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Then we substitute the above equation and Eq.(14) into Eq.(16) to arrive at μ∂2u − ∇xypv + ǫ∂2φ∇xyψv ∂z2 ∂z2 −VT F [exp(−zkφ ̃) − 1] cvk∇xy ln cvk = 0, (19) N k=1 Since the above equation is linear in terms of u, it can be integrated twice to explicitly express u(x, y, z) as where is the pressure-driven flow part of u, is the electroosmotic part, and N u=uP +uEO +uDO. k=1 P h2 − z2 v u =− 2μ ∇xyp uEO = −ǫ(φ − ζ)∇xyψv μ (20) (21a) (21b) (21c) DO VTFh2 v uk = − μ ckχk∇xy lnck is the diffusioosmotic part, and χk(x,y,z) is a dimensionless integral defined in Appendix Appendix A. Up to now, we have reduced the original N + 5 three-dimensional unknowns ck, ψ, p, u, v, w to N + 2 independent two-dimensional unknowns cvk, ψv, pv and one three-dimensional unknown φ, which largely simplifies the problem. 2.3. Depth-averaged equations In this section, we will integrate the three-dimensional model under thin channel assumption (Sec.(2.2)) in the z direction to get the depth-averaged equations. In the following, we will drop the subscripts in ∇xy to simplify the notations. First, we average the velocities (Eq.(21a)-(21c)), and arrive at ueff = uck =βPuP +βEOuEO +βDOuDO, (22) kckkk kll l where the coefficients βP , βEO, and βDO account for the difference between the averaged convective flux and kk kl theproductoftheaveragedvelocityandconcentration.PleasecheckAppendixAppendix Aforhowtocal- culate them.The velocities can be obtained from the fields of pressure, electric potential, and concentration: v uP =−kP∇pv, uEO = −kEO∇ψv, uDO = −kDO∇lncv, kkk (23a) (23b) (23c) wherekP =h2,kEO =−ǫζαEO,andkDO =VTFh2cvαDO arethepermeabilityforeachflowmechanism, 3μ μ k μkk and the dimensionless coefficients αEO and αDO are defined in Appendix Appendix A. ck 1 v Next, we define δk = cvk = h to relate ck and ck. The total averaged concentration ck can thus be decomposed to the bulk concentration cvk, and the surface excess concentration (δk − 1)cvk. The average of the NP flux (Eq.(12)) yields Jk =ueff −Dk∇ln( ck/δk)+zkψ ̃vck. (24) 6 kPDF Image | Theory of shock electrodialysis
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