ELSEVI!ER Int. .I. Miner. Process. 44-45 (1996) 373-382
Characterization of laboratory-scale tumbling mills
H.J. Steiner University of Mining and Metallurgy. Department of Mineral Processing, Leoben, Austria
According to a generally accepted principle, the data describing test conditions should be always sufficiently complete to enable the reader to repeat the experiment. However, it is not quite an exception if a technical report on or a scientific publication of results of grinding tests carried out in Iab-scale tumbling mills omits most relevant data (e.g.: total mass of the mill charge) in favour of less important information (e.g.: number and sizes of the grinding media) or of somewhat doubtful data (e.g.: volume fraction occupied by the mill charge in the grinding chamber:l.
Shortcomings of that kind may occur less frequently if the information on the conditions of the grinding tests can be checked for completeness by referring to relationships which comprise the main variables of the net power draw of the mill and of the energy transferred to the mineral load. In additi#Dn, relationships of that kind would offer the advantage of compressing the information on test conditions, and of making test results better comparable as well.
A basic relationship serving the said purpose is derived from simplified similarity considera- tions. A nominal coefficient of mill-internal friction is found suitable to normalize the results of net power measurements. The main free variables of grinding tests are combined in an index which is proportional to the specific energy consumption.
In general, lab-scale tumbling mills are characterized by a net power draw in the range of 10 W to 100 W. For example, the net power draw of the Bond-mill, recommended for the determination of the Bond-work-index, amounts to approximately 70 w.
Selected chronological data: The introduction of small tumbling mills into the mineral processing laboratories went along with the need of preparing samples for flotation tests. For this purpose, Richards and Locke (1925) and Taggart (1927) recommended a fi 203 mm X 254mm ball mill on a roller drive. The recommendation of using rods as more suitable grinding media in preparing the feed of flotation tests may be credited to
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374 H.J. Srriner/Int. J. Miner. Process. 44-45 (1996) 373-382
Holman (1928). The application of lab-scale tumbling mills for the purpose of determin- ing the resistance of mineral matter to comminution (testing for grindability) will be forever linked to the names Yancey et al. (1934), Maxson et al. (1939) and Bond (1952). The priority in sizing industrial ball mills by means of accurate measurements of the energy demand of a laboratory mill may be credited to Mittag (1954). Rose and Evans (1956) studied the power-draw of a small model-mill and established relationships on the basis of dimensional analysis. Yang et al. (1967) and Pietsch (1971) measured the power draw of lab-scale tumbling mills under various conditions of both dry and wet grinding.
2. Power draw of lab-scale tumbling mills
2.1. Power draw measurements
The ultimate goal of power draw measurements is the determination of the net power draw of the mill, i.e. the power demand of the tumbling mill charge. In order to obtain a reliable result, calibration procedures and corrections to the measurements are necessary. The difference between the power draw while tumbling the mill charge and the idling power while rotating the emptied mill does not correspond to the net power draw of the grinding chamber, since additional frictional losses are caused in the bearings by the weight of the mill charge. The appropriate procedure to determine the said additional losses are mill runs with a torque-free dummy-load accurately centered in the axis of the mill chamber. A Prony-brake or a cord-brake in the axis of the mill is quite indispensible for the purpose of calibrating the instrumentation.
Expressing the results of power-draw measurements as energy per one revolution of the mill provides the uniform basis to allow for the different kinds of corrections and eliminates the influence of small variations of the mill speed on the final result.
Torque measurements rely either on direct or on indirect effects: The first category comprises (a) the method of monitoring the counter-torque acting on the casing of the mill motor, and (b) the insertion of a planetary gear between the motor and the mill (dynamometer-method). The MERGAN-mill makes use of method (a>, while method (b) has been applied by Mittag (1954) and also by Rose and Evans (1956).
The indirect methods of torque measurements are based on monitoring the torsion of a shaft in between two couplings. The so-called torquemeters, at present representing the standard devices for measuring the torque-draw of lab-scale tumbling mills, are monitoring the torsion via its influence on the electric resistance of strain-gauges wound spirally around the shaft.
At a given torque, the angle of torsion is proportional to the ratio length of the shaft/cross section area squared. Thus, a long shaft of comparatively small diameter will produce an already optically detectable angle of torsion in the range of one to several degrees. The applicability of this principle has also been tested in the authors
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laboratory on a roller drive: The shaft, equipped with reflecting mirrors on both ends, was housed in the hollow drive roller. The torque-dependent time lag of two laser-beams reflected by the mirrors was electronically monitored.
A determination of the power draw of lab-scale tumbling mills via measurements of the electric energy input of the motor is characterized by a low expenditure on instrumentation but at the disadvantage of a rather low accuracy with comparison to the previously mentioned methods. The low accuracy is due to the low electro-mechanical efficiency of the drive system. A just modest degree of accuracy (e.g.: + 10%) at a somewhat better reproducibility can be expected if a high precision integrating meter with sufficient sensitivity (e.g.: < 500 J per one revolution of the meters rotor) is applied and calibrated on-line with the actual drive system via a Prony-brake on the axis of the tumbling mill.
3. Theoretical approach to the power draw of lab-scale tumbling mills
The objective of the following theoretical approach to the power draw of tumbling mills is not a strict analysis like the admirable one conducted by Rose and Evans (1956) but - on the basis of more simplified similarity considerations - the derivation of an equation of dimensional homogeneity and of simple structure, containing the most dominant, directly measureable variables of the power draw, and found suitable to normalize the results of measurements of the power draw of lab-scale tumbling mills.
The :starting point of the mathematical treatment is the well known theory developed by Barth (1930) and independently by Uggla (1930), describing the surface of the mill charge as an equiangular spiral resulting from the combined action of gravity, centrifu- gal force and internal friction (see Fig. 1).
It is not quite clear whom to credit the priority in equating the torque T drawn by the mill charge with the multiplication product weight F of the mill charge X horizontal displacement h of the center of gravity of the mill charge: Eq. 1. Rose and Sullivan (1957) attribute this conceptive and most valuable model to Carey and Stairmand (1952).
Eq. 2, stating proportionality between the internal mill diameter D and the horizontal displacement h of the center of gravity of the mill charge will be valid under the condition of both geometric and kinematic similarity (The said similarity criterions will be defined in the next chapter). Eq. 3 expresses a trivial fact: power P equals torque T times angular velocity o. The relation between the angular velocity w and mill speed n (= number of revolutions divided by time) is expressed by Eq. 4. Substituting the Eqs. 2 to 4 into Eq. 1 leads to Eq. 5, stating proportionality between the net power draw P and the term internal mill diameter D X weight F of the mill charge X mill speed n.
376 HJ. Steiner /ht. J. Miner. Process. 44-45 (1996) 373-382
Fig. 1. (I) Actual motion of a mill charge. (II) Equilibrium surface of the mill charge according to Barth (1930). (III) The ultimate stage of abstraction: the mill charge simplified to a circular segment.
T=Fh (1) haD (2) P=To (3) w=27rn (4) PaDFn (5)
It may be of interest to know that an identical result can be obtained by means of a simplified version of dimensional analysis without any reference to a special kind of physical model: The relation P a DnFBnr, comprising the unknown exponents (a, /3, y), is based on the empirical knowledge that the power draw is likely to depend on the free variables internal diameter D of the mill, weight F of the mill charge, mill speed n. The logarithmic version of the said relation together with the requisite of dimensional homogeneity provides a set of 3 linear equations, expressing the balance of the basic dimensions Mass [MI, Length [ ~1 and Time [T] which are coordinated to the power draw [ i14L2T- 1 and to the variables mill diameter [L], weight of the mill charge [ MLT-2] and mill speed [T-l]. The coefficients to the unknown quantities ((Y, p, 7) in the set of linear equations are listed in Table 1.
The solution of the said linear equation reads: CY = p = y = 1. This corresponds to P a DFn, as already stated in Eq. 5.
Table 1 Coefficients to the unknown quantities ((I, B, y) in the se4 of linear equations
Balance of coefftcients to: Vector element
a P Y
Mass 0 1 0 1 Length 1 1 0 2 Time 0 -2 -1 -3
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The weight F (dimension: force; SI-unit: N) of the mill charge can be expressed by the product mass M of the mill charge (determined by weighing the mill charge) X acceleration g of gravity: Eq. 6. The substitution of Eq. 6 in Eq. 5 leads to Eq. 7.
F=Mg (6) P a DMgn (7)
3.2.1. Power number C, All the variables in Eq. 7 are related to basic measurements. The coefficient of
proportionality, referring to the net power draw of the mill charge and defined by Eq. 8, represents the dimensionless power number C,.
C, = P/( DMgn)
3.2.2. Net energy input per one revolution of the mill The net power draw P divided by the mill speed n is equal to the net energy input E
divided by the number of mill revolutions U: Eq. 9. Substituting Eq. 9 in Eq. 8 yields the net energy input (E/U) per revolution of the mill, expressed as a function of the power number C,, of the internal mill diameter D, of the mass M of the mill charge and of the acceleration g of gravity: Eq. 10.
P/n = E/U (9) E/U = C, DMg (10)
4. Criterions of kinematic similarity
The power number C, derived under the assumption of both geometric and kinematic similarity may only be expected to remain fairly constant if the following parameters remain constant: Bulk volume fraction of the mill charge inside the mill chamber, frictional conditions within the mill charge and at the boundary between the charge and the shell, ratio of centrifugal acceleration to gravity acceleration at the perimeter of the mill chamber.
The ratio of centrifugal acceleration (Ro2> to gravity acceleration g at the perimeter of the mill chamber of radius R is expressed by the Froude-number according to Eq. 11. Substituting the trivial Eqs. (12, 13) in Eq. 11 yields Eq. 14 expressing the houde-num- ber as a function of mill speed n, internal diameter D of the mill and acceleration g of gravity.
Fr = Ro2/g (1) ,R = D/2 V-1 w=2Tn (13)
.Fr = 2r2n2D/g
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A more common term referring to the said ratio of acceleration is the so-called fraction (n/n,) of the critical mill speed. By agreement, the critical speed n, is coordinated to Fr = 1. Substituting Fr = 1, n = nc in Eq. 14 yields Eq. 15. Resubsti- tuting Eq. 15 in Eq. 14 yields Eq. 16, according to which the fraction (n/n,> of the critical speed equals the square root of the Froude-number. Thus, the recommendable Froude-number of l/2, i.e. centrifugal acceleration at the shell equalling l/2 of the acceleration due to gravity, corresponds to a fraction of approximately 0.707 = 70.7% of the so-called critical speed.
Fr= 1: n=nc:
n, = d~V(277-~D) (15) (n/n,) = 4% (16)
5. Range and examples of C,-values
The (net!) power numbers of lab-scale tumbling mills crowd around C, = 1. In general, they are distributed in the range C, = 0.8 to 1.3, provided the conditions with respect to the bulk volume fraction cp of the mill charge and the fraction (n/n,> of the critical speed are well chosen within recommendable limits: (cp = 0.2 to 0.4, n/n, = 0.6 to 0.8). The following examples of Cr.-values partly refer to measurements carried out by the author, partly to an analysis of published data:
Laboratory rod mill: fl 150 mm X 300 mm, 7.5 to 8 kg steel rods, (n/n,) = 0.65 to 0.75: c, = 1.11 (average result from 7 1 dry grinding tests).
Laboratory bull mill: fl 200 mm X 200 mm, 9.2 kg steel balls, (n/n,) = 0.7: CP = 1.10 (average result from 44 dry grinding tests).
The data published by Pietsch (1971) on the power draw of a fi 172 mm X 200 mm ball mill in dry grinding tests correspond to C, = 1.06 while the power draw of a fl 254 mm X 292 mm ball mill corresponded to C, = 1.09.
Pilot-scale tunbling mills: Mittag (1954) measured the power draw of a d 0.8 m X 0.5 m ball mill charged with 320 kg steel balls and rotating at about 75% critical speed. The power number coordinated to dry grinding amounted to about C, = 1.1 if a reasonable allowance is made for the mass of the powder charge (cement). The data published by Carey and Stairmand (1952) on the horizontal displacement of the mill charge correspond to C, = 0.95 to 1.4 in the range p = 0.3 $- (20%), n/n, = 0.7 f (10%) (dry grinding).
Industrial scale tumbling mills: The range of the power numbers of industrial scale tumbling mills may be derived from the various empirical power-draw-formulas pre- sented in the standard literature. For example, power numbers in the range C, = 1.1 to 0.95 result from an analysis of the equation of Rowland (1982) applied to overflow ball-mills of 2.5 to 5.5 m nominal diameter.
5.1. Description of un instructive experiment related to the power number C,
An experimental set-up of utmost simplicity is employed by the author in his lectures at Leoben to demonstrate the usefulness of similarity relationships exemplified by the
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Fig. 2. Mini model of a ball-mill powered by a cord-and-roll drive (instructive experiment)....