EC 17,3 192 Engineering Computations, Vol. 17 No. 3, 2000, pp. 192-217. # MCB University Press, 0264-4401 Received November 1998 Revised August 1999 Accepted January 2000 Numerical modelling of mass transfer in slits with semi-permeable membrane walls Võ Âtor Geraldes Chemical Engineering Department, Instituto Superior Te Âcnico, Portugal Viriato Semia Äo Mechanical Engineering Department, Instituto Superior Te Âcnico, Portugal, and Maria Norberta Pinho Chemical Engineering Department, Instituto Superior Te Âcnico, Portugal Keywords Mathematical model, Predictive techniques, Filtration, Hydrodynamics, Membranes Abstract A mathematical model to predict the concentration polarisation in nanofiltration/ reverse osmosis is described. It incorporates physical modelling for mass transfer, laminar hydrodynamics and the membrane rejection coefficient. The SIMPLE algorithm solves the discretised equations derived from the governing differential equations. The convection and diffusive terms of those equations are discretised by the upwind, the hybrid and the exponential schemes for comparison purposes. The hybrid scheme appears as the most suitable one for the type of flows studied herein. The model is first applied to predict the concentration polarisation in a slit, for which mathematical solutions for velocities and concentrations exist. Different grids are used within the hybrid scheme to evaluate the model sensitivity to the grid refinement. The 5525 grid results agree excellently for engineering purposes with the known solutions. The model, incorporating a variation law for the membrane intrinsic rejection coefficient, was also applied to the predictions of a laboratory slit where experiments are performed and reported, yielding excellent results when compared with the experiments. Nomenclature  = Constant characteristic of a given membrane-solute system (Equation 17) A h = Hydraulic permeability, m 2 s kg ±1 A n = Coefficients of the finite difference equations for nodes neighbour of P (N,S,E,W) A P = Coefficient of the finite difference equations for node P b = Osmotic coefficient, defined by b = =! A , Pa C = Dimensionless mass fraction, ! A /! A0 D AB = Binary mass diffusion coefficient, m 2 s ±1 f 0 = Intrinsic rejection coefficient of the membrane, defined by f 0 = (! Am ± ! Ap )/! Am f = Apparent rejection coefficient of the membrane, defined by f = (! A0 ± ! A P )/! A0 h = Channel half-height, m L = Dimensionless axial co-ordinate, x/h p = Pressure, Pa p* = Guessed pressure field for SIMPLE, Pa p 0 = Correction pressure field for SIMPLE, Pa Re = Circulation Reynolds number, defined by Re =  u 0 2h/ Rep = Permeation Reynolds number, defined by Re p =  v p 2h/ R = Dimensionless transverse co- ordinate, 1-y/h The current issue and full text archive of this journal is available at http://www.emerald-library.com