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More negative σ leads to less deionization at the same I ̃ but also larger overlimiting conductance, so medium σ should require the least E. Larger H leads to more water recovery and deionization at the same I ̃ but also smaller conductance, so there is a plateau of E between H = 2.7–5 mm. In contrast, larger L leads to larger conductance but less water recovery and deionization, so medium L would be the most favorable. To conclude, to obtain 99% deionization at the minimal energy cost, we need the macroporous material to have small hp and moderate σ, H and L, and operate the system under flow feed rate Q. However, note that the decrease of hp requires higher pressure at the inlet and thus more robust sealing. To scale up the output rate without increasing the pumping pressure, we increase H, increase hp, or decrease L to increase Q and keep the pressure drop ∆p (which ∼ QL/h2pH) at zero current unchanged. The specific energy consumption at two deionization levels is shown in Fig.10. So six new cases are added. Basically, as we scale up the process, the specific energy consumption increases. Among the three parameters, the scale-up by hp increases the specific energy in the least extent. When we decrease L, the current density becomes very high at 90% deionization, and transport in electrode streams get limited by diffusion, so the voltage and energy consumption becomes very large. A numerical problem occurs when we want to reach more deionization, so the result for the smallest L at 99% deionization is not shown in the figure. In the future, we aim to create more innovative designs, instead of just varying geometries, for scale-up of the process without sacrificing much of the energy consumption. 4. Conclusion This work established a depth-averaged model for shock electrodialysis with multiple ions and more realistic boundary conditions. We identified the importance of hydronium transport and electroosmotic vortices for shock ED performance in terms of conductance AND deionization, and find good consistency between the model and experiments for binary electrolyte. We also give suggestions on how to optimize the process in terms of specific energy consumption, and how to improve the flow rate without increasing pressure, based on the current prototype design. In the second part of the series paper, we will give a preliminary analysis for selective ion removal. In the future, we will also investigate the effect of adsorption and porous heterogeneity on shock ED. Acknowledgments The research was supported by a Graduate Student Fellowship awarded by the MIT Abdul Latif Jameel Water and Food Systems Lab and funded by Xylem, Inc. The authors would like to thank Mohammad Mirzadeh and Pedro de Souza for insightful discussions. Appendix A. Coefficients of the depth-averaged model In this part, we introduce how we calculate the coefficients δk, αEO, αDO, βP , βEO, and βDO. First, we kkk kl need to solve the PB equation (Eq.(15)) for φ with given cvk, σ, and h. Then we can use the φ to calculate the coefficients: where ck 1h δk = cv = h exp(−zkφ ̃)dz. k0 αEO = 1 h1−φdz, h0ζ 1 h z′′ (A.1) (A.2) (A.3) (A.4) DO 1h αk = h χk(x,y,z)dz, [exp(−zkφ ̃′)−1]dz′dz′′, z0 16 χk(x,y,z)= h2 0PDF Image | Theory of shock electrodialysis
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