Numerical study of concentration polarization in a rectangular reverse osmosis membrane channel: Permeate flux variation and hydrodynamic end effects

Concentration polarization in a rectangular reverse osmosis (RCRO) and a laboratory plate-and-frame RO (PAFRO) membrane channels were investigated using a finite-element numerical model consisting of the fully coupled governing equations for fluid dynamics and mass transfer. The model considers both the concentration dependence of model parameters and axial variation of permeate flux. In both the RCRO and PAFRO channel geometries the flux varies axially along the membrane commensurate with the coupled development of the concentration boundary layer. For the case of the RCRO channel, the effect of permeate flux was quantified via the computed feed-side mass transfer coefficient (commonly used in estimations of concentration polarization) in terms of a dimensionless correlation for the average Sherwood number, image, that reduces to the form of the classical Lévêque solution in the limit of vanishing permeate flux. Simulations of desalting in a PAFRO channel demonstrated the importance of accounting for local permeate flux variability and coupling of the hydrodynamics and mass transfer model equations in order to predict local concentration variations and in particular local maxima of concentration polarization, especially in regions of flow recirculation where scaling by sparingly water soluble mineral salts could be first triggered should their concentrations exceed saturation.