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1.3.2 Concentration Shock Dynamics in Microchannels In addition to the steady-state behavior of systems with concentration polarization, the dynamic behavior of these systems is also important, one of which is the con- centration shock. This concentration shock was investigated in theory by Mani et al [68] and in experiment by Zangle et al [121] and Kim et al [55]. Mani et al came up with two different models to describe concentration shock dynamics in a micro-nano- micro channel system. The first model is based on a one-dimensional (area-averaged) unsteady partial differential equation to describe the flow and charge transport in the system. The model uses the exact solutions of the Poisson-Boltzmann equation to determine the effect of the electric double layer (EDL) on axial transport. This model can be derived from the general ion transport equations, and the final set of area-averaged equations is ∂ ∂ ̄pee ∂∂c0 ∂t(hfc0) + ∂x[h(upf2 + u0f2 + ν2z2FEf)c0] = ∂x[hD2f ∂x ] (1.4) ∂ 2σ∂ ̄pee ∂ ∂c02D1∂σ ∂t(hfc0 − z1F )+ ∂x[h(upf1 +u0f1 +ν1z1FEf)c0 −2ν1Eσ] = ∂x[hD1f ∂x − z1F ∂x] (1.5) (1.6) u0 = ηe(z1 − z2) where f, fe, f1e, f2e, f1p, f2p, and fζ are integrating factors that arise due to area- averaging the model. Equations (1.4) and (1.5) are the area-averaged equations for transport of ions 2 and 1, respectively, in a dilute solution, equation (1.6) is the equation for conservation of mass, and equation (1.7) is the Helmholtz-Smoluchowski slip formula. This model would have to be solved numerically as done in Zangle et al [121]. The more interesting model is the second model derived by Mani et al [68], because it is simpler and analytically tractable and can therefore give a good intuition about the problem. This model is a simplified version of the area-averaged model described above. The simplification that is made is essentially that the EDLs are confined to the ∂ ∂x(hup + hu0f ) = 0 ̄ e −2kTεEfζ (1.7) ee 34PDF Image | Shock Electrodialysis for Water Purification and Electrostatic Correlations
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