Concentration polarisation in electrodialysis

The transport (flux) of cations through the membrane caused by an electrical potential difference is given by  

In these various equations J and Jbi are the electrically driven fluxes in the membrane and the boundary layer, while Jq is the diffusive flux in the boundary layer. The transport numbers of the cation in the membrane and in the boundary layer are t and tb. z is the valence of the cation (z = 1 for Na ) is the Faraday constant v is the electrical current and dc/dx is the concentration gradient in the boundary layer. 

At steady state the transport of cations through the membrane is equal to the combined electrical and diffusive flux towards the membrane in the boundary layer, i.e. 

Integration of eq. Vn - 55, assuming a constant diffusion coefficient (linear concentradon profile) and using the following boundary conditions. 

The ohmic resistance is located mainly in the boundary layer where ion depletion has occurred. Because of such depletion the resistance in the boundary layer will increase so that part of the eiecuical energy may be dissipated as heat (electrolysis of water) if the concentration becomes too low. The current density v in that layer can be obtained from eq. vn-56. 

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