## Solutiondiffusion Model and Concentration Polarization

The aim is now to calculate wp and c2 as well as An. Thus, we need three independent equations. Considering the increasing concentration c3 at the membrane surface for the calculation of An and JD, we obtain Eqs. (12.68) and (12.69) from Eq. (12.24b), respectively Eq. (12.11) and (12.12).

For the solution diffusion model (reverse osmosis and nanofiltration) wp follows according to Eqs. (12.27) and (12.68):

Equations for the concentration of the retained component c3 and the permeate c2 are given for c2 according to Eqs. (12.29) and (12.27):

12.3 Mass Transfer Resistance Mechanisms | 307 The third independent equation is given by elimination of c3 from Eq. (12.61):

After introducing Eq. (12.72) in Eq. (12.71), c3 and c2 can be eliminated:

B co

With Eqs. (12.72) and (12.73) in Eq. (12.70), we finally obtain an implicit equation for wp:

Equation (12.74) can only be solved by graphical or numerical methods after determination of kL using one of the empirical equations in Table 12.3 (see Section 12.4.1). With these solutions, c3 and c2 can be calculated using Eqs. (12.72) and (12.73).

There is, however, a correlation between the true retention coefficient Rt and c3, the concentration at the surface of the membrane according to Eq. (12.2). With Eqs. (12.70) and (12.71), we obtain the true retention coefficient Rt which can be calculated using Eqs. (12.72) and (12.73):

Explicit solutions of Eq. (12.74) are possible for some specific cases:

1. If concentration polarization does not occur, c3 = c0, exp (wp/kL) = 1 follows from Eqs. (12.60) and (12.62). Solving the quadratic equation we obtain:

2. Furthermore, if the membrane is not permeable for the solute (B = 0), the permeate specific flow rate is:

3. In the case of low feed concentration c0, then An can be neglected:

Wp c0

308 | 12 Membrane Technology in Biological Wastewater Treatment 12.3.5

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