AG – Autogenous & SAG Semi-Autogenous Mill Design Calculations

AG – Autogenous & SAG Semi-Autogenous Mill Design Calculations

Mill Sizing: After laboratory and pilot plant testing confirm the feasibility of autogenous or semi-autogenous grinding, it can be used to establish the exact grinding circuit and mill size. In the pilot plant tests, the tare power of the pilot plant mills should be determined before and after each test run. The tare power should be for the empty mill. To give the same bearing pressures as a loaded mill, weights can be bolted to the shell in a pattern to give a balanced rotation. The tare power should be deducted from the total mill power drawn when grinding at full loads so that the net power can be established and net power per ton calculated.

Experience to date in using net power per ton data from pilot plant tests indicates that production mills will require close to the same net power per ton as the pilot plant mill. To date, the Bond Work Index has not been used to determine the power required for grinding in a autogenous mill because there are factors other than Work Index (grindability) which influence mill performance. There has not been any correlation between Work Indices calculated from operating data for autogenous mills with grindability test results. Operating Work Indices have varied from slightly less than to more than twice the Work Indices obtained from grindability tests.

The net power determined from pilot plant work is increased to cover mechanical losses in the mill trunnion bearings, in the gear and pinion and in the pinionshaft bearings to obtain the power per ton at the mill pinionshaft. From this the total power required is determined and the mill or mills that will draw the required power are selected. A typical calculation is given in the following example:

Example I.

  • Feed Rate: 240 Dry STPH
  • Feed Size: All passing 8″
  • Product Size: All passing 6 mesh with 80% passing 20 mesh.
  • Net power from pilot plant: 16.5 KWH/ST
  • Mill, gear and pinion friction multiplier: 1.025
  • Mill Power required at pinionshaft = (240 x 16.5 x 1.025) ÷ 0.746 = 5440 Hp
  • Speed reducer efficiency: 98%

5440 Hp ÷ 0.98 = 5550 HP (required minimum motor output power). Therefore select a mill to draw at least 5440 Hp at the pinionshaft.

Based upon the pilot plant test results, the volumetric loading for the mill can be determined. Normally, autogenous and semi-autogenous mills are selected for a 30% volumetric loading; however, with soft ore, and conglomerates it is not always possible to build up a 30% loading. This will show up in the operation of the pilot plant test mill.

Data collected to date shows that it is usually not possible to extrapolate power draw and capacity direct from small, pilot plant size mills to production size mills. Allis-Chalmers has a proprietary equation, adapted from the kilowatt hours per ton of balls (Kwb) equation for calculating ball mill power published by Bond, to determine the power which an autogenous mill or semi-autogenous mill will draw under specified conditions or ore specific gravity, pulp density in mill, mill speed and volumetric loading.

The ore in Example I has a specific gravity of 3.4. The computer data input sheet Figure 2 shows the set of data given to the computer to finalize the mill sizing. Figure 3a and 3b gives the results of the computer calculation. The mill power at the pinionshaft for a 30% volume charge is the sum of:


Figures 3a & 3b give the power for an autogenous mill. Figures 4a and 4b are for the same size mill with a ball charge of 6% of mill volume (290 lbs. per cubic foot).


In the above example the power was calculated for a 30% volume. However, with the same sheets the power can be determined for any volumetric loading from 15% to 35%. The computer program can also be used to calculate power for larger or smaller volumetric loadings of balls.

Table I shows a comparison of calculated power and measured power for some autogenous mills.

Mill power draft is a function of the diameter to the 2.3 exponent and is directly proportional to mill length. The early history of autogenous mills in the United States and Canada has featured mills of large diameter and short length. Primary autogenous and semi-autogenous mills in Sweden and South Africa have much larger length to diameter ratios than mills in the United States and Canada. At Boliden’s Aitik plant in Sweden, the primary mills are 20 feet in diameter by 34 feet long. Some of the mills purchased recently for installation in the U.S. and Canada are trending toward larger mill length to mill diameter ratios. Experience in the next few years, hopefully, will begin to answer some of the questions on the effect of mill length to diameter ratios in production sized mills.

Once the feasibility for either autogenous or semi-autogenous primary grinding and the total grinding circuit established, and required power and mill size determined, the mill design for the circuit selected can be established.

SAG Mill Charge

Defining the charge weight and height for fully autogenous grinding mill requires allowance for the ore specific gravity (and its variations) and accounting for the water content and the fine ore which will fill the interstitial spaces between the large ore pieces. This charge weight is usually calculated for the percent filling determined from the performance of the pilot plant tests and by the design of the feed end of the mill. Generally some user-specified safety factor is provided above the normal operating conditions such as actually designing for a 35% volume rather than 30%.

The pulp density in the mill is normally greater than that of the mill discharge due to an assumed differential flow rate of solids and water through the mill. Generally, a pulp density of 75% solids is used for the fines and water filling the interstitial space in the media portion of the ore charge.

For semi-autogenous grinding, the charge weight calculation and definition is more complicated because the charge has an additional component — the ball charge.

The net charge density changes at each incremental change in total charge volume. Consider for example, a 30% total charge of ore with a 3.4 specific gravity, a 6% ball charge at 290 lbs. per cubic foot, and 75% slurry solids in the mill. The following illustrates the calculation of the net charge density:

A. Balls, fines and water weigh 343.1 lbs. per cubic foot.
B. Rock, fines, and water weigh 181.43 lbs. per cubic foot.

Proportioning “A” & “B” to the 30% total charge (6% balls and 24% coarse ore):

(6 ÷ 30) x 343. 10 = 68.6

(24 ÷ 30) x 181.43 = 145.1

Totaling the two gives 213.7 lbs. per cubic foot net at 30% total charge. (See Figure 4a for 30% charge).

It can easily be seen that at 29% or any other charge the net charge density will change. This variation lends itself to computer calculation with the increments as finite as desired. Figures 4a and b show how the total charge density varies with a comparable ore at 6% ball charge and various total mill charges.

Figures 2 and 2a illustrate the computer input form used and give an explanation of the print-out used in Figures 3, 3a, 4 and 4a.

Experience to date has indicated that the ball charges used in semi-autogenous grinding have generally been most effective in the range of 6% to 10% of the mill volume including the void spaces between the balls, i.e. ball charge volume based upon 290 lbs. per cubic foot.

A mill greater than 26′ diameter and rated greater than approximately 4,000 horsepower will most likely have conical ends. The ore charge contained within the conical ends of course must also be considered in calculating the total charge weight. Figures 3a and b, and 4a and b show a typical print-out of mill cylinder and cone charge weight for an autogenous and semi-autogenous mill. The total charge weight is the sum of cylinder and conical charge weights.

The total charge weight, as calculated in this step of the design process, will have a direct input in the calculation of head and shell stresses, the selection of the trunnion bearing sizes and, of course, influences mill power draft. The importance of this step, although seemingly fundamental and simple, cannot be underestimated.

SAG Mill Horsepower — Drive Power

When the normal operating horsepower has been defined. It can be related to a design drive horsepower.

If the mill operating horsepower is defined as “net at the shell”, then the appropriate losses for trunnion bearings, pinionshaft bearings, and gearing must be accounted for to compute the power at the pinionshaft: the “power point” upon which both gear sets and speed reducers are rated. See Example I.

The losses in hydrodynamically lubricated trunnion bearings are generally no more than 1%. Even this is a very liberal amount. Modern designed, well lubricated, properly aligned ring gears and pinions can normally be assumed to be at least 98.5% efficient. Experience has shown that the losses for properly lubricated, anti-friction type, spherical roller type pinionshaft bearings can well be included within the 98.5% efficiency of the main gear and pinions. These result in the 1.025 multiplier used in Example I.

For fully autogenous grinding the mill drives are usually selected with a rating so that the drive is slightly oversized, to allow for variations in ore specific gravity and charge level. For example, if the ore is quite uniform and normal operation is expected to be 30% charge, a drive is selected at, perhaps, a 35% charge. Drive components—such as gearing, reducer, and couplings—should be rated with the motor horsepower capability, including its service factor.

For semi-autogenous grinding the drive is usually selected for both a higher total charge and a higher ball charge than normally expected. In order to be assured of fully utilizing the mill’s total capacity capability. When selecting the drive for a mill that is being designed for both autogenous and semi-autogenous grinding (keeping in mind power and capacity requirements). It is necessary to specify the exact conditions under which the drive is selected. The exact ratio of drive rating to mill power is often a function of operator’s preference. Therefore, we will not lay down any “hard” rules at this time.

The user of the mill or his engineering consultant will normally have to make a detailed power system study in order to determine the maximum motor size which can be started “across the line” at any one time. In addition, any inrush limitations must be determined since both of these factors will have a direct effect on the drive design.

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Ore Testing for AG & SAG Grinding Mill Design

The application of autogenous and semi-autogenous grinding circuits in recent years has contributed toward substantial savings in capital and operating costs compared to conventional circuits, particularly for large scale copper and molybdenum operations.

Basis For Evaluation

The decision as to how much grinding testwork is required is more complex for autogenous or semi-autogenous (collectively read ASAG for economy) grinding than for conventional grinding. Firstly, conventional crushing and grinding circuits can be designed confidently on the basis of small-scale batch or locked-cycle tests requiring only 35 kg sample per test. Secondly, ASAG circuits can be designed using various methods each of which carries an element of risk commensurate with the experience, of the engineers involved and the amount of sample tested.

Prior to testing for autogenous or semi-autogenous grinding, it is important to have an understanding of the specific energy requirements for crushing and grinding as well as mineralogy.

The Bond Work Index is the generally accepted parameter for gauging specific energy requirements in conventional crushing and grinding. The results are highly reliable if these tests are conducted on representative samples, usually of drill core, and follow the prescribed Bond method and test equipment.

Analysis of Bond Work Index test results for crushing, rod milling and ball milling will indicate uniformity or otherwise of breakage characteristics in different size ranges. Very often one is much higher (or lower) than the others and such variations can govern the power distribution between primary and secondary stages of grinding. Also, these observations will give an indication of the potential existence of a critical size relative to autogenous or semi-autogenous grinding.

It is considered that the determination of power requirement for an ASAG mill requires the same understanding of breakage characteristics and Bond Work Index developed for conventional grinding in testing particular ore samples. In addition, however, it is necessary to know how competent run-of-mine or primary crushed ore is in the coarser sizes and in what is generally termed the “critical size” range (75,000 to 13,000 microns) when operating in the presence of a reduced ball charge (up to 12% by volume) or when grinding autogenously.

The most critical parameter governing ESAG is the 80 percent passing size of the primary circuit product, PSAG. In using this formula, lower values of ESAG can be expected for coarser product sizings and these in turn depend upon ball loading and the production of natural fines. The usual relationships found in testing competent ores, i.e. those in which breakage is predominantly across grain boundaries.

With reference to the power distribution between primary and secondary grinding stages, higher power efficiencies can be obtained with a coarser primary product sent to ball milling. There are some exceptions, but generally industrial practice has been to utilize secondary ball milling capacity to the fullest and, if necessary, install more. Ball milling capacity can be justified if the incremental decrease in specific power consumption for the primary mill, for example between 500 and 1000 microns for primary mill circuit product.

It is important therefore that a metallurgical engineer’s judgement should be obtained in such circumstances, which is independent from any given by mill manufacturers, regarding the power required for a particular operation. The design of the test program, the nature of the samples tested, and the interpretation of the results should benefit from such independent expertise.

The principal factor governing the cost of a test program is the length of time required to achieve stable conditions under which the circuit can be sampled. Until recently, several days of operation with periodic checks of load level and successive sample campaigns were required to ensure confidence in results. The application of load cells to the support frame of the test mill has in most cases considerably reduced the time taken to achieve stability. Use of this tool will enable three or four test conditions to be investigated during a 24 hour period. Exceptions are usually very hard ores for which critical sizes have to be crushed and recycled..

For single-stage circuits in which the primary mill is operated autogenously, test parameters to be investigated are feed rates, specific power consumption and, if necessary, the effect of pebble crushing in varying quantities on feed rate and specific power consumption (kWh/tonne) to produce a final product siting. Where the primary mill is semi-autogenous, the principal variables are ball charge volume and ball size distribution.

For two-stage circuits, the test parameters which should be investigated in order to establish the most power-efficient design criteria for the primary mill and the overall circuit are:

  • Screen or trommel mesh opening on the primary mill since this can influence feed rate and specific power consumption. The coarser the product prepared for secondary ball milling the more efficient will become the overall circuit as discussed previously.
  • Circulating load of screen oversize which is returned to the primary mill
  • Ball charge volume with a given ball size distribution: at least three ball charge volumes should be tested (3, 6, 9% and possibly 12%) in order to prepare curves for product size vs specific power consumption
  • Pebble crushing since this can improve the power efficiencies of both autogenous and semi-autogenous mills through the extraction and crushing of a critical size which is limiting feed rate.

The Escalante Project (750 mtpd) was designed following two stages of study. Firstly, a comparison of capital and operating costs was conducted based on empirical scale-up from Bond Work Index data. Product for cyanidation was set at 80% passing 44 microns and various drill core sections were tested for grindabllity.

Current operational results show that the SAG mill is operating at 5% ball charge level by volume and is delivering a K80 800 micron product as predicted. Power drawn at the pinion is 448 kW, 13.617 kWh/tonne (SAG mill) and 570 kW, 17.325 kWh/tonne (ball mill) when processing 32.9 mtph, for a total of 30.942 kWh/tonne or 14.6% above the conventional estimate. A ball charge volume which is lower than those tested has been selected in order to minimize wear. Obviously, the SAG mill is consuming more and the ball mill less power than was predicted by the empirical method, but the overall total is 99.5% of the predicted total.


The Hemlo gold project of Teck-Corona (1000 mtpd) provides an example of mill sizing using the empirical calculation method alone. The situation facing Teck-Corona was their inability to obtain samples for batch or continuous testing. Underground access was not possible in sufficient time to provide bulk samples. A tight construction schedule dictated that engineering proceed on the basis of information obtainable only from drill core. This meant that Bond Work Indices had to be used for mill sizing which was reviewed with mill manufacturers and other consultants.

The SAG mill was sized with 746 kW to give a 25% contingency for power swings on the calculated power. The ball mill was sized at 1120 kW. Total installed power for comminution exceeded that required for conventional grinding by a factor of 1.163 (including secondary crushing).

The power split between primary and secondary grinding stages in a two-stage circuit is a function of PSAG for maximum power efficiency. Since ball milling involves a more efficient transmission of energy for comminution compared to autogenous or semi-autogenous grinding, the selection of PSAG for ball mill feed is made at the point where it begins to cost less to grind by ball milling than it does to produce a finer product by primary grinding.

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testing for autogenous and semi-autogenous grinding a designer's point of view