Modelling NaCl concentration profiles across a cation exchange membrane in the transient regime

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A model is developed to simulate NaCl concentrations across a cation exchange membrane in the transient regime. The convectionless Nersnt-Planck equation is combined with the continuity equation to obtain a partial dierential equation in time and space. The boundary conditions at the edges of the cell for this equationare a constant bulk concentration. Furthermore, in presence of an applied potential, the potential gradient is assumed to be linear with dierent values for the gradient inside and outside the membrane. This dierent value inside the membrane is due to Donnan potentials that arise at the membrane/electrolyteinterfaces. On top of this, because of the Donnan exclusion, the concentrationsare not continuous on the interface. To account for this, the cell is splitinto three regions: the membrane and two electrolyte regions on either side of the membrane. The boundary conditions to tie these regions together, is that the ux should be continuous across the interface. The problem has the form of an initial value problem. The partial differential equation is discretized using a finite difference method and is integrated using the forward Euler method.The results show an accumulation of ions at the membrane interface closest to the positive electrode, while ions are depleted at the other interface. This accumulation of ions is too big however, with the Na+ concentration at one point reaching 12M after only 0.10 seconds, while the concentration was 1M atthe start. This is clearly not what one would expect to happen physically. Therewere, however, no errors found in the numerical method as it was possible tocorrectly deduce and predict concentration changes, based on the concentrations and ux in a certain time step. Therefore it appears that there is an error in the assumptions that were made when developing the model. It is believed that this error lies in the assumption that the potential gradients only has two different values. This assumption ignores the diffusion boundary layers on the outside of the membrane. In this diffusion boundary layer, the potential gradient has another, different value. Therefore, to obtain more accurate results, the cell should actually be split into five different regions: the membrane, two diffusion boundary layers and two electrolyte solutions.