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parameters models, this is an acceptable assumption as batch processes are usually characterised by small stacks with time constants much smaller than those of the tanks adopted for recirculation. 5.2.3 Complex approaches for non-Ohmic phenomena and mass transport models As mentioned in Section 4.3, all the contributions of the DBL to the total voltage drop (Ohmic voltage drop, diffusion potential and concentration overpotential) can be calculated as logarithmic functions of polarization coefficients. In general, in a cell pair there are four different values of polarization coefficient, one for each interface. In turn, polarization coefficients depend on current density, mass transport related to hydrodynamics, which is accounted by the Sherwood number, and transport numbers in membrane and solution (eq. (11)). Often mass transport phenomena within the channel have been characterized by experimental measurements (chronopotentiometry, EIS, limiting current density, see Section 4). Nevertheless, mass transfer characteristics have been computed by numerical simulation of the convective-diffusive transport equation of the electrolyte, predicting the Sherwood number. As mentioned in Section 5.1.1, under the local electroneutrality condition, from the Nernst– Planck equations of the two ions of a binary electrolyte the well-known convective-diffusive transport equation can be obtained [202, 236, 280, 302, 305]: 𝜕𝐶 ⃗ (40) 𝜕 𝜏 + 𝑢 ⃗ ∙ ∇ 𝐶 = 𝐷 ∇ 𝐶 where C denotes the electrolyte concentration. In many works devoted to ED (or RED) systems, ideal mass transport within the channels has been simulated by this equation [6, 185, 193, 281, 288, 291, 296, 297, 303, 304, 342, 353], which, in principle, is valid to the same extent as the Nernst–Planck equation coupled with the electroneutrality condition. However, in the models that limit the domain to one channel, this approach imposes the need to choose the boundary condition at the IEM-solution interface (uniform concentration, uniform flux, or mixed condition). Nevertheless, in typical scenarios the boundary conditions affect mass transfer coefficient only slightly [297], although a larger effect may occur when profiled membranes are adopted. An important advantage of the convection-diffusion equation is that CFD codes are suitable for the simulation of relatively complex 3-D geometries of spacer-filled and profiled-membrane channels, providing accurate spatial resolution. Under the hypothesis of fully developed conditions, the unit cell (Figure 19) has been simulated in several studies [6, 185, 193, 281, 284, 296, 297] using periodic boundary conditions and solving the convection-diffusion equation along with the Navier-Stokes and continuity equations. 60PDF Image | Electrodialysis for water desalination
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