where C is the local solute concentration, D is the solute diffusivity, Cp is the solute permeate concentration, and y is the distance normal to the membrane sur-face.[31] Integrating over the solute concentration polarization layer thickness (ds) with the appropriate boundary conditions (y = 0, C = Cm; y = ds, C = Cb) results in the classic film theory equation,

where Cm is the membrane surface solute concentration, Cb is the bulk solute concentration, Cp is the permeate solute concentration, and the ratio of solute diffusivity (D) to the solute concentration polarization layer thickness (ds) is set equal to the solute mass transfer coefficient (k). This expression can be rearranged to provide a direct calculation of the concentration polarization factor (Cm/ Cb) as

For laminar flow in a thin rectangular channel, the mass transfer coefficient is often related to a Sherwood number (Sh) through a relationship such as k = Sh — = 1.851 Re Sc dh dh L

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