Semi-analytical solutions for one- and two-dimensional pellet problems

The problem of heat and mass transfer within a porous catalytic pellet in which an irreversible first-order exothermic reaction occurs is a much-studied problem in chemical-reactor engineering. The system is described by two coupled reaction-diffusion equations for the temperature and the degree of reactant conversion. The Galerkin method is used to obtain a, semi-analytical model for the pellet problem with both one- and two-dimensional slab geometries. This involves approximating the spatial structure of the temperature and reactant-conversion profiles in the pellet using trial functions. The semi-analytical model is obtained by averaging the governing partial differential equations. As the Arrhenius law cannot be integrated explicitly, the semi-analytical model is given by a system of integrodifferential equations. The semi-analytical model allows both steady-state temperature and conversion profiles and steady-state diagrams to be obtained as the solution to sets of transcendental equations (the integrals are evaluated using quadrature rules). Both the static and dynamic multiplicity of the semi-analytical model is investigated using singularity theory and a local stability analysis. An example of a stable limit cycle is also considered in detail. Comparison with numerical solutions of the governing reaction-diffusion equations and with other results in the literature shows that the semi-analytical solutions are extremely accurate.