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the thickness of the channel, allowing us to presume it semi-infinite, and that the boundary condition on the side opposite of the boundary layer was a constant con- centration condition. Furthermore, this model assumed that all of the potential drop is across the desalted region (i.e. η ≤ α). Furthermore, he had assumed a binary symmetric electrolyte; however, this assumption is not necessary and here we hence have extended his theory to any arbitrary electrolyte. Overall, these assumptions allowed for a similarity solution of the problem. 0, if η ≤ α cb = 2z−c− = erfc(η) (3.1) cb,0 1−erfc(α) , ifη>α where α is a constant that represents the fraction of the diffusion boundary layer that is taken up by the shock, cb,0 is the inlet bulk concentration of charge, and η is the boundary layer variable, which is defined as η = √ y , where D ̄ is the harmonic 2 D ̄x/U0 mean of the ionic diffusion coefficients and U0 is the uniform applied flow velocity. Furthermore, this model allows us to define a simple expression for current (per unit width) L L V√U0 V√U0L I = jdx = − √ ̄ ν+z+e|ρs|yˆdx = − √ ̄ ν+z+e|ρs|yˆ (3.2) 0 02αDx αD where V is the voltage drop, ν+ is the co-ion mobility (in this problem we assume that ν+ = D+/kBT), z+ is the charge number of the co-ion, e is the elementary charge, ρs is the volumetric surface charge (negative in our case), and yˆ is the unit vector in the y-direction. The voltage was defined by matching conditions at η = α to be 2kBTc0 αexp(−α2) V= √ (3.3) |ρs |z ̄ π erfc(α) where z ̄ is the harmonic mean of the ionic charge numbers. We will use this model in this work to see how well a simple model works to predict experimental data. However, as we will see, none of the models mentioned so far are able to predict the increase in water recovery that we have observed in 76PDF Image | Shock Electrodialysis for Water Purification and Electrostatic Correlations
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