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absence of surface charge, the depleted region would only move due to diffusion. The current density is of the form e⃗ κs ̃ k T(j+ρs⃗u)=−β∇κb −[κb +h ]∇phi (1.14) Bp where β = D+−D− , κb = (z+ν+=z−ν−)e2cb , and κs = z+ν+e|σs|. Equation (1.14) can z+D++z−D− 2 then be plugged into equation (1.12) to obtain a single, nonlinear PDE for the bulk conductivity κb: ∂κb + ∂ [uκb + z−ν−e(κs/hp)(j + ρsu)] = ∂ [D(κb)∂κb ] (1.15) ∂t ∂x κb +κs/hp ∂x ∂x where D(κb) = D ̄(1 − z ̄(D+−D−)(κs/hp)). This PDE is a general model for porous 2z+D+(κb+κs/hp) microstructures [67]. If surface conductivity can be neglected, the PDE reduces to the classical linear convection-diffusion equation. The second term in square brackets on the left side can be interpreted as advection of surface charge due to electromigration, gradients of which are responsible for exchanges between bulk and the double layer. This PDE can now be used to describe shock behavior in microstructures, Mani and Bazant [67] also showed that propagating shocks are at least mathematically extremely stable. However, it does remain to be determined how fluid flow and shock structure affect this stability. A similar model was presented by Dydek et al [24], who then explored the behavior of the shock wave in a uniform pressure-driven crossflow without EOF in a "leaky membrane" (i.e. porous medium) sandwiched between two ion-selective elements for a binary symmetric electrolyte. They determined that the desalination shock still forms, and in fact will collide with an enrichment shock under certain circumstances, and that use of this phenomenon for desalination should be possible in this geometry. However, more importantly, this study explored the behavior of the required energy to perform this separation as a function of the surface charge, suggesting that an optimum surface charge (as compared to the feed salt concentration) exists when the total surface charge density per volume of the porous medium is roughly 1-10 % 38PDF Image | Shock Electrodialysis for Water Purification and Electrostatic Correlations
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