B.F. Gray

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.

Unified theory of explosions, cool flames and two stage ignitions. Part 2

A unified (chain-thermal) theory is applied to cool flame and critical phenomena observed in the static oxidations of many organic compounds in the low-temperature region. The cool-flame oscillations, negative temperature coefficient, lobes on the critical curve and numerous physical discontinuities often observed are reproduced in the theory and causally linked thereby. General requirements of the type of kinetic scheme necessary are found, but the theory can throw no light on the species involved.

Analysis of chemical kinetic systems over the entire parameter space - I. The Sal’nikov thermokinetic oscillator

In this series of papers we re-examine, using recently developed techniques, some chemical kinetic models that have appeared in the literature with a view to obtaining a complete description of all the qualitatively distinct behaviour that the system can exhibit. Each of the schemes is describable by two coupled ordinary differential equations and contain at most three independent parameters. We find that even with these relatively simple chemical schemes there are regions of parameter space in which the systems display behaviour not previously found. Quite often these regions are small and it seems unlikely that they would be found via classical methods. In part I of the series we consider one of the thermally coupled kinetic oscillator models studied by Sal’nikov. He showed that there is a region in parameter space in which the system would be in a state of undamped oscillations because the relevant phase portrait consists of an unstable steady state surrounded by a stable limit cycle. Our analysis has revealed two further regions in which the phase portraits contain, respectively, two limit cycles of opposite stability enclosing a stable steady state and three limit cycles of alternating stability surrounding an unstable steady state. This latter region is extremely small, so much so that it could be reasonably neglected in any predictions made from the model.

A method for the complete qualitative analysis of two coupled ordinary differential equations dependent on three parameters

In this paper, recent advances in bifurcation theory are specialized to systems describable by two coupled ordinary differential equations (ODEs) containing at most three independent parameters. For such systems, by plotting in the relevant parameter plane the locus of successively degenerate singular points, a complete description of all the qualitatively distinct behaviour of the system can be obtained. The description is in terms of phase portraits and bifurcation diagrams. Even though much use is made of existing results obtained via local analyses, the results of this technique cover the entire parameter space. Furthermore, because the information is built up in successive stages the question of whether the parameters universally unfold a given degeneracy does not arise. This can mean a major saving in effort, particularly for degenerate Hopf points. Finally if, as is often the case, the parameters appear in the system in a simple way, the procedure can be applied analytically because the variables (which will appear non-linearly) can be used to parametrize the relevant loci.

The ignition of hygroscopic combustible materials by water

Abstract The problem of autoignition by the absorption of moisture has been postulated to occur in a number of situations of high practical importance. In this article we examine this phenomenon from a theoretical point of view. The problem is formulated as a combination of classical criticality theory and the critical initial value problem as defined more recently. We shown that the phenomenon of wetting-induced ignition (WII) is possible in certain regions of parameter space, but however close the dry material may have been to criticality before wetting a finite amount of water is required to cause ignition. Equally interesting is the finding, that for a given material, WII is impossible in sufficiently small samples but possible in larger ones. This reveals an important flaw in scaling test procedures.

Disjoint bifurcation diagrams in combustion systems

It is shown that the classical non-dimensionalisation of many simple combusting systems is sufficiently nonlinear to conceal some of the differences in behaviour shown when different bifurcation parameters are varied quasi-statically to produce bifurcation diagrams. Variations of ambient or feed temperature, in particular, can produce a bifurcation diagram where the upper stable state is on a branch which is disjoint from the rest of the diagram, indicating that the combustion state cannot be extinguished by lowering this temperature to any physically attainable level.

Self-heating and drying in two-dimensional bagasse piles

This paper describes a two-dimensional model for self-heating and changes in water levels in bagasse piles of constant rectangular or triangular cross section. (Bagasse is the residue, mainly cellulose, that remains after sugar has been extracted from sugar-cane.) After milling, the bagasse has almost 50% water by weight, as hot water is used to remove the last of the sugar. The bagasse can be used as fuel in electrical power stations, but needs to be dried out before use. This paper discusses the way in which the drying out of a pile depends on the ambient conditions, and the shape and size of the pile. Accordingly, the energy equation, and equations for liquid water, water vapour and oxygen are solved numerically using the method of lines. The equations include terms describing heat conduction, diffusion of water vapour and oxygen, condensation and evaporation and an Arrhenius self-heating term. In addition, recent measurements show that there is also self-heating due to the presence of water in the bagasse, with a maximum effect near 60 °C, which is modelled by a modified Arrhenius expression. The local maximum in the heat release curve for the problem leads to approximate steady-state behaviour on short time scales that eventually is lost as the pile dries out. This interesting physical behaviour motivates an approximate analytical model for the rate at which liquid water is reduced in the pile. Analytical and numerical results are presented for a variety of pile configurations and some fairly general conclusions are drawn.

The influence of initial temperature-excess on critical conditions for thermal explosion

Abstract In the conductive theory of thermal explosion, an exothermic system is considered to be an ignition hazard only if the numerical value of the dimensionless group δ = Qσa 0 2 A exp (− E/RT a κ(RT a 2 /E) exceeds a critical value. [Here Q is the exothermicity, σ the density, a0 the half-width, A exp (− E RT a ) the rate constant at ambient temperature, and κ the thermal conductivity.] This classical approach implicitly assumes that the material is initially assembled at or near to ambient temperature. Such initial conditions represent only a subclass of the whole problem. In some situations of technical importance the reactant may be initially considerably above Ta. The present paper considers the temperature evolution in bodies subject to Frank-Kamenetskii boundary conditions but which are assembled with a positive, uniform temperature-excess. It is shown that in the usual exponential approximation, any system with δ ⩽ δcr has a critical initial temperature, above which thermal runaway occurs. Exact numerical results for the critical conditions are presented for the three class A geometries (infinite slab, infinite cylinder, and sphere). Very accurate analytical approximations are also provided. For an Arrhenius rate-law, ignition cannot occur in this way for very low δ δ ex ∼ O[ exp (−ϵ − 1 2 )] and ϵ = RT a E . A comparison is made between the predictions of this model and the critical conditions observed in an actual, practical example; the spontaneous ignition of piles of bagasse (extracted sugar cane) for which δ ≈ 0.02 ⪡ δcr.

On the determination of critical ambient temperatures and critical initial temperatures for thermal ignition

Abstract A new set of dimensionless parameters and variables is defined with a view to simplifying and clarifying the theoretical determination of the initial ambient and the critical initial temperatures in spatially uniform self-heating systems. With these new variables the determination of the critical ambient temperatures can be achieved without the iterative procedure necessary when using the traditional variables with the Arrhenius function. The determination of the critical initial temperature using the new variables highlights a significant new source of error which arises when the exponential approximation arises. There is neither necessity nor advantage in making this approximation with these new variables.

The heat release rates and cool flames of acetaldehyde oxidation in a continuously stirred tank reactor

Abstract The continuously stirred tank reactor (CSTR) technique is applied toequimolr acetaldehyde oxygen mixtures across a wide range of vessel residence times. Heat release curves are obtained in each case in the regions where stable steady state exists. In the regions of oscillatory cool flames the steady state is unstable and cannot be obtained experimentally. The oscillatory region always contains part of the region of negative temperature coefficient, as expected on theoretical grounds. At very short residence times a new effect is shown in this system, i.e., a discontinuous transition from the familiar type of cool flames to flames with quite different amplitude, frequency, and shape. In the immediate region of this bifurcation the system shows “indecision” or interspersed flames of each type for a short period. The reverse transition occurs at quite a different ambient temperature, showing hysteresis of as much as 25K. A tentative qualitative description of the nature of this bifurcation is given.