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∇·J=0 (3.12) We can then combine equations 3.4 – 3.12 to reduce the number of variables to just the three variables of interest, which are the concentration of the anion (since the concentration of the cation is easily calculated from it), the electrical potential, and the pressure (which will help us investigate the change in water recovery with current). ε−1/2∂c− +∇·−D ∇c +z−ec ∇φ−c (k ∇φ+k ∇P)=0 (3.13) p∂t −−kT− −EO D ∇ · (kEO∇φ + kD∇P) = 0 (3.14) e ∇· z−e(D+ −D−)∇c− + kT [(z+D+ −z−D−)z−ec− +D+z+ρs]∇φ +ρs(kEO∇φ+kD∇P) =0 (3.15) For ease of simulation, the dimensionless forms of these equations were used. The equations were non-dimensionalized using the dimensionless scales from Table 3.1. ε−1/2∂c ̃− +∇ ̃ ·−∇ ̃c ̃ +z c ̃ ∇ ̃φ ̃−c ̃ k ̃ ∇ ̃φ ̃+k ̃ ∇ ̃P ̃=0 p∂t ̃ −−− −EO D (3.16) (3.17) ∇ ̃ ·k ̃EO∇ ̃φ ̃+k ̃D∇ ̃P ̃=0 ̃ ̃ ̃ ̃ ̃ ̃ ̃ ∇· z− D+ −1 ∇c ̃− + z+D+ −z− z−c ̃− +D+z+ρ ̃s ∇φ ̃ ̃ ̃ ̃ ̃ ̃ Lastly, we have to briefly consider one short-coming of this model, namely that it does not satisfy Onsager symmetry. This short-coming stems from smearing out the surface charge over the volume of the pore. In reality, the charge sits mostly in 80 +ρ ̃s kEO∇φ+kD∇P =0 (3.18)PDF Image | Shock Electrodialysis for Water Purification and Electrostatic Correlations
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