Application of Petrov–Galerkin methods to transient boundary value problems in chemical engineering: adsorption with steep gradients in bidisperse solids

Abstract

Petrov–Galerkin methods are known to be versatile techniques for the solution of a wide variety of convection–dispersion transport problems, including those involving steep gradients, but have hitherto received little attention by chemical engineers. We illustrate the technique by means of the well-known problem of simultaneous diffusion and adsorption in a spherical sorbent pellet comprised of spherical, non-overlapping microparticles of uniform size and investigate the uptake dynamics. Solutions to adsorption problems exhibit steep gradients when macropore diffusion controls or micropore diffusion controls, and the application of classical numerical methods to such problems can present difficulties. In this paper, a semi-discrete Petrov–Galerkin finite element method for numerically solving adsorption problems with steep gradients in bidisperse solids is presented. The numerical solution was found to match the analytical solution when the adsorption isotherm is linear and the diffusivities are constant. Computed results for the Langmuir isotherm and non-constant diffusivity in microparticle are numerically evaluated for comparison with results of a fitted-mesh collocation method, which was proposed by Liu and Bhatia (Comput. Chem. Engng. 23 (1999) 933–943). The new method is simple, highly efficient, and well-suited to a variety of adsorption and desorption problems involving steep gradients.