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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.