Transport through a slab membrane governed by a concentration-dependent diffusion coefficient III. Numerical solution of the diffusion equation: 'early-time' and '?' procedures

Abstract 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 1 a and D 1 d 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 ), ‘ t ’ plots were constructed, the slopes of which yielded a further pair of integral diffusion coefficients: D 3 a and D 3 d . 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 1 a ≅D 0 , D 1 d ≅D(C 0 ) . Some consideration has been given to ‘weighted-mean’ (integral) diffusion coefficients.