Preprint C104/2010
Chemical reactors in porous media
Pablo CastaƱeda
Keywords: Chemical reactor | Porous media | heat losses | in-situ combustion | nonlinear boundary value problem | infinite ODE system.

We are interested in an exothermic chemical reaction occurring in air within a small region of a conductive spherical solid porous medium. The air is injected at the center of the sphere. Heat is generated near the center, and conducted to the wall of the sphere. The issue is to determine if there is ignition or extinction, depending on the predominance of reaction or conduction. We simplify the mathematical description of the system by imagining that the reaction occurs only in a region with uniform temperature located around the center of the sphere, while conduction occurs in the rest of the sphere, a surrounding shell. The system of reaction-diffusion equations reduces to a linear heat equation in the shell, coupled at the internal boundary to a nonlinear ordinary differential equation in the reaction region. This ODE can be regarded as a (nonlinear) boundary condition for the heat equation in the shell. This simplification allows making a complete analysis of the time evolution of the system. We show that, depending on physical parameters, the system admits one or three equilibria. The latter case has physical interest: the two equilibria represent attractors (``extinction'' or ``ignition'') with basins of attraction separated by the stable manifold for the third equilibrium. We utilize operator theory for the linear stability analysis, as well as fixed point theory of a nonlinear Volterra equation for the existence and uniqueness of solutions for all times. We also use a spectral decomposition to describe the evolution by means of an infinite number of coupled nonlinear ordinary differential equations, providing regularity of the solutions for general Cauchy data, as well as the nonlinear asymptotic behavior for long times. One interesting practical conclusion is that higher dimensionality of the sphere increases the probability of extinction. Another interesting conclusion is that the whole system is quite well described by a single ODE, which is a kind of ``reduced'' model for the reactor.