## how to size a ball mill -design calculator & formula

A) Total Apparent Volumetric Charge Filling including balls and excess slurry on top of the ball charge, plus the interstitial voids in between the balls expressed as a percentage of the net internal mill volume (inside liners).

B) Overflow Discharge Mills operating at low ball fillings slurry may accumulate on top of the ball charge; causing, the Total Charge Filling Level to be higher than the Ball Filling Level. Grate Discharge mills will not face this issue.

C) This value represents the Volumetric Fractional Filling of the Voids in between the balls by the retained slurry in the mill charge. As defined, this value should never exceed 100%, but in some cases particularly in Grate Discharge Mills it could be lower than 100%. Note that this interstitial slurry does not include the overfilling slurry derived from the difference between the Charge and Ball Filling.

D) Represents the so-called Dynamic Angle of Repose (or Lift Angle) adopted during steady operation by the top surface of the mill charge (the kidney) with respect to the horizontal. A reasonable default value for this angle is 32, but may be easily tuned to specific applications against any available actual power data.

The first step in mill design is to determine the power needed to produce the desired grind in the chosen ore. The most used equation, for this purpose, is the empirical Bond equation (Bond, 1960, 1961; Rowland and Kjos, 1978).

In this equation, E is the specific energy required for the grind, and F80 and P80 are the sizes in micrometers that 80% of the weight passes of the mill feed and product respectively. The parameter Wi, known as the work index of the ore, is obtained from batch bench tests first devised by Bond (1961). The power calculated on using equation 1, (Bond, 1961; Rowland and Kjos, 1978), relates to:

1) Rod milling a rod mill with a diameter of 2.44 meters, inside new liners, grinding wet in open circuit.
2) Ball milling a ball mill with a diameter of 2.44 meters, inside new liners, grinding wet in open circuit.

When the grinding conditions differ from these specified conditions, efficiency factors (Rowland and Kjos, 1978) have to be used in conjunction with equation 1. In general, therefore, the required mill power is calculated using the following equation

where n is the number of efficiency factors, EFi, used and fo is the feed rate of new ore to the mill. The power calculated from equation 2 can be looked up in published tables (Rowland and Kjos, 1978) and the correct mill size and type can be selected.

The philosophy in the development of the MRRC grinding simulation package was to build interactive software that could be used as an inexpensive means of providing a semi-quantitative check on a grinding mill design. In addition the software is designed to slot in to a general mineral processing package now undergoing development at the MRRC.

## calculate and select ball mill ball size for optimum grinding

In Grinding, selecting (calculate)the correct or optimum ball sizethat allows for the best and optimum/ideal or target grind size to be achieved by your ball mill is an important thing for a Mineral Processing Engineer AKA Metallurgist to do. Often, the ball used in ball mills is oversize just in case. Well, this safety factor can cost you much in recovery and/or mill liner wear and tear.

## how can i calculate new ball size and weight desing for ball mill - page 1 of 2

Dear All Expert,Our ball mill has got two compartments. But soon we will building roller press unit so, we will cancel first compartment. How can I calculate new ball desing (size and weight)? I know make calculation(formula) for max. ball size but I dont know formula for ball size and weight design. RegardsDeniz

Our ball mill has got two compartments. But soon we will building roller press unit so, we will cancel first compartment. How can I calculate new ball desing (size and weight)? I know make calculation(formula) for max. ball size but I dont know formula for ball size and weight design.

Dear Khan,According to my point of view:-1.Firstly look for the performance of roller press product then go for the mono chamber.2.Is the roller press with the separator or not?3.Mono chamber mill totally depend on the performance of the prigrinder.4.See the availability of roller press in comparision with the mill.3.If the performance of roller press will be too good the u can run the mill only @24-26% loading with a very good out put and @ low specific energy.4.Tonnage of grinding media u can calculate on the basis of dia.length and the filling degree of mill.Thanks Pankaj

Hi,We have a similar mill. Pregrinding with hammer crusher and mono-chamber mill. Thisis what a proposed based on literture review i did and others agree its more and lesscorrect. But remember it all depends on your mill feedsize after pregrinding.This is the raw mill not the finish mill. So Wi should be11 and density of 2.67 as normal kiln feed. I used d80 of 3500 micron to be on safe side and got max ball dia. of41 mm (use 50 mm steel balls).
Ball
[mm]
Percent [%]
80
0
70
0
60
0
50
13
40
24
30
34
25
29

We have a similar mill. Pregrinding with hammer crusher and mono-chamber mill. Thisis what a proposed based on literture review i did and others agree its more and lesscorrect. But remember it all depends on your mill feedsize after pregrinding.

This is the raw mill not the finish mill. So Wi should be11 and density of 2.67 as normal kiln feed. I used d80 of 3500 micron to be on safe side and got max ball dia. of41 mm (use 50 mm steel balls).

Hi,We have a similar mill. Pregrinding with hammer crusher and mono-chamber mill. Thisis what a proposed based on literture review i did and others agree its more and lesscorrect. But remember it all depends on your mill feedsize after pregrinding.This is the raw mill not the finish mill. So Wi should be11 and density of 2.67 as normal kiln feed. I used d80 of 3500 micron to be on safe side and got max ball dia. of41 mm (use 50 mm steel balls).
Ball
[mm]
Percent [%]
80
0
70
0
60
0
50
13
40
24
30
34
25
29

We have a similar mill. Pregrinding with hammer crusher and mono-chamber mill. Thisis what a proposed based on literture review i did and others agree its more and lesscorrect. But remember it all depends on your mill feedsize after pregrinding.

This is the raw mill not the finish mill. So Wi should be11 and density of 2.67 as normal kiln feed. I used d80 of 3500 micron to be on safe side and got max ball dia. of41 mm (use 50 mm steel balls).

## ball mill design/power calculation

The basic parameters used in ball mill design (power calculations), rod mill or anytumbling millsizing are; material to be ground, characteristics, Bond Work Index, bulk density, specific density, desired mill tonnage capacity DTPH, operating % solids or pulp density, feed size as F80 and maximum chunk size, productsize as P80 and maximum and finally the type of circuit open/closed you are designing for.

In extracting fromNordberg Process Machinery Reference ManualI will also provide 2 Ball Mill Sizing (Design) example done by-hand from tables and charts. Today, much of this mill designing is done by computers, power models and others. These are a good back-to-basics exercises for those wanting to understand what is behind or inside the machines.

W = power consumption expressed in kWh/short to (HPhr/short ton = 1.34 kWh/short ton)
Wi = work index, which is a factor relative to the kwh/short ton required to reduce a given material from theoretically infinite size to 80% passing 100 microns
P = size in microns of the screen opening which 80% of the product will pass
F = size in microns of the screen opening which 80% of the feed will pass

Open circuit grinding to a given surface area requires no more power than closed circuit grinding to the same surface area provided there is no objection to the natural top-size. If top-size must be limited in open circuit, power requirements rise drastically as allowable top-size is reduced and particle size distribution tends toward the finer sizes.

A wet grinding ball mill in closed circuit is to be fed 100 TPH of a material with a work index of 15 and a size distribution of 80% passing inch (6350 microns). The required product size distribution is to be 80% passing 100 mesh (149 microns). In order to determine the power requirement, the steps are as follows:

The ball mill motorpower requirement calculated above as 1400 HP is the power that must be applied at the mill drive in order to grind the tonnage of feed from one size distribution. The following shows how the size or select thematching mill required to draw this power is calculated from known tables the old fashion way.

The value of the angle a varies with the type of discharge, percent of critical speed, and grinding condition. In order to use the preceding equation, it is necessary to have considerable data on existing installations. Therefore, this approach has been simplified as follows:

A = factor for diameter inside shell lining
B = factor which includes effect of % loading and mill type
C = factor for speed of mill
L = length in feet of grinding chamber measured between head liners at shell- to-head junction

Many grinding mill manufacturers specify diameter inside the liners whereas othersare specified per inside shell diameter. (Subtract 6 to obtain diameter inside liners.) Likewise, a similar confusion surrounds the length of a mill. Therefore, when comparing the size of a mill between competitive manufacturers, one should be aware that mill manufacturers do not observe a size convention.

In Example No.1 it was determined that a 1400 HP wet grinding ball mill was required to grind 100 TPH of material with a Bond Work Index of 15 (guess what mineral type it is) from 80% passing inch to 80% passing 100 mesh in closed circuit. What is the size of an overflow discharge ball mill for this application?

## the grinding balls bulk weight in fully unloaded mill

In the previous article we considered the method for determining the bulk weigh of new grinding media. Determination the grinding balls bulk weigh directly operating in a ball mill becomes necessary on practice. It is done in order to accurately definition the grinding ball mass during measuring in a ball mill and exclude the mill overloading with grinding balls possibility

In this article, we will consider the technique for determining the grinding balls bulk weight in fully unloaded mill. This method used in the mills repair (armor plates replacement). The grinding balls unload from the mill into a special pit (needs to open hatches and pour the grinding balls from the drum during mill scroll). Then, need to definition maximum and minimum grinding balls diameter located in the mill. Unloaded grinding balls sorted by classes gradation by diameters. The gradation scale selected in steps of 10 mm. Sorting can be done manually (samples measured by a caliper in diameter and visually sorted by classes comparing with other balls size) or by the screen use.

The grinding balls bulk weight determined by using tabular data. The grinding balls bulk weight corresponds to the calculated average grinding balls diameter in the mill. Calculated (tabular) data of the steel grinding balls bulk weight present in the tables of the following specialized printed publications:

The calculated steel grinding balls bulk weight shown below in the table. Please note the calculated bulk weight may differ from the actual weight. This depends on several factors: the material of the grinding media, the range by geometric dimensions.

Thus, during calculating the grinding balls mass in ball mill (after measuring the mill filling degree with grinding media) needs to use the grinding balls bulk weight whose diameter was determined earlier. In this case, it may be differ from the grinding balls bulk weight loaded to the mill. The formula for calculating the grinding balls mass in ball mill is given below (we will consider the measurements process and calculations in more detail in our next articles).

The correct determination of the grinding balls bulk weight in mill allows accurately determination the mill balls feed weight. The mill balls feed weight is necessary for calculating the grinding media specific consumption and avoid mill overloading, thereby eliminating the motor load increasing possibility.

## grinding media wear rate calculation in ball mill

In the previous discussion the fact was established that the work done by a ball when it strikes at the end of its parabolic path is proportional to its weight and velocity; then, since the velocity may be considered as constant for all the balls in the mill, the work done by a ball is proportional to its weight. Since the amount of ore crushed varies as the work done upon it, it seems reasonable that the amount of steel worn from the balls varies as the work done upon them; in other words, the ball wear is proportional to the work done. But it has been shown that the work done is proportional to the weight of the ball; hence, the wear is proportional to the weight of the ball, or

in which R is the rate of wear of any ball of weight W; K is a constant and depends upon the operating conditions of the mill and the resistance of the material from which the balls are made. In order to establish the accuracy of this equation, some transformations are necessary.

After a ball-mill has been in continuous and steady operation for, say, a year, during which time balls of only one diameter have been added at regular intervals, it may be assumed that the ball charge has reached a constant working condition. In this constant state, there is no change in the charge from day to day, either as to the weight of the charge or the average diameter of the balls composing it. Any particular ball enters the mill at a maximum diameter D, and gradually wears down until it is completely worn away; the residual charge of balls, however, always remains the same.

Suppose these balls composing the residual charge to be laid out in a line, beginning with the largest and ending with the smallest. Suppose intervals to be marked off along this line so that all of the balls of diameter from Dm to D1 are in the first interval, from D1 to D2 in the second interval, D2 to D3 in the third interval, and so on. Let N1, N2, N3, etc., represent the number of balls in each interval, and let D1, D2, D3, etc., represent the mean diameters of the balls in their respective intervals.

In the constant state, the number of balls in each of these intervals does not vary, and when a new ball of diameter Dm is added to the first interval, a ball of diameter D1 passes into the second interval, and so on down the line, until in the last interval one ball is entirely worn away. Then, as time goes on, each ball of diameter Dm which was added gradually passes along the line until its diameter becomes D1 when it passes from the first interval into the second. If in time T, Nw balls of diameter Dm are added, then Nw balls will pass from each interval into the next following interval. The time T required for any ball to pass through any interval may then be expressed by the formula,

or the time required for any ball to reduce its diameter from Da to Db is equal to the total number of balls in the interval divided by the number of balls added to the intervals, or passing from the interval during this time.

If, as previously suggested, the wear varies as the weight of the ball, then R = KW = K/6SD, in which S is the weight of the material from which the balls are made and D is the diameter of the ball in question.

This formula shows the rate of wear at any instant. Since rate is always equal to the differential of space with respect to time, then R = dw/dt in which dw is the weight of material worn off in a very small interval of time dt.

By use of formula (31) it is possible to compute the actual screen analysis of the balls in the mill. If the balls are removed from the mill when they reach a certain minimum diameter, D0, formula (31) becomes,

in which Da and Db are the upper and lower limits of the diameter for any desired interval, Dm is the diameter of the balls charged to the mill, and Do is the diameter below which the balls are removed from the mill.

It should be remembered, however, that these formulas (31) and (34) hold true only provided that the balls wear down at a rate proportional to their weight. It would seem that if the percentage weight computed by formulas (31) and (34) agreed reasonably closely with the actual results secured by carefully screening a ball charge after the mill had been in operation a sufficient length of time for the charge to become steady, and if this agreement could be secured in a number of cases under different conditions, it would be convincing evidence that the ball wear varies with the weight of the ball.

In an endeavor to follow this plan, the attempt has been made to secure reliable data showing the screen analysis of ball charges which have been in steady use for a long period. It seems to be difficult to secure reliable information on this subject but the following results seem to indicate the truth of this law of ball wear.

Miami Copper Co.s plant after the mill had been operating for a year with a ball load of 14,800 lb. (6713 kg.) which was maintained by the addition of 400 lb. (181 kg.) of 2-in. (50.8-mm.) steel balls daily.

In this table, the actual per cent, weight obtained by weighing the balls is compared with the theoretical per cent, weight computed by use of formula (30). The two columns of figures are almost identical, thereby showing the accuracy of the formula and the truth of the law of ball wear.

At the Golden Cycle Mining and Reduction Co.s plant a dry crushing test was made on a 6 ft. 2 in. by 6-ft. (1.85 by 1.8-m.) Kominuter ball-mill. The mill was operated for 694 hr., during which time 4825 lb. (2188 kg.) of 5-in. (139.7-mm.) balls were added. The original ball load in the mill was 6614 lb. (3000 kg.) and the load at the end of the 694 hr. was 6338 lb. (2874.8 kg.). During this time, 590 lb. (267.6 kg.) of balls less than 3 in. (76.2 mm.) in diameter were discarded from the mill. The screen analysis of the ball charge at the end of the operation is shown in Table 20.

The slight irregularity in these results may be explained by the fact that at one time during the test, after about 400 hr., twenty-two 5-in. balls were added at one time. This may explain the irregularities at about 5-in. size.

From the two tests reported, the agreement between the actual per cent, weight and the computed per cent, weight is as close as could be expected. One was on a wet-crushing conical mill, and the other was on a dry-crushing cylindrical mill, and as far as the available data are concerned, the law of ball wear seems to be proved. It may develop however, when more data are collected, that the wear, instead of being proportional to the cube of the diameter, will be proportional to some slightly higher or lower power.

Nw = number of balls added in time T to compensate for the ball wear. Nt = total number of balls in the mill. Nt = total number of balls in any interval. N1, N2, N3, etc. = number of balls in the first, second, third, etc., intervals. D = diameter of any ball under consideration. Dm = diameter of balls added to compensate for wear. Da = mean diameter of balls in any interval. Da = diameter at beginning of interval. Db = diameter at end of interval. D1, D2, D3, etc. = diameter of balls at end of first, second, third, etc., intervals. Wt = total weight of balls in the mill, Wt = total weight of balls in any interval. w = weight of any ball. R = rate of ball wear. Rt = loss in weight of the mill charge in time T. It is equal to /6Dm SNw. T = time required for any ball to pass through any interval. S = weight of material from which balls are made.

The question finally arises as to how fine-crushing practice can be improved by the application of any of the principles that have been set forth. It is felt that the chief benefit to the mill operator will be derived from the fact that he may be better acquainted with exactly what is going on inside the ball-mill under various conditions. He should have a mental picture of the action of the charge and know better how to correct the difficulties encountered. He should also have a better idea how to proceed in order to produce any desired result.

While it is possible mathematically, as has been shown, to calculate the proper mill speed for any definite volume of charge, the size of the balls to be used must be determined experimentally. The size of balls is a most important factor in crushing, and each different condition requires balls of a different size. Whether the ore is hard or soft, coarse or fine, does not affect the proper mill speed or the volume of the charge; these depend almost entirely upon the size, and possibly to some small extent on the characteristics of the mill. But the proper size of the balls can be determined only by a careful study of the existing conditions. Experimental data must here be resorted to and the following method is recommended as a good means for determining the exact size of the balls that should be used.

Charge the mill with large balls, say of 5-in. (127 mm.) diameter. A smaller size might be better, but balls should be used that are known to be too large. An ammeter or wattmeter should be connected in the circuit of the driving motor so that the operator may keep the ball load constant by observing the power required by the mill. For maintaining this constant ball load, only balls of the same diameter as are already in the mill should be added. That is, if the test is started with all 5-in. balls, at the end of 24 hr. these balls will all be, say 4 in.; the ball load should then be restored to its original weight by adding only 4- in. balls. Thus, each time balls are added a different size must be used.

In this manner the mill will be filled with balls of approximately uniform diameter at all times. Then by keeping the records of each days run, it will be possible for the operator to determine just which size produced the best results under the conditions at hand. The ball charge should then be composed of balls as near this size as is practical and economical.

There are two methods of determining the proper size of the balls to be added at the end of each 24 hr. One method is to sample the ball charge and actually measure the balls. In some types of mills a sampler can be inserted through the discharge trunnion. If this is not possible the size of the balls can be computed, but samples should be obtained at certain intervals in order to check the computations. The method of computing the ball size comes directly from the ball-wear formula and is, as follows:

First determine the ball wear for the previous 24 hr. This may be done by obtaining a rough calibration of the power meter in the motor circuit so that any definite decrease in power indicates a given decrease in the weight of the charge. This is, then, the ball wear in pounds per day.

Then in ball-wear formula (25), T = 6.9/K Log10 Da/Db; but from (29), K = Rt/Wt. Then T = 6.9Wt/Rt Log10 Da/Db T is 1 day, Wt is the original weight of the ball charge, and Rt is the ball wear for one day. Then Log10 Da/Db = Rt/6.9Wt are all known, and it is only necessary to solve for Db, the diameter of the balls to be added. If only Rt lb. of these balls are added, then any error in computing Rt will not accumulate but will be corrected on the following day.

The above method for determining the proper size of the balls will, of course, require the careful attention of some one outside the ordinary mill crew. It also calls for a large assortment of balls of various diameters. As compensation for the trouble and expense necessary for the proper carrying out of this experiment, the operator stands a good chance of increasing the ball-mill capacity by a very considerable amount.

Once the proper size of balls is determined, the charge should be maintained so that it is composed of balls as little larger and as little smaller than the average diameter ball as is possible. Just how closely the proper ball charge can be maintained depends upon the facilities and economic conditions at the plant. The removal of the small balls, which is the main difficulty, is not so serious as it may seem at first; proper equipment makes this easy and inexpensive. It is to be hoped that the makers of ball-mills will succeed in producing a mill that will automatically discard balls of any desired diameter as rapidly as they are formed.

Another point which will bear investigation is the classifier capacity. As has been pointed out, large circulating loads seem essential for best efficiency. Classification is cheap compared with fine crushing and a classification capacity in excess of the capacity of the ball-mill is very much to be desired.

Acknowledgment is made for permission to publish data to the managers of this Mesabi Syndicate and allied interests under whose direction this work was done; to B. B. Gottsberger, General Manager of the Miami Copper Co., Miami, Ariz.; to A. L. Blomfield, General Manager of the Golden Cycle Mining and Reduction Co., Colorado Springs, Colo., who furnished valuable data; and to W. G. Swart, Fred A. Jordan and T. B. Counselman, at Duluth, who assisted in the experimental work and in the compilation of results.

## the bulk weight of grinding balls

In the grinding material process at ball mills becomes necessary to calculate the bulk weigh of used grinding media. The bulk weigh of the grinding balls is necessary to understand the grinding balls mass in the cube, the grinding balls mass loaded into the mill, the prevention of mill overloading with grinding balls, etc. These data often used in operational management of grinding process.

Consider a definition technique the bulk weight of new grinding balls. Methodological tables in specialized handbooks indicated the bulk weight of grinding balls different sizes. The indicators were calculated more than 20 years ago. Currently, these data can give an error more than 10%. This is a big error in production operational management. This error is caused by the fact that in the modern grinding balls production apply many materials (different grades of steel, white and gray cast iron) in contrast to past years. Besides, the actual diameter (size) and shape of grinding media affect the bulk weigh value.

Experts of Energosteel company have developed a common methodology together with technical specialists of Ukrainian and CIS countries mining & processing enterprises. This methodology allows to accurately determining a bulk weigh of grinding balls. Below, we give the main points that reveal the determining process the bulk weigh of grinding media (grinding balls).

To measure the bulk weigh of grinding media uses container with regular geometric shape (cube) and volume no less than 1.0 cubic meter. Pay your attention, calculations results will be more accurate if cube has large capacity. Cube capacity should not have flaws (waves, protrusions, holes, etc.). For the reliability of the results, measurements carry out with using the same capacity for all grinding media types. Measurements, calculations and commission inspection of the container carried out in the presence of responsible persons: representatives of grinding media producer and representatives of enterprise where grinding media used. Container fills by the same ways for all grinding media types.

Container fills by grinding media and lifted to a height of 10-12 cm from the concrete floor, then the crane operator disengages the lifting-lowering mechanism and container freely hits the floor. These ups-downs carry out at least 3 times. The capacity fills with grinding media to the upper edge level, after each shake. This procedure carried out at least 3 times with one grinding media type.

If its technically impossible to carry out the shake-up process (because of safety rules, lack of vibration surface, etc.), then grinding media loaded into container by a washer with 200-250 kg. The grinding media evenly distributed over the container into the hand way, after each unloaded washer. Grinding media should be evenly distributed in cube; surface of them should be smooth in the container.

The container is weighing, when it filled by grinding media as much as possible. Weighing can be done by any measuring instruments: crane scales, stationary scales, etc. It is important that weighing takes by the same measuring instruments and under the same conditions.

Based on the weighing results, needs to make an act (report) about the bulk weigh of each grinding media type. The act introduces the average value of bulk weight by 3 weighing results. The obtained result of bulk weight determination can be used in operational accounting, control of grinding process.