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region near the wall. Instead of using a Boltzmann distribution of ions, uniform wall normal concentrations in the bulk are assumed with a delta distribution of counterions near the wall representing the EDL. Further assumptions that are made are that the zeta potential is much larger than the thermal voltage (to include all the EDL charges as counterions) and that the electroosmotic velocity has a uniform profile across the channel height with zero velocity at the wall. All these assumptions lead to the conclusion that all the integrating factors are equal to one. These simplifications give the following nondimensionalized governing equation: ∂∗∗ ∂∗ I∗ 1∂∗∂x∗ ∂t∗(hc)+∂x∗(c +h∗c∗+1)=Pe∂x∗(h ∂x∗) (1.8) wheret∗ = tQ,x∗ = x ,h∗ = h ,Pe= Q,c∗ =(ν1z1−ν2z2)Fz1hrefc0,and h2ref href href D2 2ν1 z1 −σ I∗ = ( ν2z2 )Ihref . Rearranging equation (1.8) gives 2ν1z1 σQ h∗I∗ ∂c∗ c∗I∗ ∂h∗ 1 ∂ ∗∂x∗ ∂t∗(hc)+(1−(h∗c∗+1)2)∂x∗ =((h∗c∗+1)2)∂x∗ +Pe∂x∗(h ∂x∗) (1.9) ∂ ∗ ∗ When analyzing equation (1.9) using the method of characteristics, the coefficient of the convective term is the characteristic velocity. ∗ h∗I∗ U =1−(h∗c∗+1)2 (1.10) In this case, the non-dimensional velocity of the characteristics is a function of channel height, current, and concentration. The characteristic velocity can differ drastically between a microchannel and a nanochannel [68]. In fact, the characteristic velocity can easily be negative in a nanochannel, while it is still close to 1 in the microchannel. A shock can form for example when an upstream region with a positive characteristic velocity encounters a downstream region with a negative or smaller positive character- istic velocity, since overlapping characteristics would otherwise predict a multivalued function for concentration [68]. Whether the shock moves forwards or backwards is determined by the relative characteristic velocities in the two regions. In this case, the shock velocity V ∗ is given 35PDF Image | Shock Electrodialysis for Water Purification and Electrostatic Correlations
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