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

Published online by Cambridge University Press:  17 February 2009

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

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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, 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 resultsis presented, together with recommendations as to their use.

Keywords

adsorbentadsorptiondiffusioneigenfunction expansionerror functionfinite-volume fluidfractional uptakeLaplace domainLaplace transformlongtime solutionseries of negative exponentialsshort-time solutionsorption
Type
Research Article
Information
The ANZIAM Journal , Volume 48 , Issue 4 , April 2007 , pp. 503 - 521
Copyright
Copyright © Australian Mathematical Society 2007

References

[1]Carman, P. C. and Haul, R. A. W., “Measurement of diffusion coefficients”, Proc. R. Soc. bond. Ser. A Math. Phys. Eng. Sci. A222 (1954) 109–18.Google Scholar
[2]Carslaw, H. S. and Jaeger, J. C., Conduction of Heat in Solids, 1st ed. (Oxford, London, 1947).Google Scholar
[3]Crank, J., The Mathematics of Diffusion, 1st ed. (Oxford: Clarendon Press, London, 1956).Google Scholar
[4]Dyer, A. and Amin, S., “Self-diffusion of simple alcohols in heteroinic forms of LTA zeolites”, Microporous Mesoporous Mat. 46 (2001) 163–176.CrossRefGoogle Scholar
[5]Ruthven, D. M., Principles of Adsorption and Adsorption Processes (Wiley, New York, 1984).Google Scholar
[6]Valenzuela-Calahorro, C., Navarrete-Guijosa, A., Stitou, M. and Cuerda-Correa, E., “Retention of progesterone by an activated carbon: Study of the adsorption kinetics”, Adsorption-J. Int. Adsorption Soc. 10 (1) (2004) 19–28.CrossRefGoogle Scholar