Although ultrafiltration and hyperfiltration have replaced many liquid phase separation equipment, both are still considered as “non–unit operation” processes because the sizing of both equipments could not be calculated using either the equilibrium stage, or the rate–based methods. Previous design methods using the dead–end and complete–mixing models are unsatisfactory because the dead–end model tends to underestimate the membrane area, due to the use of the feed concentration in the driving force, while the complete–mixing model tends to overestimate the membrane area, due to the use of a more concentrated rejection concentration in the driving force. In this paper, cross–flow models for both ultrafiltration and hyperfiltration are developed by considering mass balance at a differential element of the cross–flow module, and then integrating the expression over the whole module to get the module length. Since the modeling is rated–based, the length of both modules could be expressed as the product of the height of a transfer unit (HTU), and the number of transfer unit (NTU). The solution of the integral representing the NTU of ultrafiltration is found to be the difference between two exponential integrals (Ei(x)) while that representing the NTU of hyperfiltration is found to be the difference between two hypergeometric functions. The poles of both solutions represent the flux extinction curves of ultrafiltration and hyperfiltration. The NTU for ultrafiltration is found to depend on three parameters: the rejection R, the recovery S, and the dimensionless gel concentration Cg. For any given Cg and R, the recovery, S, is limited by the corresponding flux extinction curve. The NTU for hyperfiltration is found to depend on four parameters: the rejection R, the recovery S, the polarization β, and the dimensionless applied pressure difference ψ. For any given ψ and R, the recovery, S, is limited by the corresponding flux extinction curve. The NTU for both ultrafiltration and hyperfiltration is found to be generally small and less than unity but increases rapidly to infinity near the poles due to flux extinction. Polarization is found to increase the NTU and hence the length and membrane area of the hollow fiber module for hyperfiltration.
Key words: Ultrafiltration; hyperfiltration; reverse osmosis; hollow fiber module design; crossflow model; number of transfer unit; height of a transfer unit
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