## 5 Concentration Polarization in Electrodialysis (ED)
### Current-Induced Concentration Gradients
The expression for \(i_{\rm lim}\) can be obtained from Equations (37), (38), when writing the concentration gradient of the counterion (1) at the membrane surface as:  
\[\left(\frac{\partial c_{1}}{\partial x}\right)_{s}=-\frac{c_{1}^{0}-c_{1\,s}}{ \delta} \tag{39}\]  
where \(\delta\) is the thickness of the Nernst DBL in the depleted solution. According to Equation (39), \(\delta\) is the distance from the membrane to the intersection point of the tangent drawn to the concentration profile at the interface with the line corresponding to the bulk solution concentration. Substituting Equation (39) into Equation (37) and setting \(c_{1s}=0\) yields:  
\[i_{\rm lim}=\frac{FDc^{0}}{\delta(T_{1}-t_{1})} \tag{40}\]  
where \(c^{0}=z_{+}c_{+}^{0}=-z_{-}c_{-}^{0}\) is the bulk electrolyte concentration in eq L\({}^{-1}\). Equation (40) was first obtained by [PERSON] in 1956 [211]. According to the classical theory [57] based on the LEN assumption, when \(i\) tends to its limiting value (\(i_{\rm lim}\)), the PD over a membrane surrounded by two DBLs tends to infinity. However, in real membrane or electrode systems, the limiting current density can be exceeded in several times [212; 213; 214; 215]. The causes of the overlimiting current are vividly discussed in literature [212; 214; 216; 217; 218; 219]. The main effects are current-induced convection (electro- and gravitational convection) and generation of additional charge carriers (e.g., water splitting at the depleted solution/membrane interface, which produces the H\({}^{+}\) and OH\({}^{-}\) ions). While current-induced convection enhances the mass transfer across the membrane, water splitting in most cases is undesirable effect causing pH changes and the loss in current efficiency. Experimental observations [212; 213; 214; 215; 216; 217; 218; 219] as well as theoretical evidences [216; 230; 231] show the dominant role of electroconvection in salt counterion overlimiting transfer in dilute solutions. Vortices, which appear under the action of electric field on the space charge, can increase the useful transfer of salt counterions in several times by mixing depleted solution [217; 230; 232; 233]. Along with these two main effects, additional effects, which may occur in some conditions, are discussed in literature. [PERSON] et al. [234] described theoretically the effect of current-induced membrane discharge. This effect can occur in a system with an AEM, which has weakly basic functional groups. In this case, OH\({}^{-}\) ions generated in water splitting at the depleted membrane interface may deprotonate these functional groups, that is, discharge them. This produces: (1) a decrease in the surface charge density, which reduces electroconvection developing as electroosmosis of both the 1 st and 2 nd kind [235; 236] and (2) a decrease in IEC, which results in increasing co-ion back diffusion through the membrane. The latter leads to a decrease in \(T_{1}\), hence, in an increase of the limiting current density according to Equation (40). Note that electroosmosis of the 2 nd kind is named also induced-charge electroosmosis [237; 238], since this effect is due to the extended space charge region [239], which emerges next to the equilibrium EDL at overlimiting currents. In the early period of ED, the use of underlimiting current was prescribed [210; 240]. However, nowadays the overlimiting current regimes are widely used in ED and especially in electro-deionisation of diluted electrolyte solutions [212; 241]. The use of such intensive current regimes allows reduction of expensive membranes' area, which can cause the reduction of overall costs of ED desalination [210]. The effect of electroconvection is relevant also in a number of nano- and microfluidic devices [194; 242] such as electrokinetic micropumps [243; 244]. Besides, intensive current regimes are applied in electrophoresis [242], electrodeposition [245], layering of colloid crystals on electrode surfaces [246] and other domains related to separation science [58; 203].