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Figure 13. Sherwood number in spacer-filled channels. (a) Lower Reynolds numbers: results from CFD simulations of fully developed flow within woven spacers with different pitch-to-height ratios (Δl/h) and flow attack angles (γ) (adapted from [281]). (b) Higher Reynolds numbers: experimental data (measurements of ilim) concerning floating eddy promoters with different pitch-to-height ratios (Δl/h) [195]. Note that the equivalent diameter to calculate Re and Sh is equal to the channel thickness h, according to the definitions in [195]. Note that, in the literature, plane (spacerless) channels have often been considered as an acceptable idealization/simplification of the channel geometry in models [195, 234, 236, 302–304] or as a reference case for comparison purposes both in models and in lab scale experiments [6, 50, 139, 178, 223, 281]. However, the actual use of plane channels in stacks at industrial scale is not allowed due to dimensional stability issues. Attention should be given also to the entry effects on mass transfer phenomena. The flow field within the entire channel can be assumed fully developed, as the Reynolds number is usually sufficiently small. Nevertheless, due to the high Schmidt number (~600 for NaCl solutions at concentration below 0.5 M), the concentration field develops in a longer entrance region [184, 189, 236, 302] in which the local Sherwood number decreases towards its fully developed value. The topic of entry effects in heat and mass transfer is known as the “Graetz-Lévêque problem” [305, 306]. In the case of relatively short plane channels (length L < 0.02uh2/D, being h the channel thickness), the correlation of the average Sherwood number in laminar conditions, as reported by several authors [139, 203, 216, 229, 231], is 𝑆h = 1.47 𝑅𝑒 𝑆𝑐 h 𝐿 (24) Further theoretical predictions along with experimental data have been also reported [48, 195, 307, 308]. Experimental data concerning spacer-filled channels show some entrance effects at very low 43PDF Image | Electrodialysis for water desalination
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