Modeling of crossflow membrane separation processes under laminar flow conditions in a tubular membrane

One of the major problems associated with membrane separation processes, which restricts the widespread application of this process in the industry is the decline in flux. This occurs due to the build up of the solute concentration near the membrane surface, which is defined as concentration polarisation. An accurate quantification of concentration polarisation as a function of process conditions is necessary to estimate the system performance satisfactorily. In order to solve the governing mass transfer equation one needs to evaluate the detailed velocity profiles in the particular system. A numerical solution of the coupled Navier-Stokes, Darcy’s law and mass transfer equations has been developed to predict the rate of growth of the concentration polarisation boundary layer along the length of a tubular membrane. The mass transport mechanisms are evaluated mathematically using the two-dimensional convective diffusion equation. A 2-D finite difference solution coded in MATLAB, has been developed. The aim of the study is to quantify the effects of various operating conditions (e.g. the axial Reynolds number, the wall Reynolds number and the Schmidt number) on the concentration boundary layer thickness during crossflow ultrafiltration.

ap C C0 Cw D d e — — — — — — Average solute particle diameter Solute concentration Feed concentration Concentration at membrane surface Diffusion coefficient Inner diameter of the tubular membrane — Membrane thickness Length of tubular membrane Pressure Inlet pressure External pressure Radius of the tubular membrane The resistance of concentration polarisation layer The resistance of membrane Radial coordinate Specific resistance of concentration polarisation layer Axial velocity Inlet average axial velocity Radial velocity Local permeation flux Inlet local permeation flux Axial coordinate Greek δp κ µ ρ ε εp — Thickness of concentration polarisation layer — Permeability — Fluid viscosity — Fluid density — Tolerance — Porosity of cake layer Dimensionless groups Re — Axial Reynolds number Rew — Wall Reynolds number Sc — Schmidt number Subscript and exponent — Dimensionless variable — Membrane surface — Inlet value