A Model for Porous-Medium Combustion

Abstract

A model of time-dependent porous-medium combustion is presented. The model is of combustion in a three-dimensional porous medium. The typical situation envisaged is the combustion of a non-deforming porous solid medium through which a gas such as air passes. The model represents conservation of mass and energy for both the gas and solid species, whilst the fluid flow is governed by Darcy's law and the ideal-gas law. This model is highly complex and requires sophisticated computer analysis. Consequently we derive a simplified model as a one-dimensional version of the equations, by a number of asymptotic considerations. Central to the analysis is the concept of the large-activation-energy limit. This limit is shown to have entirely different features from those which arise in conventional flame theory. This fact is a consequence of the two-stage reaction rate governing porous-medium combustion; the stages are first the diffusion of gas components between the gas mainstream and the reaction sites in the solid and secondly the conventional Arrhenius reaction. Thus the overall reaction rate is not proportional to the Arrhenius reaction rate, but is a rational function of it. Because of this two-stage reaction rate, the limit E→∞ has a novel result not encountered in conventional flame theory. A critical switching temperature T_c, determined by A = exp (E/T_c), where A is the pre-exponential factor in the Arrhenius reaction term, arises naturally from the large-activation-energy analysis. For temperatures beneath T_c the reaction rate is negligible whereas for temperatures above T_c the reaction is controlled by the ability of the active gas components to diffuse into or out of the reaction sites in the solid. This rate of active gas-component diffusion has been shown experimentally to be proportional to a power (approximately the square) of the gas temperature. Thus, when switched on, the rate-limiting reaction rate grows algebraically with the temperature, in contrast to the explosive exponential growth of the Arrhenius term which governs the switching process.