the mathematics of combustion

The first chapter of this volume introduces the equations of the subject,and briefly discusses such fundamental matters as the small Mach numberapproximation, deflagration waves, kinetic

THE MATHEMATICS

OF COMBUSTION

FRONTIERS IN APPLIED MATHEMATICS

H T Banks, Managing Editor

This series is intended to serve as a provocative intellectual forum on both ing or rapidly developing research areas and fields that are already of great interest

emerg-to a broad spectrum of the scientific community

The Mathematics of Combustion is the second volume in this series The first is The Mathematics of Reservoir Simulation, edited by Richard E Ewing.

Editorial Board John D Buckmaster Charles J Holland

Robert Burridge Alan LaubJack J Dongarra Robert VoigtRichard E Ewing

THE MATHEMATICS

OF COMBUSTION JOHN D BUCKMASTER, EDITOR

SiaJTL

PHILADELPHIA

1985

Copyright © 1985 by Society for Industrial and Applied Mathematics.All rights reserved.

Library of Congress Catalog Card Number: 85-50339

ISBN: 0-89871-053-7

Contributors vii

Foreword H T Banks ix Preface John D Buckmaster xi

Chapter I An Introduction to Combustion Theory

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JOHN D BUCKMASTER is Professor of Aeronautical and Astronautical Engineering, University

of Illinois, Urbana, Illinois.

J F CLARKE is Professor of Theoretical Gas-Dynamics, College of Aeronautics, Cranfield Institute of Technology, Cranfield, Bedford, England.

WILDON FICKETT is with Los Alamos National Laboratory, Los Alamos, New Mexico HERSCHEL RABITZ is Professor of Chemistry, Princeton University, Princeton, New Jersey.

F A WILLIAMS is Professor of Mechanical and Aerospace Engineering, Princeton University, Princeton, New Jersey.

Vll

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This is the second volume in the SIAM series Frontiers in Applied

Math-ematics This continuing series will focus on "hot topics" in applied

math-ematics, and will consist of, in general, unrelated volumes, each dealing with

a particular research topic that should be of significant interest to a spectrum

of members of the scientific community Distinguished scientists and appliedmathematicians will be solicited to contribute their points of view on "state-of-the-art" developments in the topics addressed

The volumes are intended to provide provocative intellectual forums onemerging or rapidly developing fields of research as well as be of value inthe general education of the scientific community on current topics In view

of this latter goal, the solicited articles will be designed to give the nonexpert,nonspecialist some appreciation of the goals, problems, difficulties, possibleapproaches and tools, and controversial aspects, if any, of current efforts

in an area of importance to scientists of varied persuasions

Each volume will begin with a tutorial article in which technical terms,jargon, etc are introduced and explained This will be followed by a number

of research-oriented summary contributions on topics relevant to the subject

of the volume We hope that the presentations will give mathematicians andnonmathematicians alike some understanding of the important role mathe-matics is playing, or perhaps might play, in what academicians often eu-phemistically call "the real world." We therefore expect each volume tocontribute some further understanding of the important scientific interfacesthat are present in many applied problems, especially those found in indus-trial endeavors

At the printing of this volume, a number of other volumes are already inprogress Volume 3 will focus on seismic exploration, and Volume 4 onemerging opportunities related to parallel computing Other topics in sci-entific computing are among those currently under consideration for futurevolumes Members of a rotating editorial board will encourage active par-

ix

mathemat-at the University of Wyoming.

H T BANKSfotrwood

Combustion is, to a large extent, a superset of fluid mechanics, but—unlike that subject—one does not find within it a rich interaction of math-ematics and experiment Without offending those (or their memory) whohave made important theoretical contributions to the subject, it can safely

be said that combustion has been (and remains) primarily an empirical ence As evidence (at least for the Western world) one need only examinethe proceedings of the biennial International Symposia organized by theCombustion Institute

sci-When theoretical work has been carried out, it has seldom exploited thefull range of mathematical tools familiar in mechanics Engineering approx-imations of an ad hoc nature are often made at different stages of the analysis,which obscures the theoretical model, and therefore the conclusions thatmay be drawn from its success or failure in emulating the physical world.Preoccupation with "reality" (as the physical world is called) leads to re-sistance—even rejection—of models that do not parallel the physical world.The notion of a mathematical universe in which one can set aside densityvariations in order to more readily explore the interaction of other mecha-nisms; a universe in which, having defined a model, one can construct abody of theoretical knowledge without further reference to the physics; auniverse which will, provided the model has been judiciously chosen, em-

ulate the physical world here and there and thus provide vivid insights into

that world; such a notion is not one that many in combustion are comfortablewith Experimental results are always of value, but mathematical conclu-sions tend to be subjected to the test of immediate physical relevance Inshort, the world of combustion is not one in which the goals and methods

of applied mathematics are widely understood

Fortunately, this situation is rapidly changing In recent years appliedmathematicians have been attracted in growing numbers to a subject whoserichness affords ample scope for their talents Asymptotic methods and bi-furcation theory have made a significant impact in the last decade And thestunning success of constant density, small heat release models in unrav-elling the mysteries of deflagration instabilities, together with other impor-

xi

PREFACE

This volume contains five chapters, each of which deals with topics inwhich mathematics plays an important role Only in a partial sense is it aself-contained exposition of the subject; it is difficult to make useful con-tributions to combustion without undergoing an extensive apprenticeship inthe relevant physics, and for this the reader must turn elsewhere The well-known book by B Lewis and G von Elbe, cited in Chapter I, is an excellent

source, as are graduate college texts such as Combustion Fundamentals by

R Strehlow

The first chapter of this volume introduces the equations of the subject,and briefly discusses such fundamental matters as the small Mach numberapproximation, deflagration waves, kinetic modelling, ignition, the hydro-dynamic model, and activation energy asymptotics Some specific problemsare described in order to convey something of the flavor of the theoreticalquestions that have been considered in recent years

The remaining chapters deal with more specific questions Combustionmodelling, especially the kinetics, is a difficult subject often complicated bypoorly known data How this affects the accuracy and reliability of the re-sults is the subject of sensitivity analysis, discussed in Chapter II This isnot a tool unique to combustion theory of course, and at the present timehas had a greater impact outside of combustion than within it Many wouldagree with the suggestion that this should change

Chapter III discusses turbulence Although laminar flames are not a ematical or laboratory artifice, turbulent combustion plays a vital role inmany situations of great practical significance Interestingly enough, tur-bulent combustion is not simply classical turbulence plus the complication

math-of exothermic reaction It can, instead, be less complicated; unique fications give it a flavor all its own

simpli-Chapters IV and V deal with compressible flows Detonation is one of theoldest topics in the field, and one of the most difficult The analysis of modelequations, simpler than the exact ones, is useful in unravelling these diffi-culties, and such an approach forms the heart of Chapter IV Chapter Vdiscusses the behavior of finite amplitude waves in combustible gases Thematerial in these two chapters is relevant to the important and poorly under-stood subject of deflagration-to-detonation transition

JOHN D BUCKMASTER

THE MATHEMATICS

OF COMBUSTION

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JOHND.BUCKMASTER

1 The fundamental equations The equations that govern the motion of

a chemically reacting gas are complex Some of this complexity is essential:the heat released by exothermic reaction is what makes a flame hot Much

is not: that each component of the reacting mixture has a different specificheat is a quantitative detail, usually swamped by other uncertainties Thus,

in our discussion of the equations governing combustion, we shall deal onlywith those ingredients that are usually important, omitting those that arelikely to be important only in special cases

The simplest way to do this is in the context of a dilute mixture, one whosefundamental physical properties (viscosity, thermal conductivity, etc.) aredefined by a single component In air mixtures, nitrogen might play this role.Mixtures which are not dilute can often be modelled by the same equations

if appropriate averages are adopted for the various properties The question

as to when and how a mixture can be represented by "single-fluid" equations

is an important one, but is not discussed in these pages

We shall start by writing down that subset of the equations that will be

so familiar to the reader with a passing acquaintance of fluid mechanics that

it needs little comment This consists of the equation of mass conservation,

that of momentum,

where 2, the stress tensor, is given by

1 Acknowledgment The author was supported by the National Science Foundation and the

Army Research Office during the time in which this chapter was written, and the editing chores for the volume were undertaken.

3

4 THE MATHEMATICS OF COMBUSTION

for a Newtonian fluid with no bulk viscosity; and the equation of state,

The molecular weight is that of the diluent or, for a nondilute mixture, may

be defined by

where «, is the number fraction of the /th component, Y f the mass fraction.The energy equation deserves a more detailed treatment since some of itsingredients might be unfamiliar to the combustion neophyte Consider a fluid

in which no irreversible processes occur The energy density is

and the energy flux density is

irre-AN INTRODUCTION TO COMBUSTION THEORY 5

diffusion fluxes These are

The diffusion velocities {V,} depend on the various concentration gradients.For a dilute gas it is appropriate to adopt a Fickian law relating V,- to the

gradient of Y f for all components but the diluent (the Mh component), sothat

Since, by definition,

we have

The separate enthalpies can be written as

where C pi is the specific heat and /z° is the heat of formation of the /th

component at the temperature T 0 2

It follows that the diffusion enthalpy flux, the second term in (10), is

2 TO is usually taken to be 298°K and the hf for elemental gases in their natural molecular

state at that temperature (e.g H 2 ,02) are zero Thus for a constant pressure process represented by

for which the initial and final temperatures are 298°K, the enthalpy of the initial mixture is zero, that of the water -57.8 kcal/mole [1] Energy equal to 57.8 kcal must be removed for every mole of water generated, in order to prevent the exothermic reaction from creating a temperature rise.

THE MATHEMATICS OF COMBUSTION

if we neglect the second term (heats of formation are large compared todifferences in specific heats); this equals

Thus the formula (7) for the energy flux must be replaced by

where

and (9) is replaced by

mixture composition

satisfies a conservation equation of the form

where p/ represents the mass increase per unit volume of the rth speciesdue to chemical reaction Since the left side of (19) contains a term

p 2^ A? DYi/Dt we can use (20) to rewrite (19) in the form

where we have used the condition 2^ P 0- This is identical to a fluid equation with the heat released by reaction appearing as a source term.Equations (1), (2), (4), (20) and (21) are adequate to describe most laminar6

singleGL-AN INTRODUCTION TO COMBUSTION THEORY 7

flames, and are an appropriate starting point for turbulence modelling ter III) Their description is completed by a discussion of the reaction terms

(Chap-2 Chemical kinetics The quantity p/ is the mass of the rth component

generated in unit time per unit volume due to chemical reaction Usuallythere will be many reactions occurring simultaneously For example, Clarke[2] has used the following system to describe the steady hydrogen/oxygendiffusion flame:

here M represents all species

For 2-body reactions the reaction rate depends on the energy of collision

Consider a Maxwellian distribution of molecules each of mass m; the number

N with speed between c0 and infinity has a temperature dependence

If only high energy collisions can trigger chemical change, the number ofmolecules that can effect this change can be estimated from the asymptoticbehavior of (23) for large c0; we have

THE MATHEMATICS OF COMBUSTION

Each of the reactions shown in (22) is reversible When the forward rate

is equal to the reverse rate the reaction is said to be in equilibrium Consider(22a) for example,

According to the law of mass action the reaction rate is proportional to thenumber density of each reactant Thus

where temperature dependent proportionality factors are not shown At

equilibrium kf and k r are equal so that

where K, although temperature dependent, is called the equilibrium con stant The full form for k f -is usually taken to be

with a similar form for k r (note that Y M = 1 since M represents all species).

Contributions to p/ come from every reaction in which the /th speciesparticipates, either as a reactant or a product The rates must be weightedaccording to the stoichiometric coefficients and the molecular weights Thus

if the reaction

consumes H2 at a rate of k molecules per unit volume per unit time, and is

the only one in which H2 and O2 participate, we have

It is apparent that a system such as (22), described by rates of the form(29), adds a great deal of complexity to the mathematical description Onlyunder exceptional circumstances is an analytical treatment possible For thisreason simplified kinetic models are often adopted Consider, for example,the combustion of methane in oxygen The essential nature of this process

is that methane and oxygen are consumed, and products and heat are erated It can, therefore, be modelled by a one-step irreversible process,represented by

gen-8

AN INTRODUCTION TO COMBUSTION THEORY V

An even simpler model can be appropriate if the reactants are supplied

as a homogeneous mixture In general one of the reactants (the deficientone) will be present in less than stoichiometric proportion and so will becompletely consumed along with a portion of the other (the surplus reactant)

If the mass fraction of the deficient component is F, the reaction can berepresented by

at a rate

The simple models represented by (31) and (33) play a central role in much

of the mathematical treatment of combustion Whether they are appropriate

or not depends, to a large extent, on the aim of the analysis We shall return

to this question later, after some elementary combustion concepts have beenintroduced For the moment we shall proceed with our discussion using aone-step irreversible model when an explicit choice for the kinetics must bemade

3 The small Mach number approximation For many flames, velocities

are small (say 100 cm/s), temperatures large (2000-3000°F) A representativeMach number Ma is then small and this leads to important simplifications

of the governing equations Except for detonation, the approach to it, oracoustic propagation, the simplified set is at the heart of all combustionmodelling

At small Mach numbers the kinetic energy is small in comparison withthe thermal energy, and so can be neglected in the energy equation (21).The viscous term V • (v • X') is likewise negligible, depending as it does onthe square of the velocity Equation (21) can therefore be replaced by

Moreover, from the momentum equation, spatial changes in /? satisfy theestimate

10 THE MATHEMATICS OF COMBUSTION

whence

Thus p has the representation

and it is p 0 that appears in the equation of state, the simplified energy

equa-tion and the reacequa-tion terms, p\ in the momentum equaequa-tion (the small pressure

gradient drives the small velocities)

With two variables (p 0 , p\) replacing one (p} it is not clear that the reduced

system is complete There is no difficulty, however: for an unbounded

prob-lem p 0 (t) is usually an assigned quantity; for a bounded problem global

con-siderations, manifest through an application of Gauss's law, relate changes

in po to the chemical activity within the volume, and the thermal fluxes at the boundary Consider the case when C p is constant Noting that

which, upon integration over the entire combustion field of volume V0, yieldsthe compatibility condition

The initial value problem for the small Mach number system has somepeculiarities, which will be identified in §12

4 The plane flame or deflagration wave We now turn our attention to

the formulation of a central problem in combustion, the so-called deflagrationwave Consider a homogeneous mixture of fuel and oxidizer This can sup-port a one-dimensional premixed flame which is wavelike in the sense that

it can propagate at constant speed with unchanging form relative to themixture The gas ahead of the wave is cold and fresh; behind it is hot andburnt

We shall adopt a coordinate frame fixed to the flame with the flow passing

at speed u at right angles to the flame in the direction of increasing x If the

AN INTRODUCTION TO COMBUSTION THEORY 11

subscript/is used to denote the unburnt state, corresponding to x —» -=c,

w/is the speed of the wave relative to the fresh mixture This is the flamespeed, to be calculated as part of the solution

For this steady one-dimensional problem mass conservation implies

With one-step kinetics (N = 2) the energy and species equations then have

the form

where pi is a function of p, Y\ and T(the density is related to the temperature

by the equation of state (4) where p is the constant p 0 ) A linear combination

of equations (42) can be formed that contains no reaction term,

A key feature of the one-step kinetic scheme represented by (31), (33) isthat reaction can only cease when the reactant is completely consumed; this

corresponds to the equilibrium state Thus Y vanishes in the burnt gas (jt— »

+ x) 3

With these facts in mind, integration of (43) from (-Qc) to ( + oc) yields a

value for T b , the temperature of the burnt gas,

this is the adiabatic flame temperature

Equations (42) can be solved numerically A convenient nondimensional

system can be defined by using the characteristic length \/MC p,4

charac-teristic temperature h ( \/C p , and then we have

3 Reaction must also vanish far upstream, which is possible only if (34) is modified by

intro-ducing a cut-off temperature T c (> Tf) below which all reaction ceases The difficulty which

forces this modification is known as the cold-boundary difficulty, and has been extensively discussed in the literature.

4 X and M, are usually temperature dependent, but here we approximate them by constants.

12 THE MATHEMATICS OF COMBUSTION

Here Le = X./Cp|xn is the Lewis number, 6 is a nondimensional activation

energy, and D is proportional to B and, in addition, depends on p 0 ; T and

x are now dimensionless variables Typical solutions are shown in Fig 1.

Later we shall discuss an analytical approach but before that we want toreturn to the matter of kinetic modelling

5 Kinetics revisited Whether or not the simple kinetics embodied in (33)

is good enough depends on the goal of the analysis The parameters at ourdisposal can always be chosen so that the theoretical flame speed coincideswith the experimental value for specific upstream conditions; this choice isnot unique The behavior of small perturbations, as in a stability analysissay, does not differ qualitatively with different choices of -y and p Thus thephysical mechanisms responsible for certain instabilities of the deflagrationwave can be satisfactorily explored by assigning convenient values to theseparameters (1 and 0, for example) Variations of flame speed with mixturestrength for fuel-lean hydrocarbon flames can be adequately fitted using (33)

On the other hand, for fuel-rich hydrocarbon flames, (33) is inadequatefor predicting flame-speed variations The two-component model (31) is sat-

isfactory for this purpose provided yi and -y2 are carefully chosen [3], butthis success is misleading since variations of flame temperature are inade-quately described As an example, consider the combustion of methane inair, represented by (31) A stoichiometric mixture contains, by weight, fourtimes more oxygen than methane For a lean mixture, equilibrium corre-sponds to the complete consumption of methane, so that a formula like (44)

is valid and, assuming air is 21% by volume O2, this has the form

where [CH4] is the mass fraction of methane This describes the transitionfrom a temperature of 293°K (room temperature) when [CH4] vanishes, to

FIG 1 Structure of the deflagration wave.

AN INTRODUCTION TO COMBUSTION THEORY 13the experimentally observed maximum of 2250°K at stoichiometry when[CH4] = 055 In practice there is a limit to how cool the flame can be; toolean a mixture gives too cool a flame, which cannot survive This lean limitoccurs at [CH4] = 03, at a temperature predicted by (46) of 1350°K, 900°Kbelow the maximum The measured limit temperature is 1500°K, 750°Kbelow the maximum, so that the error in using (46) is reasonably small.

On the rich side of stoichiometry, equilibrium for our irreversible modelcorresponds to the complete consumption of oxygen, and (46) is replacedby

where, at stoichiometry, [02] = 22 The rich limit corresponds to [O2] =.21 for which (47) predicts a flame temperature of 2180°K compared to theexperimental value of 1900°K Here the model predicts unrealistically smallvariations in temperature with mixture strength

Success of the model on the lean side can be attributed to the fact thatcomplete oxidation occurs (there is a surplus of oxygen) so that (31) rea-sonably approximates the initial and final states Failure on the rich side isdue to the fact that the oxidation is incomplete, CO and OH are present,and the temperature depends on the equilibrium of reversible reactions.Equation (31) is an inadequate representation

If this is important a more sophisticated model is required, such as [4]

where X represents the atoms C and H and the second (reversible) reaction

is assumed to be in equilibrium

6 Kinetic modelling in ignition This matter of kinetic modelling also

arises when we consider the question of auto-ignition, the genesis and olution of vigorous burning in a combustible material Auto-ignition is a self-accelerating process and one important mechanism is adequately described

ev-by one-step Arrhenius kinetics To demonstrate this, consider the spatiallyhomogeneous energy equation

with initial condition T(0) = T 0

If we ignore density and concentration changes (a realistic assumptionduring the early stages of the ignition process) and assume no algebraic

dependence of pi on T (for convenience only) then, after appropriate

seal-14 THE MATHEMATICS OF COMBUSTION

ings, (49) can be written in the nondimensional form

where, it will be recalled, 9 is a nondimensional activation energy

The heat generated by reaction raises the temperature, which increasesthe vigor of the reaction — this is the process of self-acceleration It is mostclearly seen when 0 is very large, a characteristic of combustion For then,

at early times, we can seek a solution to (50) of the form

whence

with solution

T\ is unbounded as t — » 1, implying a breakdown in the representation (51).

Self-acceleration becomes so strong that the temperature deviates cantly and rapidly from the initial value.5

signifi-The process that we have just described is known as thermal ignition Aquite different mechanism is responsible for the ignition of hydrogen at (say)400°C, 4mm/Hg pressure In this case the kinetics can be described by thefollowing system;

M represents all species Overall, hydrogen and oxygen burn to form water.The reactions have been placed in one of four categories The first of

these, (i), is called initiation', collision of molecular hydrogen with any other

5 In an inhomogeneous problem this runaway occurs in a small region and ultimately leads

to a deflagration wave travelling from the ignition point out into the unburned gas A complete treatment, valid in the limit 0 -> <*, is given by Buckmaster and Ludford [5] following work of Kassoy and (independently) Kapila Corrections which are mathematically important, but do not affect the fundamental picture, have been described in [6].

AN INTRODUCTION TO COMBUSTION THEORY 15

body generates atomic hydrogen Quantitatively this is not a significant action but it plays a crucial role in providing the seeds for the subsequentreactions

re-The second category contains the branching reactions Each atom of

hy-drogen or oxygen (the active bodies) breaks up one molecule, at the sametime generating two active bodies (OH, H or O) which, in turn, break uptwo molecules to form three active b o d i e s , This is a rapidly acceleratingprocess, responsible for ignition

The third category is propagation; the active body OH does its job and

produces a single active body

The fourth and final category is termination; H atoms are removed

Ter-mination can also occur when H atoms collide at a wall If terTer-mination isstrong enough it can overcome the branching, and ignition will not occur

In contrast to thermal ignition, the initial process here is essentially thermal since the branching/propagation reactions are weakly exothermic.The high temperatures eventually achieved come from the highly exothermicrecombination reaction (iv), which must eventually limit the number ofatomic particles that are generated

iso-A system such as (54) can be modelled in a way proposed by Zeldovich:

The first reaction has a high activation energy and no heat release, the secondhas no temperature dependence, large heat release For the ignition problem

a nonzero value of B must be specified initially as a substitute for the

ini-tiation or seeding reaction Thus, the ignition problem for this model can bedescribed, following Kapila [7], by the equations

this system is designed with the limit 0 —» °c in mind

The early development, analogous to (51) for the thermal model, is scribed by

de-16 THE MATHEMATICS OF COMBUSTION

The nature of the solution depends on the value of D If D is small,

termi-nation dominates; if D is large, branching dominates and ignition occurs

The critical value of D is the smallest root of

and for D greater than the critical TI, Y Al and Y Bl all become singular at

some critical time t c TI is logarithmically singular as in (53), but Y Al , Y Bl are much larger, behaving like (t c - t}~ 1 This signals the onset of ignition

as a process characterized in its early phases by 0(1) changes in Y A and Y B ,

small changes in T.

Just because a branching model must be used for a particular gas in order

to realistically simulate the ignition process, does not mean that a thermalmodel is inappropriate to describe the high temperature reactions sustainingdeflagration Different conditions can require different models, since reac-tions that are important under some circumstances can be unimportant underothers To demonstrate this, consider the Zeldovich model (55) describedby

these formulas are slightly different from (56) If D 2 ^> D\e~ QIT the

concen-tration of reactant B will be very small; any generated by the first reaction

is rapidly consumed by the second Indeed there will be a reactive balance

obtained formally by setting p B ~ 0 so that

and this determines Y B Thus

and the heat release, proportional to D 2 Ys, is proportional to

The results (61), (62) are equivalent to a one-step process involving reactant

A alone.

Peters [8] has carried out numerical calculations for a deflagration wave

approaching a cold wall with 0 = 20,000, T b (the adiabatic flame temperature)

equal to 2000 He found that for D 2 ID\ = 10~2 the results are as for a

one-step reaction; for D 2 ID\ = 10~8 there are significant differences

To conclude our brief and incomplete discussion of kinetics, it is to benoted that many of the advances that have been made in combustion in thelast ten years or so have been done in the context of one-step irreversiblekinetics with large activation energy Our understanding of ignition, flamestability, interaction between flames and flows, and diffusion flame struc-

AN INTRODUCTION TO COMBUSTION THEORY 17

ture, has increased significantly as a consequence of this work, but the vigor

of such developments has declined recently—there is a limit to what can bedone with but a simple model Clearly there is a need to develop mathe-matical frameworks that can cope with more sophisticated kinetic modelling,and although there have been attempts of this type, much remains to bedone

A word of caution, however Modelling should be done in a clear physicalcontext as in the case of the system (48), and the Zeldovich system (55);otherwise the effort is likely to be nothing more than a mathematical ex-ercise In a field that is dominated by empiricism and ad hoc intuitive rea-soning, it is an unfortunate fact that mathematical modelling has to contin-ually prove its worth, and failure to attend to the physics reinforces theprejudices of those who find such proofs difficult to appreciate

The kinetic modelling discussed here is necessarily simple, designed as astarting point for mathematical analysis For problems in which detailedcomplex kinetics is to be incorporated into numerical calculations, differentquestions can arise Often, reaction rates and their variations with temper-ature are not well known; orders of magnitude uncertainties are common-place What effect these uncertainties have on the results is the subject ofsensitivity analysis, discussed by Rabitz in Chapter II Sensitivity analysiscan be important to the modeller by suggesting that certain reactions areunimportant and so can be neglected

7 Acoustics and the small Mach number approximation The small Mach

number equations derived earlier preclude acoustic disturbances and yettheir effect is a subject of great importance Certain types of rocket motorinstabilities are intimately associated with such interactions Mathematicallythe problem is one of multiple scales; the length and velocity scales for theacoustic disturbance are much larger than those associated with the flamestructure, and only by accounting for this can we incorporate acoustics into

a small Mach number description

Consider the one-dimensional equations in the form

S is the entropy/

6 These are the equations needed to describe plane fast deflagrations and detonations They are the point of departure for Fickett's discussion of detonations in Chapter IV.

18 THE MATHEMATICS OF COMBUSTION

Small perburbations about a uniform chemistry-free state satisfy the tions

equa-where p, v, p, jfnow represent undisturbed variables To nondimensionalize

these equations, we take as our reference quantities pressure/?, temperature

T, density p, velocity c = VyRT (the sound speed), time X/pC p v 2 , and length

\/Ma pCpV Note that the characteristic time is the natural one, defined by

parameters that control the flame structure, but the characteristic length ismuch larger than the flame length identified in §4, if Ma is small

Equations (64) can now be written in nondimensional form:

If terms of order O(Ma2) are neglected, these are the equations of acoustics

If on the other hand v is taken as the reference velocity rather than c, and

is chosen for the length, the limit equations a

These are linearized versions of the small Mach number equations derived

in §3

In discussions of the interaction between acoustic waves and flames bothscales might have to be considered Waves propagating through flames arediscussed in Chapter V

AN INTRODUCTION TO COMBUSTION THEORY 19

8 The hydrodynamic model The deflagration wave discussed in §4 has

a thickness ~\/MCp, a length also identified in §7; typically this is less than

1 millimeter On a scale much larger than this the structure is unresolvedand the flame appears as a discontinuity of temperature, density, and ve-locity If the flame is curved it will generate a nonuniform flow field; thisflow is incompressible when the Mach number is small Thus it is necessary

to solve Euler's equations for a constant density fluid, on each side of afront across which there are connection conditions representing the con-servation of mass and momentum

If V is the speed of the front back along its normal (Fig 2), these connection

conditions are

T b is the adiabatic flame temperature and W = v nf + V is the adiabatic flame

speed, the speed of the front relative to the fresh gas Both T b and W are

specified quantities In the case of a stationary front (V = 0) we have (Fig.3)

The flow is refracted towards the normal on passage through the front.The single most important application of the hydrodynamic model is thestability analysis done independently by Darrieus and Landau They con-sidered infinitesimal perturbations of the plane flame and showed that it isunconditionally unstable The analysis will not be repeated here—it can befound in [9]

An elementary kinematic argument yields important insight into the ifestation of this instability (Fig 4) A corrugated front traveling normal toitself at a uniform speed will develop sharp ridges in those portions of thefront that are concave when viewed from the fresh gas These ridges pointtowards the burnt gas.7

man-One of the most interesting and important demonstrations of this instability

is seen when a nominally plane flame travels down a tube Such flames arestrongly curved and can be thought of as a single-celled manifestation of thehydrodynamic instability This defines one of the few interesting problems

in combustion that are purely hydrodynamical in nature, and we shall discuss

7 A full understanding of the consequences can only come from a nonlinear analysis that goes beyond the hydrodynamical model and accounts for the effects of curvature on flame speed; such treatments have recently been reviewed by Sivashinsky [10] Numerical solutions of el- egant, well-motivated model equations show that the flame breaks up into cells delineated by sharp ridges This type of structure is observed experimentally.

20 THE MATHEMATICS OF COMBUSTION

FIG 2 Hydrodynamic front.

it briefly, as a vehicle for certain ideas that are essential ingredients of thehydrodynamical problem

One effect of flame curvature is to increase the propagation speed down

the tube If this speed is V, fuel is consumed at a volumetric rate VA t where

A t is the cross-sectional area of the tube (Fig 5) This equals the flux across

the flame surface, WAf, where A/is the flame area, so that

We have already noted that the flow is refracted on passage through theflame In addition, a curved flame generates vorticity In a frame moving

with the flame so that V = 0 we can use the results (68) to calculate the

total head (//) of the fluid behind the flame This is given by the formula

The upstream vorticity can be assigned and here the natural choice is zero

so that ///is constant everywhere in the fresh mixture H b is constant oneach streamline of the burnt gas, but will vary from streamline to streamlineaccording to the variations of t// along the flame front

The generation of vorticity complicates the analysis There have beennumerical computations in which this vorticity is neglected, but this can only

FIG 3 Refraction through a stationary hydrodynamic front.

AN INTRODUCTION TO COMBUSTION THEORY 21

FIG 4 The evolution of ridges in a hydrodynamic front: (a) initial disturbance; (b) subsequent

shape.

be strictly justified for the special cases cr = 1 and <r -> °° The first choice

is an ingredient of Sivashinsky's models (small heat release) but the secondappears not to have been exploited That it is true can be seen from (70),for, as a -» oc with conditions in the fresh gas fixed, we have the order ofmagnitude estimates

whence

which is small The velocity of the burnt gas is large (0(aW)) and so its

tangential component at the front, being O(W), vanishes to leading order.

The normal component is the constant crW and the leading order pressure

is constant at the front also A global momentum balance for the burnt gasthen shows that, to leading order, the front must be plane; curvature, likethe vorticity, is a perturbation for the tube problem

FIG 5 Propagation of a flame down a tube.

22 THE MATHEMATICS OF COMBUSTION

FIG 6 Resolution of the kinematic incompatibility: (a) flame in a tube; (b) flame tip with

dead water; (c) flame tip with hydrodynamic singularity.

The nature of these perturbations has not been investigated and it might

be useful to do so since the nature of the flow is not clear in all respects.There is a difficulty at the point where the flame front intersects the wall.This intersection is not at right angles so that the refraction generates akinematic incompatibility between the burnt gas and the wall, a difficultyrecognized in [11] Zeldovich [12] has proposed the existence of a dead waterregion between the limiting streamline and the wall (Fig 6)

A similar difficulty occurs with plane flame tips, the plane version of thefamiliar Bunsen-burner flame (Fig 6), in which the line of symmetry replaces

the tube wall In the context of slender tips, Buckmaster and Crowley [13]

have described a solution without dead water in which the incompatibility

is resolved through a singularity (in turn, assumed to be resolved on a scalesmall compared to the hydrodynamical one) This solution agrees qualita-tively with observed flow fields

The dead water hypothesis for the confined flame is an attractive one, butwhy it should be appropriate there but not for the burner flame is not clear.These hydrodynamic descriptions are not uniformly valid and we know little

of the manner in which the nonuniformities should be resolved

9 Activation energy asymptotics for premixed flames In our discussion

of the hydrodynamical model we assumed that the flame speed W is a

spec-AN INTRODUCTION TO COMBUSTION THEORY 23

ified quantity It is to be calculated by solving the deflagration wave problemintroduced in §4 In this, as in most combustion problems, the reaction plays

a central role in the mathematical description In the context of one-stepirreversible kinetics the difficulties introduced thereby can be largely cir-cumvented by an asymptotic treatment, valid in the limit of infinite activationenergy This approach has been widely discussed in recent years [5], [9],[10], [14], [15], so that there is little need for another review in these pages.But neither can we ignore it, since it has come to play such an importantrole in the theory

The essential ideal of activation energy asymptotics is that, provided tants are present, the reaction rate is a maximum at points of maximumtemperature and is exponentially smaller (and so negligible) elsewhere—theArrhenius (exponential) factor is extremely sensitive to temperature In mostcontexts it then follows that reaction is confined to a thin zone called a flamesheet For the premixed flame the thickness of this zone is given by the

reac-estimate X/M C p • 1/0 where 6 is a dimensionless activation energy E/RT ref ,

where the reference temperature might be the adiabatic flame temperature

In this section we shall explore the consequences of this idea in the context

of premixed flames

On a scale defined by X/M C p the reaction zone, in the limit, has a welldefined normal We shall examine the structure of a small portion of thissheet, choosing the jc-axis to coincide locally with the normal We then have,within the sheet, the leading order balance

deduced from (45) when, for convenience, we choose -y = 1, (3 = 0 Thesheet is thin and the reaction vigorous, hence the balance between diffusionand reaction terms

It follows that

with solution

where C\ and €2 are constants This relation is valid throughout the reaction

zone

Now dT/dx can, in principle, assume 0(1) negative values immediately

behind the flame sheet, but the corresponding flame structure is known to

24 THE MATHEMATICS OF COMBUSTION

be unstable [16], so we shall suppose that dTldx is 0(1) there In the absence

of a significant falling-off in temperature as the gas leaves the sheet, reactioncan only cease when all the mixture is consumed, so that Evanishes behind

the sheet It follows that C\ vanishes and we shall define J* so that (74)

There is then a balance provided T differs by no more than an 0(0"*) amount from T*, so that within the reaction zone we may write

and

so that

On the hot side of the flame sheet, (T - T*) and the temperature gradient

both vanish, so that a single integration of (78) yields

8 The reader who wishes to identify a formal asymptotic balance should note that (T - T*)

= O(Q~ l ), d/dx = O(Q) so that 2) must be O(02 ).

AN INTRODUCTION TO COMBUSTION THEORY 25

On the cold side of the sheet the temperature drops sharply, so that theexponentials in (79) vanish It follows that the temperature gradient there isgiven by the formula

On the scale defined by \/M C p , the reaction zone, in the limit, is a surface

of discontinuity across which there is a jump in the gradients of T and Y.

Thus the reaction term in (45) can be replaced by a Dirac S-function ofstrength

10 The 8-function model and its application The formula (81) is an

asymptotic result, valid in the limit 0 —>• oo As such it should be used withappropriate asymptotic expansions beyond the flame sheet, matched withhigher order descriptions within the flame sheet if necessary The 8-functionmodel abandons this framework and simply replaces the reaction term bythe 8-function, with 0 taken as a finite parameter (although simplifications

valid when 0 is large may be adopted in the subsequent analysis) Y is set

equal to zero behind the flame sheet but no restriction is imposed on thetemperature gradient there

In view of the loose connection between the complex kinetics that properlydescribe real flames, and one-step irreversible kinetics, it is difficult to argue,

a priori, that such a model is inferior to a treatment based on rational totic analysis It has the advantage of being much simpler—certain technicaldifficulties that can arise in the asymptotic treatment are irrelevant And itcan be understood without any grasp of asymptotic theory It is, indeed,simple enough to be used as the foundation of an undergraduate textbookdealing with a great many important questions in flame theory, an oppor-tunity that is yet to be exploited We shall complete this section by applyingthe model to the deflagration wave of §4, and by a brief description of variousproblems in which either the model or its genesis has been used

asymp-Deflagration revisited We seek a solution to (45) subject to the boundary

conditions

Then with the choice ^ = 1 , ^ = 0, the reaction term is replaced by a

8-function whose strength is given by (81) with T* replaced by T b (other choices

for y and (3 lead to minor differences); this 8function can be located at x

-0

26 THE MATHEMATICS OF COMBUSTION

The equations on each side of the sheet are easily solved, so that we have

The jump in heat flux across the flame sheet has magnitude Y f so that

flame In this way the flame speed is determined, the essential goal of theanalysis The distinguished limit (77) is, in this case, a recognition of the

functional dependence of M on the activation energy.

Applications The description of the deflagration wave is the starting point

for a large number of important combustion problems, some of which willnow be described Undoubtedly others would pick different examples: thosepresented here were chosen on the basis of their importance, their intrinsicinterest, their relation to experimental results, and personal bias

(a) Stability of plane deflagration The problem of the linear stability of

the plane deflagration wave is one that is easily formulated but not easilysolved The undisturbed solution is described by simple formulas, and theperturbation equations are linear, but they have nonconstant coefficients.Only in the case of plane disturbances can this difficulty be circumvented

in a natural way The constant density model avoids the problem by coupling the fluid mechanics from the thermochemistry, adopting as thefundamental equations the set

de-where v is a specified velocity field In the case of a deflagration propagating

into a quiescent gas, v can be equated to zero everywhere The solution forplane deflagration is unchanged by this approximation, but perturbations to

it are governed by constant-coefficient equations These can be easily solved

to describe the evolution of disturbances proportional to e iky+at , where y is

measured in the plane of the undisturbed flame sheet T* is not constant and

not equal to the adiabatic flame temperature; it differs from it by a smallperturbation

The key parameter is Le, the Lewis number Indeed, we have here anexample of a Turing system in which instabilities arise because of the in-teraction of variables that diffuse at different rates The stability boundariesare shown in the wave-number, Lewis-number plane in Fig 7 for the case

0 —> oo, 0(Le - 1) = 0(1) At any wave number there is a band of stabilitywhich always includes Le = 1

AN INTRODUCTION TO COMBUSTION THEORY 27

FIG 7 Stability regions for plane deflagration.

The left stability boundary, for which a = 0, is associated with cellularflames, a common laboratory phenomenon The Lewis number is properlydefined by the diffusion coefficient of the deficient component of the mixture,and the conduction coefficient of the entire gas For light fuels such as meth-ane, fuel-lean mixtures can have values of Le small enough to trigger cellularinstabilities; for heavy fuels, the mixture must be rich

Because there is no single "most unstable" disturbance, the manifestation

of the instability is an unsteady one, despite the fact that a vanishes on thestability boundary An equation that describes weakly nonlinear disturb-ances has been derived and discussed by Sivashinsky; he and his coworkershave also considered several important extensions beyond the problem ofplane deflagration [10]

On the right stability boundary Im(a) does not vanish, so that this is sociated with pulsations or travelling waves Only rather exotic gases, such

as-as lean bromine/hydrogen mixtures, have values of Le large enough to reachthis boundary Nonlinear treatments (Hopf bifurcation) for this and relatedproblems (see (b), below) have been extensively discussed in recent years

by Matkowsky and his coworkers, mostly in the SIAM Journal on AppliedMathematics, e.g [17]

The full problem of flame stability, accounting for the fluid mechanics,remains to be solved For very small wave numbers the hydrodynamicalmodel of §8 is appropriate, and predicts unconditional instability If gravityeffects are added, long waves can be stabilized, but not short ones Thissuggests that stable plane flames can exist with gravity stabilizing the longwaves, conduction-diffusion effects (as represented by Fig 7) stabilizing theshort ones Whether this is true or not, and what shape the stability boundarywill take, are questions that could be answered numerically, but have notbeen Clavin and his coworkers, e.g [18], have explored the matter ana-lytically, using small wave-number expansions