Chemistry Reduction for Laminar Oxyfuel Combustion

Dissertation zur Erlangung des Grades Doktor–Ingenieur

der FakultätfürMaschinenbau der Ruhr–UniversitätBochum

von Valentin N04 Bomba

aus Yegue-Assi, Kamerun

Bochum 2016

Dissertation eingereicht am: 15.12.2016 Tag der mündlichen Prüfung: 01.02.2017 Erstgutachter: Prof. Dr.-Ing. B. Rogg Zweitgutachter: Prof. Dr.-Ing. W. Eifler Contents

Abstract iii

1 Introduction 1 1.1 Motivation for and Overview on the Reduction of Chemical Mechanisms 2 1.2 Previous Work ...... 5

1.2.1 Combustion in O2/CO2 Atmospheres ...... 5 1.2.2 Combustion in Pure Oxygen ...... 5 1.3 Outline of the Present Work ...... 7

2 Combustion Geometry and Governing Equations 8 2.1 Diﬀusion Flames in CounterFlow ...... 8 2.2 Higher-Order Eﬀects ...... 13

3 Thermodynamics and Molecular Transport 15 3.1 Equations of State ...... 15 3.2 Phenomenological Relationships ...... 16 3.2.1 Stress Tensor ...... 16 3.2.2 Heat-Flux Vector ...... 17 3.2.3 Diffusion Velocities ...... 17 3.3 Transport Coefficients ...... 18 3.3.1 Simple Property and Data Models ...... 18 3.3.2 Detailed Property and Data Models ...... 19 3.3.2.1 Dynamic Viscosities ...... 20 3.3.2.2 Thermal Conductivities ...... 20 3.3.2.3 Ordinary-Diffusion Coefficients ...... 22 3.3.2.4 Thermal-Diffusion Coefficients ...... 23 3.4 Heat Capacities, Enthalpies and Entropies ...... 23

4 Chemistry 25 4.1 Ozone Chemistry as an Introductory Example ...... 25

i 4.2 Phenomenological Expressions of Chemical Kinetics ...... 25 4.3 Detailed Mechanisms of Elementary Reactions ...... 27 4.4 Partial Equilibria ...... 28 4.5 Steady States Species ...... 28 4.6 Global Reactions and Global Reaction Mechanisms ...... 30

5 Discretization and Numerical Methods 32 5.1 Formulation ...... 33 5.1.1 Differential Equations ...... 33 5.1.2 Finite Differences ...... 33 5.1.3 Differential Algebraic Systems ...... 35 5.2 Numerical Methods ...... 36 5.2.1 A Modified Newton Method ...... 36 5.2.2 An Extrapolation Method ...... 37 5.3 Adaptive Selection of Grid ...... 37

6 Methods of Reduction of Detailed Reaction Mechanisms to a Skeletal Mechanism 39 6.1 Basics ...... 39 6.2 Directed Relation Graph (DRG) ...... 40 6.3 DRG-Aided Sensitivity Analysis (DRGASA) ...... 42 6.4 Unimportant-Reaction Elimination ...... 43 6.5 Reaction Elimination Based on DRGASA ...... 44 6.6 Summary of Reduction Methods ...... 44

7 Derivation of Reduced Mechanisms of Global Reactions 47 7.1 Notational Issues ...... 47 7.2 Species Splitting ...... 48 7.3 Reaction Splitting ...... 49 7.4 Reduced Stoichiometric Matrix ...... 52 7.5 Global Reaction Rates Determined Systematically ...... 54 7.5.1 Theory of Derivation ...... 54 7.5.2 Sample Derivation ...... 58 7.6 Global Reaction Rates Determined Empirically ...... 60

8 Results for Detailed Mechanisms 64

9 Results for Skeletal Mechanisms 68

ii 9.1 Exploratory Reaction-Pathway Analysis ...... 68 9.2 Oxy-Methane Skeletal Mechanism Valid Over the Entire Strain Rate Range ...... 71 9.3 Oxy-Methane Skeletal Mechanism for Diﬀusion Flames at Low Strain Rates ...... 79

10 Results for Global, Systematically Reduced Mechanisms 83 10.1 Low Strain Rates ...... 83 10.2 Entire Strain Rate Range ...... 88

11 Results for Global, Empirically Reduced Mechanisms 92

12 Summary and Outlook 98

Appendix A Skeletal Mechanisms 100 A.1 Oxy-Methane Skeletal Mechanism for Counterflow Diffusion Flames . . 100 A.2 Extremely Low Strained Oxy-Methane Skeletal Mechanism for Counter- flow Diffusion Flames ...... 102

Appendices 100

Appendix B Reactions, Their Rates, and Associated Quantities 104 B.1 Basic Reaction ...... 104 B.2 Third-Body Reaction ...... 105 B.3 Pressure-Dependent Reaction ...... 105 B.3.1 Unimolecular/Recombination Fall-Oﬀ Reactions ...... 105 B.3.1.1 Lindemann formulation ...... 106 B.3.1.2 Troe formulation ...... 106 B.3.1.3 SRI formulation ...... 107 B.3.2 Chemically Activated Bimolecular Reactions ...... 107

Bibliography 108

iii Abstract

Combustion plays an important role in many practical engineering applications such as industrial furnaces, internal combustion engines and power trains, jet engines, and power plants. Today it has become vital for further environmentally responsible development of such combustion processes – e.g., for decreasing further the formation of pollutants in and hence the emission of pollutants from these processes – that engineers and scientists have available powerful chemistry models that they can employ in spatially and physically realistic and hence complex numerical simulations. To this end, realistic but computationally manageable chemistry models are required. The present thesis makes a contribution in this respect. Specifically, the thesis is concerned with the development of chemical reaction mechanisms for use in CFD codes that are powerful in the sense that they mimic combustion chemistry accurately but at the same time are efficient and hence affordable in terms of computational costs.

The key results obtained in the research work for the present thesis include the derivation of reduced chemistry models for non-premixed combustion of hydrocarbon fuels in pure oxygen rather than air, and the development of computer tools, or methods, that make that derivation possible in an efficient manner. Specifically, a new method for reaction elimination from detailed mechanisms is developed. The results obtained with the new method, combined with traditional methods, include powerful detailed reaction mechanisms for non-premixed oxyfuel combustion and even shorter, and hence ideal for CFD applications, reduced mechanisms consisting only of 11 global reactions steps. It is shown that with the reduced mechanisms developed in the present thesis non-premixed laminar flame structures are predicted that in detail and accuracy are comparable to truly detailed – but due to their shear size not CFD-capable – mechanisms of elementary reactions.

iv Acknowledgments

Fisrt and foremost, I sincerely thank my supervisor Prof. Dr-Ing. B. Rogg for his in- sightful comments, constructive feedback and recommendations throughout the present thesis. In addition, thank you Dear Prof. Rogg for the help provided beyond the academic context.

I am deeply indebted to Prof. Dr-Ing. W. Eiﬂer for his constant support with respect to my scholarship extension requirements.

Thanks to all my colleagues, namely, Dr-Ing. Yin, Shaheen, Baik, Lien, Azizi and Xie of the chair of ﬂuid mechanics at the Ruhr-University Bochum for various form of assistance and the friendly atmosphere. Special thanks to Dr-Ing. Shaheen for the useful discussions about the directed relation graph reduction method.

Thanks are also due to the successive secretaries of our chair, namely Ms S. Hentrich, S. Oltersdorf and U. Boehm for the help provided with respect to administrative issues.

I gratefully acknowledge the ﬁnancial support of the German Academic Exchange Ser- vice (DAAD).

Special thanks to a friend, namely Father G. Chumacera and my family. Speciﬁcally, I thank my mother Albertine and my sister Sidonie for their constant encouragement.

Finally, my heartfelt thanks to my wife Cristelle and our children Daniel and Anaelle for their patience and their exceptional support throughout the time I spent abroad for my PhD thesis. I owe you so much.

v Nomenclature

Roman Symbols

Ai,b pre exponential factor in forward rate constant of reaction i

Ai,f pre exponential factor in forward rate constant of reaction i

3 Ck molar concentration of species k [mol/m ] cp mixture (mass-based) specic heat capacity at constant pressure [J/(kg K)] cpk speciﬁc heat capacity of species k at constant pressure [J/(kg K)]

D characteristic diﬀusion coeﬃcient [C.m]

T Dk thermal diﬀusion coeﬃcient of species k [kg/(m s)]

2 Dk mixture-averaged diﬀusion coeﬃcient of species k [m /s]

2 dk dipole moment of species k [m /s]

Ek error induced to global target parameters due to the elimination of the intermediate species k from the chemical mechanism

Ei,b activation energy of backward step in reaction i [J/mol]

Ei,f activation energy of forward step in reaction i [J/mol]

Fi error induced to global target parameters due to the elimination of reaction i from the chemical mechanism h enthalpy (mass-based) of the gas mixture [J/kg]

Hk molar enthalpy of species k [J/mol]

Hk molar entropy of species k [J/(mol K)] hk enthalpy (mass-based) of species k [J/kg]

vi I number of reactions in the system

IA,i contribution of reaction i to the production rate of species A

2 jk, jkα diﬀusion ﬂux vector of species k [kg/(m s)]

K number of species in the system kB Boltzmann constant

Kc,i equilibrium constant in concentration units for reaction i [depends on reaction] ki,b backward rate constant of reaction i [depends on reaction] ki,f forward rate constant of reaction i [depends on reaction]

3 Kp,i equilibrium constant in pressure units for reaction i [mol/cm ]

Lek Lewis number of species k

LeZ Lewis number of the mixture fraction

[M] total molar concentration of a mixture [depends on reaction] mk mass of the molecule [kg]

3 ωi rate of reaction i [mol/(cm s)] p pressure [Pa]

Pr constant Prandtl number q, qα heat ux vector [W/m]

Vk,Vkα diﬀusion velocity vector of species k [m/s]

R Universal gas constant [J/(kmol K)]

Rc Universal gas constant, in same units as activation energy [J/(kmol K] rAB measure for the deviation of the rate of production of species A if species B is removed from the chemical mechanism

T temperature [K] t time [s]

vii T 0 reference or standard-state temperature [K] u velocity components in the x direction [m/s]

D D Vk ,Vkα mass diﬀusion velocity vector of species k [m/s]

T D Vk ,Viα thermal diﬀusion velocity vector of species k [m/s] v velocity components in the y direction [m/s]

W molecular weight (molar mass) of the mixture [kg/mol]

Wk molecular weight (molar mass)of species k [kg/mol] x spatial coordinate [m]

Xk mole fraction of species k y spatial coordinate [m]

Yk mass fraction of species k

Z mixture fraction zki third-body eﬃciency of species k in reaction i

Greek Symbols

βi Temperature exponent in the rate constant of reaction i

χ scalar dissipation rate [1/s]

δαk unit tensor

λ mixture thermal conductivity [W/(m K)]

λk thermal conductivity of species k [W/(m K)]

µ mixture dynamic viscosity [Ns/m2]

2 µk dynamic viscosity of species k [Ns/m ]

00 νki stoichiometric coeﬃcient of product species k reaction i

0 νki stoichiometric coeﬃcient of reactant species k reaction i

νki stoichiometric coeﬃcient of species k reaction i

viii 3 ωk net mass rate of production of species k [kg/m /s]

3 ωk+ mass rate of creation of species k [kg/m /s]

3 ωk− mass rate of destruction of species k [kg/m /s]

ρ mass density [kg/m3]

σαβ stress tensor [Pa]

ταβ viscous part of stress tensor [Pa]

Superscript

0 referring to low-pressure limit of a rate constant

∞ referring to high-pressure limit of a rate constant

Subscript

0 referring to forward reaction

00 referring to backward reaction o conditions at standard-state pressure and/or temperature

T relating to thermal diﬀusion

Abbreviations

CFD Computational Fluid Dynamics

CSP Computational Singular Perturbation

DNS Direct Numerical Simulation

DRG Directed Relation Graph

DRGASA DRG-Aided Sensitivity Analysis

DRGEP DRG with Error Propagation

DRGEPSA DRG with Error Propagation and Sensitivity Analysis

ILDM Intrinsic Low Dimensional Manifold

IUPAC International Union of Pure and Applied Chemistry

ix LES Large Eddy Simulation

PDF Probability Density Function

PFA Path Flux Analysis

PSR Perfectly-Stirred Reactor

QSSA Quasi-Steady State Approximation

RANS Reynolds Averaged Navier-Stokes

SS Steady State

TSA Targeted Search Algorithm

x List of Figures

2.1 Counterflow geometries ...... 8 2.2 Mass fractions of fuel, oxidizer and major products in an opposed-jet counterflow geometry...... 10 2.3 As Fig. 2.2, but only for temperature T ...... 11 2.4 As Fig. 2.2, but only for mixture fraction Z...... 11 2.5 Mass fractions of fuel, oxidizer and major products of a diffusion flame in mixture-fraction space...... 12 2.6 As Fig. 2.5, but only for temperature T ...... 13

6.1 Flow chart of the skeletal reduction procedure ...... 45

7.1 Comparison of the forward rates of global reaction I obtained respectively with the systematic reduction procedure (red dashed line) – see Eq. (7.62) – and a corresponding ﬁtting function (green line) – see Eq. (7.63)–...... 63

8.1 Profiles of temperature and reactants for an oxy-methane diffusion flame:

100%O2 vs. 65%CH4 - 35%N2, constant scalar dissipation rate χ = 1.71 s−1; lines show numerical predictions using the GRI3.0 mechanism and the mechanisms by Glarborg and Kee and symbols show experimental results of Bennett et al. [11] ...... 65 8.2 Profiles of temperature and reactants for an oxy-methane diffusion flame: −1 100%O2 vs. 20%CH4 - 80%N2, strain rate a = 60 s ; lines show numerical predictions using the GRI3.0 mechanism and symbols show experimental results of Cheng et al. [23] ...... 66 8.3 As Fig. 8.2, but only for products...... 66

9.1 Reaction pathway analysis of C-atoms for an oxy-methane diﬀusion

flame with conditions p = 1 bar, YO2 = 1.0, YCH4 = 1.0, TF uel = −1 TOxidizer = 300 K and a strain rate a = 0.2 s in an opposing flows test bed configuration...... 69

xi 9.2 As in Fig. 9.1, but for H-atoms...... 70 9.3 As in Fig. 9.1, but for O-atoms...... 70

9.4 Comparison of predicted peak flame temperature Tmax as a function of strain rate a as obtained with the GRI3.0 mechanism and the skeletal mechanism of App. A.1...... 71 9.5 Predicted temperature profiles computed using the GRI3.0 mechanism and the skeletal mechanism of App. A.1 for an oxy-methane diffusion −1 flame: 100%O2 vs. 100%CH4 and a strain rate a = 35 000 s ...... 73 9.6 As Fig. 9.5, but only for reactants ...... 74 9.7 As Fig. 9.5, but only for products...... 74 9.8 As Fig. 9.5, but only for intermediate species O and OH...... 75 9.9 As Fig. 9.5, but only for intermediate species H...... 75 9.10 Predicted temperature profiles computed using the GRI3.0 mechanism and the skeletal mechanism of App. A.1 for an oxy-methane diffusion −1 flame: 100%O2 vs. 100%CH4 and a strain rate of 62 000 s ...... 76 9.11 Predicted mass fraction profiles of reactants computed using the GRI3.0 mechanism and the skeletal mechanism of App. A.1 for an oxy-methane −1 diffusion flame: 100%O2 vs. 100%CH4 and a strain rate a = 62 000 s . 77 9.12 As Fig. 9.11, but only for products...... 78 9.13 As Fig. 9.11, but only for intermediate species O and OH...... 78 9.14 As Fig. 9.11, but only for intermediate species H...... 79 9.15 Predicted profiles of temperature computed using the GRI3.0 mechanism and the skeletal mechanism of App. A.2 for a diffusion flame with the

conditions p = 1 bar, YO2 = 1.0, YCH4 = 1.0, TFuel = TOxidizer = 300 K and a strain rate a = 0.2 s−1...... 80 9.16 Predicted mass fraction profiles of reactants computed using the GRI3.0 mechanism and the skeletal mechanism of App. A.2 for a diffusion flame

with the conditions p = 1 bar, YO2 = 1.0, YCH4 = 1.0, TFuel = TOxidizer = 300 K and a strain rate a = 0.2 s−1...... 81 9.17 As Fig. 9.15, but only for intermediate species O and OH...... 81 9.18 As Fig. 9.15, but only for intermediate species H...... 82

10.1 Predicted proﬁles of temperature computed using the GRI3.0 mechanism

and the 9-step mechanism with the conditions p = 1 bar, YO2 = 1.0, −1 YCH4 = 1.0, TFuel = TOxidizer = 300 K and a strain rate a = 0.2 s . . . . 85 10.2 As Fig. 10.1, but only for reactants...... 85

10.3 As Fig. 10.1, but only for intermediate species H2, O and OH...... 86

xii 10.4 As Fig. 10.1, but only for intermediate species H...... 87

10.5 As Fig. 10.1, but only for intermediate species C2H2 and C2H4...... 87

10.6 Comparison of predicted peak ﬂame temperature Tmax as a function of strain rate a as obtained with the GRI3.0 mechanism (dashed line) and the 11-step mechanism (solid line)...... 88 10.7 Predicted temperature proﬁles computed using the GRI3.0 mechanism

and the 11-step mechanism for an oxy-methane diﬀusion ﬂame: 100%O2 −1 vs. 100%CH4 and a strain rate a = 62 000 s ...... 89 10.8 As Fig. 10.7, but for reactants...... 89 10.9 As Fig. 10.7, but for stable products...... 90 10.10As Fig. 10.7, but for intermediate species O and OH...... 90 10.11As Fig. 10.7, but for intermediate species H...... 91

11.1 Comparison between the empirical fitted and the original forward rate for global reaction VIII ...... 94 11.2 Comparison between the empirical fitted and the original forward rate for global reaction V ...... 95 11.3 Predicted temperature profiles computed using the GRI3.0 mechanism and the empirically fitted 11-step mechanism for an oxy-methane diffu- −1 sion flame: 100%O2 vs. 100%CH4 and a strain rate a = 5 s ...... 95 11.4 As Fig. 11.3, but for reactants...... 96 11.5 As Fig. 11.3, but for stable products...... 96 11.6 As Fig. 11.3, but for intermediate species O and OH...... 97

xiii List of Tables

7.1 Retrieval of 5-step methane-air mechanism by Chelliah et al. [18] with the 11-step oxyfuel mechanism as the starting point...... 60

11.1 Summary of the empirical ﬁtting of the forward rates of the 11-step mechanism given in Chap. 7.5.2...... 93 11.2 Summary of the empirical ﬁtting of the backward rates of the 11-step mechanism given in Chap. 7.5.2...... 93

A.1 Skeletal mechanism for oxy-methane diffusion flames ...... 101 A.2 Skeletal mechanism for extremely low strained oxy-methane diffusion flames ...... 103

xiv Chapter 1

Introduction

Combustion research plays a fundamental role in the design and optimization of combustion systems such as furnaces or burners, as well as energy conversion systems such as gas turbine combustors or internal combustion engines. In recent decades, these investigations have been significantly influenced by environmental implications of combustion processes. According to the fifth assessment report of the intergovernmental panel on climate change published in 2013, CO2 emissions from fuel combustion are a key driver of climate change [99]. In order alleviate the negative effects of CO2 emissions released to the atmosphere, new technologies – such as the so-called oxyfuel combustion – are being developed. The terminology oxyfuel combustion has been indistinctly used in the literature to refer to fuel combustion in O2/CO2 mixture [1, 13, 17, 94, 95], or alternatively, fuel combustion in pure oxygen [9, 23, 37, 69, 95, 101] rather than air.1 This technology, which constitutes one of the major current challenges in both research and technical applications, is attractive due to its benefits compared to the conventional air/fuel combustion; the benefits include lower pollutant emissions and exhaust gas volume, higher temperatures and hence higher efficiency of heat transfer and an easier CO2 sequestration [8]. High temperatures are required, e.g., in the melting furnaces of the glass industry, and they are being realized there favorably by oxyfuel combustion rather than by complicated adjustments and optimizations of the combustion processes in furnaces operated with air as oxidizer.

1 For instance, combustion in air can not be called oxyfuel combustion because in air the oxygen is strongly diluted by nitrogen.

1 1.1 Motivation for and Overview on the Reduction of Chemical Mechanisms

To be able to numerically simulate a real, technical combustion system, various mathematical models are required, e.g., models for thermodynamics, molecular transport, heat transfer and radiation, chemistry and turbulence. In reactive CFD problems – here, and below, CFD is an acronym for Computational Fluid Dynamics – combustion chemistry plays, naturally, a major role and hence adequate and – in view of the overall computational costs – powerful chemistry models are required. For the purpose of CFD, available chemistry models range from extremely simple – an example would be the so-called eddy-break up model originally suggested by Magnussen [12] – over flamelet models – examples would be the laminar flamelet model of non-premixed turbulent combustion by Peters [73] or the Bray-Moss-Libby-Champion model [15, 16] of turbulent combustion –, or straightforward but arbitrarily complex models employing detailed mechanism of elementary reactions. From the chemistry point of view, the latter models are most accurate because in these models only chemistry is modelled, whereas other models also attempt to include the effects of turbulence. On the other hand, from the computational point of view, models employing detailed mechanism of elementary reactions are the most CPU-time consuming and hence most expensive models because, for a given combustion problem, practically there is no upper bound to the number of elementary reactions to be taken into account other than the limits imposed by available computer resources. Since in practical applications computer resources are, of course, limited, this is the bottleneck for the use detailed mechanisms of elementary reactions in practical CFD simulation of reactive problems.

Thus, for the purpose of CFD, most desirable are reaction mechanisms that are as accurate as possible within the bounds of given computational resources. Specifically, if chemical mechanisms are to be employed in CFD simulations, these can be so- called detailed mechanisms, so-called skeletal mechanisms, or so-called reduced mechanisms. Detailed mechanisms consist of elementary reactions that generally enable the mechanisms to be used with confidence for a variety of problems in premixed and non-premixed combustion, in wide ranges of pressure, temperature and stoichiometry. Thus, a detailed mechanism can be arbitrarily complex – in particular, they can comprise a nearly arbitrarily big number of elementary reactions – and computer codes exist that automatically generate such mechanisms [24, 38]. In contrast, a skeletal mechanism is, by definition, the smallest possible mechanism of elementary reactions that for a given problem describes chemistry sufficiently well. Finally, a reduced mecha-

2 nism consists, by deﬁnition, not of elementary but of global reaction steps whose rates are to be computed in a special manner to be speciﬁed.

Detailed chemical kinetic mechanisms, and hence also skeletal mechanisms, contain only elementary reactions. According to the International Union of Pure and Applied Chemistry (IUPAC) 2 chemical terminology, an elementary reaction is defined as a reaction for which no reaction intermediates have been detected or need to be postulated in order to describe the chemical reaction on a molecular scale. An elementary reaction is assumed to occur in a single step and to pass through a single transition state [68]. Detailed mechanisms are generally developed under a wide range of oper- ating conditions such as high and/or low temperatures, high and low pressures and a range of equivalence ratio. Furthermore, they can sometime involve both small and larger hydrocarbons. As consequence, detailed chemical models can contain from a few dozen up to several hundred or even thousands species and up to several thousand elementary reactions. For example, the GRI3.0 mechanism [96], which was developed in the late 1990s for hydrocarbons up to C3, contains 325 reactions amongst 53 species. More recently, detailed mechanisms have been developed for larger hydrocarbons [7], real fuels such as kerozene [28] or biofuels [113]. For instance, Westbrook et al. [114] developed a comprehensive detailed mechanism for C8-C16 alkanes consisting of 2115 species and 8157 reactions. As already indicated above, the incorporation of such a large detailed mechanism in computational simulations of practical combustion systems leads to significant or prohibitive computational cost. A reduction of simulation costs can be achieved with the so-called chemical model reduction, viz., methods which identify and subsequently remove unimportant species and reactions from a starting detailed mechanism without a significant loss of accuracy. Simplified mechanisms derived with chemical model reduction techniques have the key advantages that they are smaller than the original ones and can still reproduce the main characteristics of the corresponding detailed mechanism. In the present work, only simplified mechanisms valid within the entire simulation domain are of interest; adaptive chemistry reduction and dynamic adaptive chemistry reduction are beyond the scope of the current work. Reviews on both topics are provided by Zhang [116] and by Turanyi [105].

Chemical reduction is generally carried out in two steps as follows.

(i) In a first step, a detailed mechanism is reduced to the so-called skeletal mechanism which, by definition, is still mechanism of elementary reactions. In fact the 2The International Union of Pure and Applied Chemistry (IUPAC) is a leading international chemistry society which works in fields such as chemical nomenclature and terminology, standardized methods for measurement or atomic weights.

3 skeletal reduction procedure does not alter Arrhenius parameters, reaction order and pressure-dependency of any elementary reaction. Methods such as sensitivity analysis [83, 103], path flux analysis (PFA) [100], directed relation graph (DRG) [59, 60, 61, 62, 63], DRG-aided sensitivity analysis (DRGASA) [117] and other DRG based methods such as DRG with error propagation (DRGEP) [72] or DRG with error propagation and sensitivity analysis (DRGEPSA) [70] have been devised to develop skeletal mechanisms. Different skeletal mechanisms suitable for different applications are provided online, e.g., by Lu [64], or have been reported in the literature, e.g., by Peters [78]; however, these skeletal mechanisms are derived under traditional air-firing conditions, which render most of them inap- propriate for oxyfuel combustion. Consequently, in the present work new skeletal mechanisms valid under oxyfuel conditions will be derived.

(ii) In a second step, preliminary skeletal mechanisms can be further reduced using methods such as the so-called quasi-steady state approximation (QSSA) [18, 22, 76], the so-called intrinsic low dimensional manifold (ILDM) [65], or the computational singular perturbation (CSP) [56, 57].

On completion of the chemical reduction procedure, on the basis of a skeletal mechanism the so-called global reduced mechanism is derived. In contrast to the two other chemical models just discussed, viz., detailed and skeletal mechanisms, a global reduced mechanism contains only non-elementary, i.e., global reactions. Chemical processes occurring in combustion systems are, sometimes, described with an overall global one-step reaction. Such a simple chemical model involves only reactants, namely fuel and oxidizer species, and products. For instance, Westbrook and Dryer [112] predicted flame speeds over a range of conditions using a global one-step reaction with good accuracy. However, they found that the one-step reaction model overestimates adiabatic flame temperature. Such flaws of the single-step reaction model can be eliminated by a multistep global reaction mechanism. In addition to reactants and products, a multistep global reaction mechanism involve intermediate species such as H, O, or the OH radical. The mechanisms by Jones and Lindstedt [47], Westbrook and Dryer [112] or Peters [74] are well-known examples of multistep global reduced mechanisms. In addition, a significant number of multistep global reduced mechanisms is provided online by Lu [64]. However, due to the fact that these reduced mechanisms were derived in air-firing conditions, they are not expected to produce reliable results when used under oxyfuel conditions. Frassoldati et al. [37] and Bibrzycki and Poinsot [13] reported such discrepancies when using the original 4-step mechanism by Jones

4 and Lindstedt in oxyfuel combustion. Hence, the present work intends to systematically derive new multistep global reduced mechanisms valid under oxyfuel conditions with, speciﬁcally, methane as fuel. Hence, below also the terminology oxy-methane combustion will be used.

1.2 Previous Work

1.2.1 Combustion in O2/CO2 Atmospheres

Numerous studies have focused on fuel combustion in O2/CO2 mixture in the past decades. For instance, Seepana and Jayanti [94] analysed the flame structure of oxy-methane non-premixed flames. They found that the limiting concentration of O2 in oxy-methane flames below which the flame may quench is 24.5% by volume, the rest being CO2. In addition, they reported that an oxy-methane flame mixture consisting of

32% O2 and 68% CO2 displays an equivalent flame structure as an air-methane flame. This result is in agreement with previous studies by Buhre et al. [17] and Ditaranto and Hals [32]. Andersen et al. [1] refined both the two and the four-step mechanisms by Westbrook and Dryer and Jones and Lindstedt under O2/CO2 conditions respectively. Furthermore, they validated the modified mechanisms against a reference detailed mechanism presented by Glarborg and Bentzen [40] under oxyfuel conditions. Bibrzycki and Poinsot [13] modified two previous global reduced mechanisms in order improve the agreement with detailed chemistry under oxyfuel conditions. An overview of computational fluid dynamics (CFD) modelling of combustion in oxyfuel atmosphere is provided in Sharma and Jayanti [95]. In particular, detailed and reduced mechanisms used in combustion simulations under oxyfuel conditions, viz., in

O2/CO2 and in pure oxygen atmosphere are listed in Sharma and Jayanti [95]. Re- search investigations carried out in O2/CO2 atmosphere were reviewed in Buhre [17]. Due to the fact that these investigations are beyond the scope of the present work, in- terested readers are referred to the latter review as well as references mentioned above for further details.

1.2.2 Combustion in Pure Oxygen

Throughout the rest of the current work, only fuel combustion in pure oxygen shall be termed oxyfuel combustion. Compare to fuel combustion in O2/CO2 atmosphere, a relatively limited number of experimental and numerical investigations have been focused on combustion of fuel in pure oxygen. For instance, Cheng et al. [23] experimentally and numerically investigated planar oxygen-enhanced methane counter-

5 flow flames including oxy-methane diffusion flames. They reported a good agreement between experimental and numerically-predicted species and temperature profiles obtained using the GRI3.0 mechanism for an oxy-methane diffusion flame mixture, the oxidizer stream consisting of 100% O2, and the fuel stream of 20% CH4 and 80% N2, with a strain rate of 60 s−1. Naik et al. [69] investigated counterflow diffusion flames under sooting oxyfuel conditions. They found a good agreement between measured and numerically-predicted NO concentration in regions at flames temperatures lower than 2600 K; however they reported a poorer agreement in regions at flames temperatures greater than 2600 K. Sung et al. [101] developed four global reduced mechanisms for methane oxidation with NO chemistry. They found that their 15-step global reduced mechanism, which includes 12 non-elementary reactions for methane oxidation and 3 steps for NO chemistry, reproduce the flame temperature and the peak mole fraction of CO under oxygen-enriched conditions (99% O2 and 1% N2) with a good accuracy. However, it over-predicted the peak mole fraction of NO for moderate strain rates. In addition, empirically fitted Arrhenius parameters, which generally describe reduced chemistry models implemented in computational combustion simulations, were not derived in their study. Furthermore, the reduced mechanism by Sung et al. was derived on the basis of the zero-dimensional perfectly-stirred reactor (PSR) system. It is of course assumed that a PSR system with an extensive thermodynamic parametric variations can cover the range of conditions found in both zero and one-dimensional systems considered for validation purposes. However, in the present work, a different approach has been used. Specifically, the chemical reduction procedure is directly based on one-dimensional system, namely, flamelet equations. The latter approach was for example used by Chelliah et al. [18] in the derivation of global reduced mechanisms for counterflow methane-air diffusion flames. Frassoldati et al. [37] modified the well-known 4-step mechanism by Jones and Lindstedt [47] for methane diffusion flames by adding two more reactions. In particular, they improved the overall prediction capacities of the mechanism by Jones and Lindstedt by including the intermediates species O, OH and H in the final reduced mechanism and by fitting the new reduced mechanism with data obtained from a detailed mechanism. However, the oxyfuel global reduced mechanism by Frassoldati et al. [37] was not derived through a systematic reduction approach. In addition, the reported agreement between the detailed chemistry and their new reduced mechanism can be improved.

6 1.3 Outline of the Present Work

The overall structure of the present thesis takes the form of 12 chapters, including the introduction, the summary and the outlook chapters. This introduction chapter is followed by Chap. 2, which describes the counterflow diffusion configuration considered in the present thesis. In addition, flamelet equations are presented in the this chapter. Chapter 3 contains background information about thermodynamics and molecular transport models used in the present work. Chapter 4 focuses on chemistry models. Specifically, details about detailed, skeletal and global reduced mechanisms encoun- tered in combustion chemistry are provided here. In addition, an efficient method, namely the targeted search algorithm (TSA) [102] for the automatic selection of the so-called steady state species (see Chap. 4.5 for the definition) is presented in this chapter. Chapter 5 is concerned with numerical methods.

In Chap. 6, attention is focused on the skeletal reduction methods. In particular, skeletal reduction methods used in the present work, namely, directed relation graph (DRG), DRG-Aided sensitivity analysis (DRGASA), unimportant reaction elimination are presented and discussed. In addition, a new method for reaction elimination, namely, reaction elimination based on DRGASA, is presented in this chapter. The systematic derivation of global reduced mechanisms is outlined in Chap. 7.

In Chaps. 8 to 11 the results and the key findings of this study are presented. Specifi- cally, in Chap. 8 computational results obtained using various detailed mechanisms are compared and finally a suitable detailed mechanism, from which the overall mechanism- reduction procedure can be started for oxy-methane non-premixed combustion, is selected. In Chap. 9 results for skeletal mechanisms are presented and discussed, and – finally – in Chaps. 10 and 11 results for reduced, global mechanisms the rate of whose global steps have been derived systematically and, alternatively, in an empirical manner.

Finally, Chap. 12 summarizes the ﬁndings of the present work and gives recommendations for future work. Two appendices conclude the thesis.

7 Chapter 2

Combustion Geometry and Governing Equations

2.1 Diﬀusion Flames in CounterFlow

A distinction is made between approaches that consider counterflowing streams of fuel and oxidizer, (i), between two opposed nozzles whose exits are a final distance apart and, (ii), inside the viscous region of a stagnation-point flow, i.e., inside a viscous boundary layer whose edges extend, in terms of a suitably defined similarity variable, to +∞ and −∞, respectively.

Figure 2.1: Counterﬂow geometries

Schematics of examples of such counterflow geometries are shown in Fig. 2.1. In the left picture of this figure, a counterflow diffusion flame between opposed nozzles is shown. This is a so-called opposed-flow counterflow geometry [50]. In the governing equations of the quasi-onedimensional mathematical model for this flow geometry, in the vicinity of the nozzles potential flow prevails [23, 35, 50] but inside the domain

8 bounded by the two opposed nozzles a stagnation point and hence a viscous flow region develops, in which the stagnation point is located. In this flow geometry, depending on stoichiometry and nozzle-exit conditions, a diffusion flame may stabilize anywhere, i.e., in either the viscous or non viscous region of the flow.

In the right picture of Fig. 2.1, a schematic of the so-called Tsuji counterflow geometry is shown. The geometry is named after the combustion scientist Tsuji who, with his colleague, was the first [104] to investigate experimentally diffusion flames in this geometry. Here gaseous fuel is pressed from inside a porous cylinder outside whilst a stream of oxidizer blows against the cylinder surface. Between the outer cylinder surface and approximately the dashed circle in the figure the flow is stagnating and hence viscous there. In this viscous region a boundary-layer develops along the cylinder surface, which can be described by the well known boundary-layer equations [93], even in a quasi-one dimensional similarity formulation [93]. Naturally for a boundary-layer model, at the edge of the boundary layer potential flow boundary conditions apply. Due to the model equations only diffusion flames located inside the boundary layer can be described within the framework of this model [10, 34, 89].

For either geometry, following traditional convention [93], y denotes the longitudinal and x the transversal coordinate. Also in either geometry, the flow can be planar or, alternatively, axisymmetric. In the latter geometry, x represents a radial coordinate. The velocity components in the x and y direction are denoted by u and v, respectively. Traditionally, for the stagnation-point geometry the origin of the coordinate system is selected such that the stagnation plane, where by definition v = 0, is located at y = 0. Naturally, for the opposed nozzle geometry the y-locations of the nozzle exits are fixed and hence there is no degree of freedom left to fix the position of the stagnation plane.

For the purpose of illustration, shown in Figs. 2.2 to 2.4 are profiles of fuel, oxidizer and major products, of temperature and of mixture faction for an opposed-jet diffusion flame corresponding to the left picture in Fig. 2.1. Specifically, the profiles shown in this figure are profiles along the stagnation-point streamline y, i.e., profiles as function of distance y. The profiles shown in this figure were computed and plotted with the Cosilab software [27]. Similar profiles could be computed and plotted for the a diffusion flame in the Tsuji geometry shown in the right picture of Fig. 2.1. It is readily seen from Figs. 2.2 to 2.4 that, if in Fig. 2.2 and 2.3 the space variable y is eliminated by the mixture fraction Z plotted in 2.4, then mass fractions and temperatures become functions of only Z. Such a transformation is possible because Z is monotonic in y.

9 1

CH4 CO2

O2 0.8 H2O

0.6

0.4 Mass fraction

0.2

0 0 0.002 0.004 0.006 0.008 0.01 0.012 Distance from top jet (m)

Figure 2.2: Mass fractions of fuel, oxidizer and major products in an opposed-jet counterﬂow geometry.

Peters [75, 79] could show that the essential feature of laminar diffusion flames in various geometries can be captured by making the governing equations independent of the spatial coordinates by introducing the mixture fraction Z as a new, independent space-like variable. The governing equations then take a universal form with only time t and mixture fraction Z as independent variables. Counterflow diffusion flames described or computed by such a set of governing equations are commonly termed counterflow diffusion flames in mixture-fraction space. Clearly, counterflow diffusion flames in mixture-fraction space have the advantage that the profiles of all relevant variables such as temperature and mass fractions can be computed directly as functions of the mixture fraction rather than going through the complication described above where, in postprocessing of computational results, the space variable y has to be eliminated in favor of the mixture-fraction variable Z. As examples, shown in Figs. 2.5 and 2.6 are the profiles of the mass fractions of fuel, oxidizer and major products, and the temperature, in mixture fraction space.

In the present thesis, we consider laminar counterflow diffusion flames in mixture- faction space. The governing equations of such flames, originally derived by Peters, have since been used in a great number of research works [22, 25, 66, 79, 94] and books

10 2600 2400 2200 2000 1800 1600 1400 1200

Temperature (K) 1000 800 600 400 200 0 0.2 0.4 0.6 0.8 1 1.2 Distance from top jet, cm

Figure 2.3: As Fig. 2.2, but only for temperature T .

0.8

0.6

0.4 Mixture fraction Z

0.2

0 0 0.002 0.004 0.006 0.008 0.01 0.012 Distance from top jet (m)

Figure 2.4: As Fig. 2.2, but only for mixture fraction Z.

11 1

O2 CH4 0.8 CO2 H2O

0.6

0.4 Mass fraction

0.2

0 0 0.2 0.4 0.6 0.8 1 Mixture fraction Z

Figure 2.5: Mass fractions of fuel, oxidizer and major products of a diﬀusion ﬂame in mixture-fraction space.

[36, 80]. They are

2 ∂Yk ρχ 1 ∂ Yk ρ = 2 + wk , (2.1) ∂t 2 Lek ∂Z K ∂T ρχ ∂2T 1 X ρ = 2 − hkwk . (2.2) ∂t 2 ∂Z cp k=1 In deriving Eqs. (2.1) and (2.2), low Mach-number ﬂow, spatially and temporally constant pressure, and an ideal gas mixture have been assumed. In addition to time and mixture fraction that have already been introduced above, in (2.1) and (2.2), and below, ρ denotes the mixture mass density, Yk the mass fraction of species k, k = 1, ..., K, where K denotes the number of species in the system. The quantities wk, hk, and Lek are the mass rate of production, enthalpy, and Lewis number, respectively, of species k;

T is the temperature and cp the mixture mass-based speciﬁc heat capacity at constant pressure. The quantity χ denotes the scalar dissipation rate, which can be approximated as a constant with a value equal to the value at stoichiometric conditions,

χ ≈ χst . (2.3)

12 3000

2700

2400

2100

1800

1500

Temperature (K) 1200

900

600

300 0 0.2 0.4 0.6 0.8 1 Mixture fraction Z

Figure 2.6: As Fig. 2.5, but only for temperature T .

Alternatively, χ can be approximated as a function of the mixture fraction [80], a χ ≈ ∞ exp − 2erfc−1(2Z)2 . (2.4) π

Here a∞ denotes strain rate at the edge of the boundary layer of a correspondingly described counterflow diffusion flame, and erfc is the complementary error function.

2.2 Higher-Order Eﬀects

Equations (2.1) and (2.2) can be viewed as leading-order equations resulting from, (i), an expansion of variable transport properties expressed in terms of suitably deﬁned variable Lewis numbers about constant reference values and, (ii), a similar expansion of the molar mass of the gas mixture which, traditionally, is also termed mixture molecular weight. The Lewis numbers are deﬁned as λ LeZ = (2.5) ρ cp DZ for the mixture-fraction equation, and as λ Lek = (2.6) ρ cp Dk

13 for a species conservation equation k, k = 1, ..., K, where DZ and Dk, respectively, are suitable defined averaged diffusion coefficient – for details, Chap. 3.3 should be consulted. The molar mass of the mixture, W , is given, in terms of species mole or mass fractions, Yk or Xk, respectively, by mixture

K " #−1 X Yk W = XkWk = , (2.7) Wk k=1 where Wk denotes the molar mass of species k, k = 1, ..., K.

If the second-order, variable-property terms are taken into account rather than neglected, the following equations governing species mass conservation and overall energy conservation [79] result, viz.,

2 ∂Yk ρχ LeZ ∂ Yk ρ = 2 + wk ∂t 2 Lek ∂Z 2 K 2 2 ρχ LeZ Yk ∂ W ρχ X LeZ ∂ Yj Yk LeZ ∂ W + − Y + Y 2 Le W ∂Z2 2 Le k ∂Z2 W j Le ∂Z2 k j=1 j j 1 ∂ LeZ LeZ ∂ρχ cp ∂ λ ∂Yk + 2ρχ + − 1 + ρχLeZ 4 ∂Z Lek Lek ∂Z λ ∂Z cpLeZ ∂Z 1 Yk ∂ LeZ LeZ ∂ Yk cp ∂ λ Yk ∂W + 2ρχ + ρχ + ρχLeZ (2.8) 4 W ∂Z Lek Lek ∂Z W λ ∂Z cpLeZ W ∂Z K 1 X ∂ LeZ LeZ ∂ cp ∂ λ ∂Yj − 2ρχY + (ρχY ) + ρχLe Y 4 k ∂Z Le Le ∂Z k Z λ ∂Z c Le k ∂Z j=1 j j p Z K 1 X YkYj ∂ LeZ LeZ ∂ Yk cp ∂ λ YkYj ∂W − 2ρχ + ρχ Y + ρχLe 4 W ∂Z Le Le ∂Z W j Z λ ∂Z c Le W ∂Z j=1 j j p Z and

2 K ∂T ρχ ∂ T ρχ LeZ ∂cp ∂T 1 ∂P 1 X ρ = LeZ 2 + + − hkwk ∂t 2 ∂Z 2 cp ∂Z ∂Z cp ∂t cp k=1 1 ∂LeZ ∂ρχ cp ∂ λ ∂T + 2ρχ + (LeZ − 1) + ρχLeZ (2.9) 4 ∂Z ∂Z λ ∂Z cpLeZ ∂Z K X ρχ LeZ ∂Yk Yk ∂W cpk ∂T − + 1 − , 2 Lek ∂Z W ∂Z cp ∂Z k=1 respectively. The second-order terms appearing in (2.8) and (2.9) are discussed in detail in [79].

14 Chapter 3

Thermodynamics and Molecular Transport

In this chapter the models for thermodynamics and molecular transport as used in the present research work are presented and discussed. These are the so-called detailed models, which are more realistic and accurate than the so-called simple models. The latter are thought, e.g., in undergraduate engineering classes. In the present thesis, however, both simple and detailed models are presented because both kind of models appear in the governing equations of Chap. 2. In combustion, the first unified approaches to detailed modelling of molecular transport and thermodynamic data in a manner suitable for the implementation into computer codes was initiated by Dixon- Lewis, see e.g. [33], and Warnatz, see e.g. [109]. These approaches were taken up, modified, improved and summarized by others. The models presented in Chap. 3.3.2 are essentially the ones summarized in a series of publication co-authored by Kee, see e.g. [49, 52].

Since in the framework of the present thesis only ideal-gas mixtures need to be considered, in the following only thermodynamics and transport models for such mixtures are considered.

3.1 Equations of State

Naturally, the numerical solution to a reactive-ﬂow problem must satisfy certain thermal and caloric equations of state. For the ideal-gas mixtures considered herein, the thermal equation of state is p RT = . (3.1) ρ W In addition to the quantities already deﬁned in previous chapters, here R denotes the universal gas constant.

15 For ideal-gas species, the caloric equation of state is taken in the form

Z T o hk(T ) = hk + cpk dT k = 1, ..., K ; (3.2) T o

o o here T is an arbitrary reference temperature – often taken as 298.15 K –, hk is the o value of hk at T , and cpk is the speciﬁc heat at constant pressure.

The enthalpy of the gas mixture is

K X h = Ykhk . (3.3) k=1

Summing up the products Ykcpk yields the mixture frozen speciﬁc heat at constant pressure, viz., K X cp = Ykcpk . (3.4) k=1 3.2 Phenomenological Relationships

The mathematical description of molecular transport of mixture momentum and energy, and of species mass requires, phenomenological models to be used.

Specifically, models are required for the stress tensor σ, for the heat-flux vector q, and for the species diffusion velocities V k, k = 1, ..., K. In certain situations it is more practical to work with the species diffusion flux,

jk = ρYkV k , k = 1, ..., K , (3.5) rather than with the diﬀusion velocity. Subsequently, σ, q and V k will be given in tensor notation. Although the stress tensor does not explicitly appear in the governing equations presented in Chap. 2, for the sake of completeness a short section on its is included here.

3.2.1 Stress Tensor

As usual [5, 98, 115], the stress tensor is decomposed into a pressure part and a viscous part, i.e.,

σαβ = −pδαβ + ταβ . (3.6) The viscous part of the stress tensor is taken as Newtonian,

∂uα ∂uβ 2 ∂uγ ταβ = µ( + ) − (µ − κ) δαβ . (3.7) ∂xβ ∂xα 3 ∂xγ

16 Here µ is the dynamic viscosity of the mixture which is taken as a function of pressure, temperature and species concentrations; κ is the bulk viscosity , a quantity that – following usual practise – is taken as zero.

3.2.2 Heat-Flux Vector

The heat-ﬂux vector is written as

K K K T ∂T X X X Dk Xj qα = −λ + hkjkα + RT (Vkα − Vjα) + qRα . (3.8) ∂xα WkDkj k=1 k=1 j=1

On the right hand side of (3.8), the first term represents molecular energy transport by heat conduction according to Fourier’s law, the second is a multi-component contribution to molecular energy transport, the third is the Dufour contribution, and the fourth is the radiative heat flux. In addition to the quantities already defined above, T Dk denotes the thermal diffusion coefficient of species k, and Dkj is the binary diffusion coefficient between species k and j.

Generally the Dufour contribution to the heat-ﬂux vector is negligibly small and hence neglected.

3.2.3 Diﬀusion Velocities

Assuming gravity to be the only body force, the multi-component diﬀusion equation is

K ∂Xk X XkXj 1 ∂p = (V − V ) + (Y − X ) ∂x D jα kα k k p ∂x α j=1 kj α K K T T ! 1 ∂T X X XkXj Dj D + − k . (3.9) T ∂xα ρDkj Yj Yk k=1 j=1

On the right hand side of (3.9), the first term represents ordinary mass diffusion due to species concentration gradients, the second represents pressure-gradient diffusion, and the third term represents the Soret effect, i.e., species mass diffusion due to a temperature gradient. As the Dufour effect, the Soret effect is a thermal-diffusion effect. However, in contrast to the Dufour effect, at least in some premixed flames it has been found to be non-negligibly small. Therefore, in the computer code, this effect is retained.

If pressure-gradient diﬀusion and Soret eﬀect are neglected, then (3.9) reduces to the so-called Stefan-Maxwell equations.

17 Equation (3.9) is seen to represent a linear system of equations to be solved for the diﬀusion velocities Vkα, k = 1, ..., K, assuming that all other quantities are known. Computationally this can be quite time consuming, a problem that was addressed previously by various authors [26, 43, 52, 71] who also give further references. In the present thesis a somewhat simple yet still detailed and quite powerful model is used.

Following [71] it is assumed that the diﬀusion velocity V k, k = 1, ..., K, is composed of three parts, i.e., D T V k = V k + V k + V c . (3.10)

D Here V k is the ordinary-diﬀusion velocity for which an approximation recommended by Hirschfelder and Curtiss [45, 52] is adopted, viz.,

D Dk V k = − ∇Xk , k = 1, ..., K . (3.11) Xk

Here Xk denotes the mole fraction of species k and Dk is its mixture-averaged diffusion T coefficient. A suitable expression for Dk will be given in Chap. 3.3. In (3.10), V k is the thermal diffusion velocity,

T T Dk V k = − ∇T , k = 1, ..., K , (3.12) ρYkT a nonzero value of which is included only for the light species H and H2. In (3.12), T T is the temperature and Dk is the thermal-diﬀusion coeﬃcient of species k. A formula T for the evaluation of the Dk is given in Chap. 3.3.2.4. The correction velocity V c appearing in Eq. (3.10) is determined according to refs. [26, 71].

3.3 Transport Coeﬃcients

Transport coefficients are ordinary and thermal species diffusion coefficients, species and mixture thermal conductivities, and species and mixture dynamic viscosities.

3.3.1 Simple Property and Data Models

Simple relationships between µ, λ and the Dk can be derived by assuming constant, but not necessarily equal, Lewis numbers for all species, λ Lk = = constantk , k = 1, ..., K , (3.13) ρcpDk and a constant Prandtl number, µc P r = p = constant . (3.14) λ

18 For the Prandtl number a value between 0.7 and unity may safely be assumed. Lewis numbers usually vary through a flame, particularly in non-premixed systems. There- fore, in the absence of better information often for all species equal and constant Lewis numbers of unity are assumed, an approximation that is consistent with the other simplifications presented in this section on simple models of molecular transport and thermodynamic data. Alternatively, the Lewis numbers of the light species H2 and H may be taken as approximately 0.25; in hydrocarbon flames with fuels up to ethane a Lewis number of unity for all other species may be used; in hydrocarbon flames with fuels higher than ethane Lewis numbers greater than one are appropriate for the fuel molecule, for instance Le = 2 for n-heptane, and Lewis numbers of unity for all other species except H2 and H. Yet another alternative for premixed flames is to estimate the Lewis number of species i using relationships such as

p 3 Lk ≈ 0.2 10 Wk , (3.15) where Wk is the species’ molar mass (molecular weight) in units of kg/mol.

If in (3.13), Lek, cp and λ are known, then (3.13) may be solved for the product ρDk which is required to evaluate the modified Fickian diffusion flux ∇X j = −ρD k ≈ −ρD ∇Y . (3.16) k k k k ¯ Xk In (3.16), in writing the approximate equality, variations of the molar mass of the gas mixture have been neglected. It is seen from (3.13) that if Lek, cp and λ are taken as constants, then ρDk is a constant. Alternatively, λ may be taken as a function of temperature, e.g., λ λ T ω = , (3.17) cp cp 0 T0 where T0 is a reference temperature and where the subscript 0 indicates that the respective quantity is to be evaluated at T0. In general, suitable values for the exponent ω appearing on the r.h.s. of (3.17) range from 0.5 and 1. With the Prandtl number and λ/cp known, the viscosity µ is then obtained from (3.14). Alternatively, instead of determining first λ and then µ, first µ may be determined, say from an empirical relationship such as the Sutherland law, and then λ/cp may be obtained from (3.14).

3.3.2 Detailed Property and Data Models

Whilst the Lewis numbers occur in the governing equations of Chap. 2 that are deﬁned in Eq. (3.13), also transport properties – such as the mixture thermal conductivity –

19 there appear for which more realistic and accurate models are required if numerical simulations are pursued with the aim of predicting confidently flow and flame structures, also for subsequent comparison with experimental data. As mentioned above, in combustion, the first unified approaches to detailed modelling of molecular transport and thermodynamic data in a manner suitable for implementation into computer codes was initiated by Dixon-Lewis, see e.g. [33], and Warnatz, see e.g. [109]. These approaches were taken up, modified, improved and summarized by others. The models presented in this section are essentially the ones summarized in a series of publication co-authored by Kee, see e.g. [49, 52].

3.3.2.1 Dynamic Viscosities

According to [46], the dynamic viscosity of a pure gaseous species k is calculated from √ 5 πmkkBT µk = 2 (2,2)? , (3.18) 16 πσkΩ where σk denotes the Lennard-Jones collision diameter, mk the mass of the molecule, (2,2)? and kB is the Boltzmann constant. The collision integral Ω is a known function of the reduced temperature T ? and of the reduced dipole moment δ?. The quantities T ? and δ? both include properties of the species under consideration; they are given by

? ? kBT T = Tk ≡ (3.19) k and 2 ? ? 1 dk δ = δk ≡ 3 . (3.20) 2 kσk

Here dk is the dipole moment of species k, and k is its Lennard-Jones potential well depth.

The dynamic viscosity of the mixture is calculated from the respective properties of the pure species according to

K 1X 1 µ = X µ + . (3.21) 2 k k PK k=1 k=1 Xk/µk

3.3.2.2 Thermal Conductivities

The thermal conductivity of a pure species k is calculated from

µk λk = (ftransCv,trans + frotCv,rot + fvibCv,vib) , (3.22) Wk

20 where ftrans, frot and fvib denote the species’ translational, rotational and vibrational degrees of freedom, respectively, and Cv,trans, Cv,rot and Cv,vib the respective contribution to the molar heat capacity at constant volume, Cv, of species k, k = 1, ..., K.

For a species that is a single atom there are no rotational and translational contributions to Cv. Specifically, for single atoms we have ftrans = 1/2 and frot = fvib = 0. For linear and non-linear molecules the individual degrees of freedom are given by 5 2 Cv,rot A ftrans = 1 − , (3.23) 2 π Cv,trans B ρDkk 2 A frot = 1 + , (3.24) µk π B and ρkDkk fvib = , (3.25) µk respectively. Here the quantities A and B are defined as 5 ρ D A = − k kk , (3.26) 2 µk 2 5 Cv,rot ρkDkk B = Zrot + + , (3.27) π 3 R µk where R is the specific gas constant of species k. For all species the translational part of Cv is given by C 3 v,trans = . (3.28) R 2

For a linear molecule, the molar heat capacity at constant volume, Cv, and its translational, rotational and vibrational parts are related through C 2 C C − C 5 v,rot = , v,rot = 1 , v v,vib = . (3.29) Cv,trans 3 R R 2 For a non-linear molecule, the respective relationships are C C 3 C − C v,rot = 1 , v,rot = , v v,vib = 3 . (3.30) Cv,trans R 2 R

The density ρk and the “self-diﬀusion” coeﬃcient Dkk appearing in the above equations are calculated from the single-species ideal-gas equation of state

pWk ρk = , (3.31) RT and the expression p 3 3 3 2πkBT /Wk Dkk = 2 (1,1)? (3.32) 16 pπσkΩ ,

21 respectively. The rotational collision number Zrot of species k is taken as a prescribed function of temperature T . The thermal conductivity of the gas mixture is calculated from the respective properties of the pure species according to

K 1X 1 λ = X λ + . (3.33) 2 k k PK k=1 k=1 XK /λk

3.3.2.3 Ordinary-Diﬀusion Coeﬃcients

The binary diﬀusion coeﬃcients are given by the expression

p 3 3 3 2πkBT /Wkj Dkj = 2 (1,1)? . (3.34) 16 pπσkjΩ Here 2WkWj Wkj = (3.35) Wk + Wj is the reduced molar mass for the pair of species (k, j), and σkj is the average collision (1,1)? ? diameter. The collision integral Ω depends on the reduced temperature Tkj,

? kBT Tkj = , (3.36) kj which in turn may depend on the species reduced dipole moments and polarizabilities.

Here kj is an average Lennard-Jones potential well depth. In computing the average quantities, two cases are considered, depending on whether the collision partners are polar or non-polar. For the case that the partners are either both polar or both non- polar, it is assumed that

r kj = k j , (3.37) kB kB kB 1 σ = (σ + σ ) , (3.38) kj 2 k j dkj = dkdj . (3.39)

For the case of a polar molecule interacting with a non-polar molecule, it is assumed that r pn = ξ2 p n , (3.40) kB kB kB 1 σ = (σ + σ ) , (3.41) pn 2ξ6 p n

dpn = 0 , (3.42)

22 where r 1 ? ? p ξ = 1 + αndp . (3.43) 4 n ? In the above equations, αn is the reduced polarizability of the non-polar molecule, and ? dp is the reduced dipole moment for the polar molecule, viz., s ? αn ? dp αn = 3 , dp = 3 . (3.44) σn pσp

The mixture-averaged diffusion coefficients are given explicitly in terms of the binary diffusion coefficients Dkj by 1 − Y D = k . (3.45) k PK j=1 Xj/Dkj j6=k

3.3.2.4 Thermal-Diﬀusion Coeﬃcients

The thermal diﬀusion coeﬃcient of species k is taken as

T ρYk Dk = ΘkDk , (3.46) Xk where Θk is the thermal diffusion ratio of species k, Dk the mixture-averaged diffusion coefficient of species k defined in (3.46), and Xk is the mole fraction of species k, k = 1, ..., K.

3.4 Heat Capacities, Enthalpies and Entropies

Thermodynamic properties required in the system of governing equations presented in Chap. 2, but also in many of the relationships given above for the calculation of transport properties, are heat capacities and enthalpies. We also describe here the calculation of entropies because these have been used in the present thesis to calculate equilibrium constants.

For each species k, k = 1, ..., K, polynomial curve ﬁts of the NASA type are used to calculate the thermodynamic properties. Speciﬁcally, the standard-state molar heat capacity at constant pressure given by

0 Cpk 2 3 4 = a1 + a2T + a3T + a4T + a5T , (3.47) R

23 the standard-state molar enthalpy given by

0 Hk a2 2 a3 3 a4 4 a5 5 = a6 + a1T + T + T + T + T , (3.48) R 2 3 4 5 and the standard-state molar entropy given by

0 Sk a3 2 a4 3 a5 4 = a7 + a1lnT + a2T + T + T + T . (3.49) R 2 3 4

There are two sets of the coeﬃcients a0 - a7, one for the temperature range between 300

K and 1000 K, and one for the range from 1000 K to 5000 K. The molar quantities Cpk,

Hk and Sk are related to the respective mass-based quantities through Cpk = cpkWk,

Hk = hkWk and Sk = skWk, where Wk is the molar mass of species k. The mass-based mixture frozen speciﬁc heat capacity, enthalpy and entropy are given by

K K K X X X cp = Ykcpi , h = Ykhk , s = Yksk . (3.50) k=1 k=1 i=1

24 Chapter 4

Chemistry

4.1 Ozone Chemistry as an Introductory Example

For the sake of discussion in this and later chapters of this thesis, here we reproduce the smallest existent detailed mechanism, i.e., the one that describes ozone decomposition

[44, 91]. The mechanism contains three reactions amongst ozone, O3, molecular oxygen,

O2, and atomic oxygen O, i.e.,

O3 + M = O2 + O + M (1)

O3 + O = 2 O2 (2)

2 O + M = O2 + M (3)

In reaction (1), which has a large activation energy, ozone decomposes under the in-

fluence of a third body – which is either O, O2 or O3, or a combination thereof – into molecular and atomic oxygen. The O-atoms produced in reaction (1) immediately react with other ozone molecules in reaction (2), which has a small activation energy. Reaction (3) is a recombination step in which O-atoms recombine to molecular oxygen. Reaction (1) determines the overall rate of ozone decomposition, and reaction (3) has no pronounced influence on the laminar flame speed.

4.2 Phenomenological Expressions of Chemical Ki- netics

In this thesis, the number of chemical species in a combustion system is denoted by K, and the number of elementary reactions that take place in the system by I. The (net)

25 mass rate of production of species k, k = 1, ..., K is

I X wk = Wk νkiωi . (4.1) i=1

Here νki denotes the net stoichiometric coeﬃcient of species k which can be written as

00 0 νki = νki − νki , (4.2)

0 00 where νki denotes the stoichiometric coeﬃcient of a reactant k in reaction i, and νki that of a product k in reaction i. The reaction rate of reaction i is given by

K K 0 00 Y νki Y νki ωi = ki,f Ck − ki,b Ck . (4.3) k=1 k=1

Here ki,f and ki,b denote the forward and backward rate constants, respectively, of reaction i. In the case of Arrhenius kinetics, these rate constants are given by

E βi,f i,f ki,f = Ai,f T exp − (4.4) RcT and E βi,b i,b ki,b = Ai,bT exp − , (4.5) RcT 1 respectively. In (4.4) and (4.5), T is the temperature, and Rc denotes the universal gas constant. The quantities Ai,f and Ai,b denote constants, the exponents βi,f and

βi,b characterize the pre-exponential temperature dependence of rate ωi, and Ei,f and

Ei,b denote the activation energy of the forward and backward step, respectively, of reaction i. In (4.3), ρYk Ck = (4.6) Wk denotes the molar concentration of species k, k = 1, ..., K; ρ is the mixture density and

Wk the molar mass of species k.

The reaction rates ωi as given in (4.3) are so-called Arrhenius rates that represent – in a good approximation – the true, experimentally determined rates of most elementary reactions. However, a non-negligible number of reactions exist for which the Arrhenius rates do not or not reasonably well describe the true rates such as pressure-dependent reactions or also other reactions. To present and discuss here the rate expressions for such reactions would make the text of this chapter rather unreadable. Therefore, these

1 The universal gas constant Rc in the case of Arrhenius kinetics is to be taken in the same units system as the activation energy.

26 rate expressions are summarized in App. B.

4.3 Detailed Mechanisms of Elementary Reactions

For example, the rates of the three reactions (1), (2) and (3) in the above ozone mechanism can be straightforwardly described by the phenomenological equations given in Chap. 4.2. However, for the work presented in this thesis the so-called GRI3.0 mechanism [96] was employed - at least as a starting point for the derivation of shorter mechanisms – which comprises 325 reactions amongst 53 species. To be able to describe arbitrarily large reaction mechanism by relatively simple equations eﬃciently, the following notation will be used.

Here and below, species symbols will be denoted by M, i.e., Mk is the symbol of species k for k = 1, ..., K. For instance, if the k-th species is methane, then Mk is

CH4. Similarly, here and below, reactions will be denoted by R, i.e., Ri is the equation of elementary reaction i for i = 1, ..., I. For instance, if the i-th reaction is that of the recombination of two hydrogen atoms, then Ri is 2 H + M = H2 + M; here M denotes a third body. More generally, with the notation just introduced, the i-th chemical reaction can be written as K X Ri = νkiMk = 0 , (4.7) k=1 or, alternatively, as K K X 0 X 00 νkiMk = νkiMk . (4.8) k=1 k=1 Hence, a chemical mechanism comprising I reactions can be written as

K X νkiMk = 0 i = 1, ..., I , (4.9) k=1 or K K X 0 X 00 νkiMk = νkiMk i = 1, ..., I , (4.10) k=1 k=1 respectively.

27 4.4 Partial Equilibria

A chemical reaction i is said to be in partial equilibrium if, in a good approximation,

ωi ≈ 0 and both

K K 0 00 Y νki Y νki |ωi| ki,f Ck and |ωi| ki,b Ck . (4.11) k=1 k=1 Thus, if reaction i is in partial equilibrium, then the relationship

K K 0 00 Y νki Y νki ki,f Ck = ki,b Ck (4.12) k=1 k=1 represents an equation that – for given temperature T –, relates algebraically the molar concentrations of the species participating in this reaction. Each partial equilibrium (4.12) prevailing in a given combustion system can be used to dispense with one of the species mass conservation equations (2.1), a fact which was frequently used in older derivations, typically from the late 1980s and early 1990s of systematically reduced mechanisms such as in [14, 18]. It was only in the late 1990s when computer-aided or even automatic reduction of reaction mechanisms began and therefore partial equilibria were dropped in favor of so-called steady states [19, 20, 21, 67, 101, 102]. The latter will be discussed in the next section.

4.5 Steady States Species

Prior to deﬁning the so-called steady-state species, the notions of the mass rates of creation and consumption, respectively, of a chemical species need to be introduced. For species k, the mass rate of production, which by deﬁnition is a net rate, was given in (4.1), i.e.,

I X wk = Wk νkiωi . i=1 Simple algebraic manipulations of this equation show that, alternatively, one can write

+ − wk = wk − wk , (4.13)

+ − where both wk and wk are non-negative and

I K I K 0 00 + X 00 Y νki X 0 Y νki wk = Wk νkiki,f Ck + νkiki,b Ck (4.14) i=1 k=1 i=1 k=1

28 and

I K I K 00 0 − X 00 Y νki X 0 Y νki wk = Wk νkiki,b Ck + νkiki,f Ck , (4.15) i=1 k=1 i=1 k=1

+ − respectively. wk and wk denote the so-called mass rate of creation and consumption, respectively.

For a spatially homogeneous system, species k is to be said in steady state if, in a good approximation, wk ≈ 0 and

+ − |wk| wk and |wk| wk , (4.16) k = 1, ..., K. A species that is in steady state is called a steady-state species.

For the ozone decomposition in systems with low initial ozone concentration, atomic oxygen is known to be in steady state [85, 91]. This is not the case for initial ozone concentrations that are not moderate [90]. For instance, from the ozone reaction mechanism given in Chap. 4.1 it is readily derived that

+ wO = ω1 , (4.17) − wO = ω2 + 2 ω3 . (4.18)

Thus, e.g., in a good approximation

ω1 = ω2 + 2 ω3 . (4.19)

Since, for suﬃciently low initial ozone concentration reaction (3) is negligible as well as the backward steps of reactions (1) and (2) [91], (4.19) reduces to

ω1,f = ω2,f , (4.20) the steady-state molar concentrations of O-atoms is obtained as

[O] = (k1,f /k2,f )[M] . (4.21)

The latter equation can be used to replace the species conservation equation for O- atoms. The replacement of species conservation equations by a corresponding number of suitable steady-state relationships is one of the fundaments on which both traditional and current methods for the reduction of detailed mechanisms of elementary reactions to relatively short mechanisms of global reactions are based and hence are extensively employed – also, below, in the present thesis.

29 In presence of transport phenomenon, species k is to be said in steady state if, in addition to 4.16, its concentration remain very small and the first two terms of 2.1 can be neglected compared to the reaction rates [74, 76, 77]. In the framework of the present thesis, the targeted search algorithm (TSA) developed by Tham et al.[102] is implemented to select the optimum steady-state species in a chemical mechanism. Basically, in this approach all species except the combustion reactants and products are sorted by their maximum concentration values in ascending order. Then, the steady state approximation is applied to the species with the lowest concentration and the influence this approximation on global parameter such as temperature profile is assessed. If the relative error induced to the selected global parameter, e.g. temperature profile, is less than a user-specified threshold value, then this first species is kept in the steady-state species list. Otherwise, this first species is included in the list of species to appear in the global reduced mechanism. This procedure is repeated until the relative error is greater than the user-specified threshold. Compared to the Computational Singular Perturba- tion (CSP) [57] approach, the TSA is an effective and straightforward method to select the optimum steady-state species, in particular in presence of transport phenomena.

4.6 Global Reactions and Global Reaction Mecha- nisms

For a given combustion system, a reaction is termed global if it summarizes (reasonably well) the chemical “eﬀect” of several – or even all – elementary reactions of a corresponding detailed mechanism of elementary reactions. For instance, for the detailed ozone mechanism given in Chap. 4.1, a single global reaction may be postulated, i.e.,

O3 ⇒ 1.5 O2 . (I)

Clearly, in this limiting case the detailed reaction mechanism consisting of three elementary reaction reduces to a mechanism that consists of only one global reaction step. However, most combustion systems can not be described suﬃciently well by a single global reaction. Rather, in most cases several global reactions are required for an adequate representation of the combustion chemistry. These global reactions then form a so-called global reaction mechanism.

The rates of production of chemical species that participate in a global reaction mechanism are not unique. For instance, they can – but need not – be described by Eqs. (4.1) and (4.2). It is important to note that, if this alternative applies, the expression given for the reaction rates in (4.3) requires modiﬁcation in that the reaction orders

30 are no longer identical to the stoichiometric coefficients. Specifically, the modified rate expressions are given by

K K 0 00 Y nki Y nki ωi = ki,f Ck − ki,b Ck , (4.22) k=1 k=1

0 00 where nki and nki denote empirically determined reaction orders. Of course, also the parameters of the forward and backward rate constants, ki,f and ki,b, respectively, are determined empirically.

Another alternative to formulate the rates of production of chemical species that participate in a global reaction mechanism is based on a more systematical approach, which will be presented and discussed in detail in Chap. 7.5 of the present thesis. For instance, if this systematical approach is applied to the detailed ozone mechanism given in Chap. 4.1 and the global reaction given in (I), for ozone and molecular oxygen the mass rates of production

wO3 = − WO3 ωI and wO2 = +1.5 WO2 ωI (4.23) are obtained. Here ωI denotes the rate of the global reaction (I) which can be expressed - see Chap. 7.5 - as a linear combination of the rates of elementary reactions of the ozone mechanism.

31 Chapter 5

Discretization and Numerical Methods

To tackle successfully a reactive-flow problem numerically, the proper choice of a suitable numerical method is in the computer code developed within the framework of the present thesis essential. Particularly the stiffness of the governing equations, which is introduced through the chemical source terms, makes the numerical solution of a laminar reactive-flow problem considerably more difficult than the solution of a comparable non-reactive flow problem. Therefore, three different numerical methods are implemented in the computer code developed within the framework of the present thesis each method having its particular advantages.

The use of adaptive methods, i.e., of methods that “automatically” adapt the computational mesh in some “optimal” way to the specific problem under consideration, is the desirable numerical approach to all engineering problems; it is a must, however, for the computation of reactive flows. The physical reason for this is the heat release associated with any combustion process. As a consequence of the heat release, profiles of quantities involved in combustion processes typically exhibit steep gradients and strong curvature, thus necessitating the use of adaptive methods in order to control both the temporal and spatial discretization errors.

In this chapter we ﬁrst formulate the governing equations in a general form suitable for a numerical solution. Then we discuss the numerical solution methods and, ﬁnally, the adaptive-gridding technique implemented in the computer code developed within the framework of the present thesis.

32 5.1 Formulation

5.1.1 Diﬀerential Equations

Let u denote a vector of unknown dependent variables – for the example of freely T propagating flame flame that would be u = (Y1, ..., YI , T, m) . In terms of u, the general class of differential equations solved by the computer code developed within the framework of the present thesis can be written in the form

B(u) ut = f(x, t, u, ux, uxx) , a ≤ x ≤ b, t > 0 , (5.1)

0 = ra(a, t, u(a, t), ux(a, t)) , t > 0 , (5.2)

0 = rb(b, t, u(b, t), ux(b, t)) = 0 , t > 0 , (5.3)

0 = u(x, 0) − v(x) , a ≤ x ≤ b . (5.4)

Here t the time, and x is a space-like coordinate. Depending on the specific problem under consideration, the latter can be, e.g., one of the cartesian spatial coordinates, or a similarity coordinate, or – in the case of diffusion-flame formulation in mixture- fraction space – the mixture fraction. Subscripts t and x denote differentiation with respect of the time and the space-like coordinate. The quantities a and b denote the location of the left and the right boundary, respectively, of the computational domain; B is a usually sparse coefficient matrix; f is a vector functions whose components represent the governing equations in residual form, however excluding terms involving time-derivatives of the dependent variables – the latter terms are summarized on the left hand side of (5.1).

Equations (5.2) and (5.3) represent boundary conditions on u, (5.4) initial conditions.

Thus, ra, rb and v are given vector functions of the same dimension as u and f; the function v(x) contains the given initial proﬁles of the dependent variables.

For steady problems the time derivative ut is zero, and the initial conditions (5.4) are redundant.

5.1.2 Finite Diﬀerences

The solution to a steady problem can be obtained by either solving the steady form of the governing equations directly or, alternatively, by solving a time-dependent version

33 of the problem until the steady state is approached. Since the computer code is designed for the solution of both steady and unsteady problems, it permits both the former and the latter approach to a steady solution.

Integration of system (5.1) – (5.4) with respect to time is performed in steps starting with speciﬁed proﬁles – which, preferably, should satisfy (5.1) – at time level m = 0 with t = t0 ≡ 0. Solutions to (5.1) are sought at the subsequent time levels (m = 1; t = t1), (m = 2; t = t2), and so on, with 0 = t0 < t1 < t2 < ··· < tm < ··· , where here and below the superscript m is used to identify quantities at time level m, m = 0, 1, 2, ··· .

The integration of (5.1) is considered complete if either a speciﬁed level mmax or a speciﬁed time tmax is reached.

With respect to the space-like variable x, system (5.1) – (5.4) is discretized on a mesh Mm of grid points, m m m m M = {a = x1 < x2 < ... < xN = b} . (5.5) Note that due to adaptive gridding the number of grid points may vary from one time level to another. If the spatial domain of integration is infinitely large, for example extending from minus to plus infinity, then a and b are chosen as positive and sufficiently large to ensure that the conditions at either boundary can asymptotically be satisfied. A similar remark applies in case of a semi-infinite domain.

For the first-order derivatives central and, alternatively, one-sided differences are adopted. For the example of a scalar dependent variable T at grid point j, the second-order accurate central-differencing scheme is

∂T h hj−1 hj − hj−1 hj i ≈ Tj+1 + Tj − Tj−1 , (5.6) ∂x j hj(hj + hj−1) hjhj−1 hj−1(hj + hj−1)

m m where hj = xj+1 − xj . For convenience of notation, here and below the superscript m is omitted from the hj as well as from other quantities. The ﬁrst-order accurate one-sided diﬀerencing-schemes are ∂T T − T ≈ j j−1 (5.7) ∂x j hj−1 and ∂T T − T ≈ j+1 j , (5.8) ∂x j hj respectively.

34 The second-order derivatives are approximated by ∂ ∂T 2 h i λ ≈ λj+1/2(Tj+1 − Tj)/hj − λj−1/2(Tj − Tj−1)/hj−1 , (5.9) ∂x ∂x j hj + hj+1 where λ = λ(x, t, u) is given by x + x u + u λ = λ( j−1 j , t, j−1 j ) , (5.10) j−1/2 2 2 for j = 2, ··· N. Recall that due to adaptive gridding the value of N may vary from one time level to another.

Depending upon the approach to the solution of system (5.1) – (5.4), eventually the time derivatives must be discretized as well. If after discretization Newton’s method is employed at each time step, a backward-Euler finite-difference approximation for the time-derivative is used. If, after spatial discretization, system (5.1) – (5.4) is solved with a solver suitable for the numerical integration of differential-algebraic systems, the time-derivative appearing in (5.1) – (5.4) needs not to be discretized. However, in either case a fully implicit spatial finite-difference formulation is necessary – and employed in the computer code developed within the framework of the present thesis – in order to successfully cope with the stiffness of system (5.1) – (5.4) that arises through the chemical source terms appearing in the equations governing any laminar reactive-flow problem.

5.1.3 Diﬀerential Algebraic Systems

Ultimately, the system (5.1) – (5.4) with the vector function f, ra and rb in discretized, is cast into the form of a diﬀerential algebraic system. To this end, the vector of dependent variables u is formed at each grid point n, n = 1, ..., N, and denoted by un. Then the overall vector of unknowns

T U = (u1, ..., uN ) (5.11) can be deﬁned. Similarly, upon deﬁning for each grid point n, n = 1, ..., N, the local vector of right-hand sides, fn, the overall vector of right-hand sides

T F = (f1, ..., fN ) (5.12)

35 can be defined. With these definitions, (5.1) – (5.4) can be cast into the form of a differential algebraic system of equations (DAE system) [6, 42, 55], viz.,

dU A(U) = F(U) . (5.13) dt 5.2 Numerical Methods

Implemented in the computer code developed within the framework of the present thesis is a modified Newton method for the effective solution of steady problems and an extrapolation method for the effective solution of time-dependent problems. Currently both the Newton and the extrapolation method can be used for all problems. In the following subsections the Newton method and the extrapolation method are briefly discussed.

5.2.1 A Modiﬁed Newton Method

For the solution of time-dependent problems at each time step or steady problems (not considered herein), Newton’s method can be applied to the system of nonlinear equations, G(U) = 0 , (5.14) which results from the discretization of the governing equations. In (5.14), the vector function G is deﬁned as U − Um−1 G(U) = F(U) − A(U) . (5.15) ∆t Here both U and G are taken at the time level m under consideration, and ∆t = tm − tm−1. For steady problems, the matrix A is zero. Thus, at each time level m the linear system k k+1 k k J(U )(U − U ) = −ωkG(U ), k = 0, 1, ... , (5.16)

k k is solved, where U denotes the solution after k Newton iterations, and ωk and J(U ) are the damping parameter [29] and the Jacobian matrix, respectively, based on Uk. The Jacobian is re-evaluated only periodically [97].

If applied to unsteady problems, this modified method has been found very robust and relatively weekly sensitive towards the initial profiles. If applied to steady problems, his methods has been found to converge fast, provided the initial guesses are sufficiently good. A weakness of the method is its crude first-order differencing of the time derivatives, and the availability of only a limited time-step size control. Still,

36 – for steady problems with suﬃciently good initial guess, and

– for unsteady problems with poor initial proﬁle, this is the method of ﬁrst choice.

5.2.2 An Extrapolation Method

For the effective solution of time-dependent reactive-flow problems the solver LIMEX [29, 30, 31] is implemented in the computer code developed within the framework of the present thesis. This solver is suitable for the solution of stiff differential algebraic systems.

5.3 Adaptive Selection of Grid

The procedures and criteria for the adaptive selection of grid points are of critical importance to the efficiency of the algorithms that are used for the solution of reactive- flow problems. In particular, strategies are required that place the grid points where they are needed in order to bound the local space-discretization error [39, 86, 87, 88]. In generalization of procedures outlined for the adaptive computation of steady reactive- flow problems [39], for any fixed time level m we equidistribute the mesh Mm on the m m m interval [a = x1 , xN = b] with respect to a non-negative weight function W and a constant Cm, i.e., W m is selected such that

m Z xj+1 W mdx = Cm, j = 1, ··· ,N m − 1 , (5.17) m xj where N = N m. Speciﬁcally, the weight function W m is chosen as

m m W = max Wk , (5.18) 1≤k≤2N+1 where

m |∂Uk/∂x| Wk = m , 1 ≤ k ≤ K (5.19) g |maxUk − minUk| 2 2 m |∂ Uk/∂x | WN+k = m , 1 ≤ k ≤ K, (5.20) c |max(∂Uk/∂x) − min(∂Uk/∂x)| and m m W2N+1 = d . (5.21) In (5.19) and (5.20), and below, K denotes the number of dependent variables, and “min” and “max” stand for the minimum and maximum value in the interval a ≤ x ≤ b

37 of the respective quantity, and gm and cm are positive scaling factors; their numerical values are less than unity if in (5.17) Cm = 1 is employed. In (5.21), dm is a positive constant which represents the maximum size of any interval hj. To prevent the size of adjacent mesh intervals from varying too rapidly, we require that at any time level m the mesh be locally bounded ,viz.,

−1 R ≤ hn/hn−1 ≤ R, n = 2, ··· ,N − 1 , (5.22) where R is a constant greater than one.

One of the two criteria, called grid convergence, for successful solution of a laminar- ﬂame problem, is that the number of grid points for two consecutive grids is the same, m say Nl , and that in addition

N m 1 Xl (xm − xm )2 ≤ , (5.23) N m n,l n,l−1 l n=1 where is a small positive constant. The other, called alternating grid convergence, is that repeatedly alternating insertion and removal of the same number (typically 1 or 2) of grid points occurs. Finally, we report the observation that it is advantageous to limit the number of grid points by which grids are allowed to change from one value of l to the next; typical values for this limit are 3 to 7.

38 Chapter 6

Methods of Reduction of Detailed Reaction Mechanisms to a Skeletal Mechanism

6.1 Basics

In Chap. 4, the general representation of chemistry by a detailed mechanism of elementary reactions was presented. Generally, to understand combustion chemistry in more and more detail, bigger and bigger reaction mechanisms have been developed. In particular chemists are strongly engaged in this branch of science and research. However, from the engineering point of view, relatively small yet “powerful” reaction mechanisms are needed that can be used, e.g., in CFD computations of combustion processes that take place in complicated, spatially three-dimensional geometries under complex, mostly turbulent and often transient, conditions. Here the adjective “powerful” means that a reaction mechanism must be sufficiently small to enable computations within a reasonable span of CPU time and, at the same time, sufficiently large to be able to reflect important features of the combustion or ignition process under investigation such as flame structures, flame speeds, ignition-delay times, or flame extinction. The so-called reduced mechanisms aim to satisfy both these requirements. Methods for the derivation of reduced mechanisms are presented and discussed in this chapter. Specif- ically, reduction methods are discussed and also developed that lend themselves to numerical treatment thereby enabling computationally assisted reduction procedures.

For a given combustion problem, say, the non-premixed combustion of methane in air, or in pure oxygen, a detailed mechanism of elementary reactions is not unique. Rather, the mechanism can contain more or less reactions depending upon the solution details required for the particular investigation. For instance, if the simulation of such a diﬀu- sion ﬂame needs to be accurate only for a narrow region in the entire parameter space,

39 e.g., only for low but not high pressures, for low but not high boundary temperatures, and if soot formation is not of interest for the investigator, then a relatively short detailed mechanism, i.e., a mechanism comprising perhaps only 20 or 30 rather than hundreds of chemical reactions may suﬃce. If, however, the simulations are required to yield also physically meaningful results at high pressures, say, or close to ﬂame extinction, then a substantially larger chemical mechanism is required to yield realistic results that can keep up with the accuracy of experimentally obtained data, say.

Thus, for one and the same physical combustion problem, there exist – usually – many alternative detailed reaction mechanisms of varying degrees of detail. At this point it is appropriate to define a so-called skeletal mechanism. A skeletal mechanism is defined for, (i), a given physical combustion problem under given, specified conditions, say, at low pressures remote from extinction conditions, and, (ii), for given target quantities. Target quantities could be, e.g., maximum flame temperatures or flame structures in terms of the concentration profiles of the stable species and the most important reaction intermediates. For instance, in the framework of oxyfuel combustion, the methane/ogygen diffusion flames computed in the present thesis require the prediction of reasonable maximum temperatures for all strain rates as well as reasonable

flame structures for the major species – these are the reactants CH4 and O2, the stable products H2O and CO2, as well as CO and, to a certain degree, the most important species of the hydrogen-oxygen pool, i.e., H, O and OH. Not of interest are, in the context of the present thesis, e.g. concentrations of species that can be considered soot precursors or other intermediate species considered unimportant from the engineering point of view. Coming back now to the definition of a skeletal mechanism, a skeletal mechanism can be defined as the smallest detailed mechanism of elementary reactions that, for a given combustion problem and specified target quantities still gives “reasonably accurate” results. Here the adjective “reasonably accurate” is not strictly defined, of course.

In the remainder of this chapter, methods and strategies for deriving a skeletal mechanism from a truly detailed mechanism of elementary reactions are presented and discussed.

6.2 Directed Relation Graph (DRG)

To conform to the notation of current literature [58, 59, 60, 61, 62, 63], in this section species will be denoted by capital letters A, B, and so on. The molar rate of production PI of species A in a given combustion system is – see Chap. 4.2 – i=1 νAiωi, where ωi

40 denotes the rate of reaction i, i = 1, ...I. Each term νAiωi of this sum makes a positive, negative or zero contribution to the rate of production of species A. Therefore, the sum

I X |νA,iωi| i=1 can be considered as a gross measure of the chemical activity involved in the production of species A from all I reactions. Similarly, given a species B 6= A, the sum

I X |νA,iωiδBi| i=1 where ( 1, if species B participates in reaction i , δBi = 0 otherwise , can be considered as a gross measure of the chemical activity involved in the production of species A from those reactions only in which B participates as either a reactant, a product, an inert or a third body. Hence, the coeﬃcient

PI |ν ω δ | r = i=1 A,i i Bi (6.1) AB PI i=1 |νA,iωi| is a measure for the deviation of the rate of production of species A if species B is removed from the system, i.e., if from the reaction mechanism all reactions are removed in which species B participates. Obviously, 0 ≤ rAB ≤ 1 and, the smaller rAB, the smaller is the eﬀect of B on the rate of production of species A.

In [58] and a series of publication resulting thereof [59, 60, 61, 62, 63], Lu developed a systematical strategy based on the coefficients rAB that allows the elimination of – in the sense of the above discussion – unimportant chemical species and reactions related with them. His theory was based on graph theory and, therefore, his method is commonly referred to as DRG, which is an acronym for directed relation graph – therefore, rAB is also termed a relation. Not only allows DRG the identification of individual unimportant species but also the identification of groups of species that are unimportant and hence can be removed as groups, or, if they are important, are kept as groups.

In the present work, DRG has been employed before of using several other methods to derive a skeletal mechanism from a comprehensive detailed mechanism of elementary

41 reactions. Specifically, the comprehensive detailed mechanism used herein is the so- called GRI-mechanism version 3.0 [96].1 However, DRG has well-known limitations in that – in its original or pure form – it eliminates from a reaction mechanism only those species that have a negligible effect on the production rate of other species. In doing so DRG fails to eliminate species that may have a non-negligible effect on other species’ production rates but nevertheless are irrelevant or unimportant for the specific combustion problem under consideration [117]. For instance, for a specific case – particularly in engineering applications –, the concentrations of some or even many species A may not be of interest even though their relations rAB with many species B may not be small. To be able to reduce mechanisms beyond the bounds experienced by DRG, i.e., to be able to derive even smaller skeletal mechanisms, the original DRG method has been complimented by several auxiliary reduction schemes. Such reduction schemes are presented and discussed in the remainder of the chapter. In Chap. 6.5, also a new reduction scheme developed and applied in the framework of the present thesis will be presented.

6.3 DRG-Aided Sensitivity Analysis (DRGASA)

Sensitivity analysis is a general, powerful tool for identification of parameters whose values have a sufficiently small influence on a so-called target quantity. For instance, for a homogeneous ignition process the ignition delay time τ could be selected as a target, and the influence of the values of the rate parameters of individual reactions on that target could be of interest. In this example, usually the relative sensitivity coefficients [53] ∗ (i) ki ∂τ ki ∆τ Sτ = ≈ ∗ (6.2) τ ∂ki τ ∆ki are evaluated, where – specifically – the variation of the rate constant ki is realized by a variation of the constants Ai,f and Ai,b, respectively. In the second equality of (6.2), ∗ ∆ki denotes a prescribed perturbation about τ , and ∆τ is the deviation resulting from it.

In the context of the present thesis, the temperature values Tn = T (Zn), where 0 =

Z1 < ... < ZN denote the discrete values of mixture fraction that where obtained by discretization in computer code – see Chap. 5. The parameter varied was the number of species in the reaction mechanism. Speciﬁcally, a ﬁrst run with all K species yielded

1 The GRI3.0 is a comprehensive detailed mechanism developed and optimized for natural gas ﬂames simulations. This mechanism has been used as a chemistry model in several investigations under the traditional air-ﬁring conditions [25, 48, 62, 106, 116] as well as in oxyfuel combustion [11, 23].

42 ∗ Tn , n = 1, ..., N. Then, for each subsequent run, one diﬀerent species was removed at a time, i.e., each subsequent run was done with K − ∆K := K − 1 species. Then – for each run – the resulting relative deviation

k ∗ k ∗ k k K ∆Tn K Tn − Tn Tn − Tn STn ≈ ∗ = ∗ = K ∗ (6.3) Tn ∆K Tn 1 Tn could have been calculated. In (6.3), K is – as everywhere in this thesis – the number of species in the reaction mechanism, i.e., a constant. The derivation of (6.3) helps to understand the original formulation of a somewhat different but still similar sensitivity coefficient that is usually quoted in DRG sensitivity analysis – also termed DRGASA k [117] –, and which was also evaluated, rather than the coefficients STn given in (6.3) – viz., |x − x∗| Ek := max , (6.4) x∈D x∗ where Ek is a sensitivity coefficient, or error, according to the DRGASA terminology, and x a sensitivity target such as temperature T ; D denotes a temporal or spatial domain in which discrete value of x, such as Tn, are defined.

6.4 Unimportant-Reaction Elimination

After the discussion in Chap. 6.2 of the relation rAB it is straightforward to comprehend that the so-called importance index, IA,i, deﬁned as

|ν ω | I = A,i i , (6.5) A,i PI j=1 |νA,jωj| can be interpreted as the contribution of the i-th reaction to the production rate of species A, A = 1, ..., K, i = 1, ..., I [63].

Based on a user-speciﬁed threshold for all species, reaction i is said to be unimportant and removed from the skeletal mechanism if IA,i < for all species A, i.e., A = 1, ..., K. If the reaction is important, then it is kept in the skeletal mechanism. In the computations performed in the framework of the present thesis, typical values of have been found in the range from 0.1 to 0.3. Speciﬁcally, with a threshold in this order of magnitude it was found that often up to 40 % of the reactions in a detailed mechanism of elementary reactions could be eliminated.

43 6.5 Reaction Elimination Based on DRGASA

In the work for the present thesis it was found, that DRG, DRGASA and unimportant- reaction elimination as just described in Chaps. 6.2, 6.3 and 6.4, respectively, led to a “skeletal” mechanism that was not really the smallest possible detailed reaction mechanism and hence did not deserve to name “skeletal mechanism”. Therefore, a way, or method, was sought with which a mechanism could be reduced further. It was found that a criterion similar to (6.4) is constructive, but now applied to individual reactions rather than to individual species. In other words, rather than taking out from the mechanism one species k at a time which according to (6.4) gives a sensitivity coeﬃcient or error Ek, now one reaction i is taken out of the mechanism at a time giving a sensitivity coeﬃcient or error

|x∗ − x| Fi := max , (6.6) x∈D x∗ where Fi is the error induced to target global parameters such as temperature proﬁle due to the elimination of the i-th reaction from the starting skeletal mechanism; here, the starting skeletal mechanism is the one derived after successive application of DRG, DRGASA and unimportant-reaction elimination, respectively.

In the applications considered in the present thesis, that are all related to oxyfuel combustion, it was found that the application of criterion (6.6) helps to reduce – after application of DRG, DRGASA and unimportant-reaction elimination – the reaction mechanism by another 30 to 40 % of elementary reactions. The mechanisms thus obtained could generally be considered as true skeletal mechanisms.

6.6 Summary of Reduction Methods

The four reduction methods discussed above are – usually – carried out in succession. The methods themselves and the sequence in which they are carried out are summarized in the chart shown in Fig. 6.1. We now discuss that chart taking the GRI3.0 mechanism [96] as an example for a truly detailed mechanism from which the overall reduction procedure starts. At this point it is important to note that before the overall reduction procedure is started, the mechanism has been successfully tested to describe really well the combustion problem at hand in its entire parameter range of interest. Such benchmarking can be carried out, e.g., against experimental results or against computational results obtained on the basis of an even more comprehensive, detailed mechanism of elementary reactions. In the framework of the present thesis, the bench-

44 Figure 6.1: Flow chart of the skeletal reduction procedure marking was carried out against experimental data available in the literature, and it will be described in detail in Chap. 8.

In the top left corner of Fig. 6.1, the GRI3.0 mechanism serves as input to the overall mechanism-reduction procedure after it was veriﬁed – see Chap. 8 – that it is able to represent really well non-premixed laminar oxy-methane combustion in the entire range of scalar dissipation rates – or, alternatively, strain rates – from the practically unstrained state up to extinction due to excessive strain. The GRI3.0 mechanism comprises 325 elementary reactions amongst 53 chemical species. Employing the DRG reduction method reduced the mechanism to 192 reactions amongst 32 species. The resulting mechanism can by deﬁnition of a skeletal mechanism – see Chap. 6.1 – not be called a skeletal mechanism yet because it is not yet the smallest possible mechanism of elementary reactions from which, ultimately, a global reaction mechanism consisting of global reaction steps will be derived.

The reduced detailed mechanism obtained by DRG is reduced further by employing DRGASA, yielding a mechanism of elementary reactions comprising 117 elementary reactions amongst 24 species. Subsequently, unimportant-reaction elimination is employed, leading to a mechanism of elementary reactions comprising 85 elementary re-

45 actions amongst 24 species. Last, reaction elimination based on DRGASA is employed leading to the ﬁnal detailed mechanism – the so-called skeletal mechanism – comprising 47 elementary reactions amongst 20 species. This skeletal mechanism serves as a starting point for the reduction procedures described in Chap. 7.

46 Chapter 7

Derivation of Reduced Mechanisms of Global Reactions

7.1 Notational Issues

We consider a combustion system whose chemistry is described by a skeletal mechanism – see Chap. 6.1 – comprising I chemical reactions amongst K chemical species.1 The K species in the system are made up of E elements.

The skeletal mechanism stoichiometric matrix, N, is a (K × I) matrix,

N = (νki) 1≤k≤K . (7.1) 1≤i≤I

By default – i.e., unless stated otherwise – all vectors are column vectors. For instance, the vector of species mass fractions is

T Y = (Y1, ..., YK ) , (7.2) the vector of species molecular weight is

T W = (W1, ..., WK ) , (7.3) the vector of the molar species production rates is

T ω = (ω1, ..., ωK ) , (7.4) and the vector of reaction rates is

T r = (r1, ..., rI ) . (7.5)

1 Some of the species may be inert.

47 Thus, in terms of a suitably deﬁned accumulative-convective-diﬀusive operator L, in symbolic form, the species conservation equations can be written as

L(Y ) = N r = ω . (7.6) where Y Y = , (7.7) diag(W )

For instance, for a spatially homogeneous system the species conservation equations are generally written in terms of the vector of species mass fractions Y rather than the quantity Y ; A possible choice is

T d ρY1 ρYK d T dC L(Y ) = , ..., = (C1, ..., CK ) = . (7.8) d t W1 WK d t d t

In the present thesis, for the equations (2.1) governing diﬀusion ﬂames described in mixture-fraction space, L(Y ) is chosen such that for any chemical component k

2 L(Yk) ∂Yk ρχ 1 ∂ Yk L(Y k) = = ρ − 2 , (7.9) diag(W ) ∂t 2 Lek ∂Z k = 1, ..., K.

Here and below, species symbols will be denoted by M, i.e., Mk is the symbol of species k for k = 1, ..., K. For instance, if the k-th species is methane, then Mk is

From a notational point of view, it will be convenient to introduce the sets K and I, which contain all possible species indices and reaction indices, respectively, i.e.,

K := {1, 2, ..., K} and I := {1, 2, ..., I} . (7.10)

7.2 Species Splitting

The ﬁrst step in the derivation of a reduced mechanism of global reaction steps is to partition the set K into two sets KS and KN corresponding to steady-state species and

48 non-steady-state species, respectively.2 Hence

KS = {i ∈ K | Mi is SS-species} = {i1, ..., iKS } (7.11) and

KN = {j ∈ K | Mj is not SS-species} = {j1, ..., jKN } , (7.12) respectively. Here and below, SS stands for steady-state. Note that

KS ∪ KN = K and KS ∩ KN = ∅ (7.13) and hence

KS + KN = K. (7.14)

In terms of the (K ×I) matrix of stoichiometric coeﬃcients, N, the reaction mechanism can be written as N T m = 0 , (7.15) where m is the vector of species symbols,

T m = (M1, .., MK ) . (7.16)

Thus we can write K X νki Mk = 0 , i = 1, ..., I (7.17) k=1 or X X νki Mk + νki Mk = 0 , i = 1, ..., I . (7.18)

k∈KS k∈KN

Criteria for the selection of steady-state species and hence for ﬁxing both the number

KS as well as the members of the set KS were discussed in Chap. 4.4. In the following it will be assumed that suitable steady-state species have been selected.

7.3 Reaction Splitting

Linear algebra and stoichiometry tell that in a detailed reaction mechanism – e.g., a skeletal mechanism – comprising I elementary reactions amongst K species,

II ≤ K − E (7.19)

2 Steady-state species were discussed in Chap. 4.5.

49 mutually linearly independent reactions exist – here E denotes the number of chemical elements in the system –, and, consequently,

ID = I − II ≥ I − K + E (7.20) linearly dependent reactions. The number of linearly independent reactions is greater or equal than the number of applicable steady states, i.e.,

II ≥ KS . (7.21)

Hence the partitioning of reactions as described in the following is feasible. In particular, if II = K −E and II > KS, – which generally, but not always, is the case – also the mechanism-reduction procedure described below can be carried out. In other words,

II > KS is requisite to deriving a reduced mechanism by the systematical method outlined below. The number of global reactions in such a reduced mechanism, IG, is

IG = K − KS − E. (7.22)

After, in a ﬁrst step, the K species have been partitioned as discussed in Chap. 7.2, the second step in the derivation of a reduced mechanism of global reaction steps is to partition the set of chemical reactions, I, into a subset of mutually independent 3 but undesired reactions, IU , and the complementary subset of remaining reactions,

IR = I − IU . Naturally IU ⊆ II , where II denotes the set of linearly independent reactions. Note that

IU ∪ IR = I and IU ∩ IR = ∅ . (7.23)

At this stage, we still have to explain what undesired reactions are. The number of reactions contained in the corresponding set IU is prescribed. However, since IU is not unique, criteria are needed that help to determine the “best” possible set.

In other words: The question is, which elementary reactions are best eliminated from the global reaction rates? Common sense tells that one should keep those elementary reactions whose rates have a relatively large and/or long-lasting inﬂuence on the reaction kinetics, and that, therefore, one should drop or eliminate elementary reactions whose rates have only a relatively small and/or short-lasting inﬂuence on the reaction kinetics.

A small or short-lasting inﬂuence on the rate of the global system have elementary

3 This subset is not unique.

50 reactions that in the combustion or ignition process under consideration quickly approach or even reach chemical equilibrium. Usually such reactions are termed fast. Hence, in [41] the recommendation was given that fast reactions should eliminated.

From the above discussion it can be concluded that fast – better the fastest – elementary reaction reactions in the skeletal mechanism can be taken as the undesired reactions.

The question remains what suitable methods are to determine the fastest IU reactions in the skeletal mechanism. As will become obvious below, selecting the number IU equal to KS makes the determination of rates of the global reactions in a reduced mechanism particularly simple.4 However, this particular choice is not compulsory.

For the person deriving a reduced, global mechanism to make the right selection of fast reactions, a certain knowledge of reaction kinetics is required. To assist the mechanism developer, in [20] the criterion was given that those elementary reactions should be taken as the set of independent reactions, IU , towards which the molar rates of production of the steady-state species, ωk, k ∈ KS, have a small sensitivity. Speciﬁcally,

ﬁrst for the KS steady-state species the sensitivity coeﬃcients

∂ωk , k ∈ KS and i = 1, ..., I , (7.24) ∂ki are computed. Second, the linearly independent reactions pertaining to the set IU are, step by step, selected as follows. Starting with first steady-state species in the set KS, k1, the first independent elementary reaction, i1 ∈ IU , is taken as that reaction i that for steady-state species k1 yields the smallest sensitivity coefficient. For the second steady-state species in KS, k2, the second independent elementary reaction i2 ∈ IU is taken as that reaction i that for species k2 has the smallest sensitivity coefficient.

Should that reaction already be an element of set IU , the reaction with the smallest sensitivity coeﬃcient for steady-state species k2 is selected that is not yet an element of set IU . The selection procedure continues stepping though all KS steady-state species until IU comprises IU linearly independent elementary reactions.

As will be seen in Chap. 7.5, the procedure just outlined to construct IU ensures that the fastest elementary reactions are eliminated from the global reaction rates whilst the slowest elementary reactions are kept in the global rates.

Similarly, assigning reactions to IR that by use of the sensitivity coeﬃcients discussed

4 The simplicity arises through the fact that N1 – see Eq. 7.25 – has full rank and hence N1 needs not to be subjected to a singular-value decomposition but can be inverted by similar methods such as LU decomposition or Gaussian elimination. Also, the so-called third-body tracking then becomes straightforward because, for this purpose, the simple Gaussian elimination procedure can be employed.

51 above have been found ineﬀective in contributing to the (near-)zero elementary rates of the steady-state species ensures that for the classiﬁcation of a species as steady-state species important rates are kept in the description of the global rates.5

7.4 Reduced Stoichiometric Matrix

We now derive a stoichiometric matrix for the reduced mechanism of global reactions. To this end, we assume that, by applying the methods and strategies outlined in Chaps. 7.2 and 7.3, we have found a suitable partition of the stoichiometric matrix N, viz.,

N N N = 1 2 . (7.25) N3 N4

Thus, the mechanism of elementary reactions, i.e., the skeletal mechanism, (7.15), can be written as T T T N1 N3 mS 0 = N m = T T , (7.26) N2 N4 mN where mS and mN denote the vectors of the symbols of the SS and the non-SS species, respectively – see Eq. (7.16)–. Equation (7.26) represents two vector equations for the two unknown vectors mS and mN , viz.,

T T N1 mS + N3 mN = 0 , (7.27) T T N2 mS + N4 mN = 0 . (7.28)

Since the global mechanism will contain only the non-SS species, the task now is to eliminate mS from (7.27) and (7.28). This is accomplished as follows.

From (7.27) we obtain

T T 0 = N3 mN + N1 mS T T T = N3 mN + (mS N1) T T T T = N3 mN + (mS UDV ) T T = N3 mN + VDU mS ,

T where the single value decomposition of N1, i.e UDV has been used. Thus, taking the

5 For steady-state species k – in terms of creation rate ωk+, consumption rate ωk−, and production rate ωk = ωk+ − ωk− –, the criterion eﬀectively helps to eliminate those reactions that are making the smallest contributions to the creation rate being approximately equal to the consumption rate whilst keeping both these rates large compared to the (net) production rate. In other words, the criterion ensures that only elementary rates are eliminated from the global reaction rates that are not needed for a good production-creation-consumption representation of a steady-state species, thereby ensuring that the global mechanism consistently incorporates the elimination of the steady species.

52 algebraic manipulations step by step, the following sequence of equations is obtained:

T T 0 = N3 mN + VDU mS , T T T 0 = V N3 mN + DU mS , −1 T T T 0 = D V N3 mN + U mS , −1 T T 0 = UD V N3 mN + mS , −1 T T T 0 = (VD U ) N3 mN + mS , ˜ −1 T T 0 = (N1 ) N3 mN + mS , and, ﬁnally, ˜ −1 T T mS = −(N1 ) N3 mN . (7.29) Substituting (7.29) into (7.28) yields to another sequence of algebraic equations, viz.,

T T 0 = N2 mS + N4 mN , T ˜ −1 T T T 0 = −N2 (N1 ) N3 mN + N4 mN , ˜ −1 T T 0 = −(N3N1 N2) mN + N4 mN , T ˜ −1 T 0 = (N4 − (N3N1 N2) ) mN , ˜ −1 T 0 = (N4 − N3N1 N2) mN .

Thus, the global, reduced mechanism is given by

˜ −1 T (N4 − N3N1 N2) mN = 0 . (7.30)

If IU = I − K − E − IG and hence IR = IG, then

−1 NG := N4 − N3Ne1 N2 (7.31) is the stoichiometric matrix of the reduced, global mechanism. If IU = KS and hence ˜ −1 −1 IR = I − KS, then IG < IR. In this case, N1 = N1 , and the stoichiometric matrix of the reduced, global mechanism is assembled by IG linearly independent columns of −1 N4 − N3 N1 N2 that can be selected arbitrarily.

Once the stoichiometric matrix NG is known, the reduced mechanism can be straightforwardly derived from it. For instance, if the non-steady state occurring in the reduced combustion system are CO, H2O, CO2,H2, H and O2 and if the stoichiometric matrix is [92, 107, 108]

53  −1 0 0   −1 0 2     1 0 0  NG =   . (7.32)  1 1 −3     0 −2 2  0 0 −1 then the reduced mechanism is

CO + H2O = CO2 + H2 (I’)

2 H + M = H2 + M (II’)

O2 + 3 H2 = 2 H2O + 2 H (III’)

At this stage it is important to note that the reduced mechanism is unique6 but the corresponding global reaction rates are still arbitrary. This is so because in deriving ∗ NG, the original matrix N1 has been made a diagonal matrix N1 – where we have assumed that N1 has full rank – and the original matrix N2 has been transformed to ∗ ∗ N2 = 0. N3, in general has become some none-zero matrix N3 . It is possible though ∗ not necessary to make N3 zero by suitable row manipulations involving the diagonal ∗ matrix N1 . Note that the latter manipulations do not aﬀect NG – it is in this sense ∗ that NG is unique. Since there is no obvious advantage in making N3 zero, this step is skipped here.

7.5 Global Reaction Rates Determined Systemati- cally

7.5.1 Theory of Derivation

The species conservation equations can be written as

L(Y ) = ω = N r , (7.33)

6 Of course, NG can still be subjected the usual linear combinations of columns, and rows may be interchanged.

54 where Y Y = , (7.34) diag(W ) T Y = (Y1, ..., YK ) , (7.35) T W = (W1, ..., WK ) , (7.36) T ω = (ω1, ..., ωK ) , (7.37) T r = (r1, ..., rI ) . (7.38)

Here Y is the vector of mass fractions, W the vector of species molecular weight, ω is the vector of molar species production rates, and r denotes the vector of reaction rates. The chemical source terms,

ω = N r , (7.39) are associated with the subsets KS, KN , IU and IR.

To accomplish the partitions

ω r ω = S and r = U , (7.40) ωN rR two permutation vectors are required, ps for the species and pr for the reactions, that are deﬁned as follows. The permutation vector of the species and of their production rates, ps, is deﬁned by

T T ps = (ps,1, ..., ps,K ) = (k1, ..., kS, kS+1, ..., kK ) . (7.41)

Similarly, the permutation vector of the reactions and the reaction rates, pr, is deﬁned by T T pr = (pr,1, ..., pr,I ) = (i1, ..., iU , iU+1, ..., iR) . (7.42)

By applying the two permutation vectors, the stoichiometric matric N is partitioned into7 N N N = 1 2 (7.44) N3 N4 such that ω N N r S = 1 2 U . (7.45) ωN N3 N4 rR

7 The transpose of N is T T T N1 N3 N = T T . (7.43) N2 N4

55 From (7.40) to (7.45), the size of the matrices N1 to N4, i.e., the numbers of their rows and columns, is obvious. Since for strict – or exact – steady states L(Y S) = ωS = 0,

N1 rU + N2 rR = 0 (7.46) and hence

N1 rU = −N2 rR . (7.47)

With a singular-value decomposition of N1, (7.47) becomes

T UDV rU = −N2 rR . (7.48)

Since U T U = I, this becomes

T −1 T V rU = −D U N2 rR , (7.49)

Since VV T = I, we thus obtain

−1 T −1 rU = − VD U N2 rR = − Ne1 N2 rR , (7.50) where we have deﬁned −1 −1 T Ne1 := VD U . (7.51) Thus, we can write

ωN = N3 rU + N4 rR −1 T = −N3 VD U N2 rR + N4 rR −1 T = (N4 − N3 VD U N2) rR −1 = (N4 − N3 Ne1 N2) rR . (7.52)

In general, steady states are not exact, i.e., in general L(Y S) = ωS 6= 0. Thus,

L(Y S) − N2 rA = N1 rU (7.53) or −1 −1 rU = Ne1 L(Y S) − Ne1 N2 rR . (7.54)

56 Hence

L(Y N ) = N3 rU + N4 rR −1 −1 = N3 Ne1 L(Y S) − N3 Ne1 N2 rR + N4 rR −1 −1 = N3 Ne1 L(Y S) + (N4 − N3 Ne1 N2) rR or −1 −1 L(Y N ) − N3 Ne1 L(Y S) = (N4 − N3 Ne1 N2) rR . (7.55) Obviously, (7.55) shows that the source terms for the combined variables on the l.h.s of this equation indeed do not contain the rates of the IU undesired, independent reactions. At this stage recall that IU ⊂ II . Since each component of L(Y S) is of higher order compared to each component of L(Y N ), to leading order the equation

L(Y N ) = NG rR . (7.56) results for the KN non-steady-state species, where NG is the stoichiometric matrix of the reduced mechanism as deﬁned in (7.31). The vector ωN := NG rR is its molar rate-of-production vector, where it is now understood that ωN is given as

ωN = ωN (Y N , Y S(Y N )) . (7.57)

Here the vector Y S(Y N ) is obtained from the KS steady-state relationships. Note that one or more of these relationships may be replaced by other relationships such as partial-equilibria. The latter were discussed in Chap. 4.4.

At this stage it is worthwhile to come back to the procedure used to build the set IU that was discussed in Chap. 7.3. For that purpose, let us consider the rate of production of the steady-state species, ωS. We have – see (7.46) and (7.50) –

ωS = 0 = N1 rU + N2 rR , and from there the equation

−1 rU = − Ne1 N2 rR that we use to eliminate the rates of the undesired elementary reactions, rU , from the global reaction rates. Thus, assigning rate-determining reactions to IR (their rates are the components of rR) and fast reactions to IU (their rates are the components of rU ) indeed ensures that the fastest elementary reactions are eliminated from the global reaction rates whilst the slowest elementary reactions are kept in the global rates.

57 7.5.2 Sample Derivation

The starting mechanism derived for the oxyfuel non-premixed ﬂames investigated in the present thesis is summarized in App. A.1. It was derived from the complete GRI3.0 mechanism [96] using the reduction methods presented and discussed in Chap. 6. Speciﬁcally, the skeletal mechanism comprises 47 elementary reactions amongst 20 species.

Flame-structure results obtained with, (i), the GRI3.0 mechanism, (ii), the skeletal mechanism and, (iii), the reduced mechanism derived with the systematical method just presented in Chap. 7.5.1 will be presented, compared and discussed in Chaps. 9 and 10. Since the presentation of an alternative, empirical reduction method – see the following Sec. 7.6 – will greatly beneﬁt from the presentation of the systematically derived, reduced mechanism for oxyfuel non-premixed methane ﬂames on the basis of the theoretical formulation in the previous subsection 7.5.1, this reduced oxy-methane mechanism will be presented now.

Speciﬁcally, the systematically reduced oxy-methane mechanism comprises 11 global reaction steps amongst 14 chemical species. The mechanism is

H2 + O = H + OH (I)

H + O + M1 = OH + M1 (II)

O + CH3 = H + CH2O (III)

O + CH4 = OH + CH3 (IV)

OH + CO + M2 = H + CO2 + M2 (V)

2 O + C2H2 = 2 H + 2 CO (VI)

O + O2 + C2H2 = H + OH + 2 CO (VII)

2 O + C2H4 = OH + CH3 + CO (VIII)

H + OH + M3 = H2O + M3 (IX)

H + O + CH2O = H2 + OH + CO (X)

2 CH3 + M4 = C2H6 + M4 (XI)

58 Here the eﬀective third-body concentration are deﬁned as

[M1] = [O2]

[M2] = [O]

[M3] = 0.73[H2] + 3.65[H2O] + 2.00[CH4] + 3.00[C2H6] + 1.00[other] ,

[M4] = 2.00[H2] + 6.00[H2O] + 2.00[CH4] + 1.50[CO] + 2.00[CO2] + 3.00[C2H6] + 1.00[other] .

Here the formulation of third-body concentrations [M1], [M2], [M3] and [M4] closely follows Warnatz [110, 111]. Other representations of third-body concentrations are possible [27, 51, 84], in particular for computational purposes, and are used, e.g., in App. A.1.8

For the global reaction steps (I) to (XI), the global reaction rates in terms of the elementary rates are

ωI = ω1 − ω16 − ω19 + ω21 − ω24 + ω25 − ω31 − ω33 + ω34 − ω46

ωII = ω2 + ω16 + ω17 + ω18 − ω21 − ω22 + ω24 − 2ω25 − ω27 + ω29

+ ω31 + ω32 + ω33 − 2ω34 − ω38 − ω39 − ω40 − ω41 − 2ω42 − ω44

ωIII = ω3 + ω25 − ω33 + ω34 + ω41 + ω42 + ω44 + ω46

ωIV = ω4 − ω18 + ω19 + ω27 + ω33 − ω37

ωV = ω6 + ω28 + ω47

ωVI = ω7 − ω14 − ω17 + ω24 − ω25 + ω31 + ω32 + ω33 − ω34 − ω41

− ω42 − ω43 (7.58)

ωVII = ω8 + ω14 + ω17 − ω24 + ω25 − 2ω31 − ω32 − ω33 + ω34 + ω41 + ω43

ωVIII = ω9 − ω32 + ω42

ωIX = ω15 + ω21 + ω22 + ω23 + ω25 + ω27 + ω34 + ω45

ωX = ω20 − ω24 + ω25 − ω33 + ω34 + ω42 + ω44

ωXI = ω36

The well-known 5-step methane-air mechanism by Chelliah et al. [18] can easily be retrieved from the above deﬁned 11-step oxyfuel mechanism if additional species are

8 It is important to note that different authors of third bodies and their efficiencies may refer to a different reference species. Here, following [51, 84] we adopt the convention that the third- body efficiencies are expressed as multiple of the efficiency of methane. Warnatz [111], e.g., in his mechanisms quotes third body efficiencies relative to the efficiency of molecular hydrogen.

59 assumed to be in steady state. Specifically, starting with the 11-step oxyfuel mechanism, the derivation of the 5-step methane-air mechanism is summarized in table 7.1. Table 7.1 has four columns. The first column gives the global reaction number in the 5-step methane-air mechanism by Chelliah et al. [18]. The second column shows the equivalence symbol. The third column contains linear combinations of global reactions of the 11-step oxyfuel mechanism to be considered in order to retrieve the global reaction of the 5-step methane-air mechanism appearing in the same line. The last column gives the steady state species as well as the third body considered in each case. The efficiencies of third bodies M1,M2 and M3 appearing in table 7.1 have been given in the 11-step oxyfuel mechanism.

5 − step methane− 11 − step oxyfuel Assumed steady air mechanism mechanism species and third body Ia ≡ (IV) − (I) O, OH

Ib ≡ (III) + (X) − (IX) − 2(I) O, OH, CH2O 2H + M3 is a third body II ≡ (V) − (IX) − (I) + (II) O, OH 2H + M1 + M2 + M3 is a third body III ≡ −(I) + (II) O, OH M1 is a third body IV ≡ (VII) − (VI) + 2(IX) + (II) O, OH, C2H2 +(II) − 3 − (I) + (II) 5H + 2CO + 4M1 + 2M3 is a third body

Table 7.1: Retrieval of 5-step methane-air mechanism by Chelliah et al. [18] with the 11-step oxyfuel mechanism as the starting point.

7.6 Global Reaction Rates Determined Empirically

In Sec. 7.5.2 the rates of the global reaction steps are expressed as a linear combination of elementary rates – see (7.42) for the theoretical basis and (7.58) for the example of the 11-step oxy-methane mechanism derived in the present work. It will be shown in Chap. 10, that the results for the non-premixed oxy-methane ﬂames, in terms of ﬂame structure, obtained with the global 11-step mechanism and the systematically derived global rates (7.58) are excellent. However, the algebraic relationships from which the concentrations of the steady-state species that appear in some of the elementary reac-

60 tion rates on the right-hand-side of (7.58) are prone to numerical difficulties. Therefore, the systematically derived rates are more of an academic character, i.e., they can be used for academic problems in which spatially one-dimensional simulations are sufficient and hence computing costs remain limited, but they are not suitable for the simulation of complex combustion phenomena in three-dimensional, complex flow geometries or combustion chambers and under, perhaps, turbulent conditions. Under such truly complicated but yet realistic conditions – such conditions are generally termed complex – systematically derived global mechanisms with systematically derived global reaction rates – such as the ones in Eq. (7.58) – are not successful because they inherit the disadvantageous numerical properties of the elementary rates such as stiffness. 9

For the reasons just discussed, for simulations of complex combustion phenomena – such as turbulent, reactive ﬂows in complex geometries – e.g. in realistic combustion chambers – generally global rates are preferred that are not a linear combination of elementary rates but, rather, can be expressed similarly in a form as described in Chap. 4.2 for detailed mechanisms of elementary reactions. That is, the global reaction rates are, similarly as in the elementary case, written as

K X Ri = νkiMk = 0 , (7.59) k=1 or, alternatively, as K K X 0 X 00 νkiMk = νkiMk , (7.60) k=1 k=1 where, however, now K and I denote the numbers of global species and reactions, respectively, and where the stoichiometric coeﬃcients need not to be integer numbers. Similarly, the rate of global reaction i is written as

K K 0 00 Y nki Y nki ωi = ki,f Ck − ki,b Ck , (7.61) k=1 k=1

0 00 where in contrast to (4.3) now the reaction orders nki and nki are, generally, not identical to the respective stoichiometric coeﬃcients. Formally, the rate constants ki,f and ki,b are still given by (4.4) and (4.5), respectively, but their rate parameters as well as the reaction orders take rather empirical values that generally are not supported by physical-chemistry considerations. Speciﬁcally, the global rates expressed through Eqs.

9 Stiffness can but needs not to be a mathematical property of differential equations that makes the numerical solutions of such equations particularly difficult [6, 42, 55]. In particular, reactive- diffusive equations with Arrhenius rates often are stiff [84].

61 (7.61), (4.4) and (4.5) are fitted using by either pure trial and error or by employing more systematical mathematical methods – e.g., non-linear least-square fitting [81, 82] – such that the rates of production as functions of the independent variable – in this work the mixture fraction Z – for those species that are contained in both the skeletal and the global mechanism agree as well as possible. Alternatively, for each global reaction i the parameters can be fitted such that the global rate expressed as a linear combination of elementary reactions – with the profiles of temperatures and mass fraction from the skeletal mechanism – agrees as well as possible with the empirical global rate expressed in terms of the fitted parameters.

In the present work the latter ﬁtting strategy was adopted, i.e., for each global reaction 0 i the forward and the backward rate parameters, Ai,f , βi,f , Ei,f , nik,f and Ai,b, βi,b, Ei,b, 00 nik,b, respectively, were ﬁtted such that the forward and backward rates, respectively, as functions of the mixture fraction Z agree as well with those of the systematical approach.

We now describe the ﬁtting procedure developed in the framework of the present thesis using the global reaction (I) of the 11-step mechanism given above in Chap. 7.5.2. The global reaction is

H2 + O = H + OH ,

0 0 and the parameters to be fitted for this reaction are Af , βf , Ef , nH2 and nO for its 00 00 forward rate, Ab, βb, Eb, nH and nOH for its backward rate. The fitting strategy is employed separately for the forward and the backward step, i.e., for each step five parameters have to be fitted.

As function to be ﬁtted, for the forward rate of the global reaction (I)

ωI,f = ω1,f + ω16,b + ω19,b + ω21,f + ω25,f + ω33,b + ω34,f , (7.62)

– see. Eq. (7.58) – is adopted. The corresponding ﬁtting function is

0 n n0 ω = A T βf exp (−E /R /T ) C H2 C O . (7.63) ﬁt f f c H2 O

For the fitting the Fortran subroutine mrqmin [82] is employed. Fig. 7.1 shows a comparison between ωI,f and ωfit, both as function of the mixture fraction. Here ωfit is plotted using the fitted parameters listed in the first row of Table 11.1. The remaining forward and backward rates of the global mechanism are fitted analogously leading, ultimately, to Tables 11.1 and 11.2.

62 2.5

ω I, f 2 ω fit

1.5

1 Reaction rates

0.5

0 0 0.2 0.4 0.6 0.8 1 Mixture fraction Z

Figure 7.1: Comparison of the forward rates of global reaction I obtained respectively with the systematic reduction procedure (red dashed line) – see Eq. (7.62) – and a corresponding ﬁtting function (green line) – see Eq. (7.63)–.

63 Chapter 8

Results for Detailed Mechanisms

Detailed chemical mechanisms are generally valid over a wide range of conditions such as pressure, equivalence ratio, and wide temperature ranges. Consequently, for hydrocarbon combustion, they contain from dozens to several hundred or even thousands chemical species and from hundred to several thousand elementary reactions. For example, the GRI3.0 chemical mechanism [96] contains 53 species and 325 reactions, while other common detailed mechanisms such as the mechanism by Kee [54] and the mechanism by Glarborg [40] comprise 58 elementary reactions amongst 17 species and 779 elementary reactions amongst 97 species, respectively.

The incorporation of such detailed mechanisms in simulations of complex reacting flows leads to significant computational costs. Empirically and drastically reduced kinetic schemes, however, such as the ones by Westbrook and Dryer [112] or Jones and Lindstedt [47] with 2 reactions amongst 6 species and 4 reactions amongst 7 species, respectively, were devised earlier. These mechanisms were originally derived for combustion in air and they are commonly used in CFD software such as Fluent [2], CFX [3] or Star-CD [4] and they inaccurately predict temperature the peak and CO species mole fraction profiles when used under oxy-fuel conditions [37].

To find a suitable detailed reaction mechanism from which then to start the overall reduction procedure described in Chaps. 6 and 7, oxy-methane diffusion flames have been simulated with different detailed mechanisms, namely, the GRI3.0 mechanism [96], the mechanism by Kee [54] and the mechanism by Glarborg [40]. Shown in Fig. 8.1 is a systematic comparison of numerical results obtained from computations with these three mechanisms and with existing experimental data by Bennett et al. [11]. That comparison was done with the aim of identifying the best possible mechanism from which the reduction procedure can be started. Since the experimental data were represented in physical space rather than mixture-fraction space, for the validations

64 T(K) GRI3.0

O2 GRI3.0

CH4 GRI3.0 1 T(K) Glarborg 3000

O2 Glarborg CH4 Glarborg T(K) Kee 2700

O2 Kee CH Kee 0.8 4 T(K) Experiment 2400

O2 Experiment CH Experiment 4 2100 0.6 1800

1500

Mole fraction 0.4 1200 Temperature (K)

900 0.2

600

0 300 0 0.2 0.4 0.6 0.8 1 Mixture fraction

Figure 8.1: Profiles of temperature and reactants for an oxy-methane diffusion flame: −1 100%O2 vs. 65%CH4 - 35%N2, constant scalar dissipation rate χ = 1.71 s ; lines show numerical predictions using the GRI3.0 mechanism and the mechanisms by Glarborg and Kee and symbols show experimental results of Bennett et al. [11] presented in this chapter, the Cosilab code [27] was employed.

It is seen that computational results obtained with the GRI3.0 mechanism show better agreement with this specific experimental data than the computational results obtained with the other two mechanisms. It is worthwhile to mention that flame simulations with pure methane as fuel and pure oxygen as oxidizer were not validated due to the lack of experimental data for this case. The oxyfuel diffusion flame case 100%O2 vs. 20%CH4

- 80%N2 is the only one dimensional counterflow oxyfuel flame case with experimental data validated against the GRI3.0 mechanism. One dimensional counterflow flames are needed because in the so-called flamelet approach [73, 75], a turbulent diffusion flame is described as an ensemble of locally thin one-dimensional flames embedded in the turbulent flow. Hence, it can be expected that the reduced mechanisms developed herein may later also be applied in the simulation of turbulent flames with commercial CFD codes, say.

Shown in Figs. 8.2 and 8.3 is a comparison of proﬁles as obtained computationally with the GRI3.0 mechanism and obtained experimentally by Cheng et al. [23]. It is

65 T(K) GRI 3.0

CH4 GRI 3.0

O2 GRI 3.0 1 T(K) Experiment 2700 CH4 Experiment O2 Experiment 2400

0.8 2100

1800 0.6 1500

1200

Mole fraction 0.4 900 Temperature (K)

600 0.2

300

0 0 0 0.002 0.004 0.006 0.008 0.01 0.012 Distance from top jet (m)

Figure 8.2: Profiles of temperature and reactants for an oxy-methane diffusion flame: −1 100%O2 vs. 20%CH4 - 80%N2, strain rate a = 60 s ; lines show numerical predictions using the GRI3.0 mechanism and symbols show experimental results of Cheng et al. [23]

0.25

CO2 GRI 3.0

H2O GRI 3.0 CO2 Experiment H O Experiment 0.2 2

0.15

Mole fraction 0.1

0.05

0 0 0.002 0.004 0.006 0.008 0.01 0.012 Distance from top jet (m)

Figure 8.3: As Fig. 8.2, but only for products.

66 seen from these ﬁgures that predicted major species mole fraction and temperature proﬁles match the experimental data with a good accuracy.

Based on the preliminary investigations, it was decided that the GRI3.0 reaction mechanism with 53 species and 325 elementary reactions is a suitable mechanism from which the overall reduction procedure presented and discussed in Chap. 6 can be started.

67 Chapter 9

Results for Skeletal Mechanisms

In Chap. 8 we have shown that the GRI3.0 mechanism appears to be the most suitable detailed mechanism of elementary reactions to serve as a starting point to derive a so-called skeletal mechanism. In the framework of the present thesis, two alternative skeletal mechanisms have been derived with the methods of Chap. 6. Flame-structure results computed with these skeletal mechanisms near extinction limit and at a particular low strain rate are presented and discussed in Chaps. 9.2 and 9.3. The mechanisms themselves are presented in Appendices A.1 and A.2 respectively.

9.1 Exploratory Reaction-Pathway Analysis

The choice of reactions which will constitute the skeletal mechanism can be guided by the so-called reaction-pathway analysis. Generally, in a network of the chemical species present in a given system, a reaction-pathway analysis aims at the identification of important reactions in which – on one hand – one species is produced from one or several other species and – on the other hand – in which one species is converted in one or several other chemical species. As an example, such a network of species is shown in Fig. 9.1 which represents reaction pathways in an oxy-methane counterflow diffusion flame. In the figure, the blue connecting lines of varying thickness represent the reaction pathes. The thicker such a line or reaction path, the more important the path is for the system under consideration. Reaction pathes can be defined in various ways. In the present thesis the definition by Zsely´ et al. [118] is adopted who defined a reaction path between any two species A and B as the flux of atoms of a certain kind – e.g., C-atoms – between these species, the flux itself occurring due to certain chemical reactions.1 Specifically, Fig. 9.1 is based on the flux of C-atoms, Fig. 9.2

1 The ﬂux-based formulation of reaction pathways makes sense because, physically, in each chemical reaction element conservation prevails. That is, on the reactants and the products side of each reaction the same number of atoms of a certain kind are present but they are associated with diﬀerent

68 Figure 9.1: Reaction pathway analysis of C-atoms for an oxy-methane diffusion flame with conditions p = 1 bar, YO2 = 1.0, YCH4 = 1.0, TF uel = TOxidizer = 300 K and a strain rate a = 0.2 s−1 in an opposing flows test bed configuration. on the flux of H-atoms, and Fig. 9.3 on the flux of O-atoms. The thicker the blue line between two molecules, the bigger the atom flux between them and hence the more important the reaction path. Molecules that in the figures are not connected by a blue line either have no reaction path between them or, alternatively, only a weak reaction path whose flux, or strength, fall below a certain, small threshold and therefore is not shown. To analyse the reaction pathes shown in Figs. 9.1 to 9.3 the software package Cosilab [27] has been used.

It is important to realize that species that, for all atoms in the system – here C, H, and O atoms – are not connected to other species by a reaction path do not play a role – or at least not an important role – in the underlying reaction mechanism and, therefore, can be deleted from the mechanism. Similarly, reactions can safely be deleted from the reaction mechanism that make no or only small contributions to the importance – i.e., the thickness – of the “blue” reaction pathways.

Naturally, reaction pathways based on C-atoms lend themselves to investigate the importance species and reactions involved in the chemical conversion of a hydrocarbon fuel, here CH4 to the major products CO2 and CO. Similarly, an analysis based on O-atoms is ideal for the investigation of reaction pathways leading from the oxidizer, species, namely reactants on the reactants side and products on the products side. It is in this sense that atoms ”ﬂow” from a reactant species to a product species of a chemical reaction.

69 Figure 9.2: As in Fig. 9.1, but for H-atoms.

Figure 9.3: As in Fig. 9.1, but for O-atoms.

70 here O4 to H2O, CO2 and CO. Finally, an analysis based on H-atoms is ideal for the investigation of reaction pathways leading a hydrocarbon fuel to H2O. Specifically, for the oxy-methane diffusion flames investigated in the present thesis it has been found by the reaction-pathway analysis performed, that the species C3H7,C3H8, HCCOH, C2H,

CH3OH, CH3O, CH, C and H2O2 can be eliminated from the detailed mechanism, thereby taking a step towards ﬁnding, ultimately, a skeletal mechanism.

9.2 Oxy-Methane Skeletal Mechanism Valid Over the Entire Strain Rate Range

Shown in Fig. 9.4 is the maximum temperature, Tmax, in an oxy-methane nonpremixed ﬂame as a function of the strain rate, a. In all computations presented in the chapter,

3300

3000

2700 (K)

max 2400

2100

1800

1500

1200 GRI 3.0 Peak flame temperature T 900 skeletal mechanism

600

300 0 20000 40000 60000 80000 Strain rate (1/s)

Figure 9.4: Comparison of predicted peak ﬂame temperature Tmax as a function of strain rate a as obtained with the GRI3.0 mechanism and the skeletal mechanism of App. A.1. the boundary mass fractions for methane and oxygen were unity at the fuel side and oxidizer side, respectively. The temperature at either boundary was 300 K; the pressure was 1 bar.

To generate the graph Tmax(a) approximately 30 ﬂame structure computations based

71 on the governing equations (2.1) and (2.2) were performed with the computer code developed for the present research. Specifically, steady solution to the governing equations were sought. To this end, the so-called method of continuation was used, i.e., the steady results obtained for a strain rate a were taken as the initial profiles for the subsequent computation with strain rate a + ∆a. Obviously, the smaller ∆a, the more computations are required to cover a certain range of strain rate. Note that ∆a needs to be particularly small when flame extinction is approached.

In the present research, the range of strain rate from practically zero – the corresponding flames are usually referred to a unstrained flames2 – to extinction was of interest because this range is of particular interest for the modelling of turbulent combustion in the so-called flamelet regime [73, 75].

It is seen from Fig. 9.4 that the dashed curve obtained with the GRI3.0 mechanism and the solid curve obtained with the skeletal mechanism agree very well for the en- −1 tire range of strain rates. Particularly the extinction strain rate, aq ≈ 69 646 s , is well represented by the skeletal mechanism. The latter observation suggests that this skeletal mechanism may also be useful for turbulent non-premixed flames that are simulated either with Reynolds Averaged Navier-Stokes (RANS) equations or by Large-Eddy Simulation (LES) employing a suitable flamelet model or, alternatively, by Direct Numerical Simulation (DNS). Also probability-density function (PDF) methods are candidates for employing this particular skeletal mechanism, either in a flamelet or a non-flamelet formulation.

Shown in Figs. 9.4 to 9.9 are flame-structure results in terms of temperature and selected species mass fractions for a flame strained moderately at a strain rate of 35 000 s−1. In each of these figures, profiles are shown as obtained with the GRI3.0 and the skeletal mechanism. From Fig. 9.5 it is seen that the temperature profiles agree very well.

2 So-called unstrained flames can be counterflow flames at particularly low strain rates. Also, jet diffusion flames often experience little strain and, therefore, often are in a good approximation unstrained.

72 3000 GRI 3.0 skeletal mechanism 2700

2400

2100

1800

1500

Temperature (K) 1200

900

600

300 0 0.2 0.4 0.6 0.8 1 Mixture fraction Z

Figure 9.5: Predicted temperature profiles computed using the GRI3.0 mechanism and the skeletal mechanism of App. A.1 for an oxy-methane diffusion flame: 100%O2 vs. −1 100%CH4 and a strain rate a = 35 000 s .

Shown in Fig. 9.6 is a comparison of the mass-fraction proﬁles of fuel and oxidizer. It can be seen from the graph that the agreement is excellent. Shown in Fig. 9.7 are the mass-fraction proﬁles of the major combustion products H2O, CO2 and CO. It is seen that for all three species the agreement is very good.

Shown in Figs. 9.8 and 9.9 are the mass-fraction proﬁles of the most important species of the hydrogen-oxygen radical pool, i.e., of the O and the H atoms, and of the OH radical. It is seen from these graphs that the skeletal mechanism reproduces the mass fraction proﬁles obtained with the GRI3.0 mechanism with a good accuracy.

73 1

0.8

0.6

0.4 Mass fraction

O GRI 3.0 0.2 2 CH4 GRI 3.0

O2 skeletal mechanism

CH4 skeletal mechanism

0 0 0.2 0.4 0.6 0.8 1 Mixture fraction Z

Figure 9.6: As Fig. 9.5, but only for reactants

0.35 H2O GRI 3.0 CO GRI 3.0

CO2 GRI 3.0

H2O skeletal mechanism 0.3 CO skeletal mechanism

CO2 skeletal mechanism

0.25

0.2

0.15 Mass fraction

0.1

0.05

0 0 0.2 0.4 0.6 0.8 1 Mixture fraction Z

Figure 9.7: As Fig. 9.5, but only for products.

74 0.04

0.035 O GRI 3.0 OH GRI 3.0 O skeletal mechanism 0.03 OH skeletal mechanism

0.025

0.02

Mass fraction 0.015

0.01

0.005

0 0 0.2 0.4 0.6 0.8 1 Mixture fraction Z

Figure 9.8: As Fig. 9.5, but only for intermediate species O and OH.

0.0012

H GRI 3.0 0.001 H skeletal mechanism

0.0008

0.0006 Mass fraction 0.0004

0.0002

0 0 0.2 0.4 0.6 0.8 1 Mixture fraction Z

Figure 9.9: As Fig. 9.5, but only for intermediate species H.

75 Shown in Figs. 9.10 to 9.14 are flame-structure results obtained with the GRI3.0 mechanism and the skeletal mechanism of App. A.1 for an oxy-methane nonpremixed flame at a close-to-extinction strain rate value of 62 000 s−1. Similarly to the moderately strained flame case mentioned above, it is seen from these graphs that the skeletal mechanism predictions are, overall, in good agreement with those obtained with the starting detailed mechanism, viz., the GRI3.0 mechanism. Specifically, from Fig. 9.10,

2700 GRI 3.0 Skeletal mechanism 2400

2100

1800

1500

1200 Temperature (K)

900

600

300 0 0.2 0.4 0.6 0.8 1 Mixture fraction Z

Figure 9.10: Predicted temperature profiles computed using the GRI3.0 mechanism and the skeletal mechanism of App. A.1 for an oxy-methane diffusion flame: 100%O2 −1 vs. 100%CH4 and a strain rate of 62 000 s . it is seen that the numerically predicted temperature profiles are in good a agreement except in the narrow region where the temperatures peak. The difference in peak temperature is 52.33 K, which is reasonably small provided the NOx formation is not of interest as it is the case, e.g., for the reduced mechanisms to be derived in the present thesis. Comparing Figs. 9.5 and 9.10 shows that the maximum flame temperature drops from 2857.507 K in the moderately strained diffusion flame case mentioned above – strain rate a = 35 000 s−1 – to 2633.487 K in a close-to-extinction nonpremixed flame at a strain rate of 62 000 s−1.

Shown in Fig. 9.11 are the mass fraction proﬁles reactants CH4 and O2 . It is seen from the graph that the skeletal mechanism and the GRI3.0 mechanism predictions

76 are in good agreement. Shown in Fig. 9.12 are the mass-fraction proﬁles of combustion products H2O, CO2 and CO. It is seen that for all three species, the agreement is very good. In addition, comparing Figs. 9.7 and 9.12 indicates that mass fraction of combustion products decrease with increasing strain rate values.

Shown in Figs. 9.13 and 9.14 are the mass fraction profiles of important intermediates species, namely O, OH and H species. Comparing Figs. 9.13 and 9.8 and Figs. 9.14 and 9.9 shows that differences between the skeletal mechanism predictions and the predictions obtained with the GRI3.0 mechanism are more pronounced at high strain rates, e.g. a close-to-extinction strain rate of 62 000 s−1, than they are at moderate strain rates, e.g. a moderate strain rate of 35 000 s−1. Specifically, at high strain rates peak mass fraction of intermediate species O, OH and H predicted using the skeletal mechanism are lower than the ones predicted using the starting detailed mechanism. This indicates that the oxy-methane diffusion flame computed with the skeletal mechanism quenches before the one computed with the GRI3.0 mechanism, which is consistent with data displayed in Fig. 9.4

0.8

0.6

0.4 Mass fraction

O2 GRI 3.0

CH4 GRI 3.0 0.2 O2 skeletal mechanism

CH4 skeletal mechanism

0 0 0.2 0.4 0.6 0.8 1 Mixture fraction Z

Figure 9.11: Predicted mass fraction profiles of reactants computed using the GRI3.0 mechanism and the skeletal mechanism of App. A.1 for an oxy-methane diffusion flame: −1 100%O2 vs. 100%CH4 and a strain rate a = 62 000 s .

77 0.3 H2O GRI 3.0 CO GRI 3.0

CO2 GRI 3.0

H2O skeletal mechanism 0.25 CO skeletal mechanism CO2 skeletal mechanism

0.2

0.15 Mass fraction 0.1

0.05

0 0 0.2 0.4 0.6 0.8 1 Mixture fraction Z

Figure 9.12: As Fig. 9.11, but only for products.

0.03

O GRI 3.0 0.025 OH GRI 3.0 O skeletal mechanism OH skeletal mechanism

0.02

0.015 Mass fraction 0.01

0.005

0 0 0.2 0.4 0.6 0.8 1 Mixture fraction Z

Figure 9.13: As Fig. 9.11, but only for intermediate species O and OH.

78 0.0008

H GRI 3.0 H skeletal mechanism 0.0006

0.0004 Mass fraction

0.0002

0 0 0.2 0.4 0.6 0.8 1 Mixture fraction Z

Figure 9.14: As Fig. 9.11, but only for intermediate species H.

9.3 Oxy-Methane Skeletal Mechanism for Diﬀusion Flames at Low Strain Rates

Performances of the skeletal mechanism comprising 24 reactions amongst 19 species and available in the App. A.2 are given in this section. This skeletal mechanism is derived for an oxy-methane nonpremixed flame with the conditions: p = 1 bar, YO2 = 1.0, −1 YCH4 = 1.0, TFuel = TOxidizer = 300 K and a low strain rate a = 0.2 s . Shown in Figs. 9.15 to 9.18 are flame-structure results in terms of temperature and selected species mass fraction profiles obtained with this skeletal mechanism. The skeletal mechanism predictions are validated against results obtained with the GRI3.0 mechanism.

Presented in Fig. 9.15 is a comparison of predicted temperature proﬁles obtained with the skeletal mechanism and the GRI3.0 mechanism. It is seen from this graph that the temperature proﬁles agree very well.

Shown in Fig. 9.16 is a comparison of the mass-fraction proﬁles of fuel and oxidizer. It can be seen from the graph that the agreement is perfect. Comparison of the mass- fraction proﬁles of combustion products, viz., species H2O, CO2 and CO, yields a similar perfect agreement(data not shown).

79 3000 GRI 3.0 skeletal mechanism 2700

2400

2100

1800

1500

Temperature (K) 1200

900

600

300 0 0.2 0.4 0.6 0.8 1 Mixture fraction Z

Figure 9.15: Predicted profiles of temperature computed using the GRI3.0 mechanism and the skeletal mechanism of App. A.2 for a diffusion flame with the conditions p = −1 1 bar, YO2 = 1.0, YCH4 = 1.0, TFuel = TOxidizer = 300 K and a strain rate a = 0.2 s .

Shown in Figs. 9.17 and 9.18 are the mass-fraction proﬁles of selected intermediate species, namely, O, OH and H. From these graphs, it can be seen that the mass- fraction proﬁles computed with the skeletal mechanism and the GRI3.0 mechanism perfectly agree.

80 1

0.8

0.6

0.4 Mass fraction

O GRI 3.0 0.2 2 CH4 GRI 3.0

O2 skeletal mechanism

CH4 skeletal mechanism

0 0 0.2 0.4 0.6 0.8 1 Mixture fraction Z

Figure 9.16: Predicted mass fraction profiles of reactants computed using the GRI3.0 mechanism and the skeletal mechanism of App. A.2 for a diffusion flame with the conditions p = 1 bar, YO2 = 1.0, YCH4 = 1.0, TFuel = TOxidizer = 300 K and a strain rate a = 0.2 s−1.

0.06

0.055 O GRI 3.0 OH GRI 3.0 0.05 O skeletal mechanism OH skeletal mechanism 0.045 0.04 0.035 0.03 0.025 Mass fraction 0.02 0.015 0.01 0.005 0 0 0.2 0.4 0.6 0.8 1 Mixture fraction Z

Figure 9.17: As Fig. 9.15, but only for intermediate species O and OH. 81 0.0016

H GRI 3.0 0.0014 H skeletal mechanism

0.0012

0.001

0.0008

Mass fraction 0.0006

0.0004

0.0002

0 0 0.2 0.4 0.6 0.8 1 Mixture fraction Z

Figure 9.18: As Fig. 9.15, but only for intermediate species H.

82 Chapter 10

Results for Global, Systematically Reduced Mechanisms

10.1 Low Strain Rates

The transition from a skeletal mechanism to a global reduced one is achieved by applying the steady-state approximation to a selected group of intermediates species. For an oxy-methane diﬀusion ﬂame at a low strain rate of 0.2 s−1, a 9-step mechanism can be derived from the skeletal mechanism of App. A.2 by introducing steady-state approximations for species HO2, CH2(S), CH2O, CH2, CH2CO, C2H5 and C2H6. This reduced mechanism comprises 9 global reactions amongst 12 species, namely, H2, H,

O, O2, OH, H2O, CH4, CO, CO2, CH3,C2H4 and C2H2. The reduced mechanism is

H + CH4 = H2 + CH3 (I”)

H2 + 2 H + CO = OH + +CH3 (II”)

H + O2 = O + OH (III”)

2 CH3 = H2 + C2H4 (IV”)

C2H4 = H2 + C2H2 (V”)

O + OH + C2H2 = H2 + H + 2 CO (VI”)

2 H = H2 (VII”)

H + OH = H2O (VIII”)

OH + CO = H + CO2 (IX”)

The reaction rates for the global steps (I”) to (IX”) expressed in terms of elementary

83 reaction rates are

ωI” = − ω7 + ω8

ωII” = 2 ω1 + ω11 − 2 ω14 − 2 ω19 + ω20 − ω22

ωIII” = ω4

ωIV” = ω1 − ω14 − ω19 + ω20 + ω23

ωV” = ω1 − ω14 − ω19 + ω24 (10.1)

ωVI” = ω1 + ω2 − ω14

ωVII” = − ω1 + ω5 + ω7 − ω13 + ω14 + ω19 − ω20

ωVIII” = ω6 + ω13 + ω14 + ω15 + ω22

ωIX” = ω18 − ω22

Flame-structure results obtained from numerical simulations performed using the 9- step reduced mechanism and the GRI3.0 mechanism under the conditions p = 1 bar, −1 YO2 = 1.0, YCH4 = 1.0, TFuel = TOxidizer = 300 K and a strain rate of 0.2 s are plotted and compared in Figs. 10.1 to 10.3. It can be seen from Fig. 10.1 that temperature proﬁles predictions obtained with both mechanisms perfectly agree.

Shown in Fig. 10.2 is a comparison of the mass-fraction proﬁles of reactants, i.e., of species CH4 and O2. It can be seen from the graph that the agreement is perfect. A similar agreement is observed with mass-fraction proﬁles of combustion products, viz., species H2O, CO2 and CO (data not shown).

84 3000 GRI 3.0 9-step mechanism 2700

2400

2100

1800

1500

Temperature (K) 1200

900

600

300 0 0.2 0.4 0.6 0.8 1 Mixture fraction Z

Figure 10.1: Predicted proﬁles of temperature computed using the GRI3.0 mechanism and the 9-step mechanism with the conditions p = 1 bar, YO2 = 1.0, YCH4 = 1.0, −1 TFuel = TOxidizer = 300 K and a strain rate a = 0.2 s .

0.8

0.6

0.4 Mass fraction

CH4 GRI 3.0 O GRI 3.0 0.2 2 O2 9-step mechanism CH4 9-step mechanism

0 0 0.2 0.4 0.6 0.8 1 Mixture fraction Z

Figure 10.2: As Fig. 10.1, but only for reactants.

85 Comparisons of mass-fraction proﬁles of selected intermediate species, namely, H, O,

OH, H2,C2H2 and C2H4 are given in Figs. 10.3 to 10.5. It is seen from these graphs that the proﬁles of species H, O, OH and H2 predicted with the 9-step reduced mechanism are in perfect agreement with the results obtained using the GRI3.0 mechanism. In addition, the mass-fraction proﬁles predicted with both mechanisms for species C2H2 and

C2H4 are also in perfect agreement except on the fuel side where a slight discrepancy is observed.

0.06

H2 GRI 3.0 O GRI 3.0 OH GRI 3.0

0.05 H2 9-step mechanism O 9-step mechanism OH 9-step mechanism

0.04

0.03 Mass fraction 0.02

0.01

0 0 0.2 0.4 0.6 0.8 1 Mixture fraction Z

Figure 10.3: As Fig. 10.1, but only for intermediate species H2, O and OH.

86 0.002

H GRI 3.0 H 9-step mechanism

0.0015

0.001 Mass fraction

0.0005

0 0 0.2 0.4 0.6 0.8 1 Mixture fraction Z

Figure 10.4: As Fig. 10.1, but only for intermediate species H.

C2H2 GRI 3.0

C2H4 GRI 3.0

0.04 C2H4 9-step mechanism C2H2 9-step mechanism

0.03

0.02 Mass fraction

0.01

0 0 0.2 0.4 0.6 0.8 1 Mixture fraction Z

Figure 10.5: As Fig. 10.1, but only for intermediate species C2H2 and C2H4.

87 10.2 Entire Strain Rate Range

Shown in Fig. 10.6 is the maximum flame temperature computed respectively with the GRI3.0 mechanism (dashed line) and the systematically 11-step reduced mechanism (solid line) given in Chap. 7.5.2. Inspection of the figure shows that the global reduced mechanism accurately predicts peak flame temperature over the entire range of strain rate, and that the strain-rate value at extinction is very well predicted. Comparing Fig. 10.6 to Fig. 9.4 shows that the 11-step reduced mechanism has even better prediction capabilities than the corresponding skeletal mechanism.

3300

3000

2700 (K)

max 2400

2100

1800

1500

1200 GRI 3.0 11-step mechanim Peak flame temperature T 900

600

300 0 20000 40000 60000 80000 Strain rate (1/s)

Figure 10.6: Comparison of predicted peak ﬂame temperature Tmax as a function of strain rate a as obtained with the GRI3.0 mechanism (dashed line) and the 11-step mechanism (solid line).

Figures 10.7 to 10.11 show comparisons of flame-structure results obtained with the 11-step mechanism and the GRI3.0 mechanism, respectively. Specifically the results shown in these figures are for a close-to-extinction strain rate value of 62 000 s−1. It is seen that the overall agreement between the reduced mechanism predictions and those obtained with the GRI3.0 mechanism is very good.

88 2700 GRI 3.0 11-step mechanism 2400

2100

1800

1500

1200 Temperature (K)

900

600

300 0 0.2 0.4 0.6 0.8 1 Mixture fraction Z

Figure 10.7: Predicted temperature profiles computed using the GRI3.0 mechanism and the 11-step mechanism for an oxy-methane diffusion flame: 100%O2 vs. 100%CH4 and a strain rate a = 62 000 s−1.

0.8

0.6

0.4 Mass fraction

CH4 GRI 3.0 O2 GRI 3.0 O 11-step mechanism 0.2 2 CH4 11-step mechanism

0 0 0.2 0.4 0.6 0.8 1 Mixture fraction Z

Figure 10.8: As Fig. 10.7, but for reactants.

89 0.3 CO GRI 3.0

CO2 GRI 3.0 H2O GRI 3.0

H2O 11-step mechanism 0.25 CO 11-step mechanism CO2 11-step mechanism

0.2

0.15 Mass fraction 0.1

0.05

0 0 0.2 0.4 0.6 0.8 1 Mixture fraction Z

Figure 10.9: As Fig. 10.7, but for stable products.

0.03 O GRI 3.0 OH GRI 3.0 O 11-step mechanism 0.025 OH 11-step mechanism

0.02

0.015 Mass fraction 0.01

0.005

0 0 0.2 0.4 0.6 0.8 1 Mixture fraction Z

Figure 10.10: As Fig. 10.7, but for intermediate species O and OH.

90 0.0008

0.0006 H GRI 3.0 H 11-step mechanism

0.0004 Mass fraction

0.0002

0 0 0.2 0.4 0.6 0.8 1 Mixture fraction Z

Figure 10.11: As Fig. 10.7, but for intermediate species H.

91 Chapter 11

Results for Global, Empirically Reduced Mechanisms

It has been shown in Chap. 7.5.2, Eq. (7.58), that the global rate expressions of multistep reaction mechanisms can be written as linear combinations of elementary reaction rates provided the global rates have been derived with the systematic method described in Chap. 7. However, as already discussed at length in Chap. 7.6, such systematically derived global rates are prone to numerical difficulties when used in simulations of complex reactive flows – e.g. three-dimensional or turbulent reactive flows – because the stiffness due to the elementary rates occurring in the global rates is vastly retained. Therefore, in Chap. 7.6 an empirical approach to rate expressions for the global rates was devised, in which empirical values of activation energy, pre-exponential rate factor, pre-exponential temperature dependence, global reaction orders and global third-body efficiencies are to be determined. In the present chapter, we present the results obtained for these empirical reaction parameters. Also a few typical flame-structure results obtained on the basis of the empirical rate expressions are presented and compared to corresponding results obtained with the GRI3.0 mechanism.

Summarized in Tables 11.1 and 11.2 are the rate parameters obtained with the ﬁtting procedure described in Chap. 7.6 for the forward and backward global rates, respectively, of the 11-step mechanism summarized in Eq. (7.58). The ﬁtting procedure is implemented in this chapter at a strain rate value of a = 5 s−1. Flame-structure results later shown in Figs. 11.3 to 11.6 were computed using the Cosilab code [27].

92 Global reaction A β E n0 n0 n0 f f f k1 k2 k3 I 3.375 1010 0.000 9.185 104 4.614 10−1 5.297 10−2 0.000 II 1.244 1018 0.000 −3.466 104 2.000 0.000 0.000 III 8.470 1024 −5.000 6.302 104 0.000 7.261 10−1 0.000 IV 2.541 1010 0.736 6.978 104 0.000 1.079 0.000 V 4.118 1032 −4.600 70.000 0.005 1.683 0.000 VI 2.007 1014 −1.000 1.704 104 1.456 0.000 0.000 VII 5.580 1006 1.000 2.490 104 1.138 1.575 10−1 0.000 VIII 3.092 1026 −4.006 8.677 104 0.000 9.905 10−1 0.000 IX 4.062 1013 0.000 0.000 5.653 10−1 6.503 10−1 0.000 X 1.293 1035 −6.000 5.420 103 8.253 10−1 0.000 1.099 XI 4.953 1033 −5.000 654.000 2.056 0.000 0.000

Table 11.1: Summary of the empirical ﬁtting of the forward rates of the 11-step mechanism given in Chap. 7.5.2.

Global reaction Ab βb Eb n”k1 n”k2 n”k3 I 2.347 1012 0.000 −2.762 104 1.380 0.700 0.000 II 1.731 1030 0.000 2.488 105 0.800 0.000 0.000 III 1.042 106 1.000 2.946 104 0.000 9.969 10−1 0.000 IV 1.822 108 0.000 1.776 104 0.000 1.052 0.000 V 7.937 1027 0.000 70.000 2.509 0.700 0.000 VI 1.274 109 −1.000 7.437 104 0.100 0.000 0.000 VII 3.815 1035 −5.000 1.704 104 0.005 2.674 0.000 VIII 6.166 1012 0.000 5.346 104 0.000 1.494 0.000 IX 5.433 1011 0.000 7.724 104 0.100 0.000 0.000 X 3.875 1048 1.000 5.862 104 5.598 0.000 3.153 XI 5.625 1017 0.000 6.803 104 1.000 0.000 0.000

Table 11.2: Summary of the empirical ﬁtting of the backward rates of the 11-step mechanism given in Chap. 7.5.2.

The tables have seven columns. The first column gives the global reaction number. From column two to column four, the fitted Arrhenius parameters are given, i.e., the pre-exponential factor, the temperature exponent and the activation energy. In column five to seven, the reaction orders of species participating in the respective global reaction are given.

Shown in Figs. 11.1 and 11.2 are exemplary proﬁles of rates as obtained with the skeletal and from the empirically derived rates. This two examples have been selected

93 such that Fig. 11.1, which shows perfect agreement between the two rates for global reaction VIII, is representative for most rate fittings. Figure 11.2, is representative for a fitting of modest quality. Specifically, a relative poor agreement between the rate of the backward global reaction V and the corresponding fitted rate parameters is displayed in Fig. 11.2. However, the quality of the fittings for the global rates is not crucial. Crucial is the quality of the flame-structure predictions with the empirically fitted rate parameters. Exemplarily it is shown in Figs. 11.3 to 11.6 that the temperature profiles, the mass-fraction profiles of the major species, as well as the mass-fraction profiles of H and OH predicted with the empirical rate expressions indeed agree very well with the corresponding flame-structures obtained with the GRI3.0 mechanism.

8E-05

ω VIII, f ω fit 6E-05

4E-05 Reaction rates

2E-05

0 0 0.2 0.4 0.6 0.8 1 Mixture fraction Z

Figure 11.1: Comparison between the empirical ﬁtted and the original forward rate for global reaction VIII

94 0.25

ω V, f ω 0.2 fit

0.15

0.1 Reaction rates

0.05

0 0 0.2 0.4 0.6 0.8 1 Mixture fraction Z

Figure 11.2: Comparison between the empirical ﬁtted and the original forward rate for global reaction V

3000

GRI 3.0 2700 Fitted 11-step mechanism

2400

2100

1800

1500

Temperature (K) 1200

900

600

300 0 0.2 0.4 0.6 0.8 1 Mixture fraction Z

Figure 11.3: Predicted temperature profiles computed using the GRI3.0 mechanism and the empirically fitted 11-step mechanism for an oxy-methane diffusion flame: 100%O2 −1 vs. 100%CH4 and a strain rate a = 5 s .

95 1

0.8

0.6

0.4 Mass fraction

CH4 GRI 3.0 O GRI 3.0 0.2 2 O2 Fitted 11-step mechanism CH4 Fitted 11-step mechanism

0 0 0.2 0.4 0.6 0.8 1 Mixture fraction Z

Figure 11.4: As Fig. 11.3, but for reactants.

0.7 CO GRI 3.0

CO2 GRI 3.0

H2O GRI 3.0 H O Fitted 11-step mechanism 0.6 2 CO Fitted 11-step mechanism

CO2 Fitted 11-step mechanism 0.5

0.4

0.3 Mass fraction

0.2

0.1

0 0 0.2 0.4 0.6 0.8 1 Mixture fraction Z

Figure 11.5: As Fig. 11.3, but for stable products.

96 0.06 O GRI 3.0 OH GRI 3.0 O Fitted 11-step mechanism 0.05 OH Fitted 11-step mechanism

0.04

0.03 Mass fraction 0.02

0.01

0 0 0.2 0.4 0.6 0.8 1 Mixture fraction Z

Figure 11.6: As Fig. 11.3, but for intermediate species O and OH.

97 Chapter 12

Summary and Outlook

In the present thesis, new simplified mechanisms have been developed for laminar diffusion flames under oxyfuel conditions. The reduction procedure applied in the current work uses the steady solution of laminar flamelet equations. The overall chemical reduction procedure used to develop the new simplified mechanisms, i.e. skeletal and global reduced mechanisms, is implemented in two steps.

In the first step, the GRI3.0 mechanism has been reduced to a so-called skeletal mechanism. Specifically, four skeletal reduction methods presented and discussed in Chap. 6 are applied for this purpose. It was found that the detailed mechanism of elementary reactions obtained with the first three skeletal reduction methods described in Chaps. 6.2, 6.3 and 6.4, respectively was not the smallest possible one. Hence, this chemical mechanism has been further reduced with a new, in this thesis developed, skeletal reduction approach, namely, reaction elimination based on DRGASA– see 6.5. It was found that the application of this new reduction method helps to reduce – after application of the first three reduction methods – the reaction mechanism by another appreciable 30 to 40% of elementary reactions.

Two skeletal mechanisms have been developed with the above mentioned skeletal reduction techniques and benchmarked against the starting detailed GRI3.0 mechanism. A skeletal mechanism has been derived specifically for weakly strained diffusion flames, at a particular low strain rate. Then a second skeletal mechanism has been derived for similar conditions, but for the entire range of strain rates, i.e., for strain rates of practically zero up to the extinction value. Numerical simulations results obtained using these skeletal mechanisms are given in Chap. 9. They indicate that both skeletal mechanisms reproduce the main features of the detailed mechanism with a very good accuracy. In particular, the extinction strain rate, profiles of temperature, mass- fraction profiles of major species such as reactants and combustion products, as well

98 as the most important species of the hydrogen-oxygen radical pool, i.e., of the O and the H-atom, and of the OH radical are reproduced with a good accuracy.

In the second step, skeletal mechanisms are further reduced by employing steady-state approximations. The targeted search algorithm (TSA) [102] presented and discussed in Chap. 4.5 has been implemented to effectively select the optimum steady-state species, in particular, in presence of transport phenomena. Here also two global reduced mechanisms are derived. Specifically, a 9-step and an 11-step mechanisms valid for oxy- methane diffusion flames under weak strain and, alternatively, over the entire range of strain rates have been developed. Computational results given in Chap. 10 show that the global reduced mechanisms reproduce the flame-structure of the GRI3.0 mechanism with a very good accuracy.

In Chap. 11, empirical global rate expressions are derived for the reactions of the 11- step mechanism. Specifically, the global rates ωI to ωXI of each global reaction given in Chap. 7.5.2 are expressed in terms of one activation energy, one pre-exponential rate factor, one pre-exponential temperature dependence, global reaction orders and global third-body efficiencies. Empirical fitting results are summarized ,respectively, in Table 11.1 for the forward global reactions and in Table 11.2 for the backward global reactions. Numerical simulations performed with this empirical reduced mechanism show that it reproduces the flame-structure of the GRI3.0 mechanism with a similar level of accuracy as the 11-step mechanism mentioned above.

Future investigations might extend the comprehensiveness of the derived reduced mechanisms. For instance, it could be interesting to also consider high pressure cases too thereby extending the validity of the various chemistry models to an entire range of pressures. Another further study could also incorporate the reduced mechanisms derived in the present thesis into numerical modeling and computations of turbulent non-premixed ﬂames.

99 Appendix A

Skeletal Mechanisms

A.1 Oxy-Methane Skeletal Mechanism for Coun- terﬂow Diﬀusion Flames

Here we list the skeletal mechanism obtained after successive application of four reduction methods presented in Chap. 6. This skeletal mechanism comprises 47 elementary reactions amongst the 20 species, namely, H2, H, O, O2, OH, H2O, HO2, CH3, CH4,

CO, CO2, HCO, CH2O, CH3O, C2H2, HCCO, CH2,C2H4, CH2(S) and C2H6.

The mechanism follows.

No. Elementary reaction A β Ea(cal/mole) 1 O + H2 <=> H + OH 3.870E+04 2.7000 6260.00 2 O + HO2 => OH + O2 2.000E+13 0.0000 0.00 3 O + CH3 => H + CH2O 5.060E+13 0.0000 0.00 4 O + CH4 <=> OH + CH3 1.020E+09 1.5000 8600.00 5 O + HCO => OH + CO 3.000E+13 0.0000 0.00 6 O + HCO => H + CO2 3.000E+13 0.0000 0.00 7 O + C2H2 => H + HCCO 1.350E+07 2.0000 1900.00 8 O + C2H2 => CO + CH2 6.940E+06 2.0000 1900.00 9 O + C2H4 => CH3 + HCO 1.250E+07 1.8300 220.00 10 O + HCCO => H + 2 CO 1.000E+14 0.0000 0.00 11 H + O2 + M1 <=> HO2 + M1 2.800E+18 -0.8600 0.00 12 H + 2 O2 => O2 + HO2 2.080E+19 - 1.2400 0.00 13 H + O2 + H2O <=> H2O + HO2 1.126E+19 - 0.7600 0.00 14 H + O2 <=> O + OH 2.650E+16 -0.6707 17041.00 15 H + OH + M2 <=> H2O + M2 2.200E+22 -2.0000 0.00 16 H + HO2 <=> O2 + H2 4.480E+13 0.0000 1068.00 17 H + HO2 => 2 OH 8.400E+13 0.0000 635.00 18 H + CH3(+M3) <=> CH4(+M3) 1.390E+16 - 0.5340 536.00 LOW /2.62E+033 -4.76 2440/ TROE /0.783 74 2941 6964/

100 Table A.1 – (Continued) No. Elementary reaction A β Ea(cal/mole) 19 H + CH4 <=> CH3 + H2 6.600E+08 1.6200 10840.00 20 H + CH2O => HCO + H2 5.740E+07 1.9000 2742.00 21 OH + H2 <=> H + H2O 2.160E+08 1.5100 3430.00 22 2 OH <=> O + H2O 3.570E+04 2.4000 - 2110.00 23 OH + HO2 => O2 + H2O 1.450E+13 0.0000 - 500.00 24 OH + CH2 => H + CH2O 2.000E+13 0.0000 0.00 25 OH + CH3 <=> CH2 + H2O 5.600E+07 1.6000 5420.00 26 OH + CH3 <=> CH2(S) + H2O 6.440E+17 - 1.3400 1417.00 27 OH + CH4 <=> CH3 + H2O 1.000E+08 1.6000 3120.00 28 OH + CO <=> H + CO2 4.760E+07 1.2280 70.00 29 HO2 + CH3 => OH + CH3O 3.780E+13 0.0000 0.00 30 CH2 + O2 => OH + H + CO 5.000E+12 0.0000 1500.00 31 2 CH2 => H2 + C2H2 1.600E+15 0.0000 11944.00 32 CH2 + CH3 => H + C2H4 4.000E+13 0.0000 0.00 33 CH2 + CH4 <=> 2 CH3 2.460E+06 2.0000 8270.00 34 CH2(S) + O2 => H + OH + CO 2.800E+13 0.0000 0.00 35 CH3 + O2 => O + CH3O 3.560E+13 0.0000 30480.00 36 2 CH3(+M4) <=> C2H6(+M4) 6.770E+16 - 1.1800 654.00 LOW /3.4E+041 -7.03 2762/ TROE /0.619 73.2 1180 9999/ 37 CH3 + HCO => CH4 + CO 2.648E+13 0.0000 0.00 38 HCO + H2O => H2O + H + CO 1.500E+18 - 1.0000 17000.00 39 HCO + M5 => H + CO + M5 1.870E+17 - 1.0000 17000.00 40 HCO + O2 => HO2 + CO 1.345E+13 0.0000 400.00 41 CH3O + O2 => HO2 + CH2O 4.280E-13 7.6000 - 3530.00 42 C2H4(+M4) => H2 + C2H2(+M4) 8.000E+12 0.4400 86770.00 LOW /1.58E+051 -9.3 97800/ TROE /0.7345 180 1035 5417/ 43 HCCO + O2 => OH + 2 CO 3.200E+12 0.0000 854.00 44 O + CH3 => H + H2 + CO 3.370E+13 0.0000 0.00 45 OH + HO2 <=> O2 + H2O 5.000E+15 0.0000 17330.00 46 OH + CH3 => H2 + CH2O 8.000E+09 0.5000 - 1755.00 47 CH2 + O2 => 2 H + CO2 5.800E+12 0.0000 1500.00

Table A.1: Skeletal mechanism for oxy-methane diﬀusion ﬂames

The various third bodies appearing in the mechanism are given as

[M1] = 0.00[O2] + 0.00[H2O] + 0.75[CO] + 1.50[CO2] + 1.50[C2H6]

[M2] = 0.73[H2] + 3.65[H2O] + 2.00[CH4] + 3.00[C2H6]

[M3] = 2.00[H2] + 6.00[H2O] + 3.00[CH4] + 1.50[CO] + 2.00[CO2] + 3.00[C2H6]

[M4] = 2.00[H2] + 6.00[H2O] + 2.00[CH4] + 1.50[CO] + 2.00[CO2] + 3.00[C2H6]

101 Here, in contrast to the formulation given in Chap. 7.5.2 in the main text of the present thesis – we have used the computational standard notation to express the concentrations of the third bodies [27, 51, 84]. Using this standard formulation, the generally adopted convention is that on the right-hand-side of a third body definition, in the sum, only terms relating to those species are listed explicitly whose efficiencies are different from unity. In other words, all species not mentioned explicitly in this sum have an efficiency of exactly unity.

For unimolecular/recombination fall-off reactions such as reaction number 36 of the Table A.1, after the keyword ”LOW”, are specified three Arrhenius parameters to be used in the low-pressure limit for the evaluation of the rate constant. Similarly, after the keyword ”TROE”, are specified four, or alternatively three, parameters used in the evaluation of pressure-dependent rate with the Troe model – see App. B.3.1.2 for details about the Troe formulation of pressure-dependent rates.

A.2 Extremely Low Strained Oxy-Methane Skele- tal Mechanism for Counterﬂow Diﬀusion Flames

For extremely low strained oxy-methane counterflow diffusion flame at 1 bar, a skeletal mechanism involving 24 elementary reactions amongst 19 species has been derived using the skeletal reduction procedure described in Chap. 6. The 19 skeletal species are H2, H, O, O2, OH, H2O, HO2, CH2(S), CH4, CO, CO2, CH2O, CH3, CH2, CH2CO,

C2H5,C2H6,C2H4 and C2H2.

No. Elementary reaction A β Ea(cal/mole) 1 O + H2 <=> H + OH 3.870E+04 2.7000 6260.00 2 O + C2H2 <=> CO + CH2 6.940E+06 2.0000 1900.00 3 H + O2 + H2O <=> H2O + HO2 1.126E+19 -0.7600 0.00 4 H + O2 <=> O + OH 2.650E+16 -0.6707 17041.00 5 2 H + H2O <=> H2O + H2 6.000E+19 -1.2500 0.00 6 H + OH + M2 <=> H2O + M2 2.200E+22 -2.0000 0.00 7 H + CH3(+M2) <=> CH4(+M2) 1.390E+16 -0.5340 536.00 LOW / 2.620E+33 - 4.76 2440 / TROE / 0.783 74 2941 6964 / 8 H + CH4 <=> CH3 + H2 6.600E+08 1.6200 10840.00 9 H + C2H4(+M4) <=> C2H5(+M4) 5.400E+11 0.4540 1820.00 LOW / 6.0E+41 - 7.62 6970 / TROE / 0.9753 210. 984 4374 / 10 H + C2H6 <=> C2H5 + H2 1.150E+08 1.9000 7530.00

102 Table A.2 – (Continued) No. Elementary reaction A β Ea(cal/mole) 11 H + CH2CO <=> CH3 + CO 1.130E+13 0.0000 3428.00 12 H2 + CO(+M4) <=> CH2O(+M4) 4.300E+07 1.5000 79600.00 LOW / 5.07E+27 - 3.42 84350 / TROE / 0.932 197 1540 10300 / 13 OH + H2 <=> H + H2O 2.160E+08 1.5100 3430.00 14 2 OH <=> O + H2O 3.570E+04 2.4000 -2110.00 15 OH + HO2 <=> O2 + H2O 1.450E+13 0.0000 -500.00 16 OH + CH2 <=> H + CH2O 2.000E+13 0.0000 0.00 17 OH + CH3 <=> CH2(S) + H2O 6.440E+17 -1.3400 1417.00 18 OH + CO <=> H + CO2 4.760E+07 1.2280 70.00 19 OH + C2H2 <=> H + CH2CO 2.180E-02 4.5000 -1000.00 20 CH2 + CH3 <=> H + C2H4 4.000E+13 0.0000 0.00 21 CH2 + CO(+M4) <=> CH2CO(+M4) 8.100E+11 0.5000 4510.00 LOW / 2.69E+33 -5.11 7095 / TROE / 0.5907 275 1226 5185 / 22 CH2(S) + CO2 <=> CO + CH2O 1.400E+13 0.0000 0.00 23 2 CH3(+M4) <=> C2H6(+M4) 6.770E+16 -1.1800 654.00 LOW / 3.40E+41 -7.03 2762 / TROE / 0.619 73.2 1180 9999 / 24 C2H4(+M4) <=> H2 + C2H2(+M4) 8.000E+12 0.4400 86770.00 LOW / 1.58E+51 -9.3 97800 / TROE / 0.7345 180 1035 5417 /

Table A.2: Skeletal mechanism for extremely low strained oxy-methane diﬀusion ﬂames

In the above mechanism, the eﬀective third-body concentrations are expressed in the form discussed in App. A.1, i.e., they are given by

[M2] = 0.73[H2] + 3.65[H2O] + 2.00[CH4] + 3.00[C2H6]

[M3] = 2.00[H2] + 6.00[H2O] + 3.00[CH4] + 1.50[CO] + 2.00[CO2] + 3.00[C2H6]

[M4] = 2.00[H2] + 6.00[H2O] + 2.00[CH4] + 1.50[CO] + 2.00[CO2] + 3.00[C2H6]

See App. A.1 for details about the keywords ”LOW” and ”TROE”.

103 Appendix B

Reactions, Their Rates, and Associated Quantities

Let us consider the i-th elementary reaction, i = 1, ..., I, whose rate was formulated in Eq. (4.3). For ease of reading is the rate rewritten here, i.e.,

K K 0 00 Y νki Y νki ωi = ki,f Ck − ki,b Ck . (B.1) k=1 k=1 Reaction i may be, alternatively, a so-called basic reaction, a third-body reaction or a pressure-dependent reaction. Details about reaction-rate expressions for these three types of reaction will be given in the subsequent sections.

B.1 Basic Reaction

A reaction is considered a basic reaction if it does not involve inert species and its rate is pressure independent. The rate of a basic reaction i is given by

K K 0 00 Y νki Y νki ωi = ki,f Ck − ki,b Ck . (B.2) k=1 k=1

0 00 Here Ck denotes the molar concentration of species k, k = 1, ..., K and νki and νki the stoichiometric coeﬃcients of reactants and products respectively. The quantities ki,f and ki,b denote the forward and backward rate constants, respectively, of reaction i. The latter quantities have been deﬁned in Chap. 4.3 by equations (4.4) and (4.5).

If the backward Arrhenius parameters are unknown, ki,b can be calculated using the relation ki,b = ki,f /Kc,i . Here Kc,i denotes the equilibrium constant

PK " K # k=1 νki 0 0 Patm X S H K = exp ν k − k , (B.3) c,i RT ki R RT k=1

104 0 0 1 where Sk and Hk denote the standard-state molar entropy and standard-state molar enthalpy of the k-th species respectively. These quantities can be computed using the NASA polynomials as deﬁned in Chap. 3.4 by Eq. (3.48) and (3.49) respectively.

B.2 Third-Body Reaction

A third-body reaction is a reaction which, needs an species to remove or add energy to its reactants. Such non-reacting species participating in a reaction are called third- body species. A third-body species can be any species or linear combination of species involved in a reaction mechanism. Such species are generally designated by the symbol M or by a more specific species symbol, e.g: H, if the third-body species reduces to a single species. For instance, the dissociation reaction of the ozone mechanism presented in Chap. 4.1 (O3 + M = O2 + O + M) is a third-body reaction. For such reactions, the rate expression given in (B.2) needs to be modified in order to take into account the effects third-bodies species. Specifically, the reaction rate of a third-body reaction is

K K K 0 00 X Y νki Y νki ωi = zkiCk ki,f Ck − ki,b Ck . (B.4) k=1 k=1 k=1

Here zki is the third-body eﬃciency of species k in reaction i. Default values of zki are unity.

B.3 Pressure-Dependent Reaction

The rate expressions of some reactions depend, under certain conditions, on both temperature and pressure. Such reactions are termed pressure dependent reaction. There are two categories of pressure dependent reactions in gas phase chemistry, namely, unimolecular/recombination fall-oﬀ reactions and chemically activated bimolecular reactions. Details about reaction-rate expressions for the two groups of pressure-dependent reaction are given in the next subsection.

B.3.1 Unimolecular/Recombination Fall-Oﬀ Reactions

The rate expression for an unimolecular/recombination fall-oﬀ reactions is

K K 0 00 Pri Y νki Y νki ωi = Fi ki,f Ck − ki,b Ck . (B.5) 1 + Pri k=1 k=1

1The superscript o refers to the standard-state 1 atmosphere

105 Here Pri denotes the reduced pressure; it is deﬁned as

k0,i Pri = [M], (B.6) k∞,i where [M] accounts for the third body eﬀects. k0,i and k∞,i denote the low and high pressure limit rate constant respectively. In Arrhenius form, they are deﬁned as follows:

E β0,i 0,i k0,i = A0T exp − (B.7) RcT E β∞,i ∞,i k∞,i = A∞,iT exp − (B.8) RcT

In the so-called Chemkin format2, the Arrhenius coeﬃcients for the high-pressure limit, viz., A∞,i, β∞,i and E∞,i, are given on the reaction line. The additional three rate parameters for low-pressure limit, viz., A0,i, β0,i and E0,i are speciﬁed on a reaction line containing the keyword ”LOW”.

Fi term depends on the pressure-dependent rate formulation considered. Three models of pressure-dependent reaction rate are commonly used, namely, Lindemann, Troe and

SRI formulations. These three models diﬀer only through their respective Fi functions.

B.3.1.1 Lindemann formulation

The Lindemann model is the simplest formulation of a pressure-dependent reaction rate. In this formulation, Fi is unity; hence the reaction rate is

K K 0 00 Pri Y νki Y νki ωi = ki,f Ck − ki,b Ck , (B.9) 1 + Pri k=1 k=1 here ki,f = k∞,i. More accurate models of pressure-dependent rate are given in the next sections

B.3.1.2 Troe formulation

In the Troe formulation, the Fi term is 1/ 1+(A/B)2 Fi = Fcent , (B.10) 2The Chemkin format is a standard format which is used to write chemical information in combustion chemistry. Detailed reaction mechanism such as GRI3.0 [96] used in the present work are written in Chemkin format.

106 where Fcent, A and B are T T T ∗∗ F = (1 − a) exp − + a exp − + exp − (B.11) cent T ∗∗∗ T ∗ T

A = log10 Pri − 0.67 log10 Fcent − 0.4 (B.12)

B = 0.806 − 1.1762 log10 Fcent − 0.14 log10 Pri (B.13)

The parameters a, T ∗∗∗, T ∗ and T ∗∗, are speciﬁed in this order on the reaction line ∗∗ with the keyword ”TROE”. Fcent is computed with its ﬁrst two terms if T is not provided in the parameters list.

B.3.1.3 SRI formulation

SRI form is another widely-used model of pressure-dependent rate. In this formulation, the Fi is b T X F = d T e a exp − + exp − , (B.14) i T c where 1 X = (B.15) 2 1 + log10 Pri

The parameters a, b, c, d and e in Eq. (B.14) are speciﬁed by the user. Default values for parameters d and e are used if these two coeﬃcients are not provided. By default, d = 1 and e = 0.

B.3.2 Chemically Activated Bimolecular Reactions

For such reactions, the pressure-dependent rate is

K K 0 00 1 Y νki Y νki ωi = Fi ki,f Ck − ki,b Ck , (B.16) 1 + Pri k=1 k=1 here ki,f = k0,i.

In Chemkin format, the Arrhenius parameters for the low-pressure limit rate constant k0,i, viz., A0,i, β0,i and E0,i are given on the reaction line. The other three rate coefficients for the high-pressure limit rate constant k∞,i, viz., A∞,i, β∞,i and E∞,i are specified on a reaction line containing the keyword ”HIGH”. In addition, the three choices for the Fi function are identical to the ones presented above for unimolecular/recombination fall-off reactions.

107 Bibliography

[1] J. Andersen, C. L. Rasmussen, T. Giselsson, and P. Glarborg. Global combustion mechanisms for use in CFD modeling under oxy-fuel conditions. Energy & Fuels, 23(3):1379–1389, 2009.

[2] Ansys Inc. Ansys ﬂuent software, 2011.

[3] Ansys Inc. Ansys CFX software, 2016.

[4] Ansys Inc. STAR-CD software, 2016.

[5] R. Aris. Vectors, tensors, and the basic equations of ﬂuid mechanics. Dover, New York, 1989.

[6] U. M. Ascher and L. R. Petzold. Computer methods for ordinary diﬀerential equations and diﬀerential-algebraic equations. Society for Industrial and Applied Mathematics, Philadelphia, 1998.

[7] F. Battin-Leclerc. Detailed chemical kinetic models for the low-temperature combustion of hydrocarbons with application to gasoline and diesel fuel surrogates. Progress in Energy and Combustion Science, 34(4):440–498, 2008.

[8] C. E. Baukal, JR. Introduction. In C. E. Baukal, JR., editor, Oxygen-enhanced combustion, Industrial combustion series, pages 1–23. CRC Press Taylor & Fran- cis Group, Boca Raton, 2013.

[9] C. E. Baukal, JR., editor. Oxygen-enhanced combustion. Industrial combustion series. CRC Press Taylor & Francis Group, Boca Raton, second edition edition, 2013.

[10] F. Behrendt, H. Bockhorn, B. Rogg, and J. Warnatz. Modeling of Turbu- lent CO/Air Diﬀusion Flames with Detailed Chemistry. In J. Warnatz and W. J¨ager,editors, Complex Chemical Reaction Systems, pages 376–385. Springer, Berlin/Heidelberg, 1987.

108 [11] B. A. Bennett, Z. Cheng, R. Pitz, and M. Smooke. Computational and experimental study of oxygen-enhanced axisymmetric laminar methane ﬂames. Com- bustion Theory and Modelling, 12(3):497–527, 2008.

[12] M. B.F. and H. B.H. On the mathematical modeling of turbulent combustion with special emphasis on shoot formation and combustion. In The Combustion Institute, pages 719–729, 1976.

[13] Bibrzycki, J. and Poinsot, T. Reduced chemical kinetic mechanisms for methane

combustion in O2/N2 and O2/CO2 atmosphere.

[14] R. W. Bilger, S. H. St˚arner,and R. J. Kee. On reduced mechanisms for methane air combustion in nonpremixed ﬂames. Combustion and Flame, 80(2):135–149, 1990.

[15] K. Bray, M. Champion, and P. Libby. The Interaction Between Turbulence and Chemistry. In Turbulent Reactive Flows, R. Borghi and S.N.B. Murthy (Eds.), Lecture Notes in Engineering 40, pages 541–563. Springer, New York, 1989.

[16] K. Bray, J. Moss, and P. Libby. Turbulent Transport in Premixed Flames. In J. Zierep and H. Oertel, editors, Convective Transport and Instability Phenomena, pages 389–423. Braun-Verlag, Karlsruhe, 1982.

[17] B. Buhre, L. K. Elliott, C. D. Sheng, R. P. Gupta, and T. F. Wall. Oxy-fuel combustion technology for coal-ﬁred power generation. Progress in Energy and Combustion Science, 31(4):283–307, 2005.

[18] H. K. Chelliah, K. Seshadri, and C. K. Law. Reduced kinetic mechanisms for counterflow methane-air diffusion flames. In N. Peters and B. Rogg, editors, Reduced kinetic mechanisms for applications in combustion systems, volume 15 of Lecture Notes in Physics Monographs, pages 224–240. Springer Berlin Heidelberg, Berlin, Heidelberg, 1993.

[19] J.-Y. Chen. A general procedure for constructing reduced reaction mechanisms with given independent relations. Combustion Science and Technology, 57(1- 3):89–94, 1988.

[20] J.-Y. Chen, editor. Development of reduced mechanisms for numerical modeling of turbulent combustion, 1997.

109 [21] J.-Y. Chen, R. W. Dibble, and R. W. Bilger. PDF modeling of turbulent non-

premixed CO/H2/N2 jet ﬂames with reduced mechanisms. Symposium (Interna- tional) on Combustion, 23(1):775–780, 1991.

[22] J.-Y. Chen, Liu Y., and Rogg B. CO/H2/N2/air diffusion flames: thermal radiation and transient effects. In N. Peters and B. Rogg, editors, Reduced kinetic mechanisms for applications in combustion systems, volume 15 of Lecture Notes in Physics Monographs, pages 196–223. Springer Berlin Heidelberg, Berlin, Hei- delberg, 1993.

[23] Z. Cheng, J. A. Wehrmeyer, and R. W. Pitz. Experimental and numerical studies of opposed jet oxygen-enhanced methane diﬀusion ﬂames. Combustion Science and Technology, 178(12):2145–2163, 2006.

[24] C. Chevalier, J. Warnatz, and H. Melenk. Automatic generation of reaction mechanisms for the description of the oxidation of higher hydrocarbons. Berichte der Bunsengesellschaft fr physikalische Chemie, 94(11):1362–1367, 1990.

[25] K. Claramunt, R. Consul, and Carbonell, D., Perezsegarra, C. Analysis of the laminar ﬂamelet concept for nonpremixed laminar ﬂames. Combustion and Flame, 145(4):845–862, 2006.

[26] T. Coﬀee and J. Heimerl. Transport algorithms for premixed, laminar steady- state ﬂames. Combust. Flame, 43:273–289, 1981.

[27] Cosilab. Rotexo software, gasphase-chemistry user manual, 2013.

[28] P. Dagaut and S. Gail. Chemical kinetic study of the eﬀect of a biofuel additive on jet-a1 combustion. The journal of physical chemistry. A, 111(19):3992–4000, 2007.

[29] P. Deuﬂhard. A modiﬁed Newton method for the solution of ill-conditioned systems of nonlinear equations with application to multiple shooting. Numer. Math., 22:289, 1974.

[30] P. Deuflhard, E. Hairer, and J. Zugck. One-step and extrapolation methods for differential-algebraic systems. Technical Report Preprint No. 318, Universität Heidelberg, Sonderforschungsberich 123, 1985.

[31] P. Deuﬂhard and U. Nowak. Extrapolation integrators for quasilinear implicit odes. Technical Report Preprint No. 332, Universit¨atHeidelberg, Son- derforschungsberich 123, October 1985.

110 [32] M. Ditaranto and J. Hals. Combustion instabilities in sudden expansion oxy-fuel ﬂames. Combustion and Flame, 146(3):493–512, 2006.

[33] G. Dixon-Lewis. Computer modeling of combustion reactions in ﬂowing systems with transport. In W. Gardiner, Jr., editor, Combustion Chemistry, chapter 2, pages 21–125. Springer, New York, 1984.

[34] G. Dixon-Lewis, T. David, P. Gaskell, S. Fukutani, H. Jinno, J. Miller, R. Kee, M. Smooke, N. Peters, E. Effelsberg, J. Warnatz, and F. Behrendt. Calculation of the Structure and Extinction Limit of a Methane-Air Counterflow Diffusion Flame in the Forward Stagnation Region of a Porous Cylinder. In 20th Sym- posium (International) on Combustion, pages 1893–1904, Pittsburgh, 1984. The Combustion Institute.

[35] M. Drake and R. Blint. Thermal NOx in Stretched Laminar Opposed-Flow Dif-

fusion Flames with CO/H2/N2 Fuel. Combust. Flame, 76:151–167, 1989.

[36] T. Echekki and E. Mastorakos, editors. Turbulent combustion modeling: Ad- vances, new trends and perspectives, volume v. 95 of Fluid mechanics and its applications, 0926-5112. Springer, Dordrecht, 2011.

[37] A. Frassoldati, A. Cuoci, T. Faravelli, E. Ranzi, C. Candusso, and Tolazzi D., editors. Simpliﬁed kinetic schemes for oxy-fuel combustion, 2009.

[38] C. Gaoa, J. Allena, W. Greena, and R. West. Reaction mechanism genera- tor: Automatic construction of chemical kinetic mechanisms. Computer Physics Communications, 203:212–225, 2016.

[39] V. Giovangigli and M. Smooke. Adaptive continuation algorithms with application to combustion problems. Technical Report Report ME-102-87, Yale Univer- sity, 1987.

[40] P. Glarborg and L. L. B. Bentzen. Chemical eﬀects of a high CO2 concentration in oxy-fuel combustion of methane. Energy & Fuels, 22(1):291–296, 2008.

[41] J. G¨ottgensand P. Terhoeven. Redmech-an automatic reduction program. In N. Peters and B. Rogg, editors, Reduced kinetic mechanisms for applications in combustion systems, Lecture Notes in Physics Monographs, pages 345–349. Springer Berlin Heidelberg, Berlin, Heidelberg, 1993.

111 [42] E. Hairer and G. Wanner. Solving ordinary differential equations II: Stiff and differential-algebraic problems, volume 14. Springer Berlin Heidelberg, Berlin, Heidelberg, 1996.

[43] J. Heimerl and T. Coﬀee. Transport algorithms for methane ﬂames. Combust. Sci. and Tech., 34:31–43, 1983.

[44] J. M. Heimerl and T. P. Coﬀee. The detailed modeling of premixed, laminar steady-state ﬂames. I. Ozone. Combustion and Flame, 39(3):301–315, 1980.

[45] J. Hirschfelder and C. Curtiss. Theory of propagation of ﬂames. Part I: Gen- eral equations. In Third Symposium on Combustion and Flame and Explosion Phenomena, pages 121–127, Baltimore, 1949. Williams and Wilkins Co.

[46] J. Hirschfelder, C. Curtiss, and R. Bird. Molecular theory of gases and liquids. Wiley, New York, 1954.

[47] W. P. Jones and R. P. Lindstedt. Global reaction schemes for hydrocarbon combustion. Combustion and Flame, 73(3):233–249, 1988.

[48] M. Karalus. An Investigation of lean blowout of gaseous fuel, alternatives to natural gas. Dissertation, University of Washington, Washington, 2013.

[49] R. Kee, M. Coltrin, and P. Glarbourg. Chemically reacting ﬂow. Wiley- Interscience, 2003.

[50] R. Kee, J. Miller, G. Evans, and G. Dixon-Lewis. A Computational Model of the Structure and Extinction of Strained, Opposed Flow, Premixed Methane-Air Flames. In 22nd Symposium (International) on Combustion, pages 1479–1494, Pittsburgh, 1988. The Combustion Institute.

[51] R. Kee, F. Rupley, and J. Miller. Chemkin-II: A Fortran chemical kinetics package for the analysis of gas-phase chemical kinetics. Technical Report Report SAND89- 8009, Sandia National Labs., September 1989.

[52] R. Kee, J. Warnatz, and J. Miller. A Fortran computer code package for the evaluation of gas-phase viscosities, conductivities, and diﬀusion coeﬃcients. Technical Report Report SAND-8209, Sandia National Labs., 1983.

[53] R. J. Kee, M. , Coltrin, and P. Glarborg. Chemically reacting ﬂow modeling: Theory & practice. Wiley, New York and Chichester, 2003.

112 [54] R. J. Kee, J. F. Grcar, M. D. Smooke, and J. A. Miller. A fortran program for modeling steady laminar one-dimensional premixed ﬂames.

[55] P. Kunkel and V. Mehrmann. Diﬀerential-algebraic equations. European Math- ematical Society Publishing House, Zuerich, Switzerland, 2006.

[56] S. H. Lam. Using CSP to understand complex chemical kinetics. Combustion Science and Technology, 89(5-6):375–404, 1993.

[57] S. H. Lam and D. A. Goussis. The CSP method for simplifying kinetics. Inter- national Journal of Chemical Kinetics, 26(4):461–486, 1994.

[58] T. Lu. Comprehensive reduction and application of chemical reaction mechanisms. Dissertation, Princeton University, Princeton, 2004.

[59] T. Lu and C. K. Law. A directed relation graph method for mechanism reduction. Proceedings of the Combustion Institute, 30(1):1333–1341, 2005.

[60] T. Lu and C. K. Law. Linear time reduction of large kinetic mechanisms with directed relation graph: n-heptane and iso-octane. Combustion and Flame, 144(1- 2):24–36, 2006.

[61] T. Lu and C. K. Law. On the applicability of directed relation graphs to the reduction of reaction mechanisms. Combustion and Flame, 146(3):472–483, 2006.

[62] T. Lu and C. K. Law. A criterion based on computational singular perturbation for the identiﬁcation of quasi steady state species: A reduced mechanism for methane oxidation with no chemistry. Combustion and Flame, 154(4):761–774, 2008.

[63] T. Lu and C. K. Law. Strategies for mechanism reduction for large hydrocarbons: n-heptane. Combustion and Flame, 154(1-2):153–163, 2008.

[64] Lu T. Reduced mechanisms download, 2015.

[65] U. Mass and S. B. Pope. Simplifying chemical kinetics: Intrinsic low-dimensional manifolds in composition space. Combustion and Flame, 88(3-4):239–264, 1992.

[66] F. Mauß, D. Keller, and N. Peters. A lagrangian simulation of flamelet extinction and re-ignition in turbulent jet diffusion flames. Symposium (International) on Combustion, 23(1):693–698, 1991.

113 [67] Montgomery C. J., Cremer M. A., Chen J.-Y., Westbrook C. K., and Maurice L. Q. Reduced chemical kinetic mechanisms for hydrocarbon fuels. Journal of Propusion and Power, 18(1):192–198, 2002.

[68] P. Muller. Glossary of terms used in physical organic chemistry (iupac recommendations 1994). Pure and Applied Chemistry, 66(5):1077–1184, 1994.

[69] S. V. Naik and N. M. Laurendeau. Quantitative laser-saturated fluorescence measurements of nitric oxide in counter-flow diffusion flames under sooting oxyfuel conditions. Combustion and Flame, 129(1-2):112–119, 2002.

[70] K. E. Niemeyer, C.-J. Sung, and M. P. Raju. Skeletal mechanism generation for surrogate fuels using directed relation graph with error propagation and sensitivity analysis. Combustion and Flame, 157(9):1760–1770, 2010.

[71] E. Oran and J. Boris. Detailed modelling of combustion systems. Prog. Energy Combust. Sci., 7:1–72, 1981.

[72] P. Pepiot-Desjardins and H. Pitsch. An eﬃcient error-propagation-based reduction method for large chemical kinetic mechanisms. Combustion and Flame, 154(1-2):67–81, 2008.

[73] N. Peters. Laminar diﬀusion ﬂamelet models in non-premixed turbulent combustion. Progress in Energy and Combustion Science, 10(3):319–339, 1984.

[74] N. Peters. Numerical and asymptotic analysis of systematically reduced reaction schemes for hydrocarbon ﬂames. In R. Glowinski, B. Larrouturou, and R. Temam, editors, Numerical Simulation of Combustion Phenomena, volume 241 of Lecture notes in physics, pages 90–109. Springer-Verlag, Berlin/Heidelberg, 1985.

[75] N. Peters. Laminar ﬂamelet concepts in turbulent combustion. Symposium (In- ternational) on Combustion, 21(1):1231–1250, 1988.

[76] N. Peters. Reducing mechanisms. In M. D. Smooke, editor, Reduced kinetic mechanisms and asymptotic approximations for methane-air ﬂames, volume 384 of Lecture notes in physics, pages 48–67. Springer-Verlag, Berlin and New York, 1991.

[77] N. Peters and R. J. Kee. The computation of stretched laminar methane-air diﬀusion ﬂames using a reduced four-step mechanism. Combustion and Flame, 68(1):17–29, 1987.

114 [78] N. Peters and B. Rogg, editors. Reduced kinetic mechanisms for applications in combustion systems. Lecture Notes in Physics Monographs. Springer Berlin Heidelberg, Berlin, Heidelberg, 1993.

[79] H. Pitsch and N. Peters. A consistent flamelet formulation for non-premixed combustion considering differential diffusion effects. Combustion and Flame, 114(1- 2):26–40, 1998.

[80] T. Poinsot and D. Veynante. Theoretical and numerical combustion. Edwards, Philadelphia, 2nd ed. edition, 2005.

[81] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery. Numerical recipes in Fortran 77: The art of scientiﬁc computing. Cambridge University Press, Cambridge, 2nd ed. edition, 1992.

[82] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery. Numerical recipes in Fortran 90: The art of parallel scientiﬁc computing, volume vol. 2 of Fortran numerical recipes of Fortran numerical recipes. Cambridge University Press, Cambridge, 2nd ed. edition, 1996.

[83] H. Rabitz, M. Kramer, and D. Dacol. Sensitivity analysis in chemical kinetics. Annual Review of Physical Chemistry, 34(1):419–461, 1983.

[84] R.J. Kee, M.E. Coltrin, P. Glarborg. Theory and Practice. John Wiley & Sons, United Kingdom, 2003.

[85] B. Rogg. Response and ﬂamelet structure of stretched premixed methane-air ﬂames. Combustion and Flame, 73(1):45–65, 1988.

[86] B. Rogg. Adaptive computational methods for time-dependent problems in combustion engineering. In P. G.N. and M. J., editors, NUMETA 90, Numerical Methods in Engineering, Vol.2, pages 1129–1140, London-New York, 1990. Else- vier.

[87] B. Rogg. Adaptive methods in computational ﬂuid dynamics of chemically reacting ﬂows. Computer-Methods in Applied Mechanics and Engineering 90, pages 659–670, 1991.

[88] B. Rogg. Time and space self-adaptive numerical methods for chemically reacting ﬂows. In R. Vichnevetsky and J. Miller, editors, Proc. 13th World Congress on Computation and Applied Mathematics Vol. 2, July 22-26, 1991, Trinity College,

115 Dublin (Ireland), pages 898–901. International Association for Mathematics and Computers in Simulation, 1991.

[89] B. Rogg, F. Behrendt, and J. Warnatz. Turbulent Non-Premixed Combustion in Partially Premixed Diﬀusion Flamelets with Detailed Chemistry. In 21st Sym- posium (International) on Combustion, pages 1533–1541, Pittsburgh, 1986. The Combustion Institute.

[90] B. Rogg, A. Liñán,and F. A. Williams. Deflagration regimes of laminar flames modeled after the ozone decomposition flame. Combustion and Flame, 65(1):79– 101, 1986.

[91] B. Rogg and I. S. Wichman. Approach to asymptotic analysis of the ozone- decomposition ﬂame. Combustion and Flame, 62(3):271–293, 1985.

[92] B. Rogg and F. A. Williams. Structures of wet CO ﬂames with full and reduced kinetic mechanisms. Symposium (International) on Combustion, 22(1):1441–1451, 1989.

[93] H. Schlichting. Boundary-Layer Theory. McGraw-Hill, New York, 7th edition, 1979.

[94] S. Seepana and S. Jayanti. Flame structure investigations of oxy-fuel combustion. Fuel, 93:52–58, 2012.

[95] A. Sharma and S. Jayanti. CFD modeling of oxygen-enhanced combustion. In C. E. Baukal, JR., editor, Oxygen-enhanced combustion, Industrial combustion series, pages 327–384. CRC Press Taylor & Francis Group, Boca Raton, 2013.

[96] Smith G.P., Golden D.M., Frenklach M., Moriarty N.W., Eiteneer B., Goldenberg M., Bowman C.T., Hanson R.K., Song S., Gardiner W.C. Jr., Lissianski V.V., and and Qin Z. GRI-Mech 3.0.

[97] M. Smooke. An error estimate for the modiﬁed Newton method with application to the solution of nonlinear two-point boundary value problems. J. Opt. Theory and Appl., 39:pp. 489, 1983.

[98] J. Spurk. Fluid Mechanics. Springer, Berlin, 1997.

[99] T. F. Stocker, D. Qin, G.-K. Plattner, M. Tignor, S. K. Allen, J. Boschung, A. Nauels, Y. Xia, V. Bex, and P. M. Midgley, editors. Climate change 2013: The physical science basis. Contribution of working group I to the ﬁfth assessment

116 report of the intergovernmental panel on climate change. Cambridge University Press, Cambridge, United Kingdom and New York, NY, USA, 2013.

[100] W. Sun, Z. Chen, X. Gou, and Y. Ju. A path ﬂux analysis method for the reduction of detailed chemical kinetic mechanisms. Combustion and Flame, 157(7):1298–1307, 2010.

[101] C. Sung, C. Law, and J.-Y. Chen. Augmented reduced mechanisms for no emission in methane oxidation. Combustion and Flame, 125(1-2):906–919, 2001.

[102] Y. F. Tham, F. Bisetti, and J.-Y. Chen. Development of a highly reduced mechanism for iso-octane hcci combustion with targeted search algorithm. Journal of Engineering for Gas Turbines and Power, 130(4):042804, 2008.

[103] A. S. Tomlin, T. Turanyi, and M. J. Pilling. Mathematical tools for the construction, investigation and reduction of combustion mechanisms. In M. J. Pilling, editor, Low-temperature combustion and autoignition, volume 35 of Comprehen- sive chemical kinetics, pages 293–437. Elsevier, Amsterdam and New York, 1997.

[104] H. Tsuji and I. Yamaoka. Structure analysis of counterflow diffusion flames in the forward stagnation region of a porous cylinder. Symposium (International) on Combustion, 13(1):723–731, 1971.

[105] T. Tur´anyi and A. S. Tomlin. Analysis of kinetic reaction mechanisms. Springer Berlin Heidelberg, Berlin, Heidelberg, 2014.

[106] L. Wang, Z. Liu, S. Chen, and C. Zheng. Comparison of diﬀerent global combustion mechanisms under hot and diluted oxidation conditions. Combustion Science and Technology, 184(2):259–276, 2012.

[107] W. Wang and B. Rogg. Reduced kinetic mechanisms and their numerical treatment I: wet CO ﬂames. Combust. Flame, 94:271–292, 1993.

[108] W. Wang, B. Rogg, and F. Williams. Reduced kinetic mechanisms for wet CO ﬂames. In N. Peters and B. Rogg, editors, Reduced kinetic mechanisms for applications in combustion systems, chapter 4, pages 44–57. Springer-Verlag, Berlin- Heidelberg, 1993.

[109] J. Warnatz. Inﬂuence of transport models and boundary conditions on ﬂame structure. In N. Peters and J. Warnatz, editors, Numerical Methods in Flame Propagation, pages 87–111. Vieweg Verlag, Braunschweig/Wiesbaden, 1982.

117 [110] J. Warnatz. Rate coeﬃcients in the C/H/O system. In W. Gardiner, Jr., editor, Combustion Chemistry, chapter 5, pages 197–360. Springer, New York, 1984.

[111] J. Warnatz, U. Maas, and R. W. Dibble. Combustion: Physical and chemical fun- damentals, modeling and simulation, experiments, pollutant formation. Springer, Berlin and London, 4th ed. edition, 2010.

[112] C. K. Westbrook and F. L. Dryer. Simpliﬁed reaction mechanisms for the oxidation of hydrocarbon fuels in ﬂames. Combustion Science and Technology, 27(1-2):31–43, 2007.

[113] C. K. Westbrook, C. V. Naik, O. Herbinet, W. J. Pitz, M. Mehl, S. M. Sarathy, and H. J. Curran. Detailed chemical kinetic reaction mechanisms for soy and rapeseed biodiesel fuels. Combustion and Flame, 158(4):742–755, 2011.

[114] C. K. Westbrook, W. J. Pitz, O. Herbinet, H. J. Curran, and E. J. Silke. A comprehensive detailed chemical kinetic reaction mechanism for combustion of n-alkane hydrocarbons from n-octane to n-hexadecane. Combustion and Flame, 156(1):181–199, 2009.

[115] F. A. Williams. Combustion theory: The fundamental theory of chemically reacting ﬂow systems. Combustion science and engineering series. Addison/Wesley Pub. Co, Redwood City, Calif., 2nd ed. edition, 1985.

[116] S. Zhang, I. P. Androulakis, and M. G. Ierapetritou. A hybrid kinetic mechanism reduction scheme based on the on-the-ﬂy reduction and quasi-steady-state approximation. Chemical Engineering Science, 93:150–162, 2013.

[117] X. L. Zheng, T. F. Lu, and C. K. Law. Experimental counterﬂow ignition temperatures and reaction mechanisms of 1,3-butadiene. Proceedings of the Combustion Institute, 31(1):367–375, 2007.

[118] I. G. Zsély, I. Virág,and T. Turányi. Investigation of a methane oxidation mechanism via the visualization of element fluxes. In F. Beretta, N. Sel\cuk, and M. S. Mansour, editors, Proceedings of the 4th Mediterranean Combustion Symposium, 2005.

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CURRICULUM VITAE

Valentin N°4 Bomba

Personal information

Date of birth 25.03.1979 Nationality Cameroonian Gender male

Education

2012-2017 PhD in combustion - Ruhr university Bochum

2009 Master degree with thesis in telecommunications University of Yaounde I, Cameroon

2006 Master degree with thesis in applied mechanics, University of Yaounde I, Cameroon

2002 Bachelor in physics University of Yaounde I, Cameroon

Work experience

01/05/2016 - 31.03.2017 Research staff – Chair of fluid mechanics Ruhr-University Bochum

01/04/2016 - 30.04.2016 Research Assistant – Chair of fluid mechanics, Ruhr-University Bochum

2006 - 2007 Teaching assistant, department of physics University of Yaounde I, Cameroon