## 5 Concentration Polarization in Electrodialysis (ED)
### 2D Modelling of Electrodialysis with Homogeneous Membranes  
#### 5.2.1 Governing Equations and General Boundary Conditions  
The first adequate 2D model of ED (named the hydrodynamic theory of desalination) was developed by [PERSON] and [PERSON] [247]. Later on other versions of this model was proposed, in particular, by [PERSON] et al. [248,249] and [PERSON] et al. [250], who reported a numerical solution of the problem formulated by [PERSON] and [PERSON], when taking into account the difference in the diffusion coefficients of cation and anion in solution; [PERSON] et al. developed an analytical solution for special cases [251] and accounted for the effect of ion-conducting spacer in numerical solution [252]. [PERSON] et al. [253,254] proposed a \"mesh step\" model for the effect of a net type spacer--turbulence promoter--on mass transfer in an ED channel. The role of spacer in ED desalination was also studied by a number of authors [255,260,261,262,263]. Other aspects of modelling of ED are considered in Refs. [262,263] and other contributions are reviewed by [PERSON] et al. [264]. The basic version of 2D mathematical model of ED involves [247,248,249] the extended Nernst-Planck equation (Equation (22)) in vector form, the equation expressing the current density as a sum of ion flux densities (Equation (23)), the continuity equation for the conservation of species \(i\) written for the steady state) and the LEN condition:  
\[\overset{\rightarrow}{j}_{i}=-D_{i}\left(\
abla c_{i}+z_{i}c_{i}\frac{F}{RT} \
abla\varphi\right)+c_{i}\overset{\rightarrow}{V},i=1,2 \tag{41}\]  
\[i=F\sum z_{i}j_{i} \tag{42}\]  
\[div\overset{\rightarrow}{j}_{i}=-\frac{\partial c_{i}}{\partial t}=0, \tag{43}\]  
\[\sum z_{i}c_{i}=0 \tag{44}\]  
The system of coordinates is shown in Figure 8.  
Equations (41)-(44) are written for a cation and an anion denoted by subscript \(i\) as 1 and 2, respectively. It is assumed that the fluid velocity at the surface is zero (the no-slip condition) [80]:  
\[\overset{\rightarrow}{V}(x=0,y)=0,\overset{\rightarrow}{V}(x=h,y)=0 \tag{45}\]  
The fluid flow is considered laminar and steady. For such a flow between two parallel plates (without spacer), the solution of the Navier-Stokes equation with the boundary no-slip condition gives a parabolic velocity distribution along the transverse coordinate known as the Poiseuille distribution, the normal component of flow velocity is zero:  
\[V_{x}=0,V_{y}(x)=6\overline{V}\left(\frac{x}{h}-\frac{x^{2}}{h^{2}}\right) \tag{46}\]  
where \(\overline{V}\) is the average flow velocity between the plates. The continuity of the ion fluxes through the interface (Equation (37)), which is rewritten in the form  
\[\left(\frac{\partial c}{\partial x}\right)_{x=0}=\frac{i(T_{1}^{c}-t_{1})}{ FD},\left(\frac{\partial c}{\partial x}\right)_{x=h}=-\frac{i(T_{2}^{a}-t_{2})}{ FD} \tag{47}\]  
is used as the boundary condition for both AEM and CEM. Note that the effective transport numbers of cations in the CEM (\(T_{1}^{a}\)) and anions in the AEM (\(T_{2}^{a}\)) can take into account the co-ion or H\({}^{+}\)(OH\({}^{-}\)) transport across the membrane. For this, experimental [76,265] or theoretical [266] evaluation of \(T_{1}^{c}\) and \(T_{2}^{a}\) should be carried out. The feed solution concentration at the channel entrance is set as the initial condition:  
\[c(x,0)=c_{0} \tag{48}\]  
The condition specifying the parameters of the operating mode assumes that the PD \(\Delta\phi\) between two planes passing through the centre of two nearest concentration compartments (CC) is known. This PD can be found as \(\Delta\phi=\)(\(U-\Delta\phi_{el}\))/\(n\), where \(U\) is the voltage measured between the polarizing electrodes of the ED cell, \(\Delta\phi_{el}\) is the PD on the electrodes and n is the number of cell pairs. It is assumed that any plane passing through the centre of an CC is equipotential and the current density and ionic fluxes along this plane are zero.