ANZIAM  J.  48 (2007), 503-521

Extension of a short-time solution of the diffusion equation with application to micropore diffusion in a finite system

P. D. Haynes Centre for Industrial and Applied Mathematics   School of Mathematics and Statistics   University of South Australia   Mawson Lakes, SA 5095   Australia     Paul.Haynes@unisa.edu.au S. K. Lucas Centre for Industrial and Applied Mathematics   School of Mathematics and Statistics   University of South Australia   Mawson Lakes, SA 5095   Australia

Abstract

The diffusion equation is used to model and analyze sorption, a process used in the purification or separation of fluids. For the diffusion inside a spherical porous solid immersed in a limited-volume and well-stirred fluid, Ruthven [5], Crank [3] and, for the analogous flow of heat, Carslaw and Jaeger [2] give an eigenfunction expansion solution to the diffusion equation that provides accurate long-time solutions when only a few terms are used. However, to obtain the same accuracy for short-time solutions the number of eigenfunction terms required increases exponentially. An alternative error function solution of Carman and Haul [1] is accurate for sufficiently short times but not for long times. Although their solution is well quoted ([3],[4] and [6]), Carman and Haul do not provide a derivation in their paper. This paper provides a full derivation of the short-time solution of Carman and Haul that uses only the first term of a negative exponential series in the Laplace domain. It is shown that the accuracy and range of the short-time result is improved by the inclusion of additional terms of the negative exponential series. An analysis of short-time and long-time results is presented, together with recommendations as to their use.

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