Magnetic separation of rare-earth ions: Transport processes and pattern formation

Despite the relevance of the Kelvin force in many physical and electrochemical systems involving magnetic species, the underlying convective flows are scarcely understood. For that purpose, we simulate a simplified rare-earth system in the presence of competing Kelvin and gravity forces. The results are experimentally validated using interferometry and microscopic particle image velocimetry. Based on the excellent agreement of numerical and experimental results, the underlying mechanism of ions' magnetic separation can be explained, which paves the way for a prospective application in rare-earth beneficiation.

Figure 1

(a) A rare-earth solution in a cuboid cuvette undergoes evaporation in a nonuniform field of a NdFeB magnet. The laser beams for interferometry and stereo $\mu$-PIV are indicated. (b) Density of the potential energy, Eq. (4), computed with the values given in the text. The position of the energy barrier is marked.

Figure 2

Comparison of concentration and velocity fields [(a) and (b)] and of velocity fields [(c) and (d)] between experiment (left) and simulation (right) at different times; see Fig. 3.

Figure 3

(a) Mean kinetic energy ${\overline{e}}_{\mathrm{kin}}$ of the flow vs time in the different Phases I–III. The times at which the snapshots were taken in Fig. 2 are indicated by (.). (b) Zoom into Phase I based on simulating the evolution of ${\overline{e}}_{\mathrm{kin}}$ at varying evaporation flux.

Figure 4

Space-time plots of line profiles of concentration $c\left(y\right)$ (a) and vorticity ${\omega }_x\left(y\right)$ (b) close to interface ($z=0.5$ mm) from cell rim to cell center.

Figure 5

Schematics explaining the transformation of planar concentration isolines (a) into oval isolines by virtue of the Kelvin-force-driven convection (b) leading to a vanishing angle $\theta$, or forcing, in the Kelvin-force-dominated zone. (c) Snapshots of the $\mathbf{\nabla }\times \mathit{f}=-\mathbf{\nabla }c\times \mathbf{\nabla }{e}_{\mathrm{pot}}$ at different times.