Rate-Based Design of Non-Fouled Cross-Flow Hollow-Fibre Membrane Modules for Hyperfiltration

Abstract

Although hyperfiltration has replaced many liquid phase separation equipment, it is still considered as a ‘non-unit operation’ process because the sizing of hyperfiltration equipment could not be calculated using either the equilibrium stage or the ratebased methods. Previous design methods using the dead-end and the 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 the more concentrated rejection concentration in the driving force. In this paper, a cross-flow model for hyperfiltration is 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 modelling is rate-based, the length of the module could be expressed as the product of theheight of a transfer unit (HTU) and the number of transfer units (NTU). The solution of the integral representing the NTU of hyperfiltration is found to be the difference between two hypergeometric functions. The poles of the solution represent the flux extinction curves of hyperfiltration. 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 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 fibre module for hyperfiltration.