## 5 Concentration Polarization in Electrodialysis (ED)
### 2D Modelling of Electrodialysis with Homogeneous Membranes
#### 5.2.2 Concentration Distribution and Diffusion Layer Thickness
However, similar equations can be obtained by treating the values of \(i_{\text{lim}}\) or \(\delta_{N}\) found numerically from a convection-diffusion problem where zero concentration at the wall is used as the boundary condition. In this case, the limiting current density is in excellent agreement with Equations (60), while the values of the local and average \(\delta_{N}\) are slightly (about 4%) higher than those presented by Equations (61) due to the effect of the omitted second term. Namely, instead of 0.68 and 1.02, the numerical solution gives 0.71 and 1.06, for the local and the average \(\delta_{N}\), respectively [269]. Equations (60)-(61) are verified only for \"short\" channels where \(L\leq 0.02h^{2}V_{0}/D\) (or \(Y=LD/\left(\overline{V}h^{2}\right)\leq 0.02\)) [248]. Taking into account the definition of \(i_{\text{lim }av}\) above and the fact that \(i_{\text{lim}}\) and \(\delta_{N}\) are inversely proportional to each other, Equation (40), we can arrive at the meaning of the average value of [PERSON]'s DBL thickness: \(\delta_{N\ av}^{-1}=\frac{1}{L}\int\limits_{0}^{L}\delta_{N\ loc}^{-1}(y)dy\). On the other hand, \(\delta_{N\ av}\) can be also understood as the effective DBL thickness, \(\delta_{N\ ef}\), in a channel where \(\delta_{N}\) is a function of the distance from the entrance: the substitution of \(\delta_{N\ av}\) in the Peers equation, Equation (40), gives the correct value of \(i_{\text{lim }av}\). The mass transfer rate in an ED channel at the limiting current can be also expressed in terms of the Sherwood, Reynolds and Schmidt numbers [80; 248]:  
\[Sh=1.23[Re\ Sc\ (2h/L)]^{1/3}-0.4,Sh_{av}=1.85[Re\ Sc\ (2h/L)]^{1/3}-0.4 \tag{62}\]  
where  
\[Sh=\frac{i_{\text{lim}}2h(T_{1}-t_{1})}{FDc_{0}}=\frac{2h}{\delta_{N}},Re= \frac{2h\overline{V}}{\
u},Sc=\frac{\
u}{D} \tag{63}\]\(Sh_{uv}\) is the average value of \(Sh\) in a channel of length \(L\), \(\
u\) is the kinematic viscosity. For an ED channel of a dimensionless length \(Y=\frac{LD}{h^{2}V_{0}}\) = 0.01, the local and average DBL thicknesses reach 0.23\(h\) and 0.15\(h\), respectively; the degree of desalination at the limiting current density is 10-13% (it depends on the type of electrolyte). These values are typical for \"short\" channels in cells commonly used for electrochemical studies of IEMs. For \"long\" channels, where the DBLs formed at the AEM and CEM are overlapped, it is possible to use numerically calculated \(Sh\) vs. \(Y\) curves [248]. For rough estimations of the local limiting current density using Equation (40), the effective average value of \(\delta_{N}\) can be taken equal to \(h/3\). Equations (60) agree well with the experimentally determined limiting current density in a cell specially designed [270] to satisfy the conditions required for obtaining this equation. The cell has the input and output comb-like devices, which assure the laminar flow for the average velocities from \(10^{-2}\) cm/s to 2 cm/s. The intermembrane distance (\(h=5-6\) mm) and the channel length (20 mm) give the dimensionless length \(Y\) in the interval \(10^{-5}\)-\(10^{-2}\), where the Leveque approximations (Equations (60)) are well satisfied. The possibility of knowing the theoretical limiting current density and the DBL thickness is very important, as it allows evaluation the coupled effects of concentration polarization. Indeed, the theoretical values of \(i_{\rm lim}\) and \(\delta\) are determined for the case of an \"ideal\" IEM, which does not generate current-induced convection and does not split water molecules. Hence, the comparison of the experimental and theoretical \(i_{\rm lim}\) can give some ideas, for example about the intensity of electroconvection [217]. The role of the Leveque equations for the slot cells with laminar flow is similar to the Levich equation obtained for the rotating disk electrode [271] or rotating membrane disc [225; 272; 273].