ZsjtL3T- hiD--1

INVESTIGATION OF SPARK IGNITION AND AUTOIGNITION IN METHANE AND AIR USING COMPUTATIONAL FLUID DYNAMICS AND CHEMICAL REACTION KINETICS

A numerical study of ignition processes in internal combustion engines.

by Rune Nordrik

Thesis for the degree of Doctor of Engineeing (Dr.Ing.)

Department of Marine Engineering The Norwegian Institute of Technology The University of Trondheim, Norway

(WgnaMWnON (OF IHB DOCUMENT IS (a&JWmED disclaimer

Portions of this document may be Illegible In electronic image products. Images are produced from the best available original document Preface

Preface

The arm of this thesis has been to investigate ignition phenomena in internal combustion engines by the aid of numerical simulation methods. The background for such an investigation is the increased emphasis during later years on the processes inside the combustion chamber. These processes are responsible for extracting energy from fuel and oxygen and converting the energy to mechanical available energy utilized by the piston movement. There are both economical and environmental reasons for making this process as efficient as possible. The combustion are also the origin of pollutants, which have gained increased concern due to the environmental impacts. Ignition is the onset of combustion and is therefor an important aspect of combustion research. It is the authors hope that this work will be an aid in the ongoing struggle to get a better understanding of the ignition phenomena.

This work was initiated in February 1990 when I was employed as scientific assistant at the Division of Marine Engineering at the Norwegian Institute of Technology. Since March 1991 1 have been receiving a stipend from NTNF (The Royal Norwegian Council for Scientific and Industrial Research) The stipend was administrated by MARINTEK (Norwegian Marine Technology Research Institute) through the project Environmental Friendly Diesel Engines. My working place has been at the Division of Marine Engineering during the whole study.

I would like to express my acknowledgement to my supervisor at the Division of Marine Engineering, Associate Professor Harald Valland. He gave me strong support during periods of hard struggle and apparently slow progress, but also fruitful critical comments when results was emerging. His abilities to always be available for talks and discussions was very much appreciated. I would also like to thank researcher Nils Inge Lilleheie for his guidance in the field of turbulent combustion mode lling. Thanks are also due to my student fellows in the field of combustion engineering, Hallvard Paulsen, Tor 0yvind Ask and Vilmar ,33s0y, for discussions and spiritual support. Eilif Pedersen and the other people at Preface

MARINTEK and the Division of Marine Engineering have been of great help in solving practical problems. It has been a pleasure to work with all of you.

Finally I would like to thank my wife Astrid for her understanding in times of despair, and for her encouragement in completing this work.

This thesis will be submitted for the degree

Doktor Ingenipr (Dr.Ing.) Doctor of Engineering at The University of Trondheim, The Norwegian Institute of Technology.

Trondheim, December 1993

Rune Nordrik Contents

Contents

Preface ...... i Contents ...... iii

Abstract ...... vii

Symbols ...... viii

Chapter 1 Introduction ...... 1 1.1 Background ...... 1 1.2 Some aspects of combustion reseach ...... 2 1.3 Scope of the thesis ...... 3 1.4 Framework of the thesis ...... 4

Chapter 2 Ignition mechanisms ...... 5 2.1 Spark Ignition ...... 5 2.1.1 Flame propagation in premixed gases ...... 5 2.1.2 A simple model of minimum ignition energy ...... 11 2.1.3 The spark ...... 13 2.1.4 Zero dimensional models of spark ignition ...... 15 2.1.5 Flow field simulations of spark ignition ...... 16 2.2 Autoignition ...... 19 2.2.1 Phenomenological description of auto ignition ...... 19 2.2.2 Classical models of autoignition ...... 20 2.2.3 Kinetic autoignition models...... 29 2.2.4 Two stage autoignition models...... 33 2.2.5 Flow field simulations of auto ignition ...... 39 2.3 Other ignition mechanisms ...... 43 2.3.1 Hot body ignition and prechambers ...... 43 2.3.2 Plasma jet ignitors ...... 43 2.3.3 Ignition by irradiation ...... 44 2.3.4 Shock wave ignition ...... 45 2.4 Use of computers in combustion calculations ...... 45

Chapter 3 The conservation equations for multicomponenent reacting systems...... 46 3.1 Conservation of mass...... 46

iii Contents

3.2 Conservation of species ...... 46 3.3 Conservation of momentum ...... 47 3.4 Conservation of energy ...... 48 3.5 General transport equation ...... 49

Chapter 4 Submodels of physical phenomena ...... 50 4.1 Reaction kinetics in combustion calculations ...... 50 4.1.1 The chemical production term...... 50 4.1.2 Skeletal mechanisms ...... 51 4.1.3 Reduced mechanisms ...... 54 4.1.4 No Transport of Radicals Concept ...... 58 4.1.4.1 Global four step mechanism with time integration for group 1 species ...... 59 4.1.4.2 Full skeletal mechanism with no transport of radicals...... 59 4.2 Thermodynamical and thermochemical properties ...... 62 4.2.1 The relation between pressure, density and temperature ...... 62 4.2.2 The relation between temperature and enthalpy ...... 63 4.2.3 Concentration variables and bulk properties ...... 64 4.3 Transport properties ...... 67 4.3.1 Thermal conductivity ...... 68 4.3.2 Species diffusivity...... 68 4.3.3 Viscosity ...... 69 4.4 The spark plug ...... 70 4.4.1 Spark energy input ...... 70 4.4.2 Heat loss to spark electrodes ...... 71 4.5 Constant pressure approximation ...... 74 4.6 Boundary conditions ...... 75 4.6.1 Boundary conditions for one dimensional spherical flame...... 75

Chapter 5 Solution of the equations ...... 78 5.1 Discretization methods ...... 78 5.2 Finite volume method ...... 80 5.2.1 Gauss theorem applied to the general transport equation ...... 80 5.2.2 Discretization of the general transport equation ...... 80 5.2.3 Diffusion coefficients and source terms ...... 83 5.2.4 Constant pressure approximation ...... 84 5.2.5 Modifications due to the No Transport of Radicals Concept ...... 85 5.2.5.1 Mass conservation equation ...... 85 5.2.5.2 No transport of radicals discretizing of general transport equation ...... 86 5.2.5.3 Species conservation equations ...... 87

iv Contents

S.2.5.4 Energy conservation equation ...... 88 5.2.6 Transport coefficients on control volume boundaries ...... 90 5.2.7 Source term treatment ...... 92 5.2.8 Solution of system of discretized equations ...... 93 5.2.9 Convergence criteria and time step ...... 94 5.3 Solution algorithm for ID constant pressure simulations ...... 96

Chapter 6 Performance of the No Transport of Radicals Concept and computer code ...... 99 6.1 Model assumptions ...... 99 6.2 Comparisons of burning velocity ...... 100 6.3 Comparisons of temperature and species profiles ...... 103 6.4 Importance of diffusion on flame propagation ...... 107 6.5 Computer demands and calculation stability ...... 109

Chapter 7 Simulation of spherical spark ignition ...... 110 7.1 Model assumptions ...... 110 7.2 Results and discussion ...... Ill 7.2.1 Development of the flame kernel ...... Ill 7.2.2 Heat loss and flame stretch ...... 115 7.2.3 Influence of pressure ...... 117 7.2.4 Temperature profiles ...... 120 7.2.5 Hydrogen in lean methane-air mixtures...... 122

Chapter 8 Autoignition of a methane bubble ...... 124 8.1 Model assumptions ...... 125 8.2 Results and discussion ...... 126 8.2.1 Influence of pressure and air temperature ...... 126 8.2.1.1 Ignition and combustion characteristics ...... 126 8.2.1.2 Lean combustion mode ...... 126 8.2.1.3 Near stoichiometric combustion mode ...... 133 8.2.1.4 Diffusion flame combustion mode ...... 138 8.2.1.5 Chart of combustion modes ...... 144 8.2.1.6 Ignition delay...... 146 8.2.2 Influence of bubble radius...... 147 8.2.2.1 Chart of combustion modes ...... 147 5.2.2.2 Ignition delay...... 148 8.2.3 Influence of methane temperature ...... 149 8.2.3.1 Chart of combustion modes ...... 153 8.2.3.2 Ignition delay...... 155 8.2.4 Comparison with premixed ignition delay ...... 155

Chapter 9 Turbulent gas jet autoignition ...... 159 9.1 The KTVA-II computer program ...... 160 9.1.1 The original KTVA-H code ...... 160

V

Mr ## m-mMe.':: Contents

9.1.2 Earlier extensions of the KIVA-II code...... 161 9.1.3 Extensions of the KTVA-II code to gas jet autoignition ...... 161 9.2 Turbulence models...... 162 9.2.1 Turbulent flow field model ...... 163 9.2.2 Eddy Dissipation Concept ...... 164 9.2.2.1 Steady state perfectly stirred reactor ...... 167 Q.2.2.2 Transient perfectly stirred reactor ...... 169 9.2.2.S Inclusion of the No Transport of Radicals Concept ...... 171 9.3 Simulation of turbulent gas jet autoignition ...... 172 9.3.1 Boundary and initial conditions ...... 172 9.3.2 Results and discussion ...... 173 9.3.2.1 Qualitative description of simulation results . . . 173 9.5.2.2 Ignition delay...... 174 9.3.2.3 Towards combustion of the jet...... 176

Chapter 10 Conclusions ...... 184

References...... 186

vi Abstract

Abstract

This work treats chemically reactive, multicomponent flow simulations. Ignition processes occuring in internal combustion engines are investigated.

A qualitative description of the ignition phenomena is given. Ealier work found in the litterature on both spark ignition and autoignition is discussed. Then the fundamental physical relations connected with the ignition phenomena are addressed. This includes flow field conservation equations, thermodynamics, chemical reaction kinetics, transport properties and spark modelling.

The inclusion of chemical kinetics in the flow field equations is given special attention. A method which neglect the transport of selected intermediate species is described. The method is called the No Transport of Radicals Concept and is utilized throughout the work. This consept has the advantage of reducing the computational effort while retaining the essence of physical relations. Validation of the method is provided through comparisons with flame propagation data.

A computational method is described, and the method is utilized in simulations of spark ignition in laminar premixed methane-air mixtures, as well as the autoignition process of a methane bubble surrounded by hot air. The premixed spark igniton results compare very well with experimental results from the literature. The effects of air/fuel equivalence ratio, temperature, pressure, spark energy, heat losses and addition of hydrogen is discussed. The autoignition of a methane bubble identifies the importance of diffusive and chemical processes acting together. The ignition delay times are longer than experimental values found in the literature for premixed ignition delay. The mixing process and lack of information on low temperature reactions in the skeletal kinetic mechanism is addressed as main sources of the difference.

The K3VA-II computer code is used to simulate transient turbulent methane jet autoignition. Turbulent combustion is modelled by the Eddy Dissipation Concept. The simulation gave good qualitative descriptions of the autoignition process. The computed ignition delay are somewhat longer than experimental values found in the literature. Reasons for this are discussed.

vii Symbols

Symbols

A surface area A pre-exponential constant in Arrhenius expression A reactants A constant in expression for thermal conductivity A% flame front area behind flame Ag contact area between flame and spark electrodes Aj. flame front area Afc correction factor to heat trans fer coefficient Ay flame front area in front of flame a coefficient in the final form of the discretized equations ac critical radius B branching agent, intermediate b coefficient in the final form of the discretized equations C chain carrier Cj concentration of species i in moles per volume constants in turbulent flow model c coefficient in the final form of the discretized equations cp specific heat capacity, constant pressure cPjk specific heat capacity for species k, constant pressure cv specific molar heat capacity, constant volume D low pressure stable chain carrier Di m diffusion coefficient for species i into rest of the mixture d coefficient in the final form of the discretized equations dq quenching diameter E molar activation energy in Arrhenius expression Em.„ minimum ignition energy e internal energy F fuel 4,4- parameters in propagating rate coefficients h, h c heat transfer coefficient h enthalpy

Vlll Symbols h k enthalpy for species k h° f enthalpy of formation at standard state conditions (latm, 298K) h M enthalpy for group two species I current i index number j index number K Karlovitz number (with dimensions 1/s) k index number k turbulent kinetic energy ki,kg. kinetic rate coefficients ^ backward rate coefficient kf forward rate coefficient kfau fall off correction to rate coeffisient kg gas termination rate coefficient kp propagating rate coefficient k* wall termination rate coefficient k_ high pressure limit rate coefficient L mathematical operator containing time derivative, convection and diffusion L’,L* turbulent length scale, large eddies and smallest eddies respectively Le Lewis number M third body M number of species in group two M; one mole of species i m; mass of species i m mass transfer rate (mass per total mass and time) m* mass transfer rate (mass per total mass of finestructures and time) m exponent in Arrhenius expression m fuel composition in Cn H2m n fuel composition in Cn H2m n exponent in Arrhenius expression n number of moles n count variable n number of last control volume Uj number of moles of species i n%t total number of moles O oxygen P power (effect) Pr Prandtl number

IX Symbols p consumption of mole oxygen per cycle p pressure p b pressure behind flame (burnt) p c critical pressure Pf partial pressure of fuel p u pressure in front of flame (unbumt) Q intermediate species Q heat addition Q k chemical heat release Q l heat loss %,q molar burning value of fuel R residue in convergence calculation R gas constant (R=8.3144 J/molK) R reactive intermediate R number of species in group one RH fuel molecule Rf flame radius Rj Reaction rate in mass per total mass and time unit r radius, cylindrical and spherical coordinate r constant in expression for thermal conductivity r total number of reactions S total number of species Sb velocity of flame relative to gas behind flame Sd gas velocity change due to difference in density through flame Sf velocity of flame relative to wall Su velocity of flame relative to gas in front of flame S source term Sc constant source term SP proportional source term s surface area T temperature T0 start temperature Tb temperature behind flame (burnt) Tc critical temperature T; ignition temperature Tu temperature in front of flame (unbumt) Tw wall temperature t time

x Symbols t„hnr characteristic time t; ignition delay U voltage u velocity vector Uj velocity component i ^-direction u’,u* turbulent velocity, large eddies and smallest eddies respectively V volume W,Wj molecular weight [kg/mol] XA mole fraction of reactants XF mole fraction of fuel XM mole fraction of third body X q mole fraction of oxygen x space coordinate in x-direction Xj space coordinate in x^-direction Y;,Yk massfraction of species i or k y space coordinate in y-direction z space coordinate in z-direction

Greek a branching coefficient P exponential temperature factor in Arrhenius expression Fj concentration variable for species i in moles per mass unit r* general diffusion parameter y* mass fraction occupied by the finestructure y, mass fraction occupied by the finestructure regions A, critical parameter in Frank-Kamenetski analysis e turbulent dissipation rate rio preheat zone thickness T); initiation zone thickness T|r reaction zone thickness k thermal conductivity X air/fuel equivalence ratio p k,Pe,Pr turbulent viscosities, for k, e and momentum equation respectively p absolute viscosity v kinematic viscosity vi>k stoichiometric factor for species i in reaction k

xi Symbols p density p molar density pb density behind flame (burnt) pu density in front of flame (unbumt) o k,aE constants in turbulent flow model x1 first ignition delay t2 second ignition delay Tjj viscous stress tensor TyR turbulent stress tensor t* finestructure timescale ((> general variable in transport equation % fraction of finestructure which can react to; chemical production in mass per volume and time of species i (o’,co* turbulent vorticity, large eddies and smallest eddies respectively

Mathematical max maximum of numbers in parantheses d ordinary derivative d partial derivative V gradient (grad) V- divergence (div) A small interval 5x distance between control volume centers 5ti Cronecker delta, equals one for i=j, otherwise zero Z summation operator II multiplication operator oc proportional to = very close to [ ] concentration in moles per volume unit

Accent

molar quantities time averaged quantities (turbulence models) per time unit vector ~ mass averaged quantities (Favre averaged, turbulence models) Symbols Subscript

0 start condition 0 preheat zone A reactants a air b burnt ball spherical volume c critical c contact char characteristic cons.eq. value given by the conservation equation E neighbour grid point to the east e control volum surface to the east side el electrical elec electrode ext extinction fu fuel F fuel f flame fall fall off correction (reaction coefficients) g gas 1 ignition i initiation zone i initial value ij,k count variables K chemical heat production L loss M third body m mixture min minimum ox oxygen O oxygen P current control volume p constant pressure p pressure p propagating (reaction rate) pr products

xiii Symbols q quenching r reaction zone ref reference conditions (temperature, pressure) s stoichiometric T temperature T turbulent tot total u unburat v constant volume vise viscous terms W neighbour grid point to the west w control volume surface at the west side w wall

Superscript

0 standard state (Tref, 1 atm.) 0 value in surroundings * value at previous iteration * finestructure value ’ large turbulent eddies value

XIV Chapter 1 Introduction

Chapter 1 Introduction

1.1 Background

The internal combustion engine is a vital part in order to keep our industrialized society running. In land and sea transportation it dominates totally. But also in power generation it finds widespread use. The principles of these engines dates back to 1876 when Nicolaus A. Otto developed the spark-ignition (SI) engine and to 1892 when Rudolf Diesel invented his compression-ignition (Cl) engine. During the time which has elapsed since that time there has been a continuously development towards more efficient, compact and less polluting engines. In the latest decades two major problem areas has influenced the research activity around these engines. The internal combustion engine consumes a considerable amount of the worlds limited fossil fuel reserves. It has also become a major contributor to the pollution problem. This is particularly evident in densely populated areas, but it is also a problem in global scale. Due to this challenges, combustion research has become an important activity in the latest years of engine development.

Engine manufacturers meet this challenges in different ways. The use of lean mixtures in Si-engines has the potential of lowering emissions. A change to higher compression ratios will give an increase in fuel efficiency. Such changes in engine parameters require improved engine design. The engine must be able to operate under more extreme conditions. The ignition starts the process wich converts chemically bounded energy to heat energy. This whole process is highly affected by the engine design. A stable and reliable ignition and flame development is essential in keeping emissions low and efficiency high. Uncompleted combustion leads to high levels of emissions and a drop in efficiency.

1 Chapter 1 Introduction 1.2 Some aspects of combustion reseach

Ignition and combustion are complicated phenomenas wich involve chemical reactions, ionization processes, thermodynamical processes and heat- and mass- transfer. It is often difficult to extract the impact of different mechanisms on the total process. The experimental investigations use engines, combustion bombs and other experimental equipment to investigate the effect of various conditions. Advanced measuring techniques has made it possible to get detailed information about the processes. These investigations are often expensive and they require a great deal of resources to investigate the impact of for instance change of geometry. An other investigation approach is the development of mathematical simulation models which can be run on a computer. These models are based on elementary physical considerations. When a simulation model functions properly, it is relatively easy to change the boundary conditions and investigate a wide scale of different conditions.

In this context it is important to stress that the experimental investigations search to monitor the reality while the simulation model is a model of the reality. Neither of them can claim to hold the fully correct answer. An experimental investigation depends on how good the experiment represents the actual problem, the accuracy and calibration of instruments and how the instruments affect the meassured property. A simulation model depends on that all important physical interactions are considered and that they are modelled in a way that covers correctly the whole range of property variations. Also the numerical solution of the equations requires care to minimize the discretization error. In practice both experiments and simulations suffer from a compromise between resources (time, equipment, computers) and accuracy.

Therefore it should always be comparisons of a simulation model with experimental values and vice versa. Numerical results from a model can be verified by comparing them with experimental results. On the other hand with a theoretical model it is easier to quantify the different physical interactions which influence the process. It will give knowledge about the elementary physical properties. This information is needed in constructing the experimental set-up. In this way experimental and theoretical investigations is closely connected. Their advantages and weaknesses supplement each other. Together their results can give an answer closer to the reality than one investigation alone.

2 Chapter 1 Introduction 1.3 Scope of the thesis

It is the intention of this thesis to look at the theoretical aspects of ignition and flame development. Our knowledge of the fundamental physical relations governing these processes, has reached a level which makes mathematical modelling of the ignition process possible. In this work a model founded on sound physical considerations will be established. This model will be used in simulation of spark ignition and autoignition in laminar flows as well as autoignition in turbulent flow.

The physical foundations of the model are the conservation equations for mass, species, momentum and energy. The conversion from reactants to products is described with a chemical reaction mechanism. Computer limitations make it necessary to introduce some assumptions regarding the interaction between chemical kinetics and the conservation equations. It is a substansial task to describe how to introduce a full reaction mechanism in a model capable of being used in calculations of complicated laminar and turbulent flow fields.

This text will illustrate that the status of the mathematical models, numerical solution routines and computer resources have reached an extent which make practical ignition and combustion simulations possible. It is the authors belive that techniques similar to those discussed here will become a valuable tool in constructing more efficient and less polluting engines as well as other combustion devices.

In this work methane is used as fuel. There are several reasons for choosing this fuel in a theoretical investigation. For the pressures and temperatures in question, methane is in gas phase. Then vaporization of fuel will not complicate the task. It also exists well founded reaction mechanisms which describe the conversion to products. Methane has also a practical interest since it is the main substance of natural gas.

3 Chapter 1 Introduction 1.4 Framework of the thesis

In chapter 2 a qualitative description of the ignition phenomena is given, covering both spark ignition and autoignition. Chapter 3 deals with the fundamental mathematical description of multicomponent reactive laminar flows. The submodels needed in this description are treated in chapter 4. This includes chemical kinetics and calculation of thermodynamical and transport properties. The inclusion of chemical kinetics is given special attention, and an alternative way to include reaction mechanisms is suggested.

In chapter 5 the solution of the conservation equations is discussed. Validation of the model and the computer code is given in chapter 6 by comparing steady laminar flame propagation results with results found in the literature.

The next two chapters utilize the described solution method to model ignition in laminar one dimensional flow fields. Chapter 7 discusses spark ignition of a premixed methane-air mixture, and in chapter 8 the model is extended into an autoignition model of a gas bubble of methane in surrounding hot air.

Autoignition of a transient turbulent methane jet in hot compressed air is treated in chapter 9. This task involves inclusion of chemical kinetics in turbulent combustion models. The chemical modelling principle from the laminar case is utilized to make the computation effort manageable. The simulation is two dimensional, and is accomplished by the use of a commercially available computer program.

The main conclusions is summarized in chapter 10.

4 Chapter 2 Ignition mechanisms

Chapter 2 Ignition mechanisms

There are two different main principles used in internal combustion engines. The spark-ignition (SI) engine uses a premixed charge of fuel and air in the cylinder. This mixture is ignited with an electrical spark and a flame propagates through the unbumed charge. The compression-ignition (Cl) engine injects the fuel into the compressed air in the sylinder. Due to high temperatures, fuel and air starts to bum. In this chapter we will look into the physics of these processes in some detail. Earlier work from the literature will be revealed in order to find the current status in this field of research.

2.1 Spark Ignition

2.1.1 Flame propagation in premixed gases

In an homogenous fuel-air mixture a flame may be described as a combustion front which propagates through the mixture. As fuel and oxygen react to form products, the temperature in the reaction zone is increased. The resulting temperature gradient causes heat to be conducted from the hot reaction zone into the cold gas in front of the flame. Then this gas will start to react, and a combustion wave moves into the cold gas. Figure 2.1a shows a qualitative picture of the temperature in a flame front and the location of the reaction-zone, t|r, and preheat- (or initiation-) zone, %

As a mass element moves through an adiabatic flame, the element first receives heat by conduction in the preheat zone and the temperature is increased. When the temperature has reached a temperature T;, the chemical reactions start to release chemical energy and act as an heat source. This inreases the temperature

5 Chapter 2 Ignition mechanisms

Figure 2.1. Flame front temperature and velocities. further. But now the mass element also loses heat by conduction until all the conducted heat from the preheat zone is lost. When the reactions have converted the reactants to products, the temperature has reached the adiabatic flame temperature Tb . The ignition temperature T; is has not a clearly defined value. All the processes taking part are gradually interchanging their relative influence.

In order to illustrate the flame velocity we will look at an one-dimensional planar flame. The velocity of the flame front relative to the upstream gas-velocity is defined as Su. If the upstream gas moves with a velocity Su towards the flame, we will have a steady flame position as in figure 2.1b. When the gas passes through the flame front, chemical reactions will change the composition, the temperature and thus the density of the gas. This density decrease causes a velocity change, Sd, as the gas expands through the flame. The velocity of the downstream gas relative to the flame front is defined as Sb. We see that this velocity equals

6 Chapter 2 Ignition mechanisms

sb = Su+Sd (2.1) where Sd is the velocity change due to gas expansion.

By continuity the mass flow through the flame is constant and this gives Pb SA = PuSA (2-2)

For a planar flame the flow area is constant and we get a relation between upstream and downstream flame velocity

Pb Sb = PuSu (2l3)

Combination of equation 2.1 and 2.3 yields an expression for the velocity change due to gas expansion

sd (2.4)

Now let the gas downstream of the flame be forced to rest. This is the situation when a flame moves from a wall. We can then find by vector summation in figure 2.1b, that the flame-front moves with a velocity Sb relative to the end wall. The gas upstream of the flame moves with a velocity Sb -Su. Figure 2.1c shows this. Notice that this is also the situation we get when we ignite centrally and two flames move appart from one another. In this case the propagation velocity of the flame relative to the ignition source is (from fig. 2.1 and eq. 2.1 and 2.4)

Sf - Sb - Su+Sd ——Su (2.5) Pb

The burning velocity tabulated from experimental and theoretical work is Su, and it is important to distinguish between this relative velocity and the propagation velocity of a flame.

In order to find a relationship between the burning velocity and the preheat zone thickness we can use energy conservation. The heat convected results in a temperature increase in the cold gas. Using figure 2.2 and equating heat conduction to heat accumulation in the T; plane results in

7 Chapter 2 Ignition mechanisms

Figure 2.2. Flame front with definition of preheat zone T| p.

(2.6)

Here k is the thermal conductivity, A is the flame surface and V is the volume swept out by the flame during the time At. The temperature gradient is taken to be a constant and this defines a preheat zone thickness t|0. During the time At the gas in V has been raised from a temperature Tu to T;. Equation 2.6 then gives

TrTu VTrTu K Pcp (2.7) no A At

Then V/A is the distance the flame has propagated and V/(AAt) is the burning velocity Su. The relation between this velocity and the preheat zone thickness as defined in figure 2.2 is

Su = (2.8) pCpTlo

Lewis and Von Elbe [1] comments that this is a fair approximation to to the actual process taking place. The major disadvantage is that the implications of chemical kinetics is not taken into account.

Heat conduction is the most important mechansim that leads to the propagation of a laminar combustion front. Appart from heat conduction and chemical kinetics

8 Chapter 2 Ignition mechanisms also other mechanisms make contributions. There are many chemical species taking part in a combustion reaction. The diffusion of these species will give mass transport of different species in the flame front and this will influence the propagation process. Secondary diffusion effects are species diffusion due to temperature gradients (thermal diffusion or Soret effect), and the reciprocal effect which is heat flux produced by concentration gradients (Dufour effect). These secondary effects have only limited influence.

Figure 2.1 of the planar flame shows an idealized situation. In most flames the flame surface changes over time, or the flow field is skew relative to the flame, that is the flow velocity is not normal to the flame surface. Both these situations will indroduce stretching of the flame surface. The consept of flame stretch was first introduced by Karlovitz [2] when he used this consept to describe flame extinction in veloctiy gradients. The flame stretch can be quantified by defining a Karlovitz-number (see for instance Williams [3])

K = —— (2.9) A dt

The boundary of the surface A moves tangentially to the surface at the local tangential component of the fluid velocity. According to this definition a flame kernel which is ignitied centrally and propagates spherically outwards will have a Karlowitz number:

K = —(2.10) Rf dt where Rf is the instantaneous flame radius. This shows that an outwardly propagating spherical flame is positivly stretched. The interpretation of this is that the flame surface increases with time, and thus both the total heat conduction and mass consumed in the chemical reactions will also increase.

As earlier stated flame propagation is mainly due to heat conduction. For a positivly stretched flame this means that a fast surface increase requires a relativly larger portion of the heat in the reaction zone to be conducted to the cold gas in front of the flame than in the situation with a slow surface increase. This is illustrated in figure 2.3. When the radius is small the relative growth of the flame surface is large. When the radius is large the similarity with a planar flame with constant area is evident. This should imply that it is more difficult for a flame with small radius to propagate, and that the propagating speed of the flame

9 Chapter 2 Ignition mechanisms

Figure2.3. Spherical flame, surface variations. should be lower for such a flame. This phenomena is demonstated in experimental work by Ask [4].

Diffusion of species is also affected by flame stretch. Due to differences in diffusivity of the species, the implications are not always straightforward. Law [5] gives a rewiew of the dynamics of stretched flames. Flame stretch is not a phenomena that needs seperate modelling. Its consequences are preserved by the conservation equations for reacting flows.

To investigate the pressure drop over a planar flame as in figure 2.1, we need to take the momentum into account. If we neglegt the transport of species, the equation of conservation of momentum will reduce to the well known Bernoulli equation

Pusu+Pu = Pb sb +Pb (2'n)

Together with the continuity equation 2.3, this will give the following pressure change over the flame

10 Chapter 2 Ignition mechanisms

(2.12)

Typical values for a stoichiometric methane-air flame at 1 bar are upstream density p u=l.lkg/m3, flame velocity Su= 0.37m/s and density ratio= 7. This will give a pressure chan ge (p u-p b )= 0.9Pa= 0.009mbar. This is less than 0.01 % of the absolute pressure. Such a small change may be neglected, and the process is often considered as a constant pressure process.

2.1.2 A simple model of minimum ignition energy

Kanuri [6] refers to spark ignition as forced ignition. An electrical discharge between two electrodes heats up the premixed gas between the electodes. The discharge also produces electrons and ions. This will lead to chemical reactions between the molecules in the gas. These reactions which converts reactants to products, further increase the temperature. If the energy supplied in the spark is sufficient, a stable flame front will propagate from the electrodes further into the premixed gas. In quiescent mixtures this flame will have a nearly spherical form. If the energy supplied in the spark is smaller than some critical amount the flame will rappidly die, and only a small amount of products will be left. The heat losses to the electrodes and the surrounding gas will in this case be larger than the heat released by the chemical reactions.

These considerations lead to a simple model for minimum ignition energy. If we only consider heat loss to the surrounding gas by conduction, we state that a condition for ignition is that a minimal flame has been created. A minimal flame has the dimensions of the preheat zone thickness, and the inner temperature is the adiabatic flame temperature. The amount of energy necessary to create such a zone is of order EU = AfqoPcp (Tb -Tu) (2.13) where A* is the flame front area, and T|0 is the preheat zone thickness. If we use equation 2.8 to express this in terms of burning velocity Su we get

11 Chapter 2 Ignition mechanisms

EU = Af^-(Tb -Tu) (2.14) b u

A picture of a spherical minim al flame is shown in figure 2.4. By the term total heat in figure 2.4 is ment the supplied energy plus the thermal energy released by the chemical reactions. The temperature gradient leads to heat conduction, and thermal energy is thus transported outwards due to conduction.

In a spherical minimal flame we can represent the flame front area by using the quenching distance dq. This distance is an experimental value which essentially is a measure for the smallest channel dimension a flame can propagate trough. If we take dq as the minimal flame diameter we get

Emin = ttd^OVTJ (2.15) b u

Lewis and von Elbe [1] have compared this theoretical minimum ignition energy with experimental values. They show good agreement for high temperature, fast

Figur 2.4. Model of minimal flame. Adapted from ref. [1].

12 Chapter 2 Ignition mechanisms burning mixtures of fuels with oxygen or oxygen-enriched air. For low temperature, slow burning air mixtures the theoretical values are too high. Lewis and von Elbe comment that the major reason for the disagreement at slow burning mixtures, is probably the neglection of interdiffusion of species. This process is more important as the burning velocity gets lower. Light species diffuse faster than heavy species. Methane which is ligther than oxygen, has its lowest experimental minimum ignition energy on the lean side of stoichiometric composition. This is because the fast diffusing methane molecule diffuses from the surro undings into the minimal flame and tends to enrich the mixture. Propane which is heavier than oxygen, has its lowest experimental minimum ignition energy on the rich side. Propane diffuses from the surroundings into the minimal flame faster than oxygen and shifts the composition in the flame towards stoichiometric conditions.

2.1.3 The spark

A spark is an electrical discharge between two electrodes. Bjprge [7] gives a description of the mechanisms in this complicated process. His work also contains references to detailed investigations of the processes occuring. This text will only give a short overview.

Figure 2.5 shows the voltage and current of a technical ignition system as a function of discharge time. We can divide the process into three phases, breakdown, arc discharge and glow discharge. The breakdown phase has a short duration (nanoseconds). The resistance in the gas is high, and the voltage has to be built up to high values before electron avalanches originate from the cathode. The result of the breakdown phase is a highly conducting cylindrical channel between the cathode and the anode. The channel has an extremly high temperature, more than 10,000 K, and the gas is in plasma phase. The radius of this channel is about 20 pm. When the channel is established, the voltage drops because of the lower resistance in the channel, and a self sustained discharge will be the result.

The high temperature leads to a sudden expansion of the channel. At the beginning the velocity will be as high as 1000 - 10,000 m/s. This blast wave will rapidly decrease to a shock wave after about 200 ns and further to an accoustic wave after some microseconds. The first part of the self sustained discharge is

13 Chapter 2 Ignition mechanisms

150 V) ! (500 V) MmJ) I ! (30mJ)

Figure 2.5. Voltage andcurrent of ignition systems as functions of discharge time. The three discharge modes are shown. From ref. [11]. called arc discharge and have a duration of order microseconds. The cathode emits electrons from a small area called the cathode spot. The area close to the cathode is called the cathode fall region. The anode attracts electrons. The area close to the anode is called the anode fall region, and this area is the main supplier of positive ions in the arc discharge. Between the two fall regions is a region called the positive column. This region is larger than the two fall regions and it consist of plasma. The heat transfer process from the positive column to the gas is very efficient due to radial transport of the dissociated particles found in the arc. The temperature in the arc is still high and can reach a value of 5000 to 10,000 K The main energy conversion from electical energy to heat proceeds through dissociation of gas molecules. Molecules are ionized by the electrical current and when returning to a stable mode they will reject heat.

When the current which flows in the arc becomes lower, it can no longer support the arc, and the discharge will transform into a glow discharge. The glow discharge has a duration of milliseconds, and the temperature in the positive column is lower than in the arc discharge. Hence the heat transfer process is not so efficient.

If the energy supplied in the spark is sufficient, the channel expansion will transform into an expansion of a flame kernel, and a stable flame front will propagate.

14 Chapter 2 Ignition mechanisms

The different phases convert a certain amount of the electrical energy into thermal energy in the gas. The efficiency is best for the breakdown phase with approximatly 90% energy converted. But this phase has also the shortest duration. The arc discharge converts between 30 and 50%, and the longest phase, glow discharge, converts 10 to 30% of the energy. The losses are mainly heat transfer to the electrodes.

For this reason efforts have been made to construct high voltage, short duration igniton systems. These systems are called capacitive and have a short discharge time with a larger amount of the energy released in the breakdown and arc discharge phase. Conventional systems are of an inductive type, and the lower voltage result in longer discharge times. The term composite sparks is used when the system first gives a capacitive spark discharge of short duration followed by an inductive spark discharge of longer duration.

2.1.4 Zero dimensional models of spark ignition

A large amount of publications concerning spark ignition exist. Both experimental and theoretical investigations have given insight into different aspects of this phenomena. This chapter will try to give an overview of what kind of mechanisms which are considered in various simulation investigations.

Ballal and Lefebvre [8,9,10], Maly and Vogel [11], De Soethe [12], Ziegler et al. [13] and others have carried out experimental work on electrical discharges and subsequent flame propagation or flame quenching. These works contain phenomenological information of great importance to the attempt of making simulation models of the ignition phenomena.

Pischinger and Heywood [14,15] have made a zero dimensional model of ignition by using energy conservation. They balanced the heat addition from the electrical spark and the heat loss to the electrodes to the propagation of a laminar flame front. The chemical mass-conversion was taken into account by using the flame speed. They used a lumped heat transfer coefficient of 2000 W/m2K to model the heat transfer to the electrodes. They found that that variations in heat losses and heat deposits from the spark was the main reason for cyclic variability in flame growth. These variations resulted in differences in magnitude and direction of the early flame kernel motion. Maly [16] has developed a model from similar

15 Chapter 2 Ignition mechanisms considerations, but also included the conductivity of the gas. He found that a minimum ignition radii is required for succesful ignition, independent of the actual plasma kernel shape. Both investigations have made extensive use of experimental values to confirm their results.

Adelman [17] developed a model of the spark kernel growth behind a spark induced shock wave by simulating the time dependent energy release in an ideal non-combustible gas without transport effects. His theory was based on that the spark kernel must grow to a critical size before its expansion velocity falls below a critcal rate.

2.1.5 Flow field simulations of spark ignition

Flow field simulations imply the use of all or at least some of the dynamic conservation equations for mass, species, impulse and energy. These equations have to be discretized over a solution grid. This is a substantial task and depending on the problem in question, this may be done in one, two or three spatial dimensions. The geometry is usually cylindrical or spherical. In addition a model of chemical chemistry is needed if combustion is to be considered. This model can be at different levels, ranging from simple one overall reaction models including asymptotic analyses, via reduced kinetic mechanisms, to full kinetic schemes of various sizes. Caloric and thermal gas laws must also be included. The transport properties governing the different diffusion processes have to be modelled. In order to be able to calculate all this, the problem is usually reduced by introducing various kinds of simplifications.

Champion et al. [18] have used a one step Arrhenius type expression for the overall reaction rate. By combining the energy conservation equation and the reactant specie conservation equation and neglegting the convection, they formulate a large activation energy asymptotic theory. This theory divides the problem into three regions. The burned gas region where the radius is smaller than the flame radius, a thin reactive layer around the flame radius, and the unbumed region outside the flame radius. They found that the ignition is controlled by the existence of an unstable equilibrium radius of the flame kernel. This radius is particulaiy sensitive to the diffusivity of the limiting reactant of the chemical process. In their theory they state that a successful flame propagation depends not only on that the mixture is succesfully ignited, but that this ignition

16 Chapter 2 Ignition mechanisms

leads to a spherical flame ball with diameter larger than the critical radius. They give several examples of well ignited flame kernels which collaps and extinguish before they reach a critical size. In other words sufficient energy must be supplied not only to make the chemical reactions run away, but also to expand the flame ball across this critical diameter. According to their theory it is also necessary to supply the ignition energy in an adequate way. The power of the ignition source must be larger than some critical value. They point out that an energy supply considerably larger than the •minimum ignition energy may not lead to flame propagation if the energy is supplied too slowly. Their theory is supported by experiments in the case of lean mixtures with a fuel having molecular weight greater than oxygen.

Wu et al. [19] have also emphasized the importance of a critical flame diameter. In their work they have used an overall reaction and calculated the critical diameter and the minimum ignition energy for methane-air and propane-air mixtures. They have also obtained a relationship between oxygen concentration and critical ignition pressure.

Kono et al. [20] and Ujiie et al. [21] have simulated the early flame kernel formation produced by short duration sparks. They have used a two-dimensional cylindrical solution domain with the spark gap placed centrally along the axis. The heat release by chemical reactions was neglected, and the gas consisted of a single component. Heat transfer to the spark electrodes was neglected. The spark discharge was considered instantaneous, and the starting conditions was an high temperature, high pressure channel between the electrodes. They found that this channel set up a shock wave which within 2 ps separated from the hot channel and travelled into the surrounding atmosphere. With thin electrodes (0.2 mm), the hot channel transformed initially into an ellipsoid, and later, as a consequence of the expansion away from the spark gap, it got the shape of a torus. The temperature distribution and velocity field is shown on figure 2.6. Then the ring of the torus grows, and the center of the ring extends outwards. The maximum temperature was found in the center of the ring. The torus shape was set up by a pair of vortices with opposite directions, and the center of each vortex located at the periphery of the torus ring. Thicker electrodes (e.g. 0.8 mm) reduced the maximum temperature in the torus ring, and made a region of high temperature in the center of the spark gap. Here the quenching effect is rather strong compared to the quenching effect in the torus ring which is effectivly blown away from the electrodes. Using the same ignition energy, but wider spark gap, reduced

17 Chapter 2 Ignition mechanisms

Time-0.1 msec r (mm) —► , a 10 m/sec ' ^ V h ► A ' V A A -f T > " v > 'v * Tlme-O.lmsec r (mm) ' v " A***-*-**''*'1 jjya.'lSVVVV 4 < y 4 4 V V V V V rvvvvvt'NVVVVVV nvi-mi *4 * 4 KkkkkfVY ^444444*^ ♦ ♦

m k p + A. J. J* V 4 ^ ^ >4 fetiUhJi*,?------^ < < < <4

Figure 2.6. Temperature and velocity distribution in the torus shaped expansion kernel. From ref. [21].

the outward movement of the torus ring because of the reduced spark energy per unit length. The simulations compared very well with experimental values.

In a proceeding paper Ishii et al. [22] have included an overall reaction for a stoichiometric propane-air combustion as well as the heat transfer to the electrodes in this model. The trends were the same as for the non combustion investigation. Short spark duration (2 ps) and thin electrodes (0.2 mm) resulted in a flame kernel with the shape of a torus. The main difference was higher temperatures due to the heat release in the combustion reaction. This also resulted in a larger torus ring and the inside of the ring stayed close to the spark gap for a long time instead of moving away from the electrodes. The effect of larger electrode diameter (0.4mm) was that a high temperature region was created also between the electrodes. This could be avoided by increasing the electrode spacing from 1 mm to 2 mm, or by making the tip of the electrodes like a cone with an angle of 45°. Longer spark durations (10 ps) resulted in a flame kernel with the shape of an elipsoide. The strength of the shock wave and the velocities decreased in the case of longer spark durations with the same total energy. Consequently the flame did not become a torus, and the flame kernel stayed in the spark gap were the heat loss to the electrodes is relativly large. The authors concluded that the value of minimum ignition energy was strongly dependent on the considerations made above. Short duration capacitance spark was superior in the case of thick electrodes with small electrode spacing where the heat transfer to the electrodes is of major importance. This is due to the development of a flame kernel apart from the electrode gap as in a torus. For thin electrodes much of the

18 Chapter 2 Ignition mechanisms energy in the capacitance spark is lost in the strong shock wave and the high velocities of cool inflow around the electrodes. For this reason long duration composite sparks which consists of a short conductive phase followed by a longer inductive phase, is superior in the case of thin electrodes. The authors found an optimum spark duration of 10 ps in their simulations. The experimental investigation found this optimum to be 200 ps. The difference was claimed to be mainly due to differences in spark energy.

Sher and Refael [23] have simulated methane ignition with both a single step reaction and a 18 step reaction mechanism. They assumed constant pressure, and could exclude the momentum equation. The geometry was cylindrical, but since the heat transfer to the electrodes was neglected they calculated only in the radial direction, making the assumption that there were no axial variations. The spark energy was added as heat in a gas pocket at the centerline during a specified time. The one step reaction was found to give unrealistic values in the early phase. The 18 step reaction mechanism gave a burning velocity at stoichiometric conditions which was high in the beginnin g, and decreased rapidly to a constant value of approximatly 40 cm/s after less than 0.1 ms.

2.2 Autoignition

2.2.1 Phenomenological description of auto ignition

At sufficiently high temperature and pressure, a mixture of fuel and oxygen will start to convert reactants to products. The conversion from reactants to products goes through a chain of individual reaction steps involving intermediate species. The speed of these reactions are highly dependent of temperature, and some of them are also dependent of pressure. When the temperature has been raised to an extent where the reactions convert a significant amount of chemical species, the release of chemically bounded energy will increase the temperature further in the reaction mixture. This will in turn result in that the chemical convertion rate is further increased. The increase in reaction rate is also contributed to the increase of radical concentration. Radicals are intermediate species which are very reactive, and hence of crucial importance in the reaction chain as will be shown later. This

19 Chapter 2 Ignition mechanisms self accelerating process becomes very fast when first initated, and is called autoignition or spontaneous ignition (Kanuri [6]).

If the reactants are not premixed, the auto ignition process have to take place in the mixing layer between fuel and oxygen. In this case the mixing of the reactants will play an important role for the autoigniton process. The mixing on large scale takes place by convection in for example turbulent eddies. The mixing on molecular scale is by diffusion processes. If in addition the fuel is in liquid phase, it has to be evaporated before molecular mixing and combustion takes place. In diesel engines where autoignition is utilized, the liquid fuel is injected into air which is compressed. The fuel is injected as a spray, and the breakup of the spray to smaller and smaller droplets is a very complicated process to treat mathematically. The liquid evaporates and mixes with the hot air in the cylinder. Eventually due to high temperature and pressure the chemical reactions start to convert reactants to products. In an engine the time from fuel is injected to the combustion start, the ignition delay, is an important parameter. An effort to estimate the delay will not only need a model of the chemistry, but also a description of the spray behaviour. This is a very complicated task, and is an area of extensive research. However this text will not deal with the behaviour of a spray, and the term ignition delay will be reserved for the reactant mixing and chemical initation process.

Autoignition can also occur in otto engines. When the flame front ignited by the spark propagates through the combustion chamber, the pressure and temperature will rise. This may lead to autoignition of the premixed unbumt charge in front of the flame close to the end wall. This phenomena is known as engine knock and may result in excessive wear on the engine. Since the heat release in this type of autoignition is fast compared to the heat release in the flame front, pressure waves is initiated, and may cause damage on the engine parts. In otto engines autoignition is therefore an unwanted phenomena, while diesel engines relay on this phenomena to initiate combustion.

2.2.2 Classical models of autoignition

There are two major questions concerning auto ignition. Will ignition occur, and if this is the case, how long time will it take until the onset of a combustion process?

20 Chapter 2 Ignition mechanisms

We can estimate the ignition delay by considering a homogeneous mixture with constant properties, and consider the ignition process to be a constant volume process. The heat transport is neglected and we express the chemical conversion by an overall reaction. In this way we can balance the sensible internal energy accumulation with the chemical heat release where is the fuel’s heating value.

•S ' "if (2.16)

We here assume constant mixture heat capacity, cv, and molar density, p. This should be a resonable assumption in the early stages of the process before the temperature increase is large. [F] is the concentration of fuel in [mol/m 3] and may be expressed with aid of the ideal gas law as

[F] = _P_|V| (2-17) RTl p )

The ratio pp/p is the partial pressure of the fuel divided by total pressure which equals the molefraction of fuel, XF. The chemical reaction rate in eq. 2.16 is expressed by an Arrhenius expression where O is oxygen

d[F] ATP[F]n [0]nl e\Kr (2.18) dt Equation 2.16 gives now an expression for the temperature rise.

.d-I! = -S|LATp [F]n [0]me^^ (2.19) dt pc.

If we integrate this expression from time 0 to t, and make an assumption that (RTq ZE) is considerably smaller than one, we get an expression containing a characteristic time

Tfe^'1) . !_ JL (2.20) Vo Lcbar

where

21 Chapter 2 Ignition mechanisms

t - PCyRTp-P c RT0 (2.21) AqF[F] n [0] mE

This characteristic time is a measure for how long time it takes before the temperature starts to increase rappidly. Equation 2.19 is integrated numerically in figure 2.7 with different starting temperatures and pressures. The parameter values are taken from Seery and Bowman [24] which have studied methane-air combustion. The values used are: A=119.7 (m3/mol)°" 2( 1/sK2), E=215 kJ/mol, n=-0.4, m=1.6, and R=8.3144 J/molK. The mixture has an air/fuel equivalence ratio, 1=1.1, and the mixture heat capacity and heating value of fuel are respectively: cv=21.5 J/molK and qp=800 kJ/mol. The corresponding characteristic time is calculated from eq. 2.21 and shown on the figure. The characteristic time correspond very well with experimental values from Seery and Boman. An expression on the form of equation 2.19 will always lead to ignition, although it may take extremly long

Ignition of methane-air, lambda=l .1 2100

2000 2.24 bar 1.73 bar ----- 1.85 bar 1.85 bar 1.60 bar 1900

1800

1700

1600

1500 -

0.0005 0.001 0.0015 0.002 time Is

Figure 2.7. Temperature development as function of time. The corresponding characteristic time is shown by vertical arrows.

22 Chapter 2 Ignition mechanisms time if the temperature is low. In order to investigate if ignition will occur we will have to introduce heat loss in the model.

The Semenov analysis [25] includes heat losses by an heat transfer coefficient, h, to a specified area, s. If we add this heat loss to the energy balance in equation 2.16 we get

- -5FM-ZhCr- To, (2.22)

As an example this equation is integrated numerically as a function of time with the values from eq. 2.19. The geometry is a sphere of radius 1 mm. This gives the volume and heat loss area. The heat transfer coefficient is approximated by h=x/r where k=0.08 J/smK is the heat conductivity and r is the sphere radius. Since the change in reactant species concentration is small during the igniton phase, the concentrations of fuel and oxygen is held constant throughout the integration. The result is shown in figure 2.8. If the pressure is low, the heat losses will balance the heat released by the chemical reaction, and only a slight temperature rise will be the result. Above a critical pressure the temperature rise will firstly be slow, but after some time the exponential behaviour of the reaction rate will dominate the situation. We get a thermal explosion.

These considerations lead to an ignition limit depending on temperature and pressure. In figure 2.9 the first term on the right of equation 2.22 is plotted together with the second term as a function of temperature. We see that the condition for a thermal runaway is that the exponential reaction term must always be above the linear heat loss term. On the limit, the linear term is exactly a tangent to the exponential term. We can find the critical temperature and pressure on this limit, by equating the term for reaction heat with the term for heat loss, and search the solution were the to fines have the same slope.

23 Chapter

Figure temperature [K1

1900 1850 1750 2 1650 1600 1550 1500

Ignition 2.8.

-

Temperature Figure

0.001 mechanisms

2.9. 0.002

Reaction

heat Ignition development 0.003 values

source

0.004 of heat

methane-air of

at

24

time

pressure. source in

0.005 different

Semenov

[s] linear

and 0.006 ,

lambda-1.1

pressures

heat heat

p

theory.

= 0.007

0.55

0.1

loss loss

bar bar bar Different

0.008 term.

------

starting Chapter 2 Ignition mechanisms

(“.) = qp ATP[F]n [0]me h(T c-To) (2.23)

Aj qp ATp [F] n [0] me |h(T c-To) (2.24) dT

By ass uming that the consumption of reactants during the ignition delay time is negligible, these two equations can be solved to get an expression for the critical temperature

(Te-To) (2.25)

Since (RT/E) is asumed to be considerably smaller than one, (E/RTC) will dominate over P and the equation simplifies to

CTc-To) = (2.26)

This equation has two roots. The lower root corresponds to ignition, and the • critical temperature is

T - JL-JL i 4RT° (2.27) c 2R 2RN E

If we substitute equation 2.26 in 2.23, and convert concentrations to molefractions with the aid of eq. 2.17, we get the Semenov relation

p (n+m) = hsR (1+n * m> T(2-p*n*m)JwJ (2.28) VqpAXpX^E C

In figure 2.10 this equation is plotted as critical temperature versus critical pressure for a mixture of methane-air with an air/fuel equivalence ratio equal to 1.1. Above the fine spontaneous ignition is possible, while under the line the mixture will not ignite spontaneously. We can compare this limit with the time-

25 Chapter 2 Ignition mechanisms

Ignition limit for methane-air, lambda=l.l le+09

le+08

le+07

100000

10000

1000 1000 1200 1400 1600 1800 start temperature [K]

Figure 2.10. Ignition limit from Semenov theory. integration in figure 2.8. In figure 2.8 the temperature T0 equals 1500K. The critical pressure is then found (eg. 2.27 and 2.28 or figure 2.10) to be 0.52 bar. This confirms the different behaviour of the fines for 0.5 and 0.55 bar in figure 2.8.

Frank-Kamenetski [26] considers the heat balance between conductive loss and reaction source. As shown in figure 2.9, if this balance represent a steady physical solution no ignition will occur. If there is no steady physical solution we will have a thermal runaway. The criterion for ignition is then that the steady heat balcance equation no longer have a solution. The steady heat balance in terms of conduction and reaction source for a spherical geometry may be expressed as

J2 dT) +q ATP[F]n [0]me^ = 0 (2.29) k dr2 r dr j where k is the thermal conductivity. This equation is solved under the following boundary conditions. See figure 2.11

26 Chapter 2 Ignition mechanisms

Figure 2.11. Geometry and temperature of reacting gas body.

T = T0 at r=±ac (2.30) — = 0 at r=0 dr

In order to solve the equation it is necesarry to nondimensionalize the variables and cancel some parts by order of magnitude analysis. The resulting equation is dependent of a parameter Ac. If this parameter is large, the equation has no solution. This correspond to a thermal runaway. If Ac is small, we obtain a steady state solution, and hence no ignition. The border value of delta in spherical coordinates is calculated to 3.32. (Frank-KamenetsM [26] or Kanuri [6]). Thus the ignition limit for a sphere according to Frank-Kamenetski is

fe) (2.31) 3.32 xRTf 1

We can compare this kriterion with Semenov ’s kriterion. For a sphere the surface to volume ratio is 3/r. If the heat transfer coefficient in Semenovs relation (eq. 2.28) is approximated by h=K/a c, equation 2.28 may be written

(2.32)

The agreement between Semenov, eq. 2.32, and Frank-Kamenetski, eq. 2.31, is excellent. The critical size of the reacting body, ac, may be estimated by using flame zone thickness data. If data of the chemical reaction controlling the ignition

27 Chapter 2 Ignition mechanisms phase is known, equations 2.31 and 2.32 predict ignition ranges with reasonable accuracy.

The ignition delay can be estimated by assuming that the chemical energy release per unit volume during the ignition delay time, is used to heat the unit volume from the starting temperature up to the critical temperature. Neglecting heat losses we get

_ _ (2.33) pc v(Tc-To) = qFATcp [F]n [0]me^Kr^ti

Solving for t; and substituting eq. 2.26 results in

mRTy .(*) ti = - (2.34) qp A[F]n [0]mE

This expression for t; is much the same as equation 2.21. The difference is the use of critical temperature, Tc, instead of start temperature, T0. Due to the neglection of heat losses, equation 2.34 gives best results when the conditions are well inside the ignition region in figure 2.10. Close to the ignition limit the heat losses are of greater importance, and a direct time integration like in figure 2.8 will give a better estimate of the ignition delay. Note that equation 2.34 gives the delay to the temperature has reached the critical temperature. Figure 2.8 shows that close to this critical temperature the time-dependent temperature increase is at its slowest. It takes still some time for the temperature increase to get steep.

The Semenov and Frank-Kamenetsky analyses may also be used to obtain values for the activation energy, E, for simple thermal reactions. From experimental work we can find corresponding values of Tc and p c. If we first divide equation 2.28 with T^-p+n+m then take the logarithm, we can plot ln(p cI1+m/Tc2"p+n+m ) as a function of 1/TC. This plot will be a straight fine with slope E/R.

Such experiments give different values of the activation energy depending on the start temperature in which the experiments are carried out in. At intermediate temperatures the activation energy is high. At high temperatures the activation energy for most fuels will get lower. This should imply that the chemical model consisting of only one overall reaction is not sufficient for detailed studies.

28 Chapter 2 Ignition mechanisms 2.2.3 Kinetic autoignition models

The change of activation energy with temperature is not the only evidence of the shortcommings of one overall reaction taking care of the conversion from reactants to products. The powers of concentrations, m and n, are found to be fractions and not whole numbers as they should be according to the reaction theory. In addition ignition limits do not have a such simple form as shown in figure 2.10. Experimental limits show complicated relations with one or more peninsulas, and regions of one or more cool flames which only partially convert reactants to products. Advanced measurement techniques have also identified a variety of intermediate species, particularly in the hot part of the reaction zone.

The total reaction is thus taking place through a number of individual reaction steps. These reactions involve unstable intermediates which are veiy reactive and play an important role in the total scheme. The set of individual reactions is called a kinetic reaction mechanism.

In order to illustrate the implications of such a mechanism we can postulate a general reaction mechanism consisting of 5 steps and two intermediates. It is essentially the same as proposed by Strehlow [27]. The mechanism is constructed in order to get 3 ignition limits from 5 reactions. The group of intermediates called C is very reactive while the group called D is more stable and only reacts at high pressures. The reactants are called A. M is a third body molecule which is necesarry to supply enough collision energy to the reaction, but it does not change composition during the reaction.

k, nA - C Initiation (2.35)

kg A+C - a C + products Chain branching (2.36)

kw C - D + products Linear termination (2.37)

29 Chapter 2 Ignition mechanisms

ki C+M+M - D + products Cubic termination (2.38)

kp A+D - C+D High pressure chain branching (2.39)

Reaction 2.35 splits the fuel molecule into intermediates. It is a slow reaction, but necessary to initiate the mechanism. When the concentration of C increases, reactions of type 2.36 takes over the fuel consumption. When a is larger than 1, these reactions will produce more intermediates than they consume and the result is a net increase of C. Reactions of this type are called branching chain reactions. Reaction 2.37 is a linear termination reaction. It consumes C and produces the molecule D which is stable at low pressures. Strehlow names this reaction a wall termination reaction. Reaction 2.38 is a cubic termination reaction. It needs the aid of two third body molecules to get enough energy to react. Reaction 2.39 is a chain branching reaction which only takes place at high pressures.

If we use the steady-state assumption on the chain carrier C we get

= 0 = kJA] “ -katCHA] + akg[C][A] -kg[C][M]2 -kJC] +k,[A][D] (2.40)

The concentration of C is therefore at steady state

k1[A]n +kp[A][D] [C] (2.41) kg[A](l-a) +kg[M]2+kw

If the concentration of the reactive intermediate C gets large, the production of products will also be large and the mixture is reacting. Hence we use the concentration of C as the ignition parameter. Equation 2.41 gives that C will be large when the denominator approaches zero. Be aware of that the concentration always must be positive. To investigate this we write the denominator as.

k^AKl-a) +kg[M]2 +k, > 0 (2.42)

In terms of a this becomes

30 Chapter 2 Ignition mechanisms

a < 1 + kg[M]2_lkw, (2.43) kg[A]

As long as the right hand side of 2.43 is considerably larger than a, no auto ignition will occur. When the right hand side of 2.43 approaches very close to a, the consentration of C will be large, and a spontaneous igniton of the mixture will be the result. From 2.43 we can see that this happens when the denominator of 2.43 (that is reaction 2.36, the branching chain reaction) dominates the nominator (that is reaction 2.37 and 2.38, the terminating reactions). In terms of molefractions, temperatures and pressure, equation 2.43 becomes (substituting eq. 2.17)

The right hand side of this expression aproaches a (ignition) when the temperature is high and the pressure is intermediate. Both low and high values of pressure leads to high values of the right hand side, and hence no ignition. This is shown on a pressure versus right hand side of eq. 2.44 plot for different values of temperature in figure 2.12.

The high pressure reaction 2.39 will lead to high values of the intermediate C when the pressure is high. This can be seen from equation 2.41. For this reason ignition may occur at high pressures. In figure 2.13 this high pressure limit, often called the third ignition limit, is shown on a temperature versus pressure plot together with the intermediate pressure limit due to cubic termination (second ignition limit) and the low pressure limit due to linear termination (first ignition limit) from figure 2.12.

The high pressure limit (third limit) has the same form as the thermal ignition limit from Semenov theori (eq. 2.28 and fig. 2.10). Both are declining functions of temperature when plotted against pressure. The Semeov theory gives an exponential decrease with temperature (exp(EZRT) ). Reaction 2.39 gives a similar decrease which can be seen from rearranging equation 2.41. Due to this similarity there are some discussion in the literature whether the third limit is due to thermal or kinetic restrictions. The thermal limitation due to heat loss will always

31 Chapter 2 Ignition mechanisms

Eq. 2.44

no ignition ignition for T, for To

Figure 2.12. The right handside of equation 2.44 at different temperatures.

3. limit

ignition no ignition 2. lhpit

1. limit

TTXX

Figure 2.13. First, second and third ignition limit.

32 Chapter 2 Ignition mechanisms

be present, while the existence of a low pressure stable molecule D, depends on the reaction path from reactants to products. As an example the hydroperoxyl radical, H02, is often referred to as molecule which is stable at low pressures. In mechanisms where this molecule play an important role the kinetics is believed to have an influence on the third limit. This is the case for hydrogen and most hydrocarbons. The first and second limit behaviour is as shown due to the competition between chain branching and chain terminating reactions.

2.2.4 Two stage autoignition models

From experiments using a rapid compression machine, ignition in some fuel-air mixtures has been found to take place in two stages. For a short time, t1( immediately following the compression, one or more cool flames are observed. This results in a small pressure rise and a limited consentration of intermediate products. Then after some time, x2, the pressure rise get abrupt and ignition has occured. A typical oscilloscope record of the pressure is shown in figure 2.14a. For some pressures and temperatures the total ignition delay thus consists of two separate periods. At higher temperatures ignition occurs a a single stage process as shown in figure 2.14b.

In figure 2.15 the variation of ignition delay and cool flame intensity is shown as a function of temperature for a mixture of iso-octane and air with fuel/air equivalence ratio 0.9. When the temperature is higher, the total ignition delay gets shorter. But at some temperatures this trend gets opposite. The ignition delay gets longer as the temperature is increased. This "negative" temperature dependence lasts over a range of some hundred degrees until the ignition delay again gets shorter with increasing temperature.

Halstead et al. [28,29] have proposed a general reaction mechanism consisting of 8 reactions and 5 species which they have shown to be able to predict the observed behaviour. Their model has been used to simulate auto ignition of fuel blends of iso-octane, n-heptane and toulene. This types of fuel consist of long chains of carbon atoms, and they have a large number of intermediates and products. For this reason the intermediates is collected in three groups with different properties. The explanation of the species and the groups is as follows:

33 Chapter 2 Ignition mechanisms

c D

Figure 2.14. Typical pressure record of auto igniton in a rapid compression machine, a) Two stage ignition, h) Single stage ignition at higher temperature and pressure. From ref. [28],

Figure 2.15. The variation of igniton delays and cool flame intensity with temperature during the autoignition. From ref. [28],

34 Chapter 2 Ignition mechanisms

EH fuel molecule 02 oxygen R reactive intermediate, radical B intermediate, degenerate branching agent Q intermediate

The general reaction mechanism is written

kq EH + 02 - 2R initiation (2.45)

kp R - R + products and heat propagation cycle (2.46)

fikp R -* R + B propagation forming B (2.47)

R + Q -* R + B propagation forming B (2.48)

fakp R - out linear termination (2.49)

tk, R - R + Q propagation forming Q (2.50)

kt 2R - out quadratic termination (2.51)

kB B - 2R degenerate branching (2.52)

Each reaction in this mechanism is not necessarily a fundamental reaction step, but may be the total result of a reaction cycle. Therefore the rate coefficients can not be found from known elementary reactions. The k’s and fs are found from parameter analyses together with results from experiments on a specific fuel. This mechanism constitutes the competition between a number of reactions with

35 Chapter 2 Ignition mechanisms

different activation energies and temperature dependence. The ignition becomes the result of the total supply and consumption of intermediates.

In addition to the chemical model, the thermodynamics of the mixture must also be modelled. This is done by considering a closed system where the change of total number of moles is neglected in the early ignition phase. The resulting heat balance equation is similar to equation 2.22 in the Semenov analysis. The difference is that the volume change is so large that the expansion work also has to be taken into consideration. The volume change is a function of the piston movement. The model thus consists of five coupled differential equations, one for each of the five species except fuel which is algebraically coupled to oxygen, and one differential equation for the temperature change.

= 2(kq[KH][02] +kB[B]-kJR]2)-f3kp[R] (253)

1 dn B %[R]+f^[Q][R]-kB[B] (2.54) V dt

1

1

(n o 2 "n o 2(t=0)) (2.57) n,RH + n RH(k~0) pm

dT ntptRT dV (2.58) dt cAot V dt j

In these equations n x correspond to the number of moles of the species X. The total number of moles is n^. The small amount of intermediates R, B and Q were neglected in the calculation of total mole concentration. It is assumed that two hydrogen atoms are abstracted from a combustible molecule during a single propagation cycle. Then if the initial fuel structure is Cn H2m there will be 1/m molecules of fuel removed per cycle. The overall stoichiometry is approximated by

36 Chapter 2 Ignition mechanisms assuming a constant C0/C02 ratio for the combustion process. The consumption of oxygen is p moles per cycle, depending on the air-fuel equivalence ratio, X, where

n(2-A)+m (2.59) 2m

The fuel’s heating value, q, was calculated for each specific fuel. In the energy balance (eq. 2.58), the first term on the right is due to chemical heat release and is given by

Q k = MV[R] (2.60)

The heat loss, QL, is written in Semenov form and expressed as

Q l = sh(T-T w) (2.61)

The final term in equation 2.58 is the expansion work due to piston motion. The parameters specific heat, cv, and heat transfer coefficient, h, were calculated from curve fits obtained from JANAF polynomials and experiments respectivly.

Figure 2.16 shows a typical time development of temperature and species from a simulation of two stage ignition in a rapid compression machine. During the first ignition delay the concentration of all intermediates increases until the temperature has increased with some 100 degrees Kelvin. Then a negative temperature depence of one or more of the propagating or branching reactions leads to consumption of the intermediate B to such an extent that the concentration of radicals and branching agents is decreased. The branching agent B is thus responsible for the first ignition delay. This is the end of the cool flame region, and the temperature increase is halted. The only intermediate which still increases its concentration, although slowly, is Q. During the second ignition delay, the production of Q evolves enough heat to stabilize heat losses and slowly increase the temperature until the propagating and branching reactions with positive temperature dependence overcomes the negative temperature dependent reactions. At this point temperature and all intermediate concentrations abruptly increase in magnitude and ignition has occured. The intermediate Q is responsible for the onset of an ignition after this second ignition delay time. Without a such species there would be complete extinction after the cool flame process.

37

S Chapter 2 Ignition mechanisms

TWE FROL* 6C6IN8WG OF C0» TOSOH STROKE OCrH)

Figure 2.16. Time development of temperature and intermediate concentrations in a simulation of a rapid compression machine. From ref. [29].

At elevated pressures this process has an order of milliseconds as shown in figure 2.16. When the pressure is near atmospheric the process is much slower, and the timescale is in seconds. The authors has shown that this model is capable of simulating ignition delays with such large changes in timescales.

The fitting of appropriate reaction rates is essential in Halstead et al.’s work. They have developed a fitting procedure and tabulated values for different reference fuels. Their work is primarily devoted to the knock phenomena in spark ignition engines, but since it is commonly accepted that the same processes occur in the auto ignition phase of diesel engines, their work is frequently used in this type of simulations. Their model is often referred to as the "Shell knock model".

Cox and Cole [30] have refined the model of Halstead et al. They have used a more detailed description of the chain propagating step and assembled steps involving alkylperoxy radical isomerization and oxidation. Their model consist of 10 species

38 Chapter 2 Ignition mechanisms and 15 generalized reactions. This model extension have been done in order to describe the autoignition process in terms of rational chemical kinetic interpretations of the reaction orders and temperature dependence.

By the use of e.g. hydroperoxide as the brancing agent B, and aldehydes as the intermediate Q, they have been able to use information from known elementary reactions in the description of rate coefficients.

Theobald and Cheng [31] have used Halstead et al.’s model in a simulation of spray ignition and combustion in the combustion chamber in a diesel engine. In order to extend the model to fuels of higher molecular weight typical of diesels, they first examined the sensitivity of the model for the ignition of fuel vapour mixed with air. Then they did simulations of a two dimensional axi-symmetric spray in a combustion bomb.

Rapid compression engine simulations were performed with the KIVA-code [32,33]. This code includes spray and turbulence models. The three dimensional simululation showed qualitative behaviour characteristic of experimental values. The shape of the reacting zones matched the luminous zones observed with a high speed camera. But the time development of the flame was different from experiments. The simulations gave a shorter ignition delay and a more gentle pressure rise early in the combustion. Later in the combustion the pressure rise was higher than in the experiments, and to simulate the post ignition combustion the ignition model was replaced by a one step overall reaction. In this phase the combustion was mixing controlled by the available fuel and oxygen.

2.2.5 Flow field simulations of auto ignition

Like flow field simulations of spark ignition, flow field simulations of auto ignition is a substantial task due to the large number of equations which have to be solved. Due to this fact various simplifications have been used in describing the model in question. Lutz et al. [34] have computed the dynamics of a local autoignition center. Such centers arise from inhomogenities in the bulk mixture, and they ignite earlier than the surrounding mixture. Based on this idea their model of an exothermic center considers a small, homogenous mass of reacting mixture that is surrounded by an inert mixture. This small mass is treated as a perfectly stirred reactor, and hence no transport equations is necessary in this small mass

39 Chapter 2 Ignition mechanisms

element. Only the species equations and the energy equation is considered here. The reacting center is coupled to the inert surroundings through the expansion of the reacting body, and in the inert surroundings the dynamic equations for an inert gas are used. In this way the compression wave produced by the explosion can be simulated together with complicated chemistry.

Lutz et al. have used a comprehensive reaction mechanism consisting of 150 reversible reactions involving 36 species for the reacting center. The fuels may be hydrogen, methane, acetylene, ethylene, and ethane. The constraint for the motion of the interface between reacting mixture and inert gas, is a function of the pressure in the kernel. In the derivation of this constraint they assume isentropic flow in the surroundings, which is valid for the weak shock wave generated by the kernel. In this way the only transport process over the interface is momentum transfer. It has the advantage of combining the interface motion with energy and species equations inside the reacting body. Complex chemistry are thus combined with dynamic behaviour of the gas. The geometry may be planar, cylindrical or spherical.

Figure 2.17 shows the behaviour of an exothermic center in a stoichiometric hydrogen-air mixture. After an induction time of several milliseconds, t„ the center

Pressure (aim)

W Q

(c) --- 1------1 0.1 0.2 2000 4000 0.5 t Radius (cm) Temperature (K) Energy Power

Figure 2.17. Autoignition of an exothermic center, a) The position of the kernel interface, b) Temperature and pressure, c) Chemical work, Q, and expansion work, W. d) Chemical power. From ref. [34].

40 Chapter 2 Ignition mechanisms reacts during some microseconds. The time from the end of the induction time to the maximum chemical power (fig 2.17d) is called the excitation time, te. The exitation produces a compression wave where the pressure ratio can be as high as 1.8 for acetylene, while no more than 1.2 was observed for methane. As the exitation time gets shorter the pressure ratio gets higher. The pressure in the center decreases as the pressure wave moves outwards. In figure 2.17c we can see that the chemical work is some 4 times larger than the expansion work.

The induction time is strongly dependent on temperature. For methane it decreases from 50 to 0.1 ms as the temperature increases from 1200 to 1800 K. The excitation time has only a slight decrease with increasing temperature, but as the pressure increases it has a strong decrease. For this reason they found the highest pressure ratios at high initial pressures. The induction time decreases only slight with increasing pressure. Methane has the longest induction and excitation times of the hydrocarbons in question. For all hydrocarbons, except ethane and acetylene at high temperatures, the induction time is much longer than the excitation time. For methane the difference is about two ordes of magnitude.

The authors conclude that a short excitation time may be an important parameter for accelerating the combustion in the bulk mixture. The compression waves generated by the the ignition centers heat the mixture and reduce its induction time.

Goyal et al. [35] have simulated what they call hot spot ignition. This is much the same as the autoignition center from Lutz et al. Discontinuities in the mixture, due to for example turbulence, will create non-homogeneties in temperature. Goyal et al. have used a large kinetic mechanism for the whole calculation domain. The geometry is one dimensional and may be planar, cylindrical or spherical. The fuels are hydrogen (37 reactions, 8 species) or methane (127 reactions, 18 species). Their solution algorithm is lagrangian and they use an adaptive non-uniform grid method, solved with implicit methods, in order to resolve the sharp gradients with a minimum of computer time. In other words is this a full flow field simulation which include continuity, momentum, species and energy conservation together will a large reaction mechanism.

The inital conditions are uniform except for the temperature which has a slightly higher value in the center of the geometry. The authors found that a temperature difference of 10 K was enough to initiate hot spot ignition due to the strong

41 Chapter 2 Ignition mechanisms temperature dependence of the induction time. Figure 2.18a shows the temperature development in an igniting stoichiometric hydrogen-oxygen mixture. After a short time the center of the mixture explodes and a pressure wave propagates to the right. The center temperature increases and a deflagration wave travels behind the pressure wave. This deflagration wave accelerates to a detonation wave and then catches up the pressure wave, merge with it, and forms a detonation front travelling with a speed of about 2100 m/s.

Figure 2.18b shows the temperature development in an igniting rich methane-air mixture. The induction time is longer than for hydrogen. Then after some time a deflagration front develops. In this case the deflagration wave does not transform into a detonation front. This may happen if the calculation was continued for longer time. Now the deflagration wave does not catch up the shock front and no detonation front is created. This result suggests that the chances of detonation of rich mixtures are less if the vessel diameter is small.

The authors conclude that a model of this complexity is able to explain induction, excitation, propagation as well as transitional phenomena in hot spot ignition. The processes during hot spot ignition have marked influence on the bulk autoignition process. The simulations compare well with experimental results found in the literature.

Figure 2.18. Calculated temperature profiles in an igniting mixture, a) Stoichiometric hydrogen-air. b) Rich methane-air. From ref. [35].

42 Chapter 2 Ignition mechanisms 2.3 Other ignition mechanisms

Spark ignition and autoignition have much in common. The difference is the way the mixture or part of the mixture is injected and heated, and the processes which are most important in the initiation phase. There are also other ways to initiate combustion in a mixture. Although not so common they are mentioned here for the sake of completeness.

2.3.1 Hot body ignition and prechambers

A glow plug is sometimes used to enhance autoignition in Diesel engines. This is common in prechamber configurations, where the fuel spray is injected on a hot plug in the prechamber. The glow plug may be electrically heated, especially when the engine is started from cold conditions. When the engine is running the plug is held hotter than the intake air due to its large heat capacity and long contact time with hot exhaust gases. The contact between the hot body and the gas heats up the gas and thus shortens the induction time for autoignition.

As mentioned glow plugs are often used in a prechamber which may be constructed for good mixing between fuel and air. The composition in the prechamber is rich, and partially burnt reactants will expand from the prechamber into the main combustion chamber. This movement ensure good mixing and fast combustion of the bulk mixture.

In Otto engines it is possible to place a spark plug inside the prechamber. The spark plug will ignite the rich mixture in the prechamber and this partially burnt mixture will expand into the main chamber. This is sometimes referred to as torch igniters (Boston et al. [36]).

2.3.2 Plasma jet ignitors

The gas channel between the electrodes in a conventional spark plug is heated by the discharge to such a temperature that the gas becomes plasma. Plasma consists of ions and dissociated molecules. When the plasma returns to a stable mode, the

43 Chapter 2 Ignition mechanisms

energy transfer to the gas is very efficient. The idea of a plasma jet is to arrange the electrodes in such a way that the hot plasma is injected into the unbumt mixture, and thus creates a flame kernel some distance away from the cold electrodes.

teflon insert INSULATOR CHAMBER (28 MM3) BUSHING ELECTRODE OUTER ELECTRODE

rnr/a. PLASMA PORT

GAS INLET SEAL STEEL BOOT

Figure 2.19. Plasma jet ignitor. From ref. [37].

One way to do this is shown in figure 2.19 which is taken from Orrin et al. [37]. Between the inner and outer electrodes there is a small cavity. The gas in this cavity becomes plasma when there is a discharge between the electrodes. Then the plasma will expand through the hole in the outer electrode and create a plasma jet into the unbumt mixture. The chamber may be individually supplied with suitable gases.

2.3.3 Ignition by irradiation

An electrical discharge both heats the gas and creates ions and dissociated molecules. Lee et al. [38] irradiated acetylen-oxygen, hydrogen-oxygen and hydrogen-chlorine mixture in a chamber with ultraviolet light through quartz windows. The ultraviolet light is absorbed either by the oxygen or chlorine molecule and causes these molecules to dissociate and form free radicals. When these intermediate products reaches high enough concentrations they will initiate chain reactions which eventually can lead to a local explosion. In this way Lee et al. achieved ignition in mixtures where the temperature was under the autoignition temperature.

44 Chapter 2 Ignition mechanisms 2.3.4 Shock wave ignition

In a diesel engine the fuel and air is heated by piston compression of the gas. When the temperature exceed the autoignition temperature of the respective pressure, the mixture will ignite after an ignition delay time. It is also possible to obtain compression with a strong pressure wave, a shock wave. Since the induction time for autoignition is strongly dependent of temperature, only a small temperature rise may result in autoignition. The shock wave can be created in several ways. It has already been mentioned that hot spot ignition results in pressure waves which enhance the bulk mixture ignition. In shock tube experiments the shock wave is created by bursting a thin diaphragm separating a high from a low pressure region.

2.4 Use of computers in combustion calculations

This chapter has discussed some of the work found in the literature on ignition phenomenas. Early theoretical investigations had not the possibility to use computers in order to solve the mathematical relations resulting from the physical modelling. For this reason a limited number of physical interactions could be investigated by the same model. In addition these interactions often had to be strongly idealized in order to be able to solve the resulting equations. Nevertheless gives the early investigations important informations, and shold be kept in mind when employing more sophisticated models used in computer analysis.

Some examples of computer simulations have been discussed in chapter 2. Numerical solution methods have made it possible to include more of the physics into the mathematical models. Both chemical reaction mechanisms and flow field equations can be solved by numerical simulations. They are thus more reliable for a wider range of parameter variations.

The rest of this text will be utilizing models which take chemical reaction kinetics as well as the flow field into consideration. The conservation equations for multicomponent reacting systems are the physical foundations for the flow field modelling. The next chapter is devoted to these equations.

45 Chapter 3 The conservation equations for multicomponent reacting systems

Chapter 3 The conservation equations for multicomponenent reacting systems

The laws of fluid dynamics are well established, and can be written in many equivalent ways. A common way to formulate the laws is by applying the principle of conservation to the properties mass, species, momentum and energy. The concept of conservation means that the variation of these conserved properties within a given volume, is due to the net effect of internal sources and the transport of the property over the boundary surface. The detailed derivation of the equations can be found in numerous textbooks including Kuo [39] and Hirsch (not the species equations) [40]. This chapter will only summarize the result.

3.1 Conservation of mass

The conservation of mass states that if no mass is created or destroyed, the time change of mass inside the control volume must equal the net flux of mass through the volume boundary.

^ +V-(pu) = 0 (3.1) 3t

3.2 Conservation of species

The time change of mass per volume of a species, equals the flux by convection and diffusion through the boundary, plus the net chemical production of the species inside the volume. If we neglect thermal diffusion (diffusion of species

46 Chapter 3 The conservation equations for multicomponent reacting systems resulting from a temperature gradient, also called Soret effect) and express the diffusion velocity in terms of a gradient formulation according to Pick's law we get:

|-(pY i)+V-(puY i) = V-(pD^VY,) + W; (3-2) ot

The sum of all species equations equals the mass conservation equation. Hence only S-l species equations, where S is the total number of species, are independent. The last massfraction is calculated from the fact that the sum of all massfraction equals one.

EYi = l (3-3) i=l

Since no mass is created or destroyed, the sum of all chemical production terms must equal zero.

(3.4)

Then the conservation condition that the sum of all species diffusion terms must equal zero can be seen from equation 3.1 and the sum of equations 3.2. s E(PD^VY,) = 0 (3.5)

3.3 Conservation of momentum

The momentum conservation equation states that the time change of momentum per volume in a given direction equals the convection of momentum plus momentum changes due to pressure gradients and viscous forces. In absence of body forces (i.e. gravity) this may be written

-^(PU;)+V-(PUU;) = -Jfc+V-Ty (3.6) dt 3xj

47 Chapter 3 The conservation equations for multicomponent reacting systems

The viscous stress tensor xy is with the Stokes relation for local thermodynamic equilibrium: 'auufhVl Ujj - 2(i •|li(V-u)8 Jj (3.7) &

3.4 Conservation of energy

The energy equation gives the time change of internal and kinetic energy as a function of convection of enthalpy and kinetic energy, heat flux due to temperature gradients and species diffusion, viscous work and external sources. If we neglect the Dufour effect (heat flux due to concentration gradients, reciprocal of thermal diffusion) and body forces, the energy conservation may be written

d_ fl2 ) P(e+y) +V- pfl(h+^-) KVT + p^ChjJDk aVyk) + i:ij-u|+Q (3-8) at \ ^ J k=l

This equation contains the properties internal energy, enthalpy and temperature together. It is often advantageous to express the energy equation only with internal energy and temperature or enthahlpy and temperature. This can be done by using the relation between enthalpy and internal energy

h = e + £ (3.9) P

Inserting this expression and subtracting the momentum equation (3.6) multiplied with the velocity vector to eliminate the kinetic energy terms, we obtain in terms of internal energy

, s + t3+Q (3.10) ^-(pe)+V

Or writing the equation in terms of enthalpy gives

48 Chapter 3 The conservation equations for multicomponent reacting systems

—(ph)+V-(puh) = i^+u-Vp+V' kVT + p^] (hfcD^mVyk) +Tij"~ +Q (3.11) at at k=l ) DXj

Equation 3.10 and 3.11 are common used forms of the energy equation. Further we can also express the temperature gradient in terms of internal energy gradients or enthalpy gradients. Writing the enthalpy gradient in terms of temperature and species gradients gives

Vh =VE CW = E (YkVh k +h kVYk) = E (Yk^VT)+ E (h kVYk) k=l k=l k-1 k=l (3.12) =cp VT+E(h kVYk) k=l Rearranging this expression for VT

VT = — Vh-E(h kVYk) (3.13) k=l )

Substituting in equation 3.11 gives

-^m-^)h kVyk) + Tij^+<^ at XCP k=l a^

3.5 General transport equation

Equations 3.2, 3.6 and 3.14 all conform into a transport equation for a general variable <)>. This general transport equation has the form

-^W+V-M) = V-(r*V$)+S* (3.15) at

<}> can represent any scalar, massfraction, velocity component, enthalpy and so on. is in this respect a general diffusion parameter and S$ is a general source term. Thus it may be convenient for the solution procedure to write the equations in this way.

49 Chapter 4 Submodels of physical phenomena

Chapter 4 Submodels of physical phenomena

The conservation equations for reacting systems are the framework of the mathematical modellin g of combustion. Several terms in these equations need seperate modelling. This chapter will deal with the submodels of chemical production rates, thermodynamical relations, transport properties, modelling of phenomena like spark discharge and heat loss, as well as the treatment of boundary conditions for the conservation equations. Turbulence models will be treated in chapter 9.

4.1 Reaction kinetics in combustion calculations

The chemical production rates of the different species necessitates some kind of kinetic scheme. As mentioned in chapter 2.2, the schemes may vary in size from one overall reaction to kinetic mechanisms with several hundred individual reactions. The inclusion of a great number of species poses limitations of the computability of combustion problems. On the other hand, if too few species are considered the predictive capabilites of the mechanism will suffer.

4.1.1 The chemical production term

A general reversible reaction nr. j in a mechanism consisting of r reactions may be written

50 Chapter 4 Submodels of physical phenomena

S ^ S E vijM " E vijMi (4.1) i=l i=l ^>j

Mj is one mole of species i and Vy is the stoichiometric factors of species i in reaction j.

The chemical production term for species i, coi5 can then be expressed in terms of the species consentrations C;. The units are mass of species i per volume and time.

" wHfiT * w‘£ +(Vij Vij)kbjll (4.2) at j=i k=l k=l

Wj is the molecular weight of species i.

4.1.2 Skeletal mechanisms

The first step towards reducing the number of species is to neglect unimportant reactions and intermediate species with very small concentrations in large reaction mechanisms. In this way the mechanisms are reduced to skeletal mechanisms with a limited number of species and reactions. A skeletal mechanism for methane-air combustion is shown in table 4.1. The mechanism is a standardized reference mechanism for which a consensus was reached prior to a conference in La Jolla, California in 1989 [41]. The standarization was necessary for comparisons of different approaches from several research teams. Smooke has edited a collection of papers [41] which contains 10 contributions [42-51] from different researchers all using the same premisses.

The skeletal mechanism includes 15 species (plus nitrogen) and consists of 10 reversible and 15 irreversible reactions. Reaction nr. 10 is of special interest. It describes the unimolecular decomposition of methane into methyl and hydrogen atom, and the reverse reaction. The reaction is known to be pressure dependent and is of great importance in obtaining good values for burning velocities and extiction limits. In accordance with unimolecular reaction theory the pressure dependence is expressed in Lindemann form The rate coefficients for reaction 10 in table 4.1 gives somewhat high values of the

51 Chapter 4 Submodels of physical phenomena

REACTION A P E If H + 02 —> OH + 0 2.000E+14 0 70.291 lb OH + 0 H + 02 1.575E+13 0 2.887 2f 0 + Hg OH + H 1.800E+09 1.000 36.928 2b OH + H -> 0 + H2 8.000E+10 1.000 28.284 3f Hg + OH HgO + H 1.170E+09 1.300 15.171 3b HgO + H -> H2 + OH 5.090E+09 1.300 77.772 4f OH + OH -> 0 + HgO 6.000E+08 1.300 0 4b O + HgO -> OH + OH 5.900E+09 1.300 71.249 5 H +02 + M->H02 + M a 2.300E+18 -0.800 0 6 H + H02 -> OH + OH 1.500E+14 0 4.201 7 H + H02 —> H2 + H2 2.500E+13 0 2.929 8 OH + H02 -> H20 + 02 2.000E+13 0 4.184 9f CO + OH -> C02 + H 1.510E+07 1.300 -3.171 9b C02 + H -> CO + OH 1.570E+09 1.300 93.458 lOf CH4 + (M) -> CH3 + H + (M) b 6.300E+14 0 435.136 10b CH3 + H + (M) -» CH4 + (M) b 5.200E+12 0 -5.481 Ilf CH4 + H CH3 + H2 2.200E+04 3.000 36.610 lib CH3 + Hg -» CH4 + H 9.570E+02 3.000 36.610 12f CH4 + OH CH3 + H20 1.600E+06 2.100 10.293 12b CH3 + H20 -4 CH4 + OH 3.020E+05 2.100 72.894 13 CH3 + 0 —> CHgO + H 6.800E+13 0 0 14 CH20 + H-» HCO + Hg 2.500E+13 0 16.698 15 CHgO + OH -» HCO + H20 3.000E+13 0 5.000 16 HCO + H -> CO + H2 4.000E+13 0 0 17 HCO + M CO + H + M 1.600E+14 0 61.505 18 CH3 + 02 ^ CH30 + 0 7.000E+12 0 107.328 19 CH30 + H CH20 + Hg 2.000E+13 0 0 20 CH30 + M CH20 + H + M 2.400E+13 0 120.549 21 H02 + HOg —> H2Og + 02 2.000E+12 0 0 22f HgOg + M -4 OH + OH + M 1.300E+17 0 190.372 22b OH + OH + M -> HgOg + M 9.860E+14 0 -21.213 23f HgOg + OH -4 HgO + H02 1.000E+13 0 7.531 23b HgO + HOg -> HgOg + OH 2.860E+13 0 137.193 24 OH + H + M -> HgO + M a 2.200E+22 -2.000 0 25 H + H + M->Hg + M a 1.800E+18 -1.000 0

Table 4.1. Skeletal methane-air reaction mechanism. Rate coefficients in the form: k=ATp exp(-E/RT). Units are: mole, cm3, s, K and kJ/mole. a Third body efficiencies: CH4=6.5, H%0=6.5, C02=1.5, H2=1.0, CO=0.75, O2=0.4, N2=0.4. All other species =1.0. b Lindemann form, k=k„/(l+kfaJ1/[M]) where kfaU=0.0063exp(-75312/RT).

52 Chapter 4 Submodels of physical phenomena

(4.3) [M] burning velocity and extinction limit at atmospheric conditions. Wamatz [52] have found different coefficients for this reaction and using his expression gives very good agreement with experimental values. Wamatz’s rate constants are for the forward direction of reaction 10:

, 478482 . . . . kw = 2.3E+38T' 7e (4‘4) wl0f and for the reverse direction: (.37865) (4.5) kWiob = 1.9E+36T* 7e ^

In this work the Wamatz rate coefficients for reaction 10 will be used at one atmosphere. For all other pressures, the coefficients in table 4.1 with fall off behaviour expressed in Lindemann form, will be used. The fall off expression recognizes that the rate coefficients for unimolecular decomposition or recombination (i.e. reaction lOf and 10b) approaches the high pressure limit k» at elevated pressures, and approaches the third body dependent low pressure limit kg at low pressures. Since the third body dependency is included both in the Wamatz rate and the Lindemann form, no third body concentration should be included in the rate expression (eq. 4.2). This is marked with M in brackets for reaction 10 in table 4.1.

Notice that the coefficients are given in the units cm3 and for this reason all concentrations must be in moles/cm 3. This gives a factor of 10"6 from calculations with the SI unit m3. If no third body efficiencies are given, the third body concentration is given by

[M] = P (4.6) RT

If third body efficiencies are given (marked with a in table 4.1), M is the sum of all concentrations multiplied with their respective efficiencies.

53 Chapter 4 Submodels of physical phenomena

The skeletal mechanism for methane gives excellent results for flame propagation in lean and stoichiometric mixtures, and fairly good results for rich mixtures [43]. The reason for the deviation in rich mixtures is mainly due to the omittance of the Cg-chain which will be present in noticeable concentrations in fuel rich combustion.

4.1.3 Reduced mechanisms

Peters [44] and Bilger et al. [46] have shown how it is possible to further reduce the number of species by applying steady state assumptions for selected intermediates. In this way 8 of the 15 species may be assumed in steady state, and only 7 species are left in the global reduced mechanism.

Following Peters [44] we first define a concentration variable with units moles of species i per total mass, P;: r. = (4.7) W;

Dividing the species conservation equations (eq. 3.2) with the molecular weight we may write

(4.8) where the L-operator is defined by

LOTi) = -A(pl\) +V-(puI\) - V-(pD^VT\) (4.9) ot

Using the quantity r; instead of Yi; allows us to directly compare the convective and diffusive terms between different species on a mole basis. Asymptotic analysis [45] have shown that within the reactive layer the diffusive terms are dominant compared to the convective terms. We thus concentrate on the diffusive terms.

The diffusion coefficient, Di m, for combustion in air is roughly inverse proportional to the square of the harmonic mean of the molecular weights of species i and nitrogen.

54 Chapter 4 Submodels of physical phenomena

(4.10) D:„w “ D;i,N2 N 2WjWNz

The molefraction is proportional to the concentration variable F; in equation 4.9. Peters has tabulated the maximum molefractions of all intermediate species and weighted them with the diffusion proportionality factor in equation 4.10. He concludes that the weighted molefractions essentially fall into two groups, those below and those greater than one percent.

Group 1: The intermediates which are below one percent in weighted molefraction: OH, O, H02, CH3, CH20, CHO, CH30 and H202.

Group 2: The intermediates which have a weighted molefraction greater than one percent: CO, H% and H. The main reactants and products: CH4, 02, N2, C02 and H20.

The reason for H-atoms to be in the second group although their concentration is small, is due to the large diffusion coefficient for light species.

Based on this analysis, the L-operator is neglected in comparison with the reaction rates for all species in the first group in equation 4.8. Setting the L-operator to zero involves using the steady state assumption, that is setting the time derivatives to zero, as well as neglecting transport of these species by convection and diffusion. From the reaction mechanism algebraic relations can now be found for first-group species concentrations as a function of the rate coefficients and the concentrations of the species in group two, and, unfourtunalely, other species in group one. When the algebraic expression for one steady state species includes the concentration of another steady state species, a non-linear system of algebraic expressions will occur. Such a system does not have a unique solution, and the correct root must be found among all the possible roots in the system. One way to overcome this problem is by truncation of some steady state relations.

The eight steady-state conditions can be used to eliminate eight reaction rates from the total mechanism. By sensitivity analysis Peters has found the fastest reactions that consume each steady-state species. Further he has identified four main chains within the mechanism. These chains are:

55 Chapter 4 Submodels of physical phenomena

1) The oxidation of CH4 via CH3, CELj and CHO to CO. 2) The main chain for the conversion of CO to C02. 3) The chain for the chain breaking reactions. 4) The chain for the chain branching reactions.

From this information we obtain the global four-step mechanism for methane flames:

I CH4+2H+H^O - CO+4Hg II CO+HgO * COa+Hjj IH 2H+M « Hg+M IV 02+3Hg = 2H+2HgO

The main reaction rates for these steps are, referring to table 1: con for I, ro9 for II, (06 for III and co 4 of IV. The remaining reactions are called side reactions, and their rates have to be added to the main reaction rates for each step. From summing and subtracting reactions, and comparing them which eachother, Peters found the global reaction rates to be:

&>i - ui0 + tl)ll + u12 “n = “9 (4.12) toxn = to5 ~ to 10 + to 16 "" to 18 + to 19 ~ ^21 + to23 + to24 + tl)25 toIV = tol + to6 + to 18 + to 21 ~ to23

The truncation of steady state relations in order to get explicit relations for the intermediates in group one, is based on comparison of magnitude of the reactions involved. The species OH, O and CH3 are closely related to eachoter, and Peters has used partial equilibrium in addition to truncation in order to obtain the relations:

[Oil] - WPlDfl+MHlfOal (4.13)

[O] -b-«Vb2-4ac (4.14) 2a where a = k13B, b = BD +k13(C - A), c = -AD (4.15)

56

% Chapter 4 Submodels of physical phenomena and

A = klf[H][02] +k2b [0H][H] +k4f[OH]2 B = klb[OH] +k2ftH2] +k4f[H20] (4.16) C = (kllf[H] +k12f[OH])[CHJ D = k10b [H] +kn b[Hg] +k12b [H20]

(kllf[H] + k^fOH]) [CHJ (4.17) ^ ™ k10b [H] +kllb [H2] +k12b [H20] +k13[0]

The remaining steady state solutions for the intermediates in the C-chain can then be found:

. MCHJIOJ (4.18) k19 [H] +k2o[M]

rCR Q1 _ k13[CH33[[Q3 +(k19 [H] +k20[M])[CH3O] (4.19) 2 " k14[H] +k15[OH]

[HCO] = (^Wl^tOHDtCHaO] (4.20) k16[H] +k17[M]

The steady state relation for H02 must be truncated. This leads to kg[H][Og][M] [HOJ = (4.21) (ke+k7)[H]+kg[OH]

And finally the steady state relation for H202 gives

^[HOJ2+k22b[OH]2[M] •t-k23b[H20][HQ23 (4.22) 2 21 " k22f[M] +k23f[OH]

The reduced mechanism reduces the chemistry of methane flames to four global reactions including seven species. This means that only seven species transport equations are necessary instead of 15 which is necessary with a skeletal mechanism. This has significant influence on the computational cost of a combustion problem. In steady flame situations the reduced mechanism gives results close to the skeletal mechanism. Flame velocities, extinction limits and species concentrations are calculated with reasonable accuracy. The concentrations of hydrogen atoms and molecules as well as the intermediates assumed in steady

57 Chapter 4 Submodels of physical phenomena state are somewhat overestimated in the reaction zone. In the case of hydrogen it may be argued that these concentrations also contains hydrogen that actually should be contained in steady state species. Since non of these are included in the mass-balance equation, the continuity results in that the concentrations of the seven remaining species are calculated a little high. This only becomes severe for remaining species with small concentrations.

For pressures above 10.5 atmospheres no steady state solutions were obtained (Smooke and Giovangigli [43]). This may imply that the reduced scheme not was able to single out the correct root of the non-linear system of steady state relations, and hence a non-physical solution lead to a breakdown of the computation. The reason for the scheme not to find the correct root, may be due to the fact that the relative influence of the reactions change as the conditions (temperature and pressure) change. Then the truncations of the steady state relations will no longer be valid.

Peters [44] remark that carefulness should be applied when using the reduced mechanism outside the applications for which it was derived. As an example ignition problems are mentioned. In ignition processes different elementary reactions will be important, and the truncations and partial equilibrium assumptions may no longer be valid.

4.1.4 No Transport of Radicals Concept

The truncation process in the deduction of reduced mechanisms involves the elimination of unimportant reactions. However the relative influence of the reactions change as the temperature and pressure change. This means that the correct root may not be selected when the conditions change. An alternative way to attack this problem is to only neglect the transport terms by convection and diffusion and retain the time derivative in the L-operator in equation 4.8. By doing this the conservation equations for species in group one are reduced to ordinary differential equations in terms of time, and thus decoupled from the space derivatives.

The application of the time integrated relations for the species in group one can proceed in two different ways.

58 Chapter 4 Submodels of physical phenomena

Alternative A: Global four step mechanism with time integration for group 1 species

Alternative B: Full skeletal mechanism with no transport of radicals and time integration for group 1 species

4.1.4.1 Global four step mechanism with time integration for group 1 species

It is possible to apply the global four step mechanism and simply replace the truncated steady state expressions for the first group concentrations in the reduced mechanisms with a time integration for these intermediates. The time integration may be performed by a simple stable method, for example implicit Euler. By doing this the computational cost will be approximately the same as for reduced mechanisms, but the problem with finding the correct root for different conditions is omitted because no truncations and partial equilibrium assumptions have to be made. Thus the stability problem for high pressures should be excluded. However the limitations to seven species still imply the problem that the first group concentrations have to be contained in the remaining concentrations for group two species according to mass continuity. Also the assumptions made in order to reduce the skeletal mechanism to four main chains will prevail. This alternative A is not further discussed in this text. Instead the following alternative B will be utilized.

4.1.4.2 Full skeletal mechanism with no transport of radicals

When the full skeletal mechanism is to be used, all species must be considered in the mass-balance equation. Otherwise a mass inconsistency will evolve as the chemical conversion proceeds through intermediate species with different molecular structures. This necessitates modification of the equation for conservation of mass. Also the other conservation equations will be affected by the no transport of radicals concept.

The computational gain of this method will lay in the fact that the species conservation equations for first group intermediates are reduced to ordinary differential equations in terms of time. Since they are not space dependent they

59 Chapter 4 Submodels of physical phenomena may be integrated by a simple routine.

Figure 4.1. Differential fluid element through which a fluid is flowing without transport ofR species.

Figure 4.1 is an illustration of the mass balance over an arbitrary differential fluid element. The number of intermediate species in group one is called R, while the number of intermediate and main species in group two is called M. The total number of species is then: R+M = S. Neglecting transport of R species gives that the mass inside the element contains all species, S, while the mass flux accross the boundary only contains the species M. The mass conservation equation (3.1) will then be modified to 3 M FS- + V-(p Z (Yk)u) = 0 (4.23) 5t k=l

In the species conservation equations (3.2) for species in group one, all transport terms vanish and the equations simplify to

~~"(p Y;) = CD: (4.24) at

The species conservation equations for main species and intermediate species in group two are not changed

60 Chapter 4 Submodels of physical phenomena

-^(pYj)+V-(puYi) = V-(pDi,mVYi) + (oi (4.25) at

In the momentum equation (3.6), the convection of momentum will only include the mass of M species

—(pUj)+V'(p E(Yk)uuj) = --^2-+V~cjj (4.26) dt k=i dxz

In the energy equation (3.11), the terms due to transport of species from the first group must be cancelled. This gives the equation:

a M ( m \ da. -|-(ph)+V-(pu^Ykhk) = 5B +u.Vp+V- kVT + p£ (hA^Wk) + ^-^+0(4.27) 9t k=i dt ^ k=i ) dXj

Or if the thermal diffusion is expressed in terms of enthalpy gradients (eq. 3.14) M -£

r9Uj 9uj rij - 2p (4.29) ^axj a%

The relations S.3-3.5 still apply, and we note that they may be written as R M EV£Y. = i <4-30) i=l i=l

R M £“i+£“i = o (4.31) i=l i=l

61 Chapter 4 Submodels of physical phenomena

S R M M E (pD^VYi) = £ (pD^VYi) +£ (pD^VYi) = 0+£ (pD^VY,) = 0 (432) i=l i=l i=l i=l Thus the mass conservation equation (4.23) is the sum of all species conservation equations (4.24 and 4.25), and one of the species conservation equations (4.25) should be replaced with the condition 4.30.

Comparison between the skeletal mechanism, the reduced mechanism and the No Transport of Radicals Concept (alt. B) are performed in chapter 6.

4.2 Thermodynamical and thermochemical properties

In combustion calculations two additional property relations is necessary. One expression is relating the properties temperature, pressure and density. The other expression is relating temperature to enthalpy or specific heat capacity. The thermodynamical properties h and cp may be calculated directly from molecular spectroscopic data using statistical mechanics equations. In deciding a reference value for the enthalpy of a substance, the thermochemical property enthalpy of formation is most usually selected. The thermochemical properties also take into account the chemical processes necessary to create the substance, and thus regognize the chemically bounded energy in the substance.

4.2.1 The relation between pressure, density and temperature

From experiments it is found that a pure low density gas follows the thermal gas law or equation of state

p#T P = (4.33)

This relation is known as the ideal gas law. It is valid for low pressure, high temperature conditions. Near the critical point and at extremely high pressures

62

-W Chapter 4 Submodels of physical phenomena

the deviations becomes appreciable, and a compressibility factor must be included. An alternative is to apply a more complex gas law. At the pressures and temperatures occuring in most combustion appHances the ideal gas law is a very good approximation to the real gas behaviour. The temperature in the reaction zone is far beyond critical temperatures for most substances, and the pressure is usually not so extreme that it calls for the use of compressibility considerations.

4.2.2 The relation between temperature and enthalpy

The enthalpy of a substance consist of its enthalpy of formation at some reference state added to the difference in enthalpy from the reference state to the actual state.

hpT - (hfXef + (AhW-p/r (4.34)

The pressure dependence is weak and it is often neglected in combustion calculations. Thus the enthalpy can be assumed a temperature function alone.

T h(T) = (h°)^+ / cp (T)dT (4.35) Tn, where the specific heat capacity at constant pressure is defined as

CpCO - (4.36) h and cp for different substances are tabulated in a number of sources. Among the most used is the JANAF Thermochemical Tables [52].

The use of tables in a computer program involves storing of a large number of values and the use of interpolation routines. From a computational viewpoint it is more convenient to express the temperature dependence as a polynomial. The curve fitting of mathematical expressions to tabulated values can be done in several ways.

The NASA-polynomials have found extensive use in combustion calculations. Based on the JANAF-tables and other tabulations, the NASA-polynomials are fitted with simultaneously least squaring of three properties; heat capacity,

63 Chapter 4 Submodels of physical phenomena entropy and enthalpy. The polynomials are written in the form

= a1+a2T+a3T2+a4T3 + a5T4 (4.37) R

(4.38)

Tabulations of the coefficients, a,, for a large number of species can be found in reference [53].

For intermediate species which are present only in a narrow temperature range in the reaction zone and for which the massfractions are small, it is possible to express the enthalpy as a function of temperature in a simpler way. The enthalpy and specific heat capacity are found for some reference temperature, Tm, and the enthalpy is assumed a linear function of the temperature.

Cp Cpjn Cp(T m) (4.39)

h = iv+^cr-Tj (4.40)

The reference temperature, Tm, should be choosen as the temperature where the ma ximum concentrations of intermediate species occurs.

4.2.3 Concentration variables and bulk properties

Table 4.2 gives the molecular weights of the 16 species (including N2) in the skeletal mechanism for methane-air combustion given in table 4.1.

The average molecular weight may be calculated from molefractions or massfractions by s 1 w = EPW = (4.41)

Then molefractions may be calculated from massfractions by the relation

64 Chapter 4 Submodels of physical phenomena

Species Molecular weight [kg/mol]

ch 4 0.01604276 o 2 0.0319988 n 2 0.0280134 C02 0.0440098 h 2o 0.01801628 h 2 0.00201588 H 0.00100794 CO 0.0280104 OH 0.01700734 0 0.0159994 ch 3 0.01503482 ch 3o 0.03103422 CHjO 0.03002628 HCO 0.02901834 ho 2 0.03300674 h 2o 2 0.0340147

Table 4.2. Molecular weights for species in skeletal mechanism.

A w; JW Xt = Y; (4.42) E© w, i-11 w,

And vice versa

Y. . . *ij. 1 s (4.43) EftW.) i=l

The concentrations are given by equation 2.17 and may be expressed in terms of molefractions or massfractions. The ideal gas law (4.33) can also be used to rearrange the expressions in terms of density. The different options are

65 Chapter 4 Submodels of physical phenomena

-Ep_Y - = _P_Y . (4.44) WjRT 1 Wj 1

For a multicomponent mixture of gases we calculate the average molecular weight for the mixture from equation 4.41. Then we can apply the ideal gas law (equation 4.33).

The total heat capacity and enthalpy for a mixture may be calculated on a mole basis by using s (4.45) cp

s h ECW (4.46)

Or if the properties are given on a mass basis the analogeous expressions will be s (4.47) cp - 52 (YiCpj

s h - 220%) (4.48)

When expressing the energy conservation equation in terms of enthalpy it is necessary to be able to determine the temperature from a given enthalpy, h _ , and composition. This can be done by for example a simple Newton iteration.

h(Tn)-h,cons.eq. hcrj-hcons.eq. T».i Tn - >W) c„(Tn) (4.49)

In the first iteration T0 is set to the previous time step temperature, and equations 4.37 to 4.40 and 4.45 to 4.48 are used to calculate the corresponding mixture properties.

Some solution algorithms are designed to work with transport of always positive variables. The enthalpy of formation is negative for some substances resulting in negative values for the total enthalpy. This can be remedied by adding a constant to all enthalpies of formation. The constant is choosen to be the absolute value of

66 Chapter 4 Submodels of physical phenomena the largest negative enthalpy of formation. Doing this will only change the reference point and not the relative difference between the species.

4.3 Transport properties

The formulation of transport fluxes can be done with different levels of precision. In the conservation equations for multicomponent mixtures in chapter 3, the diffusion of species is written in terms of species gradients according to Pick's law. The reason for doing this is that the detailed diffusion formulation involves the use of diffusion velocities which in the case of multicomponent mixtures can not be written in a species gradient formulation. The solution of these equations can then not be done within the framework of a general transport equation (3.15). Also the effects of thermal diffusion as well as the reciprocal Dufour effect is neglected. The implications of using a detailed description can be found in Dixon-Lewis [54]. He also describe the derivation of transport properties from elementary molecular data which is a rather tedious task.

By using Pick's diffusion formulation, the use of multicomponent diffusion coefficients is made necessary. These coefficients can be deduced from the elementary binary diffusion coefficients and forced to obey the criteria 3.5. In the description of a software package from SANDIA by Lee et al. [55] it is shown how the elementary molecular data can be used in obtaining multicomponent diffusion coefficients as well as mixture conductivity and viscosity coefficients.

Dixon-Lewis has shown how approximate equations for transport fluxes in multicomponent mixtures can be formulated. By combining results from Chapman and Cowling [56], Hirschfelder et al. [57] and others he propose a set of relations which is considerably simpler than the elementary theory. He claims that these relations should not differ more than 5-10% from the elementary theory. Still considerably computational effort is necessary for calculating the transport properties.

This text uses a very simple formulation of transport properties. The formulations is taken from Smooke [42], and was used in the joint research effort on reduced kinetic mechanisms for methane-air combustion which was described earlier in this chapter [41].

67 Chapter 4 Submodels of physical phenomena 4.3.1 Thermal conductivity

Both the specific heat capacity, cp, and the thermal conductivity, k, can be approximated by complex relations involving elementary molecular data. However the ratio between the two properties can be approximated by a simple expression. Smooke [42] uses the expression

(4.50)

The reference temperature T0 is set to 298K. Smooke has fitted the results from a detailed transport model with equation 4.50 by means of a nonlinear least square procedure. He found for lean conditions:

A = 2.58E-05-^ ms (4.51) r = 0.7

Comparisons of this relation with complex transport expressions show good agreement for lean methane-air combustion. For rich conditions the agreement is not so good, especially at low temperatures. Smooke suggest the use of a two zone model developed in a similar fashion for use in inhomogeneously lean/rich mixtures. However, for simplicity and consistency, the constants (4.51) is retained for all conditions.

4.3.2 Species diffusivity

The Lewis number is defined as the ratio of the thermal diffusivity and the mass diffusivity

K (4.52)

If the Lewis number of the different species is taken to be a constant, the calculation of multicomponent diffusivity coefficients will be dramatically simplified. Smooke used the complex transport formulation to calculate species Lewis numbers for both premixed and counterflow flames. Then he fitted them to a constant. The fitted constants show remarkably good agreement except for low

68 Chapter 4 Submodels of physical phenomena temperature regions [42]. The constants are given in table 4.3.

Species Lewis number

ch 4 0.97 o 2 1.11 n 2 1.00 CO, 1.39 H,0 0.83 h 2 0.30 H 0.18 CO 1.10 OH 0.73 0 0.70 ch 3 1.00 ch 30 1.30 ch 2o 1.28 ECO 1.27 ho 2 1.10 h 202 1.12

Table 4.3. Simplified transport model Lewis numbers. The multicomponent diffusion coefficients for diffusion of a species into the rest of the mixture is then given by rearranging equation 4.52

= —i— (4-53) pc p Lek

4.3.3 Viscosity

The Prandtl number is defined as the ratio of the kinematic viscosity and the thermal diffusivity

Pr (4.54) K

The viscosity can then be estimated by selecting an appropriate Prandtl number. For example choosing the value Pr=0.75 gives

69 Chapter 4 Submodels of physical phenomena

0.75 — (4.55)

4.4 The spark plug

The spark plug has a three dimensional geometrical form. In addition the processes in the spark discharge described in chapter 2.1.3 are complicated. In this work the inital flame kernel will be modelled as a sphere and consequently a simplified model of the heat input and heat loss to the spark plug is necessary. Figure 4.2 shows the geometry and model assumptions for the spark plug.

heat loss, contact area

heat input, spark power

Figure 4.2. Geometry andmodel assumptions of spark plug.

4.4.1 Spark energy input

Following the ideas of Pischinger and Heywood [14,15] the spark energy deposit is modelled as a heat source in a small ball shaped volume with the diameter of the spark gap distance. Heywood has meassured the electrical energy release in the positive column between the electrodes. This results in a power profile as a function of time. As a function of this power profile the heat input per volume unit will be

70 Chapter 4 Submodels of physical phenomena

S(t)el = (4.56) ’ball

In this work the power profile has been modelled as a constant during the first millisecond, and then approximated by the first quarter periode of the cosinus function as the power approaches zero. This curve, which is shown in figure 4.3, is close to the meassured values of Heywood. In this way the energy input during the short breakdown phase is neglected. Since the duration of this phase is of order nanoseconds it should be a fair approximation as long as the effects of the shock wave set up by this process is not considered. For a conventional spark plug Heywood reports that the majority of the heat input will be in the glow discharge mode which has a time duration of order milliseconds. Heywood also achieved the more efficient arc discharge mode by using spark plugs with smaller electrodes.

0.0005 0.001 0.0015 0.002 0.0025 0.003 time (s) Figure 4.3. Power profile of spark discharge.

4.4.2 Heat loss to spark electrodes

The gas loses heat by conduction to the electrodes. A detailed modelling of this phenomena would require a three dimensional modelling of the flow around the electrodes. Instead we use a Semenov type of heat loss with a variable heat loss area. The flame is considered spherical and Heywood has calculated the contact

71 Chapter 4 Submodels of physical phenomena area, A,., between the flame and the electrodes as a function of the flame radius. The contact area is thus the surface area of the electrodes contained within the spherical flame ball. Figure 4.4 shows a curve fit to the calculated values. When the flame is within the electrodes, the contact area is zero. When the spherical flame reaches the electrodes, the contact area will rapidly grow. As the flame propagates outside the electrodes the contact area is set to a constant value. The curve fit for the contact area used in figure 4.4 is

Ac = 0 for r 3 0.5E-03 Ac = 11.778r 2 - 10.111E-03r+2.111E-06 for 0.5E-03 < r s 2.0E-03 Ac = 37.000E-03r -45.00E-06 for 2.0E-03 < r s 3.0E-03 (4.57) Ac = 18.5r 2 + 148.00E-03r -211.5E-06 for 3.0E-03 < r z 4.0E-03 Ac = 84.50E-06 for 4.0E-03 < r

The heat loss per unit volume is expressed as

g _ h A(r) (Tgas ~ ^elec) (4 58) L " V(r)

The heat transfer coefficient, h c, can be estimated in several ways. Some researchers have chosen to use a constant value for h c. This value is found from

9e-05

8e-05

7e-05

6e-05

5e-05

4e-05

3e-05

0.002 0.004 0.00b 0.008 0.01 0.012 0.U14 radius (ml

Figure 4.4. Contact area for heat loss to spark plug electrodes.

72 Chapter 4 Submodels of physical phenomena either experimental investigations or theoretical considerations, and have large variations between different researchers. Heywood used a constant value between 1500 to 2000 W/m2K. Woschni and Shiling [58,59] predicted a value below 100 W/m2K, while Ko et al. [60] suggested a value between 150 and 200 W/m2K If a constant value for h c is to be used, it was found in this work that a value of 120 W/m2K for a conventional spark plug gave the best agreement between experiments and simulated results.

The use of a constant value for h c has however some drawbacks and is therefore discouraged. It is to be expected that the heat transfer coefficient has a pressure (density) dependence. Also the heat loss is at its largest in the beginning of the heat loss process. After a while the temperature gradient in the gas will flatten out as a result of the heat loss, and the amount of heat transfer will decay. This can be taken into consideration by modelling the heat transfer as transient conduction in a semi infini te gas volume. Due to the much larger heat capacity in the electrodes the electrode temperature is set constant and the surface gas temperature is set equal to the electrode temperature. This constitutes one of the boundary conditions. In the infinite end of the gas volume the temperature is set equal to the homogeneous start temperature for the gas volume. The heat conduction equation is

3T(x,t) _ k cPT(x,t) (4.59) St pcp dx 2

Solved with the boundary conditions

T(0,t) = T^ T(~t) = Ta, T(x,0) = T, (4.60) the heat flux per unit area at the electrode surface (x=0) may be expressed as (Kreith and Black [61], Arpaci [62], Carslaw and Jaeger [63] or Ko et al. [60])

.cT, ttpCp (4.61) Qelec© - ix=0 CTVTo) dx \ Tit

Then the heat transfer coefficient in equation 4.58 can be expressed as

73 Chapter 4 Submodels of physical phenomena

K£Cp (4.62) X - 7lt

The constant corrects for the neglection of convection and the fact that the gas temperature during ignition is not immediately raised to a constant value Ta at the onset of the spark. Both corrections suggest that the correction factor should be slightly higher than 1. Ko et al. [60] found that the heat transfer coefficient calculated in this way was approximately 30% lower than the experimental value. For this reason a value of A^l.4 was found appropriate in this work.

4.5 Constant pressure approximation

In chapter 2.1.1 the pressure change over a subsonic methane-air flame is approximated to 0.9Pa. This is less than 0.01% of the total pressure. Due to this fact the flame propagation process may be considered a constant pressure process.

If we consider an actual spark ignited combustion process in a bomb or an engine cylinder, the spark will set up a sharp pressure wave. Due to the spherical volume expansion this pressure wave will rapidly faint out to an accoustic wave as explained in chapter 2.1.3. Then it no longer has an important effect on the flame propagation. A detailed three dimensional simulation including the geometry of the spark plug, will have to take this pressure wave into consideration. If however the ignition process is modelled in a simplified one dimensional way as described in chapter 4.4, there is no hope of getting any detailed information of the processes occuring close to the discharge channel and within the spark electrodes. This area is modelled as a sphere with empirical heat exchange relations. Just outside the electrodes experimental investigations (Ask [4]) has shown that the flame kernel is nearly spherical. When the flame kernel has reached this size the one dimensional approach should be justified. However at this distance from the discharge channel the shock wave has decreased to an accoustic wave [16], and no longer has an pronounced effect on the flame propagation.

In a closed vessel like an engine cylinder and a combustion bomb, the pressure rise will be slow in the beginning and show a sharp increase at the end of the

74 Chapter 4 Submodels of physical phenomena

combustion process. If we only consider the first phase of the process, the pressure rise is small. Due to this it is argued that a constant pressure approximation will give a good representation in such a simplified one dimensional model of spark ignition.

During auto ignition of premixed gases the local pressure rise is significant. It is in fact a major explanation of the rapid flame propagation in such process (chapter 2.2.4 and 2.2.5). In this case the pressure variations have to be modelled. If the gases are not mixed and there is a sharp division between fuel and oxygen, the speed of the chemical conversion will be diffusion controlled. Then the chemical conversion rate will be considerably slower than in the premixed case, and consequently the local pressure rise will be much slower. For this reason it should be possible to model a stratified auto ignition process with a constant pressure approximation.

The constant pressure approximation has great implications on the system of conservation equations in chapter 3. When the pressure is set, the momentum conservation equation is excluded. This makes the solution procedure easier because the strong non-linear effect of pressure gradients on the density and velocity no longer has to be taken into consideration. It will reduce the number of equations to be solved. The solution procedure followed in this work is to calculate temperature and massfractions from from the energy- and species-conservation equations respectively. The density is calculated from the ideal gas law, and finally the velocity may be deduced from the overall mass conservation equation.

4.6 Boundary conditions

4.6.1 Boundary conditions for one dimensional spherical flame

In the spark ignition simulation the development of a flame kernel from an electrical spark is approximated by a sphecial flame growth model. The modelling of the spark plug relations is discussed in chapter 4.4. The boundary conditions for the energy and species equations are given by a zero gradient condition in the center of the sphere.

75 Chapter 4 Submodels of physical phenomena

A = 0 for r = 0 (4.63) dr

—- =0 for r = 0 (4.64) dr

The zero gradient condition is necessary for continuity of the properties and their derivatives in a point of symmetry. For the same reason the velocity is set to zero. This constitutes the boundary condition for the mass conservation equation at the center of the sphere

u = 0 for r = 0 (4.65)

For the outer radius we specify an open boundary. For a successful ignition and flame propagation, the outer boundary will be an outflow boundary due to the expansion of gases in the flame. If the flame is quenched, the heat loss to the spark plug can result in inflow through the outer boundary. In both cases since the pressure is set constant, no outer boundary changes of temperature or massfractions can occur prior to the flame front arrival to the outer boundary. The diffusion processes in absence of flame propagation, is too slow to be important on the outer boundary. When the flame reaches the outer boundary (the temperature starts to rise on the boundary) the simulation is stopped.

For an outflow condition (positive velocity) the enthalpy and massfractions on the boundary are set to the value in the first control volume inside the boundary.

k(n) - h

(4.67)

For an inflow condition (negative velocity) the enthalpy and massfraction is set to their initial value at the boundary. (4.68)

(4.69)

The mass continuity equation needs no outer boundary value. The continuity

76 Chapter 4 Submodels of physical phenomena equation for the last control volume will give the boundary velocity according to mass continuity for the whole calculation domain.

Intital conditions have to be specified for temperature, pressure, velocity and composition (massfractions) for the whole calculation domain. In the case of premixed spark ignition, addition of spark energy as described in chapter 4.4.1 will start the ignition process. For the autoignition of a methane bubble, the high air temperature together with mixing of methane and oxygen will be sufficient to cause an onset of combustion.

77 Chapter 5 Solution of the equations

Chapter 5 Solution of the equations

The solution of the set of partial differental equations describing reacting flows in chapter 3 requires special attention. In this chapter one of the solution methods, the finite volume method, will be described in some detail. Also the structure of a computer program for solving the equations will be discussed.

5.1 Discretization methods

The set of partial differential equations in chapter 3 are characterized by being nonlinear, coupled and containing variables with a wide spectrum of timescales. The momentum equation expresses a strong influence of pressure gradients on the velocities. In the species equations the reaction rates have an exponential temperature dependence. This means that they may change rapidly with even small temperature changes. The chemical conversion will have an affect on the temperature which in turn influence density and pressure. Moreover the timescale of different species can vary by several orders of magnitude.

These couplings pose challenges to the solution procedure. The literature provides a variety of different methods to solve the equation set. Hirsch [64] contains descriptions of useful methods in fluid computations. The methods can be placed in one of three main groups. These are:

Finite difference methods Finite volume methods Finite element methods

Finite difference methods discretize the equations from the form in which the

78 Chapter 5 Solution of the equations equations in chapter 3 are written. All differential expressions are written in terms of finite differences over a solution grid. The grid or mesh points are distributed along families of non-intersecting lines.

The finite volume methods applies Gauss theorem on the transport equations written in a conservative form. The theorem converts a volume integral to a surface integral. Then the equations will express the time changes in a finite volume as a sum of internal sources and the fluxes at the volume boundary. The method has the advantage of using an arbitrary mesh.

Finite element methods divides the computational domain in a number of arbitrary shaped elements. Triangular elements are common. On each element a basis function is introduced. The field variables are approximated by linear combinations of these known basis functions. Integration of the global function representing all basis functions is an essential step in the finite element method. It requires use of precise definitions of functionals and norms necessarry to minimize the difference between the physical solution and the fini te element discretization.

All methods contain advantages and drawbacks. Some are common for all methods and some are related to only one. The chose of a suitable method depends both on the problem type and the resources available for implementation of the method. In addition it will also be a question of the programmers preferences. While some numerical specialists prefer the fini te element method because of its elegant mathematical formulation, other will choose the finite volume method because of its simple physical interpretation.

In this work the finite volume method have been chosen because cf its good documentation and widespread use. It is relatively easy to program and provides a well established way to treat strongly nonlinear terms.

79 Chapter 5 Solution of the equations 5.2 Finite volume method

5.2.1 Gauss theorem applied to the general transport equation

Gauss theorem (divergence theorem) states that the volume integral of the divergence of a field variable and the unit surface normal vector pointing outwards.

f(V-)dV = $(<$>-n)dA (5.1) v A

If we take the volume integral of the general transport equation 3.15 and apply Gauss theorem on the convection and diffusion terms we get

/(—(p4>))dV +

5.2.2 Discretization of the general transport equation

Let us now consider an one-dimensional geometry shown in figure 5.1. The figure shows a control volume P with neighbouring contol volumes W (West) and E (East) with the positive x-direction to the right. The boundary between the control volumes is marked w and e. Discrete values of the general variable

The volume integrated equation 5.2 may now be written for control volume P in a one dimensional geometry. Since the geometry is orthogonal (planar or spherical), the dot product of the general variable and the unit normal surface vector will always be equal to the general variable.

80 Chapter 5 Solution of the equations

Ax

6x,w 8x -a o o

W w e

Figure 5.1. One dimensional discretization of control volume.

(^(p4>))pV +(puA) e-(puA) w = (T^VdjjA^ - (r^VA)w+S^V (5.3) at

If we apply implicit Euler time integration and l.order differencing for the gradient we get

i^£^V.(puA)A-(puA).* w - ffA).*!^-

Superscript 0 denotes previous time step value. If the general variable (j> is set to 1 and the source term S* equals 0, we obtain the mass continuity equation (3.1). In discretized form it will read o V+(puA)e - (puA)w = 0 (5.5) At Multiplying 5.5 with and subtacting it from 5.4 we obtain

\^PpV +(puA) e(4)e-4>p) -(puA) w(4)w-4)p) At (5.6) = (TA)^-(TA)w^+S,V 6xe 8x w

This is the form of the discretized finite volum equation for a general variable § that we want to use. Notice that it includes the determination of the variable at

81 Chapters Solution of the equations the boundaries e and w. These terms originate from the convection term and express that the variable at the boundary. This will however lead to the classical example of unstable solution. A simple and frequently used way to avoid the problem of unstability is to use upstream differencing. This concept uses the value of p. The reason for this will be discussed in chapter 5.2.5. The transport equation (5.6) may be written

£t((*P — bw + ^ (5.7)

upstream differencing

b = De +max(-Fe,0) (5.8)

c = Dw+max(Fw,0) (5.9)

a = b +c + ^-S^V (5.10)

d' s‘-cV^^ (5.11)

Fe = (puA)e (5.12)

II (5.13) 1

De = (5.14)

82 Chapter 5 Solution of the equations

Dw = Is-A" (5.15)

This scheme is easily expanded to two or three dimensions for orthogonal coordinates. Also the use of more advanced differencing methods are straightforward to adopt in the scheme. The details may be found in Patankar [65]. However this text will only use the described method for one dimensional geometries in planar and cylindrical coordinates with upstream differencing, and the above treatment will be sufficient.

5.2.3 Diffusion coefficients and source terms

The mass conservation equation (3.1) is adapted to the general transport equation by setting

S* = 0 (5.17)

This will make the diffusion terms equal to zero, and the only transport is by convection.

The species conservation equations (3.2), will be achieved by setting

II (5.18)

I* - pD i>m (5.19)

S* = “i (5.20)

The equation for conservation of momentum in x^-direction(3.6) is found by setting

83 Chapter 5 Solution of the equations

(5.22)

s* = -|r+s* c (5-23) axj where is the rest of the viscous stress tensor not contained in the diffusion term. The discretization for the momentum equation is done on a staggered grid where the velocities are stored on the ordinary control volume boundaries. This is done to handle the strong influence of the pressure gradient term on the velocities in a proper way. Since the pressure is stored in the center of the control volumes, its gradient will be most conveniently calculated at the boundaries. The momentum equation is excluded when the pressure is set constant, and will therefore not be discussed any further. Additional information can be found in Patankar [65].

The energy conservation equation (3.14) is obtained by setting

= h (5.24)

r* = - (5.25) cp

8+ = V- •)h kVyk) +-^+u-Vp+-r ij-^i+Q (5.26) k=i

5.2.4 Constant pressure approximation

As argued in chapter 4.5 it is possible to treat some combustion processes as if they were occuring at constant pressure. This simplification has great implications on the solution procedure. The stong nonlinear effect of pressure gradients on velocity and density are no longer considered. Patankar [65] has described a special procedure to count for these effects. Setting the pressure constant excludes the necessity for this procedure.

84 Chapter 5 Solution of the equations

Since the pressure is prescribed, the density is decided by the ideal gas law once the values for temperature and mass-fractions have been found. Then the velocity is the only unkn own in the mass conservation equation, and can be solved from this. The momentum conservation equation will be redundant. This also involves excluding the viscous effects which is of no importance in one-dimensional constant pressure flow.

5.2.5 Modifications due to the No Transport of Radicals Concept

As we have seen in chapter 5.2.2, the continuity equation is used in the derivation of the final discretization equation. The reason for this is that the implicit formulation requires the problem to be solved in an iterative manner, and the mass balance may not be satisfied during the first iterations. This can destroy the the progress towards a convergent solution of the other conservation equations, and thus the whole iteration process. To avoid this the mass balance is included in the general transport equation by multiplying the mass balance equation with the general variable and subtracting the result from the general transport equation.

Since the continuity equation is changed in the no transport of radicals concept, the final discretization equation in chapter 5.2.2 (eq. 5.6) will no longer be valid. The new form of the discretized equation will be discussed in this chapter after the discretized form of the new mass conservation equation has been derived.

The consequences of the No Trans port of Radicals Concept and constant pressure appro ximation on the flux and source terms are also discussed.

5.2.5.1 Mass conservation equation

The No Transport of Radicals Concept neglects the transport of selected species, and according to equation 4.23 the density at the control volume boundary should be modified by multiplying with

85 Chapters Solution of the equations

M pEor k) (5.27)

The boundary values of the massfractions are found from the aritmetric mean of the control volume values. The velocities are stored at the boundaries of the control volumes as in a staggered grid.

If the velocity at one boundary of the calculation domain is known (for instance zero in the center of a sphere), the only unknown in the mass conservation equation for the first control volume is the velocity at the other side of this volume. Then we can solve for this velocity explicitely. For the next control volume the same argument will then be true. One of the boundary velocities are known, and the other can be found explicitely, and so on through the whole calculation domain. The algorithm for finding the velocity at the right hand side of the control volume ,(e), when the density field and the velocity at the left hand side ,(w), is known may be expressed as

(5.28)

k=l

The values on the boundary for (ZYk)w are found using the aritmetric mean of the values at the neighbouring control volumes.

5.2.5 2 No transport of radicals discretizing of general transport equation

If the continuity equation (5.28) is multiplied with p and subtracted from the general transport equation (5.4), we will arrive at

And in the same mann er as for the full transport of all species we may write this equation as

86 Chapter 5 Solution of the equations

ap — bij)g +c^y+d (5.30) where the coefficients are for upstream differencing b = De+max(-Fe,0) (5.31)

c = Dw+max(Fw,0) (5.32)

M M a = D„ *maz(F„0) -F.£ (Yt). *D„ *max(-F»,0) *F W£ (Yt)„ k=l k=l (5.33) ^-S^V

a - (5.34)

Fe = (puA) e (5.35)

Fw = (puA) w (5.36)

r D = ——Ae (5.37) " 8%. °

Dw = ^Aw (5.38) 5xw

S.2.5.3 Species conservation equations

The conservation equations (4.24) for the species which are not transported will be ordinary differential equations in terms of time. They are solved with implicit Euler time integration to conform with the practice in the general transport equation (5.4).

87 Chapter 5 Solution of the equations

. „.v (5.39) At

This gives the expression for Y;

Yj = —((pYj)0 + WjAt) (5.40) P

It is important for the stability of the calculation to express the time change with correct slope in terms of the other variables. This is discussed in chapter 5.2.7 under source treatment.

The species conservation equations for the transported species (4.25), will be achieved by setting 4> = Yj (5.41)

r* = PDyn (5.42)

s* = “i (5.43)

These terms are identical with the terms in the full transport case, but remember that the general discretized transport equation looks different. Since the source terms are nonlinear functions of both temperature and massfractions, it is important that they are linearized in a proper way. This is discussed in chapter 5.2.7.

S.2.5.4 Energy conservation equation

The energy equation (4.28) is obtained by setting A n (5.44)

(5.45) 'p

The source term consists of the other terms in equation 4.28. When the pressure

88 Chapter 5 Solution of the equations is set constant the pressure-change dependent terms vanishes. Also the viscous dissipation term can be neglected. Applying the no transport of radicals concept requires some attention. When setting the total enthalpy as the general variable, the diffusion term arising from the temperature gradient is correctly represented, but the convection term will include the enthalpy of all species. Equation 4.28 states that only the enthalpy of the species which are transported should be used in the convection term.

There are several ways to fulfil equation 4.28. One way is to subtract the convected enthalpy arising from group one species from the energy equation and put it in the source term. Another way is to multiply Fw and Fe in equation 5.35 and 5.36 with the upstream value of the ratio of enthalpy of the transported species to the enthalpy of all species. M LOW M------(5.46) h ECW k=l

In this case the source term will be

M S, = V' ((pDk^n —)h kVyk) (—h kVyk) Q (5.47) Vk=i k=l C

A third way is to set the general variable

The surprising result of implementing one of these representations of equation 4.28, is that it gives a very good estimation of the downstream temperature and massfractions, but a relatively large underestimation of the burning velocity. The reason for this will be discussed in chapter 6.

The solution of this problem emerges from the discussion in chapter 6 when comparing the no transport of radicals concept with the reduced mechanism. In the reduced mechanism only group two species are considered in the energy balance. Doing the same assumption in the no transport of radicals assumption gives values for the burning velocity which are very close to the values obtained

89 Chapter 5 Solution of the equations with full sceletal mechanism.

The following form of the energy equation should then be used

^HphM)+V-(puhM) = at M (5.48) •^+u-Vp+V- -Vh M+£((pDk^--^)h kWk) at v cp k=1 CP where M hM = E(YA) (5.49) k=l

Equation 5.48 is easily adapted to the general transport equation by setting 4> = h M (5.50)

r* = -5- (5.51)

M 8$ = V- ]C ((PDk>m - —)hkVyk) Q (5.52) k=i ^ y

The pressure is here assumed constant.

5.2.6 Transport coefficients on control volume boundaries

The properties temperature, density and massfractions are used to calculate the transport coefficients thermal conductivity, viscosity, and species diffusivity. All of the properties are stored in the center of the control volume while the conductivity and the diffusivity are needed at the control volume boundary. This necessitates a procedure for calculating the appropriate mean value of the two control volumes.

Refering to figure 5.2, the aritmetric mean of a general variable

90 Chapter 5 Solution of the equations

6x„

. 8 < >

o o p e

Figure 5.2. Distances used in the interface calculations.

as

K 4>e 4)p+t_l<5)e (5.53) fix. ox.

It is tempting to use the same formula to calculate the transport properties, but this will not give correct values for the flux when the difference between the contol volumes is large. In the extreme case we can consider one control volume to be an insulator with zero conductivity. If the boundary is situated halfway between the two control volume centers, the aritmetric mean for the thermal conductivity will give a value of one half of the conductivity in the other neighbour control volum. Obviously the flux should be zero. We need a mean value calculation which are controlled by the lowest value of the two control volumes. Patankar [65] has shown that the harmonic mean fulfils this objective. The transport coefficients on the boundaries are thus calculated by 1 8x; T Sx; (5.54) 5xerP sxerE

91 Chapter 5 Solution of the equations 5.2.7 Source term treatment

The formulation of source terms are crucial for the success of the iterations to reach a converged solution. The source term consist of one constant part and one proportional part.

S = SC+SP4> (5.55)

The solution of an equation for one control volume is found from (5.7)

(5.56) a

Many of the variables are always poitive. Examples are the massfractions. In this work also the enthalpy is formulated in such a way that its value always will be positive. If one of these variables should take a negative value during the iteration process it will be not only be a physically unrealistic solution, it will also lead to a breakdown of the iteration process. For this reason it is of major importance to ensure that the variables never can attain a negative value. From equation 5.56 we see that this can easily be provided by making sure that the coefficients a, b, c and d always must be positive. From the definition of the coefficients (5.8 - 5.11) we find that this will be the case if Sc is always positive and SP is always negative.

When a nonlinear source term is written in the form of equation 5.55, it has to be linearized. For always positive variables like massfractions and enthalpy, it is important that the linearization obey the demand for a positive Sc and a negative SP. This may be accomplished by using the following rules.

1. Positive constant source terms are put into Sc. 2. Negative constant source terms are divided by the previous value of P, are calculated with the previous value of

Rule 4 needs further explanation. A negative source term represents a decay in the value of the variable. If the decay is a function of the variable itself, its decay slope is decided from the functional dependence (f.ex. power) of the variable. The

92 Chapter 5 Solution of the equations correct decrease slope will be the source term’s derivative with respect to the variable in that point.

If we mark the previous iteration step with a star, the source term derivative at the previous iteration step may be approximated with

fdSV = S,zS- (5.57) ldJ *

Rearranging this expression will give us an expression for the source term at the present iteration step.

S (5.58) ld

This is a linearized form of a general source term, S, where its slope is multiplied with the general variable <]) just like in equation 5.55. The next cycle of iteration will give a better estimation of * and S*. Then the slope will be better approximated and so on until convergence is reached.

As an example if the source is given as: S = 6 - 72, equation 5.58 will give: S = 6-74f = 6 - 7* 2 -14**(* - **) = 6+7** 2-14

This gives the linearization Sc = 6 + 7* 2 SP=-14*

5.2.8 Solution of system of discretized equations

Equations 5.7 or 5.30 constitutes a system of n equations, one for each control volume. The equation for control volume i may be written

a;*; = b;*;„i+Ci<|);_i+dj (5.60)

In matrix notation this will be

93 Chapter 5 Solution of the equations

al -t>l *1 V

-Cg $2 d2

-Cg a3 -b 3 4*3 dg (5.61) -Cj -bj di

_®n-l ®n-l ^n-1 ^n-l dn-l

-cn an . >=. dn

The matrix in 5.61 is a tridiagonal matrix. The equation set can be solved by the following procedure (Cheney and Kincaid [66], formulated in fortran syntax):

Eliminate all cj} starting with c%, by subtracting the previous line (i-1) multiplied with -c/a^:

do i=2, n mult = -c/a^ - mult*(-b i.1) dj = dj - mult*d i.1 enddo

The back substitution starts with finding 4>n from dn /an . Then the rest of the ^ can be found (starting with

n = dA, do i=n-l, 1, -1 i= di + (b/ai)*

5.2.9 Convergence criteria and time step

The implicit formulation requires iterations in order to arrive at a converged solution from an initial guess. In the case of time dependent calculations we start

94 Chapter 5 Solution of the equations from a very good guess, namely the previous time step. This usually necessitates only a few iterations per time step. However if the time changes are fast, the first order accuracy discretization will be too inaccurate with long time steps and many iterations. This necessitates some kind of procedure to adjust the time step in order to retain good accuracy and a limited number of iterations per time step.

With convergence of iterations is here understood the process of finding values of the variables at the new time step which is a solution to the discretized equations. The accuracy of the method is however related to how well this discretized solution fits the original differential equations. As mentioned, the method is of first order both in time and space. To ensure an accurate solution from a first order method the time step should be kept small relative to the changes. This will also keep the number of iterations per time step to a minimum.

The convergence may be monitored by calculating how well the current values of the dependent variables, , satisfies the discretization equations. From each discretization equation (5.7 or 5.30) a relative residue may be calculated by

R = ------^------1 (5.62) b^g+ctjj^+d

The coefficients a to d are calculated from the current solution of all

When the residue approches zero, convergence is achieved. In practice some small number must be chosen as the limit of the residue. When the largest value of |R | are smaller than this limit, further iterations will only give minor changes in the dependent variables. In this work convergence is said to be obtained when no relative residue exceed 1-1 O'3.

Since the computer time for calculating the coefficients a to d are much longer than the time for the equation solving routine described in chapter 5.2.8, new values of the dependent variables are calulated from the present values of the coefficients before the iteration procedure is ended. Also the minimum number of cycle iterations per time step is set to 3, even if convergence is achieved with fewer iterations.

The length of the time step is decided from three considerations. The velocity resticts the time step so that no mass can move a distance greater than the size of a control volume in one time step.

95 Chapter 5 Solution of the equations

If the number of iterations exceed 6 the time step is progressively made shorter. If the the number of iterations is 3 or 4 the time step is made longer, but not beyond the value decided by the velocity, the maximum time step or the maximum percentage of reactants which were slowed to react during each timestep.

Not more than 20% of the remaining massfractions of methane or oxygen in any control volume is allowed to react during one time step. If the net reaction rate for one of these species exceed this limit, the time step will be adjusted correspondingly. The maximum allowable time step is 5-lO^s. For a normal flame propagation this limit was never reached. From a starting value of l-10'7s, the time step stabilized on a value between 1.5-10'7s and l.O-lO^s, decided from the maximum methane consumption limit.

5.3 Solution algorithm for ID constant pressure simulations

Figure 5.3 shows a flow chart of the solution algorithm for one dimensional constant pressure calculations. First the geometry is set up. A uniform grid spacing of 1.0*10' 5m to 2.0*10 ‘sm is used. The control volume surfaces are located midway between the centers. Also the surfaces and volumes of each cell are calculated dependent on if the geometry is planar or spherical.

Then molecular data and control parameters for the different routines are set. The inital values for temperature, massfractions, pressure and velocity have to be specified. To avoid the calculation of new temperature dependent chemical rate constants for each iteration cycle, a table of rate constants as a function of temperature is created. This will save a considerably amount of computation time since the exponential function in the rate expressions is expensive to calculate. The density is calculated from the ideal gas law, and the enthalpy is calculated from NASA-p olynomials . The initialization ends with writing the start values of the variables of interest to the result file.

The time integration loop starts with adding the specified time step to the current time. Then the old time field variables (denoted with 0 in chapter 5.2) are set equal to the current field variables.

96 Chapter 5 Solution of the equations

Start

Geometry Constants Start values (Yi.T.p.u) Table of rate constants Initialization of variables Density from gas law Enthalpy from NASA-polynom Write start field to file Time step

Start of time integration loop

t-t+At

Set old time field variables equal to current time field variables Transport coefficients Chemical reaction rates Solve species conservation -X^Start of iteration loop equations for massfractions Solve energy conservation Main loop calculations equation for enthalpy Calculate temperature Density from gas law Velocity from mass conservation equation

Find new time step according to criteria

Figure 5.3. Flow chart for ID constant pressure algorithm.

97 Chapter 5 Solution of the equations

The iteration loop performs the calculation of new current field variables. Each time this loop is worked through, better estimates for the variables are found. The loop starts with the calculation of thermal conductivity and species difiusivity. Then the chemical production rates are calculated from the current temperature and composition.

The first conservation equations to be solved are the species equations. If there are S species, S-l equations have to be solved. The last massfraction is calculated from the sum of all massfractions which equals one. The equation solution procedure starts with finding the coefficents for the equation. Boundary conditions and source terms must be specified. Also interpolation routines for the control volume boundary values are needed. Then the residue (eq. 6.62) is calculated with the present values of the variables. Finally the tri-diagonal matrix solver calculates the new values of the massfractions.

The energy equation is solved in the same manner. The result is the total enthalpy for the mixture in each control volume. To find the temperature we apply the Newton iteration procedure (eq. 4.49). This procedure requires the functional dependence of specific heat capacity and enthalpy on temperature discussed in chapter 4.2.2.

The density is calculated from the ideal gas law and finally the velocity field is found from the total mass conservation equation (eq. 5.28 for no transport of radicals concept).

This ends the iteration loop. If the values have converged, and the number of iterations are equal or greater than three, the loop is left. If not, the iterations are continued until the convergence criteria are fulfilled. The new values of the variables in interest are written to the result file at specified time intervals. Finally in the time integration loop a new time step is calculated on the basis discussed in chapter 5.2.9. Then the calculation proceed from the start of the time integration loop until the specified end requirement is satisfied.

98 Chapter 6 Performance of the No Transport of Radicals Concept and computer code

Chapter 6 Performance of the No Transport of Radicals Concept and computer code

In order to verify the computation method and program code described in chapter 5, simulations of a planar laminar flame propagation in premixed methane-air was performed. Simulations were done with three different implementation methods: the full skeletal mechanis m described in chapter 4.1.2, the reduced mechanism in chapter 4.1.3 as well as the No Transport of Radicals Concept described in chapter 4.I.4.2. This was done in order to compare the performance of the No Transport of Radicals Concept with the full and reduced mechanism.

The simulations were also compared with flame propagation results found in the literature to ensure that the computer code was functioning properly.

6.1 Model assumptions

In order to compare the burning velocities obtained from the No Transport of Radicals Concept with results using reduced mechanisms and skeletal mechanisms, a one dimensional planar laminar flame propagation was simulated. The calculation domain was 6.0mm long and the mixture was ignited by inserting a heat source in a 0.02mm section in one end of the calculation domain. The amount of energy supplied by the heat source was just enough to start a flame propagation. Then a laminar flame would propagate towards the other end of the calculation domain wich was modelled as an outflow boundary. The ignition end of the calculation domain was modelled as a zero gradient closed boundary. After approximately 2.0mm the propagation speed of the flame front became constant. For this reason a total length of 6.0mm was considered to be sufficient for obtaining steady flame propagation properties.

99 Chapter 6 Performance of the No Transport of Radicals Concept and computer code

The computation method was the finite volume method described in chapter 5.2. To obtain a grid independent solution a grid spacing of 2.0-10"5m was found necessary. Most simulation results shown in this chapter were obtained using this grid spacing. At a pressure of latm and an air/fuel equivalence ratio of 1, simulations with grid spacing 1.0-10"5m were performed. The results showed only very small differences from the results using the double spacing.

One simulation had to be done for each combination of stoichiometric air/fuel factor and pressure. This required a number of simulations for obtaining a curve of for example pressure versus burning velocity. The reason for using a time dependent method for obtaining a steady flame propagation, was the minimal amount of modifications which was necessary in the computer code for spherical ignition described in chapter 5 and 7.

In order to verify the results from the computer code, some simulations was run with reduced mechansism and skeletal mechanism and compared with the results of Smooke and Giovangigli [43]. They have obtained their results by applying an arclength continuation method [43]. This is an efficient steady (not transient) solution method, which ensures that the solution smoothly changes as the parameter of interest (i.e. pressure) changes.

6.2 Comparisons of burning velocity

The burning velocity was found by multiplying the flame front propagation velocity at the end of the computation domain with the ratio of burnt to unbumt gas density as described by equation 2.5. The burnt gas density was taken at the location where the temperature had reached the adiabatic flame temperature. Because of the energy supplied to start the combustion, the temperature in the ignition end of the calculation domain would exceed the adiabatic flame temperature. Even when the minimum amount of energy was supplied only in a 1/600 part of the calculation domain, the temperature in the ignition end was signifi cantly affected due to the change in density times specific heat capacity between unbumt and burnt gas. However using the density at the adiabatic flame temperature location, was found to compensate for this.

Figure 6.1 shows the burning velocity as a function of air/fuel equivalence ratio, X, at a pressure of latm. Calculations with the no transport of radicals concept are

100 Chapter 6 Performance of the No Transport of Radicals Concept and computer code

No transport of radicals — Skeletal mechanism o Reduced mechanism + Skeletal mechanism, from ref. [43] — Reduced mechanism, from ref. [43] —

Wamatz temperature dependence for all curves.

1.2 air/fuel equivalence ratio

Figure 6.1. Calculated values of burning velocity for methane-air at 1 atm as a function of air! fuel ratio. Results from this work and litterature. performed in the range of 1=0.7 to 1=1.7. It should be pointed out that the results with air/fuel equivalence ratios below 1 are suspected to contain some uncertainties. This is because the skeletal mechanism does not contain species in the Cg-chain, which are known to be present in noticeable concentrations in rich mixtures. Compared with the results from the skeletal mechanism at 1-values of 1.0, 1.3 and 1.5 the no transport of radicals concept shows very good agreement. The calculations with reduced mechanism (1=1.0, 1.3 and 1.5) gives some underestimation of the burning velocity.

Comparison with the literature data of Smooke and Giovangigli [43] gives fairly good agreement for the skeletal mechanism. The difference is assumed to be mainly due to the different solution methods. Using a transient method to solve a steady problem necessitates the use of the adiabatic flame temperature as described earlier. However the differences are small and should verify that the the computer code functions properly.

Smooke and Giovangigli ’s simulation with reduced mechanism give an over ­ estimation of the burning velocity. This is the opposite result from the simulations

101

•' -• -Vf f. Chapter 6 Performance of the No Transport of Radicals Concept and computer code

No transport of radicals----- Skeletal mechanism, from ref. [43]----- Reduced mechanism, from ref. [43] -----

Reduced and skeletal mechanisms have correction factor for Wamatz temperature dependence near 1 atm.

pressure [atm]

Figure 6.2. Calculated values of burning velocity for methane-air at stoichiome ­ tric conditions as function of pressure. Results from this work and litterature. in the present work. Some of the reason for this disagreement may be caused by the inclusion of the decomposition reaction (lOf in table 4.1) for methane in the expression for methyl concentration. Doing this was necessary in order to obtain ignition in the time dependent simulation. This issue is not ivestigated any further in this work. The important point is that the No Transport of Radicals Concept compare well with both own and literature simulations using full skeletal mechanism.

Figure 6.2 shows the burning velocity as a function of pressure. The no transport of radicals concept is compared with results using skeletal mechanism and reduced mechanisms from ref. [43]. For the dissociation and recombination of methane (reaction 10 in table 4.1) the no transport of radicals uses a Lindemann pressure dependence with k„ described by the coefficients in table 4.1. Smooke and Giovangigli [43] found that this would give a too high burning velocity at atmospheric pressure, and they included a pressure dependent correction factor which forced the reaction coefficient for reaction 10 to approach Wamatz rate coefficient (eq. 4.4 and 4.5) near 1 atm. This is the reason for the disagreement near latm. At higher pressures the agreement between the no transport concept

102 Chapter 6 Performance of the No Transport of Radicals Concept and computer code and the full skeletal mechanism is remarkably good. Another feature is that the concept is stable for all pressures. Unlike the reduced mechanism there is no pressure limit where the simulations become unstable.

6.3 Comparisons of temperature and species profiles

Figures 6.3 to 6.8 show the space variation of different properties in a stoichio ­ metric flame at latm. The grid spacing used in the simulations was 1.0-10‘5m and the flame propagates towards the right. All figures contains plots for each of the three implementation techniques. The solid lines correspond to the no transport of radicals concept, dashed lines correspond to full skeletal mechanism and dotted lines correspond to reduced mechanism.

In order to compare the profiles they were aligned to x=0 at the point of maximum chemical energy conversion [J/m3s]. This is shown in figure 6.3. We can see that due to the neglection of transport of selected radicals this curve has a higher peak value and a more narrow space distribution than the full skeletal mechanism. When all species is allowed to diffuse, the reaction zone will consequently be distributed over a greater distance. It is difficult to observe any difference in upstream and downstream spreading, and this confirms the assumption made in chapter 4.1.3 that the diffusion terms dominate the convection terms within the reaction zone.

The temperature profiles are shown in figure 6.4. Due to the higher and narrower chemical energy conversion profile, the no transport of radicals concept gives a slightly steeper temperature profile resulting in a higher temperature in the reaction zone. Some distance downstream the temperatures approaches eachoter again. The reduced mechanism also gives a higher temperature in the reaction zone, but since no chemical energy can be bounded in the limited consentrations of first group radicals (they are not included in the mass balance), the temperature will remain a little high downstream.

Figure 6.5 shows the profiles of the reactants and main products. We can see that most of the reactants are consumed upstream of the peak value of chemical energy conversion. Most of the H20 is also produced upstream of this peak, while C02 is

103 Chapter

temperature [K] c h e m ic ae l n e r g yc o n v e r s io [ nJ/m 3 s] Figure Figure 3e+09 4e+09 le+09 2e+09 5e+09 6e+09

2200 2400 2000 1400 1600 1800 1000 1200 6 200

-0.002 -0.002 Performance

6.3. 6.4. ------tn - - - - -

Profiles Temperature -0.0015 -0.0015 1 ------of

the

of -0.001

-0.001 No

chemical ' ------

Transport profiles propagation

-0.0005 -0.0005

energy 1 ------for of distance distance atm.

Radicals 104

stoichimoetric

at conversion

1

[m] [ml

atm. Concept 0.0005 0.0005 No No

for

Skeletal Skeletal and flame Reduced transport Reduced transport stoichiometric 0.001 0.001 computer

propagation

mechanism mechanism mechanism mechanism

of of 0.0015 0.0015

code rad. rad.

------flame

0.002 0.002 at

1 Chapter 6 Performance of the No Transport of Radicals Concept and computer code

lid/line: No transport of radicals - 5 j dash line: Skeletal mechanism 1 / dot line: Reduced mechanism

0.05

-0.002 -0.0015 -0.001 -0.0005 0 0.0005 0.001 0.0015 0.002 distance [m] Figure 6.5. Profiles of reactants and main products for stoichiomtric flame propagation at 1 atm.

0.025

solid line: No transport of radicals dash line: Skeletal mechanism 0.02 dot line: Reduced mechanism -

0.015

0.01

0.005

-0.002 -0.001 0.001 0.0015 0.002 distance [m] Figure 6.6. Profiles of hydrogen molecules and atoms for stoichiometric flame propagation at 1 atm.

105 Chapter

m olefraction (m o l/m o l) raolefraction [m o l/m o l)

0.00015 6 0.0001 0.0002 Figure

Figure 0.006 0.007 0.008 0.001 0.002 0.003 0.004 0.005 0.009 5e-05 Performance o.oi -0.002

6.7. 6.8.

-0.0015 -0.0015

Profiles of Profiles

the

No -0.001 -

of 0.001 of Transport propagation propagation

first

first

-0.0005

group group of CH30

Radicals 106 distance distance

species

species at at solid solid

1 1 0

atm. atm. Concept

for

« for (m) |m] line: line: dash dash CH20 0.0005 dot dot stoichiometric stoichiometric

and No

line: No line: H202 line: line:

transport transport

computer

0.001 0.001

Skeletal Skeletal Reduced Reduced

of of code

flame

0.0015 flame

mechanism mechanism mechanism mechanism radicals radicals

0.002 0.002

-

Chapter 6 Performance of the No Transport of Radicals Concept and computer code mainly formed downstream of the peak. The latter is due to the intermediate production of CO which has it highest concentration near the peak of chemical energy conversion. The different implementation methods compare very well on the upstream side of the flame, but show some differences on the downstream side. The largest difference is for the H20 molecule where the no transport of radicals underestimates its concentration. This is also the case for the molecule showed in figure 6.6. The H-atom concentration is on the other hand very well estimated for all implementation methods. This is a very reactive radical and the success in describing the correct burning velocity depends strongly on the success of estimating this concentration correctly.

The concentrations of the first group radicals, for which transport is neglected in the concept, is shown in figures 6.7 and 6.8. For the reduced mechanism these species are considered to be in steady state and are not included in the mass balance. Their concentrations are only used to find the appropriate main reaction rates. Except for OH and O concentrations all these species have a sharp peak near the maximum chemical energy conversion location, and a very low concentration elsewhere. The species containing C-atoms have the peak almost exactly at the location, while the species containing only H- and O-atoms have the peak a little distance upstream. The agreement between the different implemen ­ tations are not so good for most of these species. This is probably the main explanation for the smaller differences between the group and main species in figures 6.5 and 6.6. It should also be noticed that for the most narrow peaks, the curves are not smooth. This becomes most apparent in the case of the no transport of radicals concept because of the narrowing of the reaction zone as discussed above. A way to remedy this is by choosing even a smaller grid spacing. However, since the burning velocity is well represented, it was not done in this work. If the species concentration is of great importance, for instance in calculating pollutant agents, a smaller grid spacing is recommended near stoichiometric conditions.

6.4 Importance of diffusion on flame propagation

In chapter S.2.5.4 the implementation of the energy conservation equation in the no transport of radicals concept was discussed. The surprising point was that if the enthalpy of the first group species was included in the total enthalpy this would result in burning velocities which was too low by approximately 70%. However the temperature profile was closer to the profile of the full skeletal

107 Chapter 6 Performance of the No Transport of Radicals Concept and computer code mechanism than those profiles shown in figure 6.4. The profiles of chemical conversion rate and species was approximately equal to the narrow profiles shown in figure 6.3 and 6.5 to 6.8.

This reflects the importance of diffusion of reactive first group species on the flame propagation process. When these species are not transported, the flame speed will consequently slow down. Both reduced mechanisms and the No Transport of Radicals Concept will suffer from this.

The effect mentioned above has somehow to be compensated. In the reduced mechanism only group two species are included in the total energy of the system. The total energy is conserved by the energy conservation equation. When group one species not are included in the energy conservation equation, their enthalpy contribution to the total energy in the system is not included. Their contribution must then be found as higher temperature of the remaining species. This can be seen from the temperature profiles in figure 6.4. In the reaction zone the reduced mechanism has a steeper temperature increase than the full skeletal mechanism. It also stabilizes on a higher temperature behind the flame. Smooke [43] found a similar behavior.

When the temperature profile has a steeper rise through the flame zone, the flame propagation will speed up. This is mainly caused by the increased amount of energy which is transported upstream due to thermal conduction. It is interesting to notice that this effect almost balance the opposite effect of species diffusion.

For this reason it was found advisable to use the same assumption in the No Transport of Radicals Concept. Chapter 5.2.S.4 discusses how only group two species are included in the total energy conservation.

The advantage of the No Transport of Radical Concept compared to the reduced mechanism, lies in the fact that the time integration of first group species and inclusion of all reaction rates allow calculations of a broader range of processes, like high pressure and ignition simulations. In addition will the inclusion of all species in the mass balance give a better approximation of the concentrations in the reaction zone and the temperature behind the flame front when conditions approach equilibrium at high temperature.

108 Chapter 6 Performance of the No Transport of Radicals Concept and computer code 6.5 Computer demands and calculation stability

The computing time of the different implementations is important when complex flow configurations are to be considered. The reduced mechanism and the No Transport of Radicals Concept gave approximately the same time consumption, while the full skeletal mechanism required considerably longer time. It is difficult to give a rational concrete time difference because of the dependence on the computer program efficiency. However the number of partial differential equations to be solved by the program is almost halfed with the application of reduced mechanism and No Trans port of Radicals Concept. The latter requires storage of all present and old time-step concentrations, and will for that reason need computer storage comparable to full skeletal mechanism.

Stability of the computations is also difficult to discuss because of its dependency of the solution method and the programming. However it was experienced that the No Transport of Radicals Concept was comparable with full skeletal mechanism in its demand for time-step limitations and error bounds with respect of stability. The reduced mechanis m was more demandin g, and required more severe limitations in order not to give unstabilities during the simulation. The reason for this may be the solution of the truncated equation system for the staty state species. The reduced mechanism would not give any solution at pressures above 10.5 bar [43].

109 Chapter 7 Simulation of spherical spark ignition

Chapter 7 Simulation of spherical spark ignition

A simple model of spark ignition is used to investigate the growth of a spherical flame kernel. As shown in figure 4.2, is the spark located in the centre of the sphere. The flame will propagate away from the spark and constitute a spherical flame. The growth of this kernel will be influenced by several processes, some of which may be investigated by a spherical symmetric model and some which demand a more detailed model. This chapter will discuss the results found with a one-dimensional spherical symmetric model and compare the simulations with experimental values from a combustion bomb.

7.1 Model assumptions

Chapter 4.4 discusses the modelling of the spark discharge and the heat loss to the electrodes. The one-dimensional model does not allow a detailed geometrical representation of the electrodes and the discharge channel because the modelling parameters are restricted to be expressed as functions of time and radius.

The initial gas is a homogenous mixture of methane and air. For all simulations in this chapter the intital temperature of the mixture is 300K. The chemical kinetic mechanism is described in chapter 4.1 and includes 15 species and 25 reactions. The mechanism makes it possible to include hydrogen in the fuel/air mixture. The full skeletal mechanism with the No Transport of Radicals Concept described in chapter 4.1.4.2 is utilized.

The gas is initially quiescent and only laminar processes is considered. Instabilities in tangential directions and turbulence models are not included. The pressure is set constant throughout the calculation. The implications of doing this

110 Chapter 7 Simulation of spherical spark ignition is discussed in chapter 4.5. The distance between the grid points is 2.0-10"5m for all calculations.

The initial composition of the gas is characterized by the air/fuel equivalence ratio, X, which is defined as the ratio of actual air mass over fuel mass to the stoichiometric air mass over fuel mass.

ma mr X = (7.1) m0

lmF Js

7.2 Results and discussion

7.2.1 Development of the flame kernel

Figure 7.1 shows the development of the temperature profile for ignition of a stoichiometric mixture. At 0.1ms after the onset of the spark the flame kernel has a radius of approximately 1mm. Due to the spark discharge the temperature is very high in the center, and the profile has a steep gradient. Since ionization reactions are not included in the reaction mechanism, the temperature in the kernel attains a higher value during the spark discharge than experimental investigations show [11]. At temperatures over 4000-5000K the gas should be in the plasma phase, and the high enthalpy of formation of ions will lead to a lower temperature than shown in figure 7.1. However when the spark discharge is furnished, the temperature will drop due to heat losses to gas and electrodes. This will reverse the ionization process and the gas state behind the flame front will return to the behaviour described by the model. As seen from figure 7.1 the flame front moves quickly away from the high temperature center and is thus little affected by the processes in the center of the sphere.

At 1.5ms the flame front has almost reached a radius of 4mm. The temperature just behind the reaction zone has dropped to a value well below the adiabatic flame temperature, and as will be shown later, the flame are now propagating very slowly. The reason for this temperature drop is heat losses both to the gas in front of the flame and to the spark electrodes.

Ill Chapter 7 Simulation of spherical spark ignition

0.lms ----- 1.5ms ----- 2.5ms 5.0ms

3000 -

2500 -

2000

1500

1000

0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 radius [m] Figure 7.1. Temperature profiles at different times of a centrally ignited, propagating stoichiometric methane-air flame at a pressure of latm.

As discussed in chapter 2.1.1, an expanding spherical flame at small radiuses suffers from stretching effects. The surface area of a sphere increases with the radius in second power. The effect of stretch on diffusion processes like heat conduction and species diffusion which leads to flame propagation, will result in lower temperature of the burnt gas behind the flame and slower flame propagation. As the radius increases, the situation will become closer to a planar flame front (figure 2.3), and the effect of stretch will be less pronounced.

In addition to heat loss to the gas in front of the flame, comes heat loss to the spark electrodes. The electrode heat loss is at its largest when the flame is close to the electrodes, that is at small radiuses. This will further strengthen the temperature drop and the slow down of flame propagation. •

At 2.5ms the effect of stretching and heat loss to the electrodes are not so strong, and the temperature behind the flame front approaches the adiabatic flame temperature as the flame moves outwards. At 5.0ms the flame front is close to the situation in planar flame propagation (figure 6.4).

112 Chapter 7 Simulation of spherical spark ignition

Figure 7.2 shows the gas velocity profile during the same ignition event. At 0.1ms the peak velocity is high due to the expansion resulting from the electrical discharge and the rapid combustion in the high temperature gas. At 1.5ms the spark discharge has no longer any effect on the peak velocity. The drop in temperature (figure 7.1) causes the chemical reaction rate to decrease and consequently the peak velocity goes through a minimum. As the temperature behind the reaction zone approaches the adiabatic flame temperature the gas velocity peak approaches a higher steady value.

As the gas temperature near the electrodes decreases due to electrode heat loss, a backflow of gas will result as the density increases. This is most pronounced at 2.5ms.

Upstream of the velocity peak the the velocity decreases. This is due to the surface increase of a sphere as the radius gets larger. Mass continuity implies that the velocity decreases as the inverse of the radius in second power. The result is that a relatively large velocity near the center will give little influence on the velocity at larger radiuses.

O.lras 1.5ms 2.5ms 5.0ms

-0.5 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 radius |m Figure 7.2. Gas velocity profiles at different times of a centrally ignited, propagating stoichiometric methane-air flame at a pressure of latm.

113 Chapter 7 Simulation of spherical spark ignition

Flame velocity, lambda=1.0 ------Experimental flame velocity, lambda=l. 0 o Flame velocity, lambda=l .3 -

Flame velocity, lambda=l .5 lambda=l .5 Q

0.002 0.004 0.006 0.008 0.012 0.014 0.016 radius [m] Figure 7.3. Flame propagation velocity as function of radius. Different air/fuel equivalence ratios. Pressure latm. Experimental valus from [4],

Figure 7.3 shows the flame velocity as a function of flame radius for different values of air/fuel equivalence ratios at a pressure of one atm. In this text the flame radius is defined as the location where the temperature has increased by 100K. The flame velocity is defined as the flame radius derivative with respect to time. All curves show a high initial value as the electical discharge ignites the mixture. Then the flame velocity goes through a minimum when the stretch and electrode heat loss effects are strong. When these effects ceases to be important, the flame velocity approaches a steady value at large radiuses. This steady value can be found by using the bu rning velocity and burnt to unbumt density ratio as described in equation 2.5. At stoichiometric conditions the flame velocity is higher than at lean conditions.

The experimental values are found from Schlieren measurements in a combustion bomb done by Ask [4]. The agreement is reasonable, especially for lean conditions. Reasons for disagreement can both be attributed to the simple modelling of the spark discharge and electrode heat loss, as well as the one dimensional modelling and restriction of the skeletal mechanism at air/fuel equivalence ratios near one. On the other hand inaccuracies may arise in the calculation of the time derivative

114 Chapter 7 Simulation of spherical spark ignition of the experimental flame radius. At stoichiometric conditions, Ask predicts the burning velocity at large radiuses to be 45cm/s. The value most commonly accepted is between 37-40cm/s [42]. The model predicts a value of about 38cm/s (figure 6.1).

7.2.2 Heat loss and flame stretch

Figure 7.4 and 7.5 give a more detailed look at the effects of respectively heat loss to electrodes and flame stretch. If we set the heat transfer coefficient between the gas and the spark electrodes to zero, but retain the spark energy we get the results shown in figure 7.4. This implies infinitely small electrodes. For comparison reasons, the results with full heat loss from figure 7.3 are also shown. We see that the pronounced minimum value of flame propagation velocity at radiuses of about 3-4mm will no longer occur. The energy supplied to the gas during the spark discharge will expand the flame front away from the critical small diameters where stretch effects are important. Since there is no heat loss to the electrodes this will take place either with only a small minimum in flame propagation velocity (1=1) or without a local minimum (1=1.3 and 1.5).

To clearly see the stretch effect at small radiuses it is necessary to reduce the spark energy input while still neglegting heat loss to the electrodes. This is done in figure 7.5 where the curves for full spark energy from figure 7.4 is also shown for comparison. Now the reduced spark energy input will lead to a pronounced reduction in the initial expansion of the flame kernel. When the discharge ends, the flame kernel has only reached a radius of about 1mm. At such small radiuses the stretch effect discussed in chapter 2.1.1 will have a strong influence on the flame propagating velocity. We will again get a pronounced minimum value, in this case around l-2mm depending on X. The spark energy supplied is 0.3mJ, which is about the smallest amount sufficient for achieving ignition. It should be remarked that not only the amount of energy, but also the spark power and duration influence the capability for a spark to ignite the mixture. However this topic is not discussed any further in this investigation. The power profile for a full spark is shown in figure 4.3. The 0.3mJ spark has a constant power of 0.3W duration for 0.5ms and then decreases with time as explained in chapter 4.4.1.

The conclusion which may be drawn from this is that the electrical discharge not only has the objective of igniting the mixture in the spark gap. A successful flame

115 Chapter 7 Simulation of spherical spark ignition

Neglecting heat loss electrodes lambdn-1 .0 ------Neglecting heat loss electrodes Neglecting heat loss electrodes Including heat loss electrodes Including heat loss electrodes Including heat loss electrodes

0.002 0.004 0.006 0.008 0.012 0.014 0.016

Figure 7.4. Flame propagation velocity as a function of radius. Different X. Pressure latm. Curves with and without heat loss to electrodes.

Spark energy 0.3mJ, lambda=l .0 ----- Spark energy 0.3mJ, lambda=1.3 ----- Spark energy 0.3mJ, lambdas! .5 ...... Spark energy 24mJ, lambda=1.0 ------Spark energy 24mJ, lambda=1.3 ------Spark energy 24mJ, lambda=l .5 -----

0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016

Figure 7.5. Flame propagation velocity as a function of radius. Different X. Pressure latm. No heat loss to electrodes. Curves for strong and weak spark.

116 Chapter 7 Simulation of spherical spark ignition propagation will also depend on the capability of the spark to expand the flame kernel away from the strong infl uence of heat loss and stretch effects at small radii. This expansion is a result both of the density ratio between hot and cold gas, as well as the fast combustion and burning velocity at elevated temperatures.

Ideally the heat loss is minimized by choosing as small electrodes as possible. This will however conflict with the durability properties of the spark plug. In practice some compromise has to be found. The stretch effect results from a geometrical property of a spherical flame itself, and for a specific fuel,A, and pressure, it can not be remedied as long as the flame propagates spherically outwards from the electrodes.

7.2.3 Influence of pressure

At very lean conditions the flame will no longer be able to propagate after it has been centrally ignited. For the full spark and heat loss model, this is shown to be the case for 1=1.6 in figure 7.6. The curve for latm. is calculated under the same

pressure- latm pressure=2Oatm

0.002 0.004 0.006 0.008 0.012 0.014 0.016 radius [m] Figure 7.6. Flame propagation velocity as a function of radius. 1=1.6. Pressure: 1 and 20atm. Including heat loss to electrodes.

117 Chapter 7 Simulation of spherical spark ignition

assumptions as the less lean curves in figure 7.3. If we increase the pressure to 20atm. the model will predict the flame to be able to propagate. However at high pressures a methane flame propagates very slowly and no experimental investigation are found which utilizes a conventional spark plug for such lean mixtures and high pressures.

The major explanation for the flame to be able to prapagate at higher pressures is probably due to the difference in pressure dependency of the total heat capacity and the heat transfer coefficient to the electrodes. The total heat capacity per volume unit is proportional to the density and thus pressure, while the heat transfer coefficient is proportional to the square root of the density (equation 4.62). As the pressure rises, relatively less energy is lost to the electrodes. Another explanation is that as the pressure rises, the thickness of the reaction zone and initiation zone (figure 2.1) becomes smaller. This is shown in the next chapter. The effect of a tinner flame zone is to shift the strong influence of stretch to smaller radiuses (figure 2.3). Then the spark energy can be sufficient to expand the flame kernel past the critical size where strething tends to extinguish the flame.

Ask [4] has performed experimental investigations on lean high pressure natural gas mixtures. The natural gas contained 97.0% methane, 2.7% ethane, 0.2% propane and some small amount of other gases. For these experiments he used two thin threads as electrodes is order to reduce the influence from the electrodes on the flame kernel to a minimum. In figure 7.7 these experiments are compared with simulations of lean methane-air ignition where the heat loss to the electrodes is neglected. The figure is a plot of the flame radius as defined in the beginning of this chapter as a function of time. The comparisons at 1 and 8atm compare fairly well. What is surprising is that at a pressure of one atmosphere the simulations predict faster flame propagation, while at 8 atmospheres the simulations predicts a slower flame propagation speed than the experiments.

The explanation of this may be due to several sources. One possibility is that methane is the only species in the natural gas for which the burning velocity decreases with pressure. For the other gases the burning velocity increases with pressure. At 8 atmospheres the small content of ethane and propane may lead to a higher burning velocity than in the case of pure methane.

The decrease in flame zone thickness with pressure, which is showed in the next chapter, may also lead to another explanation. Several researchers, f.ex. Markstein

118 Chapter 7 Simulation of spherical spark ignition 0.016

0.014

o.oi -

0.008 -

0.006 -

0.004 Flame radius, CH4, p=latm. — Experimental flame radius. Natural gas (97% CH4), p=latm. o Flame radius, CH4, p=8atm. — Experimental flame radius. Natural gas (97% CH4), p=8atm. + 0.002 Flame radius, CH4, p=20atm. Flame radius, CH4, p=40atm. — Flame radius, CH4, p=60atm. —

0.05 0.15 time [s] Figure 7.7. Flame radius as a function of time. 7^1.6. Different pressures. Neglecting heat loss to electrodes. Experimental values from [4].

[67] have found that instabilities in a flame can arise at special conditions. Sivashinsky [68] points out that some instabilities may be explained as a thermodiffusive instability. The source of this instability are termed preferential diffusion. For a methane-air mixture this means that the light methane molecule diffuses faster than the heavier oxygen molecule. Sivashinsky has shown that this can lead to instabilities in lean mixtures.

The mechanism behind this unstability may be explained by figure 7.8. The light methane molecule will diffuse from the preheat zone into the reaction zone. If a concave perturbation of the flame zone should appear, this will lead to a depletion of methane in the neighborhood of the perturbation and hence a drop in burning velocity here. The result is that the concave perturbation will grow to a groove. These instabilities will affect the propagation of the flame.

As the flame thickness decreases these instabilities will more easily arise due to the shorter distance between burnt and unbumt gas. Ask [4] has observed cellular structure on lean natural gas flame kernels above 3-4 atmospheres, and this may imply instabilities in the flame front.

119 Chapter 7 Simulation of spherical spark ignition

Preheat zone of flame

Diffusion

Diffusion

Propagation direction of flame

Figure 7.8. Rustratrion of preferential diffusion leading to instabilities. Flame zone with a developing groove.

The one dimensional model can not simulate these instabilities because they are at least two dimensional in nature. This may lead to uncertainties in the prediction of the flame propagation velocity.

At higher pressures figure 7.7 predicts a further decrease in the flame propagation speed. The decrease is not so pronounced when the pressure approaches 60atm, and it is expected that a further increase of pressure not will lead to large changes in the flame propagation velocity. This is also supported by the curves in figure 6.2 which shows the planar burning velocity as a function of pressue. The decrease in flame propagation velocity with pressure may be explained by the temperature profiles in the next chapter.

7.2.4 Temperature profiles

Figure 7.9 shows the temperature profiles at a pressure of latm for different methane-air mixtures at large radiuses. As the mixture gets leaner the profiles gets less steep. This may be explained by a drop in the chemical reaction rate for lean mixtures. Oxygen is in excess and the final temperature will be lower. The result is a slower flame propagation velocity (figure 7.3), because when the driving thermal gradient decreases this will lead to less thermal conduction necessary for preheating the gas in front of the reaction zone.

120 Chapter 7 Simulation of spherical spark ignition 2200

2000

1800

1600

1400

1200

Lambda=1.6 — Lambda= 1.5 — Lambda=1.3 — Lambdas 1.0 — Experimental values lambdas 1.6 o

0.009 0.01 0.011 0.012 0.013 0.014 0.015 radius [ml Figure 7.9. Temperature profiles at large radius. Different X. Pressure latm. Experimental values from [4].

1800

1600 -

1400 -

1200 -

1000 -

800 - Pressures latm. - Pressures 8atm. - Pressures20atm. • Pressures4 Oatm. - Pressures60atm. - Experimental values, pressures latm.

0.009 0.011 0.012 0.013 0.014 0.015 radius (ml Figure 7.10. Temperature profiles at large radius. Different pressures. 'k=1.6. Experimental values from [4].

121 Chapter 7 Simulation of spherical spark ignition

The experimental values for X=1.6 are taken from Ask [4]. He used laser Schlieren measurement to find the light deflection through a flame. Together with the known change in refractive index with density, he could obtain density and temperature profiles of the flame. There are close agreement between the experimental and simulated profiles.

Figure 7.10 shows the temperature profiles for a lean (X=1.6) mixture at different pressures. The temperature gradient gets steeper as the pressure is rised. From figure 7.7 we see that the flame propagation velocity gets slower as the pressure is rised. The steeper temperature gradient has thus the opposite effect on the flame propagation velocity as in the case of different mixture composition at constant pressure. The reason for this lies in the different dependency of pressure for total heat capacity and heat conduction. The total heat capacity per volume unit is proportional to the pressure (C=pc p ), while the heat conductivity is independent of pressure (equation 4.50). As the pressure rises, the total heat capacity per unit volume rises, while the conductivity not is changed. This implicate that it will take longer time to rise the temperature in the initiation zone in front of the flame with the same temperature gradient. The result will be a steeper temperature gradient. For methane combustion the pressure dependency of the chemical reaction rates are not capable of fully compensating for the drop in the ratio of conductivity to total heat capacity, and the result is a slower flame propagation.

7.2.5 Hydrogen in lean methane-air mixtures

In practical combustion devices like internal combustion engines the slow flame propagation velocity of lean methane-air mixtures at high pressures may pose a problem. For this reason different actions may be taken to speed up the process. Good mixing through turbulence will accelerate the process, but the turbulence may also lead to extinction if it is too intense. It has also been suggested to add some small amount of other species to the mixture in order to improve the ignition and combustion qualities. Ask [4] have experimented with both hydrogen and heavier hydrocarbon gases. The chemical kinetic mechanism used in this work allows only hydrogen to be added without major changes in the model.

Figure 7.11 shows the effect of adding 5 and 10% H2 per volume of fuel to a

122 Chapter 7 Simulation of spherical spark ignition

0.016

0.014

0.012

0.01

0.008

0.006 Flame radius, CH4 ----- Experimental flame radius, Natural gas o 0.004 Flame radius, CH4 + 5% H2 ------Experimental flame radius, Natural gas + 5% H2 + Flame radius, CH4 + 10% H2 ...... Experimental flame radius, Natural gas + 10% H2 0 0.002

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 time [s] Figure 7.11. Flame radius as a function of time. Pressure 8atm. 7c=1.6. Influence ofH2 addition. Experimental values from [4]. methane-air mixture. The air/fuel equivalence ratio is 1.6 and the pressure is 8atm. The simulations show an increase in flame propagation velocity with increasing volume fraction of hydrogen in the fuel. The increase is not so pronounced as for the experimental values where hydrogen is added to natural gas. This can be due to several of the already mentioned reasons. At. high pressures natural gas is expected to have a larger flame propagation velocity than methane. Also limitations in the chemical kinetic model, and instabilities in the flame front can influence the results.

The conclusion is nevertheless that the model predict an increase in burning velocity when hydrogen is added to a mixture of methane and air.

123 Chapter 8 Autoignition of a methane bubble

Chapter 8 Autoignition of a methane bubble

As discussed in chapter 2, autoignition is often termed as spontaneous ignition. The mixture of fuel and oxygen starts to react without any external ignition devices. It is the high temperature and pressure environment the mixture is exposed to which causes the onset of a combustion process after an inital ignition delay time. The autoignition process is utilized in diesel engines where fuel and oxygen start to react after fuel has been injected in the combustion chamber. For this reason modelling of the autoignition process is an important task in internal combustion engine research.

The spherical ignition model described in the previous chapters should also be capable of simulating autoignition. This is investigated by considering a spherical symmetric methane bubble surrounded by hot air. The autoignition process of such a bubble depends on the species mixing in the boundary region between fuel and air. Also the mixed reactants must have a sufficiently high temperature for the chemical processes to take place. This depends on the heat conduction from the hot surro unding air. All these effects are included in the model described in chapter 3 to 5.

It should be mentioned that the spherical autoignition model has clear analogies with liquid droplet autoignition. The difference is that the evaporation of the fuel is not considered. Investigation of droplet autoignition is important in order to get a better understanding of the mechanisms of diesel spray ignition and combustion. The analogy is not further discussed in this text.

124 Chapter 8 Autoignition of a methane bubble 8.1 Model assumptions

Figure 8.1 shows the geometry of the autoigniton simulation. The bubble consists of pure methane. Outside the methane bubble is air. These initial conditions will give step functions for the massfractions of CH4, 02 and N2. The temperature is specified separately for methane and air, while the pressure is set constant for the whole calculation domain. As for the spark ignition simulation in chapter 7, the geometry is spherical symmetric.

The chemical kinetic mechanism is described in chapter 4.1.2, and is used together with the no transport of radicals concept in chapter 4.I.4.2. The constant pressure solution algorithm as described in chapter 5 is utilized. As argued in chapter 4.5 this may be done when the influence of diffusion processes are strong.

Four different parameters are varied in order to investigate the behaviour of the autoignition process. These parameters are the initial values of methane and air temperatures, pressure and bubble radius.

Figure 8.1. Geometry of methane bubble autoignition.

125 Chapter 8 Autoignition of a methane bubble 8.2 Results and discussion

8.2.1 Influence of pressure and air temperature

To investigate the influence of pressure and air temperature, the radius of the methane bubble is set to 1mm. The methane temperature is initially 300K.

8.2.1.1 Ignition and combustion characteristics

The next subchapters describes in detail the different combustion modes which can result from autoignition of a methane bubble. As pressure and air temperature are varied, different combustion events are found.

At low pressure and low air temperature the result will be a pure diffusion process where methane and air is mixed without any signifi cant chemical conversion of reactants to products. The step temperature profile will simultaneosly be smoothed due to thermal conduction. The result will be methane diluted with air to such a lean mixture that ignition and combustion never take place within timescales that are interesting in this investigation.

If, at a pressure of 1 atm., the air temperature is raised to about 1200K the result will be a lean combustion mode. As the pressure or air temperature is further increased the process passes through a near stoichiometric combustion mode into a diffusion flame combustion mode. Chapter 8.2.2 gives an overview of the modes as function of pressure and air temperature.

8.2.1.2 Lean combustion mode

An example of the lean combustion mode is given in figures 8.2 to 8.5. The pressure is 1 atm., and the air temperature is initially set to 1300K. Figure 8.2a shows the temperature profile as function of radius and time. During the first 50ms we observe the result of thermal conduction between cold methane and hot air. The temperature in the center of the sphere will approach the air temperature at 1300K. The centre volume is dominated by the volume at larger radii due to the geometrical volume dependency of the radius in third power. This leads to a large

126

8

aj °fa

tirrj

6J °* Chapter 8 Autoignition of a methane bubble change in centre temperature and only a small change in the temperature at greater radii. Also the larger heat capacity of air compared to methane has an effect.

The temperature profile at 58ms shows a sudden rise in temperature at a radius of approximately 4mm. At this point autoignition has occured. A premixed flame propagates in each direction from the ignition point. As will be shown later, the mixture is lean at all radii at the time of ignition. This is the reason for the small temperature rise during combustion. Since the leanest mixture is on the outside of the ignition point, the flame travelling outwards has a slower propagation velocity than the flame travelling towards the center.

Figure 8.2b shows the chemical energy conversion profiles. The chemical energy conversion is here defined as the sum of species reaction rates multiplied with their enthalpy of formation at standard state (p=latm, T=298K).

E„ = E(wA%) (81) i=l This may be looked upon as an expression for the thermal energy released in the chemical reactions per time and volume units. The scale is logarithmic, and for readability of the figure, all values below 1.0-10s are plotted as 1.0 TO5. It can be seen that exothermic reactions are present at 56ms. Then the chemical energy conversion rapidly grows into a peak at a radius of approximately 4mm before dividing into two peaks travelling in opposite directions. The peaks show the areas where the most exothermic reactions take place. After the flame has reached the centre, the chemical energy release will decrease to a low value. The flame travelling outwards will be less exothermic as the mixture gets leaner.

Figure 8.3 shows the methane and oxygen molefractions. Notice that the figures are rotated in order to give a good view of the plotted surfaces. The orientation may be seen from the axis text. During the first 50ms only the effect of diffusion processes are evident from the figures. The gases are mixed to an air/fuel equivalence ratio around 10 in the centre of the sphere at 50ms. Then at 58ms both methane and oxygen are consumed by chemical reactions around a radius of 4mm. The air/fuel equivalence ratio at this location just before the rapid consumption of reactants starts to take place, is close to 20. At room temperatures this is outside the ignition limit, but at the actual temperature, ignition will occur. After ignition the methane on the inside of the ignition point is rapidly consumed

129

mode.

temperature

and

combustion

.

’ 1mm

Lean

t- Q " radius

latm . O is d O with

v bubble pressure

and

u methane

a 1300K

of

is

OH.

Autoignition temperature b)

radius

0. 2 8.4.

A ir

n '° CH

aCt a) 300K Figure o\ett

o o ro

A % o bubble Chapter

a) CO

70

radius [mm] 9 20 time imsl

b) CO2

131 Chapter 8 Autoignition of a methane bubble by the flame propagating towards the centre. Since the mixture on the outside is leaner, the outward propagating flame will not consume the reactants so fast. This flame can be followed for several milliseconds after the ignition.

Some more insight into the process can be gained from figure 8.4 where two of the intermediate species profiles are shown. CfLjO is an intermediate in the carbon chain. In figure 8.4a formation of this specie can be detected already after 40ms at a radius of approximately 4mm. At 54ms the increase in molefraction is significant, and at 58 ms two peaks with a local minimum close to zero between the maximum values appear. At the location of the local minimum molefraction, consumption of methane is almost completed. We can then follow the two flame fronts as they propagates in each direction.

The CHgO profiles show that reactions are taking place much earlier than can be detected from the temperature and reactant profiles. However the extent of the reactions during the early diffusion process is small.

Figure 8.4b shows the OH-radical profiles. In contrast to the CH20 profiles they show no gradually growth previous to autoignition. Between 57 and 58ms the molefraction increases from near zero to around maximum value. For this reason the OH-radical is an useful indicator of the ignition delay.

Figures 8.5a and b show the molefraction of CO and C02 respectively. CO is formed first and then oxidized to C02 through a reversible exothermic reaction which count for a significant part of the chemical conversion energy. This reaction will thus be an important part in the self accelerating autoignition process. As can be seen from figure 8.5b and 8.3a, the timescale for production of C02 is considerably longer than the timescale for consumption of CH4. Together with the fact that the reversible reaction implies a chemical equilibrium between CO and C02, this will cause the molefraction of CO not to vanish completely. However in this lean case with low product temperatures, the rest molefraction of CO will eventually be small.

HjO profiles show essentially the same pattern as C02 profiles, the difference being that HgO have a more rapid production than C02.

132 Chapter 8 Autoignition of a methane bubble 8.2.1.3 Near stoichiometric combustion mode

If the air temperature is raised and all other properties are held constant, the autoignition will appear earlier in the diffusion process. For an air temperature of 1500K autoignition will occur at a time when the mixture in the centre of the bubble is approximately stoichiometric. This is shown in figure 8.6 to 8.9.

Figure 8.6a shows the temperature profiles as function of radius and time. Thermal conduction smooths out the temperature profile until, at 7.1ms, the temperature starts to rise at a radius close to 4mm. As in the lean mode, one flame front propagates in each direction from the ignition point. The mixture is lean at the ignition point and the flame travelling towards the centre will be richer until it reaches a near stoichiometric value at the centre. The temperature in the centre after combustion is close to stoichiometric adiabatic. The second flame propagates outwards into further leaner conditions, and will consequently have a lower velocity.

The chemical energy conversion profiles in figure 8.6b gives a better view of the exothermic reaction zones. Near stoichiometric conditions at the centre, the exothermicity in the reaction zone attains a high peak value.

The reactant molefraction profiles are shown in figure 8.7a and b. During the first 7ms the diffusion mixes methane and air to an air/fuel equivalence ratio, X, around 0.8 in the centre of the sphere. At the ignition point around 4mm, the mixture is however lean with X close to 16. After ignition the reactants are rapidly consumed, mainly by the inward propagating flame. From the oxygen profiles in figure 8.7b it may be seen that due to the slightly rich mixture in the centre, all of the oxygen is consumed just after the flame has reached the centre. Diffusion of oxygen from outside will however cause oxygen to be in excess less than a microsecond later.

In figure 8.8a the molefraction profiles of the intermediate specie CH20 is shown. Already between 3 and 4ms some production of this specie can be detected. After ignition around 7ms the two flame fronts are clearly shown. The lean flame travelling outwards constitutes small peaks in the CHzO profile, while the near stoichiometric to slighly rich flame front travelling towards the centre, constitutes high peak values. The fast moving high peaks are challenges to the solution procedure. For stoichiometric to rich conditions the intermediate species

133 Chapter 8 Autoignition of a methane bubble

energy [J/m3s] b) 1.02+11 k""" 1.02+10 - 1.02+09 - 1.02+08 - 1.02+07 - 1.02+06 - 1.02+05

10 radius [mm] 8

Figure 8.6. Autoignition of a methane bubble with radius 1mm and temperature 300K Air temperature is 1500Kand pressure is latm. Near stoichiometric mode, a) Temperature, b) Chemical energy conversion.

134 <%4

'^4 QJ Chapter 8 Autoignition of a methane bubble

molefraction [mol/mol] a) CH20 0.005

0.004

0.003

10

molefraction [mol/mol]

b)OH

10 radius [mm] time [ms] Figure 8.8. Autoignition of a methane bubble with radius 1mm and temperature 300K. Air temperature is 1500Kand pressure is latm. Near stoichiometric mode, a) CH20. b) OH.

136 Chapter 8 Autoignition of a methane bubble

molefraction [mol/mol]

a) CO 0.1 r"""*'

radius [mm] 10

time [ms]

molefraction [mol/mol] b) CO.

radius [mm] time [ms]

Figure 8.9. Autoignition of a methane bubble with radius 1mm and temperature 300K. Air temperature is 1500K and pressure is latm. Near stoichiometric mode, a) CO. b) C02.

137 Chapter 8 Autoignition of a methane bubble

molefractions will show abrupt changes over small distances. As also shown in the chemical energy conservation profiles in figure 8.6b, the reaction zone are thin and highly exothermic.

The OH-radical are shown in figure 8.8b. From autoignition between 7.0 and 7.1ms it has a fast increase until the near stoichiometric flame reaches the centre. Afterwards the profiles decrease.

Figures 8.9a and b show the CO and C02 profiles respectively. CO are produced first and reaches a high value in the reaction zone at the centre due to the slighly rich conditions here. As oxygen reduces CO to C02, the molefraction of CO approaches low values. This can also be seen from figure 8.9b where the production of C02 are somewhat slow in the centre due to the necessary diffusion of 02.

The HgO profiles show the same trend as the C02 profiles, the difference being that the production of H20 are faster and thus consumes 02 prior to the production of C02. For that reason no slow production of H20 in the centre due to shortage of 02 is observed.

8.2.1.4 Diffusion flame combustion mode

If the air temperature is retained at 1500K while the pressure is raised, a different situation will result. The effect of rising the pressure is that diffusion processes will be slower while the kinetic processes will be faster.

The diffusion coefficient and thermal conduction coefficient are almost independent of pressure. The density is however proportional to the pressure, so with the same amount transported by diffusional processes, the species mixing and temperature profile smoothing will take longer time. Together with the increase in chemical reaction rate for some reactions, this results in that the autoignition process takes place earlier in the diffusion process.

Figure 8.10 to 8.13 gives an example where the pressure is set to 50atm. In figure 8.10a the temperature profiles show that after 2.3ms the temperature starts to rise at a radius of approximately 1.5mm. At this time the temperature profile still have a noticeable gradient between cold methane and hot air. In the same manner

138 Chapter 8 Autoignition of a methane bubble

temperature [K]

a)

time [ms]

radius [mm]

b) 10

8 energy [J/m; 1.0E+11 6 1.0E+10 time [ms] 1.0E+09 4 1.0E+08 1.0E+07 2 1.0E+06 0 0.5 1.5 2.5

radius [mm]

a) Temperature, b) Chemical energy conversion.

139 Chapter 8 Autoignition of a methane bubble

molefraction a)CH4 0.8 h

time [ms]

radius [mm]

molefraction

0.1 b 0.05 k time [ms]

radius [mm]

Figure 8.11. Autoignition of a methane bubble with radius 1mm and temperature 300K. Air temperature is 1500K and pressure is 50atm. Diffusion flame mode, a) CH4. b) 02.

140 Chapter 8 Autoignition of a methane bubble has the species diffusion process resulted in a mixing layer between pure methane on the inside and hot air on the outside. The flame will for that reason develop into a diffusion flame with methane on the inside and air on the outside of the hot products. After ignition the temperature peak moves somewhat outwards and the maximum temperature raises beyond the stoichiometric adiabatic temperature. The reason for this is the diffusional nature of the flame. In order to react, methane and oxygen must diffuse from each side into the already hot product zone. The exothermic reaction will then lead to a further temperature increase as the gases (mainly nitrogen and products) at this location already are hot. The slow progation of the flame outwards is due to the gas expansion during combustion.

The chemical energy conversion profiles in figure 8.10b gives an interesting picture. Some exothermic reactions are visible already after a short time. After ignition the peak divides into three separate reaction zones. The lean premixed flame travelling outwards will rapidly vanish. Then there are one reaction zone on each side of the products. The zone closest to the centre is dependent on oxygen diffusing through the products from the outside, while the flame zone on the outside of the products is dependent on methane diffusing from the inside. Since methane has a larger diffusion coefficient, the major flame zone will be at the outside of the products.

The reactant profiles are shown in figure 8.11. Only a minor part of the methane in the mixing layer reacts shortly after the autoignition. Combustion is then dependent on the diffusion of reactants from each side through the products. The oxygen profiles show that the ignition takes place on the lean side in the mixing layer where X is close to 15. Similar to the two previous combustion modes this is caused by the strong temperature dependence of the reactions. Since air is hottest, reactions will go faster on the air side of the mixing layer.

The CH20 profiles in figure 8.12a show a small production after 1ms in the mixing layer of the reactants. Then the molefraction grows rapidly during ignition around 2.3ms. After ignition the three reaction zones can be seen. The largest molefraction of CHzO in the diffusion flame, are found in the zone on the inside of the products. This is the rich flame zone wich is dependent on oxygen diffusing through the products from outside.

The ignition point can be seen clearly from the OH profiles in figure 8.12b. Between 2.3 and 2.4ms the molefraction grows from a near zero value to its

141 Chapter 8 Autoignition of a methane bubble

moleffaction [mol/mol] a) CHsO

10

time [ms]

radius [mm]

molefraction [mol/mol]

Figure 8.12. Autoignition of a methane bubble with radius 1mm and temperature 300K. Air temperature is 1500K and pressure is 50atm. Diffusion flame mode, a) CH20. b) OH.

142 Chapter 8 Autoignition of a methane bubble

Figure 8.13. Autoignition of a methane bubble with radius 1mm and temperature 300K. Air temperature is 1500K and pressure is 50atm. Diffusion flame mode,

a) CO. b) CO2 -

143 Chapter 8 Autoignition of a methane bubble maximum. After ignition the maximum molefraction is found in the reaction zone on the outside of the products.

Figure 8.13 shows the CO and C02 profiles. As have been mentioned earlier, CO is produced first and then oxidized further to C02. After ignition CO has a maximum value located between the two reaction zones where also the temperature is at its maximum value. The C02 profiles have two local maximums, one on each side of the CO maximum. The largest local C02 maximum is found on the air side. HgO profiles are similar to C02 profiles.

8.2.1.5 Chart of combustion modes

From the description of the three ignition and combustion modes, it is clear that the mode is a result of at what time in the diffusion process the chemical kinetic processes becomes important. The ignition delay depends on both processes. Figure 8.14 gives a chart of the different combustion modes for a 1mm methane bubble at 300K. The air temperature along the x-axis varies between 1200K and 2000K. The pressure along the y-axis varies between latm. and 50atm. Three different symbols are used in the figure. They indicate for which combinations of air temperature and pressure simulations are carried out. The symbol "diamond" represents lean modes, the symbol "cross" represent near stoichiometric modes and the symbol "square" represent diffusion flame modes.

At low pressures and air temperatures the diffusion processes will run for a long time before the chemical kinetic processes become significant. Autoignition will for this reason take place at a time where the mixture is lean at all radii. As pressure or air temperature is raised the chemical kinetic processes will become significant in an earlier stage of the mixing process. Autoignition takes place when the mixture in the centre of the sphere is approximately stoichiometric. At high pressures and air-temperatures the poor mixing between methane and oxygen at the time of autoignition leads to the development of a diffusion flame.

Pressure and temperature affects both diffusion and kinetic processes. Higher temperature accelerates both mixing through larger diffusion coefficients, and chemical kinetics through higher values of the reaction rates.

An increase in pressure leads to slower mixing. As mentioned in chapter 8.2.1.3,

144 Chapter 8 Autoignition, of a methane bubble

50 13 ET 1T IT

40 □

H 15 30 □ □ □ □ □ !£ DIFFUSION FLAME 1 20 \ □ □ □ B □ Q. \ \ \ \ \ \\ \\ \ \ 10 o \ \ □ □ □ □ □ \X NEAR STOICHIOMETRIC 5 o Nt- -B__ □ □ □ LEAN ~- o <> o ~=f—- 1 *i r~ f ■f- 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 air temperature [K]

Figure 8.14. Chart of combustion modes for a BOOK, 1mm methane bubble.

this is a result of the pressure dependency of density related total properties per unit volume like concentration [mol/m 3] and energy [J/m3], in contrast to the negligible pressure dependence of the transport coefficients i.e. diffusion coefficients [m2/s] and thermal conduction coefficient [J/smK]. An increase in pressure also leads to higher values of the initiating rate coeffients in the methane combustion mechanism. This is evident from the ignition delay as presented in the next chapter.

It is the combination of all these aspects which gives the picture in figure 8.14. In the case of increasing pressure it is obvious that slower mixing and faster initiating reactions will give a tendency towards diffusion flame mode. In the case of increasing temperature, both mixing and reactions will be faster. However since the reaction rates have an exponential temperature dependence, this will dominate the weaker temperature dependence for transport coefficents. The result is for that reason also a tendency towards diffusion flame mode.

145 Chapter 8 Autoignition of a methane bubble

8.2.1.6 Ignition delay

The ignition delay will be strongly affected by the diffusion processes takin g place prior to and simultaneously with the chemical kinetic processes. Ignition delay times for the simulations carried out in figure 8.14 is shown in figure 8.15. As indicated in chapter 8.2.1, the OH radical molefraction is an useful indicator of ignition. From a very low value, the OH molefraction rises abrubtly when autoignition occur. Ignition delay is thus here defined as the time when the OH molefraction exceed a value of 1.0-10"3. For very lean mixtures at the time of ignition, a value of 1.0 •10'4 or 1.0-10'5 was used.

Figure 8.15 shows that there is a strong dependency of air temperature. From 1200K to 2000K the ignition delay is reduced by about four orders of magnitude for pressures ranging from latm to 50atm. The decrease is not linear in the logartithmic plot. The curve for latm has a slightly different appearance than the curves for higher pressures. This is caused by the use of Wamatz [41,69] reaction rate for the decomposition reaction of CH4 and its reverse reaction at latm. (Chapter 4.1.2).

pressure: 1 atm. 5 atm. 10 atm. 20 atm. 30 atm. 50 atm.

1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 air temperature [K]

Figure 8.15. Ignition delay for a 1mm, 300K methane bubble.

146 Chapter 8 Autoignition of a methane bubble

The ignition delay is also generally decreasing with increasing pressure. The decrease is most pronounced at high temperatures and near atmospheric pressures.

At 1300K the ignition delay is slightly increased from 1 to 5atm. This is a consequence of using Wamatz reaction rate at latm. (eq 4.4 and 4.5), while changing to a unimolecular reaction rate with fall off behaviour (table 4.1) at 5atm. and above. A common reaction rate which perform well both at latm. and at higher pressures would be preferable.

8.2.2 Influence of bubble radius

Changing the size of the methane bubble will affect the length scales in the process. A smaller bubble will mean that the time for the mixing process to reach near stoichometric conditions in the centre will be shorter. Also heating of the cold methane by thermal conduction from the hot surrounding air will lead to a more rapid temperature increase in the centre of the sphere.

In this chapter the size of the bubble is reduced to 1/10 of the radius from chapter 8.2. This will give a methane bubble radius of 0.1mm. The methane temperature is retained at 300K.

8.2.2.1 Chart of combustion modes

The qualitative representation of the combustion modes will be the same as explained in detail in the previous chapters. The differences appear in the time and length scales. Since the resulting modes are a consequence of at what time in the diffusion process the kinetic processes becomes significant, the change of bubble size will change the combinations of temperature and pressure for which the different modes appear.

This is shown in figure 8.16. For smaller bubble radius, the lean and near stoichiometric combustion modes appear at combinations of higher pressure and air temperature. As explained above this is a result of the mixing process taking shorter time.

147 Chapter 8 Autoignition of a methane bubble

50- O O 1 T------Q------□------l \ \ \ \ \ \ \ l 1 40- l \ \ \ □ l l t \ DIFFUSION FLAME \ \ 30- o \A □ □ \ \ 2 \ \ \ \ 3 1 \ 8 \ V 20- 0 o\ \ □ □ \ X \ \ \ X / NEAR STOICHIOMETRIC o \> Q 10- □

o / z X 5- o LEAN 1- o o o % 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 air temperature [K]

Figure 8.16. Chart of combustion modes for a O.lmm, 300K methane bubble.

It may also be observed that for low pressures or air temperatures, the mixture has become lean to such an extent that OH radical formation never exceed the limit given in chapter 8.2.3. The temperature increase in these cases are also extremely low. This implies that the limit between ignition / no ignition at the elevated air temperatures in question here is not sharp. As the mixture gets leaner, the temperature increase approaches zero, and the ignition delay increases rapidly to large values. In this work no points are indicated in the combustion mode chart (figure 8.16) if the OH molefraction never exceeds 1.0-1 O'4.

S.2.2.2 Ignition delay

Ignition delays for the O.lmm methane bubble are shown in figure 8.17. They show the same characteristics as the delays for the 1.0mm bubble in figure 8.15. If the delays are compared, it is found that for the diffusion flame mode (figure 8.16), the ignition delay is almost equal for a O.lmm and 1.0mm methane bubble. For the near stoichiometric combustion mode, the ignition delay is distinct lower for a 0.1 bubble than for a 1.0mm bubble. This trend is strengthened as the mode gets leaner. An explanation of this may be that the temperature increase in the

148

"7TT Chapter 8 Autoignition of a methane bubble

pressure: 1 atm. 5 atm. h— 10 atm. -b— 20 atm. -x— 30 atm. -a— 50 atm. —

1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 air temperature [K]

Figure 8.17. Ignition delay times for a O.lmm, 300K methane bubble. centre of a small bubble dominates the dilution of the methane, and thus leads to faster ignition.

However as the OH limit for ignition is approached, that is extremly lean mixtures at the time of ignition, it may be seen that the ignition delay at latm for a O.lmm bubble is approaching the delay for a 1.0mm bubble again. At such lean conditions methane is diluted by air to such an extent that the ignition delay rapidly increases.

8.2.3 Influence of methane temperature

The previous calculations have been done with a cold methane bubble. Since the reaction rates have an exponential dependence of temperature it is reason to believe that the ignition delay will be considerably smaller if the methane initially is hotter. In this chapter the temperature of the methane bubble is set equal to the surrounding air temperature.

Figure 8.18a shows the temperature profiles for a 1mm methane bubble at a

149 Chapter 8 Autoignition of a methane bubble

temperature [K]

0 1 2.3 4 5 radius [mm]

molefraction [mol/mol] ... 6; GBgO 0.015

radius [mm] 3

time [ms] Figure 8.18. Autoignition of a methane bubble with radius 1mm and temperature 1400K Air temperature is 1400K and pressure is lOatm. Diffusion flame mode, a) Temperature, b) CH20.

150 Chapter 8 Autoignition of a methane bubble

Figure 8.19. Autoignition of a methane bubble with radius 1mm and temperature 1400K. Air temperature is 1400K and pressure is lOatm. Diffusion flame mode, a) CH4. b) 02.

151 Cknpkr 8 AxioWam

Figure 8.20. Autoignition of a methane bubble with radius 1mm and temperature 1400K. Air temperature is 1400K and pressure is lOatm. Diffusion flame mode.

a) CO. b) C02. 152 Chapter 8 Autoignition of a methane bubble pressure of lOatm. Both air and methane has initially a temperature of 1400K. At about 3.4ms the temperature starts to rise, and the combustion develops into a diffusion flame mode.

From the CH20 profiles in figure 8.18b, it can be seen that production of this intermediate species starts at an early stage. The molefraction grows gradually until it reaches a peak around 1mm at 3.5ms. At this point ignition have occured, and the molefraction decreases rapidly afterwards.

From the reactant profiles in figure 8.19a and b, we find that ignition takes place on the lean side of the mixing layer. The air/fuel equivalence ratio is close to 5 just before igntion. Since the start temperature is uniform, this must be caused by features of the chemical kinetic scheme. Implications of this are discussed further in chapter 8.5.

CO and C02 profiles are shown in figure 8.20. CO is produced first and retain a high value due to the 02 shortage in the center. The C02 molefraction has for the same reason a peak at around 1.5mm. The rest reduction of CO to C02 in the centre will depend on 02 diffusing from the surrounding air.

8.2.3.1 Chart of combustion modes

Figure 8.21 shows a chart of the combustion modes as a function of temperature and pressure for the uniform temperature case. Compared with the chart for a cold methane bubble (figure 8.14) with the same radius, it can be seen that the limit between the lean mode and the diffusion mode is slightly shifted towards higher temperatures and pressures. This is mainly due to the increased mixing speed between methane and oxygen. The diffusion coefficients are proportional to the temperature. A rise in methane temperature from 300K up to the surrounding air temperature will thus lead to enhanced mixing.

The effect of the rapid mixing leading to leaner conditions at the time of ignition, is somewhat reduced by the simultaneously increase in chemical reaction rates. As will be shown in the next chapter, the ignition delay is shorter. In other words, the autoignition tends to appear earlier in the mixing process. However as figure 8.21 shows, the effect of rapid mixing at high methane temperature dominates the effect of autoignition taking place earlier. The result is that the lean and near

153 Chapter 8 Autoignition of a methane bubble

50 Tj IT

40

E 2, 30 *1 DIFFUSION FLAME 1 20 Q.

10 0 □ E3 □ NEAR STOICHIOMETRIC 5 1 — 4 4 100011001200 130014001500 160017001800 1900 2000 2100 air and methane temperature [K]

Figure 8.21. Chart of combustion modes for a 1mm methane bubble. Methane temperature is initially equal to air temperature. stoichiometric combustion modes appear at slightly higher combinations of pressure and temperature.

Two other features should also be noted when comparing the cold and hot methane bubble simulations. For a hot methane bubble ignition was observed at HOOK and 50atm. The ignition delay for this simulation was 140ms. For the cold methane bubble, simulations was run for 1000ms without ignition being observed. This is a consequence of the enhancement of chemical reaction rates at elevated temperatures.

On the other hand ignition was not observed at 1200K and latm for the hot methane bubble. For a cold bubble ignition was observed after 572ms. At such extreme lean conditions, the enhancement in mixing velocity for a hot methane bubble causes dilution of the methane to such an extent that ignition not was observed.

154 Chapter 8 Autoignition of a methane bubble

pressure: 1 atm. -e— 10 atm. -s— 50 atm. —

1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 air and methane temperature [K]

Figure 8.22. Ignition delay for a 1mm methane bubble. Methane temperature is initially equal to air temperature.

S.2.3.2 Ignition delay

The ignition delay for the hot methane bubble simulations where methane and air initially has the same temperature, is shown in figure 8.22. When the methane temperature is set equal to the air temperature, the ignition delay has a signifi cant decrease compared with the cold methane bubble in figure 8.15.

8.2.4 Comparison with premixed ignition delay

Ignition delay times for premixed methane combustion have been reported in numerous articles. Experimental ignition delay is often expressed as one equation models on the form discussed in chapter 2.2.2 (eq. 2.21 and 2.34).

Lifshitz et al. [70] gives the expression:

155 Chapter 8 Autoignition of a methane bubble

,195000) (8 .2) t, = 3.62-10"14 [CHJ033 [02] -103e ^

Tsuboi and Wagner [71] have a similar expression:

.222000) (8.3) ti = 4.0-10-16[CH4]a32[O2]-102e m

The experiments of Eubank et al. [72] fall within the limits:

(26000±600 (8.4) ^ = (1.8±0.8) 10-14 [CHJ0-4 [OJ-10e T

All ignition delays are given in seconds. Concentrations are given in moles/cm 3. The measurements of Eubank et al. are done within a temperature range of 1200K to 1850K and with a total molar density around 3.0 103 mol/cm 3, corresponding to a pressure of about 4atm at 1600K. The onset of ignition was determined by time-resolved absorption spectroscopy of a helium-neon laser.

The simulations in chapter 8.4 with a methane bubble with initially the same

this work, 300K methane bubble -e- this work, hot methane bubble h— Lifshitz et al., premixed, stoichiometric — suboi and Wagner, premixed, stoichiometric — Eubank et al., premixed, stoichiometric .....

1200 1300 1400 1500 1600 1700 1800 1900 2000 temperature [K]

Figure 8.23. Ignition delay. Comparison between methane bubble simulations and experimental premixed values from litterature. Pressure latm.

156 Chapter 8 Autoignition of a methane bubble

1000 this work, 300K methane bubble this work, hot methane bubble 100 - Lifshitz et al., premixed, stoichiometric jboi and Wagner, premixed, stoichiometric -ubank et al., premixed, stoichiometric C/3 £ £ CD "O 1 - c o c03

0.01 -

0.001 1200 1300 1400 1500 1600 1700 1800 1900 2000 temperature [K]

Figure 8.24. Ignition delay. Comparison between methane bubble simulations and experimental premixed values from litterature. Pressure lOatm.

this work, 300K methane bubble -e- this work, hot methane bubble -t— Xx Lifshitz et al., premixed, stoichiometric — Tsuo^and Wagner, premixed, stoichiometric — %. Eubank et al., premixed, stoichiometric .....

1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 temperature [K]

Figure 8.25. Ignition delay. Comparison between methane bubble simulations and experimental premixed values from litterature. Pressure 50atm.

157 Chapter 8 Autoignition of a methane bubble

temperature as the surrounding air, showed that ignition took place at the lean side of the mixing layer. Equations S.2-8.4 support that this is most likely to happen. The ignition delay decreases with decreasing methane concentration and increasing oxygen concentration. As Eubank [72] states: "Methane has an inhibiting effect on its own ignition."

Figures 8.23 to 8.25 show the ignition delay of both cold and hot methane bubbles of 1mm compared with the premixed experimental ignition delay from expressions 8.2 to 8.4. The pressure is 1,10 and 50 atm. There is a significant decrease in ignition delay from cold methane bubble autoignition via hot methane bubble autoignition to the experimental autoignition results. This decrease is most pronounced at high pressures where the mixing process is slow. The premixed ignition delay is calculated for a stoichiometric mixture. For a lean mixture the ignition delay would have been even smaller.

It is reason to believe that some of the difference in ignition delay between bubble and premixed autoignition is caused by the mixing process. For a methane bubble ignition to occur, methane and oxygen have to be mixed to some extent proior to any reactions taking place.

It should however be remarked that the skeletal kinetic mech anis m (table 4.1) is developed to give good results for flame propagation. For this purpose the high temperature reactions in the exothermic flame zone have a major influence. Compared to a full kinetic mechanism the skeletal mechanism has neglected a number of reactions which have min or influence on the flame propagation process. These neglected reactions and sprecies may however play and important role in the early stages of the autoignition process.

Some of the difference between the simulated bubble and the experimental premixed autoignition delay, can thus be caused by the limited information in the skeletal mechanism. In future ignition simulations a kinetic mechanism developed with more emphasis on the low temperature reactions should be considered.

Nevertheless, as figures 8.23 - 8.25 show, there is good agreement between simulations and experiments. The skeletal mechanism and the No Transport of Radicals Concept model the autoignition process with a reasonable degree of accuracy.

158 Chapter 9 Turbulent gas jet autoignition

Chapter 9 Turbulent gas jet autoignition

Diesel engines injects the fuel into the hot compressed air in the cylinder. If the fuel is liquid, for example in the case of diesel fuel, this will result in a spray of droplets entering the combustion chamber. The droplets will collide, break up and eventually evaporate and mix with the surrounding air. Under normal engine operating conditions the mixture of fuel vapour and air will undergo an autoignition process due to the high temperature and pressure in the combustion chamber. Some engines are supplied with ignition devices like glow plugs which heat the air/fuel vapour mixture locally, and thus enhances ignition. This is useful for cold start and high speed engines.

For a gaseous fuel the process leading to combustion is very much the same, except for the evaporation of the fuel. The fuel is injected as a gas jet. In the case of methane it has been demonstrated in chapter 8 that this fuel requires a high temperature for the autoignition process to take place within timescales that are interesting in combustion engines. For this reason diesel engines operated on methane are supplied with an ignition enhancing device. This is normally a pilot spray of diesel, but also glow plugs are used.

Simulation of diesel spray autoignition is an extremly complicated problem. The modelling of spray development with droplet coalescence and breakup is one of the major problems. Another problem is the lack of chemical kinetic mechanisms for the diversify of diesel fuels.

When modelling methane jet autoignition there is no need for a spray model. Also well established chemical kinetic schemes for the reaction from reactants to products exists. There is however a need for turbulence models both for the flow and for the combustion. The flow is at least two dimensional and the solution of the flow field requires a more advanced computational fluid dynamical program

159 Chapter 9 Turbulent gas jet autoignition than the one dimensional program described in chapter 5. A description of the selected program and the turbulent combustion model will follow in subsequent chapters.

9.1 The KIVA-II computer program

9.1.1 The original KIVA-II code

Several program packages are capable of simulating gas jet autoignition. The selection fell on the KIVA-II computer program [73]. This computer program has been used for several years in NTH/MARINTEK research, and much work has been carried out to improve its abilities to calculate internal engine combustion.

The KIVA-II code solves the unsteady equations of motion of a turbulent chemically reactive mixture of ideal gases. This involves time dependent three dimensional solution of the mass conservation equation (3.1), species conservation equations (3.2), momentum conservation equations (3.6) and energy conservation equation (3.10 or 3.11). In addition two transport equations for turbulent properties are solved. These are the turbulent kinetic energy equation (9.4) and the turbulent dissipation equation (9.5) which together constitutes the well known k-E model for turbulence.

In addition the KIVA-II code has models for spray formation and evaporation of liquid fuel. Also routines for the influence of chemical reactions on the compostion of the fluid are available. KIVA-II contains however no model for the effect of turbulence on the mean reaction rates. In this work neither the spray model or the built in chemical reaction model are used.

The solution procedure is based on a finite volume methode called the arbitrary Lagrangian Eulerian method. This allows the calulation to be done within an arbitrary compressing or expanding calculation domain. The progam is thus particularly suitable for calculations of the fluid motion in the cylinder of a reciprocating combustion engine. The KIVA-II report [73] gives a detailed description of the mathematical models and the solution procedure used in the program.

160 Chapter 9 Turbulent gas jet autoignition 9.1.2 Earlier extensions of the K1VA-H code

Researchers at NTH/MARINTEK have included several new models in the KIVA- II code. The Eddy Dissipation Consept [74] is utilized to describe the influence of turbulence on the mean reaction rates. In its present form the combustion model included in KTVA-II assumes infini te fast chemistry and is thus totally mixing controlled. The turbulent combustion model is described in chapter 9.2. Also a transport equation model is included. This is utilized for example in soot calculations. Models for soot calculations and autoignition in diesel fuel combustion are included. The extensions are described in reference [75].

In this work only parts of the existing turbulent combustion model is used. Chapter 9.2 describes how the model is modified to include the effect of finite chemical reaction rates.

9.1.3 Extensions of the KTVA-II code to gas jet autoignition

The boundary conditions in the original KTVA-II code did not contain options for a jet entering the computation domain. This necessitated modifications in the routines for the calculation of boundary conditions. Originally the code has an option for inflow through the bottom boundary. A jet inlet at the top boundary was implemented using the bottom inflow option as a guide. This implicated changes in the boundary condition routines for velocity, species density and energy. In addition modifications in the pressure iteration routine was necessary to prevent the solution procedure to treat the jet inflow as a solid wall.

As already mentioned in the previous chapter, a turbulent combustion model including finite chemical production rates was implemented. The inclusion of chemical kinetics was necessary in order to model the autoignition which occur if pressure and temperature are high. An infinite fast chemistry model using the eddy dissipation concept had been included earlier, and in this work the model was changed to include the effects of finite reaction rates. This topic is further discussed in chapter 9.2.

161 Chapter 9 Turbulent gas jet autoignition 9.2 Turbulence models

Turbulence is often referred to as a chaotic three dimensional time dependent flow with eddies of different sizes. The flow exhibit strong diffusive and dissipative characteristics. Work is extracted from the mean flow and generates a cascade of eddies ranging from large eddies with lengthscale comparable to the dimensions of the mean flow, down to the smallest eddies responsible for molecular mixin g. Figure 9.1 illustrates the transfer of energy to smaller and smaller eddies.

If the smallest lengthscales are to be resolved by the computational grid the problem would be impossible to handle for even the largest supercomputer. For this reason mean values are used by the conservation equations, and the effects of turbulence on the mean values are calculated by the use of turbulence models.

This chapter discusses the turbulent flow field model and the turbulent combustion model to some extent. For details about the models it is referred to the references.

TURBULENT ENERGY TRANSFER MODELLING CONCEPT

■0- WORK FROM MEAN FLOW

u',L",W ' HEAT GENERATION &WORK TRANSFER u\L\ue = 2ui* HEAT r-r j. WORK I------1 . ' U". L",w = 2w*-i f==!> HEAT & WORK

UM'.W =>’HEAT

Figure 9.1 Turbulent energy transfer model.

162 Chapter 9 Turbulent gas jet autoignition 9.2.1 Turbulent flow field model

The influence of turbulence on the flow field is assumed to mainl y take place through the larger eddies denoted by the lengthscale L’ in figure 9.1. The istantaneos velocity is expressed by

Uj = tij+u- (9.1)

The velocity denoted by a tilde is the mass averaged velocity, also called the Favre averaged velocity, and is defined by

£Uj Si (9.2) P

Mass fractions, internal energy, and temperature is averaged in a similar way.

The turbulent kinetic energy is defined as the kinetic energy in the velocity fluctuations:

k = — (u^+v* 5 hW 2) (9.3) 2

The influence of turbulence on the flow field is described by the standard k-e model. See for instance Launder and Spalding [76] or Hirsch [40]. The variable k represent the turbulent kinetic energy, and the variable e represent its dissipation rate. The transport equation for k is modelled as:

—(pk)+V*(puk) = V'(p kVk)+(t?-V)-u-pe (9.4) dt

The transport equation for e is in similar fashion expressed as

-|-(pe) +V-(pue) = V-(p eVe) +C£l(if-Y)-u| -C^ (9.5)

The turbulent stress tensor TyR is analogeous to the viscous stress tensor:

x? - 2p T (9.6)

And the total stress tensor becomes

163 Chapter 9 Turbulent gas jet autoignition

(9.7)

The turbulent viscosity is found from dimensional arguments to be

n k2 P? " PC„--- (9.8)

And the eddy diffusivities in the k and e equation are respectively

Pt Pk = P +— (9.9) °k

Pc (9.10)

For a standard k-e model the constants in equations 9.4 to 9.10 are: C„=0.09 Ce1=1.45 0,2=1.92 ak=l a,=1.3 For a gas jet the value 0,2=1.79 is choosen according to Raiszadeh [77] and Rodi

Ideal gas has variable density as pressure, temperature or composition is changed. This will affect the turbulence. In KTVA-II this is effect is modeled by introducing additional terms for velocity dilatation in the k-e model.

The interaction with the flow field is given through the exchange of diffusive terms in the momentum conservation equations with the respectective turbulent terms. That is replacing p with p? and x with xT.

In the energy conservation equation the diffusive terms are replaced by the turbulent terms in a similar manner. The velocity, species massfraction, internal energy and temperature is now interpreted as Favre averaged quantities. This is described in detail in the KTVA-II report [73].

9.2.2 Eddy Dissipation Concept

The turbulent combustion model choosen is the Eddy dissipation Concept (EDC) by Magnussen et al.[79-83]. According to this concept mechanical energy is

164 Chapter 9 Turbulent gas jet autoignition transferred from the mean flow to large turbulent structures and further down to smaller structures. Most of the energy is dissipated into heat in the smallest eddies. The smallest eddies are also responsible for the molecular mixing. The chemical reactions will take place in those isolated small regions where matter are mixed on a molecular level. This regions are called the finestructure. The lenghtscale of the finestructure is denoted by *, and is closesy related to the Kolmogorov lengthscale. Figure 9.2 gives an illustration of how the turbulent combustion is assumed to take place.

As a result of the turbulent energy transfer model by Magnussen [79], the expressions for the lengthscale and the velocity scale in the finestructures will respectively be

= 1.43^ (9.11) eV4

= 1.74(ev)V4 (9.12)

The velocity scale of the larger eddies used by the flow model is found from equation 9.3 when isotropic turbulence is assumed.

u- = (—k)3'2 (9.13) 3

oxygen

oxygen

oxygen

Figure 9.2. An artists impression of turbulent combustion. From ref [84].

165

^ O'. - Chapter 9 Turbulent gas jet autoignition

According to EDC the mass fraction occupied by the finestructure can be expressed by:

l 3 f \ u* ve (9.14) Y = = 9.7 lu-J Ua J

The fins tinctures are localized in fine structure regions between larger eddies. The mass fraction of finestructure regions to total mass is expressed as:

\l/4 (9.15) U'

Figure 9.3 shows a modelling view of how the finstructures are distributed. The finestructures are considered homogeneously distributed in the finestructure regions.

From geometrical considerations the mass transfer between the finestructure and its surroundings per total mass and time unit can be expressed as:

Z \V4 m 23.6 e_ 11.08 Y^ (9.16) k k

Figure 9.3. Modellers view of the distribution of finestructures. The massfraction of one of the reactants is shown. From ref [84].

166 Chapter 9 Turbulent gas jet autoignition

If we divide this expression by the finstructure density y, we get the mass exchange per finestructure mass and time unit.

m* m (9.17) Y*

The inverse of this expression will be a typical residence time of the mass in the finestructure.

1 T (9.18) m*

9.2.2.1 Steady state perfectly stirred reactor

Figure 9.4 shows a perfectly stirred reactor (PSR). In the EDC we assume that the finestructure is a PSR. The mass transfer rate between the reactor and the surroundings for species i in a steady state PSR can be expressed per mass of the reactor as

R* = p'm'(Y° -Yj") (9.19)

Here superscript * denotes finestructure and superscript 0 denotes surroundings. If the chemistry is assumed infini te fast, the reaction will be controlled by this mass transfer rate.

Only a fraction % of the finestructure is heated enough to react. This fraction is modelled as

l+rfu X = (9.20)

Yx +max(Yfu,—) 1+rfu rfu ,

Here the subscript fu denotes fuel, ox denotes oxygen and pr denotes products. The tilde refer to mean values. The variable rfo is the mass of oxygen required to convert one mass unit of fuel to stoichiometric products. In some versions of the EDC [74] also a probability term is used for the probability of fuel and oxygen to

167 Chapter 9 Turbulent gas jet autoignition

Reoctants ^rod Products III ! 1 1 Fine 1 1 structure 1 m* —► m* —► 1 reactor ,

C?u T° 1 1 cr„ t* ■ e° 6* Surrounding fluid

Figure 9.4. Perfectly stirred reactor (PSR). coexist. However this is not used in this text.

The expression for the mean values of the massfractions will then be:

% = Y'%Y;'+(1-Y'%)Y° (9.21)

Since reactions only take place in the part of the finstructures which have high enough temperature, the mean reaction rate may be expressed as

= (9.22) Y*

The division by yx in equation 9.22 is due to the assumption that reactants are concentrated in the finestructure regions during combustion.

Combining eqations 9.19, 9.21 and 9.22 we get the final expression for the mean reaction rate per unit mass

% = - frVX—(Y.-Yi*) (9.23) Y%(l-Y'%)

If we assume infinitely fast reactions, the convention of reactants to products will be equal to the mass transfer rate of the deficient species. That is fuel if the mixture is lean, and oxygen if the mixture is rich. Thus the reaction rate for the fuel will be

168 Chapter 9 Turbulent gas jet autoignition

K = 'rX 0.24) Yx(l"Y*x) where min refers to the minimum value of the massfractions of fuel and oxygen divided by rfu:

(9.25) rfi.

The mean reaction rate for oxygen and products is then determined by the chemical stoichiometry for the reactions in question.

9.2 2.2 Transient perfectly stirred reactor

When finite chemical reaction rates are included, the reactor can no longer be considered in steady state. The mass conservation equation for species i in a transient reactor may be written as

)'^L+p1h'(Y,'-Y°) = to.- (9.26) dt

This is equation is similar to the species conservation (3.2) except for the diffusive terms.

Substituting equation 9.21 and rearranging yields

S = -Ltoi * ----^(Y.* -Yj) (9.27) dt p* (1-y‘x)

Inserting this expression for finstructure mass convertion in the mean reaction rate (eq. 9.22) we get

B; = ^4—---- (Yj* -%)) (9-28) Yx Ip* (i-y* x) J

The finstructure chemical term cof is calculated from a kinetic mechanism, for instance the one given in table 4.1 for methane. Since all the chemical reactions

169 Chapter 9 Turbulent gas jet autoignition are taking place in the reactor, the finestrucure massfractions have to be stored by the computer program until the next time step. Energy conservation has to be applied in the finestructure and the finestructure temperature is deduced from the finstructure enthalpy by an iteration routine similar to equation 4.49. The finestructure density is calculated from the finestructure massfractions and the ideal gas law. Pressure is assumed to be the same for the finestructure as for the surroundings.

The source term in the species conservation equations (3.2) is given by the mean density and the mean reaction rate in equation 9.28.

= pRi (9.29)

Before ignition no products are present. This will make the fraction of finestructure which have high enough temperature to react, % (eq. 9.20), equal to zero. For an ignition simulation this fraction have to be modified until some amount of products is made. In this work % is set equal to the fraction of mean temperature to finestructure temperature until the % calculated by equation 9.20 exceeds the value 0.1. This is ment to reflect that all of the finestructure can react when the temperature in the finestructure is equal to the mean temperature. Then as the temperature starts to rise in the finestructure due to chemical reactions, a finestructure fraction smaller than one are enough heated to react.

If the timescale for the finestructure given by equation 9.18 is shorter than the chemical time scale, the flame will be quenched. The chemical mass convertion will be unable to keep up with the mass transfer through the reactor. The result is extinction of the reactions. A chemical timescale may be calculated from the kin etic mechanism, but this is not straightforward. In this work a crude estimate of the chemical timescale is made from the experiments of Chomiac et al. [85] and calculations of Bradley [86,87] and Martenay [88]. The extinction of a methane mixture at a pressure of one athmosphere occured at a finestructure timescale of approximately lO^s. From the results in chapter 8 it is assumed that the extinction timescale is shorter by a factor of 10"1 when the pressure is raised to 50 bar. At calculations at 50 bars extinction is thus estimated by adjusting % according to:

170 Chapter 9 Turbulent gas jet autoignition

If T* is greater than 0.1 times 10'5

(9.30)

If T* is smaller than 0.1 times 10‘5

Xext = 0 (9.31)

9.2.2.3 Inclusion of the No Transport of Radicals Concept

Since all the chemical reactions are assumed to take place in the finestructure the No Transport of Radicals Concept described in chapter 4.1.4 can be utilzed together with the EDC. The mean reaction rate for the species in group two are calculated in the finestructure by equation 9.22 and used in the mean species conservation equations (3.2). The intermediate species in group one are only present in the finestructure. No mean reaction rates are calculated for these species, and mean species conservation equations are thus not necessary for group one species.

The finestructure conservation equations for group two species will only consist of the chemical rate term.

(9.32)

By applying this concept, mass conservation during chemical reactions will be taken care of in the finestructure. Mean species conservation equations only has to be solved for species in group two. No modifications of these equations are necessary to secure mass consistency.

These equations may be integrated by an ordinary differential equation solver, and are decoupled from spatial gradients. In this work the implicit Euler time integration described in chapter 5.2 has been used. The procedure requires iterations in order to arrive at a converged solution which also is mass conservative. The method is stable, but due to the mass inconsistency which can arise during the iteration procedure, it is probably not an optimum choice of Chapter 9 Turbulent gas jet autoignition solution procedure.

The chemical time scale is much shorter than the flow field timescale. For this reason the chemical routine is sybcycled with shorter timesteps than the mean calulations. This is also a necesity in order to keep mass inconsistency to a minimum when implicit Euler time integration is used.

In the EDC the finestructure is not transported. The same principle applies to species in group one which are calculated only in the finestructure. This makes the two concepts well suitable to be used in combination.

9.3 Simulation of turbulent gas jet autoignition

9.3.1 Boundary and initial conditions

The computation domain is an cylinder with radius 2.5cm and height 10cm. The calculation is symmetric around the axis. The grid is two dimensional with 150 grid points along the axis and 45 grid points in the radial direction. Along the axis the grid points are uniformly distributed. In the radial direction the grid spacing is smaller near the centre where the jet inlet are placed.

Initially the cylinder contains compressed hot air at a temperature of 1600K and a pressure of 50bar. The gas jet is injected at the top centre of the cylinder. The jet contains methane with an inlet temperature of BOOK and inlet pressure of 50bar. The velocity is 400 m/s which is close to sonic conditions (Mach number close to 0.9). The jet boundary conditions are designed to be close to critical nozzle flow with a nozzle diameter of 0.8 mm and an outlet pressure of 50bar.

The cylinder is closed with adiabatic walls. The standard wall functions in KTVA-II are applied, but since the jet does not impinge on the wall, the wall functions will have little influence.

172 Chapter 9 Turbulent gas jet autoignition 9.3.2 Results and discussion

9.3.2.1 Qualitative description of simulation results

Figure 9.6 shows the temperature in the calculation domain at different instants of time. The cold methane jet propagates downwards in the cylinder. The jet has a conical shape with a bulb on the tip. A slight temperature increase may be seen along the edge of the jet halfway between the nozzle and the jet tip after 1.2ms. The temperature increases further, and a combustion zone is developed behind the bulb. This is clearly seen at 2.0ms and 2.4ms. At 2.4ms the temperature has almost reached 2000K. The temperature increase hence starts along the edge and develops to a more intence combustion zone in the wake behind the bulb.

Figure 9.7 shows the mean massfraction of CO. At 0.8ms traces of CO may be found in the shear layer near the nozzle. The turbulence is however so intence in this area, that any major convertion of reactants can not occur. After 1.2 ms the highest CO concentration are found in the wake behind the bulb. At 1.6ms there is a layer of CO along the edge of the spay, reaching from the nozzle down to the wake behind the bulb. The largest massfractions are found in the wake. This tendency is strengthened at 2.0ms and 2.4ms. At 2.4ms an intence combustion zone is established in the wake, but reactions also take place along the jet edge closer to the nozzle.

Until 2.4ms no reaction take place at the side and in front of the bulb jet tip. It may also be noted that due to the high velocity in the centre of the jet, CO is convected downwards on the inside of the reaction zone.

The mean massfraction of the product C02 is shown in figure 9.8. A trace of this product can be seen at 1.2ms. At 1.6ms C02 is produced along the jet edge, mainly in the wake but also some distance downstream. At 2.4ms the massfraction of C02 has reached 0.03 and as shown in figure 9.6 the temperature has raised accordingly.

As discussed in chapter 8 is autoignition not an instantaneous event. The chemical convertion starts immediately after matter is mixed. This means that some chemical convertion can be detected just after the injection of methane starts. The convertion rates are however so small that any signifi cant product generation or

173 Chapter 9 Turbulent gas jet autoignition temperature increase do not occur until the mixing process and the following chemical reactions have generated sufficient amount of intermediates to supply the strongly exothermic reactions. The convertion of CO to C02 is one of the major exothermic reactions.

9 3.2.2 Ignition delay

The definition of ignition delay is not unambigous. A common indicator of experimental ignition delay, is to monitor the pressure rise in a combustion bomb. A3s0y [89] have measured the pressure rise in a combustion bomb when a methane jet is injected trough a heated cylinder into the bomb. His measurements was done with a maximum temperature of 1250K in the heated cylinder. He found that the ignition delay indicated by the pressure rise was around 1.7ms for an air temperature of 1250K.

Figure 9.5 shows the average pressure in the cylindrical calculation domain. As

5.12e+07

5.1e+07

5.086+07

0- 5.066+07

3 5.046+07

5.026+07

5e+07

4.986+07

time [ms]

Figure 9.5. Average pressure in the cylindrical calculation domain as a function of time. Injection starts at Oms.

174 Chapter 9 Turbulent gas jet autoignition may be seen from the figure, gives the simulation a pressure rise after approximately 2.0ms, that is some time after the temperature has started to increase in the combustion zone behind the bulb. The surrounding air temperature is 1600K. Since the ignition delay is decreasing strongly with increasing temperature, the experiments of v$)s0y indicate that the calculation gives a somewhat too slow ignition.

This finding is strengthened by the experiments of Fraser et al [90]. They have measured ignition delay for methane and ethane jets in a combustion bomb. The high temperature environment in the bomb was obtained by burning a premixed hydrogen, ethylene, oxygen and nitrogen mixture prior to the jet injection. They claimed that the products of this premixed combustion was air-like with an oxygen percentage of 21%. However the products also contained 3.1% carbondioxide and 9.2% water. In their paper they assumed that this extra C02 and H 20 acted like diluents and caused no problems of representing hot compressed air. Neither the jet temperature or injection velocity are reported in the paper.

For a bomb pressure of 40 atm. and a temperature of 1600K, they measured an ignition delay of 0.4ms in the case of methane jet fuel. The ignition delay was taken as the time delay from start of injection to a pressure rise in the bomb occured.

Both experimental investigations indicate that the simulation overestimates the ignition delay 2 to 5 times. It should be pointed out that the conditions in the combustion bomb experiments were not directly comparable to the simulation. In the case of /Es0y’s results, the heated cylinder in the experiments had a small radius, and the methane jet interacted with the walls of the cylinder. The shear layer on the heated cylinder did thus play an important role for the autoignition process. The temperature of the injected methane in iEs0y’s experiments was also 300K higher than in the simulation.

In the case of Fraser et al.’s experiments the premixed combustion prior to the injection would leave some amounts of C02, H20 in the combustion chamber. At the high temperatures in question, also some small amount of more reactive species like CO and H will be present. These species may influence the early reactions and cause a more rapid autoignition.

However it is likely that the simulation results give a somewhat too slow ignition.

175 Chapter 9 Turbulent gas jet autoignition

Reasons for this is probably found in the skeletal kinetic mechanism and in the turbulent combustion model.

The limited information on low temperature reactions in the skeletal mechanism has already been discussed in chapter 8.2.3 It is the authors belief that this is the major reason for the slow ignition in the simulations. As mentioned in chapter 8, the use of a kinetic mechanism with stronger emphasis on the low temperature reactions should be considered in later investigations.

Also the turbulent combustion model is subject to give deviations from a real turbulent combustion situation. In particular the modelling of the reactive finestructure fraction, %, should be considered in more detail.

The No Transport of Radicals Concept as well as the EDC depends on that intermediate products are quickly produced and consumed. Quickly here refers to chemical kinetic reactions relative to fluid motion. In a combustion situation this will be the case, but in the ignition phase the concentration growth of intermediates is slow. The transport of intermediates belonging in group one as well as the finestructure itself, is neglected. The effect of radicals being produced close to the jet inlet and transported downstream with the main flow will thus not be taken into account. This will result in that for each grid cell the increase of radical concentration is solely due to chemical reactions in that grid cell. Any contribution from upstream cells which start radical prodution earlier is neglected. This may also be a part of explaining the slow ignition in the simulations.

The qualitative picture of turbulent gas jet autoignition is believed to be well represented by the model. The gradual development of the autoignition process is simulated, and the onset of a jet combustion is obtained. There is however no doubt that further efforts both by experimental and modelling technics are needed in order to gain a better understanding of the autoignition process of a transient turbulent gas jet.

9.S.2.3 Towards combustion of the jet

After 2.4ms the temperature and product massfractions continue to rise, and the process turns into combustion of the jet. This is an important aspect with the simulation model. It is not only an ignition model, but also a combustion model.

176 Chapter 9 Turbulent gas jet autoignition

After 2.9ms the jet tip impinged at the bottom wall boundary, and the jet is no longer a free jet. Figure 9.9 shows the temperature, density, turbulent kinetic energy and dissipation at 2.9ms. The maximum temperature has now reached 2330K in the combustion zone.

The mean massfractions of methane, oxygen and hydrogen molecules at 2.9ms are shown in figure 9.10. Figure 9.11 shows the mean massfractions of hydrogen atoms, carbonmonoxide, carbondioxide and water at 2.9ms. It may be seen from these figures that the chemical conversion from reactants to products has started to spread around the bulb of the jet tip. This is particularly evident from the Hg, CO and H^O massfractions.

50 CPU hours was used on a HP9000/750 computer to reach 2.4ms in simulation time. After 30 more CPU hours the simulation time reached 2.9ms. For temperatures over 2000K the calculation timestep becomes small. For this reason the calculation proceeds slowly at high temperatures. As discussed in chapter 9.2.2.S there is room for improvements in the time integration procedure. The main scope in this work was however to establish and discuss the physical model, and this task has been fulfiled. Figure 9.6. Temperature [K] at different times. eo_3S£ O.QO?

0.00583

O.0046?

0.0035

0.OO233

0.0011?

0

Figure 9.7. Mean massfraction of CO at different times.

CO t=2.4ms times.

different

at

2 i C0 I 1 ■t i

of

t=2.0ms massfraction

Mean

9.8.

Figure t=1.6ms i t=1.2ms

180 Temperature [K]

temp_370 2.33e+03

7'f:

Figure 9.9. Temperature, density, turbulent kinetic energy and turbulent dissipation rate at 2.9ms. CO

\ 2.9ms. at

2 H and

2 0

, t CH of massfractions

Mean

9.10.

Figure

182 Massfraction of H Massfraction of CO Massfraction of C02 Massfraction of HjO

Figure 9.11. Mean massfractions of H, CO, C02 and H20 at 2.9ms. Chapter 10 Conclusions

Chapter 10 Conclusions

The knowledge of fluid flow modelling, chemical reaction kinetics and numerical solution techniques together with the capabilities of modem computer systems, make it possible to simulate ignition and flame development processes with a reasonable degree of accuracy. Such simulations give detailed informations of the physical processes occuring in ignition and combustion events. In scientific research simulations are important tools to explain experimental findings. In addition they have the ability to investigate a wide range of parameter variations with minimal additional costs. Numerical simulations are also capable of being a powerful tool in practical engine development.

The present thesis has demonstrated that a skeletal chemical kinetic mechanism for methane combustion is able to model both spark ignition and autoignition. In the case of spark ignition a simple one dimensional laminar spherical symmetric model reproduce experimental combustion bomb results with a high degree of accuracy. The effects on flame kernel development from air/fuel equivalence ratio, temperature, pressure and heat losses to spark electrodes and surrounding gas, are taken care of by the model. It has been shown that the early flame kernel is vulnerable to heat losses to the spark electrodes and surrounding gas. The spark energy provided by the electrode discharge must be sufficient to create an initial flame kernel and expand the kernel away from the electrodes. When the flame kernel has reached larger radii the flame propagation speed approaches the planar flame propagation speed, which depends on temperature, pressure and air/fuel equivalence ratio. Addition of hydrogen to the premixed methane-air mixture will enhance the flame propagation speed.

Laminar autoignition simulations of a methane bubble surrounded by hot air shows that this process depends on both diffusion and chemical kinetic processes. Different combustion modes will be the result when air or methane temperature,

184 Chapter 10 Conclusions pressure and buble size is changed. The ignition delay is greater than for experimental premixed values. This is partly explained by the mixing process between the methane bubble and the surrounding air. For autoignition simulations it is believed that the limited information on low temperature reactions in the skeletal mechanism is responsible for longer simulated ignition delay than what is found in experimental investigations. It is recommended that a kinetic mechanism with more emphasis on low temperature reactions should be used in future autoignition simulations.

Autoignition of a transient turbulent methane jet in hot compressed air is performed using the skeletal mechanism and Eddy Dissipation Consept for turbulent combustion modelling. The computations were done with the KIVA-II computer code. The simulation shows the gradual development of autoignition. Traces of reaction activity are found in the highly turbulent shear layer near the injection nozzle. Extinction processes due to high turbulence levels will however limit the reaction activity close to the nozzle. An intense combustion zone is created in the wake behind the bulb of the methane jet tip. From this zone the flame spreads around the bulb and results in combustion of the jet. The ignition delay is somewhat longer that indicated by experimental results. The discussion of this disagreement concludes that the skeletal mechanism is the most probable source of the disagreement. However, more research, both experimental and by simulations, is needed to gain a better understanding of the jet autoignition process.

A major task in this work has been to reduce the computational effort when chemical reaction kinetics is included in complex flow simulations. The No Transport of Radicals Consept in this thesis shows promising results. This concept only include the main species and the highly diffusive intermediate species in the flow field species conservation equations. The other intermediate species are not transported, and their concentrations are expressed by ordinary differential equations in terms of time. The concept reduces the computational effort while retaining the essence of physical relations. The consept is particularly suitable when combined with eddy breakup turbulent combustion models like the Eddy Dissipation Concept.

185 References

References

1 Lewis, B. and Von Elbe, G.: Combustion, flames and explosion of gases. Third edition, Academic Press Inc., 1987.

2 Karlovitz, B., Denniston, D.W., Knapschaefer, D.H. and Wells, F.E.: 4. Symposium (International) on Combustion, p 613, 1953.

3 Williams, FA.: Combustion Theory. Chap. 9 and 10, Benjamin-Cummins, Palo Alto, 1985.

4 Ask, T.0.: Ignition and flame growth in lean gas-air mixtures. Dr.Ing. thesis, The Norwegian Institute of Technology, 1992.

5 Law, C.K: Dynamics of stretched flames. 22. symposium (International) on combustion, 1988.

6 Kanuri A.M.: Introduction to combustion phenomena. Gordon and Breach Science Publishers, 1975.

7 Bj0rge, T.: Spark ignition of lean mixtures under engine like conditions. Dr.Ing. thesis, The Norwegian Institute of Technology, 1988.

8 Ballal, D.R, and Lefebvre, A.H.: The influence of spark discharge characteristics on minimum ignition energy in flowing gases. Combustion and flame, vol. 24, p. 99, 1975.

9 Ballal, D.R, and Lefebvre, A.H.: Combustion and flame, vol. 35, p. 155, 1979.

10 Ballal, D.R, and Lefebvre, A.H.: 15. symposium (International) on combustion, p. 1473, 1974.

186 References

11 Maly, R. and Vogel, M.: Initiation and propagation of flame fronts in lean CH4-air mixtures by the three modes of the ignition spark. 17. symposium (International) on combustion, p. 821,1978.

12 De Soethe, G. G.: Propagation behaviour of spark ignited flames in early stages. International conference on combustion in engineering, paper C 59/83, 1 Mech E Conference publications 1983-3, p. 93, 1983.

13 Ziegler, G., Wagner, E. and Maly, R.: Ignition of lean methane-air mixtures by high pressure glow and arc discharges. 28. symposium (International) on combustion, p. 1817, 1984.

14 Pishinger, S., Heywood, J.B.: How heat losses to the spark plug electrodes affect flame kernel develoment in an Si-engine. SAE paper 900021, 1990.

15 Pishinger, S., Heywood, J.B.: A model for flame kernel development in a spark-ignition engine. 23. symposium (International) on combustion, p. 1033, 1990.

16 Maly, R.: Ingnition model for spark discharges and the early phase of flame front growth. 18. symposium (International) on combustion, p. 1747, 1981.

17 Adelman, H. C.: A time dependent theory of spark ignition. 18. symposium (International) on combustion, p. 1333, 1981.

18 Champion, M., Deshaiel, B, Joulin, G. and Kinoshita, K: Spherical flame initiation: Theory versus experiments for lean propane-air mixtures. Combustion and flame 65, p 319, 1986.

19 Wu, Y., Huang, Z. and Xiong T.: The numerical solution of governing equations for ignition by a sphere of burned gas or spark. 20. symposium (International) on combustion, p. 151, 1984.

20 Kono, M., Niu, K., Tsukamoto T. and Ujiie, Y.: Mechanism of flame kernel formation produced by short duration sparks. 22. symposium (International) on Combustion, p 1643, 1988.

187 References

21 Ujiie, Y., Niu, K, Tsukamoto, T. and Kono, M.: Development of flame kernels produced by capacitance sparks in ignition process of combustible mixtures. JSAE Review Vol.10, No.4,1989.

22 Ishii, K, Tsukamoto, T., Ujiie, Y. and Kono, M.: Analysis of ignition mechanism of combustible mixtures by composite sparks. Combustion and flame 91, p. 153, 1992.

23 Sher, E. and Refael, S.: Numerical analysis of the early phase development of spark-ignited flames in CH4-air mixture. 19. symposium (International) on combustion, p. 251, 1982.

24 Seery and Bowman. Combustion and flame 14, p. 37, 1970.

25 Semenov, N.N.: Chemical kinetics and chain reactions. Oxford University Press, London, 1935.

26 Frank-Kamenetski: Diffusion and heat exchange in chemical kinetics. Translated from russian by N. Thon. Princeton University Press, Princeton, 1955.

27 Strehlow, R.A.: Combustion fundamentals. McGraw-Hill, 1984.

28 Halstead, M.P., Kirsch, L.J., Prothero, A. and Quinn, C.P.: A mathematical model for hydrocarbon autoignition at high pressures. Proc. Roy. Soc. Lond. A 346, p. 515,1975.

29 Halstead, M.P., Kirsch, L.J. and Quinn, C.P.: The autoignition of hydrocarbon fuels at high temperatures and pressures - Fitting of a mathematical model. Combustion and flame 30, p. 45, 1977.

30 Cox, RA. and Cole, J.A.: Chemical aspects of the autoignition of hydro carbon-air mixtures. Combustion and flame 60, p. 109, 1985.

188 References

31 Theobald, M.A. and Cheng, W.K.: A numerical study of diesel ignition. ASME, 87-FE-2, 1987.

32 Amsden, A., Ramshaw, J., O’Rourke, P. and Dukowicz, J.: KIVA: A computer program for two- and three-dimensional fluid flows with chemical reactions and fuel sprays. Los Alamos National Laboratory Report LA-10245-MS, 1985.

33 Amsden, A., Ramshaw, J., Cloutman, L. and O’Rourke, P.: Improvements and extensions to the KIVA computer program. Los Alamos National Laboratory Report LA-10534-MS, 1985.

34 Lutz, A.E., Kee, R.J., Miller, JA., Dwyer, HA., Oppenheim, A.K: Dynamic effects of autignition centers for hydrogen and C12-hydrocarbon fuels. 22. symposium (International) on combustion, p. 1683, 1988.

35 Goyal, G., Wamatz, J. and Maas, U.: Numerical studies of hot spot ignition in H2-02 and CH4-air mixtures. 23. symposium (International) on combustion, p. 1767, 1990.

36 Boston, P.M., Bradley, D., Lung, F.K-K, Vince, I.M., Weinberg, F.J.: Flame initiation in lean, quiescent and turbulent mixtures with various igniters. 20. symposium (International) on combustion, p. 141, 1984.

37 Orrin, J.E., Vince, I.M. and Weinberg, F.J.: A study of plasma jet ignition mechanisms. 18. symposium (International) on combustion, p. 1755,1981.

38 Lee, J.H.S., Knystautas, R. and Yoshikawa, N.: Acta Astronautica, 5, p. 971, 1978.

39 Kuo, K.K.: Principles of combustion. John Wiley & Sons, 1986.

40 Hirsch, C.: Numerical computation of internal and external flows. Vol. 1 and 2., John Wiley & Sons, 1990.

189 References

41 Smooke, M.D.: Reduced kinetic mechanisms and asymptotic approximations for methane-air flames. Springer-Verlag, 1991.

42 Smooke, M.D. and Giovangigli, V.: Formulation of the premixed and nonpremixed test problems. Contained in ref. 41.

43 Smooke, M.D. and Giovangigli, V.: Premixed and nonpremixed test problem results Contained in ref. 41.

44 Peters, N.: Reducing mechanisms. Contained in ref. 41.

45 Williams , F.A.: Overview of asymptotics for methane flames. Contained in ref. 41.

46 Bilger, R.W., Esler, M.B. and Stamer, S.H.: On reduced mechanisms for methane air combustion. Contained in ref. 41.

47 Seshadri, K. and Gottgens, J.: Structure of the oxidation layer for stoichiometric and lean methane-air flames. Contained in ref. 41.

48 Chelliah, H.K., Trevino, C. and Williams, FA: Asymptotic analysis of methane-air diffusion flames. Contained in ref. 41.

49 Rogg, B.: Sensitivity analysis of laminar premixed CH4-air flames using full and reduced kinetic mechanisms. Contained in ref. 41.

50 Chen J.-Y. and Dibble, R.W.: Application of reduced chemical mechanisms for prediction of turbulent nonpremixed methane jet flames. Contained in ref. 41.

190 References

51 Lam, S.H. and Goussis, D.A.: Conventional asymptotics and computational singular perturbation for simplified kinetics modelling. Contained in ref. 41.

52 JANAF Thermochemical Tables.

53 Gardiner, W.C. (editor), Burcat, A., Dixon-Lewis, G. Frenklach, M., Gardiner, W.C., Hanson, R.K, Saliman, S., Troe, J., Wamatz, J. and Zellner, R.: Combustion chemistry. Springer-Verlag, 1984.

54 Dixon-Lewis, G.: Computer modelling of combustion reactions in flowing systems with transport. Contained in ref. 53.

55 Lee, R.J., Dixon-Lewis, G., Wamatz, J., Coltrin, M.E. and Miller, J.A.: A fortran computer code package for the evaluation of gas-phase multicomponent transport properties. SANDIA report, SAND86-8256, UC-401, 1990.

56 Chapman, S. and Cowling, T.G.: The mathematical theory of non-uniform gases, 3rd ed. The University Press, Cambidge, 1970.

57 Hirschfelder, J.O., Curtiss, C.F. and Bird, R.B.: Molecular theory of gases and liquids. John Wiley and Sons, New York, 1954.

58 Woshni, G.: A universally applicable equation for the instantaneous heat transfer coefficient in the internal combustion engine. SAE paper 670931, SAE Trans., Vo. 76, 1967.

59 Shiling, K. and Woschni, G.: Experimental investigation of the instantaneous heat transfer in the cylinder of a high speed diesel engine. SAE paper 790833,1979.

60 Ko, Y. and Anderson, R.W.: Electrode heat transfer during spark ignition. SAE paper 892083, 1989.

191 References

61 Kreith, F. and Black, W.Z.: Basic heat transfer. Harper & Row Publishers, New York, 1980.

62 Arpaci, V.S.: Conduction heat transfer. Addison-Wesley, Reading, MA, 1966.

63 Carslaw, H.S and Jaeger, J.C.: Conduction of heat in solids. Clarendon Press, Oxford, 1959.

64 Hirsch, C.: Numerical computation of internal and external flows. Vol 1 and 2. John Wiley & Sons, Chichester, 1988.

65 Patankar, S.V.: Numerical heat transfer and fluid flow. McGraw Hill, New York, 1979.

66 Cheney, W. and Kincaid, D.: Numerical mathematics and computing. Brooks / Cole Publishing Company, Monterey, California, 1985.

67 Markstein, G.H.: Non-steady flame propagation. Macmillan, New York, 1964.

68 Sivashinsky, G.I.: Instabilities pattern formation and turbulence in flames. Ann. Rev. Fluid Mech., 15 p. 179-199, 1983.

69 Wamatz, J.: The mechanism of high temperature combustion of propane and butane. Comb. Sci. and Tech., Vol. 34 p. 177-200, 1983.

70 Lifshitz, A., Scheller, K, Burcat, A. and Skinner, G.B.: Combustion and Flame 16, p 311, 1971.

71 Tsuboi, T. and Wagner, H.: 15. Symposium (Int.) on Combustion, p. 883, 1974.

72 Eubank, C.S., Rabinowitz, W.C., Gardiner, W.C. and Zellner, R.E.: Shock-initiated ignition of natural gas-air mixtures. 18. Symposium (Int.) on Combustion, p.1767, 1981.

192 References

73 Amsden, AA., Rourke, P.J.O., and Butler, T.D.: KTVA-IIA computer program for chemically reactive flows with sprays. Los Alamos National Laboratory Report LA-11560-MS, 1989.

74 Lilleheie, N.I., Byggstoyl, S. and Magnussen, B.F.: Modeling and chemical reactions. Evaluation of reduced chemical mechanisms, PSR extinction limits and turbulent flame calculation. Nordic Gas-technology Centre Report, Copenhagen, 1991.

75 Lilleheie, N.I.: Implementation of turbulent combustion modification to KIVA-II. MAR3NTEK Report MT22-A92-0022, Trondheim, 1992.

76 Launder, B.E. and Spalding, B.: Mathematical models of turbulence. Academic Press, New York, 1972.

77 Raiszadeh, F. and Dwyer, H.A.: A study with sensitivity analsis of the k-e turbulence model applied to jet flows. AIAA paper 83-0285 AIAA 21st Aerospace Meeting, 1983.

78 Rodi,W: Turbulence models and their application in hydraulics. Delft, Netherlands. International Association for Hydraulics Research, 1980.

79 Magnussen, B.F. and Hjertager, B.H.: On mathematical modeling of turbulent combustion with specieal emphasis on soot formation and combustion. 16. Symposium (Int.) on Combustion, 1976.

80 Magnussen, B.F., Hjertager, B.H., Olsen, J.G. and Bhaduri, D.: Effects of turbulent structure and local concentration on soot formation and combustion in CH-diffusion flames. 17. Symposium (Int.) on Combustion, 1978.

81 Magnussen, B.F., Magnussen, P. and Lilleheie. N.I.: Eddy dissipation concept simulation of a hydrogen-air diffusion flame. Proceedings eight task leader meeting TEA, Heidelberg, 1986.

193 References

82 Byggst0yl, S. and Magnussen.B.F.: A model for flame extinction in turbulent flow. Turbulent shear flows 4. Selected papers from 4th International symposium on turbulent shear flows. Springer Verlag, 1985.

83 Lilleheie, N.I.; Byggstpyl, S. and Magnussen, B.F.: Strategy for inclusion of chemical kinetics into the eddy dissipation concept. DBA conference paper, 1987.

84 Grimsmo, B.: Numerical simulation of turbulent flow and combustion in a four stroke homogeneous charge internal combustion engine. Ph.D.-thesis 1991:83, University of Trondheim, 1991.

85 Chomiac, J. and Jarosinski, J.: Flame quenching by turbulence. Comb. Flame 48, p. 241, 1982.

86 Bradley, D. and Abdel-Gayed, R.G.: A two eddy theory of premixed turbulent flame propagation. Proc. Roy. Soc. A 1457, 1981.

87 Bradley, D., Chin, S.B., Draper, M.S. and Hankinson, G.: 16. Symposium (Int.) on Combustion, p. 1571, 1976.

88 Martenay, P.: Analytical study of the formation of nitrogen oxides in hydrogen air combustion. Comb. Sci. Tech. 1, p. 461, 1970.

89 IEs0y, V.: Glpdeflate tenning av transient gass-jet. (Hot surface ignition of transient gas jet). Project work for the MSc.-degree. Inst, for Marine Engineering, Univ. of Trondheim, NTH, 1989.

90 Fraser, RA.., Edwards, C.F. and Siebers, D.L.: The autoignition of methane and natural gas in a simulated diesel environment. 2. joint technical meeting of the Canadian and Western State Sections of the Combustion Institute, 1990.

194