A theoretical model for the Lorentz force particle analyzer

In a previous paper [X. Wang et al., J. Appl. Phys. 120, 014903 (2016)], several experimental devices have been presented, which demonstrate the efficiency of electromagnetic techniques for detecting and sizing electrically insulating particles entrained in the flow of a molten metal. In each case, a non-uniform magnetic field is applied across the flow of the electrically conducting liquid, thereby generating a braking Lorentz force on this moving medium and a reaction force on the magnet, which tends to be entrained in the flow direction. The purpose of this letter is to derive scaling laws for this Lorentz force from an elementary theoretical model. For simplicity, as in the experiments, the flowing liquid is modeled as a solid body moving with a uniform velocity U. The eddy currents in the moving domain are derived from the classic induction equation and Ohm's law, and expressions for the Lorentz force density j × B and for its integral over the entire moving domain follow. The insulating particles that are eventually present and entrained with this body are then treated as small disturbances in a classic perturbation analysis, thereby leading to scaling laws for the pulses they generate in the Lorentz force. The purpose of this letter is both to illustrate the eddy currents without and with insulating particles in the electrically conducting liquid and to derive a key relation between the pulses in the Lorentz force and the main parameters (particle volume and dimensions of the region subjected to the magnetic field).