On unsteady state moving boundary problems for non-plane geometries

Abstract

Many models for fluid-solid non-catalytic reactions in chemical engineering involve the solution of the diffusion equation for non-plane geometries subject to a moving boundary. One of the simplest cases is that in which diffusion of the gaseous reactant through the product layer controls the overall rate of the reaction. Even in this case the moving boundary introduces a non-linearity so that few analytical solutions have been attempted. The ratio of the densities of the fluid and solid reactants, β, is an important parameter in this model. In the limit as β → 0 the equations can be solved analytically. The resulting solution, referred to as the pseudo-steady state approximation, is widely used in chemical engineering especially in the design of reactors. Perturbation solutions built around this approximation are, however, known to lead to erroneous solutions in non-plane geometries. In the method developed in this paper the curvature term is retained in the equations even in the zero order approximation. The dimensionless radial width of the ash layer is used effectively as the perturbation parameter. The resulting series solution appears to converge rapidly to the exact solution even for large values of β. It is shown, incidentally, that the pseudo-steady state solution does not provide a conservative estimate of the time of reaction.