Proceedings of the 2 nd International Conference on Fluid Flow, Heat and Mass Transfer Ottawa, Ontario, Canada, April 30 – May 1, 2015 Paper No. 147 147-1 A Theoretical Analysis of Solute Transport through a Membrane Bioreactor Buntu Godongwana, Marshall Sheldon Cape Peninsula University of Technology, Department of Chemical Engineering P.O. Box 652, Cape Town 8000, South Africa godongwanab@cput.ac.za; sheldonm@cput.ac.za Deon Solomons University of Cape Town, Department of Mathematics and Applied Mathematics Private Bag X3, Ronderbosch 7700, South Africa deon.solomons@uct.ac.za 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 specie. 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 non-linear, 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 multi-variate 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 Keywords: Membrane bioreactor; Convection-diffusion equation; Implicit finite-difference; Upwind scheme; Multivariate Newton-Raphson; Regular perturbation Nomenclature B m coefficient of series solution, defined in text c substrate concentration (g dm -3 ) c 0 substrate feed concentration (g dm -3 ) C = c/c 0 dimensionless substrate concentration D AB substrate diffusivity (m 2 s -1 ) f = u 1 /u 0 fraction retentate h step-size in the r-dimension (m) i grid point index in the r-dimension j grid point index in the z-dimension J n () Bessel function of order n of the first kind K step-size in the z-dimension (m) K m saturation (or Michaelis) constant (g dm -3 )  m K dimensionless Michaelis constant L membrane effective length (m) M(a,b ,) Kummer function of the first kind Pe u = u 0 R L /D AB axial Peclet number Pe v = v 0 R L /D AB radial Peclet number r radial spatial coordinate (m) R = r/R 1 dimensionless radial spatial coordinate R L membrane lumen radius (m) u axial velocity (m s -1 ) u 0 feed axial velocity (m s -1 ) U = u/u 0 dimensionless axial velocity v radial velocity (m s -1 ) V = v/v 0 dimensionless radial velocity