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Membranes 2020, 10, 221 9 of 72 Since NbtMemb > NbtSol this implies that JMemb > JSol. This flux difference results in a depletion of iiii the ionic species in the DBL at the diluate-CEM side and by an excess of these ions in the DBL at the concentrate-CEM side. Moreover, this excess of ions at the concentrate-CEM side is intensified by the co-ions excluded by the membrane and that are trapped in the DBL. A similar phenomenon occurs at the interfaces of the anionic membranes. In a general manner, at the CEM-diluate and AEM-diluate interfaces, the respective concentrations in cations and in anions decrease, to impoverish the diluate in salts. At the same time, at the CEM-concentrate and AEM-concentrate interfaces, the anion and cation concentrations increase respectively to enrich the concentrate solution in salts. At steady state, the flux of the ionic species from the diluate solution to the concentrate solution should be constant. There is then another transport mechanism which compensates for the depletion of ionic species. This mechanism is called the diffusion. It occurs due to concentration gradients formed on both sides of the membrane. Since the flux of the ionic species through the DBL is equal to a fraction of the total quantity of equivalent that can be transported according to the Faraday’s law, this fraction is calculated by the transport number of this ion: NbtClim dI JClim = i (9) iF with JClim the flux of ions by electrotransport (in mole/m2·s), NbtClim the transport number of the ion in ii the DBL, dI = SI the current density (in A/m2 of electrode) and F the Faraday constant (96,500 C/mol). The flux of this ionic species by diffusion through the same DBL is expressed, as previously, by the Fick’s first law: (CSol − CMemb) JDiff = −Di i i (10) iδ At steady state, the ionic flux and the diffusion of ions due to the concentration gradient established between the turbulent solution and the membrane surface, the DBL, are equivalent: (CSol − CMemb) JMemb −JSol = JDiff = −Di i i (11) iiiδ d NbtMemb − NbtSol −D (CSol − CMemb) Ii i=iii (12) or where dI is the current density (in A/m2 of electrode), F the Faraday constant, Di the salt diffusion coefficient, CSol the concentration of the species i in the solution (in mole/m3), CMemb the concentration of the species i in the DBL at the interface of the membrane on the diluate side (in mole/m3), NbtMemb i and NbtSol the salt counterion effective transport number in the membrane and solution, respectively, i and δ the thickness of the DBL (in m). Then, if the mass transfer is further increased by increasing the voltage applied between the electrodes of the ED cell, this will result in an increase of the current density and of concentration gradients in the DBLs. The driving force for the diffusion (Figure 5b) is then more important. Effectively, CMemb at the diluate/CEM side will decrease resulting in an increase of the (CSol − CMemb) difference iii in the DBL. Fδ This equation is the well-known Nernst’s equation and can also be expressed as: FDi(CSol − CMemb) ii dI = − NbtMemb − NbtSol δ (13) ii iiPDF Image | Electrodialytic Processes
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