FUNDAMENTALS OF TURBULENT AND MULTIPHASE COMBUSTION ppt

Preface xix 1 Introduction and Conservation Equations 1 1.1 Why Is Turbulent and Multiphase Combustion Important?, 31.2 Different Applications for Turbulent and MultiphaseCombustion, 3 1

OF TURBULENT AND MULTIPHASE COMBUSTION

KENNETH K KUO

RAGINI ACHARYA

JOHN WILEY & SONS, INC

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Library of Congress Cataloging-in-Publication Data:

Kuo, Kenneth K.

Ragini Acharya — 1st ed.

Ragini p cm.

978-111-8-09932-2 (ebk.); 978-111-8-10767-6 (ebk.); 978-111-8-10768-3 (ebk.);

978-111-8-10770-6 (ebk.)

Ragini II Acharya, Ragini III Title.

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.

Preface xix

1 Introduction and Conservation Equations 1

1.1 Why Is Turbulent and Multiphase Combustion

Important?, 31.2 Different Applications for Turbulent and MultiphaseCombustion, 3

1.2.1 Applications in High Rates of Combustion ofMaterials for Propulsion Systems, 5

1.2.2 Applications in Power Generation, 71.2.3 Applications in Process Industry, 71.2.4 Applications in Household and Industrial Heating, 71.2.5 Applications in Safety Protections for UnwantedCombustion, 7

1.2.6 Applications in Ignition of Various CombustibleMaterials, 8

1.2.7 Applications in Emission Control of CombustionProducts, 8

1.2.8 Applications in Active Control of CombustionProcesses, 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, 111.6.1 Conservation Equations, 11

1.6.2 Transport Equations, 111.6.3 Common Assumptions Made in CombustionModels, 11

1.6.4 Equation of State, 12

1.6.4.1 High-Pressure Correction, 131.7 Definitions of Concentrations, 14

1.8 Definitions of Energy and Enthalpy Forms, 161.9 Velocities of Chemical Species, 19

1.9.1 Definitions of Absolute and Relative Mass andMolar Fluxes, 20

1.10 Dimensionless Numbers, 23

1.11 Derivation of Species Mass Conservation Equation andContinuity Equation for Multicomponent Mixtures, 231.12 Momentum Conservation Equation for Mixture, 29

1.13 Energy Conservation Equation for Multicomponent

Mixture, 331.14 Total Unknowns versus Governing Equations, 40

Homework Problems, 41

2 Laminar Premixed Flames 43

2.1 Basic Structure of One-Dimensional Premixed LaminarFlames, 46

2.2 Conservation Equations for One-Dimensional PremixedLaminar Flames, 47

2.2.1 Various Models for Diffusion Velocities, 49

2.2.1.1 Multicomponent Diffusion Velocities(First-Order Approximation), 492.2.1.2 Various Models for Describing SourceTerms due to Chemical Reactions, 542.2.2 Sensitivity Analysis, 66

2.3 Analytical Relationships for Premixed Laminar Flameswith a Global Reaction, 68

2.3.1 Three Analysis Procedures for Premixed LaminarFlames, 77

2.3.2 Generalized Expression for Laminar FlameSpeeds, 80

2.3.2.1 Reduced Reaction Mechanism for HC-AirFlame, 81

2.4.2 Governing Equation for Premixed Laminar FlameSurface Area, 94

2.4.3 Determination of Unstretched Premixed LaminarFlame Speeds and Markstein Lengths, 952.5 Modeling of Soot Formation in Laminar Premixed

Flames, 1032.5.1 Reaction Mechanisms for Soot Formation andOxidation, 104

2.5.1.1 Empirical Models for SootFormation, 106

2.5.1.2 Detailed Models for Soot Formation andOxidation, 108

2.5.1.3 Formation of Aromatics, 1092.5.1.4 Growth of Aromatics, 1102.5.1.5 Migration Reactions, 1122.5.1.6 Oxidation of Aromatics, 1132.5.2 Mathematical Formulation of Soot FormationModel, 114

Homework Problems, 124

3 Laminar Non-Premixed Flames 125

3.1 Basic Structure of Non-Premixed Laminar Flames, 1283.2 Flame Sheet Model, 129

3.3 Mixture Fraction Definition and Examples, 130

3.3.1 Balance Equations for Element Mass Fractions, 1343.3.2 Temperature-Mixture Fraction Relationship, 1383.4 Flamelet Structure of a Diffusion Flame, 142

3.4.1 Physical Significance of the Instantaneous ScalarDissipation Rate, 145

3.4.2 Steady-State Combustion and Critical ScalarDissipation 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, 1553.6.3 Coflow Diffusion Flame or Jet Flames, 1653.7 Soot Formation in Laminar Diffusion Flames, 1723.7.1 Soot Formation Model, 173

3.7.1.1 Particle Inception, 1743.7.1.2 Surface Growth and Oxidation, 1743.7.2 Appearance of Soot, 175

3.7.3 Experimental Studies by Using CoflowBurners, 176

3.7.3.1 Sooting Zone, 1783.7.3.2 Effect of Fuel Structure, 1823.7.3.3 Influence of Additives, 1833.7.3.4 Coflow Ethylene/Air Laminar DiffusionFlames, 186

3.7.3.5 Modeling of Soot Formation, 191Homework Problems, 204

4 Background in Turbulent Flows 206

4.1 Characteristics of Turbulent Flows, 2104.1.1 Some Pictures, 212

4.2 Statistical Understanding of Turbulence, 2134.2.1 Ensemble Averaging, 214

4.2.2 Time Averaging, 2154.2.3 Spatial Averaging, 2154.2.4 Statistical Moments, 2154.2.5 Homogeneous Turbulence, 2164.2.6 Isotropic Turbulence, 2174.3 Conventional Averaging Methods, 2174.3.1 Reynolds Averaging, 218

4.3.1.1 Correlation Functions, 2224.3.2 Favre Averaging, 225

4.3.3 Relation between Time Averaged-Quantities andMass-Weighted Averaged Quantities, 2274.3.4 Mass-Weighted Conservation and TransportEquations, 228

4.3.4.1 Continuity and MomentumEquations, 228

4.3.4.2 Energy Equation, 2304.3.4.3 Mean Kinetic Energy Equation, 231

CONTENTS xi

4.3.4.4 Reynolds-Stress Transport Equations, 2324.3.4.5 Turbulence-Kinetic-Energy Equation, 2344.3.4.6 Turbulent Dissipation Rate Equation, 2364.3.4.7 Species Mass Conservation

Equation, 2424.3.5 Vorticity Equation, 2434.3.6 Relationship between Enstrophy and the TurbulentDissipation Rate, 246

4.4 Turbulence Models, 247

4.5 Probability Density Function, 249

4.5.1 Distribution Function, 2504.5.2 Joint Probability Density Function, 2524.5.3 Bayes’ Theorem, 254

4.6 Turbulent Scales, 256

4.6.1 Comment on Kolmogorov Hypotheses, 2604.7 Large Eddy Simulation, 266

4.7.1 Filtering, 2684.7.2 Filtered Momentum Equations and Subgrid ScaleStresses, 270

4.7.3 Modeling of Subgrid-Scale Stress Tensors, 2744.8 Direct Numerical Simulation, 279

5.2.4 Summerfield’s Analysis (1955), 2975.2.5 Kovasznay’s Characteristic Time Approach(1956), 298

5.2.6 Limitations of the Preceding Approaches, 2995.3 Characteristic Scale of Wrinkles in Turbulent PremixedFlames, 304

5.3.1 Schlieren Photographs, 3055.3.2 Observations on the Structure of Wrinkled LaminarFlames, 305

5.4.1 Physical Interpretation of Various Regimes inBorghi’s Diagram, 311

5.4.1.1 Wrinkled Flame Regime, 3115.4.1.2 Wrinkled Flame with Pockets Regime(also Called Corrugated FlameRegime), 311

5.4.1.3 Thickened Wrinkled Flames, 3135.4.1.4 Thickened Flames with PossibleExtinctions/Thick Flames, 3145.4.2 Klimov-Williams Criterion, 314

5.4.3 Construction of Borghi Diagram, 316

5.4.3.1 Thick Flames (or Distributed ReactionZone or Well-Stirred Reaction Zone), 3185.4.4 Wrinkled Flames, 318

5.4.4.1 Wrinkled Flamelets (WeakTurbulence), 320

5.4.4.2 Corrugated Flamelets (StrongTurbulence), 322

5.5 Measurements in Premixed Turbulent Flames, 3245.6 Eddy-Break-up Model, 324

5.6.1 Spalding’s EBU Model, 3355.6.2 Magnussen and Hjertager’s EBU Model, 3365.7 Intermittency, 337

5.8 Flame-Turbulence Interaction, 3395.8.1 Effects of Flame on Turbulence, 3415.9 Bray-Moss-Libby Model, 342

5.9.1 Governing Equations, 3495.9.2 Gradient Transport, 3535.9.3 Countergradient Transport, 3545.9.4 Closure of Transport Terms, 357

5.9.4.1 Gradient Closure, 3575.9.4.2 BML Closure, 3585.9.5 Effect of Pressure Fluctuations Gradients, 3615.9.6 Summary of DNS Results, 364

5.10 Turbulent Combustion Modeling Approaches, 368

5.12 Scales in Turbulent Combustion, 376

5.13 Closure of Chemical Reaction Source Term, 380

5.14 Probability Density Function Approach to TurbulentCombustion, 381

5.14.1 Derivation of the Transport Equation for ProbabilityDensity Function, 386

5.14.2 Moment Equations and PDF Equations, 3915.14.3 Lagrangian Equations for Fluid Particles, 3925.14.4 Gradient Transport Model in Composition PDFMethod, 395

5.14.5 Determination of Overall Reaction Rate, 3975.14.6 Lagrangian Monte Carlo Particle Methods, 3985.14.7 Filtered Density Function Approach, 3985.14.8 Prospect of PDF Methods, 399

Homework Problems, 400Project No 1, 400Project No 2, 401

6 Non-premixed Turbulent Flames 402

6.1 Major Issues in Non-premixed Turbulent Flames, 4046.2 Turbulent Damk¨ohler number, 406

6.3 Turbulent Reynolds Number, 407

6.4 Scales in Non-premixed Turbulent Flames, 407

6.4.1 Direct Numerical Simulation and Scales, 4116.5 Turbulent Non-premixed Combustion Regime

Diagram, 4146.6 Turbulent Non-premixed Target Flames, 418

6.6.1 Simple Jet Flames, 419

6.6.1.1 CH4/H2/N2Jet Flame, 4206.6.1.2 Effect of Jet Velocity, 4306.6.2 Piloted Jet Flames, 432

6.6.2.1 Comparison of Simple Jet Flame andSandia Flames D and F, 448

xiv CONTENTS

6.6.3 Bluff Body Flames, 4526.6.4 Swirl Stabilized Flames, 4556.7 Turbulence-Chemistry Interaction, 4566.7.1 Infinite Chemistry Assumption, 456

6.7.1.1 Unity Lewis Number, 4576.7.1.2 Nonunity Lewis Number, 4586.7.2 Finite-Rate Chemistry, 458

6.8 Probability Density Approach for Turbulent Non-premixedCombustion, 462

6.8.1 Physical Models, 4656.8.2 Turbulent Transport in Velocity-Composition PdfMethods, 466

6.8.2.1 Stochastic Mixing Model, 4676.8.2.2 Stochastic Reorientation Model, 4686.8.3 Molecular Transport and Scalar MixingModels, 469

6.8.3.1 Interaction by Exchange with the MeanModel, 471

6.8.3.2 Modified Curl Mixing Model, 4716.8.3.3 Euclidean Minimum Spanning TreeModel, 472

6.9 Flamelet Models, 4766.9.1 Laminar Flamelet Assumption, 4776.9.2 Unsteady Flamelet Modeling, 4786.9.3 Flamelet Models and PDF, 4796.10 Interactions of Flame and Vortices, 480

6.10.1 Flame Rolled Up in a Single Vortex, 4826.10.2 Flame in a Shear Layer, 483

6.10.3 Jet Flames, 4836.10.4 K´arm´an Vortex Street/V-Shaped FlameInteraction, 484

6.10.5 Burning Vortex Ring, 4846.10.6 Head-on Flame/Vortex Interaction, 4856.10.7 Experimental Setups for Flame/Vortex InteractionStudies, 486

6.10.7.1 Reaction Front/Vortex Interaction in

Liquids, 4866.10.7.2 Jet Flames, 4876.10.7.3 Counterflow Diffusion Flames, 4886.11 Generation and Dissipation of Vorticity Effects, 492

6.14.3 Analysis of Edge Flames, 503Homework Problems, 506

Project No 6.1, 506Project No 6.2, 507Project 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, 5147.3 Homogeneous versus Multi-component/Multiphase

Mixtures, 5157.4 CFD and Multiphase Simulation, 516

7.5 Averaging Methods, 520

7.5.1 Eulerian Average—Eulerian Mean Values, 5227.5.2 Lagrangian Average—Lagrangian Mean Values, 5237.5.3 Boltzmann Statistical Average, 524

7.5.4 Anderson and Jackson’s Averaging for DenseFluidized 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, 5387.7.2 Fluid-Solid Modeling, 540

7.7.2.1 Closure Models, 5417.7.2.2 Dense Particle Flows, 5477.7.2.3 Dilute Particle Flows, 5497.8 Eulerian-Lagrangian Modeling, 550

7.8.1 Fluid-Solid Modeling, 551

7.8.1.1 Fluid Phase, 5517.8.1.2 Solid Phase, 5527.9 Interfacial Transport (Jump Conditions), 555

7.10.2.1 Markers in Fluid (MAC

Formulation), 5687.10.2.2 Volume of Fluid Method, 5697.11 Discrete Particle Methods, 573

Homework Problems, 575

8 Spray Atomization and Combustion 576

8.1 Introduction to Spray Combustion, 5788.2 Spray-Combustion Systems, 5808.3 Fuel Atomization, 582

8.3.1 Injector Types, 5828.3.2 Atomization Characteristics, 5848.4 Spray Statistics, 584

8.4.1 Particle Characterization, 5848.4.2 Distribution Function, 585

8.4.2.1 Logarithmic Probability DistributionFunction, 588

8.4.2.2 Rosin-Rammler DistributionFunction, 588

8.4.2.3 Nukiyama-Tanasawa DistributionFunction, 589

8.4.2.4 Upper-Limit Distribution Function ofMugele and Evans, 589

8.4.3 Transport Equation of the DistributionFunction, 590

8.4.4 Simplified Spray Combustion Model for Liquid-FuelRocket Engines, 591

8.5 Spray Combustion Characteristics, 5948.6 Classification of Models Developed for Spray CombustionProcesses, 602

8.6.1 Simple Correlations, 6028.6.2 Droplet Ballistic Models, 6038.6.3 One-Dimensional Models, 6038.6.4 Stirred-Reactor Models, 604

CONTENTS xvii

8.6.5 Locally Homogeneous-Flow Models, 605

8.6.6 Two-Phase-Flow (Dispersed-Flow) Models, 6058.7 Locally Homogeneous Flow Models, 605

8.8 Two-Phase-Flow (Dispersed-Flow) Models, 634

8.8.1 Particle-Source-in-Cell Model (Discrete-DropletModel), 637

8.8.1.1 Models for Single Drop Behavior, 6398.8.2 Drop Breakup Process and Mechanism, 654

8.8.2.1 Drop Breakup Process, 654

8.8.2.2 Multi-component Droplet Breakup byMicroexplosion, 659

8.8.3 Deterministic Discrete Droplet Models, 662

8.8.3.1 Gas-Phase Treatment in DDDMs, 6648.8.3.2 Liquid-Phase Treatment in DDDMs, 6668.8.3.3 Results of DDDMs, 667

8.8.4 Stochastic Discrete Droplet Models, 669

8.8.5 Comparison of Results between DDDMs andSDDMs, 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

8.9 Group-Combustion Models of Chiu, 700

xviii CONTENTS

8.11 Optical Techniques for Particle Size Measurements, 7108.11.1 Types of Optical Particle Sizing Methods, 7118.11.2 Single Particle Counting Methods, 711

8.11.2.1 Scattering Ratio Technique, 7128.11.2.2 Intensity Deconvolution Method, 7138.11.2.3 Interferometric Method (Phase-Shift

Method), 7138.11.2.4 Visibility Method Using a Laser Doppler

Velocimeter LDV, 7138.11.2.5 Phase Doppler Sizing Anemometer, 7138.11.3 Ensemble Particle Sizing Techniques, 714

8.11.3.1 Extinction Measurement Techniques, 7148.11.3.2 Multiple Angle Scattering Technique, 7148.11.3.3 Fraunhofer Diffraction Particle

Analyzer, 7158.11.3.4 Integral Transform Solutions for

Near-Forward Scattering, 7168.12 Effect of Droplet Spacing on Spray Combustion, 7178.12.1 Evaporation and Combustion of DropletArrays, 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

There is an ever-increasing need to understand turbulent and multiphase tion due to their broad application in energy, environment, propulsion, transporta-tion, industrial safety, and nanotechnology More engineers and scientists withskills in these areas are needed to solve many multifaceted problems Turbulenceitself is one of the most complex problems the scientific community faces Itscomplexity increases with chemical reactions and even more in the presence ofmultiphase flows

combus-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, andtransport properties for multicomponent mixtures As the research in laminarflames was overwhelming, he decided to develop two separate books dedicatedentirely to turbulent and multiphase combustion

Turbulence, turbulent combustion, and multiphase reacting flows have beenmajor 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 mental studies with phenomenological approaches, resulting in the development

experi-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 theoreticalmodeling and numerical simulation Experimental diagnostics, especiallynonintrusive laser-based measurement techniques, have been developed and used

to obtain accurate data, which have been used for model validation There is agreater synergy between the experimental and theoretical/numerical approaches.Due to these ongoing developments and advancements, theoretical modeling andnumerical simulation hold great potential for future solutions of problems Inthese two new books, we have attempted to integrate the fundamental theories ofturbulence, combustion, and multiphase phenomena as well as experimental tech-niques, 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 aswell as material that builds on these fundamentals This volume also can beuseful 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 eightbasic topical areas in combustion and multiphase flows, including laminarpremixed and nonpremixed flames; theory of turbulence; turbulent premixed andnonpremixed flames; background of multiphase flows; and spray atomization andcombustion 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 thefirst volume Its chapters include: solid propellant combustion, thermal decom-position and combustion of nitramines burning behavior of homogeneous solidpropellants, chemically reacting boundary-layer flows, ignition and combustion ofcombustion of single energetic solid particles, and combustion of solid particles

in multiphase flows The major reason for including solid-propellant tion here is to provide concepts for condensed-phase combustion modeling as

combus-an example Nitramines are explosive or propellcombus-ant ingredients; their sition and reaction mechanisms are also good examples for combustion behavior

decompo-of condensed-phase materials Chapters in Volume 2 focus on the applicationaspect of fundamental concepts and can form the framework for an advancedgraduate-level course in combustion of condensed-phase materials However, theselection of materials for instruction depends extirely on the interests of instruc-tors and students Although several chapters address solid propellant combustion,this volume is not a textbook for solid propellant combustion; many topics inthis 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 multiphaseflows 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, conservationequations, various models for diffusion velocities in a multicomponent gas systemwith increasing complexities, laminar flame thickness, asymptotic analyses, andflame speeds Effect of flame stretch on laminar flame speed, Karlovitz number,and Markstein lengths are also discussed in detail along with soot formation inlaminar premixed flames

Chapter 3 discusses the basic structure of laminar nonpremixed flames andprovides detailed descriptions of mixture fraction definition, balance equationsfor mixture fraction, temperature-mixture fraction relationship, and examples,since mixture fraction is a very important parameter in the study of nonpremixedflames The chapter also discusses laminar flamelet structure and equations, crit-ical scalar dissipation rate, steady-state combustion, and examples of laminardiffusion flames with equations and solutions Since pollution, specifically sootformation, has become a major topic of interest, it is also covered in this chapterwith respect to laminar diffusion flames Appendix D provides a detailed soot for-mation mechanism and rate constants that was proposed by Wang and Frenklach.Chapter 4 is devoted entirely to turbulent flows It covers the fundamentalunderstanding of turbulence from a statistical point of view; homogeneous and/orisotropic turbulence, averaging procedures, statistical moments, and correlationfunctions; Kolmogorov hypotheses; turbulent scales; filtering and large-eddysimulation (LES) concepts along with various subgrid scale models; and basicdefinitions to prepare readers for the probability density function (pdf) approach

in later chapters This chapter also includes the governing equations for pressible flows A short introduction of the direct numerical simulation (DNS)approach is also provided at the end of the chapter

com-Chapters 5 and 6 focus on the turbulent premixed and nonpremixed flames,respectively Chapter 5 consists of physical interpretation; studies for turbulentflame-speed correlation development; Borghi diagram and physical interpretation

of various regimes; eddy breakup models; measurements in premixed turbulentflames; flame-turbulence interaction (effects of turbulence on flame as well aseffect of flame on turbulence); turbulence combustion modeling approaches;Bray-Moss-Libby model (gradient and counter-gradient transport); level setapproach and G-equation for flame surfaces; and the pdf approach and closure

of chemical reaction source term In Chapter 6, the discussion focuses on majorproblems in nonpremixed turbulent combustion; turbulent Damk¨ohler numberand 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 edgeflames

The fundamentals of multiphase flows are covered in Chapter 7, which hassections on classification of multiphase flows; homogeneous versus multiphasemixtures; averaging methods; local instant formulation; Eulerian-Eulerian mod-eling; Eulerian-Lagrangian modeling; interface transport (tracking and capturing)

xxii PREFACE

methods (volume of fluid, surface fitted method, markers on interface); and crete particle methods This chapter also provides many contemporary approachesfor modeling two-phase flows

dis-Spray combustion is an extremely important topic for combustion, and Chapter

8 provides a comprehensive account of various modeling approaches to spraycombustion associated with single drop behavior, drop breakup mechanisms, jetbreakup models, group combustion models, droplet-droplet collisions, and densesprays 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 combustionbehavior, including desirable characteristics; oxygen balance; homogeneous andheterogeneous propellants; fuel binders, oxidizer ingredients, curing and cross-linking agents, and aging; hazard classifications; material characterization of solidpropellants; and gun performance parameters including thrust, specific impulse,and stable/unstable burning behavior

Chapter 2 focuses on nitramine decomposition and combustion; phase formation; and three different approaches for thermal decomposition of royaldemolition explosive (RDX) as well as gas-phase reactions This chapter alsodescribes a modeling approach for RDX combustion

trans-Chapter 3 covers the burning behavior of homogeneous (e.g., double-base) pellants, describing both the experimental and modeling approaches to study andpredict the burning rate and temperature sensitivities of common solid propellants.The transient burning characteristics of a typical homogeneous propellant is alsopresented in detail, including the Zel’dovich map technique and the Novozhilovstability parameters

pro-Chapter 4 covers reacting turbulent boundary-layer flows, a topic of researchfor the last six decades The chapter discusses the modeling approaches from1940s to the current date Graphite nozzle erosion process by high-temperaturecombustion 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 energeticparticles (such as micron-size boron and aluminum particles) including mul-tistage combustion models for cases with and without the presence of oxidelayers, kinetic mechanisms, criterion for diffusion-controlled combustion versus,kinetic controlled combustion, effect of oxidizers (such as oxygen- and fluorine-containing species), combustion of nano-size energetic particles, and their strongdependency on kinetic rates

Chapter 6 addresses the two-phase reacting flow simulation and focuses ongranular bed combustion with different solution techniques for the governingequations It also includes experimental validation of the calculated results

We would like to acknowledge the contributions of many of our combustionand 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, CornellUniversity; 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 thematerial covered in Volume 1 and Volume 2 We also want to thank ProfessorMichael Frenklach of University of California-Berkeley for providing us thedetailed information on soot formation kinetics used in Appendix D of Volume

1 We also like to thank Professor William A Sirignano of University ofCalifornia-Irvine for his valuable input on evaporation and combustion ofdroplet arrays Professor Norbert Peters of the Institut f¨ur Technische Mechanik

of Aachen, Germany, was very geneous to provide his book draft to KennethKuo while he was visiting the Pennsylvania State University His notes werevery 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 ArmyResearch Office (ARO) hosted and supported a series of his lectures The lecturematerials, which we prepared jointly, were used in the development of severalchapters 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 researchproject sponsors, since his in-depth understanding of many topics in turbulent andmultiphase combustion has been acquired through multi-year research Thesesponsors 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 ofNSWC-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 authorswould like to thank Mr Henry T Rand of ARDEC and Mr Jack Sacco of SavitCorporation for sponsoring our project on granular propellant combustion.Ragini Acharya would like to thank several professors at The PennsylvaniaState University for developing the framework and knowledge base to aid her inwriting 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 hisnumerical 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 metalizedpropellant combustion Also, Professor Luigi De Luca and his colleagues Dr.Filippo Maggi at the Polytechnic Institute of Milan for granting the permission to

Kenneth K Kuo and Ragini Acharya

University Park, Pennsylvania

INTRODUCTION AND

CONSERVATION EQUATIONS

SYMBOLS

B i Body force per unit volume in i-direction (vector) F/L3

C i Molar concentration of the ithspecies N/L3

D Ab Binary mass diffusivity for A-B system L2/t

Ea k Activation energy for the kth reaction Q/N

fi External force per unit mass on species i (vector) F/M

I Identity matrix or vector form of Kronecker delta

2 INTRODUCTION AND CONSERVATION EQUATIONS

m i Mass of the ithspecies in the mixture M

m t Total mass of a multi component gaseous mixture M

n i Number of moles of ith species in the gaseous

T◦ Fixed standard reference temperature, at 298.15 K T

u i Velocity component in ith-direction L/t

vi Velocity of ithspecies with respect to stationary

coordinate axes (vector)

L/t

Vi Mass diffusion velocity of ithspecies (vector) L/t

V∗i Molar diffusion velocity of ithspecies (vector) L/t

Z Frequency of molecular collisions of gaseous

species per unit surface area

L−2t−1

Greek Symbols

α i Thermal diffusion coefficient for species i L2/t

l Thermal conductivity or second viscosity Q/tLT or Ft/L2

μ Dynamic viscosity or first viscosity Ft/L2

μ ij Reduced mass of molecules of species i and j M

˙ i Molar rate or production of species i N/(tL3)

˙ω i Mass rate of production of species i M/(tL3)

DIFFERENT APPLICATIONS FOR TURBULENT AND MULTIPHASE COMBUSTION 3

This chapter first discusses turbulent and multiphase combustion as a majorarea of research for understanding and importance of solution of multiple chal-lenging and interesting problems related to energy, environment, transportation,and chemical propulsion, among other fields The second topic provides a sum-mary of the major conservation equations used by researchers in the combustioncommunity

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 andwood), and gases (such as natural gas composed of largely methane and otherhydrocarbons like ethane, propane, butanes and pentanes) For example, duringthe first decades of the twenty-first century more than 50% of the electricity inthe United States was generated by coal-fired furnaces This trend is expected tocontinue 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 theireffects 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 andnumerical solutions, some of these flows are still unresolved problems of ourtime The complexity of the problem increases even further with the presence ofmultiple phases

In recent years, there has been a greater move to increase combustion ciency while keeping the emissions level as low as possible We live in times inwhich energy has become a very critical commodity Therefore, it is importantthat the unresolved problems of combustion should be understood and solved.Well-trained combustion engineers and scientists are needed to engage in numer-ous challenging combustion problems This chapter provides some general back-ground about the applications of turbulent and multiphase combustion, the generalconcept of modeling, and basic conservation equations for gas-phase mixturescontaining multiple species

effi-1.2 DIFFERENT APPLICATIONS FOR TURBULENT

AND MULTIPHASE COMBUSTION

There are various applications of turbulent and multiphase combustion associatedclosely with our daily life Some of these are:

• Power generation from combustion (one example of two-phase turbulentcombustion used for energy generation from coal-fired burners can be seen

in Figure 1.1)

4 INTRODUCTION AND CONSERVATION EQUATIONS

Electricity

Gas to Stack Gas

Cleaning Condenser Electricity Pressure

Let Down

Solids Return

Gas Turbine

Air to Gasifier Air

Compressor Gas Turbine and Generator Waste Heat Recovery

Steam

Char Circulating FB Combustor

External Heat Exchanger FB: Fluidized Bed

• Household and industrial heating;

• Active control of combustion processes;

• Safety protections for unwanted combustion;

• Ignition of various condensed-phase combustible materials (like solid pellants airbags in automobiles) for safety enhancement under emergencysituation

pro-• Pollutant emission control of combustion products (about one-third of carbonemissions in the United States comes from coal-fired power plants, one-third from transportation, and the rest from the industrial, commercial, andresidential sources)

Figure 1.2 shows the distribution of total emissions estimates in the UnitedStates by source category for specific pollutants in 2008 The major air pol-lutants are particulate matter, CO, CO2, SOx, NOx, VOCs (volatile organiccompounds), NH , mercury, and lead Electric utilities contribute about 70%

DIFFERENT APPLICATIONS FOR TURBULENT AND MULTIPHASE COMBUSTION 5

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

Non-Road Mobile

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) contributeover 80% of national NH3 emissions Almost 50% of the national VOC emis-sions originate from solvent use (other processes) and highway vehicles Highwayvehicles and nonroad mobile sources (e.g., aircrafts, agricultural vehicles, ships,etc.) together contribute approximately 80% of national CO emissions Fossilfuel combustion is the primary source contributing to CO2 emissions In 2007,fossil fuel combustion contributed almost 94% of the total CO2emissions Majorsources 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 imately 33% Advance in combustion technology can lead to higher burningefficiency and less production of harmful compounds

approx-1.2.1 Applications in High Rates of Combustion of Materials

for Propulsion Systems

Many propulsion systems employ combustion of condensed phase materials togenerate 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

Transonic Region Throat

Converging Section

Diverging Section Nozzle

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

Supersonic Region

Viscous Boundary Layer

Subsonic Region

Fuel

Combustion Chamber

• 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 bycombustion The thrust of a propulsion system is proportional to the momentum

of the exhaust jet The specific impulse(I sp ), defined as the thrust per propellant

weight flow rate, is known to be proportional to the square root of the flametemperature divided by the average molecular weight of the combustion products,

as shown in Equation 1.1

DIFFERENT APPLICATIONS FOR TURBULENT AND MULTIPHASE COMBUSTION 7

More detailed description of this relationship is given in Chapter 1 of KuoAcharya, 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 generationsystems 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 hasbeen used for obtained elevated temperature conditions in the manufacturingprocess For example:

• Production of iron, steel, glass, ceramics, cement, carbon black, and refinedfuels through thermal heating processes

• Direct fabrication of ceramic materials by self-propagating high-temperaturesynthesis (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 verted to thermal energy by turbulent and multiphase combustion processes

con-• Thermal energy generated by combustion: Used for heating of residences,

factories, offices, hospitals, schools, and various types of buildings; andheating 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 variousfire 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 variouscombustion materials, for example

• 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

1.2.7 Applications in Emission Control of Combustion Products

The success of emission control of combustion products depend strongly uponthe knowledge of the turbulent and multiphase combustion with application indifferent aspects, such as:

• 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 combustionproducts

1.2.8 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 ofnanosize energetic particles

1.3 OBJECTIVES OF COMBUSTION MODELING

With significant advancements in computational power and numerical schemes inrecent 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/ormulti-phase combustible materials

• To develop predictive capability for combustion systems under various ating conditions

oper-• 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 coupling and interactions between many constituent disciplines Background inthe following areas would be very helpful for scientists and engineers to acquireand to apply to various unresolved combustion problems

• 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

1.5 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 ing equations that must be solved with multiple input parameters and initial andboundary conditions, as shown in Figure 1.4 As one can observe, there is asignificant level of coupling between the intermediate solution from governingequations and the input parameters, such as reaction mechanism, turbulence clo-sure conditions, and diffusion/transport mechanisms The major output of the

govern-10 INTRODUCTION AND CONSERVATION EQUATIONS

Boundary conditions Reaction mechanism

and kinetic data

Governing Equations:

1 Conservation equations

2 Equation of state

3 Transport equations Intermediate solution for

major variables, e.g., T, Y i

Turbulence closure considerations

Convergence criteria and numerical method

Final Output: Flame structure, flame speed, flame surface area, mass consumption rate, etc

Initial conditions and physical model of a given problem

Figure 1.4 General structure of a theoretical model.

model consists of flame structure, speed, surface area, burning rate, flow fieldstructure, and the like

A combustion problem can be solved by using different numerical approaches.Currently there are three major categories of such approaches: Reynolds averageNavies-Stokes (RANS) simulation, large-eddy simulation (LES), and directnumerical simulation (DNS) A discussion of these methods is provided inChapter 4 The effect of these different numerical approaches on the finalsolution can be seen in Figure 1.5, which shows the predicted results for adiffusion 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

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 acompromise between the two and provides excellent reliability and applicability

1.6 GOVERNING EQUATIONS FOR COMBUSTION MODELS

1.6.1 Conservation Equations

The five groups of conservation equations consist of:

1 Conservation of mass (continuity equation)

2 Conservation of molecular species (or conservation of atomic species)

3 Conservation of momentum (for each independent spatial direction)

4 Conservation of energy

5 Conservation of angular momentum

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

1.6.2 Transport Equations

Transport equations are usually required for turbulent combustion problems Theyinclude:

1 Transport of turbulent kinetic energy

2 Transport of turbulence dissipation rate (or turbulent kinetic energy pation rate)

dissi-3 Transport of turbulent Reynolds stresses

4 Transport of probability density function

5 Transport of moments such as

1.6.3 Common Assumptions Made in Combustion Models

Certain commonly used assumptions are listed below Renders must recognizethat some of these assumptions can be relaxed nowadays due to the advancements

in numerical predictive schemes and/or the availability of thermal and transportproperty data

• Reacting fluid can be treated as a continuum

• Infinitely fast chemistry (chemical equilibrium) can be applied for temperature combustion problems

high-12 INTRODUCTION AND CONSERVATION EQUATIONS

• Simple, one-step, forward irreversible global reaction can sometimes beapplied for less comprehensive models

• Ideal gas law can be used for low pressure with moderately high temperaturereacting flow problems

• Lewis, Schmidt, and Prandtl numbers may be assumed equal to 1, undercertain combustion conditions

• Equal mass diffusivities of all species was used by many researchers whenthere were no diffusivity data available

• Fick’s law of species mass diffusion can be assumed to be valid in manycircumstances

• Constant specific heats of the gas-phase species had been assumed when nothermal data were available

• Reacting solid surfaces are sometimes assumed to be energetically geneous

homo-• Uniform pressure can be assumed for the region having low-speed tion situations

combus-• 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

pV = nR u T = m R u

where Ru is the universal gas constant [= 8.3144 J/(mol K)]

Other Forms of Ideal Gas Law

GOVERNING EQUATIONS FOR COMBUSTION MODELS 13

In terms of specific volumev, the ideal gas law can be written as:

pv = RT where v = V

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 modifiedversion of the ideal gas law, expressed by Equation 1.5, except that it accountsfor the intermolecular forces that exist between molecules (represented by theterm a/υ2) and also corrects for the covolume, b, occupied by the molecules

themselves The van der Waals equation of state is:

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

If a is equal to 0, then the van der Waals equation of state is called the

Noble-Abel equation of state

p = RT

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 Waalsequation and has been shown to be very successful not only for pure substancesbut also for mixture calculations and phase equilibrium correlations The originalRedlich-Kwong equation is given as

The values of critical pressure (p c ) and critical temperature (T c ) for various

hydrocarbon fuels are listed in Kuo (2005), Appendix C

Soave-Redlich-Kwong and Peng-Robinson Equations of State The Soave’s ified RK equation or (SRK) and the Peng-Robinson equations of state are both

mod-“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

8Pc

27 64

R2

u T2

c Pc

Pc

0.42748R2

u T2.5 c PcT0.5

both the first and second partial derivatives of pressure with respect to specificvolume 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 areprovided in Table 1.1 for common cubic equations of state

The last four equations of state discussed above can be classified as cubicequations of state; that is, if expanded, the equations would contain volumeterms raised to the first, second, or third power These equations (containing two

parameters a and b) can be expressed by the following equation:

DEFINITIONS OF CONCENTRATIONS 15

4 Mole fraction X i ≡ C i /C is the molar concentration of the ith speciesdivided by the total molar concentration of the gaseous mixture or liquidsolution

Mole Numbers: Gaseous molecules and atoms are conveniently counted interms of amount of substances or mole numbers One mole (1 mol) of compoundcorresponds to 6.02252 × 1023 molecules (or atoms) Avogadro’s number (NA)

Average Molecular Weight:

The mole fractionX i and mass fractionY i are related by:X i = Y i

Mw /Mw i

,

where Mw is the average molecular weight of the multicomponent gas mixture

in the control volume It can be evaluated by:

16 INTRODUCTION AND CONSERVATION EQUATIONS

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

1.8 DEFINITIONS OF ENERGY AND ENTHALPY FORMS

Several definitions of energy are useful in the conservation equations It is veryimportant to have a clear understanding of the physical meaning and mathematicalexpression of each of these energy forms as well as their relationships with

each other Sensible internal energy of ithspecies (e s,i) can be determined with

temperature measurements; therefore, it is called sensible When the heat of formation of the ithspecies is added to the sensible internal energy, their sum isrepresented bye i as shown in Table 1.3 The total internal energy of the ithspecies

(e t,i) includes sensible, kinetic, and chemical energies The total nonchemical

energy (e tnc ,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 ith Species

chemical etnc,i = e s,i+uj uj

2

DEFINITIONS OF ENERGY AND ENTHALPY FORMS 17

The enthalpy and internal energies are related by:

The sensible internal energy is defined to satisfyh s,i = e s,i + p i /ρ i The

sen-sible internal energy for the ith species is defined as:

Since at reference temperature of 298.15 K, the sensible enthalpy is defined

to be zero, that is, h s,i

i thspecies 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 Forexample, see Kuo (2005), Chap 1

The mass-based constant-pressure heat capacities (C p,i ) of the ith species isrelated to the molar heat capacities (C m

p,i) by:

C p,i = C m

For a perfect diatomic gas:

C p,i m = 3.5R u and C p,i = 3.5R u /Mw i (1.27)

In many combustion problems, the change of C p,i with T is quite significant

in chemically reacting flows C p,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 C p increases with temperature due to an increase in the

stored internal energies of different modes, including vibrational, rotational, andtranslational modes at higher temperatures Near room temperature, the molarheat capacity of diatomic gases such as N2 and H2 are very close to 3.5R u;

however, their heat capacities increase rapidly at high temperatures

The mass-based and molar-based constant-volume specific heats are related tothe constant-pressure specific heats by:

C v,i = C p,i − R u /Mw i or C m

v,i − R u (1.28)The constant-pressure heat capacity of the mixture C p is defined by:

(1.31)The specific internal energy of the mixturee = h − p/ρ can be written as:

Table 1.4 summarizes the definitions of different from of energy and enthalpy

of the mixture containing multi-component chemical species