A turbulent flow theory of electrodialysis

Abstract

A hydrodynamic theory is presented of desalination by electrodialysis in a parallel channel system with unobstructed fully developed turbulent flow. A set of turbulent transport equations is derived for the mean salt concentration and electric field. Closure of the set is brought about by introducing a turbulent diffusion coefficient whose behavior is assumed known from empirical mass transfer considerations, and by making two different but phenomenologically reasonable assumptions about that part of the average ion flux which is due to the correlation between the concentration and electric field fluctuations. Estimates show both assumptions to lead to sensibly the same final results. From the equations and boundary conditions it is shown that there are five parameters which govern the system performance. Four are the same as for the corresponding laminar case, while the fifth measures the thickness of the concentration diffusion sublayer, the value of which is taken to be given by semi-empirical turbulent mass transfer results for high Schmidt numbers. The thickness of this diffusion layer is found to be the critical factor governing the salt removal rate. An integral method of solution is used to obtain the distributions of salt concentration and current density. A numerical quadrature is required for the complete solution. Analytic solutions are obtained in the unpolarized and fully polarized limits. Because of the very abrupt transition between these two limits, taken together the limiting analytic solutions are adequate to describe the system performance over the entire range of operating variables. For small fractional salt removal a universal correlation for the average current density embodying the five similarity parameters of the system is suggested based on the limiting solutions. Comparison of the theory is made with the experimental results of Cowan and Brown on average limiting current density and the agreement is found to be satisfactory.