Thesis Liangyu

7/30/2019 Thesis Liangyu

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The Pennsylvania State University

The Graduate School

Department of Mechanical and Nuclear Engineering

DETAILED CHEMISTRY, SOOT, AND RADIATION

CALCULATIONS IN TURBULENT REACTING FLOWS

A Thesis in

Mechanical Engineering

by

Liangyu Wang

c 2004 Liangyu Wang

Submitted in Partial Fulfillmentof the Requirements

for the Degree of

Doctor of Philosophy

May 2004


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The thesis of Liangyu Wang has been reviewed and approved* by the following:

Daniel C. HaworthAssociate Professor of Mechanical EngineeringThesis Co-AdviserCo-Chair of Committee

Stephen R. TurnsProfessor of Mechanical EngineeringThesis Co-AdviserCo-Chair of Committee

Andre L. BoehmanAssociate Professor of Fuel Science

Robert J. SantoroProfessor of Mechanical Engineering

Richard C. BensonProfessor of Mechanical EngineeringHead of the Department of Mechanical and Nuclear Engineering

*Signatures are on file in the Graduate School.


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Abstract

The present work aims at a comprehensive approach for the simulation of tur-

bulent reacting flows. In particular, it focuses on the modeling of detailed chemistry,

detailed soot formation and oxidation, and the modeling of detailed radiative heat trans-

fer in gas-phase turbulent flames. In addition, the present work centers on numerical

investigations of oxygen-enriched turbulent nonpremixed flames.

Issues that arise in calculating detailed chemistry, soot formation and oxidation,

and thermal radiation in turbulent reacting flows are reviewed and discussed. Two

detailed models of turbulent combustion are developed using state-of-the-art models of

detailed chemistry, soot, and radiation calculations in turbulent flames. One of the

models is based on an empirical description of the turbulent flow field and the other isbased on CFD modeling of the flow field.

The empirical-description-based model is an extension of Two-Stage Lagrangian

(TSL) model of turbulent jet flames. This extension includes the incorporation of a

detailed soot model and the improvement of the radiation model. The soot model is a

detailed one adopted from Appel-Bockhorn-Frenklachs soot model. The dynamics of

soot particles are described by the method of moments adapted to the TSL formulation.

The original constant-emissivity radiation model is improved by solving the radiative

transfer equation on the spatial configuration of the TSL model using the spherical

harmonic P1 method and the discrete ordinate S2 method. The gray medium assumption

is employed and the Planck-mean absorption coefficient is used to determine the radiative

properties of both gas-phase species and soot particles. With the extended TSL model,

the characteristics of soot, radiation and NOx emissions in oxygen-enriched flames are

studied.

The CFD-based model is based on an engineering CFD code (GMTEC) and it

solves the compressible flow equations on unstructured meshes. GMTEC is extended byincorporating a detailed chemistry model, a detailed soot model, and a detailed radiation

model. The detailed chemistry model is based on the use of the CHEMKIN libraries,


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and the calculations of chemistry are accelerated by using the ISAT software. The

effective use of ISAT for detailed chemistry in nonhomogeneous systems is outlined. The

detailed soot model is adopted from Frenklachs detailed soot model with the method of

moments. It is coupled to the three-dimensional CFD code through transport equations

of soot moments. Two detailed radiation models are implemented, the P1-gray model

and the P1-FSK model. Both models employ the spherical harmonic P1 method for the

solution of the radiative transfer equation on three-dimensional unstructured meshes.

The P1-gray model employs the gray medium assumption and Planck mean absorption

coefficient for radiative property evaluations. The P1-FSK model addresses the nongray

nature of the radiative heat transfer by using the full-spectrum k-distribution method.

The CFD-based comprehensive model is then exercised to simulated an oxygen-enriched

flame.

The two detailed models developed have proven to be successful in the simulation

of oxygen-enriched turbulent flames. The advantage of the TSL model is its compu-

tational economy. It is shown to be capable of predicting the general trends of soot,

radiation, and NOx emission with oxygen index, fuel type, and initial jet velocity, but it

failed to provide quantitative predictions of flame structure due to its simplistic treat-

ment of the hydrodynamics.

The advantage of the CFD-based model is its capability of performing detailed,

quantitative predictions and of capturing the strong couplings among soot, radiation,

flame structure, and NOx emissions in oxygen-enriched flames. It can be used to identify

the key sensitivities in soot and NOx formations, to study the effects of nongray gas-

phase and soot radiation, and to study the influence of mixing, fuel type, and oxygen

index on the soot formation, NOx emission, and thermal radiation characteristics of

oxygen-enriched turbulent flames. The deficiencies of the CFD-based model include the

simple turbulent combustion model, neglect of turbulent fluctuations in composition and

temperature, and the P1 approximation used to solve the RTE. These are the subjects

of ongoing research.


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Table of Contents

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii

Chapter 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Chapter 2. Detailed Chemistry Calculations . . . . . . . . . . . . . . . . . . . . 7

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Expenses in Chemistry Calculations . . . . . . . . . . . . . . . . . . 10

2.3 Strategies in Detailed Chemistry Calculations . . . . . . . . . . . . . 13

2.3.1 Simplification of the Flow Field Physics . . . . . . . . . . . . 13

2.3.2 Reduction of Chemistry . . . . . . . . . . . . . . . . . . . . . 13

2.3.3 Storage/Retrieval Scheme . . . . . . . . . . . . . . . . . . . . 15

2.4 In Situ Adaptive Tabulation (ISAT) . . . . . . . . . . . . . . . . . . 17

2.5 Turbulence/Chemistry Interactions . . . . . . . . . . . . . . . . . . . 21

2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Chapter 3. Soot Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2 Soot Formation and Oxidation: Current Understanding . . . . . . . 30

3.2.1 Soot Particle Inception . . . . . . . . . . . . . . . . . . . . . . 31

3.2.2 Soot Surface Growth . . . . . . . . . . . . . . . . . . . . . . . 32

3.2.3 Soot Particle Coagulation . . . . . . . . . . . . . . . . . . . . 32

3.2.4 Soot Particle Oxidation . . . . . . . . . . . . . . . . . . . . . 33


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3.3 Soot Formation and Oxidation: Modeling . . . . . . . . . . . . . . . 34

3.4 Method of Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.5 Soot Calculations in Turbulent Flames . . . . . . . . . . . . . . . . . 41

3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

Chapter 4. Radiation Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.2 Governing Equations for Radiative Heat Transfer . . . . . . . . . . . 46

4.3 Solution Methods for the Radiative Transfer Equation . . . . . . . . 48

4.3.1 Optically Thin Approximation . . . . . . . . . . . . . . . . . 49

4.3.2 Spherical Harmonic Method . . . . . . . . . . . . . . . . . . . 49

4.3.3 Discrete Ordinate Method . . . . . . . . . . . . . . . . . . . . 50

4.3.4 Zonal Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.3.5 Statistical Method . . . . . . . . . . . . . . . . . . . . . . . . 52

4.3.6 Hybrid Methods . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.4 Radiative Properties of Participating Media . . . . . . . . . . . . . . 53

4.4.1 Radiative Properties of Gas-Phase Species . . . . . . . . . . . 53

4.4.2 Radiative Properties of Soot Particles . . . . . . . . . . . . . 57

4.5 Turbulence-Radiation Interactions . . . . . . . . . . . . . . . . . . . 61

4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

Chapter 5. Two-Stage Lagrangian Simulations of Oxygen-Enriched flames . . . 65

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.2 The Two-Stage Lagrangian Model . . . . . . . . . . . . . . . . . . . 66

5.3 Detailed Chemical Kinetics . . . . . . . . . . . . . . . . . . . . . . . 68

5.4 Soot Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.5 Radiation Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.6 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.6.1 Performence of Radiation Sub-Models . . . . . . . . . . . . . 76

5.6.2 NOx Emissions . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.6.3 Soot Volume Fractions . . . . . . . . . . . . . . . . . . . . . . 80


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5.6.4 Flame Radiation . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.6.5 Artificially Enhanced In-Flame Soot . . . . . . . . . . . . . . 93

5.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

Chapter 6. CFD Modeling of Oxygen-Enriched Flames . . . . . . . . . . . . . . 97

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

6.2 The CFD model of flow fields . . . . . . . . . . . . . . . . . . . . . . 98

6.3 Detailed Chemistry Modeling . . . . . . . . . . . . . . . . . . . . . . 100

6.3.1 Chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

6.3.2 Turbulent Combustion Model . . . . . . . . . . . . . . . . . . 101

6.3.3 Effective Use of ISAT . . . . . . . . . . . . . . . . . . . . . . 102

6.4 Detailed Soot Modeling . . . . . . . . . . . . . . . . . . . . . . . . . 111

6.4.1 Soot Moment Transport Equations . . . . . . . . . . . . . . . 111

6.4.2 Soot Moment Source Terms . . . . . . . . . . . . . . . . . . . 116

6.4.3 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

6.5 Detailed Radiation Modeling . . . . . . . . . . . . . . . . . . . . . . 117

6.5.1 P1-Gray model . . . . . . . . . . . . . . . . . . . . . . . . . . 118

6.5.2 P1-FSK model . . . . . . . . . . . . . . . . . . . . . . . . . . 120

6.5.3 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

6.6 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 125

6.6.1 On Radiation Calculations . . . . . . . . . . . . . . . . . . . . 126

6.6.1.1 Effects of Nongray Radiation . . . . . . . . . . . . . 129

6.6.1.2 Effects of Soot Radiation . . . . . . . . . . . . . . . 134

6.6.1.3 Applicability of the P1 method to Jet Flames . . . . 136

6.6.2 Simulation of an Oxygen-Enriched Flame . . . . . . . . . . . 140

6.6.2.1 Key Sensitivities in Soot Predictions . . . . . . . . . 140

6.6.2.2 Key Sensitivities in Radiation Predictions . . . . . . 145

6.6.2.3 NOxemissions . . . . . . . . . . . . . . . . . . . . . 148

6.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149


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Chapter 7. Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 151

7.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

7.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

7.3 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

Appendix A. Detailed Reaction Mechanism . . . . . . . . . . . . . . . . . . . . . 172

Appendix B. Radiation Submodels in the TSL Model . . . . . . . . . . . . . . . 184

Appendix C. Evaluation of Planck-Mean Absorption Coefficients . . . . . . . . . 195


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List of Tables

5.1 Test Conditions of Oxygen-Enriched Flames . . . . . . . . . . . . . . . . 75

5.2 Calculated Peak Temperatures by TSL . . . . . . . . . . . . . . . . . . . 78

6.1 Statistics of Using ISAT . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

6.2 Comparisons Between ISAT and Direct Integration . . . . . . . . . . . . 109

6.3 Statistics of the Two Approaches to the Mapping Gradient Matrix Cal-

culation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

6.4 Summary of Turbulence Model Parameters . . . . . . . . . . . . . . . . 127

6.5 Case specifications for soot sensitivity study. . . . . . . . . . . . . . . . 142

6.6 Comparisons of global quantities soot models. . . . . . . . . . . . . . . 144

6.7 Comparisons of global quantities radiation models. . . . . . . . . . . . 148


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List of Figures

2.1 Interconnection between simulation code and ISAT algorithm . . . . . . 19

5.1 TSL two-reactor and reactor/diffusion-flame models. . . . . . . . . . . . 675.2 The evolution of GRI-Mech. . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.3 The performance of radiation models. . . . . . . . . . . . . . . . . . . . 77

5.4 NOx emission indices from TSL model and experiments: fuel, velocity,

and oxygen index effects. . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.5 Temperature profiles from TSL model for propane flames with 40% oxy-

gen index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.6 Axial profile of equivalent soot volume fraction from experiments for

propane flames with v0 = 21.8 m/s. . . . . . . . . . . . . . . . . . . . . . 82

5.7 Axial profile of equivalent soot volume fraction from TSL model for

propane flames with v0 = 21.8 m/s. . . . . . . . . . . . . . . . . . . . . . 84

5.8 Normalized peak equivalent soot volume fraction from experiments (curves

with symbols) and TSL model: fuel and oxygen index effects. . . . . . . 85

5.9 Normalized peak equivalent soot volume fraction from experiments (curves

with symbols) and TSL model: fuel jet velocity and oxygen index effects. 86

5.10 Radiant fractions from experiments: fuel, velocity, and oxygen index effects. 885.11 Radiant fractions from TSL model: fuel, velocity, and oxygen index effects. 89

5.12 Calculated global residence times as functions of oxygen index for differ-

ent fuels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.13 Comparisons between calculated soot and gas-phase radiation heat flux

for propane and natural gas flames with v0 = 21.8 m/s, 40% oxygen

index, and soot volume fractions increased to match experiments. . . . . 92

5.14 Calculated soot contribution to total radiation for propane flames with

peak soot volume fractions increased to match experiments. . . . . . . . 94

5.15 Effects of artificially enhanced soot on NOx emission and radiation for a

propane flame with 30% oxygen index and v0 = 21.8 m/s. . . . . . . . . 95


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6.1 Performance of ISAT in terms of CPU time and CPU time ratio . . . . 110

6.2 A control volume for derivation of moment transport equation . . . . . . 112

6.3 Computational domain for modeling oxygen-enriched turbulent jet flames 127

6.4 An artificial absorption coefficient and its planck mean . . . . . . . . . . 130

6.5 Axial profile of radiation heat flux and temperature calculated by the

two radiation models: gray vs. nongray . . . . . . . . . . . . . . . . . . 132

6.6 Radial profiles of temperature calculated by the two radiation models at

four axial locations: gray vs. nongray . . . . . . . . . . . . . . . . . . . 133

6.7 (Manipulated) distribution (axial profile left, contour right) of soot vol-

ume fraction for a propane flame with 40% oxygen index and 21.8 m/s

jet velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

6.8 Axial profile of radiation heat flux and temperature calculated by the

two radiation models: soot effects . . . . . . . . . . . . . . . . . . . . . . 137

6.9 Radial profiles of temperature calculated by the two radiation models at

four axial locations: soot effects . . . . . . . . . . . . . . . . . . . . . . . 138

6.10 Computational configuration to test the P1 approximation . . . . . . . . 139

6.11 Axial profiles of radiation heat flux: study of the P1 approximation . . . 141

6.12 Axial profiles of equivalent soot volume fraction. . . . . . . . . . . . . . 143

6.13 Axial profiles of radiant heat flux. . . . . . . . . . . . . . . . . . . . . . 146

6.14 Computed contours of flame temperature, soot volume fraction, and

species mass fractions of CO2, H2O, and NO. . . . . . . . . . . . . . . . 147


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Acknowledgments

I am most grateful and indebted to my thesis advisors, Dr. Dan Haworth, for

his guidance and for his profound knowledge in turbulent combustion modeling and in

numerical methods, and Dr. Steve Turns, for his guidance and for his profound knowledge

in combustion theory and combustion diagnostics. I am also grateful and indebted to Dr.

Michael Modest, for his profound knowledge in radiative heat transfer and in numerical

methods. I thank my other committee members, Dr. Andre Boehman and Dr. Robert

Santoro, for their insightful commentary on my work. Finally, I thank my family for

their support and encouragement.


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Chapter 1

Introduction

1.1 Motivation

Combustion has had significant impact on our daily life since the beginning of

human history. Today we depend heavily on the combustion processes that transform

the chemical energy in fossil fuels into the thermal energy that powers our society. Auto-

mobiles, aircraft, power plants, and furnaces are only a few examples where combustion

plays an important role. However, in addition to making our lives easier and better,

combustion also negatively affects our society. It threatens human lives by generating

environmental pollutant such as oxides of nitrogen (NOx), and by changing the global

climate pattern via greenhouse effects. Considering the importance of combustion and

the decreasing resource of fossil fuels, it becomes more and more critical to obtain com-

bustion processes of high fuel efficiency but low pollutant emissions.

Most combustion processes occur in a turbulent flow environment: this includes

automotive engines, gas turbine combustors, and industrial burners, among other de-

vices. Turbulent combustion is among the most challenging and important subjects in

theoretical and engineering sciences. It involves a range of complex physical and chemical

phenomena that interact, strongly and nonlinearly with one other. The major physical

processes include turbulent transport, finite-rate chemical reaction, radiative heat trans-

fer, and multiphase flow. Each process individually represents a challenging subject, let

alone the nonlinear interactions among them. The difficulties of understanding turbulent

combustion are further compounded by three-dimensional flows in complex geometries,

such as the combustion chamber of an automotive engine.

Until about 30 years ago, the study of combustion processes and the development

of combustion technologies relied almost exclusively on experimental methods. The ex-

perimental approach is to analyze and optimize the performance of a real combustion


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process by conducting experiments usually on an abstracted or small-scale version of the

real process or device, since the actual operating conditions may not be accessible for

repeated and controlled evaluations. While capable of providing the most realistic an-

swers to many combustion problems, experimental methods suffer from scaling problems,

measurement difficulties, operating costs, and time consumption [1].

Today numerical modeling plays an increasingly important role in the design

and optimization of turbulent combustion processes. In comparison with experimental

methods, numerical modeling and numerical experiments may be less expensive and take

less time than experimental programs. More importantly, they can provide information

that cannot be obtained from experiments.

Because of practical limitations of computer storage and speed, and our inability

to understand and describe mathematically the complex phenomena involved in turbu-

lent combustion, simplifications of varying degree must be employed in constructing a

turbulent combustion model. Current simulation codes for single-phase turbulent com-

bustion may be computational fluid dynamics (CFD) based, with a turbulence sub-

model such as k-, and a probability density function (PDF) based approach for turbu-

lence/chemistry interactions. A reduced mechanism for the chemistry of interest is used,

because even a moderately detailed mechanism that considers 50 chemical species would

be computationally intractable. Radiative heat transfer should be considered in many

combustion analysis, but radiation is such a complicated phenomenon that it is either

treated simply or ignored all together in current combustion simulation codes.

Detailed chemical mechanisms are required in combustion simulations to address

issues such as extinction and ignition phenomena, pollutant emissions including NOx,

soot, CO and unburned hydrocarbons, and unsteady phenomena including combustion

instabilities. Radiative heat transfer needs to be considered and treated accurately in

combustion simulations, because it often is the dominant heat transfer mode due to its

fourth power dependence on temperature. Besides being a primary mechanism for heat

transfer, radiation changes the flame temperature, affects the flow field through densitychanges, and affects NOx emissions through the thermal NO mechanism. Soot is a major

pollutant and its formation represents incomplete combustion. Soot is also an important


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industrial product and it is a strong source of radiation from flames. The prediction of

soot formation is therefore of interest in many situations.

With continuous improvements in physical understanding, numerical methods,

and computer capabilities, combustion modeling is becoming more and more sophisti-

cated. The incorporation of detailed chemical mechanisms and sophisticated radiation

models into the current combustion simulation codes is a timely research subject.

1.2 Objectives

Development of a comprehensive combustion simulation tool

Investigation of oxygen-enriched combustion

Over the past 30 years, advances in computer science and numerical methods have

made CFD, in place of wind tunnels, the primary analytical tool in solving many aero-

dynamic problems [1]. At present, however, CFD-based combustion simulation tools

remain limited in their capacity to deal simultaneously with three-dimensional time-

dependent turbulent flow, realistic chemistry and turbulence-chemistry interaction, mul-

tiphase/heterogeneous systems, radiation heat transfer and turbulence/radiation inter-

action [2, 3]. The need for experiments will remain for the foreseeable future, not only

to play a primary role in combustion investigation, but also to provide valuable data

and physical insights for model development and model validation. Nevertheless, it isexpected that in the near future combustion simulation will follow the same pattern as

observed for aerodynamics where CFD has become a dominant tool in the analysis and

design of processes and devices.

For numerical simulations to be truly useful and productive in the design and

analysis of combustion processes, a comprehensive approach has to be taken, that is, a

combustion simulation tool must take into account all the pertinent chemical-physical

phenomena and must incorporate the appropriate corresponding sub-models into the

overall solution to a combustion process of interest [3, 4]. A comprehensive combustion

simulation code should be capable of providing detailed information on a combustion

process and properties, such as temperature and pressure distributions, velocity fields,


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chemical species compositions, NOx formation, soot formation and oxidation, radiative

heat loss, and so on.

Combustion models also serve as a way of organizing our physical understanding.

A good model should capture our physical understanding of the underlying processes.

One objective of the present research is to extend the scope of current combustion

simulation tools by incorporating calculations of detailed chemistry, soot formation and

oxidation, and radiative heat transfer.

Another objective of this research is to explore oxygen-enriched combustion. The

use of oxygen-enriched air or pure oxygen as an oxidizer offers a number of advantages

in many combustion applications, such as metal heating and melting, glass melting, and

waste incineration [5]. In glass melting, for example, use of oxygen can result in reduced

particulate emissions, decreased NOx emissions, increased productivity, and fuel sav-

ings [6]. Soot formation and thermal radiation are closely coupled factors in determining

flame structure, temperature, and pollutant emissions, and have particular significance in

common oxy-fuel combustion applications. Industrial oxy-fuel burners have been devel-

oped that create low-momentum, highly luminous flames [7]. Studies have demonstrated

the benefits of this technology in glass manufacturing processes [8]. In other applications

such as aluminum melting, maintaining a relatively large convective heat transfer com-

ponent, in addition to increased luminosity, can be advantageous. In such applications,

an air-oxygen oxidizer is sometimes used rather than pure oxygen. A goal of this research

is to use the newly developed comprehensive simulation tools to investigate numerically

the effects of process variables such as oxidizer oxygen content, fuel composition, and

fuel jet velocity on soot formation, radiation, and pollutant emissions in turbulent jet

flames.

1.3 Approach

If the fuel stream and oxidizer stream are initially separated, a nonpremixed flame

can then be formed as the two streams mix and react, with the rate of reaction being

controlled by the rate of mixing. Turbulent nonpremixed flames are employed in many

practical combustion systems, principally because of the ease with which such flames


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can be controlled [9]. Examples of such combustion systems include diesel engines, gas

turbine combustors, and industrial burners. The combustion processes in these systems

can be idealized as a turbulent nonpremixed jet flame, which is an example of a canoni-

cal turbulent reacting flow. The jet flame retains the essential physical features of these

practical combustions systems while allowing detailed quantitative measurements includ-

ing initial and boundary conditions, and parametric variations of key global parameters,

such as Reynolds and Damkohler numbers [3]. Measurement and modeling of turbulent

non-premixed jet flames have been the subject of a biennial international workshop [10].

Turbulent nonpremixed jet flames are the focus of this research effort, not only

because of their importance to practical combustion systems, but also because of the

availability of a large amount of high quality experimental data. Model validation and

development will be based on two sets of experimental data: measurements for piloted

methane-air turbulent jet diffusion flames from Sandia National Laboratory [11], and

measurements for oxygen-enriched turbulent jet flames from the Propulsion Engineering

Research Center at Penn State University [12].

Two numerical simulation codes are used for this research effort: the Two-Stage

Lagrangian (TSL) model developed at Sandia National Laboratory [13] and the GMTEC

CFD code developed at General Motors [14]. The TSL model has proved to be a useful

and computationally efficient model for the representation turbulent jet diffusion flames

[15]. The most significant advantage of the TSL model is its computational efficiency,

even with detailed chemical mechanisms of more than 50 species. This advantage derives

from fact that turbulent mixing in the TSL model is treated simply by using experimental

correlations. In this research, we extend the ability of the TSL code to deal with detailed

chemistry by incorporating a soot model and by improving its original simple radiation

model.

The GMTEC code is a finite-volume CFD code that solves a system of cou-

pled nonlinear partial differential equations (pdes) for a compressible multi-component

turbulent flow. Principle pdes correspond to conservation of mass (continuity), mo-mentum, absolute enthalpy and species mass fractions. GMTEC employs an iterative

time-implicit pressure-based sequential (segregated) procedure for the solution of the


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coupled pdes. Favre-averaged dependent variables are calculated and a standard two-

equation k- model is used for turbulent closure. The conservation equations are solved

on an unstructured mesh using cell-centered variables. The discretization accuracy is

first-order in time and up to second-order in space. GMTEC was designed for hydrody-

namics of chemically reacting flows, with ports for incorporation of additional combustion

sub-models. In this research, we incorporate detail chemistry, soot, and radiation models

into GMTEC.


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Chapter 2

Detailed Chemistry Calculations

2.1 Introduction

The mathematical starting point for most nonreacting flow problems is the Navier-

Stokes equations. For compressible flows, an energy equation is required. The energy

variable often is taken to be internal energy. Supplemental to this set of conservation

equations (pdes) are the equations of state and fluid property specification. For real or

ideal gases, these equations serve as the link between fluid dynamics and thermodynamic

aspects of the flow. Together with appropriate initial and boundary conditions, these

equations describe completely a nonreacting flow.

The mathematical starting point for turbulent reacting flow problems also resides

on the Navier-Stokes equations. Reacting flows involve frequently large heat release and

transformation of chemical species. These features of reacting flows require the addition

of additional terms and equations to the standard set of Navier-Stokes equations. Using

Cartesian tensor notation, the set of conservation equations for chemically reacting flows

can be expressed as follows (e.g. [16, 17]):

mass

t+

uixi

= 0 (2.1)

momentum

ujt

+ujui

xi=

ijxi

p

xj+ gj for j = 1, 2, 3 (2.2)

speciesY

t+

Yuixi

= Jixi

+ S for = 1, 2, . . . , N s (2.3)


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absolute enthalpy

h

t+

huixi

= Jhixi

+Dp

Dt+ ij

uj

xi+ Qrad. (2.4)

In these governing equations, a Roman index denotes a component of a three-dimensional

vector(e.g., i = 1, 2, 3), a Greek index denotes a chemical species (e.g., = 1, 2, . . . , N s),

and the usual summation convention applies over repeated Roman indices within a term.

Here u denotes velocity, Y denotes the mass fractions of the Ns chemical species, and h

the absolute enthalpy. Mixture mass density is , pressure is p, body force (per unit mass)

is g, and , J,

Jh, denote, respectively, the viscous stress tensor and the molecular fluxes

of species and enthalpy. The chemical source term, S, equals the product of the molar

chemical production rate, , and molecular weight, W, for species , S = W.

The volume rate of heating due to radiation is Qrad.

Compared to the nonreacting case, this set of conservation equations includes

species transport equations with chemical source terms to address chemical transforma-

tions, and it employs absolute enthalpy instead of internal energy as the energy variable.

Chemical heat release produces high temperature in the flow field and variable density

effects. The high temperature results in thermal radiation playing a dominant role in

heat transfer. These are a few complexities arising from chemical reactions that com-

pound the already difficult solutions of the Navier-Stokes equations. Detailed discussions

on these issues can be found in references [3, 9, 17].

All of these chemistry-related computations, including radiation properties for a

participating medium, are based on the information provided by a chemical reaction

mechanism of interest, which works with a thermodynamic database and a transport

property database for the chemical species involved. These databases usually organize

the thermodynamic and transport data in terms of polynomials as functions of temper-

ature: for example, CHEMKIN database [18], JANAF Tables [19], and NASA database

[20]. A reaction mechanism is a collection of elementary (or global) reactions necessary to

describe the chemical transformation and reaction rate coefficients for each reaction; usu-

ally the latter are given in Arrhenius form. The rate coefficients are derived mainly from


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experimental measurements in shock tubes and flow reactors, from quantum-mechanical

calculations based on different theoretical approximations, or from rough estimates based

on simple collision theories. Dedicated software such as the CHEMKIN package [21] has

been developed to facilitate chemistry-related computations. Given a reaction mecha-

nism, a thermodynamic database and a transport database as inputs, the CHEMKIN

package can return information on chemical production rates, thermodynamic properties,

and molecular transport coefficients.

The number of species and the number of elementary reactions in a mechanism

determine the level of complexity in chemistry calculations. In reality, a chemical mech-

anism for the transformation of hydrocarbon fuels can include as many as thousands

of elementary reactions involving hundreds of intermediate species. The combustion of

a simple fuel like methane in air requires 325 reactions and 53 species (GRI Mech 3.0

[22]) for a satisfactory chemical description. In the case of auto-ignition of a Diesel fuel

with the typical fuel cetane, several thousand elementary reactions are required to de-

scribe the overall process [3]. In practice, the number of species tracked in combustion

simulations impacts the computer memory usage and CPU time. To make the chem-

istry calculations tractable in three-dimensional time-dependent CFD simulations even

with the largest available supercomputers, only a moderate level of chemistry (tens of

species and hundreds of elementary reactions) generally can be allowed. Broadly used in

current combustion simulations are mechanisms reduced from their detailed versions, or

even simpler one-step global mechanisms that have only three generic species: reactant,

oxidizer, and product.

However, a detailed reaction mechanism is a prerequisite for a realistic and accu-

rate prediction of chemically reacting flows. Predictions that require detailed chemistry

include flame propagation speed, flame location and size, flame extinction and ignition

phenomena, and pollutant emissions. Many important issues that arise in practical com-

bustion applications and environmental concerns can be addressed numerically only with

detailed chemistry. The simulation of engine combustors requires detailed mechanismsto address kinetically controlled phenomena [23] such as low-temperature auto-ignition,


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and to address time-dependent phenomena such as combustion instability [24] and vari-

ability [25, 26]. The use of alternative fuels and fuel additives require detailed mecha-

nisms to address fuel composition issues [27]. The application of turbulent combustion

models in the chemical industry requires detailed mechanisms to study operating char-

acteristics of chemical reacting systems [3, 28]. Stringent pollutant emission regulations

require detailed mechanisms to predict trace pollutant species such as NOx, unburned

hydrocarbons, and soot [29]. With advances in computer technology and numerical al-

gorithms, detailed chemistry calculations will eventually become an intrinsic part of any

comprehensive, reliable, and credible combustion simulation tool.

2.2 Expenses in Chemistry Calculations

It is computationally demanding to solve the complete conservation equations

including detailed chemistry even for simple two-dimensional laminar flames [30]. For

more complex flows, especially turbulent flames, the inclusion of a detailed chemistry

calculation is even more challenging. For the foreseeable future, CPU time and com-

puter memory limitations will prohibit implementations of fully detailed descriptions of

chemistry into three-dimensional time-dependent CFD simulations of combustion appli-

cations [31]. This statement is based on three factors that make chemistry calculations

expensive in CPU time and demanding in memory requirement.

The first factor is that the chemical system is usually described in terms of themass fractions of each chemical species as functions of space and time. Because of

the large differences in species diffusivities due to large differences in their molecular

weight, and because of the large differences in species chemical source terms due to

large differences in their chemical bonds, each species requires one pde (equation 2.3) for

its conservation. Therefore, if there are Ns species in the system under consideration,

usually Ns1 species conservation equations (the sum of species mass fraction is unity)

are needed to determine the chemical state of the reacting system. For even a moderately

sized chemical mechanism, this number far exceeds the number of pdes required for mass,

momentum and energy conservation.


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The differences in species molecular diffusivities mentioned above often can be

neglected in turbulent flows because of the relatively large effect of turbulent mixing.

However, the differences in chemical source terms result in great complexity and intro-

duce fundamental difficulties in the simulation of turbulent reacting flows.

The second factor that makes chemistry calculations expensive is the nonlinearity

and complexity of the chemical source terms. A chemical reaction mechanism involving

Ns species and L elementary reactions can be written as [9]

Ns=1

lX

Ns=1

lX for l = 1, 2, . . . , L (2.5)

where l and

l are the stoichiometric coefficients for the th species X and l

th

reaction. The net molar production rate, i.e., the chemical source term , for each

species can be written as

=

Ll=1

lql for = 1, 2, . . . , N s, (2.6)

where,

l = (

l

l), (2.7)

and,

ql = kflNs=1

[X]l krl

Ns=1

[X]l . (2.8)

Here [X] denotes the molar concentration of species X, kfl and krl are, respectively,

the forward and reverse rate coefficient for the lth reaction and they are related through

an equilibrium constant. The rate coefficients are commonly expressed in Arrhenius form

as,

kfl = AflTbfl exp

EA,flRuT

, (2.9)

where Afl is the pre-exponential factor, bfl is the temperature exponent, and EA,fl is the

activation energy. These parameters usually are determined by experiments. Inspection

of the rate expression shows that in order to compute the chemical source term of each


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species, all the elementary reactions in the mechanisms have to be considered, in general.

This is tedious for a mechanism with hundreds or thousands of reactions.

The third expense in chemistry calculations results from the fact that there is a

large range of chemical time scales inherent in a reaction mechanism, typically ranging

from 1010 to 1 second or more in combustion problems [32] . Mathematically, chemical

time scales can be defined by the inverses of the absolute values of the eigenvalues of

the Jacobian matrix dS/dY. Conceptually, these time scales correspond to the time

required for the species concentrations to fall from their initial values to a value equal

to 1/e times their initial value [9]. Since the chemical time scales are different from

typical mixing time scales in a combustion system, species conservation equations often

are solved by an operator-splitting method [33, 34], where the the change in composition

resulting from chemical reaction is determined for every time step by integrating a set

of odesdY

dt= S(

Y , T , p) for = 1, 2, . . . , N s. (2.10)

The ratio between the largest and smallest time scales characterizes the degree of stiff-

ness of the ode set. The smallest time scales have to be resolved in the numerical

solution, even if one is interested only in the slow processes. Otherwise the numerical

solution tends to become unstable. The wide variation in time scales makes the above

odes very stiff and severely increases the cost of solving the equations.

These three observations together illustrate why in a practical combustion calcu-lation with a moderately detailed chemical mechanism, over 90 percent of the CPU time

may be spent on chemistry calculations. This poses a serious obstacle to combustion

simulations. For example, in a finite-volume simulation of an axisymmetric laminar jet

flame with a 122-species mechanism, the CPU time for the integration of equations (2.10)

is about 0.06 second per time step on a 2.8 GHz Intel Xeon processor. If we use a mod-

erately fine mesh of 5,000 cells, then 2,000 time steps would require 10 7 integrations,

which corresponds to seven days of CPU time for just the chemistry. This illustrates

that detailed chemistry calculations are impossible or extremely expensive to implement

in three-dimensional time-dependent CFD simulations of practical turbulent combustion

systems.


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2.3 Strategies in Detailed Chemistry Calculations

Over the past decade, significant progress has been made in the ability of compu-

tational models to model both chemical kinetics and fluid mechanics in turbulent reacting

flows. Describing turbulent flames using full chemical kinetics and fully resolved fluid

dynamics is still computationally infeasible due to time and memory constraints. Various

strategies have been developed to address this issue and these strategies can be classified

into three categories:

1. Simplification of the flow field physics;

2. Reduction of chemistry;

3. Storage/retrieval schemes for using both detailed descriptions of chemistry and

fluid dynamics.

2.3.1 Simplification of the Flow Field Physics

Simplifications of the flow field physics in turbulent flames have been made possi-

ble by the dramatic developments in experimental combustion diagnostics in recent years,

particularly in the use of lasers. These developments have led to an ability to make de-

tailed measurements within turbulent flames, from which it is possible to construct a

detailed picture of the structure of the flames [35]. Based on insightful experimental

observations, models that capture the salient features of combustion fluid dynamics have

been proposed. One example is the Two-Stage Lagrangian (TSL) model [15] of turbulent

jet flames. The advantages of such models reside in the fact that they permit the use of

arbitrarily complex chemistry while retaining the main features of the flow fields. Dis-

advantages are that although the use of experimental correlations relax the constraints

imposed by computer time and memory, it precludes using these models as quantitative

simulation tools for detailed predictions of flame structure.

2.3.2 Reduction of Chemistry

The reduction of complex chemical systems has long been desired in combus-

tion simulations. Besides the driving force obtained from computational complexity in


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combustion simulations, there are the large uncertainties in determining some elemen-

tary chemical kinetic rates, which may exceed three orders of magnitude [36]. Reduced

chemistry enables the simulations of the flames that could not be simulated otherwise,

and therefore enhances the understanding of these flames and provides more accurate

predictions or more educated guesses of the flame structures. By introducing approxi-

mations, however, reduced chemistry looses the ability to describe combustion chemistry

in a full, accurate, and general manner. Therefore they fail to address many subtle is-

sues in combustion applications and sometimes can only be applied to a limited range

of thermo-chemical conditions. Nevertheless, reduced chemistry is used predominately

in current combustion simulations.

Numerous approaches have been devised for the reduction of complex chemical

systems. Here, the reduction of complex chemical systems refers not only to the re-

duction of reaction mechanisms and number of the chemical species, but also to other

methods that lower the dimension, or number of degrees of freedom, in the description of

the time evolution of a chemical system. These other methods include the rate-controlled

constrained-equilibrium (RCCE) method [37, 38], the computational singular perturba-

tion (CSP) method [39, 40], and the intrinsic low-dimensional manifolds (ILDM) method

[41, 42, 43]. Compared to reaction mechanism reductions, which are applied through-

out the computational domain and time, these other methods can be viewed as reduced

mechanisms with time- and space-varying stoichiometric coefficients, rate parameters,

and even number of reactions, although they usually do not provide reduced reaction

mechanisms in an explicit form. Therefore, mechanism reduction can be termed as

global chemistry reduction, and the other methods as local chemistry reduction.

All reduction methods aim at reducing the number of differential equations re-

quired to solve the chemical system; this usually is the bottleneck in turbulent com-

bustion calculations. The techniques frequently employed include sensitivity analysis

(used to identify the rate-limiting reaction steps), reaction flow/flux analysis (used to

determine the characteristic reaction path), and eigenvalue/vector analysis (used to de-termine the characteristic time scales and directions of the chemical reactions) [32]. In


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reaction mechanism reductions, quasi-steady-state and/or partial equilibrium assump-

tions [9] commonly are employed to identify species in steady state and reactions in

partial equilibrium; these are then used to eliminate species and/or reactions that can

be represented by algebraic expressions. Reviews on mechanism reduction can be found

in [44, 45], and the general procedure of applying this approach is described in [46, 36, 31].

In local reduction methods, CSP and ILDM are based on approaches from dynamic sys-

tem theory, while RCCE is based on the maximum entropy principle of thermodynamics.

Both CSP and ILDM employ eigenvalue analysis to identify the fast time scales of the

chemical reacting systems, the dimensions of which can then be reduced. However, CSP

aims at decoupling the slow and fast time scales to remove the stiffness of the odes (equa-

tion 2.10) of the chemical system, while ILDM aims at describing the chemical system

with only a small number of reaction progress variables. RCCE assumes that a non-

equilibrium reacting system will relax to its final equilibrium state through a sequence

of rate-controlled constrained equilibrium states that can be determined by maximizing

the entropy subject to the instantaneous values of the constraints. Thus only the rate

equations for the constraints, the number of which is much smaller than the number of

species, must be integrated. The accuracy of RCCE can be systematically improved by

adding constraints imposed by increasingly faster reactions. All these reduction methods

take a detailed mechanism as input. CSP and ILDM can be automated. For mecha-

nism reduction and RCCE, a computer program can be written to realize automated

chemistry reduction [47, 38].

2.3.3 Storage/Retrieval Scheme

The storage/retrieval schemes [48] comprise the third category of strategies in de-

tailed chemistry calculations. The basic idea of storage/retrieval schemes is that during

the course of solving the time evolution of the chemical system represented by equa-

tions (2.10), the information generated at each integration time step (such as the initial

conditions and the solutions), are stored in a specially organized table. When similar

initial conditions are encountered, instead of performing an expensive direct integration

of the odes, a retrieval or interpolation of the data in the table is performed with a CPU


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time that is much less than that of the direction integration. For this approach to be

workable, table reuse must be sufficiently high. Intuitively, this should be the case when

the system approaches equilibrium or steady state. Mathematically, this has been shown

to be true for many situations in combustion simulations with the aid of the concepts

introduced in the ILDM method [41, 48, 32]. ILDM is a useful concept for understanding

the evolution of reacting systems, in addition to providing a method for chemistry reduc-

tion. Its conclusions regarding chemical composition space establish the foundation for

various storage/retrieval methods. Together with the understanding obtained via ILDM

for reacting flows, the storage/retrieval scheme constitutes an effective solution to the

problem of using detailed chemistry in CFD simulations.

Several methods that have been developed recently fall into the storage/retrieval

category. These methods can be grouped into methods based on tabulation, such

as structured look-up table (LUT) [49], in situ adaptive tabulation (ISAT) [48], and

database on-line for function approximation (DOLFA) [50]; methods based on neural

networks [51, 52]; and methods based on orthogonal polynomials [53, 54, 55]. General

criteria by which storage and retrieval methods can be judged are the CPU time re-

quired to create the store; the memory required for the store; inaccuracy in the retrieval

(or interpolation error); the CPU time required for the retrieval/interpolation; and the

degree to which the method is generally applicable and automated [48].

Among these storage/retrieval methods, one of the most successful and promis-

ing is in situ adaptive tabulation. ISAT achieves an efficient solution of the reaction

equations through the dynamic creation of a look-up table based on direct integration

results and an accuracy controlled retrieval based on eigenvector analysis of the reacting

system, thus allowing for the implementation of detailed chemistry in turbulent reacting

flow calculations. Successful application of ISAT to a detailed chemical mechanism of

160 species and 1540 reactions has been reported [56]. ISAT will be discussed further in

section 2.4.


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2.4 In Situ Adaptive Tabulation (ISAT)

The primary concern of ISAT for reacting flow calculations is the efficient solution

of the reaction equations (2.10). The essential features of ISAT that set it apart from

other storage/retrieval methods are on-the-fly (or dynamic) tabulation of reaction data

as needed (in situ), unstructured and adaptive tabulation of data of unknown topology,

and explicit control of retrieval/interpolation error. It is these features that make ISAT

suitable to facilitate implementation of detailed chemistry, or chemistry acceleration, in

turbulent reacting flows. The basic idea and operations of ISAT are described in the

paper by Pope [48].

At any spatial location and time in a reacting flow, the thermo-chemical state, or

the composition, of the gas mixture can be characterized by the mass fractions Y( =

1, 2, . . . , N s) of Ns species, the enthalpy h or temperature T, and the pressure P. These

are called the composition variables and can be written in vector form as

= {1, 2, . . . , Ns+2}. (2.11)

There are dependences among the components of ; for example, the mass fractions sum

to unity. Therefore, if the number of independent components is assumed to be D, then

the thermo-chemical state of fluid is completely determined by

= {1, 2, . . . , D}. (2.12)

One set of values of can be considered as one point, or a vector, in the D-

dimensional composition space. All physically possible values of define the realizable

region of the composition space, which is a convex polytope in D-dimensional space. The

accessed region of the composition space is defined as the set of all compositions that

occur in a particular flame calculation; it is much smaller than the realizable region.

To illustrate this, consider a location where the temperature and pressure are 1,800 K

and 1 atm, respectively. In principle, the mass fraction of fuel species can be any value

between zero and unity; however, under these conditions, its value is almost certain to


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be zero. Therefore, the accessed region is a small subset of the realizable region, and

is an intrinsic low-dimensional manifold (ILDM). This observation forms the foundation

on which various storage/retrieval methods are based.

ISAT addresses the efficient solution of the reaction equations (2.10) over a time

step starting from the initial condition (t0) =0, where temperature and pressure are

taken to be constant without loss of generality. The composition at time t0 + t is a

unique function of 0; this is called the reaction mapping and is denoted byR(0). The

conventional approach to determine the mapping is to integrate the reaction equation

numerically using a stiff ode solver such as DVODE [57]. That approach is referred to

as direct integration, or DI. For a detailed mechanism, DI is computationally expensive

for the reasons that have been discussed in section 2.2. ISAT alleviates this problem

by building up a table during the course of the reacting flow simulation. Each table

entry corresponds to a solution evolved from a certain initial condition. As the table,

or as the intrinsic low-dimensional manifold, becomes populated, computation efficiency

is realized by solving the reaction equation though retrieving and/or interpolating from

the existing solutions instead of performing expensive DI.

The coupling of ISAT with a reacting flow simulation code is illustrated in Fig-

ure 2.1.

The ISAT algorithm works as follows. Every time the flow code needs a solution,

or a mapping, of the reaction equation, a query is sent to the ISAT module. At the

beginning, the ISAT table is empty, so a DI is performed and the resulted mapping is

forwarded to the flow code and is stored in the table together with its initial condition.

The store corresponds to a generation of a table entry, or a record, and it is termed an

add. During an add, computed and stored as well is the mapping gradient A(0), which

is the Jacobian of the mapping R(0) and defined as

Aij =dRi(

0)

dj. (2.13)

The mapping gradient contains information about the sensitivity of R to the variations

in and this information is used for two purposes. First, it establishes a neighborhood


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Reacting

Flow

Simulation

Code

ISAT

Module

Reaction

Mapping

Calculation

t, tol

0

q

0

A( ) 0R( )qR( )

0R( )

Fig. 2.1. Interconnection between simulation code and ISAT algorithm

centered on 0, where a linear interpolation suffices to provide for any point (initial

condition) in the neighborhood the reaction mapping to an accuracy specified by the flow

code. This neighborhood is called the region of accuracy (ROA). Second, it provides the

coefficients necessary to perform the linear interpolation.

For subsequent queries by the flow code with an initial condition q, ISAT at-

tempts to find a ROA that contains the query point q. Three situations can occur: if

such a ROA is found, then linear interpolation is performed and the resulting mapping

is returned. This outcome is termed a retrieve. If such a ROA is not found, then a

direct integration is performed. Based on the mapping R(q) from DI and the mapping

gradient, some ROAs are examined to determine if the linear approximation in a certain

ROA for R(q) is sufficiently accurate. If it is, then that ROA is grown to include the

query point q and this outcome is termed a grow. If not, the a new ROA is added and

the outcome is an add.

The CPU time savings of ISAT comes from mainly the economy of the retrieval/

interpolation operations compared to the direct integration. One important aspect is

how fast a ROA can be found that contains the query point q. The organization of the


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record in the table plays an important role in this regard. ISAT adopted a binary tree

[58] structure for its record organization. Each leaf of the tree is a record, which consists

of the tabulation point 0, the reaction mappingR(0), the mapping gradient

A(0),

and information on the shape of region of accuracy (consisting of a unitary matrix and

a vector). Each node of the tree is a cutting plane, which is defined by a vector and a

scalar such that as the tree is traversed from the top for a query composition q, the

information given by the vector and the scalar directs the search to either the left branch

or the right branch of the tree. In this manner, the tree is traversed quickly to find a

leaf that can be used to approximate the query point q.

The CPU time savings of ISAT is counterbalanced by the cost of add and grow

operations, since both operations include direct integration in addition to other manip-

ulations. Specifically, the fraction of total queries that result in adds and grows is a

key parameter that determines the gain of ISAT in computational efficiency over direct

integration. This fraction depends both on the problem at hand and the algorithm used

to determine the shape and the extent of a ROA added to the table. In general, ISAT

should be more efficient for state-steady cases than for transient cases, for premixed

problems than for nonpremixed problems, and for homogeneous reactants than for non-

homogeneous reactants. There are parameters that control the total number of adds and

grows, the number of trees used in the table, the error tolerance for linear interpolation,

and the scaling and transformation of the tabulation points and their mappings.

ISAT was developed originally for Lagrangian Monte Carlo PDF-based turbulent

combustion modeling, and most applications to date have been in that context. Most

applications have been limited to homogeneous systems or to statistically stationary

configurations with small to moderate-sized mechanism (fewer than 50 species) [48, 59].

In such cases, speedups compared to direct integration of up to a factor of 1,000 have

been reported. In other cases [60, 61], speedups of 10 to 100 have been found. ISAT

also can be used for grid-based CFD applications, such as finite-volume calculations of

turbulent reacting flow. A difference with respect to a particle PDF method is that thenumber of computational cells used in grid-based CFD calculations is smaller than the

number of particles used in PDF-based calculations by about two orders of magnitude.


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ISAT can also be used for large mechanisms containing more than 100 species. The

difficulty is that the table size becomes quite large and memory usage becomes an issue

(memory scales as square of the number of species). Special measures have to be taken

to work around this issue for large mechanisms. Therefore, the benefits of using ISAT for

grid-based CFD calculations and for large mechanisms are expected to be lower than for

PDF-based calculations and for medium-sized mechanisms. Still, examples with various

degrees of success have been reported [62, 56].

2.5 Turbulence/Chemistry Interactions

Most combustion applications take place in a turbulent environment. Turbulent

flow and chemical kinetics are among of the most challenging problems in non-linear

physics. The strong nonlinear interactions between turbulence and chemistry make tur-

bulent combustion even harder to understand. Turbulence/chemistry interactions have

been a central subject in the research of turbulent reacting flows.

Turbulence/chemistry interactions (TCI) arise from the fact that mixing processes

in turbulent flow are not fast compared with the rates of chemical reactions. The time

scales of chemical reactions can range from 1010 s to more than 1 s [32], while the

time scale of turbulent mixing typically is no smaller than 103 s or 104 s [48]. TCIs

arise also from the fact that large spatial and temporal variations in species composition

and temperature occur in turbulent combustion. In nonpremixed combustion, turbulentmixing creates pockets of fuel-rich and fuel-lean mixture, while in premixed combustion,

turbulent mixing creates pockets of cold unreacted and hot reacted mixture. The mean

chemical reaction rate can not be evaluated directly from the averaged values of species

composition and temperature [63].

The effect of turbulence on chemical reactions takes place through the large-scale

stirring motions of turbulence. By stretching and curvature, the large-scale motions of

turbulence enhance greatly the molecular diffusion rates of chemical species and heat,

and therefore enhance greatly chemical reactions, which occur at molecular scales and

must be balanced by molecular diffusion. The differential diffusion of chemical species

also can become important. In addition, the turbulent motions produce large variations


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in species composition and temperature, which cause the mean reaction rates to be

strongly coupled to molecular diffusion at the smallest scales of the turbulence.

The effect of chemical reaction on turbulence takes place through the large heat

release from chemical reactions. On one hand, the large heat release produces large

density variations, which influence greatly the solution of Navier-Stokes equations.The

resultant large density gradients produce a source term in the vorticity transport equa-

tion and therefore enhance the intensity of turbulence. On the other hand, the heat

release produces local dilatation, which acts as a sink term in the vorticity equation and

therefore reduces the intensity of turbulence. Furthermore, the heat release produces

high temperature regions and therefore high viscosity regions, which enhance the dif-

fusion of the vorticity in the turbulent flow field. The net effect of heat release on the

turbulent intensity depends on the specific conditions in which chemical reactions occur

in turbulent flows.

Great success has been achieved in applying the conservation equations (2.12.4)

and detailed chemistry to laminar flames. This demonstrates that our knowledge of

chemical kinetics and molecular transport processes is sufficient for accurate predictions

from first principles. Turbulent reacting flows are governed by the same conservation

equations and the same chemical kinetics (a chemical mechanism for turbulent flames

should not be different from one for laminar flames, since chemical reactions occur at

molecular level [64]). However, turbulent reacting flows are characterized by a broad

spectrum of length and time scales and a complete numerical simulation of such flows

must resolve the smallest and the largest of all these scales. Direct numerical simulation

(DNS) is an established technique for that purpose.

DNS involves solving the conservation equations directly with extremely fine

meshes to fully resolve all relevant length and time scales and to provide a sequence

of realizations of the flow that contains a vast amount of details. DNS has been ap-

plied to simple turbulent flows with simple reactions and to the study of fundamental

processes involved in turbulent flames, e.g. [65, 66]. It is a powerful research tool fromwhich much can be learned about the physics of turbulent combustion and from which


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turbulent combustion models can be developed and calibrated. However, even in nonre-

acting flows, DNS can provide useful information only for simple geometry flows at low to

moderate Reynolds numbers. In reacting flows, the composition fields introduce length

and time scales that may be much smaller than those of the velocity fields. The species

conservation equations with highly nonlinear reaction source terms are more difficult

to solve than the incompressible Navier-Stokes equations. Therefore for the foreseeable

future, DNS will not be feasible for accurate predictions of practical turbulent flames.

Even if a DNS solution of a practical turbulent flame could be obtained, the large

amount of detail in time and space would be overwhelming and of little practical interest.

In practice, it is the mean quantities, such as the mean fuel consumption rate and mean

pollutant formation rate, that typically are desired. This requires limiting the dynamic

range of length and time scales in the problem and that is accomplished by applying

various averaging techniques to the conservation equations. Favre averaging (density-

weighted averaging) is usually employed because of large fluctuations in density due to

chemical heat release. Averaging reduces greatly the number of degrees of freedom in the

problem, but it introduces unclosed terms that need to be modeled. The Favre-averaged

conservation equation are written as follows [16]:

mass

t+

uixi

= 0 (2.14)

momentum

ujt

+uj ui

xi=

uju

i

xi+

ij

xi

p

xj+ gj for j = 1, 2, 3 (2.15)

species

Yt

+Yui

xi

= Y u

ix

i

Ji

xi

+ S for = 1, 2, . . . , N s (2.16)


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enthalpy

h

t+

huixi

= hu

ixi

Jh

i

xi+

Dp

Dt+ ij

uj

xi+ Qrad. (2.17)

Here the angle bracket and the tilde operators denote the conventional Reynolds

averaging and Favre averaging, respectively. The superscript denotes fluctuations from

the Favre-averaged mean.

The effects of turbulence on chemistry calculations are embodied in the unclosed

terms: turbulent fluxes of species (Y

ui

) and energy (hui

), and the mean species

chemical reaction rates (S). The turbulent transport terms for species and enthalpy

are usually modeled based on the gradient diffusion and turbulent viscosity hypothesis

[67] for non-reacting flows and are expressed as:

species

Yu

i = t

Sct

Yxi

(2.18)

enthalpy

hui = t

P rt

h

xi. (2.19)

Here t is the turbulent viscosity estimated from a turbulence model, and Sct and

P rt are the turbulent Schmidt number for the species and turbulent Prandtl number,

respectively.

Theoretical [68] and experimental work [69] have shown that turbulent diffusiv-

ity for reacting scalars is quite different from that for nonreacting cases and further

investigations are required. Nevertheless, these hypotheses are employed ubiquitously

in practice. Alternative treatments for the turbulent fluxes that provide more accurate

results include second-order transport models and PDF methods [3].

The determination of the mean reaction rates is the central problem of practical

interest in turbulent combustion simulations. The various approaches for modeling themean reaction rates are generically referred to as turbulent combustion models.


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Turbulent combustion models can be divided into two major categories according

to the relative time scales for chemistry compared to other physical processes: fast-

or-slow-chemistry models and finite-rate-chemistry models. In each category, turbulent

combustion models are further divided into models for premixed, nonpremixed, or par-

tially premixed reactants. Although quite different in appearance, the various models

essentially attempt to represent the same physical processes. Common links among these

models have been explored, for example, by Veynante and Vervisch [63]. The wide vari-

ety of combustion model reflects the difficulties that arise in averaging the reaction rate

terms in the species equations.

Fast-or-slow-chemistry combustion models essentially ignore the interactions be-

tween turbulence and chemistry by assuming that the time scales associated with chem-

ical reactions are very short or very long in comparison with time scales for turbulent

mixing. That is, mixing or chemistry is the rate-determining process. In the fast- or

slow-chemistry limit, models are developed based on different concepts. First, a simple

and widely used approach is to express the mean reaction rate in the same functional

form as that of the unaveraged reaction rate. Because the latter usually are expressed

in Arrhenius form, this is sometimes referred to as the Arrhenius model [3]. Second,

based on the concept of a turbulence energy cascade in nonreacting flows, eddy-breakup

(EBU) models [70] and variants, such as the eddy-dissipation model [71], link the mean

reaction rate to the rate of turbulent mixing and hence to a turbulence time scale. Third,

based on the concept of laminar flamelets [72], a variety of flamelet models have been

formulated for premixed, nonpremixed, and partially premixed combustion regimes [68].

Classical flamelet models that fall into this category include the conserved scalar equilib-

rium models (CSEM) [68] for nonpremixed flames, the Bray-Moss-Libby (BML) model

[73, 74] and the coherent flame model (CFM) [75, 76] for premixed flames. These fast-or-

slow-chemistry models sometimes provide plausible predictions, but they cannot capture

the finite-reaction-rate effects that are often the focus of interest.

Finite-rate-chemistry combustion models treat chemistry more realistically byconsidering the full range of chemical time scales and by dealing explicitly with the in-

teractions between turbulence and chemistry. However, special mathematical approaches


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have to be employed to handle the strong nonlinearities in determining the mean reac-

tion rates. Statistical approaches are most suitable to this task. These include models

based on PDF methods and models based on conditional moment closure (CMC) meth-

ods. The PDF-based models can be further divided into transported PDF models, where

a modeled PDF transport equation is solved, usually by means of a Lagrangian Monte

Carlo particle-based method [77, 78, 14], and presumed PDF models, where the shape or

form of the PDF is parameterized and modeled equations are solved for the parameters

[3, 17]. Here we shall use the term PDF method to refer to a transported PDF method.

The CMC method [79, 80] considers conditional averages and higher moments of quan-

tities such as species mass fractions and enthalpy, conditional on mixture fraction (for

nonpremixed combustion) or reaction progress variable (for premixed combustion). The

concept of CMC and its applications have been reviewed in a recent paper by Klimenko

and Bilger [81].

Laminar flamelet concepts also can be applied to the modeling of finite-rate chem-

istry. Flamelet models based on a scalar G-equation for premixed combustion and

based on mixture fraction for nonpremixed flames have been developed and widely used

[68]. Another approach to account for finite-rate chemistry is the linear eddy model

(LEM), which was first formulated for nonreacting flows and later was extended to react-

ing flows [82, 83]. LEM is a method of simulating molecular mixing on a one-dimensional

domain embedded in a turbulent flow and therefore is capable of capturing molecular

effects in turbulent reacting flows.

Finite-rate-chemistry combustion models have been a key research subject in the

simulation of turbulent combustion, and the above overview offers only a brief catego-

rization of the most popular models currently in use. More detailed descriptions and

formulations can be found in many dedicated books and monographs, such as Libby and

Williams [17], Peters [68], and Haworth [3].

The above discussion is based on invoking a statistical averaging technique to re-

duce the dynamic range of scales in turbulent reacting flows. This approach is commonlyreferred to as Reynolds-averaged Navier-Stokes (RANS) modeling. With appropriate clo-

sure models, the RANS approach allows only for the determination of mean quantities,


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which may differ largely from the instantaneous values. Strong unsteady mixing effects

are observed in turbulent flames and the knowledge of steady statistical means may not

always be sufficient to describe the turbulent combustion phenomena of interest. An

alternative approach to reducing the dynamic range of scales while accounting explicitly

for the unsteady effects is a spatial filtering technique. This is the approach that is taken

in large eddy simulation (LES).

In large eddy simulation [67, 68, 84, 85], the larger energy-containing scales are

resolved explicitly while the effects of unresolved smaller scales are modeled. The distinc-

tion between the resolved large scales and the modeled small scales usually is determined

by the grid resolution that is affordable in the computation domain. The small scales

are the subgrid scales and the models sometimes are referred to as subgrid models.

Mathematically, the large and small scales are separated by filtering the instantaneous

governing equations. The resulting filtered equations contain unclosed terms that need

to be modeled. The modeled equations are solved numerically to simulate the unsteady

behavior of the large-scale motions. Compared to RANS, LES provides information on

the large resolved scales, which is valuable in many practical applications such as com-

bustion stabilities in a gas turbine combustor. As in RANS, the interactions between

turbulence and chemistry occur at unresolved scales of computation. Therefore, the

basic tools and formalism of RANS-based turbulent combustion modeling can carried

directly to LES. Most of the RANS combustion models discussed above can be modified

and adapted to LES subgrid-scale modeling [63].

2.6 Summary

Issues that arise in calculating detailed chemistry in turbulent reacting flows have

been discussed in this chapter. Detailed chemistry is critical for realistic and accurate

predictions of turbulent combustion. However, the characteristics of detailed chem-

istry make the calculations quite difficult. This is mainly due to the complexity of the

chemistry, e.g., large number of species and elementary reactions required, and the broad


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range of time scales involved, from 1010 s to more than 1 s. Different strategies to over-

come these difficulties have been developed, such as simplification of flow field descrip-

tion, chemistry reduction, and storage/retrieval schemes. One of the storage/retrieval

schemes, the in situ adaptive tabulation, has been described in detail. This is one of the

most promising strategies for implementing detailed chemistry in turbulent combustion

calculations and is the approach that has been adopted for this study. The nature of

turbulence-chemistry interactions also has been discussed. Direct numerical simulation

can provide a complete description of a turbulent reacting flow, but is not feasible in

the foreseeable future for practical combustion systems because of computational power

limitations and the huge amount of data that is generated. Averaging and filtering tech-

niques are employed to reduce the range of scales to be resolved in the flows to make

the simulation tractable. LES is capable of capturing the effects of unsteady mixing by

explicitly resolving large energy-containing scales and modeling the more homogeneous

small scales. RANS models can only provide mean quantities but are sufficient for many

practical applications. Averaging and filtering introduce unclosed terms that need to

be modeled in the governing equations. The various combustion models that have been

developed can be divided into fast-or-slow chemistry and finite-rate-chemistry models.

Fast-or-slow chemistry models are used widely in combustion simulations due to their

simplicity. Finite-rate-chemistry models provide more realistic results but need statisti-

cal tools to accommodate the random nature of turbulent flames. The PDF method with

a Monte Carlo solution scheme is considered to be one of the most promising approaches

for the modeling of finite-rate turbulent combustion.


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Chapter 3

Soot Calculations

3.1 Introduction

Most combustion processes that involve hydrocarbons produce soot. Under ideal

conditions, combustion of hydrocarbons leads mainly to carbon dioxide and water. Under

practical conditions, in locally fuel-rich regions, the combustion or pyrolysis of hydrocar-

bons generates intermediate species and radicals that lead eventually to the appearance

of soot particles.

Soot consists mainly of carbon. Other elements such as hydrogen and oxygen

are usually present in small amounts. For example, soot emitted from long-residence-

time turbulent nonpremixed flames, including toluene, benzene, acetylene, propylene,

and propane flames burning in air, have the following elemental mole ratio ranges: C:H

of 8.318.3, C:O of 58109, and C:N of 292976 [86]. Soot density is less than that of

carbon black and usually in the range of 17001800 kg/m3, depending on the porosity

of soot [87]. Soot particles are generally small, ranging in size from 5 nm to 80 nm, but

may be up to several micron in extreme cases [88]. While mostly spherical in shape, soot

particles may also appear in agglomerated chunks and even long agglomerated filaments

[86]. Experiments in diffusion flames of hydrocarbon fuels have shown that the soot

volume fraction generally lies in the range of 106108 [88]. These physical properties

of soot particles affect their optical properties, which in turn affect the accuracy of

determining soot quantities and radiation in both experiments and simulations.

Soot formation is a complicated phenomenon that involves highly coupled chem-

ical and physical processes. It is remarkable that hydrocarbon fuel molecules containing

only a few carbon atoms transform into soot particles containing millions of carbon

atoms. The study of soot processes in combustion systems has drawn great attention.

Progress has been achieved in understanding the essential features of the chemistry and


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physics. However, the understanding is incomplete and many questions persist and de-

bates continue regarding the details of soot nucleation, growth, and oxidation. This

reflects the difficulty of the problem that soot formation poses for combustion systems.

The prediction of soot formation is of interest for the following four reasons:

The formation of soot stems from incomplete combustion, which reduces the com-bustion efficiency.

Soot is a ma jor pollutant. It contains trace elements that have hazardous effects

on human health. The particles themselves also are an issue.

Industrial applications, such as furnace and heat generators, require formation

of soot to enhance the heat transfer via radiation. However, the soot has to be

oxidized before these devices release the exhaust into the environment.

Soot is an important industrial product refered to as carbon black. It finds wide

application, such as filler in tires or other materials, toner in copiers, and black

pigment in color printings.

3.2 Soot Formation and Oxidation: Current Understanding

The study of soot formation and oxidation includes experimental, theoretical, and

computational efforts.

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