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Membranes 2020, 10, 221 6 of 72 Migr Ji F Sol∆U = −Di RUTziCi ∆X (1) with JMigr the flux by electromigration of the species i (in mole/s.m2), Di the diffusion coefficient of i the species i (in m2/s), F the Faraday constant (96,500 C/mol), zi the valence of the species i, CSol the i concentration of the species i in the solution (in mole/m3), ∆U the potential difference applied at the electrodes (in V), RU the ideal gas constant (8314 J/mol.K), T the temperature of the solution (in K) and ∆X the distance between the electrodes (in m). 2. The transport by diffusion (JDiff) through the partially desalinated boundary layer of the membrane i given by the Fick’s first law: −Di(CSol − CMemb) JDiff = i i (2) ii in the diffusion boundary layer at the interface of the membrane on the diluate side (in mole/m3) and δ the thickness of the boundary layer (in m). 3. The transport through the membrane (JTot) given by the Nernst–Planck’s equation: i iδ with JDiff the flux by diffusion of the species i (in mole/s.m2), CMemb the concentration of the species i ∆C F ∆U JTot =−Di i −zi Ci +Ciν (3) i ∆X RUT ∆X with JTot the flux of the species i through the membrane (in mole/m2·s), Ci the concentration of the i species i (in mol/m3) and ν the convection velocity (in m/s). Although the Nernst–Planck’s equation did not take into account the coupling of the fluxes between species migrating through the membrane, this equation describes major factors having an effect on the transfer of ions. This theory is based on the assumption that an ion-exchange membrane is considered as a thick phase equivalent to a solution, separating two adjacent aqueous phases. To describe correctly the transport of charged species through the membranes, it is necessary to couple the Nernst–Planck equation to the electroneutrality condition: zi·Ci = 0 (4) Or, considering the transport numbers of the anions and cations through the same membrane: Nbt− + Nbt+ = 1 (5) with Nbt− the transport number of the anions and Nbt+ the transport number of the cations given by the following equation: Nbti = FJizi (6) dI With F the Faraday constant (96,500 C/mol), Ji the molar flux of the species i (in mole/s·m2), zi the valence of the species i, dI = SI the current density (in A/m2 of electrode). 4. The transport by diffusion (JDiff) through the partially mineralized diffusion boundary layer on the i other side of the membrane given, as previously, by the Fick’s first law. 5. And, finally, the transport by migration (JMigr), of the charged species from the mineralized i membrane diffusion boundary layer to the concentrate solution, given as previously by the Faraday’s law.PDF Image | Electrodialytic Processes
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