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as a function of current, which we can do by estimating that E = j/σ, where j is the current density and σ is the bulk conductivity of the salt solution [103]. Using this expression, we obtain an equation that we can use to estimate the electro-osmotic flow rate. QEOF = εζjA = εζI (2.3) μσ μσ We can now use this expression to write down an expression for the water recovery, which is defined as R = Qd/Q, where Qd is the desalinated water flow rate and Q is the total flow rate. The total flow rate is fixed by the experimental conditions; however, the desalinated flow rate is not. Let’s define the desalinated flow rate as Qd = αQ + βQEOF , where α is the split ratio expected due to splitter placement (i.e. the water recovery at zero current) and β is a factor less than one that attenuates the contribution of electro-osmotic flow to the desalinated flow, since we still expect there to be some pressure-driven recirculation within the porous medium. Using this expression in the expression for water recovery and combining this with equation 2.3, we obtain a new formula for water recovery. R=Qd =αQ+βQEOF =α+βεζI (2.4) Q Q μσQ Since we have a rough estimate of the zeta potential (about 100 mV in our porous medium [112]), we can now use the expression in equation 2.4 to rescale the data in Fig. 2-14 with the newly defined dimensionless current, εζI , which is the ratio of μσQ the electro-osmotic flow rate in the transverse direction to the total applied flow rate. Figure 2-15 shows that using this scale, we get reasonable collapse of the data onto a single straight line. In fact, if we fit the expression in equation 2.4 to this data set, we get a value of α = 0.454, which is consistent with the observed water recoveries at zero current, and a value of β = 0.092, which means that in this model only 9.2% of the electro-osmotic flow contributes to increase the flow of desalted water, whereas the rest is either retarded due to back-pressure or surface charge regulation [22] in the depleted region. Furthermore, a statistical analysis provides a reduced χ2 value of 1.22, which indicates that the line is a reasonably good fit of the data, strongly 67PDF Image | Shock Electrodialysis for Water Purification and Electrostatic Correlations
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