Shear electroconvective instability in electrodialysis channel under extreme depletion and its scaling laws

The electroconvective instability (ECI) in an electrodialysis channel under a strong electric field is studied here. The phenomenon of ECI with extreme depletion (ECI-HD) is reported; that is, the overlapping vortices cause the extreme depletion zone to propagate in the horizontal direction. Using scaling theory and direct numerical simulation, we indicate a series of features under the ECI-HD. The decrease in ion transport rate with voltage in ECI-HD is different from the enhancement in the ECI with moderate depletion (ECI-MD), which results in a unique peak in the voltage-current curve. More importantly, we reveal that the ECI is regulated by a scaling factor consisting of the electric field, hydrodynamic coupling coefficient, and Péclet number. For the ECI-HD, the scaling factor has an opposite effect on the vortex size and overlimiting current as that on the ECI-MD. The extreme depletion zone of the ECI-HD also has an uncommon diffusion self-similar dynamics. These unique scaling laws allow one to establish the quantitative bridge between the ion concentration, electric field, and vortex size by the overlimiting current.

Figure 1

Salt solution between horizontal anion-selective membrane and cation-selective membrane subject to dc electric field $E_b$ in (a) the ECI-HD and (b) the ECI-MD. The ECI-HD phenomenon is that under a strong electric field, the vortices completely impact to form fully overlapping EC vortices, emerging from a noncomplete depletion $\left(d_{\mathrm{ECI}-\mathrm{HD}}\right)$ and an extreme depletion $\left(d_H\right)$ with the Poiseuille flow. As the electric field strength increases, the extreme depletion zone propagates toward the entrance. However, under a low electric field, the ECI-MD $\left(d_{\mathrm{ECI}-\mathrm{HD}}\right)$ propagates in the normal direction, which is driven by the EC vortices. Surface plots demonstrate the fully developed concentration fields superimposed with streamlines. All snapshots are obtained in a statistically stationary state. Here, $\mathrm{Pe}=60$ and ${\kappa }_0=0.5$.

Figure 2

The unique peak of the voltage-current curve in (a) $\mathrm{Pe}=60$ and (d) $\mathrm{Pe}=20$. The relations between the length of the IDZ and the ion transport rate in (b) $\mathrm{Pe}=60$ and (c) $\mathrm{Pe}=20$ As the electric field strength increases, ECI-MD gradually transforms into ECI-HD. Note that the relation presented by the Nu in the ECI-HD and the length of the IDZ is different from the relation presented by the ECI-MD. Here, for convenience, both $d_H$ and $d_{\mathrm{ECI}-\mathrm{MD}}$ are represented by $x_d$.

Figure 3

The scaling relations of vortex size. These scalings reflect the joint mechanism of the external electric field $E_b$, hydrodynamic coupling coefficient ${\kappa }_0$, and Pe number to the vortex size. As the scaling factor ${\kappa }_0{E}_c^2/\mathrm{Pe}$ increases, the vortex size of the ECI-MD increases and the scaling relation satisfies slope $\sim 1/3$. When the vortices completely overlap, the vortex size of the ECI-HD satisfies a unique scaling law, and the slope is $\sim -1$. The transition of the two scaling laws determines the peak Nu.

Figure 4

The spatiotemporal evolution of ECI-HD. The solid line is the fitted theoretical curve, and the illustration is the spatiotemporal evolution of horizontal extreme depletion. In these spatiotemporal pictures, the time is plotted on the $x$ axis and distance is on the $y$ axis. The color map corresponds to ion concentration (blue is high and red is low). Here, $E_b=80$ and ${\kappa }_0=0.5$.