Transport through a slab membrane governed by a concentration-dependent diffusion coefficient III. Numerical solution of the diffusion equation: 'early-time' and 'root t' procedures

摘要

Using the method of finite differences, numerical solution of the diffusion equation for a slab membrane has been effected for five functional dependencies of the (differential) diffusion coefficient upon concentration. Transient-state concentration profiles corresponding with 'adsorption' and 'desorption' permeation have been employed to derive fluxes and quantities of diffusant crossing a plane perpendicular to the flow. The 'adsorption' and 'desorption' fluxes J(a)(l, t) and J(d)(0, t), were then used to construct 'early-time' plots from the slopes of which integral diffusion coefficients (D) over bar (a)(1) and (D) over bar (d)(1) were derived. Similarly, from the amounts of diffusant crossing the ingoing and outgoing faces of the membrane, Q(a)(0, t) and Q(d)(l, t), 'roott' plots were constructed, the slopes of which yielded a further pair of integral diffusion coefficients: (D) over bar (a)(3) and (D) over tilde (d)(3). For each system studied a comparison of nine integral diffusion coefficients has been made. Those derived from the slope of the 'early-time' plots gave the two ends of the integral diffusion coefficient 'spectrum', the remaining seven coefficients lying between these limits. For both the strictly-increasing and strictly-decreasing functions employed it was observed that (D) over bar (a)(1) congruent to D-0, (D) over tilde (d)(1) congruent to D(C-0). Some consideration has been given to 'weighted-mean' (integral) diffusion coefficients. (C) 2000 Elsevier Science B.V. All rights reserved. [References: 13]