R. O. Weber

Combustion waves for gases (Le = 1) and solids (Le→∞)

The traditional combustion problems of calculating flame speeds for a premixed gaseous fuel and for a premixed solid fuel are revisited using a simpler (than previously) non–dimensional temperature. It turns out to be possible to carry out asymptotic calculations for flame speed and the agreement with corresponding numerical calculations is remarkably good. In each case the uniqueness of the speed is considered using phase plane methods, with a little effort to determine the nature of the ‘cold’ critical point. Consideration of the stability of the travelling combustion wave fronts suggests a period doubling route to chaos for the premixed solid fuel (as the exothermicity is decreased) and corresponds with previous work using different non–dimensional temperature and parameters.

Evans Function Stability of Combustion Waves

In this paper we investigate the linear stability and properties, such as speed, of the planar travelling combustion front. The speed of the front is estimated both analytically, using the matched asymptotic expansion, and numerically, by means of the shooting and relaxation methods. The Evans function approach extended by the compound matrix method is employed to numerically solve the linear stability problem for the travelling wave solution.

An oscillatory route to extinction for solid fuel combustion waves due to heat losses

A numerical method is used to show that heat loss increases can lead to a period–doubling route to the cessation of propagation of solid fuel combustion. Oscillatory combustion waves are found in certain regions of the parameter space. The behaviour of these oscillatory waves becomes more complex as the heat loss is increased until extinction of the combustion reaction occurs. Large excursions in temperature, above the adiabatic temperature, are possible in the non–adiabatic case close to this extinction point.

Evans function stability of non-adiabatic combustion waves

In this paper we investigate the linear stability and properties of the planar travelling non–adiabatic combustion front for the cases of zero and non–zero ambient temperature. The speed of the front is estimated numerically using the shooting and relaxation methods. It is shown that for given parameter values the solution either does not exist or there are two solutions with different values of the front speed, which are referred to as ‘fast’ and ‘slow’. The Evans function approach extended by the compound–matrix method is employed to numerically solve the linear–stability problem for the travelling–wave solution. We demonstrate that the ‘slow’ branch of the solutions is unstable, whereas the ‘fast’ branch can be stable or exhibits Hopf or Bogdanov–Takens instability, depending on the parameter values.

Flame balls with thermally sensitive intermediate kinetics

Spherical flame balls are studied using a model for the chemical kinetics which involves a non-exothermic autocatalytic reaction, describing the chain-branching generation of a chemical radical and an exothermic completion reaction, the rate of which does not depend on temperature. When the chain-branching reaction has a large activation temperature, an asymptotic structure emerges in which the branching reaction generates radicals and consumes fuel at a thin flame interface, although heat is produced and radicals are consumed on a more distributed scale. Another model, based more simply, but less realistically, on the generation of radicals by decomposition of the fuel, provides exactly the same leading order matching conditions. These can be expressed in terms of jump conditions across a reaction sheet that are linear in the dependent variables and their normal gradients. Using these jump conditions, a reactive–diffusive model with linear heat loss then leads to analytical solutions that are multivalued for small enough levels of heat loss, having either a larger or a smaller radius of the interface where fuel is consumed. The same properties are found, numerically, to persist as the activation temperature of the branching reaction is reduced to values that seem to be typical for hydrocarbon chemistry. Part of the solution branch with larger radius is shown to become stable for low enough values of the Lewis number of the fuel.

NON-ADIABATIC COMBUSTION WAVES FOR GENERAL LEWIS NUMBERS: WAVE SPEED AND EXTINCTION CONDITIONS

This paper addresses the effect of general Lewis number and heat losses on the calculation of combustion wave speeds using an asymptotic technique based on the ratio of activation energy to heat release being considered large. As heat loss is increased twin flame speeds emerge (as in the classical large activation energy analysis) with an extinction heat loss. Formulae for the non-adiabatic wave speed and extinction heat loss are found which apply over a wider range of activation energies (because of the nature of the asymptotics) and these are explored for moderate and large Lewis number cases - the latter representing the combustion wave progress in a solid. Some of the oscillatory instabilities are investigated numerically for the case of a reactive solid.

Combustion pseudo-waves in a system with reactant consumption and heat loss

A model for the combustion of a reactant with heat loss in an infinite two-dimensional layer or an infinitely long circular cylinder is derived. Using center manifold techniques the model is shown to reduce to a one-dimensional model on an infinite region for both geometries. For an exothermic first-order reaction with Arrhenius temperature dependence, a numerical method is used to calculate solutions and comparisons are made with the no-reactant consumption case. Traveling pseudo-waves are shown to exist and their speeds determined. A first-order estimate of the reaction zone thickness, obtained by perturbation methods, is shown to be in excellent agreement with the full numerical solution.

Combustion waves: Non adiabatic

A non-adiabatic thermal model of combustion is presented. The model admits ignition and extinction waves as exact solutions. The shape of these waves and their speed are determined numerically. Phase plane equations are used to understand the uniqueness of the wavespeeds and to determine watershed criteria for initial conditions.

Combustion leftovers

Combustion processes almost never completely exhaust all of the available fuel. In this paper, we will consider three combustion scenarios (back-to-back premixed flames in stagnation point flow, travelling combustion waves, and microgravity spherical flame balls) and show how to calculate the amount of fuel which will be left over no matter how long we allow the combustion processes to continue. Mathematical biologists will be familiar with an analogue of this in disease modelling, where there are always some susceptibles left after the passage of an epidemic. In a bushfire or forest fire context, this is seen as unburnt solid fuel. The main reason for combustion leftovers is the heat loss, which can never be completely eliminated.