Copyright 2003 Scott Montgomery Martin

©Copyright 2003

Scott Montgomery Martin

The Conditional Moment Closure Method for Modeling Lean Premixed Turbulent Combustion


A dissertation submitted in partial fulfillment of the requirements for the degree of

Doctor of Philosophy

University of Washington

2003

Program Authorized to Offer Degree: Mechanical Engineering

University of Washington Graduate School

This is to certify that I have examined this copy of a doctoral dissertation by


And have found that it is complete and satisfactory in all respects, and that any and all revisions required by the final examining committee have been made.

Chair of Supervisory Committee :

______________________________________________ John C. Kramlich

Reading Committee:

______________________________________________ John C. Kramlich ______________________________________________ James J. Riley ______________________________________________ George Kosály

Date: _______________________

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University of Washington

Abstract

The Conditional Moment Closure Method for Modeling Lean Premixed Turbulent Combustion


Chair of the Supervisory Committee: Professor John C. Kramlich Department of Mechanical Engineering

Natural gas fired lean premixed gas turbines have become the method of choice for new power generation systems due to their high efficiency and low pollutant emissions.

As emission regulations for these combustion systems become more

stringent, the use of numerical modeling has become an important a priori tool in designing clean and efficient combustors. Here a new turbulent combustion model is developed in an attempt to improve the state of the art.

The Conditional Moment Closure (CMC) method is a new theory that has been applied to non-premixed combustion with good success. The application of the CMC method to premixed systems has been proposed, but has not yet been done.

The

premixed CMC method replaces the species mass fractions as independent variables with the species mass fractions that are conditioned on a reaction progress variable (RPV). Conservation equations for these new variables are then derived and solved. The general idea behind the CMC method is that the behavior of the chemical species is closely

coupled to the reaction progress variable. Thus, species conservation equations that are conditioned on the RPV will have terms involving the fluctuating quantities that are much more likely to be negligible.

The CMC method accounts for the interaction

between scalar dissipation (micromixing) and chemistry, while de-coupling the kinetics from the bulk flow (macromixing).

Here the CMC method is combined with a

commercial computational fluid dynamics program, which calculates the large-scale fluid motions.

The CMC model is validated by comparison to 2-D reacting backward facing step data. Predicted species, temperature and velocity fields are compared to experimental data with good success. The CMC model is also validated against the University of Washington’s 3-D jet stirred reactor (JSR) data, which is an idealized lean premixed combustor. The JSR results are encouraging, but not as good as the backward facing step. The largest source of error is from the turbulence models, which are inadequate for the variable density and recirculating flows modeled here.

The limitations of the

turbulence models affected the calculation of the flow statistics, which are used to calculate the variance of the RPV, the scalar dissipation and the PDF.

TABLE OF CONTENTS

Page

List of Figures…………………………………………………………………………….. v

List of Tables..............................................................……………….………………… xix

Glossary…….......................................................................……………………..……. xx

Chapter I:

Chapter II:

Introduction..........................................................................…………….... 1 1.1

Background and Justification...................……………………........... 1

1.2

Objectives and Approach........................................….....….………. 6

1.3

Organization..................................................................…..……….... 7

Overview of Current Numerical Modeling Techniques for Premixed Combustion.……………………………………..…......…………….…… 8 2.1

Introduction.......................................................................….……..... 8

2.2

Reacting Computational Fluid Dynamics..........….……………….... 8

2.2.1 Direct Numerical Simulation……………………………………..… 9 2.2.2 Large Eddy Simulation……………………………………………. 12 2.2.3 Reynolds Stress Model…………………………………………..…13 2.2.4 κ − ε Model………………….…………………….……………… 16 2.3

Methods of Computing Chemistry in Premixed Turbulent Reacting Flows………………………………………………………….…… 22

2.3.1 One Equation Combustion Models……………………………..…. 27 2.3.2 Eddy Breakup Model……………………………………………… 28 2.3.3 Flamelet Models………..…………………………………..………33 i

2.3.4 G Equation Model……………………………………………….….38 2.3.5 Probability Density Function……………………………….………41 2.4

Chemical Reactor Models……………………………………….… 49

2.4.1 Simple Reactor Models………………………………………….… 49 2.4.2 Chemical Reactor Model with PDF micromixing………………… 50 2.5

Chapter III:

Summary............……………………………………………….….. 51

The Conditional Moment Closure Model....................……………….…. 54 3.1

Introduction..........………………… ………………………..…..… 54

3.2

The Non-Premixed Conditional Moment Closure Method…....…...55

3.3

The Premixed CMC Method…………………………………….… 58

3.3.1 Derivation of the Premixed CMC Equations………………….…... 59 3.3.2 Derivation of the Volume Averaged Equations…………………… 63 3.3.3 Uniform Conditioned Species Method Formulation.………………65 3.3.4 Analysis of Error Terms……………………………………………66 3.4

Results of the Stand Alone Premixed CMC Method………..…….. 67

3.4.1 Atmospheric Pressure Results………..………………..………….. 71 3.4.2 High Pressure Results……………………………...………..…….. 81 3.4.3 Comparison to Other Models………………………..……..…….... 82 3.5

Chapter IV:

Summary..........…………………………….………………….…... 85

Fluent Reacting Flow Results..……………………………….…………122 4.1

Introduction................………………………………………...….. 122

4.2

Test Problem……………………………………………..………. 122

4.3

Backward Facing Step Results …………..……………...……..…123

4.3.1 Adiabatic Walls……………………………………………..……. 125 4.3.2 Non-Adiabatic Walls………………………………………..…….127 4.4

Summary..........……………………………….……….…….…… 128 ii

Chapter V:

Chapter VI:

Combination of CFD and The Premixed CMC Method...…………..…. 135 5.1

Introduction………………………………………………………. 135

5.2

Test Problem………………………………………….…...………135

5.3

Probability Density Function Shapes………………….…….….... 136

5.4

Backward Facing Step Results..……………………………….….137

5.5

Summary..........………………………………………....…...…… 145

Jet Stirred Reactor Results.……………………….………………..……166 6.1

Introduction……………………………...…………………...……166

6.2

Atmospheric Pressure Results ……………………………………166

6.3

High Pressure Results ………………………………….…………172

6.4

Summary ……………………………………………….…………175

Chapter VII: The Non-Adiabatic CMC Method…………………………………..…..200

Chapter VIII: Conclusions and Future Work…………………………………….…….204 8.1

Conclusions……………………………..……………………...… 204

8.2

Future Work……………………………………………………… 206

Bibliography......................................…………………………………………….…..…208 Appendix A: Derivation of the Enthalpy Conservation Equations……………………219 Appendix B: Derivation of the Reaction Progress Variable…….………………….…222 Appendix C: Derivation of the PDF Conservation Equation……..………………….. 224 Appendix D: Derivation of the Premixed CMC Equation………..…………….…..…227 Appendix E: PTURCEL Derivation………………………………………………….. 238 Appendix F: Derivation of the Premixed CMC Method, Bilger’s Method………….. 244 Appendix G: Derivation of the Averaged c Equation and its Variance…………...…..249 iii

Appendix H: Source Term For Variance Equation…………………...……………….254 Appendix I:

Additional Figures from Chapter III…………………...………………. 258

Appendix J:

Analytical Solution for 1-Step CH 4 Mechanism……………………… 271

Appendix K: Comparison of the G Equation to Models based on Reactive Scalars…. 277

iv

LIST OF FIGURES

Figure Number

Page

2.1

Premixed combustion regimes plot………………………………………………53

3.1

Illustration of conditional average………………………………..………………87

3.2

Premixed CMC with GRI2.11. Inlet temperature 300 K, pressure 1 atm., equivalence ratio 0.9 and scalar dissipation 10 1/sec.……………………………88

3.3

Premixed CMC with GRI2.11. Inlet temperature 300 K, pressure 1 atm., equivalence ratio 0.9 and scalar dissipation 1,000 1/sec.……..………………… 89

3.4

Premixed CMC with GRI2.11. Inlet temperature 300 K, pressure 1 atm., equivalence ratio 0.9 and scalar dissipation 10 1/sec, c based on temperature ratio…………………………………………………………....………………… 90

3.5

Premixed CMC with GRI2.11. Inlet temperature 300 K, pressure 1 atm., equivalence ratio 0.9 and scalar dissipation 1,000 1/sec, c based on CO 2 ratio…………………………………………………………....………………… 91

3.6

Premixed CMC with GRI2.11. Inlet temperature 300 K, pressure 1 atm., equivalence ratio 0.9 and scalar dissipation 10 1/sec.………………………….. 92

3.7

Premixed CMC with GRI2.11. Inlet temperature 300 K, pressure 1 atm., v

equivalence ratio 0.9 and scalar dissipation 1,000 1/sec.……………………..… 93

3.8


3.9

Premixed CMC with GRI2.11. Inlet temperature 300 K, pressure 1 atm., equivalence ratio 0.9 and scalar dissipation 1,000 1/sec.………………….……..95

3.10


3.11

Premixed CMC with GRI2.11. Inlet temperature 300 K, pressure 1 atm., equivalence ratio 0.9 and scalar dissipation 1,000 1/sec.………………...………97

3.12

Premixed CMC with GRI2.11. Inlet temperature 300 K, pressure 1 atm., equivalence ratio 0.9 and scalar dissipation 10 1/sec.……………………..……. 98

3.13

Premixed CMC with GRI2.11. Inlet temperature 300 K, pressure 1 atm., equivalence ratio 0.9 and scalar dissipation 1,000 1/sec.……………………...…99

3.14

Premixed CMC with GRI2.11, CH 4 versus scalar dissipation, 1/sec. Inlet temperature 300 K, pressure 1 atm. and equivalence ratio 0.9.……………..….100

3.15

Premixed CMC with GRI2.11, CO versus scalar dissipation, 1/sec. Inlet temperature 300 K, pressure 1 atm. and equivalence ratio 0.9.………………... 101

3.16

Premixed CMC with GRI2.11, OH versus scalar dissipation, 1/sec. Inlet temperature 300 K, pressure 1 atm. and equivalence ratio 0.9.…………………102 vi

3.17

Premixed CMC with GRI2.11, c equation source term versus scalar dissipation, 1/sec. Inlet temperature 300 K, pressure 1 atm. and equivalence ratio 0.9…….103

3.18

Premixed CMC with full Miller-Bowman. Inlet temperature 300 K, pressure 1 atm., equivalence ratio 0.9 and scalar dissipation 10 1/sec.…………….………104

3.19

Premixed CMC with full Miller-Bowman. Inlet temperature 300 K, pressure 1 atm., equivalence ratio 0.9 and scalar dissipation 10 1/sec.………………….…105

3.20

Premixed CMC with full Miller-Bowman. Inlet temperature 300 K, pressure 1 atm., equivalence ratio 0.9 and scalar dissipation 10 1/sec.………….…………106

3.21

Premixed CMC with full Miller-Bowman. Inlet temperature 300 K, pressure 1 atm., equivalence ratio 0.9 and scalar dissipation 10 1/sec.………………….…107

3.22


3.23


3.24

Premixed CMC with reduced Miller-Bowman. Inlet temperature 300 K, pressure 1 atm., equivalence ratio 0.9 and scalar dissipation 14 1/sec.………..…………110

3.25

Premixed CMC with reduced Miller-Bowman. Inlet temperature 300 K, pressure 1 atm., equivalence ratio 0.9 and scalar dissipation 14 1/sec.……………..……111

vii

3.26

Premixed CMC with reduced Miller-Bowman. Inlet temperature 300 K, pressure 1 atm., equivalence ratio 0.9 and scalar dissipation 14 1/sec.………………..…112

3.27

Premixed CMC with reduced Miller-Bowman. Inlet temperature 300 K, pressure 1 atm., equivalence ratio 0.9 and scalar dissipation 14 1/sec.…………..………113

3.28

Premixed CMC with reduced Miller-Bowman. Inlet temperature 300 K, pressure 1 atm., equivalence ratio 0.9 and scalar dissipation 14 1/sec.……………..……114

3.29

Premixed CMC with reduced Miller-Bowman. Inlet temperature 300 K, pressure 1 atm., equivalence ratio 0.9 and scalar dissipation 14 1/sec.………………..…115

3.30

Premixed CMC with GRI2.11, CO versus pressure, atm. Inlet temperature 603 K, scalar dissipation 350 1/sec and equivalence ratio 0.529.…………………..…..116

3.31

Premixed CMC with GRI2.11, CO 2 versus pressure, atm. Inlet temperature 603 K, scalar dissipation 350 1/sec and equivalence ratio 0.529.……………….…..117

3.32

Premixed CMC with GRI2.11, O 2 versus pressure, atm. Inlet temperature 603 K, scalar dissipation 350 1/sec and equivalence ratio 0.529.…………………..…..118

3.33

Premixed CMC with GRI2.11, NO versus pressure, atm. Inlet temperature 603 K, scalar dissipation 350 1/sec and equivalence ratio 0.529.……………………....119

3.34

Premixed CMC with GRI2.11, NO 2 versus pressure, atm. Inlet temperature 603 K, scalar dissipation 350 1/sec and equivalence ratio 0.529.……………….…..120

3.35

Premixed CMC with GRI2.11, N 2 O versus pressure, atm. Inlet temperature 603 viii

K, scalar dissipation 350 1/sec and equivalence ratio 0.529.……………….…..121

4.1

Schematic of the 2-D backward facing step reactor showing flame and recirculation zones. Note: horizontal and vertical dimensions not to scale.………………………………………………………………………...…..129

4.2

El Banhawy et al. (1983) data, equivalence ratio=0.9, axial velocity m/sec, axial velocity rms, temperature K, unburned hydrocarbons-wet, O 2 -dry, CO 2 -dry and CO-dry, species are mole percentages ( O 2 , CO 2 and CO have the UHC removed). Reprinted by permission of Elsevier Science from Premixed, Turbulent Combustion of a Sudden-Expansion Flow, by Y. El Banhawy et al. Combustion & Flame, Vol. No. 50, pp 153-165, Copyright 1983 by The Combustion Institute.……………………………………………………………130

4.3

Fluent 1-step EDM, Reynolds stress model equivalence ratio=0.9 axial velocity (m/sec), axial velocity rms (m/sec), temperature K, CH 4 -wet, O 2 -dry and CO 2 dry, species are mole percentages, ( O 2 and CO 2 have the CH 4 removed)…..…131

4.4

Fluent 3-step EDM, Reynolds stress model equivalence ratio=0.9 axial velocity (m/sec), axial velocity rms (m/sec), temperature K, CH 4 -wet, O 2 -dry, CO 2 -dry and CO-dry, species are mole percentages, ( O 2 , CO 2 and CO have the CH 4 removed)..…………………………………………………………………….…132

4.5

Fluent with built-in premixed combustion model, RSM turbulence model, equivalence ratio=0.9, axial velocity (m/sec), temperature K, progress variable c, Damköhler number and turbulent Reynolds number………………………...… 133

ix

4.6

Fluent 1-step EDM, Reynolds stress model equivalence ratio=0.9 axial velocity (m/sec), axial velocity rms (m/sec), temperature K, CH 4 -wet, O 2 -dry and CO 2 dry, species are mole percentages, ( O 2 and CO 2 have the CH 4 removed), with wall temperature = 300 K………………………………………………….……134

5.1

Fluent and CMC with GRI2.11, RSM turbulence model, equivalence ratio=0.9, axial velocity (m/sec), axial velocity rms (m/sec), temperature K, CH 4 -wet, O 2 dry, CO 2 -dry and CO-dry, species are mole percentages, ( O 2 , CO 2 and CO have the UHC removed).………………………………………………………..…… 147

5.2

Fluent and CMC with GRI2.1, Realizable turbulence model, equivalence ratio=0.9, axial velocity (m/sec), axial velocity rms (m/sec), temperature K, CH 4 wet, O 2 -dry, CO 2 -dry and CO-dry, species are mole percentages, ( O 2 , CO 2 and CO have the UHC removed)…………………………………………………… 148

5.3

Fluent and CMC with GRI2.11, RNG turbulence model, equivalence ratio=0.9, axial velocity (m/sec), axial velocity rms (m/sec), temperature K, CH 4 -wet, O 2 dry, CO 2 -dry and CO-dry, species are mole percentages, ( O 2 , CO 2 and CO have the UHC removed)………………………………………………...…………… 149

5.4

Fluent and CMC with GRI2.11, RNG turbulence model with delta-rho term, equivalence ratio=0.9, axial velocity (m/sec), axial velocity rms (m/sec), temperature K, CH 4 wet, O 2 -dry, CO 2 -dry and CO-dry, species are mole percentages, ( O 2 , CO 2 and CO have the UHC removed)………………………150

5.5

Fluent and CMC with GRI2.11, RNG turbulence model with delta-rho term and scalar dissipation increased by a factor of 10, equivalence ratio =0.9, axial x

velocity (m/sec), axial velocity rms (m/sec), temperature K, CH 4 -wet, O 2 -dry,

CO 2 -dry and CO-dry, species are mole percentages, ( O 2 , CO 2 and CO have the UHC removed)…………………………………………………………..………151

5.6a

Fluent and CMC with GRI2.11, RNG turbulence model with delta-rho term and scalar dissipation increased by a factor of 20, equivalence ratio=0.9, axial velocity (m/sec), axial velocity rms (m/sec), temperature K, CH 4 -wet, O 2 -dry, CO 2 -dry and CO-dry, species are mole percentages, ( O 2 , CO 2 and CO have the UHC removed)……………………………………………………………..………… 152

5.6b

Fluent and CMC with GRI2.11, RNG turbulence model with delta-rho term and scalar dissipation increased by a factor of 20, equivalence ratio=0.9, c, variance of c, scalar dissipation, HNO, NO, NO 2 and N 2 O mass fractions…………………153

5.6c

Fluent and CMC with GRI2.11, RNG turbulence model with delta-rho term and scalar dissipation increased by a factor of 20, equivalence ratio=0.9, CH, HCN, CN, NCO, O, H and OH mass fractions ……………………………..…………154

5.7

Fluent and CMC with GRI2.11, RNG turbulence model with delta-rho term and scalar dissipation increased by a factor of 30, equivalence ratio=0.9, axial velocity (m/sec), axial velocity rms (m/sec), temperature K, CH 4 -wet, O 2 -dry, CO 2 -dry and CO-dry, species are mole percentages, ( O 2 , CO 2 and CO have the UHC removed)……………………………………...…………………………………155

5.8

Fluent and CMC with GRI2.11, RNG turbulence model with delta-rho term and scalar dissipation increased by a factor of 20, scalar dissipation calculated with alternate method, equivalence ratio=0.9, axial velocity (m/sec), axial velocity rms xi

(m/sec), temperature K, CH 4 -wet, O 2 -dry, CO 2 -dry and CO-dry, species are mole percentages, ( O 2 , CO 2 and CO have the UHC removed)…………………156

5.8b

Fluent and CMC with GRI2.11, RNG turbulence model with delta-rho term. Scalar dissipation for conditions in Figures 5-4, 5-5, 5-6a, 5-7 and 5-8, respectively ……………………………………………………………..………157

5.9

Fluent and CMC with GRI2.11, RNG turbulence model with delta-rho term and scalar dissipation increased by a factor of 20, c equation source term decreased by 20%, equivalence ratio=0.9, axial velocity (m/sec), axial velocity rms (m/sec), temperature K, CH 4 -wet, O 2 -dry, CO 2 -dry and CO-dry, species are mole percentages, ( O 2 , CO 2 and CO have the UHC removed)……………...………158

5.10

Fluent and CMC with GRI2.11, RNG turbulence model with delta-rho term and scalar dissipation increased by a factor of 20, C3=1 in c variance equation source term, equivalence ratio 0.9, axial velocity (m/sec), axial velocity rms (m/sec), temperature K, CH 4 -wet, O 2 -dry, CO 2 -dry and CO-dry, species are mole percentages, ( O 2 , CO 2 and CO have the UHC removed)………………………159

5.11

El Banhawy et al. (1983) data, equivalence ratio=0.77, temperature K, unburned hydrocarbons-wet, O 2 -dry, CO 2 -dry and CO-dry, species are mole percentages ( O 2 , CO 2 and CO have the UHC removed). Reprinted by permission of Elsevier Science from Premixed, Turbulent Combustion of a Sudden-Expansion Flow, by Y. El Banhawy et al. Combustion & Flame, Vol No 50, pp 153-165, Copyright 1983 by The Combustion Institute…………………………….………………. 160

5.12

Fluent and CMC with GRI2.11, RNG turbulence model with delta-rho term and xii

scalar dissipation increased by a factor of 20, equivalence ratio=0.77, axial velocity (m/sec), axial velocity rms (m/sec), temperature K, CH 4 -wet, O 2 -dry,

CO 2 -dry and CO-dry, species are mole percentages, ( O 2 , CO 2 and CO have the UHC removed)…………………………………………………………………. 161

5.13

El Banhawy et al. (1983) data, equivalence ratio=0.95, temperature K, unburned hydrocarbons-wet, O 2 -dry, CO 2 -dry and CO-dry, species are mole percentages ( O 2 , CO 2 and CO have the UHC removed). Reprinted by permission of Elsevier Science from Premixed, Turbulent Combustion of a Sudden-Expansion Flow, by Y. El Banhawy et al. Combustion & Flame, Vol No 50, pp 153-165, Copyright 1983 by The Combustion Institute…………………………………………...…162

5.14

Fluent and CMC with GRI2.11, RNG turbulence model with delta-rho term and scalar dissipation increased by a factor of 20, equivalence ratio=0.95, axial velocity (m/sec), axial velocity rms (m/sec), temperature K, CH 4 -wet, O 2 -dry,

CO 2 -dry and CO-dry, species are mole percentages, ( O 2 , CO 2 and CO have the UHC removed)…………………………………………………………………. 163

5.15

Fluent and CMC with full Miller-Bowman, RNG turbulence model with deltarho term and scalar dissipation increased by a factor of 20, equivalence ratio=0.9,axial velocity (m/sec), axial velocity rms (m/sec), temperature K, CH 4 wet, O 2 -dry, CO 2 -dry and CO-dry, species are mole percentages, ( O 2 , CO 2 and CO have the UHC removed)…………………………………………………… 164

5.16

Fluent and CMC with reduced Miller-Bowman, RNG turbulence model with delta-rho term and scalar dissipation increased by a factor of 20, equivalence ratio =0.9 axial velocity (m/sec), axial velocity rms (m/sec), temperature K, CH 4 -wet, xiii

O 2 -dry, CO 2 -dry and CO-dry, species are mole percentages, ( O 2 , CO 2 and CO have the UHC removed)……………………………….………………………..165

6.1

Schematic of Jet Stirred Reactor……………………...….…………………….. 177

6.2

Fluent and CMC with GRI2.11, RSM, phi=0.606, 1.0 atm., inlet velocity 470 m/s, Tin=593, C3=100. Temperature K, N 1/sec, O 2 mole %, CO mole %, NO x ppm,

N 2 O ppm (species are on a wet basis).………………….…………………….. 178

6.3

PDF at probe location from Figure 6-2……...…………………………………..179

6.4


N 2 O ppm (species are on a wet basis). C equation source term reduced by 50%………………………………………….…………………………………..180

6.5

Fluent and CMC with GRI2.11, RSM, phi=0.606, 1.0 atm., inlet velocity 470 m/s, Tin=593, C3=100. Temperature K, N 1/sec, O 2 mole %, CO mole %, NO x ppm, N 2 O ppm (species are on a wet basis). C3 = 0.0….……………………..181

6.6


N 2 O ppm (species are on a wet basis). Variance set to zero…………………..182

6.7

Fluent and CMC with GRI2.11, RSM, phi=0.606, 1.0 atm., inlet velocity 470 m/s, Tin=593, C3=100. Temperature K, N 1/sec, O 2 mole %, CO mole %, NO x ppm, N 2 O ppm (species are on a wet basis). N reduced by 50%..………...…..183 xiv

6.8

Fluent and CMC with GRI2.11, RSM, phi=0.606, 1.0 atm., inlet velocity 470 m/s, Tin=593, C3=100. Temperature K, N 1/sec, O 2 mole %, CO mole %, NO x ppm, N 2 O ppm (species are on a wet basis). N calculated with alternate method……………………………………....…………………………………..184

6.9

Fluent and CMC with GRI2.11, RNG, phi=0.606, 1.0 atm., inlet velocity 470 m/s, Tin=593, C3=100. Temperature K, N 1/sec, O 2 mole %, CO mole %, NO x ppm, N 2 O ppm (species are on a wet basis).…………………………………..185

6.10

Fluent and CMC with GRI2.11, RNG, phi=0.606, 1.0 atm., inlet velocity 470 m/s, Tin=593, C3=100. Temperature K, N 1/sec, O 2 mole %, CO mole %, NO x ppm,

N 2 O ppm (species are on a wet basis). With delta-rho terms added to turbulence model…………………………….…………………………………….………..186

6.11


N 2 O ppm (species are on a wet basis). Variance set to zero, Sc reduced by 50% and N reduced by 80%………………...……………………………………….. 187

6.12

Fluent and CMC with GRI2.11, RSM, phi=0.529, 4.74 atm., inlet velocity 293 m/s, Tin=603, C3=100. Temperature K, N 1/sec, O 2 mole %, CO mole %, NO x ppm, N 2 O ppm (species are on a wet basis)….………………………………..188

6.13

Fluent and CMC with GRI2.11, RSM, phi=0.529, 4.74 atm., inlet velocity 293 m/s, Tin=603, C3=100. Temperature K, N 1/sec, O 2 mole %, CO mole %, NO x ppm, N 2 O ppm (species are on a wet basis). C equation source term reduced by xv

50%……………………………..…………………………………………….....189

6.14

Fluent and CMC with GRI2.11, RSM, phi=0.529, 4.74 atm., inlet velocity 293 m/s, Tin=603, C3=100. Temperature K, N 1/sec, O 2 mole %, CO mole %, NO x ppm, N 2 O ppm (species are on a wet basis). C3=0.0……………………….... 190

6.15

Fluent and CMC with GRI2.11, RSM, phi=0.529, 4.74 atm., inlet velocity 293 m/s, Tin=603, C3=100. Temperature K, N 1/sec, O 2 mole %, CO mole %, NO x ppm, N 2 O ppm (species are on a wet basis). Variance set to zero..…………...191

6.16

Fluent and CMC with GRI2.11, RSM, phi=0.529, 4.74 atm., inlet velocity 293 m/s, Tin=603, C3=100. Temperature K, N 1/sec, O 2 mole %, CO mole %, NO x ppm, N 2 O ppm (species are on a wet basis). N reduced 50%…..……………..192

6.17

Fluent and CMC with GRI2.11, RSM, phi=0.529, 4.74 atm., inlet velocity 293 m/s, Tin=603, C3=100. Temperature K, N 1/sec, O 2 mole %, CO mole %, NO x ppm, N 2 O ppm (species are on a wet basis). N increased by a factor of 10…..193

6.18

Fluent and CMC with GRI2.11, RSM, phi=0.529, 4.74 atm., inlet velocity 293 m/s, Tin=603, C3=100. Temperature K, N 1/sec, O 2 mole %, CO mole %, NO x ppm, N 2 O ppm (species are on a wet basis). N calculated with alternate method……………………………………….…………………………………..194

6.19

Fluent and CMC with GRI2.11, RNG, phi=0.529, 4.74 atm., inlet velocity 293 m/s, Tin=603, C3=100. Temperature K, N 1/sec, O 2 mole %, CO mole %, NO x ppm, N 2 O ppm (species are on a wet basis)….………………………………..195 xvi

6.20

Fluent and CMC with GRI2.11, RNG, phi=0.529, 4.74 atm., inlet velocity 293 m/s, Tin=603, C3=100. Temperature K, N 1/sec, O 2 mole %, CO mole %, NO x ppm, N 2 O ppm (species are on a wet basis). With delta-rho terms added to the turbulence model………………………...……………………………………... 196

6.21

Fluent and CMC with GRI2.11, RSM, phi=0.529, 4.74 atm., inlet velocity 293 m/s, Tin=603, C3=100. Temperature K, N 1/sec, O 2 mole %, CO mole %, NO x ppm, N 2 O ppm (species are on a wet basis). Variance = 0.0, N reduced 80% and Sc reduced 50%………………………....…………………………………..…..197

I.1

Premixed CMC with GRI2.11, O 2 versus scalar dissipation, 1/sec. Inlet temperature 300 K, pressure 1 atm. and equivalence ratio 0.9.……………..…. 258

I.2

Premixed CMC with GRI2.11, CO 2 versus scalar dissipation, 1/sec. Inlet temperature 300 K, pressure 1 atm. and equivalence ratio 0.9.…………….….. 259

I.3

Premixed CMC with GRI2.11, H 2 O versus scalar dissipation, 1/sec. Inlet temperature 300 K, pressure 1 atm. and equivalence ratio 0.9.……………..…. 260

I.4

Premixed CMC with GRI2.11, O 2 versus scalar dissipation, 1/sec. Inlet temperature 603 K, pressure 4.74 atm. and equivalence ratio 0.529.……….…. 261

I.5

Premixed CMC with GRI2.11, CO versus scalar dissipation, 1/sec. Inlet temperature 603 K, pressure 4.74 atm. and equivalence ratio 0.529.………..… 262

I.6

Premixed CMC with GRI2.11, CO 2 versus scalar dissipation, 1/sec. Inlet temperature 603 K, pressure 4.74 atm. and equivalence ratio 0.529…………... 263 xvii

I.7

Premixed CMC with GRI2.11, NO versus scalar dissipation, 1/sec. Inlet temperature 603 K, pressure 4.74 atm. and equivalence ratio 0.529.……..…… 264

I.8

Premixed CMC with GRI2.11, NO 2 versus scalar dissipation, 1/sec. Inlet temperature 603 K, pressure 4.74 atm. and equivalence ratio 0.529.……….…. 265

I.9

Premixed CMC with GRI2.11, N 2 O versus scalar dissipation, 1/sec. Inlet temperature 603 K, pressure 4.74 atm. and equivalence ratio 0.529.………….. 266

I.10

Premixed CMC with GRI2.11, CO versus scalar dissipation, 1/sec. Inlet temperature 603 K, pressure 4.74 atm. and equivalence ratio 0.529.……..…… 267

I.11

Premixed CMC with GRI2.11, NO versus scalar dissipation, 1/sec. Inlet temperature 603 K, pressure 4.74 atm. and equivalence ratio 0.529.………...…268

I.12

Premixed CMC with GRI2.11, NO 2 versus scalar dissipation, 1/sec. Inlet temperature 603 K, pressure 4.74 atm. and equivalence ratio 0.529.……..…… 269

I.13

Premixed CMC with GRI2.11, N 2 O versus scalar dissipation, 1/sec. Inlet temperature 603 K, pressure 4.74 atm. and equivalence ratio 0.529.………...…270

K.1

Schematic of thin flame showing relationship between the G parameter and the methane mass fraction.………………………………………………..……...…291

xviii

LIST OF TABLES

Number

6-1

Page

Results at sample probe for 2 cc jet stirred reactor, 1.78 msec, 1.0 atm., 593 K, inlet velocity 470 m/s, equivalence ratio = 0.606. O 2 and CO are mole percentage wet, NO x and N 2 O are ppm wet…….…...……………………...…198

6-2

Results at sample probe for 2 cc jet stirred reactor, 3.98 msec, 4.74 atm., 603 K, inlet velocity 293 m/s, equivalence ratio = 0.529. O 2 and CO are mole percentage wet, NO x and N 2 O are ppm wet….…...………………………...… 199

xix

GLOSSARY

A

area

Aj

reaction rate coefficient

A, a, b

constants

Bj

reaction rate coefficient

C, c

reaction progress variable, constant

CP

specific heat at constant pressure

D

diffusivity

Da

Damköhler number

eQ

conditioned error terms based on Q

eY

conditioned error terms based on Y

f

body force

Gζ

conditioned error term

H

height

h

enthalpy

hf

heat of formation

Ka

Karlovitz number

k

thermal conductivity

Le

Lewis number

l, L

integral length scale

lT

turbulence macro scale

MW

molecular weight

& m

mass flow rate

N

number, number of species, scalar dissipation

ns

number of species xx

P

pressure, production term or PDF

Q

conditioned mass fraction

Re

Reynolds number

R kin

kinetic limited reaction rate for EBU model

R mix

mixing limited reaction rate for EBU model

rj

reaction rate

rk

volumetric reaction rate

rf

O2 stoichiometric coefficient

Sc

source term for reaction progress variable equation

Sl , S L

laminar flame speed

Sκ

source term for k equation

Sε

source term for ε equation

SΦ

source term for scalar Φ equation

T

temperature

T

time

U

velocity component

u

fluctuation of U about its mean

V

volume

WR

non-dimensional heat loss term

X

scalar dissipation, 2N

x

spatial dimension, conditioning variable

Y

species mass fraction

y

fluctuation of Y about its mean

xxi

α

temperature ratio

δ

dirac delta function

δl

laminar flame thickness

ε

turbulent energy dissipation

Φ

scalar variable

ϕ

conditioned PDF function

Γ

diffusivity for scalar Φ

γ

conditioned PDF function

η

conditioning variable

η, λ

Kolmogorov length scale

κ

turbulent kinetic energy

µ

viscosity

υ

kinematic viscosity

υi , j

stoichiometric coefficient

ρ

density

σ

Prandtl number

σk

species mole number

τ

stress tensor

τ ex

turbulent time

τr

reactor residence time

& ω, ω

reaction rate

ξ

non-premixed conditioning variable

Ψ, φ

species concentration in PDF

ζ

premixed conditioning variable

xxii

Subscripts 0

reference value

A

activation temperature

a

generic conditioning variable

ad-u

difference between adiabatic and unburned value

f, fu

fuel

h

enthalpy

i, j

coordinate system component

kin

kinetics

LHV

Lower heating value

mix

mixing

N

number of species

Prod

products

T, t

signifies turbulent variable

u

unburned value

ζ

conditioned variable

Superscripts .

time derivative

_

Reynolds averaged value

~

Favre averaged value

’

Reynolds averaged fluctuating term

"

Favre averaged fluctuating term

i

species number

s

sensible enthalpy

xxiii

ACKNOWLEDGEMENTS

I would like to express my sincere appreciation to my advisor, Professor John Kramlich for his advice, encouragement and continual push for excellence. I would especially like to thank him for suggesting that I look at the conditional moment closure concept as a potential model for premixed turbulent combustion. In addition, I would like to thank Professors James Riley and George Kosály for their invaluable help as part of my Supervisory Committee. I would like to thank Dr. Jon Tonouchi and Dr. Dave Nicol for their valuable discussions about combustion modeling. I would also like to thank Christophe Duwig of Lund Institute of Technology for his discussions on premixed combustion modeling.

The Department of Mechanical Engineering is acknowledged for providing partial financial support through teaching and graduate service assistantships. Company provided partial tuition and computer support.

xxiv

Ford Motor

DEDICATION

To the memory of my Grandfather, a friend and inspiration to all.

xxv

1

Chapter I: Introduction

1.1

Background and Justification

Lean Premixed (LPM) Gas Turbine (GT) combustors have become a popular method to reduce the production of oxides of nitrogen (NOx = NO + NO2) in gas turbine systems with minimal negative effects on efficiency, reliability and cost (Hornsby and Norster, 1997). The NOx in these systems is exclusively formed through the fixation of atmospheric nitrogen. Here, the predominant source is the Zeldovich mechanism in which N2 is directly oxidized:

N2 + O --> NO + N N + O2 --> NO + O N + OH --> NO + H

Other mechanisms include (1) the prompt reactions in which N2 is converted to HCN by hydrocarbon radicals from the flame (with the HCN subsequently oxidized to NO), (2) the N2O mechanism in which the reaction N2 + O + M --> N2O + M is followed by N2O + O --> NO + NO, and (3) the NH mechanism where N 2 is reacted to NH by free radicals (with subsequent oxidation to NO). The rate of the critical Zeldovich mechanism is strongly favored by high temperature since considerable collision force between N2 and O is necessary to overcome the stability of the N-N bond. Thus, the key to reducing NOx formation is to reduce either the combustion temperature or the time that the reacting gases are exposed to high temperature.

In traditional diffusion flame (non-premixed) combustors, the flame generally locates itself at the stoichiometric interface, leading to a flame temperature near the

2

stoichiometric adiabatic value. Premixing the fuel and oxidant is one means of avoiding these high temperature flame sheets and the high NOx formation rates they induce. By premixing to a sufficiently fuel-lean stoichiometry, the maximum flame temperature can be controlled such that NOx formation is significantly reduced. This is the idea behind LPM combustors.

The problems of achieving a low temperature and a stable flame are, however, major challenges for the designer.

Some of the problems encountered include the

following. If the unburned mixture is not perfectly premixed, some areas will have a richer stoichiometry then the average and some will be leaner, the former resulting in pockets of higher temperatures. Due to the extreme dependence of NO x formation rates on temperature, these pockets must be avoided, and overall average temperatures kept as low as possible to minimize NOx formation.

These burners usually provide flame

stabilization by a recirculation zone within which the cold unburned charge is mixed with the hot products of combustion. This produces a complex flow field, in which, at the desired low temperatures, the combustion can be prone to stability problems. Predicting the behavior of these combustors with respect to flame stability and emissions, and selecting combustor configurations and operating conditions is both a crucial and a very difficult problem. As CO and NOx emission regulations for these combustion systems become more stringent, the ability to accurately predict these emissions during the design process has become critical.

The dominance of lean premixed gas turbine combustion systems in the market for new power generation has led to a large ongoing research effort focused on the problems listed above. This research is based on experiments, as well as on numerical modeling. The experiments are normally performed on Jet Stirred Reactors (JSR) or small-scale atmospheric pressure GT combustors. Some experiments are performed at more realistic elevated pressures, but these are very expensive, and few laboratories are

3

equipped for such demanding work. Laboratory experiments allow the measurement of velocities, the mass fractions of the stable species, and the temperature at selected locations. Interpreting these complex data to identify the exact mechanisms involved in emission formation is a very difficult problem. Also, experiments are very expensive and time consuming to use for design, especially at actual GT conditions (i.e., high pressure and full scale).

Computers have increased in memory and speed over the last decade to the point where numerical modeling of GT combustor performance has become possible. The advantages of numerical modeling include: (1) ease of comparing different designs, (2) generation of a complete description of the flow (all the species mass fractions, including intermediates and free radicals), and (3) identification of the exact mechanisms that create the emissions (which reactions, turbulence effects) (Sloan et al., 1986, Hornsby and Norster, 1997). The primary difficulty with numerical modeling is that while the fundamental differential equations that describe turbulent reacting flows are known, their solution is numerically impossible for practical problems given present computing capabilities. In order to obtain predictions for practical problems, the physics represented by the fundamental equations must be compromised through ad hoc or empirical approaches. This leads to issues of the accuracy of the predictions, and the universality of the assumptions from one application to another.

One approach to the problem is termed Chemical Reactor Modeling (CRM), an idea from the Chemical Engineering literature. In CRM, the combustor is divided into a series of elementary reactors in which perfectly-stirred reactors (PSR) are used to represent regions of strong mixing, and plug flow reactors (PFR) are used for regions of low mixing where the chemistry is allowed to evolve with time. A complex combustor can be represented by networking a series of these simple reactors together. Such a representation requires that the flow patterns be known as an input (or that they be

4

guessed). The calculation then shows how the chemistry responds to the imposed flow field. The advantage of this method is that it can accommodate detailed chemistry, but the disadvantage is that the flow field is an input rather than a calculation and the turbulence/chemistry coupling is weakly represented.

Computational fluid dynamics (CFD) approaches involve the solution of simplified versions of the fundamental equations of motion.

Reacting flows are

accommodated by including the species equations with chemical reaction source terms. The turbulence modeling and the turbulence/chemistry coupling are handled by empirical simplifications to the governing equations.

To make the calculations tractable, the

fundamental equations are usually time averaged or filtered to remove both the small scale turbulent structures that cannot be numerically resolved, and also to remove the direct calculation of the turbulent fluctuations. The contributions of the small scales and the turbulent fluctuations are then introduced as statistical correlations via empirical models. The two most common approaches are (1) to average the entire equations such that only the steady, time mean behavior is calculated (Reynolds averaged Navier-Stokes equations, RANS), or (2) to solve the unsteady equations (providing, for example, a "movie" of the large scale eddy behavior), but to filter the equations such that the small scales are not resolved (Large Eddy Simulation, LES).

In the RANS approach, the governing equations are time averaged and solved with the grid spacing larger then the smallest eddy size. The time averaging is inherently an information-losing process, so the averaged equations do not contain the same physics as the original equations. In addition, the averaging process results in a number of unclosed terms involving the averages of the products of fluctuating quantities. (The product of two fluctuating velocities is normally termed a Reynolds stress, while the product of a fluctuating velocity and a scalar, such as the temperature or a mass fraction, is termed a Reynolds flux.) These averages require ad hoc modeling to represent them in

5

terms of existing variables in the equations. A particular problem is that the chemical source terms in the species equations are often strong functions of temperature, requiring that both the mean temperature and the statistics of its fluctuations be accurately modeled. Also, because of computer limitations, severely truncated chemical kinetics mechanisms are used, called global mechanisms. These global mechanisms ignore most species and reaction steps, so they often cannot accurately predict emissions, which are a function of the intermediate and radical species.

The second major approach to the fluid mechanics (LES) involves a filtering of the fundamental equations at the smaller scales. The larger scales are left to be computed in three dimensions in a time-unsteady calculation. The dissipation of turbulent kinetic energy at the unresolved scales must be modeled. Also, the chemistry normally occurs at the smaller scales and it too must be modeled. The advantages of LES are a higher fidelity to the physics than RANS since time dependent behavior, at least at the larger scales, is retained. Also, it is believed that the turbulence at the smaller scales is more likely to be "universal" and amenable to modeling than is the case with RANS where all scales are averaged. The computational cost of LES is much higher than that of RANS, but with the progressive increase in computer capabilities, LES is expected to eventually grow into the method of choice for practical calculations.

The Conditional Moment Closure (CMC) Method is a new theory that has been applied to non-premixed combustion with good success. The application of the CMC method to premixed systems has been proposed, but has not yet been done.

The

premixed CMC method solves averaged species mass fraction equations, which are conditioned on a reaction progress variable (RPV). The general idea behind the CMC method is that the behavior of the chemical species is closely coupled to the reaction progress variable. Thus, species conservation equations that are conditioned on the RPV will have higher order moments that are much more likely to be negligible. In essence,

6

the assumption is that knowledge of the statistics of the RPV contains most of the information needed to calculate species mass fractions.

In return for this better

conditioned problem, an increase in dimensionality occurs and a new conservation equation (for the RPV) must be solved, along with a second conservation equation for the variance of the RPV. These equations bring with them new terms for the averages of the products of fluctuating quantities, and these must be modeled.

The CMC method

accounts for the influence of scalar dissipation (micromixing) on the chemistry, while decoupling the kinetics from the bulk flow (macromixing). The conditioned averages reduce the variability of the fluctuating quantities in the species equations, allowing them to be neglected with less error, hence improving the accuracy of the representation of the chemical kinetic rates. For highly turbulent flows (strong mixing) the premixed CMC equations can be simplified by assuming uniform conditional species, thereby reducing the dimensionality of the problem.

1.2

Objectives and Approach

The goals of this research are to develop a comprehensive model for LPM GT combustors. To achieve this, the following objectives are proposed:

Use the Reynolds stress and Renormalization Group turbulence models in an existing commercial CFD program to model the recirculation zone of an idealized LPM GT combustor. Wall boundary conditions are handled using the non-equilibrium wall functions. Since the focus of this research is to develop a new premixed combustion model, the known limitations of the turbulence model will be accepted.

Develop the uniform conditioned species version of the Premixed CMC Method (as opposed to the volume averaged premixed CMC method proposed by Smith, 1994), and add this to the CFD program to account for coupling between the small scale mixing

7

and the detailed chemical kinetics. Use a full methane kinetic mechanism to model emission formation.

Validate the model with experimental data for a reacting backward facing step flow, and both atmospheric pressure and high-pressure jet stirred reactor data, all burning methane.

The accuracy of the simplified version of the premixed CMC method is

investigated, and the magnitude of neglected terms of the CMC model are estimated. The accuracy of the model used for the production term of the RPV variance equation is also investigated.

A non-adiabatic version of the premixed CMC model is also proposed to model the influence of heat loss from the combustor on the reactor variables.

1.3

Organization

This dissertation starts with an overview of existing numerical modeling techniques for turbulent premixed reacting flows in Chapter 2. Chapter 3 derives the adiabatic premixed CMC method and presents solutions of the premixed CMC equations under a range of conditions. Chapter 4 presents results of a commercially available reacting CFD program with its built-in global combustion models for a backward facing step reacting flow as a benchmark for the premixed CMC results. Chapter 5 combines the premixed CMC model with the same commercial CFD program and compares numerical results to backward facing step reacting flow data. Chapter 6 compares the new model to JSR combustor data at atmospheric and elevated pressures. Chapter 7 develops a methodology to model non-adiabatic walls. Chapter 8 summarizes the results and gives recommendations for future research.

8

Chapter II: Overview of Current Numerical Modeling Techniques for Premixed Combustion

2.1

Introduction

Combustion processes are normally subdivided into premixed and non-premixed problems. In non-premixed combustion the chemistry is usually concentrated into a narrow flame zone at the stoichiometric surface that separates the fuel and oxidizer streams. The reaction rates are normally limited by the mixing rates of the fuel and oxidizer.

This allows the reactions to be modeled by relatively simple combustion

models that emphasize mixing rates rather than chemistry rates (i.e., these are based on the rate at which fuel and oxidizer are delivered to the interface rather than on the rate of the chemical kinetics). In premixed combustion the fuel/oxidizer mixing process is removed, which enhances the importance of the chemical kinetics to the problem. This results in flame zones that are thinner than those found in non-premixed combustion; in fact, the flames are usually thinner than the length scale of the smallest eddies. Thus, depending on the Reynolds and Damköhler numbers, the reaction rates can be influenced by either mixing or chemistry. The current state of the art for non-premixed combustion models is much more advanced than that for premixed combustion due to (1) the prevalence of non-premixed combustion in practice, and (2) the greater ease of modeling the mixing-dominated non-premixed problem relative to the chemistry-dominated premixed problem. This research deals exclusively with premixed combustion. The following sections provide a description of the most common methods currently used along with their advantages and disadvantages.

2.2

Reacting Computational Fluid Dynamics

There are various formulations used in Computational Fluid Dynamics (CFD) to

9

solve the Navier-Stokes equations, which along with the conservation of mass, energy and the chemical species equations, are used to calculate the reacting flow field. Four methods are described below, starting with the most fundamental approach and working towards the least computationally demanding approach that contains the most assumptions. There are other, simpler methods, but these do not solve highly turbulent reactive flow problems to the desired accuracy.

2.2.1

Direct Numerical Simulation

A complete set of governing equations that describe chemically reacting flow are shown below (Lilleheie et al., 1989). ∂ρ ∂ρU j =0 + ∂x j ∂t ∂ρU i ∂ρU i U j ∂P ∂τ ij + =− + + ρf i ∂t ∂x j ∂x i ∂x j  ∂U ∂U j  2 ∂U k − µ τ ij = µ i + δ  ∂x  3 ∂x ij ∂ x j i k   ∂ρΦ ∂ρΦU j ∂  ∂Φ  + = Γ + ρS φ ∂t ∂x j ∂x j  ∂x j 

Here ρ is the density, Ui are the velocity components, P is the pressure, fi is the body force, µ is the viscosity, Γ is the diffusion coefficient for the scalar Φ , Φ is a scalar quantity such as enthalpy or a species mass fraction (there is an equation for every specie), and Sc is the source term for that scalar.

Direct Numerical Simulation (DNS) solves these exact, time-unsteady, threedimensional equations with no modeling approximations. To achieve this the grid must

10

be fine enough to capture the smallest eddies (the Kolmogorov length scale) and large enough to cover the full computational domain. The ratio of the smallest to largest length scales is given by (Tennekes and Lumley, 1972) 3 η − ∝ Re T 4 . l

Where Re T is the turbulent Reynolds number based on the square root of the turbulent kinetic energy, the integral length scale and the kinematic viscosity. The total number of grid points required for a channel flow simulation is (Wilcox, 1993)

N DNS = (3 Re T ) 4 . 9

Similarly, the time step must be of the same order as the Kolmogorov time scale (Wilcox, 1993), which for the channel flow example is given by

∆t ≈

0.003 H . Re T u t

Where H is the height of the channel and u τ is the friction velocity. Wilcox (1993) gives an example of the immense computer power required by DNS. For an incompressible channel flow with no heat addition or chemical reactions with Re T = 180, four million grid points are required with a run time of 250 CPU hours on a Cray X/MP. For a LPM GT combustor the Re T is at least an order of magnitude larger than this, plus additional equations for the enthalpy and chemical reactions are required. Also, the characteristic length scales for the chemistry are normally smaller than the Kolmogorov scale, requiring an even finer grid than would be needed just for simulating turbulence. For this reason,

11

DNS is not feasible for modeling LPM GT combustors, and will not be so in the foreseeable future. To date, most DNS simulations have been done on incompressible flows using constant diffusivities and no chemical reactions, although some have involved lower Reynolds and Damkohler numbers with severely truncated chemistry (e.g. 1-step kinetics). Other limitations of DNS are the accuracy of the finite difference algorithms, and the very non-trivial problem of setting up appropriate boundary and initial conditions (Wilcox, 1993). Spectral solution methods are commonly used, where the equations are Fourier transformed in the stream-wise and span-wise directions. Chebychev polynomial expansions are used in directions with solid boundaries (Wilcox, 1993).

DNS has much potential in the fundamental study of turbulent reacting flows. The result of such a calculation is the time evolution of the detailed structures of the turbulence and chemistry fields, with all the important length and time scales resolved. It is therefore possible to create benchmark data sets, against which existing theories can be tested, new physics learned, and possibly new models developed, all in the absence of the modeling assumptions required in RANS and LES. In essence, DNS is a complement to laboratory data and has become a critical tool for understanding the physics of turbulent reacting flows, and for evaluating the assumptions used in turbulence models. Such work has been a significant focus at the University of Washington (see e.g., Riley et al., 1986; McMurtry et al., 1989; Mell et al., 1994; Montgomery et al., 1997; Nilsen and Kosály, 1997, 1999) as well as elsewhere (see, e.g., the review by Vervisch and Poinsot, 1998). In fact DNS has been beneficial in the understanding of turbulence characteristics, and to fine tune the turbulence models which are discussed in the following sections (Moin and Spalart, 1989)

12

2.2.2

Large Eddy Simulation

Large Eddy Simulation (LES) was originally developed by atmospheric scientists and fluid dynamists before being adopted by the turbulent combustion community. In this methodology, the larger-scale three-dimensional, time dependent motions are numerically resolved, and only the small-scale turbulence is modeled. The premise of LES is that the small-scale turbulence is more likely to be "universal" in the sense that it depends less on the large-scale geometry and flow field, and more on local flow features (e.g., local vorticity, shear, or turbulent kinetic energy). As opposed to the full-spectrum modeling needed by RANS (discussed in the next section), it is hoped that the LES turbulence models will not need to be tuned for different applications. The large eddies are directly affected by the flow boundary conditions and the flow geometry, and the direct time-unsteady calculation of these leads to a more high-fidelity representation of the flow than is possible from RANS (Wilcox, 1993). In this, LES holds the promise of overcoming some of the known shortcomings of RANS, which are discussed in Section 2.2.4.

LES has primarily been evaluated in parabolic flows that are characterized by continuous combustion, such as vertical jet flames.

(Here we define continuous

combustion as combustion without local extinction/reignition events.)

Important

challenges in these applications include the modeling of the chemistry (which takes place entirely at the subgrid scale for moderate to large Damköhler numbers), and the feedback of the heat release (dilatation) into the subgrid mixing model. Also, LES calculations should be three-dimensional to accurately capture the physics of the large-scale turbulence, even for applications such as jets that have two-dimensional time-mean symmetry. This introduces a substantial demand on computational resources relative to performing the same task by RANS.

13

The approach of implementing LES is as follows. A filter of size ∆ is applied to the equations, which usually corresponds to the cell size. Since the small scales are modeled, less grid points are required and larger time steps can be used relative to DNS. Wilcox (1993) states that LES typically requires 5 to 10% of the CPU time compared to DNS, but still requires large computer resources. LES has the same problems as DNS with numerics, and boundary and initial conditions (Wilcox, 1993). The accuracy of LES is dictated by its subgrid scale model, which has the same limitations as the models discussed in the next two sections. As computer memory and speed increase, LES is expected by many to become the most important tool in modeling turbulent combustion. The eventual goal is to apply the premixed CMC model described in this research as a subgrid-scale model for LES, although this is not proposed within the scope of the present work.

2.2.3

Reynolds Stress Model

The most common approach to making the solution of the Navier-Stokes equations (along with the equations for continuity, energy and species) more tractable is to time average the equations (Reynolds averaging). This set of equations is normally called “Reynolds Averaged Navier-Stokes” and abbreviated RANS.

Normally the

averaging is performed on the incompressible form of the equations. Favre averaging is used when the density is variable, such as in reacting flow. This results in averaged equations that are of the same form as the incompressible averaged equations, but with the Favre-averaged variables replacing the normal averages. Recovery of the normal averaged variables then requires some knowledge of the statistics of the density fluctuations. See Wilcox, (1993), Lilleheie et al., (1989) and White, (1991) for the derivation of these equations. Below is the set of incompressible equations to be solved.

14

∂U j ∂x j ∂u j ∂x j

=0 =0

 ∂U i ∂U i ∂  ∂U i 1 ∂P − uiu j  ν =− + + Uj  ∂x j ρ ∂x i ∂x j  ∂x j ∂t   ∂  Γ ∂Φ ∂Φ ∂Φ − φu j  + Sφ = + Uj  ∂x j ∂x j  ρ ∂x j ∂t 

U and Φ are mean values and u and φ are the fluctuating components. The overbar indicates the time mean value.

Due to the averaging, there are new terms in the

equations, the Reynolds stresses ρu i u j , the Reynolds fluxes ρφu j and the averaged sources Sφ . There are 6 Reynolds stresses, and for every scalar Φ there are 3 Reynolds fluxes and a source term, giving more unknowns then equations; this is termed the “turbulence closure problem”. To solve this system of equations, additional equations and models are required to express the unknowns in terms of existing variables. One method to close this system of equations is termed the Reynolds stress model (RSM), which is a second order model. It requires the solution of a set of differential equations governing the transport of the Reynolds stresses, given below in incompressible form (Sloan et al., 1986)

 ∂U i ∂u ∂u j P  ∂u i ∂u j ∂Uj   − 2ν i = − u j u k + uiuk +  + Dt ∂ x ∂ x ∂ x ∂ x ρ  ∂x k ∂x k  k k  k k  ∂u i u j P ∂  − + (δ jk u i + δ ik u j ) u i u j u k − ν ∂x k  ∂x k ρ  Du i u j

  

15

The terms on the right hand side of the equation are generation (production), dissipation, pressure strain and diffusion of the Reynolds stresses, respectfully. additional equations, but also 22 additional unknowns.

This creates 6

At this point modeling

assumptions are used to equate the higher order unknowns to known quantities. There are a variety of models in the literature, Launder et al. (1975), Lilleheie et al. (1989), Nallasamy (1986), Sloan et al. (1986) and Wilcox (1993). Currently there is no model that is considered the best for all applications.

The RSM is less computationally intensive then DNS and LES, yet provides reasonably accurate results in many applications. RSM includes the effects of streamline curvature, sudden changes in strain rate, secondary motions, etc. (Wilcox, 1993), which make this the model of choice for flows containing these characteristics. Rodi (1984) gives the Reynolds flux differential equations and their modeling assumptions.

Gradients of the dependent variables appear in the transport equations only in the rate of change, convection and diffusion terms. Hence, when these gradients can be eliminated by model approximations, these differential equations can be converted into algebraic expressions (Rodi, 1984 and Sloan et al., 1986), termed the algebraic RSM. Below is Rodi’s formulation, neglecting the buoyancy terms

uiu j ≈

2 κδij + 3

(1 − C ) κ  P γ

1  − Pii δ ij  3 

ε 1 C1 + Pii − 1 2ε ij

.

Where P is the production term from above, the C’s are constants, κ is the turbulent kinetic energy and ε is the turbulent dissipation. Equations for the last 2 variables are given in Section 2.2.4 below. Sloan et al. (1986) gives the full set of algebraic stress

16

equations in cylindrical coordinates. The algebraic RSM is becoming popular for flows with curvature and recirculation because of its improvements over simpler models with a small amount of additional computer time.

The most significant limitation to this approach is in the modeling of the averaged source terms, which for the species equations are the averaged reaction rates. Due to the large fluctuation in temperatures, and the strong dependence of the reaction rates on temperature, the average reaction rates cannot be calculated from the average temperature. Instead the fluctuating statistics must be considered, which is one goal of the approach followed in this research. The RSM applied to reacting flows is rare to date (Han and Reitz, 1995) due to the longer run times compared to the simpler models described in the following sections, so it capabilities are not well understood or documented.

2.2.4

κ − ε Turbulence Model

The most common method used to solve the Reynolds or Favre averaged NavierStokes equations along with the continuity, energy and species equations is the κ − ε turbulence model. This is the basis for most commercial CFD programs. At a minimum, turbulence is characterized by two quantities, normally a velocity and length scale, so a minimum of a two equation model is required (Taulbee, 1989). All the length and time scales are modeled, greatly reducing the number of grid points required. This gives an approximate solution to the equations. The modified Boussinesq model is used to relate the Reynolds stresses to other known quantities. The modified Boussinesq model uses a gradient diffusion term, analogous to molecular viscous shear stresses as follows (White, 1991)

17

uiu j ≈

r r  ∂U ∂U j  2 . δ ij µ t ∇ ⋅ V + ρ κ − µ t  i +  ∂x  ∂ x 3 j i  

(

)

Where µ t is the eddy viscosity (turbulent viscosity) and is a property of the flow, not of the fluid. The eddy viscosity is related to the flow using the turbulent kinetic energy and the dissipation rate by the isotropic eddy viscosity model, as follows (Wilcox, 1993 and Sloan et al., 1986)

µ t = ρ Cµ

κ2 ε

The turbulent kinetic energy and the dissipation rate are solved using differential equations derived from moments of the Navier-Stokes equations, shown below (Wilcox, 1993)

 ∂κ  ∂ (µ + µ t / σ κ )  ∂x j  ∂U i  ∂κ ∂κ  ρ + ρU j = τij − ρε + ∂t ∂x j ∂x j ∂x j  ∂ε  ∂ (µ + µ t / σ ε )  ∂x j   ∂ε ∂ε ε ∂U i  ρ + ρU j = C ε1 τij − Cε2 ρ ε + ∂t ∂x j κ ∂x j ∂x j C ε1 = 1.44, C ε 2 = 1.92.C µ = 0.09, σ κ = 1.0, σ ε = 1.3

κ=

1 uiui 2

Where σ κ is the turbulent Schmidt number and σ ε is the turbulent Prandtl number. There are other two equation models (Wilcox, 1993), but this one has become the most

18

popular because the kinetic energy equation has the turbulent dissipation term in it, so it seems natural to compute the turbulent dissipation from a separate equation (Taulbee, 1989).

The above model uses empirical constants that are based on experimental

measurements of turbulent flows. One reason that turbulence models are not always accurate is due to the accuracy and completeness of the experimental data that they are based on (Taulbee, 1989).

Other errors in the model are the assumption that the

turbulence is isotropic, that the entire energy cascade process can be described by a single length scale, and neglecting the convection effects by the mean and fluctuating velocities on the stresses (Taulbee, 1989).

The above set of equations are not appropriate for swirling and recirculating flows, because of the assumption of isotropicity in the eddy viscosity model and neglecting the turbulent shear stresses (Sloan et al., 1986). Curvature of streamlines exerts a large influence in the turbulence structure of a shear flow. Curvature affects primarily the higher-order correlations such as the turbulent Reynolds stresses by imposing additional rates of strain, which distort the Reynolds stress field (Sloan et al., 1986). The above equations can be modified by one of two methods; by a gradient or flux Richardson number correction to the source term of the dissipation equation or by modifying the constant in the isotropic eddy viscosity model. Sloan et al. (1986) gives a detailed description of most of the models proposed and discusses their strengths and weaknesses. For the above reasons, one of these corrections or the use of the algebraic RSM should be used to model current LPM gas turbine combustors.

The Reynolds flux terms also need to be modeled. The most common method is a gradient diffusion model as given below (Nikjooy et al., 1988)

− ρu iφ =

µ t ∂Φ . σ t ∂x

19

Where σ t is the turbulent Schmidt number.

The model neglects the effect of temperature and species fluctuations on reaction rates, so the reaction rates can have an error of up to 2 orders of magnitude (Pope, 1990) due to the strong dependence of reaction rates on temperature as implied by Arrhenius rate constants i.e. Sφ (T, φ) ≠ S φ ( T , φ) . Such a model can be used to estimate overall heat release, but it cannot be used to accurately calculate NO x emissions due to the high activation energy associated with thermal NO x formation. Generally a 1 to 5-step global reaction mechanism is used with this model, in the interest of conserving computer runtime and memory. A variant of the κ − ε turbulence model is the RNG-based κ − ε turbulence model which is derived from the instantaneous Navier-Stokes equations, using a mathematical technique called ‘‘renormalization group’’ (RNG) methods (Fluent, 2001). The analytical derivation results in a model with constants different from those in the standard κ − ε model, and additional terms and functions in the transport equations for κ and ε . A comprehensive description of RNG theory and its application to turbulence can be found in Yakhot and Orszag, 1986 and Yakhot et al., 1992. Another variant of the

κ − ε model is the realizable κ − ε model (Shih, 1995). The term ‘‘realizable’’ means that the model satisfies certain mathematical constraints on the normal stresses, consistent with the physics of turbulent flows. Both of these are improvements over the standard model, but still cannot model recirculating flows accurately. In addition, all of the κ − ε model variants do not account for the turbulence generated by the density gradients of reacting flows.

For internal flows like those modeled in later chapters, the density

gradients from combustion generate substantial turbulence, several authors have attempted to add terms to these models to account for this, as detailed next.

20

El Tahry (1983) made one of the earliest attempts to add terms to the κ − ε turbulence model to account for the affects of variable density. Here the two equations were re-derived using Reynolds averaging and an order of magnitude argument was used to determine which terms were important in variable density applications. This procedure gave five additional terms to the ε equation, shown below.

− C3 ρ ε

∂U i ε ∂µ µ ∂ υ ∂ε µ ∂ ρ ∂ε ε ∂υ + + ρU j − t − t ∂x i υ ∂t υ ∂x j σ ε υ ∂x j ∂x j σ ε ρ ∂x j ∂x j

The first term is one other researchers have used with the constant C3 taking values between –1 and 1. El Tahry rigorously showed that C3 = -1/3 assuming local isotropy. The second term is due to compression and the last three terms are due to combustion (thermal stratification).

Adding these five terms requires no additional models or

constants. The κ equation was re-derived in the same manner, giving the following additional terms to the right side of the equation.

′ ′ ∂u j ∂ρ′k ′u j ∂u ∂u i ∂u i ∂u i ∂ρ′k ′ ∂ρ′k ′ ′ ′ ′ ′ + p′ i − − ρ′k ′ − ρ′u i u j − ρ′u i u j + ρ′u i −uj − ∂x i ∂x j ∂x j ∂x j ∂x j ∂t ∂x j ∂t ′ ′ 2 ∂ 2u j ∂2k 2 ∂µ ′ ∂k ′ ′ ′ ∂ uj − µu i + µ′ + + µu i 2 3 ∂x i ∂x j ∂x j ∂x i ∂x j ∂x j ∂x j

These terms model the turbulent kinetic energy generation due to combustion and require models to close them, to date there has not been an experimental database generated to develop such a closure model.

El Tahry (1983) was interested in non-reacting

reciprocating engines, and did not attempt to derive models for these terms.

21

El Tahry (1990) used the same procedure as above, but using Favre averaging giving one additional term to both equations,

ρ′u ′i

µ ∂ ρ ∂P ∂P =− t ∂x i 0.7 ∂x i ∂x i

In reacting flows the density gradient will be negative, so the sign of the pressure gradient will determine whether this term adds or destroys turbulence. El Tahry states that the addition of these terms to the laminar flamelet model had some affect on the average turbulence level, but did not significantly influence the reaction rate. Jones and Whitelaw (1982) also add this term to both equations, but not the additional terms that El Tahry derives. Lilleheie et al. (1989) only add this term to the κ equation and use C3 = -1/3.

Han and Reitz (1995) state that predicting the combustion parameters are significantly influenced by the treatment of flow compressibility in the turbulence model. They go on to modify the RNG turbulence model for reacting flow modeling of an internal combustion engine. There are other attempts to improve turbulence models for reacting flows in the literature, but they will not be discussed here. A recent advancement of the κ − ε model is by replacing the linear eddy viscosity model with a non-linear model. Bianchi et al. (2002a, 2002b) compared cubic and quadratic eddy viscosity models to the linear model for incompressible internal combustion engine flows. They claim that the non-linear models are better able to capture the anisotropy and curvature affects, like the RSM without the additional computational expense. Their initial results were encouraging, though they state that further validation is required. Its possible that in the future the non-linear eddy viscosity model can be extended to compressible flow, which may be an improvement over existing compressible turbulence models.

22

This section gave a brief overview of the common methods used to model turbulent flows and the basic equations used to model the chemical reactions. DNS is the most accurate, but is computationally intractable for high Reynolds number reacting flows. LES is less computationally demanding, while still providing a good model for the turbulence, but has not been fully explored for reacting flows.

The κ − ε based

turbulence models incorporate many approximations to make them solvable for high Reynolds number flows and have performed well for boundary layer type flows, but lack some of the details required to model recirculating flows. It was shown that none of the turbulence models are fully developed and validated for variable density flows. Thus, there is no “best” model for these types of flows. Several of these models and corrections will be used and compared in later chapters, but since the goal of this work is to develop the premixed CMC model, no attempt will be made to improve these turbulence models. We now turn towards more detail on the different models used to calculate the chemical species and the chemistry/turbulence interactions.

2.3

Methods of Computing Chemistry in Premixed Turbulent Reacting Flows

The goal of this section is to review the methods used to model turbulence/chemistry interactions in premixed combustion, and give examples of how these methods have been employed to model LPM GT flows. We start by briefly describing the turbulent premixed combustion problem.

Combustion chemistry can be represented in detail by mechanisms of elementary reactions. These mechanisms include all important species and reactions that participate in an overall combustion process. For example, the GRI2.11 mechanism (Bowman et al., 1998) is a state-of-the-art fundamental mechanism that describes methane combustion with nitrogen chemistry. This consists of 48 species and 277 reactions. Note that most of

23

these reactions are termed "elementary" in the sense that they represent the results of actual collisions between pairs of gas molecules. As such, the rates at which these reactions occur can be expressed in a fundamental way in terms of the local temperature.

Many of these reactions occur between collision partners that are intrinsically stable. To break such a bond and form new species, the collision must be sufficiently violent to overcome the bond energy. Such reactions are strongly favored by high temperature where the average collision energy is increased.

The temperature

dependence of these reaction rates is usually represented by an Arrhenius function, a form that can be derived from the first principles of statistical mechanics. Many of the important reactions involved in NOx formation have strong temperature dependencies, principally due to the collision energy needed to rupture the stable N2 bond.

In the absence of strain, a premixed flame propagates into a fresh fuel/air mixture principally by upstream diffusion of heat and free radicals. These raise the temperature and radical content of the fresh gas to the point where the mixture ignites, and it’s subsequent reaction provides heat and radicals to the next element of gas. The rate at which the flame propagates into the unreacted mixture is termed the laminar flame speed, (SL). This process can be directly computed using detailed chemical kinetics, and a onedimensional version of the species and energy equations (pressure is usually assumed to be constant throughout, and continuity is used only to relate local density to local velocity).

To first order, SL is determined by the rate of the combustion reaction. Higher reaction rates lead to thinner reaction zones (i.e., less time is needed to complete the reaction). This in turn leads to steeper temperature gradients on the upstream face of the flame, leading to more heat conduction and a faster flame propagation. characteristic reaction time is closely correlated with the flame speed.

Thus, the

24

In turbulent flows the flame zone can be subjected to strain.

Imagine the

interaction of a propagating laminar flame with a series of eddies. The eddies twist and corrugate the flame front, increasing its surface area. This increases the overall reaction rate since the consumption of reactants is proportional to the product of SL and the flame area. At higher strain rates, the progressive increase in flame area provides a competing mechanism with flame propagation for the introduction of fresh material into the flame. Thus, turbulence acts to enhance the overall reaction rate in premixed flames, and models that seek to resolve the chemistry must account for both turbulence and chemistry. Note that this scenario leads to three regimes of premixed reaction rate control. First, in the absence of turbulence the reaction rate is governed by the laminar premixed flame propagation mechanism, which is largely a laminar diffusion transport process. Second, as turbulence intensity increases, the reaction rate becomes governed by the rate at which the turbulence delivers reactants to the flame zone. One way to represent this regime is as a stationary laminar premixed opposed jet flame where increases in flow rate (strain) lead directly to increases in reaction rate. Effectively the increase in surface area caused by the straining is the major mechanism for the entry of reactants into the flame, and it dominates over the simple flame propagation mechanism. Third, at very high turbulence levels, the flame structure becomes broken up and distributed. Mixing of reactants and products becomes so fast that the fundamental chemical kinetics become rate controlling.

The use of detailed chemical kinetics generally places too much of a practical burden on computing capability (i.e., one new conservation equation is required for each species added to the system).

Practical calculations generally adopt simplified,

approximate chemistry and make use of semi-empirical models to represent the turbulence/chemistry interaction. This section reviews these approaches for the various regimes of turbulent premixed combustion.

25

Before discussing the different models currently available for premixed combustion we will briefly place the phenomenological description of turbulent premixed combustion given above into its customary formal regimes.

The three regimes of

turbulent combustion are: weakly turbulent, wrinkled reaction sheets (flamelets) and distributed reactions (Borman and Ragland, 1998) also see Heywood (1988), Peters (1986 and 2000), Borghi and Destriau (1998) and Griffiths and Barnard (1995). Some authors subdivide these three regimes. The weakly turbulent flame is an extension of the laminar flame (Borman and Ragland, 1998) and is also called weak turbulence, wrinkled flames and wrinkled flamelets by different authors. Here the turbulent velocity is less than the laminar flame speed, so laminar flame propagation dominates and the turbulence just wrinkles the flame to a small degree. The flamelet regime (also called wrinkled reaction sheets, corrugated flamelets and wrinkled flames with pockets) is when the thin flame zone is wrinkled by the turbulence to a larger degree, and the turbulent velocity is larger than the laminar flame speed. The flamelet regime can be broken down into single and multiple flamelets. Single flamelets have a distinct thin flame zone that is wrinkled by the turbulence. In the multiple flamelets regime the thin flame zone becomes so wrinkled that the flamelet folds over on itself, creating more than one thin flamelet. Distributed reactions (also called broken reaction zones, perfectly stirred reactors, well stirred reactors and thickened flames) are when the turbulence is so intense that the thin flame zone completely breaks apart into many small reacting zones. This creates small lumps of reactants surrounded by burned and partially burned products. These lumps proceed downstream with the main flow and eventually become fully burned (Borman and Ragland, 1998). These premixed combustion regimes can be displayed on a turbulent Reynolds number versus Damköhler number plot (sometimes called a Borghi plot) as in Borman and Ragland (1998) and Heywood (1988) or in a slightly different format as in Peters (2000). Figure 2-1 is such a plot using the second method with the log of the ratio of the integral length scale to the laminar flame thickness (

L ) versus the log of the ratio δl

26

of the turbulence intensity to the laminar flame speed (

constant turbulent Damköhler (

stretch factor (

u′ ). Figure 2-1 shows lines of SL

SL L u ′L ), turbulent Reynolds number ( ) and Karlovitz u ′δ l ν

u ′δ l ). L is the integral length scale of the turbulence, δ l is the laminar SL λ

flame thickness, u ′ is the turbulence intensity, S L is the laminar flame speed and λ is the Kolmogorov length scale and it is assumed that λ = 0.01L . Peters (2000) has a slightly different view of the third regime, stating that when the Kolmogorov eddies become smaller than the inner layer of the flame (i.e., the reaction zone as opposed to the preheat zone) they cause the flame to break down from loss of radicals and heat, resulting in extinction. It is believed that this is referring to an open flame, such as a Bunsen burner. It will be shown in Chapter VI that jet stirred reactors, which have very high turbulence intensity remain ignited due to the intense recirculation. This is an important point that is over looked by some researchers. When discussing the regimes of turbulent premixed combustion, most only refer to the small-scale turbulence (micromixing), which directly affects the reaction rates. Normally it is assumed that the bulk flow is laminar (such as the Bunsen burner). When the small-scale turbulence is low to moderate, the bulk flow does not affect the combustion. When the small-scale turbulence becomes large the bulk flow will have a significant affect of the combustion. In the case of laminar bulk flow, intense small-scale turbulence will extinguish the flame, as Peters (2000) states. In the case of large bulk flow turbulence (recirculation), intense small-scale turbulence will not extinguish the flame, as in a jet stirred reactor. Each of these flame regimes requires a different numerical model as discussed next. The exact boundaries of the different combustion regimes are not agreed upon and the nomenclature is not consistent between researchers, especially in the well stirred/distributed reaction regime. Some researchers define the distributed reaction regime as the area where the log of the length scale ratio is less then zero, while some say it starts at the Ka = 1 line. Figure 2-1

27

has the most common definitions labeled. Figure 2-1 also shows the location of the LPM GT combustor experiments of Polifke et al. (1995) and Zimont et al. (1997) and the 16 cc atmospheric jet stirred reactor data of Steele (1995). It is seen that the Damköhler number is of order unity and the turbulent Reynolds number ranges from 130 to 1,000, all in the distributed reaction regime. The goal of this research is to model LPM GT combustors, which fall in the distributed reaction regime, so models applicable to this regime will be emphasized here.

In Figure 4-4 it will be seen that the premixed

combustion model in the Fluent (2001) CFD code gives the Damköhler number varying from 1.5 to 31.5 and the turbulent Reynolds number varies from 6 to 519 for the backward facing step case. As will be discussed in Chapter IV, this model has several deficiencies, so the Damköhler and turbulent Reynolds numbers it calculates may not be exact. But it does indicate that the backward facing step is not quite in the distributed reaction regime.

2.3.1

One Equation Combustion Models

Zimont et al. (1997) developed a one equation combustion model called the Turbulent Flame-speed Closure model, that has some similarities to the BML model described below in Section 2.3.3. Their model solves the reaction progress variable equation and closes the reaction rate source term with an empirical flame speed model that avoids the fast chemistry assumption of the BML model. They used a 3-D finite volume CFD program with the standard κ − ε turbulence model along with a single transport equation for the progress variable to model an ABB double cone burner, burning natural gas. They estimated the turbulent Reynolds number to be of order 1,000 and a Damköhler number of about 3, placing this flame in the distributed reaction regime. This model ran about twice as fast as the eddy breakup model of Polifke et al. (1995) (described in the next section). The velocity and temperature match was poor, but the authors did not determine if the error was the combustion model, the κ − ε turbulence

28

model or a combination of both. The authors state that, “the modeling of the turbulent exchange of momentum needs to be improved.”

The limitations of the approach are that it is highly empirical and it has not been shown that the same constants are valid of all conditions. Also, this model does not calculate the species mass fractions, just the global heat release and temperature.

The Fluent CFD code (Fluent, 2001) also has a one equation premixed combustion model similar to the one that Zimot et al. (1997) used. This model solves the reaction progress variable with five user input parameters to define the flame and does not solve a variance equation or require a PDF. Results from this model will be presented in Chapter IV for a backward facing step and compared to the eddy breakup model. Both of these results are used as a baseline for comparison to the CMC results in Chapter V.

2.3.2

Eddy Breakup Model

Eddy breakup models (EBU) may be used in premixed combustion CFD when the turbulent mixing rate controls the overall chemistry. In many cases this condition is met. The low computational overhead associated with this model has led to its extensive use in describing premixed combustion.

Spalding (1971, 1977) was the first to propose an eddy breakup model. Here, the reaction rates are controlled by either the chemical kinetics or the rate of eddy breakup (turbulence mixing). The theory is as follows; the reactions are assumed to take place in a very thin sheet that is smaller then the Kolmogorov scale. The rate at which the premixed flame propagates via laminar flame propagation is assumed to be slow relative to the rate at which turbulent mixing delivers unreacted material to the vicinity of the flame (e.g., the rate at which the flame is stretched by the addition of new material, as

29

described above). This means that the overall chemistry proceeds at a rate set by the rate at which unreacted material is deposited at the Kolmogorov scale by the energy cascade (Bai and Fuchs, 1995). The problem thus involves finding the rate at which material is processed by the cascade. Spalding modeled this using the velocity gradient. Since then other researchers have used the ratio of the dissipation rate to the turbulent kinetic energy, ε . Generally the mixing controlled reaction rate takes the following form (Lilliheie, κ

1989, Williams, 1985a, Libby and Williams, 1985, Jones and Whitelaw, 1982, Mason and Spalding, 1973):

R mix = C EBU y ′fu

2

ε κ

The constant is between 0.1 and 100, and normally is of order unity. (Note that the large variation of this constant for different experiments and configurations indicates a lack of “universality” for the approach, which is viewed as a modeling weakness.) The second term is the RMS of the fuel mass fraction. To account for rich pockets, some researchers use a similar relation for the mass fraction of oxygen, and use the smallest rate term as the controlling rate.

Others use the mean mass fractions (called the Eddy Dissipation

Method), instead of the RMS values, one advantage of this being reduced computational time (Magnussen and Hjertager, 1977, Lin, 1987, Nikjooy et al., 1988, Bai and Fuchs, 1993, 1994 and 1995). Some refer to this as the eddy dissipation concept (EDC):

R mix = A j ρ

 Yprod Y ε min Yfu , air , B j κ rf 1 + rf 

   

The A’s and B’s are constants for each reaction. The Y’s are the mean mass fractions of the fuel, air and products, respectively, rf is the oxygen stoichiometric coefficient. The

30

air mass fraction term above is needed to limit the reaction rate at rich locations, as noted above. The product term limits the reaction rate when hot products are mixed with fresh premixed reactants. The original models assume infinite rate chemistry (large Damköhler number), with only mixing controlling the reaction rates. When combined mixing and chemical kinetics are considered, the following modified model is used (Libby et al., 1986, Lin, 1987, Nikjooy et al., 1988, Bai and Fuchs, 1993 and 1994): R = min (R kinetic , R mix )

Where R kinectic is the Arreheneus reaction rate.

Philipp et al. (1992) accounted for the possibility of the reaction being kinetically or mixing controlled with a resistance-in-parallel approach as follows:

R=

1 1 R kinetic

+

1

.

R mix

More recently Polifke et al. (1995) developed a modified version of the above expression:

R=

1 1− η η + R kinetic R mix

l   η ≡ tanh C 0 ∇T t  T 

The tanh term is used to normalize the gradient between 0 and 1. T is the temperature

31

and l T is the turbulent macro scale. Polifke et al. found the constant to be of order 10. The rational behind the model is that in homogenous regions the temperature (and species) gradient will be small and the reaction will be kinetically controlled. Near the flame zone there will be pockets of hot products and fresh mixture, giving a large temperature gradient. Using the average temperature to calculate the kinetic rates would greatly under-predict the reaction rate, using their formulation the reaction is mixing controlled. This model is discussed in more detail below.

From the literature there is no agreed upon “best” EBU model. To allow for the widest range of conditions of premixed and non-premixed flames an EBU model must account for the kinetics as well as the mixing effects. Philipp et al. (1992) and Polifke et al. (1995) do not compare their models to the earlier model, so it is not known if it was an improvement or not.

When chemical kinetics are included, most of the literature uses a 1-step mechanism. About half of the researchers used the fast chemistry assumption, i.e. they neglected the chemical kinetic terms. When more then a one step mechanism is used, Nikjooy et al. (1988) appear to have the most sophisticated method, which they say is valid for diffusion and premixed flames. Nikjooy et al. used a 2-step mechanism, the first step oxidizes the hydrocarbon fuel to CO and H2O and the second step oxidizes the CO. They calculate the mixing and kinetic rates for both the fuel and CO oxidation rates. For each reaction step, they use the smaller of the two rates to govern the overall chemistry. Bai and Fuchs (1993) use a 4-step mechanism, but only calculate one kinetic rate for the overall reaction. They use the smaller of the kinetic rate and the mixing rates for the fuel and oxygen, as the limiting rate. Since no researcher has compared the different models, it is impossible to say which one is the best.

Polifke et al. (1995) used a 2-D axisymmetric finite-volume CFD code with a

32

κ − ε turbulence model and flux Richardson number correction to predict NOx formation in an ABB double cone burner. The turbulent Reynolds number was about 250, the Damköhler number was about unity and the Karlovitz number was > 2. A 2-step global oxidation mechanism for the methane fuel was used with the combined eddy breakup model described in the next section. A lookup table for NOx formation was developed as a function of CO, using PSR/PFR's (described later) and a 1-D model with a full kinetics mechanism. For each grid cell the local CO value was used to look up the NOx formation rate from this table, with the assumption that the local CO concentration is proportional to the free radical concentration, which drives the NOx formation rate. For the limited cases run they achieved very good agreement with experimental results, though they say that this match was to “some extent coincidental.” Polifke et al. (1995) state that “A more reliable turbulent combustion model would certainly increase our confidence in the NOx model’s qualitative predictions.” They found significant disagreement in the vicinity of the flame front for CO and temperature, compared to measurements, which is expected with a 2-step mechanism. Since the CO match was not good and the NOx prediction is based on CO, one cannot have much confidence in their NOx results. One possible explanation is that the inlet feed in the experiment may not have been perfectly premixed, while the model assumed a perfectly premixed inlet.

Hornsby and Norster (1997) used a commercial CFD program, STAR-CD, to model small LPM GT combustors. The code used the standard κ − ε turbulence model with no swirl corrections, a combined eddy breakup/kinetics model and a 2-step global methane oxidation mechanism. NOx was estimated using a correlation chart, but minimal details were provided. They claim that the temperature field was under predicted, but the temperature and CO patterns were reasonable, though no plots were shown. They state that enhanced turbulence and combustion models are needed for more accurate predictions, but would have a considerable runtime penalty.

33

Jackson et al. (1997) used the commercial CFD code, Fluent, with a standard

κ − ε turbulence model to predict the NOx in a 3-D combustor. They used C12H23 liquid fuel with a 2-step global oxidation mechanism and a NOx post processor. The reaction rate was the slowest of the kinetic and eddy breakup rates. Jackson et al. (1997) did comparisons between 4 designs, but did not compare to experimental data. They state that the κ − ε turbulence model is bad for swirling flows, but adequate for predicting qualitative trends of the recirculation zone. The authors question its ability to predict absolute values of exit emissions, but feel it will predict the correct trends.

Nicol et al. (1997) develop two 5-step global mechanisms for methane oxidation with NO formation. These mechanisms are used with the STAR-CD CFD code to model an industrial combustor.

The code is 2-D axisymmetric, uses the standard κ − ε

turbulence model with no swirl or recirculation corrections.

For reaction rates the

minimum of the eddy breakup and the kinetic values are used. No comparison is made to experimental data, but the results are suspect since only the standard κ − ε turbulence model was used.

While some of these models provided reasonably good matches to the data (Polifke et al. 1995 and Hornsby and Norster, 1997), it was partially achieved by adjusting the model constants. There is not one set of model constants that works for all the burners tested. The accuracy of these models is also limited by the global kinetic mechanisms and neglecting the temperature fluctuations. Even with these limitations, this has been the most frequently used model for LPM GT combustor modeling because of the short run times compared to other available models.

2.3.3

Flamelet Models

The simplest flamelet model is the Bray Moss Libby (BML) model, which is an

34

extension of the Bray Moss model (Bray et al., 1985; Libby and Williams, 1985 and 1993). Here the flow is broken into three zones, the fully burned mixture, the unburned mixture and an infinitely thin flame zone (smaller then the Kolmogorov scale) between these two limiting values. The flame zone has no intermediate values of temperature resolved and has no volume, which limits the model to the fast chemistry limit, i.e. the flame proceeds instantly from the unburned to fully burned condition. The progress variable is normally a non-dimensional temperature or product mass fraction (Peters, 2000). An assumed shape PDF is used to relate the instantaneous composition to the laminar flame composition.

The PDF has two spikes, at the unburned and burned

conditions, with no probability of the flame having intermediate values of the progress variable. A conservation equation is solved for the mean progress variable and its variance, which are used to calculate the PDF. The two conservation equations are the same as used in the CMC model (which will be discussed in Chapter III), but with the reaction rates based on the laminar flame speed instead of calculated from the chemical kinetics. The reaction rate is the product of the unburned density, the laminar flame speed, a flame stretch factor and the flame surface density. The last two terms are an ad hoc model to incorporate weak turbulence into a laminar model. The model also assumes constant average molecule weight. A third conservation equation for flame surface density is also required, but the current closure assumptions lack generality (Peters, 2000).

By relating the combustion to a laminar flame this model is limited in its ability to model turbulent combustion, i.e. the affects of small-scale turbulence on the reaction rates are not accounted for and it assumes the bulk flow is close to laminar. The model does show that counter gradient diffusion can be important along with the normal gradient diffusion effect.

Counter gradient diffusion comes from the gas expansion during

combustion and gradient diffusion comes from the turbulent mixing.

For weak

turbulence the BML model is useful to show the relative effects of the two scalar

35

transport effects. The goal of this research is to develop a model for highly turbulent flows, so the BML model is not appropriate.

Abou-Ellail et al. (1999) developed a 1-dimensional flamelet model based on a reaction progress variable and used a 3-step global kinetic mechanism. This model is limited to laminar bulk flow, but allows the scalar dissipation to be varied up to the extinction value. The reaction progress variable is defined as follows.

c=

YCO 2 YCO 2,ad

A Lagrangian coordinate system is attached to the reaction surface and c is in the normal direction from the flame surface.

The species mass fraction equations are

transformed to c space and solved independently of the fluid equations. This leads to a set of species conservation equations with c as the independent variable, instead of time and space, shown below (Abou-Ellail et al. Equation 8).

ρu c

N ∂Yi X ∂ 2 Yi −ρ = ρ ν in ω n ∑ ∂c 2 ∂c 2 n =1

N

ν CO 2, n ω n

n =‘

YCO 2,ad

where u c = ∑

and X = 2D

∂c ∂c ∂x i ∂x i

This equation is solved for the sensible enthalpy and all the species except CO 2 . This equation is similar to equation 3.16, which is the simplified CMC equation used in this work (described in the next chapter). The difference is the boundary conditions at c=1. The CMC model sets all the species mass fractions at c=1 to their adiabatic equilibrium

36

values. Here CO 2 is set to the equilibrium value, CO is set by the conservation of c atoms and the other species and sensible enthalpy have zero gradient at c=1. These boundary conditions may not guarantee conservation of atoms or mass. The physical meaning of the boundary conditions is likely the assumption that the post flame reaction rates taper off to zero due to heat loss without ever reaching equilibrium values. The premixed CMC model developed in the next chapter assumes intense bulk mixing, which keeps the post flame gases mixed with the burning mixture so the mixture can approach equilibrium.

The derivation of this model follows Bilger’s derivation of the CMC mode (see Bilger 1991 and 1993a, Smith 1994, Klimenko and Bilger 1999 and Appendix F), but without using conditioned variables and a different definition of c. The species mass fractions are decomposed into mean and fluctuating components, i.e. Yi = Q i + y i . Here the fluctuating terms are dropped at the beginning of the derivation because the bulk flow is laminar. It is also assumed that the conditioned species have the same functional relationship with respect to c in the entire reactor, i.e. Q(c) is the same everywhere. The most likely reason is because the Da number is large, giving fast reaction rates that are not affected by the fluid mechanics (note that the simplified CMC model developed in the next chapter makes this assumption for flows with intense mixing, so the reactions do not vary with the bulk mixing). With these assumptions, equations F5 (see Appendix F) are reduced to the following. ∂Yi ∂Q i ∂c = ∂t ∂ζ i ∂t ∇Yi =

∂Q i ∇c i ∂ζ

∇ ⋅ (ρD i ∇Yi ) =

∂Q i ∂ 2Qi ∇ ⋅ (ρD i ∇c) + ρD i ∇c ⋅ ∇c ∂ζ ∂ζ 2

37

These are Abou-Ellail et al. (1999) equations 3-5, with the notation used in this work. These relations are substituted into the species mass fraction conservation equations, equation F1. Equation F2 is used to simplify the expression and gives Abou-Ellail et al. (1999) equation 8, listed above. In the CMC derivation the above assumptions are not made at this point, the equations are transformed to c space and combined in the same manner, then the equation is conditionally averaged. At this point the full premixed CMC equations are simplified by assuming the error terms are negligible and the conditioned species mass fractions are uniform in real space (the justification for this is given in Chapter III), giving equation 3.16, which has the same form as Abou-Ellail et al. (1999) equation 8. Because of the different definition of c, the terms in Abou-Ellail et al. (1999) equation 8 may be different in magnitude, but the relative magnitudes will be the same as the CMC version.

It is interesting that very similar equations are derived for totally different flow regimes, with the c=1 boundary condition the only difference. The species equations in c space are identical for the laminar flamelet and well-stirred reactor regimes, i.e. when the bulk mixing does not affect the reactions. In these two regimes it is only the small-scale turbulence that affects the reaction rates. Peters (2000) and Veynante and Vervisch (2002) show that for non-premixed combustion the flamelet and simplified CMC equations for homogeneous flows are identical, even though the underlying physical assumptions strongly differ. Klimenko (1995 and 2001) shows that for non-premixed flows the flamelet model is a subset of the more general CMC model. It appears that this may also true for premixed flames, though more work must be done to prove this is true for all conditions. Results of this model will be compared to the CMC model used in this work in Section 3.4.3 with an explanation of the differences.

Bradley et al. (1988) added a laminar flamelet model to a 2-dimensional

38

axisymmetric CFD code to model the turbulent combustion in a jet stirred reactor burning methane and propane. A first order turbulence model was used with terms added to account for compressibility. Equations for temperature and its variance were solved and used to calculate a beta function PDF. The mean temperature was used to obtain the species and density from a look up table.

The PDF was used to account for the

temperature fluctuations. The match of the intermediates was not good in regions of high turbulence straining. To reduce the calculation complexity, the authors assume that the turbulent combustion is comprised of an array of laminar flamelets. In other words, the species are only a function of temperature, not flame straining. The authors state this is an important limitation of the model. The purpose of the present research is to develop such a model.

Bradley et al. (1994) extended the above model by using a double PDF to account for flame stretch and temperature fluctuations. This double PDF was assumed to be equal to the product of two single PDF’s. The flame stretch PDF was assumed to be Gaussian based on DNS data. The temperature PDF was assumed to be a beta function, as above. The turbulence prediction was improved by using the RSM with equations for the turbulent energy flux. One-dimensional flames were modeled. The ratio of turbulent to laminar flames speeds was in good agreement to data. The authors state that future research needs to understand the correlation between flame stretch and temperature fluctuations.

2.3.4

G equation Model

The G equation (also called the level set approach) is valid for modeling the corrugated flamelet and thin reaction zone regimes of turbulent premixed combustion. It uses a non-reacting scalar G that is only defined on the flame, dividing the burned and unburned regions. It is defined so there are no turbulent fluxes normal to the turbulent

39

flame front; thus, a model for the counter-gradient diffusion is not required (Peters, 2000). G is a 3-dimensional field variable that is only defined on the 2-dimensinal flame surface. Along with an equation for time-mean G it has an equation for its variance and the time-mean of the absolute value of the gradient of G.

This is a kinematic

representation of the flame location with an ad hoc model relating the flame displacement speed to the laminar flame speed, the flame straining and the flame curvature. Solving these three equations along with the RANS and turbulence model equations gives the mean turbulent flame location. To obtain the species and temperature across the flame another model is required to resolve the laminar flame structure. The BML model is the simplest, which assumes jump conditions across the flame from the unburned to the equilibrium conditions. If finite rate chemistry affects are to be accounted for, premixed flamelet equations must be derived. Peters (2000) derives such a model and discusses its assumptions and limitations, but does not compare it to experimental data. The idea of combining a flamelet model with the G equation seems inconsistent. Since G is only defined at G=0, using other values of G to calculate the species would appear to be inconsistent.

Peters defines G as a measure of the distance relative to the center of the flame, while the reaction progress variable, c, defined in the next chapter is a measure of how far the reaction has progressed toward fully burned. Appendix K derives the G equation and shows G is really a reactive scalar, such as c or YCH 4 , not a distance. It is shown that the G equation is the same basic equation as the one-equation, EBU/EDM and BML combustion models described above, but with a more detailed model for the flame speed. The premixed CMC model described in the next chapter solves the chemical kinetics directly without relying on ad hoc flame speed closure models.

The G equation model is only valid for flames with a well defined burning velocity where the flames do not overlap onto themselves (e.g., turbulent Bunsen-type

40

flames or turbulent flame propagation across the cylinder of a spark ignition engine). The model has twelve constants that must be determined from empirical data. It is too early to know if one set of constants will work for all applications. Peters (2000) gives a very detailed treatment of the G equation model and some applications. The G equation model assumes the reactions are not affected by the turbulence, so it is only valid for large Da flows. The G equation model will likely not work for LPM GT combustors, which do not meet the infinitely thin flame and large Da number requirements and are in the thickened and distributed flame regimes.

Nilsson (2001) combined the G equation model described above with a laminar flamelet look up table for the species to model propane reacting in a channel. The density, temperature, species mass fractions and reaction rates are calculated with a laminar flamelet model using a detailed kinetic mechanism assuming no scalar dissipation and stored in a table. The three equations from the G equation model are solved with the RANS and turbulence model equations at each CFD cell. The density and temperature are determined from the table lookup based on the value of G. G and its variance are used to calculate an assumed Gaussian shaped PDF, which accounts for flame stretch. The new density and temperature are fed back to the CFD code for the next iteration. Once steady state is achieved the same procedure is used to look up the species. The slow reaction rate species, such as NO, are calculated based on a reaction rate that is looked up from the table.

This is the first attempt to combine the G equation method with the flamelet model. It assumed the turbulent flame speed was an input, rather then calculating it. The model gave a good match of the species and temperature to the experimental data in the first half of the reactor. The match in the later part of the reactor was poor because of the poor prediction of the turbulence intensity in the flame zone. The turbulence intensity is used as an input to some of the closure models. This poor prediction by the RANS

41

models of turbulence intensity in reacting flow was also found in this work (discussed in Chapter V) and is a known limitation to these types of turbulence models. Since the G equation and flamelet models were each developed for laminar bulk flow applications, it is doubtful that the combined model will be valid in the distributed reaction regime where modern LPM GT engines operate.

2.3.5

Probability Density Function

Probably the most straightforward way to introduce the Probability Density Function (PDF) approach is by describing its implementation via Monte Carlo methods. Envision a finite computational cell in a flow field. This contains a variety of gases of different states of reactedness (i.e., values of c). We discretize the cell into a finite number of parcels (say 50), each with an assigned value of c such that all the parcels represent the PDF of c within the cell. The parcels are then allowed to do four things:

1. They can be convected from upwind cells to downwind cells. The number of parcels transported in a given time depends on the local average velocity. 2. They can be exchanged for other parcels in neighboring cells to represent the turbulent diffusion process.

The rate of exchange is based on local turbulence

parameters. 3. They can mix with other parcels within the cell. The idea here is that an isolated, non-reacting cell would eventually converge to a uniform composition due to internal mixing. A number of models exist to describe this process. As an example, the Curl model selects two parcels at random, mixes them, and assigns this average mixture composition to the two cells. The mixing intensity (i.e., frequency of the averaging process) is modeled based on turbulence parameters. 4. They can react. Between the transport-based events described under 1-3, the parcels all undergo reaction as isolated simple batch reactors in parallel.

42

The transport processes described in items 1 and 2 are based on turbulence models. The mixing model in item 3 is the most empirical part of the model in that both the mixing frequency, and the way the mixing is done, are empirical approaches. The chemistry described under item 4 is, however, exact. This is the strong point for the model. In particular, the influence of fluctuations in temperature and composition on the chemical reactions is calculated exactly in the sense that the chemistry of each portion of the PDF is calculated independently. This is just as it occurs in the laboratory.

The ideas presented above are formalized by writing balance equations for the PDF’s of the various properties. These equations include terms describing convection, turbulent diffusion, mixing, and reaction. The direct solution of these equations is often difficult, so the PDF’s are generally discretized as described above, and the Monte Carlo solution procedure is followed until the field variables approach steady state.

An

alternative to direct solution and Monte Carlo is the assumed shape PDF approach, described below.

There are several variations of the PDF approach. The single-point PDF is the simplest PDF formulation, where C is defined as a reaction progress variable. C is uniquely related to the mass fractions and enthalpy. A single-point PDF uses a 1-step chemical reaction. The PDF of C, P(C), is defined such that P(C)dC is the probability that C ( x , t ) is in the range (Tonouchi, 1996)

C−

dC dC . < C( x , t ) < C + 2 2

The species and energy equations are recast into a PDF balance (evolution) equation as follows (Borghi, 1988)

43

 ∂ ∂ ∂ [− ω(C)P(C)] − ∂ δ(C( x, t ) − C) ∂ P ( C) + u a δ ( C( x , t ) − C ) = ∂t ∂x a ∂C ∂C  ∂x a 

 ∂C  D  ∂x a

  

where x is the position, u is the velocity with α representing the 3 components, ω& ( C ) is the chemical species production term and D is the diffusion coefficient. The four terms represent the unsteady effect, convection in physical space, convection in probability space and flux in the probability space (Borghi, 1988). Borghi (1988) discusses the assumptions used in this equation. The last term is the small scale mixing term (SSMT) and must be modeled (this is the most challenging part of the model), as discussed next.

The two most common SSMT’s are the Coalescence-Dispersion (C-D) and Interaction by Exchange with the Mean (IEM) models, discussed in more detail in the Section 2.4.2 below. The PDF for the C-D model is (Curl, 1963 and Pope, 1982) ∞ ∞ 1 ∂P   = −4ωP + 4ω∫ ∫ P(Ψa )P(Ψb )δ Ψ − (Ψa + Ψb ) dΨa dΨb − ∞ − ∞ 2 ∂t  

where ω is the decay (micromixing) frequency and approximately equal to

ε . The κ

equation is integrated using finite differencing with a time step much less then the inverse of the decay frequency. When ω is zero the equation models a PFR, when ω is infinite the equation models a PSR.

The equation for the IEM model is (Kosaly, 1986) ∂P ∂ [(Ψ − φ )P] =ω ∂t ∂Ψ

44

This equation is easier to integrate then the joint PDF discussed below (Correa, 1993). The computational time increases linearly with the number of species. Pope (1982) describes several improved C-D models. Kosaly (1986) shows that the C-D, IEM and improved C-D models are all special cases of a general formulation for the molecular diffusion phenomenon model. Kosaly and Givi (1987) discusses the sensitivity of the different models.

There are three methods to solve the PDF equation, they are; to numerically integrate the evolution equation, use the Monte Carlo method and to use an assumed shape PDF.

Numerical integration can be difficult, due to peaks in the PDF.

Computationally, the integration is not intense for one specie in one or two-dimensional parabolic flow. The integration becomes almost impossible for moderately complex chemical mechanism (Borghi, 1988 and Pope, 1981).

The Monte Carlo method avoids the numeric problems discussed above (Spielman and Levenspiel, 1965). For simple problems the Monte Carlo method is inefficient compared with standard numerical methods. The computational time required for numerical integration increases exponentially with the number of species, where the Monte Carlo method only increases linearly, so for larger problems it becomes the method of choice.

The final solution method uses an assumed shape for the PDF with constraints on the first and second moments. The PDF will be different for different types of flames and at different locations within the flame. The PDF is assumed to take a given form (e.g. Gaussian or Beta function). Its shape is then uniquely specified by two parameters (central tendency and variance), which become the targets of the solution. The assumed shape makes the evolution equation easier to solve, but reduces its accuracy.

The

45

assumed shape method will only work on perfectly premixed flames and diffusion flames with very fast chemistry (Borghi, 1988).

For a single step reaction with equal

diffusivities, the reaction rate, oxygen mass fraction and temperature are expressed as functions of Yf, the fuel mass fraction. The governing equation becomes (Borghi, 1988) ∂ ∂ ~ ~ &f ( ρ U a Ya ) = (ρ u a y f ) + ρ ω ∂x a ∂x a

where Yf is the fuel mass fraction, y f is the fluctuation of the fuel mass fraction and the tilde overbar signifies Favre averaging. There is a similar expression for ~y f2 . The chemical species production rate is given as follows

Yf , max

∫ ω&

&f = ω

f

~ (Yf )P(Yf )dYf

0

with a similar expression for y f ω& f . The PDF is chosen based on three parameters, a, b and c, that meet the following conditions (Borghi, 1988)

Yf , max

1=

~

∫ P(Y

: a , b, c)dYf

f

0

~ Yf =

Yf , max

~

∫ Y P(Y f

f

: a , b, c)dYf

0

~ 2 2 Yf + ~y f =

Yf , max

∫y

2 f

~ P(Yf : a , b, c)dYf

0

A turbulence model is required for u α y f , u α y f 2 and τ ex (the turbulent time). Borghi (1988) discusses different turbulence models and possible PDF shapes. The above set of

46

equations are solved along with the mean continuity and Reynolds equations and the

κ − ε equations. The accuracy of this model is dictated by the selection of the assumed shape for the PDF. This method is mainly used for diffusion flames, where chemical equilibrium is assumed (Pope, 1990).

A joint PDF is a PDF of more then one variable. The joint PDF gives the probability of each of the variables having a given value, so the dimension of the problem is increased. There are three common types of joint PDF’s; a composition PDF as described above, but with more than one species. Since there is no information about the velocity field the κ − ε or Reynolds stress model equations are also solved. The gradientdiffusion term in the PDF is modeled (Pope, 1990).

The second type of joint PDF is the velocity-composition PDF. Here the three velocity terms are accounted for in the PDF, so all forms of convection and transport are treated exactly. The PDF provides no information on the length or time scales of the turbulence, so the equation for the dissipation is also solved.

The third common type of joint PDF is the velocity-dissipation-composition PDF. No additional equations are needed, but the PDF is more complicated to solve. The assumption in the model is that the scales of the composition field are proportional to the scales of the velocity field (Pope, 1990). Normally a Lagrangian viewpoint is used to simplify the equations. The reaction, gravity and mean pressure gradient terms are treated exactly. The fluctuating pressure gradient, molecular stresses and rate of dissipation terms require modeling (Pope, 1990).

In theory the same three solution techniques used for the single PDF can be used. But as mentioned above, the computational time for direct integration becomes prohibitively long if more then a couple of species are used. The assumed shape PDF

47

method also becomes all but impossible to solve with multiple species and it is also hard to design the assumed shape PDF for multiple species (Pope, 1990). For these reasons the Monte Carlo method is the main solution method for joint PDF equations. Pope (1990) gives the complete velocity-dissipation-composition PDF equations and estimates it would take about 280 hours of supercomputer time to solve for 40 species and 105 particles. For this reason most models are limited to about 5 species. Some researchers use a table lookup to determine the reaction rates, instead of integrating the kinetic equations.

Using a joint PDF to model the reactions is an improvement over the methods described in Section 2.3.1, but with a much larger computational requirement. Aside from the large runtimes required to solve the joint PDF equations when many reaction steps are used, the micromixing model is the weak link of this approach.

Anand et al. (1989) used the velocity-scalar joint PDF with a CFD code to model the turbulent recirculating flow over a backward-facing step. The joint PDF was solved with the Monte Carlo method based on Pope’s earlier work. The CFD code supplies the Monte Carlo code with the mean velocity and pressure field and the turbulent time scale. The Monte Carlo code supplies the Reynolds stress and turbulent kinetic energy to the CFD code. To solve the combined model, first the CFD code is solved to convergence. Then data are passed to the Monte Carlo code, which is solved to convergence and data are passed back to the CFD code. This is repeated until the overall solution converges, usually 2 or 3 complete cycles. The authors only treated non-reacting flows, with reacting flows the Monte Carlo code would also pass the density to the CFD code. This approach is slightly different then other researchers have done. Here the Reynolds stresses are solved by the joint PDF, not in the CFD code, so no turbulence model is required. The 2D model was compared to experimental data of a backward-facing step with good agreement. The reattachment length was closer to the measured value then the κ − ε

48

model, but still not very close. The general trends of the turbulence velocities and of the Reynolds stresses were well predicted, with some differences in magnitude. In some cases the κ − ε model results were closer to experiment then the joint PDF. The CFD calculations were done on an IBM 3084 requiring about 2 minutes of CPU time per iteration for about 200 iterations, the number of grid cells was not given. The Monte Carlo code was done on a CRAY-X/MP in about 45 seconds for 50 time steps and 100,000 particles. The authors plan on modifying the joint PDF to solve for the mean velocity and pressure fields and the turbulence time scale, so the CFD code is not needed, which they feel will be a better model. The authors feel that the models need improving and faster solution algorithms are needed to make this method feasible for 3-D reacting flow.

Cannon et al. (1997) combined a velocity-composition joint PDF with the Fluent CFD code and a 5-step kinetics mechanism to predict CO and NOx in LPM GT combustors. The CFD code was 2-D and used the standard κ − ε turbulence model, which the authors felt was the main cause of error. Even using constants in the κ − ε model that were tailored to swirling flow, the authors still felt the turbulent viscosity was being over predicted, severely limiting the model. An unstructured grid was used with 4302 triangular cells and about 310,000 particles. The joint PDF evolution equation was solved with the Lagrangian Monte Carlo method using the IEM model for the molecular (small scale) mixing.

In an effort to avoid numerically integrating the stiff kinetic

equations, an in-situ look-up table was generated. Based on experimental work and other numerical work the size of the table was limited to the expected range of concentrations, thus reducing the required computer storage. megabytes of storage.

This small table still required 333

The table look-up would be less accurate then solving the

equations numerically. The model was compared to results from a laboratory-scale gas turbine combustor. The model predicted the correct treads and was a reasonable match except near the recirculation zone, which the authors attributed mainly to the turbulence

49

model. No run times were given, but it is assumed to be rather computationally intensive, even for this relatively small problem.

See Borghi et al. (1986), Phillipp et al. (1992), Roekaerts (1992), Leonard et al. (1994), Meng et al. (1997) and Hsu et al. (1997) for more results using PDF methods.

2.4

Chemical Reactor Models

This section briefly describes some of the other models that are used for LPM GT combustor modeling that do not solve the Navier-Stokes equations.

2.4.1

Simple Reactor Models

Chemical Reactor Modeling (CRM) treats the combustor as a combination of PSR’s and PFR’s. A PSR assumes infinite rate mixing, so the chemical reaction rates, not the mixing, limit the rate of combustion. A PSR is zero-dimensional, so only the conservation of mass, species and energy equations are solved (Nicol, 1995). This allows for very short run times when using detailed chemical kinetic mechanisms.

A PFR is a one-dimensional reactor with the reactions taking place in independent stream tubes with negligible cross-stream mixing. Only the unsteady conservation of mass, species and energy equations are solved, see Nicol (1995) for details.

In this model, the PSR’s and PFR’s are configured such that they roughly represent the target combustor. For example, a gas turbine combustor that consists of a strong recirculation stabilized flame followed by a one-dimensional burnout region could be modeled as a PSR followed by a PFR. The volumes of the two reactors would be based on the fluid dynamics of the combustor, as established by CFD or experiment.

50

Since CRM requires the flow field to be supplied as an external input, it is not a comprehensive model. It does, however, allow the use of detailed chemistry with very little computer resources.

Since CRM does not solve the momentum equations, it is not accurate when finite-rate mixing effects are important, such as considering emission formation. (The next section describes a modification that covers this situation.) CRM has been used with good success to model Jet Stirred Reactors (JSR) when the mixing intensity is very high (Steele, 1995).

2.4.2

Chemical Reactor Model with PDF Micromixing

The Finite-Rate Mixing Model (FMM) was developed as a tool to model jet stirred reactors (JSR) by Tonouchi (1996). The FMM uses the Jain-Spalding (1966) jet entrainment model to simulate the large scale recirculating motion (macromixing) inside of the reactor.

It uses either Curl’s (1963) Coalescence-Dispersion (C-D) or the

Interaction by Exchange with the Mean (IEM) (Borghi, 1988) micromixing models to simulate the small scale mixing. The IEM model is also known as the Linear Mean Square Estimation model. The FMM models the JSR as a 2-D axisymmetric flow. This is a Monte Carlo PDF method, as described in Section 2.3.5, with the fluid mechanics and turbulence model replaced by the Jain-Spalding jet entrainment model. The FMM can also be thought of as a modified CRM to account for a non-perfectly stirred reactor. It can also be thought of as a CFD-PDF model in which the mean flow field is specified as an input. Where a PSR integrates the chemical kinetics assuming a uniform (perfectly stirred) reactor, the FMM allows different areas of the reactor to be mixed at different rates, both on the macro and micro level. The FMM is a hybrid of CRM and PDF methods for solving turbulent reacting flows.

51

Using the 17 species, 25 reaction reduced mechanism of Nicol (1995), Tonouchi (1996) was able to fairly closely reproduce the data of Steele (1995) burning methane and various CO/H2 mixtures. Using a 486/100 PC these runs required about 40 hours. By using the simplified fluid mechanics model the FMM is able to use a relatively detailed kinetics mechanism, compared to current CFD models. There are two limitations to the FMM, the first being the fact that the user must specify the recycle parameter and the micromixing frequency. The second is that the FMM is limited to modeling JSR’s. An advantage of the FMM is that it can independently vary the rates of macromixing and micromixing, to determine their effect on emissions.

2.5

Summary

This chapter described different methods of modeling turbulent premixed combustion, specifically for LPM GT combustors. The three most common methods for modeling LPM GT combustors are using a commercial CFD program with a κ − ε based turbulence model, the joint PDF method and CRM.

The limitation of the κ − ε

turbulence model approach is in calculating the mean reaction rates from the mean temperature or using an ad hoc EBU (or EDM) model, either way it is limited to small global kinetic mechanisms to keep runtimes reasonable. Another limitation of the κ − ε turbulence model and its variants is that they were originally developed for incompressible flow. There have been attempts to modify them for reacting flows, but so far this has not been successfully accomplished. The advantage of this method is its relatively short run times, which have made it the most commonly used method to date. The limitations of the joint PDF method is in calculating the correct value of micromixing and the large computer resources required. The advantage of this method is that the mean reaction rates are solved exactly. CRM modeling allows very detailed kinetic models to be used with a short run time, but limits the fluid mechanics to the PSR and PFR limits.

52

This research addresses these limitations to make an improved model for LPM GT combustors by using the premixed conditional moment closure method to model the reactions and small scale mixing. The CMC method uses instantaneous reaction rates instead of average reaction rates and does not require a micromixing value from the user, it is calculated as part of the solution. Due to the speed of the model, it allows the use of detailed chemistry with reasonable runtimes.

This model is developed in the next

chapter. Chapter V combines the premixed CMC model with a commercial CFD code and presents results. Different turbulence models are used, but no attempt is made to improve them.

53

Premixed Combustion Regimes 4 PSR

Da=1.0E-3 Da=1

3 Re=1.0E+3 Well Stirred Reactor

Log of Velocity Scale Ratio

Ka>1

Distributed Reactions

2

Thickened Flames Ka=1 * Zimot, 1997

*

1

Polifke, 1995

Ka=

N

1 N

∑ Y ( x, t ) i

3.1

i

The < > brackets denote an averaged value. The instantaneous values are defined as follows Y ( x , t ) =< Y ( x , t ) > + y( x , t )

3.2

Where Y is the instantaneous value at a given location and time, is the unconditioned average and y is the fluctuating (deviatoric) term.

The governing

conservation equations are normally based on the unconditional average values (due to the Reynolds or Favre averaging) and require closure models for the fluctuating terms. Normally first order closure models are used, which are based on the average values. If the fluctuating terms are large relative to the average values the model will be inaccurate. This is one problem with traditional CFD combustion modeling. The conditionally averaged values are defined as follows

< Y( x , t ) | x = x a >=

N

1 N

∑ Y (x, t )δ( x i

i

− xa )

3.3

i

Only values of Y are averaged that meet a given condition, in this case x = x a . A short hand notation for the conditional average is < Y | x a > . The vertical bar means that only those that meet the conditions to the right of the bar are used in the averaging. For every value of the conditioning variable, x a , there will be a different average value < Y | x a > , thus increasing the dimensionality of the problem. The instantaneous value is similar to

57

before Y( x , t ) =< Y | x a > + y( x, t )

3.4

With the correct choice of conditioning variable the fluctuating term in equation 3.4 will be smaller then in equation 3.2. In essence, by increasing the dimensionality of the problem, the fluctuating terms are reduced, which should allow more accurate prediction of the average species concentrations. Additional conditioning constraints can be added to further reduce the fluctuating terms as well as second order conditioning.

For the non-premixed method the conditioning variable is normally the mixture fraction and the conditional average is taken with the mixture fraction within an infinitesimal range of a set value. The mixture fraction is a non-dimensional conserved scalar, normally a species mass fraction. In the absence of differential diffusion, the mixture fraction is a unique descriptor of the instantaneous state of mixedness between the fuel and oxidizer (Smith, 1994). When the mixture is pure oxidizer the mixture fraction is zero, when the mixture is pure fuel the mixture fraction is unity. Crocco (1948) was the first to use a normally dependent variable as an independent variable in boundary layer theory. Klimenko (1990) has emphasized that turbulent diffusion in mixture fraction space can be modeled more rigorously than in physical space. Bilger (1991 and 1993a) observed that most fluctuations in mass fraction were associated with fluctuations of the mixture fraction. Rather than considering flame surface statistics and the laminar reactive-diffusive structure attached to the flame surface as in the flamelet model, the CMC model is based on conditional moments at a fixed location x and time t within the flow field (Peters, 2000). A simplification of the conservation equations for the remaining dependent variables results when they are transformed into the new coordinates (Bilger, 1991). For non-premixed flows the transformed energy and species equations become

58

∂ρQ ∂Q ∂ 2Q + < ρu | η > =< ρω | η > + < ρD∇ξ∇ξ | η > 2 + e Q + e y ∂t ∂x ∂ξ

3.5

Where Q ≡< Y( x , t ) | ξ = η > , is the value of the species mass fraction or enthalpy conditioned on the mixture fraction ( ξ ) being equal to a given value η (Bilger, 1991). Here ρ is the density, u is the velocity, ω is the reaction rate, D is the diffusivity and the last 2 terms are error terms. The complete derivation can be found in Smith (1994) and will not be reproduced here, the complete derivation of the premixed CMC will be given in the next section and follows the non-premixed derivation. The coefficient of the second term in equation 3.1 is the conditioned velocity and requires a model, the coefficient of the fourth term is the scalar dissipation (or micromixing frequency). The conditioned reaction rate is approximated as < ρω | η >≈ ω(Q1 , Q 2 ......Q ns , Q h ) , where ns is the total number of species and h is the enthalpy. The fluctuating components of the species and enthalpy are much smaller in conditioned space, so this will give a much better approximation of the reaction rate than the normal unconditioned methods. Experiments have shown that for some problems the conditioned variables are nearly constant in some spatial coordinates, so the spatial derivatives can be ignored, reducing the dimensionality of the problem. Smith (1994) and Klimenko and Bilger (1999) give some examples of non-premixed results.

3.3

Premixed Conditional Moment Closure Method

This section describes the Conditional Moment Closure (CMC) method for premixed combustion in general, and its adaptation for modeling Lean Premixed (LPM) Gas Turbine (GT) combustors.

The first subsection will develop the general CMC

equations for premixed combustion. The second subsection will present a simplified

59

form of the general equations for use in modeling LPM GT combustors.

One of the main differences between the premixed and non-premixed CMC methods is that the premixed version is based on a non-conserved scalar, which adds an additional term to the equations. This extra term makes the equations more difficult to solve as explained below. The RPV is a non-dimensional sensible enthalpy and is a measure of how far the reaction has progressed. RPV=0 means the mixture is in the unreacted state, RPV=1 means it is completely burned to the adiabatic equilibrium state. The species mass fractions are set at the two end points by the boundary conditions. At any intermediate value of the RPV, the amount of fuel that has burned will be a function of the scalar dissipation. The only known attempt to solve the premixed CMC equations has been Smith (1994), who was not able to obtain a converged solution.

3.3.1

Derivation of the Premixed CMC Equations

The derivation of the premixed CMC equations using Smith’s (1994) method (which follows Klimenko, 1990) starts from the conservation of species mass fraction equation given below. There are several errors in Smith’s (1994) derivation, possibly typographical errors, so the differences to his equations are pointed out during the derivation. We start with the species mass fraction conservation equation, shown below.

ρ

∂Yi + ρu ⋅ ∇Yi = ∇ ⋅ (ρD i ∇Yi ) + ρωi ∂t

3.6

This is Smith (1994) equation 3.18 and Klimenko and Bilger (1999) equation 86. Here ρ is the density (kg/m^3), Yi is the species mass fraction, u is the velocity vector (m/s), D i is the species diffusivity (m^2/s), ωi is the production rate of species Yi (1/s) and i=1,ns where ns is the number of species.

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The enthalpy conservation equation is given below, see Appendix A for the derivation.

ρ

∂h ∂P + ρu ⋅ ∇h = ∇ ⋅ (ρD h ∇h ) − ρWR + ∂t ∂t

3.7

This is Klimenko and Bilger (1999) equation 114 and Bilger (1993b) equation 15. Smith (1994) equation 3.53 assumes the pressure term is negligible. Here h is enthalpy (J/kg),

D h is the enthalpy diffusivity (m^2/s), ρWR is the heat loss due to radiation (J/m^3-s) (it is positive for heat loss) and P is the pressure (N/m^2). Klimenko and Bilger (1999) and Bilger (1993b) assume that Le=1, so all the diffusivities are equal. Below is the sensible enthalpy equation, which is derived in Appendix A.

∂h s ∂P ρ + ρu ⋅ ∇h s = ∇ ⋅ (ρD h ∇h s ) − ρWR + − ρ ∑ ω i h f ,i ∂t ∂t i

3.8

This is Bilger (1993b) equation 16 (he is missing the last minus sign), where h s is the sensible enthalpy (J/kg) and h f ,i is the species heat of formation (J/kg).

The enthalpy is

defined as follows.

ns

h ≡ h s + ∑ h f ,i Yi

3.9

i

ns

h s = ∑ ∫ Yi C pi dT i

T

T0

3.10

Equations 3.7 and 3.8 neglect Soret and Dufour effects, but otherwise are exact relations.

61

The reaction progress variable (RPV) is defined as

c( x , t ) ≡

(h s − 2h ) − (h su − 2h u ) ∆h sad −u

3.11

The u subscript signifies the unburned state.

The denominator is the difference in

sensible enthalpy between the adiabatic and unburned states, which is fixed for a given mechanism and initial conditions. This definition is somewhat arbitrary, but forces c to be a monotonically increasing function and allows for values of c greater that unity. This corresponds to the post flame gases being non-adiabatic, once the fuel has become fully burned the NO x chemistry continues while the gases cool. This definition for c allows the system to be non-adiabatic but to remain single valued (Bilger, 1993b). With heat loss the enthalpy (h) will decrease, while the unburned value remains constant, which allows c to become greater than unity. For adiabatic problems the enthalpy (h) remains constant and c becomes a function of only the sensible enthalpies, which is a common definition in the literature. Here only adiabatic systems will be considered. Chapter VII will introduce a non-adiabatic model. If the species specific heats are assumed constant then c becomes a function of only temperature (it becomes a non-dimensional temperature with values between 0.0 and 1.0). The conservation equation for c is derived in Appendix B and is shown below.

∂c ρ + ρu ⋅ ∇c = ∇ ⋅ (ρD h ∇c) + ∂t

ρWR − ρ∑ ωi h f ,i ∆h

i s ad − u

≡ ∇ ⋅ (ρD h ∇c) + ρS c

3.12

These are Smith (1994) equations 3.54 and 3.55. Klimenko and Bilger (1999) equation 9.57 and Bilger (1993b) equation 18 are similar to this except they are missing the minus sign in front of the ρ∑ ωi h f ,i term, which comes from their incorrect definition of i

62

equation 3.8. Equation 3.12 is essentially a non-dimensional energy (enthalpy) equation.

The PDF equation for c is derived in Appendix C and is shown below. ∂ < ρ | ζ > Pξ ∂t

+ ∇ ⋅ (< ρu | ζ > Pξ ) − ∇ ⋅ (< ρD h | ζ > ∇Pξ ) =

∂2 ∂ − 2 < ρD h (∇c) 2 | ζ > Pξ − (< ρSc | ζ > Pξ ) ∂ζ ∂ζ

(

)

3.13

Up to this point all of Smith’s (1994) derivations are correct. The premixed CMC equation is derived in a similar manner to give, see Appendix D. ∂Q i ∂ 2Qi + < ρu | ζ > ⋅∇Q i =< ρωi | ζ > + < ρD(∇c) 2 | ζ > ∂t ∂ζ 2 ∂Q i − < ρS c | ζ > + < e y | ζ > + < eQ | ζ > ∂ζ

< e y | ζ >=

1 ∂ ∂2 {2 [< ρD(∇c ⋅ ∇y i ) | ζ > Pζ ] − 2 [< ρD(∇c) 2 y i | ζ > Pζ ] Pζ ∂ζ ∂ζ

− ∇ ⋅ (< ρuy i | ζ > Pζ ) − < e Q | ζ >=

3.14

∂ (< ρS c y i | ζ > Pζ )} ∂ζ

1 ∂ {2 [< ρD(∇c ⋅ ∇Q i ) | ζ > Pζ ] + ∇⋅ < ρD | ζ > ∇(Q i Pζ ) Pζ ∂ζ

− Q i ∇ ⋅ (< ρD | ζ > ∇Pζ )}

The last two terms, e Q and e y , are error terms, the magnitude of these will be addressed in Section 3.3.4. At this point there are some differences between these equations and those Smith (1994) derives. Smith (1994) apparently has some typographical errors, because his equations are not consistent. This equation describes the conditional averages of the species as a function of space and c. It would nominally

63

be solved as a coupled system with equations for momentum and c (mean and variance).

3.3.2 Derivation of the Volume Averaged CMC Equations

Jet flame data suggest that the spatial derivatives in equation 3.14 can often be neglected. This is particularly true for the radial dependence in jet flames, but often it is also true for the axial dependence in jet flames with fast chemistry (e.g. hydrogen flames) (Smith 1994, Bilger 2000 and Klimenko and Bilger 1999). Note that this does not imply that mean species mass fractions are independent of axial or radial position. Instead, it implies that the mass fractions depend only on the conditioning variable (mixture fraction for non-premixed, c for premixed), and its scalar dissipation rather than on conditioning variable and the spatial position. If the spatial dependence can indeed be neglected, a substantial reduction in computational complexity can be obtained. Specifically, the behavior of the conditioned variables depends only on the conditioning variable (mixture fraction or c) and its scalar dissipation, which will vary with spatial location. Thus, the calculation of the conditioned variables can be decoupled from the flow field calculation, since the only inter-linkage will be through c and its scalar dissipation. This opens the possibility of doing a detailed chemistry calculation off line and using a lookup table to feed information back to the fluid dynamics calculation.

For premixed problems with high mixing intensities (i.e. in the distributed reaction regime) the conditioned species may be sufficiently uniform within the reactor to allow the equations to be simplified by neglecting spatial derivatives (Smith 1994) in a way similar to that done in non-premixed jet flames. This is along the lines of a partially stirred reactor, which has strong back mixing giving nearly uniform conditioned properties throughout the reactor. This is a generalization of a perfectly stirred reactor, and is similar to a lean premixed gas turbine combustor. Several authors have shown this to be true for non-premixed flames (Smith 1994, Bilger 2000 and Klimenko and Bilger

64

1999). With this assumption the equations are volume averaged to give equation 3.15 (see Appendix E for the derivation). The volume averaging starts with equation D.14a neglecting the error terms ( e Q and e y ) and assumes steady state. This equation is volume averaged with the divergence theorem applied to the inlet and outlet. The same procedure is applied to equation 3.13 then both of these equations are combined to give the conditional mean reactive scalar equation.

{{ρ}}(Qi − Qi,0 )

{Pζ* }in τr

= {{Pζ < ρωi | ζ >}} 3.15a

∂ 2 Qi ∂Q + {{Pζ < ρD(∇c) | ζ >}} 2 − {{Pζ < ρSc | ζ >}} i ∂ζ ∂ζ 2

{{ρ}}

{Pζ* }out − {Pζ* }in τr

=−

 ∂ ∂ {{< ρD(∇c) 2 | ζ > Pζ }} + {{< ρSc | ζ > Pζ }}  ∂ζ  ∂ζ 

(

)

3.15b

For a perfectly premixed inlet stream the left side of the above two equations will always be zero because the PDF will only be non-zero at c=0, and at c=0 the difference in Q’s will be zero (Smith 1994). The volume averaging will cause the error terms in equation 3.14 to become negligible. Here the scalar dissipation is also volume averaged, so only one value is used through out the reactor. This reduces the CMC database to 1dimensional, i.e. flame data are only stored for this single value of scalar dissipation. The advantage of volume averaging the CMC equations is that it eliminates the spatial variation of the variables (reducing the dimensionality of the problem), so the equations can be solved much faster. This leaves just three terms in equation 3.15a, which are the source, diffusion and convection in c space, respectfully. Bilger (1993b) derives the same equations using a different method and assumptions, see Appendix F. Smith (1994) was unable to obtain a converged solution for this system of equations stating the problem

65

appeared to be due to poor initial conditions causing negative scalar dissipation values. Smith solved the CMC equations without a CFD code, assuming a constant volume averaged mean and variance of the RPV and basing the volume averaged scalar dissipation on the volume averaged variance of the RPV.

Here the volume averaged CMC model was combined with a CFD code and gave a converged solution. The solution procedure is to solve the CFD code, calculating a volume averaged scalar dissipation, which is used with the table lookup to obtain the conditioned reaction rates and density that are used at all CFD nodes. The PDF (which is different at all CFD nodes) is used to convert these conditioned variables to real space in the CFD code. Even though the conditioned values are the same at every node, the nonconditioned values will be different at each node. No results for this model are presented here (see Martin et al., 2001 and 2002 for results), the next section will develop a model similar to this, but with less restrictions.

3.3.3

Uniform Conditioned Species Formulation

A new approach is introduced here, assuming that the conditioned species mass fractions are nearly uniform in the reactor. This will cause the second term in equation 3.14 to become negligible, eliminating the need to model the conditional velocity. This is similar to the method Smith (1994) used, but without performing the volume averaging of equation 3.14. Limiting to only steady state problems and neglecting the error terms ( e Q and e y ) gives.

∂ 2Qi ∂Q i < ρωi | ζ > + < ρD(∇c) | ζ > − < ρS c | ζ > =0 2 ∂ζ ∂ζ 2

3.16

66

This has the same form as equation 3.15a, except the scalar dissipation and reaction rates are not volume averaged. Note, the terms in Equation 3.16 are Favre averaged, the standard notation of a tilda over bar will not be used here for simplicity. The advantage of this is that when coupled with a CFD program a different value of scalar dissipation can be used in each CFD cell, while still de-coupling the CMC model from the bulk flow. This approach is still limited to flows with high mixing, but should be applicable to a wider range of conditions then equation 3.15. This is the model that will be used in subsequent chapters (also see Martin, 2003). Stand alone results using an inputted value of scalar dissipation will be presented in Section 3.4 and results of the model combined with a CFD code will be presented in Chapters V and VI.

3.3.4 Analysis of Error Terms

In the previous section the error terms in equation 3.14 were neglected, here a more detailed analysis will be performed. First the e Q term will be analyzed. This term is the macro transport of Q by molecular diffusion and it will be argued that it is negligible for high Reynolds number flows. Klimenko and Bilger (1999) argue that the Laplace operator applied to an averaged value does not increase the order of the term, so ∇⋅ < ρD | ς > ~< ρD | ς >~ 1 / Re → 0 , this is the well known argument that high Re flows

do not depend on molecular diffusivities. This says that the last two terms of e Q are negligible. It is not as clear if this argument can be applied to the first term, but with the assumption that Q is uniform within the reactor volume, the gradient of Q is zero and the first term becomes zero.

Determining the size of the e y terms is not as straight forward. These terms depend on flame surface effects, i.e. flame front proximity and topology. Klimenko and Bilger (1999) derive the CMC equations by two different methods, using different

67

assumptions. To make both derivations equivalent they state that the first two terms of e y must be of equal magnitude, so they cancel out. While this is not a detailed proof, it

gives an indication that the two terms may be negligible. Another way to evaluate these terms is from the basic premise of the CMC method, that the conditioned species mass fractions fluctuations are much smaller then their unconditioned counterparts. So if y i is small its gradient in physical space should also be small, making the first e y term negligible.

Similarly for the second and forth e y terms, y i is multiplied by the

conditioned scalar dissipation and the conditional source term, respectively. Klimenko and Bilger (1999) equation 139 propose a gradient diffusion closure for the third term, < ρuy i | ς >= − D t ∇Q i .

They state it might be a justified closure for non-premixed

flames, but that it is unknown if it is valid for premixed flames. For low Reynolds number premixed flows counter gradient diffusion can dominate and this model will give the wrong sign. When limited to high Reynolds number flows this model may at least give a reasonable approximation. In the uniform conditioned species form, the gradient of Q is zero and this term drops out. For flows with ignition and extinction, the error terms will be important and require closure models. To fully understand the importance of these terms for a given flow will require detailed experimental data, which most likely will not be available in the near future.

3.4

Results of the Stand Alone Premixed CMC Method

The above uniform conditioned species CMC equations for the conditioned species mass fractions are solved with the full GRI2.11 kinetic mechanism (Bowman et al. 1998) using the TWPBVP ordinary differential equation solver (Cash 1998). The GRI2.11 mechanism has 48 species and 277 reactions for burning methane with NO x chemistry. All cases are adiabatic so the conditioned enthalpy equation is not required. The boundary conditions are unburned values at c=0 and adiabatic equilibrium at c=1.

68

Note that these are solved with only the scalar dissipation as an input, so the goal is to look at parametric behavior of the conditioned species.

Before discussing results from the above equations it is useful to look at limiting cases. The scalar dissipation (also called the micromixing frequency or mixing rate) is defined as N =< D(∇c) 2 | ζ >

3.17

N has units of inverse time and is a measure of the mixing intensity of the system, N is inversely proportional to the mixing time.

In the limit of very large N the diffusion term of equation 3.16 dominates and the equation reduces to ∂ 2Qi =0 ∂ξ 2

3.18

Solving this gives Q i = Q i0 + (Q iad − Q 0i )c

3.19

The second term on the RHS of the equation in the slope of the species production curve, it can be thought of as the reaction rate in c space. This says that the species mass fractions change linearly from their initial values as a function of the progress variable, independent of the chemical mechanism used. The results show CH 4 , CO, H 2 O , O 2 and CO 2 mass fractions (the results are presented below) varied linearly as a function of

69

c for N=10,000 1/sec with an equivalence ratio of 0.9 and a fuel/air inlet temperature of 300 K (for N=1,000 the mass fractions are nearly linear). A large value of N gives a small Damköhler number, which is in the distributed reaction regime of the premixed combustion regime plot, Figure 2-1. Larger values of N were run giving the same results, though they are not shown here. Also the full Miller-Bowman mechanism and a 16 specie reduced Miller-Bowman mechanism were run with N=10,000 and larger, giving the same results. Other equivalence ratios also gave this linear relationship, but the mass fractions were different based on the stoichiometry. The intermediate and free radicals are not linear in c space, but are close and vary much less then at smaller values of N. Here the larger mass fractions are linear in c space, but when they are transformed back to physical space they will not necessarily be linear in space or time.

In the limiting case of very large N, the system becomes a perfectly stirred reactor (PSR) in c space. A PSR has infinitely fast mixing rates, causing it to be spatially uniform. The equation for a PSR in terms of mole number is & (σ k + σ 0k ) = rk V m

3.20

This can be written in terms of mass fractions as 0 & ∆t Yk = Yk + Y k

3.21

Where the superscript zero signifies initial conditions and Y are the species mass fractions in real space. This equation has a similar form (in real space) as equation 3.19 (in c space), both representing a PSR. The two approaches become equivalent with the additional condition that the mixing is sufficiently intense that c becomes uniform throughout the reactor volume. As N increases, approaching a PSR the PDF becomes a

70

delta function at the exit conditions, i.e. c is uniform in the reactor. Increasing N is achieved by increasing the inlet velocity of the reactor (by increasing the mass flow rate or decreasing the inlet area) which also decreases the residence time. Since the species are kinetically limited, reducing the residence time will reduce the value of c in the reactor, i.e. the delta PDF will move to smaller values of c. As the delta PDF moves below about 0.75, when the solution is transformed back to physical space, it will give a non-ignited solution. This is the blowout residence time of the reactor.

In the limit of very small N (large Damköhler number) the diffusion term of equation 3.16 becomes negligible and only the convection and source terms remain giving a first order equation. ∂Q i =< ρωi | ζ > / < ρS c | ζ > ∂ζ

3.22

Here the mixing is so slow that the reactions will burn nearly instantly, once enough heat and free radicals diffuse into the flame zone, so the overall flame is diffusion limited.

As N decreases the flame becomes a laminar diffusion flame and as N

approaches zero the flame becomes a PFR, i.e. no diffusion. A small value of N gives a large mixing time relative to the chemistry time, which makes the equations very stiff as described below. This is the opposite of the case with very large N, where the formation rates are independent of the mixing rate. For all cases between these two limiting cases the scalar dissipation and kinetics will affect the species formation rates. The term < ρS c | ζ > / < ρD(∇c) 2 | ζ > (from equation 3.16) can be thought of as a global Damköhler number for the system and the term < ρωi | ζ > / < ρD(∇c) 2 | ζ > as a Damköhler number for species i. Sc is a convective time scale of the energy release and

71

< ρD(∇c) 2 | ζ > is a diffusive time scale.

This indicates that there are multiple

characteristic time scales in turbulent premixed combustion. Bilger (2000) states that at least two chemical time scales are necessary to describe the structure of laminar premixed flames: one associated with the chain branching of radical reactions and another associated with the three-body recombination reactions.

In turbulent premixed

combustion the present work suggests that additional time scales are required, more experimental and DNS data are needed to fully understand the different time scales. Only in limiting cases can it be reduced to a single time scale, such as the low mixing regimes of the flamelet and G equation models. A general premixed combustion model must account for these different time scales and their interactions with the turbulence. Results presented at the end of Section 3.4.1 will discuss the need to use detailed kinetic mechanisms, which ties into the multiple time scales described here.

3.4.1

Atmospheric Pressure Results.

Figure 3-2 shows CH 4 , CO, CO 2 , H 2 O and O 2 conditioned mass fractions for N=10 1/sec (N is the scalar dissipation) with an equivalence ratio of 0.9, 1 atm. pressure and an inlet temperature of 300 K using the full GRI2.11 mechanism (Bowman et al., 1998). Figure 3-3 shows the same conditioned species for N=1,000 1/sec. These are the same conditions that will be used with the backward facing step results in Chapter V. It is seen that the CH 4 burns up faster producing much more CO when N=10, compared to N=1,000. The progress variable, c, is a measure of how much energy has been released during the reaction (it is approximately equal to the non-dimensional temperature). For an adiabatic system a value of c=0.5 means that half of the available chemical energy has been released. When comparing Figure 3-2 to Figure 3-3 it is seen that the former has burned up more CH 4 at c=0.5. The reason for this is with the small value of N in Figure 3-2 there is little mixing, so a significant portion of the CH 4 burns to CO and not to CO 2

72

early in the flame zone. The burning of CH 4 to CO liberates less energy then burning

CH 4 to CO 2 , so more CH 4 must be burned to release the same amount of energy at c=0.5. Once all the CH 4 is burned, at about c=0.7, the CO burns to CO 2 liberating the remaining energy. For even smaller values of N the effect becomes more pronounced (not shown here).

Another way to interpret these trends is a competition between reactions and micromixing (scalar dissipation). For small N, isolated gas particles essentially move from c=0 to c=1 as individual batch reactors. This can be viewed as approaching the thickened flame or flamelet regimes of premixed combustion. The flame is a continuous zone that does not wrap around on its self.

For this to occur there must be low

macromixing or bulk turbulence. The smallest value of N run to date was 0.04 1/sec and the equations became very stiff, requiring a finer grid in c-space and taking much longer to solve, corresponding to something like a laminar diffusion flame transitioning to a PFR.

For larger N, there is more micromixing along the c axis, giving mixing in

competition with the reactions. This tends to force everything toward a linear solution in c space, which yields less CO and more CH 4 and CO 2 at a specific value of c. For this condition to remain ignited it requires large macromixing to continuously mix hot burned products with cold un-reacted mixture. Without this large macromixing the flame would extinguish due to the large straining, as was discussed in Chapter II. In real reactors the micromixing and macromixing are not independently adjustable over the entire operating range of a reactor. The reactor geometry and inlet conditions dictate the micromixing and macromixing. The finite rate mixing model of Tonouchi (1996) allows the micromixing and macromixing to be independently adjusted. The simplified CMC model used here implicitly assumes that the macromixing is at a level that will sustain ignition with the value of micromixing used. This simplified CMC model is not valid for ignition and extinction modeling, which will most likely require second order conditioning and

73

possibly double conditioning.

At small N the premixed CMC model has some

similarities with the Euclidean Minimum Spanning Trees Model of Subramaniam and Pope (1998), where only particles with similar progress variable can mix. This is another way of saying that there is no bulk mixing, only small scale mixing like in the flamelet regime.

A PFR with a cold inlet cannot sustain ignition. As N decreases, approaching the PFR limit, the stiffness in the equations is the numerical way of saying the solution is not physically valid. For smaller values of N the equations would not solve. The GRI2.11 mechanism would give a solution for smaller values of N then the full Miller-Bowman and the reduced Miller-Bowman. At larger values of N the results look the same as the N=1,000 case, which approaches the PSR limit. Because all runs have the same boundary conditions at c=0 and c=1, the mass fractions at the boundary are the same, but in between the species are much different as N varies. These results are in agreement with the physical explanation given at the beginning of this section.

Plotting the conditioned results versus c takes out the spatial and temporal variation, so looking at these plots gives no relative flame thickness or total reaction time. It can be hypothesized that the larger values of N, approaching a PSR, would have faster reaction rates, so the flame would be thinner. As N is decreased (approaching the batch reactor or flamelet limit) the equations become much harder to solve. As discussed previously the simplified CMC equations are only valid for well-mixed flows, so there will be a point where decreasing N invalidates the equations. At this point it is not known at what value of N this takes place at, but it is at some condition when the Damköhler number becomes significantly greater than unity, this will be discussed in more detail later. Also, c does not scale linearly with time or space, so c=0.5 is not necessarily the middle of the flame, it is the point that half of the available energy has been released. The species mass fractions presented here are the conditioned values, to

74

get the total values in real space, the conditioned values are multiplied by the PDF of c and integrated across c space. This will be discussed in more detail in Chapter V.

Figure 3-4 is for the same conditions as Figure 3-2, with c defined as the nondimensional ratio of temperature, defined in equation 3.23.

c=

T − Tu Tad − Tu

3.23

This is equation 3.11 with constant specific heats and assuming the reactor is adiabatic. This shifts the data slightly to the right (higher values of c), because the specific heats are not constant in this temperature range. This means that at c=0.5 half of the energy has not been released, this occurs at about c=0.55. Near c=0.4 in the figure there is a small jump in the results due to the thermochemical data. The database is broken into two parts, one for temperatures below 1,000 K and one for temperatures above 1,000 K. At the overlap point there is a small inconsistency in the data. Figure 3-5 is the same conditions with c defined as the ratio of CO 2 , defined in equation 3.24.

c=

YCO 2 YCO 2,ad

3.24

This shifts the data significantly to the left (smaller values of c). Here the CH 4 is completely destroyed by c=0.3, while in Figure 3-2 the CH 4 is not completely destroyed until c=0.7. This will change the definition of the source term in equation 3.12 and the premixed CMC equations will need to be re-derived using this definition. This will give the same form of the equation (equation 3.16) with slightly different coefficients for the terms. The PDF’s used with these different definitions of c will have a different shape,

75

but once the conditioned species are transformed to physical space the results will be the same. This shows that it is important to use the same definition of c when comparing conditional results. Defining c in terms of the sensible enthalpy (as in equation 3.11) is more accurate because it accounts for variations in the specific heats due to temperature, which equations 3.23 and 3.24 do not. Abou-Ellail et al. (1999) use the definition in equation 3.24 in their flamelet based model, which will be discussed later.

Figures 3-6 and 3-7 show conditioned mass fractions of CH 2 O , C 2 H 4 , C 2 H 6 ,

H 2 , NO, O and OH for N=10 and 1,000, respectfully. For the small value of N there is much more CH 2 O , C 2 H 4 , C 2 H 6 and H 2 at c0.65, compared to N=1000. At large N when a molecule of CH 4 starts to burn it goes completely to CO 2 and H 2 O , before the next molecule of CH 4 ignites, so there are less intermediate species. The physical reason for this is that at large N the mixing provides contact across c space, providing free radicals that promote the rapid oxidation of intermediates from the initial breakdown of methane.

At small N, the methane

oxidation occurs as a serial process, with little influence from mixing. Here, the reaction intermediates show up as a series of species. At a given value of c, a larger N will have more CH 4 and CO 2 than a smaller value of N, while the smaller value of N will have more CO and intermediates. Figure 3-6 has two distinct groupings of species. Below c=0.65 and above. This can be explained by looking at Figure 3-2, which shows at low values of c there is significant CO just as Figure 3-6 shows significant intermediate species. This supports the above hypothesis that at small values of N the majority of the fuel burns to CO and intermediates before oxidizing to CO 2 , like a batch reactor. As the fuel is used up the intermediates are burned up quicker then the CO.

Once the

intermediates are used up, the CO starts to rapidly oxidize at about c=0.7 and the free radicals start to increase very rapidly, then they decrease as the CO in used up. At N=1,000 the NO starts to increase much sooner than with the smaller values of N. The

76

possible reasons for this and the 3 pathways for NO formation will be discussed below. Keep in mind that the PDF for the different values of N will determine the total unconditioned NO. When the fuel in a reactor becomes fully burned, i.e. c=1, the unconditioned NO mass fraction will be the same for all values of N, because the PDF becomes a delta function at C=1, but will require a different time to reach c=1. For the cases shown here, this peak value of NO is about 3,300ppm (the adiabatic equilibrium value), which is several orders of magnitude larger than what is seen in jet stirred reactor (JSR) experiments discussed in Chapter VI.

Figures 3-8 and 3-9 show conditioned mass fractions of CH 3 , C 2 H 2 , H, HO 2 and H 2 O 2 for N=10 and 1,000, respectfully. At N=1,000 there is much more CH 3 , especially at higher values of c than for the cases with smaller N. The reason for this is that for smaller N the CH 3 combines to give C 2 H 4 and C 2 H 6 and it also oxidizes to give CH 2 O . It was shown above that at smaller N there are more of these species. This again shows that the variation in micromixing causes different CH 4 oxidation pathways, giving different amounts of the intermediate species throughout the flame. The smaller N also has more H 2 O 2 at small values of c.

Figures 3-10 and 3-11 show conditioned mass fractions of CH 2 , CH 2 CO , CH 3 OH , C 2 H 5 , HNCO and NO 2 for N=10 and 1,000, respectfully. At smaller N there is more CH 3 OH then at the larger N, which is another reason there is less CH 3 at the lower N. At larger N there is much more HNCO, which is an intermediate in prompt NO formation and more NO 2 , helping to explain the much larger amount of NO at larger values of N. This shows that at higher mixing levels, as used in modern premixed gas turbine engines, more NO 2 is formed in the middle part of the flame.

77

Figures 3-12 and 3-13 show conditioned mass fractions of CH 2 OH , CH 3 O , C 2 H 3 , HCCO, HCNO, HCO, N 2 O for N=10 and 1,000, respectfully. At small values of N there is a large spike of these intermediates between c=0.6 and 0.7. This is the point where the CH 4 is burned up and the CO starts to oxidize to CO 2 . At N=1,000 there is much more HCNO and N 2 O compared to the smaller values of N.

These are

intermediates in the formation of NO.

From these plots it is seen that the scalar dissipation (micromixing frequency) has a large affect on the chemical speciation. For large values of N the CH 4 burns to CH 3 and then to CO 2 with little build up of CO and intermediates. For smaller values of N the CH 3 combines to give C 2 H 4 and C 2 H 6 and oxidizes to CH 2 O and CH 3 OH , then to CO. Once all the CH 4 is burned, the CO begins to oxidize to CO 2 . The CH 4 is gone by c=0.7 and as the CO oxidizes there are spikes in many of the smaller intermediate species. At larger values of N (more micromixing) some of the intermediates come into contact with other species ( CH 4 and other intermediates) causing them to react and combine through different pathways than for the smaller values of N. This increased micromixing prevents the intermediates from building up as is seen in the case with smaller N. At small values of N (with minimal micromixing) the reactions all take place based on the kinetic rates as isolated pockets, so a pocket of partially reacted mixture will not mix with neighbor pockets that are unburned or fully burned. Since laminar flamelet models solve the chemical kinetics assuming laminar flow, and then apply a PDF to account for the strain affects, it is doubtful that they can correctly predict the affects of micromixing (scalar dissipation) on the reaction rates. See Section 2.3.3 for a more detailed discussion on flamelet models.

The larger value of N has much more NO and NO 2 . Since the temperature is the

78

same versus c for all values of N, variation in the Zeldovich (thermal) NO formation rate is due to different mass fractions of the key species as N varies. As was discussed above at N=1,000 there are several of the intermediates associated with the nitrous and prompt NO pathways, so the extra NO also comes from these sources. The other variable that affects NO formation is time, the longer the species that create NO are present, the more NO will be formed. One would expect that at high N (approaching the PSR limit) the reactions would be quicker, giving less time to form NO, but the results are contrary to this. It is possible that this is not the case, that the larger N actually burns slower then the small N.

Figure 3-14 shows conditioned CH 4 mass fraction for a range of scalar dissipation values (N) for the same initial conditions as above. As was seen above, the

CH 4 is nearly a linear function of c at N=1,000, for larger N it becomes truly linear. At smaller values of N, the CH 4 burns up quicker and is completely gone by c=0.75. For N=0.05 (not shown here) the CH 4 is gone by c=0.7. Figures I-1 to I-3 show O 2 , CO 2 and H 2 O , respectively (see Appendix I).

Figure 3-15 shows conditioned CO mass fraction for a range of scalar dissipation values (N) for the same initial conditions as above. As N is decreased more CO is formed, peaking at about c=0.7, then rapidly decreasing. At N=0.05 and 1 (not shown) the peak CO is 0.0614. There is a small region between c=0.8 and 0.9 where the CO is not a monotonic function of N, i.e. for N=200, 400 and 600 there is more CO then at N=130, while for N=800 and 1,000 the opposite is true.

Figure 3-16 shows conditioned OH mass fraction for a range of scalar dissipation values (N) for the same conditions as above. This shows that for the smaller values of scalar dissipation the OH reaches super-equilibrium values late in the flame.

79

Figure 3-17 shows the source term for the c equation over a range of scalar dissipation values for the same conditions as above. This is also the coefficient for the convection term in equation 3.16, S c .

This term is the sum of the reaction rates

multiplied by the corresponding heat of formation, non-dimensionalized. This term is the rate energy is being added to the system (an energy source term) due to c being a nonconserved scalar. Here it is plotted in conditioned form, the PDF is used to convert it to physical space to use in the c equation. As N decreases the peak of the source term increases up to N=400, then it slowly starts to decrease. For N=0.1 (not shown) the peak is at 4,000. As N increases the peak also moves to higher values of c. The results show that between c=0.95 and c=1.0 the source term takes on small negative values with small N. This may be due to small inconsistencies in the thermochemical data used and round off error in the calculations. It is also possible that the negative values signify counter gradient diffusion, more detailed experimental data are needed to verify this. A large value (such as late in the flame) means that the flame is convectively controlled, while a small value (such as early in the flame) means that the flame is diffusively controlled. For smaller values of N (not shown) the source term becomes narrower, indicating that more of the energy release takes place in a narrow region of the flame, consistent with the flamelet regime where the reactions approach the fast reaction limit. At larger values of N the source term is non-zero over a much wider fraction of the flame zone indicating that the flame starts to burn sooner and burns more uniform over c space, like a PSR.

Figures 3-18 through 3-22 show the conditioned species mass fractions with N=10 for the same conditions as above, calculated using the full Miller-Bowman mechanism (Miller and Bowman, 1989). These show very similar mass fractions as compared to the full GRI mechanism, except there is much more CH 3 O , because there is no CH 3 OH in the Miler-Bowman mechanism. Figure 3-23 shows the source term for the c equation. It

80

is seen that the peak value is slightly higher (4.63E3 compared to 4.16E3) and at a slightly lower value of c (0.63 compared to 0.675) compared to the GRI2.11 mechanism.

Figures 3-24 through 3-29 are the same plots as above with N=14 using the reduced Miller-Bowman mechanism of Nicol (1995) with 16 species and 24 reactions. The premixed CMC equations would not converge for smaller values of N. The major species match the full mechanism remarkably well, the minor species show some difference because the reduced mechanism does not have all the minor species, so the ones it has show higher mass fractions. Figure 3-29 shows the source term for the c equation is slightly lower and peaks at a lower value of c, so it is expected it will burn sooner in the CFD program.

Cases were also run using a 1-step methane mechanism, (results are not shown). It was found that N did not affect the results. This makes sense because there are no intermediate species such as CO that can vary to allow the CO 2 to vary at a constant value of c, so the kinetics become independent of N. This highlights a limitation of DNS and LES results using 1-step kinetics, namely it does not allow the small scale mixing to affect the chemical reactions. Klimenko and Bilger (1999) found the same thing when studying DNS data with a 1-step propane mechanism, the turbulence had no effect on the conditioned reaction rates.

This ties into the discussion in the previous section of

multiple chemical time scales in premixed combustion, each interacting differently with the turbulence. In order to model the different time scales a multi-step mechanism is required. Appendix J shows analytically that a 1-step mechanism will always have a linear relationship between the species mass fractions and c. 3.4.2

High Pressure Results

Figures I-4 through I-13 show conditioned species mass fractions for a range of scalar dissipation of 5 to 50,000 for an equivalence ratio of 0.529, inlet temperature of

81

603 K and a pressure of 4.74 atm. These are conditions that will be used in Chapter VI to model a jet stirred reactor and are presented here for completeness.

Figures 3-30 through 3-35 show the affect of reactor pressure on species mass fractions for an equivalence ratio of 0.529, an inlet temperature of 603 K and a scalar dissipation of 350 1/sec for pressures of 1, 2, 4.7 and 7 atm. Figure 3-30 shows that CO is not a monotonic function of N. CO is lowest at 1 atm. And highest at 2 atm., then decreases slowly with increasing pressure. As expected, Figure 3-31 shows the opposite trend for CO 2 , with the value at 1 atm. being the largest. Figure 3-32 shows O 2 is a weak function of N, the difference in scales tends to hide the affect. Figure 3-33 shows that NO is a monotonically decreasing function of N, increasing pressure decreases NO. Figure 3-34 shows NO 2 is nearly a monotonically increasing function of N, except at 1 atm. and larger values of c. Figure 3-35 shows that N 2 O decreases with pressure up to 4.7 atm. then starts to increase again. For most values of c the NO 2 and N 2 O are at super equilibrium values.

The results presented above were solved on an 800 MHz Intel PC using the Linux operating system. The runtimes ranged from 5 seconds for large values of N to 5 minutes for small N. The difference is due to the increased stiffness as N decreases, requiring a finer solution mesh and more iterations to converge. The large values of N only required 21 grid points for the solution, while the smallest values required 783 grid points and many more iterations to converge.

3.4.3

Comparison to Other Models

The reactor conditions used in Abou-Ellail et al. (1999) were run using the simplified CMC model developed in section 3.3.3 using the equilibrium conditions for

82

the c=1 boundary conditions.

The conditions are premixed methane and air at an

equivalence ratio of 1.0 and an inlet temperature of 300 K. Abou-Ellail et al. (1999) figure 3 shows different values of CO at c=1 as scalar dissipation is varied, even though the CH 4 and CO 2 are the same, indicating that c atoms are not conserved. This calls into question the boundary conditions used, at least for the simplified global kinetic mechanisms they used. The CMC model gives the same species mass fractions at c=1 for all values of scalar dissipation, guarantying atom and mass conservation. Because of the different definitions for c, no attempt was made to convert their scalar dissipation values to the definition used in the present work. Trying a range of scalar dissipation values in the present model had a larger affect on the species then their model. This could be that they are in a range where scalar dissipation has a small affect, i.e. the scalar dissipation is very small, giving a very large Da number. This would give very fast reaction rates, independent of small scale mixing. The only variation in species in their model is near c=1, which are influenced by the c=1 boundary conditions.

As scalar dissipation

increases the H 2 O decreases at c=1, while the CO, O 2 , H 2 and temperature decrease, indicating partial extinction. They found a maximum scalar dissipation, above which the flame would completely extinguish. This is consistent with laminar flames, i.e. as the small-scale turbulence increases the flame becomes so strained that without any largescale recirculation the flame extinguishes. They did not measure the extinction value of scalar dissipation to compare with the model prediction. The CMC model used here does not have a maximum scalar dissipation value because it assumes there is intense bulk mixing to continuously mix hot products with cold reactants, i.e. in the well-stirred reactor regime.

It appears that the c=1 boundary conditions are the method to

differentiate between these to very distinct flow regimes, even though the equations are the same.

The difference between the Abou-Ellail et al. (1999) model and the one used in this work is the boundary conditions at c=1. They used Y’(c=1)=0, i.e. no gradients in the

83

species at c=1. The model developed here set the species to their equilibrium values at c=1. Looking at Figure 3-17, as N increases the range of the maximum source term moves toward c=1 and the gradient at c=1 is steep. For smaller values of N (not shown) the values of non-zero source term move toward smaller values of c and the gradient at c=1 becomes zero. Using the Y’(c=1)=0 boundary condition cannot be solved when the gradient of the source term at c=1 is non-zero. This says numerically that the flamelet extinguishes at larger values of N (scalar dissipation), which is a known fact from experimental results and matches Peters (2000). The assumption built into the AbouEllail et al. (1999) model limits it to the flamelet regime with laminar bulk flow. The CMC model used here has different assumptions, namely the bulk flow is highly turbulent, and it remains ignited with larger values of N.

Comparing Abou-Ellail et al. (1999) figure 6, which has their results plotted with c defined as a non-dimensional temperature ratio (see equation3.23), to results using the simplified CMC (with an estimated value of scalar dissipation) model shows similar results from both models for the main species (results not presented here). The OH is much better with the CMC model, most likely because of the large kinetic mechanism used, compared to their 3-step global mechanism. One advantage of the flamelet and CMC type of combustion models is they allow the use of very detailed mechanisms, so it is odd that Abou-Ellail et al. (1999) only used a 3-step global mechanism.

Rogg and Peters (1990) in their figure 3b show a schematic of a laminar flame. The figure shows that the CH 4 is completely burned up at the point of peak CO. The results from Figures 3-14 and 3-15 for small values of scalar dissipation (N) match this. As was described earlier, small N approaches the batch reactor or laminar flamelet regime. Figures 3-14 and 3-15 show that for large values of N, approaching the PSR regime, there is a much different relation between CH 4 and CO. For the larger values of

84

N shown, the CH 4 is not fully burned until c=1, while the CO peaks at about c=0.8. This shows that the flame zone in the well stirred and distributed reaction flame regimes (large N) are much different then the laminar flamelet regimes (small N). This is further evidence that combustion models based on laminar flamelets are not valid for highly turbulent flows.

Earlier it was stated that the solution of the species mass fraction equations in c space gave no indication of flame thickness. Rogg and Peters (1990) figure 3b can be used to give a qualitative relationship between c space and physical space by comparison to Figure 3-15. Figure 3-15 shows that for small N, the peak CO is at about c=0.7, then decreases rapidly to c=1.0. Rogg and Peters (1990) Figure 3b shows that the peak CO is located about 40% of the way through the zone where CO is present. The physical explanation for this is that in the early part of the flame the CH 4 is burned to CO, with only a portion going to CO 2 . The oxidation of CH 4 to CO gives off less energy then the oxidation of CO to CO 2 . So when all the fuel has been consumed (c=0.7), not all the energy has been released because of the significant amount of CO present. Based on the relative thickness of the CO zone before and after the peak CO location in physical and c space, it appears that the CH 4 to CO reaction rate is faster then the CO to CO 2 reaction rate, even though the later has a larger energy release.

3.5

Summary

In this section the premixed CMC equations are derived by two different methods. Then the volume averaged premixed CMC equations are derived based on the proposal of Smith (1994) with an explanation of when they are valid. A new approach is proposed and developed assuming uniform conditioned species within the reactor. Results were presented for two operating conditions, which will be used in later chapters where the

85

CMC model is combined with a CFD code. The important points from this chapter are the following.

At a given value of c, say c=0.5, as N varies the mass fractions of the species varies. With a small N almost all of the fuel is reacted to CO before oxidizing to CO 2 , with a large N about half of the fuel is completely reacted to CO 2 with minimal CO.

As N varies the reactor goes between the two limiting cases of a PFR and a PSR. A PFR has no diffusion and the reactions are kinetically limited. A PSR has infinite mixing rates, so the flame is again kinetically limited. As N approaches zero, the flame becomes a laminar flamelet, and eventually a PFR. A laminar flamelet has fast reaction rates relative to the diffusion, so the net burning rate is based on how fast the heat and free radicals diffuse into the flame zone. For all values of scalar dissipation between these limiting cases the micromixing will affect the reaction rates, giving different reaction pathways.

For large values of scalar dissipation the intermediate species are mixed with neighboring species and are “washed out”, preventing a build up of all intermediates except CH 3 .

Two alternative definitions for c were presented and compared to the definition used in this work.

When comparing conditioned results, they must use the same

definition of c.

It was shown that the laminar flamelet equation and the simplified CMC equation developed here are identical except for the c=1 boundary condition, but are developed with entirely different assumptions and are valid in different combustion regimes. The

86

flamelet model predicts extinction at a critical value of scalar dissipation, above which the flame will not remain ignited. The simplified premixed CMC model allows infinitely large scalar dissipation. The difference is that the CMC model assumes strong bulk mixing to continually mix hot products with the partially burned mixture to avoid extinction. This is an important distinction between the models.

Scalar dissipation and pressure have a large affect on emissions. Increasing N decreases CO while increasing NO. Increasing pressure decreases CO and NO.

The effects of different kinetic mechanisms was shown. The full GRI and full Miler-Bowman mechanisms gave very similar results, while the reduced Miller-Bowman gave small differences.

Small global mechanisms (1 to 5-steps) gave significantly

different results though data was not presented, showing the importance of detailed chemistry.

In turbulent premixed combustion multiple chemical time scales are present and must be modeled. In certain limiting cases, i.e. the flamelet regime, a single chemical time scale may be adequate. These multiple time scales may require revisiting the current method of placing turbulent premixed flames in regime diagrams, such as Figure 2-1.

87

Mixture fraction

Conditioned temperature

Temperature Mean temperature

Time

Figure 3-1: Illustration of a conditional average.

88

0.25

0.20

Species Mass Fraction

0.15

CH4 CO CO2 H20 O2

0.10

0.05

0.00 0.0

0.2

0.4

0.6

0.8

1.0

c

Figure 3-2: Premixed CMC with GRI2.11. Inlet temperature 300 K, pressure 1 atm., equivalence ratio 0.9 and scalar dissipation 10 1/sec.

89

0.25

0.20


0.15

CH4 CO CO2 H20 O2

0.10

0.05

0.00 0.0

0.2

0.4

0.6

0.8

1.0

c

Figure 3-3: Premixed CMC with GRI2.11. Inlet temperature 300 K, pressure 1 atm., equivalence ratio 0.9 and scalar dissipation 1,000 1/sec.

90

0.25

0.20


0.15

CH4 CO CO2 H20 O2

0.10

0.05

0.00 0.0

0.2

0.4

0.6

0.8

1.0

c based on Temperature

Figure 3-4: Premixed CMC with GRI2.11. Inlet temperature 300 K, pressure 1 atm., equivalence ratio 0.9 and scalar dissipation 10 1/sec. C based on temperature ratio.

91

0.25

0.20


0.15

CH4 CO CO2 H20 O2

0.10

0.05

0.00 0.0

0.2

0.4

0.6

0.8

1.0

c basd on CO2

Figure 3-5: Premixed CMC with GRI2.11. Inlet temperature 300 K, pressure 1 atm., equivalence ratio 0.9 and scalar dissipation 10 1/sec. C based on CO 2 ratio.

92

6.00E-03

5.00E-03


4.00E-03

CH2O C2H4 C2H6 3.00E-03

H2 NO O OH

2.00E-03

1.00E-03

0.00E+00 0.0

0.2

0.4

0.6

0.8

1.0

c


93

6.00E-03

5.00E-03


4.00E-03

CH2O C2H4 C2H6 3.00E-03

H2 NO O OH

2.00E-03

1.00E-03

0.00E+00 0.0

0.2

0.4

0.6

0.8

1.0

c


94

1.00E-03

8.00E-04


6.00E-04

CH3 C2H2 H HO2 H2O2

4.00E-04

2.00E-04

0.00E+00 0.0

0.2

0.4

0.6

0.8

1.0

c


95

1.00E-03

8.00E-04


6.00E-04

CH3 C2H2 H HO2 H2O2

4.00E-04

2.00E-04

0.00E+00 0.0

0.2

0.4

0.6

0.8

1.0

c


96

1.20E-04

1.00E-04


8.00E-05

CH2 CH2CO CH3OH

6.00E-05

C2H5 HNCO NO2

4.00E-05

2.00E-05

0.00E+00 0.0

0.2

0.4

0.6

0.8

1.0

c


97

1.20E-04

1.00E-04


8.00E-05

CH2 CH2CO CH3OH

6.00E-05

C2H5 HNCO NO2

4.00E-05

2.00E-05

0.00E+00 0.0

0.2

0.4

0.6

0.8

1.0

c


98

5.00E-05

4.00E-05


3.00E-05 CH2OH CH3O C2H3 HCCO HCNO HCO N2O 2.00E-05

1.00E-05

0.00E+00 0.0

0.2

0.4

0.6

0.8

1.0

c


99

5.00E-05

4.00E-05

3.00E-05 Species Mass Fraction

CH2OH CH3O C2H3 HCCO HCNO HCO N2O

2.00E-05

1.00E-05

0.00E+00 0.0

0.2

0.4

0.6

0.8

1.0

c


100

0.05

0.04

CH4 Mass Fraction

0.03 1000 800 600 400 200 130 0.02

0.01

0.00 0.0

0.2

0.4

0.6

0.8

1.0

c

Figure 3-14: Premixed CMC with GRI2.11, CH 4 versus scalar dissipation, 1/sec. Inlet temperature 300 K, pressure 1 atm. and equivalence ratio 0.9.

101 0.06

0.05

CO Mass Fraction

0.04

1000 800 600

0.03

400 200 130

0.02

0.01

0.00 0.0

0.2

0.4

0.6

0.8

1.0

c

Figure 3-15: Premixed CMC with GRI2.11, CO versus scalar dissipation, 1/sec. Inlet temperature 300 K, pressure 1 atm. and equivalence ratio 0.9.

102

5.00E-03

4.50E-03

4.00E-03

3.50E-03

OH Mass Fraction

3.00E-03 1000 800 600

2.50E-03

400 200 130

2.00E-03

1.50E-03

1.00E-03

5.00E-04

0.00E+00 0.0

0.2

0.4

0.6

0.8

1.0

c

Figure 3-16: Premixed CMC with GRI2.11, OH versus scalar dissipation, 1/sec. Inlet temperature 300 K, pressure 1 atm. and equivalence ratio 0.9.

103

6000

5000

C Equation Source Term

4000

1000 800 600

3000

400 200 130

2000

1000

0 0.0

0.2

0.4

0.6

0.8

1.0

c

Figure 3-17: Premixed CMC with GRI2.11, c equation source term versus scalar dissipation, 1/sec. Inlet temperature 300 K, pressure 1 atm. and equivalence ratio 0.9.

104 0.25

0.20


0.15

CH4 CO CO2 H20 O2

0.10

0.05

0.00 0.0

0.2

0.4

0.6

0.8

1.0

c

Figure 3-18: Premixed CMC with full Miller-Bowman. Inlet temperature 300 K, pressure 1 atm., equivalence ratio 0.9 and scalar dissipation 10 1/sec.

105 6.00E-03

5.00E-03


4.00E-03

CH2O C2H4 C2H6 3.00E-03

H2 NO O OH

2.00E-03

1.00E-03

0.00E+00 0.0

0.2

0.4

0.6

0.8

1.0

c


106 1.00E-03

9.00E-04

8.00E-04

7.00E-04


6.00E-04

CH3 C2H2 5.00E-04

H HO2 H2O2

4.00E-04

3.00E-04

2.00E-04

1.00E-04

0.00E+00 0.0

0.2

0.4

0.6

0.8

1.0

c


107

1.20E-04

1.00E-04


8.00E-05

CH2 CH2CO 6.00E-05

C2H5 HNCO NO2

4.00E-05

2.00E-05

0.00E+00 0.0

0.2

0.4

0.6

0.8

1.0

c


108 5.00E-05

4.00E-05


3.00E-05 CH2OH CH3O C2H3 HCCO HCNO HCO N2O 2.00E-05

1.00E-05

0.00E+00 0.0

0.2

0.4

0.6

0.8

1.0

c


109

6000

5000

C Equation Source Term

4000

3000

SC

2000

1000

0 0.0

0.2

0.4

0.6

0.8

1.0

c


110

0.25

0.20


0.15

CH4 CO CO2 H2O O2

0.10

0.05

0.00 0.0

0.2

0.4

0.6

0.8

1.0

c

Figure 3-24: Premixed CMC with reduced Miller-Bowman. Inlet temperature 300 K, pressure 1 atm., equivalence ratio 0.9 and scalar dissipation 14 1/sec.

111 6.00E-03

5.00E-03


4.00E-03

CH2O H2 3.00E-03

NO O OH

2.00E-03

1.00E-03

0.00E+00 0.0

0.2

0.4

0.6

0.8

1.0

c


112 1.00E-03

8.00E-04


6.00E-04

CH3 H HO2

4.00E-04

2.00E-04

0.00E+00 0.0

0.2

0.4

0.6

0.8

1.0

c


113 1.20E-04

1.00E-04


8.00E-05

6.00E-05

CH2

4.00E-05

2.00E-05

0.00E+00 0.0

0.2

0.4

0.6

0.8

1.0

c


114 5.00E-05

4.00E-05


3.00E-05

N2O

2.00E-05

1.00E-05

0.00E+00 0.0

0.2

0.4

0.6

0.8

1.0

c


115

6000

5000


4000

3000

Sc

2000

1000

0 0.0

0.2

0.4

0.6

0.8

1.0

c


116

0.03

0.02

CO Mass Fraction

0.02

1 atm. 2 atm. 4.7 atm. 7 atm.

0.01

0.01

0.00 0.0

0.2

0.4

0.6

0.8

1.0

c

Figure 3-30: Premixed CMC with GRI2.11, CO versus pressure, atm. Inlet temperature 603 K, scalar dissipation 350 1/sec and equivalence ratio 0.529.

117

0.09

0.08

0.07

CO2 Mass Fraction

0.06

0.05 1 atm. 2 atm. 4.7 atm. 7 atm. 0.04

0.03

0.02

0.01

0.00 0.0

0.2

0.4

0.6

0.8

1.0

c

Figure 3-31: Premixed CMC with GRI2.11, CO2 versus pressure, atm. temperature 603 K, scalar dissipation 350 1/sec and equivalence ratio 0.529.

Inlet

118

0.24

0.22

0.20

O2 Mass Fraction

0.18 1 atm. 2 atm. 4.7 atm. 7 atm. 0.16

0.14

0.12

0.10 0.0

0.2

0.4

0.6

0.8

1.0

c

Figure 3-32: Premixed CMC with GRI2.11, O2 versus pressure, atm. Inlet temperature 603 K, scalar dissipation 350 1/sec and equivalence ratio 0.529.

119 3.00E-03

2.50E-03

NO Mass Fraction

2.00E-03

1 atm. 2 atm.

1.50E-03

4.7 atm. 7 atm.

1.00E-03

5.00E-04

0.00E+00 0.0

0.2

0.4

0.6

0.8

1.0

c

Figure 3-33: Premixed CMC with GRI2.11, NO versus pressure, atm. Inlet temperature 603 K, scalar dissipation 350 1/sec and equivalence ratio 0.529.

120 1.80E-04

1.60E-04

1.40E-04

NO2 Mass Fraction

1.20E-04

1.00E-04 1 atm. 2 atm. 4.7 atm. 7 atm. 8.00E-05

6.00E-05

4.00E-05

2.00E-05

0.00E+00 0.0

0.2

0.4

0.6

0.8

1.0

c

Figure 3-34: Premixed CMC with GRI2.11, NO 2 versus pressure, atm. temperature 603 K, scalar dissipation 350 1/sec and equivalence ratio 0.529.

Inlet

121 1.00E-05

9.00E-06

8.00E-06

7.00E-06

N2O Mass Fraction

6.00E-06

1 atm. 2 atm.

5.00E-06

4.7 atm. 7 atm.

4.00E-06

3.00E-06

2.00E-06

1.00E-06

0.00E+00 0.0

0.2

0.4

0.6

0.8

1.0

c

Figure 3-35: Premixed CMC with GRI2.11, N 2 O versus pressure, atm. temperature 603 K, scalar dissipation 350 1/sec and equivalence ratio 0.529.

Inlet

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Chapter IV: Fluent Reacting Flow Results 4.1

Introduction

Before comparing the premixed CMC method to experimental data a commercially available CFD program was used to model the test case using the eddy dissipation model (EDM) combustion model as a benchmark for the premixed CMC method. The purpose of this step is to use a state of the art commercial code to develop a baseline for comparison to the new combustion model developed in Chapter III. Chapter V uses the same commercial CFD code with the new combustion model and the same mesh.

The CFD program used is Fluent 6.0.12, which is a general-purpose 3-

dimensional finite volume reacting flow code with adaptive grid capability (Fluent, 2001).

4.2

Test Problem

The test problem used is an experiment by El Banhawy et al. (1983) that consists of a flame stabilized by a backward facing step. Premixed natural gas (94% CH 4 ) and air are flowed across the backward facing step, which establishes a recirculation zone that stabilizes the flame. Figure 4.1 is a schematic of the reactor configuration showing the general location of the recirculation and flame zones along with the reactor dimensions. A flame sheet is anchored to the lip of the step, from which it progresses across the free portion of the duct. The flame sheet touches the wall at a specific axial location, and matching this “touching distance” is a challenging test of the various modeling approaches. The test chamber is a rectangular channel, 0.47 m long, 0.04 m high and 0.157 m wide with step heights of 0.01 and 0.02 m. The test chamber walls are watercooled. All measurements are taken at the center of the channel to ensure the flow is 2dimensional, i.e. away from side wall effects. Three equivalence ratios are tested, 0.77, 0.90 and 0.95. The Reynolds number based on the large step height, and inlet velocity

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and temperature is approximately 13,500 and the airflow rate is 125 kg/hr giving an inlet velocity of 10.5 m/sec at 300 K. Velocity measurements of non-reacting (not shown in their paper) and reacting flows are measured with a laser-Doppler anemometer. Both mean and RMS axial velocities are reported. Temperature measurements are taken with a bare thermocouple and have an error of up to 40 K due to radiative heat transfer between the thermocouple and the cold walls. Species measurements are made by extractive sampling and process gas analyzers.

The instruments included a flame ionization

detector for unburned hydrocarbons (UHC), infrared detectors for CO and CO 2 and a paramagnetic analyzer for O 2 . CO, CO 2 and O 2 samples had the UHC and H 2 O removed before being measured. The UHC samples retained the H 2 O .

This test case is chosen because it has many of the characteristics of LPM GT combustors, which have lean, premixed fuel and air burning in a flame that is stabilized by a recirculation zone. This is generally easier to model than an actual burner, which has a complex 3-dimensional, swirl induced recirculation zone and normally burns natural gas. The purpose of this research is to show the potential of the premixed CMC combustion model for modeling LPM GT combustors, so simplifications of the fluid mechanics that do not greatly affect the combustion are desirable. These simplifications reduce the errors caused by the turbulence model and reduce runtimes. The fuel is modeled as pure methane instead of natural gas. This case has a simple geometry with clean boundary conditions, reducing the possibility of numeric errors.

4.3

Backward Facing Step Results

Calculations are made with Fluent using the built in 1-step and 3-step methane kinetic mechanisms for an equivalence ratio of 0.9 and a step height of 0.02 m (expansion ratio of 2.0). Four different turbulence models ( κ − ε , RNG, realizable and RSM) and several grids are used to determine their affect on the solution. The grid resolution near

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the walls is important for turbulence models to work correctly. The non-dimensional distance to the center point of the cell adjacent to the wall (in the normal direction) is Y + , define as

Y + = ρC1µ/ 4 k 1 / 2 y / µ

4-1

where ρ is the density at that point, C µ is a model constant defined in Chapter III, κ is the turbulent kinetic energy at that point, y is the normal distance to that point and µ is the viscosity at that point. The density, viscosity and turbulent kinetic energy vary as the temperature and velocity change through out the reactor. This variation is not known a priori, so several grids are made and tested for both of the wall boundary conditions used. Grids that are coarse near the wall used the non-equilibrium wall functions as wall boundary conditions for the turbulence models, requiring Y + to be in the range of 30 – 60. Fine grids used the 2-layer wall functions as wall boundary conditions for the turbulence models, requiring Y + to be in the range of 1-5 (Fluent, 2001). All runs performed here are done with structured grids with constant normal distance to the center of the first cell along the walls. This does not always give Y + in the required range, but since the goal of this work is to develop and validate the premixed CMC combustion model, the small error along the walls is acceptable. The κ − ε turbulence model gave the poorest match of velocity and flame location (temperature and species), so no results are shown. The RNG and realizable turbulence models are better than the κ − ε , but still not as good as the RSM, so only results for the RSM are presented. As was discussed in Chapter II, the κ − ε based turbulence models assume isotropic turbulence, which is not true for this problem, so it is not surprising that they do not give a good comparison to the data. The current premixed CMC method only allows adiabatic walls while the experiment has walls that are externally cooled to approximately 300K. Fluent runs are made using the EDM with both adiabatic wall

125

boundary conditions and a constant wall temperature boundary condition (300 K) to show the influence of the wall boundary condition on the results.

4.3.1

Adiabatic Walls

In the first attempt the reaction rates are calculated using only chemical kinetics, with the result that the flame burns extremely fast. Most of the fuel is burned at the entrance to the channel. Next, the reaction rates are calculated using the EDM, as described in Chapter II, which gives a better match to the data. Figure 4-3 shows contour plots of axial velocity m/sec, the rms of the axial velocity m/sec, temperature K, unburned hydrocarbons (UHC) as mole percentage-wet, O 2 as mole percentage-dry,

CO 2 as mole percentage-dry and CO as mole percentage-dry for the 1-step EDM combustion model using a 110x20 grid with the RSM turbulence model (the O 2 , CO and

CO 2 had the UHC removed). The y-axis is expanded 200% for better viewing and to match the way the experimental data are plotted. Figure 4-2 shows the same variables from the experimental data of El Banhawy et al. (1983) using the same contour scales. The x-axis of the experimental data is in millimeters, while the modeled results are in meters. The experimental data are only plotted for the first 0.38 m of the chamber, while the computed results are presented for the entire 0.47 m length of the chamber. The reattachment location of the recirculation zone matches the data well, but the peak velocity is about 6 m/sec lower, which is explained by the temperature and species profiles. From the temperature and species plots it is seen that the modeled flame starts at about the same location as the experimental data, but burns much slower. Here the starting location of the flame is defined as the point the temperature starts to increase and the ending point is defined as the point the temperature stops increasing. The flame thickness is defined as the distance required for the flame to reach its peak temperature. Because the flame burns slower, some of the fuel exits the reactor before it is fully burned, so less energy is released within the reactor, which lowers the temperature and

126

peak velocity. The EDM combustion model is based on the turbulence of the flow and the poor match would indicate the turbulence model is inadequate. Looking at the modeled data it is seen that the turbulence intensity matches well in the recirculation zone (where the density is nearly constant), but is under predicted down stream (where there is large density changes), hence the poor match of the species. This is due to the fact that the turbulence models and their wall functions were developed for incompressible flows and are missing terms to account for variable density, as was discussed in Chapter II. Another reason for the poor match is the 1-step mechanism, which leaves out all the important intermediate steps and species. In a full mechanism, each step has its own rate, in a 1-step global mechanism there is only one rate. With only one rate in the model it is impossible to get the correct flame thickness. The model shows steep velocity gradients along the walls due to the no slip condition, which are not seen on the experimental data due to the coarseness of the velocity measuring instrument. The peak temperature of the modeled results is about 200 K higher than the experimental results because the model uses adiabatic walls, while the experimental has water-cooled walls.

This higher

temperature is only in the upper part of the reactor, the average exit temperature is lower then the data (since not all the fuel is fully burned), hence giving a lower average exit velocity. The results of the EDM model may be improved by adjusting the constants in the model. This was not pursued because there is no physical justification for this and no guarantee that these “improved” constants will be an improvement for other conditions.

The same conditions modeled with a 200x60 grid using the 2-layer wall functions for the turbulence model boundary conditions (results not presented) starts to burn a little later but burned slightly faster, reaching about the same conditions at the end of the chamber. The main difference is along the lower wall where the combustion starts near the inlet to the chamber, again showing a problem with the EDM model.

One

explanation for this is that the EDM model was developed for coarse grids using wall functions as wall boundary conditions and is not valid for very fine grids near the wall. The fine grid using the RNG turbulence model gave higher turbulence intensities with a

127

corresponding faster burning rate. The faster burning rate gave a higher temperature at the exit, causing a larger acceleration of the flow and hence larger velocities at the exit.

Figure 4-4 shows modeled results using the coarse grid and RSM turbulence model with the 3-step mechanism and the EDM model. The results look very similar to the 1-step case, with the flame starting at the same location and burning slightly faster. The faster burning flame gives slightly larger exit velocities due to the higher peak temperature.

Figure 4-5 shows modeled results using the coarse grid and RSM turbulence model with Fluent’s premixed combustion model (Fluent, 2001). The premixed model uses the RPV equation like the CMC method, but closes the reaction rate term with a flamelet type model, no variance equation is solved and a PDF of the progress variable is not used. The model gave a better match of the data then the EDM and overall a reasonable match, the main problem is that the flame burned significantly slower then the data shows. There are 8 input parameters that can be adjusted that might improve the match, these are turbulent flame speed constant, turbulent length scale constant, stretch factor coefficient, turbulent Schmidt number, adiabatic flame temperature, critical rate of strain, laminar flame speed and molecular heat transfer coefficient. This is not attempted, since it defeats the purpose of a predictive model. Another drawback of this model is that it does not provide species information, but does provide the Damköhler and turbulent Reynolds numbers.

4.3.2

Non-Adiabatic Walls

Figure 4-6 shows modeled results for the coarse grid using the RSM turbulence model and the 1-step mechanism with the walls held at 300K, to match the data. As expected, the heat loss reduces the burning rate, giving a lower temperature and an exit velocity about 4 m/sec lower. The temperature profile has steep gradients near the wall

128

(due to the heat loss) similar to the experimental data. Note: the data do not show the wall temperature as 300 K due to measurement resolution.

4.4

Summary

These cases are run for two reasons, to show the performance of a popular commercially available CFD combustion model as a baseline to compare the premixed CMC model with and to show the general differences between an adiabatic and nonadiabatic wall problem. The ability of a 1-step and 3-step EDM combustion model to model a turbulent premixed problem is very poor. This poor match is due to neglecting most of the species and reaction steps in the burning process and because the reaction rates are based solely on turbulence quantities. Since the reaction rates are tied directly to the turbulence model, all limitations of the turbulence models will adversely affect the combustion model. Also, the way the reaction rates are calculated near the wall is especially poor. It was seen that the RSM is slightly better than the other, first order turbulence models, but requires slightly longer too solve. This is the expected result based on the discussion of Chapter II, which indicates that the RSM contains more of the physics that are needed to replicate recirculating flow patterns. All the turbulence models suffer when modeling variable density flows. It was also seen that the fine grid with the 2-layer wall functions gave similar results as the coarse grid with the non-equilibrium wall functions, with the main difference near the walls. In the next chapters the coarse grid with the non-equilibrium turbulence wall boundary conditions will be used, since it has similar accuracy and shorter runtimes then the finer mesh and 2-layer wall model.

Also shown was the affect a cold non-adiabatic wall has on the bulk flow compared to an adiabatic wall. This will be discussed more in the next chapter.

129

Length, 0.47 m Recirculation zone Outlet, 0.04 m Inlet, 0.02 m

Flame zone Water cooled walls

Figure 4-1: Schematic of the 2-D backward facing step reactor showing flame and recirculation zones. Note: horizontal and vertical dimensions not to scale.

130

Velocity

Velocity RMS

Temperature

CH 4

O2

CO 2

CO

Figure 4-2: El Banhawy et al. (1983) data, equivalence ratio=0.9, axial velocity m/sec, axial velocity rms, temperature K, unburned hydrocarbons-wet, O 2 -dry, CO 2 -dry and CO-dry, species are mole percentages ( O 2 , CO 2 and CO have the UHC removed). Reprinted by permission of Elsevier Science from Premixed, Turbulent Combustion of a Sudden-Expansion Flow, by Y. El Banhawy et al. Combustion & Flame, Vol No 50, pp 153-165, Copyright 1983 by The Combustion Institute.

131

0.04 7

Y

Velocity

0.1

Y

0.03

2

0.02

16

0.2

0.3

X

0.4

20

10

1.5

1

0.5

1

0.01 0

18

11

1.5

0.04

24

16 14

10.5

0

20

11

5 10 10.

0.01

14

9

5

-00.5

0.02

0

Velocity RMS

-1

0.03

0.5

0

1

0.1

0.2

0.3

X

0.4

0.04

Temperature

1900

Y

0.03

1800

0.02

600 301

0.01 0

1000

0

600

0.1

0.2

0.3

X

0.4

0.04

CH 4

1 0.03

0.5

5

Y

3

0.02

7

8

7

0.01 0

0

0.1

8

5 8.

0.2

0.3

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5

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O2

6

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4.54 10

14

16

0.01 0

18

0

0.1

0.2

0.3

X

0.4

0.04

9.5 0.03 Y

CO 2

0.02

6

4

2

1

0.01

2

0.01 0

0

0.1

0.2

X

0.3

0.4

Figure 4-3: Fluent 1-step EDM, Reynolds stress model equivalence ratio=0.9 axial velocity (m/sec), axial velocity rms (m/sec), temperature K, CH 4 -wet, O 2 -dry and CO 2 dry, species are mole percentages, ( O 2 and CO 2 have the CH 4 removed).

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-0.5

0

0.04

Y

Velocity

.5 0-0

-1

0.03

10 .5 10

0.02

5

0.4

11

10

1.5 2

1

0.5

0.01 0

20

0.3

X

1.51

Y

0.2

2

0.03 0.02

16

14

0.1

0.04

Velocity RMS

18

11 10.5

0

18

16

14

0.01 0

10.5

11

9

7

5

0.5

1

0

0.1

0.2

0.3

X

0.4

0.04

Temperature

Y

0.03 1400

0.02

1800

600

1000

0.01

1000

301

0

0

600

0.1

0.2

X

0.3

0.4

0.3

0.4

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5

0.02

7

8

8.5

CH 4

Y

0.03

0.01 0

0

5

7

0.1

0.2

X

0.04 4

Y

0.03

O2

4.5

5 14

0.02

18

16

0.01

16

18

0

0

0.1

0.2

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0.4

0.04 8

CO 2

Y

0.03 2

0.02

1

2

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0.01

0

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X

0.3

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0.2

0.3 00.4.5

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1

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0.04

CO

6

4

0

0.1

0.2 0.4

0.5

0.1 0.3

0.2

X

0.3

0.4

Figure 4-4: Fluent 3-step EDM, Reynolds stress model equivalence ratio=0.9 axial velocity (m/sec), axial velocity rms (m/sec), temperature K, CH 4 -wet, O 2 -dry, CO 2 -dry and CO-dry, species are mole percentages, ( O 2 , CO 2 and CO have the CH 4 removed).

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14

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20 18 16 14

9 0.1

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20

5

24

30 14

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18 16

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24 24

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20

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10.5

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18 16

9 11 0.4

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1124

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20

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11

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79

24

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10 .5

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14

79

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18

5

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11

Velocity

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10 10.57

-1

-1 -1

0 0-0.5

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0.04

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0

1

0

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1800

301

Temperature

600

0

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1400

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301

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6 00

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Progress variable

1900

1000

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1

1

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0.02

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7

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19

21 21

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15 21 17

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0

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9

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Damköhler number

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13

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11

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0.04

100

0.03 100

0

0.02

30

Turbulent Reynolds number

400

200 10 0

0.01 0

0

0.1

Figure 4-5: Fluent with built-in premixed combustion model, RSM turbulence model, equivalence ratio=0.9, axial velocity (m/sec), axial velocity rms (m/sec), temperature K, progress variable c, Damköhler number and turbulent Reynolds number.

134

5

1 0 11

9 .5 10

0.01

7 0

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Y

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0.5

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0.5

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0.1

0.2

0.02

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1000301

600

0.01 0

600

1400

00 14 00 301 10

0.03

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X

600

0.04

Y

10 0.3

1

0.01

Temperature

16

14

0.1

1

0.03

16

11

0.04

Velocity RMS

20

18 14

0.02

0

9

10.5

7

0 10. 5

Y

0.03

Velocity

.5 -0

-1

0.04

1000

301 0.1

0

600

0.2

0.3

X

0.4

0.04

CH 4

Y

0.03 35

0.02

0.5

1

7

8

0.01 0

7

0

8.5 8

0.1

0.2

5 3

0.3

X

0.4

0.04 4

0.03

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10

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14

O2

18

0.02 0.01

16

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0

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0.2

18

14

0.3

X

0.4

0.04

Y

0.03

CO 2

86

0.02

4 2

1

0.01 0

0

0.1

01 0.

1

4

2

0.2

X

0.3

0.4

Figure 4-6: Fluent 1-step EBU, Reynolds stress model equivalence ratio=0.9 axial velocity (m/sec), axial velocity rms (m/sec), temperature K, CH 4 -wet, O 2 -dry and CO 2 dry, species are mole percentages, ( O 2 and CO 2 have the CH 4 removed), with wall temperature = 300 K.

135

Chapter V: Combination of CFD and The Premixed CMC Method

5.1

Introduction

Here the uniform conditioned species version of the premixed CMC method (equation 3.16) developed in Chapter III is combined with the Fluent CFD program. This is a reduced version of the full premixed CMC method, which assumes strong bulk mixing to give uniform conditioned species within the reactor.

This causes the

conditioned velocity term to drop out of the equations. The error terms discussed in section 3.3.4 are assumed negligible. The CMC equations are solved offline for a range of scalar dissipation and saved in tabular format. The simplified CMC model is validated by comparison to experimental data.

The c equation and the variance of c equation (equations G6 and G13 with H13) are added to Fluent version 6.0.12 (Fluent, 2001) as scalar equations using the built in User Defined Function (UDF) capability. User defined functions are written to calculate the diffusivity and the source terms for the two scalar equations. Every iteration Fluent passes c and its variance to the CMC function, which are used to calculate the PDF of c and the scalar dissipation for each computational cell. These are also used to lookup the density and the c equation source term for each cell from the tabular data, which are passed back to Fluent for use in the next iteration. This is continued until convergence is achieved. A structured 2-D grid of 110 axial and 20 vertical cells is used, the same as was used in Chapter IV with the RSM and RNG turbulence models and the nonequilibrium wall boundary conditions.

5.2

Test Problem

The same test case is used as was used in Chapter IV with equivalence ratios of 0.77, 0.9 and 0.95 and step heights of 0.01 and 0.02 m. The full GRI2.11 kinetic

136

mechanism (Bowman et al., 1998) is used, as was used in Chapter III. The full MillerBowman (Miller and Bowman, 1989), a 16 specie reduced Miller-Bowman (Nicol, 1995) and a 2-step global mechanism (Nicol, 1995) are also used for comparison.

5.3

Probability Density Function

A Probability Density Function (PDF) of c is required to convert between conditioned and unconditioned variables. For every cell in the CFD mesh a PDF is calculated from the current value of c and its variance using the assumption that the PDF is shaped as a beta function. Pierce (2001) was not sure the beta function PDF is good for non-conserved scalars, such as the mass fraction, but it is used here due to lack of data to determine a better PDF shape. Equation G6 is used to calculate c, which is the Favre averaged form of equation 3.12.

µ ∂ ρ ~c + ∇ ⋅ [ρ~ u~c ] − ∇ ⋅ ( t ∇~c ) = − ∂t σc

~h ] [∑ ρ ω i f ,i i

∆h sad − u

G6

Equation G13 is used to calculate the variance with the source term from Appendix H, equation H13. ~ε µ ∂ρ~c ′′ 2 + ∇ ⋅ [ρ~ u ~c ′′ 2 ] − ∇ ⋅ [ t ∇~c′′ 2 ] = C c1µ t (∇~c ) 2 − C c 2 ρ ~ ~c ′′ 2 κ ∂t σc ~ ′′h ] [ρ ~c′′ω −2

NS

∑ i

∑

i

G13

f ,i

i

∆h sad − u ~c′′ω ~ ′′h i f ,i h s ad − u

~h ~c′′ 2 NS ω i f ,i { s = C3 ~ } ∑ T ( c + α) h ad − u i

H13

137

Bradley et al. (1988) suggests that Equation H13 should change sign at c=0.7, which this model does not. They state that below about c=0.7 c′′ and ω′′ are positively correlated, since an increase in c above the mean would increase the heat release rate. While above c=0.7 the opposite is true. Figure 3-17 shows the source term for equation G6. For the smaller values of scalar dissipation the source term peaks at about c=0.7, following Bradley et al. (1988). As the scalar dissipation increases, this inflection point increases, an improved version of equation H13 should be able to model this. Equation H13 has several assumptions in the derivation that will require experimental or DNS data (which so far does not exist) to validate. For the runs presented here C 3 =100 in equation H13, unless stated other wise. In equation H13 α =

Tu , the unburned temperature was (Tad − Tu )

300 and the adiabatic temperature was 2100, giving α = 0.1667. 5.4

Backward Facing Step Results

The current formulation of the CMC method only allows for adiabatic walls, while the experimental data has water-cooled walls. The results of Chapter IV with adiabatic and non-adiabatic walls (using Fluent’s built in EDM reaction model) will be used to provide a qualitative comparison between the adiabatic CMC model and the nonadiabatic test data. Chapter VII discusses a proposed non-adiabatic CMC model.

Figure 4-2 shows contour plots of axial velocity (m/sec), the rms of the axial velocity (m/sec), temperature (K), unburned hydrocarbons (UHC) as mole percentagewet, O 2 as mole percentage-dry, CO 2 as mole percentage-dry and CO as mole percentage-dry from El Banhawy et al. (1983) for an equivalence ratio of 0.9 and a step height of 0.02 m. Figure 5-1 shows the same variables calculated from the premixed CMC method using the full GRI2.11 kinetic mechanism with the RSM turbulence model and a 110x20 grid. The velocity contours show similar reattachment locations and

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recirculation zones as the experimental data. The flow acceleration from the reactions starts slightly earlier and reaches a higher peak velocity, part of this difference is due to the adiabatic walls in the model and water-cooled walls in the experimental data. Chapter IV (Figures 4-3 and 4-6) shows that the non-adiabatic case had a peak velocity about 4 m/sec lower than the adiabatic case and started to burn slightly later. The heat loss from the walls gives a lower peak temperature, which gives a smaller density change through the flame and from continuity a smaller velocity change. Scaling the velocities seen here by this amount brings them closer to the data, though the model still over predicts the peak velocity. The turbulence intensity in the recirculation zone closely matches the data, but down stream it is much lower, as was seen in the Chapter IV results. One explanation for this is that there is no direct link from the reactions to the turbulence generation. The CMC model provides a direct link for the turbulence (smallscale mixing) to feed into the reaction rates, but there is not a source term in the turbulent kinetic energy equation for the reactions to generate turbulence, which seems physically possible. There are three ways that the reactions can effect the turbulence, dilatation (gas expansion), local pressure gradients and viscosity gradients. Dilatation is the negative of the relative rate of change in the local density due to combustion. The local pressure gradients are caused by the combustion waves and the viscosity gradients are caused by the local temperature gradients. The last two affects will increase the local turbulence. It is not known what effect the dilatation has on the turbulence in channel flows, some research suggests that it decreases the turbulence in jet flows. Current turbulence models do not account for these effects, El Tahry (1983) suggested additional terms for a turbulence model in reacting flows, but did not add them to a model or compare them to data. This topic is discussed in detail later in this chapter where corrections to the turbulence models to account for this are explored.

The peak temperature is about 200 K higher than the data, which has heat loss through he walls. In Chapter IV it was seen that the non-adiabatic walls reduced the peak temperature by about 200 K, so overall the temperature is a good match.

The

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temperature increases slightly faster in the flame then the data shows, part of this is due to no heat loss in the model, which was seen in the results of Chapter IV. The UHC (plotted here as CH 4 ), O 2 and CO 2 match the data very well, except for the slightly faster burning rate (narrower flame zone) late in the flame. The CO gradients are in the correct location, but the magnitude is high in the modeled results. The experimental data has a peak CO concentration of about 0.50%, while the calculated peak is about 3.6%. (Note the decimal points on the data plot are not very clear and the peak could be 5.0%, but this seems too high for this lean of a flame). One reason the CO may be high is from the value of scalar dissipation calculated. Figure 3-15 showed that as scalar dissipation is increased the CO mass fraction decreases. Here N was calculated at each CFD cell with the following expression. ~ε < N | ζ >≅< N >= 2 ~ ~c′′ 2 κ

5.1

Here κ , ε and the variance ( ~c′′ 2 ) are the unconditioned values and will give the unconditioned value of N. At this point there is no experimental or DNS data to relate the conditioned and unconditioned values of N, so here they are assumed equal. Alternative relations for calculating N will be explored later.

Overall the premixed CMC method modeled the experimental data very well, once the affects of the adiabatic walls are extrapolated to the non-adiabatic walls of the test data. A significant error appears to be the turbulence model and its inability to account for the turbulence generated by the reactions (see the discussion above). Other sources of errors may be from the calculation of the conditioned scalar dissipation, the shape of the beta function PDF and the source term in the variance equation. These issues will be explored next.

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Figure 5-2 shows the same conditions as Figure 5-1 using the realizable turbulence model. The turbulence intensity in the recirculation zone is higher then with the RSM, giving a closer match to the data. After the flame there is more turbulence generated then was seen in the RSM model, though still well below the experimental data. The temperature and species plots show the flame burns a little slower then the RSM model giving a closer match to the data.

Figure 5-3 is for the same conditions as Figures 5-1 and 5-2 using the RNG turbulence model. Here the turbulence intensity in the recirculation zone and the flame zone is slightly higher than the realizable model and is a better match to the data. All three turbulence models show a large turbulence intensity along the wall in the aft part of the reactor.

This is believed to be an error with the wall functions used with the

turbulence models. These wall functions were developed for incompressible flows and are not as accurate for the large density gradients seen in reacting flows. The flame zone is about the same thickness as with the realizable model and overall gives the best match to the data of the three turbulence models used. In Chapter II is was hypothesized that the RSM, being a second order model would give better results then the first order models (RNG and realizable), but for reacting flows this appears not to be the case. The RSM also takes longer to solve because of the extra equations required. All remaining cases in Chapter V use the RNG model.

Figure 5-4 is for the same conditions as Figure 5-3 with the following terms added to the right side of the κ and ε equations (Jones and Whitelaw, 1980).

Sκ = −

µ t ∂ ρ ∂P 0.8ρ 2 ∂x i ∂x i

S ε = − C1

ε µ t ∂ ρ ∂P κ 0.8ρ 2 ∂x i ∂x i

As discussed in the end of Section 2.2.4, several authors have added these terms (or a variation) to one or both turbulence equations to account for variable density effects.

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This term has not been rigorously developed in the literature and has not been proven to account for variable density affects in all reacting flow cases. It was used here to show that current turbulence models are inadequate for reacting flows and no attempt will be made here to try and improve the turbulence models.

These terms increased the

turbulence generated in the flame, but still under-predicts the magnitude by a factor of two. The flame became slightly wider with the addition of these terms, probably because the change in turbulence affected the reaction rates.

Next the calculation of the scalar dissipation was explored. Figure 5-5 is for the same conditions as Figure 5-4 (with the additional turbulence term) and the scalar dissipation (calculated by equation 5.1) increased by a factor of 10.

This further

increased the turbulence intensity in the flame, but had the adverse effect of causing the flame to start burning sooner and slightly faster (narrower flame). As discussed above, the larger values of N decreased the peak CO magnitude in the flame (3.20% versus 3.62% for Figure 5-4), but it is still significantly higher then the data.

Figure 5-6a shows the same conditions as above with N increased by a factor of 20 everywhere. This had a minimal affect of the turbulence and flame location. The peak CO was decreased to 2.84%. Figure 5-6b shows the RPV, its variance, the scalar dissipation and HNO, NO, NO 2 and N 2 O mass fractions for the same conditions as Figure 5-6a. Figure 5-6c shows CH, HCN, CN, NCO, CN, NCO, O, H and OH mass fractions.

Experimental data are not available for comparison, but this shows an

advantage of the CMC model over other simpler models, namely the ability to predict intermediate species and radicals that are important to emission production.

Figure 5-7 is for the same conditions as the previous three figures with the scalar dissipation increased by a factor of 30. This caused the turbulence generated in the flame to decrease and the flame to start burning later then the previous case. Here the peak CO increased to 3.98%. These results show that the constant in equation 5.1 is not the reason

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the model will not do a better job matching the data. Equation 5.1 may need to be a function of additional variables to correctly predict the conditioned scalar dissipation and be conditioned on c.

To develop an improved model for the conditioned scalar

dissipation will require experimental and DNS data and an improved compressible turbulence model.

Figure 5-8 is for the same conditions as Figure 5-6a (with the scalar dissipation increased by a factor of 20) with the scalar dissipation calculated with the following relationship. < N | ζ >=< D(∇c) 2 | ζ >≅< D(∇c) 2 >≅ [(∇~c ) 2 + (∇~c′′) 2 ]µ t / ρ

5.2

This created less turbulence in the flame and caused the flame to start burning later. Figure 5-8b shows the calculated scalar dissipation for Figures 5-4, 5-5, 5-6a, 5-7 and 58. The first four cases have N calculated with equation 5-1 and have their largest values at the end of the flame next to the wall. It is most likely this is due to the wall functions used in the turbulence model and is not physically realistic.

The last case has N

calculated with equation 5-2 and does not show the increase near the wall. These five cases show the effect that scalar dissipation has on the solution and highlights the need for data to develop an improved model for calculating N.

Figure 5-9 is for the same conditions as Figure 5-6a with the source term in the c equation reduced by 20%. This caused the flame to start slightly later and gives a wider flame. This would give the same effect of using a narrower PDF. As seen in Figure 317, the source term for the c equation is a strong function of c in the flame, using a wider PDF to convert the conditioned value to the unconditioned value will tend to increase the source term by using more of the higher values in the averaging. The opposite will happen with a narrower PDF.

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Figure 5-10 is the same conditions as in Figure 5-6a with the constant multiplying the source term for the variance equation equal to 1.0, all other runs had this term set to 100.0. This is the constant C3 in equation H13. Reducing this constant will reduce the variance, giving a narrower PDF. As seen in Figure 5-9 a narrower PDF causes the flame to start later and burn a little slower. Experimental and/or DNS data of conditional variables will allow the model used for the variance equation source term (developed in appendix H) to be more thoroughly evaluated and if needed improved.

Figure 5-11 shows contour plots of temperature (K), unburned hydrocarbons (UHC) as mole percentage-wet, O 2 as mole percentage-dry, CO 2 as mole percentage-dry and CO as mole percentage-dry from El Banhawy et al. (1983) for an equivalence ratio of 0.77 and a step height of 0.02 m. Figure 5-12 shows the same variables (plus axial velocity (m/sec), the rms of he axial velocity (m/sec)) calculated from the premixed CMC method using the full GRI2.11 kinetic mechanism. The model predicts the flame starting later and burning faster than the data shows, similar to what was seen in the above results.

Figure 5-13 shows contour plots of temperature (K), unburned hydrocarbons (UHC) as mole percentage-wet, O 2 as mole percentage-dry, CO 2 as mole percentagedry and CO as mole percentage-dry from El Banhawy et al. (1983) for an equivalence ratio of 0.95 and a step height of 0.02 m. Figure 5-14 shows the same variables (plus axial velocity (m/sec), the rms of the axial velocity (m/sec)) calculated from the premixed CMC method using the full GRI2.11 kinetic mechanism. This gives an overall good match with the calculated flame starting slightly later and burning a little faster. Of the three equivalence ratios used, the richest case (0.95) matched the flame starting location and thickness the best. This is because the richer mixture burns faster, giving less time for the flame zone to lose heat to the wall, even though the temperatures are higher. For both of these cases the delta-rho term was added to the turbulence equations, C3=100.0,

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the scalar dissipation was calculated using equation 5.1 and was increased by a factor of 20.

Figure 5-15 shows results for the same case run in Figure 5-6a using the full Miller-Bowman (Miller and Bowman, 1989) mechanism. It starts to burn at about the same location as the data, giving a better match than the GRI2.11 cases, but burns much faster. Overall the match is not as good as the GRI2.11 cases. Chapter III showed that both of these mechanisms gave the same major species, but here it is seen that the differences in the intermediate species have a significant affect on the overall flow fields.

Figure 5-16 shows results for the same case run in Figure 5-6a using the reduced Miller-Bowman kinetic mechanism.

The reduced Miller-Bowman (Nicol, 1995)

mechanism starts to burn sooner and burns much faster, showing that by removing minor species, radicals and reactions steps the mechanism losses much of its accuracy. Cases were also run (not shown here) using a 2-step global mechanism (Nicol, 1995). These cases burned almost instantly at the inlet to the burner, giving a completely unrealistic solution.

This indicates that using greatly reduced and global mechanisms, like in

traditional CFD and PDF methods, adversely affects the results. This instant burning rate was also seen using Fluent’s 1 and 3-step global mechanisms and solving them by purely kinetics (no EDM). The reason the 1-step mechanism burned so fast is the lack of intermediate steps. As part of the fuel in one computational cell burns, the CH 4 proceeds directly to CO 2 giving off all the available heat. This large amount of heat causes other fuel molecules within the cell to ignite, which causes more fuel to ignite. This heat also convects and diffuses to the adjacent cells causing them to also burn instantly to CO 2 . This causes the flame to only be one cell thick and to continue to move forward until it reaches the entrance of the reactor. Mathematically this is seen by a high activation energy for the 1-step mechanism and a corresponding high reaction rate. The multi-step mechanisms have a different activation energy and reaction rate for each step, giving a

145

more realistic overall reaction rate and flame thickness. The 2 and 3-step mechanisms have a similar problem. With the larger mechanisms, as the CH 4 starts to burn in one computational cell it only burns part way to CO 2 giving off only a portion of the total heat available. These intermediate species and partial increase in temperature convect and diffuse to adjacent cells where the intermediate species continue to oxidize and give off heat.

This slower liberation of heat causes the flame to be wider and causes the

upstream movement of the flame to be balanced by the incoming flow velocity. For both of these cases the delta-rho term was added to the turbulence equations, C3=100.0, the scalar dissipation was calculated using equation 5.1 and was increased by a factor of 20.

5.5

Summary

The premixed CMC method is an improvement over the global mechanisms current CFD programs use with similar runtimes. The method gives reasonable velocity, temperature and species values throughout the burner while giving results for all species involved in the reaction. Use of a reduced mechanism gave significantly poorer results. A 2-step mechanism gave even poorer results, partially explaining the poor matches (seen in Chapter 4) from 1 and 3-step global EDM mechanisms that are built into Fluent. This is one of the main advantages of the premixed CMC method, it allows the use of very detailed chemistry with a small increase in computer resources. Another advantage of the premixed CMC method is that it calculates the reaction rates directly from the kinetic relationships (accounting for the affect of temperature fluctuations on the reaction rates) and accounts for the affects of the small-scale turbulence on the reaction rates. This avoids the ad hoc modeling of the EDM model, which requires different model constants for each type of flow.

The premixed CMC model also allows the size of the small scale mixing (scalar dissipation) to be varied independently of other parameters to show its affect on the

146

reactions. This is similar to the model of Tonouchi (1996) but in this case it does so with full chemical kinetic mechanisms, which other models cannot handle.

The RSM turbulence model does not give an advantage over the RNG and realizable models at modeling the effects of variable density due to combustion. This is an area that needs further research, which will also be applicable to other combustion models, namely a source term in the turbulent kinetic energy equation to account for the turbulence generated by combustion.

While the initial results are encouraging there are issues that must be addressed with the method, once improved compressible turbulence models are developed. First, a method to calculate the conditioned scalar dissipation is needed. experimental and DNS data of conditioned variables.

This will require

The assumption that the

conditioned variables are uniform needs to be validated with data. This has been done for some non-premixed cases, but no data exists for premixed flames. Experimental and DNS data are needed to validate and improve the source term for the variance equation. Another area that requires work is on the c and variance equations to determine if they require a term to account for the Favre averaging. The size of the error terms that are dropped in the derivation require further investigation, again experimental and DNS data are required. After these issues have been addressed the next step in the development of the method is to derive a non-adiabatic model, which will more accurately model real combustors having heat loss.

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24

20 0

0.04

Velocity RMS

30

20 18

16 14

11 5

0.01

14

30

5 7 10 11

7

0.02

14 10 24

24

18

9 .5 10

Velocity

9

11

10.5

20

-0.5 0

0.03

1

0

0.1

0.2

0.3

0.4

0.04 0.03 19 0 14000

0.02

0

30 0

19 00 1 400

600

0

0.1

18

100

Temperature

1 1 0 0800 0

0.01

0.2

00

0.3

0.4

0.04 0.03 0.02

1 7

1 53

0.01 0

8.5

0

0.5

8

0.1

7

0.2

1

CH 4

0.3

0.4

0.3

0.4

0.04 0.03

O2

4. 5 128

0.02

4.5

0.01

18

5

16

0

0

0.1

4.5 4 8 12 0.2

0.04 0.03 0.02

9.5 6 1

0.001

CO 2

8

0.01 0

2

0. 00 1 0

0.1

4

9.5 6

8

1

0.2

0.3

0.4

0.04 0.03 0.02

CO

0.5 0.4 0. 0.2 00 1

0.01

0. 5

0.5

0.5

0.1

0

0.4

0.5

0

0.1

0.2

0.3

0.4

Figure 5-6a: Fluent and CMC with GRI2.11, RNG turbulence model with delta-rho term and scalar dissipation increased by a factor of 20, equivalence ratio=0.9, axial velocity (m/sec), axial velocity rms (m/sec), temperature K, CH 4 -wet, O 2 -dry, CO 2 -dry and CO-dry, species are mole percentages, ( O 2 , CO 2 and CO have the UHC removed).

153

0.04 0.03

c

0.8

0.02

1

0.4 0.6

0.01 0

0

0.2

0.8

0.1

0.2

0.3

0.4

0.04 0.03

Variance of c

0.0 0.03 1 0.01 0.02

0.02

0.04

0.01

0.02

0.1

0.03

4

0

0.0

0.01 0.01 0.02

0.03 0

0.2

0.3

0.2

0.3

0.4

0.04 0.03 30 0

0.02

10 0

100

0.01 0

300

Scalar dissipation

0

0.1

10 0 0 30

0.4

0.04

1.5E-0 7

5E-08 32.5 .5EE --0 077 5E-08

0.01 0

1.5E-07

0

0.1

1.5

0.2

5E 7 0E- 08 2.5 0.3

5 5EE--0 077

HNO

5E-08 5E -01.5E-07 8

2.5 E-07 1.5E -07

0.02

2.5E-0 7

0.03

E-0 7

0.4

0.04 0.03

NO

0.0

0.02

02

5

0.001

0.01

0

03 0.00 050.000.002 15

25 0.00

0.1

0.003

0.001 0.3

0.2

0.00

0

0.0

0.4

0.04 0.03

6

-0 6

5E-0

5E

0.01

5E-06

5E-0 6

0.02

6 -0 5E

NO 2

0

0

5E -0 6

0.1

0.2

0.3

0.4

0.04

0.1

4E-06

2E-06

06

1E

-06

0.2

3E-0

6

3E-0

3E-06

6

0.3

2E-06 E-06

0

5E-06

0

1E -

1E-06

0.01

-06 2E

0.02

1E-06

N 2O

2E-06

0.03

0.4

Figure 5-6b: Fluent and CMC with GRI2.11, RNG turbulence model with delta-rho term and scalar dissipation increased by a factor of 20, equivalence ratio=0.9, c, variance of c, scalar dissipation, HNO, NO, NO 2 and N 2 O mass fractions.

154

0.04 0.03 5E-0 8 07 1E-

1E-07

0.02

1E07 5E-08

5E-0

8

0.1

1E-0 5E-0

0

1E-07

5E-08

0

5E-08

0.01

1E-07

7 1E-0

CH

0.2

0.3

0.4

0.04

9E

-1 3

5E

13

3

3 1E-1

0.1

3E

0.2

5E-1

-1 3

-1 3

3

0

5E-13

-1 7E

0

5E

3 -1 3E

3 -1 -13 5E 7E

3E

1E 3E-13 -1 5E-13 3

0.01

5E

0.02

-1 3 5E

HCN

-13

0.03

0.3

-1 3

0.4

0.04 0.03

-08

0.1

7. 5E -0 8

2.5E-08

0.2

1E-07 1.25E-0

0

8 5E-0-08 2.5E

2.5E

0

5E-08 1E-07

7.5E-08

0.01

8 5E-0

8

5E-088 -0 5E 2. -08

CN

2.5E-0

5E

5E-08

0.02

0.3

0.4

0.04 0.03

7 -0 2E6E-0E -07 47

4E-07

0.2

1E-06

0.1

6E 6E -07 -07

8E-07

0

4E -0 7

8E-07

0

4E-07

0.01

2E-07

2E-07 2E4E-07 -0 7

0.02

4E-07

NCO

0.3

0.4

0.04

O

0.03

00 0.

0.0 00 3

04

0.02

0. 00

0

0.1

0.2

0.0.00.0 00 0000 0 43

1 00 0.0

04

3

0

0.0 00 30.0 00 0.0 4 00 1

00 0.0

0.01

0.3

0.4

0.04 0.03 1E-0

H

5

3E-05

0.02

1E

0.01

1E-05 3E-0

0.1

0.2

0.1

0.2

5

0

1E-0

0

1E -0 5 -05

0.3

5

0.4

0.04

OH

0.03 0.0 005

0.02

0 .00 0.00115

0.01 0.0

0

0

005

0.0015

0.0 01

0.3

0.4

Figure 5-6c: Fluent and CMC with GRI2.11, RNG turbulence model with delta-rho term and scalar dissipation increased by a factor of 20, equivalence ratio=0.9, CH, HCN, CN, NCO, O, H and OH mass fractions.

155

0.04

-1

5

20

5

-0 .

1100 11.5

0 5

0.02

9

18

16

11

14

24

16

5

0.1

0.04

18

20

0.2

24

7 0.4

0.3

2

4

1.5

0.03

10

10.5

30

10.5

0

14 11

30

20

0.01 0

14

Velocity

18

16

9

7

24

-1

0.03

1 .5

1 1

1.5

0.02

1

0.01

2 1.5

Velocity RMS

0

0

0.1

0.2

0.3

0.2

0.3

0.4

0.04 0.03 0.02

1400 1800

19 00 1000 600

0.01 0

300

0

1800

0.1

00 19

Temperature

140 0

0.4

0.04 0.03

CH 4

0.02

1

7

0 .5 5

8.58

0.01

3

0.5

1 7

0

0

0.1

0.2

0.3

0.4

0.04 0.03 0.02

16

5 12

8 18

0.01

5 16

0.1

12

8

0

4

0

4 4.5

4.5

O2

0.2

0.3

0.4

0.04 0.03

CO 2

0.02

9.5 8

2 4

0.01

0.0 01

1

6

9.5

8

2 4

6

0

0

0.1

0.2

0.3

0.4

0.04 0.03

0.2

0. 5

0

0.1

0.5

0.5

0.2

0. 5 02 0.3

0. 5 0.5

0. 00 1

4 0.

0.4 0.1

0.01 0

.5 00.4

0.02

0.001

CO

0.4

Figure 5-7: Fluent and CMC with GRI2.11, RNG turbulence model with delta-rho term and scalar dissipation increased by a factor of 30, equivalence ratio=0.9, axial velocity (m/sec), axial velocity rms (m/sec), temperature K, CH 4 -wet, O 2 -dry, CO 2 -dry and CO-dry, species are mole percentages, ( O 2 , CO 2 and CO have the UHC removed).

156

0.04

1 11 0.5

0 5

0.04

24

10.5

5

0

30

18 20

16

14

11

0

18

24

9

0.01

10

0.1

0.2

9 0.4

0.3

9

3

2

1.5

1.5

0.03

18 16

20

7

16

0.02

14

Velocity

24

9

10

7 10.5

14

11

5

30

-1 5 0.

-1 -

0.03

1

1

Velocity RMS

1 .5

0.02 0.01 0

1

0

0.1

0.2

0.3

0.4

0.04 0.03 18 00

0.02

Temperature

19

00

0.01 0

30

0

1400

0.1

1900

1800

1000

600

0

0.2

0.3

0.4

0.04 0.03

0.5

1

8

0.01

5

0

0.1

3

7

1

8.5

0

0.5

CH 4

3

8

0.02

0.2

0.3

0.4

0.04 0.03

O2

0.02

16

4

4.5

5 12

8

0.01

4 5

5

18

0

0.1

0.2

12

16

0

0.3

0.4

0.04 0.03

0

4 9.5

0. 0

00

8

9 .5

6 1

1

8

2

0.01

24

0.02

CO 2

0.1

0.2

0.3

0.4

0.04 0.03

0

0.1

0 .5

0.4 0.1

0.2

0.3

0 .5

0.4

0 .5

0

0 .4

0.5

1

5 0.

CO

0.5 0.4

0.4 0.2 0. 00

0.01

0.4

50 0 . .5

0.20.4

0.5

0.4

0.5

0.02

Figure 5-8: Fluent and CMC with GRI2.11, RNG turbulence model with delta-rho term and scalar dissipation increased by a factor of 20, scalar dissipation calculated with alternate method, equivalence ratio=0.9, axial velocity (m/sec), axial velocity rms (m/sec), temperature K, CH 4 -wet, O 2 -dry, CO 2 -dry and CO-dry, species are mole percentages, ( O 2 , CO 2 and CO have the UHC removed).

157

0.06 0.04

10 11.5 9 107 14

Y

0

5

20 10 16

9 0.1

0

11 24 10.5

20

30

0.02 0.01 0.01

5

16 10

18

0.03 0.02

-1 -00.5

0.05 0.03 0.04

Equation 5-1

24

10 18 11 10.5 0.2 X

14

30

20 20 0.3

0.4

0.04

10 times Equation 5-1

50

Y

0.03 0.02

150

50

0.01 0

150

0

0.1

50 0.2

X

15025 350 0 0.3

0.4

0.04

0.02

50

Y

0.03

0 25


15 025 0

50

52050 50 3

0.01

50

0

0

0.1

0.2

X

450

650

150

5050 0.32 1

0.4

0.04

760

Y

0.03


0.02

32 10 0100 0 320

0.01

100

320

0

540

0

0.1

0.2

X

0.3

0.4

0.04 0.03 300 Y

Equation 5-2

0.02

100

500

0.01

30 0

0

0.1

0.2

X

300 500 100

0.3

100

0

100 500

0.4

Figure 5-8b: Fluent and CMC with GRI2.11, RNG turbulence model with delta-rho term. Scalar dissipation for conditions in Figures 5-4, 5-5, 5-6a, 5-7 and 5-8, respectively.

158

0

5

0.02

24

20

14

10

16

30

24

20

18

5

24

20

14 0.2

10 0.1

0

1 .5

18

30

16

11

0.04

11

14

10.5 10

0.01 0

.510

5

18

Velocity

-1 0

11

-0.5

0.03

9 7

0.04

0.3

11

0.4

2 1.5

0.03 1.5

1

0.02 1.5

0.01

1

Velocity RMS

1

2

0

0

0.1

0.2

0.3

0.2

600 0.3

0.4

0.04 0.03 0.02

Temperature

600

0.01 0

18 00 19

00 1400

300

1000

0

0.1

1800

1900

0.4

0.04 0.03

CH 4

0.02

8.5 3

0.01 0

0.5

1 7

8

5

8.5

0

0.1

0.2

0.3

0.2

0.3

0.4

0.04

O2

0.03 0.02

12 5

16

4.5

0.01

5 18

8 4.5

4.5 0

0.1

12

5

0

0.4

0.04

CO 2

0.03 0.02

1

6

4

2

8

9.5

0.001

0.01

1

0

0

0.1

6

4

0.2

2

8

9.5

0.3

0.4

0.04 0.03

4 0.

0.5

0.4

0.5 0.001

0.5

0.1

0.01

0.4

0.4 0.2

0

0

0.1

0.2

0 001

0.5

0.4

0.02

.4 0.5

CO

0.3

0.5

0. 5

0.5

0.4

Figure 5-9: Fluent and CMC with GRI2.11, RNG turbulence model with delta-rho term and scalar dissipation increased by a factor of 20, c equation source term decreased by 20%, equivalence ratio=0.9, axial velocity (m/sec), axial velocity rms (m/sec), temperature K, CH 4 -wet, O 2 -dry, CO 2 -dry and CO-dry, species are mole percentages, ( O 2 , CO 2 and CO have the UHC removed).

159

14

11 1 9 0

5

.5

14

0

16

30

24

20

18

11

0.01

10

18

30

10

7

10.5

18

0.02

7

11

20

0

Velocity

5

.5

24

--10

0.03

16

0.04

10 0.1

0

16 7

14 0.2

0.3

0.4

0.04

Velocity RMS

1.5

1.5

0.03

1

1.5

1

0.02

1 0.01 0

1.5

0

0.1

0.2

0.3

0.2

0.3

2

0.4

0.04 0.03 0.02

Temperature

300

0.01

18 00 1000

600

0

0.1

18 00

0 14

0

18 14000 0

0.4

0.04 0.03 0.02

8

CH 4

0.5 5

1 3

8

7

0.01

0.5 5

8.5 0

0

0.1

1 3

8

0.2

0.3

0.4

0.04

O2

0.03 8

0.02

5 4 16 12 18

0.01

8

0

0

0.1

4.55 4

0.2

0.3

0.2

0.3

16

0.4

0.04

CO 2

0.03 0.02

8

9.5

6

0.01 0.001

0

0

4

1

2

0.1

89.5

0.4

0.04 0.03 0.02

1 00.4 .00

CO

0. 5 0. 5 0.4

0.01

0

0.1

0. 0.55 0. 0.2 00 1

0.2

0.4

0.5

0.3

0.5

0

0.10.5

0.4

Figure 5-10: Fluent and CMC with GRI2.11, RNG turbulence model with delta-rho term and scalar dissipation increased by a factor of 20, C3=1 in c variance equation source term, equivalence ratio 0.9, axial velocity (m/sec), axial velocity rms (m/sec), temperature K, CH 4 -wet, O 2 -dry, CO 2 -dry and CO-dry, species are mole percentages, ( O 2 , CO 2 and CO have the UHC removed).

160

Temperature

CH 4

O2

CO 2

CO

Figure 5-11: El Banhawy et al. (1983) data, equivalence ratio=0.77, temperature K, unburned hydrocarbons-wet, O 2 -dry, CO 2 -dry and CO-dry, species are mole percentages ( O 2 , CO 2 and CO have the UHC removed). Reprinted by permission of Elsevier Science from Premixed, Turbulent Combustion of a Sudden-Expansion Flow, by Y. El Banhawy et al. Combustion & Flame, Vol No 50, pp 153-165, Copyright 1983 by The Combustion Institute.

161

11

5

14

9 10

7 10

16

0.02

9

24

16

0

18 20

14

11 5

16

20

.5 10

0.01 0

10

30

18

14

11

5

Velocity

24

.5 -0

0

0.03

18

20

.5

-1

0

0.04

10.5

7

0.1

0.2

0.3

0.4

0.04

2

1.5

0.03

1.5 1

2

1.5

Velocity RMS

1

0.02

1.5

0.01 0

1

0

0.1

0.2

0.3

0.4

0.04 0.03 12 00

1000

0.02

Temperature

1700

0

1610 4000

60 400 0

0.01

300 0

0.1

100

0

0.2

1700

1200

0.3

0.4

0.04 0.03

CH 4

2

0.02

3

7

0.01 0

5

0.5 1

6

4

2

3

0

0.1

0.2

0.3

0.4

0.04

O2

0.03

78. 5

0.02

18

9 12

14

0

8

16

0.01

10

7..55

18 0

0.1

0.2

9

12

0.3

14

0.4

0.04

CO 2

0.03 7

0.02

4

0

7.5

0. 00 1

0.01

0

2 1

0.1

6

7 7.5

0.2

4

0.3

0.4

0.04

0..2 4

0.15

0.1

0.25

0.25

0.1 0.23

00.2 .23 .3

03 3 0..3 5 0.2

0.3

.1 0.2 5 0 0 .3

0.18

0.3

0. 25

0.15

0.2

0.35

0.2 0.23

0.1

0.4

0.3

0

0.2

0.2

0.3 0.4 0.4 5

0.23

0

1 0.18 0.

0.35 0.4

0.1

0.18 15

1 0.

0.3

0.25

0. 0.135 5 0.180.25

0.1 85

0. 4

00. 0.2 3.33 0.4

5 0.2

0.01

8 .1.51 0.30.25 00.01

0.00 0.1 1

4 0.

3 0.3 2.2 0.0 00.2.3 3 0.2

0.35

0.02

000.2.2 .33

0.03

CO

0.4


162

Temperature

CH 4

O2

CO 2

CO

Figure 5-13: El Banhawy et al. (1983) data, equivalence ratio=0.95, temperature K, unburned hydrocarbons-wet, O 2 -dry, CO 2 -dry and CO-dry, species are mole percentages ( O 2 , CO 2 and CO have the UHC removed). Reprinted by permission of Elsevier Science from Premixed, Turbulent Combustion of a Sudden-Expansion Flow, by Y. El Banhawy et al. Combustion & Flame, Vol No 50, pp 153-165, Copyright 1983 by The Combustion Institute.

163

10

10.5

18

24

20

11

10

16

14

0.02

30

0.01

16

0.04

0.

5

14

0

10

0.1

7

0.2

18

9

24 11 20

0.3

5

0.4

1.5

0

9 24

30

5

Velocity

5

16

14

.5

18 20

-1 0

7 9 11 1 0

0 -0.5

0.03

24

0.04

0.03

5 1.

1 1

1

Velocity RMS

0.02

0

2

1.5

0.01

1

0

0.1

0.2

0.3

0.4

0.04 0.03 19001800

0.02

Temperature

16000 14 00 00 30 0

0.01

2000

18 0 0

0.1

600

0.2

1000 0.3

00

0

14

0

0.4

0.04 0.03 0.02

8

CH 4

0.5

5

9

0.01

1

3

7

0

8

0.1

0.2

0.5 5

0

0.3

0.4

0.04

O2

0.03

2.5 6

0.02

10 0.01

3 18

14

2 .5

6 3

0

0

0.1

0.2

0.3

0.4

0.04

CO 2

0.03 8

0.02

6

0. 00 1

0.01

24

10

8 6

0

0.1

0.2

0.3

10

0

0.4

0.04 0.03 0.8 0.8

0.02

0.4

CO

0.01

0.8 0.8

0.4

0.7

0.0.10.7 00 1

0. 8

0.8

0. .6 80

0.2

0

0

0.1

0.7

0.2

0.6

0. 8

0.3

0.4


164

20

10

30

.5

1 14 6

11

0.01 0

1 24 14

14

30

9 11

10

14 16

11 10

0

5 7 10 .5

.5

24

0.02

-0

18

-1

Velocity

0.03

9

0.04

10.5 0

1 820

24

7 0.1

20 5

0.2

18

10 9 0.4

0.3

0.04 2 1

1

1.5

0.03 0.02

1 1.5

1

Velocity RMS

0.01 0

2 3

1

0

0.1

0.2

0.3

0.4

0.3

0.4

0.3

0.4

0.3

0.4

0.3

0.4

0.3

0.4

0.04 0.03 19

0.02

00

10

0.01

Temperature

0

00

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Chapter VI: Jet Stirred Reactor Results

6.1

Introduction

In this chapter the uniform conditioned species version of the premixed CMC model as developed in Chapter III is used to model the jet stirred reactor (JSR) data of Steele (1995). The results are also compared to the macromixing/micromixing model of Tonouchi (1996). The same PDF equations and implementation approach within the Fluent CFD program is used here as was used in Chapter V. All cases use the 2 cc volume reactor with a single 1 mm diameter inlet jet burning methane, see Figure 6-1 for a schematic of the reactor. The single jet enters the bottom of the reactor and proceeds upward to the top of the reactor, which causes the flow to deflect toward the outside of the reactor, creating a recirculation zone. This strong recirculation region continually mixes hot products with fresh fuel/air mixture. This strong mixing keeps the flame ignited even though there is very strong flame straining. Note that a planar flame would extinguish under this amount of strain. There are 4 exit holes evenly spaced in the azimuthal direction about a fifth of the way up the reactor. The species samples and temperature are taken from 2 holes evenly spaced in the azimuthal direction about two thirds of the way up the reactor. The probe location is at x=0.004 m and y=0.0143 m up from the bottom of the reactor (x is the radial direction and y is the vertical direction), see Steele (1995) for more details on the reactor and how the measurements were performed. A mesh of 1/8th of the reactor volume is used, taking advantage of reactor symmetry in the azimuthal direction.

6.2

Atmospheric Pressure Results

The first condition examined is at atmospheric pressure with an equivalence ratio of 0.606 and inlet temperature of 593 K. This is the first case run by Steele (1995) on 7/19/94 and had a residence time of 1.78 ms and inlet velocity of 470 m/s. Figure 6-2

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shows temperature K, scalar dissipation (calculated with equation 5.1) 1/sec, O 2 mole percentage, CO mole percentage, NO x ppmv and N 2 O ppmv, all species are on a wet basis. This run uses the Reynolds stress turbulence model (RSM) and the full GRI2.11 mechanism. The C3 constant in equation H13 is set to 100.0. Table 6-1 gives the measured data at the probe location and the calculated results. Table 6-1 also gives the adiabatic equilibrium values for this case and PSR results at 2 residence times for comparison. Results for other runs with the same conditions using different constants in the sub-models are also shown, which will be discussed below. For the atmospheric pressure results of Steele (1995), data are only available at the probe location, the high pressure results have radial profile data, and will be discussed in Section 6.3.

Figure 6-2 shows that the predicted temperature at the probe is 1852 K compared to 1801 from the data. Here, we address possible reasons for this discrepancy, and following this will discuss the kinds of information needed to resolve the discrepancy. The two most probable causes of this difference are (1) that the value of the scalar dissipation and (2) that the shape of the PDF of the reaction progress variable is incorrect, both of which are affected by the turbulence model. These would cause errors in the predicted value of c. The temperature is nearly a linear function of c, the non-linearity being due to the variation of specific heats with temperature. Thus, if c is too large, the temperature will be too high. C can become too large if the turbulence model gives an incorrect value of turbulent viscosity or an incorrect value of scalar dissipation (N) is used, which will give incorrect values of reaction rates, see equation G6. Figure 3-17 shows how the reaction rate source term varies with N, this plot is for a different inlet temperature and equivalence ratio, but the trends are the same for other conditions. Since ~ε N is calculated using equation 5.1 ( < N | ζ >≅< N >= 2 ~ ~c′′ 2 ), any errors from the κ turbulence model will also give errors in the calculation of N. In addition, equation 5.1 assumes that the conditioned values of N are equal to the unconditioned values, which most likely is not true, this requires detailed conditioned data to understand the

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relationship and to develop a better model. If the PDF of the progress variable is skewed to the right, it will give an unconditioned temperature that is higher than the mean of the conditioned temperatures. The way the PDF becomes skewed (unsymmetrical) is when the variance becomes large. As seen from equation G13 along with the closure model of equation H13, if the turbulence model or reaction source terms are incorrect, the variance will be wrong. Figure 6-3 shows the shape of the PDF at the probe location for c=0.9615 and a variance of 0.001008. This shows that the PDF is indeed skewed to the right. Other errors in the calculation of c could be from the Favre averaging of the c and variance equations.

When these equations were derived, the terms due to density

fluctuations were neglected. Additional experimental data are required to determine if these terms are required and to develop a model for them. This is analogous to the problems of Favre averaging the turbulence models discussed in Chapter V. Unfortunately, there are no data available for the turbulence quantities, c or the variance in the reactor, so the problems cannot be resolved at this time. The next series of calculations vary these parameters to understand their affect on the solution. Lastly, the temperature measurements are estimated to have an experimental error of at least 20 K, so the predictions may not be as far off as they first appear. The reactor is insulated, but there is still a small amount of heat loss, though it is not measured. The model assumes adiabatic walls, so it will slightly over-predict the temperature.

Figure 6-2 and Table 6-1, also show that the model slightly under-predicts the O 2 and N 2 O , over-predicts the CO, and drastically over-predicts the NO x . In section 6.3 we will show that small changes in c and N can lead to large variations in NO x . Thus, the same with the temperature noted above (i.e., errors in the turbulence model leading to errors in c, N, or the PDF of c) can influence the species predictions. This is particularly true for NO x due to its large sensitivity to c as c approached unity. Away from the probe location all the values seem reasonable except the NO x , which reaches equilibrium values in nearly half of the reactor. As seen in Figure I-11 (which is for different

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conditions), the NO is a very strong function of N for c greater then 0.9, so a small improvement in the model for N could improve this match dramatically. This may turn out to be a limitation of the model as accurate predictions of NO may require unrealistically accurate values for N.

Figure 6-4 shows results for the same conditions as Figure 6-2 with the reaction source term in equation G6 decreased by 50%. This decreased the temperature to within the experimental error of the measurement and brought the O 2 , NO x and N 2 O closer to the data, though the NO x is still off by two orders of magnitude. The CO match is not as good as before. This shows that the source term for the c equation has a significant affect on the solution.

One of the potential strengths of the CMC method over other

combustion models is a more accurate calculation of the reaction rates, so having to adjust this term to improve the match seems to negate this benefit. The sensitivity of the species mass fractions to the source term is studied here and two potential sources of error are developed. First, the conditioned source term may be accurate, but if the PDF is incorrect the unconditioned source term used in equation G6 will have an error. Second, the solution of equation G6 is tied to the turbulence model, which is known to have difficulties with variable density, recirculating flows.

Figure 6-5 is for the same conditions as Figure 6-2 with C3 in equation H13 set to zero. C3 is the constant that multiplies the reaction rate source term in the variance equation. This is the first attempt to develop a closure model for this term, and its sensitivity is explored here. This gives nearly the same solution as in Figure 6-2 with a variance of 0.00100, while it was 0.001008 in Figure 6-2. This says that the reaction rate source term in the variance equation has little affect on the solution or that the error in the other source terms of equation G13 due to the turbulence model are dominating the reaction rate source term. Again, this will require conditioned experimental data to understand and resolve.

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Figure 6-6 is for the same conditions as Figure 6-2 with the variance set to zero everywhere, giving a delta PDF at the mean. This is not a realistic scenario, but is shown as a limiting case for adjusting the variance. Most likely any error in the variance will be different throughout the reactor, because the variance is related to the turbulence parameters which vary throughout the reactor. This reduces the value of c at the probe and improves the match of all variables except CO. This says that the problems in the model are not just in the variance equation, but must also be in the N calculation or the turbulence model.

Figure 6-7 shows results for the same conditions as Figure 6-2 with N reduced 50% everywhere. While there are no data to support such a reduced value of N, it is shown to understand the sensitivity of the solution to N. This slightly improved the CO,

NO x and N 2 O matches but slightly increased the temperature error. Chapter V showed that increasing the value of N by a factor of 20 improved the match to the data everywhere, but here decreasing N helps the match at the probe location. Since the JSR has more intense recirculation then the backward facing step used in Chapter V, and the calculation of N is tied to the turbulence model, it is not surprising that the error in N seen here is different then was seen in Chapter V.

Figure 6-8 is for the same conditions as Figure 6-2 with N calculated from Equation 5.2 ( < N | ζ >=< D(∇c) 2 | ζ >≅< D(∇c) 2 >≅ [(∇~c ) 2 + (∇~c′′) 2 ]µ t / ρ ). This gives N=0.9 at the probe where Figure 6-2 gave N=13.9. This yields a slightly larger value of c and a smaller value for the variance of c, 0.9699 and 0.0008547, respectively, giving a slightly higher temperature. The four species are closer to the measured values. This had a similar affect as reducing N in Figure 6-7. Again there are no data to determine which model for N is better, but this does show the sensitivity of the solution to N.

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Figure 6-9 shows results for the same conditions as Figure 6-2 using the RNG turbulence model. This gave a lower value for c, but a larger variance, 0.9491 and 0.001664, respectively. The temperature and O 2 match are improved, but the other species are worse. In Chapter V it was seen that the RNG turbulence model gave a better match, but here there are not enough data to conclusively determine which model is best. Figure 6-10 is for the same conditions with the terms for variable density added to the RNG model, as discussed in Chapter V (see Figure 5-4). This had a negligible effect on the solution, which is different then was seen in Chapter V. One reason for the smaller affect here compared to Chapter V is the much lower equivalence ratio used here, 0.606 versus 0.9 from Chapter V, giving a smaller decrease in density. Another reason may be the type of turbulence, here there is stronger recirculation and more shearing of the inlet jet then in the backward facing step used in Chapter V.

Figure 6-11 is for the same conditions as Figure 6-2 with the variance set to zero, the reaction rate source term reduced 50% and N reduced 80%. This gave the best match overall, especially for NO x . There are no data to support these changes, but it shows that by varying several of the constants in the sub-models the solution can be improved. Because the experimental data are only available at the probe location, it is nearly impossible to determine how the above changes affect the match within the entire reactor.

This section showed that reducing c, its variance and N helped improve the match and that the turbulence model and the method of calculating N were the main causes of the errors. These changes gave a significant improvement in the match, but still not to the desired accuracy. To justify these changes and to further improve the match will require a detailed set of experimental data to determine the root cause of the mismatch. The next section will explore how varying these parameters affect the solution at higher pressures.

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6.3

High Pressure Results

This case uses the same reactor configuration as above at a pressure of 4.74 atmospheres, an equivalence ratio of 0.529, inlet temperature of 603 K and an inlet velocity of 293 m/s giving a residence time of 3.98 ms. This case is from Steele (1995) page 231 data taken on 1/21A/95. Figure 6-12 shows temperature K, scalar dissipation (calculated with equation 5.1) 1/sec, O 2 mole percentage, CO mole percentage, NO x ppm and N 2 O ppm, all species are on a wet basis. This run uses the Reynolds stress turbulence model and the full GRI2.11 mechanism with C3=100.0 in equation H13. The model predicted a temperature of 1769 K, 5 K above the measured data, well within the experimental error. This is just 3 K below the adiabatic equilibrium temperature. Even though the measured temperature and c are approaching the equilibrium values the CO and NO x are far from their equilibrium values. This is due to the small residence time of the reactor, so the CO and NO x do not have time to reach their equilibrium values. The CO in the reactor is above the equilibrium value because there is not time for all of it to oxidize. The NO x in the reactor is much lower then its equilibrium value because NO x production is a strong function of time and there is not enough time to generate large levels of NO x . In a PSR it would take several minutes for the NO x to approach its equilibrium value, far from the 4 ms residence time of the reactor. The difference between the measured and calculated species is similar to what was seen at 1 atmosphere, but larger here because c is closer to unity. As c approaches unity (its maximum value) small changes in c, its variance and N give large changes in CO and NO x , with much smaller changes in O 2 and temperature. The results from the CMC model shown in Chapter III are not a function of time. The time scale enters through N, the small scale mixing rate. If N is wrong, the time scale of the reactions will be wrong, giving errors in some of the species. Table 6-2 gives the measured data at the probe location and the calculated results. Table 6-2 also gives the equilibrium values for this case and PSR

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results for 2 residence times for comparison, as well as results from Tonouchi (1996). ~ε Tonouchi defines the scalar dissipation as < N >= 2 ~ . κ Figure 6-13 is for the same conditions as Figure 6-12 with the reaction rate source term reduced by 50%. Because the reactions are so close to their equilibrium values, reducing the reaction rate has a small affect. The variance has the largest change, increasing 50%. As discussed in section 6.2, if the PDF is incorrect, the unconditional reaction source term will be incorrect, even if the conditioned source term is correct.

Figure 6-14 is for the same conditions as Figure 6-12 with C3=0.0. As was seen in Section 6.2, this has a negligible affect on the solution because the reaction rate source term is small compared to the other source terms in the variance equation. Figure 6-15 is for the same conditions as Figure 6-12 with the variance set to zero everywhere, giving a delta function PDF at the mean. This decreases the NO x a small amount, drastically increased the CO and improved the N 2 O match.

Figure 6-16 is the same case as Figure 6-12 with N decreased 50%. This has a negligible affect, again because the reactions are so close to the equilibrium limit the small change in N had minimal affect. Figures I-7 and I-8 show that if N is reduced more, the NO x value will decrease significantly. At this large of value of c, N must be smaller then 0.3 to give NO x values close to the measured data. Without measured data it is impossible to know what the correct value of N should be.

Figure 6-17 is for the same conditions as Figure 6-12 with N increased 10% everywhere. This helps the CO match, but has little affect on the other variables. When Figure I-10 is expanded it is seen that for CO to decrease, either N must decrease or c must decrease. In this case even though N was artificially increased, the value of c at the

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probe was decreased and the variance doubled, causing the CO to increase. For the O 2 ,

NO x and N 2 O matches to improve, N or c must also decrease. Having the PDF symmetrical or skewed to the left will help improve the match of all the species, instead of the PDF being skewed to the right as seen here. Because there are an infinite number of combinations of c, N and PDF shape (set by the variance) it is impossible to determine a combination that will exactly match the experimental data, assuming a combination exists. As an example, when c at the probe is 0.99 with a symmetrical and some what narrow PDF are used a value on N between 0.1 and 1.0 would come close to matching the data within experimental error.

Figure 6-18 is for the same conditions as Figure 6-12 with N calculated using equation 5-2. This gives N=0.09, compared to 4.4 seen in Figure 6-12. C is increased closer to equilibrium, so there is no improvement in the match of the species. This highlights the fact that the problem is in more then one area, the calculation of N needs to be improved, but there are also errors calculating c and its variance, possibly due to the turbulence model.

Figure 6-19 is the same conditions as above using the RNG turbulence model and Figure 6-20 adds the terms to account for variable density to the RNG model. Neither has an affect on the solution because c is so close to unity.

Figure 6-21 is for the same conditions as Figure 6-12 with the variance set to zero, the reaction rate source term reduced 50% and N reduced 80%. This gives the best match of all the conditions run, but is not as much of an improvement as was seen in the atmospheric pressure runs in Section 6.2, Figure 6-11.

In general, adjusting the

parameters for the high pressure case had a smaller affect then with the atmospheric pressure case, though the trends were in the same direction.

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Page 243 of Steele (1995) shows axial temperature profiles at the probe. It is believed that some of the curves are miss marked, but the trends are the same for all. Through most of the reactor the temperature is nearly constant, with a small decrease near the reactor walls due to heat loss. The calculated results do not show this because the CMC model assumes adiabatic walls. In the center of the reactor, in the jet the data shows the temperature decreases. This is because the chemistry is still incomplete at this point, which the model also shows. Note that in the Steele data the radial dimension is plotted as radius squared non-dimensionalized by the maximum radius squared. Page 244 of Steele shows CO profiles at the probe height (y=0.143 m). The modeled results show the CO has a peak in the jet shear layer, then decreases significantly in the recirculation zone and remains nearly constant. This matches the measured data, though the magnitudes are off.

Page 245 of Steele shows NO x and N 2 O profiles. The

calculated NO x shows the same trends as the data, but the magnitudes are off as discussed above. The calculated N 2 O profile shows an increase in the jet, while the data shows a decrease, with the remainder of the reactor relatively constant. The calculated data shows this increase of N 2 O in just a small pocket of the jet, with the limited data it is not known if this pocket exists in the real burner and the model is just predicting it in the wrong location or if the model is totally incorrect.

6.4

Summary

This chapter describes the application of the uniform CMC model to the measured data of Steele (1995). The model predicted the general trends correctly, based on the limited data available, but over-predicted the temperature a small amount, predicted the

O 2 , N 2 O and CO with less accuracy and very significantly over-predicted the NO x . Several parameters were varied to determine their sensitivity on the solution. It was found that as c approaches unity (equilibrium) the NO x becomes very sensitive to c, the PDF shape and the scalar dissipation. A large portion of the errors are due to the

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turbulence model and the model used to calculate N. Because the JSR has less heat loss then the backward facing step used in Chapter V, the errors from neglecting heat loss in the adiabatic CMC model should be smaller.

The CMC model has the potential to predict all species much more accurately then simple models such as the EDM, which are limited to very simple global mechanisms and ad hoc determination of the model constants. In order to provide this improved species modeling the CMC method relies on the statistics of the flow field, so the results are only as accurate as the statistics, which are a function of the scalar dissipation and PDF. To improve the predictive capabilities of the CMC method more accurate models are required for the variance and conditioned scalar dissipation. Also, improved turbulence models that account for the affects of variable density due to combustion are needed. The JSR is much more challenging for the turbulence models then the backward facing step. The higher turbulence intensity, strong recirculation, etc. may be impossible for the RANS models to capture enough physics and LES may ultimately be required. To develop and test these new models a significant amount of experimental and possibly DNS data are required. This data would include velocity, turbulence parameters, a wide range of species, both conditioned and unconditioned, the scalar dissipation, the reaction progress variable and is variance.

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y 4 probe and outlet locations, evenly spaced 90 degrees around the circumference height 0.02223 m

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Figure 6-1: Schematic of jet stirred reactor.

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X

Figure 6-5: Fluent and CMC with GRI2.11, RSM, phi=0.606, 1.0 atm., inlet velocity 470 m/s, Tin=593, C3=100. Temperature K, N 1/sec, O 2 mole %, CO mole %, NO x ppm, N 2 O ppm (species are on a wet basis). C3 = 0.0.

182

1800

40 60 80 100

0.02

20

0.015

0.015

Y

Y

1800

N

2000

10600 20

140 18 000

0

0.005

0.01

000108 00 460 50 0 0

1600

0.005

800

0

20

2000

12001000

0.005

1400

Temperature

1080 0 60 40

500

1600

0.01

0.01

0

0.015

X

0

0.005

0.01

0.015

X

0.02

0. 751

1.5

0.5

8

10

5 1.501.2 .25

0.75 0.5

1 1.25

0.015 0.015

8 10

18

0.005

CO

0

0.005

0.01

0

0.015

X

1100

0.01

Y

1500 190 0

50013001700 2100 2300 23 00 2 3 210 0 2100 0 0 17 1 00 1900 15 30000 700 900

0

1

2100

1.5

Y

1

1100

2.5 50

0

1 2. 1.5 0.5 5 1.5

0

0.005

N 2O

1700 1700

1900

210 0 230

0

0 15

0.5

900 100 300 1100

0.005

1

1900 1500

NO x

0.015

0.5

0.015 700

2 300

0.01

0

130000 5 00 117

00

0.01

2.5

900

17

0.015

00

0.005

X

30

11

1300

0.02

0

2.5 0.5

0

14 8

12

16

0.005

0.25

14

O2

0.5 0.25 1 5 5 0.2 0.70.7 50.1.2 5 1 5

Y

Y

12

0.01

0.01

0.005

0.01

X

0.015

0

0

0.005

0.01

0.015

X

Figure 6-6: Fluent and CMC with GRI2.11, RSM, phi=0.606, 1.0 atm., inlet velocity 470 m/s, Tin=593, C3=100. Temperature K, N 1/sec, O 2 mole %, CO mole %, NO x ppm, N 2 O ppm (species are on a wet basis). Variance set to zero.

183

60

0.02

100

1800

1400 1600 1800

100 60 40 20

N

1200 00 0 6

0.005

1000

80

0

0.005

0.01

0

0.015

X

0

0.005

0.01

0.015

X

0.02 11.25 1.5

0.5

18 1600

0

2000 500

1200

0.005

500 02400 80

Y

Y

0.01

0.01

Temperature

80

0.015

0.015

0.015

25 .25 1. 0

Y

Y

0.01

0.01

8 10

0 25 0 25

0

0.005

0.01

0

0.015

X

0 90

Y

00

1900

0

0.005

0

0.005

0.01

X

0.015

0

12.50.5 1.5

90

2100 19 1500 00 1700 900 1300 1100 700 500

50 0 700 0 0 1300 1500 110 17000 190 2100

13

1.5 1 0.5

2100 2300 2300

0.01

2.5

Y

0.5 1 1.5 2.5

170

0

0.015

0.015

300

0.005

0.01

500

2300

NO x

00 21 1900 1700 1500 1300 1100 900 700 500 300

0.01

0.005

X

700 900 00 00 11130 150

0.015

0

0.5

0.02

25

812

20

0

CO

0.2 5 0.5

0.005

0.005

0.5

14

1 0. 0.275 0.5 5

8

16

0.75

12

O2

1.251. 15 0.5

10

0.75 0. 25

8

0.015

0

N 2O 0.005

0.01

0.015

X

Figure 6-7: Fluent and CMC with GRI2.11, RSM, phi=0.606, 1.0 atm., inlet velocity 470 m/s, Tin=593, C3=100. Temperature K, N 1/sec, O 2 mole %, CO mole %, NO x ppm, N 2 O ppm (species are on a wet basis). N reduced by 50%.

184

2000 500 4010080 20

Y

0.01

1600

0.005 500 2000 80

800

0

0

0.005

0.01

0

0.015

X

N

0010 20

1000

0.005

1200 140 18000

Temperature

60

0.01

1800

140000 12

Y

60

500

40

80100

1800 00 16

0.015

0.015

20

60

0.02

0

0.005

0.01

0.015

X

0.5

0.02 0.75

1.5

1.25

0.015

1

0.015

0.005

0.01

0

0.015

X

700100

0.715 0.25 0.5 0.751.5 0.5

0.5

Y

1.51 2.5

700

23 00

05

00

0.5 1.5

1700 1900 2101500 0

0

N 2O

0.005

2.5 1 0.5

0

0.5

210170 0 0 1900 1500 1100 500 300 10 0

Y

0.01

1.5

19

0.015

0.015

00

0.005

21

2300 1100 100700 1300 900500

NO x

0.01

2.5

0.01

300 700 100 1700 1 2300 21005010100500900 2300

0.015

0.005

X

0 900 0 13000 7 00 130 90 0 01300 17000 900 1 00 50 1100150 21 1300 300 1900

0.02

0

1 0.5

0

CO

0.25

1412 10

20

0

00.7 .5 5 1

0.5

0 25

0.005

0.005

0.25

Y

Y

8

1.25

1618

O2

0.25

8 10 12 14

0.01

0.01

0.005

0.01

X

0.015

0

0

0.005

0.01

0.015

X

Figure 6-8: Fluent and CMC with GRI2.11, RSM, phi=0.606, 1.0 atm., inlet velocity 470 m/s, Tin=593, C3=100. Temperature K, N 1/sec, O 2 mole %, CO mole %, NO x ppm, N 2 O ppm (species are on a wet basis). N calculated with alternate method.

185

80 60 20

1800

Y

80 20 60

1600 1800

1600 1400

500 80 100

0

N

620 01000 50 0 0

Y

0.005

2000

1200 01000 80

0.005

500

1400

Temperature

0 0.01 00 2

40100

0.015

0.015

0.01

40

0 10

0.02

0

0.005

0.01

0

0.015

X

0

0.005

0.01

0.015

X

0.75

0.5

0.02

0.015 1

0.015

1.150.55 0.2

8

Y

Y

0.01

0.005

0.005

CO

0

0.005

0.01

0

0.015

X

0.512.5

Y

2100

1900 21 23 2300 00 00 1500 17 00 901 01103000 19100 2100 5 00 500 900

0.005

0.5

N 2O

0.5

1300 1700 1900 230 01500 2100

2.5

Y

1 .5

0

0. 5

700 300

0

0.015

0.01

0

1100 900 500 100

0.005

0.015

70

2300 0 0 190170

0.01

NO x

0.01

X

01010 0 13 1700 0 150 0 190 1700 00 00 15

0.015

0.005

15

0.02

0

0.512.5 1.5 1 1.5

0

20

128

O2

10 14

16

0.2 0.50.7 5 15 0.2 5

8

12

0.01

0.25

10

0.50.75 1.25

0.75

2300

0.005

0.01

X

0.015

0

0

0.005

0.01

0.015

X

Figure 6-9: Fluent and CMC with GRI2.11, RNG, phi=0.606, 1.0 atm., inlet velocity 470 m/s, Tin=593, C3=100. Temperature K, N 1/sec, O 2 mole %, CO mole %, NO x ppm, N 2 O ppm (species are on a wet basis).

186

1800

0.02

1400 1600

2000 401050 0 060 280 0

Y

Y

1800

500 0.01

0.01

Temperature

0.005

N

2100

2000 100

00 0001200 1800

0.005

0

40100 60

0.015

20

80

16 0 0.015 0

0

0.005

0.01

0

0.015

X

0

0.005

0.01

0.015

X

0.75

0.02

1.5

.2 5 11

0.5 0.75

8

0.01

Y

Y

0.005

0.01

0

0.015

X

00 15 0 0 17

1700 1500

Y

230 0

1300110 9000

15 00

21023 1500 0 100 190 300 0 11 7900000 0

0.5

0.005

N 2O

0.5

1700 1500 1300 230 2100 1900 0

0.01

1.5 1

Y

0.015

0

0

1700

0

0.015

00

110

NO x

1100 700 500100 900 300

0.005

2100

00 0 00 21 2300 19017

0.015 00

0.01

X

0.5

1900

0.015300 0 0 700 19

13

0.005

2.5

0.02

0

CO

2.5 1 1.5

0

25

1 1.5 2.5

0

0.

0.5 0.2 75 1 5 0.0.25

0.005

10 8

O2

16 18

0.005

14 12

8

0.01

0.5

10

0.5

0.2

5

75 0.

1.25 1

0.015 0.015

0.005

0.01

X

0.015

0

0

0.005

0.01

0.015

X

Figure 6-10: Fluent and CMC with GRI2.11, RNG, phi=0.606, 1.0 atm., inlet velocity 470 m/s, Tin=593, C3=100. Temperature K, N 1/sec, O 2 mole %, CO mole %, NO x ppm, N 2 O ppm (species are on a wet basis). With delta-rho terms added to turbulence model.

187

0.02 20

00

40

18

0.015

0.015

40

0.005

1800

0

0.005

0.01

0

0.015

X

5002 0

2000 80 40 100

1000

0

N

10 8060 0

1601400 0

0.005

20

1800

500

1200

Temperature

0.01

Y

Y

60 80 100

1600 0.01

0

0.005

0.01

0.015

0.01

0.015

0.01

0.015

1

X

0.02

0.75

1.25 1.5

0.015

10

1 0 .5

0.015

16 18 20

1214 10

0.25 0.251.25 0.75 01.5 1 .5

0.005

0.005

0

0.75 1.5 1.25

Y

Y

5 10.0.25

O2

0.01

12

0.01

8

0

0.005

0.01

0

0.015

X

0

CO 0.005

X

0.02

100

0.015

1.15

0.01

Y

0

1.5 0.51 0.5

500

0

0 70

0

2.5

10

0.005

1.5 0.5 1

700 00 132100 1500 1700 1300 1100 900 300700 100 500

1500 1700 230 1900 110 0000 19 1100 900 900 300 500

700

0.005

2.5

Y

100

0.01

NO x

0.5

300

300

0 10

500

0.015

0.005

0.01

X

0.015

0

0

0.005

X

N 2O

Figure 6-11: Fluent and CMC with GRI2.11, RSM, phi=0.606, 1.0 atm., inlet velocity 470 m/s, Tin=593, C3=100. Temperature K, N 1/sec, O 2 mole %, CO mole %, NO x ppm, N 2 O ppm (species are on a wet basis). Variance set to zero, S c reduced by 50% and N reduced by 80%.

188

0.02

1600

60

0.015

1600

100 500

80 40 20

0.015

60

N

0500 80 102000

0

0

0.005

1400

0.005

1002000 500 10 800 40 20

1600

1200 100 8000

Temperature

20 0

Y

0.01

1600

Y

1400

0.01

0

0.005

0.01

0

0.015

X

0

0.005

0.01

0.015

X

25 0.

0.02

1

0.5 0.5

Y

0.005 0.205.2

12

4 0

0.005

0.01

0

0.015

X

0.02

23

0

0.005

23 00

1. 5

Y

0.01 1

11009001300 1500 1700 1900 2100 23019 00 1700 0 210 0

1

0.005

0.5

0.005

700100 300

NO x

1 1.5 2.5 0.5

Y

0.52.5 2.5 1.5

1 1700300 1900 2300

1

0.015

1100

0.01

0

0

0.015

00

0 002100 19 2100 170 1500

0.015

0.01

X

1.5

0

CO

5

18

0.005

10

Y

14 16

O2

0.01

12

0.01

0.25

0.20.5 5

0.75

0.015

10

0.015

0.005

0.01

X

0.015

0

0

N 2O 0.005

0.01

0.015

X

Figure 6-12: Fluent and CMC with GRI2.11, RSM, phi=0.529, 4.74 atm., inlet velocity 293 m/s, Tin=603, C3=100. Temperature K, N 1/sec, O 2 mole %, CO mole %, NO x ppm, N 2 O ppm (species are on a wet basis).

189

60

Y

Y

0.01

1400 1600

1200 0 801000

500

2000

0.01

N

60

0.005 2000 80 500 40

0

0.005

0.01

0

0.015

X

220 001001 00040

0.005

0

10040 20 80

0

1600

40

500

0.015

0.015

Temperature

20

40 0 1080

0.02

0

0.005

0.01

0.015

X

0.02 0.75

0.755 0.

10 1.2 5

25 0.

0.5 0.70.5 5 0.75 0.25

1

0.015 0.015

0.75

12 14

Y

Y

0.01

16 18

0.005

0.005

20

0.005

0.01

0

0.015

X

1500 1700 0 150 1100 170000 00 15 13 900

1.5

121 9000 1700 0

0.015

1300 1100 19 1500 00 1700 1900 23002100210190 1291000 01700 150 170 0 0 2300

0.01 1.5

2.5

Y

Y

0.01

900

0.005

0

0

0.5

N 2O

0.5

1700 700 500 300 100

0.005

0.015

2.5 1

0.015

0.01

X

230 0

0

0.005

0.5

0.0213 0

0

1.5 0.5

0

1.5 0 1 .5 1 2.5

0

NO x

CO

10

14 12

O2

0.5 0.5 0.25 0.25

10

0.01

0.005

0.01

X

0.015

0

0

0.005

0.01

0.015

X

Figure 6-13: Fluent and CMC with GRI2.11, RSM, phi=0.529, 4.74 atm., inlet velocity 293 m/s, Tin=603, C3=100. Temperature K, N 1/sec, O 2 mole %, CO mole %, NO x ppm, N 2 O ppm (species are on a wet basis). C equation source term reduced by 50%.

190

0.02 40 60 80100

20

0 0.015 50

0.015 1600 1600 1 400

500 10080 60 2040

N

0

0.005

0.01

0

0.015

X

120 1000 50 0 20 80 00

1200

1400 1600

0.005 2000 0 500 460

0

2000

0.005

1000800

Temperature

Y

Y

0.01

1200

0.01

0

0.005

0.01

0.015

X

0.02

0.015

10

0.5 0.25

Y

0.01

12

Y

14

0.01

10

0.015

2 0.50.25 555 0.0.7 0.75

1

18

0.25

0.005

0.005

16 14 12

10

0

0

CO

0.25

O2

0.005

0.01

0

0.015

X

0

0.005

0.01

0.015

X

0.02

1 2.5

1700 210019 00 000 23 11790 1500 1700 300 0 700300 1 1100 900 500 110

0.015

0

1.5

Y

2.5

1.5

N 2O

0.5 1

19 100 500 1300 00 300 1700 2100

05

10

0

0.5

0.005

0.005

NO x

1 1.5

0.01

1

1900 2100 210 21230 0 0 00 0190

0.5 2.5

00

0.01

21

Y

0.015

0.005

0.01

X

0.015

0

0

0.005

0.01

0.015

X

Figure 6-14: Fluent and CMC with GRI2.11, RSM, phi=0.529, 4.74 atm., inlet velocity 293 m/s, Tin=603, C3=100. Temperature K, N 1/sec, O 2 mole %, CO mole %, NO x ppm, N 2 O ppm (species are on a wet basis). C3=0.0.

191

0.015

1400

0.01

Y

0.01200

0.005

001000 801050 0 0

500 40

6001200 1400

2000

0.005

0

N

40 20

1600

2000

1000

Temperature

80 60100

500

Y

0.015

40 20

00 16

60 80 100

0.02

0

0.005

0.01

0

0.015

X

0

0.005

0.01

0.015

X

0.02 1

0.25 0.5 0.75 10.5 0.75

0.015

0.01

0.25

0.005

0.5 0.75 0.55 0.2

Y

Y

12

10

0.015

0.01

14 0.005

0

0.005

0.01

0

0.015

X

1

Y 0 190

0.005 0

N 2O

1

191300 0000 1700 21 210 190 170 0 00 0

0.01

00

1 90

5

2 30

1300

15 00

17 190000 19 00 19001900

Y

0

05

0

2.

0.5

0.005

0.015

1

1500 1100 900 700 100 500

NO x

00

1.5

1900

0 17

0.01

0.015

2.5

00

0.01

X

1

17

19

0.005

1500

0.015

7 00

0.02

0

1.52.5 1.52.5

0

CO

0.25

16 1012

O2

0

0.005

0.01

X

0.015

0

0

0.005

0.01

0.015

X

Figure 6-15: Fluent and CMC with GRI2.11, RSM, phi=0.529, 4.74 atm., inlet velocity 293 m/s, Tin=603, C3=100. Temperature K, N 1/sec, O 2 mole %, CO mole %, NO x ppm, N 2 O ppm (species are on a wet basis). Variance set to zero.

192

20 40

60

0.02

Y

N

0 80

Temperature

0.005

2000 80 40 500

0

000 120016 140 0 00

0.005

0

0.005

0.01

2000 40 210060850 00 0

Y

500

1600 1400 00 12

0.01

0.01

10080 40 20 60

0.015

0.015

0

0.015

X

0

0.005

0.01

0.015

X

0.02

0.5 .0.2 75 5 0.750.75 0

10

0.015

1.5

0.015

1

0.01

0.25 0.5 0. 0.7 5 5

0.005

0.005

CO

0.25

O2

14

12

10

0.01 16

Y

Y

0. 0 25.5

75 0.

16

10

18

0

0

0.005

0.01

0

0.015

X

0.02

0

0.005

0.01

0.015

X

00 23

0 101500 15 0 200 17 0

1300 230 0

170 0

0.5

0.015

1700 1900 1900 1700 2100 190 0 2300

700

1.5 1 2.5

0.015

1300 002100

0

0

N 2O

0.5

0.005

11.5 1.52.5 0.5 2.5

Y

Y

0.005

1 0.5

2300 0 1500 0 1100 130 210 900 0 00 500 10 3700

NO x

0.01

1700

0.01

0.005

0.01

X

0.015

0

0

0.005

0.01

0.015

X

Figure 6-16: Fluent and CMC with GRI2.11, RSM, phi=0.529, 4.74 atm., inlet velocity 293 m/s, Tin=603, C3=100. Temperature K, N 1/sec, O 2 mole %, CO mole %, NO x ppm, N 2 O ppm (species are on a wet basis). N reduced 50%.

193

140 161200 00

0.005

0.01

500

200 500 10

800

0

20 6 0801 00 40

0.005

0.005

0

6080100 40 2000 50 0

0

0

0.015

X

0

20

Y

0.01

1600

Y

00

0

1400

0.0120

Temperature

2000

0.015

0.015

20

1600

6080 10 0

0.02

N 20

0.005

0.01

0.015

X

10

0.02

2 0.015

Y

0.01

10

0.005

0.005

14 1216

CO

18 20

O2

14

0.01

12

Y

10

0.015

0

0

0.005

0.01

0

0.015

X

0

0.005

0.01

0.015

X

0.02

1

2300

0.5

2300 2300 2100

0.015

1.5

0.015

0.01

0.005

N 2O

0

0

0.5

100

00 500 700 1900 700 500

0.005

NO x

2300 1500 1300 1100 900

1700 0 23210 0023 00

0.5 0.5

0 230

Y

Y

1900

0.01

0.005

0.01

X

0.015

0

0

0.005

0.01

0.015

X

Figure 6-17: Fluent and CMC with GRI2.11, RSM, phi=0.529, 4.74 atm., inlet velocity 293 m/s, Tin=603, C3=100. Temperature K, N 1/sec, O 2 mole %, CO mole %, NO x ppm, N 2 O ppm (species are on a wet basis). N increased by a factor of 10.

194

40 80

0.02

100

60 500

0.015

1200 1400 1600

800

Temperature

0.005

60

1020 0 00

1000

0

N

8400

0.005

500 2060

Y

Y

0.01 20 00

0.01

100

0

80 40 20

160

0.015

0

0.005

0.01

0

0.015

X

0

0.005

0.01

0.015

X

25

0.02

0.25

10 0.5 0.75 5 1 11.2 .5

0.015

0.5 0.25 0.25 0.5

0.75 0.2 1 0.005

1216 14 10

0.005

CO

0.5

O2

0.01

12

0.01

Y

Y

10

0.75 0.25

0.015

18

0

0.005

0.01

0

0.015

X

0.015

0. 5

2 1300 100

100 9007050 0 13 0 00 1700 2100 1900 1500 190 210 0 0

2300

0.01

500 1300 210170 23000 019 19 01700 000

1

1.5

1500 1100 300

0.5 2.5

Y

1 2.5

1.5

1

0.5 1.5

Y

0.015

00

0.005

0.5

N 2O

1

0.005

700 500

NO x

0 150013 0017 01300 15 0 1100 90 300

0.01

0.01

X

90 02 0 1 7 030 00 0

0.015

0.005

2100

0.02

0

2.51.5 0.5

0

0

0

0.005

0.01

X

0.015

0

0

0.005

0.01

0.015

X

Figure 6-18: Fluent and CMC with GRI2.11, RSM, phi=0.529, 4.74 atm., inlet velocity 293 m/s, Tin=603, C3=100. Temperature K, N 1/sec, O 2 mole %, CO mole %, NO x ppm, N 2 O ppm (species are on a wet basis). N calculated with alternate method.

10 0

0.02

200 0

100 20

Y

1400

0.015

500 4080 60

2000

1200

1600

Y

0.01

N

500

100 100

1400

60

1000 800

200 10200

0.005

0.005

0

40

60 80

0 1600 0 160

0.01

0 50

160

0.015

Temperature

20

195

0

0.005

0.01

0

0.015

X

0

0.005

0.01

0.015

X

10

0.02

0.525 0.25 0.75 0.75

10

0.015

0.5 10

12

0.015

0.005

0.005

CO

0

16 14

2018

O2

10

12

14 16

0.25

0.5

Y

Y

0.01

0.01

0

0.005

0.01

0

0.015

X

0

0.005

0.01

0.015

X

0.02

2.5

1700

1

900 1500 21 00 1901700 0 2100 2300 210190 0 0

1

0.01

Y

2.5 0.5 1.5

1700 1500

0.015

0

1 0.5

0.005

N 2O

0.5

300700 500

0

1.5

1300 1100 0.005

NO x

1.5

Y

5 0.2.5

23 00 2100 00 0000 19 1515 1900 1500

0.01

1

1700

0.015

0.005

0.01

X

0.015

0

0

0.005

0.01

0.015

X

Figure 6-19: Fluent and CMC with GRI2.11, RNG, phi=0.529, 4.74 atm., inlet velocity 293 m/s, Tin=603, C3=100. Temperature K, N 1/sec, O 2 mole %, CO mole %, NO x ppm, N 2 O ppm (species are on a wet basis).

196

0.02

500 40 100 80 60

Y

20

0.005

0.01

N

20

0

0.015

X

8050200 0 810 0 0 0 10 0

1400 1600

0

0.005

100 10020 80

0

1200

0.005

2000 500

1000 800

Temperature

200 0

Y

0.01

1200

0.01

20

40

0.015

1600 0 160

0.015

0

0.005

0.01

0.015

X

0.02

10 0.5

0.015

5

0.25

0. 250.7 5

0 .7

0.015

12 0.5

10

4

Y

Y

0.01

0.01

0.2

5

0.25

1618

20

O2

0

0.25

0.005

14 10 12

0.005

CO

0

0.005

0.01

0

0.015

X

0

0.005

0.01

0.015

X

0.02

1

1700

1

1700 1900 2100

0

0

1 0.5

1.5

1.5 2.5

Y

1 .5

700100

NO x

0.005

0.01

X

0.015

0

N 2O

0.5

Y

1500 1700 21 190 00 0 2100

0.005

0.5

0.005

0.01

00 0 21230 1500 1300 1100 900 500 300

0.01

0.015

1.5 0.5 2.5

150 0

300 2000 000721 11950 1 0 1900

0.015

0

0.005

0.01

0.015

X

Figure 6-20: Fluent and CMC with GRI2.11, RNG, phi=0.529, 4.74 atm., inlet velocity 293 m/s, Tin=603, C3=100. Temperature K, N 1/sec, O 2 mole %, CO mole %, NO x ppm, N 2 O ppm (species are on a wet basis). With delta-rho terms added to the turbulence model.

197

0.02 20

80

0.015

60

0.015

20

40

0.01

500 60 80 100

Y

Y

1600 1400

0.01

Temperature 500 80 20

1002000 2040

0

0.005

0.01

0

0.015

X

40

1400 1200 1600

1000 0

N

0.005

0.005

0

0.005

0.01

0.015

X

0.02

0.25 0.5 0.75 5 1 .5 1.2 1

10

0.015 0.015

0.01

0.25 0.75 1 0.5

10 12

O2 0

1.25 1.5 0.75 1 0.5 0.25

14

0.005

16

0.005

1.25 1.5

0.25

10

Y

Y

12

0.01

0

0.005

0.01

0

0.015

X

CO

0

0.005

0.01

0.015

X

1700

13

300 700

00

0.015

0.01

1.5

1.5 2.5 1

1

0

N 2O

500

NO x

0.005

0 51

1500 700 00 1300

2.5 1

1

1100

0.005

1

00

Y

Y

15

0.01

1.5

90500 900 91100 0 70 000 1300 90 11300 0 700 3 00 15100013 500 700 0 010 1190 000 700 1300 17

00

0.015

1 0.5

500

1100 1100

100

15

1

900900

00

0.02

0

0.005

0.01

X

0.015

0

0

0.005

0.01

0.015

X

Figure 6-21: Fluent and CMC with GRI2.11, RSM, phi=0.529, 4.74 atm., inlet velocity 293 m/s, Tin=603, C3=100. Temperature K, N 1/sec, O 2 mole %, CO mole %, NO x ppm, N 2 O ppm (species are on a wet basis). Variance = 0.0, N reduced 80% and Sc reduced 50%.

198

Table 6-1: Results at sample probe for 2 cc jet stirred reactor, 1.78 msec, 1.0 atm., 593 K, inlet velocity 470 m/s, equivalence ratio = 0.606. O 2 and CO are mole percentage wet, NO x and N 2 O are ppm wet, temperature is K. C

Temp

Scalar

O2

CO

K

dissipation

mole

mole

1/sec

%

%

NO x ppmv

N 2O ppmv

Baseline

0.9615 1,852

13.9

7.99

0.32

1,742

0.36

Reduce Sc 50%

0.9276 1,812

14.8

8.1

0.576

591

0.39

C3=0.0

0.9615 1,852

13.8

7.99

0.32

1,743

0.36

Variance = 0.0

0.9529 1,843

17.3

8.00

0.374

1,489

0.40

Reduce N 50%

0.9660 1,857

6.1

7.99

0.29

1,590

0.37

Alternate N

0.9699 1,862

0.9

8.02

0.289

942

0.42

RNG model

0.9491 1,837

3.3

8.03

0.44

1,879

0.33

Modified RNG

0.9497 1,837

29

8.03

0.43

1,889

0.33

Variance=0.0, Sc 0.9302 1,815

2.7

8.1

0.555

23.7

0.39

reduce 50% and N reduced 80% Steele (1995)

-

1,801

-

8.32

0.224

6.05

0.67

Equilibrium

1.0000 1,891

-

7.59

0.0011

3,674

0.20

PSR, 1 msec

0.9200 1,787

-

7.95

0.662

13.4

0.40

PSR, 1.5 msec

0.9350 1,806

-

7.89

0.534

15.8

0.40

199

Table 6-2: Results at sample probe for 2 cc jet stirred reactor, 3.98 msec, 4.74 atm., 603 K, inlet velocity 293 m/s, equivalence ratio = 0.529. O 2 and CO are mole percentage wet, NO x and N 2 O are ppm wet, temperature is K. C

Temp

Scalar

K

dissipation mole mole % 1/sec %

O2

CO

NO x

N 2O

ppmv

ppmv

Baseline

0.9968 1,769

4.4

9.57

0.00127 2,539

0.311

Reduce Sc 50%

0.9966 1,769

6.6

9.57

0.00127 2,538

0.31

C3=0.0

0.9969 1,770

29.0

9.57

0.00127 2,539

0.31

Variance = 0.0

0.9958 1,769

6.5

9.58

0.0839

1,949

0.70

Reduce N 50%

0.9972 1,770

1.7

9.57

0.00127 2,539

0.31

Increase N by

0.9951 1,768

91.1

9.58

0.0120

2,516

0.32

Alternate N

0.9976 1,770

0.09

9.56

0.00127 2,540

0.31

RNG model

0.9970 1,770

6.0

9.57

0.00127 2,539

0.31

Modified RNG

0.9968 1,769

5.8

9.57

0.00127 2,538

0.31

Variance=0.0, Sc 0.9980 1,771

0.9

9.60

0.090

1,562

0.97

4.36

1.84

factor of 10.0

reduce 50% and N reduced 80% Steele (1995)

-

1,764

-

9.83

0.031

Equilibrium

1.0000 1,772

-

9.24

0.00122 2,459

0.30

PSR, 2 msec

0.987

1,757

-

9.42

0.156

7.83

2.1

PSR, 3 msec

0.991

1,762

-

9.40

0.118

8.70

2.1

Tonouchi (1996)

-

1,775

-

-

0.034

4.8

1.65

200

Chapter VII: Non-Adiabatic CMC Method

In the previous chapters the premixed CMC method is developed and results are presented for adiabatic systems. Here modifications to these equations are presented and discussed for non-adiabatic systems. We have not, however, numerically explored the implementations of this approach in the present work.

The first difference is in the definition of c. Equation 3.11 reproduced below is the definition suggested by Bilger (1993b).

c( x , t ) ≡

(h s − 2h ) − (h su − 2h u ) h s − h su

7.1

For adiabatic systems the enthalpy is constant and equation 7.1 reduces to the following relationship of sensible enthalpy. This is the definition used in the previous chapters.

c( x , t ) ≡

h s − h su h sad − h su

7.2

With equation 7.2 the maximum value of c is unity when the reactions proceed to the adiabatic equilibrium limit. With heat loss the reaction will not necessarily reach the equilibrium condition and as the temperature decreases due to the heat loss, c will decrease, so c is no longer a monotonically increasing function. This will cause c to become multi-valued, i.e. two values of c will have the same sensible enthalpy and temperature, but different temperature and species mass fractions. The goal is to define c to account for heat loss and to remain a monotonically increasing function. Equation 7.1

201

fits this requirement. At first this definition seems to be arbitrary, but rearranging the equation with equation A4 gives the following.

c( x , t ) ≡

∑ (Y

i,u

− Yi )h f ,i + ∫ WR ∆t h sad − h su

7.3

The first term in the numerator is the difference of the heats of formation of the reactants and products (how much energy is released by the reactions) and the second term in the numerator is the rate of heat loss integrated over the time the heat was lost. For an adiabatic system this reduces to the definition of equation 7.2. For a non-adiabatic system the maximum value of c is greater than unity and c will be a monotonically increasing function. The maximum value of c will be when the reactions proceed to the equilibrium limit adiabatically without heat loss, then the gases are cooled to their initial conditions. The amount of heat removed will be the heating value of the fuel, so a new definition of c can be written as follows.

c( x , t ) ≡

∑ (Y

i,u

− Yi )h f ,i + ∫ WR ∆t Q LHV

7.4

Here the denominator is the lower heating value of the fuel. With this definition the maximum value of c is unity and for an adiabatic system the maximum value of c will be less then unity. Both definitions will give the same solution as long as the equations are derived accordingly. Pierce (2001) suggests using a definition of c based on the entropy (though no relationship is given). Another definition based on the Gibbs free energy could be derived. It is believed that both definitions could be transformed to equation 7.4 using thermodynamic relations.

202

The CMC equation (equation 3.16) will be the same with heat loss, with the definition of Sc retaining the heat loss term WR . A similar looking equation is solved for the conditioned enthalpy.

< ρD(∇c) 2 | ζ >

∂ 2Q h ∂Q h − < ρSc | ζ > + < ρWR | ζ >= 0 2 ∂ζ ∂ζ

7.5

Where Qh is the conditioned enthalpy and the reaction rate in the first term is replaced with the heat loss rate. The c=0 boundary condition will remain the same, set to the unburned conditions. The second boundary condition must be modified, because c=1 is no longer the equilibrium condition, in fact the system will most likely never reach equilibrium with heat loss. The new boundary condition must reproduce the adiabatic

′ results when there is no heat loss. Smith (1994) suggests using Qi (c max ) = 0 , stating that the slope of species at the maximum c tends to zero due to high heat loss, i.e. the reactions stop in the post flame. In physical space the system may never reach the maximum value of c, but in c space the equations are solved over the entire possible range of c and only the values of c calculated by the CFD code will be used. Bilger ″ (1993b) suggests Qi (c = 0) = 0 , because there are no reactions at c=0. This changes the

equation type from a boundary value problem to an initial value problem. Neither author has published results using these boundary conditions, or any other boundary conditions, so it is not known if they work. Both boundary conditions were tried in this work and neither would give a converged solution. Neither boundary condition guarantees mass or atom conservation at c=1, but it is not known if that was the reason they would not converge. The problem may have been with the numerics. More research is needed to fully understand this problem.

The conditioned species are now a function of c, the scalar dissipation and the heat loss. The solution will depend on the time history of the heat loss, i.e. 1) at what value of c the heat loss starts, 2) the rate that the heat is lost and 3) how much total heat is

203

lost. Two examples are 1) no heat loss until equilibrium is reached then rapid heat loss or 2) a slow, constant heat loss starting at c=0 with equilibrium never being reached. Both cases will give drastically different final solutions. The possible solution routes are infinite with heat loss, making a table lookup impractical. It is most likely that the unsteady form of the CMC equations must be used, to account for the time history of the heat loss. For open flames where the heat loss is only by radiation it may be possible to make the heat loss a function of c and assume that the heat loss is the same everywhere in the system. For closed reactors, like the backward facing step used in Chapter V, this will not be appropriate. Generally, each location in the reactor will have a different heat loss history, so the conditioned species at each location will be different. This will invalidate the uniformed conditioned CMC model used in this work and will require the gradient of the conditioned species term to be retained, along with the conditioned velocity term that multiplies it, see equation 3.14.

The equations for c and its variance are still solved with the CFD code with the

WR term added to Sc with a model used to calculate WR . At every iteration c, the scalar dissipation, WR and the CFD code time step are passed to the unsteady CMC code, which calculates the new conditioned variables. The CMC code passes the new density and temperature back to the CFD code. A table lookup of pre-calculated CMC results cannot be used, the unsteady CMC equation must be solved at the same time as the CFD code. This will greatly increase the computational cost and will necessitate a smaller kinetic mechanism to keep runtimes reasonable.

For testing such a model it is

recommended to start with a 1-step mechanism to simplify the calculations.

204

Chapter VIII: Conclusions and Future Work

8.1

Conclusions

The CMC methodology has been successfully adapted to turbulent premixed combustion. The uniformed conditioned species version is developed and compared to 2dimensinal and 3-dimensional reacting flow data with encouraging results, see also Martin et al. (2001, 2002) and Martin (2003). The CMC equations are solved offline and used as table lookups with the solution of the fluid equations. By using the table lookup, detailed kinetics could be used with similar runtimes as current models. All research objectives were met, with the key findings listed below.

The 2-dimensional backward facing step results were very good, except for CO in the flame, which was over predicted by a factor of 6. It is believed that the main source of this error is the calculation of the conditional scalar dissipation.

The 3-dimensional jet stirred reactor results were good, but not as good as the backward facing step results. Several model constants were varied to understand their affect on the solution. Due to the limited number of variables from the experimental data, an optimum set of constants could not be obtained. NO was severely over predicted because of the extreme sensitivity of NO to the conditional scalar dissipation and the PDF near c=1. The jet stirred reactor problems shown here had much more recirculation then the backward facing step, which makes it harder for the turbulence models to predict the flow field, which is another reason the results were not as good.

The main source of error was from the turbulence models, which were unable to predict the turbulence generated by the combustion. The second order turbulence models offered no improvement over the first order models for reacting flows. The error in

205

turbulence generation directly affected the scalar dissipation and had an affect on the closure models for the c and its variance equations.

Assuming that the conditional scalar dissipation was equal to the unconditional value was most likely a poor assumption that will need further research once the appropriate data is available.

The full GRI and Miller-Bowman kinetic mechanisms gives very similar results. A 16 species, 24-step reduced mechanism based on the full Miller-Bowman burned noticeably faster and a 2-step global mechanism burned instantly, at the reactor inlet, showing the advantage of full mechanisms. In the case of a 1-step mechanism the species are a linear function of c, independent of scalar dissipation. This is also true when the scalar dissipation becomes infinite.

The uniformed conditioned species version of the premixed CMC equation is demonstrated to be the same as the 1-dimensional laminar flamelet model, with the c=1 boundary conditions being the only difference. The models are derived using different assumptions for totally different flow conditions, but arrived at the same equation. The CMC model assumes strong bulk mixing while the laminar flamelet model assumes no bulk mixing and fast reaction rates, both leading to the assumption that the conditioned species are uniform in the reactor.

The advantage of the premixed CMC model over other combustion models is, 1) it allows the use of detailed kinetic mechanisms with reasonable runtimes, 2) it accounts for the affects of small scale mixing on the reaction rates, 3) accounts for the affects of temperature fluctuations on the reaction rates, and 4) allows multiple chemical time scales.

206

A non-adiabatic model is proposed, but a converged solution was not obtained. The unsteady form of the CMC equations may be required for this application with the equations solved along with the fluids equations, i.e. a table lookup approach may not be possible.

The G equation model is shown to be an extension of the BML model, where the diffusion and reaction terms are replaced with an ad hoc flame displacement speed model. G is a coordinate transformation of c or YCH 4 .

8.2

Future Work

While the initial results of the simplified premixed CMC method are encouraging, much work is needed to extend the model to other flow conditions. Below is a list of recommended research to achieve this.

Extend the model to flows where the conditional species are not uniform, which will require the development of a model for the conditioned velocity. Experimental and DNS data such as Mantel and Bilger (1995) will be required.

Use experimental and DNS data to estimate the size of the error terms that were neglected and develop a model for the important terms.

Improved closure models will need to be developed for the turbulent scalar flux to allow for gradient and counter gradient diffusion. This term is in the reaction progress variable equation and the equation for its variance. Existing experimental data, such as Chen et al. (2000) and DNS data, such as Swaminathan et al. (1997, 2001) can be used.

207

Models for the conditioned scalar dissipation must be developed using DNS databases, such as Swaminathan and Bilger (2001a, 2001b) and available experimental data.

The accurate prediction of flame extinction and re-ignition will require one or both of the following, 1) doubly conditioned formulation, 2) second order conditioning. Cha et al. (2000, 2001) developed a doubly conditioned version of the non-premixed CMC method with encouraging improvements for model predictions. They used the scalar dissipation in addition to the mixture fraction as conditioning variables. For premixed flames, available data can be analyzed to determine the appropriate second conditioning variable, following which a model can be developed. Cha and Pitsch (2001, 2002), Kronenburg et al. (1998), Mastorakos and Bilger (1998) and Swaminathan and Bilger (1998) have developed second order CMC models for non-premixed flames. This second order conditioning uses a conditional variance of the species mass fractions to improve the closure of the reaction rate source term. For premixed combustion a similar approach may be taken, a conservation equation for the conditional variance will need to be developed based on available data. The new doubly conditioned and second order conditioning models can be compared individually to experimental and DNS data. If neither model gives satisfactory results, a combination of the two models may be needed.

Improved RANS turbulence models must be developed, both first and second order, to improve the prediction of turbulence generation from combustion. The next step would be to use an LES model, though work needs to be done to improve and validate its use for reacting flows. This should give a more accurate solution of the fluids equations, but at a large computational expense, which may limit the size of the kinetic mechanism that can be used.

208

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219

Appendix A Derivation of the Enthalpy Conservation Equations

The general form of the energy equation with no heat loss and neglecting the unsteady pressure term is

ρ

∂u ∂h ∂P + ρu ⋅ ∇h = ∇ ⋅ (k∇T) + τ ij i − ρWR + ∂t ∂x j ∂t

A1

For gases the last term is negligible. The third term is rewritten as

∇ ⋅ (k∇T ) = ∇ ⋅ ( Ckp ∇h ) = ∇ ⋅ (L e ρD h ∇h ) − ρWR +

∂P ∂t

A2

Smith (1994) and Bilger (1993b) assume the Le number is unity (this is a good approximation for gases), which gives

ρ

∂h ∂P + ρu ⋅ ∇h = ∇ ⋅ (ρD h ∇h ) − ρWR + ∂t ∂t

A3

The enthalpy is made up of the sensible enthalpy and the sum of the heats of formation.

N

h ≡ h s + ∑ h f ,i Yi i

Equation A4 is substituted into equation A3 to give

A4

220

∂ ∑ h f ,i Yi ∂h s +ρ + ρu ⋅ ∇h s + ρu ⋅ ∇[∑ h f ,i Yi ] = ∇ ⋅ (ρD h ∇h s ) + ∂t ∂t ∂P ∇ ⋅ (ρD h ∇[∑ h f ,i Yi ]) − ρWR + ∂t

ρ

∂ ∑ h f ,i Yi ∂h s + ρu ⋅ ∇h s = ∇ ⋅ (ρD h ∇h s ) − ρ − ρu ⋅ ∇[∑ h f ,i Yi ] + ∂t ∂t ∂P ∇ ⋅ (ρD h ∇[∑ h f ,i Yi ]) − ρWR + ∂t

ρ

The heats of formation are assumed constant with respect to spatial location . They have a weak temperature dependence, but since the temperature is close to uniform inside the reactor their gradients will be small, so this assumption will be a negligible error. The summations are taken outside of the derivatives to give. ∂Y ∂h s + ρu ⋅ ∇h s = ∇ ⋅ (ρD h ∇h s ) − ρ∑ (h f ,i i ) − ρu ⋅ (∑ h f ,i ∇Yi ) + ∂t ∂t ∂P ∇ ⋅ [ρD h (∑ h f ,i ∇Yi )] − ρWR + ∂t

ρ

∂Y ∂h s ρ + ρu ⋅ ∇h s = ∇ ⋅ (ρD h ∇h s ) − ∑ h f ,i [ρ i + ρu ⋅ ∇Yi − ∇ ⋅ (ρD h ∇Yi )] ∂t ∂t ∂P − ρWR + ∂t

The species conservation equation is

ρ


A5

This is substituted into the previous equation assuming constant and equal diffusivities to give.

221

ρ

∂h s ∂P + ρu ⋅ ∇h s = ∇ ⋅ (ρD h ∇h s ) − ρWR + − ∑ h f ,i ω i ∂t ∂t

A6

This is different then Bilger (1993b) equation 16, which is missing the minus sign in front of the last term.

Using Bilger (1993b) equation 16 the c equation is the same as in Bilger (1993b) and Klimenko & Bilger (1999). Using equation A6 gives the c equation the way Smith (1994) has it, which is the correct way, therefore equation A6 is used here. The CMC equation is derived in Appendix B.

Using the correct relation (equation A6) the CMC equation is consistent using the 1-step mechanism and produces the expected results, independent of the scalar dissipation.

222

Appendix B

Derivation of the Reaction Progress Variable Equation

The conservation equation for c is derived by combining equations 3.7 and 3.8 in the following form, (h s − 2h ) − (h su − 2h u ) , while neglecting the pressure terms.

[ρ

= +

∂h s ∂h ∂h s ∂h + ρu ⋅ ∇h s ] − 2[ρ + ρu ⋅ ∇h ] − [ρ + ρu ⋅ ∇h s ]u + 2[ρ + ρu ⋅ ∇ h ] u ∂t ∂t ∂t ∂t ∆h sad −u

[∇ ⋅ (ρD h ∇h s ) − ρWR − ρ∑ ωi h f ,i ] − 2[∇ ⋅ (ρD h ∇h ) − ρWR ] i

∆h sad −u

− [∇ ⋅ (ρD h ∇h s ) − ρWR − ρ∑ ωi h f ,i ]u + 2[∇ ⋅ (ρD h ∇h ) − ρWR ]u i

∆h sad −u

B1

[ρ

∂h s ∂h ∂h s ∂h − 2ρ −ρ | u +2ρ | u ] + [ρu ⋅ ∇h s − 2ρu ⋅ ∇h − ρu ⋅ ∇h s | u +2ρu ⋅ ∇h | u ] ∂t ∂t ∂t ∂t ∆h sad − u

[∇ ⋅ (ρD h ∇h s ) − 2∇ ⋅ (ρD h ∇h ) − ∇ ⋅ (ρD h ∇h s ) | u +2∇ ⋅ (ρD h ∇h ) | u ] = ∆h sad −u +

[−ρWR + 2ρWR + ρWR | u −2ρWR | u ] + [−ρ∑ ωi h f ,i + ρ∑ ωi h f ,i | u ] i

i

∆h sad − u

WR | u = ρ∑ ωi h f ,i | u = 0 , since there is no radiation or reactions in the unburned state. i

223

[h s − 2h − h s | u +2h | u ] ∂ h s − 2 h − h s | u +2 h | u ρ [ ] + ρu ⋅ ∇ ∂t ∆h sad −u ∆h sad − u [h s − 2h − h s | u +2h | u ] = ∇ ⋅ (ρD h )∇ + ∆h sad −u

∂c ρ + ρu ⋅ ∇c = ∇ ⋅ (ρD h ∇c) + ∂t

[ρWR ] − [ρ∑ ωi h f ,i ] i

∆h sad − u

[ρWR ] − [ρ∑ ωi h f ,i ] ∆h

i s ad − u

≡ ∇ ⋅ (ρD h ∇c) + ρS c

B2

224

Appendix C

Derivation of the PDF Conservation Equation The PDF conservation equation for c is derived by introducing a function γ , defined as

γ (ζ , x , t ) ≡ δ(c ( x , t ) − ζ )

C1

The PDF for c is defined as Pζ (ζ, x , t ) ≡< γ (ζ, x , t ) >

C2

Klimenko & Bilger (1999) derive 3 relationships for Dirac delta functions as follows. ∂γ ∂ ∂c = − (γ ) ∂t ∂ζ ∂t

C3

∂ ( γ∇c) ∂ζ

C4

∇γ = −

∇ ⋅ (ρD h ∇γ ) =

∂2 ∂ ( γρD h (∇c) 2 ) − ( γ∇ ⋅ (ρD h ∇c)) 2 ∂ζ ∂ζ

C5

These are Klimenko & Bilger (1999) equations 35, 37 and 38 and Smith (1994) equations 3.12, 3.13 and 3.13. Equation C3 is re-written as

ρ

∂γ ∂ ∂c = − ( γρ ) ∂t ∂ζ ∂t

C6

225

Substituting equation 3.12 into equation C6 gives

ρ

∂ ∂γ = − ( γ (∇ ⋅ (ρD h ∇c) + ρSc − ρu ⋅ ∇c)) ∂t ∂ζ

C7

Next equation C4 is multiplied by ρu ⋅ and making use of the continuity equation giving.

ρu ⋅ ∇γ = −

∂ ( γρu ⋅ ∇c) ∂ζ

C8

Combining this with equation C7 gives

ρ

∂γ ∂ + ρu ⋅ ∇γ = (− γ∇ ⋅ (ρD h ∇c) − γρSc ) ∂t ∂ζ

C9

With equation C5 this becomes

ρ

∂γ ∂2 ∂ + ρu ⋅ ∇γ − ∇ ⋅ (ρD h ∇γ ) = − 2 ( γρD h (∇c) 2 ) − ( γρSc ) ∂t ∂ζ ∂ζ

C10

This is Smith (1994) equation 3.59. Using the continuity equation this can be written in divergence form. ∂ργ ∂2 ∂ + ∇ ⋅ (ρuγ ) − ∇ ⋅ (ρD h ∇γ ) = − 2 ( γρD h (∇c) 2 ) − ( γρS c ) ∂t ∂ζ ∂ζ

C11

This is Klimenko & Bilger (1999) equation 54. This equation is averaged using the following relationship

226

< ργ >=< ρ | ζ > Pζ (ζ )

C12

This is Klimenko & Bilger (1999) equation 29. The angle brackets signify averages and Pζ (ζ ) is the PDF for the RPV, c. This gives the PDF conservation equation as

∂ < ρ | ζ > Pζ ∂t

+ ∇ ⋅ (< ρu | ζ > Pζ ) − ∇ ⋅ (< ρD h | ζ > ∇Pζ ) =

∂2 ∂ − 2 (< ρD h (∇c) 2 | ζ > Pζ ) − (< ρS c | ζ > Pζ ) ∂ζ ∂ζ

This is Smith (1994) equation 3.60 and Klimenko & Bilger (1999) equation 56.

C13

227

Appendix D

Derivation of the Premixed CMC Equation

The premixed CMC equation is derived in a similar manner as the PDF equation (see Klimenko and Bilger 1999, section 2.4.1). A function is defined in terms of the mass fraction and RPV as

ϕ(ζ, s, x, t ) ≡ δ(Yi ( x, t ) − s)δ(c( x , t ) − ζ )

D1

The joint PDF for Y and c is defined as Pζ ,s (ζ, s, x , t ) ≡< ϕ(ζ, s, x , t ) >

D2

Three relationships for Dirac delta functions are derived as in Appendix C.

ρ

∂Y ∂ϕ ∂ ∂c ∂ = − (ρϕ ) − (ρϕ i ) ∂t ∂ζ ∂t ∂s ∂t

ρu ⋅ ∇ ϕ = −

∂ (ϕρu ⋅ ∇c) − s(ϕρu ⋅ ∇Yi ) ∂ζ

− ∇ ⋅ (ρD i ∇ϕ) = −

∂2 ∂2 2 ( ϕρ D ( ∇ c ) ) − (ϕρD i (∇Yi ) 2 ) i ∂ζ 2 ∂s 2

∂2 ∂ ∂ −2 (ϕρD i (∇c ⋅ ∇Yi )) + (ϕ∇ ⋅ (ρD i ∇c)) + (ϕ∇ ⋅ (ρD i ∇Yi )) ∂s ∂ζ∂s ∂ζ

These 3 equations are summed to give

D3

D4

D5

228

ρ

∂Y ∂ϕ ∂ ∂c ∂ + ρu ⋅ ∇ϕ − ∇ ⋅ (ρD i ∇ϕ) = − (ρϕ ) − (ρϕ i ) ∂t ∂t ∂ζ ∂t ∂s

∂2 ∂ ∂ ∂2 2 − (ϕρu ⋅ ∇c) − (ϕρu ⋅ ∇Yi ) − 2 (ϕρD i (∇c) ) − 2 (ϕρD i (∇Yi ) 2 ) ∂ζ ∂s ∂s ∂ζ

D6

∂2 ∂ ∂ −2 (ϕρD i (∇c ⋅ ∇Yi )) + (ϕ∇ ⋅ (ρD i ∇c)) + (ϕ∇ ⋅ (ρD i ∇Yi )) ∂ζ∂s ∂ζ ∂s ∂ϕ ∂ ∂c + ρu ⋅ ∇ϕ − ∇ ⋅ (ρD i ∇ϕ) = − [ϕ(ρ + ρu ⋅ ∇c − ∇ ⋅ (ρD i ∇c))] ∂t ∂ζ ∂t ∂Y ∂ − [ϕ(ρ i + ρu ⋅ ∇Yi − ∇ ⋅ (ρD i ∇Yi ))] ∂s ∂t 2 ∂ ∂2 ∂2 − 2 (ϕρD i (∇c) 2 ) − 2 (ϕρD i (∇Yi ) 2 ) − 2 (ϕρD i (∇c ⋅ ∇Yi )) ∂ζ∂s ∂ζ ∂s

ρ

D7

Using equations 3.12 and 3.6 the following 2 relationships are derived.

ρ

∂c + ρu ⋅ ∇c − ∇ ⋅ (ρD i ∇c) = ∇ ⋅ (ρ(D h − D i )∇c) + ρS c ∂t

ρ

∂Yi + ρu ⋅ ∇Yi − ∇ ⋅ (ρD i ∇Yi ) = ρωi ∂t

3.12

3.6

Inserting these 2 equations into equation D7 gives

ρ

∂ϕ ∂ + ρu ⋅ ∇ϕ − ∇ ⋅ (ρD i ∇ϕ) = − [ϕ(∇ ⋅ (ρ(D h − D i )∇c) + ρS c )] ∂t ∂ζ

∂ ∂2 ∂2 ∂2 2 2 − [ϕ(ρωi )] − 2 (ϕρD i (∇c) ) − 2 (ϕρD i (∇Yi ) ) − 2 (ϕρD i (∇c ⋅ ∇Yi )) ∂s ∂ζ∂s ∂ζ ∂s

Next equation D8 is put into divergence form using the continuity equation.

D8

229

∂ ∂ρϕ + ∇ ⋅ (ρuϕ) − ∇ ⋅ (ρD i ∇ϕ) = − [ϕ(∇ ⋅ (ρ(D h − D i )∇c) + ρS c )] ∂t ∂ζ ∂ ∂2 ∂2 ∂2 − [ϕ(ρωi )] − 2 (ϕρD i (∇c) 2 ) − 2 (ϕρD i (∇Yi ) 2 ) − 2 (ϕρD i (∇c ⋅ ∇Yi )) ∂s ∂ζ∂s ∂ζ ∂s

D9

This equation is averaged using a similar expression to equation C12 to give ∂ [< ρ | ζ, s > Pζ ,s ] + ∇ ⋅ [< ρu | ζ, s > Pζ ,s ] − ∇ ⋅ [< ρD i | ζ, s > ∇Pζ ,s ] = ∂t ∂ ∂ − [< (∇ ⋅ (ρ(D h − D i )∇c) + ρS c ) | ζ, s > Pζ ,s ] − [< ρωi | ζ, s > Pζ ,s ] ∂ζ ∂s ∂2 ∂2 − 2 [< ρD i (∇c) 2 | ζ, s > Pζ ,s ] − 2 [< ρD i (∇Yi ) 2 | ζ, s > Pζ ,s ] ∂ζ ∂s −2

D10

∂2 [< ρD i (∇c ⋅ ∇Yi ) | ζ, s > Pζ ,s ] ∂ζ∂s

This is Smith (1994) equation 3.63. Equation D10 is multiplied by s and integrated from s 0 to s1 . To do this the following 2 relationships are used.

∫

s1

sPζ ,s ds =< Yi | ζ > Pζ

D11

< ρYi | ζ >≡< ρ | ζ > Q i

D12

s0

Equation D.11 is Klimenko & Bilger (1999) equation 84. Equation D12 is the Favre average of the conditioned mass fraction. Where appropriate, all terms are Favre averaged unless stated other wise, though the standard tilda over bar notation is not used.

First term of equation D10

230

∫

s1

s0

=

s

∂ ∂ s1 ∂ [< ρ | ζ, s > Pζ ,s ]ds = ∫ s < ρ | ζ, s > Pζ ,s ds = [< ρYi | ζ > Pζ ] ∂t ∂t s 0 ∂t

∂ [< ρ | ζ > Q i Pζ ] ∂t

Second term

∫

s1

s0

s∇ ⋅ [< ρu | ζ, s > Pζ ,s ]ds = ∇ ⋅ ∫ s < ρu | ζ, s > Pζ ,s ds = ∇ ⋅ [< ρuYi | ζ > Pζ ] s1

s0

Third term

∫

s1

s0

s∇ ⋅ [< ρD i | ζ, s > ∇Pζ ,s ]ds = ∇ ⋅ ∫ s < ρD i | ζ, s > ∇Pζ ,s ds = ∇ ⋅ [< ρD i ∇Yi | ζ > ∇Pζ ] s1

s0

= ∇ ⋅ [< ρD i | ζ > ∇(Q i Pζ )]

It is believed that the last step requires D i to be constant.

Fourth term ∂ [< (∇ ⋅ (ρ(D h − D i )∇c) + ρSc ) | ζ, s > Pζ ,s ]ds s 0 ∂ζ ∂ s1 ∂ s1 = s < ∇ ⋅ (ρ(D h − D i )∇c) | ζ, s > Pζ ,s ds + ∫ s < ρSc | ζ, s > Pζ ,s ]ds ∫ ∂ζ s0 ∂ζ s 0 ∂ ∂ = [< ∇ ⋅ (ρ(D h − D i )∇c)Yi | ζ > Pζ ] + [< ρSc Yi | ζ > Pζ ] ∂ζ ∂ζ

∫

s1

s

Fifth term, integrate by parts

∫

s1

s0

s

∂ [< ρωi | ζ, s > Pζ ,s ]ds = [s < ρωi | ζ, s > Pζ ,s ]ss10 − < ρωi | ζ > Pζ = − < ρωi | ζ > Pζ ∂s

231

Sixth term

∫

s1

s0

s

∂2 ∂2 2 [ < ρ D ( ∇ c ) | ζ , s > P ] ds = [< ρD i (∇c) 2 Yi | ζ > Pζ ] i ζ ,s 2 2 ∂ζ ∂ζ

Seventh term, integrate by parts ∂2 ∂ s 2 2 ∫s0 ∂s 2 [< ρD i (∇Yi ) | ζ, s > Pζ,s ]ds = s ∂s [< ρD i (∇Yi ) | ζ, s > Pζ,s ]s10 s1 ∂ −∫ [< ρD i (∇Yi ) 2 | ζ, s > Pζ ,s ]ds = 0 − [< ρD i (∇Yi ) 2 | ζ, s > Pζ ,s ]ss10 = 0 s 0 ∂s s1

s

Eighth term, integrate by parts ∂2 ∂ s1 ∂ ∫s0 ∂ζ∂s [< ρD i (∇c ⋅ ∇Yi ) | ζ, s > Pζ,s ]ds = 2 ∂ζ ∫s0 s ∂s [< ρD i (∇c ⋅ ∇Yi ) | ζ, s > Pζ ,s ]ds ∂ ∂ s1 = 2 [s < ρD i (∇c ⋅ ∇Yi ) | ζ, s > Pζ ,s ]ss10 − 2 ∫ < ρD i (∇c ⋅ ∇Yi ) | ζ, s > Pζ ,s ds ∂ζ ∂ζ s 0 ∂ = 0 − 2 [< ρD i (∇c ⋅ ∇Yi ) | ζ > Pζ ] ∂ζ s1

2s

Combining the 8 terms gives ∂ [< ρ | ζ > Q i Pζ ] + ∇ ⋅ [< ρuYi | ζ > Pζ ] − ∇ ⋅ [< ρD i | ζ > ∇(Q i Pζ )] ∂t ∂ ∂ = − [< ∇ ⋅ (ρ(D h − D i )∇c)Yi | ζ > Pζ ] − [< ρS c Yi | ζ > Pζ ]+ < ρωi | ζ > Pζ ∂ζ ∂ζ −

∂2 ∂ [< ρD i (∇c) 2 Yi | ζ > Pζ ] − 0 + 2 [< ρD i (∇c ⋅ ∇Yi ) | ζ > Pζ ] 2 ∂ζ ∂ζ

D13

232

∂ [< ρ | ζ > Q i Pζ ] + ∇ ⋅ [< ρuYi | ζ > Pζ ] − ∇ ⋅ [< ρD i | ζ > ∇(Q i Pζ )] ∂t ∂G ζ =< ρωi | ζ > Pζ + ∂ζ where ∂ G ζ ≡ 2 < ρD i (∇c ⋅ ∇Yi ) | ζ > Pζ − [< ρD i (∇c) 2 Yi | ζ > Pζ ] ∂ζ − < ρSc Yi | ζ > Pζ − < ∇ ⋅ (ρ(D h − D i )∇c)Yi | ζ > Pζ

D14

This is Smith (1994) equation 3.64, though he drops the differential diffusion term because it is small for high Reynolds number flows. At this point everything follows Smith (1994) derivation and there are no approximations. Next the instantaneous mass fraction is split into an average and fluctuating component as follows, with equal diffusivities. Yi = Q i + y i ∇Yi = ∇Q i + ∇c

∂Q i + ∇y i ∂ζ

Equation D15b comes from Bilger (1991) equation 10. equation D14b to give. G ζ ≡ 2 < ρD(∇c ⋅ ∇{Q i + y i }) | ζ > Pζ ∂ [< ρD(∇c) 2 {Q i + y i } | ζ > Pζ ] ∂ζ − < ρSc {Q i + y i } | ζ > Pζ −

D15

These are substituted into

233

G ζ ≡ 2 < ρD(∇c ⋅ ∇Q i ) | ζ > Pζ + 2 < ρD(∇c ⋅ ∇c + 2 < ρD(∇c ⋅ ∇y i ) | ζ > Pζ − −

∂Q i ) | ζ > Pζ ∂ζ

∂ [< ρD(∇c) 2 Q i | ζ > Pζ ] ∂ζ

∂ [< ρD(∇c) 2 y i | ζ > Pζ ]− < ρSc Q i | ζ > Pζ − < ρS c y i | ζ > Pζ ∂ζ

G ζ ≡ 2 < ρD(∇c ⋅ ∇Q i ) | ζ > Pζ + 2 < ρD(∇c ⋅ ∇c) | ζ > Pζ + 2 < ρD(∇c ⋅ ∇y i ) | ζ > Pζ − −

∂Q i ∂ζ

∂ [< ρD(∇c) 2 | ζ > Q i Pζ ] ∂ζ

D16

∂ [< ρD(∇c) 2 y i | ζ > Pζ ]− < ρS c | ζ > Q i Pζ − < ρS c y i | ζ > Pζ ∂ζ

Smith (1994) 3.67 is different than this, as shown below.

G ζ ≡ 2 < ρ(∇c ⋅ ∇Q i ) | ζ > Pζ + 2 < ρD(∇c) 2 | ζ > + 2 < ρD(∇c ⋅ ∇y i ) | ζ > −

∂Pζ ∂ζ

−

∂ (Q i Pζ ) ∂ζ

∂ [< ρD(∇c) 2 | ζ > Q i Pζ ] ∂ζ

D17

∂ [< ρD(∇c) 2 y i | ζ > Pζ ]− < ρS c | ζ > Q i Pζ − < ρS c y i | ζ > Pζ ∂ζ

Smith (1994) is missing the diffusivity in the first term, has the derivative different in the second term and has an additional derivative in the third term, these could just be typos. Next combine equation D14a and D16 (the differences that follow might be because the derivation here (equation D.16) doesn’t match equation D17, which is from Smith (1994)).

234

∂ [< ρ | ζ > Q i Pζ ] + ∇ ⋅ [< ρuYi | ζ > Pζ ] − ∇ ⋅ [< ρD | ζ > ∇(Q i Pζ )] = ∂t ∂Q i ∂ ∂ < ρωi | ζ > Pζ + 2 [< ρD(∇c ⋅ ∇Q i ) | ζ > Pζ ] + 2 [< ρD(∇c) 2 | ζ > Pζ ] ∂ζ ∂ζ ∂ζ +2 −

∂ ∂2 [< ρD(∇c ⋅ ∇y i ) | ζ > Pζ ] − 2 [< ρD(∇c) 2 | ζ > Q i Pζ ] ∂ζ ∂ζ

∂2 ∂ ∂ [< ρD(∇c) 2 y i | ζ > Pζ ] − [< ρSc | ζ > Q i Pζ ] − [< ρS c y i | ζ > Pζ ] 2 ∂ζ ∂ζ ∂ζ

Replace the mass fraction in the second term with equation D15 and simplify the derivatives to give.

∂ ∂ [< ρ | ζ > Pζ ]+ < ρ | ζ > Pζ [Q i ] + ∇ ⋅ [< ρu{Q i + y i } | ζ > Pζ ] ∂t ∂t ∂ − ∇ ⋅ [< ρD | ζ > ∇(Q i Pζ )] =< ρωi | ζ > Pζ + 2 [< ρD(∇c ⋅ ∇Q i ) | ζ > Pζ ] ∂ζ ∂Q ∂ ∂ ∂Q +2 i [< ρD(∇c) 2 | ζ > Pζ ] + 2 < ρD(∇c) 2 | ζ > Pζ [ i ] + ∂ζ ∂ζ ∂ζ ∂ζ ∂ ∂ ∂ 2 [< ρD(∇c ⋅ ∇y i ) | ζ > Pζ ] − { [< ρD(∇c) 2 | ζ > Q i Pζ ]} − ∂ζ ∂ζ ∂ζ

Qi

∂2 ∂ ∂ [< ρD(∇c) 2 y i | ζ > Pζ ] − Q i [< ρSc | ζ > Pζ ]− < ρSc | ζ > Pζ [Q i ] − 2 ∂ζ ∂ζ ∂ζ ∂ [< ρSc y i | ζ > Pζ ] ∂ζ

235

∂ ∂ [< ρ | ζ > Pζ ]+ < ρ | ζ > Pζ [Q i ] + ∇ ⋅ [< ρu | ζ > Qi Pζ ] + ∂t ∂t ∇ ⋅ [< ρuy i | ζ > Pζ ] − ∇ ⋅ [< ρD | ζ > ∇(Q i Pζ )] =< ρωi | ζ > Pζ + Qi

∂Q ∂ ∂ [< ρD(∇c ⋅ ∇Q i ) | ζ > Pζ ] + 2 i [< ρD(∇c) 2 | ζ > Pζ ] ∂ζ ∂ζ ∂ζ ∂ ∂Q ∂ + 2 < ρD(∇c) 2 | ζ > Pζ [ i ] + 2 [< ρD(∇c ⋅ ∇y i ) | ζ > Pζ ] ∂ζ ∂ζ ∂ζ ∂ ∂ ∂ − {Q i [< ρD(∇c) 2 | ζ > Pζ ]+ < ρD(∇c) 2 | ζ > Pζ [Q i ]} ∂ζ ∂ζ ∂ζ 2

∂2 ∂ − 2 [< ρD(∇c) 2 y i | ζ > Pζ ] − Q i [< ρSc | ζ > Pζ ] − ∂ζ ∂ζ ∂ ∂ < ρS c | ζ > Pζ [Q i ] − [< ρS c y i | ζ > Pζ ] ∂ζ ∂ζ ∂ ∂ [< ρ | ζ > Pζ ]+ < ρ | ζ > Pζ [Q i ] ∂t ∂t + Q i ∇ ⋅ [< ρu | ζ > Pζ ] + [< ρu | ζ > Pζ ∇ ⋅ Q i ]

Qi

∇ ⋅ [< ρuy i | ζ > Pζ ] − ∇ ⋅ [< ρD | ζ > ∇(Qi Pζ )] ∂Q ∂ ∂ [< ρD(∇c ⋅ ∇Q i ) | ζ > Pζ ] + 2 i [< ρD(∇c) 2 | ζ > Pζ ] ∂ζ ∂ζ ∂ζ ∂ ∂Q i ∂ [ ] + 2 [< ρD(∇c ⋅ ∇y i ) | ζ > Pζ ] + 2 < ρD(∇c) 2 | ζ > Pζ ∂ζ ∂ζ ∂ζ ∂ ∂ ∂ ∂ − Q i { [< ρD(∇c) 2 | ζ > Pζ ]} − [< ρD(∇c) 2 | ζ > Pζ ] {Q i } ∂ζ ∂ζ ∂ζ ∂ζ ∂ ∂ ∂ ∂ − [Q i ] < ρD(∇c) 2 | ζ > Pζ − < ρD(∇c) 2 | ζ > Pζ { [Q i ]} ∂ζ ∂ζ ∂ζ ∂ζ =< ρωi | ζ > Pζ + 2

∂2 ∂ − 2 [< ρD(∇c) 2 y i | ζ > Pζ ] − Q i [< ρS c | ζ > Pζ ] − ∂ζ ∂ζ ∂ ∂ < ρS c | ζ > Pζ [Q i ] − [< ρSc y i | ζ > Pζ ] ∂ζ ∂ζ

236

∂ [< ρ | ζ > Pζ ] + Q i ∇ ⋅ [< ρu | ζ > Pζ ] − Q i ∇ ⋅ [< ρD | ζ > ∇(Pζ )] ∂t ∂2 ∂ + Q i 2 [< ρD(∇c) 2 | ζ > Pζ ] + Q i [< ρSc | ζ > Pζ ] ∂ζ ∂ζ ∂ + < ρ | ζ > Pζ [Q i ]+ < ρu | ζ > Pζ ⋅ ∇[Q i ] ∂t ∂ 2 Qi ∂2 2 − < ρ D ( ∇ c ) | ζ > P [Q i ] =< ρωi | ζ > Pζ + 2 < ρD(∇c) 2 | ζ > Pζ ζ ∂ζ 2 ∂ζ 2 ∂ ∂ − < ρSc | ζ > Pζ [Q i ] + 2 [< ρD(∇c ⋅ ∇Q i ) | ζ > Pζ ] ∂ζ ∂ζ ∂ ∂ ∂ + 2 [< ρD(∇c ⋅ ∇y i ) | ζ > Pζ ] − [< ρD(∇c) 2 | ζ > Pζ ] {Q i } ∂ζ ∂ζ ∂ζ ∂Q ∂ ∂ ∂ − [Q i ] {< ρD(∇c) 2 | ζ > Pζ } + 2 i [< ρD(∇c) 2 | ζ > Pζ ] ∂ζ ∂ζ ∂ζ ∂ζ Qi

∂2 ∂ [< ρD(∇c) 2 y i | ζ > Pζ ] − [< ρSc y i | ζ > Pζ ] 2 ∂ζ ∂ζ − ∇ ⋅ [< ρuy i | ζ > Pζ ] + ∇ ⋅ [< ρD | ζ > ∇(Q i Pζ )] − Q i ∇ ⋅ [< ρD | ζ > ∇(Pζ )] −

The first 5 terms in the equation above are equation 3.13 multiplied by Q, so they cancel out and like terms are combined. ∂ [Q i ]+ < ρu | ζ > Pζ ⋅ ∇[Q i ] =< ρωi | ζ > Pζ ∂t ∂2 ∂ + < ρD(∇c) 2 | ζ > Pζ 2 [Q i ]− < ρSc | ζ > Pζ [Q i ] ∂ζ ∂ζ

< ρ | ζ > Pζ

∂ ∂2 [< ρD(∇c ⋅ ∇y i ) | ζ > Pζ ] − 2 [< ρD(∇c) 2 y i | ζ > Pζ ] − ∇ ⋅ [< ρuy i | ζ > Pζ ] ∂ζ ∂ζ ∂ ∂ − [< ρSc y i | ζ > Pζ ] + 2 [< ρD(∇c ⋅ ∇Q i ) | ζ > Pζ ] ∂ζ ∂ζ + ∇ ⋅ [< ρD | ζ > ∇(Q i Pζ )] − Q i ∇ ⋅ [< ρD | ζ > ∇(Pζ )] +2

The last three terms are rewritten in the next step.

237

∂Q i ∂ 2Qi + < ρu | ζ > ⋅∇Q i =< ρωi | ζ > + < ρD(∇c) 2 | ζ > ∂t ∂ζ 2 ∂Q i − < ρS c | ζ > + < e y | ζ > + < eQ | ζ > ∂ζ

< e y | ζ >=

∂2 1 ∂ {2 [< ρD(∇c ⋅ ∇y i ) | ζ > Pζ ] − 2 [< ρD(∇c) 2 y i | ζ > Pζ ] Pζ ∂ζ ∂ζ

− ∇ ⋅ (< ρuy i | ζ > Pζ ) − < e Q | ζ >=

D18

∂ (< ρS c y i | ζ > Pζ )} ∂ζ


− Q i ∇ ⋅ (< ρD | ζ > ∇Pζ )}

Equation D.18 is not exactly the same as Smith (1994) 3.68-3.70, as shown below. The e y and e Q terms have many small differences (some of these differences are from

equation D17 being different than equation D16), but Smith neglects these terms in the PTURCEL model so they are not important at this time. Klimenko and Bilger (1999) equations 103-105 are slightly different then the two forms shown here. ∂Q i ∂ 2Qi + < ρu | ζ > ⋅∇Q i =< ρωi | ζ > + < ρD(∇c) 2 | ζ > ∂t ∂ζ 2 ∂Q i − < ρS c | ζ > + < ey | ζ > + < eQ | ζ > ∂ζ ∂Pζ 1 ∂ < e y | ζ >≡ {2[< ρD(∇c ⋅ ∇y i ) | ζ > ] − [< ρD(∇c) 2 y i | ζ > Pζ ] Pζ ∂ζ ∂ζ

− ∇ ⋅ (< ρuy i | ζ > Pζ )} − (< ρS c y i | ζ >) < e Q | ζ >≡ [< ρ(∇c ⋅ ∇Q i ) | ζ >] + ∇ ⋅ (< ρD | ζ > ∇(Q i Pζ ))

D19

238

Appendix E

PTURCEL Derivation

The PTURCEL equations are derived starting with equations D14a and D16 (following Smith 1994) this will derive e y and e Q differently then in Appendix D to see which form of the error terms is correct.

∂ [< ρ | ζ > Q i Pζ ] + ∇ ⋅ [< ρuYi | ζ > Pζ ] =< ρωi | ζ > Pζ + ∇ ⋅ [< ρD | ζ > ∇(Q i Pζ )] ∂t ∂Q i ∂ ∂2 2 ] − 2 [< ρD(∇c) 2 | ζ > Q i Pζ ] + 2 [< ρD(∇c) | ζ > Pζ ∂ζ ∂ζ ∂ζ ∂ ∂ − [< ρS c | ζ > Q i Pζ ] + 2 [< ρD(∇c ⋅ ∇y i ) | ζ > Pζ ] ∂ζ ∂ζ ∂2 ∂ [< ρD(∇c) 2 y i | ζ > Pζ ] − [< ρSc y i | ζ > Pζ ] 2 ∂ζ ∂ζ ∂ + 2 [< ρD(∇c ⋅ ∇Q i ) | ζ > Pζ ] ∂ζ

−

Using equation 3.5 gives.

239

∂ [< ρ | ζ > Q i Pζ ] + ∇ ⋅ [< ρu | ζ > Q i Pζ ] + ∇ ⋅ [< ρuy i | ζ > Pζ ] =< ρωi | ζ > Pζ ∂t ∂ 2Qi + ∇ ⋅ [< ρD | ζ > ∇(Q i Pζ )] + 2 < ρD(∇c) 2 | ζ > Pζ ∂ζ 2 ∂Q i ∂ ∂2 [< ρD(∇c) 2 | ζ > Pζ ] − 2 [< ρD(∇c) 2 | ζ > Q i Pζ ] ∂ζ ∂ζ ∂ζ ∂ ∂ − [< ρSc | ζ > Q i Pζ ] + 2 [< ρD(∇c ⋅ ∇y i ) | ζ > Pζ ] ∂ζ ∂ζ +2

∂2 ∂ − 2 [< ρD(∇c) 2 y i | ζ > Pζ ] − [< ρSc y i | ζ > Pζ ] ∂ζ ∂ζ ∂ + 2 [< ρD(∇c ⋅ ∇Q i ) | ζ > Pζ ] ∂ζ

Rearranging gives. ∂ [< ρ | ζ > Q i Pζ ] + ∇ ⋅ [< ρu | ζ > Q i Pζ ] =< ρωi | ζ > Pζ ∂t ∂2 ∂ − 2 [< ρD(∇c) 2 | ζ > Q i Pζ ] − [< ρSc | ζ > Q i Pζ ] ∂ζ ∂ζ ∂ 2 Qi ∂Q ∂ [< ρD(∇c) 2 | ζ > Pζ ] + 2 < ρD(∇c) | ζ > Pζ +2 i 2 ∂ζ ∂ζ ∂ζ + < e y | ζ > Pζ + < e Q | ζ > 2

where ∂ ∂2 [< ρD(∇c ⋅ ∇y i ) | ζ > Pζ ] − 2 [< ρD(∇c) 2 y i | ζ > Pζ ] ∂ζ ∂ζ ∂ − ∇ ⋅ [< ρuy i | ζ > Pζ ] − [< ρSc y i | ζ > Pζ ] ∂ζ ∂ < e Q | ζ >= 2 [< ρD(∇c ⋅ ∇Q i ) | ζ > Pζ ] + ∇ ⋅ [< ρD | ζ > ∇(Q i Pζ )] ∂ζ

< e y | ζ > Pζ = 2

E1

240

These are Smith (1994) equations 8.4, 3.69 and 3.70, with some minor differences. With these corrections, Smiths (1994) equations work out, so he must have some typos. Smith’s (1994) version is as follows.

∂ [< ρ | ζ > Q i Pζ ] + ∇ ⋅ [< ρu | ζ > Q i Pζ ] =< ρωi | ζ > Pζ ∂t ∂ ∂ + [< ρD(∇c) 2 | ζ > Q i Pζ ] − [< ρS c | ζ > Q i Pζ ] ∂ζ ∂ζ + < e y | ζ > Pζ + < e Q | ζ > where < e y | ζ > Pζ = 2[< ρD(∇c ⋅ ∇y i ) | ζ >

∂Pζ

∂ζ − ∇ ⋅ [< ρuy i | ζ > Pζ ] − [< ρS c y i | ζ >]Pζ

]−

∂ [< ρD(∇c) 2 y i | ζ > Pζ ] ∂ζ

< e Q | ζ >= [< ρ(∇c ⋅ ∇Q i ) | ζ >] + ∇ ⋅ [< ρD | ζ > ∇(Q i Pζ )]

The error terms are neglected and the steady state assumption is applied. ∇ ⋅ [< ρu | ζ > Q i Pζ ] =< ρωi | ζ > Pζ −

∂2 ∂ [< ρD(∇c) 2 | ζ > Q i Pζ ] − [< ρS c | ζ > Q i Pζ ] 2 ∂ζ ∂ζ

+ 2 < ρD(∇c) 2 | ζ > Pζ

E2

∂ 2Qi ∂Q ∂ +2 i [< ρD(∇c) 2 | ζ > Pζ ] 2 ∂ζ ∂ζ ∂ζ

Equation E2 is volume averaged and the divergence theorem is applied to the inlet and outlet.

∫

A

(< ρu | ζ > Q i Pζ ) out ⋅ dA − ∫ (< ρu | ζ > Q i Pζ ) in ⋅ dA = ∫ < ρωi | ζ > Pζ dV A

−∫

V

V

∂ ∂ [< ρD(∇c) 2 | ζ > Q i Pζ ]dV − ∫ [< ρS c | ζ > Q i Pζ ]dV 2 V ∂ζ ∂ζ 2

∂ 2Qi ∂Q i ∂ + 2 ∫ < ρD(∇c) | ζ > Pζ dV + 2 ∫ [< ρD(∇c) 2 | ζ > Pζ ]dV 2 V V ∂ζ ∂ζ ∂ζ 2

E3

241

The same procedure is applied to equation 3.13 to give.

∫ (< ρu | ζ > P )

ζ out

A

⋅ dA − ∫ (< ρu | ζ > Pζ ) in ⋅ dA = A

∂ ∂ [< ρD(∇c) 2 | ζ > Pζ ]dV − ∫ [< ρS c | ζ > Pζ ]dV 2 V ∂ζ V ∂ζ 2

−∫

Multiply equation E4 by Q i and subtract it from equation E3.

∫ (< ρu | ζ > Q P ) ⋅ dA − ∫ (< ρu | ζ > P ) Q ⋅ dA − ∫ (< ρu | ζ > Q P ) ⋅ dA + ∫ (< ρu | ζ > P ) Q ⋅ dA = ∫ < ρω | ζ > P dV i

A

ζ out

i,0

A

ζ in

A

i

i

ζ

i

V

ζ in

ζ out

A

∂ ∂2 2 −∫ [< ρD(∇c) | ζ > Q i Pζ ]dV + ∫ [< ρD(∇c) 2 | ζ > Pζ ]Q i dV V ∂ζ 2 V ∂ζ 2 ∂ ∂ −∫ [< ρS c | ζ > Q i Pζ ]dV + ∫ [< ρS c | ζ > Pζ ]Q i dV V ∂ζ V ∂ζ 2

+ 2 ∫ < ρD(∇c) 2 | ζ > Pζ V

∫ (< ρu | ζ > P )

ζ in

A

∂ 2Qi ∂Q i ∂ dV + 2 ∫ [< ρD(∇c) 2 | ζ > Pζ ]dV 2 V ∂ζ ∂ζ ∂ζ

(Q i − Q i , 0 ) ⋅ dA = ∫ < ρωi | ζ > Pζ dV V

∂ ∂ ∂2 { [< ρD(∇c) 2 | ζ > Q i Pζ ]}dV + ∫ Q i 2 [< ρD(∇c) 2 | ζ > Pζ ]dV V ∂ζ ∂ζ V ∂ζ ∂ ∂ ∂ − ∫ Q i [< ρS c | ζ > Pζ ]dV − ∫ < ρS c | ζ > Pζ [Q i ]dV + ∫ Q i [< ρS c | ζ > Pζ ]dV V V V ∂ζ ∂ζ ∂ζ −∫

+ 2 ∫ < ρD(∇c) 2 | ζ > Pζ V

∂ 2 Qi ∂Q i ∂ dV + 2 ∫ [< ρD(∇c) 2 | ζ > Pζ ]dV 2 V ∂ζ ∂ζ ∂ζ

E4

242

∫ (< ρu | ζ > P )

ζ in

A

(Q i − Q i ,0 ) ⋅ dA = ∫ < ρωi | ζ > Pζ dV V

∂ ∂ ∂ ∂ −∫ {Q i [< ρD(∇c) 2 | ζ > Pζ ]}dV − ∫ {< ρD(∇c) 2 | ζ > Pζ [Q i ]}dV V ∂ζ V ∂ζ ∂ζ ∂ζ + ∫ Qi V

∂Q i ∂2 [< ρD(∇c) 2 | ζ > Pζ ]dV − ∫ < ρSc | ζ > Pζ dV 2 V ∂ζ ∂ζ

+ 2∫ < ρD(∇c) 2 | ζ > Pζ V

∫ (< ρu | ζ > P )

ζ in

A


(Q i − Q i , 0 ) ⋅ dA = ∫ < ρωi | ζ > Pζ dV V

∂ ∂ ∂ [< ρD(∇c) 2 | ζ > Pζ ]}dV − ∫ [< ρD(∇c) 2 | ζ > Pζ ] {Q i }dV 2 V ∂ζ ∂ζ ∂ζ ∂ ∂ ∂ ∂ − ∫ < ρD(∇c) 2 | ζ > Pζ { [Q i ]}dV − ∫ [Q i ] {< ρD(∇c) 2 | ζ > Pζ }dV V V ∂ζ ∂ζ ∂ζ ∂ζ

− ∫ {Q i

2

V

+ ∫ Qi V

∂Q i ∂2 [< ρD(∇c) 2 | ζ > Pζ ]dV − ∫ < ρS c | ζ > Pζ dV 2 V ∂ζ ∂ζ

+ 2 ∫ < ρD(∇c) 2 | ζ > Pζ V

∫ (< ρu | ζ > P )

ζ in

A

(Q i − Q i ,0 ) ⋅ dA = ∫ Pζ < ρωi | ζ >dV

+ ∫ < ρD(∇c) 2 | ζ > Pζ V


V

∂ Qi ∂Q i dV − ∫ < ρSc | ζ > Pζ dV 2 V ∂ζ ∂ζ 2

E5

This is exactly Smith (1994) equation 8.7, so these corrections seem to work and the earlier differences were just typographical errors. The following terms are defined.

{Pζ* } ≡

A {P < ρu | ζ >} & ζ m

E6

243

τr ≡

{{ρ}}V & m

E7

With these substitutions the conditional mean reactive scalar equation and RPV PDF conservation equations (which is developed like the equation above, but starting from equation 3.13) become

{{ρ}}(Qi − Qi,0 )

{Pζ* }in τr

= {{Pζ < ρωi | ζ >}} E8

∂ 2 Qi ∂Q + {{Pζ < ρD(∇c) | ζ >}} 2 − {{Pζ < ρSc | ζ >}} i ∂ζ ∂ζ 2

{{ρ}}

{Pζ* }out − {Pζ* }in τr

=−

 ∂ ∂ {{< ρD(∇c) 2 | ζ > Pζ }} + {{< ρSc | ζ > Pζ }}  ∂ζ  ∂ζ 

(

)

E9

244

Appendix F

Derivation of the Premixed CMC Equations – Bilger’s Method

The derivation of the premixed CMC equations using Bilger’s method starts with the species mass fraction and reaction progress variable (c) conservation equations.

ρ


∂c ρ + ρu ⋅ ∇c = ∇ ⋅ (ρD h ∇c) + ∂t

F1

[ρWR ] − [ρ∑ ωi h f ,i ] i

∆h sad − u

≡ ∇ ⋅ (ρD h ∇c) + ρS c

F2

The conditioned value of the mass fraction, Q, is defined in Favre averaged form as.

Q(ζ , x , t ) ≡

< ρ( x, t )Y( x, t ) | c( x , t ) = ζ > < ρ( x , t ) | c ( x , t ) = ζ >

F3

ζ is the sample space variable corresponding to c. Y, the mass fraction of species i, is

defined as

Yi ( x, t ) = Q i (c( x, t ), x, t ) + y i ( x, t ) = Q i (c) + y i ( x, t )

Derivatives of Y are defined as

F4

245 . ∂Yi ∂Q i ∂c ∂y i = Qi + + ∂t ∂ζ i ∂t ∂t

∇Yi = ∇Q i +

∂Q i ∇c + ∇y i ∂ζ

∂Q i ∂Q i ∇ ⋅ (ρD i ∇Yi ) = ∇ ⋅ (ρD i [∇Q i + ∇c + ∇y i ]) = ∇ ⋅ (ρD i ∇Q i ) + ∇ ⋅ (ρD i ∇c) ∂ζ ∂ζ +

F5

∂ 2Qi ∂Q ρD i ∇c ⋅ ∇c + ρD i ∇ i ⋅ ∇c + ∇ ⋅ (ρD i ∇y i ) 2 ∂ζ ∂ζ

These 3 derivatives are substituted into equation F1 to give

.

ρ(Q i + Q′i

∂Q i ∂c ∂y i + ∇c + ∇y i ) − [∇ ⋅ (ρD i ∇Q i ) + ) + ρu ⋅ (∇Q i + ∂t ∂t ∂ζ

∂Q i ∂ 2Qi ∂Q ∇ ⋅ (ρD i ∇c) + ρD i ∇c ⋅ ∇c + ρD i ∇ i ⋅ ∇c + ∇ ⋅ (ρD i ∇y i )] = ρωi 2 ∂ζ ∂ζ ∂ζ

F6

Rearranging gives . ″ ′ ∂c ρωi = ρ Q i + ρu ⋅ ∇Q i − ρD i ∇c ⋅ ∇cQ i + Q i [ + ρu ⋅ ∇c − ∇ ⋅ (ρD h ∇c)] ∂t ′ ′ − ∇ ⋅ (ρD i ∇Q i ) − ρD i ∇c ⋅ ∇Q i − Q i ∇ ⋅ [ρ(D i − D h )∇c]

+ρ

F7

∂y i + ρu ⋅ ∇y i − ∇ ⋅ (ρD i ∇y i ) ∂t

Equation F2 is combined with this to give. . ″ ′ ρωi = ρ Q i + ρu ⋅ ∇Q i − ρD i ∇c ⋅ ∇cQ i + ρQ i S c ′ ′ − ∇ ⋅ (ρD i ∇Q i ) − ρD i ∇c ⋅ ∇Q i − Q i ∇ ⋅ [ρ(D i − D h )∇c]

+ρ

∂y i + ρu ⋅ ∇y i − ∇ ⋅ (ρD i ∇y i ) ∂t

F8

246

Next take the conditional expectation of this equation for c( x, t ) = ζ to give.

.

< ρ | ζ > Q i + < ρu | ζ > ⋅∇Q i =< ρωi | ζ > + < ρD i ∇c ⋅ ∇c | ζ > Q i ′ − < ρS c | ζ > Q i + e Q + e y

″

F9

′ ′ e Q =< {∇ ⋅ (ρD i ∇Q i ) + ρD i ∇c ⋅ ∇Q i + Q i ∇ ⋅ [ρ(D i − D h )∇c]} | ζ > e y = − < [ρ

∂y i + ρu ⋅ ∇y i − ∇ ⋅ (ρD i ∇y i )] | ζ > ∂t

This is the same as equation 79 in Bilger (1991) where he simplifies the error terms to be ″ e Q = 0 and e y = ∇⋅ < ρu y i | ζ > , saying the other terms are negligible.

In Bilger

′ (1993a) he has the same equations, except e y = −∇ ⋅ {< ρu y i | ζ > p c (ζ)} / p c (ζ ) . He does not give an explanation on how the PDF is added to the equation in the paper. In Bilger (1993b) he says that when the CMC equation is multiplied by the PDF and integrated over all values of c, it should yield the mean species equation. For this to happen he states that the error terms must be as shown above. Klimenko and Bilger (1999) equations 103-110 also have the PDF in the error term, but do not explain how it is added for the premixed case.

For the non-premixed case they give a detailed

description of why this error term is used.

Smith (1994) does the derivation completely different (following Klimenko’s derivation), but gets the same CMC equation with different error terms, which are.

ey ≡

∂Pζ 1 ∂ {2[< ρD(∇c ⋅ ∇y i ) | ζ > ] − [< ρD(∇c) 2 y i | ζ > Pζ ] ∂ζ Pζ ∂ζ

− ∇ ⋅ (< ρuy i | ζ > Pζ )} − (< ρS c y i | ζ >) e Q ≡ [< ρ(∇c ⋅ ∇Q i ) | ζ >] + ∇ ⋅ (< ρD | ζ > ∇(Q i Pζ ))

F10

247

It appears that Smith has some minor typographical errors in his error terms, when the equations are derived here the equations using his method give the following.

ey =

1 ∂2 ∂ {2 [< ρD(∇c ⋅ ∇y i ) | ζ > Pζ ] − 2 [< ρD(∇c) 2 y i | ζ > Pζ ] Pζ ∂ζ ∂ζ

− ∇ ⋅ (< ρuy i | ζ > Pζ ) − eQ =

∂ (< ρS c y i | ζ > Pζ )} ∂ζ

F11


− Q i ∇ ⋅ (< ρD | ζ > ∇Pζ )}

Klimenko and Bilger (1999) say the last term in the Q error term is negligible for high Reynolds number flow. The second term of the Q error term might be negligible for the same reason.

Next the volume averaged premixed CMC equations are derived by assuming both error terms are zero and steady flow. With these simplifications the CMC equation reduces to.

″ ′ < ρu | ζ > ⋅∇Q i =< ρωi | ζ > + < ρD i ∇c ⋅ ∇c | ζ > Q i − < ρS c | ζ > Q i

F12

Next the equations is volume averaged with the divergence theorem applied to the inlet and outlet terms to give. The continuity equation is used on the first term on equation 13.

∫ ∫

A

V

(< ρu | ζ > Q i ) out dA − ∫ (< ρu | ζ > Q i ) in dA = ∫ < ρωi | ζ > dV + A

V

″

′

< ρD i ∇c ⋅ ∇c | ζ > Q i dV − ∫ < ρS c | ζ > Q i dV V

F13

248

[Q i − Q i , in ]

& m ″ ′ = {{< ρωi | ζ >}} + {{< ρD i (∇c) 2 | ζ > Q i }} − {{< ρS c | ζ > Q i }} V

F.14

The LHS term is different than Smith (1994) and will not cancel out as Bilger (1993a) has.

249

Appendix G

Derivation of the Favre averaged C Equation and its Variance

Equation G1 is the conservation equation for c in compressible form.

∂ρc + ∇ ⋅ (ρuc) = ∇ ⋅ (ρD h ∇c) − ∂t

[∑ ρωi h f ,i ] i

∆h sad −u

G1

~ + ω′′ and The Favre (mass) averaged equation is developed using c = ~c + c′′ , ω = ω u=~ u + u ′′ , to give. ∂ ( ρ + ρ′)(~c + c′′) + ∇ ⋅ [( ρ + ρ′)(~ u + u ′′)(~c + c′′)] = ∇ ⋅ [( ρ + ρ′)D h ∇( ~c + c′′)] − ∂t ~ + ω ″ )h ] [ ( ρ + ρ′)(ω

∑

i

i

G2

f ,i

i

∆h sad − u

Expand similar to the development of the Favre averaged equations to give. Neglect the diffusion terms. ∂ ρ ~c ∂ ρ c′′ ∂ρ′~c ∂ρ′c′′ + + + + ∇ ⋅ [ρ ~ u ~c ] + ∇ ⋅ [ ρ ~ uc′′] + ∇ ⋅ [ ρ u ′′~c ] + ∇ ⋅ [ ρ u ′′c′′] + ∂t ∂t ∂t ∂t ∇ ⋅ [ρ′~ u ~c ] + ∇ ⋅ [ρ′~ uc′′] + ∇ ⋅ [ρ′u ′′~c ] + ∇ ⋅ [ρ′u ′′c′′] ~ h ] [ ρ ω ″ h ] [ ρ′ω ~ h ] [ ρ′ω ″ h ] [∑ ρ ω ∑ ∑ ∑i i f ,i i f ,i i f ,i i f ,i i =− i s − i − − ∆h ad − u ∆h sad − u ∆h sad − u ∆h sad − u

G3

Next time average the equation and use the Favre average relations to cancel out like terms.

250

~h ] [∑ ρ ω i f ,i ~ ∂ρ c i ~ ~ ′ ′ ′ ′ ′ ′ ′ ′ ′ + ∇ ⋅ [ ρ u c ] + ∇ ⋅ [ ρ u c ] + ∇ ⋅ [ρ u c ] = − ∂t ∆h sad −u

∂ ρ ~c + ∇ ⋅ [ρ~ u ~c ] + ∇ ⋅ [( ρ + ρ′)u ′′c′′] = − ∂t

~h ] [∑ ρ ω i f ,i i

∆h sad − u

G4

The following relationship is used to simplify the equation.

ρu ′′c′′ = −

µt ~ ∇c σc

G5

Where µ t is the turbulent viscosity and σ c is the Schmidt number. With these equation G4 becomes.

µ ∂ ρ ~c + ∇ ⋅ [ρ~ u ~c ] − ∇ ⋅ ( t ∇~c ) = − ∂t σc

~h ] [∑ ρ ω i f ,i i

G6

∆h sad − u

This is the time averaged c equation to use in the CFD code. To get the variance equation, equation G4 is subtracted from equation G3 (the derivation is done with the incompressible equations).

∂c ′ + ∇ ⋅ [ uc′] + ∇ ⋅ [u ′c] + ∇ ⋅ [u ′c′] − ∇ ⋅ [u ′c′] = − ∂t

Equation G7 is multiplied by c′ to give.

′ [ ∑ ω i h f ,i ] i

∆h sad − u

G7

251

c′

∂c′ + c′∇ ⋅ [ uc′] + c′∇ ⋅ [u ′c] + c′∇ ⋅ [u ′c′] − c′∇ ⋅ [u ′c′] = −c′ ∂t

′ [ ∑ ω i h f ,i ] i

∆h sad − u

G8

The first five terms are rewritten as follows.

c′

∂c′ 1 ∂c′ 2 = ∂t 2 ∂t

c′∇ ⋅ [uc′] = ∇ ⋅ [uc′ 2 ] − uc′ ⋅ ∇c′ =

1 1 1 1 1 ∇ ⋅ [uc′ 2 ] + ∇ ⋅ [uc′ 2 ] − u ⋅ ∇c′ 2 = ∇ ⋅ [uc′ 2 ] + c′ 2 ∇ ⋅ u + 2 2 2 2 2

1 1 1 u ⋅ ∇c′ 2 − u ⋅ ∇c′ 2 = ∇ ⋅ [uc′ 2 ] 2 2 2 c′∇ ⋅ [u ′c] = c′(∇c ⋅ u ′ + u ′ ⋅ ∇c) = u ′c′ ⋅ ∇c

c′∇ ⋅ [u ′c′] = ∇ ⋅ [u ′c′ 2 ] − u ′c′ ⋅ ∇c′ c′∇ ⋅ [u ′c′] = ∇ ⋅ [c′u ′c′] − u ′c′ ⋅ ∇c′

For incompressible flow we get

∇ ⋅ u = ∇ ⋅ u′ = 0

With these relationships equation 8 becomes. 1 ∂c ′ 2 1 + ∇ ⋅ [uc ′ 2 ] + u ′c′ ⋅ ∇c + ∇ ⋅ [u ′c′ 2 ] − u ′c′ ⋅ ∇c′ − 2 ∂t 2 ′ [∑ c′ωi h f ,i ] ∇ ⋅ [c′u ′c′] + u ′c′ ⋅ ∇c′ = − i ∆h sad −u

G9

252

Next time average equation G9.

1 ∂ c′ 2 1 + ∇ ⋅ [ u c′ 2 ] + u ′c′ ⋅ ∇c + ∇ ⋅ [u ′c′ 2 ] − u ′c′ ⋅ ∇c′ − 2 ∂t 2 ′ [∑ c′ωi h f ,i ] ∇ ⋅ [c′u ′c′] + u ′c′ ⋅ ∇c′ = − i ∆h sad − u

G10

Multiple by 2 and drop 2 terms to give.

υ ∂ c′ + ∇ ⋅ [u c′ 2 ] − 2 t ∇c ⋅ ∇c + 2∇ ⋅ [ u ′c′ 2 ] − 2u ′c′ ⋅ ∇c′ = − ∂t σc 2

′ [∑ c′ωi h f ,i ] i

∆h sad − u

G11

The following 2 models are used.

2∇ ⋅ [u ′c′ 2 ] = −∇ ⋅ [

µt ∇c′ 2 ] σc

ε − 2u ′c′ ⋅ ∇c′ = C c 2 ρ c′ 2 κ C c1 = 2.0 / σ c = 2.86

G12

C c 2 = 2.0

This gives the same as Lilleheie et al. (1989) has (in compressible form). ~ε µ ∂ρ~c′′ 2 u~c′′ 2 ] − ∇ ⋅ [ t ∇~c′′ 2 ] = C c1µ t (∇~c ) 2 − C c 2 ρ ~ ~c′′ 2 + ∇ ⋅ [ρ~ ∂t σc κ ~ ′′h ] [ρ ~c′′ω −2

∑ i

∆h sad − u

f ,i

G13

253

These terms are the time rate of change, the transport by convection, diffusion by velocity fluctuations, production by mean gradient, dissipation by molecular diffusion and production due to reaction rate fluctuations.

254

Appendix H

Derivation of the Source Term for the Variance Equation

The source term for the conservation equation of the variance of the reaction ~ ′′h NS ~ c′′ω i f ,i progress variable is ∑ s , which was derived in Appendix G. A model is needed h ad −u i for this term to close the equation. r j = A jρ

Yj1

Y j2

T j exp(− n

MW j1 MW j2

TA , j T

A general elementary reaction rate is

)MW j1 where Y is the mass fraction.

fluctuations of reaction rate due to ρ, Y j1 , T fluctuations due to fluctuations of exp(− elementary reaction rate is rj = A jρ

Yj1

TA , j T

Y j2

n

j

are assumed small compared to

).

With this assumption the mean

T j exp(− n

MW j1 MW j2

The

can be written is simplified form as r j = B j exp(−

TA , j T

TA , j T

)MW j1 . The reaction rate

) with the temperature fluctuation

written as T ′ = T − T . The last two equations are combined to give

r j = B j exp(−

TA , j T + T′

) = B j exp[−

TA , j

1 ( )] T 1+ δ

where

δ=

T′ . T

If the temperature

fluctuations are small relative to the mean temperature a Taylor series expansion can be used to give

1 ≅ 1− δ . 1+ δ

With this the reaction rate becomes

255

r j = B j exp[− rj exp(

TA , j T

TA , j T

(1 − δ)] = B j exp[−

TA , j T

+

TA , j T

δ] = B j exp(−

TA , j T

) exp(

TA , j T

δ) =

δ)

TA , j TA , j ′ rj = rj − rj = rj exp( δ) − rj = rj [exp( δ) − 1] T T

H1

The reaction progress variable can be written as c = ~c + c′′ , which can also be written as

c=

h − hu T − Tu T + T ′ − Tu ≈ = h ad − h u Tad − Tu Tad − Tu

These two relationships give ~c = T − Tu Tad − Tu

δ=

c′′ =

T′ Tad − Tu

c′′(Tad − Tu ) T′ =~ = T c (Tad − Tu ) + Tu ~ c+

c′′ c′′ =~ Tu c +α (Tad − Tu )

H2

Combining equations H1 & H2 gives TA , j c′′ ′ r j = r j [exp( ) − 1] T ~c + α

H3

The total reaction rate is the sum of all the forward and reverse elementary reaction rates.

256

NR

NR

j

j

~ = (υ′′ − υ′ )(r − r ) . ωi = ∑ (υ′ij′ − υ′ij )(rj − r− j ) and ω ∑ ij ij j − j i

H4

Similarly the fluctuation of the total reaction rate is

NR

NR

j

j

ω′i′ = ∑ (υ′ij′ − υ′ij )( r j′ − r−′ j ) = ∑ (υ′ij′ − υ′ij )( rj [exp(

TA , − j c′′ c′′ ) 1 ] r [exp( − − ) − 1]) − j T ~c + α T ~c + α

TA , j

This is to complex, so try a simplification below. ________________________________________________________________________ Method 1

Assume,

NR

NR

NR

j

j

j

∑ υij rj′ =∑ (υij rj ) ∗ ∑ υij [exp(

c′′ ) − 1] , so ~ T c +α

TA , j

H5

NR TA , j c′′ ″ ωi = ωi ∑ υ ij [exp( ) − 1] T ~c + α j

H6

NR TA , j c′′ ″ c′′ωi = ω i c′′∑ υ ij [exp( ) − 1] T ~c + α j

H7

NS

∑

~c′′′ω ~ ′′h i f ,i

i

If (

h s ad − u

~h NR TA , j ~c′′ ω i f ,i ~ c′′∑ υ ij [exp( ) − 1]} = ∑{ s T ~c + α h ad −u i j NS

H8

~c′′ ) ≤ 1 the exponential term can be approximated with a Taylor series T ~c + α

TA , j

expansion.

257

NS

∑ i

NS

∑ i

NS

∑ i

~c′′ω ~ ′′h i f ,i h s ad − u

~c′′ω ~ ′′h i f ,i h s ad − u

~c′′ω ~ ′′h i f ,i h s ad − u

~h NR TA , j ~c′′ ω i f ,i ~ c′′∑ υ ij [1 + = ∑{ s − 1]} T ~c + α h ad − u i j

H9

ωi h f ,i ~ NR υ ij TA , j ~c′′ c′′∑ [ ]} T ~c + α h s ad − u j

H10

~ h NR ~c′′ 2 NS ω i f ,i { s ∑ ∑ υijTA, j } ~ T ( c + α) h ad − u

H11

NS

NS

= ∑{ i

=

i

j

Another simplification is

NS

∑ i

~c′′ω ~ ′′h i f ,i h s ad − u

=

~h ~c′ 2 NS ω NS NR i f ,i { } ∗ ∑ h s ad−u ∑ {∑ υij TA, j } T (~c + α) i

i

H12

j

The double summation is a constant for a given mechanism giving

NS

∑ i

~c′′ω ~ ′′h i f ,i h s ad − u

= C3

~h ~c′′ 2 NS ω { ∑ h si ad−f ,ui } T (~c + α)

H13

i

The remaining summation is the source term for the c equation, so this closes the variance equation. There are several approximations in this derivation, so it is hard to predict how accurate it will be, but it may give the correct trends.

258

Appendix I 0.25

0.20

0.15

O2 Mass Fraction

1000 800 600 400 200 130 0.10

0.05

0.00 0.0

0.2

0.4

0.6

0.8

1.0

c

Figure I-1: Premixed CMC with GRI2.11, O2 versus scalar dissipation, 1/sec. Inlet temperature 300 K, pressure 1 atm. and equivalence ratio 0.9.

259 0.14

0.12

0.10

CO2 Mass Fraction

0.08 1000 800 600 400 200 130 0.06

0.04

0.02

0.00 0.0

0.2

0.4

0.6

0.8

1.0

c

Figure I-2: Premixed CMC with GRI2.11, CO 2 versus scalar dissipation, 1/sec. Inlet temperature 300 K, pressure 1 atm. and equivalence ratio 0.9.

260 0.12

0.10

H2O Mass Fraction

0.08

1000 800 600

0.06

400 200 130

0.04

0.02

0.00 0.0

0.2

0.4

0.6

0.8

1.0

c

Figure I-3: Premixed CMC with GRI2.11, H 2 O versus scalar dissipation, 1/sec. Inlet temperature 300 K, pressure 1 atm. and equivalence ratio 0.9.

261 0.24

0.22

0.20

50000 30000 10000

0.18 O2 Mass Fraction

7000 3000 2000 1500 1000 800 0.16

600 400 100

0.14

0.12

0.10 0.0

0.2

0.4

0.6

0.8

1.0

c

Figure I-4: Premixed CMC with GRI2.11, O2 versus scalar dissipation, 1/sec. Inlet temperature 603 K, pressure 4.74 atm. and equivalence ratio 0.529.

262 0.03

0.03

0.02

50000 30000 CO Mass Fraction

10000 7000 3000 2000

0.02

1500 1000 800 600 400 100

0.01

0.01

0.00 0.0

0.2

0.4

0.6

0.8

1.0

c

Figure I-5: Premixed CMC with GRI2.11, CO versus scalar dissipation, 1/sec. Inlet temperature 603 K, pressure 4.74 atm. and equivalence ratio 0.529.

263 0.09

0.08

0.07

0.06

50000 30000 CO2 Mass Fraction

10000 0.05

7000 3000 2000 1500 1000 800

0.04

600 400 100

0.03

0.02

0.01

0.00 0.0

0.2

0.4

0.6

0.8

1.0

c

Figure I-6: Premixed CMC with GRI2.11, CO 2 versus scalar dissipation, 1/sec. Inlet temperature 603 K, pressure 4.74 atm. and equivalence ratio 0.529.

264 3.00E-03

2.50E-03

2.00E-03

50000 30000 NO Mass Fractions

10000 7000 3000 2000

1.50E-03

1500 1000 800 600 400 100

1.00E-03

5.00E-04

0.00E+00 0.0

0.2

0.4

0.6

0.8

1.0

c

Figure I-7: Premixed CMC with GRI2.11, NO versus scalar dissipation, 1/sec. Inlet temperature 603 K, pressure 4.74 atm. and equivalence ratio 0.529.

265

2.50E-04

2.00E-04

50000 1.50E-04

30000

NO2 Mass Fraction

10000 7000 3000 2000 1500 1000 800 600 400

1.00E-04

100

5.00E-05

0.00E+00 0.0

0.2

0.4

0.6

0.8

1.0

c

Figure I-8: Premixed CMC with GRI2.11, NO 2 versus scalar dissipation, 1/sec. Inlet temperature 603 K, pressure 4.74 atm. and equivalence ratio 0.529.

266 1.40E-05

1.20E-05

1.00E-05

50000 30000 10000

N2O Mass Fraction

8.00E-06

7000 3000 2000 1500 1000 800 6.00E-06

600 400 100

4.00E-06

2.00E-06

0.00E+00 0.0

0.2

0.4

0.6

0.8

1.0

c

Figure I-9: Premixed CMC with GRI2.11, N 2 O versus scalar dissipation, 1/sec. Inlet temperature 603 K, pressure 4.74 atm. and equivalence ratio 0.529.

267 3.50E-02

3.00E-02

2.50E-02

CO Mass Fractions

2.00E-02 90 70 50 30 10 5 1.50E-02

1.00E-02

5.00E-03

0.00E+00 0.0

0.2

0.4

0.6

0.8

1.0

c

Figure I-10: Premixed CMC with GRI2.11, CO versus scalar dissipation, 1/sec. Inlet temperature 603 K, pressure 4.74 atm. and equivalence ratio 0.529.

268 3.00E-03

2.50E-03

NO Mass Fraction

2.00E-03

90 70 50

1.50E-03

30 10 5

1.00E-03

5.00E-04

0.00E+00 0.0

0.2

0.4

0.6

0.8

1.0

c

Figure I-11: Premixed CMC with GRI2.11, NO versus scalar dissipation, 1/sec. Inlet temperature 603 K, pressure 4.74 atm. and equivalence ratio 0.529.

269 1.80E-05

1.60E-05

1.40E-05

NO2 Mass Fraction

1.20E-05

1.00E-05

90 70 50 30 10 5

8.00E-06

6.00E-06

4.00E-06

2.00E-06

0.00E+00 0.0

0.2

0.4

0.6

0.8

1.0

c

Figure I-12: Premixed CMC with GRI2.11, NO 2 versus scalar dissipation, 1/sec. Inlet temperature 603 K, pressure 4.74 atm. and equivalence ratio 0.529.

270 3.00E-06

2.50E-06

N2O Mass Fraction

2.00E-06

90 70 50

1.50E-06

30 10 5

1.00E-06

5.00E-07

0.00E+00 0.0

0.2

0.4

0.6

0.8

1.0

c

Figure I-13: Premixed CMC with GRI2.11, N 2 O versus scalar dissipation, 1/sec. Inlet temperature 603 K, pressure 4.74 atm. and equivalence ratio 0.529.

271

Appendix J

Analytical Solution for 1-STEP CH 4 Mechanism

Here it is shown that for a 1-step chemical kinetic mechanism, the solution of the premixed CMC equation is a linear function of c, independent of scalar dissipation (N).

For a stoichiometric mixture of methane the chemical balance equation is as follows.

CH 4 + 2O 2 + 7.546 N 2 → CO 2 + 2H 2 O + 7.546 N 2

Starting with the definition of the progress variable c for adiabatic flows (equation 3.11 with constant h) and the relationship between sensible enthalpy, enthalpy and heats of formation (equation 3.9).

c≡

h s − h su ∆h sad −u

J1

h s = h − ∑ h f ,i Yi

J2

Substitute equation J2 into equation J1.

c=

(h − ∑ h f ,i Yi ) − (h − ∑ h f ,i Yi ) u ∆h sad −u

For adiabatic flows h is constant giving.

J3

272

c=

− ∑ h f ,i Yi + (∑ h f ,i Yi ) u

J4

∆h sad −u

The f subscript signifies a heat of formation and the u subscript is for the unburned state. Expanding the summations with h f ,O 2 = h f , N 2 = 0 gives.

c=

0 − (h f ,CH 4 YCH 4 + h f ,CO 2 YCO 2 + h f ,H 2O YH 2 O ) + (h f ,CH 4 YCH 4)

∆h sad −u

J5

For a stoichiometric flow and a 1-step mechanism, define all mass fractions in terms of

CH 4 . Here all mass fractions are a linear function of the CH 4 mass fraction.

0 YCH 4 is the initial YCH 4 .

YCO 2 =

MWCO 2 0 (YCH 4 − YCH 4 ) MWCH 4

YH 20 = 2

MWH 20 0 (YCH 4 − YCH 4 ) MWCH 4

J6

J7

YO 2 = 2

MWO 2 YCH 4 MWCH 4

J8

YN 2 = 2

MWN 2 0 YCH 4 MWCH 4

J9

Substitute equations J6-J9 into equation J5.

273

  MWCO 2 MWH 2O 0 0 − h f ,CH 4 YCH 4 + h f ,CO 2 YCH h f , H 2O YCH 4 − YCH 4 + 2 4 − YCH 4  MWCH 4 MWCH 4  c=  s ∆h ad − u

(

+

)

(

)

0 (h f ,CH 4 YCH 4)

∆h sad − u

  MWCO 2 MWH 2 O 0 0 0 h f ,CO 2 YCH h f , H 2 O YCH 4 − YCH 4 − 2 4 − YCH 4  + ( h f , CH 4 YCH 4 ) − h f ,CH 4 YCH 4 − MW MW CH 4 CH 4   c= s ∆h ad − u

(

)

(

)

  MWH 2 O MWCO 2 0 0 0 h f ,H 2 O YCH h f ,CO 2 YCH 4 − YCH 4 − 2 4 − YCH 4  h f ,CH 4 YCH 4 − YCH 4 − MWCH 4 MWCH 4  c=  s ∆h ad − u

(

)

(

)

  MWCO 2 MWH 2 O h f ,CO 2 − 2 h f ,H 2O  h f ,CH 4 − MWCH 4 MWCH 4   0 c= YCH 4 − YCH 4 s ∆h ad − u

(

(

0 c = H YCH 4 − YCH 4

(

)

)

)

  MWCO 2 MWH 2O h f ,CO 2 − 2 h f ,H 2O  h f ,CH 4 − MWCH 4 MWCH 4  with H ≡  s ∆h ad − u

18 44   − 4.6 − 16 (−8.94) − 2 16 (−13.4) H= = 18.1 2.77 = 0

For a fixed equivalence ratio H is a constant. Rearranging equation J10 gives.

J10

J11

J12

274

0 YCH 4 = YCH 4 −

c H

J13

Take the 1st and 2nd derivatives with respect to c.

′ 4 =− YCH

1 H

J14

′′ 4 = 0 YCH

J15

This says that for a 1-step mechanism, the CMC equation will always give a linear relation between c and Yi , independent of the scalar dissipation. The model results support this with values of scalar dissipation ranging from 10,000,000 to 0.1 1/sec. The smaller values of scalar dissipation make the CMC equation stiffer, which requires a finer mesh and taking longer to solve.

Apply the boundary conditions to equation J13 to determine H. At c = 1 YCH 4 = 0 , equation J13 becomes

0 0 = YCH 4 −

1 0 = YCH 4 H

1 H

J15b

0 From equation J12, H=18.1, 1/H=0.0552, which is YCH , the initial methane mass 4

fraction.

275

This section showed that for a 1-step mechanism the species mass fractions are always a linear function of c. In hindsight this is an intuitive result. From the results of Chapter III it was seen that as the scalar dissipation is varied at a constant value of c (using full mechanisms), the relative portions of CH 4 , CO 2 and CO vary. With a 1-step mechanism there is no CO and there is only one pathway for the CH 4 to burn to CO 2 . Since c is a linear function of enthalpy (h) and h varies linearly with the amount of CH 4 burned, the amount of CH 4 must vary linearly with c. Since all the other species are linearly related to CH 4 for a 1-step mechanism, they must also vary linearly with c. _______________________________________________________________________ Next analyze the source term of equation 3.12, S c .

Sc =

− ∑ ω i h f ,i ∆h sad − u

Next write all the reaction rates in terms of the methane reaction rate. ωCH 4 is the reaction rate for methane

ωCO 2 = −

MWCO 2 ωCH 4 MWCH 4

ω H 20 = −2

MWH 20 ω CH 4 MWCH 4

Substitute these into equation J16 and simplify.   MWCO 2 MWH 2 O −  ω CH 4 h f ,CH 4 − ω CH 4 h f ,CO 2 − 2 ω CH 4 h f ,H 2 O  MWCH 4 MWCH 4   Sc = s ∆h ad − u

J16

276

S c = −ω CH 4

  MWCO 2 MWH 2 O  h f ,CH 4 − h f ,CO 2 − 2 h f , H 2 O  MWCH 4 MWCH 4   s ∆h ad − u

Using the definition of H from equation J11, this becomes. S c = −ω CH 4 H

J17

______________________________________________________________________ Now show that the above results are consistent. The simplified premixed CMC equation is

~ ~ ~ ~ | ζ >= 0 < ρ N > Q ′i′− < ρ Sc | ζ > Q ′i + < ρ ω i

J18

For i= YCH 4 , substitute equation J17 into equation J18 and using equations J13-J15 gives. ~ ~ 0− < − ρ ω CH 4 H | ζ > ( − 1 / H ) + < ρ ω CH 4 | ζ >= 0 ~ ~ − < ρω CH 4 | ζ > + < ρ ω CH 4 | ζ >= 0

0=0

These relations satisfy the simplified CMC equation.

Equation J18 can be directly integrated in closed form (not shown here) to give equation J13. These same relationships hold for all equivalence ratios and all fuels.

277

Appendix K

Comparison of the G Equation to Models based on Reactive Scalars

Sethian (1999) has written the only known book devoted to the subject of level set methods. In short the method treats the infinitely thin flame as a discontinuity and tracks the evolution of the boundary (the propagating interface). This assumes the flame is infinitely thin and that one is only interested in the jump conditions across the discontinuity and not the internal flame details. The diffusion and reaction rate terms in the conservation equations are replaced with an empirical flame displacement model. This is taking a Lagrangian viewpoint of the flame, looking at the flame as a uniform entity that moves through the flow field, while the individual species and reactions are ignored. The level set method replaces the Lagrangian geometric perspective with an Eulerian PDE. The level set method is the resulting initial value problem of the evolving interface. If a boundary value problem is obtained, it is termed a fast marching method. Once the PDE is developed it is solved using conventional computational methods.

The advantage of this approach is that the chemical kinetic equations are not required, which are very time consuming to solve. The limitation is that only infinitely thin flames can be solved with this technique and a model is required to relate the flame displacement speed to the flow field. This seems somewhat reversed, since it is the kinetics that drive the flame speed, which influence the flow field.

Here it is shown that the G equation (which is another name for the level set approach applied to combustion) as used in the combustion literature is the same as the reactive scalar equations (methane mass fraction, YCH 4 and reaction progress variable, c) with the reaction rate closure the only difference. The G equation is simply a coordinate transformation of c or YCH 4 , and in some cases G is the reactive scalar with no coordinate transformation. The G equation replaces the reaction kinetics and diffusion terms in the

278

conservation equations with a flame displacement speed, based on the flow field kinematics. This has some similarities to the BML and EBU models, which replace the reaction kinetics with models based on the turbulent flame speed and turbulence intensity, respectively.

This differs from the premixed CMC model, which uses

conditioned reaction rates based on the chemical kinetics.

Start by deriving the g equation following Peters et al. (1998). First the flame location is defined as the isoscalar surface. G( x, t ) = G 0

K1

G 0 is defined as the flame location, G < 0 is the unburned mixture and G > 0 is the burned mixture. G is the distance from the flame zone and is normalized by the preheat zone size, l F . G = -1.0 is at the unburned edge of the flame, prior to the reactions starting, see Figure K-1. G = 0 is the flame location where 90% of the CH 4 has been destroyed. The G=0 definition is arbitrary and is usually selected to be the point of peak CO concentration.

Differentiating equation K1 with respect to t gives the following ∂G ∂x + ∇G ⋅ ∂t ∂t

Where

∂x ∂t

=0

K2

G =G 0

is the flame displacement speed at the G = 0 isoscalar surface. This G =G 0

equation is only valid at G = 0.

279

The flame displacement speed is calculated from the CH 4 mass fraction balance equation as described next.

ρ

∂YCH 4 & CH 4 + ρv ⋅ ∇YCH 4 = ∇ ⋅ ρD(∇YCH 4 ) + ρω ∂t

K3

(This is equation 3.6) The flame surface is defined as the location where the methane mass fraction is equal to 10% of its unburned value, i.e. YCH 4 = 0.1YCH 4,u

K4

Where YCH 4,u is the unburned methane mass fraction. On the flame zone isoscalar surface the following condition must be satisfied.

ρ

∂YCH 4 ∂x + ρ∇YCH 4 ⋅ ∂t ∂t

=0

K5

YCH 4 = 0.1YCH 4 , u

Equations K3 and K5 are combined and after simplification give the following.

∂x ∂t

YCH 4 = 0.1YCH 4 , u

 ∇ ⋅ ρ D(∇YCH 4 ) + ρω & CH 4  = v0 −   ⋅n ρ ∇YCH 4   0

K6

Where the 0 subscript signifies the flame zone isoscalar surface YCH 4 = 0.1YCH 4,u , which by definition is also G = 0. This says that the rate the flame isoscalar surface moves is related to the flow field velocity, the rate of diffusion and the reaction rate of YCH 4 . The normal vector n is defined as (see Figure K-1)

280

n=

∇YCH 4 ∇G =− ∇YCH 4 ∇G

K7

and points toward the unburned mixture.

Combining equations K2 and K6 gives the following

   ∇ ⋅ ρ D(∇YCH 4 ) + ρ ω & CH 4  ∂G + ∇G ⋅  v 0 −   ⋅ n = 0 ∂t ρ ∇YCH 4   0  

Rearranging gives

 ∇ ⋅ ρD(∇YCH 4 ) + ρω & CH 4  ∂G + v 0 ⋅ ∇G =   ∇G ⋅ n ∂t ρ ∇YCH 4   0

Using equation K7 gives

 ∇ ⋅ ρD(∇YCH 4 ) + ρ ω & CH 4  ∂G + v 0 ⋅ ∇G =   (− n ∇G ) ⋅ n ∂t ρ ∇YCH 4   0  ∇ ⋅ ρD(∇YCH 4 ) + ρω & CH 4  ∂G + v 0 ⋅ ∇G = −   ∇G n ⋅ n ∂t ρ ∇YCH 4   0  ∇ ⋅ ρD(∇YCH 4 ) + ρω & CH 4  ∂G + v 0 ⋅ ∇G = −   ∇G ∂t ρ ∇YCH 4   0

K8

This is Peters et al. (1998) equation 8 and is only valid at G = 0. This can be put into another form using (from Echekki and Chen, 1999).

281

∇ ⋅ ρD(∇YCH 4 ) = ρD ∇YCH 4 κ + n ⋅ ∇(ρDn ⋅ ∇YCH 4 )

K9

Where κ = ∇ ⋅ n Combining equations K8 and K9 gives.

& CH 4   ρD ∇YCH 4 κ + n ⋅ ∇(ρDn ⋅ ∇YCH 4 ) + ρω ∂G + v 0 ⋅ ∇G = −   ∇G ∂t ρ ∇YCH 4   0

K10

Next define normal and reaction velocities as follows.  n ⋅ ∇(ρDn ⋅ ∇YCH 4 )  Vn = −   ρ ∇YCH 4  

K11

 ρωCH 4 Vr = −   ρ ∇YCH 4

K12

  

These are Peters et al. (1998) equations 12 and 13. Their equation 12 is missing the minus sign, which Gran et al. (1996) and Peters (2000) have correctly. Combining equations K10-K12 gives the following. ∂G + v 0 ⋅ ∇G = − Dκ ∇G + (Vn + Vr ) ∇G ∂t

K13

This is Peters et al. (1998) equation 10. Defining a displacement speed as S d = − Dκ + Vn + Vr

K14

282

The displacement speed has components due to the curvature of the flame (tangential diffusion), the normal diffusion and the reactions. The displacement speed is the flame speed relative to the flow field. Equation K13 can be written as ∂G + v 0 ⋅ ∇G = S d ∇G ∂t

K15

Peters et al. (1998) gives empirical relationships for the displacement speed from 2-D DNS simulations as follows. S d = C1Dκ + C 2S L , 0

K16

Where the 2 constants are different for the 2 flows they computed, where each had a different equivalence ratio. The constants were significantly different, so it appears that there is not a universal relationship for the displacement speed based on these two parameters. This agrees with the findings of Driscoll (2003) that there is not a universal relationship for the turbulent burning velocity. Equation K13 can be written in terms of an effective diffusivity and an effective burning velocity as follows. ∂G + v 0 ⋅ ∇G = − D * κ ∇G + S*L ∇G ∂t

K17

Again, this expression is only valid at the flame zone location, defined as G = 0. By defining the displacement speed as a function of only the CH 4 , this model assumes that the burning of the CH 4 is the only reaction that causes the flame to propagate. This implicitly assumes a 1-step kinetic mechanism, i.e. the CH 4 burns directly to CO 2 . This also means that G is linearly related to the CH 4 mass fraction, as discussed in Appendix J. It appears that the G variable is just a coordinate transformation of YCH 4 or c and not

283

an independent variable. Also, this model is only valid in the thin reaction zone regime of turbulent premixed combustion, where there is a definable displacement speed, it is not valid in the distributed reaction and well stirred reactor regimes. Peters (1999 and 2000) develops these same equations for the thin reaction zone regime.

Peters (1992, 1999 and 2000) develops a similar expression for the corrugated flamelet regime starting with equation K2 and uses the following relation ∂x ∂t

= v ⋅ n − SL

K18

YCH 4 = 0.1YCH 4 , u

To give ∂G + v 0 ⋅ ∇G = S L ∇G ∂t

K19

Where the burning velocity is defined as follows S L = S 0L − S 0L Lκ + Ln ⋅ ∇v ⋅ n

K20

The three terms on the right side of equation K20 are the burning velocity for an unstretched flame and the flame stretch due to curvature and flow divergence, respectively. L is the Markstein length. Equation K19 is averaged and an equation for the variance of G is derived. To close the source terms an equation for the flame surface area ratio is also required. The source terms for these equations require empirical closure models. Equation K19 has the same form as equations K15 and K17, with slightly different definitions for the displacement speed. Peters and co-workers base their G equation on the work of Williams (1985a and 1985b).

284

Gran et al. (1996) derive a similar expression to equation K15 using Yi instead of G ∂Yi + v 0 ⋅ ∇Yi = −S d ,i ∇Yi ∂t

K21

Where S d ,i is defined as in equation K14, but with a different value for every species i. The difference between equations K14 and K21 is that equation K14 is defined only at the flame location while equation K21 is valid in the entire domain. Equation K21 solves all the species, while equation K14 only solves for G, which is a transformed YCH 4 . Equation K21 has a minus sign in front of the displacement speed because the G gradients are defined to be in the opposite direction to the YCH 4 gradients. This is further evidence that the G equation is just a coordinate transformation of one of the reactive scalars. From equations K14 and K21 it is evident that ∂YCH 4 ∂G = −C ∂t ∂t

K22

∇YCH 4 = −C∇G

K23

∇YCH 4 = C ∇G

K24

Where C is a constant based on the definition of G in relation to YCH 4 . Equations K14 and K21 have the same limitation, an empirical closure model is needed for the displacement speed. Echekki and Chen (1996 and 1999) also use equation K21.

Nilsson (2001) used the G equation model with a table lookup flamelet model based on the mean and variance of G. This model uses G from the entire flow field to

285

calculate the species mass fractions. This seems inconsistent since G is only defined at G = 0. To use G outside of the G = 0 isoscalar, G must really be a reactive scalar or reaction progress variable, which are defined throughout the flow. This is consistent with the findings above, that G is really a coordinate transformation of YCH 4 or c. This model requires the turbulent flame speed to be inputted.

Dandekar and Collins (1995) state that there are two classes of level set approaches for combustion modeling, the Huygen equation (also called the G equation) and the Sivashinsky equation (also called the F equation). The difference is that the G equation neglects the effects of hydrodynamic and thermal-diffusive instabilities, while the F equation accounts for these instabilities that arise from small Lewis numbers. Below the G equation work is discussed (in addition to the work described above). The Sivashinsky equation is described in great detail in Sivashinsky (1976, 1977, 1979 and 1983), Dandekar and Collins (1995) use the model to investigate the effect of small Lewis number. Both classes of level set approaches are limited to the thin flamelet regime and to constant density.

Collins (1995) uses the G equation (Huygen equation) to model a passive flame surface in 2-D DNS. A passive flame surface assumes the density is constant across the flame. Collins acknowledges that this is physically incorrect, but the assumption makes the modeling much easier and allows the investigation of the effects of the turbulence on the simplified flame front. Collins states that the G equation is valid for the flamelet regime of premixed combustion where the flame is thin compared to the turbulence scales. This allows for separation of length and time scales, such that the mean rate of reaction is determined by the mean surface area of the flamelets to first order. Collins uses the turbulent energy spectrum to determine the flame area.

This has some

similarities to the BML model. Since the flow is incompressible the Navier-Stokes equations are reduced to one equation based on the z component of vorticity and the

286

stream function.

The G equation of Peters (1992) is transformed to the new fluid

variables.

Kerstein et al. (1988) and Aldredge (1992) derive equation K2 by different methods and develop different closures for the displacement speed, but the overall G equation concept is the same as the above works. Kerstein et al. base their G equation on the work of Williams 1985b and Osher and Sethian 1988, again assuming constant density. Yakhot (1988) uses the same G equation with the Renormalization Group Theory to derive an expression relating the flame front speed to the laminar flame speed and the rms velocity.

Ashurst (1994) develops a G equation the same as equation K19, stating that it neglects gas expansion effects. Ashurst acknowledges that this appears very artificial, but finds that it reveals concepts of turbulent combustion. “Therefore, the flamelet assumption has been exploited in order to decouple the complexity of combustion from that of turbulence." This is the same argument that Collins (1995) makes. The right side of the equation connects the flame area to the gradient of G, as a replacement for the reaction kinetics. The gradient of G corresponds to the flame surface density (flame area per volume). This follows Damköhler's idea that creation of flame area by turbulent motion causes the larger consumption rate. This idea has some similarities to the BML and Coherent Flame models, where the reaction rates are replaced with a term based on the flame surface density. Ashurst states that the temperature for a premixed flamelet is obtained by a transformation from physical space to G space.

Ashurst relates the

progress variable, c, to the fluctuation of G as follows. ∞

c = ∫ P(G ′)dG ′ −x

K25

287

Here P is the PDF. From data the PDF was calculated to give c, which was also calculated by measuring the displacement of selected G surfaces. Both methods gave the same results, which shows that G and c are related.

The Huygen equation used here is the same as used in the first section, with different closure models for the flame speed.

Vervisch and Veynante (2000) relate the reaction rate closures of several combustion models to each other, showing that the seemingly different models are at least partially related. They present the species mass fraction equation as follows.

ρ

∂Yi & i = ρS d ,i ∇Yi + ρv ⋅ ∇Yi = ∇ ⋅ (ρD i ∇Yi ) + ρω ∂t

K26

Where S d ,i is the displacement speed of the isosurface Yi . This matches Gran et al. (1996), shown here as equation K21. Comparing this to equation K15 shows that the G variable is really a reactive scalar, as discussed above. They break the current turbulent combustion models into three types, geometrical analysis, turbulent mixing and one-point statistical analysis. The geometrical analysis describes the flame as a thin isosurface and uses a model for the displacement speed to eliminate the reaction kinetics. The G equation, BML and flamelet models fall into this category. The turbulent mixing models assume a large Damköhler number, so the reactions are mixing limited, such as the EBU and EDM models. The one-point statistical analysis models use PDF’s to extract means and correlation’s, the CMC and PDF models would be included in this group.

The fact that the G equation is just a coordinate transformation of the mass fraction or progress variable equations is not a coincidence, it is an outcome of the definition of G. Since G = 0 is defined as the point where YCH 4 = 0.1YCH 4,u , G = -1.0 is

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defined as the point where YCH 4 = YCH 4, u and the displacement speed of G at G = 0 is defined to be the displacement speed of the YCH 4 = 0.1YCH 4,u isoscalar surface, G is really defined to be a function of YCH 4 or equivalently c. This will not be a linear function unless a 1-step mechanism is used (or infinite reaction rates), but G is only defined at G = 0 where it has a defined relationship with YCH 4 . G, YCH 4 and c are monotonically related.

A nice aspect of the G equation is that its flame displacement speed can be used to analyze experimental and DNS data to determine the relative effects of curvature, normal diffusion and kinetics on the overall flame speed. The drawback of the G equation method is that it does not provide a first principles closure of the reaction rates, an empirical closure model is required. Limitations of the G equation are: 1) It assumes the density is constant on both sides of the flame (some versions assume it is constant across the flame). 2) It assumes length and time scale separation, i.e. the turbulence does not effect the reaction rates (it only increases the flame front area). And 3) it assumes the flame is infinitely thin with infinite reaction rates (this is the same as using a 1-step mechanism).

In section 2.3.3 it was shown that the premixed flamelet equation is the same as the uniform premixed CMC equation. Here it is shown that the flamelet equation derived from the G equation is also the same as the uniform premixed CMC equation. The G equation version replaces S c in equation 3.16 with an ad hoc model based on the laminar flame speed.

The flamelet equation based on G from Peters (2000) equation 1.159 on page 52 is shown below.

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ρS L ∇ G

dΨi d  2 dΨi  =  ρD i ∇ G  + ωi dG dG  dG 

K27

Expand the first term on the right and rearrange.

ρS L ∇ G

(

)

(

(ρD ∇G )ddGΨ + dGd (ρD ∇G ) ddGΨ − ρS 2

i

)

2 dΨi d 2 dΨi 2 d Ψi = ρD i ∇G + ρD i ∇G + ωi dG dG dG dG 2

2

i 2

2

i

i

L

∇G

dΨi + ωi = 0 dG

with N ≡ D∇G ⋅ ∇G = D(−n ∇G ) ⋅ (− n ∇G ) = D ∇G n ⋅ n = D ∇G 2

2

K28

d 2 Ψi d ρN + (ρN ) dΨi − ρS L ∇G dΨi + ωi = 0 2 dG dG dG dG

ρN

d 2 Ψi  d + (ρN ) − ρS L ∇G  dΨi + ωi = 0 2 dG  dG  dG

Using CMC notation Ψi becomes Q i and as shown above G is really c.

d 2 Q i  dρN  dQ i ρN + − ρS L ∇ c  + ωi = 0 2 dc  dc  dc

K29

The uniform CMC equation from section 3.3.3 (equation 3.16) is

d 2Qi dQ i ρN − ρS c + ωi = 0 2 dc dc

K30

290

where ρS c =

− ρ∑ ωi h f ,i i

K31

∆h sad −u

Comparing equations K30 and K31 to equation K29 gives.

− ρS c =

ρ∑ ω i h f , i i

∆h

s ad − u

=

dρN − ρS L ∇c dc

K32

This says that the flamelet equation derived from the G equation is the same as the uniform CMC equation with the convection term replaced with a model based on the laminar flame speed. A physical description is the S c term is the summation of the reaction rates multiplied by their heat of formation, which is the rate energy is being added to the system (in non-dimensional form). The flamelet equation models this as the gradient of the product of the density and the scalar dissipation, and the laminar flame speed multiplied by the gradient of c.

The rate energy is being released is what

determines the density gradient and the laminar flame speed, so the flamelet model seems to have some physical justification. The flamelet model requires the laminar flame speed as an input (or as a function of other known data) while the CMC model calculates the chemical kinetics from first principles. Equation K32 could be used to calculate the laminar flame speed from the CMC solution.

On the bottom of page 52 Peters (2000) states that the flamelet model based on G allows the use of complex chemistry. The G equation is derived with the assumption of an infinitely thin flame, which implies the fast chemistry limit (or in other words a 1-step mechanism), so there is an inconsistency.

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burned gas

unburned gas n

preheat zone

G=-1.0 Y_CH4=Y_CH4,max

flame zone

G=0 Y_CH4=0.1Y_CH4,max

G=0.11 Y_CH4=0

Figure K-1: Schematic of thin flame showing relationship between the G parameter and the methane mass fraction.

292

Vita

Scott Martin was born and raised in Federal Way, Washington. At the University of Washington in earned a Bachelor of Science degree and a Masters of Science degree in Aeronautics and Astronautics. He served four years active duty in the United States Marine Corps and has worked for NASA and the Boeing Company. He currently lives in Michigan, working for the Ford Motor Company.

In 2003 he earned a Doctor of

Philosophy at the University of Washington in Mechanical Engineering.