The estimation of non-normalized breakage distribution parameters from batch grinding tests

Powder Techdogy-Ekvier

Sequoia SA_, Lausanne-Prmted in the Netherlands

The Ektimation of Non-Normalized

267

Breakage Distribution Parameters from Batch Grinding

Tests L. G. AUSTIN

and P_ T. LUCKIE

(ReceivedJune9. 1971)

Summary 77ne Harris hypothesis that a WeibuNpIot of grinding data shouldgive a straight Iine of slope one ifsrinding is first-order breakage

is disproved_

distribution

Due

parameter

to the shape of the B and the fact

that

it is not normalized, the slope is generaMy greater than one even for perfect first-order grti&ng_ A computation technique irdescribedforcakuIating non-normalized values of B, usxkg the results of batch grinding

tests on a closeI?

sized sample.

INTRODUCTION

In recent papers’-’ Austin has discussed the advantages of describing ball mill circuits using the concepts of specilic rates of breakage (S,) and cumulative breakage distribution (B,,,) parameters. Several papers3+ have described techniques for estimating these parameters from laboratory milling tests However, all prior work has assumed that the breakage distribution parameters are normal, and the techniques at best measure only an apparent mean set of B values. Our experience has shown that the assumption of normalized B values is not generally correct for ball milling of brittle materials_ The false assumption can lead to significant errors in mill simulation and in interpretation of experimental data This article gives a method for estimating the appropriate parameters, which is applicable to much of our experimental data_ Herbst and Fuerstem& have described a technique for determining S and B parameters based on the assumption that the first-order specitic rate of breakage S is a simple power function of the particle size breaking, S=af, and that B V&ES for small receiving sizes (x < 11)are also a simple normalized Powder Tech&

5 (1971/q

power function, B(xj_\) = Cp(~;y)~_ with z = 7; B(x 1~) is the fractional weight of material just broken from size 1 which falls less than size x, _X< J-_ Harris’-’ and Austin et aL9-‘0 have shonn that these conditions applied to breakage of a closely-sized starting fraction gives l-P(s,t)=[l-P(x-,O)]esp[-a&xftJ. s < starting size

(I)

where P(x_ t) is weight fraction less than size s after time r of grinding This means that if the percentage oversize at some specified size S, R(Z_ t) = lP(Z, t), is plotted on a double-log scale rersus time on a log scale (Weibull plot), the result vvould be a straight line of slope I_ As Harris pointed out. experimental results do indeed give a straight line but the slope is greater than one. He ascribes this to the breakdown of the first-order assumption inherent in the definition of S. Since our experimental results give good agreement with the first-order hypothesis. this sug_gests that Harris% explanation is not correct.

EXPERIMENTAL

RESULTS

A typical set of carefully performed experimental results, where the B values are calculated using Method II or III’, is shown in Fig 1. These were obtained by short _erind times of material principally in the top size, so the values are dominated by the B of this top size and the non-normalized nature of B does not significantly fleet ‘Lheresults of the BII or BIB caIcula~on_ It is clear from Fig 1 &at the B values are not normahzable ; the slope of the bottom portion of the curve remains the same, but the \due of the intercept 4 increases with decreasing particle size_ Figure 2 shows a plot of 4 G~~SL(S size: it is a

w

D20

~

g

~ v ~

g

n

or, for up~

since

of ~ t i o R

( s ~ F i g 4) a pow~ a ~ of about ~ m ~ q ~ ~ ~ a t =

Figu~ 3

~¢ B

for ~ c

~EORY

S~ R ~'° ~ °

o f S values to apply up to

(a)

NON-NORMALIZED

BREAKAGE

DKTRIBUTION

PARAMETERS

269

_&riding equation for the condition of all of the starting feed in the top size interval’ ‘_ Since this is a simulation using the first-order hypothesis with known S and B. there can be no question of experimental variability, or non-fmt-order effects. The values of S were taken as a power function (eqn. 4), with S, = 1. for varying values of a from 0.5 to 12 Equation (3) was used for the B values, with 7 equal toa;& wastakenas03andj3as3_0or5_0_Arange of 6 values were computed_

2

4

OlYENSIONLESS

hg

5. Sunulated

6

6

10

TINE

DF

GQINOING.

Wcibull plots.normald

20

30

S,1

R walun

Figure 5 shows a typical result on RosinRammler paper (Weibull plot) for normalized B’s, that is 6=0_ Because the B values are not a single straight line on log-log piots, the Weibull plots are not straight lines of slope 1. In fact, these computed results show a slight but distinct change in the slopes as time progresses. However, when one examines real experimental results on this type of plot (Fig. 6), it is apparent that slight deviations in slope cannot be detected in the presence of typical experimental variability. The slope of the computed Weibull plot is greater than one. It is concluded, then, that a perfect first-order breakage wilI give rise to a Weibull slope greater than one because of the shape of the B values cemu.ssize, even for normalized Bk This immediately disproves Harris% explanation of slopes greater than one. Pow&-

T&l_

5 (1971/72)

P I

2

Fig

7. Simulated

6

4

OIYENflDNLESS

Wciiull

TiYE

OF

8

10

GRIWDING.

2u S,t

pLts, non-norrnallzcd 3 lahrts.

Figure 7 shows another typical set of data. as a function of 6/q ie- the extent of non-normalization of the B plots The slope of the Weibull plots for the it is fine size increases as 6 increases; at a/z=0 about 1.09 and at b/a=Od it is 124. More important. the percent iess than the 20th interval size (minus 400 mesh ifthe top size were 18 x 20 mesh) at time 10 varies from 41 oA to 56% as 6 increases Thusalthough the slopes do not change by very much, the effect of non-normalizable &3’s on the amount of

270

I_ G. AUSTIN.

fines is pronounced_ Contrary to the conclusions of Herbst and Fuerstenau’, it is often not possible to get accurate results if the effect of non-normalizable B is neglected.

PARAMETER

ESTIMATION

It was hoped that computing results for ranges of the parameters involved would enable the nonnormalized B parameters to be estimated by comparing experimental results such as those of Fig. 8

SIEVE

FIN. 8. Experimental

SIZE. MICRONS

(points)and computeds~~~dlstributions

(S, = IO mill-‘, 1=0.99, j3=4.5, p=O.84, 6=0 15, f&,=0295); type I cement clinker ground in &m.-d~am ball mill; feed SIZE 2Ox2Smesh.

with the computed results. However, there are so many variables involved (LY$J,S,~,R) u-hich affect the results that graphical presentation of all possible combinations would require far too much space for this publication. Therefore a computer method was developed for estimating 6, using known experimental values of a (or a, y where necessary), S,, 4r, j3andR. Values of S and B are determined using the experimental techniques previously described4-‘*_ For cases where the experimental results plotted as in Fig. 8 give a series of parallel lines for the smaller sizes, the value of a is determined from the slope of the lines If the lines are not parallel, two or three values of S should be determined and S wxws size plotted on log-log scales: the value of a is then the Powder Technol., 5 (1971/72)

P. T. LUCKIE

slope of this line. The plot of B for the top size interval, as in Fig. 1, enables $i. y and p to be determined. The computer program operates as follows. First, the values of P as a function of size and time are computed in the normal way’ I7 assuming that B’s are normalized, using the experimental values of a, y. S,, &1 and jI_The grind time for this simulation is set to the time of one of the experimental curves of Fig. 8,3 minutes For example. In general, the computed curve will Fall below the experimental curve at the small sizes, because of the non-normalized nature of B. This computation in effect uses 6=0. A convenient control point is chosen, For example the top size of the 19 interval in Fig. 8. The pro_mm then chooses a value of6 so that the computed result passes through the experimental value at this poinf that is, P,,(3) COMP=P,,(3) MPT. The procedure For converging upon the appropriate 6 value is given in the Appendix. If necessary, 6 is determined from other times also, and a mean taken. Figure 8 shows the computed result for comparison with the experimental result For the optimum 6, 6 =0.15. Also shown are the size distributions at other grind times, using the same 6 value of course_ The simulation is good, within the limits of experimental reproducibility of the experimental results_ It should be noted that the assumptions that breakage is a simple first-order process and that S is a power function (eqn. 4) are true only For particle sizes which are less than some maximum. The value of this maximum is a Function of material, ball size and mill diameter_ Figure 4 shows a t_ypical result. It is clear that the estimates of S using eqn. (4) are radically in error ifs, is determined Fora size above 20x 25 mesh in Fig. 4. However, the computer technique Fordetermining 6 discussed here will work perfectly well in this ease providing that a real set of S values is used rather than assuming eqn. (4) to apply. CONCLUSIONS

The Harris hypothesis that a Weibull plot of grinding data should give a straight lime of slope one if grindiig i, first-order is disproved. Due to the shape of the breakage distribution B, and its nonnorn~alized nature, the slope is generaIly greater than one even for perfect first-order grinding. A computation technique is described for calculating non-normalized values of B, using the results of batch grinding tests on a closely sized sample.

NON-NORU4LIZED

BREAKAGE

DI~IBUTION

The iterations

APPENDIX

The convergence algorithm employed to determine the (5 value is a variation of the Newton Rapheson numerical approximation to the roots known as the Secant or Regula Falsi Method- This method is by no means the only seeking method available and its utilization here does not mean that it is the only approach that can be succcssfuliy employed in this application. Let 6* be the unknown value that generated the experimental control value for a given size at a given time t. Let d be the estimation of 6* that achieves the condition that the calculated cumulative value minus some slight error equals the experimental cumulative value ; i.e. P,(t,6')=

(C5e-&.JAP Ap 0

where AP = P*(r. &) - P*(t. 6*) AP,, = P,(r. 6,) - i’,(t, &) 6,CCS and &>a*

Ponder

Tedyrol,

are continued

until

AP<&. The detetminztion of 6 typically takes a compile, compute time of about 12 seconds on an IBM X0/67 machine. REFERENCES 1 I._ G Austrn. UnderstandinS ball mill sizing (in press& P_ T_ Luck and I_ G. A-in, Snnularion of grindin_r drcull& Proc la SarL ~fe&ng. 5. Ajiicm Inn. ckm &ngm Lhxrhln Augusr 19x 3 R. R Klimpcl and I- G. Austin. I-EC_ Fzmdmmruk 9 (1970) 230. 4 L_ G _4us1in and P_ T. LucLit Mc~hods for dcnzrxnina&m of brgliagc disnibnrion paramnas. Pmrdcr TccknoI_ 5

2

(197Ij72)215-222

P,(t,S)-&

where E represents the slight error uhich is acceptable to the user (0.001, for exampie). Since a plot of P,(r, 8) ~XTSUS8 is monotonically increasing, the iterative algorithm to estimate CT is &n*l=&n-

271

PARAMETERS

5 (1971/72)

7 C C Harris and _A. Chal.raxaru. Pmrdcr TcchmL 4 (I) (197G) 57. 8 C C Harris Powder Tt-chn& 3 (5) (I 970) 309. 9 I_ G. Au&n_ P_ T. Luckic and R R Kiimpl. Solu~ons of&z