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versa, and fouling [260, 318, 320–323]. The increase of non-Ohmic resistance arising from a non- uniform distribution of the flow rate can be explained by the exponential reduction of the non-Ohmic resistance with the flow rate. In fact, the increase of non-Ohmic resistance in the areas with a flow rate lower than the average is larger than the decrease generated in the other areas, thus leading to an overall increase in the total non-Ohmic losses. In this respect, wider manifolds lead to a more uniform delivery of feed solutions flow rate, with beneficial effects in terms of non-Ohmic resistances reduction [260]. Finally, large differential pressures between adjacent compartments cause solution leakages through the membrane [322], occurring especially in counter-current and cross-flow configurations [321]. 4.5.2 Flow distribution within the channels Some recent works have dealt with the flow distribution throughout a channel [259, 319, 321, 323– 327]. Kostoglou and Karabelas [324] simulated the flow distribution by modelling the spacer as a continuous porous medium filling the channel, in a study of hydrodynamics of spiral-wound elements with permeable membranes. The simplifying assumption of an isotropic medium was actually made for the simulations and the flow field in the spacer-filled channel was characterized by computational fluid dynamics (CFD) methods applied for fully developed flow (“unit cell” approach), keeping the flow attack angle fixed. Kodým et al. [325] developed a semi-empirical two-dimensional model, based on momentum balance equations for two interacting sub-layers taking into account the anisotropy of spacers (constituted by overlapped filaments) in the anisotropic form of Darcy’s law. A subsequent work [319] extended the model by including in the momentum balance equation fluid inertia, which has a significant influence in the inlet and outlet regions where sudden contractions and expansions, respectively, occur. Significant non-uniformities in the flow distribution in the proximity of the manifolds and larger pressure drops were shown. More uniform flow distributions were predicted as the flow rate decreased and the filaments of the spacer were placed more transversally with respect to the main flow direction (by letting the diagonals of the diamond spacer to vary while maintaining the main flow direction bisecting the angle between the two wires). This corresponds to higher pressure drops within the channel, which make also the distribution among channels more uniform [318, 320, 321]. However, in other cases the flow distribution within the channel is improved by less frictional geometries. For example, Güler et al. [259] visualized the dispersion of a black ink in a transparent cell equipped with a profiled membrane (Figure 16). A pillar profiled-membrane exhibited an even distribution over the membrane surface and also low pressure drops, while profiled membranes with 48PDF Image | Electrodialysis for water desalination
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