Several years ago, Davis assumed that the rate of wear of the different sizes of balls in a ball mill was directly proportional to the weight of each ball, and he evolved a formula for calculating a balanced charge. Operators have used this formula when purchasing balls for a new mill or when reloading an old one that had been emptied for repair. The formula required that the largest ball size and the size to be rejected should be determined, and after that the other sizes were set. Stress was laid on the coarsest size, and to facilitate the use of the formula many writers have made their contribution by reporting ratio of coarsest particle size to the optimum ball size. Close adherence to this ratio has prevented giving attention to sizes and amounts of particles not falling in the category of the coarsest size.

The inadequacy of the formula and the futility of extensive experimentation for ratio determinations involving the coarsest particle size only is at once obvious when it is seen that the formula did not take into account the slow grinding rate of the finer sizes of ore and the amount present. To be sure, operators who were doing very fine grinding have sometimes altered the make-up load by using some additional small balls with the big ones, but this practice has been somewhat haphazard. Too much of the work has followed the old idea that “there should be no ball present that is incapable of crushing the largest particle in the feed.”

Today operators have a keener sense of the relatively large amount of work required to finish the finest sizes, so that the insufficiency of the formula is readily seen. It would have been fortunate had the formula been devised to attract more attention to the large amount of finer but unfinished particles. The formula is excellent from the basis of balance with respect to ball wear, but the literature has contained very little about the rationing of ball sizes for the best grinding of all sizes and amounts of particles extending throughout the length of the mill. Research has submitted in this matter.

It is not denied that the coarse particles have to be crushed else no fine material would accrue, but here the fact is emphasized that when crushing to 200-mesh stress should be on the selection of balls of the right size and amount to crush, say, from 100- to 200-mesh; or, when crushing to 65-mesh, the operator should judiciously load the mill for crushing from 48- to 65-mesh. If this were done, the circulating load would be relieved of the large amount of nearly finished size, and in its stead there would be some coarser material from which the classifier could more easily remove the finished size. Opposing this idea is the fact that a coarse circulating load would be undesirable in some of the recent supplementary recovery processes. However, this objection might be met by introducing a bypassing screen at the end of the ball mill.

Optimum Size of Balls

Tests of other experimenters have been supplemented with detailed information on the optimum size of balls for grinding sized ore. Figures have been obtained that show what particular size of ball is the most efficient in crushing certain sizes of chert and dolomite. It is fortunate that this work has been done,

because it has brought out facts that would have been unsuspected otherwise. The method used here for showing what particular size of ball is best for a particular particle size of ore is to some degree unique. The reason for this is that usually such tests have been run to finish the grinding at a fine size. Those tests were as much a criterion of the work on the particle size in the finished product as of the feed, but they were not so interpreted. The tests reported in this paper are different because the first step in reduction is given the main emphasis.

As a guide in laying out this work, a mill was visualized as divided into sections. The first section had the largest media and performed the first step in grinding by reducing the particles for the second section; the second section, in turn, used smaller media to reduce the articles for the third section; and so on. This line of thought was the basis for the distribution of sizes in the ball loads already mentioned.

The ball sizes were 2.75-to 0.62-inch and the ore sizes plus 65- to plus 10-mesh. The results for chert are shown in four series in table 27 and for dolomite in five series in table 28. The ore (feed) sizes are in quotation marks because they are only nominal; their meaning is set forth in the sizing analyses under “feed.”

Any plan adopted would give but litle more than an approximation of the facts sought, owing to the difficulty in timing the grinding correctly. If it is desired to find the effect of balls grinding 20-mesh ore and the mill is loaded with 20-mesh material, the grinding time should be infinitely short, because fine particles are made as soon as the mill starts and if the run continues the test is of the comminuted products of the 20-mesh sample rather than that which was supplied for the test.

Extrapolation back to zero time would be desirable if it were possible. However, a very short period is unsatisfactory because the flaky particles, being the first to yield, would give a wrong impression of the sample as a whole. Long grinding periods would be useless because the particle size at the end of the run would be too far removed from the original particle size under investigation. A mean procedure had to be adopted.

In these tests the batch-grinding periods for the chert were about 6 minutes and for the dolomite about 2 minutes.

The surface calculations that are given must be used guardedly, else they will be misleading. The fine particle sizes are likely to be weighted too much; when the ball size for crushing 10-mesh sizes through 14-mesh is sought, the very fine sizes should be weighted with caution.

A casual examination of each series for minimum of cumulative weights in the coarse sizes of the screen analyses probably would be a fair guide to the best ball size. But this minimum, though important of consideration, is not final, because the amount, power, and time have to be taken into account. These three quantities are resolved into tons per horsepower-hour and will be applied in table 29. Before going to that table, however, the present tables may be used to bring out a fact not commonly known—balls that were too large as well as balls that were too small failed in selective grinding. In any of the series except the last one of each table, where the largest ball sizes were not large enough, the low cumulative percentage weight of the coarse sizes is in a mean position and rises with the use of larger as well as smaller balls. Hence, it is shown that balls that were too large did nonselective grinding.

Figure 4 is offered in further consideration of the nonselective grinding of balls that are off size. The data are taken from the “35-

mesh” size in table 27 and are shown in sizing diagrams. The percentage weights of the products from the largest, and the smallest balls are shown by broken lines. They are high in the upper part of the diagram. Their position shows that much of the coarse material was not reduced through 35-mesh. The solid line shows good selective work of the balls of optimum size.

In the study of these diagrams, it must be remembered that the main variables in the tests were ball size and that the tests were “timed” to give the same amount of subsieve size. The conditions imposed on the tests were entirely different from closed-circuit grinding, in which the composite feeds would have been unlike, although the new feeds might have been the same.

The nonselective grinding of the off-size balls may be explained as follows: The largest balls failed on the coarsest sizes because they did not offer a sufficient number of points of contact for the number of grains present; hence, some of the particles remained at the end of the test. Furthermore, due to the small number of points of contact of such large balls, the crushing impulse was so great that the grains that did meet it received excessive comminution and much of the subsieve size resulted.

The smallest balls had so many points of contact that the impulse at a given point was too much reduced to exert sufficient stress on the coarsest particles; hence, some of them remained without the desired reduction. However, a few that were reduced yielded grains readily comminuted by the smallest balls, and much subsieve size again resulted; hence, there was an intermediate ball size for the best work.

Table 29 will now be discussed: It is made by using the two preceding tables. It gives the amount of the coarsest size per unit of power crushed through a stated coarse but finer size. To illustrate the method of calculation, take the first test in table 27: The amount of “plus 65-mesh” crushed through 100-mesh is 89.5—63.3=26.2 parts per hundred, and by the table the ton per horsepower-hour was 0.16; hence, the tons per horsepower-hour crushed through 100-mesh was 26.2/100×0.16=0.042. Similarly, in the first test in the second series 97.3—52.0=45.3, and 45.3/100×0.186=0.084 ton per horsepower-hour through 48-mesh. Thus, table 29 has four series of tests or chert and five series for dolomite. The preferred value in each series is underscored to show what seems to be the preferable ball size. The optimum ball size for grinding closely sized particles through the limiting screen, as determined by these experiments, may be expressed in the following equation

D² = Kd

where D is diameter of ball, d is diameter of particle to be ground, and K is a constant depending on the grindability of the ore. When D and d are expressed in inches, the value of K for chert is 55 and for dolomite is 35. This formula is of the same type as that developed by Starke. He evaluated the grind through a broader range and his dimensions are in microns.

Having selected the best ball size, it will be seen by referring to tables 27 and 28 that the preferable ball size usually gave the best capacity and efficiency. Also, the preferable ball size coincides closely with the best selective grinding, the main exception being the “plus 10-mesh” series in table 28. There the preferable ball size is smaller than the size for the best selective grinding. Probably the exception is due to an error in planning the “plus 10-mesh” series; the time periods were too long and too much grinding resulted. The spread in reduction in this series was greater than in any other series. It was intended to avoid such a broad spread in reduction. In the study of the exception and the study of the sizing analyses in the other tests an attempt has been made to gain additional information by using the Gaudin “ log-log method for plotting sizing analyses, but the results were not satisfactory. It is believed, however, that the method was not expected to apply to the moderate reduction of a sized product.

Tables 27 and 28 cannot be dismissed without consideration of the variation of power throughout a test. Figure 5 is submitted for that purpose. In it the time extends from 0 to 3.5 minutes. The change in power through the grinding periods was watched in all the tests. This change is illustrated in figure 5, which deals with the “plus 20-mesh” size in table 28. In the discussion of this figure, what will be said about the relation of power to other factors is premised by the belief that the degree to which the balls nip the particles influences the power, and that when nipping is best the power will be the highest. The curve at the bottom of the figure shows that the 2.75-inch balls required less power than the other loads. The balls were too big for good nipping, and as the grinding continued they became relatively bigger and further power reduction resulted. Correlated with this is the fact that the grinding was poor in selection and unsatisfactory in capacity and efficiency. (It is not consistent to compare the numerical-values of capacities and efficiencies of one series in tables 27 and 28 with those of another series. The principles underlying the reason were mentioned under “Sillimanite balls.”)

Turning next to the deportment of the 0.62-inch balls, which were the smallest in the group, the change in power from beginning to end of the run is in a reverse order from that with the largest balls. The balls were too small for good nipping, but as comminution proceeded they became relatively larger so that nipping and power increased but did not reach the high power indicating good nipping. The selective grinding, capacity, and efficiency were again poor.

The record of the 1-inch balls is more favorable. The power was high throughout the test, indicating that a desirable mean size had been reached. The selective grinding, capacities, and efficiencies were good. This all indicates that when nipping is best the mill (when not run too fast) will do its best work. This statement is not new; the evidence is given for those who wish to weigh it.

Tests with Different Mixtures of Balls in Conical, Cylindrical, and Grid Mills

A comprehensive examination of mills that segregate the ball sizes shows that they require mixtures containing a greater number of small balls than is supplied by the Davis ball load. This deficiency was met by using the rationed ball load, in which small balls predominated. Before going ahead, the mills will be considered.

Conical mills and cylindrical mills with grids were contemplated in introducing the new loads. Hence, these mills must be discussed before showing the tests, and they must be compared with the standard cylindrical mill.

Should the ball sizes be segregated, or should they be mixed as in the standard cylindrical mill? In the metallics industry the most effective method of segregating is to place the mills in series and use succeedingly smaller balls from first to last mill in the series. In the cement industry, dividers or grids are used to divide the long mills into sections, each of which has the appropriate size of medium. Finished material is removed at the end of each section.

With the knowledge that the cone of a conical mill functions like a grid in segregating the balls with respect to size, conical mills were built and tested. The first one was only 3 feet long. A taper of 2 inches to the foot was ample to segregate the largest balls in the big end and the smallest balls in the small end. Grinding tests in this mill with a rationed ball load were compared with the old cylindrical mill loaded with the old style ball load. A decided advantage was gained by the newer practice.

A larger conical mill was built and is shown in figure 6. It was 6 feet long and had the same taper as the smaller one. The big end was 2 feet in diameter and the small end 1 foot. The ability of the mill to segregate the balls was demonstrated by tests.

Grinding tests with several types of mills and ball loads led to the conclusion that advantages that had been gained were due more to the appropriate average size of balls than to the new design of mill. It was difficult to show that the conical mills had an outstanding advantage over the cylindrical mill. The 6-foot conical mill had a disadvantage; it induced the media to drift to the big end and pile up there so much that the balls passed through the feed entrance into the scoop. For a simple remedy a grid was placed on the feed opening to retain the load. A change was made to a cylindrical mill lined with a series of identical truncated cones. The idea was suggested by C. L. Carman, of Independence, Kans.

Although the efficiency of the long cone was good, the loss in capacity induced by the taper was marked. This may be shown by the following analysis: If the last unit section with diameter D2 = 1 foot could be speeded up to the same percent critical as the first unit section with diameter D1 = 2 feet, it would have a relatively low capacity

Its capacity is further reduced because its actual percent critical is below the above-mentioned assumption by the factor

1/√D1/1/√D2

Hence, the relative capacity of the last unit section as compared with the first is

That is the last unit section has only 12 percent of the capacity of the first.

A 2- by 3-foot cylindrical mill was lined to employ the conical effect, but instead of having one cone it had three identical truncated cones, end to end, and apexing in the same direction. (See fig. 7.) Any cylindrical mill may be lined in this fashion by using liners tapered in thickness. If the liners are 2 feet long, a 12-foot mill would have six truncated cones, end to end. The mill would have the same capacity at the discharge end as at the feed end. In some way, at least, this would be an advantage over the long cone. The mill with truncated cones proved to be as good a sizing device as the long single cone, but when compared with the old cylindrical mill its advantage as a grinder was not marked.

Finally, a cylindrical mill with a grid was used. The grid was 1 foot from the feed end of a 2- by 3-foot cylindrical mill. Balls of 2.5 inches to 1 inch were placed in the feed-end sections and 0.75-inch balls in the discharge end. The grinding was moderately better than without the grid. Possibly the grid would have appeared to have more advantage if the feed had been coarser and the finishing finer. The

tests will now be shown. They were all in continuous open-circuit grinding.

In table 30, grinding to a fine size was stressed to give the extra amount of small media in the new load a chance to work advantageously. Grinding was continuous and about 74 percent of the product passed through 200-mesh. The Davis ball load in the cylindrical mill was used first; next, the rationed ball load was used in the same mill; and finally, the rationed ball load was used in a mill having a lining of truncated cones. In selecting the Davis ball load the no. 1 load was used instead of no. 2 in accordance with the old idea that all of the balls should be of a size to crush any of the particles of ore. The free migration of the ore induced by the large, interstices would be compatible with a heavy circulating load. In the cylindrical mill the work of the rationed ball load was about 60 percent better than the Davis ball load, and when the mill which was lined with truncated cones was used there was a further gain of about 5 percent. The rationed ball load left more of the coarse sizes unfinished.

In table 31 the results of five tests with different ball loads in cylindrical, grid, and conical mills are shown. The feed was coarser than was used in table 30. The grinding in test 2 with the rationed ball load, which contained 64 percent of 0.75-inch balls, was about 44 percent more efficient than with the Davis load. The power was about 11 percent higher. If Davis ball load no. 2 instead of no. 1 had been used, the divergence in grinding results would have been reduced. In test 3, in which the grid was used to segregate the different sizes of balls, a further advantage of about 4 percent in efficiency is shown. The conical mill in test 4 increased the efficiency to 58 percent more than in test 1. The efficiency with the long (6-foot) conical mill was about the same as with the short (3-foot) one.

The new ball loads undoubtedly were superior to the old one, and a comparison of the results with the conical mills and cylindrical mill showed some advantage in favor of the conical mills.

The validity of having graded sizes of balls to grind the ore in steps with ample provisions for a circulating load and removal of fines in each step cannot be denied, but without this quick removal of finished material the advantage was not great.

In the conical mills or in the grid mill, as used in these tests, it was difficult to set a correct feed rate. If the feed were too fast some of the coarse particles would pass the zone intended to grind them. Having passed that zone, they were likely to continue without being ground. Again, if the feed were too slow, energy would be wasted by making the fine particles remain too long with the coarse medium. Nonselective grinding and inefficiency would result.

The new ball loads showed about the same superiority in the plain cylindrical mill as in the other mills.

Thus far the evidence of the efficacy of a rationed load in plant operation may be questioned because, as is shown by tables 30 and 31, the coarse sizes were not reduced as much as with the Davis load. Fear was entertained lest a circulating load might develop trouble- some characteristics. Hence, closed-circuit grinding was tried.

Closed-Circuit Grinding

Rationed ball sizes were of advantage in batch and open-circuit grinding. The degree depended on the particle size of feed and product. Examination will now be made to see if the deportment of rationed sizes is satisfactory in closed-circuit grinding.

The tests were performed as shown in tables 32 and 33. In the first table dolomite B was used, and in the second the feed was chert rejects from earlier grinding tests. The procedures in the two tables have one fundamental difference; in table 32 the feed to the rationed ball load was increased on account of the extra efficiency of the rationed sizes, whereas in table 33 the feed was maintained at the same rate but the mill speed was reduced. That is, in the first table the advantage is shown by the increased amount of ore ground, and in the second the advantage is shown by the power saved. If preference is given to one of the two methods it should apply to the latter, because in it the two ball loads being compared deal with the same amount of feed, and the drag is worked under almost identical conditions. The pulp consistencies of the drag overflows were maintained at 17 percent solids.

In table 32 the drag classifier finished at a finer size when rationed sizes of balls were used. This variation is on the right side for safe conclusions about the advantage of the new ball load. With the Davis ball load, 2.95 pounds per minute were finished, and with the rationed ball load the amount was increased to 4.06 pounds perminute— an increase of 37 percent. The surface tons per hour show, an increase of 45 percent in favor of the rationed ball sizes, and the surface tons per horsepower-hour show a more moderate advantage—37 percent. The reason the advantage in capacity was greater than in efficiency is because of the difference in power in the two tests; the smaller balls required more power than the larger ones. The surface calculations are made from the part of the table marked “section 3.” There a composite feed has been calculated, so that surface calculation can be based on feed and product. However, the ultimate values would have been the same if the sizing analyses of new feed and over- flow in sections 1 and 2 had been used.

It will be seen that the circulating loads in each table are about the same, respectively. Due precaution was taken to make sure that the circulating load was balanced, about 2 hours being required after the last adjustment.

The closed-circuit set-ups are shown in figures 8 and 9. They do not include the inclined belt and weightometer formerly used. A better plan was to permit the drag sand to fall into buckets and at set intervals to pass the sand back to the new-ore belt feeder after a hurried weighing. The record of the weights obtained after decanting

superfluous water indicated the trend of the circulating load, but a more accurate estimate was made at frequent intervals by catching the ball-mill discharge in a graduate and weighing it. The weight of solids minus new feed gave the circulating load with exactness. The test was continued for a goodly period after the amount of discharge became constant.

In the final study of the products close attention was given to sizing analyses, size by size, to see that the drag sand plus the overflow balanced the ball-mill discharge.

In the two tests shown in Table 33, the overflows are nearly identical. The innovation in the manner of conducting the tests, as stated before, was to keep the new feed constant and reduce the speed of the mill containing the new ball load until the circulating load in section 2 was the same as in section 1. When the new ball load was used, the speed was reduced from 70 to 55 percent critical and the capacity was maintained. The increase in efficiency was 28 percent. The Davis ball load took 22.6 percent more power than the rationed ball load.

The comparison of different sizes of media when the mill speeds are not the same might not have been justified by the old literature, but it is justified by table 13, which shows that for speeds from 40 to 70 percent critical, inclusive, the efficiencies were almost identical when the amount of ore in the mill was the same; of course, capacity increased with speed. It is readily seen from table 33 that the capacity with the rationed ball load at 55 percent speed was about the same as with the Davis load at 70 percent speed. If the finishing could have been at 200-mesh in all the closed-circuit tests, the load of large balls would have been greatly handicapped and the load of small balls would have had a greater relative advantage. Then the difference in efficiency might have been as much as 75 percent. The grinding seems to have been a little more selective with the larger media.

By table 2 the diameter of the ball of average weight in the rationed load no. 2 was 1 inch. A load of 1-inch balls would have given about the same results but would not have permitted the study of the effect of segregation in the grid and conical mills. Furthermore, the practical application would have been doubtful. A Davis ball load with sizes from 1½ to 1½ inch would have done good work, but it would not have been representative of the old standard because some of the balls would have been too small to crush the largest particles.

The quantities obtained in these tests enable the mill man to get a vision of the amount of power required to do his grinding. Take, for example, the tests represented by section 1 in table 33, in which grinding was to flotation size by what may be called the ordinary ball load and the ore feed was almost 100-percent Tri-State chert through 8-mesh. Calculations show that the net energy input was 21 horsepower-hours per ton. One-third should be added for friction and motor losses, which would bring the motor input up to 28 horsepower-hours per ton of ore. An ore would have to be rich to justify the expenditure of so much additional power for grinding.