The existence of steady states to a combustion model with internal heating

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Introduction

In recent numerical studies [5], [6], [7], [20], [26], the authors examined the effect of a constant source of heat placed at the centre of a reactive solid X. Reactions with and without oxygen, in all class-A geometries (slab, concentric cyclinders, concentric spheres) were considered. The background and the motivation for these numerical studies was the thermal decomposition of combustible solids such a bulk powders and coal dust, occurring for example in many operations in the process control industry. Equipment such as blenders, mills and screw feeders can develop into localized sources of heat from mechanical impact or failure of a bearing. This source of heat, or ‘hot-spot’, can be sufficient to cause slow local combustion of the surrounding material, or even initiate a self-propagating combustion wave. Clearly an understanding of the conditions under which thermal decomposition occurs and the extent of the decomposition are crucial for hazard prevention. In this work we study the existence and nonexistence of steady states to the model in [5], generalized to include the combustion of gases and non class-A shapes. That is, we imagine a gas or a solid containing a source of heat placed arbitrarily in its interior—independent of the heat produced by chemical reaction (see Fig. 1).

Similar numerical studies [5], [6], [7], [20], have been carried out in, for example, [15], [27], where the authors focus on predicting the critical initial conditions for the safe storage of potentially exothermic materials; a study of spontaneous ignition. On the other hand, the character and properties of a fully developed combustion wave have been addressed in, amongst others, [14], [22], [23]. Analytical progress in describing the ignition event using the equations in [5] goes back to Liñán and Williams [19], and Kapila [16], in which the authors develop an asymptotic theory for the ignition of a solid slab subjected to a constant heat flux. The former work neglects reactant consumption and in the latter the author only addresses the gaseous problem with equidiffusion (of heat and mass).

The purpose of the present study lies in the relationship between the steady-state solutions and the critical behaviour of the original system. This is most easily demonstrated by considering a single equation with Dirichlet data, describing the evolution of temperature and which can be considered as an eigenvalue problem. If one uses the Frank–Kamenetskii approximation (reaction rate ∝eu, where u is temperature), it is well known that the spectrum of the steady-state problem is bounded above. Furthermore, for a reaction-rate coefficient smaller than the upper bound, i.e. when a steady-state exists, the solution relaxes to one of the steady states, provided the initial condition is small enough. For values greater than the upper bound the solution can become infinite in finite time; a situation referred to as blow-up. Moreover, a study regarding reactant-dependent equations has been performed in [9], [10], [11], yielding results of a similar nature. For further details see [9] and references therein. These ideas will be made more precise in the context of our problem in a future publication, in which we examine the behaviour (including probable multiplicity) of steady-state solutions as various parameters are varied. Here we concentrate solely on the question of the existence of at least one steady state.

In this work we use a reaction-rate of the type e−1/u, corresponding to the nondimensionalization used in [5], [6] and to the full ignition problem for both gases and solids. We show that steady-state solutions exist for all Damköhler numbers—a measure of the rapidity of the reaction (see [8] for details). The Frank–Kamenetskii approximation [12] in this model of the burning corresponds to a perturbation problem about a reference temperature and concentration, and is derived under certain asymptotic-limit assumptions and by assuming that the power of the heat source is asymptotically small. This situation is entirely realistic for bulk materials such as compost, moist wood chips, lead azide and ammonium nitrate, where the reactant consumption is very small. We are able to demonstrate that steady-state solutions to the perturbation problem do not exist for all finite Damköhler numbers and are able to find an upper limit of Damköhler number (above which no solution exists).

In the next section we provide some definitions necessary for the work that follows. The mathematical problem is stated in Section 3 and in Section 4 we lay the foundation for the analysis. Based on maximum and comparison principles, in Section 5 we prove that steady-state solutions of the full problem lie in an invariant set, with respect to boundary conditions. The proof of existence for the full problem then appears in Section 6. The remaining part of the paper is devoted to the perturbation problem in which the Frank–Kamenetskii approximation is used.

Section snippets

Preliminaries and notation

Cα(Ω) as the space of Hölder continuous functions in Ω⊂Rn, with exponent 0⩽α⩽1. A norm on these spaces is defined by|f|Cα(Ω)=|f|αΩsupΩ|f|+[f]αΩ,where [f]αΩsupx,y∈Ω|f(x)−f(y)|/|x−y|α. We shall include the space of continuous functions C(Ω) among the Cα(Ω) as those corresponding to α=0. Cα,k(Ω) consists of functions whose partial derivatives up to and including order k have a finite |·|αΩ norm, so a norm on these spaces is given by|f|Cα,k(Ω)=|f|k,αΩj=0ksup|β|=jsupΩ|Dβf|+sup|β|=k[Dβu]αΩfor

Mathematical model

For simplicity, and as in [5], we assume that the fully burnt material occupies the same volume and has the same characteristics as the unburnt material and undergoes reaction according to the Arrhenius law. The former assumption is quite valid for reactions that do not involve a change of phase, and the latter assumption is common in combustion modelling. A general form of the equations representing temperature T and reactant mass fraction X in a combustion process is as follows:ρCp∂T∂t′−κ2

Fixed points of the abstract problem

The existence proofs are based on locating fixed points of a completely continuous (relatively compact and continuous) map using Leray–Schauder degree deg(a,b,c), an integer measuring in a continuous fashion the number of zeros of the map. Suppose that we seek the Leray–Schauder degree of the equality f(x)=y, where f:E⊃Σ→E and y is a point in the range of f. Then a,b and c in the symbol above would respectively represent the map f, its domain Σ and the point y. The results from degree theory

A priori estimates

We shall need estimates for the classical solutions of problem , to ensure that the Leray–Schauder degree is defined—in the sense that Ø∂Σλ for λ∈[0,1]. For this we require the following result:

Theorem 5.1

Let u,v∈C2(Ω)∩C(Ω̄) be a solution of , . Then0⩽v⩽v̄andū⩽u⩽ū,where v̄=supx∈Γ1v0(x), ū=infx∈Γ1u0(x) and ū is a solution of the problem2ū=−Qωuv̄<∞onΩ,ū∂n=α>0onΓ2,ū=u0onΓ1.

Theorem 5.1 is a consequence of classical maximum principles and Lemma 5.1 below, which relates to the mixed boundary value

Existence for Type I—full reactant consumption

We now demonstrate the existence of solutions to problem , . Throughout, let the assumptions (BA) regarding Ω and ∂Ω hold. It will be necessary to extend the solution operator K, defined by (17), to one which is completely continuous. (In all that follows any extensions of K will also be labelled K). We construct such an extension by generalising K to a Sobolev space and employing imbedding properties.

Consider the linear problem2h=fonΩ,h=g1onΓ1,∂h∂n+βh=g2onΓ2,with f∈L2(Ω), g1W3/2,2(Γ1), g2W

Existence and nonexistence for Type II—small reactant consumption and weak power-sources

For many bulk materials such as moist wood chips, wool, compost, ammonium nitrate and lead azide, the approximation of no-reactant-consumption is a valid one since these systems vary little from their initial state. In such cases, we can go even further and introduce the so-called large activation energy limit Ea→∞, which leads naturally to a Frank–Kamenetskii type problem [12] (what we have called type II) for the temperature perturbation due to self heating. An interesting limit in the

Concluding remarks

For any Damköhler number, we have demonstrated the existence of steady-state solutions to a system representing the heating of a combustible material by an internal source. In contrast, steady-state solutions of the associated perturbation problem, arising from a high activation-energy asymptotics approach, exist only for Damköhler numbers below a critical value that is characterized by the principal eigenvalue of a Helmholtz problem.

In forthcoming work we will advance this study by

Acknowledgements

The first author is grateful to the EPSRC for financial support through Research Grant GR/R 22179 and to OCIAM, Oxford, UK for supporting a short visit.

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