The classic concentration polarization model describes the fluxes according to con-vective flow and retransfer of the solute, i.e. back diffusion through concentration boundary layer dependent on the concentration gradient between the bulk region and the surface of the membrane (Fig. 12.5).
One solution to the mass balance in the boundary layer with thickness 8 is obtained at steady state conditions, dJx/dz = 0. There are two parts of the total flux: the flux of the solute counter to the z-direction, -wpc, and the back diffusion of the solute, -Ddc/dz. Assuming that the permeate velocity is independent of the z-coordinate it follows that:
After differentiation of Eq. (12.45), we obtain at steady-state conditions:
dc d2 c
A possible solution of Eq. (12.46) is the exponential function:
and after differentiating Eq. (12.47): dc
dz d2c 2
Three boundary conditions are necessary to determine the constants a1, a2 and a3. According to Fig. 12.5: (1) at the boundary layer with thickness 8, the concentration is the same as in the bulk flow c0, (2) the highest concentration c3 is attained at the surface of membrane at z = 0, and finally, (3) the condition for the permeate flux of the solute is the equality of the sum of convective flux and the back diffusive flux at the membrane surface:
From Eqs. (12.48) and (12.49) in Eq. (12.46) we obtain:
and finally for the constant a3:
- wpc2 = - wpc3- D a3a2 (12.55) or with Eq. (12.54):
Finally, considering condition Eq. (12.50), then Eq. (12.47) gives:
wp c0 = a1 + (c3-c2) exp I - 8 I (12.57) respectively:
Now, all three integration constants are known and the concentration profile near membrane (0 < z < 8) follows to:
c = c0 - (c3-c2) exp | —— 8) + (c3-c2) exp (- —- z ) (12.59)
Considering Eq. (12.51), a simple result describes the dimensionless concentration polarization as a function of velocity wp and diffusion coefficient D of the retained component at the membrane surface z = 0 according to Eq. (12.59):
C0-C2 \ D The permeate flow rate wp is then given by:
with the mass transfer CoeffiCient: D
The mass transfer coefficient can be expressed by dimensionless numbers with wcf as the cross-flow rate:
These numbers and L/d result from a dimensional analysis of the problem. The determination of the relation between the Sherwood, Schmidt and Reynolds numbers as well as L/d is based on the analogy between heat and mass transport (Table 12.2). For laminar flow and simple geometric conditions, they can be calculated by solving mass and momentum balances. The relations presented are
Table 12.2 Sherwood number for different flow conditions experimentally proved for mass transport by Linton and Sherwood (1950) and Rautenbach and Albrecht (1981).
Sherwood number Flow condition
Canal with rectangular cross-section
Sh = 1.62 (Re Sc d/L)1/3 Sh = 0.023 Re7/8 Sc1/4 Sh = 0.04 Re3/4 Sc1/3
Sh = Shtube
Laminar Turbulent Re Sc dh/L > 102
proven experimentally for mass transport by convection and diffusion in tubes for laminar and turbulent flow conditions (Linton and Sherwood 1950) and in channels for laminar flow conditions (Rautenbach and Albrecht 1981).
From steady-state laminar flow in tubes of diameter d and length L as well as d/LP 1:
is valid. After calculation of kL from Eq. (12.66) and introduction into Eq. (12.61), the permeate rate follows:
We must determine not only the concentrations but also the diffusive coefficient D. A possible approximation for D is given by the Stokes Einstein equation in Eq. (5.4) as a function of absolute temperature T, solute radius R, dynamic viscosity r| and with the Boltzmann constant K (Section 5.1; Mulder 2000):
Deviations arise if D and v become dependant on concentration, as observed in the ultrafiltration of macromolecules, or when the cross flow rate rises for small values of d/L (Rautenbach 1997). This occurs particularly in ultrafiltration processes in the field of wastewater treatment if suspended solids are present which accumulate at the boundary layer.
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