the effect of ball size on mill performance

Powder Technology, 14 (1976) 71 - 79 @ Elsevier Sequoia Sk. Lausanne - Printed in the Netherlands

71

The Effect of Ball Size on Mill Performance

L G. AUSTIN

Deporlment of Materials Science. dlineral Processing Section, 7%~ Pennsylcnnia State Uniuersity. University Park. Pa. 16802 (U.S.A.)

K. SHOJI

h’isso Engineering Compnrsy Ltd.. Department of Technical Deuelopment. l-4-3, Einji-rho. Chiyodn-1:~. Toi:yo

(Japan)

P. T. LUCKIE

Kennedy Van Saun Corporation, Danviile. Pa. I7821 (bi_S_A.)

(Received September 10. 1975)

SUMMARY

The specific rates of breakage of particles in a tumbling ball mill are described by the equation Si = ax: Q(z), where Q(Z) is the probability function which ranges from 1 to 0 as particle size increases. This equation produces a maximum in S, and the particle size of the maximum is related to ball diameter by x, = k 1 d2_ The variation of a with ball diameter was found to be of the form (z = k2/d*-5_ Both kl and k2 vary with mill diameter, and simple power laws have been assumed, k, 0: Do-‘, k2 = Da6_ If it is also assumed that the mean overall values of Si for a mixture of balk is the weighted mean of Sj values for each ball size, equations are derived for calculating this mean value. As an example, the results are used in a mill simulation to show the quantitative effect of different balI mixes in a two-compartment cement mill versus a uniform mix over the whole mill.

INTRODUCTION

One of the most difficult questions to answer in the optimal design of ball milling circuits [l] is the choice of the mixture of bail sizes to be used in the mills. An empirical rule which has been used for many years [2] is

x=Kd2 (1)

where K = lo- 3 to (0.7) (10s3) for soft to hard materials; d is ball diameter in mm, x

particle sieve size in mm_ This is only a rough guide to the best size of balls to use to break a given size of particle_ In particular, of course, a spectrum of ball sizes is established in the mill [ 31 due to wear of balls and continual addition of larger ball sizes. To perform any kind of rigorous optimization of ball charge, it is necessary to be able to predict the behavior of different mixtures of balls. This paper gives a method of converting experimental data for specific rates of breakage of a material from one ball size to another, and one mill diameter to another, and also proposes a method of getting overall specific rates of breakage for a mixture of ball sizes.

ABNORMAL BREAKAGE

It has been found for many materials that the normal breakage of a given size fraction (for example, JZ or *size interval) of material in a batch ball mill can be described by a first-order law [4] t

“rate of breakage of size i” = SiWi W (2)

where Wi is the weight fraction of charge of size i, W is the total weight of charge, and Si is the specific rate of breakage. On the other hand, when a particle is too large with respect to the ball size, the result is as shown in Fig. 1, where w1 (0) is the starting fraction of the largest size present The reasons for this abnormal breakage have been discussed [ 53.

0 I 2 3 4 5

GRINDING TIME. MINUTES

Fig. 1. Experimental results of abnormal breakage of 8 x 10 mesh cement clinker in an S-in--dim. baii miI1.

This type of result can be treated as if it were the sum of a harder and softer componen:: 161, with o the fraction of harder component A, e.g..

dw (t) 0

dwl.a(t) +(1-o)

dwldt) -= dt dt dt

where

dw,(t) dt

= -%=Lwl_A(t)

dwdf) dt

= -%w1l3 (f)

The variation of specific rate of breakage with particle size for the two appsrent components is then as shown in Pig. 2. As a first approsimation, we can force-fit a single mean value of S to the curve given in Fig. 1. The method of back-calculating S values used by KeIsalI and Reid 171 in effect performs this force fitting since it assumes ah breakage is first-order_ We use a mean rate of breakage defined by the time needed to break 95% of the material in the size interval to smaller sizes. Snow [S] used the function Si/Sm = (xi/x,)= exp[-(xJx,)] to fit mean results such as those in Fig. 2. We did not find it possible to fit our resuhs with this function as its more generaI form

Si 5 QX~eXp[-(WXf)"] (4)

where Q. Q, w, n are descriptive parameters. Snow [4] also apparently assumed this fullc- tion to apply directly to a mixture of b&Is as

PARTICLE SIEVE SIZE.mm

Fig_ 2. Specific rate of breakage as a function of particle size, S-in. ball mills (afiintervals). Cement clinker Type II: l mean. A slower; Taut5 gold ore: m faster, l mean, A slower.

if the mixture behaved as a single size of ball. We did not find this approach to be valid for our data.

Gupta and Kapur 191 have suggested a met;., d for converting Si values at one bah diameter to values for another ball diameter,

using the data of KeIsaII et al. [lo] . The method we give below was developed before their paper appeared and is, in our opinion, somewhat more convenient for our purposes.

EFFECT OF BALL DIAMETER

Results of many tests suggest that the form of the S function for normal breakage in bail mi&is

Si=MF (5)

It is consequently very convenient to have a more general fitting function which reduces to this for lower values of x, to the Ieft of the maximum shown in Fig. 2. For simplicity, as few additional parameters as possible should be used in the more general function.

We have used the function

Si = &Q(Z) (6)

where z = log(x,/p)/Iog u and Q(z) is the Gaussian probability function

Q(z) = (l/G) f e*E’*r’ d(E/fl) z

'73

_.. 0 01 0.5 09

r = s,foIp

Fig. 3. Log-probability plot of y us. particle size for different ball diameters_

which can be readily calculated from well- known polynomial approximations.

From experimental values of Si and ax:, the ratio Si/axs can be determined:

(Si/aXy) = Q log xi - log p

log (3

Thus a plot of Xi uersus Si/aXf on log- probability paper should give a straight line from which p and u can be determined_ Figure 3 shows data of Kelsall et al_ [lo] for E, % and 1 in. diam. steel balls, wet grinding, piotted in this way_ The results can be fitted by a straight line over much of the range; a was found to be about 2.0, and from a log- log plot Of Si versus Xi for small Xi, CY WaS found to be 1.33. Figure 2 shows the applicability of eqn. (6) to our own results.

Accepting this function as a first approximation to the shape of the mean S function, bearing in mind that the use of mean S is in itself an approximation, ~i/dx = Q(z)ccYx~-~ - (ax Q-1/fi2.3 log (T) exp(--z2/2), hence for the maximum in S,

2_3a& log u = exp(-z%/2)lQ(X,)

where z, = log(x,/p)/log a; and x, is the particle size for maximum S for the given baii diameter. Figure 4 shows a plot of the function exp(-9/2)/Q(z) WEUS z + 2.5, which gives over the range of interest:

=na = 1.75(2.3ofi log u)~~* - 2.5

or (7)

log(X,/P) = log u [4.352(a log a)452 - 2.51

Fig. -I. Plot of esp(-z*/!?)Q(z) VS. z - 2.5.

DIAMETER OF BALL d. mm

Fig. 5. Plots of a and x, vs. ball diameter for the data of Ketsall ef aI_

PARTICLE SIEVE SIZE. “m

Fig. 6. Comparison of the predictions of eqn. (10) with experimental resmelts, for data of Kelsall et al. (a intervals)_

Knowing a and a, then, x,/cc can be determined. For the data of KeisalI et al_, x,/p = l-15_

Figure 5 shows the variation of x, with ball

7-I

TABLE 1

Values of descriptive constants for data of Fig. I?. Units of x. d. D in mm, S in min-’

Material

Goid ore

harder mean softer

Cement clinker II

harder meifn

Qi.KWtr (Kelsall EL nf. )

mf=an

Cement clinker I

mean

3 (h)(lO ) k;,(lo3) a Q kz Mill

diameter (mm)

“-50 0.86 1.79 0.65 20 206 2.75 0.95 1.99 3.56 1.23 2.36.

1.69 0.59 1.66 0.86 157 195 1.91 0.68 1.73

l-25 O-40 1x-l 1.33 570 303

2.14 O-75 ND. 0.98 104 196

*It1 = 0.2

diameter, again using the data from Kelsall et al. The results are of the form

s, = fi,d’ (8)

where for this data, kL = 1.25 X 10-‘3, mm-l. Kelsall et a[_ did not deduce this result, presumably because of the considerable scatter of the esperimental data (see Fig. 5) However, Snow [S] reported the result based on their data and also pointed out that this agrees with the rule given in eqn. (I), since the position of the maximum in the specific race of breakage corresponds to the particle size, which is broken best by that particular ball diameter_ Fi,we 5 also shows the variation of a with d:

a = k, jdl-= (9)

For units of S of mm-‘, and x, d in mm, kz = 570 for the Kelsall data. Then eqn. (6) becomes

S,(d) = (Wd=*)x?Q(xi, P, a)

where

D = k, d2/antilog(log u [4_352(Q log u)Q5* -

- 2-51) W)

Figure 6 shows the fit of this expression to Kelsali’s data; it describes the values of S to the left of the maximum quite accurately,

and the position of maximum S, but the value of the maximum fits well only for the smallest and largest sizes. Considering the inherent variability of wet ball mill tests, and the correction procedures necessary in the Kelsall-Reid technique 173 to get the values of S, the fit is considered as satisfactory. Other results are shown in Fig. 2, for cement clinker and gold-bearing ore [S] ground with 1-in_-diam_ balls_

From very limited results from an S-in.- diam. mill compared to one of 2 ft. diam. WI 9

or

k, = k,,(D/I)“’ (12)

where klo is the value of k, at unit mill diameter_ A trial and error fit of results from a 13-ft.-diam. cement mill [ 121 suggests that the exponent is 0-L Thus eqn_ (11) goes to

zr = k,,(D/l)RI d2/antilog{log oC4.352

(a log u)Q=- 2.51) (13) where nl = 0.1 to 0.2; further experimental work will be needed to decide on the correct value for n1 _

Table 1 summarizes the data values for k,, kIO and u, determined by leas~squares regres- sion analysis.

75

0 wnSEEBACH (13) 50% BALLS a AUSTIN et of (141 20% BALLS

- UC/O 159~[2:2(0.159/1,1~ 35-q

1 I I 0.1 0.2 0.3 0.4

f,.FRACTfON MILL VOLUME FILLED BY POWDER

Fig_ 7. Variation of absolute rates of breakage wirh powder filling.

MIXTURE OF BALL SIZES

Assuming independent be_havior of the balls, the mean value of S,, Si say, is

(14)

We will assume this to be true, that is, one set of biOi values only will be used. We will further assume, on the basis of rather limited experimental information, that bi.1 values do not change with mill conditions or mill diameter in the region of usual operating conditions_

where mk is the weight fraction of balls of EFFECT OF MILL DLXMETER AND OTHER

size k, or alternatively, CONDITIONS ON k,

j&Z dj”” Si(d)dm(d)

&xix,

Thus, from eqns. (lo), (11) and (12) the overall values of Si can be calculated for a given mix of balls, knowing kl. k2, CK and u.

Let t)i.i_k be the fractional breakage into size i from breakage of size j by size k balls_ The total specific rate of breakage into size i is

It is reasonably well established [ 111 that k2 will change with miII diameter according to

g bi.i.&i.kmk k=l

F .z 0~ Do-6 (16)

The variation of absolute rate of breakage in the normal breakage region with powder filling for normal values of J is shown in Fig. 7 [13,14] _ These data for steel balls can be approximately described by

s = (f.S)X (l/0.159) [2( 0_159/f,)Q”” - l] )

The total specific rate of breakage from sizej is

O-05 < f, < 0.3 (17) where f, is the fractional volume filling by powder and (f,S)* is the maximum value of Sf,, in the almost horizontal region of the plot. Assuming this relation can be used for other values of J, 2 Si.kmk

1

Hence, the mean &j,j is given by

bi,j = 5 bi.j.kSi.k Wz/ S Sj.kmk k=l k=l

(15)

4

]

00 0.1 0.2 03 04 05 06

J. FRACTION OF MILL VOLUME FILLED BY BALL BED

Fig. S. Proportional variation of powder with mill filling, based on Rose and Sullivan formula [ 15 ] _

If biej k is not a function of k, this reduces to

k2 = (f&T)* [2(0.1-59/f,)“-55 - l] (17a)

In addition, it can be reasoned that the maximum absolute rate of breakage (Sf,)* varies with ball filling J in a manner proportional to the variation of power input to the mill. The power input to a mill, based on the

i6

z I

3 L . 0 EXPERIMENTAL_

3 I - StMULAlEO i . I ,.I,,, t t I.,,.

100 1000

SIEVE SIZE,pm

Fig. 9. Rosin-Rammler plot of circuit simulation vcrsws experimental results; assuming simple power function for S.

data of Rose and Sullivan [ 153 , varies with J according to

* (J) = 3.0453 f 4.55 J’ - 20.4J3 + Y_2.9J4 (16)

where 0 (J) is proportional to mill power, or

(f,S)* 0 F(J) = J-i- 1.5 J” - 6.75s -I- 4.253’ (16a)

Figure 8 shows the proportionality between this absolute rate of breakage and J. which applies for filling conditions which give breakage in the horizontal region of Fig_ 7. The optimum value of J for normal breakage lies at about J = O-4_

Combining eqns. (17a) and (Ma) gives

kz 0 [ 2(0.159/f,)“-5” - l]

(J + 1.5J’- 6.7 J3 + 4.25 J’) (19)

Then, for a fixed value off,, k2 is proportional to F(J), as shown in Fig. 8_ Thus for low values of J the specific rate of breakage increases almost linearly with the number of balls, other factors held constant. This is in reasonable agreement with experimental results 1143 _ Equation (19) wiII probably be modified for different mills as more information on the variation of S with f, and J becomes available. Note that for filling conditions to just fill the ball interstices, that is, f, = 0_4J, the value of k2 does not vary widely with J due to compensating factors in eqn. (191.

Rates of breakage do not vary much with fraction of critical speed provided that the mill is run close to the optimum value for maximum power, which lies between 65 and

TABLE 2

Weighting factors for ball charge in a large tube mill; rnk = weight fraction

Ball size in_ mm

31.4 69 3 76 2% 63.5 2 51 1% 38 1% 31.6 1 25.4 718 22.3

First Second compartment compartment

mk mk

0.30 0.37 0.23 0.10

overall

WC

O-19 0.30 0.26 0.25

0.10 0.12 0.07 0.03 0.13 O-20 0.18 0.17

85% of critical speed depending on the size of the mil!, ball and powder filling in the mill, and the powder size.

APPLICATIONS

In a recent paper 1161 it has been shown how a mathematical model can be constructed for a large-scale cement clinker grinding circuit, using data for a lSft.-diam. by 34ft.- long, two-compartment tube mill. In the construction of the model, laboratory data on S values from an S-in. ball mill loaded with 1-in.-diam. balls was extrapolated to larger particle sizes using the simple power furl&ion of eqn. (5), Si = ax:. It was assumed tnat a single conversion factor existed between specific rates of breakage in the small mill and the overall values (mean for all ball sizes) in the big mill:

(large mill Si) = (k) (small mill Si) (26)

On this basis, the best match between experimental and computed m_fil size distributions was obtained with a value of k = 2.75, treating the mill as one continuous mill with a residence time distribution corresponding to ten equal fully-mixed reactors in series. Applying these estimates of Si to the complete mill circuit gave the size distributions shown in Fig. 9.

Equations (lo), (13), (14), (15) and (16) enable us to make better estimates of the proportional specific rates of breakage in the large mill by a more valid extrapolation to larger particle sizes, by aIIowi.ug for the mix of different ball sixes, and by scaling for mill

77

10.0 -

0.1 102 103 IO4

SIEVE SiZE.prn

Fig_ lo_ Estimation of overall values of specific rates of breakage (*intervals) in the production mill, scaled from Fig. 3 by Da6 (plotted against upper size of interval).

SIEVE SIZE. v

Fig_ 11. Rosin-RammIer plot of circuit simulation results us_ experimental results: function with maximum in S values, two-compartment mill.

diameter. The weighting factors ml are shown in Table 2, where the final column gives the values to be applied if all the balls were mixed in a single compartment. The values of the parameters determined for Bond filling conditions in the S-in. test mill for Type I cement clinker, with I-in.-diam. balls, were Q = 0.98, k,, = (0.75) (l(r3), k2 = 104. The value of J was assumed to be that shown for cement clinker in Fig_ 2, Le. 1.73. Thus applying -ns. (lo), (13), (14) and (16) gives a set of Si values as shown in Fig. 10. Using this set of data in the simulation of the mill [lS, 171 gave best agreement between computed and

experimental mill product size distributions for a conversion factor of 0.76.

Note that the conversion factor is now much closer to unity, although the ball- powder loading conditions in the small test mill do not match those in the large mill, so even if the scaling laws of eqns. (lo), (13) and (14) were exact, k would not be exactly 1.

The model was immediately taken a step closer to reality by considering the two- compartment tube mill to be equivalent to two mills in series. Since it is likely [17] that the residence time distributions of each compartment is roughly proportional to compartment length, the tube mill was considered to be equivalent to a mill of three equal reactors in series containing the first set of balls, followed by a mill of seven equal reactors in series with the second set of balls. The two sets of mean Si values are shown in Fig. 10. Again, the value of scale-up factor k was adjusted to give the best agreement between computed and experimental results, and a conversion factor of k = 0.71 was obtained.

As before, the converted values of S were used in a complete circuit simulation 1121 and gave the size distributions shown in Fig. 11. Comparing Fig_ 11 with Fig_ 9, it is seen that the influence of the maximum in S values is to allow larger sizes to pzsist longer than if

no maximum esisted, so that the match between the computed and the experimental size distributions of the mill product and recycle is considerably better for the upper sizes. In addition, the values of k for the single- compartment t’ersus the two-compartment will show directly the quantitative advantage of using a mix of larger ball diameters in the first compartment and smaller in the second instead of a complete mix through the whole mill. The reasoning is as follows:

The real mill is more efficient than the first simulation using a mix of balls because it contacts the larger feed particles with a higher proportion of larger balls_ Thus, to force the false simulation to fit the real result, an artificially higher conversion factor (of 0.76) is required. On the other hand, the second, real, simulation gives a truer estimate of O-71; the comparative difference in efficiency is approximately 0.05;0.71, or 7%. Put another way, if the S, values were corrected using the true 0.71 factor and applied to a single- compartment mill with a miu of all balls, the flow rate through the mill (and hence circuit output rate) would have to be decreased to get the desired product size distribution (by about 7%), or if the same flow rate were used the product size distribution would be coarser.

Equations are given which enable the conversion of specific rates of breakage measured for one set of ball milling conditions to values for other conditions, in particular the conversion from one ball diameter to another or a misture of balls. The position of the maximum in the specific-rate-of-breakage versus particle size plot is related to ball diameter d by x, = h, d”. This can be compared to the well-known criterion for ball size selection, that is, x = Kd’ where d is the “best” ball size to break particles of size x_ Clearly, this old criterion is crude, since the particles do not stay at size x as they move through the mill. We have demonstrated that using the developed relations combined with a mill model for a large cement mill enables us to predict that a mix of larger balls in the first compartment and smaller in the second compartment is 7% more efficient than the

same amounts of balls in a single mix throughout the mill.

For future work, the equations we have developed here will be used to establish the optimum mixture of balls to go from a given feed to a desired product, and we are currently formulating and programming this problem. This will provide a considerably more sophisticated criterion for ball selection. We will then be able to answer such questions as whether the ball mixes used in the cement mill were optimum, what would be the effect of classifying liners which classify the balls by size along the mill length, etc_

It should be noted that the equations need further esperimental verification, since they are based on quite limited data_

ACKNOWLEDGEiUEPil-S

We gratefully acknowledge the financial assistance of the National Science Foundation, Grant No. GK41901_

LIST OF SYMBOLS

a

A hi

bi.i.k

J

k

constant defined by eqns. (4), (5) or (6), min- 1 mm-- subscript to denote harder material weighted mean value of bi.i.k for the mixture of balls in the mill the primary breakage function; the fraction of sizej material which after primary breakage by size k balls appears in size i subscript to denote softer material ball diameter, mm inntemal mill diameter, e.g. mm fraction of mill volume filled by powder mavimum value of f,S; at f, = 0.16, min- r a number denoting the particle size interval being considered, e.g. 1 for the top size interval, 2 for the next smaller, etc. fraction of mill volume filled with static ball bed, based on a formal ball bed porosity of 0.4 subscript denoting ball size range, or, in eqn. (20). a scale conversion factor defined by eqn. (20)

k-1 k2

k K1° mk n Q(z) Si

i3f

Si.k

t

wiA tf)

wi(rl

W _x

*In

G,(J)

log u

constant defined by eqn. (S), mm-’ constant defined by eqn. (9), k2 = SdL5/x” = mmL5/(mm)Q mm value of kl for D = 1 the constant in eqn. (l), mm-’ weight fraction of balls of size k a constant Gaussian probability function specific rate of breakage of material in size interval i, min- r the weighted mean value of Si for the mixture of ball sizes in the mill, min- l specific rate of breakage of size i material being broken with size k balls, min-’ grind time, min weight fraction of material of type A which is of size i, at time t

weight fraction of total mill charge which is of size i, at time t total mill charge, g particle (sieve) size; for a size interval the upper size is used to represent the particle size, mm particle size for which S is a maximum for a given ball and mill diameter, mm dimensionless p ammeter log(x,/p)/ log u value of 2 for _x = x, a constant value of particle size at which Q(z) is 0.5, mm a constant, length- 1 apparent weight fraction of ith size material at t = 0 which is component A

a function of J defined by eqn. (18), proportional to mill power standard deviation of the probability function

REFERENCES

1 P_ T_ Luckie and L. G. Austin, Review introduction to the solution of the grinding equations by digital

2

3

4

5

6

7

S

9

10

11

13

13

1-I

15

16

17

79

computation, Miner. Sci. Eng., i (1972) 2-2_ W. H. Coghill and F. D. Devaney, Ball mill grinding, U. S. Bureau of Mines, T_ P. 581 (1937), p_ 47. F_ C. Bond, Grinding ball size selection. Min. Eng., 10 (1958) 592. L. G. Austin, Review introduction to the descrip- tion of grinding as a rate process, Powder Technol., 5 (1971172) 1. L_ G. Austin, K. Shoji and M. D. Ever-all, Explanation cf abnormal breakage of large particle sizes in laboratory mills, Powder Technol., 7 (lSi3) 3. W. J. Taut&, P. Meyer and L. G. Austin. Compar- ison of breakage parameters in two tumbling ball mills of different diameters, Proc. Symp. Automatic Control in Mining. Mineral and hIeta Processing, IFAC, Sydney, Australia, Aug. 1973, p- 21. D. J. Kelcall and K. J. Reid, The derivation of a mathematical model for breakage in a small continuous wet ball mill. Proc. AIChE/Chem. E. Joint hleeting. London. 1965, Sect. -l, p_ 11. R. Snow, Grinding mill simulation and scale-up of ball mills, Proc. 1st Int. Conf. Particle Technol.. IITRI, Chicago, Aug. 19’73, p_ 28. V. K. Gupta and P. C. Kapur, Empirical correlations for the effects of particulate mass and ball size on the selection parameters in the discretized batch grinding equation, Powder Technol.. 3 (19’71) 217. D. F. Kelsall, K_ J. Reid and C. J. Restarick, Powder Technol., 1 (1968) 291. L. G. Austin, Understanding ball mill sizing, Ind. Eng. Chem. Process Des. Dev., 12 (1973) 121. I, G. Austin, P_ T_ Luckie and H. M_ van Seebach. Optimisation of a cement milling circuit with respect to particle size distribution and strength development, by simulation models, 4th Eur. Symp. on Size Reduction, Nuremburg, Sept. 1955. H. M. van Seebach. Effect of vapours of organic liquids in the comminution of cement clinker in tube mills, Research Institute of the Cement Industry, Beton-Verlag GmbH, Dusseldorf, West Germany, 1969_ L. G. Austin, k N. Beattie and R_ R. Klimpel, Proc. 2nd Eur. Symp. on Size Reduction, Amsterdam, Dechema Monogr., 57 (1967) 281. H. E. Rose and R. M. E. Sullivan, Ball, Tube and Rod Mills, Chem. Pub., New York, 1958. L. G. Austin, P_ T. Luckie and D. Wightman, Steady-state simulation of a cement milling circuit, Int. J_ Miner. Process_, 2 (1975) 127. L G. Austin, P. T. Luckie and B. G. Ateya, Residence time distributions in mills, Cem. Concr. Res., l(l971) 211.

the effect of ball size on mill performance

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