Bond's Law and the Work Index

Fred Bond published his third theory of comminution in the Transactions of the AIME in 1952, and every grinding circuit sized since has passed through it. The relation is $E = 10 W_i (1/\sqrt{P_{80}} - 1/\sqrt{F_{80}})$, with the sizes in micrometres and the energy and the work index both in kilowatt-hours per tonne. The 10 is $\sqrt{100}$: Bond defined the work index as the energy to reduce material from an infinitely coarse feed down to 80 % passing 100 micrometres, and that definition is what fixes the constant.

The reason it is called the THIRD theory is that two others came first and neither fitted the whole range. Rittinger's law (1867) says the energy goes with the new SURFACE created, so with $1/x$, and it fits fine grinding best. Kick's law (1885) says it goes with the VOLUME reduction ratio, so with $\ln(F/P)$, and it fits coarse crushing best. Bond's exponent of $-1/2$ sits between them, and it was chosen because it fitted plant data over the range that matters commercially, not because it was derived from anything. Bond's law is a correlation, not a mechanism, and outside the range it was fitted over it drifts — in fine and ultra-fine grinding it drifts in the unsafe direction, UNDERSTATING the energy required. The whole apparatus of Bond efficiency factors, for mill type, circuit configuration, feed preparation and product fineness, exists precisely to patch this.

The work index is defined by ONE SPECIFIC TEST and nothing else. Bond's ball mill grindability test is a locked-cycle procedure in a 305 mm by 305 mm mill with a specified ball charge, a specified feed preparation, and a circulating load held at 250 %. $W_i$ is whatever that procedure returns. A hardness figure from a drop-weight test, a SAG mill comminution test, an abrasion index, a point load index or a UCS is a DIFFERENT QUANTITY — and several of them are quoted in the same kilowatt-hours per tonne. Substituting one for another gives an answer with the right units and no meaning, and nothing about it looks wrong. If you did not commission the Bond test yourself, find out which test produced the number before you use it.

The half power is what makes comminution expensive. Every halving of the product size costs roughly another 41 % on the term that dominates, and the cost accelerates because the feed term stops helping. That asymmetry is also why the FEED size matters so much less than the product size: $1/\sqrt{F_{80}}$ is small because $F_{80}$ is large, so coarsening the feed from 10 mm to 20 mm changes it by about 0.003 in units where the product term is around 0.1. Most of the energy is spent reaching the product size, not getting away from the feed size. Mine-to-mill programmes — spending more on explosive to give the mill a finer feed — are real and worthwhile, and they work mostly by unloading the coarse crushing and SAG stages rather than by moving this term.

Two practical notes. A work index backed out of plant data is an OPERATING work index and it is not the same thing as a laboratory Bond work index: it contains the ore's hardness AND every inefficiency in the circuit — ball charge, liner condition, classification efficiency, circulating load, mill speed and filling. Comparing the two for the same ore is one of the most useful diagnostics in a concentrator, and their ratio is what people mean by grinding efficiency. And be exact about the tonne. Work index is quoted per METRIC TONNE almost everywhere and per SHORT TON in parts of North America, and the two differ by about 10 % — enough to matter in a mill sizing and not enough to look wrong. Comminution is commonly the single largest electricity consumer on a mine site, so a 10 % error here is not academic.