Figure 10.3 Water flux related to water recovery for two concentrations of inorganic components (salts).
The rejection ratio is also expressed by the following equation:
where, Cwp, is the water concentration in the permeate. Notice that Kp • Cwp and Wp (AP - n) must be expressed in the same units. As Wp(AP - n) = F is often expressed as g or kg / cm2 or m2/ sec. Cwp must be expressed as g / cm3 or kg / m3.
The equations given so far are idealized because a good mixing on the brine side has been assumed, so that there is no concentration polarization. However, in reality salt concentrations build up at the membrane surface and a concentration gradient is established.
The increased concentration of the membrane surface raises the local osmotic pressure, so reducing the driving force. Concentration polarization is
defined as the ratio of the salt concentration at the membrane surface to the salt concentration in the stream.
Percentage water recovery Figure 10.4 Quality of product related to water recovery for two concentrations.
When the concentration adjacent to the membrane surface exceeds a critical value, the flux begins to level off with increasing driving pressure, AP. The flux is then controlled by the membrane permeability as well as the concentration polarization. This is illustrated in Fig. 10.5.
The following differential equation describes concentration polarization: F * Ci dCi
Owe dx where Owe is the water concentration in the reject.
If the membrane is impermeable to the solute, it means that Kp = 0, and equation (10.10) can be integrated to give:
Cia Owe * Ds where
Cim = the concentration of i in the fluid at the membrane surface Cia = the average concentration of I in the reject.
Equation (10.12) indicates the usual relation between flux and concentration of retained substances. The flux decreases, see equation (10.12) with increasing concentration of retained substances:
k is an overall mass transfer coefficient
Cs is the concentration of retained species adjacent to the membrane surface
Q is the concentration of the species, i, in the concentrated stream (reject)
The following description for polarization in turbulent flow has been developed:
Sc = the Schmidt number (the definition; see Section 7.4) f = the fanning friction factor p = the specific gravity of the solution.
The concentration polarization is seen to be a function of the ratio, average product flow rate to average brine velocity, the fanning friction factor and the Schmidt number. Since F/(p*v) is almost proportional to recovery, polarization is favored by high recovery. However, high recovery can be maintained at low concentration polarization by recirculating the brine. The concentration polarization can be reduced by increasing the friction factor, so promoting turbulence.
The concentrations of the ions in the waste water, the required concentrations in reject and effluent are the dominating variables in membrane processes. They determine the relation between flux and pressure according to the equations given above.
However, the temperature and pH play an important role In the durability of membranes. This is illustrated for cellulose acetate membranes in Figure 10.6. Cellulose acetate is not recommended for extreme pH values or high temperatures, but it is widely used due to its moderate costs. At extreme conditions other membranes should be chosen; see Section 10.4. The relationships for all types of membranes between on the one side the durability and on the other side pH and temperature are approximately as illustrated for cellulose acetate in Fig. 10. 6.
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