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4.2 Mass transport equations and limiting/overlimiting current conditions 4.2.1 Formulation of mass transport equations in ED and related processes Regardless of the true concentration profile, the interfacial condition expressed in eq. (10) can also be written in Nernstian form, taking into account the DBL thickness (δ) [169, 170, 173, 178, 190, 191] or the Sherwood number (Sh) [50, 174, 178, 192]: 𝜕𝐶 𝐶 − 𝐶 𝑆h 𝐶 − 𝐶 𝑖 (11) 𝜕𝑦 =± 𝛿 =± 𝑑 =−𝑧𝐹𝐷𝑇 −𝑡 where 𝐶 is the concentration at the IEM-solution interface (solution side), 𝐶 is the bulk concentration, deq is the equivalent diameter. In eq. (11) the sign + has to be considered when moving from bulk to interface, while the sign – in the opposite case. For simplicity, the difference of the transport numbers is often assumed equal to 0.5 [6, 192, 193]. Eq. (11) identifies the most common expressions of mass transfer rate in these systems, also allowing an in depth comprehension of mass transport phenomena at the boundaries. In fact, under steady state conditions, both diffusion and convection in solution from bulk to interface or vice versa compensate for the difference between the migrative fluxes across solution and membrane. The Sherwood number is defined as 𝑆h = 𝑘 𝑑 (12) 𝐷 where k is the mass transfer coefficient. From dimensional analysis, it can be found that the Sherwood number depends on the channel configuration (geometry of spacer or membrane profiles, but also active walls surface, depending on the presence of conductive spacers or profiles), the Reynolds number (Re) and the Schmidt number (Sc) [194–201]. For more details, see the mass transfer correlations discussed in Section 4.4.4. Besides concentration polarization visualization techniques, some experimental methods for the measurement/estimation of δ (and thus Sh) have been developed and tuned up, e.g. based on chronopotentiometry [74, 139, 202, 203] and EIS [204]. However, mass transfer characteristics are often evaluated by measurements of limiting current density, an important parameter controlling the operation of ED units. 4.2.2 Limiting current density in ED units The increase of current density in a stack, always accompanied by an increase in diffusive transport of ions in solution, leads to a depletion of salt in the solution at the IEMs interface of the diluate 26PDF Image | Electrodialysis for water desalination
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