# Ionic separation in electrodialysis : analyses of boundary layer, cationic partitioning, and overlimiting current

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Electrodialysis performance strongly depends on the boundary layer near ion exchange membranes. The thickness of the boundary layer has not been clearly evaluated due to its substantial fluctuation around the spacer geometry. In this study, the boundary layer thickness was defined with three statistical parameters: the mean, standard deviation, and correlation coefficient between the two boundary layers facing across the spacer. The relationship between the current and potential under conditions of the competitive transport between mono- and di-valent cations was used to estimate the statistical parameters. An uncertainty model was developed for the steady-state ionic transport in a two-dimensional cell pair. Faster ionic separations were achieved with smaller means, greater standard deviations, and more positive correlation coefficients. With the increasing flow velocity from 1.06 to 4.24 cm/s in the bench-scale electrodialyzer, the best fit values for the mean thickness reduced from 40 to less than 10 μm, and the standard deviation was in the same order of magnitude as the mean. For the partitioning of mono- and di-valent cations, a CMV membrane was examined in various KCl and CaCl₂ mixtures. The equivalent fraction correlation and separation factor responded sensitively to the composition of the mixture; however, the selectivity coefficient was consistent over the range of aqueous-phase ionic contents between 5 and 100 mN and the range of equivalent fractions of each cation between 0.2 and 0.8. It was shown that small analytic errors in measuring the concentration of the mono-valent cation are amplified when estimating the selectivity coefficient. To minimize the effects of such error propagation, a novel method employing the least square fitting was proposed to determine the selectivity coefficient. Each of thermodynamic factors, such as the aqueous- and membrane-phase activity coefficients, water activity, and standard state, was found to affect the magnitude of the selectivity coefficient. The overlimiting current, occurring beyond the electroneutral limit, has not been clearly explained because of the difficulty in solving the singularly perturbed Nernst-Planck-Poisson equations. The steady-state Nernst-Planck-Poisson equations were converted into the Painlevé equation of the second kind (P[subscript II] equation). The converted model domain is explicitly divided into the space charge and electroneutral regions. Given this property, two mathematical formulae were proposed for the limiting current and the width of the space charge region. The Airy solution of the P[subscript II] equation described the ionic transport in the space charge region. By using a hybrid numerical scheme including the fixed point iteration and Newton Raphson methods, the P[subscript II] equation was successfully solved for the ionic transport in the space charge and electroneutral regions as well as their transition zone. Above the limiting current, the sum of the ionic charge in the aqueous-phase electric double layer and in the space charge region remains stationary. Thus, growth of the space charge region involves shrinkage of the aqueous-phase electric double layer. Based on this observation, a repetitive mechanism of expansion and shrinkage of the aqueous-phase electric double layer was suggested to explain additional current above the limiting current.