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

Abstract

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.

Effectiveness Factors and Conversion in a Biocatalytic Membrane Reactor

“…The solution of Eq ( 5 ) was given by Godongwana et al . [ 13 ] as an asymptotic expansion in terms of the membrane hydraulic permeability κ : Where and F m (θ) in Eq ( 7 ) is the Kummer function: …”

“…For brevity only the zero-order approximation will be considered, the first order perturbation approximation is given in Appendix A following the procedure of Godongwana et al . [ 13 ]. Eq ( 15 ) then reduces to: …”

“…In these studies, as well as in the majority of available exact solutions [ 10 – 12 ], axial diffusion and radial convective flows are neglected and the kinetics are generally considered linear. These assumptions are not always justified [ 13 ] and are imposed with the intention of attaining closed-form expression of the transport equation. The analytical solution of the mass balance equation is not always feasible, and a number of numerical schemes have been developed for this purpose [ 14 – 20 ].…”

“…The current analysis is aimed at developing expressions of the effectiveness factor for an MBR immobilized with biofilm, based on the model developed by Godongwana et al . [ 13 ]. The asymptotic behaviour as the Thiele-modulus becomes large will be considered.…”

Continuous Biotechnological Treatment of Cyanide Contaminated Waters by Using a Cyanide Resistant Species of Aspergillus awamori

“…However, one of the biggest challenges in membrane operations is the effect of fouling and feeding mode, orientation and other mechanisms that would limit their application on a large scale. The effects of membrane modes of operations, as shown in Figure 12, have been studied using both experimental [6,21,30] and theoretical approaches [2,8,9,13]. The mode of operation is concerned with the distribution and flow of the fluid in the system, as shown in Figure 12.…”