This study is concerned with the engineering design and analysis of systems for which the basis for separation is the selective permeation of dissociated salts from aqueous feed to strip solutions through immobilized liquid membranes. Theoretical equations are derived for binary mixtures, assuming Fick's law, to model perfect-mix permeation stages, and separation cascades containing perfect-mix stages. The results should apply both to passive (ordinary) diffusion and macrocycle-mediated transport. Important stage variables include the ratio of total dissociated salt concentrations in the stage feed and strip solutions, the total concentration in the stage strip solution, the stage cut, and the ratio of permeability coefficients for the two species in the separation mixture, i.e., the ideal separation factor. The stage separation factor is a complex function of these variables, and may be much less than the ideal separation factor, depending on values of these variables. Severe stagewise diffusional limitations can exist, particularly if the feed solution must be stripped to very low concentration levels. A conceptual design for an ion permeation cascade is presented which suggests that the overall ion permeation process is quite complex and may be energy intensive. For many separations, cascading is required to achieve desired separations, even with very large ideal separation factors.
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