Bond's third theory of comminution says the energy needed to reduce a feed of 80%-passing size F80 to a product P80 is W = 10 Wi (1/√P80 − 1/√F80) (sizes in micrometres; Wi is the work index in kWh/t). The standard lab test grinds in a 12 x 12 inch mill until a steady circulating load is reached and reports the net sieve undersize per revolution, Gbp, and Wi = 49 / (P10.23 Gbp0.82 (10/√P80 − 10/√F80)) (metric constant; 44.5 for short tons).

The ball motion on the left is a 2D discrete-element toy (spring contacts, wall friction) in units where the critical speed is 1. Critical speed is Nc = 42.3/√D rpm (D in metres), where centripetal acceleration equals gravity at the shell. Below it balls cascade and cataract; above it they pin to the shell and no grinding happens.

Simplifications: size distributions use a Gates-Gaudin-Schuhmann shape (m = 0.8) rather than a measured curve; real mills need efficiency factors for wet/dry grinding, diameter and fineness. Educational model, not a plant design tool.