Phenomenological theory of reverse osmosis in macroscopically homogeneous membranes and its specification for the capillary space-charge model

Abstract

The problem of transport of binary electrolyte solutions through macroscopically homogeneous membranes under reverse osmosis conditions with arbitrary thermodynamic forces has been solved in quadratures in terms of irreversible thermodynamics. Phenomenological coefficients have been specified for the model of straight cylindrical capillaries with an arbitrary cross-section and surface forces of an arbitrary nature. These general expressions have been further specified for the space-charge model and identical capillaries with a circular cross-section. A new technique to describe the structure of strongly and moderately overlapped double electrical layers has been developed. As a result, analytical expressions for the main characteristics of charged reverse osmosis membranes have been derived at arbitrary fixed charge densities. Limiting rejection, hydrodynamic permeability, filtration potential and the rate of approaching limiting rejection have been calculated as functions of fixed charge density, pore size, binary electrolyte concentration, valency type and the ratio of cation and anion mobilities. A comparison with experiment has been carried out for model membranes having identical straight circular cylindrical pores with radius 40 or 50 Å.