A contribution to ‘ball wear and ball size distributions in tumbling ball mills’

Powder Technology, 46 (1986) 281 - 285

281

A Contribution to 'Ball Wear and Ball Size Distributions in Tumbling Ball Mills' L. A. VERMEULEN Council for Mineral Technology, Private Bag X3015, Randburg 2125 (South Africa) (Received November 20, 1985)

SUMMARY R e c e n t work by Austin and Klimpel concerning the nature o f ball wear in rotary mills is discussed. The remarkable ambiguity that the results o f a marked ball test can be accurately described by either the volume or the surface theories o f ball wear is explained, and it is concluded that such tests cannot be used for determination o f the values o f the A u s t i n - K l i m p e l parameter A, unless certain conditions are met. This important parameter can be determined from studies o f the equilibrium ball-size distributions in the grinding charges o f ball mills, and additional motivations for its measurement are presented.

INTRODUCTIO N Austin and Klimpel [1] recently derived functions to describe the size distributions of balls in tumbling mills in terms of a parameter A, which they have introduced into the Bond formulation [2, 3] of the rate of ball wear in the form dm -- ~ = kr 2+~ dT

(1)

where m(T) is the ball mass after T tons of material have been milled, r is the ball radius, and k is a 'constant' whose dimensions depend on the value of A. Since T = tF, where t is time and /~ the average feed rate, it follows that eqn. (1) is essentially a timedependent equation. Indeed, in most milling operations, one usually endeavours to keep F at a constant value. If A is zero, then eqn. (1) corresponds to the 'surface' [4] theory of ball wear. This theory predicts that the frequency size distribution of grinding elements in a mill charge 0032-5910/86/$3.50

will exhibit equal numbers of elements in all the size intervals, provided t h a t equilibrium conditions obtain and only top-size balls are fed to the mill. The theory has its origin in the idea that, in ball milling, the size reduction of the particles occurs predominantly by three-body abrasive interactions and, since it is the surfaces of the balls that are exposed to abrasion, the rate of ball wear should be proportional to r 2 and A should be zero. Vigorous corrosion [5] could also contribute to decrease of the ball masses at a rate proportional to r:. If A is unity, then eqn. (1) corresponds to the 'volume' theory [6] of ball wear. This theory predicts a hyperbolic frequency size distribution and is based on the assumption that, in ball milling, particle size reduction occurs predominantly by three-body impacrive interactions. During these interactions, the particles are stressed and fracture if the stresses are sufficiently large. As the impulses that balls deliver during collisions are proportional to their masses or their volumes, the rate of ball wear should be proportional to r a and A should be unity in eqn. (1). There can be little d o u b t that both types of interaction are prevalent in large industrial mills. Indeed, smooth and well-worn grinding elements, together with broken ones, are often found in the grinding charges of most industrial ball mills, thus providing strong support for Bond's idea that both wear mechanisms are operative in industrial ball milling. Bond [2, 3] suggested that A would have a value between zero and unity, depending on the milling conditions; values close to zero would indicate predominantly abrasive interactions and values close to unity, predominantly impactive interactions. Austin and Klimpel [ 1] and other recent work [7] have shown that A can be deter© Elsevier Sequoia/Printed in The Netherlands

282 mined from studies of ball-size distributions, and the former authors [1] have provided strong motivations for doing so. An additional factor is that such determinations can also provide a basis for the selection of an appropriate ball type for a given milling situat i o n - - a factor of considerable economic significance. Hence, if A were close to zero for balls in a given milling situation, then a switch to balls of greater hardness could be advantageous; if A were close to unity, the use of balls made of tougher material could be advantageous. It may also be noted that the relative intensity of the impactive interactions in ball milling can be increased, e.g. by increase of the mill speed, the use of larger balls, or equipping of the mill with lifter bars. On the other hand, the intensity of abrasive interactions can be enhanced, e.g. by the use of smaller balls, decrease of the mill speed, or operation with a smooth lining, and so on. Clearly, the measured values of A will provide indications, not only of the inherent characteristics of the balls but also of the milling conditions. Indeed, it might be advantageous for certain materials to be comminuted predominantly by abrasive interactions and others predominantly by impactive interactions. Austin and Klimpel [1] have noted that data given by Davis [6] lead to A = 1, and that their own data and those of Lorenzetti [8] and Vermeulen and Howat [9] yield A = 0. They have c o m m e n t e d that the 'situation is clearly confused' and have questioned the assertion of Vermeulen et al. [9] that use can be made of either the volume theory or the surface theory of ball wear to give an accurate description of the rate of ball wear in a marked-ball test. The aim of the present communication is to given a quantitative explanation of the remarkable ambiguity that each of these two theories of ball wear appears to give a very accurate description of the rate of ball wear, despite the fact that a given theory ascribes all of the wear to only one of two possible, although very different, wear mechanisms, and to show that the accurate determination of A from the results of marked-ball tests is n o t possible unless certain conditions are met. AMBIGUITY IN THE INTERPRETATION OF THE RESULTS OF A MARKED-BALLTEST Equation (1) can be integrated to yield

m~(T) = m 0 expl--~- )

A= 1

(2a)

A=0

(2b)

and ms(T) = mo(1 -- bT) 3

Equation (2b) has also been given by Austin and Klimpel [ 1 ] in the form

rs(T) = r0 1 - -

(2c)

In eqns. (2a) to (2c), the subscripts v and s refer to the volume and surface theories of ball wear respectively, ~, b, and K are constants, m 0 and r 0 are respectively the original ball mass and ball radius, and r(T) and re(T) the ball radius and ball mass after T tons of material have been milled. In the derivation of these equations, a constant shape factor was assumed. This assumption is apparently unavoidable in discussions of ball wear and ball-size distributions in milling. Attempts have been made [7, 9] to take account of this factor, and to estimate the uncertainties introduced by its use. The theoretical dependence of the ball mass (or radius) on the a m o u n t of material milled (or the time of milling) as expressed by eqns. (2a) and 2(b) can be compared directly with measurements of the ball masses in marked-ball tests. Other workers, including Austin and Klimpel [ 1], have adopted the view that, if the data are best fitted by eqn. (2a), then the volume (A = 1) theory of ball wear is applicable and, if the best fit is provided by eqn. (2b), then the surface (A = 0) theory is applicable. In the marked-ball test to which reference is made [ 9], the masses of six grades of highchromium white-iron marked balls were monitored at m o n t h l y intervals over a period of about 6 months in an 8 ft X 8ft rubberlined industrial mill fitted with lifter bars. Merensky Reef ore was ground to the required degree of fineness at a milling rate of 650 t/d. The test was discontinued when the search for sufficient numbers of the markedballs to give reasonably accurate representations of the rate of ball wear became too time-consuming. The functions given in eqns. (2a) and (2b), which describe respectively exponential and cubic decays of the ball mass with the a m o u n t of material milled (or the time of milling),

283 ~t.

® Experimental data Prediction by use of volume theory

800

follows. According to the volume theory (eqn. 2(a)), the ball mass will vary as exp(--T]r), which can be expressed as

• %°•%.

exp (-~---rT) = [exp (~--T) 13

700-

m

= 1--

~600.

T

+ ~1

(5;

-- 1/6

(;;

+ higher-order terms I 3

500-

400 0

50

100

150 '

200 '

2~0

--

to first order

(3)

Mass of material milled, kt

Fig. 1. Comparison of theory with experimental results for grade 4 grinding balls. (After Vermeulen etal. [9].) were fitted to the data. These functions make use of measured data, i.e. ball masses, rather than derived data like ball diameters or ball radii, the use of which tends to suppress important experimental scatter when the cube roots of measured data are considered. Figure 1 (from [9])gives the experimental data for grade 4 balls and graphs o f the fitted functions. The non-linear least-squares fitting procedure that was employed yielded the following values for the parameters r and b and the X2 function: r = 363.1 k t ×v2 = 24.8 X 10 -4 b = 8•21 X 10 -7 t -1 Xs~ = 14.9 × 10 -4 Substitution of this value of b into eqn. (2c) yields K = 0•67 × 10 -6 m/h, which is in good agreement with the value determined by Austin and Klimpel [1] in their re-analysis of the same data. Those authors [1] also pointed out that, for grade 4 balls, since X~2 is smaller than × 2 , these results lead to z~ = 0. They apparently discounted the results for the other ball grades and our contention [9], as shown in Fig. 1, that the exponential and cubic decays give very accurate descriptions of the decrease of the ball mass as a function of the a m o u n t of material milled (or time of milling). This ambiguity can be quantitatively explained as

The last part of eqn. (3) is identical in form to eqn. (2b), which describes the variation of ball mass in terms of the surface theory. Therefore, it can be seen that, to first order, the exponential decay is identical to the cubic decay of the ball mass with the a m o u n t of material milled. The first-order identity is valid for T very much less than 3r. In practice, because of experimental scatter and errors, it will be difficult for one to distinguish between the exponential and cubic decays if the durations of marked-ball tests correspond to less than r tons milled, i.e. if reduction of the ball mass is by a factor of less than e, the base of natural logarithms. The above is a proper, quantitative explanation of the ambiguity that either of the eqns. (2a) and (2b), although they are based on very different wear mechanisms, can be used to give accurate descriptions of the decreases in the masses of balls in milling as a function of time or of the a m o u n t of material milled• The implication is that the value of z~ cannot be determined with reasonable accuracy from the results of a marked-ball test unless the duration o f the test corresponds to at least 2 v tons milled, (i.e. mass reduction of the balls by a factor of at least 7.3). Since the data pertaining to marked-ball tests published thus far do not meet this requirement, one must discount the major conclusion of the authors concerned, namely that their results confirmed that the 'surface theory of ball wear' is the 'true' theory. However, these considerations do not detract from the value of marked-ball tests. They are very useful for the determination, in comparatively short times (i.e. less than v/F), of the relative durability and cost-

284 TABLE Ball-wear parameters as determined by the results of a marked-ball test (After Vermeulen et al. [ 9 ]) Ball grade

Volume theory (V)

Surface theory (S)

7" (kt)

Xv2 (units of 10 -4)

b

Xs 2

I 2 3 4 5

128.1 123.1 127.4 363.1 220.5

2.84 × 10-3 3.317 x 10-4 3.30 x 10-4 2.479 x 10-3 8.275 × 10-4

2.276 x 10-6 2.35 × 10-6 2.314 x 10-6 8.261 x 10-7 1.4010 X 10-6

6

415.5

3.018 x 10-4

7.485 × 10-7

1.637 1.490 5.813 1.486 3.918 7.568

effectiveness of several ball types for given milling situations. Their value is much enhanced when balls representative of the background ball charge in a mill are also marked and monitored during the test, and if the monitoring of the masses of marked balls is carried out as frequently as possible. The results obtained for the other ball grades that were tested by Vermeulen e t al. [9] also have an important bearing upon the present discussion. The Table gives the values of r, b, and ×2 that were obtained for all the ball grades under test, and shows that the volume theory gives statistically the best description of the ball consumption for three of the ball grades, whereas the surface theory is best for the others. But the balls were together in the s a m e mill over the entire period of the test! That is to say, they simultaneously interacted with the same aggressive environment in which the milling rate was about 650 t of new feed per day. For some of them, A was closer to unity than to zero whereas, for the others, it was closer to zero. This result emphasizes that the value of A is n o t a property of ball milling p e r se. Its precise value is determined partly by the milling conditions and partly by the properties of the materials from which the balls are manufactured. When various types of ball are present in the same mill, they respond to the given milling environment, i.e. the given intensities of abrasive and impactive interactions that are provided by the milling conditions (mill speed, mill diameter, liner configuration, pulp density, charge volume, etc.), by wearing in different ways by virtue of their differing chemical and metallurgical properties. Thus, in the given test, the relative durabilities of the balls, as measured by the values of r that

Statistical best fit

× 10-3 × I0-a × 10-4 × 10-3 × 10-4 x 10-4

S V V S S V

are shown in Table, varied by a factor of more than 3. The above discussion suggests that the r-values should, because the milling conditions were the same for all the ball types, be related to the material properties of the balls under consideration. This hypothesis is confirmed, for example in Fig. 2, which shows

400-

1 to 6 = GradeSin given°fmillballs under test

6¢

Trend of ball durability as a function of chromium content

4¢ /

300.

/

E 200-

,

~

2

O

100-

i'0

®

I

20

3.,/

;0

Chromium content (%)

Fig. 2. Dependence of the tonnage parameter r on chromium content. that the r-values are u n d o u b t e d l y related to the chromium content of the material from which the balls were manufactured. Of course, one cannot deduce from this figure t h a t the higher the chromium content the more durable will be the ball, because other factors like microstructure, the chromium-tocarbon ratio, hardness, etc., play significant roles.

CONCLUSIONS

The above analysis has shown that the Austin-Klimpel parameter A cannot be determined unambiguously from the results of

285

marked ball tests, unless their durations correspond to at least 2r tons milled. The conclusions of Prentice [4], Norman and Loeb [10], and Norquist and Moeller [11], w h o claimed that their results support the surface theory of ball wear, b u t whose experiments did not meet this criterion, have therefore to be treated with some reservation. Although the determination of A from the results of marked-ball tests is difficult, such tests are very valuable in that they are comparatively inexpensive, much less timeconsuming than plant tests, and can be used for the simultaneous determination of the relative durability and cost-effectiveness of several ball types. The results of such tests are greatly enhanced when the masses o f balls that are representative of the background ball charge are also monitored [9]. The value of the parameter A can be determined from studies of the ball-size distributions in conjunction with the distribution functions that have been derived by Austin and Klimpel [1]. This is because the range of ball sizes in the equilibrium grinding charge of a mill will be sufficiently large. The greater the number of size intervals, the greater will be the accuracy and, if samples of the grinding charge are drawn from the mill, they must, of course, be r e p r e s e n t a t i v e - - a difficult matter for one to ensure. It is important for one to notice that the distribution functions given by Austin and Klimpel in eqns. (14) and (15) of their paper [1] apply only to steady-state conditions in which material is milled at a constant rate and new top-size balls are fed to the mill at a constant rate to compensate for depletion of the ball charge by ball wear. The time required for a ball mill fed with a single size of balls to achieve

steady-state conditions will be approximately 3M/F(AM/AT), where M is the total mass of the grinding charge, F is the feed rate, and AM/AT is the ball consumption (kilograms per ton milled). Austin and Klimpel [1] and the present work have provided motivations for measurement of the important parameter A, the value of which should be intimately related to the material properties of the balls and the grinding conditions in tumbling ball mills [7].

ACKNOWLEDGEMENTS

This paper is published by permission of the Council for Mineral Technology (Mintek). Professor D. D. H o w a t and Dr P. T. Wedepohl are thanked for discussions and criticism. REFERENCES 1 L. G. Austin and R. R. Klimpel, Powder Technol., 41 (1985) 279. 2 F. C. Bond, Trans. AIME, 153 (1943) 373. 3 F. C. Bond, J. Chem. MetaU. Min. Soc. S. Aft. (Jan. 1943) 131. 4 T. K. Prentice, J. Chem. Metall. Min. Soc. S. Aft. (Jan. 1943) 99. 5 P. Bernutat, Zement Kalk Gips, 9 (1964) 397. 6 E. W. Davis, Trans. AIM& 61 {1919) 250. 7 L. A. Vermeulen and D. D. Howat, J. S. Afr. Inst. Min. Metall., 86 (Apr. 1986) 113. 8 J. J. Lorenzetti, Proc. 3rd Symposium on Grinding, Armco, Chile, SAMI. Viva del Mar, 1980. 9 L. A. Vermeulen, D. D. Howat and C. L. M. Gough, J. S. Aft. Inst. Min. Metall., 83 (Aug. (1983) 189. 10 T. E. Norman and C. M. Loeb, Trans. AIME, 183 (1949) 330. 11 D. E. Norquist and J. E. Moeller, Trans. AIME, 187 (1950) 712.