# Concentration polarization model by du4Gee3

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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 CJ 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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