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composition cin. Although we assume uniform surface charge σ for the charged channel, in experiments k σ may vary significantly in the macroporous material due to charge regulation with local concentration and pH. This will be considered in future work. In this paper, σ is treated like an artificial parameter for the model. All the constants used in this paper are summarized in Table 3, and parameters that vary for different models will be declared where appropriate. In the following, we show simulation results for a binary electrolyte in Sec.3.2. The transport of hydro- nium and hydroxide is also included in addition to the salt ions. 3.2. Results For this part, we choose a base case: 10mM NaCl aqueous solution at pH 7 is injected into the feed and electrode channels, and σ = −20.8 mC/m2, hp = 250 nm, H = 2.7 mm, Q = 76 μL/min, QE = 4Q, HF /(HF + HB ) = 0.45. We first present the results for the base case and identify the importance of hydronium transport and electroosmosis for shock ED. Next, we obtain results for different working conditions and compare the results with available experimental data. Finally, we investigate more parameters that have not yet been tested experimentally, and give suggestions for optimization and scale-up. In the following, we calculate current efficiency η, dimensionless overlimiting conductance κ ̃, dimension- F less fresh concentration c ̃ , water recovery ω, and specific energy consumption E to evaluate the shock ED process. The current efficiency η is defined as the current carried by the salt cations divided by the total cur- rent through the membranes. The overlimiting conductance κ is estimated by ∆I/∆V on the I-V curves over the range I ̃ = 2–4. Then it is scaled by the theoretical surface conductance κsc = z+D+|σ|ǫphioL/VT Hhp [11], where ”+” represents the salt cation, and ǫphioL is the total cross-sectional area of the charged channels or pores in the xz-plane. The fresh ion concentration cFk is defined as the flow-weighted mean concentration of the fresh stream. We set cF = cF+ to represent the fresh concentration of the binary electrolyte, and scale it by cin. The water recovery is defined as ω = Q /Q, where Q is the fresh stream flow rate. Finally, define +FF the specific energy consumption E = (IV + Q∆p)/ωQ, where ∆p is the pressure drop across the device and the pump efficiency is assumed to be 1. 3.2.1. Hydronium transport and electroosmotic vortices ̃v v Fig.3 shows the profiles of cˆNa, pH, ψ , p ̃ and uˆ for the base case. Only the feed channel is shown. As we can see, for both models, an ion-depleted zone forms near the cathode-side in the charged channel, where almost all of the potential drop occurs. In the depleted zone, the concentration of Na+ is almost constant, while that of H+ increases in the x-direction, which should be due to the downward transport of H+ released at the anode. This is the first time that pH profiles of shock ED are shown by modeling. As we can see, the pH in the charged channel is usually below 5, which indicates that the transport of OH− is negligible compared with H+, and the concentration of H+ is comparable with that of Na+ in the ion-depleted zone. More hydronium ions are produced at the anode as current increases, which results in decrease of current efficiency η through the membranes especially at the anode side, as also shown in Fig.4. We now examine the roles of electroosmosis and diffusioosmosis on shock ED. First, we calculate (||uDO|| + ||vDO||)/(||uEO|| + ||vEO||) for the velocity field obtained from the DAfull model. This value is less than 0.1%, which means that electroosmosis overwhelms diffusioosmosis. In the charged channel, the electroosmotic flow tends to be aligned with the electric field but is hindered by the membrane, which will build up a pressure near the cathode side to push the fluid either up or to the two sides to the inlet and outlet zones. Therefore, two vortices are formed near the interfaces between the charged channel and the inlet and outlet zones. The vortices bring the brine down to the depleted zone and bring the fresh water up in the inlet and outlet zones. Therefore, while the depleted zone grows monotonically in the x-direction in the DAp model, the depleted zone shrinks at the outlet in the DAfull model. This can make it extremely hard for the upper boundary of the depleted zone to pass the splitter. As current increase, most of the flow leaves the charged channel through the depleted zone, which guarantees strong desalination, but the ion diffusion from the brine zone to the outlet zone may increase the concentration of the fresh product. We quantitatively analyze deionization later. Next, we compare the dimensionless overlimiting conductance κ ̃ = κ/κsc from different models for the basic case, as shown in Fig.5. As we can see, the conductance from the model of [20] is very close to 10PDF Image | Theory of shock electrodialysis
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