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Spectral optimization and uncertainty quantification in combustion modeling

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Spectral optimization and uncertainty quantification in combustion modeling

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Content SPECTRAL OPTIMIZATION AND UNCERTAINTY QUANTIFICATION IN
COMBUSTION MODELING

by

David Allan Sheen

A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(AEROSPACE ENGINEERING)

May 2011

Copyright 2011 David Allan Sheen
ii
Dedication
To my grandfathers, Allan Armstrong and Walter Sheen, who taught me to reach to
the skies for my dreams.
iii
Acknowledgements
I thank my advisor, Professor Hai Wang, without whom this dissertation would not
have been possible. He has provided me with the foundations of my knowledge of
chemistry and challenged me to be the best I can be. With him I have learned the thrill of
discovery and the joy of searching for the unknown. For his kindness, patience, and
dedication, I am forever in his debt. I shall greatly miss the time I have spent with him.
Thanks are due as well to my committee members Professors Fokion N. Egolfopoulos,
Nelson Bickers and Roger G. Ghanem, for their wisdom and guidance.
Many thanks are due as well to my colleagues in the Combustion Kinetics Laboratory
at the University of Southern California. Enoch E. Dames has worked tirelessly to help
with preparing this dissertation, reading drafts of many chapters. I am also grateful to Dr.
Baptiste Sirjean for answering my constant questions about chemistry, to Dr. Xiaoqing
You, and Dr. Mustafa Gurhan Andac for teaching me how to perform flame simulations,
and to Dr. Aamir D. Abid, for our endless discussions. Thanks are also due to Saro
Memarzadeh, Tsutomu Shimizu, Erik Tolmachoff, and Joaquin Camacho for their
comradeship.
I am indebted to my parents, Dr. Marcia Armstrong and Tim Sheen, for their advice
and discussions, and for their unconditional support, as well as my grandmother, Sylvia
Jander, whose letters I have enjoyed so much. In addition, I am thankful to my friends,
most especially Andrea McColl, who stood by me during my long hours of hard work.
Funding for this work was provided by the United States Air Force Office of
Scientific Research and the Combustion Energy Frontier Research Center, an Energy
Frontier Research Center funded by the United States Department of Energy, Office of
Science, Office of Basic Energy Sciences. Their support is gratefully acknowledged.

iv
Table of Contents
Dedication ii
Acknowledgements iii
List of Tables vi
List of Figures vii
Abstract xiii
Chapter 1 Introduction 1
1.1 Background 1
1.2 Development of Chemical Kinetics Mechanisms 3
1.3 Uncertainty Quantification 10
1.4 Towards a Comprehensive System 12
1.5 Chapter 1 Endnotes 14
Chapter 2 Theory and Computational Methods 20
2.1 Stochastic Spectral Projection 20
2.2 Response Surface Construction by Solution Mapping 23
2.3 Solution Mapping and Model Optimization 29
2.4 Uncertainty Minimization 32
2.5 Chapter 2 Endnotes 36
Chapter 3 The Method of Uncertainty Quantification and Minimization
Using Polynomial Chaos Expansions: A Case Study 39
3.1 Introduction 39
3.2 Case Study 41
3.3 Results and Discussion 48
3.4 Conclusions 66
3.5 Chapter 3 Endnotes 68
Chapter 4 Quantitative Analysis of Hierarchical Strategies of Building
Combustion Reaction Models 71
4.1 Introduction 71
4.2 Methods 73
4.3 Results and Discussion 78
4.4 Conclusions 88
4.5 Chapter 4 Endnotes 89
v
Chapter 5 Modeling the Negative Pressure Dependence of Mass Burning
Rates of H
2
/CO/O
2
/diluent Flames 93
5.1 Introduction 93
5.2 Methods 97
5.3 Results and Discussion 102
5.4 Conclusion 118
5.5 Chapter 5 Endnotes 119
Chapter 6 Combustion Kinetic Modeling Using Multispecies Time-Histories
in Shock-Tube Oxidation of Heptane 123
6.1 Introduction 123
6.2 Methodology 124
6.3 Results and Discussion 132
6.4 Conclusion 147
6.5 Chapter 6 Endnotes 148
Chapter 7 Conclusions and Future Work 151
7.1 Summary of Conclusions 151
7.2 Future Work 153
7.3 Chapter 7 Endnotes 155
Bibliography 156
Appendices
Appendix A Supplementary Materials for Chapter 4 167
Appendix B Supplementary Materials for Chapter 6 174
Appendix C Chemical Parameters 179

vi
List of Tables
Table 3-1. Combustion property targets and uncertainties. Targets in italics were
rejected or removed from the final target set (see text). 43
Table 3-2. Matrix of active rate coefficients and their uncertainty factors. Active
parameters considered in a target response surface are marked with an X. 47
Table 3-3. Comparison of rate parameter values for uniformly-distributed (posterior
Model I) and normally-distributed (posterior Model II) rate parameters. 56
Table 3-4. Comparison of standard deviations for the prior and posterior models.
The MC column is calculated using a Monte-Carlo sampling of Eq. (2-15),
while the PCE column is a direct solution of Eq. (2-16). 64
Table 4-1 List of targets for H
2
/CO and C
2
H
4
mixtures. 76
Table 4-2. List of targets for C
3+
fuels. 77
Table 4-3. List of large alkane validation data. 78
Table 5-1. List of experimental targets. 99
Table 5-2. List of active reaction rates and optimized parameters for posterior
models. 100
Table 5-3. Active parameter matrix for Experiments 1-35. 101
Table 5-4. Active parameter matrix for Experiments 36-53 102
Table 6-1. Experimental Datasets Considered 127
Table A-1. List of rate parameters in posterior models. 167
Table A-2. Active parameters for H
2
/CO experimental targets 169
Table A-3. Active parameters for C
2
H
4
experimental targets. 170
Table A-4. Active parameters for C
3
experimental targets. 171
Table A-5. Active parameters for n-C
5
H
12
and n-C
7
H
16
experimental targets. 172
Table B-1. Active rate parameters and experimental targets
a
174
Table B-2. Experimental conditions of the target data set. 176
Table B-3. List of rate parameters in posterior models 177
Table C-1. List of Arrhenius rate coefficients for the H
2
/CO submodel. Units are
moles, cm, s, and kcal. 179
Table C-2. Rate coefficients for H + O
2
(+M) ↔ HO
2
(+M) fits. 181

vii
List of Figures
Figure 1-1. Schematic representation of the explosion limits of hydrogen. 4
Figure 1-2. The three possible reaction chains as proposed in [23]. a) Chain
termination process, resulting in a chain carrier being destroyed. b) Chain
carrying process, resulting in a single infinitely long chain. c) Chain branching
process, resulting in an infinite number of infinitely long chains. 5
Figure 1-3. Rate constant evaluations of reaction (R2) taken from [70] 10
Figure 2-1. Illustration of the application of the central limit theorem. Probability
density functions for a random variable U
M
with variance 1 (solid lines), which
is the sum of M uniform random variables, for M = [1,2,3,4]. This is compared
with the normal distribution with variance σ = 1 (dashed line). 27
Figure 2-2. Comparison of Monte Carlo sampling of the ethylene-air laminar flame
speed at equivalence ratio of 0.7 for normally-distributed rate parameters ( ,
dashed line) and uniformly-distributed rate parameters (■, solid line). The
computation was carried out using the prior model. 28
Figure 3-1. Experimental ignition delay data as described by Eq. (31). Data are
taken from Homer and Kistiakowsky [25] (

: 0.5%C
2
H
4
-3%O
2
-Ar, p
5
= 0.6-0.8
atm; : 0.5%C
2
H
4
-1%O
2
-Ar, p
5
= 0.6-0.7 atm); Baker and Skinner [29] ( :
1% C
2
H
4
-3%O
2
-Ar, p
5
= 3 atm; : 5% C
2
H
4
-3%O
2
-Ar, p
5
= 3 atm; : 0.25%
C
2
H
4
-0.75%O
2
-Ar, p
5
= 12 atm; : 2% C
2
H
4
-3%O
2
-Ar, p
5
= 3 atm; : 1%
C
2
H
4
-1.5%O
2
-Ar, p
5
= 3 atm; : 1% C
2
H
4
-6%O
2
-Ar, p
5
= 3 atm; : 0.25%
C
2
H
4
-6%O
2
-Ar, p
5
= 3 atm; : 0.25% C
2
H
4
-1.5%O
2
-Ar, p
5
= 3 atm; : 0.5%
C
2
H
4
-0.75%O
2
-Ar, p
5
=3 atm); Hidaka et al. [30] ( : 0.5% C
2
H
4
-1.5%O
2
-Ar,
C
5
= (1.93±0.05)×10
–6
mol/cm
3
; : 0.5% C
2
H
4
-4.5%O
2
-Ar, C
5
=
(1.87±0.06)×10
–6
mol/cm
3
); Hidaka et al. [24] ( :0.1% C
2
H
4
-0.6%O
2
-Ar, C
5
=
(1.61±0.06)×10
–5
mol/cm
3
; : 1% C
2
H
4
-1.5%O
2
-Ar, C
5
= (1.63±0.07)×10
–5

mol/cm
3
; : 0.5% C
2
H
4
-1.5%O
2
-Ar, C
5
= (1.58±0.07)×10
–5
mol/cm
3
; : 0.5%
C
2
H
4
-3%O
2
-Ar, C
5
= (1.59±0.09)×10
–5
mol/cm
3
); Brown and Thomas [23] ( :
1% C
2
H
4
-3%O
2
-Ar, C
5
= (1.44±0.14)×10
–5
mol/cm
3
; : 6.25% C
2
H
4
-18.7%O
2
-
Ar, C
5
= (1.92±0.22)×10
–5
mol/cm
3
); Saxena et al. [31] ( : 3.5% C
2
H
4
-
3.5%O
2
-Ar, p
5
= 2.4 atm;

: 3.5% C
2
H
4
-3.5%O
2
-Ar, p
5
= 2 atm;

: 3.5% C
2
H
4
-
3.5%O
2
-Ar, p
5
= 10 atm;

: 3.5% C
2
H
4
-3.5%O
2
-Ar, p
5
~ 20atm; –: 1.75%
C
2
H
4
-5.25%O
2
-Ar, p
5
= 2.4 atm; │: 1% C
2
H
4
-3%O
2
-Ar, p
5
= 2.4 atm;

: 0.5%
C
2
H
4
-1.5%O
2
-Ar, p
5
= 2.5 atm;

: 1.75% C
2
H
4
-5.25%O
2
-Ar, p
5
= 11.5 atm;

:
1% C
2
H
4
-3%O
2
-Ar, p
5
= 11 atm;

: 1% C
2
H
4
-3%O
2
-Ar, p
5
= 20 atm;

: 0.5%
C
2
H
4
-1.5%O
2
-Ar, p
5
= 11 atm;

: 0.5% C
2
H
4
-1.5%O
2
-Ar, p
5
= 21 atm). Lines
denote the uncertainty bound in the data. 45
viii
Figure 3-2. Variation of laminar flame speed with equivalence ratio. Left: prior
model. Right: posterior Model I. Symbols are the data from [20]. The shaded
bands indicate the 2σ model prediction uncertainty; color indicates the
probability density as indicated by the color bar, and the actual ±2σ curves are
indicated by the dashed lines. 49
Figure 3-3. Comparison of experimental (□: p
5
= 2.4 atm and 11 atm [31]; ○: p
5
= 3
atm [29]; dotted lines: C
5
= 1.44±0.14×10
–5
mol/cm
3
[23]) and computed (solid
lines: mean; dashed lines: 2σ uncertainty). Top panels: prior model; bottom
panels: posterior Model I. 50
Figure 3-4. Consistency analyses for uniformly-distributed rate parameters, leading
to posterior Model I. 53
Figure 3-5. Comparison of experimental values and 2σ uncertainties with calculated
values before and after model constraining. Top panels: uniformly-distributed
rate parameters; bottom panels: normally-distributed rate parameters. 54
Figure 3-6. Covariance matrix of Model I, expressed as
ij ij
K = Σ (see text). 58
Figure 3-7. Covariance matrix of Model II, expressed as
ij ij
K = Σ (see text). 60
Figure 3-8. Contours of joint probability density function for reactions R95, R99,
and R103 in the prior model (cube, squares) and the posterior Model I (ellipses). 62
Figure 3-9. Contours of joint probability density function for reactions R95, R99,
and R103 in the prior model (sphere, circles) and the posterior Model II
(ellipses). 63
Figure 3-10. Variation of OH and H
2
O mole fraction with mean residence time
in a simulated PSR. Lines are the model calculation and symbols are the
uncertainty calculated using Monte Carlo sampling of the uncertainty space.
Dashed line and open symbols: prior model; solid line and solid symbols:
posterior Model I. 66
Figure 4-1. Experimental (○, [20];◊, [22] ) and predicted 2σ standard deviation
bands (left panel: prior model; right panel: Model III) for the laminar flame
speeds of ethylene-air mixture at 2 atm and 298 K unburned gas temperature.
Grey scale indicates the probability density function of the uncertainty band.
The dashed lines mark the 2σ uncertainties. 79
Figure 4-2. Experimental (□: p
5
= 2.4 atm [47]; ○: p
5
= 3 atm [48]; dotted lines: C
5
=
1.44±0.14)×10
–5
mol/cm
3
[21]) and predicted (left panel: prior model; right
panel: Model III) ignition delay time behind reflected shock waves. The dashed
lines mark the 2σ uncertanties. 79
ix
Figure 4-3. Prediction uncertainties for the laminar flame speed of C
2
H
4
-air mixtures
at 1 atm and 298 K unburned gas temperature for the prior model and posterior
Models I through III. 81
Figure 4-4. Experimental and predicted C
2
H
4
-air laminar flame (a, b) and H
2
-air and
H
2
/CO-air flame speeds (c, d) for targets of Table 1. The error bars represent
the 2σ uncertainties. 81
Figure 4-5. Information index matrix K (presented as ln K) of Model III for H
2
/CO
and C
2
H
4
(see Table 4-1 for index descriptions). 83
Figure 4-6. Prediction uncertainties for the laminar flame speed of C
3
H
8
-air mixtures
at 1 atm and 298 K unburned gas temperature for the prior model and posterior
Models I through III. 85
Figure 4-7. Prediction uncertainties for the laminar flame speed of n-C
8
H
18
-air
mixtures at 1 atm and 353 K unburned gas temperature for several model cases.
Shown are the prior model, the model constrained against all H
2
/CO and C
2
H
4

experiments (Model III), extensions of Model III that are further constrained
against C
3
experiments and n-pentane experiments, and Model IV constrained
against all experiments up to n-heptane. 86
Figure 4-8. Information index matrix K (presented as ln K) of Model IV (see Table
4-2 and 4-3 for index descriptions). Highlighted sections are K indices for a) n-
pentane and n-heptane laminar flame speeds, b) n-pentane ignition delay times,
c) n-heptane ignition delay times, d) n-octane, nonane, and dodecane laminar
flame speeds, and e) n-octane ignition delay times. Indices for H
2
/CO and C
2
H
4

experiments are close to 0 and not shown. 87
Figure 4-9. Prediction uncertainties for the ignition delay times of n-C
8
H
18
-O
2
/Ar
mixtures at 2 atm. Shown are the prior model, the model constrained against all
H
2
/CO and C
2
H
4
experiments (Model III), extensions of Model III that are
further constrained against C
3
experiments and n-pentane experiments, and
Model IV constrained against all experiments up to n-heptane. 88
Figure 5-1. Probability density function of mass burning rate predicted by the prior
model for H
2
/O
2
/He mixtures (φ = 0.85, T
u
= 298 K, T
b
= 1600 K). The pair of
dashed lines bracket the 2σ standard deviation. Symbols are experimental data
taken from Ref. [14]. Lines are predictions of the models reported by Davis et
al. [10] (solid) and Konnov [17] (dash-dot) models. 103
Figure 5-2. Sensitivity coefficients for the mass burning rates predicted for H
2
/O
2
/He
mixtures (φ = 0.85, T
u
= 298 K, T
b
= 1600 K). 105
x
Figure 5-3. Probability density function of mass burning rate predicted by posterior
Model I for H
2
/O
2
/He mixtures (φ = 0.85, T
u
= 298 K, T
b
= 1600 K). The pair
of dashed lines bracket the 2σ standard deviation. Symbols are experimental
data taken from Ref. [14]. Lines are predictions of the models reported by
Davis et al. [10] (solid) and Konnov [17] (dash-dot) models. 107
Figure 5-4. Probability density function of mass burning rate predicted by posterior
Model I for H
2
/O
2
/He mixtures (φ = 1.0, T
u
= 298 K, T
b
= 1800 K). The pair of
dashed lines bracket the 2σ uncertainty. Symbols are experimental data taken
from Ref. [14] (○) and [50] (◊). 108
Figure 5-5. Probability density function of mass burning rate predicted by posterior
Model I for H
2
/O
2
/CO
2
mixtures (φ = 2.5, T
u
= 298 K, T
b
= 1800 K). The pair
of dashed lines bracket the 2σ uncertainty. Symbols are experimental data
taken from Ref. [14]. 108
Figure 5-6. Experimentally measured values and model predictions for laminar flame
speeds (left) and ignition delay times (right) used in Davis et al. [10],with
associated uncertainties. 109
Figure 5-7. Probability density function of mass burning rate predicted by posterior
Model II for H
2
/O
2
/He mixtures (φ = 0.85, T
u
= 298 K, T
b
= 1600 K). The pair
of dashed lines bracket the 2σ uncertainty. Symbols are experimental data
taken from Ref. [14]. Lines are predictions of the models reported by Davis et
al. [10] (solid) and Konnov [17] (dash-dot) models. 110
Figure 5-8. Laminar flame mass burning rates with respect to pressure for the flame
data in [14] (symbols) and calculations (lines: dashed, prior model; solid,
posterior Model II). a) He-diluted flames at φ = 1. b) Ar-diluted flames at φ =
2.5. c) CO
2
-diluted flames at φ = 2.5. d) H
2
/CO/O
2
/Ar flames at φ = 2.5. 112
Figure 5-9. Probability density function of mass burning rate predicted by posterior
Model II for H
2
/O
2
/CO
2
mixtures (φ = 2.5, T
u
= 298 K, T
b
= 1800 K). The pair
of dashed lines bracket the 2σ uncertainty. Symbols are experimental data
taken from Ref. [14]. 113
Figure 5-10. Probability density function of mass burning rate predicted by posterior
Model II (top panel) and Model I (bottom panel) for H
2
/O
2
/He mixtures (p = 1
atm, T
u
= 298 K). The pair of dashed lines bracket the 2σ uncertainty. Symbols
are experimental data taken from Ref. [50] (○) and [14] (◊). 113
Figure 5-11. Covariance matrices computed for Model I (left) and Model II (right). 115
Figure 5-12. Contour plots of probability density functions of the individual rate
parameters. Circles correspond to the prior model. Left panels: posterior
Model I; right panels: Model II. 117
xi
Figure 6-1. Uncertainty estimates for species time history measurements (Series 1i
of Table 6-1). Solid line: predictions of the prior reaction model at the nominal
temperature shown; long-dashed lines: computed uncertainty bounds due to ±10
K uncertainty in the T
5
value [3]; short dashed lines: 20% uncertainty on the
nominal mole fraction values. 128
Figure 6-2. Uncertainty estimates for species time history measurements (Series 1ii
of Table 6-1). Solid line: predictions of the prior reaction model at the nominal
temperature shown; long-dashed lines: computed uncertainty bounds due to ±10
K uncertainty (2σ) in the T
5
value [3]; short dashed lines: 20% uncertainty on
the nominal mole fraction values. 129
Figure 6-3. Uncertainty estimates for CH
3
time history measurements (Series 2 of
Table 6-1). Solid line: predictions of the prior reaction model at the nominal
temperature shown; long-dashed lines: computed uncertainty bounds due to ±10
K uncertainty (2σ) in the T
5
value [3]; short dashed lines: 20% uncertainty on
the nominal mole fraction values. 130
Figure 6-4. Experimental (solid lines, [3]) and computed (dashed lines:
nominal prediction; dots: uncertainty scatter) species time histories for Series 1i:
300 ppm nC
7
H
16
/3300 ppm O
2
/Ar, T
5
= 1495 K, p
5
= 2.15 atm (see Table 6-1).
The open circles and the corresponding error bars designate data used as Series
1 targets and 2σ uncertainties, respectively. Left panel: prior model. Right
panel: posterior Model I. 134
Figure 6-5. Experimental (solid lines, [3]) and computed (dashed lines:
nominal prediction; dots: uncertainty scatter) species time histories for Series
1ii: 300 ppm nC
7
H
16
/3300 ppm O
2
/Ar, T
5
= 1365 K, p
5
= 2.35 atm (see Table 6-
1). The open circles and the corresponding error bars designate data used as
Series 1 targets and 2σ uncertainties, respectively. Left panel: prior model.
Right panel: posterior Model I. 135
Figure 6-6. Probability density function of laminar flame speed predicted for
heptane-air mixtures (p = 1 atm, T
u
= 353 K). The pair of dashed lines bracket
the 2σ uncertainty. Panel (a) prior model, (b) posterior Model I using Series 1
data), (c) posterior model constrained by Series 1 data with a uniform
uncertainty value of ±5%, (d) posterior model IV constrained by all data of
Table 6-1. Symbols are experimental data taken from Ref. [24]. 136
Figure 6-7. Variation of the standard deviation predicted for the n-C
7
H
16
-air laminar
flame speed (T
u
= 353 K, p = 1 atm) as a function of the percentage uncertainty
in the species time-history data. Left panel: prior and posterior models
constrained with Series 1 data; right panel: impact of species time-histories
selected on posterior model. 137
xii
Figure 6-8. Experimental (solid lines, [6]) and computed (dashed lines: nominal
predictions; dots: uncertainty scatter) CH
3
time histories of Series 2i and 2iii
(see Table 6-1). The open circles and the corresponding error bars designate
data used as Series 2 targets and 2σ uncertainties, respectively. Left panel: prior
model. Right panel: posterior Model II. 139
Figure 6-9. Experimental (solid lines, [3]) and computed (dashed lines: nominal
predictions; dots: uncertainty scatter) OH time history of Series 1i (see Table 6-
1). The open circles and the corresponding error bars designate data of series 1
targets and 2σ uncertainties, respectively. Left panel: posterior model
constrained by Series 1 and 2 time histories (Model II). Right panel: posterior
model constrained against laminar flame speeds and ignition delay times
(Model III). 141
Figure 6-10. Ignition delay times of Series 3ia. Experiments: ◊,[32]; ●,[31]. Dots
represent the results of Monte Carlo sampling of predictions by Model I (top
panel) and Model IV (bottom panel). The circled data indicate the temperatures
of the ignition delay targets. 143
Figure 6-11. Ignition delay times for Series 3ib (top panel), Series 3ii (middle panel)
and Series 3iii (bottom panel). Experiments: ◊,[32]; ●,[31]. Dots represent the
results of Monte Carlo sampling of predictions by Model IV. The circled data
indicate the temperatures of the ignition delay targets. 144
Figure 6-12. Covariance matrices computed for Model I (top panel) and Model III
(bottom panel). 145
Figure 6-13. Contour plots of probability density functions of the individual rate
parameters. Circles correspond to the prior model. Left panels: Model II;
center panels: Model III; right panels: Model IV. 146

xiii
Abstract
Reliable simulations of reacting flow systems require a well-characterized, detailed
chemical model as a foundation. Accuracy of such a model can be assured, in principle, by a
multi-parameter optimization against a set of experimental data. However, the inherent
uncertainties in the rate evaluations and experimental data leave a model still characterized
by some finite kinetic rate parameter space. Without a careful analysis of how this
uncertainty space propagates into the model’s predictions, those predictions can at best be
trusted only qualitatively.
In this work, the Method of Uncertainty Minimization using Polynomial Chaos
Expansions is proposed to quantify these uncertainties. In this method, the uncertainty in the
rate parameters of the as-compiled model is quantified. Then, the model is subjected to a
rigorous multi-parameter optimization, as well as a consistency-screening process. Lastly,
the uncertainty of the optimized model is calculated using an inverse spectral optimization
technique, and then propagated into a range of simulation conditions. An as-compiled,
detailed H
2
/CO/C
1
-C
4
kinetic model is combined with a set of ethylene combustion data to
serve as an example.
The idea that the hydrocarbon oxidation model should be understood and developed in a
hierarchical fashion has been a major driving force in kinetics research for decades. How
this hierarchical strategy works at a quantitative level, however, has never been addressed. In
this work, we use ethylene and propane combustion as examples and explore the question of
hierarchical model development quantitatively. The Method of Uncertainty Minimization
using Polynomial Chaos Expansions is utilized to quantify the amount of information that a
particular combustion experiment, and thereby each data set, contributes to the model. This
knowledge is applied to explore the relationships among the combustion chemistry of
hydrogen/carbon monoxide, ethylene, and larger alkanes.
xiv
Frequently, new data will become available, and it will be desirable to know the effect
that inclusion of these data has on the optimized model. Two cases are considered here. In
the first, a study of H
2
/CO mass burning rates has recently been published, wherein the
experimentally-obtained results could not be reconciled with any extant H
2
/CO oxidation
model. . It is shown in that an optimized H
2
/CO model can be developed that will reproduce
the results of the new experimental measurements. In addition, the high precision of the new
experiments provide a strong constraint on the reaction rate parameters of the chemistry
model, manifested in a significant improvement in the precision of simulations.
In the second case, species time histories were measured during n-heptane oxidation
behind reflected shock waves. The highly precise nature of these measurements is expected
to impose critical constraints on chemical kinetic models of hydrocarbon combustion. The
results show that while an as-compiled, prior reaction model of n-alkane combustion can be
accurate in its prediction of the detailed species profiles, the kinetic parameter uncertainty in
the model remains to be too large to obtain a precise prediction of the data. Constraining the
prior model against the species time histories within the measurement uncertainties led to
notable improvements in the precision of model predictions against the species data as well
as the global combustion properties considered. Lastly, we show that while the capability of
the multispecies measurement presents a step-change in our precise knowledge of the
chemical processes in hydrocarbon combustion, accurate data of global combustion
properties are still necessary to predict fuel combustion.

1
Chapter 1 Introduction
1.1 Background
Thermodynamic theory, as put forward by Gibbs [1], has been perhaps the greatest
advance of the modern age. In particular, it has enabled great advances in theoretical
chemistry and fluid dynamics, and indeed has driven these fields for the past century. It
can tell whether a process is profitable or whether it is even possible in the first place.
However, despite its great predictive power, thermodynamics gives no information about
how fast these processes might occur. Kinetics, the study of momentum and energy
transfer between particles, is also central to the study of chemistry and fluids. Systems
dominated both by aerodynamics and chemistry are termed reacting flows. Artificial
reacting flows have existed for thousands of years, although a thorough mathematical
understanding of them has come somewhat more recently; indeed, some would argue [2]
that this understanding continues to elude us.
Mathematical modeling of reacting flows is based on the Navier-Stokes equations for
conservation of momentum and conservation of energy. Both equations contain highly
non-linear terms, which make them analytically and sometimes even computationally
intractable. There has been much work, and some success, modeling cold [3-5] and
reacting [6-11] flows by using Large Eddy Simulations (LES); small scales, those at
which energy is dissipated by turbulence and released by chemical reactions, are not, and
need not be, fully resolved. Given the still-unsatisfactory state of knowledge, work has
tended to concentrate either on cold turbulent flows or on laminar reacting flows with
geometry as simple as possible.
2
Solutions of reacting flow problems require the use of a chemical model which
specifies chemical processes and rates. Because of its simplicity, hydrogen gas has been
a primary candidate for study. At the simplest level, a combustion process can be viewed
as a one-step autocatalytic reaction, such as the reaction between hydrogen and oxygen to
form water,

o
2 2 2
2H O 2H O
r
H + → + Δ , (R1)

where
o
r
H Δ is the enthalpy of reaction. In the absence of further information, one might
assume that the reaction (R1) is the reaction which actually takes place at the molecular
level. Therefore, according to the Law of Mass Action, the rate at which the reaction
proceeds, q, may be written as

[ ] [ ]
2
2 2
H O exp
b a
E
q AT
RT
−  
=
 
 
, (1-1)

where T is the temperature in Kelvins, R is the universal gas constant, A, b, and E
a
are the
Arrhenius parameters, and [ ] denotes the molar concentration. The reaction is
exothermic, i.e.
o
0
r
H Δ < , and it is autocatalytic because the heat released by the
formation of water increases q. Early studies of H
2
oxidation showed, however, that the
oxidation process could not be described by this simple one-step mechanism. If such a
mechanism does not satisfactorily describe the process, then a new mechanism must be
proposed [12-21]. A modern comprehensive example can be seen in Mueller, et al. [22],
which requires 9 species (not including nitrogen) and 19 elementary reactions.
3

1.2 Development of Chemical Kinetics Mechanisms
1.2.1 The hydrogen mechanism: an example
Early work in H
2
oxidation kinetics by Hinshelwood and co-workers [12-14]
recognized that the behavior of H
2
/air mixtures was far more complex than could be
described by the simple one-step mechanism in (R1). In the experiments described in
these papers, a silica pressure vessel was filled with some amount of hydrogen and
oxygen gas which was then heated to some specified temperature. It was noted that H
2

exhibits what has been termed “explosion-limit” behavior, shown in Fig. 1-1. For a fixed
temperature, below a certain pressure the oxidation proceeds very slowly. Above this
pressure, the reaction is explosive, that is, it releases heat more quickly than the heat can
be removed from the reaction zone by heat transfer. This explosive behavior continues as
the pressure increases to another critical pressure, above which the reaction is no longer
explosive. Later work [15, 16] also discovered the presence of a third explosion limit.
Originally, it was argued that the reaction was catalyzed by thermally-excited H
2
O and
H
2
O
2
molecules [12], or by the hydroxyl radical [13, 14]; the lower explosion limit was
caused by destruction of these catalysts on the vessel wall. Later work suggested the
importance of free H and O atoms and HO
2
[17-21].

4

Figure 1-1. Schematic representation of the explosion limits of hydrogen.

The reaction mechanism proposed from these experiments was not a thermally-
catalyzed chain reaction but a chemically-catalyzed one. At about the same time,
Semenoff [23] developed a theory of reaction kinetics that combined the principles of
Arrhenius kinetics with the idea of an autocatalyzed chain reaction. In such a chain
reaction, there is a chain carrier which catalyzes the oxidation process; this process may
in turn produce more chain carriers. The three possible pathways are shown in Fig. 1-2,
which shows a single chain carrier being produced and then participating in a process
which either (a) eventually destroys the chain carrier, (b) produces exactly one chain
carrier for each one consumed, or (c) produces several chain carriers for each one
consumed.

p
T
Non-explosive
Explosive
5

Figure 1-2. The three possible reaction chains as proposed in [23]. a) Chain
termination process, resulting in a chain carrier being destroyed. b) Chain
carrying process, resulting in a single infinitely long chain. c) Chain
branching process, resulting in an infinite number of infinitely long chains.

For the case of H
2
, the chain reaction process follows

2
H O OH O + → + , (R2)

2
O H OH H + → + , (R3)

2 2
OH H H O H + → + , (R4)

which can be expressed as a net reaction

2 2 2
3H O 2H O 2H + → + , (R5)

which is catalyzed by a single H atom; the two additional H atoms go on to begin new
chains. At low pressures, the H atoms are consumed in surface reactions at the vessel
wall before these new chains can begin, which is the cause of the first explosion limit. At
high pressures, this chain reaction competes with
a. b. c.
H
H
HO
2

H
6

2 2
H O M HO M + + → + , (R6)

2
HO H 2OH + → , (R7)

followed by (R4) to produce H
2
O. This net reaction is the same as (R1), catalyzed by a
single H atom. This pathway is not explosive, because new chain carriers (H atoms) are
not generated. Furthermore, HO
2
is more stable than H, meaning it is more likely to be
consumed by the vessel wall, which terminates the chain reaction It can also react with
H
2
,

2 2 2 2
H HO H O H + → + , (R8)

which forms stable hydrogen peroxide, also terminating the chain. These processes cause
the second explosion limit. The third limit is controlled by the decomposition of H
2
O
2

into OH,

2 2
H O M 2OH M + → + , (R9)

followed by (R4) to continue the chain; the net reaction is the same as (R5). The
hydrogen mechanism proposed above has 7 reactions and therefore 21 Arrhenius
parameters. Given the state of knowledge by the mid-1960s, it was possible to assign
values to these parameters [24] and to use the flame propagation theories to begin
actually modeling laminar flames [25-27]. By 1977, the hydrogen mechanism was
thoroughly-documented [28].
7

1.2.2 Extension to hydrocarbon mechanisms
The techniques of developing a hydrocarbon oxidation mechanism are in principle the
same as those applied to hydrogen. Like hydrogen, hydrocarbon chemistry has been
well-documented for some time, as illustrated in Westbrook and Dryer’s comprehensive
review [29]. Furthermore, the hydrogen oxidation mechanism forms the basis of any
hydrocarbon oxidation mechanism; the overall reaction may be summarized by

2 2 2 2 2 2 2 2
1 1
CH O CH O+ O CO H O CO H O
2 2
+ → → + + → + , (R10)

where CH
2
represents a general hydrocarbon. The study of chemical kinetics, at least as
far as hydrogen and small hydrocarbons are concerned, is no longer a question of trying
to understand the basic chemistry, but merely one of assigning accurate rate parameters to
individual reactions.
It is deceptive to describe the task so simply, however, and doing so wipes a great
deal of effort and controversy under the rug. Since the work of Dixon-Lewis [24],
numerous chemical models have been published, such as for methane [30-33], methanol
[34, 35], ethane [36, 37], ethylene [38], ethanol [39], higher aliphatics [40, 41], and
aromatics [42, 43]. Once generated, these mechanisms are validated against a wide range
of experimental data, to include shock-tube ignition delay times [31, 44-47]; species
profiles in shock tubes [34, 48], laminar flow reactors [40], turbulent plug flow reactors
[22, 49], continuously-stirred reactors [50], and laminar flames [51, 52]; and laminar
8
flame speeds [53-57]. The result of all this work has been somewhat less than satisfying;
a recent review [58] denounced chemical model development as a black art and took a
very dim view of the current state of combustion modeling, pointing out that no one has
yet tried to compare the performance of the currently-available mechanisms in any
systematic manner. As an example, consider the reaction of vinyl radicals with oxygen,
which has at least two pathways,

2 3 2 2
C H O CH O HCO + → + , (R11)

2 3 2 2
C H O CH CHO O + → + . (R12)

Both of these pathways generate active radical species, but (R11) is a chain-carrying
process, whereas (R12) is a chain-branching brocess, and so the choice of which rate
expression to use for which pathway has significant effects on predictions from any
chemical model. Although these reactions, and the total consumption rate of their
reactants, have been known since 1984 [29, 59, 60], the relative contribution of each
pathway has been the subject of much study and controversy [61-66] and has not yet been
settled.

1.2.3 Effects of kinetic uncertainty on model development
In order to be generally predictive, a chemical reaction model must be complete, that
is, it must properly describe all of the reaction pathways that take place in a chemical
9
system. Even if the mechanism is complete, however, the uncertainty in the rate
coefficients describing these reaction pathways generally precludes the possibility of
predicting all combustion properties of a fuel a priori [67, 68]. Consider the case for
reaction (R2). A comprehensive review and rate parameter recommendation can be
found in the work of Baulch, et al. [69, 70]. As an example, the data used in the rate
evaluation for (R2) in [70] is shown in Figure 1-3; the uncertainty for (R2) is estimated at
a factor of 1.2 or larger. If the logarithmic sensitivity coefficient of a calculated laminar
flame speed to this reaction is about 0.4, then the uncertainty from considering (R2) alone
is at around 10%; considering additional reaction rate uncertainties could only increase
this number. Therefore, without a rigorous quantification of the effects of rate
uncertainties on the model predictions, a chemical model compiled from nothing more
than a literature review will not be expected to reproduce any particular piece of
combustion data.
Because of the appreciable uncertainties in the rate parameters of all chemical
reactions in combustion processes, kinetic models always can, and almost always do,
undergo some tuning in order to better reproduce a certain set of combustion experiments
data. This possibility has resulted in a proliferation of hydrogen and hydrocarbon kinetic
models that differ only in detail, as discussed recently [58, 71]. The amount of tuning
that goes in to a particular mechanism varies. Konnov’s Small Hydrocarbon Mechanism
[37] is not tuned at all. The systematic solution mapping and optimization technique [67,
68], which has been applied to H
2
[72], CH
4
[33], and C
3
H
8
[73] oxidation mechanisms,
represents the opposite extreme. Despite the ongoing research, if one examines the H
2

10
mechanisms that have been published in the last decade [58, 72, 74, 75], one sees that
there is still substantial disagreement regarding the precise rates at which the chemical
reactions occur, even for such a simple fuel whose chemistry is otherwise “well-known.”
These different mechanisms, however, really just represent random points within the rate
parameter uncertainty space; as long as each mechanism reproduces all experimental data
equally well, there is not any fundamental difference among them.

10
-15
10
-14
10
-13
10
-12
10
-11
0 0.5 1 1.5 2
k (cm
-3
s
-1
)
1000/T (K)

Figure 1-3. Rate constant evaluations of reaction (R2) taken from [70]

1.3 Uncertainty Quantification
1.3.1 Previous work in chemical uncertainty quantification
Few kinetic modeling studies have attempted to quantify the kinetic uncertainties.
Warnatz [69] discussed the use of local sensitivity analysis combined with published
11
uncertainties to determine the reactions which would contribute to prediction
uncertainties. McRae and coworkers advanced the method of parametric uncertainty in
the modeling of the various aspects of atmospheric chemistry [76, 77] and later hydrogen
oxidation in supercritical water [78]. Najm, Ghenam and coworkers [79-81] presented the
spectral projection methods to quantify uncertainties in reactive flow simulations.
Turanyi, Tomlin and co-workers used similar methods [82] and combined them with
Monte Carlo sampling [83, 84] to evaluate the uncertainty in laminar flame speed
predictions using the Leeds chemical kinetic model as the test case [32]. Likewise,
Frenklach and co-workers developed techniques to analyze an optimized model such as
GRI-Mech [33]. The methods they have developed later allow the residual uncertainty of
the model to be calculated [85-87], and can draw conclusions about the internal
consistency of the optimization data set [86, 88]. Klippenstein et al. [89] combined
global uncertainty screening and ab initio theoretical chemical kinetics to iteratively
improve a comprehensive CO, CH
2
O and CH
3
OH kinetic model originally proposed by
Dryer and coworkers [35].
The uncertainty propagation methods were originally developed in the work of
Wiener, Cameron, and Martin [90, 91], who brought a level of rigor to statistical
mechanics [92] by treating Brownian motion [93, 94] stochastically. Later, they were
applied to the problem of turbulent flows. These flows are fundamentally a stochastic
process with a measurable and mathematically treatable fuzziness [95, 96]. Of particular
interest is the work of Meecham and Siegal [97-100], which attempted to treat a turbulent
flow as a Gaussian process, as it should be by the Central Limit Theorem. Further work
by Ghanem expanded on the spectral approach [101, 102].
Stochastic modeling of uncertainty is particularly powerful when the source of the
uncertainty is due to a particular parameter or condition, which makes it a useful tool
12
when examining zero- and one-dimensional reacting flows, such as plug-flow reactors
and laminar flames. If the physical problem is well formulated through conservation of
mass, energy and species along with properly chosen boundary or initial conditions, the
sources of uncertainty include the reaction rate parameters and sometimes the transport
parameters. The work of McRae and co-workers [78] dealt with the propagation of
reaction rate uncertainty in the oxidation of hydrogen in supercritical water. The method
was extended by Najm, Ghanem and co-workers to laminar premixed flame problems
[79-81]. Methods have already been developed to estimate chemical kinetic rate
parameters in the presence of large amounts of data [67, 68, 73, 86-88], and work has
begun on estimating the uncertainty associated with these parameters [85, 86]. It is also
possible, once this uncertainty has been estimated, to propagate it into almost any
reacting flow simulation [78-81, 103].

1.4 Towards a Comprehensive System

In Chapter 2, the Method of Uncertainty Minimization using Polynomial Chaos
Expansions (MUM-PCE) is proposed to combine chemical model development and
uncertainty propagation into a single unified toolkit. In this method, a chemical kinetics
model and experimental database are compiled. Uncertainty quantification uses the
stochastic spectral expansion method (SSE) [78-81], which will be combined with
solution mapping to calculate prediction uncertainties in a simulation. The model is then
optimized against the experimental database, and SSE is used to quantify the uncertainty
in the model. Finally, solution mapping is used to propagate the uncertainty in the
optimized mechanism into a practical combustor. A test case for MUM-PCE is presented
13
in Chapter 3, using the USC-Mech II [104] and a set of ethylene combustion data as an
example. In Chapter 4, the hierarchical development strategy that has been employed for
model development is tested using the JetSurF 1.0 large-alkane oxidation model. A set of
H
2
/CO oxidation data is added to the ethylene experimental database in order to explore
the relationships between the two fuels. This database is then extended with a set of n-
pentane and n-heptane oxidation data, showing that a kinetic foundation can be
adequately developed using only experimental data up to pentane. Chapters 5 and 6
demonstrate the effect of adding new experiments for an existing fuel to the database. In
Chapter 5, a new set of mass burning rate measurements is added to the H
2
/CO database,
which, it is shown, can be used to generate a new, more-precise model. In Chapter 6,
species time histories during n-heptane oxidation behind reflected shock waves are added
to the n-heptane database. These measurements are shown to provide notable
improvement in the precision of the model predictions, although accurate measurements
of global combustion properties are still necessary to predict fuel combustion.
14
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15
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18
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84. A. S. Tomlin, Rel. Engng. Sys. Safety 91 (2006) 1219-1231.

85. T. Russi, A. Packard, R. Feeley, M. Frenklach, J. Phys. Chem. A 112 (2008)
2579–2588.

86. R. Feeley, P. Seiler, A. Packard, M. Frenklach, J. Phys. Chem. A 108 (2004)
9573-9583.

87. P. Seiler, M. Frenklach, A. Packard, R. Feeley, Optim. Eng. 7 (2006) 459-478.

88. R. Feeley, M. Frenklach, M. Onsum, T. Russi, A. Arkin, A. Packard, J. Phys.
Chem. A 110 (2006) 6803-6813.

89. S. J. Klippenstein, L. B. Harding, M. J. Davis, A. S. Tomlin, R. T. Skodjec, Proc.
Combust. Inst. 33 (2011) in press (doi:10.1016/j.proci.2010.05.066).

90. N. Wiener, Am. J. Math. 60 (1938) 897-936.
19

91. R. H. Cameron, W. T. Martin, Ann. Math. 48 (1947) 385-392.

92. J. W. Gibbs, Elementary principles in statistical mechanics: developed with
especial reference to the rational foundation of thermodynamics C. Scribner, New
York, 1902.

93. R. Brown, A brief account of microscopical observations made in the months of
June, July and August, 1827, on the particles contained in the pollen of plants;
and on the general existence of active molecules in organic and inorganic bodies.
1828.

94. A. Einstein, Ann. der Phys. 17 (1905) 549-560.

95. C. Canuto, M. Y. Hussaini, A. Quarteroni, T. A. Zhang, Spectral methods in fluid
dynamics. Springer-Verlag, Berlin, 1988.

96. M. O. Deville, P. F. Fischer, E. H. Mund, High Order Methods for Incompressible
Fluid Flow. Cambridge University Press, Cambridge, 2002.

97. W. C. Meecham, A. Siegel, Phys. Fluids 7 (1964) 1179-1190.

98. T. Imamura, W. C. Meecham, A. Siegel, J. Math. Phys. 6 (1965) 695-706.

99. W. C. Meecham, D. T. Jeng, J. Fluid Mech. 32 (1968) 225-249.

100. W. C. Meecham, J. Fluid Mech. 41 (1970) 179-188.

101. R. G. Ghanem, P. D. Spanos, Stochastic Finite Elements: A Spectral Approach.
Springer-Verlag, New York, NY, 1991.

102. R. G. Ghanem, P. D. Spanos, J. Eng. Mech. 117 (1991) 2351-2372.

103. D. B. Xiu, G. E. Karniadakis, J. Comp. Phys. 187 (2003) 137-167.

104. H. Wang, X. You, A. V. Joshi, S. G. Davis, A. Laskin, F. N. Egolfopoulos, C. K.
Law, USC Mech Version II. High-Temperature Combustion Reaction Model of
H
2
/CO/C
1
-C
4
Compounds. http://ignis.usc.edu/Mechanisms/USC-
Mech%20II/USC_Mech%20II.htm; 2007

20
Chapter 2 Theory and Computational Methods
2.1 Stochastic Spectral Projection
The stochastic spectral projection method [1-3] will be summarized here briefly. Let
u be the solution of a general dynamical model L(x,t); and this solution depends upon a
vector of random variables ξ ,

( ) , , = u u t y ξ , (2-1)

where y is the spatial coordinates and t is the time. The elements of ξ ξ ξ ξ are independent,
identically distributed (iid) random variables. The number and probability distributions
of these variables are chosen in a manner that depends on the system being considered, as
will be discussed in more detail later.
To apply the stochastic spectral projection, it is necessary to define a set of
orthogonal functions as the basis. If the elements of ξ are members of the Wiener-Askey
set of random variables, the basis of the projection will be one of the Askey sets of
polynomials [3], which include the Hermite and Legendre polynomials. These
polynomials are orthogonal with respect to the probability density function (PDF) of the
distribution with which they are associated. For instance, the Hermite polynomials H are
orthogonal with respect to the normal distribution and they have the property that

( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( )
2
2
2 2
E
1
E E 2
i j
i j ij
i i
H H
e d
H H
H H
ξ
ξ ξ
ξ
ξ ξ δ
ξ ξ π
∞
−
−∞
= =
∫
. (2-2)

E is the expectation operator, which is defined as ( ) ( ) ( ) ( ) E
∞
−∞
=
∫
X
f X f x p x dx for a
random variable X with PDF p
X
(X).
ij
δ is the Kronicker delta function. Assuming for the
21
moment that there is only one basis variable, ξ , the stochastic representation of u is an
expansion given by

( ) ( ) ( )
1
, , , ξ ξ
∞
=
=
∑
i i
i
u t u t H y y . (2-3)

The coefficients u
i
can be determined by multiplying both sides of Eq. (2-3) by ( )
j
H ξ
and then taking the expected value,

( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
2
1
2
1
2
E , , E ,
,
, E .
ζ
ξ ξ ξ ξ
ζ ζ ζ
ξ
∞
=
∞
∞
−
=
−∞
 
=
 
 
=
=
∑
∑
∫
j j i i
i
i i j
i
i i
H u t H u t H
u t H H e d
u t H
y y
y
y
(2-4)

By the orthogonality relation, we have

( ) ( ) ( )
( ) ( )
2
E , ,
E
ξ ξ
ξ
=
j
i
i
H u t
u
H
y
. (2-5)

The basis can be extended to multiple basis variables by introducing the multivariate
Hermite polynomials. These basis functions will be represented by
k
Ψ , and the first few
are

22

0
1 1
2 2
2
1 1
2 1 2
1
...
1
n n
n
n
ξ
ξ
ξ
ξ
ξ ξ
+
+
Ψ =
Ψ =
Ψ =
Ψ =
Ψ = −
Ψ =

Such a basis is termed a polynomial chaos (PC) and the projection of a random variable
onto this basis is its PC expansion (PCE).
The physical significance of such a multi-dimensional expansion is that two
correlated observables are related through basis variables. In principle, the number of
basis variables is infinite, as in the original work of Wiener [4] and its applications to
turbulent flows [5-8]. In general, laminar flows do not have this infinite-dimension
nature and consequently one can get away with a finite-dimension PC expansion [3, 9-
13]. In such cases, it is tempting to give the basis variables a very specific meaning. For
example, in [3] a laminar flow around a cylinder with uncertain material properties is
modeled. Therefore, there is a
1
ξ for the uncertain spring constant and a
2
ξ for the
uncertain viscoelastic damping coefficient. Similarly, for the chemical uncertainty work
in [9-12] there is aξ for each and every uncertain rate parameter.
Note that choosing a PC basis inherently ties us to a particular expression for the
distribution of ξ ξ ξ ξ. If we chose to work with the Hermite polynomials, we would have no
choice but to assume that each ξ is normally distributed. Because we do not wish to be
so constrained, all of the PC expansions used in the remainder of the paper will be
23
expressed in the form of a general polynomial, rather than in terms of the PC basis
functions.

2.2 Response Surface Construction by Solution Mapping
Response surface methodology has been applied to a wide variety of model
optimization studies in the form of the Solution Mapping (SM) [14-18]. The method
allows us to map the solutions of differential equations in the form of an expansion. The
variables of this expansion are simply the rate parameters, and the coefficients of
expansion are basically sensitivity coefficients. Each uncertain rate parameter may be
normalized into a factorial variable x
i
,

,0
ln
ln
i i
i
i
A A
x
f
= , (2-6)

where A
i
is the value of the i
th
rate coefficient, A
i,0
is its nominal value, and f
i
is its
multiplicative uncertainty factor. In this way, each rate coefficient is recast into a
parameter with the nominal value equal to 0 with uncertainty bound between –1 and 1.
For the current work, we express the model response
r
η as

( )
,0
1 1
N N N
r r i i ij i j
i i j i
a x b x x η η
= = ≥
= + + +
∑ ∑ ∑
x L, (2-7a)

and in the vector-matrix form as

,0
T T
r r
η η = + + + a x x bx L, (2-7b)

24
where N is the number of factorial variables considered in the expansion, and
,0 r
η is the
model prediction by an as-compiled reaction model. For most combustion problems, a
second-order expansion is usually sufficient if the form of
r
η is chosen properly [16].
The coefficients a
i
and b
ij
can be determined by a regression test against computational
experiments using a factorial design [19] or from the Sensitivity-Analysis-Based (SAB)
method [14]. Since a combustion reaction model usually consists of a large number of
chemical reactions, on the order of hundreds to a few thousands, a factorial design is
seemingly prohibitively expensive. In reality, because of effect sparsity [20], only a few
reaction rates, on the order of 10-40, are actually important for any given set of
experimental responses we wish to model [16].
In Bayesian terminology, the as-compiled model is the “prior” model. It represents
the state of knowledge before any global experiments are taken into account [21]. All the
other models constrained by the global experiments are therefore “posterior” models. We
shall use these terminologies hereafter in the current paper.

2.2.1 Combining solution mapping and spectral expansion methods for
uncertainty propagation.
Solution mapping and stochastic spectral expansion are particularly powerful if they
are combined together [22]. Here, each uncertain parameter x
i
is assumed to be a random
variable, independent of all the others, with uncertainty bound by [–1,1]. In chemical
kinetics problems, the uncertainty factor is often derived empirically to represent rough
uncertainty estimates in a rate parameter. Two plausible limiting cases are each x being
uniformly distributed and being normally distributed. At this point it is worthwhile to
consider what these two limiting cases physically represent. If a reaction rate parameter
is said to be normally distributed, it is implied that its value has been measured a
25
statistically significant number of times and this set of measurements has a meaningful
standard deviation (σ). In this case there is a single preferred value, and values farther
from it are considered less probable even if they are physically possible. On the other
hand, if the rate parameter is uniformly distributed, it is implied that the value has been
measured few times, only enough to obtain a rough upper and lower bound for its value.
In this case all values between the upper and lower bounds are considered to be equally
probable. In reality, of course, most of the rate constants fall between these limiting
cases. We shall examine these two limiting cases carefully in the present work. We will
show that the uncertainty in a model prediction depends on the variance of the rate
parameter but not strongly on the exact form of its distribution.
A PCE of x may be defined for each of the limiting cases. For uniform x’s, each ξ is
an iid uniform variable on –1 to 1, U(–1,1). For normal x’s, each ξ is an iid normal
variable with mean equal to 0 and variance 1, N(0,1), and the f
i
’s are treated as 2σ-
uncertainties. This is equivalent to stating that

i i
x ξ = (2-8)

for uniform x’s and

1
2 i i
x ξ = (2-9)

for normal x’s. These expressions can be propagated into Eq. (2-7) to yield stochastic
expressions for
r
η ,

( )
,0 , ,
1 1
M M M
r r r i i r ij i j
i i j i
a b η η ξ ξ ξ
= = =
≅ + +
∑ ∑ ∑
ξ (2-10)
26

for uniform x’s and

( )
,0 , ,
1 1
1 1
2 4
M M M
r r r i i r ij i j
i i j i
a b η η ξ ξ ξ
= = =
≅ + +
∑ ∑ ∑
ξ . (2-11)

for normal x’s.
It is, of course, important to understand the difference in results between
assuming uniformly and normally distributed x’s. The PDF of Eq. (2-10) is similar to the
sum of M iid uniform variables; such a PDF can be calculated by

( )
( )
( ) ( )
1
0
1
1
1 !
x
k n
E
k
M
P x x k
k M
 
  −
=
 
= − −
 
−
 
∑
(2-12)

available in [23]. Figure 2-1 shows this PDF. The variance of the basis distribution is

( )
2 1
2
1,1
1
1
2 3
σ
−
−
= =
∫
U
x
dx (2-13)

and the variance of the sum is
3
M
. Likewise, Eq. (2-11) is close to a sum of M iid normal
distributions and is itself normal; the variance of such a sum is
4
M
. The ratio of standard
deviations is 2 3 , or about 1.15. The Central Limit Theorem states that a sum of M
independent variables approaches a normal distribution if M is large. This property is
illustrated in Figure 1, which plots the PDF of Eq. (2-12) for M = 1 to 4. The PDFs are
normalized to have σ = 1 and compared with that of a normal distribution also with σ =
1. As can be seen in the figure, the relaxation to normality is very fast, and therefore the
choice of distribution of the reaction rate parameters does not make much difference,
27
other than to change the variance by about 15%. Figure 2-2 shows a real example, the
probability distributions of the flame speed of an ethylene-air mixture at the equivalence
ratio φ = 0.7, assuming either normally-distributed or uniformly-distributed reaction rate
parameters in USC Mech II [24]. In this figure, the number of rate parameters with the
coefficient a = O(1 cm/s) is about 15, and as expected the probability distribution
assuming uniformly-distributed rate parameters is very well-approximated by a normal
distribution.

0.0
0.1
0.2
0.3
0.4
0.5
-2 -1 0 1 2
Probability Density, p
Ξ
(ξ)
ξ
M = 1
M = 4
M = 3
M = 2

Figure 2-1. Illustration of the application of the central limit theorem. Probability
density functions for a random variable U
M
with variance 1 (solid lines),
which is the sum of M uniform random variables, for M = [1,2,3,4]. This is
compared with the normal distribution with variance σ = 1 (dashed line).

28
0
30
60
25 30 35 40 45 50
Count
Flame Speed, s
u
o

Figure 2-2. Comparison of Monte Carlo sampling of the ethylene-air laminar flame
speed at equivalence ratio of 0.7 for normally-distributed rate parameters ( ,
dashed line) and uniformly-distributed rate parameters (■, solid line). The
computation was carried out using the prior model.

If a more sophisticated description of the rate parameters is desired, a general PC
expansion for x can be introduced,

0
1 1
...
M M M
i i ij i j
i i j i
ξ ξ ξ
= = =
= + + +
∑ ∑ ∑
x x α β (2-14)

which can likewise be propagated into Eq. (2-7) to yield a general stochastic expression
for ( )
r
η ξ ,

( ) ( )
0 , ,
1 1
ˆ
ˆ ...
M M M
r r r i i r ij i j
i i j i
η η α ξ β ξ ξ
= = =
= + + +
∑ ∑ ∑
ξ x . (2-15)

29
The prediction uncertainty can be calculated by taking the expectation of Eq. (2-15),

( ) ( ) ( )
2 2 2
1
2 2 2
, , ,
1 1 1 1
E E
ˆ ˆ
ˆ 2
r r r
M M M M
r i r ii r ij
i i i j i
σ η η
α β β
−
= = = = +
  ≅ −  
   
= + +
∑ ∑ ∑ ∑
x ξ ξ
. (2-16)

Obviously, there must be some justification for why such a sophisticated distribution is
needed, and later sections will describe where it might come from. The point is,
however, that if the variation of the factorial variables is prescribed, the solution mapping
method can be used to explicitly calculate of the uncertainty in model predictions, and
this calculation is inexpensive compared to Monte Carlo sampling.

2.3 Solution Mapping and Model Optimization
The inverse problem, in which a model is constrained against a set of experimental
data, was the original application of SM, which is termed here SM/MO. In this problem,
a set of n experimental properties are chosen to define the applicability of the model. The
r
th
property is observed to have a certain value
obs
r
η with some standard deviation
obs
r
σ .
A simulation of this experiment is performed, and the same property is calculated to have
a value ( ) η
r
x , depending upon the rate coefficient vector x. Response surfaces are
generated for each simulated property with respect to each active rate parameter and the
difference between the model response and experimental observations is minimized.
Explicitly, this minimization is done with least-squares optimization,

( )
( )
2
obs
obs
1
min
n
r r
r
r
η η
σ
=
 
  −
 
 
Φ =
 
 
 
∑
*
x
x
x , (2-17)
30

subject to the original constraints that { } 1 1
i
x − < < , which results in an constrained
factorial vector x
*
and the corresponding rate coefficient vector k
*
. The constrained
vector k
*
represents a posterior model that best describes the experimental data given to
it. As discussed earlier, a choice can be made between uniformly- and normally-
distributed x’s, and Eq. (2-17) assumes that the x’s are uniformly distributed. An
alternative expression can be given if x’s are normally distributed, which is

( )
( )
2
obs
2
obs
1 1
min 2
n N
r r
i
r i
r
x
η η
σ
= =
 
  −
 
 
Φ = +
 
 
 
∑ ∑
*
x
x
x . (2-18)

Such an expression treats each reaction rate constant evaluation as a single experiment,
equivalent to, for example, a laminar flame speed measurement. They are also
independent of one another. As was mentioned earlier, assigning a normal distribution to
any particular rate parameter means that it has been carefully measured, which is usually
not true. For these rate parameters, the
2
2
i
x term can be omitted, replaced instead by the
original constraint { } 1 1
i
x − < < , just like in Eq. (2-17).

2.3.1 Consistency of a reaction data set
Although the solution x
*
calculated from Eqs. (2-17) or (2-18) represents the “best”
chemical model that can be developed with the available data, it is possible, indeed
likely, that some of the model predictions
( )
r
η
*
x will not agree with the corresponding
experimental observations
obs
r
η . In this case, it must be asked whether the experimental
dataset is internally consistent within the framework of the prior model. Of course, the
31
origin of inconsistency can be an erroneous data uncertainty being stated, the model
being incomplete, or both. For the time being, we shall examine the case where the
experimental data uncertainty is miss-stated. Some of the work on data collaboration [25,
26] has addressed the problem of data consistency, although the question that is answered
in these papers is whether a point exists that is compatible with all the experimental data;
it does not address the consistency of the final posterior model x
*
. In this section, a
consistency measure will be put forward that does address this issue. The effect of a
particular experiment on the posterior model is measured by the normalized scalar
product S
r
between the response surface gradient
( )
*
r
η ∇ x and the constrained factorial
vector x
*
,

( )
( )
* *
* *
r
r
r
S
η
η
∇ ⋅
=
∇
x x
x x
. (2-19)

The target that has the greatest effect on the posterior model will therefore have the
largest value of S
r
. Whether the target is consistent is measured by the magnitude of its
contribution to the objective function, F
r
,

( )
obs *
obs
2
r r
r
r
F
η η
σ
−
=
x
(2-20)

If |F
r
| > 1, the r
th
experiment is inconsistent with x
*
. If there is only one inconsistent
target, that target is removed and the model is re-constrained. If multiple such targets
exist, each target has its weighted scalar product W
r
calculated by

32
=
r r r
W S F (2-21)

The single target that has the greatest |W
r
| is removed and the model is re-constrained.
This procedure is continued until an x
*
is calculated such that all F
r
are smaller than
unity. These tests are referred to as F and W tests, respectively.
It should be noted that there are three reasons why a target might be removed. There
might be multiple measurements under a single condition which do agree with each other.
In this case, no statement is made about the validity of one data value over another; one is
selected as being the “preferred” value, with the recommendation that the others be
folded into the experimental uncertainty estimate. Alternatively, there may be multiple
measurements for a particular experiment which do not agree with each other. In this
case, ideally all but one of these measurements will not be able to be reconciled with the
other experimental targets and will be removed. The most interesting case is when there
is only one measurement under a particular condition, but this experimental measurement
cannot be reconciled with the other targets. There is no alternative measurement that the
consistency algorithm can choose, and so the algorithm will unequivocally remove this
target from consideration. However, the fact that a target has been flagged as
inconsistent could be an indication that something is wrong either with the model or with
the other experimental data, and care should be taken when interpreting this fact, as we
discussed before.

2.4 Uncertainty Minimization
Recall that earlier it was suggested that a more sophisticated expression for x might
be introduced, and in this section it will be discussed where such an expression might
come from. Consider the case of the reactions C
2
H
3
+ O
2
CH
2
O + HCO and C
2
H
3
+ O
2

33
CH
3
CHO + O, discussed extensively in [27-35]. The controversy over the rate
constants for these reactions notwithstanding, it is clear that they should be coupled
because of their relation in the C
2
H
3
+ O
2
products system. In order to determine what
this coupling might be, we wish to see what information can be extracted from the
deterministic optimization procedure. First, the objective function in Eq. (2-17) is
interpreted to represent a joint PDF for x,

( ) ( )
1
2
exp p A   = − Φ
 
X
x x , (2-22)

where A is a normalization constant. Equation (2-22) can be rewritten as

( )
2
obs
,0 obs
1
1 1
log log
2
n
T T
r r
r
r
p A η η
σ
=
 
 
− = − + + −
 
 
 
∑
a x x bx . (2-23)

The coupling among the rate parameters comes from this PDF, so we wish to find a
corresponding PC expansion as in Eq. (2-14). Consider a formulation that includes only
first order terms,

0
1
M
i i
i
ξ
=
= +
∑
* * *
x x α , (2-24)

x
*
follows a multivariate normal distribution with mean
0
*
x and covariance
matrix
T
Σ =
* *
α α . We find
*
α by linearizing the response surface in the neighborhood of
0
*
x which gives

( )
,0 0
T
r r
η η   + +
 
*
x bx a x . (2-25)
34

We then rewrite Eq. (2-23) as

( ) ( )
{ }
( )
2
obs 1
log
2
T
T T T T T
n
r
r
p
A
σ
=
  − + + + −
 
=
∑
* *
x x bxx b ax b b xa aa x x
. (2-26)

This resembles a multivariate normal distribution, which has a joint density function
( ) ( )( ) ( )
1
exp 1 2
T
p
−
 
∝ − − Σ −
 
x x μ x μ , with mean μ and covariance matrix Σ . In this
case,
0
=
*
μ x and Σ is given by

( )
( )
1
0 0 0 0 2
obs 1
1
n
T T T T T
r
r
σ
−
=
 
 
Σ = + + +
 
 
∑
* * * *
bx x b ax b b x a aa . (2-27)

It can be shown that, in the case where the x’s are normally distributed, the covariance
takes a similar form,

( )
( )
1
0 0 2
obs 1
1
2
n
T
T T T T
r
r
σ
−
=
 
 
Σ = + + + +
 
 
∑
* *
bx x b ax b b xa aa I . (2-28)

For the “real,” hybrid problem, where some rate parameters are uniformly-distributed and
some normally-distributed, those rows of I in Eq. (2-28) that represent uniformly-
distributed rate constants can be replaced by zeros. If Σ is positive-definite,
*
α can be
calculated by taking the Cholesky factorization of Σ . However, in order for this to be the
case, there must be at least as many experimental targets as there are reactions, or else
Σ will be singular and the Cholesky factorization cannot be calculated. Note that
35
Σ calculated in Eq. (2-28) is always positive-definite, so for normally-distributed reaction
rate coefficients this is not a problem. If Σ is singular, an additional assumption must be
made, which is that each experiment is independent of every other. Then, it is possible to
define a distribution, termed the target distribution, where each target has its own
i
ξ , as

( )
obs obs
r r r r
η η σ ξ = + ξ (2-29)

If x
*
is in this same basis that reproduces the target distribution, an
*
α can be found that
minimizes the difference, in the least-squares sense, between the expressions in equations
(15) and (29), giving us a new objective function,

( )
( )
( )
2
obs
, , 2
obs 1 1 1
1
ˆ
ˆ min
M M M M
r ir r i r ij
r i i j i
r
σ δ α β
σ
= = = =
 
Φ = − +
 
 
∑ ∑ ∑ ∑
*
α
α , (2-30)

which results in a posterior model
*
x whose uncertainty space is described by
*
α .
Together,
*
x and
*
α form a posterior model which most closely approximates the
observed experimental uncertainty and with reduced parameter uncertainties, though such
uncertainties become tightly coupled to each other. The a priori uncertainty in the rate
parameters is preserved by introducing a constraint that the standard deviation of the i
th

rate parameter, σ
i,
, defined by
2 2
, i r i
r
σ α =
∑
cannot exceed ½. Also, a term
( )
2
1
2 i
i
ε σ −
∑
is added to the objective function, with ε << 1. This ensures that, if the
uncertainty of a particular reaction is not well-constrained by the experimental data, the a
priori assumption of σ
i
= ½ will be preferred.

36
2.5 Chapter 2 Endnotes

1. R. G. Ghanem, P. D. Spanos, Stochastic Finite Elements: A Spectral Approach.
Springer-Verlag, New York, NY, 1991.

2. R. G. Ghanem, P. D. Spanos, J. Eng. Mech. 117 (1991) 2351-2372.

3. D. B. Xiu, G. E. Karniadakis, SIAM J. Sci. Comp. 24 (2002) 619-644.

4. N. Wiener, Am. J. Math. 60 (1938) 897-936.

5. W. C. Meecham, A. Siegel, Phys. Fluids 7 (1964) 1179-1190.

6. T. Imamura, W. C. Meecham, A. Siegel, J. Math. Phys. 6 (1965) 695-706.

7. W. C. Meecham, D. T. Jeng, J. Fluid Mech. 32 (1968) 225-249.

8. W. C. Meecham, J. Fluid Mech. 41 (1970) 179-188.

9. B. D. Phenix, J. L. Dinaro, M. A. Tatang, J. W. Tester, J. B. Howard, G. J.
McRae, Combust. Flame 112 (1998) 132-146.

10. M. T. Reagan, H. N. Najm, B. J. Debusschere, O. P. Le Maitre, M. Knio, R. G.
Ghanem, Combust. Theory Modelling 8 (2004) 607-632.

11. M. T. Reagan, H. N. Najm, R. G. Ghanem, O. M. Knio, Combust. Flame 132
(2003) 545-555.

12. M. T. Reagan, H. N. Najm, P. P. Pebay, O. M. Knio, R. G. Ghanem, Int. J. Chem.
Kinet. 37 (2005) 368-382.

13. D. B. Xiu, G. E. Karniadakis, J. Comp. Phys. 187 (2003) 137-167.

14. S. G. Davis, A. B. Mhadeshwar, D. G. Vlachos, H. Wang, Int. J. Chem. Kinet. 36
(2004) 94-106.

15. M. Frenklach, Combust. Flame 58 (1984) 69-72.

16. M. Frenklach, H. Wang, M. J. Rabinowitz, Prog. Energ. Combust. Sci. 18 (1992)
47-73.

17. Z. Qin, V. Lissianski, H. Yang, W. C. Gardiner, Jr., S. G. Davis, H. Wang, Proc.
Combust. Inst. 28 (2000) 1663-1669.
37

18. G. P. Smith, D. M. Golden, M. Frenklach, B. Eiteener, M. Goldenberg, C. T.
Bowman, R. K. Hanson, W. C. Gardiner, V. V. Lissianski, Z. W. Qin, GRI-Mech
3.0. http://www.me.berkeley.edu/gri_mech/; 2000

19. G. E. P. Box, W. G. Hunter, J. S. Hunter, Statistics for experimenters:
Introduction to design, data analysis, and model building. Wiley, New York, NY,
1978.

20. G. Box, R. D. Meyer, J. Res. Nat. Bur. Stan. 90 (1985) 495-500.

21. G. E. P. Box, G. C. Tiao, Bayesian Inference in Statistical Analysis. Wiley, New
York, NY, 1973.

22. D. A. Sheen, X. You, H. Wang, T. Løvås, Proc. Combust. Inst. 32 (2009) 535-
542.

23. S. Sadooghi-Alvandi, A. Nematollahi, R. Habibi, Statistical Papers 50 (2007)
171-175.

24. H. Wang, X. You, A. V. Joshi, S. G. Davis, A. Laskin, F. N. Egolfopoulos, C. K.
Law, USC Mech Version II. High-Temperature Combustion Reaction Model of
H
2
/CO/C
1
-C
4
Compounds. http://ignis.usc.edu/Mechanisms/USC-
Mech%20II/USC_Mech%20II.htm; 2007

25. R. Feeley, P. Seiler, A. Packard, M. Frenklach, J. Phys. Chem. A 108 (2004)
9573-9583.

26. R. Feeley, M. Frenklach, M. Onsum, T. Russi, A. Arkin, A. Packard, J. Phys.
Chem. A 110 (2006) 6803-6813.

27. C. K. Westbrook, F. L. Dryer, Prog. Energ. Combust. Sci. 10 (1984) 1-57.

28. J. Y. Park, M. C. Heaven, D. Gutman, Chem. Phys. Lett. 104 (1984) 469-474.

29. I. R. Slagle, J. Y. Park, M. C. Heaven, D. Gutman, J. Am. Chem. Soc. 106 (1984)
4356-4361.

30. P. R. Westmoreland, Combust. Sci. Technol. 82 (1992) 151-168.

31. B. K. Carpenter, J. Am. Chem. Soc. 115 (1993) 9806-9807.

32. B. K. Carpenter, J. Phys. Chem. 99 (1995) 9801-9810.

38
33. A. M. Mebel, E. W. G. Diau, M. C. Lin, K. Morokuma, J. Am. Chem. Soc. 118
(1996) 9759-9771.

34. B. K. Carpenter, J. Phys. Chem. A 105 (2001) 4585-4588.

35. T. Carriere, P. R. Westmoreland, A. Kazakov, Y. S. Stein, F. L. Dryer, Proc.
Combust. Inst. 29 (2002) 1257-1266.

39
Chapter 3 The Method of Uncertainty Quantification and
Minimization Using Polynomial Chaos Expansions: A Case
Study
3.1 Introduction
To be generally predictive, a chemical reaction model must be complete with respect
to the reaction process about which it intends to make predictions. Even if a reaction
model is complete, however, the uncertainty in the often numerous reaction rate
parameters generally precludes the reliability of a model a priori [1, 2]. Consider the
reaction responsible for the free-radical chain branching in high-temperature combustion
of any hydrocarbon fuels,

H + O
2
→ O + OH . (R1).
The uncertainty factor of k
1
has remained at a factor of 1.2 or larger for a long while
[3, 4]. Although the recent effort by Hanson, Davidson and coworkers [5] has placed a
much better bounds to k
1
(10-15%), this uncertainty remains significant. For example,
the logarithmic sensitivity coefficient of a calculated laminar flame speed to this reaction
is typically ~0.4. Then the model uncertainty expected from R1 alone is 4-6%, and
considering additional parameter uncertainties could only increase this number. This
discussion therefore illustrates the fact that a chemical model as compiled is not expected
to reproduce any particular piece of combustion data, and even if it does, there is no
mathematical reason to deduce that the model will be predictive outside of the condition
spaces in which it is validated.
While the only certainty is for the parameter uncertainties to persist well into the
future, model proliferation abounds. Since the number of elementary reactions that need
to be considered is large, there can be an almost infinite number of rate parameter
40
combinations, all of which may be “acceptable” with respect to predictions of a particular
set of combustion data. To illustrate this point, let us examine the combustion models of
hydrogen published in the last decade (e.g., [6-10]). A comparison of these models
shows that while there are still apparent disagreements regarding the choices of rate
parameters, there is no fundamental difference among them with respect to the
elementary reactions and pathways considered. These recent studies may well be viewed
as a kind of statistical sampling of the rate parameters within their respective range of
uncertainties, even though this is a rather small statistical sample.
Historically, published kinetic models almost always undergo some tuning in order to
better reproduce the combustion data considered. The method and amount of tuning
varies. Perhaps the most appropriate approach taken has been the systematic solution
mapping and optimization technique [1, 2], which has been applied to models of H
2
[8],
CH
4
[11], and C
3
H
8
[12] oxidation. The conclusion to be drawn from these studies is that
for this complex inverse problem there will never be a sufficient number of accurate
experiments to “pin” the rate parameters to a unique set. This is especially true
considering that the active set of rate parameters also evolves with the condition space of
the experiment and that all experiments have their inherent uncertainties.
The purpose of this chapter is to propose a method to combine chemical model
development and kinetic uncertainty propagation and minimization into a single unified
theoretical and computational toolkit, termed the Method of Uncertainty Minimization
using Polynomial Chaos Expansions (MUM-PCE). A simplified version of this method
was introduced in a previous study and tested against the combustion model of ethylene
[13], and more recently the time-resolved multispecies data of heptane oxidation in a
shock tube [14]. The mathematical background of MUM-PCE will be introduced and
discussed in detail. In this method, uncertainty quantification will use the stochastic
41
spectral expansion (SSE) method [15-18], which is combined with solution mapping to
calculate prediction uncertainties in a simulation. The model will then be constrained by
the experimental database, and the MUM will be used to quantify the prior and posterior
uncertainties. We show that the information from the global experiments can be used
effectively to reduce the uncertainty in model predictions of various combustion
processes and properties. Various fundamental limitations of the method are also
discussed at length.

3.2 Case Study
The methods are tested here by examining a chemical reaction model against a set of
global and detailed combustion properties and determining the resulting uncertainty. The
prior model considered is of H
2
/CO/C
1
-C
4
combustion, or USC-Mech II [19], which is
composed of 111 species and 784 elementary reactions. The model was constructed
based on a wide variety of experimental studies and ab initio calculations. This model is
used here as the prior model for further development. It should be noted that the H
2
/CO
submechanism in USC-Mech II has been previously optimized in [8]; for this study, those
rates which were optimized have been reset to their original values. Consequently, the
results in this study differ somewhat from that in [13]. The uncertainty factors for each
rate coefficient were taken in consultation with literature compilations [e.g., [3, 4]] or
from our own rate evaluations [13].
Twenty-two experiments, including the flame speed [20-22], ignition delay times [23-
25], and species concentration in a flow reactor [26] and a burner-stabilized flame [27]
were targets. They are detailed in Table 3-1. The two-standard deviation (2σ) of flame
speed is estimated to be ±2 cm/s. Flow reactor targets are the rates of ethylene
42
disappearance from 90% to 70% of the initial value and have an uncertainty factor of 1.5
or a 2σ logarithmic uncertainty of 0.4.
43

Table 3-1. Combustion property targets and uncertainties. Targets in italics were
rejected or removed from the final target set (see text).

Target Composition (mole %) T
0
p
0
Model Prediction
a

C
2
H
4
O
2
Diluent (K) (atm) Value
a
Prior
Posterior
(Model I)
Note Ref
Flame speeds S
u
0
(cm/s)
36 ± 2 [20] fls 1a-1
fls 1a-2
φ = 0.7
35 ± 2
36.8 ± 8 34.8 ± 1.4
[21]
65 ± 2 [21]
65 ± 2 [22]
fls 1b-1
fls 1b-2
fls 1b-3
φ = 1.0
63 ± 2
60.8 ± 11 62.4 ± 1.2
[20]
71 ± 2 [21]
65 ± 2 [22]
fls 1c-1
fls 1c-2
fls 1c-3
φ = 1.2
1
63 ± 2
61.6 ± 12 65.6 ± 1.5
[20]
fls 2a φ = 0.7 28 ± 2 30.6 ± 7 28.7 ± 1.2
[20]
56 ± 2 [22] fls 2b-1
fls 2b-2
φ = 1.0
53 ± 2
54.0 ± 10 55.2 ± 1.1
[20]
60 ± 2 [22] fls 2c-1
fls 2c-2
φ = 1.2
2
58 ± 2
55.1 ± 12 58.9 ± 1.1
[20]
fls 3a φ = 0.7 21 ± 2 23.6 ± 6 21.4 ± 1.4 [20]
fls 3b φ = 1.0 44 ± 2 44.2 ± 9 44.9 ± 1.3 [20]
fls 3c φ = 1.2 50 ± 2 45.2 ± 10 48.5 ± 1.4 [20]
fls 3d φ = 1.6
Air 298.2
5
22 ± 2 21.5 ± 7 22.0 ± 1.9 [20]

Ignition delay times T
5
p
5
ln (τ, μs)

ign 1a 1 3 Ar 1292 1.3 5.74 ± 0.2 6.53 ± 0.9 5.97 ± 0.5 b [23]
ign 1b 1 3 Ar 1837 2.3 2.44 ± 0.2 3.13 ± 0.4 2.91 ± 0.2 b [23]
ign 2a 0.5 1.5 Ar 1643 2.3 4.18 ± 0.6 4.66 ± 0.3 4.39 ± 0.2 c [24]
ign 2b 0.5 1.5 Ar 1443 1.9 5.60 ± 0.6 5.70 ± 0.4 5.47 ± 0.2 c [24]
ign 2c 0.5 1.5 Ar 1267 1.5 7.27 ± 0.6 6.82 ± 0.6 6.89 ± 0.3 c [24]
ign 3a 0.5 1 Ar 1745 0.7 4.80 ± 0.2 4.95 ± 0.3 4.86 ± 0.3 d [25]
ign 3b 0.5 1 Ar 2150 0.7 3.97 ± 0.2 3.67 ± 0.4 3.67 ± 0.3 d [25]

Fuel disappearance rate in
a flow reactor T p
ln (d[X]/dt,
ppm/ms )

flw 1a 0.1496 0.2263 N
2
1002 1 0.99 ± 0.4 1.45 1.06 e [26]
flw 2a 0.1436 0.7876 N
2
1037 1 2.81 ± 0.4 2.76 2.99 f [26]
flw 3a 0.1488 0.4365 N
2
961 1 1.22 ± 0.4 1.41 1.20 g [26]

Peak Mole fraction in a
premixed flat flame T
0
p

X
max

flf 1a 0.194 0.306 Ar 298.2 0.026 1.1×10
-2
1.0×10
-2
1.1×10
-2
Η
[27]
flf 1b 0.194 0.306 Ar 298.2 0.026 2.2×10
-3
2.6×10
-3
2.4×10
-3
ΟΗ
[27]

44
Table 3-1. (Continued)

a
The ± values are 2σ uncertainty.
b
Onset of CH emission.
c
Onset of CO
2
emission.
d

10% of maximum [CO] + [CO
2
].
d
0.9 → 0.7 [O
2
] conversion rate.
e
0.95 → 0.85 [C
2
H
4
]
conversion.
f
0.6 → 0.4 [C
2
H
4
] conversion.
g
0.9 → 0.7 [C
2
H
4
] conversion. .

Determining the uncertainty in the experimental ignition delay time is not as
straightforward, since each data set may contain both random and systematic errors. An
additional difficulty comes from the unevenness of the sample size. For example, the
data set reported by Brown and Thomas [23] contains a very large number of experiments
which thickly populate its temperature range. The other experimental data sets are
relatively sparse over their temperature ranges. To deal with these problems, we fit all
data considered with an empirical expression given as [28],

[ ] [ ] [ ]
2 4 2
C H O Ar exp
ω λ γ
τ
 
=
 
 
n a
T
AT
T
, (3-1)

where ω , λ, γ, n and T
a
are parameters. Figure 3-1 shows the results of such a fit for a
wide range of C
2
H
4
ignition delay data taken from [23-25, 29-31]. If the empirical
expression were an accurate representation of the physical process and there were no
systematic error in the data, the scatter in the fit would be representative of the
uncertainty of the measurements. This is, of course, not the case. Systematic differences
do exist from one shock tube to another; the subtle difference in the methods with which
the ignition delay times are defined (e.g., CH* emission, pressure gradients etc) can also
contribute the scatter in the data. At the moment, the empirical fit is the best tool we
have to analyze the data. Based on the results presented in Figure 3-1, we assigned a 2σ
uncertainty factor of 2 for all ignition delay targets considered.

45

10
-3
10
-2
10
-1
10
0
10
1
0.4 0.5 0.6 0.7 0.8 0.9 1.0
τ τ τ τ [C
2
H
4
]
-0.15
[O
2
]
1.01
[Ar]
-0.37
1000K/T
Uncertainty factor = 2

Figure 3-1. Experimental ignition delay data as described by Eq. (31). Data are taken
from Homer and Kistiakowsky [25] (

: 0.5%C
2
H
4
-3%O
2
-Ar, p
5
= 0.6-0.8
atm; : 0.5%C
2
H
4
-1%O
2
-Ar, p
5
= 0.6-0.7 atm); Baker and Skinner [29]
( : 1% C
2
H
4
-3%O
2
-Ar, p
5
= 3 atm; : 5% C
2
H
4
-3%O
2
-Ar, p
5
= 3 atm; :
0.25% C
2
H
4
-0.75%O
2
-Ar, p
5
= 12 atm; : 2% C
2
H
4
-3%O
2
-Ar, p
5
= 3 atm;
: 1% C
2
H
4
-1.5%O
2
-Ar, p
5
= 3 atm; : 1% C
2
H
4
-6%O
2
-Ar, p
5
= 3 atm; :
0.25% C
2
H
4
-6%O
2
-Ar, p
5
= 3 atm; : 0.25% C
2
H
4
-1.5%O
2
-Ar, p
5
= 3 atm;
: 0.5% C
2
H
4
-0.75%O
2
-Ar, p
5
=3 atm); Hidaka et al. [30] ( : 0.5% C
2
H
4
-
1.5%O
2
-Ar, C
5
= (1.93±0.05)×10
–6
mol/cm
3
; : 0.5% C
2
H
4
-4.5%O
2
-Ar, C
5

= (1.87±0.06)×10
–6
mol/cm
3
); Hidaka et al. [24] ( :0.1% C
2
H
4
-0.6%O
2
-Ar,
C
5
= (1.61±0.06)×10
–5
mol/cm
3
; : 1% C
2
H
4
-1.5%O
2
-Ar, C
5
=
(1.63±0.07)×10
–5
mol/cm
3
; : 0.5% C
2
H
4
-1.5%O
2
-Ar, C
5
= (1.58±0.07)×10
–5

mol/cm
3
; : 0.5% C
2
H
4
-3%O
2
-Ar, C
5
= (1.59±0.09)×10
–5
mol/cm
3
); Brown
and Thomas [23] ( : 1% C
2
H
4
-3%O
2
-Ar, C
5
= (1.44±0.14)×10
–5
mol/cm
3
; :
6.25% C
2
H
4
-18.7%O
2
-Ar, C
5
= (1.92±0.22)×10
–5
mol/cm
3
); Saxena et al.
[31] ( : 3.5% C
2
H
4
-3.5%O
2
-Ar, p
5
= 2.4 atm;

: 3.5% C
2
H
4
-3.5%O
2
-Ar, p
5

= 2 atm;

: 3.5% C
2
H
4
-3.5%O
2
-Ar, p
5
= 10 atm;

: 3.5% C
2
H
4
-3.5%O
2
-Ar,
p
5
~ 20atm; –: 1.75% C
2
H
4
-5.25%O
2
-Ar, p
5
= 2.4 atm; │: 1% C
2
H
4
-3%O
2
-
Ar, p
5
= 2.4 atm;

: 0.5% C
2
H
4
-1.5%O
2
-Ar, p
5
= 2.5 atm;

: 1.75% C
2
H
4
-
5.25%O
2
-Ar, p
5
= 11.5 atm;

: 1% C
2
H
4
-3%O
2
-Ar, p
5
= 11 atm;

: 1% C
2
H
4
-
3%O
2
-Ar, p
5
= 20 atm;

: 0.5% C
2
H
4
-1.5%O
2
-Ar, p
5
= 11 atm;

: 0.5% C
2
H
4
-
1.5%O
2
-Ar, p
5
= 21 atm). Lines denote the uncertainty bound in the data.

Active rate parameters for each target are determined from a sensitivity analysis for
that target. Table 3-2 shows the list of active pre-exponential factors as well as their
46
uncertainty factors. In addition to the experimental targets listed in Table 1, response
surfaces are calculated for a steady-state, constant-temperature (T =1400 K), constant-
pressure (p = 2 atm), ethylene-air (φ = 1) perfectly-stirred reactor (PSR) at a range of
mean residence times. The PSR is included here because of its use in approximating
practical combustors [32, 33] and is treated as a canonical, idealized combusting flow.
The response surfaces of flame speeds, species concentration in the burner stabilized
flame, and PSR are generated with an automated SAB method [34], and those of ignition
delay and species concentration gradient in flow reactor from the factorial design method
[2]. Simulations of freely propagating and burner-stabilized flames were performed
using the Sandia Premix code [35] with multicomponent transport. Ignition delay
simulations assumed adiabatic and constant volume conditions with the ignition criteria
made consistent with the experiments. Flow reactor simulations assumed constant
temperature and pressure assumptions. Steady-state PSR calculations used the Sandia
code [36] under the adiabatic or constant-temperature condition. Uncertainty
quantification and optimization were conducted using the methods described previously.
Minimization of the objective function used the constrained-optimization ZXMWD
subroutine of the International Math Subroutine Library (IMSL) [37].
47

Table 3-2. Matrix of active rate coefficients and their uncertainty factors. Active
parameters considered in a target response surface are marked with an X.
fls ign flw flf
1 1 1 2 2 2 3 3 3 3 1 4 4 4 6 6 1 2 3 1 1
No. Reaction f
i
a b c a b c a b c d a a b c a b a a a a b
1 H+O
2
↔O+OH 1.2 X X X X X X X X X X X X X X X X X X X X X
9 H+OH+M↔H
2
O+M 2 - - - - X - - X X - - -
12 H+O
2
(+M)↔HO
2
(+M) 1.2 - - - X - - X - - - - -
17 H+O
2
(+H
2
O)↔HO
2
(+H
2
O) 1.2 X - - X - - X - - - - -
20 H
2
+O
2
↔HO
2
+H 1.3 - - - - - - - - - X X X X X X - -
23 HO
2
+H↔2OH 2 X X X X X X X X X X X X X X -
25 OH+HO
2
↔H
2
O+O
2
2 - - - - - - - - - - - -
27 OH+HO
2
↔H
2
O+O
2
2 - - - - - - - - - - X X - -
38 CO+OH↔CO
2
+H 1.2 X X X X X X X X X - X X
42 HCO+H↔CO+H
2
2 X X X - X X - X X X X X X X X
46 HCO+M↔CO+H+M 4 X X X X X X X X X - X X X X X X X X X -
47 HCO+H
2
O↔CO+H+H2O 4 X X X X X X X X X - X X
48 HCO+O
2
↔CO+HO
2
2 X - - X - - X - - - X X X X X X X X X -
66 CH
2
+O
2
↔HCO+OH 2 - - - - - - - - - X - -
95 CH
3
+H(+M)↔CH
4
(+M) 2 - - X - X X X X X X X X X - -
99 CH
3
+OH↔CH
2
*+H
2
O 5 X X - X - - - - - - - -
103 CH
3
+HO
2
↔CH
3
O+OH 3 X X X X X X X X X X X X X X - -
111 2CH
3
(+M)↔C
2
H
6
(+M) 2 - - - - - - - - - - X X - -
112 2CH
3
↔H+C
2
H
5
5 - - - - - - - - - X X - -
161 C
2
H
2
(+M)↔H
2
CC(+M) 2 - - - - - - - - - - X X - -
162 C
2
H
3
(+M)↔C
2
H
2
+H(+M) 1.5 - X X - X X - X X - X X X X X
165 C
2
H
2
+O↔HCCO+H 1.5 - - - - - - - - - - X X X
184 H
2
CC+O
2
↔2HCO 3 - - - - - - - - - - X X - -
195 C
2
H
3
+H↔C
2
H
2
+H
2
3 - X X - - X - - - X X - -
201 C
2
H
3
+O
2
↔CH
2
CHO+O 4 - X X - X X - X X X X X X X X X X - X
202 C
2
H
3
+O
2
↔HCO+CH
2
O 4 - - - - - - - - - - X X X X - -
255 C
2
H
4
(+M)↔H
2
+H
2
CC(+M) 3 - - - - - - - - - - X X X - -
257 C
2
H
4
+H↔C
2
H
3
+H
2
2 - - - - - - - - - - X X -
258 C
2
H
4
+O↔C
2
H
3
+OH 3 X - - X - - X - - - X X X X X X X X - -
259 C
2
H
4
+O↔CH
3
+HCO 2 - - - - - - - - - X X X X X - -
261 C
2
H
4
+OH↔C
2
H
3
+H
2
O 2 X X X X X X X X X X X X X X X X X X - -

48
The procedure for constraining the reaction model is performed in two stages. First,
the posterior model x
0
*
is determined using the objective function in Eqs. (2-17) or (2-
18), along with a self-consistent data set using the consistency criterion in Eq. (2-21).
Then
*
α is calculated using the objective function in Eq. (2-30) or from the Cholesky
factorization in Eq. (2-28), as appropriate. The model calculated using the objective
function in Eq. (2-17), with uncertainty from Eq. (2-30), is termed posterior Model I.
Likewise, the model calculated using Eq.s (2-18) and (2-28) is termed posterior Model II.
They represent the two limiting cases for the form of the probability distribution of the
rate coefficients, namely, the uniform and normal distributions, respectively.

3.3 Results and Discussion
3.3.1 Uncertainty propagation
Before the question of uncertainty minimization can be addressed, the uncertainty in
the prior model must be evaluated and compared to that in the experimental data. We
shall perform these analyses assuming that the uncertainty in the rate parameters is
uniformly distributed. 2σ standard deviations for the prior model as well as all target
data are shown in Table 3-1. Additionally, a 2σ uncertainty band for C
2
H
4
-air laminar
flame speeds at p = 5 atm over a range of equivalence ratios is illustrated in the top panel
of Figure 3-2 using the data of Jomaas et al. as an example [20]. As seen, the underlying
uncertainty in the individual rate coefficients gives a model prediction uncertainty far
larger than the scatter in the flame speed data, and certainly larger than the estimated
uncertainty in any individual measurement. For the leanest flame tested, the 2σ
uncertainty is ±8 cm/s wide, which increases to ±10 cm/s for the stoichiometric flame and
about ±12 cm/s for the fuel-rich flames.
49

Figure 3-2. Variation of laminar flame speed with equivalence ratio. Left: prior model.
Right: posterior Model I. Symbols are the data from [20]. The shaded bands
indicate the 2σ model prediction uncertainty; color indicates the probability
density as indicated by the color bar, and the actual ±2σ curves are indicated
by the dashed lines.

The fact that model outputs are less certain for the fuel-rich flames than for the lean
flames is caused by a greater number of rate coefficients exhibiting above-noise
sensitivities. Likewise, the nominal values and 2σ uncertainty band computed for the
ignition delay times behind reflected shock waves with p
5
= 2 atm is shown in the top
panels of Figure 3-3; the experimental data sets shown are those reported by Baker and
Skinner [29], Brown and Thomas [23] and Saxena et al. [31]. Model predictions are less
certain at low temperatures, again due to a larger number of active rate parameters
impacting the prediction.

50
10
1
10
2
10
3
10
4
Ignition Delay Time, τ τ τ τ (μ μ μ μs)
p
5
~ 2 atm p
5
= 10 atm

10
1
10
2
10
3
0.4 0.5 0.6 0.7 0.8 0.9
Ignition Delay Time, τ τ τ τ (μ μ μ μs)
1000/T (K)
p
5
~ 2 atm
0.5 0.6 0.7 0.8 0.9
p
5
= 10 atm
1000/T (K)

Figure 3-3. Comparison of experimental (□: p
5
= 2.4 atm and 11 atm [31]; ○: p
5
= 3 atm
[29]; dotted lines: C
5
= 1.44±0.14×10
–5
mol/cm
3
[23]) and computed (solid
lines: mean; dashed lines: 2σ uncertainty). Top panels: prior model; bottom
panels: posterior Model I.

Several observations may be made from the above discussion. Equation (2-17)
assumes that all physically-realizable values for the rate coefficient given by, for example
[4], are equally probable. This means that any physically-realizable version of USC-
Mech II will produce a line falling within the uncertainty band presented in the top panel
51
of Fig. 3-2 and left panels of Fig. 3-3, and most of these models will not agree with the
experimental flame speed measurements or ignition delay times. Therefore, assuming [4]
to be the state of our kinetic knowledge, reaction models as-compiled should not be
considered as trustworthy, even if the nominal values of experiment and model
predictions are in close agreement.
Furthermore, the prior model uncertainty for flame speed being greater than the
experimental uncertainty means that these experiments contain information that can be
used for constraining the kinetic model, as will be discussed later. If the
obs
r
σ values are
large compared to the prior model prediction, the contribution of each experimental target
to Φ in Eq. (2-17) will be small. Therefore, the solution obtained from Eq. (2-17) is
essentially a random point within the uncertainty space. If the
obs
r
σ values are small,
each target will have a large contribution to Φ; the conclusion is that there will be a
preferred model within the uncertainty space, and therefore the experiments will have
taught us something. These observations underscore the purpose and utility for model
constraining, but the focus should be placed equally on minimizing the uncertainty of the
reaction model, as discussed in what follows.

3.3.2 Deterministic optimization and consistency analysis
Model constraining was performed in two stages, as discussed in Section 2. In the
first stage, the nominal value of the pre-exponential factor was optimized against the
experimental targets of Table 3-1, starting from the case with the uniformly distributed
rate uncertainties and using the consistency procedure discussed in Chapter 2. As was
noted, there are several reasons a target might have been removed. The multiple
measurements of fls 1c, the 1-atm C
2
H
4
/air laminar flame speed at φ = 1.2, is an example
52
of one member of the set being selected as a genuine best based on the consistency tests
(Eqs. 2-20 and 2-22). Figure 1-4 shows the state of posterior Model I by the F and W
tests. Consider the top panels of Figure 1-4, where the initial F test shows that targets fls
1c-1, fls 1c-3 and fls 2b-2 are inconsistent with all the other targets. The bottom panels
show the result of the W tests. It is clear that fls 1c-1 has the greatest effect on the
solution initially and is removed first. This is followed by fls 2b-2 and fls 1c-3, leading to
an entirely consistent set of which all F
r
< 1.
The multiple measurements of fls 1a, the 1-atm C
2
H
4
/air laminar flame speed at φ =
0.7, is an example where a choice is made between two equally-good targets. Here fls-
1a1 was removed from the final target set because it yields a larger F
r
value than fls-1a2.
It must be reiterated that the rejection of any one target does not mean that that particular
piece of data is wrong, but it does mean that the available data in that region of condition
space must be re-evaluated, along with the model prediction there.
Table 3-1 presents comparisons of the experimental data and the values predicted by
the prior and posterior models for uniformly-distributed x’s; this is also shown
graphically in the top panels of Figure 3-5. As expected, rate constraining brings model
predictions closer to the target values. For example, the mean absolute deviation of
computed flame speed from the experimental data is reduced from 2.3 cm/s to 0.7 cm/s
after optimization. Likewise, the mean, absolute deviation of ignition delay ln(τ, μs) is
reduced from 0.42 to 0.25.
53
0 1 2
1a-1
1a-2
1b-1
1b-2
1b-3
1c-1
1c-2
1c-3
2a
2b-1
2b-2
2c-1
2c-2
3a
3b
3c
3d
1a
1b
4a
4b
4c
6a
6b
2c
3a
4c
1a
1b
F
r
Flame speed
fls
Ignition delay
ign flw flf
0 1 2
3c
4c
2c
4c
F
r
0 1 2
1a-1
1a-2
1b-1
1b-2
1b-3
1c-1
1c-2
1c-3
2a
2b-1
2b-2
2c-1
2c-2
3a
3b
3c
3d
1a
1b
4a
4b
4c
6a
6b
2c
3a
4c
1a
1b
F
r

0 0.04
1a-1
1a-2
1b-1
1b-2
1b-3
1c-1
1c-2
1c-3
2a
2b-1
2b-2
2c-1
2c-2
3a
3b
3c
3d
1a
1b
4a
4b
4c
6a
6b
2c
3a
4c
1a
1b
W
r
Flame speed
fls
Ignition delay
ign flw flf
0 0.2
1a-1
1a-2
1b-1
1b-2
1b-3
1c-1
1c-2
1c-3
2a
2b-1
2b-2
2c-1
2c-2
3a
3b
3c
3d
1a
1b
4a
4b
4c
6a
6b
2c
3a
4c
1a
1b
W
r
0 0.25
1a-1
1a-2
1b-1
1b-2
1b-3
1c-1
1c-2
1c-3
2a
2b-1
2b-2
2c-1
2c-2
3a
3b
3c
3d
1a
1b
4a
4b
4c
6a
6b
2c
3a
4c
1a
1b
W
r

Figure 3-4. Consistency analyses for uniformly-distributed rate parameters, leading to
posterior Model I.
54
30
40
50
60
70
30 40 50 60 70
Prior
Posterior (model I)
Calculated (S
u
o
, cm/s)
Experimental (S
u
o
, cm/s)
3
4
5
6
7
8
3 4 5 6 7 8
Calculated (log τ τ τ τ
ign
, μ μ μ μs)
Experimental (log τ τ τ τ
ign
, μ μ μ μs)

30
40
50
60
70
30 40 50 60 70
Prior
Posterior (model II)
Calculated (S
u
o
, cm/s)
Experimental (S
u
o
, cm/s)
3
4
5
6
7
8
3 4 5 6 7 8
Calculated (log τ τ τ τ
ign
, μ μ μ μs)
Experimental (log τ τ τ τ
ign
, μ μ μ μs)

Figure 3-5. Comparison of experimental values and 2σ uncertainties with calculated
values before and after model constraining. Top panels: uniformly-
distributed rate parameters; bottom panels: normally-distributed rate
parameters.

55
3.3.3 Influence of probability distributions of rate coefficient uncertainty
So far, all discussion has been made by assuming that the rate parameter uncertainty
is uniformly distributed (Model I). It can be seen in Table 3-3 that a large number of
constrained rate coefficients reach the uncertainty bound (x = -1 or +1) for this
distribution. As noted in previous studies such as [2], there is no global minimum to Eq.
(2-17) within the uncertainty space; nearly half of the rate parameters lie on the bounds of
the space. This result is definitely unsatisfactory, as it illustrates the fact that the
available combustion data set is either too small or too inaccurate, or both. This
observation reinforces the notion that rate parameter constraining alone based on
combustion data, as is exercised here, is unlikely to produce a model that is both
predictive and physically justifiable. Attaining such a model must require both careful
experimental and/or theoretical analyses of the underlying potential energy surfaces and
rate parameters of individual reactions.
To shed light on the problem of combustion data sample size, we carry out additional
tests by assuming that the rate parameter uncertainty is normally distributed and using
Eq. (2-18) in the rate coefficient constraining (Model II). The consistency analysis tests
have virtually the same final set for Models I and II. The results for the rate coefficients
are compared in Table 3-3. As expected, all of the rate parameters now fall well within
their uncertainty bounds. A comparison of the posterior models shows that there is only
marginal difference in the predictions of target values made by Models I and II (cf, Fig.
3-5). Clearly, most of the target values have small impacts on those rate parameters that
reached their respective bounds in the prior model, at least compared to the sensitivity to
the major chain branching reaction

H + O
2
↔ OH + O. (R1)

56
As a result, pushing the rate values to their uncertainty bounds results in only small
improvements in the prediction of many experimental targets. However, if Eq. (2-18) is
used, the improvement to the model prediction is offset by the cost in altering the rate
coefficient from its nominal value.

Table 3-3. Comparison of rate parameter values for uniformly-distributed (posterior
Model I) and normally-distributed (posterior Model II) rate parameters.

Model I Model II
No. Reaction x
i
*
A
i
*
/A
i,0
x
i
*
A
i
*
/A
i,0
σ
*
i

1 H+O
2
↔O+OH –0.87 0.85 0.19 1.04 0.43
9 H+OH+M↔H
2
O+M –1 0.5 0.01 1.00 0.46
12 H+O
2
(+M)↔HO
2
(+M) 1 1.2 0.14 1.03 0.50
17 H+O
2
(+H
2
O)↔HO
2
(+H
2
O) 0.4 1.07 0.08 1.02 0.49
20 H
2
+O
2
↔HO
2
+H –0.96 0.78 0.03 1.01 0.50
23 HO
2
+H↔2OH –1 0.5 –0.22 0.86 0.46
25 OH+HO
2
↔H
2
O+O
2
a
1 2 0.05 1.04 0.50
27 OH+HO
2
↔H
2
O+O
2
a
0.29 1.22 –0.10 0.93 0.49
38 CO+OH↔CO
2
+H –1 0.83 –0.32 0.94 0.46
42 HCO+H↔CO+H
2
–0.27 0.83 –0.47 0.72 0.43
46 HCO+M↔CO+H+M 0.46 1.88 0.07 1.10 0.35
47 HCO+H
2
O↔CO+H+H2O –0.50 0.5 –0.02 0.98 0.46
48 HCO+O
2
↔CO+HO
2
–0.98 0.51 0.47 1.39 0.44
66 CH
2
+O
2
↔HCO+OH 1 2 0.12 1.08 0.49
95 CH
3
+H(+M)↔CH
4
(+M) –0.08 0.95 0.17 1.13 0.39
99 CH
3
+OH↔CH
2
*+H
2
O –0.32 0.6 –0.10 0.85 0.45
103 CH
3
+HO
2
↔CH
3
O+OH –1 0.33 –0.20 0.80 0.45
111 2CH
3
(+M)↔C
2
H
6
(+M) 1 2 –0.04 0.98 0.50
112 2CH
3
↔H+C
2
H
5
–1 0.2 0.01 1.02 0.47
161 C
2
H
2
(+M)↔H
2
CC(+M) 1 2 0.02 1.01 0.49
162 C
2
H
3
(+M)↔C
2
H
2
+H(+M) –1 0.67 0.07 1.03 0.49
165 C
2
H
2
+O↔HCCO+H 1 1.5 0.11 1.05 0.49
184 H
2
CC+O
2
↔2HCO –1 0.33 0.00 1.00 0.48
195 C
2
H
3
+H↔C
2
H
2
+H
2
–0.78 0.42 –0.35 0.68 0.46
201 C
2
H
3
+O
2
↔CH
2
CHO+O 0.63 2.4 0.18 1.29 0.44
202 C
2
H
3
+O
2
↔HCO+CH
2
O 1 4 0.42 1.8 0.47
255 C
2
H
4
(+M)↔H
2
+H
2
CC(+M) 0.51 1.76 –0.07 0.93 0.46
257 C
2
H
4
+H↔C
2
H
3
+H
2
0.55 1.46 0.13 1.09 0.50
258 C
2
H
4
+O↔C
2
H
3
+OH 0.49 1.72 –0.15 0.85 0.47
259 C
2
H
4
+O↔CH
3
+HCO 1 2 0.15 1.11 0.48
261 C
2
H
4
+OH↔C
2
H
3
+H
2
O 1 2 0.12 1.08 0.48
57

3.3.4 Uncertainty minimization
3.3.4.1 Reaction rate parameters
Uncertainty minimization was carried out using Eq. (2-28) for normally-distributed
rate parameters and Eq. (2-30) for the case of uniformly-distributed rate parameters. We
first examine the covariance matrix, which is indicative of the degree of couplings among
rate coefficients in relation to the constraining experimental targets. As will be seen, the
discussion here points to the fundamental limitation of MUM-PCE or any rate-
constraining, inverse problem methods in handling this complex chemical kinetic
problem.
For uniformly distributed uncertainties, there are 16 experimental targets whose
uncertainty was considered, while there are 31 active rate parameters. We are very deep
in the singular, mathematically ill defined region, and as such, the standard deviations of
all constrained rate coefficients are expected to extend to their entire range of uncertainty,
i.e., σ* = ½. This would imply that the experimental targets do not impose a constraint
on the uncertainty space of the reaction rates, at least when using Eq. (30). This is,
however, not the case. The covariance Σ , presented here as
ij ij
K = Σ , is shown in
Figure 3-6 for Model I. The transformation allows us to compare covariances to standard
deviations, i.e., the squared root of the diagonal elements shown in the figures. It can be
seen by an examination of the figure that off-diagonal elements of the covariance matrix
can be fairly large, indicating that the rate parameters are coupled as the model becomes
constrained by the experimental targets. The reduction in total uncertainty space comes
from this coupling. It is also this coupling that yields a feasible parameter space in which
58
the experimental targets can be reconciled by the posterior model—an issue that will be
discussed further later.

K
ij

Figure 3-6. Covariance matrix of Model I, expressed as
ij ij
K = Σ (see text).

The covariance matrix is sparse, though certain clusters of rate-coefficient coupling
jump out as being particularly interesting. For instance, the rate coefficient of the H-
radical termination reaction

H + O
2
(+M) ↔ OH + O (+M) (R12)
59

is strongly coupled to many other reactions that would impact the H atom concentration.
They include,
H
2
+ O
2
↔ H + HO
2
(R20)
HO
2
+ H ↔ 2OH (R23)
HCO + M ↔ H + CO + M (R46&47)
C
2
H
3
+ H ↔ C
2
H
2
+ H
2
. (R195)
Another example is the coupling of methyl reactions,
CH
3
+ H (+M) ↔ CH
4
(+M) (R95)
CH
3
+ HO
2
↔ CH
3
O + OH . (R103)
Likewise, the two dominant vinyl oxidation reactions are tightly coupled also,
C
2
H
3
+ O
2
↔ CH
2
CHO + O (R201)
C
2
H
3
+ O
2
↔ HCO + CH
2
O . (R202)
These “piece-wise” couplings, in fact, underscore the limitation of the rate constraining
method. Many of the couplings are similar to what would happen in individual rate
coefficient measurements under a degree of chemical isolation (e.g., R95 and R103) or in
a reaction rate theory analysis of the branching ratios of a particular reaction (e.g., R201
and R202).
In the case of normally distributed parameter uncertainty (Model II), each rate
parameter evaluation is treated effectively as an experimental target. So there are
effectively 47 experimental targets, and Σ is no longer singular. Likewise the standard
deviations of the rate coefficient are somewhat smaller (cf. Table 3), ranging from about
0.35 for the most highly-constrained rate coefficient up to ½ for the least constrained.
60
The most highly-constrained rates are those responsible for the free-radical chain
branching and termination, including reactions R1, R46, and R95. The covariances are
presented in Figure 3-7. They are much smaller because the normal distributions pin the
rate coefficient values to an extent in the least squares procedure. The impact from the
global combustion experiments on individual rate coupling is essentially weakened.

K
ij

Figure 3-7. Covariance matrix of Model II, expressed as
ij ij
K = Σ (see text).

61
Because Σ defines a multivariate normal distribution, it is possible to calculate a
joint PDF for any set of rate coefficients. The joint PDF of all coefficients is a statistical
representation of the “feasible set” described by Frenklach et al. [38]. Contours of these
functions are shown in Figure 3-8 for Model I and Figure 3-9 for Model II. The rate
parameters shown as examples are the pre-exponential factors for R95, R103 and

CH
3
+ OH ↔ CH
2
* + H
2
O . (R99)
These parameters were chosen because they show quite strong coupling as seen in Figure
3-6. In Figure 3-8, the uncertainty space of the prior model is shown by the cube, while
in Figure 3-9 it is the sphere. In both cases, the ellipsoids show the uncertainty space
after minimization. It can be seen that the standard deviation of any particular rate
coefficient is not reduced by the uncertainty minimization, but, due to the coupling
among the rate parameters, the total uncertainty space is reduced in size, and in some
cases, the reduction in the uncertainty space can be significant. It should be noted that
there are significant differences between the uncertainty spaces depending on the
assumptions made about the uncertainty distribution of the rate coefficients. These
differences are caused by the fact that, when Eqs. (2-17) and (2-18) are solved, the most
probable rate coefficient set differs, and thus the joint PDFs are therefore subject to
assumptions made about the uncertainty distribution of the rate coefficients. The
fundamental issue again goes back to the ill-defined mathematical problem being dealt
with here. Namely, with the current suite of experimental techniques to probe the
fundamental combustion properties of a fuel, we would always be facing with the
problem of a lesser number of known quantities than unknowns.
To summarize, MUM-PCE or any other rate constraining methods cannot reduce the
uncertainty of individual rate coefficients effectively, but, as a mathematical tool, it can
62
provide useful information about rate coupling in a kinetic model when it is used to
predict a set of combustion properties. Moreover, the method can generate a joint PDF
for the rate coefficients which would yield reduced uncertainties in predicted combustion
property values, as will be discussed in what follows.

Figure 3-8. Contours of joint probability density function for reactions R95, R99, and
R103 in the prior model (cube, squares) and the posterior Model I (ellipses).
63

Figure 3-9. Contours of joint probability density function for reactions R95, R99, and
R103 in the prior model (sphere, circles) and the posterior Model II
(ellipses).

3.3.4.2 Prediction of experimental targets
In all cases, the uncertainty in the combustion properties predicted by the posterior
model has been greatly reduced, a result that would not have been attained without the
MUM-PCE analysis. Table 1 shows the uncertainty of the optimized model with respect
to the flame speed and ignition delay targets for uniformly-distributed x’s. Table 3-4
64
compares the standard deviations predicted by posterior Models I and II. Additionally,
the standard deviations produced by the PC expansions are compared with Monte Carlo
sampling, and the results are very close, as is expected. The uncertainty results are also
shown graphically in Figs 3-2, 3-3, and 3-5, comparing predictions by the prior and
posterior models. The uncertainty in the model prediction after optimization is often
significantly less that the experimentally-measured uncertainty.

Table 3-4. Comparison of standard deviations for the prior and posterior models. The
MC column is calculated using a Monte-Carlo sampling of Eq. (2-15),
while the PCE column is a direct solution of Eq. (2-16).
Prior model Posterior Model I Posterior Model II
MC PCE MC PCE MC PCE
fls_1a 3.8 3.7 0.67 0.67 0.58 0.58
fls_1b 5.5 5.5 0.61 0.61 0.56 0.56
fls_1c 6.1 6.1 0.77 0.77 0.71 0.7
fls_2a 3.3 3.3 0.61 0.61 0.57 0.56
fls_2b 5.1 5 0.52 0.52 0.46 0.46
fls_2c 5.7 5.7 0.56 0.56 0.54 0.54
fls_3a 2.8 2.8 0.68 0.68 0.54 0.54
fls_3b 4.5 4.5 0.64 0.64 0.52 0.53
fls_3c 5.2 5.1 0.68 0.68 0.61 0.62
fls_3d 3.5 3.5 0.94 0.94 0.82 0.83

ign_1a 0.45 0.45 0.24 0.24 0.16 0.16
ign_1b 0.17 0.17 0.11 0.11 0.07 0.07
ign_2a 0.14 0.14 0.11 0.11 0.07 0.07
ign_2b 0.19 0.19 0.11 0.11 0.08 0.09
ign_2c 0.32 0.32 0.15 0.15 0.14 0.14
ign_3a 0.16 0.16 0.17 0.17 0.07 0.07
ign_3b 0.19 0.19 0.17 0.17 0.08 0.08

65
3.3.5 Uncertainty propagation into application
We are interested here in learning how the posterior models improve prediction
uncertainty when simulating a combustion problem outside the realm of the target set.
We use a PSR as an example for a canonical, idealized combustion problem. OH and
H
2
O mole fractions with respect to residence time for a steady-state, constant-temperature
(T =1400 K), constant-pressure (P = 2 atm), ethylene-air (φ = 1.0) PSR are shown in
Figure 3-10, along with associated uncertainties predicted by the prior model and the
posterior Model I. As can be seen, three distinct regions of the residence time can be
identified based on the behavior of the reactor. At short residence times, unburned fuel
leaves the reactor as fast as it enters; the chemistry is largely frozen. As the residence
time is increased, corresponding to a decreased flow rate, a critical point is reached where
the mixture in the reactor suddenly ignites, as evidenced by the hysteresis in the OH and
H
2
O mole fractions. Eventually, the residence time becomes sufficiently long that the
mixture has time to reach its chemical equilibrium state. Because the equilibrium
concentration does not depend on the reaction rate coefficients, the uncertainty drops to 0
in this region. On the other hand, the uncertainty of the model reaches a maximum in the
vicinity of the ignition point, where the OH uncertainty predicted by the prior model is
about one order of magnitude in each direction of the mole fraction axis. Furthermore,
the residence time of ignition varies by nearly the same amount. As can be seen from the
figure, the uncertainty in the calculated mole fractions is significantly reduced in the
posterior model. Most pronounced is the uncertainty in the residence time of ignition,
shown by the rectangles. Clearly the posterior model presents a significant reduction in
its prediction uncertainty as compared to the prior model. Overall, the OH uncertainty is
reduced from a factor of 3-4 to about a factor of 2. MUM-PCE therefore allows us to
66
characterize the reactor behavior much more reliably than compiling a model from the
literature would have.

10
-10
10
-8
10
-6
10
-4
10
-7
10
-6
10
-5
10
-4
10
-3
10
-2
OH Mole Fraction
Residence time, τ τ τ τ (s)
10
-11
10
-9
10
-7
10
-5
10
-3
10
-1
10
-7
10
-6
10
-5
10
-4
10
-3
10
-2
H
2
O Mole Fraction
Residence time, τ τ τ τ (s)

Figure 3-10. Variation of OH and H
2
O mole fraction with mean residence time in a
simulated PSR. Lines are the model calculation and symbols are the
uncertainty calculated using Monte Carlo sampling of the uncertainty space.
Dashed line and open symbols: prior model; solid line and solid symbols:
posterior Model I.

3.4 Conclusions
The Method of Uncertainty Minimization by Polynomial Chaos Expansion (MUM-
PCE) has been introduced to quantify the uncertainty in a detailed kinetic model. This
uncertainty can be propagated into the predictions of that model, thereby allowing a
direct comparison between predictions and experimental measurements. Furthermore,
the methodology allows us to use the information present in experimental measurements
to quantitatively constrain the model, thereby expressing the kinetic uncertainty in terms
of the uncertainty in the measurements. The method includes an algorithm for
67
determining the consistency of experimental targets, thereby allowing us to obtain an
optimized model based on a self-consistent experimental data set.
When using an experiment as a target for optimization, we show it to be a necessary
condition that the experimental uncertainty be less than the model prediction uncertainty.
A rigorous uncertainty analysis is therefore crucial to obtaining a physically realistic
model, since an uncertainty that is unreasonably small may result in an experiment being
removed as inconsistent, while one that is too large does not provide any information.
We note that the optimization procedure used here suffers from some of the same
limitations as earlier studies, in that it is difficult or perhaps impossible to select a unique
chemical model that can best reproduce the experimental measurements. This may be
because there is simply insufficient data or because experiments do not provide
independent constraints in the reaction rate parameter space. As a means of addressing
this problem, we have suggested using reaction rate estimates or evaluations themselves
as experimental data for this purpose.
The last point is that MUM-PCE or other similar methods cannot reduce the
uncertainty of individual rate coefficients effectively, but as a mathematical tool, it can
provide useful information about various couplings of rate coefficients in the prior model
for a given set of known combustion properties of a fuel. More importantly, the joint
PDF for the rate coefficients constrained by known combustion properties can yield
reduced prediction uncertainties both inside and outside of the experimental conditions
under which these properties were studied.

68
3.5 Chapter 3 Endnotes

1. M. Frenklach, Combust. Flame 58 (1984) 69-72.

2. M. Frenklach, H. Wang, M. J. Rabinowitz, Prog. Energ. Combust. Sci. 18 (1992)
47-73.

3. D. L. Baulch, C. J. Cobos, R. A. Cox, C. Esser, P. Frank, T. Just, J. A. Kerr, M. J.
Pilling, J. Troe, R. W. Walker, J. Warnatz, J. Phys. Chem. Ref. Dat. 21 (1992)
411-734.

4. D. L. Baulch, C. T. Bowman, C. J. Cobos, R. A. Cox, T. Just, J. A. Kerr, M. J.
Pilling, D. Stocker, J. Troe, W. Tsang, R. W. Walker, J. Warnatz, J. Phys. Chem.
Ref. Dat. 34 (2005) 757-1397.

5. Z. Hong, D. F. Davidson, E. A. Barbour, R. K. Hanson, Proc. Combust. Inst. 33
(2010) doi:10.1016/j.proci.2010.05.101.

6. J. Li, Z. W. Zhao, A. Kazakov, F. L. Dryer, Int. J. Chem. Kinet. 36 (2004) 566-
575.

7. M. O'Conaire, H. J. Curran, J. M. Simmie, W. J. Pitz, C. K. Westbrook, Int. J.
Chem. Kinet. 36 (2004) 603-622.

8. S. G. Davis, A. V. Joshi, H. Wang, F. N. Egolfopoulos, Proc. Combust. Inst. 30
(2005) 1283-1292.

9. P. Saxena, F. A. Williams, Combust. Flame 145 (2006) 316-323.

10. A. A. Konnov, Combust. Flame 152 (2008) 507-528.

11. G. P. Smith, D. M. Golden, M. Frenklach, B. Eiteener, M. Goldenberg, C. T.
Bowman, R. K. Hanson, W. C. Gardiner, V. V. Lissianski, Z. W. Qin, GRI-Mech
3.0. http://www.me.berkeley.edu/gri_mech/; 2000

12. Z. Qin, V. Lissianski, H. Yang, W. C. Gardiner, Jr., S. G. Davis, H. Wang, Proc.
Combust. Inst. 28 (2000) 1663-1669.

13. D. A. Sheen, X. You, H. Wang, T. Løvås, Proc. Combust. Inst. 32 (2009) 535-
542.

14. D. A. Sheen, H. Wang, Combust. Flame (2010) submitted.

69
15. B. D. Phenix, J. L. Dinaro, M. A. Tatang, J. W. Tester, J. B. Howard, G. J.
McRae, Combust. Flame 112 (1998) 132-146.

16. M. T. Reagan, H. N. Najm, R. G. Ghanem, O. M. Knio, Combust. Flame 132
(2003) 545-555.

17. M. T. Reagan, H. N. Najm, B. J. Debusschere, O. P. Le Maitre, M. Knio, R. G.
Ghanem, Combust. Theory Modelling 8 (2004) 607-632.

18. M. T. Reagan, H. N. Najm, P. P. Pebay, O. M. Knio, R. G. Ghanem, Int. J. Chem.
Kinet. 37 (2005) 368-382.

19. H. Wang, X. You, A. V. Joshi, S. G. Davis, A. Laskin, F. N. Egolfopoulos, C. K.
Law, USC Mech Version II. High-Temperature Combustion Reaction Model of
H
2
/CO/C
1
-C
4
Compounds. http://ignis.usc.edu/Mechanisms/USC-
Mech%20II/USC_Mech%20II.htm; 2007

20. G. Jomaas, X. L. Zheng, D. L. Zhu, C. K. Law, Proc. Combust. Inst. 30 (2005)
193-200.

21. F. N. Egolfopoulos, D. L. Zhu, C. K. Law, Proc. Combust. Inst. 23 (1991) 471-
478.

22. M. I. Hassan, K. T. Aung, O. C. Kwon, G. M. Faeth, J. Propul. Power 14 (1998)
479-488.

23. C. J. Brown, G. O. Thomas, Combust. Flame 117 (1999) 861-870.

24. Y. Hidaka, T. Nishimori, K. Sato, Y. Henmi, R. Okuda, K. Inami, T. Higashihara,
Combust. Flame 117 (1999) 755-776.

25. J. B. Homer, G. B. Kistiakowsky, J. Chem. Phys. 47 (1967) 5290-5295.

26. A. Laskin, H. Wang, 2000. Unpublished data

27. A. Bhargava, P. R. Westmoreland, Combust. Flame 113 (1998) 333-347.

28. A. Lifshitz, in: Handbook of Shock Waves III, G. Ben-Dor; O. Igra; T. Elperin; A.
Lifshitz, (Eds.) Academic Press: San Diego, 2001; pp 211-256.

29. J. A. Baker, G. B. Skinner, Combust. Flame 19 (1972) 347-350.

30. Y. Hidaka, W. C. Gardiner, Jr., C. S. Eubank, Journal of Molecular Science China
2 (1982) 141-153.
70

31. S. Saxena, M. S. P. Kahandawala, S. S. Sidhu, Combust. Flame (2010) in press
(doi:10.1016/j.combustflame.2010.10.011).

32. S. Niksa, G. S. Liu, Fuel 81 (2002) 2371-2385.

33. M. Falcitelli, S. Pasini, N. Rossi, L. Tognotti, App. Therm. Engng. 22 (2002) 971-
979.

34. S. G. Davis, A. B. Mhadeshwar, D. G. Vlachos, H. Wang, Int. J. Chem. Kinet. 36
(2004) 94-106.

35. R. J. Kee, J. F. Grcar, M. D. Smooke, J. A. Miller, A FORTRAN Program for
Modeling Steady Laminar One-Dimensional Premixed Flames, Sandia National
Laboratories: Albequerque, NM, 1986; SAND85-8240.

36. P. Glarborg, R. J. Kee, J. F. Grcar, J. A. Miller, PSR: A Fortran Program for
Modeling Well-Stirred Reactors, Sandia National Laboratories: Albequerque,
NM, 1986; SAND86-8209.

37. IMSL Library Reference Manual, 9th Edition; International Math and Science
Libraries: Houston, TX, 1982.

38. M. Frenklach, A. Packard, P. Seiler, R. Feeley, Int. J. Chem. Kinet. 36 (2004) 57-
66.

71
Chapter 4 Quantitative Analysis of Hierarchical Strategies of Building
Combustion Reaction Models

4.1 Introduction
To be predictive, a chemical kinetic model must consider numerous elementary
reactions critical to a set of combustion phenomena. The large number of reactions and
thus rate parameters, all of which are uncertain to an extent, suggests that an as-compiled
model cannot be expected to predict any particular combustion data, especially if the data
span a wide range of combustion phenomena [1-4]. Rather, this kinetic model may have
to undergo some optimization before it becomes predictive [1]. The rate parameter
uncertainties also mean that an accurate prediction of a particular piece of the data is of
limited value if the uncertainty of this prediction is not stated [4]. For example, a model
may be accurate at reproducing the laminar flame speed, but it may not be precise owing
to the uncertainty of model predictions. Likewise, fundamental combustion properties
have their own associated uncertainties. The impact of these uncertainties on model
development often went unquantified until recently [5-7].
To address these problems, we proposed the Method of Uncertainty Propagation
using Polynomial Chaos Expansions (MUM-PCE) [4]. The method extends the approach
developed by Najm, Ghanem and coworkers [5-7] by combining it with the response
surface methodology and optimization [1-3, 8, 9]. MUM-PCE is capable of propagating
rate parameter uncertainties into uncertainty of model predictions. It also enables us to
address the impact of uncertainties in fundamental combustion data on the precision of a
kinetic model. In the current paper, we use this method to address yet another question
critical to detailed reaction model development, namely the hierarchical strategy [10].
72
It has been a long standing practice to build kinetic models of hydrocarbon
combustion hierarchically. In this strategy, an H
2
/CO and small hydrocarbon (e.g.,
methane, ethylene etc) oxidation model is used as a kinetic foundation, and oxidation and
pyrolysis pathways for higher hydrocarbons are added as needed [10]. In this way, a
model may be developed consistently and, over the long run, it should lead to a self-
consistent set of rate and thermodynamic parameters that can be justified at the
fundamental level and is also capable of reconciling observations made for a wide range
of fuels and combustion conditions. Yet, at a quantitative level, it was never clear
whether rate parameters critical to all or a majority of combustion phenomena would
become accurate enough to ensure model convergence.
Consider the work of Qin et al. [3]. A kinetic model of propane combustion was
proposed by expanding its kinetic foundation, the GRI-Mech 3.0 [9], which had been
optimized against a large set of data for CH
4
oxidation. As a test for the hierarchical
model principle, a set of ignition delays and laminar flame speeds for several C
2
-C
3
fuels
was added to the original optimization data set. It was found that the model predictions
could not be reconciled with the C
2
-C
3
combustion data without relaxing the already
optimized rate parameters in the GRI-Mech, many of which are active for all C
2
-C
3
fuels
considered. Consequently, a sequential optimization approach does not ensure the
accuracy of a submodel for use as a kinetic foundation of higher hydrocarbons. That is, a
hierarchical kinetic model constrained by combustion data in a piecewise fashion does
not guarantee its predictiveness as it is expanded to include the chemistry of higher
hydrocarbons; in this case the optimized rate parameters for the oxidation of the smaller
fuel molecules do not have physical meaning. Therefore, a joint optimization that
considers a range of fuels must be taken at some point. The question that follows is
whether this approach should be exercised for all fuels of interest, and whether the impact
73
of a particular set of combustion experiments on the overall accuracy and precision of a
kinetic model can be quantified.
The purpose of this chapter is to address the above questions. MUM-PCE is
extended by defining an information index, which quantifies how much information a
combustion target provides about the model prediction of itself and other targets. The
paper uses the combustion of H
2
/CO, C
2
H
4
, and C
3
H
8
as examples, illustrates strategies
of hierarchical model optimization and uncertainty minimization, and demonstrates that a
basic understanding of the hierarchical assembly of critical combustion data set has an
important role in the process of kinetic model development and optimization.

4.2 Methods
We define the information index for a particular target within the data set. This
index, defined as,

ln
ln
j
i j obs
i
d
K
d
σ
σ
→
= . (4-1)
predicts how the precision of the i
th
target impacts the prediction uncertainty of the j
th

target. Here
j
σ is evaluated after least squares minimization. An important aspect of
i j
K
→
is that a target is considered to be strong if
i j
K
→
→ 1. For
i j
K
→
<< 1, a target
provides less than the maximal information.
We use the combustion of H
2
/CO, C
2
H
4
, C
3
H
8
n-C
5
H
12
, n-C
7
H
16
, and n-C
8
H
18
as our
test cases. The as compiled, prior model is the 194-species, 1459-reaction JetSurF 1.0
[11]. This model is in turn based on the 111-species, 784-reaction USC-Mech II [12],
developed for H
2
/CO/C
1
-C
4
hydrocarbon combustion [3, 11, 16-21]. It should be noted
that the H
2
/CO submodel has been previously optimized in [8] and subsequently updated
[21-23]. To compare the effect that each data set has on the model, those previously
74
optimized rates were reset to their trial values, and the updates are retained. Additionally,
the rate parameter for the OH + HO
2
↔ H
2
O + O
2
reaction was revised using the
expression of Baulch et al. [13] in light of the recent work of Hanson, Michael and
coworkers [14, 15]. The complete set of chemical rate parameters is presented in
Appendix C.
We consider 105 optimization targets, including laminar flame speed, shock-tube
ignition delay times, detailed species profiles in premixed flat flames and flow reactors.
Experiments for H
2
/CO and C
2
H
4
are shown in Table 4-1, and C
3
, C
5
, and C
7

hydrocarbons in Table 4-2. Conditions for n-C
8
H
18
are shown in Table 4-3. The list of
active rate coefficients, with uncertainty factors, is presented in Table A-1, with active
parameter matrices in Tables A-2 through A-4. The response surfaces of flame speeds
and species concentration in burner stabilized flames are generated with an automated
SAB method [16] and those of ignition delay and species concentration gradient in flow
reactor are from the factorial design method [1]. Premixed flame simulations were
performed using the Sandia Premix [17] with multicomponent transport, with updates
taken from [18]. Minimization of the objective function used the constrained-
optimization ZXMWD subroutine of the International Math Subroutine Library. The
optimization is performed in two steps. The optimized model x
0
*
is determined using the
objective function, equation (2-18). Then,
*
α is calculated from the Cholesky
factorization of Σ . K values are then calculated for each target pair by finite-difference.
Analysis is broken into two parts. First the relationship between the H
2
/CO and C
2
H
4
combustion data is examined, in order to present how two basis fuels can be combined to
form a quantitative chemical foundation. We consider three constrained, posterior
models. Model I considers the H
2
/CO targets only; Model II considers the C
2
H
4
targets
only; and Model III considers all 57 experimental targets for the two fuels. We then
75
proceed to extend Model III by progressively adding additional constraints from the C
3

combustion data, followed by n-C
5
H
12
and finally n-C
7
H
16
data; this final model is Model
IV. Rate parameters for all posterior models are presented in Table A-1. Constraint
from n-C
8
H
18
experiments is not considered; predictions are calculated to show the
dependence on the smaller hydrocarbons and therefore the kinetic foundation.
76

Table 4-1 List of targets for H
2
/CO and C
2
H
4
mixtures.
Index
φ
p (atm) Refs. Index Fuel (%) O
2
(%) T
5
(K) p
5
(atm) Refs.
C
2
H
4
-air flame speeds (T
0
=298 K) C
2
H
4
-O
2
-Ar ignition delay times
1 0.7 1 [19, 20] 27
f
1 3 1292 1.3 [21]
2 1 1 [19, 22] 28
g
0.5 1.5 1643 2.3 [23]
3 1.2 1 [19, 20, 22] 29
g
0.5 1.5 1443 1.9 [23]
4 0.7 2 [20] 30
g
0.5 1.5 1267 1.5 [23]
5 1 2 [20, 22] 31
h
0.5 1 1745 0.7 [24]
6 1.2 2 [20, 22] 32
h
0.5 1 2150 0.7 [24]
7 0.7 5 [20] H
2
/CO-O
2
-Ar ignition delay times
8 1 5 [20] 33
a,i
6.67 3.33 1051 1.7 [25]
9 1.2 5 [20] 34
a,i
6.67 3.33 1312 2 [25]
10 1.6 5 [20] 35
a,j
20 10 1033 0.5 [26]
H
2
/CO-air flame speeds (T
0
=298 K) 36
a,j
20 10 1510 0.5 [26]
11
a
1 1 [27-31] 37
a,j
0.5 0.25 1754 33 [32]
12
a
3 1 [27, 28, 31] 38
a,j
2 1 1189 33 [32]
13
b
1.148 1 [33] 39
a,j
2 1 1300 33 [32]
14
b
3.895 1 [33] 40
a,j
0.1 0.05 1524 64 [32]
15
c
1 1 [33] 41
l,f
12.22 1 2160 1.5 [34]
16
c
3.895 1 [33] 42
l,k
12.22 1 2160 1.5 [34]
17
d
0.49 1 [35] 43
l,k
12.22 1 2625 1.9 [34]
18
E
0.49 1 [35] 44
l,g
12.22 1 2625 1.9 [34]
H
2
-(12%O
2
in He) flame speeds (T
0
=298 K) C
2
H
4
-O
2
-N
2
Flow reactor species profiles
19 1 1 [27] 47
p
0.149 0.437 961 1 [36]
20 2.25 1 [27] 46
q
0.144 0.788 1037 1 [36]
H
2
-(8%O
2
in He) flame speeds 45
r
0.15 0.226 1002 1 [36]
21 1 15 [27] H
2
/CO-O
2
-N
2
Flow reactor species profiles
22 1.74 15 [27] 48
a,q
1.18 0.61 914 15.7 [37]
19%C
2
H
4
-31%O
2
-Ar flat flame (T
0
= 574 K) 49
a,q
1.01 0.52 935 6 [37]
23
m
1.84 0.026 [38] 50
a,q
0.5 0.5 880 0.3 [37]
24
n
1.84 0.026 [38] 51
a,s
0.95 0.49 934 3 [37]
39.7%H
2
-10.3%O
2
-Ar flat flame (T
0
= 432 K) 52
a,t
0.95 0.49 934 3 [37]
25
m
1.93 0.047 [40] 53
u,p
1.01 0.517 1038 1 [39]
26
o
1.93 0.047 [40] 54
u,s
1 0.494 1038 3.5 [39]
55
u,v
1 0.494 1038 3.5 [39]
56
u,q
1 0.482 1038 6.5 [39]
a
H
2
only.
b
H
2
:CO=95:5.
c
H
2
:CO=1:1.
d
H
2
:CO=54:46.
e
H
2
:CO=17:83.
f
Onset of CH emission.
g

Onset of CO
2
emission.
h
10% of maximum [CO] + [CO
2
].
i
onset of p rise.
j
onset of OH omission.
k

Maximum [O].
l
H
2
:CO=0.05:12.17.
m
maximum H mole fraction.
n
maximum OH mole fraction.
o

maximum O mole fraction.
p
0.9-to-0.7 O
2
fractional conversion rate.
q
0.6-to-0.4 fuel fractional
conversion rate.
r
0.9-to-0.6 fuel fractional conversion rate.
s
0.95-to-0.75 fuel fractional conversion
rate.
t
0.9-to-0.4 fuel conversion time.
u
Moist CO oxidation with 0.65% H
2
O in the unreacted mixture.
v
0.95-to-0.75 fuel conversion time.
77
Table 4-2. List of targets for C
3+
fuels.
Laminar flame speeds in air Ignition delay times in Ar
Index φ Index Fuel % O
2
% T
5
(K) p
5
(atm)
(T
0
=298 K, p = 1 atm) a-C
3
H
4
a
[43]
C
3
H
8
[41, 42] 9 1 2 1894 2.1
1 0.8 10 1 2 1477 2.1
2 1 11 1 4 1331 2.1
3 1.2 12 1 4 1776 2.1
4 1.6 13 1 8 1256 2.1
C
3
H
6
[41, 42] 14 1 8 1754 2.1
5 0.8 p-C
3
H
4
a
[43]
6 1 15 1 2 1865 3.5
7 1.2 16 1 2 1379 3.5
8 1.6 17 1 4 1689 3.5
18 1 4 1287 3.5
19 1 8 1594 3.5
20 1 8 1295 3.5
(T
0
=353 K, p = 1 atm)
n-C
5
H
12
[44] n-C
5
H
12

b
[45]
1 0.8 9 0.25 4 1533 1.7
2 1 10 0.25 4 1263 1.7
3 1.2 11 0.5 4 1489 1.9
4 1.4 12 0.5 4 1267 1.9
13 0.5 4 1449 3.5
14 0.5 4 1261 3.5
n-C
7
H
16
[44] n-C
7
H
16
c
[46]
5 0.8 15 0.4 0.44 1486 1
6 1 16 0.4 0.44 1429 1
7 1.2 17 0.4 0.44 1701 2
8 1.4 18 0.4 0.44 1383 2
19 0.4 0.44 1475 4
20 0.4 0.44 1395 4
21 0.4 0.22 1701 1
22 0.4 0.22 1499 1
23 0.4 0.88 1499 1
24 0.4 0.88 1300 1
a
Onset of p rise.
b
Onset of CH emission.
c
Onset of OH emission.
78
Table 4-3. List of large alkane validation data.
Laminar flame speeds in air Ignition delay times in Ar
Index φ Index Fuel % O
2
% T
5
(K) p
5
(atm)
(T
0
=353 K, p = 1 atm) a-C
3
H
4
a
[43]
n-C
8
H
18
37 0.16 4 1434 2
25 0.8 38 0.16 4 1252 2
26 1 39 0.32 4 1455 2
27 1.2 40 0.32 4 1265 2
28 1.4 41 0.32 4 1421 3.63
42 0.32 4 1273 3.63
(T
0
=403 K, p = 1 atm)
n-C
9
H
20

29 0.8
30 1
31 1.2
32 1.4
n-C
12
H
26

33 0.8
34 1
35 1.2
36 1.4

a
Onset of OH emission.
4.3 Results and Discussion
To illustrate the utility of the MUM-PCE method, we present in Fig. 4-1 the 2σ
uncertainty bands predicted for C
2
H
4
-air laminar flame speeds at 2 atm pressure as a
function of the equivalence ratio, comparing the as compiled model and the fully
optimized model (III), along with experimental data. The same bands are likewise shown
in Fig. 4-2 for selected ignition delay times of C
2
H
4
-O
2
-argon mixtures. As seen, the
precision of the prior model is rather poor. It predicts an uncertainty of ±10 cm/s for the
flame speed of ethylene at the stoichiometric condition, despite the close agreement
between the nominal prediction and experimental data. The precision may be enhanced,
as shown in the bottom panels of Figs. 4-1 and 4-2, by least squares minimization of the
model prediction and its precision against the target set of Table 4-1, leading to the
optimized Model III.

79

Figure 4-1. Experimental (○, [20];◊, [22] ) and predicted 2σ standard deviation bands
(left panel: prior model; right panel: Model III) for the laminar flame
speeds of ethylene-air mixture at 2 atm and 298 K unburned gas
temperature. Grey scale indicates the probability density function of the
uncertainty band. The dashed lines mark the 2σ uncertainties.
10
1
10
2
10
3
10
4
0.4 0.5 0.6 0.7 0.8 0.9
Ignition Delay Time, τ τ τ τ (μ μ μ μs)
1000/T (K)
10
1
10
2
10
3
0.4 0.5 0.6 0.7 0.8 0.9
Ignition Delay Time, τ τ τ τ (μ μ μ μs)
1000/T (K)

Figure 4-2. Experimental (□: p
5
= 2.4 atm [47]; ○: p
5
= 3 atm [48]; dotted lines: C
5
=
1.44±0.14)×10
–5
mol/cm
3
[21]) and predicted (left panel: prior model;
right panel: Model III) ignition delay time behind reflected shock waves.
The dashed lines mark the 2σ uncertanties.

80
We explore next the question whether a hierarchical submodel (H
2
/CO) imposes
proper constraints on a “full” model (C
2
H
4
). Again, consider the prediction uncertainties
σ
r
for C
2
H
4
-air laminar flame speeds at 1 atm, computed at several equivalence ratios and
shown in Fig. 4-3. As seen, the uncertainty of Model I is almost as large as that of the
prior model. Recall that Model I is constrained by the H
2
/CO data only. The results thus
indicate that, quantitatively, constraining the H
2
/CO submodel has little effect on model
precision for the combustion properties of ethylene. As expected, the most effective
constraints on the predictive uncertainty of ethylene flame speed come from the
combustion data of ethylene themselves. For φ = 1, Model II gives σ
r
=0.8 cm/s, reduced
from ~5.5 cm/s of the prior model. This result is further underscored in Fig. 4-4, which
compares the model predictions and their 2σ uncertainties with the experimental
observations for all flame speed targets of Table 4-1. Clearly, the H
2
/CO target set
provides little constraint on the nominal values and their precision for ethylene, and vice
versa. On the other hand, Model III, being constrained by both sets of data, gives model
uncertainties similar to those constrained by each individual target set.

81

Figure 4-3. Prediction uncertainties for the laminar flame speed of C
2
H
4
-air mixtures at
1 atm and 298 K unburned gas temperature for the prior model and
posterior Models I through III.

20
30
40
50
60
70
20 30 40 50 60 70
Prior Model
Model I
Model Prediction (cm/s)
Measured S
u
o
(cm/s)
(a)

20 30 40 50 60 70
Model II
Model III
20
30
40
50
60
70
Measured S
u
o
(cm/s)
(b)
50
100
150
200
250
300
50 100 150 200 250 300
Prior Model
Model II
Measured S
u
o
(cm/s)
(c)
50 100 150 200 250 300
Model I
Model III
50
100
150
200
250
300
Measured S
u
o
(cm/s)
(d)

Figure 4-4. Experimental and predicted C
2
H
4
-air laminar flame (a, b) and H
2
-air and
H
2
/CO-air flame speeds (c, d) for targets of Table 1. The error bars
represent the 2σ uncertainties.

The fact that the H
2
/CO and C
2
H
4
target sets are orthogonal to each other is not
surprising in retrospect. H
2
/CO combustion depends strongly on the rates in the H-atom
dominated branching mechanism, whereas in C
2
H
4
combustion a broader set of reactions
involving CH
3
, C
2
H
3
, and others also become critical. The ignition delays of ethylene,
for example, are more sensitive to the reactions of C
2
H
3
than to chain branching due to H
atom attack on molecular oxygen. Therefore, the constraint on the main chain branch
82
processes provided by the H
2
/CO targets does not have very much effect on the precision
for C
2
H
4
combustion.
The factors just discussed may be quantified by the information indices defined in Eq.
4-1. Figure 4-5 shows the component values for K, computed from Model III. The grey
intensity quantifies the impact of the i
th
target on the prediction of j
th
target. The first
thing to note is that the information matrix K is largely diagonal. The greatest influence
that most experiments have is on predicting themselves. Several critical conclusions may
be made by inspecting Fig. 4-5. First, similar experimental targets often provide a great
deal of information about each other, yet dissimilar targets provide almost no constraints
on each other. For example, the C
2
H
4
flame speeds show cross-constraints among each
other, but they provide little help for improving the model precision for ignition delay
times. Second, the more detailed combustion responses, e.g. the species concentrations
in premixed flames, have surprisingly little impact on improving the model precision for
almost all types of targets. It may be noted that this observation is rather conditional,
since the information index measures the impact of a particular experiment, and this
impact is strongly related to the precision of that experiment. Hence, the somewhat
greater
i j
K
→
values for the flame speed are attributed, at least in part, to the rather small
uncertainties associated with the experimental values reported in the literature.
Conversely, the small
i j
K
→
values associated with the flat flames are mostly because the
precision of these experiments is still poor. In other words, the information index is not a
measure of the value of a type of experiment. Rather, it is sensitive to the precision of a
datum, and thus measures the impact of that datum.

83
s
u
o
(C
2
H
4
)
s
u
o
(H
2
/CO)
flat flames
τ τ τ τ
ig
(C
2
H
4
)
τ τ τ τ
ig
(H
2
/CO)

Figure 4-5. Information index matrix K (presented as ln K) of Model III for H
2
/CO and
C
2
H
4
(see Table 4-1 for index descriptions).

Elements of the information matrix may be summed to show how a particular H
2
/CO
experiment impacts the precision of an optimized model for predicting a specific set of
data. For example, summing over all j elements for C
2
H
4
targets, a new index L may be
defined as the impact of the i
th
target on precision of an optimized model for predicting
ethylene combustion properties,

2 4
2 4
C H
C H
i i j
L K
→ →
=
∑
.
Likewise, the impact of all H
2
/CO targets on the precision of Model III for predicting
ethylene combustion may be quantified by

2 2 4 2 4
2
H /CO C H C H
H /CO
i
L L
→ →
=
∑
.
It is noted that
2 2 4
H /CO C H
L
→
and
2 4 2
C H H /CO
L
→
are both near zero, and hence the H
2
/CO data
are almost orthogonal to C
2
H
4
data, and vice versa.
84
Extrapolating the current observation, one may conclude that the accuracy and
precision of a reaction model developed for a given fuel are entirely determined by the
availability of accurate and precise combustion data for that fuel. This is, however, not
entirely true. As Fig. 4-3 shows, the prediction precision in C
2
H
4
flame speeds can be
enhanced, though only slightly, by considering the H
2
/CO data. Yet for an H
2
/CO
measurement to improve the prediction precision for C
2
H
4
, the precision (and accuracy)
of current experimental methods needs be improved drastically.
The orthogonality between H
2
/CO and C
2
H
4
is beneficial to hierarchical model
development for higher hydrocarbons. Consider the prediction precisions for propane-air
flame speeds shown in Fig. 4-6. It is seen that the reduction in predicted uncertainty
provided by the H
2
/CO data (Model I) is somewhat greater than that provided for
ethylene predictions; the C
2
H
4
targets provide a similar improvement for predicting
propane flame speeds (Model II). Model III, derived from all H
2
/CO and C
2
H
4
targets
being considered in the optimization, gives a somewhat more precise prediction for the
propane flame speed, with σ
r
=2-3 cm/s, compared to σ
r=
2-4 cm/s as predicted by the as
compiled model. This represents a 25% reduction in the flame speed uncertainty, which
is still substantially more than the roughly ±1 cm/s 1σ uncertainty normally assumed for
flame speed measurements. The current results show that by constraining the H
2
/CO and
C
2
H
4
submodels collectively, the resulting model ought to be somewhat predictive for
this somewhat higher hydrocarbon, although not nearly as predictive as constraining it
against itself.
85

Figure 4-6. Prediction uncertainties for the laminar flame speed of C
3
H
8
-air mixtures at
1 atm and 298 K unburned gas temperature for the prior model and
posterior Models I through III.

As suggested by Qin et al. [3], a quantitative, hierarchical kinetic foundation may
need to be constrained by combustion data of just several small hydrocarbons. This
statement can be directly tested by optimizing the model against a small set of large-
alkane combustion data. Prediction precision for n-C
8
H
18
/air flame speeds are shown in
Fig. 4-7. The reduction in uncertainty between the prior model and Model III is much
larger than that of C
3
H
8
, although σ
r
is still larger than the assumed 1 cm/s for
obs
r
σ .
Adding constraints from C
3
chemistry reduces σ
r
by only a small amount. It can be seen,
however, that once constraints from n-C
5
H
12
combustion data is added, σ
r
is reduced
below
obs
r
σ , and the further addition of n-C
7
H
16
data has only a small effect on σ
r
. Once
the model has been constrained by the n-C
7
H
16
combustion data, σ
r
is small enough that
n-octane combustion data should not add any further constraint to the model. This effect
is not limited to n-octane, as evidenced by the K indices for the n-C
5
H
12
, C
7
H
16
, C
8
H
18
,
C
9
H
20
, and C
12
H
26
flame speed data shown in Fig. 4-8. The predicted flame speed
86
precisions of the three higher-hydrocarbon fuels are all strongly constrained by the
measurements of n-C
5
H
12
flame speed, indicating that an adequate kinetic foundation for
general alkane oxidation can be constrained by considering fuels only up to C
5
.

Figure 4-7. Prediction uncertainties for the laminar flame speed of n-C
8
H
18
-air mixtures
at 1 atm and 353 K unburned gas temperature for several model cases.
Shown are the prior model, the model constrained against all H
2
/CO and
C
2
H
4
experiments (Model III), extensions of Model III that are further
constrained against C
3
experiments and n-pentane experiments, and Model
IV constrained against all experiments up to n-heptane.

87

Figure 4-8. Information index matrix K (presented as ln K) of Model IV (see Table 4-2
and 4-3 for index descriptions). Highlighted sections are K indices for a)
n-pentane and n-heptane laminar flame speeds, b) n-pentane ignition delay
times, c) n-heptane ignition delay times, d) n-octane, nonane, and
dodecane laminar flame speeds, and e) n-octane ignition delay times.
Indices for H
2
/CO and C
2
H
4
experiments are close to 0 and not shown.
For some experiments, a well-quantified kinetic foundation would be sufficient, as
has already been shown for flame speeds. In other cases, the foundation may prove to be
insufficient. For instance, the modeling of benzene emissions from engines likely is not
strongly constrained by the n-C
5
H
12
oxidation data. An example can be seen in the
prediction precision for n-C
8
H
18
/O
2
/Ar ignition delay times, shown in Fig. 4-9. Here, the
prediction uncertainties are still somewhat larger the
obs
r
σ values once constraints from n-
C
7
H
16
are considered. In Fig. 4-8, likewise, it can be seen that the L values for ignition
delay predictions between different sets of fuels, for instance
5 12 7 16
-C H -C H n n
L
→
, are close to
0. In this case, therefore, the C
5
foundation is insufficient to constrain these predictions,
and it is profitable to consider experimental measurements of n-C
8
H
18
ignition delay
time.
a
b
c
d
e
88
0
0.1
0.2
0.3
0.4
0.5
Prior Model III III + C
3
III + C
5
Model IV
σ (ln τ)
T
5
= 1252 K
T
5
= 1433 K
Model

Figure 4-9. Prediction uncertainties for the ignition delay times of n-C
8
H
18
-O
2
/Ar
mixtures at 2 atm. Shown are the prior model, the model constrained
against all H
2
/CO and C
2
H
4
experiments (Model III), extensions of Model
III that are further constrained against C
3
experiments and n-pentane
experiments, and Model IV constrained against all experiments up to n-
heptane.

4.4 Conclusions
The MUM-PCE method was extended to determine how much information each
experiment provides about every other in a multi-parameter kinetic model optimization
problem. An information index was defined and then used to show that fundamental
combustion data for H
2
/CO and C
2
H
4
provide very little constraint on each other.
However, the constraints provided by both data sets notably improved the prediction
precision for C
3
H
8
combustion. As the model is constrained by experimental data from
progressively larger hydrocarbon fuels, a convergence is eventually reached. Hence, the
hierarchical principle to kinetic model development is greatly benefited by the use of a
quantitative approach, such as MUM-PCE, that will provide a quantitative closure to the
problem.
89
4.5 Chapter 4 Endnotes

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47-73.

2. M. Frenklach, Combust. Flame 58 (1984) 69-72.

3. Z. Qin, V. Lissianski, H. Yang, W. C. Gardiner, Jr., S. G. Davis, H. Wang, Proc.
Combust. Inst. 28 (2000) 1663-1669.

4. D. A. Sheen, X. You, H. Wang, T. Løvås, Proc. Combust. Inst. 32 (2009) 535-
542.

5. M. T. Reagan, H. N. Najm, B. J. Debusschere, O. P. Le Maitre, M. Knio, R. G.
Ghanem, Combust. Theory Modelling 8 (2004) 607-632.

6. M. T. Reagan, H. N. Najm, R. G. Ghanem, O. M. Knio, Combust. Flame 132
(2003) 545-555.

7. M. T. Reagan, H. N. Najm, P. P. Pebay, O. M. Knio, R. G. Ghanem, Int. J. Chem.
Kinet. 37 (2005) 368-382.

8. S. G. Davis, A. V. Joshi, H. Wang, F. N. Egolfopoulos, Proc. Combust. Inst. 30
(2005) 1283-1292.

9. G. P. Smith, D. M. Golden, M. Frenklach, B. Eiteener, M. Goldenberg, C. T.
Bowman, R. K. Hanson, W. C. Gardiner, V. V. Lissianski, Z. W. Qin, GRI-Mech
3.0. http://www.me.berkeley.edu/gri_mech/; 2000

10. C. K. Westbrook, F. L. Dryer, Prog. Energ. Combust. Sci. 10 (1984) 1-57.

11. B. Sirjean, E. Dames, D. A. Sheen, X. You, C. J. Sung, A. T. Holley, F. N.
Egolfopoulos, H. Wang, S. S. Vasu, D. F. Davidson, R. K. Hanson, H. Pitsch, C.
T. Bowman, A. Kelley, C. K. Law, W. Tsang, N. P. Cernansky, D. Miller, A.
Violi, R. P. Lindstedt, A high-temperature chemical kinetic model of n-alkane
oxidation, JetSurF version 1.0.
http://melchior.usc.edu/JetSurF/JetSurF1.0/Index.html; 2009

12. H. Wang, X. You, A. V. Joshi, S. G. Davis, A. Laskin, F. N. Egolfopoulos, C. K.
Law, USC Mech Version II. High-Temperature Combustion Reaction Model of
H
2
/CO/C
1
-C
4
Compounds. http://ignis.usc.edu/Mechanisms/USC-
Mech%20II/USC_Mech%20II.htm; 2007

90
13. D. L. Baulch, C. J. Cobos, R. A. Cox, C. Esser, P. Frank, T. Just, J. A. Kerr, M. J.
Pilling, J. Troe, R. W. Walker, J. Warnatz, J. Phys. Chem. Ref. Dat. 21 (1992)
411-734.

14. Z. Hong, S. S. Vasu, D. F. Davidson, R. K. Hanson, J. Phys. Chem. A 114 (2010)
5520-5525.

15. N. K. Srinivasan, M.-C. Su, J. W. Sutherland, J. V. Michael, B. Ruscic, J. Phys.
Chem. A 110 (2006) 6602-6607.

16. S. G. Davis, A. B. Mhadeshwar, D. G. Vlachos, H. Wang, Int. J. Chem. Kinet. 36
(2004) 94-106.

17. R. J. Kee, J. F. Grcar, M. D. Smooke, J. A. Miller, A FORTRAN Program for
Modeling Steady Laminar One-Dimensional Premixed Flames, Sandia National
Laboratories: Albequerque, NM, 1986; SAND85-8240.

18. P. Middha, H. Wang, Combust. Theory Modelling 9 (2005) 353-363.

19. F. N. Egolfopoulos, D. L. Zhu, C. K. Law, Proc. Combust. Inst. 23 (1991) 471-
478.

20. G. Jomaas, X. L. Zheng, D. L. Zhu, C. K. Law, Proc. Combust. Inst. 30 (2005)
193-200.

21. C. J. Brown, G. O. Thomas, Combust. Flame 117 (1999) 861-870.

22. M. I. Hassan, K. T. Aung, O. C. Kwon, G. M. Faeth, J. Propul. Power 14 (1998)
479-488.

23. Y. Hidaka, T. Nishimori, K. Sato, Y. Henmi, R. Okuda, K. Inami, T. Higashihara,
Combust. Flame 117 (1999) 755-776.

24. J. B. Homer, G. B. Kistiakowsky, J. Chem. Phys. 47 (1967) 5290-5295.

25. R. K. Cheng, A. K. Oppenheim, Combust. Flame 58 (1984) 125-139.

26. A. Cohen, J. Larsen, Report BRL 1386, 1967;

27. S. D. Tse, D. L. Zhu, C. K. Law, Proc. Combust. Inst. 28 (2000) 1793-1800.

28. D. R. Dowdy, D. B. Smith, S. C. Taylor, A. Williams, Proc. Combust. Inst. 23
(1991) 325-332.

91
29. F. N. Egolfopoulos, C. K. Law, Proc. Combust. Inst. 23 (1991) 333-340.

30. C. M. Vagelopoulos, F. N. Egolfopoulos, C. K. Law, Proc. Combust. Inst. 25
(1994) 1341-1347.

31. O. C. Kwon, G. M. Faeth, Proc. Combust. Inst. 25 (2001) 590-610.

32. E. L. Petersen, D. F. Davidson, M. Rohrig, R. K. Hanson, 20th International
Symposium on Shock Waves (1996) 941-946.

33. I. C. McLean, D. B. Smith, S. C. Taylor, Proc. Combust. Inst. 25 (1994) 749-757.

34. A. M. Dean, D. C. Steiner, E. E. Wang, Combust. Flame 32 (1978) 73-83.

35. C. M. Vagelopoulos, F. N. Egolfopoulos, Proc. Combust. Inst. 25 (1994) 1317-
1323.

36. A. Laskin, H. Wang, 2000. Unpublished data

37. M. A. Mueller, T. J. Kim, R. A. Yetter, F. L. Dryer, Int. J. Chem. Kinet. 31 (1999)
113-125.

38. A. Bhargava, P. R. Westmoreland, Combust. Flame 113 (1998) 333-347.

39. T. J. Kim, R. A. Yetter, F. L. Dryer, Proc. Combust. Inst. 25 (1994) 759-766.

40. J. Vandooren, J. Bian, Proc. Combust. Inst. 23 (1991) 341-346.

41. C. M. Vagelopoulos, F. N. Egolfopoulos, Proc. Combust. Inst. 27 (1998) 513-519.

42. S. G. Davis, C. K. Law, Combust. Sci. Technol. 140 (1998) 427-449.

43. H. Curran, J. M. Simmie, P. Dagaut, D. Voisin, M. Cathonnet, Proc. Combust.
Inst. 26 (1996) 613-620.

44. C. Ji, E. Dames, Y. L. Wang, H. Wang, F. N. Egolfopoulos, Combust. Flame 157
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371.
92

47. S. Saxena, M. S. P. Kahandawala, S. S. Sidhu, Combust. Flame (2010) in press
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93
Chapter 5 Modeling the Negative Pressure Dependence of Mass
Burning Rates of H
2
/CO/O
2
/diluent Flames
5.1 Introduction
The H
2
/CO/O
2
oxidation mechanism is a crucial topic in combustion research.
Hydrogen has been proposed as an alternative to hydrocarbon fuels in engines [1].
Synthesis gas, containing mixtures of H
2
and CO, is burned in integrated gasification
combined cycle (IGCC) turbines as an alternative to conventional coal technologies [2].
Furthermore, oxidation models for H
2
/CO form the core of all hydrocarbon oxidation
models. As a result, the reaction rate constants in this submodel are among the most
thoroughly documented in the combustion field [3-13].
Recently, Ju, Dryer and coworkers [14] examined the laminar burning rates of
H
2
/CO/O
2
/diluent mixtures over a wide range of equivalence ratios and, more
importantly, under pressures up to 25 atm. Measurements of the mass burning rates were
carefully made using a stretch-corrected spherical flame method and the data are reported
with clearly defined uncertainties and known adiabatic flame temperature T
b
. They
observed substantial differences between predictions of literature models [8-10, 15-18]
and the experimental data, and in most cases, the model predictions fall outside of the
experimental uncertainty. Among the seven models tested, the predictions themselves
also show significant scatters, up to a factor of three at high pressures. In particular, a
large number of the reaction models were not able to predict the negative pressure
dependence of mass burning rates towards high pressures. Burke et al. [14] attributed the
variation among the model predictions to the fact that the combustion kinetics of
hydrogen remains poorly characterized.
Fundamentally, it is well known that hydrogen has two distinct burning regimes with
a sharp crossover, known as the extended second explosion limit [5, 19]. Even at
94
temperatures where all mixtures are explosive, there is a kinetic competition [5] between
hydrogen chain branching via
H + O
2
↔ OH + O (R1)
and chain propagation via
H + O
2
(+M) ↔ HO
2
(+M). (R12-17)
HO
2
then reacts with H, proceeding through the chain propagation step
HO
2
+ H ↔ 2OH (R23)
or the termination channel
HO
2
+ H ↔ H
2
+ O
2
. (R20)
The location in temperature-pressure space of the extended second limit is determined by
a critical branching ratio among the rates of these four reactions, defined in [14] as

[ ]
20 23 5
20 12
2
1
2 M
k k k
k k
+
= . (5-1)
The temperature-pressure space of the flames studied straddles the extended second limit,
so there will be some point where the mixture transitions from high-density to low-
density behavior. At low pressures, the crossover point is at a low temperature far
upstream of the flame region, so flame speed is affected mostly by heat release rate. As
the pressure increases, the crossover temperature also increases and the crossover point
intrudes into the flame. This causes a reduction in the H atom concentration, and thereby
the reactivity and flame speed.
While the fundamental cause for the negative pressure dependence of the mass
burning rates in H
2
/CO/O
2
/diluent flames has been well explained [14], quantitative
95
prediction of this dependence remains problematic. Revisions of the reaction model of Li
et al. [8] led to only moderately improved predictions; and the optimized model of Davis
et al. resulting from an earlier, rudimentary parameter optimization study [10] also does
not predict the pressure dependence well. It was proposed that the model and
experimental discrepancy may be caused by nonlinear effects in the collision stabilization
of the HO
2
complex, leading to higher order effects in the rate coefficient of the H + O
2

combination reaction (R12).
The scatter in the predictions by the published models at high pressures is also
disturbing, especially considering that the H
2
/CO submodel is the kinetic foundation for
any hydrocarbon combustion. At the moment, it is not clear whether the divergence is
due to rate parameters being too uncertain to achieve convergence or because of
fundamental differences in the choices of the reaction pathways and rate parameters
chosen in these models.
The purpose of this chapter is to shed further light on the aforementioned problems.
Specifically, we wish to address whether the inability of the models to reproduce the
experimentally observed pressure dependence of the mass burning rates can be caused by
factors other than the inaccuracy in the description of reaction R12. To answer this
question, we shall first address a related question, namely, is the scatter from model
predictions the result of the remaining rate parameter uncertainty? If the answer is yes,
then the experimental data of Burke et al. [14] would be very useful for exploring the
underlying rate uncertainties, for constraining the model parameters, and to determine
those rate parameters which require further, independent study.
We approach the problem using the Method of Uncertainty Minimization by
Polynomial Chaos Expansion (MUM-PCE) presented in Chapter 2. Previously, a
simplified version of this method was introduced and tested against an ethylene
96
combustion model [20]. The method allows us to propagate the kinetic uncertainty of a
chemical reaction model into prediction uncertainty of a particular combustion property.
It also enables us to examine the rate coefficient coupling when the prior model is
constrained by a set of experiments, and in doing so, to generate a posterior model with
reduced prediction uncertainties. As was discussed in Chapter 3, given the current state
of kinetic knowledge and the accuracy of the combustion experimental data, MUM-PCE
is not able to reduce the uncertainty in the individual rate parameter, but it is capable of
reducing the global uncertainty of the model prediction through identification of a
“feasible” parameter set [21, 22] for a set of the experimental data. In MUM-PCE, this
feasible set is represented by a joint probability distribution function of the rate
parameters. The approach taken here is not a simple optimization procedure [10, 18, 23,
24]. Rather, it relies on a rigorous mathematical, statistical approach to obtain useful
information from the experiment.
Similar approaches have been proposed and tested in recent years, extending the
localized sensitivity analysis of Warnatz [25] and others. For example, McRae and
coworkers proposed the method of parametric uncertainty in kinetic modeling of
atmospheric chemistry [26, 27] and in supercritical hydrogen oxidation [28]. Najm,
Ghenam and coworkers [29-31] used the spectral projection methods to quantify
uncertainties in reacting flow simulations. Turanyi and co-workers expanded the local
sensitivity analysis [32] and later combined it with global Monte-Carlo-based methods
[33-36] to estimate the parametric uncertainty in laminar flame speed predictions [37].
Likewise, Tomlin and co-workers have employed high-dimensional model representation
(HDMR) [38] to uncertainty analysis. Very recently, Klippenstein et al. [39] combined
uncertainty screening with theoretical chemical kinetics to improve a comprehensive
C
1
H
x
O
y
model proposed earlier by Dryer and coworkers [13].
97
We use a revised H
2
/CO submodel of USC-Mech II [40] as representative of the
current best estimates for the reaction rate expressions; the experimental knowledge is
represented by the Davis data set [10] augmented by the high-pressure flames from Burke
[14]. We then use our MUM-PCE approach to constrain the model and estimate the
remaining kinetic uncertainty.

5.2 Methods
The as compiled prior model used here is the H
2
/CO submodel of USC-Mech II [40],
which consists of 14 species and 43 reactions. This submodel is based on the Davis
H
2
/CO model [10] with updates from [41-43]. This hydrogen oxidation model is nearly
the same as the others in common use, as discussed in the introduction, with the
significant difference being the rate coefficient of reaction

OH + HO
2
↔ H
2
O + O
2
. (R27)
Following the recent experimental study in [44], we adopt the rate expression of Li et al.
[13]. For reference, the complete set of chemical rate parameters is presented in
Appendix C. The rate uncertainties were estimated from the compilation of Baulch et al.
[12] and in some case, from our own rate evaluations. They are presented in Table 5-2
for active rate parameters.

5.2.1 Computational details
The complete data set considered here is the data set of Davis et al. [10] appended by
a subset of the flame measurements of Burke et al. [14]. The combined data set is
tabulated in Table 5-1. Three data sets and corresponding models are considered here.
98
The prior model is the revised H
2
/CO subset extracted from USC-Mech II, as discussed
before. Posterior Model I is constrained by the original Davis data set. The uncertainty
calculated for this model is effectively the posteriori uncertainty in the Davis model [10].
Model II is constrained by the Davis data set plus the 18 high-pressure flame values from
Burke et al. [14].
Active parameters for each experiment were identified using one-at-a-time sensitivity
analysis and are listed, along with the posterior model parameters, in Tables 5-2, 5-3, and
5-4. Response surfaces for laminar flame speeds and flat-flame species profiles are
determined using SAB [45], while those for ignition delay times and flow reactor species
profiles use a central-composite factorial design [24, 46]. Premixed flame simulations
were performed using Sandia Premix [47], with multicomponent transport, and updates
taken from [48]. Minimization of objective function used the constrained-optimization
ZXMWD subroutine of the International Math Subroutine Library [49].

99
Table 5-1. List of experimental targets.
Index φ
H
2
/(H
2
+
CO), %
O
2
(%)-
diluent
p
atm Refs. Index
H
2
%
CO
%
O
2

%
T
K
p
atm Refs.
Laminar flame speeds (T
0
= 298 K) Ignition delay times (H
2
/CO-O
2
-Ar)
1 1 21-N
2
1 [50-54] 15
a
6.67 3.33 1051 1.7 [55]
2 3 21-N
2
1 [50, 51, 53] 16
a
6.67 3.33 1312 2 [55]
3 1.15 95 21-N
2
1 [56] 17
b
20 10 1033 0.5 [57]
4 3.90 95 21-N
2
1 [56] 18
b
20 10 1510 0.5 [57]
5 1 50 21-N
2
1 [56] 19
b
0.5 0.25 1754 33 [58]
6
3.89
5
50 21-N
2
1 [56] 20
b
2 1 1189 33 [58]
7 0.49 54 21-N
2
1 [59] 21
b
2 1 1300 33 [58]
8 0.49 17 21-N
2
1 [59] 22
b
0.1 0.05 1524 64 [58]
9 1 12-He 1 [50] 23
c
0.05 12.17 1 2160 1.5 [3]
10 2.25 12-He 1 [50] 24
c
0.05 12.17 1 2160 1.5 [3]
11 1 8-He 15 [50] 25
c
0.05 12.17 1 2625 1.9 [3]
12 1.74 8-He 15 [50] 26
d
0.05 12.17 1 2625 1.9 [3]
Flat flame species profiles (T
0
= 432 K) Flow reactor species profiles (H
2
/CO-O
2
-N
2)

13
e
1.93 17-Ar 0.047 [60] 27
f
1.18 0.61 914 15.7 [5]
14
g
1.93 17-Ar 0.047 [60] 28
f
1.01 0.52 935 6 [5]
Mass burning rates (T
0
= 298 K) [14] 29
f
0.5 0.5 880 0.3 [5]
36 0.85 8-He 15 1600
h
30
i
0.95 0.49 934 3 [61]
37 0.85 8-He 25 1600
h
31
j
0.95 0.49 934 3 [61]
38 1 6.7-He 25 1600
h
32
k,l
1.01 0.517 1038 1 [61]
39 1 7.5-He 10 1700
h
33
k,l
1 0.494 1038 3.5 [61]
40 1 7.5-He 25 1700
h
34
k,m
1 0.494 1038 3.5 [61]
41 1 8-He 10 1800
h
35
k,l
1 0.482 1038 6.5 [61]
42 1 8-He 20 1800
h

43 2.5 8.3-Ar 25 1600
h

44 2.5 9.5-Ar 25 1700
h

45 2.5 11-Ar 25 1800
h

46 2.5 50 9-Ar 15 1600
h

47 2.5 50 9-Ar 25 1600
h

48 2.5 10 8.3-Ar 25 1600
h

49 2.5 21.5-CO
2
25 1500
h

50 2.5 24.5-CO
2
25 1600
h

51 2.5 27.5-CO
2
25 1700
h

52 2.5 30.8-CO
2
10 1800
h

53 2.5 30.8-CO
2
25 1800
h

a
Onset of p rise.
b
Onset of OH omission.
c
Maximum [O].
d
Onset of CO
2
emission.
e
Maximum H mole
fraction.
f
0.6-to-0.4 fuel fractional conversion rate.
g
Maximum OH mole fraction.
h
Adiabatic flame
temperature of the unburned mixture, K.
i
0.95-to-0.75 fuel fractional conversion rate.
j
0.9-to-0.4 fuel
conversion time
k
Moist CO oxidation with 0.65% H
2
O in the unreacted mixture.
l
0.6-to-0.4 CO conversion
rate.
m
0.95-0.75 CO conversion time.
100
Table 5-2. List of active reaction rates and optimized parameters for posterior models.
Model I Model II
f x* A*/A
0
x* A*/A
0

1
H+O
2
↔O+OH 1.2 0.32 1.06 0.10 1.02
2
O+H
2
↔H+OH 1.3 -0.29 0.93 0.26 1.07
3
OH+H
2
↔H+H
2
O 1.3 -0.64 0.85 0.33 1.09
4
2OH↔O+H
2
O 1.3 0.00 1.00 0.08 1.02
5
2H+M↔H
2
+M 2 0.07 1.05 0.10 1.07
6
H
2
2 0.00 1.00 0.00 1.00
7
H
2
O 2 0.12 1.08 -0.26 0.84
9
H+OH+M↔H
2
O+M 2 0.63 1.54 -0.62 0.65
10 O+H+M↔OH+M 2 0.13 1.09 -0.07 0.95
12
H+O
2
(+M)↔HO
2
(+M) 1.2 -0.53 0.91 -0.85 0.86
13
H
2
1.2 0.00 1.00 -0.02 1.00
14
O
2
1.2 0.00 1.00 0.02 1.00
15 CO 1.2 0.00 1.00 0.05 1.01
16
CO
2
1.2 0.00 1.00 0.11 1.02
17
H
2
O 1.2 0.12 1.02 0.63 1.12
18 Ar 1.2 -0.16 0.97 -0.14 0.98
19 He 1.2 0.00 1.00 0.14 1.01
20
H
2
+O
2
↔HO
2
+H 1.3 0.14 1.04 0.89 1.26
21
2OH(+M)↔H
2
O
2
(+M) 1.5 0.00 1.00 0.00 1.00
22
HO
2
+H↔O+H
2
O 3 0.07 1.08 0.02 1.03
23
HO
2
+H↔2OH 2 -0.14 0.91 -0.13 0.91
24
HO
2
+O↔OH+O
2
2 -0.08 0.95 -0.09 0.94
25
2HO
2
↔O
2
+H
2
O
2
a
2 0.04 1.03 -0.02 0.99
26
2HO
2
↔O
2
+H
2
O
2
b
2 -0.16 0.89 -0.06 0.96
27
OH+HO
2
↔H
2
O+O
2
2 0.43 1.35 0.80 1.74
28
H
2
O
2
+H↔HO
2
+H
2
2 0.02 1.01 0.05 1.04
29
H
2
O
2
+H↔H+H
2
O 5 -0.16 0.77 -0.02 0.98
33
CO+O(+M)↔CO
2
(+M) 2 -0.02 0.99 -0.31 0.80
34
CO+OH↔CO
2
+H
b
1.2 -0.34 0.94 -0.29 0.95
35
CO+OH↔CO
2
+H
a
1.2 -0.08 0.99 -0.05 0.99
36
CO+O
2
↔CO
2
+O 3 -0.02 0.98 -0.05 0.95
37
CO+HO
2
↔CO
2
+OH 1.5 -0.15 0.90 -0.10 0.93
38
HCO+H↔CO+H
2
4 0.00 1.00 -0.08 0.89
42 HCO+M↔CO+H+M 4 0.01 1.02 -0.01 0.99
43
HCO+H
2
O↔CO+H+H
2
O 4 0.00 1.00 -0.40 0.58
a
Sum of two rate constants, low temperature expression.
b
Sum of two rate constants, high temperature
expression.
101

Table 5-3. Active parameter matrix for Experiments 1-35.
Laminar Flame Speeds
Flat
Flame Ignition Delay Time Flow Reactor
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
1 x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x
2 x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x
3 x x x x x x x x x x x x x x x x x x x x x x x x x x x x x
4 x x x x x x x x x x x
5 x x x x x x x x x x x x x
6 x x
7 x x x x x x x x x x
9 x x x x x x x x x x x x x x x x x x x x x x
10 x x x x x x x x x x x x
12 x x x x x x x x x x x x x x
13 x x x x x x x x x x x x
14 x x x x x x x x
17 x x x x x x x x x x x x x x x x x
18 x x x x x x x x x x x x
19 x x x x x x
20 x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x
21 x x x x x
22 x x x x
23 x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x
24 x x x x x x x x x x x x x x x
25 x x
26 x x x x x x
27 x x x x x x x x x x x x x x x x x x x x x x x x x x x
28 x x x x x x
29 x x
33 x x x x x x x x x
34 x x x x x x x x x x x x x x
35 x x x x x x x x x x x x x x
36 x x x x
37 x x x
38 x x x x x
42 x x x

102
Table 5-4. Active parameter matrix for Experiments 36-53
Mass Burning Rates
36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53
1 x x x x x x x x x x x x x x x x x x
2 x x x x x x x x x x x x x x x x
3 x x x x x x x x x x x x x x x
4 x x x x
5 x x x
7 x x x x x x x x x x x
9 x x x x x x x x x x x x x x x x x x
10 x x x x x x
13 x x x x x x x x
15 x x
16 x x x x x
17 x x x x x x x x x x x x x x x x x x
18 x x x x x x
19 x x x x x x x
20 x x x x x x x x x x x x x x x x x x
21 x x
23 x x x x x x x x x x x x x x x x x x
24 x x x x x x x
27 x x x x x x x x x x x x x x x x x x
34 x x x x x x x x
38 x x x
42 x
43 x

5.3 Results and Discussion
We shall present the results of MUM-PCE analysis of the H
2
/CO reaction kinetic
model and its underlying uncertainty here, progressing from the prior model to the two
posterior models. These models were constrained initially by the original, lower-pressure
dataset of Davis et al. [10], and finally by the entire dataset including mass burning rate
data up to 25 atm. Burke et al. [14] presented the evidence that the H
2
/CO oxidation
mechanism is poorly characterized by the existing models. In fact, the wide scatter in the
model predictions is consistent with and can be expressed by the underlying kinetic
103
parameter uncertainty. Figure 5-1 shows the 2σ standard deviation and the probability
distribution of the prediction uncertainty of the prior model for the mass burning rates of
H
2
/O
2
/He mixtures at φ = 0.85 and pressures from 1 to 25 atm; this is the same mixture
as Experiments 36 and 37 in Table 5-1. These predictions are compared with the
measurements reported by Burke et al. [14] and Tse, et al. [50]. Nominal predictions of
the models reported by Davis et al. [10] and Konnov [17] are also included in Fig 5-1 for
comparisons, representing roughly the range of model predictions [14]. It can be see that
all of model predictions fall within the uncertainty bound of the current model, bracketed
by the two dashed lines representing the ±2σ band. For this reason, the available model
predictions are essentially a limited statistical sampling of the kinetic parameter
uncertainty we are facing today for the oxidation kinetics of hydrogen.

Figure 5-1. Probability density function of mass burning rate predicted by the prior
model for H
2
/O
2
/He mixtures (φ = 0.85, T
u
= 298 K, T
b
= 1600 K). The
pair of dashed lines bracket the 2σ standard deviation. Symbols are
experimental data taken from Ref. [14]. Lines are predictions of the
models reported by Davis et al. [10] (solid) and Konnov [17] (dash-dot)
models.
The prediction uncertainty grows larger towards higher pressures. This expanded
uncertainty originates from the greater sensitivity of the burning rate to the reaction rates,
104
as shown in Fig. 5-2. Since the uncertainty factor of the rate parameter is assumed to be
independent of pressure, the prediction uncertainty must increase. This explains, to a
large extent, a greater degree of divergence of the literature models towards high
pressures.
The above observation illustrates the fact that the historical model development
process, applied independently by individual groups, has resulted in many different
H
2
/CO oxidation models [8-10, 15-17, 40], all validated against similar sets of
experimental data. With very few exceptions, the reaction rate expressions used in these
models are the same within the claimed uncertainty bounds, usually 20-30%, so the
models are fundamentally the same. Quantitatively, the flame propagation experiments
included in the data set are primarily conducted at atmospheric pressure, since it has not
been feasible until recently to measure flame speeds under engine-relevant conditions (up
to 30 atm or so) [50]. As a result, when these various H
2
/CO models are extrapolated to
higher pressures, they exhibit large variation depending on which of the models is used,
as shown by Burke et al. [14].
At a more detailed level, reactions directly impacting the second explosion limit
feature heavily on the sensitivity chart (Fig. 5-2), along with the reactions giving primary
heat-release. At the ambient pressure, the strongest sensitivity comes from
OH + H
2
↔ H
2
O + H , (R3)
which controls the heat release rate and the re-generation of the H atom. As pressure
increases, the burning rate becomes sensitive to the temperature along the extended
second explosion limit. Consequently, the magnitude of the sensitivity coefficients
increases for the major chain branching reaction
H + O
2
↔ OH + O

, (R1)
105
and the increase in the sensitivity coefficients leads to an increased response to rate
parameter uncertainties and thus an expanded 2σ uncertainty band.

0 5 10 15 20 25
-1.0
-0.5
0.0
0.5
1.0
H+O
2
↔O+OH
HO
2
+H↔2OH
OH+H
2
↔H+H
2
O
O+H
2
↔H+OH
OH+HO
2
↔H
2
O+O
2
H+OH(+M)↔H
2
O(+M)
H
2
+O
2
↔HO
2
+H
H+O
2
(+He)↔HO
2
(+He)
H+O
2
↔HO
2
(+H
2
O)
Pressure (atm)
Logarithmic Sensitivity
Coefficient

Figure 5-2. Sensitivity coefficients for the mass burning rates predicted for H
2
/O
2
/He
mixtures (φ = 0.85, T
u
= 298 K, T
b
= 1600 K).
The above results underscore the need for high-quality, high pressure data, since the
current state of knowledge in the hydrogen oxidation kinetics, though qualitatively
understood, is far from being accurate under these conditions. Clearly, the reason for the
observed discrepancies among the models is that the low-pressure H
2
/CO flame data used
to optimize the models provide a weak constraint on the model. In the data set used by
106
Davis et al. [10] for their model optimization, the experimental uncertainty at 1 atm is
slightly less than the prior model uncertainty, so optimizing against only atmospheric-
pressure flame speed data would provide limited constraint to the optimized model at
high pressures. For this reason, the high-pressure data of Burke et al. are particularly
useful because they provide significantly stronger constraints to the model parameters
simply because of enlarged sensitivity of the experimental responses to the kinetic
parameters, as will be discussed below.
Consider Fig. 5-3, which shows the uncertainties of the mass burning rates of
H
2
/O
2
/He flames at φ = 0.85 predicted by posterior Model I, which was constrained by
the dataset used by Davis et al. [10] only, without considering the high-pressure data of
Burke et al. [14]. Importantly, predictions of Model I agree with the experimental
measurements, although the uncertainty band size of the model is larger than the
experimental counterpart. The results obtained at φ = 1.0 exhibit a similar behavior, as
seen in Figure 5-4. The difference in the uncertainty size indicates that the high-pressure
data of Burke et al. [14] embeds critical kinetic information unavailable in the mostly
low-pressure dataset, as will be discussed later.
Importantly, the uncertainty band size of the posterior Model I is reduced
significantly as compared to that of the prior model (cf, Figs. 5-1 and 5-2). Hence, the
mostly low-pressure dataset used earlier does impose some constraints for the high-
pressure mass burning rate. Note that the Konnov model falls outside of the 2σ
uncertainty band of Model I. Had the author compared his model to the 15-atm flame
speed measurements of [50], the problem would perhaps have been resolved before the
model was proposed.

107

Figure 5-3. Probability density function of mass burning rate predicted by posterior
Model I for H
2
/O
2
/He mixtures (φ = 0.85, T
u
= 298 K, T
b
= 1600 K). The
pair of dashed lines bracket the 2σ standard deviation. Symbols are
experimental data taken from Ref. [14]. Lines are predictions of the
models reported by Davis et al. [10] (solid) and Konnov [17] (dash-dot)
models.
Next, we address some of the other experimental datasets. We show in Fig. 5-5 the
experimental and computed mass burning rates of H
2
/O
2
/CO
2
mixtures predicted with
posterior Model I for φ = 2.5. The experimental measurements show a maximum in the
burning rate at 15 atm similar to that shown in Figure 5-3. However, the predictions
made by posterior Model I does not capture this negative pressure dependence.
Moreover, the model uncertainty band is outside the experimental uncertainty bar. A
similar but less pronounced effect can be seen in Fig. 5-3.

108

Figure 5-4. Probability density function of mass burning rate predicted by posterior
Model I for H
2
/O
2
/He mixtures (φ = 1.0, T
u
= 298 K, T
b
= 1800 K). The
pair of dashed lines bracket the 2σ uncertainty. Symbols are experimental
data taken from Ref. [14] (○) and [50] (◊).

Figure 5-5. Probability density function of mass burning rate predicted by posterior
Model I for H
2
/O
2
/CO
2
mixtures (φ = 2.5, T
u
= 298 K, T
b
= 1800 K). The
pair of dashed lines bracket the 2σ uncertainty. Symbols are experimental
data taken from Ref. [14].
The results just discussed suggest that posterior Model I is overly constrained by data
inconsistent with measurements reported in [14]. To identify these inconsistent
measurements and examine whether the observed negative pressure dependence is within
the realm of the current kinetic uncertainty space, we applied MUM-PCE to the entire
dataset, by appending the data used in [10] by Burke et al.’s mass burning rate data [14].
109
In the process, we examined internal consistency of the data by utilizing the consistency
test developed in Chapter 2. The results show that among the experimental targets listed
in Table 1-1, targets 1, 3, 5, 9, 36, 40, and 52 were rejected, and the remaining targets
provide the basis for posterior Model II. The thus constrained rate coefficients and their
corresponding factorial variable values are shown in Table 5-2.
We first address the performance of posterior Model II with respect to the
experimental data. The predictions of the prior and posterior models are compared with
the experimental values in Fig. 5-6. It can be seen that the greatest impact of model
constraining by the method employed here is in the reduced uncertainty of the model
after it is constrained by the experimental dataset. A smaller benefit lies in the nominal
predictions, which are somewhat closer to the experiment.

10
100
10 100
Prior model
Posterior model II
Predicted value (cm/s)
Measured value (cm/s)
1
2
3
4
5
6
1 2 3 4 5 6
Predicted value (ln τ τ τ τ, μ μ μ μs)
Measured value (ln τ τ τ τ, μ μ μ μs)

Figure 5-6. Experimentally measured values and model predictions for laminar flame
speeds (left) and ignition delay times (right) used in Davis et al. [10],with
associated uncertainties.

The posterior Model II does predict the negative pressure dependence of the mass
burning rate. As shown in Fig. 5-7, the uncertainty band shows the rise-then-fall
110
behavior as observed experimentally. The uncertainty band is much smaller than either
the prior model or the posterior Model I, especially towards high pressures. The model
uncertainty also overlaps with the experimental counterpart. It was found that as
expected the greatest constraint to the model parameter comes from the measurement at
25 atm, φ = 0.85.

Figure 5-7. Probability density function of mass burning rate predicted by posterior
Model II for H
2
/O
2
/He mixtures (φ = 0.85, T
u
= 298 K, T
b
= 1600 K). The
pair of dashed lines bracket the 2σ uncertainty. Symbols are experimental
data taken from Ref. [14]. Lines are predictions of the models reported by
Davis et al. [10] (solid) and Konnov [17] (dash-dot) models.
Predictions of the mass burning rates before and after model constraining are
presented in Figure 5-8 and compared with experiments. With a few exceptions, the
nominal prediction of the model falls within the experimental uncertainty. The
predictions for the CO
2
-diluted flames still require some improvements, although the
basic trends in the experiments are still reproduced by the model. The remaining
problem may lie in the uncertainty assignment for the rate coefficient of reaction

H + O
2
(+CO
2
) ↔ HO
2
(+CO
2
)

. (R16)
111
In the current analysis, the uncertainty factor of k
26
is assigned a value of 1.2 uniformly
across the pressure range. In fact, there is relatively little kinetic information available
for the third body effect; and the uncertainty of the pressure fall-off parameters are
largely unknown. A better prediction of the negative pressure dependence of the mass
burning rate will have to wait until the kinetic parameters of the aforementioned reaction
becomes better known.
Predictions of the two posterior models for the mass burning rates of H
2
/O
2
/He flames
at 15 atm are shown in Fig. 5-10 as a function of the equivalence ratio. The numerical
results are compared with the experimental data taken from [14] and [50]. These
experiments provide the strongest constraint on Model II. As seen, the model uncertainty
is equal to the experimental uncertainty at φ = 1.0. As expected, the prediction
uncertainty of posterior Model II is considerably narrower than that of Model I,
indicating the strong constraint provided by the high-pressure experiments.

112
0.0
0.1
0.2
0.3
0 5 10 15 20 25 30
6.7%
7.5%
8.0%
a)
Mass burning rate (g cm
-2
s
-1
)
O
2
/ (O
2
+ He)
0.0
0.5
1.0
1.5
2.0
0 5 10 15 20 25 30
8.3%
9.5%
11%
b) O
2
/ (O
2
+ Ar)
0.0
0.5
1.0
1.5
2.0
0 5 10 15 20 25 30
Mass burning rate (g cm
-2
s
-1
)
Pressure (atm)
21.5%
24.5%
27.5%
30.8%
c) O
2
/ (O
2
+ CO
2
)
0.0
0.5
1.0
0 5 10 15 20 25 30
10%
50%
100%
d)
Pressure (atm)
H
2
/ (H
2
+ CO)

Figure 5-8. Laminar flame mass burning rates with respect to pressure for the flame data
in [14] (symbols) and calculations (lines: dashed, prior model; solid,
posterior Model II). a) He-diluted flames at φ = 1. b) Ar-diluted flames
at φ = 2.5. c) CO
2
-diluted flames at φ = 2.5. d) H
2
/CO/O
2
/Ar flames at φ =
2.5.
113

Figure 5-9. Probability density function of mass burning rate predicted by posterior
Model II for H
2
/O
2
/CO
2
mixtures (φ = 2.5, T
u
= 298 K, T
b
= 1800 K). The
pair of dashed lines bracket the 2σ uncertainty. Symbols are experimental
data taken from Ref. [14].

Figure 5-10. Probability density function of mass burning rate predicted by posterior
Model II (top panel) and Model I (bottom panel) for H
2
/O
2
/He mixtures (p
= 1 atm, T
u
= 298 K). The pair of dashed lines bracket the 2σ uncertainty.
Symbols are experimental data taken from Ref. [50] (○) and [14] (◊).
114

As was noted from Fig. 5-6, the prior model generally overpredicts flame speeds,
ignition delay times, and flow reactor fuel consumption rates in the Davis data set. In
order for Model I to reproduce the flow reactor data, therefore, k
12
(effectively the N
2

chaperone efficiency for R12) is reduced substantially, by nearly half its uncertainty
bound. Likewise, k
1
is increased to reconcile the ignition delay times, coupled with a
slight decrease in k
18
. These changes increase the flame speed simulations. To
compensate, R3, R27, and
O + H
2
↔ H + OH. (R2)
must have their rates reduced considerably, while the reaction
H + O
2
+ M ↔ H
2
O + M (R9)
has its rate increased. R20 and R23 also see their rates changed, but by small amounts as
would be expected since the low-pressure flame speed simulations are only weakly
sensitive to these rates. The opposite effect is seen in Model II, however. As can be seen
from Fig. 5-8, the prior model overpredicts the high-pressure flame speeds by so much
that R20 sees a rate increase of over 25%, nearly as much as the uncertainty allows.
Likewise the rate coefficient for,
H + O
2
(+H
2
O) ↔ HO
2
(+H
2
O)

. (R17)
is increased substantially. In order to preserve agreement with the flow reactor, ignition
delay, and low-pressure flame data, rate coefficients for R2 and R3 must now be
increased instead of decreased.
As discussed earlier, the covariance matrix Σ defines a multivariate normal
distribution for the factorialized rate parameters. We present a section of this covariance
115
matrix for the two posterior models Fig. 5-11. The rate constants chosen are those used
in the extended-second-limit formulation, k
1
, three chaperone efficiencies for R12 (k
17
,
H
2
O; k
18
, Ar; and k
19
, He), k
20
and k
23
. Additionally, k
27
and k
34
are included, as the
coupling shown in Fig. 5-11 indicates that they must also play a strong role in the
explosion limit kinetics. There is a strong coupling between k
1
and k
18
in Model I. When
the data from [14] are added, there is considerable new constraint imposed on the model,
as evidenced by the increased inter-parameter coupling and by the decrease in standard
deviation for several rate constants, notably k
1
and k
23
, in Model II.

Figure 5-11. Covariance matrices computed for Model I (left) and Model II (right).

As stated above, the covariance uniquely defines a joint PDF with respect to the
reaction rate parameters. Level surfaces of the PDF are ellipsoids whose principle axes
are oriented along the eigenvectors of Σ and whose extent in those directions are
determined by its eigenvalues. For the prior model, all of the eigenvalues are ¼,
corresponding to the 2σ uncertainty of 1 for each factorial variable. In the case of Model
II, combustion simulations have similar sensitivity vectors, since combustion is always
accelerated by increasing chain branching and heat release rates and damped by
increasing chain termination rates. Two-dimensional projections of this eight-
116
dimensional PDF are shown in Fig. 5-12. Sections are shown for both Models I and II. It
is seen that the PDF of Model II generally lies within the space of Model I, demonstrating
the new information made available by the high-pressure mass burning rate data [14]. In
particular, it can be seen that k
17
, k
19
,

and

k
27
share a strong cross-coupling in Model II
that is absent from Model I.

117

Figure 5-12. Contour plots of probability density functions of the individual rate
parameters. Circles correspond to the prior model. Left panels: posterior
Model I; right panels: Model II.
118

5.4 Conclusion
The oxidation reaction kinetics of H
2
/CO were examined in detail using the recently
proposed Method of Uncertainty Minimization using Polynomial Chaos Expansions. A
set of recently-published laminar mass burning rate measurements of H
2
/CO flames [14]
was combined with the experimental data used in the earlier work of Davis et al. [10].
This combined data set was used to develop an improved version of the H
2
/CO oxidation
model, and to improve our quantitative understanding of the kinetic uncertainties in the
H
2
/CO oxidation.
It was found that the uncertainty in mass burning rate predictions increases
significantly with pressure, thereby demonstrating the utility of high-pressure flame
measurements. In general, the prediction uncertainty of the prior model is sufficiently
large that every experimental measurement and every model proposed recently is
individually consistent with each other. Hence, the experimental-model discrepancy
reported by Burke et al. is simply a consequence of the inaccuracy in our current state of
kinetic knowledge.
The final, constrained H
2
/CO oxidation model is able to reproduce a wide segment of
the new high-pressure mass burning rate measurements. To accomplish this, the method
proposes an increased rate coefficient of the reaction OH + HO
2
↔ H
2
O + O
2
by a factor
of 1.75, as well as that of HO
2
+ H ↔ H
2
+ O
2
by 1.25. Other reaction rates are altered
by smaller factors in order to reconcile lower-pressure measurements. The uncertainty in
the model predictions under many conditions is decreased by up to a factor of 3 as
compared to the prior model due to the constraint placed by the high-pressure mass
burning rates.

119

5.5 Chapter 5 Endnotes

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47-73.
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26. A. H. Falls, G. J. McRae, J. H. Seinfeld, Int. J. Chem. Kinet. 11 (1979), 1137-
1162.
27. M. Koda, G. J. Mcrae, J. H. Seinfeld, Int. J. Chem. Kinet. 11 (1979), 345-449.
28. B. D. Phenix, J. L. Dinaro, M. A. Tatang, J. W. Tester, J. B. Howard, G. J.
McRae, Combust. Flame 112 (1998), 132-146.
29. M. T. Reagan, H. N. Najm, B. J. Debusschere, O. P. Le Maitre, M. Knio, R. G.
Ghanem, Combust. Theory Modelling 8 (2004), 607-632.
30. M. T. Reagan, H. N. Najm, R. G. Ghanem, O. M. Knio, Combust. Flame 132
(2003), 545-555.
31. M. T. Reagan, H. N. Najm, P. P. Pebay, O. M. Knio, R. G. Ghanem, Int. J. Chem.
Kinet. 37 (2005), 368-382.
121
32. T. Turanyi, L. Zalotai, S. Dobe, T. Berces, Phys. Chem. Chem. Phys. 4 (2002),
2568-2578.
33. J. Zádor, I. G. Zsély, T. Turányi, Rel. Engng. Sys. Safety 91 (2006), 1232-1240.
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Chem. A 109 (2005), 9795-9807.
35. I. G. Zsely, J. Zador, T. Turanyi, Proc. Combust. Inst. 30 (2005), 1273-1281.
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38. T. Ziehn, A. S. Tomlin, Int. J. Chem. Kinet. 40 (2008), 742-753.
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2
/CO/C
1
-C
4
Compounds. http://ignis.usc.edu/Mechanisms/USC-
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41. R. Sivaramakrishnan, A. Comandini, R. S. Tranter, K. Brezinsky, S. G. Davis, H.
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122
49. IMSL Library Reference Manual, 9th Edition; International Math and Science
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2
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2
/O
2
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123
Chapter 6 Combustion Kinetic Modeling Using Multispecies Time-
Histories in Shock-Tube Oxidation of Heptane
6.1 Introduction
Recently, highly accurate multispecies time-history data have been made available for
shock tube oxidation of hydrocarbons [1-4]. The species quantified include the parent fuel
molecule, C
2
H
4
, H
2
O and CO
2
, and radical species, including the CH
3
and OH radicals. The
measurements enable us to follow the time sequence of events occurring in a highly-complex
reaction process, from initial fuel breakdown, the formation of intermediates, radical build up
during and after the induction period, finally to the formation of the final combustion
products. As a lifelong effort of Professor Hanson, culminating advances of laser diagnostics
in shock tubes and their applications in combustion reaction kinetics, the multispecies time-
history technique has formed the basis for critical testing of chemical reaction models (e.g.,
[3, 4]) and will undoubtedly be valuable to guiding future model development.
An important aspect of this experimental advance is the ultra small uncertainties
associated with the measurements. In many cases, they are as small as 5% in the absolute
concentration (one–standard deviation). Hence, these multispecies data are expected to
contain rich kinetic information that may be utilized beyond their basic usage for model
validity. In fact, they impose quantitative constraints on the parameters of a model and its
predictions well beyond the conditions under which the data are collected. In this paper, we
use the Method of Uncertainty Minimization by Polynomial Chaos Expansions (MUM-PCE)
[5] to analyze and compare the current uncertainties in our kinetic knowledge against several
sets of species time-history data for n-C
7
H
16
oxidation at high temperatures [3, 6]. These
data are utilized to show how they impact the accuracy and precision of a reaction model in
predicting global combustion properties, including the laminar flame speed (heat release rate)
and ignition delay. We further address the question about the impact of the multispecies
124
time-history data on coupled uncertainties of kinetic rate parameters. Lastly, we demonstrate
how the global combustion property measurements may be utilized in conjunction with these
data as an inverse problem to obtain a reaction kinetic model that is accurate and precise. n-
C
7
H
18
combustion is used as a test case here not only because it has aroused substantial
interest due to its wide use as a model compound to understand the combustion of gasoline
and jet fuels but also because its combustion kinetics are relatively well understood among
liquid hydrocarbon fuels (e.g. [7, 8]).
6.2 Methodology
6.2.1 Reaction model
The chemical kinetic model used here is based on JetSurF 1.0 [9], which consists of 196
species and 1462 reactions. JetSurF was developed as a collaborative effort directed at better
understanding the combustion chemistry of jet fuels through a surrogate approach. It
describes the high-temperature pyrolysis and oxidation of n-alkanes up to n-dodecane. An
earlier version of the model was discussed in You et al. [10]. USC Mech II [11] was used as
the H
2
/CO/C
1
-C
4
kinetic foundation with model details provided in a series of earlier
publications [12-23]. The reaction model is largely untuned except for the H
2
/CO chemistry,
which was based on a comprehensive optimization study conducted earlier [12]. A wide
range of validation tests have been conducted and documented online [9, 11] or elsewhere
[24], considering over 140 sets of global and detailed combustion data. In the current study,
the rate parameters of the H
2
/CO subset were restored to their nominal, as-compiled values.
Additionally, the rate parameter for the OH + HO
2
↔ H
2
O + O
2
reaction was revised using
the expression of Baulch et al. [25] in light of the recent work of Hanson, Michael and
coworkers [26, 27]. Extrapolating their kinetic rates show little evidence of the peculiar drop
125
in the rate coefficient near 1000 K reported in [28, 29]. The complete set of reaction rate
parameters is presented in Appendix C.
The reaction model described above is referred to as the unconstrained, or prior, model,
since all rate coefficients were taken as they are from the literature. Uncertainty factors for
Arrhenius pre-factors in the H
2
/CO/C
1
-C
4
subset (i.e., USC-Mech II) are given in the
supplemental materials of [5]. These factors were taken in consultation with literature
compilations (e.g., [25, 30]) or from our own evaluations. The pre-factors for all reactions
not considered in USC Mech II are assumed to have an uncertainty factor of 3. The
uncertainty in the activation energy or the pressure fall-off parameters is not considered in
the present work. Such uncertainties inevitably exist in reaction models currently available,
and in many cases, their impact on the overall uncertainty of the model and its predictions
can be rather large. This nonlinear uncertainty effect will be considered in a future work.

6.2.2 Experimental datasets
The data considered here are two sets of C
2
H
4
, OH, H
2
O and CO
2
time histories (Series 1i
and 1ii) [3] and three CH
3
time histories (Series 2i-iii) [6], all of which were determined for
stoichiometric shock tube oxidation of highly diluted n-C
7
H
16
in oxygen and argon. The
experimental conditions are summarized in Table 6-1 and described in detail in Table B-2.
Additionally, two types of global combustion properties of n-heptanes were considered,
ignition delay times and laminar flame speeds. They provide useful information concerning
the impact of multispecies time-histories on the model predictions of global combustion
responses. Series 3 contains five sets of ignition delay data for three n- C
7
H
16
-O
2
-Ar
mixtures [31, 32], spanning the equivalence ratio range of 0.5-2.0 and pressures from 1 to 4
atm. Series 4 is a set of laminar flame speed data at the atmospheric pressure [24].
126
Uncertainty in the ignition delay time is estimated at ± 20% based on a regression of the data.
Laminar flame speed uncertainty is estimated at ±2 cm/s. Again, the conditions of the last
two series of experiments are listed in Table 6-1.
It should be noted that the model posterior uncertainty space depends strongly on the
uncertainties of the experimental measurements. In [3], the measurement uncertainty in
[OH] is ± 3%, but the precision of the measurement is confounded somewhat by the
uncertainty in the post-shock temperature T
5
. To illustrate this point, we show in Figs. 6-1
and 6-2 the nominal predictions of the OH, H
2
O, CO
2
, and C
2
H
4
profiles using the prior
model for Series 1i and 1ii experiments, respectively. The solid lines designate calculations
at the nominal temperature of the experiments; the long-dashed lines represent those
calculated with ± 10 K uncertainty (2σ-standard deviation) in T
5
. It can be seen that in
regions of large gradients the species measured can be influenced to a greater extent by the
uncertainty in the temperature than in the concentration itself. In light of this result, we
assigned a uniform uncertainty of ± 20% for Series 1i over the entire time period considered
and for Series 1ii during the induction period, as shown by the short-dashed lines in both
figures. For the lower temperature experiment (Series 1ii: T
5
= 1365 K), the temperature
uncertainty of ± 10 K impact greatly the species concentration after ignition. The uncertainty
in the measured profiles after 800 μs can be as large as a factor of 3. Henceforth the 2σ-
standard deviations for those measurements correspond to an uncertainty factor of 3.
As stated in Ref. [6], the uncertainty in the CH
3
concentration is ± 20%. Simulation
shows that the ± 10 K uncertainty in T
5
impacts the concentrations of CH
3
only in the later
part of the reaction leading to its complete destruction (Fig. 6-3). Hence, we assigned all
CH
3
profiles a uniform uncertainty of ± 20%.
127

Table 6-1. Experimental Datasets Considered
Model constraints
Series Data type/conditions
a
I

II III IV
1 OH, CO
2
, H
2
O, C
2
H
4
time histories [3]
i) 300 ppm nC
7
H
16
/ 3300 ppm O
2
/ Ar
b

T
5
= 1494 K, p
5
= 2.15 atm
ii) 300 ppm nC
7
H
16
/ 3300 ppm O
2
/ Ar
c

T
5
= 1365 K, p
5
= 2.35 atm
×
× ×

2
CH
3
time histories [6]
500 ppm nC
7
H
16
/ 5500 ppm O
2
/ Ar
i) T
5
= 1395 K, p
5
= 1.72 atm
ii) T
5
= 1461 K, p
5
= 1.67 atm
iii) T
5
= 1545 K, p
5
= 1.61 atm

× ×

3
Ignition delay
i) 0.4% nC
7
H
16
/ 4.4 % O
2
/ Ar
a) T
5
= 1429-1486 K, p
5
= 1 atm [31, 32]
b) T
5
= 1383-1503 K, p
5
= 2 atm [31, 32]
c) T
5
= 1394-1474 K, p
5
= 4 atm [32]
ii) 0.4% nC
7
H
16
/ 2.2 % O
2
/ Ar
T
5
= 1499-1700 K, p
5
= 1 atm [31, 32]
iii) 0.4% nC
7
H
16
/ 8.8 % O
2
/ Ar
T
5
= 1300-1499 K, p
5
= 1 atm [31, 32]

× ×

4
Laminar flame speeds (nC
7
H
16
/air) [24]
T
u
= 353 K, p = 1 atm

× ×
a
The uncertainty of T
5
in Series 1 and 2 is ±10 K (2-C deviation).
b
Actual T
5
and p
5
values: 1494 K, 2.155 atm
(OH & CO
2
), 1502 K, 2.262 atm (H
2
O), 1506 K, 2.36 atm (C
2
H
4
).
c
Actual T
5
and p
5
values: 1378 K, 2.326 atm
(OH), 1375 K, 2.282 atm (CO
2
), 1368 K, 2.86 atm (H
2
O), 1358 K, 2.436 atm (C
2
H
4
).
128
10
-5
10
-4
10
-3
10
-5
10
-4
10
-3
Mole Fraction
Time (s)
C
2
H
4
H
2
O
OH
CO
2
300 ppm nC
7
H
16
/ 3300 ppm O
2
/ Ar
T
5
= 1494 K, p
5
= 2.15 atm

Figure 6-1. Uncertainty estimates for species time history measurements (Series 1i of
Table 6-1). Solid line: predictions of the prior reaction model at the nominal
temperature shown; long-dashed lines: computed uncertainty bounds due to
±10 K uncertainty in the T
5
value [3]; short dashed lines: 20% uncertainty on
the nominal mole fraction values.
129
10
-6
10
-5
10
-4
10
-3
10
-4
10
-3
Mole Fraction
Time (s)
300 ppm nC
7
H
16
/ 3300 ppm O
2
/ Ar
T
5
= 1365 K, p
5
= 2.35 atm
C
2
H
4
H
2
O
OH
CO
2

Figure 6-2. Uncertainty estimates for species time history measurements (Series 1ii of
Table 6-1). Solid line: predictions of the prior reaction model at the nominal
temperature shown; long-dashed lines: computed uncertainty bounds due to
±10 K uncertainty (2σ) in the T
5
value [3]; short dashed lines: 20%
uncertainty on the nominal mole fraction values.

130
10
-5
10
-4
10
-3
Mole Fraction
500 ppm nC
7
H
16
/ 5500 ppm O
2
/ Ar
T
5
= 1545 K, p
5
= 1.61 atm

10
-5
10
-4
10
-3
Mole Fraction
500 ppm nC
7
H
16
/ 5500 ppm O
2
/ Ar
T
5
= 1461 K, p
5
= 1.67 atm

10
-5
10
-4
10
-3
10
-5
10
-4
10
-3
Mole Fraction
Time (s)
500 ppm nC
7
H
16
/ 5500 ppm O
2
/ Ar
T
5
= 1395 K, p
5
= 1.72 atm

Figure 6-3. Uncertainty estimates for CH
3
time history measurements (Series 2 of Table
6-1). Solid line: predictions of the prior reaction model at the nominal
temperature shown; long-dashed lines: computed uncertainty bounds due to
±10 K uncertainty (2σ) in the T
5
value [3]; short dashed lines: 20%
uncertainty on the nominal mole fraction values.

6.2.3 Computational details
The shock tube problem was solved using the Sandia ChemKin code [33] with a constant
density model for both species time histories and ignition delay times. All ignition delay data
considered here were measured by the onset of OH* emission. Computationally, the ignition
point was determined from the maximum d[OH*]/dt. Because the underlying problem is
neither constant density nor constant pressure, the current approximation is expected to yield
reasonably good comparison between model and experiment.

Premixed flame simulations
131
were performed using Sandia Premix [34], with multicomponent transport, with updates
taken from [35, 36].
Active parameters for each experiment were determined using one-at-a-time sensitivity
analysis and are listed in Table B-1. Parameters were screened by ranking the reaction
prefactors in order of their uncertainty-weighted sensitivity coefficients, defined as
=
,
ln
w ik ik k
S S f where
ik
S is the sensitivity coefficient of the i
th
experiment to the k
th
rate
parameter, and choosing those for which
, w ik
S was less than 10% of the largest for any given
experiment. This procedure was chosen because
, w ik
S is a measure of the uncertainty
contribution of each reaction to that particular experiment. Response surfaces for laminar
flame speeds and species profiles were determined using the SAB method [37], while those
for ignition delay times were determined using a central-composite fractional factorial design
of resolution VI [38]. Minimization of the objective function used the constrained-
optimization ZXMWD subroutine of the International Math Subroutine Library. Calculation
of α α α α* used the SLATEC Common Mathematical Library.
As we discussed earlier, in Bayesian terminology, the untuned, base model is the “prior”
model since it denotes the state of knowledge before any global experiments are taken into
account [39]. All the other models are therefore “posterior” models. To examine the impact
of the Stanford multispecies time history data on model accuracy and precision, we
constrained the prior model by the Series 1 data [3] only, yielding posterior Model I. Next,
we considered both Series 1 and 2 data, yielding Model II. To compare the impact of
multispecies time history data and that of global combustion properties (ignition delay and
laminar flame speeds), we constrained the base model for data in Series 3 and 4 only. This
produced Model III. Lastly, we considered both multispecies time history data and the
combustion properties, which yielded a fully constrained, Model IV. A summary of the
132
posterior models is presented in Table 6-1 and the rate parameters of these models are
presented in Table B-3.

6.3 Results and Discussion
To illustrate the utility of the multispecies time-history data, we plot in Figs. 6-4 and 6-5
comparisons of measured and computed profiles for Series-1 experiments. The dashed lines
and scattered dots represent nominal predictions and the results of Monte Carlo sampling of
model uncertainties for the prior model (left panels) and posterior Model I (right panels). As
it is seen, the nominal prediction of the prior model is in good agreement with the data, but
the precision of the model prediction is poor. For example, while the nominal prediction for
the OH mole fraction falls within the uncertainty of the experiment over a wide range of
reaction times, the uncertainty of the model prediction is at least a factor of 2, as judged by
the width of the scatter before the OH model fraction reaches the plateau region where its
concentration is governed by thermodynamic equilibrium. Hence, the results shown in the
left panels of Figs. 6-4 and 6-5 illustrate the fact that the prior model is accurate but not
always precise. The fact that the model uncertainty is larger than the equivalent experiment
shows that these data will be useful to constrain the uncertainty of the prior model.
Indeed, after the chemical model is constrained using just a few data points on the
profiles (marked by the open symbols), we observed not only a significantly narrowed
uncertainty band in the predictions, but an even better agreement between the data and
nominal prediction for these species profiles, as exhibited on the right panels of Figs. 6-4 and
6-5. The impact of these multispecies time-history data on the model precision is depicted in
Fig. 6-6. The top plot (a) shows the probability density function and the 2-σ standard
deviation of laminar flame speeds of n-C
7
H
16
-air mixtures predicted by the prior model.
Again, the results may be best described as: the nominal prediction by the model is accurate,
133
but the precision is rather poor. In particular, the 2-σ standard deviation can be larger than
10 cm/s for a flame speed value of 50 cm/s. In panel (b), we present the results of Model I.
The two sets of species data alone reduce the uncertainty of the predicted flame speeds by
almost a factor of 2. Clearly this result illustrates that the value of the multispecies
measurement is well beyond the usage of model validation.
134
10
-5
10
-4
10
-3
10
-5
10
-4
10
-3
Mole Fraction
C
2
H
4
10
-5
10
-4
10
-3
C
2
H
4

10
-5
10
-4
10
-3
10
-5
10
-4
10
-3
Mole Fraction
OH
10
-5
10
-4
10
-3
OH

10
-5
10
-4
10
-3
10
-5
10
-4
10
-3
Mole Fraction
H
2
O
10
-5
10
-4
10
-3
H
2
O

10
-5
10
-4
10
-3
10
-5
10
-4
10
-3
Mole Fraction
CO
2
Time (s)
10
-5
10
-4
10
-3
CO
2
Time (s)

Figure 6-4. Experimental (solid lines, [3]) and computed (dashed lines: nominal
prediction; dots: uncertainty scatter) species time histories for Series 1i: 300
ppm nC
7
H
16
/3300 ppm O
2
/Ar, T
5
= 1495 K, p
5
= 2.15 atm (see Table 6-1).
The open circles and the corresponding error bars designate data used as
Series 1 targets and 2σ uncertainties, respectively. Left panel: prior model.
Right panel: posterior Model I.
135

10
-6
10
-5
10
-4
10
-3
10
-5
10
-4
10
-3
Mole Fraction
C
2
H
4
10
-5
10
-4
10
-3
5 5
C
2
H
4

10
-6
10
-5
10
-4
10
-3
10
-5
10
-4
10
-3
Mole Fraction
OH
10
-5
10
-4
10
-3
OH

10
-6
10
-5
10
-4
10
-3
10
-5
10
-4
10
-3
Mole Fraction
H
2
O
10
-5
10
-4
10
-3
H
2
O

10
-6
10
-5
10
-4
10
-3
10
-5
10
-4
10
-3
Mole Fraction
CO
2
Time (s)
10
-5
10
-4
10
-3
CO
2
Time (s)

Figure 6-5. Experimental (solid lines, [3]) and computed (dashed lines: nominal
prediction; dots: uncertainty scatter) species time histories for Series 1ii: 300
ppm nC
7
H
16
/3300 ppm O
2
/Ar, T
5
= 1365 K, p
5
= 2.35 atm (see Table 6-1).
The open circles and the corresponding error bars designate data used as
Series 1 targets and 2σ uncertainties, respectively. Left panel: prior model.
Right panel: posterior Model I.
136
Laminar Flame Speed, s
u
(cm/s)
○
(a)
(b)
(c)
(d)
Laminar Flame Speed, s
u
(cm/s)
○
Laminar Flame Speed, s
u
(cm/s)
○
Laminar Flame Speed, s
u
(cm/s)
○
(a)
(b)
(c)
(d)

Figure 6-6. Probability density function of laminar flame speed predicted for heptane-air
mixtures (p = 1 atm, T
u
= 353 K). The pair of dashed lines bracket the 2σ
uncertainty. Panel (a) prior model, (b) posterior Model I using Series 1 data),
(c) posterior model constrained by Series 1 data with a uniform uncertainty
value of ±5%, (d) posterior model IV constrained by all data of Table 6-1.
Symbols are experimental data taken from Ref. [24].

We digress here to discuss the impact of the accuracy of the species measurement on
laminar flame speed predictions. In the left panel of Fig. 6-7, we show the standard deviation
of predicted flame speeds at two equivalence ratios (φ = 1.0 and 1.4) as a function of the
137
hypothetical experimental uncertainties in the species measurement. The 1/(2σ
obs
) value of 0
represents the prior knowledge in which Series 1 data do not exist. As seen, while the
current data with ±20% uncertainty lowers the model uncertainty notably, further improved
accuracies in the data do not appear to offer significant benefits. In fact, if a ±5% uncertainty
is assigned to all species concentration (i.e., neglecting the uncertainty in T
5
), the model can
become ill-constrained, as seen in panel (c) of Fig. 6-6. Although the experimental flame
speeds still lie within the 2σ uncertainty band of the prediction, the nominal prediction
becomes notably lower than the experimental data. Hence, the analysis here shows that an
even more careful uncertainty analysis is critical to the utilization of highly precise data.

1
2
3
4
0 5 10 15 20
1/(2σ σ σ σ
obs
)
φ φ φ φ = 1.0
σ σ σ σ
s
(cm/s)
u
o
φ φ φ φ = 1.4
20% 10% 7% 5%
Uncertainty in Species Value, 2σ σ σ σ
obs
2
3
4
0 5 10 15 20
1/(2σ σ σ σ
obs
)
20% 10% 7% 5%
Uncertainty in Species Value, 2σ σ σ σ
obs
CH
3
(Series 2) only
OH (Series 1) only
All multi-species (Series 1 & 2)

Figure 6-7. Variation of the standard deviation predicted for the n-C
7
H
16
-air laminar flame
speed (T
u
= 353 K, p = 1 atm) as a function of the percentage uncertainty in
the species time-history data. Left panel: prior and posterior models
constrained with Series 1 data; right panel: impact of species time-histories
selected on posterior model.

138
The prior model can also be constrained by the CH
3
profiles published earlier [6] (Series
2 experiments). As examples, we show in Fig. 6-8 the comparison of experimental data of
two such profiles and the predictions of both the prior model and posterior Model II. Again
the prior model considered here is accurate but not precise (the left panels of Fig. 6-8), but it
can be constrained by the measurements leading to a notably precise model (right panels).
Here we considered only the data (marked by the open symbols) beyond the first peaks in the
CH
3
profiles. In general, the peak methyl concentration is reached around or before 10 μs
reaction time. Notable disagreement exists between the measurement and both the prior and
posterior models for Series 2i. Attempts to include the data prior to CH
3
peaking yielded
inconsistency among datasets considered for the prior model used. Hence, the results,
especially that shown in the upper-right panel of Fig. 6-8 suggest that either the reaction
model misses critical reactions important to the very early stage of the fuel break down or it
is unable to capture the mixture condition within the first few microseconds of reflected
shock passage. This problem will be examined in a future work.

139
10
-5
10
-4
10
-3
Mole Fraction
500 ppm nC
7
H
16
/ 5500 ppm O
2
/ Ar
T
5
= 1395 K, p
5
= 1.72 atm
(Series 2i)

10
-5
10
-4
10
-3
10
-5
10
-4
10
-3
500 ppm nC
7
H
16
/ 5500 ppm O
2
/ Ar
T
5
= 1545 K, p
5
= 1.61 atm
(Series 2iii)
Mole Fraction
Time (s)
CH
3

10
-5
10
-4
10
-3
Time (s)

Figure 6-8. Experimental (solid lines, [6]) and computed (dashed lines: nominal
predictions; dots: uncertainty scatter) CH
3
time histories of Series 2i and 2iii
(see Table 6-1). The open circles and the corresponding error bars designate
data used as Series 2 targets and 2σ uncertainties, respectively. Left panel:
prior model. Right panel: posterior Model II.
Regardless, the methyl time histories have similar impact on the precision of the flame
speed predictions. As shown in the right panel of Fig. 6-7, the ±20% uncertainty in the CH
3

mole fraction reduces the uncertainty of the predicted flame speeds by 50%; a ±10%
uncertainty yielded even better results. Also shown in the same figure is the result of a
similar analysis using the OH profile only. Clearly, the OH profiles provided stronger
constraints to the model uncertainty that the CH
3
profiles. The cause is quite obvious, in that
the OH profiles contain information both in the induction-period radical chain branching
process and the heat release rates after the induction period. Of course, considering all
species profiles of Series-1 and 2 experiments gives the most stringent constraint on the
flame speed prediction.
140
The results presented above also confirm that the prior model can reconcile all species
profile data considered and that the two species data sets are consistent with each other
within the framework of the prior model used, except for the very early stage of the CH
3

profile. As shown in the left panel of Fig. 6-9, consideration of both sets of species data in
the least-squares minimization scheme led to a posterior model (Model III) equally accurate
and precise for prediction of the OH profile as compared to the case in which the CH
3

profiles were excluded from the minimization. In the right panel of Fig. 6-9, we show the
Series 1i OH prediction for Model III, which was constrained by the global combustion
properties, including the flame speed and ignition delay. It can be seen that the scatter in the
predicted OH concentration is marginally smaller than that predicted with the prior model (cf,
Fig. 6-4), but the nominal concentrations predicted for OH are significantly lower than the
experiments in the post-induction period. Hence, as one may expect, while detailed species
time-histories present notable improvements in the precision of model predictions against
global combustion properties, these global properties provide little constraints on the detailed
evolution of key species concentrations during n-C
7
H
16
oxidation.
Combining the multi-species time history data with global combustion properties led to a
posterior model fully constrained by the current knowledge considered, posterior Model IV.
As shown in Fig. 6-6(d), the precision of the predicted flame speed is now either almost the
same as or better than the experimental uncertainties. The results illustrate that a combined
consideration of the detailed, multi-species profiles in conjunction with the global
combustion properties is necessary for deriving a highly accurate and precise model. Hence,
highly accurate measurements of global combustion properties remain valuable in the
development of fundamental reaction mechanisms.

141
10
-5
10
-4
10
-5
10
-4
10
-3
Mole Fraction
Time (s)
300 ppm nC
7
H
16
/ 3300 ppm O
2
/ Ar
T
5
= 1495 K, p
5
= 2.15 atm
10
-5
10
-4
10
-3
Time (s)
300 ppm nC
7
H
16
/ 3300 ppm O
2
/ Ar
T
5
= 1495 K, p
5
= 2.15 atm

Figure 6-9. Experimental (solid lines, [3]) and computed (dashed lines: nominal
predictions; dots: uncertainty scatter) OH time history of Series 1i (see Table
6-1). The open circles and the corresponding error bars designate data of
series 1 targets and 2σ uncertainties, respectively. Left panel: posterior model
constrained by Series 1 and 2 time histories (Model II). Right panel: posterior
model constrained against laminar flame speeds and ignition delay times
(Model III).

Figure 6-10 shows the comparison of model accuracy and precision of ignition delay
times of a stoichiometric n-C
7
H
16
-O
2
-Ar mixture (Series 3ia, p
5
= 1 atm). The top panel
corresponds to the prior model and the bottom panel represents Model IV. Figure 6-11
shows the ignition delay times for Series 3ib, 3ii, and 3iii. As seen, Model IV gives
significantly improved accuracy and precision for the experimental data over the entire
temperature range. Interestingly, Model IV is unable to reconcile with the experimental data
at the elevated pressure of 2 atm (φ = 1) and under the fuel rich condition of φ = 2 (p
5
= 1
atm). In particular, both sets of the data show a pronounced curvature for T
5
> ~1500 K; the
model predictions fall out of the uncertainty band of the data (defined by the two solid lines)
and show notably shorter ignition delay times than the experiment. These results suggest a
142
certain physical deficiency of the model above 1500 K, which cannot be accounted for with
the parameter uncertainty in the prior model. While the chemistry details responsible for the
discrepancy will be discussed in a future work, it is noted that the above results in fact
highlight the usefulness of MUM-PCE, in that the analysis using this method can isolate the
model uncertainties due to the existence of physical deficiency in a reaction model from
those due to imprecise knowledge in the rate parameters.
Lastly, we discuss the different aspect of model constrains imposed by the multi-species
time history data and by the global combustion properties. Fig. 6-12 presents the covariance
matrices Σ computed for Model I (top panel) and Model III (bottom panel). Though there is
a tremendous, joint constraining effect from both types of data, the constraint from the
species profiles is predominantly imposed on reactions of radical chain branching and those
involving CH
x
radicals, propene, and the parent fuel breakup. Flame speeds and ignition
delay appear to constrain more strongly the heat release and chain branching reactions.
Figure 6-13 presents contour plots of the probability density functions of three individual rate
constants, where the left, center and right columns represent Model II (species profiles),
Model III (global combustion properties) and Model IV. As seen, the rate parameter of the
initial C-C fission reaction nC
7
H
16
→ pC
4
H
9
+ nC
3
H
7
is not constrained by the global
properties, but it is constrained notably by the species profiles. This constraint turns out to
have a large effect on the joint rate parameter uncertainty space. As the right panels show,
the size of the original joint uncertainty space of the three rate parameters is reduced
significantly, by as much as a factor of 4 to 8, when both detailed species time history and
global combustion property data are used to constrain the model parameter uncertainty.

143

10
2
10
3
10
4
Ignition Delay, τ τ τ τ (μ μ μ μs)
0.4% nC
7
H
16
/ 4.4 % O
2
/ Ar (p
5
= 1 atm)

10
2
10
3
10
4
0.60 0.65 0.70 0.75
1000 K/T
Ignition Delay, τ τ τ τ (μ μ μ μs)

Figure 6-10. Ignition delay times of Series 3ia. Experiments: ◊,[32]; ●,[31]. Dots
represent the results of Monte Carlo sampling of predictions by Model I (top
panel) and Model IV (bottom panel). The circled data indicate the
temperatures of the ignition delay targets.
144

10
2
10
3
10
4
0.60 0.65 0.70 0.75
Ignition Delay, τ τ τ τ (μ μ μ μs)
0.4% nC
7
H
16
/ 4.4 % O
2
/ Ar (p
5
= 2 atm)

10
2
10
3
10
4
0.58 0.60 0.62 0.64 0.66 0.68 0.70
Ignition Delay, τ τ τ τ (μ μ μ μs)
0.4% nC
7
H
16
/ 2.2 % O
2
/ Ar (p
5
= 1 atm)

10
1
10
2
10
3
10
4
0.60 0.65 0.70 0.75 0.80
0.4% nC
7
H
16
/ 8.8 % O
2
/ Ar (p
5
= 1 atm)
1000 K/T
Ignition Delay, τ τ τ τ (μ μ μ μs)

Figure 6-11. Ignition delay times for Series 3ib (top panel), Series 3ii (middle panel) and
Series 3iii (bottom panel). Experiments: ◊,[32]; ●,[31]. Dots represent the
results of Monte Carlo sampling of predictions by Model IV. The circled data
indicate the temperatures of the ignition delay targets.
145

Figure 6-12. Covariance matrices computed for Model I (top panel) and Model III (bottom
panel).
146

Figure 6-13. Contour plots of probability density functions of the individual rate
parameters. Circles correspond to the prior model. Left panels: Model II;
center panels: Model III; right panels: Model IV.

147
6.4 Conclusion
A set of data containing the recently reported multispecies time-histories during n-
heptane oxidation behind reflected shock waves were utilized along with global combustion
properties (flame speed and ignition delay) to analyze and improve the accuracy and
precision of a modified JetSurF mechanism. The results may be summarized as follows:
For n-heptane combustion the kinetic uncertainty in JetSurF is substantially larger than
the experimental data set considered in the current work. In particular, the reaction model is
shown to be accurate in predicting the experiments, but the precision of the model is rather
poor given the kinetic parameter uncertainty in the model.
The high-precision species measurements were shown to provide a strong constraint on
the model predictions of species profiles, as well as global combustion properties. In some
cases, the prediction precision was improved by as much as a factor of 3. Strong constraints
were imposed on the joint uncertainty space of the rate parameters of radical chain
branching/heat release reactions, as well as the parent-fuel breakup, CH
x
and propene
reactions. On the other hand, global combustion properties (laminar flame speeds and
ignition delay times) were shown to provide constraints more on reactions involved in heat
release.
Lastly, while the high-precision species measurements in shock tubes are proven to be
critical toward future kinetic model development, global flame property measurements are
still necessary to achieve highly accurate and precise predictions for fuel combustion.

148
6.5 Chapter 6 Endnotes

1. R. K. Hanson, Proc. Combust. Inst. 33 (2010) in press.
(doi:10.1016/j.proci.2010.09.007).

2. D. F. Davidson, R. K. Hanson, Shock Waves 19 (2009) 271-283.

3. D. F. Davidson, Z. Hong, G. L. Pilla, A. Farooq, R. D. Cook, R. K. Hanson,
Combust. Flame 157 (2010) 1899-1905.

4. D. F. Davidson, Z. Hong, G. L. Pilla, A. Farooq, R. D. Cook, R. K. Hanson, Proc.
Combust. Inst. 33 (2010) in press (doi: 10.1016/j.proci.2010.05.104).

5. D. A. Sheen, X. You, H. Wang, T. Løvås, Proc. Combust. Inst. 32 (2009) 535-542.

6. D. F. Davidson, M. A. Oehlschlaeger, R. K. Hanson, Proc. Combust. Inst. 31 (2007)
321-328.

7. H. J. Curran, P. Gaffuri, W. J. Pitz, C. K. Westbrook, Combust. Flame 114 (1998)
149-177.

8. M. Chaos, A. Kazakov, Z. Zhao, F. L. Dryer, Int. J. Chem. Kinet. 39 (2007) 399-414.

9. B. Sirjean, E. Dames, D. A. Sheen, X. You, C. J. Sung, A. T. Holley, F. N.
Egolfopoulos, H. Wang, S. S. Vasu, D. F. Davidson, R. K. Hanson, H. Pitsch, C. T.
Bowman, A. Kelley, C. K. Law, W. Tsang, N. P. Cernansky, D. Miller, A. Violi, R.
P. Lindstedt, A high-temperature chemical kinetic model of n-alkane oxidation,
JetSurF version 1.0. http://melchior.usc.edu/JetSurF/JetSurF1.0/Index.html; 2009

10. X. You, F. N. Egolfopoulos, H. Wang, Proc. Combust. Inst. 32 (2009) 403-410.

11. H. Wang, X. You, A. V. Joshi, S. G. Davis, A. Laskin, F. N. Egolfopoulos, C. K.
Law, USC Mech Version II. High-Temperature Combustion Reaction Model of
H
2
/CO/C
1
-C
4
Compounds. http://ignis.usc.edu/Mechanisms/USC-
Mech%20II/USC_Mech%20II.htm; 2007

12. S. G. Davis, A. V. Joshi, H. Wang, F. N. Egolfopoulos, Proc. Combust. Inst. 30
(2005) 1283-1292.

13. S. G. Davis, C. K. Law, H. Wang, Combust. Flame 119 (1999) 375-399.

14. S. G. Davis, C. K. Law, H. Wang, J. Phys. Chem. A 103 (1999) 5889-5899.

149
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16. A. Laskin, H. Wang, C. K. Law, Int. J. Chem. Kinet. 32 (2000) 589-614.

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151
Chapter 7 Conclusions and Future Work
7.1 Summary of Conclusions
It is usually observed that a chemical reaction model, as compiled, will not predict a wide
range of combustion data a priori, unless a detailed analysis of the kinetic uncertainty in the
model and of the model predictions is performed. In this dissertation, the Method of
Uncertainty Minimization using Polynomial Chaos Expansions (MUM-PCE) was introduced
to quantify the uncertainty in the predictions of a detailed kinetic model, allowing a direct
comparison between predictions and experimental measurements. It was shown how the
information present in the experimental measurements can be used to quantitatively constrain
the model, and how the kinetic uncertainty can then be expressed in terms of the uncertainty
of the global measurements. The methodology was applied to a database with known
inconsistent measurements and, through the use of its consistency algorithm, allowed
conclusions to be drawn about which measurements could be reconciled with other
experimental measurements. It should be noted here that the MUM-PCE methodology has
begun to attract attention within the combustion field [1, 2].
Reaction kinetics in combustion chemistry has been called an ill-defined problem due to
the large number of rate parameters that make up a chemical reaction model and the relative
sparsity of experimental data against which the model can be compared. MUM-PCE was
developed in an attempt to address this ill-defined nature, although it is shown to be limited
by a small experimental dataset with respect to the number of free rate parameters. As a
means of addressing this problem, we have suggested using reaction rate estimates
themselves as experimental data for this purpose. Furthermore, when using an experiment as
a target for optimization, it was shown to be a necessary condition that the experimental
uncertainty be less than the model prediction uncertainty. A rigorous uncertainty analysis is
therefore crucial to obtaining a physically realistic model, since an uncertainty that is
152
unreasonably small may result in an experiment being removed as inconsistent, while one
that is too large does not provide any information.
The long-standing practice for hydrocarbon oxidation model development has been to
compile models in a hierarchical fashion, beginning from H
2
/CO oxidation and building
through methane, ethylene, and so on to larger hydrocarbons. In principle, this methodology
should lead to a self-consistent set of kinetic and thermochemical parameters. However, it
has never been clear whether the hierarchical structure would eventually result in a
converged foundation. An information index was included in MUM-PCE to address this
concern. Fundamental combustion data for H
2
/CO and C
2
H
4
were shown to provide very
little constraint on each other, indicating that both sets of data are needed in order to define
the kinetic foundation. However, these data sets provide a significant constraint on
predictions for C
3
H
8
combustion, as well as n-pentane and n-heptane. It is then shown that a
suitable kinetic foundation for n-alkane oxidation can be established by applying constraint
from data up to n-pentane.
With the kinetic foundation established, the effect of introducing additional experiments
to the database was addressed. In the first case, a new set of laminar mass burning rate
measurements in H
2
/CO flames was combined with the pre-existing H
2
/CO experimental
database. These new measurements were used to develop an improved H
2
/CO submodel for
the USC-Mech II which is able to reproduce a wide range of the new high-pressure burning
rate measurements. Furthermore, the precision of the model predictions was increased
substantially, with strong constraints on the high-pressure HO
2
chemistry. However, it was
found that the new model is unable to reproduce some of the older flame speed
measurements, and there are still some of the new measurements that cannot be reconciled.
These results indicate that, if the new burning measurements are correct, there is still
considerable work to be done in properly characterizing hydrogen combustion.
153
In the second case, a set of multispecies time histories in n-heptane oxidation in shock
tubes was presented. Unlike the previous case of H
2
/CO burning rates, measurements of this
type had only recently been performed with great precision, and it was expected that they
would be able to improve the accuracy and precision of a modified JetSurF model. It was
found that the prediction precision in the species time histories was improved by almost a
factor of 3, while that of the global combustion properties was improved by a factor of 2.
The time-history measurements were found to provide strong constraints on parent-fuel
breakup, CH
x
and propene reaction rates, while the global combustion properties were
limited to heat release rates. It should be noted, however, that the global flame property
measurements are still necessary to properly characterize fuel combustion.

7.2 Future Work
The dissertation here has shown that the comprehensive model development approach,
MUM-PCE, is possible and can produce useful results. It is hoped that this methodology will
represent a change in the way that chemical models are developed. The next step in the
implementation of the method is to move to a web-based architecture. In this
implementation, a user wishing to perform simulations would choose a particular fuel such as
H
2
, CH
4
, or iso-octane, and a particular application, such as combustion at elevated pressures
with NO
x
formation. The software will then dynamically generate a chemical model and
suggest several optimization targets suitable for the user’s desired application. Crucially,
such an implementation would also allow the user to propose experiments, adding the results
of simulations to the experimental database. In this way, an experimenter could explore the
effect of a particular experiment on the model, thereby determining how precisely the
experiment must be performed in order to apply a significant constraint.
154
The studies addressed here only addressed n-alkanes up to n-heptane. This leaves a large
set of potential jet fuel surrogate components still to be characterized, including branched
alkanes, cyclic alkanes, and aromatics. The JetSurF project is intended to be a
comprehensive experimental database for these types of fuels, along with a chemical model.
Furthermore, an understanding of soot formation will also be crucial for a complete
understanding of combustion processes. As such, MUM-PCE can applied to guide the
continued development of the JetSurF model.
These expected modeling studies will require an extensive experimental database in order
to properly constrain the model. It is still, however, an open question of how such a database
should be built. A conclusion presented here was that a suitable kinetic foundation for n-
alkane flame speeds need only consider fuels up to pentane. Ignition delay times, on the
other hand, appear to converge far less rapidly. The question remains, therefore, what
experiments can be suitably described by the n-pentane foundation, which may be described
by an expanded foundation, and which will not converge at all. For instance, the combustion
of aromatic compounds such as toluene and xylene will likely need to include experiments on
benzene oxidation to constrain their kinetic foundations. Detailed species time histories, due
to their sensitivity to parent-fuel breakup, will likely have no such foundation. As has been
shown, MUM-PCE can provide a direct answer to this question, guiding not only the
development of the model itself but also the compilation of the experimental database.

155
7.3 Chapter 7 Endnotes

1. M. Sander, R. I. A. Patterson, A. Braumann, A. Raj, M. Kraft, Proc. Combust. Inst. In
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2. R. H. West, R. A. Shirley, M. Kraft, C. F. Goldsmith, W. H. Green, Combust. Flame
156 (2009) 1764-1770.

156
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167
Appendix A Supplementary Materials for Chapter 4

Table A-1. List of rate parameters in posterior models.
Model I Model II Model III Model IV
f
c
x* A*/A
0
x* A*/A
0
x* A*/A
0
x* A*/A
0

1 H+O
2
↔ O+OH 1.2 0.26 1.05 0.13 1.02 0.34 1.06 0.37 1.07
2 O+H
2
↔ H+OH 1.3 -0.25 0.94 0.00 1.00 -0.19 0.95 -0.39 0.90
3 OH+H
2
↔ H+H
2
O 1.3 -0.64 0.84 0.00 1.00 -0.74 0.82 -0.65 0.84
4 2OH ↔ O+H
2
O 1.3 -0.03 0.99 0.00 1.00 0.00 1.00 -0.03 0.99
5 2H+M ↔ H
2
+M 2 0.07 1.05 0.00 1.00 0.06 1.04 0.24 1.18
6 2H+H
2
↔ 2H
2
2 0.00 1.00 0.00 1.00 0.01 1.01 0.01 1.01
7 2H+H
2
O ↔ H
2
+H
2
O 2 0.12 1.09 0.00 1.00 0.19 1.14 0.17 1.13
9 H+OH+M ↔ H
2
O+M 2 0.51 1.43 -0.09 0.94 0.31 1.24 0.43 1.35
10 O+H+M ↔ OH+M 2 0.10 1.07 0.00 1.00 0.07 1.05 0.18 1.14
12 H+O
2
+M ↔ HO
2
+M 1.2 -0.52 0.91 0.10 1.02 -0.57 0.90 -0.48 0.92
13 H+O
2
+H
2
↔ HO
2
+H
2
1.2 0.00 1.00 0.00 1.00 -0.04 1.00 -0.04 1.00
17 H+O
2
+H
2
O ↔ HO
2
+H
2
O 1.2 0.15 1.03 0.15 1.03 0.27 1.05 0.30 1.05
18 H+O
2
+Ar ↔ HO
2
+Ar 1.2 -0.15 0.97 0.00 1.00 -0.15 0.97 -0.15 0.98
19 H+O
2
+He ↔ HO
2
+He 1.2 0.23 1.04 0.00 1.00 0.27 1.05 0.23 1.04
20 H
2
+O
2
↔ HO
2
+H 1.3 0.16 1.04 0.01 1.00 0.12 1.03 0.26 1.07
22 HO
2
+H ↔ O+H
2
O 3 0.04 1.05 0.00 1.00 0.00 1.00 0.26 1.32
23 HO
2
+H ↔ 2OH 2 -0.11 0.93 -0.32 0.80 -0.15 0.90 -0.21 0.87
24 HO
2
+O ↔ OH+O
2
2 -0.10 0.93 0.00 1.00 -0.09 0.94 -0.05 0.97
25 2HO
2
↔ O
2
+H
2
O
2
2 0.04 1.03 0.00 1.00 0.05 1.03 0.03 1.02
26 2HO
2
↔ O
2
+H
2
O
2
2 -0.16 0.89 0.00 1.00 -0.15 0.90 -0.26 0.83
27 OH+HO
2
↔ H
2
O+O
2
2 0.43 1.34 0.33 1.25 0.47 1.39 0.14 1.10
28 H
2
O
2
+H ↔ HO
2
+H
2
2 0.03 1.02 0.00 1.00 0.02 1.02 -0.01 0.99
29 H
2
O
2
+H ↔ OH+H
2
O 5 -0.16 0.77 0.00 1.00 -0.18 0.75 -0.09 0.86
33 CO+O(+M) ↔ CO
2
(+M) 2 -0.03 0.98 0.00 1.00 -0.05 0.97 0.15 1.11
34 CO+OH ↔ CO
2
+H 1.2 -0.29 0.95 -0.41 0.93 -0.41 0.93 -0.51 0.91
35 CO+OH ↔ CO
2
+H 1.2 -0.05 0.99 0.00 1.00 0.04 1.01 -0.11 0.98
36 CO+O
2
↔ CO
2
+O 3 -0.02 0.98 0.00 1.00 -0.02 0.97 -0.01 0.99
37 CO+HO
2
↔ CO
2
+OH 2 -0.15 0.90 0.00 1.00 -0.15 0.90 -0.17 0.89
38 HCO+H ↔ CO+H
2
2 0.00 1.00 -0.15 0.90 -0.21 0.87 -0.61 0.66
41 HCO+OH ↔ CO+H
2
O 3 0.00 1.00 0.03 1.03 0.02 1.02 -0.26 0.75
42 HCO+M ↔ CO+H+M 4 0.04 1.06 0.37 1.66 0.23 1.38 -0.45 0.54
43 HCO+H
2
O ↔ CO+H+H
2
O 4 0.00 1.00 -0.07 0.91 -0.07 0.91 0.13 1.19
44 HCO+O
2
↔ CO+HO
2
2 0.00 1.00 0.24 1.18 0.18 1.13 -0.09 0.94
48 CH+H ↔ C+H
2
2 0.00 1.00 0.00 1.00 0.00 1.00 0.30 1.23
51 CH+H
2
↔ CH
2
+H 3 0.00 1.00 0.03 1.03 0.02 1.03 -0.24 0.77
53 CH+O
2
↔ HCO+O 10 0.00 1.00 0.32 2.09 0.31 2.06 0.51 3.23
62 CH
2
+O
2
↔ HCO+OH 2 0.00 1.00 0.07 1.05 0.06 1.04 -0.40 0.76
76 CH
2
*+O
2
↔ H+OH+CO 2 0.00 1.00 0.00 1.00 0.00 1.00 0.21 1.16
168
Table A-1 (Continued)
Model I Model II Model III Model IV
f
c
x* A*/A
0
x* A*/A
0
x* A*/A
0
x* A*/A
0

91 CH
3
+H(+M) ↔ CH
4
(+M) 2 0.00 1.00 0.11 1.08 0.17 1.12 -0.24 0.85
92 CH
3
+O ↔ CH
2
O+H 2 0.00 1.00 0.12 1.09 0.12 1.09 1.00 2.00
93 CH
3
+OH(+M) ↔ CH
3
OH(+M) 5 0.00 1.00 0.25 1.50 0.23 1.45 -0.08 0.88
95 CH
3
+OH ↔ CH
2
*+H
2
O 5 0.00 1.00 -0.13 0.81 -0.08 0.88 0.02 1.02
99 CH
3
+HO
2
↔ CH
3
O+OH 3 0.00 1.00 -0.38 0.66 -0.41 0.64 -0.45 0.61
107 2CH
3
(+M) ↔ C
2
H
6
(+M) 2 0.00 1.00 0.00 1.00 0.00 1.00 0.33 1.26
108 2CH
3
↔ H+C
2
H
5
5 0.00 1.00 0.00 1.00 -0.01 0.99 -0.44 0.49
109 CH
3
+HCCO ↔ C
2
H
4
+CO 10 0.00 1.00 -0.09 0.82 -0.11 0.77 -0.29 0.52
150 HCCO+H ↔ CH
2
*+CO 2 0.00 1.00 -0.05 0.97 -0.04 0.97 -0.23 0.85
152 HCCO+O
2
↔ OH+
2
CO 5 0.00 1.00 -0.04 0.94 -0.05 0.92 0.31 1.64
158 C
2
H
3
(+M) ↔ C
2
H
2
+H(+M) 1.5 0.00 1.00 0.05 1.02 0.06 1.02 -0.44 0.84
161 C
2
H
2
+O ↔ HCCO+H 1.5 0.00 1.00 0.06 1.03 0.07 1.03 0.10 1.04
180 H
2
CC+O
2
↔ 2HCO 3 0.00 1.00 0.08 1.09 0.07 1.08 0.33 1.44
188 CH
2
CO+OH ↔ HCCO+H
2
O 10 0.00 1.00 -0.13 0.74 -0.13 0.74 -0.17 0.68
191 C
2
H
3
+H ↔ C
2
H
2
+H
2
3 0.00 1.00 -0.15 0.85 -0.12 0.88 1.00 3.00
192 C
2
H
3
+H ↔ H
2
CC+H
2
5 0.00 1.00 -0.09 0.87 -0.07 0.89 0.24 1.48
194 C
2
H
3
+O ↔ CH
3
+CO 3 0.00 1.00 0.05 1.06 0.04 1.05 -0.05 0.95
195 C
2
H
3
+OH ↔ C
2
H
2
+H
2
O 3 0.00 1.00 0.01 1.02 0.01 1.01 -0.22 0.79
197 C
2
H
3
+O
2
↔ CH
2
CHO+O 4 0.00 1.00 0.32 1.56 0.31 1.53 -0.53 0.48
198 C
2
H
3
+O
2
↔ HCO+CH
2
O 4 0.00 1.00 0.24 1.39 0.24 1.40 0.27 1.44
216 CH
2
CHO+H ↔ CH
3
+HCO 5 0.00 1.00 0.09 1.16 0.09 1.16 0.07 1.12
252 C
2
H
4
+H(+M) ↔ C
2
H
5
(+M) 3 0.00 1.00 -0.02 0.98 -0.02 0.98 -0.73 0.45
253 C
2
H
4
+H ↔ C
2
H
3
+H
2
2 0.00 1.00 0.09 1.07 0.09 1.06 0.45 1.37
254 C
2
H
4
+O ↔ C
2
H
3
+OH 3 0.00 1.00 0.08 1.09 0.10 1.12 0.10 1.12
255 C
2
H
4
+O ↔ CH
3
+HCO 2 0.00 1.00 0.01 1.01 0.00 1.00 -0.89 0.54
257 C
2
H
4
+OH ↔ C
2
H
3
+H
2
O 2 0.00 1.00 0.28 1.21 0.33 1.26 1.00 2.00
263 C
2
H
4
+CH
2
* ↔ aC
3
H
5
+H 5 0.00 1.00 -0.04 0.94 -0.03 0.96 -0.26 0.66
264 C
2
H
4
+CH
3
↔ C
2
H
3
+CH
4
5 0.00 1.00 0.00 1.00 0.00 1.00 0.35 1.77
287 C
3
H
3
+H ↔ pC
3
H
4
10 0.00 1.00 0.00 1.00 0.00 1.00 0.04 1.09
288 C
3
H
3
+H ↔ aC
3
H
4
10 0.00 1.00 0.00 1.00 0.00 1.00 -0.07 0.84
293 C
3
H
3
+HO
2
↔ pC
3
H
4
+O
2
3 0.00 1.00 0.00 1.00 0.00 1.00 -0.04 0.96
308 aC
3
H
4
+H ↔ aC
3
H
5
2 0.00 1.00 0.00 1.00 0.00 1.00 -0.09 0.94
326 pC
3
H
4
+CH
3
↔ C
3
H
3
+CH
4
2 0.00 1.00 0.00 1.00 0.00 1.00 0.06 1.04
328 aC
3
H
5
+H(+M) ↔ C
3
H
6
(+M) 3 0.00 1.00 0.00 1.00 0.00 1.00 -0.27 0.74
337 aC
3
H
5
+HO
2
↔ OH+C
2
H
3
+CH
2
O 3 0.00 1.00 0.00 1.00 0.00 1.00 0.62 1.98
362 C
3
H
6
+H ↔ C
2
H
4
+CH
3
2 0.00 1.00 0.00 1.00 0.00 1.00 -0.09 0.94
363 C
3
H
6
+H ↔ aC
3
H
5
+H
2
2 0.00 1.00 0.00 1.00 0.00 1.00 0.06 1.04
367 C
3
H
6
+O ↔ C
2
H
3
CHO+
2
H 1.3 0.00 1.00 0.00 1.00 0.00 1.00 -0.23 0.94
372 C
3
H
6
+OH ↔ aC
3
H
5
+H
2
O 2 0.00 1.00 0.00 1.00 0.00 1.00 -0.13 0.91
376 C
3
H
6
+CH
3
↔ aC
3
H
5
+CH
4
1.4 0.00 1.00 0.00 1.00 0.00 1.00 0.06 1.02
169
Table A-1 (Continued)
Model I Model II Model III Model IV
f
c
x* A*/A
0
x* A*/A
0
x* A*/A
0
x* A*/A
0

413 C
3
H
8
+H ↔ H
2
+iC
3
H
7
3 0.00 1.00 0.00 1.00 0.00 1.00 0.02 1.02
556 1-C
4
H
8
+H ↔ C
3
H
6
+CH
3
5 0.00 1.00 0.00 1.00 0.00 1.00 -0.71 0.32
815 PXC
5
H
11
(+M) ↔ C
2
H
4
+nC
3
H
7
(+M) 2.5 0.00 1.00 0.00 1.00 0.00 1.00 -0.17 0.85
836 pC
4
H
9
+CH
3
↔ NC
5
H
12
2.5 0.00 1.00 0.00 1.00 0.00 1.00 0.03 1.03
881 C
6
H
12
+H ↔ S
2
XC
6
H
11
+H
2
2.5 0.00 1.00 0.00 1.00 0.00 1.00 -0.16 0.86
952 S
2
XC
7
H
15
(+M) ↔ nC
3
H
7
+C
4
H
81
(+M) 2.5 0.00 1.00 0.00 1.00 0.00 1.00 -0.08 0.93
953 S
2
XC
7
H
15
(+M) ↔ C
6
H
12
+CH
3
(+M) 2.5 0.00 1.00 0.00 1.00 0.00 1.00 -0.28 0.77
976 pC
4
H
9
+nC
3
H
7
↔ NC
7
H
16
2.5 0.00 1.00 0.00 1.00 0.00 1.00 -0.07 0.94
978 NC
7
H
16
+H ↔ SXC
7
H
15
+H
2
2.5 0.00 1.00 0.00 1.00 0.00 1.00 -0.06 0.94
979 NC
7
H
16
+H ↔ S
2
XC
7
H
15
+H
2
2.5 0.00 1.00 0.00 1.00 0.00 1.00 0.09 1.08
980 NC
7
H
16
+H ↔ S
3
XC
7
H
15
+H
2
2.5 0.00 1.00 0.00 1.00 0.00 1.00 -0.01 0.99

Table A-2. Active parameters for H
2
/CO experimental targets
Laminar Flame Speeds
Flat
Flame Ignition Delay Time Flow Reactor.
11 12 13 14 15 16 17 18 19 20 21 22 25 26 33 34 35 36 37 38 39 40 41 42 43 44 48 49 50 51 52 53 54 55 56
1 x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x
2 x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x
3 x x x x x x x x x x x x x x x x x x x x x x x x x x x x x
4 x x x x x x x x x x x
5 x x x x x x x x x x x x x
6 x x
7 x x x x x x x x x x
9 x x x x x x x x x x x x x x x x x x x x x x
10 x x x x x x x x x x x x
12 x x x x x x x x x x x x x x
13 x x x x x x x x x x x x
14 x x x x x x x x
17 x x x x x x x x x x x x x x x x x
18 x x x x x x x x x x x x
19 x x x x x x
20 x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x
21 x x x x x
22 x x x x
23 x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x
24 x x x x x x x x x x x x x x x
25 x x
26 x x x x x x
27 x x x x x x x x x x x x x x x x x x x x x x x x x x x
28 x x x x x x
29 x x
33 x x x x x x x x x
34 x x x x x x x x x x x x x x
35 x x x x x x x x x x x x x x
36 x x x x
37 x x x
38 x x x x x
42 x x x
170

Table A-3. Active parameters for C
2
H
4
experimental targets.

Laminar Flame Speeds
Flat
Flame Ignition Delay Time
Flow
Reactor.
1 2 3 4 5 6 7 8 9 10 25 26 27 28 29 30 31 32 33 45 46 47
1 x x x x x x x x x x x x x x x x x x x x
9 x x x x x x x x x
12 x x
17 x x x x x
20 x x x x x x
23 x x x x x x x x x x x x x x x x x
27 x x x x x x x x x x x x x x x
34 x x x x x x
38 x x x x x x x x x x x x x x x x x
41 x x x x x x x x x
42 x x x x x x x x x x x x x x x x x x x x x
43 x x x x x x x x x
44 x x x x x x x x x x x x
51 x x x x x x x
53 x x x x x x x
62 x x x x x x x x x x
63 x x x
76 x x x x x x x x
91 x x x x x x x x x x x x x x x x
92 x x x
93 x x x x x x x
95 x x x x x x x x x x x x x x x x x
99 x x x x x x x x x x x x x x x x
108 x x x x x x x x x x x x x x
109 x x x x x x
150 x x x x x x
152 x x x x x x x x x x x x
158 x x x x x x x x x x x x x x
161 x x x x
180 x x x x x x x x x x
188 x x x x x
191 x x x x x x x x x x x x x x x x
192 x x x x x x x x x x x x x x x
194 x x
195 x x x x x x x x x
197 x x x x x x x x x x x x x x x x x x x x
198 x x x x x x x x x x x x
216 x x x
252 x x x
253 x x x x x
254 x x x x x x x x x x x x x x x x x x x x
255 x x x x x x x x x x x x x x x x x x x
257 x x x x x x x x x x x x x x x x x x x x x
263 x x x

171

Table A-4. Active parameters for C
3
experimental targets.
Laminar Flame Speed Ignition Delay Time
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
1 x x x x x x x x x x x x x x x x x x x x
9 x x x x
12 x
17 x x
23 x
26 x x
27 x x x x x
34 x x x x x x x x
38 x x x x x x x
42 x x x x x x x x x x x x
43 x x x x x x
44 x x x x x
62 x x x x x
91 x x x x x x x x
92 x x x x
95 x x x x x x x x x x
99 x x x x x x x x x x x
107 x x x x x
152 x x x x x x
158 x x x
161 x x x x x x
173 x x x x x x x x x x x x
191 x x x
197 x x
279 x
287 x x x x x x x x
288 x x x x
290 x x x
292 x x
293 x x x x x x x x
326 x x x x
328 x x x x x
362 x
363 x x x x x x x
367 x
372 x x x x x x
413 x x x
718 x x x x x

172
Table A-5. Active parameters for n-C
5
H
12
and n-C
7
H
16
experimental targets.

Laminar flame speed Ignition Delay Time

Pentane Heptane Pentane Heptane
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
1 x x x x x x x x x x x x x x x x x x x x x x x x
2 x x x
3 x x x x
9 x x x x x
12 x x
17 x x x x
20 x x x x x x x x x x x x x x x x x x
21
23 x x x x x x x
26 x
27 x x x x x x x x x x x x x x x x x
34 x x x x x x x
35
38 x x x x x x x x x x x x x x
41 x x x
42 x x x x x x x x x x x x x x x x x x
43 x x x x x x x x
44 x x x x x x x x x x
48 x
53 x
62 x x x x x x x x
63 x
76 x x x x
91 x x x x x x x x x x x x x x x x x x x x
92 x x x x x x x x x x x x x x
95 x x x x x x x x x x x x x x x x x
99 x x x x x x x x x x x x x x x x x x x x x
107 x x x x x x x x x x x x
108 x x x x x x
150 x x x
152 x
158 x x x x x x x x x x x x x x x
161 x x x x x x
191 x x x x x x x x x
192 x x x x
197 x x x x x x x x x x x x x x x x x x x x x
198 x x x x x
252 x x x x x
253 x x x x x
254 x x x x x x x x x x x x x
255 x x x x x x x x x x x x
257 x x x x x x x x x x x x x x x x x x x x x
264 x x x x
308 x x x x x x
328 x x x x x x x x x x x x x x x x x x x x x
337 x x x x x x
362 x x x
363 x x x x x x x x x x x x x x x x x x x x x x
372 x x x x x x x x x x x x x x x x x x x x
556 x x x x x x x x x x
815 x x x x
173
Table A-5 (Continued)
Laminar flame speed Ignition Delay Time
Pentane Heptane Pentane Heptane
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
836 x x x x x x
840 x x x x x x x
881 x
952 x
953 x
976 x x x x x x x x x x
978 x x x x x x x x
979 x x x x x x
980 x

174
Appendix B Supplementary Materials for Chapter 6
Table B-1. Active rate parameters and experimental targets
a

Flame
Speed
Ignition Delay Time OH Species
History
CO 2 H 2O History CH 3 History
ia ib ic ii iii i ii i i ii i ii iii
a b c d a b a b c a b a b a b a b c a b c d a b a b c a b c a b c a b a b
1 H+O 2 ↔ O+OH
x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x 0 0 0 x 0 x x
9 H+OH+M ↔ H 2O+M
x x x 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
17 H+O 2+H 2O ↔ HO 2+H 2O
x x 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
18 H+O 2+Ar ↔ HO 2+Ar
x x x 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x x x 0 0 0 0 x x 0 x 0 0 0 0 0 0 0 0 0
20 H 2+O 2 ↔ HO 2+H
x x x x 0 x x x x x x x x x x 0 x x 0 0 0 0 0 x 0 0 x x 0 0 0 0 0 0 0 x x
21 2OH(+M) ↔ H 2O 2(+M)
x 0 0 0 0 0 0 0 0 0 0 0 0 0 x 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
23 HO 2+H ↔ 2OH
x x x 0 0 0 0 x 0 0 0 0 0 x 0 x x x x x 0 0 x x x 0 x x 0 0 0 0 0 0 0 x x
27 OH+HO 2 ↔ H 2O+O 2
x x 0 0 0 x x 0 x x x 0 0 x x 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
32 H 2O 2+OH ↔ HO 2+H 2O
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
34 CO+OH ↔ CO 2+H
x x 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x 0 0 0 0 0 0 0 0 0
38 HCO+H ↔ CO+H 2
x x x x 0 0 x x 0 0 0 x 0 x 0 0 0 x 0 0 0 0 0 0 0 0 0 x 0 0 0 0 0 0 0 0 x
41 HCO+OH ↔ CO+H 2O
x x 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
42 HCO+M ↔ CO+H+M
x x x x 0 0 x x 0 x 0 x 0 x x 0 0 x 0 0 0 0 0 0 0 0 x x 0 0 0 0 0 0 0 0 x
43 HCO+H 2O ↔ CO+H+H 2O
x x x x 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
44 HCO+O 2 ↔ CO+HO 2
x 0 0 0 0 0 0 0 x 0 x 0 0 x 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x 0
51 CH+H 2 ↔ CH 2+H
0 0 0 0 0 0 0 0 0 0 0 x 0 0 0 x x x 0 0 0 0 0 x 0 0 x x 0 0 0 0 0 0 0 x x
53 CH+O 2 ↔ HCO+O
0 x x x 0 0 0 x 0 0 0 0 0 0 0 0 0 x 0 0 0 0 0 0 0 0 x x 0 0 0 0 0 0 0 x x
62 CH 2+O 2 ↔ HCO+OH
0 0 0 x 0 0 0 x 0 0 0 x x 0 0 x x x x x 0 0 x x x 0 0 x x 0 0 0 0 0 0 x x
63 CH 2+O 2 ↔ CO 2+2H
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x 0 0 0 0 0 0 0 0 x x x 0 0 0 0 0 0 0 0
91 CH 3+H(+M) ↔ CH 4(+M)
x x x x 0 x x x x x x x x x 0 x x x x x x 0 x x x x x x x 0 x 0 x x x x x
92 CH 3+O ↔ CH 2O+H
0 0 0 0 0 x x x x x 0 x x x x x 0 0 x x 0 0 x x x 0 x 0 x 0 0 0 x x x x x
93 CH 3+OH(+M) ↔ CH 3OH(+M)
x 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x x x x x 0 0 x x x x x 0 x 0 0 0 0 0 0 x 0
95 CH 3+OH ↔ CH 2*+H 2O
x x x x 0 x x x x x x 0 x x x x x x x x x 0 x x x x x x x x x x x x x x x
99 CH 3+HO 2 ↔ CH 3O+OH
x x x x 0 x x x x x x 0 x x x 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x 0
107 2CH 3(+M) ↔ C 2H 6(+M)
0 0 0 0 x x 0 0 x x x 0 x 0 x 0 0 x 0 0 0 0 0 0 0 0 0 0 x 0 x x x x x x x
108 2CH 3 ↔ H+C 2H 5
x x x x x 0 0 0 0 0 0 0 x 0 0 0 0 x x 0 0 0 x 0 0 0 0 0 0 0 x 0 0 x 0 x x
152 HCCO+O 2 ↔ OH+2CO
x x x x 0 0 0 x 0 0 0 0 0 0 0 0 0 x 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
158 C 2H 3(+M) ↔ C 2H 2+H(+M)
0 0 x x x 0 0 0 0 0 0 x 0 0 0 0 0 x 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x
191 C 2H 3+H ↔ C 2H 2+H 2
x x x x 0 0 x x 0 0 0 x x 0 0 0 0 x 0 0 0 0 0 0 0 0 x x 0 0 0 0 0 0 0 0 x
192 C 2H 3+H ↔ H 2CC+H 2
0 x x x 0 0 0 x 0 0 0 x 0 0 0 0 0 x 0 0 0 0 0 0 0 0 x x 0 0 0 0 0 0 0 0 x
197 C 2H 3+O 2 ↔ CH 2CHO+O
x x x x x x x x x x x x x x x 0 0 0 0 x 0 0 0 0 0 0 0 0 x 0 0 0 x 0 x x x
198 C 2H 3+O 2 ↔ HCO+CH 2O
0 0 0 x x 0 0 0 0 0 0 0 0 0 x 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
252 C 2H 4+H(+M) ↔ C 2H 5(+M)
x x x x 0 0 0 0 0 0 0 0 0 0 0 x 0 0 x 0 0 0 x 0 0 0 0 0 0 0 x 0 0 x 0 x x
254 C 2H 4+O ↔ C 2H 3+OH
x 0 0 0 0 x x x x x x x x x x x x x 0 x x 0 x x 0 x x x x 0 0 0 0 0 0 x x
255 C 2H 4+O ↔ CH 3+HCO
0 0 0 0 0 x 0 0 x x x x x 0 0 0 x x 0 0 0 0 0 0 0 0 x x 0 0 0 0 x x x x x
257 C 2H 4+OH ↔ C 2H 3+H 2O
x x x 0 0 x x 0 x x x x 0 x x x x x x x x 0 x 0 0 0 0 0 x 0 0 0 0 0 x x 0
308 aC 3H 4+H ↔ aC 3H 5
0 0 0 0 x x x 0 0 x 0 0 x 0 0 x x x 0 x x 0 0 x 0 x x 0 x 0 0 0 x 0 0 0 x
328 aC 3H 5+H(+M) ↔ C 3H 6(+M)
x x x x x x x x x x x x x x x x x x x x x x x x x x x x x 0 0 0 x 0 0 x x
362 C 3H 6+H ↔ C 2H 4+CH 3
x
363 C 3H 6+H ↔ aC 3H 5+H 2
x x x x X x x x x x x x x x x x x x 0 x x 0 0 x x x x x x 0 0 0 x 0 0 x x
175

Table B-1. (Continued)
Flame
Speed
Ignition Delay Time OH Species
History
CO 2 H 2O History CH 3 History
ia ib ic ii iii i ii i i ii i ii iii
a b c d a b a b c a b a b a b a b c a b c d a b a b c a b c a b c a b a b
372 C 3H 6+OH ↔ aC 3H 5+H 2O x x 0 0 0 x x 0 x x x 0 x x x x x x 0 x x 0 0 0 0 0 x 0 x 0 0 0 0 0 0 x x
556 C 4H 81+H ↔ C 3H 6+CH 3 x x x x 0 x x 0 x x x 0 x 0 x x x x x x x 0 x x x x x 0 x 0 x x 0 x 0 x x
797 SXC 5H 9 ↔ C 5H 8-14+H 0 0 0 0 0 0 0 0 x 0 0 0 0 0 x 0 0 0 x x 0 0 0 0 x 0 0 0 0 0 0 0 0 0 0 x 0
801 C 5H 10 ↔ C 2H 5+aC 3H 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x x 0 0 0 0 x 0 0 0 0 0 0 0 0 0 0 0 0
815 PXC 5H 11(+M) ↔ C 2H 4+nC 3H 7(+M) 0 0 0 0 x 0 0 0 0 0 0 0 0 0 0 0 0 0 x 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
875 C 6H 12 ↔ aC 3H 5+nC 3H 7 0 0 0 x 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x 0 0 0 0 0 0
881 C 6H 12+H ↔ S 2XC 6H 11+H 2 x 0 x x 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
890 PXC 6H 13(+M) ↔ C 2H 4+pC 4H 9(+M) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x x
951 SXC 7H 15(+M) ↔ pC 4H 9+C 3H 6(+M) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x 0 0 0 0 0 x 0 0 0 0 0 x 0 0 x 0 x 0
952 S2XC 7H 15(+M) ↔ nC 3H 7+1-C 4H 8(+M) 0 0 0 x 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x 0
953 S2XC 7H 15(+M) ↔ C 6H 12+CH 3(+M) x 0 0 x 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x 0 0 0 0 0 0 0 0 0 0 0 x 0 0 0 0 0 0
974 PXC 6H 13+CH 3 ↔ NC 7H 16 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x 0 0 x 0 0 0 0 0 x 0 0 0 x 0 x 0 0 x 0 x x
975 PXC 5H 11+C 2H 5 ↔ NC 7H 16 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x x x x x 0 0 x x x x x 0 x 0 0 x 0 x x x x
976 pC 4H 9+nC 3H 7 ↔ NC 7H 16 0 0 0 x x x x x x x x x x x x x x x x x x x x x x x x x x 0 x x 0 x x x x
977 NC 7H 16+H ↔ PXC 7H 15+H 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x x x x x 0 0 x x x x x x x 0 x 0 0 0 0 x x
978 NC 7H 16+H ↔ SXC 7H 15+H 2 0 0 0 x x x x x x x x 0 x x x x x x x x x 0 x x x x x x x 0 x x 0 x 0 x x
979 NC 7H 16+H ↔ S2XC 7H 15+H 2 0 0 0 0 0 x x x x x x 0 x x x x x x x x x 0 x x x x x x x 0 x x 0 x x x x
980 NC 7H 16+H ↔ S3XC 7H 15+H 2 0 0 0 0 x 0 0 0 0 0 0 0 0 0 0 0 x x x 0 x 0 x x 0 0 x 0 x 0 x x 0 x 0 0 x
981 NC 7H 16+O ↔ PXC 7H 15+OH 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
982 NC 7H 16+O ↔ SXC 7H 15+OH 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x 0 0 0 x 0 x 0 0 0 0 0 0 0 0 0 0 0 0
983 NC 7H 16+O ↔ S2XC 7H 15+OH 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x 0 0 0 0 0 x 0 0 0 0 0 0 0 0 0 0 0 0
984 NC 7H 16+O ↔ S3XC 7H 15+OH 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x 0 0 0 0 0 x 0 0 0 0 0 0 0 0 0 0 0 0
985 NC 7H 16+OH ↔ PXC 7H 15+H 2O 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x 0 0 x x 0 0 0 0 x x 0 0 x 0 0 0 0 0 0 0 0
986 NC 7H 16+OH ↔ SXC 7H 15+H 2O 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x x 0 0 0 0 x 0 0 0 x 0 0 0 0 0 0 0 0
987 NC 7H 16+OH ↔ S2XC 7H 15+H 2O 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x 0 0 x x 0 0 0 0 x x 0 0 x 0 x 0 0 0 0 0 0
988 NC 7H 16+OH ↔ S3XC 7H 15+H 2O 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x 0 0 0 0 0 0 0 0 0 x 0 0 0 0 0 0 0 0

a
See Table B-2 for the experimental conditions of the targets.

176
Table B-2. Experimental conditions of the target data set.

Laminar flame speed - Equivalence ratio
a 0.7
b 1
c 1.2
d 1.4
Ignition delay time - Temperature (Kelvins)
Series ia ib ic ii iii
a 1485 1666 1474 1701 1499
b 1428 1503 1394 1499 1300
c 1383
Time History - Time (microseconds)
Species CO
2
Series i ii i i ii i ii iii
a 30 30 300 100 100 30 30 30
b 100 100 1000 300 300 100 100 100
c 300 300 1000 300
d 1000
OH H
2
O CH
3

177
Table B-3. List of rate parameters in posterior models
Model I Model II Model III Model IV
x* A*/A
0
x* A*/A
0
x* A*/A
0
x* A*/A
0

1 H+O 2 ↔ O+OH 0.04 1.01 -0.24 0.96 0.13 1.02 -0.21 0.96
9 H+OH+M ↔ H 2O+M 0.00 1.00 0.00 1.00 -0.08 0.95 -0.28 0.82
17 H+O 2+H 2O ↔ HO 2+H 2O 0.00 1.00 0.00 1.00 0.00 1.00 -0.01 0.99
18 H+O 2+Ar ↔ HO 2+Ar 0.00 1.00 0.00 1.00 0.00 1.00 0.01 1.00
20 H 2+O 2 ↔ HO 2+H 0.02 1.01 0.04 1.01 0.00 1.00 0.07 1.02
21 2OH(+M) ↔ H 2O 2(+M) 0.00 1.00 0.00 1.00 0.00 1.00 0.00 1.00
23 HO 2+H ↔ 2OH -0.13 0.91 -0.13 0.91 -0.02 0.99 -0.10 0.93
27 OH+HO 2 ↔ H 2O+O 2 0.00 1.00 0.00 1.00 0.02 1.01 -0.02 0.99
32 H 2O 2+OH ↔ HO 2+H 2O 0.00 1.00 0.00 1.00 -0.01 0.99 -0.03 0.97
34 CO+OH ↔ CO 2+H -0.16 0.97 -0.17 0.97 0.06 1.01 0.00 1.00
38 HCO+H ↔ CO+H 2 -0.04 0.97 -0.10 0.94 -0.06 0.96 -0.24 0.84
41 HCO+OH ↔ CO+H 2O 0.00 1.00 0.00 1.00 0.00 1.00 0.00 1.00
42 HCO+M ↔ CO+H+M 0.05 1.08 0.07 1.10 -0.17 0.79 -0.31 0.65
43 HCO+H 2O ↔ CO+H+H 2O 0.00 1.00 0.00 1.00 0.11 1.08 0.40 1.32
44 HCO+O 2 ↔ CO+HO 2 0.00 1.00 -0.07 0.95 0.07 1.05 0.01 1.01
51 CH+H 2 ↔ CH 2+H 0.21 1.26 0.31 1.41 -0.03 0.97 0.27 1.34
53 CH+O 2 ↔ HCO+O 0.09 1.22 0.11 1.28 0.00 1.00 0.06 1.16
62 CH 2+O 2 ↔ HCO+OH -0.51 0.70 -0.46 0.73 0.07 1.05 -0.48 0.72
63 CH 2+O 2 ↔ CO 2+ 2H -0.07 0.95 -0.13 0.92 0.02 1.02 -0.02 0.99
91 CH 3+H(+M) ↔ CH 4(+M) 0.02 1.02 -0.06 0.96 -0.08 0.95 -0.10 0.94
92 CH 3+O ↔ CH 2O+H 0.03 1.02 0.41 1.32 0.01 1.00 0.39 1.31
93 CH 3+OH(+M) ↔ CH 3OH(+M) 0.74 3.30 0.77 3.43 0.00 1.00 0.70 3.09
95 CH 3+OH ↔ CH 2*+H 2O 0.35 1.75 0.44 2.04 0.06 1.10 0.42 1.95
99 CH 3+HO 2 ↔ CH 3O+OH 0.00 1.00 -0.05 0.94 0.01 1.01 -0.13 0.86
107 2CH 3(+M) ↔ C 2H 6(+M) -0.01 0.99 -0.08 0.94 -0.12 0.92 -0.02 0.99
108 2CH 3 ↔ H+C 2H 5 0.34 1.72 0.42 1.97 -0.10 0.85 0.20 1.39
152 HCCO+O 2 ↔ OH+2CO 0.08 1.13 -0.03 0.95 0.00 1.00 -0.05 0.93
158 C 2H 3(+M) ↔ C 2H 2+H(+M) 0.07 1.03 0.20 1.08 0.03 1.01 0.14 1.06
191 C 2H 3+H ↔ C 2H 2+H 2 -0.08 0.92 -0.10 0.89 -0.10 0.89 -0.18 0.82
192 C 2H 3+H ↔ H 2CC+H 2 -0.08 0.88 -0.10 0.86 0.00 1.00 -0.10 0.85
197 C 2H 3+O 2 ↔ CH 2CHO+O 0.04 1.06 -0.20 0.76 0.01 1.01 -0.24 0.72
198 C 2H 3+O 2 ↔ HCO+CH 2O 0.00 1.00 0.00 1.00 -0.06 0.92 0.03 1.04
252 C 2H 4+H(+M) ↔ C 2H 5(+M) -0.21 0.79 -0.42 0.63 0.05 1.06 -0.28 0.73
254 C 2H 4+O ↔ C 2H 3+OH -0.36 0.67 -0.31 0.71 -0.04 0.96 -0.31 0.71
255 C 2H 4+O ↔ CH 3+HCO 0.11 1.08 -0.38 0.77 -0.09 0.94 -0.47 0.72
257 C 2H 4+OH ↔ C 2H 3+H 2O 0.10 1.07 0.11 1.08 -0.06 0.96 0.15 1.11
308 aC 3H 4+H ↔ aC 3H 5 0.15 1.11 0.23 1.18 0.08 1.06 0.45 1.37
328 aC 3H 5+H(+M) ↔ C 3H 6(+M) -0.17 0.83 -0.24 0.77 -0.06 0.94 -0.22 0.79
362 C 3H 6+H ↔ C 2H 4+CH 3 0.00 1.00 0.00 1.00 0.01 1.01 0.04 1.03
363 C 3H 6+H ↔ aC 3H 5+H 2 -0.29 0.82 -0.22 0.86 -0.03 0.98 -0.20 0.87
372 C 3H 6+OH ↔ aC 3H 5+H 2O -0.17 0.89 -0.16 0.90 0.00 1.00 -0.23 0.85
556 C 4H 81+H ↔ C 3H 6+CH 3 0.09 1.16 0.12 1.21 0.08 1.14 0.21 1.41
797 SXC 5H 9 ↔ C 5H 8-14+H 0.00 1.00 0.00 1.00 0.00 1.00 0.00 1.00
801 C 5H 10 ↔ C 2H 5+aC 3H 5 0.05 1.04 0.05 1.05 0.00 1.00 0.07 1.06
815 PXC 5H 11(+M) ↔ C 2H 4+nC 3H 7(+M) -0.03 0.97 -0.03 0.98 0.01 1.01 0.01 1.01
875 C 6H 12 ↔ aC 3H 5+nC 3H 7 0.00 1.00 -0.03 0.97 0.00 1.00 -0.02 0.98
881 C 6H 12+H ↔ S 2XC 6H 11+H 2 0.00 1.00 0.00 1.00 -0.03 0.98 -0.07 0.94
890 PXC 6H 13(+M) ↔ C 2H 4+pC 4H 9(+M) 0.00 1.00 -0.04 0.96 0.00 1.00 -0.07 0.94
951 SXC 7H 15(+M) ↔ pC 4H 9+C 3H 6(+M) 0.12 1.12 0.09 1.09 0.01 1.01 0.10 1.09
178

Table B-3. (Continued)
Model I Model II Model III Model IV
x* A*/A
0
x* A*/A
0
x* A*/A
0
x* A*/A
0

952 S2XC 7H 15(+M) ↔ nC 3H 7+1-C 4H 8(+M) -0.03 0.97 0.00 1.00 0.02 1.01 0.06 1.06
953 S2XC 7H 15(+M) ↔ C 6H 12+CH 3(+M) 0.09 1.09 0.00 1.00 -0.03 0.97 -0.06 0.95
974 PXC 6H 13+CH 3 ↔ NC 7H 16 0.14 1.13 0.20 1.20 0.00 1.00 0.38 1.42
975 PXC 5H 11+C 2H 5 ↔ NC 7H 16 0.08 1.08 0.00 1.00 0.00 1.00 0.02 1.02
976 pC 4H 9+nC 3H 7 ↔ NC 7H 16 0.32 1.34 0.36 1.39 -0.06 0.95 0.21 1.21
977 NC 7H 16+H ↔ PXC 7H 15+H 2 0.04 1.04 -0.06 0.94 0.00 1.00 -0.09 0.92
978 NC 7H 16+H ↔ SXC 7H 15+H 2 0.06 1.05 0.04 1.04 0.01 1.01 -0.09 0.92
979 NC 7H 16+H ↔ S2XC 7H 15+H 2 0.01 1.01 -0.13 0.89 0.06 1.06 -0.09 0.92
980 NC 7H 16+H ↔ S3XC 7H 15+H 2 0.07 1.06 0.14 1.14 0.01 1.01 0.13 1.13
981 NC 7H 16+O ↔ PXC 7H 15+OH 0.03 1.03 0.05 1.05 0.00 1.00 0.05 1.05
982 NC 7H 16+O ↔ SXC 7H 15+OH 0.05 1.05 0.06 1.06 0.00 1.00 0.08 1.07
983 NC 7H 16+O ↔ S2XC 7H 15+OH 0.09 1.09 0.10 1.10 0.00 1.00 0.12 1.11
984 NC 7H 16+O ↔ S3XC 7H 15+OH 0.07 1.06 0.07 1.07 0.00 1.00 0.09 1.08
985 NC 7H 16+OH ↔ PXC 7H 15+H 2O 0.20 1.20 0.10 1.10 0.00 1.00 0.13 1.13
986 NC 7H 16+OH ↔ SXC 7H 15+H 2O 0.25 1.26 0.16 1.16 0.00 1.00 0.21 1.21
987 NC 7H 16+OH ↔ S2XC 7H 15+H 2O 0.14 1.13 0.01 1.01 0.00 1.00 0.04 1.03
988 NC 7H 16+OH ↔ S3XC 7H 15+H 2O 0.11 1.11 0.09 1.08 0.00 1.00 0.11 1.10

179
Appendix C Chemical Parameters

The H
2
/CO submodel used in this dissertation is presented in Table C-1. Modifications
are as stated in the text. As noted in the text, the third-body coefficients for the H + O
2

(+M) ↔ HO
2
(+M) reaction are included as active parameters. This requires that the
reaction be split into individual reactions for each major species and rate coefficients be
fit to fixed pressure. These rate coefficients are presented in Table C-2.

Table C-1. List of Arrhenius rate coefficients for the H
2
/CO submodel. Units are moles,
cm, s, and kcal.
A b E
a

H + O
2
↔ O + OH 2.65 · 10
16
-0.70 17041.00
O + H
2
↔ H + OH 3.87 · 10
04
2.70 6260.00
OH + H
2
↔ H + H
2
O 2.16 · 10
08
1.50 3430.00
OH + OH ↔ O + H
2
O 3.57 · 10
04
2.40 -2110.00
H + H + M ↔ H
2
+ M 1.00 · 10
18
-1.00 0.00
Enhanced third-body efficiencies H
2
0.00
H
2
O 0.00
CO
2
0.00
Ar 0.63
He 0.63
H + H + H
2
↔ H
2
+ H
2
9.00 · 10
16
-0.60 0.00
H + H + H
2
O ↔ H
2
+ H
2
O 6.00 · 10
19
-1.30 0.00
H + H + CO
2
↔ H
2
+ CO
2
5.50 · 10
20
-2.00 0.00
H + OH + M ↔ H
2
O + M 2.20 · 10
22
-2.00 0.00
Enhanced third-body efficiencies H
2
2.00
H
2
O 6.30
CO 1.75
CO
2
3.60
Ar 0.38
He 0.38
O + H + M ↔ OH + M 4.71 · 10
18
-1.00 0.00
Enhanced third-body efficiencies H
2
2.00
H
2
O 12.00
CO 1.75
CO
2
3.60
Ar 0.70
He 0.70

180
Table C-1. (Continued)
O + O + M ↔ O
2
+ M 1.20 · 10
17
-1.00 0.00
Enhanced third-body efficiencies H
2
2.40
H
2
O 15.40
CO 1.75
CO
2
3.60
Ar 0.83
He 0.83
H + O
2
(+M) ↔ HO
2
(+M)
a

H
2
+ O
2
↔ HO
2
+ H 7.40 · 10
05
2.40 53502.00
OH + OH ( +M ) ↔ H
2
O
2
( +M ) k
inf
7.40 · 10
13
-0.40 0.00
k
0
1.34 · 10
17
-0.58 -2293.00
Troe centering 0.73 94.00 1756.00 5182.00
Enhanced third-body efficiencies H
2
2.00
H
2
O 6.00
CO 1.75
CO
2
3.60
Ar 0.70
He 0.70
HO
2
+ H ↔ O + H
2
O 3.97 · 10
12
0.00 671.00
HO
2
+ H ↔ OH + OH 7.08 · 10
13
0.00 295.00
HO
2
+ O ↔ OH + O
2
2.00 · 10
13
0.00 0.00
HO
2
+ HO
2
↔ O
2
+ H
2
O
2
1.30 · 10
11
0.00 -1630.00
HO
2
+ HO
2
↔ O
2
+ H
2
O
2
4.20 · 10
14
0.00 12000.00
OH + HO
2
↔ H
2
O + O
2
2.89 · 10
13
0.00 -500.00
H
2
O
2
+ H ↔ HO
2
+ H
2
1.21 · 10
07
2.00 5200.00
H
2
O
2
+ H ↔ OH + H
2
O 2.41 · 10
13
0.00 3970.00
H
2
O
2
+ O ↔ OH + HO
2
9.63 · 10
06
2.00 3970.00
H
2
O
2
+ OH ↔ HO
2
+ H
2
O 2.00 · 10
12
0.00 427.00
H
2
O
2
+ OH ↔ HO
2
+ H
2
O 2.67 · 10
41
-7.00 37600.00
CO + O ( +M) ↔ CO
2
( +M ) k
inf
1.80 · 10
10
0.00 2384.00
k
0
1.55 · 10
24
-2.79 4191.00
Enhanced third-body efficiencies H
2
2.00
H
2
O 12.00
CO 1.75
CO
2
3.60
Ar 0.70
He 0.70
CO + OH ↔ CO
2
+ H 7.05 · 10
04
2.10 -355.70
CO + OH ↔ CO
2
+ H 5.76 · 10
12
-0.70 331.80
CO + O
2
↔ CO
2
+ O 2.53 · 10
12
0.00 47700.00
CO + HO
2
↔ CO
2
+ OH 3.01 · 10
13
0.00 23000.00

181
Table C-1. (Continued)
HCO + H ↔ CO + H
2
1.20 · 10
14
0.00 0.00
HCO + O ↔ CO + OH 3.00 · 10
13
0.00 0.00
HCO + O ↔ CO
2
+ H 3.00 · 10
13
0.00 0.00
HCO + OH ↔ CO + H
2
O 3.02 · 10
13
0.00 0.00
HCO + M ↔ CO + H + M 9.35 · 10
16
-1.00 17000.00
Enhanced third-body efficiencies H
2
2.00
H
2
O 0.00
CO 1.75
CO
2
3.60
HCO + H
2
O ↔ CO + H + H
2
O 1.12 · 10
18
-1.00 17000.00
HCO + O
2
↔ CO + HO
2
1.20 · 10
10
0.80 -727.00
a
Explicit pressure dependence. See Table C-2.

Table C-2. Rate coefficients for H + O
2
(+M) ↔ HO
2
(+M) fits.
A b E
a
p
H + O
2
+ M ↔ HO
2
+ M 5.04 · 10
19
-1.391 22.71 0.03 atm
5.22 · 10
19
-1.397 94.56 1 atm
5.56 · 10
19
-1.404 136.28 2 atm
6.65 · 10
19
-1.427 233.01 5 atm
8.66 · 10
19
-1.459 360.24 10 atm
1.09 · 10
20
-1.486 466.67 15 atm
1.85 · 10
20
-1.55 716.39 30 atm
H + O
2
+ H
2
↔ HO
2
+ H
2
5.04 · 10
19
-1.391 22.71 0.03 atm
5.22 · 10
19
-1.397 94.56 1 atm
5.56 · 10
19
-1.404 136.28 2 atm
6.65 · 10
19
-1.427 233.01 5 atm
8.66 · 10
19
-1.459 360.24 10 atm
1.09 · 10
20
-1.486 466.67 15 atm
1.85 · 10
20
-1.55 716.39 30 atm
H + O
2
+ O
2
↔ HO
2
+ O
2
3.79 · 10
19
-1.391 20.81 0.03 atm
3.86 · 10
19
-1.395 82.11 1 atm
4.04 · 10
19
-1.401 116.59 2 atm
4.64 · 10
19
-1.418 195.74 5 atm
5.72 · 10
19
-1.443 300.05 10 atm
6.90 · 10
19
-1.466 388.39 15 atm
1.09 · 10
20
-1.521 602.02 30 atm

182

Table C-2. (Continued)
H + O
2
+ CO ↔ HO
2
+ CO 6.04 · 10
19
-1.391 24.04 0.03 atm
6.34 · 10
19
-1.398 103.74 1 atm
6.84 · 10
19
-1.408 150.94 2 atm
8.44 · 10
19
-1.433 260.84 5 atm
1.14 · 10
20
-1.47 404.74 10 atm
1.48 · 10
20
-1.501 523.79 15 atm
2.60 · 10
20
-1.569 796.26 30 atm
H + O
2
+CO
2
↔ HO
2
+ CO
2
1.20 · 10
20
-1.39 30.14 0.03 atm
1.37 · 10
20
-1.408 150.94 1 atm
1.58 · 10
20
-1.425 227.25 2 atm
2.29 · 10
20
-1.47 404.74 5 atm
3.68 · 10
20
-1.527 626.36 10 atm
5.20 · 10
20
-1.569 796.26 15 atm
9.19 · 10
20
-1.636 1133.51 30 atm
H + O
2
+ H
2
O ↔ HO
2
+ H
2
O 4.91 · 10
20
-1.381 6.00 0.03 atm
7.94 · 10
20
-1.438 217.44 1 atm
1.23 · 10
21
-1.491 387.83 2 atm
3.25 · 10
21
-1.606 775.39 5 atm
7.86 · 10
21
-1.71 1185.35 10 atm
1.16 · 10
22
-1.753 1436.72 15 atm
1.22 · 10
22
-1.749 1797.85 30 atm
H + O
2
+ He ↔ HO
2
+ He 6.59 · 10
18
-1.191 15.84 0.03 atm
6.45 · 10
18
-1.19 57.52 1 atm
6.56 · 10
18
-1.193 79.65 2 atm
6.98 · 10
18
-1.201 129.31 5 atm
7.77 · 10
18
-1.214 194.33 10 atm
8.60 · 10
18
-1.226 249.99 15 atm
1.13 · 10
19
-1.259 389.59 30 atm
H + O
2
+ Ar ↔ HO
2
+ Ar 6.59 · 10
18
-1.191 15.84 0.03 atm
6.45 · 10
18
-1.19 57.52 1 atm
6.56 · 10
18
-1.193 79.65 2 atm
6.98 · 10
18
-1.201 129.31 5 atm
7.77 · 10
18
-1.214 194.33 10 atm
8.60 · 10
18
-1.226 249.99 15 atm
1.13 · 10
19
-1.259 389.59 30 atm

Abstract (if available)

Abstract Reliable simulations of reacting flow systems require a well-characterized, detailed chemical model as a foundation. Accuracy of such a model can be assured, in principle, by a multi-parameter optimization against a set of experimental data. However, the inherent uncertainties in the rate evaluations and experimental data leave a model still characterized by some finite kinetic rate parameter space. Without a careful analysis of how this uncertainty space propagates into the model's predictions, those predictions can at best be trusted only qualitatively.

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Asset Metadata

Creator Sheen, David Allan (author)

Core Title Spectral optimization and uncertainty quantification in combustion modeling

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Aerospace Engineering

Publication Date 01/25/2011

Defense Date 12/07/2010

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag kinetic modeling,OAI-PMH Harvest,optimization,uncertainty quantification

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Wang, Hai (committee chair), Bickers, Nelson Eugene, Jr. (committee member), Egolfopoulos, Fokion N. (committee member), Ghanem, Roger Georges (committee member)

Creator Email sheen.david@gmail.com,sheen@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-m3626

Unique identifier UC1155282

Identifier etd-Sheen-4250 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-437126 (legacy record id),usctheses-m3626 (legacy record id)

Legacy Identifier etd-Sheen-4250.pdf

Dmrecord 437126

Document Type Dissertation

Rights Sheen, David Allan

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Repository Name Libraries, University of Southern California

Repository Location Los Angeles, California

Repository Email cisadmin@lib.usc.edu