∂ + ∂x

∂T l ∂x

−ρ

N

Cp,k Yk Vk

k=1

∂T ∂x

small N ˙ + + Q + ρ Yk fx,k Vk k=1 Viscous No extemal

dissipation heat source small

small

(2.3)

Body force work

where

N

ω˙ T = −

k=1

and hk = hs,k +

h0f,k

=

T Tref

ω˙ k hk ,

Cp,k dT +

(2.4)

h0f,k

(2.5)

Chemical Enthalpy

Sensible Enthalpy

Note that the term ω˙ T is different from ω˙ T , which is deﬁned by ω˙ T = −

N k=1

ω˙ k h0f,k

(2.6)

Under steady-state conditions, ρu = constant = ρu SL

(2.7)

d (ρ (u + Vk ) Yk ) = ω˙ k dx

(2.8)

CONSERVATION EQUATIONS FOR 1-D PREMIXED LAMINAR FLAMES

N dT dT dT d − ρ ρCp u l Cp,k Yk Vk = ω˙ T + dx dx dx dx

49

(2.9)

k=1

In the last set of equations, there are N + 1 coupled ordinary differential equations, which should be solved. The source term in Equation 2.8 for the k th chemical species can be given by: ω˙ k = Mwk

M

vk,i

−

vk,i

Ai T

ni

i=1

Eai exp − Ru T

N j =1

Xj p Ru T

v

j,i

(2.10)

There are various levels of approximations for diffusion velocity and reaction kinetics. In real systems, the laminar ﬂames contain multiple species involved in multiple elementary reactions. It is a challenging task to address each of these terms with appropriate complexity in terms of both the concept and solution. We ﬁrst discuss the various models for modeling the species diffusion velocity.

2.2.1

Various Models for Diffusion Velocities

An excellent survey of the diffusion velocities was conducted by Hilbert et al. (2004). Historically, the importance of molecular diffusion for combustion problems was recognized as early as 1949 by Hirschfelder and Curtiss. In general, expressing diffusion velocities in a multicomponent system is not an easy task and can be very costly in terms of central processing unit (CPU) time in practical simulations. In this section, different levels of approximation currently in use are presented, starting from the more general expression down to the simplest models. 2.2.1.1

Multicomponent Diffusion Velocities (First-Order Approximation) The diffusion velocity of the k th species in the mixture can be expressed in the general form: N Xk Vk = − Dkj dj − DkT (∇T /T ) (2.11) j =1

where the vectors dj account for gradients of mole fraction and pressure as well as the difference in the body forces of various species.

ρ N dj = ∇Xj + Xj − Yj ∇p/p + Yj Yk fk − fj k=1 p

(2.12)

In Equation 2.11, Dkj is the multicomponent diffusion coefﬁcient, and it is a function of all state variables (e.g., temperature, pressure, and species concentrations). Therefore, it can be stated that Dkj is a component in the N × N matrix. The term DkT in Equation 2.11 is the thermal diffusion coefﬁcient of k th species. The last term of Equation 2.11, where the temperature gradient appears, is known

50

LAMINAR PREMIXED FLAMES

as the Soret effect (or thermodiffusion). This effect accounts for the diffusion of mass due to temperature gradients and tends to drive lighter molecules toward hotter regions and heavier molecules toward the colder regions of the ﬂow. It is often neglected in simple models because it is relatively expensive in terms of computing time. It is nevertheless known to be important, in particular for hydrogen combustion and more generally for cases where very light radicals like H or H2 play a major role (de Charentenay and Ern, 2002 Ern and Giovangigli, 1998). Soret and Dufour effects have in fact the same origin and are reciprocal. Nevertheless, all published results seem to agree that the impact of the Soret effect is considerably higher than that of the Dufour effect in combustion simulations. Equation 2.11 for the diffusion velocity requires the evaluation of the transport coefﬁcients, which are nonlinear functions of the local composition, temperature, and pressure of the mixture. This evaluation can be very expensive in terms of CPU time. Those transport coefﬁcients in multicomponent mixtures are not given explicitly by kinetic theory of gases. The Stefan–Maxwell–Boltzmann equations have been used to evaluate the multicomponent diffusion matrix, yielding a transport linear system of size N × N under general assumptions, obtained by Ern and Giovangigli (1994, 1995, 1996). These authors have shown that the diffusion matrix D can be expanded in terms of convergent series. This series yields approximate expressions for the transport coefﬁcients through truncation of higher-order terms. Different levels of approximation of multicomponent transport properties can then be found, depending on the order at which the expansion of the diffusion matrix is truncated. The advantage of this approach and of the resulting algorithms is that they keep a symmetric formalism, leading to transport linear systems of reduced size, better iterative techniques, and improved vectorization (Ern and Giovangigli, 1996). In most cases, pressure-induced diffusion is neglected and external forces act identically on all species. In this case, Equation 2.12 simply becomes: dj = ∇Xj

(2.13)

First-Order Approximation Keeping the two ﬁrst terms of the convergent series mentioned earlier for the expansion of the diffusion matrix D leads to the most accurate expression of the binary diffusion coefﬁcients (Ern and Giovangigli, 1996 Giovangigli, 1999). In this case, cross-diffusion, and Soret and Dufour effects are taken into account. A computer library called EGLIB has been developed by Ern and Giovangigli (1996) to determine the diffusion matrix with ﬁrst-order approximation. For already expensive computations (e.g., direct numerical stimulation [DNS] with detailed chemistry), the overhead in CPU time by this model seems globally acceptable (Charentenay and Ern, 2002). The developers of EGLIB even claim that, on a vector computer, this ﬁrst-order approximation runs as fast as the zeroth-order approximation. Hirschfelder-Curtiss (or Zeroth-Order) Approximation In the model of Hirschfelder, Curtiss, and Bird (1954), an approximate diffusion coefﬁcient

CONSERVATION EQUATIONS FOR 1-D PREMIXED LAMINAR FLAMES

51

for k th species against the mixture is calculated by this equation: ⎛

N

∗ = (1 − Yk ) ⎝Mw Dkm

j =1,j =k

⎞−1

Yj ⎠ Mwj Dkj

(2.14)

Another form of Equation 2.14 is: 1 − Yk

∗ = Dkm

N k=1,j =k

(2.15)

Xk /Dkj

This is based on the next relationship between Xk and Yk : Xk = Yk

Yk /Mwk Mwmixture = N Mwk (Yk /Mwk )

or Yk =

k=1

Xk Mwk N k=1

Xk Mwk

,

k = 1, 2, . . . , N (2.16)

In Equation 2.15, Dkj is the binary diffusion coefﬁcient. It depends on species pair properties, pressure, and temperature. It has been proved that this approximation is equivalent to keeping only the ﬁrst term D [0] of the series expansion of the diffusion matrix D (Ern and Giovangigli, 1994 Giovangigli, 1999), therefore, it also is called zeroth-order approximation. Note that the binary diffusion coefﬁcient Dkj is different from the elements in the diffusion matrix D. The binary diffusion coefﬁcient Dkj is determined when there are only two species present whereas the elements in the diffusion matrix D are determined when all N species are present in the mixture. This model accounts only for diagonal terms in the diffusion matrix D and does not consider cross-diffusion, Soret, and Dufour effects. Due to this approximation, mass conservation is no longer ensured, and the species equation could result in a nonphysical condition indicating that molecular diffusion could result in violation of the mass conservation equation, that is, N

Yk Vk = 0

(2.17)

k=1

To overcome this problem, the diffusion velocity of the k th species is split into two parts: (2.18) Vk = V∗k + Vc where V∗k is a predictor term and Vc a corrector term. By introducing dk = ∇Xk , the predictor term is given by: ∗ dk Xk V∗k = −Dkm

(2.19)

52

LAMINAR PREMIXED FLAMES

and the correction velocity Vc is calculated so that, if all species equations are summed, the mass conservation is recovered: Vc = −

N

Yk V∗k

(2.20)

k=1

It is also possible to add an explicit term to account for thermodiffusion, with the thermal diffusion ratio χk∗ deﬁned in such a way that the thermal diffusion ∗ velocity is given by −Dkm χk∗ ∇T T . The predictor term is then given by: ∗ ∗ dk − Dkm χk∗ Xk V∗k = −Dkm

∇T T

(2.21)

and the correction velocity is calculated by using Equation 2.20. In Equation 2.21, χk∗ is the dimensionless thermal diffusion ratio deﬁned in the following equation: (2.22) χk∗ ≡ ρ/ C 2 Mwk Mwj DkT /Dkj where C is the molar concentration of the mixture, and the thermal diffusion coefﬁcient DkT of the k th species has units of ρDkj . Note that the thermal diffusion coefﬁcient DkT is different from thermal diffusivity α ≡ l/ρCp . Thermal diffusivity is associated with heat conduction whereas the thermal diffusion coefﬁcient gives a measure of the mass diffusion process due to temperature gradient. This level of approximation is the one classically used in the TRANSPORT library of the CHEMKIN code, developed by Kee, Dixon-Lewis, Warnatz, Coltrin, and Miller (1986), and in similar programs (Mass and Warnatz, 1989). It is also available in the EGLIB package (Ern and Giovangigli, 1996). Recent work in combustion modeling uses this level of approximation but generally neglects the thermal diffusion term, which eliminates the need for determining the χk∗ values and thus increases the computational efﬁciency. Hilbert et al. (2004) determined that the consideration of thermal diffusion with the zeroth-order approximation of diffusion coefﬁcient increases the CPU time by a factor of 3 compared to the case when it is not considered. Fick’s Law Many combustion codes use a simpliﬁed approach for the expression of diffusion velocities based on a Fick’s law approximation. In this approach, the diffusion velocity of the k th species in the mixture is written in the form:

Yk Vk = −Dkm ∇Yk

(2.23)

where Dkm is the diffusion coefﬁcient of the k th species against the mixture. Note that this diffusion term is now based on a gradient of mass fraction and not the mole fraction, which corresponds to the exact diffusion equation. The major issue in using this approach is determing and deﬁning the diffusion coefﬁcient Dkm . An easy solution can be obtained while considering highly diluted ﬂames. In most

CONSERVATION EQUATIONS FOR 1-D PREMIXED LAMINAR FLAMES

53

cases, the diffusion coefﬁcient of the k th species in the mixture is approximated by the diffusion coefﬁcient of k th species in a diluent like N2 . Otherwise, explicit expressions for Dkm are required to apply the Fick’s law. For example, the diffusion coefﬁcient can be deduced by using a prescribed Lewis number, as explained in the next subsection. Apart from the relatively standard case where all values of Dkm are equal, this approach obviously does not ensure mass conservation (Poinsot and Veynante, 2005). A correction velocity is therefore required. There is no real consensus in the literature concerning how Dkm should be evaluated when Fick’s law is adopted. For many authors, it corresponds to an even simpler model in which all diffusion coefﬁcients Dkm are equal (Dkm = D) and ρD = constant. This is a very special case for the evaluation of the diffusion coefﬁcient. In this simple version, the term ρD∇Yk can be substituted directly into the species equation for the diffusion term. The species diffusion velocities are not required to be solved since the terms associated with them are substituted byρD∇Yk . In other words, the total number of unknown variables is substantially reduced. For example, the one-dimensional steady-state species equation (Equation 2.8) can be written as: d (ρuYk − ρD∇Yk ) = ω˙ k dx

(2.24)

This approximation is very popular in theoretical studies. It is very simple and has the advantage that it directly ensures mass conservation without the correction velocity. Again, if Dkm is not equal for all species, the correction velocity is needed. Constant Lewis Number Approach By using the mass, momentum, and energy transport properties (Dkm , νk , and αk ) of the k th species, three dimensionless numbers can be deﬁned

1. Schmidt number (Sck ) of the k th species is deﬁned as the ratio between viscous diffusion (sometimes called diffusion of momentum) of the mixture and molecular diffusion of the k th species: Sck ≡

ν μ = Dkm ρDkm

(2.25)

2. Prandtl number (Pr) of the mixture is deﬁned as the ratio of diffusive transport of momentum and energy: Pr ≡

μCp ν = α l

(2.26)

3. Lewis number (Lek ) of the k th species is deﬁned as the ratio of thermal diffusivity of the mixture and molecular diffusivity of the k th species: Lek ≡

α Sck l = = Dkm ρCp Dkm Pr

(2.27)

54

LAMINAR PREMIXED FLAMES

Based on the previously deﬁned dimensionless numbers, a simple and attractive approach to express the diffusion coefﬁcient Dkm is to assume that the Lewis number for the k th species is constant. This means that if the thermal diffusivity of the mixture is known, the diffusion coefﬁcients for all species can be determined by the next equation by using the prespeciﬁed Lewis number for all species: Dkm =

l

ρCp Lek

(2.28)

This approach is quite simple and allows accounting the differential diffusion effects between species, while offering the possibility to mathematically manipulate the conservation equations. If all Lek are not equal, a correction velocity must be added to ensure mass conservation. Note that this way of computing the coefﬁcients Dkm can be combined either with mass fraction gradients (standard Fick’s law, as presented earlier) or with mole fraction gradients, similar to more realistic transport models. Unity Lewis Number Approach Even simpler is the unity Lewis number approach, which is based on the assumption that all Lewis numbers Lek are constant and equal to 1 (i.e., that each species has the same diffusion coefﬁcient as the thermal diffusivity of the mixture). This assumption is very useful in simple cases, especially when only two reacting species are considered. This approach permits the analytical solutions of simple ﬂame structures. Furthermore, it is a very easy task to implement such an approach in code, and it does not increase noticeably computing times. As a consequence, most existing turbulent combustion models are based on this assumption. However, this assumption can lead to large discrepancies with respect to the local ﬂame structure and thermochemical properties when compared to more realistic transport models. All different approaches for modeling multicomponent diffusion process and the corresponding deﬁnitions of diffusion velocities are summarized in Table 2.1. 2.2.1.2 Various Models for Describing Source Terms due to Chemical Reactions In general practice, several input parameters are required for the chemically reacting system. These parameters include a set of elementary chemical reactions, the kinetic parameters of these reactions, and the thermochemical properties of chemical species. This information is called the reaction mechanism for a combustion problem. The speciﬁcation of reaction mechanism is a very important step in the solution of the ﬂame structure and ﬂame speed. In this section, various approaches to represent the chemical reactions in a reacting system are discussed. The aim is to express the chemical source terms ω˙ k in the species conservation equations.

CONSERVATION EQUATIONS FOR 1-D PREMIXED LAMINAR FLAMES

55

TABLE 2.1. Different models for Diffusion Velocities (modiﬁed from Hilbert, Tap, El-Rabii, and Th´evenin, 2004) Level of Approximation Multicomponent diffusion velocities (ﬁrst order)

HirschfelderCurtiss approximation with thermal diffusion (zeroth-order + thermal)

Diffusion Velocity

Diffusion Coefﬁcient

Xk Vk = N − Dkj dj − DkT (∇T /T )

Binary diffusion coefﬁcients are given by a ﬁrst-order approximations D [1] of the diffusion matrix Dkj . Soret effect, Dufour effect, and cross-diffusion effects are considered.

j =1

∇P with dj = ∇Xj + Xj − Yj P

ρ N Yj Yk fk − fj ; + k=1 P very often, dj = ∇Xj ∇T Xk Vk = −Dk∗ dk − Dk∗ χk∗ + Vc T N with dk = ∇Xk , Vc = Dk ∇Yk k=1

HirschfelderCurtiss approximation (zeroth-order)

Xk Vk = −Dk∗ dk + Vc

Fick’s law

Yk Vk = −Dk ∇Yk ,

with dk = ∇Xk , Vc =

k=1

Correction velocity: N Dk ∇Yk Vc = k=1

Constant Lewis number approach

Unity Lewis numbers

N

Dk = l/ ρCp Lek

Dk = αth,k = l/ρCp

Dk ∇Yk

Approximate diffusion coefﬁcient Dk∗ of k th species in the mixture corresponds to the zeroth-order approximation D [0] of the diffusion matrix Dkj . A correction velocity Vc is needed to ensure mass conservation. Approximate diffusion coefﬁcient Dk∗ of k th species in the mixture corresponds to the zeroth-order approximation of the diffusion matrix Dkj . A correction velocity Vc is needed to ensure mass conservation. One diffusion coefﬁcient Dk for k th species. A correction velocity Vc is needed to ensure mass conservation if Dk = constant. Each species can have a separate assigned value of Lewis number; mass diffusion coefﬁcients are related to thermal diffusivity of the mixture. All species have the same diffusivity as thermal diffusivity, Lek = Le = 1.

56

LAMINAR PREMIXED FLAMES

All chemical models used in combustion share the same description of elementary chemical reactions, based on an Arrhenius law, leading to a rate coefﬁcient expressed as: Ea n (2.29) k = AT exp − Ru T where A is the pre-exponential factor, Ea is the activation energy, Ru is the universal gas constant (Ru = 8.3144 J/mole-K), n is the temperature exponent, and T is the temperature. The values of A, Ea , and n are speciﬁed for each elementary reaction separately. Based on this expression, different levels of approximation can be deﬁned to describe the kinetics. Comprehensive Reaction Scheme In the comprehensive reaction scheme, one takes into account all the single-step reactions involving all the chemical species present in the conﬁguration of interest in order to build a reaction mechanism (see, e.g., Lindstedt, 1998, Warnatz, 1992; Warnatz, Maas, and Dibble, 1996). Consider a complete chemical system composed of N species reacting through Nr reversible elementary equations. Each elementary reaction can be written in the form: kfi N N vik Mk −→ vik Mk (for i = 1 . . . Nr ) (2.30) ←− k=1

kbi

k=1

where Mk denotes the k th species. For each i th reaction, the following relation between molar stoichiometric coefﬁcients is veriﬁed (see Kuo, 2005, Chaps. 1, 2), corresponding to the global mass conservation by each elementary reaction: N

vik Mwk =

N

vik Mwk

(2.31)

(for i = 1 . . . Nr )

(2.32)

k=1

k=1

which can also be written: N

vik Mwk = 0

k=1

where

vik ≡ vik − vik

(2.33)

The stoichiometric coefﬁcients also satisfy linear relations associated with the conservation of all individual chemical elements. The reaction rate of the i th elementary reaction (RRi ) is deﬁned as: RRi = kfi

N k=1

v

Ck ik − kbi

N k=1

v

Ck ik

(2.34)

CONSERVATION EQUATIONS FOR 1-D PREMIXED LAMINAR FLAMES

57

where Ck denotes the molar concentration of the k th species and the forward and backward constants (kfi , kbi ) of the i th elementary reaction are expressed via a semiempirical Arrhenius law in the form: Eai ni ki = Ai T exp − (2.35) Ru T The forward and backward constants of the reaction are linked by the equilibrium constant KCi : kf (2.36) KCi ≡ i kbi Thermochemical analysis gives the next expression for the equilibrium constant: N Hi0 Si0 pk k=1 vki K Ci = (2.37) exp − Ru T Ru Ru T The parameters Si0 and Hi0 correspond to entropy and enthalpy changes during the transition from reactants to products for an i th elementary reaction, respectively. The term pk corresponds to the partial pressure of k th species in the mixture. These quantities are obtained from tabulations based on experimental measurements. The mass production rate of the k th species is the sum of all contributions from all the elementary reactions: ω˙ k = Mwk

Nr

vki (RR)i

(2.38)

i=1

If there is net consumption of the k th species in the reacting system, then ω˙ k is a negative number. In a reacting system, some species are consumed and others are produced, but the net production or consumption of total mass should be equal to zero. From the conservation of total mass, it can be stated that: N

ω˙ k = 0

(2.39)

k=1

For numerical simulations of reacting ﬂows, a chemical reaction scheme must be speciﬁed. This implies that all chemical species involved in all elementary reactions should be known along with their respective Arrhenius parameters before the computation can be performed. In the combustion community, the CHEMKIN format (Kee, Rupley, and Miller, 1989) has become a practical standard. In this formulation, reactions are listed using a prescribed format, along with the values for Ai (in CGS units), ni and Ea,i (in cal/mol). Backward rates are computed using Equations. 2.36 and 2.37. More complex expressions—as for example, Lindemann or Troe formulation (Warnatz, Maas, and Dibble, 1996),

58

LAMINAR PREMIXED FLAMES

may be found in particular to describe the pressure dependency of rate coefﬁcients (such as fall-off curves, Kuo, 2005, Chap. 2). For example, the use of a comprehensive kinetic scheme to describe the chemistry requires about 8 species and 40 irreversible elementary reactions for hydrogen/oxygen combustion and around 50 species and a few hundred chemical reactions for methane/air combustion. More complex fuels, such as like n-decane or cetane, require several hundred species and several thousand elementary reactions. As stated before, ideally a transport equation should be solved for each of these species in order to accurately describe the physical and chemical processes occurring during combustion. It is thus clear why very few complete reaction mechanisms have been used in the numerical simulations of the reacting ﬂows. The computing costs and memory requirements are huge, and it is almost impossible to use such complete mechanisms for multidimensional simulations. The exceptions are reactive processes involving ozone, which can be described with very few species but are not of wide interest for the combustion community. Some publications use complete hydrogen/oxygen reaction schemes, since hydrogen/oxygen reaction is the simplest reacting system, involving “only” 8 species. In case the mixture contains N other chemical species and reactions must be considered in order to describe NOx production. Note that H2 /O2 reaction scheme are generally useful in investigating some practical ﬂames since they are part of the overall reaction scheme (see Kuo, 2005, Chapter 2, Fig. 2.30). Syngas (CO/H2 mixture) combustion has also been considered, since it involves less than 20 species for an almost complete description. Scientists and engineers who are working with practical applications would like to work at least with methane and more often with natural gas or n-decane, which are complex compounds involving approximately 100 to 1,000 intermediate chemical species. In such cases, complete reaction mechanisms are impossible to use for turbulent reacting ﬂows, since even the one-dimensional simulations are extremely demanding on the CPU time. Several techniques have been developed to reduce complete mechanisms to a simpler subset. Following the review article by Hilbert et al. (2004), four categories of chemistry models having different levels of complexity are deﬁned. 1. Complete mechanism (or detailed reaction mechanism). This category corresponds to the case where all kinetic processes have been taken into account. This is a demanding task, and there are considerable uncertainties in this case. This name will be used when no simpliﬁcation has been carried out on purpose and when researchers have done their best to consider all existing reactions. For the simulation of turbulent ﬂames discussed in later chapters, detailed reaction mechanism will typically be limited to ozone, hydrogen, CO, and some methane combustion. 2. Reduced mechanism. Starting from a complete mechanism, reduced mechanisms can be derived. Reduced mechanisms correspond to the case where researchers purposely reduce the complexity of the chemical reactions

CONSERVATION EQUATIONS FOR 1-D PREMIXED LAMINAR FLAMES

59

considered in the complete mechanism, while retaining the main reaction pathways corresponding to the speciﬁc operating conditions. Sensitivity analysis (Kuo, 2005, Chap. 2) has been used extensively to determine reduced reaction mechanism from a complete reaction mechanism (Warnatz, Maas, and Dibble, 1996). Typically, reduced mechanisms still consider between 20% to 50% of the species that are considered by the complete mechanism. Usually reduced mechanisms consider 5 to 20 species as seen in combustion literature. Studies relying on reduced reaction mechanisms do not (and cannot) claim a perfect quantitative accuracy, because reduced reaction mechanisms always correspond to an explicit simpliﬁcation of the chemical processes. Reduced reaction mechanisms are sometimes called skeletal mechanisms in the literature. This level always corresponds to a case where chemical processes still are directly described by a set of elementary reactions, which corresponds to a strongly reduced mechanism. In this sense, skeletal mechanisms constitute the lower limit of reduced mechanisms. 3. Semiglobal mechanism. If the skeletal reaction mechanism is further simpliﬁed, a semiglobal mechanism is obtained. The semiglobal reaction mechanism typically considers less than 5 reactions involving 5 to 10 species and neglects most chemical pathways. Studies relying on semiglobal mechanisms are useful to produce qualitative trends and still take into account the existence of some important intermediate radicals. Semiglobal mechanisms often lead to solving complementary nonlinear equations associated with steady-state and/or partial equilibrium assumptions (see Table 2.2). 4. Single-step mechanism. Finally, a further reduction step leads to single-step mechanisms, where none of the intermediate radicals is considered. Singlestep mechanisms are used mainly for obtaining analytical solutions and are less common in numerical studies. This classiﬁcation is useful but should not be considered universal. In fact, the boundaries between the different categories are not perfectly clear-cut. As mentioned, few simulations of turbulent ﬂames rely on complete reaction mechanisms. The main techniques employed to reduce a complete mechanism and yield a detailed, or ultimately a semiglobal or single-step mechanism, are now listed. Recently, several other techniques have been developed to reduce the computational cost for solving the ordinary differential equations resulting from the reactions. Some of these techniques are: computational singular perturbation (CSP) by Goussis and Lam (1992), Lam and Goussis (1994), repro-modeling (Tur´anyi, 1994), in-situ adaptive tabulation (ISAT) (Pope, 1997), piecewise reusable implementation of solution mapping (PRISM) (Bell et al., 2000), and artiﬁcial neuralnetworks (ANN) (J. Chen et al., 2000). Example of One-Dimensional Premixed H2 /O2 Laminar Flame Solution In general practice, several input parameters are required for the chemically reacting system.

60

LAMINAR PREMIXED FLAMES

TABLE 2.2. Comparison of Various Techniques for Reducing Chemical Reaction Schemes Classical Techniques

Recent Techniques

Quasi-Steady state analysis (QSSA)

Computational The basic strategy is to singular separate chemical time perturbation scales into slow and fast (CSP) method groups. (Goussis and The effects of various Lam, 1992, species and elementary Lam and reactions on the rates of Goussis, 1994) change of species mass fraction and temperature are continuously monitored by massively parallel computer programs to achieve optimum representation of the reacting system. No a-prior; elimination of species or elementary reaction is required. Intrinsic lowIntrinsic low-dimensional dimensional manifolds (mathematical manifolds spaces) are identiﬁed in (ILDM) the composition space. method (Maas These manifolds and Pope, correspond to a 1992) description of the complete reaction system by a much smaller number of coordinates. After tabulation, these manifolds are used in the reacting ﬂow simulation, where only the coordinates and not all the chemical species have to be solved for. All the intermediate species are still available throughout the computation by using look-up tables.

The net production or consumption rate of very reactive species is equal to 0, (i.e., ω˙ I,k = 0). By using above assumption, concentrations of these intermediate species can be written in terms of other major species. A detailed example of this method is given in Kuo (2005, Chap. 2).

Partial The reaction rates of a equilibrium subset of forward analysis and backward elementary reaction rates are assumed to be extremely fast. Algebraic relationship can be derived for the concentration of the species in these reactions. This assumption is more applicable for relatively high-temperature cases.

CONSERVATION EQUATIONS FOR 1-D PREMIXED LAMINAR FLAMES

61

TABLE 2.2. (continued ) Classical Techniques

Recent Techniques

Sensitivity analysis

Rate-controlled constrained equilibrium (RCCE) method (Youseﬁan, 1998)

Rate-limiting steps are identiﬁed by perturbing input kinetic parameters to check if the ﬁnal product concentrations are affected by this exercise. The information gained by this procedure is used to identify and eliminate less important reactions and generate a reduced mechanism.

An alternative method sharing ideas with both CSP and ILDM methods.

These parameters include a set of elementary chemical reactions, the kinetic parameters of these reactions, and the thermochemical properties of chemical species. This information is called the reaction mechanism for a combustion problem. The speciﬁcation of reaction mechanism is a very important step in the solution of the ﬂame structure and ﬂame speed. For premixed H2 /O2 ﬂame, a reaction mechanism proposed by O’Conaire, Curran, Simmie and Pitz, (2004) is shown in Table 2.3. This reaction mechanism consists of 19 reversible elementary reactions, as listed in the table. The heats of formation, entropy values, and temperature dependent speciﬁc heat capacity of all chemical species involved in the reaction system are listed in Table 2.4. A popular computer code for the numerical simulation of steady, onedimensional, laminar premixed ﬂame is called Premix (Kee, Grear, Smooke, and Miller, 1985). O’Conaire et al. (2004) used the Premix code to model steady, adiabatic freely propagating (expanding spherical) ﬂame speeds and to determine concentration proﬁles of major and intermediate species in a burner-stabilized ﬂame. The transport properties were determined by using the standard Chemkin transport package with the inclusion of thermal diffusion coefﬁcients. Mixture-averaged transport properties were employed. Several other modeling research groups, such as the Resources Research Institute and the University of Leed prefer to use the multicomponent transport option. Lawrence Livermore National Laboratories researchers use the mixture-averaged transport properties. A useful reference to determine the thermophysical and transport properties of

62

a

14

10 11 12 13

9

h

f ,g

5 6b 7c 8d ,e

1 2 3 4 0.00 2.67 1.51 2.02 −1.40 −0.50 −1.00 −2.00 −0.41 0.60 0.00 0.00 0.00 0.00 0.00 0.00

H2 /O2 dissociation/recombination reactions ˙ +H ˙ +M H2 + M = H 4.57 × 1019 ˙ ˙ 6.17 × 1015 O + O + M = O2 + M ˙ +H ˙ + M = OH ˙ +M O 4.72 × 1018 ˙ + OH + M = H2 O + M H 4.50 × 1022

˙2 Formation and consumption of HO ˙ ˙ H + O2 + M = HO2 + M 3.48 × 1016 ˙ ˙ H + O2 = HO2 1.48 × 1012 ˙ ˙ HO2 + H = H2 + O2 1.66 × 1013 ˙2 +H ˙ + OH ˙ ˙ = OH HO 7.08 × 1013 ˙2 +O ˙ = OH ˙ + O2 HO 3.25 × 1013 ˙ = H2 O + O 2 ˙ 2 + OH 2.89 × 1013 HO

Formation and consumption of H2 O2 ˙ 2 + HO ˙ 2 = H2 O2 + O2 HO 4.2 × 1014 ˙ 2 + HO ˙ 2 = H2 O2 + O2 HO 1.3 × 1011

n 1014 104 108 106

× × × ×

H2 /O2 chain reactions ˙ + OH ˙ ˙ + O2 = O H ˙ ˙ ˙ O + H2 = H + OH ˙ + H2 = H ˙ + H2 O OH ˙ + H2 O = OH ˙ + OH ˙ O

1.91 5.08 2.16 2.97

A

Reaction

11.98 −1.629

−1.12 0.00 0.82 0.30 0.00 −0.50

105.1 0.00 0.00 0.00

16.44 6.292 3.43 13.4

Ea

and and and and

Hampson Hampson Hampson Hampson

1986 1986 1986 19861

Hippler, Troe, and Willner, 1993 Hippler, Troe, and Willner, 1993

Mueller and Schefer 1998 Cobos, Hippler, and Troe, 1985 Mueller, Kim, Yetter, and Dryer, 1999 Mueller, Kim, Yetter, and Dryer, 1999 Baulch et al., 1994 Baulch et al., 1994

Tsang Tsang Tsang Tsang

Pirraglia, Michael, Sutherland, and Klemm, 1989 Sutherland et al., 1986 Michael and Sutherland 1988 Sutherland, Patterson, and Klemm, 1990

Ref.

TABLE 2.3. H2 /O2 Reaction Mechanism of O’Conaire et al. (2004) (units: cm3 , mol, s, kcal, K)

63

˙ + OH ˙ +M H2 O2 + M = OH ˙ ˙ H2 O2 = OH + OH ˙ ˙ = H2 O + OH H2 O2 + H ˙2 ˙ = H2 + HO H2 O2 + H ˙ = OH ˙ + HO ˙2 H2 O2 + O ˙ = H2 O + H O ˙2 H2 O2 + OH ˙ = H2 O + H O ˙2 H2 O2 + OH 1.27 2.95 2.41 6.03 9.55 1.0 5.8

× × × × × × × 1017 1014 1013 1013 106 1012 1014 0.00 0.00 0.00 0.00 2.00 0.00 0.00

45.5 48.4 3.97 7.95 3.97 0.00 9.56

Warnatz et al., 1985 Brouwer et al., 1985 Tsang and Hampson, 1986 Tsang and Hampson, 19862 Tsang and Hampson, 1986 Hippler and Troe 1992 Hippler and Troe 1992

b

Efﬁciency factors are H2 O = 12.0; H2 = 2.5. Efﬁciency factors are H2 O = 12; H2 = 2.5; Ar = 0.83; He = 0.83. c Efﬁciency factors are H2 O = 12; H2 = 2.5; Ar = 0.75; He = 0.75. d Original pre-exponential A factor is multiplied by 2 here. e Efﬁciency factors are H2 O = 12; H2 = 0.73; Ar = 0.38; He = 0.38. f Troe parameters: reaction 9, a = 0.5, T*** = 1.0 × 10−30 , T* = 1.0 × 10+30 , T** = 1.0 × 10+100 ; reaction 15, a = 0.5, T*** = 1.0 × 10−30 , T* = 1.0 × 10+30 . g Efﬁciency factors are H2 = 1.3; H2 O = 14; Ar = 0.67; He = 0.67. h Reactions 14 and 19 are expressed as the sum of the two rate expressions. i Efﬁciency factors are H2 O = 12; H2 = 2.5; Ar = 0.45; He = 0.45; 1 The pre-exponential factor A reported here is the result of original value multiplied by 2.0 (uncertainty factor). 2 The pre-exponential factor A reported here is the result of original value multiplied by 1.25 (uncertainty factor).

a

16 17 18 19h

15i ,f

64

LAMINAR PREMIXED FLAMES

TABLE 2.4. Heat of Formation @298.15 K (Kcal/mol), Entropy @ 300 K, and Constant Pressure Speciﬁc Heat as a Function of temperature (cal/mol-K) Source: From O’Conaire, Curran, Simmie, and Pitz, (2004).

Speciﬁc heat capacity, C p Species

H298 f

˙ H ˙ O ˙ OH H2 O2 H2 O ˙2 HO H2 O2 N2 Ar He

52.098 59.56 8.91 0.00 0.00 −57.77 3.00 −32.53 0.00 0.00 0.00

K

S 300 K

300 K

400 K

500 K

800 K

27.422 38.500 43.933 31.256 49.050 45.154 54.809 55.724 45.900 37.000 30.120

4.968 5.232 6.947 6.902 7.010 8.000 8.349 10.416 6.820 4.900 4.970

4.968 5.139 6.992 6.960 7.220 8.231 8.886 11.446 7.110 4.900 4.970

4.968 5.080 7.036 6.997 7.437 8.446 9.465 12.346 7.520 4.900 4.970

4.968 5.016 7.199 7.070 8.068 9.223 10.772 14.294 7.770 4.900 4.970

1000 K 1500 K 4.968 4.999 7.341 7.209 8.350 9.875 11.380 15.213 8.280 4.900 4.970

4.968 4.982 7.827 7.733 8.721 11.258 12.484 16.851 8.620 4.900 4.970

a number of gaseous and liquid species is http://webbook.nist.gov/chemistry/ ﬂuid/. Readers can also refer to Kuo (2005, Appendix A) for methods for evaluating the thermal and transport properties of individual species and mixtures in terms of baseline method, high-pressure correction, and mixing rules. The calculated ﬂame speeds by O’Conaire et al. (2004) for steady, one-dimensional, laminar premixed hydrogen-oxygen-air system are shown in Figure 2.4. In addition to the reaction mechanism proposed by O’Conaire et al. (2004), the ﬂame speeds were also calculated for reaction mechanisms proposed by each of the four other research groups. These groups include Mueller et al. (1999); the University of Leeds’ methane oxidation mechanism, version 0.5; the Gas Research Institute’s mechanism, version 3.0 (GRI-Mech 3.0); and Konnov’s mechanism, version 0.5, 2000. The methane oxidation mechanism includes a hydrogen oxidation mechanism since hydrogen is the simplest component in the hierarchical structure of the reaction mechanisms (Kuo, 2005, Chap. 2). This comparison shows that these reaction mechanisms are very different from each other in both the number of elementary reactions and the reaction rate constant expressions. As readers can see from Figure 2.4, the maximum ﬂame speed occurs at equivalence ratio greater than 1 (fuel-rich mixture). The major combustion products dissociate into the simpler molecules, and additional fuel is required to react with the products of dissociation reactions. Thus, the maximum reaction rate occurs when the mixture is fuel-rich, thereby leading to occurrence of maximum ﬂame speed at equivalence ratio higher than 1. Saxena and Williams (2006) have also tested a reaction kinetic scheme (30-step mechanism with 11 species) for combustion of hydrogen and carbon monoxide with oxygen in a laminar premixed ﬂame. Their calculated laminar burning speeds also have shown close agreement with the measured data from various groups. Readers are encouraged to refer to the original paper for more details on reaction mechanism and numerical procedure.

CONSERVATION EQUATIONS FOR 1-D PREMIXED LAMINAR FLAMES

p = 1 atm T0 = 298 K

3 Frame speed, SL, m/s

65

2

1

1

2 3 4 Equivalence ratio, (φ)

5

Figure 2.4 Comparison of calculated H2 /O2 /air ﬂame speeds versus equivalence ratio with the measured data, p = 1 atm, T o = 298 K ( Takahashi Mizomoto, and Ikai, 1983; Tse, Zhu, and Law, 2000; Dowdy, Smith, and Taylor, 1990; + Aung, Hassan, and Faeth, 1997; Iijima and Takeno, 1986; —— O’Conaire et al., 2004; -·-·-·- Mueller et al., 1999, ------ Leeds version 0.5; ······ GRI-Mech 3.0; –··–··– Konnov version 0.5 (modiﬁed from O’Conaire et al., 2004).

Vandooren and Bian (1990) performed a set of premixed H2 /O2 /Ar ﬂat ﬂame burner experiments and measured species concentrations as a function of distance above the burner at an equivalence ratio of 1.91 and a pressure of 35.5 Torr. O’Conaire et al. (2004) used the experimental ﬂame temperature proﬁle measured by Vandooren and Bian (1990) to model the steady one-dimensional laminar premixed ﬂame. The comparison of experimental data and the calculated results by using O’Conaire et al.’s reaction mechanism is shown in Figure 2.5. There is reasonable agreement between the calculated and measured major species distribution proﬁles, especially in the region away from the burner surface. However, there is some discrepancy between the calculated and experimental oxygen proﬁle in the preheating zone of the ﬂame front close to the burner. The maximum concentrations of water in both measured and calculated results are identical. The comparison of the measured intermediate species, such as H, O, and OH concentration proﬁles, with those predicted by O’Conaire et al. (2004) shows that the calculated radical concentrations are higher than the measured values. The computed proﬁle for the intermediate species shows similar distributions as the measured data, but the concentrations are overestimated by up to a factor of 2 for H and OH radicals in the ﬂame and postﬂame zones and by a factor of 5 for the H atom in the same region. O’Conaire et al. (2004) suggested that the discrepancy between the measured and calculated OH proﬁles with double peaks in the OH proﬁle near the burner surface could be attributed to the radical quenching effect in the vicinity of the burner surface. Their reaction mechanism did not consider the radical quenching effect encountered in the burner-stabilized hydrogen ﬂames.

LAMINAR PREMIXED FLAMES

0.1

p = 35.5 torr

0.3

Species mole fraction

Species mole fraction

66

H2O 0.2

H2

0.1 O2 0.0 0.0

0.5 1.0 1.5 2.0 Distance above burner (cm) (a) Major species H2, O2, H2O

H

p = 35.5 torr

0.01 O 1E−3 OH 1E−4 0.0

2.5

0.5 1.0 1.5 2.0 Distance above burner (cm) (b) Minor species H, O, OH

2.5

Figure 2.5 Comparison of calculated species proﬁles in an 39.7% H2 + 10.3% O2 + balance Ar at 39.9 torr burner-stabilized ﬂame with the experimental data of Vandooren and Bian (1990). (modiﬁed from O’Conaire et al. (2004)).

2.2.2

Sensitivity Analysis

Sensitivity analysis is a way to understand quantitatively how the solution to a model depends on parameters in the model (Kuo, 2005, Chap. 2). The solution to the model could refer to laminar ﬂame speed, ﬂame temperature, species mole fraction, and so on. The sensitivity analysis techniques have been developed over the years by Stewart and Sorenson (1976) and Saito and Scriven (1981). Sensitivity analysis is a very useful tool in interpreting the results of a ﬂame model and helps in identifying the key mechanisms of a reacting system. Generally, the sensitivity of unknown variables χi with respect to the parameters pj is calculated by a Jacobian Sij shown by Equation 2.40. Sij ≡

∂ ln χi ∂ ln pj

(2.40)

Equation 2.40 is written in a general form of an i × j matrix. It will be a vector if users wish to know the sensitivity of one single unknown parameter with respect to the parameters of interest. For example, the sensitivity of laminar ﬂame speed SL with respect the speciﬁc reaction rate constants kj can be written as: ∂ ln SL Sj ≡ (2.41) ∂ ln kj O’Conaire et al. (2004) carried out a sensitivity analysis of their H2 /O2 reaction mechanism for the laminar ﬂame speed of a freely propagating ﬂame for equivalence ratios ranging from 1.0 to 3.0. This was accomplished by multiplying the rate constant of each reaction by 2 and calculating the new “perturbed” ﬂame speed for the kinetic mechanism perturbed by the reaction in question. Their results of sensitivity analysis are shown in Figures 2.6 and 2.7 for

67

CONSERVATION EQUATIONS FOR 1-D PREMIXED LAMINAR FLAMES

H + O2 = O + OH O + H2 = H + OH OH + H2 = H + H2O H2 + M = H + H + M H + OH + M = H2O + M H + O2 + M = HO2 + M HO2 + H = H2 + O2 HO2 + H = OH + OH

φ = 1.0

HO2 + OH = H2O + O2

φ = 1.4 φ = 3.0 −15

−10

−5 Percent Sensitivity

0

5

Figure 2.6 Flame speed sensitivity analysis of a freely propagating H2 /O2 /Air ﬂame at 1 atm (modiﬁed from O’Conaire et al., 2004).

H + O2 = O + OH OH + H2 = H + H2O H2 + M = H + H + M H + OH + M = H2O + M HO2 + H = H2 + O2 HO2 + H = OH + OH

φ = 1.0

HO2 + OH = H2O + O2

φ = 1.5 φ = 2.5 −15

−10

−5 Percent Sensitivity

0

5

Figure 2.7 Flame speed sensitivity analysis of a freely propagating H2 /O2 /Air ﬂame at 5 atm (modiﬁed from O’Conaire et al., 2004).

two different pressure levels. The term “percent sensitivity” is same as Sj given by Equation 2.41 and is also referred to as the normalized sensitivity. The physical meaning of Sj is the percentage change in ﬂame speed with respect to percentage change in j th speciﬁc reaction rate constant. As shown in Table 2.3, the H2 /O2 /M reaction mechanism proposed by O’Conaire et al. (2004) consists of 19 reversible elementary reactions. They

68

LAMINAR PREMIXED FLAMES

performed sensitivity analysis of each of the 19 H2 /O2 reactions—not only for the freely propagating ﬂame but also for comparing their results with the shock tube and ﬂow reactor experimental data. Based on their study, they identiﬁed that the solution to the ﬂame model is sensitive with respect to only these nine reactions. With reaction number corresponding to these in Table 2.3. 1. 2. 3. 4. 8. 9. 10. 11. 17.

H + O2 = O + OH O + H2 = H + OH OH + H2 = H + H2 O O + H2 O = OH + OH H + OH + M = H2 O + M H + O2 + M = HO2 + M HO2 + H = H2 + O2 HO2 + H = OH + OH H2 O2 + H = H2 + HO2

It can be seen from Figures 2.6 and 2.7 that the percent sensitivities for different elementary reactions could have either positive or negative values. The absolute value of the percent sensitivity represents the degree of inﬂuence of that speciﬁc reaction on the laminar ﬂame speed. The negative percent sensitivity means that any increase in the speciﬁc reaction rate constant could reduce the laminar ﬂame speed, and vice versa. It is also worthwhile to note that the importance of a particular elementary reaction could change with pressure. For example, the elementary reaction O + H2 = H + OH shows sensitivity to ﬂame speed at 1 atm pressure but does not exhibit the same behavior at 5 atm pressure in stoichiometric and fuel-rich conditions. Similarly, the reaction H + O2 + M = HO2 + M is not sensitive to ﬂame speed at 5 atm. However, the reaction H + O2 + M = HO2 + M plays an important role in the explosion diagram in a vessel for H2 /O2 reaction system (Kuo, 2005, Chap. 2).

2.3 ANALYTICAL RELATIONSHIPS FOR PREMIXED LAMINAR FLAMES WITH A GLOBAL REACTION

Even though today’s trend in studying combustion problems is to incorporate complex chemical reactions with detailed considerations of transport properties, there are several reasons to address the analytical relationships associated with premixed laminar ﬂames based on the consideration of their global reactions. The major reasons are given next. • Understanding the interdependency among various parameters in a premixed laminar ﬂame is useful from the point of view of theoretical concepts. At the beginners’ level, solution of a simpliﬁed problem with only a one-step

ANALYTICAL RELATIONSHIPS FOR PREMIXED LAMINAR FLAMES

69

forward reaction could prove to be a good exercise for gaining an insight into the fundamental understanding. The one-dimensional steady-state premixed laminar ﬂame is still a complex problem due to inclusion of multiple species and multiple reactions, with all species having different thermal and transport properties. Addressing the reaction chemistry and diffusion process for each of these species proves to be a signiﬁcant effort. • Sometimes when a chemically reacting ﬂow ﬁeld is very complex, treating every aspect is impossible. The detailed chemical reactions could be avoided by considering the local heat release by the global reaction across the ﬂame. In this case, more effort could be placed on the ﬂow structure simulation. Of course, the validity of this simpliﬁcation depends the relative time scales of chemistry and ﬂow-ﬁeld development. • Functional relationships between the operating parameters, such as initial temperature, reactant concentration, and pressure, and solution parameters, such as laminar ﬂame speed, adiabatic ﬂame temperature, and so on, can be established readily. These relationships can provide a fundamental knowledge of the combustion systems. The next theoretical analysis follows F. A. Williams (2005) and Poinsot and Veynante (2005). A steady one-dimensional premixed laminar ﬂame model can be simpliﬁed by making four assumptions: 1. There is only one irreversible reaction. 2. All species have same constant speciﬁc heat. 3. All species have same constant mass diffusivity so that all Lewis numbers are equal. 4. The Lewis number is equal to 1. By using these four assumptions, the source terms related to the reaction chemistry can be derived by the next procedure. The single step irreversible global reaction can be represented by: N i=1

kf

νi Mi −→

N

νi Mi

(2.42)

i=1

Derivation of one-step forward chemistry conservation equations is given next. Recall the speciﬁc reaction rate can be written as: β

kf = AT β exp (−Ea /Ru T ) = AT exp (−Ta /T )

(2.43)

The rate of reaction of i th species can be expressed as:

ω˙ i = νi − νi Mwi ε˙ V = νi Mwi ε˙ V

(2.44)

70

LAMINAR PREMIXED FLAMES

In Equation 2.44, ε˙ V represents the progress rate of the single-step global reaction per unit volume. The rate of heat release term ω˙ T in the energy equation in terms of temperature becomes: ⎛ ⎞ ω˙ T = −

Q L3 t

N

hof,i ω˙ i i=1 M Q L3 t M

⎜ ⎟ N ⎜ ⎟ ⎜ o ⎟ = − ε˙ V ⎜hf,i Mwi νi ⎟ ⎜ ⎟ [N] ⎠ 1 i=1 ⎝ M Q M

L3 t

(2.45)

N

where ω˙ T is positive number for exothermic reactions. In combustion problems, the overall objective is to produce heat. Therefore, ω˙ T also can be written in terms of the molar heating value of the fuel (Qm F , which logically should be a positive number). This is shown in Equation 2.46. ω˙ T = |νF | Qm ε˙ V F

Q L3 t

[N]

Q N

1 L3 t

(2.46)

In the combustion process, the fuel is always consumed so that νF = νF − νF and ω˙ F are always negative quantities. By comparing Equations 2.45 and 2.46, we have: |νF | Qm F =−

N

hof,i Mwi νi or Qm F =−

i=1

N

N

νi o νi hof,i Mwi Hf,i = |νF | νF

i=1 i=1 o =Hf,i

(2.47) The heating value of fuel per 1 kg of fuel QF (positive value for exothermic reaction see Table 2.5) is deﬁned as: N Qm νj o F QF = = Hf,j MwF MwF νF

(2.48)

j =1

TABLE 2.5. Values of Molar (QM F ) and Mass-Based (QF ) Heating Values for Some Fuels and Values of Stoichiometric Ratio(s) Source: Modiﬁed From Poinsot and Veynante (2005).

Global reaction CH4 + 2O2 → CO2 + 2H2 O C3 H8 + 5O2 → 3CO2 + 4H2 O 2C8 H18 + 25O2 → 16CO2 + 18H2 O 2H2 + O2 → 2H2 O

Qm F (kJ/Mole)

QF (kJ/kg)

802 2060 5225 241

50100 46600 45800 120500

s=

νO MwO νF MwF 4.00 3.63 3.51 8.00

YFST 0.055 0.060 0.062 0.028

71

ANALYTICAL RELATIONSHIPS FOR PREMIXED LAMINAR FLAMES

The heat release source term ω˙ T is linked to the fuel consumption rate ω˙ F as: ω˙ T = − QF ω˙ F (+)

(+)

(2.49)

(−)

The governing equations for the 1-D steady laminar premixed ﬂames were shown in Equations. 2.7 to 2.9. By using the assumptions described in this section, those equations can be written as: ρu = constant = ρ1 u1 = ρu SL dYF dYF d ρD + ω˙ F ρu SL = dx dx dx

(2.50) (2.51)

=ρu

dT dT d − QF ω˙ F l ρu SL Cp = dx dx dx =ρu

(+)

(2.52)

(−)

By integrating Equations 2.50 to 2.52 from x = −∞ to x = +∞, we get: ρu SL YF,u = − or

∞

ω˙ F dx −∞ (−)

= F

(2.53)

(+)

SL = F / ρu YF,u

(2.54)

In Equation 2.54, F is the total fuel consumption rate per unit area of the ﬂame. Similarly, the overall energy balance can be obtained by integrating the energy equation: ρu Cp SL (Tb − Tu ) = −QF

∞

−∞

ω˙ F dx = QF F

(2.55)

By eliminating F from Equations 2.54 and 2.55, we have: Cp (Tb − Tu ) = QF YF,u

or

Tb = Tu + QF YF,u /Cp

(2.56)

The parameter Tb is the adiabatic ﬂame temperature and YF,u represents the mass fraction of the fuel in the unburned mixture. The overall energy balance for adiabatic case can also be written as: Cp (Tu − T0 ) +

Sensible enthalphy of reactants

N

N hof,i Yi,u = Cp (Tb − T0 ) + hof,i Yi,b

i=1 i=1 Sensible enthalphy

Chemical enthalphy of reactants

of products

Chemical enthalphy of products

(2.57)

72

LAMINAR PREMIXED FLAMES

where T0 = 298.15 K. Equation 2.57 can also be written as: Cp (Tb − Tu ) =

N

hof,i Yi,u − Yi,b

(2.58)

i=1

Adiabatic flame temperature (K)

This equation represents the conversion of chemical energy into thermal energy by the combustion process. From Equation 2.56, readers can see that the adiabatic ﬂame temperature Tb of a laminar premixed ﬂame is determined only by the heating value of fuel and speciﬁc heat of the gaseous mixture. Similarly, from Equation 2.54, readers can observe that the laminar ﬂame speed depends only on the total fuel consumption rate per unit area, initial concentration of fuel, and density of the unburned fuel-oxidizer mixture. Therefore, to have a higher adiabatic ﬂame temperature, changing the reaction rates would not result in anything constructive. Similarly, to change the laminar ﬂame speed, one needs to focus on the reaction rate parameters, such as pre-exponential factor and/or activation energy. As stated earlier, this analysis was much simpliﬁed in order to establish a physical understanding of premixed laminar ﬂames. It would be interesting to compare the predictions from simpliﬁed analyses with the full-chemistry model with temperature-dependent speciﬁc heats for all chemical species. It also would be useful to examine the effect of assumption 2 with constant speciﬁc heats for all chemical species on the calculated adiabatic ﬂame temperature. This type of comparison for a premixed propane-air ﬂame at initial temperature of 300 K was made by Poinsot and Veynante (2005), as shown in Figure 2.8. This ﬁgure shows that the predicted adiabatic ﬂame temperature (Tb ) as a function of φ from the simpliﬁed problem is in reasonable agreement with the solution from the other two cases.

Full chemistry and variable Cp One-step chemistry and variable Cp One-step chemistry and constant Cp

2400 2200 2000 1800 1600

Tu = 300K

1400 0.5

0.6

0.7

0.8 0.9 Equivalence ratio φ

1.0

1.1

1.2

Figure 2.8 Adiabatic ﬂame temperature for an atmospheric propane-air ﬂame using various assumptions (modiﬁed from Poinsot and Veynante, 2005).

ANALYTICAL RELATIONSHIPS FOR PREMIXED LAMINAR FLAMES

73

The fuel mass fraction and temperature can be related by performing this exercise: Let us deﬁne two reduced parameters Y and , one for the fuel mass fraction and another for the temperature, respectively. Y ≡ YF /YF,u ≡

(2.59)

Cp (T − Tu ) (T − Tu ) = QF YF,u (Tb − Tu )

(2.60)

By this reduction, we have two parameters that range from 0 to 1. The variable Y is 1 in the unburned mixture, and it is 0 in the burned gas. The variable is 0 in the unburned mixture, and it is 1 in the burned gas. The species conservation equation (Equation 2.51) and the energy conservation equation (Equation 2.52) can be rewritten in terms of these reduced variables. The equations for Y and have similar from and they are shown below: dY d dY ρu SL (2.61) = ρD + ω˙ F/YF,u dx dx dx d l d d (2.62) = − ω˙ F/YF,u ρu SL dx dx Cp dx By adding these two equations and assuming Le = l/(ρCp D) = 1, we can get: ρu SL

d d d ρD ( + Y ) ( + Y ) = dx dx dx

(2.63)

The solution to this homogenous ordering differential equation (ODE) is: +Y =1

(2.64)

This simple algebraic relationship implies that and Y are not independent quantities; only one differential equation needs to be solved in order to obtain the variable distribution in the ﬂame zone. The reduced temperature varies from 0 to 1 from the unburned side of the ﬂame to the burned side of the ﬂame zone; therefore, it also can be regarded as a reaction progress variable. The source term for a k th chemical species is given by Equation 2.10. For a single-step forward reaction the global reaction can be given kf

νF F + νO O −→ νP P

(2.65)

In case of a very lean mixture, the oxidizer mass fraction remains constant, and the reaction rate is dependent on the fuel mass fraction alone. In such case, the fuel reaction rate is a function of fuel mass fraction only. Therefore, vF = 0

74

LAMINAR PREMIXED FLAMES

and vF = 1. The reaction source term in Equation 2.62 for the fuel can be given by: Ea (2.66) ω˙ F = −AT n YF ρ exp − Ru T Using Equation 2.60, T can be written in terms of as: T = Tu + (Tb − Tu )

(2.67)

By substituting T into Equation 2.66 and replacing YF by YF,u Y = YF,u (1 − ), we have: Ea ω˙ F = A [Tu + (Tb − Tu )]n ρYF,u (1 − ) exp − Ru [Tu + (Tb − Tu )] (2.68) Equation 2.62 then becomes: d l d d = + A [Tu + (Tb − Tu )]n ρu SL dx dx Cp dx −Ta × ρ(1 − ) exp (2.69) Tu + (Tb − Tu ) In order to gain a clearer insight into the structure of the laminar ﬂames, the source term in the -equation can be rewritten in a slightly different form. Let us introduce two new parameters, α and β, which are deﬁned as:

(2.70) α ≡ (Tb − Tu )/Tb = QF YF,u / Cp Tb β ≡ αTa /Tb

(2.71)

The parameter α is a measure of the amount of heat released in the ﬂame. The parameter β is the product of heat release term and the activation energy of the reaction. F. A. Williams (1985) used asymptotic analysis and these two parameters to express the source term given by Equation 2.72. This expression is very useful for comprehensive numerical computations in laminar ﬂames. By substituting the α and β parameters into the -equation, we get: d d = ρu SL dx dx

l d

Cp dx

=−ω˙ F /YF,u

−β(1 − ) + A[T ] ρ (1 − ) exp (−β/α) exp 1 − α(1 − )

n

exp(−Ta /T )

(2.72) For Tu = 300 K , typical laminar ﬂame values are given in Table 2.6. Flame 1 has been used for numerical simulations of a turbulent combustion (to be discussed in Chapter 5), in some cases the chemical parameters are modiﬁed from a true

75

ANALYTICAL RELATIONSHIPS FOR PREMIXED LAMINAR FLAMES

TABLE 2.6. Typical Values for α and β in Premixed Flames

Flame 1 Flame 2

Tb /Tu

Ea (kJ/mole)

Ta /Tu

α

4 7

110 375

44.10 150.35

0.750 0.857

β 8.269 18.41

laminar ﬂame in order to allow for an easier computation. Flame 2 is for typical premixed hydrocarbon-air laminar ﬂames. The maximum value of the reaction rate can be determined from the following procedure: The pressure of the ﬂow ﬁeld can be considered to be constant. The density ρ can be considered as a function of by the next equation of state relationship: Tu 1−α ρ = ρu (2.73) = ρu T 1 − α (1 − ) The expression in Equation 2.73 is then substituted into Equation 2.68. By differentiating Equation 2.68 with respect to and setting the derivative equal to 0, we get the value of corresponding to maximum reduced reaction rate as: ω˙ max = 1 −

1 1 ≈1− ∵α β α+β β

(2.74)

The maximum reaction rate is then obtained as: 1−α β |ω˙ F |max = ρu YF,u AT n (2.75) exp −1 − β α

A plot of reduced reaction rate −ω˙ F / ρu YF,u A versus the reduced temperature () is shown in Figure 2.9 for a typical premixed laminar ﬂame of hydrocarbon-air mixture. The values of α and β are selected for such mixture (see Flame 2 in Table 2.6) so that the reduced reaction rate is a function of only. From the plot shown in Figure 2.9, the dependency of reaction rate on the temperature can be seen. In order to examine the effect of β on reduced reaction rate, a plot of reduced reaction rate versus for 4 different values of β ranging from 2.45 to 18.41 is shown in Figure 2.10. The reaction rate has been normalized by the maximum value in each case. The value of α is kept constant at 0.857 for all four cases. From this plot, a lower value of activation energy (which corresponds to a lower value of β) gives a broader distribution of the fuel reaction rate whereas the distribution gets narrower as the activation energy is increased with the peak shifting to a higher value of i.e., higher temperature. This plot shows that increasing the value of activation energy leads to a narrower reaction zone. Numerically, this means that the problem becomes stiffer as the heat release and reaction rates are limited to very small region. In such case, a ﬁner mesh will be required for resolution of the source terms in species and energy equations.

76

LAMINAR PREMIXED FLAMES 16 [(1 − α )/β ]exp[−(α + β )/α ]

12 10 8 6

1 − 1/(α + β )

−ω F/ρu AYF,u × 1013

Reduced Reaction Rate

14

4 2

α = 0.857, β = 18.41, n = 0

0 0

0.5

1

Reduced Temperature, Θ ≡ (T − Tu)/(Tb − Tu)

Figure 2.9 Variations of the reduced reaction rate −ω˙ F / ρu YF,u A versus ≡ (T − Tu )/(Tb − Tu ).

Normalized Reaction Rate ω F ω F max

1.2 a = 0.857 for all cases b: 18.41 9.82 4.91 2.45

1 0.8 0.6 0.4 0.2 0

0

0.2

0.4

0.6

0.8

1

Reduced Temperature, Θ ≡ (T − Tu)/(Tb − Tu)

Figure 2.10 Variations of the normalized reaction rate versus reduced temperature for various values of β.

The premixed laminar ﬂame can be divided into two separate zones. The premixed reactants are heated mainly by convection and conduction processes in the lower-temperature zone (called zone 1 or preheat zone). In this zone, there is a lesser amount of heat release than in the higher temperature zone (called zone 2 or reaction zone). The temperature at the interface between the two zones can be regarded as the ignition temperature of the reactants. The reaction zone thickness in the -space can be approximated by roughly doubling the region between the peak reaction rate and the value of equal to 1. Since α is much

ANALYTICAL RELATIONSHIPS FOR PREMIXED LAMINAR FLAMES

77

smaller than β, the zone 2 thickness can be approximated as 2/β in the space. The preheat zone thickness in the space is then equal to (1 − 2/β) for zone 1. Diffusion of chemical species is prevalent in both zone 1 and zone 2.

2.3.1

Three Analysis Procedures for Premixed Laminar Flames

An important parameter in the laminar ﬂame study is the determination of laminar ﬂame speed (SL ). There have been several classical approaches to obtain an analytical expression for SL as a function of thermo-chemical properties and operating conditions of the premixed laminar ﬂames. One of such theories was proposed by Russian scientists Zel’dovich, Frank-Kamenetskii, and Semenov in 1940s. A detailed discussion of this approach can be found in Kuo (2005, Chap. 5). In this approach, the premixed ﬂame was divided into two zones; preheat zone and reaction zone (see Figure 2.11). It was assumed that the terms associated with chemical reactions are negligible in comparison with the other terms in the preheat zone. Also, it was assumed that the convective heat transfer is negligible in comparison to the heat release term due to chemical reactions in the reaction zone. By using these assumptions and applying the heat-ﬂux balance at the interface between preheat and reaction zones, this expression for laminar ﬂame speed was obtained: Tb l 2 1 ω˙ F dT (2.76) I ; where I ≡ SL = ρu Cp (Tb − Tu ) au Tu

In Equation 2.76, au is the number density of reactant molecules in the unburned state. This theory does not give very accurate results, but a major conclusion can be derived from the expression. Equation 2.76 shows that the

Normalized Reaction Rate ω F ω F max

1 Zone 1: Preheat zone thickness, dpr (Θ) ≈ 1− 2 β 0.5

Zone 2: Reaction zone thickness, dr (Θ) ≈ 2 β 0

Figure 2.11

0

0.5 Reduced Temperature, Θ ≡ (T − Tu)/(Tb − Tu)

1 1 β β

1

Description of major zones in a premixed laminar ﬂame.

78

LAMINAR PREMIXED FLAMES

laminar ﬂame speed is proportional to the square root of thermal diffusivity of the unburned reactants and the reaction rate, as shown by the Equation 2.77. SL ∝

u αth (RR)Tb

(2.77)

The expression for ω˙ F is given in Equation 2.66 for evaluation of parameter I. An integration of ω˙ F over temperature T yields kinetic parameters A and Ea in the expression for I , which is why the reaction rate evaluated at the ﬂame temperature (RR)Tb is in the square root of Equation 2.77. In the asymptotic analysis of Zel’dovich and Frank-Kamenetskii (ZFK), a spatial variable ξ was used to replace the physical distance x . ξ was deﬁned as: x ρu SL Cp ρu SL Cp d d dx ⇒ = (2.78) ξ≡ l dx l dξ 0 By substituting this transformation in Equation 2.72 the becomes d 2 d − ω˙ = dξ dξ 2

(2.79)

In Equation 2.79, ω˙ is a reduced reaction rate parameter and is a dimensionless ﬂame parameter; they are deﬁned by: β (1 − ) (2.80) ω˙ ≡ (1 − ) exp − 1 − α (1 − ) and ≡−

ρl ρu2 SL2 Cp

AT n e−β/α

(2.81)

Thermal conductivity and density are functions of temperature. Therefore, in order to simplify the integration of Equation 2.79, ZFK assumed that the group of parameters ρ lAT n = constant = ρu lu ATun . By substituting this relationship into Equation 2.81, the ﬂame parameter, can be expressed as: =−

u αth ρu lu ATun −β/α e = ATun e−β/α 2 ρu2 SL Cp SL2

(2.82)

By solving Equation 2.79, a simple solution for was obtained by ZFK as: = 0.5β 2

(2.83)

By substituting this simple result in Equation 2.82, the expression for laminar ﬂame speed SL was derived as: SL =

1 β

u 2αth ATun e−β/α =

Ru Tb2 u 2αth ATun e−Ea /Ru Tb Ea (Tb − Tu )

(2.84)

ANALYTICAL RELATIONSHIPS FOR PREMIXED LAMINAR FLAMES

79

F. A. Williams (1985) derived this expression for the ﬂame parameter: 2 = 0.5β 2 1 + (3α − 1.344) β

(2.85)

The term “other than unity” in the square brackets in Equation 2.85 comes from carrying the ZFK approximate asymptotic analysis to the second order by formal asymptotics. By substituting this result in Equation 2.82, the expression for laminar ﬂame speed SL can be derived as: 1 SL = β

u 2αth ATun e−β/α

≈0 1 (1.344 − 3α) 1+ + O 2 β β

Ru Tb2 (1.656Tb − 3Tu ) u n −E /R T a u b 1− = 2αth ATu e Ea (Tb − Tu ) (Tb − Tu ) (Ea /Ru Tb )

(2.86)

The explicit expressions for the laminar ﬂame speed may change with the complexity of formulation. However, the qualitative result remains the same as expressed by Equation 2.77. A simpler expression for the source term in Equation 2.62 used by Poinsot and Veynante (2005) is: ω˙ F = −ρu YF,u (RR) (1 − ) H ( − | ω˙ max )

(2.87)

where H is the Heaviside function and |ω˙ max corresponds to the critical temperature at which: |ω˙ max = 1 − 1/β. Using this expression facilitates an easier analytical solution of the premixed laminar ﬂame equations. Poinsot and Veynante (2005) also assumed that the thermal conductivity was constant at a value corresponding to the unburned gas temperature (i.e., l = lu ). The ﬂame speed expression obtained by their solution is: u αth 2A β SL = where RRTb = RRTb exp − β (β − 1) β α Ru Tb2 u −Ea /Ru Tb 2Aαth e = Ea (Tb − Tu )

1 Ea (Tb − Tu )/Ru Tb2 − 1

(2.88)

Once again, the qualitative result from this analysis is same as expressed by Equation 2.77. From this exercise, readers can recognize that asymptotic and analytical techniques require assumptions for transport coefﬁcients such as thermal conductivity. A summary of various methods, transport assumptions, and ﬂame speed expressions is shown in Table 2.7.

80

LAMINAR PREMIXED FLAMES

TABLE 2.7. Summary of Flame Speed Expressions, Transport Assumptions, and Reaction Rate Expressions for Lean Flames Fuel reaction rate, ω˙ F

Model ZFK

β

−AT n e− α ρYF,u ×

(1 − ) e

β(1−) − 1−α(1−)

Thermal conductivity, l n−1 T lu Tu

β

Williams −AT n e− α ρYF,u ×

(1 − ) e

β(1−) − 1−α(1−)

lu

Poinsot −ρu YF,u RRTb × and (1 − ) ×

Veynante H − |ω˙ max

2.3.2

T Tu

Flame speed, S L ! u 2αth ATun e−β/α β

n−1 !

u 2αth ATun e−β/α (1.344 − 3α) 1+ β β

lu

u αth RRTb β (β − 1)

Generalized Expression for Laminar Flame Speeds

Mitani (1980) considered the next global reaction and developed a more general expression for handling Le = 1 reactants: kf

νF F + νO O −→ P

(2.89)

The fuel reaction rate is given by: ω˙ F =

−νF

MwF AT

n

ρYF MwF

nF

ρYO MwO

nO

Ea exp − Ru T

(2.90)

In Equation 2.90, the exponents nF and nO are not the same as stoichiometric coefﬁcients νF and νO . The equivalence ratio is related to stoichiometric ratio s by: YF,u /YO,u YF,u =s φ= , YO,u YF,u /YO,u st where

1 = YF,u /YO,u st

s=

YO,u YF,u

or φ=

st

=

mO,u mF,u

MwO YF,u νO νF MwF YO,u

st

=

νO MwO νF MwF

(2.91)

ANALYTICAL RELATIONSHIPS FOR PREMIXED LAMINAR FLAMES

81

After assuming the product ρ lAT n = constant = ρu lu ATun , the laminar ﬂame speed for a fuel-lean ﬂame was derived as: " # n +n −1 # 2lb ν ν /ν nO ρ nF +nO AT n YF,u F O LenF LenO $ F b O F O F b SL = nF +nO −1 nF +nO +1 2 ρu Cp MwF β ! β × G(nF , nO ) exp − (2.92) 2α where the G function is deﬁned by: G(nF , nO ) =

∞ 0

y

nF

φ−1 y+β LeO

nO

e−y dy

(2.93)

Even though Equation 2.92 is more general than those given in Table 2.7, it gives only reasonable accuracy for fuel-lean ﬂames. For fuel-rich ﬂame, the indices of F and O in Equation 2.92 can be exchanged to obtain an expression for laminar ﬂame speed. However, the accuracy is also quite poor from such an expression, as can be seen by comparing the calculated results with experimental data. In general, Equation 2.92 has similar accuracy for both fuel-rich and fuellean ﬂames. Also, note that all of the expressions shown in Table 2.7 give identical results for stoichiometric ﬂame. Since it was assumed that the product ρ lAT n = constant, the group ρb lb ATbn in Equation 2.92 can be replaced by ρu lu ATun , and the equation can be rewritten as: " # # 2αth,u ν ν /ν nO AT n LenF LenO ρ Y nF +nO −1 b F,u u F O F O F SL = $ nF +nO +1 Mw F β ! β × G(nF , nO ) exp − (2.94) 2α Based on the fuel consumption rate consideration, the consumption speed Scons is deﬁned as the speed at which the ﬂame burns the reactants. For a 1-D planar ﬂame, it can be evaluated from: ∞ 1 Scons = − ω˙ F dx (2.95) ρu YF,u −∞ This is one of the alternative methods to determine the laminar ﬂame speed SL . 2.3.2.1 Reduced Reaction Mechanism for HC-Air Flame It is generally recognized that the global reaction considered in asymptotic analysis is a major simpliﬁcation. In order to achieve more accurate ﬂame speed

82

LAMINAR PREMIXED FLAMES

calculations, Westbrook and Dryer (1981) proposed reduced reaction scheme for HC-Air ﬂames for a wide range of equivalence ratios. These reduced schemes have been helpful to achieve closer agreement between the calculated results and measured laminar ﬂame speeds. These reduced reactions schemes lead to next expression for the fuel reaction rate, where nF is the overall reaction order with respect to the fuel, which could have fractional values. −β(1 − ) ω˙ F nF n = −A [T ] ρ(1 − ) exp (−β/α) exp (2.96) YF,u 1 − α(1 − )

exp(−Ta /T )

The lower values of nF lead to a better comparison with the experimental data; however, they also lead to increased numerical stiffness since the reaction zone becomes thinner. The effect of nF is shown in Figure 2.12. Westbrook and Dryer (1981) have suggested nF = 0.1 and nO = 1.65 for lean C3 H8 /air ﬂame. 2.3.3

Dependency of Laminar Flame Speed on Temperature and Pressure

The direct implication from the generalized expression for laminar ﬂame speed given by Equation 2.94 is: SL ∝

nF +nO −1

αth,u (ρb )

(2.97)

Due to the fact that αth,u ∝ 1/P and ρb ∝ P , we have

P SL (P ) = SL (P0 ) P0

nF +nO −2 2

(2.98)

0.20 nF 0.25 0.5 0.75 1

0.16 −ω F ρu AYF,u × 1010

Reduced Reaction Rate

0.18

0.14 0.12 0.10 0.08 0.06 0.04

a = 0.857, b = 18.41, n = 0

0.02 0.00

0

0.5 1 Reduced Temperature, Θ ≡ (T − Tu)/(Tb − Tu)

Figure 2.12 Variations of reduced reaction rates −ω˙ F / ρu YF,u A versus for various values of nF .

83

ANALYTICAL RELATIONSHIPS FOR PREMIXED LAMINAR FLAMES

Temperature changes are difﬁcult to assess because the exponential term in Equation 2.94 is associated with Tb , which is difﬁcult to determine if only simple reaction chemistry is taken into account. Test data usually are expressed in a simple power law:

P ηP T ηT SL (P , Tu ) = SL P0 , Tu,0 (2.99) P0 Tu,0 The values for ηP and ηT for methane/air mixtures and propane/air mixtures are given in Table 2.8. These values were determined by experimental correlations by Gu, Hag, Lawes, and Woolley (2000) (for methane/air ﬂames) and by Metghalchi and Keck (1980) (for propane/air mixtures). The experimentally measured laminar ﬂame speeds for stoichiometric methane-air ﬂames and propane-air ﬂames at various pressures are shown in Figure 2.13a and b, respectively. As we can see, laminar ﬂame speed decreases with increasing pressure but it increases if the initial temperature of the unburned mixture is increased.

TABLE 2.8. Experimentally Correlated Pressure and Temperature Exponents for Laminar Flame Speeds of Premixed Hydrocarbon-Air Mixtures Source: Metghalchi and Keck (1980).

SL P0 , Tu,0 (M/S )

Fuel Methane (φ = 0.8) Methane (φ = 1) Methane (φ = 1.2) Propane (φ = 0.8 to 1.5)

0.8 0.6 0.4 0.2 (a) 0.0 300

ηP

2.105 1.612 2.000 2.18 − 0.8(φ − 1)

–0.504 –0.374 –0.438 −0.16 − 0.22(φ − 1)

1.0

φ = 1, p = 1 atm φ = 1, p = 5 atm φ = 1, p = 10 atm

Flame speed (m/s)

Flame speed (m/s)

1.0

0.259 0.360 0.314 0.34 − 1.38 × (φ − 1.08)2

ηT

0.8

φ = 1, p = 1 atm φ = 1, p = 5 atm φ = 1, p = 10 atm

0.6 0.4 0.2

METHANE/AIR 350 400 450 Fresh gas temperature (K)

500

(b) 0.0 300

PROPANE/AIR 350 400 450 Fresh gas temperature (K)

500

Figure 2.13 Experimental ﬁts for speeds of stoichiometric methane/air (Gu et al., 2000, left) and propane/air ﬂames (Metghalchi and Keck, 1980, right) (modiﬁed from Poinsot and Veynante, 2005).

84

LAMINAR PREMIXED FLAMES

2.3.4

Premixed Laminar Flame Thickness

Knowledge of ﬂame thickness is useful (and perhaps necessary) for numerical calculations as it gives an estimation of the computational domain and enables the selection of a suitable mesh resolution. The thickness of a given laminar premixed ﬂame can be deﬁned based on: • Temperature distribution across the ﬂame • Distributions of chemical species across the ﬂame • Scaling law considerations Some of these deﬁnitions do not require information about the burned mixture or calculated results and can give an estimation of laminar ﬂame thickness based on the properties of unburned mixture and laminar ﬂame speed. To estimate the thickness of a premixed laminar ﬂame using hydrocarbon fuels, the next simple expression can be obtained from the dimensional analysis. This characteristic ﬂame thickness is based on thermal and transport properties of unburned mixture due to high activation energy associated with hydrocarbon fuels, which have thin reaction zone thickness. δ≈

αth,u l = SL ρu Cp SL

(2.100)

The use of unburned mixture properties is convenient because these properties are known for a speciﬁc reactant mixtures in comparison to any other point in the premixed laminar ﬂame. Equation 2.100 also can be obtained from the ﬂame Reynolds number. The ﬂame Reynolds number for laminar ﬂames is deﬁned as: ρu SL δ μu

Ref ≡

(2.101)

if Ref ∼ O(1) and Pr = 1 then, SL δ SL δ = =1 νu αth,u

and

δ=

αth,u SL

(2.102)

Another deﬁnition of premixed laminar ﬂame thickness was proposed by Blint (1986) to account for the temperature effect on the ﬂame thickness. This deﬁnition is: 0.7 αth,u Tb 0.7 Tb δL,B = 2 = 2δ (2.103) SL Tu Tu A more relevant ﬂame thickness is based on temperature proﬁles, that is, δL,slope =

Tb − Tu

max ∂T

(2.104)

∂x

The total thickness δL,total can be deﬁned for to change from 0.01 to 0.99. But δL,slope is more useful for mesh resolution. Figure 2.14 describes these two laminar ﬂame thicknesses.

85

ANALYTICAL RELATIONSHIPS FOR PREMIXED LAMINAR FLAMES

Reduced Temperature, Θ ≡ (T − Tu)/(Tb − Tu)

1

Burned Products

Unburned Gas Mixture 0

x

δ L, slope δ L, total

Figure 2.14

Two deﬁnitions of premixed laminar ﬂame thickness.

Physically, the laminar ﬂame thickness based on temperature proﬁles is most useful deﬁnition. However, it requires a laminar ﬂame solution beforehand. In addition to the δL,slope , a reaction zone thickness (δr ) is also required. This thickness is deﬁned by the next equation: δr =

δL,slope β

(2.105)

A summary of various ﬂame thicknesses is given in Table 2.9. Note that in the ﬂame zone, certain radicals may exist over distances much smaller than δL,slope ; thus, mesh resolution must take this effect into consideration. TABLE 2.9. Flame Thickness Deﬁnitions and Their Usefulness for Meshing Source: modiﬁed from Poinsot and Veynante (2005).

Thermal wave thickness Global (total) thickness

∂T δL,slope = (Tb − Tu )/max ∂x

δL,total = distance from = 0.01 to 0.99 αth,u Diffusive thickness δ = SL Tb 0.7 Blint thickness δL,B = 2δ Tu δL,slope Reaction thickness δr = β

Best deﬁnition for meshing in numerical work Not very useful Not precise, too small Close to δL,slope ; useful Smallest physical dimension

86

2.4

LAMINAR PREMIXED FLAMES

EFFECT OF FLAME STRETCH ON LAMINAR FLAME SPEED

A premixed ﬂame in a multidimensional conﬁguration or in a nonuniform ﬂow ﬁeld could be subjected to various physical interactions. For example, curvature of the ﬂame or strain rate resulting from the non-uniform ﬂow ﬁeld could result in increased surface area of the laminar ﬂame. This process is called ﬂame stretch. Readers are referred to Ch. 5 of Kuo (2005) for a detailed discussion on the dynamic analysis of stretched premixed laminar ﬂames. For convenience of readers, a quick introduction to the effect of ﬂame stretch on the laminar ﬂame behavior is provided in the following section. 2.4.1

Deﬁnitions of Stretch Factor and Karlovitz Number

The surface of a premixed ﬂame propagating in a nonuniform ﬂow ﬁeld is subjected to strain and curvature effects, which causes change in the ﬂame surface area with time. Karlovitz (1953) initiated the study of stretched premixed ﬂames and showed the importance of aerodynamic stretching on the stability of ﬂames. The ﬂame stretch factor (κ) is deﬁned as the percentage change of ﬂame surface area with respect to time and is mathematically represented by: κ≡

1 d(δA) 1 dA = δA dt A dt

(2.106)

Physically, the effect of stretch on the ﬂame is to reduce the thickness of the ﬂame front; therefore, stretch affects the ﬂame speed. The reduction of ﬂame thickness obviously changes the ﬂame structure through its coupled effect with mass and thermal diffusion. The concept of ﬂame stretch can be applied to develop a more in-depth understanding of the several physical processes associated with laminar ﬂame studies in the areas of ﬂame stabilization, laminar ﬂame speed determination, ﬂammability limits, and even modeling of turbulent ﬂames. Markstein (1964b) also considered the effect of ﬂame stretch on the unsteady ﬂame propagation in a premixed gas mixture. During last few decades, several important papers (Buckmaster, 1979; Matalon, 1983; Chung and Law, 1984; Law 1988; Candel and Poinsot, 1990; have elucidated the mathematical relationships and physical processes associated with ﬂame stretch behavior. We can always decompose any arbitrary velocity into two parts with one part tangent to the ﬂame surface (vt ) and the other part normal to the ﬂame surface, that is v = vn + vt = (v · n)n + vt (2.107) The unit normal vector n perpendicular to the ﬂame surface can be deﬁned as: n≡−

∇ |∇|

(2.108)

where represents the nondimensional temperature deﬁned in Equation 2.60. Based on this deﬁnition, it can be seen that the unit normal n always points from

EFFECT OF FLAME STRETCH ON LAMINAR FLAME SPEED

87

the burned gas to the unburned gas. There are multiple choices for deﬁning the ﬂame surface location. From the basic deﬁnition of laminar ﬂame speed (SL ), it should correspond to the iso-level = 0. If we consider the attainment of fully reacted state as the ﬂame location, then it corresponds to the iso-level = 1. Generally, the reactions take place in a region of ﬁnite thickness where shows considerable gradient. The point of maximum temperature gradient also could be taken as the location of ﬂame surface. In such case, the ﬂame surface cannot be deﬁned as an iso-level since the temperature at the maximum -gradient locations could have different values of . Another reasonable approach to deﬁne a ﬂame surface location could be the = |ω˙ max corresponding to the location where the reaction rate is maximum (refer to Figure 2.10 earlier in the chapter). The corresponding to the demarcation between preheat zone (zone 1) and the reaction zone (zone 2) can also be considered as the location of the ﬂame surface (see Figure 2.11). Depending on the ﬂame surface location deﬁnition used by the researchers, the value at the ﬂame surface can be regarded as = fs . Among these choices, the last three deﬁnitions (especially the one based on maximum have the most physical signiﬁcance than the ﬁrst two deﬁnitions. A displacement speed Sd of the ﬂame surface is deﬁned as: Sd ≡

1 ∂ 1 D 1 ∂ ∇ = −v · n = +v· |∇| Dt |∇| ∂t |∇| |∇| ∂t

(2.109)

≡Sa

where Sa stands for the absolute speed of the ﬂame surface. The last term (v · n) on the right hand side of Equation 2.109 represents the component of ﬂow velocity normal to the ﬂame surface. If an unstretched planar ﬂame surface location is taken to be the same as the surface corresponding to iso-level = 0 (i.e., unburned mixture), then the displacement ﬂame speed (Sd ) is same as the unstretched laminar ﬂame speed (SL0 ). In all other cases, the displacement ﬂame speed Sd can be related to the unstretched laminar ﬂame speed by the density ratio as: ρu Sd = SL0 (2.110) ρ Now let us consider a ﬂame surface A(t) moving with the local velocity w. (Note that each spatial point has its own velocity). The local ﬂame surface velocity w (shown in Figure 2.15) can be given as the vector sum of local ﬂuid velocity v and the laminar ﬂame displacement speed Sd in the direction normal to the local ﬂame surface by this equation: w = v + Sd n

(2.111)

Then, the rate of change of the ﬂux of a vector G across the ﬂame surface is given by Equation 2.112, based on the Reynolds transport theorem. ∂G d G · n dA = + w · ∇G − G · ∇w + G∇ · w · n dA (2.112) dt A(t) A(t) ∂t

88

LAMINAR PREMIXED FLAMES Curved flame surface at time t + δ t Unburned Gases

v

Curved flame surface at time t Burned Gases

w

Sdn n

Figure 2.15 Description of various velocities associated with a propagating curved laminar ﬂame front.

By setting G = n, we have: ∂n d dA = + w · ∇n − n · ∇w + n∇ · w · n dA dt A(t) A(t) ∂t

(2.113)

After substituting Equation 2.113 into Equation 2.106, we have: κ≡

1 d(δA) = −nn : ∇v + ∇ · v + Sd ∇ · n δA dt Sd ∂vi 1 Dρ = −ni nj − ± ∂xj ρ Dt R Term 1

where

Term 2

(2.114)

Term 3 for cylindrical ﬂame

Term 1 = Associated with strain rate tensor. Term 2 = Associated with dilatation. Term 3 = Associated with ﬂame curvature. For a cylindrical ﬂame with radius R,we can show that ∇ · n = (1/r) ∂(rnr )/∂r = ±1/R where nr = ±1, the positive sign corresponding to the outwardly propagating ﬂame and the negative sign corresponds to the inwardly propagating ﬂame. For illustration of cylindrical ﬂames, see Figure 2.16.

Detailed steps between Equation 2.113 and Equation 2.114 can be found from Kuo 2005, Chap. 5.

Similarly, for a spherical ﬂame, ∇ · n = 1/r 2 ∂(r 2 nr )/∂r = ±2/R. Generally, the term ∇ · n is the curvature of the ﬂame front, and it is related to the radius of curvature of the ﬂame surface. It is given in terms of ﬂame surface curvature radii R 1 and R 2 (see Figure 2.17) by the next equation: 1 1 ∇ ·n=± + (2.115) R1 R2

EFFECT OF FLAME STRETCH ON LAMINAR FLAME SPEED

Unburned Gases n

Burned Gases

Burned Gases

89

Unburned Gases r

r

R

R v f = Sd n

n vf = Sdn

(a) Inwardly propagating flame vf = dR/dt < 0

(b) Outwardly propagating flame vf = dR/dt > 0

Figure 2.16 Two cylindrical premixed laminar ﬂames propagating in different directions in absence of any external ﬂow ﬁeld. n Unburned Gases η v

R1

Burned Gases R2

Figure 2.17 2005).

Deﬁnition ﬂame radii of curvature (modiﬁed from Poinsot and Veynante,

Again, the positive sign corresponds to an outwardly moving ﬂame and vice versa. There are several examples of stretching in the premixed ﬂames, as shown in Figure 2.18. The ﬂame shown in Figure 2.18c is stabilized by the recirculating zones developed in the immediate downstream of the V-gutter ﬂame holder. The energy transfer from the recirculating zone to the unburned gas provides the anchoring effect on the ﬂame. In several practical combustors, the ﬂame upstream the V-gutter can be turbulent when the Reynolds number based on the width of the V-gutter is greater than the critical value. In such case, the ﬂame surface location (as shown by the reaction rate contours in Figure 2.18d) can be highly wrinkled. It is evident that the ﬂame can be stretched by the combined effect of strain, volume expansion of the ﬂuid (dilatation), and the curvature of the ﬂame, which arise from the nonuniformities of the ﬂow and the ﬂame propagation into the unburned mixture. By considering the two independent unit vectors ν and η tangent to the ﬂame surface, we have these relations: n = ν × η ≡ eν × eη

(2.116)

90

LAMINAR PREMIXED FLAMES

Burned Gas

n

Unburned Gas Unburned Gas

n Unburned Gas

Burned Gas

z

z r

Recirculating zone

Burned Gas

V-gutter flame holder

r

(a) Forward-propagating flame to stagnation plane

(b) Rearward-propagating flame from stagnation plane

(c) Anchored premixed flame from a V-gutter flame holder

Flow direction

V-gutter flame holder

(d) Reaction rate contours in an anchored premixed flame from a V-gutter flame holder based on the 3D LES blowout analysis for propane-air mixture (modified from Smith et al., 2007)

Figure 2.18 Several examples of stretched premixed ﬂames.

In tensor notations, we have: νj νi + ηj ηi = δj i − nj ni

(2.117)

Equation 2.114 can be rewritten in tensor notation as:

∂vi ∂ni κ = −nn : ∇v + ∇ · v + Sd ∇ · n = δij − ni nj + Sd ∂xj ∂xi

(2.118)

Substituting Equation 2.117 into Equation 2.118, we have: κ≡

1 d (δA) = (νν + ηη) : ∇v + Sd ∇ · n δA dt

(2.119)

The ﬁrst term on the right-hand side of Equation 2.119 corresponds to the strain in the plane parallel to the local ﬂame surface and can be written in terms of a tangential divergence operator ∇t deﬁned as: ∇t · v ≡ (νν + ηη) :∇v = −nn : ∇v + ∇ · v

(2.120)

The tangential operator ∇t applied on the normal vector n can be expressed as:

∂ni ∇t · n = (νν + ηη) : ∇n = δij − ni nj ∂xj = δij

∂ni 1 ∂ni n ∂ni ∂ni i − ni nj = − nj = ∇ · n ∂xj ∂xj ∂xi 2 ∂xj

EFFECT OF FLAME STRETCH ON LAMINAR FLAME SPEED

Therefore,

∇t · n = ∇ · n

91

(2.121)

Substituting Equations 2.120 and 2.121 into Equation 2.119, we have: κ = ∇t · v + Sd (∇t · n)

(2.122)

The ﬁrst term on the right-hand side of Equation 2.122 is ﬂame stretch due to the nonuniformity of the local ﬂow ﬁeld; the second term on the that side is the ﬂame stretch due to the local curvature of the ﬂame surface. Another approach (C. K. Law, 2006) for calculating the stretch factor is by this following equation: 1 d (δA) δA (t + δt) − δA (t) 1 lim = δA dt δA (t) δt→0 δt

(2.123)

δA (t) = eν × eη dν dη = (dν dη) en = (dν dη) n

(2.124)

κ≡ where

At time (t + δt), the position vector r (ν, η, t + δt) will be: r (ν, η, n, t + δt) = r (ν, η, n, t) +

dr δt = r (ν, η, n, t) + wδt dt

(2.125)

A differential of position vector can be written as: dr (ν, η, n, t + δt) = dr (ν, η, n, t) + dwδt ∂w ∂w ∂w dν + dη + dn δt = dr (ν, η, n, t) + ∂ν ∂η ∂n

(2.126)

Another form of dr can be expressed as: dr = dνeν + dηeη + dnen

(2.127)

Comparing Equations 2.126 and 2.127, we can see that the components of dr (ν, η, n, t + δt) in ν and η directions can be given as: ∂w dνδt ∂ν ∂w dηδt dη (t + δt) = dη (t) + eη · ∂η dν (t + δt) = dν (t) + eν ·

(2.128) (2.129)

Therefore, the local ﬂame surface area at time (t + δt) is: δA (t + δt) = eν dν (t + δt) × eη dη (t + δt) ∂w ∂w = eν + δt dν (t) × eη + δt dη (t) ∂ν ∂η

(2.130)

92

or

LAMINAR PREMIXED FLAMES

∂w ∂w δA (t + δt) = eν + δt × eη + δt dν dη ∂ν ∂η

(2.131)

The change in local surface area in the time interval δt is: d (δA) = [δA (t + δt) − δA (t)] · n

(2.132)

By substituting Equations 2.124 and 2.131 into Equation 2.123, we get: ∂w ∂w ∂w ∂w + × eη · n = eη · + eν · (2.133) κ = eν × ∂η ∂ν ∂η ∂ν Equation 2.133 uses, the cyclic law of triple scalar product [(a × b) · c = (b × c) · a = (c × a) · b]. There is yet another form to express the stretch factor. The tangential operator ∇t applied to the local ﬂame velocity w gives:

∂eν ∂ (eν · w) ∂ eη · w ∂w ∂w ∂eη +w· ∇t · w = + = eν · + eη · + ∂ν ∂η ∂ν ∂η ∂ν ∂η (2.134) and ∂eν ∂eη ∂en + =− = − (∇ · n) n (2.135) ∂ν ∂η ∂n By substituting Equaion 2.135 and Equation 2.133 into Equation 2.134, we have: ∂w ∂w + eη · − w · (∇ · n) n = κ − w · (∇ · n) n ∇t · w = eν · ∂ν ∂η or κ = ∇t · w + w · (∇ · n) n = ∇t · w + (w · n) (∇ · n)

(2.136)

If we decompose the local ﬂame velocity into normal and tangential components as: w = wt + wn = wt + (w · n) n (2.137) the tangential component of the local ﬂame velocity is same as the tangential component of the local ﬂuid velocity, that is, wt = vt

(2.138)

Furthermore, ∇t · w = ∇t · wt . By using Equation 2.138, ∇t · w = ∇t · vt . By substituting this relationship into Equation 2.136, we have: κ = ∇t · vt + (w · n) (∇ · n)

(2.139)

EFFECT OF FLAME STRETCH ON LAMINAR FLAME SPEED

93

Equation 2.139 shows that a ﬂame can be subjected to two sources of stretch. The ﬁrst term on the right-hand side represents the effect of ﬂow nonuniformity along the ﬂame surface. The second term represents the stretch experienced by the nonstationary ﬂame due to the combined effect of normal component of ﬂame velocity w and curvature of the ﬂame represented by ∇ · n. These effects are known as aerodynamic straining, ﬂame motion, and ﬂame curvature effects, respectively. If the ﬂow is perpendicular to the ﬂame surface (i.e., vt = 0 or v × n = 0), then the ﬁrst term on RHS of Equation 2.139 does not exist. The tangential component of ﬂow velocity vt can also be written as: vt = n × (v × n)

(2.140)

Vector multiplication identity [A × (B × C) = B(A · C) − C(A · B)] given in Appendix A can be used to verify the last expression for vt . Also note that the operator ∇ can be split into two perpendicular components, that is, ∇ = ∇t + ∇n By deﬁnition, ∇n · vt = 0 ⇒ ∇t · vt = ∇ · vt . Therefore, Equation 2.139 can be rewritten as: κ = ∇ · [n × (v × n)] + (w · n) (∇ · n) = −n · ∇ × (v × n) + (w · n) (∇ · n) (2.141) This equation is identical to the expression derived by Matalon (1983). The Karlovitz number (Ka) is deﬁned as the dimensionless ﬂame stretch factor. Physically, the Karlovitz number is the ratio of two time scales: the residence time of ﬂuid passing through the unstretched ﬂame and the time scale associated with the ﬂame stretch factor. The residence time of ﬂuid passing through the unstretched ﬂame can be written in terms of the ﬂame thickness based upon thermal diffusion processes and mass (δ) and laminar ﬂame speed (SL0 ) of an unstretched ﬂame. The time scale related to the ﬂame stretch is the inverse of the stretch factor. Therefore, we have: Ka ≡

δ Residence time when crossing the unstretched ﬂame κ= Time scale associated with ﬂame stretch SL0

(2.142)

For convenience, the Karlovitz number can be written as a sum of two parts due to the contribution from strain and curvature, Ka ≡ Kas + Kac where Kas ≡

δ (νν + ηη) : ∇v SL0

and

Kac ≡

(2.143) δ Sd ∇ · n SL0

(2.144)

94

LAMINAR PREMIXED FLAMES

For a cylindrical ﬂame surface, Kac = ±

δ Sd δ SL =± 0 R SL R SL0

(2.145)

The positive sign corresponds to an outwardly moving ﬂame and vice versa.

2.4.2

Governing Equation for Premixed Laminar Flame Surface Area

In this section we follow Candel and Poinsot’s procedure in utilizing the Reynolds’ transport theorem and several previously mentioned mathematical relationships to derive a balance equation for premixed laminar ﬂame area. This equation describes the rate of change of the ﬂame surface density under transient conditions. It is useful not only useful for laminar ﬂames but also can be applied to turbulent ﬂame studies in using the ﬂamelet concept. Now let us ﬁrst deﬁne the ﬂame surface density as . ≡ δA/δV

(2.146)

The Reynolds transport theorem is utilized to determine the time derivative of the volume integral of a scalar function f . The boundary of the volume (i.e., the ﬂame surface) is moving at velocity w(x(t)). The instantaneous volume and surface area of this ﬂuid element are is V(t) and S(t), respectively. Therefore, d ∂f fdV = f w · n dA (2.147) dV + dt V (t) V (t) ∂t S(t) By substituting f = 1 into Equation 2.147 and utilizing the divergence theorem, we have: d dV = ∇ · w dV (2.148) dt V (t) V (t) For a ﬂuid element δV, Equation 2.148 becomes: 1 d(δV ) =∇ ·w δV dt

(2.149)

By differentiating ﬂame surface density with respect to time, we have: 1 dA 1 dV 1 d = − = [−nn : ∇w + ∇ · w] − [∇ · w] = −nn : ∇w dt A dt V dt (2.150) By substituting w = v + Sd n in the above equation, we have: 1 d = −nn : ∇v − n · ∇Sd dt

(2.151)

95

EFFECT OF FLAME STRETCH ON LAMINAR FLAME SPEED

Also, the Lagrangian derivative of can be decomposed into these components: d ∂ ∂ (2.152) = + w · ∇ = + v · ∇ + Sd n · ∇ dt ∂t ∂t By substituting Equation 2.151 into Equation 2.152, we have: ∂ (−nn : ∇v − n · ∇Sd ) = + v · ∇ + Sd n · ∇

∂t d = dt

or

∂ + ∇ · (v) = −(nn : ∇v − ∇ · v) − n · ∇(Sd ) ∂t

(2.153)

By using Equation 2.120, Equation 2.153 also can be written as: ∂ + ∇ · (v) = [(νν + ηη) :∇v] − n · ∇ (Sd ) ∂t

(2.154)

The governing equation for the ﬂame surface area can also be written in term of ﬂame area per unit mass, af , which is deﬁned as: af ≡ /ρ

(2.155)

By substituting Equation 2.155 into Equation 2.154, we have:

∂ ρaf + ∇ · ρvaf = ρ [(νν + ηη) : ∇v] af − n · ∇ ρSd af ∂t

(2.156)

Using the ﬂame stretch factor given by Equation 2.119, Equation 2.156 becomes:

∂ ρaf (2.157) + ∇ · ρ (v + Sd n) af = ρκaf ∂t A related topic to these equations called G-equation is given in Section 11 of Chapter 5.

2.4.3 Determination of Unstretched Premixed Laminar Flame Speeds and Markstein Lengths

Following an early idea of Markstein (1964), both asymptotic theories by Matalon and Matkowsky (1982) and Clavin (1985) as well as the experimental measurements have suggested a linear relationship between ﬂame speed and ﬂame stretch

96

LAMINAR PREMIXED FLAMES

factor. This dependence is characterized by a coefﬁcient of the order of the ﬂame thickness, which has come to be known as the Markstein length. Sd ( = 0) = SL0 − LM κ

(2.158)

SL = SL0 − LM κ

(2.159)

or

where Sd ( = 0) SL SL0 κ LM

= = = = =

ﬂame displacement speed at = 0 stretched laminar ﬂame speed unstretched one-dimensional laminar ﬂame speed ﬂame stretch factor Markstein length

Usually length is a positive quantity. However, the Markstein length LM could be a negative number (as we will see later). The Markstein length is a parameter with the same units as length; therefore, it has been referred as such by the combustion community. Readers should not get confused by this terminology, since length is a positive quantity in common usage. Positive ﬂame stretch implies that the ﬂame surface area increases (κ > 0); negative ﬂame stretch means that the ﬂame surface area decreases (κ < 0). Physically, two major effects (heat loss and mass diffusion of reactants), could inﬂuence the ﬂame speed due to ﬂame stretching. For ease of discussion, let us consider the case of a lean mixture with positive ﬂame stretching. When the surface area of the ﬂame increases, the rate of heat conduction from hot burned gas toward the cold unburned gases increases. The rate of mass diffusion of reactant species from the cold unburned region into the hot burned region also increases when the ﬂame stretch is positive. Since the mixture is lean, the transport properties and concentration of fuel dominate the reaction in the ﬂame zone. If the Lewis number of fuel (LeF ) is greater than 1, the thermal effect supersedes the mass diffusion effect. In such case, the heat loss from the ﬂame is larger than the heat generated due to increased inﬂux of reactant gases. Therefore, the laminar ﬂame speed decreases due to ﬂame stretching (see Figure 2.19a and b). The structure of premixed ﬂames in general ﬂow ﬁelds has been described using a multiscale approach where one length scale, δ, characterizes the diffusive ﬂame thickness and the other, , is a hydrodynamic length that characterizes the “size” of the ﬂame. For example, the diameter of a combustor could be the characteristic size of a ﬂame in that combustor. Typically δ is much smaller than . Viewed on the hydrodynamic length scale , the ﬂame may be regarded as a surface of density discontinuity, advected and distorted by the ﬂow. In the vicinity of the ﬂame front, the combustion ﬁeld is described by a local analysis where lengths are measured in units of δ. On this scale, the ﬂame structure is quasi-steady and quasi-one-dimensional, with variations occurring primarily along the normal direction to the ﬂame surface, but with a parametric

EFFECT OF FLAME STRETCH ON LAMINAR FLAME SPEED

Hot Burned Gas Mass diffusion

Hot Burned Gas Mass diffusion n

Thermal diffusion

n

SL

Thermal diffusion

Cold Unburned Gas

Cold Unburned Gas

z

z r

(a) κ > 0, LeF > 1, SL <

97

SL

r 0 SL

0

(b) κ > 0, LeF < 1, SL > SL

Figure 2.19 Schematic description of the effect of stretch factor and Lewis number on the laminar ﬂame speed of a stretched ﬂame.

dependence on local transverse velocities and on the ﬂame front curvature. If a one-step overall reaction with high activation energy is adopted for the chemistry, the ﬂame temperature is found to remain close to its adiabatic value, Ta = Tu + Q/Cp , where Tu is the temperature of the unburned gas, Cp is the speciﬁc heat (at constant pressure), and Q is the heat released per unit mass of either fuel (if the mixture is lean; Q = QF YF,u ) or oxidizer (if the mixture is rich Q = QO YO,u ). The ﬂame speed, however, may differ substantially from the laminar ﬂame speed and is given by Equations 2.158 or 2.159. Kwon, Tseng, and Faeth (1992) suggest that the Markstein length LM is proportional to the characteristic ﬂame thickness δ because both are representative of the scale of distances over which the diffusion of mass and heat occurs in the ﬂame. This assumption leads to the dimensionless parameter Markstein number (Ma) deﬁned as: LM (2.160) Ma ≡ δ Substituting Equations 2.160 and 2.142 into Equation 2.159 gives the dimensionless relationship: LM κ LM κδ SL =1− 0 =1− = 1 − Ma Ka 0 δ SL SL SL0 or SL = SL0 (1 − Ma Ka)

(2.161)

A slightly different deﬁnition of the Karlovitz number has led to the use of stretched laminar ﬂame speed, as shown in the next equation. Ka∗ =

δ κ SL

(2.162)

98

LAMINAR PREMIXED FLAMES

With this deﬁnition, the stretched laminar ﬂame speed can be related to the unstretched ﬂame speed by these equation: SL0 SL LM κ LM κδ LM κδ = + =1+ =1+ SL SL SL δSL δ SL

(2.163)

SL0 = 1 + Ma Ka∗ SL

(2.164)

or

By either Equation 2.161 or 2.164, it is apparent that the ﬂame speed is related to two dimensionless parameters namely the Karlovitz and Markstein numbers. In terms of the Markstein lengths corresponding to the laminar ﬂame displacement speed and consumption speed, Clavin and Joulin (1983) gave the analytical expressions for Markstein numbers (Mad and Mac ) for a lean, single-step ﬂame with variable density and constant viscosity as: Tb −Tu Tu Tu Tb ln(1 + x) Tb 1 Ma = ln + β (LeF − 1) dx Tb − T u Tu 2 Tb − T u 0 x (2.165) Tb −Tu Tu 1 Tu ln (1 + x) dx (2.166) Mac = β (LeF − 1) 2 Tb − Tu 0 x d

The term β is deﬁned as: β≡

Tb − T u Tb

Ta Tb

(2.167)

The sign of the displacement Markstein length controls the stability of the laminar ﬂame fronts. According to F. A. Williams (1985), negative Markstein lengths create a natural intrinsic front instability leading to cell formation. Generally, measurement of Markstein lengths has proved to be difﬁcult. Unlike the phenomenological approach of Markstein, the asymptotic approach, derived systematically from the conservation equations (Clavin and Joulin, 1983; Matalon and Matkowsky, 1982) and provides an explicit expression for the Markstein length in terms of the physico-chemical parameters including the effective Lewis number of the deﬁcient reactant (i.e., fuel in case of lean mixtures) and the global activation energy of the chemical reaction. The theory has been found useful by experimentalists whose raw data from ﬂame speed measurements correlates well with the linear dependency of ﬂame speed on the stretch factor. Some have used this fact to extract values of the laminar ﬂame speed from measurements on stretched ﬂames (Wu and Law, 1985; Yamaoka and Tsuji, 1985) by extrapolating the data to zero stretch. Others (Aung, Hassan, and Faeth 1997; Bradley, Gaskell, and Gu, 1996; Dowdy, Smith, and Taylor 1990; Kwon, Tseng, and Faeth, 1992; Tseng, Ismail, and Faeth, 1993) have directly measured the

EFFECT OF FLAME STRETCH ON LAMINAR FLAME SPEED

99

Markstein length to quantify stretch effects and incorporate them in numerical simulations on turbulent premixed ﬂames. The asymptotic theory assumes that the resulting Markstein length depends only on a single Lewis number of fuel for lean mixtures and on Lewis number of the oxidizer in rich mixtures. The values of Markstein length that are calculated from these expressions can be very different for lean and rich mixtures, especially for the heavy hydrocarbon/air mixtures for which the fuel Lewis number can be quite large. Predictions based on this theory are clearly not valid at nearstoichiometric conditions. Indeed, experiments have shown that the Markstein length can vary signiﬁcantly as a function of stoichiometric value of the mixture. Although the experimental conﬁgurations and ﬂame surface locations used to measure ﬂame speeds differ between groups, their data exhibit similar qualitative trends. In general, the Markstein length tends to increase monotonically with equivalence ratio for hydrogen-air and methane-air mixtures and tends to decrease monotonically for other hydrocarbon-air mixtures. These trends have been found to persist, for example, in measurements made on spherically expanding ﬂames and on ﬂames in stagnation point ﬂows (Wu and Law, 1985). Tseng, Ismail, and Faeth (1993) used an outward-propagating spherical ﬂame in order to investigate the effects of stretch on laminar ﬂame speed. Their experimental apparatus is a windowed test chamber that is “near spherical” with a volume of 0.011 m3 and a 260-mm cross-sectional diameter at its center. For ignition, they used electrodes extending from the top and bottom of the chamber to two tungsten wires. The ﬂame is assumed to be a spherical deﬂagration wave propagating away from the ignition source. The motion of the unburned gas was considered negligible. Also, the time variations of unsteady ﬂame thickness were regarded as negligible. To minimize the transient effects of ﬂame thickness, it was necessary to have δ/rf 1. In order to simplify the multicomponent diffusion of species in the mixture, the binary diffusivity of fuel into the diluent (i.e., nitrogen) was considered in the equations. The stretch factor κ can be calculated by measuring the instantaneous ﬂame radius, rf (t), and its time derivative: κ=

1 dA 2 d = rf (t) A dt rf dt

(2.168)

The laminar ﬂame speed can be determined by using: SL =

ρb drf ρu dt

(2.169)

The measured ﬂame radius as a function of time is shown in Figure 2.20a and the deduced laminar ﬂame speeds versus ﬂame radius are shown in Figure 2.20b for equivalence ratios in the range from 0.8 to 1.8 for the propane/air laminar premixed ﬂames. The Markstein number for each data point can be determined by using Equation 2.164. It was found that the conditions where the Markstein number was 0 or negative (Ma ≤ 0), the ﬂames were neutral or unstable. The

100

LAMINAR PREMIXED FLAMES 0.4

32 1.4

1.6

1.8

24

UNSTABLE

0.3 UNSTABLE

φ = 1.4

16

0.2

1.8 φ = 1.2 1.0 1.3

0

SL (m/s)

8 rf (mm)

1.6

0.8

0.1

1.2 1.3

0.3

24

1.0

STABLE 0.8

16

0.2

8

0.1

STABLE

PROPANE /AIR @ 1 atm

PROPANE /AIR @ 1 atm 0

0

10

20 t (ms)

30

(a) Flame radius versus time

40

0.0

0

8

16 rf (mm)

24

32

(b) Laminar flame speed versus radius

Figure 2.20 Measured ﬂame radius versus time and the deduced laminar ﬂame speeds for propane/air mixtures at various equivalence ratios (Tseng et al., 1993).

ﬂames for unstable preferential-diffusion conditions developed irregular surfaces at larger radii. These data points are represented by closed circles in Figure 2.20a and b. For the positive Markstein number conditions (Ma > 0), the ﬂames were stable; those data points are represented by open circles in these ﬁgures. The measured laminar ﬂame speeds of Tseng et al (1993) as a function of equivalence ratio for four hydrocarbon-air ﬂames were in general agreement with the experimental data measured in earlier works by various researchers. The relationship between preferential-diffusion/stretch interactions and the stability of thin ﬂames can be easily seen from Equations. 2.164 and 2.168. When the Markstein number is negative, the laminar burning velocity increases as the ﬂame stretch (or Ka* ) increases through Equation 2.164. Any wrinkles developed on the spherical ﬂame surface behave as per the ﬂame stretching at local points. If the local ﬂame surface is concave toward the combustion products, then the ﬂame stretch is positive, resulting in a positive Ka* . The consequence of the local stretch in such case will be an increase in the local laminar ﬂame speed resulting in growth of the bulged surface. For the local ﬂame surface that is convex toward the combustion products, the ﬂame stretch is negative, resulting in a negative Ka* . The consequence of the local stretch in such case will be a

EFFECT OF FLAME STRETCH ON LAMINAR FLAME SPEED

101

reduction in the local laminar ﬂame speed, resulting in slower displacement of the ﬂame surface at this location. As a result, the ﬂame surface will become more wrinkled. Thus, the negative Markstein number leads to an unstable ﬂame front. Conversely, if the Markstein number is positive, the laminar burning velocity decreases with increasing stretch (Ka* ), and similar bulges in the ﬂame surface become smaller so that the ﬂame is stable to preferential-diffusion effects. It is worthwhile to point out the reason for the increase of laminar ﬂame speed with radius for the stable ﬂames. When the ﬂame radius increases, the slope of the rf versus time curve increases; however, the stretch factor decreases by Equation 2.168 since the increase in the slope is much less pronounced than the increase in the ﬂame radius. This implies that the Ka* also decrease with ﬂame

3.0

3.0 METHANE/AIR @ 1 atm φ SYMBOL φ = 1.35

2.5 2.0

1.2

0 SL 1.5 SL

1.0 1.1

ETHANE /AIR @ 1 atm

0.6 0.7 0.9 1.0 1.1 1.2 1.35

2.5 2.0 0 SL 1.5 SL

0.9

1.0

0.7

1.0

0.5

0.6

0.5

0.0 0.0

0.1

0.2

0.3

0.4

0.0 0.0

1.4

0.1

1.1 0.8 1.2

φ = 1.0

1.6

0.2

Ka*

Ka*

(a)

(b)

3.0

φ SYMBOL 0.8 1.0 1.1 1.2 1.4 1.6

0.3

0.4

3.0 PROPANE/AIR @ 1 atm

ETHYLENE /AIR @ 1 atm

2.5

2.5 φ = 0.8

2.0

2.0

0 SL 1.5 SL

0 SL 1.5 SL

1.2,1.0 1.3 1.4

1.0

1.6 1.8

0.5 0.0 0.0

0.1

0.2

φ SYMBOL 0.8 1.0 1.2 1.3 1.4 1.6 1.8

0.3

φ = 1.0

0.8 1.2 1.3 0.9 1.4

1.0 0.5

0.4

0.0 0.0

0.1

0.2

Ka*

Ka*

(c)

(d)

φ SYMBOL 1.6 0.8 0.9 1.8 1.0 1.2 1.3 1.4 1.6 1.8

0.3

0.4

Figure 2.21 Laminar burning velocity as a function of Karlovitz number and equivalence ratio for (a) methane/air mixtures; (b) ethane/air mixtures; (c) propane/air mixtures; and (d) ethylene/air mixtures (Tseng, Ismail, and Faeth, 1993).

102

LAMINAR PREMIXED FLAMES

radius. Since the Markstein number is positive, then the stretched laminar ﬂame speed (SL ) should increase as the ﬂame radius increases, ﬁnally approaching the unstretched laminar ﬂame speed SL0 at larger ﬂame radius (or smaller Ka* ) by Equation 2.164.

Based on the measured laminar ﬂame speeds and Ka* , the plots of SL0 /SL versus Ka* for various equivalence ratios were constructed by Tseng et al. (1993), as shown in Figure 2.21a to d for methane, ethane, propane, and ethylene/air

ﬂames, respectively. These plots show a linear relationship between SL0 /SL and Ka* . Thus, the Markstein number can be deduced from the slope of these

8

8

6

6

4

4 CORRELATION CORRELATION

Ma

2

Ma

2 0 −2 −4 −6 0.0

0 −2

METHANE/AIR @ 1 atm SOURCE STABLE UNSTABLE PRESENT DATA TAYLOR (1991)

0.5

1.0 φ

1.5

−4 2.0

−6 0.0

ETHANE/AIR @ 1 atm SOURCE STABLE UNSTABLE PRESENT DATA TAYLOR (1991)

0.5

(a) 8

6

6

4

4

−4 −6 0.0

0

SOURCE STABLE UNSTABLE PRESENT DATA TAYLOR (1991) PALM-LEIS & STREHLOW (1969) FRISTROM (1965)

0.5

1.0 φ (c)

CORRELATION

Ma

Ma −2

2.0

2

CORRELATION PROPANE/AIR @ 1 atm

0

1.5

(b)

8

2

1.0 φ

−2 −4 1.5

2.0

−6 0.0

ETHYLENE /AIR @ 1 atm SOURCE STABLE UNSTABLE PRESENT DATA TAYLOR (1991)

0.5

1.0 φ

1.5

2.0

(d)

Figure 2.22 Markstein number as a function of equivalence ratio for (a) methane/air mixtures; (b) ethane/air mixtures; (c) propane/air mixtures; and (d) ethylene/air mixtures (Tseng, Ismail, and Faeth, 1993).

MODELING OF SOOT FORMATION IN LAMINAR PREMIXED FLAMES

103

7 C8H18

Markstein number

5 C3H8 3

C2H5OH CH3OH

1 CH4 −1 0.0

H2 0.4

0.8

1.2

1.6

2.0

φ

Figure 2.23 Effect of equivalence ratio on Markstein number for various hydrocarbon/air ﬂames and H2 /air ﬂame (Bechtold and Matalon, 2001).

plots. It can also be seen that the Markstein number is independent of Ka* but strongly dependent on the fuel type and equivalence ratio. If the Karlovitz number is too large, ﬂame extinction could occur due to excessive stretching. However, in the cases measured by Tseng, Ismail, and Faeth (1993), this range of Ka* is not close to extinction conditions. On these plots, the open symbols (stable ﬂames) represent the cases with positive Markstein numbers and the closed symbols represent the cases with negative Markstein numbers (unstable ﬂames). The deduced Markstein lengths for each of the four hydrocarbon air ﬂames are shown in Figure 2.22 as a function of equivalence ratio. The four plots show interesting behavior. Among these four ﬂames, methane/air mixture shows a positive slope on the Markstein number versus the equivalence ratio plot. Unlike other hydrocarbon-air fuels, methane/air mixture shows unstable ﬂames at fuellean conditions. This behavior is similar to that of H2 /O2 /N2 ﬂame as observed by Kwon, Tseng, and Faeth (1992). This can be explained qualitatively using classical phenomenological theories of preferential-diffusion instability. Another plot showing the dependence of the Markstein number on equivalence ratio for various hydrocarbon air ﬂames and H2 /air mixture is shown in Figure 2.23. Again, it can be clearly seen that CH4 /air and H2 /air ﬂames show a trend opposite the rest of the hydrocarbon/air ﬂames.

2.5 MODELING OF SOOT FORMATION IN LAMINAR PREMIXED FLAMES

Environmental concerns have motivated the combustion community to study the pollutant formation in hydrocarbon air ﬂames; particularly soot formation. Most

104

LAMINAR PREMIXED FLAMES

fundamental studies of soot-generation phenomena in ﬂames are performed in laminar ﬂames due to complexity and limited understanding of turbulent reacting ﬂows. In turbulent ﬂames, the regions involving signiﬁcant soot reactions are thin, and the soot generation process is also an unsteady process, thereby posing problems for both measurements and numerical simulations. Therefore, the use of steady laminar ﬂames for studying the soot-generation processes in ﬂames is a very attractive and fundamental alternative.

2.5.1

Reaction Mechanisms for Soot Formation and Oxidation

Soot formation during the combustion of complex hydrocarbon fuels has been studied extensively in the past. A general mechanism of soot formation is shown in Figure 2.24. In order to explain the soot formation process and soot oxidation pathways, let us consider the combustion of a heavy hydrocarbon, C12 H23 , in a combustor. The vast majority of the fuel undergoes oxidation to ﬁnal products, indicated by the bold arrow in Figure 2.24. However, a portion of the fuel in fuelrich zone in the premixed ﬂames (or diffusion ﬂames) follows the reaction path to soot via pyrolysis reactions. These reactions break down the fuel molecules into unsaturated radicals and intermediates like acetylene (C2 H2 ), known as soot precursors. These soot precursors then react to form small polycyclic aromatic hydrocarbons (PAH). The growth of the PAH species continues through reactions in which acetylene is added to aromatic ring structures. These PAH collide with each other and form larger PAH, until they become liquid particles known as incipient soot particles. These incipient soot particles continue to grow by reactions with acetylene and smaller PAH, a process known as surface growth. They also collide and coalesce, resulting in formation of soot particles. These individual soot particles, known as primary soot particles, typically range in size from 30 to 50 nm regardless of the fuel used or the combustion system; the processes that limit primary particles to this size range are not known. The primary particles collide and stick together to form complex, convoluted clusters that Surface Growth and Coalescence Pyrolysis C12H23

CxHy H H2 C2H2

Inception

Coagulation

PAH C 2H 2 Competing Oxidation Reactions

Oxidation

CO2, CO, H2O

Figure 2.24 Soot formation process and oxidation pathways during combustion of hydrocarbon air mixtures.

MODELING OF SOOT FORMATION IN LAMINAR PREMIXED FLAMES

105

Fractal dimension = 1.8

Figure 2.25 Commonly observed soot particles in an ethene-oxygen premixed ﬂame (modiﬁed from Chakrabarty, 2009).

Toxic Material Metals Secondary Sulfate and Nitrate Organic Carbon Compounds Elemental Carbon Core

Figure 2.26 Structure of a single soot particle from a diesel engine exhaust (not to scale) (modiﬁed from http://www.catf.us/diesel/dieselhealth/faq.php?site=0).

can be several hundred nanometers in size. All along the reaction path to the ﬁnal soot particles, competing oxidation processes are occurring that consume the growth species, PAH, and the soot particles themselves. These oxidation processes mostly occur through reactions involving highly reactive radical species such as the hydroxyl radical (OH) and oxygen atoms. A micrograph showing the morphological structure of soot in an ethene-oxygen premixed fuel-rich ﬂame is shown in Figure 2.25. The composition of a single soot particle from a diesel engine exhaust is shown in Figure 2.26. In term of the underlying science, soot formation may be viewed as being comprised of four major processes, as described by Frenklach (2002): (1) homogeneous nucleation of soot particles; (2) particle coagulation; (3) particle surface reactions (growth and oxidation; the two competing reaction processes); and

106

LAMINAR PREMIXED FLAMES

(4) particle agglomeration. These major processes were also discussed earlier by Calcote (1981), Haynes and Wagner (1981), Glassman (1988), Kennedy (1997), and Palmer and Cullis (1965), Richter and Howard (2000). Kennedy made an extensive review of models of soot formation and oxidation and classiﬁed the models into three categories: (1) empirical correlations; (2) semiempirical approaches that solve rate equations for soot formation with some input from experimental data, and (3) detailed models that solve the rate equations for elementary reactions that lead to soot formation. The review by Kennedy (1997) discusses soot formation in both premixed and diffusion ﬂames. Some of these ﬂames are also turbulent diffusion ﬂames; however, the detailed models are all associated with laminar ﬂames. In this chapter, the discussion of soot formation and oxidation is focused on laminar premixed ﬂames. The soot formation in laminar diffusion ﬂames are covered in the next chapter. 2.5.1.1

Empirical Models for Soot Formation

Calcote and Manos (1983) studied the effect of molecular structure on the onset of soot formation in both premixed and diffusion ﬂames and found great similarity. They deﬁned a “threshold soot index” (TSI) that ranks fuels from 0 to 100 (0 = least sooty) and is independent of the particular experimental apparatus in which the data were obtained. The introduction and utilization of TSI can be beneﬁcial to use all of the literature data to interpret molecular structure effects of a fuel on the onset of soot formation and thus arrive at rules for predicting the effect of molecular structure for compounds that have not yet been measured or to correlate the results from one experimental system with another. In premixed ﬂames, the lower the critical carbon-to-oxygen ratio (C/O) or the lower the critical equivalence ratio φc , the greater the tendency of the fuel to soot. The critical equivalence ratio is deﬁned as an equivalence ratio above which the onset of soot formation begins. In the hydrocarbon air premixed ﬂames studied, the value of φc is generally greater than 1, implying that sooting occurs in mostly fuelrich mixtures. Therefore, TSI should be deﬁned by a parameter that reﬂects the correlation of incipient sooting with the oxidative chemistry of the fuel and not with the transport properties due to the measurement apparatus. Such a parameter would be the critical equivalence ratio φc . The TSI is deﬁned as: TSI = a − bφc

(2.170)

where parameters a and b are apparatus-dependent constants and φc is the reported results from different experiments. The critical equivalence ratio for an identical fuel could be two different values in two different operating conditions. TSI is independent of the test equipment and conditions. Readers interested in the way a and b constants were determined should refer to Calcote and Manos (1983). The effect of number of carbon atoms in hydrocarbon fuels on TSI is shown in Fig. 2.27. The effect of weight percentage of hydrogen in different families of hydrocarbon fuels on TSI is exhibited as various regions in Fig. 2.28 by Calcote and Manos, (1983).

MODELING OF SOOT FORMATION IN LAMINAR PREMIXED FLAMES

107

100 C

C

90 C

80

C

C

C

C

C

C

C

C

Tendency to Soot. TSI

70 N Alkanes 60 50

N Alkenes

40 30 20 10 0

1

C

C

2

3

4

5 6 7 8 Number of Carbon Atoms

9

10

11

12

Figure 2.27 Effect of number of carbon atoms on the TSI of various hydrocarbons (Calcote and Manos, 1983).

Threshold Soot Index, TSI

100

75

Aromatics

Alkanes

Cyclic Alkenes

50

CH4

Alkynes 25

Alkenes C2H2

0

5

10 15 20 25 Weight percentage of hydrogen in hydrocarbon fuels

Figure 2.28 Effect of weight percent hydrogen on the TSI of various hydrocarbons in premixed ﬂames (modiﬁed from Olson and Pickens, 1984).

Earlier, Street and Thomas (1955) reported this qualitative, relative ordering for the tendency to soot of hydrocarbons in premixed ﬂames: acetylene < alkenes < isoalkanes < n-alkanes <

monocyclic aromatic < naphthalenes. hydrocarbons

Gill and Olson (1984) adopted the deﬁnition of TSI in Equation 2.170 to predict soot thresholds for fuel mixtures by using the individual TSIs of each

108

LAMINAR PREMIXED FLAMES

component. The mixing rule that they selected for premixed ﬂames was Xj (1.1)TSIj (2.171) (1.1)TSImix = j

where TSIj and X j are TSI value and mole fraction of the j th component in the fuel mixture. Many measurements have been made of the critical C/O ratio at the onset of sooting in premixed ﬂames. Takahashi and Glassman (1984) attempted to collapse premixed ﬂame data by considering an effective equivalence ratio, ψ = [C + H/2)]/O. Their analysis of the experimental results for the critical ψ at the onset of soot led them to conclude that the sooting tendency of a fuel was a function of its ﬂame temperature, the C/H ratio, and the number of C atoms but not the structure of the fuel. They considered a wide range of aliphatic and aromatic fuels that were burned in a premixed ﬂame at a ﬁxed value of the adiabatic ﬂame temperature. The critical ψ value was plotted as a function of the number of C-C bonds in which double carbon bonds were counted as 2 and triple bonds were counted as 3. This simple analysis yielded an excellent correlation of the experimental data. 2.5.1.2 Detailed Models for Soot Formation and Oxidation The gas-phase mechanism describes the pyrolysis and oxidation of the fuel molecules and the main combustion reactions, which form carbon dioxide and water. The detailed gas-phase reaction mechanism for formation and growth of polycyclic aromatic hydrocarbons (PAH) consists of 527 reactions and 99 chemical species (Wang and Frenklach, 1997). The reactions and their forward rate coefﬁcients are given in Appendix D. The formation and growth of linear aliphatic and PAH is described by a detailed chemical mechanism up to a molecule called pyrene. Pyrene is a consisting of four fused benzene rings, resulting in a ﬂat aromatic system. Further growth of the aromatic molecules is assumed to follow the H-abstraction-C2 H2 -addition (HACA) reaction sequence (Frenklach and Wang, 1991). This reaction mechanism was extended by the implementation of a new vinyl oxidation submechanism. This reaction class includes all those acetylene addition processes that show an intermediate H-radical migration, as described in Frenklach and Carmer (1999). The particle inception step is modeled through the coalescence of PAHs. The dynamics of the solid soot phase includes coagulation, PAH addition to the soot surface as well as heterogeneous surface reactions. The reactions of gas-phase species with the soot surface determine the rate of growth or shrinkage of the particles, which depends on the surrounding gas mixture composition and temperature. The particles are oxidized by OH-radicals and O2 , whereas they grow by the addition of acetylene, which is identiﬁed by Bockhorn (1994) to be the main growth species in laminar premixed ﬂames.

MODELING OF SOOT FORMATION IN LAMINAR PREMIXED FLAMES

109

2.5.1.3 Formation of Aromatics As discussed by Frenklach (2002), the transition of gas-phase species to solid particles is probably the least understood part of the soot formation process. In the initial reaction model (Frenklach, Clary, Gardiner, and Stein, 1985), “soot” was deﬁned as the mass accumulated in PAH species above a certain size; in other words, the transition from gaseous species to solid particles was assumed to take place as the result of a purely chemical growth. While this deﬁnition basically accounts for the amount of soot mass, it greatly underpredicted particle sizes. In the follow-up work (Wang and Frenklach, 1997), the model of nucleation was expanded. At a critical size, PAH species begin to stick to each other during collision, thus forming PAH dimers. PAH dimers collide with PAH molecules, forming PAH trimers, or with other dimmers, forming PAH tetramers, and so on, all of this taking place while individual PAH species keep increasing in size via molecular chemical growth reactions. In this manner, the PAH clusters evolve into solid particles. The formation of dimers was assumed to mark the emergence of the “solid” particle phase in Wang and Frenklach’s model. The essence of this model is the accumulation of particle mass via chemical reactions with gaseous precursors occurring simultaneously with the growth of particle size by collisions among PAH molecular species and clusters. Such a model provided a consistent transition from gas-phase aromatics chemistry to particle surface reaction and aggregation dynamics. Therefore, there are two mechanisms for soot formation (chemical growth and molecular collisions), described in the subsequent paragraphs. Let us consider PAH species growing chemically at the edges while forming clusters when colliding. We can envision two regimes. The ﬁrst is realized when the chemical growth is limited by a high degree of reaction reversibility. Under these conditions, the reaction pathways of aromatic growth are switched quickly to an acetylene-based HACA and the growing PAH species are dominated by compacting most thermodynamically stable species (stabilomers), as only those can survive such conditions; other chemical species with weaker bonding will decompose. The stabilomers form physically bound clusters upon collision. This regime seems to be able to describe soot formation in higher-temperature combustion. (Appel, Bockhorn, and Wulkow, 2000; Yoshihara, Kazakov, wang, and Frenklach, 1994). Markatou, Wang, and Frenklach, 1993; Frenklach and Wang, 1991; Wang, Sung, and Law, 1996; Mauss and Bockhorn, 1995; Balthasar, Heyl, Mauss, Schmitt, and Bockhorn 1996. The other regime is realized when the formation of chemical bonds is not counterbalanced heavily by fragmentation and hence there is no thermodynamic limitation on the structural form that the growing molecule may assume. This leads to the formation of a variety of chemical structures as a result of molecular collisions between different gaseous partners. Having a diverse molecular base may well lead to a network of aromatic-aliphaticlinked structures.

110

LAMINAR PREMIXED FLAMES

Both paths for soot formation occur in real cases. Readers should understand that these two models of nucleation should not be considered two separate models but different realizations of fundamentally the same underlying mechanism. It also follows that various patterns of behavior, exhibiting different regimes under different conditions spanning between the two extremes should be anticipated. 2.5.1.4 Growth of Aromatics The HACA mechanism is a popular way to describe the growth of aromatics. It consists of two repetitive reaction sequences of two principal steps: (1) abstraction of a hydrogen atom from the reacting hydrocarbon by a gaseous hydrogen atom, followed by (2) addition of a gaseous acetylene molecule to the radical site formed, Ai + H → Ai− + H2 (R2.1)

Ai− + C2 H2 → Products

(R2.2)

Here Ai is an aromatic molecule with i peri-condensed rings and Ai− is its radical. The term “peri-condensed” refers to an aromatic compound in which three or more rings share common carbon atoms. The essence of the two-step feature is described next. The ﬁrst step (R2.1) activates a molecule to further growth by converting it to a radical. Of course, this can be accomplished in many different ways, and many such possibilities have been tested. It turns out that under the conditions of typical shock-tube and ﬂame experiments (Frenklach and Warnatz, 1987), which provide most current training experimental data, the H-abstraction step typically dominates the process. The key feature of the ﬁrst step of HACA is its reversibility. The reverse steps can be the reverse direction of the H abstraction itself or other reactions, such as the combination with a gaseous H atom, Ai− + H2 → Ai + H (R2.3) Ai− + H → Ai

(R2.4)

The contribution of reaction (R2.4) as compared to the reaction (R2.3) increases with pressure and molecular mass (i.e., as the rate coefﬁcient of R2.4 approaches its high-pressure limit). The reversibility of the acetylene addition step, reaction R2.2 or, more precisely, the degree of its reversibility is what determines whether this step will contribute to molecular growth. Based on calculations of Frenklach and coworkers, reaction R2.2 results in entropy loss, which makes this reaction highly reversible. The reaction R2.2 was modiﬁed by the addition of a hydrogen atom as a product (shown by reactions R2.5 and R2.6), which have less entropy loss than reaction R2.2. These reactions are still reversible: Ai− + C2 H2 → Products + H (R2.5)

MODELING OF SOOT FORMATION IN LAMINAR PREMIXED FLAMES

Ai− + C2 H2 Ai C2 H + H

111

(R2.6)

The recovery of entropy in a reaction is not sufﬁcient to stop it from becoming reversible. In addition to the entropy, the recovery of energy should be large. In such case, the reaction becomes more irreversible, thereby resulting in the formation of stabilomers. The coupling between the thermodynamic resistance of the reaction reversibility and the kinetic driving force is the basis of the HACA model. The critical role of this coupling for aromatic growth became apparent from the very beginning of the kinetic simulations, as shown by Frenklach et al. (1985). Among numerous possible reaction pathways, the one that emerges as dominant pathway is not necessarily the shortest one or the one that does not require costly reactivation, but the one with the largest reaction afﬁnity (Prigogine and Defay, 1954): that is, with the smallest thermodynamic resistance. The latter is attained most effectively in the case of PAH growth by a repetitive reactivation by hydrogen atoms while hopping from one island of stability to another. This thermodynamics-kinetics coupling explains the reason for overcoming the energy barrier (Stein and Fahr, 1985) to PAH growth and the appearance of stabilomer sequences during the growth (Bockhorn, Fetting, and Wenz, 1983). Acetylene is obviously not the only species that can be envisioned to propagate the growth of aromatic rings. Several such proposals have appeared, including those involving methyl, propargyl, and cyclopentadienyl. The focus of these proposals is on the resonantly stabilized structure of the reacting radicals, which, in a general sense, follows the suggestion of Glassman (1979) that hydrocarbons with conjugated structures and their derivatives are critical intermediates to soot nucleation. Indeed, although the initiation of aromatics formation may proceed through different reactions speciﬁc to the fuel used and conditions applied, numerical simulations indicate that such initial pathways quickly relax to the acetylene-addition path. For instance, in the pyrolysis of benzene, the aromatics growth is initiated by the formation of biphenyl, • +

+

H

(R2.7) but this following growth proceeds via acetylene addition: • + C2H2

+ H

(R2.8) The same pattern appears under different conditions and fuels as well as in ﬂames with increased stoichiometry and pressure.

112

LAMINAR PREMIXED FLAMES

2.5.1.5 Migration Reactions Recent theoretical investigations have revealed new reaction pathways for aromatic ring growth, such as the enhanced formation of ﬁve-member aromatic rings: H •

H •

H C2H2

H H

H •

(−H)

(R2.9) R2.10 depicts enhanced formation of six-member aromatic rings: H H •

• H

C2H2

H H H •

(−H)

(R2.10) R2.11 depicts interconversion of ﬁve- and six-member rings: •

C2H2 •

•

(R2.11)

113

MODELING OF SOOT FORMATION IN LAMINAR PREMIXED FLAMES

R2.12 shows migration of the cyclopenta ring along zigzag aromatic edges: •

•

(R2.12) All of these pathways have one critical mechanistic feature in common: The reaction pathway is induced or assisted by hydrogen atom migration. The kinetics and thermodynamics of such migration were investigated by Moriarty, Brown, and Frenklach (1999) for the reaction R2.13 at several levels of quantum ab initio theory. • •

(R2.13)

The most reliable level of theory results in reaction rates sufﬁciently fast for these reactions to play a role in high-temperature aromatic chemistry. For large aromatic structures, such as condensed multiring soot precursors or those developed at the edges of the soot particle surface, the hydrogen migration step opens an additional and somewhat faster channel for reaction pathways R2.9 and R2.10, with the result of at least doubling the rate of cyclization. For some aromatics, the H migration may be ineffective because when they react, there could be excessive energy barriers to rotation (Bittner and Howard, 1981 and the H migration may be ineffective because when they react, there could be excessive energy barriers to rotation Moriarty and Frenklach, 2000) or even may run in reverse, toward ring fragmentation since the equilibrium constant of reaction R2.13 is signiﬁcantly lower than unity (∼0.2 at 1500 K) (Moriarty, Brown, and Frenklach, 1999). •

•

(−H)

2.5.1.6

•

(R2.14)

Oxidation of Aromatics

Parallel to aromatics growth is aromatics oxidation. The primary mechanism seems to be the oxidation of aromatic radicals by O2 ; oxidation by OH is rather unimportant, as deduced from numerical simulations of laminar premixed ﬂames (Frenklach and Warnatz, 1987). The largest effect in the oxidation of aromatics occurs at the very beginning of the aromatics growth, at the phenyl stage. This

114

LAMINAR PREMIXED FLAMES

is due to the rapidly decreasing concentration of O2 in fuel-rich environments sustaining aromatics growth. According to the HACA model, the growth is governed by the H atom production, but H atoms also destroy O2 ; hence, as the aromatics growth progresses, it should be accompanied by the depletion of O2 . The latter observation may explain why soot inception usually appears in the vicinity (in time or space) of the main combustion zone, in an environment rich in H atoms and lean in O2 molecules. The mechanism of soot oxidation by OH is still poorly understood. Oxidation of aromatics removes carbon mass from further growth. However, even more important is the removal of mass at earlier stages—those preceding the aromatics formation. Numerical simulations identify oxidation of C2 H3 as the key point of branching between carbon growth and carbon oxidation. The effect of oxidation at this small-molecule level is twofold. On one hand, it diverts carbon mass from further growth. On the other hand, added in relatively small quantities in high-temperature pyrolytic environments, molecular oxygen actually promotes formation of soot by building up the radical pool, and speciﬁcally H atoms. The latter phenomenon, identiﬁed in experimental shocktube studies (Frenklach et al., 1985) and parallel computational analysis, recurs repeatedly in analyses of diffusion ﬂames (see Pels Leusden and Peters, 2000). 2.5.2

Mathematical Formulation of Soot Formation Model

A detailed model of soot formation and oxidation should treat both gas and solid phases separately and should also provide information about the interactions between them. For laminar premixed ﬂames, the prediction of soot formation and oxidation with a kinetic model requires knowledge of the concentrations of the relevant chemical species and ﬂame temperature. Therefore, the conservation equations for the energy and mass fractions of the corresponding chemical species should be solved. Appel, Bockhorn, and Wulkow (2001) performed a theoretical study on the evolution of soot particle size distributions in an acetylene/oxygen/argon laminar premixed ﬂame. For a one-dimensional steady-state laminar premixed ﬂame, the gas-phase governing equations are: Ns Ns dT dT d dT ρuCp l Cp,i ρYi Vi − hm ˙ i − q˙r = − i ω dy dy dy dy i=1 i=1 dYi d ρu = − (ρYi Vi ) + ω˙ i Mwi dy dy

(2.172) (2.173)

where Yi = mass fraction of the i th species Vi = diffusion velocity of the i th species ω˙ i = chemical production rate of the i th species with the molecular weight Mwi Cp,i = heat capacity of the i th species hi = molar enthalpy of the i th species q˙r = radiation heat ﬂux.

MODELING OF SOOT FORMATION IN LAMINAR PREMIXED FLAMES

115

The chemical rate terms also include the rates of formation and destruction of chemical species due to heterogeneous reactions with the soot surface. The mechanism for the heterogeneous surface reactions reads as shown next. Csoot i H + H Csoot i + H2 soot

C

iH

+ OH C

soot

i

+ H2 O

Csoot i + H → Csoot i H soot

C

i

+ C2 H 2 → C

soot

C

iH

+ OH → C

soot

i+1 H

i−1

(R2.16) (R2.17)

+H

Csoot i + O2 → Csoot i−1 + 2CO soot

(R2.15)

+ HCO + CH

(R2.18) (R2.19) (R2.20)

Since the soot particles grow mainly by the addition of an acetylene molecule; therefore, the smallest difference between two soot particles is the mass of two carbon atoms. The rate of acetylene consumption by surface growth of soot can be expressed as: MwC2 H2 ω˙ C2 H2 = −αs kR18 XC2 H2 · χC Soot ·S (2.174) NA where χC Soot = fraction of radical sites on the surface of the soot particles αs = fraction of the surface area S that is available for chemical reactions (Appel et al., 2000) NA = Avogadro number The particles are assumed to be spherical so that the surface area of a soot particle with i number of monomers (i.e., C = C units) can be determined by Equation 2.175. 3m1 2/3 2/3 Si = 4π i (2.175) 4πρSoot where m1 is the mass of a monomer unit, which is a C2 -group. The diameter of these particles is given by 3m1 1/3 1/3 i (2.176) di = 2 4πρSoot The heat loss due to radiation of soot particles (q˙r ) can be modeled by using the temperature variation of the radiation coefﬁcient of soot which gives the simple expression q˙r = Cfv T 5 (2.177) where C is a constant and fv represents the soot volume fraction. The solution of Equations 2.172 and 2.173 for a number of chemical species prescribed by the chemical mechanism proposed by Frenklach et al. (1995) and Mebel, Diau,

116

LAMINAR PREMIXED FLAMES

Lin, and Morokuma (1996) can form the framework for the simulation of the dynamics of the PAH and the soot particle ensemble. As mentioned earlier, the surface growth mechanism considers acetylene to be the main growth species. Therefore, the size distribution of the soot particles is discretized by the mass of two carbon atoms. From this proposition, a set of balance equations for the soot particle number densities of the different particle sizes can be derived as: d (Ni /ρ) d 1 dT d (Ni /ρ) d ρu ρDi + 0.55u = Ni + S (Ni ) (2.178) dy dy dy dy T dy In Equation 2.178, the second term on the right-hand side describes the transport of soot particles due to thermophoresis. Thermophoresis is phenomenon observed when a mixture of two or more types of particles is subjected to a temperature gradient and it induced motion of these particles. It is similar to the Soret effect in gas mixtures. In laminar premixed ﬂames, the region of steep gradients in temperature exhibits low soot particle number densities, so that the effect of thermophoresis on the soot particle ensemble is small. The diffusion coefﬁcient of soot particles in the single-size classes is denoted by Di . Seinfeld (1986) showed that the diffusion coefﬁcient of an aerosol of very small particles is proportional to d −2 , where d is the diameter of the particles. The term S(Ni ) represents the source of particles Ni (which are built up from i number of monomer units) from all underlying chemical and physical processes of the model for soot formation. Soot particles can consist of about 107 carbon atoms; therefore, the number of size groups for the soot particles and their balance equations would be of the same order of magnitude. This makes it impossible to solve such a large system of partial differential equations (PDEs). An efﬁcient way of deriving information from this set of differential equations was introduced by Frenklach and Harris (1987). This approach is commonly known as method of moments. The deﬁnition of the statistical moments of the particle size distribution function (PSDF) is given as: Mr =

∞

i r Ni

(2.179)

i=1

Equation 2.179 leads to a new set of balance equations for the r th moment of the size distribution: Soot d MrSoot d M(r−2/3) d ρu = ρD1 dy ρ dy dy ρ

1 dT Soot d 0.55u + + S MrSoot (2.180) Mr dy T dy

MODELING OF SOOT FORMATION IN LAMINAR PREMIXED FLAMES

117

The method of moments is a strong numerical tool for the computation of particle number densities and soot volume fractions. However, the full details about the soot particle and the PAH size distributions are not provided, and the impact of different models or the underlying physics on the PSDF is hard to quantify. From the numerical point of view, furthermore, the method of moments has the disadvantage that a closed formulation of the equations of the moments is often not possible. These problems can be solved by a logarithmic Lagrangian interpolation. (see Frenklach and Harris, 1987). From these considerations, a numerical method that supplies the full particle size distribution at a high resolution is a necessary component for the modeling of soot formation. A high resolution is the major computational problem of the sectional method of Gelbard and Seinfeld (1978). Wulkow (1996) formulated a discrete hp-Galerkin (see Solin, Segeth, and Dolezel, 2003) method for polymerization reactions. It is a powerful algorithm for the calculation of different distributions up to any polymer degree or particle size. For modeling of soot formation, the algorithm allows the solution of the system of differential equations describing the dynamics of particles of speciﬁc sizes in a Lagrangian coordinate. dNi = f (N1 , N2 , . . . . . . . . Nm ) i = 1, 2, . . . M dt

(2.181)

In Equation 2.181, Ni is the number density of particles, which are built up from i number of monomer units. The algorithm approximates the size distributions by a multilevel hp-Galerkin method with a work-oriented reﬁnement strategy. The particle size axis is subdivided into several intervals and on each j of these intervals an expansion Ni approximates the size distribution function of N. j pl j Ni j = akl tkl (i) (2.182) Il

k=0

In Equation 2.182, j is the number of the level and l is the number of the interval of size distribution. The polynomials tkl (i) are discrete Chebyshev polyj nomials of the degree k . The number of expansion coefﬁcients pl may differ from interval to interval such that the size distribution can be resolved by varying grid and order. A crucial feature of this type of approximation is that the number of degrees of freedom is low. This is particularly important in the presence of the coagulation operator, because the evaluation of a convolution sum with an arbitrary kernel function is very costly. Coagulation is described by the Smoluchowski equation (Smoluchowski, 1917) i−j ∞ dNi 1 βj,i−j Nj Ni−j − βi,j Ni Nj (2.183) = dt 2 j =1

i=1

118

LAMINAR PREMIXED FLAMES

Here, βi,j is the size-dependent frequency factor, which can be determined from the formula of Brownian motion: 2 3m1 1/6 6kb T 1/2 1 1 1/3 · (2.184) βi,j = 2.2 + · i + j 1/3 4πρs ρs i j In the work of Appel, Bockhorn, and Wulkow (2001), the mass density of soot was assumed to be ρs = 1.8 g/cm3 (Frenklach and Wang, 1991), and m 1 indicates the mass of a monomer unit, which is a C2 -group. The hp-discretization of the size classes is combined with a time discretization by means of the Rothe method (Rothe, 1930). This technique was introduced by Bornemann (1991) for parabolic differential equations and was transferred and extended to poly-reaction kinetics in Wulkow (1996). For an approximation of the solution size distribution function N(t + τ ) between the time from t to t + τ , (N1 ), the semi-(linear)-implicit Euler scheme can be applied with the predictor steps given as: (I − τ A) N = τf (ϕ) N1 = ϕ + N

(2.185)

and the correction step given as: 1 (I − τ A) η1 = − τ 2 Af (ϕ) 2 2 1 N = N + η1

(2.186)

such that no additional linear system has to be treated (Deuﬂhard, 1985). In these two sets of equations, the vector N contains information about the PSDF (i.e., the number densities Ni of the size class i ), I is the identity matrix, the matrix A contains the derivatives (dNi /dNj ), which describe the dynamics of the system, and f (ϕ) is the source vector on the right-hand side of Equation 2.181. The calculated major gas phase species and temperature proﬁles for an acetylene/oxygen/argon premixed ﬂame (Appel, Bockhorn, and Wulkow, 2001) with the experimental proﬁles (Bockhorn, Fetting, and Wenz, 1983) are compared in Figure 2.29. With the exception of the mole fraction of CO2 , the agreement between simulation and experiment is excellent. CO2 is overpredicted by about 20%. The calculated temperature ﬁts the measured ones as well. It can be seen in Figure 2.29 that the ﬂame temperature decreases with higher heights above the burner. In the simulation, this effect was attained by modeling the radiation as given in Equation 2.177. In Figure 2.30, the measured mole fractions of the higher linear hydrocarbons, benzene, and naphthalene are compared with the predicted results from numerical simulation. The agreement is acceptable, although there is a general shift of

MODELING OF SOOT FORMATION IN LAMINAR PREMIXED FLAMES

119

2000 T

1800

10−1

1400 H2 O2 H2O CH4 CO CO2 C2H2

1200 1000

10−2

Mole fraction

Temperature/ K

1600

800 600

0

2

6 4 Height above burner/cm

10−3 10

8

Figure 2.29 Comparison of calculated and measured proﬁles of ﬂame temperature and mole fractions of major combustion species in an acetylene/oxygen/argon ﬂame (Appel, Bockhorn, and Wulkow, 2001). 10−2 C4H2

Mole fraction

10−3

C6H2

Benzene

10−4

C2H4 10−5 Naphthalene

10−6

0

2

4

6

8

10

Height above burner/cm

Figure 2.30 Comparison of calculated and measured proﬁles of mole fractions of larger linear aliphatic and aromatic hydrocarbons (Appel, Bockhorn, and Wulkow, 2001).

calculated results to the left-hand side in comparison with the measured data. The reason for this could be the distortion of the ﬂame by the sampling probe in the experiments. The quality of description of the chemical reactions in the gas phase is crucial for the simulation of soot formation, because heterogeneous surface growth reactions, which are dependent on the hydrogen atom, are the most important sources

120

LAMINAR PREMIXED FLAMES

of soot mass in low-pressure premixed ﬂames. The results given in Figures 2.29 and 2.30 demonstrate that the model for the gas-phase reactions gives a good description of the structure of the studied ﬂame. These results increase the conﬁdence of their model to study the PAH growth and soot particle dynamics. The PAH formation, growth, and oxidation mechanisms are given next. PAH Formation C14 H9 + C2 H2 → PAH8 + H

(R2.21)

where PAH8 refers to pyrene (C16 H10 ) that consists of eight monomer (C=C) units. PAH Growth PAHi + H → PAHi · + H2 (R2.22) PAHi + OH → PAHi · + H2 O

(R2.23)

PAHi · + H2 → PAHi + H

(R2.24)

PAHi · + H2 O → PAHi + OH

(R2.25)

PAHi · + H → PAHi PAHi · + C2 H2 → PAHi+1 + H

(R2.26) (R2.27)

PAH Oxidation PAHi · + O2 → PAHi−1 + Products

(R2.28)

PAHi + OH → PAHi−1 + Products

(R2.29)

PAHi + PAHj → Sooti+j

(R2.30)

PAHi · + PAHj → Sooti+j

(R2.31)

PAHi + Sootj → Sooti+j

(R2.32)

Particle Inception

PAH Addition

The effects of particle inception and PAH addition to the surface on the evolution of the PAH size distribution are shown in Figure 2.31. The three plots in this ﬁgure demonstrate the predicted results from three different PAH formation (and oxidation) mechanisms. Figure 2.31a shows the predicted results when only PAH growth and oxidation (from reactions R2.21 to R2.29) are considered without the inclusion of particle inception and PAH addition to soot. Figure 2.31b shows the predicted results when PAH growth and oxidation as well as particle inception are included. Figure 2.31c shows the predicted results with the full model for PAH formation, oxidation, inception, and addition to soot. In Figure 2.31a, large PAH molecules are formed due to the growth reactions. After a few milliseconds, the oxidation reactions no longer compete with the PAH growth reactions because the concentration of molecular oxygen and hydrogen

MODELING OF SOOT FORMATION IN LAMINAR PREMIXED FLAMES 2.5 1012

121

8 ms 10 ms 12 ms 14 ms

2.0 1012

16 ms

1.5 1012 1.0 1012 4 ms

0.5 1012

6 ms

2 ms

0

(a) PAH-number density, (cm−3 C-atom−1)

11

3.0 10

4 ms

2.5 1011

6 ms

2.0 1011 1.5

8 ms

1011

1.0 1011

10 ms 12 ms 14 ms 16 ms

0.5 1011 2 ms

0

(b) 2.0 1011 1.5

4 ms

1011

1.0 1011

6 ms

0.5 1011 0

8 ms

0

10

10 ms 12 ms 14 ms 16 ms

2 ms 20 30 40 50 Number of carbon atoms (c)

60

70

Figure 2.31 Calculated PAH size distributions with different model assumptions (modiﬁed from Appel, Bockhorn, and Wulkow 2001).

radicals are low. As the reaction time increases, the peak shifts toward the PAHs with higher number of carbon atoms. In Figure 2.31b, the PAH size distribution is totally different. The PAHs are removed from the distribution due to soot particle formation. This effect is more pronounced with increasing PAH concentrations. After 12 milliseconds (ms) soot formation competes with the PAH formation reaction (R2.21). In this case, the maximum PAH number density is about one order of magnitude lower that those shown in Figure 2.31a. During the reaction time between 10 and 16 ms, larger PAHs with up to 20 carbon atoms are formed.

122

LAMINAR PREMIXED FLAMES

3.0×1011

2.5×10−8

2.5×1011

2.0×10−8

2.0×1011

1.5×10−8

1.5×1011 1.0×10−8

1.0×1011

0.5×10−8

0.5×1011 0 0

10

20 30 Reaction time, ms

40

Soot volume fraction, fv

Total soot particle number density, cm−3

After the decay of the growth rates, the PAH formation reaction again dominates the shape of the PAH distribution, so that the peak of PAH number density is at 16 to 20 carbon atoms. The effect of PAH consumption due to a transition into the solid soot phase is more pronounced when the PAH surface addition (i.e., reaction R2.32) is taken into account. In Figure 2.31c, all submodels in the overall PAH reaction model are considered. The rate of PAH soot-collisions is dependent on the number density and the size of the soot particles in the ﬂame. Therefore, the PAH distribution is rather narrow during the entire reaction time for this case. From these simulations, the inﬂuence of the single processes—that is, PAH formation, PAH growth, particle inception, and PAH addition to the soot surface—on the evolution of the PAH size distribution can be understood. Particle inception and PAH formation seem to have the largest impact on PAH dynamics, whereas the growth of the PAHs is relatively insigniﬁcant in the investigated ﬂame. The addition of PAHs to the soot surface is more important for the shape of the soot particle size distribution because it competes with particle inception. Bockhorn, Fetting, and Wenz (1983) and Wenz (1983) measured the concentration proﬁles of a number of aromatic species in the ﬂame by using molecular beam sampling and coupled gas chromatography/mass spectrometry. Appel, Bockhorn and Frenklach (2000) compared their calculated results using method of moments and discrete hp-Galerkin methods with these experimental data. This comparison in terms of total soot particle number density and the soot volume fraction at various reaction times is shown in Figure 2.32. It can be seen that the particle inception model is capable of reproducing the number of particles in the studied ﬂame. It can also be seen that both numerical methods generate quantitatively comparable results with experimental data.

0 50

Figure 2.32 Comparison of experimental data for the total particle number densities and the soot volume fractions. The dashed lines represent results from the method of moments and the solid lines represent result using discrete hp-Galerkin method (modiﬁed from Appel, Bockhorn, and Wulkow, 2001).

Particle number density, cm −3 nm−1

MODELING OF SOOT FORMATION IN LAMINAR PREMIXED FLAMES 109 4 ms 6 ms 108 8 ms 10 ms 107 12 ms 14 ms 106

123

40 ms 45 ms 50 ms

105 104 5

10

15

20

25

30

35 ms 30 ms 25 ms

22 ms 20 ms 18 ms

16 ms

2 ms

103

Particle diameter, nm

Figure 2.33 Evolution of the soot particle size distributions (modiﬁed from Appel, Bockhorn, and Wulkow, 2001).

Figure 2.33 shows the evolution of soot particle size distributions with various reaction times as calculated by Appel, Bockhorn, and Wulkow (2001). Particle inception and coagulation dominate the evolution of soot particle formation and growth in the premixed ﬂame. Based on Appel et al.’s work, surface growth (i.e., PAH addition to soot) affects the soot particle size distribution during the reaction time between 10 to 16 ms. In this duration, the soot particle diameter increase mainly by heterogeneous surface reactions of soot with acetylene (shown by reaction R2.18). The width of the distribution increases rapidly during this process, as shown in Figure 2.33. After the narrow surface growth zone, coagulation becomes the dominant source for the evolution of the PSDF. A comparison of the computed particle size distributions (Appel, Bockhorn, and Wulkow, 2001) with experimentally determined distributions (Heddrich, 1986; Bockhorn et al., 1988, and Bockhorn, Fetting, and Wenz, 1983) is shown in Figure 2.34. The experimental results were obtained from molecular beam sampling of the soot in the premixed acetylene/oxygen/argon ﬂame and TEM micrographs of the soot particles. The relative particle number density is the ratio of the number of particles of a certain size and the total number of examined particles in the experiments. The comparison between calculations and measurements were made at four different heights above the burner exit plane (i.e., 14, 20, 30, and 40 mm). At lower heights above the burner, the calculated soot particle size distributions show smaller particles than experimental data. There is better agreement between the calculated and measured size distributions at 30 and 40 mm above the burner. Readers may wish to review the results from a joint volume-surface-hydrogen model for analyzing the effects of temperature on the formation of soot particles in a paper published by Blanquart and Pitsch (2009).

14 mm

40 30 20 10 0 0

2

4

6

8

8 107 6 107 5 107 4 107 3 107 2 107 1 107 0

10 12

1.5 107

30 25 20 15

20 mm 1.0 107 0.5 107

10 5 0

0 0

4

2.0 106 1.5 106 1.0 106 0.5 106 0

0

4 8 12 16 20 24 28 32 Particle diameter, nm

Relative particle number density, nm−1

30 mm

1.5 106

25

40 mm

20

1.0 106

15 10

0.5 106

5 0

0 0

4

8

Relative particle number density, cm−3 nm−1

2.5 106

35 30 25 20 15 10 5 0

12 16 20 24 28

Particle diameter, nm

Relative particle number density, cm−3 nm−1

Relative particle number density, nm−1

Particle diameter, nm

8

Relative particle number density, cm−3 nm−1

50

Relative particle number density, nm−1

LAMINAR PREMIXED FLAMES

Relative particle number density, cm−3 nm−1

Relative particle number density, nm−1

124

12 16 20 24 28

Particle diameter, nm

Figure 2.34 Comparison of experimental (bar charts) and calculated soot particle size distributions (solid line) at different heights above burner exit plane (modiﬁed from Appel, Bockhorn and Wulkow, 2001).

HOMEWORK PROBLEMS

1.

Derive the following important relationship from the premixed laminar ﬂame theory SL ∝ α (RR)Tf

2.

Comment on various diffusion velocity expressions given in Table 2.1. Also, discuss the relative merits of each model.

3. Plot the reduced reaction rates −ω˙ F / ρu YF,u A and normalized reaction rates versus reduced temperature ≡ (T − Tu )/(Tb − Tu ) by using α = 0.80 and β = 15.0. 4.

Derive the three fuel reaction rate (ω˙ F ) expressions corresponding to the models of ZFK, Williams, and Poinsot and Veynante shown in Table 2.7.

5.

Derive Equation 2.92 for the laminar ﬂame speed.

6.

What are the typical dependencies of laminar ﬂame speed on pressure and temperature, if the overall order of reaction is second order?

7.

Discuss various ﬂame thickness deﬁnitions given in Table 2.9.

8.

Derive the expression for stretch factor κ given in Equation 2.118, and discuss the physical interpretation of each term.

9.

Show that the Karlovitz number associated with curvature for a cylindrical ﬂame surface can be given by Equation 2.145.

10.

What is the deﬁnition of threshold soot index (TSI)?

3 LAMINAR NON-PREMIXED FLAMES

SYMBOLS

Symbol a akj a0 b D Da F f fv Hk L Mk N Nk Ns Q

Description

Dimension

Strain rate, see Equation 3.17 Number of atoms of the j th element in the k th molecular species Location of the virtual origin of the fuel jet, see Figure 3.20 Height of the fuel injection slot, see Figure 3.20 Average diameter of soot particles Damk¨ohler number, deﬁned in Equations 3.1 and 3.87 Dimensionless stream function, F = γ f Dimensionless stream function, deﬁned by Equation 3.144 Soot volume fraction, see Equation 3.167 Total enthalpy of the k th species, deﬁned by Equation 3.43 Length of the diffusion ﬂame, see Equation 3.166 The k th moment of number density, deﬁned in Equation 3.176 Number of soot particles Number of moles of the k th chemical species Total number of chemical species Thermal energy gained by chemical reactions, positive for an exothermic reaction, see Equation 3.54

1/t —

Fundamentals of Turbulent and Multiphase Combustion Copyright © 2012 John Wiley & Sons, Inc.

Kenneth K. Kuo and Ragini Acharya

L L L — — — — Q L M /L3 k

#/L3 N — Q

125

126

LAMINAR NON-PREMIXED FLAMES

Symbol R s Wj y Z Ze Zj

Description

Dimension

Radius of porous cylinder, see Figure 3.12 and Equation 3.114 Stoichiometric ratio, deﬁned in Equations 2.91 and 3.4 Atomic weight of the j th element Stretched dimensionless temperature, see Equation 3.81 Mixture fraction, deﬁned in Equation 3.2, also see Equations 3.11 and 3.13 Zel’dovich number, deﬁned by Equation 3.83 Mass fraction of the j th element, deﬁned in Equation 3.24

L — M/N — — — —

Greek Symbols Thermal diffusivity deﬁned by Equation 3.65 L2 /t Empirical steric factor for growth of soot particle, — see Equation 3.174 β Dimensionless activation energy deﬁned in — Equation 3.88 Dimensionless element mass fraction parameter — β* deﬁned by Equation 3.37 γ Dimensionless parameter deﬁned by Equation 3.88 — or a constant in Equation 3.157 — (Z)F Dimensionless ﬂamelet thickness deﬁned by Equation 3.102, ≈ Zst see Figure 3.8 δ Dimensionless parameter deﬁned by Equation 3.92, — representing a scaled Damk¨ohler number ε Small dimensionless parameter of O(reaction zone — width), see Equation 3.80 ζ Conserved scalar, deﬁned in Equation 3.10 — η Similarity coordinate, deﬁned in Equation 3.122 — μ Dynamic viscosity Ft/L2 or M/(Lt) Normalized k th number density moments, deﬁned in M(k-1) μk Equation 3.177 ξ2 and ξ3 Spatial coordinates in tangential directions to the L ﬂamelet surface τ Time coordinate of ﬂamelet equations 3.71 and 3.74 t φ Equivalence ratio, deﬁned by Equation 3.18 — χ Instantaneous scalar dissipation rate, deﬁned by 1/t Equation 3.72 Dimensionless temperature parameter, deﬁned in — Equation 3.88 ψ Stream function, deﬁned in Equation 3.116 M/(Lt) s Number density of the surface sites for particle #/L2 growth, see Equation 3.175 α αst

127

SYMBOLS

Symbol

Description

ψCsoot−H

Dimension

Number of sites per unit area of soot particle, see Equation 3.175 Dimensionless parameter related to mixture fraction by Equation 3.158

ω

#/L2 —

Subscripts b fuel i ox q st u

Burned fuel Ignition Oxidizer Quenching Stoichiometric condition Unburned

Flames of gaseous fuel are commonly divided into two classes, premixed ﬂames and non-premixed (also called) diffusion ﬂames. In the premixed ﬂame, the reactants (fuel and oxidizer) are premixed before entering the reaction zone. If the compositions (the kinds of fuel and oxidizer and their mixture ratio) and the physical conditions (the pressure and temperature) of the combustible gases are speciﬁed, the compositions, pressure, and temperature of the burned gas in the ﬁnal combustion state and the characteristics of the ﬂame (e.g., the ﬂame speed or the burning velocity) can be determined as discussed in Chapter 2. Unlike a laminar premixed ﬂame, mixing of reactants has a predominant role in the non-premixed ﬂames. Also, the rate of burning and ﬂame propagation direction cannot be deﬁned easily, since the structure of a diffusion ﬂame depends strongly on the ﬂow conﬁguration of the reactants. In the diffusion ﬂame, the fuel and the oxidizer are initially separated, and the reactants mix in the same region where burning occurs. In the diffusion ﬂames, combustion occurs in the vicinity of the interface between the fuel and the oxidizer. In most diffusion ﬂames, the burning process depends more on the rate of mixing than on the rates of the chemical processes involved. A very important dimensionless number associated with the diffusion ﬂames is known as the Damk¨ohler number (Da). Prior to understanding the diffusion ﬂames, it would be useful for readers to know the deﬁnition of the Damk¨ohler number. The Da is a dimensionless number that relates chemical reaction time scale to the mixing phenomena occurring in a system. It is named after German chemist Gerhard Damk¨ohler (1908–1944). In continuous chemical processes, the general deﬁnition of the Da is: Rate of chemical reaction Rate of mixing by species diffusion td Characteristic time associated with mixing = = Characteristic time associated with chemical reaction tch

Da ≡

(3.1)

This parameter appears in various sections of this chapter and other chapters associated with turbulent diffusion ﬂames.

128

3.1

LAMINAR NON-PREMIXED FLAMES

BASIC STRUCTURE OF NON-PREMIXED LAMINAR FLAMES

As mentioned, a non-premixed ﬂame is produced when fuel and oxidizer are not mixed prior to entering into the combustion zone. The mixing of fuel and oxidizer occurs by diffusion (simultaneously with combustion); thus, the non-premixed ﬂames are also known as diffusion ﬂames. In combustion and ﬂuid dynamics, advection plays a major role. However, diffusion and advection are two different physical processes. Advection is associated with the bulk motion, whereas mass diffusion is driven by the concentration gradients of different chemical species. In the non-premixed ﬂames, diffusion process is the major driver for mixing of reactants in the ﬂame zone. Diffusion velocities have been discussed in great detail in Chapter 2 (Section 2.1). Advection also plays an important role in determining the ﬂame shape and structure. A classic example of diffusion ﬂame is shown in Figure 3.1, depicting a fuel jet burning in quiescent ambient air. The ﬂame itself induces entrainment of ambient air into the ﬂame zone due to buoyancy. Another common example of laminar diffusion ﬂame is a candle ﬂame. The parafﬁn of the candle ﬁrst melts due to radiative heating from the ﬂame to the candle, driven by capillary forces into the wick where it evaporates to become parafﬁn vapor, which serves as the gaseous fuel in the combustion process. In the diffusion ﬂame, the chemical reaction rate is generally much faster than the diffusion rates of the gaseous reactants (i.e., the characteristic chemical reaction time tch is much smaller than the characteristic diffusion time td , corresponding to a high Damk¨ohler number case). Consequently, (1) the chemical reaction occurs in a narrow zone near the interface between the gaseous fuel and the oxidizer; (2) the concentrations of fuel and oxidizer are very low in the reaction zone (where most of the products are generated); and (3) the combustion rate is controlled by the rate at which fuel and oxidizer ﬂow into the reaction zone. Therefore, in the limiting case where the chemical reaction rate is considered to be inﬁnitely fast (i.e., where the ratio of the characteristic diffusion time td to

Air

Air

Fuel

Figure 3.1 Sketch of laminar diffusion ﬂame based on a fuel jet burning in ambient air with no crosswind.

FLAME SHEET MODEL

129

the characteristic chemical reaction time tch is inﬁnitely large), we can conclude three things: 1. The reaction zone becomes inﬁnitesimally thin (i.e., the combustion zone becomes a ﬂame sheet). 2. At the ﬂame surface, the fuel and oxidizer consumption rates are in stoichiometric proportion. 3. If the diffusion rates of transport of fuel and oxidizer to the ﬂame surface are in stoichiometric proportion, then the concentrations of fuel and oxidizer become zero at the ﬂame surface, and no oxidizer is present on the fuel side of the ﬂame and no fuel on the oxidizer side of the ﬂame.

3.2

FLAME SHEET MODEL

The ﬁrst successful detailed analysis of a diffusion ﬂame by this ﬂame sheet model was given by Burke and Schumann in 1928 (discussed in Kuo, 2005, chap. 6). They analyzed the problem of fuel issuing from a cylindrical tube into air ﬂowing in a wider concentric cylindrical duct and of a laminar diffusion ﬂame being established at the mouth of the inner tube. In this case, the ﬂame either extends to the wall of the duct (the underventilated case) or converges to the axis (the overventilated case), and the ﬂame height (the axial distance from the mouth of the tube to the point at which the ﬂame reaches either the wall of the duct or the axis) can be deﬁned. In this way, Burke and Schumann obtained fairly accurate predictions of the ﬂame’s height and shape. Thus, the ﬂame sheet model proved to be very useful for the study of laminar diffusion ﬂames. Since then, many combustion problems with laminar diffusion ﬂames have been studied by using the ﬂame sheet model with the assumption that the chemical reaction rate in the diffusion ﬂame is inﬁnitely fast. In fact, this ﬂame sheet model has been used extensively for theoretical studies on the burning of a fuel droplet and on ﬂow and heat-transfer problems in laminar diffusion ﬂames (e.g., the reacting boundary-layer problems). This technique has enabled researchers to calculate the location and shape of the ﬂame, the rate of heat transfer to a solid or liquid surface, and the burning rate of solid or liquid fuel, and to obtain other important information about the laminar diffusion ﬂame structure. However, several important problems concerning the diffusion ﬂame cannot be elucidated on the ﬂame sheet model. In the diffusion ﬂame, the combustion rate is controlled by the rate at which fuel and oxidizer diffuse to the reaction zone, and this combustion rate increases with the diffusion rate, which is strongly dependent on the particular ﬂow conﬁguration, especially on the convective ﬂow velocity. As the ﬂow velocity is increased, the characteristic advection (or ﬂow) and the characteristic diffusion times become shorter. For this reason, the chemical reaction time cannot be neglected when compared with the diffusion time. If the ﬂow rates of fuel and oxidizer into the reaction zone of a diffusion ﬂame increase further and exceed a critical value, the reaction abruptly ceases because

130

LAMINAR NON-PREMIXED FLAMES

the chemical reaction cannot keep pace with the supply of fuel and oxidizer. The extinction of the diffusion ﬂame occurs as a consequence. For example, if the air ﬂow velocity exceeds a critical value, then the diffusion ﬂame around the leading face of a liquid-fuel droplet suspended in an air stream is extinguished, and the envelope ﬂame is suddenly converted into a wake ﬂame. It appears that the extinction of the diffusion ﬂame, which is observed frequently due to increased air ﬂow velocity, results from the limitations of the chemical reaction rates. This ﬂame extinction cannot be explained by the ﬂame sheet model, because it assumes an inﬁnitely fast reaction rate. This fact indicates two things: 1. It is necessary to consider the ﬂame structure of the ﬁnite reaction zone for the diffusion ﬂame in a high air ﬂow rate conditions. 2. The critical air ﬂow velocity corresponding to ﬂame extinction may be considered as a measure of the overall reaction rate in the ﬂame zone. In order to address these complexities, ﬂamelet models have been developed by various researchers. An important parameter in the ﬂamelet model is the mixture fraction Z , which represents the mass fraction of fuel (both burned and unburned) in the mixture. Sometimes a symbol f is used to represent the mixture fraction (e.g., Kuo, 2005). Modeling of diffusion ﬂames requires an understanding of the mixture fraction. Therefore, we will discuss the mixture fraction before describing modeling approaches.

3.3

MIXTURE FRACTION DEFINITION AND EXAMPLES

The mixture fraction is an extremely useful variable in the study of non-premixed combustion. “Mixture fraction” is deﬁned as the ratio of mass of mixture having its origin in the fuel stream with the mass of mixture in a control volume. Z≡

Mass of material in the local control volume having its origin in fuel stream Mass of mixture in local control volume (3.2)

In case of a hydrocarbon fuel (Cx Hy ) burning with pure oxygen, the products will consist of many species, such as H2 O, CO2 , CO, OH, H, H2 , O, CH4 , C2 H2 , and others. In a given control volume in the ﬂow ﬁeld, there could be unburned portions of the fuel species Cx Hy . If the mass fractions of all species in the control volume are known, the amount of the burned fuel can be determined from the combustion product species. For example, 1/9 of the mass of water vapor comes from the fuel, 3/11 of the mass of CO2 comes from the fuel, 3/7 of the mass of CO comes from fuel, 1/17 of the mass of OH comes from fuel, and the full amount of H, H2 , and CH4 are from fuel. By adding all these amounts of mass originally coming from the fuel, we have the burned amount of fuel in the control volume. Adding this amount with the unburned hydrocarbon in the control volume, we will be able to determine the numerator in Equation 3.2. By

131

MIXTURE FRACTION DEFINITION AND EXAMPLES

dividing this number by the total mass in the control volume, the mixture fraction Z can be determined. As shown in the following sections, there are several inherent advantages in introducing the mixture fraction. The mixture fraction is a conserved scalar variable, which means that a conservation equation for the mixture fraction will have no source term. Usually chemical source terms for species conservation equations are highly nonlinear and difﬁcult to resolve. The location of the ﬂame surface is determined where the mixture fraction is equal to its stoichiometric value. Due to this fact, the governing equations for the diffusion ﬂames can be transformed into a new coordinate system using the mixture fraction as the independent variable normal to the local ﬂame surface. Therefore, the mixture fraction is a very important parameter to understand. Usually a diffusion ﬂame is described as a two-feed system—that is, fuel stream (subscript 1) with mass ﬂow rate m ˙ 1 and oxidizer stream (subscript 2) with mass ﬂow rate m ˙ 2 . The mixture fraction represents the mass fraction of the fuel stream in the mixture (see Figure 3.2). For a hydrocarbon air combustion, the global reaction can be written as: νF Cx Hy +νO 2 O2 +νN 2 N2 → νCO CO2 + νH 2 O H2 O +νN 2 N2 2 Oxidizer

Fuel

Diluent

Products

(3.R1)

Diluent

Since the total mass in the system is independent of the chemical reaction (while the total number of moles is not), the relation between the incremental changes of mass fractions is: dYO2 dYF dYk = = = constant νk Mwk νF MwF νO2 MwO2 Deﬁne a stoichiometric ratio s as: s≡

MwO2 νO 2

(3.3)

νF MwF

(3.4) st

Z = Zst

Air m· 2 Z = 0

n2

Fuel m· 1 Z = 1

n1

Air m· 2 Z = 1

n2 Main heat release zone of diffusion flame

dZ/dn = 0, dh/dn = 0

dZ/dn = 0, dh/dn = 0

dZ/dn = 0, dh/dn = 0

Figure 3.2

A diffusion ﬂame produced from a two-stream fuel-air system.

132

LAMINAR NON-PREMIXED FLAMES

Using deﬁnition of s and Equation 3.3, Equation 3.5 can be obtained: sdYF = dYO2 or d sYF − YO2 = 0 or sYF − YO2 = constant

(3.5)

The species conservation equations for fuel and oxidizer can be written as:

∂ ∂YF ∂ρui YF ∂ρYF = (3.6) ρDF + ω˙ F + ∂t ∂xi ∂xi ∂xi

∂YO2 F ∂ρYO2 ∂ρui YO2 ∂ = (3.7) ρDO2 + ω˙ O2 + ∂t ∂xi ∂xi ∂xi Since fuel and oxidizer react in stoichiometric ratio, ω˙ O2 = s ω˙ F

(3.8)

By multiplying Equation 3.6 with s and subtracting Equation 3.7 from the resulting equation, we have:

∂ρζ ∂ ∂ζ ∂ρui ζ = ρD (3.9) + ∂t ∂xi ∂xi ∂xi where (3.10) ζ ≡ sYF − YO2 It was also assumed that the diffusivities of fuel and oxidizer are equal. From Equation 3.9, it can be seen that ζ is a conserved scalar since the governing equation of ζ does not have a source term. Let us deﬁne a normalized ζ variable as: ζ − ζO2 ,u (3.11) Z≡ ζF,u − ζO2 ,u ζO2 ,u = −YO2 ,2

ζF,u = sYF,1

(3.12)

Substituting the deﬁnition of ζ from Equation 3.10 into Equation 3.11 and utilizing Equation 3.12, we can obtain the mixture fraction as a variable that couples the local mass fractions of the fuel and the oxygen: Z=

sYF − YO2 + YO2 ,2 sYF,1 + YO2 ,2

(3.13)

From Equation 3.13, we can see that Z = 1 in the fuel stream (i.e., YF = YF,1 ; YO2 = 0) and Z = 0 in the oxidizer stream (i.e., YF = 0; YO2 = YO2 ,2 ). It can be observed that the variable Z will also satisfy the governing equation for conserved scalar ζ given by Equation 3.9. For a local control volume with stoichiometric condition for fuel and oxidizer, we have: YO2 =s (3.14) YF st

MIXTURE FRACTION DEFINITION AND EXAMPLES

133

Therefore, the corresponding stoichiometric value of the mixture fraction is:

YO2 ,2 sYF,1 −1 = 1+ (3.15) Zst = sYF,1 + YO2 ,2 YO2 ,2 If the fuel and oxidizer streams contain only fuel and oxidizer (without any diluents), respectively, then YF,1 = 1; YO2 ,2 = 1, and Zst = 1/ (1 + s). The surface deﬁned by Z = Zst should represent the ﬂame surface, since the fuel and oxidizer would be in stoichiometric ratio on this surface. This information can be utilized to establish a relationship between species mass fractions and mixture fraction, as demonstrated in the remaining part of this section. If Z < Zst in a local control volume, then the mixture is considered fuellean in that speciﬁc location. The mixture in the region is considered burned (subscript b) when all the fuel in the region is consumed (i.e., YF,b = 0 in the burned region). By using the mixture fraction deﬁnition given by Equation 3.13 and the stoichiometric mixture fraction deﬁnition given by Equation 3.15, we can calculate the remaining oxygen mass fraction in the burned gas as:

Z (3.16) , Z < Zst YO2 ,b = YO2,2 1 − Zst Similarly, if Z > Zst in the local control volume, then the mixture is considered fuel-rich. In such a case, the mixture in the region is considered burned when all oxygen in that region is consumed (i.e., YO2 ,b = 0). This leads to: YF,b = YF,1

Z − Zst , 1 − Zst

Z > Zst

(3.17)

Note that the concept of mixture fraction is deﬁned and demonstrated for a twostream system. If more than two streams are entering into the combustion zone, then multiple mixture fraction deﬁnitions would be required. This complexity will make the mixture fraction variable less attractive for modeling diffusion ﬂames. In such case, a different approach based on elemental mass fraction conservation can be utilized. The equivalence ratio in terms of supply rates of fuel and oxidizer in a twostream system can be deﬁned as: φ≡

YF,1 /YO2 ,2 sYF,1 YF,u /YO2 ,u = = YO2 ,2 YF,u /YO2 ,u st YF,1 /YO2 ,2 st

(3.18)

Using this relationship, the local mixture fraction given by Equation 3.13 can be expressed as:

YO2 sYF − YO2 + YO2 ,2 1 YF = − +1 (3.19) Z= φ sYF,1 + YO2 ,2 YF,1 YO2 ,2 (1 + φ) Note that the φ in Equation 3.19 is not an equivalence ratio based on local mass fractions but is deﬁned by Equation 3.18 in terms of the mass fractions of fuel

134

LAMINAR NON-PREMIXED FLAMES YF,1 = 1

YF,u

YD, 2 YO2, 2

YO2,u YD,u

0

Z

1

Figure 3.3 Mass fraction of mixture components in terms of mixture fraction (unburned gas). YF,1 = 1

YD,2

YP,b

YD,b

YF,b

YO2,2

YO2,b 0

Z = Zst

Figure 3.4 Mass fractions of various components in terms of mixture fraction (burned gas).

and oxidizer in their respective supply lines. In terms of the mixture fraction Z , the mass fractions of fuel and oxidizer in unburned mixture can be represented by two linear relationships, as shown in Figure 3.3. In this plot, the fuel stream has pure fuel (i.e., YF,1 = 1), and the oxidizer stream contains a diluent (e.g., nitrogen). The mass fractions of fuel, oxidizer, and products in the burned mixture are shown in Figure 3.4. The product mass fraction YP is obtained by Equation 3.20. YP = 1 − YF + YO2 + YD

(3.20)

where YD is the mass fraction of the diluent. 3.3.1

Balance Equations for Element Mass Fractions

A more general method for deﬁning the mixture fraction is by using the concept of elemental mass balance. The mass of molecular species in a reacting system may change due to chemical reactions; however, the mass of atomic species

MIXTURE FRACTION DEFINITION AND EXAMPLES

135

remains conserved. The mass of the j th atomic element in a reacting system of N species can be expressed as: mj =

N s

akj Nk Wj

(3.21)

k=1

where

akj = number of atoms of the j th element in the k th molecular species Nk = number of moles of the k th species Wj = atomic height of the j th element In terms of mole fraction, Equation 3.21 can be written as:

m Nk = Xk Mw

(3.22)

where m is the total mass of all species in the mixture and Mw is the average molecular weight of the mixture. By substituting Equation 3.22 into Equation 3.21, we have: mj =

N s

akj Xk Wj

k=1

m Mw

=

N s k=1

Yk akj Mwk

Wj m

(3.23)

The mass fraction of j th element is: N s mj Yk Wj akj Zj ≡ = m Mwk

(3.24)

k=1

The signiﬁcance of element mass fractions lies in the fact that they remain conserved during combustion. By this analogy, mixture fraction and element fractions serve the same purpose: All are conserved scalars. The species conservation equation is written as: ρ

∂Yk ∂ ∂Yk + ρVk,i Yk = ω˙ k + ρui ∂t ∂xi ∂xi

(3.25)

Equation 3.25 is multiplied by akj /Mwk and summed over all species (k = 1, Ns ), and the element mass balance conservation equation is obtained as: ∂Zj ∂Zj ∂ ρ + + ρui ∂t ∂xi ∂xi

N s akj Yk k=1

Mwk

ρVk,i Wj = 0

(3.26)

Note that the source term automatically vanishes since elemental mass does not change during combustion process. This implies that the elemental mass fraction remains conserved during combustion. Assuming that all binary mass diffusivities

136

LAMINAR NON-PREMIXED FLAMES

are equal (i.e., Dk,m = D), the balance equation for the elemental mass fraction can be written as:

∂Zj ∂Zj ∂Zj ∂ ρ = ρD (3.27) + ρui ∂t ∂xi ∂xi ∂xi where the diffusion velocity for the k th species in i th direction is: Vk,i = − Therefore, N s akj Yk k=1

Mwk

ρVk,i

D ∂Yk Yk ∂xi

(3.28)

N s akj ∂Yk ρD Wj = − Wj Mw ∂xi k=1 k Ns ∂Zj akj Yk ∂ = −ρD Wj = −ρD ∂xi Mwk ∂xi

(3.29)

k=1

The element mass fractions for a hydrocarbon fuel (Cx Hy ) air system (shown in reaction 3-R1) can be obtained by using Equation 3.24. N s Yk Zj = Wj akj (3.30) Mwk k=1

The major elements of interest in such a case are hydrogen (H), carbon (C), and oxygen (O). In the unburned mixture, Yproducts = 0. Therefore, ZC = x

WC YF,u , MwF

ZH = y

WH YF,u , MwF

ZO = 2

WO YO ,u = YO2 ,u (3.31) MwO2 2

For the burned mixture with an equivalence ratio other than 1, WC WC YF,b + YCO2 ,b , MwF MwCO2 WH WH ZH = y YF,b + 2 YH O,b MwF MwH2 O 2 WO WO WO ZO = 2 YO2 ,b + 2 YCO2 ,b + YH O,b MwO2 MwCO2 MwH2 O 2 ZC = x

(3.32)

For the burned mixture with stoichiometric fuel-oxygen ratio: WC YCO2 ,b , MwCO2 WH ZH = 2 YH O,b MwH2 O 2 WO WO ZO = 2 YCO2 ,b + YH O,b MwCO2 MwH2 O 2 ZC =

(3.33)

MIXTURE FRACTION DEFINITION AND EXAMPLES

As discussed earlier, in a stoichiometric mixture, νO YO2 ,u 2 MwO2 = =s YF,u st νF MwF

137

(3.34)

Therefore, the next function reduces to 0 at stoichiometric conditions: YO ,u YF,u − 2 =0 MwF νO2 MwF

νF

(3.35)

By using Equation 3.31, we can obtain the next expressions for mass fractions of fuel and oxidizers in the unburned mixture: ZC ZH YF,u = = , MwF xWC yWH

YO2 ,u = ZO

(3.36)

Let us deﬁne a parameter β*: as a combination of element mass fractions of C, H, and O: β∗ ≡

ZC ZH 2ZO + − νF xWC yWH νO2 MwO2

(3.37)

Since ZC , ZH , and ZO are independently conserved scalars, their linear combination deﬁned by parameter β ∗ should also be a conserved scalar. Under stoichiometric conditions, the conserved scalar β ∗ reduces to 0. This is why it is deﬁned as such. This characteristic of β ∗ is similar to the mixture fraction. In order to limit the value of β ∗ between 0 and 1, β ∗ can be normalized as: Z≡

β ∗ − β2∗ β1∗ − β2∗

(3.38)

where the subscripts 1 and 2 correspond to fuel and oxidizer stream. A normalized β ∗ behaves exactly like the mixture fraction. Therefore, it is also called the mixture fraction. Substituting the expressions for β, β1 , and β2 in Equation 3.38, we have: Mw (ZC /xWC ) + (ZH /yWH ) + 2νF YO2 ,u − ZO / νO O 2 2 (3.39) Z= ZC,1 /xWC + ZH,1 /yWH + 2νF YO2 ,2 / νO2 MwO2 Equation 3.39 may appear complex, but it is a very simple way to express mixture fraction in terms of element mass fractions of C, H, and O. These quantities could be measured by laser-spark spectroscopy type of diagnostics. Therefore, element mass fractions could be more useful in experimental measurement of mixture fractions. One implication of this analysis is that the element mass fraction of the j th element, Zj , can be written as a linear function of mixture fraction, Z : Zj = Zj,2 + Z Zj,1 − Zj,2 (3.40)

138

LAMINAR NON-PREMIXED FLAMES

This relationship is a useful concept. Other parameters, such as temperature, can also be expressed in a similar manner, as discussed in the next section. 3.3.2

Temperature-Mixture Fraction Relationship

The ﬁrst law of thermodynamics can be written in terms of enthalpy as: ˆ = dH − Vdp δQ

(3.41)

where

δ Qˆ = heat addition from the surroundings to the system H = total enthalpy of the system V = volume of the system p = pressure in the system In a multicomponent system with Ns number of species, the total enthalpy of the system can be written as the mass-weighted sums of the total enthalpy of all species: Ns Yk H k (3.42) H = k=1

where Hk is the total enthalpy of the k th species, deﬁned as the sum of its heat of formation and sensible enthalpy: T Hk =

0 Hf,k

+

mk Cp,k dT

(3.43)

Tref

In terms of their speciﬁc values, the ﬁrst law of thermodynamics can be rewritten as: δ qˆ = dh − vdp (3.44) In a multicomponent system, the speciﬁc enthalpy of the system is the massweighted sum of the speciﬁc quantities of all species: h=

Ns

Yk hk

(3.45)

k=1

The speciﬁc enthalpy is a function of temperature. It can also be written in terms of chemical enthalpy (i.e., enthalpy of formation) and sensible enthalpy as: T hk =

h0f,k

+

Cp,k dT

(3.46)

Tref

where Cp,k is the constant-pressure speciﬁc heat of k th species and h0f,k is the speciﬁc enthalpy of formation of k th species at reference state Tref . Usually the reference temperature is chosen to be Tref = 298.15 K.

MIXTURE FRACTION DEFINITION AND EXAMPLES

139

Let us consider the ﬁrst law for an adiabatic system (δ qˆ = 0) at constant pressure (dp = 0). From Equation 3.44, we then have dh = 0, which may be integrated from the unburned to the burned state as: hu = hb or

Ns

Yk,u hk,u =

k=1

(3.47)

Ns

Yk,b hk,b

(3.48)

k=1

Substituting Equation 3.46 into Equation 3.48, we have: ⎞ ⎞ ⎛ ⎛ Tb Tu N N s s ⎜ ⎟ ⎜ ⎟ o o ⎝Yk,u hf,k + Cp,u,k Yk,u dT⎠ = ⎝Yk,b hf,k + Cp,b,k Yk,b dT⎠ k=1

k=1

Tref

Tref

(3.49)

or Ns

Yk,u −

Yk,b hof,k

k=1

Tb =

Tu Cp,b dT −

Tref

where Cp ≡

Ns

Cp,u dT

(3.50)

Tref

Cp,k Yk

(3.51)

k=1

For a one-step global reaction, the left-hand side of Equation 3.50 may be calculated by integrating Equation 3.3 as: ν Mwk Yk,u − Yk,b = YF,u − YF,b k νF MwF

(3.52)

Therefore, Ns

Yk,u −

k=1

Ns YF,u − YF,b = νk Mwk hof,k νF MwF

Yk,b hof,k

(3.53)

k=1

The thermal energy gained by chemical reactions (Q) is deﬁned as: Q=−

Ns k=1

νk Mwk hof,k

=−

Ns

o νk Hf,k

where

νk = νk − νk

(3.54)

k=1

Note that Q is a positive quantity for exothermic reactions and a negative quantity for endothermic reactions. For simplicity, let us set Tu = Tref and assume Cp,b to be approximately constant. For combustion in air, the contribution of nitrogen is dominant in

140

LAMINAR NON-PREMIXED FLAMES

calculating Cp,b . At temperatures around 2,000 K, its speciﬁc heat is approximately 1.30 kJ/kg K. The value of Cp is somewhat larger for CO2 and somewhat smaller for O2 , while that for H2 O is twice as large. A ﬁrst approximation for the speciﬁc heat of the burned gas for lean and stoichiometric mixtures is then Cp = 1.40 kJ/kg K. Assuming that Cp is a constant (i.e., Cp,b = Cp,u = Cp ), the adiabatic ﬂame temperature can be determined from Equations 3.50 and 3.54 as: Tb − Tu = −

Q YF,u − YF,b Cp νF MwF

(3.55)

For a lean or stoichiometric mixture (YF,b = 0 and νF = 0), we have: Tb − Tu =

QYF,u Cp νF MwF

(3.56)

For a rich mixture, we replace Equation 3.52 by: ν Mwk Yk,u − Yk,b = YO2 ,u − YO2 ,b k νO2 MwO2

(3.57)

Using this equation for the left-hand side of Equation 3.50, we get: Tb − Tu = −

Q YO2 ,u − YO2 ,b MwO2

Cp νO 2

(3.58)

For a fuel-rich or stoichiometric mixture, there will be complete consumption of = 0). Then oxygen (YO2 ,b = 0 and νO 2 Tb − Tu =

QYO2 ,u Cp νO MwO2 2

(3.59)

If the fuel and oxidizer are mixed (either partially or totally mixed) and there is no combustion (i.e., unburned state), the mass fractions of fuel and oxidizer can be related to the local mixture fraction by these equations: YF,u = ZYF,1

YO2 ,u = (1 − Z) YO2 ,2

(3.60)

These relationships can be substituted into Equations 3.56 and 3.59: Tb (Z) = Tu (Z) +

QYF,1 Z Cp νF MwF

Tb (Z) = Tu (Z) +

QYO2 ,2 (1 − Z) for Z ≥ Zst Cp νO MwO2 2

for Z ≤ Zst

(3.61)

MIXTURE FRACTION DEFINITION AND EXAMPLES

141

Tb, st

Tb

Tb

T

T2

Tu T1 Zst

0

Figure 3.5

Z

1

Relationship between Adiabatic ﬂame temperature and mixture fraction.

The local adiabatic ﬂame temperature is plotted over mixture fraction in Figure 3.5. The maximum temperature at Z = Zst is calculated from either one of Equations 3.61 as: Tb,st = Tu (Zst ) +

YF,1 Zst Q YO ,2 (1 − Zst ) Q = Tu (Zst ) + 2 Cp νF MwF Cp νO2 MwO2

(3.62)

The unburned mixture temperature should be a mass average of the fuel-stream and oxidizer-stream temperatures, that is, Tu (Z) = ZT1 + (1 − Z) T2 = T2 − Z (T2 − T1 )

(3.63)

For the combustion of a pure fuel (YF,1 = 1) in on oxidizer stream with YO2,2 = 0.232 and Tu,st = 300 K, values for Tb,st are given in Table 3.1 by using Cp = 1.40 kJ/kg K. TABLE 3.1. Stoichiometric Mixture Fractions and Flame Temperatures for Hydrocarbon Air Mixtures Source: modiﬁed from Peters (2000).

Fuel Type CH4 C2 H6 C2 H4 C2 H2 C3 H8

Zst

Tf,st = Tb,st [K]

0.05496 0.05864 0.06349 0.07021 0.06010

2263.3 2288.8 2438.5 2686.7 2289.7

142

3.4

LAMINAR NON-PREMIXED FLAMES

FLAMELET STRUCTURE OF A DIFFUSION FLAME

The concept of laminar ﬂamelet has a signiﬁcant role in formulation of turbulent combustion. If the chemistry is fast (when compared to the transport processes such as convection and diffusion), the reactions occur in asymptotically thin layers embedded in the turbulent ﬂow. These layers are called ﬂamelets or laminar ﬂamelets. If the relevant chemical reactions are order(s) of magnitude faster than the transport processes, a chemical equilibrium can be assumed. This situation occurs in many practical combustion systems, including gas turbine combustors and internal combustion engines. Under this condition, it can be stated that combustion takes place in a thin layer in the vicinity of a surface where the mixture fraction is equal to the stoichiometric value (Z = Zst ) and the local mixture-fraction gradient is very high. The structure of a laminar ﬂamelet can be determined by the conservation equations for mixture fraction and temperature. As shown in Section 3.3, the transport equation for mixture fraction can be written as:

∂Z ∂ ∂Z ∂Z ρ = ρD (3.64) + ρuj ∂t ∂xj ∂xi ∂xi This equation does not have a chemical source term. The diffusion coefﬁcient D in this equation should be determined by models discussed in Section 2.2.1 of Chapter 2. However, for simplicity, it is set equal to the thermal diffusivity, α: D=

l

ρCp

=α

(3.65)

This equality implies that the Lewis number (Le) is assumed to be unity. The conservation equation for temperature can be written as:

∂T ∂T ∂ ∂T ∂p = ρCp D + ω˙ T + ρCp + ρCp uj (3.66) ∂t ∂xj ∂xi ∂xi ∂t In the process of obtaining this equation, spatial pressure gradients and dissipation terms have been neglected by assuming low Mach number ﬂow; however, the temporal pressure change ∂p/∂t has been retained (see Equation 1.141 in Chapter 1). The interdiffusion terms have also been considered small and thus neglected in Equation 3.66. The heat capacity Cp is assumed constant for simplicity. Note that in Chapter 1, several energy equations are given with source terms as ω˙ T , ω˙ T , or ω˙ T . If heat capacity is assumed constant, the source term can be written simply as ω˙ T . The mixture fraction Z can be obtained as a function of xi and t from the solution of Equation 3.64 along with the boundary conditions shown in Figure 3.2. Then the surface of the stoichiometric mixture can be determined from Equation 3.67: Z (xi , t) = Zst

(3.67)

FLAMELET STRUCTURE OF A DIFFUSION FLAME

143

Since combustion occurs in a thin layer around a surface deﬁned by Equation 3.67, it is convenient to transform Equations 3.64 and 3.66 in a coordinate system, which has mixture fraction as one of the coordinates. Therefore, let us introduce a local orthogonal coordinate system x1 , x2 , x3 , t attached to the surface of a stoichiometric mixture, where x1 is in the direction normal to the surface Z = Zst and x2 and x3 are tangent to the surface at a local point. In order to introduce mixture fraction in this system, the coordinate x1 is replaced by the mixture fraction Z ; x2 , x3 , and τ are retained as such. The new coordinate system is then (Z, x2 , x3 , τ ). The new time coordinate is same as the old time coordinate (τ = t). In order for the new coordinate system to be orthogonal, the coordinate Z is deﬁned as locally outward normal to the surface Z = Zst . This is a coordinate transformation of the Crocco type. The next transformation rules can be used: ∂ ∂ ∂Z ∂ = + ; ∂t ∂τ ∂t ∂Z ∂ ∂Z ∂ ∂ = + ∂xk ∂ξk ∂xk ∂Z

∂Z ∂ ∂ = ; ∂x1 ∂x1 ∂Z

(3.68)

where (k = 2, 3)

where ξ2 = x2 and ξ3 = x3 . By using the preceding transformation rules, the governing equation for temperature T can be expressed as a function of mixture fraction Z :

∂T ∂T ∂T ρ + u3 + u2 ∂τ ∂ξ2 ∂ξ3 3

3 ∂Z 2 ∂ 2 T ∂Z ∂ 2 T ∂Z ∂ 2 T ∂ 2T − ρD +2 +2 + ∂xi ∂Z 2 ∂x2 ∂Z∂ξ2 ∂x3 ∂Z∂ξ3 ∂ξk2 i=1 k=2 −

∂ (ρD) ∂T ∂ (ρD) ∂T ω˙ T 1 ∂p − = + ∂x2 ∂ξ2 ∂x3 ∂ξ3 Cp Cp ∂τ

(3.69)

If the ﬂamelet is thin in the Z direction (i.e., normal to the plane), an order-ofmagnitude analysis can show that the second derivative with respect to Z should be the dominating term on the left-hand side of Equation 3.69. All other terms containing spatial derivatives in x2 and x3 directions (or ξ2 and ξ3 directions) can be neglected—that is, ∂2 ∂2 ∂Z 2 ∂ξk2

where (k = 2, 3)

(3.70)

This is equivalent to the assumption that the temperature gradients normal to the ﬂame surface are much larger than those in tangential direction. The timederivative terms in Equation 3.69 are signiﬁcant only in case of sudden changes, such as ﬂame quenching or extinction. By retaining the time-derivative terms and assuming that the ﬂamelet is relatively thinner in the direction normal to its

144

LAMINAR NON-PREMIXED FLAMES

surface, we can neglect the terms containing the spatial derivatives in ξ2 and ξ3 directions; therefore, the ﬂamelet structure can be written as a one-dimensional time-dependent temperature equation:

∂T ω˙ T χ ∂ 2T (3.71) = − ∂τ 2 ∂Z 2 ρCp where χ is called the instantaneous scalar dissipation rate, and it is deﬁned by the next equation:

3

∂Z 2 ∂Z 2 ∂Z 2 ∂Z 2 (3.72) = 2D + + χ ≡ 2D ∂xi ∂x1 ∂x2 ∂x3 i=1

The dimension of χ is 1/s. The scalar dissipation rate is an important parameter for analysis of diffusion ﬂames. By incorporating mixture fraction Z as a coordinate, we have included the inﬂuence of transport processes in the direction normal to the surface of the stoichiometric mixture in Equation 3.71. Solutions of the ﬂamelet equations are shown for several example problems in the later sections. In the limit χ → 0, equations for the homogeneous reactor are obtained: ω˙ T dT = dt ρCp

(3.73)

The governing equations for mass fractions can also be transformed in terms of mixture fraction coordinate: ⎡ ⎤

∂ ∂ρ ∂Z ∂ρui ∂Yk ⎣ ∂Z ∂Z ⎦ ∂Yk ρ + ρD ρ + Yk + + ρui − ∂τ ∂t ∂x ∂Z ∂t ∂x ∂x ∂x i i i i 2

∂Z ∂Z ∂ Yk = ω˙ k − ρD ∂xi ∂xi ∂Z 2 Thus,

ω˙ k χ ∂ 2 Yk ∂Yk = − (3.74) ∂τ 2 ∂Z 2 ρ The transformed species equation is nearly identical to the energy equation for ﬂamelet shown in Equation 3.71. A typical solution of the ﬂamelet equations for a methane air diffusion ﬂame with simple four-step chemistry is shown in Figure 3.6. The reduced four-step reaction mechanism was given by Peters (1985) and Peters and Williams (1987) and is shown next. (3.R2) CH4 + 2H + H2 O = CO + 4H2 CO + H2 O = CO2 + H2 (3.R3) H + H + M = H2 + M (3.R4) O2 + 3H2 = 2H + 2H2 O (3.R5)

FLAMELET STRUCTURE OF A DIFFUSION FLAME

145

Oxidation layer (ΔZ)ε Chemically inert region Inner layer (ΔZ)δ T Yi

T CH4

O2 T∞ 0

Zst

T0 1.0

Z

Figure 3.6 Comparison of the ﬂame structure of a ﬂamelet model with that of a ﬂamesheet model shown by dash lines with a ﬂame surface where mixture fraction is equal to stoichiometric value (modiﬁed from Seshadri and Peters, 1988).

The equal sign in these reactions indicates that they are considered global reactions. This mechanism was derived using steady-state assumptions for the intermediates OH, O, HO2 , CH3 , CH2 O, and CHO and partial equilibrium of the reactions H2 + OH = H + H2 O and OH + OH = O + H2 O. Hydrogen atom H was not assumed to be in steady state because of its high mass diffusivity. In Figure 3.6, the dotted lines represent the Burke-Schumann solution for the diffusion ﬂame problem, and the solid curves represent the solution from the ﬂamelet model. The Burke-Schumann solution assumes a one-step global reaction CH4 + 2O2 = CO2 + 2H2 O. As mentioned earlier, the Burke-Schumann solution is also known as the ﬂame sheet model, having the ﬂame location at Z = Zst . In the ﬂamelet model as shown in Figure 3.6, the region where chemical reactions occur consists of a thin inner layer of thickness (Z)δ located slightly on the fuelrich side (Z > Zst ) and a thin H2 − CO oxidation layer with thickness O[(Z)ε ] located on the fuel-lean side (Z < Zst ). Beyond the inner layer, the rich side is chemically inert because all radicals are consumed by the chemical reaction. The comparison of the diffusion ﬂame structure with that of a premixed ﬂame shows that the rich part of the diffusion ﬂame corresponds to the upstream preheat zone of the premixed ﬂame. 3.4.1

Physical Signiﬁcance of the Instantaneous Scalar Dissipation Rate

The instantaneous scalar dissipation rate, χ, is a very useful parameter to demonstrate ﬂame stability and quenching. A stoichiometric value of the scalar

146

LAMINAR NON-PREMIXED FLAMES

dissipation rate corresponds to the stoichiometric mixture fraction, that is, χst = χ(Zst ). Physically, it can be interpreted as a characteristic diffusion rate or inverse of a characteristic diffusion time. The inverse of χst is also proportional to the Damk¨ohler number, Da. Therefore, increasing Da implies a decreasing χst or increasing χst−1 . The relationship between Da and χst is shown in Equation 3.75. Da ↑⇒ td ↑ or tchem ↓⇒

Rate of heat loss due to thermal diffusion ↓ ⇒ Tmax ↑ Rate of heat production due to chemical reactions ↑

χst ↑⇒ td ↓ ⇒ Rate of heat loss due to thermal diffusion ↑⇒ Tmax ↓ ⇓ Da ↑⇒ χst−1 ↑ or χst ↓ ⇒ Tmax ↑

(3.75)

An asymptotic solution of the diffusion ﬂamelet equations was obtained by Fendell (1965) and Li˜na´ n (1974) for a counterﬂow diffusion ﬂame with a singlestep Arrhenius-type irreversible reaction in the limit of large activation energies. This solution is obtained in the form of maximum temperature versus the χst−1 (or Da) and it looks like an S-shaped curve (see Figure 3.7). Burning of the ﬂamelet corresponds to the upper branch of the curve (called the intensely burning state branch). Certain physical processes (e.g., ﬂame stretching) can result in increasing scalar dissipation rate. Due to this process, the maximum temperature decreases until the ﬂamelet is quenched at a critical value of χst = χq because heat loss from the inner layer of the ﬂamelet to both sides can no longer be

Intensely burning state branch Tmax Q (Quenching) Physically unrealistic branch

Chemically frozen state Weakly reacting state branch

Daq

I (Ignition)

DaI

Da

Figure 3.7 S-shaped curve showing maximum temperature in diffusion ﬂame as a function of the inverse of the scalar dissipation rate at stoichiometric mixture (modiﬁed from Peters, 2000).

FLAMELET STRUCTURE OF A DIFFUSION FLAME

147

balanced by heat production due to chemical reactions. At this point, the ﬂame is quenched. On the S-shaped curve, it is indicated by the abrupt fall at state Q to the lower branch. The lower branch of the S-shaped curve starts with a chemically frozen state, corresponding to inﬁnite scalar dissipation rate (or Da = 0). It may correspond to an inﬁnitely long chemical reaction time or inﬁnitely high scalar dissipation rate, resulting in no chemical reactions or an inﬁnitely fast heat loss rate. From this chemically frozen state, if χst−1 (or Da) is increased, some chemical reactions may occur, although not sufﬁcient to produce a self-sustained ignition. When the χst−1 (or Da) reaches a state I (which corresponds to the ignition of mixture), the mixture jumps to the upper branch to reach an intensely burning state. An example for autoignition in non-premixed systems is the ignition and combustion in a diesel engine. Here interdiffusion of the fuel from the diesel spray with the surrounding hot air leads to continuously decreasing mixture fraction gradients and therefore to decreasing scalar dissipation rates. This corresponds to a shift on the lower branch of the S-shaped curve up to point I, where ignition occurs. The middle branch of the S-shaped curve corresponds to the region where the maximum temperature decreases as the Da increases. This phenomenon is physically unrealistic, as increasing the Da means that either the chemical reaction rates are higher (and, therefore, higher heat production rate) or the rate of heat loss due to diffusion is decreasing; either of these conditions should result in an increasing maximum temperature. Note that the Tmax versus Da plot is an S shaped relationship only for high-activation energy cases; for low-activation energy cases, there may not be any obvious Q and I states or they may merge into a single state. The jump conditions at Q and I states are akin to the hysteresis characteristic of the physicochemical process. The parameters χq and Daq can be interpreted as a global kinetic quantity describing nonequilibrium effects in diffusion ﬂames. It is the critical value where ﬁnite-rate kinetics just balance heat loss by diffusion. In this sense it is equivalent to the ﬂame velocity in premixed ﬂames, which also represents a global kinetic quantity. Both quantities are in principle interrelated but care must be taken to use the equivalent chemistry model for both premixed and diffusion ﬂames. 3.4.2

Steady-State Combustion and Critical Scalar Dissipation Rate

If the unsteady terms are neglected in Equations 3.71 and 3.74, these governing equations become a pair of ordinary differential equations that describes the structure of a steady-state ﬂamelet normal to the surface of stoichiometric mixture. They can be solved for general reaction rates either numerically or by asymptotic analysis. By using the assumption of steady-state burning, an analytical expression for the critical scalar dissipation rate (χq ) can be obtained. For simplicity, we assume complete combustion, where the chemical reaction is conﬁned to an inﬁnitely thin sheet around Z = Zst , Cp is treated as constant, and the temperature is a piecewise linear function of Z as shown in Equations 3.61 to 3.63. We also assume a one-step reaction chemistry with large activation energy and constant pressure. By using Equation 3.62, the heat of reaction can be written

148

LAMINAR NON-PREMIXED FLAMES

in terms of maximum temperature Tb,st or simply Tst , unburned gas temperature, stoichiometric mixture fraction Zst , and mass fraction of fuel in stream 1 as: (Tst − Tu ) νF MwF Q = Cp YF,1 Zst

(3.76)

The rate of reaction for a single-step reaction under the assumption that the reaction rate is ﬁrst order with respect to both fuel and oxygen can be written as:

ρYO2 ρYF Ea (3.77) ω˙ = −B exp − MwO2 MwF Ru T where B is the pre-exponential constant. Therefore, N

R Qj ω˙ T Q = ω˙ j = ω˙ (since NR = 1) Cp C Cp j =1 p

νF (Tst − Tu ) Ea 2 Bρ YO2 YF exp − =− YF,1 Zst MwO2 Ru T

(3.78)

Substituting Equation 3.78 and the other simplifying assumptions (like steadystate and equal values of mass diffusivity) into Equation 3.71, we have:

2BνF (Tst − Tu ) Ea d 2T = ρY Y exp − (3.79) O2 F χYF,1 Zst MwO2 Ru T dZ2 As mentioned earlier, the combustion takes place in a thin layer around the surface Z = Zst . In order to obtain a better understanding of physical processes in this thin region, let us introduce a stretched coordinate, ζ deﬁned as: ζ = (Z − Zst ) /ε

(3.80)

where ε is a small dimensionless parameter of the order of the width of the reaction zone. Since it appears in the denominator of Equation 3.80, it can also be viewed as an expansion parameter for coordinate Z . The mixture temperature and the mass fractions of fuel and oxygen are expanded around Zst as: ⎫ T = Tst − ε (Tst − Tu ) y ⎪ ⎬ YF = YF,1 [Z − (1 − ε) Zst ] = YF,1 ε (Zst + ζ ) (3.81) ⎪ ⎭ YO2 = YO2 ,2 [(1 − Z) − (1 − ε) (1 − Zst )] = YO2 ,2 ε [(1 − Zst ) − ζ ] where y is the stretched dimensionless temperature. The exponential term in the reaction rate can be expanded as:

Ea Tst Ea Ea exp (−εyZe) (3.82) = exp − = exp − exp − Ru T Ru Tst T Ru Tst

149

FLAMELET STRUCTURE OF A DIFFUSION FLAME

The Zel’dovich number is deﬁned as: Ze ≡

Ea (Tst − Tu ) Ru Tst2

(3.83)

This expansion is obtained from the next steps:

Tst (Tst − Tu ) (Tst − Tu ) (Tst − Tu ) −1 y =1+ε y + H.O.T ≈ 1 + ε y = 1−ε T Tst Tst Tst (3.84)

Ea Ea Tst ε (Tst − Tu ) y = exp − 1+ exp − Ru Tst T Ru Tst Tst ⎞ ⎛ (3.85)

⎜ Ea Ea (Tst − Tu )⎟ ⎟ ⎜ = exp − exp⎜−εy ⎟ ⎠ ⎝ Ru Tst Ru Tst2 Ze

If all other quantities in Equation 3.79 are expanded around their value at the stoichiometric ﬂame temperature—that is, if we substitute Equations 3.80 to 3.83 into Equation 3.79, we get: d 2y = −2Daε3 [(1 − Zst ) − ζ ] (Zst + ζ ) exp (−εyZe) dζ 2

(3.86)

Da is deﬁned as: Y Bρst νO 2 F,1

Ea exp − Da ≡ χst MwF (1 − Zst ) Ru Tst

Bρst νF YO2 ,2 Ea exp − = (3.87) χst MwO2 Zst Ru Tst

To further simplify Equation 3.86 for analytical solution, let us deﬁne the next transformation: ⎫ = 2y (1 − Zst ) Zst − γ ζ ⎪ ⎬ γ = 2Zst − 1 (3.88) ⎪ ⎭ β = Ze/ [2Zst (1 − Zst )] Substituting Equation 3.88 into Equation 3.86, we get the next simple form: # $ d 2 = Da ε3 2 − ζ 2 exp −βε ( + γ ζ ) 2 dζ

(3.89)

There are evidently two ways to deﬁne the expansion parameter ε: The requirement of :

1 Da1/3 1 βε = 1 ⇒ ε = β

Da ε 3 = 1 ⇒ ε =

Similarly, the requirement of :

(3.90) (3.91)

150

LAMINAR NON-PREMIXED FLAMES

The ﬁrst condition is called a large Da expansion and the second is called a high-activation energy expansion. To use both deﬁnitions of ε, let us introduce a parameter δ in this way: (3.92) δ = Da/β 3 By using Equation 3.90, we have: ε=

1 1 = 1/3 δ β Da1/3

(3.93)

By using Equation 3.93, both Equations 3.90 and 3.91 are interrelated. The parameter δ can be called a scaled Damk¨ohler number that is scaled by β 3 . Substituting Equation 3.92 into Equation 3.89, we get this simple form: $ # d 2 = 2 − ζ 2 exp −δ −1/3 ( + γ ζ ) dζ 2

(3.94)

To understand the ﬂame quenching, let us discuss the ﬂame temperature equation shown in Equation 3.86. If the Zel’dovich number is so high that its product with ε is nonnegligible, the exponential term on the right-hand side of Equation 3.86 becomes dominant. This means that the reduced temperature (y) will decay exponentially, and the ﬂame will come closer to quenching. A high Zel’dovich number corresponds to a high activation energy and a high value of β. Therefore, even if the Da is high, a high value of activation energy can lead to ﬂame instability. Physically, high-activation energy means a lower reaction rate, which could result in slower production of heat during the combustion process. A reduction in the Da means that the rate of transport of energy out of the ﬂame zone is increased. This loss of energy from the ﬂame combined with slower production of heat results in quenching of the ﬂame. In such a case, the scaled Damk¨ohler number δ attains a value δq and the critical scalar dissipation rate can be obtained by this procedure: δ = δq

⇒

Da = Daq = δq β 3

(3.95)

Substituting the expressions for Da and parameter β into the last equation, we have:

Y Bρst νO Ea 2 F,1 (3.96) = δq Ze3 / [2Zst (1 − Zst )]3 exp − χst MwF (1 − Zst ) Ru Tst or χst = χq =

Y Bρst νO 2 F,1

Ze3 MwF δq

or χq = Da

8Zst3 (1

Ea − Zst ) exp − Ru Tst 2

8χst Zst3 (1 − Zst )3 χ = Da st 3 3 δq β δq Ze

(3.97)

(3.98)

TIME AND LENGTH SCALES IN DIFFUSION FLAMES

151

(ΔZ)F

Tst δ→∞ δq T

δ→∞ 0

Zst

Z

1

Figure 3.8 Effect of scaled Damk¨ohler number δ on temperature and fuel mass fraction proﬁles plotted versus mixture fraction Z for diffusion ﬂamelet (modiﬁed from Peters, 1992).

Based on Equation 3.94, the characteristic proﬁles for the temperature versus mixture fraction Z are schematically shown in Figure 3.8. There are two limiting proﬁles for the solution of this equation. If δ → ∞, then an equilibrium solution is obtained; this corresponds to the Burke-Schumann solution (ﬂame sheet model). The other limiting proﬁle is obtained if δ → δq , which corresponds to the critical scalar dissipation rate χq . Any solution below this proﬁle is unstable, and the ﬂamelet would be extinguished. Therefore, the solution for ﬂame temperature can be obtained only between these two limits. 3.5

TIME AND LENGTH SCALES IN DIFFUSION FLAMES

For a premixed laminar ﬂame, the characteristic time √ √ associated with the ﬂame can be written as tF = α/SL2 since SL ∝ α · RR ∝ α/tF . According to Peters (1991), the premixed ﬂame time can be related to the critical scalar dissipation rate of a diffusion ﬂamelet by this equation: tF = Zst2 (1 − Zst )2 /χq

(3.99)

In a diffusion ﬂame, there are at least two time scales: one time scale can be associated with the ﬂow or velocity gradient (∂ui /∂xj ), and the other time scale tch can be associated with the rate-determining chemical reaction. At ﬂame extinction conditions, both time scales should be equal by physical interpretation of ﬂame extinction condition. The chemical time scale of a diffusion ﬂame at extinction can be related to the characteristic ﬂame time of a premixed ﬂame, which in turn corresponds to the premixed laminar ﬂame speed SL . This indicates that there is a fundamental relation between a premixed ﬂame and a diffusion

152

LAMINAR NON-PREMIXED FLAMES

ﬂame at extinction. In a diffusion ﬂame at extinction, the heat diffusion out of the reaction zone toward the lean and the rich sides balances the heat generation by the chemical reactions. In a premixed ﬂame, however, the heat conduction toward the unburned mixture is also balanced by the heat generation by the chemical reactions for a particular burning speed. It can be seen that these two processes are equivalent and thus underline the similarity between a premixed ﬂame and a diffusion ﬂame at extinction. The fundamental difference between a diffusion ﬂame and a premixed ﬂame is that a diffusion ﬂame can exist at lower scalar dissipation rates and therefore at lower characteristic ﬂow times. A premixed ﬂame, however, is controlled by the burning velocity, which is an eigenvalue of the problem. Therefore, combustion in diffusion ﬂames offers an additional degree of freedom: varying the ratio of the transport to the reactive time, represented by the Da deﬁned in Equation 3.87 as long as χst is smaller than χq . This allows non-premixed ﬂames to be more controllable and stable than premixed ﬂames. It is also one of the reasons why diesel engines, which operate in the non-premixed regime, are more robust and less dependent on fuel quality than spark ignition engines, where fuel and air are premixed before ignition. Equation 3.99 may now be used to calculate chemical time scales for diffusion ﬂames near quenching. The inverse complementary error function erfc−1 (2Zst ) is 1.13 for methane air ﬂames with Zst = 0.055 and 1.34 for H2 air ﬂames with Zst = 0.0284. (The inverse complementary error function is obtained as a solution of unsteady one-dimensional ﬂamelet model. See the next section.) Extinction of the H2 air diffusion ﬂame occurs at a strain rate of 14,260/s and that of the CH4 air ﬂame at 420/s. This leads to tch = 0.0064 ms for hydrogen air diffusion ﬂames and to tch = 0.29 ms for methane air diffusion ﬂames. The latter estimate is of the same order of magnitude as tch for stoichiometric premixed methane air ﬂames. For premixed methane air ﬂames, the velocity gradient at extinction is 2275/s, which is signiﬁcantly higher than that of methane air diffusion ﬂames. This comparison shows that while the chemical time scales at extinction in diffusion ﬂames and premixed ﬂames are comparable, the velocity gradient for extinction is signiﬁcantly higher (by a factor of 7) for premixed ﬂames. This implies that a diffusion ﬂame extinguishes more easily than a premixed ﬂame in terms of the imposed strain rate (or velocity gradient). Physical interpretation of this phenomenon is that the inner structure of a diffusion ﬂame can lose heat to both the lean and the rich sides, whereas a premixed ﬂame can lose heat only to its preheat zone. Unlike premixed ﬂames, diffusion ﬂames do not have a well-deﬁned velocity scale, such as the laminar burning speed, by which a characteristic length scale could be deﬁned (such as the premixed ﬂame thickness). However, it is possible to use the velocity gradient (gv ) and mass diffusivity (D) for deﬁning a characteristic length scale for diffusion ﬂames. The inverse of velocity gradient may be interpreted as a ﬂow time. Based on dimensional analysis, the diffusion ﬂame thickness lF can be deﬁned as: % Dref lF = (3.100) gv

EXAMPLES OF LAMINAR DIFFUSION FLAMES

153

Here the mass diffusivity D should be evaluated at a suitable reference condition (e.g., at stoichiometric condition). Assuming a one-dimensional mixture fraction proﬁle in y-direction for the unsteady mixing layer, the diffusion ﬂame thickness in mixture fraction space may be deﬁned as:

∂Z lF (3.101) (Z)F = ∂y F where (∂Z/∂y)F is the mixture fraction gradient normal to the ﬂamelet surface. This ﬂame thickness includes the reaction zone and the surrounding oxidation layer (see Figure 3.8). By combining Equation 3.101 with Equations 3.100 and 3.72, the next equation is obtained: % χref (3.102) (Z)F = 2gv where χref represents the scalar dissipation rate at the reference condition (e.g., stoichiometric condition), and the mixture fraction gradient along the ﬂame surface is considered much smaller than the gradient in the normal direction. It was shown by Peters (1991, 2000) for a counterﬂow diffusion ﬂame that if the reference condition is the stoichiometric condition, then (Z)F will be of the order of 2Zst when Zst is small. 3.6 3.6.1

EXAMPLES OF LAMINAR DIFFUSION FLAMES Unsteady Mixing Layer

In order to obtain the physical interpretation of the instantaneous scalar dissipation rate, let us consider a two-dimensional laminar mixing layer initially separated by a splitter plate between the fuel and oxidizer streams (Figure 3.9). Since the gradients in the y-direction are much stronger than those in the x -direction, the ﬂame zone can be described by the one-dimensional governing equations, as shown next.

∂Z ∂ ∂Z ρ = ρD (3.103) ∂t ∂y ∂y

∂T ω˙ T ∂T ∂ ρD + (3.104) ρ = ∂t ∂y ∂y Cp

u

y

oxidizer T = T2, Z = 0 fuel T = T1, Z = 1

Figure 3.9

x u

Mixture fraction proﬁles in ﬂow downstream of a splitter plate.

154

LAMINAR NON-PREMIXED FLAMES

It is also assumed that pressure is constant with respect to time. Therefore, the time derivative of pressure is not included in Equation 3.104. The initial and boundary conditions for this problem are: t = 0 : Z = 1, t > 0 : Z = 1,

T = T1 for y < 0, T = T1 for y → −∞,

Z = 0, T = T2 for y > 0 Z = 0, T = T2 for y → +∞ (3.105) 2 2 Let us assume that ρ D = ρ D ref and introduce the next similarity coordinate: y 1 ρ −1/2 η = (Dref t) dy; τ = t (3.106) 2 ρref 0

Substituting this coordinate into the governing equation in place of y, we obtain the next set of equations. τ

∂Z 1 ∂Z ∂ 2Z − η = ∂τ 2 ∂η ∂η2

(3.107)

τ

ω˙ T ∂T 1 ∂T ∂ 2T +τ − η = 2 ∂τ 2 ∂η ∂η Cp

(3.108)

The mixture fraction solution obtained Peters (2000) is: Z=

1 erfc (η) 2

(3.109)

where erfc is the complementary error function. The error function and the complementary error function are deﬁned by the next equation: 2 erf (η) = √ π

η e

−u2

du

and

0

2 erfc (η) = 1 − erf (η) = √ π

∞

e−u du 2

η

(3.110) The instantaneous scalar dissipation rate can be obtained in the next equation:

∂Z χ(Z, t) = 2D ∂y

2 =

1 exp −2 [η(Z)]2 2πt

(3.111)

For small values of Z (large values of χ), the complementary error function may 1 be replaced by π − /2 η−1 exp −η2 such that χ can be expressed as: $2 # χ (Z, t) = 2Z 2 erfc−1 (2Z) /t

(3.112)

where erfc−1 is the inverse (not the reciprocal) of the complementary error function. With a ﬁxed value of Z, the scalar dissipation rate is therefore inversely

EXAMPLES OF LAMINAR DIFFUSION FLAMES

155

proportional to time. The inverse complementary error function and inverse error function are given by the next equations. √ 1 π − (Z − 1) − π (Z − 1)3 + O [Z − 1]5 erfc−1 Z = 2 12 √ 3 πZ π Z+ (3.113) + O Z5 erf−1 Z = 2 12 These results are also applicable to ﬂamelets in turbulent non-premixed combustion.

3.6.2

Counterﬂow Diffusion Flames

In this section we discuss another important conﬁguration of non-premixed ﬂames called counterﬂow diffusion ﬂames that have fuel and oxidizer ﬂowing from opposite directions. The ﬂame geometry is relatively simple for theoretical formulation. Counterﬂow diffusion ﬂames are used experimentally very often because they represent one-dimensional diffusion ﬂame structures. The counterﬂow diffusion ﬂames can be classiﬁed into two groups: (1) the counterﬂow diffusion ﬂame between two opposed gaseous jets: fuel jet and the oxidizer jet, and (2) the counterﬂow diffusion ﬂame established in the forward stagnation region of a porous burner immersed in a uniform oxidizer ﬂow. Further, the ﬂames are subdivided into four types (Figure 3.10): 1. The three-dimensional or ﬂat counterﬂow diffusion ﬂame established between two opposed jets from circular tubes or rectangular nozzles (Type I ﬂame) (Otsuka and Niioka, 1972; Potter and Butler, 1959). 2. The ﬂat counterﬂow diffusion ﬂame established between two opposed matrix burners ejecting individual reactants (Type II ﬂame) (Pandya and Weinberg, 1963). 3. The counterﬂow diffusion ﬂame established in the forward stagnation region of a spherical or hemispherical porous burner (Type III ﬂame) (Simmons and Wolfhard, 1957, Spalding, 1953). 4. The counterﬂow diffusion ﬂame established in the forward stagnation region of a cylindrical porous burner (Type IV ﬂame) (Tsuji and Yamaoka, 1967). For over 50 years, these four types of counterﬂow diffusion ﬂames have been used to study the overall reaction rates for various combinations of different fuels and oxidizers and the detailed structure and reaction mechanism of the laminar diffusion ﬂame. Figure 3.11 is an example of a particle-streak picture of the stagnation ﬂow with diffusion ﬂame (Type IV ﬂame in Figure 3.10). The particles were introduced into the air stream such that they can pass through the ﬂame and the stagnation

156

LAMINAR NON-PREMIXED FLAMES Air

Oxidant

Matrix Flame

Flame Glass Beads

Fuel

Fuel

TYPE I

TYPE II

Fuel Porous Cylinder

Porous Sphere Fuel Flame Air

TYPE III

Flame

Air TYPE IV

Figure 3.10 Classiﬁcation of counterﬂow diffusion ﬂames into four types (Tsuji, 1982).

point can be seen. As shown, the stagnation point is on the fuel side of the ﬂame. The blue ﬂame location is on the air side of the stagnation point. This shows that the ﬂow ﬁeld of the counterﬂow diffusion ﬂame established in the forward stagnation region of a porous cylinder is very simple. In this conﬁguration, the ﬂow velocity of oxidizer is generally much higher than that of the fuel. Although the ﬂame is not ﬂat in shape, it is a two-dimensional ﬂame, and the stagnation point generally lies between the cylinder surface (shown by the dotted curve) and the blue ﬂame. Moreover, the experimental procedure to establish and control the ﬂame over a wide range of ﬂow velocities of fuel and oxidizer is relatively easy. Also, since the ﬂow ﬁeld in the forward stagnation region of the cylinder is very stable, the ﬂame is quite stationary and stable, which makes it suitable for a detailed study of the ﬂame structure (Tsuji, 1982). The extinction limits as well as the ﬂow ﬁeld structure, temperature, and stable-species concentration ﬁelds of this ﬂame were studied in detail by Tsuji and Yamaoka using methane, propane, and city gas as fuel. With the appropriate conditions of the uniform airstream velocity Vox and the fuel injection velocity Vfuel , a thin laminar two-dimensional blue ﬂame is established at some distance from the cylinder surface in the forward stagnation region. As the fuel

EXAMPLES OF LAMINAR DIFFUSION FLAMES

157

Porous cylinder surface

Fuel: Propane

Blue flame

Oxidizer: Air (a) C3H8(g) R

boundary layer edge stagnation point

air

Blue flame

x,u y,v

(b)

Figure 3.11 (a) Particle-streak picture of counterﬂow propane/air diffusion ﬂame in forward stagnation region of porous cylinder (R = 1.5 cm, Vox = 80 cm/s, Vfuel = 8.6 cm/s) (modiﬁed from Tsuji and Yamaoka, 1971); (b) sketch of the Tsuji ﬂame and the ﬂow directions (modiﬁed from Peters, 1992).

injection velocity was decreased or the air-stream velocity was increased, the ﬂame approached the cylinder surface, subsequently blowing off from the stagnation region and converting into a so-called wake ﬂame. When a hydrocarbon fuel or a mixed fuel including hydrocarbon (called city gas) was used with small Vox and comparatively large Vfuel , the ﬂame thickness increased remarkably and the ﬂame showed a luminous yellow inner zone (fuel side) and a blue outer zone (air side). The blow-off limits of methane/air, propane/air, and city gas/air ﬂames are presented in Figure 3.12, where the dimensionless fuel-injection rate (−fw ) is deﬁned by: 1 −fw = Vfuel /Vox (ReR /2) /2 (3.114) where ReR ≡ Vox R/ν. The mean kinematic viscosity ν was based on average temperature across the boundary layer in the stagnation region when a stable ﬂame is established. The dimensionless fuel-injection rate at blow-off limits is small over a wide range of the stagnation velocity gradient (2Vox /R), as shown in Figure 3.12.

158

LAMINAR NON-PREMIXED FLAMES

0

1

10

102 2Vox /R, 1/s

Blow-off

−fw 1

Stable Blue Flame

CH4 C3H8 CITY GAS

Flame with Luminous Yellow zone

2

103

104

Figure 3.12 Flame-stability diagrams for counterﬂow diffusion ﬂame in the forward stagnation region of a porous cylinder (R = 1.5 cm; fuel: propane, methane, and city gas) (modiﬁed from Tsuji and Yamaoka, 1971).

Notice that the abscissa is a log-scale, whereas the ordinate is a linear scale with a small interval (0 to 2). When the stagnation velocity gradient approaches a critical value, the fuel-injection rate at blow-off limits increases steeply and there exists a critical stagnation velocity gradient (2Vox /R)crit , beyond which the ﬂame could not be stabilized. Tsuji and Yamaoka proposed that the blow-off of the ﬂame at the lower fuel-injection rates is due mainly to thermal quenching of the ﬂame near the cylinder surface. In contrast, the blow-off near the critical stagnation velocity gradient is caused by chemical limitations on the combustion rate in the ﬂame zone. This blow-off mechanism is different from the blow-off caused by thermal quenching. Figure 3.13 shows the measured temperature distribution plotted against a dimensionless distance η for a methane/air counterﬂow diffusion ﬂame. The dimensionless distance η is deﬁned as: 1

η ≡ (2 ReR ) /2

y R

(3.115)

where y is the normal distance from the cylinder surface. As shown in the plot, the maximum temperature (called diffusion-ﬂame temperature) corresponds to the luminous ﬂame zone, and the temperature decreases rapidly toward both the fuel and air sides. A small dent in the temperature distribution can be seen near the inner edge (fuel side) of the luminous ﬂame zone. This dent is generally observed in the laminar diffusion ﬂame of hydrocarbon fuels; it is caused by the pyrolysis of the fuel in this region. Figure 3.14 shows the concentration distributions of various stable species measured with a quartz microprobe and a gas chromatograph. In this ﬁgure, the

EXAMPLES OF LAMINAR DIFFUSION FLAMES

159

Temperature,°C

1500

1000

500 Luminous Flame Zone

0

0

1

2

3

4

5

h

Figure 3.13 Temperature proﬁle in methane/air diffusion ﬂame (2Vox /R = 100 1/s, −fw = 1.5) (modiﬁed from Tsuji and Yamaoka, 1971). 100 N2 O2

CH4 10−1

Mole Fraction

H2 O H2 CO2

10−2 C2H2

CO

C2H4 10−3

C2H6 Luminous Flame Zone

10−4

0

1

2 h

3

4

Figure 3.14 Species concentration proﬁles in methane/air diffusion ﬂame (2Vox /R = 100 1/s, −fw = 1.5) (modiﬁed from Tsuji and Yamaoka, 1971).

160

LAMINAR NON-PREMIXED FLAMES

mole fraction of each species is plotted against the dimensionless distance η. The methane concentration decreases rapidly toward the luminous ﬂame zone, and methane disappears almost completely at the outer edge (air side) of the luminous ﬂame zone. The oxygen concentration decreases rapidly from the air side toward the luminous ﬂame zone, but there is always some oxygen on the fuel side of the ﬂame. It is well known that small amounts of oxygen strongly increase the rate of pyrolysis of hydrocarbons, so it is quite conceivable that this same mechanism operates on the fuel side of the counterﬂow diffusion ﬂame. Apart from the counterﬂow diffusion ﬂame geometry, it has also been observed in the conical methane/air diffusion ﬂame that there is always some oxygen in the ﬂame cone. Therefore, the diffusion of small amounts of oxygen into the fuel side through the ﬂame zone is a general characteristic of a laminar diffusion ﬂame of hydrocarbon and air. This process is independent of the type of ﬂame. Carbon dioxide and water vapor, which are the ﬁnal combustion products of the diffusion ﬂame; and hydrogen and carbon monoxide, which have considerably higher concentrations among the intermediate products, have their maximum concentrations in the luminous ﬂame zone. Their concentrations decrease toward both fuel and air sides of the ﬂame, but these species exist over a considerably wider region in the fuel and air sides. Various intermediate hydrocarbons (C2 H2 , C2 H4 , C2 H6 ), which are the pyrolysis products of fuel, are found at low concentrations on the fuel side of the ﬂame, and these hydrocarbons disappear almost completely at the outer edge of the luminous ﬂame zone. Finally, the nitrogen concentration decreases monotonically toward the cylinder surface, but it has a ﬁnite value at the cylinder surface. Thus, the structure of a laminar diffusion ﬂame, especially the chemical structure of a hydrocarbon/air ﬂame, is very complicated. This ﬁnding demonstrates that the real ﬂame structure of diffusion ﬂames can be quite different from that of the ﬂame sheet model. For opposing jet with Type I conﬁguration, the diffusion ﬂame can have a ﬂat-shaped geometry, as shown in the sketch in Figure 3.15. The fuel could be either gas phase or liquid phase. As shown on the right-hand side of this ﬁgure, the fuel vapor is generated from a pool of liquid fuel. The stream function for the frictionless planar compressible stagnation-point ﬂow (i.e., away from the shear layer close to the stagnation point, see Figure 3.15) can be deﬁned from the next equations: ∂ψ ∂ψ = ρu, = −ρv ∂y ∂x

(3.116)

where y → +∞ is the oxidizer side and y → −∞ is the fuel side. In this conﬁguration, we have oxidizer and fuel, which have different densities. If we assume that the origin of oxidizer and fuel streams are far from the stagnation point (y → ±∞) and the stagnation point is the origin (x = 0, y = 0), the x – and y – velocities along with the mixture fraction for the oxidizer in a region away from the viscous shear layer close to the stagnation plane can be written as: y → ∞:

v = −ay,

u = ax,

Z=0

(3.117)

161

EXAMPLES OF LAMINAR DIFFUSION FLAMES Oxidizer

Oxidizer Nozzle

Nozzle

Flame

Flame

y

y x

x Stagnation Plane

Nozzle

Stream Line

∇

Gaseous Fuel

Pool of Liquid Fuel

Figure 3.15 Sketch of two experimental setups for producing ﬂat counterﬂow diffusion ﬂames (modiﬁed from Peters, 1992).

The parameter a in Equation 3.117 is the strain rate since it represents the velocity gradients of x– and y– velocities in x– and y– directions, respectively. Therefore, a also can be interpreted as the strain rate parallel to the stagnation plane (i.e., x-direction). Equal stagnation point pressure for both streams requires that the velocities on the fuel side should have the next form: & & y → −∞: v = − ρox /ρfuel ay, u = ρox /ρfuel ax, Z = 1 (3.118) The pressure gradient in the x -direction can be related to the velocity gradient by ∂p/∂x = −ρu∂u/∂x, where (∂u/∂x) = a outside the shear layer; therefore, ∂p/∂x = −ρa 2 x. The governing equations for a planar diffusion ﬂame are presented next. ∂(ρu) ∂(ρv) + =0 ∂x ∂y

∂p ∂u ∂u ∂ ∂u =− μ +v + ρ u ∂x ∂y ∂x ∂y ∂y

∂Z ∂Z ∂ ∂Z ρ u +v = ρD ∂x ∂y ∂y ∂y

(3.119) (3.120) (3.121)

Now we introduce the similarity coordinates [ξ(x), η(x, y)]: ξ = x,

1 η= 2

a (ρμ)ox,ref

1/2 y 0

ρ dy

(3.122)

162

LAMINAR NON-PREMIXED FLAMES

This coordination transformation is known as the Dorodnitsyn-Howarth transformation described in Stewartson’s Book (1964). The nondimensional stream function can be written as: f ≡

ρv ψ # $ = −# $1/ ρox,ref x aμox,ref (ρμ)ox,ref a 2

(3.123)

The normalized tangential velocity: df = axf dη 1/ ρox,ref v=− aνox,ref 2 f ρ

u = ax

(3.124)

Let us deﬁne the Chapman-Rubesin parameter C as: C≡

ρμ (ρμ)ox,ref

(3.125)

By assuming that ρ 2 D = ρ 2 D ox,ref , upon coordinate transformation, Equations 3.120 and 3.121 can be written as:

df df ρox,ref d C +f (3.126) + − f 2 = 0 dη dη dη ρ

d C dZ dZ +f =0 (3.127) dη Sc dη dη The Schmidt number is deﬁned as: Sc ≡

μ ρD

(3.128)

The boundary conditions in the new coordinate system are: Z=0 f = 1, & η → −∞: f = ρox,ref /ρfuel,ref , Z = 1

η → +∞:

(3.129)

With C = 1, the next solution can be obtained from solving Equations 3.126 and 3.127 with boundary conditions in Equation 3.129:

η 1 (3.130) Z = erfc √ 2 2 The instantaneous scalar dissipation rate is:

∂Z χ(Z, t) = 2D ∂y

2 =

a exp − [η(Z)]2 π

(3.131)

EXAMPLES OF LAMINAR DIFFUSION FLAMES

For small values of Z , χ can be written as:

Sc χ(Z) = 2 af 2 Z 2 C

163

(3.132)

The scalar dissipation rate in Equation 3.132 has the same Z -dependence as the unsteady mixing layer and the two-dimensional ﬂow behind a splitter plate (see Equation 3.112). This observation highlights a common feature between these two ﬂow conﬁgurations in terms of the scalar dissipation dependency on the mixture fraction. Figure 3.16 shows the calculated temperature and species mass fractions in a counterﬂow CH4 /air diffusion ﬂame plotted versus a mixture fraction at two Stretched Z coordinate

Scale Split

2250 T 2000 1750

a = 100/s a = 400/s

1500 χ

1250 1000 750 500 300

0.2 0.4 0.6 0.8 1.0 Z

0.00 0.04 0.08 0.12 0.16 (a) 0.25

Scale Split

1.00

Mass Fractions

0.20

0.75

0.15 0.10

0.50

CH4

O2

0.05

0.25

a = 400/s 100/s

0.00

0.00 0.0

0.1

0.2

0.6

1.0 Z

(b)

Figure 3.16 Distributions of temperature and species mass fractions of CH4 /air counterﬂow diffusion ﬂame at two different stain rates, showing inﬂuence of strain rate a (modiﬁed from Peters, 2000).

164

LAMINAR NON-PREMIXED FLAMES

different strain rates of 100/s and 400/s, corresponding to χst equal to 4/s and 16/s, respectively (Peters and Kee, 1987). At the higher strain rate of 400/s, the peak temperature is 200 K lower than the peak temperature at strain rate of 100/s due to the higher scalar dissipation rate, χ. From the mass fraction proﬁles in Figure 3.16(b), it also can be observed that the leakage of oxygen toward the fuel-rich zone also increases by a factor of approximately 2.5 for high-strain-rate case. Note that there is no leakage of fuel through the reaction zone. A higher strain rate leads to an increasing velocity gradient, which results in reduced residence time in the inner layer (i.e., the fuel consumption layer in Figure 3.6). Because of this, less oxygen is consumed in the inner layer, thereby resulting in a higher amount of leakage of oxygen into the fuel-rich zone. This process in turn reduces the temperature in the inner layer due to coupling between reaction rate and temperature. Reaction rates are highly temperature dependent, and they are reduced even further due to lower temperature until the diffusion ﬂame approaches quenching condition. This means that there are two mechanisms for ﬂame quenching: (1) a thermal mechanism due to the high scalar dissipation rate resulting in great heat loss from the ﬂame zone and (2) a chemical mechanism due to leakage of oxygen from the inner layer to the fuel-rich zone resulting in reduced reaction rates and lower temperature in the fuel consumption layer. The distributions of products mass fractions versus the mixture fraction for a CH4 /air counterﬂow diffusion ﬂame are shown in Figure 3.17 for two different strain rates. There is surprisingly little difference in the mass fraction proﬁles of intermediates and products as a (and χ) is increased. In Figure 3.18, a plot of the calculated peak ﬂame temperatures as a function of 1/a is shown. The inverse of the strain rate is proportional to a characteristic ﬂuid dynamic residence time and consequently to the Damk¨ohler number. As the inverse of strain rate (1/a) increases (i.e., a decreases), the Da also increases. Da ∝ td ∝ tresidence ∝ (1/a) ⇓ (1/a) ↑⇒ Da ↑

(3.133)

The plot shown in Figure 3.18 constitutes the upper branch of the characteristic S-shaped curves discussed earlier. As the residence time is decreased, the maximum temperature drops until a steady-state ﬂame can no longer exist. This point is the extinction limit (Q) and corresponds to a point of vertical tangency on the curve. Experimentally, the extinction limit for these ﬂame conditions occurs at a = 330/s (Tsuji and Yamaoka, 1971). Peters and Kee (1987) calculated a stable ﬂame at a = 400/s and found the extinction strain rate at a = 450/s. The calculated results for this ﬂame with a short mechanism involving 40 reactions (Miller et al., 1984) showed the ﬂame extinction between 330/s and 350/s, which is much closer to the experimental value obtained by Tsuji and Yamaoka. Here, the effect of the reaction mechanism on predicted extinction limit is demonstrated. With both 4-step reduced mechanism and the more detailed 40-reactions mechanism, the calculated ﬂame temperature is very good, except at the extinction limit. This

EXAMPLES OF LAMINAR DIFFUSION FLAMES

Scale Split

0.20 a = 100/s Mass Fractions

165

0.15 CO2

0.10

H2O CO

0.05

H2

H 0.00

0.00 0.04 0.08 0.12 0.16 0.20 0.4 0.6 0.8 1.0 Mixture Fraction (a)

Scale Split

0.20 a = 400/s Mass Fractions

0.15

0.10

H2O CO2

0.05 H

CO H2

0.00 0.00 0.04 0.08 0.12 0.16 0.20 0.4 0.6 0.8 1.0 Mixture Fraction (b)

Figure 3.17 Distributions of product species mass fractions of CH4 /air counterﬂow diffusion ﬂame at a strain rate (a) a = 100/s and (b) a = 400/s (modiﬁed from Peters, 2000).

could be explained as due to the fact that at lower temperatures close to the extinction point, the partial equilibrium approximation of atomic oxygen radical O in the four-step mechanism becomes less accurate. Another plot of ﬂame temperature versus the inverse of stoichiometric scalar dissipation rate is shown in Figure 3.19. This plot shows remarkable similarity with the plot shown in Figure 3.18 and also represents the upper branch of the S-curve. This is due to the fact that (1/a) ∝ χ −1 ∝ χst−1 . 3.6.3

Coﬂow Diffusion Flame or Jet Flames

For non-premixed combustion, a jet ﬂame is the most common ﬂow conﬁguration. There are two conﬁgurations of interest: one is a coaxial ﬂow of fuel jet in the center and air jet as the outer ﬂow (see Figure 3.20a); the other conﬁguration is

166

LAMINAR NON-PREMIXED FLAMES 2050

Peak Temperature (K)

2000

4-step reduced mechanism 40-step short reactions a = 100/s

1950

a = 200/s

1900 a = 300/s a = 350/s

1850 1800 1750 0.000

a = 400/s

0.002

0.004

0.006 1/a (s)

0.008

0.010

0.012

Figure 3.18 Calculated peak ﬂame temperature as function of inverse of strain rate for CH4 air counterﬂow diffusion ﬂame (modiﬁed from Peters and Kee, 1987).

Peak Temperature (K)

2100

1900

1700 Numerical results (Peters) Experimental data (Seshadri and Peters)

1500

1300 0

0.2

0.4

0.6

0.8

1

χst−1(s)

Figure 3.19 Peak ﬂame temperature as function of inverse of scalar dissipation rate (modiﬁed from Peters, 2000).

when fuel jet is issued from a cylindrical port (or a rectangular slot) and air is entrained into the fuel jet from surroundings (see Figure 3.20b). The slot burner is also known by Wolfhard-Parker slot burner (see Wolfhard, 1952). Usually the jet velocity is so large that the jet becomes turbulent, and turbulent mixing determines the mixture fraction ﬁeld. Here we want to derive a similarity solution for a laminar plane jet issuing from a slot burner. In order to obtain a similarity solution, buoyancy must be neglected. Here we extend the classical solution for a constant density planar jet to nonconstant density jet ﬂames. The stream line pattern and the velocity proﬁles are shown in Figure 3.20b. The jet is issued from

167

EXAMPLES OF LAMINAR DIFFUSION FLAMES air

b

Z = Zst

y

x

fuel

flame length L

a0 virtual origin of jet air fuel Coaxial Jet Flame Burner Schematic (a)

air 2-D Planar Jet Flame Schematic (b)

Figure 3.20 Two conﬁgurations of laminar diffusion ﬂame jets (part b modiﬁed from Peters, 2000).

a small slot (with a height of b and width of unity) from a wall and entrains the surrounding air due to viscous effects. The maximum velocity at the center of the jet decreases as the distance from the wall is increased. For simplicity, we will ignore the boundary layer on the surface of the wall. Thus, the entrained air can enter into the jet only from the normal direction. We assume the boundary-layer assumptions to be valid and the pressure in the ﬂow ﬁeld to be constant. The stream function for the two-dimensional ﬂow ﬁeld is deﬁned as: ρu =

∂ψ ∂y

ρv = −

∂ψ ∂x

(3.134)

The stream function automatically satisﬁes the continuity equation. The simpliﬁed governing equations in the shear layer without the effect of buoyancy are: ∂ (ρu) ∂ (ρv) + =0 ∂x ∂y

∂u ∂ ∂u ∂u +v = μ ρ u ∂x ∂y ∂y ∂y

∂Z ∂ ∂Z ∂Z +v = ρD ρ u ∂x ∂y ∂y ∂y

(3.135) (3.136) (3.137)

Since mixture fraction and temperature have a direct relationship (as shown in earlier sections), there is no need to include the energy equation in the formulation. The boundary conditions for this conﬁguration are: y → 0:

v = 0,

∂u/∂y = 0,

∂Z/∂y = 0

y → ∞:

u = 0,

∂u/∂y = 0,

∂Z/∂y = 0,

Z=0

(3.138)

168

LAMINAR NON-PREMIXED FLAMES

By combining the momentum equation with the continuity equation by integrating the momentum equation with respect to y from 0 to ∞, we get: ∂ ∂x

# $∞ ∂u ∞ ρu2 dy + ρuv 0 = μ ∂y 0 0 ∞

(3.139)

=f (y)

By applying the boundary conditions to Equation 3.139, we get: ∂ ∂x

∞

∞ ρu dy = 0

⇒

2

0

ρu2 dy = f (y)

(3.140)

0

The integral on the left-hand side of Equation 3.139 and 3.140 must be independent of x ; therefore, it should be invariant in the x -direction. Thus, it can be equated to the momentum of the jet at x = 0, that is, ∞ ρu2 dy = ρ0 u20 b

(3.141)

0

where b is the height of the slot. Similarly, the mixture fraction equation can be integrated to establish that the integrated mass ﬂow rate is independent of x , that is, ∞ ∞ ∂ ρuZ dy = 0 ⇒ ρuZ dy = ρ0 u0 b (3.142) ∂x 0

0

where ρ0 , u0 are the density and x -direction velocity component of the fuel stream at x = 0, respectively. Let us introduce the next similarity coordinates (ζ, η):

y u0 1/3 ρ 1 dy (3.143) ζ = x + a0 , η = 3 ν0 ζ 2 ρ0 0

where a0 is the distance between x = 0 and the apparent origin of the jet (also called virtual origin, shown in Figure 3.20b). Let us also deﬁne the next function as a dimensionless stream function: f (η) =

ψ

1/3 ρ0 ν02 u0 ζ

(3.144)

Using this function in Equation 3.134, we get: 1 u= 3

u20 μ0 ρ0 ζ

1/3 f

(3.145)

169

EXAMPLES OF LAMINAR DIFFUSION FLAMES

A general transformation rule from the x−y to ζ −η coordinate system is: ∂ζ ∂ ∂η ∂ ∂ ∂η ∂ ∂ = + = + ∂x ∂x ∂ζ ∂x ∂η ∂ζ ∂x ∂η ∂η ∂ ∂ = ∂y ∂y ∂η

(3.146)

Introducing the preceding transformation into the momentum equation, we get the left-hand-side (LHS) and right-hand-side (RHS) terms as: LHS: RHS:

∂u ∂η ∂ψ ∂u ∂ψ ∂u 1 μ0 u0 ∂η 2 ∂u + ρv = − =− f + ff ρu ∂x ∂y ∂y ∂η ∂ζ ∂ζ ∂η 9 ζ ∂y

∂ ∂u 1 μ0 u0 ∂η ∂ ∂f μ = C (3.147) ∂y ∂y 9 ζ ∂y ∂η ∂η

where C is the Chapman-Rubesin parameter, and it is deﬁned as: C≡

ρ2ν ρμ = 2 ρ0 μ0 ρ0 ν0

(3.148)

By equating both LHS and RHS of Equation 3.147, we get the next ordinary differential equation: f 2 + ff + Cf = 0 (3.149) By integrating Equation 3.149 once with respect to η, we get: ff + Cf = 0

(3.150)

Boundary conditions are transformed as: η = 0:

f = 0,

η → ∞:

f = 0

f = 0 (3.151)

By assuming C = 1 and by introducing F ≡ f/2γ , ξ ≡ γ η, where γ is a constant, we integrate Equation 3.150 to obtain the next equation with the integration constant set to 1. (3.152) F + F2 = 0 A further integration yields the next solution. F = tanh ξ =

1 − exp (−2ξ ) 1 + exp (−2ξ )

(3.153)

170

LAMINAR NON-PREMIXED FLAMES

To obtain a solution for the constant γ , let us use Equation 3.141 for momentum balance. An expression for x -velocity component u can be obtained as: ∂F = 2γ 2 1 − tanh2 ξ ∂ξ 1/3 2 2 u20 μ0 u= γ 1 − tanh2 ξ 3 ρ0 ζ

f = 2γ 2

(3.154)

Substituting u in Equation 3.141, we get: 4 3 γ μ0 u0 3

∞

1 − tanh2 ξ

2

dξ = ρ0 u20 b

(3.155)

2 1 − F 2 dF = 3

(3.156)

0

where

∞

1 − tanh ξ 2

2

∞ dξ =

0

0

This leads to: γ3 =

9 ρ0 u0 b 9 = Re0 8 μ0 8

(3.157)

where Re0 is the jet exit Reynolds number. A solution for mixture fraction also can be obtained by a similar analysis. Let us introduce the next variable: 1 ω=Z α

ρ0 u0 ζ μ0

1/3 (3.158)

Substituting this equation in Equation 3.137, we have: f ω + f ω +

C ω Sc

=0

(3.159)

where the Schmidt number, Sc, = ν/D. Integrating the last equation once, we get: fω +

C ω =0 Sc

(3.160)

By substituting C from Equation 3.150 in Equation 3.160, we can establish the next relationship between ω and f : 1 d ln ω d ln f = dη Sc dη

(3.161)

EXAMPLES OF LAMINAR DIFFUSION FLAMES

171

Integrating the last equation yields: Sc ω = f

(3.162)

For C = 1, this leads to the next solution for the mixture fraction:

1/3 $Sc # 2 Sc μ0 1 − tanh2 (γη) Z = α 2γ ρ0 u0 ζ

(3.163)

In order to determine the constant α, the solution for the mixture fraction can be substituted into Equation 3.142. We get: Sc α 2γ 2

1

1 − F2

Sc

dF =

4 2 γ 9

(3.164)

0

For the case of unity Schmidt number, we get α = 1/3. In such a case, the mixture fraction ﬁeld is proportional to the velocity ﬁeld, that is, Z=

u u0

(3.165)

Equation 3.163 can be used to determine the ﬂame length of jet diffusion ﬂame. By setting Z = Zst and η = 0, we get: Sc 3

3

2Sc α 2γ 2 α 9 μ0 μ0 −a = −a L= √ Re0 Zst ρ0 u0 Zst ρ0 u0 8

(3.166)

The structure of diffusion ﬂame has been studied extensively for fuel jet issuing from a cylindrical tube into a quiescent oxidizer or an oxidizer ﬂow. The similarity solution for this conﬁguration is given in Kuo (2005, Chap. 6). In the experimental studies of the two-dimensional planar jet diffusion ﬂames, not only have the temperature and stable-species-concentration proﬁle been measured, but a detailed study of the structure of the reaction zone of a ﬂat diffusion ﬂame has been made with the aid of measurements on the emission and absorption spectra of the ﬂame zone. These ﬁne experimental results have given us valuable information on the structure of a laminar diffusion ﬂame. It is well known, however, that there is a dead space near the rim of the burner where the ﬂame is not established due to the loss of both heat and active radicals to the wall. Therefore, the direct interdiffusion of fuel and oxygen occurs in this dead space, producing a small region of premixed gases at the base, which holds the ﬂame. Consequently, the characteristics and structure of this type of diffusion ﬂame are affected by both this premixed ﬂame and by the wedge of combustion products that grows thicker as we go farther away from the burner rim. Therefore, the laminar diffusion ﬂame on the burner port is thought to be unsuitable for the study of certain fundamental processes of the diffusion ﬂame.

172

3.7

LAMINAR NON-PREMIXED FLAMES

SOOT FORMATION IN LAMINAR DIFFUSION FLAMES

The soot formation processes in premixed ﬂames were discussed in Chapter 2. In this section, we discuss soot formation in context of the diffusion ﬂames. The formation and emission of soot by combustion processes pose grave problems. Soot emission from a practical combustion appliance reﬂects poor combustion conditions and a loss of efﬁciency. Not only would it contribute to reduced atmospheric visibility, but it would also increase the particulate fallout. The soot emissions have been associated with carcinogenic polycyclic aromatic hydrocarbons (PAHs). In addition, the reduction in ﬂame temperature due to radiative heat losses can affect the ﬂame height and other temperature-dependent processes, such as the formation of NOx. Similarly, when the substantial fractions of fuel carbon are converted to soot, it leads to temporary removal of gas-phase carbon, which can shift the local H2 /H2 O and CO/CO2 conversion ratios and affect the local temperature distribution in the ﬂame. The vast majority of large-scale industrial combustion applications utilize nonpremixed ﬂames, including gas-turbine engines, furnaces, and diesel engines in automobiles. Temperatures in such systems lie between 1,500 and 2,500 K, and there is generally sufﬁcient oxygen available for the substantial combustion of the fuel. The total amount of soot formed under these conditions is usually very small compared to the amount of carbon present in the consumed fuel. Under these conditions, the time available for the soot formation is of the order of a few milliseconds. In practical devices operating within design conditions, soot should oxidize nearly to completion prior to the burner exit. Unfortunately, temporary changes in operating conditions (as well as turbulent ﬂuctuations) can momentarily cause the equivalence ratio in the oxidative part of the ﬂame to be reduced drastically. As the local equivalence ratio in a non-premixed ﬂame varies, the propensity to form soot changes. Low equivalence ratio can allow soot particles to pass through the ﬂame unconsumed by oxygen. Even if all of the soot formed in the diffusion ﬂame were oxidized to CO, temporarily localized equivalence ratio reductions might inhibit complete oxidation to CO2 , resulting in, for example, high gaseous exhaust emissions (Puri, Santoro, and Smith, 1994). Thus, the study of sooting ﬂames remains important from practical, economic, environmental, and safety standpoints as well as for purely scientiﬁc interests in the fundamental soot formation and emission control processes. Although most of the diffusion ﬂames in practical combustion systems and ﬁres are essentially turbulent, detailed direct study of these ﬂames is not always possible because of the intermittency and the short residence times involved. However, known similarities in laminar and turbulent diffusion ﬂames (Cavaliere and Ragucci, 2001; Moss, Stewart, and Young, 1995) are used in simpliﬁed analysis of turbulent ﬂames via laminar ﬂamelet concepts. Due to this similarity, the study of laminar diffusion ﬂames can provide a tractable ﬂame model for real combustion systems. For this reason, the more easily controlled laboratory experiments are performed in laminar diffusion ﬂames and shock tubes. One

SOOT FORMATION IN LAMINAR DIFFUSION FLAMES

173

disadvantage of shock tubes is that they have a very short residence time as compared to practical ﬂames. Therefore, coﬂow laminar diffusion ﬂame burners have been used mostly to study soot formation processes.

3.7.1

Soot Formation Model

A large body of scientiﬁc literature on the chemical mechanisms of soot formation in non-premixed ﬂames exists. Soot formation occurs in four major stages: (1) soot particle inception, (2) surface growth, (3) coagulation/agglomeration, and (4) oxidation. The soot growth occurs concurrently with coagulation and agglomeration. Coagulation is the process by which small particles coalesce to form larger primary particles, and agglomeration is the process by which multiple primary particles line up in tandem to form larger structures resembling a string of pearls. Soot particles could be transported toward the ﬂame front where they pass through an oxidation region in which the mass of soot is decreased by oxidation reactions with gas-phase molecules. Any soot not completely oxidized is released from the ﬂame envelope as “smoke.” Traditional modelers generally agree that a simpliﬁed model of particulate formation in diffusion ﬂames should account for all of these processes (Lindstedt, 1994). This paradigm is based on the classical view of soot formation in diffusion ﬂames where incipient soot particles with diameters on the order of several nanometers form in slightly fuel-rich regions of the ﬂame by inception or nucleation. These particles then undergo surface growth, perhaps by the (1) hydrogen abstraction by C2 H2 addition (HACA) mechanism (Frenklach and Wang, 1990) or the (2) addition of PAHs. Since HACA is initiated by H· atoms, it is likely to be most important in the main ﬂame reaction zone where H· atom concentrations generally exceed equilibrium values. The second mechanism for surface growth through PAH addition becomes important in hydrocarbon-rich regions (away from the main reaction zone) where the H· atom concentrations are low. These competing mechanisms are distinctly different because the HACA mechanism (to be discussed in detailed later in the chapter) is controlled by heterogeneous surface reactions, whereas the PAH addition mechanism is controlled by homogeneous gas-phase reactions. These chemistry issues are complex and still far from being totally resolved. Fortunately, from an engineering standpoint, the problem of soot formation in non-premixed ﬂames can be tackled due to the inherent simplicity of these types of ﬂames. The location of the ﬂame front and peak temperature is controlled by the stoichiometric conditions of the reactants rather than by complex chemical kinetics for the premixed ﬂame. Overall heat release rates for non-premixed ﬂames are controlled by diffusion of reactants to a thin ﬂame region separating the fuel-rich zone and the oxidizer-rich zone. Since the soot formation/oxidation times in non-premixed ﬂames are much longer than the main reaction times, only the characteristic diffusion times and soot formation/oxidation times need to be considered, and the chemical reactions generally can be considered as instantaneous. This corresponds to a high-Da condition.

174

LAMINAR NON-PREMIXED FLAMES

3.7.1.1

Particle Inception

The ﬁrst condensed phase material arises from the fuel molecules via their oxidation and/or pyrolysis products. Such products typically include various unsaturated hydrocarbons, particularly acetylene and its higher analogs (C2n H2 ), and PAHs. These species are relatively stable with respect to decomposition to the elements, and they are more kinetically stable compared to the parafﬁns and even the oleﬁnes. These two types of molecules often are considered the most likely precursors of soot in ﬂames. For example, naphthalenyl grows into pyrenyl through sequential acetylene addition, H-atom elimination, H-atom abstraction, and a second addition of acetylene followed by ring closure. The reaction can be written as C10 H7 + 3C2 H2 C16 H9 + 2H + H2 . This sequence can continue to form yet larger PAH structures with the overall balance of C10 H7 + 3nC2 H2 C10+6n H7+2n + 2nH + nH2 (Lautenberger, de Ris, Dembsey, Barnett, and Baum, 2005). The gas-phase species condense to form the ﬁrst recognizable soot particles (which often are called nuclei, although this term should be used with caution because of its connotations of physical condensation ˚ and the formation phenomena). These ﬁrst particles are very small (δ < 20 A), of even large numbers of them involves a negligible soot loading in the region of their formation, which generally is conﬁned to the more reactive regions of the ﬂame (i.e., in the vicinity of the primary reaction zone). 3.7.1.2

Surface Growth and Oxidation

Surface growth and oxidation is the mechanism by which the bulk of the solidphase material is generated. Surface growth involves the attachment of gas-phase species to the surface of the particles and their incorporation into the particulate phase. Some qualitative trends in this process can be seen in Figure 3.21, where

log10 Mw 7 Soot

6 5 4 3

C27H27 C24H16

2

C2H4 1 0.7

C4H2

C2H2

0.6

0.5

Polyacetylenes

C24H17

C6H6

C2H6

Figure 3.21

Polycylic Aromatics

Paraffins

0.4

0.3

C8H2 0.2

0.1

0

XH

Pathways to soot formation (modiﬁed from Homan, 1978).

SOOT FORMATION IN LAMINAR DIFFUSION FLAMES

175

the logarithm of the molecular weight of a species is plotted against its hydrogenatom mole fraction, XH . Soot particles generally have XH between 0.1 and 0.2. From this plot, it can be seen that neither polyacetylenes nor PAHs have the molecular weight closer to soot for the XH range between 0.1 and 0.2 (as shown by the different trends of soot, PAHs, and polyacetylenes on this plot). Therefore, it can be concluded that soot is not made of only polyacetylenes or only PAHs. This means that the surface growth also occurs by condensation of chemical species with higher hydrogen content than these two types of species, followed by dehydrogenation, or a combination of both. In addition to polyacetylenes, some polycyclics and saturated platelets (e.g., C27 H27 ) are major condensing species for surface growth. Soot formation is characterized by the soot volume fraction fV ∼ (cm3 soot/ cm3 ), the number density of soot particles, N (cm−3 ), and the size of the particles, D. The particles also possess a size distribution, but usually this is relatively narrow and mostly an average size is considered. The quantities ∼fV , N , and D are mutually dependent, and any two of these three parameters are sufﬁcient to characterize the system. For spherical particles, fV =

NπD 3 6

(3.167)

Surface growth reactions lead to an increase in the amount of soot (∼fV ), but the number density of particles (N ) remains unchanged by this process. The opposite is true for growth by coagulation, where particles collide and coalesce, thereby decreasing N , while fV remains constant. Particle growth (increasing D) is the result of simultaneous surface growth reaction and coagulation. These stages of particle generation and growth constitute the soot-formation process. This process often is followed by soot oxidation in which the soot is burned in the presence of oxidizing species to form gaseous products, such as CO and CO2 . The eventual emission of soot from any combustion device will depend on the balance between these processes of formation and burnout. 3.7.2

Appearance of Soot

Soot generated in combustion processes is not uniquely deﬁned. It normally looks black and consists mainly of carbon, but it is quite different from graphite. Apart from carbon, soot particles also contain up to 10 moles of hydrogen, and even more when they are young. A good deal of this hydrogen can be extracted in organic solvents where it appears mostly in condensed aromatic ring compounds. Sometimes tarlike material is emitted from combustors; these emitted materials look glassy and have black, brown, or even yellow color. Such materials are the quenched intermediates of the soot formation process. Sometimes they result as a condensation product of the heavy hydrocarbons formed during combustion or fuel droplets, which have passed through the combustion zone more or less unburned. Finally, they may be normal soot particles on which heavy hydrocarbons have condensed, as is the case with diesel smoke.

176

LAMINAR NON-PREMIXED FLAMES

(a)

(b)

Figure 3.22 (a) SEM image of the butadiene soot aggregates; individual, solid, spherical particles, 50 to 70 nm in diameter; (b) TEM image of butadiene soot aggregates, 30 to 50 nm in diameter, arranged in branching clusters (Penn, Murphy, Barker, Henk, and Penn, 2005).

The Scanning electron micrograph (SEM) and transmission electron micrograph (TEM) images of soot (displayed in Figure 3.22) show that the basic units of soot are spherical or nearly spherical particles with diameters in the range of 30 to 50 nm, which corresponds to more than 1 million carbon atoms. These particles are often called elementary soot particles. These elementary particles are aggregated together to form the straight or branched chains as shown in the ﬁgure. The elementary soot particles exhibit a size distribution, usually not far from log-normal distribution. However, the particles collected from a wide variety of combustion processes (under normal operating conditions), such as furnace ﬂames, piston engines, gas-turbine combustors, or premixed ﬂames, do ˚ in average diameter. Early not differ much in size, being typically 200 to 400 A work with X-ray diffraction has indicated that within a particle, there are randomly arranged domains of graphitelike parallel layers. The spacing between these layers is somewhat larger than in graphite. Figure 3.23 shows the TEM image of soot particles generated from a biodiesel engine at three different time intervals. The particle size distribution also is shown for these time intervals. These plots indicate that the size distribution is nearly log-normal; however, the mean soot diameter reduces as the time increases. 3.7.3

Experimental Studies by Using Coﬂow Burners

It is apparent from the foregoing qualitative discussion of soot formation in diffusion ﬂames that a number of competing processes, most simply identiﬁed as surface growth and soot oxidation (or burnout), are involved. Therefore, investigation of overall effects such as ﬂame height, smoke point, or even soot yield is unlikely to lead to a clear picture of how, and where, soot is formed in a diffusion ﬂame. A fuel’s smoke point corresponds to the maximum height of its laminar diffusion ﬂame burning in air at which soot is not released from the ﬂame tip. It is a unique measure of a hydrocarbon fuel’s sooting propensity and has long been used by

177

Counts

10

20

Figure 3.23

0

50

100

150

200

250

300

Dp(nm)

40

50

60

Counts 0

20

40

60

80

100

120

140

160

10

20

30

(b)

40

50

60

Ave. Dp = 26.7 nm

B100 soot oxidized

Dp(nm)

(a)

40% Burnoff (30 min)

0

20

40

60

80

100

10

20

30 Dp(nm)

40

50

60

Ave. Dp = 23.7 nm

B100 soot oxidized upto 75% burn off

75% Burnoff (50 min)

˚ (Song, et al., 2006). Transmission electron micrographs of soot particles of mean diameter 300 A

30

Ave. Dp = 30.7 nm

B100 soot

Initial (0 min)

Counts

178

LAMINAR NON-PREMIXED FLAMES

aviation engineers as an empirical measure of a fuel’s relative sootiness (ASTM Standard D1322-75). A higher smoke point indicates low sooting propensity. Three types of laminar diffusion ﬂame burners are commonly used by soot researchers: the coﬂow, counterﬂow, and Wolfhard–Parker burners. Wolfhard and Parker (1950) conducted some early work on soot formation and developed the rectangular slot burner named after them to examine methane and ethyleneoxygen ﬂames. At this point, it is worthwhile to mention an important difference between a Burke-Schumann–type diffusion ﬂame and a sooting ﬂame. Burke and Schumann (1928) predicted the position of the ideal ﬂame front (i.e., Z = Zst ) based on a simple model of constant thermal diffusivity for all species and a Lewis number equal to unity in a steady-state ﬂow ﬁeld. One of the major contributions of this theory has been the successful prediction of the visible heights of a variety of axisymmetric ﬂames when appropriate diffusion coefﬁcients are assumed. This model has been extended successfully to describe ﬂames on a variety of coﬂow burners with different dimensions. This is the point where a real diffusion ﬂame with soot production differs from the ideal diffusion ﬂame of Burke-Schumann. The visible height of a sooting ﬂame is not necessarily the same as the Burke-Schumann diffusion ﬂame height. The problem is that the ﬂame luminosity depends both on the overall production of soot and on its overall removal by oxidation. The soot production process occurs within the ﬂame and usually is controlled by thermal and species diffusivities. Signiﬁcant burnout of the soot particles in the laminar situation can begin only after the fuel is largely consumed—this burnout is, at least partly, chemically controlled, so that high concentrations of soot cause an increase in visible ﬂame height. Whether the ﬂame smokes or not depends on whether the soot, once formed, has enough time to burn out before radiation losses and diffusion of fresh cold air quench its oxidation process. Roper, Smith, and Cunningham (1977) report that this can be expected to occur when the ratio of soot oxidation length to the Burke-Schumann diffusion ﬂame height reaches a value around 1, regardless of the nature of the fuel. The relative locations of the soot formation, soot particle growth, and soot oxidation zones in the diffusion ﬂame above a coﬂow burner can be seen in Figure 3.24. All the soot particles that are formed in the ﬂame may not be oxidized, and some may break out of the ﬂame, resulting in formation of soot wings near the ﬂame tip. 3.7.3.1

Sooting Zone

In general, the local diffusion ﬂame surface region consists of an oxidant-rich side and a fuel-rich side, separated by a reaction zone and the peak concentration of hot products. At a given height above the burner, the O2 concentration falls steadily from the oxygen side to the fuel side, and this species is in low concentration at the position where soot formation occurs. Similarly, the OH concentration, which reaches a maximum near the position of maximum temperature (corresponding approximately to the Burke-Schumann ﬂame zone), decreases rapidly toward the fuel-rich side. Although the absolute OH concentrations reported have been questioned, this chemical species is also in low concentration in the soot-forming zone.

SOOT FORMATION IN LAMINAR DIFFUSION FLAMES

179

Soot "breakthrough"

Soot oxidation zone

Soot "wings"

Soot particle growth zone

Soot particle inception zone

Air

Fuel

Air

Air

Fuel

Air

Figure 3.24 Soot formation, particle growth, and soot oxidation zones in laminar diffusion ﬂames (modiﬁed from Turns, 1996).

The sooting zone itself occurs at a short distance (millimeters) on the fuelrich side, close to the region of maximum temperature. It is characterized by its luminosity and by a strong absorptivity. A weaker absorption extends far into the relatively cool, unburned fuel, but this cannot be attributed to a particle phase alone, for which the absorbance of small particles varies as 1/l (l = wavelength). A similar “pyrolysis absorption” has been observed in ﬂow-tube pyrolysis experiments on a variety of pure hydrocarbons (Parker and Wolfhard, 1950). A detailed spectroscopic examination of this region has shown generalized absorption maxima at l = 210, 300, 380, and 450 nm, further supporting the interpretation that the absorption deep into the fuel side is not due to soot. The soot particle number density is highest (N > 1012 cm−3 ) nearest the reaction zone and decreases sharply into the fuel-rich side. Particles are formed in a fuel-rich mixture near the reaction zone where temperatures and radical concentrations are still high enough, the same as in the ﬂame zone of a premixed sooting ﬂame. The peak particle formation rates are 1014 to 1015 cm−3 s−1 . The soot volume fraction, particle size distribution, and number density proﬁles for an ethylene–oxygen ﬂame from a Wolfhard-Parker burner are shown in Figure 3.25. The particle number density increases monotonically to exceed 1012 cm−3 at the edge of the sooting region nearer the oxidizer-rich side. The arrow marked “ﬂame surface” indicates the position of the stoichiometric fuel air interface at this height. An important structural feature of all typical hydrocarbon/air ﬂames now comes into play to limit new particle formation to a narrow zone fairly close to the main reaction zone. Because 1 mole of fuel requires more than 1 mole of oxidant to complete combustion, the stoichiometric fuel-air interface moves

LAMINAR NON-PREMIXED FLAMES

fv.108 [cm3/cm3]

180

h = 40 mm Flame surface

200 100

4

300 200

2 100 0

0

2 4 6 8 Distance from keep the burner center [mm]

˚ Soot particle radius [A]

N.10−10 [cm−3]

0

0

Figure 3.25 Soot loading, particle size, and number density proﬁles across sooting region in early stages of laminar ethylene/air diffusion ﬂame established on a Wolfhard-Parker burner (modiﬁed from Haynes and Wagner, 1980).

outward from the dividing streamline (the streamline emanating from the fuel air partition of the burner) into the air side. Streamlines from the air side passing through the ﬂame zone receive increasingly higher fuel concentration at greater distances from the burner. However, away from the ﬂame zone, the temperatures and radical concentrations soon fall to values too low to promote new particle generation, and the particle formation stage is complete, although the soot loading is still negligible. (In very large ﬂames or in easily pyrolyzed fuels, lower-temperature pyrolysis leads to soot; operating on a time scale of tens and hundreds of milliseconds could conceivably become important in the interior regions, away from the main reaction zone.) The newly formed particles follow the streamlines into the fuel-rich interior of the ﬂame. At the same time, thermophoresis reinforces this effect by driving the particles deeper into the cooler fuel side, away from the hot reaction zone. Under these conditions, oxidation of the particles can hardly be signiﬁcant, but surface growth occurs readily and the soot volume fraction increases rapidly. Too far into the interior, even surface growth ceases, and the soot loading falls off again. During this growth phase, the particles are coagulating and their number density is falling. Particle size increases both by coagulation and by surface growth. Samples withdrawn from diffusion ﬂames often have been analyzed to examine their chemical structure. The results are of direct relevance to an understanding of the process involved in sooting ﬂames as well. A variety of fuels have been investigated, including CH4 (Smith and Gordon, 1956, 1959; Tsuji and Yamaoka, 1969), C2 H4 C3 H8 , (Dearden and Long, 1968; Tsuji and Yamaoka, 1969) n-pentane (Gollahalli and Brzustowski, 1973), n-hexane (Kern and Spengler, 1970), n-heptane (Aldred, Patel, and Williams, 1971; Kent and Williams,

181

SOOT FORMATION IN LAMINAR DIFFUSION FLAMES 1

40

Hexane

Luminous envelope

Ethylene

30 7

2

20

1

Height (mm)

6

More Fraction

Hexane

fV × 10

15 10 8

Methane Propylene

Acetylene

10

Propane

−4

10−8 0

Toluene Phenylacetylene Styrene

Naphthalene Fluoranthrene Phenanthrene Anthracene Benzpyrene Benzperylene Chrysene Coronene Anthanthrene

10

0

8

1016

Benzene

Pyrene

10−6

Blue zone

Soot Concentration

Butene Pentene

16

24

32

1014

Soot Concentration (cm−3 at S.T.P.)

2 4 6

Burner wall

1018

Hydrogen

10−2

1

1012 40

Height above burner (mm)

Figure 3.26 Soot volume fraction distribution in n-hexane/air diffusion ﬂame and measured species as function of distance above burner (modiﬁed from Haynes and Wagner, 1981, based on the original work of Kern and Spengler 1970).

1975), and various alcohols. In all cases, the parent fuel decomposes rapidly as it approaches the ﬂame zone and a variety of hydrocarbon products arise, as shown in Figure 3.26 for measurements along the axis of an n-hexane-air ﬂame. Typically, the main pyrolysis products are C2 H2 , C2 H4 , CH4 , and C3 H6 . Benzene is also an important product in this region, and it is here that the polycyclic aromatics arise, albeit in low concentrations (<10 ppm). There has been little attempt to relate the type and concentrations of these hydrocarbon degradation products to the formation of soot in the ﬂame. Kern and Spengler (1970) noted that C2 H2 and C6 H6 seemed most likely to persist into the carbon-forming region and that either of these species is present in concentrations sufﬁcient to account for the amount of soot appearing as these species disappear (Figure 3.26). However, when benzene is added to a methane/oxygen diffusion ﬂame, it breaks down (as indicated by the disappearance of its characteristic absorption) long before it reaches the carbon-forming zone. Similarly, the benzene in a benzene and hydrogen/air ﬂame has almost completely disappeared before the onset of sooting. Smith and Gordon (1956) found little correlation between sooting rates and the measured concentration of benzene or acetylene. Thus, ethanol diffusion ﬂames produce only a weak yellow tip, whereas the propanols (n- and iso-) yield copious amounts of soot, even though the concentrations of benzene (∼ = 0.1%) and of acetylene (1–2%) are similar in all cases, as are the temperatures. Cole and Minkoff (1957) measured the C2 H2 absorption at various points in ﬂames of methane and ethylene with oxygen on a Wolfhard-Parker burner. They found large amounts of C2 H2 in all cases, particularly just to the fuel side of the soot formation zone. However, increases in the C2 H2 absorption brought about by

182

LAMINAR NON-PREMIXED FLAMES

addition of C2 H2 to the fuel did not change the luminosity of the ﬂame, and they concluded that C2 H2 is not a precursor of soot. In contrast to this result, Dearden and Long (1968) did ﬁnd a dependence of their soot formation rates on the amount of acetylene present in the pyrolysis region of C2 H4 and C3 H8 ﬂames. The behavior of the oxidant is also of interest. The O2 concentration in the air side decreases toward the ﬂame zone, but now it seems established beyond doubt that some O2 penetrates into the unburned fuel region without being consumed. Convective effects in the ﬂame stabilization region, where reaction intensities are low, may be involved. The products of combustion CO, CO2 , H2 , and H2 O have also been measured (e.g., Smith and Gordon, 1954, 1956; Dearden and Long, 1968; Kern and Spengler, 1970). The concentrations of CO2 and H2 O peak in the region close to the maximum temperature while CO and H2 have their maximum concentrations more on the fuel-rich side. As in the case of O2 , these species persist far into the fuel-rich region. 3.7.3.2

Effect of Fuel Structure and Pressure

The reciprocal value of the height at which a ﬂame on a given burner starts to smoke and the limiting smoke-free fuel ﬂow rate have been used to gauge the tendency of different fuels to smoke. This tendency increases from parafﬁn to mono-oleﬁnes and di-oleﬁnes, benzenes, and naphthalenes. Generally, the smaller the molecule, the greater the resistance to smoke emission. Similarly, more compact isomers, such as branched chain forms, are more prone to sooting. Acetylene produces soot copiously in the diffusion ﬂame. Similar results have been obtained from ﬂame luminosity measurements (Coward and Woodhead, 1949) although this method strictly requires the same temperatures in different ﬂames to enable correct comparisons to be made. Also, at high soot concentrations, the emissivity approaches unity, and the luminosity becomes independent of the amount of soot. Glassman and Yaccarino (1980) have conﬁrmed the importance of fuel structure in determining the smoking tendency and have shown that the C/H ratio of the fuel is not a major parameter for sooting tendency of a fuel, although this quantity is used to correlate results from practical systems. Acetylene shows a weaker sooting tendency than cyclohexane when it is added in small amounts (not enough to alter the ﬂame temperature signiﬁcantly) to a hydrogen/air diffusion ﬂame. In the same experiment, Tesner Snegiriova, and Knorre (1971) and Tesner et al. (1971) found an increasing soot yield of larger particles for the series of fuels including cyclohexane, benzene, toluene, and naphthalene. A similar technique was applied earlier by Scully and Davies (1965, 1966), who injected a large number of hydrocarbon/nitrogen mixtures (1:3 mole ratio) axially into the turbulent combustion of a fuel-rich city gas/air ﬂame at about 1,300 K. Their measurements of the yield of soot also point to the tendency of aromatic rings to produce soot under predominantly pyrolytic conditions. If atoms like O, S, and N are contained in or adjacent to the ring structure, these hetero-atoms strongly inhibit soot formation, presumably by favoring ring rupture. These largely qualitative descriptions of sooting tendency agree well with the detailed shock-tube pyrolysis results of Buckendahl (1970), Geck (1975), Fussey, Gosling, and Lampard (1978), and Maryasin and Nabutovskii (1970).

SOOT FORMATION IN LAMINAR DIFFUSION FLAMES

183

The effect of aliphatic fuels on the propensity for soot emission varies as: CH4 ∼ C2 H6 < C3 H8 << C2 H4 <<< C2 H2

(3.168)

Aromatic fuels have a high sooting propensity. Qualitatively, soot formation in turbulent diffusion ﬂames shows many similarities to that process in premixed laminar ﬂames. The effect of pressure on soot formation in diffusion ﬂames has been investigated over a wide range of conditions. Generally speaking, low-pressure conditions reduce soot formation (Milberg, 1959) while high-pressure conditions promote it (Schalla and McDonald, 1955). 3.7.3.3

Inﬂuence of Additives

The effects of addition of various gaseous substances to the fuel stream also have been investigated. Simple dilution by inert gases, such as Ar, He, and N2 , generally decrease the tendency to sooting. If sufﬁcient diluent is added, soot luminosity can be suppressed altogether (Arthur and Napier, 1955), possibly because of the substantial temperature reductions in the ﬂame zone. Dilution is qualitatively similar to a reduction in pressure, and some suppression of soot formation due to this effect can also be expected. The addition of hydrogen to the fuel also suppresses soot emission. The effect of hydrogen addition on soot suppression is slightly greater than that of inert dilution. When CO2 or H2 O is added to the fuel, there is a considerable reduction in soot-forming tendency; a concentration of 45% CO2 completely suppresses luminosity in methane/air diffusion ﬂames (Arthur and Napier, 1955). Similarly, replacement of the N2 in the air by CO2 has been observed to give rise to nonsmoking ﬂames for oxygen index (OI) between 0.17 (extinction) < OI < 0.45. The OI is deﬁned as oxygen ﬂow rate/(oxygen ﬂow rate + inert gas ﬂow rate). McLintock (1968) has suggested that the inﬂuence of CO2 and H2 O is exerted primarily in the soot oxidation zone, where these species presumably promote soot burnout. A similar mechanism has been proposed for SO2 , which is even more effective in opposing soot formation than CO2 and H2 O. The soot suppression capacity is determined in terms of the maximum fuel ﬂow rate that can be supplied without appearance of smoke. The ethylene ﬂow rates in Figures 3.27 and 3.28 are the maximum ﬂow rates without smoke. It can be seen that SO2 can support signiﬁcantly higher ethylene ﬂow rates (without smoke) as compared to, for example, helium (see Figure 3.27). However, studies of the effects of a variety of additives on the critical fuel ﬂow rate have shown that, for the most part, reductions in sooting tendency are brought about thermally. Thus, such diverse additives as He, Ar, N2 , H2 O, CO2 , and SO2 are all equally effective when considered on the basis of their heat capacities, as shown in Figure 3.28. All the different curves, shown in Figure 3.27, collapse into one single curve in Figure 3.28. Some additives promote soot formation. Foremost among these are the halogens, particularly bromine (Garner, Long, Graham, and Badakshan, 1957; Ibiricu and Gaydon, 1964; Ray and Long, 1964; Schug, Manheimer-Timnat, and Yaccarino, 1980). It has been suggested that these species act by catalyzing radical

184

LAMINAR NON-PREMIXED FLAMES

Ethylene Flow Rate, cm3/s

8

7

6

5

SO2 CO2 H 2O N2 Ar He

4

3

0

1

2

3

4

Diluent Flow Rate,

5 cm3

6

7

/s

Figure 3.27 Effect of various additives to fuel in suppressing soot, as determined by increase in fuel ﬂow rate at smoke point with increasing additive ﬂow (modiﬁed from Haynes and Wagner, 1981 based upon the original work of Schug, Manheimer-Timnat, and Yaccarino, 1980).

recombination, thus neutralizing excess OH radicals that could otherwise oxidize soot or soot precursors. Ignition promoters such as organic nitrites have been reported either to have no effect on soot emissions or to enhance them strongly. The effect of oxygen addition to the fuel is complex. In ethylene ﬂames, small additions of oxygen result in pronounced increased soot emissions. For overventilated ﬂames, the maximum yield is achieved at an oxygen/carbon ratio in the mixture around 0.4 (Chakraborty and Long, 1968a, 1968b; Wright, 1974). On an underventilated Wolfhard-Parker burner where there is limited subsequent oxidation of the carbon once formed, the soot yield increases continuously with further oxygen addition even up to the theoretical premixed sooting limit of O/C = 1. This effect of increasing yield is not purely thermal as it is far greater than that obtained at the same maximum ﬂame temperature produced by oxygen enrichment of the air (Dearden and Long, 1968). It has been suggested that the presence of oxygen accelerates the pyrolysis (Dearden and Long, 1968) or polymerization reactions (Wright, 1974) occurring in the ethylene—at 800 K, oxygen accelerates the pyrolysis without signiﬁcantly altering the product distribution. If the oxygen is added as air, the effects of such processes apparently are outweighed by those due to inert dilution (by N2 ), and a net reduction in soot emissions is observed (Wright, 1974). In the case of other fuels, both soot promotion and inhibition have been observed as the result of oxygen addition. Jones and Rosenfeld (1972) concluded that ethylene is the exception and that, for fuels such as propane, butane, and even propylene, oxygen suppresses soot emissions. In contrast to this result, others (Schug, Manheimer-Timnat, and Yaccarino, 1980; Wright, 1974) have found enhancement to be the rule when oxygen is added to

SOOT FORMATION IN LAMINAR DIFFUSION FLAMES

185

9 Pure C2H4 He CO2 SO2 Ne Ar

Ethylene Flow Rate, cm3/s

8

7

6

5

4

4

5

6

7

8

9

10

11

12

13

Heat Content of Total Flow, cal/s

Figure 3.28 The same data shown in Figure 3.27 plotted against total sensible heat content of fuel + additive stream at average ﬂame temperature (modiﬁed from Haynes and Wagner, 1981, based upon the original work of Schug, Manheimer-Timnat, and Yaccarino, 1980).

ethane, propane, butane, propylene, and benzene diffusion ﬂames. Enhancement of the sooting rate also has been observed by others (Chakraborty and Long, 1968a, 1968b) for ethane, whereas the opposite was observed in the case of propane. Probe analyses invariably indicate the presence of some oxygen on the fuel side of a diffusion ﬂame, presumably as a result of convective effects in the stabilization region near the burner. Judging by the oxygen addition results, this oxygen may have signiﬁcant effects on soot formation in the ﬂame; these effects that depend (to some degree) on the burner geometry. Additions of other hydrocarbons to the fuel also have been examined. The more the soot yield is increased, the greater the natural tendency of the additive to produce soot in a diffusion ﬂame environment. By far the most striking effects of additives are those exhibited by various metals, some of which are used commercially as smoke suppressants in liquid fuel combustion. This aspect has been reviewed by Salooja (1972), who concluded that, although different additives are more effective under different conditions, manganese is generally very effective while iron and other transition metals, such as nickel, cobalt, and copper, can signiﬁcantly reduce smoke in

186

LAMINAR NON-PREMIXED FLAMES

some circumstances. Barium, which is widely used as a diesel smoke suppressant, appears to be very effective at higher concentrations (0.5% by weight of the fuel), but at lower concentrations it is less effective than other additives. The mode of action of these sorts of additives is unknown, but there is evidence that nickel, on being incorporated into the soot phase, chemically catalyses the gasiﬁcation of soot. A similar oxidation catalysis has been proposed for the action of manganese (Fenimore, Jones, and Moore, 1957; Fenimore and Jones, 1967; Friswell, 1972). Early studies by Bartholome and Sachsse (1949) showed that these metals could greatly reduce or even suppress soot emission from a diffusion-type partial combustor. They proposed that the metals promote ionization of the soot particles, thus inhibiting their coagulation such that they are more easily oxidized. Addecott and Nutt (1969) found a weak correlation between measured ion concentration and smoke reduction. They proposed that more readily ionized additives reduce the concentration of “ionic precursors” by charge transfer, Cn Hm+ + M = Cn Hm + M+

(3.R6)

thereby suppressing soot formation. Based on the considerable evidence for a neutral mechanism (nonionic) of soot formation, the inﬂuence of metal additives has also been described in terms of neutral chemistry. Thus, Cotton et al. (1971) proposed that the activity of the alkaline earths is due to their known ability to catalyze the radical equilibrium H2 O H + OH. If OH is suppressed below its equilibrium concentration in the sooting zone by rapid reaction with soot and unburned hydrocarbons, the metal could improve the supply of OH and thereby promote soot oxidation. Csoot + OH → CO + 1/2 H2

(3.R7)

Cotton, Friswell, and Jenkins (1971) supported this interpretation by showing that the anti-soot effect is quantitatively related to the ability of such widely varying substances as Ba, SO2 , and NO to catalyze the radical equilibrium. 3.7.3.4

Coﬂow Ethylene/Air Laminar Diffusion Flames

Although the counterﬂow (opposed jet) and Wolfhard–Parker burners produce almost one-dimensional ﬂames, they may suffer from stability problems at elevated pressures. In addition, in counterﬂow burner ﬂames, the stagnation point location is very critical, and it can vary with the choice of ﬂow conditions. Coﬂow burner ﬂames are radially axisymmetric ﬂames with demonstrated stability at high pressures. For this reason, a signiﬁcant portion of research work on soot formation and oxidation has been conducted on coﬂow laminar diffusion ﬂames. A coﬂow burner operating at atmospheric pressure conditions was used by Santoro, Semerjian, and Dobbins (1983) to study laminar diffusion ﬂames burning

SOOT FORMATION IN LAMINAR DIFFUSION FLAMES

187

ethylene (C2 H4 ) and ethane (C2 H6 ). A laser scattering/extinction system was used to obtain data on the soot particle ﬁeld, which is a commonly used experimental technique to study soot formation/oxidation processes in such ﬂames. Measurements were obtained with ethylene (also called ethene) as the fuel for various fuel ﬂow rates. With ethene as the fuel, three ﬂow rate conditions were studied: one in which soot was not observed to emit from the ﬂame tip, a second in which soot emission was restricted to an annular ring near the luminous ﬂame edge (referred to as wings as shown in Figure 3.24), and, ﬁnally, a condition in which soot particles were emitted across the entire ﬂame region. These three cases are referred to as nonsooting (NS), incipient sooting (IS), and sooting (S) conditions. However, in all cases, the ﬂame was observed to have a strong yellow luminosity, which is indicative of soot formation. The terms just used refer only to whether soot actually was emitted from the ﬂame tip. The results reveal that the ﬂame can be broadly divided into two regions, a region of growth where soot formation processes dominate, and a second region where oxidation processes are dominant. Measurements show that soot is ﬁrst observed to form closer to the burner rim in an annular region inside the main reaction zone. At higher locations, this annular region widens until the entire ﬂame is observed to contain soot particles. Soot volume fraction, number density, and particle size measurements are shown in Figures 3.29 and 3.30 for the ethylene/air ﬂames. As mentioned earlier, soot particle diameter shows a distribution (mostly log-normal). For this reason, soot particle diameter is reported as an average value, calculated by Equation 3.169: ⎡ ∞

Dpq

1 ⎤ p−q

⎢ P (D)D p dD ⎥ ⎥ ⎢ ⎥ ⎢0 ⎥ ⎢ =⎢ ∞ ⎥ ⎥ ⎢ ⎣ P (D)D q dD ⎦

(3.169)

0

where P(D) is the particle size probability density function such that the quantity P(D)dD represents the fraction of all particles contained in the interval D to D + dD, from which it follows that: ∞ P (D)dD = 1

(3.170)

0

This deﬁnition corresponds to the moment ratio of p th and q th order. For example, D63 is the mean diameter corresponding to the ratio of the sixth and third moments. Another well-known parameter called the Sauter mean diameter is actually D32 (i.e., the mean diameter corresponding to the ratio of the third and second moments); they represent the average volume and average surface areas of particles, respectively. Similarly, D30 represents the volume-averaged diameter.

188

LAMINAR NON-PREMIXED FLAMES

7 × 1011

ψ 0.034 IS 0.061 NS 0.127 S 0.170 IS 0.202 NS 0.222 S

1011

1010

Particle diameter, D63 (nm)

(a) ψ 0.034 IS 0.061 NS 0.127 S 0.170 IS 0.202 NS 0.222 S

100

10 5 (b) ψ

Soot volume fraction, fv

0.034 IS 0.061 NS 0.127 S 0.170 IS 0.202 NS 0.222 S

0.00 2.00 Radial position (mm) NS: nonsooting IS: incipient sooting S: sooting

(c)

4.00

6.00

8.00

ψ: dimensionles height above the burner exit (defined by Eg. 3.172)

Figure 3.29 Measured radial distributions of (a) number density, (b) particle diameter, and (c) soot volume fraction in three ethylene/air diffusion ﬂames at various axial locations in lower region of ﬂame (modiﬁed from Santoro, Semerjian, and Dobbins, 1983).

SOOT FORMATION IN LAMINAR DIFFUSION FLAMES

Number density, N (cm−3)

1011

189

η 0.318 S 0.332 NS 0.339 IS

1010

109

Particle diameter, D63 (nm)

(a) 90 80 70 60 50

η 0.318 S 0.332 NS 0.339 IS

40 30

20 (b)

Soot volume fraction, fv

10−5

η 0.318 S 0.332 NS 0.339 IS

10−6

10−7

10−8 −4.00

−3.00

−2.00

−1.00 0.00 1.00 Radial Position (mm)

2.00

3.00

4.00

(c)

Figure 3.30 Measured radial distributions of (a) number density, (b) particle diameter, and (c) soot volume fraction in three ethylene/air diffusion ﬂames at various axial locations in upper region of ﬂame (modiﬁed from Santoro, Semerjian, and Dobbins, 1983).

190

LAMINAR NON-PREMIXED FLAMES

Santoro, Semerjian, and Dobbins (1983) used D63 as the average soot diameter; they determined this parameter from laser scattering, and the scattering cross section is proportional to the sixth moment of particle diameter. A relationship between D63 and D30 was established by Graham and Robinson (1970). They found that (D63 /D30 )3 to be equal to 2.0. The soot volume fraction can be related to D30 or D63 by: π π (3.171) fV = ND330 = ND363 6 12 The soot volume fraction proﬁles show that measurable amounts of soot are ﬁrst observed in the annular region of the ﬂame. The soot volume fraction steadily increases in this annular region, reaching a maximum between η = 0.17 and 0.22. The central region of the ﬂame is ﬁrst observed to contain soot particles at η = 0.10, which is signiﬁcantly higher than that of the annular region. The dimensionless height η is deﬁned as: η≡z where

D ˙ Q

(3.172)

z = height above the burner exit, D = mass diffusivity of fuel ˙ = volumetric ﬂow rate of the fuel Q The particles in the central region then undergo rapid growth at subsequent heights, although never reaching the levels seen in the annular region. Above this region (η > 0.24), fV can be observed to decrease for all the ﬂames as oxidative processes begin to dominate. The extent and rate of the soot particle oxidation is differentiated by the soot-emitting characteristics of the ﬂame (see Figure 3.30a—that is, nonsooting (NS), incipient sooting (IS), or sooting (S). The particle size and number density measurements reveal that the regions of maximum fV coincide with regions of relatively large particle size and nearly constant number density (∼1011 cm−3 ). The occurrence of particle sizes as large as 60 nm in the lower portion of the ﬂame indicates that strong nucleation, coagulation, and/or growth are coexistent in this annular region. An interesting feature of the lower part of the ﬂame (η = 0.034) is the occurrence of a minima in N at nearly the same radial position as the maxima of fV and D63 (see Figure 3.29c). A possible explanation in this case is that intense formation and coagulation cause the maximum volume fraction and particle size and the minimum number concentration to coincide. Other diffusion ﬂame studies in Wolfhard-Parker–type burners (described earlier in this chapter) report that the number density is observed to reach a maximum slightly inside the maximum temperature region and to decrease as the centerline is approached. In the annular region higher in the ﬂame, the particles are observed to grow to a maximum D63 of 120 nm while N remains near 1010 cm−3 . The region of smaller particles (D63 ∼ 10 nm) but larger number concentration (N ∼ 1011 cm−3 ) moves closer to the centerline, eventually ﬁlling the central core of the ﬂame. As η increases, this central region shows a large increase in particle size with an accompanying decrease in number density as the soot volume fraction increases.

SOOT FORMATION IN LAMINAR DIFFUSION FLAMES

191

In the region where particle oxidation is observed, the growth of the particles ceases. Figure 3.30b reveals that the particle sizes along the centerline are very similar for all three ﬂames (D63 ∼ 80 nm). However, the number density decreases in a manner analogous to the soot volume fraction. Particle size and number density decrease with increasing radius within the central core of the nonsooting and incipient sooting ﬂames. Particle size and number density in the oxidizing region decrease with increasing height for all ﬂames. However, the size and number density at a particular η are ordered so that the sooting ﬂame has the largest values and the nonsooting the smallest (Figure 3.30). This relation between the various ﬂames can also be seen in the soot volume fraction measurements. The integrated soot volume fraction is a measure of the total amount of soot present in a cross-sectional plane of radius R at a height corresponding to η. R = 2π fV (r, η) r dr (3.173) 0

The integrated soot volume is shown in Figure 3.31a for all three ﬂames as well as that of an ethane/air coﬂow diffusion ﬂame. Figure 3.31b shows a plot of the soot volume fraction at the maximum in the annular region and along the center line as a function of η. The soot volume is observed to grow steadily with only small differences among the three ﬂames. However, for η > 0.16, the trend shows distinct differences among the ﬂames. The transition from soot being completely oxidized to the case where only the annulus region survives and later to the case when the entire ﬂame tip exhibits soot emission can be clearly seen. The evolutionary relation described previously between the annulus and central regions of these ﬂames is also illustrated. The ethane (C2 H6 ) diffusion ﬂame shown in Figure 3.31a is a nonsooting ﬂame. Comparison of the particle ﬁelds in the ethane and ethylene (C2 H4 ) ﬂames indicates that the soot volume fraction and number density are consistently lower in the ethane ﬂame. However, the largest particle sizes were observed in the annular region (D63 ∼ 100 nm) and in the center region of the ﬂame (D63 ∼ 70 nm), which are very close to those observed in the ethylene ﬂame. Another difference is illustrated in Figure 3.30a, where it is observed that the onset and maxima in the soot volume fraction occur later in the ethane ﬂame as compared to the ethylene ﬂame. However, once the formation and growth processes cease, both ﬂames exhibit a rapid oxidation zone. 3.7.3.5

Modeling of Soot Formation

Soot modeling should include models for inception, surface growth, oxidation, and coalescence of soot particulates. Effects of thermal radiation, particle scrubbing of gas-phase growth, and oxidation species should also be included in a soot model. A successful model for soot formation should be able to predict the soot yield in a diffusion ﬂame or the smoke point of a diffusion ﬂame. Soot production is also inextricably linked to the ﬂame structure through its impact on ﬂame temperatures via radiative heat loss. Predictions of smoke point height remain among the greatest challenges facing modelers. Advances are hampered

Integrated Soot Volume Fraction, Φ

192

LAMINAR NON-PREMIXED FLAMES

10−6

10−7 C2H4 NS C2H4 IS C2H4 S C2H6 NS 10−8 (a)

Soot Volume Fraction, fv

10−5

10−6 NS Annulus (max) IS Annulus (max) S Annulus (max) NS Center Line IS Center Line S Center Line

10−7

0.00

0.06

0.12

0.18 0.24 0.30 Dimensionless Height, η

0.36

0.42

0.48

(b)

Figure 3.31 Local and integrated soot volume fraction as function of η in ethylene/air and ethane/air ﬂames: (a) integrated soot volume fraction ; and (b) comparison of centerline and annulus (maximum) local soot volume fractions fV ; and (b) (modiﬁed from Santoro, Semerjian, and Dobbins, 1983).

by uncertainties in the fundamental rates of processes such as oxidation or surface growth. Nevertheless, some useful predictions of soot formation can be obtained from relatively crude models. Soot models can be divided into three broad categories: (1) empirical correlations, (2) semiempirical models, and (3) detailed models. A list of models in all three categories is given in Table 3.2.

193

Smoke number/soot mass concentration

Concentrations of 15 sizes of carbon clusters

Aviation turbine fuel

Exhaust soot concentration

Jensen and Wilson (1975)

Diesel

Exhaust soot concentration

Soot mass fraction and particle number density Soot mass fraction and particle number density Soot mass fraction

Diesel

Molar soot concentration

Edelman and Harsha (1978) Khan, Greeves, and Probert (1971) Mehta and Das (1992) Rizk and Mongia (1990, 1991)

Fairweather et al. (1992) Fairweather, et al. (1992) Honnery and Kent (1992)

Kerosene

Concentration of carbon clusters

Hudson and Heicklen (1968)

Isopropyl nitrate

Ethane and ethylene

Propane

Natural gas

Flamelet library based on detailed chemistry Flamelet library based on detailed chemistry Equilibrium for gas phase; empirical rates for soot formation and oxidation 12-step gas phase reaction mechanism

—

—

Arrhenius expressions for soot formation and oxidation —

—

Chemistry Model

Semiempirical models

—

Empirical models

Fuel

Major Variables

Reference

(continued overleaf )

Rocket motor

Turbulent diffusion ﬂame Turbulent diffusion ﬂame Laminar diffusion ﬂames

Gas turbine engine

Diesel engine

Diesel engine

Pyrolysis in isothermal ﬂow reactor Gas turbine engine

Application

TABLE 3.2. Summary of Modeling Approaches for Soot Formation in Diffusion Flames (modiﬁed from Kennedy, 1997)

194

Soot mass fraction and particle number density

Mean soot volume fraction

Mean soot with model of Hiroyasu Nuclei and soot particle number densities (particle mass assumed) Soot mass fraction and number density

Kennedy et al.(1996)

Kollman et al. (1994)

Kouremenos et al. (1990) Kyriakides, Dent, and Mehta (1986) Leung, Lindstedt, and Jones (1991) Makel and Kennedy (1994)

Moss, Stewart, and Young (1995) Said, Garo, and Borghi (1997)

Soot volume fraction

Kennedy et al. (1990)

Soot volume fraction and particle number density Soot precursor mass fraction; soot mass fraction

Soot volume fraction

Major Variables

Reference

TABLE 3.2. (continued )

Equilibrium

Flamelet library Single-step fuel oxidation

C2 H4 /O2 /Ar Ethylene/air

111-step C1 -C3 hydrocarbon mechanism

—

Constrained equilibrium (scalars function of mixture fraction and enthalpy-radiation heat loss included) Equilibrium

Constrained equilibrium (scalars function of mixture fraction and enthalpy-radiation heat loss included) Short mechanism for C1 and C2 chemistry

Chemistry Model

Ethylene/air

C2 H4 /O2 /N2

Diesel

Diesel

Ethylene/air

Ethylene/air

Ethylene/air

Fuel

Inverse laminar diffusion ﬂame (air into fuel) Laminar diffusion ﬂame Laminar and turbulent diffusion ﬂame

Counterﬂow laminar diffusion ﬂame

Direct injection diesel engines

Diesel engine

Laminar ethylene/air diffusion ﬂame (1 atm) Turbulent diffusion ﬂame (1 atm)

Laminar ethylene/air diffusion ﬂame (1 atm)

Application

195

Species mass fractions; soot number, size and volume fraction Species mass fractions; soot number, size and volume fraction C2 H2 -O2 -Ar (12 kPa)

C2 H2 -O2 -Ar (12 kPa); C2 H4 -O2 -Ar (101 kPa)

Soot yield

Frenklach and Wang (1994) Frenklach and Wang (1990)

Frenklach and Wang (1994)

C2 H4 -O2 -Ar (1 atm); C2 H2 (12 kPa); C3 H8 (15 kPa) 1.09% acetylene/Ar

Species and soot mass fractions

Colket and Hall (1994)

32% acetylene/ 68% N2 in air

Detailed models

Detailed kinetics of acetylene pyrolysis; PAH growth

Flamelet model for thermochemistry as well as source terms for soot volume fraction Detailed kinetics up to benzene; various soot growth models Detailed kinetics of acetylene pyrolysis; PAH growth Detailed kinetics of acetylene pyrolysis; PAH growth

Flamelet library for density; temperature and soot precursor concentration

—

Acetylene/air

Ethylene/air

Flamelet library

Methane/air

Soot volume fraction

Soot particle number density and volume fraction

Soot volume fraction and particle number density Concentration of radical nuclei and soot particles

Balthasar et al. (1996)

Syed, Stewart, and Moss (1990) Tesner, Snegiriova, and Knorre, 1971; Tesner et al., 1971 Young and Moss (1995)

(continued overleaf )

Laminar, premixed ﬂames

Laminar, premixed ﬂame

Shock tube

Laminar, premixed ﬂames

Laminar diffusion ﬂame

Turbulent ethylene diffusion ﬂame (1 atm)

Turbulent diffusion ﬂames Laminar diffusion ﬂame

196

Yoshihara et al. (1994)

Mauß et al. (1994)

Markatou, Wang, and Frenklach (1993) Mauß et al. (1994)

Detailed gas phase kinetics; modiﬁed Frenklach-Wang, PAH growth, and soot growth Reduced mechanism for fuel/air reaction; Frenklach and Wang for soot

Ethane, ethylene, and acetylene/air Acetylene/air/O2

C2 H2 /O2 /Ar

88% CH4 + 6% C2 H2 + 6% C3 H8 /air

Detailed gas phase kinetics; modiﬁed Frenklach-Wang, PAH growth, and soot growth

C2 H4 /O2 /N2 ; methane/air

Soot mass fractions and particle number density Mole fractions of gas phase species and soot; critical equivalence ratio Gas phase mass fractions; moments of soot aerosol distribution; soot volume fraction

Gas phase mass fractions; moments of soot aerosol distribution; soot volume fraction Exhaust soot mass fraction

Detailed kinetics for aliphatics and PAH

Ethylene/air

Species mass fractions; soot volume fractions

Kazakov, Wang, and Frenklach (1995) Lindstedt (1994)

Chemistry Model 102 reactions for aliphatic chemistry and benzene formation Detailed gas phase kinetics, Frenklach-Wang for PAH and soot 292-step kinetics

Methane/air

Species and soot mass fractions

Hall, Smooke, and Colket (1996)

Fuel

Major Variables

Reference

TABLE 3.2. (continued )

Direct injection diesel engine

Laminar, premixed ﬂame; postﬂame soot directed against counterﬂow of pure oxygen (1 atm) Laminar premixed ﬂames at p = 9−18 kPa

Laminar counterﬂow and coﬂow ﬂames Laminar, premixed ﬂames (1 atm)

Laminar, counterﬂow diffusion ﬂames (1 atm) Laminar, premixed ﬂames (10 bar)

Application

SOOT FORMATION IN LAMINAR DIFFUSION FLAMES

197

Most of the empirical correlations are based on experimental data obtained from gas turbine engines and diesel engines. The correlations that are based on a phenomenological description have found wide use in the research community. The semi-empirical soot models are the next level of soot modeling, and they attempt to incorporate some aspects of the physics and chemistry of the phenomenon, as opposed to a correlation (albeit with physical insight) of experimental data. This approach usually leads to the development of rate equations for reactions of soot precursors and soot particles with a simple description of the chemistry. Both of these categories rely on empirical inputs for soot nucleation, growth, and oxidation rates, and they are limited inherently to speciﬁc conditions. They cannot be applied to conditions far from those under which the rates were measured. Hence, they cannot be extended easily to different fuels or different pressures. A complete description of the kinetics of PAH and soot growth is required for generality. Detailed models couple the dynamical equations for particle production with the ﬂow and gaseous species conservation equations. Frenklach and Wang (1994), Yoshihara, et al. (1994), and Wang and Frenklach (1997) have worked on the development of detailed soot models over the last 20 years. Their model (1997) has reached the greatest level of sophistication of any existing model. It treats the full details of phenomena, from the initial pyrolysis of fuel to the nucleation of soot particles, their growth and coagulation, and their ultimate oxidation. Most of the attention in the early phases of model development was directed to simulate soot formation in laminar premixed ﬂames. Yoshihara et al. (1994) applied the model to the industrially important problem of soot formation and burnout in a diesel engine. Frenklach et al. (1994) considered the dominant route for the formation of the ﬁrst aromatic ring from an aliphatic fuel to occur via vinyl addition to acetylene, leading eventually to cyclization via the interaction of an unpaired electron with a triple carbon bond. The aromatic ring then grew through the so-called HACA route (H abstraction-C2 H2 addition) in which thermodynamically stable species such as acenaphthalene and coronene were formed and were subsequently reactivated by H atom abstraction (see Figure 3.32). Reactive coagulation of PAH could also be important (i.e., the fusing of one or more PAH to form a larger entity). This mechanism may be particularly important when the parent fuel is an aromatic compound, such as benzene. The computational model consists of three logical parts: 1. Initial PAH formation, which includes a detailed chemical kinetic description of acetylene pyrolysis and oxidation, formation of the ﬁrst aromatic ring, and its subsequent growth to a prescribed size. 2. Planar PAH growth, comprised of replicating-type growth of PAHs beyond the prescribed size. 3. Spherical particle formation and growth, consisting of coagulation of PAHs formed in part 2 followed by the growth of the resulting particles by coagulation and surface reactions. The detailed chemical reaction mechanism for the initial PAH formation is a 337-reaction, 70-species mechanism (Frenklach and Wang, 1990). Appendix D

198

LAMINAR NON-PREMIXED FLAMES

H(•H2)

• C H (H) 2 2

H

H

C

C H(−H2) H

C

C

•

Benzene C2H2 C

C

C

C •

•

H(−H2)

C2H2(−H)

H

H

C2H2 • Large polycyclic aromatic hydrocarbons (PAHs)

C2H2(−H) H

Figure 3.32 H-abstraction-C2 H2 -addition (HACA) mechanism for polycyclic aromatic hydrocarbon (PAH) formation (modiﬁed from Law, 2006, based on work of Frenklach and Wang, 1994). • (−H) H(−H2) H Benzene (−H)

C2H2(−H)

H

•

H

• (−H)

H(−H2) H

•

(−H)

Large polycyclic aromatic hydrocarbons (PAHs)

Figure 3.33 Alternative polycyclic aromatic hydrocarbon (PAH) growth mechanism (modiﬁed from Law, 2006, based on work of Frenklach and Wang, 1994).

shows the PAH formation reaction and also the reaction mechanism associated with the other two steps (i.e., growth and soot oxidation), thereby making the complete gas-phase reaction mechanism with 527 reactions and 99 species (Wang and Frenklach, 1997). The second part of a soot model, PAH growth (see Figure 3.33), describes the planar growth of PAHs via the HACA reaction sequence. A soot particle is taken to be formed when two PAH molecules combine to form a dimer (formed

SOOT FORMATION IN LAMINAR DIFFUSION FLAMES

199

when two molecules are connected by a weak bond). Other key reactions in PAH growth are: n-C4 H3 + C2 H2 → C6 H5 (Phenyl)

(3.R8)

n-C4 H5 + C2 H2 → C6 H6 (Benzene) + H

(3.R9)

ArylH + H → Aryl· + H2 Aryl· + H → ArylH Aryl + C2 H2 → Aryl-C2 H + H Aryl· + O2 → Products

(3.R10) (3.R11) (3.R12) (3.R13)

“Aryl” refers to any benzene-derived aromatic compound (e.g., phenyl, tolyl, xylyl). Beginning with the dimers, forming clusters is assumed to be “solid phase” and allowed to add and lose mass by several surface reactions: k1

Csoot,i -H + H Csoot,i · + H2 k−1 k2

Csoot,i · + H → Csoot,i -H k3

Csoot,i · + C2 H2 → Csooti+1 -H + H

(3.R14) (3.R15) (3.R16)

Finally, the soot products are formed by oxidation with O2 and OHa according to: k4

Csoot,i · + O2 → products k5

Csoot -H + OH → products

(3.R17) (3.R18)

where Csoot,i -H represents an armchair site on the soot particle surface and Csoot,i · represents the corresponding radical. This mechanism is adopted based on the postulate that the HACA reaction sequence is responsible for high-temperature growth of all forms of carbonaceous materials. The rate of growth of a particle is formulated as: ω˙ g = kCg αst ψs · (S × n)

where k Cg αst ψs S n

= = = = = =

(3.174)

per-site rate coefﬁcient concentration of the growth species empirical steric factor number density of the surface sites surface area of a spherical particle number density of particles of a given size class.

In Equation 3.174, it is assumed that the soot particle could be represented as a sphere with a diameter that was based on its mass and the density of bulk soot (taken as 1,800 kg/m3 ). The surface area of the particle was multiplied by ψs to account for the number of surface radicals per unit surface area. This factor

200

LAMINAR NON-PREMIXED FLAMES

could be derived from a steady-state assumption for Csoot · applied to Reactions 3.R14 to 3.R17 so that: ψs =

k1 XH ψC -H k−1 XH2 + k2 XH + k3 XC2 H2 + k4 XO2 soot,i

(3.175)

The number of sites per unit area of soot particle ψCsoot,i -H can be estimated on the basis of dimensions of PAH rings and distance between PAH layers in soot to have a value of about 2.3 × 1019 to 2.9 × 1019 m−2 . Soot was oxidized due to the reaction of O2 with a radical site and by the attack of OH on the soot surface, as shown before in reactions 3.R17 and 3.R18. Neoh, Howard, and Saroﬁm (1981) used the above model in an experimental study and employed a reaction probability of OH with soot to be 0.13. The products of O2 and OH reaction with soot were not given explicitly because they did not play an important role in the chemistry for the conditions studied. However, this issue can be important in terms of the structure of the ﬂame, as pointed out by Kennedy (1997). The gas-phase products of soot oxidation in a heavily laden diffusion ﬂame may participate in the overall ﬂame chemistry in an important way. Current understanding of the details of OH reactions with soot is still deﬁcient. The particle dynamics associated with the evolution of soot particles undergoing simultaneous nucleation, coagulation, and surface reactions described can be accounted for via the method of linear lumping in which equations are derived for the change in moments of the distributions of number density and particle size. The k th moment of number density is deﬁned as: k

M = where

∞

mki Ni

(3.176)

i=1

M k = k th moment of number density mi = mass of PAH species of class i Ni = number density of soot particles

The size moments (μk ) are deﬁned as normalized number density moments: μk =

Mk M1

(3.177)

The technique of chemical lumping provides a mathematically rigorous description of the growth process to an inﬁnite size PAH, which can be schematically represented as: Al → Al+1 → Al+2 → . . . . . . . . → A∞ (3.178) where Al represents an aromatic species containing l fused rings. The PAH species formed in Equation 3.178 are then allowed to coagulate—that is, all the Ai (i = l, l + 1, . . . ∞) collide with each other forming dimers; the dimers, in

SOOT FORMATION IN LAMINAR DIFFUSION FLAMES

201

turn, collide with Ai , forming trimers, or with other dimers, forming tetramers, and so on. The coalescence reactions can be treated as irreversible, having sticking coefﬁcients of unity. Frenklach and Wang (1990) used a size-independent enhancement factor of 2.2, based on the results of Harris and Kennedy (1988), to calculate collision frequencies. An equation for the mass fraction of soot in the laminar diffusion ﬂame can be written as:

∂ ∂Ysoot ∂ 1 ∂T D ∂ Ysoot + ρD + 0.54μ ρuj Ysoot = (ρsoot fV ) ∂xj ∂xj ∂xj ∂xj T ∂xj Dt (3.179) The term on the left-hand side accounts for convective mass transfer of soot out of the local control volume. The ﬁrst term on the right-hand side accounts for soot diffusion, the second accounts for thermophoresis, and the ﬁnal term is the net source term. Thermophoresis is a physical phenomenon caused by the temperature gradient in the gas phase. An aerosol (like a soot particle) in the temperature gradient ﬁeld experiences a force in the direction of decreasing temperature. This occurs since the aerosol particles are bombarded by gaseous molecules continuously. Since the hot-side gas molecules have higher momentum than those in the cold side, the aerosol is veered toward the cold side. Balthasar et al. (1996) implemented the chemistry model of Frenklach and Wang (1994) with the ﬂuid dynamic model and predicted soot formation in a nitrogen-diluted axisymmetric ethyne/air-diffusion ﬂame, which was also experimentally investigated by Gomez, Littman, and Glassman (1987). The different contributions to the source term of soot volume fraction related to the mixture fraction are plotted in Figure 3.34a–c. These results correspond to a height of 7 mm above the burner and a scalar dissipation rate of 10/s, as can be seen from Figure 3.34d. The scalar dissipation rate is a function of height above the burner. Soot is formed mainly by heterogeneous surface growth reactions occurring at values of mixture fraction (Z ) between 0.23 and 0.60. The maximum surface growth rate is found close to the stoichiometric mixture fraction at Zst = 0.267, where the concentration of H radicals is high and the temperature is about 1,800 K. The soot growth rates span over a wider range of mixture fraction (from Z = 0.267 to Z = 0.6). For lower values of Z , the temperature increases, and the reverse reactions of the carbon addition prevent a further surface growth so that surface growth becomes slightly negative. The contribution of particle inception to the source term of the soot volume fraction is minor. The soot is oxidized in the fuel-lean area of the ﬂamelet between Z = 0.1 and 0.2. The calculated proﬁles of rates of the soot growth process for several scalar dissipation rates are shown in Figure 3.35. At high scalar dissipation rates (between 4/s up to 10/s) and large mixture fractions (small heights above the burner), the maximum surface growth rate decreases with decreasing dissipation rate. This is caused by the increasing temperature, resulting in higher rates of the reverse reactions of carbon addition. At lower scalar dissipation rates (between 0.5/s up to 1.5/s corresponding to increasing height above the burner) and small mixture

202

LAMINAR NON-PREMIXED FLAMES 120

(dfv/dt)sg/fv [s−1]

(dfv/dt)pi [s−1]

2 10−6 1.5 10−6 1 10−6 5

10−7

400 350 300 250 200 150 100 50 0

0.2

0.3 0.4 0.5 Mixture Fraction Z (a)

0

0.1

0.2

0.3

0.4

0.5

0.6

80 60 40 20 0 0.1

0.6

Scalar Dissipation Rate [s−1]

−1

(dfv/dt)ox/fv [s ]

0 0.1

100

0.2 0.3 0.4 0.5 Mixture Fraction Z (b)

100

10

0

Mixture Fraction Z (c)

5

10

15

20

25

Axial Distance [mm] (d)

Figure 3.34 Distributions of (a) particle inception rates, (b) normalized surface growth rates, (c) oxidation rates, and (d) scalar dissipation rate (Balthasar et al., 1996).

120 c = 10

(dfv/dt)sg/fv [s−1]

100

c=7 c=4

80

c = 1.5 c = 0.7

60

c = 0.5 40 20 0 0.25

0.30

0.35

0.40 0.45 0.50 Mixture Fraction Z

0.55

0.60

0.65

Figure 3.35 Distributions of surface growth (sg) rate for various scalar dissipation rates (modiﬁed from Balthasar et al., 1996).

SOOT FORMATION IN LAMINAR DIFFUSION FLAMES

203

fractions, the maximum surface growth rates increase again with decreasing dissipation rate, since the maximum temperature of the ﬂamelet decreases because of radiative heat losses. The resulting proﬁles of particle inception (pi) rate, surface growth (sg) rate, and oxidation (ox) rate at the centerline of the acetylene/air-diffusion ﬂame are shown in Figure 3.36. Particle inception starts at 10 mm and has a maximum value of 1.4 × 10−5 /s at 18 mm height. Surface growth appears after the ﬁrst particles are formed. The slope of surface growth rate proﬁle, ﬁrst increases up to a height of 20 mm, then decreases again between 20 and 25 mm. At 25 mm, the scalar dissipation rate reaches the value of 1.5/s. Above this axial position, the slope of the surface growth rate proﬁle increases again up to its maximum

2 10−5

(dfv/dt)φ [s−1]

4 10−4

(dfv/dt)pi (dfv/dt)ox

3 10−4

(dfv/dt)sg 1 10−5

2 10−4 1 10−4 0

0

10

20 30 40 50 Axial Distance [mm]

0 70

60

Figure 3.36 Centerline proﬁles of particle inception (pi), surface growth (sg), and oxidation (ox) rates (Balthasar et al., 1996).

Axial Distance [m]

0.040 4.0

3.0

0.020

2.0 1.0

0.5

0.000 −0.004

0.000 Radial Distance [m]

0.004

Figure 3.37 Two-dimensional proﬁles of soot volume fraction fV × 106 [cm3 /cm3 ] (Balthasar et al., 1996).

204

LAMINAR NON-PREMIXED FLAMES

of 32 mm. At this height, the local mixture fraction is higher than 0.22 and surface growth reactions suddenly stop. Beyond this distance, soot is oxidized by fast heterogeneous reactions mainly with OH. A two-dimensional contour plot of the soot volume fraction is shown in Figure 3.37. Soot appears ﬁrst at a radial distance of 4 mm, and the maximum soot volume fraction is found on the centerline. HOMEWORK PROBLEMS

1.

Show that in a laminar diffusion jet, the lines of constant axial velocity component u and the mixture fraction f can be speciﬁed by the following equation: )% *

u0 d0 16 x * 3 Redo do uo + r=√ − 1 where Redo = 32 x u ν 3 Redo where u0 and d0 are the axial velocity and diameter of the fuel jet at the exit of the injector.

2.

Show that under the thin ﬂamelet hypothesis, the species conservation equation for Yk in terms of time and spatial coordinates (t, xi ) can be written in the coordinate of mixture fraction Z and t in the following form.

∂Yk ∂Z ∂Z ∂ 2 Yk 1 ∂ 2 Yk ρ = ω ˙ + = ω˙ k + ρD ρχ k ∂t ∂xi ∂xi ∂Z 2 2 ∂Z 2 ∂Z ∂Z . where χ represents the scalar dissipation rate, deﬁned as χ ≡ 2D ∂x i ∂xi

3.

Show that the mass fractions of fuel and oxidizer and the mixture temperature in the Burke-Schumann ﬂame structure can be written in terms of mixture fraction by the following equations: On the fuel side (Z > Zst ): YF (Z) = ZYF0 + (Z − 1)

YO0 Z − Zst ; YO (Z) = 0 = YF0 s 1 − Zst

T (Z) = ZTF0 + (1 − Z)TO0 +

QYF0 1−Z Zst CP 1 − Zst

On the oxidizer side (Z < Zst ):

Z YF (Z) = 0; YO (Z) = YO0 1 − Zst T (Z) = ZTF0 + (1 − Z)TO0 +

QYF0 Z CP

HOMEWORK PROBLEMS

205

4.

If the fuel is the diluted propane (60% C3 H8 and 40% N2 by mass) and the oxidizer is air, what is the overall chemical reaction for stoichiometric reac tion with one mole of C3 H8 ? Calculate the values of YF0 , YO0 , νF , νO , s (mass stoichiometric ratio), φs (equivalence ratio related to s), and Zst (mixture fraction at stoichiometric condition).

5.

What is the deﬁnition of mixture faction (f or Z ), and why is this parameter often used for modeling the species variation in the non-premixed reacting ﬂow ﬁeld?

6.

Prove the transformation rules shown in Equation 3.68 for Crocco-type transformation. Hint: Start with a transformation of coordinate system (x1 , x2 , x3 , t) into (ξ1 , ξ2 , ξ3 , τ ) and then set ξ1 = Z, (ξ2 = x2 , ξ3 = x3 ), and τ = t, where Z = Z(x1 , x2 , x3 , t).

7.

Show that for the one-step single reaction given by {1 kg of fuel (F )} + {s kg of oxidant (O)} → {(1 + s) kg of product (P )} with no oxidant in the fuel feeding stream and no fuel in the oxidant feeding stream, the stoichiometric mixture fraction can be written in the form given by Equation 3.15:

YO2 ,2 sYF,1 −1 Zst = = 1+ sYF,1 + YO2 ,2 YO2 ,2

4 BACKGROUND IN TURBULENT FLOWS

SYMBOLS

Symbol

Description

Dimension

CD Cd Cij Cε1 Cε2

Constant used in Equation 4.149 Constant used in Equations 4.257, 4.260, 4.264 Cross-term stresses, deﬁned by Equation 4.239 Constant = 1.44 Constant = 1.92

Cε1 and Cε3

Coefﬁcients for turbulence closure, see Equation 4.127 Turbulence coefﬁcient = 0.9 Diffusivity Operator for material derivative or total derivative Speciﬁc rate of transport and diffusion of momentum, see Equation 4.102 Dissipation rate of turbulent energy per wavenumber Turbulent energy per κ wavenumber Structure function i th order Distribution function, deﬁned by Equation 4.159 Fourier transform of a function f (x ) Convolution of two functions f and g Filter function

Cμ D D/Dt D ij D(κ) E (κ) Fi Fu F (κ) f ∗g G 206

Fundamentals of Turbulent and Multiphase Combustion Copyright © 2012 John Wiley & Sons, Inc.

— — F/L2 — — — — L2 /t 1/t L2 /t3 QL/t L3 /t2 L2 /t2 — arb. units — arb. units

Kenneth K. Kuo and Ragini Acharya

207

SYMBOLS

Symbol Gr h I k L Lij Lij l lDI lIE lm l0 l PK lT Mij n Pij Pr P (U ) p q q˙j R ReL R ij Ru v Sij (Tij )sgs T TrT t

Description

Dimension

Grashof number deﬁned in Equation 4.2 Speciﬁc enthalpy Turbulence intensity, deﬁned in Equations 4.46 and 4.47 Turbulent kinetic energy (TKE), see Equations 4.103 and 4.104 Characteristic length scale Leonard stresses, deﬁned by Equation 4.238 Germano identity Length scale Length scale corresponding to the boundary between energy dissipation range and inertial subrange Length scale corresponding to the boundary between inertial subrange and energy containing range Mixing length Integral length scale of turbulence Characteristic length Eulerian integral length scale Speciﬁc rate of mass ﬂux contribution for momentum, see Equation 4.102 Temperature exponent in the viscosity law Speciﬁc rate of production of momentum, see Equations 4.101 and 4.102 Probability, see Equations 4.154 and 4.158 Probability density function of sample space variable U, see Equation 4.161 Pressure Any ﬁeld variable, see Equation 4.5 Heat ﬂux via conduction Two-point, one-time correlation function, see Equation 4.37 Reynolds number = ReL ≡ ρU L/μ Reynolds stress tensor, see Equations 4.29 and 4.240 Single-point double correlation coefﬁcient Strain rate tensor (symmetric portion) Subgrid test scale tensor Temperature Rate of energy transfer between scales Time

— Q/M — L2 /t2 L F/L2 F/L2 L L

L

L L L L L2 /t3 — L2 /t3 — — F/L2 varies Q/(L2 t) — — F/L2 — 1/t M/L-t2 T Q/t t

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BACKGROUND IN TURBULENT FLOWS

Symbol

Description

Dimension

U u, v, w

Resolved component of a variable U Velocity components in x1 , x2 , and x3 directions, respectively Root mean square velocity ﬂuctuation ≡ turnover velocity of integral scale eddies Kolmogorov velocity scale Space vector Mass fraction of k th species

arb. units L/t

urms uη x, or xi Yk

Greek Symbols α Dummy variables α ij Deviatoric part of subgrid test scale tensor (Tij )sgs β ij Deviatoric part of stress tensor (Tij )sgs γ Ratio of speciﬁc heats Filter width δ Grid spacing δ ij Kronecker’s Delta 1 Residual tensor = Lij − Lkk δij − Cd αij −βij εij 3 ε Turbulence dissipation rate, see Equation 4.118 ζ Arbitrary or dummy variables ςen Enstrophy η Kolmogorov length scale T Taylor microscale kn Wavenumber for n th eddy size l Thermal conductivity μ Dynamic viscosity ν Kinematic viscosity πij Speciﬁc rate of redistribution of momentum, see Equation 4.102 ρ Density σq Standard deviation of variable q, see Equation 4.12 Constant = 1.3 σε σκ Constant = 1.0 τ Time scale Stress tensor, see Equation 4.29 τij τturb or τt Characteristic turbulent time scale ≈k/ε or τ0 Kolmogorov time scale τη φ Indication function, see Equation 4.157

L/t L/t L —

arb. units F/L2 F/L2 — L L — F/L2 L2 /t3 arb. units 1/t2 L L 1/L Q/(tLT) M/(Lt) or Ft/L2 L2 /t L2 /t3 M/L3 Same as q — — t F/L2 t L/t —

SYMBOLS

Symbol ωij ωk ω˙ k i

209

Description

Dimension

Strain rate tensor (anti-symmetric portion) Vorticity in k th direction Rate of production of k th species per unit volume Angular velocity in i th direction

1/t 1/t M/L3 -t 1/t

Overhead Symbols — Reynolds-averaged mean ∼ Favre-averaged mean ∧ (∧) over any variable indicates that the quantity is ﬁltered Superscripts

Subscripts D DI I IE rms sgs T 0

Reynolds-averaged ﬂuctuations Favre-averaged ﬂuctuations

Dissipation Boundary between dissipation range and inertial subrange Associated with inertial subrange Boundary between inertial subrange and energy containing range Root mean square Subgrid-sale Transfer Associated with integral scale

Abbreviations arb. units Arbitrary units

In preceding chapters, we reviewed some important concepts for laminar ﬂames, either premixed or non-premixed. In early days, laminar ﬂows were visualized as ﬂows where ﬂuid layers slide over one another. The term “laminar” originates from the word “lamina,” which means layer. In laminar ﬂows, a ﬂuid particle from a given layer stays in that layer and does not move across layers (or laminae). Laminar ﬂows are also known as streamlined ﬂows. In technical terms, laminar ﬂow is a ﬂow regime, which is deﬁned by a high rate of diffusion of momentum by viscous forces and a low rate of advection of momentum in the ﬂow. The velocity, temperature, and concentration proﬁles measured in laminar ﬂow with a high-sensitivity instrument are continuous since the ﬂuid particles do not cross over the streamlines and therefore do not show any sudden jumps. Laminar ﬂows can be steady or unsteady.

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BACKGROUND IN TURBULENT FLOWS

Transition to turbulence arises via ﬂow instabilities in the laminar ﬂow as the Reynolds number or Grashof number is increased. The Reynolds number represents the ratio of the force balancing inertia per unit area of the ﬂuid element and the viscous stress. The Grashof number (Gr) is deﬁned as the product of Reynolds number and the ratio of buoyancy force per unit area over the viscous stress. The explicit form for these two dimensionless parameters is given next. ReL ≡ GrL ≡

ρU L ρU 2 Force balancing inertia/Area = = μ Viscous stress μU L

(4.1)

Buoyancy force/Area ρ 2 gβ (Ts −T∞ )L3 ρgβ (Ts −T∞ )L ρU 2 = = ×ReL μ2 Viscous stress μU L μU L (4.2)

In these expressions, the proper choice of characteristic length L, velocity U , and temperature Ts is crucial for correct estimate of these dimensionless numbers for a given problem. At sufﬁciently high Reynolds numbers under forced convection conditions or high Gr numbers under natural convection conditions, the ﬂow becomes turbulent. In turbulent ﬂow, the ﬂuid particles are able to move across the streamlines since there is momentum transfer occurring at scales that are continuous. In practice, these scales often are visualized as eddies. Therefore, turbulent ﬂows are also described as ﬂows that contain eddies of every scale (bounded by the device scale). The ﬂuid particles are able to cross the streamlines since the rate of momentum diffusion due to viscous forces is not able to suppress the rate of momentum advection at scales larger than the molecular scales. Thus, in turbulent ﬂow, a ﬂuid particle can have velocity (and momentum) in any direction. In another words, turbulent ﬂows are characterized by their ability to transport and mix the ﬂuid much more effectively than laminar ﬂows. For example, laminar ﬂow in a pipe would experience perturbations from inlet ﬂow pressure ﬂuctuations. When viscous force damps out these perturbations, the ﬂow can remain laminar. However, beyond a certain ﬂow rate (or inlet pressure ﬂuctuation), the viscous forces can no longer suppress perturbations. (Note that increasing the ﬂow rate means increase in the Reynolds number.) Thus the ﬂow perturbations are ampliﬁed as the Reynolds number is increased. This phenomenon is called ﬂow transition from laminar to turbulent. The instability results in transition of the laminar ﬂow to turbulent ﬂow. The transitional regime between the laminar and turbulent ﬂows is when the ﬂow could be laminar with sporadic burst of turbulence. Not all instabilities lead to a transition from laminar to turbulent ﬂows. The amplitude and nature of perturbations are very important in governing the ﬂow transition from laminar to turbulent. 4.1

CHARACTERISTICS OF TURBULENT FLOWS

In a turbulent ﬂow, the velocity ﬁeld is random. First, let us deﬁne the word “random” following Pope (2000). Let us consider a ﬂow experiment that can be

CHARACTERISTICS OF TURBULENT FLOWS

211

repeated several times under a speciﬁed set of conditions. Now consider an event A, such as A ≡ {u1 < 10 m/s}, where u1 is a speciﬁed velocity component at a speciﬁed position and time (measured from the initiation of the experiment). If event A occurs each time, then A is a “certain” or “sure” event. If event A never occurs or cannot occur, then A is “impossible” event. The third possibility is that A may or may not occur. In such a case, event A is called a random event, and the variable u1 is called the random variable. A random variable does not have a unique value in every repetition of the same experiment. Turbulent ﬂows have random nature, yet they can be deﬁned by the deterministic Navier-Stokes and other conservation equations. In order to understand this contradiction, we have to understand that at high Reynolds numbers, turbulent ﬂow ﬁelds display acute sensitivity to perturbations in initial conditions, boundary conditions, and material properties. Deterministic equations exhibit acute sensitivity to initial conditions; as a consequence, their solutions are unpredictable (or random). This is why deterministic equations such as Navier-Stokes equation can give random turbulent ﬂow ﬁeld solutions when the initial conditions, boundary conditions, or material properties are perturbed. As explained by Tennekes and Lumley (1972), it is not easy to give a precise deﬁnition of turbulence. It is useful to list some of the characteristics of turbulent ﬂows: 1. Irregularity. All turbulent ﬂows exhibit irregularity or randomness (see Figures 4.1 and 4.2). By “randomness,” we mean that the local ﬁeld variables in a turbulent ﬂow cannot be given as one single value but are given in terms of mean and statistical properties. Due to this, statistical methods are applied for analysis of turbulent ﬂows. 2. Large Reynolds number. Turbulent ﬂows always occur at high Reynolds numbers. The transition to turbulence occurs due to growth of instability in laminar ﬂows when the Reynolds number becomes sufﬁciently large. The instability is related to the interaction of viscous terms and nonlinear inertia terms in the equation of motion. 3. Continuum. Turbulence is a continuum phenomenon, governed by the equations of ﬂuid mechanics. Even the smallest scales occurring in a turbulent ﬂow are ordinarily far larger than any molecular length scale. 4. Turbulent transport. In a turbulent ﬂow, there is a more rapid mixing and increased rates of momentum, heat, and mass transfer than laminar ﬂows due to transport at scales that are larger-than-molecular scales. 5. Dissipation. Turbulent ﬂows are always dissipative. Viscous shear stresses perform deformation work, which increases the internal energy of the ﬂuid at the expense of the kinetic energy of the turbulence. For stationary turbulent ﬂows, a continuous supply of energy is required to make up for these viscous losses. The term “stationary turbulent ﬂow” is deﬁned in a later section. 6. Three-dimensional vorticity ﬂuctuations. Turbulence is rotational and threedimensional. It is characterized by high levels of ﬂuctuating vorticity. The

212

BACKGROUND IN TURBULENT FLOWS

ﬂuctuation of a ﬁeld variable (e.g., vorticity) is deﬁned as the difference between its instantaneous value and its mean. For this reason, vorticity dynamics plays an essential role in the description of turbulent ﬂows. 7. Turbulent ﬂows are ﬂows. Turbulence is not a property of ﬂuids but of ﬂuid ﬂows. Most of the dynamics of turbulence is the same in all Newtonian ﬂuids, whether they are liquids or gases, if the Reynolds number of the turbulence is large enough; the major characteristics of turbulent ﬂows are not dominated by the molecular properties of the ﬂuid in which the turbulence occurs. 4.1.1

Some Pictures

The photograph in Figure 4.3 shows horizontally and vertically aligned wakes close to the grid. Far downstream, the wakes interact and produce a nearly homogeneous turbulence condition. Consider an incompressible, Newtonian ﬂow with constant viscosity and density. The continuity equation for the ﬂow-ﬁeld is: ∂ui =0 ∂xi

(4.3)

The Navier-Stokes equation for such system in the i th direction is: ρ

∂ui ∂t

Fluid acceleration

∂uj ρui ∂x i

∂p =− + ∂x i

Non linear advection term

Pressure gradient

+

∂ 2 ui μ ∂xi xj

(4.4)

Viscous force term (second order)

The nonlinear terms in momentum equation 4.4 are essential to the dynamics of turbulent ﬂows. The nonlinear advection term in the momentum equation is important in generation and maintenance of turbulence.

Cylinder Wake region

Figure 4.1 Visualization of turbulent wake of a cylinder at ReD = 10,000 (modiﬁed from van Dyke, 1982, based on the original work of Corke and Nagib, 1979).

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213

Circular Cylinder

1/16" perforated plate with 3/4" square holes

Figure 4.2 Visualization of turbulent wake of a cylinder at ReD = 2000 (modiﬁed from Werl´e and Gallon, 1972).

Figure 4.3 Generation of trubulence by grid at ReD = 2000 (modiﬁed from van Dyke, 1982, based on the original work of Corke and Nagib, 1979).

4.2

STATISTICAL UNDERSTANDING OF TURBULENCE

Turbulent ﬂow is deﬁned as a chaotic process. In a turbulent ﬂow ﬁeld, at any point of time and space, the ﬁeld variables, such as velocity, show ﬂuctuations (i.e., differences from a local mean value). Turbulent ﬂow has large number of degrees of freedom in space. Turbulent ﬂow is not experimentally reproducible in detail because it is extremely sensitive to perturbations in the initial and/or boundary conditions and would require inﬁnite precision to obtain the same initial and boundary conditions in duplicating test conditions. Due to the extreme sensitivity of turbulent ﬂows to initial/boundary conditions, the ﬁeld variables

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BACKGROUND IN TURBULENT FLOWS

cannot be reproduced exactly at any point of space and time; however, turbulent ﬂows are statistically reproducible in terms of mean values and probability density distributions (deﬁned in section 4.5). The fundamental conservation equations that apply to laminar ﬂows also apply to the turbulent ﬂows. Unlike the laminar ﬂows, the major physical variables in turbulent ﬂow can be decomposed into ﬂuctuating and average components. In order to obtain spatial proﬁles of the averaged physical properties, an averaging procedure is performed for the governing equations for turbulent ﬂows. The solution of Navier-Stokes equations gives ﬁeld variables rather than the statistics of the ﬁeld variables. Navier-Stokes equations are averaged, and new equations are formed in terms of statistical terms (e.g., mean velocity components, mean of cross product of ﬂuctuating velocity components, etc.). Statistical averaging of conservation equations generates new unknowns. For example, the cross product of ﬂuctuating velocity components in x1 and x2 directions is a new unknown (called Reynolds stress) obtained after averaging of the momentum equations. Thus, statistical equations are not closed as they require heuristic approximations or closure models for these new unknowns. 4.2.1

Ensemble Averaging

Ensemble averaging is the most general averaging technique and therefore does not require ﬂow to be stationary or homogeneous (deﬁned in a later section). q e =

N 1

{q (x, t)}n N

where q = f (x, t)

(4.5)

n=1

where q is any ﬁeld variable. Stationary ﬂows are deﬁned in this way. If all statistics of a turbulent ﬂow ﬁeld are invariant under a shift in time, then the ﬂow ﬁeld is called statistically stationary or simply stationary. The term “stationary turbulent ﬂow” does not mean that the ﬂow ﬁeld is steady with time. The turbulent ﬂow is then steady globally but unsteady locally. Like geophysical processes and detonation processes, turbulent ﬂows are not stationary; however, certain turbulent ﬂows can be simulated as stationary ﬂows. Equation 4.5 these made assumptions: • Each ensemble is the same except for inﬁnitesimal differences in initial and/or boundary conditions. • N must be large enough so that the average is independent of this number. The average value is called the expected value, q e , and ﬂuctuation velocity q is deﬁned as: (4.6) q (x, t) ≡ q (x, t) − q e (x, t)

STATISTICAL UNDERSTANDING OF TURBULENCE

4.2.2

215

Time Averaging

Time averaging of any ﬁeld variable q is obtained as: 1 q t = q = tP

t0 +tP

q (x, t) dt

where q = f x, t; tP

(4.7)

t0

where the time period (tP ) should be long enough that q t is independent of tP . Time averaging is suitable for statistically stationary turbulent ﬂows. If a turbulent ﬂow is stationary, then ensemble and time averaging are same. In a stationary turbulent ﬂow, for example, the time-averaged mean is independent of shift in time: q t (x, t) = q t (x, t + t1 ) = q t (x, t + t2 ) = q t (x) 4.2.3

(4.8)

Spatial Averaging

Spatial averages along lines, over an area, or over a volume are deﬁned as: 1 q s = q (x1 , t) dx1 = f (x1 , t) (4.9) L q s =

q s =

1 V

1 A

L

q (x1 , x2 , t) dx1 dx2 = f (x1 , x2 , t)

(4.10)

A

q (x, t) dx1 dx2 dx3 = f (x1 , x2 , x3 , t) = f (x, t)

(4.11)

V

where the subscript s is used to represent “spatial” average. Spatial averages are most useful when the turbulent ﬂow ﬁeld is homogeneous. 4.2.4

Statistical Moments n The ensemble average of q is called the n th moment of q . Practically, up to fourth moments are measured. Physical interpretation of these moments is given next. 2 Variance ≡ σq2 = q (4.12) e

The variance is a measure of the deviation of a set of data from its mean. The standard deviation (σq ) is the square root of the variance. The more spread apart the data, the higher the standard deviation.

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BACKGROUND IN TURBULENT FLOWS

Skewness ≡

3 q σq3

e

(4.13)

Skewness quantiﬁes asymmetry from the any distribution (e.g., normal, beta, log-normal distribution, etc.) in a set of statistical data. Skewness can come in the form of negative skewness or positive skewness, depending on whether data points are skewed to the left (negative skew) or to the right (positive skew) of the data average. 4 q e Flatness or Kurtosis ≡ (4.14) σq4 Kurtosis is a measure of whether the data are peaked or ﬂat relative to a normal distribution. Data sets with high kurtosis tend to have a distinct peak near the mean, decline rather rapidly, and have heavy tails. Data sets with low kurtosis tend to have a ﬂat top near the mean rather than a sharp peak. These parameters are more meaningful when discussed in the context of probability density functions, which are discussed in detail later section in this chapter.

4.2.5

Homogeneous Turbulence

In this subsection, we introduce two terms: statistical homogeneity and homogeneous turbulence. A ﬁeld is called statistically homogeneous when all ‘statistics’ of a ﬁeld variable (e.g., variance of velocity and other moments) is invariant under a shift in position. Thus, if the coordinate system is transformed by displacing the origin by X, then the mean velocity and higher moments of velocity do not change in a statistically homogeneous ﬁeld. Therefore, the mean velocity is uniform. Usually the mean velocity is purposely reduced to zero with an appropriate choice of reference frame at the mean ﬂuid velocity. The term “homogeneous turbulence” is slightly different from “statistical homogeneity,” and it is less restrictive. “Homogeneous turbulence” refers to the homogeneity of the ﬂuctuating component of ﬁeld variables (e.g., velocity), not the mean of the ﬁeld variable itself. If the moments of ensemble average of the n ﬂuctuations q e are independent of spatial variables, then the turbulence is called homogeneous—that is: ∂ n q e=0 ∂xi

(4.15)

n If q e depends on position, then the turbulence is called inhomogeneous. Homogeneous turbulence simpliﬁes the governing equations since the spatial gradients of statistical moments of ﬂuctuating ﬁeld variables are zero. Usually, under homogeneous turbulence conditions, the gradients of mean ﬁeld variables are nonzero but uniform (see Pope, 2000). The term “uniform” should not be

CONVENTIONAL AVERAGING METHODS

217

interpreted as “constant.” Here uniformity implies “uniform distribution,” which means that the probability density function distribution of the gradient of the mean ﬁeld variable is constant. Therefore, the mean velocity proﬁle in homogeneous shear ﬂow could have a constant gradient in a certain turbulent ﬂow ﬁeld experiment but a different value gradient constant if the experiment is repeated. The homogeneous turbulence deﬁnition requires the statistics of the cross products of ﬂuctuating components to be invariant with displacement in coordinates. For example, if u1 and u2 are velocity components in two different directions, then the following is true in homogeneous turbulence: ∂ u u =0 ∂xi 1 2 e

(4.16)

Spectral analysis of homogeneous turbulence is possible, allowing for quantitative interpretation of even the smallest turbulence scales. Thus, homogeneous turbulence often is studied with direct numerical simulation (DNS). (DNS is discussed later in this chapter.) Unfortunately, no real turbulent ﬂows are fully homogeneous. An example of homogeneous turbulence is grid-generated turbulence. By passing a uniform stream of ﬂuid through a rectangular grid with constant mesh spacing, a nearly homogeneous turbulent ﬂow ﬁeld can be generated. Turbulent ﬂows can be called statistically one-dimensional or two-dimensional if the statistics of the ﬂow ﬁeld variables change signiﬁcantly only in one or two directions and are invariant in the other direction(s). Note that all three components of velocity vary in all three directions and time; thereby the turbulent ﬂow ﬁeld is always three-dimensional. By deﬁnition, turbulence is three-dimensional; only the statistics are independent of some coordinate directions.

4.2.6

Isotropic Turbulence

Isotropy is another important statistical property. If the statistics of ﬂuctuating components of the ﬁeld variable and the products of ﬂuctuating components are invariant under rotations and/or reﬂections of the coordinate system, the isotropy is called statistical isotropy. “Isotropy in turbulence” means that it does not have a preferred direction and it has rotational symmetry. According to Kolmogorov’s hypothesis of local isotropy, the small-scale turbulent motions are statistically isotropic at sufﬁciently high Reynolds numbers. (This also is discussed in more detail in a later section.)

4.3

CONVENTIONAL AVERAGING METHODS

Two different averaging procedures are commonly used: (1) conventional time averaging (also called Reynolds averaging) and (2) mass-weighted time averaging (also called Favre averaging). Reynolds averaging is more popular for constant

218

BACKGROUND IN TURBULENT FLOWS

density ﬂows whereas Favre averaging is used for ﬂows with varying density, such as turbulent ﬂames. Both techniques have been used in the combustion ﬁeld. The detailed averaging procedures and the relationships between the quantities averaged by these procedures are described in the next subsections. 4.3.1

Reynolds Averaging

In the conventional time-averaging procedure, the mean of a quantity q(x, t) in a stationary turbulent ﬂow ﬁeld is deﬁned by: t0 + t 1 q (x) = lim q (x, t) dt (4.17) t→∞ t t 0 Physical quantities for stationary mean ﬂows can now be decomposed into two parts: the “mean” and the “ﬂuctuation”: uj (xi , t) = uj (xi ) + uj (xi , t) ,

p (xi , t) = p (xi ) + p (xi , t) ,

T (xi , t) = T (xi ) + T (xi , t) ρ (xi , t) = ρ (xi ) + ρ (xi , t) ,

ht (xi , t) = ht (xi ) + ht (xi , t) , etc.

(4.18)

where the subscript t in ht represents “total” and ht is “total enthalpy” or “stagnation enthalpy.” The average of ﬂuctuating quantities, is zero, that is, ρ = ht = h = T = 0

(4.19)

In addition, if f and g are two ﬁeld variables, then by Reynolds averaging, we have: f +g =f +g (4.20) f =f f g = f g,

fg = f g + f g

df df = ds ds f ds = f ds

(4.21) (4.22) (4.23) (4.24)

where s denotes any one of the independent variables x , y, z , and t. The conservation equations can be expressed in terms of mean and ﬂuctuation correlation quantities. Let us ﬁrst consider the Reynolds averaged momentum equation for an incompressible ﬂuid ﬂow: ∂ ∂p ∂ui ∂ui ∂ui + − ρui uj + f i (4.25) =− μ + uj ρ ∂t ∂xj ∂xi ∂xj ∂xj

CONVENTIONAL AVERAGING METHODS

219

where f i is the mean body force in the i th direction and −ρui uj is called the turbulent stress tensor or Reynolds stress tensor. Expanding and rearranging Equation 4.25, we get the incompressible equation in the three basic directions: ∂u ∂p ∂u ∂u ∂u ∂u2 ∂u v ∂u w ρ = − +∇ (μ∇u)+fx −ρ +u +v +w + + ∂t ∂x ∂y ∂z ∂x ∂x ∂y ∂z (4.26) ∂p ∂v ∂v ∂v ∂v ∂u v ∂v 2 ∂v w = − +∇ (μ∇v)+fy − ρ ρ +u +v +w + + ∂t ∂x ∂y ∂z ∂y ∂x ∂y ∂z (4.27) ∂w ∂w ∂w ∂u w ∂v w ∂w2 ∂w ∂p ρ +u +v +w = − +∇(μ∇w)+fz −ρ + + ∂t ∂x ∂y ∂z ∂z ∂x ∂y ∂z (4.28) The last three terms in these equations are related to the Reynolds stress tensor components. In matrix form, these components for incompressible ﬂow are:

R ij Reynolds stress tensor

⎛ τxx = τij T = ⎝ τxy τxz Turbulent stress tensor

τxy τyy τyz

⎛ ⎞ ρu2 τxz ⎜ τyz ⎠ = − ⎝ ρu v τzz T ρu w

ρu v ρv 2 ρv w

⎞ ρu w ⎟ ρv w ⎠ ρw2 (4.29)

In index notation (or Einstein notation), the Reynolds averaged momentum equation can be written as: ρ

∂ui uj ∂ui ∂ ∂p ∂ui ∂ui + + fi =− μ −ρ + uj ∂t ∂xj ∂xi ∂xj ∂xj ∂xj ⎡ ∂ = ∂xj

⎤

⎢ ⎥ ⎢ ⎥ ∂uj ∂ui ⎢ ⎥ + −ρui uj ⎥ + f i ⎢ −pδij + μ ⎢ ∂xj ∂xi ⎥ ⎣Mean pressure Reynolds stress⎦ stress tensor

Mean viscous stress tensor tensor

(4.30) A loose comparison can be made between the equations for mean velocity in a turbulent ﬂow ﬁeld and those for a laminar ﬂow ﬁeld. The momentum equations appear to contain similar terms except for the terms related to the Reynolds stress tensor in the equations for mean velocity in the turbulent ﬂow.

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BACKGROUND IN TURBULENT FLOWS

Generally speaking, Reynolds stresses far outweigh the viscous components; consequently, the latter can be omitted in many actual cases with a reasonable degree of approximation. $ $ $ $ $ ∂ui ∂uj $ $ $ $ $ (4.31) ρ $ui uj $ μ $ + ∂xj ∂xi $ This last condition is not applicable in the proximity of a solid surface or toward certain boundaries of the ﬂow. In such regions, the mean velocity gradients grow larger, and there are very thin layers close to these boundaries where the viscous effects become comparably large. These viscous layers are related to the turbulent boundary layer; the viscous sublayers exist within a turbulent boundary layer. There are three independent off-diagonal terms of the Reynolds stress tensor. By drawing an analogy between the Reynolds stress tensor and the viscous stress tensor in a Newtonian ﬂuid, a turbulent eddy viscosity assumption is applied to represent turbulence (Boussinesq approximation, 1877). Under such an assumption, an analogy is made between the mean rate of transfer of momentum by turbulent ﬂuctuations, expressed by Reynolds stress, and that by the chaotic motion of molecules of a gas according to classical kinetic theory of viscosity. In kinetic theory, viscous stresses arise at a microscopic level from the mean transfer of momentum by random molecular motion, thereby leading to Newtonian macroscopic behavior. Similarly, under the eddy-viscosity approximation, we assume that the turbulent Reynolds stress has a Newtonian expression in terms of the mean velocity gradients. With this assumption, the Reynolds stress tensor is often written as: ∂uj ∂ui R ij = −ρui uj ≡ ρνT + (4.32) = τij T ∂xj ∂xi where ν T is a coefﬁcient called the turbulent eddy viscosity (analogous to the kinematic viscosity). This last model poses a problem since it implies: ∂ui 2 2 2 ui ui = u1 + u2 + u3 = −2νT =0 (4.33) ∂xi The right-hand side of Equation 4.33 is zero by the continuity equation for incompressible ﬂow. The approximation made in Equation 4.32 implies that there is no turbulence related to the diagonal terms in the Reynolds stress tensor. In order to correct this problem, the next approximation is used: ∂uj 1 ∂ui + (4.34) −ui uj ≡ − ui ui δij + νT 3 ∂xj ∂xi This model for turbulent stress tensor is used in practice rather than the eddy viscosity approximation of Equation 4.32. The turbulent eddy viscosity ν T may vary with position and time. It must be modeled. There are several different

CONVENTIONAL AVERAGING METHODS

221

models for turbulent eddy viscosity, and they are discussed later. Even after replacing Eq. 4.32 by Eq. 4.34, the latter still fails when considering a parallel shear ﬂow. In such ﬂow, the components of τ ij are zero unless i =1 and j =2 (and vice-versa). For the nonzero component, we have: ∂u1 −u1 u2 = νT (4.35) ∂x2 Equation 4.35 implies that the turbulent shear stress should be zero where the mean ﬂow velocity in the x1 direction has maxima with respect to x2 . It has been demonstrated by Mathieu (1959) in a wall jet (which is approximately close to the parallel shear ﬂow) that the locations of maximum mean velocity and zero turbulent shear stress do not coincide with each other. From the previous discussions, it appears that turbulent eddy viscosity approximation has signiﬁcant defects when compared with experimental data. Prandtl and von K´arm´an attempted to improve the eddy viscosity approximation by allowing the Reynolds stress tensor to depend on higher derivatives of the mean velocity, in the manner of a Taylor series. Such an approach makes the description less local, and the results remain unsatisfactory. Therefore, such models are treated only in a historical context. The correlation u v is in general different from zero. The sign of u v is expected to be negative for a planar shear ﬂow with u = u (y), v = 0, w = 0, and ∂u ∂y > 0 (see Figure 4.4), and it can be shown in this way. Consider the ﬂuid particle motion at a distance y from surface with a positive transverse ﬂuctuating velocity (v >0) and moved up to a higher level by turbulent eddy motion. From a statistical point of view, a ﬂuid particle that travels upward as a result of turbulence ﬂuctuation arrives at an upper layer from a region where a lower mean axial velocity component u prevails. This ﬂuid particle essentially retains its original axial momentum; therefore, a positive v associated with this incremental ﬂuid motion introduces a negative u to the upper layer. Similarly, a negative v introduces a positive u to the lower layer. Thus, the sign of u v is negative. This explanation is based on the mixing length hypothesis. We may thus expect that the time-averaged u v is not only different from zero but in fact negative for shear ﬂows. This is also expressed by stating that

Δy y

Δy

u (y)

u + ∂ u Δy ∂y

v′(+)↑ u v′(−)↓

u − ∂ u Δy ∂y

u

Figure 4.4

Explanation of cross-product of ﬂuctuating velocities.

222

BACKGROUND IN TURBULENT FLOWS

there exists a correlation between the longitudinal and transverse ﬂuctuations of velocity at a given point. The correlation functions are deﬁned as the ratio of a covariance to the product of standard deviations of velocity components. 4.3.1.1 Correlation Functions Correlation functions are useful indicators of dependencies of various ﬂuctuating velocity components on each other as a function of interval between them in time or space. The correlation functions can be used to assess the distance required between sample points for the values to be effectively uncorrelated. In addition, they can form the basis of rules for interpolating values at points for which there are measurements. Various correlation functions are deﬁned here. The single-point double correlation coefﬁcient, Ru v , is deﬁned as: uv Ru v ≡ % % (4.36) u2 v 2

where u and v are measured at the same spatial location. A two-point correlation function provides statistical information on the spatial structure of the ﬂow ﬁeld. A simplest two-point, one-time correlation function can be deﬁned by measures at point x and time t the axial velocity u(x, t), and simultaneously at a second point (x+r, t) with distance r apart from the ﬁrst one, having velocity u(x+r, t). Then the correlation between these two velocities is deﬁned by the average: (4.37) R (x, r, t) = u (x, t) u (x + r, t) For homogeneous isotropic turbulence, the location x is arbitrary and r may be replaced by its absolute value r = |r|. For this case, the normalized correlation function R (r, t) is deﬁned as: R (r, t) ≡ R (r, t) u2 (t) = u (t) u (r, t) u2 (t) (4.38) A different form of the two-point double correlation for the points separated by a displacement r can be deﬁned as: υ′ (x + r, t) R (r) =

r

u′ (x, t) u′ (x + r, t) u′ (x, t)2

u′ (x + r, t)2

1

u′ (x, t)

2

u′ (x + r, t)

(4.39)

where u (x, t) and u (x + r, t) are functions of time at point 1 and 2 locations, respectively. For stationary turbulent ﬂows, the correlation function is independent of time. Similarly, two-point triple correlation can be deﬁned as: u (x, t) u (x + r, t) v (x + r, t) R (r) ≡ % (4.40) % % u (x, t)2 u (x + r, t)2 v (x + r, t)2

223

CONVENTIONAL AVERAGING METHODS

1

1

lim R (r) → 1 ⏐r⏐→ 0 lim R (r) → 0 ⏐r⏐→ ∞

R

Results of measurements taken behind a screen of mesh width lmesh in a region of isotropic turbulence are plotted against y/lmesh.

R

0

0 0

0.2

0.4 r

0.4 0.6 y/lmesh

(a)

(b)

0.6

0.8

0

1

0.2

0.8

1

Figure 4.5 A typical relation between R and (a) physical distance r and (b) physical distance = y/l mesh .

If the turbulent ﬂow is stationary, homogeneous, and isotropic, then a correlation coefﬁcient is a function of r or a dimensionless distance y/l mesh (see Figure 4.5). The correlation function has zero slope at r = 0 or y = 0. As shown in these ﬁgures, the value of R (r) approaches unity as r approaches 0, and it reduces when the two points are separated by a very small distance r. With increasing distance, R (r) decreases. For large distances between the two points, there is less information passing from one to the other and therefore they do not contribute much to the correlation. It is necessary to specify a characteristic length, which is a measure of the magnitude of turbulent length scales. For example, in a turbulent shear ﬂow, this can be experimentally achieved by measuring the correlation coefﬁcient of x -direction velocity ﬂuctuations at two points A and B separated by a transverse distance y: uA (x, t) uB (x + y, t) RAB (y) ≡ % (4.41) % 2 2 uA (x, t) uB (x + y, t) An explicit functional form for the correlation functions can be derived as a function of turbulence parameters. This derivation is discussed later in this chapter after we introduce the turbulence scales and turbulence models. The correlation coefﬁcient decreases rapidly to zero with increasing r and the quantity, which is a characteristic scale of turbulence and is given by an Eulerian integral length scale lT as: lT =

∞

R (r) dr

(4.42)

0

lT is a measure of the length scale at which there is some correlation between the components of velocity ﬂuctuations.

224

BACKGROUND IN TURBULENT FLOWS

The correlation functions between the same ﬂuctuating velocity component at a ﬁxed spatial location but measured at two different times can also be deﬁned. The correlation coefﬁcient R ∗ (t) can be deﬁned as: u (x, t) u (x, t + t) ∗ R (t) ≡ % (4.43) % u (x, t)2 u (x, t + t)2 ∗ (also The characteristic time scale of turbulence or the mixing time tmixing called the integral time scale) is deﬁned by: ∞ ∗ tmixing ≡ R ∗ (t) dt (4.44) 0

The characteristic length scale can be scaled by the product of root-mean∗ square value of the ﬂuctuating velocity urms and the mixing time tmixing : ∗ lT∗ = urms tmixing

(4.45)

There are multiple length scales for turbulent ﬂow. The largest is the dimension of the ﬂow ﬁeld; the smallest is the diffusive scale due to molecular viscosity. The integral scale of turbulence can be regarded as the distance an eddy moves through before dissolving and losing its identity. This integral length scale is called the mixing length. We discuss other scales of turbulence besides the integral length scale later. The intensity of turbulence is another important parameter characterizing turbulent ﬂows. The intensity I is a measure of the time-averaged velocity ﬂuctuations in the ﬂow. For isotropic turbulence, turbulence intensity can be deﬁned as: % % % % 2 2 u u2 u2 u i 3 1 2 I≡ (4.46) = = =& |u| |u| |u| ui ui where |u| is the magnitude of mean ﬂow velocity. For nonisotropic turbulence, the turbulence intensity is deﬁned as: ' ' 1 1 2 2 2 2 2 2 u + u + u u + u + u 1 2 3 1 2 3 3 3 I≡ = % (4.47)

|u| u2 + u2 + u2 1

2

3

The time-averaged conservation equations can be obtained by applying the Reynolds averaging procedure. However, there are two major drawbacks when applied to the compressible ﬂow ﬁeld. The Reynolds averaged continuity equation for the compressible ﬂow is: ∂ρ ui ∂ρui ∂ρ + =0 + ∂t ∂xi ∂xi

(4.48)

CONVENTIONAL AVERAGING METHODS

225

1. There are extra terms (∂ρ ui /∂xi for i = 1, 2, 3) in Equation 4.48 that need to be modeled. 2. Most sampling probes measure values that correspond to the mass-weighted concentrations rather than time-average concentrations. For these reasons, mass-weighted averaging procedures were highly recommended by Favre (1965), Laufer and Ludloff (1970), Bilger (1976a, 1980), and others. By applying Favre averaging, the governing equations for compressible turbulent ﬂow can be obtained in same form as those for the incompressible turbulent ﬂow. 4.3.2

Favre Averaging

A mass-weighted mean velocity is deﬁned as: ( ui ≡

ρui ρ

(4.49)

The velocity may then be written as ui (xi ) + ui (xi , t) ui (xi , t) ≡ (

(4.50)

where ui (xi , t) is the superimposed velocity ﬂuctuation. Upon multiplying Equation 4.50 by ρ(xi , t) and decomposing the velocity ui into two parts, we have

ρui = ρ ( ui + ui = ρ( (4.51) ui + ρui By time-averaging Equation 4.51, we get ρui = ρ( ui + ρui

(4.52)

From the deﬁnition of ( ui given by Equation 4.49, it follows that ρui = 0

(4.53)

Similarly, we can decompose the static enthalpy, static temperature, and total enthalpy as: h(xi , t) = ( h(xi ) + h (xi , t) (4.54) T (xi , t) = T((xi ) + T (xi , t)

(4.55)

ht (xi , t) = ( ht (xi ) +

(4.56)

ht (xi , t)

where 1 ρui ui ρT ( ρh ( ( 1 ui( T( ≡ ui + , h≡ , ht ≡ h + ( ρ ρ 2 2 ρ

(4.57)

1 1 ρui ui ui ui + ui ui − ht = h + ( 2 2 ρ

(4.58)

226

Also,

BACKGROUND IN TURBULENT FLOWS

ρT = ρh = ρht = 0

(4.59)

The expression for ( ht can be obtained by the next procedure. Multiplying both sides of ht = h + 1/2ui ui by ρ, and introducing the deﬁnition of ui given by Equation 4.49 into the resulting expression, we can write 1 1 ρht = ρh + ρ( ui ( ui + ρui ui 2 2

(4.60)

Since ρht = ρ( ht and ρh = ρ( h, Equation 4.60 can be written as: 1 1 ρui ui ( ht = ( h+ ( ui + ui( 2 2 ρ

(4.61)

Also, we can follow the procedure given next to show the validity of Equation 4.59:

2 1 1 1 ht + ht = ( h + h + ( h + h + ( ui ui + ui ui ui + ( ui + ui = ( ui( ht = ( 2 2 2 (4.62) By substituting the expression for ( ht into the expression from Equation 4.61, we get 1 1 ρui ui ht = h + ( ui ui + ui ui − (4.63) 2 2 ρ The relationships derived in Equations 4.50 to 4.60 are convenient for deducing various quantities through turbulence measurements. Some physical quantities measured by different diagnostic instruments correspond to the mass-weighted averaged quantity; others are truly the time-averaged quantities. For example, hot-wire anemometry measured average velocity is Favre-averaged velocity while the laser Doppler velocimetry (LDV) measured average velocity is the Reynoldsaveraged velocity. In hot-wire anemometry, the quantities measured at low speeds are the ﬂuctuations of ρui and T; those measured at supersonic speeds are the ﬂuctuations of ρui and of a quantity that is very close to the total enthalpy. Hot-wire anemometers use a very ﬁne wire (on the order of several micrometers) heated to some temperature above the ambient. Gas mixtures ﬂowing past the wire have a cooling effect on the wire. Since the electrical resistance of most metals is dependent on the temperature of the metal (e.g., rhenium), a relationship can be obtained between the resistance of the wire and the ﬂow velocity. Since the density effect is always included in the heat-transfer rates between the gas and the hot wire, the measured average velocity represents the Favre-averaged velocity. The measurement of gas velocities based the LDV technique by using entrained particles to cross over the fringe pattern generated by two coherent laser beams, however, is not associated with gas density. Hence, the measured velocities are Reynolds-averaged quantities. variables, including pressure (p), stress tensor (τij ), heat-ﬂux vector

Several q˙j , and density (ρ) do not require mass-weighted averaging, since the gas density effect is inherently included in the measurements of these variables.

CONVENTIONAL AVERAGING METHODS

227

For example, the mechanism for pressure sensing is by continuous bombardment of gas molecules onto the pressure-sensing surface of the pressure gauge. As the gas density gets higher, the frequency of collisions of molecules on the pressure-sensing surface also increases. Therefore, the gas density effect is already included in the measured pressure. The stress tensor components are always associated with product of dynamic viscosity (= ρ × ν) and the strain rate. Thus, the density effect is also included in the stress tensor components. For this reason, the conventional decomposition is applied for p, τij , and ρ: p = p + p , τij = τ ij + τij ,

and

ρ = ρ + ρ

(4.64)

According to Fourier’s law of heat conduction, the local heat-ﬂux depends on the product of gas thermal conductivity and the temperature gradient. From the kinetic theory of gases, the gas thermal conductivity is governed by the product of the average random molecular velocity, the mean free path, constant volume speciﬁc heat (Cv ), and the number density of the molecules (and therefore the gas density); in other words:

l = urandom Cv n∗ 3 (4.65) For this reason, the conventional decomposition is used for q˙i : q˙i = q˙ i + q˙i

(4.66)

4.3.3 Relation between Time Averaged-Quantities and Mass-Weighted Averaged Quantities

A relationship between ui and ( ui can be established as follows. Using Equation 4.59, we can write a general form ρφ = (ρ + ρ ) φ = 0

(4.67)

where φ can be u, h, ht , or T . This equation can be rearranged to give: φ = −

ρ φ ρ

(4.68)

( + φ and rearranging, we get Taking the time average of φ = φ ( − φ = −φ φ

(4.69)

(− φ = ρ φ φ ρ

(4.70)

Hence,

Recalling the two separate averaging schemes, φ can be written as φ = φ + ( + φ . Multiplying both sides of that expression by ρ and applying time φ = φ averaging, we get ( + ρφ = ρφ + ρφ ρφ

(4.71)

228

BACKGROUND IN TURBULENT FLOWS

If we recall that ρφ = 0 and ρφ can be written as (ρ + ρ ) φ = ρ φ , we can combine this expression and Equation 4.71 and use Equation 4.70 to get: (− φ = ρ φ = ρ φ φ ρ ρ

(4.72)

Equation 4.72 gives the difference between the Favre and Reynolds averaged quantities in terms of the density-variable correlation parameter. 4.3.4

Mass-Weighted Conservation and Transport Equations

The well-known Navier–Stokes equations of motion for a compressible, viscous, heat-conducting ideal gas can be written in this form:

∂ρ ∂ ρuj = 0 + ∂t ∂xj

∂τij ∂ ∂ ∂p + Momentum equations: ρui uj = − (ρui ) + ∂t ∂xj ∂xi ∂xj (with negligible body force) Continuity equation:

(4.73)

(4.74)

∂ ∂ ∂p ∂ ρuj ht = − ui τij − q˙j + (ρht ) + ∂t ∂xj ∂t ∂xj (4.75)

Energy equation:

where the stress tensor τij , heat-ﬂux vector q˙j , and total enthalpy ht , are given by: ∂uj 2 ∂ui ∂uk δij + μ + (4.76) τij = μ − μ 3 ∂xk ∂xj ∂xi q˙j = −l

∂T ∂xj

1 ht = h + ui ui 2

(4.77) (4.78)

4.3.4.1 Continuity and Momentum Equations If we substitute Equations 4.64 into Equations 4.73 and 4.74, we obtain

∂ ∂ ρ + ρ + ρ( uj + ρuj = 0 ∂t ∂xj

(4.79)

and

∂

∂p ∂p ∂τij ∂ − + uj +ρuj ( ui +ρui ( uj +ρui uj = − ρ( ui( ρ( ui +ρui + ∂t ∂xj ∂xi ∂xi ∂xj (4.80)

CONVENTIONAL AVERAGING METHODS

229

Taking the time average of the terms appearing in these equations, we obtain the Favre-averaged continuity and momentum equations for compressible turbulent ﬂow:

∂ ∂ρ + ρ( uj = 0 (4.81) ∂t ∂xj I

II

The physical meaning of each term can be given in this way: ∂ρ ∂t

∂ ρ( uj II: ∂xj I:

Average rate of change of mass per unit volume Advection of average mass ﬂux per unit volume out of control volume

In terms of material derivative, ( ∂( uj Dρ ∂ρ ∂ρ = −ρ = +( uj Dt ∂t ∂xj ∂xj The Favre-averaged momentum equation is: ρ

⎛

(4.82)

⎞

(ui

D( ∂p ∂ ⎜ ∂ ∂ ⎟ ui ) + ρ( ui ( + uj = − = (ρ( ⎝ τ ij − R ij ⎠ Dt ∂t ∂xj ∂xi ∂xj III IV

(4.83)

II

I

Note that the Favre-averaged material derivative is deﬁned as: ( D ∂ ∂ = +( uj Dt ∂t ∂xj

(4.84)

Similarly, the time-averaged material derivative is deﬁned as: D ∂ ∂ + + uj Dt ∂t ∂xj

(4.85)

The physical meaning of each term can be given in this way:

ui ∂ ∂ρ( ρ( ui ( uj + ∂t ∂xj ∂p II: − ∂xi 1 III: τ ij = 2μ Sij − Skk δij 3 IV: R ij I:

Average rate of change of momentum per unit volume Average pressure gradient force per unit volume Average viscous stress tensor Mean Reynolds stresses due to turbulent diffusion of momentum

230

BACKGROUND IN TURBULENT FLOWS

The Reynolds stress tensor R ij is:

R ij = ρui uj − ρ( ui ( ui uj − ( ui( ui uj uj = ρ uj = ρui uj = ρ

(4.86)

By substituting Equation 4.86 in Equation 4.83, the Favre-averaged momentum equation can also be written as:

∂ ∂p ∂ ∂ ui ) + ρ( ui ( + τ ij −ρui uj (4.87) uj = − (ρ( ∂t ∂xj ∂xi ∂xj II

I

III

IV

4.3.4.2 Energy Equation If we substitute the expressions given by Equations 4.50, 4.64, and 4.56 into Equation 4.75 and use Equation 4.78, we obtain

∂ ( ∂ ( ht + ρht uj uj + ρht ( uj + ρuj ( ρ ht ( ρ ht + ρht + ∂t ∂xj

∂ ∂ = p+p + ui τij − q˙j ∂t ∂xj

(4.88)

After subtracting the mechanical energy (( ui × Favre-averaged momentum equation) from the total enthalpy equation, we have

∂ ( ∂ ( h + ρuj h uj + ρuj ( ρ h( uj + ρh( ρ h + ρh + ∂t ∂xj

∂

∂ q˙j ∂ ∂ui p+p + ( p + p + τij − = uj + uj ∂t ∂xj ∂xj ∂xj

(4.89)

Taking the time average of the terms appearing in the equations, we obtain the mean energy equations in terms of static enthalpy: ∂ ( ∂p ∂ ∂p ∂p ∂ ( ρh + ρ h( uj = + uj + −q˙j − ρh uj +( uj ∂t ∂xj ∂t ∂xj ∂xj ∂xj I

II

III

∂u ∂( ui + τ ij + τij i ∂xj ∂xj

IV

V

(4.90)

VI

The physical meaning of each term is as follows: ∂ ( ∂ ( ρh + ρ h( uj Average rate of change of ρ( h per unit volume ∂t ∂xj ( Dp ∂p ∂p Rate of change of mean pressure on a “ﬁctitious” II: = +( uj ∂t ∂xj Dt mean ﬂuid particle moving with mean ﬂow I:

CONVENTIONAL AVERAGING METHODS

∂p ∂xj ∂ IV: − q˙ ∂xj j ∂ −ρh uj V: ∂xj III: uj

VI: τ ij

∂u ∂( ui + τij i ∂xj ∂xj

231

Work due to pressure and velocity ﬂuctuation Gradient of net ﬂux of heat due to conduction Turbulent transport of ρh Viscous dissipation

4.3.4.3 Mean Kinetic Energy Equation By considering the scalar product of ( uj and the mean momentum equation for ( ui , ∂ ∂ ∂ ∂p ( ui ) + ui ( + τ ik − ρui uk uk ) = − uj (4.91) (ρ( (ρ( ∂t ∂xk ∂xi ∂xk

uj , and the scalar product of ( ui and the mean momentum equation for (

∂ ∂ ∂ ∂p ( ρ( uj + ρ( uj ( + τ jk − ρuj uk uk = − ui ∂t ∂xk ∂xj ∂xk

(4.92)

and adding these two equations and rearranging, we obtain

∂p ∂p ∂ ∂ ∂ ρ( ui ( −( ui +( uj τ ik − ρui uk uj + uj ( uk = − ( ρ( ui( uj ∂t ∂xk ∂xi ∂xj ∂xk ∂ +( ui (4.93) τ jk − ρuj uk ∂xk For i = j , Equation 4.93 becomes ( ( D ∂p ∂τ ik ∂ ui ui ( ρ = −( ui +( ui −( ui ρui uk Dt 2 ∂xi ∂xk ∂xk I

II

III

(4.94)

IV

Terms I through IV in Equation 4.94 have these physical meanings: ( ( ui ( D ui I: ρ Dt 2 ∂p II: −( ui ∂xi ∂τ ik III: ( ui ∂xk

Rate of change of kinetic energy of mean ﬂuid particles Flow work done by mean pressure gradient forces Work by mean viscous stresses

232

BACKGROUND IN TURBULENT FLOWS

IV:

∂ ρu u ∂xk i k ∂ ( ui ρui uk =− ∂xk ∂( ui +ρui uk ∂xk

−( ui

Spatial transport of mean kinetic energy by turbulent ﬂuctuations Production of kinetic energy in the mean ﬂow by Reynolds stress tensor acting on mean strain rate tensor

The momentum and energy equations introduce correlations involving the second moments. At the highest level of turbulent closure, the analysis of compressible, turbulent ﬂow ﬁelds through either physical or numerical studies requires equations describing the transport of the second moments, such as Reynolds stresses, in order to close the mean ﬂow equations e.g., Equations 4.87, 4.90, and 4.94. The turbulent stress (or Reynolds stress) transport equations lead directly to the turbulent kinetic energy equation, which is especially useful in understanding the underlying ﬂow energy budget through an analysis of the terms. 4.3.4.4 Reynolds-Stress Transport Equations Let us consider the scalar product of uj with the momentum equation for ui : ∂τik ∂ ∂ ∂p + (4.95) uj (ρui uk ) = − (ρui ) + ∂t ∂xk ∂xi ∂xk

and the scalar product of ui with the momentum equation for uj :

∂τjk ∂ ∂ ∂p + ui ρuj uk = − ρuj + ∂t ∂xk ∂xj ∂xk The sum of these two equations is

∂τjk ∂p ∂p ∂τik ∂ ∂ − ui + uj + ui ρui uj uk = −uj ρui uj + ∂t ∂xk ∂xi ∂xj ∂xk ∂xk

(4.96)

(4.97)

Using Equation 4.50, we can write Equation 4.97 as

*

* ∂ ) ∂ ) uj + uj + ρ ( ui + ui ( uj + uj ( uk + uk ρ ( ui + ui ( ∂t ∂xk

∂p

∂τik

∂τjk ∂p − ( ui + ui + ( uj + uj + ( ui + ui =− ( uj + uj ∂xi ∂xj ∂xk ∂xk (4.98) where τik = τ ik + τik and τjk = τ jk + τjk Taking the time average of this equation, we obtain ∂ ∂ ρ( ui ( ρ( ui ( uk ρui uj +( ui ρuj uk +( uj ρui uk +ρui uj uk uj +ρui uj + uj ( uk +( ∂t ∂xk = −( uj

∂τ jk ∂τjk ∂p ∂p ∂p ∂p ∂τ ik ∂τik −uj −( ui −ui +( uj +uj +( ui +ui ∂xi ∂xi ∂xj ∂xj ∂xk ∂xk ∂xk ∂xk (4.99)

233

CONVENTIONAL AVERAGING METHODS

Subtracting Equation 4.93 from Equation 4.99 and rearranging (with the understanding that τjk contains ρ through μ), we obtain this Reynolds-stress transport equation: ( ij ∂τjk ∂τ DR ∂( uj ∂p ∂p ∂ ρui uj uk − uj − ui + uj ik + ui − ρui uk =− Dt ∂xk ∂xi ∂xj ∂xk ∂xk ∂xk − ρuj uk

∂( ui ∂( uk − ρui uj ∂xk ∂xk

(4.100)

An alternative form of the Reynolds stress equation is: ρ

∂R (

D ∂ ij ( u uk R ij = ρPij + u+ + i j = Dt ∂t ∂xk Production

+

ρij

Redistribution

− ρεij + ρMij Dissipation

Mass ﬂux contribution

ρDij

(4.101)

Transport & Diffusion

where

)

* ∂( uj ∂( ui ρPij = − R ik + R kj wkj + R kj ( wik = − R ik ( Skj − ( Sik + ( ∂xk ∂xk , , ∂uj ∂uj ∂u ∂u i i = p ρij = p + + ∂xj ∂xi ∂xj ∂xi

∂uj ∂ui ∂u = τik + τjk i ∂xk ∂xk ∂xk ∂xk ∂τ jk ∂τ ik ∂p ∂p ρMij = ρ ui − − + ρ uj ∂xj ∂xk ∂xi ∂xk ⎪ ⎪ ⎪ ∂τ ∂τ ∂p ∂p jk ik ⎪ ⎪ = ρ ui − − + ρ uj ⎪ ⎪ ∂xj ∂xk ∂xi ∂xk ⎪ ⎪ ⎪ . / ⎪ ⎪ ∂ ⎪ ⎪ ρDij = − ρui uj uk + δik p uj + δjk p ui − τik uj + τjk ui ⎪ ⎪ ∂xk ⎪ ⎪ ⎡ ⎤ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎥ ⎪ ∂ ⎢ ⎥ ⎪ ⎢ ⎪ ρ 0u u u 0 + δ p u + δ p u − τ u + τ u = − ik jk i j k j i ik j jk i ⎦ ⎪ ⎪ ⎣ ⎪ ∂xk ⎪ ⎭ ρεij = τik

∂uj

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

+ τjk

Turbulent transport

Viscous diffusion

(4.102) g + f g = f g) (Note: f g = f g − f ( With the exception of the production term, all these terms require modeling. Where possible, these higher-order correlations are also written in terms of their Reynolds-averaged counterparts by explicitly factoring out the mean density variation. Although the equation retains a form-invariance with its incompressible counterpart, the physical meaning relative to the Reynolds variable

234

BACKGROUND IN TURBULENT FLOWS

counterparts was changed. The fundamental source of this change was the assimilation of a mass ﬂux contribution into the relationship between the Favre and Reynolds-averaged variables. In the closure of these higher-order correlations, it is beneﬁcial to rewrite them in terms of Reynolds-averaged variables so that adaptations from their incompressible counterparts are more apparent. Although turbulent stress transport models are more suitable for turbulence modeling, two-equation closures have been used for turbulence closure: closures based on the compressible, turbulent kinetic energy per unit mass (k ) and the isotropic form of the destruction term or turbulent energy dissipation rate per unit mass, (ε = εii /2). 4.3.4.5 Turbulence-Kinetic-Energy Equation The Favre averaged turbulence kinetic energy (ρ × ( k) per unit volume is deﬁned in Equation 4.103. The turbulence kinetic energy per unit mass (( k ≡ 0u i ui /2) is deﬁned in Equation 4.104, and it represents the average ﬂuctuating kinetic energy per unit mass. This term (( k) is also equal to 1/2ρui ui /ρ, as shown in Equation 4.104. In this text, we deﬁne two terms, Favre averaged TKE and Favre averaged turbulence kinetic energy per unit mass (( k), as:

1 R ii R 11 + R 22 + R 33 ρ ×( k ≡ ρui ui = = 2 2 2 1 ρui ui 1 ( k ≡ u+ i ui = 2 2 ρ

Often called k without the (()

(4.103)

(4.104)

For i = j , Equation 4.100 becomes the TKE equation: ( 1 ∂τ ∂p ∂( ui 1 ∂( uk ∂ 1 D ρui ui = − uk ρui ui −ui + ui ik −ρui uk − ρui ui Dt 2 ∂x 2 ∂x ∂x ∂x 2 ∂xk k i k k I

II

III

IV

V

VI

(4.105)

The physical meanings of the terms of Equation 4.105 are: ( 1 D I: ρu u Dt 2 i i ∂ 1 II: uk ρu u ∂xk 2 i i ∂p ∂xi ∂τ IV: ui ik ∂xk III: −ui

Rate of change of turbulence kinetic energy (TKE) of a mean ﬂuid particle Advective transport of TKE by ﬂuctuating motion due to turbulence ﬂuctuations Work due to pressure and velocity ﬂuctuations Viscous stress work effect on TKE generated due to ﬂuctuation motion (associated with turbulent transport and dissipation terms)

CONVENTIONAL AVERAGING METHODS

1 ∂( ui V: − ρui uk 2 ∂xk VI: −ρui ui

235

Production of TKE due to mean ﬂow velocity gradient and Reynolds stresses

∂( uk ∂xk

Production of TKE due to dilatation from mean ﬂuid motion

The TKE equation per unit mass can be written in terms of ( k (or simply k ) as: ( ( 1 ∂τ Dρk ∂p ∂( ui D ∂ 1 ρui ui = − uk + ui ik − ρui uk = ρui ui − ui Dt Dt 2 ∂xk 2 ∂xi ∂xk ∂xk ∂( uk 1 − ρui ui 2 ∂xk

(4.106)

In Equation 4.106, the following expression can be substituted for the ﬂuctuating stress tensor: 1 1 1 τik = 2μ Sik − Sll δik + 2μ Sik − Sll δik + 2μ S ik − S ll δik 3 3 3 1 − 2μ Sik − Sll δik (4.107) 3 An alternative form of the TKE per unit mass (k ≡ τii 2) equation is: ρ

( Dk = ρP + Dt Production

ρ

−

Redistribution

ρε + Dissipation

ρM

+

Mass ﬂux contribution

ρD

(4.108)

Transport & Diffusion

where ρP =

ρPii ∂( ui Ski = −R ik( = −R ik 2 ∂xk

ρ =

∂u ρii = p i 2 ∂xi

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

∂uj ρεii = τik Ski = τik 2 ∂xk ∂τ ik ∂p ρMii ρM = − = ρ ui 2 ∂xi ∂xk ⎡ ρε =

ρD =

ρDii ∂ ⎢ ⎢ ρ 0ui ui uk 0 + δik p u − =− i 2 ∂xk ⎣ 2 Turbulent transport

⎤

τik ui Viscous diffusion

⎥ ⎥ ⎦

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(4.109)

All the terms on the right-hand side of Equation 4.108, including the production term, require closure. The production term requires a speciﬁcation of

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BACKGROUND IN TURBULENT FLOWS

the Reynolds stress tensor, which is equivalent to specifying the Reynolds stress anisotropy. The turbulent kinetic energy can also be interpreted as the isotropic part of the Reynolds stress tensor, and it is governed by the transport equation Equation 4.108. For the production, turbulent transport, and viscous diffusion terms, closure is achieved most often through variable density extensions of the incompressible forms. For the production term, the usual two-equation closure is to assume an isotropic eddy viscosity model for the stress tensor of this form: δij δij R ij = 2ρk + 2μt ( Skk (4.110) Sij − ( 3 3 Isotropic part

where where

Anisotropic part

μt = ρCμ kτturb

(4.111)

μt = turbulent viscosity, τturb = characteristic turbulent time scale (which could be deﬁned as k/ε for k-ε model)

coefﬁcient Cμ is equal to 0.09, which is extracted from the proportionality assumption between the turbulent shear stress and turbulent kinetic energy assumed for the equilibrium layer of the two-dimensional boundary layer ﬂows. We can verify that one-half of the trace of the Reynolds stress tensor is equal to the turbulent kinetic energy, that is: R ii R 11 + R 22 + R 33 (4.112) = = ρk 2 2 The turbulent transport and viscous diffusion term in Equation 4.109 are modeled as: ∂ μt ∂k ρD = μ+ (4.113) ∂xk σk ∂xk where σk is treated as a turbulent Prandtl number for diffusion of turbulent kinetic energy and its value varies around 1.0. 4.3.4.6 Turbulent Dissipation Rate Equation The turbulent dissipation rate is deﬁned as:

ρε ≡

ρεii = τij Sij = τij Sij 2

(4.114) ,

∂uj

-

∂u where the ﬂuctuating velocity strain rate tensor + i . Similarly, ∂xi ∂xj , ∂u 1 ∂uj + i . Sij ≡ 2 ∂xi ∂xj By substituting τij from Equation 4.107, we have: 2 2 2 ρε = 2μSij Sij − μSkk Sll +2μ Sij S ij − μ Skk S ll +2μ Sij Sji − μ Skk Sll 3 3 3 (4.115) Sij

1 ≡ 2

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CONVENTIONAL AVERAGING METHODS

The contributions from the ﬂuctuating viscosity are most often neglected in the development of the mean conservation equations; however, their role in deﬁning the dissipation rate can be assessed further. In results from the cold-walled channel ﬂow simulations of Huang, Coleman, and Bradshaw (1995), the term in Equation 4.115 comprising the product of the ﬂuctuating viscosity and velocity gradient correlation with the mean strain rate made a signiﬁcant contribution (from 6% to 16% depending on wall temperature) only to the total dissipation rate in close proximity to the wall. In contrast, the term involving the third-order ﬂuctuations was small throughout the ﬂow, according to Gatski and Bonnet (2009). In view of this observation, the model developments are focused only on the terms involving the mean viscosity. The turbulence dissipation rate term can be further decomposed as: ⎫ ∂uk ∂ui 2 2 ⎪ ⎪ ρε ≈ 2μSij Sij − μSkk Sll = 2μwij wij + 2μ − μSkk Sll ⎪ ⎪ ⎪ 3 ∂xi ∂xk 3 ⎪ ⎪ ⎪ ⎪ ⎪ ∂u ∂u 2 ⎪ k i ⎪ ⎪ = μωi ωi + 2μ − μSkk Sll ⎬ ∂xi ∂xk 3 (4.116) ⎤ ⎡ ⎪ ⎪ ∂ u u ⎪ k l ∂ ⎣ 4 ⎪ ⎪ = μωi ωi + 2μ − 2 uk Sll ⎦ + μSkk Sll ⎪ ⎪ ⎪ ∂xk ∂xl 3 ⎪ ⎪ ⎪ ρεhom ⎪ ⎪ ρεdilat ⎭ (or solenoidal ρεinhom

dissipation rate)

where

,

∂ui − , ∂xi ∂xj ⎧ ⎨ 1 if indices are even permutation permutation tensor εijk ≡ −1 if indices are odd permutation ⎩ 0 if two or more indices are equal

1 ωi ≡ εijk wij = εijk 2

∂uj

The next relations were used in the derivation of Equation 4.116: ∂ 2 ui uk ∂uk ∂ui ∂ = −2 u S + Sii Skk ∂xi ∂xk ∂xi ∂xk ∂xk k ii ∂u ∂u Sik Ski = wik wki + i k ∂xk ∂xi

(4.117)

The contribution from εhom is also called the solenoidal dissipation rate (i.e., divergence free or constant density) since it is directly analogous to an incompressible counterpart, εinhom , which is an “inhomogeneous” contribution, and εdilat , which is the “dilatation” dissipation rate. The inhomogeneous contribution, εinhom , consists of a term associated with the gradient of the second moment of ﬂuctuating velocity and a term that includes the ﬂuctuating dilatation. The term involving the second moment of ﬂuctuating velocity can be related to the turbulent stress ﬁeld in Favre variables and can be assumed to be a known quantity. This turbulent stress gradient term should then dominate any contribution

238

BACKGROUND IN TURBULENT FLOWS

due to the ﬂuctuating dilatation. In homogeneous turbulent ﬂows, both εhom and εdilat are present, and their relative importance can be assessed from numerical simulations. If the ﬂow is incompressible, then the turbulence dissipation rate is (since ∂ui ∂xi = 0): ∂2 ε ≈ 2νSij Sij = νωi ωi + 2ν u u (4.118) ∂xk ∂xl k l εhom εinhom =0 in homogeneous turbulence

The transport equation for solenoidal turbulent dissipation rate εhom can be written in this way: ( hom Dε = Dt

Pε1

+

Production of dissipation due to ∂( ui /∂xj

+ Dε + Viscous diffusion

where

Pε2 + Production due to ∂( ωi /∂xj

Tεc Compressible turbulent transport

Pε3

− + Tε

Production by vortex stretching from ∂ui ∂xj

+

Bε Baroclinic term due to ∂ρ /∂xj and ∂p/∂xi

Viscous destruction

Turbulent transport

( εhom Dμ + Fε + μ Dt Force term due to ∂τik ∂xk

(4.119)

Mean viscosity variation

/ ⎫ . ωi ωj S ij + ωi Sij ωj − ωi ωi S ij + ωi Sjj ωi ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂ω ⎪ ⎪ i ⎪ −2μ uk ωi ⎪ ⎪ ⎪ ∂xk ⎪ ⎪ / . ⎪ ⎪ ⎪ ⎪ 2μ ωi ωj Sij − ωi ωi Sjj ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ωl ∂ ∂τik ⎪ ⎪ ⎪ 2μ εlij ⎪ ⎪ ∂xk ρ ∂xj ⎪ ⎪ ⎪ ⎪ ∂ ⎬ −μ uk ωi ωi ∂xk ⎪ ⎪ ⎪ ⎪ ⎪ ωl ∂ ∂τik ⎪ ⎪ εlij 2μ ⎪ ⎪ ∂xk ρ ∂xj ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ μωi ωi Skk ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ωl ⎪ ∂ρ ∂p ⎪ ⎪ −2μ 2 εlij ⎪ ⎪ ρ ∂xj ∂xi ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂τik ∂ρ ωl ⎪ ⎪ 2μ 2 εlij ⎭ ρ ∂xk ∂xj

Pε1 = 2μ Pε2 = Pε2 = = Tε = Dε = Tεc = Bε = Fε =

(4.120)

CONVENTIONAL AVERAGING METHODS

239

Both Bε and Fε are contributions due to forces on a volume element that are normal to the density gradient. Finally, the last term on the right of Equation 4.119 requires no modeling and is simply the variation of the mean viscosity. Transport equations for the inhomogeneous and dilatation turbulent dissipation rates (εinhom and εdilat , respectively) are not required since these terms can be related to the Reynolds stress transport equation and equation of motion, respectively. Equation 4.119 presents a formidable challenge to any attempts at developing a rigorous closed transport equation. However, with this form of the εhom equation, it is logical to construct a model equation in a manner analogous to that used for incompressible ﬂows. Many such models exist in the incompressible literature consisting of source, sink, and diffusion terms. The next model is an example of some of the closure issues and strategies involved (Gatski and Bonnet, 2009). Pε1

=

−Cε1

ui uj

, ( Sji − Cε1 ρε( Sjj τt

where τt =

k ε

(4.121)

In this model, Cε1 and Cε1 are closure coefﬁcients. The ﬁrst term is an extension of an incompressible model, and the second term is necessary to properly account for dilatation effects resulting from bulk compressions and expansions. Omission of the mean dilatation term causes the model to incorrectly predict the decrease of the integral length scale for isotropic expansion and the increase for isotropic compression (Lele, 1994). Coleman and Mansour (1991b) developed an exact model for the mean dilatation term in Equation 4.121 for rapid spherical (isotropic) compressions of incompressible turbulence and applied it (Coleman and Mansour, 1991a) to the case of compressible turbulence. In the latter case, pressure-dilatation effects need to be considered as well. In the former case, an exact solution to the evolution equations resulting from rapid distortion theory (RDT) can be used as a modeling Mansour, (1991a) ) guide. Coleman and * derived an expression for Cε1 as Cε1 = 1 + 3n (γ − 1) − 2Cε1 3 where n is the temperature exponent in the viscosity law. The second source term Pε2 is neglected in many two-equation models, but as Rodi and Mansour (1993) have shown for incompressible ﬂow and Kreuzinger, Friedrich, and Gatski (2006) have shown for compressible ﬂow, it is of comparable size to the combination of the other source terms and the turbulent diffusion term. The form proposed by Rodi and Mansour (1993) is generalized and adapted to the compressible case as: 1 ∂ ( 1 ∂ ( 2 2 Pε = −μτt ωj ωp Cε (ρk) Sil − εijl ( Sik − εipk ( ∂xl 2 ∂xk 2 ∂ (ρk) 1 2 ( ωq (4.122) +Cε Sik − εiqk ( 2 ∂xk

where Cε2 and Cε2 are closure coefﬁcients. The sink term in Equation 4.122 is represented by the combination of the vortex stretching term Pε3 and the

240

BACKGROUND IN TURBULENT FLOWS

destruction term . These two terms are assumed to have the same asymptotic behavior and combine to form a sink term approximated as: Pε3 − = −Cε3

ε τt

(4.123)

The turbulent transport and viscous diffusion terms can be combined, as in the incompressible case, for a model of the form ∂ε 1 ∂ μδkj + τt Cε ρ 0uk uj 0 (4.124) Dε = ρ ∂xk ∂xj This model is simpliﬁed for two-equation models as: 1 ∂ ∂ε μ μ+ t Dε = ρ ∂xk σε ∂xk where

μt = ρCμ kτt = ρCμ k 2 ε

(4.125)

(4.126)

The constants Cμ is closure coefﬁcient (=0.09), and σε (≈ 1.3) is a dissipation rate Prandtl number. These closure models have been adapted from their incompressible forms for the production, destruction, and transport-diffusion terms. Readers are referred to the books by Pope (2000) Mathieu and Scott (2000) and Launder and Sandham (2000), and related references for state-of-the-art overviews of incompressible modeling. Most incompressible models for the ε-equation do not include any contribution from Equation 4.122 and are composed only of the modeled terms Equations 4.121, 4.123, and 4.124 or 4.125 with closure coefﬁcients given by the next values, depending on the calibration: Cε1 ≈ 1.44, Cε3 ≈ 1.92, Cε ≈ 0.15, Cμ ≈ 0.09

(4.127)

It is clear that if these are the only terms included in an εhom − equation, it is essentially a variable density extension of the incompressible form. These models have been broadly used for simulation of wall-bounded ﬂows in zero-pressure gradient boundary layer ﬂows. In order to achieve closure of terms attributed to compressibility effects—that is, the terms Tεc , Bε and Fε in Equation 4.120—Kreuzinger, Friedrich, and Gatski (2006) performed an a priori assessment of the these terms in channel, boundary layer, and mixing-layer ﬂows. In the channel ﬂow (cold wall) and boundary layer ﬂow (adiabatic wall), the peak amplitude of these terms was less than 10% of the production/destruction terms. There was, however, an increase with Mach number in the range examined (1:5 < M < 3:2), although the relative contribution remained small. In the mixing-layer case, the compressible turbulent

CONVENTIONAL AVERAGING METHODS

241

transport term Tεc and the force from the viscous stress gradient term Fε were found to be negligible in cases with and without density differences across the layer. However, in the case with density gradients, Kreuzinger, Friedrich, and Gatski, (2006) found that the baroclinic term increased to a level comparable with the production term. It should be cautioned that in the absence of shear, the baroclinic term has been found (e.g., Pirozzoli and Grasso, 2004) not to affect the dynamic balance in the εhom -equation. As a ﬁrst step in gaining more insight into the baroclinic term, Bε can be rewritten as: ⎡ , ⎛ ⎞⎤ n ∞ 2

2μ ρ ∂ρ ∂p ∂ρ ∂p ρ + εlij ⎝ωl Bε ≈ − 2 ⎣εlij ωl (−1)n 2 + 2 ⎠⎦ ∂xj ∂xi ∂xj ∂xi ρ ρ ρ n=1 (4.128) This equation can be approximated by the ﬁrst term (Kreuzinger, Friedrich, and Gatski, 2006). Krishnamurty and Shyy (1997) have partitioned this term into three parts given by: , - , - , 2μ 2μ ∂ρ ∂p ∂p ∂ρ ∂ρ ∂p =− 2 εlij ωl + εlij ωl Bε ≈ − 2 εlij ωl ∂xj ∂xi ∂xi ∂xj ∂xj ∂xi ρ ρ - , ∂ρ ∂p (4.129) + εlij ωl ∂xj ∂xi From an order-of-magnitude analysis, they concluded that the second term on the right in the equation is the dominant one. Aupoix (2004) analyzed these same contributions and concluded that, for compressible mixing layers, the ﬁrst term dominates in light of the absence of pressure gradient effects in such ﬂows. Ignored the third term involving the higher-order correlation. Both studies Contrary to both of these studies, Kreuzinger, Friedrich, and Gatski, (2006) found from an a priori analysis of DNS mixing-layer data that the higher-order triplecorrelation term dominated. It is interesting that even with the same starting point for the baroclinic term, rather diverse models have been proposed. For example, Krishnamurty and Shyy (1997) retained the mean pressure gradient term in Equation 4.129 and proposed a model that included the mass ﬂux. In contrast, Aupoix (2004) argued that in mixing layers, the mean pressure is constant so that the baroclinic term is proportional to the mean density gradient, and coupled this with the assumption of isentropic ﬂow so that the ﬂuctuating pressure gradient term could be related to the mean velocity gradient. Unfortunately, the proposed model was speciﬁed only for two-dimensional ﬂow. A third example is the proposal made by Kreuzinger, Friedrich, and Gatski (2006), who developed a model for the triple-correlation term. They estimated the ﬂuctuating pressure-gradient term from spectral analysis of free shear ﬂows as well as isotropic turbulence and the ﬂuctuating density gradient term from mixing-length

242

BACKGROUND IN TURBULENT FLOWS

ideas. These three proposed and Bonnet, 2009). ⎧ , ⎪ ρ uj ∂p ⎪ −1 ⎪ τt ⎪ ⎪ ⎪ ρ ∂xj ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ux ∂ρ Bε ∝ k 3/2 ∂( ⎪ ⎪ ∂y ∂y ⎪ ⎪ ⎪ ⎪ 7 7 ⎪ ⎪ 7 1 ∂ρ 7 ⎪ ⎪ 7 ⎪ ⎩ τt−1 k 3/2 77 ρ ∂x 7 j

models can be summarized as shown next (Gatski

Krishnamurty and Shyy (1997)

Aupoix (for 2-D boundary layer) (2004)

(4.130)

Kreuzinger, Friedrich, and Gatski (2006)

It is apparent from this discussion that while variable density extensions to incompressible dissipation rate models sufﬁce for many of the unknown correlations in the εhom equation, models for terms directly associated with compressibility have yet to be ﬁnalized. Although a method of analysis and assessment has been presented here with extensive utilization of DNS data to obtain rational closures for these models, additional simulation studies with broader parameter ranges need to be assessed. Most RANS applications utilizing εhom (as well as other variables) did not include the terms directly related to the compressibility effects. The terms in the RANS equations are required to properly account for any variable density effects and yet be able to produce solutions capable of yielding correct mean ﬁeld variables. Huang, Bradshaw, and Coakley (1992, 1994) recognized this problem for compressible wall-bounded ﬂows and provided a rational modiﬁcation to the incompressible, solenoidal dissipation rate equation (as well as other potential scale equations such as speciﬁc dissipation rate ω ≡ ε/k). This formulation was extended by Catris and Aupoix (2000) and generalized to a variety of two-equation models. Readers can refer to Gatski and Bonnet (2009) for more description of closure models for ε-equations for high-speed compressible turbulent ﬂows. 4.3.4.7 Species Mass Conservation Equation The species mass conservation equation based on Favre averaging can be shown to have this form: , (k ∂Yk ∂ ( ∂ ∂Y ∂ ∂ ( ρ Yk + ρ Yk( Dρ + Dρ − ρYk ui + ω˙ k ui = ∂t ∂x ∂x ∂x ∂x ∂x i i i i i I II

III

IV

V

(4.131)

Physical meanings of these terms are: I: Rate of increase of k th chemical species (storage term) II: Net rate of k th species transported out of the control volume by advection III: Rate of increase of k th species by mass diffusion, where D is the mass diffusion coefﬁcient

243

CONVENTIONAL AVERAGING METHODS

IV: Rate of increase of k th species by turbulent scalar transport V: Rate of generation (or consumption) of k th species by chemical reactions. 4.3.5

Vorticity Equation

Turbulent ﬂows are rotational, and they contain vorticity that has intense, small-scale random variations in both space and time. The vorticity vector (ω) is deﬁned as: ω ≡∇ ×v (4.132) Vorticity can be physically related to the rotational velocity of a small ﬂuid particle at some point in the ﬂow. The angular (or rotational) velocity vector () of a ﬂuid particle is related to the vorticity vector as: 1 1 ω = ∇ ×v (4.133) 2 2 The vorticity equation can be derived by taking the curl of the linear momentum equation. Readers are encouraged to perform this derivation as an exercise. The vorticity equation in terms of angular velocity vector for compressible ﬂuid ﬂow is: ∂ D = + u ∇ · Dt ∂t . / =

Rate of increase of angular velocity of a ﬂuid particle

⎡Rate of increase ⎢of angular ⎣ velocity at point

Advection term

⎤ ⎥ ⎦

a

= · ∇u + ∇ · (ν∇ · ) Vortex stretching term

.

/ Viscous dissipation

−(∇ · u)

1 1 × ∇p − ∇ 2 ρ

. / Effect of dilatation onvorticity ⎡Vorticity generation ⎤ ⎥ ⎢due to pressure ⎣gradient in a medium⎦ of varying density

(4.134) For incompressible ﬂow, the vorticity equation reduces to the next equation: ∂ + u · ∇ = · ∇u + ν∇ 2 ∂t

∂i ∂ui ∂i ∂ 2 i = j +ν + uj ∂t ∂xj ∂xj ∂xj ∂xj (4.135) Tornados are examples of very large vortexes. Stretching a vortex tube along its axis will make the ﬂuid inside the tube rotate faster and decrease its diameter in order to maintain its angular momentum constant. This is same as angular momentum conservation law in solid mechanics. A well-known example are iceskaters who turn faster as they bring their hands near the body, and vice versa. An example in ﬂuid mechanics is the bathtub vortex that rotates faster and becomes smaller as it goes from the ﬂuid surface to the exit drain. Batchelor and Townsend (1949) found that dissipation occurs in isolated regions of concentrated vorticity at the smallest turbulent scales. They showed or

244

BACKGROUND IN TURBULENT FLOWS

Figure 4.6 Small-scale vortex tubes (worms) in turbulent ﬂow (http://www.warwick.ac. uk/∼masbu/turb_symp/worms3.jpg).

that for high-Reynolds-number turbulent ﬂows, the molecular mixing and dissipative processes are concentrated in isolated regions. The total volume of these isolated regions is a very small fraction of the volume of the ﬂuid. These severely intermittent small-scale structures that evolve at different locations are known as vortex tubes or worms. These vortex tubes can be regarded as numerous ribbons in the ﬂow ﬁeld (see Figure 4.6). The vortex worms are jumping from one set of locations to other locations based on their local maximum stretching conditions. To fully describe the mixing and dissipation events in these concentrated high-vorticity regions, these small-scale phenomena must be considered. The instantaneous locations of the vortex tubes for a chemically reacting turbulent ﬂow are of great interest to turbulent combustion (discussed in later chapters). As described by Chomiak (1979), the problem was then studied from two points of view: (a) statistical modeling of the energy cascade of turbulent ﬂuctuations and (b) consideration of the hydrodynamic vorticity production mechanism due to stretching of vortex lines. Kuo and Corrsin (1971) have experimentally shown that the severely intermittent small-scale structures are evolved at different locations. The vorticity ﬂuctuations in turbulent ﬂows can have much higher magnitudes than the mean vorticity and are randomly oriented in any direction. The vorticity vector can be decomposed (e.g, by Reynolds averaging) as: = +

or

i = i + i

(4.136)

By substituting this decomposition expression into Equation 4.135 and applying time averaging, we get the equation for mean vorticity as: ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ∂i ∂ui ∂i ∂ ⎨ ∂ 2 i + uj = j − i uj − ui Qj + ν (4.137) ∂t ∂xj ∂xj ∂xj ⎪ ∂xj ∂xj ⎪ ⎪ ⎪ ⎪ ⎪ ⎩Turbulent Turbulent⎭ Unsteady term

Advection term

Mean ﬂow stretching

advection

stretching

Viscous diffusion

245

CONVENTIONAL AVERAGING METHODS

An equation for the square of mean vorticity can be obtained by multiplying Equation 4.137 by i and rearranging the terms in the resulting equation. This equation is shown next along with the physical meaning of each term. ⎧ ⎫ ⎪ ⎪ 1 1 ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ ∂ i i ∂ i i ∂ui ∂ 2 2 + uj = i j −i i uj − ui j ∂xj ∂xj ∂xj ⎪ ⎪ ⎪ ⎪ ∂t ⎪ ⎪ ⎩ ⎭ Unsteady effect

Mean ﬂow advection

Mean ﬂow stretching

Turbulent advection

Turbulent stretching

1 ∂ 2 i i ∂i ∂i +ν 2 −ν ∂xj ∂xj ∂xj ∂xj Viscous transfer

(4.138)

Viscous dissipation

The equation for ﬂuctuating vorticity can be obtained by subtracting the Equation 4.137 from Equation 4.135: ∂i ∂i ∂u ∂ui ∂i = j i + j − uj + uj ∂t ∂xj ∂xj ∂xj ∂xj 8 9 ∂ 2 i ∂ i uj − j ui + j ui − i uj + ν − ∂xj ∂xj ∂xj

(4.139)

Equation 4.139 can be multiplied by the ﬂuctuating vorticity to obtain an equation for mean-squared vorticity ﬂuctuation. 1 1 ∂ i i ∂ i i ∂u ∂ui ∂i 2 + uj 2 = i i + j i i − i uj ∂xj ∂xj ∂xj ∂xj ∂t

I. Unsteady effect

+

II. Mean ﬂow advection

∂u i j i ∂xj

∂ − ∂xj

IV. Turbulent stretching of ﬂuctuating vorticity

III. Mean ﬂow/turbulence coupling

1 ∂ 2 i i 1 u +ν 2 2 j i i ∂xj ∂xj V. Transfer terms due to inhomogeneity

∂ ∂i −ν i ∂xj ∂xj

VI. Viscous dissipation to damp out vorticity ﬂuctuations

(4.140) Equation 4.140 is also known as the enstrophy equation. The term “enstrophy” (ςen ) is deﬁned as: ςen ≡

1 1 1 i i = i i + i i 2 2 2

(4.141)

At high Reynolds numbers, terms III and V are much lower than terms IV and VI. Therefore, Equation 4.140 sometimes can be simpliﬁed (see Mathieu and Scott, 2000).

246

BACKGROUND IN TURBULENT FLOWS

Researchers (e.g., see Gorski and Bernard, 1996) have used the enstrophy equation with the turbulence kinetic energy equation as a two-equation model to compare with another popular approach for turbulence modeling called the turbulence kinetic energy and turbulence dissipation rate (k-ε) equations. There is a strong relationship between the enstrophy equation and the turbulence dissipation rate equation, which is discussed in the next section. 4.3.6

Relationship between Enstrophy and the Turbulent Dissipation Rate

For inhomogeneous turbulent ﬂows, ε is related to ςen by: ∂ 2k ∂ ε − εijk u = 2ςen + ν ∂xj ∂xj ∂xi j k

(4.142)

where εijk is the permutation tensor. For homogeneous turbulence, the last two terms on the right-hand side are zero. Thus we have: ε = 2ςen ν

(4.143)

This simple algebraic relationship indicates a close connection between the enstrophy and turbulent dissipation rate equations. The turbulent dissipation rate equation for incompressible ﬂows can be derived as: ∂uj ∂uj ∂ui u u ∂ui ∂u ∂ 2 ui ∂u ∂u ∂u ∂ε ∂ε = −2ν −ε i k −2νuk i −2ν i i k +uj ∂t ∂xj ∂xi ∂xk ∂xk k ∂xk ∂xj ∂xk ∂xj ∂xk ∂xj ∂xj , , 2 2 ∂ ui ∂ ∂ 2ε ∂ ∂p ∂uk 2 ∂ui ∂ui −ν +ν −2ν uk −ν ∂xk ∂xj ∂xj ∂xk ∂xj ∂xj ∂xj ∂xj ∂xj ∂xk (4.144) It can be seen that the turbulent enstrophy and turbulent dissipation rate equations have many similarities. However, the turbulent enstrophy equation does not have the terms involving pressure, which is difﬁcult to model. Gorski and Bernard (1996) have shown that several production terms in the enstrophy equation can be analyzed through formal vorticity transport analysis. In this manner, the production terms can be accounted more accurately than those of the ε equation. If the enstrophy equation is used along with the k -equation to model turbulent ﬂows, the production terms in the enstrophy equation affect the energy budget while having no inﬂuence on the mean velocity and turbulent kinetic energy. Because of this, improved modeling of the enstrophy dissipation term is necessary before the new production terms can have a beneﬁcial impact on the computed ﬂow ﬁelds. Gorski and Bernard (1996) have also shown that for low Reynolds number channel ﬂow, a dissipation model with an enstrophy equation can have more accurate representations of the individual terms in the enstrophy balance, thereby leading to improved predictions of turbulent parameters.

TURBULENCE MODELS

4.4

247

TURBULENCE MODELS

The classical approach to model turbulent ﬂows is based on single-point averages of the Navier-Stokes equations. These are commonly called Reynolds Averaged Navier-Stokes (RANS) equations. Even though the k − ε two-equation model based on the RANS equations has been very popular in engineering design calculations, especially for shear ﬂows; however, its success is limited especially for ﬂows involving recirculation zones. For variable density ﬂows, the NavierStokes equations are written in divergence forms in Equations 4.73 and 4.74. The Favre-averaged forms of these equations are given by Equations 4.81 and u 0 is unknown and represents the 4.83. The Reynolds stress tensor − ρ 0u i j ﬁrst closure problem for turbulence modeling. It is possible to derive equations for the six components of the Reynolds stress tensor. In these transport equations, several terms that are not closed can appear, such as triple correlations. These socalled Reynolds stress models have been presented for variable density ﬂows—for example, by W. Jones (1994) and W. Jones and Kakhi (1996). Although Reynolds stress models contain a more complete description of the physics, they have not been broadly used in combustion. One way to treat the Reynolds stress tensor is to deﬁne the eddy viscosity νt by introducing this relationship: uk 2 ∂( 2 + ( − ρ ui uj = ρνt 2Sij − δij k − δij ρ( 3 ∂xk 3 ∂( uj ∂( ui 2 ∂( 2 uk = ρνt + k (4.145) − δij − δij ρ( ∂xj ∂xi 3 ∂xk 3 In this equation, δij represents Kronecker’s delta function, which is deﬁned such that δij = 1 for i = j and δij = 0 for i = j. The eddy viscosity model assumes that the turbulence is isotropic. The turbulence may become isotropic at smaller scales, but it is not necessarily the case at larger scales. The kinematic k) and eddy viscosity (ν t ) is related to the Favre average turbulent kinetic energy (( its local dissipation rate (( ε ). There are multiple models for the eddy viscosity, three of which are explained next. Zero-equation model (Prandtl mixing length model):

No transport equation is solved with conservation equations. The turbulent viscosity is evaluated using the next equation 7 7 (4.146) S7 μt = ρ2m 7( 7 7 where m is the mixing length and 7( S7 is the magnitude of the Favre-averaged strain-rate tensor, which is deﬁned as: ∂( uj 1 ∂( ui ( Sij ≡ + (4.147) 2 ∂xj ∂xi

248

BACKGROUND IN TURBULENT FLOWS

One-equation model of Prandtl-Kolmogorov:

The turbulent viscosity (μt ) is modeled by using the next algebraic relationship √ c ≈ 0.55 (4.148) μt = cρPK k, where PK is a characteristic length, based on empirical relationships. The turbulent kinetic energy k at any spatial location is determined from the solution of its governing equation (i.e., the k -equation). The value of constant c was determined to produce correct behavior in the log-law region of the shear ﬂow. Two-Equation model of Prandtl-Kolmogorov:

At high ε is proportional to Reynolds numbers, the turbulent dissipation rate ( k 3/2 lm (i.e., ( ε = CD k 3/2 lm ). The proportionality constant CD is related to the constant c by CD = c3 ≈ 0.166. By replacing lPK in Equation 4.148 with lm as a function of the dissipation rate and turbulence kinetic energy, we have: νt = cCD

( ( ( k 3/2 &( k2 k2 k = c4 = Cμ , ( ε ( ε ( ε

where Cμ ≈ 0.09

(4.149)

This relationship is known as the Prandtl-Kolmogorov equation, where local values of the Favre averages variables ( k and ( ε are obtained by solving both kand ε−equations shown in Equations 4.150 and 4.151. The governing equations for these two turbulence quantities have been derived earlier in Equations 4.106 and 4.144, respectively. In terms of the Reynolds stress tensor, the modeled k-ε equations can be written as shown next. The turbulent kinetic energy equation: , ∂( k k ui νt ρ ∂( ∂( k ∂ ∂( ρ uj = −ρ u+ −ρ( ε (4.150) + ρ( i uj ∂t ∂xj ∂xj σk ∂xj ∂xj Scalar transport

Production

Dissipation

The turbulent dissipation rate equation: ∂( ε ∂ νt ∂( ( ε ∂( ( ε2 ∂( ε ε ui ρ uj = ρ − Cε1 ρ u+ − Cε2 ρ + ρ( i uj ( ∂t ∂xj ∂xj σε ∂xj k ∂xj ( k Scalar transport

Production

(4.151)

Dissipation

In the standard k-ε model, the constants σk = 1.0. σε = 1.3, Cε1 = 1.44, and Cε2 = 1.92 are generally used. In the model, in Equation 4.151, the following assumption was used, that is: 7 7 7 ∂( 7 7 ∂( uj 77 ui 7 7 (4.152)

μ ρ 7u+ u + i j 7 ∂x ∂xi 7 j

PROBABILITY DENSITY FUNCTION

249

In fully developed turbulence, the Reynolds stress tensor can be as much as 2 orders of magnitude higher than the mean viscous stress tensor. Modeling the Reynolds stress tensor as a product of eddy viscosity and gradient of mean velocity is also known as the gradient diffusion model, based upon Boussinesq approximation. ∂( ui 0u i uk 0 = −νt ∂xj

(4.153)

A more detailed discussion of additional terms in the Favre averaged turbulent kinetic energy equation can be found in Libby and Williams (1994). The k-ε twoequation model is very popular due to its simplicity and cost effectiveness. Many industrial codes still rely on the k-ε model. The k-ε model is based on equations where the turbulent transport is diffusive and therefore is more easily handled by numerical methods than the closure using Reynolds stress transport equations. This is probably the most important reason for its wide use in many industrial codes. Rotta (1972) has shown that by integrating the two-point correlation equations over the correlation coordinate r, an equation for the integral length scale can be derived. This equation is used in the k -l model. The l -equation or kl equation was applied by Rodi and Spalding (1970) to turbulent jet ﬂows. From this model and from the algebraic relation between l (integral length scale), k, and ε, a governing equation for ε can be derived. A similar approach has been used by Oberlack (1998) to derive an equation for the dissipation tensor that is needed in Reynolds stress models. The dissipation rate ε plays a fundamental role in turbulence theory. The eddy cascade hypothesis states that it is equal to the energy transfer rate from the large eddies to the smaller eddies and therefore is invariant within the inertial subrange of turbulence (to be discussed later in this chapter).

4.5

PROBABILITY DENSITY FUNCTION

Traditional turbulence models, including two-equation models and the Reynolds stress transport model (second-moment closures) are based on Reynolds averaging techniques and yield modeled equations for statistical moments. In comparison to these models, probability density function (pdf) methods achieve closure through a modeled transport equation for the one-point, one-time pdf of certain ﬂuid properties in a turbulent ﬂow. The advantage of pdf methods is that both convection and chemical reaction terms do not require a modeling assumption, and they can be solved exactly by a pdf transport equation. The tremendous amount of statistical information contained in the pdfs obviously provides a fuller description of turbulent ﬂows than two-equation models or second-moment closures. Before we introduce the pdf, we deﬁne probability (Pr ) in this way. Let us consider a continuous distribution of a random variable u, each of which has

250

BACKGROUND IN TURBULENT FLOWS

been given a serial number α, lying between zero and unity (α ∈ [0, 1]) Then the probability that u < c (a ﬁxed constant) is deﬁned as Number of experiments in which u < c Total number of experiments

(4.154)

Pr {u < c} = fraction of [0,1] on which u < c

(4.155)

Pr {u < c} = Also,

If, for example, the variation u with respect to α is as shown in Figure 4.7, then the probability corresponding to this proﬁle is: Pr {u < c} = l1 + l2

(4.156)

Let us deﬁne an indicator function φ(α) for the set of points on which u < c: : 1 for u < c φ (α) ≡ (4.157) 0 otherwise We can then write

Pr {u < c} =

4.5.1

1

φ (α) dα

(4.158)

0

Distribution Function

The distribution function Fu (U ) of u is deﬁned as the probability Pr of ﬁnding a value of u < U : (4.159) Fu (U ) = Pr (u < U ) where U is the so-called sample space variable associated with the random variable u. The probability of ﬁnding a value of u in a certain interval U− < u < U+ is given by: Pr (U− < u < U+ ) = Fu (U+ ) − Fu (U− )

(4.160)

u

c

0

Figure 4.7

l1

l2

a

Example of variation of u with respect to α.

PROBABILITY DENSITY FUNCTION

251

The pdf of U is now deﬁned as

P (U ) ≡

dFu (U ) dU

or

dFu (U ) = P (U ) dU

(4.161)

It follows that P (U ) dU is the probability of ﬁnding u in the range U ≤ u < U + dU . If the possible realizations of u range from −∞ to +∞ it follows that +∞ P (U ) dU = 1 (4.162) −∞

In turbulent ﬂows, the pdf of any random variable depends on the position x and time t. These functional dependencies are expressed by P (U ; x, t). The semicolon indicates that P is the probability density in U-space and a function of x and t. In stationary turbulent ﬂows, it is independent of t; and for homogeneous turbulent ﬂow ﬁelds, it is independent of x. Since turbulent ﬂow ﬁeld is a random process, we will not distinguish between the random variable u and the sample space variable U , and write the pdf as P (u; x, t) in the following section.

n Section 4.2.4, we deﬁned statistical moments as the ensemble average of In u as the n th moment of u . Here we deﬁne the statistical moments associated with the pdf of the variable u. The pdfs represent the weighting factor for a certain value of a random variable. For example, if a random variable u (which can have only two values, either a or b) has a higher probability of having the value of a than having the value of b, it would be incorrect to calculate the mean of u as an average of a and b. In this case, the respective probabilities of a and b should be used as a weighting factor for calculating the mean. Therefore, u = aPr (u = a) + bPr (u = b) ,

where

Pr (u = a) + Pr (u = b) = 1 (4.163)

For a general case, u=

i

ui Pr (u = ui ) =

+∞ ui Fu (ui ) = uP (u) du

i

(4.164)

−∞

We can write ﬁrst moment (or mean) of u as: +∞ u (x, t) = uP (u; x, t)du −∞

(4.165)

In certain cases of turbulent ﬂows, the velocity ﬁeld may have a normal distribution. In such case, the pdf of velocity component u can be written by Equation 4.166: 2 1 − (u−u) P (u) = √ e 2σ 2 (4.166) σ 2π The normal distribution is also known as Gaussian distribution.

252

BACKGROUND IN TURBULENT FLOWS

Variance is the second statistical moment, and it is the sum of the squared distances from the mean times the probability of being at that distance. Higherorder moments, skewness (asymmetry), and kurtosis (peakedness) are similarly deﬁned, with the distances, raised to the third and fourth power, respectively. Once the pdf of a given variable is known, its n th moments can be deﬁned by the next equation: +∞ u (x, t)n = un P (u; x, t) du (4.167) −∞

Here the bar on u n denotes the average (or mean) value, sometimes also called expectations, of u n . Similarly, the mean value of any continuous function (u) can be calculated from: +∞ (x, t) = (u)P (u; x, t) du (4.168) −∞

The n th central moments around the mean are deﬁned by: )

*n

u (x, t) − u (x, t)

=

+∞

−∞

(u − u)n P (u; x, t) du

where the second central moment (called variance) can be written as +∞ *2 ) u (x, t) − u (x, t) = (u − u)2 P (u; x, t) du −∞

(4.169)

(4.170)

If we decompose the random variable u into its mean and the ﬂuctuating part u by Reynolds averaging, u (x, t) = u (x, t) + u (x, t)

(4.171)

where u = 0 by deﬁnition, the variance is found to be related to the ﬁrst and second moment by: u2 = (u − u)2 = u2 − 2uu + u2 = u2 − u2 4.5.2

(4.172)

Joint Probability Density Function

The Reynolds decomposition method (see Equation 4.18) traditionally was used for obtaining governing equations from the Navier-Stokes equations for turbulent ﬂows to solve for the ﬁrst and second moments of the ﬂow variables. Since the three velocity components and pressure depend on each other through the solutions of the Navier-Stokes equations, they are correlated with each other. To quantify these correlations, it is convenient to introduce a joint probability density function of the random variables. A joint pdf, P (u, υ; x, t), is the probability

253

PROBABILITY DENSITY FUNCTION

density of the simultaneous events u(x, t) = U and υ(x, t) = V . At each point in space and time, the joint pdf function contains a complete statistical description of u and υ, but it does not contain two-point information (i.e., it provides information at each point separately but no joint information at two or more separate points). Thus, it does not give any information about the frequency or length scale of the ﬂuctuations. The pdf of u, P (u), may be obtained from the joint pdf by integration over all possible realizations of υ, that is, +∞

P (u) =

P (u, υ) dυ

(4.173)

−∞

In this context, the pdf of u is called the marginal pdf. The mean of any function of u and υ can be determined from their joint pdf as: +∞ +∞ ψ (u, υ) = ψ (u, υ) P (u, υ) dudυ

(4.174)

−∞ −∞

In nearly homogeneous turbulence, experiments have shown that the joint pdf of velocity and a scalar variable such as mixture fraction is a joint normal distribution. However, in inhomogeneous ﬂows, especially turbulent reactive ﬂows, pdfs can be far from normal. If both random variables in a joint pdf and any linear combination of these two random variables are also normal distributions, the distribution is called a joint normal distribution. The joint normal distribution of two random variables u and υ is given by Equation 4.175. z − 1 2 (4.175) P (u, υ) = e z(1−ς ) & 2 2πσu συ 1 − ς where z ≡

2ς (u − u) (υ − υ) (υ − υ)2 u υ (u − u)2 − + and ς = R = . uυ σu2 σu συ συ2 σu συ

The correlation between u and υ (also called covariance) is given by: u υ

+∞ +∞ = (u − u) (υ − υ) P (u, υ) dudυ

(4.176)

−∞ −∞

This can be illustrated by a so-called scatter plot of u versus υ. If a series of instantaneous realizations of u and υ are plotted as points in a graph of u and υ, these points will scatter within a certain range. The means u and υ are the average positions of the points in u and υ directions, respectively. The correlation coefﬁcient u υ (σu συ ) can be related to the slope of the average straight line through the data points.

254

4.5.3

BACKGROUND IN TURBULENT FLOWS

Bayes’ Theorem

A joint pdf of two variables always can be written as a product of a conditional pdf of one variable times the marginal pdf of the other:

P (u, υ; x, t) = P ( u| υ = V ; x, t) P (υ; x, t)

or

Puυ (U, V ; x, t) = P ( U | υ = V ; x, t) P (V ; x, t)

(4.177)

In this equation, the conditional pdf P ( u| υ = V ; x, t) describes the probability density of a random variable u, conditioned at an event υ = V , where U and V are called the corresponding sample space variables for the random variables u and υ. If u and υ are not correlated, they are called statistically independent. In that case, the joint pdf is equal to the product of the marginal pdfs:

P (u, υ; x, t) = P (u; x, t) P (υ; x, t)

or

Puυ (U, V ; x, t) = P (U ; x, t) P (V ; x, t)

(4.178)

Applying this condition in Equation 4.176 and then integrating, it is easily seen that u υ vanishes if u and υ are statistically independent, as shown in Equation 4.179. u υ

+∞ +∞ = (u − u) P (u) du (υ − υ) P (υ) dudυ = u υ = 0 −∞

(4.179)

−∞

However, in turbulent ﬂows, u υ is interpreted as one of the Reynolds stress components, which is nonzero in turbulent shear ﬂows. The conditional mean of u, for a ﬁxed value of υ, is deﬁned by: u| υ =

+∞ −∞

uP (u|υ)du

(4.180)

As a consequence of the nonlinearity of the Navier-Stokes equations, several closure problems arise. These are related not only to correlations between velocity components among each other and the pressure, but also to correlations between velocity gradients (such as dissipation terms) and correlations between velocity gradients and pressure ﬂuctuations (such as pressure-strain correlations). The statistical description of gradients requires information from adjacent points in physical space. Thus, two-point correlations must be introduced. Very important aspects in the statistical descriptions of turbulent ﬂows are therefore related to two-point correlations. As described earlier, for ﬂows with large density gradients (e.g., ﬂow with heat release due to combustion), often it is convenient to introduce a densityweighted average velocity ﬂuctuations, called the Favre average, by decomposing the instantaneous u(x,t) into ( u(x,t) and u (x, t) by Equation 4.50. Not only in the

PROBABILITY DENSITY FUNCTION

255

momentum equations, but also in the conservation equations for the energy and chemical species, the respective convective and advection terms are dominant for high Reynolds number ﬂows. Since these terms contain products of the dependant variables and the density, Favre averaging is preferable to Reynolds averaging. For instance, the average of the product of density ρ with the velocity components u and υ would lead to four terms by using the Reynolds averaging technique: ρuυ = ρ u υ + ρu υ + ρ u υ + ρ υ u + ρ u υ Using Favre averaging, the equation becomes

υ + ρυ ( u + ρu υ ρuυ = ρ ( u + u ( υ + υ = ρ( u( υ + ρu(

(4.181)

(4.182)

Here ﬂuctuations of the density do not appear. Taking the time average of Equation 4.182 leads to two terms only: υ ρnυ = ρ( u( υ + ρ u

(4.183)

This expression is much simpler than Equation 4.181 and formally has the same structure as the conventional average of uυ for constant density ﬂows: uυ = u υ + u υ

(4.184)

In turbulent reacting ﬂows with a high Reynolds number, viscous and diffusive transport terms are of less importance than the turbulence transport terms and usually can be neglected; thus, the Favre averaging technique also presents fewer difﬁculties than the Reynolds averaging technique. A Favre pdf of u can be derived from the joint pdf P (ρ, u) as: ρmax ρmax ρ( P (u) = ρ P (ρ, u) dρ = ρ P ( ρ| u) P (u) dρ ρmin

= P (u)

ρmin

ρmax

ρmin

ρ P ( ρ| u) dρ = ρ| u P (u)

(4.185)

Multiplying both sides of Equation 4.185 by u and integrating yields: +∞ +∞ ρ| u uP (u)du ρ u( P (u) du = (4.186) −∞

−∞

u = ρu. The Favre mean value of u therefore is deﬁned which is equivalent to ρ( as: +∞ ( u( P (u) du u≡ (4.187) −∞

In general, for steady mean ﬂow, the Favre average of any quantity q can be obtained as: 1 ( q (f ) ( P (f ; x) df (4.188) q (x) ≡ 0

256

BACKGROUND IN TURBULENT FLOWS

For the reactive variables (e.g. temperature T and species mass fraction Yi ), the Favre-averaged values are given by the following equations. T((x) ≡

1

((f ; x) df , T (f ) P

0

4.6

(i (x) ≡ Y

0

1

((f ; x) df Yi (f ) P

(4.189)

TURBULENT SCALES

Turbulence is considered a random process with a very large number of spatial degrees of freedom. Turbulence contains a wide range of different scales in terms of velocity, length, and time. The existence of a continuum of different space and time scales sometimes is taken as conﬁrmation of the presence of turbulence. The dynamics of turbulence involves all scales; these scales coexist and are superimposed with each other on smaller scales living inside the larger scales. Although turbulence is unpredictable in detail, its statistical properties can be reproduced using various modeling tools. One such method is to model turbulent ﬂow by using Fourier series. The underlying concept of such modeling is the Fourier theorem, which states that any analytic signal on a bounded domain can be expanded in a Fourier series. Therefore, an artiﬁcial or “kinematic” turbulent-like velocity component u along a line within a three-dimensional ﬁeld of isotropic ﬂuctuations can be constructed using a Fourier series representation given by Equation 4.190 as: N

u (x) = an sin (κn x + ψn ) (4.190) n=1

Each term in Equation 4.190 is called a Fourier mode, with an called the Fourier coefﬁcient and ψn called the phase angle for the Fourier mode. Both an and ψn are random numbers. This is a useful representation for turbulence; it is a series in which each term represents a contribution with a well-deﬁned length scale ln , where the wave number κn = 2π/ln . Scaling analysis plays a vital role in understanding turbulent ﬂows. For a ﬂow with a zero mean velocity component, the ﬂuctuating velocity is the same as the velocity u. A plot of ﬂuctuating velocity, u, is shown in Figure 4.8. The turbulent velocity u can also be represented as a sum of an inﬁnite number of sine waves, each having a different wavelength ln (or wave number κn ), as per Equation 4.190. Even though turbulence has an inﬁnite number of scales, analysis of nonreacting turbulent ﬂow has been conducted for three major regions. Before discussing more about the turbulent length scales, let us discuss the concept of energy cascade. This concept proposes that most of the turbulent kinetic energy is generated by external forces or hydrodynamic instabilities at the large scales of turbulence. These large scales are of the order of integral scales of turbulence (l0 ), and they are known as energy-containing scales. Therefore, the ﬁrst region in the energy spectrum consists of the largest eddies where turbulence kinetic energy (k ) is

TURBULENT SCALES

257

0.3

Fluctuating Velocity, u′ (m/s)

0.2 0.1 0 −0.1 −0.2 −0.3 −0.4

0

0.1

0.2

0.3 0.4 0.5 Distance, x (m)

0.6

0.7

0.8

Figure 4.8 Velocity ﬂuctuation with distance showing inﬁnite number of spatial scales in turbulence.

generated by the mean ﬂow gradients. These scales are coupled with the mean ﬁeld, and they depend on the turbulence generation mechanisms; for example, external forces such as the pressure gradient force or hydrodynamic instabilities such as the Kelvin-Helmholtz, Rayleigh-Taylor, or Richtmyer-Meshkov instability. The large-scale eddies have energy proportional to u20 and a time scale τ0 ∼ l0 /u0 , where u0 is the velocity scale associated with the largest eddies; therefore, the rate at which energy enters the cascade is

TrI ∼

u20 u3 = 0 l0 /u0 l0

(4.191)

The second region is associated with the intermediate-length scales, which transfer the energy to smaller scales via inviscid nonlinear mechanisms with no inﬂuence of the ﬂuid’s molecular viscosity. This region is known as inertial range (or inertial subrange). Since the energy is transferred without any losses due to viscosity, the rate of energy transfer (TrT ) is equal to the rate of energy entering the cascade (TrI or P ). The last region contains the smallest scales in which the viscous effects become important and where the kinetic energy is dissipated into heat. This region is often called the dissipation range. If an equilibrium condition persists in the cascade, the rate of energy dissipation (εD ) at the smallest scales should be equal to the rate of energy transfer to these scales. Thus, , u3 TrI TrT εD = ε ∼ O 0 (4.192) l0

258

BACKGROUND IN TURBULENT FLOWS

Energy-containing range

Universal equilibrium range Inertial subrange

Dissipation subrange

(viscous effect negligible; inertia effect important)

(viscous dissipation of k into heat)

(energy contained in anisotropic large eddies)

Energy transfer to smaller scales

Dissipation, ε

h

Figure 4.9

Production, P

lDI

lIE

(≈ 70h)

(≈0.1l0)

l0

L

Energy cascade and various length scales (modiﬁed from Pope, 2000).

At this point, it is appropriate to discuss three different Kolmogorov hypotheses that explain the above-mentioned scales. Kolmogorov introduced this theory for homogeneous isotropic turbulence in 1941; thus, it is also known as the K41 theory. The ﬁrst hypothesis describes the fact that the large-scale eddies are anisotropic and are affected by the boundary conditions of the ﬂow; the motions at the small scales can be considered isotropic. The second hypothesis states that the statistics of small-scale motion can be described in terms of kinematic viscosity ν and the turbulent dissipation rate ε. The third hypothesis states that the statistics of intermediate scale can be described by ε alone. These three ideas are described in more detail next. Pope (2000) has deﬁned the demarcation between the three regions as lIE —, the boundary between the energy-containing range and the inertial subrange, and lDI —, the boundary between the energy dissipation range and the inertial subrange. With the addition of these two length scales, the inertial subrange is between lDI < l < lIE . Figure 4.9 shows the concept of three regions: 1. Kolmogorov’s hypothesis of local isotropy. “At a sufﬁciently high Reynolds number, the small-scale turbulent motions are statistically isotropic.” The dissipation rate ε is determined by the energy transfer rate TrT , so that these two rates are nearly equal (i.e., ε ≈ TrT ). The major implication of this hypothesis is that state of the small scales is determined by kinematic viscosity ν and the rate of energy transfer from the large scales. 2. Kolmogorov’s ﬁrst similarity hypothesis. “In every turbulent ﬂow at sufﬁciently high Reynolds number, the statistics of the small-scale motions have a universal form that is uniquely determined by ν and ε.” By using dimensional analysis, the smallest-length scale in turbulence was deﬁned by Kolmogorov as: 1/4 η ≡ ν3 ε

(4.193)

TURBULENT SCALES

259

Similarly, the velocity and time scales corresponding to this Kolmogorov scale are deﬁned as: uη ≡ (εν)1/4

(4.194)

τη ≡ (ν/ε)1/2

(4.195)

The Reynolds number based on the Kolmogorov scales is unity (i.e., ηuη /ν = 1). In terms of these Kolmogorov scales, the dissipation rate can be written as:

2 (4.196) ε = ν uη /η = ν/τη2 The Kolmogorov scales depend on the Reynolds number of the ﬂow. By using this relationship with the deﬁnition of Kolmogorov scales, the ratios of the Kolmogorov scales to the integral scales can be determined as: −3/4 −3/4 (4.197) η l0 ∼ Rel0 ∼ ReL −1/4 −1/4 uη u0 ∼ Rel0 ∼ ReL (4.198) −1/4 −1/4 τη τ0 ∼ Rel0 ∼ ReL (4.199) Note that the large-scale length L is typically of the same order of the integral scale of turbulent ﬂuctuations, l 0 . Thus, for any high Reynolds number ﬂow, the Kolmogorov scales can be determined by the these equations. Based on Kolmogorov’s hypotheses, it is understood that the turbulence of such ﬂows is statistically isotropic at the Kolmogorov scales. Therefore, all high Reynolds number turbulent ﬂows are statistically identical at the small scales. 3. Kolmogorov’s second similarity hypothesis. “In every turbulent ﬂow at sufﬁciently high Reynolds numbers, the statistics of the motions of scales between the Kolmogorov and the inertial scales have a universal form that is uniquely determined by ε and it is independent of ν.” For eddies in the inertial subrange, with size , the velocity scale and time scale for these eddies are formed from ε and : u () = (ε)1/3 = uη (/η)1/3 ∼ u0 (/0 )1/3 = function of ν

1/3 τ () = 2 /ε = τη (/η)2/3 ∼ τ0 (/0 )2/3 = function of ν

(4.200) (4.201)

By equating the energy transfer rate (kinetic energy per eddy turnover time) with the dissipation, rate it follows that this quantity is independent of the size of the eddies within the inertial subrange. For the inertial range, extending from the integral scale l 0 to the Kolmogorov scale η, ε is the only dimensional quantity apart from the correlation coordinate r that is available for the scaling of correlation function R(r,t). The next discussion leads to the derivation of turbulent kinetic energy spectrum as a function of wave number kn .

260

4.6.1

BACKGROUND IN TURBULENT FLOWS

Comment on Kolmogorov Hypotheses

The Kolmogorov hypotheses imply that, at sufﬁciently high outer-scale Reynolds numbers, the distribution of velocity differences at a scale r: (a) is statistically isotropic and (b) depends only on the local length scale r, dissipation rate ε, and kinematic viscosity ν if r is much smaller than the outer scale l 0 . Kolmogorov further hypothesized that within an inertia-dominated subset of scales r much larger than the viscous scales η (but still much smaller than l 0 ), the distribution of δu is independent of ν, implying no direct frictional effect at these scales. The Kolomogorov hypotheses were formulated for stationary high Reynolds number turbulence in equilibrium at all scales, although the underlying theory is applied broadly. The direct implications of these hypotheses is that there will be no inﬂuence of outer energy-dominated scales on the inner dissipation-dominated scales at large-scale separations (i.e., high Reynolds numbers) and that no direct energy transfer can take place between these disparate scales. This, in turn, implies that energy is transferred from the large-scale motions l 0 to the small-scale motions η within the “local” scale interactions (Onsager, 1979) through an inertia-dominated range of intermediate scales within negligible dissipation. The rate of dissipation should therefore scale on large-scale velocity and time scales in equilibrium turbulence (i.e., ε ∼ u20 τ0 ). Brasseur and Wei (1994) have shown that even though the cumulative effects of the distance triadic interactions may be dominated by the cascading effects of local-to-nonlocal triadic interactions, large–small scale dependence is always present in principle. The triadic interactions are the most fundamental block of energy transfer process from a scale with wavenumber κa due to its interactions with scales of wavenumbers κb and κc that form a triangle with κa , which means that κc = κa − κb (see Yeung, Brasseur, and Wang, 1995). As a consequence, nonstationary, nonequilibrium turbulence can be expected to deviate from large–small scale independence and local isotropy in high Reynolds number turbulence. Yeung, Brasseur, and Wang (1995) have pointed out that the classical Kolmogorov cascade can be momentarily interrupted by non-Kolmogorov direct interactions between the large energy-containing scales and the small dissipative scales (where mixing and chemical reaction occurs) by nonequilibrium inﬂuence of the large scales. These interactions are also called long-range processes due to the long separation between the scales. The dynamic inﬂuence of longrange interactions at the small scales is to redistribute vorticity and energy at those scales. The dynamic interactions between the large and small scales lead to anisotropy at the Kolomogorov scales. Although there is no direct energy transfer from the large scales to the small scales, it cannot be generally assumed that the small scales are not affected by the large scales; as a result, the small scales are always isotropic. This observation varies from the K41 theory of statistical large–small scale independence in the asymptotic limit. However, the deviation from the local isotropy is a matter of degree that depends on a variety of interacting factors (see Brasseur and Wei, 1994 for details). If the Kolmogorov hypotheses are viewed as approximations

TURBULENT SCALES

261

of reality, then, in practice, the approximations may be good in many instances. However, in cases such as nonstationary and nonequilibrium turbulent ﬂows where the long-range dynamic processes (and the structure and consequence of these processes is understood) exist in a ﬂow ﬁeld, then the hypotheses may break down. Such deviation is not unusual in combusting systems; the most obvious example is constant-pressure combustion, such as in diesel engines. These long-range couplings have a strong effect on scalar reactants and therefore can directly affect the initiation of combustion. For homogeneous turbulence, we have u (x, t)2 = u (x + r, t)2 = u2 (t)

(4.202)

Therefore, u (x, t) u (x + r, t) u (x, t) u (x + r, t) u (x, t) u (x + r, t) = % = R(r, t) ≡ % % % u (t)2 u (x, t)2 u (x + r, t)2 u (t)2 u (t)2 (4.203) The second-order structure function is deﬁned by: F2 (r, t) ≡ (u (x, t) − u (x + r, t))2 = u (x, t)2 + u (x + r, t)2 − 2u (x, t) u (x + r, t) ≈ 2u2 (t) [1 − R(r, t)] For isotropic turbulence, the second-order structure function can be written simply as F 2 (r,t), which has the dimension of [m2 /s2 ]. Recall that ε has the dimension of [m2 /s3 ]; therefore, we can express F 2 (r,t) by the following equation using dimensional analysis. F2 (r, t) = C (εr)2/3

(4.204)

where C is a universal constant (=1.5) called the Kolmogorov constant in the limit of large Reynolds numbers. In the case of homogenous isotropic turbulence, the velocity ﬂuctuations in the three coordinate directions are equal to each other. The turbulent kinetic energy is: k = υ · υ /2 = 3u2 /2

(4.205)

Using this equation, we can utilize the two expreasions of F 2 (r,t) to obtain R (r, t) = 1 −

3C (εr)2/3 4k

(4.206)

This correlation function is plotted as the dotted line in Figure 4.10 (Peters, 2000). Based on the plot shown in Figure 4.10, there are characteristic size eddies, which contain most of the kinetic energy. At these scales, there is still a relatively strong correlation R(r,t) before it decays to zero. The length scale of these eddies

262

BACKGROUND IN TURBULENT FLOWS 1 R(r,t)

1

3C (εr)2/3 4 k

0 η

l0

r

Figure 4.10 Normalized two-point velocity correlation for homogeneous isotropic turbulence as a function of distance r between the two points (from Peters, 2000).

is called the integral length scale l 0 and is deﬁned by equating the area in the shaded region in Figure 4.10 with the area under the correlation curve: ∞ l0 (t) = R (r, t) dr (4.207) 0

Let us denote the root-mean-square (rms) velocity ﬂuctuation by: urms = υ =

& 2k/3

(4.208)

which represents the turnover velocity of integral scale eddies. The turnover time of these eddies is then proportional to the integral time scale: τ0 =

k ε

(4.209)

For very small values of r, only very small eddies ﬁt into the distance between x and x+r. The motion of these small eddies is inﬂuenced by viscosity, which provides an additional dimensional quantity for scaling. The Taylor length scale T is an intermediate scale between the integral and the Kolmogorov scale. It is deﬁned by replacing the average velocity ﬂuctuation gradient in the deﬁnition of the dissipation in Equation 4.116 by υ /T . This leads to the deﬁnition ε = 15νυ 2 /2T

(4.210)

Here the factor 15 originates from considerations for isotropic homogenous turbulence. It is seen that T is proportional to the product of the turnover velocity of the integral scale eddies and the Kolmogorov time. Therefore, T may be interpreted as the distance over which large eddy ﬂuctuations interact with the Kolmogorov eddy in the vortex tube during its turnover time tη ≡ (ν/ε)1/2 (see Figure 4.11). As a somewhat artiﬁcially deﬁned intermediate scale, it has less direct physical signiﬁcance than η in turbulence or in turbulent combustion. We

TURBULENT SCALES

263

u′= Velocity fluctuations

η T

u′

u′

T

u′

= Taylor microscale

η = Kolmogorov length scale T

u′ η

Figure 4.11 Physical interpretation of Taylor microscale (after Tennekes and Lumley, 1972).

will see, however, that similar Taylor microscales may be deﬁned for nonreactive scalar ﬁelds that are useful in the interpretation of the mixing process. Let us now deﬁne a discrete sequence of eddies within the inertial subrange by n = 0 2n ≥ η, n = 1, 2, . . . . A Fourier transform of the isotropic two-point correlation function leads to a deﬁnition of the kinetic energy spectrum E (κ), which is the density of kinetic energy per unit wavenumber κ. The wavenumber κ n is inversely proportional to the eddy size n as: κn ∝ −1 (4.211) n The kinetic energy u2n at scale n is then u2n ∼ (εn )2/3 ∝ ε 2/3 κn−2/3

(4.212)

The kinetic energy density in wave-number space is proportional to E (κ) ≡

du2 ∼ ε2/3 κ −5/3 ; at κ = κn , E (κn ) = Cε 2/3 κn−5/3 dκ

where C = 1.5 (4.213)

Equation 4.213 is known as the κ −5/3 law for the kinetic energy spectrum in the inertial subrange. The wavenumbers corresponding to the integral length scale and Kolmogorov length scale are κl0 and κη In addition to these wavenumbers, κIE and κDI are deﬁned as the beginning and end of the inertial subrange. The turbulent kinetic energy spectrum (normalized by its maximum value) is shown in Figure 4.12. The spectrum attains a maximum at a wavenumber that corresponds to the integral scale, since eddies of the integral length scale contain most of the kinetic energy. There is a cut-off due to viscous effects at the Kolmogorov length scale η. Beyond this cut-off length scale (in the range called the dissipation range), the energy per unit wavenumber decreases exponentially due to viscous effects. The turbulent kinetic energy in the wavenumber range κa , κb ) is: κb E (κ) dκ (4.214) k (κa , κb ) = κa

264

BACKGROUND IN TURBULENT FLOWS

101

Energycontaining range

Inertial subrange

10−1

Dissipation range

k−5/3

E(k)/(E(k))max

10−3 10−5 10−7 10−9 10−11 10−13 10−15 100

Figure 4.12 ber κ.

kI

0

101

kIE

kDI kη

102 103 104 Wavenumber k (1/m)

105

Dimensionless turbulent kinetic energy spectrum as a function of wavenum-

The turbulence dissipation rate ε from motions in the range (κa , κb ) is: κb 2νκ 2 E (κ) dκ (4.215) ε (κa , κb ) = κa

From Kolmogorov’s ﬁrst similarity hypothesis, it is implied that the spectrum is a universal function of ε and ν. In addition to the energy spectrum, a study of the dissipation rate spectrum can be very useful in understanding the various turbulent scales and their contributions. For a turbulent ﬂow with an integral length scale of 0.75 m and a Reynolds number of ∼134,000, the spectra of kinetic energy and dissipation rate per unit wavenumber are shown in Figure 4.13 on four different plots. These spectra are normalized for better comparison of their behavior in various wavenumber ranges. As described, the turbulent kinetic energy per unit wavenumber E (κ) decreases with wavenumber by κ −5/3 law (i.e., E (κ) ∼ κ −5/3 ) in the inertial subrange. The turbulence dissipation rate per unit wave number D (κ) increases with wave number by κ 1/3 law (i.e., D (κ) ∼ κ 1/3 ). The slope of these curves should be linear on a log-log scale within the inertial subrange (since they vary by power law) as shown in Figure 4.13a. When these spectra are plotted on a linear-log scale, the peaks of these spectra are very distinctly separated, as shown in Figure 4.13b. While the peak for E (κ) is located very close to the wavenumber κl0 , the peak for D (κ) is located much closer to κDI on a linear-log scale as shown in Figure 4.13b. The more detailed plots shown in Figure 4.13c and d on a linear-linear scale show that E (κ) is concentrated in a very narrow range

265

TURBULENT SCALES

E(k)/(E(k))max or D(k)/(D(k))max

E(k)/(E(k))max or D(k)/(D(k))max

1.2 101

k1/3

10−1

k−5/3

10−3 10−5

E(k) D(k)

10−7 10−9 10−11 10−13 10−15 100

kI0 kIE 101

kDI

kη

102 103 104 Wavenumber k (1/m)

105

E(k) D(k)

1 0.8 0.6 0.4

kI0

0.2 0 100

101

1

E(k)/(E(k))max or D(k)/(D(k))max

E(k)/(E(k))max or D(k)/(D(k))max

1.2 E(k) D(k)

0.8 0.6 0.4

0 100

κIE

0

105

(b)

1.2

κI

κη

102 103 104 Wavenumber k (1/m)

(a)

0.2

κDI

kIE

κD

max

1

E(k) D(k)

0.8 0.6 0.4 κDI

0.2 0

101 102 Wavenumber k (1/m)

(c)

103

0

5 103 1 104 1.5 104 Wavenumber k (1/m)

2 104

(d)

Figure 4.13 Normalized E(κ) and D(κ) spectra with wavenumber (a) log-log scale, (b) linear-log scale, (c) linear-linear scale in a large eddy range, and (d) linear-linear scale in a dissipation range.

whereas D (κ) is distributed over a wide range. It seems to suggest that resolving the smallest scales is pertinent where viscous dissipation is a major factor. The turbulent kinetic energy (TKE) upto a wave number κ is the area under the curve of E(κ) from κ = 0 to κ see Figure 4.13b; therefore, the TKE is mainly contributed by the energy containing scales upto κIE . Kinetic energy by different wavenumber ranges. In other words, it shows the contribution of various length scales to the TKE. This analysis is also true for the dissipation rate ε. The TKE and dissipation rate distribution with wavenumber are shown on a semi-log plot in Figure 4.14. This ﬁgure shows that the slope of the normalized TKE curve increases sharply up to κIE then dramatically decreases between κIE and κDI and subsequently reduces to zero after κDI . This implies that the smaller wave numbers (or larger length scales) make the largest contribution to the TKE, whereas the larger wavenumbers (κDI or smallest scales) make no contribution. There are intermediate length scales (corresponding to the wavenumber region between κIE and κDI ) where the TKE contribution is much smaller but nonzero.

266

BACKGROUND IN TURBULENT FLOWS

k(k)/(k(k))max or ε(k)/(ε(k))max

1.2 k

1

ε 0.8 0.6 0.4 0.2 kI

kIE

kDI

kη

0

0 100

101

102 103 Wavenumber k (1/m)

104

105

Figure 4.14 Variation of normalized kinetic energy and dissipation rate with wavenumber on semi-log scale.

The normalized dissipation rate curve shows an entirely different behavior. As shown in Figure 4.14, the slope of this curve is zero up to κIE , slightly increases after κIE and before κDI , dramatically increases between κDI and κη , and reduces to zero after κη . This indicates that most of the dissipation takes place in the smaller length scales range corresponding to wave numbers κDI and κη . There is almost no dissipation occurring at large length scales, and nonzero dissipation (much lower than smallest scales) takes place in the intermediate length scales. Since neither dissipation nor TKE addition is evident at wave numbers larger than κη, a grid resolution based on this wave number should be sufﬁcient to resolve every detail of the turbulent ﬂow. 4.7

LARGE EDDY SIMULATION

The large eddy simulation (LES) technique has been developed to address the anisotropic turbulence behavior of the ﬂow by resolving intermediate spatial and temporal scales, which are small enough to simulate dynamics of large eddies explicitly while the smaller eddies are treated by homogeneous isotropic turbulence modeling. By using proper low-pass ﬁltering, scales lower than a selected x (or simply ) are eliminated and suitable equations for large scales are developed for solving the ﬂow property variations in both time and spatial variables. LES is more accurate and reliable than RANS models for ﬂows involving unsteadiness at large scales (e.g., ﬂow over bluff bodies, which involves unsteady separation and vortex shedding). Strictly speaking, there are two different ﬁltering processes in the LES approach: ﬁlter dimension and grid spacing. The turbulence processes occurring at scales smaller than the grid spacing (also known as grid

267

LARGE EDDY SIMULATION

cut-off ﬁlter) cannot be resolved in any case, and they are always modeled. These scales are called subgrid scales (SGS). For scales smaller than the resolved scales (i.e., x), SGS models are used. The second scale is associated with the scale used by the LES ﬁlter (δ), to be discussed, which can be different from the grid cut-off ﬁlter ( ). In practice, a ﬁlter scale is chosen to coincide with the grid spacing. Subgrid scales are same as subﬁlter scales (SFS; i.e., = δ), so there is no need to deﬁne separate SFS and SGS models. The concept behind the LES method is demonstrated in Figure 4.15 in both physical and spectral space. In Figure 4.15a, the subgrid scales are shown in physical space. In Figure 4.15b, two cut-off wave numbers are shown in Fourier space; one is the desirable cut-off wavenumber, which would cover a majority of the energy spectrum, and, therefore, only the dissipation is required to be modeled. However, in practice, this wavenumber could demand very ﬁne resolutions and therefore computationally be more expensive. The actual cut-off wavenumber also is shown in this ﬁgure; it usually is signiﬁcantly lower than the desirable cut-off wavenumber. A ﬁltering operation is performed, to decompose the major variables into the ) and the residual (or subgrid scale, SGS) sum of the resolved component (U component (u ) or ﬂuctuation. (x, t) + u (x, t) U (x, t) = U

(4.216)

This decomposition is known as Leonard’s decomposition. The governing equations are derived for evolution of resolved major variables (density, temperature, velocity, etc.), and these equations contain residual components (e.g., the residual stress tensor [called the SGS stress tensor in the

Resolved regions 100

E(k)/(E(k))max

10−5

Subgrid

Resolved

10−10 10−15 10−20

Cutoff wavenumber, kc

10−25

Scale Δ

Subgrid regions

(a) Physical space

10−30 100

101 102 103 Wavenumber k = 2π/Δ (b) Spectral space

104

Figure 4.15 Description of large eddy simulation methodology (the actual cut-off wavenumber represents current state-of-the-art simulations) (part a from Sagaut, 1998).

268

BACKGROUND IN TURBULENT FLOWS

E(k)

E(k)

E(k)

+

= k

k

k

Total

Modeled

Resolved (a) Reynolds averaged Navier-Stokes (RANS)

E(k)

E(k)

E(k)

= k Total

+

k Resolved (b) Large eddy simulation (LES)

k Modeled

Figure 4.16 Decomposition of energy spectrum in solution associated with (a) RANS and (b) LES (symbolic representation) (modiﬁed from P. Sagaut, 1998).

momentum equation]. These residual components must be modeled to achieve closure (e.g., the residual stress tensor is modeled by the eddy-viscosity model). The modeled ﬁltered equations are solved numerically for the resolved ﬂow properties. Major differences between RANS and LES can be demonstrated by comparing the two sets of plots shown in Figure 4.16. 4.7.1

Filtering

In signal processing, a ﬁlter is a function or procedure that removes unwanted parts of a signal. The concept of ﬁltering and ﬁlter functions is very important in large eddy simulations. One particularly elegant method of ﬁltering is by taking the Fourier transform of a signal into spectral space, performing the ﬁltering operation in the spectral space, and ﬁnally transforming the ﬁltered signal back into the physical space by taking the inverse Fourier transform. The Fourier transform of a function is given by ∞ F (κ) =

f (x) e−2π iκx dx

(4.217)

−∞

The inverse Fourier transform can return the function in the frequency space back to the physical space—that is: ∞ f (x) = −∞

F (κ) e2π iκx dκ

(4.218)

LARGE EDDY SIMULATION

269

Let us discuss the mathematical operation called convolution. A convolution is an integral that expresses the amount of overlap of one function g as it is shifted over another function f . It therefore “blends” one function with another. The convolution is sometimes also known by its German name, faltung (folding). Convolution of two functions f and g over a ﬁnite range [0, ζ ] is given by Equation 4.219: ς ) * f ∗ g (ς ) ≡ f (ξ ) g (ς − ξ ) dξ (4.219) 0

A ﬁltering operation is a convolution of a variable (e.g., a ﬂow ﬁeld variable φi ) with the ﬁlter function (e.g., G); therefore, it is an important step in order to develop an understanding of a large eddy simulation. The convolution integral of φi in the next equation was proposed by Leonard in 1974: i (x, t) = φ

t V

0

φi x , t G x − x , t − τ ; , δτ dx dτ

(4.220)

where x and x are three-dimensional position vectors of common origin, with 0 < τ < t and V representing the total system volume associated with the ﬂuid. The ﬁlter function G quantiﬁes the inﬂuence of subgrid scale (sgs) dynamics at remote points (x ,τ ) on ﬁltered values at (x,t). The ﬁlter function G must satisfy the next normalization condition: t

G x − x , t − τ ; , δτ dx dτ = 1 (4.221) V

0

For a one-dimensional case with only spatial ﬁltering, the ﬁltering operation introduced by Leonard in 1974 can be written as: (x) = U

∞

−∞

G (r) U (x − r, t) dr

(4.222)

In this case, the normalization condition becomes: ∞ G (r, x) dr = 1

(4.223)

−∞

Let us deﬁne Favre ﬁltering for a variable density ﬂow as: (i (x, t) = ρφ

t 0

V

ρφi x , t G x − x , t − τ ; , δτ dx dτ

(4.224)

; (i = ρφi φ ρ

(4.225)

By this deﬁnition, ;i (i = ρφ ρφ

or

270

BACKGROUND IN TURBULENT FLOWS

TABLE 4.1. Several Filter Functions in Physical Space and Spectral Space Type Gaussian ﬁlter Sharp cut-off ﬁlter Box ﬁlter (or grid ﬁlter)

Physical Space 1/2 6 6r 2 exp − π 2 2 sin (πr/ ) πr 1 1 H − |r| 2

Spectral Space 2 2 κ exp − 24 H (κc − |κ|) where κc ≡ π/

1 1 sin κ κ 2 2

Then a variable can be decomposed as: (i (x, t) + φi (x, t) φi (x, t) = φ

(4.226)

In a large-eddy simulation, three major ﬁlter functions often are employed: the Gaussian ﬁlter, the sharp cut-off ﬁlter, and the box ﬁlter. Their equations in both physical and spectral spaces are shown in Table 4.1. The advantage of performing the ﬁltering operation in the spectral space comes from the fact that the Fourier transform turns convolution into a multiplication operation. F (f (x) ∗ g (x)) = F (f (x)) F (g (x)) (4.227) Similarly, the Fourier transform turns a multiplication into a convolution—that is:

F (f (x) g (x)) = F (f (x)) ∗ F (g (x)) (4.228) * The symbol f ∗ g (y) denotes convolution of f and g. Convolution is more often taken over an inﬁnite range, )

)

*

∞

∞ f (ξ ) g (y − ξ ) dξ =

f ∗ g (y) ≡ −∞

g (ξ ) f (y − ξ ) dξ

(4.229)

−∞

The application of these three ﬁlters on the energy spectrum is shown in Figure 4.17. 4.7.2

Filtered Momentum Equations and Subgrid Scale Stresses

The momentum equation in the xi direction for a compressible ﬂuid can be written as:

∂uj ∂ ∂ui ∂ ∂ ∂p +μ + ρui uj = − (ρui ) + ∂t ∂xj ∂xi ∂xj ∂xj ∂xi Dilitation ≈ 0 2 ∂ ∂uk δij μ− + μ ∂x 3 ∂xk j

(4.230)

LARGE EDDY SIMULATION

271

100

E(k) /(E(k))max

10−5 10−10

Unfiltered Box filter Gaussian filter Cutoff filter

10−15 10−20 10−25 10−30 100

Figure 4.17

101

102 103 Wavenumber k = 2π/Δ

104

Application of various ﬁlters on the energy spectrum.

After applying the ﬁltering operator on this equation, we get:

∂ ∂ ∂ ∂p +μ ρ ui uj = − ρ ui ) + (; ∂t ∂xj ∂xi ∂xj

∂ uj ∂ ui + ∂xj ∂xi

(4.231)

The ﬁltered momentum equation contains a nonlinear term, ρ ui uj . In order for Equation 4.231 to be useful, it should be expressed in terms of a ﬁltered quantity ui and subgrid scale quantity ui . Since we have density as a variable in these equations, we will use Favre-ﬁltered quantities instead of regular ﬁltered quantities. In general, a nonlinear term in the governing equations due to any random variable φi and velocity component uj can be decomposed as:

(i + ρφ (i (i + φi ( ( uj + uj = ρ φ uj + ρuj φ uj + ρφ ρ φi uj ≡ ρ φ i( i uj (4.232) If the random variable φi is substituted by velocity component ui in Equation 4.232, we have:

ρ ui uj ≡ ρ ( ( ( ( ui + ui uj + uj = ρ uj + ρu ui + ρu uj + ρu ui j( i( i uj

(4.233)

( ( Adding/subtracting the term ρ uj on the right-hand side of this equation, we ui have ( ( ( ( ( ( uj + ρ uj − ρ uj + ρu ui + ρu uj + ρu ui ui ui ρ ui uj = ρ j( i( i uj

(4.234)

272

BACKGROUND IN TURBULENT FLOWS

Substituting this expression in Equation 4.231, we have the ﬁltered momentum equation as: ∂ τ ij sgs ∂ uj ∂ ∂ ui ∂ ∂p ∂ uj = − ρ( ui ( +μ + (4.235) − ρ ui ) + (; ∂t ∂xj ∂xi ∂xj ∂xj ∂xi ∂xj By using Equation 4.225, we can write the ﬁltered momentum equation as: ∂ τ ij sgs ∂ uj ∂ ∂ ui ∂ ∂p ∂ ρ u(i + ρ( +μ + (4.236) − uj = − ui ( ∂t ∂xj ∂xi ∂xj ∂xj ∂xi ∂xj The subgrid scale tensor τij sgs is: ( ( uj = Lij + Cij + Rij τij sgs = ρ ui uj − ρ ui ( ( ( ( ( ( uj − ρ uj + ρu u u = ρ ui ui + ρu (4.237) j i i j + ρui uj ; ( ( ( ( ( ( ( ( u+ − Lij ≡ ρ uj − ρ uj = ρ u u ui ui u i j i j ; ; + + ( ( ( ( Cross-term stresses: Cij ≡ ρu + ρu ρ u + u = u u u u j i i j j i i j

(4.239)

; + Rij ≡ ρu i uj = ρ ui uj

(4.240)

Leonard stresses:

Reynold’s stresses:

(4.238)

In Equations 4.238–4.240, the next relationships were used based on the definition of Favre ﬁltered quantities from Equation 4.225: < u (4.241) ui uj = ρu i j ρ or ρui uj = ρ ui uj Similarly,

; ( ui = ρ u+ ui ; ρu uj = ρ uj ui ρu j( j( i(

(4.242)

( ( ( uj = ρ uj ρ u+ ui i(

(4.243)

Also,

Let ω and ψ be two random variables. If an operator satisﬁes the next properties, it is called a Reynolds operator: ω + ψ = ω + ψ

aω = a ω

where a is a constant

ω ψ = ω (ψ) = > ∂ ω

∂ω = where s could be xi or t ∂s ∂s = > ω (xi , t) dx1 dx2 dx3 dt = ω (xi , t) dx1 dx2 dx3 dt

(4.244)

LARGE EDDY SIMULATION

Any Reynolds operator will have these identities: ω

= ω , ω =0

273

(4.245)

If the ﬁlter is a Reynolds operator, by applying these identities on Leonard stresses and cross-term stresses, we have: ; (+ ( (i ( ( ( ( (4.246) φ − φ φ u u uj − uj = 0 Lij = ρ i j i j = ρ φ i( ,

; + (i − φ Cij = ρ uj φ uj i(

-

0 0 ( ( =0 φi =ρ ( ui φ i + ( uj

(4.247)

In this section, the overhead hat ( ˆ ) over any variable (or product of variables) indicates that the quantity is ﬁltered. It does not represent time-averaged quantity as discussed in Reynolds decomposition. It should be understood that LES ﬁltering is very different from RANS—that is: ? ∂φ ∂φ = ∂xi ∂xi

and

= 0 φ

(4.248)

A comparison of ﬁltered and uniﬁltered velocity components and their corresponding ﬂuctuating quantities is shown in Figure 4.18. A good reference for large eddy simulation of incompressible ﬂows is the book by Sagaut (1998). 4 u1

uˆ 1

3 u1, uˆ 1 2

1

u1′′

Δ u′′1, uˆ 1′′ 0 uˆ 1′′ −1 0

2

4

6

8

10

x

Figure 4.18 Comparison of ﬁltered and unﬁltered velocity components and their corresponding ﬂuctuating quantities (modiﬁed from Pope, 2000).

274

4.7.3

BACKGROUND IN TURBULENT FLOWS

Modeling of Subgrid-Scale Stress Tensors

The subgrid-scale stress tensors have been modeled by using the eddy viscosity concept and the Boussinesq approximation based on the analogy between the interaction of small eddies and the perfectly elastic collision of molecules (i.e., molecular viscosity). Eddy viscosity models work for scales. The

small most commonly known eddy viscosity model for the term τij sgs in LES is the Smagorinsky model (1963), after Joseph Smagorinsky, who used this model for geophysical ﬂow calculations. The ﬁrst major result in LES was given by Lilly (1992), who showed that the constant used in the Smagorinsky model should have a universal value, not a tuning constant. For this reason, the model also is called Smagorinsky-Lilly model. The subgrid scale stress is given by Equation 4.249. ( ( ui uj − ρ uj = μsgs ui τij sgs = ρ where Sij =

1 2

∂ uj ∂ ui + ∂xj ∂xi

∂ uj ∂ ui + ∂xj ∂xi

≈ 2 ρ νsgs Sij (4.249)

The eddy viscosity νi is modeled by the next formula: 7 7 νsgs = (Cs )2 7S 7

1

where = (Volume) 3

and

7 7 & 7S 7 = 2Sij Sij

(4.250)

where the term represents the grid size and volume refers to the grid volume. Lilly showed that the value of Cs should have a universal value of 0.17 under certain assumptions. However, the value of this constant varies between 0.1 and 0.3 depending on the problem. For a case of isotropic homogeneous turbulence, Clark et al. (1979) used Cs = 0.2 whereas Deardorff (1973) used Cs = 0.1 for a two-dimensional planar channel ﬂow. Numerical simulations of shear ﬂow have used a value of Cs ∼0.1 to 0.12 (Meneveau, Lund, and Cabot, 1996; O’Neil, Driscoll, and Malmberg, 1997). The Smagorinsky model is very simple to implement; however, its major drawback is the inability to have a universal constant value for Cs and thus represent correctly different turbulent ﬁelds in rotating or sheared ﬂows, near-solid walls, or transitional regimes with one eddy viscosity equation. To overcome this deﬁciency, dynamic subgrid scale models were proposed by Germano and co-workers (1991). In this model, known as the Germano model, the model coefﬁcient Cs is computed dynamically as the calculation progresses rather than being an a priori input. The model is based on an algebraic identity between the subgrid-scale stresses at two different ﬁltered levels and the resolved turbulent stresses. The subgrid-scale stresses obtained using the Germano model vanish in laminar ﬂow and at a solid boundary and have the correct asymptotic behavior in the near-wall region of a turbulent boundary layer. The results of large-eddy simulations of transitional and turbulent channel ﬂow using the Germano model are in good agreement with the direct simulation data.

275

LARGE EDDY SIMULATION

In this approach, a two-stage ﬁltering is applied. For the purposes of this work, we deﬁne two ﬁltering operators: one is the grid ﬁlter, denoted by an overhead ˆ , and the other is called a test ﬁlter, shown as: x, x dx (x) = φ x G (4.251) φ

φ (x) = φ x G x, x dx (4.252) The ﬁlter width of the test ﬁlter is assumed to be larger than that of the grid ﬁlter (i.e., the test ﬁlter corresponds to a coarser mesh than the grid ﬁlter). A description of grid ﬁltered volume and test ﬁltered volume is shown in = GG. By applying the grid ﬁlter to the momentum Figure 4.19. Finally, let G equations one obtains the following ﬁltered equations of motion: ∂ τ ij sgs ∂ uj ∂ ui ∂ ∂ ∂ ∂p uj = − ρ( ui ( +μ + (4.253) − ρ ui ) + (; ∂t ∂xj ∂xi ∂xj ∂xj ∂xi ∂xj ( ( ui uj − ρ The subgrid-scale tensor is: τij sgs = ρ uj ui

Now apply becomes: ∂ ; ρ ui + ∂t

to Equation 4.253, and the ﬁltered momentum equation G ⎛ ⎞ ∂ Tij sgs,t ∂ u ∂ ∂ ∂ p u ∂ j i ⎝ ⎠− ( ( u i uj = − ρ +μ + ∂xj ∂xi ∂xj ∂xj ∂xi ∂xj (4.254)

where

( ( u i uj ui uj − ρ Tij sgs,t = ρ

(4.255)

This quantity is called the new subgrid-test scale tensor. Finally, consider the resolved turbulent stress: ( ( (j (i u u i uj − ρ Lij = Tij sgs,t − τ ij = ρ ui uj − ρ ui uj + ρ u sgs

( ( ( ( uj − ρ u i uj = ρ ui

Test-filtering volume

(i–1,j)

(4.256)

(i,j+1)

(i,j)

Grid-filtering volume (i+1,j)

Δ (i,j–1) Δ

Figure 4.19 Grid-ﬁltering and test-ﬁltering volumes in a two-dimensional coordinate system (modiﬁed from Wei and Brasseur, 2010).

276

BACKGROUND IN TURBULENT FLOWS

This equation is called Germano identity. The resolved turbulent stresses are representative of the contribution to the subgrid-scale stresses by the scales whose length is intermediate between the grid ﬁlter width and the test ﬁlter width (i.e., the small resolved scales). The Germano identity can be exploited to derive more accurate SGS stress models by determining, for example, the value of the Smagorinsky coefﬁcient most appropriate to the instantaneous state of the ﬂow. The test ﬁlter and grid ﬁlter are associated with a characteristic length scale

and with > . Numerical tests have shown that an optimal value of

a test ﬁlter is twice of that of the grid ﬁlter (i.e., = 2 ). The two subgrid scale tensors (τ ij )sgs and (Tij )sgs,t can be modeled by using the same constant Cd for both ﬁltering levels. In this modeling approach, the tensors (τ ij )sgs and (Tij )sgs,t are replaced by their deviatoric parts so that the trace of these tensors is zero: d 1 (4.257) Tij sgs,t − (Tkk )sgs,t δij = Tij sgs,t = Cd αij 3 d 1 τij sgs − (τkk )sgs δij = τij sgs = Cd βij 3

(4.258)

In these equations, αij and βij represent the deviatoric parts of the subgrid tensors without the constants. Note that both tensors have been modeled by using a common constant Cd . This assumption is equivalent to the assumption of scale invariance on both subgrid ﬂuxes and the ﬁlter. Substituting these two equations in the Germano identity shown in Equation 4.256, we have: d d 1 = Cd αij − Cd βij (4.259) Lij − Lkk δij ≡ Ldij = Tij sgs,t − τ ij sgs 3 Once again, the tensor Lij has been replaced by its deviatoric part Ldij because we are dealing with a zero-trace subgrid viscosity modeling. Equation 4.259 cannot be used directly since the second term on the right-hand side uses the constant CD through a ﬁltered product. In order to simplify this, another approximation is made:

Cd βij = Cd β ij

(4.260)

This approximation is equivalent to the assumption that the constant CD is

By constant over an interval, which is equal to the test ﬁlter cut-off length . substituting Equation 4.260 into Equation 4.259, we have: 1 d Lij − Lkk δij ≡ Lij = Cd αij − β ij (4.261) 3 To determine the constant CD , let us deﬁne a residual εij as: 1 εij = Lij − Lkk δij − Cd αij − β ij 3

(4.262)

LARGE EDDY SIMULATION

277

Minimization of the residual tensor εij would lead to six independent relations, which can provide six different values for the constant Cd . Lilly (1992) proposed the next least-squares method: ∂ Eij Eij = 0 ∂Cd Cd =

mij Ldij mkl mkl

(4.263)

where mij ≡ αij − β ij

(4.264)

This procedure may yield a negative value of Cd or an unbounded value of Cd . These issues are potential causes of instability in the numerical solutions. There are three major techniques to overcome this difﬁculty. The ﬁrst method is by taking a statistical average of the constant CD in the directions of statistical homogeneity (in time or local in space). The averaging procedure can be performed in two nonequivalent ways: by averaging the numerator and denominator of Equation 4.264 separately: mij Ldij Cd = (4.265) mkl mkl

or by averaging the quotient: @ Cd =

mij Ldij mkl mkl

A (4.266)

In these two equations, the operator refers to the ensemble averaging. The second method is the application of arbitrary bounds on Cd , called clipping. The constant Cd should satisfy the next two conditions: ν + νsgs ≥ 0

(4.267)

Cd ≤ Cmax

(4.268)

The ﬁrst condition ensures that the total resolved dissipation ε = νSij Sij − τij Sij remains positive or zero. The second condition establishes an upper bound. In practice, Zang, street, and Koseff (1993) found that C max is of the order of 0.04. This procedure is called the dynamic subgrid scale model because the computed value of coefﬁcient Cd is automatically adapted to the local state of the ﬂow. The dynamic behavior of the square root of the coefﬁcient Cd is shown in Figure 4.20 from a numerical simulation of freely decaying isotropic turbulence. Note that during the initial stage of the calculations, the coefﬁcient is smaller; in the latter stage, the value of this coefﬁcient approaches a constant value. This is due to the fact that during the initial stage, the energy spectrum is not fully developed, and later the numerical solution achieves a fully self-similar state. This initial transient behavior corresponds to the physical evolution of turbulence.

278

BACKGROUND IN TURBULENT FLOWS

√Cd

0.19 0.18 0.17 0.16 0.15 0.14 0.13 0.12 0.11 0.1 0.09 0.08 0.07 0.06 0.05 0.04

0

1

2

3

4

5 6 Time

7

8

9

10

Figure 4.20 Time history of the square root of the dynamic constant in LES of freely decaying isotropic turbulence (modiﬁed from Garnier et al., 1999).

Despite these constraints, the dynamic model is inconsistent. A ﬂow is considered fully resolved at the Kolmogorov length scale η. Therefore, a dynamic procedure is considered consistent if and only if: lim Cd = Cd [η] = 0

→η

(4.269)

If this condition is not satisﬁed, then the dynamic model is called inconsistent. Numerical experiments have shown that the Germano-Lilly procedure is not consistent because it returns a nonzero value of the constant Cd when the grid ﬁlter approaches the Kolmogorov length scale. Work has been conducted for over 20 years to improve the subgrid-scale treatment. More recently, a dynamic procedure for closure of the subgrid-scale model by using a single-level test ﬁlter was proposed by You and Moin (2007). The dynamic procedure of determining the model coefﬁcient is based on the “global equilibrium” between the subgrid-scale dissipation and the viscous dissipation. This concept was earlier utilized by Park and Lee (2006) for dynamic closure of the subgrid-scale model proposed by Vreman (2004). The Vreman model had a ﬁxed coefﬁcient to model the subgrid-scale viscosity; Park and Lee (2006) used a dynamic-coefﬁcient model. All three models are consistent since they predict zero subgrid-scale viscosity at the Kolmogorov scale. Large-eddy simulation of boundary-layer ﬂows has serious deﬁciencies near the surface when a viscous sublayer either does not exist (rough walls) or is not practical to resolve (high Reynolds numbers). Y. Zhou, Brasseur, and Juneja (2001) have shown that the near-surface errors arise from the poor performance of algebraic subﬁlter scale (SFS) models at the ﬁrst several grid

DIRECT NUMERICAL SIMULATION

279

levels, where integral scales are necessarily underresolved and the turbulence is highly anisotropic. In underresolved turbulence, eddy viscosity and similarity SFS models create a spurious feedback loop between predicted resolved-scale (RS) velocity and modeled SFS acceleration, and are unable to capture SFS acceleration and RS–SFS energy ﬂux simultaneously. To break the spurious coupling in a dynamically meaningful manner, Zhou, Brasseur, and Juneja introduced a new modeling strategy in which the grid-resolved subﬁlter velocity is estimated from a separate dynamical equation containing the essential inertial interactions between SFS and RS velocity. This resolved SFS-called the RSFS velocity is then used as a surrogate for the complete SFS velocity in the SFS stress tensor. Zhou and coauthors also tested the RSFS model by comparing the LES of highly underresolved anisotropic buoyancy-generated homogeneous turbulence with a corresponding direct numerical simulation. The new model successfully suppresses the spurious feedback loop between RS velocity and SFS acceleration and greatly improves model predictions of the anisotropic structure of SFS acceleration and resolved velocity ﬁelds. Unlike algebraic models, the RSFS model accurately captures SFS acceleration intensity and RSSFS energy ﬂux, even during the nonequilibrium transient, and properly partitions SFS acceleration between SFS stress divergence and SFS pressure force. In a more recent work, Wei and Brasseur (2010) described the resolution of a problem that has plagued large-eddy simulation of the high Reynolds number turbulent boundary layer since it was ﬁrst applied to this situation: the inability to predict the law-of-the-wall (e.g., the logarithmic layer) correctly with LES. (This is not the case with DNS.) The authors showed the reasons for this anomaly in their study and developed a strategy to resolve the problem. The interesting and important point is that it is not the SFS model for SFS stress alone that is the problem; rather, an integration of model parameters and grid design is at the heart of the problem. The most important ﬁnding of their study is the establishment of three nondimensional critical parameters that must be exceeded to design wall-bounded LES properly. For details, readers are encouraged to refer to the original paper. 4.8

DIRECT NUMERICAL SIMULATION

In direct numerical simulation (DNS), the conservation equations are numerically solved without any turbulence models by resolving whole range of spatial and temporal scales of the turbulence. All the spatial scales of the turbulence must be resolved in the computational mesh, from the smallest dissipative Kolmogorov scale η up to the integral scale l 0 , which are associated with the motions containing most of the kinetic energy. However, even for nonreacting ﬂows, estimates show that the number of grid points required to resolve all the length scales is 9/4 proportional to Rel1 which means that even for a moderate Reynolds number of Rel0 = 104 , the grid points needed for a DNS is N = 109 . This requirement along with the fact that to obtain data for statistical analysis sufﬁcient time evolution of the ﬂow ﬁeld must be simulated makes DNS nearly impossible for even

280

BACKGROUND IN TURBULENT FLOWS

nonreacting moderate Reynolds number ﬂows in the foreseeable future. Currently, most DNS studies are conﬁned to simple ﬂow geometries and to low Rel 0 (O(103 )) ﬂows (Menon, 2004). The extension of DNS to reacting ﬂows is even more problematic. In ﬂows of practical interest, the ﬂame structure can be very thin (e.g., in the premixed ﬂamelet regime the ﬂame thickness ±δL can be orders of magnitude smaller than the smallest turbulent scale, η). Thus, even when the Kolmogorov scale is resolved, thin ﬂame structures cannot be resolved. Two approaches have been attempted to circumvent this limitation: the use of modiﬁed chemistry to artiﬁcially thicken the ﬂame in order to resolve it (Baum et al., 1994; Collin et al., 2000) and the use of the thin ﬂame model where the ﬂame front is tracked without resolving it (Kerstein, Ashurst, and Williams, 1988). Signiﬁcant insight into ﬂame-turbulence interactions have been obtained by using these approaches. However, high-Re ﬂames, as in real devices, remain beyond the scope of DNS. Due to restrictive computational resources (processing time and memory size), DNS has been applied mainly for understanding of basic combustion phenomena in simple geometric conﬁgurations. The results from DNS calculations can supply useful information for turbulence modeling required by other numerical approaches. Direct numerical simulation has been used as a tool that allows numerical experiments to be carried out in situations where experimental studies cannot be conducted easily and accurately. The information from DNS can be used to verify an LES code. When reduced to the low Reynolds number ﬂow in simpler geometries, an accurate LES code should give the same results as DNS code. Most of the early DNS was performed on stationary homogeneous turbulence; however, recently its use has been extended to the simulation of inhomogeneous turbulence.

HOMEWORK PROBLEMS

1.

Use the Reynolds averaging procedure to derive the x-momentum equation for turbulent compressible reacting ﬂuid ﬂow in the rectangular Cartesian coordinate system. Also derive the momentum equation in the Favreaveraged form.

2.

Derive the vorticity equation 4.134, and discuss the differences of the vorticity equation in two-dimensional and three-dimensional ﬂow conditions.

3.

The variation of the mass fraction of a certain species, Y , with respect to time is given by the following sinusoidal function, Y =Y +

1 (Y+ − Y− ) sin (2πωt) 2

HOMEWORK PROBLEMS

281

where ω is frequency. Show that the distribution function F (Y ) and the probability density function P (Y ) must follow the forms given by the following equations: ⎧ 1 if Y > Y+ ⎪ ⎪ , ⎪ ⎨ 1 1 −1 Y − Y F (Y ) = + sin if Y− ≤ Y ≤ Y+ ⎪ Y+ − Y ⎪ ⎪ 2 π ⎩ 0 if Y− > Y ⎧ 0 ⎪ ⎪ ⎨ 1/π √ P (Y ) = ⎪ (Y − Y ) (Y − Y− ) ⎪ ⎩ 0 + 4.

if Y > Y+ if Y− ≤ Y ≤ Y+ if Y− > Y

Use the β –pdf deﬁned as:

P (f, xi ) = aδ f − f + + (1 − a) δ f, f −

2 given below to solve the two coefﬁcients and the deﬁnitions of f( and f+ + 2 a and b in terms of f( and f : 1 ( Pf (f, xi )df f ≡ 0

2 ≡ f+

0

1

2 f − f( P (f, xi ) df

5.

Convert Equation 4.105 into the ﬁnal form of the turbulence kinetic-energy equation 4.108. List all the required assumptions.

6.

Derive the turbulence dissipation rate equation 4.119 from the instantaneous and the mean momentum equation. List all necessary assumptions in order to obtain the exact form of this equation.

7.

Consider a low-speed ﬂow situation, and derive the scalar ﬂux equation for u2 θ by the Reynolds averaging procedure, where θ is a scalar. The density ﬂuctuations can be mostly ignored except that the body force ﬂuctuation due to density ﬂuctuation cannot be ignored; we can use Bi = ρ gi in the i th direction and θ ρ = −αc ρ θ

282

8.

9.

BACKGROUND IN TURBULENT FLOWS Derive the scalar ﬂux equation for u+ i θ by Favre averaging procedure, where θ is a scalar. In this case, consider the chemically reacting ﬂow to be compressible.

What is the general approach of the large eddy simulation (LES)?

10.

Compare the roles of different turbulence scales, especially in terms of the large integral scale versus the smallest length scale.

11.

Describe the use of direct numerical simulation (DNS) for turbulence simulation. What is the limitation of DNS approach? When can DNS approach be adopted?

12.

Describe the turbulence energy cascade concept of Richardson and Kolmogorov’s similarity hypotheses in terms of different ranges and subranges governed by various length scales.

13.

Show that under the assumption of isotropic turbulence, the dissipation rate per unit of mass (ε) deﬁned by ,

∂uj ∂ui ε=ν + ∂xj ∂xi

-

∂uj

∂u1 can be written as ε = 15ν ∂xi ∂x1

2 .

Based on the deﬁnition of the Taylor microscale (T ), it is related to the root mean square velocity ﬂuctuation (urms ) by C BD D ∂u 2 E 1 . T = urms ∂x1 Therefore, the dissipation rate per unit mass can be expressed as ε = 15ν

u2 rms . 2T

5 TURBULENT PREMIXED FLAMES

SYMBOLS

Symbol A A, B c CEBU CEBU Cp Cu D Da Dk DT E (κ) e* F1 , F2 F/O G(xi ) h k Ka Ka Kz LM

Description Elemental areas of a laminar ﬂame Model constants in Equation 5.58 Reaction progress variable Constant in Spalding’s EBU model, Equation 5.55 Constant in Spalding’s EBU model, Equation 5.54 Speciﬁc heat Flame curvature Diameter Damk¨ohler number Thermal diffusivity of k th species Turbulent diffusivity Turbulence energy as a function of wavenumber Modiﬁed ﬂame thickness Functions Fuel-oxidizer ratio (by mass ﬂow rates usually) Level set function Speciﬁc enthalpy Turbulent kinetic energy (TKE) δ 1 dA Karlovitz number ≡ L SL A dt Karlovitz number based upon fuel consumption layer thickness Kovasznay number ≡ τc /τm Markstein length, as deﬁned in Equation 5.130

Fundamentals of Turbulent and Multiphase Combustion Copyright © 2012 John Wiley & Sons, Inc.

Kenneth K. Kuo and Ragini Acharya

Dimension L2 — — — — Q/(MT) 1/L L — L2 /t L2 /t — L — — — Q/M L2 /t2 — — — L

283

284

Symbol lC lf lG lm lo lT m, n Ma

Mwa n NB p

P Pφ PrT q˙rad Rc Red Rel0 rf Sd SL Sn and Sr St Sct sjk Ta Tb

Tk

Tu Tm

TURBULENT PREMIXED FLAMES

Description

Dimension

Obukhov-Corrsin scale, deﬁned by Equation 5.159 Length scale of wrinkles, see Equation 5.30 and Table 5.2 Gibson scale ≡ SL3 /ε, deﬁned by Equation 5.157 Mixing length scale ≡ (εtF3 )1/2 , see Equation 5.158 Integral length scale Average eddy diameter Numbers Markstein number (with respect to unburnt mixture), see Mad based upon laminar ﬂame displacement speed in Equations 5.132 and 2.165; see Mac based upon laminar ﬂame consumption speed shown in Equation 2.166 Molecular weight of a th species Unit normal vector, see Equations 5.123, 5.143 τ SL Bray number ≡ , see Equation 5.108 2ξ u Pressure Production term used in Equation 5.98, deﬁned by Equation 5.101 Probability density function of φ, deﬁned by Equation 5.176 Turbulent Prandtl number ≡ vT /αT Radiation heat ﬂux Radius of curvature Reynolds number based upon diameter d Reynolds number based on lo Flame front radius Displacement speed of thin reaction zone Laminar ﬂame speed Fluctuating quantities in the G-equation, see Equations 5.149–5.151 Average velocity of displacement of combustion wave by turbulent motion Turbulent Schmidt Number ∂uj 1 ∂ui Fluctuating rate of strain ≡ + , see 2 ∂xj ∂xi Equation 5.99 Activation temperature Temperature of the burned gases Turbulent transport of TKE, see Equation 5.100 Temperature of the unburned gases Mean Temperature

L L L L L L — —

M/N — — F/L2 L2 /t3 — — Q/L2 -t L — — L L/t L/t L/t L/t — 1/t K T L3 /t3 T T

285

SYMBOLS

Symbol 2 T

t tF utf u ufg , vfg , and wfg urms v w x xf Y Ze

Description

Dimension

Favre-averaged mean of square of temperature ﬂuctuations (variance) Time Flame crossing time ≡ δL /SL Velocity of thickened ﬂamelet Fluctuating velocity based on Reynolds’ decomposition Reynolds-averaged velocity ﬂuctuations for velocity components generated by ﬂame in x -, y-, and z -directions, respectively Root mean square (rms) ﬂuctuations of turbulent velocity Local ﬂow velocity vector Flame surface velocity vector x-coordinate Location of the ﬂame front, see Equation 5.124 Mass fraction Zel’dovich number [≡Ea (Tb − Tu ) / Ru Tb2 ]

T2

Greek Symbols α Thermal diffusivity α, β, γ Coefﬁcients in Equation 5.67 αT Turbulent thermal diffusivity δc Fuel consumption layer thickness δf Thickness of thin reacting front δL Laminar ﬂame thickness δr Thickness of the reaction zone δT Turbulent ﬂame thickness δt Thickness of turbulent ﬂame brush δtf Thickness of thickened ﬂamelet δ(x) Dirac-delta function of x ε Turbulent kinetic energy dissipation rate ζ Vorticity η Kolmogorov length scale Non-dimensional temperature [≡ (T − Tu )/(Tb − Tu )], see Equation 5.55 θ Angle 1 dA in Equations 5.38 and 5.39 κ Stretch factor ≡ A dt κ Wavenumber Taylor microscale of turbulence T l Thermal conductivity lT Turbulent thermal conductivity ν Kinematic viscosity

t t L/t L/t L/t

L/t L/t L/t L L — — L2 /t — L2 /t L L L L L L L — L2 /t3 1/t L — Radians 1/t 1/L L Q/(LTt) Q/(LTt) L2 /t

286

TURBULENT PREMIXED FLAMES

Symbol νrj νT ξ ρ τ τb τc τf τik τκ or τη τL τm τo τR τr τT τu χc ω˙ T∗ ω˙ θ∗ ω˙ F,rj ω˙ T

Description

Dimension

Stoichiometric coefﬁcients for reactants in j th reactions Turbulent kinematic viscosity Efﬁciency function ≈ 1 Density Heat release factor deﬁned in Equation 5.64a Time period in which a thermocouple is exposed to burned gases, see Equation 5.29 Chemical reaction time Time scale of wrinkles Stress tenser Kolmogorev time scale Reaction time based on Laminar ﬂame speed (= δL /SL ) Characteristics aerodynamic time ≡ T /urms Characteristics time scales of eddy turnover time Characteristics time scales of residence in laminar ﬂamelet Reaction time Reaction time based on turbulent ﬂame speed (= δT /ST ) Time period in which a thermocouple is exposed to unburned gases, see Equation 5.28 ∂c ∂c Mean scalar dissipation rate, χ c ≡ 2D ∂xk ∂xk Mean rate of energy production Mean reaction rate given by Spalding’s EBU model, Eq 5.57 Average fuel consumption rate, Equation 5.58 Source term in temperature equation 5.142, derived from rate of energy production

— or N L2 /t — M/L3 — t

Subscripts amb Ambient avg Average b Burned F Fuel f Flow or ﬂame fg Flame-generated L Laminar max Maximum O Oxidizer P Products

t t F/L2 t t t t t t t t 1/t Q/L3 -t 1/L3 -t M/L3 -t MT/L3 -t

SYMBOLS

Symbol rj T tf u

Description

287

Dimension

j th reaction Turbulent Thickened ﬂamelet Unburned

Superscripts Reynolds-averaged ﬂuctuations Favre-averaged ﬂuctuations — Reynolds-averaged mean ∼ Favre-averaged mean

Acronyms CGD Counter gradient diffusion EBU Eddy breakup GD Gradient diffusion

Premixed ﬂames in laminar ﬂows were discussed in Chapter 2. In this chapter, we discuss the physics of premixed ﬂame propagation in turbulent ﬂows, ﬂame structure, various modeling approaches, and their applications and deﬁciencies. Unlike laminar ﬂames, turbulent ﬂames often are accompanied by noise and rapid ﬂuctuations of the ﬂame envelope. For a laminar premixed ﬂame, it is possible to deﬁne a ﬂame speed (SL ) that, within reasonable limits, is independent of the experimental apparatus. It would be equally desirable to deﬁne a propagation velocity for turbulent ﬂames that would be independent of the experimental apparatus and depend only on the fuel-air ratio and some transport properties, such as l, ν, and D corresponding to the laminar case. However, this is not possible, because transport properties of turbulent ﬂames are functions of the ﬂow rather than the ﬂuid. At some stoichiometric ratios, the turbulent thermal diffusivity1 can be several orders of magnitude higher than the molecular thermal diffusivity (i.e., lt l, νt ν, Dt D). Thus, the theoretical concepts for turbulent ﬂames are not as well deﬁned as they are for laminar ﬂames. Unlike laminar ﬂames, the turbulent ﬂame surfaces are usually very complex, and it is difﬁcult to locate the instantaneous ﬂame surfaces. Due to the enhanced transport properties, the turbulent ﬂame speed (ST ) is much greater than the laminar ﬂame speed (SL ) (i.e., ST SL ). Unlike laminar premixed ﬂames, turbulent premixed ﬂames are usually quite thick. The turbulent ﬂame speed ST increases as the turbulence level increases (i.e., root-mean-square [rms] turbulence intensity increases and/or turbulent integral length scale increases). 1

Conventionally, two terms are used to separate turbulent transport properties from their laminar counterparts: “turbulent,” referring to the apparent transport property based on the scales of turbulence, and “effective,” referring to the sum of the turbulent and molecular transport properties. The transport properties include viscosity, thermal diffusivity, and mass diffusivity.

288

TURBULENT PREMIXED FLAMES

The problem of the propagation of premixed ﬂames in a turbulent ﬂow can be divided into two major sub-problems: (1) the effects of the turbulence on ﬂame speed, ﬂame thickness, and ﬂame structure; and (2) the effects of the ﬂame on turbulence. Turbulent ﬂames have been studied for more than 70 years. Most research into premixed turbulent combustion has focused on the ﬁrst subproblem. Research by several schools of combustion has led to a few semi-empirical theoretical analyses or models on the mechanism of turbulent premixed combustion. The pioneering work on this topic was performed by Damk¨ohler (1940), who proposed that the turbulent reaction zone is formed of wrinkled or broken laminar ﬂames pulsating as a result of the inherent nature of ﬂow turbulence. Turbulence augments transport rates within the reaction zone when the scale of eddies present in the ﬂow is small compared with the laminar ﬂame thickness (δL ); it increases the ﬂame surface area (Af ) by wrinkling when the turbulence scale is large. Damk¨ohler considers the large scales of the turbulent eddies to wrinkle the ﬂame front, thereby increasing its burning surface area and the rate of reactant consumption. If the smallest scales of the turbulence are larger than the ﬂame-front thickness, it is hypothesized that the instantaneous ﬂame front will propagate locally at the laminar burning velocity. The turbulent ﬂame will then propagate at a speed equal to the laminar burning velocity multiplied by the ratio of the wrinkled instantaneous ﬂame surface area to its projected area. This model is known as the wrinkled ﬂame model. Similar to the relationship given in Equation 2.77 between laminar ﬂame speed and thermal diffusivity in Chapter 2, small-scale turbulence (i.e., turbulent eddies smaller than the laminar ﬂame-front thickness) is hypothesized to increase the propagation speed of the instantaneous ﬂame front by a factor equal to the ratio of the apparent turbulent diffusivity of the small scales to the molecular diffusivity raised to the one-half √ power (i.e., ST ∝ αT RR). Damk¨ohler’s paradigm is considered to embody the hypothesis that the turbulent ﬂame speed and ﬂame structure can be related to correlations and classiﬁcations dependent on a small number of parameters (e.g., Reynolds number). The ﬂame speed is (a) independent of the Reynolds number when Red ≤ 2,300, (b) proportional to the square root of the Reynolds number when 2,300 ≤ Red ≤ 6,000, and (c) proportional to the Reynolds number when Red ≥ 6,000.2 Obviously, regions (b) and (c) are inﬂuenced by turbulent eddy sizes, and hence the measured ﬂame speeds depend on geometry and ﬂow conditions. Summerﬁeld et al. (1955) suggested another physical model in light of their own experimental ﬁndings, which regards the turbulent ﬂame as a region of distributed reactions occurring in sequence along the ﬂow lines. This concept is different from Damk¨ohler’s wrinkled ﬂame model. Either of these earlier models seems to cover a wide range of ﬂame conﬁgurations under different ﬂow velocities and Reynolds number. Either model— wrinkled laminar ﬂame and disintegration of the ﬂame front into the region of distributed reactions—is possible depending, on the magnitude of the Reynolds 2

Note that these values are device dependent. For a tube, the geometric dimensions of the tube are ﬁxed and the real dependence is on the velocity and not Red .

PHYSICAL INTERPRETATION

289

number and other dimensionless parameters (discussed later in this chapter). Further, a “mixed” model may be possible in the intermediate region. The possibility of the simultaneous existence of two models in the intermediate region is also supported by Povinelli and Fuhs (1960), who have shown that small- and large-scale eddies can occur simultaneously in the ﬂow ﬁeld and are responsible for distributed and wrinkled laminar ﬂame, respectively. Kovasznay (1970) proposed a criterion by which the model of turbulent combustion might be assessed. This criterion is based on the crossover from the characteristic chemical reaction time to the characteristic aerodynamic mixing time by turbulent eddies. The effect of turbulence on the ﬂame speed has been reviewed in many publications including in the two edited volumes of Turbulent Reacting Flows by Libby and Williams (1994) as well as in the textbooks by Lewis and Von Elbe (1961); Shchetinkov (1965); Williams (1985, 2000); Kuo (1986); Chomiak (1990); Kuznetsov and Sabel’nikov (1990); Warnatz, Mass, and Dibble (2006); Poinsot and Veynante (2005); Peters (2000); Glassman and Yetter (2008); and others; in the review papers by Chomiak (1979); Borghi (1985, 1988); Clavin (1985, 1994, 2000); Libby (1985); Pope (1985, 1994, 1997); Williams (1985d, 2000); Peters (1986); Daneshyar and Hill (1987); Law (1988); G¨ulder (1990); Bradley (1992, 2002); Ashurst (1994); Heywood (1994); Bray (1995, 1996); Poinsot Candel, and Trouv´e (1995); Poinsot (1996); Candel et al. (1999); Klimenko and Bilger (1999); Bilger (2000); Law and Sung (2000); Renard et al. (2000); Kerstein (2002); Lipatnikov and Chomiak (2002, 2005); Veynante and Vervisch (2002); Dinkelacker (2003); Bilger et al. (2005); Kadowaki and Hasegawa (2005); Janicka and Sadiki (2005); Westbrook et al. (2005); Pitsch (2006); Driscoll (2008); and in many other papers of a narrower scope.

5.1

PHYSICAL INTERPRETATION

The effects of turbulence on ﬂame propagation can be understood in terms of effects of turbulent intensity and scales—that is, the Kolmogorov scale (η) and the integral length scale (0 ) or geometric length scale, L]. Ballal and Lefebvre (1976) have examined the effect of turbulence on ﬂame structure and propagation speed. Whether the turbulence results in wrinkled ﬂame or distributed reaction zones depends on the turbulent intensity and scales. As shown in Figure 5.1, proceeding from left to right, the ﬁrst diagram represents Region 1, where turbulence intensity is low. As both η and 0 are larger than δL , the turbulent eddies only cause ﬂame wrinkling. The top photograph in Figure 5.2 is typical of those obtained under these conditions. The ﬂame has a relatively smooth appearance, with the surface comprising an agglomeration of round swellings that gradually grow in size as the ﬂame expands downstream. The second diagram in Figure 5.1 corresponds to the maximum wrinkling of the ﬂame, which occurs when η = δL . The third diagram shows that when η < δL , part of the turbulence energy (E expressed as a function of the wavenumber, κ) is contained in eddies within

290

TURBULENT PREMIXED FLAMES Region 1

Region 2 E(κ )

E(κ )

0

fresh mixture

dL h

l0

0 h = dL

l0

Region 3 E(κ )

0

h

dL l0

E(κ )

0 h

dL l0

burned gases

turbulence intensity increasing

Figure 5.1 Diagrams illustrating the effect of turbulence energy E(κ) distribution on ﬂame shape and ﬂame structure (modiﬁed from Ballal and Lefebvre, 1976).

(a)

(b)

Figure 5.2 Stoichiometric propane air ﬂames under conditions of low and high turbulence (a) u = 31 cm/s (low turbulence intensity) and (b) u = 305 cm/s (high turbulence intensity) (modiﬁed from Ballal and Lefevbre, 1976).

SOME EARLY STUDIES IN CORRELATION DEVELOPMENT

291

the ﬂame, and a smaller amount is available for ﬂame wrinkling. Under these conditions (Region 2), the wrinkling is less pronounced, but the disruption created by the entrainment and combustion of numerous small pockets of fresh mixtures within a thickened reaction zone is manifested as a roughening of the ﬂame surface. At the higher levels of intensity encountered in Region 3, the wrinkling effect is further diminished, and almost all of the available energy is embodied in the multitudinous small eddies. An increase in turbulence intensity is accompanied by a decrease in Kolmogorov length scale η (for a ﬁxed integral length scale 0 ). Thus, at very high levels of turbulence, all the small eddies are too small to produce any noticeable wrinkling of the ﬂame surface. Under these conditions, the combustion zone may be regarded as a fairly thick matrix of burned gases interspersed with eddies of unburned mixtures (see Figure 5.1). It is clear from this ﬁgure that the concept of a continuous, coherent ﬂame surface is no longer realistic and that combustion is sustained almost solely by reactions taking place at the interfaces formed between the combustion products and the eddies of fresh mixtures. As shown in Figure 5.1, it is also no longer feasible to regard combustion as being conﬁned within fairly well-deﬁned boundaries at or near a contiguous ﬂame surface. Rather, combustion is seen as a process that occurs in depth as eddies of unburned mixture are gradually and sometimes violently consumed during their passage through a thickened reaction zone. Within each eddy, the burning rate is enhanced by the ﬂow of heat and active species (e.g., radical species) from the enveloping ﬂame front. In some instances, depending on the properties of the mixture and the turbulence scale, the acceleration of chemical reactions within an individual eddy may proceed to such an extent that eventually combustion occurs almost simultaneously throughout its volume, ahead of the advancing ﬂame. In this manner, pressure pulsations are produced, which lacerate and rupture the ﬂame surface (as shown in the lower photograph of Figure 5.2). These pulsations give rise to the noise that characterizes high intensity turbulent ﬂames and also provide a rational explanation for the second subproblem of turbulent combustion known as ﬂame-generated turbulence. 5.2

SOME EARLY STUDIES IN CORRELATION DEVELOPMENT

Of major interest is the prediction of turbulent ﬂame speed, which determines, for example, ﬂame travel times in spark-ignition (SI) engines and the length of ﬂames on Bunsen-type burners. The Damk¨ohler paradigm suggests that, in problems of practical interest, the degree of ﬂame wrinkling would be the dominant effect and that it should be largely a kinematic effect dependent on the ratio of a turbulent velocity scale to the laminar burning velocity. The paradigm became focused on an ideal turbulent ﬂame that was one-dimensional in the mean and was statistically stationary in a reference frame moving with the turbulent ﬂame “brush.” As in laminar ﬂows, it was assumed that angled ﬂames could be treated by considering propagation in the direction normal to the mean ﬂame front (see Figure 5.3).

292

TURBULENT PREMIXED FLAMES

Fresh gas

Burned gas

Arbitary wavy flame front SL

Flame front after Δ t time

Figure 5.3 An exaggerated view of turbulent ﬂame front (modiﬁed from Karlovitz, Denniston, and Wells, 1951).

5.2.1

Damk¨ohler’s Analysis (1940)

Although in very early days Mallard and Le Chatelier (1883) recognized that turbulence affects burning velocities, systematic investigations of turbulent ﬂames began only in 1940 with Damk¨ohler’s classical theoretical and experimental study. Figure 5.4a shows the Bunsen-burner measurements of the ﬂame speed at various Reynolds numbers by Damk¨ohler. He found that the ﬂame speed is (a) independent of the Reynolds number when Red ≤ 2300, (b) proportional to the square root of the Reynolds number when 2300 ≤ Red ≤ 6000, and (c) proportional to the Reynolds number when Red ≥ 6000. Obviously, items (b) and (c) are inﬂuenced by turbulence; hence the measured ﬂame speeds depend on geometry and ﬂow. Williams and Bollinger (1948) carried out experiments to test Damk¨ohler’s proposed equations with different tube diameters and different hydrocarbon fuels (C3 H8 , C2 H4 , and C2 H2 ). Their experimental results are shown in Figure 5.4b. From visual observation, the general shape of the curve from their measured data shows higher curvatures near the lower Reynolds number range. As the Reynolds number increases, the curvature is lower. It also can be observed that besides the Reynolds number, the burner diameter and the fuel type have signiﬁcant inﬂuence on the measured turbulent burning speed (ST ).

293

SOME EARLY STUDIES IN CORRELATION DEVELOPMENT

C2H2

Turbulent Flame Speed, ST (cm/s)

300

5 Smaller integral length scale l0

ST /u0 or ST /SL

4

Larger integral length scale l0

3

250

200

SL of C2H2 (147 cm/s) SL of C2H2 (70 cm/s)

150

C2H4

100 C3H8

2 50 1 SL of C3H8 (45 cm/s) 0

0

4

8

12

Red × 10−3 (a)

16

20

0

0

10,000

20,000

Burner diameter (mm) 6.35 9.53 15.87 28.58

30,000

40,000

Reynolds number of Pipe Flow, Red (b)

Figure 5.4 (a) Effect of Reynolds number on turbulent ﬂame speed (modiﬁed from Damk¨ohler, 1940) and (b) variation of turbulent burning velocity with Reynolds number (modiﬁed from Bollinger and Williams, 1948).

In the range 2,300 ≤ Red ≤ 6,000, the integral length scale and mixing length are much smaller than the ﬂame-front thickness. In this Red range, the intermediate turbulent scales enhance the intensity of transport processes within the combustion wave. Under these circumstances, transport of heat and species is due to the turbulent diffusivity but not the molecular diffusivity. When Red ≥ 6,000, the integral scale of turbulence is much larger than the laminar ﬂame thickness. These larger eddies do not increase the diffusivities as the small eddies do, but they distort the otherwise “laminar” ﬂame front, as shown in Figure 5.3. The inﬂuence of these folds in the ﬂame front increases the ﬂame surface area per unit cross section of the tube of the Bunsen burner. As a consequence, the apparent ﬂame speed is increased without any change in the instantaneous local ﬂame structure itself. Damk¨ohler (1940) state that for the turbulent ﬂows with Reynolds number between 2,300 to 6,000, any incrense in the Reynolds number simply increases the transport properties in the wave. He investigated these changes as a function of Reynolds number in the following manner. For a laminar ﬂame, the ﬂame speed SL is proportional to the square root of the product between thermal

294

TURBULENT PREMIXED FLAMES

diffusivity (α) and reaction rate (RR) as given by Equation 5.1. √ SL ∝ αRR

(5.1)

It is reasonable to expect the speed of the turbulent ﬂame ST to be ST ∝ αT RR

(5.2)

where αT is the turbulent thermal diffusivity. Thus αT ST ≈ SL α

(5.3)

If both the turbulent Prandtl number (PrT ≡ νT /αT ) and the Prandtl number based on molecular transport properties are approximately equal to 1, then Equation 5.3 becomes νT ST ≈ (5.4) SL ν For pipe ﬂow, the ratio of turbulent kinematic viscosity (also known as eddy viscosity, νT ) and molecular kinematic viscosity (ν), is related to the Reynolds number as: νT /ν ≈ 0.01 Red . Therefore, ST ≈ 0.1 Red SL

(5.5)

This equation indeed predicts the trend of Damk¨ohler’s small-scale (i.e., small burner diameter) burning speed measurement. However, Equation 5.2 has a serious drawback: as νT → 0, ST → 0 instead of approaching SL , which can be easily ﬁxed by replacing αT with αT + α. In the case of larger burner diameter and low-intensity turbulence, the ﬂame will be wrinkled, but the molecular transport properties will remain the same; therefore, the laminar-ﬂame speed will remain constant. Since for constant SL , the ﬂame area is proportional to the ﬂow velocity, it is expected that the increase in ﬂame surface area due to turbulence is proportional to urms . In addition, νT is proportional to the product of intensity and mixing length (which can be considered constant), urms is proportional to νT . Also νT ∝ Red for constant ν, so we have ST ∝ area ∝ (magnitude of ﬂuctuation) ∝ νT ∝ Red (5.6) SL It is not surprising, therefore, that certain experimental results appear to correlate as: (5.7) ST = A Red + B where A and B are constants. This relationship describes Damk¨ohler’s burning speeds quite satisfactorily. In summary, Damk¨ohler’s approach states that for

295

SOME EARLY STUDIES IN CORRELATION DEVELOPMENT

given turbulence intensity, sufﬁciently increased ﬂow rate merely wrinkles a premixed laminar ﬂame without signiﬁcantly modifying its internal structure. Of course, this pioneering work does not cover the full story of turbulent combustion. 5.2.2

Shchelkin’s Analysis (1943)

Shchelkin (1943) considered the importance of the times (τ ) associated with the turbulence. For laminar ﬂames,

α l l SL ∝ ∝ ∝ (5.8) τr ρCp τr τr For turbulent ﬂames, he proposed a similar relationship,

l + lT l lT lT = 1+ = SL 1 + ST ∝ τr τr l l

(5.9)

where τr is the reaction time. Shchelkin also considered large-scale, low-intensity turbulence. He assumed that surfaces are distorted into cones whose base area was proportional to the square of the average eddy diameter lT , as shown in Figure 5.5. The height of the cone is proportional to urms and to the time during which an element of the combustion wave is associated with an eddy motion in the direction normal to the wave. This time, then, can be taken as equal to lT /SL . Shchelkin then proposed that the ratio of ST to SL equals the ratio of the average cone area to the average cone base. From the geometry, AC = AB where

4h2 1+ 2 lT

1/2 (5.10)

AC = area of the cone AB = area of the base h = cone height, which can be represented by h = urms t = urms

lT SL

(5.11)

AC lT AB

Figure 5.5

h=

u′rms lT SL

A conical ﬂame front formed from distortion of a ﬂat ﬂame front.

296

TURBULENT PREMIXED FLAMES

Therefore,

ST = SL 1 +

2urms SL

2 (5.12)

For large values of urms /SL , Equation 5.12 reduces to the one developed by Damk¨ohler: ST ∝ urms . 5.2.3

Karlovitz, Denniston, and Wells’s Analysis (1951)

In order to further advance the understanding of turbulent burning velocities, Karlovitz, Denniston, and Wells (1951) conducted more experimental studies. They compared the predictions of the theory with turbulent burning velocity measurements and discovered that the turbulent ﬂame itself generates additional turbulence. They proposed a set of equations to explain the production of turbulence by the turbulent ﬂame and to calculate of the intensity of ﬂame generated turbulence. These authors suggested that to obtain the total turbulent burning velocity ST , the normal burning velocity SL has to be added to the velocity St (deﬁned as the average velocity of the displacement of the combustion wave by the turbulent motion), (5.13) ST = SL + St where the velocity St is given by three different equations, depending on the turbulence level ⎧ 1/2 ⎪ for very strong turbulence 2SL urms ⎪ ⎪ ⎪ ⎪ ⎨2S u 1/2 1 − S /u L rms L rms (5.14) St = 1/2 ⎪ × 1 − exp −u /SL for intermediate turbulence ⎪ rms ⎪ ⎪ ⎪ ⎩ urms for very weak turbulence The ﬂame-generated (fg) turbulent intensity can be calculated as u2 fg

+

2 υfg

+

2 wfg

=

ρu −1 ρb

2 SL2

(5.15)

In case of isotropic turbulence, the ﬂame-generated turbulent intensity can be calculated as 2 1 ρu 2 2 2 ufg = υfg = wfg = − 1 SL2 (5.16) 3 ρb By dividing Equation 5.13 by SL , considering the case of a low level of turbulence intensity, and replacing urms /SL by an empirical constant K1 × ST /SL , we have αT ST = 1 + K1 (5.17) SL α

SOME EARLY STUDIES IN CORRELATION DEVELOPMENT

297

Karlovitz’s work was modiﬁed and expanded by other researchers. For butane-air ﬂames, Wohl et al. (1953) found ST β urms = 1 + K2 U + 0.01 (5.18) SL U where U is the approach velocity in the burner tube, and K2 and β depend only on the fuel-air ratio.

5.2.4

Summerﬁeld’s Analysis (1955)

Summerﬁeld et al. (1955) considered the time scale associated with turbulent ﬂames to be different from that for laminar ﬂames. They suggested evaluating the ratio of ST to SL as ν /τ ST = T T (5.19) SL ν/τL The new feature is that the reaction time is affected by the turbulence. The reaction time can be represented by τL =

δL δT ,τ = SL T ST

(5.20)

where δ is the ﬂame thickness. This approach considers the reaction zone to be extended in the turbulent case. Conceptually, it considers SL to be a measure of the chemical kinetics alone. With the expressions for τ , it follows that: ST2 νT ST /δT = 2 vSL /δL SL

(5.21)

After cross-multiplying and rearranging, we have SL δL ST δT = νT ν

(5.22)

Experimental data on laminar ﬂames give

Thus

SL δL ≈ O (1) ν

(5.23)

SL δL S T δT = ≈ O (1) νT ν

(5.24)

so that ST may be determined from the turbulent ﬂame thickness and eddy viscosity. This relationship was developed for small-scale turbulence. It is also known as the distributed reaction model, as mentioned in the introduction.

298

5.2.5

TURBULENT PREMIXED FLAMES

Kovasznay’s Characteristic Time Approach (1956)

The assumption that the turbulent ﬂame consists of a continuous laminar ﬂame is explicit in many of the earlier qualitative studies of turbulent combustion. Following the work of Damk¨ohler and Shchelkin, many other researchers utilized the same concept of the wrinkled ﬂame, considered the breakup of the ﬂame surface more extensively and thus derived more complicated formulae. However, information obtained on many turbulent ﬂames, particularly those in which smallscale, high-intensity turbulence was evident, showed that laminar ﬂamelets do not exist. The failure of this wrinkled-ﬂame concept for certain turbulent premixed ﬂames led others to consider a distributed reaction zone. Somewhat later, more precise experiments on how turbulence affects ﬂame structure led to the proposal that a series of possible mechanisms describe the effect of turbulence on the combustion zone. One of these postulations, given by John and Mayer (1957), says that the mechanisms can be interpreted in terms of a characteristic chemical time τc and a characteristic aerodynamic time τm . The chemical reaction time is deﬁned as: τc =

δL SL

(5.25)

where δL is the thickness of the laminar ﬂame. τc increases as either the pressure or the chemical activity, or both, is lowered. The aerodynamic time τm is deﬁned as: T τm = (5.26) urms where T is the Taylor microscale of turbulence discussed in Chapter 4. A dimensionless number called the Kovasznay number (Glassman, 1977) can be formed from both these times: τc (5.27) τm Weak turbulence urms small and τm τc merely wrinkles the ﬂame front. τm , in this case, can be regarded as being inversely proportional to the velocity gradient characteristic of the ﬂow approaching the ﬂame front; and τc is inversely proportional to the laminar ﬂame speed. Stronger turbulence (τm ≈ τc ) disrupts the laminar ﬂame front, with τm and τc thereby losing their signiﬁcance as reciprocal velocity gradients. In this case, τm is the mean lifetime of an eddy and τc the time for a chemical reaction in these combustible pockets. Still stronger turbulence (τm τc ) shows its effects by locally diluting and preheating the initial centers of deﬂagration. In the limit when mixing time is negligible in comparison with chemical reaction time, strong turbulence results in homogeneous reacting mixtures. In this context, this limit is sometimes called the perfectly stirred reactor. Kz ≡

SOME EARLY STUDIES IN CORRELATION DEVELOPMENT

299

In addition, many research groups have developed turbulent ﬂame speed correlations in terms of laminar ﬂame speed and turbulence intensity (urms ), occasionally with both urms and Reynolds number ReL . Some of these correlations are summarized in Table 5.1.

5.2.6

Limitations of the Preceding Approaches

Based on a review paper by Bilger, Pope, Bray, and Driscoll (2005), turbulent ﬂame speed correlations have been developed from experimental data over the past 60 years, and this work is continuing. Extensions of the turbulent paradigm discussed earlier include dependence on other parameters, such as the turbulence length scale and Reynolds number, laminar ﬂame thickness, volumetric expansion ratio, and effects of the nonunity Lewis number. Many authors have claimed success. These correlations have limited applications. Results seem to be particularly sensitive to the ﬂow conﬁguration. There have been efforts to overcome the difﬁculties speciﬁc to turbulent premixed ﬂames, which include the determination of “canonical” ﬂame geometries to use for model validation and an unambiguous deﬁnition of turbulent burning velocity. Researchers have agreed that premixed turbulent ﬂames should be separated into the oblique category (i.e., rod-stabilized V-ﬂames), the envelope category (i.e., Bunsen ﬂames that form an envelope about all of the reactants) and the “unattached” category (which includes low-swirl ﬂat ﬂames and counterﬂow ﬂames that do not attach to burner hardware). • Envelope ﬂames are generated by anchoring the ﬂames at the rim of the burner. The turbulent ﬂame brush burns toward the center and merges to form an envelope over the premixed gases. The premixed gases cannot escape without burning due to moderate ﬂow and turbulence levels. Therefore, open ﬂame tip and local ﬂame extinction are unlikely to occur. Most studies have used pilot ﬂames to extend the test matrix because the burner rim is not a very effective ﬂame stabilizer (see Figure 5.6a, b). • Oblique ﬂames are generated by a ﬂame holder at the center of the burner. The turbulent ﬂame brush interacts with the incident turbulence and grows thicker downstream of the stabilizer. The size of the ﬂame holder is kept to minimum so as to reduce its inﬂuence on the developing turbulent ﬂame brush. Larger stabilizers generally are used for investigating the contributions of the stabilizer wake (i.e., shear turbulence) to ﬂame structure and blow-off limits (see Figure 5.6c, d). • Unattached ﬂames do not require ﬂame stabilizers. They can be sustained in divergent ﬂows by virtue of the propagating nature of premixed ﬂames. The turbulent ﬂame brush is locally normal to the approach ﬂow and free to respond to the incident turbulence without being restrained or pinned down at the ﬂame attachment point (see Figure 5.6e, f).

300

TURBULENT PREMIXED FLAMES

TABLE 5.1. Summary of Turbulent Flame Speed Correlations Source: Modiﬁed from Bilger, et al. (2005).

Correlation urms

ST =1+ SL SL ST 1/2 ≈ ReL SL

2 1/2 ST 2urms = 1+ SL SL ST ≈ Re0.24 L SL ST urms = 2.1 SL SL ST =1+ SL

urms SL

Major Assumptions

Reference

Continuous wrinkled laminar ﬂame δL 1

Damk¨ohler (1940)

For small-scale, high-intensity turbulence

Damk¨ohler (1940)

Continuous wrinkled laminar ﬂame δL 1 urms < SL

Shchelkin (1947)

ReL > 100 and SL /urms → 0

Abdel-Gayed and Bradley (1977)

For urms /SL > 3.9 and conﬁned ﬂames

Libby, Bray, and Moss (1979)

Formulation of kinematic aspects of ﬂame wrinkling and subsequent inﬂuences on turbulent ﬂame speed

Clavin and Williams (1979)

Isotropic turbulence

Clavin and Williams (1979)

Simpliﬁed model for turbulence characterized by a single length scale and a single velocity scale

Klimov (1983)

Fractal ﬂame surface with outer and inner cutoffs

Gouldin (1987)

Fractal ﬂame surface with outer cutoff and inner cutoff Gibson scale

Peters (1988)

Formulation through dynamic renormalization group method

Yakhot (1988)

Pair-exchange model

Kerstein (1988)

2

1/2 ST u2 = 1+ SL SL2 0.7 ST urms = 3.5 SL SL ST 1/4 = ReL SL ST urms = SL SL 2 urms /SL ST = exp SL (ST /SL )2 0.5 ST urms = Re1.5 L SL SL

SOME EARLY STUDIES IN CORRELATION DEVELOPMENT

301

TABLE 5.1. (continued ) Correlation ST =1+ SL

u 5.3 rms SL0.5

1/2 ST urms 1/4 = 1 + 0.6 ReL SL SL ST urms =C SL SL ST SL 3/4 = 6.4 urms urms 2 1/2 ST urms ≈ 1+ SL SL ST urms = 1.26 + 0.38 SL SL

ST =1+ SL

urms SL

4/3

ST urms = 2.53 SL SL

Major Assumptions

Reference

Experimental curve ﬁt

Liu and Lenze (1989)

Isotropic turbulence, urms → 0 ⇒ ST /SL → 1

G¨ulder (1990)

C = 2.42 for zero heat release; C = 7.25 for large heat release

Bray (1990)

Experimental curve ﬁt

G¨ulder (1990)

Interpretation of physical picture of Clavin and Williams

Kerstein and Ashurst (1992)

In the framework of a nonlinear model, which incorporates the Landau-Darrieus instability mechanism

Cambray and Joulin (1994)

Mean passage rate of a propagating interface, subject to random advection or random variation of local propagation speed, is investigated analytically and computationally

Kerstein and Ashurst (1994)

Experimental curve ﬁt

Bedat and Cheng (1995)

Each category is associated with different boundary conditions and has a different ﬂame-wrinkling process. For example, extensive merging of ﬂamelets occurs near the tip of a Bunsen ﬂame, but this degree of merging does not occur in counterﬂow geometry. Therefore, it is recommended that any scaling relation that is determined (either from computations or experiments) for one category should be identiﬁed only with that category and not applied to all categories. None of these ﬂow conﬁgurations meets the ideal paradigm of onedimensionality in the mean. Some also are not statistically stationary in a framework moving with the turbulent ﬂame brush. Insight into the limitations of application of the ideal paradigm in turbulent ﬂows can be gained from considering its mother paradigm—that for premixed laminar ﬂames. Ideal behavior there is not exhibited where a characteristic length scale of the

302

TURBULENT PREMIXED FLAMES Axisymmetric Envelope Flame Also known as conical flame.

(a) Plane-Symmetric Oblique Flame A V-flame stabilized by a small rod is most common rendition of a plane-symmetric laboratory oblique flame. (c) Unattached Flames in Impinging Flows The divergent flow generated by flow impingement on a stagnation plate allows the flame to position itself at a short distance upstream of the stagnation point. (e)

Plane-Symmetric Envelope Flame Flame is generated in a rectangular shaped burner (or slot burner) with flame brushes originating at two opposite edges. To preserve the “envelope” features, the two remaining sides of the burner need to be confined. (b) Axisymmetric Oblique Flame A small bluff body or pilot flame generates an axisymmetric oblique flame that shapes like an inverted cone. (d) Unattached Flame in SwirlGenerated Divergent Flows Low swirl produces a divergent flow with the swirling motion confined to the flow periphery. In the center region, the turbulent flame brush is swirl free. (f)

Figure 5.6 Photographs of envelope, oblique, and unattached ﬂames (after R. K. Cheng, 2004).

ﬂow—ﬂame-ball diameter or distance from a wall—is of the same order of magnitude as the laminar ﬂame thickness or the strain rate in the ﬂow is comparable with the inverse of the convective time scale of passage through the laminar ﬂame. Furthermore, initially planar ﬂames often become highly distorted due to thermo-diffusive instabilities. Accordingly, we should expect that turbulent ﬂame speeds should strongly depend on similar considerations. Flames stabilized at ﬁxed locations (such as Bunsen-burner types, V-ﬂames, etc.) should be sensitive to convective time scales corresponding to passage through a normal turbulent ﬂame brush. Low-swirl ﬂames, counterﬂow ﬂames, and stagnation ﬂames have mean strain rates that correspond to the mean convective time through the turbulent ﬂame brush. There has been much evidence that the external pressure gradients which occur in such ﬂows affect the turbulent burning velocity to a signiﬁcant extent. This is probably because the conditional momentum balance for burned and unburned gases is crucial in determining the conditional velocities, which are closely related to the turbulent burning velocity. Turbulent ﬂame brushes corresponding to different ﬂow conﬁgurations are shown in Figure 5.7. In each case the turbulent ﬂame brush is deﬁned as a relatively thick zone consisting of thin reaction zones that separate unburned reactants from burned products and that are transported by the turbulent eddies randomly. The thickness of turbulent ﬂame brush is designated by δt , and the thickness of

SOME EARLY STUDIES IN CORRELATION DEVELOPMENT

Flamelet

Flame brush

U

x

Flame brush

Flamelet

U

U

U

303

x r

y

(a)

(b)

Wall x

x Flame brush

Flamelet

Flame brush

Flamelet

d

w u

u v

r

v r

Jet exit

Jet exit (c)

(d)

Figure 5.7 Flame brush and ﬂamelet thicknesses in various types of turbulent premixed ﬂames: (a) Bunsen or conical ﬂame, (b) V-shaped ﬂame, (c) impinging jet ﬂame, and (d) swirl-stabilized ﬂame (after Lipatnikov and Chomiak, 2010).

thin reacting front is designated by δf . The thickness of the turbulent brush is controlled by the random motion of the front around its mean position under the inﬂuence of turbulent eddies. The ﬂame front often is assumed to have the same local structure as a stretched laminar ﬂame. Sometimes the ﬂame front is called a ﬂamelet. Generally, “ﬂame front” and “ﬂamelet” are two words that have been used to describe the burning zone in turbulent ﬂames. “Flame front” is a more general term; “ﬂamelet” is speciﬁcally used in the context of stretched ﬂames. To deﬁne the turbulent burning velocity properly, Cheng and Shepherd (1991) propose using the consumption speed rather than the displacement speed, which has been reported in the past. Consumption speeds are determined by deﬁning a control volume—which could be a local (Cheng and Shepherd, 1991) or global (Filatyev, Driscoll, and Carter, 2005)—and computing or measuring the mass ﬂow rate of reactant that crosses all boundaries, including the mass ﬂow rate associated with density–velocity correlations. “Displacement speed” is deﬁned as the velocity of the wave (which is the velocity of an iso-surface of the mean

304

TURBULENT PREMIXED FLAMES

reactedness) with respect to the gas velocity ahead of the wave, in the direction normal to the wave. It is difﬁcult to deﬁne exactly where the gas velocity ahead of the wave should be determined, and this leads to large uncertainties. Thus, the use of a displacement speed is not recommended.

5.3 CHARACTERISTIC SCALE OF WRINKLES IN TURBULENT PREMIXED FLAMES

The effect of the turbulence scale of unburned gas on the scale of wrinkles of premixed turbulent ﬂames was directly observed experimentally by Yoshida and Tsuji (1982) through Schlieren photography. The characteristic scale of wrinkles was found to be much larger than that of the unburned-gas turbulence, which was produced by a grid or wire gauze. Throughout their study, the authors made extensive use of thermocouples in studies of turbulent premixed ﬂames; the integral length scale of the temperature ﬂuctuations was measured, which corresponds to the mean diameter of unburned and burned gas lumps. This technique was applied to determine the time scale of wrinkles of the ﬂame front, from which the length scale was deduced. The burner consisted of a thin-walled tube with inside diameter of 10 mm, through which a lean propane-air mixture ﬂowed. Considering the catalytic effect of the thermocouple and the ﬂame stability limit of the burner, the equivalence ratio of the mixture was determined to be 0.724. The equivalence ratio of the unburned gas was ﬁxed at 0.68, whereas the mean velocity u varied from 4 to 8 m/s. The homogeneous gas mixture passed through a settling chamber, a contoured nozzle, and a turbulence-producing grid. The relative intensity of turbulence at the burner exit, 50 mm downstream of the grid, was 2.0%. The details of each grid and turbulence parameters are presented in Table 5.2. An eight-port pilot burner was placed concentric to the burner tube to stabilize the main ﬂame

TABLE 5.2. Turbulence-Producing grids, Turbulence Parameters, and Length Scale of Wrinkles Source: Modiﬁed from Yoshida and Tsuji (1982).

uavg (m/s)

urms (m/s)

l0 (mm)

lf (mm)

G-1a

4.0 8.0

0.12 0.15

2.46 1.41

— 4.14–5.59

G-4b

4.0 8.0

0.36 0.45

1.54 1.79

4.67–5.48 5.23–5.75

Grid no.

l0 : Integral scale of turbulence. lf : Length scale of ﬂame wrinkles.

a 40-mesh

wire gauze. perforated plate with mesh size 3.0 mm and hole diameter 1.5 mm.

b 1-mm-thick

CHARACTERISTIC SCALE OF WRINKLES IN TURBULENT PREMIXED FLAMES

305

at the burner rim. The turbulence characteristics of the grids were determined by a hot-wire anemometer and checked by a laser D¨oppler velocimeter. In order to measure velocity proﬁles with the laser anemometer, the burner was mounted on a traversing mechanism, while the optical system of the laser anemometer was set up on a platform. The temperature measurements were made with an uncoated 50 μm Pt–Pt: 13% Rh thermocouple and time-resolved temperature signals were obtained up to 3 kHz. Note that the values given in Table 5.2 (urms /uavg ) are relatively low compared to practical combustion systems. 5.3.1

Schlieren Photographs

Schlieren photography has been very useful in recognizing the existence of wrinkled laminar ﬂames. Figure 5.8 shows Schlieren photographs taken during the experimental study just discussed. As can be seen from this ﬁgure, for the case of uavg = 8 m/s, the G-1 ﬂame appears to have a continuous wavy ﬂame front. For the G-4 ﬂame, increasing the turbulence intensity causes a more distorted ﬂame front and decreases the ﬂame length, which corresponds to an increase in the turbulent burning velocity. When uavg is reduced to 4 m/s, the wrinkles are still observed in the G-4 ﬂame, but they disappear in the G-1 ﬂame, which has a nearly laminar appearance (Figure 5.8c) even though the unburned gas is turbulent. Note that there is very low wrinkling due to urms /SL < 1. 5.3.2

Observations on the Structure of Wrinkled Laminar Flames

Yoshida and Tsuji (1982) studied the thermal structure of wrinkled laminar ﬂames by using a ﬁne-wire thermocouple. Figure 5.9 shows the mean and ﬂuctuating

(a)

(b)

(c)

(d)

Figure 5.8 Typical Schlieren photographs: (a) G-1, uavg = 8 m/s; (b) G-4, uavg = 8 m/s; (c) G-1, uavg = 4 m/s; and (d) G-4, uavg = 4 m/s (after Yoshida and Tsuji, 1982).

306

TURBULENT PREMIXED FLAMES 1600 x/D 5 10 15

T, °C

T

T ″2

1200

600

800

400

400

200

0

0

0.1

0.2 R/D

0.3

T ″2, °C

0.4

Figure 5.9 Distributions of mean and ﬂuctuating temperatures for G-1, uavg = 8 m/s (after Yoshida and Tsuji, 1982).

temperature distributions for the G-1 ﬂame with uavg = 8 m/s. In the upstream region of the turbulent ﬂame (small x /D), no heat release is observed near the ﬂame axis. Moving radially outward, an abrupt increase in both mean and ﬂuctuating temperatures occurs on the unburned side of the turbulent-ﬂame zone. The ﬂuctuating temperature reaches a maximum near the center of the ﬂame zone, and then decreases on the burned side, whereas the mean temperature increases monotonically in the ﬂame zone and then reaches a temperature plateau. In the downstream region of the turbulent ﬂame (large x /D), the mean and ﬂuctuating temperatures are high even on the ﬂame axis, due to the closure of the ﬂame zone at the top. The maximum ﬂuctuating temperature is nearly constant, almost independent of the downstream location. This is one of the characteristic features of wrinkled laminar ﬂames. Figure 5.10 shows typical probability density functions (pdf) measured at x /D = 10 for the same ﬂame at various radial locations. From unburned to burned side (i.e., from R/D = 0 to R/D = 0.3), the probability that the thermocouple is exposed to the unburned gas decreases whereas that of burned gas increases. The probability of ﬁnding unburned or burned mixture at the center of the ﬂame zone is nearly the same.

CHARACTERISTIC SCALE OF WRINKLES IN TURBULENT PREMIXED FLAMES

307

6

PDF (1/°C × 103)

4

2

0.2 0.1 0

0

2000

1000 T (°C)

R/D

0

Figure 5.10 Probability density functions for G-1, uavg = 8 m/s at x/D = 10 (after Yoshida and Tsuji, 1982). tb1

tb2

tb3

340 °C/div

Tb Tm Tu 1 msec/div tu1

tu2

Figure 5.11 Typical time record of temperature signal at a ﬁxed location, showing the unburned-burned intermittency (after Yoshida and Tsuji, 1982).

5.3.3

Measurements of Scales of Unburned and Burned Gas Lumps

Figure 5.11 shows the time records of the temperature signal at the center of the ﬂame zone. In this ﬁgure, the high and low signal levels correspond to the burned and unburned gas lumps respectively. With regard to the temperature measurements, it can be assumed that the temperature corresponds to a burned

308

TURBULENT PREMIXED FLAMES

gas lump if it is higher than the algebraic mean of high and low levels [Tm = (Tb + Tu )/2] and to an unburned gas lump if it is lower than Tm . Therefore, the mean period during which the thermocouple is exposed to unburned or burned gas can be determined as: 1 τu = lim τun (5.28) n→∞ n 1 τbn (5.29) τb = lim n→∞ n In the work of Yoshida and Tsuji (1982), n was taken to be 150 to 300, depending on the signal condition. For the case of G-1 ﬂame with uavg = 4 m/s, the Schlieren photograph (Figure 5.8c) shows the perturbed laminar ﬂame appearance. The length scale of ﬂame wrinkles in a fully developed wrinkled laminar ﬂame is known to have an amplitude that is much larger than the laminar ﬂame thickness. Although the distortion of the laminar ﬂame front is found to grow slightly downstream, the amplitude of the length scale associated with perturbation is comparable with the laminar ﬂame thickness. The observed pdf proﬁle for this case was found to be different from the other three cases; it had only one peak centered at the mean temperature, even in the middle of the ﬂame zone. For the other three cases, well-deﬁned bimodal pdfs were obtained. This observation highlights the difference between the laminar and turbulent premixed ﬂames. Figure 5.12 shows the time scales of unburned and burned gas lumps for G-1 and uavg = 8 m/s. From unburned to burned side, the time scale of the burned 1.6

τu, τb (msec)

1.2 x/D τu τb 5 10 15 0.8

0.4

0

0

0.1

0.2 R/D

0.3

0.4

Figure 5.12 Time scales of unburned and burned gas for G-1, uavg = 8 m/s (after Yoshida and Tsuji, 1982).

CHARACTERISTIC SCALE OF WRINKLES IN TURBULENT PREMIXED FLAMES

309

gas lumps increases rapidly from near zero to very large values in the turbulent ﬂame zone, whereas the time scale of unburned gas lumps decreases from very large values to zero. Near the tip of the turbulent ﬂame zone, the ﬂame closure makes both time scales of unburned and burned gas lumps ﬁnite even on the axis. Similar trends were observed for the other two cases. The time scale of wrinkles is determined at the point where the scales of unburned and burned gas lumps are equal. At this point (i.e., at the center of turbulent ﬂame zone), the ﬂuctuating temperature is at its maximum. The length scale of wrinkles (lf ) is usually the wavelength of the wrinkles, assuming that the wrinkles have a wavelike appearance. The exact conversion of the time scale into the length scale requires the propagating velocity of wrinkles along the ﬂame front. However, as discussed before, the propagating velocity can be neglected, and the transport velocity is assumed to be the axial unburned gas velocity. Therefore, the time scale of wrinkles is associated with the length scale of wrinkles (lf ) and the mean axial velocity of the ﬂame (uavg ) and it can be expressed by the following equation: τf ≈

lf

(5.30)

uavg

Figure 5.13 shows distributions of time scales of wrinkles of the ﬂame front. For uavg = 8 m/s, the time scale falls in the range of from 0.45 to 0.62 ms. Even though the G-1 ﬂame is much longer than the G-4 ﬂame, in the downstream region of the turbulent ﬂame, the time scale of wrinkles for both ﬂames is about 0.6 ms. Although the characteristics of turbulence generated by G-1 and G-4 grids were different, as shown in Table 5.2, the time scale is somewhat unaffected by the upstream turbulence. For uavg = 4 m/s, however, the time scale it falls into the range from 0.86 to 1.01 ms and is roughly twice that for uavg = 8 m/s. From Figure 5.13, it is clear that the time scale of wrinkles tends to increase slightly with the downstream distance in all three cases.

1.0 G−4, uavg = 4m/s τf (ms)

0.8 G−4, uavg = 8m/s 0.6 G−1, uavg = 8m/s

0.4

0

5

10

15

x/D

Figure 5.13 Time scales of wrinkles of the ﬂame front (after Yoshida and Tsuji, 1982).

310

TURBULENT PREMIXED FLAMES

10 u (G-4)

8

8

6

u, u' × 10 (m/s)

u, u' × 10 (m/s)

u (G-1) u' (G-4)

4 u' (G-1) 2 0

−0.4

−0.2

0 R/D (a)

0.2

0.4

u (G-1)

6

u (G-4) 4 u' (G-4) 2 0

u' (G-1)

−0.4

−0.2

0 R/D (b)

0.2

0.4

Figure 5.14 Mean and ﬂuctuating velocity distributions of unburned gas at burner exit (a) uavg = 8 m/s, umax = 9.27 m/s and (b) uavg = 4 m/s, umax = 5.43 m/s.

5.3.4

Length Scale of Wrinkles

Figure 5.14 shows the radial distributions of mean and ﬂuctuating velocities at the burner exit. The velocities are uniform except for the pipe-wall boundary layer, where the mean velocity decreases whereas the ﬂuctuating velocity increases due to the wall shear stress. For uavg = 8 m/s, the length scale determined falls in the range from 4.14 to 5.75 mm for the G-1 grid; for uavg = 4 m/s, it ranges from 4.67 to 5.48 mm. Several conclusions can be gathered from Yoshida and Tsuji’s experimental investigation: 1. The length scale of wrinkles is around 5 mm, almost independent of the upstream turbulence and the unburned gas velocity. The integral length scale is between 1.41 and 2.46 mm for both G-1 and G-4 ﬂames. 2. The Schlieren photographs show that the ﬂame length decreases with an increase in the ﬂuctuating velocity, resulting in an increase of the turbulent burning velocity. For the wrinkled laminar ﬂame, the turbulent burning velocity is determined by the ﬂame area, which increases with the ﬂuctuating velocity. 3. Periodicity of the temperature signal was found. This fact suggests that the wrinkles that appear upstream are transported downstream along the ﬂame front. 5.4 DEVELOPMENT OF BORGHI DIAGRAM FOR PREMIXED TURBULENT FLAMES

As discussed in Section 2.6, it is not easy to correlate turbulent ﬂame speed with ﬂow parameters. If the turbulent ﬂame speeds cannot be easily correlated by such

DEVELOPMENT OF BORGHI DIAGRAM FOR PREMIXED TURBULENT FLAMES

311

a simple paradigm as that of Damk¨ohler, an attempt should be made to capture the structure of the turbulent premixed ﬂame. In order to consider the dependencies of ﬂame speed and ﬂame structure, appearance, and so on, on turbulence, a regime plot would be useful. Therefore, diagrams based on the mapping of ﬂame structure onto a two-dimensional plot with axes of relative velocity scales versus relative length scales (turbulent versus laminar) have been constructed. Sometimes Damk¨ohler number versus Reynolds number are also used as axes coordinates. These regime diagrams are also known as Borghi diagrams, since Borghi was one of the ﬁrst researchers to plot them. Regions identiﬁed on these diagrams are given such names as “wrinkled ﬂamelets,” “corrugated ﬂamelets (or wrinkled ﬂamelets with pockets),” “distributed reaction zone,” “thin reaction zone,” “quenched reaction zone (or broken ﬂamelets),” and so on. This is often done on the basis of scaling arguments and on experimental observations as well as direct numerical stimulation (DNS) for two-dimensional laminar ﬂame-vortex interactions. A typical Borghi diagram is shown in Figure 5.15a. The classical Borghi diagrams have been drawn for the ideal academic case (single-step chemistry with high activation energy; no heat losses; Le = 1; equal molecular diffusivities of reactants; steady and spatially uniform characteristics of an unburned mixture). An attempt to modify this diagram by accounting for experimental contributions to turbulent combustion theory is shown in Figure 5.15b.

5.4.1 5.4.1.1

Physical Interpretation of Various Regimes in Borghi’s Diagram Wrinkled Flame Regime

This ﬁrst regime has been identiﬁed by Damk¨ohler as the wrinkled ﬂame regime. This regime occurs when turbulence intensity is low, while all length scales including the Kolmogorov length scale η are larger than δL . Due to this, the turbulence has the effect of wrinkling only the ﬂame front. The turbulent ﬂame, when seen instantaneously, displays a thin quasi-laminar front that is wrinkled and lengthened by the turbulent movements. As long as urms or abbreviated as u is small with respect to SL (i.e., u /SL < 1), the motion of the instantaneous ﬂame front is governed by ﬂame propagation and not by advection of the mixture. As a consequence, the ﬂame front is much less wrinkled and shows some cusps toward the unburned gases. This phenomenon is also shown in region 1 of Figure 5.1. Of course, this thin front is ﬂuctuating, and the mean turbulent ﬂame appears thicker than the laminar case. 5.4.1.2 Wrinkled Flame with Pockets Regime (also Called Corrugated Flame Regime)

When the rms of ﬂuctuating velocity, u , become larger with respect to SL , the ﬂame front resembles more a ﬁlament of ﬂuid particles (similar to a material surface in a turbulent ﬂow). It is possible then that pockets of fresh or burned gases would appear (as emphasized by Shchelkin) due to large-scale interactions of two elements of the instantaneous ﬂame front (i.e., turbulent ﬂuctuations and

312

TURBULENT PREMIXED FLAMES 3

Turbulent Reynolds no.: ReT = u′ l0 /v where u′ represents u′rms

Thickened flames with possible extinction (u′ >> SL, Ka > 1, Da > 1 )

Thick flames (l0 < δL)

log10(u′/SL)

2 = Da

(u′ > SL , Ka > 1, Da < 1)

1

Re

T

=

1 Wrinkled flames Ka = with pockets (u′ > SL, Ka < 1)

1

0

−1 −1

Thickened wrinkled flames

1

Wrinkled flames (u′ < SL)

Laminar flames (ReT < 1) 0

1

2

3

log10(l0 /δL) (a)

log10(u′/SL)

2

Flamelet quenching

Thickened reaction zone

1

Moderate turbulence (wrinkled, stretched, thickened, and locally quenched flamelets)

0

Weak turbulence (hydrodynamically unstable flamelets)

Laminar flames −1 −1

Circles and triangles show results of experimental investigations of flame quenching by intense turbulence performed by Chomiak and Jarosinski (1982) and by Abdel– Gayed and Bradley (1985).

0

1 log10(l0 /δL)

2

3

(b)

Figure 5.15 Regimes of premixed turbulent combustion: (a) Borghi diagram and (b) modiﬁed diagram (modiﬁed from Lipatnikov and Chomiak, 2002).

ﬂame propagation). A rough estimation of the ratio u /SL above which these pockets could be seen can be obtained. Let us consider, for example, a pocket of fresh gases; it is created by the interactions of large-scale turbulent movements; consequently its length scale is l0 , and the time between the birth of two such pockets is l0 /u . However, this

DEVELOPMENT OF BORGHI DIAGRAM FOR PREMIXED TURBULENT FLAMES

313

pocket is consumed by the laminar ﬂame surrounding it in a time of the order of l0 /SL . If l0 /SL is longer than l0 /u , we can consider that these pockets will remain present. This corresponds to the inequality u /SL > 1. If we are interested next in pockets of burned gases, we have to consider not only the size of this pocket but also its distance from the continuous ﬂame front, which can be decreased by the increase in laminar ﬂame propagation toward the unburned mixture. The rest of the reasoning is identical to the wrinkled ﬂames and shows again that these pockets can remain separated from the continuous wrinkled ﬂame front if u /SL > 1. The turbulence intensity u is also represented by the square root of turbulent kinetic energy, k . 5.4.1.3 Thickened Wrinkled Flames Considering again a turbulence with l0 > δL and higher turbulence intensity u , the inﬂuence of the turbulence on the ﬂame fronts becomes more complicated. It can be summarized in this way.

1. The interactions between ﬂame front and turbulent eddies occur more and more frequently, since, in a given volume, the ﬂame fronts are very close together and are present in greater numbers. 2. The instantaneous ﬂame front is stretched to a greater degree due to the velocity gradient imposed on it by the effect of turbulence. It is known that intense stretching can extinguish laminar ﬂames. 3. As the turbulent Reynolds number (ReT ) increases, the Kolmogorov scale η becomes smaller and smaller, and there arises a condition when η = δL even when the integral scale of turbulence is greater than the reactivediffusive laminar ﬂame thickness (i.e., l0 > δL ). This fact implies that the smallest turbulent eddies can inﬂuence the transport properties in the local ﬂame zone structure. At this point, the Kolmogorov scale may be too small or have velocities that are too small (compared to the ﬂame speed) to affect the ﬂame efﬁciently. This concept was theorized by Poinsot and Veynante (2005). The Kolmogorov scales and the ﬂame scales (speed and thickness) are linked by the same relations with the kinematic viscosity (i.e., ηu /ν δL SL0 /ν 1). Kolmogorov vortices are the most efﬁcient in terms of induced strain rate but due to the viscous dissipation, they have a short lifetime and therefore may have only limited effects on the combustion. This process is complex but generally results in thickening of the ﬂamelets while retaining the wrinkled shapes. 4. The frequencies involved in the turbulence become higher and higher and the quasi-laminar fronts ﬁnd it more and more difﬁcult to survive, as their own characteristic frequency is governed by the inverse of chemical reaction time scale, 1/τc . The equality η = δL can give us an upper limit for the ‘wrinkled ﬂame with pockets’ regime. Increasing u over the η = δL curve (equivalent to Ka = 1

314

TURBULENT PREMIXED FLAMES

curve, to be discussed in the next subsection), we enter the domain where the ﬂamelets are modiﬁed by the turbulence in a complicated way. It is obvious from dimensional arguments that η = δL also corresponds to τc ∝ τK , where τK is the Kolmogorov time deﬁned as τK = (k 3/2 / l0 ν)−1/2 . Thus, η = δL could also be seen as a limit when the highest turbulence frequency 1/τK is greater than 1/τc . In addition, detailed studies of stretched laminar ﬂames reveal that they can be extinguished when the stretching rate, κ (a measure of the velocity gradient to which the ﬂame is subjected), becomes larger than the reciprocal of an extinction time (which is proportional to τc ); then η = δL could also represent, with a multiplicative constant, the limit where local and instantaneous extinctions of ﬂamelets could occur in a turbulent ﬂame, even though the measure of the highestvelocity gradient is just 1/τK . The ﬂamelets are always wrinkled, as long as the thickness of the modiﬁed ﬂamelets (e * ) is less than the integral length scale, l0 . The modiﬁed ﬂamelet thickness e * is related to the turbulence parameters as: e∗ ∝

k 3/2 τc3 /l0

(5.31)

The regime of thickened-wrinkled ﬂame is shown in Figure 5.15a, between the curves η = δL and e∗ = l0 , which turns out to be equivalent to the condition τR = τO , where τR and τO are the characteristic time scales for residence in laminar ﬂamelet and eddy turnover time (τO = l0 /u ), respectively. This condition also corresponds to the Damk¨ohler number equal to unity. The turbulent Damk¨ohler number is deﬁned as the ratio of these two time scales: Da ≡

5.4.1.4

τ Eddy turnover time l0 /u l0 SL = = O = laminar ﬂame time τR δL /SL u δL

(5.32)

Thickened Flames with Possible Extinctions/Thick Flames

When u is so high that this limit is crossed over, ﬂamelets can no longer be distinguished; the reacting turbulent mixture no longer possess ﬂamelet-type small-scale structure. In Figure 5.15a, the regime where l0 is larger than δL is called thickened ﬂame and the regime where l0 is smaller than δL is called thick ﬂames. In thick ﬂames, all turbulence length scales are smaller than δL . These regimes have also been called the distributed combustion regime by Damk¨ohler and shown by Ballal and Lefebvre in 1975 (see Figure 5.1).

5.4.2

Klimov-Williams Criterion

To answer whether a laminar ﬂame can exist in a premixed turbulent ﬂow, Klimov (1963) and Williams (1975) showed, from solutions of the laminar ﬂame equations in an imposed shear ﬂow, that a propagating laminar ﬂame may exist

315

DEVELOPMENT OF BORGHI DIAGRAM FOR PREMIXED TURBULENT FLAMES

only if the Karlovitz number (Ka) deﬁned as a dimensionless stretch factor is less than a critical value of the order of unity. The Karlovitz number is deﬁned as: Residence time for crossing an unstretched ﬂame δL 1 dA Ka ≡ = SL A dt Characteristic time for ﬂame stretching

(5.33)

where δL , SL , and A represent the thickness, speed, and elemental areas of a laminar ﬂame, respectively. As discussed in Chapter 4, the Kolmogorov length scale (η) and integral length scale (l0 ) are related through the turbulent Reynolds number as: η −3/4 ∼ Rel0 l0 where Rel0 ≡

u l0 ν

(5.34)

l0 ≈

and

k 3/2 ε

(5.35)

The Taylor microscale (T ) and the integral length scale (l0 ) are related through the turbulent Reynolds number as: −1/2

T /l0 ∼ Rel0

(5.36)

By taking square of Equation 5.34 and dividing Equation 5.36 by it, we get: Rel0 T u T ∼ ⇒ 2 = 2 η l0 η ν

or

ν u = 2 T η

(5.37)

It is known that the percentage of elemental area variation under the turbulent ﬂow condition can be approximated by κ≡

1 dA ∼ u = A dt T

(5.38)

where u and T represent the turbulence intensity and Taylor microscale. By substituting Equation 5.37 into Equation 5.38, we can obtain: κ≡ and

1 dA ∼ ν 1 = 2 = A dt η τη δL τ δ ν = τR ⇒ Ka = R = L 2 = SL τη SL η

2 2 δL ν δL ≈ SL δ η η L

≈1

(5.39)

316

TURBULENT PREMIXED FLAMES

where τη is the time scale associated with the Kolmogorov length scale η [also deﬁned in Equation 4.195 in Chapter 4] and τR is the residence time in crossing an unstretched laminar ﬂame. According to the Klimov-Williams criterion, the existence of laminar ﬂamelets in premixed turbulent ﬂow requires that: δL η,

which is equivalent to Ka 1

(5.40)

This Klimov-Williams criterion is a sufﬁcient condition for this regime (wrinkled ﬂamelets with and without pockets), but it occurs in a substantially wider domain, as discussed next. 5.4.3

Construction of Borghi Diagram

After analyzing the various regimes in the Borghi diagram, let us establish relationships among the Reynolds number, Karlovitz number, turbulent ﬂuctuating velocity, and laminar ﬂame parameters. δL δ u Rel0 u ν ν u = = Re = Rel0 = L Rel0 (5.41) l0 SL SL Rel0 SL u l0 SL δL l0 l0 ∴ log10 Also,

u SL

≈1

= log10 Rel0 − log10

l0 δL

3 1/4 δL ν ν u3 where η = , ε = Ka = SL η2 ε l0 1 3 1/4 νl0 ν 2 ν l0 ⇒ η2 = ∴η= 3 u u u l0

(5.42)

(5.43) (5.44)

Now the Karlovitz number can be written as: 1 1 δL u u l0 2 δ u = = L (5.45) Rel0 2 SL l0 ν SL l0 3/2 1 − 1 2 − 1 l0 l u l0 2 u l u 0 0 = Ka or = Ka Rel0 2 = Ka SL δL δL δL SL SL δL (5.46) u l0 2 1 log10 = log10 (Ka) + log10 (5.47) SL 3 3 δL δ ν Ka = L SL (νl0 /u )

u l0 ν

1 2

By using Equations 5.42 and 5.47, we can construct the regime diagram for constant Ka and Rel0 . The regime diagram proposed by Peters (2000) is shown in Figure 5.16. This diagram is simpler than the original Borghis diagram with a few additions. Another regime diagram for premixed turbulent ﬂames is shown

DEVELOPMENT OF BORGHI DIAGRAM FOR PREMIXED TURBULENT FLAMES

317

103 Broken reaction zones (or distributed reaction zones) =1 Ka Δ h=Δ on iteri Thin reaction zones s cr m a i h < dL ill ov-W Klim Re Corrugated =1 h = dL Ka T = flamelets 1 (u′ > SL, Ka < 1, dL < η)

u′/SL

102

10

1 Wrinkled flames (u′ < SL, dL < η)

Laminar flames (ReT < 1)

−0.1 −0.1

1

10

102

103

104

l0 /δL

Figure 5.16 Regime diagrams of premixed turbulent combustion (modiﬁed from Peters, 2000).

= /S

L

Single Sheets

u′

106

=

1

(Wrinkled Flames)

/S

L

104

u′

Damköhler Number Da based on Integral Scale l0

Weak Turbulence

10 −

2

108

102

Multiple Sheets (Fractal)

(Corrugated Flames)

δL η= tion eac R s in Th Zone

Broken Flamelet 100

10−2

l0

/δ

= 1 Distributed Reaction Zones

100

L

102 104 106 108 Turbulence Reynolds Number ReT

Figure 5.17 Regime diagrams of premixed turbulent combustion (modiﬁed from F. A. Williams, 2000).

in Figure 5.17, which is based on the concepts of Damk¨ohler and described by Williams (2000). Another turbulent Karlovitz number based on fuel consumption layer thickness (δc ) is deﬁned as: 2 2 δL δc δc2 δc = 2 Ka where ≡ (5.48) Ka = 2 = η δL η2 δL

318

TURBULENT PREMIXED FLAMES

TABLE 5.3. Summary of Relative Magnitudes of Length Scales for Various Types of Turbulent Premixed Flames Type of Flame Wrinkled ﬂame Severely wrinkled ﬂame with thin reaction zones Flamelets in eddies with thin reaction zones Distributed reaction zone

Relative Magnitudes of Length Scales δL < η η < δL < T T < δL < l0 l0 < δL

The fuel consumption layer thickness is discussed in Chapter 2. Thin ﬂame zones exist between Ka = 1 and Ka = 1, where η is between δc and δL —that is: δc η δL (5.49) This classiﬁcation is based on the assumption that the fuel consumption zone is much thinner than the laminar ﬂame thickness. A summary of relative magnitudes of length scales corresponding to different types of ﬂames is given in Table 5.3. Two limiting regimes are emphasized in all the regime diagrams: thick ﬂames (or distributed reaction zones) and wrinkled ﬂames. Physical descriptions of these two regimes are given next. 5.4.3.1 Thick Flames (or Distributed Reaction Zone or Well-Stirred Reaction Zone) In a thick ﬂame regime, the chemical reactions controlling heat release proceed in a combustion zone distributed over all the turbulent length scales. When discussing this regime, it is a common practice to (1) model the effects of turbulence on combustion as enhancement of heat and mass transport, and (2) utilize the classic expression by Damk¨ohler for turbulent ﬂame speed. This regime is associated mainly with small-scale, high-intensity turbulence (u SL ), and l0 < δL commonly is considered to be a sufﬁcient condition for this regime. This limit is unlikely to be reached in laboratory or industrial burners, unless special arrangements are made. In the limiting case, a thick ﬂame regime may exist at Da < 1 or Da 1, but for a more realistic case with non-adiabatic conditions and without equal diffusivity for all species in the ﬂame; the existence of this regime is questionable since quenching may occur. Summerﬁeld, Reiter, Kebely, and Mascolo (1955) have reported certain experimental data that indirectly indicate the existence of this regime; these early observations are still unique in the ﬁeld. Due to the lack of deﬁnitive experimental data that prove the existence of thick ﬂames, this regime is considered hypothetical. 5.4.4

Wrinkled Flames

The second limiting regime is the wrinkled ﬂame regime (also known as reaction sheet or ﬂamelet regime), in which the chemical reactions controlling heat

DEVELOPMENT OF BORGHI DIAGRAM FOR PREMIXED TURBULENT FLAMES

319

Full Fresh-Mixture Boundary wf Full Burned Product Boundary

Flame Speed

SL Burned Products

ST SL

AT

Macroscopic Turbulent Flame Surface

Unburned gas mixture Sponge Region (a)

Premixed Fresh Gas (b)

Figure 5.18 (a) Schematic description of wrinkled turbulent premixed ﬂame (modiﬁed from Kikuta, Nada, and Miyauchi, 2004) and (b) superimposed picture of 50 images of the extracted ﬂame front location of a wrinkled methane air turbulent premixed ﬂame (Kobayashi et al., 1996).

release are conﬁned to thin, highly-wrinkled, convoluted, and strained interfaces separating unburned reactants from burned products. These interfaces are commonly called ﬂamelets and are assumed to have the same local structure as perturbed laminar ﬂames. This regime is associated mainly with large-scale, weak and moderate turbulence. (See the schematic description in Figure 5.18a and superimposed boundaries of 50 images of a wrinkled ﬂame in Figure 5.18b. This regime often is subdivided into two subregimes. The ﬁrst, characterized by u < SL , is called the weak, wrinkled ﬂame, or wrinkled ﬂamelet subregime. It is commonly associated with simply connected, weakly perturbed ﬂamelets. Borghi (1985) has stressed that the smoothing effects of ﬂame propagation on the ﬂamelet surface keep the ﬂame from becoming corrugated. The second subregime—called wrinkled ﬂame with pockets or corrugated ﬂamelets—is associated with highly convoluted, multiple connected ﬂamelets, including the pocket formation. Let us discuss these two subregimes in context with the experimental data. The composition of fuel-air mixtures is characterized by the fuel type and the equivalence ratio φ, as given in Figure 5.19a. In this plot, the curves show the third-order polynomial ﬁt of experimental data points. For propane-air mixtures, solid and dashed curves are associated with Le > 1 and Le < 1, respectively.

320

TURBULENT PREMIXED FLAMES

5

ST (m/s)

4 3 2

ST (m/s)

CH4, f = 1.67 C3H8, f = 1.25 C3H8, f = 0.83 C3H8, f = 0.71 C2H6, f = 1.0 H2, f = 0.71 Le<1 Le>1

f 0.7 0.8 0.9 5 1.1 4 1.3 3 1

0

2

4 u′(m/s)

6

8

0

C3H8/air mixture

f = 0.8

0

u′max

u′max u′max

4

8 u′(m/s)

(a)

u′max 12

16

40

2.4

0.1MPa 0.5MPa 1.0MPa 2.0MPa 3.0MPa

30 ST /SL

ST (m/s)

f = 0.9

(b)

3.2

1.6 0.8 0.0

f = 1.1 f = 1.3

f = 0.7

2

1 0

Le 2.03 1.98 1.97 0.88 0.85

10

CH4 + 2O2 + 7.17N2 CH4 + 2O2 + 6.56N2 + 1.25He CH4 + 2O2 + 4.78N2 + 4.88He 0

1

2 u′(m/s)

3

20

Reactants: Lean CH4-air mixture 4

0

0

(c)

5

10

15

u′/SL (d)

Figure 5.19 Measured dependencies of turbulent ﬂame speed on rms turbulent ﬂuctuation velocity u by (a) Karpov and Severin (1980); (b) Abdel-Gayed, Bradley, and Al-Khishali (1984); (c) Kido et al. (1989); and (d) Kobayashi et al. (1996).

Figure 5.19b shows the experimental data for propane/air mixtures measured by Abdel-Gayed et al. (1984) at various equivalence ratios. The shading in the plot shows quenching regions. The measured data of Kido et al. (1989) are shown in Figure 5.19c. The curves in this plot show the second-order polynomial ﬁt of experimental points. The measured data of Kobayashi et al. (1996) for a lean (φ = 0.9) methane air mixture at different pressures is given in Figure 5.19d. 5.4.4.1

Wrinkled Flamelets (Weak Turbulence)

On its face, the basic physical mechanism of the inﬂuence of turbulence on combustion is the same for both subregimes. It is the increase in the ﬂamelet surface area by turbulent eddies, although the surface geometry looks very different in these two subregimes. Figure 5.19a to c show no changes in the functional behavior of ST in the range of u ≈ SL . This observation supports the same nature of

DEVELOPMENT OF BORGHI DIAGRAM FOR PREMIXED TURBULENT FLAMES

321

the governing mechanism both at u < SL and u > SL0 (unstretched laminar ﬂame speed). However, most of the experimental data shown in Figure 5.19a to c correspond to u > SL0 and hence are not indicative of the behavior of ﬂame speed at weak turbulence. At this point, it is worthwhile to discuss the Darrieus-Landau (DL) instability, which is speciﬁc for weakly turbulent combustion. Darrieus-Landau instability The DL instability is due to hydrodynamic effect (see Figure 5.20). Suppose the initially ﬂat ﬂame front is disturbed slightly to have a curvature. Due to this curvature, the unburned ﬂow velocity at the ﬂame surface, uu,f , is higher (lower) than the laminar ﬂame speed, which remains constant since it is a chemical property of the mixture—that is, uu,f > SL for concave bulge (uu,f < SL for convex bulge). Because of this difference in velocity, the convex region of the ﬂame front has the tendency to travel farther into the unburned mixture and the concave region travels farther into the burned mixture. This could cause the ﬂame to be more curved. (The convex and concave regions become more protruded.) As a result, the amplitude of the disturbance increases and causes further increases in the velocity perturbation. Kobayashi and associates, experiments with turbulent ﬂames (shown in Figure 5.19d) have demonstrated that by increasing pressure (i) the DL instability becomes stronger and (ii) a substantially higher speed of turbulent ﬂame propagation is obtained (the speed increased by a factor 3 to 4 in their experiments). These results have been interpreted as an additional increase in the turbulent ﬂame velocity produced by the DL instability of ﬂamelets. The effect of pressure and turbulence intensity u was correlated by the next equation:

ST ∝ SL0

m

p pamb

u SL0

n (5.50)

In this equation, both m and n are positive numbers, and m was found to be equal to 0.4. To explain the effect of pressure on hydrodynamic DL instability,

uu = SL

uu,f < SL SL

uu = SL

uu,f > SL

Flame surface

SL

Figure 5.20 Schematic showing the mechanism of the Darrieus-Landau instability (modiﬁed from Lipatnikov and Chomiak, 2010).

322

TURBULENT PREMIXED FLAMES

Zel’dovich et al. (1985) and Chomiak (1990) performed stability analysis of Markstein’s equation, LM SL = SL0 1 − (5.51) Rc where LM is the Markstein length and Rc is the radius of curvature of the ﬂame. This stability analysis is out of the scope of this book. It shows that the ﬂame is stable with respect to short-wavelength (κ > κn ) perturbations but unstable with respect to long-wavelength (κ < κn ) perturbations, where κn is the neutral wave number. Kobayashi et al. (1996) and Kobayashi and Kawazoe (2000) also determined that κn scales as 1/δL , the domain of the instability (κ < κn ) is expanded by pressure since as pressure increases, δL decreases. This effect “may play an important role in the rapid increase in ST /SL0 with increasing u /SL0 at small u /SL0 in a high-pressure environment” Lipatnikov and Chomiak (2005). As p increases, κn increases; therefore, the domain of instability (from 0 up to κn ) is enlarged. The expansion factor is deﬁned as the ratio of unburned fuel mixture density to that of the burned gas (i.e., ≡ ρu /ρb ). If is close to 1, the propagating ﬂame front does not inﬂuence the external turbulence, which simpliﬁes the problem of ﬂame dynamics considerably. However, in reality, the expansion factor could be as large as 5 to 10, and the ﬂame interaction with the turbulent ﬂow may be quite strong. Such interaction leads to the thermal-diffusive instability (different from the DL instability). The theoretical studies by Cambray, Joulain, and Joulin (1994) have also come to the same conclusion: Realistically, large thermal expansion of the burning gases and the DL instability lead to a considerably higher turbulent ﬂame speed. Kuznetsov and Sabel’nikov (1990) have also reviewed the importance of ﬂame instabilities in turbulent combustion. Interested readers are referred to their book and to a review paper by Lipatnikov and Chomiak (2002) for more discussion on the determination of weak turbulence in premixed ﬂames. 5.4.4.2

Corrugated Flamelets (Strong Turbulence) When u is markedly higher than SL , the role played by the instabilities is reduced, and the basic physical mechanism of the inﬂuence of turbulence on combustion consists of the increase in the ﬂamelet surface area by turbulent eddies. As a result, the ﬂame speed is substantially increased by u , as indicated by most of the measured data shown in Figure 5.19. At high turbulence intensity, the measured dependencies of ST on u , presented in Figure 5.19a to c, change qualitatively; in particular, a maximum of the ST (u ) curve, followed by a decrease in ﬂame speed, is observed. These changes in the behavior of ST (u ) imply that some other physical mechanisms control highly turbulent combustion. Two different physical mechanisms are commonly emphasized.

1. The smallest eddies are assumed to be able to penetrate into the preheating zone of ﬂamelets, to thicken the zone, and to intensify the heat and mass

DEVELOPMENT OF BORGHI DIAGRAM FOR PREMIXED TURBULENT FLAMES

323

transfer inside it. Such a penetration is possible only if the preheating zone thickness is larger than the scale of the smallest eddies (i.e., the Kolmogorov length scale). Borghi (1974) found that penetration can occur if Ka = (δL /η)2 > 1. This mechanism causes an increase in local wrinkled ﬂamelet thickness. Zimont (1979) proposed a theory that the eddies can penetrate into the laminar ﬂamelets when η < δL , which can modify the structure and speed of ﬂamelets by thickening them. The thickening is counteracted by combustion and a universal burning structure. This universal burning structure has been called thickened ﬂamelets by various researchers. The thickness of this structure (δtf ) is larger than δL but much smaller than l0 . The local burning rate per unit area of the thickened ﬂamelet surface inside this structure is controlled by small-scale turbulence. By assuming (a) δtf to be in inertial subrange of the Kolmogorov turbulence spectrum, and (b) the local velocity and thickness of thickened ﬂamelet to be functions of mass diffusivity and characteristic time of chemical reaction in this structure, Zimont (1979) utilized dimensional analysis and estimated that: δtf ∼ l0 Da−3/2

and

utf ∼ u Da−1/2

(5.52)

These relationships can be deduced from the following relationships and Equation 4.195 4/3

Dtf ∼ ε1/3 δtf ,

1/2 utf ∼ Dtf /τc ,

1/2 δtf ∼ Dtf τc

(5.53)

2. Based on this estimate, Borghi (1985) suggested across the line Da = 1 (in Figure 5.15a), the regime of thickened wrinkled ﬂames changes into the regime of thickened ﬂames, in which the chemical reactions controlling heat release proceed in a thick distributed reaction zone comparable to the integral length scale. Ka = 1 and Da = 1 are shown in the classical Borghi diagram (see Figure 5.15a). There are several different criteria for penetration of small eddies into the ﬂame; however, understanding under what conditions small-scale eddies are able to change the local burning rate markedly is more important than discussing the conditions by which such eddies can penetrate into ﬂamelets. From this viewpoint, two issues are worth noting. 1. Within the framework of the thermal theory of a planar, one-dimensional laminar ﬂame, the chemical reactions controlling heat release occur in a thin reaction zone, the thickness of which is δr = δT in the asymptotic case of high activation energy. 2. Variations in the diffusivity in the preheating zone are of secondary importance, whereas an intensiﬁcation of heat and mass transfer in the much thinner reaction zone is required to change the combustion rate markedly.

324

TURBULENT PREMIXED FLAMES

Within the framework of such an asymptotic, planar, purely one-dimensional model, the penetration of turbulent eddies into the reaction zone rather than into the preheating zone can limit the wrinkled ﬂame regime. Such phenomena can occur if δr > η. Often this condition is considered to be the equivalent of the well-stirred reactor condition.

5.5

MEASUREMENTS IN PREMIXED TURBULENT FLAMES

A discussion of experimental studies and methods is crucial to understanding the underlying physics on ﬂame-turbulence interaction. As noted, the experimental data are particularly sensitive to ﬂame geometry. Therefore, ﬂames are divided into four major groups, as shown in Figure 5.6 and 5.7. In addition to these four major groups, two more types of ﬂame geometry have been studied by researchers: other unconﬁned ﬂames, which do not ﬁt in the four categories, and conﬁned ﬂames. Some of the data in Table 5.4 are marked with an asterisk (*). This indicates that some of the data required to calculate Reynolds number, Da, or Ka were not reported initially in the papers of a research group, but that the same research group reported the missing data in subsequent publications.

5.6

EDDY-BREAK-UP MODEL

Before we focus our attention to the eddy-break-up (EBU) model, let us consider the interaction of a turbulent eddy with a ﬂame front and distortion of a stabilized ﬂame front due to a single turbulent eddy (as shown in Figure 5.21). At higher turbulence levels (i.e., case c), the turbulent eddy and ﬂame-front interaction results in the formation of a fold; at lower turbulence levels, it results only in the formation of a wrinkled ﬂame surface (i.e., cases a and b). Now consider the case of an anchored premixed ﬂame shown in Figure 5.22. A stream of premixed gaseous fuel and air ﬂows steadily through a duct. In the center of the duct, there is a bluff body serving as a ﬂame holder, and the ﬂame spreads obliquely across the duct into the unburned stream, ultimately consuming all the reactants. Although the eye perceives a continuous and thick sheet of ﬂame, high-speed photography reveals that the reaction zone is highly convoluted, with many isolated pockets of hot and cold gas mixtures, which are isolated from the main stream. Several investigations of this phenomenon have been performed by G. C. Williams, Hottel, and Scurlock (1949); Solntsev (1961); Wright and Zukoski (1962); Howe, Shipman, and Vranes (1963); and others. The most striking feature of the experimental ﬁndings has been that the angle of spread of the ﬂame is almost unaffected by the experimental conditions: The distance from the symmetry plane to the ﬂame edge is equal to about 0.1 times the distance from the ﬂame holder, regardless of the mixture ratio, the approach velocity, the initial temperature of the mixture, and the level of free-stream turbulence; that is the angle θ is not a function of x , F/O, U , T0 , and urms .

325

Shepherd and Moss (1982, 1983) Chandran, S. Komenath, and Stable (1984)

Moss (1980) Yanagi and Mimura (1981) Tanaka and Yanagi (1983) 3.6

2.5 1

CH4 /air, φ = 0.6

C3 H8 /air, φ = 1.6

C3 H8 /air, φ = 0.8

0.4–3 2.1–4.8 0.6–3.4 0.22 15 0.5 0.5 10

C3 H8 /air, φ = 0.6–0.9 CH4 /air, φ = 0.7–1 C3 H8 /air, φ = 0.68 LPG/air CH4 /air, φ = 0.64

7.5

0.4

0.2

1.3–7.0 0.32

10

4

20 22

1.4 Same 0.5–1.9 11–59 0.08–0.67 14–54 Same

Bunsen ﬂame, fully developed pipe ﬂow

Bunsen ﬂame, grid-generated turbulence Bunsen ﬂame

Bunsen ﬂame Bunsen ﬂame

Same Bunsen ﬂame

Bunsen ﬂame, grid-generated turbulence Same

CH4 /air, φ = 0.8 C3 H8 /air, φ = 0.68

Bunsen Flames

0.7–2.1

Burner

CH4 /air, φ = 0.8, 0.9

Rel0

Yoshida and G¨unther (1980) Yoshida (1981) Yoshida and Tsuji (1982) Yoshida (1986) Yoshida (1988)

Ka

Lean C3 H8 /air

Da

Yoshida and Tsuji (1979)

u /SL0

Mixture

Reference

Source: Modiﬁed from Lipatnikov and Chomiak (2010).

TABLE 5.4. Summary of Experimental Studies of Turbulent Premixed Flames

(continued overleaf )

LDA, thermocouple Pitot probe

LDA, Mie-scattering

LDA, thermocouples

Mie scattering, LDA LDA, thermocouples

Thermocouples Thermocouples

LDA, thermocouples LDA, thermocouples

Thermocouples

LDA, thermocouples

Technique

326

Same

Burner

0.5–1.7 0.3–3.0 0.8–3.2

0.3–2.0 20–190 0.03–0.6 20–150 Same 0.7–1.8

C3 H8 /air, φ = 0.8–1.4 CH4 /air, φ = 0.7–1.0

CH4 /air, φ = 0.7–1.0

CH4 /air, φ = 0.7–1.0

CH4 /air, φ = 0.7

Furukawa et al. (2002) Deschamps et al. (1992) Smallwood et al. (1995) Ghenai, Chauveau, and G¨okalp (1996) Ghenai and G¨okalp (1998)

13–32 0.15–0.57 23–55 Same

0.5*

C3 H8 /air, φ = 1.1

Furukawa, Noguchi, and Hirano (2000)

725

10

Rel0

2

0.41

0.3

Ka

LDA, Rayleigh scattering, Pitot probe LDA, Mie scattering

Technique

2-point Rayleigh scattering 2-point Rayleigh scattering

Bunsen ﬂame gridgenerated turbulence 7–60 0.14–1.4 70–98 Same LDA, Mie scattering 70, 115 0.12, 0.07 70 20–67 0.1–0.38 41–130 Bunsen ﬂame fully Rayleigh scattering developed pipe ﬂow Electrostatic 45* 0.3* 240* Bunsen ﬂame fully microprobes, LDA developed pipe ﬂow, air co-ﬂow at lower velocity 420* 0.03* 160* Same LDA, 3-element electrostatic microprobes 100–420 0.03–0.2 160–360 Same Same 27–220 0.07–0.45 30–250 Bunsen ﬂame gridRayleigh scattering generated turbulence 9–42 0.3–0.9 60–230 Same Mie scattering

66

11

Da

C3 H8 /air, φ = 1.1

1.1–3.7 CH4 /air, φ = 0.7–1.0 C2 H4 /air, φ = 0.65, 0.75 1,0.8 CH4 /air, φ = 0.7–1.0 0.83–2.0

2.9

C2 H4 /air, φ = 0.6

Cheng and Shepherd (1987) Cheng and Shepherd (1991) Boukhalfa and G¨okalp (1988) Furukawa, Okamoto, and Hirano (1996)

0.8

CH4 /air, φ = 1.0

Waldherr, de Groot, and Strable (1991)

u /SL0

Mixture

Reference

TABLE 5.4. (continued )

327

25

0.42

108

Bunsen ﬂame

LPG/air, φ = 0.7

Kalt, Frank, and Bilger LPG/air, φ = 0.6–1.0 (1998)

Frank, Kalt, and Bilger CH4 /air, φ = 0.61–1.3 (1999)

O’Young and Bilger (1997)

0.6–37 1.4–150

CH4 /air, φ = 0.56–0.8 0.05–35

1.5–4.2* 225–450* Same

60–2440 Bunsen ﬂame grid-generated turbulence 2.4 60 0.35 450 Bunsen ﬂame grid-generated turbulence 2.4–8.8 2.2–63* 0.35–9.6* 300–480* Piloted Bunsen ﬂame, grid-generated turbulence 2.4–8.5 6.3–65* 0.4–4* 350–640* Same

5–10*

4–8*

CH4 /air, φ = 1.0

225–750* Same

4–13* 3.4–10*

CH4 /air, φ = 1.0 1.5–8.1

6–13* 3.4–7.3* 2.6–8.1* 350–750* Same

350–750 Piloted jet plane

CH4 /air, φ = 1.0

2.6–8.1

Mansour, Chen, and Peters (1998) Chen and Mansour (1998) Chen and Mansour (1999) Buschmann et al. (1996)

3.4–7.3

6–13

CH4 /air, φ = 1.0

4–7.3 5.7–10* 1.5–3.5* 224–400* Piloted jet plane

1.8

0.5–1.4 45–256 0.03–0.024 70–128 Same

Chen, and Peters (1996)

Gagnepain, Chauveau, CH4 /air, φ = 0.65–0.8 and G¨okalp (1998) Dumont, Durox, and CH4 /air, φ = 1.5 Borghi (1993) Mansour et al. (1992) CH4 /He/air, φ = 1.0

(continued overleaf )

PIV, OH LIF

PIV, OH LIF

2-sheet Rayleigh scattering

2-sheet Rayleigh scattering LDA, line Raman/Rayleigh /PLIF OH, Rayleigh scattering CH LIF, Rayleigh scattering 2-sheet Rayleigh scattering OH-PLIF, Rayleigh Scattering OH LIF, Rayleigh scattering

LDA, Mie scattering, Rayleigh scattering, LDA

328

Mixture

Burner

350

Same

Bill et al. (1981), Bill, Talbot, and Robben 1982 Namazian, Talbot, and Robben (1984) Cheng and Ng (1983) Cheng (1984)

C2 H4 /air, φ = 0.66–0.75 0.9–1.4 C2 H4 /air, φ = 0.7 1

90*

Same

26–82 0.12–0.45 75–140 Same 75 0.12 75 Same

0.2*

2-point Rayleigh scattering LDA LDA

50*

C2 H4 /air, φ = 0.6 1.2*

LDA, Rayleigh scattering

V-shaped Flames

Conditioned PIV 3-D cinema stereoscopic PIV

OH LIF, Rayleigh scattering 2-sheet Rayleigh scattering OH LIF, 2-sheet Rayleigh scattering OH LIF, 2-sheet Rayleigh scattering PIV, OH LIF Mie scattering, OH LIF Stereo PIV, Rayleigh scattering

Technique

C2 H4 /air, φ = 0.55–0.75 0.7–1* 50–107* 0.09–0.2* 33–90* V-shaped ﬂame, gridgenerated turbulence

Pfadler et al. (2008) CH4 /air, φ = 0.57–0.9 Steinberg, Driscoll, and CH4 /air, φ = 0.6, 0.7, Ceccies (2008) 1.35

0.5–5.2 370–840 Same 0.3–1.7 360–833 9–11 375–715 Same

0.5

1.1–2.4 25–170 0.1–0.55 190–520 Bunsen ﬂame 0.9–15 2–50 0.1–10 40–470 Bunsen ﬂame, gridgenerated turbulence 0.3–2.9* 4–140* 0.03–1.7 16–49* Bunsen ﬂame with stabilizing ring, grid generated turbulence 0.6–2.6 3–59 0.09–1.8 16–49 Same 2.3, 3.8 1.8, 5 1.2, 3.4 38 2D slot Bunsen burner

36

Rel0

0.5–5.2 370–840 Same

Ka

Chen and Bilger (2000) H2 /air, φ = 0.45–0.7 G¨ulder et al. (2000), C3 H8 /air, φ = 0.8–1 (2007) Pfadler et al. (2005) CH4 /air, φ = 0.6–1

2.7

5–52

Da

3–10 5–52 2.5–5.9 15–77 12–15 1.9–2.7

LPG/air, φ = 0.7

3–10

u /SL0

Chen and Bilger (2002) CH4 /air, φ = 0.65–1 LPG/air, φ = 0.65–1 Chen and Bilger (2004) H2 /air, φ = 0.325

Chen et al. (2002)

Chen and Bilger (2001) CH4 /air, φ = 0.65–1

Reference

TABLE 5.4. (continued )

329

Soika, Dinkelacker, CH4 /air, φ = 0.5–0.8 and Leipertz (1998) 1.6–9

0.09

82

0.3–9

0.1

66

1.3–44

0.09

50

56, 48 0.1, 0.2 14–60 0.14–0.48 70, 115 0.12, 0.07 6–120 0.04–1.5

1–3 11–78* 0.14–1 2.4, 7 2, 17* 0.56, 4.8* 0.9, 1.5 35, 75 0.12, 0.19 0.9, 1.5 35, 75 0.12, 0.19 1–1.6 34–106 0.1–0.31

C2 H4 /air, φ = 0.6, 0.8 CH4 /air, φ = 0.6, 0.8 CH4 /air, φ = 0.83 C2 H4 /air, φ = 0.7 C2 H4 /air, φ = 0.75–0.8

134

200

59

22

32, 84 34–70 70 25–76

V-shaped ﬂame, gridgenerated turbulence V-shaped ﬂame, gridgenerated turbulence V-shaped ﬂame, gridgenerated turbulence V-shaped ﬂame, gridgenerated turbulence

(continued overleaf )

2-sheet Rayleigh scattering

LDA, Rayleigh scattering LDA, Rayleigh scattering OH PLIF. Rayleigh scattering CARS

LDA

V-shaped ﬂame, gridgenerated turbulence Same

LDA, Mie scattering LDA, Mie scattering

LDA

2-point Rayleigh scattering 2-point Rayleigh scattering LDA

LDA

Same Same

93–140 Same 93* 75, 103 Same 75, 103 110 Same

35, 75 0.12, 0.19 75, 103 Same 35, 75 0.12, 0.19 75, 103 17–78 0.14–0.28* 117–131* Same

0.9, 1.5 0.9, 1.5 1–2.4

CH4 /air, φ = 0.83 C2 H4 /air, φ = 0.7 C2 H4 /air, φ = 0.6–0.8

CH4 /air, φ = 0.98 0.24, 0.42 1–2 CH4 /air, φ = 0.7–1.0 C2 H4 /air, φ = 0.65, 0.75 1, 0.8 0.4–3 CH4 /air, φ = 0.75–1 C3 H8 /air, φ = 0.8–1 C2 H4 /air, φ = 0.6, 0.7 0.6 Gouldin and Dandekar CH4 /air, φ = 0.8 (1984) Rajan, Smith, and C3 H8 /air, φ = 0.75 0.8 Rombach (1984) Azzazy, Daily, and CH4 /air, φ = 0.6, 0.75 Namozian (1986) Bradley et al. (1992) CH4 /air, φ = 1.1 0.65

Cheng, Talbot, and Robben (1984) Namazian, Talbot, and Robben (1984) Namazian, Shepherd, and Talbot (1986) Cheng and Shepherd (1986) G¨okalp, Shepherd, and Cheng (1988) Cheng et al. (1989) Cheng and Shepherd (1991) Dandekar and Gouldin (1982)

330

1.3, 1 0.8–1.5 32–43 0.14–0.3 0.24, 0.42 56, 48 0.1, 0.2 1 56 0.15

0.5–0.7 110–250 0.03–0.12 1.1–2. 17–60 0.14–0.48 0.5–1.0 70–260 0.03–0.12 0.6

CH4 /air, φ=0.79, 0.9

CH4 /air, φ = 0.9–1.0 CH4 /air, φ = 0.98 CH4 /air, φ = 1

C2 H4 /air, φ = 0.75–1

CH4 /air, φ = 0.75–1.0 C2 H4 /air, φ = 0.65–1 CH4 /air, φ = 0.89

Cho et al. (1988) Cheng et al. (1989) Shepherd, Cheng, and Goix (1990), Shepherd, Cheng, and Talbot (1992) Shepherd, Cheng, and Goix (1990) Cheng and Shepherd (1991) Li, Libby, and Williams (1994)

18, 32* 0.3, 0.2*

Rel0

Burner

70–90 70

70

40–90 32, 84 70

40*

LDA, Mie scattering LDA Impinging ﬂame grid-generated turbulence

Mie scattering

LDA, Mie scattering LDA, Mie scattering Mie scattering

LDA

PIV, Rayleigh scattering Crossed-plane Rayleigh scattering Rayleigh scattering, acetone PLIF

LDA, thermocouple

Technique

Same

Same

Impinging ﬂame gridgenerated turbulence Same Same Same

V-shaped ﬂame, gridgenerated turbulence V-shaped ﬂame, gridgenerated turbulence V-shaped ﬂame, gridgenerated turbulence 53 and 101 V-shaped ﬂame, gridgenerated turbulence

Impinging Jet Flames

0.2–2

Cho et al. (1986)

1.3

CH4 /air, φ = 0.7 5–40

1–3.5

0.5

CH4 /air, φ = 0.7

0.9–0.44

CH4 /air, φ = 0.53–0.69

Ka

CH4 /air, φ = 0.7–0.75

Da

Ghenai, Gouldin, and G¨olkap (1998) Most, Dinkelacker, and Leipertz (2002) Knaus, Sattler, and Gouldin (2005) Robin et al. (2008)

u /SL0

Mixture

Reference

TABLE 5.4. (continued )

331

CH4 /air, φ = 0.9

Escudie, Aaddur, and Brun (1999)

CH4 /air, φ = 0.6–0.8

B´edat and Cheng (1995) Plessing et al. (2000) Cheng et al. (2002)

Shepherd et al. (2002) CH4 /air, φ = 0.7 Kortschik, Plessing and CH4 /air, φ = 0.7 Peters (2004)

CH4 /air, φ = 0.7 CH4 /air, φ = 0.7

CH4 /air, φ = 0.6–1.0 C2 H4 /air, φ = 0.5–0.75

Cheng (1995) 5–100 0.06–1.6

12, 48 0.6, 0.13 60 0.15

Impinging ﬂame in swirling ﬂows Low-swirl burner, grid-generated turbulence 30–100 Same

60,36 80

9–55

5–25 5–25

3–17 3–17

1.6–19 700–3,700 Same 1.6–19 700–3,700 Same

0.2–1.0 33–420 0.01–0.22 26–85 2.3–21 3–80 0.3–13 490–1,600 Low-swirl burner, turbulence generator 5–25 3–17 1.6–19 700–3,700 Same 3–15 8–34 0.6–6 480–2,100 Same

0.5–3

CH4 /air, φ = 0.8, 1 C2 H4 /air, φ = 0.65

Chan et al. (1992)

Swirl-stabilized Flames 2, 0.75 1

C3 H8 /air, φ = 0.88 H2 /air, φ = 0.46,1.33

12–65 0.09–0.25

Impinging ﬂame grid-generated turbulence 0.28–0.84 23–69 0.13–0.65 75–224 Impinging ﬂame grid-generated turbulence 1.3–6.2 2–22 0.3–5 44–96 Impinging ﬂame grid-generated 1.2–5.4 2–37 0.2–4.3 43–134 turbulence 0.75–5.5 2–28 0.2–4.6 21–94 Same

0.7–0.9

Chen et al. (2008)

Chen and Bilger (2005) CH4 /air, φ = 0.75–1

Kalt, Chen, and Bilger CH4 /air, φ = 0.73–1.0 (2002) LPG/air, φ = 0.73–1

CH4 /air, φ = 0.6–1.3

Stevens, Bray, and Lecordier (1998)

(continued overleaf )

LDA, Rayleigh scattering PIV, OH PLIF LDA, OH PLIF, Mie scattering Rayleigh scattering OH PLIF, 2-sheet Rayleigh Scattering

LDA, Mie scattering

LDA, Mie scattering

OH PLIF, 2-sheet Rayleigh scattering PIV, Mie scattering

PIV, OH PLIF

LDA

PIV, Mie scattering

332

Mixture

u /SL0 Da

Ka

CH4 /air, φ = 0.7–1.0

Moreau and Boutier (1976)

Troiani et al. (2009)

Hartung et al. (2008)

280

0.6

Preheated CH4 /air, φ = 0.8

∼10

12, 8

12

9.4 0.04

2.7

0.3–0.6 16–60 0.04–0.15

2-sheet Rayleigh scattering LDA

Technique

130

1,000

6

Conﬁned oblique ﬂame in LDA a shear layer

PIV, Rayleigh scattering OH PLIF, stereoscopic PIV OH PLIF, PIV

Unconﬁned oblique ﬂame, LDA, Rayleigh scattering grid generated turbulence Unconﬁned ﬂame behind a LDA, thermocouples disk Propagating planar ﬂames LDA

32–1,300 Low-swirl burner with bluff body 530 Swirl-stabilized V-shaped Conditioned PIV ﬂame

Bluff-body stabilized ﬂame C2 H4 /air, φ = 0.55–0.7 15–40 7–30 2.4–14 4,700–9,000 Bluff-body stabilized ﬂames CH4 /air, φ = 0.67–1.12 3.8–9.1 12–70 0.55–3.2 1,500 Bluff-body stabilized ﬂames Conﬁned Flames

Heitor, Taylor, and CH4 /air, φ = 0.79 White law (1987) Videto and Santavicca C3 H8 /air, φ = 1.0 (1990) Most et al. (2002) CH4 /air, φ = 0.7, 0.8

Gulati and Driscoll (1986, 1988)

Burner

840, 1,160 Low-swirl burner

Rel0

Other Unconﬁned Flames

O’Young and Bilger LPG/air, φ = 0.7 4.5, 6.2 32, 23 0.9, 1.5 (1997) Schneider, Dreizler and CH4 /air, φ = 0.833–1.0 4–16 0.6–20 0.45–21 Janika (2005) Pfadler et al. (2007) CH4 /air, φ = 0.67–1 5.6–15 1.7 ÷1.3 1.8–14

Reference

TABLE 5.4. (continued )

333

Preheated CH4 /air, φ = 0.46–0.6

CH4 /air, φ = 1

130

0.16

410

Conﬁned ﬂame behind ﬂame-holder Ducted ﬂame, rearward facing step Same

1.2–8.4 8–104 0.12–6.6 81–1,680 Conﬁned V-shaped ﬂame in a shear layer 11.5 7 5.5 1,500 Opposed jet burner 8–16 14–24 1.8–5.1 1,950–5,100 Opposed jet burner 2 30 0.42 160 Conﬁned V-shaped ﬂame, grid-generated turbulence 8 23 1.7 1,600 Flame expansion in a tube with obstacles 23–34 1.9–3.5 12–25 1,870–2,250 Swirl-stabilized ﬂame in a double-corner burner

1.5

0.04–3

OH LIF, 2-sheet Rayleigh scattering

LDA

Thermocouples LDA, Mie scattering

LDA, thermocouples

LDA, Mie scattering

LDA, Mie scattering

Thermocouples

Same LDA Same LDA, CARS 12–1,000 Conﬁned V-shaped ﬂame, LDA pilot burner Same Same

PIV = Particle Image Velocimetry; OH LIF = OH Laser-Induced Fluorescence; LDA = Laser Doppler Anemometry. CARS = Coherent Anti-Stokes Raman Spectroscopy. PLIF = Planar Laser-Induced Fluorescene.

Lindstedt and Sakhitharan (1998) Dinkelacker et al. (1998)

Shepherd, Moss, and C3 H8 /air, φ = 1.2 Bray (1982) Shepherd and Moss C3 H8 /air, φ ∼ 1 (1983) Katsuki et al. (1988), CH4 /air, φ = 0.6, 0.65 (1990) Yoshida (1988) C3 H8 /air, φ = 0.82 Yoshida et al. (1992) C3 H8 /air, φ = 0.95 Veynante et al. (1996) C3 H8 /air, φ = 0.9

Same Same Same Same C3 H8 /air, 0.52–6.6 6–80 φ = 1.0, p = 0.2 atm Ballal (1979) C3 H8 /air, φ = 0.65 and 0.2–16 1.0, C2 H2 /air, H2 /air, φ = 1.0, p = 0.2 atm Lewis and Moss (1979) C3 H8 /air, φ = 1.0

Moreau (1981) Magre et al. (1988) Ballal (1978, 1979)

334

TURBULENT PREMIXED FLAMES

Flam ont

ont

ont

e fr

e fr

e fr

Flam

Flam Unburned gas

Burned gas

Unburned gas

Burned gas

Unburned gas

u′ = 1 SL0 3

(a)

Burner rim

Burner rim

Burner rim u′ = 0.1 SL0 3

Burned gas

u′ = 10 SL0 3

(b)

(c)

Figure 5.21 Distortion of stabilized ﬂames during passage of single eddies (modiﬁed from Scurlock and Grover, 1953). fold formation at flame edge

θ

Recirculation region

Flame “edge”

Velocity

fuel + air

Rod-shaped flame holder

Temperature

Oxygen concentration

Figure 5.22 Schematic structure of the ﬂame edge in a combustor (modiﬁed from Spalding, 1971).

EDDY-BREAK-UP MODEL

5.6.1

335

Spalding’s EBU Model

Several theoretical models describing the interaction phenomenon between turbulence eddy and ﬂame front structure have been developed; one is Spalding’s eddy-breakup model (Spalding, 1971b, 1976), which was developed in 1971 and was modiﬁed further in 1976. Spalding found that when the chemical reaction rate in a premixed turbulent ﬂame was assumed to be an Arrhenius-type function of the local time-averaged properties, results obtained disagreed from the experimental data. However, when the local volumetric reaction rate was assumed to be a function of the rate of break-up of lumps of unburned mixture, the results agreed better with the experimental data. Based on this observation, spalding proposed that the importance of chemical kinetics in the determination of the local consumption rate of premixed gases was considered. The rate of consumption of the premixed reactants should depend more on the rate of breakup of lumps of unburned fuel/air mixture. The rate of breakup of lumps is controlled by turbulence. Therefore, the EBU model stresses the importance of turbulence effects over chemical kinetics in the determination of fuel consumption rate (or heat release rate). Spalding suggested that turbulent combustion processes are best understood by focusing attention on coherent bodies of gas mixture, which are squeezed and stretched during their travel through the ﬂame. This suggestion leads to an expression for the rate of reaction that can be used together with a suitable scheme for solving the relevant differential equations in order to predict turbulent premixed ﬂame phenomenon. Spalding (1970, 1971b) proposed that the mean rate of energy production in the reactions (ω˙ T∗ ) under the EBU concept is: ε ∗ ω˙ T = CEBU ρ (5.54) T 2 k This expression also can be obtained from dimensional arguments, without invoking the particular physical arguments (which are probably simplistic) Spalding originally used. In terms of a nondimensional temperature ≡ (T − Tu )/(Tb − Tu ), the EBU model can be written as: ε ∗ ω˙ = CEBU ρ (5.55) 2 k 2 is the variance of temwhere is the nondimensionalized temperature and perature ﬂuctuation. Physically, the temperature ﬂuctuation could be interpreted as the measure of unmixedness (not to be confused with intermittency) between the relatively cold unburned pockets of fuel/air mixture and the hot burned products. The ratio of turbulent kinetic energy and turbulent dissipation rate (k/ε) can be regarded as the characteristic turbulent mixing time. The EBU constant CEBU is a model parameter. For an inﬁnitely thin ﬂame, 2 2 − 2 = ρ − ˜ =ρ ˜2 =ρ ˜ − ˜ 2 = ρ ˜ 1− ˜ ρ

(5.56)

336

TURBULENT PREMIXED FLAMES

2 = ˜ was used. This simpliﬁcation was made based In the above equation, ˜ on the argument that can take a value of either 0 or 1 for fresh unburned gas mixture or burned products (also see Poinsot and Veynante, 2005). The ﬁnal EBU model for the mean reaction rate is: ε ∗ ˜ 1− ˜ ω˙ = CEBU ρ (5.57) k There is a mismatch between Equations 5.57 and 5.55 as a square root is missing from Equation 5.57. The reason why the square root was removed from the ﬁnal EBU model stems from various physical and mathematical problems during model implementation (see Poinsot and Veynante, 2005). Certain commercial codes have adopted Spalding’s EBU model as one of the options for turbulent premixed ﬂame calculations in the early days. 5.6.2

Magnussen and Hjertager’s EBU Model

Magnussen and Hjertager (1977) also proposed an EBU model that was different from, although related to, Spalding’s model. Magnussen and Hjertager’s model was developed by assuming that in most technical applications, the chemical reaction rates are fast compared to the mixing. Thus, the reaction rate is governed by the rate of intermixing of fuel- and oxygen-containing eddies (i.e., by the rate of the dissipation of eddies). For such a case, the EBU model can be written: ˜O ˜P ε Y Y 3 ,B (5.58) ω˙ F kg/m · s = Aρ min Y˜F , k ν 1+ν where

ω˙ F = the average fuel consumption rate (The tilde indicates Favre averaging.) A and B = experimentally determined constants of the model ε = turbulence dissipation rate k = turbulent kinetic energy Y = mass fraction ν = stoichiometric coefﬁcient for the overall reaction written on mass basis

The product dependence for the reaction rate is a deviation from the pure fast chemistry assumption, since the assumption here is that without products the temperature will be too low for reactions. In many recent studies, the EBU is used in an extended form, which allows for the use of ﬁnite-rate chemistry. In this version, the EBU model can be written: ˜O Y ε ε 3 (5.59) , ω˙ kinetic,rj ω˙ F,rj kg/m · s = min Aρ Y˜F , Aρ k k νrj

INTERMITTENCY

where

337

ω˙ F,rj = fuel consumption rate νrj = stoichiometric coefﬁcients for reactants (per mole of fuel) in j th reaction subscript rj = that the j th reaction is considered.

This form implies that it is possible to take multistep reaction kinetics into account with the modiﬁed EBU model. One weakness of this model is that the kinetically controlled reaction rate is calculated using mean quantities (i.e., mean concentrations and a mean temperature). Despite the limitations of this model, encouraging results have been reported using this form of the EBU model (Magel et al., 1995). Although the EBU model is an ad hoc model, Duclos, Veynante, and Poinsot (1993) have pointed out that in the premixed case, a number of ﬂamelet models reduce to an EBU form when there is a local equilibrium between production and dissipation of ﬂame surface density. Several commercial engineering software, based on computational ﬂuid dynamics (CFD) use the EBU model of Magnussen and Hjertager (1977). One reason for its widespread use is that it is easy to implement. The model usually is implemented in such a way that a multistep reaction mechanism can be utilized. The EBU model has applications in turbulent premixed ﬂames as well as in turbulent nonpremixed ﬂames, as discussed in Chapter 6.

5.7

INTERMITTENCY

The phenomenon of intermittency in shear ﬂows was discovered by Corrsin (1943) and Townsend (1948), who documented two different types of signals from a hot-wire probe placed near the outer edge of a free ﬂow (a jet in Corrsin, 1943, and a wake in Townsend, 1948). The signals showed either sharply or smoothly varying velocity ﬂuctuations. Corrsin and Kistler (1955) and Townsend (1976) further investigated this phenomenon and proposed that, in free turbulent ﬂows, there is a thin wrinkled interface (called the entrainment interface or viscous superlayer) that separates the turbulent ﬂow characterized by a large vorticity and the ambient, essentially irrotational ﬂow. The two types of signals were recorded in these two types of ﬂows, respectively. Townsend, observing that there is a relatively sharp division between regions of fully developed turbulence and those of nearly laminar ﬂow, postulated that the regions of fully developed turbulence existed for a sufﬁciently long time to allow the establishment of local isotropy over approximately the whole region. The ﬂow picture is of an intermittent distribution of turbulence with the sharp and distinct boundaries between regions of nearly laminar motion and each individual region of turbulent motion where local isotropy was established. The existence of local isotropy in the turbulent wake of a cylinder has been veriﬁed, but allowance must be made for the intermittent character of the ﬂow. The ﬂow alternates irregularly between

338

TURBULENT PREMIXED FLAMES

Reactants Flame front

uu

Products c=0

c=1

Figure 5.23 Schematic representation of the structure of a premixed turbulent ﬂame under conditions of intense turbulence (modiﬁed from Libby, 1989).

laminar and turbulent ﬂow, but within each patch of turbulent ﬂow, local isotropy is established. The aerothermochemical picture of the structure of a turbulent premixed ﬂame in intense turbulence is shown in Figure 5.23. This schematic diagram corresponds to the shredding of reacting surfaces into packets of products surrounded by and growing in a background of reactants. Similarly, the packets of reactants are surrounded by and consumed in the background of products. At this point, it is useful to introduce the vorticity equation, since turbulent ﬂows are always rotational: ∂ζ 1 + (u · ∇) ζ = (ζ · ∇) u + ν∇ 2 ζ −ζ (∇ · u) + 2 ∇ρ × ∇ρ ρ ∂t 1 2 3

(5.60)

4

where ζ = ∇ × u.

Due to the viscous diffusion of vorticity (term 2 in Equation 5.60), any irrotational ﬂow region in the vicinity of a vortex acquires vorticity, which is then ampliﬁed by the stretching caused by the turbulent ﬂow (term 1 in the Equation). More detailed discussion of vorticity is given in Chapter 4. An important feature of turbulent premixed combustion is the intermittency of unburned and burned mixture that are separated by a thin, wrinkled (or corrugated) instantaneous ﬂame front, the thickness δf of which is much less than the thickness δt of a turbulent ﬂame brush.

FLAME-TURBULENCE INTERACTION

339

Parallel to this phenomenon, another, internal intermittency of turbulent ﬂows was discovered by Batchelor and Townsend (1949), who documented that the instantaneous dissipation rate 2νsij sij intermittently attained values much larger than the mean dissipation rate ε. Interested readers are referred to books by Monin and Yaglom (1975) and by Pope (2000) and review papers by Bilger (2004) and Sreenivasan (2004) for more detailed discussion of intermittency in incompressible ﬂows. In summary, three types of intermittency have been observed: (1) turbulent/laminar ﬂow at the edge of shear ﬂow; (2) internal intermittency in fully turbulent ﬂows; and (3) burned/unburned gas intermittency in premixed turbulent ﬂames.

5.8

FLAME-TURBULENCE INTERACTION

As discussed in the introduction, ﬂame-turbulence interaction has two aspects: the effect of turbulence on the ﬂame and the effect of combustion on the turbulent ﬂow ﬁeld. The effects of turbulence on ﬂame propagation are associated either with the calculation of mean reaction rate (ω˙ c ) or with evaluating the turbulent burning velocity (ST ). The functional dependencies of these two parameters can be hypothesized as: uT l0 ST = uT F1 , (5.61) SL δL uT uT l0 ρu ω˙ c = ρu F2 , , Yi , τ where τ ≡ −1 (5.62) l0 SL δL ρb where uT represents the rms turbulent burning velocity of ST . This velocity may differ signiﬁcantly from the rms velocities associated with turbulent ﬂuctuations ˜ 1/2 ]. The rms (u or urms ) or turbulent kinetic energy [i.e., (ρui ui /ρ)1/2 or (2k/3) turbulent velocity is calculated with contributions from both burned and unburned portions of the mixture, whereas the effects of turbulence on ﬂame propagation often are associated only with the upstream turbulence just ahead of the ﬂame front. The term τ in Equation 5.62 is called the heat release factor, governed by the density change from unburned to the burned conditions. The functions F1 and F2 cannot be speciﬁed in a general form; however, various researchers have attempted to model these functions with reasonable success. The effects of the premixed ﬂame on turbulence can be divided into direct and indirect categories. The indirect effects are associated with the ﬂame-induced changes in the magnitudes of some terms in the governing equations. The direct effects are mainly associated with the heat release in the ﬂame, and the pressure drop across the combustion zone. Due to the heat release, gas density decreases within the instantaneous ﬂame front, and the variations in density affect ﬂow

340

TURBULENT PREMIXED FLAMES

velocity by virtue of mass conservation. Such effects are not restricted to the ﬂame but also may be pronounced ahead of the ﬂame. For example, perturbations propagate at extremely high speed in an incompressible ﬂow. The perturbation of such a ﬂow at a ﬂame front instantly disturbs the ﬂow far ahead of the ﬂame. Flame-generated disturbances may be produced by velocity gradients induced by the pressure drop across the combustion zone associated with the density decrease of the gases upon passage through the ﬂame front. In most instances such disturbances are believed to be of importance in distorting the ﬂame. The combustion pressure drop causes the low-density burned gases to accelerate more rapidly than the unburned gases. The uniform ﬂow is thus converted upon passage through the combustion zone into a nonuniform ﬂow with large velocity gradients near the ﬂame front. These gradients are potential sources of turbulence. Under conditions where the Reynolds number of the ﬂow is sufﬁciently high, all of the energy loss due to mixing of the burned gas with unburned gas near the combustion zone might result in increase in turbulence. Another effect of the ﬂame on turbulence via temperature can be seen in Figure 5.24. On this plot, the instantaneous ﬂame front location has been shown by a solid dark line, which is designated by G = 0 surface. The G-equation model is an important topic in turbulent premixed combustion; it is discussed later in this chapter. From this direct numerical stimulation (DNS) result (at a relatively low Reynolds number Rel0 ≈ 820 and with homogeneous isotropic turbulence) of Treurniet, Nieuwstadt, and Boersma (2006), it can be seen that the vorticity drops behind the ﬂame because of low dynamic viscosity (a consequence of the high temperature in the burned region). Unburned region

Flame surface designated by G=0

0.8

0.6 z

Burned region

0.4

0.2

2.0

2.5

3.0

3.5

x ζiζi 0

10

20

30

40

50

60

70

80

√

90

100

Figure 5.24 DNS results for the modulus of the vorticity ζi ζi , plotted in an instantaneous slice in the (x, z )-plane (modiﬁed from Treurniet, Nieuwstadt, and Boersma, 2006).

FLAME-TURBULENCE INTERACTION

5.8.1

341

Effects of Flame on Turbulence

Scurlock (1953) was a pioneering researcher who put forward a hypothesis of ﬂame-generated turbulence. By analyzing experimental results from conﬁned ﬂames stabilized behind bluff bodies, he assumed that large mean velocity gradients caused by the acceleration of hot combustion products generated additional turbulence, which markedly exceeded the approach stream turbulence. Scurlock assumed that ﬂame-generated turbulence could increase turbulent ﬂame speed ST similar to the unperturbed turbulence ﬂuctuating velocity. Therefore, the sum of the ﬂame-generated ﬂuctuating velocity (ufg ) and unperturbed turbulent ﬂuctuating velocity should be used to determine the turbulent ﬂame speed. This model reasonably described the early experimental data on turbulent ﬂame speeds reported by Williams and Bollinger (1949) and Karlovitz, Denniston, and Wells (1951), with a substantial contribution by ﬂame-generated turbulence to the calculated ST . Karlovitz, Denniston, and Wells also reported results of measurements of the speeds of Bunsen ﬂames stabilized in fully developed turbulent pipe ﬂows. The ﬂames were enveloped by a coaxial air ﬂow of comparable velocity to reduce turbulence generation in the shear layer. The measured ﬂame speeds were much higher than the values of ST yielded by the models developed early by Damk¨ohler (1940); Shchelkin (1943); and Karlovitz, Denniston, and Wells (1951). To explain this quantitative difference, Physically, the ﬂow velocity normal to an instantaneous ﬂame front increases from SL to SL (1 + τ ) (in the coordinate framework attached to the ﬂame front) due to mass conservation. Since the ﬂame front is wrinkled, the orientation of the “ﬂame-induced” velocity τ SL ﬂuctuates so that it can be decomposed into the average and random ﬂame-induced velocities, with the latter quantity being ﬂamegenerated rms turbulent velocity. The ﬂame-generated turbulence diffuses against the ﬂow before the ﬂame front, increases ﬂame-front surface distortions, and, therefore, accelerates ﬂame propagation. Karlovitz estimated ufg to be much higher than the rms turbulent velocity of the approach ﬂow under the conditions of their measurements. As noted by Sokolik (1960), the rationale for introducing the concept of ﬂamegenerated turbulence was the discrepancy between calculated and measured ST . For instance, Richmond et al. (1957) applied a burner identical to that used by Karlovitz, Denniston, and Wells (1951) but improved the method of measuring turbulent ﬂame speed. Richmond et al. showed that Karlovitz’s method substantially overestimated ST under the measurement conditions because Karlovitz, Denniston, and Wells did not take into account the cylindrical curvature of the burner and the divergence of the mean ﬂow in the unburned mixture. Many other measurements of turbulent ﬂame speeds performed in the 1950s and 1960s yielded values of ST comparable with u and thus did not imply that ﬂamegenerated turbulence played a substantial role in the burning rate. However, a few researchers, such as Kozachenko (1962), invoked the hypothesis of ﬂamegenerated turbulence to explain high ﬂame speeds that they measured. Scurlock’s and Karlovitz’s hypotheses called for experimental investigations of turbulence in premixed ﬂames. Measurements that used hot-wire anemometers were restricted

342

TURBULENT PREMIXED FLAMES

to unburned gas. Gross (1955) did not observe a signiﬁcant increase in the turbulent intensity ahead of an open V-shaped ﬂame. Jensen (1956) investigated conﬁned ﬂames stabilized behind a rod and reported an increase in velocity ﬂuctuations in the upstream ﬂow in the combustion case. He associated this increase with acoustic phenomena. Modeling of the effects of a premixed ﬂame on turbulence is still a major challenge to the combustion community. Despite the abundance of particular results, the general understanding of these effects is poor in comparison with the general understanding of the effects of turbulence on ﬂames. Many important issues relevant to the former effects are on the periphery of scientiﬁc discussions, and some basic problems are not discussed in contemporary literature. For example, what quantities should be used to characterize turbulence in a premixed ﬂame? A review paper by Lipatnikov and Chomiak (2010) lists several issues regarding this question. Here we summarize the physical mechanisms of ﬂamegenerated turbulence, experimental results obtained by several researchers, and the unresolved questions pertaining to this important topic. • Pressure perturbations generated in a laminar premixed ﬂame may enhance velocity perturbations in the approach ﬂow and generate vorticity in the burned mixture, even though the vorticity ﬁeld in the unburned mixture remains laminar. This mechanism describes the generation of ﬂame-induced turbulence in formerly laminar ﬂow. This analogy can be extended to describe the ﬂame-generated turbulence in a turbulent premixed ﬂame. • The difference in the instantaneous normal ﬂow velocities immediately ahead and behind a ﬂame front contributes to the turbulence in the burned mixture after averaging over random orientations of the ﬂame fronts. Then the ﬂame-generated turbulence diffuses to the unburned mixture (the Karlovitz model). • Intermittency and preferential acceleration of the burned mixture by the mean pressure gradient within a free turbulent ﬂame can cause countergradient diffusion and increase the apparent energy [k˜ = (ρui ui /2ρ)1/2 ]. 5.9

BRAY-MOSS-LIBBY MODEL

The ﬁrst model of premixed turbulent combustion that straightforwardly addressed the unburned-burned intermittency was developed by Prudnikov (1960, 1967) and Prudnikov, Volynski, and Sagalorich (1971). In contemporary literature, the unburned-burned intermittency is commonly described using the approach developed by Bray, Moss, and Libby (1977). A new variable was introduced at that time, called the combustion progress variable, c. Originally, the combustion progress variable was introduced as the product mass fraction when only two species are identiﬁed as reactant (mass fraction 1-c) and product (mass fraction c). The combustion progress variable describes the

BRAY-MOSS-LIBBY MODEL

343

thermochemical state of the mixture at any point in space and time. By this deﬁnition, the state c = 0 denotes reactants (unburned state); c = 1 indicates products (burned state); and any intermediate values between 0 and 1 denote gas-mixture with temperatures and composition between those of reactants and products, involving the partially burned mixture. For example, the average pressure of the system can be determined as: p = ρRu T

1−c c + Mwu Mwb

(5.63)

Under idealized conditions and with some assumptions, the reaction progress variable can also be expressed as: T − Tu ρu − ρ ≈ Tb − T u ρu − ρb

c≈

or

c=

γp γp,b

(5.64)

In these equations, the subscripts u and b represent unburned and burned mixture, respectively. Equation 5.64 can also be written as T = 1 + τ c, Tu

ρ Tu 1 = = ρu T 1 + τc

τ≡

and

qc γu cp T u

(5.64a)

where τ is the heat release parameter, it is also true that τ = (Tb /Tu ) − 1 = (ρu /ρb ) − 1 = (1 − rp )/re , where rp is the density ratio of ρb /ρu . The Bray-Moss-Libby (BML) model introduced the bimodal approximation of the joint probability density function (PDF) that characterizes the probability of ﬁnding a value of the combustion progress variable c at spatial location x and at instant t. The pdf P (c; x, t) is expressed in three parts, representing pockets of unburned reactants and fully burned products, with probabilities α(x, t) and β(x, t), respectively, and a partially burned mixture with probability γ (x , t).

P (c; x, t) = α (x, t) δ (c) + β (x, t) δ (1 − c) + γ (x, t) f (c; x)

unburned gases

burned gases

(5.65)

partially burned mixture

In Equation 5.65, δ is Dirac-delta function and α(t, x ) and β(t, x ) are the probabilities of ﬁnding the unburned and burned mixture, respectively, at point x at instant t (see Figure 5.25). The function f (c; x ) satisﬁes this condition: t

f (c; x, t) dc = 1.0 0

(5.66)

344

TURBULENT PREMIXED FLAMES α (x,t) β (x,t)

P (c;x,t)

γ (x,t) f (c;x,t)

0

Figure 5.25

1

c (x,t)

Bimodal probability distribution function for reaction progress variable, c.

The function f can be considered as a continuous function of product mass fraction. From the characteristics of a pdf, it can be shown readily that the coefﬁcients α, β, and γ must satisfy this relationship: α (x, t) + β (x, t) + γ (x, t) = 1

(5.67)

The central assumption of BML approach is that the ﬂow conditions and chemical system are such that the chemistry is “fast”; thus, the intermediate values of c associated with the strength γ (x,t) are relatively rare. Based on this assumption, the probability γ (x,t) of ﬁnding intermediate states of a gas mixture is much less than unity at any point x at any instant t. The bimodal approximation is therefore based on this assumption. In the fast-chemistry or thin-ﬂame combustion regime, in which a turbulent ﬂame brush consists of unburned and fully burned gases separated by thin burning zones, it is found that γ ∼ O(1/Da), where Da is the Damk¨ohler number representing the ratio of turbulent mixing time to the chemical reaction time. The novel conceptual viewpoint following from this formulation implies that all mean quantities are the sum of three contributions: the conditional means in reactants and products, weighted by their probabilities, α and β, respectively, together with a burning zone contribution weighted by γ . The mean values of quantities, such as velocity or progress variables, are simply weighted averages of their conditional means in reactants and products and are independent of γ . Reaction rates and reactant concentration gradients are zero everywhere, except in the thin-ﬂame reaction zones, so their means are proportional to γ . For high Da ﬂow, γ 1. Therefore,

P (c; x, t) = α (x, t) δ (c) + β (x, t) δ (1 − c)

unburned gases

burned gases

(5.68)

BRAY-MOSS-LIBBY MODEL

345

3000

Voltage (arbitary units)

2500

2000

1500

1000

500

0 360

364

368

372 Times (ms)

378

380

384

(a) c or Θ 1

time

0 (b)

Figure 5.26 Intermittency between fresh and fully burned gases at a location x in the reaction zone: (a) instantaneous Rayleigh-scattering signal recorded from a lean (φ = 0.8) methane/air turbulent Bunsen ﬂame (modiﬁed from Deschamps et al., 1992) and (b) schematic of the intermittency function.

where α (x, t) + β (x, t) = 1

(5.69)

The intermittency between unburned and fully burned gases at a given location x in the reaction zone can be expressed by the graph shown in Figure 5.26. The bimodal pdf distribution of the progress variable c has been veriﬁed by the DNS results of Chakraborty and Cant (2009), as shown in Figure 5.27. The measured temperature pdfs for methane/air turbulent Bunsen ﬂames at various Da are shown in Figure 5.28. Other measured pdfs for temperature ﬂuctuations at several radial and axial locations are shown in Figure 5.29. These studies (DNS and experimental) conﬁrm the existence of bimodal pdfs in turbulent premixed ﬂames.

346

TURBULENT PREMIXED FLAMES 0.3 0.25

Probability

0.2 0.15 0.1 0.05 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Combustion progress variable

Figure 5.27 Combustion progress variable pdf obtained by DNS (modiﬁed from Chakraborty and Cant, 2009). 0.02

Probaility density function

ReΛT = 31.6

(a) 0.01

(b) (c)

0

0

340

680

1360 1020 Temperature (°C)

1700

2040

Figure 5.28 Temperature pdfs obtained from methane air turbulent Bunsen ﬂames at Damk¨ohler number (a) = 0.77, (b) = 2.18, and (c) = 4.59 (modiﬁed from Yoshida, 1988).

By assuming a bimodal pdf for the reaction progress variable, we decompose many terms into contributions coming from differences between reactants and products. This simpliﬁes closure so that with a few modeling assumptions, the BML approach can predict four things: 1. Turbulent kinetic energy increases through the ﬂame brush at higher heat release rates. 2. Mean dilatation and dissipation reduce turbulent kinetic energy.

BRAY-MOSS-LIBBY MODEL

347

19 18 16

4 PDF × 103

15

x/D = 17 5

14

3

13

2 1 0

500 0

1000

T (°C) 1700

(a) 0.95 0.85

0.55 0.50 0.45 0.35 0.30

PDF × 103

R/D = 0.75 15

10

5 500 0

1000 T (°C) 1700

0

(b)

Figure 5.29 Measured probability density function of temperature (a) axial and (b) radial (Yoshida and Tsuji, 1979).

3. The mean pressure gradient is a source of turbulent kinetic energy. 4. Countergradient transport of scalars prevail in the ﬂame brush under some conditions. The original BML model considered the mean pressure gradients only, although the omission of the ﬂuctuating pressure is questionable and has been included in later studies.

348

TURBULENT PREMIXED FLAMES

The mean value of any quantity can be obtained from Equation 5.70: 1

q= 0

where qu ≡

1

qP (c; x, t) dc = α (x, t) qu + β (x, t) qb

qδ (c) dc

0

and

qb ≡

1

qδ (1 − c) dc

(5.70)

(5.71)

0

Based on these deﬁnitions, the ratio of the mean density of the mixture can be calculated as: 1

ρ (x, t) = 0

ρ P (c; x, t) dc = α (x, t) ρu + β (x, t) ρb =

(1 + ατ ) ρu (1 + τ )

(5.72)

Recall that τ is the heat release factor, deﬁned as τ = (ρu /ρb ) − 1 = (Tb /Tu ) − 1. The Favre averaged progress variable can be obtained from: c˜ (x, t) ≡

ρc ρu = ρ ρ

1 0

ρu β (x, t) c P (c; x, t) dc = 1 + τc ρ (1 + τ )

(5.73)

The time-averaged mean progress variable can be expressed as: 1

c= 0

cP (c; x, t) dc = α (x, t) cu + β (x, t) cb = β (x, t)

(5.74)

Note that at unburned state cu = 0 and at burned state cb = 1. The Favre-averaged progress variable c˜ is related to the time-averaged mean progress variable according to: c (5.75) c˜ (x, t) ≡ 1 + (1 − c) τ This also yields a simple expression for the density ratio in terms of τ and c: ˜ 1 ρ (x, t) = ρu 1 + τ c˜

(5.76)

An extension of the methodology introduced by Equation 5.68 is to consider the joint pdf of the progress variable and any velocity component, for example, the ﬂame normal velocity u in x -direction. This may then be written as:

P (u, c; x, t) = α (x, t) δ (c) P (uu ; x, t) + β (x, t) δ (1 − c) P (ub ; x, t)

(5.77)

BRAY-MOSS-LIBBY MODEL

349

where P (uu ; x, t) and P (ub ; x, t) are conditional pdfs of the velocities in the unburned and burned mixture, respectively. Multiplying Equation 5.77 by the product of ρu and integrating over u and c spaces, we obtain the Favre averaged velocity as: 1

1 u˜ (x, t) = ρ

∞

ρuP (u, c; x, t) dudc = (1 − c) ˜ uu (x, t) + c˜ ub (x, t) (5.78) 0 −∞

where uu and ub are conditional mean velocities in the unburned and burned mixtures, respectively. Ten additional assumptions used in BML model are discussed next. 1. Only two chemical species are considered, “reactant” (mass fraction 1–c) and “product” (mass fraction c), which are deﬁned for a combustible mixture of given stoichiometry and which may include diluents. 2. A single-step, irreversible chemical reaction, reactant→product, occurs at a rate described by a global reaction-rate expression. 3. Reactant and product species are treated as ideal gases. 4. The speciﬁc heat at constant pressure, Cp , of both reactant and product is the same and is a constant. 5. Thermal-diffusion and pressure-diffusion effects are ignored, while normal binary diffusion is represented through Fick’s law. 6. The Lewis number [Le ≡ l/(ρCp D)] is equal to unity for both reactant and product. 7. The ﬂow within the ﬂame occurs at a Mach number much less than unity, so that the terms in the energy-balance equation representing effects of pressure changes and viscous dissipation may be ignored. 8. Pressure ﬂuctuations are assumed to be of small intensity and are ignored. 9. The ﬂow is adiabatic 10. The ﬂow far upstream of the combustion zone is steady, one-dimensional, and uniform in all of its properties. These assumptions are used in present-day simulations of turbulent premixed ﬂames too, although the one-dimensionality assumption has been relaxed for simulation of turbulence.

5.9.1

Governing Equations

The transport equation for the progress variable can be derived from either the species equation or the energy equation: ∂ ∂c ∂ ∂ (5.79) ρD + ω˙ c (ρui c) = (ρc) + ∂t ∂xi ∂xi ∂xi

350

TURBULENT PREMIXED FLAMES

Other equations are: ∂ρ ∂ (5.80) + (ρui ) = 0 ∂t ∂xi ∂ui ∂uj 2 ∂uk ∂ ∂p ∂τij ∂ + where τij = μ + − δij ρui uj = − (ρui ) + ∂t ∂xj ∂xi ∂xj ∂xj ∂xi 3 ∂xk (5.81) N ∂p ∂Yk ∂ ∂ ∂h ∂ +ρ D k hk ρuj h = + ρα − q˙rad,loss (ρh) + ∂t ∂xj ∂t ∂xi ∂xi ∂xi k=1

(5.82) Favre averaged equations for the mean and the variance of the reactive scalars can be derived by decomposing c into a Favre mean and a ﬂuctuation component: c = c˜ + c

(5.83)

After substituting Equation 5.83 into Equation 5.79 and taking the average, we have: ∂ c˜ ∂ ∂ ∂c ∂ c˜ ρ =− ρui c + ω˙ c + ρD (5.84) + ρ u˜ i ∂t ∂xi ∂x ∂x ∂xi i i A

B

C

Similarly, Equation 5.81 becomes: ρ

∂ u˜ i ∂ u˜ i ∂p ∂ ∂τ ik =− − ρui uk + + ρ u˜ k ∂t ∂xk ∂xi ∂xk ∂xk

(5.85)

The terms A, B , and C are called turbulent scalar transport, mean chemical source term, and molecular transport, respectively. The molecular transport term is usually small in comparison with the other two terms, both of which require modeling. B has received considerable attention in recent years, and various models have been derived and incorporated into practical codes for turbulent combustion. Most often, pdf modeling is used to close the source term B , which is discussed in Sections 5.13 and 5.14. The A, however, has received considerably less attention and generally is described with a simple classical gradient eddyviscosity model (also known as gradient diffusion model): μt ∂ c˜ ρui c = ρ ui c = − Sct ∂xi

(5.86)

Both theoretical and experimental research (Bray et al. 1981; Bray, Moss, and Libby 1982; Shepherd, Moss, and Bray, 1982) have found evidence of countergradient transport in some turbulent ﬂames. Countergradient transport means that in these ﬂames, the turbulent ﬂux ρ u ˜ i have the i c and the gradient ∂ c/∂x same sign, which is opposite of the prediction of Equation 5.86. This leads to the

351

BRAY-MOSS-LIBBY MODEL

possibility that turbulent viscosity should be negative. Details of this issue are discussed in a next subsection. In short, this effect generally is due to the different effects of pressure gradients on cold reactants and hot products. Studies based on direct numerical simulations of turbulent premixed ﬂames without externally imposed pressure gradients (Rutland and Cant, 1994; Trouv´e and Poinsot, 1994) have conﬁrmed that both gradient and countergradient transport are possible. The quantities ρa b , where a and b are certain characteristics of the turbulent ﬂow, are called second moments. Balance equation for Reynolds stresses are: ρui uj ∂ ∂ ρui uj ρ + ρ u˜ i ∂t ρ ∂xi ρ ∂ u˜ j ∂τj k ∂ u˜ i ∂τik ∂ = −ρui uk − ρui uk − ρui uj uk +uj + ui ∂xk ∂xk ∂xk ∂xk ∂xk I

∂p

II

III

∂p

∂p ∂p −uj − ui −uj − ui ∂xi ∂xj ∂xi ∂xj

(5.87)

V

IV

Summation of Equation 5.87 leads to the Favre-averaged turbulent (TKE) ˜ equation (or k-equation): ρ

∂ u˜ j ∂τj k ∂ ˜ ∂ ∂ ˜ ∂p ∂p − ρuk k +uj −uk −uk k + ρ u˜ i k = −ρuj uk ∂t ∂xi ∂xk ∂xk ∂x ∂x ∂x k k k I

II

III

IV

V

(5.88) where

k˜ ≡

ρuk uk

. 2ρ The turbulence scalar transport is also a second moment. The balance equation for turbulent scalar transport is: ρui c ∂ ρui c ∂ ρ + ρ u˜ k ∂t ρ ∂xk ρ A

B

∂ ∂ c˜ ∂ u˜ i ∂τik = −ρui uk −ρuk c − ρu u c +c ∂xk ∂xk ∂xk i k ∂x k

i1

i

i2

ii

iii

∂c ∂ ∂p ∂p + ui −c −c + ui ω˙ c ρD ∂xk ∂xk ∂xi ∂xi iv

v

vi

vii

(5.89)

352

TURBULENT PREMIXED FLAMES

The right-hand side of Equation 5.89 for the ﬂux ρui c contains seven terms. Term i is closed if the Favre-averaged Reynolds stresses ρui uk are evaluated by solving their transport equations. In order to close the term vi , Equation 5.90 can be used: c ≡ (c − c) ˜ = ρτ c(1 ˜ − c)/ρ ˜ u = (ρu − ρ) (1 − c)/ρ ˜ u

(5.90)

The remaining ﬁve terms require additional equations to be closed. These are transport term ii , cross-dissipation terms iii and iv , term v (involving the ﬂuctuating pressure gradient), and reaction term vii . Similarly, the balance equation for scalar ﬂuctuations can be written as: ∂ 2 ∂ c˜ ∂ ∂ 2 u˜ k ρc2 = −2ρuk c ρc + − ρuk c ∂t ∂xk ∂xk ∂xk +

2c ω˙

c

+

2c

∂ ∂xk

∂c ρD ∂xk

(5.91)

The last term on the right-hand side of the above equation can be written as: 2c

∂ ∂xk

∂c ρD ∂xk

! " ∂c ∂ c˜ ∂ ∂c − ρχc =2 ρDc − 2ρD ∂xk ∂xk ∂xk ∂xk

(5.92)

where χ c is the mean scalar dissipation, and it is deﬁned by the next equation: χ c ≡ 2D

∂c ∂c ∂xk ∂xk

(5.93)

At high Reynolds numbers, the ﬁrst and second terms on the right-hand side of Equation 5.92 often are ignored and, thus, the last term on the right-hand side of Equation 5.91 is approximately equal to the scalar dissipation associated with −ρχc . At this point, we can again see the difference between the two aspects of ﬂame-turbulence interaction. • Modeling the effects of turbulence on combustion is associated with closing the term B in Equation 5.84. • Modeling the effects of combustion on turbulence is associated with closing terms ii to vii in Equation 5.89. The difference in the number of unclosed terms associated with the effects of turbulence on combustion (a single term B in Equation 5.84) and with the

353

BRAY-MOSS-LIBBY MODEL

effects of combustion on turbulence (sixth terms, ii to vii ) clearly shows the overwhelming complexity of the latter problem. In nonreacting constant-density ﬂows, where wa is the weight percentage of an admixture (a compound formed by mixing), Equations 5.87, 5.88, and 5.89 reduce to: ∂ ∂ ui wa + u˜ k uw ∂t ∂xk i a 2 ∂ 2 ui ∂w a ∂ui ∂ 1 ∂p ∂ wa − uk wa − ui uk wa +νwa +Du − w = −ui uk i 2 ∂xk ∂xk ∂xk ρ a ∂xi ∂xk2 ∂xk i

ii

iii

iv

v

(5.94) ∂ ∂ ui uj + u˜ k uu ∂t ∂xk i j =

−uj uk

−

1 ρ

2 ∂ 2 uj ∂ ∂ui ∂uj ∂ ui − ui uk − u u u +ν uj 2 + ui 2 ∂xk ∂xk ∂xk i j k ∂xk ∂xk I

uj

II

III

∂p ∂p − ui ∂xi ∂xj

(5.95)

IV

∂ 2 uj 1 ∂ ∂uj ∂ ∂ ∂k k = −uj uk − uk k +νuj − uk p + u˜ k 2 ∂t ∂xk ∂xk ∂xk ρ ∂x ∂x k k I

II

III

(5.96)

IV

In these equations, terms ii to v would represent the effect of mixing on turbulence.

5.9.2

Gradient Transport

By using the equality given by Equation 5.97, Equation 5.96 reduces to Equation 5.98: ∂ 2 uj ∂ uj uk sjk − 2sjk sjk = 2 (5.97) ∂xj ∂xk2 ∂k ∂k ∂ Tk = +P−ε + u˜ k ∂t ∂xk ∂xk

(5.98)

354

TURBULENT PREMIXED FLAMES

The ﬂuctuating rate of strain is deﬁned as: ∂uj 1 ∂uk + sjk ≡ 2 ∂xj ∂xk

(5.99)

The turbulent transport of the TKE is deﬁned as: 1 2

Tk ≡ uj uj uk +

uk p − 2νuj sjk ρ

(5.100)

The term shown in Equation 5.100 is also known as turbulent ﬂux of turbulent kinetic energy. The production term is deﬁned as: ∂uj 1 ∂uk P ≡ −uj uk Sjk where Sjk ≡ + (5.101) 2 ∂xj ∂xk The dissipation term is: ε ≡ 2νsjk sjk ≥ 0

(5.102)

In order to close the production term, a gradient transport approximation usually is applied. The gradient transport approximation is a ﬁrst-order turbulence closure approximation that assumes that turbulent ﬂuxes of any variable are associated with the local gradient of that mean variable, analogous to molecular transport. ∂uj ∂uk 2 uj uk = −νt + (5.103) + kδkj ∂xj ∂xk 3 This local turbulence closure approach assumes that turbulence consists of only small eddies, causing diffusionlike transport. In Equation 5.103, the kinematic ﬂux uj uk is modeled as being equal to an eddy viscosity times the transverse gradient of mean velocity. This theory is also called eddy-viscosity theory.

5.9.3

Countergradient Transport

It is a common practice in turbulent combustion to employ the gradient transport assumption for reactive scalars. The turbulence correlation of velocity ﬂuctuation with reacting scalar ﬂuctuation then takes this form: ρui c = ρ ui c = −ρDT

∂ c˜ ∂xi

(5.104)

with the turbulent diffusivity DT being controlled by turbulence characteristics— for instance, DT ∝ k 2 /ε > 0 within the framework of the k–ε model. Recall Equation 5.70. By setting q = ρc in that Equation, it can be shown that ρc = αρu cu + βρb cb = βρb . Since ρc = ρ c, ˜ we have βρb = ρ c. ˜

BRAY-MOSS-LIBBY MODEL

355

By substituting q = ρ Equation 5.70, we have: ρ = αρu + βρb = αρu + ρ c˜ or αρu = ρ (1 − c). ˜ Therefore, using Equations 5.69, 5.74, and 5.70, we can shown that q = (1 − c) qu + cqb

and

q˜ = (1 − c) ˜ qu + cq ˜ b

(5.105)

By substituting q = ρui c in Equation 5.105, we can get: ρui c = ρ ui c = ρ c˜ (1 − c) ˜ ui,b − ui,u

(5.106)

Libby and Bray (1980) analyzed Equation 5.106, applied it to a statistically planar, one-dimensional ﬂame propagating from right to left (i.e. ∂ c/∂x ˜ > 0 with x pointing from left to right or unburned to burned), and noted that across the ﬂame, we have ub > uu and ρui c > 0, which is contrary to the gradient transport approximation shown by Equation 5.104. Moss (1980) and Yanagi and Mimura (1981) experimentally conﬁrmed the observation of countergradient scalar transport in turbulent Bunsen ﬂames. In turbulent ﬂames, the signs of ρui c and ∂ c/∂x ˜ may be the same (i.e., the transport of the scalar c may occur in the direction of an increasing c). The simplest explanation of this phenomenon is that since pressure decreases from the reactant to the product side of a free ﬂame, the pressure gradient causes a stronger acceleration of the hot, low-density products than of the cold, highdensity reactants. The differential acceleration results in the preferential motion of the hot eddies toward the product side, and, therefore, to countergradient transport (which is sometimes called pressure-driven transport in order to highlight this physical mechanism). In the combustion literature, countergradient scalar transport sometimes is called countergradient diffusion (CGD). For the sake of brevity, we use this abbreviation and also gradient diffusion (GD). However, it is worth emphasizing that the physical mechanisms that cause countergradient scalar transport are totally different from the physical mechanism of turbulent diffusion. Thus, from a basic point of view, CGD does not seem to be the proper term. Veynante et al. (1997) have given the next expression to account for both gradient transport and countergradient transport: ui c = −2ξ u + τ SL c˜ (1 − c) ˜

(5.107)

where u is the rms turbulence velocity ﬂuctuation and ξ is an efﬁciency function. The turbulent scalar ﬂux may be viewed as the sum of two contributions acting in opposite directions, one induced by turbulent motions and the other by thermal expansion. Then the turbulent transport is analyzed in this way: For a sufﬁciently high turbulence level, the ﬂame is unable to impose its own dynamics to the ﬂow ﬁeld, and the turbulent transport is of the gradient transport type for the reacting scalar c. When the turbulence level remains low, however, the thermal expansion due to heat release dominates the process of turbulent scalar transport,

356

TURBULENT PREMIXED FLAMES

and the ﬂame is able to impose its own dynamics leading to a countergradient turbulent transport. Countergradient turbulent transport occurs when u i c is positive. Equation 5.107 can be used to derive a criterion delineating gradient and countergradient regimes. Such criterion indicating the presence of gradient or countergradient transport in atmospheric ﬂames has been derived by Veynante, Trouv´e, Bray, and Mantel (1997). This criterion leads to a reduced number called the Bray number, deﬁned by: τ SL NB ≡ (5.108) 2ξ u The term ξ is an efﬁciency function of order unity, introduced by Veynante et al. (1997) to take into account the reduced ability of small turbulent vortices to affect the ﬂame front. This function is plotted on Figure 5.30 as a function of the length-scale ratio l0 /δL . Countergradient transport is indicated by NB > 1 while gradient transport is indicated by NB < 1. This criterion is well veriﬁed by DNS results, as shown in Figure 5.31. The DNS results for the combustion regime diagram are plotted as a function of the velocity ratio, u /SL , and length-scale ratio, l0 /δL . Also plotted are the DNS results from the Center for Turbulence Research (CTR) at Stanford University for τ = 2.3 and τ = 3. As the turbulence is decaying in the CTR simulation, CTR results are displayed as an almost vertical line. The symbols ◦ (τ = 3) and (τ = 6) correspond to the DNS results from the Centre de Recherche en Combustion Turbulente (CRCT) in France. Filled symbols denote gradient and open symbols denote countergradient turbulent transport. The transition criterion NB ≡ τ SL /2ξ u = 1 separating CGD (below) from GD (above) is plotted for τ = 3 and τ = 6.

Efficiancy function

1.2 1.0 0.8 0.6 0.4 0.2 0 0

5

10

15

20

Length-scale ratio, l0 /δL

Figure 5.30 A DNS-based estimate of the efﬁciency function ξ introduced by Veynante et al. (1997) to take into account the reduced ability of small turbulent vortices to affect the ﬂame front.

BRAY-MOSS-LIBBY MODEL

7 6 5 4 3

DISTRIBUTED REACTION

= Re

t et

nso Poi

10

al

0

Velocity ratio, u′/SL

100

357

2

s

am

FLAMELET COMBUSTION

illi v-W o im

Kl

10 CTR

7 6 5 4 3

G

t=6

CG

t=3

G

2 Re =1

CG

0

1

CR

1

10 100 Length scale ratio, l0 /dL

1000

Figure 5.31 Premixed turbulent combustion diagram (modiﬁed from Veynante et al., 1997).

5.9.4

Closure of Transport Terms

The problem of closing the transport terms II (i.e., ρui uj uk ) and ii (i.e., ρui uj c) in Equations 5.87 and 5.89, respectively, consists of modeling the third moments. 5.9.4.1

Gradient Closure

In early papers, Bray et al. (1976, 1977, 1979) used the gradient transport closure—Equation 5.104 for the transport term II in Equation 5.87. Other researchers also have used the gradient transport approximation of the third moments (Bailly, Karmed, and Champion, 1997; Bigot, Champion, and Garret-Bruguieres, 2000; Bradley, Gaskell, and Gu, 1994; Karmed, Champion, and Bruel, 1999; Lindstedt and Vaos, 2006), using these equations: ∂ k˜ ρui uj uk = −Cs ρuk ul ε˜ ∂xl ρui uj c

∂ k˜ = −Cc ρuj ul ε˜ ∂xl

ρui uj

ρ ρui c ρ

(5.109) (5.110)

where Cs and Cc are constants and typically have values between 0.15 and 0.22. The use of the gradient transport approximation for the third moments has no basic justiﬁcation in ﬂames, where a similar approximation for the second moments is not applicable. Driscoll and Gulati (1988) have documented the countergradient transport of TKE in an oblique unconﬁned ﬂame. The DNS data by Chakraborty and Cant (2009b) do not support Equation 5.110, as shown in

358

TURBULENT PREMIXED FLAMES Gradient diffusion approch (GD model): ∼ k ∂ u ″c″ rui″uj″c″ = −Cs ∼ ruj″uk″ i ∂xk ε 0.3 0.2

ru1″u1″c″/ruS 2L

Binary-Moss-Libby (BML) model:

DNS GD 0.25 × BML CC

0.25 0.15

rui″uj″c″ = rc(1 − c )(1 − 2c )(tSL)2 Chakraborty and Cant (CC)’s model: ∼ k ∂ u ″c″ rui″uj″c″ = − Cs ∼ ruj″uk″ i ∂xk ε

0.1 0.05 0

+c (1 − c )(1 − 2b1c) ×

−0.05 −0.1 −0.15

rui″c″

2

a rc(1 − c ) 1 ui″uj″

a1 = 0.5, b1 = 0.7 0

0.2

0.4

∼c

0.6

0.8

1

Figure 5.32 Comparison of DNS data with three models for the third moment (modiﬁed from Chakraborty and Cant, 2009).

Figure 5.32. At the present time, none of the models is able to correctly predict the third moments, calculated by using DNS; however, these models can predict the general trend for the third moment. 5.9.4.2

BML Closure To develop a more consistent approach, Libby and Bray (1980, 1981) have proposed closing the transport terms ii in Equation 5.89 and II in Equation 5.87 using the BML model. This approach models the third moments ρui uj c by the next equations:

ρui uj c = ρ c˜ (1 − c) ˜ where

# $ ρui c · ρuj c + (1 − 2c) ˜ ui uj − ui uj (5.111) b u ρ c˜ (1 − c) ˜

ρui c · ρuj c ρ c˜ ui uj = ρui uj − ρ (1 − c) ˜ ui u j − b u ρ c˜ (1 − c) ˜

(5.112)

To complete the model, a submodel of the Reynolds stresses (ui uj )u conditioned on unburned mixture is required. To propose such a submodel for a statistically steady, planar, one-dimensional ﬂame, Libby and Bray initially assumed: u2 u20 u = u2 u2u 0

(5.113)

359

BRAY-MOSS-LIBBY MODEL

where the subscript 0 designates the ﬂame leading edge. However, this expression yields a rapid decay of u2 u with a downstream distance, whereas the experimental data by Moss (1980) indicated an increase in u2 u in the same direction. To correct the model, Bray, Libby, Masuya, and Moss (1981) proposed using the next closure, where K3 = 0.1. 2 2 2 u2 b − uu = K3 ub − uu

(5.114)

Masuya and Libby (1981) used a similar closure for modeling conditioned rms velocity tangential to the mean ﬂame surface. Experimental data obtained by Cheng and Shepherd (1987) from V-shaped ﬂames showed variations in K3 ranging from −0.011 to 0.102 and do not lend strong support to Equation 5.114, which they claimed was a satisfactory ﬁrst approximation. Anand and Pope (1987) reported that the numerical results obtained by solving a balance equation for a joint PDF differ signiﬁcantly from Equation 5.114. Measurements by Driscoll and Gulati (1988) do not support this closure either. Libby (1985) improved the closure is Equation 5.114 as shown next. # $ 2 c˜ u2 u2 + (1 − Kr ) × u2 + 1 − Kp u2 b − uu = − 1 − Kp ∞

0

0

(5.115)

where Kp = 0.5 and Kr = 0.5 are constants, and indices 0 and ∞ designate the leading and trailing edges of ﬂame brush. Driscoll and Gulati (1988) have shown 2 that this closure yields a decrease in u2 b − uu with c˜ in reasonable agreement with their experimental data. Heitor, Taylor, and Whitelaw (1987) have claimed that their experimental results support neither Equation 5.114 nor Equation 5.115. Bray, Moss, and Libby (1985) have proposed using this closure:

ui uj

ρui uj − ui uj = 1 − Kij 1 c˜ + Kij 0 − 1 (1 − c) ˜ b u ρ

(5.116)

where Kij 0 > 0 and 0 < Kij 1 < 1 are constants. This closure was used by Cant and Bray (1989) with K110 = 1.1 and K111 = 0.9. The measurements by Driscoll and Gulati (1988) do not support Equation 5.116. Now let us consider the third moment in the Reynolds stress equation—that is term II in Equation 5.87. The BML model for this term is: ρui uj uk = ρ (1 − c) ˜ ui uj uk + ρ c˜ ui uj uk u

ρui c

b

ρuj c

· ρuk c 2

· ρ c˜ (1 − c) ˜ $ $ # # + ui uj − ui uj ρuk c + ui uk − ui uk ρuj c b u b u $ # + uj uk − uj uk ρui c (5.117) + (1 − 2c) ˜

b

u

360

TURBULENT PREMIXED FLAMES

To close the third moments ρui uj uk using this model, we have to know not only the conditioned Reynolds stresses (ui uj )b and (ui uj )u but the conditioned third moments: (ui uj uk )u and (ui uj uk )u . Initially, Libby and Bray (1981) ignored these moments by assuming that the conditioned PDFs Pu (u) and Pb (u) are nearly Gaussian so that (u3 )u = (u3 )b = 0. Cheng and Shepherd (1987) have obtained small values of u3 and v 3 from V-shaped ﬂames and have shown that their results support this assumption. Gulati and Driscoll (Gulati and Driscoll, 1986, 1988; Driscoll and Gulati, 1988) have documented the Gaussian velocity PDFs Pu (u) conditioned on unburned mixture, whereas the form of Pb (u) was found to be non-Gaussian. In the simulations by Anand and Pope (1987), the conditioned third moments (u3 )u and (u3 )b differed signiﬁcantly from zero and were comparable with magnitude of u3 at c˜ > 0.6. (See Figure 5.33). Bray et al. (1981) invoked a gradient transport approximation to evaluate the conditioned third moments as follows: ˜ ρul uk ρui uj ∂ k (5.118) ˜ ui uj uk + ρ c˜ ui uj uk = −Cs (1 − c) u b ε˜ ρ ∂xl ρ where Cs is a positive constant. Cant and Bray (1989) used this submodel with Cs = 0.11. From the most recent studies of turbulent premixed combustion by DNS, it appears that state-of-the-art closure for the third moments is yet to be developed. The BML models can predict the correct trends for turbulent ﬂame speed; however, they do not predict the magnitudes and distributions of these moments correctly.

~

6.0

u″3

3.0 u′3 u 0.0 u′3 b

−3.0

−6.0 0.0

0.2

0.4

0.6

0.8

1.0

∼c

Figure 5.33 Unconditional and conditional third moments of ﬂuctuating velocity calculated by Anand and Pope (1987) for τ = 7 or Tb = 8Tu (modiﬁed from Anand and Pope, 1987).

BRAY-MOSS-LIBBY MODEL

5.9.5

361

Effect of Pressure Fluctuations Gradients

When applying Equations 5.87 and 5.89 to turbulent premixed ﬂames, the most difﬁcult and still unresolved problem consists of closing term IV in Equation 5.87 and term v in Equation 5.89 that involve the ﬂuctuating pressure gradient. Initially, Bray et al. (1982) ignored these terms. Later researchers took these terms into account by invoking models developed for constant-density ﬂows. Before Bray and Libby’s work, Borghi and Dutoya (1978) had done this to account for the ﬂuctuating concentration–pressure gradient term v in Equation 5.89. The problem of closing the ﬂuctuating velocity–pressure gradient term IV in Equation 5.87 in turbulent ﬂames was ﬁrst addressed by Jones (1980), and many subsequent models followed his proposal. Figure 5.34 shows the contribution of various terms in the k -equation to the kinetic energy budget. As can be seen from this plot, the pressure dilatation term is very large in comparison to other terms, and it is a dominant term in the TKE Equation 5.88. Another aspect of the effect of pressure gradients on turbulent premixed combustion relates to the gradient transport model. A favorable pressure gradient (i.e., a pressure decrease from unburned to burned region) is found to decrease ﬂame wrinkling, ﬂame brush thickness, and turbulent ﬂame speed. It also promotes countergradient turbulent transport. Adverse pressure gradients, however, tend to increase ﬂame brush thickness and turbulent ﬂame speed and promote classical pressure dilatation

Kinetic Energy Budget

0.2

0.1

pressure work

0 mean dilatation

−0.1

transport

viscous terms −0.2

0

0.2

0.4 0.6 Mean Progress Variable, c

0.8

1

Figure 5.34 Normalized terms in the TKE equation for a planar turbulent premixed ﬂame (modiﬁed from Zhang and Rutland, 1995). TABLE 5.5. Input Conditions for DNS Study by Veynante and Poinsot (1997) Case A B C D E F

u /SL0

l0 /δL

External Mean Pressure Gradient

5 5 5 2 2 2

3.5 3.5 3.5 3.5 3.5 3.5

0 Less favorable Favorable 0 Less adverse Adverse

362

TURBULENT PREMIXED FLAMES 0.4

1.5

0.3 1.0

0.2 0.1

0.5

0 −0.1

0

−0.2 −0.3

−0.5 0

0.2 0.4 0.6 0.8 Mean progress variable, c

1.0

0

0.8 0.2 0.4 0.6 Mean Progress variable, c

1.0

(b) Zero External Pressure Gradient

(a) Favourable External Pressure 0.8 0.3 0.4 0.2 0 −0.2 −0.4 −0.6 0

0.2 0.4 0.6 0.8 Mean Progress variable, c (c) Adverse External Pressure

1.0

Term of Equation 5.89 A B i1 i2 ii iii iv v vi vii unbalanced

Figure 5.35 Variations of different terms appearing in the turbulent-scalar ﬂux (a) with a favorable pressure gradient (Case C), (b) without a pressure gradient (Case D), and (c) with an adverse pressure gradient (Case F) (modiﬁed from Veynante and Poinsot, 1997). (Note: Different cases are labeled in Table 5.5.)

gradient turbulent transport. Veynante and Poinsot (1997) studied the large pressure gradients on a premixed turbulent ﬂame by using DNS (see Table 5.5). If buoyancy forces are present, there could be an additional or external pressure gradient, which could be either a favorable or an adverse pressure gradient in the ﬂame. The mean pressure gradients have been shown to affect the turbulent-scalar transport term such as ρ ui c . Readers may recall that the gradient transport model for the turbulent-scalar transport requires that ui c /(∂ c/dx ˜ i ) < 0. For a positive ∂ c/dx ˜ (i.e., from unburned to burned gases), gradient transport will occur if i ui c < 0. A typical DNS evaluation of terms appearing in the turbulent-scalar ui c (see Equation 5.89) is shown in Figure 5.35a–c for transport budget of ρ Cases C, D, and F, respectively. Therefore, if any term in this equation contributes to make ρ ui c negative, it promotes gradient transport, and vice versa. These ﬁgures also display the imbalance (i.e., the difference between the sums of the right- and left-hand-side terms) that was found when numerically closing

BRAY-MOSS-LIBBY MODEL

363

the turbulent-scalar transport budget. Generally, there is some imbalance in the governing equation caused by inherent numerical errors involved in the simulations and in the postprocessing of the data. The budget of the transport equation for Case C is presented in Figure 5.35a. The favorable mean pressure gradient acts to promote countergradient turbulent transport from term vi of Equation 5.89. The ﬂuctuating pressure gradient term (v) tends to counterbalance term vi. The mean pressure term vi is balanced by the sum of the three contributions: the cross-dissipation term (iii + iv), the pressure ﬂuctuation term (v), and the source term due to gradients represented by i1 . It can be clearly observed that the pressure terms (both mean and ﬂuctuating pressure gradients) are not negligible in Equation 5.89. The DNS results for Case D without an external pressure gradient are shown in Figure 5.35b. It can be seen that dissipation terms iii and iv are of the same order, and they promote gradient transport. Pressure terms v and vi and the velocity-reaction rate correlation (vii) strongly act to promote countergradient transport. The two source terms due to mean progress variable gradient (i1 ) and mean velocity gradient (i2 ) tend to decrease the turbulent ﬂuxes and therefore promote gradient transport. Again, the ﬂuctuating pressure term (v) in Equation 5.89 cannot be ignored. A similar analysis is shown for Case F, where, due to the imposed adverse pressure gradient, the turbulent-scalar transport becomes of gradient type, as indicated in Figure 5.35c. As expected, the mean pressure gradient term tends to promote gradient transport. Again, the ﬂuctuating pressure term (v) is not negligible and acts to counterbalance the mean pressure gradient term (vi). In fact, the combined term (v) + (vi) is largely negative and corresponds to gradient transport. The reaction-velocity correlation term (vii) acts to promote countergradient transport. In summary, the external mean pressure gradient, which could be imposed on a turbulent premixed ﬂame due to gravity or due to pressure drop along the duct through which the ﬂame propagates, can affect the turbulent-scalar transport by making it either gradient or countergradient type. An adverse external mean pressure gradient promotes gradient transport whereas a favorable mean pressure gradient promotes countergradient transport. The ﬂuctuating pressure gradient tends to counterbalance the effects of the mean pressure gradient (Veynante and Poinsot, 1997). The turbulent ﬂame speed ST (normalized by laminar ﬂame speed) and turbulent ﬂame brush thickness (normalized by laminar ﬂame thickness) is shown as a function of reduced time (normalized by the characteristic residence time across a laminar ﬂame) in Figures 5.36 and 5.37, respectively. A favorable pressure gradient (i.e., ∂p/∂x < 0) leads to a thinner turbulent ﬂame brush and a lower turbulent ﬂame speed. The decrease in ST could be as high as 30%. An adverse pressure gradient (i.e., ∂p/∂x > 0), however, induces an increase in ﬂame brush thickness and a higher turbulent ﬂame speed. The symbols in Figure 5.36 represent the calculated turbulent ﬂame speed by using Libby’s model (1989), which produces very reasonable trends.

364 Turb. Flame Speed, ST/SL

0

TURBULENT PREMIXED FLAMES

2.2

0

Case A Case B Case C

2.2

Case D Case E Case F

1.8

1.8

1.4

1.4

1.0

1.0 0

0.5

1.0

1.5

2.0

2.5

0

3.0

1

2

3

Reduced time, tSL0 δL0

Reduced time, tSL0 δL0

(a)

(b)

4

Figure 5.36 Turbulent ﬂame speed plotted as a function of reduced time for (a) high turbulence intensity cases (u /SL0 = 5) and (b) lower turbulence intensity cases (u /SL0 = 2) (modiﬁed from Veynante and Poinsot, 1997). (Note: Different Cases are labeled in Table 5.5.) 0

Flame brush thickness

δT δL 16

8

Case A Case B Case C

12

Case D Case E Case F

6

8

4

4

2

0

0 0

0.5

1.0

1.5

Reduced time,

2.0 tSL0

2.5 δL0

(a)

3.0

0

1

2

Reduced time,

3 tSL0

4

δL0

(b)

Figure 5.37 Turbulent ﬂame brush thickness plotted as a function of reduced time for (a) high turbulence intensity cases (u /SL0 = 5) and (b) lower turbulence intensity cases (u /SL0 = 2) (modiﬁed from Veynante and Poinsot, 1997). (Note: Different Cases are labeled in Table 5.5.)

5.9.6

Summary of DNS Results

The DNS results are very useful to examine the validity of various models, such as BML and gradient transport models. Table 5.6 summarizes DNS studies for the convenience of readers. Several DNS studies reported in this table used single-step single-reactant chemical kinetics, although a few studies also use more detailed reaction kinetics. The DNS results obtained by considering different reaction kinetic schemes could be used to understand the effect of combustion chemistry on turbulence, although it is hard to ﬁnd a conclusive result with the data shown in the table. Also, most DNS studies have been performed for relatively low Reynolds numbers compared to the practical combustion systems. Therefore, more DNS studies are required to develop detailed understanding of this complex phenomenon. The terms u0 and l0,i represent the initial ﬂuctuating velocity and the integral length scale, respectively, in decaying turbulence. The term δL,slope represents the thickness of preheat zone given by Equation 2.104 (see Section 2.3.4 in Chapter 2).

365

2D

Single-step, single-reactant

2D

Veynante and Poinsot (1997)

4-step reduced chemistry, Tu = 800 K CH4 /air, φ = 1.0 Single-step, single-reactant

3D

2D

Single-step, single-reactant

3D

Rutland and Cant (1994) Zhang and Rutland (1995) Echekki and Chen (1996)

Veynante et al. (1997)

Single-step, single-reactant

2D

Baum et al. (1994)

Single-step, single-reactant Complex preheated H2 /O2 /N2 , φ = 0.35–1.3

Single-step, single-reactant Single-step, single-reactant

3D

3D

2D

Dimension Chemistry

Trouv´e et al. (1994)

Ashurst, Peters, and Smooke. (1987) El Tahry, Rutland, and Ferziger (1991)

Reference

Source: Modiﬁed from Lipatnikov and Chomiak (2010).

Turbulence

Variable, τ = 3

Variable, τ = 3

Decaying u0 /SL = 2–10, l0,i /δL = 11 Decaying u0 /SL = 2, 5, l0,i /δL,slope = 3.5

Random ﬂow ﬁeld u /SL = 1, l0 /δL = 10 Constant Decaying u0 /SL = 2.3–7.6, l0,i /δL,slope = 0.6–0.9 Variable Decaying u0 /SL = 10, l0,i ∼ δL,slope Variable Decaying u0 /SL = 1.2–3.2, 31, l0,i /δL = 1.3–4.3 Variable, τ = 2/3 u0 /SL ∼1.4, l0 /δL,slope ∼ 7.8 Variable, τ = 2/3, Decaying u0 /SL ∼ 1, 1.5 l0,i /δL,slope ∼ 7 Variable Decaying u0 /SL = 4.2, l0,i /δL,slope = 3.4

Constant

Density

TABLE 5.6. Summary of DNS Results for Turbulent Premixed Flames

(continued overleaf )

Ret = 75, 190, Da = 14, 5.6, Ka = 0.6, 2.5

Ret = 22–110, Da = 1.1–5.5, Ka = 0.9–10

Ret = 57, Da = 29, Ka = 0.26 Ret = 39–53, Da = 54–64, Ka = 0.11 Ret = 135, Da = 8, Ka = 1.5

Ret = 70, Da = 0.7, Ka = 12 Ret = 275–926, Da = 0.3–185, Ka = 0.09–56

Ret = 10, Da = 10, Ka = 0.3 Ret = 6.5–33, Da = 0.6–1.3, Ka = 2–10

Numbers

366

3D

3D

Chakraborty et al. (2008)

Single-step, single-reactant

Single-step, single-reactant

Single-step, single-reactant

3D

Jenkins et al. (2006)

Single-step, single-reactant

Complex, CH4 /air, φ = 1.59 Single-step, single-reactant

3D

2D, 3D

Th´evenin et al. (2002)

Single-step, single-reactant

Chakraborty and Cant (2004) Chakraborty and Cant (2005, 2006)

3D

Nishiki et al. (2002, 2003, 2006)

Complex, H2 /air, φ = 1.0, Tu = 700 K

2D

3D

Tanahashi et al. (2000, 2002, 2004)

Single-step, single-reactant

Domingo and Bray (2000)

3D

Dimension Chemistry

Boger et al. (1998)

Reference

TABLE 5.6. (continued ) Turbulence

Variable, τ = 0.5–6

Decaying u0 /SL = 4, 10, l0,i ≈ δL,slope Variable Decaying u0 /SL = 0.85–3.4, l0,i /δL,slope = 0.85–3.4 Variable, τ = Stationary, decays along 1.5,4,6.53 streamline, u0 /SL = 0.9–1.3, l0,i /δL,slope = 16–22 Variable Decaying, u0 /SL = 17, l0,i /δL,slope = 2.4 Variable, τ = 3 Decaying, u0 /SL = 1.5–2.85, l0,i /δL,slope = 2–9.2 Variable, τ = 3 Decaying, u0 /SL = 7.2, l0,i /δL = 2.4 Variable, τ = 2–4 Decaying, u0 /SL = 3.9–7.2, l0,i /δL,slope = 2.3–4.1 Variable, τ = 2–4 Decaying, u0 /SL = 4–12, l0,i /δL = 2.4 Variable, τ = 3 Decaying, u0 /SL = 7.6, l0,i /δL,slope = 2.4

Density

Ret = 41, Da = 0.6, Ka = 12

Ret = 25–75, Da = 0.2–0.6, Ka = 3.2–16.6

Ret = 45, Da = 0.9, Ka = 7.7 Ret = 24–70, Da = 0.6–1.1, Ka = 4.4–11

Ret = 74, Da = 0.23, Ka = 37. Ret = 17–57, Da = 4.2–11, Ka = 0.5–1, 6

Ret = 140 Da = 12–200, Ka = 0.06–1. Ret = 95.5, Da = 55–115, Ka = 0.08–0.18.

Ret = 17.6, 44, Da = 0.44, 1.1, Ka = 3.8, 9

Numbers

367

Treurniet, Nieuwstadt, and Boersma (2006) Sankaran et al. (2007)

van Oijen and Bastiaans et al. (2005) van Oijen and Bastiaans et al. (2005) Domingo et al. (2005)

Chakraborty et al. (2008, 2009) Hawkes and Chen (2006)

Single-step, single-reactant

Complex, CH4 /air, φ = 0.52

Complex, CH4 /air, φ=1

Complex, CH4 /air, φ=1

Complex, CH4 /air, φ=1

G-equation

Complex CH4 /air, φ = 0.7, Tu = 800 K

3D

2D

3D

3D

2D

3D

3D Variable

Variable, τ = 0–5

Variable

Variable

Variable

Variable

Variable, τ = 4.5

Decaying, u0 /SL = 4.1, 41, l0,i /δL = 4 V-ﬂame ﬂow, u /SL = 1.25–3.75, l0 /δL = 10 Stationary, decays along streamline u /SL = 0.16–1 Bunsen ﬂame, u /SL = 3, l0 /δL,slope = 1

l0,i /δL,slope = 0.85 Decaying, u0 /SL = 3, l0,i /δL = 45

Decaying, u0 /SL = 7.5, l0,i /δL,slope = 2.45 Decaying, u0 /SL = 28.5,

Ret = 7.5, Da = 1.8, Ka = 1.5

Ret = 200–800

Ret = 18–55, Da = 2.9–8.6, Ka = 0.5–2.6

Ret = 23–230, Da = 1,0.1, Ka = 5, 150

Ret = 180, Da = 15, Ka = 0.9

Ret = 47, Da = 0.6, Ka = 11 Ret = 40, Da = 0.05, Ka = 125

368

5.10

TURBULENT PREMIXED FLAMES

TURBULENT COMBUSTION MODELING APPROACHES

In addition to the BML theory, two other major approaches have been developed recently for modeling turbulent premixed ﬂames. These approaches can be identiﬁed as a geometrical analysis of ﬂames via either the G-equation model or the ﬂame surface density (-equation) approach and a statistical description of turbulent combustion via a probability density function approach. These approaches are discussed in the next sections. In the geometrical analysis, a turbulent ﬂame is described as a geometrical surface. This approach traditionally requires the thin-ﬂame assumption, which means that the ﬂame surface is thin compared to all other length scales in the turbulent reacting ﬂow. Following this view, scalar ﬁelds represented by the reaction progress variable (c) are studied, and ﬂame surfaces are deﬁned by the iso-value surfaces like c = c* , where c* can be a prespeciﬁed value between 0 and 1. The ﬂame surface can be considered as an interface between unburned and burned gases. Another approach in this analysis is to consider the ﬂame surface area per unit volume, also known as ﬂame surface density (). In Chapter 2 Section 4, a derivation of the -equation is shown for laminar premixed ﬂames. The next section discusses extension of this concept into turbulent ﬂow is discussed. In the statistical approach, the properties of scalar ﬁelds are treated as random variables. The mean values and correlations are then extracted via the knowledge of pdfs. An introduction to the pdf is given in Chapter 4 Section 4.5. The transport equations for velocity and velocity-composition joint pdfs are derived from the governing equations, and few terms in these equations require closure via models. The pdf approach is applicable to both premixed and nonpremixed turbulent ﬂames; this approach is discussed in Section 5.14 here and also in Chapter 6. Predictions of radicals and intermediate species, such as OH, or pollutants, such as CO, require the description of the detailed ﬂame structure (i.e., intermediate states between fresh and burned gases in premixed ﬂames). Even though Gﬁeld and ﬂame surface density of (-equation) need some statistical treatments, initially they are based on a geometrical view describing the ﬂame as a thin interface. In pdf methods, this assumption is not required, and the statistical properties of intermediate states within the ﬂame front can be resolved.

5.11 GEOMETRICAL DESCRIPTION OF TURBULENT PREMIXED FLAMES AND G-EQUATION

Although the governing equations in terms of the progress variable (c or c) ˜ are the most general approach in turbulent premixed combustion, there are complexities associated with this equation in the form of countergradient diffusion and the source term. To simplify this issue, an alternative approach called level set function has been proposed to track the evolution of ﬂame surface. The level set function G is a nonreactive scalar, which deﬁnes the ﬂame surface. A governing equation for level set function can describe the evolution of the ﬂame surface.

GEOMETRICAL DESCRIPTION OF TURBULENT PREMIXED FLAMES

369

Since the level set function is a nonreactive scalar, it does not have any source terms. Therefore, it avoids the complications associated with the closure of source terms. However, the applicability of level set function requires the ﬂame to be a surface, which means that the thin ﬂame assumption should be applicable (e.g., wrinkled ﬂamelet regime or corrugated ﬂamelet regime). The level set method was introduced initially by Sethian (1982, 1996, 1999). It is a mathematical theory for tracking the evolution of interfaces and surfaces. A general level set function is a mapping function, which is deﬁned as: G (xi ) = yk

where xi ⊂ Rn , yk ⊂ Rm

G : Rn → Rm

such that

(5.119)

where Rn and Rm represent n- and m-dimensional spaces of real numbers, respectively. In turbulent premixed ﬂames, we encounter a three-dimensional space for the domain space and one-dimensional space for the target space (the m-dimensional space in Equation 5.119). A level set is the set of all possible solutions of Equation 5.119. In the context of turbulent premixed ﬂames, the level set function describes a surface like a ﬂame surface; and the level set is the set of all points that lie on this surface. Normally, the ﬂame surface is deﬁned as: G (xi ) = G0

where xi ⊂ R3 , G0 ⊂ R1

(5.120)

where G0 could be a constant real number such as 0 or any other real number. However, for a given combustion problem, it has only one value—that is, one equation. The turbulent premixed ﬂame can be considered to have a demarcation between the unburned and the burned regions, separated by the ﬂame surface (i.e., G < 0 is the unburned region and G > 0 is the burned region, as shown in Figure 5.38). The level set function G is dependent on xf (t) and time, and it is deﬁned as: G xf (t), t = 0

y

(5.121)

G (x,t) = G0 dx

n dxn G < G0

Unburned Zone

−n G > G0

Burned Zone x

Figure 5.38 The iso-scalar surface deﬁning the ﬂame front (modiﬁed from Peters, 2000).

370

TURBULENT PREMIXED FLAMES

By differentiating Equation 5.121 with respect to time and by applying the chain rule, we obtain the equation 5.122: ∂G dxf + · ∇G = 0 ∂t dt

(5.122)

As shown in the laminar premixed ﬂames in Chapter 2, a unit normal is deﬁned as a vector that points in the direction of the unburned region from the ﬂame front. The unit normal n is deﬁned in terms of the conserved scalar G as: n=−

∇G |∇G|

(5.123)

The negative sign in this equation is required since the unit normal always points toward the unburned region, but the value of G increases from unburned to burned region, thereby resulting in a positive gradient from unburned to burned region. If xf is the location of the ﬂame front within the ﬂow ﬁeld, then its rate of propagation through the ﬁeld can be deﬁned as dxf dt

= w = v + SL n

(5.124)

where w = ﬂame surface velocity v = local ﬂow velocity at the ﬂame front SL = laminar ﬂame speed A similar equation has been shown in Equation 2.111 in Chapter 2 and demonstrated in Figure 2.15. Substituting Equation 5.124 in 5.122 as well as utilizing ∇G = −n |∇G| yields what is referred to in the combustion literature as the G-equation: ∂G (5.125) + v · ∇G = SL |∇G| ∂t The G-equation contains a local unsteadiness and advection term on the lefthand side of the equation and a propagation term with the laminar ﬂame speed on the right-hand side. Note that there is no diffusion term in Equation 5.125. The values used for v and SL are those deﬁned at the surface G = Go . The scalar G is uniquely deﬁned only at the ﬂame surface, where it is set equal to Go . A distance xn , which is equal to the distance from the ﬂame front in the normal direction, can be deﬁned by introducing its differential increase toward the burned region of the mixture. By using the deﬁnition of the unit normal deﬁned in Equation 5.123, we get: ∇G dxn = −n · dx = · dx (5.126) |∇G|

371

GEOMETRICAL DESCRIPTION OF TURBULENT PREMIXED FLAMES

In this equation, d x is a differential vector pointing from the ﬂame front to the surrounding ﬂow. If a static or frozen G-ﬁeld is considered, then a differential increase in the value of G would correspond to dG = ∇G · dx

(5.127)

Substituting Equation 5.127 into Equation 5.126 deﬁnes the relationship between dxn and dG: dG (5.128) dxn = |∇G| The scalar function G is not uniquely deﬁned away from the ﬂame surface; therefore, these expressions for dxn are needed to resolve distance from the instantaneous ﬂame surface. By deﬁnition, the level set approach is applicable to the ﬂamelet regime (wrinkled ﬂamelet and corrugated ﬂamelet regimes on the combustion regime diagram shown in Figure 5.16). However, a level set approach has been tailored to describe the high-turbulence combustion regimes like thin reaction zone and broken reaction zone regimes. In these two regimes, the reaction occurs at multiple locations in the ﬂame, thereby increasing the ﬂame thickness with respect to the turbulence scales, even though the reaction zone remains thinner than the Kolmogorov scale. For this reason, it is difﬁcult to assume the ﬂame as a surface and apply the level set function to describe the propagation of this type of ﬂame. A modiﬁed level set approach has been suggested by Peters (2000) for these combustion regimes; it is discussed later in this Chapter. However, let us ﬁrst discuss the level set approach for ﬂamelet regimes.

5.11.1

Level Set Approach for the Corrugated Flamelets Regime

Recall that the laminar ﬂame speed is affected by ﬂame stretching. Chapter 2 presents an expression for laminar ﬂame speed for stretched ﬂames. In turbulent premixed ﬂames, especially in the ﬂamelet regimes, a similar concept can be applied to account for the effect of stretching on the laminar ﬂame speed: ⎡ ⎤ SL = SL0 − LM κ = SL0 − LM ⎣

(νν + ηη) ∇v Effect of strain+dilatation

+

⎦

SL0 ∇ · n Effect of ﬂame curvature

(5.129) where SL SL0 LM v

= = = =

stretched laminar ﬂame speed unstretched laminar ﬂame speed Markstein length local ﬂow velocity

372

TURBULENT PREMIXED FLAMES

Equation 2.119 in Chapter 2 gives the deﬁnition of the stretch factor κ. Note that we use SL0 instead of the displacement speed Sd since we are determining the speed of ﬂame propagation in the unburned mixture, and in such cases the displacement speed is the same as the laminar ﬂame speed. Let us deﬁne Sr ≡ (νν + ηη) ∇v and Cu ≡ ∇ · n and rewrite Equation 5.129 as: SL = SL0 − LM Sr − SL0 LM Cu

(5.130)

The ﬂame curvature Cu can be deﬁned in terms of the level set function G: ∇G ∇ 2 G − n · ∇ (n · ∇G) Cu = ∇ · n = ∇ · − (5.131) =− |∇G| |∇G| The ratio of Markstein length and diffusive laminar ﬂame thickness (LM /δ or LM /δL ) is known as the Markstein number. An expression for the Markstein number with respect to the unburned mixture is given by: Tu Tb LM 1 Tb Ma = ln + Ze (LeF − 1) = δ Tb − T u Tu 2 Tb − T u

Tb −Tu Tu

ln (1 + x) dx x 0 (5.132) This expression is same as Equation 2.165 in Chapter 2, except that the Zel’dovich number Ze is used instead of the symbol β. Ze is deﬁned by: d

Ze ≡ Ea (Tb − Tu )/Ru Tb2

(5.133)

where Ea is the activation energy and Ru is the universal gas constant. Equation 5.132 for the Markstein number was obtained by Clavin and Joulin (1983) and Matalon and Matkowsky (1982) with the assumption of a single-step reaction with high activation energy, constant transport properties, and a constant heat capacity. By substituting the expression for laminar ﬂame speed for stretched ﬂames in Equation 5.125, we have: ∂G + v · ∇G = SL0 |∇G| − LM Sr |∇G| − SL0 LM Cu |∇G| ∂t

(5.134)

DL =Markstein Diffusivity

When the expression of Cu from Equation 5.131 is substituted into Equation 5.134, a second-order derivative of G is introduced into the G-equation. This second-order term introduces the diffusive effect and prevents the formation of singularities in the form of cusps in the G-ﬁeld that could result from a constant value of SL0 . The effects of strain rate and ﬂame curvature on quasi-laminar ﬂames have been well studied and there exist a large body of literature on the subject. An important ﬁnding is that curvature and strain have similar effects on the laminar

GEOMETRICAL DESCRIPTION OF TURBULENT PREMIXED FLAMES

373

ﬂame speed and can be grouped together in the concept of ﬂame stretch. Once again, the effect of ﬂame stretch has been thoroughly studied. The contribution of the Lewis number to the effect of ﬂame stretch is signiﬁcant, as discussed in Chapter 2. Here, the value of Lewis number also can affect the mathematical nature of the G-equation (see Equation 5.134 and 5.132). The Markstein length can become negative if the Lewis number is sufﬁciently small with respect to unity. If this is the case, it can lead to diffusive-thermal instabilities. Conversely, a stabilizing effect on the ﬂame front is produced by a Lewis number greater than unity. In the G-equation (see Equation 5.134), these two diffusive and thermal effects present themselves as follows. When LM > 0 (the stabilizing condition), the Gequation is similar to a Hamilton-Jacobi equation with a parabolic second-order differential operator. However, when LM < 0, the G-equation becomes ill-posed, thereby leading to instabilities. One example of using the G-equation can be demonstrated by examining the curvature effect on the ﬂame speed. Let us now consider a spherical ﬂame propagating outward in the radial direction. Since the bulk motion of the burned products inside the spherical ﬂame can be treated as zero, the second term on the left-hand side of Equation 5.134 can be discarded. In addition, the second term on the right-hand side of Equation 5.134 also can be discarded, since there is no strain rate effect for a perfectly symmetric spherical ﬂame. Due to the fact that we are using the burned gases as our reference medium, we should use the laminar ﬂame speed with respect to burned products. Let us deﬁne the laminar ﬂame speed with respect to the burnt medium as SL,b . Note that we have not used the superscript 0, which is used to deﬁne unstretched ﬂame speeds. This is due to the fact that a spherical ﬂame is subjected to stretching induced by changing radius of curvature, known as the curvature effect in the Equation 5.129. The displacement velocity of ﬂame front can be written as: drf dt

= SL,u + vu = SL,b + vb

(5.135)

By mass continuity, we have: ρu SL,u = ρb SL,b

or

ρu

drf dt

− vu

= ρb

drf dt

− vb

By realizing that vb = 0, Equation 5.136 becomes: ρu − ρb drf vu = ρu dt Also, SL,b =

drf dt

and

SL,u =

ρb ρb drf SL,b = ρu ρu dt

(5.136)

(5.137)

(5.138)

374

TURBULENT PREMIXED FLAMES

The G-equation then becomes: ) ) ) ∂G ) 2LM,b ∂G ∂G 0 ) ) + = SL,b ) ∂t ∂r ) rf ∂r

(5.139)

In Equation 5.139, the Markstein length is based on the burned gas properties, and it is different from the expression shown in Equation 5.132. For an outwardly propagating spherical ﬂame, the ﬂame curvature can be derived to be equal to 2/rf and ∇G always will be a negative quantity due to the fact that the value of G decreases across the ﬂame surface in the positive r direction. The G-function can be written as: G (r (t)) = rf (t) − r (5.140) By this deﬁnition, G < 0 when r > rf and G > 0 when r < rf , indicating that the burned products are inside the instantaneous spherical ﬂame surface and the unburned gases are outside of it. By substituting Equation 5.140 into Equation 5.139, we have: drf 2LM,b 0 (5.141) = SL,b 1 − dt rf A similar equation was derived by Clavin (1985) for outwardly propagating spherical ﬂames. This equation describes the instantaneous location of the ﬂame front during its propagation. If rf < 2LM,b , then there will be no physically meaningful solution for Equation 5.141. This fact indicates that there should be a minimum radius of the ﬂame kernel for the ﬂame propagation to occur. It also should be noted that the value of 2LM,b /rf term is more inﬂuential to the rate of increase of rf due to the curvature effect. 5.11.2

Level Set Approach for the Thin Reaction Zone Regime

In order for the G-equation shown in Equation 5.134 to be applicable, a welldeﬁned laminar ﬂame speed is necessary. In the thin reaction zone regime, the presence of eddies within the preheat zone of the ﬂame can cause the ﬂame surface to be not easily deﬁned. It also is difﬁcult to deﬁne the laminar ﬂame speed in such cases. Therefore, the conventional G-equation cannot be applied to the system; instead, a modiﬁed approach is required to deﬁne a G-equation in this combustion regime. The thin reaction zone within the ﬂame is the location where the majority of the chemical reactions occur; therefore, the maximum heat release takes place in this region. The location of the reaction zone can be deﬁned by a surface given by the equation: T (x,t) = TR , where TR is the temperature in the inner layer. This equation is similar to the level set function shown in Equation 5.120, although it deﬁnes only the reaction zone instead of the ﬂame surface. Also, T is

GEOMETRICAL DESCRIPTION OF TURBULENT PREMIXED FLAMES

375

not a nonreactive scalar like G. The governing equation for temperature is given as: ∂T ρ (5.142) + ρv · ∇T = ∇ · (ρα∇T ) + ω˙ T ∂t The similarity between the iso-scalar surface deﬁned by T (x,t) = TR and the iso-scalar surface obtained from the G-equation lies in the fact that both scalars can be used to deﬁne a unit outward normal vector. The outward unit normal to the inner layer is given as: ) ∇T )) n=− (5.143) |∇T | )T =TR By performing the Lagrangian differentiation of the equation T (x,t) = TR , we have: ∂T dx + · ∇T = 0 (5.144) ∂t dt Let us deﬁne the velocity dx/dt in a similar manner as the propagation velocity of ﬂame surface. dx (5.145) = v + Sd n dt where v is the ﬂow velocity at the thin reaction zone and Sd can be deﬁned as the displacement speed of the thin reaction zone. By comparing Equations 5.144, 5.145, and 5.142, we can obtain an expression for the displacement speed of thin reaction zone as: ! ") ∇ · (ρα∇T ) + ω˙ T )) Sd = (5.146) ) ρ |∇T | T =TR In order to formulate a G-equation to describe the location of the reaction zone, the surfaces deﬁned by G = G0 and T = TR must coincide. In such a case, the normal vectors deﬁned by both Equations 5.143 and 5.123 also should be the same. By using these conditions in Equations 5.123 and 5.122, we get: ! " ∂G ∇ · (ρα∇T ) + ω˙ T |∇G| + v · ∇G = (5.147) ∂t ρ |∇T | The term on the right hand side can be split into three terms that individually represent the effects of diffusion normal to the surface, reaction, and curvature. The ﬁrst term on this side can be written as: ∇ · (ρα∇T ) = n · ∇ (ραn · ∇T ) − ρα |∇T | ∇ · n

(5.148)

Equation 5.147 can be rewritten as: ∂G + v · ∇G = ∂t

S |∇G| n

+ Sr |∇G| −

αC |∇G| u

Effect of diffusion normal to reaction zone

Reaction effect

Flame curvature effect

(5.149)

376

TURBULENT PREMIXED FLAMES

where

n · ∇ (ραn · ∇T ) ρ |∇T | ω˙ T Sr = ρ |∇T |

Sn =

Cu = ∇ · n

(5.150) (5.151) (5.152)

It should be noted that the sum of Sn and Sr is not equal to SL ; rather it represents a ﬂuctuating quantity that connects the G-equation and the governing equation for temperature. The DNS studies conducted by Peters (1999) found that the sum of Sn and Sr is of the same order of magnitude as the laminar burning speed. Equation 5.149 is also written in this form: ∂G + v · ∇G = SL,s |∇G| − αCu |∇G| ∂t

(5.153)

=Sn +Sr

We have described the two different formulations for ﬂamelet regime (wrinkled and corrugated ﬂamelets) and thin reaction regime; now let us compare the Gequations for these two regimes. The G-equation for the ﬂamelet regime contains one additional term associated with the strain rate. According to Peters (2000), the strain rate term can be ignored based on an order-of-magnitude analysis. The velocity SL,s is of the same order of magnitude as the unstretched laminar ﬂame speed, SL0 . If the Markstein length is of the same order of magnitude as the diffusive laminar ﬂame thickness, then the Markstein diffusivity DM is same as the thermal diffusivity α. Therefore, a common G-equation for all three combustion regimes can be written as: ρ

∂G + ρv · ∇G = ρSL0 |∇G| − (ρα) Cu |∇G| ∂t

(5.154)

This equation can be used as a starting point for turbulent combustion modeling. All the terms have been multiplied by density so that Favre averaging can be easily applied to all terms in the equation. The term ρSL0 represents the mass ﬂux through a planar steady ﬂame and should be constant. Since the term (ρα) is a weak function of temperature and is also independent of the pressure, it can be approximated as constant, calculated at temperature TR . 5.12

SCALES IN TURBULENT COMBUSTION

As discussed in Chapter 4 and in the beginning of this chapter, turbulent ﬂows involve continuous scales. Some of the representative scales are the Kolmogorov length scale, the Taylor microscale, and the integral length scale. In reacting ﬂows such as turbulent premixed ﬂames, additional length scales emerge that are related to the reaction zone, oxidation layer, preheat zone, inner layer, consumption layer,

SCALES IN TURBULENT COMBUSTION

377

H2-CO Oxidation layer O(v) C2H4-CH2O Consumption layer O(m) Inner layer O(d) T

Preheat zone O(l)

or T0

n-C7H16

CO

Yi

H2 C2H4 H2 Consumption layer O(e) CH2O

x

Figure 5.39 Several thin reaction zones embedded within each other in the asymptotic structure of a premixed n-heptane ﬂame (modiﬁed from Peters, 2009; Seshadri, 1996).

and overall ﬂame thicknesses. A schematic description of these thicknesses for an n-heptane/air ﬂame is shown in Figure 5.39. This ﬁgure is for laminar premixed ﬂames but also can be applied to turbulent premixed ﬂames. The ordering of the thicknesses of these layers is δ < μ < ε < ν < 1. The preheat zone thickness is of the order of the ﬂame thickness δL . The parameters δ, μ, ε and ν represent the ratios of the thicknesses of these respective layers to the ﬂame thickness, δL . Each of these thicknesses can be characterized by the laminar ﬂame speed and the time scales associated with each of these zones in a ﬂame. The inherent assumption for this relationship is the existence of laminar ﬂamelets in the turbulent premixed ﬂames. SL =

δL lδ lμ lε lν = = = = tF tδ tμ tε tν

(5.155)

In Equation 5.155, tF , tδ , tμ , tε , and tν are the characteristic time scales associated with the length scales δL , lδ , lμ , lε , and lν through SL . The time scales for the layers associated with reactions are determined based on the rates of reaction of the controlling reactions in these zones. The length, velocity, and time scales associated with turbulent eddies are related to the mean dissipation rate as shown next: ε∼

vn3 v2 l2 ∼ n ∼ n3 ln tn tn

(5.156)

This relationship is applicable to all turbulence scales in the inertial subrange. In theory, the interaction between combustion and turbulence could occur at all scales at which both processes can take place. Unlike nonreacting turbulent ﬂows,

378

TURBULENT PREMIXED FLAMES

which have continuous velocity scales, there is only one velocity scale for the premixed combustion, the laminar burning velocity SL . Therefore, all the reaction layer thicknesses are related to their respective chemical time scales by this velocity scale. As long as the ﬂame thickness is smaller than the Kolmogorov scale (δL < η), the ﬂame thickness can be assumed to represent the entire ﬂame structure with respect to the scales of turbulence. Recall that the condition in Equation 5.156 corresponds to the corrugated ﬂamelet and wrinkled ﬂame regimes in the combustion regime diagram (see Figure 5.16). The length scale for turbulenceﬂame interaction in these regimes should be deﬁned by the turbulence cascade equation shown in Equation 5.156 with laminar burning velocity used as a velocity scale. The ﬂame thickness should not be used as a length scale for the turbulence-ﬂame interaction. Such a length scale is called the Gibson scale (lG ), and it is deﬁned by the next equation: lG ≡

SL3 ε

(5.157)

The physical meaning of Gibson scale is interpreted in terms of ﬂame-eddy interaction. Smaller eddies of size less than the Gibson scale will have turnover velocity (vn ) smaller than SL ; therefore, such eddies will not be able to wrinkle the ﬂame front. The turnover velocity of the large eddies of size greater than lG is higher than SL ; therefore, these eddies can push the ﬂame front around, causing a substantial corrugation or wrinkling. Following Equation 5.156, the logarithm of the eddy turnover velocity versus the logarithm of length scale is plotted in Figure 5.40. Since the Gibson scale can be represented by an eddy length scale with the turnover velocity equal to SL , we can locate the Gibson scale on this plot by using vn = SL . The relative magnitudes of various length scales can be seen from this plot. Note that the laminar ﬂame thickness is not

ln υn

υ′ υn = (eln)1/3 sL υη

lF

η

lG

l0

ln ln

Figure 5.40 Eddy turnover time versus the length scale for corrugated/wrinkled ﬂamelet regimes (modiﬁed from Peters, 2009).

SCALES IN TURBULENT COMBUSTION

379

same as the Gibson length scale. Also, the laminar ﬂame thickness is smaller than the Kolmogorov scale (for the corrugated ﬂamelet and wrinkled ﬂame regimes) whereas the Gibson scale is larger than the Kolmogorov scale since the laminar ﬂame speed is higher than the Kolmogorov velocity scale. In case of higher turbulence levels, such as the thin reaction zone in the combustion regime diagram, the Kolmogorov length scale is smaller than ﬂame thickness. Therefore, let us deﬁne a mixing length scale lm that is associated with the ﬂame-crossing time tF by the next relationship: 1/2 lm = εtF3

where

tF ∼

δL SL

(5.158)

The mixing length scale can be interpreted as the size of an eddy that has a turnover time equal to the time needed to diffuse scalar quantities over a distance equal to the ﬂame thickness δL . By this deﬁnition, the eddy of size lm can interact with the advancing reaction front and can transport preheated ﬂuid from a region of thickness δL in front of the reaction zone over a distance corresponding to its own size lm . Eddies with sizes much smaller than lm also can accomplish this task, but since their size is smaller, their actions are covered by eddies of size lm . Eddies of sizes larger than lm have longer turnover times and they will corrugate the broadened ﬂame structure at scales larger than lm . The physical interpretation of lm is therefore that of the maximum distance that the preheated ﬂuid can be transported ahead of the ﬂame. Following Equation 5.156, the logarithm of the eddy turnover time versus the logarithm of length scale is plotted in Figure 5.41. Since the mixing scale can be represented by a time scale equal to tF , we can locate the mixing scale on this plot by using tn = tF . The relative magnitudes of various length scales can be seen from this plot. The mixing length scale is larger than the ﬂame thickness in

ln tn

τ tn = (ln2/e)1/3 tF tη

lG

η = lC

lF

lm

l0

ln ln

Figure 5.41 Eddy turnover time versus the length scale for thin reaction zone regimes (modiﬁed from after Peters, 2009).

380

TURBULENT PREMIXED FLAMES

this case. This plot also shows an additional length scale. This length scale, lC , is called Obukhov-Corrsin scale, and it is deﬁned by: 1/4 (5.159) lC ≡ D 3 /ε This length scale represents the lower cut-off scale of the scalar spectrum for the thin reaction zone regime. If we assume ν = D, the Obukhov–Corrsin scale lC is equal to the Komogorov length scale η. Therefore, Gibson scale lG is physically relevant in the corrugated ﬂamelet/wrinkled ﬂames regimes, and the mixing scale lm is physically relevant to the thin reaction zone regime. 5.13

CLOSURE OF CHEMICAL REACTION SOURCE TERM

As mentioned in Section 5.9, the source term ω˙ c in the governing equation for the reaction progress variable, c, must be modeled (see Equation 5.84) to close the Bray-Moss-Libby model. In order to determine the inﬂuence of chemistry on the scalar time scale and reactive ﬂux, we must consider the reaction rate. The exponential term in the Arrhenius Equation can be expanded into a series as shown in Equation 5.160. Then, by averaging procedures, several new terms will be obtained. Expanding the exponential term leads to: x2 x3 xn + + ....... + + ... 2! 3! n! Ea Ta where, x = − =− Ru T T ex = 1 + x +

(5.160)

By taking average of the exponential term, we have: ex = 1 + x +

x2 x3 + + . . . . . . . higher moments + . . . 2! 3!

(5.161)

This procedure requires a large number of higher moments to be considered before the exponential terms may converge. Using Equation 5.161 for the exponential term, we can show that the source term ω˙ c can be expressed as: Ta ˜ ˜ ω˙ c = kf YF YO exp − T˜ ⎞ ⎡ ⎛ 2 Y T Y Y T Y T T T a a ⎣1 + F O + ℘1 ⎝ F + O ⎠ + ℘2 Y˜F Y˜O T˜ Y˜F T˜ Y˜O T˜ T˜ T˜ 2 ⎞ ⎤ ⎛ 2 2 3 Y T Ta ⎝ YF T T ⎠ + ℘3 Ta + ℘1 + O + . . . . .⎦ 2 2 ˜ ˜ ˜ ˜ ˜ ˜ T YF T YO T T T˜ 3 (5.162)

PROBABILITY DENSITY FUNCTION APPROACH

381

where ℘i (Ta /T) are the i th polynomials of Ta /T. This expansion is valid if the ratio T /T < 1, and a good approximation requires a large number of moments to be retained. For this reason, this approach is not very useful. Alternative approaches, such as PDF methods described in the next section, are in wide use.

5.14 PROBABILITY DENSITY FUNCTION APPROACH TO TURBULENT COMBUSTION

Pdf modeling is in broad use in turbulent combustion, including the source term closure problem discussed in Section 5.13. Some background information on the mathematics of probability has been discussed in Section 4.5 of Chapter 4. Haworth (2010) has presented comprehensive review of progress in the pdf approach for turbulent reacting ﬂow. It is common to encounter the term “PDF method” in combustion literature. The term refers to an approach based on solving a modeled transport equation for the one-point, one-time Eulerian joint pdf of a set of variables that describe the hydrodynamic and/or thermochemical state of a reacting medium: that is, a transported pdf method. Turbulence closure based on the solution of a modeled transport equation for the joint pdf of the velocity components originated with the work of Lundgren (1967, 1969). Dopazo and O’Brien (1974b, 1976) were the ﬁrst to consider a modeled equation for the pdf of a set of scalar variables that describe the thermochemical state of a reacting medium (a composition joint pdf) to model mixing and chemical reaction in turbulent reacting ﬂows. pdf methods subsequently were developed and elucidated by Dopazo (1976, 1979), O’Brien (1980), Janicka, Kolbe, and Kollmann, 1979; Pope, 1984; and Borghi, 1985 and others. The relationship between particle models and pdf methods was established by Pope (1994). Before we derive the transport equation for pdf (or P ), let us brieﬂy review the governing equations for a multispecies reacting mixture. The conservation equations for mass, momentum, and species are: ∂ρ ∂ρui =0 + ∂t ∂xi ∂τij ∂ρuj ui ∂ρuj ∂p =− + + ρgj + ∂t ∂xi ∂xj ∂xi

(5.163) (5.164)

where τij = μ

∂uj ∂ui + ∂xj ∂xi

2 ∂ul δij − μ 3 ∂xl

∂ρYk ∂Jk,i ∂ (ρui Yk ) =− + ω˙ k for kth species = 1,2, .., N + ∂t ∂xi ∂xi

(5.165)

382

TURBULENT PREMIXED FLAMES

where Jk,i is the relative mass ﬂux of species to the bulk ﬂuid motion by molecular diffusion. The energy equation given in terms of the enthalpy can be written as: ∂J h ∂ρh ∂ (ρui h) Dp ∂ui = − i + τij + + + q˙rad ∂t ∂xi ∂xi ∂xi Dt

(5.166)

where the enthalpy h contains both chemical and sensible enthalpies and Jih is the molecular ﬂux of enthalpy. ⎧ ⎡ ⎤⎫ T ⎪ ⎪ N ⎬ ⎨ ⎢ ⎥ h= Cp,k (Tα ) dTα ⎦ (5.167) Yk ⎣h0f,k + ⎪ ⎪ ⎭ k=1 ⎩ Tref

The pressure can be calculated from the equation of state: p = p (ρ, h, Y1 , Y2 , Y3 , . . . . . . YN ) = p (ρ, h, Y)

(5.168)

where Y is written in a vector form and consists of species mass fractions of N species. The caloric equation of state can be written as: h = h (p, T , Y1 , Y2 , Y3 , . . . . . . YN ) = h (p, T , Y)

(5.169)

For an ideal gas mixture, p = ρRT, R = Ru /Mw. Let us consider a combustion system involving N species and L elementary reactions such that: N

kf,l

νlk Mk

k=1

−→ ←− kb,l

N

νlk Mk

(l = 1, 2, . . . L)

The chemical source term for k th species can be written as: ⎧ ⎤⎫ ⎡ L ⎨ N N ⎬ 3 3 ν ν ⎣ Cβl,β − kb,l Cβl,β ⎦ ω˙ k = Mwk νlk − νlk kf,l ⎩ ⎭ l=1

(5.170)

k=1

β=1

(5.171)

β=1

The speciﬁc reaction rate constants usually follow the Arrhenius reaction rate form. The forward speciﬁc reaction rate constant for the l th reaction can be written as: kf,l = Af,l T bf,l exp −Ea,f,l /Ru T (5.172) This set of equations is complete. In principle, solutions can be obtained for velocity, density, species mass fractions, temperature, and so on as a function of spatial variables and time (i.e., ui = ui (x, t)). However, in practice, this is

PROBABILITY DENSITY FUNCTION APPROACH

383

rarely feasible, even on a high-speed computer. Essentially, these highly nonlinear coupled equations are characterized by extreme sensitivity to small variations in initial and boundary conditions and by a broad dynamic range of spatial and temporal scales. It is not feasible to specify initial and boundary conditions to the precision required for solving the system of conservation equations just summarized. Therefore, it is necessary to reduce the dynamic range of scales inherent in the original conservation equations. This can be accomplished by averaging and ﬁltering the equations. Averaging or ﬁltering of nonlinear equations gives rise to new terms (as demonstrated in Section 5.13), which must be modeled. A probabilistic description, in which the dependent variables (ρ, ui , Yk , etc.) are treated as random variables, provides an appropriate framework for analysis and modeling. In turbulent reacting ﬂows, the principle difﬁculty arises from averaging or ﬁltering the chemical source terms in the species and enthalpy equations. The ensemble average of the reaction rate ω˙ k cannot be determined based on the ensemble averages of pressure, temperature, and the species mass fractions. ω˙ k = ω˙ k (p , T , Y)

(5.173)

In this equation, the operator refers to the ensemble averaging. In the pdf method, ensemble averaging is replaced by probabilistic means; therefore, denotes the probabilistic mean and is called simply the mean or the expected value. For any function, φ = φ (x, t) , φ = φ (x, t) can be deﬁned in terms of the pdf of φ, Pφ (ψ; x, t), where ψ is the sample space variable corresponding to the random variable φ and Pφ is the probability density in ψ-space. Properties of Pφ include: Pφ (ψ; x, t) ≥ 0 (5.174) ∞

Pφ (ψ; x, t) dψ = 1

(5.175)

Pφ (ψ; x, t) dψ = Probability {φ (x, t) < ψ1 }

(5.176)

−∞ ψ1

−∞

It is useful to consider the joint pdf of two or more random variables, u = u(x,t) and φ = φ (x, t)—say, Puφ (V , ψ; x, t); then ∞

Pu (V ; x, t) =

Puφ (V , ψ; x, t) dψ

(5.177)

−∞ ∞

Pφ (ψ; x, t) =

Puφ (V , ψ; x, t) dV −∞

(5.178)

384

TURBULENT PREMIXED FLAMES

The conditional pdf of u given that φ (x, t) = ψ is:

Pu|φ (V |ψ; x, t) ≡ Puφ (V , ψ; x, t)/Pφ (ψ; x, t)

(5.179)

Similarly, the conditional pdf of φ given that u(x, t) = V is:

Pφ|u (ψ|V ; x, t) ≡ Puφ (V , ψ; x, t)/Pu (V ; x, t)

(5.180)

Consider Q(u, φ), a function of random variables u and φ. The probabilistic mean of Q is: ∞

∞

Puφ (V , ψ; x, t) Q (V , ψ) dVdψ

Q = Q (u, φ) = −∞ −∞ ∞

=

⎡

Pφ (ψ; x, t) ⎣

−∞

⎤

∞

Pu|φ (V |ψ; x, t) Q (V , ψ) dV⎦ dψ

−∞

≡Q(u,φ)|φ=ψ

(5.181) where Q (u, φ) |φ = ψ is called the conditional expectation of Q, given that φ (x, t) = ψ. The probabilistic means of time and spatial derivates have these properties: 4 5 5 4 ∂Q ∂ Q ∂Q ∂ Q = (5.182) = and ∂t ∂t ∂xi ∂xi Under low Mach number conditions, the mixture mass density and chemical source terms are functions of h, Y, and a reference pressure p0 , which is either a constant or at most a function of time (i.e., p0 = p0 (t)): ρ = ρ (p0 , h, Y) ;

ω˙ k = ω˙ k (p0 , h, Y)

(5.183)

For low Mach number ﬂows, the viscous dissipation term in the enthalpy equation can be ignored. In the enthalpy equation, Dp/Dt ≈ 0 when the reference pressure is constant. If we also ignore radiation, then both the species and enthalpy equations have similar forms. The governing equations can be written as: ∂ρui ∂ρ =0 + ∂t ∂xi Duj ∂τij ∂p ρ + + ρgj ≡ ρAj =− Dt ∂xj ∂xi ρ

Dφk ∂Jk,i + ρ S˙k ≡ ρk =− Dt ∂xi

(5.184) (j = 1, 2, 3)

(for k = 1, 2, . . . , N + 1)

(5.185) (5.186)

PROBABILITY DENSITY FUNCTION APPROACH

385

where φk (x, t) is a vector of N +1 composition variables (N species mass fractions plus enthalpy) such that φk = Yk where k = 1, 2, . . . N φN +1 = h

(5.187)

In this set of Equations 5.184 to 5.186 Aj represents the acceleration in the j th direction and k represents the source term (chemical + molecular transport) for the composition vector. The term S˙k is related to ω˙ k in this way: ρ S˙k = ω˙ k

if k = 1, 2, . . . N

ρ S˙k = 0

if k = N + 1

(5.188)

The source term for the enthalpy equation is zero because enthalpy includes both the sensible and the chemical energy: The ﬂux vector Jk,i for the enthalpy equation can be obtained for k = N + 1 as: JN +1,i = Jh,i

(5.189)

Knowledge of the joint pdf of the composition variables, φk (x, t) [or, in vector notation, (x,t)] should provide sufﬁcient information to determine the mean mixture density and mean species chemical production rates. The expected values of density and production rates can be written as: ∞

ρ (ψ) P (ψ; x, t) dψ

ρ = ρ (x, t) =

(5.190)

−∞

6 7 1 S˜˙ k = ρ S˙k /ρ = ρ

∞

ρ (ψ) S˙k (ψ) P (ψ; x, t) dψ

(5.191)

−∞

We have discussed the closure problem for the chemical source term, and Equation 5.191 represents the solution of such a problem using pdf methods. For an arbitrary detailed, skeletal, or reduced chemical mechanism, the mean chemical source term is closed by this equation. The tilde denotes a Favre- or mass-averaged mean quantity. We can also introduce a Favre pdf: P (ψ; x, t) ≡ ρ (ψ) P (ψ; x, t)/ρ

(5.192)

such that for any Q = Q (): ∞

˜ =Q ˜ (x, t) = Q

Q (ψ) P (ψ; x, t) dψ −∞

(5.193)

386

TURBULENT PREMIXED FLAMES

Similarly, a joint pdf of velocity also can be deﬁned. This joint pdf can be used to address the closure problems for the Reynolds stress tensor. The Favre-averaged form of the Reynolds stress tensor is: ui uj −ρui uj = −ρ

(5.194)

The term ui uj in the Reynolds stress can be written as: ∞

ui uj

=

Pu (V; x, t) dV (Vi − u˜ i ) Vj − u˜ j

(5.195)

−∞

Recall that the ﬂuctuations about the Favre-averaged mean and the conventional mean are: (5.196) ui = ui − u˜ i ; and ui = ui − ui Finally, by considering the velocity-composition joint pdf, we can determine any one-point joint statistics of the velocities and compositions. For example, the turbulent scalar ﬂux is: ∞ u i φk

∞

=

Pu (V, ψ; x, t) dVdψ (Vi − u˜ i ) ψk − φ˜ k

(5.197)

−∞ −∞

Up to this point, we have discussed the function Q with a single variable: the composition vector . Now let us consider Q as a function of two variables: Q = Q(u, ). The expected value of Q can then be given as: ∞

∞

Q (V, ψ) Pu (V, ψ; x, t) dVdψ

Q =

(5.198)

−∞ −∞

The Favre-averaged mean of the function Q can be given as: ∞

∞

˜ = Q

Q (V, ψ) Pu (V, ψ; x, t) dVdψ

(5.199)

−∞ −∞

5.14.1 Derivation of the Transport Equation for Probability Density Function

Essentially, most turbulent combustion models invoke a pdf at some point. One of the approaches is to derive a transport equation for pdf and solve to obtain the statistical properties of the turbulent ﬂow. There are different approaches to deriving and writing pdf transport equations; for example, those of Pope (1985)

387

PROBABILITY DENSITY FUNCTION APPROACH

and Lundgren (1969). Lundgren’s work (1969) used a ﬁne-grained pdf. Here, we have adopted the approach used by Pope to derive a transport equation for a velocity-composition joint pdf. The derivation for other pdfs, such as composition pdfs and velocity-composition-dissipation pdfs, is similar. Let us consider Q = Q(u, ), which is an almost arbitrary function. The Lagrangian derivative of Q can be written as: ρ

∂ (ρQ) ∂ (ρui Q) DQ ∂Q ∂Q = =ρ + ρui + Dt ∂t ∂xi ∂t ∂xi

(5.200)

This equation useds the continuity equation, 5.184. By averaging Equation 5.200: 5 4 5 4 5 4 ∂ (ρQ) ∂ (ρui Q) ∂ ∂ DQ ρQ + ρui Q = + (5.201) = ρ Dt ∂t ∂xi ∂t ∂xi The expected value of left-hand-side term can be obtained as: 4 5 DQ ∂ ρ = ρ (ψ) Q (V, ψ) Pu (V, ψ; x, t) dVdψ Dt ∂t ∂ ρ (ψ) Vi Q (V, ψ) Pu (V, ψ; x, t) dVdψ + ∂xi

(5.202)

Since Pu is the only parameter written as a function of t and x in the integrand of Equation 5.202, this equation can be rewritten as: " 4 5 ! DQ ∂ Pu ∂ Pu dVdψ (5.203) ρ = Q (V, ψ) ρ (ψ) + ρ (ψ) Vi Dt ∂t ∂xi In Equation 5.203, ψ and Vi are treated as independent variables. The expected value of the right-hand-side terms can be obtained via the following procedure. By chain rule: DQ ∂Q Dui ∂Q Dφk = + Dt ∂ui Dt ∂φk Dt

where k = 1, 2, . . . N + 1 and i = 1, 2, 3 (5.204)

By taking average of the product of density and Equation 5.204, we have: 5 4 5 4 5 4 5 4 5 4 ∂Q Dui ∂Q Dφk ∂Q ∂Q DQ = ρ + ρ = ρ Ai + ρ k (5.205) ρ Dt ∂ui Dt ∂φk Dt ∂ui ∂φk Consider the ﬁrst term on the right-hand side of Equation 5.205: ) 5 5 4 4 ∂Q (u, ) )) ∂Q Ai = Ai ) V, ψ Pu dVdψ ρ ρ ∂ui ∂ui =

ρ (ψ)

∂Q (V, ψ) Ai | V, ψ Pu dVdψ ∂Vi

(5.206)

388

TURBULENT PREMIXED FLAMES

Using the differentiation of the product of two terms by parts, we have: 4

5

∂Q ρ Ai = ∂ui

≡Int

∂ ρ (ψ) Q (V, ψ) Ai | V, ψ Pu dVdψ ∂Vi

−

Q (V, ψ)

∂ ρ (ψ) Ai | V, ψ Pu dVdψ ∂Vi

(5.207)

Let us consider the term labeled “Int”; by carrying out the integration with ψ, we have: 7 ∂ 6 ρQAi |V Pu dV (5.208) Int = ∂Vi This term is the integral of a divergence in V-space and can be rewritten using the divergence theorem as a surface integral at the boundary of V-space (V→∞). For well-behaved functions Q and pdf’s Pu , this integral is equal to zero 6(Pope,7 1985). In other words, if Q is a monotonic function as |V| → ∞ and ρQAi exists as a ﬁnite quantity, then Int = 0. Therefore, 4 5 ∂Q ρ Ai = − ∂ui

Q (V, ψ)

∂ ρ (ψ) Ai | V, ψ Pu dVdψ ∂Vi

(5.209)

Q (V, ψ)

∂ ρ (ψ) k | V, ψ Pu dVdψ ∂ψ k

(5.210)

Similarly, 5 4 ∂Q k = − ρ ∂φk

By combining the results from Equations 5.209 and 5.210 for 5.205, we have: 8

4 5 DQ ρ =− Dt

Q (V,ψ)

9 ρ (ψ) Ai | V, ψ Pu dVdψ (5.211) + ∂ψ∂ ρ (ψ) k | V, ψ Pu ∂ ∂Vi

k

By subtracting Equation 5.203 from Equation 5.211, we have: 0=

Q (V,ψ) ⎧ ⎨ρ (ψ) ∂ Pu + ρ (ψ) Vi ∂ Pu + ∂t ∂xi × ⎩+ ∂ ρ (ψ) k | V, ψ Pu ∂ψ k

∂ ∂Vi

⎫ ρ (ψ) Ai | V, ψ Pu ⎬

⎭

dVdψ (5.212)

PROBABILITY DENSITY FUNCTION APPROACH

389

For Equation 5.212 to hold for any arbitrary function Q, the terms in {} must be equal to 0. Thus, ρ (ψ)

∂ ∂ Pu ∂ Pu =− ρ (ψ) Ai | V, ψ Pu + ρ (ψ) Vi ∂t ∂xi ∂Vi ∂ − ρ (ψ) k | V, ψ Pu ∂ψ k

Derivation of the Favre pdf transport equation is essentially the same: ∂ ρ ∂ ρ Vi Pu Pu ∂ ρ Ai | V, ψ =− Pu + ∂t ∂xi ∂Vi ∂ ρ k | V, ψ − Pu ∂ψ k

(5.213)

(5.214)

Substituting the deﬁnitions of Ai and k into Equation 5.214, we have: Pu Pu ∂ ρ ∂ ρ Vi + ∂t ∂xi 5 ! 4 ) " ) ∂τj i ∂ 1 ∂p ) ρ =− + + ρgi ) V, ψ Pu − ∂Vi ρ ∂xi ∂xj 5 ! 4 ) " ) ∂ 1 ∂Jk,i ρ − − + S˙k )) V, ψ Pu (5.215) ∂ψ k ρ ∂xi The pressure gradient can be decomposed in mean and ﬂuctuating components as: ∂ p ∂p ∂p = + (5.216) ∂xi ∂xi ∂xi Therefore, 4

) 5 5 4 ) 1 ∂p )) 1 ∂p )) ∂ p V,ψ = − V,ψ + − ρ ∂xi ) ρ (ψ) ∂xi ∂xi )

(5.217)

An equation governing the mean pressure ﬁeld can be derived by taking the divergence of the mean momentum equation and invoking the mean continuity equation: ⎧ ∂τ ∂p ˜0 ⎨ρA0i ≡ ∂xij − ∂x + ρgi ρ 2 ρ 2 p 2 ρ ∂ A i u˜ i u˜ j ∂ ∂ ∂ j i = − + where ⎩ρ A˜ 0 ≡ ∂τ ij + ρ g˜ ∂xi ∂xi ∂t 2 ∂xi ∂xj ∂xi i

∂xj

i

(5.218) This equation involves the mean density and one-point statistics of the velocity ﬁeld, thus conﬁrming that the mean pressure ﬁeld can be closed in terms of the

390

TURBULENT PREMIXED FLAMES

velocity-composition joint pdf, Pu . The body force vector gi is independent of u and ; therefore, gi | V,ψ = gi

(5.219)

The chemical source terms S˙k are known functions of the composition variables; thus: 7 6 ) S˙k ) V,ψ = S˙k (ψ) (5.220) Substituting Equations 5.217, 5.219, and 5.220 into Equation 5.215, we obtain: ∂ ρ Pu ∂ ρ Vi Pu ∂ ρ S˙k Pu ρ ∂ p ∂ Pu + + + ρ (ψ) gi − ∂t ∂xi ∂ψ k ρ (ψ) ∂xi ∂Vi ) 5 II " 5 4 III ) !4 I ! " ρ ∂ ∂τj i )) ∂Jk,i )) ∂p 1 ∂ V, ψ Pu + ρ V, ψ Pu = − ρ (ψ) ∂Vi ∂xi ∂xj ) ∂ψ k ρ (ψ) ∂xi ) V

IV

(5.221) The above pdf transport equation contains several terms. Physical interpretation of these terms is given below: I: Transport in physical space due to advection by mean ﬂow and turbulence II: Transport in composition space due to the chemical reaction III: Transport in velocity space due to the body force and mean pressure gradient IV: Effects of the ﬂuctuating pressure gradient and viscosity V: Effects of the molecular diffusion of species and enthalpy Since density and reaction rate are known functions of the composition vector and the mean pressure gradient can be determined from the joint pdf, the terms I, II, and III can be accounted for exactly, without any approximation. The terms on the right-hand side of Equation 5.221 represent transport in velocity space by the viscous stresses and by the ﬂuctuating pressure gradient and transport in composition space by the molecular ﬂuxes. Before the equation can be solved for Pu , the conditional expectations appearing in these terms must be determined or approximated. Thus, the terms IV and V require modeling. The Favre-averaged composition pdf P (ψ; x, t) can be obtained from the Pu (V, ψ; x, t) by integrating Favre-averaged velocity-composition joint pdf over velocity space: P (ψ; x, t) =

Pu (V, ψ; x, t) dV

(5.222)

PROBABILITY DENSITY FUNCTION APPROACH

391

Integration of Equation 5.221 over velocity space yields the transport Equation P (ψ; x, t): 5.233 for the composition pdf ∂ ρ S˙k ∂ ρ u˜ i P P P ∂ ρ + + ∂t ∂xi ∂ψ k Mean-ﬂow advection

Chemical reaction

6 ) 7 ∂ ∂ ρ ui ) ψ =− P + ρ ∂xi ∂ψ k Turbulent diffusion

!

) 5 " 4 1 ∂Jk,i )) ψ P ρ (ψ) ∂xi )

(5.223)

Turbulent mixing

In Equation 5.223, the effects of chemical reaction and advection by the mean velocity can be accounted for exactly without any approximation; turbulent diffusion and mixing by turbulent velocity ﬂuctuations require modeling. By including radiation, the transport equation for the joint pdf of velocity and composition is given by: Pu ∂ ρ Vi Pu ∂ ρ S˙k Pu ρ ∂ ρ ∂ p ∂ Pu + + + ρ (ψ) gi − ∂t ∂xi ∂ψ k ρ (ψ) ∂xi ∂Vi ) ) 5 5 4 !4 " ! " ) ) ρ ∂ ∂Jk,i ) ∂p ∂τj i ) 1 ∂ V, ψ Pu + ρ V, ψ Pu = − ρ (ψ) ∂Vi ∂xi ∂xj ) ∂ψ k ρ (ψ) ∂xi ) ) 7 ρ ρ 6 ∂ ∂ ˙ rad,em ˙ rad,ab ) V, ψ + δk Pu −δk Pu Q Q ∂ψ k ρ (ψ) ∂ψ k ρ (ψ) (5.224) In Equation 5.224, δk is 0 for k = 1,2, . . . N and δk = 1 for k = N + 1.

5.14.2

Moment Equations and PDF Equations

Equations for any one-point, one-time statistics of the composition ﬁeld can be P ); Equation 5.193. Equations for any oneobtained from the composition pdf ( point, one-time joint statistics of the velocity and/or composition ﬁelds similarly Pu ) using Equation 5.198. can be obtained from the velocity-composition pdf ( Integration of the velocity-composition pdf equation over velocity-composition space, or integration of the composition pdf equation over composition space, will yield the mean continuity equation. The ﬁrst moments of velocity and composition vectors are the expected values or mean values of velocity and composition ﬁelds, respectively (see Equation 4.14 in Chapter 4). The derivation of moment equations is shown in Chapter 4 by using Reynolds averaging or Favre averaging. To relate the pdf equations with the original moment equations, an equation for mean momentum can be derived by multiplying the velocity-composition pdf equation by V and integrating over velocity-composition space. The mean species mass fraction and

392

TURBULENT PREMIXED FLAMES

mean enthalpy equations can be derived similarly by multiplying the velocitycomposition (or composition) pdf equation through by the composition vector ψ and integrating over velocity-composition (or composition) space. The resulting equations from this procedure should be the same as the mean momentum, composition, and enthalpy equations. The second moments of velocity and composition vectors are the turbulentscalar ﬂux and Reynolds stress tensors. The transport equation for the evolution of turbulent scalar ﬂux can be derived by multiplying Equation 5.214 by (Vi − u˜ i )(ψk − φ˜ k ) and integrating over the velocity-composition space to obtain the transport equation for u i φk . D u φ : Dt i k

∞

∞

Pu dVdψ (5.225) (Vi − u˜ i ) ψk − φ˜ k Equation for

−∞ −∞

Similarly, D φ φ : Dt k l

∞

ψk − φ˜ k

ψl − φ˜ l

P dψ Equation for

(5.226)

−∞

The second-central-moment equations often are useful in developing models for pdf methods and for comparing results obtained using pdf methods with those obtained using simpler models. As discussed by Haworth (2010), the derivation of the moment equations based on the pdf equation has several advantages over traditional methods. The pdf equation is established as the point of departure for turbulent combustion modeling. Probability density functions play a central role in almost all turbulent combustion models. The preceding derivation technique establishes the connections among the instantaneous governing PDEs, pdf equations, and moment equations. It elucidates the physics of the terms that require modeling. Furthermore, it minimizes confusion between formal manipulations of the governing equations and the particular techniques used to approximate mean quantities in experimental measurements (e.g., ensemble averaging, time averaging, spatial averaging, or mass-weighted averaging).

5.14.3

Lagrangian Equations for Fluid Particles

A relationship between ﬂuid particles and pdf transport equations can be established immediately by noting that A and are respectively the time rate of change of velocity and composition following a ﬂuid particle. The position, velocity, and composition (xi∗ , u∗i , φk∗ ) of a ﬂuid particle evolve by these equations: Dx∗i = u∗i Dt

(5.227)

PROBABILITY DENSITY FUNCTION APPROACH

1 ∂ p∗ Du∗i ∗ ∗ + ai,p = A∗i = gi − ∗ + ai,vis Dt ρ φk ∂xi ∗ Dφk∗ ∗ − δk q˙rad,em , φk + δk q˙rad,ab = ∗k = S˙k φk∗ + θk,mix Dt where δk = 0 for k = 1, . . . N and = 1 for k = N + 1

393

(5.228) (5.229)

where superscript* = any particle a mean quantity mean value evaluated at the particle location with superscript∗ = (e.g., ∂ p∗ /∂xi = ∂ p (x∗ (t) , t)/∂xi ∗ ai,p = a particle acceleration due to the ﬂuctuating pressure gradient ∗ ai,vis = particle acceleration due to molecular viscosity (velocity mixing) = increment in enthalpy due to radiative absorption q˙rad,abs ∗ = increment in composition due to molecular θk,mix diffusion (scalar mixing) ∗ The mean density ∗ at the particle location ρ can be used instead of the particle density ρ φk that appears in the mean pressure gradient term in the particle velocity equation (Equation 5.228). A generalized Langevin model (GLM), in a form proposed by Pope (1985), can be used for modeling the position and velocity in the preceding particle equations:

dx∗i = u∗i dt du∗i = gi −

1 ∂ p∗ ∗ ρ φk ∂xi

(5.230) dt + Gij u∗i − u˜ j dt + (C0 ε)1/2 dWi

(5.231)

In these equations, d is used instead of D, but the meaning remains the same. It has been assumed that the composition vector and density can be modeled by using some other techniques. Equation 5.231 is a linear Markov model for the velocity of a ﬂuid particle. It is analogous to the Langevin equation for the velocity of a particle undergoing Brownian motion. In this model, Gij is based on a second-order tensor function of local one-point velocity statistics, and C0 > 0 is a model constant. The value C0 = 2.1 was determined by considering the evolution of the thermal wake behind a line source in grid turbulence (Anand and Pope, 1985). The last two terms involving Gij and Wi in Equation 5.231 provide closure models for the ∗ ∗ terms ai,p dt and ai,vis dt in Equation 5.228. The last term on the right-hand side of Equation 5.231 represents a random walk in velocity space, and the quantity Wi (t) is an isotropic Wiener process. The ε parameter in this last term represents the turbulent dissipation rate. Properties of diffusion processes, Langevin

394

TURBULENT PREMIXED FLAMES

equations, Wiener processes, and related quantities are reviewed in Pope (2000, Appendix J.) Some key properties of Wi (t) are given next. dWi = 0

and

6

7 dWi dWj = dtδij

(5.232)

The increment d W is a joint normal random vector with zero mean and with covariance dt δij . In a numerical implementation, dWi could be approximated as dWi = ηi t 1/2 , where ηi is a vector of three independent standardized Gaussian (zero mean, unit variance) random variables and t is the computational time step. To maintain the Markovian nature of the system, dWi must be independent of its coefﬁcients in Equation 5.231. A review of the use of Langevin equations and closely related discrete Markov chains or random walks for modeling turbulent ﬂows can be found in the introduction of Haworth and Pope (1986a). The velocity pdf equation corresponding to Equation 5.231 is a Fokker-Planck equation: ∂ (ρ Pu ) ∂ (ρui Pu ) ∂ p ∂ Pu + ρgi − + ∂t ∂xi ∂xi ∂Vi = −Gij

1 ∂ 2 ρ Pu ∂ ρ Vj − u˜ j Pu + C0 ε ∂Vi 2 ∂Vi ∂Vi

(5.233)

The terms on the right-hand side of Equation 5.233 are the closure models that are implied by Equation 5.231 for the terms involving the conditional expectations of ∂τij /∂xj and of ∂p /∂xi on the right-hand side of Equation 5.231. A general model for the second-order tensor Gij , as a function of local mean quantities, is given by Haworth and Pope (1986): Gij = f

uk ul , ∂ui /∂xj , ε

(5.234)

This form is based on the physical reasoning of the evolution of turbulence kinetic energy, modeling of second-order terms such as viscous stress, rapid-distortion limit, as well as the mathematical constraints involving tensor properties. This functional form can be rigorously justiﬁed only in simpliﬁed ﬂows. In homogeneous ﬂows, the mean velocity and the Reynolds stresses provide a complete one-point statistical description of the ﬂow ﬁeld since the velocity pdf has a joint normal distribution. In statistically stationary ﬂows, the current values of all Eulerian statistics are equivalent to need to include their past values; for this reason’ Equation 5.234 does not any ﬂow history. Experimental observations have suggested that in general ﬂows, Gij would depend upon the velocity ﬁeld within an integral length scale of a given location, the history of the ﬂow within one integral time scale of the current time, and the Reynolds number of the ﬂow (Veynante et al., 1997). For homogeneous ﬂows, the only relevant velocity ﬁeld statistics are the mean velocity gradients and the Reynolds

395

PROBABILITY DENSITY FUNCTION APPROACH

stresses. Therefore, a functional form of Gij can be derived in terms of these two parameters. For example, a model for Gij can be proposed that is linear in the mean velocity gradients and the Reynolds stresses (or equivalently in the normalized anisotropy tensor bkl ): 1 1 ∂ uk Gij = α1 δij + α2 bij + Hij kl τ τ ∂xl where

6 7 1 bij ≡ ui uj /uk uk − δij , 3 τ = uk uk /2ε = characteristic turbulent time scale

(5.235)

(5.236) (5.237)

Hij kl = β1 δij δkl + β2 δik δj l + β3 δil δj k + γ1 δij bkl + γ2 δik bj l + γ3 δil bj k + γ4 bij δkl + γ5 bik δj l + γ6 bil δj k

(5.238)

As can be easily notice, this model has several coefﬁcients. Some are determined by applying model constraints (see Haworth and Pope, 1986,) Section IV and the remaining ones can be established through calibration with respect to experimental and/or DNS data. Several reduced versions of the model have also been considered. For example, the simpliﬁed Langevin model (SLM) is given by: 1 3 Gij = − (5.239) + C0 ωδij 2 4 where the frequency ω is deﬁned as: ω=−

; : d ln ui uj dt

(5.240)

This form corresponds to the case where Gij is isotropic and independent of the mean velocity gradients and Reynolds stresses but depends on the rate of decay of Reynolds stresses. The model corresponds to Rotta’s linear-return-toisotropy model for the Reynolds stresses, where the value of the Rotta constant is CR = (3C0 + 2)/2. With C0 = 2.1 (the standard value), CR = 4.15. This is higher than the standard value of CR = 1.5 but is close to the value CR = 4.5 that has been found to give good results for free shear ﬂows in the absence of an explicit model for the rapid-pressure terms (i.e., the terms involving the mean velocity gradients). A stochastic Lagrangian model for ﬂuid particle acceleration was proposed by Pope (1985).

5.14.4

Gradient Transport Model in Composition PDF Method

In the composition pdf methods, transport by turbulent velocity ﬂuctuations must be modeled. (Note that no such modeling is required in the velocity-composition

396

TURBULENT PREMIXED FLAMES

pdf methods.) A gradient transport hypothesis often is used in pdf methods to m model this term as (using the symbol =): 6 ) 7 ! " P m ∂ ∂ ρ ui ) ∂ P ∂ 2 P P ∂T ∂ = − + (5.241) T = T ∂xi ∂xi ∂xi ∂xi ∂xi ∂xi ∂xi where the apparent turbulent transport coefﬁcient T is: T = cμ ρ σT−1 k 2 /ε

(5.242)

The standard values of cμ and σT are 0.09 and 0.7, respectively. This implies gradient transport for the turbulent scalar ﬂuxes: ∂ φ˜ k m = − ρ φ T k ui ∂xi

(5.243)

A ﬂuid particle model that is consistent with gradient transport for the composition pdf is: ∗ ∗1/2 m ∗ ∗= −1 ∂T dxi u˜ i dt + ρ dt + 2 ρ−1 T dWi (5.244) ∂xi where a vector Wiener process Wi is used. A molecular transport coefﬁcient can be added to the apparent turbulent transport coefﬁcient where warranted, although, strictly speaking, that effect corresponds to the J term in the composition pdf equation rather than to the term involving turbulent velocity ﬂuctuations. The composition pdf equation can be written in the next form using a gradient diffusion model for turbulent diffusion term: ∂ ρ P˜ ∂ ρ S˙k P˜ ∂ ρ u˜ i P˜ + + ∂t ∂xi ∂ψk $ ∂ cφ ∂ # ∂ P˜ T ψk − φ˜ k ρ P˜ = + (5.245) ∂xi ∂xi 2τT ∂ψk Model for turbulent diffusion

Turbulent mixing

where cφ is a model constant with standard value of 2.0 and the turbulent time scale τT ≡ k/ε. 2 The moment equations for the composition variables (φ˜ k and φ k ) can be derived from the composition pdf equation by multiplying it by composition variable vector and integrating in the composition space. The resulting equations for ﬁrst and second moments are: ∂ ρ φ˜ k ∂ ρ u˜ i φ˜ k ∂ ∂ φ˜ k = (5.246) T + ρ S˙ k + ∂t ∂xi ∂xi ∂xi

PROBABILITY DENSITY FUNCTION APPROACH

2 ∂ ρ φ k ∂t

+

2 ∂ ρ u˜ i φ k ∂xi

397

2 ∂φ ∂ φ˜ k ∂ φ˜ k ∂ T k + 2T = ∂xi ∂xi ∂xi ∂xi 2 ˜ ˜ − c ρ φk (5.247) + 2 ρ S˙ k φk − S˙ k φ k φ τT

In these equations, the standard k -ε model has been invoked for turbulence closure. However, this is not absolute requirement, since a Reynolds stress turbulence model could also be used. If a velocity-composition pdf were used, then 1 k˜ = Pu (V, ψ; x, t) dψdV (5.248) (Vi − u˜ i ) (Vi − u˜ i ) 2 2 After Equations 5.246 and 5.247 are solved for φ˜ k and φ k , their local values can be used to determine the shape of local pdf by the next two equations: ∞

ψ P (ψ; x) dψ

φ˜ k (x) =

(5.249)

−∞

2 φ k (x) =

∞

2

˜ ψ −

P (ψ; x) dψ

(5.250)

−∞

where ψ is the sample space variable (e.g., species mass fraction or enthalpy). 5.14.5

Determination of Overall Reaction Rate

At sufﬁciently high Damk¨ohler numbers, the local structure of a turbulent ﬂame may essentially be that of a laminar ﬂame subjected to the same aero-thermochemical conditions. In such cases, the turbulent ﬂame can be viewed as an ensemble of small laminar ﬂamelets. The consumption rate per unit ﬂame surface area (Sc ) can be computed by using a simple model of laminar planar stagnation point ﬂow with detailed chemistry. The results are stored in a ﬂamelet library where laminar consumption rate per unit ﬂame surface area (Sc ) is tabulated for a range of stretch rates (κ) at a given equivalence ratio and initial temperature. The mean ﬂamelet consumption speed over the turbulent ﬂame surface can be calculated from: ∞ Sc (κ) P (κ) dκ

Sc s =

(5.251)

0

In a real turbulent ﬂow, ﬂame interaction and extinction mechanisms must be taken into account. The mean reaction rate ω˙ can be determined as a product

398

TURBULENT PREMIXED FLAMES

of ﬂame surface density , gas density ρ0 , and local consumption rate per unit of ﬂame area Sc s : ω˙ = ρ0 Sc s (5.252) 5.14.6

Lagrangian Monte Carlo Particle Methods

The modeled composition pdf equation is an integral-differential equation in N + 4 independent variables along with time. Solving by conventional methods for large N is not practical. Monte Carlo methods provide an alternative. A Monte Carlo method is a technique that involves using random numbers and probability to solve problems. The central idea of this technique is to solve the modeled particle equations developed for a large number of computational particles. The 2 expected values of quantities like (φ˜ k , φ k ) are extracted by ensemble averaging over all the particles. The statistical error in this case generally scales by N −1/2 . For computations where the Favre-averaged composition pdf various in x and t, hybrid Lagrangian particle/Eulerian mesh (LPEM) methods can be used. There are a number of computational issues associated with such a technique, including Lagrangian/Eulerian consistency, particle number density control, particle tracking through unstructured mesh, estimations of mean quantities from noisy particle data, and others (Subramaniam and Haworth, 2000).

5.14.7

Filtered Density Function Approach

An alternative to the probabilistic (Reynolds-averaged) approach for reducing the wide dynamic range of scales inherent in the instantaneous governing equations at high Reynolds number and/or high Da is to ﬁlter the equations in space and/or in time. This is the basis for large-eddy simulation (LES) of turbulent reacting ﬂows (see Section 4.7 in Chapter 4). The usual practice is to apply spatial ﬁlters only. In LES, the dynamics of scales that are larger than the ﬁlter width (the resolved scales) are captured explicitly while the effects of subgrid scale ﬂuctuations are modeled. This is in contrast to Reynolds averaging, where the effects of all turbulent ﬂuctuations about the local mean at all scales must be modeled. Therefore, it can be expected that LES should be advantageous compared to Reynolds averaging in accuracy and in generality. However, there are signiﬁcant differences between LES for nonreacting ﬂows (where most of these arguments have been made) and for reacting ﬂows. In constant-density ﬂows with highReynolds-number hydrodynamic turbulence, without chemical reactions and any walls, the rate-controlling processes are determined by the resolved large scales. However, in chemically reacting turbulent ﬂows, the essential rate-controlling processes (molecular transport and chemical reaction) occur at the smallest (usually unresolved) scales. In such cases, the LES-based models that are required

PROBABILITY DENSITY FUNCTION APPROACH

399

to parameterize the effects of subgrid-scale ﬂuctuations on local ﬁltered reaction rates in terms of resolved-scale quantities must be essentially the same as the Reynolds averaging-based models that are required to parameterize the effects of turbulent ﬂuctuations on local mean reaction rates in terms of mean quantities. In this sense, the arguments favoring LES over Reynolds averaging are less compelling in the case of turbulent reacting ﬂows. Nevertheless, a wide and rapidly growing body of evidence demonstrates quantitative advantages of LES in modeling studies of laboratory ﬂames and in applications to gas-turbine combustors, Internal Combustion engines, and other combustion systems. The analogue of a pdf method in the LES context is known as ﬁltered density function (FDF) method. The use of pdf-based approaches for subgrid-scale modeling in LES was suggested by Givi (1989). The FDF formulation was proposed by Pope (1990), and a transport equation for a composition FDF was derived and modeled by Gao and O’Brien (1993). A comparison of LES/FDF approach with Reynolds averaging/PDF approach is given in a review paper by Haworth (2010).

5.14.8

Prospect of PDF Methods

The pdf methods enable modeling and solution of an equation that governs the evolution of the one-point, one-time pdf for a set of variables that determines the local thermochemical and/or hydrodynamic state of a reacting system. The current emphasis has been on composition pdf and velocity-composition pdf methods for low-Mach-number reacting ideal-gas mixtures. Signiﬁcant progress has been made in pdf methods since Pope’s 1985 paper. At that time, pdf methods were used mainly as academic research tools. Nowadays, pdf methods have become a mainstream turbulent combustion modeling approach for laboratory-scale ﬂames and for device-scale applications. Probability density function methods offer compelling advantages for modeling chemically reacting turbulent ﬂows. In particular, they provide an elegant and effective resolution to the closure problems that arise from averaging or ﬁltering the highly nonlinear chemical source terms and terms that correspond to other one-point physical processes (e.g., radiative emission) in the instantaneous governing equations. Complex interactions among hydrodynamic turbulence, gasphase chemistry, soot, liquid fuel sprays, and thermal radiation can be captured in a natural and direct manner using pdf methods. For these reasons, pdf-based models have been successful where other approaches have not. Examples include their ability to capture strong turbulence–chemistry interactions even in ﬂames with strong local extinction/reignition and to capture strong turbulence–radiation interactions in luminous ﬂames. The major advantage of pdf methods stems from the applicability of same physical models across a broad range of aero-thermo-chemical conditions and

400

TURBULENT PREMIXED FLAMES

conﬁgurations without speciﬁc tuning for each application. By following a systematic approach, users can avoid tuning a model for one physical process (e.g., chemical kinetics) to compensate for deﬁciencies in modeling a different physical process (e.g., turbulence-chemistry interactions). Currently pdf methods have been applied primarily to reacting ideal-gas mixtures using single turbulencescale models; however, multiple-physics, multiple-scale information also has been incorporated. While pdf methods have been mostly applied to atmospheric pressure, laboratory-scale, statistically stationary, nonluminous, nonpremixed ﬂames with relatively simple fuels, the methods can be applied to high-Da cases and to low-to-moderate-Da systems, to premixed ﬂames as well as to nonpremixed and partially premixed ﬂames, and to practical combustion devices and to laboratory-scale ﬂames. It is anticipated that pdf-based methods will be adopted even more broadly through the twentyﬁrst century to address important combustion-related energy and environmental issues. HOMEWORK PROBLEMS

1.

Show that the characteristic thickness (δtf ) and velocity (utf ) of a thickened turbulent premixed ﬂame can be related to the Damk¨ohler number, integral length scale, and turbulence intensity by Equations 5.52 and 5.53.

2.

Discuss the countergradient and gradient transport approximations with respect to each other.

3.

Obtain G-equation (Equation 5.125) by using the level-set method.

PROJECT NO. 1

Consider the mixing and combustion of two turbulent streams of premixed gases (streams A and B have the same F/O ratio but different velocities and temperatures) in a constant-area duct. List the major governing equations you would use. State the boundary conditions, method of closure, and basic assumptions.

UA = 550 m/sec Stream A TA = 600 K UB = 120 m/sec TB = 2000 K Stream B

Mixing layer

PROJECT NO. 2

401

PROJECT NO. 2

To determine whether a laminar ﬂame can exist in a premixed turbulent ﬂow, Klimov and Williams showed, from solutions of the laminar ﬂame equations in an imposed shear ﬂow, that a propagating laminar ﬂame may exist only if the following stretch factor κ is less than a critical value of the order of unity: κ≡

δL 1 dA SL A dt

(1)

where δL , SL , and A represent the thickness, speed, and elemental areas of laminar ﬂame, respectively. It is also known that the percentage of elemental area variation can be approximated by 1 dA ∼ u = A dt T

(2)

where u and T represents the turbulence intensity and Taylor microscale. (i) Use the listed equations and any other necessary relationships to show that the stretch factor κ can be expressed as κ∼ =

δL η

2 (3)

where η is the Kolmogorov microscale. By setting κ equal to 1, the following criterion for laminar ﬂame to exist in a premixed turbulent ﬂow can be obtained. δL η (4) This is called Klimov-Williams criterion; when it is satisﬁed, wrinkled laminar ﬂames may occur. (ii) Construct a dimensionless intensity (u /SL ) versus Reynolds number map to show different regimes of turbulent premixed ﬂames. Sketch lines corresponding to constant values (e.g., 0.1, 1, 10, 100, 1000) of η/δL , Da, and l/δL . Note these deﬁnitions of various Reynolds numbers: Rel ≡ They are related by

u l u T u η , ReT ≡ , Reη ≡ ν ν ν Re4η ∼ = Re2T ∼ = Rel

Also indicate the special line that represents the Klimov-Williams criterion. (iii) Discuss physical meanings for each regime.

6 NON-PREMIXED TURBULENT FLAMES

SYMBOLS

Symbol A a D Da f Le k l ld N r r 1/2 Re S SL u T Wj x x Y Z z Zj 402

Description

Dimension

Arrhenius factor (for a reaction of order m) Strain rate, see Equation 3.17 Thermal diffusivity Damk¨ohler number, deﬁned in Equations 6.1 and 6.2 Mixture fraction Lewis number Turbulent kinetic energy Length scale of turbulence or ﬂame thickness Diffusive layer ﬂame thickness deﬁned in Equation 6.12 Number Radial coordinate Local half width of the radial jet (see Figure 6.28) Reynolds number Segregation factor used in Section 6.5 Laminar ﬂame speed Velocity component Temperature Atomic weight of the j th element Spatial coordinate Grid spacing Mass fraction Mixture fraction, deﬁned in Equation 3.2; also see Equations 3.11 and 3.13 Differential diffusion parameter deﬁned by Equation 6.37 Mass fraction of the j th element, deﬁned in Equation 3.24 Fundamentals of Turbulent and Multiphase Combustion Copyright © 2012 John Wiley & Sons, Inc.

(N/L3 )1-m /t 1/t L2 /t — — — L2 /t2 L L — L L —

Kenneth K. Kuo and Ragini Acharya

L/t L/t T M/N — L — — — —

403

SYMBOLS

Symbol (Z)r (Z)F Greek δ ε ρ η ξk τ φ χ

Pv Puv ω˙ F ν

Description Dimensionless reaction zone thickness deﬁned by Equation 6.9 Dimensionless ﬂamelet thickness deﬁned by Equation 3.102, ≈ Zst see Figure 3.8

Symbols Thickness Turbulent dissipation rate Gas density Kolmogorov length scale deﬁned in Equation 6.21 Normalized elemental mixture fraction Time scale Equivalence ratio, deﬁned by Equation 3.18 Instantaneous scalar dissipation rate, deﬁned by Equation 3.72 Marginal probability density function of v =V (also see Equation 4.177 in Chapter 4) Joint probability density function of sample space variables u and v Reaction rate Kinematic viscosity

Subscripts ch Chemical D Diameter d Diffusive layer thickness f Flame F Fuel L Laminar o Integral O2 Oxygen q Quenching m Mixing r Reaction zone st Stoichiometric t Turbulent t,m Turbulent, Mixing η Kolmogorov Others

∼

Fluctuation quantity using Reynolds averaging technique Fluctuation quantity using Favre averaging technique Favre averaged quantity

Dimension — —

L L /t M/L3 L — t — 1/t 2 3

— — 1/(L3 t) L2 /t

404

NON-PREMIXED TURBULENT FLAMES

Non-premixed (or diffusion) combustion occurs in all systems where fuel and oxidizer are not perfectly premixed before entering the combustion chamber. Because many practical combustion devices operate with non-premixed ﬂames in the presence of turbulent ﬂow, modeling of non-premixed turbulent combustion has become a central issue in understanding combustion systems. Examples are gas turbine and diesel engines; oil-, gas-, and pulverized coal-ﬁred boilers and furnaces; chemical lasers; rocket exhaust plumes; and ﬁres. In this chapter, the scope is limited to gaseous turbulent non-premixed ﬂames with emphasis on fundamental understanding of the combustion chemistry, turbulent mixing processes, and their interaction. Understanding of the turbulent non-premixed ﬂames is helpful for development or improvement of many practical combustion systems.

6.1

MAJOR ISSUES IN NON-PREMIXED TURBULENT FLAMES

Turbulence is characterized by the presence of a continuum of length and time scales in the ﬂow ﬁeld (see Chapter 4). Non-premixed ﬂames in a turbulent environment are different from the non-premixed laminar ﬂames due to turbulencechemistry interaction and mixing affected by turbulence. In addition, the effect of combustion on turbulence is also an important issue. Turbulent ﬂows have a large spectrum of length and time scales. The largest scales are based on the physical size of the combustion chamber, whereas the smallest scales represent the dissipation of turbulence energy by viscosity. The elementary chemical reactions also have a broad spectrum of time scales. Depending on the overlap between turbulence time scales and chemical times, strong or weak interactions between chemistry and turbulence may occur (Bray, 1996). In the non-premixed combustion, the fuel and oxidizer are supplied separately, which results in their entrainment by the larger-scale eddies. This process creates pockets of fuel-rich and fuel-lean mixture while stretch of iso-concentration surfaces enhances the local rate of mixing between these pockets. In certain cases, mixing at the molecular scale (micro-mixing) can be faster than the chemical reaction. The resulting heat release leads to volume expansion, which modiﬁes the turbulent ﬂow. Figure 6.1 identiﬁes some of the resulting interactions between combustion chemistry and the turbulent ﬂow. The chemical reaction occurs at the molecular scale, the turbulent mixing (or micro-mixing) is a small-scale mixing process that commences after ﬁnite volumes of ﬂuid are entrained and brought into contact by large-scale eddies. A good analogy of this behavior can be seen in Figure 4.15 of Chapter 4, where the large-scale eddies contain most of the turbulent kinetic energy and this energy is dissipated at the smallest scales while the intermediate scales simply transfer this energy from larger-scale eddies to the smallest-scale eddies. Stretching and curvature of iso-concentration surfaces by turbulent motion can greatly enhance molecular mixing, and the rates of heat and individual chemical species diffusion can then become important. This is how the small-scale mixing (or micromixing) is linked closely to the microscale structure of the turbulent ﬂame. In some speciﬁc circumstances, this local behavior of turbulent non-premixed ﬂames may be similar to the structure of a laminar ﬂame. In this

MAJOR ISSUES IN NON-PREMIXED TURBULENT FLAMES

Non premixed turbulent combustion, involving a continuum of length and time scales

Entrainment of fuel and oxidizer by large-scale eddies leading to incomplete mixing

Heat release leading to volume expansion, shear, and pressure changes thereby affecting turbulent flow

Stretch of iso-concentration surfaces, enhancing micro-mixing and diffusion, allowing chemical reactions to occur

405

Figure 6.1 Flow diagram illustrating the turbulence-chemistry interactions (modiﬁed from Bray, 1996).

case, a deterministic relationship between instantaneous values of thermochemical variables and their gradients can be established. In other cases, a more random behavior is to be expected. Since the description of non-premixed turbulent combustion requires an understanding of simultaneous turbulent mixing and combustion processes, the modeling of these ﬂames represents a major scientiﬁc challenge, a task that has been the subject of numerous studies (Bilger, 1976a, 1989; Peters, 1986). A turbulent combustion model should be composed of four components: a chemical kinetic model, a turbulent ﬂow model, and two interaction models—one for the effects of combustion on the turbulent ﬂow and the other for the inﬂuence of turbulent ﬂuctuations on the mean rate of chemical reaction. The stabilization and ignition of non-premixed turbulent ﬂames and the special role of edge-ﬂames (ﬂames formed near the physical boundary or edge separating the fuel and oxidizer zones) in these processes are the major topics of interest. Thus, the major research elements in turbulent non-premixed combustion studies are: • Turbulence-chemistry interaction • Mixing models and chemistry models • Flame-vortex interactions • Flame quenching • Flame instability • Partially premixed ﬂames • Edge-ﬂames In premixed turbulent ﬂames, general attempts to characterize the turbulence and combustion interaction are to determine mean heat-release rate in terms of a turbulent burning velocity. This approach requires extreme caution. Due to the thickness of the ﬂame brush, experimental values of the turbulent burning velocity from curved, strained, or transient turbulent ﬂames (Abdel-Gayed, Bradley, and Lawes 1987; Shepherd and Kostiuk, 1994) are sensitive to the choice of the location within the brush at which this velocity is to be evaluated. Comparison between theory and experiment therefore requires special care. It is important to note that in the non-premixed combustion, there is no concept of burning velocity.

406

NON-PREMIXED TURBULENT FLAMES

A popular approach to model the turbulent non-premixed combustion process is to represent the turbulent ﬂame as a collection of reaction-diffusion layers (or ﬂamelets) that are continuously displaced and stretched by the ﬂow. The properties of the internal structure of the thin layers and their possible quenching have been the subject of many fundamental studies. The other approach is based on a one-point statistical description of the ﬂame and does not invoke the ﬂamelet assumption. This is the case for models introducing probability density functions (pdf). In pdf models, chemical effects are included in a closed form, and the stumbling block is the closure of turbulent micro-mixing. Direct Numerical Stimulation (DNS) has emerged as a standard tool to study micromixing related problems in turbulent non-premixed ﬂames. 6.2

¨ TURBULENT DAMKOHLER NUMBER

The choice of suitable parameters to describe turbulence-chemistry interaction is an open question for non-premixed turbulent ﬂames. The Damk¨ohler number (Da) is a strong candidate; it is deﬁned as the ratio of a characteristic ﬂow time to a characteristic chemical time. Even when using a single-step forward reaction assumption, chemical times may vary greatly inside a given ﬂame, and some care must be taken to choose relevant expressions for Da. This may be accomplished by using activation-energy asymptotics (Li˜na´ n, 1974). This technique leads to analytical expressions for all quantities characterizing a laminar counterﬂowing fuel and oxidizer diffusion ﬂame and to a proper Da expression. A generic classiﬁcation of premixed ﬂames is commonly proposed by using two ratios: a ratio of characteristic speeds and one of characteristic lengths (Borghi, 1988). However, the diffusion ﬂames are usually characterized by the Da only, as shown by various DNS studies. Due to the lack of any reference length scales or time scales in non-premixed combustion, a chemical time scale can be compared with a characteristic turbulence time to deﬁne the Da. This number may be derived from activation-energy asymptotics, for example. The turbulent Da is deﬁned as Dat ≡

Turbulence time scale τt = Chemical time scale τch

(6.1)

The turbulence time scale could be either the integral time scale (τ0 ), turbulent mixing time scale (τm ), or Kolmogorov time scale (τη ). Typically, either the integral time scale or the turbulent mixing time scale is used as the turbulence time scale. Therefore, we can deﬁne the turbulent Da as: l0 /u τ0 Dat,0 = = (6.2) τch δF /SL where u = turbulence intensity; l0 = integral length scale τ0 = integral time scale.

SCALES IN NON-PREMIXED TURBULENT FLAMES

407

Similarly, a Da based on turbulent mixing time can be deﬁned as Dat,m = where the χ˜ st is deﬁned as

τm 1 = τch (δF /SL ) χ˜ st

2 ρ χ˜ st = 2ρD |∇Z|2Z=Zst ≈ 2ρD |∇Z |2Z=Zst = 2 ρD ∇Z Z=Zst

(6.3)

(6.4)

where Z is the mixture fraction and Z is the mixture fraction ﬂuctuations (see Sections 3.3 and 3.4 in Chapter 3 for description of mixture fraction and scalar dissipation rates). The Da based on the mixing time scale Equation 6.3 corresponds to the local non-premixed ﬂame structure; in contrast, the deﬁnition based on the integral time scale (τ0 ) corresponds to the global structure of the turbulent non-premixed ﬂame. The laminar ﬂame speed SL , the laminar ﬂame thickness δF , and the residence time across the ﬂame (or chemical time) τch are related to each other by this relationship: (6.5) τch = δF /SL where SL δF /ν ≈ 1. The integral length scale of the energy-containing large eddies is deﬁned by this equation: u3 k 3/2 (6.6) l0 = ε ε where ε is the turbulent dissipation rate. Some researchers prefer to work in terms 2 )–Da. This alternative deﬁnition of the Da of mixture fraction ﬂuctuations (Z is given as: τt 1 Dat,Z = (6.7) 2 τch Z 6.3

TURBULENT REYNOLDS NUMBER

It is natural to choose the Reynolds number as a second dimensionless group. Instead of mean ﬂow velocity, it should be deﬁned in terms of turbulence intensity (u ), integral length scale (l0 ), and laminar viscosity (ν): u l0 k 1/2 k 3/2 k2 Ret ≡ = (6.8) ν ν ε νε 6.4

SCALES IN NON-PREMIXED TURBULENT FLAMES

Unlike premixed ﬂames, diffusion ﬂames do not have a well-deﬁned reference length, and multiple choices are possible to deﬁne such lengths (Bilger, 1988;

408

NON-PREMIXED TURBULENT FLAMES

Bray and Peters, 1994; Cook and Riley 1996; Lee, 1994). However, certain length scales can be extracted from the understanding of non-premixed combustion processes. As shown in Chapter 3, the mixture fraction ﬁeld is a key variable in non-premixed ﬂame structure, as the width of the reaction zone lr is independent of the ﬂow ﬁeld in mixture fraction space but not in physical space. A diffusion ﬂame contains a diffusive layer in which a reaction zone is embedded, as shown in Figure 6.2. Let us consider a one-dimensional mixture fraction proﬁle in the y-direction. For laminar diffusion ﬂames, the relationship between ﬂame thickness in mixture fraction space (ZF ) and the physical space (lF ) was shown in Equation 3.101 in Section 3.5 of Chapter 3. In terms of the local reaction zone thickness in a turbulent non-premixed ﬂame, this equation can be rewritten as: lr =

(Z)r ∂Z ∂y Z=Zst

(6.9)

The scalar dissipation rate may be estimated in terms of the mixture fraction as follows (also see Equation 3.72 in Chapter 3): ρ χ˜ = ρχ = 2ρD |∇Z|2

(6.10)

We assume that in a non-premixed turbulent ﬂame, the mean thickness of the mixing layer is of the same order as the integral length scale of turbulence; that is k˜ 3/2 ˜ −1 ∇ Z ∼ ε˜ Oxidizer Z=0

(6.11)

Z = Zst

lt hk

ld Fuel Z=1

lr

Figure 6.2 Sketch of a non-premixed turbulent ﬂame (modiﬁed from Vervisch and Poinsot, 1998).

SCALES IN NON-PREMIXED TURBULENT FLAMES

409

The diffusive layer thickness ld adjacent to the reactive zones may be evaluated by: ⎞ 1/2 ⎛ 1 ⎠ (6.12) ld ∼ ⎝ ∇Z 2 It is implied by the last equation that scalar dissipation χ˜ quantiﬁes the intensity of turbulent mixing, which operates at scales of the order of magnitude of ld . ˜ may As a consequence, the gradient of the mean value of mixture fraction (∇ Z) be neglected in comparison with the gradient of the mixture fraction ﬂuctuation (∇Z ).

2 2 ˜ 2 ρ χ˜ = 2 ρD ∇ Z + ρD ∇Z + 2∇ Z˜ · ρD∇Z 2ρD ∇Z = ρ χ˜ f

(6.13) where χ˜f is the averaged scalar dissipation rate associated with mixture fraction ﬂuctuations. By extending this analysis for a one-dimensional case to an isotropic turbulence case (i.e., in all three directions), the total scalar dissipation rate may be modeled as: 2 2 ∂Z ∂Z ρ χ˜ = 3 × 2 ρD = 3 × 2 ρD (6.14) ∂y ∂y In the case of a simple planner ﬂame, y can be regarded as the instantaneous coordinate perpendicular to the local ﬂame surface. Considering the dissipation in the y-direction, the right-hand-side term in Equation 6.13 is evaluated by a simple linear relaxation model given as: 2 ε˜ Z 2 2 χ˜f = 2D ∇Z = cD = cD Z τt k˜

(6.15)

where cD is a model constant of the order unity. This relationship implies that the turbulence dissipation time and the scalar dissipation time are approximately equal. 2 1 k˜ Z = (6.16) χ˜f cD ε˜ By combining Equations 6.9, 6.13, and 6.16, we can show that the mean reaction zone thickness can be expressed by the next equation 6.17, when the mixture fraction gradient is in the y-direction only (Z)r lr = 1/2 2 Z

2D k˜ cD ε˜

1/2 (6.17)

410

NON-PREMIXED TURBULENT FLAMES

˜ ε ) 1/2 has the length dimension; when the local mixture is The quantity (D k/˜ ˜ the value of lr is reduced to a smaller subjected to the large-scale strain rate ε˜ /k, thickness. Considering Equation 6.12, the typical diffusion layer thickness can be related to the mean scalar dissipation rate of the mixture fraction conditioned on the stoichiometric mixture fraction surface as: ld ∼

Dst χ˜ st

1/2 (6.18)

The average of the mixture fraction gradient in the last equation is a conditional average on the stoichiometric ﬂame surface. The ratio between the turbulence and diffusion length scales is then given by: 1 1/2 1/2 k˜ 2 1/2 1/2 χ˜ st /2 k˜ 3/2 l0 k˜ 2 2 Sc=1 2 ∼ = Re Z ∼ Z Z t ld Dst ε˜ νst ε˜ (Dst ε˜ )1/2 (6.19) In this expression, it is assumed that the Schmidt number is equal to 1 and the turbulent Reynolds number is given by Equation 6.8, conditioned on stoichiometric properties, based on average parameters k˜ and ε˜ . By a similar procedure, the ratio of the Kolmogorov scale to the diffusion thickness scale is given by: 1/2 2 η Z ∼ (6.20) 1/2 ld Ret where the Kolmogorov length scale is given by (also see Equation 4.161 in Chapter 4): 3 1/4 ν (6.21) η≡ ε For a single-step forward reaction νF F + νO O → P, it can be shown that the reaction layer thickness is related to the diffusion layer thickness by: lr ∼ ld (Dat )−1/νtot

(6.22)

where νtot = νO + νF + 1 (Vervisch and Poinsot, 1998). The ratio of Kolmogorov scale to reaction layer thickness is then given by: η ∼ lr

2 Z

1/2 Ret

1/2 (Dat )1/νtot

(6.23)

Let us revisit the turbulent Da, which is deﬁned as a ratio of a mixing time to a chemical time. Based on the physical reasoning just presented, the mixing

SCALES IN NON-PREMIXED TURBULENT FLAMES

411

time can be estimated as the inverse of the scalar dissipation calculated at the stoichiometric surface: 1 τm = (6.24) χst The laminar Da is given by: Da =

1 τch χst

(6.25)

The turbulent Da involving mixture fraction ﬂuctuations is given by Equation 6.7. There is another way to express the turbulent Da: by expressing the mixture fraction ﬂuctuations in terms of a segregation factor S . The segregation factor S gives an indication on the degree of mixing: If S is close to 1, the ﬂow is composed of pockets of fuel-rich and fuel-lean mixtures. If S is close to 0, the reactants are almost perfectly mixed. The Da thus becomes: Dat,Z,S =

τt 1 τ Z˜ 1 − Z˜ ch S 1

(6.26)

A high level of mixture fraction ﬂuctuations leads to a decrease of Dat and even to a local extinction. This is in agreement with the fact that high values of Z induce a thin mixing zone, leading to small values of ld . Let us discuss the time scales associated with turbulent non-premixed combustion. For multistep chemistry, it may be convenient to replace τch by τcϕ based on the value of the critical scalar dissipation rate at quenching, χq (see Figure 3.7 in Chapter 3). The Kolmogorov time τη of the small scales is deﬁned from the dissipation energy of the turbulence ε and the kinematic viscosity ν. The ratio of the energy of the turbulence k˜ to ε˜ is used to estimate the turbulence time scale τt . The Da derived from asymptotics incorporates the properties of the reaction zone and of the mixture fraction: Dat,η is used to compare Kolmogorov time with τcϕ : τη Dat,η ≡ = χq τη (6.27) τcϕ where τη =

ν 1/2 ε

(6.28)

Equation 6.2 is also shown in Equation 4.163 in Chapter 4. A summary of length and time scales as well as various Da deﬁnitions are given in Table 6.1.

6.4.1

Direct Numerical Simulation and Scales

All scales in turbulent motion are supposed to be resolved by DNS. Therefore, the governing equations must capture the length and time scales that represent chemical reaction, viscous dissipation, and diffusion processes. In other words,

412

NON-PREMIXED TURBULENT FLAMES

TABLE 6.1. Length and time scales in turbulent non-premixed combustion (modiﬁed from Vervisch and Veynante, 1999) Scale

Combustion

Turbulence

Dimensionless quantities

Time

Simple chemistry: Chemical time τch determined from asymptotics for one-step formulation

Integral time: ˜ ε) τ0 = (k/˜

Turbulent Damk¨ohler number: Dat,0 = τ0 /τch

Scalar dissipation time: τm = (1/χ˜ st )

Dat,m = τm /τch Dat,η = τη /τch Dat,m = τm /τcϕ

Kolmogorov time:

Dat,z =

Complex chemistry: τcϕ = χq−1

τη = (ν/ε)

Length Diffusive thickness: ld |∇Z|−1 Z=Zst (D˜ st /χ˜ st )1/2

=

Reaction thickness: −1/νtot lr ld Dat,m

1/2

Integral scale: l0 k˜ 3/2 /˜ε Kolmogorov scale: 1/4 η ν 3 /ε

τt 1 2 τch Z

Turbulent Reynolds number: k2 u l0 Ret = ν νε Length scale ratios: 1/2 2 Re l0 / l d = Z t l0 / lr = (l0 / ld )(Dat,m )1/νtot where νtot = νF + νO + 1 1/2 −1/4 2 Ret η/ ld = Z η/ lr = (η/ ld )(Dat,m )1/νtot

a grid (or mesh) used for DNS should be able to capture the length scales, such as the reaction zone thickness (lr ), the Kolmogorov microscale (η), and the diffusive characteristic lengths (ld ). In practice, once the Kolmogorov microscale is resolved, the diffusive length scale also is resolved, except for scalars with very large Schmidt numbers (in which case the diffusive length scales could be smaller than the Kolmogorov microscale) (Lesieur 1990). This can be shown by dividing the Kolmogorov length scale expression in Equation 6.21 by the diffusive length scale expression in Equation 6.18. Consider a grid in which there are N number of grid points in one direction. The reaction zone thicknesses can be resolved if x < lr /Nr where Nr is the number of points required to resolve the reaction zone property variations and x is the characteristic mesh size. The second condition on the mesh size can be obtained by the Kolmogorov length scale. In order for the mesh to capture the Kolmogorov scale, it is required that x < η. The number of integral length scales embedded within the computational domain is N0 . The number N0 is an important parameter quantifying the possible impact of large scales on the ﬂame. The number of grid points in one direction in the characteristic mesh can be

413

SCALES IN NON-PREMIXED TURBULENT FLAMES

obtained from the next equation: N=

N0 l0 x

(6.29)

By using the length scale ratios summarized in Table 6.1, we can obtain these two inequalities: 4/3 N (6.30) Ret < N0 1/2 1/ν 2 Ret Z Dat,m tot <

N N0 Nr

(6.31)

The resulting limitations for DNS of non-premixed turbulent ﬂames are summarized in Figure 6.3. Despite the fact that DNS cannot simulate the entire range covered by typical laboratory jet ﬂame experiments, this diagram shows that representative conditions corresponding to certain real ﬂames (with relatively low Reynolds number and Damk¨ohler number) can be approached using DNS. In this diagram, the jet ﬂame domain corresponds to typical laboratory experiments based on the works of Dibble, Long, and Masri (1986); Masri, Bilger, and Dibble (1988b), and T.S. Cheng, Wehrmeyer, and Pitz (1991). Recently, Chen and coworkers have performed large DNS on a massive supercomputer named “Jaguar,” at Oak Ridge National Lab. Readers can reference the report by Im et al. (2009) and the technical article by Oefelein, Chen, and Sankaran, 2009.)

Dat,0

104

102

Condition on chemistry Δx < lr/Nr

Domain of laboratory jet flames

2D DNS

Nr: number of points required to resolve the reaction zone N0: number of integral length scales within the computational domain N: total number of grid points in one direction Δx: mesh size h: Kolmogorov length scale lr: reaction zone thickness l0: integral length scale For 3D calculations: N = 500, N0 = 5, Nr = 10

1

For 2D calculations: N = 1000, N0 = 5, Nr = 10

Possible domain for 2D and 3D DNS

Condition on turbulence Representative conditions: Δx < h Z = 0.5 and vF = vO = 1

10−2 1

Z′ 2 ≈ Z(1−Z) 10

1

10

2

3

10

4

10

Ret

Figure 6.3 Limitations of direct numerical simulation (DNS) calculations in a Dat,0 -Ret diagram (modiﬁed from Vervisch and Poinsot, 1998).

414

NON-PREMIXED TURBULENT FLAMES

6.5 TURBULENT NON-PREMIXED COMBUSTION REGIME DIAGRAM

The identiﬁcation of combustion regimes in non-premixed cases is complicated by the lack of characteristic length and velocity scales. The ﬂame structure is inﬂuenced by the velocity ﬁeld and the mixing rate. The ﬂame front is wrinkled and affected by turbulence. Regime classiﬁcation is based on the comparison between the chemical time τch and a turbulent time τt (which has multiple deﬁnitions, as discussed in Section 6.4). A reasonable approach should be to utilize the shortest turbulent time for comparison with the chemical time. Generally, the chemical times are much shorter than the turbulent times (Poinsot and Veynante, 2005). As discussed earlier in this chapter, the turbulent Da based on turbulent mixing time, i.e., Dat,m can be a reasonable choice for a dimensionless number in the study of turbulent non-premixed combustion. The second such parameter could be the turbulent Reynolds number. A combustion regime diagram for turbulent non-premixed combustion can be established in terms of Ret and Dat,m . If the chemical reactions are much faster than the other processes (fast chemistry), then the chemical times are lower and the Da is large. In addition, if these chemical reactions are irreversible, then the ﬂame can be assumed to be very thin (i.e., lr ld ), and it can be identiﬁed as a laminar ﬂame element called ﬂamelet. For large chemical times (ﬁnite rate chemistry and low Da), or with reversible chemical reactions, the reaction zone thickness lr could be of the same order as the diffusive length scale ld . In such cases, the ﬂame structure could be very different from laminar ﬂame structure, and unsteady effects could be expected. As shown in Equation 6.7, high mixture fraction ﬂuctuations also lead to a low Da number. For a low Da number case (large chemical time, high level of mixture fraction ﬂuctuations), ﬂame extinctions could occur. Physically, the effect of low or high Da number can be interpreted in terms of mixture fraction ﬂuctuations. If the mixture fraction ﬂuctuations are high, then the reaction zones are separated and individual non-premixed ﬂamelets may exist; otherwise the reaction zones are connected. In order to distinguish the separated reaction regime from the connected reaction zone regime (or ﬂamelet regime), we may consider the ratio of the Kolmogorov length scale to the reaction zone thickness, which was initially proposed by Bilger (1988a). When the ratio (η/ lr ) is greater than unity, the reaction zone is embedded in the Kolmogorov length scale and the ﬂamelet regime prevails. This criterion leads to the next inequality: 1/2 νtot /2 Ret Existence of ﬂamelet regime, when Dat,m > (6.32) 2 Z In the ﬂamelet regime, if the segregation factor S is close to 1, reaction zones develop between unmixed reactant pockets and the ﬂame can be well deﬁned. If S < 0.5, the ﬂamelets merge. The criterion for existence of nonﬂamelet regime 1/2 νtot /2 can be similarly deﬁned as: Ret Dat,m < (6.33) 2 Z

415

TURBULENT NON-PREMIXED COMBUSTION REGIME DIAGRAM

In this regime, the ﬂame becomes sensitive to unsteadiness induced by turbulent mixing. The ﬂamelet regime can be further divided into two subregimes. 1. If S < 0.5, the mean reaction zone is thickened by the turbulent ﬁeld. 2. If S > 0.5, local extinctions appear and grow as S approaches unity. Based on this analysis, a combustion regime diagram for turbulent nonpremixed combustion can be constructed as shown in Figure 6.4. In using this classiﬁcation, readers should remember that the analysis relies on intuitive considerations and uses approximations. For instance, turbulence is described solely in terms of the Kolmogorov scale, and the unsteady effects are neglected. The limits between regimes deﬁne a trend and should not be interpreted as real frontiers between regimes. The analysis relies on a single chemical time scale while real ﬂames feature a range of time scales. This fact as well as the use of a parameter S for describing the level of mixing makes the physical quantities used in this diagram not easy to evaluate and therefore difﬁcult to use in practice. Another non-premixed combustion diagram is proposed in Poinsot and Veynante (2005) based on two characteristic ratios: (1) a length scale ratio of integral (l0 ) and diffusive scales (ld ) and (2) a time scale ratio of the turbulence integral characteristic time (τ0 ) and the chemical time (τch ). A relationship between the Damk¨ohler number based on the mixing time scale Equation 6.3 (corresponding to the local non-premixed ﬂame structure) and that based on integral time scale (τ0 ) (corresponding to the global structure of the turbulent non-premixed ﬂame) can be established: Dat,0 =

τt τt τη = τch τη τch

Assumptions

≈

τt τm τt = Dat,m ∼ Ret Dat,m τη τch τη

(6.34)

Flamelet regime

Dat,m

Laminar flames

es) zon n o cti rea 0.5 ted , S ~ c e nn = l r h (Co es) zon on ets) i t c ea ck ed r (po Z′′ 2 increase arat , S ~ 1 p e (S = l r h Non-flamelet regime (Strong effects of unsteadiness)

Daq Quenching 1

Ret

Figure 6.4 Non-premixed combustion diagram (modiﬁed from Vervisch and Veynante, 1999).

416

NON-PREMIXED TURBULENT FLAMES

The assumptions in Equation 6.34 are that the turbulence is homogeneous and isotropic such that the ﬂame is strained solely due to the turbulent eddies of the Kolmogorov scale, and any possible compression or viscous dissipation by Kolmogorov scale is ignored. In such a speciﬁc case, the diffusive thickness (ld ) and time scale (1/χ˜ st ) are assumed to be controlled by the Kolmogorov length and time scales (η and τη ), that is, τm ≈ 1/χ˜ st ≈ τη ,

ld ≈ η

(6.35)

The time scale ratio (τt /τη ) can be related to the turbulent Reynolds number as: τt (k/ε) ≈ = τη (ν/ε)1/2

k2 νε

≈

Ret

(6.36)

Let us construct a combustion diagram based on Equation 6.34. On a log-log Dat,0 − Ret plot, constant Damk¨ohler number mixing Dat,m corresponds to lines of slope 1/2. If the chemistry is fast (high Damk¨ohler number), then the laminar ﬂamelet assumption (LFA) can be implemented. This condition may be expressed as Dat,m > DaLFA t,0 . If the chemical reaction times (low Da) are very long, then ﬂame extinction may occur. This condition may be expressed as Dat,m > Daext t,0 . Between these two limits, unsteady effects are dominant. These conditions are expressed in Figure 6.5. In addition to the previous analysis, Peters (2000) also has given a turbulent non-premixed combustion regime diagram as a log-log plot of the two ratios: (1) a ratio of mixture fraction ﬂuctuation conditioned on stoichiometric value and the ﬂame thickness in mixture fraction space (Zst /|Z|F ), and (2) a ratio of time scales of critical scalar dissipation rate for quenching and the mean Dat,0

Flamelet regime A

LF

a t,0

Da t,m

=D

Laminar flames

ts ffec y e aext,t0 d a D ste Un a t,m = D

Quenching Dat,m > Daext t,0

1

Figure 6.5 1994).

Ret

Non-premixed combustion diagram (modiﬁed from Cuenot and Poinsot,

TURBULENT NON-PREMIXED COMBUSTION REGIME DIAGRAM

417

scalar dissipation rate conditioned on stoichiometric value (χq /χ˜ st ). Recall that the inverse of the scalar dissipation rate represents a time scale; all of these quantities have been deﬁned either earlier in this chapter or Chapter 3. The physical reasoning for constructing a combustion diagram is described next. • When the mixture fraction ﬂuctuations are large such that Zst > |Z|F , the ﬂuctuations in the mixture fraction space extend to sufﬁciently lean and rich mixtures. Due to this, the diffusive layers surrounding the reaction zones are separated. • For small mixture fraction ﬂuctuations (i.e., Zst < |Z|F ), when there is intense mixing or partial premixing of the fuel stream, the mixture fraction ﬂuctuations are small. In such cases, the diffusive layers separating the reaction zone are not too far apart, and ﬂame zones stay connected. Therefore, the criterion Zst = |Z|F represents a demarcation between the two regimes. • If the mixture fraction ﬂuctuations are smaller than the reaction zone thickness, even the reaction zones are very close to each other, and they can be considered to be connected. This means that the mixture fraction zone is nearly homogeneous. On the combustion diagram, the line Zst = |Z|r has a slope of –1/4 (Seshadri and Peters, 1988). • Flame extinction will occur when the mean scalar dissipation rate is equal to or greater than the critical scalar dissipation rate for quenching. This criterion can be represented by a vertical line on the combustion regime diagram, represented by χ˜ st = χq . The turbulent non-premixed combustion regime diagram based on this reasoning is shown in Figure 6.6. Z′st ΔZ F 10

Separated flamelets Z′st = ΔZ

F

0.1

Flame quenching

1 Connected flame zones Z′st = ΔZ

Connected reaction zones Z′st < ΔZ r 0.01 1

Figure 6.6

10

100

r

χq χ∼st

Non-premixed combustion regime diagram (modiﬁed from Peters, 2000).

418

6.6

NON-PREMIXED TURBULENT FLAMES

TURBULENT NON-PREMIXED TARGET FLAMES

Turbulent non-premixed combustion in real systems could have complex boundary conditions and an inability to implement multiple diagnostic techniques. Because of these issues, a complex system may not be very useful to study fundamental processes such as turbulence, chemical kinetics, radiation, and pollutant formation as well as their interaction with each other in turbulent non-premixed ﬂames. Therefore, research burners are required to perform experimental studies and to obtain detailed data for these processes to facilitate the development of mathematical models for turbulent non-premixed combustion. A well-designed research burner must not only hold a ﬂame that satisﬁes the speciﬁc objectives of the experiment; it also must have well-deﬁned boundary conditions. A common research procedure is to isolate physical processes associated with a given practical application and to study each process separately or in simple combinations with each other. In a real system, however, all of the fundamental physical processes—turbulence, chemical kinetics, thermal radiation, and pollutant formation—interact together in a complex geometry. A research burner has simple boundary conditions and two, possibly three, of the listed complications present in the ﬂames. For example, a laminar ﬂame is suitable for studying chemical kinetics, soot formation, and thermal radiation. Nonsooting turbulent ﬂames may be tailored to study many aspects of combustion, such as chemistry interactions with turbulence, effects of complex recirculating ﬂows, and various fuel mixtures. Over the last decade, many burner geometries for turbulent non-premixed ﬂames were investigated. Generally, researchers should be able to use any burner that has a simple geometry and well-deﬁned boundary conditions to obtain useful information. However, it would be easier for collaborative comparisons of experimental data and model predictions among the research community to have a set of well-documented and relatively simple ﬂames that could serve as benchmark cases. A good source of documentation for standard ﬂames is the International Workshop on Measurement and Computation of Turbulent Non-premixed Flames (TNF Workshop) helmed by Sandia National Laboratories. A major aim of the TNF Workshop is the establishment of extensive experimental data sets from standard ﬂames, which can be used for the veriﬁcation and improvement of mathematical models. These target ﬂames have been identiﬁed and divided into four categories in the order of increasing complexity: (1) simple jet ﬂame burner, (2) piloted jet ﬂame burner, (3) bluff-body burner, and (4) swirl stabilized burner. These ﬂames are shown in Figure 6.7. One of the most modeled and best-understood ﬂames is Sandia piloted ﬂame (Sandia ﬂame D). The simple jet ﬂame mentioned in the target ﬂames is also called DLR (also known as the German Aerospace Center) simple jet ﬂames of CH4 /H2 /N2 . The reason for studying these target ﬂames is to improve the understanding of the effectiveness of various models in addressing combustion phenomena. A crucial question in these studies is the accuracy and reproducibility of measurements and the reliability of ﬂow conditions. Therefore, experimental

TURBULENT NON-PREMIXED TARGET FLAMES

(a) DLR jet flame

(b) Sandia piloted flame

(c) Sydney bluff body flame

419

(d) Sydney swirling flame

Figure 6.7 Photographs of four different turbulent non-premixed target ﬂames (from Sandia TNF proceedings www.sandia.gov/TNF/abstract.html).

or computational investigations should compare two or more independent measurements in the same ﬂame performed at different laboratories. 6.6.1

Simple Jet Flames

Simple jet ﬂames are the simplest and most common burner on which the original studies of turbulent non-premixed ﬂames by Hawthorne, Weddell, and Hottel (1949) and Wohl, Kapp, and Gazley (1949) were made. A simple jet ﬂame consists of a simple stream of fuel jet surrounded by a coﬂowing stream of air. The fuel stream issues either from a simple pipe, in which case the ﬂow at the jet exit plane is turbulent and similar to fully developed turbulent pipe ﬂow, (Dibble, Rambach, and Hollenbach, 1981; Drake et al., 1981; Drake, Pitz, and Shyy, 1986) or from a well-contoured nozzle, in which case the ﬂow is laminar and undergoes a transition to become turbulent as it leaves the burner (Kent and Bilger, 1973; Starner, 1983). The transition to turbulence and associated instabilities have been the subject of many research articles (Masri, Starner, and Bilger 1984; Takahashi and Goss, 1990). This burner also has been used to study the effects of differential diffusion (explained later) on the ﬂame structure (e.g., Smith et al., 1995). At very low jet velocities, the ﬂames are laminar; hence soot formation and thermal radiation can be studied. At higher jet velocities, hydrocarbon ﬂames tend to blow off and cannot be anchored on the simple jet ﬂame burner. However, if the coﬂowing air velocity

420

NON-PREMIXED TURBULENT FLAMES

is very low, a lifted ﬂame can be stabilized at some distance downstream of the burner’s exit plane. There are different theories on the mechanism of lifted ﬂame stabilization. The early theory of Vanquickenborne and Van Tiggelen (1966) proposed that the ﬂame is stabilized at the location where the turbulent gas velocity is equal and opposite to the counter-propagating turbulent ﬂame speed. Peters and Williams (1983a) theorized that in the region where the ﬂame is stabilized at some distance away from the burner exit, the fuel and oxidizer are not fully mixed; they proposed another theory based on the laminar ﬂamelet concept. The ﬂamelets are locally extinguished when the local scalar dissipation rate exceeds a critical limit. The ﬂame is believed to be stabilized in the region where the heat released due to burning, and heat losses due to scalar dissipation are balanced. Broadwell, Dahm, and Mungal (1984) developed a theory based on the entrainment of air into the fuel jet by large-scale motion. According to this theory, a ﬂame will blow-off if the ratio of a mixing time scale over a chemical time scale (i.e., the Da) decreases below a critical value. The generalizability of all three approaches remains questionable, and these issues have been thoroughly reviewed by Pitts (1988). A more likely and realistic view is that the reactants in the stabilization zone are partially premixed. Using this concept, M¨uller, Breitbach, and Peters (1994) combined the approaches based on laminar premixed ﬂame propagation and non-premixed ﬂame quenching to develop a formulation for the burning velocity. 6.6.1.1 CH4 /H2 /N2 Jet Flame A standard burner for simple jet ﬂames uses methane as a fuel component and ambient air as an oxidizer. Hydrogen is added to stabilize the ﬂame, without changing the simple ﬂow ﬁeld of the round jet. There are several quantities of interest in these ﬂames, and multiple diagnostic techniques, particularly Raman scattering, are required to obtain a complete characterization of the ﬂame. Nitrogen is added to this ﬂame to provide dilution, which decreases thermal radiation and therefore improves the signal quality of the spontaneous Raman scattering technique. The measurements were made by three different research groups at Sandia-USA, DLR-Stuttgart of Germany, and Darmstadt University of Technology-Germany. A summary of quantities of interest and diagnostic method used for their measurement is shown in Table 6.2. Simple Jet Flame Burner Geometry The burner consists of a straight stainless steel tube (l = 350 mm, I.D. = 8 mm) with a thinned rim at the exit. The coﬂow nozzle has a diameter of 140 mm at the DLR and is 300 mm square at Sandia. The burner can be moved vertically and horizontally to change the measuring position, which is determined by the optical arrangement. All three burners have following ﬂow conditions:

• Nozzle: D = 8.0 mm inner diameter, tapered to a sharp edge • Fuel composition: 22.1% CH4 , 33.2% H2 , 44.7% N2 • Stoichiometric mixture fraction: fstoic = 0.167

TURBULENT NON-PREMIXED TARGET FLAMES

421

TABLE 6.2. Summary of experimental methods for CH4 /H2 /N2 simple jet ﬂames Quantities of Interest

Diagnostic Technique

Organization

Joint probability density functions (JPDFs) of mixture fraction, temperature, and major species concentrations

• Single-pulse point-wise

DLR-Stuttgart of Germany

Distributions of OH, CH, and NO and temperature ﬁelds

Planar laser-induced ﬂuorescence (PLIF) and planar Rayleigh scattering yielding 2D distributions

Sandia National LaboratoryUSA

Laser Doppler anemometry (LDA)

Darmstadt University of TechnologyGermany

These minor species are especially important for the understanding of the turbulence-chemistry interaction in turbulent ﬂames and for the assessment of the quality of CFD calculations. Velocity distributions

spontaneous Raman/Rayleigh scattering for the determination of the temperature and major species concentrations; • 2D Rayleigh scattering for the determination of instantaneous temperature distributions; and • 2D LIF measurements of OH; CH, and NO for the visualization of the structures within the ﬂame.

• Adiabatic ﬂame temperature: Tad = 2,130 K • Coﬂow: 0.3 m/s, 292 K, 0.8% mole fraction H2 O • Flow conditions: Ujet = 42.2 ± 0.5 m/s; ReD = 15,200; Ujet = 63.2 ± 0.8 m/s; ReD = 22,800 The ambient pressure during the measurements was 953 mbar at the DLR and 990 mbar at Sandia. Due to the difference in pressure, the total mass ﬂow rates were slightly different for the same exit velocity. In the Sandia experiment, the coﬂow was fed with ﬁltered room air with a water vapor concentration of 0.8%. At the DLR, the coﬂowing air was dry, but for x /D > 30, the ambient air could also reach the ﬂame. Here x is the axial distance from the nozzle and D is the internal diameter of the tube. In both cases, the coﬂow velocity was 0.3 m/s. In order to study the inﬂuence of Reynolds number, two simple jet ﬂames were studied with the same fuel composition, burner, and boundary

422

NON-PREMIXED TURBULENT FLAMES

conditions but with two different exit velocities. Compared to the ﬂame with ReD = 15,200, the mass ﬂow rate of the second ﬂame was increased by 50%, resulting in a Reynolds number of 22,800. At the higher Reynolds number, the ﬂame was close to blow-off. It should be noted that by increasing the jet velocity, the Damk¨ohler number decreases since the aerodynamic time decreases at higher jet velocities. This may have a stronger inﬂuence than the Reynolds number. Flame Characterization Measurements were performed at six heights above the nozzle at x /D = 5, 10, 20, 40, 60, and 80. At each height a radial proﬁle consisting typically of 15 measuring positions was obtained on one-half side of the ﬂame. Only one side was measured because symmetry was assumed. Additionally, an axial proﬁle from x /D = 2.5 to x /D = 100 was measured. At each location, 400 single-pulse measurements were performed, from which the joint pdfs of temperature and species concentrations were deduced. Figure 6.8 depicts the proﬁles of the mean temperatures determined from Raman and Rayleigh scattering and rms (root mean square) ﬂuctuations of these quantities for x /D = 5, 40, and 60. Both mean temperatures agree well for most of the measuring locations, except inside the reaction zone, where they differ by at most 65 K. Near the fuel jet exit (x /D = 5), the temperature proﬁle has room temperature in the center of the fuel jet and a maximum mean temperature of 1,634 K. At x /D = 60, the maximum mean temperature is located at the axis and the temperature proﬁles become ﬂatter with bell-shaped distribution. The highest mean temperature at this axial location was measured to be 1,926 K. The correlation between temperature and mixture fraction is displayed in the scatter plots of Figure 6.9 for the downstream locations x /D = 5, 20, and 60. Each symbol represents the result of a single-shot measurement, and the three lines show the calculated results from laminar non-premixed ﬂame corresponding to three different assumptions: (1) adiabatic ﬂame with equilibrium chemistry, (2) strain rate a = 20 s−1 and unity Lewis number, and (3) strain rate a = 200 s−1 and nonunity Lewis number. The effect of the Reynolds number can be observed by comparing the plots on the right to those on the left. An apparent difference is observed at x /D = 5, where the ﬂame with ReD = 22,800 exhibits low temperatures more frequently than the turbulent ﬂame at a lower Reynolds number of 15,200. There are two possible explanations for the low temperatures:

1. Spatial averaging—that is the probe volume (the local volume at which spectroscopic measurements were made) could contain a mixture of different gases, one part with f ≈ 0.05, T ≈ 1,200 K and another part with f ≈ 0.45, T ≈ 1,200 K, which would result in an average of f ≈ 0.25, T ≈ 1,200 K. This effect should be less pronounced in the ﬂame with ReD = 15,200 because at lower Reynolds numbers, the range of turbulence scales decreases (see Equation 4.165 in Chapter 4), resulting in less likelihood of strong ﬂuctuations. 2. At higher Reynolds numbers, the probe volume often has only partially reacted mixture due to local ﬂame extinction. At axial locations close to

423

TURBULENT NON-PREMIXED TARGET FLAMES 0.20

2000

x/D = 60

0.18

DLR Sandia

1600

~ f

0.14 0.12 0.10 0.08 0.06 0.04

x/D = 60

1000 800 600

T′′ rms

0

−2

0

2

4 6 8 Radial Podition r/D

10

12

−2

14

0.35

0

2

4 6 8 Radial Podition r/D

10

12

14

2000 DLR Sandia

0.30 0.25

1600

x/D = 40 ~ f

0.20 0.15 0.10

1400

~ T

1200

x/D = 40

1000 800 600 400

f′′rms

0.05

DLR Sandia

1800

Temperature/K

Mixture Fraction

1200

200

0.00

T′′ rms

200 0

0.00 0

2

4 6 Radial Podition r/D

8

10

0

1.0

2

4 6 Radial Podition r/D

8

10

2000 DLR Sandia

0.8 0.7

~ f

0.6

x/D = 5

0.5 0.4 0.3

f′′rms

0.2

1600 1400 1000 800 600 400 200

0.0

0 0.5 1.0 Radial Podition r/D

x/D = 5 ~ T

1200

0.1 0.0

DLR Sandia

1800

Temperature/K

0.9 Mixture Fraction

~ T

1400

400

f′′rms

0.02

DLR Sandia

1800

Temperature/K

Mixture Fraction

0.16

1.5

T′′ rms 0.0

0.5 1.0 Radial Podition r/D

1.5

Figure 6.8 Comparison of radial proﬁles of f˜, f , T˜ , and T at x /D = 5, 40, and 60 for ReD = 15,200 (modiﬁed from Meier, Duan, and Weigand, 2005).

the jet exit, the difference between the two Reynolds numbers is stronger, and the effects of spatial averaging cannot be discounted. From the species scatter plots (to be discussed), it can be shown that the second explanation is responsible for the greater ﬂuctuations in case of a higher-Reynolds-number turbulent ﬂame. The comparison of turbulent non-premixed ﬂame measurements with the laminar ﬂame calculations allows an assessment of the thermochemical state and the inﬂuences of ﬂame stretch and Lewis number in the combustion processes. At x /D = 5, the measured results are not in agreement with adiabatic equilibrium

424

NON-PREMIXED TURBULENT FLAMES

1500

exp.(Sandia) adiab. equil. a = 10 s–1, Le = 1 a = 200 s–1, Le ≠ 1 ReD = 15,200 x/D = 60

1000

500

1500

ReD = 22,800 x/D = 60 1000

500 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mixture Fraction

1500

exp.(Sandia) adiab. equil. a = 10 s–1, Le = 1 a = 200 s–1, Le ≠ 1 ReD = 15,200 x/D = 20

1000

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mixture Fraction

500

1500

500

exp.(Sandia) adiab. equil. a = 100 s–1, Le = 1 a = 400 s–1, Le ≠ 1 ReD = 15,200 x/D = 5

1000

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mixture Fraction fstoich. 2000

Temperature/K

Temperature/K

1500

ReD = 22,800 x/D = 20

1000

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mixture Fraction

2000

exp. adiab. equil. a = 50 s–1, Le = 1 a = 400 s–1, Le ≠ 1

2000

Temperature/K

Temperature/K

2000

exp. adiab. equil. a = 50 s–1, Le = 1 a = 200 s–1, Le ≠ 1

2000

Temperature/K

Temperature/K

2000

1500

exp.(Sandia) a = 400 s–1, Le = 1 a = 400 s–1, Le ≠ 1 ReD = 22,800 x/D = 5

1000

500

500 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mixture Fraction

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mixture Fraction

Figure 6.9 Scatterplots of temperatures in comparison to laminar ﬂame calculations at different Reynolds numbers (modiﬁed from Meier et al., 2000).

calculations since the comparison with adiabatic equilibrium should be applied only for fuel-lean and near-stoichiometric mixtures, not beyond stoichiometric mixture fraction. It is interesting that the strained laminar ﬂame calculations with Le = 1 do not match the experimental results well, except for very fuel-rich mixtures: for f ≥ 0.5 for ReD = 15,200 and f ≥ 0.7 for ReD = 22,800. For lower mixture fractions, the calculated temperatures are too high, especially for ReD = 22,800. A possible explanation for this is described next. In a laminar ﬂame with Le < 1, the mass transport is faster than the heat transport; slower thermal transport results in accumulation of heat, thereby increasing the temperature. The differences between the turbulent non-premixed jet ﬂame

TURBULENT NON-PREMIXED TARGET FLAMES

425

data and the laminar ﬂame calculations are lower for high strain rates (a = 400 s−1 ) because ﬂame stretch reduces the temperature. The calculations with Le = 1 are in better agreement with the measured results for f ≈ 0.15–0.5 with the best ﬁt obtained for a strain rate of a = 100 s−1 for ReD = 15,200 and a = 200–400 s−1 for ReD = 22,800. The ﬂame stretch increases with the Reynolds number; therefore, the strain rate at which the calculations match with the experimental data in ReD = 22,800 case is higher than that for the case of ReD = 15,200. For f < fstoich , the experimental results lie between the curves with Le = 1 and Le = 1, indicating that models that ignore differential diffusion cannot be expected to predict the near-ﬁeld measurements accurately. At the start of the jet (x /D = 5), the turbulent ﬂow ﬁeld is laminarized in both the ﬂame region and the air-side of the reaction zone (Bergmann et al, 1998). The increase of kinematic viscosity due to heat release in the reaction zone results in laminarization of the turbulent ﬂow near the reaction zone in the neighborhood of the jet exit. This phenomenon explains why the effects of molecular diffusion and Lewis number are important, especially for ReD = 15,200. Differential Diffusion. The relative importance of molecular diffusion and turbulent transport can be investigated further by calculating the differential diffusion parameter, which is deﬁned as:

z ≡ ξH − ξC where ξH ≡

ZH − ZH,2 ZH,1 − ZH,2

and

ξC ≡

(6.37) ZC − ZC,2 ZC,1 − ZC,2

(6.38)

where ZH , and ZC are elemental mass fractions for hydrogen and carbon. In Equation 6.38, subscripts 1 and 2 refer to fuel and air streams, respectively. As discussed earlier, the differential diffusion is not the only parameter to stress the signiﬁcance of molecular diffusion. Molecular diffusion can be important even when there is no differential diffusion. A comparison of ξH and ξC at axial locations x/D = 5 and 20 is shown in Figure 6.10a and b respectively. In the absence of differential diffusion effects, the measured values should coincide with the dashed line, which has a slope of 1. However, the elemental mixture fraction ξH exceeds the elemental mixture fraction ξC at x/D = 5 except for very rich mixtures measured around the ﬂame axis (where both ξH and ξC are equal to 1 by deﬁnition). The largest deviations are observed for the elemental mixture fraction ξC ∼ 0.1 corresponding to a radial position on the air side not too far from the reaction zone, indicated by fstoich . Also shown in these plots is the result of a calculation for a strained laminar counterﬂow diffusion ﬂame including differential diffusion. The agreement between this calculation and the measured values from the turbulent jet ﬂame is reasonably close for x/D = 5, even though the geometry and ﬂow ﬁeld of the jet ﬂame are quite different from a counterﬂow diffusion ﬂame. The differences between ξH and ξC can be explained qualitatively by the high diffusivity

426

NON-PREMIXED TURBULENT FLAMES 1

0.8 ξH

1

ReD = 15,200, x/D = 5

0.6

ξH

fStoich. 0.4

ReD = 15,200, x/D = 20

0.8

opposed flame calc. (J.-Y, Chen)

fStoich.

0.6 0.4

equal diffusivities

0.2

0.2 (10 pulses condit. averaged)

(10 pulses condit. averaged) 0

0 0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

ξC

ξC

(a)

(b)

0.8

1

Figure 6.10 Element mixture fractions of hydrogen and carbon (modiﬁed from Bergmann et al., 1998).

of H2 compared to the other molecular constituents: Diffusion of H2 in the radial direction is faster than CH4 so the fuel close to the reaction zone is enriched by H2 ; thus, ξH becomes larger than ξC . As a consequence, the burned gas composition should contain more H2 O and less CO2 than the case for equal diffusivities of all species. At x/D = 20 (see Figure 6.10b), the differences between ξH and ξC are smaller but still show the same trends. Therefore, it can be concluded that differential diffusion effects should have a signiﬁcant inﬂuence on the gas composition and temperature. To understand the intense ﬂuctuations in the ﬂame temperature and possibility of local ﬂame extinction, let us discuss the scatter plots of the hydroxyl radical, shown in Figure 6.11. A signiﬁcant difference between the two ﬂames at different Reynolds numbers can be seen at x /D = 5, where the OH mass fractions from the ﬂame with ReD = 22,800 exhibit larger ﬂuctuations in comparison to ReD = 15,200. Also, a greater number of samples with near-stoichiometric mixture fraction have very low or zero OH concentrations for the ﬂame with ReD = 22,800. This reduction of OH mass fraction to zero implies local ﬂame extinction. Apart from the partially reacted mixtures, the measurements show OH concentrations signiﬁcantly larger than the equilibrium values (the dotted line), especially near the jet exit. Considering that OH concentrations decay toward equilibrium via slow three-body recombination reactions, the large OH mass fractions are not unusual and are also in agreement with the strained laminar ﬂame calculations. Calculations including Lewis number effects generally yield OH concentrations that are too large, and a reasonable ﬁt can be achieved only at x /D = 5, by applying an unrealistic low strain rate of a = 5 s−1 . The calculations with Le = 1 are in better agreement with the measured data, although the strain rates are quite low. From these results, we can see that in the near ﬁeld of the turbulent ﬂame jet at higher Reynolds numbers (such as ReD = 22,800), local ﬂame extinction reduces the temperature and concentrations of combustion products in the ﬂame.

TURBULENT NON-PREMIXED TARGET FLAMES 0.006

0.006 ReD = 15,200 x/D = 20

exp. adiab. equil.

0.005

–1

a = 5 s , Le ≠ 1 a = 25 s–1, Le = 1

0.004

–1

a = 5 s , Le = 1

0.003 0.002

OH Mass Fraction

OH Mass Fraction

0.005

0.001

exp. adiab. equil. a = 5 s–1, Le ≠ 1 a = 25 s–1, Le = 1

0.004

a = 5 s–1, Le = 1

0.003 0.002

0.000 0.0

0.1

0.2 Mixture Fraction

0.3

0.4

0.0

0.006

0.006

ReD = 15,200 x/D = 5

exp. adiab. equil.

0.005

–1

a = 5 s , Le ≠ 1 a = 100 s–1, Le = 1

0.004

a = 25 s–1, Le = 1 a = 5 s–1, Le = 1

0.003 0.002 0.001

OH Mass Fraction

OH Mass Fraction

ReD = 22,800 x/D = 20

0.001

0.000

0.005

427

0.1

0.2 Mixture Fraction

ReD = 22,800 x/D = 5

0.3

0.4

exp. adiab. equil. a = 100 s–1, Le ≠ 1 a = 5 s–1, Le ≠ 1

0.004

a = 25 s–1, Le = 1

0.003 0.002 0.001

0.000

0.000 0.0

0.1

0.2 Mixture Fraction

0.3

0.4

0.0

0.1

0.2 Mixture Fraction

0.3

0.4

Figure 6.11 Comparison of measured OH mass fractions with laminar ﬂame calculations (modiﬁed from Meier et al., 2000).

Near the jet exit, the comparison with strained laminar ﬂame calculations shows that the thermochemical state of the ﬂames cannot be described satisfactorily with one Lewis number and one value of strain rate. Differential diffusion effects play an important role in some mixture fraction regions and are more pronounced in the slower jets. The inﬂuence of Lewis number can be analyzed further by comparing the scatterplot of the heaviest molecule, CO2 , with the laminar ﬂame calculations with unity and nonunity Lewis numbers (see Figure 6.12). This comparison shows that the measured values lie between the ﬂame calculations with Le = 1 and Le = 1. They could be matched slightly better by the Le = 1 calculation and a strain rate a > 400 s−1 . Neither calculation can adequately ﬁts the experimental results. At axial locations farther from the jet exit, the correlation between CO2 and f compares better with laminar ﬂame calculations with adiabatic equilibrium chemistry. It is almost impossible to image the fuel constituents (H2 , CH4 , N2 ) directly by laser-induced ﬂuorescence (LIF) techniques in ﬂames, so the fuel is often doped with NO (∼70 ppm) and then the LIF from NO can be imaged. To interpret the resulting LIF images correctly, quenching of the NO ﬂuorescence has to be considered. Within the pure fuel, the NO ﬂuorescence is only very weakly quenched (e.g., at a rate of Q ≈ 5 × 107 s−1 ). However, entrainment of the

428

NON-PREMIXED TURBULENT FLAMES 0.10 ReD = 15,200 x/D = 5

Mass Fraction

0.08

exp.(Sandia) adiab. equil. a = 25 s–1, Le = 1 a = 400 s–1, Le = 1 a = 50 s–1, Le ≠ 1

0.06

0.04 CO2 0.02

0.00 0.0

0.1

0.2

0.3

0.4 0.5 0.6 0.7 Mixture Fraction

0.8

0.9

1.0

Figure 6.12 Comparisons of measured CO2 mass fractions with laminar ﬂame calculations (modiﬁed from Meier et al., 2000). 160

No seeded into fuel

OH distribution

CH distribution

natural NO

temperature

max 120

T [K]

x (mm)

Intensity

2000

80

1000

0 40 min

0 −20

0

+20

Figure 6.13 LIF intensity and temperature distributions each composed of four singlepulse images. (In the temperature image, the area from x = 0 to 20 mm has been omitted in order to avoid errors due to stray light from the burner nozzle) (modiﬁed from Bergmann et al., 1998.)

effective quenchers, such as H2 O and CO2 , into the fuel/ NO mixture strongly enhances quenching. The left frame of Figure 6.13 shows the LIF distribution of seeded NO for the downstream region from x/D = 0 to 20 (composed of four single-pulse images). The bright regions represent pure fuel whereas the contours of the dark regions reﬂect the entrainment of partially burned gas into the fuel jet. It can be seen that the boundary between light and dark zones is

TURBULENT NON-PREMIXED TARGET FLAMES

429

curved and wrinkled due to turbulent structures generated in the shear layer. From x ∼ 40 mm (i.e., x/D = 5), downstream combustion products often are present on the ﬂame axis. Above x ∼ 80 mm, pockets of pure fuel no longer exist. In general, these images reﬂect the decay of the fuel jet and the decay of turbulence in the inner core of the jet. The corresponding single-pulse images of OH are shown in the next frame of Figure 6.13. OH is formed in the reaction zones where its concentration is higher than the equilibrium value, and it is generally not observed below T ∼ 1,200 K. The OH ﬂuorescence distribution thus reﬂects the position and shape of the reaction zones and the high-temperature regions. The OH distribution in the ﬁgure shows that the reaction zones are thin and rather laminarlike up to a downstream position of x ∼ 40 mm. When this behavior is compared with the fuel-jet behavior (the image of NO seeded into fuel), it appears to be contradictory since in the fuel-jet image, the turbulence is quite evident, even at the downstream locations. This behavior of the OH mass fraction distribution can be explained by the inﬂuence of the heat release within the reaction zone on the ﬂow ﬁeld. As the temperature rises, the viscosity is strongly increased, resulting in a decrease in the local Reynolds number. As a consequence, the ﬂow is laminarized in this region (Takagi, Shin, and Ishio 1980; Everest et al., 1995). This effect is also the main reason why differential diffusion is pronounced in the immediate vicinity of the jet exit. Above x ∼ 40 mm (x/D = 5), the reaction zones are more wrinkled and the laminarization is rapidly reduced. Furthermore, the OH regions become thicker with increasing downstream location and are predominantly broad in the upper part of the ﬂame (see also Figure 6.14). It also can be seen from the OH distributions in Figure 6.13 that reaction zones are not always continuous. Gaps in the reaction zone can be observed in the start region of the jet (e.g., at x ∼ 10 mm and x ∼ 70 mm in Figure 6.13). The obvious explanation for this behavior is ﬂame extinction. A closer look at the results from the single-pulse Raman measurements (see Figures 6.9 and 6.11) supports this interpretation. The

OH distribution

CH distribution

max

Intensity

300

250 min

x (mm)

275

225 −40

−20

0

−40

−20

0

Figure 6.14 Two-dimensional LIF distributions from OH and CH at medium heights in the ﬂame, each frame composed of three single-pulse images (modiﬁed from Bergmann et al., 1998).

430

NON-PREMIXED TURBULENT FLAMES

temperature scatter plots in Figure 6.9 show that a few samples (from f ∼ 0.1 to 0.25) have temperatures far too low for those gas compositions (i.e., the gas has only partially reacted). Based on both the Raman and the two-dimensional LIF measurements, it is more likely that local ﬂame extinction occurs in the region close to the jet exit. CH radicals are an intermediate species in the reactions between fuel and oxidizer and are found in considerable amounts only in the reaction zone itself. Figure 6.13 displays the CH distributions in the start region of the jet. The small amount of background present over the whole area is due to Rayleigh scattering. In this region, the CH distribution is very similar to the OH distribution, although the width of the CH distribution (in the radial direction) is smaller. An evaluation of the thickness (based on full width at half maximum [FWHM], given by the distance between points on the curve at which the function reaches half its maximum value) of the ﬂuorescence zones at x/D = 5 from 60 single-pulse exposures revealed a mean value of 0.8 mm for CH and 1.0 mm for OH. Three-dimensional effects on the two-dimensional cut through the ﬂame should be of minor importance at this downstream position, where the ﬂame front is predominantly parallel to the ﬂame axis, and thus have not been taken into account. The result is in agreement with general knowledge about reaction mechanisms: CH is present only on the fuel-rich side of the reaction zone, whereas OH also is present on the fuel-lean side (Smooke et al., 1992). Farther downstream, the similarity between the CH and OH distributions vanishes, as can be seen from Figure 6.14. Here, the CH ﬂuorescence still occurs in thin layers, in contrast to the OH distributions (the bright spots in the CH image are due to Mie scattering from dust) (Nguyen and Paul, 1996; Schefer et al., 1992). Starner et al., 1992. The small width of the CH distributions supports the concept of combustion taking place in ﬂamelet-like reaction zones throughout the entire ﬂame. The ﬂuorescence distribution from the combustion-generated NO is seen in the fourth frame of Figure 6.13. From a comparison with a well-documented H2 /N2 jet diffusion ﬂame (Meier, Vyrodor, Bergmann, and Stricker, 1996), the absolute NO level is estimated to be 50 to 100 ppm in the lower part of the ﬂame x /D < 30 and 100 to 150 ppm in the upper part. Prompt and thermal NO are formed mainly in the reaction zones where high temperatures and superequilibrium (i.e., greater than those obtained from equilibrium assumption) radical concentrations are present; some thermal NO also is built in high-temperature regions outside the reaction zone (J.Y. Chen and Kollmann, 1992; Turns, 1995). Accordingly, the NO distributions reﬂect roughly (but not linearly) the temperature distribution, with the additional effect of growing NO levels with increasing downstream position where NO accumulates in the exhaust gas. 6.6.1.2

Effect of Jet Velocity

The Rayleigh intensity distributions from ﬂames with four different exit velocities (Ujet = 10.5, 21, 42, and 63 m/s corresponding to ReD = 3800, 7600, 15200, and 22800) in the region from x /D = 0.5 to 20 are shown in Figure 6.15. It can be seen that the width of the ﬂame zone near the jet exit station decreases from

431

TURBULENT NON-PREMIXED TARGET FLAMES ReD = 3,800

ReD = 7,600

ReD = 15,200

ReD = 22,800

160

max

80

x (mm)

Intensity

120

40 min 4

Figure 6.15 Two-dimensional Rayleigh intensity distributions for four different jet exit velocities (nozzle diameter: D = 8mm) (modiﬁed from Bergmann et al., 1998).

ReD = 3,800 to 15,200. This behavior is explained by considering that the time for molecular diffusion is longer at slower ﬂow velocities; thus, the diffusion path in a radial direction is also longer, leading to an extended ﬂame zone. For ReD = 22,800, this trend still can be recognized very close to the nozzle x < 20 mm, but farther downstream, transport by turbulent diffusion prevails over molecular diffusion and the ﬂame zone tends to increase in width. Looking at the transition from the ﬂame zone to the surrounding air, the images show that the shape of this transition changes with ReD . For ReD = 3,800, the boundary of the ﬂame is smooth up to x = 160 mm; for ReD = 7,600, it becomes slightly curved above x = 80 mm; for ReD = 15,200, it is curved and wrinkled above x = 40 mm; and for ReD = 22,800, it is wrinkled above x = 20 mm. This illustrates how the laminarization of the ﬂow in the reaction zone is reduced with an increasing degree of turbulence. The axial proﬁles of H2 O and CO2 in the turbulent non-premixed jet ﬂame follow the temperature proﬁles, and H2 and CH4 follow the mixture fraction proﬁles. There are signiﬁcant deviations between the ﬂames at two different Reynolds numbers in the O2 proﬁle for x /D ≥ 50, where the ﬂame with the lower ReD has a stronger entrainment of air. The axial proﬁles of OH and NO, displayed in Figure 6.16, show that the OH qualitatively follows the temperature proﬁle (in a nonlinear way) with a slightly higher maximum for the higher ReD . The maxima of the NO proﬁles are at nearly the same downstream position as those of the OH proﬁles and have a mean mass fraction of 6.04 × 10−5 for ReD = 15,200 and 4.86 × 10−5 for ReD = 22,800. Usually piloted ﬂames are used to prevent lifting or extinguishing of methane ﬂames at high Reynolds numbers. Such ﬂames have the drawback of complicating the ﬂow ﬁeld or chemistry and, hence, the mathematical simulation.

432

NON-PREMIXED TURBULENT FLAMES 0.0010 0.0009

OH, ReD = 22800

0.0008 Mass Fraction

0.0007

OH, ReD = 15200

NO (×10) ReD = 15200

0.0006 0.0005

NO (×10) ReD = 22800

0.0004 0.0003 0.0002 0.0001 0.0000 0

20

40

60 80 Axial Position x/D

100

120

Figure 6.16 Axial proﬁles of the Favre mean values of OH and NO (Meier and Barlow, et al., 2000).

6.6.2

Piloted Jet Flames

Owing to the ﬂame extinction and blow-off of simple jet ﬂames at high Reynolds number and low Damk¨ohler number, piloted jet ﬂames are used to study the effect of Reynolds number and Damk¨ohler number on turbulence-chemistry interaction. The piloted jet burner produces a simple streaming ﬂow and uses a heat source from a set of premixed ﬂames to stabilize the main jet to the burner’s exit plane. The burner consists of an axisymmetric jet centered in an annulus in which a number of premixed ﬂames are stabilized on a ﬂame holder. The burner is centered in an unconﬁned coﬂowing stream of air (see Figure 6.17). The unburned pilot gases are a premixed mixture that, when burned to completion, produce combustion products having the same C/H and O/H ratio as that of a stoichiometric mixture of the main fuel and air. If the fuel mixture used in the pilot is different from the jet fuel, the adiabatic temperature of the pilot ﬂame is likely to be different from that of the main fuel mixture. This is believed not to be a major problem. It is accounted for in the computation in pdf methods but generally not in the ﬂamelet models. The boundary conditions are well speciﬁed. If the initial conditions are not measured directly, laminar ﬂow can be assumed at the jet exit plane for the pilot gases and fully developed turbulent pipe ﬂow can be assumed for the central fuel jet. Hot gases from the pilot stabilize the jet ﬂame to the burner, regardless of the fuel ﬂow rate in the main jet. At high enough jet velocities, this forces ﬂames extinction to occur farther downstream in a region where turbulent mixing time scales become of the same order of magnitude as the chemical time scales. This makes the burner extremely useful for studying the effects of the interaction between turbulence and chemistry in ﬂames without the added complications

TURBULENT NON-PREMIXED TARGET FLAMES

433

Figure 6.17 Piloted jet ﬂames (Sydney ﬂame burner, C2 H2 /H2 premixture used in pilot ﬂames) (modiﬁed from http://www.sandia.gov/TNF/DataArch/FlameD/SandiaPilotDoc21 .pdf).

of soot formation and thermal radiation. Flame reignition may occur farther downstream of the extinction zone where turbulent mixing rates are less intense. Piloted ﬂames of hydrocarbon fuels, while generally blue and visibly soot free in the upstream regions, may become yellow and loaded with soot farther downstream. The length of the blue ﬂame zone depends on the fuel and on how close the ﬂame is to blow-off. The stability characteristics of piloted ﬂames of various fuel mixtures have been reported in various papers (Barlow and Frank, 1998; Roomina and Bilger, 2001). The parameters that control the ﬂame stability are the fuel jet velocity and the stoichiometry of the pilot and of its ﬂow rate. The mixture fraction in the pilot stream is kept lean (φ = 0.77) to prevent overheating of the burner tip. The simplicity of the ﬂow and the existence of a fully turbulent region of the ﬂame where the chemical kinetic effects are signiﬁcant are important reasons why, this burner is considered to be an ideal case for the testing and development of computer models. The existence of the pilot stream can be accounted for easily, and the pilot ﬂame gases have no noticeable inﬂuence on the ﬂame composition in the region where extinction occurs. There are six standard piloted jet ﬂames with ﬁxed nozzle diameter and increasing jet velocities, corresponding to jet Reynolds numbers based on nozzle diameter D from 1,100 (laminar) to 44,800 (turbulent with signiﬁcant localized extinction). These ﬂames are known as Sandia Flame A, B, C, D, E, and F. The ﬂow parameters of these standard ﬂames are summarized in Table 6.3. These jet ﬂames use a mixture of 25% CH4 and 75% air (by volume) in the fuel jet and are stabilized on the piloted burner. At the ﬂow conditions given in Table 6.3, the fuel-rich premixed chemistry is too slow to signiﬁcantly affect ﬂame structure, and the mixing rates are high enough such that the ﬂame behaves

434

NON-PREMIXED TURBULENT FLAMES

TABLE 6.3. Flow parameters for standard piloted jet ﬂames (modiﬁed from Barlow et al., 2005) Flame

ReD

Ujet -cold (m/s)

Upilot -burnt (m/s)

Ucoﬂow (m/s)

A B C D E F

∼1,100 ∼8,200 ∼13,400 ∼22,400 ∼33,600 ∼44,800

2.44 18.2 29.2 49.6 74.4 99.2

None 6.8 6.8 11.4 17.1 22.8

0.9 0.9 0.9 0.9 0.9 0.9

as a diffusion ﬂame, which has a single reaction zone near the stoichiometric mixture fraction and no indication of signiﬁcant premixed reaction in the fuelrich CH4 /air mixtures. The advantage of using a premixture of CH4 and air in the fuel-rich jet is that this mixture signiﬁcantly reduces the problem of ﬂuorescence interference from soot precursors, allowing improved accuracy in the scalar measurements. Partial premixing with air also reduces the ﬂame length and produces a more robust ﬂame than pure CH4 or nitrogen-diluted CH4 . Consequently, the ﬂames may be operated at reasonably high Reynolds number with little or no local extinction, even with a modest pilot. The pilot is a lean (φ = 0.77) mixture of C2 H2 , H2 , air, CO2 , and N2 with the same nominal enthalpy and equilibrium composition as methane/air at this equivalence ratio. The ﬂow rates of the main jet and the pilot are scaled in proportion for the ﬂames from C to F, so that the energy release of the pilot is approximately 6% of the main jet for each ﬂame. The ﬂames are unconﬁned. These ﬂames have three features that have proven to be both interesting and challenging for modelers. 1. The cases with the highest Reynolds number (ﬂames D, E, and F; jet-exit Reynolds numbers 22,400, 33,600, and 44,800) have increasing probability of localized extinction in the lower portion of the jet ﬂame, followed by reignition and complete burning farther downstream. 2. The formation of nitric oxide is sensitive to both radiative transfer effects and detailed hydrocarbon kinetics on the fuel-rich side. 3. Because the fuel jet is partially premixed with air, there is potential for signiﬁcant chemical reaction in fuel-rich mixtures. Some models have overpredicted the progress of reaction under fuel-rich conditions. Let us revisit the mixture fraction deﬁnition even though it has been discussed at great length in Section 3.3 of Chapter 3. A mixture fraction Z can be calculated following the method of Bilger et al. (1990), but with oxygen excluded from the expression: 2 ZC − ZC,2 /WC + ZH − ZH,2 /2WH (6.39) Z≡ 2 ZC,1 − ZC,2 /WC + ZH,1 − ZH,2 /2WH

435

TURBULENT NON-PREMIXED TARGET FLAMES

where Z = elemental mass fraction, W = atomic weight, subscripts 1 and 2 = main jet and coﬂowing air stream, respectively. The partial premixing of fuel and air leads to boundary conditions for ZO that are relatively close, causing excessive noise in the mixture fraction if oxygen is included in the calculation. Therefore, oxygen is not included in the calculation of mixture fraction from Equation 6.39. Let us also revisit the deﬁnition of the conditional probability density function and the conditional means. If two random variables u and v have a joint distribution, then the conditional pdf of random variable u conditioned on the event v = V describes the probability density of u when v is at a ﬁxed value:

Pu ( u| v = V ; x, t) =

Puν (u, V ; x, t) Pν (V ; x, t)

(6.40)

where Puv is the joint pdf of sample space variables u and v and Pv is the marginal probability density function of v = V [also see Equation 4.177 in Chapter 4]. The conditional mean of u, for a ﬁxed value of v , is deﬁned by: +∞ u| v = V = uPu ( u| v = V ; x,t) du

(6.41)

−∞

The joint pdfs and conditional means of various scalars in piloted jet ﬂames are discussed in this section. Flame Characterization Scatter plots of temperature versus mixture fraction for the Sandia piloted ﬂames are shown in Figure 6.18. The solid lines in these plots represent the calculated temperature for a counterﬂow laminar diffusion ﬂame with a strain rate a = 50 s−1 , which are included to facilitate visual comparisons. These scatter plots provide a qualitative indication of the probability of local extinction, which is characterized by samples with strongly depressed temperatures. At x /D = 15 (Figure 6.18a), there is no evidence of extinction in Flame B. Flames C and D have very small probability of local extinction, and extinction becomes more important in Flames E and F. At x /D = 30 (Figure 6.18b), the scatter plots for Flames C, D, and E all show a low probability of extinguished samples, indicating a reignition process. In contrast, Flame F still has high probability of local extinction at x /D = 30, and the characteristic bimodality of the temperature distribution in lean and near-stoichiometric samples is evident. The measured conditional pdfs (cpdf) of various species mass fractions conditioned on mixture fraction in Flame A (from x /D = 5 and x /D = 10) and in Flames B through F (from x /D = 15) are shown in Figure 6.19. At the latter axial station, local extinction is most probable in the highly strained cases. The progression of cpdfs of H2 O mass fraction in the turbulent ﬂames with increasing

T (K)

T (K)

0 0.0

500

1000

1500

2000

0

500

1000

1500

2000

0

500

1000

1500

2000

2500

0.2

0.4 0.6 0.8 Mixture Fraction

Flame C x/D = 15

Flame B x/D = 15

Flame A x/D = 5

(a)

0.0

0.2

0.4 0.6 0.8 Mixture Fraction

Flame F x/D = 15

Flame E x/D = 15

Flame D x/D = 15

1.0

0 0.0

500

1000

1500

2000

0

500

1000

1500

2000

0

500

1000

1500

2000

2500

0.2

0.4 0.6 0.8 Mixture Fraction

Flame C x/D = 30

Flame B x/D = 30

Flame A x/D = 5

(b)

0.0

0.2

0.4 0.6 0.8 Mixture Fraction

Flame F x/D = 30

Flame E x/D = 30

Flame D x/D = 30

1.0

Figure 6.18 Scatter plots of measured temperature at (a) x/D = 5 in the laminar Flame A and at x/D = 15 in the turbulent Flames B to F; and (b) at x/D = 5 in the laminar Flame A and at x/D = 30 in the turbulent Flames B to F (modiﬁed from Barlow and Frank, 1998).

T (K)

T (K) T (K) T (K)

436

437

TURBULENT NON-PREMIXED TARGET FLAMES 0.25

A10 0.20 Conditional PDF

Conditional PDF

0.20

0.25

A5 B C D

0.35 < f < 0.45 x/D = 15

0.15 F

0.10

E

D

0.05 0.00 0.000

0.30 < f < 0.40 x/D = 15

A10 A5 B C D E

0.15 F

0.10

E 0.05

0.100

0.050

0.00 0.000

0.150

YCO

2

Conditional PDF

0.30

0.00 0.000

0.28 < f < 0.36 x/D = 15

A5 A10 B

C

F

0.40 Conditional PDF

A5 A10 B C D E F

0.40

0.10

D E

0.001

0.002 0.003 YOH

0.004

A10

0.30

A5

B

F C

0.20 E

D

0.00 0.000

0.005

0.020

0.040 YCO

0.060

0.080

0.48 < f < 0.58 B x/D = 15 A10

A5

0.20 E

D

C

0.10

0.001

0.002 YH

2

0.003

0.004

Conditional PDF

0.40 F

Conditional PDF

0.43 < f < 0.53 x/D = 15

0.10

F

E

0.30

0.00 0.000

0.150

2

0.50

0.50

0.20

0.100

0.050

YH O

0.30

0.33 < f < 0.41 x/D = 15 B A10

F E

0.20

D

C

0.10 0.00 0 0 10

2 10−5

4 10−5 YNO

6 10−5

8 10−5

Figure 6.19 Conditional pdfs of measured mass fractions of H2 O, CO2 , OH, CO, H2 , and NO from x/D = 15 in turbulent ﬂames B to F and from the laminar jet ﬂame at x/D = 5 (Flame A5) and x/D = 10 (Flame A10). Mixture fraction intervals used in forming cpdfs are given in each graph (modiﬁed from Barlow and Frank, 1998).

Reynolds number involves an increase in the population of extinguished samples without a signiﬁcant change in H2 O mass fraction for reacted samples (i.e., all peaks are at the same H2 O mass fraction ∼ 0.11). In contrast, the cpdfs of CO2 mass fraction show a progressive decrease in the most probable value of CO2 mass fraction (i.e., peaks shift toward a lower value of CO2 mass fraction). The cpdfs of OH mass fractions show a broadening and shift toward higher average mass fractions as the Reynolds number increases. This trend is clearly present before local extinction becomes important, and it cannot be attributed to

438

NON-PREMIXED TURBULENT FLAMES

reignition. For CO, the main effect of increased turbulence is a broadening of cpdfs relative to Flame A. This broadening is mainly in the direction of lower mass fractions, but the trend is less obvious than that for OH. At x /D = 15 in the turbulent ﬂames, the cpdfs of H2 mass fraction display a nonmonotonic response to increasing Reynolds number. The curve for Flame B is similar to that of Flame A. H2 mass fraction tends to be higher in Flame C but decreases signiﬁcantly in Flames E and F. The cpdfs of NO mass fraction show mass fractions decreasing as the Reynolds number increases; this may be primarily an effect of residence time. From these results, it is difﬁcult to discern the relative importance of convective residence time, local scalar dissipation rate, transport effects, and various branches of the NO production and reburn chemistry (Barlow and Frank, 1998). The cpdfs of H2 O mass fraction in Flames B through E are nearly identical, with the most probable values of H2 O mass fraction being about 10% higher than that of Flame A. The cpdf for Flame F is bimodal, but the portion corresponding to reacted samples (i.e., H2 O mass fraction ∼ 0.11) aligns with the other turbulent ﬂames (B–E). The cpdfs of CO2 mass fraction in Flames B to E show a most probable mass fraction about 10% lower than that of Flame A. These differences in major species are believed to reﬂect the increasing importance of turbulent transport relative to molecular diffusion as Reynolds number and stream-wise distance are increased (Smith et al., 1995). A similar effect on mass fractions of H2 O and CO2 is observed in strained laminar ﬂame calculations (J.-Y. Chen, 1989) when all diffusivities are set equal to the thermal diffusivity (i.e., Le = 1). This transport effect also is believed to affect the mass fractions that are seen in the cpdfs of H2 mass fraction at x /D = 30. A comparison of laminar ﬂame calculations at the same strain rate (a = 100 s−1 ) shows the peak mass fraction increasing by 10% for H2 O, decreasing by 9% for CO2 , and increasing by 60% for H2 when equal diffusivities are speciﬁed instead of the full detailed model for diffusivities (see Table 2.1 of Chapter 2). The cpdfs of OH mass fraction at x /D = 30 in Flames B to F continue to show higher mass fractions at peaks than in Flame A. This also may be related to molecular transport effects, because the speciﬁcation of equal diffusivities in the a = 100 s−1 laminar ﬂame calculation causes a 25% increase in peak OH mass fraction. However, interpretation is complicated by the fact that the most probable values of OH mass fraction in the cpdfs for Flame A are lower than predicted for full-transport laminar ﬂames. It is possible that radiative loss and radical recombination contribute to the observed reduction in OH levels with respect to the adiabatic counterﬂow diffusion ﬂame. The cpdfs of CO in Flames B to E are broader than those in Flame A but are centered at roughly the same mass fraction. The experimental setup for measurements of mixture fraction at several radial and axial locations and subsequent deduction of scalar dissipation rates from these measurements are shown in Figure 6.20. This setup combines line imaging of Raman scattering, Rayleigh scattering, and laser-induced CO ﬂuorescence (CO LIF) to obtain measurements of temperature and the major species concentrations, which enable the calculations of mixture fraction and scalar dissipation rates.

TURBULENT NON-PREMIXED TARGET FLAMES CO

Rotating-wheel shutter

230 nm reflector

Beam profile PM tube CCD μ-scope objective 532 nm lens reflector PCX mm Dichroic 500 combiner pe sco tele UV

co

ld

484 nm filter

ce

ll

eam 50 mm mb 32 n PCX lens 5 230 nm reflector m bea nm 230tor nua atte UV

es

UV

plat

Flame location

27 cm imaging spectrograph

Dichroic splitter 484 nm filter f/4 Raman camera

Periscope

439

f/2 achromats

Rayleigh camera 532 nm filter mechanical shutter

CO

ca

me

ra

Joule meter and photodiode

Figure 6.20 Experimental diagnostics setup for simultaneous measurements of Raman scattering, Rayleigh scattering, and CO LIF (modiﬁed from Barlow and Karpetis, 2004).

Four frequency-doubled Nd:YAG lasers are used in the Raman/Rayleigh system of the diagnostics system. Dichroic optics successively combines each frequency-doubled beam (532 nm) with the 1064 nm beam of the next laser in the series, before it passes through the doubling crystal. Collimating telescopes in three of the lasers allows for matching of the waists of the four beams. Dielectric breakdown in the probe volume is mitigated by separating the laser pulses in time by 150 ns and by using a three-leg pulse stretcher, which extendes each pulse to about 83 ns (FWHM). Alignment of the multiple Nd:YAG laser beams onto a common axis is accomplished by using seven motor-driven mirror mounts, one for each laser and one in each leg of the pulse stretcher. The combined beam is focused into the test section using a 500 mm lens. Pulse energy can be measured by using a thermoelectric Joule-meter. The experiments can be conducted using roughly 1.2 J/pulse in the probe volume. Raman scattered light is collected by using a custom-designed pair of 150 mm diameter achromats (with aperture settings of f/2 and f/4), which are mounted face-to-face, yielding a magniﬁcation of 2. (An achromat is a lens that can bring two wavelengths into focus in the same plane). The f/4 lens matches the entrance aperture of an imaging spectrometer (SPEX 270M), which is ﬁtted with a custom-built high-speed rotating mechanical shutter. This shutter provides a 9-μs gate (FWHM), rejecting ﬂame luminosity and allowing the use of a nonintensiﬁed CCD array detector. The shutter and associated electronics also provide timing signals for the lasers and cameras. A 16-bit, back-illuminated, cryogenically cooled CCD (Roper Scientiﬁc CryoTiger, 1300 × 1340 pixels) is used for detection of the Raman spectrum. The

440

NON-PREMIXED TURBULENT FLAMES

performance of such cameras is superior to that of intensiﬁed cameras with regard to quantum efﬁciency, noise, and dynamic range. The image of the laser beam was rotated by a periscope to align with the vertical entrance slit. A grating of 588 grooves/mm disperses the Stokes Raman spectrum (550–700 nm) along the horizontal axis of the CCD detector. The diameter of the probe volume is determined by the ∼0.28 mm (1/e2 ) waist of the combined laser beam, the image of which (0.56 mm after magniﬁcation) ﬁts through the 0.8-mm rotating slit. To minimize readout noise and hence improve the overall signal-to-noise ratio, CCD pixels are binned on chip to form superpixels. Each Raman superpixel corresponds to 0.2044 mm along a ∼7.3 mm segment of the laser beam with a total of 36 superpixels along the spatial dimension of the CCD. The CCD detector allows for arbitrary speciﬁcation of the location and width of superpixel columns, and 14 columns are deﬁned to collect photons corresponding to the rotationalvibrational Raman bands of the major species (CO2 , O2 , CO, N2 , CH4 , H2 O, H2 ) and interferences or background at selected spectral locations. The Rayleigh imaging system uses two matched achromats (82 mm diameter, 300 mm ﬂ), and the collected light is focused onto a back-illuminated CCD detector (Roper Scientiﬁc Spec-10 400B, 400 × 1340 pixels) through a 532 nm band-pass ﬁlter. Gating is provided by a mechanical shutter (∼33 ms). The Rayleigh scattering image is binned 1 × 3, with the three-pixel dimension being in the horizontal (laser beam) direction and corresponding projected length of 0.0596 mm in the measurement volume. The 20 μm resolution of the Rayleigh image in the vertical direction is used to monitor the position and width of the 532 nm beam on a shot-to-shot basis. A motor drive on the vertical stage of the mount for the ﬁnal focusing lens is used to maintain the average beam center within 10 μm of its reference location on the Rayleigh CCD detector. The Raman superpixel columns are adjusted to align with this beam position. Such accurate alignment is important for Raman measurements because vertical movement of the laser beam corresponds to a shift in the spectral dimension on the Raman CCD detector. Excitation of CO follows the methods developed for multiscalar point measurements at Sandia National Lab (e.g., Barlow et al., 2001). A seeded Nd:YAG laser pumps a tunable dye laser, and the dye laser beam is frequency doubled and then mixed with the Nd:YAG fundamental to reach 230 nm. Laser pulse energy is measured by using a photodiode, and the combination of a l/2 plate and Glan laser prism is used to attenuate pulse energy and maintain it in the range (∼2.0 mJ/pulse) that yields a nearly linear dependency between laser energy and a CO ﬂuorescence signal along the full length of the imaged region. The exponent for power dependency of the ﬂuorescence signal is between 1.1 and 1.2 along the length of the probe volume. The exponent reduces the importance of collisional quenching relative to photo-ionization as a loss mechanism in the ﬂuorescence equation and thereby reduces the inﬂuence of uncertainties in collisional quenching cross sections. The CO imaging system uses the same front collection lens as the Rayleigh system. A dichroic beam splitter in the collimated region reﬂects CO ﬂuorescence from the B1 + (ν = 0) → A1 + (ν = 1) band (∼484 nm) through another matched achromat. An interference ﬁlter centered

TURBULENT NON-PREMIXED TARGET FLAMES

441

at 484 nm (10 nm FWHM) passes the CO ﬂuorescence signal onto an intensiﬁed CCD camera (from Andor) with 512 × 512 pixels. The imaged region is divided into 40 superpixels of 0.1905 mm (8 × 8 binning) along the length of the laser beam. Fourteen superpixels are used in the vertical direction, with the outer pixel rows being used for background subtraction. The three camera systems are aligned, and their magniﬁcations are determined using a target of shim stock, with a series of laser-drilled holes, placed in the measurement plane, and back-illuminated with appropriate wavelengths. In the section 6.6.2.1, several measurements from this system are discussed to establish an understanding of the scalar structure (e.g., mixture fraction, scalar dissipation rate, temperature, etc.) of turbulent jet diffusion ﬂames and to estimate the effects of differential diffusion, local extinction, reignition, and turbulent transport. Conditional means of measured species mass fractions show trends that suggest an evolution in the relative importance of molecular diffusion and turbulent transport in piloted ﬂames. This effect is illustrated in Figure 6.21, which compares

−3

0.15

H2

H2O CO2

0.05

1 10−3

CO 0.2

0.4 0.6 Mixture Fraction (a)

0.8

YCO2, YH2O or YCO

0.15

5 10 Flame E, x/D = 45 a = 50/s

ReD = 33,600 H2

H2O CO2

4 10−3 3 10−3 −3

2 10

0.10 0.05 0.00 0.0

0 100 1.0

−3

0.25 0.20

−3

3 10

2 10−3

0.10

0.00 0.0

−3

4 10

YH2

0.20

5 10 Flame B, x/D = 15 ReD = 8,200 a = 50/s

YH2

YCO2, YH2O or YCO

0.25

1 10−3

CO 0.2

0.4 0.6 Mixture Fraction (b)

0.8

0 100 1.0

Figure 6.21 Measured conditional means of species mass fractions (symbols) compared with calculated results for laminar counterﬂow diffusion ﬂame with full molecular transport (dashed lines) or equal diffusivities with Le = 1 (solid lines) (modiﬁed from Barlow et al., 2005).

442

NON-PREMIXED TURBULENT FLAMES

conditional mean values of measured mass fractions for Sandia Flame B and Flame E with results of steady counterﬂow laminar diffusion ﬂame calculations. The dashed lines show results that include differential molecular diffusion (i.e., detailed model for diffusivities of all species), while the solid curves show results with equal diffusivities (all species diffusivities set equal to the thermal diffusivity, Le = 1). In both calculations the strain rate is equal to 50 s−1 . Figure 6.21a shows measurements at x /D = 15 in Flame B (ReD = 8,200). At this upstream location in this transitional jet ﬂame, strong effects of differential diffusion can be observed since the measured conditional mean mass fractions are well approximated by a laminar ﬂame calculation with full transport. In contrast, the measured conditional means at x /D = 45 in Flame E (ReD = 33,600) align with a laminar calculation that applies equal diffusivities. No local extinction was observed by Barlow et al. (2005) in either data set shown in Figure 6.21. As the Reynolds number is increased from Flame B to Flame E, the scalar dissipation increases and the relative importance of turbulent transport (stirring at scales larger than the diffusive scale) is expected to have increasing importance with respect to molecular diffusion, such that the effects of preferential diffusion of low-molecular-weight species on the measured mass fractions become less signiﬁcant as the Reynolds number increases. This is an important point to note since in the modeling of turbulent ﬂames, it is common to evoke the assumption that all species have equal diffusivities. As the Reynolds number increases, the ﬂame should evolve from a scalar structure that is strongly affected by molecular transport (at a low Reynolds number) into a scalar structure where turbulent transport appears to be dominant. It should also be noted that the past work on laboratory-scale jet ﬂames of various fuels has shown that the differential molecular diffusion effects due to non equal diffusivities of chemical species can be strong in low-Reynolds-number turbulent ﬂames (Barlow and Carter, 1996a; Drake, Lapp, and Penney, 1982; Smith et al., 1995). These effects are particularly important in the near ﬁeld of jet ﬂames that use H2 as a fuel component and have a jet Reynolds number in the neighborhood of 10,000 or lower. Modeling studies have shown that measured features of such jet ﬂames can be reproduced only by accounting for differential diffusion in the models (Chen et al., 1996; Kronenburg and Bilger, 2001; Pitsch, 2000). When diffusivities are set equal (solid lines) in laminar ﬂame calculations, the peak mass fraction of H2 increases signiﬁcantly, those of H2 O and CO increase only modestly, and that of CO2 decreases slightly. The H2 mass fraction shows the greatest difference between the two calculated transport scenarios at same strain rate. Therefore, the assumption of equal diffusivities should overpredict H2 mass fraction for Flame B whereas it can be more accurate for high-Reynoldsnumber ﬂames. Figure 6.22 shows the conditional mean values of the differential diffusion parameter z and elemental mixture fractions of hydrogen and carbon at x /D = 10 in a laminar piloted jet diffusion ﬂame. The differential diffusion parameter z in the piloted jet ﬂames B to F at x /D = 15 and 30 are shown in Figure 6.23. At this upstream location x /D = 15 in Flame B (transitional case with ReD = 8,200), the scalar structure of the reaction

TURBULENT NON-PREMIXED TARGET FLAMES 0.15

1.0 Flame A: x/D = 10 0.8 ξC

0.05

0.6

ξH

0.00

0.4

−0.05 −0.10

0.2

z 0.2

0.4

0.6

ξH, ξC

z = ξH - ξC

0.10

−0.15 0.0

443

0.8

0.0 1.0

ξ

Figure 6.22 Differential diffusion parameter z and elemental mixture fractions of hydrogen and carbon in laminar ﬂame (Sandia Flame A) (modiﬁed from Barlow et al., 2005).

zone is essentially that of fully diffusive laminar ﬂamelets with small inﬂuence of turbulent transport. As the Reynolds number is increased, the magnitude of z decreases. At x /D = 15, where signiﬁcant local extinction occurs in the higher ReD ﬂames, part of the decrease in z is due to the increase in mass fraction of CH4 , which does not contribute to z . It is also apparent from the ﬁgure that the magnitude of z decreases in each ﬂame when we move downstream from x /D = 15 to 30. In Flame F, there is still signiﬁcant local extinction at x /D = 30, and the presence of CH4 causes the measured magnitude of z near stoichiometric to be small. The pdfs of mixture fraction at three axial stations (x /D = 15, 30, and 45) for Sandia Flame D are shown in Figure 6.24 for multiple radial locations. At the axial station closest to the jet exit, the pdf at r = 0 peaks near a high-mixture fraction (∼0.9) and drops to zero for mixture fraction less than 0.6. This behavior is expected since at the jet axis and close to the jet exit, the mixture is dominated by fuel. At large radial distance (e.g., r = 15 mm), the mixture composition is mainly dominated by oxidizer; therefore, Pz has a distribution centered at Z = 0. Note that the area under each pdf curve is equal to 1 as speciﬁed by Equation 4.130. At the radial locations between 0 and 15 mm, the Pz has lower peaks but broader distribution. At the axial stations x /D = 30 and 45, peak of Pz at r = 0 shifts toward lower mixture fraction values while the peak of Pz at the outer radii shift toward slightly higher mixture fraction values, with respect to their peaks at x /D = 15. These trends imply that there is transport of fuel ingredients in radial direction as x increases. The Reynolds and Favre averaged mean and standard deviations of mixture fractions at various axial and radial locations in Sandia Flame D are given in Table 6.4. As x /D increases, the mean values at lower radial locations decreases; at higher radial locations, the mean values tend to increase. This is due to radial transport of the fuel as axial distance increases. Measured centerline proﬁles of Favre-averaged mixture fraction and temperature in Flames C, D, E, and F are shown in Figure 6.25. There are clear differences among these proﬁles, which are believed to result from effects of local extinction on the mixing process, in addition to any effects on structure that might be

444

NON-PREMIXED TURBULENT FLAMES 0.15

0.15 Flame B ReD = 8,200 x/D = 15

0.10 Zst

−0.05

−0.10

−0.10

−0.15

−0.15

0.15

0.15 Flame C ReD = 13,400 x/D = 15

0.10

0.05

Zst

0.00

−0.05

−0.10

−0.10

−0.15

−0.15 0.15 Flame D ReD = 22,400 x/D = 15

0.10 0.05

Flame D ReD = 22,400 x/D = 30

0.10 0.05

Zst

0.00

z

z

Zst

0.00

−0.05

0.15

Zst

0.00

−0.05

−0.05

−0.10

−0.10

−0.15

−0.15

0.15

0.15 Flame E ReD = 33,600 x/D = 15

0.10 0.05

Flame E ReD = 33,600 x/D = 30

0.10 0.05

Zst

0.00

z

z

Flame C ReD = 13,400 x/D = 30

0.10

z

z

0.05

Zst

0.00

−0.05

−0.05

−0.10

−0.10

−0.15

−0.15

0.15

0.15 Flame F ReD = 44,800 x/D = 15

0.10 0.05

0.05

Zst

0.00

−0.05

−0.10

−0.10 0.4

0.6

Mixture Fraction

0.8

1.0

Zst

0.00

−0.05

0.2

Flame F ReD = 44,800 x/D = 30

0.10

z

z

0.00

−0.05

−0.15 0.0

Zst

0.05

0.00

z

z

0.05

Flame B ReD = 8,200 x/D = 30

0.10

−0.15 0.0

0.2

0.4

0.6

0.8

1.0

Mixture Fraction

Figure 6.23 Conditional mean and standard deviations of the differential diffusion parameter, z , at x /D=15 and x /D=30 in Sandia Flames B, C, D, E, and F (modiﬁed from Barlow et al., 2005).

445

TURBULENT NON-PREMIXED TARGET FLAMES 12

12 Flame D r=0 r = 3 mm x/D = 15 r = 6 mm r = 9 mm r = 12 mm r = 15 mm

10

Pz

8 6

8 6

4

4

2

2

0 0.0

0.2

0.4

0.6

Mixture Fraction

0.8

10 r=0 r = 3 mm r = 6 mm r = 9 mm r = 12 mm r = 15 mm r = 21 mm r = 27 mm

10

0 1.0 0.0

Flame D x/D = 30

Flame D r=0 x/D = 45 r = 9 mm r = 18 mm r = 27 mm

8 6 4 2

0.2

0.4

0.6

Mixture Fraction

0.8

0 1.0 0.0

0.2

0.4

0.6

0.8

1.0

Mixture Fraction

Figure 6.24 PDFs of mixture fraction in Sandia Flame D as a function of x and r (modiﬁed from Barlow et al., 2005).

associated with the Reynolds number alone. For x /D between 10 and 25, the rate of decay of the centerline mixture fraction increases with the Reynolds number. Local extinction in these piloted ﬂames occurs at these axial locations. The probability of local extinction is highest in Flame F, which is close to blow-off, and the centerline decay is dramatically faster in Flame F than in the lower-Reynolds-number ﬂames. For x /D between 25 and 50, the trend is reversed such that the centerline decay of mixture fraction is slowest in Flame F and fastest in Flame C. It is possible that these trends result from extinction followed by reignition with heat release inhibiting mixing of fuel and air. The puzzle comes from the fact that there does not seem to be a similar trend in the velocity measurements (Frank, Barlow, and Lundquist, 2000) or in the centerline proﬁles of mixture fraction from model calculations of these ﬂames. Figure 6.26 shows the conditional means of temperature and scalar dissipation rates and demonstrates that the density-weighted conditional mean scalar dissipation rate [χr = 2Dm (∂Z/∂r)2 ] is dependent on the radial location. The scalar dissipation rate should decay to zero at both the air and the fuel streams since the mixture fraction variations become much smaller at these locations. The mixture diffusivity Dm is calculated by the next formula: 2 Dm cm2 /s = −0.12013 + 0.74818 T (K)/1000 + 1.1631 T (K)/1000 (6.42) Equation 6.42 is a curve ﬁt to the mixture-averaged diffusivity from a calculation of an counterﬂow, laminar, partially premixed CH4 /air ﬂame with a strain parameter a = 25 s−1 . First consider results at x /D = 15 (Figure 6.26b). The conditional scalar dissipation has a double-humped structure with local minimum or plateau just to the rich side of the stoichiometric mixture fraction, Zst = 0.351. When data from the full radial proﬁle are used, the largest scalar dissipation values are found under fuel-rich conditions near Z = 0.5. The most interesting feature in Figure 6.26b is that the relative values of the local maxima change with radial location. For data from the inside (richer) end of the probe volume (r = 5.9 mm, solid curve in Figure 6.26b), the lean-side maximum is greater than the rich-side

446

Z

0.903 0.834 0.618 0.341 0.155 0.069 —

(mm)

0 3 6 9 12 15 —

r

0.063 0.091 0.163 0.161 0.100 0.063 —

Z 2

σZ or

x/D = 7.5

0.914 0.859 0.660 0.319 0.112 0.043 —

Z˜

0.057 0.079 0.164 0.179 0.091 0.048 —

σ˜ Z or 2 Z 0 3 6 9 12 15 27

(mm)

r

0.652 0.627 0.554 0.446 0.345 0.250 0.063

Z 0.113 0.125 0.146 0.157 0.151 0.128 0.054

Z 2

σZ or

x/D = 15

0.676 0.653 0.579 0.452 0.321 0.208 0.043

Z˜ 0.106 0.119 0.151 0.173 0.166 0.130 0.042

σ˜ Z or 2 Z 0 — — 9 — 18 27

(mm)

r

0.387 — — 0.338 — 0.228 0.133

Z

0.101 — — 0.110 — 0.105 0.081

Z 2

σZ or

x/D = 30

0.379 — — 0.318 — 0.192 0.101

Z˜

0.107 — — 0.119 — 0.106 0.071

σ˜ Z or 2 Z

TABLE 6.4. Reynolds and Favre averaged mean and standard deviations of mixture fractions at various axial and radial locations in Sandia ﬂame D (modiﬁed from Barlow et al., 2005)

447

TURBULENT NON-PREMIXED TARGET FLAMES

0.8

2000

~ T

x Flame: C D E F

0.6

1500 1000

0.4

xstoic = 0.351 500

0.2 0.0

~ T(K)

Mixture Fraction

1.0

0

10

20

30

40 50 x/D

60

70

80

0

Figure 6.25 Centerline proﬁles of Favre-averaged mixture fraction and temperature in Flames C, D, E, and F (modiﬁed from Barlow et al., 2005).

2100

~ T

1800

Z,r(K)

cr

~ T

Z,r(s

−1)

r = 4.0 mm r = 5.2 mm r = 6.5 mm r = 7.7 mm r = 8.9 mm

1500

rst = 7.35 mm

1200 900 cr

600 300 0.0

0.2

0.4

0.6

120

2100

100

1800

80

1500

60

1200

40

900

20

600

0 1.0

0.8

~ T

Z,r(K)

cr

~ T

60

rst = 9.02 mm

40

0.2

0.4

0.6

(a) x/D = 7.5

(b) x/D = 15

cr

~ T

1800 1500

−1 Z,r(s )

0.8

0 1.0

30

r = 7.5 mm r = 10 mm r = 12.5mm

25

rst = 12.5 mm

20 15

1200

10

900 cr

600 300

20 10

300 0.0

Z,r(K)

50 30

cr

Mixture Fraction

~ T

−1)

r = 5.9 mm r = 8.4 mm r = 10.9 mm

Mixture Fraction

2100

Z,r(s

0.0

0.2

0.4

0.6

5 0.8

0 1.0

Mixture Fraction (c) x/D = 30

Figure 6.26 Density-weighted conditional means of temperature and radial (onedimensional) scalar dissipation χr , at various axial and radial locations in Flame D. The Favre mean stoichiometric radius, rst , is given in each graph. Each curve represents data from 1 mm segments (ﬁve superpixels) centered at positions listed in the legends (modiﬁed from Barlow et al., 2005).

448

NON-PREMIXED TURBULENT FLAMES

maximum. For data from the outside (leaner) end of the probe volume (r = 10.9 mm, short-dash curve), the rich-side maximum is greater than the lean-side maximum. A possible physical explanation for the trends in Figure 6.26b has to do with the dynamics of turbulent eddies as they transport ﬂuid of a given mixture fraction into radial locations corresponding to different mean values of mixture fraction. For example, when a fuel-rich sample (with relatively high axial velocity) is transported radially outward toward a lean region, a strong interaction should occur. Eddies of differing velocity and mixture fraction collide, producing higher-than-average scalar dissipation. This might contribute to the peak in scalar dissipation near Z = 0.5 for the data at r = 10.9 mm (short-dash curve). Similarly, when a relatively fuel-lean sample (with relatively low velocity) is transported radially inward to a richer region, there is again strong interaction that produces higher-than-average scalar dissipation. This might contribute to the peak in scalar dissipation near Z = 0.35 for the data at r = 5.9 mm (solid curve). Large-eddy simulation (LES) of this ﬂame or DNS of a reacting shear ﬂow with reasonably a high Reynolds number could provide insights into such turbulent ﬂow dynamics. It is interesting that the conditional mean temperature curves for ﬂame D at x /D = 15 (Figure 6.26b) are separated for conditions 0.2 < Z < 0.4 but fall close together for conditions 0.4 < Z < 0.6. This may reﬂect the combined inﬂuence of convective and local time scales on reactive scalars. On average, the convective time scale increases with increasing radius as the stream-wise velocity decreases. However, from Figure 6.26b we know that the trends in scalar dissipation are not so simple. For Z < 0.4, the solid curve (smaller radius) has the highest scalar dissipation rate (or shortest local time scale). This fact tends to reinforce effects of the shorter convective time scale, producing a signiﬁcant radial dependence of conditional mean scalars on convective time scale for Z < 0.4 (the lean side). For Z > 0.4, the short-dash curve (larger radius) has the highest scalar dissipation, and this should counteract the effect of the longer convective time scale that applies on average at larger radius. The cross-stream dependence of conditional scalar dissipation rate is lower at x /D = 30 (Figure 6.26c). Trends are similar to those at x /D = 15, but the differences between radial locations may not be signiﬁcant compared to the statistical uncertainty in these measurements. In contrast, the cross-stream dependence is large at x /D = 7.5 (Figure 6.26a). The maximum scalar dissipation rate (found near Z = 0.55) more than doubles when moving from r = 4.0 mm to r = 7.7 mm. There is also a shift of the peak location from the fuel-rich side to slightly on the fuel-lean side of the mean stoichiometric position. This strong cross-stream dependency of the conditional scalar dissipation rate may be signiﬁcant for the physics and modeling of local extinction, which begins in this region of the jet ﬂame. 6.6.2.1

Comparison of Simple Jet Flame and Sandia Flames D and F

Simple jet ﬂames are unstable at high ReD ; piloted jet ﬂames are more stable, and they reignite upon local extinction. In order to check the accuracy of the nearﬁeld development of the jet ﬂame predicted by numerical simulations, a detailed

TURBULENT NON-PREMIXED TARGET FLAMES

449

knowledge of ﬂow-ﬁeld characteristics and scalar measurements depicting the mixing and chemical kinetics is required. Schneider et al. (2003) conducted an experimental study on the three ﬂames by using laser Doppler velocimetry (LDV) for measurements of the ﬂow ﬁeld structure. These three ﬂames are the target ﬂames and are of great interest to the modeling community for purposes of model development and validation. The ﬂow parameters of these jet ﬂames are summarized in Table 6.5. Figure 6.27 shows the radial proﬁles of normalized mean axial velocity, turbulent kinetic energy, and the Reynolds stress component 1 mm (x /D = 0.14) above the corresponding jet exits. These quantities are normalized on the respective mean axial centerline velocities. The self-similarity of the proﬁles is quite obvious for the inner core issued from the burner. The differences in the piloted and the simple jet ﬂame are apparent for radial positions ranging from r ≈ 4 to 9 mm. Highest levels of velocity ﬂuctuations are observed in the shear layer. Anisotropy of turbulence is observed for these three ﬂames as shown by the nonzero distribution of the Reynolds stress component u ν . In the pilot region of Flame D and Flame F (i.e., between r ≈ 4 and 9 mm), turbulence levels are relatively low. In addition, no distinctive shear stress is observed in the crossover from the pilot to the coﬂow region, contrary to observations for axial downstream position of x /D = 45 shown in Figure 6.28. This observation might be caused by the fact that shear-induced turbulence just starts to evolve at the nozzle exit. Figure 6.28 compares radial distributions in the far ﬁeld at x /D = 45 (Flame D and Flame F) and x /D = 40 (DLR ﬂame), respectively. The radial dimension is normalized by the local half-width of the radial jet (r 1/2 ). The local half-width is the radial location where the mean velocity is equal to one-half of its value at the axis. These ﬂames appear to be self-similar when their axial velocity proﬁles are compared, despite the fact that at this axial position the local half-width of each ﬂame is slightly different. A similar behavior was also observed for radial proﬁles at locations farther downstream. The anisotropic nature of turbulence within these ﬂames can be observed from the Reynolds stress distributions. Among the three ﬂames, the Sandia Flame D has highest degree of turbulent ﬂuctuations (the TABLE 6.5. Flow parameters for jet ﬂames Flame Type Simple Jet Flame (DLR Flame) Dj et = 8 mm Piloted Jet Flame (Sandia Flame D) Dj et = 7.2 mm Piloted Jet Flame (Sandia Flame F) Dj et = 7.2 mm

Reactant Composition

ReD

Ujet -cold Upilot -burnt Ucoﬂow

33.2% H2 , 22.1% CH4 , and 44.7% N2 25% CH4 , 75% air

15,200

42.15

N/A

0.3

∼22,400

49.6

11.4

0.9

25% CH4 , 75% air

∼44,800

99.2

22.8

0.9

450

NON-PREMIXED TURBULENT FLAMES

u⁄ucL,0

1.0 0.8

Piloted jet flame (D), uCL,0 = 62.95 m/s

0.6

Piloted jet flame (F), uCL,0 = 125.71 m/s Simple jet flame (DLR), uCL,0 = 51.55 m/s

0.4 0.2 0.0

0.012

k ⁄u2cL,0

0.010 0.008 0.006 0.004 0.002 0.000

u′v′⁄u2cL,0

0.003 0.002 0.001 0.000 −0.001 0.000

0.002

0.004

0.006

0.008 r [m]

0.010

0.012

0.014

0.016

Figure 6.27 Radial proﬁles of normalized axial velocity (u), turbulent kinetic energy (k), and Reynolds stress component (u v ) 1 mm above jet exit (modiﬁed from Schneider et al., 2003).

normalized turbulent kinetic energy is highest), which is interesting since Sandia Flame F has the highest Reynolds number. To exemplify second-order effects more clearly, the shear stress component u ν is also shown in Figure 6.28. In Flame D and the DLR ﬂame, the highest shear stresses appear in the vicinity of r ∼ r 1/2 . For Flame F, this maximum is shifted inward by a factor of approximately 0.3r 1/2 toward the centerline and peaks around r ∼ 0.7r 1/2 . This shows the severe impact of ﬁnite chemistry on general ﬂow-ﬁeld characteristics and also implies that the scaling laws are not sufﬁciently developed to predict the second-order effects, especially the inﬂuence of ﬁnite-rate chemistry effects. Another interesting observation comes from the axial distributions of turbulent kinetic energy and mean velocity. These distributions are shown in Figure 6.29. The ﬁgure also shows the corresponding normalized local half-width of the jet (r 1/2 ), indicating the spreading rate of each jet ﬂame. The local half-widths of these three ﬂames exhibit differences at the downstream locations. If we compare only Flame D and Flame F (these two ﬂames have the same initial

451

TURBULENT NON-PREMIXED TARGET FLAMES

1.0

u⁄ucL,0

Piloted jet flame (D), uCL,0 = 62.95 m/s 0.8

Piloted jet flame (F), uCL,0 = 125.71 m/s

0.6

Simple jet flame (DLR), uCL,0 = 51.55 m/s

0.4 0.2 0.0 0.08

k ⁄u2cL,0

0.06 0.04 0.02 0.00 0.024

u′v′⁄u2cL,0

0.020 0.016 0.012 0.008 0.004 0.000 0

1

2 r/r½

3

4

Figure 6.28 Radial proﬁles of normalized axial velocity (u), turbulent kinetic energy (k), and Reynolds stress component (u v ) at dimensionless axial distance of x /D = 45 (Flame D and F) and x /D = 40 (DLR simple jet ﬂame), respectively (modiﬁed from Schneider et al., 2003).

gas composition), self-similar axial proﬁles have been shown by Tacina and Dahm (2000) with inﬁnitely fast chemistry assumption. The experimental data of Schneider et al. (2003) contrasts with their observation, since Flame D shows a higher degree of spreading than Flame F. This contrast indicates that the ﬁnite chemistry effects can impact the ﬂow ﬁeld by reducing the jet width. Next we discuss the centerline axial velocity distributions for these three ﬂames. For axial locations x /D > 15, the locally extinguishing ﬂame (Flame F) shows a slower decay compared to the other two stable ﬂames. The axial velocity decay characteristics are inﬂuenced mainly by the local degree of heat release. A slower decay of axial velocity is caused by a higher temperature rise due to heat release resulting in larger gas expansion. In the case of locally extinguishing ﬂames that exhibit reignition downstream (i.e., Flame F), heat release

452

NON-PREMIXED TURBULENT FLAMES 5 4 3 r½ /D 2 1

Piloted jet flame (D), uCL , 0 = 62.95 m/s Piloted jet flame (F), uCL , 0 = 125.71 m/s

0 1.00

Simple jet flame (DLR), uCL , 0 =51.55 m/s

0.75 u/uCL , 0

0.50 0.25 0.00

0.012

k/u2CL

0.008 0.004 0.000 0

10

20

30

40

50

60

70

x/D

Figure 6.29 Axial proﬁles of local half-width (r 1/2 ), mean axial velocity, and turbulent kinetic energy along centerline (modiﬁed from Schneider et al., 2003).

is higher at a given axial location than the nonextinguishing conﬁgurations. The axial proﬁles of the turbulent kinetic energy show an initial increase, a distinct maximum, and a subsequent decay. The initial increase is attributed to velocity gradients (mainly in the radial direction) whereas the decay is due to dissipative losses. The steepness of the initial rise as well as the axial position of the maximum value is dependent on the Reynolds number and fuel composition, respectively. In accordance with the slower axial velocity decay, Flame F shows the peak turbulence farthest downstream. This indicates that ﬂame extinction and reignition downstream can change the turbulence ﬁeld by inﬂuencing local heat release.

6.6.3

Bluff Body Flames

The bluff-body burner is a useful tool to study turbulent non-premixed ﬂames with recirculating ﬂows. The burner geometry is simple, its boundary conditions

453

TURBULENT NON-PREMIXED TARGET FLAMES

Gravity

Jet-like region

Recirculation zones

are well deﬁned, and it has a stable ﬂame for a wide range of coﬂow and jet conditions. The bluff-body burner, like the piloted burner, provides a ﬂame suitable for the study of turbulence-chemistry interactions. Bluff-body burners also bear a great similarity to practical combustors used in many industrial applications. The burner consists of a straight tube centered in a coﬂowing stream of air conﬁned by a larger cylinder that is mounted on a wind tunnel. It generally has a circular bluff-body with an oriﬁce at its center for supplying the main fuel (see Figure 6.30). This geometry is a suitable compromise as a model problem because it has some of the complications associated with practical combustors while preserving relatively simple and well-deﬁned boundary conditions. A complex ﬂow pattern is formed downstream of the face of the bluff-body, where a recirculation zone produces sufﬁcient hot gases that stabilize the ﬂame. At sufﬁciently high fuel velocity, the jet ﬂow penetrates through the recirculation zone and forms a jetlike ﬂame farther downstream, which is similar to the piloted jet ﬂame. The jet ﬂame can be extinguished in a region downstream of the recirculation zone where turbulence is well developed and the ﬁnite-rate chemistry effects are signiﬁcant. The ﬂame may reignite farther downstream where turbulent mixing rates are lower. Therefore, it can be concluded that both piloted and bluff-body jet ﬂames consist of three main zones: stabilization, extinction, and reignition zones. Bluff-body burners have been used with a range of bluff-body diameters (Db ) and fuel jet diameters (Djet ). Typically, the fuel jet diameter is constant at 3.6 mm,

UCF

Circular bluff body

UCF

Recirculation zone

Bluff-body

Central fuel jet

Ujet

Neck zone

Djet (a)

Db

(b)

Figure 6.30 (a) Bluff-body burner; (b) conﬁguration of the cylindrical bluff-body burner (modiﬁed from Kempf, Lindstedt, and Janicka, 2006).

454

NON-PREMIXED TURBULENT FLAMES

but data sets exist for smaller diameters. Stability characteristics of these ﬂames are given in terms of the fuel jet velocity (Ujet ) and external ambient coﬂow velocity (UCF ), which has a strong effect on mixing rate downstream of the neck region. Higher velocities lead to higher gas momentum around the burner and bluff-body, leading to more signiﬁcant “pinching” at the neck. The free stream turbulence in the coﬂow is around 2%. The length of the recirculation zone is about one bluff-body diameter. The addition of H2 or CO to the CH4 fuel is intended to produce a recirculation zone that is clean from soot. In pure CH4 ﬂames, the recirculation zone generally has soot or soot precursors that are convected downstream, where they interfere with the Raman signals. Partially premixing the methane with air reduces the ﬂame’s tendency to soot, and this may lead to a much cleaner recirculation zone. The three zones of the bluff-body stabilized ﬂame (Sydney burner) are shown in the temperature proﬁles predicted by an LES code (Hahn et al., 2007) in Figure 6.31. The simulation was carried out using the explicit, multigrid version of the LES solver FASTEST. The LES was started using RANS results as an initial solution and an elliptical smoothed multiblock grid with approximately 2.08 million grid points was used. These results show stabilization, ﬂame extinction, and later reignition in the bluff-body ﬂame.

Temperature [K]

time

Figure 6.31 Simulation of the Sydney bluff-body stabilized hydrogen/methane (1:1) diffusion ﬂame with air as coﬂowing gas, Ujet = 108 m/s, Ucf = 35 m/s based on the work of Janicka and coworkers, modiﬁed from http://www.ekt.tu-darmstadt.de/ekt/galerie_1/ galerie_2.de.jsp, 2007.

455

TURBULENT NON-PREMIXED TARGET FLAMES

6.6.4

Swirl Stabilized Flames

The swirl burner is a simple extension of the bluff-body burner. It offers simple design with well-deﬁned boundary conditions, yet it produces complex swirling ﬂows similar to those found in practical combustors. Cross-sectional views of the swirl burner and photographs of air supply inlets are shown in Figure 6.32. The swirl air is supplied through an annulus of 60 mm inner diameter that is added

Top view on wind-tunnel & burner

+U +V Ue

Us

Uj

U5

∅60 ∅50 ∅3.6

All dimensions in millimeters

Axial air

Ue A

130

A

Tangential air Annular seeder

Alignment screws (×3)

Wind tunnel Honeycomb mesh

Tangential air 15°

Axial air

Coarse mesh

Fuel supply

Figure 6.32 Drawing and details of Sydney swirl burner (modiﬁed from Masri et al., 2004).

456

NON-PREMIXED TURBULENT FLAMES

to the bluff-body. The annulus provides swirled air from tangentially arranged inlets at the base of the burner. This swirler adds two additional ﬂuid-dynamical parameters: the ﬂow rate of the swirled air and its swirl number. For a limited number of fuel compositions, this addition leads to a four-dimensional parameter space. Signiﬁcant experimental and computational research has been performed on this burner (Barlow et al., 2005).

6.7

TURBULENCE-CHEMISTRY INTERACTION

Turbulent ﬂows and combustion chemistry are characterized by a broad spectrum of time and length scales. The larger scales of the turbulence are directly linked to the physical size of the combustor, whereas the smaller scales represent the dissipation of turbulent energy by viscosity. When the reactants are supplied in separate streams, the entrainment of fuel and oxidizer by large-scale eddies leads to incomplete mixing. Then micromixing mechanisms, acting at smaller scales, bring fuel and oxidizer into contact in the reaction zone where products are formed. The elementary chemical reactions also have a broad spectrum of time scales. Depending on the overlap between turbulence time scales and chemical times, strong or weak interactions between chemistry and turbulence may appear (Bray, 1996). Because the description of non-premixed turbulent combustion requires an understanding of simultaneous turbulent mixing and combustion processes, the modeling of these ﬂames is a scientiﬁc challenge, which has been the subject of numerous studies. Hypotheses formulated by various researchers to construct models for non-premixed turbulent ﬂames can be organized into three major groups: 1. Assumption of inﬁnitely fast chemistry (chemical equilibrium) 2. Finite-rate chemistry, including a coupling between diffusion and reaction (the ﬂamelet assumption) 3. Finite-rate chemistry with a separated treatment of diffusion from reaction (pdf methodology)

6.7.1

Inﬁnite Chemistry Assumption

In combustion systems, reaction often occurs in very thin layers relative to the characteristic ﬂow scales. In a ﬁrst approximation, the chemistry can be assumed to be inﬁnitely fast. Then the reaction is conﬁned to mixtures near stoichiometric condition, where fuel and oxidizer are consumed rapidly with heat being released within a thin zone. Based on the work of Burke and Schumann (1928), the assumption of inﬁnitely fast chemistry has been widely used for the

TURBULENCE-CHEMISTRY INTERACTION

457

description of diffusion-controlled combustion. Indeed, large- and small-scale mixing of fuel and oxidizer are strongly sensitive to the exothermic nature of combustion. Using inﬁnitely fast chemistry, DNS can focus, for example, on heat-release effects modifying transport properties (Higuera and Moser, 1994) because of the decrease of the Reynolds number resulting from the increase of kinematic viscosity with temperature. A lower Reynolds number facilitates DNS calculations. Correlations involving density and pressure ﬂuctuations also can be studied for modeling purposes (Bilger, 1989). There are two bifurcations of this approach (e.g., Vervisch and Poinsot, 1998) the unity and the nonunity Lewis number, described next. 6.7.1.1 Unity Lewis Number According to the Burke-Schumann equilibrium condition, YF = 0 on the air side of the diffusion ﬂame and YO = 0 on the fuel side. As a consequence, the reaction rate is equal to zero in both convective and diffusive regions located on each side of the reaction zone; detailed kinetics are not necessary to describe the burning rate, which is fully controlled by diffusion. At this point, it is important to distinguish between burning rate and chemical reaction rate. The “burning rate” refers to the consumption rate of the fuel; the “chemical reaction rate” is the rate at which 1 mole of fuel is completely burned by oxidizer. The burning rate depends on the rate of supply of the fuel to the reaction zone; therefore, it is a function of both the rate of fuel supply via diffusion and rate of chemical reaction of fuel. Since the chemical reaction rate is assumed to be inﬁnitely fast, only the rate of diffusion of fuel to the reaction zone determines the burning rate. The chemistry can be modeled by a one-step reaction cast in the form νF YF + νO YO → νP YP , where YF , YO , and YP are the mass fractions of the fuel, the oxidizer, and the products, respectively, and νi is the molar stoichiometric coefﬁcient. If we assume that the Lewis numbers for all species are equal to unity (Lei = 1), then for inﬁnitely fast chemistry, the description of the diffusion ﬂame is reduced to solutions of combinations of species mass fraction and temperature that are conserved through reaction zones (Kuo, 2005, chap.6; Li˜nan and Williams 1993a, 1993b; Williams, 1985). The elimination of the chemical source terms ω˙ i = 0 results from stoichiometric relationships between species involved in the chemical reaction. It is also possible to deﬁne those conserved scalars from conservation of chemical elements. In this context, the mixture fraction Z is a conserved scalar usually introduced to characterize mixing between fuel and oxidizer (see Equation 3.13 in Chapter 3). The mixture fraction is equal to zero in the pure oxidizer stream and to unity in pure fuel. Within the inﬁnitely fast chemistry hypothesis, the knowledge of Z is sufﬁcient to determine species and ﬂame temperature.

458

NON-PREMIXED TURBULENT FLAMES

6.7.1.2

Nonunity Lewis Number

The structure of diffusion ﬂames with inﬁnitely fast chemistry also may be studied for those with nonunity Lewis numbers. DNS is of interest in this case because it allows the study of differential diffusion effects on ﬂames because their results do not explicitly depend on chemical details. The appropriate formalism to account for these preferential diffusion effects has been introduced and used in DNS of jets and mixing layers by Li˜na´ n et al. (1994). To deal with nonunity Lewis numbers, two different mixture fractions must be used; Z (deﬁned in Chapter 3), and ZL deﬁned by: YF /YF,1 − YO2 /YO2, 2 + 1 ZL = ( + 1) where

≡

ZLst =

LeO2 LeF

1 ( + 1)

YF,1 φ, φ ≡ s YO2 ,2

and

Zst =

and

s≡

MwO2 νO 2

(6.43)

νF MwF

= st

1 YF /YO2 st (6.44)

1 + (φ 1)

(6.45)

In these equations, subscripts 1 and 2 refer to the pure fuel and pure oxidizer streams. Piecewise linear relations exist between the mixture fractions and chemical species (shown in Table 6.6). In the case of inﬁnitely fast chemistry, those relations together with continuity, momentum, and energy equations permit numerical simulations that account for differential diffusion effects of temperature and concentration ﬁelds.

6.7.2

Finite-Rate Chemistry

In certain practical systems, the inﬁnitely fast chemistry hypothesis cannot be invoked everywhere. Such conditions exist in the process of ignition in chemically TABLE 6.6. Piecewise relations for inﬁnitely fast chemistry for non-unity Lewis number (modiﬁed from Vervisch and Veynante, 1999) Oxidizer Side Z < Zst and ZL < ZLst

Fuel Side Z > Zst and ZL > ZLst

Z = ZL (ZSt /SLSt )

Z = (1 − ZSt )(ZL − ZLSt )/(1 − ZLSt ) + ZSt

YF = 0

YF = YF,1 [(Z − Zst )/(1 − Zst )]

YO2 = YO2 ,2 [1 − Z/Zst ]

Y O2 = 0

T = (Tf − TO2 ,2 )Z/Zst + TO2 ,2

T = (TF,1 − Tf )[(Z − Zst )/(1 − Zst )] + Tf

TURBULENCE-CHEMISTRY INTERACTION

459

reacting systems, near-stability regions of a turbulent ﬂame, and in the regions of high-velocity gradients. Extinction controls some of these conditions in the turbulent non-premixed ﬂames. For this reason, the characterization of diffusion ﬂames from the inﬁnitely fast chemistry situation to the quenching limit is of great importance in turbulent non-premixed combustion. In a steady diffusion ﬂame, the heat loss by diffusion and convection is balanced by the heat released in the reaction zone. Quenching is observed when heat losses are greater than the production of heat by chemical reactions. For fast chemistry (but not necessarily inﬁnitely fast), the reaction zone thickness remains small relative to all ﬂow scales (the ﬂamelet assumption). The ratio of the convective term to the diffusive term is on the order of uj δr /D. When this ratio is small, the convective effect may be neglected. For unity Lewis numbers and constant physical properties, the species conservation equation in the vicinity of the stoichiometric surface becomes: 2 2 2 d Yi ω˙ i ∼ −ρD ∇ Yi = −ρD |∇Z| (6.46) dZ 2 When the reaction zone thickness δr is smaller than all ﬂow scales, the mixture fraction dissipation rate (or scalar dissipation rate χ) and the second-order derivative of the function Yi = Yi (Z ) in the vicinity of the stoichiometric conditions Z = Zst are the two key quantities controlling the burning rate of fuel (Bilger, 1976). The mixture fraction dissipation rate quantiﬁes the intensity of mixing (Dopazo, 1994) and incorporates ﬂuid dynamical properties of the reactive ﬂow. The scalar dissipation rate χ is especially correlated with the velocity gradient evaluated in the tangential plane of the iso-Zst surface. This velocity gradient deﬁnes the in-plane strain rate −ni nj ∂ui /∂xj + (∂ui /∂xi ) where ni is the normal vector to the iso-Zst surface (Nomura and Elghobashi, 1992). Hence, a mechanical or ﬂow time scale may be deﬁned from the value of the scalar dissipation rate taken at the stoichiometric surface. For a given ﬂow, the internal structure of the diffusion ﬂame depends on the time needed to consume the reactants. To estimate a chemical time scale, the reaction rate of the fuel can be rewritten in this form: νO Ta νO2 +νF νF 2 ω˙ F = νF ρ YF YO2 A exp − (6.47) Tad where A is the pre-exponential factor and Ta is the activation energy of the reaction (see Equation 2.10 in Chapter 2). According to asymptotic analysis (Cuenot and Poinsot, 1996; Li˜na´ n, 1974), a general expression for the Damk¨ohler number (Da) can be written as: Da = 16φ

νO

ν +νF −1 2 ρ O2 (1

− Zst )

2

2 Tf,ad

ν

Ta QYF,1 /Cp

Ta · A exp − χst Tf,ad (6.48)

O2 +νF +1

460

NON-PREMIXED TURBULENT FLAMES

The thermal energy gained by chemical reactions is Q, which is positive for exothermic reactions (see Equation 3.54 in Chapter 3). The response of the maximum temperature in the counterﬂow laminar diffusion ﬂame versus this Damk¨ohler number is the upper part of the well-known S -shape curve (see Section 3.4.1 in Chapter 3). From Equation 6.48, few observations can be made. If the chemistry is too slow or χst is too large, the chemical process is unable to keep up with the large heat losses and inﬂux of reactants, and ﬂame quenching occurs. Therefore, extinction can be deﬁned as a limiting situation where reactants proceed through the stoichiometric zone without burning. This situation corresponds to the quenching point shown in the S -curve shown in Figure 3.7. For a particular fuel, Damk¨ohler number are functions of the mixture fraction and its dissipation rates at stoichiometric condition. Therefore, once the mixture fraction ﬁeld is known, the particular Da of the diffusion ﬂame is obtained together with an extinction criterion. Temperature and mass fractions may be expressed as functions of Z and Da (Li˜na´ n, 1974). Asymptotic study of laminar counterﬂow diffusion ﬂame with nonunity Lewis number effects and total enthalpy variation with reactant leakage have been studied (Cuenot and Poinsot, 1996; Kim and Williams, 1997). These results have suggested that diffusion-reactive layers have two stable states: burning and quenched. The transition between these two extreme cases corresponds to an intermediate and unstable partially burning situation. Depending on the positive or negative value of the propagation speed of the partially burning front, either a fully burning or fully quenched stable state propagates along the stoichiometric line through imbalance among diffusion, convection, and reaction. These propagation phenomena are more likely to occur at the edge of the reaction zone where heat losses could be at the maximum. Therefore, the dynamics of edge ﬂame is a key issue in non-premixed turbulent ﬂames. Edge ﬂames are discussed in a later section in this chapter. Mixing time for a scalar ζ (which could be mixture fraction or species mass fraction) is deﬁned as the ratio of variance of the scalar and mean dissipation rate of the scalar: ζ 2 τm,ζ ≡ (6.49) χζ Note that the mixing time scale sometimes is deﬁned by multiplying Equation 6.49 by 1/2. In the results presented in the following section however, the mixing time was deﬁned by Equation 6.49, without 1/2. The dissipation rate for the scalar is deﬁned as: χζ = 2Dζ ∇ζ · ∇ζ

(6.50)

The integral time scale is deﬁned next. This time scale represents mixing due to large-scale turbulence: k˜ τ0 ≡ (6.51) ε˜

TURBULENCE-CHEMISTRY INTERACTION

461

The ratio of integral time scale and the scalar mixing time scale can be deﬁned as: δf δf rt,ζ ≡ τ0 dy τm,ζ dy (6.52) 0

0

where y is the distance in the transverse direction from the jet centerline and δf is the y-location where the mean mixture fraction is less than 0.05. Integration across the y direction allows convenient presentation of the results as a single time-scale ratio and reduces statistical scatter. The DNS calculations have shown how detailed diffusive transport and chemistry effects can inﬂuence the mixing of reactive scalars (e.g., species mass fractions). Scalars with nonunity Schmidt numbers can potentially have different mixing time scales. Figure 6.33 shows the mixing time-scale ratio for the H, H2 , and CO2 mass fractions versus a dimensionless time normalized by the jet characteristic time (Djet /Ujet ) along with the mixture fraction time-scale ratio. These molecules have different diffusivities: H is the most diffusive while CO2 is the least. The ﬁgure clearly shows there is an effect of diffusivity on the mixing time scales. The most diffusive scalar H has a maximum time-scale ratio of 6.2 while CO2 has a maximum time-scale ratio of 1.6. The difference between these values, however, is less than the difference in the diffusivities. The results indicate that differential diffusion effects may need to be incorporated within mixing models, at least at moderate Reynolds numbers. This conclusion could possibly be Reynolds number dependent, and a parametric study of Reynolds numbers ultimately will be required to determine any such dependence. The strong interplay between reaction and molecular mixing in non-premixed ﬂames also can affect mixing time scales. Figure 6.34 shows the mixing timescale ratios for HO2 , H2 O2 , O, and OH. Initially, these time scales are ordered according to diffusivity, but in the intermediate time, the time scales of O and OH

rz rH2 rH rCO2

Timescale ratio rf

H 6 4

H2

2 Z

CO2 0

5

10

15

Time/(Jet Time)

Figure 6.33 DNS calculated integral to scalar time-scale ratio for H, H2 , and CO2 mass fractions and mixture fraction versus dimensionless time (modiﬁed from Hawkes, Sankaran, Sutherland, and Chen, 2005).

462

NON-PREMIXED TURBULENT FLAMES rz rO rOH rHO2 rH2O2

Time-scale ratio rf

6 H2O2 4

HO2

2

0

O, OH Z 5

10

15

Time/(jet Time)

Figure 6.34 DNS calculated integral to scalar time-scale ratio for H2 O2 , HO2 , O, and OH mass fractions and mixture fraction versus time (modiﬁed from Hawkes et al., 2005).

increase and those of HO2 and H2 O2 decrease, and the hierarchy of diffusivity no longer holds. This is a result of an interesting interplay between reaction and molecular mixing. These ﬁndings show the importance of considering the interplay of diffusion and reaction, particularly when strong ﬁnite chemistry effects are involved. 6.8 PROBABILITY DENSITY APPROACH FOR TURBULENT NON-PREMIXED COMBUSTION

PDF methods are most valuable in situations where either chemical kinetics or turbulence may be rate controlling. In practice, they have been applied primarily to non-premixed systems with weak coupling between molecular mixing and chemical reaction. The numerical treatment associated with this process is called operator-splitting algorithms. However, there are no intrinsic limitations in terms of the use of pdf methods. They have been applied to premixed combustion and to ﬂamelet regimes where molecular transport and chemical reaction are tightly coupled. In practical combustion systems, ideal premixed or non-premixed ﬂame is not always available; rather, a partially premixed ﬂame is more realistic. Therefore, in contrast to the idealized diffusion ﬂame, the mixture fraction is not sufﬁcient to deﬁne the local instantaneous thermochemical state of the reacting mixture in a turbulent ﬂow. Thus, it is useful to consider joint statistics of multiple physical quantities. The derivation of transport equations for the joint pdf has been given in Section 5.14 of Chapter 5. That formulation made no assumptions about the chemistry, mixing, or geometry. Therefore, the same equations are applicable to premixed, non-premixed, and partially premixed systems in any regime of turbulent combustion. Restrictions to speciﬁc combustion regimes are introduced in the modeling stage. Laboratory-scale, atmospheric pressure non-premixed turbulent jet ﬂames are the single conﬁguration to which pdf methods (and many other turbulent combustion models) have been applied most often. The scatter plots of temperature T

PROBABILITY DENSITY APPROACH

463

versus mixture fraction Z obtained from laser-based measurements for two piloted non-premixed methane air turbulent jet ﬂames (Sandia Flames D and F) at a ﬁxed spatial location (x /D = 30, r/D = 1.67) (Barlow et al., 2005) were shown in Figure 6.18. These ﬂames have been discussed in Section 6.6 of the current chapter. In these ﬂames, the stoichiometric value of the mixture fraction has been increased (compared to its value for pure methane fuel and air oxidizer) to Zst = 0.351 by diluting the fuel jet with air, and the ﬂame continues to burn as a non-premixed ﬂame. At a ﬁxed measurement location, there are very few samples for Z < 0.1 or Z > 0.7. The fuel-jet and pilot velocities (hence ReD and Da) are different for these two ﬂames; the fuel-jet Reynolds number increases from 22,400 for Flame D to 44,800 for Flame F, while the Damk¨ohler number (the more relevant parameter) decreases from Flame D to Flame F. Each point in the scatter plot corresponds to a single, instantaneous point measurement where the local temperature and species mass fractions (CH4 , O2 , N2 , H2 O, CO2 , CO, H2 , OH, and NO) are measured simultaneously; the local mixture fraction then is determined from the species mass fractions. Due to the scatter in data, there is no unique value of T for each value of mixture fraction Z . Moreover, the level of variation in the values of T for a given value of Z is greater for Flame F than for Flame D. These notions can be quantiﬁed by introducing joint pdfs. For concreteness, let us consider joint statistics of T and Z and introduce the joint pdf of temperature and mixture fraction PT ,Z (τ, ψ; x, t), where τ and ψ represent the sample space variables for temperature and mixture fraction, respectively. This quantity can be determined operationally by using a two-dimensional analogue of the binning and limiting procedure. Here PT ,Z (τ, ψ; x, t) has these properties:

PT ,Z (τ, ψ; x,t) ≥ 0 ∞

∞

(6.53)

PT ,Z (τ, ψ; x, t) dτ dψ = 1

(6.54)

PT ,Z (τ, ψ; x,t) = 0 for ψ < 0 and ψ > 1

(6.55)

−∞ −∞

The dimensions of the joint pdf are τ −1 ψ −1 . For the joint pdf of temperature and mixture fraction, the dimension is inverse temperature, since ψ is dimensionless. The joint pdf varies with spatial location x and time in the ﬂame. The pdfs of temperature and of mixture fraction (the “marginal pdfs”) can be determined from the joint pdf:

PZ ( ψ| τ = T ; x, t) =

∞

PT ,Z (τ, ψ; x, t) dτ

(6.56)

PT ,Z (τ, ψ; x, t) dψ

(6.57)

−∞

PT ( τ | ψ = Z; x, t) =

∞ −∞

464

NON-PREMIXED TURBULENT FLAMES

The mean of any function Q = Q(T , Z ) = Q[T (x, t), Z (x, t)] = Q(x, t), is given by ∞ ∞ Q (x,t) = Q (x, t)PT ,Z (τ, ψ; x,t) dτ dψ (6.58) −∞ −∞

In practice, measured and computed pdfs are rarely compared directly. Instead, mean quantities (typically only a few lower-order moments) are used as the basis for comparisons between experimental measurements and pdf-based modeling studies. The beneﬁts of accounting explicitly for the inﬂuence of turbulent ﬂuctuations then can be seen by comparing the results of pdf-based models with those from models that do not account for turbulent ﬂuctuations or that treat them in an oversimpliﬁed manner. An example is provided in Figure 6.35. The mean temperature and mean O2 mass-fraction proﬁles computed using a pdf method are compared with proﬁles computed using a model that ignores turbulent ﬂuctuations altogether (the local mean chemical source terms are computed based on the local mean species composition and temperature) for Sandia Flame D. These models are discussed in the following sections. Here it is important to note that a simple one-step, ﬁnite-rate global methane air chemical mechanism has been used. Therefore, a good quantitative agreement with experimental data should not be expected. Nevertheless, this example serves to illustrate the beneﬁts of accounting for turbulent fluctuations in species composition and temperature using a pdf method. A model that ignores turbulent ﬂuctuations grossly overpredicts the local mean heat-release rate and overpredicts the local mean temperatures by as much as 2400

2400

x/Djet = 15

2100

1200

1800 ~ T (K )

1500

1500 1200

1500 1200

900

900

900

600

600

600

300

300

0

1

2

3

4

5

6

7

0

1

2

3

r/Djet

4

5

6

7

300

0.2

0.2

0.15

0.15

0.15

~ YO2

0.1 x/Djet = 15

1

2

3

4 r/Djet

2

3

5

6

7

0.1

5

6

7

0.1 x/Djet = 30

0.05

8

4

~ YO2

0.2

~ YO2

0.25

0

1

r/Djet

0.25

0

0

r/Djet

0.25

0.05

x/Djet = 45

2100

1800 ~ T (K )

~ T (K )

1800

2400

x/Djet = 30

2100

0

0

1

2

3

4 r/Djet

5

6

7

x/Djet = 45

0.05

8

0

0

1

2

3

4

5

6

7

8

r/Djet

Figure 6.35 Radial proﬁles of mean temperature (upper row) and mean O2 mass fraction (lower row) at three axial locations in Sandia Flame D. (Modiﬁed plots: computed curves by Zhang, 2004, and measured data from Barlow and Frank, 1998).

465

PROBABILITY DENSITY APPROACH

several hundred degrees Kelvin. Results are signiﬁcantly improved with a pdfbased model, despite the oversimpliﬁed chemistry. 6.8.1

Physical Models

The derivation of the transport equation for the velocity-composition joint pdf and the composition pdf was shown in Section 5.14.1 of Chapter 5. Since the same equations are valid for any combustion system, we can use them for nonpremixed combustion also. These equations are shown in 6.59 and 6.60. The Favre-averaged velocity-composition joint pdf, Pu , and composition pdf, P , have been deﬁned earlier in Section 5.14 of Chapter 5. Velocity-Composition pdf Equation ∂ ρ P˜ u ∂ ρ Vi P˜ u ∂ ρ S˙k P˜ u + + ∂t ∂xi ∂ψk Time variation of joint pdf

+

=

Transport in physical space due to advection by mean ﬂow and turbulence

Transport in composition space due to chemical reactions

ρ ∂ p ∂ P˜ u ρ (ψ) gi − ρ (ψ) ∂xi ∂Vi

Transport in velocity space due to body force and mean pressure gradient ! ρ ∂ ∂τji ∂p ˜ V, ψ Pu − ρ (ψ) ∂Vi ∂xi ∂xj

∂ + ρ ∂ψk

! ∂Jk,i 1 ˜ V, ψ Pu ρ(ψ) ∂xi

Effects of molecular diffusion of species and enthalphy

Effects of ﬂuctuating pressure gradient and viscosity

# ρ ρ " ˙ rad,em P˜ u − δk ∂ ˙ rad,ab V, ψ P˜ u Q Q ρ (ψ) ∂ψk ρ (ψ) Effects of radiative emissions Effects of radiative absorption (6.59)

∂ + δk ∂ψk

Composition pdf Equation ∂ ρ P˜ ∂ ρ u˜ i P˜ ∂ ρ S˙k P˜ + + ∂xi ∂ψk ∂t Time variation of pdf

Mean-ﬂow advection

Chemical reaction

" # ∂ ∂ ˙ ρ ρ ui ψ P˜ + =− ∂xi ∂ψk Turbulent diffusion

+ δ ρ

∂ ∂ψk

˙ ˜ Q rad,em P

ρ (ψ)

Effects of radiative emissions

! 1 ∂Jk,i ˜ ψ P ρ(ψ) ∂xi

Turbulent mixing

# ∂ " −1 ˙ − δk ρ Qrad,ab ψ ρ P˜ ∂ψk Effects of radiative emissions

(6.60)

466

NON-PREMIXED TURBULENT FLAMES

In equations 6.59 and 6.60, δk is a ﬂag that is equal to 0 for k = 1,2, . . . N and 1 for k = N +1. The terms to be modeled in a velocity-composition pdf equation are those corresponding to viscous stresses, the ﬂuctuating pressure gradient, scalar molecular ﬂuxes, and radiative absorption. In a composition pdf equation, the terms to be modeled are those corresponding to turbulent velocity ﬂuctuations, scalar molecular ﬂuxes, and radiation. Because these are one-point/one-time methods, turbulence scale information must be provided. These models can be summarized as follows: • Viscous stresses and the ﬂuctuating pressure gradient in velocitycomposition methods • Fluctuating velocity in composition methods • Scalar molecular ﬂuxes associated with the turbulent mixing term in both velocity–composition and composition methods. In addition the next issues require special attention in connection with pdf methods: • • • •

6.8.2

Flamelets in pdf methods Wall modeling High-speed ﬂows Turbulence scale speciﬁcation in the RANS/PDF and LES-ﬁlter density function (FDF) contexts and an alternative scale speciﬁcation

Turbulent Transport in Velocity-Composition pdf Methods

Transport of the pdf in physical space by advection, including transport by turbulent velocity ﬂuctuations, appears in closed form in velocity-composition pdf equations; this is the second term on the left-hand side of Equation 6.59. Models are required to represent the effects of the viscous stresses and the ﬂuctuating pressure gradient; these are the ﬁrst terms on the right-hand sides of the same equation. In the Lagrangian frame of reference, this corresponds to modeling particle accelerations or forces. Guidance for the construction of models can be taken from examination of the behavior of ﬂuid particles in turbulent ﬂows (e.g., from theory or DNS) and from examination of the behavior of Eulerian onepoint statistics (e.g., the implied Reynolds-stress equation). For example, one important requirement is that these models should not affect the mean velocity with the exception of the τij , which is negligible at high Re; there are no terms corresponding to the ﬂuctuating pressure gradient in the mean momentum equation. In the Lagrangian frame of reference, “particle interaction” models were used early in the development of velocity pdf and velocity-composition pdf methods (Haworth and Pope, 1986a). These models include the stochastic mixing and stochastic reorientation models. In these models, pairs of notional ﬂuid

PROBABILITY DENSITY APPROACH

467

particles are selected at random following statistical sampling rules, and the velocity of each particle is changed in a manner that leaves the mean velocity unchanged while (in homogeneous decaying turbulence) causing the turbulent kinetic energy to decay and the Reynolds stresses to tend toward isotropy, consistent with experimental observations. For a constant-density homogeneous system, the models/algorithms for N equal-mass particles can be written as follows. In a stochastic mixing model, the decaying turbulence and return to isotropy are accomplished whereas the stochastic reorientation model accomplishes return to isotropy, thereby keeping the turbulent kinetic energy unchanged. In both models, the mean velocity ﬁeld remains unchanged. 6.8.2.1

Stochastic Mixing Model

Following Haworth (2009), the probability of a pair of particles interacting in a time interval dt is Nωdt, where N is the total number of particles and ω is the turbulence frequency deﬁned as the ratio of turbulence dissipation rate to the turbulent kinetic energy (ε/k ). For each interaction event, two particles (denoted by p and q) are selected at random without replacement. The end-oftime step velocity for each of the two particles is set to the average of their beginning-of-time step velocities: 1 (p) u (t) + u(q) (t) ≡ u(pq) (t) = u(pq) (t + dt) 2 (6.61) In the case of non–equal-mass particles, a mass-weighted average would be used. This model leaves the average particle velocity vector u(pq) unchanged, and the magnitude of the velocity difference u(pq) [u(pq) (t) ≡ |u(p) (t) − u(q) (t)|] decreases toward zero. The mean velocity therefore is not affected, and it can be shown that the time rate of change of turbulence kinetic energy k and Reynolds stresses ui uj evolve according to the next equation: u(p) (t + dt) = u(q) (t + dt) =

dk = −ε, dt

% $ d ui uj dt

# " = −ω ui uj

where ω ≡ ε/k

(6.62)

The time variation of the turbulent kinetic energy of a ﬂuid particle in Equation 6.62 can be obtained from Equation 4.150 by considering decaying turbulence with no production terms due to the absence of mean velocity gradients and zero scalar transport. The rate of change of Reynolds stress in Equation 6.62 can be obtained with the weak-equilibrium assumption proposed by Rodi (1975). Under this assumption, the Reynolds stress ui uj can be decomposed as: $ % ui uj # 2 ui uj = k = k 2bij + δij k 3

"

(6.63)

468

NON-PREMIXED TURBULENT FLAMES

where bij is the normalized Reynolds stress anisotropy, deﬁned as: $ % ui uj 1 bij = − δij 2k 3

(6.64)

Therefore, small in $ $ % $ % % weak % $ equilibrium u ui uj dk u u u d ui uj #ε # " " i j i j dk d ≈ = +k = − ui uj = − ω ui uj dt k dt dt k k dt k (6.65) 6.8.2.2 Stochastic Reorientation Model The probability of a pair of particles interacting in a time interval dt is CR Nω dt, where CR is a model constant. For each interaction event, two particles (denoted by p and q) are selected at random without replacement. The end-of-time step velocity for each of the two particles is speciﬁed as:

1 u(p) (t + t) = u(pq) (t) + ηu(pq) (t) 2 1 u(q) (t + t) = u(pq) (t) − ηu(pq) (t) 2

(6.66)

In these equations, the term u(pq) is the average beginning-of-time step velocity for the pair, u(pq) (t) is the beginning-of-time step velocity magnitude difference for the pair, and η is a unit vector of random orientation. In the case of non–equalmass particles, mass-weighting should be used. By adding and subtracting the last two equations, it can be seen that this model leaves u(pq) and u(pq) unchanged. Thus, the mean velocity and the turbulent kinetic energy remain unchanged. The Reynolds stresses evolve according to this equation: % $ ! d ui uj " # 2 (6.67) = − (CR − 1) ω ui uj − kδij dt 3 Equation 6.67 corresponds to Rotta’s linear return-to-isotropy model for the Reynolds stresses (Rotta, 1951a, b). The value CR = 1.5 was suggested in the Reynolds-stress closure proposed by Launder, Reece, and Rodi (1975). Modern models for Lagrangian particle velocities usually are based on stochastic differential equations (SDEs): speciﬁcally, a stochastic diffusion process represented by a Langevin equation (see Section 5.14.3 in Chapter 5). In this model, each ﬂuid particle evolves independently; interactions between particles are weak and occur only through mean quantities that are estimated using particle properties. SDE-based models have desirable analytic properties and advantages in practical implementation compared to particle-interaction models. For these reasons, particle-interaction models are no longer widely used for particle velocities in velocity pdf or velocity-composition pdf methods; however, they remain popular for scalar-mixing models.

PROBABILITY DENSITY APPROACH

6.8.3

469

Molecular Transport and Scalar Mixing Models

Modeling mixing in particle implementations of pdf methods involves prescribing the evolution of stochastic/conditional particles in composition space such that they mimic the change in the composition of a ﬂuid particle due to mixing in a turbulent reactive ﬂow. Molecular mixing is the single most important modeling question in the pdf approach. Within the context of this framework, many different strategies have been adopted, including: • Interaction by exchange with the mean (IEM) (Villermaux and Devillon, 1972) • Linear mean-square estimation (LMSE) (Dopazo and O’Brien, 1974a) • Modiﬁed curl (MC) (Dopazo and O’Brien, 1976; Janicka, Kolbe, and Kollmann, 1979) • Euclidean minimum spanning tree (EMST) (Subramaniam and Pope, 1998) A common element in all of these approaches is a mixing time scale. Note that the model predictions are dependent on the choice of the time scale, and different choices are appropriate for different problems. Calculations have been performed by different groups, in particular especially by Tang, Xu, and Pope (2000); by Xu and Pope (2000); and by Lindstedt, Louloudia, and V´aos (2000). Normally, the time scale is assumed to be the same for each different scalar, and of the same order of magnitude as the large-scale turbulence time scale (k /ε). In ﬂames, differential diffusion and the strong interplay with mixing and reaction might degrade these assumptions. It is difﬁcult to assess these assumptions directly in a posteriori tests, and measurements of reacting scalar mixing are not yet possible. The rate of scalar variance decay in high-Reynolds-number turbulence is controlled by the large scales. In the absence of a model for the scalar gradients or energy transfer in the scalar spectrum, it is reasonable to assume that the scalar variance decay rate is proportional to the mean turbulent frequency. The evolution of the scalar pdf, however, is determined by turbulent motions down to the smallest scales. Therefore, the scalar pdf evolution is difﬁcult to model without explicit characterization of the small scales. Accounting for the effect of the small scales is also the natural way to include the effects of species diffusivity and differential diffusion. However, the description of these processes necessitates a higher level of closure and more computational expense. A mixing model should have three most desirable qualities: 1. Mean scalar quantities should not change as a result of mixing. 2. Scalar variances should decay at the correct rate. 3. Scalar quantities should remain bounded (e.g., mass fractions should remain between zero and unity and should add to unity). Other desirable properties are that the mixing models should be consistent with the linearity. Also, the independence properties should cause the pdf of

470

NON-PREMIXED TURBULENT FLAMES

conserved scalars to relax to a joint normal Gaussian distribution in statistically homogeneous systems and should not violate the concept of locality in composition space. The concept of linearity and independence of scalar variables has been detailed in an article by Pope (1982b). Most mixing models do not address differential diffusion and do not include a direct inﬂuence of chemical reaction on mixing. Independence During the mixing of a set of conserved passive scalars (e.g., mixture fraction or certain combinations of species mass fraction or temperature), the evolution of the statistics of any one of these scalar ﬁelds ζα is unaffected by the statistics of any of the other scalar ﬁelds ζβ (where β = α). This is the independence principle proposed by Pope (1983a). The implication of this concept for particle models of multiscalar mixing is that the evolution of the particle property of one scalar ζα should not depend on the particle properties (or statistics) of any of the other scalars ζβ (where β = α). For moment models of multiscalar mixing, the implication is that the evolution of any statistics involving one scalar ζα should not depend on the statistics of any of the other scalars ζβ (where β = α). Localness A mixing model must be local in composition space. For particle interaction models, this means that particles should mix with other particles in their immediate neighborhood in composition space. The reason for the failure of certain mixing models in the diffusion ﬂame study is that they are not local in composition space, resulting in mixing of particles across the reaction zone. Particle interaction mixing models are characterized for the evolution of a conserved scalar ﬁeld ζk , and they can be written in this form: N p dζk 1 & k q Mpq ζk =− dt w(p)

(6.68)

q=1

where N equals the total number of notional particles. The matrix Mijk represents the interaction between particles p and q for the k th scalar. In practice, the same matrix describes the interaction for all the scalars, but this is not required based on the linearity principle described by Pope (1983). The linearity principle requires that Mijk must be independent of k , at least for conserved passive scalars with equal diffusivities. The term w(p) is the weighting factor applied to particle p. For localness, the bandwidth of the interaction matrix should be small. For easier understanding, let us consider a single scalar in a onedimensional space. In such a case, the particles are numbered in order of their distance from each other. For example, particle numbered 100 is much farther from particle numbered 1 than the particle numbered 50. Therefore, the difference in the indices of particles represents the distance between them. Thus, a smaller bandwidth represents mixing of particles with particles only in their immediate neighborhood. For the independence property of a mixing model, the off-diagonal terms in the interaction matrix should be zero. This simple concept of ordering becomes harder to implement with multiple scalars.

PROBABILITY DENSITY APPROACH

471

6.8.3.1 Interaction by Exchange with the Mean Model The simplest model for molecular mixing in pdf methods is the interaction by exchange with the mean (IEM) model (Villermaux and Devillon, 1972). The linear mean-square estimation (LMSE) model by Dopazo and O’Brien (1974) is nearly equivalent to the IEM model. In the IEM model, particle compositions relax toward the local mean composition on the time scale τζ of Equation 6.49. The molecular mixing term in both velocity-composition joint pdf and composition pdf equations can be modeled as: ˜k ψ − ζ k ∂Jk,i 1 ∂Jk,i 1 V, ψ = ψ = − ρ (ψ) = − ρ (ψ) Cζ ω ψk − ζ˜k ∂xi ∂xi 2 τm,ζ 2 (6.69) The particle composition time rate of change vector can be modeled as: ψk − ζ˜k 1 1 ∗ (6.70) =− = − Cζ ω ψk − ζ˜k θk,mix 2 τm,ζ 2

Similarly, the scalar dissipation rate is modeled as: χζ =

1 ζk2 1 = Cζ ωζk2 2 τm,ζ 2

(6.71)

In these equations, Cζ is a model constant. The value of Cζ is usually taken as 2.0, although different values have been used depending on the speciﬁc formulation and model. Ideally, Cζ should be a universal constant. Note that the mixing time scale used in Equation 6.71 used 1/2 multiplied to the ratio of scalar variance to the mean scalar dissipation rate, which is different from the deﬁnition of mixing time given in Equation 6.49. IEM possesses the three most essential characteristics of a mixing model. However, IEM preserves the shape of the pdf (rather than causing the pdf to relax toward a Gaussian) and does not include an explicit dependence on velocity in the case of a velocity-composition joint pdf. Variants that include a dependence on velocity have been proposed by Pope (1998) and Fox (1996a); the interaction by exchange with the conditional mean (IECM) model was introduced in 1998 and is reviewed by Viswanathan and Pope (2008). In a numerical implementation for inhomogeneous systems, it is important to evaluate the local mean in a way that ensures that the mean does not change as a result of mixing. Other extensions to IEM have been proposed using analysis based on Fokker-Planck equations for the velocity-composition pdf by Heinz (2003). 6.8.3.2 Modiﬁed Curl Mixing Model The simplest of the direct particle interaction models is Curl’s model (1963) or the coalescence-dispersion model. This pair-exchange model is essentially the same as the stochastic mixing model that was discussed in Section 6.8.2.1 for particle

472

NON-PREMIXED TURBULENT FLAMES

velocities. The probability of a pair of particles interacting in a time interval dt is now Cζ Nω dt, and the particle velocity components in Equation 6.61 are replaced by particle scalar compositions. The scalar-dissipation-rate model that is implied by Curl’s model is the same as the one that is implied by IEM (Equation 6.69). In this case, particle compositions vary with time in a discontinuous manner. Curl’s model also possesses the three most essential characteristics of a mixing model. In contrast to IEM, the shape of the scalar PDF is not preserved by Curl’s model, although it still does not relax toward a Gaussian distribution. In fact, Curl’s model yields noncontinuous pdfs. Modiﬁed versions that yield continuous (but still non-Gaussian) distributions have been devised by introducing an additional random parameter that controls the extent of mixing on each interaction event; this variant is often referred to as the modiﬁed Curl (MC) mixing model (Dopazo and O’Brien, 1976; Janicka, Kolbe, and Kollman, 1979). The MC mixing model is a particle interaction model. The pairs of model particles p and q with equal weights are randomly selected from the ensemble, and their compositions are changed over the time interval dt by the next equations: 1 q p p p ζk (t + dt) = ζk (t) + η ζk (t) − ζk (t) 2 1 q q q p ζk (t + dt) = ζk (t) − η ζk (t) − ζk (t) 2

(6.72)

where η is a random number uniformly distributed in (0, 1). In this way, the model mimics the particle molecular diffusion process. The number of particles to be mixed at each time step is determined by Mitarai, Riley, and Kos´aly (2005) as Nmix = 3ω(t)Ndt, where N is the total number of model particles. A variant where particle compositions evolve continuously was proposed by Hsu and Chen (1991). Many other variations of the basic model have been proposed. These include age biasing, where the probability of a particle being selected for mixing depends on the time from its last participation in a mixing event (Pope, 1982a). Age biasing has been shown to result in distributions that relax to near Gaussian for passive scalars in homogeneous systems. Explicit dependencies on particle velocities also can be included for velocity-composition PDF methods (Pope, 1985; Bakosi, Franzese, and Boybeyi, 2008). For the case of unequal-weight particles, an implementation of MC has been developed by Nooren et al. (1997). This implementation conserves the mean and causes the correct variance decay. However, the effect on the pdf depends on the distribution of particle weights. 6.8.3.3 Euclidean Minimum Spanning Tree Model The Euclidean minimum spanning tree (EMST) mixing model (Subramaniam and Pope, 1998) is a more complicated particle interaction model, designed to overcome shortcomings of simpler turbulent mixing models, such as the IEM

PROBABILITY DENSITY APPROACH

473

and the modiﬁed Curl models. In addition to the particle composition, the model involves a state variable. The model consists of three steps. 1. Based on the values of the state variable, “live” particles that are to be mixed are selected. 2. The EMST is formed to connect live particles so that the total length of the tree is minimum. 3. Pairs of particles denoted by p and q connected by the nth branch of the tree mix over the time interval dt by q p p p ζk (t + dt) = ζk (t) + bBn dt ζk (t) − ζk (t) q q q p ζk (t + dt) = ζk (t) − bBn dt ζk (t) − ζk (t)

(6.73)

Here the value of Bn is determined based on the position of the branch in the tree (i.e., the closer to the center of the tree, the larger the value of Bn ). Coefﬁcient b is determined so that the desirable mixing frequency (i.e., ω), is obtained. EMST possesses the three most essential characteristics of a mixing model mentioned earlier and has the property of localness. It does not, however, satisfy the independence and linearity properties. Both IEM and Curl’s model (including variants) do not perform well at high Damk¨ohler numbers. Their main drawback is that they are not local in composition space. In non-premixed combustion with fast chemistry and initial equilibrium condition, the reaction is conﬁned to reaction sheets. This drawback can result in the unphysical mixing of cold fuel and oxidizer across the reaction surface. Such behavior is observed because the IEM (and other) mixing models violate the physics of mixing: namely, that it is the composition ﬁeld in the neighborhood (in physical space) of a ﬂuid particle that inﬂuences its mixing. Since the composition ﬁelds encountered in reality are smooth, this neighborhood in physical space corresponds to a neighborhood in composition space. It follows that if a mixing model is to perform satisfactorily in a diffusion ﬂame test problem (which is representative of the coupling of reaction and mixing in non-premixed combustion), the model should reﬂect the fact that the change in composition due to mixing is inﬂuenced by the neighborhood in composition space. This motivates inclusion of a new principle that mixing models should satisfy: localness in composition space. For any turbulent reactive ﬂows, there exists a region in composition space called the realizable region within which any point corresponds to a possible composition value that a ﬂuid particle may attain. Points in composition space that lie outside the realizable region correspond to compositions that cannot occur and have no physical meaning (e.g., if the composition variables represent species mass fractions, these may correspond to negative mass fractions or mass fractions greater than unity). Consequently, mixing models should preserve the boundedness of compositions. The goal of this

474

NON-PREMIXED TURBULENT FLAMES

modeling effort is to provide a phenomenological description of mixing (without explicit representation of the physical processes associated with the small scales) while satisfying the important principles of boundedness and localness. In this case, molecular transport and chemical reaction are tightly coupled, and combustion may correspond to a ﬂamelet regime. In ﬂamelet combustion, reactive scalar gradients are steepened by reaction, and it is not appropriate to express the scalar dissipation rate χζ solely as a function of the local scalar variance (see Equation 6.71), k , and ε since these are integral-scale quantities. The standard models can allow particles to mix across the reaction zone without burning. In a non-premixed system with fast chemistry, for example, a particle corresponding to pure fuel may mix with a particle corresponding to pure oxidizer yielding two particles that correspond to mixed but unreacted fuel and oxidizer. For numerical simulation of the above-mentioned particle-based pdf models, operator-splitting algorithms are used, which worsen this effect (Haworth, 2009). The high-Damk¨ohler-number problem has been recognized for many years, and several approaches have been proposed and tested to address it. Pope (1985) noted that for laminar ﬂamelet combustion, the molecular mixing and chemical source terms in the pdf equation can be combined, and the combined term appears in a closed form. Turbulent premixed ﬂames in ﬂamelet and distributed reaction regimes have been studied using this approach (Pope and Anand, 1984; Anand and Pope, 1987). A reaction-zone conditioning scheme based on the value of a mixture fraction was proposed for non-premixed ﬂames in T.H. Chen, Goss, Taney, and M. Kolaitis (1989); Dibble and Lucht (1989). A third approach for obtaining tighter reaction/diffusion coupling in particle-interaction mixing models is to bias the pairing of particles for mixing such that particles that are close to one another in composition space are more likely to mix than particles that are far apart in composition space. This is the notion of locality that was mentioned earlier. An early effort in this direction was the ordered pairing approach proposed in Norris and Pope (1991). EMST is a more general procedure developed by Subramaniam and Pope (1998). Recently, Meyer and Jenny 2009a and 2009b have proposed a parameterized scalar proﬁle (PSP) mixing model, which is able to provide the joint statistics of mixture fraction and scalar dissipation rate. If thin reaction zones are present, mixture fraction and scalar dissipation rate determine the composition and thermal state in the ﬂow domain. This mixing model is based on the argument that in the transported pdf context, the commonly used mixing models do not provide such statistics even though pdf methods have been applied with some success for the simulation of ﬂames involving local extinction. Figure 6.36 shows the scatter plots of temperature against mixture fraction Z , which are obtained with τres = 2 × 10−3 s, τmix /τres = 0.35, and unity equivalence ratio. The lines in the scatter plots correspond to chemical equilibrium. Note from Figure 6.36a that the reaction zone in mixture fraction space is from about 0.24 to about 0.5. The scatter below the equilibrium line in the reaction zone corresponds to incompletely burned ﬂuid particles or extinguished ﬂuid particles. Figure 6.36 shows the qualitatively different behavior of the three mixing models. For the IEM model, Figure 6.36a is consistent with this picture: Particles

PROBABILITY DENSITY APPROACH 2500

2500 IEM Z′ = 0.23

1500

1000

500

500 0

0.2

0.4

0.6

0.8

0

1

0

0.2

0.4

0.6

Z

Z (b)

0.8

1

12 10

EMST Z′ = 0.25

2000 1500 1000

P =1 atm Inside the reactor, reaction occurs and the mean thermochemical properties are assumed to be statistically spatially homogeneous, but the fluid is imperfectly mixed at the molecular level.

IEM MC EMST

8 PDF

T (K)

oxidant stream (N2 and O2, 79:21 by volume, T = 300 K).

(a) 2500

6 4

500 0

Two inflow streams are the fuel stream (H2 and N2, 1:1 by volume, T = 300 K) and the

1500

1000

0

Simulation conditions: MC Z′ = 0.23

2000 T (K)

T (K)

2000

475

2 0 0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

Z

Z

(c)

(d)

0.8

1

Figure 6.36 Scatter plots of temperature versus mixture fraction with (a) IEM, (b) MC, (c) EMST models, with the solid line corresponding to chemical equilibrium, and Z representing the rms ﬂuctuations of Z and (d) corresponding pdfs of the mixture fraction (modiﬁed from Ren and Pope, 2004).

corresponding to composition values outside the reaction zone relax to the mean composition and are drawn away from their initial condition on the equilibrium line; particles in the reaction zone react close to their equilibrium values due to fast reactions. It is clear that particles do not all lay close to the equilibrium line, and the model fails to reproduce the expected physical behavior in this case by producing some results close to the extinction point (i.e., the low values of T ) due to unphysical micromixing process. Figure 6.36b shows that the MC model mixes cold fuel with cold oxidant to produce cold, nonreactive mixtures that are within the reaction zone in mixture-fraction space. Clearly, this is physically incorrect in this case. Figure 6.36c shows that all compositions given by the EMST model for this case are close to equilibrium. So Figure 6.36 shows that the EMST mixing model produces the expected physical behavior, whereas the IEM model and MC model do not. The corresponding mixture fraction pdfs are also shown in Figure 6.36d, and they are also quite different from each other. Mixing models should remain an active area of research for some time, since they are a crucial element in pdf methods in both RANS and LES approaches, and current models have several well-appreciated shortcomings. There are also questions to be answered about the performance of the existing models. In nonpremixed turbulent ﬂames, whether ﬁnite-rate chemical effects are signiﬁcant depends on the Damk¨ohler number, Da. At high Da, simple models based on equilibrium chemistry or steady laminar ﬂamelets can be successful. But as

476

NON-PREMIXED TURBULENT FLAMES

Da decreases; departures from equilibrium and ﬂamelet behavior become pronounced, and local extinction and eventually global extinction can occur. In nonpremixed turbulent combustion, extinction and ignition events are associated with large and small values of the scalar dissipation, respectively. Scalar dissipation rates are inversely proportional to the Damk¨ohler number (see Equation 3.75). An ideal mixing model should be able to accurately calculate the molecular mixing irrespective of scalar dissipation rate or the range of Damk¨ohler number.

6.9

FLAMELET MODELS

The concept of ﬂamelet modeling for diffusion ﬂames has been discussed at length in Section 3.4 of Chapter 3. Here, we discuss the application of this concept in turbulent non-premixed combustion. The fundamental idea behind ﬂamelet models is the separation of the numerical solution of the turbulent ﬂow and distribution of scalars (e.g., species mass fraction, mixture fraction, scalar dissipation rate, etc.) in space/time from the solution of the chemistry. Turbulence times are very large in comparison with molecular collision times, and there is no a priori reason why a detailed chemical kinetic mechanism for a turbulent ﬂame should be different from a mechanism that is appropriate for a laminar ﬂame under similar thermochemical conditions. Unfortunately, this amount of detail is almost always impractical in turbulent combustion calculations. Many models are unable to accommodate more than a single, global irreversible rate expression. Some go further and assume the chemistry to be inﬁnitely fast so that reactants, or unburned mixture and hot products, react instantaneously when they are mixed. Applications of systematically reduced reaction schemes in turbulent combustion have also been performed (Peters and Rogg, 1993; Taing, Masri, and Pope, 1993). Another way to incorporate detailed chemical kinetics is through laminar ﬂamelet models (Bray and Peters, 1994; Peters, 1986) in which kinetic information is parameterized in terms of laminar ﬂame properties, such as the laminar burning velocity, Markstein number, ﬂame stretch at extinction, and others. The structure of a laminar ﬂame represents a balance between chemical reaction and molecular diffusion. It follows that, if laminar ﬂame structures occur in turbulent combustion, the mean reaction rate must be inﬂuenced by molecular diffusion processes and hence by the small-scale structure of the turbulent ﬂame. Reaction and diffusion are then coupled. Applying the ﬂamelet approach, transport equations for the moments of a conserved variable (i.e., mixture fraction) are solved. Turbulent mean values of the mass fractions of chemical components can then be calculated by using a presumed pdf of the mixture fraction, whose shape is determined by its statistical moments. Besides the pdf, the only requirement of the model is that there exists locally a unique relation between the mixture fraction and all scalar quantities, such as the species mass fractions and enthalpy. A universal coordinate transformation of the governing equations using the mixture fraction as independent coordinate yields the required relationship between the species concentrations

FLAMELET MODELS

477

and the mixture fraction in terms of the ﬂamelet equations. If is assumed that the ﬂamelet is relatively thinner in the direction normal to its surface, we can express the ﬂamelet equation in one-dimensional time-dependent form:

χ ∂ 2T ∂T − Z ∂τ 2 ∂Z 2

=

ω˙ T ρCp

(6.74)

where χZ is called the instantaneous scalar dissipation rate, and it is deﬁned by the next equation: χZ ≡ 2D

3 & ∂Z 2 i=1

∂xi

=2D

∂Z ∂x1

2 +

∂Z ∂x2

2 +

∂Z ∂x3

2 (6.75)

The dimension of χZ is 1/s. The scalar dissipation rate is an important parameter for analysis of diffusion ﬂames. By incorporating mixture fraction Z as a coordinate, we have included the inﬂuence of transport processes in the direction normal to the surface of the stoichiometric mixture in Equation 6.74. Solutions of the ﬂamelet equations are shown for several example problems in Section 3.4 of Chapter 3. At sufﬁciently high Reynolds and Damk¨ohler number, the local structure of a turbulent ﬂame may essentially be that of a laminar ﬂame subjected to the same aero-thermo-chemical conditions (see Figure 6.5). Flamelet models have been formulated for premixed, non-premixed, and/or partially premixed combustion regimes. There are two major approaches in order of increasing complexity: 1. Laminar ﬂamelet assumption. The local structure of the turbulent ﬂame corresponds to that of a steady-state strained one-dimensional laminar nonpremixed ﬂame subjected to the same instantaneous scalar dissipation rate as in the turbulent ﬂow. 2. Unsteady ﬂamelet modeling. The local structure of the turbulent ﬂame corresponds to that of an unsteady one-dimensional laminar non-premixed ﬂame subjected to the same aero-thermo-chemical history as in the turbulent ﬂow. 6.9.1

Laminar Flamelet Assumption

Although laminar ﬂamelet models in non-premixed combustion have been applied successfully for predictions of mean values for temperature and species concentrations (Haworth, Drake, and Blint, 1988; Rogg, Li˜na´ n, and Williams, 1986) and formation of pollutants such as NOx (Sanders, Sarh, and G¨okalp 1997) and soot (Moss, 1966), some aspects of laminar ﬂamelet modeling in turbulent combustion are still unclear; for instance, the modeling of transient effects has not been well understood as yet. The radiation can be neglected and that the ﬂame structure is hardly inﬂuenced by transient effects in case of a statistically steady

478

NON-PREMIXED TURBULENT FLAMES

turbulent, nitrogen-diluted hydrogen air diffusion ﬂame (Pitsch, Fedotov, and Ihme, 2004). However, for predictions of slow processes, such as the formation of NO, unsteady effects have to be considered. In LFA, the transient term in the ﬂamelet equations is considered negligible and the resulting ﬂamelet equation reduce to: d 2T 2ω˙ T + =0 (6.76) 2 dZ ρCp χZ 6.9.2

Unsteady Flamelet Modeling

The deﬁciency of the steady-state ﬂamelet approach is due to the strong decay of the scalar dissipation rate along the jet axis and also by radiative heat losses. The scalar dissipation rate, which can inﬂuence the ﬂamelet solution signiﬁcantly, decreases with x −4 along the axis, where x is the distance from the jet exit. Therefore, a ﬂamelet that is transported downstream has to undergo strong changes. It has been shown experimentally and in simulations of unsteady ﬂamelets (Barlow and Chen, 1992; Mauß, Keller, and Peters, 1990) that the chemical ﬂamelet structure cannot follow rapid changes of the scalar dissipation rate instantaneously. Therefore, transient effects have to be included for to describe diffusion ﬂame stabilization. There are two characteristic times in unsteady ﬂamelet approach. The unsteady ﬂamelet can be calculated as a function of the ﬂamelet time in the Lagrangian frame of reference, which can be related to the distance from the jet exit by the next equation: x 1 τﬂamelet ≡ dx (6.77) u (x )|Z=Z ˜ st 0

The other characteristic time is the mixing time associated with scalar dissipation rate of the mixture fraction, τm,Z : Z 2 τm,χ ≡ (6.78) χZ The mixing time (sometimes also called diffusion time) is the time required to exchange mass and energy over the ﬂame thickness in the mixture fraction space. If the mixing time is smaller than the ﬂamelet time, the ﬂamelet can follow the changes in the scalar dissipation rate rapidly and the unsteady term in the ﬂamelet equations can be considered negligible. Pitsch, Fedotov, and Ihme (1998) determined that this condition is applicable up to 30 jet exit diameter in a piloted jet ﬂame (called an H3 ﬂame at Darmstadt University). Farther downstream, the scalar dissipation rate rapidly decreases, and the diffusion time becomes larger than the Lagrangian ﬂamelet time. In this region, the unsteady term should be retained in the ﬂamelet equation to correctly predict the NOx formation process, even though the rest of the chemical reactions are already in chemical equilibrium. The results of the numerical simulations by Pitsch, Fedotov, and Ihme (1998) with implementation of the unsteady ﬂamelet model and LFA in comparison with

479

FLAMELET MODELS 2000

3500

1500 ~ XOH

500

0

1000 500

0

20

40

60

Temperature (K)

Temperature (K)

2000 1000

1200

2000

800 600

~ XOH

200

1

2

3

30 NO Mole Fraction [ppm]

~ YH O 2

~ YO 2

~ YH 2

0.05

~ 0.2 × XNO

25 Post-processed unsteady flamelet (dash-dot line)

20 15 10 5 0

0 0

20

40

5

(b)

0.2

0.1

4

r/Djet

(a)

Mass Fraction

1000 0

0

x/Djet

0.15

3000

x/Djet = 20

1000

400

0 100

80

4000

1400

OH Mole Fraction [ppm]

2500

OH Mole Fraction [ppm]

~ T

1500

5000

~ T

1600 3000

60

80

x/Djet

(c)

100

0

20

Unsteady flamelet with thermal NO 40 60 x/Djet

80

100

(d)

Figure 6.37 Numerical results of Favre-averaged mean values of (a) T˜ and X˜ OH along the centerline; (b) radial distribution of T˜ and X˜ OH , (c) Y˜H2 , Y˜H2 O , and Y˜O2 at r = 0; and (d) X˜ NO at r = 0; (solid lines) unsteady ﬂamelet modeling, (dashed lines) LFA, (symbols) experimental data (modiﬁed from Pitsch, Fedotov, and Ihme, 2004).

the experimental data of Favre-averaged values of temperature, mass fractions of H2 , O2 , and H2 O, and mole fractions of OH and NO are shown in Figure 6.37. Comparison of these results indicates that transient effects and radiation are not important for predictions of OH concentrations in the investigated ﬂame.

6.9.3

Flamelet Models and PDF

The accommodation of ﬂamelets in pdf methods has been performed in various ways. One approach is to solve a modeled pdf transport equation for a reduced set of composition variables (e.g., a mixture fraction and/or a small number of reaction progress variables) and to relate the local thermochemical state to this reduced set using precomputed or interactive canonical laminar ﬂamelets. For example, a transported velocity-mixture fraction joint pdf and a presumed scalar-dissipation-rate pdf were used by Haworth, Drake, Pope, and Blint (1988) with a precomputed library of stretched laminar diffusion ﬂames to model a non-premixed turbulent CO/H2 /N2 –air ﬂame. The approach can be extended to account for unsteady laminar ﬂamelets.

480

NON-PREMIXED TURBULENT FLAMES

TABLE 6.7. Summary of modeling approaches for non-premixed turbulent combustion (modiﬁed from Bilger et al., 2005) Paradigm

Chemistry

Mixing Treatment

Mixing controlled combustion Laminar ﬂamelet

Fast for major species

Scaling laws/second-order closures

Precalculated tables/ Lagrangian PDEs Already closed/reduced mechanisms Closure in terms of conditional moments Closure in terms of conditional means

Laminar counter-ﬂow/second-order closure Mixing models

PDF approach Conditional moment closure Multiple mapping conditioning

PDF integrals Mapping closure

A second approach is to combine a standard ﬂamelet model and a pdf method into a hybrid model. A hybrid model was devised for premixed and partially premixed turbulent combustion. In this approach, a laminar ﬂamelet model governs the turbulent ﬂame propagation and primary heat release, while a composition pdf or velocity-composition pdf method can be used in front of and behind the primary ﬂame. A reaction progress variable can be carried on each computational particle to mark whether that particle is in front of or behind the ﬂame, and the changes in particle properties across the ﬂame can be taken from a precomputed laminar ﬂamelet library. A third approach is to explicitly build the ﬂamelet structure into the pdf formulation and modeling. To do this, the chemical source term and the molecular transport term in the pdf transport equation can be combined into a single term. In the case of a composition pdf, see Equation 6.60. A fourth approach has been implemented by Hulek and Lindstedt (1996). In this approach, the molecular transport term is not combined with the chemical source term. Instead, only the chemical source term is taken from the underlying laminar ﬂame structure, and a binomial Langevin mixing model (Valino and Dopazo, 1991) is used. (The generalized Langevin model has been discussed in Section 5.14.3 of Chapter 5). This mixing model should not be used in an approach where the small-scale gradients associated with laminar ﬂamelets are explicitly combined with the source term (Haworth, 2009). A summary of the modeling approaches for turbulence mixing and chemistry is given in Table 6.7. 6.10

INTERACTIONS OF FLAME AND VORTICES

Flame-vortex interactions constitute a basic problem in the analysis of turbulent combustion. Turbulence is an organized motion at the largest scale, which is superposed on a ﬁne-grain random background of ﬂuctuations in the small

INTERACTIONS OF FLAME AND VORTICES

481

scales as established by theoretical concepts and experimental evidence. Some examples include, experiments carried out on plane shear layers (G.L. Brown and Roshko, 1974) and studies on jets and wakes (Cantwell, 1986). Turbulence can be described as a collection of vortices of every size and strength. Therefore, turbulent combustion can be viewed as a process dominated by the continuous distortion, stretching, production, and dissipation of the ﬂame surface by vortices of different scales. Thus, ﬂame-vortex interaction can be utilized as a model for investigation of turbulent combustion. Flame-vortex interactions often govern the combustion rate and lead to combustion instabilities. For example, vortex roll-up is found to be one of the major driving mechanisms of combustion instabilities. In the case of supersonic combustion, mixing can be enhanced by generating vortices that entrain the fresh fuel into the hot oxidizer stream ﬂowing into the combustor (Marble, 1994). Vortices are observed in free ﬂames or when a ﬂame travels down a duct and encounters bluff obstacles. Studies in turbulence carried out during the last 40 years indicate that mixing is controlled to a great extent by vortex motion, especially by the large-scale vortices developing in the highly sheared regions of the ﬂow. The elementary interaction between a vortex and a ﬂame thus appears as a key process in the description of turbulent combustion. Vortex structures also arise when a ﬂame traveling in a duct interacts with bluff obstacles, a mechanism that may lead to signiﬁcant levels of ﬂame acceleration. A starting vortex is also observed when jets or plumes are formed by a sudden injection or expansion of a mass of gases into a quiescent medium. The jet features a characteristic mushroom vortex cap. When dealing with supersonic ﬂows, early experiments on supersonic combustion showed that natural mixing was slow and that combustors could not operate in the supersonic range without some method of mixing enhancement. One possible scheme for improving the rate of span-wise mixing relies on vorticity created by the interaction between weak shock waves and the hydrogen stream. This process has been investigated extensively in relation to developments of supersonic ramjet combustors (Marble, 1994). The mushroom shapes indicate the interface between the injected fuel and the oxidizer ﬂow. Some typical devices featuring ﬂame/vortex interactions are shown in Figure 6.38. In this ﬁgure, part a shows vortex generation due to highfrequency “screech” instability (Rogers and Marble, 1956); part b shows lowfrequency instabilities in backward-facing step geometry (Keller et al., 1982); part c shows low-frequency instabilities of a dump combustor with an abrupt change in combustor contour; part d shows large-scale vortices in a premixed shear layer (Ganji and Sawyer, 1980); part e shows vortex-driven instability in a multiple-jet dump combustor (Poinsot et al., 1987); part f shows vortex-driven oscillation in a single-jet dump combustor (Yu, Trouv´e, and Candel, 1991); part g shows organized vortex motion in a premixed ducted ﬂame modulated by plane acoustic wave (Yu, Trouv´e, and Candel, 1991); part h shows control of a dump combustor using organized vortices (Yu, Trouv´e, and Candel, 1991); and part i shows vortex motion during the injection phase of a pulse combustor (Barr et al., 1990).

482

NON-PREMIXED TURBULENT FLAMES

(a)

(b)

(c)

inlet plane Premixed reactants hot products inlet

window (e)

(d)

spark plug

nozzle (f) combustion chamber tail pipe

fuel + air flame stabilizer

flapper valves

(g)

(i)

(h)

Figure 6.38 Vortex structures in various combustors (after Renard, Th´evenin, Rolon, and Candel, 2000).

A basic issue in ﬂame/vortex interactions studies is to deﬁne the ﬂow conﬁgurations. Theoretical studies are restricted to simple geometries that are not always close to reality, but they give insight into the basic mechanisms. An experimental setup should allow an easy optical access and a good repeatability. Various conﬁgurations corresponding to different applications are summarized in the next subsections. 6.10.1

Flame Rolled Up in a Single Vortex

Flames rolled up in a single vortex structure growing along the interface between fuel and oxidizer (or fresh and burnt gases) constitute a fundamental geometry (Figure 6.39). It corresponds to a situation where analytical studies are possible. These conﬁgurations also have been simulated for validation purposes. They have led to interesting information concerning the dynamics of ﬂame ignition Fuel (or fresh gas)

Vortex Flame front Oxidizer (or burned gas)

Figure 6.39 2000).

Flame rolled up in single vortex structure (modiﬁed from Renard et al.,

INTERACTIONS OF FLAME AND VORTICES

483

Flame front Fuel (or fresh unburned gas)

Oxidizer (or partially burned gas)

Figure 6.40 Flame-vortex interaction in a developing shear layer (modiﬁed from Renard et al., 2000).

and spreading. In this conﬁguration, the velocity ﬁeld is considered to be caused solely by the vortex structure, and the computations give the time evolution of the ﬂame.

6.10.2

Flame in a Shear Layer

As a further step, the interaction between ﬂames developing in a two-dimensional shear layer and the vortical structures naturally responsible for the growth of this layer has been investigated (Figure 6.40). In this case, the spatial evolution of the ﬂame is of special interest. Several vortical structures are present simultaneously. Mutual interactions between these structures and their interactions with ﬂame increase the complexity of the ﬂame-vortex interaction in the turbulent ﬂow ﬁeld. This situation is closer to realistic conditions. There have been numerous investigations of coherent motion in reactive shear layers.

6.10.3

Jet Flames

Organized vortex motion has been studied in the case of jet ﬂames (Figure 6.41). As the evolution of large-scale structure is governed by Kelvin-Helmholtz instabilities, the frequency, mutual interaction, and energy distribution among various

Oxidizer (or partially burned gas)

Flame front

Fuel (or fresh unburned gas)

Oxidizer (or partially burned gas)

Figure 6.41 Flame-vortex interaction in a jet ﬂame (modiﬁed from Renard et al., 2000).

484

NON-PREMIXED TURBULENT FLAMES

length-scales is controlled by the initial conditions of the ﬂow. Combustion processes strongly modify the instability mechanisms, but coherent structures similar to those observed in nonreacting ﬂows may be generated in an annular jet diffusion ﬂame by forcing it at a preferred mode frequency. The same instability process that ampliﬁes the axisymmetric disturbances and generates these structures ampliﬁes higher modes of instability and introduces directional dependence in the ﬂow. In the case of an axisymmetric conﬁguration, the azimuthal instability modes, which may be ampliﬁed and interact with each other, can form an inﬁnite set. This makes the analysis of the modulated jet ﬂame conﬁguration a challenge to study. In such cases, the inﬂuence of the vortical structures on the developing ﬂame has usually been the subject of investigation. 6.10.4

K´arm´an Vortex Street/V-Shaped Flame Interaction

The previous conﬁgurations are not easily described in terms of vortex structures, which interact with the ﬂame front. The interaction takes place while the vortices develop. The vortex pattern is inﬂuenced by the presence of the ﬂame as it is being generated. This problem can be avoided by considering the interaction of a V-shaped ﬂame with a K´arm´an vortex street (Figure 6.42). The vortex street develops externally, and the vortex row subsequently interacts with the ﬂame. 6.10.5

Burning Vortex Ring

One early experiment on ﬂame/vortex interactions concerns the propagation of a ﬂame front in vortical structure of premixed reactants (Figure 6.43). This

Kármán vortex street

Vortex generator

Flame front unaffected by vortex generator

Flame stabilizer

Figure 6.42 Interaction of V-shaped ﬂame with K´arm´an vortex street (from Renard et al., 2000).

INTERACTIONS OF FLAME AND VORTICES

485

Vortex ring

Mushroom-shaped flame front

Fuel (or fresh gas)

Figure 6.43 Development of ﬂame rolled up in a vortex ring (modiﬁed from Renard et al., 2000).

conﬁguration may be used in both premixed and non-premixed cases for investigation of ﬂame/vortex interaction.

6.10.6

Head-on Flame/Vortex Interaction

The need to study a very simple conﬁguration is well fulﬁlled in a head-on interaction between an initially ﬂat ﬂame and a counterrotating vortex pair or a vortex ring. This initially ﬂat (or approximately ﬂat) ﬂame is a one-dimensional unstrained ﬂame, a counterﬂow ﬂame, or a V-shaped ﬂame (Figure 6.44). The Fuel (or fresh unburned gas) Flame front

Flame front

Flame front Oxidizer (or partially burnt gases)

Vortex pair

Vortex pair (a)

Fuel (or fresh gas) (b)

Figure 6.44 Interaction between (a) a ﬂat ﬂame and a counterrotating vortex pair or toroidal vortex ring structure and (b) a V-shaped ﬂame and a vortex pair (from Renard et al., 2000).

486

NON-PREMIXED TURBULENT FLAMES

time evolution of the ﬂame front has been investigated in great detail, with particular emphasis on ﬂame structure, extinction limits, pocket formation, effects of vortex size, and strength.

6.10.7

Experimental Setups for Flame/Vortex Interaction Studies

The geometries shown in the previous subsections are for conceptual understanding of ﬂame–vortex interaction. The practical ﬂame–vortex interactions are described in terms of ﬂame conﬁgurations such as jet ﬂames, counterﬂow diffusion ﬂames, and the like. In this section, we discuss two such setups to explain practical cases. 6.10.7.1

Reaction Front/Vortex Interaction in Liquids

The earliest experimental conﬁguration analyzing ﬂame/vortex interaction dynamics in the non-premixed case was a liquid phase setup involving liquid acid/ base reactions (Karagozian and Nguyen, 1988). An aluminum-framed water tank with Plexiglas walls allows optical access. The vortex generator is set in the lower portion of the tank, consisting of two side plates forming a convergent two-dimensional nozzle. A thin aluminum plate separates the base solution from the slightly lighter acid solution. A vortex dipole can be formed by the introduction of base liquid pumped from external tanks through the lower ﬁll valves. This vortical structure of base liquid that ﬁlls the lower part of the tank is pushed into the acid liquid (see Figure 6.45). During the vortex propagation into the acid solution, the interface between the acid and base, which is established Optical Scanner 5 W Argon Laser

Laser Sheet Dilute Acid Solution

2.44 m Side Plates

Valve

Valve

Base Solution

Valve Turbulence Management System

Figure 6.45 Experimental setup for studying liquid reaction/vortex interactions (modiﬁed from Karagozian, Suganuma, and Strom, 1988).

INTERACTIONS OF FLAME AND VORTICES

487

initially at the exit of the side plates, is stretched and rolled up by the vortex pair. Flow visualization was achieved with planar LIF of a dye mixed with the acid solution. 6.10.7.2

Jet Flames

Jet ﬂames have been extensively studied by Chen and coworkers (1986–1991). The burner is composed of a fuel nozzle located at the center of a coaxial jet assembly, which is mounted in a small vertical combustion tunnel (Figure 6.46). The annular air jet has a divergent section and ﬂow straightener, which consists of a honeycomb and ﬁne mesh screens, to provide low-turbulence surrounding air for the fuel jet. Experiments were carried out using Mie scattering on TiO2 particles, providing a marker for the interface of the cool air and H2 O(g) combustion product outside the ﬂame and the cool fuel/H2 O(g) product interface inside the ﬂame. To achieve this, TiCl4 was added to both the dry air stream and the dry propane fuel ﬂow. Titanium tetrachlorine reacts with water produced in the reaction zone and yields ultraﬁne particles of TiO2 and HCl molecules. In this conﬁguration, vortices are generated from hydrodynamic instabilities. These instabilities appear only at speciﬁc excitation frequencies. It is interesting to force other frequencies with a loudspeaker to scan a broader range of frequencies. This experiment was carried out by Strawa and Cantwell (1985) in a facility capable of enclosing a combusting ﬂow at elevated pressures. The air is injected in the test section while a nozzle in the center of the section supplies fuel. The fuel jet was subjected to a periodic ﬂuctuation. By forcing the jet in a range of frequencies encompassing its unforced natural frequency, it was possible to produce a periodic controllable ﬂow. Gutmark, Schadow, and Wilson (1987) studied a conﬁguration based on the same principle using planar LIF on OH

Laser sheet

Dryer

H 2O

TiCl4

Annular jet D = 254 mm Screen Honeycomb Fuel Jet Tubes D = 11 mm D = 22.5 mm Nozzle D = 10 mm

Dryer Air

TiCl4

Fuel

Figure 6.46 Experimental setup for studying jet diffusion ﬂames (modiﬁed from Chen and Roquemore, 1986).

488

NON-PREMIXED TURBULENT FLAMES

radicals. Hsu, Tsai, and Raju (1993) focused on the effect of acoustically driven excitations on the co-annular jet conﬁguration performing Mie scattering and OH ﬂuorescence. Hancock, Schauer, Lucht, Katta, and Hsu (1996) measured temperature using CARS techniques. Mueller and Schefer (1998) also built a two-dimensional conﬁguration based on previous experiments. 6.10.7.3

Counterﬂow Diffusion Flames

The experimental setup to study the interaction of counterﬂow diffusion ﬂame with vortex rings is shown in Figure 6.47. This interaction is, however, of greatest interest in studies of turbulent combustion. This experimental setup was established by Rolon, Aguerre, and Candel (1995), and it consists of a counterﬂow diffusion ﬂame burner and a vortex generator. A steady non-premixed counterﬂow ﬂame of air and hydrogen diluted with nitrogen is ﬁrst established. This counterﬂow is surrounded by two nitrogen curtains to suppress external disturbances and prevent the appearance of a diffusion ﬂame around the fuel stream. A vortex ring is generated from a tube installed in the lower combustor nozzle and impinges on the ﬂame such that the reaction zone forms an envelope around the vortex and its wake. The axial velocity ﬁeld could be measured by laser D¨oppler anemometry (LDA), and direct visualizations of the reaction zone-vortex structure could be conducted by using Mie scattering (Th´evenin et al., 1996), spontaneous emission imaging, and planar laser-induced ﬂuorescence (PLIF) on OH radicals to follow the ﬂame evolution (Renard, Rolon, Th´evenin, and Candel, 1999). A steady ﬂame is ﬁrst established. Then, at a selected instant t0 , a toroidal vortex is impulsively injected by the action of the piston. The vortex ring rapidly accelerates toward the ﬂame. At the same time, the shutter of the ICCD camera is opened. At a well-deﬁned instant after the shutter has opened, a single shot of the laser sheet is emitted. The image is transferred and stored on a magneto-optical disk. The ﬂame surface may be deduced from the OH ﬂuorescence images because OH has been shown to be a good tracer of the reaction zone. N2 curtain Fuel

ICCD camera

fuel filter laser sheet oxidizer

vortex ring oxidizer

N2 curtain

actuator

strained diffusion flame

cylindrical lens

spherical lens

Figure 6.47 Experimental setup for studying vortex ring/counterﬂow diffusion ﬂame interactions (modiﬁed from Rolon, Aguerre, and Candel, 1995).

INTERACTIONS OF FLAME AND VORTICES

489

Vortex ﬂame interaction may cause ﬂame extinction. The effect of vortices on ﬂames is local. Some vortices may induce higher strain rates on the local ﬂame surface. This effect can result in local ﬂame quenching due to a decrease in reaction rates. Unlike the inﬂuence of vortices on ﬂame, the equivalence ratio affects the ﬂame globally. Vortex structures also may appear as a result of ﬂow instability. There is growing evidence that combustors operating in regimes of oscillation are driven by organized vortices. In many cases, the ignition and subsequent reaction of these structures constitute the sustaining mechanism by which energy is fed into the oscillation. Vortex roll-up often governs the transport of fresh reactants into burning regions. This process determines the rate of reaction in the ﬂow and the amplitude of the pressure pulse associated with the vortex burn-out. The undisturbed strained laminar ﬂame is shown in Figure 6.48a. The ﬂame surface is slightly curved due to buoyancy. The region of interest is close to the stagnation point (where the interaction occurs), which can be considered to be planar. Figure 6.48b shows the vortex impinging on the ﬂame. This ﬂame is lean enough (φ = 0.5) to be quenched even by a very weak vortex. The vortex thereafter widens the hole due to quenching, passes through the ﬂame, and stretches the edges of the hole. This is due to the inner ﬂow produced at the center of the vortex ring. As the vortex advances in the upward direction, it lowers the diameter of the hole by stretching the ﬂame upward (see Figure 6.48c). As the rotational velocity is about twice the speed of the vortex, the ﬂame front is convected through the vortex ring and then rolled up around the vortex core. Simultaneously, the reaction rate is enhanced. Species diffusion and heat release then prevent the ﬂame front from rolling up more than one turn (see Figure 6.48d). Finally, the truncated cone formed by the crossing of the vortex through the ﬂame reconnects with the outer reacting ring left behind by the vortex (see Figure 6.48e). Examination of the OH images in this case has shown (Renard et al., 1998) that moderate strain rates ﬁrst enhance the reaction rate; further increase in strain rates can lead to local ﬂame extinction when a critical strain rate value is exceeded. For the stoichiometric case with slightly higher strain rate of 55 s−1 , a set of recorded ﬂame/vortex interaction images is shown in Figure 6.49. No quenching occurs in this case because the ﬂame is not lean and the vortex is relatively weak, leading to no extinction during the interaction period. As a consequence, the increase of ﬂame surface area, which begins at t = 35 ms, is quite smooth and steep (Figure 6.49b–c). At t ∼ 48 ms, the vortex ring impinges on the upper nozzle. Since the ﬂame is richer than the previous ones, the ﬂame front envelops the vortex without extinguishing. Hence, for times greater than 46 ms, the measurements of ﬂame surface area are no longer meaningful. After t = 65 ms, the distorted part of the ﬂame in the upper injector nozzle is probably extinguished and a small decrease occurs in the ﬂame surface area. The ﬂame is distorted only by the crossing of the vortex (Figure 6.49b). Also, there is an enhancement of the reaction rate in the upper part of the foot of the mushroom while the reaction rate is reduced at the bottom of the mushroom (Figure 6.49c).

490

NON-PREMIXED TURBULENT FLAMES

region of interest buoyancy effects

buoyancy effects

(a) t = 35 ms extinction

vortex ring (b) t = 43 ms vortex ring stretch

stretch

diameter lowering (c) t = 49 ms vortex ring roll-up enhancement

(d) t = 55 ms vortex ring

reconnection

reconnection

(e) t = 61 ms

Figure 6.48 Flame-vortex interaction for φ = 0.5, aeq = 40 s−1 , vortex propagation speed UT = 1.4 m/s (after Renard et al., 1999).

The vortex ring attempts to roll up the hat of the mushroom while the wake of the vortex induces a thickening of the foot side structure (Figure 6.49d). The vortex is not strong enough to alter the top of the ﬂame signiﬁcantly; thus, the mushroom slowly breaks up. A thickening of the wall of the mushroom foot occurs while the foot section decreases. This could be due to the region behind the vortex where the products stagnate, which would explain why the widening of the mushroom sides increases with the equivalence ratio.

INTERACTIONS OF FLAME AND VORTICES

491

region of interest buoyancy effects

buoyancy effects (a) t = 30 ms

distortion

vortex ring

(b) t = 40 ms vortex ring

enhancement

lowering (c) t = 44 ms vortex ring attempt to roll up thinning

(d) t = 48 ms

vortex ring slow breakup (e) t = 52 ms

Figure 6.49 Flame-vortex interaction for φ = 1.0, aeq = 55 s−1 , vortex propagation speed UT = 2.4 m/s (after Renard et al., 1999).

In the case of φ = 0.5, the incoming vortex starts interacting with the ﬂame at t = 40 ms. The ﬂame surface area starts to increase, but extinction quickly occurs at t = 43 ms since the ﬂame is very lean. This results in a fall of ﬂame surface area to its initial value (see Figure 6.50). When the vortex travels into the fuel ﬂow, the ﬂame is stretched and rolled up (Figure 6.48c–d). The ﬂame surface area increases very quickly to a value of about three times. At this moment, the ﬂame has its maximum roll-up (Figure 6.48e) and stretching made it very weak, which means that it can no longer be entrained by the vortex. As a consequence,

492

NON-PREMIXED TURBULENT FLAMES 10

Flame Surface Area (cm2)

9

Bold segments indicate arrival of vortex at upper nozzle

8 7 6 5 4 3 2 25

30

35

40

45

50

55

60

65

70

75

80

Time (ms)

Figure 6.50 Measurements of ﬂame surface area (solid line: φ = 0.5, aeq = 40 s−1 , UT = 1.4 m/s; dashed line: φ = 0.7, aeq = 80 s−1 , UT = 3.5 m/s; dashed-dotted line: φ = 1.0, aeq = 55 s−1 , UT = 2.4 m/s) (after from Renard et al., 1999).

a second extinction occurs at the top of the ﬂame. Consequently, the increase in the ﬂame surface area ceases. The fall of surface area after this time is due to the ﬂame extinction. Four conclusions can be drawn from these observations: 1. The ﬂame surface area increase does not depend on the vortex speed (which is about equal to the rotational velocity) only. The residence time is also very important: A fast vortex will stretch a ﬂame essentially by lengthening, whereas a slow vortex does this essentially by roll-up. 2. Equivalence ratio is important: A lean ﬂame is quenched more easily than the stoichiometric one. For the latter case, the ﬂame was rolled up and lengthened to a higher degree. 3. Local extinction at the top of the ﬂame is associated with the decrease in ﬂame surface area. 4. Both equivalence ratio and the strain rate can inﬂuence the ﬂame/vortex interaction process and the time variation of the ﬂame surface area.

6.11

GENERATION AND DISSIPATION OF VORTICITY EFFECTS

It is usually believed that vorticity is generated by baroclinic effects (see the last term in Equation 4.134) and decays by thermal expansion, which can result in volume expansion. Chen et al. (1997) made a theoretical analysis of experimental results on jet ﬂames. It appeared that three regimes of vorticity generation may be distinguished. In the fuel-lean regime or outside the luminous ﬂame,

NON-PREMIXED FLAME–VORTEX INTERACTION COMBUSTION DIAGRAM

493

vorticity production due to baroclinic torque is balanced by expansion-induced dissipation. Inside the luminous ﬂame, both the volumetric expansion and baroclinicity behave like sink terms in the vorticity balance, except in a subregime close to the ﬂame sheet, where the volumetric expansion acts as a source term. In the postﬂame region, vorticity generated by the baroclinic torque may be increased or decreased by volumetric expansion that may be a source term in the core region or a sink term in the outer region. The vorticity production in the postﬂame regime is qualitatively similar to that found in a buoyant jet; conversely, opposite effects are expected for a negatively buoyant jet.

6.12 NON-PREMIXED FLAME–VORTEX INTERACTION COMBUSTION DIAGRAM

A regime diagram for the non-premixed ﬂame vortex interaction is shown in Figure 6.51. These regimes reﬂect the relative importance of the physical phenomena that play a major role in this conﬁguration involving dissipation, straining, curvature, wrinkling, roll-up, and quenching. The horizontal axis of this diagram corresponds to the ratio of the outer diameter of the vortex ring DH to the initial thermal ﬂame width δf . The vertical axis is associated with the ratio of the characteristic ﬂame time tc to the propagation time of the vortex through the ﬂame δf /UT . The ﬂame time, tc , is not easy to deﬁne. The Reynolds number is given as Revortex = (DH UT )/ν. Zone I corresponds to values of the vortex with Revortex < 1. Such vortices are dissipated immediately by viscosity and therefore do not inﬂuence the ﬂame. Zone

10

(III)

aq

(VIII) a = D D

(IV)

1 UTtc df

Da

(V)

0.1 0.01

Da

(II)

(VII) ap D = (VI)

Re

0.001

Re

0.01

vor

tex

vor

tex

(I)

0.1

1

au =D

=R =1

ec

Zones I, II, III: no effect of vortices or weak effect Zone IV: moderate curvature and wrinkling effects Zone V: thickening of reaction zones, strong curvature effects, and wrinkling Zone VI: wrinkling effects and strong roll-up of reaction zones, no extinction Zone VII: pockets of oxidizer are formed, which burn in the fuel part of the domain Zone VIII: extinction along the centerline

10

DH /df

Figure 6.51 Regime diagram for non-premixed turbulent ﬂame vortex interactions. The quadrangle in dashed-line quadrangle represents the zone accessible with the experimental setup (modiﬁed from Th´evenin et al., 2000).

494

NON-PREMIXED TURBULENT FLAMES

II corresponds to the part of the domain between Rec > Revortex > 1. Th´evenin, Renard, Fiechtner, Gord, and Rolon (2000) found that the effect of vortices on the ﬂame was negligible even when the Revortex was much higher than 1, close to a cutoff value called Rec . This cutoff value was calculated as Rec = 23. In zone III, the Reynolds number of the vortices is larger than 1, but these vortices do not survive sufﬁciently long to perturb the ﬂame before being dissipated by viscosity. This corresponds to the following condition: 2 DH < tc ν

(6.79)

A characteristic Damk¨ohler number Dad to account for the thermal-diffusion effects can be deﬁned as: δf2 Dad ≡ (6.80) αtc By introducing the Prandtl number and the Damk¨ohler number, the limit of this zone corresponding to the ratio of vortex size to ﬂame width can be deﬁned by 2 setting tc = DH /ν as: Pr 1/2 DH = (6.81) δf Dad The line given by Equation 6.81 is the limit between zone III and zone IV. In zone IV, DH /δf < 2 and, therefore, curvature effects are important. However, extinction is not observed because the vortex ring is not large enough to separate the two sides of the induced wrinkle when DH /δf < 0.5. For the same reason, the increase in ﬂame surface area cannot be very large, since the produced ﬂame surface remains in the vicinity of the symmetry axis. Zone V is limited in the horizontal direction by the vertical lines DH /δf = 0.5 on the left and DH /δf = 2 on the right. Therefore, in this zone curvature effects are still important. However, since the dimensions of the vortices are of the same order of magnitude as the ﬂame width, the reaction zone thickens. Wrinkling effects are also important. Due to the curvature effects and presence of stronger vortices, extinction can occur in this zone, even though the corresponding vortices are small. A PLIF image illustrating the thickening of the reaction zone is shown in Figure 6.48c. In zone VI, wrinkling effects become strong in comparison with zone II, and curvature effects do not play an important role since the vortices are much larger than the ﬂame thickness. However, an important roll-up of the ﬂame is observed to participate in the increase in ﬂame surface area. No quenching is observed in this zone. The PLIF image illustrating this roll-up is shown in Figure 6.48d. Zone VII is separated from zone VI by a slanted line Da = Dap , where Dap is deﬁned as the ratio of the characteristic time of oxidizer pocket formation tp to the ﬂame time tc . The pinch-off of the ﬂame front behind the vortex is essential for generating an oxidizer pocket, and it occurs when the vortex characteristic time [(δf /UT ) × (DH /δf )] is shorter than the time tp associated with the

NON-PREMIXED FLAME–VORTEX INTERACTION COMBUSTION DIAGRAM

495

pocket formation. This condition corresponds to (UT tc )/δf >(DH /δf )(1/Dap ). The wrinkling effect of the vortices on the ﬂame-surface area is still strong in this zone, as in zone VI. Curvature effects are still negligible, but roll-up is again considerable. In zone VII, this roll-up eventually leads to the formation of pockets of oxidizer, which are surrounded by an active reaction zone. The theoretical line Da = Dau runs through zones II, V, and VII. Dau is associated with the unsteady effects. Unsteady effects will appear when the vortex time scale DH /UT is too small compared to the chemical time scale tc , implying that the ﬂame cannot adjust fast enough to hydrodynamic excitations. Essentially, the unsteady effects occur when the ﬂow time scale is too small compared to the chemical time scale. For this reason, the ﬂame does not respond rapidly enough to the fast temporal evolution of the scalar dissipation rate and loses its laminar ﬂamelet structure; the maximal temperature becomes larger than the asymptotic value, and the total reaction rate drops below its asymptotic value accordingly. Finally, zone VIII is separated from zone VII by the slanted line Da = Daq , where Daq is the Damk¨ohler number associated with ﬂame extinction. Based on asymptotic analysis, Cuenot and Poinsot (1994) have given the next expression for the extinction Damk¨ohler number as: ! ! 2 2 1 − Zst (1 − Zst ) e νF Daq = · 1− (1 + νO ) 2νF νO ! 2 π exp 2Ez2 where

Ez ≡ erf −1 (Zst )

(6.82)

Therefore, extinction due to strain occurs in zone VIII, since Da < Daq . The results shown in Figure 6.52 correspond to this zone.

t = 16 ms

t = 18 ms

t = 20 ms

t = 22 ms

(a) t = 16 ms

t = 18 ms

t = 20 ms

t = 22 ms

(b)

Figure 6.52 Flame extinction due to strain produced by ﬂame-vortex interaction (Zone VIII). (a) OH PLIF images showing perturbation of reaction zone by vortex causing extinction. (b) Rayleigh images showing changes in temperature ﬁeld of reaction zone (modiﬁed from Th´evenin et al., 2000).

496

NON-PREMIXED TURBULENT FLAMES

(a) 8.3 ms

(b) 8.5 ms

(c) 9.0 ms

(d) 9.4 ms

Figure 6.53 Flame-vortex interaction from superimposed OH PLIF and PIV measurements showing extinction in annular region, followed by reignition (modiﬁed from Th´evenin et al., 2000).

The results shown in Figure 6.53 are associated with the boundary between zones V, VII, and VIII. As long as Da > Daq , extinction does occur. In this ﬁgure, extinction occurs in the annular region. An earlier version of the regime diagram based on ﬂame vortex interaction was developed by Cuenot and Poinsot (1994); it is also given in the book by Poinsot and Veynante (2005).

6.13 FLAME INSTABILITY IN NON-PREMIXED TURBULENT FLAMES

Flame instability appears in different forms and at different scales. Spontaneous oscillations of an otherwise stable propane or methane jet diffusion ﬂame occur when the fuel concentration increases and the whole ﬂame expands and contracts at a frequency of a few Hertz (F¨uri, Papas, and Monkewitz, 2000). A nominally planar ﬂame in an upward uniform ﬂow of a combustible mixture takes on a cellular appearance when varying the mixture composition, with cells of 0.5 to 1 cm in size (Markstein, 1964). The surface of a large expanding ﬂame of magnitude 5 to 10 m in diameter becomes spontaneously rough when it reaches a critical size and takes on a pebbled appearance with small ripples of approximately 10 to 50 cm covering its surface (Lind and Whitson, 1977). Sustained pressure ﬂuctuations of acoustic nature have been observed in combustion chambers where unsteady combustion occurs. These instabilities can be detrimental when they occur in practical systems. For example, they can create conditions that may cause damage and mechanical failure to the combustion device. In other situations, however, they may be favorable for enhancing mixing and increasing burning rates. Lifting the ﬂame base off the burner has the advantage of avoiding thermal contact between the ﬂame and the rim (edge planes) as well as enhancing mixing in the dead space. The disadvantage is that the resulting edge ﬂame is subjected to instabilities and possible blow-off. The primary mode is oscillations, but unlike diffusion ﬂames, which oscillate in a direction normal to the reaction sheet, an edge ﬂame moves back and forth along the stoichiometric surface, and the oscillations decay farther downstream along the trailing diffusion ﬂame.

FLAME INSTABILITY IN NON-PREMIXED TURBULENT FLAMES

497

One of the earliest known instability studies in diffusion ﬂames was reported by Gardside and Jackson (1951), who observed that when a hydrogen-air jet ﬂame was diluted with N2 or CO2 , the ﬂame surface often was comprised of triangular cells in the shape of a polyhedron. Later, Dongworth and Melvin (1976) observed that the base of a hydrogen-oxygen diffusion ﬂame on top of a splitter-plate burner, which is normally straight, takes on a cellular appearance when the ﬂow rate is sufﬁciently high and the reactants are diluted in N2 or Ar but not in He. Chen et al. (1992) also reported the occurrence of cellular structures on slot-burner hydrocarbon air diffusion ﬂames diluted with SF6 but not with N2 , CO2 , or He. Similarly, Ishizuka and Tsuji (1981) observed the formation of stripes or elongated cells on the surface of N2 -diluted hydrogen-oxygen counterﬂow diffusion ﬂames in the cross-ﬂow direction. The experimental record of ﬂame oscillations, which is another form of instability, includes condensedphase fuels (Chan and T’ien, 1978), candle and large suspended fuel droplets in a microgravity environment (Nayagam and Williams 1998; Ross, Sotos, and T’ien, 1991), jet and spray diffusion ﬂames (F¨uri, Papas, and Monkewitz 2000; Golovanevsky, Levy, Greenberg, and Matalon, 1999), and ﬂame spreading over liquid beds (Ross, 1994). The nature of the oscillation in each of these experiments is quite different. The droplet ﬂame exhibits radial oscillations. The jet ﬂame expands and contracts as a whole during a cycle. For the microgravity candle ﬂame, the edge moves back and forth along its hemispherical surface. Similarly, in ﬂame spreading, the oscillations are seen primarily near the leading edge, decaying along the trailing diffusion ﬂame. Instabilities in diffusion are driven mainly by diffusive-thermal effects and are not hydrodynamically driven; thermal expansion plays a secondary role. The Burke-Schumann solution of complete combustion is unconditionally stable. Similarly, the ﬂame for the equi-diffusion case, LeF = LeO = 1, is stable for all Da. The instabilities occur as a result of differential (i.e., nonunity Lewis numbers) and preferential (i.e., unequal Lewis numbers) diffusion. when the Damk¨ohler number is sufﬁciently low or Daq ≤ Da < ∞, where Daq denotes the conditions, where steady burning can no longer be sustained and the ﬂame is extinguished because of a low-enough temperature and excessive reactant leakage, Da* represents conditions where a marginally stable state can be achieved. The main consequence of nonunity Lewis numbers on the combustion ﬁeld is that they yield temperature and concentration proﬁles that are nonsimilar. A volumetric heat loss that affects the temperature but not the concentration ﬁelds also generates nonsimilar proﬁles and thus promotes diffusive-thermal instabilities. For example, Cheatham and Matalon (1996) found that ﬂame oscillation can be triggered by appreciable heat losses even for unity Lewis numbers, and the instability is enhanced by heat losses when the conditions already favor oscillations (Kukuck and Matalon, 2001). Oscillations in a bandwidth of 1 to 3.5 Hz were observed when a fraction of the fuel supplied in a jet ﬂame was introduced in the liquid phase in the form of droplets (Golovanevsky et al., 1999). Cellular structures, oscillations, and other competing modes of instabilities are driven primarily by diffusive-thermal effects. Although thermal expansion

498

NON-PREMIXED TURBULENT FLAMES

has a marked inﬂuence on ﬂame instability, it does not play as crucial a role in non-premixed ﬂames as it does with premixed ﬂames. This fact provides justiﬁcation for the constant-density assumption for stability analysis of nonpremixed ﬂames. When density variations are accounted for, these results are only slightly modiﬁed with thermal expansion acting to further stabilize or destabilize the ﬂame, depending on the mode of instability (Matalon, 2009). Even with the constant-density approximation, the ﬂame stability analysis is a complicated problem due to the large number of parameters involved. These include the two Lewis numbers, LeF and LeO , associated with the fuel and oxidizer, respectively; the initial equivalence ratio φ deﬁned as the fuel-to-oxidizer mass supplied in the respective streams normalized by their stoichiometric proportions; and the ﬂow conditions characterized by a Damk¨ohler number Da deﬁned as the ratio of the residence time in the ﬂame zone to the chemical reaction time tc . Figure 6.54 shows various patterns that are likely to be observed at the instability threshold in the Lewis numbers parameter plane. The dashed vertical line separates regions of relatively lean (to the left) and rich (to the right) mixtures and shifts to the right as the initial equivalence ratio decreases. Stationary cells occur in the region below the solid curve shown in the stability diagram that encompasses the domain 0 < LeF < 1, 0 < LeO < 1. This region stretches out to include a wider range of LeO when reducing φ, implying that leaner mixtures are more susceptible to cell formation. Characteristic cell size l at the onset

LeF High-frequency instability 2 w~ t −1. R , tR ~ l R D

Competing instability modes Pulsations 2 w~ t−1. D , t D ~ lD D

1

Stationary cells l ~ lD

High-frequency instability l ~ lR

1

Oscillatory cells

LeO

Figure 6.54 Diagram illustrating various modes of instability in fuel and oxidizer Lewis numbers parameter plane. The dashed vertical line separates regions of relatively lean (left) and rich (right) mixtures; the solid line represents the complete combustion under stoichiometric conditions (modiﬁed from Matalon, 2009).

FLAME INSTABILITY IN NON-PREMIXED TURBULENT FLAMES

499

of the instability are in the range from 3lD to 12lD , depending on the mixture strength and the ﬂow rate. The diffusion length scale lD is deﬁned as α/Uc , where Uc is the average fuel-supply velocity. For conditions associated with the CO2 diluted hydrogen-oxygen conﬁned ﬂame, the cells could be 0.5 to 2 cm wide. The smaller cells are expected near the extinction limit (i.e., when the marginal stability Damk¨ohler number is near Daext ), in which case the theoretical analysis must incorporate small-wavelength perturbations comparable to the reaction zone thickness that evolve on the chemical reaction time, tc . Stationary cells are formed when the two Lewis numbers are less than 1 (LeF < 1, LeO < 1) but can also result when one of the Lewis numbers is near, or even slightly above, unity, provided the other is less than 1. The characteristic cell size is given by l = 2π/k ∗ , where k * is the wave number of the most ampliﬁed disturbance. Typical cell formation time scales according to the diffusion time tD . For fuel-lean systems, k ∗ ≈ 0.5/ lD to 4/ lD , which yields cells 0.3 to 2 cm wide, depending on the characteristic speed. But for near-stoichiometric conditions and in slightly fuel-rich mixtures, the cells are much smaller and scale on the reaction zone thickness lR . Disturbances now intrude in the reaction zone, and a separate analysis that incorporates small-wavelength perturbations evolving on the characteristic chemical reaction time tc is required (Buckmaster et al. 1983; Kim 1997; Kim and Lee, 1999). These high-frequency modes, also referred to as fast time instabilities, are limited to conditions that are very near the extinction limit (i.e., Da∗ ≈ Daq ). In contrast, ordinary cells are predicted to occur over a wider range of ﬂow rates, with Da∗ − Daq = O(1), and are more likely to be observed in practice. Planar pulsation occurs in the upper right corner of the stability diagram and is associated with sufﬁciently large Lewis numbers. When both Lewis numbers are larger than 1 (LeF > 1, LeO > 1), the preferred mode of instability is planar pulsations with the ﬂame moving back and forth in a direction perpendicular to its surface. In fuel-rich systems, the frequency of oscillations scales on the inverse of the diffusion time and is estimated at 1 to 6 Hz by Kukuck and Matalon (2001); for lean mixtures, the onset occurs near the extinction limit as a high-frequency mode that scales on the inverse of the reaction time. For example, oscillations were observed for propane air ﬂames diluted in nitrogen when φ ≥ 1.32, in which case LeF varied from 1.1 to 1.8 with LeO ≈ 1, but were not observed for φ ≤ 0.76 despite the large fuel Lewis number LeF = 1.86, which resulted from the N2 -dilution. Since the dashed vertical line shifts to the left as φ increases, planar oscillations are favored in relatively fuel-rich mixtures. Although cellular structures and planar pulsations are the predominant forms of instability, other possible patterns may exist in the transition regions between various domains or for extreme values of the parameters. Oscillating cells result when LeF < 1 and LeO is sufﬁciently large from competing modes of comparable and/or disparate scales. Mixed modes of instability were noted in jet diffusion ﬂames in the form of traveling or rotating cells (Lo Jacono, Papas, and Monkewitz, 2003), and in the ﬂat ﬂame in form of transverse oscillations or traveling waves of long wavelength (Papas et al., 2004).

500

NON-PREMIXED TURBULENT FLAMES Vortical cell

Chemical reaction sheet

Figure 6.55 Vorticity ﬁeld in vicinity of cellular ﬂame at instability threshold, calculated for density ratio around 3 (modiﬁed from Matalon, 2009).

As mentioned earlier, thermal expansion does not play a major role in ﬂame instability of non-premixed ﬂames. The main effect of thermal expansion is the change in the degree of instability and thereby shifting the marginal state, Da* . Thermal expansion plays different roles on the various modes of instability. The induced ﬂow due to thermal expansion near a cellular structure involves regions of concentrated vorticity upstream of the reaction surface with extrema near the crests and troughs (see Figure 6.55). The solid curve in this ﬁgure denotes the reaction sheet. The concentrated vortical motion near the crests and troughs enhances the transport of fuel to these regions, thus sustaining the cellular structure. The vortical motion enhances the transport of fuel to these regions, which sustains the corrugated structure. The onset of cellular ﬂames occurs at a value Da* signiﬁcantly higher than that predicted by using a constant-density model, which implies that a wider range of physical states are susceptible to cell formation. Thermal expansion, however, reduces the growth rate associated with planar pulsation. Since the ﬂame remains planar and its back-and-forth motion is toward denser ﬂuid, thermal expansion has a damping inﬂuence on the oscillations. The marginal state Da* is signiﬁcantly lower than the value predicted by using a constant-density model, which implies that a narrower range of physical states is susceptible to planar pulsation. Overall, although thermal expansion has a marked inﬂuence on the dynamics of diffusion ﬂames, it does not play a crucial role for diffusion ﬂames as it does for premixed ﬂames. 6.14

PARTIALLY PREMIXED FLAMES OR EDGE FLAMES

In practical combustion devices, pure non-premixed or pure premixed ﬂame is an exception; for the most part, partially premixed ﬂame is encountered. The edge ﬂame is a fundamental structure, associated with partially premixed conditions; thus, it has characteristics of both premixed and non-premixed ﬂames. The edge ﬂame has a tribrachial structure (also referred to as a triple ﬂame) that consists of a highly curved premixed ﬂame with lean and rich branches and a diffusion ﬂame trailing behind consuming the remaining unburned reactants. Flames with edges occur in many forms, and several examples are sketched in Figure 6.56. A representative edge ﬂame formed in the wake of splitter plate separating two parallel

501

PARTIALLY PREMIXED FLAMES OR EDGE FLAMES (a)

(b)

Diffusion flame

(d) DF

Edge-flame

RPF

Fuel rich

LPF

Fuel lean

Oxidizer

Fuel

Oxidizer

(c)

Edge-flames (e)

LPF

DF

DF Fuel

RPF LPF RPF

Figure 6.56 Schematics of edge ﬂames in various ﬂow conﬁgurations (RPF: rich premixed ﬂame; LPF: lean premixed ﬂame; DF: diffusion ﬂame) (modiﬁed from Law, 2006; Matalon, 2009; and Vervisch and Poinsot, 1998).

streams, one containing fuel and the other oxidizer, is shown in Figure 6.56b. Partial mixing occurs behind the tip of the plate, and combustion takes place in the stratiﬁed medium, once the mixture is ignited. Figure 6.56c shows a ﬂame spreading over a fuel bed, solid or liquid. Most of the fuel ﬂux (nonuniform) from the bed is consumed in a diffusion ﬂame that is nominally one-dimensional, but reaction is negligible at the temperature of the bed surface so there is a dead space between the ﬂame and the bed, and the ﬂame has an edge. Figure 6.57 shows photograph of an edge ﬂame stabilized at the trailing edge of a splitter plate separating coﬂowing streams of nitrogen-diluted methane and air.

6.14.1

Formation of Edge Flames

Ignition and extinction of diffusion ﬂames must structures to transition from nonburning to burning Understanding the behavior of the structures called for explaining partial extinction phenomena and for

involve speciﬁc ﬂame states (and vice versa). edge ﬂames is relevant developing ignition and

Fuel

Splitter plate

Oxidizer

Figure 6.57 Edge ﬂame stabilized at trailing edge of splitter plate separating coﬂowing streams of nitrogen-diluted methane and air (modiﬁed from Kioni et al., 1993).

502

NON-PREMIXED TURBULENT FLAMES

turbulent ﬂame stabilization (M¨uller, Breitbach, and Peters, 1994). In turbulent ﬂows, development of non-premixed ﬂames depend on intense mixing, which transports the necessary energy to complete the combustion, or on multiplication of intense reaction zones (edge ﬂames) interacting with turbulence. Within a turbulent ﬂow, the diffusion ﬂame is continuously distorted and stretched by velocity ﬂuctuations inducing inhomogeneities in the mixing of the reactants. Because a local increase of the mixture fraction gradient (or scalar dissipation rate χ) induces larger heat losses, one consequence of turbulent ﬂame stretching is the appearance of local extinction leading to edge ﬂames (Figure 6.58). When traveling along the stoichiometric line from low values of χ to the quenching limit, the local size of the mixing layer decreases, iso-concentration surfaces of fuel and oxidizer are brought together, and the degree of premixing of the ﬂow locally grows, whereas the reactive activity ﬁrst increases and then quenching occurs. Therefore, we should expect to see intense combustion in a partially premixed regime at the edges of diffusion ﬂames (Figure 6.58).

6.14.2

Triple Flame Stabilization of Lifted Diffusion Flame

A generic picture for edge ﬂames is the triple ﬂame developing when a laminar diffusion ﬂame is lifted downstream from a burner exit. In this situation, combustion starts in a zone where fuel and oxidizer have been mixed in stoichiometric proportion. Because of the imbalance between diffusion of heat and chemical reaction, the resulting premixed kernel tends to propagate toward fresh

Quenching zone

Burning zone

Quenching zone

c > cq

c < cq

c > cq

Oxidizer Z=0 dm Mixinglayer thickness Fuel Z=1

Oxidizer

Stoichiometric line Z = Z st

Edge-flames

Fuel c > cq

c < cq

Figure 6.58 Sketch of conditions for development of partially premixed edge ﬂames (modiﬁed from Vervisch and Poinsot, 1998).

PARTIALLY PREMIXED FLAMES OR EDGE FLAMES

503

y

Lean premixed flame Stoichiometric line Oxidizer Fuel

Oxidizer-rich region

x

Fuel-rich region

Tip of triple flame

Rich premixed flame

Figure 6.59 Propagating triple ﬂame in mixing layer without velocity shear (modiﬁed from Ruetsch, Vervisch, and Li˜na´ n, 1995).

gases, and it contributes to the stabilization of a trailing diffusion ﬂame. In a mixing layer, the stoichiometric premixed kernel evolves to a rich, partially premixed ﬂame in the direction of the fuel stream, while a lean partially premixed ﬂame develops on the oxidizer side (Figure 6.59). These two premixed ﬂames are curved because their respective propagation velocities decrease when moving away from the stoichiometric condition. The overall structure is composed of the two premixed ﬂames and of the diffusion ﬂame, and it is usually called a triple ﬂame. Using DNS with single-step chemistry, Ruetsch, Vervisch, and Li˜na´ n (1995) showed that burning velocities take larger values for triple ﬂame than for planar premixed ﬂames. This increase results from the deviation of the ﬂow upstream from the triple ﬂame induced by heat release.

6.14.3

Analysis of Edge Flames

Depending on the velocities of the fuel and oxidizer streams, the ﬂame may be either attached to or lifted off away from the splitter plate. The attached ﬂame is a diffusion ﬂame that separates a region of primarily fuel from a region where there is mainly oxidizer. When the ﬂame is lifted, it assumes the tribrachial structure, which consists of lean and rich premixed segments with a diffusion ﬂame trailing behind. The standoff distance of the edge ﬂame depends on ﬁve parameters: 1. 2. 3. 4. 5.

The The The The The

mean ﬂow rates of the fuel and oxidizer streams thickness of the boundary layer diffusivities of the reactants in terms of LeF and LeO mixture equivalence ratio heat release by the chemical reaction (thermal expansion).

504

NON-PREMIXED TURBULENT FLAMES

In the absence of preferential diffusion (LeF = LeO ) and when the fuel and oxidizer are supplied in stoichiometric proportions, the edge ﬂame is symmetric with respect to the axis and the trailing diffusion ﬂame remains parallel to the plate; otherwise, it leans toward one of the two sides approximately along the stoichiometric surface. The dependence of the ﬂame standoff distance on the Da is shown in Figure 6.60 for a wide range of Lewis numbers. The standoff distance xw is deﬁned as the location where the reaction rate reaches its maximum value. For large Da, the ﬂame is attached to the plate; it lifts off when increasing the ﬂow rate, or decreasing Da, and moves away from the plate (see Figure 6.60). For sufﬁciently small values of Le (below approximately 1.2), the solution is multivalued. In this case, marginal stability coincides with the blow-off point Da = Daq , with the unstable solution corresponding to the larger value of xw . An edge ﬂame can be stabilized near the plate only for ﬂow rates corresponding to Daq < Da < ∞; for lower values of Da, it is blown off by the ﬂow. For larger values of Le (above approximately 1.4), the solution is monotonic within the computational domain. The edge ﬂame, however, cannot always be stabilized near the plate. There is a range of unstable states corresponding to Da∗1 < Da < Da∗2 where the ﬂame undergoes spontaneous oscillations with the edge moving back and forth along the axis and dragging the trailing diffusion

Le = 0.8 Le = 1 10

Le = 1.2

5

Unstable flame

Le = 1.5

4

Stable flame

3

1

Standoff distance x w

Oscillatory flame

2

8

Blow-off condition

Le = 1.45 50

6

100 150

Marginally stable state

Le = 1.5 4

Le = 1.7

2

0 100

101

102

103

Damköhler number Da

Figure 6.60 Response curves showing dependence of ﬂame standoff distance xw on Damkohler number Da for the adiabatic case (modiﬁed from Matalon, 2009).

PARTIALLY PREMIXED FLAMES OR EDGE FLAMES 10

Stable state Unstable state Marginally stable state Extinction states

Le = 1.2 Le = 1.1

Standoff distance x w

8

505

Le = 1 Le = 0.7

6

Le = 0.9

4 2

0

b≡

0

0.0005 0.0015 0.001 Heat loss parameter b

radiative heat losses chemical heat released in reaction

0.002

Figure 6.61 Response curves showing dependence of ﬂame standoff distance xw on heat-loss parameter b for ﬁxed value of Damk¨ohler number (modiﬁed from Matalon, 2007).

ﬂame behind it. (See the two dashed curves bounded by two closed circles in Figure 6.61.) The oscillations along the sheet are weakened downstream and are completely damped at sufﬁciently large distances. When the combustion ﬁeld is asymmetric, the edge of the ﬂame is characterized by two coordinates, xw and yw . In this case, oscillations are associated with both coordinates varying periodically in time, and the edge ﬂame moves back and forth along a surface that coincides approximately with the stoichiometric surface. The dependence of xw on the heat-loss parameter b is shown in Figure 6.61 for a speciﬁed value of Da. Although the standoff distance xw increases with increasing b, the ﬂame can be stabilized near the tip of the splitter plate only when heat losses are relatively small. When appreciable or b > bc , the ﬂame undergoes spontaneous oscillations. The critical value bc increases with decreasing Le and precedes the blow-off point even for Le below unity. Hence, radiative losses could trigger ﬂame oscillations; this can occur even when the Lewis numbers are equal to or slightly less than unity. These explanations are based on the oscillations resulting from diffusivethermal instabilities. Experimentally observed oscillations that occur in complex situations may be driven by more than one factor. Often they are associated with buoyancy effects (Won et al., 2002), with the gas-phase circulation created ahead of the ﬂame for ﬂame spread over liquid pools (Schiller and Sirignano, 1996), or with Marangoni instability (Higuera and Garcia-Ybarra, 1998). Edge ﬂames also may occur in premixed systems displaying various forms of instabilities including cellular structures, as discussed by Buckmaster (2002).

506

NON-PREMIXED TURBULENT FLAMES

HOMEWORK PROBLEMS

1.

Show that for the one-step irreversible reaction {1 kg of fuel (F )} + {s kg of oxidant (O)} → {(1 + s) kg of product (P)} ,the Favre-averaged mass fractions of fuel, oxidizer, and product can be written as: 1 ∗ ˜ YF = Y (f − fst )P˜ (f ; x) df fst

! 2 J 2 ˜ − f / f f = YFe f˜ + αc Y ∗ f 1 st Y˜O = sY ∗

fst

(f − fst )P˜ (f ; x) df ! 2 J 2 ˜ = Y0e f˜ + αc rY ∗ f − f / f f 1 st 0

' ∗ ˜ YP = (s + 1) Y (1 − fst )

fst

f P˜ (f ; x) df + fst

1

fst

0

( ˜ (1 − f ) P (f, x) df

! e ∗ 2 2 ˜ ˜ = YP f − αc (r + 1) Y f J1 fst − f / f

where the function J1 is deﬁned as ! 2 ˜ J1 fst − f / f ! ! fst 2 2 ˜ ˜ ˜ P (f ; x) df − H fst − f fst − f / f ≡ (fst − f ) / f 0

where H is the Heaviside function and 1 YF 1 Y ∗ ≡ YF 1 + YO2 = s 1 − fst 2.

Derive the Favre average form of mixture fraction f˜ equation.

PROJECT NO. 6.1

Consider a solid-fuel ramjet (SFRJ) having two combustor geometries shown in the next drawings. Subsonic air inlet

Subsonic air inlet Solid Fuel Configuration I: Axial Inlet

Solid Fuel Configuration II: Axial Inlet with Bypass

PROJECT NO. 6.3

507

The solid-fuel grain provides a part of the walls for the combustion chamber. A sudden expansion at the port of the air inlet (axial, with or without bypass) can be used to provide ﬂame stabilization. A turbulent boundary layer develops and includes a diffusion-controlled ﬂame between the fuel-rich zone near the wall and the oxygen-rich central core. Due to the diffusion ﬂame, heat is transferred by convection to the solid surface, causing fuel vaporization. The fuel regression rate can be represented by this Arrhenius form: Ea r˙ = A exp − Ru Tw The fuel pyrolysis and hydrocarbon combustion of the fuel vapor with air can be approximated by a four-step process: Cx Hy → Cx Hy−2 + H2 x y−2 O2 → xCO + H2 2 2 CO + 12 O2 → CO2

Cx Hy−2 +

H2 + 12 O2 → H2 O a) Formulate the three-dimensional model for a quasi-steady, subsonic ﬂow including ﬁnite-rate chemical kinetics. b) List all your assumptions. (Note: To simplify this problem, assume radiation heat transfer to be negligible.) c) Sketch your anticipated axial distributions of u, YO2 , YCx Hy , and T along the centerline of conﬁguration I.

PROJECT NO. 6.2

Demonstrate the use of either a conserved scalar approach or a two-variable approach by presenting an example in turbulent diffusion ﬂame. Please give: (a) a problem statement including the source of reference, (b) procedures used in the treatment, (c) major results obtained, and (d) your comments or merits and drawbacks of the method you select to describe. PROJECT NO. 6.3

Consider a cylindrical furnace (with length of L and radius of r3 ) heated by an axisymmetric turbulent diffusion ﬂame jet burning in a constant pressure (10 atm) condition. The thermal energy of the diffusion ﬂame comes from the combustion of oxygen and methane, supplied through a concentric injector with the oxygen at the center port (radius r1 ) and methane gas supplied from

508

NON-PREMIXED TURBULENT FLAMES

the annular port (see Figure shown below). The combustion process takes place in a quasi-steady-state manner under a fuel-rich condition (with equivalence ratio equal to 1.5) by considering that the mean ﬂow properties are time independent. The methane ﬂow rate is considered to be supplied at m ˙ CH4 . The ﬂame zone (ﬂame brush) can be considered as a source of heat. The wall temperature of the furnace is maintained at a ﬁxed level of temperature (T = Tw ). In order to predict the chemically reacting ﬂow ﬁeld, consider these items in your formulation: a) State the fundamental assumptions adopted for model formulation. b) For RANS type of formulation with a multi-equation model, specify the governing equations (conservation equations and transport equations given in cylindrical coordinates) needed to solve for chemical species concentration, temperature, density, and velocity distributions throughout the entire ﬂow ﬁeld. (In the real-world problem, the speciﬁc reaction rate constants for the 42 elementary reactions associated with methane oxidation are given in Kuo, 2005 chapter 2, where 16 different chemical species are considered.) However, global chemical reaction can be assumed for simpliﬁcation. c) Specify turbulence closure considerations and the necessary boundary conditions. d) Sketch your anticipated results in terms of the proﬁles of temperature, species, and velocity components at a ﬁxed z station. L

Turbulent diffusion flame brush

r3 O2 CH4

rf

r1 r2 r1

T=Tw

r4

z

zf

7 BACKGROUND IN MULTIPHASE FLOWS WITH REACTIONS

SYMBOLS

Symbol

Description

Dimension

Area L2 Frame-indifferent acceleration, deﬁned by Equation 7.45 L/t2 Acceleration of one particle L/t2 Fluid properties (density, viscosity, speciﬁc heat or Multiple thermal conductivity), see Equation 7.118 Drag coefﬁcient — Cd db Average size of the dispersed phase conglomerates, see L Equation 7.42 Average local particle diameter L dp D Characteristic conﬁguration length used in Figure 7.6 L D(x − xf ) Distribution function centered around xf — g(r) Weighting function deﬁned in Equations 7.15, 7.16, and 1/L3 7.17 fL Lower order ﬂux, see Equation 7.134 — fH Higher order ﬂux, see Equation 7.135 — f or F Function — fp Particle density function deﬁned in Equation 7.12 — F External force (such as gravitational or electromagnetic L/t2 force) acting on the particles per unit mass of particles g Gravitational acceleration L/t2 G Distance function, see Equation 7.116 — Speciﬁc enthalpy of i th phase Q hi A a as b

Fundamentals of Turbulent and Multiphase Combustion Copyright © 2012 John Wiley & Sons, Inc.

Kenneth K. Kuo and Ragini Acharya

509

510

BACKGROUND IN MULTIPHASE FLOWS WITH REACTIONS

Symbol I Ii J k m Mk nk ni n N p pk R, r r S or s Sp S˙ψ u V∞ Vf ∞ Vs∞ v t Tgr x, y x or x X xnM

Description

Dimension

Identity matrix Interfacial momentum transfer between the liquid and the gas associated with i th physical process Impulse Stiffness coefﬁcient Limiters Mass Interphase momentum exchange between the phases deﬁned in Equation 7.37 Unit normal associated with phase k Directional cosines associated with unit normal n Number Number Pressure Bulk pressure in phase k Radial coordinate Radial coordinate vector Surface Surface identiﬁed by subscript p Interfacial source of propertyψ Velocity vector in x direction Volume of the local space including solid and ﬂuid, deﬁned in Equation 7.18 Volume occupied by ﬂuid in a local volume Volume of all points occupied by solid matter Velocity vector in y direction Time speciﬁc kinetic energy of the velocity ﬂuctuations (or granular temperature), see Equation 7.70 Spatial coordinate Grid spacing Spatial position vector Material position vector, deﬁned in Equation 7.9 Lagrangian coordinates of a marker M at time n

—

Greek Symbols α Shortest distance from the plane to the origin, deﬁned in Equation 7.137 β interphase drag constant, deﬁned in Equation 7.55 δ Dirac delta function δn Solid angle associated with the change in the direction of n due to relative motion of the particles

M-L/t — M M/L2 -t2 — — — — M/L-t2 M/L-t2 L L L2 — Unit of ψ/t L/t L3 L3 L3 L/t t T L L L L L L ML−3 t−1 Steradians

511

SYMBOLS

Symbol

Description

Dimension

εs ρ μg μs μfriction η σ12 θ

Correction factor, see Equation 7.56 Gas density Dynamic viscosity of gas phase Solids viscosity Frictional solid viscosity Damping coefﬁcient Surface tension coefﬁcient, see Equation 7.103 Collisional transfer contribution, deﬁned in Equation 7.63 Void fraction deﬁned by Equation 7.17 Average particle velocity or kinetic energy of particles Any point property of the solid phase Any point property of the ﬂuid phase-for example, pressure p, one of the three components of the velocity, or one of the nine components of the stress tensor. Volume of ﬂuid function Shear stress tensor of the phase k Rotational velocity Rotational velocity vector for i th particle Incremental value of variable F Gradient operator with respect to spatial coordinate Gradient operator with respect to velocity coordinate

— M/L3 M/L-t M/L-t M/L-t

φ ξ ξ(x, t) ψ(x, t)

τk ωi

DF

∇ ∇v

Subscripts c drag ext f friction g hist int k lift M m p r s

Collision Viscous drag External Fluid phase Frictional Gas phase History or Basset force Interphase Phase k lift generated by the transverse force caused by rotational strain Marker Mass Particle Relative Solids phase

— — L/t or L2 /t2 — Multiple

— M/L-t2 1/t 1/t Units of F 1/L t/L

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BACKGROUND IN MULTIPHASE FLOWS WITH REACTIONS

Symbol t turb

Dimension

Tangential Turbulent

Others

Description

Local mean values averaged over volume Favre averaged ﬂuctuation quantity Reynolds averaged ﬂuctuation quantity

Multiphase ﬂows with and without chemical reactions are important for the optimum design and safe operations of a wide variety of engineering systems. Multiphase ﬂow phenomena can be observed in many engineering systems. Some comprehensive references in this area are the books by Drew and Passman (2005), Kolev (2005), Ishii and Hibiki (2006). This chapter a provides basic understanding of the thermo-ﬂuid dynamic theory of two-phase ﬂow and discusses the numerical treatment of multiphase ﬂows.

7.1

CLASSIFICATION OF MULTIPHASE FLOW SYSTEMS

In the context of ﬂuid mechanics, multiphase ﬂows can be deﬁned simply as any ﬂuid ﬂow system consisting of two or more distinct phases ﬂowing simultaneously in mixture, having some level of phase separation at a scale much larger than the molecular level. Multiphase ﬂows can exist in many different forms. Two-phase ﬂows can be classiﬁed according to the state of the different phases: gas-solid ﬂows, liquid-solid ﬂows, or gas-liquid ﬂows. Gas-solid ﬂows deal with the motion of suspended solid in the gas phase. Depending on the particle number density, such ﬂows can be characterized as either dilute or dense. When the particle number density is relatively small, particle-particle interactions are not important. Such ﬂows are referred to as dilute gas-particle ﬂows, which are governed predominantly by the surface and body forces acting on the particles. For the special case of very dilute gas-particle ﬂows with small particles, the solid particles act as tracers, and they do not contribute in altering the gas ﬂow. When the particle number density is sufﬁciently large, the motion of solid particles is affected by particle-particle interactions. In dense gasparticle ﬂows, collisions between the solid particles can signiﬁcantly inﬂuence the movement of these particles in the gas phase. In bounded ﬂow domains, the motion of solid particles following impact on the boundary walls is also affected by the surface characteristics and material properties. This effect is different from the free ﬂight of solid particles in a gas stream. Gas-particle ﬂows can be referred to as dispersed ﬂows in which the solid particles constitute the dispersed phase. Liquid-solid ﬂows consist of the transport of solid particles in liquid ﬂow. More appropriately called liquid-particle ﬂows or slurry ﬂows, they can also

513

CLASSIFICATION OF MULTIPHASE FLOW SYSTEMS

be categorized as dispersed ﬂows in which the liquid represents the continuous phase. In comparison the liquid and solid phases are mainly driven by and respond to pressure gradients in the same manner, as gas-particle ﬂows, since the density ratio between phases is normally lower than that in the gas-solid ﬂows and the drag between phases is signiﬁcantly high in liquid-solid ﬂows. Of signiﬁcant concern is the sedimentation behavior of solid particles in the liquid ﬂow, which is strongly governed by the size of particles of the dispersed phase and the ﬂow conditions of the continuous phase. Gas-liquid ﬂows can, in principle, assume several different conﬁgurations. One example is the motion of bubbles in a liquid ﬂow, while another is the motion of liquid droplets in a gas stream. These two examples can also be categorized as dispersed ﬂows. For the ﬁrst example, the liquid is taken as the continuous phase and the bubbles are considered as discrete constituents of the dispersed phase. For the second example, the gas is considered as the continuous phase and the droplets as the ﬁnite ﬂuid particles of the dispersed phase. Since bubbles or droplets can deform freely within the continuous phase, they can take on different geometrical shapes: spherical, elliptical, distorted, toroidal, cap, and so on. In addition to dispersed ﬂows, gas-liquid ﬂows often exhibit other complex interfacial structures, called separated ﬂows and mixed or transitional ﬂows. Figure 7.1 summarizes various conﬁgurations that can be found for gasliquid ﬂows (Ishii and Hibiki, 2006). The transitional or mixed ﬂows denote the transition between the dispersed ﬂows and separated ﬂows, which obviously is characterized by the presence of both of these ﬂows. The change of interfacial structures occurs through the presence of bubble-bubble interactions due to coalescence and breakup and any existing phase change process. Free surface ﬂows, which are complicated by the presence of well-deﬁned interfaces, belong to the class of immiscible liquid ﬂows. Strictly speaking, such Flow transition

Dispersed flows

Bubbly

Droplet

Gas bubbles in liquid

Liquid droplets in liquid

Mixed or Transitional flows

Cap, Slug, or Bubbly Churn-turbulent annular

Gas pocket in liquid or gas

Droplet Bubbly droplet annular annular

Separated flows

Film

Annular

Gas core Gas core Liquid film Liquid core Gas with with droplets in gas and gas film bubbles and liquid Gas film Gas core and in liquid droplets in gas liquid film film with and liquid film with gas film bubbles gas core

Figure 7.1 Classiﬁcation of gas-liquid two-phase ﬂows (modiﬁed from Ishii and Hibiki, 2006).

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Physical System

Mathematical System

Modeling

Physical laws & class of materials

Mathematical laws, general and constitutive axioms

Variables, field equations, constitutive equations

Interfacial Conditions

Figure 7.2 Schematic representation of basic procedures used to obtain a mathematical model for a physical system (modiﬁed from Ishii and Hibiki, 2006).

ﬂows do not fall into the class of two-phase ﬂows. For practical purposes, they can be treated as two-phase mixtures. In contrast to the classiﬁcations of twophase ﬂows, free surface ﬂows, which comprise mainly of gas and liquid ﬂows, usually are treated with both phases considered as continuous. Similarly, the process of freezing or solidiﬁcation may also be considered another special case of a two-phase mixture. Here the liquid and solid regions can be treated separately and then coupled through appropriate kinematic and dynamic conditions at the interface. Three-phase gas-liquid-solid ﬂows are also encountered in a number of engineering applications of technical relevance. For this particular class of multiphase ﬂows, the solid particles and gas bubbles can be treated as the discrete constituents of the dispersed phase coﬂowing with the continuous liquid phase. The coexistence of three phases signiﬁcantly complicates the ﬂuid ﬂow due to an array of phenomena associated with particle-particle, bubble-bubble, particle-bubble, particle-ﬂuid, and bubble-ﬂuid interactions modifying the physics of the ﬂow. A schematic diagram showing the basic procedure for modeling the complex multiphase ﬂows for a given physical system is shown in Figure 7.2. More detailed discussions are provided in the following sections. 7.2

PRACTICAL PROBLEMS INVOLVING MULTIPHASE SYSTEMS

Many practical problems involving multiphase ﬂows are broadly featured in a range of modern technological industries as well as in the environment we live in. Examples of multiphase ﬂows based on the different classiﬁcations are listed next. Gas-Particle Flows • Natural sand storms, volcanoes, avalanches • Biological aerosols, dust particles, smoke (ﬁnely soot particles), rain droplets, mist formation • Industrial pneumatic conveyers, dust collectors, ﬂuidized beds, solid propellant rockets, pulverized solid particles, spray drying, spray casting, granular beds, interior ballistics

HOMOGENEOUS VERSUS MULTI-COMPONENT/MULTIPHASE MIXTURES

515

Liquid-Solid Flows • Natural sediment transport of sand in rivers and sea, soil erosion, mud slides, debris ﬂows, iceberg formation • Biological blood ﬂow • Industrial slurry transportation, ﬂotation, ﬂuidized beds, water jet cutting, sewage treatment plants Gas-Liquid Flows • Natural ocean waves • Biological blood ﬂow • Industrial boiling water and pressurized water nuclear reactors, chemical reactor desalination systems, boilers, heat exchangers, internal combustion engines, liquid propellant rockets, ﬁre sprinkler suppression systems Liquid-Liquid Flows • Industrial emulsiﬁers, fuel-cell systems, microchannel applications, extraction systems Gas-Liquid-Solid Flows • Industrial air lift pumps, ﬂuidized beds, oil transportation In all the listed systems, the complex nature of multiphase ﬂows in contrast to single-phase ﬂows is due to the existence of dynamically changing interfaces, signiﬁcant discontinuities of the ﬂuid properties, and complicated ﬂow ﬁeld near the interface. When one or both of the phases becomes turbulent, interactions between the turbulent eddies and the interfacial structures and exchanges between individual phases introduce additional complexities to the ﬂow phenomena. The physics of multiphase ﬂow is also multiscale in nature. It is therefore necessary to account for the cascading effects of the various ﬂow physics at three different scales: 1. Large ﬂow structures encompassing the different individual phases within the ﬂuid ﬂow at the device scale 2. Local structural changes due to agglomeration/coalescence and breakage processes of discrete constituents at the mesoscale 3. Motion of discrete constituents within the continuum ﬂuid at the microscale

7.3 HOMOGENEOUS VERSUS MULTI-COMPONENT/MULTIPHASE MIXTURES

The materials that are deﬁned as homogeneous usually satisfy this criterion: Each part of the material has the same response to a given set of stimuli as all

516

BACKGROUND IN MULTIPHASE FLOWS WITH REACTIONS

of the other parts. An example of such material is pure water. Formulation of governing equations describing the behavior of such homogeneous materials is well understood and is described in several chapters in this book. Many other materials, both manufactured and occurring in nature, are not homogeneous. Such materials often are called mixtures or composites. These materials can exhibit a dichotomy of behaviors. Some inhomogeneous materials are solutions with components that have different response to heat, force, or other external stimuli. Others are composites—that is, the mixing of these substances occurs at a larger scale. For example, human blood, as a ﬁrst approximation, is a suspension of red blood cells in ﬂuid plasma. When this material ﬂows in large vessels, it may be treated as a homogeneous material. In smaller vessels, however, the size of the red blood cells is appreciable with respect to the size of the vessels, and blood cannot be treated as a homogeneous material. In even smaller vessels, the red blood cells have the same size as the vessels; in such cases, yet another type of theoretical treatment is needed. Generally, in order to use a continuum description for multicomponent materials, we attempt to consider that the smallest dimension of the boundaries of a boundary value problem is larger than a typical particle or pore. When this is the case, the idea of a continuum mixture pertains when each particle of the continuum is given a mathematical structure to account for the gross phenomena associated with the heterogeneous nature of the mixture. However, this is not the only way to justify a continuum description. In a later section, we describe various averaging methods to substantiate this idea. Note that a continuum model cannot describe the motions of individual particles or the ﬂow in individual pores. Just as the continuum is an idealization of nature, appropriate to the description of materials on some level of resolution and not to others, the multicomponent mixture theory is an idealization, giving an appropriate description on some level of detail but not on other factors, such as individual particle velocity and surface characteristics. Generally, the terms “multicomponent mixture” and “multiphase systems” are used interchangeably, for example by Drew and Passman (1999). Some groups differentiate between multicomponent and multiphase systems based on the scale of the components. If the components are mixed on a microscopic scale where the mixture can be represents by a bulk density or viscosity, the mixture can be treated as multicomponent. If the different components are mixed on resolvable scales, the mixture should be treated as multiphase.

7.4

CFD AND MULTIPHASE SIMULATION

Computational ﬂuid dynamics (CFD) has become an integral part of engineering design and analysis. For most engineering purposes, it is not necessary or desirable to resolve all details pertaining to turbulent ﬂuctuations, such as in direct numerical simulation (DNS) and large eddy simulation (LES) approaches and the microlevel evolution of the interfaces separating the multiple phases that coexist

CFD AND MULTIPHASE SIMULATION

517

simultaneously in the ﬂuid ﬂow. Owing to the complexities of interfaces and resultant discontinuities in ﬂuid properties as well as to physical scaling issues, it is customary to apply a macroscopic formulation based on some sort of averaging process. By modeling the multiphase ﬂows through a continuum formulation instead, such multiphase models are at the forefront of providing designs with greater efﬁciencies and signiﬁcant production output. Some examples depicting a range of systems where the subject of multiphase ﬂow has immense importance are described next. Consider the engineering system of a spray dryer. The process of spray drying turns a liquid feed, which is atomized, subjected to hot gases, and dried, into the form of a powder. Through the use of turbulence models that entail solving for the turbulent kinetic energy and dissipation of kinetic energy, the effective conservations equations based on proper averaging are solved in conjunction with the turbulent transport equations to handle the turbulent nature of the ﬂow in the continuous phase on an Eulerian mesh. For this particular system, a slurry or concentrated mixture is introduced at the top of the dryer in which the atomizer is of a pressure nozzle type. Hot gases are fed at the top and move downward through the dryer. The dried powder is collected at the bottom and removed as the ﬁnal product. The droplet behavior of the spray normally is modeled via the Lagrangian approach, which allows tracking the history of these droplets. Signiﬁcant variation of drying histories and regions of droplet recirculation or wall impingement can be identiﬁed. Such a computational model allows parametric studies to be carried out to improve the efﬁciency of current designs by optimizing the drying air and feed characteristics as well as the different atomizers and their respective locations within the system and regaining optimum conditions if changes are introduced to the feed composition, gas temperature, and the like. Consider another important engineering system in the form of a cyclone separator. Here the resulting vortex motion in the separator causes the solids to migrate toward the wall due to the centrifugal acceleration and then fall down under gravity into the accumulator vessel situated beneath the separator. Through the use of an advanced turbulence model such as the Reynolds stress model, the effective conservation equations are solved for the continuous gas phase. The determination of powder ﬂow can be simulated using the stochastic Lagrangian model. Such a proposed model provides a feasible way of studying the effects of variables related to operational conditions, cyclone geometry, and particle properties, which is important to the optimum design and control of the cyclone process. Concerning the design and construction of hydraulic systems in rivers, multiphase ﬂow calculations into the complicated nature of river ﬂows in open channels have been performed successfully by Nguyen and Nestmann (2004) to address many navigation issues and ﬂood control management in rivers. For the problem concerning the turbulent ﬂow over Lisdorf ﬂood gate in the Saar River in Germany, free surface tracking method and the volume of ﬂuid (VOF) method were employed to predict the water-free surface of the river ﬂow. Utilizing

518

BACKGROUND IN MULTIPHASE FLOWS WITH REACTIONS

Lisdorf-3D

1.000 0.950 0.900 0.850 0.800 0.750 0.700 0.650 0.600 0.550 0.500

y z

Lisdorf-3D

x

Free Surface Tracking Method

y z

x

Volume of Fluid (VOF) Method

Figure 7.3 Simulation of a river ﬂow with emphasis on prediction of the free surface shape of water (modiﬁed from Nguyen and Nestmann, 2004).

the single ﬂuid representation of the multiﬂuid formulation and a standard two-equation model in the form of the turbulent kinetic energy (k ) and dissipation of kinetic energy (ε) to account for the ﬂow turbulence, Nguyen and Nestmann (2004) have pointed out the advantages and disadvantages associated with each method in calculating the free surface. Figure 7.3 depicts the deformed free surface proﬁles predicted at t = 120 s by the free surface tracking and VOF methods. The free surface tracking method is a front tracking method, in which the free surface moving with the water ﬂow is explicitly determined. For the VOF method, the free surface is predicted implicitly through the advection of the volume fraction in a substantially larger computational domain, which extends almost twice the height of the domain used for the free surface tracking method. The shape of the free surface from both methods was found to be comparable at t = 120 s, but after t = 120 s, the free surface tracking method broke down because of strong warped elements appearing at the free surface while the VOF method continued unabated. These examples represent merely the tip of the iceberg in many exciting multiphase applications across different engineering systems. Enormous multiphase modeling efforts also are being invested in the nuclear, oil, gas, and petrochemical industries. Recent advances in multiphase modeling capabilities and increasing computational power have allowed us to incorporate more sophisticated models to better resolve the transport of complex multiphase ﬂow phenomena. Currently, a greater emphasis is placed in fostering the next stages of development and application of models in resolving turbulent multiphase ﬂows of technical relevance by direct numerical simulation. Doing this implies calculating such ﬂows at sufﬁciently high spatial and temporal resolutions, which is subject to the availability of computational resources. The computational domain

CFD AND MULTIPHASE SIMULATION

519

Liquid flow

Large eddies

Solid particles

Small eddies

Gas bubbles

Slug bubbles

Figure 7.4 A schematic representation of a gas-liquid-solid turbulent ﬂow (modiﬁed from Yeoh and Tu, 2010).

requires resolution of the largest and the smallest turbulent eddies and the exact location of the interfaces separating different phases that coexist within the ﬂow to be determined through suitable microlevel evolutionary tracking methods. Alternatively, the LES based approach is also used. In this approach, the structure of the turbulent ﬂow is viewed as the distinct transport of large- and small-scale motions, as illustrated in Figure 7.4. On this basis, the large-scale motion is directly simulated on a scale as the underlying computational mesh will allow; the small-scale motion is modeled accordingly. Since the large-scale motion is generally much more energetic and by far the most effective means of transport of the conserved properties than the small-scale ones, such an approach, which treats the large eddies exactly but approximates the smaller scales, is a viable one for turbulent modeling. In comparison to the multiﬂuid model, large computational resources are required for DNS. The computational requirement for LES is intensive but still not as costly when compared to DNS. A schematic drawing highlighting the trade-off between the computational effort and the modeling complexity of different approaches is shown in Figure 7.5. For the multiﬂuid model, calculations generally are performed for physical processes that occur at length scales larger than the integral length scale (x > l0 ), which are captured by the effective transport equations; those occurring at length scales smaller than the resolved length scale require modeling—additional Reynolds and scalar stress terms appearing within the averaged equations must be modeled. For DNS, numerical calculations

520

BACKGROUND IN MULTIPHASE FLOWS WITH REACTIONS Modeling complexity

Computational effort

High

DNS LES

Multi-fluid or Multiphase models

Low Kolmogorov length scale η

Integral length scale 0

t

Figure 7.5 Representation of the trade-off between the computational effort and the modeling complexity of different approaches (modiﬁed from Yeoh and Tu, 2010).

are performed on the length scales smaller than the Kolmogorov length scale (i.e., x < η). 7.5

AVERAGING METHODS

It is well established in continuum mechanics that the conceptual models for single-phase ﬂow of a gas or of a liquid are formulated in terms of ﬁeld equations describing the conservation laws of mass, momentum, energy, and the like. These ﬁeld equations are then complemented by appropriate constitutive relationships, such as the constitutive equations of state, stress, and chemical reactions, which specify the thermodynamic, transport, and chemical properties of a given constituent material (e.g., of a speciﬁed solid, liquid or gas). It is to be expected, therefore, that the conceptual models describing the steady-state and dynamic characteristics of multiphase or multicomponent media should also be formulated in terms of the appropriate ﬁeld equations and constitutive relationships. However, the derivation of such equations for the ﬂow of structured media is considerably more complicated than for strictly continuous homogeneous media for single-phase ﬂow. In order to appreciate the difﬁculties in deriving balance equations for structured—inhomogeneous—media with interfacial discontinuities, recall that in continuum mechanics, the ﬁeld theories are constructed on integral balances of mass, momentum, and energy. Thus, if the variables in the region of integration are continuously differentiable and the Jacobian transformation between material and spatial coordinates exists, then the Euler-type differential balance can be obtained by using the Leibnitz’s rule; more speciﬁcally, however, the Reynolds’ transport theorem allows us to interchange differential

AVERAGING METHODS

521

and integral operations. The rule for differentiation under the integral sign is named after Gottfried Leibniz and states:

d dy

x2 (y)

x2 (y)

f (x, y) dx = x1 (y)

x1 (y)

dx2 dx1 ∂ f (x, y) dx + f (x2 (y), y) − f (x1 (y), y) ∂y dy dy (7.1)

In multiphase or multicomponent ﬂows, the presence of interfacial surfaces introduces great difﬁculties in the mathematical and physical formulation of the problem. From the mathematical point of view, a multiphase ﬂow can be considered as a ﬁeld that is subdivided into single-phase regions with moving boundaries separating the constituent phases. The differential balance of mass, momentum, and energy holds for each subregion. It cannot be applied, however, to the set of these subregions in the normal sense without violating the conditions of continuity. From the point of view of physics, the difﬁculties encountered in deriving the ﬁeld and constitutive equations appropriate to multiphase ﬂow systems stem from the presence of the interface and from the fact that both the steady and dynamic characteristics of multiphase ﬂows depend on the interfacial structure of the ﬂow. For example, the steady-state and the dynamic characteristics of dispersed two-phase ﬂow systems depend on the collective dynamics of solid particles, bubbles, or droplets interacting with each other and with the surrounding continuous phase. In the case of separated ﬂows, these characteristics depend on the structure and wave dynamics of the interface. In order to determine the collective interaction of particles and the dynamics of the interface, it is necessary to describe ﬁrst the local properties of the ﬂow and then to obtain a macroscopic description by means of appropriate averaging procedures. For dispersed ﬂows, it is necessary to determine the rates of nucleation, evaporation or condensation, motion and disintegration of single droplets (bubbles), as well as the collisions and coalescence processes of droplets (or bubbles). For separated ﬂows, the structure and the dynamics of the interface greatly inﬂuence the rates of mass, heat, and momentum transfer as well as the stability of the system. For example, the performance and ﬂow stability of a condenser for space application depend on the dynamics of the vapor interface. Similarly, the rate of droplet entrainment from a liquid ﬁlm and, therefore, the effectiveness of ﬁlm cooling, depend on the stability of the vapor-liquid interface. It can be concluded from this discussion that in order to derive the ﬁeld and constitutive equations appropriate to structured multiphase ﬂow, it is necessary to describe the local characteristics of the ﬂow. From that ﬂow, the macroscopic properties should be obtained by means of an appropriate averaging procedure. The formulation based on the local instant variables results in a multiboundary problem with the positions of the interfaces unknown. In such a case, the mathematical difﬁculties encountered in obtaining solutions are prohibitively great and computationally intractable. To appreciate these difﬁculties, recall that even in single-phase turbulent ﬂow without moving interfaces, it has not been possible

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BACKGROUND IN MULTIPHASE FLOWS WITH REACTIONS

to obtain exact solutions expressing local instant ﬂuctuations. Overwhelming difﬁculties encountered in the local instant formulations stem from: 1. Existence of the multiple deformable moving interfaces with their motions being unknown. 2. Existence of the ﬂuctuations of variables due to turbulence and to the motions of the interfaces. 3. Signiﬁcant discontinuities of properties at interface. The ﬁrst effect causes complicated coupling between the ﬁeld equations of each phase and the interfacial conditions, whereas the second effect inevitably introduces a statistical characteristic originated from the instability of the NavierStokes equations and of the interfacial waves. The third effect introduces huge local jumps in various variables in space and time. Since these difﬁculties exist in almost all two-phase ﬂow systems, an application of the local instant formulation to obtain a solution is severely limited. For a system with a simple interfacial geometry, as in the case of problem involving a single or several bubbles or of a separated ﬂow, local instant formulation has been extensively used and useful information have been obtained. As most two-phase ﬂow observed in practical engineering systems have extremely complicated interfacial geometry and motions, it is extremely hard to solve for local instant motions of the ﬂuid particles. By proper averaging, the mean values of ﬂuid motions and properties can be obtained that effectively eliminate local instant ﬂuctuations. The averaging procedure can be considered as low-pass ﬁltering, excluding unwanted highfrequency signals from local instant ﬂuctuations. However, it is important to note that the statistical properties of these ﬂuctuations inﬂuencing the macroscopic phenomena should be taken into account in a formulation based on averaging. Various methods of averaging can be applied to two-phase ﬂow in particular. Depending on the basic physical concepts used to formulate thermal-hydraulic problems, averaging procedures can be classiﬁed into three main groups: 1. Eulerian averaging 2. Lagrangian averaging 3. Boltzmann statistical averaging They can be further divided into subgroups based on a variable with which a mathematical operator of averaging is deﬁned. The next sections summarize the classiﬁcations and the deﬁnitions of various averaging. 7.5.1

Eulerian Average—Eulerian Mean Values

Function : F = F (t, x) 1 F (t, x) dt Temporal mean : t t

(7.2) (7.3)

AVERAGING METHODS

1 Spatial mean : R Volume averaging :

523

R

1 V

F (t, x) dR

(7.4)

V

F (t, x) dV

(7.5)

1 F (t, x) dA A A 1 F (t, x) dC Line averaging: C C

(7.6)

Area averaging :

Statistical mean or ensemble averaging :

(7.7) N 1 Fn (t, x) N

(7.8)

n=1

The most important and widely used group of averaging in continuum mechanics is the Eulerian averaging, because it is closely related to human observations and most instrumentation. The basic concept underlining this method is the timespace description of physical phenomena. In the Eulerian description, the time and space coordinates are taken as independent variables, and various dependent variables express their changes with respect to these coordinates. Since the standard ﬁeld equations of continuum mechanics adapt to this description, it is natural to consider averaging with respect to these independent variables (i.e., the time and the space). Furthermore, these averaging processes are basically integral operators; therefore, they have the effect of smoothing out instant or local variations within a domain of integration. 7.5.2

Lagrangian Average—Lagrangian Mean Values

Function : F = F (t, X); X = X(x, t) 1 Time (temporal) mean : F (t, X) dt t t Statistical mean or ensemble averaging :

(7.9) (7.10)

N 1 Fn (t, X) N

(7.11)

n=1

where x and X are the spatial and the material coordinates, respectively. Note that the true time or statistical averaging is deﬁned by taking the limit t → ∞ or N → ∞, which is possible only conceptually. The material coordinates can be considered as the initial positions of all the particles; thus, if X is ﬁxed, it implies the value of a function following a particle. The Lagrangian mean values are directly related to the Lagrangian description of mechanics. As the particle coordinate X displaces the spatial variable x of the Eulerian description, this averaging is naturally ﬁtted to a study of the dynamics of a particle. If our interest is focused on a behavior of an individual particle rather than on the collective mechanics of a group of particles, the Lagrangian average is important

524

BACKGROUND IN MULTIPHASE FLOWS WITH REACTIONS

and useful for analyses. The Lagrangian time average is taken by following a certain particle and observing it over some time interval. In contrast to the mean values just explained the Eulerian and the Lagrangian statistical mean values are based on a statistical assumption, since they involve a collection of N similar samples denoted by Fn with n = 1, 2, N. It is useful to consider a time averaging as a ﬁltering process that eliminates ﬂuctuations. Similar samples can be considered as a group of samples that have time mean values of all the important variables within certain ranges of deviations. In this case, the time interval of the averaging and the ranges of deviations deﬁne the ﬂuctuations; thus, statistical averaging depends on the time interval or the number of samples. 7.5.3

Boltzmann Statistical Average

Particle density function : fp = fp (x, ξ, t) ψ(ξ )fp dξ Thermodynamic or ﬂow properties : ψ(t, x) = fp dξ

(7.12) (7.13)

where ξ is the average particle velocity or kinetic energy of particles. For a steady-state ﬂow based on time averaging, random sampling over a time domain can constitute a proper set of samples as it is often done in experimental measurements. In this case, the time averaging and the statistical averaging are equivalent. There are many other factors to consider. Difﬁculties arise when the constitutive equations are studied in connection with experimental data. True statistical averaging involving an inﬁnite number of similar samples is possible only conceptually; it cannot be realized. Thus, if it is considered alone, the ensemble averaging faces two difﬁculties: choosing a group of similar samples and connecting the experimental data to a model. The Boltzmann statistical averaging with a concept of the particle number density is important when the collective mechanics of a large number of particles are in question. As the number of particles and interactions between them increase, the behavior of any single particle becomes so complicated and diversiﬁed, it is not practical to solve for each particle. In such a case, the behavior of a group of many particles increasingly exhibits some particular characteristics that are different from a single particle as the collective particle mechanics becomes a governing factor. It is well known that Boltzmann statistical averaging applied to a large number of molecules with an appropriate mean-free path can lead to ﬁeld equations that closely resemble that of the continuum mechanics. This is achieved by ﬁrst writing the balance equation for the particle density function, which is known as the Boltzmann transport equation (Cercignani, 1988). Then it is necessary to assume a form of the particle interaction term as well as stochastic characteristics of the particle density function. The Boltzmann transport equation is given as: δfp ∂fp (7.14) + v · ∇fp + F · ∇v fp ≡ ∂t δt Collision

AVERAGING METHODS

525

where the term F = external force (such as gravitational or electromagnetic force) acting on the particles per unit mass of particles. ∇v = gradient operator with respect to velocity coordinate. The quantity fp dx dv = probable number of particles in the position range d x about the spatial location x and with velocities in the range d v about the velocity v at time t. The term on the right-hand side of Equation 7.14 represents the rate of change of particle number density due to particle-particle collision. Equation 7.14 has a form similar to the standard conservation equations of mass, momentum, and energy in continuum mechanics. 7.5.4

Anderson and Jackson’s Averaging for Dense Fluidized Beds

For ﬂuidized beds with dense small particles, Anderson and Jackson (1967) introduced a volume averaging procedure in Eulerian framework by using a weighting function g. The basis of their work was the replacement of point mechanical variables, such as ﬂuid velocity, ﬂuid pressure, or velocity of solid matter at a speciﬁed point within a particle, by local mean variables obtained by averaging the point variables over regions that contain many particles but are still small compared with the scale of “macroscopic” variations from point to point in the system. The weighting function g(r) deﬁned for r > 0 and t > 0 has these ﬁve properties: 1. g(r) ≥ 0 for all r, and g decreases monotonically with increasing r. 2. g(r) possesses derivatives of all orders for each value of r. 3. V∞ g (n) (r) dV exists for all values of n where g (n) is the n th derivative of g, r denotes the distance from a point in three-dimensional space, and V∞ represents the volume of whole space (including solid and ﬂuid regions). 4. The weighting function is normalized so that g(r) dV = 1 (7.15) V∞

5. A radius r0 of the weighting function is deﬁned such that: r0

∞ g(r)r dr = 4π

g(r)r 2 dr =

2

4π

1 2

(7.16)

r0

0

A local void fraction φ is deﬁned at a time t and location x as: φ(x, t) = g(x − y) dVy Vf ∞

(7.17)

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BACKGROUND IN MULTIPHASE FLOWS WITH REACTIONS

where Vf ∞ (t) is the volume occupied by the ﬂuid at time t and dVy is an elemental volume in the neighborhood of point x. If the set of all points occupied by solid matter at time t (i.e., all points lying in particles) denoted by Vs∞ (t), then V∞ = Vf ∞ + Vs∞ (7.18) g(x − y) dVy = g(x − y) dVy − g(x − y) dVy 1 − φ(x, t) = 1 − Vf ∞

=

Vs∞

V∞

Vf ∞

g(x − y) dVy

(7.19)

Let ψ(x, t) denote any point property of the ﬂuid phase—for example, pressure p, one of the three components of the velocity, or one of the nine components of the stress tensor. In the same way, let ξ(x, t) denote any point property of the solid phase. Then the local mean values of ψ and ξ are deﬁned by: Vf ∞ (t) g(x − y)ψ(y, t) dVy Vf ∞ (t) g(x − y)ψ(y, t) dVy ψ(x, t) = = (7.20) φ(x, t) Vf ∞ g(x − y) dVy Vs∞ (t) g(x − y)ξ(y, t) dVy V (t) g(x − y)ξ(y, t) dVy ξ(x, t) = = s∞ (7.21) 1 − φ(x, t) Vs∞ g(x − y) dVy The spatial derivatives of the mean quantities can be expressed by using differentiation by parts, Gauss divergence theorem, and the relationship ∂g(x − y)/∂xk = −∂g(x − y)/∂yk : ∂ ∂ [φ(x, t) ψ(x, t) ] = g(x − y) ψ(y, t) dVy ∂xi ∂y i Vf ∞ (t) − ψ(y, t)g(x − y)ni dSy (7.22) Sf (t)

where

Sf (t) = surface bounding the ﬂuid phase at time t; Sf (t) can have a number of disjoint parts, including the surface Sf ∞ bounding the whole system and Sp , the surfaces of the separate solid particles at time t ni = outward normal at the surface Sy .

Therefore, ψ(y, t)g(x − y)ni dSy = Sf (t)

Sf ∞ (t)

−

ψ(y, t)g(x − y)ni dSy

p→∞ Sp (t)

ψ(y, t)g(x − y)ni dSy

(7.23)

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Now, provided the shortest distance from point x to the surface Sf ∞ is considerably larger than the distance r0 over which the weighting function has nonzero value, the ﬁrst term on the right-hand side of Equation 7.23 will be considerably smaller than the second term. Neglecting this term and substituting Equation 7.23 into Equation 7.22, we have: ∂ ∂ [φ(x, t) ψ(x, t) ] = g(x − y) ψ(y, t) dVy ∂xi ∂y i Vf ∞ (t) + ψ(y, t)g(x − y)ni dSy (7.24) p→∞ Sp (t)

The time derivative of the mean quantities can be expressed as: ∂ ∂ [φ(x, t) ψ(x, t) ] = g(x − y)ψ(y, t) dVy ∂t ∂t Vf ∞ (t)

(7.25)

Using Leibnitz’s rule, the right-hand side of Equation 7.25 can be written as: ∂

ψ(y, t)g(x − y)vi ni ds g(x − y)ψ(y, t) dVy + Vf ∞ (t) ∂t Sf (t) The second term in the above expression can be approximated for integration over the surface Sp , as already explained. Also note that the integrand of the ﬁrst term in the same equation is equal to g(x − y)

∂ ψ(y, t) ∂t

By using these two relationships, Equation 7.25 can be expressed as: ∂ ∂ [φ(x, t) ψ(x, t) ] g(x − y) ψ(y, t) dVy = ∂t ∂t Vf ∞ (t) + ψ(y, t)g(x − y)vi ni dSy p→∞ Sp (t)

(7.26)

The average interface surface area per unit volume can be considered as: (1 − φ) Sp p→∞ gdS = (7.27) Vp p→∞ Sp p→∞

The surface average of a gas-phase property ψ is deﬁned as: Sp gψdS p→∞ ψ S-avg ≡ Sp gdS p→∞

(7.28)

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BACKGROUND IN MULTIPHASE FLOWS WITH REACTIONS

In general, the surface-average values are quite different from the overall average values. The ﬂuctuating properties ψ are introduced as: ψ = ψ + ψ

(7.29)

If it is assumed that ψ = ψ , we get

Finally,

ψ =0

(7.30)

φψ = φ ψ + φ ψ

(7.31)

In certain cases, the quantity φ ψ can be assumed to be zero. In order to study two-phase ﬂow systems, many of these averaging methods have been used by various researchers. The applications of averaging can be divided into two main categories: 1. To deﬁne averaged properties and then to correlate experimental data 2. To obtain usable ﬁeld and constitutive equations that can be used to predict macroscopic processes The most elementary use is to deﬁne mean properties and motions that include various kinds of concentrations, density, velocity, and energy of each phase or of a mixture. These properly deﬁned mean values then can be used for various experimental purposes and to develop empirical correlations. The choice of averaging and instrumentation is closely coupled since, in general, measured quantities represent some kinds of mean values themselves. Both the Eulerian time and spatial averaging are used frequently, because experimentalists incline to consider two-phase mixtures as quasi-continuum. Furthermore, Eulerian time-and spatial-averaged quantities are usually the easiest mean values to measure in ﬂuid ﬂow systems. However, when a particular ﬂuid particle is distinguishable and traceable, as in the case of a bubbly or droplet ﬂow, the Lagrangian mean values are also measured. It is only natural that these mean values are obtained for stationary ﬂow ﬁelds that can be considered to have steady-state characteristics in terms of mean values. Various correlations are then developed by further applying the statistical averaging among different data. This is the standard method of experimental physics to minimize errors. Before we proceed to the second application of averaging, we discuss brieﬂy two fundamentally different formulations of the macroscopic ﬁeld equations: the two-ﬂuid model and the mixture (drift-ﬂux) model. The two-ﬂuid model is formulated by considering each phase separately. Consequently, it is expressed by two sets of conservation equations of mass, momentum, and energy. Each of these six ﬁeld equations (with momentum equations in vector form) invariably has an interaction term coupling the two phases through jump conditions (described in Section 7.9). The mixture

AVERAGING METHODS

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(or drift-ﬂux) model is formulated by considering the mixture as a whole. Thus, the mixture model is expressed in terms of three conservation equations of mass, momentum, and energy with one additional diffusion equation, which accounts for the concentration gradients in the mixture. However, a proper mixture model should be formulated in terms of correctly deﬁned mixture quantities. The drift-ﬂux model is an example of a mixture model that includes a diffusion model, a slip ﬂow model, and an homogeneous ﬂow model. Hibiki and Ishii (2003) and Hibiki, Takamasa, and Ishii (2004) have found the drift-ﬂux model to be the most effective mixture model; it is highly developed for normal gravity (Ishii, 1977) and for microgravity conditions for practical applications. The second and more important application of these averaging methods is to obtain the macroscopic two-phase ﬂow ﬁeld equations and the constitutive equations in terms of mean values. Here, again, the Eulerian spatial and time averaging have been used extensively by various authors, although Eulerian or Boltzmann statistical averaging has also been applied. By using the Eulerian volumetric averaging, signiﬁcant contributions have been made by Zuber (1964), Zuber, Staub, and Bijwaard (1964), Wundt (1967), Delhaye (1968), and Slattery (1972) for the establishment of a three-dimensional model of highly dispersed ﬂows. These analyses were based on a volume element that contains both phases. Moreover, it was considered to be much smaller than the total system of interest; thus, its main applications are for highly dispersed ﬂows. It has long been realized that the Eulerian area averaging over a cross section of a duct is very useful for engineering applications, since ﬁeld equations reduce to a one-dimensional model. By area averaging, the information on changes of variables in the direction normal to the main ﬂow is basically lost. Therefore, the transfer of momentum and energy between the wall and the ﬂuid should be expressed by empirical correlations or by simpliﬁed models that replace the exact interfacial conditions. Even in single-phase ﬂow problems, the area-averaging method is widely used because its simplicity is highly desirable in many practical engineering applications. For example, the use of the wall friction factor or the heat transfer coefﬁcient is closely related to the concept of area averaging. A good review of single-phase ﬂow area averaging and macroscopic equations that correspond to the open-system equations in thermodynamics can be found in Bird, Stewart, and Lightfoot (1960), Whitaker (1968), and Slattery (1972). The boundary-layer integral method of von K´arm´an is also an ingenious application of the area averaging. Furthermore, numerous examples of area averaging can be found in the literature on lubricating ﬁlms, open channel ﬂow, and shell theories in mechanics. However, in applications to two-phase ﬂow systems, many authors used phenomenological approach rather than mathematically exact area averaging; thus, the results of Martinelli and Nelson (1948), Kutateladze (1952), Levy (1960), Brodkey (1967), and Wallis (1969) are in disagreement with each other, and none of them is complete (KocamustafaoguUari, 1971). The rational approach to obtain a one-dimensional model is to integrate single-phase differential ﬁeld equations over the cross-sectional area. A rigorous

530

BACKGROUND IN MULTIPHASE FLOWS WITH REACTIONS

derivation of one-dimensional mixture ﬁeld equations with an additional diffusion equation—the drift-ﬂux model—was carried out by Zuber, Staub, and Bijwaard (1964) and Zuber (1967). Their results show a signiﬁcant similarity with the ﬁeld equations for heterogeneous chemically reacting single-phase systems. The latter had been developed as the thermo-mechanical theory of diffusion based on the interacting continua, which have the equal probability of existing at all spatial locations and at all instants but having two different velocities. Numerous authors have contributed to this theory, including Fick (1855), Stefan (1871), von K´arm´an (1950), Prigogine and Mazur (1951), Hirschfelder et al. (1954), Truesdell and Toupin (1960), and Truesdell (1969). A similar result obtained by Maxwell (1867) from an entirely different method of the kinetic theory of gas mixtures should also be noted here. Eulerian time averaging, which has been widely applied in analyzing a singlephase turbulent ﬂow, is also used for two-phase ﬂow. In applying the time averaging method to a mixture, many authors coupled it with other space averaging procedures. Important contributions have been made by Russian researchers (Teletov, 1945; Frankl, 1953; Teletov, 1957; Diunin, 1963), who have used the Eulerian time-volume mean values and obtained three-dimensional ﬁeld equations. An analysis based on the Eulerian time averaging alone was apparently initiated by Vernier and Delhaye (1968); however, they did not provide a detailed study leading to a mathematical formulation. Panton (1968) derived the mixture model by performing the time integration ﬁrst and then volume integration. His analysis was more explicit in its integration procedures than the works by the Russian researchers, but both results were quite similar. In a technical article by Ishii (1971), a two-ﬂuid model formulation including the surface source terms was obtained by using the time averaging alone, then the area averaging over a cross section of a duct was carried out. Ishii identiﬁed all the constitutive equations as well as boundary conditions, which should be speciﬁed in a standard onedimensional two-phase ﬂow model. Drew (1971) also performed an extensive study by using an Eulerian multiple mixed averaging procedures. In his analysis, two integrals over both space and time domains were taken in order to smooth out higher-order singularities. These multiple integral operations are equivalent to the continuum assumption. The averaging procedure should not be considered as a pure mathematical transformation, since the constitutive model can be developed only based on the continuum assumption. Readers may wish to refer to Delhaye (1969, 1970), who provides various models based on Eulerian space averaging as well as a comprehensive review on the subject. Eulerian time averaging is particularly useful for turbulent or dispersed two-phase ﬂow for two-phase ﬂows (Ishii, 1975, 1977; Ishii and Mishima, 1984). In these ﬂows, the transport processes are highly dependent on the local ﬂuctuations of variables about the mean. Therefore, the constitutive equations are applicable to time-averaged expperimental data. This conclusion is also supported by the standard single-phase turbulent ﬂow analysis. An extensive study using Eulerian statistical averaging was carried out by Vernier and Delhaye (1968) in which they reached to an important conclusion that

AVERAGING METHODS

531

under stationary ﬂow condition, the ﬁeld equations from true time averaging (i.e., the temporal averaging with t → ∞) and the ones from statistical averaging are identical. Boltzmann statistical averaging has also been used by several authors (Buevich, 1969; Buyevich, 1972; Culick, 1964; Kalinin, 1970; Murray, 1954; Pai, 1971) for highly dispersed two-phase ﬂow systems. In general, the particle density functions are considered, then the Boltzmann transport equation for the functions can be used. Kalinin (1970) assumed that the particle density functions represent the expected number of particles of a speciﬁed mass and velocity, whereas Pai (1971) considered the radius, velocity, and temperature as the arguments of the functions. A simpliﬁed version of Maxwell’s equation of transport for each phase has been obtained from the Maxwell-Boltzmann equation by integrating over the arguments of the particle density function except time and space variables. Since it involves assumptions on the distributions as well as on inter-particle and particle-gas interaction terms, the results are not general but represent a special kind of continuum. Three different methods and views of mechanics of mixtures can be represented in a local sense: 1. The Eulerian time or statistical averaging applied to two-phase mixtures 2. The thermo-mechanical theory of diffusion based on two continua 3. The Boltzmann statistical averaging applied to gas mixtures or to highly dispersed ﬂows The ﬁrst theory considers the mixture to be essentially a group of single-phase regions bounded by interfaces, whereas in the second theory the two components coexist at the same point and time. In contrast to these two theories, which are based on continuum mechanics, the last theory is based on statistical expectations and probability. The more signiﬁcant point is that if each transfer term of these models is interpreted correctly, the resulting ﬁeld equations have very similar forms. A preliminary study using ensemble cell averaging was carried out by Arnold, Drew, and Lahey (1989), where they derived turbulent stress and interfacial pressure forces due to pressure variations over the surface of nondistorting bubbles for an idealized inviscid bubbly ﬂow. They discussed deﬁciencies inherent in spatial averaging techniques and recommended ensemble averaging for the formulation of two-ﬂuid models of two-phase ﬂows. Zhang and Prosperetti (1994a) derived averaged equations governing a mixture of equal spherical compressible bubbles in an inviscid liquid by the ensemble averaging method. They concluded that the method was systematic and general because of no ad hoc closure relations required and suggested that the method might be applied to a variety of thermo-ﬂuid and solid mechanics situations. Zhang and Prosperetti (1994b) extended this method to the case of spheres with a variable radius. Zhang (1993) summarized the other applications to heat conduction and convection, Stokes ﬂow, and thermocapillary process. Readers may wish to refer to Prosperetti (1999), which provides some considerations regarding the modeling of disperse

532

BACKGROUND IN MULTIPHASE FLOWS WITH REACTIONS

multiphase ﬂows by averaged equations. Kolev (2002) presented a two-phase ﬂow formulation mostly for development of safety analysis codes based on the multiﬁeld approach. The Lagrangian averaging to two-phase ﬂow systems is a useful approach for particulate ﬂow. However, in general, it faces considerable difﬁculties due to diffusion and phase changes. For particulate ﬂow without phase changes, the Lagrangian equation of the mean particle motion can be obtained in detail for many practical cases. Thus, the Lagrangian description of singleparticle dynamics often is used as a momentum equation for a particulate phase in a highly dispersed ﬂow (Carrier, 1958; Zuber, 1964). Many analyses on the bubble rise and terminal velocity use the Lagrangian time averaging implicitly, particularly when the continuum phase is in the turbulent ﬂow regime. Following Anderson and Jackson (1967), Slattery (1967, 1990), and Whitaker (1967), a control volume is allocated to every point in space, and the thermodynamic and ﬂow properties averaged for each velocity ﬁeld are assigned to the center of the control volume. Since there is a control volume associated with every point in space, a ﬁeld of average values can be generated for all thermodynamic and ﬂow properties. The ﬁeld of average properties is therefore smooth, and spatial derivatives of these averaged properties exist. Consider the control volume Vol shown in Figure 7.6, which is a ﬁxed number. Each phase in this control volume occupies a certain volume, which changes with time, whereas Vol remains unchanged with time. Let us consider the liquid phase with multiple droplets, which has a representative characteristic length D (e.g., droplet size). This length is much larger than the molecular mean free path for either liquid or gas. The size of the computational region (scr) of interest is much larger than the size of the representative characteristic length D and also larger than the spatial changes in the ﬂow parameters of interest. The choice of the size of the control volume, which is of order of Vol1/3 , has a major impact on the meaning of the averaged values. From the various cases possible, we consider two: 1. The size of the control volume is larger than the characteristic conﬁguration length D of the ﬁeld l ; that is, 1

molecular mean free path D ≈ Vol 3 scale of the computational region Vol1/3

Δy D Δx Scale of computational region

Figure 7.6 Comparison of possible scales of local volume averaging, scale of the measuring devices, scale of the computational region, and the global ﬂow dimensions (modiﬁed from Kolev, 2007).

LOCAL INSTANT FORMULATION

Ug Ul

533

Ug Ul (a)

(b)

Figure 7.7 Schematic diagram of (a) smooth stratiﬁed ﬂow and (b) wavy stratiﬁed ﬂow (modiﬁed from http://drbratland.com/PipeFlow2/chapter1.html).

This choice of scales is useful for dispersed ﬂows with ﬁne particles, as shown in Figure 7.6. 2. The size of the control volume is comparable to the characteristic length Lc of either the liquid or gas phase; that is, 1

molecular mean free path Lc ≈ Vol 3 size of the computational region This choice of scales is useful for direct numerical simulation or for simulation of ﬂow pattern with stratiﬁcation (see Figure 7.7).

7.6

LOCAL INSTANT FORMULATION

The singular characteristic of two-phase or of two immiscible mixtures is the presence of one or several interfaces separating the phases or components. Examples of such ﬂow systems can be found in a large number of engineering systems as well as in a wide variety of natural phenomena. The understanding of the ﬂow and heat-transfer processes of two-phase systems has become increasingly important in mechanical, chemical, aerospace, and nuclear engineering as well as in environmental and medical science. In analyzing two-phase ﬂows, we ﬁrst follow the standard method of continuum mechanics. Thus, a two-phase ﬂow is considered a ﬁeld that is subdivided into single-phase regions with moving boundaries between phases. The standard differential balance equations hold for each subregion with appropriate jump and boundary conditions to match the solutions of these differential equations at the interfaces. Hence, in theory, it is possible to formulate a two-phase ﬂow problem in terms of the local instant variable, namely, F = F (x, t). This formulation is called a local instant formulation in order to distinguish it from formulations based on various methods of averaging. Such a formulation would result in a multiboundary problem with the interface positions being unknown due to the coupling of the ﬁelds and the boundary conditions. Indeed, mathematical difﬁculties encountered by using this local instant formulation can be considerable; in many cases, they may be insurmountable. However, there are two fundamental advantages in the local instant formulation. The ﬁrst is the direct application to study separated ﬂows, such as ﬁlm, stratiﬁed, annular and jet ﬂow. The formulation can be used in such cases to study pressure drops, heat transfer, phase changes, the dynamics and stability of

534

BACKGROUND IN MULTIPHASE FLOWS WITH REACTIONS

an interface, and the critical heat ﬂux. In addition, other applications of this approach are to problems of single or several bubble dynamics, the growth or collapse of a single bubble or droplet, and solidiﬁcation and melting. The local instant formulation also is useful as a fundamental base of the macroscopic twophase ﬂow models using various averaging procedures. When each subregion bounded by interfaces can be considered as a continuum, the local instant formulation is mathematically rigorous. Consequently, two-phase ﬂow models can be derived from this formulation by appropriate averaging methods. Thus, if each subregion bounded by interfaces in two-phase systems can be considered as a continuum, the validity of local instant formulation is evident. By accepting this assumption, we can derive ﬁeld equations, constitutive laws, and interfacial conditions. An interface is a singularity in a continuous ﬁeld since two different values (corresponding to each phase) for each ﬂow property can exist at the interface. The balance at an interface that corresponds to the ﬁeld equation is called a jump condition (described in Section 7.9). As shown in Section 7.1, multiphase ﬂows exist in various forms. To properly treat these different types of multiphase ﬂows, suitable modeling strategies are required to speciﬁcally tailor and accommodate the variety of phenomena in which all phases are allowed to interact within the ﬂow system. From a computational ﬂuid dynamics perspective, the trajectory and two-ﬂuid models are commonly applied in the treatment of different multiphase ﬂows. Nevertheless, there are considerable differences in the application of these models. Consider the example of a ﬂow system where there are distinct small entities, such as ﬁnite solid particles, liquid drops, or gaseous bubbles (the disperse phase), distributed in the volume of a carrier medium, which can be treated as continuous phase. In the trajectory models, the motion of the transported disperse phase is determined by tracking the motion of either the actual particles or an ensemble of representative particles. The details of the ﬂow around each of the particles are affected by the drag, lift, and other forces that could signiﬁcantly act on and alter the trajectory of these particles. Additionally, the thermal history and mass exchange due to phase change can also be recorded during the course of motion. In the two-ﬂuid model, the transported dispersed phase is treated as another continuous phase intermingling and interacting with the carrier medium. This approach ignores the discrete nature of the disperse phase and thereby approximates its effects on the carrier medium like another ﬂuid acting on the continuous phase in the ﬂow system. Conservation equations of mass, momentum, and energy are developed and solved for each phase. Inherently, the adoption of averaging is necessary to characterize the properties of the continuous and disperse phases. In contrast, consider another example where the identiﬁcation of interfaces needs to be precise for a ﬂow system having two or more continuous phases of different ﬂuids. This category of multiphase ﬂows can normally be treated via two modeling approaches. First, the formulation may be simpliﬁed by solving the equations governing single-phase ﬂuid ﬂow in separate phases and coupling them through appropriate kinematic and dynamic conditions at the interface. The

LOCAL INSTANT FORMULATION

535

geometries of each phase are deformed continuously by the explicit tracking of the shape and position of the interface. Second, the so-called mixture formulation can be adopted, which entails solving the mixture conservation equations and treating different phases as a single ﬂuid with variable properties. For the particular system where two ﬂuids coexist, the interface is determined implicitly, and changes in the properties are accounted for by the introduction of an additional transport equation for the volume-averaged phase indicator function, described in Section 7.7. By introducting a phase indicator equation to the governing equations and by performing averaging for each phase, the equations in averaged form governing the conservation of mass, momentum, and energy are derived in the interpenetrating continua framework. Either volume-averaging or ensemble averaging can be employed to formulate the averaged conservation equations since both of them result essentially in the same form of equations. The basic governing equations for different models and a suitable averaging procedure lead to the development of the averaged conservation equations applicable to multiphase ﬂows, particularly in two-ﬂuid models. Since almost all practical ﬂows are invariably turbulent in nature, the conservation equations must be described via time or mass-weighted averaging. The physical aspects of the boundary conditions and their appropriate mathematical statements pertinent for multiphase ﬂows are also required since the appropriate numerical form of the physical boundary conditions is strongly dependent on the particular mathematical form of the governing equations and numerical algorithm used. As shown in Figure 7.8, various modeling approaches to two-phase ﬂows yield different levels of understanding and results about these ﬂows. These models can be used together to obtain a required closure models that are used in other modeling approach. As shown in this block diagram, all modeling approaches require certain empirical input except DNS, which can be computationally expensive (with current technology and computational capability) for simulation of full-scale engineering devices.

Larger geometry

Two-fluid Models Drag, pressure and viscosity closures

Simulation of two-phase flows at engineering scales Bench scales simulation Dispersion coefficients, etc.

Discrete Particle Models Collision model + drag closures

Effective particle-particle interactions Inter-particle stress and friction

DNS (lattice Boltzmann model, immersed boundary model) No model required

Effective particle-fluid interactions Drag force

Smaller scales

Phenomenological Models Several closures required

Figure 7.8 Various models for two-phase ﬂows and their respective output to be used in other models (modiﬁed from van der Hoef et al., 2006).

536

7.7

BACKGROUND IN MULTIPHASE FLOWS WITH REACTIONS

EULERIAN-EULERIAN MODELING

There are a number of sources for the derivation of governing equations for multiphase systems, but the inherent assumptions in the different derivations constrain the types of multiphase ﬂow to which they can be applied. There is a lot of research on the closure models for these governing equations. Eulerian-Eulerian modeling is essentially a two-ﬂuid modeling approach, where both phases are treated as continua and two sets of conservation equations based on the continuum hypothesis are derived for each phase separately. In the Eulerian framework, there are different formulations for ﬂuid-ﬂuid ﬂows and ﬂuid-solid ﬂows. The differences between these formulations lie in the fact that in the ﬂuid-solid ﬂows, the points in each particle can be fully correlated to each other within the particle. This is different from the case with the droplets or bubbles in ﬂuid-ﬂuid ﬂows. For applications where the gradient of the volume fraction is small, the differences between ﬂuid-ﬂuid and ﬂuid-solid ﬂows are small. However, when the gradient of the volume fraction plays an important role—for instance, in cluster formation in dilute ﬂows—it is believed that the two formulations will quantitatively differ.

7.7.1

Fluid-Fluid Modeling

In a ﬂuid–ﬂuid formulation, both phases can be averaged over a ﬁxed volume (Ishii, 1975). This volume is relatively large compared to the size of individual molecules. Before deriving the averaged equation, the phase indicator function χk (x, y, z, t) is ﬁrst introduced to distinguish the phases that are present within the ﬂuid ﬂow. By deﬁnition, ⎧ ⎪ ⎪ ⎪ ⎪ ⎨

if the point (x, y, z) is in k th phase at time t if the point (x, y, z) is in j th phase at time t χk (x, y, z, t) = where j = k ⎪ ⎪ ⎪ < 1 if the point (x, y, z) is at the interface between 0 < χ ⎪ k ⎩ the two phases (7.32) 1 0

It was demonstrated by Drew and Passman (1999) that by following the motion of the interface moving at velocity uint , the material derivative of χk (i.e., Dχk /Dt) must be equal to zero. Thus, ∂χk ∂χ Dχk = + uj,int k = 0 Dt ∂t ∂xj

(7.33)

In the two-ﬂuid framework, χ1 = 1 and χ2 = 1 describe the two distinct ﬁelds of different phases that are separated by an interface. From Equation 7.33, it can be observed that both partial derivatives of χk vanish away from the interface (since χk assumes a constant value of either 1 or 0). Noted that if mass transfer

EULERIAN-EULERIAN MODELING

537

persists across the interface from one ﬂuid to the other, the interface moves not only by advection but also by the amount of mass being transferred between the phases. In such a case, the interface velocity is not equivalent to the neighboring velocities. Averaging over this function leads to the volume fraction of both phases, 1 φk = χk (x, t) dV (7.34) V V

where V is the averaging volume (i.e., local control volume). It is easy to observe that when the value of the phase indicator function is 1 (or 0), the value of volume fraction is also 1 (or 0), indicating that the local control volume is occupied only by a single phase. However, if an interface is present in the control volume, then both phases are present in that volume. In such a case, the calculated value of volume fraction from Equation 7.34 is also between 0 and 1. Since both the continuous and dispersed phases are ﬂuids, they are treated in the same way in the averaging process. The continuity equation for the k th phase is: ∂φk ρk (7.35) + ∇ · (φk ρk uk ) = 0 ∂t The momentum balance equations for the k th phase is: ∂φk ρk uk + ∇ · (φk ρk uk uk ) = −∇ · (φk pk ) + ∇ · (φk τ k ) + φk ρk g + Mk ∂t (7.36) where k can be the number designated to either phase and M is the interk phase momentum exchange between the phases, with Mk = 0. The interphase momentum transfer is deﬁned as Mk = −

1 (pk nk − nk τ k ) Lj

(7.37)

j =k

where 1/Lj = interfacial area per unit control volume, pk = bulk pressure in phase k, τk = shear stress. In Equation 7.37, mass transfer between the phases is not included. By adding and subtracting the average pressure and average shear stress at the interface in the right-hand side of the equation, we have: Mk =

1 pk,int − pk nk − pk,int nk − nk · τ k,int − τ k + nk · τ k,int Lj j =k

(7.38)

538

or where

BACKGROUND IN MULTIPHASE FLOWS WITH REACTIONS

Mk = I + pk ∇φk + (pk,int − pk )∇φk − (∇φk ) · τ k,int

(7.39)

I = sum of the form drag j =k L1j [(pk,int − pk )nk ] and viscous drag j =k L1j [−nk · (τk,int − τ k )] p k,int = average pressure of phase k at the interface τ k,int = average shear stress at the interface

The last term on the right-hand side of Equation 7.39 represents the interfacial shear stress, which is important in a separated ﬂow. According to Ishii (1975), the term (pk,int − pk ) plays a signiﬁcant role only when the pressure is very different from that at the interface (e.g., in stratiﬁed ﬂows). For many applications, both interfacial shear stress and pressure difference negligible, and Mk can be determined from: Mk = I + pk ∇φk

(7.40)

7.7.1.1 Closure Models In the Eulerian–Eulerian modeling framework (sometimes called the two-ﬂuid approach), both the continuous and dispersed phases are considered as continuous media. These models incorporate two-way coupling, which is especially important for high-volume fraction ﬂows. A drawback of these models is that they need complex closure relations. The interfacial momentum transfer between the liquid and the gas includes a number of force contributions,

I = Idrag + Im + Ihist + Iturb + Ilift where

(7.41)

Idrag = form and viscous drag Im = added mass force that is an inertial force caused by relative acceleration Ihist = history or Basset force (a viscous force caused by relative acceleration) Iturb = effect of turbulent ﬂuctuations on the effective momentum transfer Ilift = lift force that denotes the transverse force caused by rotational strain, velocity gradients, or the presence of walls (Enwald, Peirano, and Almstedt, 1996)

The drag force represents the mean interphase momentum transfer coming from the local perturbations induced by the dispersed phase: ρk 3 Cd Idrag = φj ρj |ur |ur (7.42) ρj 4 db

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539

where db = average size of the dispersed phase conglomerates (e.g., bubbles) Cd = drag coefﬁcient, ur = local relative velocity between the dispersed phase and the surrounding ﬂuid ﬂow. In the Equation 7.42, index j represents the phase that is not the k th phase. This expression for drag is valid for relative dilute systems. As the dispersed fraction increases, the effect of mutual hindrance between the dispersed phase conglomerates plays an increasing role, and this effect is not taken into account. Expressions to determine the drag at large dispersed phase fractions are highly empirical in nature and applicable to only a limited number of systems. The added mass force takes into account the inertial forces due to the relative acceleration between the phases. A general equation for this force is given by Im = φj ρk Cm

Dj Dk uk − uj Dt Dt

(7.43)

where Cm is called the added mass coefﬁcient, and it is a function of the volume fraction (Lamb, 1932). The history (or Basset) force is a viscous force due to the relative acceleration between the two phases. Most often, this force is ignored in continuum modeling, and there is no general agreement of its formulation even for the single-bubble case. Drew and Lahey (1993) give an expression for the Besset force combined with the lift force as: Ihist

9 = φ dp k

ρj μj π

t 0

a(x, t) dτ √ t −τ

(7.44)

where the appropriate frame-indifferent acceleration is given by: a(x, t) =

Dj Dk uk − uj Dt Dt

− (uj − uk ) × (∇ × uj )

(7.45)

The effect of turbulence on interphase momentum transfer is largely unknown. An important effect of turbulence comes in the form drag and viscous drag. The relative velocity, vr , in this equation should contain the averages of ﬂuctuating velocity, sometimes referred by as the turbulent drift velocity. This type of model typically leads to dispersive forces (∼∇φj ). The lift force represents the transverse force due to rotational strain, velocity gradients, or the presence of walls. A general equation for the lift force caused by rotational strain is given by Ilift,r = φj ρk Clift,r (uj − uk ) ×

(7.46)

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BACKGROUND IN MULTIPHASE FLOWS WITH REACTIONS

and the equation for the lift force caused by velocity gradients is given by Ilift,u = φj ρk Clift,u (uj − uint ) × (∇ × uj )

(7.47)

where uint = interface velocity Clift,r = lift force coefﬁcient associated with rotational strain Clift,u = lift force coefﬁcient associated with velocity gradients For a laminar, inviscid ﬂow, the value of Clift,r can be determined to be 0.5. In literature, a wide range of values can be found for the lift force coefﬁcient, as the correlation shown in Equation 7.47 is derived from inviscid ﬂow around a single sphere. Tomiyama et al. (2002) have performed experiments of single bubbles in simple shear ﬂows and have found positive and negative values for the lift force coefﬁcient, depending on the speciﬁc bubble characteristics. A number of important closures are required in the Eulerian framework for ﬂuid-ﬂuid models. These may not be most accurate at high-volume fractions of dispersed phase.

7.7.2

Fluid-Solid Modeling

For the ﬂuid-solid Eulerian closure models, the kinetic theory from ﬁrst principles is used to derive the Boltzmann and Enskog equations (van Wachem and Almstedt, 2003). The resulting equations give a powerful basis to implement further closure models (e.g., for ﬂuid-phase turbulence interactions with particles). A particle collision model can be employed that takes into account the inelastic nature of particles but neglects particle rotation. The probability correlation between two colliding particles due to the ﬂow of the ﬂuid is ignored, but it is expected that this is an important phenomenon, especially in dilute ﬂows where the ﬂuid phase plays a dominant role. The ﬁnal closures arising from the kinetic theory of granular ﬂow are presented in the following sections, which have been used with a fair amount of success in ﬂuid-solid calculations. The ﬂuid-solid formulation developed by Anderson and Jackson is discussed in Section 7.5, under “Anderson and Jackson’s Averaging for Dense Fluidized Beds.” The differences between the ﬂuid-ﬂuid and ﬂuid-solid momentum balance equations are shown in Table 7.1. The continuity equations are the same for both approaches. When the gas-phase shear stress plays an important role, these differences may be signiﬁcant where volume fraction gradients are large (e.g., near interfaces). Some authors employ liquid-liquid governing equations to describe gas-solid ﬂows, and the impact of the discrepancy between the two sets of governing equations depends on the application (van Wachen et al., 2001).

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541

TABLE 7.1. Comparison of the Momentum and Continuity Equations for Fluid-Fluid and Fluid-Solid Modeling Approaches Source: Modiﬁed from van Wachem and Almstedt (2003).

Continuity Equations for Both Phases in Fluid-Fluid Model or Fluid-Solid Model Phases k ∂ φk ρk ∂ φj ρj + ∇ · φk ρk uk = 0; + ∇ · φj ρj uj = 0 and j ∂t ∂t without mass transfer at interface

(7.48)

Momentum Equations for Fluid-Fluid Model Phase k ∂φk ρk uk + ∇ · (φk ρk uk uk ) = −φk ∇ (p ) + ∇ · (φk τ k ) + φk ρk g + I ∂t (7.49) Phase j ∂φj ρj uj = −φj ∇ (p ) + ∇ · φj τ j + φj ρj g − I + ∇ · φj ρj uj uj ∂t (7.50) Momentum Equations for Fluid-Solid Model Gas ∂φg ρg ug + ∇ · φg ρg ug ug = −φg ∇ (p ) + φg ∇ · τ g + φg ρg g − I phase ∂t (7.51) ∂φs ρs us Solid + ∇ · (φs ρs us us ) = −φs ∇ (p ) + φs ∇ · τ g + ∇ · (τ s ) phase ∂t (7.52) − ∇ (ps ) + φs ρs g + I

7.7.2.1 Closure Models Interphase Momentum Transfer In dilute ﬂows, the interphase momentum transfer due to form and viscous drag is modeled by the drag on a single particle in an inﬁnite ﬂuid, with the same equation as in ﬂuid–ﬂuid ﬂow: ρg 3 Cd |ur | ur Idrag = φs ρg (7.53) ρs 4 dp where Cd = drag coefﬁcient dp = average local particle diameter ur = relative local velocity between the ﬂuid and the solid phase

Although most authors take this relative velocity as the difference of the local ﬂuid and the solid velocities, this is formally not correct; the undisturbed turbulent ﬂuid velocity should be used instead. This is discussed later on.

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BACKGROUND IN MULTIPHASE FLOWS WITH REACTIONS

In more dense ﬂows, the form drag and viscous drag are generally combined in one empirical parameter, the interphase drag constant β, in the modeling of the momentum transfer between the two phases. The interphase momentum transfer is then written as (7.54) Idrag = βur The interphase drag constant β can be obtained experimentally from pressure drop measurements in ﬁxed, ﬂuidized, or settling beds. Ergun (1952) performed measurements in ﬁxed liquid–solid beds at packed conditions to determine the pressure drop. Based on this model, Gidaspow (1994) proposed the drag model: ⎧ 7 φs ρs |ur | φs2 μs ⎪ ⎪ ⎪ + if φs > 0.2 150 ⎪ 2 ⎨ 4 dp (1 − φs ) dp (7.55) β= ⎪ ⎪ |u | ρ − φ φ 3 (1 ) s s s r ⎪ ⎪ (1 − φs )−2.65 otherwise ⎩ Cd 4 dp Wen and Yu (1966) have performed settling experiments of solid particles in a liquid over a wide range of solid volume fractions and have correlated their data and that of others for solid concentrations applying the Richardson and Zaki (1954) correction factor 0.01 ≤ φs ≤ 0.63: β=

3 (1 − φs ) φs ρg |ur | Cd (1 − φs )−2.65 4 dp

(7.56)

Kinetic Theory of Granular Flow The transport coefﬁcients of the solid phases must account for gas-particle interactions and particle-particle collisions. The application of kinetic theory to model the motion of a dense collection of nearly elastic spherical particles is based on an analogy to the kinetic theory of dense gases (Chapman and Cowling, 1970). The interaction of solid particles due to collisions is modeled by the kinetic theory of granular (i.e., particle) ﬂow (Jenkins and Savage, 1983). One important difference between kinetic theory for a classical dense gas and that for a rapidly deforming granular material is that in the granular material, an inhomogeneity of the mean ﬂow is necessary to force the collisions and to drive the velocity ﬂuctuations. The temperature of a dense gas can also be inﬂuenced by the addition of heat throughout its interior or over its surface. There are, of course, situations involving granular materials in which the boundary can drive the ﬂuctuations independently of the mean deformation or in which energy may be put into the ﬂuctuations directly throughout the volume in the absence of mean motion. The second important difference between the kinetic theories for a dense gas and a rapidly sheared granular material is that collisions between the particles of a granular material involve a loss of energy. A granular temperature, θ, is deﬁned to represent the speciﬁc kinetic energy of the velocity ﬂuctuations or the translational ﬂuctuation energy resulting from the

EULERIAN-EULERIAN MODELING

543

particle velocity ﬂuctuations. The granular temperature equation can be expressed in terms of production of ﬂuctuations by shear, dissipation by kinetic and collisional heat ﬂow, dissipation due to inelastic collisions, production due to ﬂuid turbulence or due to collisions with molecules, and dissipation due to interaction with the ﬂuid (Gidaspow, 1994). In kinetic theory for granular ﬂow, collisions are considered binary and instantaneous. These assumptions are considered: • Particles are smooth and spherically symmetrical. Therefore, the force that one particle exerts on the other particle (and vice versa) is in the direction along the line joining their centers. • Rotation is not considered. • The effect of any external force that acts on the particle during collision can be neglected compared to the dynamic effect of the collision. With these assumptions, the velocities before and after a collision have deﬁnite values, denoted ui , uj before the collision and ui , uj after the collision. The details of the collision itself are not relevant; only the relationship between the initial and the ﬁnal velocities should be known. Let us consider a volume element dV centered at a location r and containing N(r, t) dV particles, where N is the number density of particles. Then the ensemble average of any property ψ of a single particle can be deﬁned as: 1 ψ = (7.57) ψf up , r; t dup N where f is the single-particle velocity distribution function deﬁned such that f (up , r, t) dup dr is the probable number of particles that at time t are located in the volume element from radius r to r + d r and have velocities lying in the range (up , up + dup ). Similarly, a distribution function for a pair of (j ) (j ) (j ) (i) (i) (i) particles i and j , h(u(i) p , r ; up , r ; t), is deﬁned such that h(up , r ; up , (j ) (i) (i) (j ) r(j ) ; t) dV dV dup dup is the probability of ﬁnding a pair of particles in the volume elements dV (i) and dV (j ) centered on the points r(i) and r(j ) having (j ) (j ) (j ) (i) (i) velocities within the ranges u(i) p , up + dup and up , up + dup . The rate of change of Nψ may be expressed (Reif, 1965) as: dup ∂ψ Fext ∂ψ ∂ Nψ = N Dψ − ∇ · Nup ψ + ζc where Dψ = = ∂t dt ∂up m ∂up (7.58) In Equation 7.58, Fext is the external force acting on a particle of mass m and ζc is the rate of increase of the ψ per unit volume due to particle-particle collisions. Let us consider binary collisions among hard, smooth, and inelastic particles of uniform diameter, dp . At the time of a collision between particles i and j , we take the center of the particle j to be located at position r and the center of particle i to be at r − dp n, where n is the unit vector along the line of centers of the two particles (see Figure 7.9).

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BACKGROUND IN MULTIPHASE FLOWS WITH REACTIONS

mj

u(j) p

ur

dp

u(i) p n

r r − dpn

mi

Origin

Figure 7.9

Schematic description of a pair of colliding solid particles with diameter dp .

Let us examine the requirements for a collision to occur between particles i (j ) and j . Particle j is located at position r(j ) and is traveling with a velocity up . (j ) (i) Before the collision, the relative velocity of the two particles is ur ≡ (up − up ). In time δt prior to the collision, particle i moves by a distance ur δt relative to particle j . For particle i to collide with particle j in a time interval δt in such a way that, at collision, the line connecting their centers r(j ) − r(i) = dp n is within the solid angle δn (where δn is the solid angle associated with the change in the direction of n due to relative motion of the particles), it is necessary that the center of the second particle (i.e., particle i ) must lie in a collision cylinder of volume dp2 δn (ur · n)δt (Chapman and Cowling, 1970, section 16.2). Another condition for a possible collision is as follows. If n is the normal unit vector lying in the direction of the vector joining the center of the two particles (see Figure 7.9), ur · n > 0 implies that the particles are moving toward each other. The number (j ) of collisions, when the particle j lies within the volume dV and when u(i) p , up , (j ) (j ) (j ) − and n are within the ranges dup , dup , and dn , is [dp2 (ur · n)δt]h(u(i) p ,r (j ) (j ) (i) (j ) dp n; up , r ; t)δn δup δup δV , where δV represents the incremental volume associated with the change in position vector r. After the collision, the gain in the property ψj of particle j is ψj∗ − ψj , where ψ ∗ and ψj refer to values after and before the collision. Considering only particles that are about to collide (i.e., for any pair of particles that have ur · n > 0), the rate of increase of ψ per unit volume due to collisions can be expressed as: ζc =

dp2

(j ) (j ) (i) (j ) − dp n; u(i) ψj∗ − ψj (ur · n) h u(i) p ,r p , r ; t dn dup dup

ur ·n > 0

(7.59) By interchanging the roles of the colliding particles by exchanging the subscripts i and j and replacing n by –n: ζc = dp2 ur · n > 0

) (i) (i) (i) (j ) (i) ψi∗ − ψi (ur · n) h u(j p , r + dp n; up , r ; t dn dup dup

(7.60)

EULERIAN-EULERIAN MODELING

545

After expanding the pair-distribution function in a Taylor series and rearranging, we can write: (i) (j ) (i) (j ) (i) ) (j ) h u(i) − dp n; u(j p , r ; up , r + dp n; t = h up , r p ,r ;t 2 1 3 1 + dp n · ∇ − dp n · ∇ + dp n · ∇ + . . . . 2 6 (i) (j ) (i) (7.61) × h u(i) p , r ; up , r + dp n; t By substituting Equation 7.61 in Equation 7.60, we get: ζc = −∇ · θ + χ

(7.62)

where the collisional transfer contribution θ is: θ=−

dp3 2

1 2 1 ψi∗ − ψi (ur · n) n 1 − dp n · ∇ + dp n · ∇ + . . . . 2 6

ur ·n > 0

(i) (j ) (j ) (j ) × h u(i) + dp n; t dn du(i) p , r ; up , r p dup

(7.63)

and the “source-like” contribution χ is: χ=

dp2 2

ψ1∗ + ψ2∗ − ψ1 − ψ2 (ur · n)

ur ·n > 0

(i) (j ) (j ) (i) (j ) × h u(i) p , r − dp n; up , r ; t dn dup dup

(7.64)

Considering the particles to be smooth but inelastic with a coefﬁcient of restitution e, with 0 ≤ e ≤ 1: u∗r · n = −eur · n (7.65) where the superscript * represents the property after the collision. By considering the conservation of momentum for the system of the two colliding particles, we have: (j ) ∗(i) ∗(j ) mu(i) p + mup = mup + mup

(7.66)

By using Equations 7.65 and 7.66, the translational kinetic energy change during a collision can be shown to be: E =

(i)2 1 ∗(i)2 1 )2 )2 − up + u(j = − m 1 − e2 (ur · n)2 (7.67) m up + u∗(j p p 2 4

546

BACKGROUND IN MULTIPHASE FLOWS WITH REACTIONS

By taking ψ to be m, mup , and 12 mu2p respectively, we can obtain Equations 7.68 and 7.69: dρs (7.68) = −ρs ∇ · up where ρs = mp N = φs ρp dt where ρs = bulk mass density of particles based on the total volume of a local element φs = local volume fraction of all solid particles ρp = mass density of an individual particle d up ρs = ρs b − ∇ps dt

(7.69)

where b = body force per unit mass ps = solids pressure “Solids pressure” is a new term and should not be confused with gas pressure. Solids pressure can be considered to be a measure of the rate of change of momentum of particles due to their streaming motion or free streaming. If we draw an analogy to the kinetic theory of gases, then solids pressure can be viewed as the pressure exerted by the particles on the walls of a control volume containing these particles. Solids pressure is a function of the granular temperature, the solids volume fraction, and the restitution coefﬁcient of the granular phase. The governing equation for the granular temperature is: 3 dTgr 1 2 1 ρs = −ps I : ∇up − ∇ · q − where Tgr ≡ up , ≡ −χ mup · up 2 dt 3 2 (7.70) where Tgr = speciﬁc kinetic energy of the velocity ﬂuctuations (or granular temperature) q = ﬂux of ﬂuctuation energy = collisional rate of dissipation per unit volume I = identity tensor Three major mechanisms affect the momentum and energy of solids. 1. In the dilute part of the particle ﬂow, particles can randomly ﬂuctuate and translate. The stress and energy associated with this motion is called kinetic. Solids pressure is the equivalent stress associated with this motion. 2. At higher concentration of particles, particles can collide with each other (with very short contact time) in addition to the free-streaming translational

EULERIAN-EULERIAN MODELING

547

motion. This gives rise to additional stress and energy loss term called collisional. 3. At very high concentration (more than 50% in volume), particles start to endure prolonged sliding and rubbing contacts, which gives rise to a totally different form of energy loss and stress, named frictional. The kinetic theory of granular ﬂows accounts only for kinetic and collisional mechanisms. There are additional constitutive relations for the frictional part. The solids pressure tensor in Equation 7.69 and 7.70 has two parts, a kinetic part (pk ) and a collisional part (pc ): pk I = ρs up up ps = pk + pc where (7.71) pc I = mup θ Similarly, the ﬂux of ﬂuctuation energy has two parts, a kinetic part (qk ) and a collisional part (qc ): 1 qk = ρs up2 up 2 (7.72) q = qk + qc where 1 2 θ qc = mu 2 p Equations 7.68 to 7.70 have the same form as the typical hydrodynamic equations for ﬂuids. However, the pressure in these equations is the solids pressure and it is not the usual pressure exerted by the gas phase. Numerous studies have shown the ability of the kinetic theory approach for modeling (e.g., Cao and Ahmadi, 1995; Benyahia, Syamlat, and O’Brien, 2005; Ding and Gidaspow, 1990; Goldschmidt, Coopers, and Swaaij, 2001; Hrenya and Sinclair, 1997; Pain, Mansoorzadeh, and de Oliverira, 2001; Johansson, van Wachem, and Almstedt, 2006; Patil, van Sint Ann Land, and Kuipers, 2005a, Reuge et al., 2008; Sinclair and Jackson, 1989; and Sun and Battaglia, 2006). 7.7.2.2

Dense Particle Flows

In dense particle ﬂows, such as ﬂuidized beds, the ﬂuctuating velocity of the ﬂuid phase and its correlation with the properties of particles is negligible compared to particle-particle interactions (i.e., collisions and friction) and the mean ﬂuid-particle velocity coupling (drag). Therefore, the ﬂuid phase is often modeled as laminar, and correlations of ﬂuid property with the ﬂuctuating particle velocities are not considered. At high solid volume fraction, sustained contacts between particles occur. The resulting frictional stresses must be considered in the description of the solid-phase stress. Zhang and Rauenzahn (1997) suggested that particle collisions cannot be assumed to be momentary when the solid volume fraction is very high, which was an assumption used in the kinetic theory. There are several approaches in the literature for modeling the frictional stress for densely packed particles. Typically, the frictional stress τfriction is written in

548

BACKGROUND IN MULTIPHASE FLOWS WITH REACTIONS

an incompressible Newtonian form with equivalent frictional solid pressure and frictional solid viscosity μfriction : T (7.73) τ friction = pfriction I + μfriction ∇up + ∇up Notice that Equation 7.73 for the frictional stress in particle ﬂows has the same form as viscous stress in the incompressible Newtonian form of the momentum conservation equation for ﬂuids. However, the pressure tensor does not have a negative sign. Most of the literature in this area has this convention and the constitutive equations for frictional solids pressure. An appropriate expression for the total stress for dense particle systems should be a sum of frictional stress and the stress estimated from the kinetic theory of particle ﬂows. Therefore, for φs > φs,min (where φs,min is the minimum solid volume fraction above which the bed is considered densely packed), we have: ps,total = ps + pfriction

(7.74)

μs,total = μs + μfriction

(7.75)

Johnson and Jackson (1987) propose a semiempirical equation for the frictional pressure, pfriction : C μs,total − μfriction 1 (7.76) pfriction = C3 C μs,total,max − μs,total 2 where C1 , C2 , and C3 are empirical material constants. This expression is valid for φs > φs,min , where φs,min is the solids volume fraction at which frictional stresses become important. The order of magnitude of the frictional pressure and the empirical constants associated with Equation 7.76 have been determined by various authors (see Figure 7.10). As shown in this ﬁgure, there is an extremely high level of dependence on the choice of empirical constants. The frictional viscosity can be related to the frictional pressure by the linear law proposed by Coulomb (1776) or Schaeffer (1987). μfriction =

φs

1/6

∂up,1 ∂x1

−

∂up,2 ∂x2

2

pfriction sin αfriction ∂up,1 2 ∂up,2 2 ∂u + ∂x1 + ∂x2 + + 1/4 ∂xp,1 2

∂up,2 2 ∂x1 (7.77)

where αfriction is the friction angle. The stresses in the dense regime predicted by frictional stress models are typically much larger than those predicted by kinetic theory. For densely packed granular bed combustion, please refer to Chapter 6 in Applications of Turbulent and Multiphase Combustion (Kuo and Acharya, 2012).

EULERIAN-EULERIAN MODELING

549

1060 Ocone et al. (1993) Johnson and Jackson (1987) Johnson et al. (1990) Syamlal et al. (1993)

Normal stress, [Pa]

1040

1020

100

1020 0.5

0.52

0.54 0.56 0.58 0.6 Solids volume fraction [−]

0.62

0.64

Figure 7.10 Normal frictional stresses with empirical constants from various researchers (modiﬁed from van Wachem and Almstedt 2003).

7.7.2.3

Dilute Particle Flows Kinetic theory is very helpful for modeling of many types of complex gas-solid ﬂows. Pita and Sundaresan (1991), however, showed that this model exhibits a very strong, unrealistic degree of sensitivity to the coefﬁcient of restitution, e. To illustrate this sensitivity, Figure 7.11 shows the solids volume fraction in a one-dimensional vertical pipe ﬂow for e = 1.0 and 0.99. Although the prediction

Solids volume fraction [−]

0.12 e = 0.99 e = 1.0

0.1 0.08 0.06 0.04 0.02 0

0

0.2

0.4

0.6

0.8

1

r/R [−]

Figure 7.11 Axial solids volume fraction proﬁle for two different values of the particle restitution coefﬁcient (modiﬁed from Sinclair and Jackson, 1989).

550

BACKGROUND IN MULTIPHASE FLOWS WITH REACTIONS

is fairly good for e = 1, as shown by Sinclair and Jackson (1989), this value of e is unrealistic. Lower values of e give a completely incorrect prediction of the location of the solids in the tube. Ljus (2000) has shown that in horizontal gas-particle pipe-ﬂow, without applying kinetic theory, the modeling results lead to all the solids falling to the bottom of the pipe. One important physical process that was not considered in the dilute gas-solid ﬂow models by Ljus (2000) and Sinclair and Jackson (1989) is gas-phase turbulence. While the magnitude of the gas-phase turbulence is negligible in dense gas-solid ﬂow, it may play a major role in more dilute ﬂows. Several authors have performed research to study gas-phase turbulence terms that are present in the diffusion term after closing them with an eddy viscosity model. Some of these models were successful in predicting extremely dilute ﬂows (e.g., Elgobashi and Abou-Arab, 1983; Louge, Mastorakos, and Jenkins, 1991). However, at slightly more dense ﬂows, when the drag between the gas and the solids becomes important, the gas-phase turbulence becomes relatively insigniﬁcant and the unrealistic solids volume fraction contours may appear. Elgobashi and Abou-Arab (1983) performed Reynolds decomposition of the Eulerian two-ﬂuid equations and found a large number of terms arising from this exercise. Hrenya and Sinclair (1997) modeled three of the time-averaged terms arising from this decomposition with the eddy mixing length gradient assumption. They found signiﬁcant improvement in the results, and their predictions of solids volume fraction contours were closer to the experimental data. Time averaging gives rise to an additional term in the continuity equation, which may introduce problems in the numerical implementation. Apart from this problem, time averaging also introduces scaling in the eddy viscosity coefﬁcient with the volume fraction. When using Favre averaging instead of Reynolds averaging, the additional terms arise in the momentum equations. The most important is the correlation between the ﬂuctuating particle velocity and the ﬂuctuating particle volume fraction. Balzer and Simonin (1993) used a different expression for the interphase momentum transfer. In their model, they used the undisturbed local ﬂuid velocity instead of averaged gas velocity. The different term, called the turbulent drift velocity (which accounts for particle dispersion due to transport by the large ﬂuid eddies), can be modeled with a dispersion coefﬁcient and the gradients of the solids and ﬂuid volume fraction (also see Simonin, Deutsch, and Minier, 1993). The advantage of using turbulent drift velocity is that it provides an order of magnitude estimate for the coefﬁcients in the model as well as it dependency on other ﬂow properties.

7.8

EULERIAN-LAGRANGIAN MODELING

In the Lagrangian framework, two common methods are used in ﬂuid-solid modeling: the hard-sphere approach and the soft-sphere approach. In the hard-sphere approach, particle collisions are assumed binary and instantaneous, such as the

551

EULERIAN-LAGRANGIAN MODELING

collision between two billiard balls. This model can be appropriate for dilute ﬂows, but for dense ﬂows, particle collisions are far from binary and instantaneous (as also pointed in Section 7.7.2.2). An alternative to the hard-sphere approach is the soft-sphere approach, where particles can overlap and particle interactions can be enduring. This is modeled by a slider-spring-dashpot model, with associated friction, spring, and damping coefﬁcients. Unfortunately, when the physical values for these coefﬁcients are used, the equations become stiff and numerically hard to solve. 7.8.1

Fluid-Solid Modeling

With increasing computational power, discrete particle models, or Lagrangian models, have become a very useful tool to study the hydrodynamic behavior of particulate ﬂows. In these models, the equations of motion based on Newton’s law are solved for each individual particle, and a collision model is applied to handle particle encounters. Such a modeling approach for particles is combined with an Eulerian model for the continuous phase to simulate freely bubbling and circulating ﬂuidized beds. 7.8.1.1 Fluid Phase The motion of the ﬂuid phase is calculated from the averaged ﬂuid phase– governing equations as presented in Section 7.7. The momentum equation for the ﬂuid phase is: ∂ φg ρg ug (7.78) + ∇ · φg ρg ug ug = −φg ∇p + φg ∇ · τ g + φg ρg g ∂t

−

Np k=1

where

Vs,k us,k β τg

= = = =

N

p Vs,k β ug − us,k δ x − xs,k Vs,k

k=1

volume of k th individual solid particle velocity of k th individual solid particle interphase drag constant with the dimension (ML−3 t−1 ) the stress tensor for the compressible ﬂuid phase, and it is deﬁned as:

T 1 T − μg Tr ∇ug + ∇ug I τ g = μg ∇ug + ∇ug 3

(7.79)

The last term in the momentum balance equation 7.78 represents the interphase momentum transfer between the ﬂuid phase and each k th individual particle, and δ represents the Dirac delta function. The last term ensures that the interphase momentum transfer is taken into account in the ﬂuid-phase momentum equation only at the location of the corresponding particle.

552

BACKGROUND IN MULTIPHASE FLOWS WITH REACTIONS

A problem of this Lagrangian-Eulerian approach is the appropriate length scale of the averaging procedure. In the Eulerian-Eulerian approach, the length scales of the averaged ﬂuid phase and particle phase are equal, and the so-called subgrid behavior of the particles is described by the kinetic theory of granular ﬂow. In the Lagrangian-Eulerian approach, the length scale of the ﬂuid phase is larger than the length scale of the particle phase. The information of ﬂuid-induced movement of particles as well as particle-induced movement of ﬂuid cannot be transferred between the phases on the ﬂuid-phase length scale or individual particle scale. Hence, a computational cell in which a small cluster of particles is present is penetrated by the ﬂuid phase, similar to a ﬁxed porous medium, and the ﬂuid phase does not discriminate between homogeneously distributed particles or clustered particles within one cell. In reality, the ﬂuid-phase “dodges” the particle clusters. Particle clustering due to the local ﬂuid ﬂow (“microscale” clustering) is thus not captured by the Lagrangian-Eulerian approach. This shortcoming of the outlined model in the particle-ﬂuid phase coupling should be kept in mind when attempting to use this simulation method. The so-called true direct numerical simulations can be carried out to solve the actual ﬂuid ﬁeld around each particle, but this method is extremely expensive computationally and can be performed only for a very limited number of particles. 7.8.1.2 Solid Phase Solid particles can be considered as inelastic spheres. In a Lagrangian approach, the path of each individual particle is calculated. The calculation of the paths of the particles consists of two steps:

1. Calculation of the particle motion 2. Treatment of the collision of a particle with another particle The motion of individual particles is completely determined by Newton’s second law of motion. The forces acting on each particle, next to collisions, are gravity and the traction force of the ﬂuid phase on the particle. Thus, the momentum equation describing the acceleration of the particle is: ms as = ms g + Vs ∇ · τ g − Vs ∇p + β where as Vs τg p β φs

= = = = = =

Vs ug − us φs

(7.80)

acceleration of one particle volume of an individual particle ﬂuid-phase stress tensor local gas pressure interphase momentum transfer (or drag) constant local solids volume fraction

To describe the collisions of particles, two types of approaches are possible— the hard-sphere approach and soft-sphere approach—which are described in the following subsections.

EULERIAN-LAGRANGIAN MODELING

553

Hard-Sphere Approach In the hard-sphere approach, collisions between particles are assumed to be binary and instantaneous. The velocities of the particles emerging from a collision are calculated by considering the balance of linear momentum and angular momentum in the collision. During a collision, energy is stored in elastic deformations associated with both the normal and the tangential displacements of the contact point relative to the center of the sphere. Because the release of this energy may affect the rebounding of hard spheres signiﬁcantly, coefﬁcients of restitution associated with both the normal and tangential components of the velocity at the point of contact should be taken into account. This model can be employed for both particle-particle and particle-wall collisions. Let us consider two colliding spheres with diameters d1 and d2 , masses m1 and m2 , and centers located at r1 and r2 . The unit normal along the line joining the centers of two spheres is n = (r1 − r2 )/|r1 − r2 |. During the collision, sphere 2 exerts an impulse J on sphere 1. Prior to the collision, the spheres have translational velocities up,1 and up,2 and angular velocities ω1 and ω2 . The corresponding velocities after the collision are denoted by an asterisk (* ). The velocities before and after collision can be related by Equations 7.81 and 7.82:

m1 u∗p,1 − up,1 = −m2 u∗p,2 − up,2 = J 2I2 2I1 ω1 − ω 1 = − ω2 − ω2 = −n × J d1 d2

(7.81) (7.82)

where, for example, I = md2 /10 is the moment of inertia about the center of a homogeneous sphere. In order to determine the impulse J, the relative velocity ur at the point of contact is deﬁned: ur = up,1 − up,2 −

1 1 d1 ω1 + d2 ω2 × n 2 2

(7.83)

With the above equations, the contact velocities before and after the collision are given by: 7 1 1 1 5 1 ∗ u r − ur = + + J− n (J · n) (7.84) 2 m1 m2 2 m1 m2 The coefﬁcient of restitution, e, characterizes the incomplete restitution of the normal component of ur : u∗r · n = −eur · n (7.85) where 0 ≤ e ≤ 1. In collisions that involve sliding, the sliding is assumed to be resisted by Coulomb friction, and the tangential and normal components of the impulse are related by the coefﬁcient of friction μ: n × J = μ (n · J)

(7.86)

554

BACKGROUND IN MULTIPHASE FLOWS WITH REACTIONS

where μ ≥ 0. Combining Equations 7.84, 7.85, and 7.86 provides an expression for the impulse transfer in the case when the collision is sliding: J(sl) =

(1 + e) (n · ur ) n + μ (1 + e) cot ϑ [ur − n (ur · n)] 1/m1 +1/m2

(7.87)

where ϑ is the angle between ur and n and the superscript “sl ” denotes that the collision involves sliding. With small ϑ, the collision is sliding, and as ϑ increases, the sliding stops when n × u∗r = −ξ (n × ur )

(7.88)

where 0 ≤ ξ ≤ 1 is the tangential coefﬁcient of restitution. Equivalently, Equation 7.88 can be written as: 2 (1 + ξ ) cot ϑ0 = (7.89) 7 (1 + e) μ Collisions with ϑ ≥ ϑ0 involve sticking, not sliding. In this case, the impulse is found by combining Equations 7.84, 7.85, and 7.88: J(st) = −

(1 + e) (ur · n) n + (2/7) (1 + ξ ) [ur − n (ur · n)] 1/m1 + 1/m2

(7.90)

In this expression, the superscript “st” indicates that the collision does not involve sliding but sticking. The three parameters e, μ, and ξ are taken to be constant and independent of the velocities. Collisions with a ﬂat wall are treated by considering the wall as a particle with inﬁnite mass and with the appropriate wall values of e, μ, and ξ . Soft-Sphere Approach In the soft-sphere approach, the particle interactions are modeled through a potential force. This model for contact forces was ﬁrst proposed by Cundall and Strack (1979). The soft-sphere approach is appropriate for those cases when two particles deform on collision. This deformation is described by the overlap displacement of two particles in the soft-sphere model. The larger the overlap displacement, the larger the repulsive force. In this particle-particle interaction model, the particles lose kinetic energy. When two particles slide due to a normal force, a frictional force should be considered. Considering these forces, the soft-sphere model is composed of three mechanical elements: spring, dashpot, and friction slider (see Figure 7.12). The spring simulates the effect of deformation, and the dashpot simulates the damping effect. The slider simulates the sliding force between two particles. The effects of these mechanical components on particle motion appear through these parameters: the stiffness k , the damping coefﬁcient η, and the friction coefﬁcient μ. The normal component of the forces (Fn,ij ) acting during particle contact is given by the sum of forces modeled by a spring and a dashpot, Fn,ij = −kn,ij Dn,ij − ηn,ij ur · n n (7.91)

INTERFACIAL TRANSPORT (JUMP CONDITIONS) Spring

555

Dashpot

mj

mi Slider

Figure 7.12 A spring-damper-dashpot system for soft-sphere model for particle-particle collision (modiﬁed from van der Hoef et al., 2006).

where Dn,ij is the normal overlap displacement between particles i and j , and ur is the relative velocity between the two particles. The tangential component of the contact force Ft,ij acting during particle interaction is given by the sum of forces modeled by a spring and a dashpot or by a spring and a slider, depending on the magnitude of the ratio between the normal and tangential component, which physically indicates if a particle is sliding or not. ⎧ ⎪ ⎨−kt,ij D − ηt,ij Jij −kt,ij D − ηt,ij ≤ μ Fn,ij Ft,ij = (7.92) Jij ⎪ −kt,ij D − ηt,ij > μ Fn,ij ⎩−μ Fn,ij Jij where Jij is the slip velocity at the point of contact and μ is the friction coefﬁcient that indicates when the interaction between two particles is considered sliding. The force on each particle in the system consists of the normal and tangential forces caused by the respective particle overlap. The stiffness coefﬁcient k and the damping coefﬁcient η may be related to physical particle properties by means of Hertzian contact theory used by Tsuji, Kawaguchi, and Tanaka (1993) and the displacement theory of Mindlin and Deresiewicz (1953). The time-step that can be taken in hard-sphere collision dynamics is governed by the successive time between collisions. In dense systems, this time-step may be very small, leading to very long computational times. The time-step in soft-sphere collision dynamics is governed by the stiffness of the normal and tangential forces. Unfortunately, the Hertzian contact theory predicts very high stiffness, which leads to very small possible time-steps in denser suspensions. Tsuji, Kawaguchi, and Tanaka (1993) suggested using a lower value for the stiffness coefﬁcient. According to Tsuji, Kawaguchi, and Tanaka (1993), Hoomans et al. (1996), and Crowe, Sommerfeld, and Tsuji (1998), the physical impact of this is low, but the exact meaning or limits of this are unknown.

7.9

INTERFACIAL TRANSPORT (JUMP CONDITIONS)

The standard differential balance equations are applicable to each phase up to an interface but not across it. At an interface between the two phases, properties are discontinuous, although mass, momentum, and energy ﬂuxes must

556

BACKGROUND IN MULTIPHASE FLOWS WITH REACTIONS

remain conserved. A particular form of the balance equation should be used at an interface in order to take into account the singular characteristics—namely, the sharp changes (or discontinuities)—in various variables. By considering the interface as a singular surface across which the ﬂuid density, energy, and velocity suffer jump discontinuities, the so-called jump conditions have been developed. These conditions specify the exchanges of mass, momentum, and energy ﬂuxes through the interface and stand as matching conditions between two phases; thus, they are indispensable in two-phase ﬂow analyses. Furthermore, since a solid boundary in a single-phase ﬂow problem also constitutes an interface, various simpliﬁed forms of the jump conditions are in frequent use. Because of the importance of jump conditions, their derivation and physical signiﬁcance are discussed next. The interfacial jump conditions without any surface properties were ﬁrst put into general form by Kotchine (1926) as the dynamical compatibility condition at shock discontinuities, although special cases had been developed earlier by various researchers. It can be derived from the integral balance equation by assuming that it holds for a material volume with a surface of discontinuity. Various authors (Delhaye, 1968; Kelly, 1964; Scriven, 1960; Slattery, 1964; and Standart, 1964) have attempted to extend Kotchine’s theorem. These include the introduction of interfacial line ﬂuxes, such as the surface tension, viscous stress, and heat ﬂux, or surface material properties. There are several approaches to the problem, and the results of the various authors are not in complete agreement. Delhaye (1974) presented a detailed discussion of this subject and a comprehensive analysis that shows the origins of various discrepancies among previous studies. A particular emphasis is directed to the correct form of the energy jump condition and of the interfacial entropy production. The ﬂux balances of mass, species, and energy across a gas-solid interface for a burning solid boundary is also given in Kuo (2005, Chap. 3) Two kinds of jump conditions can be derived: primary and secondary. Primary jump conditions are derived directly from the global balance laws written for these fundamental quantities: mass, linear momentum, angular momentum, total energy, and entropy. Secondary jump conditions are derived from the primary ones. They are written for these quantities: mechanical energy, internal energy, enthalpy, and entropy. These jump conditions reduce to the shock equations obtained either in inviscid ﬂuid (Serrin, 1959) or in weakly dissipative ﬂuid (Germain, 1964; Germain and Guiraud, 1960, 1961). The canonical form of jump conditions has been given as:

(7.93) ρψ (u − uint ) + Fψ · nk = S˙ψ where

ψ Fψ n ˙ Sψ

= = = =

conserved quantity its molecular or diffusive ﬂux unit normal interfacial source of ψ

INTERFACIAL TRANSPORT (JUMP CONDITIONS)

557

The mass efﬂux from the k th phase is deﬁned as: S˙k ≡ ρk (uk − uint ) · nk , k = 1, 2

(7.94)

where nk is the unit external to the component k . If there is no storage or accumulation of mass at an interface, the averaged interfacial mass balance constraint (jump condition) can be given by 2

[ρk (uk − uint ) · nk ] = 0

(7.95)

k=1

Note that the outward normal vector n1 = −n2 and the contact discontinuity velocity uint are common for the both sides of the interface. Equation 7.95 simply states that there is no storage of mass at the interface; hence phase changes are pure exchanges of mass between the two phases. Interfaces at which we have the next condition are called impermeable: u1 · n1 = uint · n2

(7.96)

Such surfaces may be solid-ﬂuid or ﬂuid-ﬂuid interface. Surfaces at which interface velocity is zero are called impermeable ﬁxed surfaces: u1 = uint = 0

(7.97)

When there is mass transfer across the interface, then next relation can be obtained between the velocity components normal to the interface and the density: 1 1 − (7.98) m ˙ 1 (u1 − u2 ) · n1 = ρ1 ρ2 If there is no mass transfer across the contact discontinuity (interface), then the relations Equations 7.99 to 7.101 shown in are obtained: ρ1 u1 − ρ2 u2 ρ1 − ρ2 ρ2 (u1 − uint ) = (u2 − u1 ) ρ1 − ρ2 ρ1 (u2 − uint ) = (u2 − u1 ) ρ1 − ρ2 uint =

(7.99) (7.100) (7.101)

These relationships are also known as shock discontinuities for the mass. For the jump condition related to the momentum balance, let us consider Figure 7.13. In the general case, where mass is transferred from one to the other phase by any physical process (e.g., evaporation, burning, etc.), the interface movement

558

BACKGROUND IN MULTIPHASE FLOWS WITH REACTIONS

in space are also controlled by the amount of mass transferred between the ﬁelds. In such cases, the interface velocity uint is not equal to the neighboring ﬁeld velocities. The mass ﬂow rate m ˙ 1 ≡ ρ1 (u1 − uint ) · n1 enters the interface control volume and exerts the force ρ1 u1 (u1 − uint ) · n1 per unit surface area on it. Note that this force has the same direction as the pressure force inside phase 1. Similarly, there is a reactive force ρ2 u2 (u2 − uint ) · n2 exerted per unit surface area on the control volume around the interface (shown by the dashed lines in Figure 7.13) by the leaving mass ﬂow rate. Assuming that the control volume moves with the normal component of the interface velocity, we obtain this force balance: 2

−ρk uk (uk − uint ) · nk + (τ k − pk I) · nk = Fint,σ

(7.102)

k=1

In Equation 7.102, the term Fint.σ represents the interfacial momentum source, which is the contribution to the total force on the mixture speciﬁcally due to the surface tension at the interface. For a constant surface-tension coefﬁcient, this singular force can be expressed as Fint,σ = σ12 κ1 n1 + ∇1 σ12 where

σ12 κ n ∇1

= = = =

(7.103)

surface tension coefﬁcient mean curvature of the interface unit normal to the interface such that n = n1 = −n2 gradient in the surface coordinates.

Equation 7.102 is the general form of the interfacial momentum jump condition. It is convenient to rewrite this equation by using the mass jump condition at the interface:

(7.104) ρ1 (u1 − uint ) (u2 − u1 ) − (p1 − p2 ) I + (τ 1 − τ 2 ) · n1 = Fint,σ Interface

p2 Field 2

ε

(u1 − uint)·n1

n1

(u2 − uint)·n2

0 uint n2

u1 uint u2

p1 Field 1

Figure 7.13

Momentum balance across the interface (modiﬁed from Kolev, 2005).

INTERFACIAL TRANSPORT (JUMP CONDITIONS)

or

m ˙ 1 (u2 − u1 ) − (p1 − p2 ) I · n1 + (τ 1 − τ 2 ) · n1 = Fint,σ

559

(7.105)

The projection of this force to the normal direction is obtained by scalar multiplication of Equation 7.105 with the unit vector n1 . The result is: m ˙ 1 (u2 − u1 ) · n1 − (p1 − p2 ) + [(τ 1 − τ 2 ) · n1 ] · n1 = Fint,σ · n1

(7.106)

Using the mass conservation at the interface, we have an important force balance normal to the interface: 2 1 1 − − (p1 − p2 ) + [(τ 1 − τ 2 ) · n1 ] · n1 = Fint,σ · n1 (7.107) m ˙1 ρ2 ρ2 From this analysis, we can see that neglecting all the forces except those caused by pressure and interfacial mass transfer results in the surprising conclusion that during the mass transfer, the pressure in the denser ﬂuid is always larger than the pressure in the lighter ﬂuid independently of the direction of the mass transfer (Delhaye, 1981, p. 52, Equation 2.64). For the limiting case of no interfacial mass transfer and dominance of the pressure difference, the velocity of the interface can be expressed as a function of the pressure difference and the velocities in the bulk of the ﬁelds. uint = u1 −

(p2 − p1 ) ρ1 (u2 − u1 )

(7.108)

This velocity is called contact discontinuity velocity. Replacing the discontinuity velocity with Equation 7.99, we obtain: (u1 − u2 )2 =

(ρ1 − ρ2 ) (p1 − p2 ) ρ1 ρ2

(7.109)

For the case ρ1 ρ2 , we have the expected result that the pressure difference equals the stagnation pressure at the side of the lighter medium (p1 − p2 ) = ρ2 (u1 − u2 )2

(7.110)

The interfacial energy jump condition can also be derived similarly by considering the interface to be nonmaterial (i.e., it does not accumulate mass or energy). By considering various forms of energy ﬂuxes at the interface (shown in Figure 7.14), we have: ⎤ ⎡ ρk (ek + 1/2uk · uk ) uk − uk,int − lk ∇Tk + q˙ radk 2 ⎦ ⎣ =0 (7.111) + ρk hik Yik Vik − uk − uk,int . (τ k − pk I) · nk k=1

i=1,N

560

BACKGROUND IN MULTIPHASE FLOWS WITH REACTIONS −(u1 − uint)·(τ1 − p1I)·n1

−(u2 − uint)·(τ2 − p2I)·n2

Field 1

Field 2 n2

n1

ρ1(e1 + 1/2u1.u1)(u1 − uint)·n1 − λ1(∇T1)·n1 qrad1·n1

T ρ1 hi1Yi1Vi1·n1 i = 1, N

ρ2(e2 + 1/2u2.u2)(u2 − uint)·n2 − λ2(∇T2)·n2 qrad2·n2

ρ2 hi2Yi2Vi2·n2 i = 1, N

u1

u2

uint Interface

Figure 7.14

Energy balance across the interface moving with velocity uint .

The ﬁrst term on the right-hand side of Equation 7.111 is the interfacial energy due to mass exchange across the interface, while the second term characterizes the ﬂux of heat being transferred into the k th phase from the other phases normal to the interface. The last two terms on the right-hand side of the equation depict the interfacial work done due to the pressure and extra stresses acting on the interface. In terms of enthalpy, the interfacial energy jump condition can be written as: ⎡ ⎤ ρk (hk + 1/2uk · uk ) uk − uk,int − lk ∇Tk + q˙rad 2 k ⎣ ⎦ · nk = 0 (7.112) + ρk hik Yik Vik − uk − uk,int · τk k=1

i=1,N

Using the jump conditions for mass, can be simpliﬁed as: ⎡

⎤ ρ1 ρ2 h1 − h2 + 1/2 (u1 · u2 − u2 · u2 ) (u2 − u1 ) ρ1 − ρ2 ⎢ ⎥ ⎢ τ2 ρ1 ρ2 τ1 ⎥ ⎢ + q˙ ⎥ · n1 ˙ cond2 + − (u2 − u1 ) · ⎢ cond1 − q ρ1 − ρ2 ρ2 ρ1 ⎥ ⎣ ⎦ + q˙ rad1 − q˙ rad2 ⎛ ⎞ 2 + ρk ⎝ hik Yik Vr,ik ⎠ · nk = 0 (7.113) k=1

i=1,N

In this equation, Vr,ik is the diffusion velocity of the i th species of the k th phase relative to the moving interface. This velocity can be evaluated based on the concentration gradient evaluated at the interface according to the formula given

INTERFACE-TRACKING/CAPTURING

561

in Table 2.1 of Chapter 2. Similarly, the conduction heat ﬂuxes are calculated based on the temperature gradient at the interface, wherever it is located. By substituting Equation 7.98 in Equation 7.113 and by ignoring species diffusion from one phase into the other, we have: q˙ cond1 − q˙ cond2 + q˙ rad1 − q˙ rad2 · n1 m ˙ 1 = τ2 τ1 1 − h1 − h2 + /2 (u1 · u1 − u2 · u2 ) + n1 · · n1 ρ2 ρ1 −

(7.114)

If there is no mass transfer across the interfacial contact discontinuity, then heat conduction and radiation are the only mechanisms transferring energy across. In the simple case of no heat conduction at both sides of the interface and zero stress tensors, the energy jump condition simpliﬁes to: h1 − h2 + 1/2 (u1 · u1 − u2 · u2 ) = 0

7.10

(7.115)

INTERFACE-TRACKING/CAPTURING

Interfacial multiphase ﬂows are encountered often both in nature and in industries. Processes such as extraction, chemical reaction, mass transfer, separation, spray, and combustion of condensed phase materials, among others, involve interfacial ﬂows. Understanding the basic hydrodynamic phenomena associated with such processes requires a proper and precise deﬁnition of the interface between two phases. The basic hydrodynamic phenomena include droplet evaporation, droplet breakup, droplet burning, bubble formation and transport, and coalescence of bubbles. A detailed computation of immiscible-ﬂuid and free-surface ﬂows requires an accurate representation of the interface separating the two ﬂuids. Immiscible-ﬂuid ﬂows are commonly encountered in nature and in industrial applications, in processes involving separation, extraction, mixing, and chemical reactions. Free-surface ﬂows, such as water waves and splashing droplets, are encountered in nature and industrial processes. These ﬂow problems include phenomena like ﬂuid coalescence and breakup, which further increase the need for an accurate and sharp interface deﬁnition. A number of techniques to track the interface have been developed in the last few decades. The most important techniques are shown in Table 7.2. The methods for computating free surfaces and ﬂuid interfaces between two immiscible ﬂuids on an arbitrary Eulerian mesh normally are classiﬁed into two categories: surface methods (interface tracking or ﬁtting) and volume methods (interface capturing). For surface methods, the interface can be tracked explicitly, either by marking it with special marker points (particles) or by attaching it to a mesh surface,

562

BACKGROUND IN MULTIPHASE FLOWS WITH REACTIONS

TABLE 7.2. Summary of Surface Tracking Methods Source: Modiﬁed from Gopala and van Wachem, 2008.

Methods

Pros

Cons

Front tracking

• Extremely accurate and robust • Accounts for substantial topological changes at the interface

• Mapping of interface mesh onto Eulerian mesh • Dynamic remeshing required • Merging and breakage of interfaces requires subgrid model

Level set

• Limited accuracy • Conceptually simple and easy to implement • Loss of mass (volume) but can be mitigated with special • Flexible treatment of surface treatments or meshing phenomena

Shock capturing

• Standard implementation • Requires ﬁne grids • Advection schemes available • Limited discontinuities

Marker particle

• Extremely accurate and robust • Accounts for substantial topological changes at interface

• Computationally expensive • Redistribution of marker particles required

SLIC VOF • Conceptually simple (simple line • Straightforward extension to interface 3D calculation, volume • Merging and breakage of of ﬂuid) interfaces occurs automatically

• Numerically diffusive • Limited accuracy

PLIC VOF • Relatively simple and (piecewise line accurate interface • Merging and breakage of calculation, volume interfaces occurs of ﬂuid) automatically

• Difﬁcult to implement in 3D • Extension to boundary-ﬁtted grids very difﬁcult

Compressive VOF (volume of ﬂuid)

• Requires very low Courant numbers for good accuracy

• Relatively simple and accurate • Easy to implement to boundary-ﬁtted grids • Merging and breakage of interfaces occurs automatically

INTERFACE-TRACKING/CAPTURING Δx Fluid 1

563

Δy

Fluid 1

Fluid 2 Fluid 2 Surface Method

Volume Method

Figure 7.15 Description of surface and volume methods for interface tracking/capturing (modiﬁed from Gopala and van Wachem, 2008).

which is then forced to move with the interface (see Figure 7.15). For volume methods, the ﬂuids on either side of the interface are, in general, marked by either particles of negligible mass or an indicator function. Thus, the exact position of the interface is not known explicitly, and special techniques are needed to reconstruct the well-deﬁned interface. These techniques are deﬁned in next sections. For tracking a small number of interfaces (few droplets or bubbles in a ﬂuid-ﬂuid ﬂow), volume methods are very good. Regardless of which method is employed, the essential features to model free surface and ﬂuid interface properly include a scheme to describe the shape and location of a surface, an algorithm to evolve the shape and location with time, and application of free surface boundary conditions at the surface.

7.10.1

Interface Tracking

With surface methods, the interface is represented by special marker particles. The marker particles move with the ﬂuid, and the interface is reconstructed from the location of the particles. Interpolation is used to approximate the points between these particles, usually using a piecewise polynomial. The advantage of this approach is that the interface position is known throughout the ﬂow ﬁeld and remains sharp as it is advected across the domain. This enables the accurate calculation of the interface curvature, which is needed for the inclusion in interface tracking method of the surface tension force. Limitations arise while simulating coalescence and breakup of the interface surface, as the particles might tend to move either apart or very close to each other, leading to lower resolution of the interface. The results reported are accurate but expensive. Also, the reconstruction of the interface can be troublesome, especially when breakup or coalescence of the interface occurs. Several surface methods exist, two of which are explained next. In front-tracking methods (Unverdi and Tryggvason, 1992), the interface is tracked explicitly on a ﬁxed Eulerian mesh by marking the interface with a set

564

BACKGROUND IN MULTIPHASE FLOWS WITH REACTIONS

of connected massless marker particles. The local velocities are used to advect these massless particles in a Lagrangian manner. The method is sensitive to the spacing between the marker particles (i.e., when the particles are far apart, the interface is not well resolved; when they are too close, the curvature is overestimated). Therefore, it is necessary to add or delete marker particles dynamically. Also, difﬁculties arise when multiple interfaces interact with each other, as in coalescence and breakup requiring a proper subgrid model. In level-set methods (Osher and Sethian, 1988), the interface is deﬁned as a zero-level set of a distance function from the interface. To distinguish between the two ﬂuids on either side of the interface, a negative sign is attached to the distance function for one of the ﬂuids. The distance function G is a scalar property and is advected with the local ﬂuid velocity by solving the scalar advection equation: ∂G + uint · ∇G = 0 ∂t

(7.116)

Level-set methods are conceptually simple and relatively easy to implement, yielding accurate results when the interface is advected parallel to one of the coordinate axis. However, in ﬂow ﬁelds with appreciable vorticity or in cases where the interface is signiﬁcantly deformed, level-set methods suffer from loss of mass. Some descriptions of the level set technique are also given in Chapter 5. 7.10.1.1 Markers on Interface (Surface Marker Techniques) The basic idea behind the markers on interface approach is to explicitly track an interface on a ﬁxed mesh by marking the interface with a set of connected marker particles of negligible mass (see Figure 7.15). By allocating a sufﬁcient amount of markers on the interface, these markers are moved according to the local advection velocity ﬁeld. During the calculation, the positions or Lagrangian coordinates of each marker source xnM can be obtained by the numerical integration from some initial position x0M source at time t = 0 according to this equation:

xnM

t =

x0M

+

uM dt

(7.117)

0

where uM are equivalent to the ﬂuid velocities in the Eulerian mesh at the timedependent location of the marker. Surface markers (SM) allow the capture of details of interface motion on scales much smaller than the mesh spacing in the Eulerian mesh. A modiﬁed version of the original marker and cell (MAC) method was developed by Chen, Cowan, and Grant (1991) by using only SM rather than markers distributed throughout the ﬂuid as proposed in the original MAC formulation. Known as the SM method, new marker movement and cell reﬂagging techniques are presented through the use of only one row of markers along the free surface. The reﬂagging of cells along the region adjacent to the free surface is carried out

INTERFACE-TRACKING/CAPTURING

565

in each computational calculation. Through the consideration of only SM, these new techniques result in a signiﬁcant reduction of the computer time and storage required to solve transient, free surface ﬂuid-ﬂow problems in comparison to those of the SMAC method. With the subsequent improvement made to the SM method via the introduction of an even ﬁner mesh of cells to better treat the pressure near the free surface, especially for ﬂow problems that involve multivalued free surfaces and breaking ﬂuid fronts, the method was further developed into the surface marker and micro-cell (SMMC) method (Chen et al., 1997). In SMMC, the smaller (micro) cells are employed only near the free surface, while the regular (macro) cells are used throughout the computational domain. Note that the advancement of the free surface is accomplished by the use of SM, while the discrete representation of the free surface for the purpose of the application of pressure boundary conditions is achieved via the micro-cells. Further considerations of this method as discussed in Chen et al. (1997) include physically motivated new procedures to carefully approximate the momentum ﬂuxes and to ensure that only physically meaningful velocity and pressure information is used to move the SM and advect the free surface, particularly for converging ﬂuid fronts. The interface remains sharp throughout the calculation. Another front-tracking method was developed by Unverdi and Tryggvason (1992). In this method the Lagrangian interface, represented by a set of connected line segments, is explicitly tracked and is used to reconstruct a representation of the ﬂuid property ﬁelds on an Eulerian mesh. To avoid introducing disturbances of length scale equal to the mesh by having the properties jump abruptly from one grid point to the next, the interface purposefully is not kept completely sharp but rather given a ﬁnite thickness of the order of the mesh size. In this transition zone, the ﬂuid properties change smoothly from the value on one side of the interface to the value on the other side. This artiﬁcial thickness is a function of the mesh size used, which does not change during the calculations; therefore, no numerical diffusion is introduced. Equations for the ﬂuid property ﬁelds can be written for the entire domain using an indicator function If (x, t), which is a scalar. This function has values of zero and unity to indicate the respective phases of the two-phase ﬂuid ﬂow. Values of the ﬂuid property ﬁelds at every location are then given by b (x, t) = 1 − If (x, t) b1 + If (x, t) b2

(7.118)

where b(x, t) refers to the ﬂuid properties (density, viscosity, speciﬁc heat, or thermal conductivity) being evaluated in space and time, and b1 and b2 are the ﬂuid properties corresponding to the two different phases. The indicator function can be written in the form of an integral over the whole domain (t) with the interface (t) as If (x, t) = δ x − x dVint (7.119) Vint (t)

566

BACKGROUND IN MULTIPHASE FLOWS WITH REACTIONS

where δ(x – x ) is a delta function which has a value of unity when x = x and zero everywhere else. Taking the gradient of the indicator function and transforming the volume integral into an integral over the interface yields ∇If = nδ x − x dSint (7.120) Sint(t)

where n is the unit normal vector on the interface. The divergence of Equation 7.120 leads to a Poisson equation to be solved for the indicator function: 2 ∇ If = ∇ · nδ x − x dSint (7.121) Sint (t)

Hence, the indicator function can be reconstructed by solving this Poisson equation, in which the right-hand side is a function of the known interface position at time t. Once the indicator function is determined, the ﬂuid property distribution ﬁeld can be calculated according to Equation 7.121. A distribution function is employed to approximate the delta function in the Equation, which deﬁnes the fraction of the interface quantity (such as the difference of ﬂuid properties of two phases, and surface tension) distributed to a nearby grid point across the artiﬁcial thickness of the front. The sharp jump of the indicator function thus is spread among the nearby grid points. A gradient ﬁeld (Gf = ∇If ) is generated that is nonzero within the ﬁnite thickness of the interface but zero everywhere else. The discrete form of the gradient function Gf is given according to Unverdi and Tryggvason (1992) as: Gf = D x − xf nf ∇Sf (7.122) f

where nf is the unit normal vector at an interface element with an area sf whose centroid is xf . The distribution function D(x − xf ) in Equation 7.122 is the form adopted by Peskin (1977) in which ⎧ π α * x − xf , if x − xf < 2 ⎨(4)−α 1 + cos 2 D x − xf = i=1 ⎩ 0 otherwise (7.123) From Equation 7.123, represents the Eulerian mesh spacing and α = 2, 3 (two and three dimensions, respectively). This function also can be used to interpolate ﬁeld variables from the background mesh to the interface front as uf = D x − xf u (x) (7.124) f

INTERFACE-TRACKING/CAPTURING

567

Subsequently, the interface is advected in a Lagrangian fashion by integrating dxf = uf dt

(7.125)

The advantage of this approach is that the interface position is known throughout the numerical calculation as it travels across the Eulerian mesh. In this sense, this approach alleviates the computational effort required for the interface curvature and its subsequent implementation for inclusion of the surface tension force. One major disadvantage of this approach is that it is sensitive to the spacing between the markers. When the markers are far apart, the interface will not be well resolved. If they are too close, local ﬂuctuations in the new positions of these markers can give rise to a very high interface curvature resulting in strong surface tension forces. As the interface evolves, the surface markers may not retain their spacing throughout the calculation, and it is necessary to redistribute the markers dynamically by either adding or deleting them during the calculation. This requirement thereby necessitates a continuous renumbering of the marker position in order to keep them in sequential order for the calculation of the interface curvature, which imposes a restriction especially on the prediction of merging or rupturing interfaces. Another disadvantage of this method is that there is no simple way of ordering the markers on surfaces in three dimensions. There may be regions where surfaces are expanding and no markers ﬁll that space. Without prior knowledge of the surface conﬁguration, there appears to be no way of adding the necessary required markers; that is, global marker distribution is somewhat difﬁcult. 7.10.1.2

Surface-Fitted Method

In the surface-ﬁtted approach, a mesh surface rather than markers is attached to the interface, and the interface curvature and position are thus known throughout the numerical calculation. Application of surface-ﬁtted methods is generally motivated by (a) a reduction in the computer storage needed for the interface markers, (b) always ensuring a sharp interface, and (c) avoidance of partially ﬁlled cells (or empty cells in when modeling free surface ﬂow between a liquid and void). Because the mesh and ﬂuid are allowed to move together, the mesh automatically tracks the free surface. Each mesh system in the respective ﬂuid domains now conforms to the shape and structure of the interface. Since the free surface interface is a boundary between the respective ﬂuid domains, accurate prescription of the boundary conditions of the interface is effectively realized. As the interface moves incrementally in time, the critical factors of this approach are the efﬁciency and stability of the numerical algorithm. The need to maintain a well-deﬁned mesh (whether body-ﬁtted mesh or unstructured mesh) throughout the calculation is an essential feature, which requires the application of a range of grid generation techniques to construct the appropriate surface mesh and the volume mesh in each computational calculation. In particular, the surface mesh of the free surface may become irregular due to uneven distribution and

568

BACKGROUND IN MULTIPHASE FLOWS WITH REACTIONS

unconstrained movement as each grid point on the interface is moved as time advances. An improper surface mesh generally degrades the numerical computation of the ﬂuid ﬂow. To ensure numerical stability and convergence of the numerical solution, it may be necessary to apply techniques that can capture the curvature information and redistribute curvature accordingly the grid points on the interface. With large amplitude motion, it also may be imperative to continuously regenerate the internal volume mesh encapsulating the ﬂuid domain, which evoke more complexities to the use of such a method in handling multiphase problems. Another shortcoming of this approach is that it can be applied only if the interface is not subjected to large deformation, since the method can lead to a signiﬁcant distortion of the internal volume mesh. Nevertheless, the major limitation of this method is that it cannot accommodate an interface that breaks apart or intersects with other surfaces. 7.10.2

Interface Capturing

A number of volume methods exist; two such volume methods are explained here. 7.10.2.1 Markers in Fluid (MAC Formulation) In the marker and cell (MAC) method of Harlow and Welch (1965), massless marker particles are scattered initially to identify each material region in the calculation. These markers are used primarily to distinguish the boundary between ﬂuids and do not participate directly in the calculations. They are used to track the trajectories of ﬂuid elements in a Lagrangian manner. These particles are transported in a Lagrangian manner along with the two phases in the ﬂow. These methods were developed initially for tracking the time history of a droplet falling into a stationary ﬂuid. Their presence in a computational cell indicates the presence of the marked material. The material boundary is reconstructed using the marker particle densities in the mixed cells with marker particles of two or more phases. Those cells containing markers are surface cells (S). All other cells with markers classiﬁed as full (F) cells are considered to be ﬁlled with ﬂuid. Usually, four types of cells are deﬁned in the MAC method in addition to full and surface cells: empty cells (E), which do not contain any ﬂuid, and boundary cells (B), which lie on the boundary of the computational domain. B cells play a static role by deﬁning the position and curvature of the ﬁxed boundary. These criteria are applied to each ﬂuid in the simulation, and the interface cells are identiﬁed as the cells that are S cells or F cells for more than one ﬂuid at the same time. Figure 7.16 shows an example of cell ﬂagging in a two-dimensional case with the consideration of two different ﬂuids. There are no empty cells or boundary cells shown in this ﬁgure since it does not show the entire domain. During each time step, the positions or coordinates of all the markers are obtained through the application of Equation 7.117. The evolution of surfaces is achieved by moving the markers with locally interpolated ﬂuid velocities. Note that special considerations are required to deﬁne the ﬂuid properties in newly

INTERFACE-TRACKING/CAPTURING

Figure 7.16

F1

F1

F1

F1

F1

F1

F1

F1

F1

S12

F1

F1

F1

F1

S12

S12

S12

S12

F1

S12 S12

F1

S12

S12

F2

F2

S12

F2

F2

S12 S12 S12

F2

F2

F2

F2

F2

F2

F2

F2

F2

F2

F2

F2

F2

F2

F2

F2

F2

F2

F2

F2

F2

F2

F2

F2

F2

F2

F2

F2

F2

F2

F2

F2

F2

F2

F2

F2

F2

F2

F2

F2

569

Schematic illustration of cell ﬂagging in a two-dimensional case.

ﬁlled cells and to cancel the values in cells that are emptied. The application of free surface boundary conditions consists of assigning the gas pressure to all surface cells. Also, velocity components are assigned to all locations on or immediately outside the surface in order to approximate conditions of zero-surface shear stress. One reason for the extraordinary success of the MAC formulation is that the markers do not track surfaces directly; rather they track the ﬂuid volumes. Surfaces are simply the boundaries of the volumes. In this sense, surfaces may appear, merge, or disappear as the volumes break apart or coalesce like the complex phenomena of wave breaking. This particular approach has been adopted by Daly (1969) to study the effect of surface tension on interface stability. The inclusion of surface tension forces extends the method to a wider class of ﬂow problems. Although the approach is readily extendable to three-dimensional computation, the already-considerable storage and computer time involved increase signiﬁcantly in accommodating a large number of necessary markers. Typically, an average of about 16 markers in each cell is required for accurate tracking of surfaces undergoing large deformations. The MAC formulation is therefore predominantly restricted to two-dimensional simulations. Another limitation is that the method cannot evaluate regions involving converging or diverging ﬂows. For example, when the ﬂuids are pulled into long convoluted strands, the markers may no longer be good indicators of the ﬂuid conﬁguration. If the markers are pulled substantially apart, unphysical voids may develop in the ﬂuid ﬂow, and an unphysical prediction of the free surface ﬂow may result. Marker particle methods are extremely accurate and robust and can be used to predict the topology of an interface subjected to considerable shear and vorticity in the ﬂuids sharing the interface. However, this method is computationally expensive due to the requirement of many particles, especially in three dimensions. Moreover, difﬁculties arise when the interface stretches considerably, which requires the addition of fresh marker particles during the ﬂow simulation. 7.10.2.2 Volume of Fluid Method The volume of ﬂuid (VOF) method is one of the best-known methods and was ﬁrst proposed by Hirt and Nichols (1981). In the VOF method, the ﬂuid location

570

BACKGROUND IN MULTIPHASE FLOWS WITH REACTIONS

is recorded by employing a volume-of-ﬂuid function, or color function, which is deﬁned as: = 1 ⇒ control volume is ﬁlled only with phase 1 = 0 ⇒ control volume is ﬁlled only with phase 2 0 < < 1 ⇒ interface present In the VOF algorithm, the color function is discontinuous over the interface, which facilitates the calculation of the properties of each of the phases and makes it possible to present an accurate numerical scheme for solving the color transport equation. In the volume of ﬂuid method, the ﬂow equations are volumeaveraged using an averaging volume smaller than the bubbles/drops used in the simulations. Considering only two phases, without mass exchange, and volumeaveraging the mass and momentum equations, three cases are encountered, as shown in Figure 7.17. In cases 1 and 2, averaging picks out either of the two phases. In such a case, we have: ∂ ρk (7.126) + ∇ · ρk uk = 0 ∂t ∂ ρk uk (7.127) + ∇ · ρk uk uk = ∇ · τk − pI + ρk g ∂t In case 3, averaging picks out a piece of the interface and both phases. In such a case, we have: ∂ ρ1 (7.128) + ∇ · ρ1 u1 = 0 ∂t ∂ (1 − ) ρ2 (7.129) + ∇ · (1 − ) ρ2 u2 = 0 ∂t ∂ ρ1 u1 + ∇ · ρ1 u1 u1 = ∇ · (τ1 − p1 I) + ρ1 g ∂t 1 + (τ1 − p1 I) · n1 dA (7.130) Vcell Aint

Volume of a computational cell is used to perform volume averaging Case 1: Phase 1 (Λ = 1) Case 2: Phase 2 (Λ = 0) Case 3: Phase 1 and 2 (0 < Λ < 1)

Figure 7.17 Averaging volume compared with the bubble or droplet volume (modiﬁed from Gopala and van Wachem, 2008).

INTERFACE-TRACKING/CAPTURING

∂ (1 − ) ρ2 u2 + ∇ · (1 − ) ρ2 u2 u2 ∂t = ∇ · (1 − ) (τ2 − p2 I) + (1 − ) ρ2 g 1 + (τ 1 − p1 I) · n2 dA Vcell Aint

(7.131)

(−σ1 · n1 − σ2 · n2 )dA =

Jump conditions :

571

Aint

Fint,σ dA

(7.132)

Aint

The scalar is the property of the ﬂuid (e.g., volume fraction) with which it moves. Its evolution is governed by the simple advection equation: ∂ + ∇ · uint = 0 ∂t

(7.133)

One of the critical issues with the VOF method is the discretization of the advection term in Equation 7.133. Lower-order schemes, such as the ﬁrst-order upwind method, smear the interface due to numerical diffusion, and higher-order schemes are unstable and result in numerical oscillations. Thus, it is necessary to derive advection schemes that can keep the interface sharp and produce monotonic proﬁles of the color function. Over the years, several volume advection techniques for ﬁnite volume and ﬁnite difference meshes have been proposed by many researchers: These include Noh and Woodward’s simple line interface calculation (1976), Hirt and Nichols’ donor–acceptor scheme (1981), Youngs’ method (1982), the ﬂux-corrected transport (FCT) method by Boris and Book (1973), and Ubbink’s compressive interface capturing scheme for arbitrary meshes (CICSAM) (1997) and Lagrangian piecewise linear interface construction method. Some of these methods are brieﬂy described. Flux-Corrected Transport FCT is based on the idea that a suitable combination of upwind and downwind ﬂuxes can be formulated that eliminates both the diffusiveness of the upwind scheme and the instability of the downwind scheme. The idea of adjusting ﬂuxes calculated with a higher-order (nonmonotonic) advection scheme to improve the monotonicity of the ﬁnal result was introduced by Boris and Book (1973) and was generalized and extended to multidimensions by Zalesak (1979). The method involves several stages of calculation. First, an intermediate value of , namely * , is determined using a lower-order monotonic (and hence diffusive) advection scheme. The scheme for solving the one-dimensional version of Equation 7.133 (for mesh cell i ) is symbolically written as:

∗i = ni −

t L L fi+ 1/ − fi− 1 /2 2 x

(7.134)

572

BACKGROUND IN MULTIPHASE FLOWS WITH REACTIONS

where f L represents the lower-order ﬂux. The ﬂux at any surface (i − 1/2) or (i + 1/2) is fi+ 1/2 = (uint )i+ 1/2 and fi− 1/2 = (uint )i− 1/2 . An antidiffusive ﬂux is introduced that is deﬁned as the difference between higher-and lower-order ﬂux approximations. Therefore, it can be written (e.g., at i + 1/2) as: H L f ad1 = fi+ 1/ − fi+ 1/ i+ /2

2

(7.135)

2

Application of the entire antidiffusive ﬂux results in the unstable higher-order ﬂux being used; thus correction factors (limiters) are introduced that limit the antidiffusive ﬂuxes. The ﬁnal step of ﬂux-corrected transport algorithm is to apply the antidiffusive ﬂuxes with the correction factors and obtain the values of the color function at the new time: t n+1 i+ 1/2 f ad1 − i− 1/2 f ad1 (7.136) = ∗i − i i+ /2 i− /2 x For detailed description of FCT, please refer to the book by Oran and Boris (1987). Lagrangian PLIC Method This Lagrangian volume of ﬂuid method based on piecewise linear interface construction (PLIC) was proposed by van Wachem and Schouten (2002). Figure 7.18 shows a computational cell with an interface separating the two ﬂuids (i.e., phases 1 and 2). The interface itself is deﬁned by the local volume fraction of one of the two ﬂuids and the normal vector n (= n1 , n2 , n3 ); the values of ni are the directional cosines of the unit normal. This interface between the two phases

e3

n

c3 Phase 2

B

Phase 1

o

c2

C

e2

c1 A

e1

Figure 7.18 Volume of a computational cell intersected by an interface ABC (modiﬁed from Gopala and van Wachem, 2008).

DISCRETE PARTICLE METHODS

573

is propagated by the local ﬂuid ﬂow along the interface. This method involves two steps: 1. Reconstruction of the Interface. As shown in Figure 7.18, e1 , e2 , and e3 are the three Cartesian directions, and c1 , c2 , and c3 are the lengths of the orthogonal computational grid cell. The general equation for a plane in three dimensions is given by: ni xi = α (7.137) where α is the shortest distance from the plane to the origin. The volume of the original prism is α 3 /(6n1 n2 n3 ). To obtain only the volume lying under the interface within the computational cell, we need to subtract the two volumes of triangular prisms that protrude outside of the original prism (OABC). The volume of these prisms is (1 − ni xi /α)3 if α > ni xi . However, this results in the volume of the small triangular prism in front of the ﬁgure being subtracted twice; hence this volume should be added again. The volume of this small prism is (1 − ni ci /α − nj cj /α)3 where i = j and if α > ni ci + nj cj . 2. Lagrangian propagation of the interface. The Lagrangian propagation of the interface can be best described by the change in interface equation 7.137 due to the movement of the ﬂow. This results in two contributions: (1) change in the values of α and ni due to the ﬂuid ﬂow leading to movement of the interface within the computational cell, and (2) change in the values of ci due to the movement of the rectangular sides of the volume, thus shifting the origin to the interface. In the closely related level-set algorithm, a color function is also employed, but this function is continuous and has no direct physical meaning. The local volume fraction is translated from the local value or gradient of the color function. An advantage of the level-set algorithm is its simplicity to compute derivatives of the color function, required, for instance, to calculate the curvature of the surface. A disadvantage of this approach is that the numerical representation of the transport equation to determine the values of the color function is prone to numerical error and leads to a loss or gain of mass when calculating the local volume fractions. In VOF methods, the color function is a semi-discontinuous function. However, the accurate calculation of the curvature of the interface, by determining the derivative of the color function, is difﬁcult from a numerical point of view. Readers interested in a more detailed description of other VOF methods may refer to Gopala and van Wachem (2008). 7.11

DISCRETE PARTICLE METHODS

The phenomena associated with the effective gas-particle interaction (drag forces), particle-particle interactions (collision forces), and particle-wall interactions are

574

BACKGROUND IN MULTIPHASE FLOWS WITH REACTIONS

not well understood. The major difﬁculty is the large separation of scales: The largest ﬂow structures can be of the order of meters, yet these structures are directly inﬂuenced by details of the particle-particle and particle-gas interactions, which take place on the scale of millimeters or even micrometers. To describe the hydrodynamics of the gas and particle phase, continuum-(Eulerian) and discrete(Lagrangian) type models have been developed. To model gas-solid two-phase ﬂows at different scales, we can choose appropriate combinations of the gas- and solid-phase models, provided that a four-way coupling is used either directly or effectively, depending on the scale of the simulation domain. The basic idea is that the smaller-scale models, which take into account the various interactions (ﬂuid-particle, particle-particle) in detail, are used to develop closure laws, which can represent the effective “coarse-grained” interactions in the larger-scale models (see the works by van der Hoef and coauthors, 2004, 2005, 2006). Note that in principle it is not guaranteed that all correlations between small-scale and large-scale processes can be captured by effective interactions. However, experience has shown that in many cases the main characteristics of gas-solid ﬂows can be well described by the use of closure relations. In this section, the intermediate level of modeling is discussed brieﬂy. Its connection to the other two levels of modeling (i.e., DNS and TFM or Euler-Euler model based on the kinetic theory of granular ﬂow) were highlighted earlier in this chapter (see Figure 7.2). That block diagram shows a schematic representation of the four different models, including the information that is abstracted from the simulations, which is incorporated in higher-scale models via closure relations with the aid of experimental data or theoretical results. DPMs have been used for a wide range of applications involving particles (e.g., the book by Ristow, 2000) ever since they were ﬁrst proposed by Cundall and Strack (1979). A major difference between their method and traditional DPM models is that a detailed description of the gas-phase dynamics is required, in order

Full-scale fluidized bed simulation by 1 phenomenological models

Discrete particle methods for 3 particles

3 1

2 4 Two-fluid models for a selected section

Finer grid for gas-phase solution

4

2

Figure 7.19 Graphic representation of the multilevel modeling scheme (modiﬁed from van der Hoof et al., 2006).

HOMEWORK PROBLEMS

575

to describe the interaction between the particles and the ﬂuidizing gas. The coupling of the DPM with a ﬁnite volume description of the gas phase based on the Navier-Stokes equations was ﬁrst reported in the open literature by Tsuji, Kawaguchi, and Tanaka (1993) and later by Hoomans et al. (1996) for the soft-sphere model and the hard-sphere model, respectively. (These are also discussed earlier in this chapter.) A multilevel modeling scheme for DPMs is shown in Figure 7.19. The arrows represent a change of model. On the left is a ﬂuidized bed on a life-size scale that can be modeled with the aid of phenomenological models, a section of which is modeled by the two-ﬂuid model (see enlargement), where the shade of gray of a cell indicates the solid-phase volume fraction. On the right, the same section is modeled using discrete particles. The gas phase is solved on the same grid as in the two-ﬂuid model. The bottom graph shows the most detailed level, where the gas phase is solved on a grid much smaller than the size of the particles

HOMEWORK PROBLEMS

1.

Comment on the modeling complexity versus computational cost of engineering problems associated with multiphase ﬂows. Refer to Figure 7.5.

2.

Obtain Equation 7.39 from Equation 7.37 by decomposition of pressure and stress into mean and instantaneous components.

3.

Obtain Equation 7.70 for granular temperature by using the steps shown in Section 7.7.2.1.

4.

Derive the impulse transfer equation shown in Equation 7.87 by using Equations 7.84 through 7.86. Also show that by using appropriate conditions, Equations 7.89 and 7.90 can be obtained.

5.

Obtain Equation 7.99 by using Equation 7.98 in the absence of mass transfer across the contact discontinuity.

6.

Derive the interface energy ﬂux balance shown in Equation 7.111.

7.

Comment on the relative merits of various interface tracking and interface capturing methods discussed in this chapter.

8 SPRAY ATOMIZATION AND COMBUSTION

SYMBOLS

Symbol

Description

B Ci

Dimension

Transfer number Parameters in turbulence model i = μ, ε1 , ε2 , g1 , g2 D Diffusivity dp or D Particle diameter E Total energy per unit mass of mixture em Internal energy per unit mass of mixture f Mixture fraction F Body force FEm Rate of energy generation per unit volume of mixture Time constant of momentum transfer F (qp) g Square of mixture-fraction fluctuations ga Acceleration of gravity G(qp) Time constant for energy transfer hc Convective heat-transfer coefficient hv Heat of vaporization (q) Ji Barycentric diffusion motion of component q k Kinetic energy of turbulence K (q) Rate of q generation by reaction in gaseous phase (s) Kp Rate of s generation by breakup or coalescence of other particle species

576

Fundamentals of Turbulent and Multiphase Combustion Copyright © 2012 John Wiley & Sons, Inc.

— — L2 /t L Q/M Q/M — F Q/L3 t 1/t — L/t2 1/t Q/L2 Tt Q/M M/L2 t L2 /t2 M/L3 t M/L3 t

Kenneth K. Kuo and Ragini Acharya

577

SYMBOLS

Symbol

Description

Dimension

Weighted mean velocity in r-direction Turbulent viscosity Viscosity of q in mixture Turbulent Prendtl/Schmidt number (φ = ρ, k, g, f ) Boundary-layer thickness Particle relaxation time Generic property (scalar) or void fraction of a two-phase mixture Emissivity Dissipation rate of turbulence kinetic energy Mass-flux fraction of i th species Volume fraction (solid) of species s Total rate of generation to particulate phase per unit volume Rate of generation of q per unit volume Deformation tensor Dilation

L/t L2 /t L2 /t — L t —

Greek Symbols v0 μt (q) μm σφ δ τp φ ε εi φs (q) ( m)ij θm

— L2 /t3 — — M/L3 t M/L3 t 1/t 1/t

Diacritical —

Time-averaged mean

Superscripts

(q) (p) (s), (r)

Turbulent fluctuating component based on time averaging Particular component of mixture Components of mixture other than q Additional phases among particles

Subscripts ∞ f p s rs m

Ambient condition Liquid fuel Particle of particle phase Particle of droplet surface Droplet radius Mixture

This chapter covers two major topics: first, spray atomization and later spray combustion. The single fuel droplet vaporization and combustion processes are given in the Kuo (2005, Chap. 6); therefore, they are not repeated in this book. Spray evaporation and combustion have broad industrial applications in power production, jet and rocket propulsion, material processing, and pollution control.

578

8.1

SPRAY ATOMIZATION AND COMBUSTION

INTRODUCTION TO SPRAY COMBUSTION

In view of convenience in transport and flexibility in storage of liquid fuels, spray combustion processes have been utilized in many engineering applications, including energy sources for propulsion and transportation systems, electrical power generation in power plants, waste disposal and energy recovery in incinerators, and furnaces for material processing purposes. A significant portion of the total energy demand has been met by the combustion of liquid fuels injected as spray into the combustion chamber. In the propulsion and transportation areas, spray combustion has been used in various engines, including liquid rocket engines (rocket motors), diesel engines (cars and trucks), gas turbine combustors (aircraft), hybrid rocket motors (space launch vehicles), and ramjets (air-breathing propulsion systems). For efficient combustion to occur, intimate mixing of fuel and air is a necessity; therefore, the study of the mixing process in an evaporating spray is important and constitutes an essential part of spray combustion studies. In some special cases, mixing can be separated from combustion, but most often combustion of the spray proceeds concurrently with mixing, due to the time required for the fuel droplets to evaporate and to move into the oxidizer-rich zone to achieve suitable mixing. In view of the intricate and coupled evaporation and burning processes, spray combustion can be considered one of the most challenging engineering topics. Spray combustion occurs in liquid rocket engines, gas turbine engines, diesel engines, industrial furnaces, and so on. Due to these varied applications, the establishment of predictive models for spray combustion processes is important to reduce the cost of development by trial-and-error methods. Numerous studies of spray combustion and associated processes have helped designers establish criteria to design efficient and stable combustors, determine rates of heat transfer to combustion-chamber surfaces, and examine the formation of pollutants such as soot, unburned hydrocarbons, (NOx ), and CO. Nitrogen oxides (NOx ) include various nitrogen compounds like nitrogen dioxide (NO2 ), nitric oxide (NO), and nitrous oxide (N2 O). It is known that NOx formation is strongly dependent on temperature, and significant reductions in NOx emissions can be achieved by reducing maximum flame temperatures. Also, it has been found from various investigations that conditions most favorable for soot formation occur when fuelrich zones have strong temperature gradients. The basic method of reducing soot formation in spray combustion is not only to reduce temperature gradients in fuel-rich zones but also to reduce the size of zones where strong temperature gradients and fuel-rich concentration can arise. In order to achieve these goals, the flow properties in combusting sprays must be determined. A realistic analytical model of a combusting spray must involve consideration of such diverse phenomena as the hydrodynamic characteristics of injection and spray formation, the transport characteristics of individual droplets, the twophase turbulent flow of spray, chemical reactions in a turbulent environment, and interactions of radiation with flame chemistry and turbulence. Some of the earlier reviews that touched on these aspects are summarized in Table 8.1 by Faeth (1977), who made significant contributions to the study of spray combustion.

INTRODUCTION TO SPRAY COMBUSTION

579

TABLE 8.1. Reviews of Spray Combustion Processes Source: Modified after Faeth, 1977.

Topic

References Injection Processes Chigier (1977), Henein (1976), Williams (1973), Hedley et al. (1971), Harrje and Reardon (1962), Reitz and Bracco (1976), Putnam et al. (1957), Cramer and Baker (1967), Tate (1969), Ranz (1956), De Juhasz (1969), Giffen and Muraszew (1953), Orr (1966), Levich (1962), Altman et al. (1960), Barr`ere et al. (1960), Browing (1958), Lewis (1963), Fraser (1957)

Injector configurations, spray breakup, drop-size correlation and distributions, initial spread rates

Single-Drop Processes Chigier (1977), Williams (1973), Williams (1968), Williams (1976), Hedley et al. (1971), Williams (1962), Williams (1965), Harrje and Reardon (1962), Sirignano and Law (1976), Odgers (1975), Putnam et al. (1957), Cramer and Baker (1967), Penner (1957), Lambiris et al. (1963), Barr`ere et al. (1960), Kumagai (1957), Bellan and Cuffel (1983), Bellan and Harstad (1988), Kuo (1996a), Sirignano (1999)

Transient effects, ignition, evaporation and combustion, convection effects, drag, extinction

Spray Processes Profiles of mean and turbulent quantities, spread rates, models

Chigier (1977), Caretto (1976), Mellor (1976), Henein (1976), Williams (1973), Williams (1968), Hedley et al. (1971), Williams (1962), Williams (1965), Harrje and Reardon (1962), Odgers (1975), Putnam et al. (1957), Cramer and Baker (1967), Lambiris et al. (1963), Spalding (1955), Barr`ere et al. (1960), Lewis (1963), Kumagai (1957), Caretto (1973), Lefebvre (1964), Sirignano (1983), Kuo (1996b), Sirignano (1999)

Pollutants in Sprays Formation of hydrocartons (HC), NOx , CO, and soot

Caretto (1976), Mellor (1976), Henein (1976), Williams (1976), Odgers (1975), Lefebvre (1975), Caretto (1973), Warnortz, Mass, and Dibble (1999), Watanabe et al. (2006)

Experimental Techniques Spray size distributions, velocities, and temperatures; droplet burning

Chigier (1977), Williams (1973), Williams (1968), Harrje and Reardon (1962), Putnam et al. (1957)

580

SPRAY ATOMIZATION AND COMBUSTION

Therefore, a substantial portion of the discussion of spray combustion covered in this chapter has been drawn from several excellent review papers by Faeth (1977, 1979, and 1983) and many papers published by him and his coworkers. For liquid rocket engine design, readers are referred to the book edited by Huzel and Huang (1992), and further updated and enlarged by a group of 12 experts of Rocketdyne Division of Rockwell International in the propulsion field. Some major advancements in spray combustion theoretical studies and experimental measurements were summarized in a two-volume set of spray combustion books edited by Kuo (1996a, b). Important subjects and associated concepts from these two volumes are extracted and presented in this chapter, emphasizing the fundamental aspects of spray atomization and combustion with experimental measurements. These fundamental background information and results can serve as complementary material to the book by Huzel and Huang (1992). In general, spray combustion processes have been studied for years to achieve more economical use of fuels, better control of pollutants in combustion products, and longer life of engineering devices. However, due to the complex nature of the spray atomization and combustion processes, many practical devices were designed based on the trial-and-error approach, which is very expensive. Over the last two decades, researchers have made significant improvements in nonintrusive diagnostic methods (laser-based techniques, X-ray radiography, high-resolution imaging, phase Doppler particle analysis, planar laser-induced fluorescence, etc.) that enable more detailed observations and measurements of spray atomization and combustion processes. The development of supercomputers with large memories and high-speed processors enabled theoreticians to formulate and numerically solve comprehensive models with more detailed consideration of physical and chemical processes involved in spray combustion. Many areas of spray combustion research have attained major advancements in recent years. Some of these areas include drop-size measurements, liquid jet breakup mechanisms, characterization of dense spray behavior, supercritical evaporation and combustion phenomena, numerical solution of comprehensive models with complex chemical kinetics, quantitative measurements of sprays using modern experimental techniques, and externally induced excitation on atomization processes. Some of the important advances in these areas are included later in this chapter. In addition, certain important features have been extracted from a very useful book by Sirignano (1999 [1st ed.], 2010 [2nd ed.]) on fluid dynamics and transport of droplets and sprays as part of the background material of this chapter. Important results from many recent publications on spray combustion have also been included.

8.2

SPRAY-COMBUSTION SYSTEMS

Sprays are burned in various ways, and each method poses different problems for the development of a reliable spray model. Several cases that are typical of the range of configurations encountered in practice are summarized by Faeth (1977), as shown in Table 8.2. In the prevaporizing system (the first case in the table),

SPRAY-COMBUSTION SYSTEMS

581

TABLE 8.2. Different Types of Spray Combustion Systems Source: After Faeth, 1977, adapted from Kuo, 1986.

Application Pre-vaporizing system: afterburners, lean combustors, carburetors, ramjets Liquid-fueled rocket engines

Gas-turbine combustors

Industrial furnaces

Diesel engines

Configuration

Independent Variablesa Structure z

Steady non-burning

z

Steady, more or less premixed

z, r

Steady, diffusion flame

x , y, z

Steady, diffusion flame

t, x , y, z

Transient, diffusion flame, ignition characteristics needed

a Simplest realistic approximation: all systems are axisymmetric or three-dimensional near the injector.

582

SPRAY ATOMIZATION AND COMBUSTION

the spray is injected into a heated air stream. The drops are almost completely evaporated before reaching the flame. Typical examples of such a configuration are afterburners and the carburetors of spark ignition engines. One-dimensional models can generally provide useful results in this case, except near the injector or when only a limited number of injectors are employed. The two-phase portion of the flow is usually noncombusting. In liquid-fueled rocket engines (the second case in Table 8.2), both fuel and oxidizer are injected from one end, providing a more or less premixed combustion system. In many designs, one-dimensional flow dominates most of the flow field, yielding relatively simple models for performance predictions. Near the injector face, and when only a few injectors are used, mixing effects are important, and more complex models must be employed. The gas-turbine combustor can be divided into three zones: a primary zone where liquid is injected into an air stream to form a nearly stoichiometric mixture of reactants in a two-phase flow, a secondary zone where combustion is completed, and a dilution zone where the combustion products are mixed with air to reduce the temperature of the flow to levels acceptable for expansion through the turbine. Since the fuel and air are not extensively premixed before combustion, the flame has the characteristics of a diffusion flame in which mixing of fuel and oxygen strongly influences the rate of reaction. One-dimensional models are not suitable for this configuration, although lumped-parameter models have been used as crude approximations. Industrial furnaces are qualitatively similar to gas turbines, although their unsymmetric configurations are more common and buoyancy effects are usually important due to the large physical size of the injector and relatively low gas velocities. The reason for using large injectors in industrial furnaces is to achieve more uniform heating in the furnace (see Kuo, 2005, Chap. 6). It is generally accepted that diesel engines (the last case in Table 8.2) represent the most difficult modeling problem. The process is primarily a diffusion flame; however, it is transient, and fuel impingement on surfaces can be important. The flow is three-dimensional, and prediction of ignition characteristics is necessary, since the combustion process is intermittent. It is important to note that for the cases illustrated in Table 8.2, the designation “premixed” or “diffusion flame” should be interpreted as only a general indication of the dominant behavior. For two-phase combusting flows, portions of the flow may be burning in a very different manner from other portions, depending on the effectiveness of the mixing process. This is because the mixing and combustion processes are closely coupled in two-phase reacting flows.

8.3 8.3.1

FUEL ATOMIZATION Injector Types

The performance of a spray combustion system has a crucial bearing on the design of the injector. An injector can be evaluated according to the distribution of drop sizes it produces, the angle of the spray, and the nature of the spray pattern [i.e., whether the spray pattern completely fills the outermost boundaries

FUEL ATOMIZATION

583

of the spray [full cone] or has a region relatively free of drops, along the axis of the injector [hollow cone]. However, the flow conditions and properties of the gas within the combustion chamber also influence the spray pattern. In general, injectors can be classified into two major categories: 1. Pressure-atomizing injectors. Only the liquid passes through such injectors, and atomization is achieved by virtue of a significant pressure drop across the injector, as the name suggests. 2. Twin-fluid injectors. Liquid fuel atomization is aided by flow of highvelocity gas through injector passages. TABLE 8.3. Various Types of Injector Systems Source: After Faeth, 1977, adapted from Kuo, 1986.

Type

Configuration

Structure

Application

Pressure-Atomizing Injectors Plain orifice

Hollow cone

Diesel engines

Pintle nozzle

Full cone or multiple cones

Diesel engines, gas turbines

Swirl nozzle (spill type) return

Hollow cone

Furnaces, gas turbines

Impinging jet

Fan spray

Rocket engine

Internal mixing

Full or hollow cone

Furnaces, gas turbines

External mixing

Hollow cone

Furnaces, gas turbines

Twin-Fluid Injectors

584

SPRAY ATOMIZATION AND COMBUSTION

Although twin-fluid injectors involve additional complexity, they can achieve much finer atomization than pressure-atomizing injectors, particularly at low fuel flow rates during off-design operation. Some typical injector types used in combustion systems are listed in Table 8.3. 8.3.2

Atomization Characteristics

Theoretical modeling and numerical evaluation of sprays require information on the distribution of droplet sizes and velocities produced by the injector. The spray formation process, however, complicates this specification, since it involves complicated processes, such as breakup of the primary jet, secondary droplet breakup, and collisions between drops. Three other difficulties are associated with the specification of the required atomization characteristics: 1. Spray characterization is normally performed under cold conditions; drop vaporization and the influence of combustion gases on the breakup process itself can alter these characteristics (Dombrowski, Horne, and Williams 1974). 2. Complete distributions of size and velocity as a function of position in the dilute portion of the spray are rarely available (Mellor, Chigier, and Beer, 1970). 3. Measurement of two-phase flow conditions near the injector exit is extremely difficult and challenging Because of these difficulties, average spray characteristics at some downstream location from the injector exit are used in many circumstances. 8.4 8.4.1

SPRAY STATISTICS Particle Characterization

According to F.A. Williams (1965), an understanding of the mechanism of spray combustion requires knowledge of (a) the burning mechanism of individual particles, (b) the statistical methods for describing groups of particles, and (c) the manner in which these groups modify the behavior of the gas in flow systems. The shape of liquid droplets may be considered to be spherical when these two conditions are met: Condition 1: The droplet collision, agglomeration, and microexplosion effects are small. Condition 2: The Weber number is low: Weg < 5. The gas-phase Weber number, Weg , is defined as: 2 ρg vp − vg D Dynamic force = Weg ≡ σs Surface-tension force

(8.1)

SPRAY STATISTICS

where ρg D σ s parameter vp − vg

= = = =

585

density of the gas diameter of the droplet surface tension force difference in the magnitude of resultant velocities between and particle (droplet) and gas

Similarly, the liquid-phase Weber number is defined as: 2 ρl vp − vg D Wel ≡ σs

(8.2)

where ρl is the density of the liquid. Although there is no actual slip between the gas and droplet at the droplet surface, the parameter vp − vg is often called the slip velocity. Condition 1 requires that the volume occupied by the condensed phase be much less than the total spatial volume. This means that the spray must be diluted so that the droplets collide with each other so seldom that collision-induced oscillations are viscously damped to negligible amplitude for most droplets. This condition is usually valid for hydrocarbon spray combustion systems. The degree of deformation of a liquid droplet caused by the so-called slip velocity between the droplet and gas depends on the Weber number. When Weg < 5, droplets are nearly spherical, and the droplet size can be adequately specified by a single parameter D, the diameter of the particle. As the Weber number increases, droplets can deform and eventually break up at high Weber numbers. There are three droplet breakup regimes based on the study of Borisov et al. (1981): parachute type, stripping type, and explosion type. These regimes are governed by the magnitudes of Weg and Weg Re−0.5 D . As described in Kuo (2005, Chap. 6) in which the Weber number is defined differently by a factor of 2. Readers should be aware of the existence of multiple definitins of Weber numbers in the literature. 8.4.2

Distribution Function

Let us now define the distribution function for a dilute spray as fj (r, x, v, t). The statistical meaning of the mathematical product of fj (r, x, v, t) dr dx dv represents the probable number of particles of chemical composition j in the radius range from r to r + dr located in the spatial range of d x centered at x with velocities in the range of d v around v at time t. Here d x and d v (shown in Figure 8.1) are the three-dimensional elements of physical space and velocity space, respectively. The dimension of fj (r, x, v, t) is (Number of particles)/[LL3 (L/t)3 ]. If the velocity dependence of the distribution function is not of primary interest, another distribution function Gj can be defined such that (8.3) Gj ≡ fj dv, j = 1, 2, . . . , M where Gj represents the number of droplets of the j th component per unit volume per unit range of radius. Similarly, we can integrate over the physical space to

586

SPRAY ATOMIZATION AND COMBUSTION dx1 dx2 dx3

x2

v2

x

dr

v

r

O

x1

dv1 dv2 dv3 v1

x3 v3 RADIUS VARIATION

PHYSICAL SPACE

VELOCITY SPACE

Figure 8.1 Length element in radius variation and volume elements in physical and velocity spaces.

define a distribution function Fj for droplet size (radius or diameter). If only one type of liquid (a single component) is present, the subscript j can be ignored. Therefore, the simplest drop-size distribution function can be written as F(D), which represents the fraction of particles per unit diameter range about D. Dropsize distribution curves obtained by Mellor, Chigier, and Beer. (1970) for a water spray in an air stream are shown in Figure 8.2. Overall spray characteristics are represented by distribution curves that can be given in terms of the cumulative percentage of droplet number, surface area, or volume as a function of droplet diameter. Figure 8.2, besides showing the 20 18

90

MB

E

8

ARITHMETIC MEAN DIAMETER = 76.61 μm

6

SAUTER MEAN DIAMETER = 109.2 μm

ΔD N

60

UM VO L

,%

12 10

ΔN

70

NU

14

100

80

ER

16

50 40 30

4

20

2

10

0

CUMULATIVE VOLUME AND NUMBER, %

100 DISTRIBUTION FUNCTION F(D)

0 0

20

40

60

80

100 120 140 160 180 200 220 240 DROP SIZE, D, μm

Figure 8.2 Drop-size distribution curves for a water spray in an air stream, measured by Mellor Chigier, and Beer (1970). (adapted from Kuo, 1986)

SPRAY STATISTICS

587

distribution function F (D), illustrates several of these methods of spray description for water injection through a swirl-type pressure atomizer into a moving air stream (Mellor Chigier, and Beer 1970). In many mass-transfer and flow processes, it is desirable to work only with average diameters instead of the complete drop-size distribution. Therefore, the droplet size frequently is represented by a mean or median diameter. However, there are at least half a dozen to choose from. A general expression for a mean diameter Djk in terms of the distribution function F (D) or dN/dD can be written as:

Djk

(j −k)

Dmax

Dmin Dmax

=

Dmin

Dj

dN dD dD

dN D dD dD

(8.4)

k

where j and k are integers. A more general form of the mean of the droplet diameter raised to a power (j − k) can be calculated from: ∞ j (j −k) D F (D) dD (8.5) ≡ 0∞ k Djk 0 D F (D) dD which yields D10 as the average (arithmetic) droplet diameter, D20 as the diameter of a droplet whose surface area times the total number of droplets equals the total area of the spray, and D30 as the volume-average diameter. The Sauter mean diameter (SMD), D32 , is the diameter of a droplet whose ratio of volume to surface area is equal to that of the entire spray. It is commonly used to represent the size of an equivalent mono-dispersed spray for approximate analysis of evaporation and combustion. A summary of the popular mean diameters and their fields of application is given in Table 8.4. An accurate knowledge of the drop-size distribution is a prerequisite for fundamental analysis of the transport of mass or heat or the separation of phases in spray combustion systems. Numerous distribution functions have been proposed to correlate drop size in a spray. General features of four important size distribution functions are given in the next sections. Among them, the TABLE 8.4. Popular Mean Diameters and Their Fields of Application k

j

Order k+j

0 0 0 1 1 2 3

1 2 3 2 3 3 4

1 2 3 3 4 5 7

Name

Field of Application

Linear Surface Volume Surface diameter Volume diameter Sauter De Brouckere

Comparisons, evaporation Absorption Hydrology Adsorption Evaporation, molecular diffusion Combustion, mass transfer, and efficiency studies Combustion equilibrium

588

SPRAY ATOMIZATION AND COMBUSTION

Rosin-Rammler distribution (1933) and Nukiyama-Tanasawa distribution (1940) are the best known. 8.4.2.1 Logarithmic Probability Distribution Function Drop diameters can be graded in size ranges exponentially rather than linearly, by defining D y ≡ ln (8.6) D30

Then the volume distribution equation can be written as: dVl = φl (y) dy

(8.7)

where φl (y) is the volume fraction of droplet material in. When sets of spray data are plotted on this basis, they ordinarily give a fairly symmetrical distribution about a certain value of y that is close to a single maximum of the curve. A good guess for φl (y) based on statistical analysis is √ 2 2 the normal distribution function (δ/ π)e−δ y . Then we have δ dVl 2 2 = √ e−δ y dy π

(8.8)

where δ is the distribution factor and has to be fitted to the data by trial and error. The general expression for the mean diameter of this distribution can be written as: 2 (8.9) Djk = D30 e(k+j −6)/4δ 8.4.2.2 Rosin-Rammler Distribution Function The volume fraction distribution equation is of the form

where

dVl δD δ−1 −(D/Dref )δ e = δ dD Dref

(8.10)

Vl = percentage volume of droplets (in the total volume of droplets) with diameter less than or equal to D Dref = characteristic size (= D30 ) δ = distribution width factor

The expression for the mean diameter is:

Djk

(j −k)

(j − 3) +1 δ (k − 3) +1 δ

= (Dref )(j −k)

where is the conventional gamma function.

(8.11)

SPRAY STATISTICS

589

8.4.2.3 Nukiyama-Tanasawa Distribution Function From their experiments with air atomization, Nukiyania and Tanasawa (1940) obtained an empirical expression for drop-size volume distribution in this form:

dVl b6 /δ δ D 5 e−bD = dD (3/δ)

(8.12)

where = gamma Function δ = distribution width factor b = size parameter, which has a dimension of D −δ The equation for the mean diameter is:

Djk

(j −k)

=b

(j + 3)/δ

(k + 3)/δ

−(j −k)/δ

(8.13)

8.4.2.4 Upper-Limit Distribution Function of Mugele and Evans It can be shown that both the Rosin-Rammler and the Nukiyama-Tanasawa distribution functions have this generic form:

F (D) = aDp exp −bDn

(8.14)

where a, p, b, and n are empirical parameters. Distributions of this type place no restriction on the size of the largest drop that can exist in a spray. Therefore, extrapolation of distributions given in Equation 8.14 can yield values of D32 greater than any observed droplet diameter in the spray in some cases, and the use of this equation also can cause significant underestimation of spray evaporation rates. In fact, all three distribution functions considered so far fail to place an upper limit on the drop diameter. Mugele and Evans (1951) have proposed a distribution function that imposes an upper limit. Basically, it is a modification of the logarithmic probability distribution function. The dimensionless parameter y is defined as: aDs y ≡ ln s (8.15) Dm − D s or y ≡ ln where

D (Dm − Dvmd ) Dvmd (Dm − D)

a = dimensionless constant s = positive integer (e.g., 1, 2, or 3) D s = simple function of the drop size (e.g., diameter, surface, or volume) Dm = maximum drop size

(8.16)

590

SPRAY ATOMIZATION AND COMBUSTION

s So Dm − D s is a measure of the “size deficiency” or reduction from maximum drop size. Dvmd in Equation 8.16 represents the volume median diameter, D30 . The expression for the Sauter mean diameter (SMD, D32 ) can be calculated from

D32 =

Dm 1 + ae−4δ2

(8.17)

The volume distribution can be calculated from dVl (D) δ = √ exp −δ 2 y 2 dD π

(8.18)

where δ is the distribution width parameter. In the study of Mugele and Evans, all four distribution functions were critically compared with experimental data. Four general conclusions were reached: 1. The logarithmic probability distribution function is so far the best one for calculating mean diameters. It predicts equality of means of the same order (e.g., D30 = D21 , D40 = D31 ), and this agrees well with experimental data. 2. The Nukiyama-Tanasawa distribution function sometimes gives a completely wrong trend for the volume distribution when the parameters are calculated from numerical distribution data. Uncertainty in δ leads to uncertainty in the mean diameter. 3. Unreasonable values of mean diameters sometimes result from the RosinRammler distribution function. 4. The upper-limit distribution function (ULDF) predicts mean diameters as well as the logarithmic probability density function (pdf) and agrees quite well with experimental data. The ULDF may be capable of eventual interpretation in terms of fundamental mechanisms. Aside from the distribution function, there are numerous correlations for various average and median drop-size parameters. These are specific to various injector designs but generally have the form (Faeth, 1977) ρudjet μu ρ∞ μ∞ u∞ Djk L = fjk , φl , , , , , (8.19) djet djet μ σs ρ μ u 8.4.3

Transport Equation of the Distribution Function

According to F.A. Williams (1959, 1965), an equation governing the time rate of change of the distribution function fj mentioned in Section 8.4.2 can be derived phenomenologically by using reasoning analogous to that employed in the kinetic theory of gases. Let us define the next quantities: Rj ≡ (dr/dt), the rate of change of the size r of a particle of kind j at (r, x, v, t)

SPRAY STATISTICS

591

Fj ≡ (dv/dt), the force per unit mass on a particle of kind j at (r, x, v, t) ˆ j , the rate of increase of fj with time due to particle formation, agglomQ eration, or coagulation (from smaller particles) or destruction (from larger particles), as in liquid droplet breakup j , the rate of increase of the distribution function caused by collisions with other particles (These collisions must occur sufficiently seldom that the aerodynamic contributions to j are separable from those in Fj ) −∇x · (vfi ), the increase of fj due to the motion of particles into and out of the spatial element dx by virtue of their velocity v −∇f · Fj fj , the increase of fj in the velocity element dv because of the acceleration Fj Adding the changes in fj , we have ∂fj ∂ ˆ j + j =− Rj fj − ∇x · vfj − ∇v · Fj fj + Q ∂t ∂r

(8.20)

for j = 1, 2, . . . , M, where M is the total number of different kinds of particles classified according to their chemical composition. Equation 8.20 is called the transport equation of the distribution function or the spray equation (F.A. Williams, 1959, 1965). Under many circumstances, the intensity of burning is comparatively low in the vicinity of the atomizer, and most of the combustion takes place in regions where particle interactions and sources are of no more than secondary importance. Since we focus our attention on the burning process, we may legitimately neglect ˆ j and j . If a steady process is considered, then ∂fj /∂t = 0. Hence for many Q combustion problems, the steady-state equation of motion of a statistical ensemble of particles of type j is ∂ Rj fj + ∇x · vfj + ∇v · Fj fj = 0 ∂r

(8.21)

This equation is coupled with the hydrodynamic equations of motion of the gas through the variables Rj and Fj , which depend on the local properties of the fluid. However, if the spray is sufficiently dilute, the statistical fluctuations in the gas properties induced by the spray can be neglected and accurate average fluid-flow variables can be used. Then the motion of the gas can be described by the ordinary fluid dynamic equations with appropriate boundary conditions determined by average spray properties.

8.4.4 Simplified Spray Combustion Model for Liquid-Fuel Rocket Engines

As mentioned in Section 8.2, the spray combustion processes occurring in liquidfuel rocket engines can sometimes be treated with a steady or quasi-steady

592

SPRAY ATOMIZATION AND COMBUSTION

approximation. Also, the reactant mixtures can be considered as premixed. The general flow characteristics are quasi–one-dimensional. Under these idealized conditions, simplified spray combustion models have been developed to study the combustion efficiency of liquid-fuel rocket-engine combustors with variable cross-sectional area. Probert (1946) initially investigated this problem, and later the model was refined by Tanasawa (1954), F.A. Williams (1958, 1962), Tanasawa and and Tesima (1958). By neglecting the velocity dependence of the distribution function, Equation 8.21 reduces to ∂ Rj Gj + ∇x · vj Gj = 0, ∂r

j = 1, 2, . . . , M

(8.22)

where Gj is given in Equation 8.3 and the bar denotes an average over all velocities, that is, 1 Rj = (8.23) Rj fj dv, j = 1, 2, . . . , M Gj and vj =

1 Gj

vfj dv,

j = 1, 2, . . . , M

(8.24)

Equation 8.22 can be rewritten in the form of Equation 8.25 to take into account of the variation of the cross-sectional area A(x) along the combustor: 1 ∂ ∂ Rj Gj + Avj Gj = 0, ∂r A ∂x

j = 1, 2, . . . , M

(8.25)

where vj is the x -component of vj and the quantities Rj , Gj , and vj are assumed to be essentially independent of the spatial coordinates normal to x . These quantities may be regarded as averages over the cross-sectional area. It is known from the droplet vaporization correlation that the dependence of Rj on droplet size may be expressed by the relation Rj = −

χj , r αj

j = 1, 2, . . . , M

(8.26)

where 0 ≤ αj ≤ 1 and χj represents a positive constant independent of r. Note that αj = 1 corresponds to the d 2 -evaporation law mentioned in Kuo (2005, Chap. 6). In view of the dependence of vj on droplet size, we generally can expect that the velocity distributions may differ for particles of different sizes. However, when the droplets are very small, they can be totally entrained by the gas. Under this circumstance, vj is independent of r. Using these approximations or assumptions, Equation 8.25 can be integrated. The solution of the distribution function

593

SPRAY STATISTICS

Gj by integrating Equation 8.25 was given by F.A. Williams (1965), and the expression obtained is α j A0 vj,0 r Gj,0 γj , j = 1, 2, . . . , M (8.27) Gj = Avj γj where γj ≡ r

αj +1

x

+ αj + 1

0

χj dx vj

1/(αj +1) ,

j = 1, 2, . . . , M

(8.28)

The subscript 0 in Equation 8.27 designates parameters at the station x = 0. The most interesting parameter in the study of liquid-fuel rocket engines is the combustion efficiency for a chamber of length L from the size distribution at position L. Let Qj denote the heat of reaction (heat released) per unit mass of material evaporated from a droplet of kind j, and let ρl,j be the density of the liquid droplets of kind j . The mass per unit volume of the spray of kind j is therefore ∞ 4 3 3 πr ρl,j Gj dr 0

and the corresponding mass flow rate is Avj times this integral term. The total amount of heat released per second by the j th spray fuel between the injector and position L is: ∞ ∞ 4 3 4 3 Qj A0 vj,0 3 πr ρl,j Gj,0 dr − Avj 0 3 πr ρl,j Gj dr 0

Since the maximum possible heat release from burning all the droplets is M

∞

Qj A0 vj,0 0

j =1

4 3 3 πr ρl,j Gj,0 dr

the combustion efficiency at x = L can be calculated from ∞ ∞ M 4 4 3 3 Qj A0 vj,0 3 πr ρi,j Gj,0 dr − A0 vj 3 πr ρl,j Gj dr ηc =

j =1

0

M j =1

0

∞

Qj A0 vj,0 0

By using Equation 8.27, the combustion efficiency can be written as M ∞ 3 r αj r Gj,0 γj dr Qj ρl,j vj,0 γj 0 j =1 ηc = 1 − ∞ M r 3 Gj,0 (r) dr Qj ρl,j vj,0 j =1

(8.29)

4 3 3 πr ρl,j Gj,0 dr

0

(8.30)

594

SPRAY ATOMIZATION AND COMBUSTION

γj in Equation 8.30 can be calculated from Equation 8.28 by setting the upper limit of the integral on the right-hand side of Equation 8.28 equal to L. Equation 8.30, together with Equation 8.28, gives the combustion efficiency for an arbitrary initial droplet size distribution Gj,0 (r). 8.5

SPRAY COMBUSTION CHARACTERISTICS

Although combustion of liquid fuel droplets in a spray is inherently governed by the diffusion of fuel vapor and oxidizer species, both premixed- and diffusionflame theories have been applied to spray combustion problems. Premixed-flame theories in general should have very limited applications to spray combustion; however, sometimes they can be applied to spray combustion in liquid rocket engines, when fuel and oxidizer streams are introduced to achieve a high degree of mixing. Premixed-flame theories can also be applied to study the stabilization of spray flames on flame holders and occasionally for the development of detonation waves in spray fields, as the most conservative condition. Figure 8.3 shows a schematic comparison of temperature and reaction-rate distributions in premixed gaseous flames and a spray combustion flame in a

FLOW DIRECTION YO

INNER EDGE OF LUMINOUS ZONE

REACTION RATE

YF YP T Z PREMIXED LEAN GASEOUS FUEL COMBUSTION

FLAME DROP FLOW DIRECTION

T

D REACTION RATE

Z DROP COMBUSTION IN A CHANNEL FLOW

Figure 8.3 Comparison of premixed gaseous fuel/oxidizer combustion with spray combustion (modified from Faeth, 1977).

SPRAY COMBUSTION CHARACTERISTICS

595

one-dimensional model. In both cases, a simple one-step chemical reaction Fuel + O2 → Products was assumed. The distribution of concentration and temperature profiles in the premixed gaseous flame is shown on the left portion of this figure. In the combustion of fuel droplets, we must consider the size and volatility of the droplet. For very small droplets of a high-volatility fuel, droplet evaporation may be completed in the heat-up process, so that the flame structure is only mildly influenced by the effect of the two-phase flow. Burgoyne and Cohen (1954) conducted a classic experiment using mono-disperse tetralin (C10 H12 ) sprays, showing that droplet sizes less than 10 μm resulted in flame speeds that are not appreciably different from those in pure gas. They also showed that the mass burning rate decreases as the initial drop size increases. This is due mainly to the increased time required to evaporate the fuel. For larger drop sizes and nonvolatile fuels, the fuel evaporation rate decreases significantly, with a region of combustion surrounds each droplet. The end of the flame zone then nearly corresponds to the disappearance of the droplet. A simplified one-dimensional model of this process has been developed and presented by F.A. Williams (1965). General agreement between his theoretical results and the experimental data of Burgoyne and Cohen (1954) was obtained. However, we must be extremely cautious in using the theory of premixed flames to treat spray combustion problems, due to the inherently nonpremixed nature of the spray. Diffusion flame theories can definitely be applied to many spray combustion problems, since the fuel vapor evaporated from the droplet surface has to mix with the ambient oxidizer before chemical reaction can occur. The background material on diffusion flames surrounding a single droplet as well as in a jet has been covered extensively in Kuo (2005, Chap. 6). Readers are encouraged to become familiar with this material before proceeding with the following discussions. In the study of spray diffusion flames, Mizutani, Yasuma, and Katsuki (1976) obtained radial profiles of the temperature, velocity, and droplet mass flux at various axial stations of a spray flame stabilized in a heated air stream (see Figure 8.4). In their experimental setup, pressure-atomized injection was used with a moderate swirl in the fuel stream, so that a full-cone spray was obtained. The two-phase flow was sampled by a specially designed sampling probe, and the liquid fraction was determined by separation and weighing. Although full-cone spray was established, the spray had some hollow-cone characteristics: Near the injector, the maximum droplet mass flux occurred at some distance away from the axis of the spray. The flame front, determined from the appearance of blue light, is located about 10 mm from the injector exit. Near the centerline, the temperature levels are low and the droplet concentrations are relatively high; hence the droplets are evaporating, and no significant combustion occurs in this cool central region. However, in the outer regions of the spray, fuel droplets are actively involved in the combustion process. The zone bounded by the maximum temperature was found to be slightly smaller than the spray cone. They concluded

596

SPRAY ATOMIZATION AND COMBUSTION

2

4

VELOCITY (m/s) 4 2 4

2

2

4

750

1250

TEMPERATURE (°C)

RADIAL DISTANCE (mm)

750 1250

750 1250 0

4

8

750 0

4

1250 8

0

4

8

LIQUID FLUX (Kg/m2–s)

30

20

10

SPRAY CONTOUR

NOZZLE 10

20 AXIAL DISTANCE (mm)

30

40

Figure 8.4 Radial profiles of temperature, velocity, and drop flux in spray flame (after Mizutani, Yasuma, and Katsuki, 1976, adapted from Kuo, 1986).

that spray flames were stabilized in a high-temperature stream, principally by the flame propagation mechanism. The flame front therefore approaches the injector exit far more closely than would he expected from the ignition-delay data. Khalil and Whitelaw (1976) studied spray combustion of kerosene fuel and employed several injectors to achieve sprays with different drop sizes [4.5 < Sauter Mean Diameter (D32 ) < 100 μm]. Measurements were made of droplet velocity, velocity fluctuations, and number density using a laser Doppler velocimeter. Values of mean temperatures were measured with a Pt: 40% Rh-Pt: 20% Rh fine-wire thermocouple of bead diameter 180 μm. Values of the rms temperature fluctuation were determined with a thermocouple of bead diameter 40 μm and Pt: 30% Rh and Pt: 60% Rh wires. Contours of isovelocity, isoturbulence, and isotherms for one of their hollow-cone spray flames with SMD = 45 μm are shown in Figure 8.5. The outline of the reaction zone is indicated by the velocity and temperature distributions. It roughly corresponds to the cone angle of the injector. Along the axis, temperatures and velocities are relatively low near the injector, suggesting that portions of the spray are evaporating in a relatively cool environment, with a major reaction zone along the periphery of the spray. This observation is similar to those of Mizutani, Yasuma, and Katsuki (1976). With larger droplet sizes and cone angles, the cool central region expands both axially and radially, suggesting relatively strong influences of the injector on the combustion process. Khalil and Whitelaw also made an attempt to examine the effect of twophase flow by studying the corresponding gas flame with mass velocity and cone angle similar to the spray. The assumptions of fast chemical reaction and a clipped Gaussian probability distribution of scalar fluctuations were used in their

597

SPRAY COMBUSTION CHARACTERISTICS U' CONTOURS OF [ ~rms ] u

15

14 12 m/s

15 m/s 14 12 10

5

5

20% 15%

15% 20% 30% 40% 50%

1600K 1400K

10%

1200K 5

10

5

0

1800K

10

600K 30% 20%

10

20

1 1 400K 0K 10300K 00 K

10 16

CONTOURS OF [T] x D

x D

x D 15 16 m/s 10

80

~ CONTOURS OF [u]

0 30 r D

5%

10

20

30 r D

0

10

20

30 r D

Figure 8.5 Contours of velocity, turbulence, and temperature for a kerosene spray flame with SMD = 45 μm (modified from Khalil and Whitelaw, 1976).

30 2000 T 20 U, m/s

1500 T, K

U 1000 10 500 0

0

10.0

5.0 x /D

Figure 8.6 Centerline distribution of velocity and temperature in kerosene spray flame with SMD = 45 μm: curves, simulated gas jet flame; ∇, measured values of U ; ◦, measured values of T (after Khalil and Whitelaw, 1976, adapted from Kuo, 1986).

theoretical calculations based on a two-equation turbulence model. Figure 8.6 gives the comparison of the centerline velocity and temperature distributions of the spray flame and the corresponding gaseous jet simulation. It indicates that the gaseous jet simulation attains a maximum temperature before the spray flame. The maximum centerline temperature of the gaseous jet flame simulation is less than that of the spray flame, but as can be seen from Figure 8.7, the maximum temperatures in two different spray flames were achieved at some distance from the centerline. This situation stems from the characteristic of hollow-cone spray flow caused by the angle of the spray and the corresponding gaseous jet flame simulation. Figure 8.7 indicates that the radial profiles of spray flames cannot be

598

SPRAY ATOMIZATION AND COMBUSTION 2000

2000 x/D = 3.64

x/D = 7.15 1500 T, K

T, K

1500

1000

500

1000

500

0

1.0

2.0 r/D

3.0

0

1.0

2.0

3.0

r/ D

Figure 8.7 Radial profiles of temperature: curves, simulated gas jet flame; •, SMD = 45 μm; ◦, SMD = 100 μm (after Khalil and Whitelaw, 1976, adapted from Kuo, 1986).

accurately simulated by corresponding gaseous jet flames. Even with fine sprays (SMD = 45 μm), two-phase effects must be considered in modeling. Onuma and Ogasawara (1974) conducted experiments on spray flames of axial jets of kerosene under low-turbulence conditions. Droplet and temperature distributions, flow velocity, and gas composition were measured in the flame of an air-atomizing burner. They found that the region where the droplets exist is limited to a small area above the burner nozzle. From the correlation between various distributions measured, they concluded that most of the droplets in the flame do not burn individually with envelope flames but that fuel vapor from the droplets forms a cloud and burns like a gaseous diffusion flame. Under the same conditions (using the same apparatus and changing only the fuel from liquid kerosene to gaseous propane), the spray combustion flame was found to be very similar in structure to the turbulent gas jet diffusion flame. Figure 8.8 shows various distributions along the flame axis obtained for two types of flames: spray flame vs. gaseous jet diffusion flame. It is quite obvious that there is a qualitative resemblance between the two cases. However, there was a slight difference. In the gas diffusion flame, chemical reactions occurred slightly downstream as compared with the spray flame. This is due to the higher initial flow velocity of propane than that of kerosene. In the spray flame, the temperature drop beyond 40 cm downstream of the injector is slightly steeper than that in the gas diffusion flame. This difference is probably caused by the higher emissivity and radiation cooling rate for the spray flame. Despite these differences, the two flames are still very similar in various property distributions. This similarity provides experimental support for group combustion theories developed by Chiu and his coworkers (1977, 1978, 1981, 1982) to be discussed in a later section.

SPRAY COMBUSTION CHARACTERISTICS

599

20

1500

20

10 T

O2

V

10

0

0

10

1000

0

CO CO2 N

500

5 HC

0 0

20

O2 MOLE %

2000

HC CO CO2 MOLE %

20

50

TEMPERATURE T °C

NUMBER OF DROPLETS N

FLOW VELOCITY V m/s

Liquid kerosene spray flame

0 60

40

DISTANCE FROM THE NOZZLE TIP cm (a)

10

5

0

20

2000

O2 25

10

1500 O2 V

T C3H8 0

O2 MOLE %

50

0

1000 TEMPERATURE T °C

C3H8, HC, CO, CO2 MOLE %

15

FLOW VELOCITY V m/s

Propane gas jet flame

CO2

CO

500 HC 0 0

40

20

60

DISTANCE FROM THE NOZZLE TIP cm (b)

Figure 8.8 Various distributions along flame axis in: (a) spray flame and (b) gas jet diffusion flame (after Onuma and Ogasawara, 1974, adapted from Kuo, 1986).

600

SPRAY ATOMIZATION AND COMBUSTION

Onuma, Ogasawara, and Inoue (1976) extended their earlier investigation in similar experiments to determine whether the similarity between the spray flame and the turbulent gas jet diffusion flame also exists in nitric oxide formation, and whether their earlier conclusions on similarity can be applied to the flames of heavy oil with low volatility. They observed that the radial profile of NO concentration in a kerosene flame exhibited two peaks symmetrical with respect to the flame axis. These peaks coincide approximately with the peaks of the temperature profile and with the positions where the local equivalence ratio is unity. This tendency is the same as in turbulent gas jet diffusion flames, and so it was concluded that the spray flame is also similar to the turbulent gas jet diffusion flame in the NO formation process. A heavy-oil spray flame was also experimentally compared with a kerosene flame under the same conditions. The results showed that the shapes and measured profiles of various quantities were almost the same for both flames. This finding confirms that the heavy-oil spray flame does not differ significantly in structure from the kerosene flame, even though the heavy oil had somewhat larger drop sizes. Styles and Chigier (1976) have also conducted measurements in a kerosene spray flame using an air-blast atomizer. Their photographs revealed an initial dense spray region with no combustion, surrounded by a gaseous diffusion flame. General agreement with the results of Onuma and Ogasawara (1974) and Onuma, Ogasawara, and Inoue (1976) was obtained. An overall picture of the spray combustion process that emerges from these studies was presented by Faeth (1977) as shown schematically in Figure 8.9. This figure illustrates the relative locations of the cold core region, reaction zone, spray boundary, and jet boundary of a coaxial spray diffusion flame. It is generally understood that the spray leaving the injector is highly nonuniform, with smaller droplets at the periphery and larger droplets near the centerline. Near the injector exit, the velocity difference (slip velocity) between the droplets and the surrounding gas is greatest, and momentum of the liquid is transferred to the gas over an extended axial distance. The small droplets around the periphery of the spray exchange momentum with the gas and cause the spray jet to entrain surrounding gas. These small droplets also evaporate rapidly to provide fuel vapor that is consumed near the outer portion of the turbulent diffusion flame. In general, beyond the maximum reaction zone, small droplets tend to follow the flow and may be largely confined to fuel-rich eddies as shown in the insert of Figure 8.9. These small droplets may evaporate significantly before high concentrations of oxygen appear in their immediate surroundings. Large droplets, however, can travel a considerable distance before evaporation and combustion processes begin; this is due to the relatively high inertia of these large droplets. With increasing distance from the injector exit, the centerline temperature increases due to combustion of the spray, causing more rapid evaporation of large droplets close to the centerline. According to Faeth (1977), measurements from various experiments show that the disappearance of droplets is closely correlated with positions of maximum temperature, in both the radial and axial directions. A number of investigations have been conducted to study these phenomena under

SPRAY COMBUSTION CHARACTERISTICS

601

SPRAY BOUNDARY

REACTION ZONE T

YO

YF

Vg

Vd

JET BOUNDARY T YO

YF ENTRAINED GAS Vd

DROPS

COLD CORE

Vg

Figure 8.9 Schematic representation of coaxial spray diffusion flame (after Faeth, 1977, adapted from Kuo, 1986).

more complex conditions akin to those in gas-turbine combustors and furnaces. The general observation is that the bulk of the fuel evaporation occurs in relatively cool, low-oxygen-concentration regions. However, this condition may have some exceptions. Faeth (1977) concluded in his review paper that the understanding of spray combustion in various practical systems shows that although in some cases spray combustion can be modeled by ignoring the details of spray evaporation and treating the system like a gaseous diffusion flame, in many circumstances, such simplifications are unwarranted and the turbulent two-phase flow must be considered. Improved injector characterization methods, more information on droplet transport characteristics in turbulent flow, and continued development of more

602

SPRAY ATOMIZATION AND COMBUSTION

complete two-phase turbulent models are needed. In particular, these four aspects need further consideration: 1. Although one-dimensional models are acceptable methods for the design of liquid rocket engines, information is lacking on three-dimensional effects near the injector, liquid jet breakup processes, transient effects, the sensitivity of model predictions to operation parameters, and injector characteristics in hot recirculating flows. 2. Additional work is necessary to firmly establish the limitations of the locally homogeneous flow models (discussed later in this chapter). 3. A consistent method of estimating drop characteristics in two-phase flows is needed. Systematic studies are required to resolve all the transport effects. 4. Criteria for drop ignition and the presence of stable envelope flames need more attention. The effect of convection on burning drops is also of substantial importance and needs to be studied further. The next section describes various spray models developed and recent advances in spray combustion processes. 8.6 CLASSIFICATION OF MODELS DEVELOPED FOR SPRAY COMBUSTION PROCESSES

In view of the two-phase nature of the turbulent reaction processes involved in spray combustion, there are numerous difficulties associated with the development of predictive models. Models with different basic assumptions and levels of sophistication have been proposed, ranging from simple correlations to complex turbulent two-phase reacting models. This text describes classes and representative samples of such models. 8.6.1

Simple Correlations

Numerous investigators have summarized experimental results by simple powerlaw correlations (e.g., Bahr, 1953; Rao and Lefebvre, 1976), yielding expressions for the percentage of fuel evaporated as a function of pressure, temperature, air velocity, injector characteristics, and distance from the injector. Empirical correlations have also been developed for the rate of spread of the spray (Bahr, 1953). For diesel engines, empirical correlations have been developed to relate the rate of injection and the rate of heat release. Such correlations are limited to the specific engine and injector considered in the study. For the purpose of air-pollution control, a number of correlations for NO emissions from specific engines have been developed. Some success was obtained for those correlations by considering the characteristics of drop evaporation, mixing rate of fuel and air, and the like in order to define characteristic times that suggest appropriate forms of correlation for specific operating conditions.

CLASSIFICATION OF MODELS DEVELOPED FOR SPRAY COMBUSTION PROCESSES

8.6.2

603

Droplet Ballistic Models

In droplet ballistic models, the ambient gas temperature and velocity are assumed to be constant, and the effect of entrainment and cooling of the ambient gases by the spray is ignored. The spray characteristic is determined solely by processes associated with individual droplets. A more detailed description of this type of model is given in Section 8.4.3. As can be seen clearly from the definition of the distribution function fj and the transport equation of fj , poly-disperse injected sprays are conveniently handled using Equation 8.20. An example of this type of model is the pioneering analysis of Probert (1946) mentioned in Section 8.4.4, in which the evolution of the size distribution for a drop was considered under the assumption of no relative motion between the ambient gas and fuel droplets. The effects of convection and drag for mono-disperse injected sprays have been studied by Natarajan and Ghosh (1975). Westbrook (1976) developed a numerical solution technique for the spray equation 8.20 and applied it to thin sprays injected into a combustion chamber. The effects of independent variations of a large number of system parameters were studied. These parameters include initial spray dispersion, amount and type of gas swirl, gas density, injection timing, chamber geometry, initial droplet size distribution, injection velocity, drag coefficient, vaporization rate, injector aperture size, droplet specific gravity, and direction of injection. The calculations were made for three-dimensional and transient conditions, neglecting the source terms Qj and j in Equation 8.20. While this is a useful calculation, neglecting the effect of the spray in modifying gas temperatures and velocities limits the application of the results to general spray combustion problems. Bracco (1974) used this type of model to study the spray combustion of ethanol drops in oxygen in a rocket motor of constant crosssectional area. His model adopts a Nukiyama-Tanasawa distribution function of initial drop radii, a Stokes drag equation, and either a modified Priem-Heidmann (1960) or a modified Spalding (1953) drop vaporization-rate equation. Bracco found that his calculated results reproduced accurately the steady-state data on the engine tested. The Priem-Heidmann and Spalding vaporization-rate equations, which were suggested for single-fuel droplets vaporizing and burning in infinite oxidizing media, were found to overestimate the vaporization rate of the droplets within the spray. 8.6.3

One-Dimensional Models

In one-dimensional models, interactions between liquid and gas phases are considered, however, the complexities of droplet diffusion in turbulent gas streams are usually ignored. As mentioned before, for liquid-fuel rocket engines with fine sprays, one-dimensional spray combustion models are broadly utilized, especially in earlier studies. In their studies of spray drying processes, Dickinson and Marshall (1968) used a one-dimensional model to determine the effect of the spray size distribution function. They assumed that the effects of drag and droplet evaporation on gas velocity are negligible and the droplet temperature was constant. In a later study, Law (1975) improved this analysis by allowing for gas velocity and temperature changes due to evaporation.

604

SPRAY ATOMIZATION AND COMBUSTION

In a simulation of droplet evaporation in a Wankel engine, Bracco (1973) solved the transient, one-dimensional spray equation together with the gas-phase energy equation, assuming no spatial pressure gradients. In his analysis, convective heating of droplets during evaporation was considered. However, the effect of transient heating of droplets was ignored. None of these models has been critically evaluated by comparison with experimental data. In order to study the combustion efficiency of a variety of rocket engines, Priem and Heidmann (1960) developed a one-dimensional model assuming constant gas temperature. Droplet heat-up process and drag were considered, but burning around individual droplets was ignored. Although droplet shattering was not considered in the model, the condition for droplet breakup was found when a critical Weber number, Wec , was exceeded. Mador and Roberts (1974) applied the stream-tube approach to the analysis of gas-turbine combustors. In their analysis, exchange of mass and heat by turbulent mixing is allowed. Although the model gave some encouraging results in the prediction of exhaust emissions, the validity of the model for predicting other characteristics of spray combustion needs to be examined. In general, one-dimensional models are comprehensive but still involve a substantial amount of empiricism. Experience with these models has led to optimized selection of physical properties and empirical parameters. These models can be helpful in design, but this in no way implies that one-dimensional models are satisfactory in all aspects for predicting spray combustion processes.

8.6.4

Stirred-Reactor Models

In order to assist in the design of certain combustion chambers for the evaporation and burning of fuel droplets, some engineers have employed simplified analyses based on the concept of stirred reactors. In this approach, the treatment of recirculating flow patterns and many detailed two-phase reacting-flow phenomena are significantly simplified by considering the so-called well-stirred or plug-flow reactors. Swithenbank, Poli, and Vincent (1973) developed a model of this type based on interconnected and partially stirred reactors, whose individual performance was computed utilizing energy-balance principles. Their model was developed to predict performance variables, such as blowoff stability limits, combustion efficiency, combustion intensity, and overall pressure loss. Turbulence levels within the combustor were also calculated to determine the noise output, ignition condition, and heat-transfer rates. Satisfactory agreement was obtained between the predicted and measured overall stability loops from a high-intensity gas turbine–type combustor by Swithenbank et al. Courtney (1960) applied a stirred-reactor analysis to study rocket-engine combustion processes. Munz and Eisenklam (1976) modeled combustion processes in a high-intensity spray combustion chamber in terms of the flow configuration of stirred and plug-flow reactor elements. The evaporation of a spray with polysized droplets followed by homogeneous pyrolysis and complex chemical reactions was calculated in order to determine the trends in NO emission.

LOCALLY HOMOGENEOUS FLOW MODELS

605

Although various researchers and designers have used this type of model as a tool to guide their combustion chamber design, the concept and application of stirred reactors have severe limitations. The diffusion processes of droplets and gaseous species are strongly coupled with the combustion processes. Also, the fuel vapors from the droplet surfaces have to mix well with the oxidizer before significant chemical reactions can occur. Therefore, this type of model must be carefully examined before it is applied. 8.6.5

Locally Homogeneous-Flow Models

In an earlier work, Thring and Newby (1953) recommended that the length of a turbulent spray flame could be estimated by assuming locally homogeneous flow (LHF), which ignores the slip effect between the condensed and gas phases. Under this assumption, the two phases are assumed to be in dynamic and thermodynamic equilibrium (i.e., at each point in the flow, they have the same velocity and temperature and are in phase equilibrium). The spray is essentially equivalent to a gas jet having the same momentum and stoichiometric conditions. Note that the LHF condition may be regarded as the limiting case of a spray consisting of infinitely small droplets. In their study of evaporation of liquid sprays at supercritical conditions, Newman and Brzustowski (1971) employed the locally homogeneous flow assumption in the analysis. Atomization, mean velocities, and temperatures were predicted quite well. Khalil and Whitelaw (1976) used a second-order turbulence closure and LHF approximation in studying kerosene spray flames. As discussed in Section 8.5, they found that their solutions were not completely satisfactory even for small spray droplets SMD = 45 μm. More extensive discussion of LHF models is given in Section 8.7. 8.6.6

Two-Phase-Flow (Dispersed-Flow) Models

The two-phase-flow (dispersed-flow) model is the most logical approach in studying spray combustion, since effects of finite rates of transport between the two phases are included in the analysis. A systematic development of turbulence models for two-phase flows with the consideration of concentration, temperature, and velocity gradients began in late 1970s. The development of two-phase flow models is of great importance, since LHF models are severely limited to the condition when the drops in the spray are extremely small. Various dispersed-flow models have been proposed and solved. More extensive discussion of dispersed-flow models is given in Section 8.8. 8.7

LOCALLY HOMOGENEOUS FLOW MODELS

The basic premise of LHF models is that rates of transport between the phases are fast compared to the rate of development of the flow field as a whole. This approximation requires that at each point in the flow, all phases have the same

606

SPRAY ATOMIZATION AND COMBUSTION

velocity and are in thermodynamic equilibrium. The LHF requirement is most easily met by a gas phase dispersed in a continuous liquid phase, due to the relatively low inertia and thermal capacity of bubbles, as in the experiments of Avery and Faeth (1974). For sprays (dispersed liquid phase and continuous gaseous phase), the LHF approximation is most appropriate when the drop sizes are small, the densities of the phases are almost the same, and the rate of development of the process itself is slow (Faeth, 1983). The justification of LHF models for some spray combustion processes is given by the observations made by Onuma and Ogasawara (1974); Onuma, Ogasawara, and Inoue (1976); Komiyama, Flagan, and Heywood (1976); and Styles and Chigier (1976). They showed that there exist striking similarities between the structure of flames fueled with gases and with well-atomized sprays having maximum drop number densities for 10 < SMD < 20 μm. There are several advantages of LHF models. First, they require minimum information concerning injector characteristics, since initial drop size and velocity distributions play no role in the computations. It is well known that injector properties are difficult to obtain: The liquid jet breakup and droplet formation processes in the dense spray are still far from being understood. The second advantage of LHF models is the saving in computer time. Computations for sprays are nearly identical to those for single-phase flows. A third advantage is that LHF models require far fewer empirical constants than discrete droplet models in dispersed flows. According to Faeth (1983), LHF models provide a reasonable first estimate of the extent and character of a spray process. They are useful in giving a lower bound on the size of the spray. LHF models can also provide an indication of potential process improvements by enhancing atomization prior to testing. 8.7.1

Classification of LHF Models

There are various kinds of LHF models of spray combustion; some are formulated by integral approaches (e.g., those of Thring and Newby, 1953; Newman and Brzustowskl, 1971; Shearer and Faeth, 1977), and some employ higher-order turbulence models calibrated using measurements in combusting and noncombusting gas flows (e.g., Khalil and Whitelaw, 1976; Khalil, 1978; Mao, Szekely and Faeth, 1980, 1981; Shearer, Tamura, and Faeth, 1979; A summary of LHF models of spray has been compiled by Faeth (1983) and is given in Table 8.5. According to Reynolds (1980), besides integral models, there are two major classes of turbulence models: full-field modeling (FFM) and large-eddy simulation (LES). The FFM method uses partial differential equations to describe the change in certain averaged quantities. These average quantities or variables can be grouped into two categories: 1. Mean flow properties, such as the velocity, mixture fraction, and temperature 2. Turbulence parameters, such as the turbulence kinetic energy k , dissipation rate ε, square of the mixture-fraction fluctuation g, and turbulent stress components

LOCALLY HOMOGENEOUS FLOW MODELS

607

TABLE 8.5. Summary of Locally Homogeneous Flow Models of Sprays (modified from Faeth, 1977) Flow Date Researcher(s) Configurationa Modelb

Experimentc

Assessment

1953 Thring and Newby

Qualitative agreement for flame length

1971

Good agreement for spray boundary

1977

1976

1978

1979

Combustion of Integral, Axisymmetric, steamboundary-layer EQ, atomized parabolic combustion spray in air, 0.1 MPa Evaporation of Integral, Newman and Axisymmetric, pressureboundary-layer EQ, Brzusatomized parabolic evaporation towski liquid near its critical point, 6–9 MPa Combustion of Integral, Shearer and Axisymmetric, pressureFaeth boundary-layer EQ, atomized parabolic combustion spray, no swirl, SMD = 30 μm (est.), 0.1–9 MPa Combustion of k-ε-g, Khalil and Axisymmetric, pressureEQP, Whitelaw swirling flow atomized elliptic with spray, swirl, recirculation SMD = and 45,100 μm, combustion 0.1 MPa Combustion of k-ε-g, Khalil Axisymmetric, pressureMEBU, swirling flow atomized elliptic with spray, no recirculation swirl, SMD and = 45,100 combustion μm, 0.1 MPa Shearer, Evaporation of k-ε-g, Axisymmetric, Tamura, air-atomized boundary-layer EQP, spray, no parabolic evaporation and Faeth swirl, in still air, SMD = 29 μm, 0.1 MPa

Spray and flame boundaries underestimated by 30%–50%, poor estimation of flow width

Rate of development of process overestimated

Improved prediction in some cases

Rate of development of process overestimated, predicted mean velocities and mixture fraction 20%–40% below measurements near the injector

(continued overleaf )

608

SPRAY ATOMIZATION AND COMBUSTION

TABLE 8.5. (continued ) Flow Date Researcher(s) Configurationa Modelb

Experimentc

Assessment

Combustion of Rate of k-ε-g, 1980 Mao, Szekely, Axisymmetric, development of air-atomized and Faeth boundary-layer EQP, process spray, no parabolic combustion overestimated, swirl, in still flame length air, SMD = underestimated 35 μm, 0.1 by 20% MPa Combustion of Rate of k-ε-g, 1981 Mao, Waka- Axisymmetric, development of pressureboundary-layer EQP, matsu, and process atomized parabolic combustion Faeth overestimated, spray, no spray length swirl, in still underestimated air, SMD = 30 μm (est.), by 20% 3–9 MPa a All

cases shown here are steady. implies local thermal equilibrium; EBU implies use of the eddy-breakup model to estimate fuel concentration in conjunction with local thermal equilibrium; MEBU is the same as EBU except for an empirical modification for drop combustion; EQP implies use of local thermal equilibrium with pdf for mean properties. c Pressures are those of the spray environment. b EQ

As described in Chapters 4 to 6, turbulence models must be constructed for various terms appearing in the governing equations for turbulence closure. When these models are constructed, contributions from all scales of turbulent motion must be considered. This procedure introduces some difficulties, since large-scale aspects of turbulent flows usually are anisotropic while processes at small scales approach isotropy. The LES approach involves completing calculations of the time-dependent, three-dimensional structure of the turbulent flow with specified initial conditions to reflect randomness. In this approach, modeling is necessary only for turbulence scales smaller than the computational grid spacing. It is well known that the small scale of turbulence is nearly universal in character and can be modeled more reliably than situations where the full range of turbulence scales must be modeled (Reynolds, 1980). Most LHF spray models have employed the FFM approach. Either Reynolds (time) averaging or Favre (mass-weighted) averaging can be used for FFM spray models. The latter is even more suitable for compressible flows. Detailed discussions for these averaging procedures are given in Chapter 4. In general, when Reynolds averaging is used, the governing equations can reach the Favre-averaged form only when a number of terms involving density fluctuations, which may not be negligible, are ignored. The solution of Reynolds-averaged equations then corresponds to a Favre-averaged solution,

LOCALLY HOMOGENEOUS FLOW MODELS

609

without computation of mass averages by solution of equations involving correlations of density and other variables. Since this is only one of many approximations in turbulent-flow models, Reynolds averaging has generally been adopted for LHF spray models due to its computational convenience, even though either method can be applied. The discussions in this text are limited to the Reynolds-averaged model equations.

8.7.2

Mathematical Formulation of LHF Models

There are several major components of the theoretical model, including basic assumptions, equation of state, conservation equations, turbulent transport equations, boundary conditions, and physical input correlations and constants. 8.7.2.1

Basic Assumptions

Six assumptions are usually made for LHF models, to yield a simple form of the theoretical formulation: 1. Transport coefficients of all species and heat are the same (i.e., Dt = αt = νt ). 2. The combustion process is adiabatic. 3. Radiation, viscous dissipation, and kinetic energy are negligible. 4. Molecular rates of reaction are infinitely fast, so local thermodynamic equilibrium is maintained. 5. The mean flow is steady and axisymmetric. 6. Due to the computational convenience and extensive validation of the twoequation model, the following isotropic-turbulent-viscosity expression is assumed to be valid: μt = Cμ ρ

k2 ε

(8.31)

Local thermodynamic equilibrium, when combined with the assumption of equal species diffusion coefficients, implies that the local state of the mixture is completely specified by pressure, velocity, mixture fraction, and temperature or total enthalpy. The equal diffusion coefficient assumption, when coupled with the assumption that radiation, viscous dissipation, and kinetic energy can be neglected, implies that the local state of the mixture can be completely specified by pressure and mixture fraction alone. 8.7.2.2 Equation of State The relationships among mixture enthalpy, composition, temperature, density, and mixture fraction can be regarded as equations of state. Each type of spray considered requires a separate equation of state. Under low enough pressures, all gases may be assumed to be ideal gases. Considering Ns species in the flow, the

610

SPRAY ATOMIZATION AND COMBUSTION

mass fraction for the i th species in the mixture can be written as Yi = Yi0 f + Yi∞ (1 − f ) ,

i = 1, . . . , Ns

(8.32)

where subscripts 0 and ∞ designate the injector-exit and far-field conditions, respectively. Since each species may exist in gaseous and/or liquid state, we have Yi = Yfi + Ygi (8.33) The enthalpy of the mixture can be expressed as: h = h0 f + h∞ (1 − f )

(8.34)

In terms of mass fractions of all species in their possible states, the enthalpy can also be written as: Ns

h= Yfi hfi + Ygi hgi , i = 1, . . . , Ns (8.35) i=1

When the mixture temperature used is at the appropriate value, these two enthalpy expressions should yield equal values. The density of the mixture can be given as: Ns

−1 ρ= (8.36) Yfi vfi + Ygi vgi , i = 1, . . . , Ns i=1

where vfi and vgi are the partial specific volumes of species i in the liquid and gas phases. Given the relationships between the enthalpy and density of each species, the local temperature, and partial pressure of each species, Equations 8.32 to 8.36 are sufficient to describe the composition, temperature, and density of the mixture for a specified mixture composition. The relative mass fractions of the gas and liquid phases of the i th species can be obtained from Equation 8.33 and the requirement that the chemical potential of each species must be the same in both phases. EXAMPLE 8.1 Specify the necessary equation-of-state relationships for a pure air jet injected into a stagnant bath of water at 298 K and 101 kPa at the plane of injection. Solution An equation of state for a two-phase air-water system can be formulated in this way. The spray can be considered isothermal, and the air can be assumed to behave as an ideal gas. The effect of water vapor can be considered negligible. Also, we can assume that no air is dissolved in the water. Under these assumptions, the mass fraction of air and water can be found from

LOCALLY HOMOGENEOUS FLOW MODELS

611

Equation 8.32: YA = f YW = 1 − f

(1) (2)

The density of the two-phase mixture in the jet can be obtained from Equation 8.36: f 1 − f −1 ρ= + (3) ρA ρW where the density of both the water and the air can be assumed to be constant. The air density can be computed assuming that it is an ideal gas. The plot of YA , YW , and ρ versus mixture f fraction is shown in Figure 8.10. It is 1000

ρ , Kg/m3

100

10

1 1.0

MASS FRACTION

0.8

0.6

0.4

YA

YW

0.2

0.0 0.0

0.2

0.4

0.6

0.8

1.0

MIXTURE FRACTION, f

Figure 8.10 Dependence of density and mass fractions of air and water on mixture fractions (equation of state for the air-water system, after Shearer and Faeth, 1977, adapted from Kuo, 1986).

612

SPRAY ATOMIZATION AND COMBUSTION

interesting to note that the mass fractions of the air and water vary linearly with the mixture fraction. However, the variation of density with mixture fraction is nonlinear. EXAMPLE 8.2 Specify the necessary equation-of-state relationships for air-atomized injection of Freon-11 as an evaporating spray into a stagnant air at 298 K and 101 kPa. Solution The mass fraction of Freon-11 can be determined from Equation 8.32: YF = YF 0 f

(1)

where YF 0 is known from the injector exit condition. The mass fraction of air, which is the complementary part, is obviously YA = 1 − YF

(2)

The enthalpy of the two-phase mixture can be determined from Equation 8.34: h = h0 f + hA∞ (1 − f )

(3)

The enthalpy of air is given by: hA = Cp,A (T − Tref )

(4)

where the reference temperature Tref is 298 K. The enthalpy of Freon-11 in the liquid and gas phases can be expressed as hFf = Cp,Ff (T − Tref )

(5)

hFg = Cp,Fg (T − Tref ) + hF, fg(Tref )

(6)

where hF, fg(Tref ) is the heat of vaporization of Freon-11 at the reference temperature Tref . The enthalpy at the injector exit is the sum of the contributions due to Freon-11 and the injected air: h0 = YA0 hA0 + YF 0 hF 0

(7)

The gas phase consists of both Freon-11 vapor and air. Assuming ideal-gas behavior, the total pressure is equal to the sum of the partial pressures of the individual components:

613

LOCALLY HOMOGENEOUS FLOW MODELS

p=

pi = pA + PFg

(8)

When the liquid is present in the spray and is at equilibrium with the vapor, the partial pressure of Freon-11 in the gas phase must equal the vapor pressure. The vapor pressure of Freon-11 can be correlated by the next expression: log10 pFg = A −

B T

(9)

where A = 6.7828 and B = 1416.1 if T is in K and p is in kPa. Then, since mole fractions in the gas phase are proportional to the partial pressures, the mass fraction of Freon-11 in the gas phase is YFg =

YA pFg MwF pA MwA (1 − pFg /pA )

(10)

YFf = YF − YFg

(11)

then

The enthalpy of the mixture can also be determined from Equation 8.35: h = YA hA + YFf hFf + YFg hFg

(12)

As long as the mixture is saturated, the mixture temperature and composition can be calculated at any mixture fraction using Equations (1) to (12). A temperature is estimated and the partial pressure of Freon vapor is determined from Equation (9). The mass fractions and mixture enthalpy are then calculated from Equations (10) to (12). The enthalpy computed in this manner is compared with the enthalpy determined directly from Equation (3). An interval-halving procedure can be employed to adjust the temperature until the two calculated values of the mixture enthalpy agree. Once all the liquid has vaporized, YFf = 0 and the mass fractions of air and Freon-11 vapor can be determined directly from YFg = YF 0 f

(13)

YA = 1 − YFg

(14)

The mixture temperature can then be determined from Equations (3), (4), (6), and (12). The mixture density can be determined from ρ=

YFf YFg YA + + ρFf ρFg ρA

−1 (15)

614

SPRAY ATOMIZATION AND COMBUSTION

The density of the Freon-11 liquid can be taken as a linear function of temperature. The air and vapor densities can be obtained from the ideal-gas expression at the temperature and total pressure of the mixture. The equation-of-state relationships for the evaporating Freon-11 spray in an air-atomized jet into a stagnant air is shown in Figure 8.11. The total mass fractions of air and Freon-11 are linear as shown in the plot in the exhibit. The presence of liquid, however, causes nonlinear behavior of mixture temperature and density. The physical properties in the computation are: MwF = 137.37 kg/kg-mole, Cp,Fg = 0.674 kJ/kg K, Cp,Ff = 0.879 kJ/kg K, hF,fg R = 171.17 kJ/kg, ρFf = 2143.7 − 2.235T kg/m3

ρ, Kg/m3

T, K

300

260

10

0 1.0

YA

MASS FRACTION

0.8

YF

0.6 YFf 0.4

0.2 YFg 0.0 0.0

0.2

0.4

0.6

0.8

1.0

f

Figure 8.11 Equation of state for evaporating Freon-11 spray (after Shearer and Faeth, 1977, adapted from Kuo, 1986).

LOCALLY HOMOGENEOUS FLOW MODELS

615

8.7.2.3 Conservation Equations With the equation of state defined for the two-phase mixture, the subsequent LHF model formulation reduces to the conservation equations, which are the same as for single-phase flows. For the case of a spray directed into an infinite stagnant medium, the flow can be assumed axisymmetric and the boundary-layer approximations are applicable. The next conservation equations are limited to cases of stationary mean flow, with axisymmetric geometries. Continuity Equation

The generalized form of the continuity equation is ∂ (8.37) ρ ui + ρ ui = 0 ∂xi

Considering an order-of-magnitude analysis for boundary layers where the thickness is very small compared to the axial distance, we assume x, u, ρ∼O(1)

and

r∼O(δ)

(8.38)

where the symbol O designates the order of magnitude. The generalized continuity equation 8.37 can be expressed for the 2-D axisymmetric flow as: (a)

(c)

(b)

(d)

∂ ∂ 1 ∂ 1 ∂ ρu + rρ v = 0 (ρ u) + (rρ v) + ∂x ∂x r ∂r r ∂r

(8.39)

The Boussinesq approximation can be employed to model the density-velocity correlations appearing in terms (b) and (d): μt ∂ρ σρ ∂x μt ∂ρ −ρ ρ v = σp ∂r

−ρ ρ u =

(8.40) (8.41)

where μt is the “turbulent viscosity” and σρ is the turbulent Prandtl/Schmidt number. Substituting Equations 8.40 and 8.41 into Equation 8.39, applying the order-of-magnitude analysis using Equation 8.38, and setting σρ = 1, we have (a)

(b)

(c)

(d)

μt ∂ρ ∂ 1 ∂ ∂ μt ∂ρ 1 ∂ r + =0 (ρ u) − (rρv) − ρ ∂x r ∂r r ∂r ρ ∂r ∂x ∂x 1

μt

v/δ

(8.42)

μt /δ 2

where the order of magnitude of each term is indicated. In order for Equation 8.42 to be nontrivial, terms (c) and (d) must be retained. This implies that v ∼ O(δ) μt ∼ O δ 2

(8.43) (8.44)

616

SPRAY ATOMIZATION AND COMBUSTION

Therefore, term (b) can be ignored with respect to the other terms, and Equation 8.42 reduces to ∂ 1 ∂ ◦ rρ v = 0 (ρ u) + ∂x r ∂r where

(8.45)

◦

ρ v = ρ v + ρ v

(8.46)

The combination of mean and fluctuating terms given by Equation 8.46 is a convenient formulation for the remainder of the analysis. Momentum Equation The time-averaged momentum equations valid for nonuniform property flow in a generalized form may be expressed as (a) ⎡ (b) ⎤ (c) (d) ∂u ∂p ∂ ⎢ i ⎥ ρuj + ρ uj = (ρ ∞ − ρ) gai − − ⎣ρui uj + ρ ui uj + uj ρ ui ⎦ ∂xj ∂xi ∂xj

⎤ ⎢ ⎥ ⎥ ∂u ∂u ∂uj ∂ui ∂ ⎢ ∂u j ⎥ ⎢ 2 ∂ui i δij + μ + + − 23 μ ∂xji δij + μ + ⎥ ⎢− 3 μ ⎢ ∂xj ∂xj ∂xj ∂xi ∂xj ∂xi ⎥ ⎦ ⎣ ⎡

(f )

(e)

(g)

(h)

(8.47) (See Hinze, 1975, for detailed derivation.) All of the terms involving molecular viscosity can be ignored for a turbulent free-jet flow. For a two-dimensional axisymmetric case, the momentum equation in the axial direction can be expressed as: (a)

(b)

(d)

(c) ∂ ∂u ∂u ρ u + ρv ρu u = gax (ρ ∞ − ρ) − ∂x ∂r ∂x (e)

(g)

(f)

1 ∂ ∂ 1 ∂ uρ u − − rρ u v − rvρ ν r ∂r ∂x r ∂r (h)

(i)

(j )

(k)

∂ 1 ∂ ∂u ∂u − ρuu − rρ u v − ρ u − ρ v ∂x r ∂r ∂x ∂r

(8.48)

As mentioned before, the Boussinesq approximation (Equations 8.40 and 8.41) can be used to model terms (f), (g), (j), and (k). Gosman, Lockwood, and Syed (1976) suggest these expressions for the Reynolds-stress terms:

617

LOCALLY HOMOGENEOUS FLOW MODELS

∂u 2 μt ∂ρ u + ∂x 3 ρ ∂x

ρ u u = −2μt

ρ

u v

= −μt

∂u ∂v + ∂r ∂x

(8.49)

(8.50)

Only the first term on the right-hand side of Equation 8.49 is significant, so to simplify the formulation, we consider it alone. Also, it is obvious that the first term on the right-hand side of Equation 8.50 is greater than the second, hence the latter can be dropped. The triple correlation terms (h) and (i) in Equation 8.48 may not be negligible; nevertheless, they are ignored in this analysis for lack of suitable correlating expressions. From the previous discussion, Equation 8.48 may be written with the order of magnitude of the terms indicated: (a)

(b)

1

1

(d)

(c) ∂ ∂u ∂u ∂u ρ u + ρv = gax (ρ ∞ − ρ) + 2 μt ∂x ∂x ∂x ∂r 1 δ2

(e)

(f )

(g)

δ2

δ

∂u 1 ∂ ∂ 1 ∂ μt ∂ρ μt ∂ρ rμt rv + + u + r ∂r ∂r ∂x σρ ∂x r ∂r σρ ∂r 1

μt + σρ

(j )

∂ρ ∂u μt + ∂x ∂x σρ

(k)

∂ρ ∂u ∂r ∂r

(8.51)

1

δ2

Due to their higher order, terms (d), (f), (g), and (j) may be eliminated, and the equation is reduced to the form ρu

∂u 1 ∂ ◦ ∂u + ρv − ∂x ∂r r ∂r

rμt

∂u ∂r

= gax (ρ ∞ − ρ)

(8.52)

Conservation of Mixture Fraction The mixture fraction Z (sometimes denoted f ) was defined in Chapter 2 so having zero value in a pure oxidant stream and unity in a pure fuel stream. The balance equation for f (Gosman, Lockwood, and Syed, 1976) may be expressed as:

(e) ⎛ (b) ⎞ (c) (d) ∂f ∂ ⎜ ∂ ∂f ⎟ ρ uj + ρ uj =− ρDf ⎝ρ uj f + ρ uj f + uj ρ f ⎠ + ∂xj ∂xj ∂xj ∂xj (a)

(8.53)

618

SPRAY ATOMIZATION AND COMBUSTION

Ignoring term (e) and putting the equation in cylindrical coordinates for the two-dimensional axisymmetric case gives (a)

(b)

(d) (c) ∂ 1 ∂ ∂f ∂f ρu ρuf − + ρv =− rρ v f ∂x ∂r ∂x r ∂r (f ) (g) (e) ∂ 1 ∂ ∂ − uρ f − ρuf rvρ f − ∂x r ∂r ∂x (j ) (i) (h) 1 ∂ ∂f ∂f rρ v f − ρ u − ρ v (8.54) − r ∂r ∂x ∂r

The triple correlation terms (g) and (h) are again ignored, as in the momentum equation. It has been shown by Gosman, Lockwood, and syed (1976) and other investigators that the term ρ f can be approximated by this integral: 1 2 dρ ρf = (8.55) f − f P (f ) df df f 0 where P (f ) is the pdf, which is unknown. They hence chose to ignore this term, as is done in the analysis here. This assumption is usually acceptable. However, for the case of an air jet injected into water, a large value of (dρ/df)f may appear at the edge of the jet (where f ≈ 0), and ρ f is not negligible. For the sake of simplicity, this term is disregarded. The Boussinesq approximation for any scalar quantity φ can be written as −μt ∂φ ρ φ u = (8.56) σφ ∂x and ρ φ v =

−μt ∂φ σφ ∂r

(8.57)

which can be used to model terms (c) and (d) of Equation 8.54, with φ = f . Equations 8.40 and 8.41 can be used to model terms (i) and (j). Therefore, Equation 8.54 can be rewritten as: (c)

(d)

(a) (b) ∂ 1 ∂ ∂f ∂f μt ∂f μi ∂f ρu + ρv = + r r ∂r σf ∂r ∂x ∂r ∂x σf ∂x 1 1

μt + σρ

δ2 (i)

μt ∂ρ ∂f + ∂x ∂x σρ δ2

1

(j )

∂ρ ∂f ∂r ∂r 1

(8.58)

LOCALLY HOMOGENEOUS FLOW MODELS

619

Ignoring higher-order terms and combining terms (b) and (j), we can express Equation 8.58 in this form: ∂f 1 ∂ μt ∂f ◦ ∂f ρu + ρv − r =0 (8.59) ∂x ∂r r ∂r σf ∂r 8.7.2.4 Turbulent Transport Equations In order to achieve turbulence closure, several turbulent transport equations must be considered. Turbulent Kinetic Energy (k) Based on the discussion in Chapter 4 Section 4.3, the generalized form of the turbulent-kinetic-energy equation can be written as:

∂k 1 ∂ ◦ ∂k ρu + ρv − ∂x ∂r r ∂r

μt ∂k r σk ∂r

= μt

∂u ∂r

2 − ρε

(8.60)

Equations for the Dissipation Rate (ε) and the Square of the Mixture Fraction Fluctuation (g) These turbulent transport equations are presented following Faeth (1983), after employing a treatment similar to that used for the previously discussed conservation equations:

2 ε2 ∂u ε − Cε2 ρ = Cε1 μt k ∂r k 2 μt ∂g ∂f ∂g εg 1 ∂ ◦ ∂g r = Cg1 μt ρu − Cg2 ρ + ρv − ∂x ∂r r ∂r σg ∂r ∂r k ∂ε 1 ∂ ◦ ∂ε ρu + ρv − ∂x ∂r r ∂r

μt ∂ε r σε ∂r

(8.61)

(8.62)

where Cε1 , Cε2 , Cg1 , and Cg2 are all constants of the turbulence model. Note: The symbol g should not be confused with the gravitational acceleration. It is defined mathematically as: g ≡ f 2

or

2 g ≡ f%

(8.63)

It can be seen that the final forms of all these turbulent transport equations and conservation equations are similar, and so we can define an operator D(φ) as shown next, where φ = f , u, k , ε, or g: ∂φ 1 ∂ μt ∂φ ◦ ∂φ D(φ) = ρ u + ρv − r (8.64) ∂x ∂r r ∂r σφ ∂r Using this operator, the conservation and turbulent transport equations can now be represented by: ∂ρ u 1 ∂ ◦ + rρv = 0 ∂x r ∂r

(8.65)

620

SPRAY ATOMIZATION AND COMBUSTION

D (u) = gax (ρ∞ − ρ) D f =0 2 ∂u D(k) = μt − ρε ∂r 2 ε2 ∂u ε − Cε2 ρ D(ε) = Cε1 μt k ∂r k 2 εg ∂f − Cg2 ρ D(g) = Cg1 μt ∂r k

(8.66) (8.67) (8.68) (8.69) (8.70)

The turbulence constants are assigned these values: Cμ = 0.09

Cε1 = 1.44

Cg1 = 2.8

σk = 1.0

σε = 1.3

σf = 0.7

σg = 0.7

σY i = 0.7

For constant-density flows, Cε2 = Cg2 = 1.89. For variable-density flows, Cε2 = Cg2 = 1.84. 8.7.2.5 Boundary Conditions The boundary conditions for these equations are

r = 0, and r = ∞,

∂φ =0 ∂r

(8.71)

φ=0

(8.72)

Besides these, several other conditions must be specified to give (a) the mass flow rate of the material injected, (b) thermodynamic state and thrust at the injector, and (c) distributions of k and ε at the injector exit. 8.7.2.6 Solution Procedures Following Faeth (1983), the governing equations under steady-state and axisymmetric conditions are summarized in Table 8.6. The governing equations in this case can be written in this general form, using φ as the generalized dependent variable: 1 ∂ ∂ μt ∂φ 1 ∂ μt ∂φ ∂ + r + Sφ (8.73) (ρ uφ) + (rρ vφ) = ∂x r ∂r ∂x σφ ∂x r ∂r σφ ∂r

The variable H is the total enthalpy of the mixture including sensible, chemical, and kinetic energies:

(8.74) H ≡ Yi hi + 12 u2 + v2 + w2

LOCALLY HOMOGENEOUS FLOW MODELS

621

TABLE 8.6. Source Terms in Equation 8.73 for Swirling Flows Source: Modified from Faeth, 1983.

φ

Sφ

Remarks

1

0 ∂u 1 ∂ ∂p ∂ ∂v μt + μt r − ∓ ag ρ ∂x ∂x r ∂r ∂x ∂x ∂u 1 ∂ 2μt v ρ w2 ∂p ∂ ∂v μt + μt r − 2 + − ∂x ∂r r ∂r ∂r r r ∂r ρ v 1 ∂μt μt + − + w r2 r r ∂r

—

u v w k ε f H

Gk − ρε ε (Cε1 Gk − Cε2 ρε) k 0 Srad ⎡

g

∂f Cg1 μt ⎣ ∂x

Yi

C i Rf

2

+

+ sign for vertical upward flow, *3 — — *1 —

∂f ∂r

— Dilation and shear work terms ignored

2 ⎤ ⎦ − Cg2 ρ ε g k

— —

2 2 2 2 w 2 ∂u ∂u ∂v ∂ ∂v 2 *1: Gk = μt 2 ∂x + ∂r + rv + r ∂r + ∂r + ∂x + ∂w ∂x r *2: Turbulence model constants are assigned these values: Cε2 = 1.92, Cg1 = 2.8, Cg2 = 2.0, Cμ = 0.09, Cε1 = 1.44, σk = 0.9, σε = 1.22, σf = σH = σg = σY i = 0.9. *3: a g represents the gravitational acceleration.

where ◦

hi ≡ hf,i +

T

Cp,i dT

(8.75)

Tref

The source term Srad in Tables 8.6 and 8.7 represents the radiation contribution to the increase of H . The term Rf in the source term of Y i represents the rate of reaction, which is discussed later. In many practical situations, some simplifications of governing equations and source terms can be made from the information listed in Table 8.6. Three separate cases of interest are defined in Table 8.8. In case 1, local chemical equilibrium with no heat losses is assumed (Khalil and Whitelaw, 1976; Mao, Szekely, and Faeth, 1980; Mao, Wakamatsu, and Faeth, 1981; Shearer, Tamura, and Faeth, 1979). The quantities solved for in the conservation and transport equations are u, v, w, f , k, ε and g.

622

SPRAY ATOMIZATION AND COMBUSTION

TABLE 8.7. Source Terms in Equation 8.76 for Non-Swirling Flows Source: Modified from Faeth, 1987.

φ

Sφ

1 u

0 ±ag (ρ∞ − ρ) 2 ∂u − ρε μt ∂r 2 ∂u ε Cε1 μt − Cε2 ρε k ∂r

k ε f

0

H

Srad

g

Cg1 μt

Yi

C i Rf

Remarks — + sign for vertical upward flow — — —

∂f ∂r

2

Dilation and shear work terms ignored ε − Cg2 ρ g k

— —

*1: Turbulence model constants are assigned these values: Cμ = 0.09, Cε1 = 1.44, Cg1 = 2.8, σk = 1.0, σε = 1.3, σf = σg = σY i = 0.7 For constant density flows, Cε2 = Cg2 = 1.89; for variable density flows, Cε2 = Cg2 = 1.84.

For nonswirling flows, the governing equation 8.73 can be simplied to this form: ∂ μt ∂φ 1 ∂ 1 ∂ ◦ r + Sφ (8.76) rρv φ = (ρ uφ) + ∂x r ∂r r ∂r σφ ∂r The source terms for Equation 8.76 are given in Table 8.7. Other scalar properties such as ρ, T , Y i , are found by a stochastic procedure involving the selection of a general form for the pdf of the mixture fraction. Since the specification of the pdf requires information on g, this approach often is called the k-ε-g procedure. As mentioned in Chapter 4, the method was initially suggested by Spalding (1971, 1976) and was subsequently developed and applied to flames by Lockwood and Naguib (1975), Bilger (1976), and many others. Once the local values of f , g, k, and ε are solved from the governing equations, the mean value of any scalar propertyθ can be determined from the integral 1

θ=

θ(f )P (f ) df

(8.77)

0

where P (f ) is the probability density function for f . This function has been measured for a number of flows by Bilger (1976), Kent and Bilger (1977), Kennedy and Kent (1979), and Moreau (1981). Various functions have been suggested for these measurements, including clipped Gausian, beta functions, incomplete beta functions, and rectangular waves, among others.

LOCALLY HOMOGENEOUS FLOW MODELS

623

The β-pdf was utilized by Richardson, Howard, and Smith (1953) and Jones and McGuirk (1980). It can be written, for 0 < f < 1, as:

P (f, xi ) =

f a−1 (1 − f )b−1 f a−1 (1 − f )b−1 = 1 a−1 (1 − f )b−1 df B(a, b) 0 f

(8.78)

where a and b can be determined explicitly from Equations 8.81 and 8.82, to be discussed later, using the values of f and g sloved from Equations 8.67 and 8.70, respectivefully. The integral in the denominator is called beta function B (a,b), which can be expressed in terms of several gamma functions, such as B(a, b) = (a) (b)/ (a + b), where (a) ≡ (a − 1)!. Gosman, Lockwood, and Salooja (1979) noted that the results are not particularly sensitive to the functional form used, although incomplete beta functions provide a compuationaly convenient formulation. The incomplete beta function defined next is a generalized beta function: B (ξ ; a, b) =

ξ

f a−1 (1 − f )b−1 df

(8.79)

0

For ξ = 1, the incomplete beta function coincides with the beta function. Lockwood and Naguib (1975) proposed and used a clipped Gaussian distribution: ⎧ * 2 + 0 ⎪ 1 1 f − μ ⎪ ⎪ exp − df if f = 0 ⎪ ⎪ 1/2 ⎪ 2 σ σ (2π) −∞ ⎪ ⎪ ⎪ * ⎪ + ⎨ 1 f −μ 2 1 P (f, xi ) = exp − if 0 < f < 1 (8.80) ⎪ 2 σ σ (2π)1/2 ⎪ ⎪ ⎪ * ⎪ 2 + ∞ ⎪ ⎪ 1 1 f − μ ⎪ ⎪ exp − df if f = 1 ⎪ ⎩ 2 σ σ (2π)1/2 1 The distribution is represented by the Gaussian function for the range 0 < f < 1, but the tails of the distribution are represented by δ-functions at f = 0 and 1. The most probable value of the distribution, μ, and the variance, σ 2 , were determined ˜ according to Equations 8.81 and 8.82. Both by the value of f (orf˜) and g (or g) the Reynolds- and Favre-averaged quantities of f and g are defined: f ≡

1

f P (f, xi ) df

0

or

f˜ ≡

1

,(f, xi ) df fP

0

2 f − f P(f, xi ) df

1

g ≡ g ≡ f 2 = 0

or

2 = g˜ ≡ f%

(8.81) 1

2 ,(f, xi ) df f − f˜ P

0

(8.82) For a given system pressure, θ(f ) represents the known state relationship described in Section 8.7.2.2. Figure 8.12 is an illustration of the state relationships for a

624

SPRAY ATOMIZATION AND COMBUSTION

TEMPERATURE

0.8

1200 0.4

0.0

400 1.0

0.8

n-PENTANE SPRAY BURNING IN AIR 2600

C3H8

N2

CO2 H2O

0.0

MASS FRACTION

0.2

CO

O2

0.2

H2 0.4

0.6

100 1800

TEMPERATURE

10 DENSITY

1000 1

.1

200 1.0

0.6

0.4

1000 0.1 MPa

0.8

1.0

DENSITY (kg/m3)

DENSITY

TEMPERATURE (k)

1.2

1600

800

MASS FRACTION

1.6

3 DENSITY (kg/m )

TEMPERATURE (k)

COMBUSTING PROPANE JET 2000

PENTANE LIQUID PENTANE VAPOR NITROGEN

.8 .6

HYDROGEN CARBON MONOXIDE

OXYGEN CARBON DIOXIDE

.4 .2

WATER VAPOR

0 0

.2

.4

.6

.8

MIXTURE FRACTION

MIXTURE FRACTION

(a)

(b)

Figure 8.12 Scalar properties as function of mixture fraction for (a) propane gas jet and (b) n-pentane spray burning in air at atmospheric pressure (after Mao, Szekely, and Faeth, 1980, adapted from Kuo, 1986).

propane jet burning in air and a n-pentane spray combustion in air at atmospheric pressure, obtained by Mao, Szekely, and Faeth (1980). These plots provide the temperature, density, and mass fractions of major species as a function of mixture fraction. The nonlinear variation of properties with f is evident. In order to determine the mean values of various properties, the stochastic method is preferable. The pdfs employed in various studies are most often characterized by two parameters, generally associated with the most probable value (μ) and the variance (σ ) for the clipped Gaussian distribution function or with a and b for incomplete beta pdf. Equations 8.81 and 8.82 are two implicit equations that can be solved to determine the two parameters in the preselected pdf. θ (f ) can then be integrated using the state relationship in Equation 8.77 to obtain ρ, T , Y i , and so on. For case 2 of Table 8.8, the assumption of local chemical equilibrium is relaxed. Because of this relaxation, the state of the mixture can no longer be fixed solely by the mixture fraction f . One additional scalar transport equation must be solved. This equation is usually selected to be the partial differential equation for Y i , where i represents an important species involved in the chemical reaction. In order to solve for Y i , the rate of reaction of species i must be specified. Spalding’s eddy-breakup model of turbulent reaction, described in Chapter 5,

625

LOCALLY HOMOGENEOUS FLOW MODELS

TABLE 8.8. Summary of LHF-Model approximations Source: Modified from Faeth, 1983.

Major Assumptions 1. 2. 3. 4. 5.

LHF flow approximation Equal exchange coefficients of all species and heat High Reynolds number, negligible contribution of laminar transport k -ε turbulence model, no effect of buoyancy on turbulence properties Reynolds averaging ignoring density fluctuation terms, i.e., using Favre-averaging type of equations without computation of mass averaged properties

Additional Assumptions 6. 7. 8. 9.

Adiabatic flow Negligible radiation Low Mach number Local chemical equilibrium

Case 1

Case 2

Case 3

Transport Equations Solved Mean quantities

u, v, w, f

u, v, w, f , Y f

Turbulent quantities

k, ε, g

k, ε, g or gYF

u, v, w, f , H , radiation transport equation k, ε, g

was developed precisely to give the reaction rate of fuel species. The reaction rate according to Spalding (1971) can be written as ε Rf = −CR ρ gY F k

(8.83)

where CR is a constant having a value on the order of unity and gY F is the square of the fuel mixture-fraction fluctuation. The governing equation for gY F is identical to the g equation in Table 8.6. Along the same line of thought, Magnussen and Hjertager (1977) proposed the next expression for Rf , which has been applied to various studies of premixed combustion in furnaces: .

Y O2 ρ YP , A Rf = Min Aρ Y F , Aρ 1 + (O/F )st (O/F )st

/

ε k

(8.84)

where A ≈ 4, A ≈ 2, and Min represents the minimum value among the group of parameters in the parentheses. Other expressions for Rf have been proposed by various researchers, including, Borghi (1974); Bray and Moss (1977); Khalil (1978); Gosman, Lockwood, and Salooja (1979); and Libby and Bray (1980, 1981).

626

SPRAY ATOMIZATION AND COMBUSTION

After the solution of the governing equations has been obtained, mean values of various scalar quantities, such as T , ρ, Y i , can be computed by stochastic averaging procedures. For chemically nonequilibrium processes (finite-rate chemistry), a parameter describing the degree of reactedness is necessary in addition to the mixture fraction for the construction of a formula to obtain the average values of the listed scalar quantities. Lockwood (1977) defined the reactedness parameter as

r

r ≡ YYi,bi −−YYi,ui,u ,

i = F, O, or P

(8.85)

where u and b designate completely unburned and burned conditions. The reactedness parameter was introduced in Kuo (2005, chap. 6 Figure 6.9). For a single-step forward chemical reaction, is independent of the major species chosen. It can, however, be extended to more complicated chemical reactions. Local mean properties depend on the fluctuations of both f and . In order to simplify the model, the f and fluctuations are assumed to be uncorrelated f = 0 . The time-averaged value of a fluid property θ(f, r) at any spatial point is then given by 1 1 θ= θ(f, )P (f )P ( ) df d (8.86)

r

r

r

0

0

r

r

r

r

where P (f ) is found from f and g, using an assumed functional form for the pdf as described in case 1. The value of P ( ) is determined from :

r

r

P( ) = (1 − ) δ( ) + δ(1 − )

r

r r r

r

(8.87)

where δ represents the Diracδ-function. Note that the preceding computational procedure to obtain mean scalar properties involves the basic assumption of negligible effect of fluctuations of f and . When property variations are not linear in f and , this simplification is questionable. In case 3 of LHF models, local chemical equilibrium is assumed but radiation transfer is not negligible. In order to consider radiative heat losses, the equation of radiation transfer must also be solved to yield the source term Srad in the total-enthalpy equation (see Table 8.6). As pointed out by Faeth (1983), a number of approximations must be used in the solution of the radiative-transfer equation since radiative transport involves complicated integro-differential equations. Readers are referred to the work of Elghobashi and Pun (1974); Bilger (1976); and Gosman, Lockwood, and Salooja (1979).

r

r

8.7.2.7 Comparison of LHF-Model Predictions with Experimental Data LHF models have been compared with several combusting and noncombusting two-phase flows. For noncombusting flows, the model solutions of various

LOCALLY HOMOGENEOUS FLOW MODELS

627

researchers are compared with a variety of single- and two-phase flows as shown in the next list: 1. Constant-density single-phase jet. Comparison was made with data of Becker, Hottel, and Williams (1967); Hetsroni and Shearer, Tamura, and Faeth (1979); Sokolov (1971); and Wygnanski and Fiedler (1969). 2. Variable-density single-phase jet. Comparison was made with data of Corrsin and Uberoi (1950); and Shearer, Tamura, and Faeth (1979). 3. Air jet into water. Comparison was made with data of Tross (1974). 4. Evaporating Freon-11 spray in air. Comparison was made with data of Shearer, Tamura, and Faeth (1979). It was found that the predicted radial profiles of u/uc , f /f c , and u v /u2c are all in reasonably good agreement with data (see Figures 8.13 to 8.15). The model predicts similarity in these coordinates over the range of data; therefore, only a single theoretical curve is shown in each case. The comparisons of uc /u0 and f c /f 0 versus x/d between theory and experiments are shown in Figure 8.16 and Figure 8.17. As anticipated, the experimental data and theoretically predicted results on mean axial velocities and mixture fraction along the centerline decrease as the axial distance increases. The data shown in these figures cover a range of ratios of initial jet fluid density to ambient fluid density from 0.0012 to 6.88. As can be seen, the predicted centerline velocities are 10% to 20% lower than the measurements, while the predicted centerline mixture fractions are 40% below the measurements. The temperature measurements also approached ambient conditions more slowly than predicted. According to Faeth (1983), the density ratio of the spray was found to be similar to the variable density jet, which was predicted quite well; therefore, variable density effects are not a prime source of error. Finite interphase transport rates are the major difficulty as will be shown later by means of drop-life-history predictions in this flow. For combusting sprays, the model of Mao, Szekely, and Faeth (1980) was used to predict three combusting sprays: 1. n-propane gas jet burning in air at 1 atm 2. Air-atomized liquid n-pentane spray burning in air at 1 atm 3. Pressure-atomized liquid n-pentane spray burning in air at pressures of 3 to 9 MPa As shown in Figure 8.18, the agreement between predicted and measured mean axial velocities and temperatures along the centerline is excellent. Predictions of both velocity and Reynolds stress are in good agreement with measurements (see Figures 8.19 and 8.20). It is interesting to note that due to large density variation in combusting sprays; radial profiles of velocity and Reynolds stress at

628

SPRAY ATOMIZATION AND COMBUSTION 1.0 ISOTHERMAL SINGLE-PHASE JET 0.8

0.6

0.4

0.2 THEORY 0.0 1.0

VARIABLE-DENSITY SINGLE-PHASE JET

0.8

u/uc

0.6

0.4

0.2

THEORY

0.0 1.0

SYMBOL

0.8

x/d

SOURCE

35

HETSRONI AND SOKOLOV

40

WYGNANSKI AND FIEDLER

170

SHEARER, TAMURA, AND FAETH

0.6 340 510

0.4 THEORY 0.2 EVAPORATING SPRAY 0.0 0.00

0.08

0.16 r/x

0.24

0.32

Figure 8.13 Radial profile of mean axial velocity for various single- and two-phase noncombusting jets (modified from Faeth, 1983).

LOCALLY HOMOGENEOUS FLOW MODELS

629

1.0 SYMBOL 0.8

x/d

SOURCE

28

BECKER et al.

170 340 510

0.6

0.4

0.2

SHEARER, TAMURA, AND FAETH

ISOTHERMAL SINGLE-PHASE JET THEORY

0.0 1.0

VARIABLE-DENSITY SINGLE-PHASE JET

0.8

f/fc

0.6

0.4

0.2

THEORY

0.0 1.0

EVAPORATING SPRAY

0.8

0.6

0.4

0.2

0.0 0.00

THEORY

0.08

0.16 r/x

0.24

0.32

Figure 8.14 Radial profile of mean mixture fraction for various single- and two-phase noncombusting jets (modified from Faeth, 1983).

630

SPRAY ATOMIZATION AND COMBUSTION 0.02

ISOTHERMAL SINGLE-PHASE JET 0.01 THEORY

0.00 0.02

u′v′/uc2

VARIABLE-DENSITY SINGLE-PHASE JET 0.01 THEORY

0.00 0.02

SYMBOL

x/d 50 60 70 170 340 510

0.01

0.08

WYGNANSKI & FIELDER SHEARER, TAMURA, AND FAETH

EVAPORATING SPRAY

THEORY 0.00 0.00

SOURCE

0.16 r/x

0.24

0.32

Figure 8.15 Radial profile of Reynolds stress for various single- and two-phase noncombusting jets (modified from Faeth, 1983).

631

LOCALLY HOMOGENEOUS FLOW MODELS

THEORY SYMBOL i ii ii iii iv v

SOURCE

TYPES OF FLOW

HEATED AIR JET ISOTHERMAL AIR JET ISOTHERMAL AIR JET VARIABLE DENSITY JET EVAPORATING SPRAY AIR-WATER JET

CORRSIN SHEARER ET AL. WYGNANSKI & FIEDLER SHEARER ET AL. SHEARER ET AL. TROSS

uc/uo

1.0

i 0.1

iv iii

ii

v 0.01

1

10

100 x/d

Figure 8.16 Axial variation of mean centerline axial velocity for various single- and two-phase noncombusting jets (modified from Faeth, 1983).

various axial stations are not similar. Predicted radial profiles of T at various axial locations are in good agreement with experimental data, as shown in Figure 8.21. As shown in Figure 8.22, the predicted species concentration distributions are in reasonable agreement with experimental data. Comparisons of theoretical results with the data on air-atomized n-pentane (liquid) spray burning in air obtained by Mao, Szekely, and Faeth (1980) are given in Figures 8.23 to 8.25. Results indicate that: 1. The spray develops more slowly than predicted. 2. The calculated T max points are closer to the injector than measured. 3. For x/d > 300, good agreement was obtained, since it is beyond the region where drops are present.

632

SPRAY ATOMIZATION AND COMBUSTION

THEORY SYMBOL i ii iii iv v

TYPES OF FLOW

SOURCE

HEATED AIR JET ISOTHERMAL AIR JET VARIABLE-DENSITY JET EVAPORATING SPRAY AIR-WATER JET

CORRSIN BECKER SHEARER ET AL. SHEARER ET AL. TROSS

1.0

ii 0.1 fc/fo

i

iv iii 0.01

0.001 v 1

10

100 x/d

Figure 8.17 Axial variation of mean centerline mixture fraction for various single- and two-phase noncombusting jets (modified from Faeth, 1983).

4. Good agreement is obtained between predicted and measured radial profiles of u/uc and u v /u2c . 5. There are large discrepancies between predicted and measured mean temperature profiles near the injector, where the overestimation of the rate of development of the flow is most pronounced.

LOCALLY HOMOGENEOUS FLOW MODELS

633

10.0 COMBUSTING PROPANE JET DATA SYMBOL CENTERLINE DATA TEMPERATURE VELOCITY THEORY

1800

uc/uo

1400

0.1

TEMPERATURE (K)

1.0

1000

600

0.01

10

100

1000

x/d

Figure 8.18 Axial variation of mean axial velocity and temperature for an n-propane gas jet burning in air at atmospheric pressure (modified from Mao, Szekely, and Faeth, 1980).

A comparison of predicted and measured spray boundaries of pressureatomized liquid n-pentane spray burning in air at p = 3, 6, and 9 MPa is shown in Figure 8.26. Both predictions and measurements indicate that the extent of the spray boundary is reduced as the combustor pressure increases. However, the theory overestimates the magnitude of the reduction. Predicted spray lengths are 10% to 20% less than the measurements. Khalil and Whitelaw (1976) compared their predictions with measurements on an open combusting liquid kerosene spray with swirl. The results, given in Figures 8.6 and 8.7, show that: 1. The u distributions along the centerline of the spray show good agreement between the prediction and measurements. 2. The predicted maximum temperature location along the centerline is closer to the injector than measured. This also indicates the overestimation by the LHF model of the development of the flow.

634

SPRAY ATOMIZATION AND COMBUSTION 1.0 COMBUSTING PROPANE JET

0.8

DATA THEORY

0.6

x/d = 510

0.4 0.2 1.0 0.8

x/d = 340

0.6 0.4 u/uc

0.2 1.0 0.8 x/d = 170

0.6 0.4 0.2 1.0 0.8

x/d = 74.5

0.6 0.4 0.2 0.0

0.04

0.08

0.12 r/x

0.16

0.20

Figure 8.19 Radial variation of mean axial velocity for an n-propane gas jet burning in air at atmospheric pressure (modified from Mao, Szekely, and Faeth, 1980).

3. The results for the spray flame having an SMD of 45 μm are approximated much better by the theory than the results for the flame having a larger SMD of 100 μm. Khalil (1978) was able to achieve better predictions for a portion of the data by using an eddy-breakup (EBU) model. He introduced additional empirical parameters to account for heterogeneous combustion of drops. The usefulness of the extension is limited, since the approach cannot truly account for the effect of relative velocity between the drops and gas. 8.8

TWO-PHASE-FLOW (DISPERSED-FLOW) MODELS

As mentioned in Section 8.6.6, the two-phase-flow model is the most logical one for the simulation of spray combustion problems, as such models specifically treat the finite rates of exchange of mass, momentum, and energy between the

TWO-PHASE-FLOW (DISPERSED-FLOW) MODELS

635

COMBUSTING PROPANE JET

0.020

DATA THEORY 0.010

x/d = 510

0.00 0.020 x/d = 340

u′v′/ uc

2

0.010

0.00 0.020 x/d = 170 0.010

0.00 0.020 x/d = 74.5 0.010

0.00

0.04

0.08

0.12 r/x

0.16

0.20

Figure 8.20 Axial variation of Reynolds stress for n-propane gas jet burning in air at atmospheric pressure (modified from Mao, Szekely, and Faeth, 1980).

liquid and gas phases. However, due to limitations on computer storage and cost of computation, researchers developing discrete droplet models of sprays have made limited attempts to model the details of the flow field around individual drops. Therefore, the exchange processes between phases must be modeled independently. Usually a set of empirical correlations for droplet drag and heat and mass transfer is employed. In general, there are three different approaches in dispersed flow analyses for evaporating and combusting sprays, which are briefly introduced next. 1. Particle-source-in-cell model (PSICM), or discrete-droplet model (DDM). Finite numbers of groups of particles are used to represent the entire spray. The motion and transport of representative samples of discrete drops are tracked through the flow field using a Lagrangian formulation, while a Eulerian formulation is used to solve the governing equations for the gas phase.

636

SPRAY ATOMIZATION AND COMBUSTION

COMBUSTING PROPANE JET SYMBOL

1800

x/d 74.5 170 340 510

TEMPERATURE (k)

1400

THEORY x/d 74.5 170 340 510

1000

600

0.00

0.04

0.08

0.12 r/x

0.16

0.20

Figure 8.21 Radial variation of mean temperature for n-propane gas jet burning in air at atmospheric pressure (after Mao, Szekely, and Faeth, 1980, adapted from Kuo, 1986).

The effect of droplets on the gas phase is taken into account by introducing appropriate source terms in the gas-phase conservation equations. 2. Continuous droplet model (CDM). As discussed in Section 8.4.2, the distribution function fj (r, x, v, t) can be used to evaluate the statistical distributions of the drop temperature, concentration, and so on. The transport equation 8.20 for fj is solved along with the gas-phase conservation equations to provide all properties of the spray. As with the DDM, the governing equations for the gas phase must also include appropriate source terms. 3. Continuum-formulation model (CFM). The motion of both drops and gas is treated as if it was interpenetrating continua. A continuum formulation of the conservation equations for both phases is used to model spray combustion and evaporation problems. In this approach, the governing equations for the two phases are similar; however, there are many difficulties in

637

TWO-PHASE-FLOW (DISPERSED-FLOW) MODELS DATA SYMBOL SPECIES O2 N2 × 1/2 CO2 CO H2O C3H8 H2 THEORY

MASS FRACTION

0.50 0.40

COMBUSTION PROPANE JET x/d = 74.5

0.50

1/2 N2 0.30 O2

CO 0.20

C3H8

0.00 0.00

0.04

0.08

0.12

0.16

0.40

COMBUSTION PROPANE JET x/d = 170

1/2 N2 0.30 O2 0.20 CO

CO2 H2 O

H2

0.10

DATA SYMBOL SPECIES O2 N2 × 1/2 CO2 CO H2O C3H8 THEORY

0.60

MASS FRACTION

0.60

CO2 H2 O

0.10 C3 H8 0.00 0.00

0.20

0.04

0.08

0.12

0.40

0.50

1/2 N2 0.30 O2 0.20 CO2

0.10 CO 0.00 0.00

0.04

0.12

0.20

0.40

0.16

0.20

COMBUSTION PROPANE JET x/d = 510

DATA SYMBOL SPECIES O2 N2 × 1/2 CO2 CO H2O THEORY

1/2 N2

0.30 O2 0.20 0.10

H2O

0.08

0.60

MASS FRACTION

0.50 MASS FRACTION

COMBUSTION PROPANE JET x/d = 340

DATA SYMBOL SPECIES O2 N2 × 1/2 CO2 CO H2O THEORY

0.60

0.16

r/x

r/x

0.00 0.00

CO2

0.04

H2O

0.08

r/x

0.12

0.16

0.20

r/x

Figure 8.22 Radial variation of mean species concentrations for n-propane gas jet burning in air at atmospheric pressure (modified from Mao, Szekely, and Faeth, 1980).

describing the droplet heat-up process, turbulent stresses, and turbulent dispersion of droplets. Each of these approaches is described in the next sections. 8.8.1

Particle-Source-in-Cell Model (Discrete-Droplet Model)

In the PSICM or DDM approach, the entire spray is divided into many representative samples of discrete drops whose motion and transport through the flow field are found using a Lagrangian formulation in determining the drop life history, while a Eulerian formulation is used to solve the governing equations for

638

SPRAY ATOMIZATION AND COMBUSTION 10.0 COMBUSTING N-PENTANE JET DATA SYMBOL CENTERLINE DATA TEMPERATURE VELOCITY THEORY

1800

uc/uo

1400

0.1

TEMPERATURE (K)

1.0

1000

600

0.01

10

100

1000

x/d

Figure 8.23 Axial variation of mean axial velocity and temperature for n-pentane liquid spray burning in air at atmospheric pressure (after Mao, Szekely, and Faeth, 1980, adapted from Kuo, 1986).

the gas phase. The representative works are those of Crowe (1974, 1978); Crowe Sharma, and Stock (1977); and Jurewicz, Stock, and Crowe (1977) on aircraft gas-turbine combustors; Alpert and Mathews (1979) on commercial sprinkler systems for fire safety; El Banhawy and Whitelaw (1980) on modeling spray combustion in furnaces; Gosman and Johns (1980) and Butler et al. (1980) on reciprocating direct-injection stratified-charge (DISC) engines; Bruce, Mongia, and Reynolds (1979); Mongia and Smith (1978); and Swithenbank, Turan, Felton (1980) on gas-turbine combustors; Anderson et al. (1980) on spray evaporation in premixed prevaporizing passages; Gosman and coworkers (1980, 1981) on spray combustion in cylindrical furnaces; Solomon (1984) on evaporating sprays; and Shuen, Chen, and Faeth (1983) and Shuen (1984) on dilute particle-laden turbulent gas jets. A comprehensive theoretical model should include both gas-phase analysis and liquid-droplet-phase analysis. To provide readers with an overall picture of

639

TWO-PHASE-FLOW (DISPERSED-FLOW) MODELS 1.0 COMBUTION n-PENTANE JET

0.8

DATA THEORY

x/d = 510

0.4

COMBUTION n-PENTANE JET

0.020

DATA THEORY

0.6

x/d = 510

0.010

0.2 1.0

0.000

0.8

0.020

x/d = 340

0.6

x/d = 340

0.4

0.010 u′v′/ uc

u/uc

2

0.2 1.0 0.8

0.000 0.020

x/d = 170

0.6 0.4

x/d = 170 0.010

0.2 1.0

0.000

0.8

0.020 x/d = 74.5

0.6 0.4

x/d = 74.5 0.010

0.2 0.0

0.04

0.08

0.12

0.16

0.20

0.000 0.00

0.04

0.08

0.12

r/x

r/x

(a)

(b)

0.16

0.20

Figure 8.24 Radial variation of (a) mean axial velocity and (b) Reynolds stress for spray of liquid n-pentane burning in air at atmospheric pressure (after Mao, Szekely, and Faeth, 1980, adapted from Kuo, 1986).

the structure of the DDM, Figures 8.27 and 8.28 show the block diagrams for gas-phase and liquid-phase analysis, respectively. The basic elements required in the gas-phase analysis, shown in Figure 8.27, include basic assumptions, transport equations for mean-flow variables, turbulence closure consideration, and radiative transport equations. For each of these elements, subelements and their components are also described. Figure 8.28 describes the basic elements required for the liquid-phase analysis: basic assumptions, drop-type classification, and drop-life history. Subelements are also shown as branches to the main elements. Depending on the consideration of the effect of turbulent fluctuations on particle motion and the method of treatment of the velocity differences between the phases, DDMs are further subdivided into deterministic discrete-droplet models (DDDMs) and stochastic discrete-droplet models (SDDMs). 8.8.1.1 Models for Single Drop Behavior Assumptions As shown in Figure 8.28, the behavior of individual drops in a spray must be examined to determine the size, velocity, temperature, and composition of individual drops as a function of position in the spray. This section describes a basic model of drop processes that employs features most frequently

640

SPRAY ATOMIZATION AND COMBUSTION

COMBUSTING n-PENTANE JET DATA Symbol x/d 165 170 340 510

1800

Temperature (k)

1400

1000 Theory x/d 74.5 170 340 510 600

0.00

0.04

0.08

0.12 r/x

0.16

0.20

Figure 8.25 Radial variation of mean temperature for spray of liquid n-pentane burning in air at atmospheric pressure (after Mao, Szekely, and Faeth, 1980, adapted from Kuo, 1986). Measured Predicted

9 MPa 40

2r/d

0 6 MPa 40

0 3 MPa 40

0

0

40

80 x/d

120

160

Figure 8.26 Predicted and measured spray boundaries for pressure-atomized liquid npentane sprays burning in high-pressure air (after Mao, Szekely, and Faeth, 1981, adapted from Kuo, 1986).

TWO-PHASE-FLOW (DISPERSED-FLOW) MODELS

Basic Assumptions

641

Dimensionality Steadiness of the Mean Flow Adiabaticity Reaction Mechanism and Formulation

Gas Phase Analysis

Transport Equations for Mean Flow Variables e.g., φ, u v, Ht, YF

Turbulence Closure Consideration

Droplet Source Term Consideration Mean Thermodynamic State Property Calculations Pollutant Production Consideration

Diffusion Controlled Reaction Rates Use of PDF Equation of State

Initial and Boundary Conditions

Algebraic Closure

Transport Equations for Turbulence Quantities e.g., k, ε, g

Radiation Transport Equations

Chemical kinetics Controlled Reaction Rate

Turbulence Modeling and Approximations

Turbulence Constants and Empirical Input Source Terms for k, ε, g Equations. due to Particle-Gas Interaction

Initial and Boundary Conditions

Figure 8.27 Block diagram showing the structure of gas-phase analysis in discretedroplet models (after Kuo, 1986).

adopted for DDMs of sprays in the form of dispersed flows (defined in Figure 7.1) and that have been subjected to experimental evaluation. Less commonly treated effects (e.g., dense spray phenomena, drop ignition, drop dispersion, etc.) are discussed later. Let us first discuss 15 common assumptions used in most drop models. 1. The drop is assumed to be spherical . It is well known that moving drops deform; however, existing correlations for drag and convective heat transfer incorporate this effect implicitly, treating the drop as an equivalent sphere. 2. The spray is assumed to be dilute. Under this assumption, both drop collision effects and the effect of adjacent drops on drop transport rates are ignored (i.e., drag and convective heat transfer correlations for drops having infinite spacing are employed without correction). 3. The flow around the drop is assumed to be quasi-steady (i.e., the flow can immediately adjust to local boundary conditions and drop size at each instant of time). The appropriate characteristic times for development of 0 the gas-phase flow field are either dp vp − v or dp2 /(α or D), whichever is smaller. The first characteristic time is usually controlling for moving drops, yielding characteristic times on the order of 0.1 to 10 μs (for drop diameters of 10 to 100 μm, relative velocities of 10 to 100 m/s). In contrast,

642

SPRAY ATOMIZATION AND COMBUSTION

particle diffusion by turbulent eddies Basic Assumptions

particle size distribution particle interaction independent variables for identifying each class

Drop Type Classification

determination of injector exit condition joint distributions of size, velocity, direction, and position

Liquid Phase Analysis

transient heat-up process rate of evaporation of drops

Drop Life History

rate of transport of drops

drop breakup drop collision initial and boundary conditions

Figure 8.28 Block diagram showing structure of liquid-phase analysis in discrete-droplet models (after Kuo, 1986).

drop lifetimes in a spray are on the order of 1 to 10 ms, which is 2 to 3 orders of magnitude greater, justifying the quasi-steady flow assumption. For motionless drops, the second characteristic time is controlling, indicating flow development times comparable to drop lifetimes at high pressures (10 to 100 atm). However, motionless drops are rarely of interest for practical sprays, preserving the utility of the quasi-steady flow assumption. 4. The radial regression rate of the liquid surface due to the evaporation of liquid is ignored . This assumption is related to assumption 3. The approximation is valid for a moving drop, as long as the liquid surface regression rate is small in comparison to the relative velocity of the drop. This is generally the case, except when the liquid droplet is very near the thermodynamic critical point. Such conditions are not encountered frequently. Surface regression rates are more important for motionless drops, and the assumption should be reexamined when such cases are of interest. 5. Effects of drag and forced convection are represented by empirical correlations. This is necessary since accurate treatment of flow around spheres is impractical due to excessive computation requirements.

TWO-PHASE-FLOW (DISPERSED-FLOW) MODELS

643

6. Gas-phase transport is based on mean ambient properties, and the effect of turbulent fluctuations is usually ignored . This assumption is adequate as long as the intensity of fluctuations is small and all fluctuating parameters needed to compute transport rates vary linearly. For example, mixture fraction is the primary fluctuating quantity in a combusting spray while temperature, which is a prime variable for drop heat transfer, varies nonlinearly with f near the stoichiometric mixture ratio (see Figure 8.12). Since the practice is widespread, this assumption has been adopted generally. Researchers should reexamine the validity of this assumption for their simulations. 7. The liquid surface is assumed to be in thermodynamic equilibrium with negligible temperature jump across the interface due to finite rates of evaporation. Furthermore, the effect of surface tension is neglected when determining phase equilibrium at the liquid surface. These assumptions are generally satisfactory for spray analysis (e.g., atmospheric pressure and above for drops having diameters greater than 1 μm). Bellan and Summerfield (1978) present extensive results concerning surface equilibrium properties of small drops. 8. The pressure is assumed to be constant and equal to the local mean ambient pressure. This approximation is also satisfactory, except for small drops at low pressures. 9. The Soret effect is neglected and the Dufour effect is neglected in the heat flux equation. Errors incurred by these approximations are on the order of 10%; therefore, complicating the model by including these phenomena is rarely warranted. 10. Radiation between the drop and its surroundings is neglected . Convective heat transfer rates of drops in sprays are high, which reduces the relative importance of radiation. Another factor helping to justify this approximation is that gaseous radiation bands generally do not coincide with absorption bands of most liquid fuels (Berlad and Hibbard, 1952). Radiation is more important when there is significant continuum radiation from hot surfaces and soot, or where there are absorbing particles present in the liquid (e.g., coal slurries). When radiation must be considered, however, the extension of the model simulation is straightforward. 11. Oxidation and decomposition processes are neglected in the adjacent flow field surrounding the drop. A drop in an oxidizing atmosphere can be: (a) ignited with an oxidizing zone completely surrounding the drop (envelope flame); (b) ignited with the oxidizing zone stabilized in the wake (wake flame); or (c) not ignited, but with reaction being completed in the bulk gas phase. For the latter two cases, this approximation is accurately represented; however, the presence of an envelope flame enhances decomposition of the fuel vapor. If the drop is a liquid monopropellant (e.g., a hydrazine or nitrate ester), decomposition rates are enhanced; this causes the increase of transport rates. However, for nonmonopropellants, decomposition results in reduced transport rates. The importance of reaction effects can be characterized by the

644

SPRAY ATOMIZATION AND COMBUSTION

local Damk¨ohler number (Da). Liquid droplets are usually confined to cool portions of the spray flow, tending to yield relatively long chemical reaction time and low (Da). Therefore, assuming negligible reaction in the close vicinity of drop flow field is a reasonable first approximation. 12. The gas-phase Lewis number, Le, is often assumed to be unity in drop models. However, predictions of drop behavior at high pressures are known to be influenced by values of Le. Furthermore, the Le of high-molecularweight materials, often encountered in analysis of sprays, departs significantly from unity. Therefore, adequate values of Le should be used in the analysis. 13. The properties of the gas-flow field are assumed to be constant at each instant of time. An effective binary diffusivity, specific heat, and molecular weight are used for all species. Properties are evaluated at an average condition, defined as: Yi,avg = ξ Yi,surf,gas + (1 − ξ ) Y i

and

Tavg = ξ Tsurf,gas + (1 − ξ ) T (8.88)

where ξ represents the average property factor, which is an empirical parameter selected to obtain agreement between model predictions and existing measurements. Since property variations are large, particularly in combusting sprays, calibration of ξ in this manner reduces uncertainties of drop computations pending development of reliable methods for treating variable property effects for flow around drops. According to Faeth (1983), the value of ξ was determined as 0.9 through a detailed droplet model calibration process. When this value is used, the calculated droplet gasification rates from models predicting drop-life histories at wet-bulb conditions match closely with the measured data. At the wet-bulb state, all the heat reaching the drop is utilized for the heat of vaporization of the evaporating liquid, and the drop temperature remains essentially constant (aside from slight changes in the wet-bulb temperature due to differences in the effect of convection on heat and mass transfer rates when the Le number is not unity). 14. Chemical reaction is ignored in the liquid phase. This approximation is analogous to the low Da assumption of the gas phase, except that a more appropriate residence time is the drop lifetime, which is longer than gasphase residence times. Since droplet temperatures are usually below boiling or critical temperatures, a low Da condition prevails. Exceptional cases can be found (e.g., liquid reactions were observed in heavy oil drops by Masdin and Thring, 1962). However, this approximation is acceptable for most spray model simulations. 15. Transport processes within the liquid phase have characteristic times, dp2 / (αl or Dl ) and dp /ucirculation . These times are comparable to drop lifetimes,

TWO-PHASE-FLOW (DISPERSED-FLOW) MODELS

645

since liquid diffusivities and circulation velocities are low. Therefore, a quasi-steady approximation analogous to the gas phase is not appropriate. Prakash and Sirignano (1980) have shown that internal circulation within moving drops is a relatively ineffective agent for mixing and severely complicates estimations of liquid phase transport.

Gas Phase Analysis The governing equations for mass and heat transfer are discussed by Bird, Stewart, and Lightfoot (1960), F.A. Williams (1985); and Kuo (2005). For the spherically symmetric and quasi-steady conditions, with a system consisting of N species, the equations have this form: Conservation of mass:

d 2 r ρv = 0 dr

(8.89)

Conservation of species: dYi d r 2 ρvYi − ρD = 0; dr dr

i = 1, 2, . . . Ns

(8.90)

Conservation of energy: dT d r 2 ρvCp T − Tref − l =0 dr dr

(8.91)

where Ns

Yi = 1

(8.92)

i=1

Integration of Equation 8.89 yields ˙ = constant r 2 ρv = m/4π

(8.93)

where m ˙ is the net rate of mass flow from the droplet surface. This includes both the mass flux of the fuel species due to the bulk velocity of the mixture at the surface and the fuel mass flux caused by the outward diffusion of fuel species at the droplet surface (see Kuo, 2005, chap. 6). The mass transfer rate of each species can be conveniently represented by the mass flux fraction, defined as (F.A. Williams, 1985) m ˙ i = εi m, ˙

i = 1, 2, . . . .Ns

(8.94)

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SPRAY ATOMIZATION AND COMBUSTION

By definition Ns

εi = 1

(8.95)

i=1

however, the values of εi can be greater or less than zero, depending on whether the species i is evaporating or condensing at the liquid droplet surface. Specification of the εi depends on the liquid transport model being used and is discussed later. The droplet surface heat-transfer coefficient on the gas-phase side is defined as: 1 dT h= l T − Ts (8.96) dr s,g and the boundary conditions are: r = rp :

T = Ts , Yi = Yi,s,g , i = 1, 2, . . . , Ns

r = ∞ : T = T,

Yi = Y i ,

i = 1, 2, . . . , Ns

(8.97)

where the local flow conditions are represented by mean values. Integrating Equation 8.90 and applying the boundary conditions of Equation 8.97 yields this expression for the net mass transfer rate: Y i − εi m ˙ (8.98) = ln , i = 1, 2, . . . Ns, Redp ≈ 0 2πdp ρD Yi,s,g − εi Integrating Equation 8.91 in the same manner gives this expression for the heattransfer coefficient: 1 mC ˙ p mC ˙ p hdp = (8.99) exp − 1 , Redp ≈ 0 l πdp l 2πdp l The quantity on the left-hand side of Equation 8.99 is the Nusselt number when m ˙ → 0, Nu = hdp /l → 2 from Equation 8.99, which is the familiar value for a sphere in the absence of convection and mass transfer. Net outward mass transfer rates reduce the value of the Nusselt number, similar to the effect of blowing. Governing Equations and Interphase Relationships for Single-Droplet Behavior Under convective stream, the momentum equation or the equation of particle motion must be considered. Similarly, appropriate expressions for drag coefficient and the convection corrections must be determined. To study the dynamics of particle motion and to determine particle trajectories, the general equation of particle motion for a spherical particle can be written as shown in Equation 8.100 (Faeth, 1983). This equation is also called the B-B-O equation, as it includes

647

TWO-PHASE-FLOW (DISPERSED-FLOW) MODELS

effects studied by Bassett, Boussinesq, and Oseen, described by Soo (1967). π π 3 Dvp = dp2 ρCD v − vp v − vp dp ρp 6 Dtp 8 Inertial force of sphere

−

Drag force on sphere including skin friction and form drag

π 3 ∂p π d v − vp dp nr + dp3 ρCI dtp 6 ∂r 12 Force on sphere due to static pressure gradient

Force on sphere due to inertia of adjacent fluid being displaced by its motion (virtual - mass term)

t + 32 dp2 (πρμ)1/2 CB tp0p

(d/dτ )(v−vp )

(tp −τ )1/2

Bassett force to account for effects of deviation of the flow from steady flow pattern around the sphere

dτ + Fe External or

(8.100)

body-force term (e.g., gravity)

where nr is a unit vector in the positive radial direction and the time derivative is taken after the particle motion. ∂ D ∂ = + vp Dtp ∂t ∂r

(8.101)

In most cases, the effect of virtual mass is ignored. Also, the Magnus lift forces due to droplet spin are ignored in the consideration of equation of motion. At low Reynolds numbers, the drag coefficient (CD ) of the flow around an accelerating particle is related to Redp by: ρ v − vp dp |vrel | dp 24 ; where Redp ≡ CD = = (8.102) Redp μ ν The standard drag coefficient for solid spheres is usually employed in calculations on dilute sprays (Faeth, 1983). An expression proposed by Putnam (1961) can be given as: ⎧ ⎤ ⎡ 2/3 ⎪ Re ⎪ dp ⎨ 24 ⎣ ⎦ for Redp < 1000 1+ (8.103) CD = Redp 6 ⎪ ⎪ ⎩ 0.44 for Redp > 1000 The next expressions of Dickerson and Schuman (1960) with a broader Reynolds number range yield similar results to Equation 8.103: ⎧ 0.84 for Redp < 80 ⎨27/ Redp (8.104) CD = ⎩0.271 Re0.217 for 80 ≤ Re ≤ 104 dp dp

648

SPRAY ATOMIZATION AND COMBUSTION

The coefficients CI and CB in Equation 8.100 at low Reynolds numbers can be set equal to unity (i.e., CI = CB = 1), according to Soo (1967). Therefore, accelerating particles at low Reynolds numbers have a drag coefficient independent of the rate of acceleration while the coefficient of the virtual mass term is identical to the value found for inviscid flow around a sphere. At higher Reynolds numbers, most investigators adopted the practice to retain the original form of the B-B-O equation; however, the coefficients are assumed to be functions of the Reynolds number and the acceleration number (see, e.g., Hamilton and Lindell, 1971; or Odar and Hamilton, 1964). ⎛ ⎞ ⎜ ρ |v | d ⎟ dp d rel p ⎜ ⎟ |v | , Cj = Cj ⎜ ⎟, rel 2 dt ⎝ ⎠ μ |v | p rel

j = I, B

(8.105)

acceleration number

Drop-life-history computations show that Redp are generally less than 104 in sprays. Typical values of acceleration numbers for drops can be estimated if we consider a drop injected into a motionless gas and neglect all terms in Equation 8.100 except the drag term, yielding dp D |vrel | = 0.75 ρ/ρp CD 2 Dt |vrel | p

(8.106)

where the density ratio (ρ/ρp ) for most sprays is on the order of 10−3 . For Redp > 1, Equation 8.106 suggests that the acceleration number is less than 10, with the largest values encountered at low Reynolds numbers and high pressures. For this range of Reynolds and acceleration numbers, it has been found that CD can be represented by the standard drag curve for steady flow around spheres; CI and CB are relatively independent of the acceleration number and can be approximated by their values at low Reynolds numbers (i.e., CI = CB = 1). With the equation of motion established, it is now of interest to examine conditions where the usual simplified form of the B-B-O equation given next is valid for spray droplets. Dvp π = − dp2 ρCD vp − v vp − v + Fg Dtp 8

π ρbf dp3 6 (8.107) The mass of the liquid particle m is evaluated based on the bulk liquid density ρbf , which is determined from the equation of state for the bulk liquid temperature and composition. In Equation 8.107, Fg is the body force due to gravitation. The arrival of Equation 8.107 is based on this simplifications from the B-B-O equation. First of all, the pressure gradient term appearing in Equation 8.100 can be neglected since mean static pressure gradients are generally small for spray processes. Turbulent fluctuations can also contribute to the pressure gradient term, but this effect can be neglected under the assumptions just stated. Faeth (1983) m

where

m=

TWO-PHASE-FLOW (DISPERSED-FLOW) MODELS

649

examined the order of magnitude of the terms in Equation 8.100 and reported that the virtual mass and Bassett terms provide only a small contribution to the drag at atmospheric pressure. At higher pressures, the coefficients CI and CB approach unity, suggesting a greater need to include them. A mitigating factor, however, is that Reynolds numbers in sprays tend to increase with pressure as well, causing the coefficient of the drag term to become larger throughout much of the lifetime of a drop in a spray. Therefore, for most sprey jets, there is no need to include the virtual mass and Bassett terms in Equation 8.100. Each case, however, should still be evaluated independently. Useful formulas for the multiplicative correction for heat and mass transfer, proposed by Faeth and Lazar (1971), are: 1/2

0.278 Redp Pr1/3 hc =1+ 1/2 (hc ) Redp =0 1 + 1.232/ Redp Pr4/3

(8.108)

1/2

m ˙ ˙ Redp =0 (m)

0.278 Redp Sc1/3 =1+ 1/2 1 + 1.232/ Redp Sc4/3

(8.109)

where (hc )Redp =0 and (m) ˙ Redp =0 are found from: (hc )Redp =0 dp l

=

mC ˙ p /πdp l exp mC ˙ p /2πdp l − 1

(8.110)

and (m) ˙ Redp =0 can be computed from Equation 8.98, in which εi is determined from Equation 8.94. The life history of each class is then computed throughout the flow field. The drop position of a given particle of a group of particles can be calculated from

t

xp = xp0 +

vp dt

(8.111)

0

where xp0 is the initial location of the drop. The instantaneous velocity is determined by solution of the equation of motion given in Equation 8.107. In terms of temperature and concentration profiles inside the droplet, there are three limiting cases, as shown in Figure 8.29. Case A corresponds to the thin-skin model, utilizing the assumption that only an infinitely thin surface layer is heated and has the composition changes required by phase equilibrium, while the bulk of the liquid remains at its initial state. Case B corresponds to uniform temperature model, involving the assumption of infinite thermal diffusivity. In this case, the drop temperature is spatially uniform but time varying. Case C corresponds to uniform state model, extending the concept of temperature uniformity to species concentrations. In this case, both temperature and composition are spatially uniform but time varying.

650

SPRAY ATOMIZATION AND COMBUSTION SURFACE LAYER T

T

T0

T0

T0

YA

YA

YFf

YAf

YAf YF 0

YA

YFf

YFf

0

T

rP THIN SKIN CASE A

r

YAf YF

0 0

rP

r

UNIFORM TEMPERATURE CASE B

YF

0 0

r

rP UNIFORM STATE CASE C

Figure 8.29 Sketch of three different drop surface layer cases and corresponding distributions of T and Yi (modified from Faeth, 1983).

Case A: Thin-Skin Model . In this case, the temperature and composition of the liquid phase remain at the injected condition, while the liquid surface has different properties and composition, as dictated by equilibrium and transport requirements. Prediction of transport rates involves 3Ns + 3 unknowns: Yi,s,f , Yi,s,g and εi , for i = 1, 2, . . . Ns, as well as Ts , h, and m. ˙ With these quantities known, conservation of mass of the drop yields dm = −m ˙ dt

(8.112)

The composition of the bulk liquid is fixed; therefore, the mass flux fractions of the bulk liquid species must be equal to their bulk liquid mass fractions. Furthermore, the nonbulk liquid species (air or ambient gas mixture) only penetrate an infinitely thin surface layer and its mass flux fraction is zero. Therefore, εi = Yi,b,f ,

i = 1, 2, . . . , Ns − 1

(8.113)

There is no bulk heating for this case; therefore, all the energy reaching the surface of the drop is utilized to provide the energy required to gasify the evaporating material, Ns πdp2 h T − Ts = εi m ˙ hi,s,g − hi,b,f (8.114) i=1

If radiation is negligible, the total heat transfer coefficient h is equal to hc , shown in Equations 8.108 and 8.110. The phase equilibrium requirements at the liquid

TWO-PHASE-FLOW (DISPERSED-FLOW) MODELS

651

surface are: Ns

i=1

Yi,s,f =

Ns

Yi,s,g = 1

(8.115)

i=1

Fi,s,f = Fi,s,g ,

i = 1, 2, . . . , Ns − 1

(8.116)

where the fugacities of Equation 8.116 are determined from the equation of state. Equations 8.95, 8.98, 8.99, 8.108, 8.109 and 8.113 to 8.116 provide the 3Ns + 3 equations needed to define surface conditions and transport rates. Given initial conditions and mean properties of the flow, the solution proceeds by numerical integration of Equations 8.107, 8.111, and 8.112. This analysis is simplified considerably if the bulk liquid is a pure material and solubility can be ignored. Denoting the bulk liquid fuel species as F , Equation 8.98 becomes m ˙ Redp =0 1 − YF = ln (8.117) 2πdp ρD 1 − YF,s,g Equation 8.114 is also simplified, as shown, πdp2 h T − Ts = m ˙ hF,s,g − hF,s,f

(8.118)

while YF,s,g is determined, knowing the liquid surface temperature and the pressure, from the vapor pressure characteristics of the liquid YF,s,g = f (Ts , p)

(8.119)

The concentrations of the remaining species at the liquid surface are needed for the property evaluations of Equation 8.88. These concentrations are found by noting that the argument of the logarithm of Equation 8.98 must be the same for all species. Therefore, since εi = 0 for i = F , we have 2 (8.120) Yi,s,g = Y i 1 − YF,s,g 1 − Y F , i = F Case B: Uniform Temperature Model . In this case, only the composition of the bulk liquid remains at the injected condition, while the surface composition varies as required by local conditions. The temperatures of the surface and bulk liquid are the same and are known at each instant. Prediction of transport rates involves 3Ns + 2 unknowns: Yi,s,f , Yi,s,g , and εi , for i = 1, 2, . . . . . . .Ns, as well as h 2 ˙ and m ˙ (or m ˙ = m/πd p ). The formulation of the uniform temperature model is very similar to the thin-skin model. The main difference is that Equation 8.114 is replaced by the equation of conservation of energy for the drop Ns dTp εi m ˙ hi,s,g − hi,b,f = h T − Tp − ρf Cp,f dp /6 dt

i=1

(8.121)

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SPRAY ATOMIZATION AND COMBUSTION

Equations 8.95, 8.98, 8.99, 8.108, 8.109, 8.113, 8.115, and 8.116 remain unchanged and are sufficient to determine the 3Ns + 2 transport parameters. In this case, Equations 8.107, 8.111, 8.112, and 8.121 must be integrated numerically. The formulation of the uniform temperature model when only a single fuel species is present in the liquid is also similar to the thin-skin model, except that Equation 8.114 is replaced by

dTp ρf Cp,f dp /6 ˙ hF,s,g − hF,s,f = h T − Tp − m dt

(8.122)

Case C: Uniform State Model . In this case, the composition and temperature of the bulk liquid and the surface are the same and are known at each instant. Prediction of transport rates involves 2Ns + 2 unknowns: Yi,s,g , and εi , for i = 1, 2, . . . , Ns, as well as h and m. ˙ Equations 8.95, 8.98, 8.99, 8.108, 8.115, and 8.116 are sufficient to determine all these unknowns. Conservation of species in the liquid phase yields dmi ˙ = −εi m, dt

i = 1, 2, . . . , Ns

(8.123)

Conservation of energy is correctly given by Equation 8.121. The mass fraction of each species in the liquid phase is computed in the conventional manner, knowing the value of mi . In this case, Equations 8.107, 8.111, 8.121, and 8.123 must be integrated numerically. When only a single species is present in the liquid fuel, the uniform state model is identical to the uniform temperature model. Drop-Life Histories in Sprays Predicted drop-life histories along the centerline of the air atomized n-pentane spray burning in air at 3 MPa by Faeth (1983) are illustrated in Figure 8.30. Both the thin-skin and uniform temperature liquidphase transport models at drop surface are illustrated since there are significant differences in the predictions of these two limiting cases. Due to the high-pressure condition, real-gas effects and the solubility of the ambient gas in the liquid drops were considered in the computations. Drop velocity, diameter, and bulk (uniform temperature model) or surface (thin-skin model) temperatures are illustrated for drops having initial diameters of 10 and 100 μm. Flow properties obtained from the LHF model are also illustrated in the figure, including mean velocity, temperature, liquid fuel mass fraction, fuel vapor mass fraction, and oxygen mass fraction. Both predictions illustrated in Figure 8.30 indicate that only drops having initial diameters less than 10 μm have velocities and temperatures nearly equal to the flow and disappear near the spray boundary estimated by the LHF model (cf. Figure 8.26). Since the test spray contained drops larger than 10 μm in diameter, these findings indicate that finite drop transport rates are significant for the test conditions and the LHF model would be expected to overestimate the rate of development of the flow. The drop velocity predictions for the two liquid drop surface models illustrated in Figure 8.30 are similar. The relatively small injector diameter results in a large rate of deceleration of the flow; therefore, there is significant values of vp − v

TWO-PHASE-FLOW (DISPERSED-FLOW) MODELS

1.0

up/uo, u/uo

.6

653

Flow properties Critical temperature Drop properties: thin-skin model

up/uo, 10 µm 100 µm

.4

u/uo

uniform temperature model Vanishing point of drop by evaporation

.2 dp/dpo 1.0 YFf

10 µm

100 µm

.6

dp/dpo, Yi

.4

.2

.1

YO2 YFg

.06 .04

TEMPERATIRE (K)

2000

T

1000

Ts, 10 µm 100 µm

CRIT. TEMP. 600 400

100 µm

Tp, 10 µm 200

20

40

60

100 x/d

200

400 600

1000

Figure 8.30 Predicted drop-life histories along centerline of n-pentane spray burning in air at p = 3 MPa (after Faeth, 1983).

for the 100 μm diameter drop in the region just downstream of the potential core. In contrast to velocity, drop temperature and diameter predictions are significantly different for the two liquid diffusivity limits. Drop surface temperatures are relatively independent of size for the thin-skin model; therefore, only a single surface temperature line is shown for this case. However, the uniform temperature model indicates that the rate of heating decreases for the larger drop size. Predictions for both liquid phase models indicate that drop temperatures tend to follow the temperature of the flow near the injector while drop temperatures farther downstream approach a constant value even though flow temperatures are increasing. Steady evaporation at a constant wet-bulb temperature is never achieved, however, since

654

SPRAY ATOMIZATION AND COMBUSTION

flow properties are changing continuously. The uniform temperature model predicts relatively low liquid surface temperatures in the region just downstream of the potential core. Fuel vapor concentrations are relatively high in this region, causing condensation of fuel vapor on the drops. The combined effect of condensation and reduced liquid densities due to heating cause predicted drop diameters for the uniform temperature model to increase for a time. In contrast, the thin-skin model provides for no bulk heating and yields sufficiently high surface temperatures so that condensation does not occur, resulting in a monotonically decreasing drop diameter as distance from the injector increases. The critical temperature of the liquid varies slightly with distance from the injector due to changes in the composition of the flow. Predicted drop temperatures remain well below the local critical temperature for both liquid models. These differences in the two models indicate that better methods of treating liquid phase transport are needed to accurately treat high-pressure sprays. The results illustrated in Figure 8.30 indicate that only large droplets having initial diameters greater than 100 μm would penetrate the flame zone and reach the region where oxygen concentrations are significant. Usually the drops simply evaporate in the cool core of the spray, yielding fuel vapor, which oxidizes farther downstream in the flame zone. Computations conducted for the pressure atomized n-pentane spray burning in air at 6 MPa gave results similar to those illustrated in the figure. Even though this pressure is almost twice the critical pressure of pure n-pentane (3.369 MPa; see Kuo, 2005), the drops did not reach the thermodynamic critical point. This behavior is observed since the mole fraction of fuel vapor at the drop surface was significantly less than unity, even when bulk heating was nearly complete. Furthermore, the presence of dissolved gases tends to raise the critical pressure of the liquid. Past work on critical conditions for individual drops, either evaporating in heated gases or burning in an oxidizing environment, also indicates that pressures on the order of twice the critical pressure of the pure liquid are required to reach the critical point (Canada and Faeth, 1975; Reid, Prausnitz, and Sherwood, 1977). 8.8.2

Drop Breakup Process and Mechanism

8.8.2.1 Drop Breakup Process In this section, we consider aspects of drop shattering, with particular reference to the combusting high-pressure sprays illustrated in Figure 8.30. As discussed in of Kuo (2005, chap. 6), several dimensionless parameters are important in controlling the droplet deformation and breakup processes. Since the relative velocity between the drop and ambient gas stream is closely associated with droplet deformation, the Reynolds number based on relative velocity is definitely pertinent to the drop breakup. In addition, the Weber number, (We) another controlling parameter, is defined as: 2 ρg vp − v dp Aerodynamic force Weg = = (8.124) 2σs Surface tension force

where σs is the surface tension of the droplet. Note that this definition of We has a factor of 2 in the denominator, which is different from the Weg used in

TWO-PHASE-FLOW (DISPERSED-FLOW) MODELS

655

many other studies. The definition of We in Equation 8.124 is physically more vp − v2 /2 and the meaningful, since the dynamic pressure is represented by ρ surface tension pressure is represented by σs / dp /4 . Therefore, the ratio of their corresponding forces gives the We expression shown above. As reported by Borisov et al. (1981) from their compilation of regimes reported by various researchers, there are three droplet breakup regimes: parachute type (bag type), stripping type, and explosion type. These regimes are defined not only by the shown in Table 8.9. ranges of the We but also by the product of Weg Re−0.5 d Past studies of drop breakup generally have subjected drops to abrupt changes of relative velocity vp − v, observing the occurrence, the mode, and the time required for breakup. The two photographs in Figure 8.31a show the formation of bag-shape liquid tributyl phosphate from an original spherical droplet due to gas penetration. Figure 8.31b shows that at higher Weg , the spherical droplet is deformed into parachute-shape bodies with a short cylindrical liquid zone near the center (called the stamen). At a later time, both the thin bag and thin parachute break up into fine droplets due to characteristic bag-shape instability. The parachute-shape body is also called umbrella-shape body or bag-and-stamenshape body. Figure 8.32 shows the formation of parachute-shape water droplets in an air stream flowing from left to right before breaking up into fine droplets due to characteristic bag-shape instability. Figure 8.33 shows a major redistribution of the liquid mass into a fragmenting sheet extending out with a significant radial velocity. Particle stripping from the edge of the liquid sheet is quite evident. TABLE 8.9. Three Droplet Breakup Regimes Drop Breakup Type

Range of Governing Parameters

Parachute type (bag type)

4 ≤ Weg ≤ 20 0.1 ≤ Weg Re−0.5 ≤ 0.8 d

Stripping type

10 ≤ Weg ≤ 104 0.5 ≤ Weg Re−0.5 ≤ 10 d

Explosion type

103 ≤ Weg ≤ 105 10 ≤ Weg Re−0.5 ≤ 100 d

Physical Processes Droplet flattens perpendicular to flow and forms a shroud or a bag (or a parachute). Liquid around the forward stagnation point of drop is deflected by flow-piercing effect. The bag expands and eventually shatters into a number of small drops due to bag-shaped instability. (see Figures 8.31, 8.32, and 8.34) The surface layer is torn off the droplets (stripping or shearing) A significant fraction of very fine droplets appears. (see Figures 8.33 and 8.34) The droplets formed are significantly smaller size than the original drops.

656

SPRAY ATOMIZATION AND COMBUSTION

Gas stream

Gas stream

(a) Weg = 3.5

(b) Weg = 6.5

Figure 8.31 Development of (a) bag-type and (b) parachute-type liquid tributyl phosphate from isolated droplets with initial diameter of 3.5 mm (modified after Theofanous et al., 2007). Water

Air

t = 969 ms

t =1528 ms

t = 1622 ms

Figure 8.32 Photographs of water droplets at 969, 1528, and 1622 μs after they were injected into air stream flowing from left to right with Weg = 32.5; these pictures show the development of parachute-shape bodies from flattened droplets (modified from Van Dyke, 1982).

Gas stream

Figure 8.33 Photographs showing liquid tributyl phosphate drop breakup due to stripping (or shearing) at high Weber number (Weg = 1, 250) in a supersonic flow stream with Mach number = 3 (modified from Theofanous et al., 2007).

Pilch and Erdman (1987) made an extensive study of acceleration-induced droplet breakup processes. They defined five different mechanisms of accelerationinduced breakup of liquid drops. Data on acceleration-induced fragmentation of liquid drops were collected from the literature and summarized in Figure 8.34. This figure contains brief descriptions of the major physical processes involved in each mechanism. Pilch and Erdman (1987) defined a critical We (Wec ) that is related

657

CASTASTROPHIC BREAKUP 175 > Weg

WAVE CREST STRIPPING 175 = Weg

SHEET STRIPPING 50 = Weg = 175

BAG-AND-STAMEN BREAKUP 25 = Weg = 50

DEFORMATION BAG GROWTH

BAG BREAKUP 6 = Weg = 25

VIBRATIONAL BREAKUP Weg = 6

BAG BURST

RIM BREAKUP

• Large-amplitude, long-wavelength waves finally penetrate the drop, creating several large fragments before wave crest stripping can significantly reduce the of the mass of the droplet. Catastrophic breakup leads to a multistage process in which fragments, and fragments of fragments, are subject to further breakup.

• At still higher Weg, large-amplitude, small-wavelength waves are formed on the windward surface of the deformed drop. The wave crests are continuously eroded by the action of the flow field over the surface of the drop.

• This breakup mechanism is distinctly different from the above breakup mechanisms. No bags are formed; instead, a thin sheet is continuously being stripped from its periphery to form small drops. A coherent residual drop exists during the entire breakup process.

• Breakup mechanism has features similar to bag breakup; however, a column of liquid (stamen) is formed along the drop axis parallel to the approaching flow. The bag bursts first; rim and stamen disintegrate later. This process is sometimes referred to as “umbrella breakup” or “claviform breakup”.

• Breakup process analogous to the bursting of soap bubbles blown from a soap film attached to a ring. The thin hollow bag eventually bursts, forming a large number of small fragments; the rim disintegrates into a small number of large fragments.

• This mechanism occurs at small Weg; oscillations may develop at natural frequency of drop. When oscillation amplitude increases, the drop may decompose into a few large fragments. Breakup time is longer than those of other mechanisms; thus, this mechanism is not usually considered in drop breakup studies.

Figure 8.34 Droplet acceleration-induced breakup mechanisms (modified from Pilch and Erdman, 1987).

FLOW

(c)

FLOW

(b)

FLOW

(a)

FLOW

(c)

FLOW

(b)

FLOW

(a)

658

SPRAY ATOMIZATION AND COMBUSTION

to the Ohnesorge (Oh) number defined next. It is obvious that the effect of liquid viscosity is reflected by the Oh number: μl

Oh =

ρl σs dp

0.5 =

Viscous force Surface tension force

(8.125)

where μl and ρl are the liquid viscosity and density. The term of “critical Weber number (Wec )” has been used generally in the literature to denote the conditions for the beginning of the bag-type breakup (the onset of bag-type breakup). This means that Wec is the lowest value of We sufficient to complete the bag-type breakup. Wec is related to the Oh number by: Wec = 6 1 + 1.077 Oh1.6 (8.126) This correlation is called Brodkey’s correlation which is also shown in Figure 8.35. Readers must recognize that the We defined by Equation 8.124 is different from the We used by Brodkey (1969) and Pilch and Erdman (1987) by a factor of 2. Therefore, the coefficient in Equation 8.126 is 6 instead of 12 in the original correlation. A dimensionless characteristic time of drop breakup by Rayleigh-Taylor or Kelvin-Helmholtz instabilities is given by: 3 tbk vp − v ρ (8.127) τbk = dp ρl For a fixed Oh number, the time required to initiate drop breakup decreases continuously with increasing We. Wolfe and Anderson (1964) found that large

CRITICAL WEBER NUMBER: Wec

35 BRODKEY'S CORRELATION

30 25 20 15

Gas-liquid system data from Haas (1964), Hanson and Domich (1956), Hanson et al. (1983), Hassler (1970, 1971) and Hinze (1949) Liquid-Liquid system data fom Li and Fogler (1978)

10 5 0 10−4

10−3

10−2 10−1 OHNESORGE NUMBER: Oh

100

101

Figure 8.35 Relationship between critical Weber number and Ohnesorge number (modified from Pilch and Erdman, 1987).

TWO-PHASE-FLOW (DISPERSED-FLOW) MODELS

659

drop viscosity (Oh > 0.1) delays initiation of breakup without altering the observed breakup mechanism. A simple empirical correlation was proposed that adequately represents the required time to initiate droplet breakup. −0.25 τbk = 1.598 Weg − 6 1 + 2.2Oh1.6

(8.128)

Again, the coefficients in Equation 8.128 have been modified to match with the We definition given in Equation 8.124. A similar correlation has been proposed by Gel’fand, Gubin, Kogarko, and Palamarchuk. (1975). Ranger and Nicholls (1969) suggested the following expression for breakup time in the stripping mode: τbk

= tbk vp − v

√ ρ/ρl = 3.5 − 5.5 dp

(8.129)

where the range of τbk results from different test conditions and a value of 4.0 is recommended in the absence of other information. From Equation 8.129 and various droplet breakup mechanisms, we can see that combusting sprays involve considerable uncertainties. Property selection rules, especially for combustion at high pressures, have not been well established for breakup phenomena. Figure 8.36 shows the breakup phenomena of a water droplet hit by a shock wave in air, simulated numerically by Terashima and Tryggvason (2009). After the shock passes, large recirculation regions are formed behind the drop. The pressure difference between the upstream-facing part of the drop and the wake flattens the drop, eventually leading to thin ligaments pulled from the side. Although first stretched directly outward, eventually the ligaments are carried downstream by the flow, thus stripping water from the original drop. It is worthwhile to note that the drop breakup phenomena by shock wave disturbances are very different from the acceleration-induced mechanisms. If a shock wave exists in the given combustor, both types of breakup mechanisms may need to be considered. In general, the We varies more gradually in combusting sprays than in the shock wave disturbance situations, which often are used to establish the empirical expressions. 8.8.2.2

Multi-component Droplet Breakup by Microexplosion

As indicated by Law (1982, 2006), sudden violent fragmentation is an interesting phenomenon that may occur during multicomponent droplet combustion (e.g., decane-dodecane droplet). The occurrence of this explosive combustion event, called a microexplosion, offers interesting potential in optimizing charge preparation. For example, current designs of spray systems emphasize producing optimum droplet size distributions such that the droplets are both large enough for penetration into the combustor interior but also small enough for rapid gasification. However, if microexplosion can be controlled to occur after penetration is achieved, then rapid gasification need not be a primary concern in designing spraying systems. In this manner, large-scale mixing can be achieved through spraying and penetration, with somewhat larger droplets, which is then followed

660

SPRAY ATOMIZATION AND COMBUSTION

(a) t = 20msec

(b) t = 98msec

(c) t = 202msec

(d) t = 300msec

(e) t = 404msec

(f) t = 455msec

(g) t = 506msec

(h) t = 550msec

(i) t = 600msec

Figure 8.36 Evolution of water droplet in air due to shock impact (after Terashima and Tryggvason, 2009).

by instant gasification and local mixing through microexplosion. Microexplosion also may improve the utilization of synthetic and less refined fuels, which generally have higher boiling point ranges. Thus, with microexplosion the fuel volatility becomes less crucial in affecting complete gasification within the combustor. The basic mechanism responsible for this microexplosion event for miscible multicomponent mixtures is the diffusional entrapment of the volatile components in the inner core region of the droplet. Law found that shortly after initiation of gasification, the multicomponent droplet surface becomes more concentrated with the less-volatile, high-boiling-point (e.g., dodecane with Tb at 216.3◦ C), components. Essentially, the concentration boundary layer develops at the droplet surface by the less volatile fuel component. Since the droplet temperature is controlled by its surface composition, the droplet temperature can attain a high

TWO-PHASE-FLOW (DISPERSED-FLOW) MODELS

661

value corresponding to the more abundant, higher-boiling-point component at the surface; therefore, it can reach high values above the Tb of the more volatile fuel component. In the meantime, the droplet core has a higher concentration of the more volatile, low-boiling-point (e.g., decane with Tb at 174.2◦ C) components. Thus, it is conceivable that the volatile components in the droplet interior can be heated beyond the local boiling point in the surface region and hence start to accumulate a substantial amount of superheated fuel vapor, as shown in Figure 8.37a. Therefore if the droplet temperature is sufficiently high, then this condition can lead to intense internal pressure build-up due to the onset of homogeneous nucleation and thereby the violent fragmentation of the droplet. Experimentally, microexplosion has been observed frequently (Lasheras, Fernandez-Pello, and Dryer, 1980; C. H. Wang and Law, 1985; C. H. Wang, Liu, and Law, 1984). Lasheras, et al. (1980) showed that this event can occur in less than 0.2 ms after a droplet was introduced into a combustor. Since this mechanism depends critically on the existence of a steep concentration gradient, it is most likely to occur for a droplet with minimum amount of internal circulation. Figure 8.37b shows the flame streak of a stream of freely falling droplets terminated by microexplosion. Since the droplet size is typically much smaller than the flame size, the “explosion ball” near the end of the streak is evidently showing the microexplosion event. Theoretical assessment of the potential occurrence of microexplosion was performed by Law (1978) through his solution of the

d2-law gasification after initial heating period of multi-component droplet

Development of concentration layer for less volatile fuel species near suface Droplet Flame Streak

Development of pressurized zone in inner core containing more volatile fuel species by continued heating Microexplosion caused by violent fragmentation of fuel droplet (a)

Explosion Ball (b)

Figure 8.37 Microexplosion of multicomponent fuel droplet: (a) physical interpretation of major processes; (b) streak photograph of a free falling droplet showing the microexplosion phenomena (modified from Law, 2006a, 2006b).

662

SPRAY ATOMIZATION AND COMBUSTION

temperature and species distributions within the droplet. Homogeneous nucleation was found to initiate at a radial location where the temperature exceeds the local concentration-weighted limit of superheated material. His theoretical study showed three distinctive properties. 1. Microexplosion can occur only if the volatilities of the components are sufficiently different and their initial concentrations lie within an optimum range. It is necessary to have less volatile components with high boiling temperature to drive up the droplet temperature and the volatile components to facilitate internal nucleation. 2. Second, since the droplet center has the highest concentration of the more volatile components while the droplet surface has the highest temperature, homogeneous nucleation should initiate somewhere between these two locations. 3. The occurrence of microexplosion is more likely to happen at high pressures, due to the increase of Tb with p. These three distinctive properties have all been experimentally verified. In particular, it was shown that the optimum composition of a two-component mixture for enhanced microexplosion is around 50% to 50% and that nucleation is initiated close to the droplet center and thereby has the maximum effect in shattering the droplet. Note that the limits of superheat of many liquids are about 90% of their respective critical temperatures, according to Blander and Katz (1975). 8.8.3

Deterministic Discrete Droplet Models

Dispersed flow models of multiphase flow specifically treat the finite-rate exchange of mass, momentum, and energy between the two phases. Most existing models generally average over processes that occur on scales comparable to the drop size (i.e., very limited attempt is made to accurately model the details of the flow field around or within individual drops due to practical limitations of computer storage and computation costs). Therefore, the exchange processes between phases must be modeled independently, usually by employing empirical expressions for drop drag and heat and mass transfer. Processes within the liquid phase are also simplified by employing one of the limiting cases described in Figure 8.29 (usually case B). Most dispersed flow analyses of evaporating and combusting sprays employ the particle source in cell model (PSICM) or discrete droplet model (DDM), described briefly in the beginning of Section 8.8.1. This model involves dividing the spray into representative samples of discrete drops whose motion and transport are tracked through the flow field, using a Lagrangian formulation similar to the drop-life history computations discussed earlier. This procedure corresponds to a statistical (Monte Carlo) computation for liquid properties since a finite number of particles are used to represent the entire spray. An Eulerian formulation is employed to solve the governing equations for the gas phase. The effect of drops on the gas phase is considered by introducing appropriate source terms in the gas phase equations of motion. Generally it

TWO-PHASE-FLOW (DISPERSED-FLOW) MODELS

663

is found that the number of particle classes required to achieve a satisfactory representation of the spray is not excessive. The formulation is also convenient for considering a relatively complete representation of drop transport processes. In deterministic discrete droplet model (DDDM) formulation, the effects of relative velocity between gas and particle and finite interphase transport rates are considered, but effects of drop dispersion by turbulence and effects of turbulence on interphase transport rates are ignored. Droplets are assumed to interact only with the mean gas motion. As mentioned by Shuen, Chen, and Faeth (1983), in the DDDM formulation, particles follow deterministic trajectories found by solving their Lagrangian equations of motion. Spray models of this type usually employ the standard drag coefficient for spheres and ignore virtual mass and Bassett forces. These approximations are appropriate for high-void fractions and high liquid-to-gas density ratios. It is generally assumed that the spray is dilute in these models. This implies that although particles interact with the gas phase, they do not interact with each other. Therefore, droplet collisions are ignored, and empirical correlations determined for single drops in an infinite medium are used to estimate interphase transport rates. The volume fraction for the liquid phase is assumed to be negligibly small in most models. Due to experimental limitations, most measurements of drop size and velocity distributions generally have been made at some distance from the injector. This so-called initial station is usually in the dilute portion of the spray. Thus, the practical application of models is limited to the dilute region of the spray. Both experimental and theoretical studies of the dense spray region are needed to understand the jet breakup process and to accurately specify the flow conditions at a selected station near the injector exit. In the DDDM structure, the droplet flow at a station downstream of the injector exit is divided into a finite number of drop classes. Each drop class is assigned an initial diameter, velocity, direction, temperature, concentration, position, and time of injection. The basic assumptions and general structure of DDDM of sprays is given in Table 8.10. Readers interested in detail comparison of various DDDMs of sprays are referred to the review paper by Faeth (1983). The main

TABLE 8.10. Basic Assumptions and General Structure of Deterministic Discrete Droplet Models (DDDM) of Sprays Basic Assumptions

Gas-Phase Radiation Variables Models

Steady, 2D, u, v, T , Y i , None, 4 dilute k, ε, g flux, or spray, no 6 flux droplet breakup

LiquidPhase Variables

Number of Drop Classes

Interfacial Transport Considera- Application tions

Drag corre- Gas turbine xp , vp , Tp , Ranging combuslation, dp or m from 5 ˙p tors, mass to about DISC transfer, 1,200 engines, and furnace, energy etc. transfer rates

664

SPRAY ATOMIZATION AND COMBUSTION

limitation to the practical use of dispersed flow models involves gaining experience with them and properly calibrating both their numerous internal features and injector properties so that reliable predictions can be obtained. The applications considered by the models involve both evaporating and combusting sprays. Cases of evaporating sprays include drop vaporization in ducts and in the prevaporizer passages of aircraft gas turbine combustors, commercial sprinkler systems for the control of fires, and the evaporation period of sprays injected into reciprocating direct-injection-stratified-charge (DISC) engines. Models of combusting sprays have been constructed for gas turbine combustors, DISC engines, and furnaces. Most models developed are for two-dimensional axisymmetric cases under steady flow conditions. Limited three-dimensional models also have been developed. In principle, there is no fundamental difficulty involved with extending these models to three-dimensional transient flow. It is generally assumed that the spray is dilute in these models, which implies that although particles interact with the gas phase, they do not interact with each other. Therefore, drop collisions are ignored in most DDDMs. Transport expressions determined for single drops in an infinite medium (see Equations 8.103 to 8.110) are used to estimate interphase transport rates (i.e., drop spacing is taken to be infinitely large). Consistent with these assumptions, the volume fraction for the liquid phase is assumed to be negligibly small in most models. 8.8.3.1 Gas-Phase Treatment in DDDMs All DDDMs employ an Eulerian formulation for gas motion and a statistical Lagrangian formulation for particle motion. This approach involves dividing the liquid flow at the injector exit (or some other location where drop properties are assumed known) into a finite number of drop classes. Each drop class is assigned an initial diameter, velocity, direction, temperature, concentration, position, and a time of injection (for transient sprays). The life history of each class is then computed throughout the flow field, generally following procedures similar to those described in Section 8.8.1.1. The computation for any class is terminated when the drops either disappear due to gasification or pass out of the flow field. Most gas-phase models are generally restricted to dilute sprays and employ k-ε or k-ε-g models of turbulence. The governing equations for the gas phase are identical to those presented in Section 8.7.2.6 (see Equations 8.73 to 8.75 and Table 8.6), except that new source terms must be added to represent interactions with the liquid phase. The solution of the gas-phase-governing equations employs a finite-difference grid of computational cells. The source terms due to drops for the Eulerian gas-phase solution are determined by computing changes in drop properties as they traverse each computational cell. For the k th drop class, the number of drops proceeding along the class trajectory per unit time, n˙ k , is known from the boundary condition at the injector exit. Then the exchange rate between the phases, for any generic property φ, is given by the following equation:

Sφi,j =

Nd

k=1

n˙ k (mk φk )in − (mk φk )out i,j

(8.130)

TWO-PHASE-FLOW (DISPERSED-FLOW) MODELS

665

where Nd = total number of drop classes mk = mass of the drops in k th drop class i , j = computational cell The parameter φ corresponding to different dependent variables having drop source terms is summarized in Table 8.6. Generally drop source terms are provided in the equations for conservation of mass (φ = 1), momentum (v), energy (H ) and injected species (Y F or f ). Most DDDMs employ a k-ε model of turbulence. When a higher-order turbulence model is used, drop source terms should appear in the governing equations for k , ε, and g as well. Most DDDMs neglect these effects, mainly because models for predicting turbulence production and dissipation by particle motion are not well developed. This procedure is acceptable at the limit of a dilute spray where such effects should be small due to low particle densities. Combusting sprays are generally modeled by assuming that drops simply evaporate with no envelope flames present. The fuel vapor is assumed to react subsequently in the gas phase with rates determined in the same manner as for gas flames. Therefore, the drops within the spray are assumed to act simply as distributed sources of fuel vapor. El Banhawy and Whitelaw (1980) employed a statistical formulation for turbulent reaction using a k-ε-g turbulence model with an assumed pdf for the mixture fraction. Mongia and Smith (1978), Boyson and Swithenbank (1979), and Gosman and Johns (1980) employed a rate taken as the smaller of either a global Arrhenius rate or an eddy breakup turbulent reaction rate. Gosman and Johns (1980) employed the Magnussen and Hjertager (1977) expression given in Equation 8.84 for the eddy breakup reaction rate, while the other models employ the Spalding (1971, 1976) expression of Equation 8.83. Mongia and Smith (1978) used the most complex reaction mechanism of the cases shown, involving a two-step reaction that considers the formation of CO as an intermediate. The global kinetic parameters for this model were obtained by fitting predictions to measurements in a gas turbine combustor. Only Butler et al. (1980) attempted pollutant predictions within a DDDM. This involves predictions of nitric oxide (NO), using an extended Zel’dovich mechanism with Arrhenius rate constants and ignoring the effects of turbulence. Methods for predicting pollutants for turbulent gas flames are highly developed nowadays, and these procedures could be adopted for DDDMs with little difficulty. Several DDDMs treat radiation, following procedures developed for turbulent gas flows. The multiflux methods (specifically 4 and 6 flux) developed by Gosman and Lockwood (1973) generally have been employed. The effect of absorption and scattering by drops usually is ignored; this approximation is appropriate only for dilute sprays. Similar to the use of Arrhenius expressions in the reaction models, mean properties have been employed in the transport equations for radiation, ignoring the effect of turbulence. Naturally, this practice is questionable due to the nonlinear characteristics of the radiation heat flux. The computer algorithms used to solve the gas-phase equations of motion in DDDMs generally are adapted from single-phase flow models. The steady-flow elliptic codes TEACH and SIMPLE, developed at Imperial College (Patankar, 1975; Pun and

666

SPRAY ATOMIZATION AND COMBUSTION

Spalding, 1977), are widely available. The ICED-ALE code developed at Los Alamos (Amsden and Hirt, 1973; Amsden, and Cook Hirt, 1974; Norton and Ruppel, 1976) and RPM developed by Gosman et al. (1980) provide a EulerianLagrangian grid scheme that is useful for the transient recirculating flows with moving boundaries encountered for sprays within reciprocating engines. 8.8.3.2

Liquid-Phase Treatment in DDDMs

The drop models employed by researchers for computations of droplet behavior fall into two categories: transient heat-up models and two-stage models. Transient heat-up models are similar to the formulation of the drop-life-history computations discussed in Section 8.8.1.1 and block diagram in Figure 8.28. The liquid phase is treated by assuming a uniform liquid temperature at each instant of time (case B of Figure 8.29). The rate of mass transfer then is determined using the difference in fuel vapor concentration between the surface of the drop and its local surroundings as the driving potential. This procedure correctly represents the fact that a drop within a spray has no fixed wet-bulb temperature due to the continuous change in the properties of its surroundings. The presence of envelope flames generally is ignored, and empirical expressions, found for single drops, are used to represent drop drag and the effect of forced convection on drop heat and mass transfer rates. Some researchers consider the two-stage drop models to involve computation of drop heat-up to a fixed wet-bulb temperature without any evaporation as the first stage. In the second stage, the evaporation process is driven by the temperature difference between the drop and its surroundings. The wet-bulb temperature is taken to be the boiling temperature of fuel at the chamber pressure of the spray. Effects of drag and forced convection are estimated using empirical relations for single drops, similar to the more complete transient heat-up models. Two-stage drop models simplify computations, but their ability to represent drop processes within a spray accurately is questionable. In high-temperature surroundings containing low concentrations of fuel vapor, drop wet-bulb temperatures approach the boiling temperature at near-atmospheric pressure. However, drops spend much of their lifetime in the cool core region of the spray, where gas temperatures are relatively low and fuel vapor concentrations are high. In this region, it is relatively inaccurate to estimate mass transfer rates by a heat transfer–controlled driving potential. This approach also requires ad hoc assumptions concerning wet-bulb temperatures when the pressure of the spray exceeds the critical pressure of the fuel. Based on the findings of various researchers, ignoring envelope flames appears to be the most realistic approximation in cases where premixed combustion does not dominate the process. Drop transport rates are strongly influenced by the selection rules used to determine average properties. This procedure should be established by calibration of the calculations using measurements for individual drops. Unfortunately, most spray modelers did not do this. Many DDDMs assume that the drops follow their fixed trajectories by ignoring turbulence-induced particle diffusion. Gosman and Ioannides (1981) found that effects of drop diffusion on spray predictions are small in comparison to uncertainties concerning the proper initial conditions of the spray. Anderson et al. (1980) considered other features important to the dynamics of drops in sprays, including drop breakup and drop collisions.

TWO-PHASE-FLOW (DISPERSED-FLOW) MODELS

667

Drop trajectory calculations are undertaken for several drop classes when using the DDDM approach. Each class is identified by its initial size, velocity, direction and position at the exit of the injector, and time of injection, as well, if the process is transient. In many DDDM simulations, known spray properties were limited to SMD and spray angle, which is not sufficient to specify the injector exit condition properly since the joint distributions of size, velocity, direction, and position are required. Given the SMD, Simmons (1977) provided a useful relationship for determining the drop-size distribution for various injector types. However, adequately estimating initial velocities and directions is less satisfactorily resolved, requiring the use of relatively ad hoc assumptions. Adequately specifying injector exit conditions constitutes the major impediment to the proper use and evaluation of DDDMs of sprays. 8.8.3.3 Results of DDDMs Butler et al. (1980) reported predictions of the transient spray processes encountered in reciprocating internal combustion engines. Figure 8.38 illustrates predicted spray-tip position as a function of time following the initiation of injection

90

P = 0.1 MPa 1 MPa

80

3 MPa

Measured 70

5 MPa

Computed Penetration, mm

60

50

40

Scaled cross section of spray at t = 3 ms P = 3 MPa

30

20

10

0

0

1

2

3 Time, t, ms

4

5

6

Figure 8.38 Spray-tip penetration as a function of time at different pressures. Predictions by Butler et al. (1980), measurements by Hiroyasu and Kadota (1974) (modified from Butler et al., 1980).

668

SPRAY ATOMIZATION AND COMBUSTION

from a single-orifice injector. The predictions are compared with the measurements of Hiroyasu and Kadota (1974) with an encouraging degree of agreement. However, these computations involved significant uncertainties with respect to the specification of injector exit conditions and a numerical grid that was too coarse to resolve the near-injector region properly. Some typical results of the computations of Boyson et al. (1981) are given in Figure 8.39. The fuel spray, which is of the hollow-cone type with an included angle of 60◦ , is introduced from an axially located fuel nozzle. The RosinRammler equation [MD = exp(−D/D∗ )n ] was used to represent a typical fuel spray with different mass fractions for various size groups, where D∗ = 60 μm and n = 2.2 were taken. The D* value was obtained experimentally using a Malvern particle sizing instrument. A total of 20 droplet size ranges and 18 angular injection locations within the representative 60◦ sector of the combustor can were used to construct the complete spray cone. This figure summarizes details of the operating conditions for the combustor. Predicted mean velocity distributions and drop trajectories are shown in the cross-sectional views of the combustor in both x-y plane and radial plane. The flow field near the inlet region shows a strong recirculation zone. The computational grids are able to resolve the main features of the dilution jets; however, they are too coarse to indicate the cooling effects of the spray near the injector exit. For this computation condition,

(a)

(b) −2

Total air flow rate = 2.125 × 10 kg/s Total fuel flow rate = 6.25 × 10−4 kg/s Fuel/Air ratio = 2.94 × 10−2 Air temperature at inlet = 351 K Swirl number = 0.8

Initial velocity of droplets = 20 m/s Air velocity through injection holes = 151 m/s Air velocity through swirler = 27.6 m/s

Figure 8.39 Predicted (a) velocity distributions and (b) drop trajectories for combusting hollow-cone spray in cylindrical gas turbine combustion chamber (modified from Boyson, Ayers, Swithenbank, and Pan 1981).

TWO-PHASE-FLOW (DISPERSED-FLOW) MODELS

669

a large number of drops strike the combustor wall, and some undergo deflection by the dilution jets, as shown in Figure 8.39b. Boyson et al. (1981) also studied a 45◦ included angle spray injected into the hot flow. They found that a much smaller fraction of the droplets are seen to hit the walls of the combustion chamber. They showed that the three-dimensional flow field in the combustor has a strong influence on the trajectories of individual droplets and vice versa. 8.8.4

Stochastic Discrete Droplet Models

Although the DDDM described in the cast section considers the relative velocity (interphase slip) effect between particles and the continuous phase, the effects of turbulent fluctuations on particle motion are ignored. Several initial stochastic discrete droplet models (SDDM) have been developed by Yuu et al. (1978); Gosman and Ioannides (1981); and Shuen, Chen, and Faeth (1983a,b) to treat both slip and the effects of droplet dispersion by turbulent eddies. Yuu et al. (1978) employed empirical correlations of turbulence intensities and length scales for their calculations of particle dispersions in jets. Gosman and Ioannides (1981) took a more comprehensive approach and used a k-ε model for predicting both flow properties and dispersion. Shuen, Chen, and Faeth (1983a,b) used an approach similar to that of Gosman and Ioannides (1981) and modified their method for evaluating the turbulent-eddy lifetime. They also made detailed comparisons of theoretical results with many sets of particle-laden jet data obtained by Laats and Frishman (1970a,b), Yuu et al. (1978), and Levy and Lockwood (1981). Shuen, Chen, and Faeth (1983a,b) found that the solutions of SDDMs agree very closely with experimental data on particle dispersion in turbulent round jets. A discussion of their comparisons is given in Section 8.8.5 after the presentation of their SDDMs. In the consideration of governing equations for the continuous phase, several basic assumptions can be introduced under certain flow conditions. For example, when the particle mass loading in the jet is sufficiently small (<0.5 %), the particles can be assumed to have negligible effect on mean and turbulent gas-phase properties. For low Mach numbers (<0.3) at the exit of the injector, density variations, the kinetic energy of the mean flow, and viscous dissipation can be neglected with small error. Also, for high jet Reynolds numbers (≥ 104 ), molecular transport can be ignored in comparison with turbulent transport. Under these assumptions, the governing equations for the continuous phase are identical to the DDDM, except that additional source terms must be included to account for the transport from the dispersed particles. Table 8.11 shows these additional source terms for the governing equations for the continuous phase, described initially in Equation 8.76. In the simulations of dilute sprays with ρp /ρ > 200, Shuen, Chen, and Faeth (1983) considered the virtual mass, Bassett forces, and Magnus forces to be negligible. They also ignored gravitational forces except in some special cases. The simplified B-B-O equation of motion for droplets was integrated with particle trajectory equation using a second-order Runge-Kutta algorithm. The trajectories of at least 1000 particles were computed and averaged to obtain dispersion

670

SPRAY ATOMIZATION AND COMBUSTION

TABLE 8.11. Source Terms in Governing Equations for the Continuous Phase due to Transport of Dispersed Particles. Source: Modified from Faeth, 1987.

φ

S pφ S pm =

1

Nd

Remarks

n˙ i mi /Vj

i=1

u˜

S pu

1

mi upi + Vj = n˙ i mim ag 1 − ρ/ρp tpi i=1 Nd

f˜

S pm

k

uSpu − uS pu

ε

g

−2Cε3 μt kε

∂S pu ∂r

S pm represents the rate of increase of gaseous mass by summing contributions from i th particle group passing the j th computational cell with volume of Vj · n˙ i is the particle number per time. S pu represents the rate of increase of gaseous momentum by summing contributions from i th particle group passing the j th computational cell with volume of Vj . Rate of increase of mixture fraction due to particle evaporation. Source terms for k -equation are related to the particle source term in the streamwise momentum equation. Source terms for ε-equation are also related to the particle source term in the streamwise momentum equation. Source terms for g-equation are related to the particle mass source term in the continuity equation.

,pm − f˜ S pm 2 fS

properties. Shuen, Chen, and Faeth used a method similar to that of Gosman and Ioannides (1981) for tracking the interaction of the particles as they interacted with a succession of turbulent eddies, each of which was assumed to have constant flow properties. A particle was assumed to interact with an eddy for a time taken as the smaller of either the eddy lifetime, te , or the transit time, tt , required for the particle to traverse the eddy. These times were estimated by assuming that the characteristic size of an eddy is the dissipation length scale 3/4

e =

Cμ k 3/2 ε

(8.131)

The characteristic time scale used by Shuen, Chen, and Faeth (1983a) was evaluated found by e (8.132) te = 1/2 2 k 3

TWO-PHASE-FLOW (DISPERSED-FLOW) MODELS

671

A dispersed-phase element was assumed to interact with an eddy as long as the relative displacement of the element with respect to the eddy did not exceed the characteristic eddy size, e , and the time of interaction did not exceed the characteristic eddy lifetime, te . Thus, the limit with respect to eddy size also incorporated isotropic turbulence ideas (e.g., only the total relative displacement was evaluated to determine the interaction time, with no concern for different eddy dimensions in the different coordinate directions). The transit time of a particle was found using the linearized equation of motion for a particle in a uniform flow, e (8.133) tt = −τr ln 1 − τr v − vp where v − vp is the velocity at the start of the interaction and τr is particle relaxation time defined as: 4ρp dp (8.134) τr ≡ 3ρCD v − vp When e > τr v − vp , Equation 8.133 has no solution. This condition can be interpreted as the eddy having captured a particle, so that the interaction time becomes te (Faeth, 1983; Gosman and Ioannides, 1981). The stochastic method generally requires an estimate of the mean and turbulent properties of the continuous phase. Particle trajectories are then calculated using random sampling to determine the instantaneous properties of the continuous phase, similar to a random-walk calculation. Mean dispersion rates are obtained by averaging over a statistically significant number of particle trajectories. A detailed review of the transport and combustion processes in sprays was given by Faeth (1987). 8.8.5

Comparison of Results between DDDMs and SDDMs

For a particle-laden round jet, Yuu et al. (1978) injected an air jet containing fly-ash particles (with dp = 20 μm and ρp = 2000 kg/m3 ) into still air. Information needed to estimate initial conditions for DDM simulation was not reported and had to be estimated. The jet nozzle was designed to provide uniform exit properties; therefore, the assumed initial condition was taken to be slug flow by Shuen, Chen, and Faeth (1983a,1983b) and Faeth (1987), except for a shear layer having a thickness = 1% of the jet exit radius at the passage wall. In the uni˙ 0 / ρ0 /πd 2 , f 0 = 1, form region, properties were specified in this way: u0 = 4m k0 = (0.02u0 )2 , ε0 = 2.84 × 10−5 u30 /d, and g0 = 0. Quantities u0 and f 0 were assumed to be linear in the shear layer. Initial values of k , ε, and g in the shear layer were found by solving their transport equations while ignoring convection and diffusion terms. Mean particle properties at the jet exit were computed using particle-trajectory computations based in the particle/gas mixing system and the nozzle geometry. Particles were assumed to be distributed uniformly across the jet exit.

672

SPRAY ATOMIZATION AND COMBUSTION 1.2 Uo = 50m/s, dp = 20mm SYMBOL x/d DATA 20–50 PREDICTION 20 PREDICTION 50 .8

C/Cc

LHF

DDDM Upo = 30m/s

SDDM Upo = 30m/s

.4

0

0

.04

.08

.12

r/x

Figure 8.40 Comparison of measured particle concentrations in a particle-laden round jet by Yuu et al. (1978) and calculated results from Shuen, Chen, and Faeth (1983b) (modified from Faeth, 1987).

The significance of turbulent dispersion in particle-laden jets can be seen from the results in Figure 8.40, from Shuen, Chen, and Faeth (1983b). A portion of the particle concentration measurements of Yuu et al. (1978) is illustrated along with LHF, DDDM, and SDDM predictions of the flows. Only the range of streamwise positions where data were measured was reported; therefore, predictions are illustrated for the limits of this range. Estimated initial particle velocities also appear on the figure. The rate of particle spread is overestimated using the LHF analysis, since effects of relative velocities between the phases v − vp are ignored. Ignoring the relative velocity causes the particle response to turbulent fluctuations, the mechanism of turbulent dispersion, to be overestimated. It also reduces streamwise particle velocities in the flow field, and the increased residence time causes further overestimation of particle spread rates. The DDDM analysis is seen to underestimate particle spread rates in Figure 8.40. In this case, particle spread is caused only by the initial radial velocities of the particles and by drag in the radial direction from the mean radial velocity of the gas phase. Both of these velocities are small in comparison to the fluctuating gas-phase radial velocities that are responsible for turbulent dispersion. Furthermore, since the radial velocities of particles eventually are dominated by gas-phase radial velocities, particles tend to accumulate in regions where v = 0. In contrast to the LHF and DDDM predictions, SDDM predictions are in reasonably good agreement with the measurements illustrated in the figure. Additional comparisons between predictions and these measurements,

TWO-PHASE-FLOW (DISPERSED-FLOW) MODELS

673

yielding the same conclusion, can be found in Shuen, Chen, and Faeth (1983b). This agreement suggests that both finite interphase transport rates and turbulent dispersion were important for this flow. Results just discussed were limited to relatively low particle loadings. This condition implies that while the gas flow influences particle dispersion, the effect of the particles on the structure of the continuous phase is small (e.g., the test conditions have emphasized one way coupling from the gas to the particles). In contrast, the measurements of Laats and Frishman (1970a, 1970b) involve relatively large particle loadings, resulting in significant effects of particles on the structure of the continuous phase. Initial conditions were not measured for these flows and had to be estimated. A constant-area pipe was used for an injector. In the absence of other information, fully-developed pipe flow was assumed at its exit by Shuen, Chen, and Faeth (1983b). The value of f 0 was taken to be unity by definition while u0 was obtained from the conventional power law expressions for pipes allowing for the variation of the power with Reynolds number. Initial values of k0 and ε0 were also obtained from Hinze (1975) for fully developed pipe flow in the same Reynolds number range as the experiments. The quantity g0 was set to zero by definition. For lack of other information, it was assumed that there was negligible relative velocity between the phases at the jet exit. Particle concentrations were assumed to be uniform at the jet exit. Predictions and measurements of centerline gas velocity distribution and meen particle mas flux distributions for the Laats and Frishman’s tests are illustrated in Figures 8.41 and 8.42 respectively. As noted earlier, the nonlinear SDDM designation refers to the present SDDM analysis. The linear SDDM designation refers to a preliminary version of the DDM analysis, based on the prescription of particle/eddy interaction time of Gosman and Ioannides (1981). As illustrated in Figure 8.41, predicted and measured mean gas velocities along the axis for both the air jet and the particle-laden jets are in agreement with the measured data, and the effects of initial velocity changes are small. The gas flow is influenced only by the particles at higher loadings—greater than 0.3—where the presence of particles tends to reduce the rate of decay of centerline velocity. Predicted and measured particle mass fluxes are illustrated in Figure 8.42. The results are for x/d = 28.5. The large particles used in these tests have significant inertia; therefore, finite interphase transport rates are significant, and the LHF analysis does not perform very well. The SDDM predictions are more satisfactory except at the highest particle loadings, similar to Figure 8.41. It was suggested that this could be due to initial relative velocities of the large particles, which were ignored, as noted earlier. For example, reduction of the particle velocities at the jet exit by 30% from the gas velocity essentially would match predictions and measurements for the particle loading of 1.4 in Figure 8.42. Note that Magnus forces and effects of turbulence modulation, were not considered in the predictions of Shuen, Chen and Faeth (1983b). These effects could be potential sources of discrepancies between predictions and measurements at high loading ratios. The measurements of particle-laden jet in a constant-area pipe by Levy and Lockwood (1981) include mean and fluctuating phase velocities, deduced from

674

SPRAY ATOMIZATION AND COMBUSTION

Uo = Upo (M/S) Data 29–60 SDDM prediction 60 SDDM prediction 29 Gas Jet 29–60 (dp(mm);LOADING RATIO) = (32;1.4)

1.0 .5 1.0

= (32;0.77)

Uc/Uo

.5 1.0 .5

= (32;0.3)

1.0 = (72;0.3)

.5 1.0 GAS JET

.5 0

0

5

10

15

20 x/d

25

30

35

40

Figure 8.41 Comparison of gas velocities in particle-laden round jets measured by Laats and Frishman (1970a, 1970b) and predictions by Shuen, Chen, and Faeth (1983b) using a nonlinear SDDM analysis (modified from Faeth, 1987).

laser Doppler anemometry (LDA) data, and are shown in Figure 8.43. Initial conditions were not reported for these tests and had to be estimated by Shuen, Chen, and Faeth (1983b) in their predictive work. The initial conditions were estimated assuming fully developed flow, following the same procedure used for the measurements of Laats and Frishman (1970a, 1970b) as last described. These tests involved rather large particles; therefore, the initial mean velocities of the particles at the jet exit were estimated by carrying out particle trajectory computations for the flow in the pipe. Particle concentrations across the exit of the pipe were assumed to be constant. The predicted results using SDDM analysis are compared with the measured data in Figures 8.43 and 8.44. Predictions of streamwise velocity fluctuations were obtained assuming isotropic turbulence (e.g., u2 = 2k/3). If levels of anistropy usually observed in jets were assumed

TWO-PHASE-FLOW (DISPERSED-FLOW) MODELS

675

Uo = Upo (M/S) x/d = 28.5 Data 29–60 60 NSDDM MODEL 29 NSDDM MODEL LSDDM MODEL 29 LHF MODEL 29–60 (dp(mm);LOADING RATIO) = (17;0.3)

1.0 .5 1.0 GP/GP CL

= (80;0.3) .5 1.0 = (32;0.3) .5 1.0 = (32;0.77) .5 1.0 = (32;1.4) .5 0

0

.02

.04 r/x

.06

.08

Figure 8.42 Comparison of mean particle mass fluxes in particle-laden round jets measured by Laats and Frishman (1970a, 1970b) and predictions by Shuen, Chen, and Faeth (1983b) using a nonlinear (N) and linear (L) SDDM analysis and LHF model (modified from Faeth, 1987).

(e.g., u2 = k), predictions would be 20% higher. Both predictions and measurements indicate relatively small effects of particle size and loading ratio on the gas properties illustrated in Figure 8.43. Figure 8.44 shows that predictions of mean particle velocities were in good agreement with measurements at all conditions. Predictions of particle velocity fluctuations, however, are underestimated except for the smallest particles. This behavior was attributed to effects of the particle/gas mixing and injection system, which was a worm gear particle feeder followed by a short length of pipe. Such an arrangement can introduce relatively high particle velocity fluctuations at the jet exit. Since large particles exchange a relatively small amount of momentum as they pass to the measuring station in this flow, these fluctuations would be

676

SPRAY ATOMIZATION AND COMBUSTION 1.0 dp (mm)

.8

1060 725 540 400 215 215

U/UCL

.6

UCL Up O CO L LOADING SDDM RATIO (m/s) (m/s) DATA PREDICTION 3.5 2.42 2.24 1.22 1.14 2.33

CLEAN AIR

25.3 21.3 21.3 21.3 21.3 21.3

7.8

14.5

25.3

.4

.2

0

(u′2)½/UCL

.25

.20

.15

.10

.05

0

.05

.10

.15 r/x

.20

.25

.30

Figure 8.43 Comparison of predicted mean and fluctuating gas velocities in particleladen jets (by Shuen, Chen, and Faeth, 1983b) and measured data (by Levy and Lockwood, 1981) (modified from Faeth, 1987).

preserved. In contrast, small particles interact with the flow field to a greater degree, so that effects of the injection system are damped. For lack of other information, initial particle velocity fluctuations were ignored in the computations by Shuen, Chen, and Faeth (1983b); therefore, the results are satisfactory only for small particles where effects of initial conditions are less persistent. The combusting spray experiments of Shuen, Solomon, and Faeth (1986) involved ultra-dilute conditions throughout the flow. Initially mono-disperse methanol drops were injected vertically upward at the base of a methane-fueled diffusion flame burning in still air. The methane flame has been studied

TWO-PHASE-FLOW (DISPERSED-FLOW) MODELS

677

Up/UpCL

1.0

.9

.8 LOADING UC LO RATIO (mm) (m/s) 1060 3.50 25.3 725 2.42 540 2.24 21.3 400 1.22 215 1.14 21.3 215 2.33 21.3 dp

.7

CO L

(m/s) 7.8

SDDM PREDICTION

8.1 14.5 14.5

.2

½

(u′P2 ) /uP CL

.3

DATA

Up

.1

0

0

.02

.04

.06

.08

1.0

r/x

Figure 8.44 Comparison of predicted mean and fluctuating particle velocities in particleladen jets (by Shuen, Chen, and Faeth, 1983b) and measured data (by Levy and Lockwood, 1981) (modified from Faeth, 1987).

extensively by Jeng and Faeth (1984), who established predictive methods for the flow using the conserved-scalar formalism in conjunction with the laminar-flamelet approximation. The methanol drops only perturbed this flow; therefore, their environment was well known throughout the flame. Mean and fluctuating drop velocities were measured using LDA; drop sizes were measured using flash photography; and drop number fluxes were measured using Mie scattering. Drop histories to any point in the flow vary due to effects of turbulence; therefore, drop sizes are not mono-disperse at any point other than the exit. This effect was not considered in the measurements: Drop properties simply were averaged over all sizes at each point. Computations were averaged in the same manner so that predictions and measurements could be compared. Only SDDM predictions are reported here. For this ultra-dilute flow, drop properties are controlled entirely by interphase transport rather than by mixing of the flow as a whole. Initial conditions for SDDM analysis were measured at one injector diameter from the burner exit. Mean streamwise velocity and all three components of velocity fluctuations were measured for the continuous

678

SPRAY ATOMIZATION AND COMBUSTION

phase. The measured rate of decay of k and the value u yielded ε. At the exit, f˜ = 1 and g = 0 by definition. Mean and fluctuating streamwise and radial drop velocities and the drop number flux distribution were measured; but mean radial drop velocities were small and were estimated. Depending on flame stability considerations, a fuel drop vaporizing in an oxidizing environment can have three flame configurations: 1. The envelope flame configuration, where the drop is completely surrounded by a diffusion flame that is consuming its vapor 2. A side- or wake-flame configuration, where the flame does not exist near the forward stagnation point but stabilizes at a point along the periphery or in the wake of the drop 3. The evaporation conditions, where the drop diffusion flame do not exist at all Drop transport rates are highest for the envelope flame condition, side flames are infrequently observed since they have limited ranges of stability, while wake flames and the evaporation configuration yield nearly the same drop transport rates. Thus, it is necessary to establish the presence or absence of envelope flames in order to estimate drop transport rates in combusting sprays. The possible existence of envelope flames around each drop is a controversial matter for analysis of combusting sprays. Envelope flames are clearly not possible in fuel-rich regions, but they could be present when drops interact with fuel-lean eddies. Szekely and Faeth (1983) studied drops supported at various positions in a turbulent diffusion flame to learn about this issue. They found that differences in transport rates between evaporating and combusting drops were relatively small (less than 10%–20%) until the mean fuel-equivalence ratio (the local fuel-air ratio divided by the stoichiometric fuel-air ratio) of the flame environment dropped below φ = 0.9. There are substantial uncertainties in drop-life history calculations in flames due to the wide variation of transport properties encountered. Therefore, calculations used in the SDDM analysis have been calibrated using measurements based on drops supported in the postflame region of a flat-flame burner (Shuen, Solomon, and Faeth, 1986). Measurements of mean gas-phase (time-averaged) velocity and mean (particle-averaged) drop velocity along the axis of the dilute combusting spray are illustrated in Figure 8.45. Drop velocities for both sprays tested are shown along with SDDM predictions. Predicted gas velocities are Favre-averaged; however, differences between time- and Favre-averaged mean velocities are not very large. Gas velocities are greater than drop velocities at the burner exit but decrease rapidly due to mixing with the surrounding fluid. Near the injector, drops have significant inertia, and their velocities increase only gradually due to momentum exchange with the surrounding gases. Near the tip of the flame (x/d ∼ 120), drops become small and rapidly approach gas velocities. The SDDM analysis predicts these trends reasonably well. Predicted (DDDM and SDDM analyses) and measured mean drop number fluxes (both time averages) are illustrated in Figure 8.46. The initially larger drops

TWO-PHASE-FLOW (DISPERSED-FLOW) MODELS

679

200.0 dpo (mm) DATA THEORY

Faver-averaged ~ gas velocity, Uc

100.0 80.0

Uc UPc 105 UPc 180

60.0

VELOCITY (m/s)

40.0

20.0

10.0 8.0 6.0 4.0

2.0

1.0

1

2

4

6

8 10

20 x/d

40

60 80100

200

400 600

Figure 8.45 Comparison of calculated (Favre-averaged) gas-phase and particle velocities along the axis of round, ultra-dilute, round combusting sprays with the measured data (modified from Faeth, 1987).

have wider profiles even though they are less responsive to turbulent dispersion. This wider calculated profiles for larger drops are caused by their ability to penetrate farther into the flame zone before evaporating. SDDM predictions provide the same ordering of spread rates and are in fair agreement with the measurements. DDDM predictions yield incorrect ordering of the spread rates and are not very effective, similar to the flows considered earlier. Measured (time and particle averages) and predicted (SDDM analysis) phase velocity fluctuations along the axis are illustrated in Figure 8.47. As discussed, their gas velocity fluctuations were computed using the normal levels of anisotropy found near the axis of turbulent jets, while drop velocity predictions result directly from the SDDM analysis. Predicted gas-phase velocity fluctuations are Favre-averaged while the measurements are time-averaged quantities. Favre averages underestimate timeaveraged fluctuating velocities in flames by as much as 50% near the flame tip (Faeth and Samuelsen, 1986). This effect, plus neglect of turbulence/buoyancy interactions, is probably responsible for the underestimation of gas-phase velocity fluctuations in Figure 8.47, particularly near the tip of the flame. Particle velocity fluctuations plotted in the figure show very high levels of anisotropy, much larger than predicted. While radial particle velocity fluctuations are predicted reasonably well in Figure 8.46, streamwise drop velocity fluctuations are substantially

680

SPRAY ATOMIZATION AND COMBUSTION 1.0 0.8

DATA

dpo = 105 mm

105

0.6

180

dpo = 180 mm

THEORY

0.4

SDDM DDDM

0.2 x/d = 20

0.0 1.0

dpo = (mm)

dpo = 105 mm

0.6

.

.

n′ / n′c

0.8

0.4

dpo = 180 mm

0.2 x/d = 50

0.0 1.0 0.8

dpo = 180 mm

0.6 0.4 0.2 0.0 0.00

x/d = 100 0.08

0.16 r/x

Figure 8.46 Comparison of calculated (Favre-averaged) drop number flux distributions in round, ultra-dilute, round combusting sprays with the measured data (modified from Faeth, 1987).

underestimated, as shown in Figure 8.47, probably due to the assumption of isotropic turbulence when eddy properties are selected for SDDM analysis. Use of Favre-averaged velocities in the simulation also causes underestimation of time-averaged velocity fluctuations, as noted earlier. Near the burner exit, drop velocity fluctuations are small in comparison to the gas phase, due to drop inertia. At the end of drop lifetime (x/d ca.90 − 120), however, the remaining small drops can respond rapidly and approach flame properties. Apte, Gorokhorski, and Moin (2003) utilized the large-eddy simulation (LES) technique to predict reacting multiphase flows in practical combustors involving complex physical phenomena of turbulent mixing and combustion dynamics.

TWO-PHASE-FLOW (DISPERSED-FLOW) MODELS

681

0.3 DATA THEORY ∼ u′/uc ∼ v′/uc

0.2

0.1

0.0 0.3 DATA THEORY ∼ u′p/upc ∼ v′p/upc

0.2

dpo = 105 mm

0.1

0.0 0.3 DATA THEORY ∼ u′p/upc ∼ v′p/upc

0.2

dpo = 180 mm

0.1

0.0 1

4

10

40

100

400

x/d

Figure 8.47 Fluctuating phase velocities along the axis of round, ultra-dilute round combusting sprays (modified from Shuen, Solomon, and Faeth, 1986).

682

SPRAY ATOMIZATION AND COMBUSTION

They developed a computational tool based on particle-tracking schemes capable of performing high-fidelity multiphase flow simulations with models to capture liquid-sheet breakup, droplet evaporation, droplet deformation, and drag. An Eulerian low-Mach number formulation on arbitrary shaped unstructured grids was used to compute the gaseous phase. The dispersed droplet phase was solved in a Lagrangian framework by tracking a large number of particles on the unstructured grid. The interphase mass, momentum, and energy transport were modeled using two-way coupling of gas-phase solution at grid points and particles. They compared their predictions with the spray evaporation data of Sommerfeld and Qiu (1998). In their experimental setup, heated air was supplied through an annulus with the hollow-cone spray nozzle mounted in the center of the injector. Isopropyl alcohol was used as a liquid because of its high evaporation rate. Measurements were taken for different flow conditions, by varying air flow rate, air temperature, and liquid flow rate. Phase-Doppler anemometry (PDA) was employed to obtain the spatial variation of the droplet size spectrum in the flow field and to measure droplet size-velocity correlations. From these local measurements, profiles of droplet mean velocities, rms velocity fluctuations, and droplet mean diameters were obtained by averaging over all droplet size classes. In the immediate downstream region (at x = 0 and 25 mm), the profiles of the droplet mass flux show the two peaks associated with a hollow-cone spray nozzle, and the spray is spreading according to the cone angle of 60◦ . Due to the recirculation region downstream of the center body, the droplet mass flux becomes negative in the core of the spray at x = 25 mm. Farther downstream, the spreading of the spray is hindered because of the entrainment of the annular air jet, and the maximum of the droplet mass flux moves toward the centerline. Downstream of x = 50 mm, the droplet mass flux is continuously decreasing, and at x = 400 mm, most of the liquid has evaporated. Figure 8.48 shows the comparison of the calculated liquid phase statistics with the experimental data. The droplet size distributions show a typical conical hollow-cone spray near the injector with large-size droplets accumulating on the outer edge of the spray and small droplets in the center core region. These droplets evaporate and their mean diameter decreases farther downstream. Their predicted mean and rms values of the axial velocity mean axial liquid mass flux, and mean and rms of droplet diameter at different axial locations are in good agreement with the experimental data of Sommerfeld and Qiu (1998). In their simulation, the initial temperature of the liquid was 313 K, which is below the boiling point (355 K) of isopropyl alcohol. Since the temperature of the surrounding hot air is only 60 K above the initial temperature of the liquid, there were no reactions involved in this spray. 8.8.6

Dense Sprays

8.8.6.1 Introduction Some aspects of dense sprays (or called dense dispersed jets) considered thus far are based largely on observations near the boundaries of dilute dispersed jets. In

683

TWO-PHASE-FLOW (DISPERSED-FLOW) MODELS

50 R, mm

25 0 –25 –50 –75 0 10 20 Axial Velocity X/R = 0.786 75

80 60 40 20 0 –20 –40 –60 –80

50 R, mm

25 0 –25 –50 –75 0 500 1000 Mass Flux, kg/m2 -s

0 500 1000 Mass Flux, kg/m2 -s

X/R = 0.786 50

R, mm

25

80 60 40 20 0 –20 –40 –60 –80

80 60 40 20 0 –20 –40 –60 –80

0 200 Mass Flux, kg/m2 -s

X/R = 1.56

5 10 15 Axial Velocity X/R = 6.25

60

60

40

40

40

40

20

20

20

20

0

0

0

–20

–20

–20

–40

–40

–40

–40

50 Diameter, mm

–60 0

25 50 75 Diameter, mm

50 25 0 –25 –50 –75

–60 0

0 20 40 60 Mass Flux, kg/m2 -s X/R = 12.5 50

0

–60 0

0 10 20 Axial Velocity X/R = 12.5 75

X/R = 9.375

60

rms up

–75

X/R = 6.25

–20

50 100 Diameter, mm

–50

0 50 Mass Flux, kg/m2 -s

0

0

0 –25

80 60 40 20 0 –20 –40 –60 –80

60

mean up

25

0 10 20 Axial Velocity X/R = 9.375

–25 –50

50

0 100 Mass Flux, kg/m2 -s

X/R = 3.125

X/R = 12.5 75

0 –20 –40 –60 –80 0

0 10 20 Axial Velocity X/R = 3.125

0 10 20 Axial Velocity X/R = 1.56

X/R = 9.375 80 60 40 20

80 60 40 20 0 –20 –40 –60 –80

80 60 40 20 0 –20 –40 –60 –80

80 60 40 20 0 –20 –40 –60 –80

X/R = 6.25

X/R = 3.125

X/R = 1.56

X/R = 0.786 75

50 Diameter, mm

–60 0

25

mean dp

0 –25

rms dp

–50 20 40 Diameter, mm

0

40 20 Diameter, mm

Figure 8.48 Comparison of simulation results with spray evaporation experimental data of Sommerfeld and Qiu (1998) with hot air at T = 373 K injected through the annulus and liquid isopropyl alcohol injected from a nozzle mounted at the center of the coaxial injector: mean and rms of droplet axial velocity, axial mass flux of the liquid fuel, mean and rms of droplet diameter (modified from Apte et al., 2004).

this section, we consider additional information obtained by direct observation and analysis of dense dispersed jets. Aspects of dense dispersed flows are treated in more detail in reviews by Griffen and Muraszew (1953); Harrje and Reardon (1972); Lefebvre (1980, 1986); Elkotb (1982); Bracco (1983, 1985); Drew (1983); Faeth (1983); Sirignano (1983); Chigier and Reitz (1996); and S. P. Lin (1996). Some of these articles are published in the two volumes of AIAA Progress Series edited by Kuo (1996a,b). The scope of the discussion here is limited primarily to dense sprays produced by round pressure-atomizing injectors. Bracco and associates (Bracco, 1985; Reitz and Bracco, 1982; Wu, Santavicca, and Bracco, 1984) have studied this flow configuration; therefore, a good portion of the material in this section has been drawn from their work. Naturally, this omits direct consideration of the wide variety of pressure and air-atomizing injectors encountered in practice. However, many fundamental aspects of the dense region of sprays near the injector are similar for all injection systems. In this text, we first treat general background concerning dense-spray processes. This involves breakup regimes, flow patterns, breakup processes, and collisions between drops.

684

SPRAY ATOMIZATION AND COMBUSTION Droplet collisions Evaporation/ droplet coalescence mixture formation areodynamic interaction

Needle

Wall impingement

Blind hole

Injection hole Primary breakup

Primary breakup Secondary breakup Jet penetration length S

Cavitation

Turbulence

Flow inside nozzle

Injection spray

Figure 8.49 Spray formation in full-cone diesel injector and various physical processes leading to jet breakup (modified from Baumgarten, 2006).

8.8.6.2

Background

A schematic diagram of a full-cone diesel injector is shown in Figure 8.49. The diesel injectors are pressure-atomized injectors with injection pressures up to 200 MPa. The liquid jet velocity at the injector exit could be higher than 500 m/s. The jet breakup occurs via atomization mechanism in this case. Once the jet leaves the injector, it starts to break up into a conical spray. There are two stages of breakup: primary and secondary. Primary breakup results in large ligaments and droplets that form a dense spray near the nozzle. In high-pressure injection, cavitation and turbulence (generated inside the injection holes) are the major mechanisms for primary breakup (see inset in figure 8.49). Secondary breakup occurs after the primary breakup region resulting in subsequent breakup of existing droplets into smaller droplets and ligaments. The aerodynamic forces due to relative velocity between droplets and surroundings are the major reasons for breakup in this region. Breakup Regimes for Steady Injection of Liquid into Stagnant Gas One of the first things that must be determined for a multiphase flow is the breakup regimes. Atomizer flows involve breakup regimes of liquid jets as well as various flow patterns. Figure 8.50 shows four main breakup regimes of liquid jet by Chigier and Reitz (1996). These include: (a) Rayleigh breakup, (b) first wind-induced breakup, (c) second wind-induced breakup, and (d) atomization regimes. Rayleigh breakup is caused by surface tension effects. It occurs many jet diameters from the injector exit and yields a stream of drops having diameters larger than the jet diameter. First and second wind-induced breakups are due to instabilities caused by the relative motion of the gas and liquid, stabilized to some extent by surface tension. First wind-induced breakup occurs many diameters from the jet exit and yields drop diameters ranging from the jet diameter to about 1 order of magnitude smaller. In the second wind-induced regime, drop sizes are smaller than the jet

TWO-PHASE-FLOW (DISPERSED-FLOW) MODELS

685

(b) (c)

(d)

(a)

Figure 8.50 Four jet breakup regimes: (a) Rayleigh regime; (b) first wind-induced regime; (c) second wind-induced regime; (d) atomization regime (injection into stagnant gas) (modified from Chigier and Reitz, 1996).

diameter, and the breakup starts at short distance downstream of the nozzle exit. The atomization regime is characterized by jet breakup immediately at the jet exit. Flow in the atomization regime yields drops whose average diameter is much smaller than the jet diameter. Only the wind-induced and atomization breakup regimes lead to a dense spray region, unless an array of injectors is used. A schematic diagram describing the characteristic behavior of the four breakup regimes is shown in Figure 8.51. The dependency of the maximum liquid intact-core length with the liquid injection velocity is shown in Figure 8.52a. On this plot, the four breakup regimes are also indicated. As shown in Figure 8.52b, the four regimes can also be represented on a map constructed by the Reynolds number and Ohnesorge number. Ranz (1958) prescribes criteria for these breakup regimes for round liquid jets based on the Weber numbers of the liquid or gas of the flow defined by Equation 8.1. The criteria for wind-induced breakup are Wel ≥ 8, 0.4 < Weg < 13 while the criteria for the atomization regime are Wel > 8, Weg > 13. These assessments are only approximate; for example, Miesse (1955) found transition to the atomization regime at Weg > 40. Identification of a flow regime is also the subjective opinion of an individual investigator about the appearance of a physical process. Thus, some investigators (e.g., Chigier and Reitz, 1996) identify two wind-induced breakup regimes, while others are satisfied with only one. Finally, factors other than Weg and Wel have been found to influence breakup regime boundaries (e.g., injector Reynolds numbers, ambient flow [cross-flow] properties, injector lengthto-diameter ratio, density ratio of the phases and cavitation within the injector, etc.)

686

SPRAY ATOMIZATION AND COMBUSTION

L1

L1

L1 L2

L2

(a) Rayleigh Regime

(b) First Wind-Induced Regime

L2

L2

(c) Second Wind-Induced Regime

(d) Atomization Regime

L1 = undisturbed lenght of the liquid jet L2 = Maximum lenght of the liquid core-jet

Ohnesorge number, Oh

Rayleigh

Torda (1973 air at 30 atm)

ed

uc

10−1

ed

ind

uc

nd

ind

wi

Rayleigh breakup

Miesse (1955)

nd

nd

100

wi

wi

Atomization

nd

First Wind induced

rst

Rayleigh

Second Wind induced

Ohnesorge (1936)

co

Turbulent flow

Fi

101

Transition range

Laminar flow

Se

Maximum liquid intact core length, L2

Figure 8.51 Schematic diagrams for describing the characteristic behavior of jet breakup in four different break-up regimes for steady injection into stagnant air (modified from Leipertz, 2005; based on the original work of Fath, Munch, and Leipertz, 1997, and Badock, Worth, and Tropea 1999).

atomization Diesel injection

10−2 10−3 100

Liquid jet velocity (a)

101

102 103 Red

104

105

jct

(b)

Figure 8.52 (a) Relationship of maximum liquid intact-core length with the injection velocity (b) breakup regimes on an Oh-Red map for a steady injection of liquid jet into stagnant air (modified from Fath, Munch, and Leipertz, 1997).

The process illustrated in Figure 8.53 begins with an all-liquid flow in a passage and finally evolves to a dilute spray in the downstream region after the flow development in the dense spray region. As the liquid leaves the injector, it develops into a churn flow pattern. This region includes the all-liquid core and other irregularly shaped liquid elements near the axis of the flow. The term “churn flow” is drawn from usage for multiphase flow in tubes, where it refers to a flow pattern involving large irregular volumes of the ultimately dispersed flow near the axis. In this sense, the churn flow descriptor is appropriate; however,

687

TWO-PHASE-FLOW (DISPERSED-FLOW) MODELS Dense spray

Dilute spray

Liquid flow

Injector

Liquid core

Figure 8.53 Sketch of the near-injector region of a pressure-atomized spray for atomization breakup conditions (modified from Faeth, 1987).

criteria for flow regime transitions to and from the churn flow pattern of tube flow are naturally different. O’Rourke and Bracco (1980) describe churn flow for sprays as a region where the volume fraction of liquid is greater than the gas, so that the liquid cannot be considered to be dispersed in the gas phase. Rather, large liquid elements are present in churn flow, including the all-liquid core. The momentum exchange capabilities of the gas phase in this region is relatively limited, since gas density and its volume fraction are small; therefore, relative velocities between the phases are small in normal situations where streamwise pressure gradients are negligible. Thus, the large liquid elements are relatively stable in this region. As the void fraction continues to increase by mixing, the large liquid elements pass into slower-moving gas where they become unstable and break up into ligaments and drops. This signals the onset of the dense-spray flow pattern. In the dense-spray region, liquid volume fractions are relatively high; there is a wide diversity of drop sizes, shapes, and velocities; and effects of collisions are probably significant. As the void fraction continues to increase with axial distance, potential effects of collisions become small, and the liquid elements become small enough to be approximated as spheres. We then enter into the dilute-spray region. Observations of pressure-atomized sprays in the atomization regime do not reveal the features of the churn and dense-spray flow patterns, since these regions are obscured by drops present in the dilute region. However, other evidence has been obtained suggesting the presence of a contiguous liquid core extending some distance from the injector. Hiroyasu, Shimizu, and Arai (1982) and Arai, Shimizu, and Hiroyasu (1985) measured the electrical resistance between the injector exit and a screen across the flow that could be traversed in the streamwise direction. It was assumed that the current carried by an unconnected region of the flow would be small, yielding the length of the contiguous liquid core. Chehroudi, et al. (1985) repeated the measurements using an extrapolation procedure to find the end of the contiguous core. A correlation of both these results yields the next expression: 1/2 Llc /djet = Cc ρl /ρg

(8.135)

688

SPRAY ATOMIZATION AND COMBUSTION

where Llc is the length of the contiguous liquid core and Cc is a constant in the range of 7 to 16, the lower value being due to the more recent measurements. Hiroyasu, Arai, and Shimizu (1991) proposed the next expression for the breakup length of a liquid jet in a diesel engine injector for a broad-spray regime: 0.05 Lpa 0.13 ρl 0.5 Llc rc pamb = 7 1 + 0.4 2 djet djet djet ρg ρl Vjet

(8.136)

where rc = the injector corner radius, Lpa = the injector passage length, and Vjet = the jet exit velocity Breakup Regimes for Steady Injection of Liquid into a Coflowing Gas Coflowing gas is frequently used in the twin-fluid injectors to enhance the atomization process and maintain the quality of atomization over a wide range of liquid flow rates. High gas velocities are generated by high-pressure gas passing through annular space surrounding the liquid jet at the center. The coflowing gas velocity transfers momentum to the liquid interface. The relative velocity between gas and liquid flows causes stretching, destabilization, and flapping of the liquid jet. Chigier and Reitz (1996) have performed a comprehensive review of several coaxial gas-liquid jets operating under a wide range of gas and liquid mass flow rates (Redjet ∼ 200−20,000; Weg ∼ 0.001−600). Based on their study, the coaxial jet breakup can be categorized into three major modes: (1) Rayleigh type, (2) membrane type, and (3) fiber type. Some representative images of these three modes and their descriptions are shown in Figures 8.54, 8.55, and 8.56, respectively. The Rayleigh-type jet breakup mode can be divided into two subgroups: axisymmetric mode (Weg < 15) and nonaxisymmetric mode (15 < Weg < 25) as shown in Figure 8.54. All three modes can be divided into two submodes: (a) pulsating jet disruption as the normal submode of atomization and (b) superpulsating jet disruptions that are connected to an extremely high periodic change between low- and high-density regions in the spray (150 < Weg < 500). Various coaxial jet breakup modes are summarized in Figure 8.57 as a function of liquid Reynolds number and aerodynamic Weber number. The liquid Reynolds number is defined as:

Redjet =

ρl vjet,ldjet μl

(8.137)

The aerodynamic Weber number is defined as: 2 ρg djet vjet,l − vjet,g Weg = 2σs

(8.138)

689

TWO-PHASE-FLOW (DISPERSED-FLOW) MODELS

Rayleigh-Type Breakup Mode • Mean drop diameter~Order of jet diameter • Max drop diameter~2×Jet diameter • Jet disintegrates into drops without formation of membrane or liquid fiber ligaments Non-axisymmetric Mode (15 < Weg < 30) Axisymmetric Mode (Weg < 15)

ul (m/s) ug (m/s)

4.35 32.4

3.37 32.4

1.02 22.9

0.55 22.9

5.0 45.8

0.643 32.4

0.225 32.4

Figure 8.54 Images of coaxial air and liquid jet breakup by the Rayleigh-type breakup mode (modified from Chigier and Reitz, 1996).

As shown in Figure 8.57, the fiber-type breakup mode occurs at very high gasphase. If the ratio of Reynolds- number to the square root of Weber number is lower than 100 (i.e., Redjet / Weg < 100), then the superpulsating submode dominates the jet breakup process. This breakup mode is desirable to achieve a high degree of atomization with fine droplets. For example, the atomization of Liquid Oxygen (LOX) in the space shuttle main engine (SSME) occurs by fiber-type liquid jet breakup mode. The intact-core length of coaxial jets was studied by Eroglu and Chigier (1991) and produced the next correlation: 0.6 Llc = 0.5 Redjet djet

Weg 2

−0.4 (8.139)

Woodward et al. (1996) (see Kuo, 1996, chap. 8) utilizied real-time X-ray radiography to study the jet breakup processes from a coaxial injector with a similar geometry to a single SSME injector element, with liquid in the center and gas in the annular space. The exit area ratio of the gas to liquid flow is 2.5. They observed continuous core region from the X-ray image. They have developed the next correlations: 0.68 −0.22/Z Llc = 0.0095 Redjet Weg djet

ρl ρg

0.36/Z (8.140)

690

SPRAY ATOMIZATION AND COMBUSTION Membrane-Type Ligaments Breakup Mode

• • • •

At higher flow rates, the round jet develops into a thin liquid sheet (i.e., membrane), very similar to thin liquid sheet. Kelvin-Helmholtz waves are formed, they break up into drops. Jet disintegration is completed within one or two wavelengths. Drop diameters (inclusing the max drop diameter) are considerably smaller than the jet diameter. (25 < Weg < 70)

ul (m/s) ug (m/s)

0.946 45.8

1.4 45.8

Figure 8.55 Images of coaxial air and liquid jet breakup by the membrane-type breakup mode (modified from Chigier and Reitz, 1996).

where djet is the diameter of the liquid jet and Z represents the ratio of acoustic velocity of the gaseous fuel simulant to that of nitrogen, that is, - - γR Z≡ / γR (8.141) gaseous fuel

N2

An example of numerical simulation of the dense spray region conducted by Apte, Gorokhovski, and Moin (2003) for simulating the experiments by Sommerfeld and Qiu (1998) is shown in Figure 8.58. The core breakup region can also be seen from the calculated solution. 8.8.6.3 Jet Breakup Models Multiple atomization models for diesel fuel sprays currently are widely used for breakup simulation. The breakup models can be divided into two groups:

1. Primary breakup takes place in the region near the injector. a. Cavitation and turbulence-based model b. Wave breakup model (or Kelvin-Helmholtz [KH] breakup model)

TWO-PHASE-FLOW (DISPERSED-FLOW) MODELS

691

c. Blob-injection method d. Sheet atomization model for hollow-cone sprays 2. Secondary breakup occurs farther downstream in the spray due to aerodynamic forces. a. Reitz-Diwakar (RD) model b. Taylor-analogy breakup (TAB) model c. Kelvin-Helmholtz (KH) breakup model d. Rayleigh-Taylor (RT) breakup model The classic breakup models, such as TAB, RD, and wave breakup model do not distinguish between the two processes (Dukowicz, 1979). The parameters of these models are usually tuned to match experimental data farther downstream Fiber-Type Breakup Mode • • • • • •

Even higher airflow rates than the Rayleigh-type mode or membrane-type mode of breakup. Fiber-shaped ligaments are formed, and they peel off the jet. Liquid core is accelerated and develops waves. Breakup may occur via nonaxisymmetric Rayleigh mechanism or by breakup into ligaments. Drop diameter is an order of magnitude smaller than the jet diameter. Superpulsating model occurs when a low mass flow rate is associated with a very high velocity of the atomizing gas. (100 < Weg < 500)

Pulsating Disintegration Submode [normal submode, Rel /(Weg) 0.5 > 100]

Superpulsating Disintegration Submode [Extremely high periodic change between low- and high-density regions in the spray]

Figure 8.56 Images of coaxial air and liquid jet breakup by the fiber-type breakup mode (modified from Chigier and Reitz, 1996).

692

SPRAY ATOMIZATION AND COMBUSTION

Liquid Reynolds Number

105

Axisymmetric Rayleigh-type disintegration Nonaxisymmetric Rayleigh-type disintegration Membrane-type disintegration Membrane-type region Fiber-type disintegration Fiber-type region Superpulsating submode

104

103 Rayleigh-type region Superpulsating region

102 10−3

10−2

10−1 100 101 Aerodynamic Weber Number

102

103

Figure 8.57 Map of various regions of coaxial air and liquid jet breakup modes (modified from Chigier and Reitz, 1996).

Transvers Coordinate, y

YF

1.0E-03 1.6E-03 2.6E-03 4.1E-03 6.6E-03 1.0E-02 1.7E-02 2.7E-02 4.3E-02 6.9E-02

2 1

Hot Air

0

Liquid Jet

−1

Hot Air

−2 0

5

10

Longitudinal Coordinate, x

Figure 8.58 Snapshots of droplets superposed on contours of instantaneous fuel mass fraction. Conditions correspond to the gas-liquid coflowing jet breakup experiment by Sommerfeld and Qiu (1998) (modified from Apte et al., 2003).

in the region of the secondary breakup. Originally these parameters are supposed to depend only on nozzle geometry; in reality they also account for numerical effects. Other models, such as KH and RT, treat the primary breakup region separately (Dukowicz, 1979). Hence, in principle they offer the possibility to simulate both breakup processes independently. The correct values for the additional set of parameters, however, are not easy to determine due to the lack of experimental data for the primary breakup region. Despite the sometimes tedious tuning of these model parameters, the use of breakup models is generally advantageous

TWO-PHASE-FLOW (DISPERSED-FLOW) MODELS

693

compared to the initialization of measured droplet distributions at the nozzle orifice. In the first approach, the droplets are simply initialized with a diameter equal to the nozzle orifice (blob injection), and the droplet size distribution automatically evolves from the subsequent breakup processes. The latter approach by considering detailed breakup process gives satisfying results only as long as injection pressure and droplet Weber numbers are low. Nozzle geometry and injection pressure influence the development of cavitation regions and turbulence inside the nozzle, which affect the atomization of high-speed jets (see Figure 8.49). These effects are captured by cavitation and turbulence-based phenomenological models. Cavitation bubbles and turbulence can cause numerous bubbles to be present in the dense core (as shown in Figure 8.59). Readers can refer to Stiesch (2003, chap. 5) for more information on this model. Taylor Analogy Breakup (TAB) Model The original TAB model was developed by O’Rourke and Amsden (1987), based on G. I. Taylor’s model (1963). This model compares an oscillating-distorting droplet to a spring-mass system where the aerodynamic force on the droplet, the liquid surface tension force, and the liquid viscosity force are analogous, respectively, to the external force acting on a mass, the restoring force of a spring, and the damping force. The distortion parameter y that represents the departure of the drop from its spherical shape divided by the drop radius is calculated by solving a spring-mass equation of the form: ρl d 2 y 5 dy 2 8 + = + (8.142) ρg dt2 Redjet ,l dt Weg 3 Dense Spray Regime cavitation bubbles surface and gas bubbles turbulence eddies waves

Dilute Spray Regime droplets

flow change caused by nozzle geometry

intact core disturbances caused by cavitation bubbles and turbulence eddies

dense core

ligaments

clusters

spray breakup forced by aerodynamic interactions

Figure 8.59 Physical processes associated with spray breakup models in the atomization regime (modified from Fath, Munch, and Leipertz, 1997 or Badock, Wirth, Fath, and Leipertz, 1999).

694

SPRAY ATOMIZATION AND COMBUSTION

where ρt is the liquid density and ρg is the gas density. If the value of y exceeds unity, the droplet breaks up into smaller droplets with radius specified in given distributions. A modified version of TAB model has been used by Ibrahim, Yang, and Przekwas (1993) for prediction of drop breakup and distortion. The predictions from this modified TAB model agreed with the experimental drop breakup results of Krzeczkowski (1980). The TAB model has also been used to account for the effect of drop distortion on drop drag coefficient by A. B. Liu, Mather, and Reitz (1993) and Kong and Reitz (1996). Wave Breakup Model The wave breakup model considers the unstable growth of Kelvin-Helmholtz (KH) waves at a liquid-gas interface due to so-called K-H instabilities, which occur when there is an intense shear motion of two fluids flowing alongside each other. This phenomenon and the resulting surface waves are also shown in the primary breakup region in Figure 8.60. The wave breakup model is also known as the Kelvin-Helmholtz model. Significant contributions for the development of this model have been made by Reitz and Diwakar (1986). A blob with surface waves is shown in Figures 8.60 and 8.61. The wave breakup Λ

η = η0 . exp[Ωt]

2B0Λ r

2a

Liquid blob with surface waves x

Gas

Figure 8.60 Schematic growth of surface perturbations in the wave breakup model (modified from Rietz and Diwakar, 1987). Wave breakup 170μm

λ

h

50 μm

Figure 8.61 Photographs showing surface waves and breakup induced by them (modified from Leipertz, 2005).

TWO-PHASE-FLOW (DISPERSED-FLOW) MODELS

695

model includes the effects of liquid inertia, surface tension, and viscous and aerodynamic forces on liquid jets. The theory offers a reasonably complete description of the breakup mechanisms of low-speed liquid jets. For high-speed jets, however, the initial state of the jet at the nozzle exit appears to be more important and less understood, and the linear stability analysis involved in the “wave” model may not be sufficient. The KH instability breakup model assumes a column of liquid of infinite extent in the direction issuing from a circular orifice into a stationary incompressible gas. The stability of the liquid surface to linear perturbations examined by the “wave” breakup theory leads to a dispersion equation. From the solution of a general dispersion equation, maximum growth rate of an initial perturbation of negligible amplitude and its wave number of the surface wave are given as: 3 0.34 + 0.38 · Weg 1.5 ρl a = (8.143a) KH · σs (1 + Oh) 1 + 1.4 · TP0.6 1 + 0.45Oh0.5 1 + 0.4TP0.7 KH (8.143b) = 9.02 0.6 a 1 + 0.87 We1.67 g where Oh =

Wel 0.5 , Taylor parameter, TP = Oh × Wel 0.5 Rel

(8.144)

The fastest-growing wave with the highest frequency KH /2π would be the most probable wave on the liquid surface. Liquid breakup was modeled by claiming that new drops of radii rchild are formed from a parent drop or blob of radius a. This definition contains a constant B0 , which is equal to 0.61. ⎧ B0 KH ≤ a B0 KH ⎪ ⎪ ⎪ ⎪ ⎡ ⎨ 1/3 3π a 2 vg (8.145) rchild = 2 ⎢ KH ⎪ B > a, one-time only min ⎪ ⎣ 0 KH 1/3 ⎪ ⎪ 3a 2 KH ⎩ 4

In Equation 8.145, it is assumed that small droplets are formed with droplet sizes proportional to the wavelength of the fastest-growing or most probable unstable surface wave; it is also assumed that the jet disturbance has frequency KH /2π (i.e., a droplet is formed each period equal to 2π/KH ). Blob Injection Model Liquid injection was simulated using the “blob” injection method of Reitz and Diwakar (1987), in which the liquid is introduced at the nozzle exit as blobs that have a characteristic diameter equal to the nozzle exit diameter (see Figure 8.62). The frequency of the addition of new blobs is related to the fuel injection rate by mass conservation and also by assuming constant density of the liquid fuel. Immediately after the injection, the K-H instabilities described by the wave model start to grow on the blob surface, such that small

696

SPRAY ATOMIZATION AND COMBUSTION

Injection nozzle

Blobs

Droplets

Liquid fuel

Figure 8.62 Schematic of blob injection model developed by Reitz and Diwakar (1987) (modified from Jiang et al., 2010).

secondary droplets are sheared off the blob surface. The blob injection method does not make a distinction between atomization and drop breakup within the dense spray near the nozzle. The rate of change of drop radius in a parent blob that contains N0 drops due to breakup was described with the next submodel. The initial radius at the initial time is a(t0 ) = a0 . The drop breakup time τ correlation is described as shown. (a − rchild ) da =− for rchild ≤ a dt τ B1 a τKH = 3.726 KH KH

(8.146) (8.147)

New blobs of product drops (once enough drops have accumulated) are added to the computations for further breakup. The condition for “enough drops” is attained if the mass of the liquid to be removed from the parent blob is at least 3% of the average mass of the injected blob and the number of drops in the newly created blob is at least the same number of drops in the parent blob. This arrangement is satisfactory because the drops are smaller in the new product blob. Mass is conserved in the parent blob as the size is diminished, but the original drop number in the parent blob is restored once a new product blob is formed. For the second part of the drop radius definition, the waves have wavelengths that approach the size of the blob or values which are longer than the characteristic size of the blob. The parent blob is completely replaced by a new blob containing drops with the size given by the drop radius definition. The appropriate number of drops is included based on mass conservation. This occurs after time τ . This procedure is allowed to occur only once for each injected parcel to prevent unrealistic increases in size. The new product blob conditions and velocity magnitude remain the same as the parent blob conditions. The value of B1 in the time constant in Equation 8.147 was chosen based on best-fit plots at high-speed inviscid liquid breakup conditions. The data were at high relative velocity flow conditions. B1 is regarded as an adjustable model constant. The value can change at other jet breakup regimes because certain quantities have a different influence, such as initial magnitude disturbance level, flow turbulence,

TWO-PHASE-FLOW (DISPERSED-FLOW) MODELS

697

and nozzle design effects. There is some uncertainty regarding the exact value of the breakup time constants used in these formulations. Several models use various values of B1 ranging between 1.73 and 20. Spray modeling has been simulated by a computer code called KIVA. The word “kiva” refers to a Pueblo ceremonial chamber that is usually round and underground and entered from above by means of a ladder through the roof. The name of the code is an analogy for a combustion chamber or engine cylinder. KIVA is a multidimensional engine combustion computer code that computes interactions between spray drops and gas and accounts for the phenomena of drop breakup, drop collision and coalescence, and the effect of drops on the gas turbulence. It solves the three-dimensional equations of transient chemically reactive fluid dynamics and those for the dynamics of an evaporating liquid spray. Combined KH and RT Secondary Breakup Model It has been observed by various researchers that the wave or KH breakup model is not suffiecient to predict the secondary breakup of liquid jet. Secondary breakup occurs farther from the injector tip, after a certain length called the jet breakup length. In order to address this issue, the KH model has been combined with RT breakup model based on the popular Rayleigh-Taylor linear instability theory. Many researchers have tried to develop hybrid models that include two breakup mechanisms for the secondary breakup. Patterson and Reitz (1998) suggested that the reason for droplet breakup is the result of competition between the KH instability and RT instability, and modified the KIVA code with the KH-RT hybrid model. Hwang, Liu, and Reitz (1996) performed double-pulsed photography of the catastrophic breakup regime in a diesel spray. They found that at very high air velocities, the drop undergoes intense deceleration due to the drag forces. These conditions favor the development of RT instabilities, which can develop if the fluid acceleration has an opposite direction to the density gradient. By using double-pulse photography, they were able to determine the acceleration of the drops and estimate the wavelengths and growth rates predicted by the RT theory for comparison. The comparison between predicted and measured wavelengths from their photographs indicated that the longer-wavelength waves seen in the catastrophic regime were RT waves. The wavelengths predicted by the KH instability tend to be much shorter. RT instabilities are found when a liquid-gas interface (i.e., a density discontinuity) is accelerated toward the low-density gas (C. F. Taylor, 1960). This problem is analyzed by considering the unstable growth of the infinitesimal surface disturbances on an interface of incompressible fluids. Solutions to the linearized hydrodynamic equations yield a dispersion equation, which can be solved to give the wavelength and growth rate of the most unstable surface waves:

RT =

3/2 1/2 −gt ρl − ρg 2 √ ρl + ρg 3 3σs

(8.148)

698

SPRAY ATOMIZATION AND COMBUSTION

In Equation 8.148, gt is given by the next equation: gt = g · nj + acc · nj

(8.149)

where g = gravitational acceleration acc = droplet acceleration nj = unit vector tangent to the droplet trajectory Note that the droplet actually decelerate due to drag forces and, therefore, acc is in opposite direction of nj : 2πCRT κRT 1/2 −gt ρl − ρg = 3σs

RT = κRT

(8.150)

(8.151)

In Equations 8.150 and 8.151, κRT is the wave number and CRT is an adjustable constant, which is determined by the comparison between measured results. CRT is also called the breakup time constant. The wavelength RT is compared to the distorted droplet’s diameter, and if the wavelength is smaller than the droplet diameter, RT waves are assumed to be growing on the surface of the droplet. The amount of time that the waves grow is tracked and compared to a breakup time, which is assumed to be: Cτ τRT = (8.152) RT where Cτ is determined to be 1.0 in the region of breakup length and increases up to 9.0 in the secondary breakup region. After the breakup time has elapsed, the parent droplet is broken up into a collection of smaller drops that have radii: rchild =

πCRT κRT

(8.153)

Xin et al. (1998) used the value of 0.3 for CRT . More details of the model are described by Su et al. (1996). Close to the injector nozzle where the droplet velocities are highest, the KH breakup is usually the governing mechanism; the RT breakup becomes more dominant or both mechanisms are important farther downstream. A schematic diagram of the combined KH-RT model for secondary breakup (in the catastrophic breakup regime) is shown in Figure 8.63. As shown, the aerodynamic force on the drop flattens the drop into the shape of a liquid sheet. The decelerating sheet breaks into large-scale fragments by means of RT instability. Apparently, much shorter wavelength KH waves originate at the edge of these fragments, and these waves are stretched to produce ligaments, which then break up into micrometer size drops (Liu and Reitz, 1993).

TWO-PHASE-FLOW (DISPERSED-FLOW) MODELS

699

Air jet RT waves KH waves

λ

Drops

Λ

Product drops

Figure 8.63 Schematic diagram of breakup mechanism in catastrophic breakup regime involving both Rayleigh-Taylor (RT) and Kelvin-Helmholtz (KH) waves (modified from Hwang, Liu, and Reitz, 1996).

8.8.6.4

Impinging Jet Atomization

Impinging jets have been used in many applications to atomize the liquid fuels or liquid oxidizers, especially for rocket engines. A variety of impinging jet injectors have been designed and utilized, including triplets, like doublets, and unlike doublets. The breakup processes shown in Figure 8.64 occur when the two liquid jets impinge at angle 2θ with the same velocity. Przekwas (1996) indicated that for like doublets, the breakup of the liquid sheet formed by the impact of two equal cylindrical jets is due to cardioid waves at low Weber number defined as (Wel = ρl U2 R/σs , where R = jet radius) according to the work of J.C. Huang (1970). At high Wel , the sheet breakup is by the growth of K-H instability waves. Huang reported that the transition between the low- and highWe breakup regimes lies approximately between 500 < Wel < 2000. The initial sheet thickness hi as a function of angular position φ is given in this form: hi =

βd R sin θ βd (1−φ/π) e e βd − 1

(8.154)

where θ is half of the included impingement angle (see Figure 8.64b). The wave decay factor βd in Equation 8.154 can be evaluated from the next equation: cos θ =

e βd + 1 1 eβd − 1 1 + (π/βd )2

(8.155)

Ibrahim and Przekwas (1991) proposed that the thickness hi should decrease as 1/r, with r being the distance from the point of impact. The thickness h of the attenuating sheet at any radial position r and angle φ can be calculated by: h R/ sin θ = hi r

(8.156)

700

SPRAY ATOMIZATION AND COMBUSTION

Ligaments

Lb φ Impingement Point 2θ

rb

π −α 2

(a) u 2R dφ r + dr U 2θ

φ

S

r X

2R u

Stagnation Point

h

(b)

Figure 8.64 (a) Schematic of a typical impinging jet spray and (b) schematic and notation for the breakup of a sheet formed by impinging jets (modified from Przekwas, 1996).

8.9

GROUP-COMBUSTION MODELS OF CHIU

In very dilute sprays with low-volatility fuels, individual drop burning is highly possible. The formulation and solution for a single fuel drop are given in Kuo (2005, chap. 6), and combustion models for dilute sprays are given in the earlier sections of this chapter. In this section, the concept of group combustion of drop clouds is introduced. This concept is especially useful for dense-spray combustion in gas-turbine engines, industrial furnaces, and diesel engines. In a combustor with dense sprays, it is not practical to follow all the particles present particularly in dense spray zones. Therefore, the average quantities related to the drops are

GROUP-COMBUSTION MODELS OF CHIU

701

considered. These include the average number of particles present in a particular volume (i.e., the number density, average distance between the droplets, and average drop diameter). Different models have been proposed. A series of group-combustion models was developed by Chiu and coworkers (1977, 1978, 1981, 1982, 1983a, 1983b) for studying the combustion behavior of particle clouds of liquid-fuel sprays. Their models were partly based on the experimental observations of Onuma and Ogasawara (1974) that there is structural similarity between two different flames; one burning liquid kerosene and the other a gaseous fuel. Onuma and Ogasawara suggested that the rate-determining step in spray combustion of kerosene was mixing between fuel vapor and air rather than the vaporization of individual drops as conventionally thought. (Note, however, that for fuels of lower volatility and in larger droplets, drop vaporization still is one of the controlling factors in spray combustion.) The second body of experimental evidence for group-combustion models is from to Chigier (1981) and coworkers. In their studies of pressure-atomized jet and air-blastatomized spray flames, they observed regions of low oxygen concentration and low temperature within the spray. The existence of these regions causes the displacement of flame toward the outer boundaries of the spray. According to the group-combustion theory of Chiu and coworkers, the collective behavior of drops in liquid sprays forms a fuel-rich mixture in the core region of the spray. This mixture is nonflammable at the spray core, due to insufficient air penetration. The radial transport of gaseous fuel by convection and diffusion leads to the formation of flammable mixtures at some distance from the centerline of the spray. There flammable mixtures burn as gaseous diffusion flames. As drops move beyond the dense core region of the spray, the separation distances between drops increase and the drop size is reduced, so that the air concentration increases. Under these conditions, some liquid drops may burn individually with flames inside the spray boundary, while other drops may burn as groups. In general, the core region consists of drops vaporizing in atmospheres of low oxygen concentration, while the outer region may contain drops burning with multidrop flames (Chigier, Okpala, and Green, 1983). In group-combustion models, the collective behavior of drops is accounted for by a simultaneous analysis of an inner heterogeneous region and an outer homogeneous gas-phase region. No detailed formulation of the group combustion of liquid droplets is given here. Interested readers should consult the original papers by Chiu and coworkers. 8.9.1

Group-Combustion Numbers

Spray-combustion models are classified according to group-combustion numbers. The group-combustion number of the second kind, G, is defined as ratio of the total heat-transfer rate between two phases to the rate of heat of vaporization associated with the diffusing fuel vapor, that is, Rate of heat exchange between two phases Rate of heat of vaporization D∞ G1 G≡ R∞ U∞

G≡

(8.157) (8.158)

702

SPRAY ATOMIZATION AND COMBUSTION

where the group-combustion number of the first kind, G1 , is defined as G1 ≡

2 2π ll rl0 n0 R∞ T0 1/2 2 + 0.6 Redl0 Pr1/3 ρ∞ D∞ Lv

(8.159)

where D∞ and ρ∞ = mass diffusivity and density of the gas in the undisturbed environment R∞ = radius of the boundary of the two-phase zone Lv = latent heat of vaporization per unit mass ll = thermal conductivity of the liquid. r10 the reference radius of the droplet. The number density of the drops at the reference condition, n0 , is given by: n0 ≡

N

(8.160)

4 3 3 πR∞

where N is the total number of droplets present in the cloud. After substituting Equation 8.160 into Equation 8.159, we have r10 ll T 0 1/2 3 G1 = 2 N 2 + 0.6 Redl0 Pr1/3 (8.161) R∞ ρ∞ D∞ Lv In 1977 paper Chiu and Liu (1997) redefined the dimensionless parameter G1 as G1 ≡

2 4π ll rl0 n0 R∞ 1/2 1 + 0.276 Red Sc1/3 l0 ρ∞ D∞ Cp

(8.162)

This new dimensionless number of the first kind gives G the next physical meaning: G≡

Rate of heat exchange between two phases Rate of energy transport by convection

Using Equation 8.162 and the expression for n0 , we can show that rl0 1/2 1/3 G1 = 3 1 + 0.276 Redl0 Sc Le N R∞

(8.163)

(8.164)

or, alternatively,

G1 = 3 1 +

1/2 0.276 Redl0 Sc1/3

Le N

2/3

rl0 di

(8.165)

where Le is the Lewis number and di is the interdroplet separation. The dimensionless parameter G or G1 represents the degree of interaction between the two phases and serves to differentiate strong and weak interactions.

703

GROUP-COMBUSTION MODELS OF CHIU

8.9.2

Modes of Group Burning in Spray Flames

As shown in Figure 8.65, four group-combustion modes of a droplet cloud are possible. In sprays where G > 102 , external sheath burning occurs, which consists of an inner nonvaporizing droplet cloud surrounded by a vaporizing droplet layer with the flame at a “standoff” distance from the spray boundary. High-G sprays usually have large group-burning rates and low core temperatures. For marginally high-G sprays (G > 1), external group combustion prevails. The spray zone consists of an inner vaporizing cloud with a standoff diffusion flame from the boundary of the droplets. For 10−2 < G < 1, the mode of combustion is internal

VAPORIZING DROPLET LAYER Ox

Ox F

F P

P

P

P

EXTERNAL FLAME NON-VAPORIZING DROPLET CLOUD

VAPORIZING DROPLETS

EXTERNAL GROUP COMBUSTION WITH "SHEATH VAPORIZATION" AT HIGH G (a)

EXTERNAL GROUP COMBUSTION WITH A "STAND OFF" FLAME (b)

INDIVIDUAL DROPLET BURNING Ox F

P

P

P

P

Ox

P F BOUNDARY OF THE SPRAY MAIN FLAME

VAPORIZING DROPLETS

FLAME OF INDIVIDUAL DROPLET

INTERNAL GROUP COMBUSTION WITH THE MAIN FLAME LOCATED WITHIN THE SPRAY BOUNDARY (c)

FLAME OF INDIVIDUAL DROPLET SINGLE DROPLET COMBUSTION AT LOW G (d)

Figure 8.65 Four group-combustion modes of a droplet cloud (modified from Chiu, Kim, and Croke, 1982).

704

SPRAY ATOMIZATION AND COMBUSTION INTERNAL GROUP COMBUSTION MULTI-DROPLET COMBUSTION DIFFUSION FLAME SPRAY BOUNDARY EXTERNAL GROUP COMBUSTION

POTENTIAL CORE

EVAPORATING DROPLETS

TURBULENT BRUSH FLAME

Figure 8.66 Schematic of liquid-fuel spray group combustion (adapted from Chiu and Croke, 1981, adapted from Kuo, 1986).

group combustion. In this mode, the main flame locates within the spray boundary, while individual drop burning occurs in the outer regions of the spray. For very low values of G(< 10−2 ), the mode becomes individual droplet combustion. A schematic diagram of liquid-fuel spray group combustion is shown in Figure 8.66. In this figure, Chiu and Croke (1981) subdivided the spray flame into many zones: potential core, external group-combustion zone with evaporating droplets, turbulent envelope diffusion flame at spray core boundary, multidroplet combustion zone with internal group-combustion behavior, and turbulent brush flame. They used their group-combustion theory to conduct predictive calculations for a C10 H14 spray flame. Their predicted temperature and concentration profiles indicate that the flame is stabilized near the spray boundary. They also reported that a relative minimum in temperature profile occurs on the axis of the spray, where the fuel vapor concentration has a maximum. Sprays with smaller G have a smaller flame standoff distance and thus have a higher average gas temperature in the spray core. Higher temperature in the fuel-rich core, in turn, increases the rate of pyrolysis of heavier hydrocarbons into lighter hydrocarbons, leading to soot and particulate formation by petroleum coking. Besides the group-combustion number G, the fuel-air mass-density ratio is also important in determining the relative location of the envelope flame to the spray boundary. The fuel-air mass-density ratio can be defined as: 3 β ≡ 43 πrl0 n0 ρl /ρ∞ (8.166) where ρl is the density of the liquid droplet. At a fixed G, an increase in β results in shift of the flame toward the spray boundary, which is due primarily to reduction in the average group burning rate. According to the calculations of Chiu and Croke (1981), for G ∼ 10−2 , β ∼ 0.1, the flame is stabilized near the spray boundary. For G = 0.5, β must be increased to 100 for the flame to approach the spray boundary. They found that in order to have external combustion, the

GROUP-COMBUSTION MODELS OF CHIU

705

value of G must be greater than a critical group-combustion number, Gc , which varies with β in this manner: β (mass density ratio) Gc (critical group combustion number)

1 9 × 10−3

10 6 × 10−2

100 7 × 10−1

Following the explanation given by Chigier (1983), high-β sprays are characterized by high jet speeds and larger drops. This leads to shorter residence time for a fixed spatial distance and poor vaporization characteristics with resultant lower group burning rates. For a fixed β, high-G sprays have a higher group burning rate and are characterized by relatively densely populated small drops that are readily vaporized. High-G sprays have larger flame radii and hence a larger flame area and thus a higher group burning rate. Chiu and Croke (1981) found in their studies of a cloud of C10 H14 , drops that the group envelope flame was stabilized on the boundary of the drop cloud for G = 1.36. As G decreases, the envelope flame penetrates into the drop cloud and divides the cloud into two zones, a strongly interacting zone located inside the group envelope flame (see Figure 8.65c) and a weakly interacting zone established between the envelope flame and the boundary of the cloud. In the strongly interacting zone, the drops vaporize, and the vapor produced is consumed at the group envelope. Drops in a weakly interacting zone burn with an envelope flame surrounding each droplet. Their calculated group combustion modes are shown in Figure 8.67, where N , the total number of drops, is plotted as a function of the dimensionless separation distance S for four different modes of group combustion shown on the figure. The separation distance S is defined as: S = 0.05

di /rl 1/2

1 + 0.276 Redl0 Pr1/3

(8.167)

where di is the average spacing between the neighboring droplets. Although the group-combustion models developed so far include many important aspects of spray combustion, numerous improvements could be incorporated into the model, such as finite thickness of the envelope flame, radiative heat-transfer effects, particle dispersion by turbulence, droplet synergetic phenomena, and spray vaporization. Some of these areas are being actively studied by Chiu and coworkers. Sirignano (2010) has cited two major shortcomings in the group-combustion theory. The first shortcoming is that the theory does not account for the fact that the Nusselt number and the vaporization law for each droplet depends on the spacing between droplets (with exceptions). A second shortcoming is that a quasisteady process was assumed, which does not consider transient droplet heating or unsteady gas-phase conduction across the particle clouds. Due to the large difference between the time scale for droplet heating and the time and length scales for conduction across the cloud, the unsteady effects can become significant. The current group-combustion theory should be viewed as a base for further studies rather than as complete. Until now, all approaches to group-combustion theory

706

SPRAY ATOMIZATION AND COMBUSTION 1010 109 108 107

Vaporizing Droplet Lever Non Vaporizing Droplet

Oxidizer Fuel

N: TOTAL NUMBER OF DROPLETS

106 10

5

10

4

G = 102

EXTERNAL SHEATH Products COMBUSTION

10−1 10−2

103 Rf

Flame

Oxidizer Products Rl

Fuel Rb

102

Rb

EXTERNAL GROUP COMBUSTION

DROPLETS

Oxidizer Flame

SINGLE-DROPLET COMBUSTION

Products

INTERNAL GROUP COMBUSTION

TRANSITION BAND

Oxidizer Products Rb

102

S = 0.05

103

104

105

106 107 108

di /r l 1/ 2

1 + 0.276 Redl0 Pr1/ 3

Figure 8.67 Group combustion modes for droplet clouds (modified from Chiu, Kim, and Croke, 1982).

have been based on infinite chemical-kinetic rates. Obviously, finite rates should produce significant quantitative differences. 8.10

DROPLET COLLISON

Some background material for droplet collision has been described in Chapter 7. The droplet collision can be in the form of droplet-droplet or droplet-wall collision. Collision processes involve the consideration of conservation of energy, momentum, and angular momentum. The droplet loses kinetic energy as it deforms. The strain leads to viscous dissipation and conversion of a part of mechanical energy into thermal energy. The droplet surface energy increases due to deformation. Surface energy can be regarded as a potential energy. The increase in surface energy in early phases of collision can lead to recoil and rebound in later phases through the conversion of surface energy back to kinetic energy. Momentum balance occurs through a force imposed on the droplet by other droplets or by the wall in collision as the droplet loses its velocity and rebounds in a different direction. When the two droplets collide at

DROPLET COLLISON

707

an angle (not a head-on collision), a torque would be imposed on them at the moment of collision. In such a case, conservation of angular momentum must be considered. Similarly, when a droplet collides with a wall at an angle other than 90 degrees, a torque would act on the droplet, and conservation of angular momentum should be considered.

8.10.1

Droplet-Droplet Collisions

Droplet-droplet collisions have been discussed in detail by Sirignano (2010). In this section, a we summarize certain aspects of particle-particle collision. Collision among the droplets seems to have a low probability in a spray in which the droplets are moving in a parallel direction or along divergent paths. In addition, O’Rourke and Bracco (1980) proposed that droplet coalescence could be an important process in the dense spray region near the injector orifice. Based on this argument, they developed a comprehensive model that accounted for collision and coalescence of the droplets. This model was used by Bracco (1985) and Reitz (1987); the results of these studies showed that the average droplet size increases with the downstream distance from the injector. In addition to droplet coalescence, several mechanisms can also cause droplet diameter to increase with downstream distance. These mechanisms include the facts that (1) smaller droplets vaporize faster leaving the larger ones; (2) condensation occurs in the cold, vapor-rich, dense-spray region near the injector; and (3) longer-wavelength disturbance on the jet will take longer Langragian time to grow and to yield droplets with larger size. Collision between two droplets is possible in other spray environments where droplet streams are oriented in intersecting directions. Weak collision, such as grazing, can occur. Stronger collision can lead to permanent coalescence; coalescence with vibration that can lead to further breakup into two smallest droplets, shattering into many smaller droplets, or even bouncing. No models have considered three or more droplet collisions. A number of fundamental experimental studies of two droplets in a collision configuration have been made. Qian and Law (1997) have identified five distinct regimes for collision in a plot of droplet 2RρU 2 /σ and impact parameter χ/2R, where χ is the shortest distance at the time of impact between the tangent lines to the trajectories of two equal-size droplets of radius R, relative velocity U , and surface tension coefficient σ . Clearly, only χ ≤ 2R is interesting. Figure 8.68 indicates the result. Regime I involves coalescence of two droplets after minor deformation. During the final instances of the approach, the thin gas film between the two droplets is squeezed and discharged in sufficient time for the two droplets to merge under the force of the surface tension. In regime II, the We based on droplet diameter is high and the allowed time for discharge of the gas film is short. Due to insufficient time for discharge, the surface tension cannot act to coalesce the droplets; thus, rebound or bounce takes place. In regime III, the higher approach velocity forces the discharge to occur in sufficient time, resulting in coalescence. In comparison with regime I, substantial deformation occurs in regime III

708

SPRAY ATOMIZATION AND COMBUSTION

(I) Coalescence (V) Off-center seperation

(V) Off-center seperation x/2R

x/2R

(II) Bouncing

(I) Coalescence (III) Coalescence (I)

(IV) Near head-on separation

(IV) Near head-on separation We (a)

We (b)

Figure 8.68 Schematic of various collision regimes of (a) hydrocarbon droplets in 1 atm and water droplets at elevated pressures, and (b) water droplets at 1 atm and hydrocarbon droplets at reduced pressures (modified from Qian and Law, 1997).

before coalescence. Regime IV occurs for small values of the impact parameter (head-on and near-head-on collisions) and is characterized by a temporary coalescence followed by separation. The We and impact parameter are high, and enough energy remains in the internal liquid flow to overcome the surface tension and separate into two larger droplets plus a smaller satellite droplet. Regime V occurs for the off-center collision and involves temporary coalescence followed by eventual separation. Rotation of the coalesced molecules takes place from the conservation of angular momentum. A satellite droplet and two larger droplets result from the separation. Nobari, Jan, and Tryggvason (1996) considered head-on collisions with planar symmetry conditions while Nobari and Tryggvason (1996) examined three-dimensional collisions; they presented their predicted results, showing the same phenomena as seen in Figure 8.69a.

8.10.2

Droplet-Wall Collision

The motivating application in droplet-wall collision is material processing for which solidification of the molten material in the droplet is a vital factor. Droplet wall calculation starts with the droplet sticking the wall, unlike the case of dropletdroplet collision. Fukai et al. (1995) performed experimental and computational studies of initially spherical droplets hitting the surface at right angles. They used water droplets and wall surfaces with various degress of wettability. The wettability of a liquid is defined as the contact angle between a droplet of the liquid in thermal equilibrium on a flat horizontal surface. Their results were sensitive to the wetting model (Mittal, 2009). They also specified the contact angle at the circular lining representing the meeting of the three phases: liquid, surrounding gas, and solid wall. The contact angle is the angle between the tangent to the liquid-gas interface (at the solid wall surface) and the solid wall. The results showed that the contact line moved in time during the process. Its

709

DROPLET COLLISON Time (ms) Regime I

Time (ms) Regime II

0

Time (ms) Regime III

0

Time (ms) Regime IV

0

0

0.50

0.05

0.10

1.05

1.10

1.15

0.15

0.20

0.25

0.60

0.40 1.45 0.75

2.05

0.85

2.65

1.40

U(m/s)

1.50

1.05

4.20 0.14

0.64 0

U(m/s)

1.45 1.24

1.0 mm

2.30 0

1.0 mm

(a) Time (ms) Regime I

Time (ms) Regime II

Time (ms) Regime III

Time (ms) Regime IV

0

0

0

0

0.55

0.10

0.10

0.10

0.85

0.20

0.25

0.25

1.80

0.30

1.05

0.35

2.45

0.40

1.70

0.45

3.05

0.85

0.70 4.95

4.80 U(m/s)

0.14

U(m/s)

0.64 0

1.0 mm

1.20 2.30

2.51 0

1.0 mm

(b)

Figure 8.69 Photographic images showing (a) representative head-on collisions in regimes I to IV and (b) representative off-center collisions in regimes I to III and V (modified from Law, 2006).

710

SPRAY ATOMIZATION AND COMBUSTION

radius increased during the early phase of the impact and later decreased possibly with subsequent oscillation. Clearly, in early collision, kinetic energy is converted to surface energy and through viscous dissipation to heat. Later, surface energy caused some recoil and converted back into kinetic energy. The maximum radius of the splat (here “splat” refers to the droplet structure after impact on the surface) decreased as the contact angle when the spreading stage increased. The rate of change of splat height during the early collision was directly proportional to the impact velocity. H. Liu, Lavernia, and Rangel (1994) considered a droplet hitting a wall with solidification occurring during the impact. They showed that during the solidification stage, the liquid flow could separate temporarily from the wall surface, creating gas bubbles that result in pores for the solidified material. Zhao, Poulikakis, and Fukai (1996) considered heat transfer by means of convection and conduction in the splat droplet; solidification was not considered. They extended the work of Fukai et al. (1993, 1995) by adding axisymmetric energy equation to the system. They found that temperature profile was truly two-dimensional and also that liquid mass tended to accumulate at the periphery of the splat. Delplanque and Rangel (1998) compared the experimental data and computed results from simpler models. In both cases with and without solidification, the agreement between experiments and numerical simulation was satisfactory. A simplified model assumed that the splat was formed immediately and maintained the shape of a cylindrical disk with time-varying radius and height. 8.10.3

Interacting Droplet in a Many-Droplet System

Practical sprays contain a statistically large number of droplets, and they behave differently from dilute sprays as the nature of interface exchange is different. As droplet spacing decreases, the collective interaction of droplets induces the vapor cloud accumulation to retard heat transfer and momentum exchange with the surrounding gas, thereby modulating the thermal, chemical, and dynamic conditions in the immediate ambience of the droplet. Collective interaction is classified into short-range interaction by which laws of interfacial exchange and state of droplets are modulated. Long-range interaction changes do not govern the laws of interfacial exchange but do govern the state of the droplet. A droplet under the effect of short-range interaction is classified as an “interacting droplet” in contrast to the “isolated droplets” that are not subject to such interaction. An early model of droplets was proposed by Tishkoff (1980), who examined the effect of interactions of droplets in an infinite droplet array. 8.11 OPTICAL TECHNIQUES FOR PARTICLE SIZE MEASUREMENTS

Characterization of droplet size distribution has been employed for applications such as the study of fuel sprays in combustion systems. Optical methods are very useful because of the particle size (and possibly velocity) are measured, and these methods are non-intrusive, meaning no physical probe is inserted into spray.

OPTICAL TECHNIQUES FOR PARTICLE SIZE MEASUREMENTS

711

Optical Particle Size Measurements Imaging Techniques

Photography

Holography

Automatic Image Analysis

Nonimaging Techniques

Single Particle Counting

Ensemble Particle Sizing

• Scanning ratio techniques

• Extinction measurement techniques

• Intensity deconvolution method • Interferometric methods • Visibility method using laser doppler velocimeter (LDV) • Phase Doppler sizing anemometer (PDSA)

Figure 8.70

8.11.1

• Multiple angle scattering measurement techniques • Fraunhofer diffraction particle analyzer • Integral transform solutions for nearforward scattering

Classification of different types of optical particle size measurements.

Types of Optical Particle Sizing Methods

As shown in Figure 8.70, imaging techniques include photography, holography, and automatic image analysis. These techniques are relatively straightforward and can be found in general literature. Nonimaging techniques, which are based on light scattering, include single particle counters and ensemble techniques. Koo and Hirleman (1996) summarized multiple nonimaging optical techniques for measurement of particle size distributions. Some of the nonimaging techniques are briefly discussed next. 8.11.2

Single Particle Counting Methods

For dilute sprays, scattered light can be used to count single particles passing through a small sampling volume, illuminated by a laser beam. Discrete size distribution can be obtained from individual particle measurements. These counters can also be used for velocity measurements. In order for particle to be detected by single particle counting (SPC), it should produce a scattering signal that exceeds the noise threshold limit of the instrument. This implies that the detected scattered power (Psca ) must exceed certain minimum threshold signal for the instrument (i.e., Psca > Pmin ). Generally, laser beam intensity has Gaussian distribution around the focus. This requirement creates two fundamental problems: trajectory ambiguity and the size-selective sampling bias problem. Trajectory ambiguity. Scattered power from a small particle passing close to the beam focus can be the same as the power scattered by a larger particle traversing an off-axis point of the beam. Therefore, this problem must be resolved

712

SPRAY ATOMIZATION AND COMBUSTION

in order to ensure accurate size measurements. Early particle counters confined flow to a nearly uniform region near the beam focus, but this solution is not acceptable for in situ applications. Size-selective sampling bias. Particles with larger scattering cross sections have a greater probability of being optically sampled because they can pass farther from the laser beam focus and still scatter adequate power to be detected. Therefore, an inherent sampling bias is created favoring particles with a larger scattering cross-section. Correction to measured distribution can be estimated by knowing sampling probability as a function of diameter. Sampling Probability. Particle population can be characterized with a distribution function F (D). Ideally, the distribution function F (D) is constructed over infinite time. Accuracy of F (D) based on finite sampling depends on a representative, unbiased sample of the population and accurate size measurement of particles in a finite sample. 8.11.2.1 Scattering Ratio Technique The scattering ratio technique was developed by Gravatt (1973). Using the ratio of power scattered into two detection apertures at different angles, this method avoids the trajectory ambiguity problem but still suffers from the selective sampling bias problem. This problem can be handled by modeling detection probability for detectors and correcting the measured distribution. A schematic of the device used by Hirleman and Moon (1982) for solving trajectory ambiguity problem is shown in Figure 8.71. The size selective sample

SINGLE-PARTICLE COUNTER OPTICAL FIBERS

LASER SENSITIVE VOLUME θi

ri

MIRROR LENS

Δri

ri

Δri Δθi RECEIVING LENS

ANNULAR IRISES PHOTOMULTIPLIER TUBES TO DATA ACQUISITION

END VIEW ANNULAR IRISES (TWO SHOWN)

Figure 8.71 Schematic of multiple ratio counter used as one of the single particle counter techniques (modified from Hirleman and Moon, 1982).

OPTICAL TECHNIQUES FOR PARTICLE SIZE MEASUREMENTS

713

bias problem was handled by modeling the ratio counters, in which the optical sampling area was calculated and used to back out the particle size distribution function F (D) from measured rate of appearance of a give sized particle. 8.11.2.2

Intensity Deconvolution Method

The intensity deconvolution technique was developed by Holve and Self (1979). This approach is also known as particle counting sizing velocimeters (PCSVs). The complete scattered signal spectrum is collected over an experiment, then it is mathematically deconvolved to obtain F (D). That is, the actual particle size is never explicitly known; only the product of irradiance and cross section is known. 8.11.2.3

Interferometric Method (Phase-Shift Method)

The interferometric technique was developed by Taubenblatt and Batchelder (1991). Instead of using total light or angular distribution of scattered light, phase shift is measured using a special interferometer. Phase shift is related to particle size but independent of position in the laser beam; thus the method avoiding the trajectory ambiguity problem. If both phase shift and extinction are measured, particle size and refractive index can be determined. 8.11.2.4

Visibility Method Using a Laser Doppler Velocimeter

The visibility technique using a Laser Doppler Velocimeter (LDV) was developed by Farmer (1972). Essentially a modification of LDV was carried out to yield particle size information as well as velocity data. This technique used the contrast or visibility in the LDV signals to provide a measure of particle size. The advantage of the visibility method is its simultaneous measurement of particle velocity and size. However, this method has limited use due to difficulties with dynamic range in particle size. 8.11.2.5

Phase Doppler Sizing Anemometer The phase Doppler sizing anemometer (PDSA) technique was developed by a number of researchers, including Farmer (1972), Durst and Zare (1976), Fl¨ogel (1981), and Bachalo and Houser (1984). This approach is similar to the LDV system. An interference fringe pattern was produced by particle scattering two cofocused laser beams. As a particle (e.g., a liquid droplet) passes through the probe volume, the fringes are swept passed the detectors. The phase lag between detector signals is a measure of the fringe spacing and, hence, particle size. Durst and Zare (1976) used a double-element photodiode with elements 2 mm apart (see PDSA setup in Figure 8.72) to obtain sphere diameter information using phase measurement between the two detected signals. Essentially, particle size is related to phase shift between two detectors. One area of concern is the accuracy for small particles (e.g. 0.5 to 1 μm particles).

714

SPRAY ATOMIZATION AND COMBUSTION

BEAM EXPANDER LASER

γ

BEAM SPLITTER

DOUBLE PHOTODIODE 1 2

TEST-SECTION

Figure 8.72 Schematic of phase Doppler sizing anemometer (modified from Durst and Zare, 1976).

8.11.3

Ensemble Particle Sizing Techniques

Particle size distributions are determined from analysis of aggregate light scattering or extinction properties of a large number of droplets. This method generally requires simpler systems than those required for SPC. These measurements inherently contain less information in ensemble measurements than SPC, since data are taken for many drops in large volume (without velocity data). In most cases, a form of the size distribution is assumed and best-fit parameter are estimated by measurements. 8.11.3.1 Extinction Measurement Techniques One of the extinction measurement techniques was developed by Dobbins and Jizmagian (1966). This method is based on fact that the amount of light removed from an incident beam is directly related to extinction cross-sections of droplets in the path. Sauter mean diameter (SMD or D32 ) can be determined from a single measurement with certain assumptions. Use of multiple wavelengths allows refractive index and concentrations to be determined. Ideally, the wavelengths used must roughly bracket the droplet sizes. 8.11.3.2 Multiple Angle Scattering Technique The multiple angle scattering technique was developed by Dobbins (1963). It measures light that is scattered by variations in refractive index due to the presence of particles. The intensity variation with scattering angle can he evaluated from the following equation: 4 5

2

2 I (θ) /Iinc = dp2 /16 α 2 2J1 (αθ) /αθ + 4m2I / m2I − 1 (mI + 1) + 1

(8.168) where = = = = l= mI = J1 =

Iinc I (θ) dp α

incident irradiance light intensity function of the scattering angle θ particle diameter πdp /l wavelength of the incident beam refractive index of particle relative to the surrounding medium Bessel function of the first kind and order unity

OPTICAL TECHNIQUES FOR PARTICLE SIZE MEASUREMENTS

715

1.0

0.1 Relative Illumination, I(θ)

Polydisperse Spray

0.01

0.001

0

2

4

6

8

Beam-Spread Parameter, pD32 θ/λ

Figure 8.73 Mean theoretical illumination profile for polydispersed sprays with sizes distributed according to the upper-limit distribution function (modified from Dobbins, Crocco, and Glassman, 1963).

There exists a convenient relationship of I (θ) with (πD32 θ/l), as shown in Figure 8.73. This technique is useful for particle sizes ranging from 10 to 250 microns and has the advantage of simplicity and low cost. The original optical system developed by Dobbins, Crocco, and Glassman (1963) is shown in Figure 8.74 for the multiple angle scattering measurement technique. Rizkalla and Lefebre (1975) have carried out particle size measurements in a spray by using this technique. 8.11.3.3

Fraunhofer Diffraction Particle Analyzer

The Fraunhofer diffraction particle analyzer is one of the most common instruments. It was utilized by Swithenbank et al. (1976) and many others for droplet size measurements. For measuring the ensemble characteristic of a spray, the most widely used nonintrsive techniques developed for droplet size distribution

716

SPRAY ATOMIZATION AND COMBUSTION

Condenser Lenses

Collimating lens Collimating aperture

Injector

Photomultiplier Receiving lens

Intensity distribution I(θ)

Light source

Spray

Aperture Filter

Pinhole aperture

Figure 8.74 Schematic drawing of the light-scattering system (modified from Dobbins, Crocco, and Glassman, 1963). TRANSFORM LENS

θ = r/f

DETECTION PLANE r

LASER θ

BEAM EXPANDER PARTICLE SPATIAL FILTER FIELD

f

Figure 8.75 Schematic of a laser light diffraction particle size sizing instrument (from Koo, 1987).

is the Fraunhofer diffraction-based Malvern particle analyzer. This method is based on the diffraction theory developed by the German scientist Joseph Fraunhofer (1787–1826) that is valid when particle sizes are large compared to the beam wavelength. A several-milliwatt He-Ne laser is expanded and collimated into a beam of several mm in diameter. After passing through a particle field, a Fourier transform lens converts angular displacement θ into spatial displacement r. The spacing of interference rings formed is dependent on the particle sizes. The generalized schematic of a laser light diffraction particle sizing instrument is shown in Figure 8.75. When a parallel beam of light interacts with a droplet, a diffraction pattern is formed, which depends on the size of the droplet. Malvern System The photodetector consists of 31 semicircular photodiode rings, and it is used to detect light distribution. Each ring is sensitive to small particular ranges of droplet sizes. This method allows quick data analysis. Droplet size range is dependent on instrument configuration, but the overall measurement range is from 0.1 to 2000 microns. 8.11.3.4 Integral Transform Solutions for Near-Forward Scattering Essentially a more exact mathematical model of near-forward scattering is

∞ ia (θ) = Iinc /k

2 0

J12 (αθ) / θ 2−a α b−2 Fb (α) dα

(8.169)

EFFECT OF DROPLET SPACING ON SPRAY COMBUSTION

717

where a and b are instrument parameters. The integral transform methodology first detects the near-forward light signature ia (θ) and then inverts the data to determine particle size distribution Fb (α). Koo and Hirleman (1996) suggested using an integral transform solution to provide the analytical expression for Fb (α). A fundamental problem of these techniques is the generation of monodisperse, spherical droplets of known size and concentration. Many different calibration standards are available by using polystyrene latex spheres, glass microspheres, droplet generators, photomask calibration reticles, polydisperse sprays, and others. In summary: • No single technique can be used for all applications. • PDSA has been used widely for droplet sizing in spray combustion. • The Fraunhofer diffraction analyzer (such as from Malvern) is very popular for ensemble particle size characterization. • The integral transform solutions for near-forward scattering are effective.

8.12 EFFECT OF DROPLET SPACING ON SPRAY COMBUSTION 8.12.1

Evaporation and Combustion of Droplet Arrays

In order to study the effect of spacing between droplets on drag coefficient, heat-transfer rate, and mass transfer rate, research has been done on specialized sets of droplets. Three classifications of the interaction of droplets in a gaseous environment were given by Sirignano (1983): 1. Droplet arrays. Few interacting droplets are considered with a specified ambient gaseous environment. 2. Droplet groups. Many droplets are considered, but the gaseous conditions far from the cloud are specified but not coupled with the droplet calculation. 3. Sprays. Many droplets are considered but contrary to the droplet groups, the gas field calculation in the domain is coupled to the droplet calculation. To understand the distinctions among arrays, groups, and sprays, a cloud or collection of droplets must be visualized as occupying a certain volume. The primary ambient gas conditions are defined as those conditions in the gas surrounding the cloud of droplets. Each droplet has a film of gas surrounding it. The local ambient conditions are defined as the gas properties at the edge of the film but within the volume of the cloud. This definition becomes imprecise when the droplets overlap. In such cases, the local ambient conditions are replaced by the average properties of the gas in the droplet neighborhood. In the work by Sirignano et al. (1996), a single stream or a few neighboring streams of droplets can be considered as a transition from arrays to groups. The stream is actually a special case of the array because its geometrical configuration

718

SPRAY ATOMIZATION AND COMBUSTION

is precisely measurable in an experiment or prescribed in a computation. Unlike the group, a statistical description of average spacing is not necessary. In particular, for droplet group theory, a number density of the droplet is considered. If a periodic behavior is exhibited within the stream, an array involving only a few droplets sometimes can be studied to predict stream behavior. Droplet arrays and streams are useful to study because they allow the development of data and insights for phenomena on the scale of the spacing between droplets, information that is very difficult to obtain experimentally or computationally. Under the supposition that the behavior of a droplet is affected primarily by the nearest neighboring droplet, information obtained from array studies is very relevant to understanding the more complex behavior of groups and sprays. The earliest work on drop interactions was conducted by Twardus and Brzustowski (1977), who analyzed two burning droplets of equal size. Stefan convection and forced convection were not considered, and the spacing between droplet centers (Ds = 2h) was used as parameter of the study. As shown in Figure 8.76, there is a critical value of the ratio of the distance between droplet centers to the droplet radius above which the droplets burn with one envelope flame. The critical value depends on the particular stoichiometry. The vaporization rate remains diffusion controlled, and the rate of diffusion and therefore the vaporization rate decreases as the droplet spacing decreases. In the limit as droplets come into contact, the vaporization rate becomes a factor of ln 2 (=0.693) of the value for two distant isolated droplets. Chiang and Sirignano (1993) extended the two-tandem-droplet calculation based on a transient axisymmetric finite-difference analysis with a grid generation

2r0 h Ds(t) r0 is the initial radius of droplets; all droplets have equal radii at time = 0

h/r0 = 8.5

r1(t = 0) = r2(t = 0) = r0 2r0 h

h/r0 = 8.587

Figure 8.76 Effect of droplet spacing on burning of two droplets (modified from Sirignano, 1983, redrawn from Twardus and Brzustowski, 1977).

EFFECT OF DROPLET SPACING ON SPRAY COMBUSTION

719

GAS-PHASE VELOCITY VECTORS LIQUID-PHASE STREAM FUNCTION

Radius of lead droplet is r1 (t) Radius of downstream droplet is r2 (t) 0.0

r1

Both droplets have same initial radii of r0

r2

−1.7

−3.4

Ds(t) 16.2 9.8 12.5 GAS AND LIQUID-PHASE VORTICITY

19.1

Contour Interval: 2.36E-01 Min: -4.49E+00 Max: 2.23E-01

Figure 8.77 Gas-phase velocity vector, liquid-phase stream function, and vorticity contours for both phases in two-tandem-droplet case. Time = 3.00, Re1 = 80.31, Re2 = 85.84, initial R1 [= r1 (0)/r0 ] = 1.00, initial nondimensional spacing [Ds (0)/r0 ] = 3.71 (modified from Sirignano, 2010).

scheme for two vaporizing droplets moving in tandem and accounting for variable thermophysical properties. Figure 8.77 shows the gas-phase velocity vector, liquid phase stream function, and vorticity contours for both phases. Chaing and Sirignano (1993) obtained correlations for instantaneous drag coefficients, Nusselt number, and Sherwood number with other dimensionless parameters. A linear regression model was used to fit over 3,000 data points. The correlations are normalized by the isolated droplet correlations. For the lead droplet: CD1 = 0.877 Re0.003 (1 + BH )−0.040 (Ds /r0 )0.048 (r2 /r1 )−0.098 m CD iso Nu1 = 1.245 Re−0.073 Pr0.150 (1 + BH )−0.122 (Ds /r0 )0.013 (r2 /r1 )−0.056 m m Nuiso Sh1 = 0.367 Re0.048 Sc0.730 (1 + BM )0.709 (Ds /r0 )0.057 (r2 /r1 )−0.018 (8.170) m m Shiso where 0 ≤ BH ≤ 1.06, 0 ≤ BM ≤ 1.29, 11 ≤ Rem ≤ 160, 0.68 ≤ Prm ≤ 0.91, 1.47 ≤ Scm ≤ 2.50, 2.5 ≤ (Ds /r0 ) ≤ 32 and 0.17 ≤ [r2 (t)/r1 (t)] ≤ 2.0 In these equations, Rem is Reynolds number base on droplet-gas relative velocity, droplet radius, free-stream gas density, and average gas-film viscosity.

720

SPRAY ATOMIZATION AND COMBUSTION

For the downstream droplet: CD2 = 0.549 Re−0.098 (1 + BH )0.132 (Ds /r0 )0.275 (r2 /r1 )0.521 m CD iso Nu2 = 0.528 Re−0.146 Pr−0.768 (1 + BH )0.356 (Ds /r0 )0.262 (r2 /r1 )0.147 m m Nuiso Sh2 = 0.974 Re0.127 Sc−0.318 (1 + BM )−0.363 (Ds /r0 )−0.064 (r2 /r1 )0.857 m m Shiso (8.171) where 0 ≤ BH ≤ 2.52, 0 ≤ BM ≤ 1.27, 11 ≤ Rem ≤ 254, 0.68 ≤ Prm ≤ 0.91, and 1.48 ≤ Scm ≤ 2.44. These correlations should be useful for considering the spacing effect between adjacent droplets. HOMEWORK PROBLEMS

1.

In the estimation of the rate of reduction of diameter of a droplet traveling within a dilute spray flame, these assumptions usually can be made: a. The temperature variation and oxygen concentration are low within the spray. b. No interaction takes place between droplets. c. Le = Sc = Pr = 1. d. The initial velocity of the droplets, Ud0 , is equal to the discharge velocity of the fuel. e. The initial temperature of the fuel is close to its boiling point. Using these assumptions and the d 2 -evaporation law, develop two ordinary differential equations describing the rates of change of droplet velocity and diameter.

2.

Derive the following transport equation for the turbulence kinetic energy k of a continuous phase that contains numerous dispersed fuel droplets. 2 1 ∂ 1 ∂ ∂u μt ∂k ∂ ◦ − ρε + uSpu − uS pu rρv k = r + μt (ρ uk) + ∂x r ∂r r ∂r σk ∂r ∂r Hints: Start with the gas-phase momentum equation: S pui ∂ 2 ui ∂ ui uj = v + ∂xj ∂xj ∂xj ρ where S pui represents the particle source term in the stream-wise momentum equation. After obtaining the k-equation in the index form, transform it into

721

HOMEWORK PROBLEMS

cylindrical coordinates and use the modeling approach of Gosman et al. (1981) or Sheun et al. (1983) for the thin shear-layer approximation. 3.

Derive the following transport equation for the turbulence dissipation rate ε of a continuous phase that contains numerous dispersed fuel droplets. ∂ 1 ∂ 1 ∂ ◦ rρv ε = (ρ uε) + ∂x r ∂r r ∂r

r

− Cε2 ρ

μt ∂ε σk ∂r

+ Cε1 μt

ε k

∂u ∂r

2

ε ∂S pu ε2 − 2Cε3 μt k k ∂r

Use the same hints as for Problem 2, except consider the transport equation for velocity fluctuation. 4.

Consider an isotropic turbulent flow of a continuous liquid that entrains many dispersed immiscible liquid droplets. Let the droplet diameter of the dispersed liquid be d, the pressure inside the droplet be Pd , and the pressure of the continuous fluid be p. a. Show that the static equilibrium condition of this mixture is Pd − p =

4σ d

where σ is the surface tension of the liquid droplet residing in the continuous liquid. b. Consider the droplet breakup process under the condition that inertial forces dominate over viscous forces. What is the relationship among the dynamic pressure fluctuation (caused by turbulence), surface tension, and the maximum diameter of a stable droplet? 5.

Use Problem 4, but consider that the droplet breakup is caused by viscous forces rather than inertial forces. What is the relationship among fluctuation velocity, surface tension, and the viscosity of the two liquids?

6.

Perform an order-of-magnitude analysis of conservation equations of mass, momentum, and mixture fraction of axial symmetric jet. Hint: Refer to Section 8.7.2 where the locally homogeneous flow assumption is valid.

APPENDIX A

USEFUL VECTOR AND TENSOR OPERATIONS

It is beneﬁcial for readers of this book to be familiar with vector and tensor operations. In this appendix, a scalar is represented with italic type and a vector is denoted with boldface type. A vector is deﬁned by both a magnitude and a speciﬁc direction in space. The vector can be represented in terms of three linearly independent components in the x1 , x2 , and x3 directions. The unit vectors in these three directions are e1 , e2 , and e3 , respectively. Thus, the vector V is represented as the sum of three component vectors, that is, V = v1 e1 + v2 e2 + v3 e3 = V1 + V2 + V3

(A.1)

The vector has a magnitude, which can be determined from its components V = |V| =

v21 + v22 + v23

(A.2)

The vector direction is determined by the relative magnitudes of v 1 , v 2 , and v 3 as shown in Figure A.1. Any unit vector in the direction of vector A can be deﬁned from the next equation: A eA ≡ |A| The dot product (also known as scalar product) of two vectors A and B is deﬁned as: A · B = |A| |B| cos θAB Fundamentals of Turbulent and Multiphase Combustion Copyright © 2012 John Wiley & Sons, Inc.

Kenneth K. Kuo and Ragini Acharya

723

724

USEFUL VECTOR AND TENSOR OPERATIONS x3

e3

V e2

e1

V2 = n2e2

V3 = n3e3

x2 V1 = n1e1

x1

Figure A.1 Vector components in the Cartesian coordinate system.

The cross product (also known as vector product) of two vectors A and B is deﬁned as: A × B = |A| |B| sin θAB e⊥A&B A.1

VECTOR ALGEBRA

A+B=B+A

(A.3)

sA = As

(A.4)

(s + p)A = sA + pA

(A.5)

s(A + B) = sA + sB

(A.6)

A·B=B·A

(A.7)

(A · B)C = A(B · C)

(A.8)

s(A · B) = (sA) · B = A · (sB) = (A · B)s

(A.9)

A · (B + C) = A · B + A · C e1 A × B = −B × A = A1 B1

(A.10) e2 A2 B2

e3 A3 B3

s(A × B) = (sA) × B = A × (sB) = (A × B)s

(A.11)

(A.12)

(A + B) × C = (A × C) + (B × C)

(A.13)

A × (B + C) = (A × B) + (A × C)

(A.14)

A × (B × C) = B(A · C) − C(A · B)

(A.15)

(A × B) × C = B (A · C) − A (B · C)

(A.16)

DERIVATIVES OF THE UNIT VECTORS

725

Thus, A × (B × C) = (A × B) × C

A1 A · (B × C) = B · (C × A) = C · (A × B) = B1 C1 (A × B) = |A| |B| − (A · B) 2

2

2

A2 B2 C2

A3 B3 C3

2

(A.17) (A.17a)

If A and B are nonzero vectors and parallel to each other, then A×B=0

(A.18)

If A and B are nonzero vectors and perpendicular to each other, then A·B=0

(A.19)

A.2 ALGEBRA OF UNIT VECTORS

In an orthogonal coordinate system, the unit vectors e1 , e2 , and e3 are perpendicular to one other. Therefore, e1 · e1 = e2 · e2 = e3 · e3 = 1,

(A.20)

e1 · e2 = 0,

(A.21)

e2 · e3 = 0,

e3 · e1 = 0

e1 × e1 = e2 × e2 = e3 × e3 = 0.

(A.22)

e1 × e2 = e3 ,

(A.23)

e2 × e3 = e1 ,

e3 × e1 = e2 .

A.3 DERIVATIVES OF THE UNIT VECTORS A.3.1

Cartesian Coordinate System

∂ex =0 ∂x ∂ey =0 ∂x ∂ez =0 ∂x A.3.2

∂ex =0 ∂y ∂ey =0 ∂y ∂ez =0 ∂y

∂ex =0 ∂z ∂ey =0 ∂z ∂ez =0 ∂z

Cylindrical Coordinate System

er = ex cos θ + ey sin θ, eθ = −ex sin θ + ey cos θ, ez = ez

(A.24)

726

USEFUL VECTOR AND TENSOR OPERATIONS z

ez eq z

er y

r

q x

Figure A.2

Vector components in the cylindrical coordinate system.

The unit vectors for the cylindrical coordinate system are shown in Fig. A.2. ∂ez = 0, ∂z ∂er = 0, ∂z ∂eθ = 0, ∂z A.3.3

∂ez = 0, ∂r ∂er = 0, ∂r ∂eθ = 0, ∂r

∂ez =0 ∂θ ∂er = eθ ∂θ ∂eθ = −er ∂θ

(A.25)

Spherical Coordinate System

The unit vectors for the spherical coordinate system are shown in Figure A.3. er = ex sin θ cos φ + ey sin θ sin φ + ez cos θ, eθ = ex cos θ cos φ + ey cos θ sin φ − ez sin θ, eφ = −ex sin φ + ey cos φ, where, 0 ≤ θ ≤ π and 0 ≤ φ ≤ 2π

z er

q f

ef

r

y eq

x

Figure A.3 Vector components in the spherical coordinate system.

DOT PRODUCTS

727

u3

h3du3 e3 dr h2du2 e2

u2

h1du1 e1

u1

Figure A.4 Vector components in the curvilinear coordinate system.

∂er = 0, ∂r ∂eφ = 0, ∂r ∂eφ = 0, ∂r A.3.4

∂er ∂er = eθ , = eφ sin θ ∂θ ∂φ ∂eθ ∂eθ = −er , = eφ cos θ ∂θ ∂φ ∂eφ ∂eφ = 0, = −er sin θ − eθ cos θ ∂θ ∂φ

(A.26)

Curvilinear Coordinate System

dr =

∂r ∂r ∂r du1 + du2 + du3 =h1 du1 e1 + h2 du2 e2 + h3 du3 e3 ∂u1 ∂u2 ∂u3

The vector components of dr for the curvilinear coordinate system are shown in Figure A.4. e2 ∂h1 ∂e1 =− − ∂x1 h2 ∂x2 ∂e2 1 ∂h1 = e1 , ∂x1 h2 ∂x2 ∂e3 1 ∂h1 = e1 , ∂x1 h3 ∂x3

e3 ∂h1 ∂e1 1 ∂h2 ∂e1 , = e2 , = h3 ∂x3 ∂x2 h1 ∂x1 ∂x3 ∂e2 e3 ∂h2 e1 ∂h2 ∂e2 =− − , = ∂x2 h3 ∂x3 h1 ∂x1 ∂x3 ∂e3 1 ∂h2 ∂e3 e1 ∂h3 = e2 , =− ∂x2 h3 ∂x3 ∂x3 h1 ∂x1

1 ∂h3 e3 h1 ∂x1 1 ∂h3 e3 h2 ∂x2 e2 ∂h3 − h2 ∂x2

where h1 , h2 , and h3 are called scale factors. A.4 DOT PRODUCTS

The scalar product (dot product) of two vectors produces a scalar.

(A.27)

728

A.4.1

USEFUL VECTOR AND TENSOR OPERATIONS

Cartesian Coordinate System

A · B = |A| |B| cos θAB = Ax Bx + Ay By + Az Bz A.4.2

Cylindrical Coordinate System

A · B = Az Bz + Ar Br + Aθ Bθ A.4.3

(A.30)

Curvilinear Coordinate System

A · B = A1 B1 + A2 B2 + A3 B3 A.5

(A.29)

Spherical Coordinate System

A · B = Ar Br + Aθ Bθ + Aφ Bφ A.4.4

(A.28)

(A.31)

CORSS PRODUCTS

The vector product (cross product) of two vectors produces a vector. In general, for a three-dimensional orthogonal coordinate system, e1 e 2 e 3 where A ≡ A1 e1 + A2 e2 + A3 e3 A × B = A1 A2 A3 and B ≡ B1 e1 + B2 e2 + B3 e3 B1 B2 B3 A.5.1

Cartesian Coordinate System

A × B = (Ay Bz − Az By )ex − (Ax Bz − Az Bx )ey + (Ax By − Ay Bx )ez (A.32) A.5.2

Cylindrical Coordinate System

A × B = (Ar Bθ − Aθ Br )ez − (Az Bθ − Aθ Bz )er + (Az Br − Ar Bz )eθ A.5.3

(A.33)

Spherical Coordinate System

A × B = (Aθ Bφ − Aφ Bθ )er − (Ar Bφ − Aφ Br )eθ + (Ar Bθ − Aθ Br )eφ (A.34)

GRADIENT OF A SCALAR

A.5.4

729

Curvilinear Coordinate System

A × B = (A2 B3 − A3 B2 )e1 − (A1 B3 − A3 B1 )e2 + (A1 B2 − A2 B1 )e3 (A.35)

A.6 DIFFERENTIATION OF VECTORS

∂ ∂A ∂B (A + B) = + ∂x ∂x ∂x ∂A ∂B ∂ (A · B) = B · +A· ∂x ∂x ∂x ∂ ∂A ∂B (A × B) = ×B+A× ∂x ∂x ∂x

(A.36) (A.37) (A.38)

Chain rule can be applied to any vector A that is a function of spatial coordinates x 1 , x 2 , and x 3 such that A = A (x 1 , x 2 , x 3 ). Therefore, dA =

∂A ∂A ∂A dx1 + dx2 + dx3 ∂x1 ∂x2 ∂x3

(A.39)

A.7 GRADIENT OF A SCALAR

When a scalar ﬁeld S is a function of independent spatial coordinates x 1 , x 2 , and x 3 such that S = S(x1 , x 2 , x 3 ), the gradient of such scalar ﬁeld is a vector. This operation is described in different coordinate systems as explained follows.

A.7.1

Cartesian Coordinate System

∇S =

A.7.2

(A.40)

Cylindrical Coordinate System

∇S =

A.7.3

∂S ∂S ∂S ex + ey + ez ∂x ∂y ∂z

∂S ∂S 1 ∂S ez + er + eθ ∂z ∂r r ∂θ

(A.41)

Spherical Coordinate System

∇S =

∂S 1 ∂S 1 ∂S er + eθ + eφ ∂r r ∂θ r sin θ ∂φ

(A.42)

730

A.7.4

USEFUL VECTOR AND TENSOR OPERATIONS

Curvilinear Coordinate System

∇S = grad S = A.8

1 ∂S 1 ∂S 1 ∂S e1 + e2 + e3 h1 ∂x1 h2 ∂x2 h3 ∂x3

(A.43)

GRADIENT OF A VECTOR

The gradient of a vector is a second-order tensor. Velocity is one such vector, and its gradient is called strain rate, which is a second-order tensor. Strain rate is an important parameter in the solution of the Navier-Stokes equations. A.8.1

Cartesian Coordinate System

⎛ ∂V

x

⎜ ∂x ⎜ ⎜ ∂Vx ∇V = ⎜ ⎜ ∂y ⎜ ⎝ ∂V x

∂z A.8.2

A.8.3

∂Vy ∂x ∂Vy ∂y ∂Vy ∂z

Cylindrical Coordinate System ⎛ ∂Vz ∂Vr ⎜ ∂z ∂z ⎜ ⎜ ∂V ∂V z r ⎜ ∇V = ⎜ ∂r ⎜ ∂r ⎜

⎝ 1 ∂Vz 1 ∂Vr Vθ − r ∂θ r ∂θ r

∂Vz ⎞ ∂x ⎟ ⎟ ∂Vz ⎟ ⎟ ∂y ⎟ ⎟ ∂V ⎠

(A.44)

z

∂z ⎞ ∂Vθ ⎟ ∂z ⎟ ⎟ ∂Vθ ⎟ ⎟ ∂r ⎟

⎟ 1 ∂Vθ Vr ⎠ + r ∂θ r

(A.45)

Spherical Coordinate System ⎛

⎞ ∂Vr ∂Vθ ∂Vφ ⎜ ⎟ ∂r ∂r ∂r ⎜

⎟

⎜ 1 ∂V ⎟ V 1 ∂V V 1 ∂V r θ θ r φ ⎜ ⎟ − + ∇V= ⎜ ⎟ ⎜ r ∂θ ⎟ r r ∂θ r r ∂θ ⎜

⎟

⎝ ⎠ 1 ∂Vφ Vφ 1 ∂Vφ Vr Vθ 1 ∂Vr Vφ − − cot θ + + cot θ r sin θ ∂φ r r sin θ ∂φ r r sin θ ∂φ r r (A.46)

A.9

CURL OF A VECTOR

The curl of a vector V is a measure of rotation of this vector in a given coordinate system. The curl of vector V is also a vector, where V is a function of independent spatial coordinates x 1 , x 2 , and x 3 such that V = V(x 1 , x 2 , x 3 ).

DIVERGENCE OF A VECTOR

Cartesian Coordinate System

∂Vy ∂Vy ∂Vz ∂Vz ∂Vx ∂Vx ∇ ×V= − ex − − ey + − ez ∂y ∂z ∂x ∂z ∂x ∂y

731

A.9.1

(A.47)

A.9.2

Cylindrical Coordinate System

∂Vθ ∂Vr 1 ∂(rVθ ) ∂Vr 1 ∂Vz ∂Vz ez − er + eθ ∇ ×V= − − − r ∂r ∂θ ∂z r ∂θ ∂z ∂r (A.48)

A.9.3

Spherical Coordinate System

1 1 ∂(rVφ ) ∂(Vφ sin θ) ∂Vθ 1 ∂Vr ∇ ×V= er − − − eθ r sin θ ∂θ ∂φ r ∂r sin θ ∂φ

1 ∂(rVθ ) ∂Vr eφ + (A.49) − r ∂r ∂θ

A.9.4

Curvilinear Coordinate System

h 1 e 1 h2 e 2 h3 e 2 ∂ ∂ 1 ∂ (A.50) ∇ × V = curl V = h1 h2 h3 ∂x1 ∂x2 ∂x3 h V h V h V 1 1 2 2 3 3

1 1 ∂(h3 V3 ) ∂(h2 V2 ) ∂(h3 V3 ) ∂(h1 V1 ) ∇ ×V= − − e1 − e2 h2 h3 ∂x2 ∂x3 h1 h3 ∂x1 ∂x3

1 ∂(h2 V2 ) ∂(h1 V1 ) + − (A.51) e3 h1 h2 ∂x1 ∂x2 A.10

DIVERGENCE OF A VECTOR

The divergence of a vector is a scalar. A.10.1

Cartesian Coordinate System

∇ ·V= A.10.2

∂Vy ∂Vx ∂Vz + + ∂x ∂y ∂z

(A.52)

Cylindrical Coordinate System

∇ ·V=

∂Vz 1 ∂ (rVr ) 1 ∂Vθ + + ∂z r ∂r r ∂θ

(A.53)

732

USEFUL VECTOR AND TENSOR OPERATIONS

A.10.3

Spherical Coordinate System

1 ∂ r 2 Vr 1 ∂ (Vθ sin θ) 1 ∂Vφ ∇ ·V= 2 + + r ∂r r sin θ ∂θ r sin θ ∂φ A.10.4

Curvilinear Coordinate System

∇ · V = div V = A.11

(A.54)

∂ (h2 h3 V1 ) ∂ (h3 h1 V2 ) ∂ (h1 h2 V3 ) 1 + + h1 h2 h3 ∂x1 ∂x2 ∂x3

(A.55)

DIVERGENCE OF A TENSOR

The divergence of a second-order tensor produces a vector. In tensor notation (or index notation), a tensor is written as: τ ≡ τ = τij ei ej The divergence operator is written as: ∇ = ei

∂ ∂xi

Therefore, in tensor notation, the divergence of a tensor is given as: ∇ · τ = ei

∂τij ∂τij ∂ · τij ei ej = ei · ei ej = ej ∂xi ∂xi ∂xi

Using Einstein’s summation, the above can be written as:

∂τij ∂τ1j ∂τ2j ∂τ3j ∇ ·τ = ej = + + ej ∂xi ∂x1 ∂x2 ∂x3 In matrix form, the above can be written as: ⎛ τ11 ⎜ ∇ · τ ≡ ∇ · τ = e1 ∂x∂ 1 e2 ∂x∂ 2 e3 ∂x∂ 3 ⎝τ21

τ12 τ22

τ13

⎞

⎟ τ23 ⎠

τ31 τ32 τ33 ∂τ12 ∂τ13 ∂τ22 ∂τ23 ∂τ21 ∂τ11 + + + + e1 + e2 = ∂x1 ∂x2 ∂x3 ∂x1 ∂x2 ∂x3 ∂τ31 ∂τ32 ∂τ33 + + + e3 ∂x1 ∂x2 ∂x3

Tensor notation: ∇ ·τ =

∂τij ej = ∂xi

∂τ1j ∂τ2j ∂τ3j + + ∂x1 ∂x2 ∂x3

ej

DIVERGENCE OF A TENSOR

A.11.1

Cartesian Coordinate System ∂τyx ∂τxy ∂τyy ∂τyz ∂τxx ∂τxz ex + ey ∇·τ = + + + + ∂x ∂y ∂z ∂x ∂y ∂z ∂τzy ∂τzx ∂τzz + + + ez ∂x ∂y ∂z

733

(A.56)

A.11.2

Cylindrical Coordinate System ∂τzz 1 ∂τθz ∂τrz 1 ∂τrθ τθθ 1 ∂ 1 ∂ ∇·τ = + (rτrz ) + ez + + (rτrr ) + − er ∂z r ∂r r ∂θ ∂z r ∂r r ∂θ r ∂τθz ∂τrθ 1 ∂τθθ 2τrθ + (A.57) + + + eθ ∂z ∂r r ∂θ r

A.11.3

Spherical Coordinate System 1 ∂ 2 ∂ 1 1 ∂τrφ τθθ + τφφ er ∇ · τ = 2 (r τrr ) + (τrθ sin θ) + − r ∂r r sin θ ∂θ r sin θ ∂φ r 1 ∂ 2 ∂ 1 1 ∂τθφ τrθ τφφ cotθ eθ + 2 (r τrθ )+ (τθθ sin θ)+ + − r ∂r r sin θ ∂θ r sin θ ∂φ r r 1 ∂ 2 1 ∂τθφ 1 ∂τφφ τrφ 2τθφ cotθ + τ ) + (r + + + eφ rφ r 2 ∂r r ∂θ r sin θ ∂φ r r (A.58)

A.11.4

Curvilinear Coordinate System ⎫ ⎧ ∂ ∂ ∂ 1 ⎪ ⎪ ⎪ (h2 h3 τ11 ) + (h3 h1 τ21 ) + (h1 h2 τ31 ) ⎪ ⎬ ⎨ h1 h2 h3 ∂x1 ∂x2 ∂x3 e1 ∇ ·τ = ⎪ ⎪ τ31 ∂h1 τ22 ∂h2 τ33 ∂h3 τ ∂h1 ⎪ ⎪ ⎭ ⎩ + 12 + − − h1 h2 ∂x2 h1 h3 ∂x3 h1 h2 ∂x1 h1 h3 ∂x1 ⎫ ⎧ ∂ ∂ ∂ 1 ⎪ ⎪ ⎪ (h2 h3 τ12 ) + (h3 h1 τ22 ) + (h1 h2 τ32 ) ⎪ ⎬ ⎨ h1 h2 h3 ∂x1 ∂x2 ∂x3 e2 + ⎪ ⎪ τ12 ∂h2 τ33 ∂h3 τ11 ∂h1 τ ∂h2 ⎪ ⎪ ⎭ ⎩ + 23 + − − h2 h3 ∂x3 h2 h1 ∂x1 h2 h3 ∂x2 h2 h1 ∂x2 ⎫ ⎧ ∂ ∂ ∂ 1 ⎪ ⎪ ⎪ ⎪ (h h τ ) + (h h τ ) + (h h τ ) 2 3 13 3 1 23 1 2 33 ⎬ ⎨ h1 h2 h3 ∂x1 ∂x2 ∂x3 + e3 ⎪ ⎪ τ23 ∂h3 τ11 ∂h1 τ22 ∂h2 τ ∂h3 ⎪ ⎪ ⎭ ⎩ + 31 + − − h1 h3 ∂x1 h3 h2 ∂x2 h3 h1 ∂x3 h3 h2 ∂x3 (A.59)

734

USEFUL VECTOR AND TENSOR OPERATIONS

A.12

LAPLACIAN OF A SCALAR

The Laplacian operator is deﬁned as: ∇ 2 ≡ ∇ · ∇ = ei

∂2 ∂2 ∂2 ∂2 ∂2 ∂ ∂ ∂2 ·ei = ei · ei 2 = 2 = = 2+ 2+ 2 ∂xi ∂xi ∂xi ∂xi ∂xi ∂xi ∂x1 ∂x2 ∂x3

Laplacian of a scalar quantity is also a scalar. A.12.1

Cartesian Coordinate System

∇ 2S = A.12.2

(A.60)

Cylindrical Coordinate System

∇ 2S = A.12.3

∂ 2S ∂ 2S ∂ 2S + 2 + 2 2 ∂x ∂y ∂z

∂ 2S 1 ∂ + 2 ∂z r ∂r

r

∂S ∂r

+

1 ∂ 2S r 2 ∂θ 2

(A.61)

Spherical Coordinate System

∇ 2S =

1 ∂ r 2 ∂r

r2

∂S ∂r

+

1 ∂ r 2 sin θ ∂θ

sin θ

∂S ∂θ

+

∂ 2S r 2 sin θ ∂φ 2 1

2

(A.62)

A.12.4

Curvilinear Coordinate System

1 ∂ h2 h3 ∂S ∂ h3 h1 ∂S ∂ h1 h2 ∂S ∇ 2S = + + h1 h2 h3 ∂x1 h1 ∂x1 ∂x2 h2 ∂x2 ∂x3 h3 ∂x3 (A.63)

A.13

LAPLACIAN OF A VECTOR

The Laplacian of a vector is a vector. It can be expressed in this vector identity: ∇ 2 V = ∇(∇ · V) − ∇ × (∇ × V) A.13.1

(A.64)

Cartesian Coordinate System

∇ 2V =

2 ∂ Vy ∂ 2 Vy ∂ 2 Vy ∂ 2 Vx ∂ 2 Vx ∂ 2 Vx + + + + + e ey x ∂x 2 ∂y 2 ∂z2 ∂x 2 ∂y 2 ∂z2 2 ∂ Vz ∂ 2 Vz ∂ 2 Vz + + + (A.65) ez ∂x 2 ∂y 2 ∂z2

LAPLACIAN OF A VECTOR

Cylindrical Coordinate System 2

∂ Vz 1 ∂ ∂Vz 1 ∂ 2 Vz ∇ 2V = + ez r + ∂z2 r ∂r ∂r r 2 ∂θ 2 2

∂ Vr ∂ 1 ∂(rVr ) 2 ∂Vθ 1 ∂ 2 Vr + + − + er ∂z2 ∂r r ∂r r 2 ∂θ 2 r 2 ∂θ

2 ∂ Vθ 1 ∂ 2 Vθ ∂ 1 ∂(rVθ ) 2 ∂Vr + 2 eθ + + + 2 ∂z2 ∂r r ∂r r ∂θ 2 r ∂θ

735

A.13.2

(A.66)

A.13.3

Spherical Coordinate System

⎤ ⎡ 1 ∂(r 2 Vr ) 1 ∂ ∂Vr 1 ∂ ∂ 2 Vr + 2 sin θ + ⎢ ∂r r 2 ∂r r sin θ ∂θ ∂θ r 2 sin2 θ ∂φ 2 ⎥ ⎥ er ∇ 2V = ⎢ ⎣ 2 ∂(Vθ sin θ) 2 ∂Vφ ⎦ − 2 − 2 r sin θ ∂θ r sin θ ∂φ

⎤ ⎡ 1 ∂ 1 ∂ 1 ∂(Vθ sin θ) 1 ∂ 2 Vθ 2 ∂Vθ r + + ⎢ r 2 ∂r ∂r r 2 ∂θ sin θ ∂θ r 2 sin2 θ ∂φ 2 ⎥ ⎥ eθ +⎢ ⎣ 2 ∂Vr 2 cos θ ∂Vφ ⎦ + 2 − 2 2 r ∂θ r sin θ ∂φ

⎤ ⎡ 1 ∂ 1 ∂ 1 ∂(Vφ sin θ) 1 ∂ 2 Vφ 2 ∂Vφ r + + ⎢ r 2 ∂r ∂r r 2 ∂θ sin θ ∂θ r 2 sin2 θ ∂φ 2 ⎥ ⎥ eφ +⎢ ⎣ 2 2 cos θ ∂Vθ ⎦ ∂Vr + + 2 2 r sin θ ∂φ r 2 sin2 θ ∂φ (A.67)

A.13.4

Curvilinear Coordinate System ⎛ ⎡ ⎞⎤ h2 ∂(h1 V1 ) ∂(h3 V3 ) ∂ − ⎢1 ∂ ⎟⎥ 1 ⎜ ∂x3 ∂x1 1 h3 ⎜ ∂x3 h ⎟⎥e1 ∇ 2V = ⎢ ⎣ h1 ∂x1 (∇·V) + h2 h3 ⎝ ∂ h3 ∂(h2 V2 ) ∂(h1 V1 ) ⎠⎦ − − ∂x2 h1 h2 ∂x1 ∂x2 ⎛ ⎡ ⎞⎤ h3 ∂(h2 V2 ) ∂(h1 V1 ) ∂ − ⎢1 ∂ ⎟⎥ 1 ⎜ ∂x1 ∂x2 2 h1 ⎜ ∂x1 h ⎥e2 ⎟ +⎢ ⎣ h2 ∂x2 (∇·V) + h1 h3 ⎝ ∂ h1 ∂(h3 V3 ) ∂(h2 V2 ) ⎠⎦ − − ∂x3 h2 h3 ∂x2 ∂x3 ⎛ ⎡ ⎞⎤ h1 ∂(h3 V3 ) ∂(h2 V2 ) ∂ − ⎢1 ∂ ⎟⎥ 1 ⎜ ∂x2 ∂x3 2 h3 ⎜ ∂x2 h ⎥e3 ⎟ +⎢ (∇·V) + ⎝ ⎣ h3 ∂x3 ∂ h2 ∂(h1 V1 ) ∂(h3 V3 ) ⎠⎦ h1 h2 − − ∂x1 h1 h3 ∂x3 ∂x1 (A.68)

736

USEFUL VECTOR AND TENSOR OPERATIONS

A.14

VECTOR IDENTITIES

In the next equations, φ and ψ are assumed to be continuous, differentiable scalars. The vectors V, A, and B are also assumed to be continuous and differentiable. ∇ (φ + ψ) = ∇φ + ∇ψ

(A.69)

∇ (φψ) = ψ (∇φ) + φ (∇ψ)

(A.70)

∇ (A · B) = (B · ∇) A + (A · ∇) B + B × (∇ × A) + A × (∇ × B) (A.71) ∇ · (A + B) = ∇ · A + ∇ · B

(A.72)

∇ · (φV) = (∇φ) · V + φ(∇ · V)

(A.73)

∇ · (A × B) = B · ∇ × A − A · ∇ × B

(A.74)

or ∇ · (A × B) = B · (∇ × A) − A · (∇ × B)

(A.75)

∇ × (φV) = (∇φ) × V + φ(∇ × V)

(A.76)

∇ × (A + B) = ∇ × A + ∇ × B

(A.77)

∇ × (A × B) = (B · ∇)A − B(∇ · A) − (A · ∇)B + A(∇ · B)

(A.78)

∇ · (∇ × V) = 0

(A.79)

∇ × ∇φ = 0

(A.80)

∇ 2 φ = ∇ · ∇φ

(A.81)

∇ V = (∇ · ∇)V 2

(A.82)

∇ × (∇ × V) = ∇(∇ · V) − ∇ V 2

(A.83)

1 ∇(V · V) − V × (∇ × V) 2 (A · ∇)B = 1/2 [∇(A · B) − ∇ × (A × B) − B × (∇ × A)

(V · ∇)V =

−A × (∇ × B) − B(∇ · A) + A(∇ · B)] ! " |A|2 A · ∇A = ∇ + (∇ × A) × A 2 ∇ · (AB) ≡ ∇ · (A; B) = B (∇ · A) + A · (∇B)

(A.84)

(A.85) (A.86) (A.87)

Consider the identity tensor I and its vector operations. We have these identities: ds · I = ds (A.88) ∇ · (pI) = ∇p ∇r = I

(A.89) (A.90a)

GAUSS DIVERGENCE THEOREM

737

where r is the position vector, r = x1 e1 + x2 e2 + x3 e3 ∇ ·r=3

(A.90b) (A.91)

∇ ×r=∇ ×∇

2

r 2

=0

∇ · (τ × r) = (∇ · τ ) × r + τ × ∇r I : ∇q = ∇ · q

(A.92) (A.93) (A.94)

Table A.1 gives the vector and tensor notations for certain quantities.

A.15

GAUSS DIVERGENCE THEOREM

Consider a control volume (CV) that is enclosed by the control surface (CS), with the local surface orientation described by the outward normal unit vector n. The Gauss divergence theorem relates the volume integrals to the surface integrals by the next equation: ###

## ## ∇ · (ρU) dV = ρU · n dS = ρU · dS

CV

CS

(A.95)

CS

In this equation, ρ is a scalar and U is a vector. The physical interpretation of the divergence theorem is that, in the absence of the source or sink, the vector ﬁeld ρU within a control volume can change only by having it ﬂow into or away from the control volume through its bounding control surface. Therefore, the volume integral of the divergence of the vector inside the control volume is equal to the net ﬂux of the ρU through its control surface. The Gauss theorem for a scalar ﬁeld ψ is as: ### CV

## (∇ψ) dV = ψn dA

(A.96)

CS

The Gauss theorem is also applicable to a second-order tensor ﬁeld, such as that of the stress tensor: ### ## (A.97) (∇ · ρτ ) dV = ρτ · n dA CV

CS

738

USEFUL VECTOR AND TENSOR OPERATIONS

TABLE A.1. Vector and Tensor Notations for Useful Quantities Vector Notation → a or − a or − a → τ or τ or τ a·b a×b ∇

Tensor Notation

Quantity

ai e i τij ei ej a i bi εij k ai bj ek ∂ ei ∂xi

Vector Tensor Scalar Vector Vector

∇s

∂s ei ∂xi

Vector

∇a

∂aj ei ej ∂xi

Tensor

∂ai ∂xi

Scalar

∂ak ei ∂xj

Vector

∇ ·a ∇ ×a

εij k

∇2

∂2 ∂xi ∂xi

Scalar

∇ ·τ

∂τij ej ∂xi

Vector

a·∇

aj

∂ ∂xj

Scalar

∂bi ei ∂xj

Vector

∂s ∂xj

Scalar

a · ∇b a · ∇s a × ∇s a × ∇b ∂b + a · ∇b ∂t

aj

aj

εij k aj

∂s ei ∂xk

∂bl ei el ∂xk

∂bi ∂bi ei + aj ∂t ∂xj εij k aj

Vector Dyadic or Tensor Vector

a · (∇ · τ )

aj

∂τij ∂xi

Scalar

τ : ∇a

τij

∂ai ∂xj

Scalar

SUBSTANTIAL DERIVATIVE OF A VECTOR

A.16

739

TOTAL OR MATERIAL OR SUBSTANTIAL DERIVATIVE

The substantial derivative is also known as total derivative or material derivative. It is known as the material derivative because of its application in describing the time rate of change of the properties of a ﬂuid or material particle in the Lagrangian frame of reference, which follows the motion of the ﬂuid particle. It is called the total derivative since it includes the derivatives with respect to time and all spatial variables. A.16.1

Cartesian Coordinate System

D ∂ ∂ ∂ ∂ = + vx + vy + vz Dt ∂t ∂x ∂y ∂z or

∂ D ∂ ∂ ∂ ∂ ∂ + v2 + v3 = = + v1 + vi Dt ∂t ∂x1 ∂x2 ∂x3 ∂t ∂xi

A.16.2

(A.100)

Curvilinear Coordinate System

v2 ∂ v3 ∂ D ∂ v1 ∂ + + = + Dt ∂t h1 ∂x1 h2 ∂x2 h3 ∂x3 A.17

(A.99)

Spherical Coordinate System

D ∂ ∂ vθ ∂ vφ ∂ = + vr + + Dt ∂t ∂r r ∂θ r sin θ ∂φ A.16.4

(A.98b)

Cylindrical Coordinate System

D ∂ ∂ ∂ vθ ∂ = + vz + vr + Dt ∂t ∂z ∂r r ∂θ A.16.3

(A.98a)

(A.101)

SUBSTANTIAL DERIVATIVE OF A VECTOR

In vector form, the substantial derivative of a vector is deﬁned as DV ∂V ∂V ≡ + (V · ∇) V ≡ + V · (∇V) Dt ∂t ∂t In index or tensor notation, Equation A.102a can be written as:

∂vi ∂vi ∂ ∂vi Dvi ei vi ei = ei ≡ ei + vj ej · ej ei + v j Dt ∂t ∂xj ∂t ∂xj

(A.102a)

(A.102b)

740

USEFUL VECTOR AND TENSOR OPERATIONS

A.18

SYMMETRIC TENSORS

A familiar second-order symmetric tensor is the stress tensor, which can be written as: ⎛ ⎞ ⎛ ⎞ τ11 τ12 τ13 τii τij τik ⎜ ⎟ ⎜ ⎟ τ ≡ ⎝τ21 τ22 τ23 ⎠ or τ ≡ ⎝ τji τjj τjk ⎠ (A.103a) τ31

τ32

τ33

τki

τkj

τkk

In Equation A.103a, i , j , and k are the indices that correspond to the unit vectors ei , ej , and ek . Each of the above-mentioned tensor components has two unit vectors associated with it, shown in the next form, although this notation is rarely used: ⎛ ⎞ τii ei ei τij ei ej τik ei ek ⎜ ⎟ τ = ⎝ τji ej ei τjj ej ej τjk ej ek ⎠ (A.103b) τki ek ei τkj ek ej τkk ek ek The incremental force acting on an inﬁnitesimal plane can be determined by taking the dot product of the outward unit normal vector of such plane with the stress tensor. This can be shown as: dF = (dAn) · τ ⎛

or dF = dA (n · τ ) = dA ni ei

nj ej

(A.104) τii ei ei

⎜ nk ek ⎝ τji ej ei τki ek ei

τij ei ej τjj ej ej τkj ek ej

or

τik ei ek

⎞

⎟ τjk ej ek ⎠ τkk ek ek (A.105a)

$ % dF = dA (ni τii + nj τji + nk τki )ei (ni τij + nj τjj + nk τkj )ej (ni τik + nj τjk + nk τkk )ek (A.105b) The d F is also known as the stress vector and is expressed in the Cartesian coordinate system. In the cylindrical coordinate system, the stress tensor can be expressed as: ⎛

τrr

τrθ

τrz

⎞

⎜ τ ≡ ⎝τθr

τθθ

⎟ τθz ⎠

τzr

τzθ

τzz

(A.106)

The outward normal unit vector of any plane can be written as: n = nr er + nθ eθ + nz ez

(A.107)

DIRECTION COSINES

So the inﬁnitesimal stress vector, d F, can be written as: ⎞ ⎛ τrr er er τrθ er eθ τrz er ez

⎜ ⎟ dF = dA nr er nθ eθ nz ez · ⎝τθr eθ er τθθ eθ eθ τθz eθ ez ⎠ τzr ez er τzθ ez eθ τzz ez ez

741

(A.108)

The stress vector can be shown in the column-vector form as: dFT = (dAn · τ )T = τ T · nT dA

(A.109)

Since the stress tensor is symmetric, τ T = τ . Therefore, the stress vector can be represented in column-vector form as: ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ dFr er τrr er er τrθ er eθ τrz er ez nr er ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ (A.110) ⎝dFθ eθ ⎠ = dA ⎝τθr eθ er τθθ eθ eθ τθz eθ ez ⎠ · ⎝nθ eθ ⎠ dFz ez

τzr ez er

τzθ ez eθ

τzz ez ez

nz ez

In either case, carrying out the matrix-vector multiplication provides this expression for the stress vector: ⎡ ⎤ (nr τrr + nθ τθr + nz τzr )er ⎢ ⎥ dF = ⎣ + (nr τrθ + nθ τθθ + nz τzθ )eθ ⎦ dA (A.111) + (nr τrz + nθ τθz + nz τzz )ez

A.19

DIRECTION COSINES

Direction cosines are used to deﬁne the direction of a vector in an orthogonal coordinate system. They play an essential role in coordinate transformations. As shown in Figure A.5, the position vector r for a point P in a Cartesian coordinate system (x 1 , x 2 , x 3 ) has three components in each direction.

x3

γ

r β

α r2 = r2e2

P(x1,x2,x3) r3 = r3e3 x2 r1 = r1e1

x1

Figure A.5 Direction cosines of a position vector in the Cartesian coordinate system.

742

USEFUL VECTOR AND TENSOR OPERATIONS

The position vector r can be expressed in terms of unit vector in the three directions as: r = r1 + r2 + r3 = r1 e1 + r2 e2 + r3 e3 (A.112) By using the Pythagorean theorem, we get the magnitude of vector r as: |r| =

r12 + r22 + r32

(A.113)

The unit vector in the direction of this position vector is: r1 r2 r3 r = e1 + e2 + e3 = e1 cos α + e2 cos β + e3 cos γ |r| |r| |r| |r|

(A.114)

where the three angles α, β, and γ describe the direction of the vector with respect to the three orthogonal coordinate axes. The direction cosines are deﬁned as the cosines of these angles. The length of r can be projected onto each coordinate axis as r1 = |r| cos α,

r2 = |r| cos β,

r3 = |r| cos γ

(A.115)

Using Equation A.113, we have the next trigonometric identity among the direction cosines: cos2 α + cos2 β + cos2 γ = 1 (A.116) The cosine of the angle between any two vectors, say A and B, can be written in terms of the pair-wise products of the direction cosines of the two vectors: cos θAB = cos αA cos αB + cos βA cos βB + cos γA cos γB

(A.117)

Thus, if A and B are orthogonal, cos αA cos αB + cos βA cos βB + cos γA cos γB = 0

(A.118)

which often proves to be a useful relationship in coordinate transformations. When A has unit length (i.e., a unit vector n), the direction cosines are the components of the unit vector: n = n1 e1 + n2 e2 + n3 e3

(A.119)

where n1 = cos α,

n2 = cos β,

n3 = cos γ

(A.120)

Of course, it must also follow that n21 + n22 + n23 = 1

(A.121)

COORDINATE TRANSFORMATIONS

743

For cylindrical coordinate system, if we are concerned with a small displacement in the θ-direction, then the curvature of the θ-axis does not need to be accounted for determination of direction cosine. Thus, we have: n2r + n2θ + n2z = 1 A.20

(A.122)

COORDINATE TRANSFORMATIONS

Consider that an orthogonal coordinate system O (x 1 , x 2 , x 3 ) is given an arbitrary rigid rotation. As shown in Figure A.6, the new coordinate system is called O*(x1∗ , x2∗ , x3∗ ). In the coordinate system O(x 1 , x 2 , x 3 ), a point P has coordinates (r 1 , r 2 , r 3 ); in the new frame O*(x1∗ , x2∗ , x3∗ ), the point P has coordinates (r1∗ , r2∗ , r3∗ ). The cosine of angle between i th axis of O and j th axis of O* is nij , where i , j =1, 2, 3. Therefore, the ni1 , ni2 , ni3 are the direction cosines of Oi in the new coordinate system O* and n1j , n2j , n3j are the direction cosines of O*j in the original coordinate system. Thus, we have nine direction cosines, which form this matrix: ⎛ ⎞ n11 n12 n13 ⎜ ⎟ (A.123) N = nij = ⎝n21 n22 n23 ⎠ n31

n32

n33

Therefore, vector rj (which is the projection of vector r on the j th axis in the original coordinate system) can be related to the new coordinate system as: rj = n1j e∗1 + n2j e∗2 + n3j e∗3

(A.124)

The position vector r can therefore be written as: r = N · r∗

(A.125)

x3 x3*

P r or r*

x2* x2

x1

Figure A.6

x1*

Transformation of the coordinate systems by rotation.

744

USEFUL VECTOR AND TENSOR OPERATIONS

Alternatively, vector r∗i (which is the projection of vector r* on the i th axis in the new coordinate system O*) can be related to the original coordinate system as: r∗i = ni1 e1 + ni2 e2 + ni3 e3 (A.126) The coordinates of point P in the coordinate system are represented by position vector r*, and they can be related to the original coordinate system by: r∗ = NT · r

(A.127)

Substituting Equation A.125 into Equation A.127, we can show that the transpose of the direction-cosine matrix is also its inverse, that is NT = N−1 , by observing that

(A.128) r∗ = NT · r = NT · N · r∗ = NT · N · r∗ or

NT · N = I or

NT = N−1

(A.129)

Equations A.128 and A.129 provide following identities among direction cosines. From the diagonal elements of the product NT · N, we get: n2i1 + n2i2 + n2i3 = 1, n21j + n22j + n23j = 1,

(A.130)

From the off-diagonal terms of the product NT · N, we get: ni1 np1 + ni2 np2 + ni3 np3 = 0,

i = p

n1j n1l + n2j n2l + n3j n3l = 0,

j = l

(A.131)

This condition expresses the fact that in the original coordinate system, the axes are mutually orthogonal; so are the axes in the new coordinate system. In addition to representing a vector in a rotated coordinate system, the matrix of direction cosines can also be used to transform a tensor (e.g., the viscous stress tensor) into the new coordinates system, as shown next: τ ∗ = NT τ N

(A.132)

Assume that the stress tensor in the original and new coordinate systems is represented by τ and τ ∗ , respectively. Let us consider a surface that has an outward normal unit vector n in the original coordinate system and the same vector denoted n∗ in the new coordinate system. In this case, the stress vector on the surface can be represented as F=τ ·n

and

F∗ = τ ∗ · n∗

(A.133)

745

PRINCIPAL AXES OF STRESS AND NOTION OF ISOTROPY

By using Equation A.127, the two stress vectors can be related by the directioncosine matrix as: F∗ = NT · F (A.134) Substituting F = τ · n into Equation A.134 and using F∗ = τ ∗ · n∗ from Equation A.133, we get: F∗ = NT · (τ · n) = τ ∗ · n∗

(A.135)

Substituting the relationship n∗ = NT · n into Equation A.135 and rearranging the terms in this equation, we have:

NT · τ · n = τ ∗ · NT · n

or

NT · τ = τ ∗ · NT

(A.136)

Right-multiplying both sides by matrix N and using Equation A.129 yields: τ ∗ = NT τ N

(A.137)

Left-multiplying both sides by matrix N and using Equation A.129 yields: τ = Nτ ∗ NT A.21

(A.138)

PRINCIPAL AXES OF STRESS AND NOTION OF ISOTROPY

The total stress tensor acting on a ﬂuid particle has two parts: One part is due to the hydrostatic pressure that acts normal to all surfaces; the other is the stresses due to the ﬂuid viscosity. This can be shown as: σij = −pδij + τij

(A.139)

The diagonal components of the stress tensor (σ11 , σ22 , σ33 ) are known as normal stresses and the off-diagonal components (σ12 , σ13 , σ21 , σ23 , σ31 , σ32 ) are called shear stresses. For linear or Newtonian ﬂuids, the total stress tensor (σij ) is symmetric since the magnitude of the stress is proportional to the strain rate. Thus, there are only six independent stress components in the stress tensor and σij = σji . For the non-Newtonian ﬂuids, there are nonlinear effects such as stress-couples, which will result in nonsymmetric stress tensor at a point. It is possible to ﬁnd a new orthogonal coordinate system by rotation of the original coordinate axes such that the shear stresses in the total stress tensor vanish in the new coordinate system and only the diagonal components remain nonzero. The axes for such a rotated coordinate system are called the principal axes, and the diagonal components are called the principal stresses. The planes normal to each of the principal axes are called principal planes, where the corresponding

746

USEFUL VECTOR AND TENSOR OPERATIONS

stress vector F is parallel to the corresponding principal axis n and there are no shear stresses. Thus, F = ln

(A.140)

where l is the constant of proportionality and in this particular case corresponds to the magnitudes of the normal stress vectors or principal stresses. We know that the stress vector can be expressed as: F = σ · n; therefore, we have σ · n = ln

(A.141)

This expression can be rewritten as (σ − lI) · n = 0

(A.142)

where I is the identity matrix. This is a homogenous system (i.e., the righthand side is equal to zero) of three linear equations where n are the unknowns. To obtain a nontrivial (nonzero) solution for n, the determinant matrix of the coefﬁcients must be equal to zero. Thus, σ11 − l σ12 σ13 σ22 − l σ23 = 0 |σ − lI| = σ21 (A.143) σ31 σ32 σ33 − l Expanding the determinant leads to the next characteristic equation: |σ − lI| = l3 − I1 l2 − I2 l − I3 = 0

(A.144)

where I1 = σ11 + σ22 + σ33 = σkk 2

2 2 I2 = σ12 + σ23 + σ31 − (σ11 σ22 + σ22 σ33 + σ33 σ11 ) σ11 σ12 σ13 I3 = det σ = σ21 σ22 σ23 σ31 σ32 σ33

(A.145) (A.146) (A.147)

I 1 , I 2 , and I 3 are the ﬁrst, second, and third invariants of the stress tensors, respectively. It means that their values remain unchanged regardless of the orientation of the coordinate axes. The characteristic equation A.144 has three real roots because the symmetric stress tensor has real elements. These three roots (also called the eigenvalues of the stress tensor) are named the principal stresses. For each eigenvalue, a nontrivial solution for the corresponding normal vector n can be obtained by solving the next equation A.148 for three directions: (σ − li I) · ni = 0

i = 1, 2, 3

(A.148)

PRINCIPAL AXES OF STRESS AND NOTION OF ISOTROPY

747

The solution of Equation A.148 would give a vector ni , but it may not have a magnitude of unity. For ni to be a direction-cosine vector, it must be a unit vector, which imposes an additional constraint on the solution of Equation A.148. In order to make the solution of Equation A.148 a unit vector, a simple method can be employed. Let us arbitrarily select one component of vector ni to be unity; for instance, choose n1i = 1. Then normalize each component of the vector ni by its magnitude. This operation yields a new vector n i , which has same direction ni but has unity for its magnitude. The components of this normal vector are: 1 n2i n3i n 1i = , n 2i = , n 3i = 12 + n22i + n23i 12 + n22i + n23i 12 + n22i + n23i (A.149) The components of these normal vectors (n 1i , n 2i , n 3i ) should satisfy Equation A.150:

2

2 n 2 (A.150) 1i + n2i + n3i = 1 Thus, the unit vectors n i are the direction-cosine vectors of the principal axes. Since the new coordinate system that corresponds to these principal axes must also be an orthogonal system, further relationships must be satisﬁed among the unit-vector solutions: n i · n j = 0 and n i × n j = n k , (A.151) where i , j , and k correspond to the three coordinate directions. If all three directions are determined separately, whether the solution produces an orthogonal coordinate system can be checked via the above relationships (Equation A.151). Alternatively, once two directions have been determined, the third follows from the above relationships. In the coordinate system corresponding to the principal axes, the stress tensor has this form: ⎛ ⎞ l1 0 0 ⎜ ⎟ σ = ⎝ 0 l2 0 ⎠ (A.152) 0

0

l3

where the values of l1 , l2 , l3 are determined by solving Equation A.152. Since invariants do not depend on the orientation of coordinate system, they can be written as: I1 = l1 + l2 + l3 (A.153) I2 = − (l1 l2 + l2 l3 + l3 l1 ) l1 0 0 I3 = 0 l2 0 = l1 l2 l3 0 0 l3

(A.154) (A.155)

748

USEFUL VECTOR AND TENSOR OPERATIONS

The general constitutive relationship between total stress tensor and strain-rate for Newtonian ﬂuids is:

∂uj 2 ∂ui ∂uk

δij + μ + (A.156) σij = −pδij + τij = −pδij + μ − μ 3 ∂xk ∂xj ∂xi where μ and μ are bulk and dynamic viscosities of the ﬂuid, respectively. For the normal stresses in an incompressible ﬂuid, Equation A.156 reduces to: σii = −pδii

1 (σ11 + σ22 + σ33 ) = −p 3

or

(A.157)

In the coordinate system with principal axes, Equation A.157 can be written as:

l1 + l2 + l3 = −3p

(A.158)

An isotropic ﬂuid is such that simple direction stress acting in it does not produce a shearing deformation. Isotropy means that there is no internal sense of direction within the ﬂuid particle, so a normal stress should not produce any differential motion in planes parallel to its line of action. A.22

REYNOLDS TRANSPORT THEOREM

Consider a continuous function F (x, t) associated with a ﬂuid contained in a control volume V(t) that is moving with the ﬂuid and let Q(t) be deﬁned as: ### F (x, t) dV (A.159) Q (t) = CV

We want to calculate the material derivative of Q (t), but we cannot take the differential inside the volume integral since the volume is also changing with respect to time. Let us say that at time t = 0, the ﬂuid particle is at a position described by vector ξ , and at time t, its position vector can be expressed by x(ξ , t). At time t = 0, the volume of ﬂuid particle is dV 0 , and at time t, the volume is dV . These two can be related by Equation A.160: dV =

∂ (x1 , x2 , x3 ) dξ1 dξ2 dξ3 = JdV0 ∂ (ξ1 , ξ2 , ξ3 ) & '( ) & '( ) dV0

(A.160)

J

The Jacobian matrix J can also be written as: ∂x1 ∂x1 ∂x1 ∂ξ1 ∂ξ2 ∂ξ3 ∂x2 ∂x2 ∂x2 J ≡ ∂ξ1 ∂ξ2 ∂ξ3 ∂x3 ∂x3 ∂x3 ∂ξ ∂ξ2 ∂ξ3 1

(A.161)

REYNOLDS TRANSPORT THEOREM

749

It can be shown that the material derivative of J with respect to time is related to the diversion of velocity vector v: DJ = (∇ · v) J Dt

(A.162)

Note that the material derivative is deﬁned as: D ∂ ≡ +v·∇ Dt ∂t

(A.163)

We can now take material derivative of Equation A.159 and use the above relationships (Equation A.163) in the next operations: ### ### $ % D D D F (x, t) dV = F x (ξ , t) , t JdV0 Q (t) = Dt Dt Dt V (t)

### =

J V0

### = V0

### = V (t)

V0

DF DJ dV0 +F Dt Dt

DF + F (∇ · v) JdV0 Dt DF + F (∇ · v) dV Dt

(A.164)

Using the deﬁnition of material derivative in Equation A.164, we have: ### ### ∂F D F (x, t) dV = + ∇ · (F v) dV (A.165) Dt ∂t V (t)

V (t)

By applying Green’s theorem to the second integral on the right-hand side of Equation A.165, we have: ### ### ## ∂F D F (x, t) dV = F v · n dS (A.166) dV + Dt ∂t V (t)

V (t)

S(t)

In Equation A.166, the ﬁrst term on the right-hand side represents the time rate of change of property F in the instantaneous control volume. The second term represents the ﬂux associated with property F at the surface of the control volume. Consequently, the velocity v in this term is the local surface velocity. Thus, the Reynolds transport theorem represented by Equation A.166 states that the rate of change of any extensive property F of the ﬂuid in the control volume is equal to the time rate of change of F within the control volume and the net rate of ﬂux of the property F through the control surface.

APPENDIX B

CONSTANTS AND CONVERSION FACTORS OFTEN USED IN COMBUSTION

Universal Gas Constant Ru = 8.3144

kJ ft-lbf Btu = 1,545.4 = 1.9872 kmole · K lbm -mole · R lbm -mole · R

= 1.9872

cal erg atm · liter = 0.08206 = 8.3144 × 107 g-mole · K g-mole · K g-mole · K

= 83.144

bar · cm3 atm · cm3 g · cm = 82.057 = 84,786.85 f mole · K g-mole · K g-mole · K

= 0.729

atm · ft3 psia · ft3 J = 10.716 = 8.3144 lbm -mole · R lbm -mole · R g-mole · K

Dimensional Conversion Factor of Gravity lbm · ft g · cm kg · m =1 =1 m 2 2 2 lbf · s dyne · s N·s slug · ft g · cm kg · m =1 = 980.665 = 9.80665 m 2 lbf · s2 gf · s2 kgf · s

gc = 32.174

Gravitational Acceleration g = 9.80665 m/s2 = 32.17405 ft/s2 Fundamentals of Turbulent and Multiphase Combustion Copyright © 2012 John Wiley & Sons, Inc.

Kenneth K. Kuo and Ragini Acharya

751

752

CONSTANTS AND CONVERSION FACTORS OFTEN USED IN COMBUSTION

Avagadro’s Number ˜ = 6.02252 × 1023 NA = N Planck’s Constant h = 6.625 × 10−34

molecules g-mole

J·s molecule

Stefan-Boltzmann Constant W erg = 5.6699 × 10−5 2 4 ·K cm · s · K4 Btu cal = 0.1714 × 10−8 2 = 1.35514 × 10−12 2 4 cm · s · K4 ft · hr · R

σ = 5.6699 × 10−8

m2

Boltzmann’s Constant K = 1.38 × 10−23

J K · molecule

Atomic Mass Unit ma = 1.660540 × 10−27 kg Proton Mass mp = 1.672623 × 10−27 kg Electron Mass me = 9.109389 × 10−31 kg Electron Charge e = 1.602177 × 10−19 Coulombs Speed of Light c = 2.997925 × 108 m/s Work/Energy Conversion Factor ft-lbf g · cm J = 778 = 42,664.9 f Btu cal Mass Units 1 kg = 2.2046226 lbm 1 Ton (short) = 2000 lbm = 907.185 kg, 1 Ton (long) = 2240 lbm = 1.016 Metric Ton 1 Ton (Metric) = 1000 kg = 2204.62 lbm

CONSTANTS AND CONVERSION FACTORS OFTEN USED IN COMBUSTION

753

Energy Units 1 cal = 4.18400 J = 4.184 × 107 erg = 0.003968 Btu 1 J = 1 N · m = 1 W · s = 107 erg = 0.737562 lbf -ft 1 Btu = 1054.4355 J Pressure Units 1 Pascal (Pa) = 1 N/m2 = 1.4504 × 10−4 psi = 9.8692 × 10−6 atm = 1.0197 × 10−5 kgf /cm2 1 atm = 1.01325 × 105 N/m2 = 176 mm Hg = 760 torr 1 bar = 105 N/m2 = 0.1 MPa 1 psi = 0.0689476 bars = 0.00689476 MPa Temperature Units T(K) = T(R)/1.8 T(F) = T(R) − 459.67 ◦

◦

T( C) = [T( F) − 32]/1.8 = T(K) − 273.15 For temperature difference, T, 1 K = 1 ◦ C = 1.8 R = 1.8 ◦ F Force Units 1 N = 1 kgm · m/s2 = 100000 dynes = 0.2248089 lbf = 0.10197162 kgf Length Units 1 m = 39.370079 inches (in.) = 3.2808399 ft = 1.0936133 yard 1 m = 100 cm = 1 × 106 μm = 1 × 1010 Angstrom 1 ft = 30.48 cm Velocity Units 1 m/s = 3.6 km/hr = 3.28084 ft/s = 2.23694 miles/hr Speciﬁc Volume Units 1 m3 /kg = 16.01846 ft3 /lbm 1 cm3 /g = 1 L/kg

754

CONSTANTS AND CONVERSION FACTORS OFTEN USED IN COMBUSTION

Density Units 1 kg/m3 = 0.06242797 lbm /ft3 Thermal Conductivity Units 1 W/m-K = 1 J/s-m-K = 0.577789 Btu/hr-ft-R Heat Flux Units 1 W/m2 = 0.316998 Btu/hr-ft2 Heat Capacity or Speciﬁc Entropy Units 1 kJ/kg-K = 0.238846 Btu/lbm-R Heat Transfer Coefﬁcient Units 1 W/m2 -K = 0.17611 Btu/hr-ft2 -R Viscosity Units 1 centipoise = 0.001 Ns/m2

B.1

10−18 1015 1012 109 106 103 102 101 10−1 10−2 10−3 10−6 10−9 10−12 10−15 10−18

PREFIXES

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

APPENDIX C

NAMING OF HYDROCARBONS

TABLE C.1. First Ten Saturated Straight Chain Hydrocarbons C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

methane

from Greek “wood”

ethane

from Greek “to burn”

propane

from Greek “fat”

n-butane

from Latin “butter”

n-pentane

penta

five

pentagon

n-hexane

hexa

six

hexagon

n-heptane

hepta

seven

n-octane

octa

eight

n-nonane

nona

nine

n-decane

deca

ten

octopus, octave

decade

The first ten saturated straight chain hydrocarbons Fundamentals of Turbulent and Multiphase Combustion Copyright © 2012 John Wiley & Sons, Inc.

Kenneth K. Kuo and Ragini Acharya

755

756

C

C

C

C

C

main chain

main chain

C

C

position of groups

multiplier of group name

group name

2,4-dimethylheptane

what

how many

where

C

Example

C

ethane

2

n-pentane n-hexane n-heptane

5 6 7

n-decane

decyl

nonyl

octyl

heptyl

hexyl

pentyl

butyl

propyl

ethyl

Multiplier*

deca

nona

octa

hepta

hexa

penta

tetra

tri

di

mono (rarely used)

*Multipliers can be used with any group–not just the one with the same value of n. Thus tetramethyl and dihexyl are valid parts of names.

n-nonane

9 10

n-octane

n-butane

4

8

propane

3

1

Number, n

Name of Group with n Carbon Atoms methyl

Naming Organic Hydrocarbons Name of Chain with n Carbon Atoms methane

TABLE C.2. Name of Chain, Group, and Multipliers

757

b. n = 4

b. n = 3

C C

C C

RC CH RC CR

CnH2n-2

acetylene

ethyne

b. C 10H8

a. C6H6

R

b. naphthalene

a. benzene

b. naphthalene

*IUPAC = International Union of Pure and Applied Chemistry

Single C-H bonds Functional & Single Group C-C bonds

Groups

H2C=CH2 RCH=CH2 RCH=CHR R2C=CR2

CnH2n a. n = 2

c. 1, 2-butadiene

b. 2-butene

c. 1, 3-butadiene a. ethene

b. 2-butene

a. benzene

CH3

CH3 R

b. C 7H5N3O6

a. C7H8

b. TNT

a. toluene

b. trinitro toluene

NO2

CH3

CH3

a. toluene

b.

a.

NO2

Toluenes

a. ethene

b.

C

C C

C

C

C

a.

Aromatics

O2N

HC CH

Alkynes

c. CH2=CHCH=CH2

b. CH3CH=CHCH3

a. CH2=CH2

Alkenes

CnH2n+2 a. n = 2

b. propane

a. ethane

b. propane

a. ethane

b. CH3CH2-CH3

RH Species (R) and Functional

Remaining

Using

Formula

General Formula

Common Name

IUPAC* Name

Specific Examples

a. CH3-CH3

Alkanes

C OH

R-OH

b. C 2H5OH

a. CH3OH

a. methyl alcohol b. ethyl alcohol

b. ethanol

a. methanol

b. CH3CH2OH

a. Ch3OH

Alcohols

TABLE C.3. Families of Organic Compounds and Special Function Groups

OR′

C O

O

O R C

CH3COC2H5

O

O

CH3COCH3 b.

a.

b. ethyl acetate

a. methyl acetate

b. ethyl acetate

a. methyl acetate

b. C4H8O2

a. C3H6O2

Esters

b.

a.

O CH3CH

O HCH

Aldehydes

b. ethanal

H

H N

H N R H

RNH2 R2NH R3N

b. C 2H5NH2

a. CH3NH2

b. ethylamine

O C H

R CH

H

O

b.CH 3CHO

a. HCHO

b. acetaldehyde

a. methylamine a. formaldehyde

b. ethylamine

a. methylamine a. formaldehyde

b. CH3CH2NH2

a. CH3NH2

Amines

R

R

O C

C O

CH3COOH3

acetone

propanone

O CH3CCH3

Ketones

COOH

OH

O

O C OH

R C

b. C 6H5COOH

a. CH3COOH

b. benzoic acid

a. acetic acid

a. ethanoic acid b. benzoic acid

b.

O CH3COH

a.

Carboxylic Acids

APPENDIX D

DETAILED GAS-PHASE REACTION MECHANISM FOR AROMATICS FORMATION

The detailed reaction mechanism consists of 527 reactions and 99 chemical species. The reactions and their forward rate coefﬁcients are given in Table D.1. For a limited number of PAH species, there exist radical isomers that differ by the position of the radical site in the aromatic structure. Because such radical isomers often have different thermochemical properties and exhibit different reactivity, they are differentiated in the reaction mechanism given in this appendix. In such case, an additional integer is attached to the species name which speciﬁes the radical position. For example, A3 -i is a phenanthryl radical with the unpaired electron located at the i th carbon atom. The numbering system follows the IUPAC nomenclature. The reverse rate coefﬁcients were calculated via equilibrium constants. The thermodynamic data were taken from GRI-Mech 1.2. It is useful for readers to learn about Troe’s falloff formula. The speciﬁc rate constant (k ) for a unimolecular reaction is both temperature and pressure dependent. It has been observed that k becomes independent of pressure at high values and it falls off at lower pressures. At lower pressures, k is proportional to pressure. The rate constants at two limiting pressures are known as high-pressure limit rate constant (k∞ ) and low-pressure limit constant (k0 ). In order to account for the fall-off effects at lower pressures, Troe (1983) proposed the following formula: 1 1 k(T , p) = F (T , p) ÷ + (D.1) k0 (T , p) k∞ (T ) where F (T , p) is known as the broadening factor. This broadening factor is expressed as: ⎧

2 ⎫ ⎨ ⎬ log10 k0 (T , p)/k∞ (T ) + c log10 F (T , p) = log10 Fc (T ) ÷ 1 + ⎩ ⎭ N − d log10 k0 (T , p)/k∞ (T ) + c (D.2) Fundamentals of Turbulent and Multiphase Combustion Copyright © 2012 John Wiley & Sons, Inc.

Kenneth K. Kuo and Ragini Acharya

759

760

DETAILED GAS-PHASE REACTION MECHANISM FOR AROMATICS FORMATION

where c, N , and d are empirical expressions and constants given as follows: c = −0.4 − 0.67log10 Fc (T ),

N = 0.75 − 1.27log10 Fc (T ),

d = 0.14 (D.3)

The parameter Fc is known as central broadening factor. It is given as following: Fc (T ) = (1 − a) exp −T /T ∗∗∗ + a exp −T /T ∗ + exp −T ∗∗ /T (D.4) where a, T ∗∗∗ , T ∗∗ , and T ∗ are ﬁtting parameters in Equation D.4. In order to specify mathematical expressions for speciﬁc reaction rate constants k0 and k∞ , let us consider conversion of reactant R into product P via a twostep chemical reaction process. In the ﬁrst step, reactant R collides with another molecule M to form an energized molecule R*. In the second step, the energized molecule R* may undergo a unimolecular reaction to form product P. These reactions can be expressed as follows: k1,f

R + M R∗ + M k2,f k2

R∗ −→ P At the high pressure limit, k1,f k∞ = k2 k1,b At low-pressure limit,

k0 = k1,f

PM Ru T

(D.5)

(D.6)

where PM is the partial pressure of molecule M. TABLE D.1. Reaction Mechanism Source: Modiﬁed from Wang and Frenklach, 1997.

No. Reactionsa

k = A T n exp(−Ea /R u T ) b Comments/ A n Ea References

Reactions of H2 /O2 1 H + O2 O + OH 8.30(+13) 14.413 2 O + H2 H + OH 5.00(+04) 2.67 6.29 3 OH + H2 H + H2 O 2.16(+08) 1.51 3.43 4 OH + OH O + H2 O 3.57(+04) 2.40 −2.11 5 H + H + M H2 + M 1.00(+18) −1.0 6 H + OH + M H2 O + M 2.20(+22) −2.0 7 O + H + M OH + M 5.00(+17) −1.0 8 O + O + M O2 + M 1.20(+17) −1.0 9 H + O2 + M HO2 + M 2.80(+18) −0.86 10 OH + OH (+M) 7.40(+13) −0.37 H2 O2 (+M) 2.30(+18) −0.9 −1.7

Frenklach et al., 1995 Frenklach et al., 1995 Frenklach et al., 1995 Frenklach et al., 1995 c, Frenklach et al., 1995 d, Frenklach et al., 1995 e, Frenklach et al., 1995 f, Frenklach et al., 1995 g, Frenklach et al., 1995 k ∞ , Frenklach et al., 1995 k o /[M ], h

761

DETAILED GAS-PHASE REACTION MECHANISM FOR AROMATICS FORMATION

TABLE D.1. (continued ) No. Reactions

a

k = A T n exp(−Ea /R u T ) b A n Ea a = 0.7346, T *** = 94, T * = 1756, T ** = 5182 Reactions of HO2 3.97(+12)

11

HO2 + H O + H2 O

12

HO2 + H O2 + H2

2.80(+13)

1.068

13

HO2 + H OH + OH

1.34(+14)

0.635

14

HO2 + O OH + O2

2.00(+13)

15

HO2 + OH O2 + H2 O

2.90(+13)

−0.5

16

HO2 + HO2 O2 + H2 O2

1.30(+11)

−1.63

17

HO2 + HO2 O2 + H2 O2

4.20(+14)

12.0

18

H2 O2 + H HO2 + H2

19

H2 O2 + H OH + H2 O

1.00(+13)

20

H2 O2 + O OH + HO2

9.63(+06)

21

H2 O2 + OH HO2 + H2 O

1.75(+12)

0.32

22

H2 O2 + OH HO2 + H2 O

5.80(+14)

9.56

23

Reactions of CO/CO2 CO + O + M CO2 + M 6.02(+14)

3.0

24

CO + OH CO2 + H

0.07

25

CO + H2 (+M) CH2 O(+M) 4.30(+07)

0.671

Reactions of H2 O2 1.21(+07) 2.0

5.2 3.6

2.0

4.76(+07) 1.228 1.5

5.07(+27) −3.42

4.0

79.6 84.35

26

CO + O2 CO2 + O

a = 0.9320, T = 197, T * = 1540, T ** = 10.3 2.50(+12) 47.8

27

CO + HO2 CO2 + OH

1.50(+14)

Comments/ References i Frenklach 1995 Frenklach 1995 Frenklach 1995 Frenklach 1995 Frenklach 1995 Frenklach 1995 Frenklach 1995 Frenklach 1995 Frenklach 1995 Frenklach 1995 Frenklach 1995 Frenklach 1995

et al., et al., et al., et al., et al., et al., et al.,

et al., et al., et al., et al., et al.,

j, Frenklach et al., 1995 Frenklach et al., 1995 k ∞ , Frenklach et al., 1995 k o /[M ], h

***

23.6

i Frenklach et al., 1995 Frenklach et al., 1995 (continued overleaf )

762

DETAILED GAS-PHASE REACTION MECHANISM FOR AROMATICS FORMATION

TABLE D.1. (continued ) No. Reactions

a

k = A T n exp(−Ea /R u T ) b Comments/ A n Ea References

28

C + OH CO + H

Reactions of C 5.00(+13)

29

C + O2 CO + O

5.80(+13)

30

CH + H C + H2

Reactions of CH 1.10(+14)

31

CH + O CO + H

5.70(+13)

32

CH + OH HCO + H

3.00(+13)

33

CH + H2 CH2 + H

1.10(+08)

34

CH + H2 O CH2 O + H

5.71(+12)

35

CH + O2 HCO + O

3.30(+13)

36

CH + CO(+M) HCCO(+M) 5.00(+13)

0.576

1.79

1.67 −0.755

2.69(+28) −3.74

1.936

a = 0.5757, = 237, T * = 1652, T ** = 5069 3.40(+12) 0.69

Frenklach et al., 1995 Frenklach et al., 1995 Frenklach et al., 1995 Frenklach et al., 1995 Frenklach et al., 1995 Frenklach et al., 1995 Frenklach et al., 1995 Frenklach et al., 1995 k ∞ , Frenklach et al., 1995 k o /[M ], h

T ***

37

CH + CO2 HCO + CO

38

Reactions of HCO HCO + H(+M) CH2 O(+M) 1.09(+12) 0.48 1.35(+24) −2.57

−0.26 1.425

39

HCO + H CO + H2

a = 0.7824, T = 271, T * = 2755, T ** = 6570 7.34(+13)

40

HCO + O CO + OH

3.00(+13)

41

HCO + O CO2 + H

3.00(+13)

42

HCO + OH CO + H2 O

5.00(+13)

43

HCO + M CO + H + M

1.87(+17) −1.0

i Frenklach et al., 1995 k ∞ , Frenklach et al., 1995 k o /[M ], h

***

17.0

i Frenklach et 1995 Frenklach et 1995 Frenklach et 1995 Frenklach et 1995 e, Frenklach 1995

al., al., al., al., et al.,

763

DETAILED GAS-PHASE REACTION MECHANISM FOR AROMATICS FORMATION

TABLE D.1. (continued ) No. Reactions

a

k = A T n exp(−Ea /R u T ) b Comments/ A n Ea References

44

HCO + O2 CO + HO2

45

Reactions of CH2 (triplet methylene) 2.50(+16) −0.8 CH2 + H(+M) CH3 (+M)

7.60(+12)

0.4

3.20(+27) −3.14

1.23

46

CH2 + O HCO + H

a = 0.68, T *** = 78, T * = 1995, T ** = 5590 8.00(+13)

47

CH2 + OH CH2 O + H

2.00(+13)

48

CH2 + OH CH + H2 O

1.13(+07)

2.0

3.0

49

CH2 + H2 H + CH3

5.00(+05)

2.0

7.23

50

CH2 + O2 CO2 + H + H

1.32(+13)

51

CH2 + HO2 CH2 O + OH

2.00(+13)

52

CH2 + C C2 H + H

5.00(+13)

53

CH2 + CO(+M) CH2 CO(+M) 8.10(+11)

1.5

0.5

4.51

2.69(+33) −5.11

7.095

54

CH2 + CH C2 H2 + H

a = 0.5907, T = 275 T * = 1226, T ** = 5185 4.00(+13)

55

CH2 + CH2 C2 H2 + H2

3.20(+13)

56

CH2 + N2 CH2 + N2

57

CH2 * + Ar CH2 + Ar

9.00(+12)

58

CH2 * + H CH + H2

3.00(+13)

59

CH2 * + O CO + H2

1.50(+13)

Frenklach et al., 1995 k ∞ , Frenklach et al., 1995 k o /[M ], h i Frenklach et al., 1995 Frenklach et al., 1995 Frenklach et al., 1995 Frenklach et al., 1995 k, Frenklach et al., 1995 Frenklach et al., 1995 Frenklach et al., 1995 k ∞ , Frenklach et al., 1995 k o /[M ], h

***

*

Reactions of CH2 * 1.50(+13)

0.6 0.6

i Frenklach et al., 1995 Frenklach et al., 1995 Frenklach 1995 Frenklach 1995 Frenklach 1995 Frenklach 1995

et al., et al., et al., et al.,

(continued overleaf )

764

DETAILED GAS-PHASE REACTION MECHANISM FOR AROMATICS FORMATION

TABLE D.1. (continued ) No. Reactions

a

k = A T n exp(−Ea /R u T ) b Comments/ A n Ea References

60 CH2 * + O HCO + H

1.50(+13)

61 CH2 * + OH CH2 O + H

3.00(+13)

62 CH2 * + H2 CH3 + H

7.00(+13)

63 CH2 * + O2 H + OH + CO

2.80(+13)

64 CH2 * + O2 CO + H2 O

1.20(+13)

65 CH2 * + H2 O(+M) CH3 OH(+M) 2.00(+13) 2.70(+38) −6.300

3.1

66 CH2 * + H2 O CH2 + H2 O

a = 0.1507, T = 134, T * = 2383, T ** = 7265 3.00(+13)

67 CH2 * + CO CH2 + CO

9.00(+12)

68 CH2 * + CO2 CH2 + CO2

7.00(+12)

69 CH2 * + CO2 CH2 O + CO

1.40(+13)

Frenklach et al., 1995 Frenklach et al., 1995 Frenklach et al., 1995 Frenklach et al., 1995 Frenklach et al., 1995 k ∞ , Frenklach et al., 1995 k o /[M ], l

***

Reactions of CH2 O 70 CH2 O + H(+M) CH2 OH(+M) 5.40(+11) 0.454 1.27(+32) −4.820

3.6 6.53

a = 0.7187, T = 103, T * = 1291, T ** = 4160 5.40(+11) 0.454 2.6

i Frenklach 1995 Frenklach 1995 Frenklach 1995 Frenklach 1995

et al., et al., et al., et al.,

k ∞ , Frenklach et al., 1995 k o /[M ], l

***

71 CH2 O + H(+M) CH3 O(+M)

2.20(+30) −4.8

5.56

72 CH2 O + H HCO + H2

a = 0.758, T = 94, T * = 1555, T ** = 4200 2.30(+10) 1.05 3.275

73 CH2 O + O HCO + OH

3.90(+13)

74 CH2 O + OH HCO + H2 O

3.43(+09)

i k ∞ , Frenklach et al., 1995 k o /[M ], l

***

1.18

i Frenklach et al., 1995 3.54 Frenklach et al., 1995 −0.447 Frenklach et al., 1995

765

DETAILED GAS-PHASE REACTION MECHANISM FOR AROMATICS FORMATION

TABLE D.1. (continued ) No. Reactions

k = A T n exp(−Ea /R u T ) b Comments/ A n Ea References

a

75

CH2 O + O2 HCO + HO2

1.00(+14)

76

CH2 O + HO2 HCO + H2 O2

1.00(+12)

77

CH2 O + CH CH2 CO + H

9.46(+13)

78

CH3 + H(+M) CH4 (+M)

40.0

Frenklach et al., 1995 8.0 Frenklach et al., 1995 −0.515 Frenklach et al., 1995

Reactions of CH3 1.27(+16) −0.63

0.383

2.477(+33) −4.760

2.44

a = 0.783, T *** = 74, T * = 2941, T ** = 6964 8.43(+13)

79

CH3 + O CH2 O + H

80

CH3 + OH(+M) CH3 OH(+M) 6.30(+13) 2.70(+38)

−6.3

3.1

81

CH3 + OH CH2 + H2 O

a = 0.2105, = 83.5, T * = 5398, T ** = 8370 5.60(+07) 1.6 5.42

82

CH3 + OH CH2 * + H2 O

2.50(+13)

83

CH3 + O2 O + CH3 O

2.68(+13)

28.8

84

CH3 + O2 OH + CH2 O

3.60(+10)

8.94

85

CH3 + HO2 CH4 + O2

1.00(+12)

86

CH3 + HO2 CH3 O + OH

2.00(+13)

87

CH3 + H2 O2 CH4 + HO2

2.45(+04)

88

CH3 + C C2 H2 + H

5.00(+13)

89

CH3 + CH C2 H3 + H

3.00(+13)

90

CH3 + HCO CH4 + CO

2.65(+13)

91

CH3 + CH2 O CH4 + HCO

3.32(+03)

k ∞ , Frenklach et al., 1995 k o /[M ], h i Frenklach et al., 1995 k ∞ , Frenklach et al., 1995 k o /[M ], l

T ***

2.47

2.81

5.18

5.86

i Frenklach 1995 Frenklach 1995 Frenklach 1995 Frenklach 1995 Frenklach 1995 Frenklach 1995 Frenklach 1995 Frenklach 1995 Frenklach 1995 Frenklach 1995 Frenklach 1995

et al., et al., et al., et al., et al., et al., et al., et al., et al., et al., et al.,

(continued overleaf )

766

DETAILED GAS-PHASE REACTION MECHANISM FOR AROMATICS FORMATION

TABLE D.1. (continued ) a

No. Reactions

k = A T n exp(−Ea /R u T ) b Comments/ A n Ea References

92

CH3 + CH2 C2 H4 + H

4.00(+13)

93

CH3 + CH2 * C2 H4 + H

1.20(+13)

94

CH3 + CH3 (+M) C2 H6 (+M)

2.12(+16) −0.97

0.62

1.77(+50) −9.67

6.22

−0.57

a = 0.5325, = 151, T * = 1038, T ** = 4970 4.99(+12) 0.1 10.6

Frenklach et al., 1995 Frenklach et al., 1995 k ∞ , Frenklach et al., 1995 k o /[M ], h

T ***

95

CH3 + CH3 H + C2 H5

96

Reactions of CH3 O/CH2 OH CH3 O + H(+M) CH3 OH(+M) 5.00(+13) 8.60(+28) −4.0

3.025

97

CH3 O + H CH2 OH + H

a = 0.8902, = 144, T * = 2838, T ** = 45569 3.40(+06) 1.6

98

CH3 O + H CH2 O + H2

2.00(+13)

99

CH3 O + H CH3 + OH

3.20(+13)

i Frenklach et al., 1995 k ∞ , Frenklach et al., 1995 k o /[M ], l

T ***

100 CH3 O + H CH2 * + H2 O

1.60(+13)

101 CH3 O + O CH2 O + OH

1.00(+13)

102 CH3 O + OH CH2 O + H2 O

5.00(+12)

103 CH3 O + O2 CH2 O + HO2

4.28(−13)

7.6

−3.53

104 CH2 OH + H(+M) CH3 OH(+M) 1.80(+13) 3.00(+31) −4.8

3.3

105 CH2 OH + H CH2 O + H2

a = 0.7679, = 338, T * = 1812, T ** = 5081 2.00(+13)

106 CH2 OH + H CH3 + OH

1.20(+13)

i Frenklach et al., 1995 Frenklach et al., 1995 Frenklach et al., 1995 Frenklach et al., 1995 Frenklach et al., 1995 Frenklach et al., 1995 Frenklach et al., 1995 k ∞ , Frenklach et al., 1995 k o /[M ], l

T ***

i Frenklach et al., 1995 Frenklach et al., 1995

767

DETAILED GAS-PHASE REACTION MECHANISM FOR AROMATICS FORMATION

TABLE D.1. (continued ) No. Reactions

k = A T n exp(−Ea /R u T ) b Comments/ A n Ea References

107 CH2 OH + H CH2 * + H2 O

6.00(+12)

108 CH2 OH + O CH2 O + OH

1.00(+13)

109 CH2 OH + OH CH2 O + H2 O

5.00(+12)

110 CH2 OH + O2 CH2 O + HO2

1.80(+13)

0.9

111 CH4 + H CH3 + H2

Reactions of CH4 6.60(+08) 1.62

10.84

112 CH4 + O CH3 + OH

1.02(+09)

1.5

8.6

113 CH4 + OH CH3 + H2 O

1.00(+08)

1.6

3.12

114 CH4 + CH C2 H4 + H

6.00(+13)

115 CH4 + CH2 CH3 + CH3

2.46(+06)

116 CH4 + CH2 * CH3 + CH3

1.60(+13)

a

2.0

Frenklach 1995 Frenklach 1995 Frenklach 1995 Frenklach 1995

8.27 −0.57

Reactions of CH3 OH 1.70(+07) 2.1 117 CH3 OH + H CH2 OH + H2

4.87

118 CH3 OH + H CH3 O + H2

4.20(+06)

2.1

4.87

119 CH3 OH + O CH2 OH + OH

3.88(+05)

2.5

3.1

120 CH3 OH + O CH3 O + OH

1.30(+05)

2.5

5.0

121 CH3 OH + OH CH2 OH + H2 O 1.44(+06)

2.0

−0.84

122 CH3 OH + OH CH3 O + H2 O

6.30(+06)

2.0

1.5

123 CH3 OH + CH3 CH2 OH + CH4 3.00(+07)

1.5

9.94

124 CH3 OH + CH3 CH3 O + CH4

1.5

9.94

125 C2 H + H(+M) C2 H2 (+M)

1.00(+07)

Reactions of C2 H 1.00(+17) −1.0 3.75(+33) −4.8

1.9

Frenklach 1995 Frenklach 1995 Frenklach 1995 Frenklach 1995 Frenklach 1995 Frenklach 1995 Frenklach 1995 Frenklach 1995 Frenklach 1995 Frenklach 1995 Frenklach 1995 Frenklach 1995 Frenklach 1995 Frenklach 1995

et al., et al., et al., et al.,

et al., et al., et al., et al., et al., et al.,

et al., et al., et al., et al., et al., et al., et al., et al.,

k ∞ , Frenklach et al., 1995 k o /[M ], h (continued overleaf )

768

DETAILED GAS-PHASE REACTION MECHANISM FOR AROMATICS FORMATION

TABLE D.1. (continued ) No. Reactions

k = A T n exp(−Ea /R u T ) b Comments/ A n Ea References

a

126 C2 H + O CH + CO

a = 0.6464, T *** = 132, T * = 1315, T ** = 5566 5.00(+13)

127 C2 H + OH H + HCCO

2.00(+13)

128 C2 H + O2 HCO + CO

5.00(+13)

129 C2 H + H2 H + C2 H2

4.90(+05)

130 HCCO + H CH2 * + CO

1.5 2.5

0.56

Reactions of HCCO 1.00(+14)

131 HCCO + O H + CO + CO

1.00(+14)

132 HCCO + O2 OH + CO + CO 1.60(+12) 133 HCCO + CH C2 H2 + CO

5.00(+13)

134 HCCO + CH2 C2 H3 + CO

3.00(+13)

135 HCCO + HCCO C2 H2 + CO + CO

1.00(+13)

0.854

Reactions of C2 H2 136 C2 H2 + H(+M) C2 H3 (+M) 5.60(+12)

2.4

3.80(+40) −7.27

7.22

137 C2 H2 + O HCCO + H

a = 0.7507, T = 98.5, T * = 1302, T ** = 4167 1.02(+07) 2.0 1.9

138 C2 H2 + O C2 H + OH

4.60(+19) −1.41

139 C2 H2 + O CH2 + CO

1.02(+07)

2.0

1.9

140 C2 H2 + OH CH2 CO + H

2.18(−04)

4.5

−1.0

141 C2 H2 + OH HCCOH + H

5.04(+05)

2.3

13.5

i Frenklach et al., 1995 Frenklach et al., 1995 Frenklach et al., 1995 m Frenklach 1995 Frenklach 1995 Frenklach 1995 Frenklach 1995 Frenklach 1995 Frenklach 1995

et al., et al., et al., et al., et al., et al.,

k ∞ , Frenklach et al., 1995 k o /[M ], h

***

28.95

i Frenklach 1995 Frenklach 1995 Frenklach 1995 Frenklach 1995 Frenklach 1995

et al., et al., et al., et al., et al.,

769

DETAILED GAS-PHASE REACTION MECHANISM FOR AROMATICS FORMATION

TABLE D.1. (continued ) No. Reactions

k = A T n exp(−Ea /R u T ) b Comments/ A n Ea References

142 C2 H2 + OH C2 H + H2 O

3.37(+07)

2.0

14.0

143 C2 H2 + OH CH3 + CO

4.83(−04)

4.0

−2.0

a

Reactions of CH2 CO/HCCOH 5.00(+13) 144 CH2 CO + H HCCO + H2

8.0

145 CH2 CO + H CH3 + CO

1.13(+13)

3.428

146 CH2 CO + O HCCO + OH

1.00(+13)

8.0

147 CH2 CO + O CH2 + CO2

1.75(+12)

1.35

148 CH2 CO + OH HCCO + H2 O 7.50(+12)

2.0

149 HCCOH + H CH2 CO + H

1.00(+13)

Reactions of C2 H3 150 C2 H3 + H(+M) C2 H4 (+M) 6.08(+12) 0.27 1.40(+30) −3.86

0.28 3.32

Frenklach et al., 1995 Frenklach et al., 1995 Frenklach 1995 Frenklach 1995 Frenklach 1995 Frenklach 1995 Frenklach 1995 Frenklach 1995

et al., et al., et al., et al., et al., et al.,

k ∞ , Frenklach et al., 1995 k o /[M ], h

a = 0.7820, T = 207.5, T * = 2663, T ** = 6095 i 3.00(+13) Frenklach et al., 1995 3.00(+13) Frenklach et al., 1995 5.00(+12) Frenklach et al., 1995 1.66(+14) −0.83 2.54 Bozzelli and Dean, 1993 2.50(+12) 0.057 0.95 20, 90 torr, Bozzelli and Dean, 1993 1.24(+13) −0.120 1.696 760 torr 1.64(+21) −2.780 2.523 20, 90 torr, Bozzelli and Dean, 1993 8.60(+21) −2.970 3.32 760 torr ***

151 C2 H3 + H C2 H2 + H2 152 C2 H3 + O CH2 CO + H 153 C2 H3 + OH C2 H2 + H2 O 154 C2 H3 + O2 C2 H2 + HO2 155 C2 H3 + O2 C2 H3 O + O

156 C2 H3 + O2 HCO + CH2 O

(continued overleaf )

770

DETAILED GAS-PHASE REACTION MECHANISM FOR AROMATICS FORMATION

TABLE D.1. (continued ) No. Reactions

k = A T n exp(−Ea /R u T ) b Comments/ A n Ea References

a

Reactions of C2 H4 157 C2 H4 (+M) H2 + C2 H2 (+M) 8.00(+12) 0.44 7.00(+50) −9.31

88.77 99.86

a = 0.7345, T = 180, T * = 1035, T ** = 5417 1.08(+12) 0.454 1.82

k ∞ , Frenklach et al., 1995 k o /[M ], h

***

158 C2 H4 + H(+M) C2 H5 (+M)

1.20(+42) −7.62

6.97

159 C2 H4 + H C2 H3 + H2

a = 0.9753, T = 210, T * = 984, T ** = 4374 1.33(+06) 2.53 12.24

160 C2 H4 + O CH3 + HCO

1.92(+07)

1.83

0.22

161 C2 H4 + OH C2 H3 + H2 O

3.60(+06)

2.0

2.5

162 C2 H4 + CH3 C2 H3 + CH4

2.27(+05)

2.0

9.2

i k ∞ , Frenklach et al., 1995 k o /[M ], h

***

Reactions of C2 H5 163 C2 H5 + H(+M) C2 H6 (+M) 5.21(+17) −0.99

1.58

1.99(+41) −7.08

6.685

164 C2 H5 + H C2 H4 + H2

a = 0.8422, T *** = 125, T * = 2219, T ** = 6882 2.00(+12)

165 C2 H5 + O CH3 + CH2 O

1.32(+14)

166 C2 H5 + O2 C2 H4 + HO2

8.40(+11)

167 C2 H6 + H C2 H5 + H2

3.875

Reactions of C2 H6 1.15(+08) 1.9

7.53

168 C2 H6 + O C2 H5 + OH

8.98(+07)

1.92

5.69

169 C2 H6 + OH C2 H5 + H2 O

3.54(+06)

2.12

0.87

170 C2 H6 + CH2 * C2 H5 + CH3

4.00(+13)

171 C2 H6 + CH3 C2 H5 + CH4

6.14(+06)

−0.55 1.74

10.45

i Frenklach 1995 Frenklach 1995 Frenklach 1995 Frenklach 1995

et al., et al., et al., et al.,

k ∞ , Frenklach et al., 1995 k o /[M ], h i Frenklach et al., 1995 Frenklach et al., 1995 Frenklach et al., 1995 Frenklach 1995 Frenklach 1995 Frenklach 1995 Frenklach 1995 Frenklach 1995

et al., et al., et al., et al., et al.,

771

DETAILED GAS-PHASE REACTION MECHANISM FOR AROMATICS FORMATION

TABLE D.1. (continued ) k = A T n exp(−Ea /R u T ) b Comments/ A n Ea References

a

No. Reactions

172 HCCO + OH C2 O + H2 O

Reactions of C2 O 3.00(+13)

173 C2 O + H CH + CO

5.00(+13)

174 C2 O + O CO + CO

5.00(+13)

175 C2 O + OH CO + CO + H

2.00(+13)

176 C2 O + O2 CO + CO + O

2.00(+13)

177 CH2 CO + H C2 H3 O 178 179 180 181

C2 H3 O C2 H3 O C2 H3 O C2 H3 O

+ + + +

Reactions of C2 H3 O 5.40(+11) 0.454

H CH2 CO + H2 O CH2 O + HCO O CH2 CO + OH OH CH2 CO + H2 O

1.00(+13) 9.60(+06) 1.00(+13) 5.00(+12)

1.83

Miller and 1992 Miller and 1992 Miller and 1992 Miller and 1992 Miller and 1992 1.82

0.22

Melius, Melius, Melius, Melius, Melius,

k 177 = 0.5 × k ∞,158 n k 179 = 0.5 × k 160 n n

Additional Reactions of C1 Hx and C2 Hx Species 5.00(+13) n 182 CH3 + HCCO C2 H4 + CO 183 CH3 + C2 H C3 H3 + H 2.41(+13) Tsang and Hampson, 1986 184 CH4 + C2 H C2 H2 + CH3 1.81(+12) 0.5 Tsang and Hampson, 1986 185 C2 H2 + CH C3 H2 + H 3.00(+13) Warnatz et al., 1982 186 C2 H2 + CH2 C3 H3 + H 1.20(+13) 6.62 Bohland et al., 1986 187 C2 H2 + CH2 * C3 H3 + H 2.00(+13) n 188 C2 H2 + CH3 a-C3 H4 + H 2.87(+21) −2.74 24.8 20, 90, torr, Dean and Westmoreland, 1987 5.72(+20) −2.36 31.5 760 torr 189 C2 H2 + CH3 p-C3 H4 + H 1.00(+13) −0.53 13.4 20, 90 torr, Dean and Westmoreland, 1987 2.72(+18) −1.97 20.2 760 torr 190 C2 H2 + C2 H C4 H2 + H 9.60(+13) o (continued overleaf )

772

DETAILED GAS-PHASE REACTION MECHANISM FOR AROMATICS FORMATION

TABLE D.1. (continued ) No. Reactions

a

191 C2 H2 + C2 H n-C4 H3

192 C2 H2 +

193 C2 H2 +

194 C2 H2 +

195 C2 H2 +

196 C2 H4 + 197 C2 H4 + 198 C2 H2 + 199 C2 H4 + 200 C2 H3 + 201 C2 H3 + 202 C2 H3 + 203 C2 H3 + 204 C2 H3 + 205 C2 H3 +

k = A T n exp(−Ea /R u T ) b Comments/ A n Ea References 1.10(+30) −6.30

2.79

20 torr, Wang, 1992 1.30(+30) −6.12 2.51 90 torr 4.50(+37) −7.68 7.10 760 torr C2 H i -C4 H3 4.10(+33) −7.31 4.60 20 torr, Wang, 1992 1.60(+34) −7.28 4.83 90 torr 2.60(+44) −9.47 14.65 760 torr C2 H3 C4 H4 + H 5.00(+14) −0.71 6.7 20 torr, Wang and Frenklach, 1994 4.60(+16) −1.25 8.4 90 torr 2.00(+18) −1.68 10.6 760 torr C2 H3 n-C4 H5 1.10(+32) −7.33 6.2 20 torr, Wang and Frenklach, 1994 2.40(+31) −6.95 5.6 90 torr 9.30(+38) −8.76 12.0 760 torr C2 H3 i -C4 H5 2.10(+36) −8.78 9.1 20 torr, Wang and Frenklach, 1994 1.00(+37) −8.77 9.8 90 torr 1.60(+46) −10.98 18.6 760 torr C2 H C4 H4 + H 1.20(+13) Tsang and Hampson, 1986 C2 H3 1,3-C4 H6 + H 7.40(+14) −0.66 8.42 20 torr, p 1.90(+17) −1.32 10.60 90 torr 2.80(+21) −2.44 14.72 760 torr HCCO C3 H3 + CO 1.00(+11) 3.0 Miller and Bowman, 1989 O2 C2 H3 + HO2 4.22(+13) 60.8 q H2 O2 C2 H4 + HO2 1.21(+10) −0.596 Tsang and Hampson, 1986 HCO C2 H4 + CO 2.50(+13) n CH3 C2 H2 + CH4 3.92(+11) Tsang and Hampson, 1986 C2 H3 1,3-C4 H6 7.00(+57) −13.82 17.6 20 torr, p 1.50(+52) −11.97 16.1 90 torr 1.50(+42) −8.84 12.5 760 torr C2 H3 i -C4 H5 + H 1.50(+30) −4.95 13.0 20 torr, p 7.20(+28) −4.49 14.3 90 torr 1.20(+22) −2.44 13.7 760 torr C2 H3 n-C4 H5 + H 1.10(+24) −3.28 12.4 20 torr, p 4.60(+24) −3.38 14.7 90 torr 2.40(+20) −2.04 15.4 760 torr

DETAILED GAS-PHASE REACTION MECHANISM FOR AROMATICS FORMATION

773

TABLE D.1. (continued ) No. Reactions

k = A T n exp(−Ea /R u T ) b Comments/ A n Ea References

a

206 C3 H2 + O C2 H2 + CO

Reactions of C3 Hx 6.80(+13)

207 C3 H2 + OH HCO + C2 H2

6.80(+13)

208 C3 H2 + O2 HCCO + CO + H 5.00(+13) 209 210 211 212 213

C3 H2 C3 H2 C3 H2 C3 H2 C3 H3

+ + + + +

CH C4 H2 + H CH2 n-C4 H3 + H CH3 C4 H4 + H HCCO n-C4 H3 + CO H (+M) a-C3 H4 (+M)

5.00(+13) 5.00(+13) 5.00(+12) 1.00(+13) 3.00(+13) 1.40(+31)

−5.0

−6.0

a = 0.5, T = 2000, T * = 10, T ** = 10000 214 C3 H3 + H (+M) p-C3 H4 (+M) 3.00(+13) 1.40(+31) −5.0 −6.0

Warnatz et al., 1982 Warnatz et al., 1982 Miller and Melius, 1992 n n n n k ∞, r k o /[M ], e

***

215 C3 H3 + O CH2 O + C2 H

a = 0.5, T *** = 2000, T * = 10, T ** = 10000 2.00(+13)

216 C3 H3 + OH C3 H2 + H2 O

2.00(+13)

217 C3 H3 + OH C2 H3 + HCO 218 C3 H3 + O2 CH2 CO + HCO

4.00(+13) 3.00(+10)

C3 H3 + HO2 a-C3 H4 + O2 C3 H3 + HO2 p-C3 H4 + O2 C3 H3 + HCO a-C3 H4 + CO C3 H3 + HCO p-C3 H4 + CO C3 H3 + CH i -C4 H3 + H C3 H3 + CH2 C4 H4 + H i -C4 H5 + H C3 H3 + CH3 C3 H3 + CH3 (+M) 1,2-C4 H6 (+M)

1.00(+12) 1.00(+12) 2.50(+13) 2.50(+13) 5.00(+13) 2.00(+13) 2.00(+13) 1.50(+13)

219 220 221 222 223 224 225 226

227 C3 H3 + C3 H3 → A1 †

2.60(+58) −11.94

2.878

2.0

i k ∞, r k o /[M ], e i Miller and Bowman, 1989 Miller and Bowman, 1989 n Slagle and Gutman, 1988 n n n n n n n k ∞, s k o /[M ]

9.77

a = 0.175, T *** = 1340, T * = 60000, T ** = 9770 1.00(+11) 1.00(+12) 2.00(+12)

i 20 torr, see text 90 torr 760 torr

(continued overleaf )

774

DETAILED GAS-PHASE REACTION MECHANISM FOR AROMATICS FORMATION

TABLE D.1. (continued ) k = A T n exp(−Ea /R u T ) b Comments/ A n Ea References

a

No. Reactions

H C3 H3 + H2 O CH2 CO + CH2 OH C3 H3 + H2 O C2 H C2 H2 + C3 H3 H C3 H3 + H2 H CH3 + C2 H2

1.15(+08) 2.00(+07) 5.30(+06) 1.00(+13) 1.15(+08) 1.30(+05)

1.9 1.8 2.0

7.53 1.0 2.0

1.9 2.5

7.53 1.0

234 p-C3 H4 + OH C3 H3 + H2 O 235 p-C3 H4 + C2 H C2 H2 + C3 H3

3.54(+06) 1.00(+13)

2.12

0.87

Reactions of C4 H and C4 H2 236 C4 H + H (+M) C4 H2 (+M) 1.00(+17) −1.0 3.75(+33) −4.8

1.9

228 229 230 231 232 233

237 238 239 240 241

a-C3 H4 a-C3 H4 a-C3 H4 a-C3 H4 p-C3 H4 p-C3 H4

+ + + + + +

C4 H + C2 H2 C6 H2 + H C4 H + O C2 H + C2 O C4 H + O2 HCCO + C2 O C4 H + H2 H + C4 H2 C4 H2 + H n-C4 H3

242 C4 H2 + H i -C4 H3 243 C4 H2 + O C3 H2 + CO

244 245 246 247 248 249 250

C4 H2 C4 H2 C4 H2 C4 H2 C4 H2 C4 H2 C4 H2

+ + + + + + +

OH H2 C4 O + H OH C4 H + H2 O CH C5 H2 + H CH2 C5 H3 + H CH2 * C5 H3 + H C2 H C6 H2 + H C2 H C6 H3

251 H2 C4 O + H C2 H2 + HCCO

a = 0.6464, T *** = 132, T * = 1315, T ** = 5566 9.60(+13) 5.00(+13) 5.00(+13) 1.5 4.90(+05) 2.5 0.56 1.70(+49) −11.67 12.80 3.30(+50) −11.80 15.01 1.10(+42) −8.72 15.30 4.30(+45) −10.15 13.25 2.60(+46) −10.15 15.50 1.10(+30) −4.92 10.80 2.70(+13) 1.72

6.60(+12) 3.37(+07) 2.0 5.00(+13) 1.30(+13) 2.00(+13) 9.60(+13) 1.10(+30) −6.30 1.30(+30) −6.12 4.50(+37) −7.68 5.00(+13)

252 H2 C4 O + OH CH2 CO + HCCO 1.00(+07)

2.0

253 H2 C4 O + O CH2 CO + C2 O

1.9

2.00(+07)

k 228 = k 232 t u n k 232 = k 167 Hidaka et al., 1989 k 234 = k 169 n k ∞ , k 236 = k 125 k o /[M ], e

i k 237 = k 190 k 238 = k 126 k 239 = k 128 k 240 = k 129 20 torr, p 90 torr 760 torr 20 torr, p 90 torr 760 torr Homann and Wellmann, 1983 −0.41 Perry, 1984 14.0 k 245 = k 142 n 6.62 k 247 = k 186 n k 249 = k 190 2.79 20 torr, k 250 = k 191 2.51 90 torr 7.10 760 torr 3.0 Miller and Melius, 1992 2.0 Miller and Melius, 1992 0.2 n

DETAILED GAS-PHASE REACTION MECHANISM FOR AROMATICS FORMATION

775

TABLE D.1. (continued ) a

No. Reactions

k = A T n exp(−Ea /R u T ) b Comments/ A n Ea References

Reactions of C4 H3 and C4 H4 3.70(+61) −15.81 1.00(+51) −12.45 4.10(+43) −9.49 n-C4 H3 + H i -C4 H3 + H 2.40(+11) 0.79 9.20(+11) 0.63 2.50(+20) −1.67 n-C4 H3 + H C2 H2 + C2 H2 1.60(+19) −1.60 1.30(+20) −1.85 6.30(+25) −3.34 i -C4 H3 + H C2 H2 + C2 H2 2.40(+19) −1.60 3.70(+22) −2.50 2.80(+23) −2.55 n-C4 H3 + H C4 H4 1.10(+42) −9.65 1.10(+42) −9.65 2.00(+47) −10.26 i -C4 H3 + H C4 H4 4.20(+44) −10.27 5.30(+46) −10.68 3.40(+43) −9.01 n-C4 H3 + H C4 H2 + H2 1.50(+13) i -C4 H3 + H C4 H2 + H2 3.00(+13) n-C4 H3 + OH C4 H2 + H2 O 2.50(+12) i -C4 H3 + OH C4 H2 + H2 O 5.00(+12) i -C4 H3 + O2 7.86(+16) −1.80 HCCO + CH2 CO n-C4 H3 + C2 H2 l -C6 H4 + H 3.70(+16) −1.21

254 n-C4 H3 i -C4 H3

255

256

257

258

259

260 261 262 263 264 265

54.89 51.00 53.00 2.41 2.99 10.80 2.22 2.96 10.01 2.80 5.14 10.78 7.00 7.00 13.07 7.89 9.27 12.12

20 torr, p 90 torr 760 torr 20 torr, p 90 torr 760 torr 20 torr, p 90 torr 760 torr 20 torr, p 90 torr 760 torr 20 torr, p 90 torr 760 torr 20 torr, p 90 torr 760 torr k 260 = 0.5 × k 151 k 261 = k 151 k 262 = 0.5 × k 153 k 263 = k 153 Slagle et al., 1989

11.1

20 torr, Wang and Frenklach, 1994 90 torr 760 torr 20 torr, Wang and Frenklach, 1994 90 torr 760 torr 20 torr, Wang and Frenklach, 1994 90 torr 760 torr 20 torr, Wang and Frenklach, 1994 90 torr 760 torr

266 n-C4 H3 + C2 H2 n-C6 H5

1.80(+19) −1.95 2.50(+14) −0.56 6.00(+33) −7.37

13.2 10.6 13.7

267 n-C4 H3 + C2 H2 A1 -‡

4.10(+33) −7.12 2.70(+36) −7.62 2.30(+68) −17.65

13.7 16.2 24.4

9.80(+68) −17.58 9.60(+70) −17.77 268 n-C4 H3 + C2 H2 c-C6 H4 + H 1.90(+36) −7.21

26.5 31.3 17.9

3.50(+41) −8.63 6.90(+46) −10.01

23.0 30.1

(continued overleaf )

776

DETAILED GAS-PHASE REACTION MECHANISM FOR AROMATICS FORMATION

TABLE D.1. (continued ) No. Reactions

k = A T n exp(−Ea /R u T ) b Comments/ A n Ea References

a

269 C4 H4 + H n-C4 H5

4.20(+50) −12.34

12.5

270 C4 H4 + H i -C4 H5

1.10(+50) −11.94 1.30(+51) −11.92 9.60(+52) −12.85

13.4 16.5 14.3

2.10(+52) −12.44 4.90(+51) −11.92 H n-C4 H3 + H2 6.65(+05) 2.53 H i -C4 H3 + H2 3.33(+05) 2.53 OH n-C4 H3 + H2 O 3.10(+06) 2.0 OH i -C4 H3 + H2 O 1.55(+06) 2.0 O p-C3 H4 + CO 2.70(+13) C2 H3 l -C6 H6 + H 7.40(+14) −0.66

15.5 17.7 12.24 9.24 3.43 0.43 1.72 8.42

1.90(+17) −1.32 2.80(+21) −2.44

10.60 14.72

271 272 273 274 275 276

C4 H4 C4 H4 C4 H4 C4 H4 C4 H4 C4 H4

+ + + + + +

277 n-C4 H5 i -C4 H5

278

279

280

281 282 283 284 285 286 287

Reactions of C4 H5 and 1,3-C4 H6 1.30(+62) −16.38 49.6

4.90(+66) 1.50(+67) n-C4 H5 + H i -C4 H5 + H 1.00(+36) 1.00(+34) 3.10(+26) 1,3-C4 H6 i -C4 H5 + H 8.20(+51) 3.30(+45) 5.70(+36) 1,3-C4 H6 n-C4 H5 + H 3.50(+61) 8.50(+54) 5.30(+44) n-C4 H5 + H C4 H4 + H2 1.50(+13) i -C4 H5 + H C4 H4 + H2 3.00(+13) n-C4 H5 + OH C4 H4 + H2 O 2.50(+12) i -C4 H5 + OH C4 H4 + H2 O 5.00(+12) n-C4 H5 + O2 4.16(+10) → C2 H4 + CO + HCO i -C4 H5 + O2 7.86(+16) CH2 CO + C2 H3 O n-C4 H5 + C2 H2 + M 4.50(+26) n-C6 H7 + M

−17.26 −16.89 −6.26 −5.61 −3.35 −10.92 −8.95 −6.27 −13.87 −11.78 −8.62

−3.28

10.2

n-C4 H5 + C2 H2 n-C6 H7

−1.27

2.9

1.10(+14)

55.4 59.1 17.5 18.5 17.4 118.4 115.9 112.4 129.7 127.5 123.6

2.5 −1.8

20 torr, Wang and Frenklach, 1994 90 torr 760 torr 20 torr, Wang and Frenklach, 1994 90 torr 760 torr k 271 = 0.5 × k 159 v k 273 = 0.5 × k 293 k 274 = 0.5 × k 294 k 275 = k 243 20 torr, k 276 = k 197 90 torr 760 torr 20 torr, Wang and Frenklach, 1994 90 torr 760 torr 20 torr, p 90 torr 760 torr 20 torr, p 90 torr 760 torr 20 torr, p 90 torr 760 torr k 281 = 0.5 × k 151 k 282 = k 151 k 283 = 0.5 × k 153 k 284 = k 153 Gutman et al., 1991 k 286 = k 264 20, 90 torr, Wang and Frenklach, 1994 760 torr

DETAILED GAS-PHASE REACTION MECHANISM FOR AROMATICS FORMATION

777

TABLE D.1. (continued ) No. Reactions

k = A T n exp(−Ea /R u T ) b Comments/ A n Ea References

a

288 n-C4 H5 + C2 H2 + M c-C6 H7 + M

5.20(+25) −4.21

4.0

n-C4 H5 + C2 H2 c-C6 H7 289 n-C4 H5 + C2 H2 l -C6 H6 + H

5.00(+24) −5.46 5.80(+08) 1.02

4.6 10.9

290 n-C4 H5 + C2 H2 A1 + H

2.10(+15) −1.07

4.8

+ + + + +

1.60(+16) −1.33 1.33(+06) 2.53 6.65(+05) 2.53 6.20(+06) 2.0 3.10(+06) 2.0 7.40(+14) −0.66

5.4 12.24 9.24 3.43 0.43 8.42

1.90(+17) −1.32 2.80(+21) −2.44

10.60 14.72

291 292 293 294 295

1,3-C4 H6 1,3-C4 H6 1,3-C4 H6 1,3-C4 H6 1,3-C4 H6

296 297 298 299 300 301

1,2-C4 H6 1,2-C4 H6 1,2-C4 H6 1,2-C4 H6 1,2-C4 H6 1,2-C4 H6

302 303 304 305 306 307 308

C5 H2 C5 H2 C5 H2 C5 H3 C5 H3 C5 H3 C5 H3

+ + + + + + +

+ + + + + +

H n-C4 H5 + H2 H i -C4 H5 + H2 OH n-C4 H5 + H2 O OH i -C4 H5 + H2 O C2 H3 C6 H8 + H

Reactions of 1, 2-C4 H6 H 1,3-C4 H6 + H 2.00(+13) H i -C4 H5 + H2 1.70(+05) 2.5 H a-C3 H4 + CH3 2.00(+13) O CH2 CO + C2 H4 1.20(+08) 1.65 O i -C4 H5 + OH 1.80(+11) 0.7 OH i -C4 H5 + H2 O 3.10(+06) 2.0

4.0 2.49 2.0 0.327 5.88 −0.298

Reactions of C5 H2 and C5 H3 OH → C4 H2 + H + CO 2.00(+13) CH C6 H2 + H 5.00(+13) O2 H2 C4 O + CO 1.00(+12) OH C5 H2 + H2 O 1.00(+13) CH C6 H2 + H + H 5.00(+13) CH2 l -C6 H4 + H 5.00(+13) O2 H2 C4 O + HCO 1.00(+12)

Reactions of C6 H and C6 H2 309 C6 H + H (+M) C6 H2 (+M) 1.00(+17) −1.0 3.75(+33) −4.8

310 C6 H2 + H C6 H3

n z n z z z n n n n n n n k ∞ , k 309 = k 125 k o /[M ]h , e

1.9

a = 0.6464, T *** = 132, T * = 1315, T ** = 5566 4.30(+45) −10.15 13.25 2.60(+46) −10.15 1.10(+30) −4.92

20, 90 torr, Wang and Frenklach, 1994 760 torr 20, 90, 760 torr, Wang and Frenklach, 1994 20, 90 torr, Wang and Frenklach, 1994 760 torr k 291 = k 159 w x y 20 torr, k 295 = k 197 90 torr 760 torr

15.50 10.80

i 20 torr, k 310 = k 242 90 torr 760 torr

(continued overleaf )

778

DETAILED GAS-PHASE REACTION MECHANISM FOR AROMATICS FORMATION

TABLE D.1. (continued ) No. Reactions

a

C6 H + O C4 H + C2 O C6 H + H2 H + C6 H2 C6 H2 + O C5 H2 + CO C6 H2 + OH → C2 H+C2 H2 +C2 O 315 C6 H2 + OH C6 H + H2 O

311 312 313 314

k = A T n exp(−Ea /R u T ) b A n Ea

Comments/ References

5.00(+13) 4.90(+05) 2.70(+13) 6.60(+12)

k 311 k 312 k 313 k 314

3.37(+07)

2.0

14.0

k 315 = k 142

2.80

20 torr, k 316 = k 257 90 torr 760 torr 20 torr, k 317 = k 259 90 torr 760 torr k 318 = k 151 k 319 = k 153 n

3.70(+22) −2.50 2.80(+23) −2.55 4.20(+44) −10.27

5.14 10.78 7.89

5.30(+46) −10.68 3.40(+43) −9.01 3.00(+13) 5.00(+12) 5.00(+11)

9.27 12.12

3.30(+44) −10.04

18.8

322 l -C6 H4 + H A1 -

2.60(+43) −9.53 5.90(+39) −8.25 3.60(+77) −20.09

18.1 15.6 28.1

323 l -C6 H4 + H c-C6 H4 + H

4.70(+78) −20.10 1.70(+78) −19.72 2.20(+47) −9.98

29.5 31.4 24.0

9.70(+48) −10.37 1.40(+54) −11.70 324 l -C6 H4 + H C6 H3 + H2 6.65(+06) 2.53 325 l -C6 H4 + OH C6 H3 + H2 O 3.10(+06) 2.0 326 c-C6 H4 + H A1 1.20(+77) −18.77

27.0 34.5 9.24 0.43 36.3

1.00(+71) −16.88 2.40(+60) −13.66

34.2 29.5

318 C6 H3 + H C6 H2 + H2 319 C6 H3 + OH C6 H2 + H2 O 320 C6 H3 + O2 → CO + C3 H2 + HCCO 321 l -C6 H4 + H n-C6 H5

327 n-C6 H5 A1 -

k 126 k 129 k 243 k 244

0.56 1.72 −0.41

Reactions of C6 H3 and C6 H4 316 C6 H3 + H C4 H2 + C2 H2 2.40(+19) −1.60

317 C6 H3 + H l -C6 H4

= = = =

2.5

Reactions of C6 H5 and l -C6 H6 1.30(+62) −15.94 35.8 1.30(+59) −14.78 5.10(+54) −13.11

35.6 35.7

20 torr, Wang and Frenklach, 1994 90 torr 760 torr 20 torr, Wang and Frenklach, 1994 90 torr 760 torr 20 torr, Wang and Frenklach, 1994 90 torr 760 torr k 324 = k 292 k 325 = k 294 20 torr, Wang and Frenklach, 1994 90 torr 760 torr 20 torr, Wang and Frenklach, 1994 90 torr 760 torr

DETAILED GAS-PHASE REACTION MECHANISM FOR AROMATICS FORMATION

779

TABLE D.1. (continued ) No. Reactions

a

k = A T n exp(−Ea /R u T ) b Comments/ A n Ea References

328 n-C6 H5 c-C6 H4 + H

2.70(+65) −15.93

59.7

329 n-C6 H5 + H i -C6 H5 + H

1.50(+64) −15.32 1.30(+59) −13.56 2.40(+11) 0.79

61.5 62.0 2.41

330 n-C6 H5 + H C4 H4 + C2 H2

9.20(+11) 0.63 2.50(+20) −1.67 1.60(+19) −1.60

2.99 10.80 2.22

331 i -C6 H5 + H C4 H4 + C2 H2

1.30(+20) −1.85 6.30(+25) −3.34 2.40(+19) −1.60

2.96 10.01 2.80

332 n-C6 H5 + H l -C6 H6

3.70(+22) −2.50 2.80(+23) −2.55 1.10(+42) −9.65

5.14 10.78 7.00

333 i -C6 H5 + H l -C6 H6

1.10(+42) −9.65 2.00(+47) −10.26 4.20(+44) −10.27

7.00 13.07 7.89

5.30(+46) −10.68 3.40(+43) −9.01 1.50(+13) 3.00(+13) 2.50(+12) 5.00(+12) 4.16(+10)

9.27 12.12

n-C6 H5 + H l -C6 H4 + H2 i -C6 H5 + H l -C6 H4 + H2 n-C6 H5 + OH l -C6 H4 + H2 O i -C6 H5 + OH l -C6 H4 + H2 O n-C6 H5 + O2 → C4 H4 + CO + HCO 339 i -C6 H5 + O2 7.86(+16) −1.80 → CH2 CO + CH2 CO + C2 H 340 l -C6 H6 + H + M n-C6 H7 + M 2.90(+17) −0.52 334 335 336 337 338

2.5

20 torr, Wang and Frenklach, 1994 90 torr 760 torr 20 torr, k 329 = k 255 90 torr 760 torr 20 torr, k 330 = k 256 90 torr 760 torr 20 torr, k 331 = k 257 90 torr 760 torr 20 torr, k 332 = k 258 90 torr 760 torr 20 torr, k 332 = k 259 90 torr 760 torr k 334 = 0.5 × k 151 k 335 = k 151 k 336 = 0.5 × k 153 k 337 = k 153 k 338 = k 285 k 339 = k 264

1.0

l -C6 H6 + H n-C6 H7 341 l -C6 H6 + H + M c-C6 H7 + M

1.50(+16) −1.69 1.70(+28) −4.72

1.6 2.8

l -C6 H6 + H c-C6 H7 342 l -C6 H6 + H A1 + H

4.70(+27) −6.11 8.70(+16) −1.34

3.8 3.5

2.00(+18) −1.73

4.5

20, 90 torr, Wang and Frenklach, 1994 760 torr 20, 90 torr, Wang and Frenklach, 1994 760 torr 20, 90 torr, Wang and Frenklach, 1994 760 torr (continued overleaf )

780

DETAILED GAS-PHASE REACTION MECHANISM FOR AROMATICS FORMATION

TABLE D.1. (continued ) No. Reactions 343 344 345 346

l -C6 H6 l -C6 H6 l -C6 H6 l -C6 H6

+ + + +

a

k = A T n exp(−Ea /R u T ) b Comments/ A n Ea References

H n-C6 H5 + H2 H i -C6 H5 + H2 OH n-C6 H5 + H2 O OH i -C6 H5 + H2 O

6.65(+05) 3.33(+05) 6.20(+06) 3.10(+06)

347 n-C6 H7 c-C6 H7

Reactions of C6 H7 and C6 H8 4.10(+24) −7.11

348 n-C6 H7 A1 + H

349

350

351

352 353 354 355 356 357 358 359 360 361

2.53 2.53 2.0 2.0

12.24 9.24 3.43 0.43 3.9

3.60(+27) −7.54 1.20(+31) −7.95 8.40(+21) −4.22

5.8 8.9 11.3

−4.86 −4.99 −8.09 −8.18 −10.72 −13.86 −12.32 −7.62 −17.01 −15.64 −10.54

13.4 15.5 19.2 21.8 15.1 21.0 19.3 11.0 24.0 23.2 14.7

8.80(+24) 3.20(+26) n-C6 H7 + H i -C6 H7 + H 4.00(+41) 1.60(+42) 2.40(+49) i -C6 H7 + H C6 H8 1.20(+60) 1.40(+55) 1.80(+39) n-C6 H7 + H C6 H8 8.70(+69) 6.70(+65) 5.60(+48) n-C6 H7 + H l -C6 H6 + H2 1.50(+13) i -C6 H7 + H l -C6 H6 + H2 3.00(+13) n-C6 H7 + OH l -C6 H6 + H2 O 2.50(+12) i -C6 H7 + OH l -C6 H6 + H2 O 5.00(+12) n-C6 H7 + O2 4.16(+10) → 1,3-C4 H6 + CO + HCO i -C6 H7 + O2 7.86(+16) → CH2 CO + CH2 CO + C2 H3 C6 H8 + H n-C6 H7 + H2 1.33(+06) C6 H8 + H i -C6 H7 + H2 6.65(+05) C6 H8 + OH n-C6 H7 + H2 O 6.20(+06) C6 H8 + OH i -C6 H7 + H2 O 3.10(+06)

2.5 −1.80 2.53 2.53 2.0 2.0

363 A1 + H A1 - + H2 364 A1 + OH A1 - + H2 O

= = = =

0.5 × k 159 0.5 × k 292 k 293 k 294

20 torr, Wang and Frenklach, 1994 90 torr 760 torr 20 torr, Wang and Frenklach, 1994 90 torr 760 torr 20 torr 90 torr 760 torr 20 torr 90 torr 760 torr 20 torr 90 torr 760 torr k 352 = 0.5 × k 151 k 353 = k 151 k 354 = 0.5 × k 153 k 355 = k 153 k 356 = k 285 k 357 = k 264

12.24 9.24 3.43 0.43

Reactions of benzene (A1 ) and phenyl (A1 -) 6.60(+25) −5.41 −5.3 362 A1 + H c-C6 H7 4.80(+30) −6.54 2.00(+38) −8.32 2.50(+14) 1.60(+08) 1.42

k 343 k 343 k 345 k 346

−0.9 6.4 16.0 1.45

k 358 k 359 k 360 k 361

= = = =

k 159 k 292 k 293 k 294

20 torr, Wang and Frenklach, 1994 90 torr 760 torr Kiefer et al., 1985 Baulch et al., 1992

DETAILED GAS-PHASE REACTION MECHANISM FOR AROMATICS FORMATION

781

TABLE D.1. (continued ) No. Reactions

a

365 A1 - + H (+M) A1 (+M)

k = A T n exp(−Ea /R u T ) b Comments/ A n Ea References 1.00(+14) 6.60(+75) −16.3

7.0

a = 1.0, T *** = 0.1, T * = 585, T ** = 6113

k ∞, p k o /[M ], e i

Formation and reactions of phenylacetylene (A1 C2 H) 2.30(+68) −17.65 24.4 20 torr, 366 n-C4 H3 + C4 H2 A1 C2 Hk 366 = k 267 9.80(+68) −17.58 26.5 90 torr 9.60(+70) −17.77 31.3 760 torr 367 A1 + C2 H A1 C2 H + H 5.00(+13) n 368 A1 - + C2 H2 n-A1 C2 H2 7.70(+40) −9.19 13.4 20 torr, Wang and Frenklach, 1994 9.90(+41) −9.26 15.7 90 torr 7.00(+38) −8.02 16.4 760 torr 369 A1 - + C2 H2 A1 C2 H + H 7.50(+26) −3.96 17.1 20 torr, Wang and Frenklach, 1994 9.90(+30) −5.07 21.1 90 torr 3.30(+33) −5.70 25.5 760 torr 370 A1 C2 H + H n-A1 C2 H2 1.00(+54) −12.76 17.2 20 torr, Wang and Frenklach, 1994 1.20(+51) −11.69 17.3 90 torr 3.00(+43) −9.22 15.3 760 torr 371 A1 C2 H + H i -A1 C2 H2 1.00(+54) −12.76 17.2 20 torr, k 371 = k 370 1.20(+51) −11.69 17.3 90 torr 3.00(+43) −9.22 15.3 760 torr 372 A1 C2 H + H A1 C2 H* + H2 2.50(+14) 16.0 k 372 = k 363 373 A1 C2 H + H A1 C2 H- + H2 2.50(+14) 16.0 k 373 = k 363 374 A1 C2 H + OH A1 C2 H* + H2 O 1.60(+08) 1.42 1.45 k 374 = k 364 375 A1 C2 H + OH A1 C2 H- + H2 O 1.60(+08) 1.42 1.45 k 375 = k 364 376 A1 C2 H- + H (+M) 1.00(+14) k ∞ , k 376 = k 365 A1 C2 H (+M) 6.60(+75) −16.3 7.0 k o /[M ], e

377 A1 C2 H* + H (+M) A1 C2 H (+M)

a = 1.0, T *** = 0.1, T * = 585, T ** = 6113 1.00(+14) 6.60(+75) −16.3

7.0

i k ∞, k 377 = k 365 k o /[M ], e (continued overleaf )

782

DETAILED GAS-PHASE REACTION MECHANISM FOR AROMATICS FORMATION

TABLE D.1. (continued ) No. Reactions

a

k = A T n exp(−Ea /R u T ) b Comments/ A n Ea References a = 1.0, T *** = 0.1, T * = 585, T ** = 6113

i

Formation and reactions of phenylvinyl (A1 C2 H2 ) and styrene (A1 C2 H3 ) 7.90(+11) 6.4 Fahr and Stein, 378 A1 + C2 H3 A1 C2 H3 + H 1989 379 A1 - + C2 H4 A1 C2 H3 + H 2.51(+12) 6.19 Fahr and Stein, 1989 380 A1 - + C2 H3 A1 C2 H3 1.90(+48) −10.52 17.5 20 torr, p 3.90(+38) −7.63 12.9 90 torr 1.20(+27) −4.22 7.2 760 torr 381 A1 - + C2 H3 i -A1 C2 H2 + H 1.80(+31) −4.63 31.7 20 torr, p 5.80(+18) −1.00 26.8 90 torr 8.50(−02) 4.71 18.4 760 torr 382 A1 - + C2 H3 n-A1 C2 H2 + H 1.50(+32) −4.91 35.5 20 torr, p 5.10(+20) −1.56 31.4 90 torr 9.40(+00) 4.14 23.2 760 torr 383 A1 C2 H3 i -A1 C2 H2 + H 1.20(+46) −9.07 118.3 20 torr, p 3.80(+37) −6.55 114.2 90 torr 5.30(+27) −3.63 109.3 760 torr 384 A1 C2 H3 n-A1 C2 H2 + H 1.90(+54) −11.39 130.2 20 torr, p 1.30(+44) −8.36 125.4 90 torr 1.10(+32) −4.77 119.5 760 torr 385 A1 C2 H3 + H A1 C2 H3 * + H2 2.50(+14) 16.0 k 385 = k 363 386 A1 C2 H3 + OH 1.60(+08) 1.42 1.45 k 386 = k 364 A1 C2 H3 * + H2 O 387 A1 C2 H3 * + H (+M) 1.00(+14) k ∞ , k 387 = k 365 A1 C2 H3 (+M) 6.60(+75) −16.3 7.0 k o /[M ], e

388 A1 C2 H3 + H n-A1 C2 H2 + H2

a = 1.0, T *** = 0.1, T * = 585, T ** = 6113 6.65(+06) 2.53 12.24

389 A1 C2 H3 + H i -A1 C2 H2 + H2

3.33(+05)

2.53

9.24

390 A1 C2 H3 + OH n-A1 C2 H2 + H2 O 3.10(+06) 391 A1 C2 H3 + OH i -A1 C2 H2 + H2 O 1.55(+06) 392 n-A1 C2 H2 + H A1 C2 H + H2 1.50(+13)

2.0 2.0

3.43 0.43

393 i -A1 C2 H2 + H A1 C2 H + H2 394 n-A1 C2 H2 + H i -A1 C2 H2 + H

3.00(+13) 2.30(+37) −6.00 1.20(+25) −2.42 9.90(+04) 3.37 395 n-A1 C2 H2 + OH A1 C2 H + H2 O 2.50(+12)

396 i -A1 C2 H2 + OH A1 C2 H + H2 O

5.00(+12)

35.2 30.5 22.0

i k 388 = 0.5 × k 159 k 389 = 0.5 × k 292 k 390 = k 294 k 391 = k 295 k 392 = 0.5 × k 151 k 393 = k 151 20 torr, p 90 torr 760 torr k 395 = 0.5 × k 153 k 396 = k 153

DETAILED GAS-PHASE REACTION MECHANISM FOR AROMATICS FORMATION

783

TABLE D.1. (continued ) No. Reactions

k = A T n exp(−Ea /R u T ) b Comments/ A n Ea References

a

Formation and reactions of naphthalene (A2 ) 397 A1 C2 H + C2 H2 A2 -1

5.20(+72) −18.11

33.9

2.00(+72) −17.74 1.10(+62) −14.56 5.50(+32) −5.46

36.6 33.1 27.6

−4.59 −1.67

26.0 18.8

2.30(+58) −12.87

44.6

5.20(+64) −14.54 5.70(+64) −14.41 2.00(+82) −20.23

52.2 57.0 36.9

2.00(+75) −18.06 6.90(+63) −14.57 401 A1 C2 H)2 + H naphthyne + H 3.90(+74) −16.91

34.5 29.9 53.7

2.70(+76) −17.32 1.90(+73) −16.30 3.30(+65) −16.79

58.2 60.9 37.4

5.90(+61) −15.42 4.90(+52) −12.43 403 A1 C2 H + C2 H A1 C2 H)2 + H 5.00(+13) 404 A1 C2 H3 * + C2 H2 A2 + H 2.10(+15) −1.07 1.60(+16) −1.33 405 n-A1 C2 H2 + C2 H2 A2 + H 2.10(+15) −1.07

36.5 33.0

1.60(+16) −1.33 2.50(+14) 2.50(+14) 1.60(+08) 1.42 1.60(+08) 1.42 1.00(+14) 3.8(+127) −31.434

5.4 16.0 16.0 1.45 1.45

*

398 A1 C2 H* + C2 H2 A1 C2 H)2 + H

4.80(+29) 1.80(+19) 399 A1 C2 H* + C2 H2 naphthyne + H 400 A1 C2 H)2 + H A2 -1

402 naphthyne + H A2 -1

406 407 408 409 410

A2 + H A2 -1 + H2 A2 + H A2 -2 + H2 A2 + OH A2 -1 + H2 O A2 + OH A2 -2 + H2 O A2 -1 + H (+M) A2 (+M)

6.0 6.6 4.8

18.7

a = 0.2, T = 123, T * = 478, T ** = 5412

20 torr, Wang and Frenklach, 1994 90 torr 760 torr 20 torr, Wang and Frenklach, 1994 90 torr 760 torr, Wang and Frenklach, 1994 20 torr, Wang and Frenklach, 1994 90 torr 760 torr 20 torr, Wang and Frenklach, 1994 90 torr 760 torr 20 torr, Wang and Frenklach, 1994 90 torr 760 torr 20 torr, Wang and Frenklach, 1994 90 torr 760 torr n 20, 90 torr, aa 760 torr 20, 90 torr, k 405 = k 290 760 torr k 406 = k 363 k 407 = k 363 k 408 = k 364 k 409 = k 364 k ∞, p k o /[M ], e

***

411 A2 -2 + H (+M) A2 (+M)

1.00(+14) 9.5(+129) −32.132

18.8

i k ∞, p k o /[M ], e (continued overleaf )

784

DETAILED GAS-PHASE REACTION MECHANISM FOR AROMATICS FORMATION

TABLE D.1. (continued ) No. Reactions

k = A T n exp(−Ea /R u T ) b Comments/ A n Ea References

a

a = 0.87, T *** = 493, T * = 118, T ** = 5652 412 A2 -1 + H A2 -2 + H

8.80(+58) −11.68 6.50(+45) −7.90 2.40(+24) −1.81

61.0 55.5 45.3

i 20 torr, p 90 torr, p 760 torr, p

Formation and reactions of ethynylnaphthalene (A2 C2 H) 413 A2 + C2 H A2 C2 HA + H 414 A2 + C2 H A2 C2 HB + H 415 A2 -1 + C2 H2 A2 C2 H2

5.00(+13) 5.00(+13) 4.50(+39)

−8.71

14.3

416 A2 -1 + C2 H2 A2 C2 HA + H

3.40(+43) 1.70(+43) 1.40(+22)

−9.56 −9.12 −2.64

18.2 21.1 17.4

417 A2 C2 HA + H A2 C2 H2

9.10(+24) −3.39 1.30(+24) −3.06 1.90(+50) −11.63

20.4 22.6 16.2

3.30(+51) −11.72 5.90(+46) −10.03 1.50(+13) 2.50(+12)

18.9 19.1

2.50(+14) 2.50(+14) 1.60(+08)

1.42

16.0 16.0 1.45

k 420 = k 363 k 421 = k 363 k 422 = k 364

1.60(+08)

1.42

1.45

k 423 = k 364

418 A2 C2 H2 + H A2 C2 HA + H2 419 A2 C2 H2 + OH A2 C2 HA + H2 O 420 A2 C2 HA + H A2 C2 HA* + H2 421 A2 C2 HB + H A2 C2 HB* + H2 422 A2 C2 HA + OH A2 C2 HA* + H2 O 423 A2 C2 HB + OH A2 C2 HB* + H2 O 424 A2 C2 HB* + H (+M) A2 C2 HB (+M)

n n 20 torr, Wang and Frenklach, 1994 90 torr 760 torr 20 torr, Wang and Frenklach, 1994 90 torr 760 torr 20 torr, Wang and Frenklach, 1994 90 torr 760 torr n n

k ∞ , k 424 = k 410

1.00(+14) 3.8(+127) −31.434

18.7

a = 0.2, T = 123, T * = 478, T ** = 5412

k o /[M ], e

***

425 A2 C2 HA* + H (+M) A2 C2 HA (+M)

1.00(+14) 9.5(+129) −32.132 18.8 a = 0.87, T *** = 493, T * = 118, T ** = 5652

i k ∞ , k 425 = k 411 k o /[M ], e i

Formation and reactions of phenanthrene (A3 ) 426 A2 C2 HB* + C2 H2 A3 -1

5.20(+72) −18.11

33.9

2.00(+72) −17.74 1.10(+62) −14.56

36.6 33.1

20 torr, k 426 = k 397 90 torr 760 torr

DETAILED GAS-PHASE REACTION MECHANISM FOR AROMATICS FORMATION

785

TABLE D.1. (continued ) No. Reactions

k = A T n exp(−Ea /R u T ) b Comments/ A n Ea References

a

427 A2 C2 HB* + C2 H2 A2 C2 H)2 + H ab

5.50(+32) −5.46

27.6

428 A2 C2 H)2 + H A3 -1

4.80(+29) −4.59 1.80(+19) −1.67 2.00(+82) −20.23

26.0 18.8 36.9

429 A2 C2 HA* + C2 H2 A3 -4

2.00(+75) −18.06 6.90(+63) −14.57 5.20(+72) −18.11

34.5 29.9 33.9

2.00(+72) −17.74 1.10(+62) −14.56 5.50(+32) −5.46

36.6 33.1 27.6

4.80(+29) −4.59 1.80(+19) −1.67 2.00(+82) −20.23

26.0 18.8 36.9

2.00(+75) −18.06 6.90(+63) −14.57 A2 C2 HA + C2 H A2 C2 H)2 + H 5.00(+13) A2 C2 HB + C2 H A2 C2 H)2 + H 5.00(+13) A3 + H A3 -1 + H2 2.50(+14) A3 + H A3 -4 + H2 2.50(+14) A3 + OH A3 -1 + H2 O 1.60(+08) 1.42 A3 + OH A3 -4 + H2 O 1.60(+08) 1.42 A3 -1 + H (+M) A3 (+M) 1.00(+14) 4.0(+148) −37.51

34.5 29.9

430 A2 C2 HA* + C2 H2 A2 C2 H)2 + H 431 A2 C2 H)2 + H A3 -4

432 433 434 435 436 437 438

16.0 16.0 1.45 1.45 20.6

a = 1, T = 1, T * = 145, T ** = 5633

20 torr, k 427 = k 398 90 torr 760 torr 20 torr, k 428 = k 400 90 torr 760 torr 20 torr, k 429 = k 397 90 torr 760 torr 20 torr, k 430 = k 398 90 torr 760 torr 20 torr, k 431 = k 399 90 torr 760 torr n n k 434 = k 363 k 435 = k 363 k 436 = k 364 k 437 = k 364 k ∞, p k o /[M ], e

***

439 A3 -4 + H (+M) A3 (+M)

440 A3 -1 + H A3 -4 + H

i

1.00(+14) 2.1(+139) −34.80 18.4 a = 1, T *** = 1, T * = 171, T ** = 4993

i

1.70(+72) −15.22 9.30(+58) −11.45 3.80(+40) −6.31

20 torr, p 90 torr 760 torr

77.2 71.1 61.8

k ∞, p k o /[M ], e

Formation and reactions of pyrene (A4 ) 441 A3 + C2 H A3 C2 H + H 442 A3 -4 + C2 H2 A3 C2 H2

5.00(+13) 6.70(+45) −10.55

21.2

6.50(+53) −12.59 8.00(+61) −14.50

26.9 34.8

n 20 torr, Wang and Frenklach, 1994 90 torr 760 torr (continued overleaf )

786

DETAILED GAS-PHASE REACTION MECHANISM FOR AROMATICS FORMATION

TABLE D.1. (continued ) No. Reactions

k = A T n exp(−Ea /R u T ) b A n Ea

a

443 A3 -4 + C2 H2 A3 C2 H + H 8.00(+17)

−1.21

22.6

444 A3 -4 + C2 H2 A4 + H

3.40(+12) 1.20(+26) 4.00(+23)

0.34 −3.44 −3.20

19.7 30.2 14.4

445 A3 C2 H + H A3 C2 H2

8.90(+24) −3.56 3.30(+24) −3.36 5.20(+47) −11.05

15.9 17.8 14.7

446 A3 C2 H + H A4 + H

1.40(+56) −13.21 1.90(+64) −15.12 6.80(+26) −4.07

21.0 29.3 9.5

447 A3 C2 H2 A4 + H

4.20(+27) −4.25 9.00(+38) −7.39 7.30(+48) −11.86

10.9 20.7 28.1

448 A4 + H A4 - + H2 449 A4 + OH A4 - + H2 O 450 A4 - + H A4

6.30(+59) −14.70 2.00(+63) −15.28 2.50(+14) 1.60(+08) 1.42 1.00(+14)

36.9 43.2 16.0 1.45

Comments/ References 20 torr, Wang and Frenklach, 1994 90 torr 760 torr 20 torr, Wang and Frenklach, 1994 90 torr 760 torr 20 torr, Wang and Frenklach, 1994 90 torr 760 torr 20 torr, Wang and Frenklach, 1994 90 torr 760 torr 20 torr, Wang and Frenklach, 1994 90 torr 760 torr k 448 = k 363 k 449 = k 364 n

Formation and reactions of biphenyl (P2 ) 451 A1 + A1 - = P2 + H 452 A1 + A1 - P2 -H 453 P2 + H P2 -H 454 A1 - + A1 - P2 455 A1 - + A1 - P2 - + H 456 P2 P2 - + H 457 P2 + H P2 - + H2

5.60(+12) −0.074 1.50(+14) −0.45 1.10(+23) −2.92 8.10(+36) −8.62 2.20(+36) −8.21 3.70(+32) −6.74 1.16(+41) −9.51 2.40(+40) −9.06 6.82(+35) −7.37 3.80(+31) −5.75 6.10(+25) −4.00 2.00(+19) −2.05 7.00(+23) −2.33 8.60(+13) 0.50 2.30(−01) 4.62 9.00(+37) −6.63 8.10(+31) −4.79 1.10(+25) −2.72 2.50(+14) 0.0

7.55 8.92 15.89 9.13 9.92 9.87 10.83 11.57 11.23 7.95 5.59 2.90 38.54 34.82 28.95 119.58 117.12 114.27 16.0

20 torr, p 90 torr 760 torr 20 torr, p 90 torr 760 torr 20 torr, p 90 torr 760 torr 20 torr, p 90 torr 760 torr 20 torr, p 90 torr 760 torr 20 torr, p 90 torr 760 torr k 457 = k 363

DETAILED GAS-PHASE REACTION MECHANISM FOR AROMATICS FORMATION

787

TABLE D.1. (continued ) No. Reactions

a

k = A T n exp(−Ea /R u T ) b Comments/ A n Ea References

458 P2 + OH P2 - + H2 O 459 P2 - + C2 H2 A3 + H

1.60(+08) 1.42 4.60(+06) 1.97 Benzene oxidation 460 A1 + O C6 H5 O + H 2.20(+13) 461 A1 + OH C6 H5 OH + H 1.30(+13) 462 A1 - + O2 C6 H5 O + O 2.10(+12) 463 C6 H5 O CO + C5 H5 464 C6 H5 O + H (+M) C6 H5 OH (+M)

1.45 7.3

k 458 = k 364 k 459 = k ∞,442

4.53 10.6 7.47

Baulch et al., 1992 Baulch et al., 1992 Lin and Lin, 1986, see text Lin and Lin, 1986 k ∞ , Davis et al., 1996 k o /[M ], e

2.50(+11) 2.50(+14)

43.9

1.00(+94) −21.84

13.9

a = 0.043, T *** = 304, T * = 60000, T ** = 5896 465 C6 H5 O + H CO + C5 H6 466 C6 H5 O + O → HCO + 2C2 H2 + CO 467 C6 H5 OH + H C6 H5 O + H2

3.00(+13) 3.00(+13)

n n

1.15(+14)

12.4

468 C6 H5 OH + O C6 H5 O + OH 2.80(+13) 469 C6 H5 OH + OH 6.00(+12) C6 H5 O + H2 O 470 C5 H5 + H (+M) C5 H6 (+M) 1.00(+14)

7.35

4.40(+80) −18.28

13.0

a = 0.068, T *** = 401, T * = 4136, T ** = 5502 471 C5 H5 + O n-C4 H5 + CO

1.00(+14)

472 C5 H5 + OH C5 H4 OH + H 473 C5 H5 + HO2 C5 H5 O + OH

5.00(+12) 3.00(+13)

474 C5 H6 + H C5 H5 + H2

2.20(+08)

475 C5 H6 + O C5 H5 + OH

1.80(+13)

476 C5 H6 + OH C5 H5 + H2 O

3.43(+09)

i

1.77

1.18

He, Mallard, and Tsang 1988 Emdee, 1992 He, Mallard, and Tsang 1988 k ∞ , Davis et al., 1996 k o /[M ], e i

Emdee, Brezinsky, and Glassman, 1992 n Emdee, Brezinsky, and Glassman, 1992 3.0 Emdee, Brezinsky, and Glassman, 1992 3.08 Emdee, Brezinsky, and Glassman, 1992 −0.447 Emdee, Brezinsky, and Glassman, 1992 (continued overleaf )

788

DETAILED GAS-PHASE REACTION MECHANISM FOR AROMATICS FORMATION

TABLE D.1. (continued ) No. Reactions

k = A T n exp(−Ea /R u T ) b Comments/ A n Ea References

477 C5 H5 O n-C4 H5 + CO

2.50(+11)

a

478 479 480 481 482 483 484

C5 H5 O + H CH2 O + 2C2 H2 C5 H5 O + O CO2 + n-C4 H5 C5 H4 OH C5 H4 O + H C5 H4 OH + H CH2 O + 2C2 H2 C5 H4 OH + O CO2 + n-C4 H5 C5 H4 O CO + C2 H2 + C2 H2 C5 H4 O + O CO2 + 2C2 H2

3.00(+13) 3.00(+13) 2.10(+13) 3.00(+13) 3.00(+13) 1.00(+15) 3.00(+13)

PAH Oxidation by 2.18(−04) 485 A1 C2 H + OH → A1 - + CH2 CO 486 A1 C2 H)2 + OH 2.18(−04) → A1 C2 H- + CH2 CO 487 A2 C2 HA + OH → A2 -1 + CH2 CO 2.18(−04) 488 A2 C2 HB + OH → A2 -2 + CH2 CO 2.18(−04) 489 A3 C2 H + OH → A3 -4 + CH2 CO 2.18(−04) 490 A1 C2 H + OH → C6 H5 O + C2 H2 1.30(+13) 491 A1 C2 H3 + OH → C6 H5 O + C2 H4 1.30(+13) 492 A1 C2 H)2 + OH → C4 H2 + C6 H5 O 1.30(+13) 493 A2 + OH 1.30(+13) → A1 C2 H + CH2 CO + H 494 A2 C2 HA + OH 1.30(+13) → A1 C2 H + H2 C4 O + H 495 A2 C2 HB + OH 1.30(+13) → A1 C2 H + H2 C4 O + H 496 A3 + OH 6.50(+12) → A2 C2 HB + CH2 CO + H 497 A3 + OH 6.50(+12) → A2 C2 HA + CH2 CO + H 498 A3 C2 H + OH 6.50(+12) → A2 C2 HA + H2 C4 O + H 499 A3 C2 H + OH 6.50(+12) → A2 C2 HB + H2 C4 O + H 500 A4 + OH → A3 -4 + CH2 CO 1.30(+13) 501 A1 C2 H + O → HCCO + A1 -

43.9

48.0

78.0 OH 4.5 4.5 4.5 4.5 4.5

PAH Oxidation by O 2.04(+07) 2.0

502 A1 C2 H)2 + O 2.04(+07) 2.0 → HCCO + A1 C2 H503 A1 C2 H3 + O → A1 - + CH3 + CO 1.92(+07) 1.83 504 A2 C2 HA + O → HCCO + A2 -1 2.04(+07) 2.0

Emdee, Brezinsky, and Glassman 1992 n n Emdee, 1992 n n Emdee, 1992 n

−1.0 −1.0

k 485 = k 139 k 486 = k 139

−1.0 −1.0 −1.0 10.6 10.6 10.6 10.6

k 487 = k 139 k 488 = k 139 k 489 = k 139 k 490 = k 461 k 491 = k 461 k 492 = k 461 k 493 = k 461

10.6

k 494 = k 461

10.6

k 495 = k 461

10.6

k 496 = 0.5 × k 461

10.6

k 497 = 0.5 × k 461

10.6

k 498 = 0.5 × k 461

10.6

k 499 = 0.5 × k 461

10.6

k 500 = k 461

1.9

k 501 = k 137 + k 139 k 502 = k 137 + k 139 k 503 = k 160 k 504 = k 137 + k 139

1.9 0.22 1.9

DETAILED GAS-PHASE REACTION MECHANISM FOR AROMATICS FORMATION

789

TABLE D.1. (continued ) No. Reactions

k = A T n exp(−Ea /R u T ) b Comments/ A n Ea References

505 A2 C2 HB + O → HCCO + A2 -2

2.04(+07) 2.0

1.9

2.20(+13) 2.20(+13) 2.20(+13) 2.20(+13) 2.20(+13) 2.20(+13) 1.10(+13)

4.53 4.53 4.53 4.53 4.53 4.53 4.53

513 A3 + O → A2 C2 HB + CH2 CO

1.10(+13)

4.53

514 A3 C2 H + O → A2 C2 HA + H2 C4 O

1.10(+13)

4.53

515 A3 C2 H + O → A2 C2 HB + H2 C4 O

1.10(+13)

4.53

2.20(+13) PAH Oxidation by O2 2.10(+12)

4.53

k 505 = k 137 + k 139 k 506 = k 460 k 507 = k 460 k 508 = k 460 k 509 = k 460 k 510 = k 460 k 511 = k 460 k 512 = 0.5 × k 460 k 513 = 0.5 × k 460 k 514 = 0.5 × k 460 k 515 = 0.5 × k 460 k 516 = k 460

7.47

k 517 = k 462

2.10(+12)

7.47

k 518 = k 462

2.10(+12)

7.47

k 519 = k 462

a

506 507 508 509 510 511 512

A1 C2 H + O → C2 H + C6 H5 O A1 C2 H3 + O → C2 H3 + C6 H5 O A1 C2 H)2 + O → C6 H5 O + C4 H A2 + O → CH2 CO + A1 C2 H A2 C2 HA + O → A1 C2 H)2 + CH2 CO A2 C2 HB + O → A1 C2 H)2 + CH2 CO A3 + O → A2 C2 HA + CH2 CO

516 A4 + O → A3 -4 + HCCO

517 A1 C2 H* + O2 → l -C6 H4 + CO + HCO 518 A1 C2 H- + O2 → l -C6 H4 + CO + HCO 519 A1 C2 H3 * + O2 → l -C6 H6 + CO + HCO 520 n-A1 C2 H2 + O2 → A1 - + CO + CH2 O 521 A2 -1 + O2 → A1 C2 H + HCO + CO 522 A2 -2 + O2 → A1 C2 H + HCO + CO 523 A2 C2 HA* + O2 → A2 -1 + CO + CO 524 A2 C2 HB* + O2 → A2 -2 + CO + CO 525 A3 -4 + O2 → A2 C2 HB + HCO + CO 526 A3 -1 + O2 → A2 C2 HA + HCO + CO 527 A4 - + O2 → A3 -4 + CO + CO †

1.00(+11) 2.10(+12) 2.10(+12) 2.10(+12) 2.10(+12) 2.10(+12) 2.10(+12) 2.10(+12)

n 7.47 7.47 7.47 7.47 7.47 7.47 7.47

k 521 = k 462 k 522 = k 462 k 523 = k 462 k 524 = k 462 k 525 = k 462 k 526 = k 462 k 527 = k 462

A1 represents benzene A1 - represents phenyl (benzene minus one hydrogen atom) a Reactions with the sign “” are reversible; those with “→” are irreversible. b The units are mol, cm, s and kcal. Numbers in parentheses denote powers of 10. The reverse rate coefﬁcients were calculated via equilibrium constants. c Third-body enhancement factors: H2 = 0.096T 0.4 , H2 O = 60T −0.25 , CH4 = 2, CO2 = 550T −1.0 , C2 H6 = 3, and Ar = 0.63. d Third-body enhancement factors: H2 = 0.73, H2 O = 3.65, CH4 = 2.0, C2 H6 = 3.0, and Ar = 0.38. e Third-body enhancement factors: H2 = 2, H2 O = 6, CH4 = 2, CO = 1.5, CO2 = 2, C2 H6 = 3, and Ar = 0.7. (continued overleaf ) ‡

790

DETAILED GAS-PHASE REACTION MECHANISM FOR AROMATICS FORMATION

TABLE D.1. (continued ) f Third-body

enhancement factors: H2 = 2.4, H2 O = 15.4, CH4 = 2, CO = 1.75, CO2 = 3.6, C2 H6 = 3, and Ar = 0.83. g Third-body enhancement factors: O = 107T −0.86 , H O = 3.35T 0.1 , CO = 0.75, CO = 1.5, 2 2 2 C2 H6 = 1.5, N2 = 134T −0.86 , and Ar = 0.25T 0.06 . h [M ] = β C , where βi are the third-body enhancement factor with its value given in comment i i i e and Ci the concentration of species i . i Troe’s broadening factor, F (T ) = (1 – a) exp(−T /T *** ) + a exp(−T /T * ) + exp(−T ** /T ). c j Third-body enhancement factors: H = 2, O = 6, H O = 6, CH = 2, CO = 1.5, CO = 3.5, 2 2 2 4 2 C2 H6 = 3, and Ar = 0.5. k Products modiﬁed from HCO + OH in Frenklach et al. (1995) to CO + H + H (see text). l [M ] is deﬁned in comment h. Third-body enhancement factors: H = 2, H O = 6.0, CH = 2, 2 2 4 CO = 1.5, CO2 = 2, and C2 H6 = 3.0. m Evaluated based on the rate data reported in Koshi et al. (1992) and Farhat et al. (1993). n Estimated. o Average of the rate coefﬁcients reported in Koshi et al. (1992) and Farhat et al. (1993). p See appendix of Wang and Frenklach. (1997). q The rate coefﬁcient expression was taken from Tsang and Hampson (1986) with 3.2 kcal/mol added to the activation energy to account for the difference in f H o 298 of C2 H3 between the present study and Tsang and Hampson (1986). r k ∞ was obtained by averaging the rate coefﬁcients reported in Homann and Wellman (1983) and Braun and Unkohoff (1989). k o and F c were obtained by ﬁtting the high-temperature data reported in Wu and Kern (1987) and Hidaka et al. (1989). s Assumed to be equal to that of C2 H3 + CH3 (+M) C3 H6 (+M). k ∞ , k o , and the falloff parameters were obtained by ﬁtting the RRKM results of Tsang and Hampson (1986) with <E down > = 600 cm−1 . [M ] is deﬁned in comment h. The third-body enhancement factors are given in comment e. t Estimated based on a theoretical study (Hammond et al., 1990). u The rate coefﬁcient expression was obtained by ﬁtting the rate data reported in Liu et al. (1988), assuming that the temperature exponent is equal to 2 (see text). v The A factor was assumed to be equal to half of that for k 271 and E reduced by 3 kcal/mol to account for the energy difference between n-C4 H3 and i -C4 H3 (see text). w The A factor was assumed to be equal to half of that for k 291 and E reduced by 3 kcal/mol to account for the energy difference between n-C4 H5 and i -C4 H5 (see text). x The A factor was assumed to be equal to that of k 294 and E increased by 3 kcal/mol to account for the energy difference between n-C4 H5 and i -C4 H5 (see text). y The rate coefﬁcient expression was obtained by ﬁtting the rate data reported in Liu et al. (1988), assuming that the temperature exponent is equal to 2 (see text). z The rate coefﬁcients were assumed to be equal to those of the analogous reactions of C3 H6 . The latter were taken from Tsang (1991). aa The rate coefﬁcient expression is the same as that of the analogous reaction 290, but with 1.2 kcal/mol added to the activation energy to account for the energy barrier difference between C2 H2 addition to n-C4 H5 and to phenyl (Wang and Frenklach, 1994). ab The molecule A2 C2 H)2 is composed of a benzene molecule with two of its adjacent hydrogen atoms replaced by ·C≡CH functional groups.

791

◦ f H298 (kcal/mol)

68.5 6.1 106.5 83.1 45.2 44.0 192.9 112.0 54.6 127.1 119.1 68.0 85.4 77.4 26.3 39.3 165.3 135.4 11.0

Speciesa

C2 O C2 H3 O C3 H2 C3 H3 a-C3 H4 p-C3 H4 C4 H C4 H2 H2 C4 O n-C4 H3 i -C4 H3 C4 H4 n-C4 H5 i -C4 H5 1,3-C4 H6 1,2-C4 H6 C5 H2 C5 H3 C5 H4 O

55.68 64.28 56.22 61.48 54.89 49.49 63.47 61.18 66.43 67.98 70.18 66.57 69.46 68.46 66.40 69.71 63.69 70.53 67.14

◦ S298 (cal/mol K)

Source: Modiﬁed from Wang and Frenklach, 1997.

TABLE D.2. Thermochemical Properties

10.3 13.9 13.2 15.8 14.1 14.6 16.0 17.7 17.3 17.7 19.9 17.5 18.8 18.1 18.3 19.3 19.9 21.0 18.4

300 11.7 20.5 17.0 19.5 19.8 19.7 18.9 21.9 21.8 23.3 24.5 24.5 26.5 26.0 27.3 27.5 26.0 27.0 28.9

500 13.7 23.4 21.6 25.0 28.0 27.7 22.4 26.6 28.7 30.1 30.6 33.7 37.1 37.0 40.5 40.1 32.8 34.9 43.3

14.6 26.4 24.1 27.5 32.0 31.8 24.1 29.0 31.5 33.4 33.6 38.0 42.3 42.2 46.8 45.6 35.4 38.5 49.0

C ◦ p (T) (cal/mol K) 1000 1500 15.2 28.2 25.8 29.2 34.2 34.0 25.2 30.5 33.3 35.0 35.2 40.2 45.1 45.2 50.2 49.5 36.9 40.6 51.8

2000 15.4 29.1 26.7 30.1 35.3 35.2 25.7 31.2 34.2 35.9 36.0 41.3 46.4 46.5 51.8 51.3 37.6 41.7 53.4

2500

1994 1994 1994 1994

1994

(continued overleaf )

Kee et al., 1987 Burcat et al., 1990 Kee et al., 1987 Kee et al., 1987 Kee et al., 1987 Kee et al., 1987 b c Kee et al., 1987 Wang and Frenklach, d Wang and Frenklach, Wang and Frenklach, Wang and Frenklach, Wang and Frenklach, Burcat et al., 1990 Kee et al., 1987 Kee et al., 1987 Burcat et al., 1990

References/ comments

792

C5 H5 C5 H5 O C5 H4 OH C5 H6 C6 H C6 H2 C6 H3 l -C6 H4 c-C6 H4 n-C6 H5 i -C6 H5 C6 H5 O C6 H5 OH l -C6 H6 c-C6 H7 n-C6 H7 i -C6 H7 C6 H8 A1 A1 A1 C2 H*

Species

a

63.5 19.3 19.2 31.9 250.4 169.5 174.7 123.5 106.0 140.9 132.9 11.4 −23.0 81.8 49.9 99.3 91.3 40.2 78.6 20.0 133.2

◦ f H298 (kcal/mol)

TABLE D.2. (continued )

66.79 73.53 74.08 65.49 75.97 74.35 78.76 76.82 67.85 80.50 78.77 73.56 75.24 78.91 72.00 82.08 79.57 79.83 68.62 64.40 78.03

◦ S298 (cal/mol K)

18.4 21.6 23.0 18.2 22.0 24.8 24.9 25.2 19.3 26.2 24.9 22.7 24.9 25.8 21.7 27.3 25.5 26.7 18.5 19.6 26.5

300 29.6 33.5 34.5 30.2 27.0 30.7 32.3 34.0 30.0 35.9 35.0 35.7 38.6 36.9 35.7 39.0 37.7 39.7 30.6 32.7 40.4

500 42.8 48.1 47.6 45.8 32.1 37.0 41.1 44.8 43.3 48.3 48.0 52.2 55.6 51.7 54.5 55.2 54.7 58.5 46.3 50.1 57.5

48.1 53.7 52.9 52.1 34.2 39.8 44.9 49.5 48.7 53.9 53.8 58.5 62.4 58.4 62.6 62.7 62.6 67.2 52.8 57.5 64.4

51.1 57.0 55.9 55.8 35.5 41.5 46.7 52.0 51.6 57.0 57.2 62.3 66.5 62.2 67.2 67.2 67.1 72.4 56.4 61.7 68.5

C ◦ p (T) (cal/mol K) 1000 1500 2000 52.7 58.7 57.6 57.7 36.3 42.4 47.8 53.1 52.7 58.3 58.2 64.3 68.7 63.6 68.7 68.7 68.7 74.0 57.8 63.2 69.5

2500 Burcat et al., 1990 Burcat et al., 1990 Burcat et al., 1990 Burcat et al., 1990 e f d Wang and Frenklach, Wang and Frenklach, Wang and Frenklach, d Kee et al., 1987 Kee et al., 1987 Wang and Frenklach, Wang and Frenklach, Wang and Frenklach, d Wang and Frenklach, Wang and Frenklach, Wang and Frenklach, Wang and Frenklach,

References/ comments

1994 1993, 1994 1993, 1994 1993, 1994

1994 1994 1994

1994 1994 1994

793

A1 C2 HA1 C2 H n-A1 C2 H2 i -A1 C2 H2 A1 C2 H3 * A1 C2 H3 A1 C2 H)2 NAPHTHYNE A2 -1 A2 -2 A2 A2 C2 HA* A2 C2 HB* A2 C2 HA A2 C2 HB A2 C2 H2 A2 C2 H2 A3 -1 A3 -4 A3 A3 C2 H A3 C2 H2 A4 -

Species

a

132.7 73.8 94.5 86.5 92.4 35.4 129.0 119.7 94.7 94.3 35.8 149.6 149.2 90.6 89.6 112.3 176.2 108.5 107.5 49.6 109.1 130.4 112.4

◦ f H298 (kcal/mol)

TABLE D.2. (continued )

78.06 76.41 86.02 81.30 84.52 81.33 84.50 82.17 83.13 83.33 80.26 91.34 91.68 91.21 91.28 100.77 102.67 97.30 97.05 95.74 105.28 112.84 99.56

◦ S298 (cal/mol K)

26.5 27.9 28.0 28.0 26.5 28.5 35.5 30.6 30.4 30.8 31.8 38.5 38.5 39.7 39.9 40.1 49.2 42.9 42.7 43.9 52.2 52.0 46.8

300 40.5 43.1 43.6 43.8 42.3 45.4 52.5 48.4 49.7 49.8 52.1 59.7 59.8 62.1 62.1 62.9 73.0 69.2 69.2 71.4 81.5 82.1 76.1

500 57.6 61.8 63.9 64.6 63.4 68.2 72.7 70.9 74.5 74.5 78.4 85.8 85.8 89.7 89.7 92.1 101.6 102.8 102.8 106.7 118.1 120.4 113.0

64.6 69.5 72.5 73.5 72.5 78.0 80.8 79.8 84.4 84.4 89.2 96.1 96.1 100.8 101.0 104.2 113.1 116.1 116.1 120.8 132.5 135.9 127.4

68.1 73.7 77.7 78.5 77.1 83.8 85.7 85.1 90.2 90.2 95.4 101.9 102.1 107.4 106.6 111.1 118.7 123.8 123.9 129.1 140.7 144.5 134.4

C ◦ p (T) (cal/mol K) 1000 1500 2000 69.8 75.2 79.1 80.2 79.3 85.4 86.7 86.2 91.6 91.6 97.1 103.6 103.4 108.8 109.3 113.2 121.3 125.6 125.5 130.9 142.9 147.2 137.7

2500

1994

1994

1994

1994

1994

1994 1994 1994

(continued overleaf )

Wang and Frenklach, 1993, Wang and Frenklach, 1993, Wang and Frenklach, 1993, d g g Wang and Frenklach, 1993, Miller and Melius, 1992 Wang and Frenklach, 1993, g Wang and Frenklach, 1993, h, i h, j h, k h, l Wang and Frenklach, 1994 h g Wang and Frenklach, 1993, g Wang and Frenklach, 1993, Wang and Frenklach, 1994 g

References/ comments

794

53.9 102.0 43.4 79.0

◦ f H298 (kcal/mol)

95.79 96.70 93.75 100.12

◦ S298 (cal/mol K)

48.7 37.1 38.2 39.7

300 79.2 61.0 63.1 65.9

500 117.3 91.6 95.4 99.8

132.2 103.9 108.6 113.7

140.7 111.1 116.5 121.6

C ◦ p (T) (cal/mol K) 1000 1500 2000 142.7 113.0 118.3 124.1

2500

Wang and Frenklach, 1993, 1994 g g h

References/ comments

thermodynamic data for O, O2 , H, H2 , OH, H2 O, HO2 , H2 O2 , C, CH, CH2 , CH2 * , CH3 , CH4 , CO, CO2 , HCO, CH2 O, CH2 OH, CH3 O, CH3 OH, C2 H, C2 H2 , C2 H3 , C2 H4 , C2 H5 , C2 H6 , CH2 CO, HCCO, and HCCOH were taken from the GRI-Mech 1.2 (Frenklach et al., 1995). b H ◦ f 298 was calculated with ethylenic C–H bond dissociation energy (at 298 K) of 132.9 kcal/mol, consistent with the assignments for C2 H2 and C2 H. The ◦ S 298 and C ◦p data were taken from Wang (1992). ◦ ◦ c H ◦ f 298 was taken from Stull et al. (1969). The S 298 and C p data were taken from Burcat et al. (1990). d H ◦ values of the resonantly stabilized radicals were calculated based on an ethylenic C–H bond dissociation energy (at 298 K) of 111.1 kcal/mol (consistent f 298 ◦ with the assignments of C2 H3 and C2 H4 ) minus 8 kcal/mol of resonance stabilization energy (see text). The S 298 and C ◦p data were obtained using the method described in Wang and Frenklach (1994). e Derived from the data of C H and C H and a “complete additivity” relation, P (C H) = 2P (C H)–P (C H). 2 4 6 4 2 f Derived from the data of C H and C H and a “complete additivity” relation, P (C H ) = 2P (C H )–P (C H ). 2 2 4 2 6 2 4 2 2 2 ◦ g f H ◦298 was taken from Wang and Frenklach (1993). The S 298 and C ◦p data were obtained using the method described in Wang and Frenklach (1994). ◦ ◦ h H ◦ f 298 was calculated using the AM1-GC method described in Wang and Frenklach (1993). The S 298 and C p data were obtained using the method described in Wang and Frenklach (1994). i 1-ethynyl-2-naphthyl. j 2-ethynyl-1-naphthyl. k 1-ethynylnaphthalene. l 2-ethynylnaphthalene. Note: A1 , A2 , A3 , and A4 are described in Appendix D, Table D.1.

a The

A4 P2 -1 P2 P2 -H

Species

a

TABLE D.2. (continued )

APPENDIX E

PARTICLE SIZE–U.S. SIEVE SIZE AND TYLER SCREEN MESH EQUIVALENTS

In the multiphase combustion area, we often encounter unburned and partially burned particles of different sizes. In the United States, these sizes are often expressed in a standard measured quantity in terms of either U.S. Sieve Size or Tyler Screen Mesh. Sieving or screening is a method of separating a mixture of particles (or grains) into two or more size fractions (see Tables E.1 and E.2). The over size particles are trapped above the screen while undersize particles can pass through the screen. Sieves can be used in stacks, to divide samples up into various size fractions and hence determine particle size distributions. Sieves and screen usually are used for larger particle sizes, d p ≥ 37 μm (0.037 mm).

Fundamentals of Turbulent and Multiphase Combustion Copyright © 2012 John Wiley & Sons, Inc.

Kenneth K. Kuo and Ragini Acharya

795

796

PARTICLE SIZE–U.S. SIEVE SIZE AND TYLER SCREEN MESH EQUIVALENTS

TABLE E.1. Standard U.S. Sieve Sizes and Tyler Mesh Sizes U.S. Sieve Size

Tyler Mesh Size

— — No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No.

2 1/2 mesh 3 mesh 3 1/2 mesh 4 mesh 5 mesh 6 mesh 7 mesh 8 mesh 9 mesh 10 mesh 12 mesh 14 mesh 16 mesh 20 mesh 24 mesh 28 mesh 32 mesh 35 mesh 42 mesh 48 mesh 60 mesh 65 mesh 80 mesh 100 mesh 115 mesh 150 mesh 170 mesh 200 mesh 250 mesh 270 mesh 325 mesh 400 mesh

3 1/2 4 5 6 7 8 10 12 14 16 18 20 25 30 35 40 45 50 60 70 80 100 120 140 170 200 230 270 325 400

Opening (mm)

Opening (in)

8.00 6.73 5.66 4.76 4.00 3.36 2.83 2.38 2.00 1.68 1.41 1.19 1.00 0.841 0.707 0.595 0.500 0.420 0.354 0.297 0.250 0.210 0.177 0.149 0.125 0.105 0.088 0.074 0.063 0.053 0.044 0.037

0.312 0.265 0.233 0.187 0.157 0.132 0.111 0.0937 0.0787 0.0661 0.0555 0.0469 0.0394 0.0331 0.0278 0.0234 0.0197 0.0165 0.0139 0.0117 0.0098 0.0083 0.0070 0.0059 0.0049 0.0041 0.0035 0.0029 0.0025 0.0021 0.0017 0.0015

797

1Reference:

4 3

5 3 2

2

2

1

6 54 3

5 3 2

10−1

(1mm)

0.001

3 4 56 8

10−5

5 3 2

2

2

2

0.1

Carbon Black

Solar Radiation

Visible

Atmospheric Dust Sea Salt Nuclei

2,500

10 1,250

3 4 56 8

5,000

2

Spray Dried Milk

6 54 3

5 3 2

0.01

2

2

0.1

6 5 4 3 2

2

2

3 4 56 8

6 5 4 3

6543

10−8

10−6

2

2

10−9

2

Flotation Ores

Pulverized Coal

10

2

10−10

2 6 54 3

65 4 3

100

3 4 56 8

10−8

2

60

65

2

1,000

(1mm.)

Beach Sand

4

4 3

3

¼″

¼″

+Fumishes average particle diameter but no size distribution. ++Size distribution may be obtained by special calibration.

2

10−11

2 6 54 3

65 4 3

(1mm.)

1,000

3 4 56 8

10−9

2

2

½″

2

¾″

¾″

2

(1cm.)

10,000

2

3

2 3

1″

1″

10−11

6 5 4 3 3 4 56 8

10−12

65 4 3

10−10

Machine Tools (Micrometers, Calipers, etc.)

Visible to Eye

Sieving

6

6 ½″

(1cm.)

10,000 3 4 56 8

Microwaves (Radar, etc.)

40 20 12 U.S. Screen Mesh 50 30 16 8

Hydraulic Nozzle Drops

2

2

35 20 10 Tyler Screen Mesh 48 28 14 8

3 4 56 8

Fertilizer, Ground Limestone

Cement Dust

6 5 4 3

65 4 3

3 4 56 8

10−7

1 Particle Diameter, microns (m)

5 3 2

10−5

3 4 56 8

10−7

5 3 2

10−4

100

100

Far Infrared

325 230 170

400 270 200 140

325 250 170

400 270 200 150

Nebulizer Drops Lung Damaging Pneumatic Dust Nozzle Drops Red Blood Cell Diameter (Adults): 7.5m ± 0.3m Human Hair Bacteria

Alkali Fume

100 3 4 56 8

Plant Spores Pollens Milled Flour

Ground Talc

Contact Sulfuric Mist Paint Pigments Insecticide Dusts

625

2

Fly Ash Coal Dust

Sulfuric Concentrator Mist

Near Infrared

Theoretical Mesh (Used very infrequently)

10,000

3 4 56 8

Particle Diameter, microns (m) 1

Electroformed Impingers Sieves Ultramicrosocope Microscope Electron Microscope Centrifuge Elutriation Ultracentrifuge Sedimentation Turbidimetry++ X-Ray Diffraction+ Permeability+ Adsorption+ Scanners Light Scattering++ Nuclei Counter Electrical Conductivity

Combustion Nuclei

Aitken Nuclei

2

Rosin Smoke Oil Smokes Tobacco Smoke Metallurgical Dusts and Fumes Ammonium Chloride Fume

Ultraviolet

1,000

3 4 56 8

Zinc Oxide Fume Colloidal Silica

Viruses

100

10−3

3 4 56 8

10−6

5 3 2

10−2

#Molecular diameters calculated from viscosity data at 0°C.

0.01 3 4 56 8

Gas Molecules

X-Rays

N2 CH4 SO2 CO H2O C4H10 HCl

0.0001

In Water at 25°C.

In Air at 25°C. 1 atm.

2

Ångström Units, Å

10

0.001

(1mm)

3 4 56 8

O2 CO2 C4H4 H2 F2 Cl2

2

Modified from the CRC Handbook of Chemistry and Physics, 83rd Edition 2002-2003, pp.15-31

*Strokes-Cunningham factor included in values given for air but not included for water

Particle Diffusion Coefficient, cm2/sec.

Methods for Particle Size Analysis

Typical Particles and Gas Dispersoids

Electromagnetic Waves

Equivalent Sizes

1

0.0001

TABLE E.2. Particle Characteristics for Different Sizes

BIBLIOGRAPHY

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INDEX

A Acceleration number, 648 Activation energy, 1 Aerodynamic time, 286, 298 Anderson and Jackson, 525, 532, 540 Angular (or rotational) velocity vector, 243 Anisotropy, 236, 260 Appearance of soot, 175–177 Arrhenius factor or parameter, 43, 57 Arrhenius law, 56, 57 Asymptotic Analysis, 74, 78–79, 81 Atomic mass unit, 750 Averaging methods, 516, 520–531, 534 Avogadro’s number, 2, 15, 22, 750 B Baroclinic term, 238, 241 Bassett, Boussinesq, and Oseen (B-B-O) Eq., 646–648, 669 Binary mass diffusivity, 1, 21 Bipropellant liquid rocket, 5, 6 blob injection model, 695–696 BML (Bray-Moss-Libby) model, 343, 347, 358, 359, 360 Boltzmann constant, 1, 22, 750 Fundamentals of Turbulent and Multiphase Combustion Copyright © 2012 John Wiley & Sons, Inc.

Boltzmann statistical averaging, 522, 524, 529, 531 Borghi diagram, 310, 311, 312, 313, 315, 316, 317, 319, 323 Boussinesq approximation, 220, 249, 274 Box ﬁlter, 270–271 Bray number, 284, 356 Breakup time constant, 697–698 Broadening factor, 756, 787 Bulk viscosity, 2, 12, 29 Burke-Schumann solution, 144, 151 C Canonical ﬂame geometries, 299 Centrifugal force, 31 CFD and multiphase simulation, 516–519 Chapman-Rubesin parameter, 162, 169 Characteristic time approach, 298 Characteristics of turbulent ﬂows, 210–213 Chemical equilibrium, 11 Chemical reaction time, 286, 289, 298, 313, 344 Coal-ﬁred burner, 3, 4 Kenneth K. Kuo and Ragini Acharya

869

870

INDEX

Coaxial jet breakup, 688 axisymmetric mode, 688–689 ﬁber-type, 689, 691–692 membrane-type, 690–692 non-axisymmetric mode, 689 Rayleigh-type, 688–689, 691–692 Coﬂow diffusion ﬂame, 165–171 Combustion efﬁciency, 592–594, 604 Complete mechanism, 58–59 Comprehensive reaction scheme, 56 Compressible mixing layers, 240–241 Conditional mean, 254 Conservation equations, 3, 11, 16, 26, 38, 40, 48, 54, 69, 98, 114 Constitutive relationship, 29, 31 Continuity equation, 212, 220, 224, 228 Continuous droplet model, 636 Continuum-formulation model, 636 Conversion factor of gravity, 749 Convolution, 206, 269–270 Coordinate transformations, rotational, 742 Coriolis force, 33 Correction velocity, 26, 27, 41, 44, 52–55 Correlation functions single-point correlation, 207, 222 two-point correlation, 207, 222, 249, 254, 262–263 Corrugated ﬂamelets, 311, 317, 319, 322, 372, 376 Counterﬂow diffusion ﬂames, 155–165, 193–196 Countergradient diffusion, 342, 354, 355, 368 Critical group combustion number, 705 Critical pressure, 13 Critical scalar dissipation rate, 147–150 Critical temperature, 13 Critical Weber number, 604, 658 Cross products of vectors, 727, 728

Cross-term stresses, 205, 272–273 Curl of a vector, 729, 730 Cutoff wavenumber, 267 D d2 evaporation law, 592, 720 Damk¨ohler number, 125–128, 146, 149–150, 152, 164, 174, 283, 311, 314, 317, 346, 397, 400 Damk¨ohler’s analysis, 292 Damk¨ohler’s paradigm, 288, 291 Darrieus-Landau instability, 321, 322 Deﬂagration-to-detonation transition (DDT), 7 Dense ﬂuidized beds, 525, 540 Dense particle ﬂows, 547 Density units, 752 Diesel engines, 3, 5, 578, 581–583, 602, 700 Diffusion velocity, 49, 52–55, 136 Diffusion velocity of the k th species, 42, 49, 51–52, 114 Dilatation turbulent dissipation rates, 239 Dilute particle ﬂows, 549 Dilute spray, 585, 647, 663–665, 669, 686–687, 693, 700, 710–711, 720 Dilute-spray region, 687 Dimensionless separation distance, 705 Direct numerical simulation (DNS), 10, 217, 241–242, 279–280, 282 Directional cosines, 740 Discrete particle methods, 573–574 Discrete-droplet model (or particle-source-in-cell method) deterministic discrete-droplet models (DDDM), 639, 663–637, 669, 671–672, 678–680 stochastic discrete-droplet models (SDDM), 639, 669, 671–680 Displacement speed, 284, 303, 304, 372, 375

INDEX

Dissipation of turbulence kinetic energy, 208, 258 Dissipative scales, 260 Distributed reaction zone and model, 289, 298, 311, 317, 318, 323 Distribution function, 205, 250, 281 Divergence of a tensor, 731, 732 Divergence of a vector, 730, 731 Donor-acceptor methods, 571 Dorodnitsyn-Howarth transformation, 162 Dot products of vectors, 726, 727 Drag coefﬁcient, 603, 646–648, 663, 694, 717, 719 Drop surface layer treatment, 649–650, 655 thin skin model, 649–654 uniform state model, 649, 652 uniform temperature model, 649, 651–654 Droplet accelaration-induced breakup, 657 Droplet arrays, 717–718 Droplet breakup process and regimes, 654–662 explosion type, 585, 655 parachute type (or bag type), 585, 655–656 stripping type, 585, 655 Droplet breakup regimes, 585, 655 Droplet breakup type, 585, 655 Droplet collision, 584, 663, 684, 706–708 droplet-droplet, 706–707 droplet-wall, 706, 708 Drop-life histories in sprays, 652 Dufour heat ﬂux, 34 Dynamic subgrid scale model, 274, 277 Dynamic viscosity, 2, 29 E Eddy cascade hypothesis, 249 Eddy turnover time, 259

871

Eddy viscosity, 220–221, 236, 247, 249, 268, 274, 279 Eddy viscosity models, 274 Eddy-breakup model (EBU model), 283, 324, 335, 336, 337 Magnussen and Hjertager, 336, 337 Spalding’s, 283, 286, 335, 336 Effect of turbulence on ﬂame structure and speed, 289, 290, 313 Electron charge, 750 Electron mass, 750 Element mass fractions, 134–137 Energy cascade, 207, 244, 256, 258, 282 Energy conservation equation, 33, 36, 37, 38 Energy containing range, 207, 209, 258 Energy equation, 228, 230–232, 234, 246, 248, 249, 281 Energy units, 751 Ensemble averaging, 214, 277 Ensemble cell averaging, 531 Enstrophy equation, 245–246 Enthalpy of formation, 17 Envelope ﬂames, 287, 299, 302, 341 Equation of state, 10, 11, 12, 13, 14, 40, 41 cubic equation of state, 13, 14 high pressure correction, 13 ideal gas law, 12, 13 Noble-Abel equation of state, 13 Peng-Robinson equation of state, 13, 14 Redlich-Kwong equation of state, 13, 14 Soave-Redlich-Kwong equation of state, 13, 14 Van der Waals equation of state, 13, 14 Equivalence ratio, 15 Error function, 152, 154 Eulerian averaging, 522–523 Eulerian statistical averaging, 530 Eulerian time averaging, 530

872

INDEX

Eulerian volumetric averaging, 529 Eulerian-Eulerian modeling, 536–549 Eulerian-Lagrangian modeling, 550–553 Extinction limit, 164 F Families of organic compounds, 755 Favre averaging, 217, 218, 225, 242, 255, 282 Favre ﬁltering, 269 Favre pdf, 255 Fick’s law, 52–55 Fick’s law of species mass diffusion, 12, 21, 22, 23, 25, 27, 40 Filtered momentum equations, 270 Filtered-density function, 398 Filtering operation, 267–270 First moment (or mean property), 251 Flame generated turbulence, 291, 296, 341, 342 Flame sheet, 129, 130, 160 Flame stretch, 44, 86–89, 91, 93–101 Flame surface density, 45, 94 Flamelet, 285, 286, 287, 298, 303, 311–323, 337, 357, 369, 371, 376–380, 397 Flamelet structure, 142–145 Flamelets, 142, 155 Flame-stability diagrams, 158 Fluctuating pressure-gradient term, 241 Fluid-ﬂuid modeling, 536 Fluid-solid modeling, 540–541, 550–551 Flux-corrected transport, 571–572 Force units, 751 Formation of aromatics, 109 Forward and backward constants of reactions, 57 Frequency of molecular collision, 2, 22 Front tracking method, 518, 563, 565 Fuel-oxidant ratio, 15

Full cone, 583, 595, 684 Full-ﬁeld modeling (FFM), 606 G g-1 ﬂame, 305, 306, 308, 309 g-4 ﬂame, 305, 309, 310 Gas turbine engines, 4, 5, 7 Gas-turbine combustors, 581, 601, 604, 638 Gauss divergence theorem, 42, 736 Gaussian ﬁlter, 270 Generalized beta function, 623 Generalized Langevin model, 393 g-equation, 284, 340, 367, 368, 370, 372, 373, 374, 375, 376, 400 Germano identity, 207, 276 Germano model, 274 Germano-Lilly procedure, 278 Gibson scale, 284, 300, 378–380 Gradient diffusion model, 249 Gradient of a scalar, 728, 729 Gradient of a vector, 729 Gradient transport, 353–356, 360–364, 396, 400 Grashof number, 207, 210 Gravitational acceleration, 749 Grid ﬁlter, 270, 275–276, 278 Group-combustion models, 700–705 Group-combustion number of the second kind, 701 Growth of aromatics, 110 H H-abstraction-C2 H2 -addition (HACA) mechanism, 108–111, 114, 173, 197–199 Hard-sphere approach, 550–553 Heat capacity, 18 constant-pressure, 17, 18 constant-volume, 18 Heat capacity units, 752 Heat ﬂux units, 752 Heat release by chemical reactions, 34 Heat release factor, 286, 339, 348 Heat transfer coefﬁcient units, 752

INDEX

873

Hirschfelder and Curtiss approximation, 27, 28, 29, 41 Hirschfelder–Curtiss (or zeroth-order) approximation, 49–50, 55 Hollow cone, 583, 595–597, 668, 682, 691 Homogeneous turbulence, 212, 216–217, 238, 246, 253, 261, 274, 279, 280 Homogeneous versus multi-component/multiphase mixtures, 515 Hydrocarbon fuels, 3, 13

sensible, 16, 17, 34 sensible plus chemical, 16, 19, 34, 35, 36, 37 total, 16, 19, 35, 36, 37 total non-chemical, 16, 19, 35, 36, 37 Interphase momentum transfer, 537–539, 541–542, 550–552 Intrinsic low-dimensional manifolds (ILDM) method, 60–61 Isotropic turbulence, 217, 222–224, 241, 258, 261–262, 266, 277–278, 282

I Inertial subrange, 207, 209, 249, 257–259, 263–264 κ-5/3 law, 263–264 Inﬂation of airbags, 8 Inhomogeneous turbulent dissipation rates, 237, 239, 246 Injector systems, 583 external mixing, 583 impinging jet, 583, 699–700 internal mixing, 583 pintle nozzle, 583 plain oriﬁce, 583 pressure-atomizing, 583–584, 683 swirl nozzle, 583 twin-ﬂuid, 583–584, 688 Instantaneous dissipation rate, 339 Intact-core length, 685–686, 689 Integral length scale, 207, 223–224, 239, 249, 262–264, 284, 287, 289, 291, 293, 304, 310, 314, 315, 323, 364, 376, 394, 400 Interdiffusion heat ﬂux, 34 Interface-capturing, 561, 568, 571, 575 Interface-tracking, 561–571 Interfacial transport (jump conditions), 555–561 Intermittency, 307, 335, 338, 339, 342, 345 Internal energy

J Jet ﬂames, 165–171 Joint probability density function, 252 K K41 theory, 258, 260 Karlovitz number, 43, 86, 93, 97–98, 101, 103, 124, 283, 315, 316, 317 Karlovitz, Denniston, Wells’s analysis, 292, 296, 341 Kelvin-Helmholtz (KH) breakup mode, 691 Kelvin-Helmholtz instability, 257 Kinetic energy equation, 34 Kinetic energy spectrum, 259, 263–264 Kinetic Theory of Granular Flow, 540, 542, 547, 552, 574 k − l model, 249 Klimov-Williams criterion, 314, 316, 317, 400 Kolmogorov hypotheses, 258, 261 Kolmogorov length scale, 208, 259, 262–263, 278–280, 285, 291, 311, 315, 316, 323, 376, 379 Kovasznay number, 283, 298 kurtosis (or ﬂatness), 216, 252 L Lagrangian averaging, 522, 532 Lagrangian PLIC, 572

874

INDEX

Laminar ﬂame speed, 44, 46, 66, 68–69, 72, 77–102, 284, 286, 287, 288, 294, 298, 299, 321, 364, 370, 371, 372, 373, 374, 376, 377, 379 Laminar ﬂamelets, 298, 316, 323, 377, 397 Laminar-to-turbulent ﬂow transition, 210 Langevin equations, 393, 394 Laplacian of a scalar, 733 Laplacian of a vector, 733, 734 large eddy simulation (LES), 266–268, 273–274, 278–280, 282 Length units, 751 Leonard stresses, 207, 272–273 Leonard’s decomposition, 267 l-equation, 249 Level set function, 283, 368, 369, 371, 372, 374 Level-set method, 564 Lewis number (Le), 23, 24, 27, 43, 53–55, 69, 96–99 Limitations, 299, 301, 337 Linear Markov model, 393 Liquid jet break up regimes, 580, 602, 606, 689–692 atomization regimes, 684 ﬁrst wind-induced breakup regime, 684–686 Rayleigh breakup regime, 684, 686 second wind-induced breakup regime, 684–686 Liquid rocket engines, 3, 5, 6 Liquid-fueled rocket engines, 581–582 Local instant formulation, 522, 533–535 Local isotropy, 217, 258, 260 Locally-homogeneous ﬂow (LHF) model, 602, 605–606, 608–609, 615, 625–626, 633, 652, 672–673, 675

Local-to-nonlocal triadic interactions, 260 Low-pass ﬁltering, 266 M Magnus lift force, 647 Markers in ﬂuid (MAC formulation), 568 Markers on interface, 564 Markstein’s equation, 322 Mass concentration, 14 Mass conservation equation (or continuity equation), 11, 23, 25, 27, 41 Mass diffusion velocity, 2, 19, 20 Mass ﬂux vector, 25 Mass fraction, 2, 14, 15, 16, 36 Mass production rate of the kth species, 57 Mass rate of production of species i, 2 Mass units, 750 Mass weighted conservation equations, 228–230 Mass weighted transport equations, 231–242 Mathematical Formulation of Soot Formation Model, 114 Mean diameter of a droplet Sauter mean diameter, 586–587, 590, 596, 714 volume median diameter, 590 Mean free path, 2, 22 Mean kinetic energy equation, 231 Measurements in premixed turbulent ﬂames, 324 Mixing length, 207, 221, 224, 241, 247 Mixing length scale, 284, 379 Mixture fraction, 126–127, 130–148, 151, 153–154, 160, 163–168, 170–171, 194, 201–202, 204–205 Molar concentration, 1, 14, 15, 16, 22, 26 Molar diffusion velocity, 2, 20

INDEX

Molar ﬂux vector, 1, 2, 20, 21, 23, 26 Molar rate of production of species i, 2, 26 Molar-average velocity, 1, 2, 19, 20 Mole fraction, 2, 15, 16, 21 Momentum advection, 210 Momentum conservation equation, 29, 30, 31, 32, 34, 40 Momentum diffusion, 210 Momentum equation, 212, 213, 214, 218, 219,228, 229, 230, 231, 232, 243, 255, 268, 270, 271, 272, 275, 280, 282 Multicomponent diffusion coefﬁcient, 49 Multicomponent diffusion velocities (ﬁrst-order approximation), 49 Multiphase ﬂow systems, 512–513, 521 N Name of chain, group, and multipliers, 754 Nanosize energetic particles, 8 Natural gas, 3, 7 Navier-Stokes equation, 211, 212, 214, 228, 247, 252, 254 Neutral wave number, 322 Non-Kolmogorov direct interactions, 260 Non-premixed laminar ﬂames, 128 Numerical simulation (DNS), 217, 279, 282 O Oblique, 299, 302, 324, 332, 357 Obukhov-Corrsin scale, 284, 380 Ohnesorge (Oh) number, 658 One-dimensional premixed H2 /O2 laminar ﬂame solution, 59 Overall ﬂame thicknesses, 377 P Particle inception, 173–174, 179, 201–203 Particle size, 792

875

Particle sizing methods, 711 Peakedness, 252 Perfectly stirred reactor, 298 Planck’s constant, 750 Pollutant emission control, 4 Polycyclic aromatic hydrocarbons (PAHs)., 172–175, 195–200 formation, 104–105, 108–109, 111, 116–117, 120–123 Prandtl number (Pr), 12, 23, 24, 43, 53 Preﬁx deﬁnitions, 752 Premixed ﬂame, 287, 288, 298, 299, 300, 303, 304, 308, 311, 318, 319, 322, 324, 335, 336, 337, 338, 339, 341, 342, 345, 349, 351, 361, 363, 368, 369, 370, 371, 376, 377, 400 Premixed laminar ﬂame thickness, 44, 84–85 Premixed laminar ﬂames, 45–48, 68, 72, 77, 86, 89 Pressure units, 751 Primary breakup, 684, 690, 692, 694 Principal axes of stress, 744 Probability, 207, 214, 216, 217, 249, 250, 251, 254, 281 Probability density function (pdf), 11, 207, 216, 217, 249, 252, 281 Bayes’ theorem, 254 conditional pdf, 254 Gaussian distribution (or normal distribution), 216, 251, 253 joint normal distribution, 253 marginal pdf, 253, 254 nth central moments ,252 Production term, 233, 235, 236, 241, 246 Progress variable, 283, 342–346, 348, 349, 362, 363, 368, 380 Proton mass, 750 Pulsating disintegration submode, 691 Q Quenched reaction zone, 311

876

INDEX

R Ramjets, 5, 6 Random event, 211, 227 Random motion molecular velocity, 22 RANS equations, 242, 247 Rapid distortion theory, 239 Rate controlled constrained equilibrium (RCCE) method, 42, 61 Rayleigh-Taylor (RT) breakup, 691 Rayleigh-Taylor (RT) instability, 257 Reactedness, 304 Reactedness parameter, 626 Reaction mechanisms for aromatics formation, 756 Reaction rate of the i th elementary reaction, 44, 56 Reciprocating engines, 5, 7 Reduced mechanism, 58, 59, 61 Regime diagrams, 311, 317, 318 Reitz-Diwakar (RD) model, 691 Relative mass diffusion velocity, 19–21 Relative molar diffusion velocity, 19–21 Residual tensor, 208, 277 Resolved-scale, 279 Reynolds average Navier-Stokes (RANS) simulation, 10 Reynolds averaging, 217, 218, 224, 244, 249, 252, 255, 280, 281 Reynolds operator, 272, 273 Reynolds stresses and transport equations, 220, 228, 229, 232, 233, 235, 236, 238, 239, 247, 249 Reynolds transport theorem (RTT), 747, 748 Reynolds’ decomposition method, 252 Richtmyer-Meshkov instability, 257

Root-mean-square (rms) velocity ﬂuctuation, 262 RSFS model, 279 S Sample space variable, 208, 250, 251, 254 Saturated straight chain hydrocarbons, 753 Scalar dissipation rate, 126, 144–147, 150–154, 162–166, 201–204 Scale of wrinkles, 284, 286, 304, 305, 309, 310 Scattering ratio techniques, 712 Schlieren photography, 304, 305 Schmidt number, 42, 44, 53 Schmidt number, Sc, 12, 23, 24 Secondary breakup, 684, 691–692, 697–698 Semiglobal mechanism, 42, 59 Sensitivity analysis, 59, 61, 66–68 SFS models, 267, 278, 279 SGS models, 267 Sharp cut-off ﬁlter, 270 Shock-to-detonation transition (SDT), 7 Simpliﬁed Langevin model, 395 Single particle counting (SPC) methods, 711, 714 Single-step mechanism:, 42, 59 Skewness, 216, 252 Smagorinsky model (Smagorinsky-Lilly model), 274 Soft-sphere approach, 550–552, 554 Solenoidal turbulent dissipation rate, 238 Solid propellant rocket motors, 3, 4, 5, 6 Soot formation, 103–123 Soot formation in laminar diffusion ﬂames, 172–204 Soot formation model, 173 Soot oxidation, 104, 114

INDEX

Soot volume fraction, 125, 175, 179–181, 187–192, 194–196, 201, 203–204 Sooting ﬂames, 172, 180, 191 Space thrusters, 6 Spatial averaging, 215 Special function groups, 755 Species mass conservation equation, 23, 25, 27, 28, 242 Speciﬁc dissipation rate, 242 Speciﬁc enthalpy, 18 sensible, 16, 17, 34 sensible plus chemical, 16, 19, 34, 35, 36, 37 total, 16, 19, 35, 36, 37 total non-chemical, 16, 19, 35, 36, 37 Speciﬁc entropy units, 752 Speciﬁc impulse, Isp, 6 Speciﬁc volume units, 751 Speed of light, 750 Splitter plate, 153, 163 Spray combustion, 577–582, 584–585, 587, 591–592, 594–606, 624, 634, 636, 638, 700–701, 705, 717,719 Spray drop distribution function logarithmic probability distribution function, 588–590 Nukiyama-Tanasawa distribution, 588–590, 603 Rosin-Rammler distribution, 588, 590 upper-limit distribution function, 589–590, 715 S-shaped curve of Tmax vs. Da, 146–147 Standard deviation, 208, 215, 222 Stationary turbulent ﬂow, 212, 214, 215, 218, 223, 251 Statistical moments, 216, 249, 251 Statistical understanding, 213, 215 Stefan-Boltzmann constant, 750 Stirred-reactor models, 604

877

Strain rate, 125, 152, 161, 163–166 Strain rate tensor, 1, 29 Stream function, 125–126, 160, 162, 167–168 Stress tensor, 207, 208, 219, 220, 226, 227, 228, 229, 230, 232, 235, 236, 247, 248, 249, 267, 268, 274, 279 total, 2, 29, 42 viscous, 2, 42 Stretch factor, 86, 91–93, 97–99, 101 Subﬁlter scales, 267 Subgrid scale stresses, 274 Subgrid scales, 267, 269, 270, 271, 272, 274,275, 276, 277, 278 Summerﬁeld’s analysis, 297 Superpulsating disintegration submode, 691 Surface growth and oxidation, 174, 203 Surface marker techniques, 564 Surface methods, 531, 563 Surface-ﬁtted method, 567 T Taylor analogy breakup, 691, 693 Taylor length scale (or Taylor microscale), 208, 262, 263, 281 Taylor microscale, 285, 298, 315, 376, 401 Temperature units, 751 Temperature-mixture fraction relationship, 138 Test ﬁlter, 275, 276, 278 Thermal conductivity units, 752 Thermal diffusion coefﬁcient, 2 Thermal diffusion coefﬁcient of kth species, 43, 49, 52, 61 Thick ﬂames, 312, 314, 315, 318, 319, 324 Thin reaction zone, 284, 302, 311, 317, 318, 323, 371, 374, 375, 377, 379, 380 Threshold soot index (TSI), 106–108

878

INDEX

Time and length scales in diffusion ﬂames, 151 Time averaging, 215, 218, 225, 227, 244 Total (material) derivative, 738 Total resolved dissipation, 277 Transport equations, 10, 11, 41 Transport equation for probability density function, 386 Transported PDF method, 381 Triple-correlation, 241 Troe’s falloff formula, 756 Turbulence, 3, 9, 10, 11, 12, 206–208, 210, 212–217, 220224, 226, 234, 237, 238, 241, 245–249, 253, 255–261,263, 264,266, 267, 274, 277–282 strong, 296, 298, 322, 399 weak, 296, 298, 312, 317, 319–322 Turbulence closure, 9, 10 Turbulence dissipation rate, 11 Turbulence intensity, 207, 224 Turbulence kinetic energy (TKE) equation, 234, 246, 248, 256, 281 Turbulence kinetic energy, k, 11 Turbulence models, 223, 247, 249, 279 Turbulence models, one-equation model, 248 Turbulence models, two-equation model of Prandtl-Kolmogorov, 240, 248, 249 Turbulence models, zero-equation model (or Prandtl mixing length model), 247 Turbulence-ﬂame interaction, 322, 378, 397 Turbulent burning velocity, 293, 296, 299, 302, 303, 305, 311, 339, 340 Turbulent dissipation rate equation, 236–242, 246, 248 Turbulent eddy viscosity, 220, 221, 247, 248

Turbulent ﬂame brush, 285, 299, 301, 302, 338, 344, 346, 347, 359, 361, 363, 364 Turbulent ﬂame speed, 286–288, 291, 293, 299, 300, 302, 310, 318, 320, 322, 341, 360, 361, 363, 364 Turbulent Karlovitz number, 317 Turbulent Prandtl number, 284, 294 Turbulent Prandtl/Schmidt number, 615 Turbulent Reynolds stresses, 11 Turbulent scales, 243, 256, 257–265 Turbulent transport, 211, 231, 233–236, 238, 240, 249 Turbulent transport properties, 287 Two-phase-ﬂow models (or dispersed-ﬂow models), 605, 634–699 Two-stage ﬁltering, 275 U U.S. sieve size (Tyler screen mesh), 792, 793 Unattached, 299, 302 Universal gas constant, 749 Unsteady mixing layer, 153, 163 V Variance (2nd statistical moment), 215, 216, 222, 233, 252, 253, 276 Vector algebra, 723, 724 Vector and tensor notations, 737 Vector identities, 735, 736 Velocity units, 751 Virtual mass effect, 647–649, 663, 669 Viscosity units, 752 Viscous diffusion, 233, 235, 236 Viscous stress gradient, 241 VOF, 517–518, 562, 569–571, 573 Volatile organic compounds (VOCs), 4, 5 Volume methods, 561, 563, 568 Volume of ﬂuid, 511, 517, 518, 562, 569–570, 572

INDEX

879

Vortex stretching, 238, 239, 243 Vortex tubes (worms), 244 Vorticity equation, 243, 280 Vorticity ﬂuctuation, 211, 244, 245 Vorticity vector, 243, 244

Wolfhard-Parker slot burner, 166 Work/energy conversion factor, 750 Wrinkled ﬂames, 288, 289, 312, 313, 317, 318, 323, 324, 337, 338, 341, 369, 378–380, 400

W Wavenumber, 207, 208, 260, 263, 264, 265–267, 271 Weber number, 584, 585, 604, 654, 656, 658, 685, 688–689, 692–693, 699 Wiener process, 393, 394, 396

Z Zel dovich number, 126, 149, 150 Zel dovich number, 285, 372 Zero-trace subgrid viscosity model, 276 Zimont model, 323 β-pdf, 623