Analysis of concentration polarization in reverse osmosis and nanofiltration: zero-, one-, and two-dimensional models

Reverse osmosis and nanofiltration are membrane-based methods that remove solutes from solvent, for instance they remove salts from water (desalination). In these methods, an applied pressure is the driving force for solvent to pass the membrane, while most of the solutes are blocked. Very important in the theory of mass transport is the concentration polarization layer (CP layer), which develops on the upstream side of the membrane. Because of the CP layer, the solvent flux through the membrane is reduced while leakage of solutes through the membrane increases, and both these effects must be minimized. So it is very important to understand and describe the nature of the CP layer accurately, especially to find a good estimate of the CP layer mass transfer coefficient, $k$. This is also important for the accurate characterization of membranes in a test cell geometry. We theoretically analyze the structure of the CP layer using three levels of mathematical models. First, we present a modification of an equation for $k$ by Sherwood et al. (1965) and show that it works very well in a zero dimensional model. Second, we evaluate a one-dimensional model that is more accurate, which can incorporate any equation for the flow of solvent and solutes through the membrane, and which also makes use of the new modified Sherwood equation. Finally, we fully resolve the complete channel in a two-dimensional geometry, to validate the lower-order models and to illustrate the structure of the CP layer. The overall conclusion is that for typical test cell conditions, the modified Sherwood equation can be used to characterize the CP layer, also when solvent flux through the membrane changes between inlet and outlet of the test cell. Furthermore, the one-dimensional model accurately describes solute removal (for instance water desalination) not just in a short test cell but also in a longer module.