Basic Equations:

The process parameters as defined above depend on several factors such as the physical structure and the chemical nature of the membranes, pressure, temperature, nature and concentration of solute and a flow rate of the feed salt solution along the membrane surface etc. A nature of the dependence of those factors on the procedures performance are analysed and a number of quantitative expressions are derived. A detailed description of their mathematical aspect is beyond the scope of this unit. A simplified expression for water flux and solute separation based on solution-diffusion mechanism is given below.

Within the solution-diffusion mechanism, a permeation of solvent as well as solute by the membrane arises because of the dissolution of the permeating species within accordance along with an equilibrium distribution law and diffusion through membranes in response to pressure and concentration gradient. An expression for water flux is given by

J_w   = D_w C_{w  w}  (P -Π) / RT ?x

where P and Π are respectively, the applied pressure and osmotic pressure of the feed solution, ?x is the thickness of the membrane, D_w  and C_w , respectively refer to diffusion coefficient and concentration of water in the membrane phase expressed as a volume ratio and which is a dimensionless quantity.

The terms  _w, R, T refer to partial molar volume of water, gas constant and temperature, respectively. ,

The quantity D_w C_{w   w} / RT?x   can be considered a membrane constant and denoted by A whose magnitude depends on the physical structure as well as the chemical nature of the membrane and the temperature of the feed. Accordingly, Eq. can be written as

J_w   = A (P - Π)

Eq. also indicates that the observed water flux across a membrane not only depends on the value of membrane constant A, but also on the applied pressure and osmotic pressure of the feed which is a function of solute molarity, its nature and temperature. For provide applied pressure, if the solute attentiveness in the feed is higher, its osmotic pressure will be higher and consequently, the observed water flux will be lower. Instead, for a given feed solution with a definite feed concentration, higher applied pressure will improve the observed water flux across the membrane. It follows that for separating pure water from salt solution having higher solute concentration, such as sea water, a higher operating pressure is required to maintain the same driving force whereas lower operating pressures are sufficient for feed solutions of lower solute concentrations.