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frits that were used in our experiments have previously been used as electro-osmotic pumps [112]. This hypothesis is supported by the fact that a simple calculation of the electro-osmotic flow velocity at higher currents, ignoring any backpressure, predicts a velocity that is comparable to the applied velocity [113]. This hypothesis is further corroborated by the fact that the water recovery went up with current, meaning that electro-osmotic flow was towards the cathode within the frit, which is actually what is known to occur at near-neutral pH, though existing models would predict pressure- driven recirculation of this flow and hence no change in water recovery. However, this hypothesis requires that there is electro-osmotic surface convection behind the desali- nation shock [25, 85] that can dissipate the built up pressure in vortices and hence will be able to redirect the net electro-osmotic flow into the cathode-side (desalinated) outlet. This mechanism has been postulated in previous publications [22, 70] to be the main mechanism for overlimiting conductance in both microchannels and porous media. In order to further test this hypothesis we would like to replot the data in Fig. 2-14 with a scale for current that is based on electro-osmotic flow to determine whether we get good collapse of the data onto one curve. To do this, we need to derive an explicit equation of water recovery as a function of current. We first set out to estimate the magnitude of the electro-osmotic flow based on the Helmholtz-Smoluchowski forumula for electro-osmotic flow uEOF = εζE (2.1) μ where μ is the viscosity of the medium, E is the electric field, ζ is the zeta potential, and ε is the permittivity of the medium. We first want to modify this formula by converting the velocity into a flow rate by assuming that the electric field is only in the transverse direction in our system, which, while only an approximation, may capture the right scalings and hence we can simply multiply by the cross-sectional area A. QEOF = εζEA (2.2) μ Lastly, since we are applying a constant current, we need to express the electric field 65PDF Image | Shock Electrodialysis for Water Purification and Electrostatic Correlations
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