How mill speed, filling and lifters shape the tumbling charge and the power draw.
-Motion regime
-Critical speed (rpm)
-Operating speed (rpm)
-Froude number (w^2 R / g)
-Drive torque (kN m per m)
-Power (kW per m of length)
-Charge kinetic energy (kJ)
-Balls in free flight
-Balls simulated
The science
A tumbling mill grinds ore with a charge of steel balls and rock lifted by the rotating shell. The shell speed is usually quoted as a fraction of the critical speed, where a particle at the top just stops pressing on the shell: w_c = sqrt(g / (R - r)), which for a point mass is N_c = 42.3 / sqrt(D) rpm with D in metres. The Froude number Fr = w^2 (R - r) / g equals 1 at the critical speed.
Below about 80 % of critical the charge rolls down its free surface (cascading); between roughly 80 and 100 % balls are thrown through the air and land on the toe (cataracting); beyond 100 % the charge is pinned to the shell (centrifuging). Here the regime label combines the speed fraction with the measured share of balls in free flight.
The simulation is a discrete-element model in two dimensions: each disc obeys m a = F_gravity + sum F_contact, with normal force k delta + c v_n and Coulomb friction limited by mu F_n. The drive torque is the sum of wall and lifter forces times their lever arms, and power is P = tau w. All values are per metre of mill length because the model is a slice.
Honest simplifications: a 2D slice (no axial flow, no ball-ball size classes beyond a spread of radii), soft contacts with a lower stiffness than steel, no ore breakage, no slurry, and no liner wear. Torque and power are qualitative and should not be used to size a motor.