Hong Life CalculatorsUnderstanding the grinding energy calculator (Bond's law) — concept and principle
Crushing and grinding are the most energy-hungry steps in mineral processing. Bond's law estimates how much energy is needed to reduce an ore from one size to another, and is the standard starting point for sizing ball mills and rod mills. This calculator returns the specific energy in kWh per tonne and the mill power for a given throughput.
How it is calculated
- Specific energy: W = 10 × Wi × (1/√P80 − 1/√F80), in kWh/t.
- Mill power: P = W × throughput, in kW when throughput is in t/h.
Wi is the Bond work index of the material (kWh/t), F80 the size in micrometres through which 80% of the feed passes, and P80 the same for the product.
The principle
Two older laws bracketed the problem: Rittinger linked energy to new surface area (proportional to 1/size), and Kick linked it to the size reduction ratio (a logarithm). In 1952 Fred Bond proposed a "third theory" in between: energy is proportional to 1/√size, reflecting crack length as particles break. The work index is defined so that W = Wi when a material is ground from a very large size down to P80 = 100 µm, since 10 × Wi × (1/√100 − 0) = Wi. Wi comes from a standard laboratory grindability test.
Worked example
- Defaults: Wi = 14 kWh/t, F80 = 10,000 µm, P80 = 100 µm, throughput = 100 t/h.
- 1/√100 = 0.1 and 1/√10,000 = 0.01, so the difference is 0.09.
- W = 10 × 14 × 0.09 = 12.6 kWh/t.
- Mill power = 12.6 × 100 = 1,260 kW.
| Bond work index (kWh/t) | Specific energy (kWh/t) |
|---|---|
| 3.5 | 3.15 |
| 7 | 6.3 |
| 10.5 | 9.45 |
| 14 | 12.6 |
| 17.5 | 15.75 |
| 21 | 18.9 |
| 28 | 25.2 |
How to read the result
Note how the fine end dominates: going from 10 mm to 100 µm costs 12.6 kWh/t, but grinding the same ore further to 50 µm would add another 5.8 kWh/t for a much smaller size change. Soft materials have low work indices and hard, competent ores high ones, so the same circuit can draw very different power on different ores. Use the result as the base power at the mill pinion, then apply the efficiency factors used in mill design (for example for open or closed circuit, fine product or oversize feed) before choosing a motor.
Common mistakes
- Mixing units: Bond's original index was in kWh per short ton; multiply by 1.1023 to get kWh per metric tonne.
- Using sizes in millimetres; F80 and P80 must be in micrometres for the constant 10 to apply.
- Using a work index measured for ball milling to estimate rod milling or crushing energy.
Frequently asked questions
What is a typical Bond work index?
It varies widely with the material; many common ores fall somewhere around 10 to 20 kWh/t, but only a laboratory test on your ore gives a design value.
Why use the 80% passing size?
It is a single, reproducible point on the particle size distribution that represents the bulk of the material better than the maximum size.
Does Bond's law apply to very fine grinding?
It becomes less accurate below a few tens of micrometres, where other models and specific tests are used.
Sources: Bond, F. C. (1952). The third theory of comminution. Transactions of the AIME, 193, 484–494. · Wills, B. A. & Finch, J. A. Wills' Mineral Processing Technology. Butterworth-Heinemann.