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Membranes 2021, 11, 697 6 of 13 On the microchannel portion of the lower boundary, (i) the electric potential of the lower boundary is 0; (ii) the velocity of the fluid is constant u1 = (0, u1 ); (iii) the concen- tration of Na+ is C2,1, the concentration of Cl− is the opposite number of the sum of other four cations: Φ=0, U=u1, C2 =C2,1, C5 =−∑CiZi, i=1, 2, 3, 4 (8) where ci and Zi represent the variable concentration and the valence of ion i. On the red dashed boundary, (i) a voltage V is applied on the membrane to generate electric field; (ii) the velocity of the fluid is the constant velocity (u2 = (0, u2 )); (iii) the con- centration of Cl− at the upper boundary is Cm; (iv) fluxes of cations across the membrane are zero. The corresponding equations can be expressed as: Φ = V, U = u2, C5 = Cm, Ji · n = 0, i = 1, 2, 3, 4 (9) The other black lines and black dashed lines are the microchannel walls, (i) no-slip condition for fluid velocity; (ii) impermeability to all anions and cations: U= 0, Ji · n= 0, i = 1, 2, . . . , 5 (10) 2.5. Numerical Methods Simulations were carried out using COMSOL Multiphysics software (version 5.6) on a Dell workstation (Precision 7920) equipped with an Intel Xeon processor (Gold 6128) and 112 GB of RAM. Steady-state simulations were used for all studies. Solution convection was modeled with the “Creeping Flow” interface. Moreover, the “Transport of Diluted Species” and the “Electrostatics” interfaces were coupled to solve the “Nernst–Planck–Poisson” equation; 3,833,759 quadrilateral elements were utilized for meshing. Near the membrane region, extremely fine meshes are used to ensure sufficient solution accuracy. To obtain the highly nonlinear solution under high electric potential, we needed to start from low voltage and sweep the high voltage parameter. Initially, the down boundary should be set as no flux for Li+, Mg2+, and K+ to converge the simulation. After the system is stable, we set it out to facilitate ion exchange between the chamber and the reservoir below. As shown in Figure 1b, in the simulation model, we kept the horizontal microfins to analyze the average ion fluxes of the outflow because the value taken on the boundary will greatly affect the calculation result. However, we ignore the vertical ones as they do not affect the results. To analyze the behavior of the separation system, we started with the setting of a particular parameter. Then, we studied the effects of two critical operational parameters, the voltage V and velocity u, and clarified how these parameters affected system performance to prove the feasibility of the proposed ion separation method. 3. Results and Discussions In this simulation, the geometric parameters of the chamber are L = 90 μm, H = 31.5 μm (see Figure 1b). The length of the microfins channel is Lc = 3 μm, and the width is Hc = 1.5 μm, while the distance between the channels is Hb = 1.5 μm. The length of membrane segments is Lm = 1.5 μm, Lb = 1.5 μm. Using a simplified brine consisting of only five ions (Li+, Na+, K+, Mg2+, Cl−) [31]. After diluting the raw brine, we selected the following typical concentrations. The con- centrations of ions (the left inlet boundary) are: C1,0 = 0.001 mM, C2,0 = 0.125 mM, C3,0 = 0.04 mM, C4,0 = 0.04 mM, C5,0 = 0.211 mM. The ion concentrations of the lower edge are C2,1= 0.375 mM, C5,1 = −∑ CiZi, i = 1, 2, 3, 4, where NaCl enters the chamber from the lower boundary as a supplement buffer to maintain electrical neutrality. In the ideal simplified model of the ion-selective membrane, the results of fixed voltage and fixed counter ion concentration are accurate in most cases, especially in the case of high voltage and/or high charge density [36]. At the membrane boundary, the voltage V is fromPDF Image | Brines Based on Free Flow Ion Concentration Polarization
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