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. 1. Introduction Since the first uses of hollow-fiber membrane bioreactors (MBRs) to immobilize whole cells were reported in the early 1970s, this technology has been used in as wide ranging applications as enzyme production to bone tissue engineering. One of the current research areas of interest into biofilm-attached membrane bioreactors (MBRs) is the development of cost-effective and environmentally friendly methods of producing various primary and secondary metabolites from bacterial, fungal, and yeast cells. These include: manganese and lignin peroxidase, secreted by the fungus Phanerochaete chrysosporium [1, 2]; actinorhodin, a noncommercial antibiotic produced by the filamentous bacterium Streptomyces coelicolor [3]; glutamic acid, an ingredient in flavour enhancers of meats and vegetables, secreted by the bacterium Corynebacterium glutamicum [4]; ethanol, extracted from the yeast Saccharomyces cerevisiae [5]; and many others. With the exception of ethanol, these bioproducts are generally classified as products of intermediate value [6]. It has been reported that bioreactor productivity, in the production of these types of products, greatly impacts on the product cost [7]. The productivity of biofilm-attached MBRs is determined in large by the biomass growth, and one of the most important factors that influence biomass growth is the availability and transport of nutrients through the bioreactor [8, 9]. The momentum transfer of solutes through MBRs has been thoroughly studied, from a theoretical and experimental perspective, for a number of configurations [10–15]. Similarly, the mass transfer has received considerable attention [8, 9, 16–22]. With the exception of the models developed
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