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definition of the concentration polarisation gradient. In fact, under certain hypotheses (for more details see Section 5.1.1), the flux 𝐽⃗ of type-i ions can be expressed as ⃗ ⃗ ⃗ (7) 𝐽 = − 𝐷 ∇ 𝐶 − 𝑧 𝐹 𝐷 𝐶 ∇ 𝜑 + 𝐶 𝑢 ⃗ where Di is the ionic diffusion coefficient, Ci is the concentration, zi is the valence, F is Faraday’s constant, 𝜑 is the electric potential, and 𝑢⃗ is the velocity vector. Hence, the total flux is given by the sum of diffusive, migrative and convective flux. The ion diffusion-migration flux can also be expressed as a chemical diffusion-Ohmic conduction flux [62], so that, for a strong binary electrolyte, it can be written as: 𝑡 𝚤⃗ ( 8 ) 𝐽 = −𝐷∇𝐶 + 𝑧𝐹 where D is the electrolyte diffusion coefficient, ti is the migration transport number and 𝚤⃗ is the current density. At the membrane-solution interface, the mass balance under steady state conditions is obtained by equating the flux on the solution side (eq. (8)) with that on the membrane side. Considering only the component normal to the membrane surface (y coordinate) one has: ⃗ ⃗ −𝐷𝜕𝐶 + 𝑡𝑖 =𝑇𝑖 𝜕𝑦 𝑧𝐹 𝑧𝐹 (9) where 𝑇 is the integral transport number within the membrane [62, 173, 174] accounting for both ionic diffusion and migration (i is positive if directed towards the positive y axis). Therefore, the following boundary condition can be written: 𝜕𝐶=− 𝑖 𝑇−𝑡 (10) 𝜕𝑦 𝑧𝐹𝐷 In the literature, this boundary condition is often given by a less rigorous expression where 𝑇 is replaced by the migration transport number within the membrane 𝑡 [168–170, 175–178], but the approximation is legitimated by the fact that diffusion inside the IEM is often negligible. Moreover, commercial membranes normally have high permselectivity, i.e. the counter-ion transport number is close to 1 in the membrane, while being close to 0.5 in solution (NaCl). As a result, a diffusive flux roughly equal to 50% of the current density (i) divided by ziF, i.e. equal to the conductive transport for each ion, is established at the IEM-solution interface. Thus, salt diffusion (of both co- and counter-ions due to electroneutrality) between the solution bulk and the interface takes place, in the same direction of migration for counter-ions and in the opposite direction for co-ions. Salt depletion occurs at the IEM-diluate channel interface, while salt enrichment occurs at the IEM- concentrate channel interface, affecting the electric potential profile as shown in Figure 6 (b). 24PDF Image | Electrodialysis for water desalination
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