Dirichlet problem for convection–diffusion–reaction inside a permeable cylindrical porous pellet

Abstract

The present article deals with the study of convection and diffusion coupled with either zero or first order reaction inside a permeable circular cylindrical porous pellet under oscillatory flow. Unsteady Stokes equations are used for the flow outside the permeable porous pellet and Darcy’s law is used inside the pellet. We use the stream function approach in order to solve the hydrodynamic problem. Then the convection–diffusion–reaction problem is formulated and solved analytically for both zero order and first order rate of nutrient uptake. The Dirichlet boundary condition, which can be achieved by neglecting the external mass transfer resistance, is used at the surface of permeable porous pellet. Also in case of zero order, an optimality criterion, which is a relationship between the Peclet number and the Thiele modulus, is proposed to avoid the starvation everywhere inside the pellet. Based on this criterion, classification is done in order to identify the regions of nutrient sufficiency and starvation. A comparison is also made with nutrient transport inside a spherical porous pellet. It is observed that in case of zero order, for a fixed combination of other parameters, spherical pellet demands a higher value of Thiele modulus compared to the cylindrical pellet in order to force starvation. Moreover, in case of first order reaction, one does not witness starvation zones either in cylindrical pellet or in spherical pellet.