Dense Media Cyclone - 2D Particle Tracks

The feed enters tangentially at v_in = sqrt(2 p / rho_m) and spins as a semi-free vortex v_t(r) = v_in (R/r)^0.8, giving a centrifugal acceleration a = v_t^2 / r of hundreds of g. A particle of density rho_p and size d is pushed outward by the net force (rho_p - rho_m)(pi/6) d^3 a against drag 1/2 rho_m C_D (pi/4) d^2 v^2 with the Schiller-Naumann law C_D = 24/Re (1 + 0.15 Re^0.687), while the inward flow carries it toward the vortex finder at u_r = Q_o / (2 pi r H). The ratio of settling to inward speed grows with radius, so the decision is made at the wall: the cut density is where the settling speed equals u_r(R).

Random turbulent dispersion spreads the sharp cut into the S-shaped Tromp curve; the probable error is Ep = (rho_75 - rho_25)/2. The clean-coal ash is computed from density with a two-component mix of coal (1300 kg/m3) and mineral matter (2600 kg/m3).

Honest simplifications: the medium is treated as a homogeneous Newtonian fluid with an effective viscosity (a real magnetite medium segregates and is non-Newtonian), a single flow split, a two-dimensional cross-section, and no air core or conical section.

• Compare Light medium and Heavy medium: the cut density follows the medium, trading yield for ash.
• Use Very turbulent to widen the partition curve and raise Ep.
• Lower the pressure: the cut moves away from the medium density, especially for fine particles.