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the electric double layers and hence experiences much less convection due to viscous drag than is predicted in this model. To ameliorate that situation, we have to modify Equations 3.15 and 3.18 by replacing ρs with ρeff = kEO and ρ ̃s with ρ ̃eff = k ̃EO , kD k ̃D respectively in the last term of each equation in order to make the coefficient for the streaming current equal to that for the electro-osmotic flow. However, it is important to note that we only make this substitution in this one place and do not make this substitution for the modified electroneutrality condition in Equation 3.8. Hence our final set of dimensionless equations is ε−1/2∂c ̃− +∇ ̃ ·−∇ ̃c ̃ +z c ̃ ∇ ̃φ ̃−c ̃ k ̃ ∇ ̃φ ̃+k ̃ ∇ ̃P ̃=0 p∂t ̃ −−− −EO D (3.19) (3.20) (3.21) ∇ ̃ ·k ̃EO∇ ̃φ ̃+k ̃D∇ ̃P ̃=0 ̃ ̃ ̃ ̃ ̃ ̃ ̃ ∇· z− D+ −1 ∇c ̃− + z+D+ −z− z−c ̃− +D+z+ρ ̃s ∇φ ̃ ̃ ̃ ̃ ̃ ̃ 3.2.2 Boundary Conditions The system has eight boundary segments (see Figure 3-1) for which we need to specify boundary conditions. Furthermore, the system has one internal boundary in domain III. We will only apply a potential or current in Domain II. Domains I and III are intended to allow inlet and outlet flows to develop properly and to also facilitate application of inlet and outlet boundary conditions. Wall Boundary Conditions (Boundaries 1, 3, 5, 7, & 9) Boundaries 1, 3, 5, and 7 are solid outside walls and as such no fluxes are possible through these boundaries. As such, we have that F ̃− = u ̃ = J ̃ = 0. In terms of the scalar variables used in equations 3.16 – 3.18, these boundary conditions are ∂c ̃− = ∂ y ̃ ∂P ̃ = ∂φ ̃ = 0. The same conditions are true at boundary 9, which is the flow splitter ∂ y ̃ ∂ y ̃ 81 + ρ ̃eff kEO∇φ + kD∇P = 0PDF Image | Shock Electrodialysis for Water Purification and Electrostatic Correlations
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