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

Using the method of finite differences, numerical solution of the diffusion equation for a slab membrane has been effected for five functional depende ncies of the (differential) diffusion coefficient upon concentration. Trans ient-state concentration profiles corresponding with 'adsorption' and 'deso rption' permeation have been employed to derive fluxes and quantities of di ffusant crossing a plane perpendicular to the flow. The 'adsorption' and 'd esorption' fluxes J(a)(l, t) and J(d)(0, t), were then used to construct 'e arly-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 th e amounts of diffusant crossing the ingoing and outgoing faces of the membr ane, 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) ove r bar (a)(3) and (D) over tilde (d)(3). For each system studied a comparison of nine integral diffusion coefficient s has been made. Those derived from the slope of the 'early-time' plots gav e the two ends of the integral diffusion coefficient 'spectrum', the remain ing seven coefficients lying between these limits. For both the strictly-in creasing 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) diffusi on coefficients. (C) 2000 Elsevier Science B.V. All rights reserved.