# Variables Affecting Grinding Mill Power

The study of grinding is a study of cause and effect. The pre-ponderance of the literature is on the latter—effect. It seems that the study began at the end, with effect, and worked backward without going back far enough to reach the cause. The cause is energy input. In this paper it is paramount and is expressed as horsepower. Power is one of the most enlightening single factors in the study of grinding. In plant operations gross power input to the ball-mill unit is the quantity of economic interest to the operator, but from the scientific aspect the study must be made of the net power expended within the mill. The difference in these two values is motor inefficiency and dead (friction) load.

Repeated dead-load tests have been made in the Rolla laboratory, and a similar test was made in a concentrator. This was done by packing the work load centrally in the mill so that it revolved like a flywheel.

In the laboratory two 220-volt direct-current motors were used interchangeably. One was 3-horsepower shunt-wound and the other was 5-horsepower compound-wound. A typical example of the calculations required for efficiency and output has been given in an earlier paper.

All powers recorded in this paper are net power unless otherwise stated.

## Critical Speed & Power Speed Curves

Critical speed has been defined and dealt with by many writers. It is the speed at which the centrifugal and centripetal forces, acting on an infinitely small particle traveling on and with the breast of the mill, are equal. Then the particle will not leave its position. It becomes part of a “flywheel.” The phenomenon is independent of specific gravity. Obviously, on account of their greater axis of revolution, small particles of ore will centrifugalize at lower speeds than large balls. Balls never centrifugalize at speeds indicated by the formula

c.s. 54.19/√r

where r is the radius of the mill in feet, because, first, they are not in finitely small and, second, they slip and may never attain the speed of the breast of the mill. Hence, the term “theoretical critical speed” is best when a safeguard in expression is required. In spite of the lack of a close analogy between the deportment and the formula, critical speed is a valuable anchor in comparing mills of different diameters. Some critics take the term too seriously. In this paper it will be designated as “percent critical”, or “percent speed.”

In power-speed curves the percent speed that is plotted is the speed of the mill in accord with the formula, although the speed of the balls may lag far behind. Tumbling tests with a wire screen on the end of the mill will indicate the lag. The investigator need not worry about undue lifting of the balls by the screen; if he selects a mill 2 feet long with a sheet-iron head on one end and a screen on the other, and uses a flashlight to view the interior he will see that the lifting at each end is a little higher than at the middle. That is, the charge sags at the midway position, but the sagging is not enough to make examination confusing. On account of the lifting effects of the ends, a mill 2 feet long is expected to take a little more than half as much power as a mill 4 feet long. Correction for length was made in a recently developed formula for power, but other variables, such as the amount of ore in the mill, were too intangible to be corrected.

Too much emphasis cannot be laid on the fact that the peak of a power curve comes at a lower speed than that which centrifugalizes the outer circle of balls. The power peak comes when the outer circle of balls strikes the breast of the mill in what may be called the 8 o’clock position. The power curve drops with regularity when the impingement is higher or lower. An extra amount of keying would cause a higher impingement but, on the other hand, slipping would reduce the elevation. Because of moderate action, power is below maximum when the impingement is below the 8 o’clock position and because of excessive action, which results in the giving back of much of the energy power, is again below maximum when the impinging is above the 8 o’clock position.

There is nothing in the power-speed curve of a big mill to indicate the speed at which centrifugalizing begins, but squirrel-cage tests have shown that maximum power is consumed at a much lower speed than is necessary to centrifugalize the balls.

Scope of Work

The work that follows is in two divisions: (1) Experimental evidence on important points in fine grinding and (2) a study of innovations in fine grinding.

1. The first includes the following: Relation of specific gravity of grinding media to power and type of grind, effect of particle size on power, different pulp densities, different sizes of discharge openings, comparison of batch and continuous open-circuit grind, various speeds and ore charges, rod-mill tests at various speeds, various diameters of rods, comparison of large balls and small rods, comparison of dry grinding and wet grinding, different volumes of balls with different amounts of ore, balls of different hardness (such as Ni-hard balls compared with softer balls), examination of tetrahedrons, pebbles, and sillimanite balls, and laboratory mills compared with a plant mill.
2. Under “innovations” are included concentric drums in cylindrical mills, rationing the amount of the different ball sizes, segregation of ball sizes by means of grids and conical mills, and making some of the mills conical by a lining of identical truncated cones apexing in the same direction. Closed-circuit grinding was employed to study the different ball loads.

## Effect of Particle Size and Mill Power

The influence that particle size may have on power is illustrated in table 5. There the variation in comminution was moderate but the power varied almost 10 percent. For this information five batch tests were run with durations of 2.5 5, 10, 15, and 20 minutes, respectively. The first column of sizing analyses shows the mean of two products—the feed at the beginning and the product after 2.5 minutes; it was derived by taking arithmetical averages, sieve by sieve. For the second column the sizing analyses for the 2.5-minute and the 5-minute tests were averaged similarly. This method was carried through, so that the last column shows the sizing analysis that could be expected if the 15- to 20-minute period were half-finished; that is, the sizing analysis is what could be expected if the test had been stopped at 17.5 minutes.

In the first test, about half of the product was on 14 mesh, and in the last test about the same amount was on 48 mesh. Power was highest during the first period, indicating that the coarser particle sizes keyed the balls more than the finer particles. The investigation was not carried far enough to use coarser ore and determine what size had the maximum keying effect.

The speed, being only 60 percent of critical, was slow enough for the keying to affect the power curve as
noted. If the speed had been high, the power curve might have been at its peak in the last tests. Then the effect of the coarser grains would have been to key the balls as before and cause them to be cataracted so high that the power would have been reduced; that is, they would have impinged above the 8-o’clock position. Hence, the accompanying variables must be taken into account in each investigation. The variation of power with particle size will come up again for discussion in figure 5.