5. Power Requirements for Size Reduction

An important consideration in design of size reduction flowsheets is the energy required for size reduction. Size reduction means generation of surface area from massive, typically hard rock. Size reduction machines are typically quite energy inefficient. Much of the energy put into a machine goes into running the machine itself, vibration and noise. Improving energy utilization is an ongoing field of research. Vibration control and engineering is a significant issue in size reduction mills.

Energy required to break up a homogeneous (all one mineral), unit weight of rock is approximately proportional to 1/d^1/2 where d is the output product particle diameter (F.D. Bond). Bond’s empirical correlations are approximate, but do provide an estimate of the energy required to break up a material to specified particle size. Bond took the particle size to be P80 (d80 in Bond’s terminology) which means, again, that 80% of the mass of ground particles has a diameter of less than the d80 size. So if d80 = 100 μm, then 80% of the mass is comprised of particles with diameter <100 μm; the other 20% is larger than 100 μm.

To reduce particles from some initial size d_I to a final size d_O requires a power input:

\[W = W_O – W_I= \frac{K}{d_O^{1/2}} – \frac{K}{d_I^{1/2}}\]

W_O and W_I are the energy requirements to break up the rock from infinite size (or just very large) to specified size and K is a proportionality constant that is specific to each material; note that d is in μm.

To reduce the size from very large to 100 m:

\[W= \frac{K}{100^{1/2}} – \frac{K}{\infty^{1/2}}= \frac{K}{10}\]

This is designated as Wi and K = 10Wi. In general then:

\[W= 10W_i \left(\frac{1}{d_O^{1/2}} – \frac{1}{d_I^{1/2}}\right)\]

W_i is the Bond work index in kWh/T (kilowatt hours per ton; 1 ton = 2000 lb = 907.2 kg). It is a measure of the energy required to break up a ton of a specified material from a starting size to an output size (as d80). Variability in minerals and in rock strengths of ±20% results in substantial variability in Wi values for a given material, and hence the numbers have substantial uncertainty. Some typical values for a few materials are shown in Table 7.

Table 7. Bond work index values for several minerals and materials

Values of the Bond work index can be determined by standard lab tests. Different size reduction machines have different Bond work index values for a given material. Multiplying W in kWh/T by the throughput in T/h gives the power requirement in kW. Although there is significant uncertainty the Bond equation and work indices are used in practice for design purposes.

While crushing and grinding are different types of processes the energy requirement for size reduction is determined by the material and to some extent the machine. Energy utilization efficiency is better in crushers and poorer in grinding mills, manufacturers can correlate the energy requirement with the right size of machine or mill. As a somewhat rougher estimate, the equation and data in Table 7 can be used to estimate the energy requirements for size reduction over several orders of magnitude.