Solution-diffusion-film model for the description of pressure-driven trans-membrane transfer of electrolyte mixtures: One dominant salt and trace ions

The solution-diffusion-film model previously used for the description of rejection of single salts by NF membranes is extended to the case of electrolyte mixtures consisting of a single dominant salt and (any number of) trace ions. An especially simple analytical solution is obtained in the approximation of constant (composition-independent) membrane permeabilities with respect to the dominant salt and trace ions. A quasi-analytical protocol is developed for the correction of observable trace rejections for concentration polarization within the scope of film model. It is demonstrated that the trans-membrane transfer of trace ions is essentially influenced by the electric fields arising due to the differences in the membrane permeabilities with respect to the cations and anions of dominant salt. Accordingly, information on these permeabilities can be extracted from experimental data on the rejection of trace ions as a function of trans-membrane volume flow. The membrane permeability to the trace ions themselves can also be estimated. This is clearly advantageous over just a single value of membrane permeability to the salt obtainable from the interpretation of traditional measurements with single salts within the scope of the same model. This novel theoretical approach is used for the interpretation of experimental data on the rejection of various dominant single salts and trace ions by a commercial NF membrane in a cross-flow setup with a rectangular spacer-filled feed channel. In this way, quantitative information could be obtained on the membrane permeabilities with respect to several single ions (Na+, Cl−, Ca2+, SO42−) being parts of dominant salts as well as traces. This information can be useful for the optimization of nanofiltration of mixed electrolyte solutions in various applications.