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however, requires that the active membrane he on the interior of the hollow fiber tubes. It is much easierto apply a uniform membrane film to the exterior surface of the tubes, but to take advantage of such an

arrangement the feed must be applied to the shell side, on which some deviation from ideal plug flow isinevitable. Such deviations can, however, be minimized by good design; so this arrangement is in factwidely used in commercial systems.8.2.1 Effect of Flow Pattern

Since the effective separation factor is reduced by back pressure (Eq. 8.10), the flow pattern has apronounced effect on the performance of a membrane system. This may be clearly shown by calculatingthe purity-recovery profiles for different flow schemes. As in any mass transfer process, countercurrent

flow maximizes the average driving force and therefore provides the most efficient arrangement. It isrelatively easy to achieve a reasonable approximation to plug flow on the high-pressure side, but this ismuch more difficult on the low-pressure side because of the wide variation in the gas velocity (from closeto zero at the closed end to a significant value at the permeate exit). If the Pressure ratio is large,deviations from plug flow on the low-pressure side have a relatively minor effect on performance,

provided that plug flow is maintained on the high-pressure side. The operation of many membranemodules, particularly those of the hollow fiber type, is therefore well represented by the cross-flow

model, which assumes plug flow on the high-pressure side with perfect mixing on the low-pressure side[Figure 8.7(b)].The worst case from the point of view of process efficiency is perfect mixing on both sides of the

membrane. This provides a useful limiting case for assessing the effect of flow pattern on performance,tail in general one would try to avoid this condition en an operating system.

8.3 Calculation of RecoveryPurity Profiles

8.3.1 Mixed Flow

The fractional recovery (R) is defined simply as the fraction of the less Permeable species that emerges inthe raffinate product stream:(8.12)for a well-mixed system the mole fractions x2and y2in the raffinate and permeate streams are relatedthrough Eq. 8.2. The separation factor is constant throughout the system and is given by Eq. 8.10 withx= x2. Calculation of the recoverypurity profile is therefore straightforward, requiring only the

combination of an overall mass balance for the less permeable species:

with Eqs. 8.2, 8.10, and 8.12.

8.3.2 Cross-Flow

The calculation is slightly more complex for the cross-flow case, since it is necessary to account for thevariation of partial pressure with position on the high-pressure sld. For the ideal cross-flow systemsketched in Figure 8.6(h), a differential mass balance for the more rapidly diffusing species gives:

whereL is the (local) molar flow rate on the high-pressure side. The local Concentrations x and y on thehigh- and low-pressure sides of the membrane are related by Eq. 8.2. Substitution in Eq. 8.14 and

rearranging yields

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which may be integrated from the inlet (x x,) to any arbitrary exists mole fraction (x2):

Combining Eqs. 8.12 and 8.16, we obtain: