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

P. D. Haynes, S. K. Lucas
2007 ANZIAM journal (Print)  
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
more » ... e 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. 2000 Mathematics subject classification: primary 76R50; secondary 35K20, 80A20.
doi:10.1017/s1446181100003187 fatcat:xvmcle7iv5hg7mrfvi3xamadyu