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Reverse osmosis (RO) is a method to desalinate water with membranes and anapplied pressure. Very important in the theory of mass transport in RO is theconcentration polarization (CP) layer, which develops on the upstream side ofthe membrane because of a combination of salt convection and diffusion. Becauseof the CP-effect, the salt concentration at the membrane surface is higher thanin the channel, and this increases the osmotic pressure there, and thustransmembrane water flux is reduced (the osmotic pressure acts against waterflux), while salt leakage through the membrane increases. So it is veryimportant to understand and describe the CP-layer accurately. We analyze aone-dimensional geometry, which is of relevance for a typical lab-scale ROsetup using small membrane coupons where the solution on the feed side of themembrane is stirred. For this geometry, the standard film layer approach isoften used that assumes a stagnant film layer of a defined thickness, whichhowever does not exist in reality. We set up a model without that assumptionbut including refreshment of solution because of the flow of water along themembrane due to stirring. We show that the `exponential law' for the CP-layerthat is predicted by the the film model, also applies for this more accuratemodel. We further improve the model by including the activity coefficient ofsalt ions, as described by the Bjerrum theory that is based on ion-ionCoulombic interactions. We evaluate the original linearized Bjerrum theory aswell as an extended Bjerrum equation that is valid up to 1.5 M saltconcentration. We show how including this activity correction leads to areduction of the diffusional driving force at high concentration, and thus thesalt concentration at the membrane further increases. However, the effect canbe easily included by reducing the CP-layer mass transfer coefficient by afixed percentage.

General validity of the exponential law for the effect of concentration  polarization in reverse osmosis in a stirred-cell geometry, including an  activity correction for 1:1 salt solutions