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Due to the small numerical value of the permittivity of water, the N-P-P model is mathematically classified as a singularly perturbed problem [245, 328]. Several techniques have been developed for one-dimensional simulations [203, 245, 263, 328–332] and also for two-dimensional simulations solving the fully coupled N-P-P and Navier-Stokes equations, providing an analysis of the electrokinetic instability that characterizes overlimiting transport phenomena [189, 210, 245, 246, 333–339]. As the energy required to charge a macroscopic system is very high and the double layer is confined in a very narrow region, when transport phenomena are investigated at higher scale and below ilim, electrolytic solutions are assumed to be electrically neutral and Poisson’s equation is replaced by the electroneutrality condition. Several works based on the Nernst–Planck approach and local electroneutrality condition have been presented for IEM-based processes (ED/RED) [185, 187, 202, 224, 236, 251, 298, 303, 304, 327, 340–346]. Note that by combining the N-P equations and the mass balances of the two ions of a binary electrolyte with the local electroneutrality condition, the convective-diffusive transport equation is obtained (see Section 5.2.3). Actually, several models based on the N-P approach solve this equation within the fluid domain for calculating the concentration field. In the N-P based models, some simplifying assumptions are usually done: only 1-D (cross-membrane) or 2-D (axial + cross-membrane) simulations have been carried out, convection has been considered only in some cases [185, 187, 236, 298, 303, 327, 340, 341, 343, 345, 346], while all the components of the cell pair have been simulated only in a few works [187, 298, 327, 341, 343], and the presence of either spacers or membrane profiles has rarely been included [185, 187, 298]. For the sake of completeness, we mention that the applicability of the Nernst–Planck approach has some limitations [347]. In concentrated solutions, each ion is surrounded not only by solvent molecules but also by other ions. In such a situation, short-range interactions become more important, thus additional frictional (interaction) forces are present. In other words, the accurate description of transport processes in concentrated solutions requires more transport coefficients and a more rigorous approach, which can be offered by the Stefan–Maxwell equation. Few examples of application of this approach on ion exchange systems can be found in the literature [301, 348, 349]. However, the N-P model is by far the most adopted due to its simplicity and robustness under a wide range of typical operating conditions of ED and related systems. 5.1.2 Process models based on the Nernst–Planck equation In order to step from the sole Nernst–Planck mathematical description of transport phenomena to the formulation of a process simulation tool, some of the above mentioned works compute also global 53PDF Image | Electrodialysis for water desalination
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