Exact solution to mass transfer in Berman flow: Application to concentration polarization combined with osmosis in crossflow membrane filtration

Concentration polarization affects numerous systems of membrane separation, and combined with osmosis, it can cause substantial reductions in permeation. We establish an exact solution to the conservation law of a solute advected by Berman flow. This flow is characteristic of reverse osmosis or nanofiltration. The resulting concentration polarization is then combined with the osmosis (counter-) effect. For large Péclet number of permeation, it yields a rigorous support to the semi-empirical “film” model, and accounts for the limit flux phenomenon. The main results are summarized in a simple diagram that relates three different Péclet numbers, and show that polarization combined with osmosis can withstand operating pressure almost totally.