Engineering & Fluid Mechanics
Bond’s Third Theory of Comminution Formula: Work Index (Wi) & Ball Mill Power
Calculate specific grinding energy W = 10 * Wi * (1/SQRT(P80) - 1/SQRT(F80)) in kWh/tonne and pinion shaft power (kW) from Fred C. Bond’s Work Index Wi and 80%-passing feed/product sizes in micrometers.
Between Rittinger’s new-surface law (d^-1) and Kick’s volume law (ln d): Bond’s 1952 inverse-square-root law!
Fred C. Bond (1952, Transactions of AIME — Allis-Chalmers) resolved decades of mineral-processing debate between Rittinger (1867) and Kick (1885) by showing that crack propagation energy in brittle ores scales with inverse square root of particle size: W = 10 * Wi * (1 / SQRT(P80) - 1 / SQRT(F80)), where W is specific pinion energy (kWh/metric tonne), Wi is the laboratory Bond Ball/Rod Mill Work Index (kWh/t, defined as energy to reduce ore from infinite feed size to 80% passing 100 micrometers—hence the factor of 10 = SQRT(100)), and F80, P80 are 80%-passing sieve apertures in micrometers (um)!
For a copper-porphyry ore with Wi = 14.2 kWh/t ground from F80 = 12,000 um (12 mm) to P80 = 150 um at throughput M_dot = 240 t/h, specific grinding energy W = 10.298 kWh/t and mill pinion power P = 2,471.5 kW—jumping to W = 15.100 kWh/t (+46.6% energy!) when regrinding finer to P80 = 75 um!
| Grinding Circuit Target (Wi = 14.2 kWh/t, F80 = 12,000 um) | Specific Energy W (kWh/t) | Mill Pinion Power at 240 t/h | Flotation Liberation Trade-off |
|---|---|---|---|
| Primary Rougher Grind P80 = 150 um | 10.298 kWh/t | 2,471.5 kW | Standard Rougher Liberation |
| Fine Regrind Target P80 = 75 um | 15.100 kWh/t | 3,624.1 kW | +46.6% Power for Fine Sulfide Liberation |
| Coarser Product P80 >= Feed F80 | Non-Comminution! | "Invalid Bond Grinding Sizes" | Requires 0 < P80 < F80 |
Bond Specific Comminution Energy W (kWh/t) = 10 * Wi * (1/SQRT(P80) - 1/SQRT(F80))
Evaluates Bond’s third-theory specific grinding work (kWh/t) from Work Index Wi (B2) and 80%-passing feed F80 (B3, um) and product P80 (B4, um).
=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0, B4 >= B3, B5 <= 0), "Invalid Bond Grinding Sizes", ROUND(10 * B2 * (1 / SQRT(B4) - 1 / SQRT(B3)), 3))Ball Mill Pinion Shaft Power Draw (kW) = Tonnage (t/h) * Specific Energy W
Multiplies circuit throughput (B5, tonnes/hour) by Bond specific energy W to size mill motor/pinion power in kilowatts.
=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0, B4 >= B3, B5 <= 0), "Invalid Bond Grinding Sizes", ROUND(B5 * 10 * B2 * (1 / SQRT(B4) - 1 / SQRT(B3)), 1))Try the example
Use the original values or change them to test your understanding.