## Mathematical Description of Concentration Polarization

The velocity profile of a fluid flowing in a channel is not constant across the thickness of the channel, because of friction at the fluid-channel surface interface. The fluid velocity decreases as the distance from the channel surface decreases. The same phenomenon occurs in the channels of a membrane module, and the resulting velocity gradient adjacent to the feed side of the membrane is characteristic of all membrane processes. To facilitate mass transfer analysis, the velocity gradient is usually represented by a step function, and it is assumed that a stagnant boundary layer exists adjacent to the membrane. Any component permeating the membrane must first pass through the boundary layer as illustrated in Figure 1.

Although the boundary layer is stagnant in the direction of the feed bulk flow, the boundary layer is subject to convective flow perpendicular to the membrane surface which is generated by the permeate flux. The convective transport of a component into the boundary layer from the bulk solution is given by the product vp • cp, where vp (cm s "1) is the convective velocity and cb (gcm~3) is the concentration in the bulk of the feed. The rate at which the same component leaves the boundary layer is vp • cp, where cp (g cm-3) is the permeate concentration. In general, if separation is achieved, cp does not equal cb, and the convective flows into and out of the boundary layer, generate a mass imbalance. This imbalance then forms a concentration gradient in the boundary layer, and the concentration gradient increases until diffusion of the component down the concentration gradient is sufficient to restore mass balance in the boundary layer.

At steady state, the sum of convective and diffusive transport in the boundary layer equals the amount permeated through the membrane. This steady state is expressed for each component by the equation:

where D (cm2s-1) is the diffusion coefficient, x (cm) is the coordinate perpendicular to the membrane surface and jw (gcm~2s_1) is the mass flux of i permeating through the membrane.

In liquid-phase separations (including pervapora-tion) concentrations are typically expressed as a weight fraction, wl = cjp where p (g cm-3) is the density of the liquid. Assuming that the density of the feed is constant in the boundary layer:

and assuming that the feed density is equal to the density of the permeate:

where wp (gg_1) is the weight fraction of i in the permeate and JWt (g cm~2 s_1) is the combined mass flux of all components permeating the membrane. Combining eqns [2] and [3] and eliminating the density p gives:

which, integrated over the thickness § (cm) of the boundary layer, yields the polarization equation:

Figure 1 Schematic of the boundary layer adjacent to the membrane surface. If cp > cb: component is enriched in permeate. If cp < cb: component is depleted in permeate.

where wm and wb are the weight fractions of 1 at the membrane surface and in the bulk of the feed, respectively, and kbl = D/ô (cms-1) is the mass-transfer coefficient in the boundary layer.

In gas-separation applications, concentrations are typically expressed as mole fraction n,, which is equal

Wm - Wp to the volume fraction, assuming the gas mixtures are ideal. Starting again with eqn [1], the mole fraction nl can be substituted for ci by using:

vP = Jot • T/(Pf • 273) elimination of the term pf • 273/T gives:

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).

Integrating eqn [10] in the same way as eqn [4] gives:

nb - np where 22 400 (cm3 (STP) mol1) is the molar volume of an ideal gas, T (K) is the gas temperature, Mi (g mol"1) is the molecular weight of i, pf (bar) is the feed gas pressure, and 273 K is the standard temperature. Also, the volume flux J V (cm3 (STP)cm~2 s"1) can be substituted for the mass flux JW using:

Elimination of the term M,/22 400 gives:

where nm and nb are the mole (or volume) fraction of i at the membrane surface and in the bulk of the feed.

Eqns [5] and [11] describe the concentration profiles that develop in the boundary layer, as illustrated in Figure 2. Any component enriched in the permeate will be depleted in the boundary layer and any component depleted in the permeate will be enriched in the boundary layer.

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