A Simple Model of Reverse Osmosis

Illustration 8.6 A Simple Model of Reverse Osmosis We set as our task here the replacement of the rigorous PDE model of reverse osmosis by a simple and approximate treatment, which makes use of certain empirical findings. Given the uncertainty of the parameters used in even the most rigorous models, this is no more than what good engineering sense dictates. 

Our starting point is the basic flux equation for water transport through membranes driven by hydrostatic and osmotic potentials, a process alluded to in Chapter 1 (see Table 1.2). We have 

The osmotic pressure 7t is a colligative property, i.e., a property that depends on the particle molarity and is for infinitely dilute solutions given by the van t Hoff equation  

This equation holds well for the dilute solutions involved in nanofiltrahon and for brackish water. Even for sea water, with Cn 10 mol/m it predicts an osmotic pressure of 7t = 10 x 8.31 x 298 = 2.48 MPa = 24.6 atm, compared to an experimental value of 23 atm (Table 8.3). We adopt this equation to replace osmotic pressure by concentration and make the further assumption that the osmotic pressure in the permeate is zero, i.e., that the membrane is 100% effective in rejecting salt and the bulk concentration Cf, is constant, as was deduced in Illustration 5.2 for laminar entry region flow. We can then combine the two equations and factor out the constant Q to obtain 

This expression opens a way to introduce Brian s equation (Equation 8.8d) to eliminate the unfaiown polarization modulus C /Q. We have 

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Illustration 8.8 A Simple Model of Reverse Osmosis and Ultrafiltration... 

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