Mathematical aspects of the combustion of a solid by a distributed isothermal gas reaction

Abstract

When a diffusing gas reacts isothermally with an immobile solid phase, the resulting equations form a semilinear system consisting of a parabolic partial differential equation for the gas concentration coupled with an ordinary differential equation for the solid concentration. Existence and uniqueness proofs are given which include the important case of nonlipschitzian reaction rates such as those of fractional-power type. Various qualitative features of the solution are studied: approach to the steady state; monotonicity in time; and dependence on initial conditions, on the porosity, and on the geometry. The relationship between the original problem and the pseudo-steady-state approximation of zero porosity is investigated. When the solid reaction rate is nonlipschitizian, there is a conversion front separating a fully converted region adjacent to the boundary and a partially converted interior core. Estimates are given for the time to full conversion. If the gas reaction rate is nonlipschitzian the gas may not at first fully penetrate the solid. Estimates are given for the time at which full penetration occurs.