Factors Determining the Extent of Concentration Polarization

The ratio of the concentration of a component at the membrane interface to the concentration in the bulk of the feed is called the 'concentration polarization modulus' and is a measure of the influence of concentration polarization on the separation process. The following expression for the modulus can be obtained from eqn [5]:

exp(vp/kbi)

Figure 2 Schematic of the concentration polarization phenomenon. The concentration profiles in the boundary layer result from the separation achieved by the membrane. The type of concentration profile formed depends on the value of wp relative to wb (or np relative to nb).

where Eo = wp/wm is the intrinsic enrichment achieved by the membrane (and equal to the actual enrichment if concentration polarization were absent). An equation equivalent to eqn [12] but expressed in mole fractions can be derived from eqn [11].

Eqn [12] allows the concentration polarization modulus to be calculated as a function of vp/kbl for different values of the intrinsic enrichment factors, Eo. The ratio vp/kbl is a Peclet number and is a measure of the influence of convection relative to the influence of diffusion in the boundary layer. The results of this calculation are shown in the very informative Figure 3, which confirms that the concentration polarization modulus is smaller than 1 (boundary layer depletion) if the permeating compound is enriched in the permeate and larger than 1 (boundary layer build-up) if the permeating compound is depleted in the permeate. The concentration polarization modulus increasingly deviates from unity as the ratio vp/kbl increases, that is, as the flux through the membrane increases or as the turbulence nm— np

Figure 3 Concentration polarization modulus, wm/wb, as function of vp/k fora range of values of the intrinsic enrichment factor Eo. Lines calculated through eqn [12]. This figure shows that compoundsthat are enriched by the membrane (Eo > 1) are more affected by concentration polarization than compounds that are rejected by the membrane (Eo < 1).

Figure 3 Concentration polarization modulus, wm/wb, as function of vp/k fora range of values of the intrinsic enrichment factor Eo. Lines calculated through eqn [12]. This figure shows that compoundsthat are enriched by the membrane (Eo > 1) are more affected by concentration polarization than compounds that are rejected by the membrane (Eo < 1).

of the feed fluid decreases. At high values for the ratio Vp/kbi, the concentration polarization modulus, wm/wb, approaches the limiting value 1/Eo. At this point, the boundary layer completely negates the separation power of the membrane permeation step. The concentration polarization modulus also increasingly deviates from unity as the intrinsic enrichment increasingly deviates from unity, that is, as the separation power of the membrane increases.

A striking feature of Figure 3 is the asymmetry with respect to enrichment and rejection. For example, when the term vp/kbl has a value of 10_1, concentration polarization is essentially nonexistent for a component rejected by the membrane with an intrinsic enrichment Eo of 10~4. On the other hand, concentration polarization is very severe for a component enriched by the membrane with an intrinsic enrichment Eo of 104. The reason for this asymmetry is that the concentration polarization effect is generated by the difference in concentration between the permeate and the feed, wp — wb = wb(E — 1), where E = wp/wb is the actual enrichment factor. It is clear that the absolute value of wp — wb is significantly larger if E > 1 than if E < 1.

A second feature of the calculations shown in Figure 3 is that the concentration polarization modulus values are independent of the bulk concentration, wb. This means that at a constant enrichment factor, E, the influence of concentration polarization is the same, no matter whether the component is present in the feed at a concentration of one part per hundred, one part per million, or one part per billion. Thus, concentration polarization does not necessarily affect components present at low concentrations more than components present at higher concentrations. The primary requirement for significant concentration polarization effects is a high value for the enrichment factor, E. However, because E has an upper bound equal to 1/wb, a low feed concentration is a secondary requirement for severe concentration polarization effects. This confirms an empirical rule long held by membrane separation practitioners.

Solar Panel Basics

Solar Panel Basics

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