Derivation of film theory coupled Spiegler-Kedem-Katchalsky model for membrane separation and purification

In this article, we will derive the differential equations considering the concentration polarization (\textsc{cp}) in membrane separations and purifications, which involve convective-diffusive equations of the concentration polarization layer and the Spiegler-Kedem-Katchalsky (\textsc{skk}) model of the membrane layer. Analytical solutions of these differential equations are straightforward and given. The film theory coupled \textsc{skk} model can be used to evaluate the degree of concentration polarization, and predict flux and rejection rates.