Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 810843, 8 pages

http://dx.doi.org/10.1155/2015/810843

## A Finite-Difference Solution of Solute Transport through a Membrane Bioreactor

^{1}Department of Chemical Engineering, Cape Peninsula University of Technology, P.O. Box 652, Cape Town 8000, South Africa^{2}Department of Mathematics and Applied Mathematics, University of Cape Town, Private Bag X3, Rondebosch 7700, South Africa

Received 26 December 2014; Revised 10 March 2015; Accepted 12 March 2015

Academic Editor: Sergio Preidikman

Copyright © 2015 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.

#### Abstract

The current paper presents a theoretical analysis of the transport of solutes through a fixed-film membrane bioreactor (MBR), immobilised with an active biocatalyst. The dimensionless convection-diffusion equation with variable coefficients was solved analytically and numerically for concentration profiles of the solutes through the MBR. The analytical solution makes use of regular perturbation and accounts for radial convective flow as well as axial diffusion of the substrate species. The Michaelis-Menten (or Monod) rate equation was assumed for the sink term, and the perturbation was extended up to second-order. In the analytical solution only the first-order limit of the Michaelis-Menten equation was considered; hence the linearized equation was solved. In the numerical solution, however, this restriction was lifted. The solution of the nonlinear, elliptic, partial differential equation was based on an implicit finite-difference method (FDM). An upwind scheme was employed for numerical stability. The resulting algebraic equations were solved simultaneously using the multivariate Newton-Raphson iteration method. The solution allows for the evaluation of the effect on the concentration profiles of (i) the radial and axial convective velocity, (ii) the convective mass transfer rates, (iii) the reaction rates, (iv) the fraction retentate, and (v) the aspect ratio.

#### 1. Introduction

Membrane bioreactors (MBRs) are finding increasing use in the production of primary and secondary metabolites such as amino acids, antibiotics, anticancer drugs, and tissue cells [1–3]. This technology is favoured by recent trends towards environmentally-friendly technologies, particularly because MBRs do not require additives, function at moderate operating conditions, and reduce by-product formation [1]. The efficiency of MBRs is dependent mainly on the transport of solutes through the bioreactor, and this is influenced by biochemical, geometric, and hydrodynamic parameters [2, 4]. This paper considers the numerical solution of the convection-diffusion equation for solute transport through a fixed-film MBR. This analysis is important for simulation of the performance (i.e., efficiency and effectiveness) of the bioreactor. The governing equation for mass transport of solutes through the bioreactor is the convection-diffusion equation with Monod kinetics [5] as follows:where is the local substrate concentration, and are the axial and radial velocity components, respectively, is the substrate diffusion coefficient, is the maximum rate of reaction, and is the saturation (or Michaelis) constant. Equation (1) is made dimensionless by introducing the following variables:Equation (1) then becomeswhere the axial and radial Peclet numbers are, respectively, defined asThe velocity profiles, and , in (3) are solutions of the and -components of the Navier-Stokes equations, respectively [6]:where is the dimensionless hydrostatic pressure which is a function of the membrane hydraulic permeability and Re and Fr are the Reynolds and Froude numbers, respectively. When the membrane hydraulic permeability is much smaller than unity, (3) reduces towhere The fraction retentate, , is defined as the ratio of the outlet to the inlet axial velocity ( for the dead-end mode and for the closed-shell mode), and is the dimensionless transmembrane pressure. The corresponding boundary conditions areBoundary condition 1 (B.C.1) corresponds to a uniform inlet substrate concentration; B.C.2 corresponds to cylindrical symmetry at the center of the membrane lumen; B.C.3 corresponds to continuity of the substrate flux at the lumen-matrix interface [7]. The solution of (6) is based on the following general assumptions: (i) the system is isothermal; (ii) the flow regime is laminar and fully developed; (iii) the fluid is Newtonian and homogenous and has constant physical and transport properties; (iv) the membrane hydraulic permeability is constant.

A schematic of the MBR is shown in Figure 1. The MBR consists of a single hollow-fibre, made of surface modified polysulphone, encased in a glass bioreactor. The membranes are asymmetric and characterized by an internally skinned and externally unskinned region of microvoids. The nutrient solution is supplied by a peristaltic pump and permeates from the lumen-side to the shell-side of the MGR. The microorganism is immobilised on either the lumen-side or the shell-side of the MGR. Humidified air is supplied on the shell-side, and two pressure transducers are fitted at the inlet and outlet of the MGR.