Traveling Waves for Distributed Gas-Solid Reactions

Abstract

We investigate traveling waves for distributed gas-solid reactions with a powerlaw reaction rate. In this model the diffusing gas reacts with an immobile solid phase, leading to coupled partial and ordinary differential equations. We consider both the case of positive porosity and the pseudo-steady-problem with zero porosity. The wave profile consists of a pair ( u, w) where u is the gas concentration and w is the solid concentration. As boundary conditions we take w(− ∞) = 0, u(+ ∞) = 0, w(+ ∞) = 1, as is appropriate for a wave traveling from left to right while consuming the solid as it moves. Our principal result, obtained by phase-plane methods, is that to each velocity v > 0 there corresponds a unique wave profile, modulo translations. If the power of the solid reaction rate is less than 1, a conversion front exists behind which the solid is fully consumed ( w ≡ 0); if the power of the gas reaction rate is less than 1, the gas does not fully penetrate the solid, leading to a penetration front ahead of which u ≡ 0. We provide estimates for the location of these fronts and analyze their local behavior. Our results both agree with and extend those of Bobisud who studies similar equations in a different context and on a semi-infinite interval.