## 5 Concentration Polarization in Electrodialysis (ED)
### ED with Heterogeneous Membranes
Similar to Equation (49), the total PD in an ED cell pair (\(\Delta\varphi\)) is presented as the sum of three contributions:  
\[\Delta\varphi=\Delta\varphi_{a}+\Delta\varphi_{DC}+\Delta\varphi_{c} \tag{70}\]  
where \(\Delta\varphi_{DC}\) is the PD within the desalination compartment limited by \(x=0\) at the left and \(x=h\) at the right, \(\Delta\varphi_{a}\) is the PD across the AEM and a half of the concentration compartment, \(\Delta\varphi_{c}\) is a similar quantity for the CEM half-cell (Figure 11). Figure 11: Schematic distribution of the electric potential. The variations within quasi-equilibrium interfacial EDLs are shown as vertical lines. \(c_{2s}\) and \(c_{1s}\) relate to the positions of counterion minimum concentration near the AEM and the CEM, respectively; \(c_{1m}\) and \(c_{2m}\) show the position of the points where the counterion concentration profile passes from the solution to the membrane phase. To find \(\varphi(0,y)\) and \(\varphi(h,y)\) as functions of the normal local current density (\(i_{x}\)) and \(\Delta\phi\), it is assumed [214] that between the point referred to the membrane surface facing the CC and the point on the surface facing the DC where the counterion concentration is set (\(x\) = 0 for the CEM and \(x\) = \(h\) for the AEM, Figure 11), there is the membrane bulk and two interfacial quasi-equilibrium EDLs [83; 214]. The PD between these two points is the sum of the ohmic PD and two Donnan PDs. For the sake of simplicity, concentration polarization at the membrane interfaces with enriched solution is not taken into account, hence, everywhere in the concentrating compartment the concentration is set equal to \(c_{0}\). Under these assumptions, the PD between the plane passing through the centre of the CC and \(x\) = 0, \(\Delta\varphi_{c}\) and the PD between the plane passing through the centre of the CC and \(x\) = \(h\), \(\Delta\varphi_{a}\), read as follows [214]:  
\[\Delta\varphi_{c} =-\frac{1}{2}R_{ms}i_{x}\ (0,y_{cond})-\frac{RT}{z_{1}F}\ln\frac{c_{1m} }{c_{0}}=-\frac{1}{2}R_{ms}i_{x}\ (0,y_{cond})-\frac{RT}{F}\ln N_{c} \tag{71}\] \[\Delta\varphi_{a} =-\frac{1}{2}R_{ms}i_{x}\ (h,y_{cond})-\frac{RT}{z_{2}F}\ln\frac{c_{0} }{c_{2m}}=-\frac{1}{2}R_{ms}i_{x}\ (h,y_{cond})-\frac{RT}{F}\ln N_{a} \tag{72}\]  
where \(R_{ms}\) is the total ohmic (membrane+solution) resistance of the system involving an AEM and a CEM as well as a concentration compartment. Equations (71) and (72) are written only for the conductive regions on the CEM surface. The second terms on the left-hand side of Equations (71) and (72) involve the total interfacial PD. Thus, for the CEM, we have  
\[\Delta\varphi_{c}^{Don}=-\frac{RT}{z_{1}F}\ln\frac{\tilde{c}_{1}}{c_{0}}-\frac {RT}{z_{1}F}\ln\frac{c_{1m}}{\tilde{c}_{1}}=-\frac{RT}{F}\ln\frac{c_{1m}}{c_{ 0}}=-\frac{RT}{F}\ln N_{c} \tag{73}\]  
Setting \(\phi\) = 0 in the middle of the left-hand concentration compartment and taking into account that \(\varphi(0,y)=\Delta\varphi_{c}\) and \(\Delta\varphi=\varphi(h,y)+\Delta\varphi_{a}\), we find from Equations (71) and (72):  
\[\left(\frac{\partial c_{1}}{\partial x}+\frac{F}{kT}z_{1}c_{1}\frac{\partial\varphi }{\partial x}\right)(h,y)=0,\left(\frac{\partial c_{2}}{\partial x}+\frac{F}{kT }z_{2}c_{2}\frac{\partial\varphi}{\partial x}\right)(h,y)=-\frac{i_{x}(0,y)}{ D_{2}F}, \tag{66}\]0  
The positions of the cross-sections for which the profiles are calculated are shown in Figure 12; the first cross-section is at 550 \(\upmu\)m from the entrance; the intermembrane distance \(h\) = 1000 \(\upmu\)m; the average flow velocity of solution in the channel between the membranes is 0.08 cm\(\cdot\)s\({}^{-1}\); the length of the conductive surface regions is fixed: \(a\) = 10 \(\upmu\)m.  
The part of the concentration profile close to the surface changes essentially when passing from one cross-section to another along the \(y\)-axis (Figure 13).