## Transport Equations Incorporating Concentration Polarization

As pointed out in the previous sections, concentration polarization primarily affects membrane permeation by the change in composition at the membrane interface relative to the bulk of the feed mixture. To calculate the effect of concentration polarization on flux and separation, the transport equation for the membrane can be combined with eqn [5] or eqn [11] to arrive at a set of equations that predict the permeate flux and composition.

Ultrafiltration, nanofiltration and reverse osmosis are membrane processes in which a solute is separated from a solvent using a solute-rejecting membrane. Typically the permeate is essentially pure solvent, free of the solute. A simple but very effective transport equation developed for this situation is given below.

The pure solvent flux /W,lvent (gcm~2s_1) of the membrane is given by:

/solvent AP/Rm

where AP (bar) is the pressure difference applied across the membrane and Rm (barcm2sg_1) is the membrane resistance to the solvent. When a solute is present, the driving force for permeation is reduced by the osmotic pressure difference between the feed at the membrane interface and the permeate, Anm (bar), therefore:

Eqn [14] is called the 'osmotic pressure model', in which the osmotic pressure is a measure of the ther-modynamic work required to produce solvent from a solvent-solute mixture. Assuming that the permeate solute concentration is neglible:

where a is a constant and n is an exponent equal to approximately 1 for low-molecular-weight solutes,

I 10

I 10

 / Pure solvent Increasing feed ff concentration, wb and/or â€”â– â€”' decreasing f yS^ mass-transfer _________ coefficient, K^râ€” Applied pressure, AP 5 10 15 Applied pressure, AP and: the membrane resistance, which also has been confirmed experimentally.

## Solar Panel Basics

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