Coupled model of film theory and the Nernst–Planck equation in nanofiltration

A mathematical model is proposed as a contribution to the modelling of the concentration polarization phenomenon occurring during increasing transmembrane pressure in membrane processes. The developed model is based on the extended Nernst–Planck equation (to describe the transport in membrane pores) and film theory equation (to describe the transfer mechanism in the polarization layer). It is characterized by two transfer parameters: solute permeability, Ps, and reflection coefficient, σ, and which allows the estimation of the thickness of the boundary layer, δ. Solute concentration at the membrane surface, Cm, and concentration profiles in the polarization layer are also estimated. The model is applied for the prediction of Ca(NO3)2, Cu(NO3)2 and Cd(NO3)2; ZnCl2 retention in nanofiltration and the influence of feed concentration on concentration polarization phenomenon were studied. The validation of the established models was obtained by the analysis of the experimental results resulting from treatment by nanofiltration of a synthetic solution with and without concentration polarization which determined the three principal parameters: reflexion coefficient σ; the solute membrane permeability, PS; and the layer polarization thickness, δ.