## 5 Concentration Polarization in Electrodialysis (ED)
### 2D Modelling of Electrodialysis with Homogeneous Membranes
#### 5.2.2 Concentration Distribution and Diffusion Layer Thickness  
Numerical solution of the convection-diffusion problem presented by Equations (45)-(49) and (58) gives the distribution of concentration \(c(x,y)\), current density \(i(y)\) and electric potential \(\phi(x,y)\) in the space between an AEM and CEM under the condition that the overall PD per cell pair \(\Delta\phi\) is known and the geometric parameters of the cell as well as the average flow velocity are set. In the version presented above, the variation of concentration for the sake of simplicity is taken into account only in the DC since namely this compartment determines the behaviour of the system in ED desalination: the concentration variations in the CC only slightly affect the overall system resistance. The concentration profiles in the DC computed at different distances from the cell entrance are shown in Figure 9.  
It can be seen that the concentration profiles are not symmetric, which is due to the difference in the transport numbers of cation and anion leading to a difference in the concentration gradients at the AEM and CEM, according to Equation (47) used as the boundary condition for the problem. Despite the difference in the concentration gradients, the thickness of the Nernst DBLs (determined through  
Figure 9: Concentration profile in the DC of an ED cell computed (at = 0.2 V) using the basic version of convection-diffusion model [247; 248; 249] at different dimensionless distances from the channel entrance \(Y=yD/\left(\overline{V}h^{2}\right)\): \(1-0,2-0.001,3-0.02,4-0.2\). AEM and CEM refer to an AEM and a CEM, respectively. the intersection point of tangents to the concentration profiles, Equation (39)) at a AEM and CEM are the same, and, in the first approximation, do not depend on the applied voltage [248]. When a sufficiently great PD is applied, the concentration at the surface of the membrane, for which the difference of counterion transport numbers \((T_{i}-t_{i})\) is maximum, reaches a nearly zero value, the average current density for this membrane (the CEM in the case of NaCl) attains the limiting value, Equation (40). Subsequent growth of \(\Delta\phi\) does not lead to an increase in the current density. When the current density is limiting, the problem (45)-(49) and (58) can be simplified. Instead of the boundary condition of the second kind [Equation (47), expressing the concentration gradient at the membrane surface], the boundary condition of the first kind, the concentration equal to zero at this surface, can be used:  
\[c(0,y)_{y>0}=0 \tag{59}\]  
This simplified boundary value problem can be solved approximately. Indeed, an asymptotic solution was obtained by [PERSON] in the case of the diffusion-convection heat transfer [267]. Adapting this solution to diffusion-convection mass transfer gives [80]:  
\[i_{\text{lim }loc}(y)=\frac{FD_{0_{0}}}{h(T_{1}-t_{1})}\bigg{[}0.978\Big{(} \frac{h^{2}V_{0}}{yD}\Big{)}^{1/3}-0.2\bigg{]},i_{\text{lim }av}(L)=\frac{FD_{0_{0}}}{h(T_{1}-t_{1})}\bigg{[}1.47\Big{(}\frac{h^{2}V_{0}} {LD}\Big{)}^{1/3}-0.2\bigg{]} \tag{60}\]  
where \(i_{\text{lim }loc}\) and \(i_{\text{lim }av}\) are the local and the average over the channel length (\(L\)) current densities, respectively; \(i_{\text{lim }av}=\frac{1}{L}\int\limits_{0}^{L}i_{\text{lim }loc}(y)dy\), where \(y\) is the distance from the channel entrance. Combining Equations (60) with the Peers equation, Equation (40), yields the following equations for the local, \(\delta_{N\ loc}\) and the average, \(\delta_{N\ av}\), values of the Nernst DBL thickness, \(\delta_{N}\), in the case where the second term (\"0.2\") in brackets on the right-hand side is neglected [248; 268]:  
\[\delta_{N\ loc}=1.02h\bigg{(}\frac{LD}{h^{2}V_{0}}\bigg{)}^{\frac{1}{3}},\delta _{N\ av}=0.68h\bigg{(}\frac{LD}{h^{2}V_{0}}\bigg{)}^{\frac{1}{3}} \tag{61}\]  
As it can be seen from Equations (60), the relative contribution of the \"0.2\" term on the right-hand side decreases with decreasing the channel length \(L\).