Rejection of single salts versus transmembrane volume flow in RO/NF: thermodynamic properties, model of constant coefficients, and its modification

Abstract

The general features of dependences of single salt rejection on the transmembrane volume flow in RO/NF are analysed thermodynamically for a macroscopically homogeneous membrane. At no concentration polarisation (CP), the rejection is shown to monotonely approach a limiting value irrespective of the rejection mechanism. The popular approximation of constant transport coefficients introduced by Spiegler and Kedem yields correct asymptotics of rejection at both small and large transmembrane volume flows. Therefore, it often works relatively well also at intermediate volume flow. However, when the transport coefficients essentially depend on the salt concentration, that approximation is not suitable for a really quantitative interpretation of experimental data. That is demonstrated for the model of charged membrane in the limiting case of excluded coions. The analytical solution obtained in this case is semi-empirically extended to the cases when the dependence of transport coefficients on salt concentration is not that pronounced. The applicability of new semi-empirical equation is checked by the comparison with model numerical calculations as well as some experimental data. It is shown that, in principle, the new approach may work essentially better than the classical Spiegler–Kedem method, especially in the mode of considerable CP, which often occurs in membrane modules used for practical separations. However, to make a real use of potentialities of new approach, much more careful and purposeful bench-scale measurements of salt rejection have to be performed than currently common.