Modeling and analysis of flame stretch and preferential diffusion in premixed flames de Swart, J.A.M.

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1 Modeling and analysis of flame stretch and preferential diffusion in premixed flames de Swart, J.A.M. DOI:.6/IR6597 Published: //29 Document Version Publisher s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication: A submitted manuscript is the author's version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. The final author version and the galley proof are versions of the publication after peer review. The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication Citation for published version (APA): Swart, de, J. A. M. (29). Modeling and analysis of flame stretch and preferential diffusion in premixed flames Eindhoven: Technische Universiteit Eindhoven DOI:.6/IR6597 General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. Users may download and print one copy of any publication from the public portal for the purpose of private study or research. You may not further distribute the material or use it for any profit-making activity or commercial gain You may freely distribute the URL identifying the publication in the public portal? Take down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Download date: 25. Dec. 28

2 Modeling and analysis of flame stretch and preferential diffusion in premixed flames

3 This research was financially supported by the Dutch Technology Foundation STW (EWO.5874). Copyright c 29 by J.A.M. De Swart All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form, or by any means, electronic, mechanical, photocopying, recording, or otherwise, without prior permission of the author. De Swart, Joseph A.M. Modeling and analysis of flame stretch and preferential diffusion in premixed flames. Eindhoven University of Technology, 29 Subject headings: combustion / premixed flames / flame stretch / preferential diffusion / flamelet modeling / direct numerical simulations A catalogue record is available from the Eindhoven University of Technology Library. ISBN: Reproduction: University Press Facilities, Eindhoven, The Netherlands.

4 Modeling and analysis of flame stretch and preferential diffusion in premixed flames PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven, op gezag van de rector magnificus, prof.dr.ir. C.J. van Duijn, voor een commissie aangewezen door het College voor Promoties in het openbaar te verdedigen op dinsdag 29 september 29 om 6. uur door Joseph Adrianus Maria De Swart geboren te Roosendaal en Nispen

5 Dit proefschrift is goedgekeurd door de promotor: prof.dr. L.P.H. de Goey Copromotoren: dr.ir. R.J.M. Bastiaans en dr.ir. J.A. van Oijen

6 Contents Summary ix Introduction. Introduction Objective Outline Equations, concepts and definitions 7 2. Combustion process General governing equations Definitions Flame stretch and preferential diffusion 2 3. Flame stretch Thermo-diffusive effect and preferential diffusion The combined effect of flame stretch and preferential diffusion Flame stability Laminar premixed flames One-dimensional framework Non-stretched mass burning rate Mass burning rate of stretched flames Sensitivity of the unstretched mass burning rate m il to changes in h and Z j The influence of stretch on the enthalpy and element mass fractions Accuracy of the model for m il Effects of hydrogen addition Conclusions v

7 vi CONTENTS 5 FGM vs. single step chemistry Definition of the chemical source term approaches Flame-vortex interaction Turbulent flames Conclusions Stretch and preferential diffusion in premixed flames 5 6. One-dimensional flames with detailed chemistry Including preferential diffusion in FGM Application of a one-dimensional manifold Application of a two-dimensional manifold Conclusions Conclusions 69 References 75 Samenvatting 8 Dankwoord 85 Curriculum Vitae 89

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10 Modeling and analysis of flame stretch and preferential diffusion in premixed flames Summary In many conventional power plants, electricity is generated by burning natural gas in a gas turbine. Modern dry low-nox combustion systems are efficient and have low emissions. Usually these combustion systems are operated in lean premixed combustion mode. In order to be able to operate a gas turbine as efficient as possible, to reduce emissions and to control flame stability, thorough understanding of the combustion process is required. The conversion rate of fuel in premixed combustion is determined by the burning speed. Under the highly turbulent conditions, as encountered in gas-turbine combustion, one can define a turbulent burning speed. The turbulent burning speed depends on the laminar burning speed, which strongly depends on the fuel. With developments towards a sustainable energy supply and the reduction of greenhouse gases, the use of alternative fuels is an option. In this context there is progress towards the application of clean coal technology and the use of biomass. Both methods can result in the production of large amounts of hydrogen that can be used as a fuel. Replacing a part of the natural gas by hydrogen influences the chemical processes significantly, because hydrogen is relatively light and highly diffusive. This has an impact not only on the direct conversion rate of the fuel mixture but also on the stability of the flame which can be responsible for an increased integral conversion rate. The phenomenon associated with the fact that all species have different diffusion rates, is referred to as preferential diffusion. In methane/air flames this effect is very small, but when hydrogen is present this effect is larger. Interactions of preferential diffusion and flame stretch, caused by e.g. turbulent flow, complicate things further. This makes it very worthwhile to study flame behavior of premixed stretched flames with hydrocarbon/hydrogen mixtures as a fuel. Numerical simulation is a powerful tool to gain insight in these complex combustion processes and can be used to predict what will happen under certain conditions. In the past only ad-hoc turbulent chemistry closures were used in which these processes could not be taken into account to determine the turbulent conversion rate.

11 x SUMMARY For fundamental investigations in which the interaction between small scale flow phenomena and the flame are important, all scales in the flow have to be resolved. Therefore, Direct Numerical Simulation (DNS) is the necessary tool to study turbulent flames. A drawback of DNS is that it is a computationally expensive method. In principle the chemical kinetics are taken into account by solving an equation for each species in the chemical system (detailed chemistry). This results in a very large and stiff system of equations and therefore the combination of DNS and detailed chemistry is hardly feasible. Many studies use single-step chemistry in combination with DNS to limit the computational cost. The main drawback of using single-step chemistry is that the complex effects due to multi-component flame behavior can not be taken into account. In this study, we investigate how the chemical reduction method Flamelet-Generated Manifolds (FGM) can be used to capture the chemical kinetics. In the FGM-approach the overall reaction progress is taken into account by solving an equation for one (or a few) reaction control variables. The FGM-approach reduces the number of equations to be solved and reduces the stiffness of the system of equations. Based on the reaction control variables, different quantities can be looked up in a flamelet database and can be used in the simulation. The combination of DNS and FGM is computationally feasible. Moreover, with a successful method the step towards statistical modeling with Large Eddy Simulations (LES) and Reynolds-Averaged Navier-Stokes (RANS) methods can be made for practical design calculations. The goal of the study presented in this thesis is to investigate the effects of flame stretch and preferential diffusion on the burning velocity of premixed flames. This will be done by using a DNS-FGM code. Detailed chemistry (which is feasible for one-dimensional cases) and single-step chemistry will be used to support results and to show differences between the different simulation approaches. By using detailed chemistry in one-dimensional stretched flames, it is found that not only the fuel but all chemical species in the system contribute to the flame (in)stability. The diffusivity of each species has an influence on the sensitivity of the mass burning rate to flame stretch. The sum of all these individual contributions results in the total sensitivity of the mass burning rate on the flame stretch. The response of the mass burning rate to flame stretch determines whether a stable or an unstable flame is found. In case of methane/air flames, different contributions cancel and this results in a small net effect. The resulting sensitivity in ethane/air and propane/air flames is larger, the mass burning rate decreases more with the same flame stretch than a methane/air flame. Adding hydrogen to the fuel causes a decreased sensitivity of the mass burning rate to flame stretch. As a first step in the application of FGM in a two-dimensional and three-dimensional computational domain, methane combustion is considered. In methane combustion

12 SUMMARY xi there are no preferential diffusion effects. First we focus on differences between a single-step chemistry approach and a D-FGM (where only a single reaction control variable is used to look up the chemical source term). In the single-step chemistry approach the chemical source term distribution is functionally fixed and tuned to get a physical behavior. This is done in such a way that the correct flame thickness and flame speed are retained. With the application of FGM no tuning is required. It is shown that globally, for the applied coherent perturbations, no large differences exist between an FGM-approach and a single-step approach. In all flame-vortex interactions, the flame surface area and the mass consumption are slightly larger in the FGM-approach. In the turbulent flames small differences (positive and negative) are seen in the flame surface area and the mass consumption as well, however the differences are more or less random with time. Then, hydrogen is added to the fuel mixture and preferential diffusion effects cause local variation in stoichiometry and flame temperature. In order to describe the combustion reaction accurately, information on the local perturbed conditions is required. The FGM-method is adjusted in such a way that an extra equation is solved that describes these local perturbations. It is clear that the effects can not be described by D methods, either single-step chemistry or a D-FGM. By application of a 2D-FGM, in which a constant flame stretch profile is taken into account, it is found that the results are much more accurate than a D-FGM simulation (or equivalently D single-step chemistry). This is concluded on the basis of a comparison in which several stretch fields along flamelet paths in the multi-dimensional solution are substituted in a D solver using detailed chemistry. It is concluded that the DNS- 2D-FGM numerical code is an efficient way to capture preferential diffusion effects to first order accuracy, taking (constant) stretch into account. This requires the use of two control variables. Adding hydrogen to the fuel results in a larger laminar mass consumption as well as in an increased turbulent consumption due to an increase in the flame surface area. The dimension of the FGM-database can be extended to gain accuracy, by taking stretch rate fluctuations through the flame front into account. In that case three control variables are needed. Progress is made in DNS with reduced chemistry for different hydrocarbons and methane/hydrogen mixtures under perturbed flow conditions, yielding insight in the flame behavior. This is essential for model closure in simulations of flame behavior in practical applications. The step towards application in LES/RANS for more practical large scale design calculations needs further research, especially in the area of the behavior of (subgrid) fluctuations in the presence of instabilities.

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14 CHAPTER ONE Introduction. Introduction Most of the world s energy comes from the combustion of fossil fuels and it is expected that combustion of fossil fuels will also be the main source of energy in the next few decades [33]. Many electricity plants use gas turbines to burn natural gas, converting chemical energy into electricity and heat. A point of concern in burning fossil fuels is that pollutants that are emitted, can result in global warming (e.g., caused by emission of CO 2 ) and acid rain (e.g., caused by emission of NO x ). The emission of NO x can be reduced by lowering the temperature. This used to be done using water injection, but more recently dry low-no x combustion systems are used. In these dry low-no x systems premixed combustion takes place under fuellean conditions to reduce the temperature and thus the emission of NO x. The power output of a gas turbine depends on the fuel conversion rate, which can be increased by burning the fuel under turbulent conditions. However, when operating the combustion system in lean premixed mode the flame movement has to be controlled because flashback and blow-off can be dangerous and have to be avoided. Emission of CO 2 can be decreased by using alternative ( green ) fuels. In so-called clean coal technology CO 2 is captured and stored. Another option is the use of biomass, where it is possible to gasify the biomass yielding a mixture of H 2 and CO. Subsequently, the water-gas shift reaction can be used to convert CO and H 2 O into CO 2 and H 2. So, the use of biomass can results in large amounts of hydrogen and instead of burning methane in a gas turbine, a mixture of methane and hydrogen can be burned. In this thesis the focus will be on accurate combustion modeling of methane and mixtures of methane and (biomass-derived) hydrogen. To ensure a high efficiency and minimum pollution and to control flame stability, we need to understand what happens in these combustion processes. From a physical point of view, a complex interaction between flame and flow

15 2 INTRODUCTION takes place. Turbulent vortices, so-called eddies, cause the flame to be curved and stretched. These deformations may have a large impact on the flame structure and the flame behavior. On the other hand, the flame influences the flow, mainly by expansion. This interaction between flow and combustion chemistry will be investigated in this study. This interaction is made even more complex by the fact that all species and heat have a different diffusion rate (preferential diffusion). When the diffusion of mass is much larger (light species) or smaller (heavy species) than the diffusion of heat, local stoichiometry and flame temperature change. These local changes have an effect on the local fuel conversion rate, which can result in an unstable flame [77]. For methane/air flames the diffusion velocities of the chemical components and heat do not differ much and preferential diffusion hardly influences the flame. When hydrogen (highly diffusive) is present, it is essential to include preferential diffusion in flame modeling. A thorough theoretical understanding is the basis for improving industrial applications in which combustion takes place. One way to understand the different processes and phenomena in combustion, is numerically simulating flames. A large benefit of numerical simulation is that no expensive set-up has to be built, which is the case with experimental research. Also, it is relatively easy to change domain sizes, inflow-conditions, etc. A disadvantage of numerical simulations is that one needs to provide the models to be used in the simulation. Also, numerical simulations can be expensive in terms of CPU-time. Therefore, a large part of the combustion community is focussing on finding accurate and efficient numerical models to simulate combustion. This is also the goal of this research. The interaction between small scale flow structures and the flame is important. Therefore we want to solve the flow up to the smallest scale (Direct Numerical Simulations, DNS). DNS is relatively expensive because a fine grid is required to solve all scales in the flow. Moreover, in order to model the chemical kinetics in detail (thus very accurately) in a numerical simulation of combustion processes, all species have to be taken into account. This means that a transport equation has to be solved for each chemical component. In this way, the individual diffusion rates of all species are included, accurately describing preferential diffusion. Solving a transport equation for each species is referred to as detailed chemistry. However, a detailed chemical reaction scheme consists of many species and reactions, resulting in a large number of equations. All these chemical reactions have a different time scale and this results in a very stiff system of equations. The combination of DNS with detailed chemistry is hardly feasible. To reduce the computational cost, a simplified chemical model can be used. Often, single-step chemistry is used (e.g. [4 8]), where only a single reaction step is taken into account involving only reactants and products. This greatly reduces the computational cost, but multi-species effects can not be taken into account. Other reduction techniques are the Intrinsic Low-Dimensional Manifolds (ILDM) method [53] and the Computational Singular Perturbation (CSP) method [49]. In these methods, the rela-

16 .2 OBJECTIVE 3 tively fast species are assumed to be in quasi steady state. A drawback of these methods is that only chemical time scales are considered, which means that these methods are not accurate in colder regions where transport and chemical kinetics are equally important. In this study, the chemical reduction technique Flamelet-Generated Manifolds (FGM) [59] is used. The basic assumption of FGM is that a three-dimensional flame structure can be considered as an ensemble of one-dimensional laminar flame solutions. It is used that almost all transport takes place in the flame-normal direction. These one-dimensional flame solutions, so-called flamelets, are computed a- priori using detailed models for the chemistry and diffusion. These flamelets stored in a flamelet database and can be retrieved later. The manifold is parameterized by controlling variables. An important feature of this method is that steady flame solutions, including convection, diffusion and chemistry, are found..2 Objective The objectives of this study are to investigate:. the individual contribution of different species and flame temperature (or equivalently elemental composition and enthalpy) to preferential diffusion effects. In a one-dimensional context, detailed analyses are performed to show that many species and flame temperature play a role in preferential diffusion. This is done for a number of hydrocarbon fuels as well as for methane/hydrogen mixtures. The change of the local conversion rate (caused by flame stretch and preferential diffusion) is used to show the influence of the fuel mixture on flame stability. 2. the influence of flame stretch and chemical source term distribution on the burning velocity of premixed flames in a two- or three-dimensional domain, but without preferential diffusion. This is done by assuming equal mass and heat diffusion coefficients. In order to have a more realistic flame stretch field, flame-vortex interactions are investigated in two- and three-dimensional computational domains. Both a singlestep (Arrhenius) chemistry approach is used as well as FGM. The results of these simulations are used for comparison with simulations where preferential diffusion is taken into account. 3. the influence of preferential diffusion on the local mass burning rate and flame stability. Simulating methane/hydrogen fuel mixtures in flame vortex interactions reveals preferential diffusion effects. An extra equation is solved that describes the changes in local stoichiometry and flame temperature. The manifold database is now two-dimensional and local changes in stoichiometry and flame

17 4 INTRODUCTION temperature are taken into account when retrieving information from the manifold database. The local mass burning rate, which can be found from the local density and the local flame speed, changes due to flame stretch and preferential diffusion and this affects the flame stability. 4. how the amount of hydrogen in the fuel has an impact on preferential diffusion. The results of simulations of three methane/hydrogen mixtures, having a different amount of hydrogen, are compared. It becomes clear what the impact of the hydrogen in the fuel is on the flame behavior. Again, local mass burning rates are recovered to study the resulting flame stability. It is difficult to compare the numerical results presented in this thesis to experimental work. In the simulations presented in this thesis, we are interested in the fundamental flame behavior. Therefore, no interaction with boundaries is taken into account. Comparison to other numerical results is possible. DNS is known to be very accurate for modeling the flow, provided that the grid resolution is sufficient. Grid studies have shown that this is the case in the simulations presented in this thesis. The (DNS) results using reduced chemical approaches will be compared to simulations where detailed chemistry is used to model the chemical kinetics. Detailed chemistry is considered to be very accurate. Also, the results obtained with FGM are compared to results obtained using single-step (Arrhenius) chemistry. Many others use singlestep chemistry and it is shown that FGM can result in a gain in accuracy of the flame solutions that are found..3 Outline First, the governing equations and several concepts will be discussed in chapter 2. Flame stretch and preferential diffusion are explained in detail in chapter 3. In chapter 4, flame stretch and preferential diffusion effects are investigated in stretched laminar one-dimensional flames, using flamelet code CHEMD [2, 8] in combination with detailed chemistry. In this chapter, the theory of De Goey and Ten Thije Boonkkamp [32] is adjusted to describe the stretched mass burning rate at the inner layer position using a flame stretch rate and only information from the flame solution of an unstretched flame. After investigating mixtures of different hydrocarbons and air, hydrogen addition is discussed. It is shown that many species have a contribution in the preferential diffusion effect and that taking the lean species (fuel) only is an incomplete approach. It is also found that adding hydrogen influences the response of the mass burning rate to flame stretch. The resulting mass burning rate affects the flame stability. Then, in chapter 5, the DNS-code SENGA will be introduced. This code will be used to simulate flame-vortex interaction a in two-dimensional computational domain and turbulent combustion in a three-dimensional computational domain. SENGA

18 .3 OUTLINE 5 has been adapted to be able to model the chemical kinetics using FGM [59]. The numerical code, especially the new FGM-module, will be validated in this chapter. Both this FGM-implementation as well as a single-step (Arrhenius) chemical model will be used and results will be compared. Results are shown of flame-vortex interaction as well as results of turbulent premixed combustion with equal diffusion coefficients for mass and heat to exclude preferential diffusion. It is shown that the increase in mass burning rate and increase in flame surface area have aa approximately linear relation. The mass burning rate of a stretched flame can be found from the dimensionless flame stretch rate and the mass burning rate of an unstretched flame. In chapter 6, the approach is to extend the FGM-implementation to a moredimensional database. This is required for capturing preferential diffusion effects. Again the adjustments will be validated and results will be shown of flame vortex interaction in a two-dimensional computational domain and turbulent premixed combustion in a three-dimensional computational domain, including preferential diffusion effects. It is shown that when local deviations are captured, accurate mass burning rates are found. The local stoichiometry and flame temperature are taken into account by solving only one extra equation. It is shown that this extra equation is able to account for local changes fairly accurate. Also, it is shown that single step methods (like D-FGM of single-step Arrhenius chemistry) are not able to capture preferential diffusion due to its multi-species character. It is relatively easy to solve an equation for the species mass fraction of NO as well. Increasing the hydrogen content results in more NO over the outflow boundary, which is related to a higher flame temperature. Finally, a discussion will follow.

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20 CHAPTER TWO Equations, concepts and definitions In this chapter the basics of a general combustion process will be discussed. In order to be able to describe the physics involved a number of definitions and concepts are required. These will be discussed in this chapter as well. 2. Combustion process A combustion reaction is an exothermal chemical reaction where fuel is oxidized. Usually a source of ignition is required and after ignition the flame produces enough heat to keep itself burning. A combustion process consists of a large number of elementary chemical steps. First the fuel breaks up into smaller molecules that will react with each another. Intermediates are formed and finally these react to form different product molecules. It is difficult to keep an overview on this large set of chemical reactions and therefore often only the global reaction is given where intermediate species have been left out. A typical example is the global reaction of combustion of methane with air, CH 4 + 2O 2 CO 2 + 2H 2 O (2.) Nitrogen present in air has been left out, since it does not participate in the basic reaction. In this thesis we will restrict ourselves to fully premixed flames. It is assumed that fuel and oxidizer are perfectly mixed, which means that the time scale at which a reaction occurs is the chemical time scale rather than the mixing (hydrodynamical) time scale. In premixed flames an unburnt side and a burnt side of the flame can be distinguished, separated by a thin flame front. The unburnt and burnt positions are located relatively far away from the flame front and all gradients are (very close to) zero at the unburnt and burnt position. In figure 2., profiles are shown of temperature, density, chemical source term and of 7

21 8 2 EQUATIONS, CONCEPTS AND DEFINITIONS ρy [kg/m 3 s], T [K], ρ 3 [kg/m 3 ] s [m] x 3 (a) Chemical source term of the progress variable (full line), temperature (dashed line) and density (dotted line). Yi [ ] s [m] x 3 (b) Profiles of of mass fraction of CH 4 (full line), O 2 (dashed line), CO 2 (dash-dotted), H 2 O (dotted) and intermediate H 2 ( 4, thin line). Figure 2.: Profiles of temperature, density, chemical source term and different species in an adiabatic one-dimensional laminar, premixed flame calculated using detailed chemistry. different species of a one-dimensional, laminar premixed flame. These profiles come from a numerical simulation, based on a lean methane/air flame (equivalence ratio φ =.7), where detailed chemistry was used. The equivalence ratio is defined as the ratio of the actual fuel/air ratio to the stoichiometric fuel/air ratio. If the equivalence ratio is equal to one, the combustion is stoichiometric. If it is smaller than one, the combustion is lean with excess air, and if it is larger than one, the combustion is rich with incomplete combustion. Coordinate s is the coordinate perpendicular to the flame surface. The flame front is positioned around the maximum of the chemical source term, around s =.6 3 m in figure 2.. The unburnt situation is approximately found where s < 3 m, while the burnt side is approximately located where s > 2 3 m. More information on the flame structure is given in section 2.3. It can be seen that as the flame front propagates (to the left in figure 2.), the reactants (CH 4 and O 2 ) are consumed and products (CO 2 and H 2 O) are formed. Intermediate species (e.g. H 2 ) are created first and then consumed. Also the temperature rises from the unburnt inlet temperature to the burnt temperature and the density decreases. It can also be seen that, since this is a lean flame, oxygen is present in the burnt mixture. The maximum of the chemical source term is positioned close to the burnt side and heat is released during the chemical reactions. Looking in more detail (figure 2.), it can be seen that the global chemical reaction, Eq. (2.), is a result of a large number of elementary reactions. For example, in figure 2.(b) the mass fraction of H 2 is seen while this is not present in global reaction (2.).

22 2. COMBUSTION PROCESS 9 These elementary reactions can be written as N s i= ν lia i N s i= ν lia i for l =,..., N r, (2.2) with ν i and ν i the molar stoichiometric coefficients of species i in reaction l, N r the number of reactions, N s the number of species and A i the chemical symbol for species i. In order to describe the complete combustion process accurately, the chemical kinetics of all of these elementary reactions have to be known and taken into account. The chemical source term of species i, indicating the mass of species i produced per unit volume per unit time, depends on all reactions in which species i is involved, l= N r ρ i = M i (ν li ν li)q l, (2.3) in which ρ i represents the chemical source term of species i in [kg/m 3 s] (and not the time derivative of the density), M i is the molar mass of species i and Q l is the reaction rate for elementary reaction l, Q l = k f l N s i= N s [A i ] ν li k b l [A i ] ν li. (2.4) i= which expresses that the total reaction rate depends on a formation reaction rate (first term on right hand side) and a consumption reaction rate (second term on right hand side). The rate of formation and consumption depends on reaction rate constants, the molar concentration and the stoichiometric coefficients. The derivation of Q l is discussed in more detail in [76]. In this expression, [A i ] = ρy i /M i represents the molar concentration of species A i, k is the reaction rate coefficient and superscripts f and b refer to forward (going from left to right in Eq. (2.2)) and backward (going from right to left in Eq. (2.2)) reactions, respectively. The reaction rate (of a forward or backward reaction), k, is usually written in an Arrhenius form, ( ) k = A c T β Ea exp, (2.5) R T with A c the pre-exponential constant, β the temperature exponent, E a the activation energy and R is the universal gas constant. Constant E a is often referred to as the R activation temperature. The backward reaction rates, k b l, can be computed from the forward rates and the equilibrium constants K cl = k f l /kb l [47]. Using the forward reaction rate and the equilibrium constant yields more accurate results than using an expression like Eq. (2.5) of the species involved in the reaction [59, 8]. The equilibrium constants are well defined by thermodynamic properties. Tables, in which data (A c,β,e a ) of a collection of elementary reactions are stored are called chemical reaction mechanisms. A mechanism that is often used for simulations of methane/air

23 2 EQUATIONS, CONCEPTS AND DEFINITIONS flames is GRI3., which is compiled by the Gas Research Institute []. The GRI3. chemical mechanism comprises 325 reactions among 53 chemical species and is generally seen as a relatively large and accurate mechanism. In this study, this detailed mechanism will be used in the simulation of laminar flames. In the turbulent flames we will use single-step chemistry or use a reduced chemical model (FGM, [59]) based on GRI General governing equations The reacting fluid-dynamical system can be described by transport equations of the fluid dynamics and of the chemical composition. The non-reacting part can be described by equations for mass density, velocity and energy. For chemically reacting systems, such as combustion, we describe the evolution of the mass of the individual chemical species additionally. The system of equations is completed by equations of state that give pressure and enthalpy. A number of unclosed terms appear in these transport equations. These terms are closed with models and this is discussed in section The fluid dynamics are described by the following governing equations. For mass it holds that ρ t + (ρu) =, (2.6) with ρ the mass density of the mixture, u the flow velocity in three dimensions and t the time. For momentum it holds that ρu t + (ρuu) = ρg P, (2.7) with g the gravitational acceleration field, P the stress tensor, which is defined as P = pi τ, with p the thermodynamic pressure, I the unit tensor and τ the stress tensor. Energy conservation can be written as ρe t + (ρue) = ρu g q Pu, (2.8)

24 2.2 GENERAL GOVERNING EQUATIONS where e is the energy density, which can be written as e = û + 2 u 2, û being the internal energy density and 2 u 2 the kinetic energy density and q is the heat flux vector. These equations (Eqs (2.6), (2.7) and (2.8)) complete the system of equations for nonreacting flow. Energy conservation can be replaced by an equation that describes enthalpy or temperature. In the numerical code CHEMD [2, 8], which is used in the simulations presented in chapter 4, conservation of enthalpy is solved rather than energy. The enthalpy is defined as h = û + p/ρ and enthalpy conservation then can be written as ρh t + (ρuh) = ρu g q + τ : ( u) + dp dt, (2.9) with dp := p/ t + u p, where the second right hand side term is neglected in dt CHEMD [2, 8]. The first right hand side term is zero when the flame is not enclosed by domain boundaries. The chemical system is described by conservation of mass of all chemical species, ρ i t + (ρ iu i ) = ρ i, (2.) with ρ i the mass density of species i, u i the velocity of species i and ρ i the chemical source term of species i, representing the formation or destruction by chemical reactions. The creation or destruction of species i is directly coupled to the chemical mechanism. The chemical source term, as described in Eq. (2.3) appears in the transport equation of species i (Eq. (2.)). The velocity of species i can be written as u i = u + U i, with U i the diffusion velocity of species i. The diffusion velocity is the deviation of the velocity of species i from the bulk flow velocity u. Using the definition of the species velocity and using that Y i = ρ i /ρ, for species mass fraction, Eq. (2.) can also be witten as (ρy i ) t + (ρuy i ) = (ρu i Y i ) + ρ i. (2.) Writing the species velocity u i as the sum of the bulk gas flow velocity u and the diffusion velocity U i is convenient because now we explicitly see a time-dependent term, convection, diffusion and a chemical source term. The diffusion U i will be modeled as is explained in section More information on the derivation of the conservation equations of a reacting flow can be found in [7, 89] Models to close the equations In order to close the system of equations, a number of models is required. Pressure and enthalpy can be deduced from mass density, temperature and species mass fractions, using a thermal and caloric equation of state. Transport models are used for

25 2 2 EQUATIONS, CONCEPTS AND DEFINITIONS the viscous stress tensor τ, the diffusion velocity U i and heat flux q. Thermal and caloric equation of state The thermal equation of state relates the pressure to density, temperature and the species mass fractions. In this system of equations, density, temperature and species mass fractions are primitive variables and from these variables the pressure follows. All species present in the flow are considered to behave like an ideal gas. The ideal gas law is used and for the partial pressure p i this reads p i = n i R T, (2.2) with n i the molar concentration of species i and T is the temperature of the mixture. The thermodynamic pressure p follows from summing over all partial pressures p i, p = N s p i = i= N s i= ρr T Y i M i, (2.3) M where it was used that n i = nx i = ny i M i = ρ Y i M i. Here, n is the molar concentration of the total gas mixture, X i is the species mole fraction and M is the mean molar mass. Enthalpy follows from a summation over species enthalpies h i, which depend on temperature and specific heat using the caloric equation of state h = N s i= T Y i h i, with h i = h re f i + c p,i (T )dt, T re f (2.4) where h is the total enthalpy of the mixture. Parameter c p,i is the specific heat of species i at constant pressure. The enthalpy of species i consists of an enthalpy of formation h re f i at temperature T re f and a thermal part (integral part of Eq. (2.4)). Usually, c p,i is tabulated in polynomial form [46], 5 c p,i R = b ni T n. (2.5) n= The specific heat of the mixture is the average specific heat of all species weighed with the species mass fractions c p = N s i= Y i c p,i. (2.6)

26 2.2 GENERAL GOVERNING EQUATIONS 3 Transport models Assuming that the medium behaves as a Newtonian fluid, the viscous stress tensor can be modeled using Stokes law of friction for a compressible mixture [75], τ = µ ( u + ( u) T 23 ) ( u)i, (2.7) where µ is the dynamic viscosity of the mixture. The diffusion velocity field U i can be found from solving the so-called Stefan- Maxwell equation [9], X i = N s X i X j (U j U i ), (2.8) j= D i j where the influence of pressure gradients and temperature gradients (Soret effect) on the diffusion velocity have been neglected [87]. In this equation X i = Y i M/M i is the mole fraction of species i and D i j is the binary mass diffusion coefficient of species i into species j. However, solving system of equations (2.8) is computationally very expensive. Therefore, a simplified approach is used. Instead of taking into account the diffusion of each species into all other species, we use Fick s law [54] that describes the diffusion of each species in the mixture, U i = D im Y i Y i. (2.9) In this equation, D im is the mixture-averaged diffusion coefficient. If air is the oxidizer, the simplified approach as given in Eq. (2.9) performs well because nitrogen is abundant [8]. The heat flux is modeled using an expression that contains the heat transport due to conduction and mass diffusion [89], N s q = λ T + ρ U i Y i h i, (2.2) i= where λ is the thermal conductivity of the mixture. Using Eqs. (2.4), (2.6) and (2.9), the heat flux can be rewritten to q = λ h λ N s ( ) c p c p h i Y i, (2.2) i= Le i where Le i is the Lewis number of species i. The Lewis number is a dimensionless number that describes the ratio between thermal diffusivity (λ/c p ) to the species

27 4 2 EQUATIONS, CONCEPTS AND DEFINITIONS mass diffusivity (ρ/d im ), Le i = λ ρd im c p. (2.22) If λ/c p and ρd im are equal, the diffusion rate coefficients of mass and heat are equal and the Lewis number is equal to one. When the Lewis number is not equal to one, caused by different diffusion coefficients for mass and heat, species and heat are redistributed and locally there is more or less mass and/or heat. It is very difficult to say what the mixture looks like depending on the Lewis number. In chapter 3 preferential diffusion is explained in more detail and the results of non-unit Lewis numbers are shown and discussed in chapter 4. Finally, the thermal conductivity and the dynamic viscosity are modeled by [79] and λ/c p = (T/298K).69 (2.23) µ/c p =.67 8 (T/298K).5. (2.24) The unit of λ/c p is [kg/m s] and the unit of µ/c p is [kg 2 /J m s]. In general the mixture averaged dynamic viscosity and conductivity depend on temperature and composition of the mixture in a complex way. The models as given in Eqs. (2.23) and (2.24) are found by fitting the results of methane/air flame simulations with more complex models. The models presented above perform well if the conditions are comparable to the fitted simulations (fuel/air with abundant nitrogen) and reduce the computational cost. The stoichiometric ratio does not have a large influence on the model coefficients. In case of hydrogen addition later in this thesis, the coefficients in these models (Eqs. (2.23) and (2.24)) are slightly adjusted. Inserting the expression for the heat flux vector (Eq. (2.2)) into the conservation equations for enthalpy and species yields ( ) ( ρh λ t + (ρuh) = λ h + c p c p N s i= ) ( )h i Y i Le i +ρu g+τ : ( u)+ dp dt (2.25) and (ρy i ) t + (ρuy i ) = ( ) λ Y i + ρ i. (2.26) Le i c p When rewriting the equations in this format, it becomes clear that there are Lewiseffects (or preferential diffusion effects) in the equations for enthalpy and species mass fractions.

28 2.3 DEFINITIONS 5 Conservation of species can be rewritten into conservation of elements taking the proper linear combination of Eq. (2.26), (ρz j ) t + (ρuz j ) = ( ) ( λ N λ s ( ) ) Z j + c p c p w ji Y i, (2.27) i= Le i where the influence of preferential diffusion explicitly appears through the term that includes the Lewis numbers Le i. This is useful for analysis further on in this thesis. 2.3 Definitions In order to be able to discuss the behavior of the flame we define a number of parameters. In this section the definitions of flame thickness, inner layer and inner layer position, consumption rate, flame velocity and reaction progress variable are given. A laminar premixed one-dimensional flame solution of a lean methane/air mixture is used to illustrate the different properties. In a three-dimensional domain, one-dimensional flamelets can be tracked by following the gradient of the progress variable and the definitions given here can be applied to these one-dimensional flame solutions. This means that these properties can change with position in a twodimensional or three-dimensional domain. When a property in unburnt or burnt state is referred to, the value is taken relatively far away from the reaction layer, where a (approximately) constant value of this property is found Flame thickness The thickness of a flame can be evaluated in a number of different ways [7]. Each definition represents a different type of thickness and results in a different value. The thermal flame thickness is commonly used and will also be used in the remainder of this thesis. This flame thickness is defined as δ f := (T b T u ) T/ s max, (2.28) where T is the temperature and s is the coordinate perpendicular to the plane of the flame front. Subscripts u and b indicate the unburnt and burnt conditions, respectively, and at these location all variables no longer change. Property T/ s max represents the absolute maximum of the one-dimensional gradient profile of temperature over the flame-normal position s. Figure 2.2 shows the temperature of a laminar one-dimensional premixed methane/air flame (φ =.7), where the maximum temperature gradient is taken to define the flame thickness δ f.

29 6 2 EQUATIONS, CONCEPTS AND DEFINITIONS 2 T b 5 T [K] 5 T u δ f s[m] x 3 Figure 2.2: Temperature as a function of flame-normal position s. Flame thickness δ f shown as defined in Eq. (2.28). is Inner layer and inner layer position A premixed one-dimensional flame can be divided in a number of zones. When traveling from the unburnt to the burnt side of the flame, first the preheat zone is encountered. Then, the inner reaction layer and the oxidation layer follow. In the preheat zone almost no chemical reactions take place. However, the unburnt premixed gas mixture is preheated by the heat released by the flame. This is indicated by pre in figure 2.3. The unburnt mixture starts to break up into smaller molecules and recombines to form other molecules in the inner layer, indicated by il in figure 2.3. Finally, the slow oxidation reactions are completed in the oxidation layer. Here the final products are formed. The oxidation layer is indicated by ox in figure 2.3. The inner layer and the oxidation layer form the reaction layer. In order to do analysis in later sections of this thesis, we want to pinpoint the inner layer position. The position of the inner reaction layer depends on the definition used. Different definitions can be found in [7]. Here, the inner layer position is defined as that position in the flame where the chemical source term reaches a maximum (indicated by the dash-dotted line in figure 2.3). This choice is not crucial, because the different choices lead to positions in the flame that are very close to each other Reaction progress variable A reaction progress variable indicates the progress of the global chemical reaction. The scaled reaction progress variable has a value of zero in the unburnt mixture and goes to one in the fully burnt mixture. This parameter can be used to look up tabulated information from a database. The reaction progress variable has to be

30 2.3 DEFINITIONS ρy [kg/m 3 s], T [K] 5 5 il pre ox s[m] x 3 2 ρy [kg/m 3 s], T [K] Y[ ] Figure 2.3: Profiles of the chemical source term of the progress variable (full line) and temperature (dashed line) as a function of position s (left) and as a function of the scaled mass fraction of CO 2 (right). monotonously increasing or decreasing, in order to be uniquely defined. The scaled mass fraction of CO 2 can be convenient, since the formation of CO 2 is relatively slow compared to other species. The gradient of the mass fraction of CO 2 stretches out over a relatively wide range in physical space, which means that it is easier to look up an accurate value of a variable in the flamelet database, Y = Y CO 2 Y CO2,u. (2.29) Y CO2,b Y CO2,u This definition is used in chapter 5, but different choices are possible. In chapter 6, where hydrogen is present in the fuel, a progress variable is defined as Y = Y CO 2 M CO2 + Y H 2 O Y H2 X H2 ( X H2 ) Y CH 4 (2.3) M H2 O M H2 M CH4 and this is scaled between zero and one by Y = Y Ymin Ymax Ymin. (2.3) The maximum and minimum value of Y are located at the unburnt and burnt side of a one-dimensional flame profile, relatively far away from the reaction layer. A more elaborate discussion on the choice of this reaction progress variable is presented in chapter 6. In later chapters, flamelets are tracked and analyzed. For this flamelet analysis, a reaction progress variable is required with a narrow chemical source term. The mass fraction of methane, Y CH4, is used for this. A separate reaction progress variable

31 8 2 EQUATIONS, CONCEPTS AND DEFINITIONS is defined to be used in the flamelet analysis, Y CH4 = Y CH 4 Y CH4,u. (2.32) Y CH4,b Y CH4,u It is important to note that Y is used in the numerical simulations, while Y CH4 is used in the flamelet analysis only. Although many variables can be used as a reaction progress variable, the choice does influence the accuracy of the solution. The choice of progress variable will be substantiated in the different chapters where they are used Consumption rate and flame speed The consumption rate is the rate of fuel consumed by the flame per unit flame surface area and per unit time. The flame surface is defined as the iso-surface with a progress variable equal to the inner layer progress variable. The consumption rate can be derived from the kinematic equation of a flame surface and a conservation equation for the reaction progress variable. The kinematic equation describes the motion of a flame surface, dy dt := Y t + u f Y =. (2.33) A local summation of laminar flame speed s L n f and flow velocity u determines the velocity of the flame surface u f u f = u + s L n f, (2.34) in which n f is the unit vector to the flame surface pointing towards the unburnt gases, n f = Y Y. Substituting Eq. (2.34) into Eq. (2.33) yields Y t + u Y s L Y =. (2.35) Conservation of the progress variable Y can be written as follows, ρ Y t + ρu Y (ρu YY) = ρ Y. (2.36) Combining Eqs. (2.35) and (2.36) yields ρs L Y = (ρu Y Y) + ρ Y. (2.37) In other publications this property (ρs L ) can be referred to ρs d, with s d the displacement speed. Defining the local mass consumption rate as m := ρs L, it can be seen that the local