A dense medium cyclone sorts particles by density in a suspension of fine magnetite. The feed enters tangentially at v = sqrt(2 dP / rho) (Bernoulli) and creates a centrifugal field a = v^2 / R, tens of times gravity, so a particle of density rho_p in a medium of density rho_m is pushed outward if rho_p > rho_m and inward if lighter. Floats leave through the vortex finder (overflow), sinks through the apex (underflow).

The magnetite itself also settles outward at the Stokes speed v_s = (rho_mag - rho_m) d^2 a / (18 mu). This segregation makes the underflow denser than the overflow, so the real cut density d50 is not the nominal medium density; a thin, unstable medium (low viscosity) or coarse magnetite widens the gradient and degrades the separation. The sharpness is described by the Tromp partition curve P(rho) = 1 / (1 + exp(-1.0986 (rho - d50) / Ep)), where the probable error Ep is the density distance between 50 % and 75 % partition, and the imperfection is Ep / (d50 - 1).

Simplifications: the segregation, cut and Ep relations are a teaching model calibrated to typical coal-washing numbers (R = 0.2 m, residence time 0.35 s), not to a specific machine; no particle-size effects, no turbulence, no medium rheology beyond one viscosity value.