Analytical solutions of reaction–diffusion phenomena by porous cylindrical pellets with finite length

Abstract

Analytical solutions were derived to predict transient change of intra‐particle concentration inside cylindrical pellets with finite length immersed in an infinitely large medium. After the derivation of steady‐state concentration by eigenfunction expansion, a transient solution was obtained by the separation of variables. Effects of the Thiele modulus (Φ) and aspect ratio of the pellet (H/R) were studied by calculating the average intra‐particle concentration at a stationary state. The effectiveness factor (η) was also affected by H/R and Φ, since irreversible first order kinetics was assumed. As Biot number (Bi) increased, both intra‐particle concentration and η increased due to the decrease of the mass transfer resistance of the surrounding film. Good agreement between analytical and numerical solutions was confirmed for both steady‐state and transient solution by comparing the analytical results with the numerical solutions by finite element method. To extend the results to a batch reactor with finite volume, Duhamel's theorem was applied by assuming time‐dependent boundary conditions to reflect the change in bulk concentration as a function of time. A method was proposed to measure intra‐particle diffusivity for batch adsorber by setting Φ = 0. To investigate the effect of H/R, η could be included in material balance of batch, continuously stirred tank reactor (CSTR), and fixed bed reactors to predict reactor performance assuming a pseudo‐steady state. First‐order reaction by egg‐shell catalysts as well as nonlinear reaction with two reactants could be solved numerically assuming finite cylindrical pellets. By solving coupled differential equations of material and energy balances, dynamic catalysis was confirmed from the enhancement of conversion, assuming forced oscillation of inlet temperature in CSTR.