  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.
