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International Journal of Chemical Engineering
Volume 2010, Article ID 738482, 12 pages
Research Article

A Solution of the Convective-Diffusion Equation for Solute Mass Transfer inside a Capillary Membrane Bioreactor

1Department of Chemical Engineering, Cape Peninsula University of Technology, P.O. Box 652, Cape Town 8000, South Africa
2Department of Mathematics and Applied Mathematics, University of Cape Town, Private Bag, Ronderbosch 7700, South Africa

Received 5 November 2009; Revised 23 March 2010; Accepted 27 April 2010

Academic Editor: Jose C. Merchuk

Copyright © 2010 B. Godongwana et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


This paper presents an analytical model of substrate mass transfer through the lumen of a membrane bioreactor. The model is a solution of the convective-diffusion equation in two dimensions using a regular perturbation technique. The analysis accounts for radial-convective flow as well as axial diffusion of the substrate specie. The model is applicable to the different modes of operation of membrane bioreactor (MBR) systems (e.g., dead-end, open-shell, or closed-shell mode), as well as the vertical or horizontal orientation. The first-order limit of the Michaelis-Menten equation for substrate consumption was used to test the developed model against available analytical results. The results obtained from the application of this model, along with a biofilm growth kinetic model, will be useful in the derivation of an efficiency expression for enzyme production in an MBR.