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Concentration polarization model The concentration profile of solute molecules close to the membrane surface is shown in Fig. 1. At steady state, a material balance of solute molecules in a control volume within the concentration polarization layer yields the following differential equation: dC (1) J CJ C D v v p 0 dx Integrating this with boundary conditions (C = Cw at x = 0; C = Cb at x = b), we get Cw C p J v k ln (2) C C b p k (= mass transfer coefficient) = D / b This equation is known as the concentration polarization equation for partially rejected solutes. For total solute rejection, i.e., when Cp = 0, the equation reduces to C (3) J k ln w v C b Membrane Back diffusion dC = D ----- dx Cw Fig. 1 Jv Bulk feed Permeate Cb Cp Concentration polarization layer = b C x When the solute concentration at the membrane surface reaches the saturation concentration for the solute (Cs), or the gelation concentration of the macromolecule (Cg), there can be no further increase in Cw. Thus C C (4) J v k ln s C k ln g C b b This is referred to as the gel polarization equation. It indicates that when Cw equals Cs (or Cg) the permeate flux is independent of the TMP. In the pressure independent region, the permeate flux for a given feed solution is only dependent on the mass transfer coefficient. For a particular mass transfer coefficient the pressure independent permeate flux value is referred to as the limiting flux (Jlim). According to the gel polarization model, the existence of this limiting flux is a consequence of gelation of the solute at the membrane-solution interface.
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