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spacers, so that higher fluid velocities are adopted in tortuous paths in order to maintain polarization phenomena under control [10, 45, 49]. The flow regime within spacer-filled channels is steady for Re values roughly up to 200-300, then changes gradually (possibly through periodic flow) towards turbulent conditions, starting at Re ≈ 1000 [39, 195, 279, 281, 299, 300]. Therefore, we can assert that the flow regime in ED units is typically steady or with incipient unsteadiness in many cases, and it reaches turbulent conditions only in some rare cases [194]. 4.4.4 Correlations for mass transfer coefficients The characterisation of mass transfer phenomena in spacer-filled channels has been often performed by means of correlations linking the Sherwood number to the main geometrical and operating parameters [48, 178, 195, 199, 224, 234, 278–280, 301]. The use of dimensionless numbers offers the advantages of an easy scalability of results and easy implementation/processing of results (input/output) in modelling tools [6, 174, 193]. However, also correlations for ilim have been often reported [39, 177, 178, 209, 221, 226–228]. In most cases data are fitted by power laws, such as 𝑆h = 𝑎𝑅𝑒𝑆𝑐 (22) 𝑖 = 𝑑𝐶𝑢 (23) Note that the exponent of Re in eq. (22) corresponds to the exponent of u in eq. (23). In some cases, b was found to be close to 0.5 [48, 177, 199, 209, 224, 226, 228, 234, 279], while in other cases b ranged from ~0.13 to ~1 [178, 195, 221, 278, 280, 301]; however, much more complex trends in a larger range of Re values have also been reported [39, 195], and modified correlations have been proposed [227]. It is worth highlighting that power laws and relevant coefficients’ values may be suitable only in a narrow range of Reynolds numbers [6, 281], while ED stacks can operate in a relatively large Re range involving different flow regimes. A log-log chart of Sh vs. Re exhibits a horizontal asymptote at very low Re (creeping flow), where mass transfer in spacer-filled channels may be worse than in the empty channel, as shown in Figure 13 (a). Then, one or more inflections at higher Re follow, up to an oblique asymptote (power law) in the turbulent regime, as shown in Figure 13 (b). The effect of the fluid properties, represented by the Schmidt number (Sc), was evaluated in [48, 279] and the exponent c in eq. (22) was found to be 1/3. This value has often been considered valid, but, in general, the effect of Sc can be different as Re and/or the spacer geometry vary [6, 197, 200, 271]. 42PDF Image | Electrodialysis for water desalination
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