## 5 Concentration Polarization in Electrodialysis (ED)
### ED with Heterogeneous Membranes  
Competition between homogeneous and heterogeneous IEMs lasts for decades. Although the conductivity and permselectivity of a heterogeneous IEM is generally lower than these parameters of a homogeneous membrane, the great advantage of heterogeneous IEMs is that they are less costly. The surface of heterogeneous membranes includes conductive and non-conductive areas of the size from several \(\upmu\)m to several tens of \(\upmu\)m. The so called homogeneous membranes (usually prepared the paste method via copolymerization [9]) also contain non-conductive inclusions (e.g., PVC binder), however, the size of these inclusions generally less than 1 \(\upmu\)m (except the threads of backing cloth). For instance, the Neosepta(r) AMX and CMX membranes have PVC particles with a diameter of about 100 nm. The presence of non-conductive areas leads to the accumulation of the electric current lines on the conductive regions and to the curvature of these lines. On one hand, this \"funnel effect\" [274] increases membrane concentration polarization and water splitting rate, while reducing the limiting current density [274]. On the other hand, tangential electric field enhances electroconvection and mass transfer [232; 275; 276; 236]. From what has been said before, the electrical heterogeneity of the membrane surface is subject to optimization. Too low surface fraction of membrane conductive area (\(\gamma\)) will lead to very high concentration polarization, which could be remediated by electroconvection. However, one can imagine that a certain heterogeneity of the surface could effectively enhance electroconvection, while only slightly increase concentration polarization. Simulations made by [PERSON] et al. [233] and [PERSON] et al. [272] predict that the rate of electroconvection and, as a consequence, the rate of mass transfer pass through a maximum with increasing \(\gamma\) at a fixed voltage. According to [233], the maximum is close to \(\gamma\) = 0.5; according to [272], it is close to \(\gamma\) = 0.8. The both models are based on the same governing equations (the Nernst-Planck-Poisson-Navier-Stokes equations). Perhaps, the difference is due to the fact that [PERSON] et al. [233] do not take into account the forced flow between the membranes, while [PERSON] et al. [272] do. Another important lever for enhancing electroconvection and mass transfer is geometric heterogeneity (surface undulation, waviness, formation of special patterns). However, this option is still poorly studied, which can be understood in view of the wide variety of possible geometric shapes and sizes. A deeper understanding of the peculiarities of ion and water transport at electrically and geometrically heterogeneous surface is needed for targeted optimization of membrane surface parameters. Later on we will consider some aspects of mathematical modelling of ion transport at heterogeneous membrane surface. Effect of Electrical Surface Heterogeneity on the Limiting Current Density and Diffusion Layer Thickness  
Consider an ED channel, which is formed by a homogeneous AEM (which may have areas on their surface that differ in conductivity but the size of these areas is less than 1 \(\upmu\)m) and a heterogeneous CEM whose surface contains alternating conductive and non-conductive areas, whose size is of the order of 10 \(\upmu\)m or more. Figure 10 gives a 2D schematic presentation of such a channel. The length of the conductive region on the surface is denoted as \(2b\), the distance between the centres of two nearest conductive regions (the spatial period of surface heterogeneity) is \(2l\). As earlier (Section 5.2), we assume that a binary electrolyte solution with initial concentration \(c_{0}\) flows between the membrane, the flow being laminar and there is Poiseuille velocity distribution, Equation (46). A PD \(\Delta\varphi\) is applied between two planes passing through the centres of the concentration compartments adjacent to the central DC.