Wall Related Lean Premixed Combustion Modeled with Complex Chemistry

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Chemical Reaction Engineering

Department of Chemical Engineering and Technology Royal Institute of Technology

TRITA-KET R 164 ISSN: 1104-3466 ISBN: 91-7283-391-2 ISRN KTH/KET/R-164-SE

Wall-related Lean Premixed Combustion

Modeled with Complex Chemistry

Johan Andrae

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Abstract

Increased knowledge into the physics and chemistry controlling emissions from flame-surface interactions should help in the design of combustion engines featuring improved fuel economy and reduced emissions.

The overall aim of this work has been to obtain a fundamental understanding of wall-related, premixed combustion using numerical modeling with detailed chemical kinetics. This work has utilized CHEMKIN, one of the leading software packages for modeling combustion kinetics.

The simple fuels hydrogen and methane as well as the more complex fuels propane and gasified biomass have been used in the model. The main emphasis has been on lean combustion, and the principal flow field studied is a laminar boundary layer flow in two-dimensional channels. The assumption has been made that the wall effects may at least in principle be the same for laminar and turbulent flames.

Different flame geometries have been investigated, including for example autoignition flames (Papers I and II) and premixed flame fronts propagating toward a wall (Papers III and IV). Analysis of the results has shown that the wall effects arising due to the surface chemistry are strongly affected by changes in flame geometry. When a wall material promoting catalytic combustion (Pt) is used, the homogeneous reactions in the boundary layer are inhibited (Papers I, II and IV). This is explained by a process whereby water produced by catalytic combustion increases the rate of the third-body recombination reaction: H+O2+M ⇔ HO2+M. In addition, the water produced at higher pressures increases the rate of the 2CH3(+M) ⇔ C2H6(+M) reaction, giving rise to increased unburned hydrocarbon emissions (Paper IV).

The thermal coupling between the flame and the wall (the heat transfer and development of the boundary layers) is significant in lean combustion. This leads to a slower oxidation rate of the fuel than of the intermediate hydrocarbons (Paper III).

Finally in Paper V, a well-known problem in the combustion of gasified biomass has been addressed, being the formation of fuel-NOx due to the presence of NH3 in the biogas. A hybrid catalytic gas-turbine combustor has been designed, which can significantly reduce fuel-NOx formation.

Keywords: wall effects, premixed flames, flame quenching, numerical modeling,

CHEMKIN, boundary layer approximation, gasified biomass, fuel-NOx, hybrid catalytic combustor.

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Sammanfattning

En ökad förståelse av de fysiska och kemiska processer som kontrollerar emissioner som uppstår vid väggnära förbränning kan bidra i utformningen av förbränningsmotorer med bättre bränsleekonomi och minskade utsläpp.

Det huvudsakliga syftet med detta arbete har varit att få en grundläggande förståelse för väggnära förbränning med hjälp av numerisk modellering med komplex kemi. Arbetsverktyg har varit CHEMKIN, som är en av de ledande komersiella programvarorna för modellering av kemiskt reagerande flöden.

Modellbränslen har varit relativt enkla gaser i form av vätgas och metan och mera komplexa bränslen i form av propan och förgasad biomassa. Tyngdpunkten i beräkningana ligger på mager förbränning och den huvudsakliga flödesgeometrin som studeras är en tvådimensionell laminär gränskiktsströmning. Här görs antagandet att den väggnära förbränningen sker med försumbar inverkan av turbulent strömning.

Olika typer av flamgeometrier behandlas i form av ’auto-ignition’ flammor (artikel I och II) och förblandade flammor som propagerar mot en vägg (artikel III och IV). När resultaten analyseras framgår det att de väggeffekter som ytkemin ger upphov till starkt påverkas av ändring i flamgeometri. Då ett väggmaterial som ger upphov till katalytisk förbränning (Pt) används, retarderas den homogena förbränningen i gränsskiktet (artikel I, II och IV). Detta förklaras av att det vatten som produceras genom den katalytiska förbränningen ökar hastigheten på tredjekropps-reaktionen H+O2+M ⇔ HO2+M. Vid högre tryck ökar det katalytiskt bildade vattnet också hastigetheten på reaktionen 2CH3(+M) ⇔ C2H6(+M), vilket ger upphov till ökade utsläpp av kolväten (artikel IV).

Den termiska kopplingen mellan flamman och väggen, d v s värmeöverföring och utvecklingen av gränskikten är betydande vid mager förbränning. Detta leder till en långsammare oxidation av bränslet jämfört med intermediära kolväten (artikel III). Slutligen i artikel V behandlas ett välkänt problem vid förbränning av förgasad biomassa: bränsle-NOx bildningen p g a den ammoniak som finns i bränngasen. En katalytisk hybridbrännkammere för tillämpning i gasturbiner designas där en betydande minskning av bränsle-NOx uppnås.

Nyckelord: väggeffekter, förblandade flammor, flamutsläckning, numerisk

modellering, CHEMKIN, gränsskiktströmning, biogas, bränsle- NOx, katalytisk hybridbrännkammare.

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The work presented in this thesis is based on the following papers, referred to by their Roman numerals. The papers are appended at the end of the thesis.

I. Andrae, J., and P. Björnbom, “Wall Effects of Laminar Hydrogen Flames over Platinum and Inert Surfaces,” AIChE J., 46, 1454 (2000).

II. Andrae, J., P. Björnbom, and L. Edsberg, ”Numerical Studies of Wall Effects with Laminar Methane Flames,” Combust. Flame, 128, 165 (2002).

III. Andrae, J., P. Björnbom, L. Edsberg, and L-E. Eriksson, ”A Numerical Study of Sidewall Quenching with Propane/Air Flames,” Proc. Combust. Inst., 29, to appear (2002).

IV. Andrae, J., P. Björnbom, L. Edsberg, and L-E. Eriksson, ”Kinetic and Transport Effects of Pressurized Methane Flames in a Boundary Layer,” submitted to Combust. Flame (2002).

V. Andrae, J., P. Björnbom, and P. Glarborg, ”A Design Concept to Reduce Fuel NOx in Catalytic Combustion of Gasified Biomass,” submitted to AIChE J. (2002).

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Below are listed works presented at conferences and symposia not included in this thesis.

1. Andrae, J., and P. Björnbom, ”Numerical Investigation of Wall Effects for Laminar Hydrogen Flames over Platinum and Inert Surfaces,” Oral Presentation, AIChE Annual Meeting, Dallas, Texas, USA, Oct. 31-Nov. 5 (1999).

2. Andrae, J., P. Björnbom, and L. Edsberg, ”Numerical Studies of Laminar Methane Flames in a Boundary Layer,” Work in Progress Poster,

Twenty-Eighth International Symposium on Combustion, Edinburgh, Scotland, July

30-Aug. 4 (2000).

3. Björnbom, P., J. Andrae, L. Edsberg, and D. Papadias, ”Modeling the Kinetics of Catalytic Monolith Reactors,” Invited Plenar Lecture, Proceedings of the

Ninth International Symposium on Heterogeneous Catalysis, Eds. L. Petrov,

Ch. Bonev, and G. Kadinov, Varna, Bulgaria, Sept. 23-27, p. 3 (2000).

4. Andrae, J., P. Björnbom, and L. Edsberg, ”Modeling Wall Effects of Laminar Methane Flames in a Boundary Layer,” Poster Presentation, AIChE Annual

Meeting, Los Angeles, California, USA, Nov. 12-17 (2000).

5. Andrae, J., P. Björnbom, and L. Edsberg, ”Numerical Studies of Wall Effects of Laminar Methane Flames,” Proceedings of the First Biennial Meeting of the

Scandinavian-Nordic Section of the Combustion Institute, Gothenburg,

Sweden, April 18-20, p. 199 (2001).

6. Andrae, J., P. Björnbom, and P. Glarborg, ”Reduction of Fuel NOx in Catalytic Combustion of Gasified Biomass,” Work in Progress Poster, Twenty-Ninth

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To my family

“There is no substitute for hard work.”

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Nomenclature (cgs units)

i

A

Pre-exponential factor in the rate constant of the ith reaction, unit depends on reaction.

p

c

Mixture specific heat capacity, erg/(g · K).

k p

c

Specific heat capacity of species k, erg/(g · K).

kj

D

Multicomponent diffusion coefficient, cm2/s.

D

_k_,_m Mixture diffusion coefficient, cm2_/s.

T k

D

Thermal diffusion coefficient, g/(cm · s).

i

E

Activation energy in the rate constant of the ith reaction, cal/mole.

g

Acceleration of gravity, cm/s2.

o k

H

Standard state molar entropy of the kth species, erg/mole.

k

h

Specific enthalpy, erg/g.

I

Total number of reactions.

y k

j

Mass flux in the y direction, g/(cm2 · s).

fi

k

Forward rate constant of the ith reaction, unit depends on reaction.

ri

k

Reverse rate constant of the ith reaction, unit depends on reaction.

i c

K

Equilibrium constant in concentration units for the ith reaction, unit depends on reaction.

i p

K

Equilibrium constant in pressure units for the ith reaction, unit depends on reaction.

K

Total number of species.

g

K

Total number of gas-phase species.

s

K

Total number of surface species.

f s

K

Number of first surface species.

l s

K

Number of last surface species.

[

M

]

Concentration of third body, mole/cm3.

p

Pressure, dyne/cm2.

atm

P

Pressure of one standard atmosphere, dyne/cm2.

R

Universal gas constant, erg/(mole · K).

c

R

Universal gas constant in the same units as activation energy, cal/(mole ·K).

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k

s&

Chemical production rate of species by surface reaction,

mole/(cm2 ·s).

o k

S

Standard state molar entropy of the kth species, erg/(mole·K).

T

Temperature, K.

Temperature of the unburned gas, K.

u

T

Temperature at the wall, K.

w

T

Fluid velocity in the x direction, cm/s.

u

Fluid velocity in the y direction, cm/s.

v

x k

V

Diffusion velocity of species k in the axial direction, cm/s.

y k

V

Diffusion velocity of species k normal to wall, cm/s. Molecular mass of species , g/mole.

k

W

k

W

Mean molecular weight of the mixture, g/mole.

x

Distance along principal flow direction, cm.

k

X

Mole fraction of species k.

y

Cross-stream coordinate, cm.

k

Y

Gas-phase species mass fractions.

k

X

Gas-phase species mole fraction.

[ ]

X

k Molar concentration species k, mole/cm

3_.

k

Z

Site fraction of species k.

Greek letters ki

α

Enhanced third-body efficiency of the kth species in the ith reaction.

i

β

Temperature exponent in the rate constant of the ith reaction.

i

γ

Sticking coefficient of the ith surface reaction.

λ

Thermal conductivity, erg/(cm·sec·K).

k

σ

Number of sites occupied by species k.

ki

ε

Coverage parameter, cal/(mole · K).

ki

η

Coverage parameter.

µ

Viscosity, g/(cm·s). ki

µ

Coverage parameter.

ρ

Mass density, g/cm3. o

Γ

Standard-state surface site density, mole/cm2.

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'

ki

υ

Stoichiometric coefficient of the kth reactant species in the ith reaction.

''

ki

υ

Stoichiometric coefficient of the kth product species in the ith reaction.

k

ω

_&

Chemical production rate of species k by gas-phase reaction, mole/(cm3 · s).

ψ

Stream function, g/(cm · s). 1 erg = 10-7 J

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1 Introduction

Combustion of fossil fuels is the most important method of generating energy in the world today. Other sources of energy such as solar, wind and nuclear energy still account for less than 20% of total energy consumption. Fossil fuel combustion is used for transportation, power generation, heating and process engineering. The combustion of fossil fuels is expected to remain a key technology in the foreseeable future, while at the same time world energy production is projected to increase by 60% over the next 20 years (EIA, 2000). It is widely known that combustion not only generates heat, which can be converted into power, but also produces pollutants such as oxides of nitrogen (NOx), soot, and unburned hydrocarbons (UHC). Increasingly stringent regulations are forcing manufacturers of automobiles and power plants to reduce polluting emissions, thereby reducing their negative impact on the environment. In addition, unavoidable emissions of CO2 are believed to contribute to global warming. Improving the efficiency of the combustion process and thereby increasing fuel economy will reduce these emissions. The use of renewable fuels (for example gasified biomass) will eliminate the net emission of CO2 into the atmosphere, but have other negative environmental impacts that must be addressed, such as the formation of fuel-NOx. Improvements in efficiency and decreases in pollutant formation in combustion systems are only possible through improving our understanding of the fundamental processes taking place during combustion.

The interaction between combustible gases and chamber walls is one of the classic problems in combustion science (see von Kàrmàn & Millàn, 1953; Kurkov & Mirsky, 1969; Ferguson & Keck, 1977; Westbrook Adamczyk & Lavoie, 1981; Hocks, Peters & Adomeit, 1981; Blint & Bechtel, 1982; Wichman & Bruneaux, 1995). Combustion is strongly influenced by the presence of walls, which can for example cause flame fronts to quench. The effects of cool walls on unburned hydrocarbon emissions in combustion engines has been studied for quite some time (see Daniel, 1956; Sloane & Schoene, 1983; Kiehne, Matthews & Wilson, 1986; Vlachos, Schmidt & Aris, 1994b; Hasse, Bolling, Peters & Dwyer, 2000). Moreover, the flame has a significant effect on the flow in the vicinity of the wall as well as on the heat flux to the wall (Vosen, Greif & Westbrook, 1984; Lu, Ezekoye, Greif & Sawyer, 1990; Ezekoye, Greif & Lee, 1992; Ezekoye & Greif, 1993; Popp & Baum, 1997).

1.1 Lean combustion

To reduce the fuel consumption in cars and thereby the emission of CO2, there is a growing trend toward the increased use of lean combustion. However, the use of excess air lowers the flame temperature, which increases the significance of the effects of the combustor walls, for instance those of thermal quenching (Vlachos, Schmidt & Aris, 1993; Vlachos, Schmidt & Aris, 1994a).

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In a modern direct-injection spark-ignition (DISI) engine, the charge is stratified near the spark plug, such that the flame initiated at the spark propagates through a partially premixed heterogeneous mixture. This means that the charge is diluted in other parts of the cylinder for an overall lean combustion. In the direct vicinity of the cold wall flame, propagation is quenched due to radical depletion—radicals are required to sustain the flame—in the cold near-wall zone by recombination reactions and by heat losses. The radicals can also diffuse to the wall and be heterogeneously quenched. The result is that some of the charge of fuel and oxidizer in each combustion cycle is left unburned at the walls. This charge may be either transported from the viscous sublayer to the main flow of the flame gases and be oxidized, or discharged into the exhaust gases as residual hydrocarbon emissions.

An additional problem is that the treatment of the exhaust gas by catalytic conversion does not work well because the molecules in the exhaust gas do not "add up" as they do in the three-way-catalyst concept in traditional engines working with stoichiometric fuel/air mixtures. Due to the air-excess, the oxygen consumes the reduction matter—normally unburned hydrocarbons from the main flow—and there is not enough to reduce all the NOx. Lean-burn engines therefore have difficulties matching the output levels of three-way-catalyst engines, especially with regard to NOx. The challenge for the automotive engineer is therefore how to modify the combustion process where possible to find a catalyst that can selectively reduce NOx to N2 in the presence of some reduction matter (Heck & Farrauto, 1995).

Another engine concept featuring lean combustion is homogenous-charge-compression-ignition, a hybrid between the spark-ignition engine and the diesel engine. The charge is very lean (2 < λ < 7 where λ = 1/φ = [Xair/Xfuel/[Xair/Xfuel]stoichiometric is the excess air ratio), which is too lean for spark ignition. Instead there is a spontaneous autoignition due to compression. The lean and homogeneous charge keeps the combustion temperature low, which leads to extremely low levels of NOx and particulate emissions. On the other hand, the CO and hydrocarbon emissions can be significant (Christensen, Johansson & Einewall, 1997; Christensen, Hultqvist & Johansson, 1999).

The effect of the combustion chamber wall is also an important issue in lean-burn gas-turbines used for energy production, for example the lean-burnout of CO in a cooled boundary layer flow (Correa, 1992). In contrast to the spark-ignition engine, the wall temperature is several hundred degrees higher and the process is stationary. The wall effects in such combustors can therefore be expected to differ from those in a lean-burn spark-ignition engine (the effects of surface chemistry or adsorption are for example not completely understood).

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1.2 Scope of this work

Due to limitations in our current understanding, it is important to gain deeper insight into the physics and chemistry controlling emissions resulting from flame-surface interactions. This increased knowledge should help in the design of combustion engines featuring improved fuel economy and reduced emissions. In this context, mathematical simulations can play an important role in accelerating the understanding of near-wall combustion. This is due to the complex nature of the wall effects in combustion, including for example: compressible fluid flow coupled to convection, diffusion and chemical reactions both in the gas-phase and on the surface (see Fig. 1.1). In addition, it is difficult to perform measurements close to walls.

Transport of momentum, heat and species Stefan velocity Adsorption, surface reactions, desorption Gas-phase reactions Diffusion, heat conduction Gas Wall

Figure 1.1 Different processes that have to be modeled to understand near-wall

combustion.

The overall aim of this work has been to obtain a fundamental understanding of wall-related premixed combustion by numerical modeling with detailed chemical kinetics. The work utilized CHEMKIN, a leading software package for modeling combustion kinetics. The model used the simple fuels hydrogen and methane, and the more complex fuels propane and gasified biomass. Emphasis has been on lean combustion and the principal flow field is a laminar boundary layer flow in two-dimensional channels. The assumption has been made that the wall effects may at least in principle be the same for laminar and turbulent flows. Using laminar models instead of turbulent ones for the gas-phase transport-equations strongly reduces the time needed for a single computation, and at the same time enables the use of very detailed chemical reaction mechanisms. There is usually a trade-off between the use of complex chemistry and

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turbulence models (see Fig. 1.2). This is mainly due to the fact that resolution of the small scales in turbulent flows demands far more grid points than does the analogues laminar flow. Furthermore, unlike laminar flow solutions, the Navier-Stokes solution to turbulent flows is itself time-dependent, and in any case, there is not a steady solution (Warnatz, Maas & Dibble, 1996).

Chemistry complexity 3D-CFD simulation

Flow model complexity

2D laminar model

Figure 1.2 Illustration of the usual trade-off between the use of complex chemistry and

flow model complexity.

The first part of this thesis treats the basic formalism of gas-phase and surface kinetics, using the combustion of hydrogen as an example. The modeling of near-wall combustion is then described together with the mathematical treatment of a chemically-reacting boundary layer flow. Finally the software package CHEMKIN®, used to accomplish a numerical solution for the reacting flow model is described, which concludes this part of the thesis.

The second part of the thesis deals with autoignition flames. Uniform reactor inlet conditions are specified, with temperatures above the autoignition temperature for the specified stoichiometry. Premixed hydrogen-air flames have been modeled in Paper

I, and the fuel/air ratio, wall material, and carrier gas have all been varied in order to

determine the relative importance of the chemical wall-effects compared to the thermal wall-effects in lean combustion. In Paper II methane has been used as the fuel in the model, and the inlet conditions were changed somewhat from those used in Paper I. The effects of changing fuel/air ratio, wall material and pressure have all been discussed.

The third part of the thesis deals with premixed flame fronts. The inlet conditions vary strongly as a function of distance from the wall and a premixed flame has been modeled that propagates towards a wall under the influence of a boundary layer flow. In Paper III, propane has been used as the fuel in the model and the effects of fuel/air ratio, pressure and wall temperature on the flame propagation and emissions

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have all been discussed. Paper IV investigates the influences of different fuel/air ratios and wall boundary conditions on the propagation of pressurized methane flames. A comparison has been made between these results and those obtained in Paper II.

Finally in the last part of the thesis, Paper V treats a somewhat different but nevertheless related area, namely combustion of gasified biomass in a hybrid catalytic combustor for gas-turbine applications. Paper V discusses different operations of the combustor for reducing Fuel-NOx, which is a major hurdle when using gasified biomass as a fuel in combustion applications. To model the combustion in the gas-turbine combustor, detailed homogenous and heterogeneous chemical kinetic schemes have had to be used.

The author’s contribution to the above papers has been: (i) setting up the reacting flow problems, (ii) performing the modeling and simulation work, and (iii) playing the major role in reporting on the work and drafting the manuscripts. The other authors of these papers have acted as academic advisors, involving for example providing inspiration, ideas and as scientific reviewers of manuscripts.

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2 Fundamentals of Chemical Kinetics

This chapter deals with the basic formalism of gas-phase and surface kinetics. To illustrate the detailed mechanisms of chemical kinetics that are used to describe combustion processes, both the gas-phase and catalytic combustion of hydrogen have been described in detail.

2.1 Gas-phase chemistry: oxidation of hydrogen in air

The net reaction or overall reaction for the combustion of a stoichiometric mixture of hydrogen and air (an exothermic reaction with heat of combustion of -571.7 kJ/mole) is described by the equation:

2 H2 (g) + O2 (g) + 3.762 N2 (g) → 2 H2O (g) + 3.762 N2 (g). (2.1)

Often these overall reactions are bound by complicated rate laws, where the reaction orders of the individual reactants may be: (i) non-integers, (ii) negative and (iii) dependent on the time and reaction conditions. However, detailed investigations show that water is not produced by one single collision between the three reacting molecules; instead, the overall reaction is a consequence of a large number of

elementary reactions. Resolution of the elementary reactions can be a difficult and

time-consuming task. The formation of water for example, can be described by 38 elementary reactions (19 reversible reactions), as shown in Table 2.1. This mechanism was used in Paper I. For the hydrocarbons, which are more complex than hydrogen, hundreds or even thousands of elementary reactions are needed to describe their combustion. The largest mechanisms are normally reduced, and the larger the number of species used in these reduced mechanisms, the more exactly the chemistry can be represented. Although the oxidation mechanism of hydrogen in air is a fairly simple system compared to that of a hydrocarbon fuel, it does contain the chemistry of the radicals H, O and OH that are very important in combustion processes.

A summary of the experimental techniques used for the determination of elementary reactions can be found in Warnatz et al. (1996). Experiments are often carried out in: (i) isothermal vessels filled with reactants, and the time behavior of the concentrations is measured, or (ii) in flow reactors where the spatial profiles of the concentrations provide information about their time behavior. Usually the reactive species such as H and O have to be produced as reactants. This may be accomplished thermally by using high temperatures (for example dissociation by heating in a shock

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tube), or by a microwave discharge. High dilution with noble gases (He and Ar) slows down the reaction of the reactive species with themselves.

Table 2.1 The gas-phase reaction mechanism for hydrogen-air.

No. Reaction Aa βa _Ea [mol,cm,s] [-] [kJ/mol] 1. H+O2=O+OH 5.1×1016 _{-0.82 69.1} 2. H2+O=H+OH 1.8×1010 _1.00 _37.0 3. H2+OH=H2O+H 1.2×109 _1.30 _15.2 4. OH+OH=H2O+O 6.0×108 _1.30 _0.0 5. H2+O2=OH+OH 1.7×1013_0.00_200.0 6. H+OH+M=H2O+M* 7.5×1023 _{-2.60 0.0} 7. O2+M=O+O+M 1.9×1011_0.50_400.1 8. H2+M=H+H+M** 2.2×1012_0.50_387.7 9. H+O2+M=HO2+M*** 2.1×1018 _{-1.00 0.0} 10. H+O2+O2=HO2+O2 6.7×1019 _{-1.42 0.0} 11. H+O2+N2=HO2+N2 6.7×1019 _{-1.42 0.0} 12. HO2+H=H2+O2 2.5×1013 _0.00 _2.9 13. HO2+H=OH+OH 2.5×1014 _0.00 _7.9 14. HO2+O=OH+O2 4.8×1013 _0.00 _4.2 15. HO2+OH=H2O+O2 5.0×1013 _0.00 _4.2 16. HO2+HO2=H2O2+O2 2.0×1012 _0.00 _0.0 17. H2O2+M=OH+OH+M 1.2×1017_0.00_190.7 18. H2O2+H=HO2+H2 1.7×1012 _0.00 _15.7 19. H2O2+OH=H2O+HO2 1.0×1013 _0.00 _7.5

a_{The rate constants are formulated with Equation 2.9.} Third-body enhancements factors:

* H2O /20.0/.

** H2O /6.0/ H/2.0/ H2 /3.0/.

*** H2O /21.0/ H2 /3.3/ O2 /0.0/ N2 /0.0/.

Source: Giovangigli & Smooke (1987).

The concentration measurement used should be sensitive and fast. Methods used include mass spectroscopy, electron spin resonance, optical spectroscopic methods and gas chromatography.

Using elementary reactions has many advantages: (i) the reaction order of elementary reactions is always constant, and in particular is independent of time and the experimental reaction conditions, and (ii) can be determined easily. All one has to do is

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to look at the molecularity of the reaction. Only three possible values of the reaction molecularity are observed, and these are described as follows.

Unimolecular reactions describe the rearrangement or dissociation of a molecule:

A → products. (2.2)

These have a first-order time behavior. When the initial concentration is doubled, the reaction rate is also doubled.

Bimolecular reactions are the most common reaction type found (see Table 2.1). These

proceed according to the following reaction equations:

A + B → products, (2.3)

or

A + A → products. (2.4)

Bimolecular reactions always follow a second-order rate law. The doubling of the concentration (doubling of the pressure) quadruples the reaction rate.

Trimolecular reactions are usually recombination reactions and obey a third-order law:

A + B + C → products, (2.5)

A + A + B → products, (2.6) or

A + A + A → products. (2.7)

In general, the molecularity equals the order of the elementary reactions, and the rate laws can be easily derived. For an elementary mechanism, the rate of formation (mole/cm3 · s) of a species k, is given by summation over the rate equations of all the elementary reactions Ias follows:

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[ ]













₋

−

=

∑ ∏ ∏ = = = '' 1 ' 1 1 , ,,

₎

(

K ki k k ri ki K k k fi I i ki ki k

k

X

k

X

υ υ

υ

ω

_&

. (2.8)

In Equation 2.8, the superscript symbol ' indicates forward stoichiometric coefficients

(

υ

_ki), while '' indicates reverse stoichiometric coefficients for species k in reaction . The symbol

[ ]

i k

X

(

=

Y

_k

ρ

/

W

_k

)

is the molar concentration of species

k

. This expression can also be modified to take into account arbitrary reaction orders and third-body reactions.

The temperature dependence of the rate constant ( ) in reaction

i

, is a synthesis of (i) the theory of the Swedish chemist Arrhenius (Arrhenius, 1889), (ii) the collision theory and (iii) the theory of activated complex:

f

k

) / ( E_i R_cT i i fi

A

T

e

k

=

β − , (2.9)

where is the constant of proportionality between the concentrations of the reactants and the rate at which they collide,

i

A

i

β

is the temperature exponent, the activation energy, the universal gas constant in the same units as the activation energy, and

i

E

c

R

f

denotes that it is a forward reaction.

The reverse rate constants are determined by the forward rate constant and the equilibrium constant: i c fi ri

K

k

=

. (2.10)

The equilibrium constants are calculated from the thermodynamic properties of the species: ∑ =       ₋













=

K k ki ki atm i p i c

RT

P

K

1 ' '' _υ υ , (2.11)

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











∆

₋

∆

=

RT

H

R

S

K

o i o i i p

exp

, (2.12)

(

)

R

S

R

S

o k K k ki ki o i _∑ =

−

=

∆

1 ' ''

υ

(2.13) and

(

)

∑ =

−

=

∆

K k o k ki ki o i

RT

H

RT

H

1 ' ''

υ

, (2.14).

where refers to the change that occurs in passing completely from reactants to products in the i

∆

th_{reaction. The standard state is defined as}

atm

P

= 1 atm (Kee, Rupley, Meeks & Miller, 1996). An alternative way is to explicitly state the reverse rate coefficients.

2.1.1 Interpretation of the elementary reaction mechanism

The combustion of H2 in air proceeds through a chained-branched explosion

when a fuel/air mixture within the flammability limits (the leanest or richest concentrations that will self-support a flame) comes into contact with an ignition source (Glassman, 1996). Chain-branched explosions are characterized by a period of ignition delay, during which the radical pool (O, OH and H) builds up (Warnatz et al., 1996). This means that during the first phase of the explosion, the temperature will not increase much, because the energy released from the fuel and the oxidizer is stored in the free radicals. The concentration of radicals therefore increases with time, and either the fuel or the oxidizer concentration lowers. At a certain point in time, radical recombination or chain-termination reactions become faster than chain-branching reactions and the energy is finally released. The concentration of radicals decreases and the temperature increases until the thermodynamic equilibrium of the system is finally reached. Examples of different types of chain reactions in the reaction mechanism

outlined in Table 2.1 are: • initiation (No. 5, forward reaction),

• chain-branching (No. 1, forward reaction),

• propagation (Nos. 2, 3 & 4 forward reactions), and • termination (No. 8, backward reaction).

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More free bonds are created than consumed in the chain-branching reaction and the initiation reaction. However, in the initiation reaction the reactants contain no free bonds. In the propagation reaction one free bond is consumed and one free bond is created. In the termination reaction two free bonds are consumed, and the chain is terminated due to a recombination of the two radicals.

In the recombination reactions, more energy is released by bonding two molecules than the product molecule can carry, and a third species M is required that can carry some of the energy. The third species M provides the energy required to split the product back to reactants in the reverse reaction. In reaction Nos. 6, 7, 8, 9 and 17, M denotes an unspecified molecule. The concentration of this third body is calculated from the sum of concentrations of all the species in the gas times their factor of effectiveness to carry the energy that is released or required, according to:

[ ]

∑

[ ]

=

K k ki

X

k

M

1

α

(2.15)

where

α

_ki is the enhancement factor for species

k

in reaction . If all the species in the mixture contribute equally as third bodies (

i

ki

α

= 1), then the concentration of the third body M equals the total concentration of the mixture.

Equation 2.8 then changes to:

[ ]













₋

−

=

∑ ∏ ∏ = = = '' 1 ' 1 1 , ,,

₎

(

M

k

X

k

X

υ υ

υ

ω

_&

. (2.16)

In reaction Nos. 6, 8 and 9 in Table 2.1, the enhancement factors for some molecules differ from unity in the mixture. Clearly water has a strong third-body efficiency. As shown in next section, an important feature of the third-body reactions is their pressure dependence. At lower pressures, these reactions are relatively slow, but as the pressure increases, they become increasingly important and compete with the two-body reactions.

2.2 Pressure-dependent reactions

Under certain conditions, some reaction rate expressions depend on pressure as well as temperature. An example of such a reaction is the recombination of methyl

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radicals, which is part of all reaction mechanisms for hydrocarbons, and is shown below:

2CH3 (+ M) = C2H6 (+M). (2.17)

This is a so-called third-body-recombination-fall-off reaction. At the high-pressure limit, the reaction can be described as 2CH3 = C2H6. At the low-pressure limit, a third body is required to carry some amount of energy liberated in the recombination reaction. When the reaction is at either limit, solely temperature-dependent rate-expressions are applicable. However when the pressure and temperature are such that the reaction falls between these limits, the rate expressions are more complicated. The simplest approach is the model developed by Lindemann (1922). Arrhenius parameters are required for both the high (

k

_∞) and the low (

k

_o) pressure-limit cases:

)

/

exp(

E

R

T

A

k

o _o _c o o

=

−

β _, _(2.18) and

)

/

exp(

E

R

T

A

k

∞ _∞ _c ∞ ∞

=

−

β . (2.19) The rate constant at any pressure is taken to be:

F

P

k

r r













+

=

_∞

1

, (2.20)

where the reduced pressure is given by

[ ]

∞

=

k

M

k

P

o r . (2.21)

The limiting behavior (

F

= 1) becomes:

- high-pressure limit

[

M

]

→ ∞ and

k

= k

_∞, and - low-pressure limit

[

M

]

→ 0 and

k

=

k

_o

[ ]

M

.

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The Lindemann model itself is a simplified model, and other approaches involve more complex descriptions of

F

.

The reaction expressed in Equation 2.17 competes with other bimolecular steps, such as 2CH3 (+M) = C2H5 + H (+M). This reaction—which is endothermic—occurs through the same energetically excited species C2H6* as the reaction expressed in Equation 2.17: 2CH3 → C2H6* , (2.22) C2H6* → 2CH3 , (2.23) C2H6* → C2H5 + H , (2.24) and C2H6* +M → C2H6 +M . (2.25) As the pressure increases, deactivating collisions between C2H6* and other molecules cause the rate coefficient of C2H6 formation to increase. At the same time, these deactivating collisions preclude the dissociation of C2H6* into C2H5 + H.

2.3 Surface chemistry

The formalism of the surface chemistry is very similar to that of the gas-phase reactions, via resolution into elementary reactions (Coltrin, Kee & Rupley, 1991; Warnatz, 1992). The net chemical production rate of species by surface reaction (in mole/cm2·s) is defined as:

[ ]













₋

−

=

∑ ∏ ∏ = = = '' 1 ' 1 1 , ,,

₎

(

k

X

k

X

s

_&

υ

υ υ , (2.26)

where

I

is the total number of elementary reversible (or irreversible) surface reactions. For adsorbed species, the concentrations are given in moles/cm2 by:

[ ]

X

k

=

Z

k

Γ

/

σ

k, (2.27)

where

Z

_kis the site fraction of species

k

,

Γ

the surface site density, and

σ

_kthe number of sites occupied by species

k

. In the surface chemistry used in this thesis, the species reside in only one surface phase.

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The rate coefficients for adsorption, desorption and surface reactions can also depend on surface coverage. This coverage dependence is taken into account by an additional factor leading to the following rate expression for the rate coefficient of the forward reaction

k

_f : [ ]

_{[ ]}

[ ]











 −











 −

=

∏ =

R

T

Z

T

R

E

T

A

k

c k ki ki k l s K f s K k k Z ki c i i i fi

ε

µ η β

_exp

₁₀

_exp

, (2.28)

where

η

_ki,

µ

_ki, and

ε

_kiare the coverage parameters for species

k

in reaction

i

, and is the site fraction. The product of Equation 2.28 covers only those surface species that are specified as contributing to the coverage modification.

k

Z

Reverse rate constants can be determined from Equation 2.10 where:

( )

∏ = − − ∑ = − − ∑ =

Γ













=

Ksl f s K k ki ki k l s K f s K k ki ki o ki g K k ki atm i p i c

T

R

P

K

( '' ' ) ) ' '' ( ) ' 1 '' ( υ υ υ υ υ υ

σ

. (2.29)

The thermodynamics of gas-phase reactions are well known. Utilizing Equation 2.10 and evaluating is the normal manner of describing reverse reaction rates when the forward reaction rate is known. However for adsorbed species on a surface, the thermodynamics are not well known. The reversible reactions can then be split into two irreversible reactions, for example the adsorption or desorption of a species.

c

K

For some simple surface-reaction mechanisms, it is convenient to specify the adsorption step in terms of a sticking coefficient. This is a common method for characterizing the transfer of a molecule from the gas to the surface. The sticking coefficient is the probability of a specific molecule hitting the surface and adsorbing onto it instead of bouncing back into the gas phase, and is expressed as follows:

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[ ]

k k n k fi i

W

T

R

X

k

t

bombardmen

of

rate

adsorption

of

rate

Π

Γ

=

2 γ

. (2.30)

The functional form of the sticking coefficient is taken to be:

[

b_i c_i R_cT

]

i i

a

T

e

/

,

1 max

−

=

γ

, (2.31)

where and are unitless, and the units of

c

are compatible with . The sticking coefficient must lie between 0 and 1 to make physical sense. The expression for the relation between the sticking coefficient and the rate constant becomes:

i

a

b

_i _i

R

_c k n i fi

W

RT

k

Π

Γ

=

2

1 γ

, (2.32)

where is the sum of the stoichiometric coefficients for the number of reactant surface-species. The sticking coefficient can also depend on the surface coverage, according to Equation 2.28. For large sticking coefficients—nearing 1—the velocity distribution becomes skewed. Species whose random motion carries them close to the surface have a high probability of staying there, which causes a non-Maxwellian velocity distribution that in turn alters the net species flux near the surface. Motz and Wise (1960) provided the following correction factor for the rate coefficient:

n

k n i i fi

W

RT

k

Π

Γ













−

=

2

1

2 /

1 γ

γ

. (2.33)

Table 2.2 shows the surface mechanism for hydrogen oxidation on platinum used in Paper I. The oxidation of H2 on Pt is a well-studied phenomenon, both experimentally and theoretically (Hellsing, Kasemo, Ljungstrom, Rosén & Wahnstrom, 1987; Ljungstrom, Kasemo, Rosén, Wahnstrom & Fridell, 1989; Williams, Marks & Schmidt,

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1992; Warnatz, Allendorf, Kee & Coltrin, 1994; Rinnemo, Deutschmann, Behrendt & Kasemo, 1997; Aghalayam, Park & Vlachos, 2000a).

Table 2.2 The surface-reaction mechanism for hydrogen oxidation on Pt.

Reaction no. Aa [cm,mol,s] Ea [kJ/mol] 1. H2 + Pt(s) = H2(s) 0.05b 0.0 2. H2(s) + Pt(s) = H(s) + H(s) 7.50×1022 15.6 3. O2 + Pt(s) = O2(s) 0.023b 0.0 4. O2(s) + Pt(s) = O(s) + O(s) 2.50×1024 0.0 5. H(s) + O(s) = OH(s) + Pt(s) 3.70×1021 _19.3 6. H(s) + OH(s) = H2O(s) + Pt(s) 3.70×1021 0.0 7. OH(s) + OH(s) = H2O(s) + O(s) 3.70×1024 100.5

8. H + Pt(s) = H(s) 1.00b 0.0 9. O + Pt(s) = O(s) 1.00b 0.0 10. H2O + Pt(s) = H2O(s) 0.75b 0.0 11. OH + Pt(s) = OH(s) 1.00b 0.0 12. H2O2 + 2Pt(s) = OH(s) + OH(s)# 1.00b 0.0 13. HO2 + 2Pt(s) = OH(s) + O(s)# 1.00b 0.0

Source: Warnatz et al. (1994).

Pt(s) denotes a free surface platinum site and X(s) is an adsorbed species. All reactions are assumed to be reversible.

a _{The rate constants are formulated with Equation 2.9.} b_{The sticking coefficients are formulated with Equation 2.31.} Site density of the Pt-wall is = 2.71Γ ×10-9_mole/cm2

#_{Reaction Nos. 12 and 13 have been added to the original mechanism.}

The mechanism is based on laser-induced-fluorescence (LIF) measurements of desorbed OH, and involves the following steps:

• dissociative adsorption of H2 and O2 (reaction Nos. 1-4),

• the adsorbed atoms collide while attached to the surface, forming first OH and H2O in surface reactions (reaction Nos. 5-7), and

• finally desorption of H2O takes place (reaction No. 10).

The adsorption and desorption of radicals is accounted for in the remaining reactions. The type of elementary gas-phase and surface-reaction mechanisms described in this chapter can be very useful when coupled to reactive flow models for modeling the coupling between homogeneous and heterogeneous combustion.

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3 Modeling of Near-wall Combustion

Wall effects in combustion present a complicated problem for analysis since many species and still more reactions are involved. The most complex situation is encountered when chemical interactions are established between the gas and the wall, such as in catalytic combustion. The simplest possible situation is that of a chemically-inert wall, in which the coupling between the gas and the wall is mainly of a thermal nature due to heat transfer, and of viscous shear due to momentum transfer. There are no completely inert walls though, and the possibility of active radical destruction must also be considered.

Factors that influence the wall effects include the temperature decrease across the thermal boundary layer, and the destruction of radicals in the boundary layer, which is more pronounced at lower temperatures. In turbulent combustion, additional complexities such as the scale of the turbulence across the turbulent boundary layer and the properties of the laminar sublayer may influence the flame-wall interaction. To understand more about the wall effects, it is desirable to separate one of those factors and to elucidate its relative importance.

In order to explain the quenching phenomenon, model calculations with rigorous fluid mechanics and transport coefficients are required. It is inadequate however to use global chemistry, as it does not account for the role of the radical pool (see Section 2.1), and the intermediate hydrocarbons have been found to be significant during quenching and must be accounted for in the kinetic model (Hocks et al., 1981; Kiehne et al., 1986; Hasse et al., 2000). Although lean combustion of simpler hydrocarbons like methane have limited pathways for forming intermediate hydrocarbons, detailed chemistry schemes are needed to describe the interaction between transport and fuel oxidation in the gas and on the surface. The results are further improved when the effect of thermal diffusion (Soret effect) is considered (Popp & Baum, 1997). Thermal diffusion denotes the diffusion of mass caused by temperature gradients.

Studies of maximum wall heat fluxes have shown that the following two generic cases must be considered for laminar premixed flames (see Fig. 3.1).

• Stagnation or head-on quenching: when the premixed flame propagates towards the wall, and where the flame stops at a certain distance from the wall.

• Sidewall quenching: when the premixed flame propagates along the wall. The quenching distance is larger than for head-on quenching, and the wall heat flux slightly lower.

Numerical modeling with complex chemistry has mostly used the stagnation configuration in either the transient case of a flame freely propagating towards a wall (Westbrook et al., 1981; Popp, Smooke & Baum, 1996; Popp & Baum, 1997; Hasse et al., 2000) or the stationary case with head-on stagnation point flow (Vlachos et al., 1993; Vlachos, Schmidt & Aris, 1994a; Bui, Vlachos & Westmoreland, 1996;

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Egolfopoulos, Zhang & Zhang, 1997; Aghalayam, Bui & Vlachos, 1998; Forsth, Gudmundson, Persson & Rosén, 1999).

a b Flame zone

Flame zone

Wall Wall

Stagnation configuration Sidewall configuration

Figure 3.1 Typical flow configurations used to model near-wall combustion.

Furthermore, sidewall configuration has been dealt with by von Kármán and Millán (1953) and Blint and Bechtel (1982), however not specifically using complex chemistry. The emphasis in this thesis has therefore been placed on the sidewall configuration.

3.1 Near-wall turbulence

Turbulent flows are characterized by a wide range of spatial and temporal scales. The largest scales are given by the size of flow boundaries and by the flow velocity— the integral length scale (lo). The smallest scales are at the level where turbulence is

dissipated by viscosity. At this so-called Kolmogorov length scale (lk), the time taken

for an eddy to rotate half a revolution is equal to the diffusion time across the diameter (lk). The time scales for different chemical reactions can overlap with the turbulent time

scales.

Almost all practical flows in industry and the environment are turbulent. The understanding and modeling of turbulent phenomena that occur near walls is a formidable task. The treatment of near-wall turbulence for non-reacting flows is still a difficult problem and quite often the limiting factor in making practical predictions. Wall problems become even more critical in chemically-reacting flows. In practice,

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very little is known about the effects of turbulence during flame-wall interaction. In turbulent flows, the mixing between the fuel and the oxidant is drastically enhanced, and in general the heat and mass transfer rates would be higher in a turbulent boundary layer than in a laminar boundary layer. In order for computations with turbulent reactive flows to be accurate, these must respect the smallest scales of length and time, which often strain storage and computational efficiency past practical limits. In internal combustion engines, the length scales of wall quenching are much smaller than those of the turbulence (that is, the diffusion and molecular transport is in general faster than the turbulent transport).

k l

Figure 3.2 At the Kolmogorov length scale, the time taken for an eddy to rotate

half a revolution is equal to the diffusion across the diameter lk .

The laminar sublayer is usually taken to be the region where eddies are completely absent, however there is evidence that eddies do occasionally penetrate right up to the surface. The situation is therefore highly complex, and any attempt to model the process must involve simplifying assumptions.

3.2 Laminar models

Although the chemical reactions in both an internal combustion engine and a gas turbine combustor take place in a turbulent flow field, there will always be a laminar sublayer close to the wall, where the turbulence has died out and momentum transferred due solely to viscous shear. This suggests that results obtained for laminar models are also valid and can be compared with results obtained from turbulent models. As the quench layer is usually within the laminar sublayer, and the quench distances in engines can be well correlated to the distances obtained for laminar flame-quench calculations (Ferguson & Keck, 1977), the use of laminar models has been generally accepted. In this thesis, the numerical modeling work done on homogeneous premixed combustion influenced by walls has been carried out exclusively using laminar models in the sidewall configuration (see Fig. 3.1). The following section presents a description of the

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mathematical treatment of a two-dimensional chemically-reacting boundary layer over a flat wall.

3.2.1 Mathematical description of chemically-reacting boundary layer flow

Two-dimensional boundary layer flow is illustrated in Fig. 3.3. When modeling a chemically-reacting boundary layer, fluid flow pattern, temperature and species fields are calculated from conservation equations for mass, momentum, energy and species. Reaction rates using factors such as the molecular weights, concentrations and stoichiometries for specific species in all reactions are used in each species-conservation equation to balance the mass transfer from one species to another.

y

Boundary layer

x Wall

Figure 3.3 Two-dimensional boundary layer flow over a flat wall.

Elliptic approach (Navier-Stokes equations)

The laminar, multicomponent, chemically-reacting Navier-Stokes equations involve few assumptions, and represent the most general approach. In a two-dimensional Cartesian coordinate system, and at steady state, they can be expressed as follows (after Bird, Stewart & Lightfoot, 2002).

Continuity:

0 =

∂

+

∂

y

v

x

u

ρ

. (3.1)

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Axial momentum:

0

3

2

2 =

+

























∂

+

∂

−

























∂

+

∂

−

∂

−

∂

+

∂

+

∂

g

x

v

y

u

y

v

x

u

x

u

x

p

y

u

v

x

u

ρ

µ

ρ

. (3.2) Cross-flow momentum:

0

3

2

2 _

+

=























∂

+

∂

−

∂

−

























∂

+

∂

−

∂

+

∂

+

∂

g

y

v

x

u

y

v

y

u

x

v

x

y

p

y

v

x

v

u

ρ

µ

ρ

. (3.3) Thermal energy: k k g K k k y k k x k k g K k pk p

W

h

y

T

V

Y

x

T

V

Y

c

y

T

y

x

T

x

y

p

v

x

p

u

y

T

v

x

T

u

c

ω

ρ

λ

ρ

&

∑ ∑ = =



_

−









∂

+

∂

−













∂

+













∂

+

∂

+

∂

=













∂

+

∂

1 1 . (3.4) Species continuity: g k k y k k x k k k k

K

k

W

V

Y

y

V

Y

x

y

Y

v

x

Y

u

,

1

0 )

(

)

(

=

−

∂

+

∂

+

∂

+

∂

_ρ

_ω

ρ

_&

. (3.5)

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State:

W

T

R

p

=

ρ

. (3.6)

For reactive walls, the variation of surface coverage—the fraction of surface sites covered by species

k

—is calculated from the surface reaction rates according to the following expression:

s k k

s

_k

_K

dt

dZ

,

1 =

Γ

= &

. (3.7)

Equation 3.7 simply states that at steady state the surface composition does not change. In some senses this could be considered a boundary condition (possibly complex) on the gas-phase system, however because the surface composition is determined as a part of the solution, Equation 3.7 should be considered to be a part of the system of governing equations. All the variables are described in the Nomenclature section at the beginning of the thesis.

Parabolic approach (Boundary layer approximation)

Pursuing solutions based on the Navier-Stokes equations for a propagating flame—fluid flow equations coupled to the diffusion and energy equations with finite-rate chemistry—is a formidable task, even for a two-dimensional stationary system. Therefore it may be desirable to first undertake some simplifications of the fluid mechanics. Prandtl first addressed the boundary layer approximations around 1904 (Prandtl, 1961) and these well-known approximations are still applied widely today in fluid mechanics. Compared with an elliptical approach, the validity of the boundary layer equations depends on the inlet velocity and the fuel/air ratio (Mantzaras, Appel & Benz, 2000). As the flow rate in the channel increases (high Reynolds numbers) and the fuel/air ratio (φ) decreases, the boundary layer approximations become increasingly valid. Compared with the diffusive transport, under these conditions the convective transport is dominant in the principal flow direction (

x

). As a result, all the second derivatives in the

x

direction are eliminated. It can be shown by order-of-magnitude arguments that the cross-flow momentum equation is reduced to the simple statement that there can be no pressure variation across the channel (Bird, Stewart & Lightfoot, 2002).

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Continuity:

0 =

∂

+

∂

y

v

x

u

ρ

. (3.8) Axial momentum:

0 =

+

∂

+













∂

−

∂

+

∂

g

x

p

y

u

y

u

v

x

u

ρ

µ

ρ

. (3.9) Cross-flow momentum:

0 =

∂

y

p

. (3.10) Thermal energy:

0

1

∂

+

1

=

∂

+













∂

−

∂

+

∂

∑ ∑ = = k g K k g K k k k k p y k k p p

h

W

y

T

c

V

Y

y

T

y

T

c

v

x

T

c

u

ω

ρ

λ

ρ

&

. (3.11) Species continuity:

0 )

(

−

=

∂

+

∂

+

∂

k k y k k k k

_Y

_V

_W

y

Y

v

x

Y

u

ρ

ω

ρ

_&

. (3.12) g

K

k

=

1 ,

,

The state expressed by Equation 3.6, and the surface species conservation expressed by Equation 3.7 have to be added to the system. In horizontal channels, the term

ρ

g

in Equation 3.9 can be omitted when the channel height is sufficiently small, as in this thesis. The independent variables are denoted

x

and , and the dependent variables are denoted

y

,

u

v

T

Y

_k

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elliptical to parabolic. This represents a huge simplification, leading to much more efficient computational solution (Mantzaras et al., 2000; Raja, Kee, Deutschmann, Warnatz & Schmidt, 2000).

The boundary layer approximations for the transport equations have been used throughout this work. In Paper II, the validity of the boundary layer approximations for the flame position was investigated. It was found that in most cases the effect of axial-diffusive transport was small, although at lower pressures (higher flame velocities) and higher concentrations, the parabolic approach should be used with caution.

3.2.2 Boundary conditions

Turning now to the energy equation, either the temperature (see Equation 3.13 below) or a zero heat flux (adiabatic) condition (see Equation 3.14) is specified at the solid wall: w

T

=

(3.13) or

=

0 ∂

∂

y

T

. (3.14)

In addition, the axial velocity (

u

) is assumed to be zero at the wall (the non-slip condition). For chemically inert walls, the following zero-gradient Neumann boundary condition is used for each species:

0 =

∂

y

Y

_k . (3.15)

The boundary condition at the wall becomes relatively complex in the presence of heterogeneous surface reactions. The convective and diffusive mass fluxes of gas-phase species at the surface are balanced by the production (or depletion) rates of gas-phase species by surface reactions, as represented in the following expression:

k k ky k y k

Y

V

v

s

W

j

=

ρ

(

+

)

=

_&

,

k

=

1 ,

,

K

_g . (3.16) In the above equation, the gas-phase diffusion velocity of species

k

in the direction normal to the wall is given by:

Wall Related Lean Premixed Combustion Modeled with Complex Chemistry