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and 1 h z2 h 0 1 − h 2 e x p ( − z k φ ̃ ) d z P u P c k βk = P = 2δ , (A.5) (A.6) (A.7) u ·ck 3 k βEO= k uEOc 1 h(1− φ)exp(−zkφ ̃)dz k =h 0 ζ uEO ·ck uDOck , βDO= l kl DO αEOδk 1 h exp(−zkφ ̃)χldz ul ·ck =h 0 . αDOδ References [1] M. A. Shannon, P. W. Bohn, M. Elimelech, J. G. Georgiadis, B. J. Mar ̃ıas, A. M. Mayes, Science and technology for water purification in the coming decades, Nature 452 (7185) (2008) 301–310. doi:10.1038/nature06599. [2] F. Fu, Q. Wang, Removal of heavy metal ions from wastewaters: A review, Journal of Environmental Management 92 (3) (2011) 407–418. doi:10.1016/j.jenvman.2010.11.011. URL http://dx.doi.org/10.1016/j.jenvman.2010.11.011 [3] Y. Kim, W. S. Walker, D. F. Lawler, Competitive separation of di- vs. mono-valent cations in electrodialysis: Effects of the boundary layer Water Research 46 (7) (2012) 2042–2056. doi:10.1016/j.watres.2012.01.004. URL http://dx.doi.org/10.1016/j.watres.2012.01.004 [4] B. Van Der Bruggen, A. Koninckx, C. Vandecasteele, Separation of monovalent and divalent ions from aqueous solution by electrodialysis and nanofiltration, Water Research 38 (5) (2004) 1347–1353. doi:10.1016/j.watres.2003.11.008. [5] A. Mani, M. Z. Bazant, Deionization shocks in microstructures, Physical Review E - Statistical, Nonlinear, and Soft Matter Physics 84 (6) (2011) 1–13. doi:10.1103/PhysRevE.84.061504. [6] D. Deng, E. V. Dydek, J. H. Han, S. Schlumpberger, A. Mani, B. Zaltzman, M. Z. Bazant, Overlimiting current and shock electrodialysis in porous media, Langmuir 29 (52) (2013) 16167–16177. doi:10.1021/la4040547. [7] D. Deng, W. Aouad, W. A. Braff, S. Schlumpberger, M. E. Suss, M. Z. Bazant, Water purification by shock electrodialysis: Deionization, filtration, separation, and disinfection, Desalination 357 (2015) 77–83. doi:10.1016/j.desal.2014.11.011. URL http://dx.doi.org/10.1016/j.desal.2014.11.011 [8] S. Schlumpberger, N. B. Lu, M. E. Suss, M. Z. Bazant, Scalable and Continuous Water Deionization by Shock Electro- dialysis, Environmental Science and Technology Letters 2 (12) (2015) 367–372. doi:10.1021/acs.estlett.5b00303. [9] M. Z. Bazant, E. V. Dydek, D. Deng, A. Mani, Method and apparatus for desalination and purification, U.S. Patent No. 8, 801, 910 (2014). [10] M. Z. Bazant, E. V. Dydek, D. Deng, A. Mani, Desalination and purification system, U.S. Patent No. 8, 999, 132 (2015). [11] E. V. Dydek, B. Zaltzman, I. Rubinstein, D. S. Deng, A. Mani, M. Z. Bazant, Overlimiting current in a microchannel, Physical Review Letters 107 (11) (2011) 1–5. doi:10.1103/PhysRevLett.107.118301. [12] V. V. Nikonenko, A. V. Kovalenko, M. K. Urtenov, N. D. Pismenskaya, J. Han, P. Sistat, G. Pour- celly, Desalination at overlimiting currents: State-of-the-art and perspectives, Desalination 342 (2014) 85–106. doi:10.1016/j.desal.2014.01.008 . URL http://dx.doi.org/10.1016/j.desal.2014.01.008 [13] E. V. Dydek, M. Z. Bazant, Nonlinear dynamics of ion concentration polarization in porous media: The leaky membrane model, AIChE Journal (2013). doi:10.1002/aic.14200. [14] S. Nam, I. Cho, J. Heo, G. Lim, M. Z. Bazant, D. J. Moon, G. Y. Sung, S. J. Kim, Experimental verification of overlimiting current by surface conduction and electro-osmotic flow in microchannels, Physical Review Letters 114 (11) (2015) 1–5. doi:10.1103/PhysRevLett.114.114501 . [15] A. Yaroshchuk, Over-limiting currents and deionization shocks in current-induced polarization: Local-equilibrium analysis, Advances in Colloid and Interface Science 183-184 (2012) 68–81. doi:10.1016/j.cis.2012.08.004. URL http://dx.doi.org/10.1016/j.cis.2012.08.004 [16] I. G. Wenten, K. Khoiruddin, M. A. Alkhadra, H. Tian, M. Z. Bazant, Novel ionic separation mechanisms in electrically driven membrane processes, Advances in Colloid and Interface Science 284 (2020) 102269. doi:https://doi.org/10.1016/j.cis.2020.102269. URL http://www.sciencedirect.com/science/article/pii/S0001868620305388 [17] K. M. Conforti, M. Z. Bazant, Continuous ion-selective separations by shock electrodialysis, AIChE Journal (July) (2019) 1–6. doi:10.1002/aic.16751. [18] M. A. Alkhadra, K. M. Conforti, T. Gao, H. Tian, M. Z. Bazant, Continuous Separation of Radionuclides from Contaminated Water by Shock Electrodialysis, Environmental Science and Technology 54 (2019) 527–536. doi:10.1021/acs.est.9b05380 . [19] M.A.Alkhadra,T.Gao,K.M.Conforti,H.Tian,M.Z.Bazant,Small-scaledesalinationofseawaterbyshockelectrodialysis, Desalination 476 (November 2019) (2020) 114219. doi:10.1016/j.desal.2019.114219. URL https://doi.org/10.1016/j.desal.2019.114219 17 l kPDF Image | Theory of shock electrodialysis
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