Eddy-Current Separator

What is modelled

A magnetic rotor with many poles spins inside the head drum of a conveyor, so a piece on the belt sees a rapidly alternating field. In a conductor this induces eddy currents whose own field opposes the drum field (Lenz), and since the field falls off quickly with distance the piece is pushed outwards. A non-conductor feels nothing and just falls. The size of the push depends on the ratio of the particle radius to the skin depth, through the complex polarizability of a conducting sphere.

Equations

• Skin depth δ = √(2 / (μ0 σ ω)), ω = 2π np · rpm / 60.
• Sphere polarizability α = −3/2 [1 − 3/z2 + 3 cot z / z], z = (1 + i) a/δ, with Re α → −4x4/105 for small and −3/2 for large x = a/δ.
• Field B(z) = B0 e−z/λ, λ = R / np; push per unit mass |Re α| B2 / (2 μ0 ρ λ).
• Flight under gravity plus this radial push; landing where the path reaches the collection bed.

Checks run in the page

• The full polarizability against its small and large limits.
• The ballistic landing point of a non-conductor.
• The force against the gradient of the interaction energy and the energy gained by a pushed particle.
• The logarithmic slope of the force against conductivity in the two regimes.

Honest simplifications

The particle is a sphere in a field that is uniform over its size and decays as a single exponential, the belt thickness and friction are ignored, the force is always radial, rotation and tumbling of the piece are left out, the field amplitude is a given surface value and the moment of the induced currents is the steady-state value. Real separators are tuned experimentally and depend on particle shape.

• Press Slow rotor and watch the aluminium fall short of the splitter.
• Press Poor conductor: stainless steel is barely pushed.
• Press Fine particles: the small ones are pushed less and land closer.