Bounds for moving boundary problems with two chemical reactions

Abstract

HEAT-DIFFUSION moving boundary problems are usually referred to as Stefan problems and are nonlinear owing to the two boundary conditions imposed at the moving front. The “shrinking core” model for fluid-solid reactions is an important moving boundary problem which occurs in various chemical engineering applications. In this paper we examine the model considered by Krishnamurthy and Shah [a], which arises from an essentially instantaneous fluid-solid reaction (giving rise to a moving reaction front), together with a slower pseudofirst order reaction occurring in the region behind the reaction front. This problem arises in the oxydesulphurization of coal which contains both organic and inorganic sulphur. The oxidation of the former, and carbon is very slow whilst that of the latter is very rapid. As is usual with moving boundary problems, the motion of the boundary, in this case a reaction front, is of primary interest. We develop a unified approach enabling bounds on the motion of the reaction front to be obtained for planar, cylindrical and spherical geometries, with or without mass transfer at the surface. By generalizing the method employed in Dewynne and Hill [2] the problem is reduced to a pair of integrodifferential equations, enabling the motion of the reaction front to be formally integrated. This formulation and the physically obvious inequalities 0 s c c 1 for the nondimensional concentration enable simple upper and lower bounds to be obtained for the motion of the reaction front. Following the method used in Hill [5], the pseudo-steady state concentration is shown to be an upper bound on the actual concentration, and this gives rise to an improved upper bound on the motion of the reaction front. Moreover, following the method used in Hill and Dewynne [7] the lower bound is improved by utilizing a known upper bound on the speed of the reaction front. We are thus able to obtain a number of useful bounds on the motion of the reaction front and for the time taken for the rapid reaction to extinguish itself. The results given here extend and generalize Hill [6], which gives some of these bounds for the spherical geometry only. We consider a slab of width a, an infinite circular cylinder of radius a, or a sphere of radius a, consisting of an inert solid matrix in which various solid reactants are supported. The structure is porous, allowing a fluid to diffuse into the inert matrix. At time t* = 0 the surface rx = a is subjected to a concentration cg of a fluid reactant. which is held at c0 thereafter. The fluid is assumed to react instantaneously with one of the solid reactants, giving rise to a reaction front Z?“(P), whilst in the region between the surface and reaction front a second slower reaction occurs. We assume this second slow reaction is pseudo-first order with respect to the