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.