Journal of M embrane Science, 79 (1993) 133- 158 Elsevier Science Publishers B.V., Amsterdam 133 Phenomenological theory of reverse osmosis in macroscopically homogeneous membranes and its specification for the capillary space-charge model zyxwvutsrqponmlkjih A.E. Yaroshchuk and S.S. Dukhin A. V. Dumanskij Institute of Colloid and W ater Chemistry, Ukrainian Academy of Sciences, Vernadskij au. 42,252142 Kiev (Ukraine) (Received March 9,199l; accepted in revised form November 11,199l) A bstract The problem of transport of binary electrolyte solutions through macroscopically homogeneous mem- branes under reverse osmosis conditions with arbitrary thermodynamic forces has been solved in quad- ratures 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 ar- bitrary 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 ap- proaching 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 A. Keywords: theory; reverse osmosis; space-charge model zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIH Introduction The classical monograph by Schloegl [ zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDC 1] al- ready contained a general solution of the prob- lem of reverse osmosis of binary electrolyte so- lutions in terms of irreversible thermodynamics under linear conditions. Under nonlinear con- ditions the solution was obtained by Spiegler Correspondence to: A.E. Yaroshchuk, A.V. Dumanskij In- stitute of Colloid and Water Chemistry, Ukrainian Acad- emy of Sciences, Vernadskij av. 42,252142 Kiev, Ukraine. and Kedem in terms of continuous irreversible thermodynamics in the particular case of con- stant phenomenological coefficients [ 21. In 1969 Elata [3] generalized their approach, al- lowing a dependence of phenomenological coefficients on solute concentration and hydro- static pressure but assuming the variations to be small, so that, in expansions of the coeffi- cients in the Taylor series with respect to de- viations from mean values, terms of higher or- ders could be neglected. As a result the main system was completed by equations for the first 0376-7388/93/$06.00 0 1993 Elsevier Science Publishers B.V. All rights reserved.