Effect of concentration polarization on membrane performance

We want to obtain an expression to relate the solvent flux (typically water) to measurable quantities. In general, we need to include the effect of osmotic pressure, especially for reverse osmosis applications. 

Specific correlations for concentration polarization using flat sheet membranes have been derived [15]. 

An important limit in concentration polarization is complete rejection R= 1, Cp = 0). For organics, this can lead to gel layer formation which controls the permeation rate (Vp). For this limiting case, the equation for concentration polarization becomes 

Mass transfer correlations have been derived that are used for membranes of various configurations [16]. For flow through lubes, the following are used (see Appendices A and B for details on dimensionless numbers and mass transfer correlations, respectively). 

V = feed velocity d = tube diameter b = channel height. 

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