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quantities as the voltage-current relationship and the electric power of the stack. The most simplified models are 1-D (cross-membrane) [304, 342, 344], simulate spacerless channels and predict the stack performance by lumped parameters. In other cases, variations along the stack length are taken into account in 2-D (axial + cross-membrane) by simulating the entire channel from inlet to outlet (stack performance predicted by distributed parameters along the axial direction). In some cases fluid dynamics is explicitly simulated by the Navier-Stokes and continuity equations [187, 340, 341], while other models assume either a developed flow field with parabolic velocity profile (Hagen-Poiseuille equation) [303], or even a flat profile [345]. Differently, in Gurreri et al. [298] a periodic portion of a single cell pair was simulated, thus allowing a very accurate spatial resolution, suitable for channels with either non-conducting spacers or membrane profiles, while keeping the total number of elements compatible with acceptable RAM requirements and computing time. Tado et al. [343] simulated the spacer as a porous domain (Darcy’s law) and integrated the governing equations over the channels width, thus obtaining a 1-D axial model. Enciso et al. [327] performed 3-D simulations of a stack with spacerless channels by a simplified approach. Some models have simulated the membranes as a domain with Ohmic behaviour, i.e. by neglecting the effect of concentration variations (secondary current distribution) [187, 327, 341], and in some models membranes were not even included in the computational domain [340, 343], but were modelled by imposing appropriate boundary conditions. Conversely, various simulation tools have explicitly simulated the membranes by the Nernst–Planck equation, taking into account the concentration variation [298, 303, 304, 342, 344, 345]. Water transport has been simulated only in a few cases [343, 345]. Clearly, the complete simulation including the membranes in the computational domain implies that all the coefficients characterizing the transport through the membranes are known. Finally, several models [187, 298, 327, 340, 341] were implemented in multi-physics modelling platforms [350] based on the Finite Element Method (FEM). As a matter of fact, this general overview shows that, to the best of our knowledge, all models that simulate mass transport by the Nernst–Planck equation have been mostly focused on the membrane modelling, while only little attention has been paid to the modelling of the overall process behaviour for simulation and design purposes. As a consequence, hydrodynamics and associated phenomena of concentration polarization have been taken into account only by a simplified approach. These models have been implemented by either solvers of differential equations or FEM-based software, which are devised for the simulation of simple geometries and would suffer from a prohibitively large memory requirement for the simulation of complex, accurately discretized, 3-D geometries even in the case of a small periodic portion of a cell pair. For example, Gu et al. [273] performed FEM simulations of a 3-D periodic domain of a spacer-filled channel discretized with 1.4-2.9 million elements, simulating 54PDF Image | Electrodialysis for water desalination
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