The permeate flux, F, through a semipermeable membrane is given by: Ds * Cw * V
Ds = the diffusion coefficient
Cw = the concentration of water
V = the molar volume of water
AP = the driving pressure (see Fig. 10.2)
R = the gas constant d = thickness of membrane
The equation (10.3) indicates that the water flux is inversely proportional to the thickness of the membrane. These terms can be combined with the coefficient of water permeation, Wp, and equation (10.3) reduces to:
For the solute flux, Fs, the driving force is almost entirely due to the concentration gradient across the membrane, which leads to the following equation (Clark, 1962):
dx d where
C'i = the concentration of species, i, within the membrane AC'i = concentration difference measured across the membrane
This equation can be restated in terms of the concentration of the solution, Ci, on either side of the membrane, incorporating the so-called distribution coefficient, Kd, which is a constant for the membranes generally used (Lonsdale et al„ 1965):
d where Kp is termed the coefficient of permeability.
Wp and Kp are both characteristics of the particular membrane type. As seen from equations (10.4) and (10.7), the water flux depends on the net pressure difference, while the solute flux depends only on the concentration. Therefore, as the feed water pressure increases, water flow through the membranes increases, while the solute flow is approximately constant. Consequently the amount and quality of purified water increase as the net driving pressure is increased, but the quality of the water decreases as the feed water solute concentration increases, with a constant pressure, because of an increase in osmotic pressure. As ever more water is extracted from the waste water, the solute concentration becomes higher and the water flux falls. Figures 10.3 and 10.4 illustrate these relations. The water flux as a function of the water recovery and at a fixed pressure is shown in Fig. 10.3 for two different salinities. The variation of the water quality with recovery is shown in Fig. 10.4. As can be seen, the water quality decreases with increasing feed salinity and increasing recovery. This is the problem touched upon as point 1 on p. 339.
The rejection ratio in reverse osmosis, R, is defined on the basis of the following equation:
Ci = the concentration of the species, i, in the concentrated stream (reject) Cpi = the concentration of i in the permeate (product).
0 20 40 60 80 100
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