Modeling and Numerical Simulation of Partially Premixed Flames

Ilona Zimmermann

Universitat¨ der Bundeswehr Munchen¨ Fakultat¨ fur¨ Luft- und Raumfahrttechnik

Thema der Dissertation: Modeling and Numerical Simulation of Partially Premixed Flames

Verfasser: Ilona Zimmermann

Promotionsausschuss:

Vorsitzender: Prof. Dr. sc. techn. J¨orn Sesterhenn 1. Berichterstatter: Prof. Dr. rer. nat. Michael Pﬁtzner 2. Berichterstatter: Prof. Wolfgang Polifke, Ph.D.

Tag der Pr¨ufung: 21.07.2009

Mit der Promotion erlangter akademischer Grad: Doktor der Ingenieurwissenschaften (Dr. Ing.)

Neubiberg, den 16.11.2009

Acknowledgements

This work was carried out at the University of Armed Forces Munich, Institute for Thermodynamics. I would like to express my sincere thank to Professor M. Pfitzner for his guidance and encouraging supervision during the entire project. He offered me the possibility to gain an insight to many different subjects. I am greatful to Professor Polifke who agreed to assist my work as co-examiner. I would also like to thank my colleagues of the Institute for Thermodynamics for interesting discussions, and my colleagues at ANSYS for the fantastic working atmosphere. Furthermore I would like to thank my friends who supported all the time.

Danksagung

Ein ganz besonderer Dank gilt natürlich meiner Familie fürdie stetige Unterstützung auf meinem Lebensweg und die Liebe die sie mir schenken. Vor allem mein Mann Mario gab mir stets den notwendigen Rückhalt und hat mich immer wieder auf meinem Weg bestärkt.

Abstract

The demands on motor vehicles are steadily increasing. They have to be powerful and competitive but also environmentally friendly. This is not only due to the tight pollutant emission regulations, but also due to public interests. Although alternatives to internal combustion engines like hybrid or electro vehicles are under investigation and are already used for certain applications, it will take several decades until a wider use is possible. Thus conventional internal combustion engines have to be improved further. There is still potential like downsizing with supercharger, variable valve stroke or variable compression. But to be able to further improve the internal combustion engines, a deeper understanding of the combustion process is necessary. As experiments give only limited information, computational fluid dynamics (CFD) is a very powerful tool to gain more knowledge of the processes occurring within the engine and to improve the processes. In industry, the Eddy Dissipation Model is still a commonly used combustion model. As this is a very simple model, it is not successful to describe all the complex processes occurring in internal combustion engines. So this work focuses on the development of a suitable combustion model for direct injection engines in industrial applications. In such applications, combustion occurs under partially premixed conditions. The model presented is based on a flamelet concept with an additional transport equation for the reaction progress variable to account for the partially premixed regime. Different approaches are compared to close the source term of the reaction progress variable transport equation. The main effort has been spent for the correct description of the laminar flame speed, which changes with pressure, temperature and mixture composition, which strongly vary in internal combustion engines. As the laminar flame speeds are derived in the context of premixed combustion, the validity range of the correlations is not wide enough for diffusion flames. So they are extended to the flammability limits, for which correlations are derived within this work to account for the pressure and temperature dependence. To also account for the turbulence effects, an effective laminar flame speed is introduced, used in pre-integrated tables, which also depends on the mean mixture fraction and mixture fraction fluctuation. The developed model is validated on a hydrogen jet flame, on a piloted methane- and heptane-air flame, and on a methane-air swirl burner. Especially for the hydrogen jet flame, and the piloted methane flame, a detailed set of experimentally obtained variables is available, like species mass fractions and temperature distributions. In comparison to the Eddy Dissipation Model, which is based on the assumption of a single step reaction, it could be seen that the Eddy Dissipation Model is not reliable enough to account for the complex processes occurring in internal combustion engines, whereas the flamelet model together with the reaction progress variable and the effective laminar flame speed shows promising and reliable results.

Contents

1 Introduction 1

2 Fundamentals of Turbulent Reactive Flows 5 2.1 Characteristic Variables ...... 5 2.2 Flame Types ...... 7 2.3 Laminar Flame Speed ...... 13 2.4 Flammability Limits ...... 14

3 Basic Equations for Modeling of Non Reactive Flows 17 3.1 Instantaneous Equations ...... 17 3.2 Turbulent Flows ...... 19 3.3 Turbulence Models ...... 22

4 Modeling of Turbulent Reactive Flows 27 4.1 Eddy Dissipation Model ...... 31 4.2 The Laminar Flamelet Concept ...... 32 4.3 The Reaction Progress Variable ...... 38 4.4 Partially Premixed Flames ...... 41

5 H2 Jet Flame 44 5.1 Computational Domain and Boundary Conditions ...... 45 5.2 Inﬂuence of Chemistry Table ...... 47 5.3 Choice of Turbulence Model ...... 51 5.4 Implementation of Laminar Flame Speed ...... 52 5.5 Results and Discussion ...... 59

6 Sandia D Flame 69 6.1 Computational Domain and Boundary Conditions ...... 70 6.2 Inﬂuence of Chemistry Table ...... 71 6.3 Choice of Turbulence Model ...... 75 6.4 Implementation of Laminar Flame Speed ...... 75 6.5 Results and Discussion ...... 84

7 Low Swirl Burner 97 7.1 Computational Domain and Boundary Conditions ...... 98 7.2 Results and Discussion ...... 99

8 Conclusion and Outlook 110 8.1 Conclusion ...... 110 8.2 Outlook ...... 111 A The Presumed β-pdf Approach 113

B Flammability Limits 118 B.1 Hydrogen ...... 118 B.2 Methane ...... 120

C Combustion Chemistry 122 C.1 Methane Mechanism ...... 122 C.2 Hydrogen Mechanism ...... 131

D Additional Figures 132 D.1 H2 Jet Flame ...... 132 D.2 Sandia D Flame ...... 134

E Material Properties of Hydrogen 135 Nomenclature

Latin Symbols

A flame front surface area m2 c reaction progress variable – cp heat capacity at constant pressure J/(kg K) D diffusion coefficient m2/s G stretch factor – g gravity m/s2 H total enthalpy J h specific enthalpy J/kg k turbulent kinetic energy m2/s2 l length m m mass kg P probability density function p pressure P a R specific gas constant J/(kg K) s stoichiometric mass ratio – sl laminar flame speed m/s st turbulent flame speed m/s T temperature K t time s U diffusion velocity m/s u velocity m/s W molecular weight kg/mol X mole fraction – x space coordinate m Y mass fraction – Z mixture fraction – Greek Symbols

χ scalar dissipation rate 1/s ε turbulent dissipation rate m2/s3 λ heat conductivity W/(Km) λ air excess ratio – µ dynamic viscosity kg/(ms)

µt turbulent viscosity kg/(ms) ν kinematic viscosity m2/s ν0, ν00 stoichiometric coeﬃcients; forward, backward reac- – tion ω source term or turbulent frequency φ equivalence ratio – ρ density kg/m3

τc chemical time scale s

τf ﬂow time scale s

τfl ﬂame time scale s

τη Kolmogorov time scale s 2 τij stress tensor N/m

Abbreviations

CFD Computational Fluid Dynamics CPU Central Processing Unit DNS Direct Numerical Simulation JANAF Joint Army, Navy, and Air Force (USA) LES Large Eddy Simulation RANS Reynolds Averaged Navier Stokes Equation

Dimensionless Numbers

Da Damk¨ohlernumber Ka Karlovitz numer Le Lewis number Re Reynolds number Indices c chemical F fuel i species or space direction j species k species

O2 oxidant q quenching st stoichiometric t turbulent u unburnt

Chapter 1 Introduction

Combustion is the oldest and most important energy conversion method worldwide. In 2006, the total world primary energy demand was 11,741 millions of metric tons of oil equivalent (Mtoe). Figure 1.1 shows the development of the world energy demand recorded by the International Energy Agency (IEA) [168]. Since start of recording in 1971 the energy demand has already doubled, and is still steadily increasing. 81 % of the energy conversion for domestic heating, power generation and transportation is due to the combustion of either oil, coal or gas. Energy sources like nuclear, hydro, solar or wind energy still play a minor role. Thus combustion of fossil fuels also remains a key technology in the foreseeable future. Besides the generation of heat for conversion to secondary energy like thermal, mechanical or electric energy, the combustion also produces pollutants like oxides of nitrogen (NOx) or sulfur, which are responsible for acid rain and eﬀects on health. Pollutants like carbon dioxide are in the discussion to be responsible for global warming.

Figure 1.1: World energy demand since 1971 [168]; **Other: geothermal, solar, wind, heat, etc

Therefore, especially in the transportation sector, climate protection and saving resources has moved increasingly into public interests. Future automotive engines for example have to fulfill several requirements demanded by political conditions, social trends as well as the wish for competitive products in the global surroundings. Public discussions about climate change and lobbyists lead to increased public consciousness concerning costs and environment protection. Additionally the energy industry is dependent on the availability of fuel and thus is dependent on the few countries producing oil. In the foreseeable future the resources will decrease and the mineral oil output will reduce. The legislator also influences the public opinion and decisions as well as the requirements for automotive engines by introducing tax benefits, driving limitations, 2 CHAPTER 1. INTRODUCTION consumption and emission regulations. The development of the emission regulations for pollutants is given in table 1.1 for petrol and diesel [169].

Emission EURO 1 EURO 2 EURO 3 EURO 4 EURO 5 EURO 6 [g/km] 1993 1997 2000 2005 2010 2014 Petrol CO 3.16 2.2 2.3 1 1 1 HC+NOx 1.13 0.5 HC 0.2 0.1 0.1 0.1 NOx 0.15 0.08 0.06 0.06 Diesel CO 3.16 1 0.64 0.5 0.5 0.5 HC+NOx 1.13 0.7 0.56 0.3 0.23 0.17 NOx 0.5 0.25 0.18 0.8 Particles 0.18 0.08 0.05 0.025 0.005 0.005

Table 1.1: Limitations for pollutant emissions according to EU exhaust gas regulations

These tight regulations and the continued demand for powerful cars which are also environmentally friendly is the main challenge of today’s design of internal combustion engines. Combustion engines will also be in the future of great importance as the development of alternative driving concepts did not yet reach all the necessary requirements. For examples hybrid vehicles are very heavy and are too expensive. Vehicles with electric drive are also very heavy and additionally have a short range as the storage batteries are still in a development status. Furthermore the infrastruc- ture for these alternatives is missing. The use of these concepts is only possible in combination with commonly used internal combustion engines. So in order to reduce emissions and save resources, the efficiency of combustion processes has to be improved. As the development for internal combustion engines is already very mature, it is necessary to get a deeper understanding of these combustion processes. To be able to fulfill the limitations, complex systems are necessary with complex injection processes. This has resulted in the introduction of new variable fueling technology. Instead of a single injection in Diesel engines, multiple injections per cycle are now used. Variable valve timing technology, variable turbocharging systems or even systems that allow varying engine compression ratios are also options to control the combustion process. As fossil fuels face some disadvantages, like limited power dynamics and unsatisfactory power-weight ratios, the use of hydrogen for internal combustion (IC) engines offers promising solutions [60]. One benefit of this fuel is the reduced carbon dioxide emissions and negligible NOx formation, as the main combustion product is water vapor. Combustion anomalies such as pre-ignition, backfiring and knocking can be avoided, which allows compression ratios in the range of diesel engines, high turbocharging and and downsizing measures. Furthermore it can be produced from water and a renew- able energy source like solar or wind power. The cryogenic storage and supercharging systems provide the ability of a high power density of spark-ignition engines. In order to reduce development time, engine manufacturers are continuously search- ing improvements of the development process. Since in experimental configurations the optical access is limited, efficient modeling and simulation becomes more and more important. Simulations offer the advantage of supplying information that is not available from experiments due to e.g. missing optical access. Very complex laser diagnostics 3 are necessary to obtain detailed information. The simulation time improvements due to good parallelization and reliable model development together with increasing computer power, leads to its use in an early stage of the engine design process. Having such detailed information at this state already avoids the number of prototype parts. The simulation results can also be used to calibrate simpler low-dimensional models that in turn can be used for defining control algorithms. This work focuses on combustion model development for partially premixed flames. In order to benefit from the advantageous features of both, the premixed combustion where fuel and oxidizer are completely mixed before entering the combustion chamber, and the non premixed combustion where fuel and oxidizer enter separately, most technical applications work under partially premixed conditions. If fuel and oxidizer enter separately the combustion chamber, and mix due to turbulence such that a stratified mixture exist at the moment when it is ignited, it is called partially premixed combustion. In automotive applications, turbulent flame propagation in a stratified mixture occurs for example in diesel engines. Liquid fuel is injected directly into the cylinder in several injections, early enough in the cycle so that the liquid fuel is able to vaporize and partially mix with the air before the mixture is auto-ignited. In modern direct injection spark ignition engines also an inhomogeneous mixture is ignited. The charge is stratified such that the flame initiated at the spark propagates through a partially premixed inhomogeneous mixture. A further example is the hydrogen direct injection. It is injected into the cylinder when the valves are closed in the beginning or later in the compression stroke, where it starts mixing with the surrounding air until it is ignited. It may also still be injected during the combustion process by multiple injections. In all of the examples the fuel-air mixture in the cylinder is partially premixed at the time of ignition. The fuel in partially premixed regions is rapidly consumed then combustion takes place under non-premixed conditions. Partial premixing thus leads to situations in which flames have characteristics of both, premixed and non-premixed flames. In the case of local extinction of a diffusion flame, intermixing of unreacted fuel and oxidizer is possible. Thus these two combustion regimes of partially premixed and non-premixed flames have to be covered by the developed combustion model. Due to the new variable fueling technology, the engine got more flexibility. Com- bined with electronic control, the engine design and calibration process has become more difficult. In order to achieve even small improvements, the combustion process has to be determined more precisely. A simple model, like the Eddy Dissipation model which assumes one-step chemistry and fast chemistry, will no longer be able to ful- fil the demands in the future. Nevertheless, simulations can not require too much computational time in order to remain competitive. So the use of Direct Numerical Simulation (DNS) or Large Eddy Simulation (LES) tools, especially in the combination with combustion, is limited due to the computational cost and thus is still not suitable for the simulation of internal combustion engines. That is why this work focuses on the optimization of combustion models for direct injection internal combustion engines using the Reynolds Averaged Navier Stokes (RANS) approach. In this work, the Com- putational Fluid Dynamic (CFD) tool ANSYS CFX is used. The goal of this work is the development and implementation of the improved com- 4 CHAPTER 1. INTRODUCTION bustion model with respect to partially premixed flames. Furthermore, it is validated with experimental data and compared to existing models commonly used in industry. The validation also shows the advantages and disadvantages of the model. The work is structured as follows: Chapter 2 gives an overview of the fundamentals of turbulent reactive flows, where some characteristic variables and combustion regimes for non-premixed and partially premixed flames are presented. The variables influencing the laminar flame speed and flammability limits, necessary for the developed model, are also discussed. Chapter 3 presents the basic equations for turbulent flows. Due to limited computer performance, it is necessary to apply an averaging procedure to obtain the Reynolds averaged Navier Stokes equations. This chapter also addresses the turbulence models used within this work. The modeling of turbulent reactive flows is discussed in chapter 4, where a review of combustion models for turbulent non-premixed and premixed flames is given, and the used and newly introduced combustion models are briefly discussed. Within chapters 5, 6 and 7, three different flames are examined, a hydrogen jet flame, a methane and heptane jet flame and a methane low swirl burner. A detailed description of the experimental setup is given. Different sources for the laminar flame speed and flammability limits for hydrogen, methane and heptane are compared and correlations for the temperature and pressure dependent flammability limits are derived. Chapter 8 gives a conclusion and an overview of the main findings of this work and provides an outlook with recommendations for further work. Chapter 2 Fundamentals of Turbulent Reactive Flows

Turbulent reactive flows involve a wide range of different processes related to thermodynamics, chemical kinetics, fluid mechanics and transport processes. Concerning combustion, several chemical species react in multiple chemical reactions. In the following variables characterizing the reacting mixture are presented. Chapter 2.2 gives an overview of the different regimes that may occur in different applications. As within this work, special attention is paid on the laminar flame speed as well as on the flammability limits, these two parameters are described in separate chapters 2.3 and 2.4, respectively.

2.1 Characteristic Variables

The mole fraction Xi is deﬁned as the number of moles of species i in relation to the total number of moles in the mixture. It can be related as

W Xi = Yi (2.1) Wi

where W is the mean molar mass, Wi is the molar mass of species i and the mass fraction Yi is deﬁned as the ratio of the mass m of species i to the whole mass of the mixture m

m Y = i (2.2) i m

The total sum of all species mass fractions, respectively mole fractions, is equal to one. Species diffusion can be caused by concentration gradients, temperature gradients, also called Soret effect, or pressure gradients. In general, the latter two are neglected [155]. So the diffusion velocities Ui can be expressed by the Stefan-Maxwell equation [51]

Ns ∂Xi X XiXj = (2.3) ∂x D (U − U ) i j=1 ij j i

where the binary diffusion coefficient Dij is independent of the mixture composition. It can be seen in equation 2.3 that the diffusion velocity of a species depends on the 6 CHAPTER 2. FUNDAMENTALS concentration gradients of all other species, which makes the evaluation of this equation very CPU-hour-intensive. Later on in this work, this model is referred to under the name ‘complex’. To reduce the computational cost, the diffusion velocities can also be modeled by a Fick-like expression [99]

∂Yi UiYi = −Dim (2.4) ∂xi

with the mixture averaged diffusion coefficient Dim describing the diffusion of species i in the mixture. With the definition of the Lewis number

λ Lei = (2.5) ρDimcp the species diﬀusion ﬂuxes can also be expressed as

λ ∂Yi ρUiYi = − (2.6) Leicp ∂xi

where λ is the thermal conductivity of the mixture, ρ the density, and cp is the speciﬁc heat capacity at constant pressure of the mixture

N Xs cp = Yicpi (2.7) i=1

the heat capacities cpi are tabulated in polynomial form for various species. For more detailed information about solving equations for the diﬀusion velocities, please refer to [115]. If fuel and oxidizer are completely consumed, and only combustion products are formed, the combustion occurs under stoichiometric conditions. If there is still fuel available after the combustion process, it is called rich and in the case that there is still oxidizer it is called lean. The global stoichiometric reaction describing the combustion of any hydrocarbon fuel can be expressed as

0 0 00 00 νF CmHn + νO2 O2 → νCO2 CO2 + νH2O H2O (2.8)

0 0 0 00 00 where νF = 1, νO2 = (m + n/4)ν , νCO2 = m and νH2O = n/2 are the forward and backward stoichiometric coeﬃcients. 2.2. FLAME TYPES 7

The stoichiometric mass ratio s, which is deﬁned as the minimum mass of oxidant per unit mass of fuel needed for complete combustion, can then be calculated as

0 νO2 WO2 s = 0 (2.9) νF WF

The index F refers to fuel, and Wi is the corresponding molecular weight of species i. Additionally, the mixture of fuel and air can be described by the so called air excess ratio λ, which relates the fraction of fuel and air in a mixture to the stoichiometric mass ratio. Its reciprocal is called equivalence ratio φ. Both can be expressed as follows

1 Y Y 1 Y λ = = O2,u F,u = O2,u (2.10) φ (YO2,uYF,u)st s YF,u

The index u refers to the initial conditions in the unburnt mixture and the index st refers to stoichiometric conditions.

2.2 Flame Types

Depending on the provision of the reactants and on the combustion process, the flame types can be classified in two basic flame types, the premixed and the non-premixed flames. Premixed combustion occurs if fuel and oxidizer are mixed before they enter the reaction zone. In the non-premixed combustion regime fuel and oxidizer are injected separately from each other. In that case fuel and oxidizer are mixed during the combustion process. When flame-fronts occur in the transition of non-premixed and premixed combustion, the combustion process is called partially premixed.

Non-Premixed Flames In non-premixed flames, fuel and oxidizer are injected separately from each other into the combustion chamber. As non-premixed flames do not exhibit inherent propagation speeds they cannot flash back. That is why non-premixed injection is often used due to safety reasons. As diffusion is the rate controlling process in these flames, a common expression is also diffusion flame. The flame fronts are more complex than for premixed flames, as they cover the whole range of equivalence ratio from 0 for pure air to ∞ for pure fuel. Combustion takes place in the vicinity of stoichiometric mixture composition. Non-premixed flames can be found in early diesel engines, aircraft gas turbines or H2-LOx rocket combustion engines to name a few. The time needed for convection and diffusion, both being responsible for turbulent mixing, is typically much larger than the time needed for most of the combustion reactions to occur. So turbulent mixing and chemical reactions are the rate limiting processes. They are related through the dimensionless Damköhlernumber Da, where 8 CHAPTER 2. FUNDAMENTALS

the timescale for the dissipation of turbulent ﬂuctuations τt is related to a characteristic timescale τc for the chemical reaction system.

τ Da = t (2.11) τc

Poinsot et al. [124] introduced a regime diagram for non-premixed flames, presented in figure 2.1, depending on the Damköhler number and the turbulent Reynolds number Ret, which describes the ratio of turbulent inertial forces to molecular viscous forces

u0 l Re = t (2.12) t ν

0 where u is the velocity ﬂuctuation, lt the integral length scale and ν the ﬂow kinematic viscosity.

Figure 2.1: Regime diagram for non-premixed combustion [124]

For large Damköhlernumbers, i.e. for fast chemistry, the flame is very thin, and the reactive layer is thinner than the diffusion layer, which is assumed to be equal to the Kolmogorov size, which is the size of the smallest eddies. This region is defined as the Flamelet region as the flame occurs as laminar flame elements. The Flamelet region is bounded where the flame Damköhlernumber Dafl is equal to the Damköhler number of the steady laminar flamelet assumption DaLF A. The flame Damköhlernumber compares the molecular diffusion and chemical time scales. It is only a function of the local flame structure and does not take vortex time scales into account τ Dafl = f (2.13) τc

where τf , the ﬂow time scale, can be estimated with the scalar dissipation rateχ ˜st as 2.2. FLAME TYPES 9

−1 τf ≈ (˜χst) (2.14)

The Damköhlernumber can be related to the flame Damköhler number according to Poinsot [124] as follows

p fl Da ≈ 2 RetDa (2.15)

For a longer chemical time scale, the thickness of the reactive layer becomes equal to the Kolmogorov length scale. In this case, a departure from laminar flame structures and unsteady effects is expected. For large chemical scales and small Damköhlernum- bers extinction occurs. The regime of unsteady effects and extinction is divided where the flame Damköhler number is equal to the Damköhler number at extinction Daext. As the diffusion flame thickness ld is assumed to be equal to the Kolmogorov length 3/4 scale lη it follows lt/ld ≈ Ret . The regime of turbulent non-premixed flames has been discussed in several papers [18][33][92][118]. Compared to premixed flames it is difficult to classify a diffusion flame because the local flame scales, like local flame thickness and speed, depend on the local flow conditions such as local strain rates. So a second well established regime diagram derived by Peters [121] is presented in figure 2.2. He plots the ratio of the mixture p 002 fraction variance Zg and the diffusion thickness (4Z)F against the time scale ratio of the scalar dissipation rate at quenching χq to the conditional Favre mean scalar dissipation rateχ ˜st. The mixture fraction Z is a conserved scalar, which is 1 in the fuel stream, and 0 in the oxidizer stream and will be discussed briefly in chapter 4.2. The scalar dissipation rate χ can be interpreted as the inverse of a characteristic diffusion time. For χq > 1, three regions can be detected, the separated flamelet region, the con- χst nected flame zones and the connected reaction zones. p 002 The line Zg/(4Z)F = 1 separates two regimes, the separated flamelet and the connected flame zones. In the separated flamelet regime, the mixture fraction fluctuations are so large that they extend to sufficiently lean and rich mixtures. So the p 002 diffusion layers and thus the reaction zones are broken up. For Zg/(4Z)F < 1, the mixture fraction variance is too small to be able to break up the flame zones. In that case there might be intense mixing or partial premixing. p 002 Figure 2.2 shows an additional line with Zg/(4Z)R = 1, where the mixture fraction fluctuations are equal to the thickness of the reaction zone, which separates the connected flame zones and the connected reaction zones. Within the connected reaction zones regime, the fluctuations are smaller than the reaction zone thickness and are not able to disconnect the reaction zone. For values of χq/χ˜st < 1 the diagram shows the regime of flame extinction. 10 CHAPTER 2. FUNDAMENTALS

Figure 2.2: Regime diagram for non-premixed combustion [121]

Premixed Flames

In premixed flames, fuel and oxygen are already completely mixed before combustion takes place. Compared to diffusion flames, high temperatures occur and less soot is produced. The flame front separating the unburnt and the burnt gas state propagates due to heat and radical diffusion into the unburnt mixture. A characteristic quantity of premixed flames is the laminar flame speed sl, described in chapter 2.3. For example spark ignition engines or household burners operate under premixed conditions. Several researchers [3][17][125][120] introduced regime diagrams for turbulent premixed combustion, also. The classical regime diagram for premixed combustion has been introduced already in 1985 by Borghi [17]. A modification of this diagram has been published by Peters [120] in 1999. Since they are both very similar, only the regime diagram of Peters is shown in figure 2.3.

The abscissa marks the ratio lt/lfl as a measure of the size of the turbulent eddies 0 that interact with the flame. The ordinate depicts u /sl as a measure for the turbulence intensity of the flow. Under the condition that the diffusivities for all species are equal, and unity Schmidt number Sc = ν/D = 1, where D is the diffusion coefficient, the flame thickness lfl and the flame time τfl can be expressed as follows

D D lfl = , τfl = 2 (2.16) sl sl

Thus equation 2.12 and 2.11 for the turbulent Reynolds number and Damk¨ohler number respecitvely, can be rewritten to 2.2. FLAME TYPES 11

Figure 2.3: Regime diagram for premixed combustion [120]

0 u lt Ret = (2.17) sl lfl

sl lt Da = 0 (2.18) u lfl

Furthermore, two turbulent Karlovitz numbers are distinguishing features in the regime diagram for premixed combustion (figure 2.3). The first one is defined as the ratio of the flame time scale τfl to the Kolmogorov time scale

2 2 τfl lfl uη Ka = = = 2 (2.19) τη lη sl

where τη, lη and uη are the Kolmogorov time, length and velocity scales.

According to [121], the second Karlovitz number relates the inner layer thickness lδ to the Kolmogorov length scale

2 lδ 2 Kaδ = 2 = δ Ka (2.20) lη with δ being the nondimensional thickness of the inner layer. 12 CHAPTER 2. FUNDAMENTALS

The regime diagram can now be separated in ﬁve regions:

• Laminar flames: In this region, only laminar, flat flames occur. It is separated from the turbulent region by the line Ret = 1

• Wrinkled flamelets: 0 The wrinkled flamelet regime lies in the region of u /sl < 1 and Ret > 1, where the laminar burning velocity still dominates the turbulent velocity fluctuations. Therefore the turbulent eddies within the flow field are not able to deform the flame front and only small wrinkling of the flame front can be observed. Since in industrial applications a higher turbulence intensity is required to enhance the combustion process, this regime is not of practical interest.

• Corrugated flamelets: 0 u /sl ≥ 1 and Ka < 1 limit the region of corrugated flamelets. The smallest eddies of size lη are larger than the flame front thickness, and are thus able to corrugate the flame front. The flame is not perturbed by the turbulent fluctuations, and can be locally considered as quasi laminar.

• Thin reaction zones: V V The thin reaction zones regime is defined by Ret > 1 Ka > 1 Kaδ < 1. The condition Ka > 1 implies, that the smallest eddies of the Kolmogorov size lη are smaller than the laminar flame thickness lfl. So they are able to penetrate into the reactive-diffusive flame structure, but they are still larger than the inner layer thickness lδ, which results from the criterion Kaδ < 1. The turbulent eddies, penetrating into the preheat zone, enhance the transport of chemical species and temperature. As they are not able to penetrate into the inner layer, the chemical reactions are essentially not influenced by turbulence.

• Broken reaction zones: The last region is the broken reaction zone. It is only bounded by Kaδ ≥ 1. Eddies, smaller than the inner reaction zone, penetrate into this zone. Due to the enhanced heat loss towards the preheat zone there occurs local extinction of the ﬂame. Fuel and oxidizer are able to mix without chemical reaction.

Partially Premixed Flames In technical applications usually neither pure premixed nor pure non-premixed combustion occurs. In the main applications it is common practice to make use of the advantages of both and avoid their drawback, resulting in partially premixed combustion. They can avoid for example the high pollutant emissions of diffusion flames, while still operating under safe conditions. Partially premixed flames occur for example, when fuel and oxygen enter the combustion separately, and partially mix by turbulence such that combustion takes place in a stratified medium when it is ignited. Examples for partially premixed flames are aircraft gas turbines or direct injection internal combustion engines. This work focuses on the combustion model development 2.3. LAMINAR FLAME SPEED 13 for partially premixed flames. For example in the case of DI internal combustion engines, fuel enters the combustion chamber during the compression stroke and starts mixing with the air. When ignition occurs near top dead center, the fuel has partially mixed with the air and the flame propagates through a stratified mixture.

2.3 Laminar Flame Speed

As already mentioned in chapter 2.2, the laminar flame speed sl is the most important parameter governing the regime of premixed combustion. For partially premixed flames it is also of growing interest, as can be seen within this work. The laminar flame speed depends on the initial conditions of the mixture, its composition, its pressure and its temperature and is defined as the velocity at which the flame propagates normal to itself and relative to the flow into the unburnt mixture. The experimentally obtained laminar flame speeds of hydrogen, methane and heptane are shown in figures 5.8, 6.7 and 6.8 respectively, as well as the correlations derived from the experiments or chemical kinetics calculations. It can be seen that the values of the laminar flame speed show a substantial scatter. This can be attributed to the fact that no experimental set-up exists which is able to measure a one-dimensional, planar, adiabatic, steady, unstrained laminar flame for which sl is defined. Additionally some published experimental results did not take stretch effects into account (e.g. [75] or [107]). Stretch effects caused by curvature of strain rate have an influence on the flame temperature and thus on the laminar flame speed. For small to moderate stretch rates, this can be expressed as follows [43]

sl,s = sl,u − LK (2.21) where the indices s and u refer to the stretched and unstretched laminar flame speed sl respectively. L is the Markstein length and K the stretch rate. Depending on the sign of L, the laminar flame speed can be increased or decreased. If the flame is stable, the Markstein length is positive and the laminar flame speed is decreased. For a negative Markstein length the flame is unstable and quickly develops cellular structures. Stretch-free laminar flame speeds for hydrogen are presented by Verhelst [150]. He also showed that the stretched laminar hydrogen flame at ambient conditions has a higher laminar flame speed, which he addressed to thermodiffusively unstable flames or flame acceleration due to cellularity. A further difficulty is to obtain measurements at strain rates low enough, that a linear extrapolation to a strain rate of zero leads to accurate results. Tien et al. [145] showed that the linear extrapolation of the nonlinear dependence of the laminar flame speed on the strain rate may lead to over-predictions of sl by 5% to 15%. A further experimental uncertainty to mention is the flame cooling. In order to fix the flame at a certain position, the flame is often stabilized on a burner, which is only possible when the flow velocity on the unburnt side is smaller than the laminar flame speed. This is due to heat loss of the flame to the burner and does not fulfill the adiabatic condition any more. Additionally, depending on the measurement technique, researchers use different definitions of the laminar flame speed. It can be 14 CHAPTER 2. FUNDAMENTALS based on the entrainment velocity of the unburned mixture as follows

1 dmu sl = − (2.22) Aρu dt

where A is the flame front area, ρu the density of the unburnt mixture and the fraction dmu/dt is the rate of entrainment of the unburnt mixture into the flame front. The laminar flame speed can be also expressed by the rate of production of the reacted gas as

1 dmb sl = (2.23) Aρu dt

Although both equations 2.22 and 2.23 lead to the same result for the ideal case of a one-dimensional, stretch free, planar flame, they are different for non-planar flames due to the finite flame thickness. Within the recent years the calculation of 1D planar flames has also become a popular method to obtain the laminar flame speed. Calculated laminar flame speeds for hydrogen, methanol and hydrocarbon fuels have been presented by Peters et al. [122]. The influence of heat loss due to radiation has been addressed by Kennel et al. [83]. Recent publications by Verhelst [150] for hydrogen, or by Hermanns [71] for methane- hydrogen-air mixtures directly compared measurements and simulations.

2.4 Flammability Limits

The flammability limits are defined as the concentration limits, where a homogeneous mixture of a combustible gas and oxidizer is still able to self support a flame. Different experimental setups are presented in literature to measure the flammability limits. A summary of different combustible gases and liquids measured with different methods is given by Coward et al. [47]. Measurements of flammability limits are performed either in a spherical vessel or bomb, or in a tube which can be mounted horizontally or vertically. In the case of vertical mounting, the mixture in the tube can be ignited at the top or at the bottom. For a mixture with an upward propagating flame ignited at the bottom, the flammability limits are wider than those of a downward propagating flame ignited at the top. Jarosinski et al. [77] determined experimentally the lean limit extinction mechanism for upward and downward propagating flames, which is also described in detail by Van den Schoor [147]. In the case of downward propagating flames, the flame is extin- guished near the walls due to heat loss. Natural convection causes the cold combustion products to be carried ahead of the flame front until the mixture is diluted so much by the combustion products that the flame extinguishes. Due to gravity, the upward propagating flame is faster and has a convex shape towards the unburned mixture, than the downward propagating flame, which has a nearly flat shape. Therefore the heat loss is larger for a downward propagating flame causing earlier extinction which narrows the flammable range. For a horizontal mounting of the tube, the flammability 2.4. FLAMMABILITY LIMITS 15 limits lie in between the ones detected with the upward and downward propagating flame. The influence of the different experimental setups can be seen in Appendix B where the flammability limits for hydrogen, methane and heptane in air are listed for different setups. The flammability limits are not only influenced by the direction of flame propagation. If the size of the vessel becomes too small, for example when the radius of the tube is too small, the limits are narrowed as the walls reduce the flame temperature [141]. This phenomenon can also be seen in appendix B for the flammability limits of a horizontal propagating methane/air flame. Additionally the ignition energy must be high enough to initiate a flame. If it is too high, the flame propagates faster as a result of the heat provided by the ignition source and not the heat liberated by the flame itself. Thus the flammability limits are broadened. A further influence on the flammability limit is the definition of the criteria at which a flame is able to propagate or not. If the flame has to propagate over a distance of 0.1 m, the flammability limits are wider than for a flame which has to propagate over a distance of 1.5 m. An increase in temperature broadens the flammable range, as less heat has to be supplied from the burning layer. The flammability limits vary nearly linearly with temperature. Van den Schoor [147] determined three mechanisms that are responsible for flame extinction and four mechanisms of flame propagation which have an influence on the flammability limits.

Mechanisms that lead to flame extinction:

• Chemical kinetics: When the mixture fraction of a combustible mixture reaches the flammability limits, the flame temperature decreases, which has an effect on the chain branching and chain terminating reactions, whereas the chain terminating reactions are less sensitive on temperature. Thus the concentration of flame carriers is reduced which leads to flame extinction.

• Heat loss: When the heat loss to the surroundings becomes too large compared to the heat production within the reaction zone, the ﬂame might extinguish.

• Flame stretch: Increasing the stretch rate leads to a thinner ﬂame. The residence time reduces and only incomplete reaction is achieved. This lowers the ﬂame temperature which in turn leads to extinction.

Mechanisms of flame propagation:

• Flame stretch: The flame stretch has a strong effect on the laminar burning velocity and on the flame temperature, which in turn has an influence on the flammability limits. 16 CHAPTER 2. FUNDAMENTALS

• Natural convection: A flame kernel will be created, when a near-limit mixture in quiescent atmosphere is ignited. During the kernel growth, it is also moving upwards which causes flow strain. With increasing flow strain, the probability of flame extinction also increases. The effect of pressure on the flow strain is negligibly small, whereas with increasing temperature the flow strain is decreased.

• Flame front instabilities: For flames close to the flammability limits, the most important flame front instability is the diffusive-thermal instability, which leads to cellular flame fronts. These cellular structures have a smaller radius of curvature which in turn leads to a higher flame stretch, reducing the flammable range.

• Preferential diffusion: The effects of preferential diffusion becomes particularly important for cellular flames which occur close to the flammability limits. The reactant with the higher diffusivity reaches the cells more quickly, so the cells are enriched with the higher diffusive species which causes a shift in composition.

Up to now numerical methods still fail to predict the ﬂammability limits correctly [149][147]. Experimental data from literature are used within this work to derive correlations for the pressure and temperature dependent ﬂammability limits. Chapter 3 Basic Equations for Modeling of Non Reactive Flows

The governing equations describing the flow of a mixture are based on the assumption that the fluid is regarded as continuum. This means that the smallest volume element is large compared the mean molecular distance, and the mean free path is small compared to the characteristic length of the flow field.

3.1 Instantaneous Equations

The flow field of a mixture can be described by partial differential equations for the conservation of mass, momentum (also known as Navier-Stokes equations), species mass fraction of species k and energy. With Einsteins’s summation convention they can be written in Cartesian coordinates.

• Continuity equation ∂ρ ∂ + (ρuj) = 0 (3.1) ∂t ∂xj

• Momentum equation

∂ ∂ ∂p ∂τij (ρui) + (ρujui) = − + + ρgi (3.2) ∂t ∂xj ∂xj ∂xj

If the ﬂow is non-buoyant, the term ρgi = 0.

• Species mass fraction ∂ ∂ (ρYk) + (ρ(uj + Vk,j)Yk) =ω ˙ k (3.3) ∂t ∂xj

The source term due to chemical reaction is given here for the sake of complete- ness. It is Zero for non-reactive ﬂows, and will be discussed in detail in chapter 4.

• Energy equation

∂ ∂ ∂p ∂qj ∂ (ρH) + (ρujH) = − + (ujτij) + SE (3.4) ∂t ∂xj ∂t ∂xj ∂xj H is the total enthalpy, which is deﬁned as the sum of static enthalpy h and kinetic energy of the ﬂuid. qj the molecular transport of energy, and SE is a source or sink term for example due to radiative heat losses. 18 CHAPTER 3. BASIC EQUATIONS

In the above equations, ρ is the ﬂuid density, ui is the velocity in direction xi, p is the static pressure and τij the viscous stress tensor, which can be expressed for Newtonian ﬂuid as

∂ui ∂uj 2 ∂ul τij = µ + − δij (3.5) ∂xj ∂xi 3 ∂xl where µ is the laminar dynamic viscosity and δ denotes the Kronecker delta. The molecular transport of energy is deﬁned by:

K ∂T X ∂Yk qj = −λ + ρDkhk (3.6) ∂xj ∂xj k=1

λ is the conductivity, Dk the diﬀusion coeﬃcient and hk the enthalpy of species k.

To close the system of diﬀerential equations, the ideal gas law relates the density to the pressure and temperature T :

K X Yk p = ρRT = ρ

K X h = Ykhk (3.8) k=1

The speciﬁc enthalpy of a species k can be deﬁned as the sum of chemical (heat of formation given at a reference temperature T0) and sensible enthalpy

Z T 0 hk = ∆hf,k + cp,k dT (3.9) | {z } T0 chemical | {z } sensible

cp is the heat capacity at constant pressure. In the case of an ideal gas, it is only a function of the temperature. 3.2. TURBULENT FLOWS 19 3.2 Turbulent Flows

Turbulence is a three-dimensional, time-dependent, chaotic motion of vortices. Thus, a turbulent flow is characterized by a large spectrum of time and length scales. As turbulence occurs at high Reynolds numbers nearly all kinds of flows in nature and in technical applications are turbulent. Turbulence is irregular, dissipative and mixing intensive so that the transport of scalar properties is much faster than the molecular transport. The turbulent flow consists of movements with different lengthscales, in- teracting with each other. For the flow dynamic, the so called eddies represented by different dimensions, are important. The dimension of these eddies have an upper and a lower bound. The upper bound is given by the geometrical dimensions, the lower bound is given by the viscosity of the fluid. The viscosity of a fluid is able to dissipate vortices and velocity fluctuations only at very small lengthscales. Thus it avoids the formation of infinitely small eddies as the energy of very small eddies is transformed to heat. The smallest eddies to occur in a flow field are the so called Kolmogorov eddies with a length scale lη, a timescale τη and a velocity scale uη. With the viscosity ν and the dissipation rate ε they are defined as

ν3 1/4 l = (3.10) η ε ν 1/2 τ = (3.11) η ε 1/4 uη = (νε) (3.12)

For turbulence simulations, the Direct Numerical Simulation (DNS) approach is the most accurate, as the instantaneous transport equations (3.1 - 3.4) are solved directly without any modelling or averaging. Due to computer storage capacities and performance limitations, the application of DNS is restricted to small Reynolds numbers. The cells of the numerical grid have to have a size comparable to the Kolmogorov length scale of the ﬂow, to be able to resolve the smallest eddies. Introducing ε ≈ u3/l into equation 3.10, the ratio of the length scale can be estimated as:

3/4 3/4 l/lη ∼ (ul/ν) = Re (3.13)

Additionally, for turbulent reactive flows, the mesh has even to be fine enough to resolve the inner structure of the flame. Nowadays, DNS studies are only possible on small geometries and flows with a reasonably small Reynolds number, but it is a valuable tool for the understanding of turbulent flames as its resolution allows a deep analysis of isolated crucial combustion phenomena. An alternative approach is the Large Eddy Simulation (LES). Three dimensional, unsteady turbulent motions are directly simulated down to a certain dimension and the small eddies are modeled. Because the small turbulent structures are modeled, a 20 CHAPTER 3. BASIC EQUATIONS coarser mesh may be used which reduces the computational cost compared to DNS. Al- though the importance of this method concerning combustion is growing in the field of research, especially for cases with large scale unsteadiness like bluff body flows, strong swirl or unsteady sound generation, the numerical costs are still too high for practical applications. As the focus of this work is the simulation of direct injection, internal combustion engines, the Reynolds averaged Navier Stokes equations (RANS) is the most suitable method. All the turbulent structures are modeled, which allows an even coarser grid than for the LES approach. In most applications, the turbulent fluctuations are of less interest than the mean values of the flow field. With RANS it is possible to simulate statistically stationary flows. The main idea of this approach goes back to the averaging of Osborn Reynolds [133]. He suggested splitting all variables ψ into a mean ψ¯ and a fluctuating ψ0 value

ψ(x, t) = ψ¯(x, t) + ψ0(x, t) (3.14)

Forming the mean on the left and right side of this equation, the mean value of the ﬂuctuation disappears ψ0 = 0. Further rules for the Reynolds averaging method are

a¯ =a ¯

a + b =a ¯ + ¯b

ab¯ =a ¯¯b (3.15) ab =a ¯¯b + a0b0

∂a ∂ = a¯ ∂x ∂x

The Reynolds averaging has the disadvantage that it results in an unclosed quantity 0 0 0 of the form ρ ujui. Additionally, in several processes like combustion, large density ﬂuctuation occur, which makes the use of Favre averages, which is a density weighted averaging procedure, more suitable. It can be written as

ψ = ψe + ψ00 (3.16)

Again the mean value of the ﬂuctuation disappears ψf00, and the mean value is expressed with

ρψ ψe = (3.17) ρ¯ 3.2. TURBULENT FLOWS 21

Using the Favre strategy to average the ﬂow variables, the instantaneous balance equations 3.1 - 3.4 can be derived as follows:

• Continuity equation ∂ρ¯ ∂ = (¯ρu˜j) = 0 (3.18) ∂t ∂xj • Momentum equation

∂ ∂ ∂p¯ ∂τ¯ij ∂ 00 00 (¯ρu˜i) + (¯ρu˜ju˜i) = − + − ρ¯u]i uj +ρg ¯ i (3.19) ∂t ∂xj ∂xi ∂xj ∂xj • Species mass fraction

∂ ∂ ∂ 00 00 (¯ρYek) + (¯ρu˜jYek) = − Vk,jYkρ¯u]j Yk + ω˙ k (3.20) ∂t ∂xj ∂xj • Energy equation ∂ ∂p¯ ∂ (¯ρHe) − + (¯ρu˜jHe) = ∂t ∂t ∂xj K ! (3.21) ∂ ∂Te X ∂Yek 00 00 ∂ λ + ρD¯ khk − ρ¯u]j h + (ujτij) + SE ∂xj ∂xj ∂xj ∂xj k=1 On the right hand side of these equations, there occur unclosed terms due to the averaging procedure. The unknown quantities are the:

00 00 Reynolds Stresses u]i uj The Reynolds stresses arise from the non-linear convective term in the non-averaged equations. They are closed by the applied turbulence model (chapter 3.3), which ranges from a zero-equation Prandtl mixing length model to Reynolds Stress Models where transport equations for all components of the Reynolds stress tensor and the dissipation rate are solved.

00 00 00 00 Species u]j Yk and Enthalpy u]j h Turbulent Fluxes The turbulent ﬂuxes also result from the non-linear convective term of the un- averaged equation. The most common approach to close the turbulent ﬂuxes is the gradient transport assumption for the species:

00 00 µt ∂Yek ρ¯u]j Yk = − (3.22) Sct,k ∂xi and for the enthalpy: ˜ 00 00 µt ∂h ρ¯u]j h = − (3.23) Sct ∂xi 22 CHAPTER 3. BASIC EQUATIONS

where the turbulent viscosity µt is estimated from the turbulence model and Sct(,k) is the turbulent Schmidt number (of species k). This assumption is simple to implement and has the advantage to increases the stability of CFD codes. The diffusive contribu- tion is added to the laminar diffusion term. In a weak turbulent flame, counter-gradient transport may occur, where this assumption is not valid any more.

Species Chemical Reaction Rate ω¯˙ k The modeling of the mean reaction rate is discussed in chapter 4. It is the main goal of most turbulent ﬂame studies.

The laminar diﬀusive ﬂuxes for species or enthalpy as well as the pressure-velocity correlation are in general neglected.

3.3 Turbulence Models

In order to close the Reynolds stress terms of the averaged equations, a large number of turbulence models have been developed, where some selected models, especially the models used within this work, are presented in the following. They can be distinguished between eddy viscosity models and Reynolds stress models

Eddy Viscosity Models Within the eddy viscosity models, the Reynolds stresses are related to the mean velocity gradient using the Boussinesq hypothesis

00 00 ∂u˜i ∂u˜j 2 ∂u˜l 2 ρ¯u]i uj = −µt + − δij + ρk¯ (3.24) ∂xj ∂xi 3 ∂xl 3 where k denotes the turbulent kinetic energy, which is deﬁned as the variance of ﬂuc- tuations in velocity:

3 1 X k = u]00u00 (3.25) 2 i i i=1

The turbulent viscosity (or eddy viscosity) µt requires further modeling.

Zero- and one-equation models The Prandtl mixing length model [131] represents a zero-equation model. The turbulent viscosity, expressed by an algebraic equation, contains the velocity gradient. Therefore no additional transport equation has to be solved. 3.3. TURBULENCE MODELS 23

For the Prandtl-Kolmogorov one-equation model, a balance equation for the turbulent kinetic energy is solved. The Spalart-Allmaras [171] model solves a transport equation for the kinematic eddy viscosity. For the Prandtl mixing length model, and for the Prandtl-Kolmogorov model, empirical relations have to be given. They have no physical foundation and therefore they can not be applied universally.

Two-equation models The most widely used turbulence models are the two equation models, as they are a good compromise between computational accuracy and numerical eﬀort. For these models, two additional transport equations are solved. k − ε model The most popular two-equation model, which is still commonly used, is the k − ε model introduced by Jones and Launder [80]. The relation of turbulent kinetic energy k and dissipation ε to turbulent viscosity is given by the equation

k2 µ = C ρ¯ (3.26) t µ ε

where Cµ = 0.09 is a model constant. For the turbulent kinetic energy and dissipation rate, two additional transport equations are solved with the assumption of local isotropy in the ﬂow:

∂ ∂ ∂ µt ∂k (¯ρk) + (¯ρu˜ik) = µ + + Pk − ρε¯ (3.27) ∂t ∂xi ∂xi σk ∂xi

2 ∂ ∂ ∂ µt ∂ε ε ε (¯ρε) + (¯ρu˜iε) = µ + + Cε1 Pk − Cε2ρ¯ (3.28) ∂t ∂xi ∂xi σε ∂xi k k

where Cε1 = 1.44, Cε2 = 1.92, σk = 1.0 and σε = 1.3 are the default model constants. The turbulent production Pk due to viscous and buoyant forces is modeled as

00 00 ∂u˜i Pk = −ρ¯u]i uj (3.29) ∂xj

The k − ε model with the default model constants behaves fairly well for a wide range of wall-bounded and free shear flows. In addition it is very cheap, so it is still today a very popular model. But it is also well known to over-predict the spreading rate of axi-symmetrical jet flows. Flow separation is predicted too late and the amount of separation is underpredicted as a result. 24 CHAPTER 3. BASIC EQUATIONS k − ω model A further popular two-equation model is the k − ω model. Compared to the k − ε model, it does not involve the complex non-linear damping functions near the wall. Thus it is more accurate and more robust, especially concerning the prediction of flow separation. With this approach, the turbulent viscosity is expressed by the turbulent kinetic energy and the turbulent frequency ω as

k µ = ρ (3.30) t ω

Analogously to the k−ε model, two transport equations are solved for the turbulent kinetic energy and the turbulent frequency according to Wilcox [157]

∂ ∂ ∂ µt ∂k (¯ρk) + (¯ρu˜ik) = µ + + Pk − ρC¯ µkω (3.31) ∂t ∂xi ∂xi σk ∂xi

∂ ∂ ∂ µt ∂ω ω 2 (¯ρω) + (¯ρu˜iω) = µ + + Cω1 Pk − Cω2ρω¯ (3.32) ∂t ∂xi ∂xi σω ∂xi k

where the default model constants are σk = 2.0, σω = 2.9, Cω1 = 5/9 and Cω2 = 0.075. The calculation of the source term Pk is given in equation 3.29. Although this model behaves quite well for ﬂow separation, the dissipation in free shear ﬂows is underpredicted.

SST model To overcome the deficiencies of both, the k − ε model, which behaves well for free shear flows, and the k − ω model, which shows good performance in the near wall region, Menter introduced the SST model [100]. This model blends between the k − ω model near the wall and the k − ε model in the free stream flow. To combine these two models, the differential equations of the k − ε model are transformed to a k − ω formulation and a blending function F1 is introduced, where the Wilcox formulation is multiplied by F1 and the transformed k − ε formulation by (1 − F1). The modified transport equations are [170]

∂ ∂ ∂ µt ∂k (¯ρk) + (¯ρu˜ik) = µ + + Pk − ρC¯ µkω (3.33) ∂t ∂xi ∂xi σk3 ∂xi 3.3. TURBULENCE MODELS 25

∂ ∂ (¯ρω) + (¯ρu˜iω) = ∂t ∂xi ∂ µt ∂ω 1 ∂k ∂ω ω 2 µ + + (1 − F1)2¯ρ + CSST 1 Pk − CSST 2ρω¯ ∂xi σω3 ∂xi σω2ω ∂xi ∂xi k (3.34)

The coeﬃcients of this model are linear combinations of the corresponding coeﬃcients of k − ω model and the transformed k − ε model

Φ = F1Φk−ω + (1 − F1)Φtrans.k−ε (3.35)

To accounted for the transport of turbulent shear stress from which the overprediction of eddy-viscosity results, a limiter is introduced to the eddy-viscosity

a1k µt = ρ (3.36) max (a1ω, SF2)

where F2 is the second blending function restricting the limiter to the wall boundary layer, S is an invariant measure of the strain rate and a1 = 0.31 is a model constant.

Reynolds Stress Model The main disadvantage of the Boussinesque hypothesis is the assumption of isotropic turbulence. To overcome this drawback, a second category of turbulence models, the Reynolds stress models, was introduced. Either algebraic expressions or transport 00 00 equations are solved for the individual stress componentsρ ¯u]i uj and for the dissipation rate. Thus seven additional equations have to be solved in a 3D ﬂow simulation. Although the exact turbulence production term and the inherent modeling of stress anisotropies should make the Reynolds stress models more suited for complex ﬂows where sudden changes in mean strain rate occur, practice shows that they do not often give better predictions than the eddy viscosity transport equation and have the disadvantage of additional numerical cost and are less robust as more transport equations have to be solved. For the low swirl burner test case examined in chapter 7, the Baseline (BSL) Reynolds stress model is used. Similar to the SST model it blends between an ε- formulation in the outer region and an ω-formulation close to the wall. The equations for the Reynolds stresses can be written as follows:

∂ ∂ 2 0 ∂ µt ∂τij (ρτij) + (ulρτij) = −ρPij + β ρωkδij − ρΠij + µ + (3.37) ∂t ∂xl 3 ∂xl σ∗ ∂xl 26 CHAPTER 3. BASIC EQUATIONS

The ε based transformed transport equation results to

∂ ∂ (ρω) + (ρu˜iω) = ∂t ∂xi (3.38) ω 2 ∂ µt ∂ω 1 ∂k ∂ω α3 Pk − β3ρω + µ + + (1 − F1)2ρ k ∂xi σω3 ∂xi σ2ω ∂xi ∂xi

Here the coeﬃcients are also blends between the ε and ω based models. A detailed description of the models used within this work can be found in [170], and an overview of several turbulence models is given by Wilcox [157]. Chapter 4 Modeling of Turbulent Reactive Flows

Although it is possible to describe reacting flows with exact governing equations, it could already be seen in chapter 3 that even modeling of nonreactive flows with DNS is not possible for the simulation of reacting flows within practical devices. In addition to the transport equations for nonreactive flows, the transport equation for each species needs to be solved. As can be seen in figure 4.1 a wide range of turbulent and chemical timescales is involved in combustion modeling. The chemical processes occur on a molecular level and occur on a wide range of time scales, which span several orders of magnitude. Fast chemical processes occurring on a time scale of nano seconds correspond to reactions which reach equilibrium conditions, whereas NO-formation happens very slowly. So the chemical timescales span a much larger range than the time scales of the fluid flow like turbulent mixing time, diffusion or heat transfer, which depends on the flow conditions and geometrical dimensions. As a result, the coupling of turbulence and reaction makes the system of partial differential equations very stiff.

Figure 4.1: Physical and chemical timescales in turbulent combustion

To be able to simulate combustion in industrial applications, the Reynolds averaged Navier-Stokes equations are still the best choice. In these equations, the chemical source term appears in unclosed form and needs to be modeled. Depending on the way how the fuel and oxidizer are injected into the combustion chamber, several combustion models have been developed for nonpremixed and premixed flames. It is possible to include effects from both sides into a model for partially premixed flames. In the following, an overview of existing models is given for non-premixed and premixed flames. Furthermore, a more detailed description of the combustion models still commonly used in industry is given. These models are also used within this work for 28 CHAPTER 4. TURBULENT REACTIVE FLOWS comparison purposes. The proposed combustion model developed especially for partially premixed internal combustion engines is presented in section 4.4.

Models for Non-Premixed Combustion During the last decades, several non-premixed combustion models have been developed. A fundamental basis has been introduced by Burke and Schumann [35] in 1928. It is an analytical solution of the reacting structure of a diffusion flame, which is often called ’flame sheet model’. The assumption of this solution is infinitely fast and irreversible chemistry. The complete combustion process is described by a single step reaction. The flame front thickness is assumed to be so small that it may be treated as a geometrical surface. The reaction occurs when fuel and air are in stoichiometric proportion (mixed-is-burned). If there are no heat losses, the chemical equilibrium state is a function of mixture fraction. In this model infinitely fast and reversible chemistry is assumed. Especially at stoichiometry this assumption is different to the Burke-Schumann limit as dissociation of the products is accounted for. Therefore fuel and oxidizer can coexist, provided that the temperature is high enough. Most algorithms to calculate the chemical equilibrium are based on the minimization of Gibbs free energy, which is used to compute the species mole fraction. Examples for the equilibrium calculation of an H2/Air system can be found in [119] and [137]. The assumption of fast chemistry (infinite Damköhlernumber) is no longer valid with the Laminar Chemistry model, which is also called Finite Rate Chemistry model. Finite rate chemistry becomes important for finite Damköhlernumber. The reaction zone broadens around Z = Zst, the stoichiometric mixture fraction, which results in a more complex flame structure. In the Laminar Chemistry model, all conservation equations have to be considered. If the mixing time is much faster than the chemical reactions, the reactants can be considered as well mixed and the effect of turbulent fluctuations can be ignored. The finite chemical effects of a non-premixed coflow hydrogen-air diffusion flame have been examined by Magre et al. [98]. The Eddy Dissipation Model derived by Magnussen and Hjertager [97] is based on a single step reaction and fast chemistry assumption. As it is still industrial standard, it is also used within this work for comparison reasons. Therefore a detailed description can be found in chapter 4.1. The models mentioned up to now have the disadvantage that they do not prop- erly take the turbulence-chemistry-interaction into account. Thus several models have been introduced based on the presumed pdf (probability density function) approach of conserved scalars. The Flamelet Concept, described in detail in chapter 4.2, is based on this approach. A variety of models have been developed which are based on a presumed pdf for the mixture fraction and a transport equation for a reaction progress variable. The first to mention is the ”ModèleIntermittent Lagrangien” (MIL). This model is supposed to be valid in the thickened wrinkled flame regime and was first introduced by Borghi et al. [19] and improved by Gonzalez et al. [64] and Obounou et al. [114]. Borghi tried to simulate the interaction between turbulence and chemical reaction, particularly where chemical and turbulent timescales are in the same order of magnitude. For this, he takes the whole spectrum of turbulent timescales into account, instead of a single mean 29 timescale. This model assumes a high activation energy, which is applicable in combustion. The local instantaneous consumption rate is determined by the fuel mass fraction on the lean side, and by the oxidizer mass fraction on the rich side of the mixture. As mixture fraction pdf, a β-function is used, and for the pdf distribution of timescales an exponential function is assumed. A further model, which uses the presumed pdf for the mixture fraction and reaction progress variable models is the PEUL (probabilistic Euler Lagrangian) model [18][153]. The turbulent flow field is calculated with an Eulerian balance equation for the mean velocity, turbulent kinetic energy and dissipation rate. The Lagrangian tracking of a fluid particle only in composition space and not in physical space describes the chemical reaction. Schlatter [137] applied this method for hydrogen-air and methane-air jet flames. The approach of Conditional Moment Closure was independently proposed by Kli- menko [84] and Bilger [14]. Both give a detailed review of this model in [85]. The assumption behind this model is that most of the fluctuations in the scalar quantities can be associated with the fluctuations of a key quantity. Instead of using conventional averages, the reactive scalars are conditioned on the mixture fraction. For premixed combustion a conditioning on the progress variable is proposed [85]. Means or averages are the first moments, second moments are variances and co-variances of the fluctuations about the averages, and third moments are triple correlations between the fluctuations. Due to the conditioned description of the transport equations, the unclosed terms can be closed by modeling the higher level moments in terms of the lower moments. Roomina et al. [135] and Bradley et al. [26] applied this method to a methane-air jet flame. Pope [129] discusses in detail different pdf transport equation models, whereas Möbus[108] and Möbus et al. [109] examined the scalar and the joint pdf transport equation methods for a fast hydrogen jet flame. Lindstedt et al.[93] examined a methane-air jet flame with the joint pdf approach. Within these models, a transport equation for the pdf itself is solved. Compared to the presumed pdf method, the shape of the pdf is not restricted, and the reaction term is closed as it depends only on local quantities. Closure is needed for molecular mixing and turbulent convection terms into the pdf. The joint pdf approach includes the flow parameters and thermochemical parameters. In the composition pdf approach only the pdf of the thermochemical variables is solved for, the flow field is computed with RANS, using a turbulence model for closure.

Models for Premixed Combustion As for the non-premixed flames, several models have been developed for premixed combustion. The Eddy Break Up (EBU) model proposed by Spalding [142], closes the chemical source term of the species transport equation. It assumes a one-step reaction with infinitely fast chemistry, where turbulent mixing is the rate determining process, and an infinitely thin flame. The Bray-Moss-Libby (BML) model, introduced by Bray and Moss [32] and further improved by Bray et al. [29][31] is based on the assumption of a one-step irreversible reaction between fresh gases and combustion products. Further assumptions of this model are incompressible flow, constant chemical properties and unity Lewis number. 30 CHAPTER 4. TURBULENT REACTIVE FLOWS

Thus, the reaction progress can be described with a single scalar variable c, which is 0 for fresh gases and 1 for burnt gases. The pdf of the progress variable is assumed to consist of two delta functions, one at c = 0 and at c = 1. Additionally a Favre averaged transport equation for the mean progress variable is solved, neglecting molecular diffusion. Among several existing flame surface density models, namely the Cant, Pope and Bray (CPB) model [37], the Mantel and Borghi (MB) model [101], the Cheng and Diringer (CD) model [41] or the Choi and Huh (CH) model [42], the Coherent Flame Model (CFM) based on the ideas of Marble and Broadwell [102] is the most popular one. Under the flamelet assumption based on flame surface area, the main idea of all these models is to obtain the mean reaction rate from a product of the flame surface density and the local consumption rate per unit of flame area. Furthermore, a transport equation for the flame surface density is solved. The CFM model has been further improved by Trouvéet al. [146] and Colin et al. [45] to the Extended Coherent Flame Model (ECFM). An adaption of the ECFM model to hydrogen internal combustion engines has been presented by Knop et al. [87]. The mean chemical source term depends only on the turbulent time scale and thus operates in the condition of infinitely fast chemistry. The so called G-equation model describes the propagation of the flame front. The G-equation was first introduced by Williams [158]. This flamelet model is a kinematic approach based on a level-set formalism for the nonreactive scalar G instead of the reaction progress variable c. An iso-scalar surface which divides the flow field in two regions is defined as G(x, t) = G0. If the point x is located in the burnt gas, then: G > G0, if it is located in the fresh gas G < G0. Although the choice of G0 is arbitrary, it is fixed for a certain combustion simulation [140]. For RANS modeling, a transport equation for the Favre mean and the variance of G is solved. This model for premixed combustion is equivalent to the Flamelet concept for non-premixed combustion described in chapter 4.2, where a tansport equation for the mean and variance of the mixture fraction is solved. The G-equation model is described in detail in [121]. As well as for the non-premixed case, the basic ideas for pdf transport equation combustion modeling are the same for premixed combustion. This equation is solved using a Lagrangian particle tracking method called Monte-Carlo method. Although the Monte-Carlo method decreases the complexity of this approach, still a large number of particles per grid cell is needed, which makes it difficult to apply to industrial applications as it is very time consuming. A new Monte-Carlo joint PDF approach has been introduced by Brandt et al. [23][24][25], where the representative ensemble of particles is generated in a preprocessing step with a biased random number generator. A more detailed summary of non-premixed and premixed combustion models with respect to internal hydrogen combustion engines is given in [70]. 4.1. EDDY DISSIPATION MODEL 31 4.1 Eddy Dissipation Model

Still a popular combustion model in industrial applications is the Eddy Dissipation Model (EDM) derived already more than 30 years ago by Magnussen and Hjertager [97]. It is an extension to non-premixed combustion of the Eddy Break Up (EBU) model proposed by Spalding [142] and is based on the assumption of a single step reaction and fast chemistry. In addition to the common conservation and transport equations for cold ﬂows, a transport equation for each species has to be solved [124]

∂ (ρYj) ∂ (ρuiYj) ∂ ∂Yj + = ρDj +ω ˙ j (4.1) ∂t ∂xi ∂xi ∂xi

where ui is the velocity component in i direction, Dj is the diffusion coefficient, Yj the species mass fraction andω ˙ j is the reaction rate of species j. During the mixing process, the turbulent eddies follow a cascade process from the largest eddies down to molecular scales. The Eddy Dissipation Model assumes that fuel and oxygen are in separate eddies and the cascade process controls the rate of combustion. So the process which determines the reaction rate is the mixing process on the smallest length scale. As fuel and oxygen occur as fluctuating intermittent quantities, the fluctuations and the mean concentrations of the species can be related. The local rate of combustion is determined by the equation that gives the lowest reaction rate [124]

! ! 1 YeO YeP ε YeO YeP ρω˙ F = Cmagρ min YeF , , β ≈ Cmagρ min YeF , , β (4.2) τt ν 1 + ν k ν 1 + ν

ω˙ F fuel mean burning rate τt turbulent mixing time k turbulent kinetic energy ε dissipation rate Cmag, β model constants ν stoichiometric oxygen to fuel mass ratio

The indices F , O and P of the mass fraction Y denote fuel, oxidizer and products respectively. The rate of combustion is related to the turbulent mixing time scale and thus with the turbulent kinetic energy and the dissipation rate. Additionally the EDM limits the rate of combustion to the minimum mean concentration of reacting species. This model is still commonly used as it has the advantage that it can be implemented easily, needs low computational resources and is very robust, which is especially important in industrial applications with complicated geometries, complicated flow fields and large grids. But this model does not include any effects of chemical kinetics and it overestimates the reaction rate where the flow field is highly strained. Additionally, the results depend strongly on the model constants, which have to be fitted to each case and cannot easily be extrapolated to new operating conditions. 32 CHAPTER 4. TURBULENT REACTIVE FLOWS 4.2 The Laminar Flamelet Concept

In order to uncouple chemical kinetics from turbulence, Peters [116][117] introduced the laminar flamelet concept. It is based on the assumption that the flame structure of the local mixture fraction is determining the main characteristics of the combustion process. Thus a diffusion flame is considered to consist of an ensemble of laminar diffusion flamelets, which are stretched and wrinkled by the flow field. Due to this stretching and wrinkling, the molecular transport in the flamelet and thus the chemical reaction is affected. The reaction is assumed to take place within these flamelets in a thin reactive-diffusive layer. As long as this thin sheet, also called fuel consumption or inner layer, is smaller than the Kolmogorov eddies, it can be regarded as laminar. If the Kolmogorv eddies become smaller than the inner layer, they are able to destroy the structure. The location of the reaction layer is defined by the fuel consumption layer, which for simplicity is located at stoichiometric condition, so that the flame surface is defined as the surface of stoichiometric mixture fraction Z(x, t) = Zst. The mixture fraction Z is a conserved scalar. For a system with an oxidizer stream and a fuel stream, it can be defined by the mass fractions of fuel (index 1) and oxidizer (index2)

Y Z = 1 (4.3) Y1 + Y2

In the fuel stream, the mixture fraction is equal to 1, and in the oxidizer stream it is equal to 0. The stoichiometric mixture fraction is deﬁned by

−1 ν YF,1 Zst = 1 + (4.4) YO,2

YF,1 is the mass fraction of the fuel in the fuel stream, whereas YO,2 denotes the mass fraction of oxidizer in the oxidizer stream. The stoichiometric oxygen-to-fuel mass ratio is given by ν. The stoichiometric mixture fraction for H2/air is 0.0284, for methane/air 0.055 and for C3H8/air ﬂames it is 0.0601. According to [121], the mixture fraction can be related to the well known equivalence ratio φ as follows

Z 1 − Z φ = st (4.5) 1 − Z Zst

In order to take non equilibrium effects in the combustion model into account, Peters [116] introduced the flamelet equations, which reduces the three dimensional problem in physical space to a one dimensional in mixture fraction space. The detailed chemistry calculations done in a preprocessing step thus do not need to be coupled with the turbulent flame calculation. 4.2. THE LAMINAR FLAMELET CONCEPT 33

Figure 4.2: Surface of a stoichiometric mixture in a turbulent jet diﬀusion ﬂame [121]

As can be seen in figure 4.2, to obtain the flamelet equations, a coordinate transformation of Crocco type has to be performed using the mixture fraction as independent coordinate. At a definite time at least one coordinate, e.g. x1, has to be normal to the surface of stoichiometric mixture. The origin of the coordinate system can move together with the surface or independently, and has no influence on the results. For the coordinate transformation, x1 is replaced by the mixture fraction Z, whereas the other coordinates do not change

t, x1, x2, x2 → τ, Z, Z2,Z3 (4.6) with the transformation rules follows [116]

∂ ∂ ∂Z ∂ = + ∂t ∂τ ∂t ∂Z ∂ ∂Z ∂ = (4.7) ∂x1 ∂x1 ∂Z ∂ ∂ ∂Z ∂ = + (k = 2, 3) ∂xk ∂Zk ∂xk ∂Z

When x1 can be uniquely expressed by Z, the transformation is valid. Then laminar flamelets exist in the turbulent flow field. So the analysis can be performed within an infinite region around the reaction zone. The coordinate transformation results in the flamelet equations, where the species mass fraction Yi and temperature T are transformed into mixture fraction space. The flamelet equation for the species mass fraction can be written as [170]

2 ∂Yi 1 χ ∂ Yi ρ − ρ 2 =ω ˙ i (4.8) ∂t 2 Lei ∂Z 34 CHAPTER 4. TURBULENT REACTIVE FLOWS

The convective derivatives have disappeared. The ﬂamelet equations for the temperature follows to

5 3 1 2 4 z }| { z}|{ z }| { z }|2 { z }| { N ∂T 1 ∂p χ ∂ T χ ∂T ∂cp X 1 χ cp,i ∂Yi ∂T ρ − − ρ − ρ − ρ ∂t c ∂T 2 ∂Z2 2c ∂Z ∂Z 2 Le c ∂Z ∂Z p p i i p (4.9) N ˙ 1 X q˙R hchem + m˙ h − = c i i c c p i p p | {z } |{z} | {z } 6 7 8

The individual terms of this equation have the following meaning:

1. instationary temperature change

2. instationary pressure change

3. heat diffusion with scalar dissipation rate as diffusion coefficient

4. convection in mixture fraction space

5. convection in mixture fraction space

6. chemical source term

7. radiation

8. enthalpy per unit time

The terms 4 and 5 in equation 4.9 describe the convection in mixture fraction space, which results from differential diffusion in physical space between temperature and mixture fraction. N is the number of chemical species, ρ the density,q ˙R the radiative heat loss, cp,i the heat capacities of species i at constant pressure,ω ˙ i the chemical production rates hi the enthalpies and Lei the Lewis number of species i. The Lewis number is defined as

λ Lei = (4.10) ρcpDim

with the heat conductivity of the mixture λ and the molecular diffusion coefficient Dim. The instantaneous scalar dissipation rate χ [1/sec], which is an essentially non- equilibrium parameter and can be interpreted as the inverse of a characteristic diffusion time, couples the flamelet equations with the flow field in physical space 4.2. THE LAMINAR FLAMELET CONCEPT 35

∂Z 2 χ = 2DZ (4.11) ∂xj

with the thermal diffusivity DZ . The influence of convection and diffusion, normal to the surface of stoichiometric mixture fraction due to the transformation, is also included in the scalar dissipation rate.

As can be seen in appendix D, where the temperature distribution for H2/air and methane/air flames as function of the mixture fraction is shown for different scalar dissipation rates, the maximum flame temperature reaches the adiabatic flame temperature for χ → 0. When the scalar dissipation rate increases, the solution is more and more influenced by finite rate chemistry and the increasing heat loss across the flame sheet. In general when flamelet models are used in combustion simulations, the pre- integrated results are stored as a pre processing step in flamelet libraries. Therefore, they are not dependent on the flow field, and the scalar dissipation rate χ(Z) needs modeling before its use within the flamelet equations. According to [116] it can be approximated as a function of mixture fraction diffusivity

a χ = exp{−2[erfc−1(2Z)]2} (4.12) π with the assumption of constant density and diﬀusivity. erfc is the complementary error function. As CFD codes provide only an unconditioned, averaged scalar dissipation rate, the above equation can also be expressed as follows

exp{−2[erfc−1(2Z)]2} f(Z) χ(Z) = χ(Z ) ≡ χ(Z ) (4.13) st −1 2 st exp{−2[erfc (2Zst)] } f(Zst)

For this equation the deﬁnition of the mean scalar dissipation rate at stoichiometric mixture χst is needed, and can be calculated as

ε˜ 002 Cχ Zg f(Zst) χ(Z ) = k˜ (4.14) e st R 1 ˜ 0 f(Z)P (Z) dZ

˜ where Cχ = 2 and P (Z) is the Favre averaged probability density function. A further approach to obtain flamelet tables for a combustion simulation is to eval- uate a laminar, counterflow diffusion flame (see figure 4.3). With this geometrical setup it is possible to obtain a one-dimensional diffusion flame structure, which can also be mapped into mixture fraction space. Within this work, the code CHEM1D [172] is used for this purpose.

In the counterflow configuration, the flame is stabilized near the stagnation plane and can be at first considered as two-dimensional planar or axially symmetric. In the 36 CHAPTER 4. TURBULENT REACTIVE FLOWS

Figure 4.3: Counterflow diffusion flame following u is the velocity in x or radial direction, and v is the velocity in y or axial direction. With the ansatz u = Ux, the velocity u is related to the gradient U of the velocity u and the direction x. Therefore, the governing equations can be expressed as follows: Continuity d(ρv) + (j + 1)ρU = 0 (4.15) dy

Momentum dU d dU ρv = −ρU 2 + ρ a2 + µ (4.16) dy ∞ dy dy

Mixture fraction dZ d dZ ρv = ρD (4.17) dy dy dy

Reactive scalars dψ d dψ ρv i = ρD i + ω (4.18) dy dy i dy i

For planar configurations j = 0, and for axially symmetric configurations j = 1. a is the strain rate. The scalar dissipation rate is already given in equation 4.11. With the assumption of sufficiently large velocities, far away from the stagnation plane, a potential flow may be assumed. Thus the boundary conditions for the oxidizer stream (y → ∞) are v∞ = −ay, U∞ = a and Z = 0. For the fuel side (y → −∞) 1/2 the boundary conditions are v−∞ = −(ρ∞/ρ−∞) ay, U−∞ = aρ∞/ρ−∞ and Z = 1. A 4.2. THE LAMINAR FLAMELET CONCEPT 37 detailed description of the laminar counterflow flame can also be found in [121]. Within the flamelet generation code implemented in ANSYS CFX 11 [170], only a single value for the Lewis number of the single species may be provided by the user, which leads in most cases to the assumption of unity Lewis number for all species. The CHEM1D code has the advantage, that it also accounts for the species differential diffusion. Three different models are available. A Lewis number can be provided for each species according to ANSYS CFX 11. The ’Mixtureaveraged’ model and the ’Complex’ model calculate the diffusion velocities based on assumption described in chapter 2. The Favre mean species mass fraction can then be calculated with a probability density function P˜

Z 1 ˜ ˜ Yi = Yi(Z)P (Z)dZ (4.19) 0

Commonly a β-pdf shape is used for P˜(Z, x, t). The pre-integration of flamelet tables is done with CFX-RIF [170] automatically. The tables which are generated with CHEM1D [172] still have to be pre-integrated, which is done within this work with the calculation of the β-pdf function described in detail in appendix A. Within the framework of the flamelet model, instead of solving the transport equation for each species, only the transport equations for the Favre mean mixture fraction and its variance are solved. Assuming that all species have equal diffusivities, the transport equation for the Favre mean mixture fraction can be written as [170]

" # ∂(¯ρZe) ∂(¯ρu˜jZe) ∂ µt ∂Ze + = µ¯ + (4.20) ∂t ∂xj ∂xj σZ ∂xj

where µ is the molecular dynamic viscosity, µt the turbulent viscosity and σZ is a model coeﬃcient. As the mixture fraction is a conserved scalar, its transport equation contains no source term. The transport equation for the mixture fraction variance Zg002 can be modeled as follows [170]

2 002 002 " 002 # ! ∂(¯ρZg) ∂(¯ρu˜jZg) ∂ µt ∂Zg µt ∂Ze + = µ¯ + + 2 − ρ¯χ (4.21) ∂t ∂x ∂x σ ∂x σ ∂x e j j Zg002 j Z j where σ and C are model coeﬃcients. The ﬁrst term on the right hand side rep- Zg002 χ resents the production of variance, whereas the last term models its dissipation. The instantaneous scalar dissipation rate χe is modeled as

ε˜ 002 χ = Cχ Zg (4.22) e k˜ 38 CHAPTER 4. TURBULENT REACTIVE FLOWS withε ˜ and k˜ being the dissipation rate and the turbulent kinetic energy, respectively. ˜ As the species mass fractions Yi are previously stored in a flamelet table, they can be obtained within the CFD simulation by coupling the CFD code with the flamelet libraries as shown in figure 4.4, adapted from [121]

Figure 4.4: Code structure of the ﬂamelet model

The properties Favre mean mixture fraction, mixture fraction variance and scalar dissipation rate are calculated from the CFD code. From these values, the pre- integrated mean values for the species mass fractions are then obtained by look-up in the ﬂamelet library.

4.3 The Reaction Progress Variable

In addition to the conservation equations together with the equations of state, the transport equation for the Favre mean mixture fractions and its variance, the transport equation for the Favre mean progress variablec ˜ has to be solved for the improved combustion model presented in chapter 4.4. Similar ideas have, for example, been used by Domingo et al. [52] for partially premixed flamelets in a Large Eddy Simulation (LES) for non-premixed lifted flames and a similar model where the flamelet model is coupled with a pdf transport equation model has been used by Demiraydin et al. [50] for turbulent non-premixed methane/air flames. The progress variable is defined as the normalized mass fraction of products and describes the progress of reaction within a flamelet. By definition c = 1 for completely burned mixtures and c = 0 for unburned mixtures. ˜ The mean species mass fraction Yi can be approximated as

Yei = (1 − c˜) Yei,u +c ˜Yei,b (4.23) where the index u and b denote the unburned and burned values of the species mass fraction. The species mass fraction of the burned state Yei,b is derived from the ﬂamelet model in CFX. The mass fraction for the unburned state can be obtained by linear blending of fuel and oxidizer composition. 4.3. THE REACTION PROGRESS VARIABLE 39

The transport equation of the reaction progress variable has the following form [28]

∂ (¯ρc˜) ∂ (¯ρu˜jc˜) ∂ 00 00 + = − ρ¯ugj c + w˙c (4.24) ∂t¯ ∂xj ∂xj

00 00 whereas the turbulent scalar transport termρ ¯ugj c is modeled with a gradient transport assumption:

∂ (¯ρc˜) ∂ (¯ρu˜jc˜) ∂ νt ∂c˜ + = − ρ¯ · + w˙c (4.25) ∂t¯ ∂xj ∂xj Sct ∂xj

Sct = 0.9 is the turbulent Schmidt number and νt is the turbulent viscosity.

To close the source term w˙c in equation 4.25, several models are examined within this work, which are presented in the following.

Zimont The turbulent flame speed closure model was first introduced by Zimont [164] and further improved by Zimont et al. [166][165]. The model assumes that the combustion front consumes fuel at a turbulent flame speed st. The flame brush width of the combustion front increases in time and space. This is the so called Intermediate Steady Propagation (ISP) combustion regime. The source termw ˙c of the transport equation for the reaction progress variable (equation 4.25) is modeled by

w˙c = ρust|∇c˜| (4.26)

where ρu is the density of the unburned, unreacted mixture. The turbulent ﬂame speed is then deﬁned according to [165] as

03/4 1/2 −1/4 1/4 st = AGu sl au lt (4.27) where A is a model constant (for hydrogen combustion it is equal to 0.61) and u0 is the turbulent velocity intensity. The turbulent integral length scale lt can be obtained 3/2 from the turbulent kinetic energy and its dissipation rate lt = k /ε. au is the thermal diffusivity of the unreacted mixture. Wrinkling and straining of the flame front caused by the turbulent velocity field are included in the expression for the turbulent flame speed, which is also dependent on changes in operating pressure, fuel concentration, composition and temperature in a physically meaningful way. The stretch factor G models the extinction and accounts for reduction of the flame velocity due to large strain rate. It can be obtained by integrating the log-normal distribution of the turbulent dissipation rate [164] 40 CHAPTER 4. TURBULENT REACTIVE FLOWS

( ) 1 r 1 ε σ G = erfc − ln cr + (4.28) 2 2σ εe 2

In this equation σ is the standard deviation of the turbulent dissipation rate ε

lt σ = µstrln (4.29) lη

where the default value of the model constant µstr is 0.28.

The critical dissipation rate εcr can be obtained from the kinematic viscosity ν and the critical velocity gradient gcr

2 εcr = 15νgcr (4.30)

If the velocity gradient is smaller than this critical value, differently strained flamelets are reduced to unstrained ones, where no quenching occurs. If it is larger, the flamelets are highly strained. For ε > εcr extinction of the flamelets occurs. The model has been successfully applied to a hydrogen/air jet diffusion flame by Zimmermann et al. [163][154].

Bradley As second correlation for the turbulent flame speed, the correlation of Bradley et al. [27] was examined. It is based on a large number of experiments performed to measure the turbulent flame speed for different fuels and conditions, which are presented by Abdel-Gayed et al. [4]. These measurements were performed using the double-kernel technique, where two kernels are ignited at the same time and meet each other in homogeneous, isotropic stationary turbulence. A Schlieren technique is used to detect the flame kernels. The turbulent flame speed is derived from the time span until the kernels collide. Bradley et al.[27] validated their correlation in the range of 0.01 < Ka Le < 0.63, where Ka is the Karlovitz stretch factor, for which they obtained the following equation

st −0.3 0 = 0.88(Ka Le) (4.31) uk

0 uk is the eﬀective cold mixture root mean square (r.m.s.) turbulent velocity acting on a ﬂame. 4.4. PARTIALLY PREMIXED FLAMES 41

For the Karlovitz number, the deﬁnition derived by Abdel-Gayed et al. [1][2] is used

0 2 u −0.5 Ka = 0.157 Ret (4.32) sl

When time approaches infinity t → ∞, the effective cold mixture r.m.s. turbulent velocity can be approximated as u0 [95], which results in the following expression for the turbulent flame speed

−0.3 00.55 0.6 0.15 −0.15 st = C Le u sl lt ν (4.33) with C = 0.88 · 0.157−0.3.

Lindstedt-Vaos´ Lindstedt and Váos [94] propose to model the source term of equation 4.25, the transport equation for the reaction progress variable, using a correlation for the flame surface density Σ, based on an assumption of fractal dimension of the flame surface

3/4 sl ε˜ w˙c = ρustΣ = CRρu c˜(1 − c˜) (4.34) ν1/4 k˜

Depending on the experimental configuration, the model constant CR lies between 1.25 and 1.95. For example, the model constant with a value of 1.25 is adapted to the database of Abdel-Gayed [2], whereas the value of 1.95 is correlated to the database of Gulati et al. [68]. Lindstedt and Váos [94] tuned the constant to 2.6. Bray et al. [30] investigated a methane flame with promising results although a 0 higher value of CR was needed to fit the data at high u . Brandl et al. [24] also successfully applied this model at higher pressures, where he modeled the turbulent scalar 00 00 transport termρ ¯ugj c with a first order gradient transport assumption.

4.4 Combustion Model for Partially Premixed Flames

In a direct injection internal combustion engine for automotive applications, cold fuel is injected into the cylinder during the compression stroke, often in multiple injections. Finally the mixture is ignited. The compression causes a rise of pressure and temperature of the mixture, which results in a temperature diﬀerence between pure air, fuel/air mixture and injected (cold) hydrogen. Thus, the inﬂuence of temperature and pressure of the unreacted mixture is very important and has to be taken into account 42 CHAPTER 4. TURBULENT REACTIVE FLOWS

in the flamelet tables. The effect on combustion products like H2O or CO2 or radicals like O2 and OH are demonstrated for a hydrogen/air flame in chapter 5.2, figure 5.4 and for a methane/air flame in chapter 6.2, figure 6.3. Although the mass fractions of the radicals are small, they have a strong influence on the pollutant prediction. In the models to close the source term of the reaction progress variable transport equation, the laminar flame speed sl occurs within all mentioned correlations (Zimont, Bradley and Lindstedt-Váos). As could be seen in chapter 2.3, and can be seen for hydrogen, methane and heptane combustion in chapters 5.4 and 6.4, the laminar flame speed is also strongly dependent on temperature and pressure, as well as on the mixture fraction or equivalence ratio, respectively. As the laminar flame speed was especially used in the past for premixed combustion, only a constant laminar flame speed was generally applied. With the extended use for partially premixed flames, also the dependence on mixture fraction is applied successfully in [161][162][163][154][136]. To describe the combustion process in a direct injection gasoline engine, Kech et al. [81] used a similar approach as presented in this work. It is also based on a combination of mixture fraction and reaction progress variable. The laminar flame speed was calculated in dependence on the mean mixture fraction and its variance as follows:

2/3 s¯ = s P Z, Z002 C (4.35) l(Z˜) l,Zst Zst e g Zst

where CZst = 0.9. This is only an approximation of the detailed description presented below. Polifke et al. [126] developed an extended turbulent flame speed closure model for inhomogeneously premixed combustion, where the influence of model parameters on mixture fraction is examined in detail. Although the laminar flame speed covered the whole mixture fraction range also occuring in non premixed combustion, it is only dependent on the mean mixture fraction, and not on its variance. In the framework of LES, Domingo et al. [52] showed that equation 4.25 neglects terms expressing fluxes in mixture fraction space. This approach was also picked up by Duwig et al. [55] and Durand [54]. The right hand side of equation 4.25 then contains two extra terms. The first term contains the scalar dissipation rate and represents the mixing effect. It can be neglected when the fuel mass fraction in the burned state is a linear function of the mixture fraction. In the model considered in this work, this is only the case near stoichiometry and thus can be neglected compared to the other terms. The second term is the cross-scalar dissipation rate defined between the progress variable and the mixture fraction. When the length scale of the mixture fraction variation is large compared to the flame thickness, this term can also be neglected, which was done in this work. The mixture fraction dependence of the laminar flame speed can be obtained by two options, either by previously calculating the laminar flame speed for different mixture fractions, temperatures and pressures with a chemistry code like CHEMKIN [82] or CHEM1D [172], or by using one of the several published correlations available for different kinds of fuel [107][86][111]. Using a chemistry code has the disadvantage that a large range of mixture fraction, pressure and temperature combinations has to be calculated in a previous step for internal combustion engines, which is very time consuming. Due to the analytical expressions of the laminar flame speed correlations, this dependence can be accounted for very quickly. But all expressions have the dis- 4.4. PARTIALLY PREMIXED FLAMES 43 advantage, as they are developed for premixed combustion cases, that they are not valid in the whole mixture fraction range for diffusion flames with 0 ≤ Z ≤ 1. For non-premixed flames Z = 1 for fuels, and Z = 0 for the oxidizer. To extend the range of validity of the different correlations for laminar flame speed, within this work, the last valid points of the flame speed correlation on the lean or rich side, respectively, are connected linearly with the lower or upper flammability limits. Outside the flammability limits the laminar flame speed is so small that the flame is quenched immediately, as soon as a perturbation occurs. So a value of 0 m/s can be assumed for the laminar flame speed outside the flammable range. The previous optimizations widen the range of validity of any correlation for the laminar flame speed to the whole range occurring in diffusion flames. The correlations are dependent on pressure, temperature and equivalence ratio (or mixture fraction). As commercial CFD codes solve time-averaged equations, only the Favre averaged mean mixture fraction and its variance are known. The time-averaged source term in the transport equation for the reaction progress variable in partially premixed combustion contains a mixture fraction dependence within the laminar flame speed. Additionally to take the turbulence effects on the time averaged source term into account, a probability density function (pdf) P is used. The resulting effective laminar flame speed eZ,e Zg002 used in the correlations is defined by:

Z s˜ (Z, Z002) = s (Z)P (Z; x, t) dZ (4.36) l e g l eZ,e Zg002

Thus the effective laminar flame speed also depends on the Favre averaged mixture fraction and its variance. To be able to use the effective laminar flame speed efficiently within a CFD code, a table has to be generated for the laminar flame speed depending on pressure, temperature, Favre mean mixture fraction and mixture fraction variance. Examples of the laminar flame speed correlation extension and table generation are given in chapter 5 for hydrogen and in chapter 6 for methane and heptane combustion. It is common practice in non-premixed combustion simulations to use a β-pdf shape [76]

Γ(a + b) P˜(Z; x, t) = Za−1(1 − Z)b−1 (4.37) Γ(a)Γ(b)

where Γ is the gamma function, and α and β can be expressed by the mean mixture fraction and its variance. The implemented β-pdf approach within this work is explained in detail in appendix A. Chapter 5

H2 Jet Flame

For evaluation purposes, the ’H2 Jet Flame’ of the TNF workshop [167] has been selected as a test case for the non-premixed hydrogen diffusion flame model. It offers the advantage of well defined boundary conditions and a relatively simple flow field. The hydrogen jet flame has been experimentally examined by Barlow and Carter [10][8][9] and Flury [62] with a helium dilution of 0 %, 20 % and 40 %, which offers a good data set for comparisons with numerical methods. Thus it has been used by several researchers for model evaluation in the framework of Large Eddy Simulations (LES) and RANS simulations. Near the injection nozzle, the flame is characterized by thin mixing and reaction layers, whereas downstream the burnt gases occupy a large region of the domain and the reaction layer is broad with near equilibrium regions at the flame tip. Due to the relatively fast combustion chemistry of hydrogen, there is no local extinction at the injection velocities considered. Additionally, there occurs no soot formation.

Figure 5.1: Hydrogen Jet Flame

The following chapters give an overview of the computational domain and boundary conditions derived from the experimental setup(chapter 5.1), followed by exami- nation of the influence of the Lewis number (chapter 5.2), the comparison and choice 5.1. BOUNDARY CONDITIONS 45 of a suitable turbulence model (chapter 5.3), and the influence of the laminar flame speed (chapter 5.4). The results, where also different closure models are compared, are discussed in chapter 5.5. The presented simulations are performed using the CFD (computational fluid dynamics) tool ANSYS CFX [170].

5.1 Computational Domain and Boundary Conditions

The hydrogen jet in both configurations is injected from a central tube with an inner diameter of 3.75 mm and a wall thickness of 0.55 mm. In the case of Barlow and Carter [9] the burner was centered at the exit of a vertical prismatic wind tunnel contraction with a cross sectional area of 0.3 m × 0.3 m. The air coflow at the experimental setup of Flury [62] is surrounded by a vertical wind tunnel with a hexagonal base and a diameter of 0.6 m. A comparison of both setups with LDV (Laser Doppler Velocimetry) measurements has shown no difference [62]. The radial positions where experimental data are available are marked in red in figure 5.1. Using the CFD program CFX, it is not possible to solve 2D flow fields. To be able to solve a ’quasi 2D’ case, a plane with the dimension 0-150 mm in radial direction and 0-800 mm in axial direction has been rotated by three degrees to create a 3D case. To be able to use a structured grid, the axis has been replaced by a small tube (diameter 0.02 mm). Calculations with more cells in the third direction showed negligible differences, so a width of one cell in circumferential direction is sufficient and reduces calculation time. In the domain with 156,000 nodes and 236,193 elements, the grid has been refined near the visible flame. A grid refinement study showed that the solution is grid independent. A symmetry boundary condition has been chosen for the rotated surfaces. At the “axis tube” and the radial boundary an adiabatic no slip wall has been placed. The air coflow has a velocity of 1 m/s initialized with a constant profile, where the mean velocity varies over the radius by about 1.3 % and the turbulence intensity is about 10 %. The inlet temperature is set to 298 K. Figure 5.2(a) shows the normalized velocity profile measured with 20 % and 40 % helium dilution. Using pure hydrogen, no measurements of the velocity profile exist, since it was not possible to seed the fuel due to security reasons [137]. The velocity was normalized in both cases with the maximum velocity Umax measured. In the case of a dilution of 20 % helium, the maximum jet velocity is 302.49 m/s, with a dilution of 40 % helium the maximum jet velocity is 263.21 m/s. The Reynolds number of the hydrogen jet is 10,000. Figure 5.2(a) shows an additional velocity inlet profile calculated with the power law

1 r n u(r) = 1 − U (5.1) R max

1 A fully turbulent velocity proﬁle can be obtained by the so called 7 -rule, where n = 7, but the proﬁle representing the measurements is obtained with n = 12 [113], so the 46 CHAPTER 5. H2 JET FLAME

(a) Normalized radial inlet velocity (b) Turbulent kinetic energy

Figure 5.2: Profiles of velocity and turbulent kinetic energy for different dilutions of Helium, 2 20 % Helium dilution, 4 40 % Helium dilution, - power law with n = 12 measured profile does not represent fully developed turbulent tube flow. The measured inlet profile for the turbulent kinetic energy of the radial inlet profile with a dilution of 20 % and 40 % Helium is shown in figure 5.2(b). The turbulent kinetic energy can be expressed as

1 k ≡ u0 u0 (5.2) 2 i i

For the dissipation rate ε no measurements are available. To determine the relationship between Taylor and Kolmogorov scales, Pope [130] deﬁnes the lengthscale L characterizing the large eddies as

k3/2 L ≡ (5.3) ε

CFX [170] advises to use 30 % of the inner diameter of the tube as the choice of the length scale yielding L = 0.3 · 3.75mm. With this assumption, the dissipation rate ε can be evaluated. Calculations with the Eddy Dissipation Model and different length scales, ranging from 1L, 0.5L and 0.3L, for the calculation of the dissipation rate, used as inlet boundary condition, show nearly no difference in the behavior of the achieved results (see figure 5.3). This has also been seen by Schlatter [137] who performed several tests with different levels of k and ε and obtained only negligible sensitivity of the results. The inlet temperature of the fuel is 298 K. As the modification of hydrogen is not known, hydrogen is assumed to be normal hydrogen (see appendix E), which consists of 25% para and 75% ortho hydrogen. The CFD simulations were performed using the High Resolution discretization scheme and double precision. Depending on the local flow field, the High Resolu- tion discretization scheme blends between a first and a second order advection scheme. 5.2. INFLUENCE OF CHEMISTRY TABLE 47

Figure 5.3: Inﬂuence of diﬀerent length scales on the simulation results

The convergence criterion is 1e-6 for the residuum of all transport equations. Further boundary conditions like the choice of the turbulence model, the ﬂamelet table and laminar ﬂame speed are given in the following chapters 5.2-5.4

5.2 Influence of Chemistry Table

Within a DI engine, the cold fuel is injected into the cylinder during the compression stroke. This is often done in multiple injections until the mixture is ignited. The compression causes a rise of pressure and temperature of the mixture, which results in a temperature difference between pure air, hydrogen/air mixture and injected (cold) hydrogen. As described in chapter 4.4, the change in unburned temperature and pressure has an influence on the flamelet table. The laminar flamelet tables were generated with CHEM1D [172]. The β-pdf described in appendix A is used to generate flamelet tables which take the turbulence effects into account. They are needed for the Flamelet model, and the Flamelet + Re- action Progress Variable model. The Conaire hydrogen chemical reaction mechanism [46] used contains 10 species. It is listed in appendix C.2. For this flame, pressure and temperature variations as they occur in internal combustion engines are negligible. Therefore flamelet tables were generated with a pressure of 1 bar and a temperature of fuel and oxidizer of 298 K. 10 laminar flamelet tables ranging from a strain rate of a = 10 1/s to the maximum strain rate depending if differential diffusion effects are taking into account or not are used. 100 points are distributed non-uniformly along the mixture fraction axis with a refinement of resolution near stoichiometry.

Temperature and Pressure Dependence Figure 5.4 shows the inﬂuence of pressure and temperature on the mass fractions of H2O, O and OH as a function of mixture fraction Z. Temperatures and pressures range from ambient conditions of 1 bar and 300 K to elevated conditions of 50 bar and 900 K. The laminar ﬂamelet tables have been generated with a strain rate of 10 1/s. 48 CHAPTER 5. H2 JET FLAME

(a) YH2O

(b) YO (c) YOH

Figure 5.4: Temperature and pressure dependence of species mass fractions of H2O, O and OH

In addition, the stoichiometric mixture fraction is plotted. It can be seen that changes in temperature and pressure have little influence on the mass fraction of H2O in contrast to the strong effect on the mass fractions of the mi- nor species O and OH. The influence on H2O occurs mainly at stoichiometric mixture fraction. The maximum mass fraction drops with increasing temperature and increases with increasing pressure. It can also be expected that the influence on the global flow field is nearly negligible since the mass fraction of H2O represents the main combustion product of hydrogen, and so temperature and density are closely related to the H2O concentration. Figures 5.4(b) and 5.4(c) clearly show changes in the mass fraction of O and OH. With increasing temperature both mass fractions increase strongly and with increasing pressure the maximum decreases. Increasing temperature also broadens the mixture fraction range where the minor species occur. Increasing pressure decreases the mixture fraction range. As the mass fractions of O and OH are very small, they only exert little influence 5.2. INFLUENCE OF CHEMISTRY TABLE 49 on the main flow field, but as can be seen from the Zel’dovich mechanism [160], these radicals have a large effect on the NO production and thus on the pollutant prediction.

O + N2 ↔ NO + N

N + O2 ↔ NO + O (5.4) N + OH ↔ NO + H

Therefore if only the flow field is of interest, it is sufficient to use one flamelet table at one pressure and temperature, but as soon as an accurate prediction of NO is of interest, it is necessary to use pressure and temperature dependent flamelet tables.

Differential Diffusion Flamelet tables for the hydrogen jet flame have been generated with the laminar counterflow flame of CHEM1D [172] at 298 K and 1 bar with a strain rate of 10 1/s. As reaction mechanism, the Conaire mechanism [46] has been selected. As described in chapter 2.1, there are three options available to describe the diffusion process: complex, mixture averaged and with user supplied Lewis numbers for all species. Figure 5.5 shows a comparison of these three methods where the Lewis number for all species is set to unity. Additionally the chemical equilibrium (dark blue dots) obtained with Cosi- lab [173] and the stoichiometric mixture fraction (red vertical dashed line) are shown. It can clearly be seen, that the simulations with the options ’complex’ and ’mixture averaged’ result with a maximum temperature of 2540 K in a higher value than the adiabatic flame temperature with 2390 K. As expected, the maximum temperature using unity Lewis numbers lies slightly below the adiabatic flame temperature. These super-equilibrium temperature levels have also been observed for turbulent H2/N2/air jet diffusion flames [106] and have been investigated on laminar counterflow flames and turbulent H2/N2/air flames with direct numerical simulations (DNS) [63]. Such a significant difference between unity and non-unity Lewis number has also a strong influence on the mixing process, which can already be seen in chapter 5.3 and 5.5. Gicquel et al. [63] examined in detail the effects which lead to the super-equilibrium temperature and found that the local mixture when taking differential diffusion effects into account is less strongly diluted with inert species than the mixture obtained with unity Lewis numbers. Whereas the amount of hydrogen and oxygen determines the heat release, the inert species act as heat ‘sink’, since the inert species have to be heated without providing energy themselves.

Comparison with Temperature Scatter Plots Whether the consideration of differential diffusion effects is really important for this hydrogen jet flame can be estimated from experimental scatter plots. Figure 5.6 shows the experimental temperature scatter plot at y/L = 1/8 compared to the flamelet tables at different strain rates between 10 and 5000 1/s. It can be seen that flamelet 50 CHAPTER 5. H2 JET FLAME

Figure 5.5: Differential diffusion effects tables with unity Lewis number fit better the experiment than the one which takes differential diffusion effects (Complex) into account. Thus, the flame is not influenced by this effect. Within the HyICE project, Colin [44] showed that for IC engines differential diffusion effects can be neglected already for Reynolds numbers above 100. With increasing strain rate a, the maximum temperature decreases. Simulations with CHEM1D [172] yield quenching at a strain rate of a = 11627 1/s. At this strain rate, the stoichiometric scalar dissipation rate is χst = 145 1/s. Thus, the experiment is far from quenching. A full set of scatter plots at all measured downstream positions is given in appendix D.1.

Figure 5.6: Temperature distribution at y/L = 1/8 5.3. CHOICE OF TURBULENCE MODEL 51 5.3 Choice of Turbulence Model

The spreading rate of round jets, known as the ’round-jet/plane-jet anomaly’ is over predicted by most turbulence models. It is defined as the value of radial distance to axial distance where the velocity reaches half of the centerline value [157]. Different approaches have been developed to improve the prediction [128][105][110]. Dally et al. [48] propose to use either the k-ε model or the Reynolds Stress model with a modification of Cε1 = 1.6 (standard value: Cε1 = 1.44). Due to the greater computational cost of the Reynolds Stress model, the k-ε model has been chosen here. Calculations have been performed with different values of Cε1 (1.44, 1.5 and 1.6). The value of 1.5 shows the best fit to the experimental data of the mixture fraction field, which is shown in figure 5.7. Figure 5.7(a) shows the mean mixture fraction plotted along axial direction normalized with the visible flame length Lvis = 675 mm. As the mixture fraction is a conserved scalar it is only influenced by the flow field. Therefore it is suited to com- pare different turbulence models. Figure 5.7(b) shows the isolines of the stoichiometric mixture fraction. The experimental stoichiometric flame length is Lst = 475 mm. Although the standard values of the k-ε model predict the stoichiometric flame length well, it is in poor agreement with the experiment close to the axis. Thus the following calculations were performed with Cε1 = 1.5. With this choice, one slightly underpre- dicts the mixture fraction at the first measurement point at y/Lvis = 0.125 (y/L = 1/8) and slightly overpredicts the next measurement point y/Lvis = 0.25 and the measurement points closer to the jet tip.

(a) Mixture fraction over y/Lvis (b) Stoichiometric mixture fraction proﬁles

Figure 5.7: Inﬂuence of diﬀerent values of Cε1 52 CHAPTER 5. H2 JET FLAME 5.4 Implementation of Laminar Flame Speed

Within the laminar flamelet plus reaction progress variable model presented here, three source term closure models for the reaction progress variable transport equation are compared: The Zimont model [166][165], the Lindstedt-Váos model [94] and the Bradley correlation [27]. Within all three models, the laminar flame speed has to be provided. Commonly, a constant value is used. When premixed flames are studied, the laminar flame speed is in general provided as a function of the mixture fraction or the equivalence ratio. Within this work, it is additionally provided as a function of the mean mixture fraction and mixture fraction variance to account for turbulence effects.

Figure 5.8: Laminar ﬂame speed of hydrogen/air mixture for diﬀerent experiments and correlations at ambient temperature and pressure

Figure 5.8 shows a comparison of different experiments and correlations for the laminar flame speed at ambient temperature and pressure. The symbols denote experiments (dark blue rhombe: Verhelst [150], red squares: Dowdy et al. [53], green delta: Koroll et al. [88], yellow circle: Egolfopoulos et al. [59], bright blue rhombe: Aung et al. [7], blue line: simulation with CHEM1D [172], green line: correlation of Iijima and Takeno [75], orange line: correlation of Knop et al. [86]). As can be seen, the maximum value of the laminar flame speed is not reached at stoichiometric conditions for a hydrogen flame, as is the case for many hydrocarbon fuels, but on the rich side for hydrogen-air combustion. The double kernel measurements of Koroll [88] give higher burning velocities than the stretch-free measurements by [150][53][59][7]. The simulation with CHEM1D and the chemistry of Conaire [46] shows good agreement especially with the experiment of Dowdy [53]. The correlation of Knop [86] agrees well with the experiments, especially at the lean side. As was demonstrated by Zimmermann et al. [161] the laminar flame speed correlation used has a noticeable influence on the results. In the following details of the correlations used are presented. 5.4. IMPLEMENTATION OF LAMINAR FLAME SPEED 53

Correlation of Milton and Keck [107]

α β Tu p sl = Sl0 (5.5) T0 p0

with Sl0 = 2.17m/s, α = 1.26, β = 0.26, p0 = 1atm, T0 = 298K.

The domain of validity is 0.5atm < p < 7atm, and 198K ≤ Tu ≤ 550K at a stoichiometric equivalence ratio φ = 1. Thus, this correlation does not take the changes in equivalence ratio into account. The laminar ﬂame speed for this test case is sl = 2.163m/s. Correlation of Iijima and Takeno [75]

α p Tu sl = Sl0 1 + β ln (5.6) p0 T0 with 2 3 Sl0 = 2.98 − (φ − 1.70) + 0.32(φ − 1.70) [m/s] α = 1.54 + 0.026(φ − 1) β = 0.43 + 0.003(φ − 1)

p0 = 1atm, T0 = 291K

The domain of validity is 0.5 ≤ φ ≤ 4.0, 0.5atm < p < 25atm and 291K ≤ Tu ≤ 500K Correlation of Knop et al. [86]

α β Tu p sl = Sl0 (5.7) T0 p0

The correlation domain is separated in two regions

• 0.25 ≤ φ ≤ 1.90

5 4 3 2 Sl0 = −1.7460φ + 10.8087φ − 25.5368φ + 26.9161φ − 9.4727φ + 1.1588

• 1.90 < φ φ − 1.90 S = S (φ = 1.90) + [1.08 − S (φ = 1.90)] l0 l0 l0 5.05 − 1.90 with

β = −0.9791φ2 + 1.7829φ − 0.6323 α = 1.62 5 p0 = 1.013 · 10 P a

T0 = 298 K 54 CHAPTER 5. H2 JET FLAME

The validity domain of this correlation is 0.25 ≤ φ ≤ 5.0, 1 bar < p < 10 bar and 298 K ≤ Tu ≤ 550 K

As under real engine conditions the ’unburned’ temperature Tu and pressure p are not constant, several researchers [65][151][150][152][34] have examined the influence on the laminar flame speed for hydrogen combustion. Figure 5.9 shows the results obtained by Verhelst [151] for lean mixtures. With increasing pressure, the laminar flame speed decreases, and with increasing temperature it increases.

Figure 5.9: Temperature and pressure dependence of the laminar ﬂame speed [151]

Extension of the laminar flame speed Because all the correlations for the laminar flame speed have been derived in the context of premixed combustion, the validity domain does not cover the whole range of diffusion and partially premixed flames. In order to amend this problem, an approach is presented in chapter 4.4 to overcome this problem using the pressure and temperature dependent flammability limits. Since the mean laminar flame speed is not only dependent on the Favre mean mixture fraction, but also on its variance, the improvements possible using a pre integrated laminar flame speed table are demonstrated in chapter 5.5. The hydrogen limits of flammability in air are 4 vol. % and 75 vol. % [38] at ambient temperature and pressure. That means that the flammable range lies in the range of fuel/air equivalence ratios of 0.099 ≤ φ ≤ 7.135 which corresponds to a mixture fraction range of 0.0029 ≤ Z ≤ 0.1726. As can be seen in figure 5.8 where the correlation of Iijima & Takeno [75] is plotted, the correlations, generally derived for premixed combustion, don’t cover the whole flammable range, and thus are not applicable in the whole range of diffusion flames, with 0 ≤ Z ≤ 1, usually producing unphysical results outside the given domain of validity (see figure 5.10). In diffusion flames Z = 1 for fuels, and Z = 0 for oxidizer, as both reactants are injected in non-premixed form. The 5.4. IMPLEMENTATION OF LAMINAR FLAME SPEED 55

Figure 5.10: Temperature and pressure dependence of the laminar ﬂame speed [151] equivalence ratio φ can be expressed by the mixture fraction Z and its stoichiometric value Zst as

Z 1 − Z φ = st (5.8) 1 − Z Zst

The term involving Zst is constant for a given combination of fuel and oxidizer. For hydrogen-air combustion, the stoichiometric mixture fraction is equal to Zst = 0.0285. For pure fuel φ → ∞ and for pure oxidizer φ = 0. Figure 5.11 shows the temperature dependent flammability limits of hydrogen. The experiment (red dots) has been performed by White [156] with a downward propagating flame in a 25 mm in diameter tube. An increase of the oxidizer temperature broadens the flammability limits because less heat has to be supplied from the burning layer. The flammability limits vary nearly linearly with temperature. As the flammability limits are also influenced by the size of the vessel and the direction of flame propagation, appendix B shows the values for different vessel sizes, and in the following, correlations are derived for a downward propagating flame and an assumed upward propagating flame. The correlations for the upward propagation are assumed, as only experimental results at ambient temperature are available. The green lines in figure 5.11 represent the correlation derived for the lower flammability limit, and the blue line for the upper flammability limit. With the assumption of an upward propagating flame, the flammability limits at ambient temperature and pressure are higher for the hydrogen lean side, and lower for the hydrogen rich side. Finally the flammability limits with a broader flammable range are used for the correlated curves for the laminar flame speed. The following expressions give the temperature dependence of the lower and upper flammability limit of hydrogen in air, where LF L0 and UFL0 denote the lower and upper flammability limit of hydrogen at ambient temperature and pressure respectively. The correlation of flammability limits obtained from experimental results of a downward propagating flame [156] marked with dashed lines can be written as 56 CHAPTER 5. H2 JET FLAME

Figure 5.11: Temperature dependent ﬂammability limits of hydrogen

T − T 0 LF L = LF L0[vol.%] 1 − (5.9) T,experiment 1200 [K]

T − T 0 UFL = UFL0[vol.%] 1 + (5.10) T,experiment 2700 [K]

The approximations for the lower and upper flammability limits assuming an upward propagating flame, marked with solid lines in figure 5.11, are

T − T 0 LF L = LF L0[vol.%] 1 − (5.11) T,corrected 500 [K]

T − T 0 UFL = UFL0[vol.%] 1 + (5.12) T,corrected 2800 [K]

The index 0 refers to a reference state at T 0 = 293 K and p = 1 bar. The pressure dependent flammability limits are shown in figure 5.12. The grey solid line [13] and the grey solid line with filled dots [144] show the experimental results obtained with a downward propagating flame, and the solid line with no filled dots [16] shows the experimental results obtained within a spherical vessel. The different results obtained from the measurements cannot be only attributed to experimental uncertainties. Moreover, they result from different interpretations. Bone et al. [16] (solid line with no filled dots) defined the criterion for the flammability limits to be 100 % combustion, whereas Berl et al. [13] defined the flammability limits for 80 % combustion. 5.4. IMPLEMENTATION OF LAMINAR FLAME SPEED 57

Figure 5.12: Pressure dependent ﬂammability limits of hydrogen

As can be seen, with increasing pressure, the flammability limits are at first narrowed and with further increasing pressure they are widened. Berl et al. [13] ascribe this phenomenon to the highly caloric ignition. When the silver wire fuses, heat quantity is supplied to the system. It has a stronger influence at low pressures, hence a low mass in the volume. The higher the pressure the smaller the influence of this phenomenon. The dashed lines show the correlations adapted to the experiment of Berl et al. [13], and the solid lines show again the approximations assuming an upward propagating flame. The green lines denote the lower flammability limits, and the blue lines the upper flammability limits. Due to the fact that the flammability limits are at first narrowed and with further increasing pressure widened, the correlations are split up to two ranges for the lower flammability limit as well as for the upper flammability limit. The experiments performed with a downward propagating flame can be approximated by

• Upper ﬂammability limit

– 0 atm ≤ p < 9.22 atm p − 285 UFL = (5.13) p1,experiment −4

– 9.22 atm ≤ p < 220 atm

−3 UFLp2,experiment = 2.17017 lnp − 6.48263 · 10 p + 63.753 (5.14) 58 CHAPTER 5. H2 JET FLAME

• Lower ﬂammability limit

– 0 atm ≤ p < 18.6844 atm p + 31 LF L = (5.15) p1,experiment 4.578 – 18.6844 atm ≤ p < 220 atm

−2 −4 2 −7 3 LF Lp2,experiment = 11.5476−4.0747·10 p+1.8864·10 p −3.0605·10 p (5.16)

The pressure and temperature dependent lower and upper ﬂammability limits can be approximated as

0 0 LF L(p, T ) = LF Lp T + LF LT − LF L (5.17)

0 0 UFL(p, T ) = UFLp T + UFLT − UFL (5.18)

The pressure and temperature dependent flammability limits are used to extend the range of validity of the laminar flame speed, and therefore of the correlation. Thus, as model parameters used in the simulations, at first a constant laminar flame speed with 2 m/s derived from the correlation of Milton and Keck [107] equation 5.5 was used. Then a mean mixture fraction dependent laminar flame speed where the result derived with CHEM1D are connected linearly to the flammability limits was used. With this function an integration was performed to obtain a laminar flame speed table dependent also on the mixture fraction variance. Figure 5.13(a) and 5.13(b) show the mean mixture fraction dependent laminar flame speed, and the mean mixture fraction and mixture fraction variance dependent effective laminar flame speed, respectively.

002 (a) sl(Ze) (b) sl(Z,e Zg)

Figure 5.13: Boundary conditions for the eﬀective laminar ﬂame speed 5.5. RESULTS AND DISCUSSION 59 5.5 Results and Discussion

This chapter compares different commonly used combustion models and the improved combustion model with the experimental data of the hydrogen jet flame described at the beginning of chapter 5 and 5.1. Compared (in the following figures) are the Eddy Dissipation Model (EDM, grey solid line), the Flamelet model (Flamelet, green solid line) and the partially premixed model with different implementations of the laminar flame speed. As source term closure models, the Zimont model [165][166] (Zim., solid line) with a constant laminar flame speed of 2 m/s (dark blue dotted line) derived from the correlation of Milton and Keck [107], a mean mixture fraction dependent laminar flame speed extrapolated to the flammability limits (sl(Z), orange solid line), and the laminar flame speed table dependent on the mean mixture fraction and mixture fraction variance (slTable, cyan solid line). Furthermore, the source term closure models together with the laminar flame speed table, the Bradley model [27] (Bra., bright green dashed line) and the Lindstedt-Váos model [94] (Lin., purple dashed line) are compared. Laminar flame speed tables with unity Lewis number (Le) and taking differential diffusion (dd, solid lines with squares) effects into account are also shown in all the following figures of this chapter. The rms values of the experiment are given as error bars. The following results are plotted as functions of the radial distance at different downstream positions at y/L = 1/8 (y/D = 23), y/L = 3/8 (y/D = 68), y/L = 1/2 (y/D = 90) and y/L = 1/1 (y/D = 180) where L is the visible flame length and D the diameter of the hydrogen injector. For a visualization of compared radial positions please refer to figure 5.1. r is the radial distance and R is the radius of the hydrogen injector (R = 1.875 mm).

Mixture Fraction For the experimental values, according to the deﬁnition of Bilger et al. [15] the mixture fraction is deﬁned as

0.5 (YH − Y2H ) /WH + 2 (YO − Y2O) /WO ZBlgr = (5.19) 0.5 (Y1H − Y2H ) /WH + 2 (Y1O − Y2O) /WO

where YH and YO are the hydrogen and oxygen mass fractions in the measured sample, the indices 1 and 2 denote the mass fractions of hydrogen and oxygen in the fuel stream and the coflow stream respectively. WH and WO are the atomic masses of hydrogen and oxygen. The measurements of the mixture fraction based on validation flames have an uncertainty of 5.2 % [10]. Figure 5.14 shows the mean mixture fraction distribution at different radial downstream positions. All in all, the difference between the models is very small. The main difference can be seen close to the injector, at y/L = 1/8. All models under-predict the experimental mixture fraction close to the axis, which is a result of the turbulence model selected (see also figure 5.7). The pure Flamelet model predicts the highest mixture fraction close to the axis, followed by the Flamelet model together with a transport equation for the reaction progress variable and a constant laminar flame 60 CHAPTER 5. H2 JET FLAME

(a) y/L = 1/8 (b) y/L = 3/8

Figure 5.14: Mixture fraction distribution at diﬀerent radial downstream positions

speed, a mean mixture fraction dependent laminar flame speed sl and the effective laminar flame speed table. For r/R ¿ 2, the difference between the simulation models is negligibly small. Between r/R = 3 and r/R = 5 there is good agreement between simulation and experiment. From r/R = 6, the experiment is slightly over predicted, which is, as already discussed, influenced by the turbulence model. As could already be seen in figure 5.7, the difference between the models using flamelet tables accounting for differential diffusion or not is negligible until y/L = 3/8, where the models using flamelet tables generated with the assumption of unity Lewis number predict a slightly smaller mixture fraction than the flamelet tables accounting for differential diffusion. At y/L = 1/1, the mixture fraction is over predicted by all the models over the entire radial range. This will have a strong effect on the temperature distribution since in this (lean) region, the temperature very sensitively depends on the mixture fraction. Regarding again figure 5.5 the temperature distribution has a high gradient on the lean side. Considering the point at the axis, the experiment gives a mixture fraction 5.5. RESULTS AND DISCUSSION 61 of 0.014, and the simulation a mixture fraction of 0.02. Assuming unity Lewis number and a strain rate of 10 1/s, the corresponding temperatures are 1604 K and 2463 K, which is a difference of 859 K. Of course the resulting temperature also depends on the scalar dissipation rate of the considered point. As the mixture fraction is a conserved scalar, it could already be seen in chapter 5.3 that it mainly depends on the flow field and thus on the selected turbulence model. So the low sensitivity of this quantity on the combustion model was to be expected.

Figure 5.15: Mixture fraction variance along axial direction

Figure 5.15 shows the mixture fraction variance at the axis of the hydrogen jet flame. Close to the nozzle exit, the variance is close to zero, strongly increases to a maximum of 0.17 and decreases again below 0.001 for y/L ¿ 0.2. Where the variance is close to 0, nearly no difference between the effective laminar flame speed table and the mean mixture fraction laminar flame speed is expected. In that case the pdf reaches a Dirac delta peak and as can be seen in figure 5.13(b), the laminar flame speed is nearly the same as for sl(Z) (figure 5.13(a)).

Mass Fraction of Hydrogen Figure 5.16 shows the mass fraction of hydrogen plotted as a function of the radial distance at different downstream positions. All in all, poor agreement is achieved with the models considering differential diffusion effects, which was already expected, since the scatter plots show, that differential diffusion effects are unimportant in this flame. At y/L = 1/8 the best agreement with the experimental values is obtained with the Eddy Dissipation model. Considering the models with unity Lewis number, the Flamelet model shows slightly better agreement close to the axis. But already from r/R = 1 all of the Flamelet models again show negligible difference. Close to the axis they under predict the experiment due to the turbulence model. The simulated mixture fraction is smaller than the one of the experiment (see figure 5.14(a)), so also the 62 CHAPTER 5. H2 JET FLAME

(a) y/L = 1/8 (b) y/L = 3/8

Figure 5.16: Mass fraction of H2 at different radial downstream positions predicted mass fraction of hydrogen is smaller. Good agreement is achieved between r/R = 3 and 6 whereas they slightly over predict the experiment between r/R = 7 and 8, where the EDM also shows very good agreement. At the further downstream positions (figure 5.16(b) - 5.16(d)) the EDM and all the Flamelet models assuming unity Lewis number show no difference any more, as the fluctuations are smaller, too. At all positions, excellent agreement with the experiment is achieved. At y/L = 1/1 (figure 5.16(d)) hydrogen is completely consumed.

Mass Fraction of Oxygen

The mass fraction profiles of O2 predicted by the different combustion models compared to the experimental data are given in figure 5.17. Again poor agreement is achieved taking differential diffusion effects in the flamelet tables into account (dd), but this is also an effect of the selected turbulence model, where it also showed worse agreement (see figure 5.7) than the tables generated with unity Lewis number (Le). 5.5. RESULTS AND DISCUSSION 63

(a) y/L = 1/8 (b) y/L = 3/8

Figure 5.17: Mass fraction of O2 at diﬀerent radial downstream positions

At y/L = 1/8 between r/R = 5 and 10, the profiles predicted by all models are somewhat shifted to the oxygen side, which can be an effect of the turbulence model used, but may also be an effect due to the averaging process of the experimental data, which shifts the sampling probes to the rich side [10]. In this region there is negligible difference between the models using flamelet tables with unity Lewis number. Between r/R = 5 and 8, the Eddy Dissipation model shows slightly worse agreement than the other models, between r/R = 8 and 10 it shows slightly better agreement. Close to the axis at 0 < r/R < 5, the EDM model as well as the laminar Flamelet models do not predict the increased value of oxygen. The only model that reflects the experiment near the axis is the Flamelet + Reaction Progress Variable Model, where the best agreement with the experimental results is achieved with laminar flame speed accounting for the dependency on mixture fraction and also on mixture fraction variance. Regarding the results obtained with the Zimont model, best agreement is achieved using the laminar flame speed table, where the effective laminar flame speed depends on the mean mixture fraction and its variance, followed by the laminar flame speed which depends only on the mean mixture fraction. Worst agreement shows the simulation using a constant 64 CHAPTER 5. H2 JET FLAME value for the laminar flame speed. The Bradley model and the Lindstedt-Váos model used with the laminar flame speed table show negligible difference, but show worse agreement than the Zimont model in that region. In this region close to the nozzle, the accuracy of the experimental measurement is not known. The measurements were performed with a Raman scattering technique. Partly the signal of O2 may be due to temperature dependent crosstalk from H2 rota- tional Raman scattering. Although Barlow performed a correction for this crosstalk, it my not be accurate enough [113].

(a) whole computational domain

(b) EDM (c) Flamelet (d) Flamelet + RPV

Figure 5.18: Comparison of diﬀerent combustion models near the injector exit

Figure 5.18 shows a comparison of the distributions of oxygen mass fraction and temperature obtained with the different combustion models close to the nozzle exit. The temperature distribution is plotted on the left side and the O2 mass fraction dis- 5.5. RESULTS AND DISCUSSION 65 tribution on the right side of the pictures. Figure 5.18(a) shows the results of whole computational domain obtained with the Flamelet model and unity Lewis number where the black box denotes the zoomed region presented in figures 5.18(b)-5.18(d). These figures show the results of the Eddy Dissipation Model, the Flamelet model and the Flamelet + Reaction Progress Variable Model, where the Zimont model with the laminar flame speed table was used. With the Eddy Dissipation model, in contrast to the other models, no air diffuses towards the centerline. As combustion is controlled by the species that deliver the lowest reaction rate (see equation 4.2), the air is burned before it is able diffuse so close to the centerline. For the other models, the mass fraction is dependent on the profile of the probability density function (described in appendix A) which in turn depends on the mixture fraction and its variance. In shear layers and also close to the nozzle, the turbulent kinetic energy is high and thus the mixture fraction variance, which influences the shape of the pdf. Thus near the injector, pure air can exist near the centerline. Figure 5.17(b) shows the results at a downstream position of y/L = 3/8. Again all the models using a flamelet table with unity Lewis number show nearly no difference. Between r/R = 8 and 18, the experiment is slightly under predicted, then it shows good agreement until r/R = 24, where the experiment is over predicted. Worse agreement is achieved with the Eddy Dissipation model which strongly under predicts the experiment between r/R = 8 and 18. For larger radial distance it shows the same behavior as the other models. The results obtained with tables accounting for differential diffusion strongly under predict the experiment over the whole radial range. The same behavior of the different combustion models can be seen in figure 5.17(c), but this time the experiment is under predicted over nearly the whole range. Only from r/R = 26 the simulation shows good agreement. At y/L = 1/1 (figure 5.17(d)) all models completely under predict the experimental values, but the profile is identical.

Mass Fraction of Water

Figure 5.19 shows the mass fraction distribution of H2O at different axial positions. Near the centerline at y/L = 1/8 best agreement is achieved with the Eddy Dissi- pation Model, the Flamelet model and the Flamelet + RPV model with a constant laminar flame speed and a flamelet table with unity Lewis number. The Bradley and the Lindstedt-Vaós model using a laminar flame speed table under predict the experimental values, but still show better agreement than the Zimont model using sl(Ze) and 002 sl(Z,e Zg). From r/R = 4, the models using a flamelet table with unity Lewis number again show nearly no difference. Although the maximum concentration is under predicted by all of the models, the position of the maximum is reproduced well. Only the maximum value of the EDM model and the models using flamelet tables calculated with differential diffusion effects, is shifted to the oxygen side. The last ones again show all in all the worst agreement at all downstream positions. But with increasing radial distance the EDM model shows slightly better agreement than the other models which in turn results in a different overall shape than obtained by the experiment. At the following downstream positions (figure 5.19(b) - 5.19(d), all the flamelet models (unity Lewis number) show no difference, as well as the flamelet models (differential diffusion). At y/L = 3/8 the first slightly under predict the experimental values 66 CHAPTER 5. H2 JET FLAME

(a) y/L = 1/8 (b) y/L = 3/8

Figure 5.19: Mass fraction of H2O at different radial downstream positions until r/R = 8. Between r/R = 8 and r/R = 20, they show excellent agreement. With further increasing radial distance, they under predict the experimental results. The EDM model also slightly under predicts the experiment close to the axis, but where the other models show good prediction, it strongly over predicts the experiment until it shows the same behavior as the other models from r/R = 19 onwards. The models accounting for differential diffusion in the flamelet tables at first under predict the experiment and from r/R = 10 strongly over predict it. At y/L = 1/2, all the flamelet models (unity Lewis number) and the EDM model show good agreement with the experiment until r/R = 5, where the flamelet models start to slightly, and the EDM strongly over predicts the experiment. Again, the flamelet models (differential diffusion) strongly over predict the experiment. At the position of the visible laminar flame length (figure 5.19(d)) all the combustion models completely over predict the experiment, whereas the profile is equivalent, which is a result of the under prediction of the mixture fraction at this station. 5.5. RESULTS AND DISCUSSION 67

Temperature Distribution

(a) y/L = 1/8 (b) y/L = 3/8

Figure 5.20: Temperature distribution at diﬀerent downstream positions

Figure 5.20 shows the temperature profiles of the hydrogen diffusion jet flame. Again, the radial profiles at different downstream positions are compared whereas the same behavior as before can be seen. The main difference between the different combustion models can only be seen at y/L = 1/8. Best agreement for small r/R values is achieved with the EDM, the Flamelet and the Flamelet + RPV with sl = 2 m/s. The Zimont models using sl(Ze) and a laminar flame speed table, as well as the Bradley and Lindstedt-Váos model, slightly under predict the temperature close to the centerline. The models using differential diffusion flamelet tables over predict the experimental results. From r/R = 4 the flamelet (Le) models again show negligible difference, and the results compared to the experiment are slightly shifted to the oxygen side, but the maximum temperature is well reproduced. The EDM model strongly over predicts the maximum temperature, but shows slightly better agreement from r/R = 8. The flamelet models (dd) strongly over predict the maximum temperature and show worst agreement with increasing radial distance. 68 CHAPTER 5. H2 JET FLAME

Figure 5.20(b) shows the results at y/L = 3/8. All the models using the unity Lewis number flamelet table show excellent agreement close to the centerline until r/R = 8 where they start to slightly over predict the temperature. The Eddy Dissipation Model also shows good agreement until r/R = 6 where it starts to strongly over predict the temperature until it shows the same behavior as the other models from r/R = 20 onwards. The models using flamelet tables accounting for differential diffusion show good agreement at the centerline, then slightly under predict the experiment until they strongly over predict the experiment from r/R = 11. At y/L = 1/2 (figure 5.20(c)), the simulation results are also shifted to the oxygen side. The maximum temperature is well predicted by the models using flamelet tables with unity Lewis number. Worse agreement is achieved with the EDM model, and worst agreement with the combustion models using flamelet tables with differential diffusion. The same behavior of the combustion models can be seen in figure 5.20(d). Al- though all the models completely over predict the experimental results, the models (Le) show better agreement than the EDM model. The worst agreement show again the models (dd). As could already be seen in figure 5.7, the flame length in the simulations is longer than in the experiment, which is a result of the over prediction of mixture fraction ratio at this station (see also the discussion of the mixture fraction distribution). As already mentioned, the mixture fraction distribution depends mainly on the turbulence model used, thus some of the difference between predicted and measured temperature can be attributed to the turbulence model.

Summary Summing up the results of the comparison of different combustion models with the experimental data of a hydrogen jet diffusion flame, it could be seen, that the main difference between experiment and simulation is due to the turbulence model selected. The Eddy Dissipation Model shows the worst behavior of all, as it assumes a one step reaction which especially results in a strong overprediction of the maximum temperature. The Flamelet model accounting for differential diffusion effects within the flamelet tables showed different behavior to the models assuming unity Lewis number. This is also an effects of the turbulence model. Using the same turbulence model, the flame is longer, which can be accounted to the different position and and a higher value of the maximum flame temperature. As Chapter 5.2 showed, differential diffusion effects in this flame are of minor importance within the flamelet tables. The main difference between the models using a flamelet table can be seen at positions where high fluctuations in mixture fraction occur. Especially the increased oxygen distribution close to the injector exit could be predicted only with the burning velocity model using an effective laminar flame speed table or a mean mixture fraction dependent laminar flame speed. Chapter 6 Sandia D Flame

As second validation test case, a further experimentally well characterised flame of the TNF Workshop [167] has been selected. It is a piloted methane/air diffusion flame examined experimentally by Barlow and Frank [11] and Schneider [138], where all in all six flames with different Reynolds number have been evaluated. The range is from laminar cases with Re = 1100 (Sandia A) to turbulent flames with high Reynolds numbers close to extinction (Re = 44800, Sandia F). The burner has been developed by Masri et al. [104] and consists of an axissymmetric jet surrounded by a pilot placed in a coflowing air stream. The pilot of this burner is a heat source of hot gases produced by small premixed flames in order to stabilize the main jet at the burner exit. Within this study, only the Sandia D flame is examined with a Reynolds number of 22400 with a small degree of extinction. With increasing Reynolds numbers the probability of local extinction also increases in the region close to the injector exit. There is a reignition and complete combustion further downstream.

Figure 6.1: Sandial D Flame

The following chapter gives an overview of the computational domain and the used boundary conditions in the simulations (chapter 6.1). Since also heptane is examined in this study, the following chapters 6.2 and 6.4 show the inﬂuence of both, methane and heptane, on the chemistry table and the laminar ﬂame speed, respectively. Chap- 70 CHAPTER 6. SANDIA D FLAME ter 6.3 shows the choice of the turbulence model, and chapter 6.5 gives the results of the simulations compared to the experiment.

6.1 Computational Domain and Boundary Conditions

The fuel, which consists of a mixture of 25 vol. % CH4 and 75 vol. % dry air, resulting in a mass fraction for methane of YCH4 = 0.156 and air of YAir = 0.844, is injected through a central nozzle with a diameter of 7.2 mm. Within this work also a hypho- thetical heptane flame with the same stoichiometry (Zst = 0.351) is examined, so the mass fraction of heptane at the inlet is YC7H16 = 0.175389 and the mass fraction of air consists of YN2 = 0.632480 nitrogen, and YO2 = 0.192131 oxygen. The pilot annulus inner diameter is 7.7 mm and its outer diameter is 18.2 mm. The wall thickness of the outer annulus is 0.35 mm. The coflowing air is unconfined whereas the wind tunnel exit of the experiment has the dimensions of 300 mm × 300 mm. Figure 6.1 shows the computational domain as well as the radial positions, where experimental data are available. As already mentioned in chapter 5.1, it is not possible to perform real 2D simulations with CFX. So again a ’quasi 2D’ case is generated by rotating the symmetry plane by three degrees with one cell in the third direction around the y-axis. The symmetry plane has the dimensions 0 to 600 mm in radial direction and 0 to 1000 mm in axial direction. As the air coflow is unconfined, a larger radial distance is used than the air wind tunnel exit to be sure that the wall won’t influence the flame shape. In order to be able to use a structured grid, the axis has been replaced by a small tube with a diameter of 0.4 mm. The resulting computational domain consists of 54,792 nodes and 27,068 elements, the grid has been refined near the visible flame. A grid refinement study showed that the solution is grid independent. On the rotated surfaces, a symmetry boundary condition is used. At the radial boundary and at the ”axis tube” an adiabatic slip wall has been chosen. The ’additional’ air coflow has a mean inlet velocity of 0.1 m/s, a temperature of 291 K and a pressure of 0.993 atm. The velocity profile from the wind tunnel measurements is given in figure 6.2. The turbulent kinetic energy and eddy dissipation are calculated as in equations 5.3 and 5.2. The velocity distribution of the lean pilot and the main jet is also given in figure 6.2. The temperature of the pilot is 1880 K and the mixture fraction is 0.27 (φ = 0.77). The temperature of the main jet is 294 K. The CFD simulations were performed using the High Resolution discretization scheme and double precision. The convergence criterion of the residuum of all transport equations is 1e-6. Additional boundary conditions like the choice of the turbulence model, the used flamelet tables and the used effective laminar flame speed are given in the following chapters 6.2-6.4. 6.2. INFLUENCE OF CHEMISTRY TABLE 71

Figure 6.2: Inlet velocity boundary conditions

6.2 Influence of Chemistry Table

As explained in chapter 5.2, the unburned temperature and pressure changes extremely within a DI engine. This is due to the strongly transient behavior resulting from the compression, which generates a rise of pressure and temperature of the mixture which, in turn, results in a temperature difference between pure air, the fuel/air mixture and the injected cold fuel. The laminar flamelet tables used within the simulations of the Sandia D flame are generated with CHEM1D [172] for the methane combustion simulations, and with CFX-RIF [170] for the simulations with heptane. For the methane simulations, laminar flamelet tables for unity Lewis number as well as tables accounting for differential diffusion effects are generated using the GRI3.0 [174] mechanism for 15 strain rates ranging from a = 10 1/s to the maximum strain rate, which is different if differential diffusion effects are taken into account or not. 100 points with a refinement close to the stoichiometric point are distributed along the mixture fraction range. With the β-pdf described in appendix A, the flamelet tables are modified to also obtain the influence of the mean mixture fraction and mixture fraction variance.

Temperature and Pressure Dependence To examine the influence of temperature and pressure on the flamelet tables, figure 6.3 shows the mass fractions of H2O, CO2, O2 and OH as function of mixture fraction for pure methane injected into air. Figure 6.4 shows the results of pure gaseous heptane injected into air. The unburned temperatures are 300 K, 600 K and 900 K, and the pressures are 1 bar and 50 bar. In addition, the stoichiometric mixture fraction is indicated, which is 0.0548 for methane and 0.0618 for heptane combustion. Note that in figures showing the mass fractions of H2O and CO2 (figures 6.3(a),6.3(b),6.4(a),6.4(b)) the mixture fraction range is from 0 to 1, and in the figures showing the mass fractions of O and OH (figures 6.3(c),6.3(d),6.4(c),6.4(d)) it is from 0 to 0.2. It can be seen for both the methane and heptane cases, that for the major species like H2O and CO2 changes in temperature and pressure have a stronger effect on the 72 CHAPTER 6. SANDIA D FLAME

(a) Dependence of YH2O (b) Dependence of YCO2

Figure 6.3: Temperature and pressure dependence of a methane/air ﬂame

mass fractions than in the hydrogen case (see figure 5.4). For H2O, the maximum value is slightly shifted to the fuel side for all considered cases and is higher for the methane air flame. There is nearly no influence on the lean side. The maximum value of H2O mass fraction of the methane case is nearly independent on pressure and temperature changes, but on the rich side the mass fraction increases with decreasing temperature and decreases with increasing pressure. In the case of CO2, the influence of temperature and pressure is considerably stronger for methane and heptane. Again, there can be seen no difference on the lean side, but the maximum mass fraction value increases with increasing pressure and decreases with decreasing temperature. It is higher for the heptane flame compared to the methane flame. On the rich side a different behavior for methane and heptane can be observed. For methane the mass fraction of CO2 decreases with increasing temperature and decreases with increasing pressure. For heptane, the mass fraction also decreases with increasing temperature, but regarding the pressure behavior, it can be seen that for 300 K the mass fraction increases with increasing pressure, for 600 K the mass fractions at 1 bar and 50 bar are nearly the same, and for 900 K they decrease for increasing pressure. As H2O and CO2 are the 6.2. INFLUENCE OF CHEMISTRY TABLE 73

(a) Dependence of YH2O (b) Dependence of YCO2

Figure 6.4: Temperature and pressure dependence of a heptane/air flame main combustion products, and thus the temperature and density are closely related to these concentrations, it can be expected that they have also an influence on the global flow field. The effect of temperature and pressure on the minor species O and OH is presented in figures 6.3(c) and 6.3(d) for methane and in figures 6.4(c) and 6.4(d) for heptane. Here the influence is even stronger than for the major species. For both mass fractions and cases, the mass fractions increase with increasing temperature and decrease with increasing pressure. With increasing temperature, the mixture fraction range is broadened. With increasing pressure, the mixture fraction range is narrowed. As the mass fractions of O and OH are small, they have only little influence on the main flow field, but as can be seen again from the Zel’dovich mechanism [160] (see equation 5.5), the radicals have a strong effect on the NO production and thus on the pollutant prediction. In cases where the unburned temperature and pressure changes, like in the DI engine, it is necessary to use temperature and pressure dependent flamelet tables, as they influence both, the global flow field as well as the pollutant prediction. 74 CHAPTER 6. SANDIA D FLAME

Differential Diffusion

Figure 6.5: Temperature distribution at y/D = 15

Barlow et al. [12] examined the complete piloted CH4/air jet flame series (Sandia A-F) to identify the importance of molecular diffusion and turbulent transport. These flames span a Reynolds number range from 1100 to 44800. So they cover a range from laminar flames (Sandia A) to transitional jet flames (Sandia B) to turbulent flames (Sandia D-F). They found that even the Sandia B flame (Re = 8,200) shows strong effects of differential diffusion. With increasing Reynolds number and with increasing downstream position, the diffusive transport of the scalar structure is more and more dominated by the turbulent transport. They also stated that in flames D and E the differential diffusion effects are negligible, which is also confirmed by figure 6.5. It shows a scatter plot at x/D = 15 obtained from the experimental data together with the flamelet tables generated with CHEM1D for the Sandia D flame with the GRI3.0 [174] mechanism. As described in chapter 2.1 there are three options to describe the diffusion process. Here the option of unity Lewis number is used (Le, solid lines) with a strain rate close to Zero and a strain rate of 1227 1/s, where differential diffusion effects are neglected, and the option ”complex” (solid lines with square symbols) is used with a strain rate close to Zero, which accounts for differential diffusion effects. Additionally the position of the stoichiometric mixture fraction is indicated (Zst = 0.351). With the assumption of differential diffusion, the maximum temperature is shifted to the lean side and is lower compared to the assumption of unity Lewis number. But it can be seen that the assumption of unity Lewis number better reproduces the shape of the scatter plots. Additional comparisons at different downstream positions can be found in appendix D.2. Because in most industrial combustion applications the Reynolds numbers typically involved are even higher than in these laboratory flames, it might be expected that it is appropriate to assume equal diffusivities. 6.3. CHOICE OF TURBULENCE MODEL 75 6.3 Choice of Turbulence Model

The Sandia D flame as well as the hydrogen jet flame, discussed in chapter 5, are both featuring round jet whose spreading rate is over predicted by most turbulence models (see chapter 5.3). Again the k-ε model is used for the simulations due to the low computational cost. As well as for the hydrogen jet, simulations have been performed with Cε1 = 1.44 (standard value), 1.5 and 1.6. Figure 6.6 shows in figure 6.6(a) the mean mixture fraction plotted along axial direction normalized with the nozzle diameter d = 7.2 mm. Additionally the position of stoichiometric mixture fraction is plotted. The mixture fraction is suitable for the choice of turbulence model, as it is a conserved scalar and mainly influenced by the flow field. Figure 6.6(b) shows the isolines of the stoichiometric mixture fraction. The solid lines represent Flamelet simulations with flamelet tables generated with unity Lewis number (LE), and the solid lines with square symbols represent simulations with Flamelet tables which account for differential diffusion effects (CO). The visible flame length of the Sandia D flame is Lvis = 482 mm, and the stoichiometric flame length is Lst = 338 mm. The results show that simulations with Cε1 = 1.5 produce best agreement with the experiment. The difference of flamelet tables with differential diffusion effects and unity Lewis number is not as large as for the hydrogen jet flame.

(a) Mixture fraction over y/D (b) Stoichiometric mixture fraction proﬁles

Figure 6.6: Inﬂuence of diﬀerent values of Cε1

6.4 Implementation of Laminar Flame Speed

To close the source term of the transport equation of the reaction progress variable, the models considered here contain the laminar flame speed, which has to be provided. The following section shows a comparison of measurements and derived correlations for the laminar flame speeds of methane and heptane. As the correlations also only 76 CHAPTER 6. SANDIA D FLAME cover a limited mixture fraction range, the extension to the temperature and pressure dependent flammability limits is shown. Figure 6.7 shows a comparison of different experiments and correlations for the laminar flame speed of methane air mixtures at ambient temperature and pressure for different values of equivalence ratio. The symbols denote the results obtained by experiments (green squares: Bosschaart et al. [20], mauve rhombes: Egolfopoulos et al. [58], green crosses: Andrews et al. [6], red deltas: Iijima and Takeno [75]). The correlations are given by solid lines, whereas the red line shows the correlation of Iijima and Takeno [75], equation 6.2, the orange line shows the correlation of Mülleret al. [111], equation 6.3, and the blue line shows the results obtained with the program CHEM1D [172]. Compared to the laminar flame speed of hydrogen (see figure 5.8), the maximum value is much smaller for methane fuel. Although the maximum value is also shifted slightly to the rich side, the shift is not as strong as for the hydrogen case. Figure 6.8 shows the laminar flame speed for heptane/air mixtures at ambient temperature and pressure. The symbols denote the experimental results obtained by Davis et al. [49] (green squares), Kwon et al. [90] (mauve rhombes) and Huang et al. [73] and [74] (red deltas). The red and blue line show the results of computations performed by Huang et al. [73] and Held et al. [72], respectively. The orange line again shows the correlation of Mülleret al. [111], equation 6.3. In the following, the correlations for the laminar flame speed of methane and heptane are given.

Figure 6.7: Laminar ﬂame speed of Methane/Air mixtures for diﬀerent experiments and correlations at ambient temperature and pressure 6.4. IMPLEMENTATION OF LAMINAR FLAME SPEED 77

Figure 6.8: Laminar ﬂame speed of Heptane/Air mixtures for diﬀerent experiments and correlations at ambient temperature and pressure

Correlation of Gu et al. [67] The correlation derived by Gu et al. is only valid for methane air mixtures.

α β Tu pu sl = Sl0 [m/s] (6.1) T0 p0 with

φ Sl0 α β 0.8 0.259 2.105 -0.504 1.0 0.360 1.612 -0.372 1.2 0.314 2.000 -0.438

The validity domain of this correlation is between 300 and 400 K and between 0.1 and 1.0 bar. This correlation has the disadvantage, that it changes the constants with equivalence ratio. Thus in order to be able to use it within simulations, a correlation of the constants dependent on the equivalence ratio would have to be derived.

Correlation of Iijima and Takeno [75] Iijima and Takeno examined the eﬀects of temperature and pressure on the burning velocity of hydrogen and methane. The derived correlation of hydrogen is given in equation 5.6, the correlation for methane is given in the following equation:

α p Tu sl = Sl0 1 + β ln [cm/s] (6.2) p0 T0 78 CHAPTER 6. SANDIA D FLAME with

2 3 Sl0 = 36.9 − 210(φ − 1.12) + 335(φ − 1.12) [cm/s] α = 1.60 + 0.22(φ − 1) β = −0.42 − 0.31(φ − 1)

p0 = 1atm, T0 = 291K

The domain of validity is 0.8 ≤ φ ≤ 1.3, 0.5 atm < p < 30 atm and 291 K ≤ Tu ≤ 500 K

Correlation of Müller et al. [111] Mülleret al. derived correlations for lean to stoichiometric hydrocarbon/air and methanol/air flames for a wide range of pressures and temperatures from numerically calculated mixtures.

0 n 0 m Tu Tb − T sl = A T YF,u 0 (6.3) T Tb − Tu

The indices u and b denote the unburned and burned state. The inner layer temperature T 0 represents the crossover temperature between chain-branching and chain-breaking reactions. The constants m and n are given in table 6.1. YF,u is the mass fraction of fuel in the unburned gas. The function A (T 0) only depends on thermodynamic and kinetic properties and is given as:

G A T 0 = F exp − (6.4) T 0 where F and G are constants given in table 6.1. With the assumption that the inner layer temperature T 0 does not depend on the equivalence ratio, Mülleret al. [111] derived a correlation for T 0 for different pressures such that the best fit for the burning velocities is obtained. The constants for B and E are also given in table 6.1.

E p = B exp − (6.5) T 0

Mülleret al. defined the burnt gas temperature Tb as the adiabatic flame temperature at chemical equilibrium and approximated it for lean flames with the following equation:

2 3 Tb = aTu + b + cφ + dφ + eφ (6.6) 6.4. IMPLEMENTATION OF LAMINAR FLAME SPEED 79

Tables 6.1 and 6.2 contain the constants and coefficients for calculating the laminar flame speed and the adiabatic flame temperature.

Fuel B [bar] E [K] F [cm/s] G [K] m n CH4 3.1557e8 23873.0 2.21760e1 -6444.27 0.565175 2.5158 n − C7H16 1.7000e6 17508.0 7.95600e3 912.00 0.52 2.30

Table 6.1: Approximation constants for burning velocity

Fuel a b [K] c [K] d [K] e [K] Le CH4 0.627 1270.15 -2449.0 6776 -3556 0.91 n − C7H16 0.490 758.7 -277.8 4269 -2642 2.056

Table 6.2: Coeﬃcients for calculating the adiabatic ﬂame temperature and values for the Lewis number

Further constants to calculate the laminar ﬂame speed are available for C2H2, C2H4, C2H6, C3H8, CH3OH, i − C8H18 in [111]. The validity domain of the temperatures lies between 298 K and 800 K, for the pressures between 1 bar and 40 bar and for the fuel-to-air equivalence ratio φ between 0.6 and 1.

Extension of the laminar flame speed Also the correlations derived for methane and heptane don’t cover the whole flammable range occurring in diffusion flames. Thus the correlations are extrapolated linearly to the temperature and pressure dependent flammability limits which lies in the range of 5 vol. % and 15 vol. % [47] for methane in air. This corresponds to a mixture fraction range of 0.0283 ≤ Z ≤ 0.0888. For heptane only few data for the flammability limits are available from literature, which are also only measured at the reference state at ambient temperature and pressure. The lower and upper flammability limits of heptane are given for different measurement techniques in table 6.3. For methane a broader range of measurements has been performed in the past. Appendix B shows the influence of different measurement techniques and propagation directions on the flammability limits at ambient temperature and pressure. Also temperature and pressure dependent flammability limits are available. Figure 6.9 shows the temperature dependent lower (left) and upper (right) flammability limits given for different equivalence ratios φ measured with different techniques. The blue lines show the results obtained by White [156] with a tube of 25 cm in diameter for a downward propagating flame. The orange line shows the lower flammability limit measured by Burrell et al. [36] for a downward propagating flame in an explosion pipette with a volume of 100 cm3. The green line with circles gives the results of Taffanel et al. [143] for a downward propagating flame. The results for both, the lower and upper flammability limit of a downward propagating flame, obtained by Mason et al. [103] are marked 80 CHAPTER 6. SANDIA D FLAME

Diameter LFL [%] UFL [%] Ref.

Upward Propagation of Flame 2 [inch] 1.10 6.70 [47] 5.3 [cm] 2.26 - [57]

Downward Propagation of Flame - 1.1 - [47]

Vessel - 1.0 6.0 [61]

Table 6.3: Flammability limits of heptane with red lines. The bright blue line and the green line with squares shows the upper ﬂammability limits obtained by Van den Schoor [147][148] using a bomb of 200 mm inner diameter and Vanderstraeten et al. [149] obtained with a closed spherical reaction vessel with a volume of 8 dm3, respectively.

Figure 6.9: Temperature dependent ﬂammability limits of methane

In the following, temperature dependent flammability correlations are obtained from the experiments. Assuming a downward propagating flame, marked with dashed lines in figure 6.9, the correlations for the lower flammability limit and upper flammability limit are as follows:

T − T 0 LF L = LF L0[vol.%] 1 − (6.7) T,experiment 1720 [K]

T − T 0 UFL = UFL0[vol.%] 1 + (6.8) T,experiment 1400 [K] 6.4. IMPLEMENTATION OF LAMINAR FLAME SPEED 81

The lower flammability limit correlation is approximated using the experimental results of Burrell et al. [36], and the upper flammability limit is correlated with the experiment of White [156]. Assuming the widest possible flammable range, which is in general obtained with an upward propagating flame (see appendix B), the lower flammability limit becomes:

T − T 0 LF L = LF L0[vol.%] 1 − (6.9) T,corrected 1390 [K]

The upper flammability limit is correlated on the results of Vanderstraeten et al. [149] who used a spherical vessel for the experiments and obtained a larger upper flammability limit than previous measurements with upward propagating flames. Thus the UFL is correlated as

T − T 0 UFL = UFL0[vol.%] 1 + (6.10) T,experiment,bomb 1200 [K]

The correlations derived with the assumption of widest possible ﬂammable range are marked with solid lines in ﬁgure 6.9. The index 0 within equations 6.7 to 6.10 refer to a reference state at T 0 = 293 K and p = 1 bar.

Figure 6.10: Pressure dependent ﬂammability limits of methane

Figure 6.10 shows the influence of pressure on the flammability limits. Whereas the lower flammability limit changes only slightly, the upper flammability limit changes remarkably. All measurements of downward propagating flame shows at first an increase of the LFL with pressure and above a certain value, which cannot be defined exactly from the present data available, it decreases. This behavior cannot be seen for 82 CHAPTER 6. SANDIA D FLAME the UFL. Except for the data measured by Bone et al. [16] (lilac curves) all the other measurements show an increase of the UFL with increasing pressure. Above 35 bar the slope of the linear increase in pressure changes abruptly [149]. The measurements of Mason et al. [103] are marked in red, those of Terres et al. [144] are marked in orange and those of Berl et al. [13] are marked in dark blue. Vanderstraeten et al. [149] derived the following correlation for the pressure dependent upper flammability limit

" # p p 2 UFL = UFL0[vol.%] 1 + 0.0466 − 1 − 0.000269 − 1 p,experiment,bomb p0 p0 (6.11)

This correlation is marked in green in ﬁgure 6.10 together with their measurements in a spherical vessel (solid green line with squared symbols). It can be seen that this correlation gives only reasonable results up to a pressure of about 80 bar, above which it changes the slope and the UFL shows unphysical behavior and strongly decreases. Thus the following correlation is derived assuming a maximum possible ﬂammable range

0 UFLp,max = UFL [vol.%] + 1.5 lnp + 0.2 p (6.12)

With the assumption of a downward propagating ﬂame, a correlation following the measurements of Terres et al. [144] and Berl et al. [13] is derived

0 UFLp,downward = UFL [vol.%] + 0.1192 p (6.13)

In contrast to the upper flammability limit, the lower flammability limit slightly increases at first up to approximately a pressure of 35 bar. Then it decreases with increasing pressure. Thus the correlation for the lower flammability limit is divided in two parts, the increasing and decreasing part.

• Downward propagation: Assuming downward propagation, the correlation is derived in accordance to the measurements of Berl et al. [13], Terres et al. [144] and Mason et al. [103].

– 0 bar ≤ p ≤ 35 bar

0 LF Lp1,downward = LF L [vol.%] + 0.0574 p (6.14) 6.4. IMPLEMENTATION OF LAMINAR FLAME SPEED 83

– 35 bar < p < 200 bar p LF L = LF L| [vol.%] + 0.3 · 1 − (6.15) p2,downward 35 bar 35 bar

• Assumption of maximum possible range: The correlation under the assumption of a maximum possible range is derived in accordance to the measurements of Bone et al. [16] with the latest ﬁndings derived by Van den Schoor [147]

– 0 bar ≤ p ≤ 30 bar

0 LF Lp1,max = LF L [vol.%] + 0.005 p (6.16)

– 30 bar < p < 200 bar p LF L = LF L| [vol.%] + 0.3 · 1 − (6.17) p2,max 35bar 35 bar

The resulting dependence on pressure and temperature can be found in equations 5.17 and 5.18. In order to take the influence of turbulence on the laminar flame speed into account, an effective laminar flame speed table is generated using the pre integrated β-pdf approach (appendix A) to obtain the influence of the Favre mean mixture fraction and mixture fraction variance on the effective laminar flame speed. The resulting table for methane can be seen in figures 6.11(b). The compared values for the laminar flame speed are a constant value of 0.42 m/s and a mean mixture fraction dependent laminar flame speed derived using the correlation of Müller et al. [111] (equation 6.3) and extended linearly to the flammability limits at ambient temperature and pressure. Also for the effective laminar flame speed table for methane and heptane, dependent on the mean mixture fraction and the mixture fraction variance, the correlation of Mülleret al. [111] is used in combination with the extension to the flammability limits. Figure 6.11(a) and 6.11(b) show the dependence of the effective laminar flame speed on the mean mixture fraction and on the mixture fraction variance, respectively. Of course for an internal combustion engine also the influence of pressure and temperature on the laminar flame speed has to be taken into account. 84 CHAPTER 6. SANDIA D FLAME

002 (a) sl(Ze) (b) sl(Z,e Zg)

Figure 6.11: Boundary conditions for the eﬀective laminar ﬂame speed

6.5 Results and Discussion

This section describes the results obtained with the improved combustion model compared to the standard flamelet model and experimental data. The experiment is marked with red symbols. The bars used in the figures reflect the rms degree of fluctuation. The simulation results obtained with the Flamelet model are marked with green lines, whereas the flamelet table generated with unity Lewis number is marked with the solid green line (Flamelet, Le) and the table taking differential diffusion effects into account is marked with a green line and square symbols (Flamelet, dd). For clarity, the comparison of unity Lewis number and differential diffusion effects are only plotted for the Flamelet model. The burning velocity model is used only with flamelet tables generated with unity Lewis number. There the influence of the choice of the laminar flame speed is examined. As flame speed closure model the Zimont model [165][166] is used with a constant laminar flame speed of 0.42 m/s (dark blue solid line: Zim., Le, sl=const.), a mean mixture fraction dependence (orange solid line: Zim., Le, sl(Z)) and an effective laminar flame speed dependent on mean mixture fraction and variance (bright blue solid line: Zim., Le, slTable). Additional further source term closure models together with the laminar flame speed table are compared; the Bradley model [27], bright green dashed line with delta symbols (Bra., Le, slTable) and the Lindstedt-Váos model [94], purple dashed line with delta symbols (Lin., Le, slTable). The dark blue dashed line (C7H16, Zim., Le, slTable) show the results obtained for heptane with the Zimont model, flamelet tables with unity Lewis number and a laminar flame speed table. The results are compared as function of the radial distance at different downstream positions at y/D = 15, 30, 45 and 60, where D = 7.7 mm is the inner diameter of the main jet injector, and r is the radius. The positions are marked in figure 6.1. 6.5. RESULTS AND DISCUSSION 85

Mixture Fraction In the analysis of experimental data, the deﬁnition of Bilger et al. [15] was applied to calculate the mixture fraction

0.5 (YH − Y2H ) /WH + 2 (YC − Y2C ) /WC ZBlgr = (6.18) 0.5 (Y1H − Y2H ) /WH + 2 (Y1C − Y2C ) /WC whereas the subscript H denotes hydrogen, and C carbon. The subscript 1 denotes the fuel inlet and 2 the oxidizer inlet. WH and WO are the atomic masses of hydrogen and oxygen.

(a) y/D = 15 (b) y/D = 30

Figure 6.12: Mixture fraction distribution at diﬀerent radial downstream positions

Figure 6.12 shows the mean mixture fraction distribution at different radial downstream positions. At y/D = 15 (figure 6.12(a)) close to the axis, best agreement is achieved with the Flamelet model where flamelet tables are used taking differential diffusion effects into account. Flamelet tables with unit Lewis number predict a slightly 86 CHAPTER 6. SANDIA D FLAME higher mixture fraction than the experiment. The same results are obtained with the burning velocity model with constant laminar flame speed. The Zimont model using a laminar flame speed table slightly under predicts the experiment close to the axis. 002 There the Zimont model with sl(Z) and the Bradley model with sl(Z,e Zg) show the same behavior and under predict the experiment even more. Worst agreement shows the Lindstedt-Váos model with a laminar flame speed table, which strongly under predicts the mixture fraction close to the axis. From r/D = 1 all the models show nearly the same behavior and over predict the experimental results. The over prediction is mainly a result of the turbulence model, which could already be seen in figure 6.6, where the stoichiometric mixture fraction is shifted further to the oxygen side than in the experiment. In figure 6.12(b) at y/D = 30, all models under predict the experiment close to the axis whereas the Zimont model using a table for sl shows good agreement with the Reynolds averaged experimental results followed by the Zimont model with sl(Z), the Zimont model with sl = const., which again shows equivalent results as the Flamelet model with unity Lewis number, and the Bradley model. The Lindstedt-Váos model followed by the Flamelet model with differential diffusion tables results in the lowest mean mixture fraction close to the axis. At r/D = 1.5, all the models again show negligible difference and over predict the experiment, which was already expected regarding figure 6.6. At y/D = 45 (figure 6.12(c)) the mean mixture fraction is over predicted by all models over the whole range. Between r/D = 0 and 2.5, best agreement is achieved by Flamelet model (Le = 1), Lindstedt-Váos model and the burning velocity model (flamelet + reaction progress variable) with constant laminar flame speed. Worst agreement shows the model with sl(Ze). The other models are in between. But with regard of the rms values of the mixture fraction, plotted as error bars, the models show good agreement with the experiment. Between r/D = 2.5 and 6 the difference between the models is negligible, whereas the flamelet model with differential diffusion flamelet tables predicts a slightly higher mixture fraction than the other models. The same behavior can also be seen at y/D = 60, with the difference that close to the axis the models over predict the experiment but reach them with increasing radial distance.

The experimental stoichiometric flame length of the methane flame is Lst = 338 mm, whereas the simulated stoichiometric flame length is 345 mm and for the heptane flame using the same turbulence and combustion models it is 403 mm. So the heptane flame is longer than the methane flame, which is reflected in figure 6.12. For all axial distances, the mixture fraction predicted by the heptane flame is higher than for the methane flame. Having a look at equation 2.8, which describes the global stoichiometric reaction, it can be seen that for the heptane flame with

C7H16 + 11.0O2 + 11.0 · 3.762N2 → 7CO2 + 8H2O + 11.0 · 3.762N2 (6.19) compared with the methane reaction

CH4 + 2.0O2 + 2.0 · 3.762N2 → CO2 + 2H2O + 2.0 · 3.762N2 (6.20) 6.5. RESULTS AND DISCUSSION 87

much more air has to be supplied for a stoichiometric reaction. As mixing occurs on the same time level, the ﬂame becomes longer for the heptane ﬂame.

Figure 6.13: Mixture fraction variance along axial direction

The axial distribution of the mixture fraction variance of the Sandia D flame can be seen in figure 6.13. As the maximum variance of 0.0145 is reached at about y/D = 30, it can be expected, that the main difference between the combustion models using an effective laminar flame speed table, and the combustion models using sl(Z), can be seen at this position close to the axis.

Mass Fraction of Methane The major species methane is presented in figure 6.14 at various downstream locations. At y/D = 15 all models under predict the measured mass fraction. Best agreement shows the Zimont model with a table for sl followed by the unity Lewis number flamelet model and constant laminar flame speed. These two models again show very similar behavior. Slightly lower values are predicted by the flamelet table accounting for differential diffusion and the Bradley model. The lowest mass fraction for methane and thus also the worst agreement is predicted by the Lindstedt-Váosmodel. From r/D = 1.5 onward the models show similar behavior and good agreement with the experiment. At y/D = 30 (figure 6.14(b)) a stronger difference between the models can be iden- tified. As already the mixture fraction of the simulations show a lower value, also the methane mass fraction is under predicted. Best agreement is achieved with sl(Ze). The Bradley model and the Lindstedt-Váos model, which are nearly equivalent, show a lower mass fraction followed by the two models, which predict very similar results, the Flamelet with differential diffusion flamelet tables and the Zimont model with a sl table. The Flamelet model with unity Lewis number flamelet table shows smaller 88 CHAPTER 6. SANDIA D FLAME

(a) y/D = 15 (b) y/D = 30

Figure 6.14: Mass fraction of CH4 at different radial downstream positions values than the one with differential diffusion flamelet tables. Again, this one shows the same results as the burning velocity model with constant laminar flame speed. At y/D = 45 and 60 all the methane is nearly consumed in the experiment as well as in the numerical results. As methane is only a combustion product within the heptane combustion, the values are of course lower. At y/D = 15 the mass fraction at first increases until r/D ≈ 0.75 and then decreases until r/D = 1.5. For larger radial distance no methane is predicted. At y/D = 30 the maximum mass fraction is predicted close to the axis and decreases to zero until r/D = 2. For y/D = 45 the maximum methane mass fraction is 1.3e-3 and for y/D = 60 it is 3.8e-5. 6.5. RESULTS AND DISCUSSION 89

Mass Fraction of Oxygen

(a) y/D = 15 (b) y/D = 30

Figure 6.15: Mass fraction of O2 at diﬀerent radial downstream positions

An even stronger influence of the different combustion models can be seen in the oxygen mass fraction distribution, which is plotted at different downstream positions in figure 6.15. At y/D = 15 (figure 6.15(a)) between r/D = 0 and 1 the model with sl(Ze) and the Bradley model show nearly the same behavior and agree well with the Favre averaged experimental results. Although the Lindstedt-Váos model shows the same behavior close to the axis it starts to over predict the values in this region with increasing radius. The Zimont model with an effective laminar flame speed table shows good agreement with the Reynolds averaged results. Close to r/D = 0 it can be seen, that the Flamelet model accounting for differential diffusion within the flamelet tables shows higher values than the unity Lewis number flamelet tables, but both under predict the experimental measurements. There is negligible difference between the Flamelet model, and the burning velocity model with constant laminar flame speed. The difference between the Zimont and Flamelet models decreases with increasing radius. For r/D > 1 these models under predict the experiment. The minimum value 90 CHAPTER 6. SANDIA D FLAME is nearly equal for these models and below 0.05. Although the Bradley model under predicts the experiment, too, its minimum value is larger. The Lindstedt-Váos model even over predicts the minimum oxygen mass fraction obtained by experiments. With further increasing distance the models show more and more the same behavior and the oxygen mass fraction is predicted lower than in the experiment.

At y/D = 30 all the models under predict the measurements although they show good agreement at radii larger than r/D = 5. The increased H2O value close to the axis is predicted by the Zimont model using sl(Ze) and the Lindstedt-Váosmodel very well; also the Bradley model, which at first shows smaller values and then larger ones shows the increased value of oxygen close to the axis. The Zimont model with a effective laminar flame speed table strongly under predicts the experiment close to the axis. The flamelet model with tables accounting for differential diffusion effects predicts higher values in this region than the one generated with the assumption of unity Lewis number, which shows the same behavior as the burning velocity model with constant laminar flame speed. Between r/D =1 and 4 the flamelet model with a differential diffusion flamelet table shows smaller values for the oxygen mass fraction than the other models, which show nearly similar results.

Between 0 < r/D < 2, at y/D = 45, (figure 6.15(c)), good agreement with the Reynolds averaged experimental results is achieved with the Flamelet (flamelet table with unity Lewis number) model, with the burning velocity model with constant laminar flame speed and with the Lindstedt-Váosmodel, followed by the Zimont and the Bradley model with a laminar flame speed table, which show negligible difference in that region. Slightly smaller values are predicted with sl(Ze). The Flamelet model accounting for differential diffusion within the flamelet tables predicts the lowest O2 mass fraction. Between 2 < r/D < 3.5 there is negligible difference between the models, except the Flamelet model with differential diffusion, but all under predict the measurement results. Between 3.5 < r/D < 8 the simulations tend to show good agreement with the experiment.

The same behavior of the models compared to each other can also be seen in figure 6.15(d) at y/D = 60. Close to the axis all models under predict the experimental values. Above r/D = 3 there is negligible difference between the models, and they agree well with the experiment, except the Flamelet model with differential diffusion, which gives smaller values. The reason for the lower prediction of the oxygen mass fraction at y/D = 45 and 60 is the higher prediction of the mixture fraction (see figure 6.12), which in turn is an effect of the turbulence model.

Similar trends in the mass fraction distribution of oxygen for the heptane flame. At y/D = 15, starting at a slightly smaller value than the equivalent model used for methane combustion close to the axis, decreases strongly and predicts a much smaller value than for methane, whereas the minimum is reached for a smaller radius, and increases strongly again until it shows the same behavior as the methane flame. The same shape behavior can be seen in figure 6.15(b) for y/D = 30. For y/D = 45 and 60 the shape is also the same as for the methane models and the experimental results, but again the mass fraction predicted for small radii is lower than for the methane flame. 6.5. RESULTS AND DISCUSSION 91

(a) y/D = 15 (b) y/D = 30

Figure 6.16: Mass fraction of H2O at diﬀerent radial downstream positions

Mass Fraction of Water Figure 6.16 shows the mass fraction distribution of the product species water. At y/D = 15, close to the axis, very good agreement with the measurements is achieved for the burning velocity models using the Zimont, Lindstedt-Váos and Bradley model for source term closure and an effective laminar flame speed table and sl(Ze). The Zimont model using sl(Ze) and the Bradley model correspond well with the Favre averaged measurements, whereas the other two models show better agreement with the Reynolds averaged experimental results. Near the axis worst agreement is achieved with the two flamelet models and the partially premixed model with constant laminar flame speed. All models except the Lindstedt-Váos model over predict the maximum mass fraction of water and also the position is shifted to the oxygen side. So above r/D = 1 the simulations over predict the mass fraction of hydrogen, whereas burning velocity models using a laminar flame speed table or sl(Ze) show slightly better agreement than the flamelet model or the partially premixed model with constant laminar flame speed. The flamelet model with differential diffusion flamelet tables predicts the highest H2O mass fraction. 92 CHAPTER 6. SANDIA D FLAME

A bit further downstream at y/D = 30 all models over predict the water mass fraction until r/D = 4.5. The measurements show an increase of water until a maximum value is reached at about r/D = 1.25, where it starts to decrease again. This shape is only predicted by the Zimont model with sl(Ze), the Bradley and the Lindstedt-Váos model with a laminar flame speed table and with flamelet tables which take the effects of differential diffusion into account. The position of the maximum is represented best by the Bradley and Lindstedt-Váos model. Although the Zimont model with constant laminar flame speed and a laminar flame speed table as well as the Flamelet model (Le) also show a slight increase close to the axis, it is not as strongly developed as for the other models. Above r/D = 2.5 there is negligible difference between the models expect again for the Flamelet model with differential diffusion flamelet tables, which predicts higher values. Figure 6.16(c) shows the mass fraction of water in radial direction at y/D = 45. Nearly all model show a relatively good agreement with the Reynolds averaged measurement results. Close to the axis, the Flamelet model with unity Lewis number together with the Zimont model with constant laminar flame speed predicts the lowest values, whereas the Zimont model with sl(Ze) predicts slightly higher values. The other models lie in between. The Favre averaged results are lower. For r/D > 0.5, the Flamelet model with the flamelet table which accounts for differential diffusion strongly over predicts the results. For r/D > 6.5 the other models under predict the measurements. Except for the Flamelet model with differential diffusion, all models show nearly the same behavior for r/D > 2. The same behavior of the models can be seen at y/D = 60 in figure 6.16(d), although at this position even the Reaynolds averaged results are over predicted for small radiuses. This changes with r/D > 7, where also the Favre averaged measurements are under predicted.

Also for the mass fraction of H2O the predicted shape is the same using heptane instead of methane. The maximum value at y/D = 15 is reached for a smaller radial distance, which is the same at y/D = 30. At y/D = 15 near the position of highest mass fraction, the prediction lies in between the calculation of the methane flame with the Lindstedt-Váos model and the other models. Closer to the axis and with increasing radial distance as well as at further downstream positions, a smaller mass fraction is predicted over the whole radial direction than for the methane flame. This is also a result of the reaction mechanism. Having again a look at equations 6.19 and 6.20, it can also be seen, that the ratio of moles of CO2 to H2O is 7:8 for heptane, and 1:2 for methane. Thus less hydrogen mass fraction is predicted for the heptane. In the following it can be seen that this is vice versa for CO2, where the heptane/air flame shows a higher mass fraction for carbon dioxide than the methane/air flame.

Mass Fraction of Carbon Dioxide As second combustion product carbon dioxide is examined at different downstream positions along radial direction in figure 6.17. Figure 6.17(a) shows the results of measurement and simulation at y/D = 15. Between r/D = 0 and 1 the Zimont model with sl(Ze), the Lindstedt-Váos model and the Bradley model show the same behavior; also the Zimont model with the laminar flame speed table and the Lindstedt-Váosmodel 6.5. RESULTS AND DISCUSSION 93

(a) y/D = 15 (b) y/D = 30

Figure 6.17: Mass fraction of CO2 at diﬀerent radial downstream positions

as well as the Flamelet model (unity Le) and the Zimont model (sl = const.). Between r/D = 0 and 1.5, the first mentioned models slightly under predict the measurements, the second at first also slightly under predict the measurement, but then show good agreement, and the third mentioned slightly over predict the measurements close to the axis, but with increasing distance also show good agreement. The maximum value is over predicted by all models except the Lindstedt-Váos model which predicts nearly the same value as the Favre averaged experimental results, and are shifted to a larger radial distance, whereas the Flamelet model (Flamelet, dd) predicts the highest mass fraction. For r/D > 1.5 all models over predict the experiment. Figure 6.17(b) shows the radial distribution at y/D = 30. Close to the axis between 0 ≤ r/D < 1.2, the lowest mass fraction of CO2 is predicted by the Zimont model with sl(Ze). The Bradley and Lindstedt-Váos model correspond very well with the Favre averaged experimental data. Higher mass fraction is predicted by the Zimont model followed by the Zimont model with constant laminar flame speed and the Flamelet model with unity Lewis number. Both show also negligible difference. The Flamelet 94 CHAPTER 6. SANDIA D FLAME model which accounts for differential diffusion effects within the flamelet table predicts the highest values, even for the maximum mass fraction. At the maximum all other models show negligible difference. Compared to the experiment it is over predicted and shifted to the oxygen side. Above r/D = 3.5, the Flamelet model (Flamelet, dd) predicts slightly lower values than the other models, and the Lindstedt-Váos model shows slightly higher values. At y/D = 45 (figure 6.17(c)) the same behavior of the models compared to each other as in figure 6.17(b) can be seen. Although close to the axis the difference is smaller, and all models slightly over predict the experimental results. With increasing radial distance the simulation again is shifted to the oxygen side, whereas again the Flamelet model (Flamelet, dd) predicts a slightly lower mass fraction of carbon dioxide. Further downstream at y/D = 60 (figure 6.17(d)), the model behavior compared to each other changes. Although the Favre averaged experimental results are over predicted between ≤ r/D < 8.5, the simulation only slightly over predicts the Reynolds averaged results above r/D = 4. Between r/D = 0 and r/D = 3.5, the lowest mass fraction is derived with Flamelet model with unity Lewis number, and the highest value is predicted by the Zimont model with the laminar flame speed dependant on the mean mixture fraction. The other models lie in between, but they are all close together. Above r/D = 4 the models again show negligible difference except the Flamelet model (differential diffusion), which show the same behavior as at the other downstream positions.

Comparing the heptane flame to the methane flame, the predicted CO2 mass fraction is much higher at all downstream positions. With increasing axial and radial distance, the predicted mass fraction of the heptane flame reaches the same values as the methane flame.

Temperature Distribution Examining the temperature distribution at different downstream positions, it can be seen that at y/D = 15 (figure 6.18(a)), the Zimont model with sl(Ze) and the model using the Bradley correlation for the turbulent flame speed show nearly the same behavior. At first the temperature is slightly over predicted, whereas with increasing radial distance they show good agreement with the Favre averaged experimental results. But the maximum value differs for both and is predicted at a larger radius than within the experiments, whereas the maximum temperature is higher for the Zimont model than for the Bradley model. Also the Zimont model using the effective laminar flame speed table agrees quite well with the experiment close to the axis, and with the Reynolds averaged results with increasing radius. The Lindstedt-Váos model predicts slightly higher values for small radius and lower values with increasing radius. Again, the maximum temperature is shifted to the oxygen side, as also the stoichiometric mixture fraction is shifted to a larger radius as can be seen in figure 6.12(a), which in turn is due to the turbulence model, but the maximum value corresponds quite well to the experiment. Worst agreement close to the axis is achieved with both flamelet models and the Zimont model with constant laminar flame speed. With increasing radius after the maximum temperature has been reached, the experiment is over predicted by all the models, whereas the Zimont model with sl(Ze) predicts the lowest, and the Flamelet 6.5. RESULTS AND DISCUSSION 95

(a) y/D = 15 (b) y/D = 30

Figure 6.18: Temperature distribution at diﬀerent radial downstream positions

(Flamelet, dd) models predicts the highest values. All other models lie in between. Figure 6.18(b) shows the results at y/D = 30. Close to the axis best agreement is achieved with the Zimont model with sl(Ze), followed by the Lindstedt-Váos model, Bradley model, Flamelet model where differential diffusion effects are accounted for in the flamelet tables, the Zimont model using the effective laminar flame speed table, the Zimont model with sl = constant and the Flamelet model with unity Lewis number, which predicts the highest values. The difference between the last to models is negligibly small. All models over predict the temperature and the maximum is again shifted to a larger radius. For the Flamelet model (Flamelet, dd) the maximum temperature is even slightly larger and shifted further to a larger radius. This behavior remains the same when the temperature is decreasing, whereas the other models predict slightly smaller values and show only small differences whereas again the Zimont model with sl(Ze) predicts the lowest and the Lindstedt-Váosmodel the highest temperature. The other models lie in between. The difference between the models becomes even smaller at y/D = 45 (figure 6.18(c)). 96 CHAPTER 6. SANDIA D FLAME

Only the Flamelet model which accounts for differential diffusion effects within the flamelet tables predicts a higher temperature. The experimentally measured temperature is lower than the simulated temperature. Having a closer look at the point e.g. r/D = 2.2, the experiment results in a mixture fraction of 0.22, and the Zimont model with the laminar flamelet table predicts a mixture fraction of 0.27 (see figure 6.12(c)). Having an additional look at figure D.2(d) in appendix D.2, it can be seen that the temperature strongly increases from Z = 0.22 to 0.27 of several hundreds of Kelvin. The same behavior of the models with respect to the experiment as at y/D = 45 can be seen for y/D = 60. All models again over predict the temperature, whereas the model which accounts for differential diffusion within the flamelet tables again results in the highest values. Close to the axis the flamelet model with unity Lewis number shows the lowest temperature, and the Zimont model with sl(Ze) shows the highest. For the heptane flame, the temperature predicted close to the axis at y/D = 15, is approximately the same as for the methane flame, but the maximum temperature is much higher, whereas with increasing radius from the temperature is approximately the same as for the simulated methane flame. The same behavior can be seen for y/D = 30, although the difference of the maximum temperatures is smaller. At y/D = 45, the heptane flame shows nearly the same temperature distribution at the methane flame calculated with the flamelet model with differential diffusion, but shifts towards the results of other models for values larger than r/D = 3. At y/D = 60 the temperature close to the axis is higher than for the methane flame, whereas it reaches the same temperature closer to the air region.

Summary As could already be seen for the hydrogen jet flame, the agreement of the simulation with the experimental results of the piloted methane/air diffusion flame is strongly dependent on the selected turbulence model. The mixture fraction is a conserved scalar and thus is only influenced by the flow field. But all mass fraction are dependent on the correct prediction of the mixture fraction. This can be seen e.g. for oxygen, where the mass fraction is predicted too low as the mixture fraction is predicted too high at the positions y/D = 45 and 60. The overprediction of water above y/D = 30 is a further effect of the mixture fraction (see figure 6.6). For Z > Zst, the simulated value is smaller than in the experiment. In that case, more fuel has been consumed and more products like water are produced. For Z < Zst, the simulated mixture fraction is higher than the experiment. Thus more oxygen has been consumed which results in a higher prediction of the products. The main difference between the combustion models can again be seen in regions, where the variance of the mixture fraction is high. The laminar flame speed was either kept constant in the whole flow field, or was only dependent on the mean mixture fraction or also on the mixture fraction variance. As for low fluctuations the pdf used to generate the effective laminar flame speed table degenerates to a dirac delta peak, the difference between sl(Z) and the effective laminar flame speed is negligible. The main influence can be seen in regions with a high fluctuation. Chapter 7 Low Swirl Burner

As third validation case a premixed flame generated by a low swirl burner has been selected, examined experimentally by Gregor [66] and Petersson et al. [123], which is similar to the design used by Cheng et al. [40][78]. The premixed flame is stabilized by the swirling flow field and operates in the thin reaction zone regime (see figure 2.3). It has already been investigated numerically by Nogenmyr et al. [112], who simulates the flame with a LES based level-set G-equation approach as a premixed flame model.

Figure 7.1: Low swirl burner

A lean premixed methane/air mixture enters through four nozzles, equally distributed in circumferential direction. From there it moves upward into the burner and passes two perforated plates, after which the flow is divided in two flow streams. The inner stream passes a perforated plate. The surrounding stream passes the swirl generator, which consists of eight curved vanes and an exit angle of 37◦. The nozzle exit has a diameter of 50 mm. Due to the design of the burner, no recirculation zone is formed and the flame is broadened by the swirl. Its main characteristics is that a 2D turbulent flame exists in the core, and the outer region offers the opportunity to investigate a stratified, lean premixed flame with local extinction due to the overly lean mixture. Thus the local equivalence ratio and the mixture fraction vary spatially. With this varying mixture fraction, also the laminar flame speed varies. The flame is stabilized about 0.63 diameters above the burner, where the turbulence intensity is high and the flow speed is low. Figure 7.1 shows the positions, where experimental 98 CHAPTER 7. LOW SWIRL BURNER data of the temperature distribution are available. In the next chapter an overview over the computational domain and boundary conditions is given, followed by a discussion of the results. Correlations for the laminar flame speed and flammability limits have already been discussed in chapter 6.4. Due to the limited experimental database available, only the influence of the different options to define the laminar flame speed within the burning velocity model are examined, as this is the aim of this work.

7.1 Computational Domain and Boundary Conditions

An air coflow of diameter 600 mm surrounds the burner assembly with a velocity of approximately 0.5 m/s to avoid dust particles from the environment entering the measurement domain. The swirling mixture of methane and air enters the domain with an equivalence ratio φ of 0.62 through a nozzle with an inner diameter of 50 mm. The details of the burner design are not taken into account in the simulations, as the exact design data are not available. Petersson [123] describes the flame as stable and symmetric. Although previous studies [139] showed a precessing vortex core for high swirl flows, this behavior was not noticed for this low swirl burner configuration. Thus in order to reduce computational effort, the assumption and usage of a ’quasi 2D’ case for the combustion CFD simulation is allowed. To be able to solve this ’quasi 2D’ flame, a plane with the dimension of 0-300 mm in radial and 0-600 mm in axial direction has been rotated by three degrees to obtain a 3D case with one element in the third direction. The rotation axis has been replaced by a tube of 0.02 mm radius. The computational domain with 119,200 nodes and 59,052 elements has been refined in the vicinity of the nozzle exit and the experimental flame bowl. For the axis wall and the opposite outer wall, an adiabatic free-slip wall has been placed. On the rotated surface periodic boundary conditions are used. As inlet condition, the mean velocity and velocity fluctuations measured 2 mm downstream of the exit of the nozzle by Petersson et al. [123], shown in figure 7.2 are provided. The mean profiles are quite symmetric, there is no swirl at the center, and the rms values reach a minimum at the centerline. The co-flowing air has a velocity of about 0.5 m/s. The fuel/air mixture as well as the surrounding air are injected at ambient temperature and pressure. The turbulent kinetic energy k and dissipation rate ε are calculated according to equation 5.2 and 5.3, respectively. As turbulence models, the SST and the BSL Reynolds Stress model are compared. As there exists only free slip walls, the SST model reduces to the k − ε turbulence model, where the transport equation for the dissipation rate is transformed to the ω based transport equation (see chapter 3.3). The combustion model used is the laminar flamelet model together with an additional transport equation for the reaction progress variable, using the Zimont model to close the source term. As discretisation scheme again the High Resolution scheme is selected as for the Sandia D flame and the hydrogen jet flame. The generated flamelet tables assume unity Lewis number. 7.2. RESULTS AND DISCUSSION 99

(a) Normalized radial inlet velocity (b) Turbulent kinetic energy

Figure 7.2: Proﬁles of mean velocity and rms values

7.2 Results and Discussion

This chapter shows the results obtained with the improved combustion model. The effects of using a constant laminar flame speed of 0.42 m/s (dark blue solid line in the following figures) is compared to the laminar flame speed evaluated with the mean mixture fraction (orange solid line) and the effective laminar flame speed table with a dependence on the mean mixture fraction and its variance. As the injected mixture consists again of methane and air, the same correlations and flammability limits as for the Sandia D flame (see chapter 6.4) are used to obtain the mixture fraction dependent laminar flame speed and the laminar flame speed table. In the literature (Gregor [66]), only the temperature distribution at different downstream locations marked in figure 7.1 is available. Thus in the following only the combustion models are compared to each other for the distribution of mixture fraction and of mass fractions of methane, oxygen water and carbon dioxide. The figures show the comparison at axial positions of y = 15, 30, 40 and 50 mm.

Temperature Distribution The temperature distributions resulting from different laminar flame speed descriptions are compared to the experimental results in figure 7.3. Compared to the hydrogen jet flame (chapter 5) and the Sandia D flame (chapter 6), the difference in using a constant laminar flame speed or a laminar flame speed which also depends on the mean mixture fraction is even more obvious. The choice of constant laminar flame speed gives completely unrealistic results. The low swirl burner produces a lifted flame, which cannot be predicted with a constant laminar flame speed, which can be seen especially at y = 15 mm in figure 7.4(a). While the other models and the experiment still show ambient temperature, the temperature predicted with the model using constant laminar flame speed strongly increases. Also at all downstream positions the behavior of this model is unphysical. 100 CHAPTER 7. LOW SWIRL BURNER

(a) y = 15 mm (b) y = 30 mm

Figure 7.3: Temperature distribution at diﬀerent radial downstream positions

At y = 30 mm (figure 7.3(b)) the models using the SST turbulence model based on sl(Ze) and the effective laminar flame speed table show reasonable agreement with the experiment, although the experiment does not show a temperature maximum at r/R = 0. It predicts at first a lower temperature which increases with increasing radius up to approximately r/R = 0.5 where it reaches its maximum temperature and then decreases again to ambient temperature. According to Gregor [66], close to the axis, the temperature tends to be under predicted due to the estimated, mean Rayleigh scatter cross-section, whereas it is rather over predicted further outside. Having an additional look at the experimentally determined temperature standard deviation in figure 7.4, it can be seen in figure 7.4(b) that it is about 400 K close to the axis, increases to nearly 600 K and drops to negligible values with increasing radial distance. Thus it can be said that the simulated results are in fairly good agreement with the experiment. Comparing the simulation results, the combustion model using the effective laminar flame speed table predicts slightly higher temperatures close to the axis than the mean mixture fraction dependent laminar flame speed sl(Ze). This difference decreases with 7.2. RESULTS AND DISCUSSION 101 increasing radius. The model using the BSL turbulence model and sl(Ze) is lifted higher than the other flames, thus there is still no temperature increase at y = 30 mm position.

(a) y = 15 mm (b) y = 30 mm

Figure 7.4: Temperature standard deviation at diﬀerent radial downstream positions

The same behavior of the SST models can also be seen at y = 40 mm. The combustion models using sl(Ze) and a laminar flame speed table predict the maximum temperature close to the axis, the first model again predicts a slightly lower temperature. The experiment has its maximum temperature of 1400 K again at r/R = 0.5. In that region, the standard deviation is about 450 K (see figure 7.4(c)). Close to the axis the experimental temperature is 950 K, and the standard deviation is 600 K. So it can be said, while the experimental maximum temperature are at different positions, the simulation agrees reasonably with the experiment. The BSL model also shows an increase in temperature. Although it is much too low compared to the experiment, it predicts a lower temperature close to the axis and increases to a maximum at r/R = 0.45. At y = 50 mm (figure 7.3(d)), the experiment does not show such a strong lowered temperature distribution close to the axis. But it also increases from approximately 102 CHAPTER 7. LOW SWIRL BURNER

1420 K at the axis to 1540 K at r/R = 6, whereas the simulations with the SST turbulence model show their maximum temperature again at the axis where again the simulations with an effective laminar flame speed table predict higher temperatures than using sl(Ze). The maximum temperature obtained using a constant laminar flame speed is much too high, and the shape is again unphysical. The shape of the temperature distribution using the BSL model for r/R < 0.8 corresponds well with the experiment, although the temperature is still under predicted, which is due to the fact that the flame is lifted more in this simulation and thus reaches these temperature values at further downstream position. The experimentally obtained standard deviation shows again high values, which range between 400 K close to the axis, decrease and increase again to a maximum value of 550 K in the shear layer. The experimental data are obtained from Gregor [66]. The temperature distribution presented by Petersson et al. [123] at a downstream position of 39.9 mm and 35.4 mm show a slightly different behavior. There the maximum temperature is also at the axis and the difference of the temperature at the axis to the maximum temperature is not so large, more similar to the results obtained in the simulations with the SST turbulence model. Although Petersson et al. [123] and Gregor [66] examine the same low swirl burner at Re = 20000 and φ = 0.62, it is not clear if the boundary conditions provided for the simulation are really the same, as there is no comparison available. Due to the complexity of the BSL Reynolds Stress turbulence model, where the turbulence is no longer isotropically, seven additional transport equations for the Reynolds stresses and the turbulent frequency are solved. Thus a very low timestep of the order of 1e-6 s is used compared to the SST model, where a timestep of 1e-2 s is used. Thus it takes a very long time to obtain a converged solution. As the main aim of this work is the combustion modeling, the BSL Reynolds stress model has only been used for the simulation with sl(Ze), as there is also no reliable experimental data-set available as for the hydrogen and methane jet flames examined in chapters 5 and 6. Although Pe- tersson [123] describes the flame as stable and symmetric, it is not really clear if there isn’t a small PVC which induces additional turbulence which has to be accounted for in a 3D unsteady simulation.

Mixture Fraction Figure 7.5 shows the mixture fraction profiles of the low swirl burner. As at y = 15 mm only the flame with constant laminar flame speed is burning, it predicts a higher mixture fraction for r/R > 0.6 than the others, which show the same mixture fraction distribution as for the still cold mixture. All models using the SST turbulence model show the same values, the BSL turbulence model predicts a higher mixture fraction for r/R > 0.7. The difference in the mixture fraction distribution increases for the two combustion models sl(Ze) and an effective laminar flame speed table and the SST turbulence model with increasing axial distance (figures 7.5(b) to 7.5(d)). In agreement with the temperature distributions, the model with the effective laminar flame speed table predicts a slightly higher mixture fraction. This difference increases with increasing radial distance. The BSL model predicts higher values for the mixture fraction with increasing radius and increasing downstream position. 7.2. RESULTS AND DISCUSSION 103

(a) y = 15 mm (b) y = 30 mm

Figure 7.5: Mixture fraction distribution at diﬀerent radial downstream positions

Figure 7.6 shows the mixture fraction variance of the low swirl burner. A maximum of only 1e-4 is reached close to the injector exit. The mixture fraction in the whole domain reaches only very small values. Therefore the probability density function nearly reaches a dirac delta peak everywhere and it follows that the looked up values of the effective laminar flame speed table are close to sl(Z). In the following, only a small difference can be seen between slT able and sl(Z).

Mass Fraction of Methane The mass fraction distribution of methane at different downstream positions is given in figure 7.7. At y = 15 mm (figure 7.7(a)) there is no difference between the combustion models using sl(Ze) and the effective laminar flame speed table using the SST turbulence model as combustion did not yet take place. The value is the highest in the vicinity of the axis. Due to the swirl it mixes with the surrounding air and decreases in the range around the nozzle diameter. With the BSL turbulence model, the mixing 104 CHAPTER 7. LOW SWIRL BURNER

Figure 7.6: Mixture fraction variance process is stronger and a higher value of methane is predicted for r/R > 0.8. As the ﬂame is not lifted when using the model using constant laminar ﬂame speed, methane is already consumed where it mixes with the air. Thus this model predicts a lower CH4 mass fraction than the other models with increasing radial distance.

As the flame is not yet burning with the BSL model and sl(Ze) at y = 30 mm (see figure 7.7(b)), the shape of the CH4 mass fraction distribution is nearly the same as at y = 15 mm, except that it is more broadened. Using constant laminar flame speed, the flame shape is completely different, thus it predicts still the inlet condition mass fraction of methane close to the axis, which decreases rapidly to 0 at r/R = 1 where it is already completely consumed. As the temperature is the highest for the both SST models (sl(Ze) and effective laminar flame speed table), which is a sign that the mixture is already burning at this position, the mass fraction of methane is lower in the vicinity of the axis, and increases with radial distance until r/R = 0.7, where it decreases again. As with the effective laminar flame speed table the temperature is higher than for sl(Ze), the mass fraction of CH4 is lower in the range between the axis and r/R ≈ 1, which can also be seen at the further considered downstream positions.

Mass Fraction of Oxygen Figure 7.8 shows the mass fraction distribution of oxygen. At y = 15 mm (figure 7.8(a)), the radial mass fraction distribution is the same for all models as no oxygen is consumed yet, except the combustion model using constant laminar flame speed, where oxygen is already consumed in the shear layer region. At y = 30 mm (figure 7.8(b)) the constant laminar flame speed model shows again the same behavior, whereas the minimum ’peak’ is broadened, as the flame is also 7.2. RESULTS AND DISCUSSION 105

(a) y = 15 mm (b) y = 30 mm

Figure 7.7: Mass fraction of methane at different radial downstream positions broadened with increasing axial distance. As the BSL model is not yet burning, the oxygen distribution is the one of a cold mixture, so the shape is nearly the same as at y = 15 mm. For the SST models with sl(Ze) and an effective laminar flame speed table, oxygen is consumed close to the axis, so they predict a lower value there until r/R < 0.6. More oxygen is consumed by the model with an effective laminar flame speed table. This can also be seen in figure 7.8(c), where these two models again predict the lowest oxygen mass fraction close to the axis up to r/R = 1. The mass fraction is also lowered with the BSL model, although the value is still much higher. At about r/R = 0.4 it shows a minimum and increases again to the value of the surrounding air. In the vicinity of the axis, the model with sl = const. shows the highest value and decreases rapidly where is has a broad minimum in the shear layer until it increases again, which is also the case at y = 50 mm. This can be seen in figure 7.8(d). There the oxygen distribution compared to the position of y = 40 mm broadens and lowers for all models. Neglecting the model with 106 CHAPTER 7. LOW SWIRL BURNER

(a) y = 15 mm (b) y = 30 mm

Figure 7.8: Mass fraction of oxygen at diﬀerent radial downstream positions constant laminar ﬂame speed, the BSL model even predicts lower values than the SST models in the region of r/R = 0.8 to 1.5.

Mass Fraction of Water Having a look at the mass fraction distribution of water in figure 7.9, it can be seen again, that only the model with constant laminar flame speed is burning at y = 15 mm (figure 7.9(a)), and shows an increased value of the combustion product. As all the other models predict a lifted flame as obtained in the experiment, no water is still produced. As this is also the case for the BSL model at y = 30 mm (figure 7.9(b)), it shows no increased value of water, compared to the other models. The highest mass fraction is obtained with a constant laminar flame speed. The SST models with sl(Ze), which predicts a slightly lower mass fraction than the one with a laminar flame speed table predict the highest combustion product close to the axis and decrease to 0 until r/R = 0.8. 7.2. RESULTS AND DISCUSSION 107

(a) y = 15 mm (b) y = 30 mm

Figure 7.9: Mass fraction of water at diﬀerent radial downstream positions

At y = 40 mm (see figure 7.9(c)) even the model with constant laminar flame speed predicts an increased value at the axis. So does the BSL model with sl(Ze), which predicts its maximum at r/R = 0.45. As in that case the flame temperature is still much smaller than for the comparable SST model, it predicts a much smaller mass fraction of water. The SST model using a laminar flame speed table produces a slightly higher mass fraction than the one with sl(Ze). For both, the maximum is reached at the axis and drops down to 0 at r/R = 1.1. At y =50 mm (figure 7.9(d)) the predicted water mass fraction is even higher, except for the case with constant laminar flame speed, where the maximum is only broadened. The BSL model predicts lower values than the comparable SST model with sl(Ze) for r/R < 0.8. For r/R > 0.8 it predicts slightly higher values. The SST model with an effective laminar flame speed table again predicts slightly higher values than the SST model with sl(Ze). Both have their maximum at the axis and decrease to 0 at r/R = 1.5. 108 CHAPTER 7. LOW SWIRL BURNER

Mass Fraction of Carbon Dioxide

(a) y = 15 mm (b) y = 30 mm

Figure 7.10: Temperature distribution at diﬀerent radial downstream positions

Exactly the same behavior of the models can be seen for the combustion product carbon dioxide given at different radial downstream positions in figure 7.10. At y = 15 mm (figure 7.10(a)) of course the only model which predicts an increased value due to the fact that the flame is already burning is the constant laminar flame speed model. All the other do not result in an increased value as expected. At y = 15 mm (figure 7.10(b)), which is also expected, only the flame simulated with the BSL model does not burn, and thus this model is also not able to predict an increased mass fraction. Similar to the results for water, the SST model with sl(Ze) predicts a slightly lower increased value of CO2 than the model with a laminar flame speed table. Their maximum CO2 concentration is reached at the axis and reduces to 0 at r/R = 0.7.

The maximum CO2 mass fraction predicted with these two models is approximately 7 times higher at y = 40 mm than at y = 30 mm. There, also an increased mass fraction is obtained with the BSL model, but with a mass fraction of 0.009 it is still 7.2. RESULTS AND DISCUSSION 109 much lower than for the SST model where a mass fraction of 0.071 is obtained. As discussed previously, the constant laminar flame speed model results in an unrealistic behavior with a minimum at the axis and a maximum plateau between 0.8 < r/R < 1.2. As could be seen in the previous discussions, this maximum plateau broadens with increasing axial distance. Thus it is even broader at y = 50 mm, as can be seen in figure 7.10(d). The BSL model reaches a value of 0.068 at the axis where the comparable SST model with sl(Ze) predicts a value of 0.091. The SST model with an effective laminar flame speed results in a slightly higher mass fraction. The BSL model predicts again an increase to a maximum at r/R = 0.43. All three models do not predict an increased mass fraction for CO2 for r/R > 1.7.

Summary Summarizing the results of the lean methane/air low swirl burner, not only the choice of the turbulence model has a strong effect on the results, but also the combustion model. It could be seen that using a constant laminar flame speed in that case ignited the mixture already at the injector exit, and was not able to predict a lifted flame. Using sl(Z) or an effective laminar flame speed table, the laminar flame speed is almost 0 at this position compared to the assumption of a constant laminar flame speed everywhere in the flow field. As the mixture fraction variance (see figure 7.6) is below 1e-4 everywhere in the flow field, the difference between sl(Z) and an effective laminar flame speed table is only small. Chapter 8 Conclusion and Outlook

With the aim of the automobile industry to fulﬁll the EU exhaust gas regulations and still remain competitive, it is necessary to exactly understand what is happening within the combustion chamber of an internal combustion engine. Experiments are very expensive and provide limited information, as it is hardly possible to measure all necessary quantities to get a deeper understanding of the combustion process and thus to be able to further improve it. Thus combustion CFD becomes a more and more important tool to be able to further improve the processes in internal combustion engines. To remain competitive, time is an important factor for the development process, which still often disqualiﬁes the LES approach. Thus the focus of this work lied on the combustion model improvement for RANS simulations.

8.1 Conclusion

Most real combustion processes do not occur under perfect conditions like realized in perfect non-premixed or premixed laboratory flames. In direct injection, internal combustion engines the fuel is injected into the combustion chamber during the compression stroke, where it starts mixing with the air. Thus it is able to partially premix with the surrounding air until it is ignited near the top dead center. A further option is the use of multiple injections where partly mixed regions and burned regions occur in the combustion chamber and non-premixed fuel is injected. So pure non-premixed or premixed combustion models are not applicable to this situation. As fuel is still injected non-premixed into the combustion chamber, a model which has its origin from diffusion flames has been used as basis, namely the flamelet model where transport equations for the mean mixture fraction and the mixture fraction variance are solved. The mixture fraction is equal to 1 in the fuel stream, and 0 in the oxidizer stream. To be applicable to both the partially premixed and non-premixed combustion regimes, an additional transport equation for the reaction progress variable is solved, which is defined as 0 for the unburned mixture and 1 for the burned mixture. To be able to close the source term of the reaction progress transport equation, different approaches are compared, the flame speed closure model developed by Zimont [164], a further turbulent flame speed correlation developed by Bradley et al. [27] and the Lindstedt and Váos [94] model. In internal combustion engines, pressure and temperature change locally and tem- porally and thus the laminar flame speed as well as the flammability limits change as well. As the correlations for the laminar flame speed were derived in the context of premixed flames, their region of applicability does not span the whole mixture fraction range of 0 ≤ Z ≤ 1 occurring in non-premixed and partially premixed flames. Outside their validity domain, the correlations show extremely unphysical behavior. So the laminar flame speed is extrapolated linearly to zero at the temperature and pressure 8.2. OUTLOOK 111 dependent flammability limits. For the flammability limits, correlations are derived for the pressure and temperature dependence of methane, heptane and hydrogen flames. Furthermore to take the turbulence effects into account, a pdf integration has been performed to obtain an effective laminar flame speed table. Zimmermann et al. [161] compared different correlations for the laminar flame speed for a hydrogen diffusion flame. Within this work, three test cases are examined with the commercial CFD code AN- SYS CFX. The first is the hydrogen jet flame examined experimentally by Barlow and Carter [10][8][9] and Flury [62], as second test case the Sandia D flame was selected, a methane/air piloted jet flame, where experiments have been performed by Barlow and Frank [11] and Schneider [138]. The Sandia D flame was also examined hypothetically using a heptane/air mixture. The third case is a low Swirl burner where a lean mixture of methane and air is injected, examined by Gregor [66] and Petersson et al. [123]. In all these cases two models show promising results: The ones using the mean mixture fraction dependent and to the flammability limits extended laminar flame speed, and the effective laminar flame speed table. It could be seen that the Eddy Dissipation Model or the assumption of constant laminar flame speed is in no case sufficient to predict partially premixed flames.

8.2 Outlook

As especially for the low swirl burner the differences between the model results and the experiments are clearly to see, a more detailed experimental database is necessary. The choice of an appropriate turbulence model is difficult as not enough data are available. So the comparison of the combustion models with the experiment is difficult as it is not clear which turbulence model predicts the correct cold flow field. In addition, the behavior of the species mass fraction and mixture fraction could not be compared as no experimental data are available. According to Petersson et al. [123], who examined the low swirl burner, countergradient diffusion associated with thermal expansion and the acceleration of fluid passing the flame front dominates in this flow. Within this work, countergradient diffusion was not taken into account. In a first step it would be interesting to see, if this effect also dominates in internal direct injection combustion engines, and if this is the case how strong is the influence of this effect? As could also be seen in chapter 5, where the hydrogen jet flame was examined and in chapter 6 where the Sandia D flame was examined, accounting for differential diffusion in the flamelet tables may have a strong effect on the results, but within the CFD simulation only unity Lewis number is assumed. So this is a further interesting subject to examine the effect of differential diffusion within the simulations. A further step would also be to find an experimental configuration with a detailed set of information about velocity, turbulence, mass fractions of several species and temperature, where the pressure and temperature dependence plays an important role. Within direct injection internal combustion engines these two values strongly change, as could be seen in chapters 5.2 and 6.2. They also have a strong influence on the minor species which are important for pollutant prediction. 112 CHAPTER 8. CONCLUSION AND OUTLOOK

As flamelet tables for non-premixed cases do not take effects from premixed flames into account and vice versa, a further interesting subject to examine would be the combination of both. For example the combination of a counterflow diffusion flame as used within this work combined with a flamelet generated manifold [115] could be interesting to cover the whole range of partially premixed flames. A good choice of the turbulence model is also an important factor to fit the experimental results. The difference in computational effort and time is steadily decreasing between URANS, LES or SAS simulations, thus an extension of the developed model could also be an interesting further step. As the simulation of a real engine is time consuming, and only few experimental data are available for combustion model development, a suitable experimental setup is necessary to cover the most important features occurring in internal combustion engines. The mixture fraction field should be well characterized, and the turbulence level should be high. Furthermore the flame should also be able to propagate, and should not be purely diffusive. Appendix A The Presumed β-pdf Approach

Already in 1895, Reynolds [133] postulated the existence of statistical mean values. He proposed to decompose the variable ψ, which can be the component of the velocity vector or any other scalar quantity, into an averaged ψ and a ﬂuctuating value ψ00

ψ = ψ + ψ0 (A.1)

The largest ﬂuctuations and thus the largest variance occur at intermittency. So the maximum variance depends on the mean value as follows:

max ψ02 = max (ψ − ψ)(ψ − ψ) = ψ(1 − ψ) (A.2)

The probability density function (pdf) P (ψ) is a measure for the probability that any quantity ψ takes values between ψ and ψ + dψ. It is positive and continuous, and fulﬁlls the standardization condition so that

Z P (ψ)dψ = 1 (A.3)

The Reynolds averaging procedure has several drawbacks [89]:

• Extra term in continuity equation which has to be modeled • This term contains a mean mass interchange across the mean streamline • Concentration measurements are mostly mass-weighted instead of time-averaged • The resulting Favre averaged equations have a simpler form

Thus it is common practice to use the Favre averaging procedure. The mass weighted quantity ψ˜ can be deﬁned as

ρψ ψe ≡ (A.4) ρ and the mean value can be derived from a Favre-pdf P˜ as:

Z 1 ψe(x, t) = ψ(Z)P˜(Z; x, t)dZ (A.5) 0 114 APPENDIX A. THE PRESUMED β-PDF APPROACH

In order to take the turbulence effects into account, the reactive scalars can be related to the mean mixture fraction and its variance using a probability density function. Several methods have been proposed to presume the shape of a probability density function for non-premixed as well as for premixed combustion in advance. Several pdf assumptions are summarized in [89]. As the results of the calculation of turbulent flames seem to be insensitive to the exact shape of the pdf [79], the β-pdf approach is commonly used as it has the advantage that it contains only two parameters a and b. Despite of this simplicity, it is able to predict different forms of the pdf’s [134] and can be written as

Za−1(1 − Z)b−1 P˜(Z; x, t) = (A.6) R 1 a−1 b−1 0 Z (1 − Z) dZ where the two non-negative parameters a and b, which are dependent of the mean mixture fraction Ze and its variance Zg002 can be expressed as

a = Zg,e b = (1 − Ze)g, (A.7) with Ze(1 − Ze) g = − 1 ≥ 0 (A.8) Zg002

As could be seen in equation A.2, the largest mixture fraction variance that can occur depends on the mean mixture fraction as Zg002 = Ze(1 − Ze). So g is always greater than or equal to zero. In mathematics, the denominator of equation A.6 is called in general beta-function. Using the relationship between the beta and gamma functions [5], the β-pdf can be rewritten as

Γ(a + b) P˜(Z; x, t) = Za−1(1 − Z)b−1 (A.9) Γ(a)Γ(b)

The beta and gamma functions have been implemented as described in [132]. Despite the advantages, the numerical integration of the β-pdf encounters singularity and overflow problems. Only few publications address these issues. Chen et al. [39] and Liu et al. [96] describe how they solved these problems numerically. The influence of the parameters a and b on the pdf shape can be seen on figure A.1(a). Depending on the values of a, b and g, the β-pdf can be subdivided in different shapes: a>1, b>1 If the mixture fraction variance approaches small values (see figure A.1(a), yellow area), g increases and with it a and b. In that case, no singularities occur and the pdf can have a peak anywhere between 0 < Z < 1. Its maximum value occurs at

1 Z = (A.10) max 1 + (b − 1)/(a − 1) 115

(a) Variation of the parameters a and b (b) Representative shapes of the β-pdf

Figure A.1: Inﬂuence of the parameters a and b

When g is very large, which occurs at very small variances (see figure A.1(a), thin red area), and thus a or b is very large, overflow may occur. The form of the pdf is close to that of a delta function. The peak occurs at Z = Ze. So if a = b the peak is at Ze = 0.5. If Ze > 0.5 then it follows that a > b and the peak moves towards the fuel side. If Ze < 0.5 then it follows that a < b and the peak moves towards the oxidiser side. To overcome the overflow problem, Chen et al. [39] proposed if a or b exceeds a limit of 500, then this component a0 or b0 is set to this number and the same ratio of a0/b0 is kept as for the original values as shown in the following equation

b b0 = a0, a0 = 500 (A.11) a

a a0 = b0, b0 = 500 (A.12) b

This equation has the disadvantage that overﬂow may already occur for smaller values for a and b than 500. In addition it might alter the shape of the pdf, as a0 or b0 can be reduced to a value smaller than one. Liu et al. [96] proposed the following equations to overcome the overﬂow problem:

a0 − 1 − Z (a0 − 2) b0 = max , a0 = 500 (A.13) Zmax

b0 − 1 − Z (b0 − 2) a0 = max , b0 = 500 (A.14) Zmax if a or b is very large, respectively. 116 APPENDIX A. THE PRESUMED β-PDF APPROACH

Depending on the implementation of the gamma function, overflow occurs when g exceeds a certain value. Within this work g has to be smaller than 171.6. If g exceeds this value it is set to 171.6, and a and b are calculated as in equation A.7. As this occurs only at such small variances, the values of a and b cannot be reduced to a smaller value of one, and thus the pdf cannot change its shape. Additionally with this method the ratio is kept constant, and the pdf shape is qualitatively preserved. a<1, b>1 That case is marked bright blue in figure A.1(a). It occurs at moderate mixture fraction variances and for mean mixture fractions smaller than 0.5. As can be seen in figure A.1(b), the pdf develops a singularity at the oxidizer side (Z = 0). It decays rapidly towards the fuel side. a>1, b<1 The green area in figure A.1(a) represents the case a > 1 and b < 1. The corresponding qualitative pdf shape is plotted in figure A.1(b). This time the singularity appears on the fuel side and decays rapidly to the oxidizer side. Such pdf shapes occur at moderate variance but this time for mean mixture fractions larger than 0.5. a<1, b<1 The pdf in this region has two singularities, one at the oxidizer side at Z = 0, and one at the fuel side at Z = 1 (figure A.1(b)). There the pdf approaches infinity, and the mixture fraction variance is large. This region is colored dark blue in figure A.1(a). Liu et al. [96] calculated the mean density ρ with the β-pdf approach

R 1 ρ(Z)Za−1(1 − Z)b−1 dZ ρ = 0 (A.15) R 1 a−1 b−1 0 Z (1 − Z) dZ

To overcome the singularity problems for a < 1 and/or b < 1 they introduced a polynomial function of the mixture fraction given as

ρ ρ(Z) = ref (A.16) Pj=0 j J djZ

In order to perform the numerical integration, they considered the numerator and denominator separately and introduced the following approximations:

• Oxidizer side: Z → 0 ⇒ (1 − Z)b−1 ≈ 1

• Fuel side: Z → 1 ⇒ Za−1 ≈ 1 117

Thus for the numerator and denominator follows

Z 1 a Z 1−ε b a−1 b−1 ε a−1 b−1 ε ρ(Z)Z (1 − Z) dZ ≈ ρox + ρ(Z)Z (1 − Z) dZ + ρfuel (A.17) 0 a ε b

Z 1 εa Z 1−ε εb Za−1(1 − Z)b−1 dZ ≈ + Za−1(1 − Z)b−1 dZ + (A.18) 0 a ε b where ε is a small parameter. As it is diﬃcult to describe other variables like the species mass fractions or the whole range of the laminar ﬂame speed with polynomials, the integration cannot be performed analytically and has to be approximated numerically. Due to this, numerical uncertainties occur and the species concentrations do not sum up to unity any more (see equation A.3). So the pdf should be integrated and normalized before the integration to obtain the mean value. Within this work, the same approximations used by Liu et al. [96] are used to overcome the singularity problem.

Z 1 Za−1(1 − Z)b−1 Z ε Za−1(1 − Z)b−1 = R 1 a−1 b−1 R 1 a−1 b−1 0 0 Z (1 − Z) dZ 0 0 Z (1 − Z) dZ Z 1−ε Za−1(1 − Z)b−1 + R 1 a−1 b−1 ε 0 Z (1 − Z) dZ Z 1 Za−1(1 − Z)b−1 (A.19) + R 1 a−1 b−1 1−ε 0 Z (1 − Z) dZ

Z 1−ε Za−1(1 − Z)b−1 = εa + + εb R 1 a−1 b−1 ε 0 Z (1 − Z) dZ Appendix B Flammability Limits

B.1 Hydrogen

Dimensions of tube [cm] Firing end Limits [%] Content of Diameter Length Lower Higher aqueous vapor

Upward Propagation of Flame 7.5 150 closed 4.15 75.0 half-saturated 5.3 150 open 4.19 74.6 dried 5.3 150 open 4.12 74.3 dried 5.3 150 open 4.17 74.8 dried 5.0 150 closed 4.15 74.5 half-saturated 5.0 180 open 4.00 72.0 dried 4.8 150 open 4.00 73.8 dried 4.5 80 closed 3.90 - - 2.5 150 open 4.20 - dried 2.5 150 closed 4.25 73.0 half-saturated 2.5 96 closed 4.10 72.8 - 2.2 45 closed 3.90 73.0 saturated 1.6 96 closed 4.22 71.2 - 0.8 96 closed 5.10 67.9 -

Downward Propagation of Flame 21.0 31 open 9.30 - saturated 8.0 37 closed 8.90 68.8 saturated 7.5 150 closed 8.80 74.5 half-saturated 7.0 150 closed - 74.5 - 6.2 33 open 8.50 - saturated 6.0 120 open 9.45 - partly dried 6.0 120 closed 9.30 - partly dried 6.0 120 - - 68.0 partly dried 5.3 150 open 9.00 - dried 5.0 150 closed 9.00 74.0 half-saturated 5.0 65 open 8.90 71.2 - 4.0 50 closed 8.80 - saturated 2.5 150 closed 9.40 71.5 half-saturated 2.5 150 open 9.70 - saturated 2.2 - closed 6.80 69.3 - 2.2 45 closed 9.10 73.0 - 1.9 40 closed 9.45 66.4 saturated B.1. HYDROGEN 119

Dimensions of tube [cm] Firing end Limits [%] Content of Diameter Length Lower Higher aqueous vapor 1.9 40 closed 9.45 65.25 saturated 1.6 30 closed 7.70 72.6 dried 1.4 20 closed 9.80 63.0 dried

Horizontal Propagation of Flame 7.5 150 closed 6.50 - half-saturated 5.0 150 closed 6.70 - half-saturated 2.5 150 closed 7.15 - half-saturated 2.5 150 open 6.20 - saturated 2.5 - open - 71.4 - 0.9 150 open 6.70 65.7 saturated

Spherical Vessel or Bomb Capacity [cm3] - closed 9.2 - saturated - closed 8.5 67.5 saturated 1000 closed 8.7 75.5 saturated 810 closed 5.0 73.5 - 350 closed 4.6 70.3 saturated 35 closed 9.4 64.8 saturated

Table B.1: Flammability limits of hydrogen [47] 120 APPENDIX B. FLAMMABILITY LIMITS B.2 Methane

2[!h]

Dimensions of tube [cm] Firing end Limits [%] Content of Diameter Length Lower Higher aqueous vapor

Upward Propagation of Flame 10.2 96 closed 5.00 15.00 dry 7.5 150 closed 5.35 14.85 half-saturated 6.5 33 open 5.45 13.5 saturated 6.0 200 closed 5.40 14.8 small 5.3 150 open 5.26 14.3 dry 5.0 50 closed - 15.11 - 5.0 150 open 5.40 14.25 half-saturated 5.0 180 open 5.24 14.02 dry 5.0 180 open 5.33 13.80 saturated 4.7 100 open 5.3 14.3 small 4.0 100 closed 5.5 14.1 - 2.7 - open 5.28 - dry 2.5 150 open 5.5 - saturated 2.5 150 closed 5.80 13.20 half-saturated 2.25 125 open 5.48 - nearly dry

Downward Propagation of Flame 8.0 37 closed 5.9 12.9 saturated 7.5 150 closed 5.95 13.35 half-saturated 6.2 33 open 6.3 - saturated 6.1 120 - 6.1 13.0 partly dry 6.0 200 closed 6.0 13.4 small 5.0 50 closed 5.80 13.38 - 5.0 150 closed 6.12 13.25 half-saturated 5.0 125 open 5.85 - dry 4.0 100 closed 6.1 13.3 - 2.7 - open 5.84 - dry 2.5 150 open 6.1 - saturated 2.5 150 closed 6.30 12.80 half-saturated 2.25 125 open 6.41 - nearly dry 2.2 - closed 5.6 13.6 - 1.9 40 closed 6.1 12.8 saturated 1.9 40 6.15 12.0 saturated 0.815 - open 6.25 - dry 0.515 - open 7.03 - dry B.2. METHANE 121

Dimensions of tube [cm] Firing end Limits [%] Content of Diameter Length Lower Higher aqueous vapor

Horizontal Propagation of Flame 7.5 150 closed 5.40 13.95 half-saturated 6.0 200 closed 5.4 14.3 small 5.0 150 closed 5.65 13.95 half-saturated 5.0 50 closed 5.39 14.28 - 4.0 100 closed 5.6 13.9 - 2.7 - open 5.64 - dry 2.5 150 open 5.85 13.3 saturated 2.5 150 closed 6.20 12.90 half-saturated 2.25 125 open 6.04 - nearly dry 2.0 40 closed 5.59 13.31 - 0.90 300 open 7.8 11.6 saturated 0.81 300 open 8.3 10.9 saturated 0.72 300 open 8.4 10.6 saturated 0.56 300 open 8.4 10.6 saturated 0.45 300 open limits coincide: 9.95 saturated

Spherical Vessel or Bomb Capacity [cm3] Ignition 2500 central 5.77 - saturated 2000 central 5.6 14.8 dry 100 above 5.5 14.0 - 35 side 6.0 12.6 dry 35 side 5.9 13.1 saturated

Table B.2: Flammability limits of methane [47] Appendix C Combustion Chemistry

For the usage of Chem1d [172], chemistry tables have to be provided similar to the ones used by Chemkin [82]. In these tables, the chemical reactions are listed together with the pre-exponential factor A, the temperature exponent n and the activation energy Ea. The specific reaction rate constant k can be calculated with the Arrhenius law (Arrhenius 1889): E k = AT n exp(− a ) (C.1) RuT where T is the temperature and Ru the universal gas constant. If the reactions are uni-, bi- or trimolecular, the factor AT n has different physical meanings [155]. For unimolecular reaction, the reciprocal value of A represents a mean lifetime of a reactive molecule. For bi- or trimolecular reactions it represents the collision frequency. The exponential term is called the Boltzmann factor, which specifies the fraction of collisions with energy levels greater than the activation energy.

C.1 Methane Mechanism

5 elements: O, H, C, N, Ar 53 species: H2, H, O, O2, OH, H2O, HO2, H2O2, C, CH, CH2, CH2(S), C, CH4, CO, CO2, HCO, CH2O, CH2OH, CH3O, CH3OH, C2H, C2H2, C2H3, C2H4, C2H5, C2H6, HCCO, CH2CO, HCCOH, N, NH, NH2, NH3, NNH, NO, NO2, N2O, HNO, CN, HCN, H2CN, HCNN, HCNO, HOCN, HNCO, NCO, N2, AR, C3H7, C3H8, CH2CHO, CH3CHO

3 Reaction A[mol, cm , s] n Ea[cal/mol]

H + O2 ↔ O + OH 1.915E14 0.00 1.644E04 2O + M ↔ O2 + M 1.200E + 17 −1.000 0.00 O + H + M ↔ OH + M 5.000E + 17 −1.000 0.00 O + H2 ↔ H + OH 3.870E + 04 2.700 6260.00 O + HO2 ↔ OH + O2 2.000E + 13 0.000 0.00 O + H2O2 ↔ OH + HO2 9.630E + 06 2.000 4000.00 O + CH ↔ H + CO 5.700E + 13 0.000 0.00 O + CH2 ↔ H + HCO 8.000E + 13 0.000 0.00 O + CH2(S) ↔ H2 + CO 1.500E + 13 0.000 0.00 O + CH2(S) ↔ H + HCO 1.500E + 13 0.000 0.00 O + CH3 ↔ H + CH2O 5.060E + 13 0.000 0.00 O + CH4 ↔ OH + CH3 1.020E + 09 1.500 8600.00 O + CO(+M) ↔ CO2(+M) 6.020E + 14 0.000 3000.00 O + CO ↔ CO2 1.800E + 10 0.000 2385.00 continued... C.1. METHANE MECHANISM 123

3 Reaction A[mol, cm , s] n Ea[cal/mol] O + HCO ↔ OH + CO 3.000E + 13 0.000 0.00 O + HCO ↔ H + CO2 3.000E + 13 0.000 0.00 O + CH2O ↔ OH + HCO 3.900E + 13 0.000 3540.00 O + CH2OH ↔ OH + CH2O 1.000E + 13 0.000 0.00 O + CH3O ↔ OH + CH2O 1.000E + 13 0.000 0.00 O + CH3OH ↔ OH + CH2OH 3.880E + 05 2.500 3100.00 O + CH3OH ↔ OH + CH3O 1.300E + 05 2.500 5000.00 O + C2H ↔ CH + CO 5.000E + 13 0.000 0.00 O + C2H2 ↔ H + HCCO 1.350E + 07 2.000 1900.00 O + C2H2 ↔ OH + C2H 4.600E + 19 −1.410 28950.00 O + C2H2 ↔ CO + CH2 6.940E + 06 2.000 1900.00 O + C2H3 ↔ H + CH2CO 3.000E + 13 0.000 0.00 O + C2H4 ↔ CH3 + HCO 1.250E + 07 1.830 220.00 O + C2H5 ↔ CH3 + CH2O 2.240E + 13 0.000 0.00 O + C2H6 ↔ OH + C2H5 8.980E + 07 1.920 5690.00 O + HCCO ↔ H + 2CO 1.000E + 14 0.000 0.00 O + CH2CO ↔ OH + HCCO 1.000E + 13 0.000 8000.00 O + CH2CO ↔ CH2 + CO2 1.750E + 12 0.000 1350.00 O2 + CO ↔ O + CO2 2.500E + 12 0.000 47800.00 O2 + CH2O ↔ HO2 + HCO 1.000E + 14 0.000 40000.00 H + O2 + M ↔ HO2 + M 2.800E + 18 −0.860 0.00 H + 2O2 ↔ HO2 + O2 2.080E + 19 −1.240 0.00 H + O2 + H2O ↔ HO2 + H2O 11.26E + 18 −0.760 0.00 H + O2 + N2 ↔ HO2 + N2 2.600E + 19 −1.240 0.00 H + O2 + AR ↔ HO2 + AR 7.000E + 17 −0.800 0.00 H + O2 ↔ O + OH 2.650E + 16 −0.6707 17041.00 2H + M ↔ H2 + M 1.000E + 18 −1.000 0.00 2H + H2 ↔ 2H2 9.000E + 16 −0.600 0.00 2H + H2O ↔ H2 + H2O 6.000E + 19 −1.250 0.00 2H + CO2 ↔ H2 + CO2 5.500E + 20 −2.000 0.00 H + OH + M ↔ H2O + M 2.200E + 22 −2.000 0.00 H + HO2 ↔ O + H2O 3.970E + 12 0.000 671.00 H + HO2 ↔ O2 + H2 4.480E + 13 0.000 1068.00 H + HO2 ↔ 2OH 0.840E + 14 0.000 635.00 H + H2O2 ↔ HO2 + H2 1.210E + 07 2.000 5200.00 H + H2O2 ↔ OH + H2O 1.000E + 13 0.000 3600.00 H + CH ↔ C + H2 1.650E + 14 0.000 0.00 H + CH2(+M) ↔ CH3(+M) 1.040E + 26 −2.760 1600.00 H + CH2 ↔ CH3 6.000E + 14 0.000 0.00 H + CH2(S) ↔ CH + H2 3.000E + 13 0.000 0.00 H + CH3(+M) ↔ CH4(+M) 2.620E + 33 −4.760 2440.00 H + CH3 ↔ CH4 13.90E + 15 −0.534 536.00 H + CH4 ↔ CH3 + H2 6.600E + 08 1.620 10840.00 H + HCO(+M) ↔ CH2O(+M) 2.470E + 24 −2.570 425.00 H + HCO ↔ CH2O 1.090E + 12 0.480 −260.00 continued... 124 APPENDIX C. COMBUSTION CHEMISTRY

3 Reaction A[mol, cm , s] n Ea[cal/mol]

H + HCO ↔ H2 + CO 7.340E + 13 0.000 0.00 H + CH2O(+M) ↔ CH2OH(+M) 1.270E + 32 −4.820 6530.00 H + CH2O ↔ CH2OH 5.400E + 11 0.454 3600.00 H + CH2O(+M) ↔ CH3O(+M) 2.200E + 30 −4.800 5560.00 H + CH2O ↔ CH3O 5.400E + 11 0.454 2600.00 H + CH2O ↔ HCO + H2 5.740E + 07 1.900 2742.00 H + CH2OH(+M) ↔ CH3OH(+M) 4.360E + 31 −4.650 5080.00 H + CH2OH ↔ CH3OH 1.055E + 12 0.500 86.00 H + CH2OH ↔ H2 + CH2O 2.000E + 13 0.000 0.00 H + CH2OH ↔ OH + CH3 1.650E + 11 0.650 −284.00 H + CH2OH ↔ CH2(S) + H2O 3.280E + 13 −0.090 610.00 H + CH3O(+M) ↔ CH3OH(+M) 4.660E + 41 −7.440 14080.0 H + CH3O ↔ CH3OH 2.430E + 12 0.515 50.00 H + CH3O ↔ H + CH2OH 4.150E + 07 1.630 1924.00 H + CH3O ↔ H2 + CH2O 2.000E + 13 0.000 0.00 H + CH3O ↔ OH + CH3 1.500E + 12 0.500 −110.00 H + CH3O ↔ CH2(S) + H2O 2.620E + 14 −0.230 1070.00 H + CH3OH ↔ CH2OH + H2 1.700E + 07 2.100 4870.00 H + CH3OH ↔ CH3O + H2 4.200E + 06 2.100 4870.00 H + C2H(+M) ↔ C2H2(+M) 3.750E + 33 −4.800 1900.00 H + C2H ↔ C2H2 1.000E + 17 −1.000 0.00 H + C2H2(+M) ↔ C2H3(+M) 3.800E + 40 −7.270 7220.00 H + C2H2 ↔ C2H3 5.600E + 12 0.000 2400.00 H + C2H3(+M) ↔ C2H4(+M) 1.400E + 30 −3.860 3320.00 H + C2H3 ↔ C2H4 6.080E + 12 0.270 280.00 H + C2H3 ↔ H2 + C2H2 3.000E + 13 0.000 0.00 H + C2H4(+M) ↔ C2H5(+M) 0.600E + 42 −7.620 6970.00 H + C2H4 ↔ C2H5 0.540E + 12 0.454 1820.00 H + C2H4 ↔ C2H3 + H2 1.325E + 06 2.530 12240.00 H + C2H5(+M) ↔ C2H6(+M) 1.990E + 41 −7.080 6685.00 H + C2H5 ↔ C2H6 5.210E + 17 −0.990 1580.00 H + C2H5 ↔ H2 + C2H4 2.000E + 12 0.000 0.00 H + C2H6 ↔ C2H5 + H2 1.150E + 08 1.900 7530.00 H + HCCO ↔ CH2(S) + CO 1.000E + 14 0.000 0.00 H + CH2CO ↔ HCCO + H2 5.000E + 13 0.000 8000.00 H + CH2CO ↔ CH3 + CO 1.130E + 13 0.000 3428.00 H + HCCOH ↔ H + CH2CO 1.000E + 13 0.000 0.00 H2 + CO(+M) ↔ CH2O(+M) 5.070E + 27 −3.420 84350.00 H2 + CO ↔ CH2O 4.300E + 07 1.500 79600.00 OH + H2 ↔ H + H2O 2.160E + 08 1.510 3430.00 2OH(+M) ↔ H2O2(+M) 2.300E + 18 −0.900 −1700.00 2OH ↔ H2O2 7.400E + 13 −0.370 0.00 2OH ↔ O + H2O 3.570E + 04 2.400 −2110.00 OH + HO2 ↔ O2 + H2O 1.450E + 13 0.000 −500.00 OH + H2O2 ↔ HO2 + H2O 2.000E + 12 0.000 427.00 continued... C.1. METHANE MECHANISM 125

3 Reaction A[mol, cm , s] n Ea[cal/mol]

OH + H2O2 ↔ HO2 + H2O 1.700E + 18 0.000 29410.00 OH + C ↔ H + CO 5.000E + 13 0.000 0.00 OH + CH ↔ H + HCO 3.000E + 13 0.000 0.00 OH + CH2 ↔ H + CH2O 2.000E + 13 0.000 0.00 OH + CH2 ↔ CH + H2O 1.130E + 07 2.000 3000.00 OH + CH2(S) ↔ H + CH2O 3.000E + 13 0.000 0.00 OH + CH3(+M) ↔ CH3OH(+M) 4.000E + 36 −5.920 3140.00 OH + CH3 ↔ CH3OH 2.790E + 18 −1.430 1330.00 OH + CH3 ↔ CH2 + H2O 5.600E + 07 1.600 5420.00 OH + CH3 ↔ CH2(S) + H2O 6.440E + 17 −1.340 1417.00 OH + CH4 ↔ CH3 + H2O 1.000E + 08 1.600 3120.00 OH + CO ↔ H + CO2 4.760E + 07 1.228 70.00 OH + HCO ↔ H2O + CO 5.000E + 13 0.000 0.00 OH + CH2O ↔ HCO + H2O 3.430E + 09 1.180 −447.00 OH + CH2OH ↔ H2O + CH2O 5.000E + 12 0.000 0.00 OH + CH3O ↔ H2O + CH2O 5.000E + 12 0.000 0.00 OH + CH3OH ↔ CH2OH + H2O 1.440E + 06 2.000 −840.00 OH + CH3OH ↔ CH3O + H2O 6.300E + 06 2.000 1500.00 OH + C2H ↔ H + HCCO 2.000E + 13 0.000 0.00 OH + C2H2 ↔ H + CH2CO 2.180E − 04 4.500 −1000.00 OH + C2H2 ↔ H + HCCOH 5.040E + 05 2.300 13500.00 OH + C2H2 ↔ C2H + H2O 3.370E + 07 2.000 14000.00 OH + C2H2 ↔ CH3 + CO 4.830E − 04 4.000 −2000.00 OH + C2H3 ↔ H2O + C2H2 5.000E + 12 0.000 0.00 OH + C2H4 ↔ C2H3 + H2O 3.600E + 06 2.000 2500.00 OH + C2H6 ↔ C2H5 + H2O 3.540E + 06 2.120 870.00 OH + CH2CO ↔ HCCO + H2O 7.500E + 12 0.000 2000.00 2HO2 ↔ O2 + H2O2 1.300E + 11 0.000 −1630.00 2HO2 ↔ O2 + H2O2 4.200E + 14 0.000 12000.00 HO2 + CH2 ↔ OH + CH2O 2.000E + 13 0.000 0.00 HO2 + CH3 ↔ O2 + CH4 1.000E + 12 0.000 0.00 HO2 + CH3 ↔ OH + CH3O 3.780E + 13 0.000 0.00 HO2 + CO ↔ OH + CO2 1.500E + 14 0.000 23600.00 HO2 + CH2O ↔ HCO + H2O2 5.600E + 06 2.000 12000.00 C + O2 ↔ O + CO 5.800E + 13 0.000 576.00 C + CH2 ↔ H + C2H 5.000E + 13 0.000 0.00 C + CH3 ↔ H + C2H2 5.000E + 13 0.000 0.00 CH + O2 ↔ O + HCO 6.710E + 13 0.000 0.00 CH + H2 ↔ H + CH2 1.080E + 14 0.000 3110.00 CH + H2O ↔ H + CH2O 5.710E + 12 0.000 −755.00 CH + CH2 ↔ H + C2H2 4.000E + 13 0.000 0.00 CH + CH3 ↔ H + C2H3 3.000E + 13 0.000 0.00 CH + CH4 ↔ H + C2H4 6.000E + 13 0.000 0.00 CH + CO(+M) ↔ HCCO(+M) 2.690E + 28 −3.740 1936.00 CH + CO ↔ HCCO 5.000E + 13 0.000 0.00 continued... 126 APPENDIX C. COMBUSTION CHEMISTRY

3 Reaction A[mol, cm , s] n Ea[cal/mol]

CH + CO2 ↔ HCO + CO 1.900E + 14 0.000 15792.00 CH + CH2O ↔ H + CH2CO 9.460E + 13 0.000 −515.00 CH + HCCO ↔ CO + C2H2 5.000E + 13 0.000 0.00 CH2 + O2 → OH + H + CO 5.000E + 12 0.000 1500.00 CH2 + H2 ↔ H + CH3 5.000E + 05 2.000 7230.00 2CH2 ↔ H2 + C2H2 1.600E + 15 0.000 11944.00 CH2 + CH3 ↔ H + C2H4 4.000E + 13 0.000 0.00 CH2 + CH4 ↔ 2CH3 2.460E + 06 2.000 8270.00 CH2 + CO(+M) ↔ CH2CO(+M) 2.690E + 33 −5.110 7095.00 CH2 + CO ↔ CH2CO 8.100E + 11 0.500 4510.00 CH2 + HCCO ↔ C2H3 + CO 3.000E + 13 0.000 0.00 CH2(S) + N2 ↔ CH2 + N2 1.500E + 13 0.000 600.00 CH2(S) + AR ↔ CH2 + AR 9.000E + 12 0.000 600.00 CH2(S) + O2 ↔ H + OH + CO 2.800E + 13 0.000 0.00 CH2(S) + O2 ↔ CO + H2O 1.200E + 13 0.000 0.00 CH2(S) + H2 ↔ CH3 + H 7.000E + 13 0.000 0.00 CH2(S) + H2O(+M) ↔ CH3OH(+M) 1.880E + 38 −6.360 5040.00 CH2(S) + H2O ↔ CH3OH 4.820E + 17 −1.160 1145.00 CH2(S) + H2O ↔ CH2 + H2O 3.000E + 13 0.000 0.00 CH2(S) + CH3 ↔ H + C2H4 1.200E + 13 0.000 −570.00 CH2(S) + CH4 ↔ 2CH3 1.600E + 13 0.000 −570.00 CH2(S) + CO ↔ CH2 + CO 9.000E + 12 0.000 0.00 CH2(S) + CO2 ↔ CH2 + CO2 7.000E + 12 0.000 0.00 CH2(S) + CO2 ↔ CO + CH2O 1.400E + 13 0.000 0.00 CH2(S) + C2H6 ↔ CH3 + C2H5 4.000E + 13 0.000 −550.00 CH3 + O2 ↔ O + CH3O 3.560E + 13 0.000 30480.00 CH3 + O2 ↔ OH + CH2O 2.310E + 12 0.000 20315.00 CH3 + H2O2 ↔ HO2 + CH4 2.450E + 04 2.470 5180.00 2CH3(+M) ↔ C2H6(+M) 3.400E + 41 −7.030 2762.00 2CH3 ↔ C2H6 6.770E + 16 −1.180 654.00 2CH3 ↔ H + C2H5 6.840E + 12 0.100 10600.00 CH3 + HCO ↔ CH4 + CO 2.648E + 13 0.000 0.00 CH3 + CH2O ↔ HCO + CH4 3.320E + 03 2.810 5860.00 CH3 + CH3OH ↔ CH2OH + CH4 3.000E + 07 1.500 9940.00 CH3 + CH3OH ↔ CH3O + CH4 1.000E + 07 1.500 9940.00 CH3 + C2H4 ↔ C2H3 + CH4 2.270E + 05 2.000 9200.00 CH3 + C2H6 ↔ C2H5 + CH4 6.140E + 06 1.740 10450.00 HCO + H2O ↔ H + CO + H2O 1.500E + 18 −1.000 17000.00 HCO + M ↔ H + CO + M 1.870E + 17 −1.000 17000.00 HCO + O2 ↔ HO2 + CO 13.45E + 12 0.000 400.00 CH2OH + O2 ↔ HO2 + CH2O 1.800E + 13 0.000 900.00 CH3O + O2 ↔ HO2 + CH2O 4.280E − 13 7.600 −3530.00 C2H + O2 ↔ HCO + CO 1.000E + 13 0.000 −755.00 C2H + H2 ↔ H + C2H2 5.680E + 10 0.900 1993.00 C2H3 + O2 ↔ HCO + CH2O 4.580E + 16 −1.390 1015.00 continued... C.1. METHANE MECHANISM 127

3 Reaction A[mol, cm , s] n Ea[cal/mol]

C2H4(+M) ↔ H2 + C2H2(+M) 1.580E + 51 −9.300 97800.00 C2H4 ↔ H2 + C2H2 8.000E + 12 0.440 86770.00 C2H5 + O2 ↔ HO2 + C2H4 8.400E + 11 0.000 3875.00 HCCO + O2 ↔ OH + 2CO 3.200E + 12 0.000 854.00 2HCCO ↔ 2CO + C2H2 1.000E + 13 0.000 0.00 N + NO ↔ N2 + O 2.700E + 13 0.000 355.00 N + O2 ↔ NO + O 9.000E + 09 1.000 6500.00 N + OH ↔ NO + H 3.360E + 13 0.000 385.00 N2O + O ↔ N2 + O2 1.400E + 12 0.000 10810.00 N2O + O ↔ 2NO 2.900E + 13 0.000 23150.00 N2O + H ↔ N2 + OH 3.870E + 14 0.000 18880.00 N2O + OH ↔ N2 + HO2 2.000E + 12 0.000 21060.00 N2O(+M) ↔ N2 + O(+M) 6.370E + 14 0.000 56640.00 N2O ↔ N2 + O 7.910E + 10 0.000 56020.00 HO2 + NO ↔ NO2 + OH 2.110E + 12 0.000 −480.00 NO + O + M ↔ NO2 + M 1.060E + 20 −1.410 0.00 NO2 + O ↔ NO + O2 3.900E + 12 0.000 −240.00 NO2 + H ↔ NO + OH 1.320E + 14 0.000 360.00 NH + O ↔ NO + H 4.000E + 13 0.000 0.00 NH + H ↔ N + H2 3.200E + 13 0.000 330.00 NH + OH ↔ HNO + H 2.000E + 13 0.000 0.00 NH + OH ↔ N + H2O 2.000E + 09 1.200 0.00 NH + O2 ↔ HNO + O 4.610E + 05 2.000 6500.00 NH + O2 ↔ NO + OH 1.280E + 06 1.500 100.00 NH + N ↔ N2 + H 1.500E + 13 0.000 0.00 NH + H2O ↔ HNO + H2 2.000E + 13 0.000 13850.00 NH + NO ↔ N2 + OH 2.160E + 13 −0.230 0.00 NH + NO ↔ N2O + H 3.650E + 14 −0.450 0.00 NH2 + O ↔ OH + NH 3.000E + 12 0.000 0.00 NH2 + O ↔ H + HNO 3.900E + 13 0.000 0.00 NH2 + H ↔ NH + H2 4.000E + 13 0.000 3650.00 NH2 + OH ↔ NH + H2O 9.000E + 07 1.500 −460.00 NNH ↔ N2 + H 3.300E + 08 0.000 0.00 NNH + M ↔ N2 + H + M 1.300E + 14 −0.110 4980.00 NNH + O2 ↔ HO2 + N2 5.000E + 12 0.000 0.00 NNH + O ↔ OH + N2 2.500E + 13 0.000 0.00 NNH + O ↔ NH + NO 7.000E + 13 0.000 0.00 NNH + H ↔ H2 + N2 5.000E + 13 0.000 0.00 NNH + OH ↔ H2O + N2 2.000E + 13 0.000 0.00 NNH + CH3 ↔ CH4 + N2 2.500E + 13 0.000 0.00 H + NO + M ↔ HNO + M 4.480E + 19 −1.320 740.00 HNO + O ↔ NO + OH 2.500E + 13 0.000 0.00 HNO + H ↔ H2 + NO 9.000E + 11 0.720 660.00 HNO + OH ↔ NO + H2O 1.300E + 07 1.900 −950.00 HNO + O2 ↔ HO2 + NO 1.000E + 13 0.000 13000.00 continued... 128 APPENDIX C. COMBUSTION CHEMISTRY

3 Reaction A[mol, cm , s] n Ea[cal/mol] CN + O ↔ CO + N 7.700E + 13 0.000 0.00 CN + OH ↔ NCO + H 4.000E + 13 0.000 0.00 CN + H2O ↔ HCN + OH 8.000E + 12 0.000 7460.00 CN + O2 ↔ NCO + O 6.140E + 12 0.000 −440.00 CN + H2 ↔ HCN + H 2.950E + 05 2.450 2240.00 NCO + O ↔ NO + CO 2.350E + 13 0.000 0.00 NCO + H ↔ NH + CO 5.400E + 13 0.000 0.00 NCO + OH ↔ NO + H + CO 0.250E + 13 0.000 0.00 NCO + N ↔ N2 + CO 2.000E + 13 0.000 0.00 NCO + O2 ↔ NO + CO2 2.000E + 12 0.000 20000.00 NCO + M ↔ N + CO + M 3.100E + 14 0.000 54050.00 NCO + NO ↔ N2O + CO 1.900E + 17 −1.520 740.00 NCO + NO ↔ N2 + CO2 3.800E + 18 −2.000 800.00 HCN + M ↔ H + CN + M 1.040E + 29 −3.300 126600.00 HCN + O ↔ NCO + H 2.030E + 04 2.640 4980.00 HCN + O ↔ NH + CO 5.070E + 03 2.640 4980.00 HCN + O ↔ CN + OH 3.910E + 09 1.580 26600.00 HCN + OH ↔ HOCN + H 1.100E + 06 2.030 13370.00 HCN + OH ↔ HNCO + H 4.400E + 03 2.260 6400.00 HCN + OH ↔ NH2 + CO 1.600E + 02 2.560 9000.00 H + HCN(+M) ↔ H2CN(+M) 1.400E + 26 −3.400 1900.00 H + HCN ↔ H2CN 3.300E + 13 0.000 0.00 H2CN + N ↔ N2 + CH2 6.000E + 13 0.000 400.00 C + N2 ↔ CN + N 6.300E + 13 0.000 46020.00 CH + N2 ↔ HCN + N 3.120E + 09 0.880 20130.00 CH + N2(+M) ↔ HCNN(+M) 1.300E + 25 −3.160 740.00 CH + N2 ↔ HCNN 3.100E + 12 0.150 0.00 CH2 + N2 ↔ HCN + NH 1.000E + 13 0.000 74000.00 CH2(S) + N2 ↔ NH + HCN 1.000E + 11 0.000 65000.00 C + NO ↔ CN + O 1.900E + 13 0.000 0.00 C + NO ↔ CO + N 2.900E + 13 0.000 0.00 CH + NO ↔ HCN + O 4.100E + 13 0.000 0.00 CH + NO ↔ H + NCO 1.620E + 13 0.000 0.00 CH + NO ↔ N + HCO 2.460E + 13 0.000 0.00 CH2 + NO ↔ H + HNCO 3.100E + 17 −1.380 1270.00 CH2 + NO ↔ OH + HCN 2.900E + 14 −0.690 760.00 CH2 + NO ↔ H + HCNO 3.800E + 13 −0.360 580.00 CH2(S) + NO ↔ H + HNCO 3.100E + 17 −1.380 1270.00 CH2(S) + NO ↔ OH + HCN 2.900E + 14 −0.690 760.00 CH2(S) + NO ↔ H + HCNO 3.800E + 13 −0.360 580.00 CH3 + NO ↔ HCN + H2O 9.600E + 13 0.000 28800.00 CH3 + NO ↔ H2CN + OH 1.000E + 12 0.000 21750.00 HCNN + O ↔ CO + H + N2 2.200E + 13 0.000 0.00 HCNN + O ↔ HCN + NO 2.000E + 12 0.000 0.00 HCNN + O2 ↔ O + HCO + N2 1.200E + 13 0.000 0.00 continued... C.1. METHANE MECHANISM 129

3 Reaction A[mol, cm , s] n Ea[cal/mol]

HCNN + OH ↔ H + HCO + N2 1.200E + 13 0.000 0.00 HCNN + H ↔ CH2 + N2 1.000E + 14 0.000 0.00 HNCO + O ↔ NH + CO2 9.800E + 07 1.410 8500.00 HNCO + O ↔ HNO + CO 1.500E + 08 1.570 44000.00 HNCO + O ↔ NCO + OH 2.200E + 06 2.110 11400.00 HNCO + H ↔ NH2 + CO 2.250E + 07 1.700 3800.00 HNCO + H ↔ H2 + NCO 1.050E + 05 2.500 13300.00 HNCO + OH ↔ NCO + H2O 3.300E + 07 1.500 3600.00 HNCO + OH ↔ NH2 + CO2 3.300E + 06 1.500 3600.00 HNCO + M ↔ NH + CO + M 1.180E + 16 0.000 84720.00 HCNO + H ↔ H + HNCO 2.100E + 15 −0.690 2850.00 HCNO + H ↔ OH + HCN 2.700E + 11 0.180 2120.00 HCNO + H ↔ NH2 + CO 1.700E + 14 −0.750 2890.00 HOCN + H ↔ H + HNCO 2.000E + 07 2.000 2000.00 HCCO + NO ↔ HCNO + CO 0.900E + 13 0.000 0.00 CH3 + N ↔ H2CN + H 6.100E + 14 −0.310 290.00 CH3 + N ↔ HCN + H2 3.700E + 12 0.150 −90.00 NH3 + H ↔ NH2 + H2 5.400E + 05 2.400 9915.00 NH3 + OH ↔ NH2 + H2O 5.000E + 07 1.600 955.00 NH3 + O ↔ NH2 + OH 9.400E + 06 1.940 6460.00 NH + CO2 ↔ HNO + CO 1.000E + 13 0.000 14350.00 CN + NO2 ↔ NCO + NO 6.160E + 15 −0.752 345.00 NCO + NO2 ↔ N2O + CO2 3.250E + 12 0.000 −705.00 N + CO2 ↔ NO + CO 3.000E + 12 0.000 11300.00 O + CH3 → H + H2 + CO 3.370E + 13 0.000 0.00 O + C2H4 ↔ H + CH2CHO 6.700E + 06 1.830 220.00 O + C2H5 ↔ H + CH3CHO 1.096E + 14 0.000 0.00 OH + HO2 ↔ O2 + H2O 0.500E + 16 0.000 17330.00 OH + CH3 → H2 + CH2O 8.000E + 09 0.500 −1755.00 CH + H2(+M) ↔ CH3(+M) 4.820E + 25 −2.80 590.0 CH + H2 ↔ CH3 1.970E + 12 0.430 −370.00 CH2 + O2 → 2H + CO2 5.800E + 12 0.000 1500.00 CH2 + O2 ↔ O + CH2O 2.400E + 12 0.000 1500.00 CH2 + CH2 → 2H + C2H2 2.000E + 14 0.000 10989.00 CH2(S) + H2O → H2 + CH2O 6.820E + 10 0.250 −935.00 C2H3 + O2 ↔ O + CH2CHO 3.030E + 11 0.290 11.00 C2H3 + O2 ↔ HO2 + C2H2 1.337E + 06 1.610 −384.00 O + CH3CHO ↔ OH + CH2CHO 2.920E + 12 0.000 1808.00 O + CH3CHO → OH + CH3 + CO 2.920E + 12 0.000 1808.00 O2 + CH3CHO → HO2 + CH3 + CO 3.010E + 13 0.000 39150.00 H + CH3CHO ↔ CH2CHO + H2 2.050E + 09 1.160 2405.00 H + CH3CHO → CH3 + H2 + CO 2.050E + 09 1.160 2405.00 OH + CH3CHO → CH3 + H2O + CO 2.343E + 10 0.730 −1113.00 HO2 + CH3CHO → CH3 + H2O2 + CO 3.010E + 12 0.000 11923.00 CH3 + CH3CHO → CH3 + CH4 + CO 2.720E + 06 1.770 5920.00 continued... 130 APPENDIX C. COMBUSTION CHEMISTRY

3 Reaction A[mol, cm , s] n Ea[cal/mol]

H + CH2CO(+M) ↔ CH2CHO(+M) 1.012E + 42 −7.63 3854.0 H + CH2CO ↔ CH2CHO 4.865E + 11 0.422 −1755.00 O + CH2CHO → H + CH2 + CO2 1.500E + 14 0.000 0.00 O2 + CH2CHO → OH + CO + CH2O 1.810E + 10 0.000 0.00 O2 + CH2CHO → OH + 2HCO 2.350E + 10 0.000 0.00 H + CH2CHO ↔ CH3 + HCO 2.200E + 13 0.000 0.00 H + CH2CHO ↔ CH2CO + H2 1.100E + 13 0.000 0.00 OH + CH2CHO ↔ H2O + CH2CO 1.200E + 13 0.000 0.00 OH + CH2CHO ↔ HCO + CH2OH 3.010E + 13 0.000 0.00 CH3 + C2H5(+M) ↔ C3H8(+M) 2.710E + 74 −16.82 13065.0 CH3 + C2H5 ↔ C3H8 0.9430E + 13 0.000 0.00 O + C3H8 ↔ OH + C3H7 1.930E + 05 2.680 3716.00 H + C3H8 ↔ C3H7 + H2 1.320E + 06 2.540 6756.00 OH + C3H8 ↔ C3H7 + H2O 3.160E + 07 1.800 934.00 C3H7 + H2O2 ↔ HO2 + C3H8 3.780E + 02 2.720 1500.00 CH3 + C3H8 ↔ C3H7 + CH4 0.903E + 00 3.650 7154.00 CH3 + C2H4(+M) ↔ C3H7(+M) 3.00E + 63 −14.6 18170.0 CH3 + C2H4 ↔ C3H7 2.550E + 06 1.600 5700.00 O + C3H7 ↔ C2H5 + CH2O 9.640E + 13 0.000 0.00 H + C3H7(+M) ↔ C3H8(+M) 4.420E + 61 −13.545 11357.0 H + C3H7 ↔ C3H8 3.613E + 13 0.000 0.00 H + C3H7 ↔ CH3 + C2H5 4.060E + 06 2.190 890.00 OH + C3H7 ↔ C2H5 + CH2OH 2.410E + 13 0.000 0.00 HO2 + C3H7 ↔ O2 + C3H8 2.550E + 10 0.255 −943.00 HO2 + C3H7 → OH + C2H5 + CH2O 2.410E + 13 0.000 0.00 CH3 + C3H7 ↔ 2C2H5 1.927E + 13 −0.320 0.00 Table C.1: Skeletal mechanism for methane (GRI3.0 [174]) C.2. HYDROGEN MECHANISM 131 C.2 Hydrogen Mechanism

4 Elements: H, O, N, Ar 10 Species: H, H2, O, O2, OH, H2O, N2, HO2, H2O2, AR

3 Reaction A[mol, cm , s] n Ea[cal/mol] H + O2 ↔ O + OH 1.915E14 0.00 1.644E04 O + H2 ↔ H + OH 5.080E04 2.67 6.292E03 OH + H2 ↔ H + H2O 2.160E08 1.51 3.430E03 O + H2O ↔ OH + OH 2.970E06 2.02 1.340E04 H2 + M ↔ H + H + M 4.577E19 −1.40 1.044E05 O2 + M ↔ O + O + M 4.515E17 −0.64 1.189E05 OH + M ↔ O + H + M 9.880E17 −0.74 1.201E05 H2O + M ↔ H + OH + M 1.912E23 −1.83 1.185E05 H + O2(+M) ↔ HO2(+M) 3.482E16 −0.411 −1.115E03 H + O2 ↔ HO2 1.475E12 0.60 0.000E00 HO2 + H ↔ H2 + O2 1.660E13 0.00 8.230E02 HO2 + H ↔ OH + OH 7.079E13 0.00 2.950E02 HO2 + O ↔ OH + O2 3.250E13 0.00 0.000E00 HO2 + OH ↔ H2O + O2 2.890E13 0.00 −4.970E02 H2O2 + O2 ↔ HO2 + HO2 4.634E16 −0.35 5.067E04 H2O2 + O2 ↔ HO2 + HO2 1.434E13 −0.35 3.706E04 H2O2(+M) ↔ OH + OH(+M) 1.202E17 0.00 4.550E04 H2O2 ↔ OH + OH 2.951E14 0.00 4.843E03 H2O2 + H ↔ H2O + OH 2.410E13 0.00 3.970E03 H2O2 + H ↔ H2 + HO2 6.025E13 0.00 7.950E03 H2O2 + O ↔ OH + HO2 9.550E06 2.00 3.970E03 H2O2 + OH ↔ H2O + HO2 1.000E12 0.00 0.000E00 H2O2 + OH ↔ H2O + HO2 5.800E14 0.00 9.557E03

Table C.2: Skeletal mechanism for hydrogen combustion (Conaire [46]) Appendix D Additional Figures

D.1 H2 Jet Flame

(a) y/L=1/8 (b) y/L=1/4

(e) y/L=5/8 (f) y/L=3/4

(g) y/L=1/1

Figure D.1: Laminar ﬂamelet tables for diﬀerent strain rates and temperature scatter plots 134 APPENDIX D. ADDITIONAL FIGURES D.2 Sandia D Flame

(a) y/D=7.5 (b) y/D=15

(e) y/D=60 (f) y/D=75

Figure D.2: Laminar ﬂamelet tables for diﬀerent strain rates and temperature scatter plots Appendix E Material Properties of Hydrogen

This chapter gives a short overview of the most important low temperature material properties of hydrogen. Figure E.1 shows the p-T-phase diagram for hydrogen. For comparison, also the properties of oxygen and nitrogen are shown, as within DI engines, hydrogen reacts with air.

Figure E.1: p-T-diagram of hydrogen, nitrogen and oxygen

The sublimation line, starting at T = 0 K and ending at the triple point, which is at T = 13.957 K and and p = 0.072 bar for hydrogen, represents the equilibrium between solid and gas. The saturation line separates gas and liquid, and ends at the critical point which has a temperature of 33.19 K and a pressure of 13.15 bar for normal hydrogen. The phases solid and liquid are divided by the melting line. At the triple point, sublimation line, saturation line and melting line intersect each other. All three phases gas, solid and liquid are in equilibrium at this point. At the critical point the critical isotherm shows a horizontal tangent in a p-v-diagram. Above this point a distinction between liquid and gas is impossible. A specialty of hydrogen molecules is, that two variants of cryogenic hydrogen can coexist, which are varying in their physical properties especially for temperatures lower than 400 K. Thus additional care has to be taken to describe the temperature dependence of properties like the speciﬁc heat capacity at constant pressure cp, the enthalpy h or the entropy s correctly [21][154]. At ambient temperatures normal hydrogen exists of a mixture of 75% ortho and 25% para hydrogen. Para and ortho hydrogen diﬀer by the orientation of the nuclear spin of their atoms. The spin indicates the angular momentum of the elementary particles of an atom. Ortho hydrogen, which cannot exist in pure form shows a parallel spin, whereas para hydrogen can exist in pure form and shows an anti parallel spin. 136 APPENDIX E. MATERIAL PROPERTIES OF HYDROGEN

The following equilibrium relation shows the diﬀerence in energy:

o − H2 p − H2 + 0.08 kJ (E.1)

At cryogenic conditions hydrogen is practically in the form of pure para modiﬁcation. The conversion without catalysts from one modiﬁcation into the other is a very slow process.

Figure E.2: Comparison of para- and ortho hydrogen at low temperature

Figure E.2 shows the dependence of heat capacity at constant pressure on temperature, which is very different between ortho and para hydrogen. For low heat capacities only little heat has to be supplied to increase the temperature by for example 1 K. Thus for para hydrogen much more heat has to be supplied to increase the temperature than for ortho hydrogen. Figure E.2 also shows a comparison of different datasets. The data are from collected from the NIST database [91] (triangle symbols), experiments performed by Woolley at al. [159]. The NASA polynomials (named ’Chemkin 300K - 1000K’ are only defined in two temperature intervals, T = [300; 1000] K and T = [1000; 5000] K. The temperature range covered by the polynomial of Poling [127] T = [50; 1000] K are also not valid below 400 K. Polynomials for para and ortho hydrogen have been derived by Heckelmann [69] to cover also the low temperature region. They are of the form

c0 p = a + a T + a T 2 + a T 3 + a T 5 (E.2) R 1 2 3 4 5

The parameters a1 - a5 for the description of para and ortho hydrogen are listed in table E.1 and E.2. Any desirable mixture of para and ortho hydrogen can be calculated as follows 137

c p,mixture = [xa +(1−xa )]+[xa +(1−xa )]T +...+[xa +(1−xa )]T 4 (E.3) R 1,p 1,o 2,p 2,o 5,p 5,o where x is the mixing ration and the indices p and o denote para and ortho hydrogen. 138 APPENDIX E. MATERIAL PROPERTIES OF HYDROGEN 5[ a5 [ a4 [ a3 [ a2 5[ a5 [ a4 [ a3 [ a2 1[-] a1 1[-] a1 K K K K K K K K − − − − − − − − 5 4 3 2 5 4 3 2 ] ] ] ] ] ] ] ] 2.53089749E+00 -6.82026417E-07 -2.32990853E-03 0.00000000E+00 2.52743134E+00 -4.00656669E-05 -1.06086924E-03 3.26016701E-09 5.93580974E-05 1.25208511E-06 5 100K - 15K 5 100K - 15K 3.54758580E+00 -6.94176619E-07 -2.79571528E-02 -2.43228586E-06 6.40147402E-10 2.41220803E-04 4.93627689E-09 2.76284287E-04 1.96633848E-02 4.67957385E-01 0K-215K - 100K 0K-215K - 100K al .:Zr rsueplnmasfrotohydrogen ortho for polynomials pressure Zero E.2: Table al .:Zr rsueplnmasfrpr hydrogen para for polynomials pressure Zero E.1: Table 0.00000000E+00 -8.63395259E-10 -3.64074380E-04 -5.15450302E-05 9.36490571E-07 5.72180284E-02 9.06972890E-01 4.16471877E-08 2.13817387E-02 5.27077701E-01 1K-320K - 215K 1K-320K - 215K 2 2 [!h] [!h] 6.60039418E+00 -9.78768963E-08 -2.46460408E-02 1.23210313E+00 -2.47625018E-11 -4.59862737E-05 4.85912328E-11 7.37735298E-05 5.54904388E-08 1.67916125E-02 2K-500K - 320K 2K-500K - 320K 2.34430289E+00 -7.37602861E-12 -1.94779168E-05 2.34430289E+00 -7.37602861E-12 -1.94779168E-05 2.01569677E-08 7.98042491E-03 2.01569677E-08 7.98042491E-03 0K-1000K - 500K 0K-1000K - 500K 2.93283057E+00 -6.88796128E-16 -1.46400566E-07 2.93283057E+00 -6.88796128E-16 -1.46400566E-07 1.54098505E-11 8.26598029E-04 1.54098505E-11 8.26598029E-04 00 5000K - 1000K 00 5000K - 1000K Bibliography

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