Computer prediction of stationary states of membranes from differential permeabilities

Discussions of fluxes and forces across ion-exchange membranes have frequently been couched in the terms of non-equilibrium thermodynamics in its linear form. The value of many such treatments has been restricted because it is well known that the phenomenological coefficients, which describe the behaviour of such a membrane in a macroscopic experiment, are functions of the concentrations outside the membrane and of the concentration profiles within the membrane. As a result the empirical coefficients appear to be functions of the forces. This is in conflict with the requirements of linear theory. p]Despite these limitations on the non-equilibrium approach as applied directly to global or macroscopic flux data, it has been shown that the local or microscopic behaviour of ion-exchange membranes obeys the linear approximation well over usefully wide ranges of the three gradient forces: activity, electric potential and pressure. By using an appropriate analysis of sufficient experimental data it has been found possible to extract the local phenomenological permeability coefficients of a cation exchange membrane as functions of local composition. From these coefficients, resistance and friction coefficients have been calculated and they have revealed much concerning the mechanism of interaction between the various mobile ionic components, water and the membrane material. p]It is desirable to be able to predict from the data on the concentration-dependent permeabilities how a macroscopic membrane of known dimensions would perform under a given set of external constraints e.g. when a predetermined current is passed through the membrane placed between particular external solutions as in electrodialysis. p]The problem at once arises that the fluxes and profiles are strongly interdependent and mean permeabilities cannot be chosen on a simple basis. In order to overcome this difficulty a computational procedure has been developed which enables the final steady state under given constraints to be evaluated by an iterative procedure. p]The membrane is regarded as a hypothetical series array of slices of equal thickness. The number of slices has to be chosen sufficiently large that the difference in composition across any single slice is small enough for its behaviour to be described by the local coefficients appropriate to the mean composition of the slice. The computation then adjusts the profiles across the whole membrane until each slice is in interfacial equilibrium with its neighbours and the imposed constraints are satisfied. The required fluxes and forces are then evaluated without difficulty. Useful and informative by-products from the computation are the steady profiles of the intensive variables in the membrane. These are not accessible to direct measurement. p]The procedure has been tested by calculating the osmotic and salt diffusion fluxes and the membrane potentials when the membrane separates different solutions at zero current and pressure difference. Although the computation requires the inversion of a 60 x 60 square matrix in most cases, the results have been found to agree satisfactorily with those obtained experimentally. p]The procedure has then been used to generate fluxes when an electric current is passed between different solutions as in electrodialysis. The data show how the membrane properties in electrodialysis depend on the current density, external concentrations and membrane thickness in a more precise way than has been available hitherto. The membrane profiles are found to exhibit a very pronounced dependence on the current density.