Features of a tube flame

Both inside and outside the laboratory it is commonly observed that a premixed flame can be stabilized at the mouth of a tube through which the mixture passes. Such a flame, usually conical in shape though not necessarily so, can be conveniently divided into three parts: the tip, the base (near the rim of the tube), and the bulk of the flame in between.

Elementary considerations of the flame speed and the nature of the flow explain the conical shape (see Figure 8.2 and the accompanying discussion). Simple hydrodynamic arguments provide salient features of the associated flow field, as we shall see in section 2; additional details are outlined in section 6.

An understanding of the nature of the combustion field in the vicinity of the rim is crucial in questions of existence and stability of the flame. Gas speeds near the tube wall are small because of viscous effects, so that if the flame could penetrate there it would be able to propagate against the flow, traveling down the tube in a phenomenon known as flashback. In actuality, the flame is quenched at some distance from the wall through heat loss by conduction to the tube (for a stationary flame); this prevents it from reaching the low-speed region. Such quenching enables unburnt gas to escape between the flame and the wall through the so-called dead space, a phenomenon that is described mathematically in section 5.

Existing combustion books are primarily phenomenological in the sense that explanation, where provided, is usually set in an intuitive framework; when mathematical modeling is employed it is often obscured by ad hoc irrational approximation, the emphasis being on the explanation of existing experimental results. It is hardly necessary to add that the philosophy underlying such texts is scientifically legitimate and that they will undoubtedly stay in the mainstream of combustion science for many years to come. Nevertheless, we are of the opinion that there is need for texts that treat combustion as a mathematical science and the present work is an attempt to meet that need in part.

In this monograph we describe, within a mathematical framework, certain basic areas of combustion science, including many topics rightly covered by introductory graduate courses in the subject. Our treatment eschews sterile rigor inappropriate for a subject in which the emphasis has been physical, but we are deeply concerned with maintaining clear links between the mathematical modeling and the analytical results; irrational approximation is carefully avoided. All but the most fastidious of readers will be satisfied that the mathematical conclusions are correct, except for slips of the pen.

Although the material covered inevitably reflects our special interests and personal perspectives, the entire discussion is connected by a singular perturbation procedure known as activation-energy asumptotics. The description of reacting systems characterized by Arrhenius kinetics can be simplified when the activation energy is large, corresponding to an extreme sensitivity to temperature.

Nonuniformities

An understanding of the response of a premixed flame to nonuniformities in the gas flow is important in many technological situations. To sustain a flame in a high-velocity stream the turbine engineer must provide anchors, and these generate strong shear. The designer of an internal combustion engine is concerned with the burning rate in the swirling flow of the mixture above the piston. Turbulence is ubiquitous; then the flame is subject to highly unsteady shear and strain. These situations are extremely complicated and it is unlikely that mathematical analysis will ever provide detailed descriptions; those must be left to empirical studies augmented by extensive numerical computations. Nevertheless, analysis of the response of a flame to a simple shear, for example, can provide useful insight into the interaction mechanism in more complex situations.

Moreover, there are simple circumstances in which such an analysis has direct significance. A burner flame is subject to shear in the neighborhood of the rim, and its quenching depends on the local character of that shear. A flame immersed in a laminar boundary layer experiences both shear (due to velocity variations across the layer) and strain (due to streamwise variations) and its quenching will depend on their local values.

The problem

Chapter 4 was concerned with the steady combustion of the gases produced by vaporization of a linear condensate at its surface through pyrolysis or evaporation. The results were characterized by response curves of burning rate versus pressure (represented by the Damköhler number). If the applied pressure varies in time, then so also must the burning rate; the nature of the dependence is examined in this chapter.

The effect of variations in pressure on solid pyrolysis has received considerable attention because of its relevance to the stability of solid-propellant rocket motors. Acoustic waves bouncing around the combustion chamber will impinge on the propellant surface and thereby generate fluctuations in the burning rate. These fluctuations will affect the reflected wave which, it is argued, might have a larger amplitude than the incident wave. If so, the transfer of energy (provided it is greater than losses through dissipation and other mechanisms) implies instability.

Our discussion will focus on the response of a burning condensate (solid or liquid) to an impinging acoustic wave. Mathematically we must deal with the disturbance of a steady field containing large gradients; consequently, a frontal attack on the governing equations is not feasible. Six regions can be distinguished: condensate, preheat zone, flame, burnt gas, entropy zone, and far field; without rational approximation, the discussion soon degenerates into either a numerical or ad hoc analysis (or both).

Approach

The development of the equations governing combustion involves derivation of the equations of motion of a chemically reacting gaseous mixture and judicious simplification to render them tractable while retaining their essential characteristics. A rigorous derivation requires a long apprenticeship in either kinetic theory or continuum mechanics. (Indeed, the general continuum theory of reacting mixtures is only now being perfected.) We choose instead a plausible, but potentially rigorous, derivation based on the continuum theory of a mixture of fluids, guided by experience with a single fluid. Ad hoc arguments, in particular the inconsistent assumption that the mixture itself is a fluid for the purpose of introducing certain constitutive relations, will not be used.

Treating the flow of a reacting mixture as an essentially isobaric process, the so-called combustion approximation, is a safe simplification under a wide range of circumstances if detonations are excluded. But the remaining simplifications, designed as they are solely to make the equations tractable, should be accepted tentatively. They are always revocable should faulty predictions result; for that reason they are explained carefully. Nevertheless, whosoever is primarily interested in solving nontrivial combustion problems, as we are, can have the same confidence in the final equations as is normally placed in the equations of a non-Newtonian fluid, for example.

Prologue

The propagation speed of a plane deflagration wave is extraordinarily sensitive to changes in the flame temperature. The result (2.22) shows that, for fixed D and Js, an O(1) change in Tb produces an exponentially large change in the burning rate. More modestly, an O(θ–1) change in Tb produces an O(1) change in flame speed; it is perturbations of such a magnitude that concern us in this chapter.

Such a change may be engineered for the unbounded flame by an O(θ−1) change in Tf, an elementary example that is not of great importance either mathematically or physically. A much more interesting example is cooling by heat loss through the walls of a uniform duct along which the flame is traveling. It is well known that flames cannot propagate through very narrow passages (a key safety principle where explosive atmospheres are involved), and this can be adequately explained by such a heat-loss mechanism, as we shall see.

Perturbations of the same magnitude can also be produced by changes in the size of the duct that occur over distances O(θ). Now there will be slow variations in the combustion field developing on a time scale O(θ). Such slow variations can even be self-induced by residual perturbations of the initial conditions (on that time scale) in the absence of boundary perturbations. In all such cases an obvious conjecture is that the flame velocity is not close to the unperturbed value.

Diffusion flames

Earlier chapters have been concerned with flames for which the reactants are supplied already mixed. When two reactants are initially separate and diffuse into each other to form a combustible mixture, the term diffusion flame applies. A Bunsen burner with its air hole closed supports a diffusion flame between the gas supplied through the tube and the surrounding oxygen-rich atmosphere. A candle supports a vapor diffusion flame, so called because the fuel is produced by liquefaction and subsequent evaporation of the wax caused by the heat of the flame.

Premixed and diffusion flames share certain features, but there are important differences. For example, there is no unlimited plane flame with fuel supplied far upstream, oxidant far downstream. The only bounded solutions of the chemistry-free equations behind the flame sheet are constants, so that the oxidant fraction would be constant there, with no mechanism to generate the necessary flux towards the reaction zone. Supplying the fuel at a finite point upstream does not change this picture. Moreover, since cylindrical flames are geometrically attenuated versions of plane flames (cf. section 7.2), there is no cylindrical diffusion flame either.

To gain insight into the nature of diffusion flames we seek a simple onedimensional configuration that has some physical reality. One possibility is to introduce the oxidant at a finite point in the plane flame; details of that choice have been worked out by Lu (1981).

The flame as a hydrodynamic discontinuity

The plane premixed flame discussed in Chapters 2 and 3 is an idealization seldom approximated, since in practice the flame is usually curved. A Bunsen burner flame is inherently so; but even under circumstances carefully chosen to nurture a plane state, instabilities can precipitate a multidimensional structure. Such flames have been extensively studied using a hydrodynamical approach (Markstein 1964, p. 7); a brief description of it will provide an introduction to our subject.

On a scale that is large compared to its nominal thickness λ/cpMr, the flame is simply a surface across which there are jumps in temperature and density subject to Charles's law (as appropriate for an isobaric process). Deformation of the surface from a plane is associated with pressure variations in the hydrodynamic fields of the order of the square of the Mach number (see section 1.5). These small pressures jump across the surface to conserve normal momentum flux. Because Euler's equations for small Mach number hold outside the flame (cf. the end of section 3), the temperature and density do not change along particle paths; so that for a flame traveling into a uniform gas the temperature and density ahead are constant and the flow is irrotational. The flow behind is stratified, however, since flame curvature generates both vorticity and nonuniform temperature jumps. Variations in temperature from the adiabatic flame temperature are usually neglected everywhere (a matter we shall treat later); but vorticity generation cannot be ignored so that, even though Euler's equations apply behind the flame, the flow is not potential there.

Models of mixtures

In general, two-reactant flames can be classified as diffusion or premixed. In a premixed flame the reactants are constituents of a homogeneous mixture that burns when raised to a sufficiently high temperature. In a diffusion flame the reactants are of separate origin; burning occurs only at a diffusion-blurred interface.

Both kinds of flames can be produced by a Bunsen burner. If the air hole is only partly open, so that a fuel-rich mixture of gas and air passes up the burner tube, a thin conical sheet of flame (typically blue-green in color) stands at the mouth; this is a premixed flame. Any excess gas escaping downstream mixes by diffusion with the surrounding atmosphere and burns as a diffusion flame (typically blue-violet). If the air hole is closed, only the diffusion flame is seen; if all the oxygen is removed from the atmosphere (but not from the air entering the hole), only the premixed flame is observed. (Yellow coloring due to carbon-particle luminosity may also be seen.)

A premixed flame can also be obtained by igniting a combustible mixture in a long uniform tube; under the right conditions, a flame propagates down the tube as a (more or less) steady process, called a deflagration wave. The Bunsen burner brings such a wave to rest by applying a counterflow and stabilizes it with appropriate velocity and thermal gradients.

Synopsis

Ignition has been discussed in three contexts: burning of a linear condensate (Chapter 4), spherical diffusion flames (Chapter 6), and spherical premixed flames (Chapter 7). Based upon the S-shaped response curves determined by steady-state analyses it is argued that the burning rate will jump from a weak, almost extinguished, level to a vigorous one as the pressure (and, hence, the Damköhler number) is increased through some critical value (corresponding to the lower bend of the S). The occurrence of an intrinsically unsteady phenomenon, appropriately called ignition, is thereby inferred from results inherent in the steady state.

Such analyses have a fairly long history in combustion investigations, the simplest example being the thermal theory of spontaneous combustion identified with the name of its originator, Frank-Kamenetskii (1969). Section 2 presents a mathematical version of that theory, which shows an early appreciation of activation-energy asymptotics (though not in the formal sense of this monograph).

Even though steady-state analyses are undoubtedly useful, descriptions of the unsteady state are also needed; providing them is the main goal of this chapter. The three contexts already considered are too complicated for our purposes; we concentrate instead on a simpler problem containing the essential feature of ignition: the evolution of a deflagration wave from an unburnt state.

Consider a thermally insulated enclosure containing a mixture at sufficiently low temperature for the reaction to be very weak. Since the heat released cannot escape, the temperature must rise, albeit very slowly.

Cylindrical flames

Although it has not been studied to the same extent as the plane premixed flame, the cylindrical premixed flame is (in principle) almost as easy to produce. The reacting mixture is supplied through the surface of a circular cylinder and is induced to flow radially by means of end plates sufficiently close together. The flame then forms a coaxial cylinder that can be observed through the end plates, which should be transparent and good thermal insulators.

Analytically the cylindrical flame lies between the plane and spherical flames. The structure of its reaction zone is the same as that of the plane flame, with temperature constant beyond, so that there is none of the curvature effect of the spherical flame. Like the spherical flame it does not exhibit the cold-boundary difficulty: the mixture must be introduced at a finite radius, which can be so small that a line source is effectively formed. In their attempt to treat curved flames Spalding & Jain (1959) use plane-flame analysis on the spherical flame, where it is never valid, and neglect the cylindrical flame, where it is always valid. The results, however, are qualitatively correct (Ludford 1976).

Sections 2 and 3 show how cylindrical geometry modifies a premixed flame (Ludford 1980). For simplicity we shall consider a single reactant under going a first-order decomposition and take ℒ = 1. Spherical premixed flames, which have many similarities to spherical diffusion flames, will be treated in sections 4–6.

Scope

Flame stability is a very broad subject; both the mechanisms and manifestations of instability assume many forms. A whole monograph could be devoted to it if a less analytical approach than ours were adopted. It has been studied since the 1777 observations of Higgins (1802) on singing flames, a phenomenon involving interaction between an acoustic field and an oscillating flame of the kind described in Chapter 5. Such interactions were considered to be merely laboratory curiosities for many years, but lately they have assumed technological importance in the development of rocket motors and large furnaces. This type of instability is well understood qualitatively (Chu 1956). Other combustion phenomena thought to be due to instability are, in general, poorly understood; in some cases instability is merely suspected of playing a role but no certain evidence is yet available. The following examples are representative of the abundance of instability phenomena in combustion.

The stability of burner flames depends on an appropriate interaction between the flame and its surroundings, in particular, a flux of heat from the flame to the burner rim. The role of this flux in anchoring the flame and preventing blow-off has already been mentioned in Chapter 9.

Propagation limits are important questions in the study of premixed flames, with steady, sustained combustion being possible only for a certain range of the fuel to oxidant ratio. If a mixture is too rich or too lean it will not burn, and instability may well be involved.