Analytical solution of solute concentration for the systems of dialysis coupled with ultrafiltration in cross-flow membrane modules was obtained by the method of perturbation, and thus separation efficiencies could be calculated for various operating and design conditions. The overall mass transfer coefficient was assumed to be constant and the concentration polarization and pressure drop on both compartments were neglected in solving this problem. Moreover, the mean values of permeation flux and sieving coefficient were taken and were treated as constant. It was found that increasing the flow rate in retentate phase is more beneficial to the mass transfer than increasing in dialysate phase. The performance of dialysis can be improved significantly by the effect of ultrafiltration, especially for the systems with low mass transfer coefficient. The present analysis could be applied to all kinds of configurations, such as parallel-flow, radial-flow and spiral-wound types, but with the limitations of laminar flow and low viscosity and high diffusivity of the fluids.
