Materials Chemistry and Pl?ysics, 25 ( 1990) 3 I 35 21

C~~~I~UTION ENERGETICS PART III : BOND'S LAW AND BREAKAGE ENERGY

S. Tarasiewicz and P. Radziszewski

Departement de genie mecanique, Universite Laval, Ste. FOY.

@Gbec, GIK 7~4 Canada

Received June 7, 1989; accepted September 12, 1989

ABSTRACT

Based on a so-called comminution energetics approach, a developed breakage

energy function is modified to include an experimentally determinable comminution

law. In re-evaluating the classical eomminution law, such as Bond's, as an inte-

gral element in a quasi-fundamental breakage energy function, we combine in a

natural way any energy lost in grinding as well as, the specific energy rate of

change into the determination of material size evolution. The comminution law

thus completes the comminution energetics approach as both a fundamental law of

comminution and a practical tool to optimizing real and/or conceptual grinding

systems.

INTRODUCTION

The grinding process, characterized by an effort spent on a material to change

it into another smaller material size distribution, has existed for centuries.

Serious study on the comminution process was initiated with the Rittinger law of

grinding (1867), followed by Kick (1885) and later by Bond (1952) [1,21. All of

these laws relate the total energy consumed in grinding to particle size reduction.

Despite a certain success in sizing ballmills for given contexts, the comminution

laws fall short in determining an answer to the question : "How can a comminu-

tion process be optimized ?" Further, with the advent of the batch grinding

equation, these comminution laws have been relegated to a consequence of the

batch grinding equation and unapplicable to cases of non-standard feed data !21.

In this work, we will attempt to determine if such comminution laws, fore-

runners to the comminution energetics approach, are applicable in the modifica-

tion of the wholly theoretical breakage energy function into a quasi-fundamental

0X4-0584/90/$3.50 OEIsevierSequoia/Printed in The Netherlands

22

breakage energy function. In developing such an aspect to the breakage energy

function, we hope to alleviate two problems in the practical application of the

bren function to real systems. These are :

(i) the tensile strength, Young's Modulus, particle shape and material fracture

mechanism per energy level used are difficult, although not impossible, to

experimentally determine for heterogenous material;

(ii) presently, studies determining the amount of energy lost due to plastic

deformation of grinding media and liners, heat generated through friction,

kinetic energy dissipated in breakage, as well as noise created in grinding

are rare; any quantization of such lost energy would undoubtedly be diffi-

cult to determine experimentally.

GRINDING LAWS

The three laws of Rittinger, Kick and Bond can all be derived from the follow-

ing relationship, proposed by Charles (1957), between energy consumed and mate-

rial size C.51 :

&=-kg Yn

Setting the exponent n to 1, 11 or 2 upon integration defines the laws of Kick,

Bond and Rittinger respectively.

Of these laws, Bond's third law of comminution Cl,21 has developed the largest

following as a result of the Bond Work Index Wi. This index can easily be deter-

mined using an experimental procedure for any material. In such a way, the third

law can be used to determine the energy requirement to grind a given material to

desired size range. This of course provides the design parameter for ballmill

sizing to the desired application. The Bond law as described by cl,21 :

eg = 10 Wi ( -L - -?. ) _:

a Y2 Yl

will be used in the subsequent analysis.

3~~GE ENERGY MODEL

The fully developed and generalized bren function including the energy lost

term is defined as 131 : . *

gffY)= _ m’(Y) - Q(Y)l*st Y-l m'(e)

dt e(Y)

de(y), E(Y)in X m(y) I

+x -

Y=Yo e(y)

de(o)+ E(afin-Q(a)lost

dt m(o) I

Y'Yo (3)

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Redefining the specific energy term e(y), which describes the surface energy

per unit mass of nriterial, as specific energy consumed eB(y), which describes

the energy consumed per unit mass of material in particle size y, allows for the

following modification to the bren function :

@’ (Y) = _ g_(Y) _ dt e,(y) I 1 y-1 E(Y)in f c

m(y) U'Yo Y>Yo

(4)

This modification to the bren function integrates the energy lost term into

the specific energy consumed term which can be determined experimentally. It

also includes the specific energy rate of change term. That is if it can be

assumed that local energy consumed in grinding material of size interval Y is

defined by a comminution law, such as Bond's then the modified bren function

becomes easily identifiable using a standardized experimental procedure.

MODEL ~~RI~ICATIO~

Using batch grinding equation data for quartz and limestone 123, material size

evolution will be compared with that determined by the modified bren function

using the Bond law and work indexes for quartz and limestone 121. The test mill

simulated the standard laboratory ballmill (Annex I) [2,41. The initial feed

size distribution for both quartz and limestone is found at t = 0. Size evolu-

tion results for quartz is found in Fig. 1 and for limestone in Fig. 2.

Using the Bond law of comminution results in a faster grind than what was to

be expected of this empirical parameter. Many reasons may be possible, but the

one that seems most plausible is that defining the Bond law as a feed size pass-

ing a certain percentage ground to a product size passing a certain percentage

does not inherently determine the energy spent on a specific size interval. Upon

correcting the Bond law with a factor of 100, further simulation showed that the

ground product using the modified bren function converged to the ground product

as determined by the batch equation (Fig. 1 and Fig. 2) in the same magnitude of

time (25 time units).

In determining the correction factor, one must keep in mind a number of facts:

(f) the fracture mechanism plays a major role in how fast a feed size evolves

in time 13,41;

(ii) the Bond law of comminution, as well as Rittinger and Rick, were developed

for a global 'black box' understanding of grinding and not a local break-

age phenomena;

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(iii) a number of important variables were not available on the laboratory test

mill operation (specifically material charge friction factors and power

draught) which plays a major role in the determination of energy spent in

grinding C3,61.

All of these factors can greatly affect the precision of the calculated results,

which of course underlines the need for adequate identification methods.

z II 10 102 103

Sieve size y [p-d

Fig. 1. Quartz size evolution

‘2 c -I batch 1 C

3 1, equotibn

1

102 103

Sieve size y [p-N Fig. 2. Limestone size evolution.

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CONCLUSION

Earlier, it was noted that comminution laws, such as Bond's, were relegated

to a consequence of the batch grinding equation. Analysis of results obtained

through a modified, quasi-fundamental, breakage energy function provides the

basis on which a simple conclusion may be made : comminution laws are an integral

element in the application of the modified bren function to the real world.

Although present forms of comminution laws are not adequate to provide the neces-

sary confidence in their immediate application, they do confirm that the direc-

tion is very promising, Future work will undoubtedly provide an easily determin-

albe comminution law that will complete the comminution energetics approach as

both a fundamental law of comminution and a practical tool to optimizing real

and/or conceptual grinding systems.

REFERENCES

1 G.C. Lowrinson, Crushing and Grinding, Butterworths, London, 1974.

2 L.G. Austin, R.R. Klimpel and P.T. Luckie, Process Engineering of Size Reduc-

tion : Ball Milling, Sot. Min. Eng., New York, 1981r.

3 S. Tarasiewicz and P. Radziszewski, Mater. Chem. Phys., 25 (1990) 1.

4 S. Tarasiewicz and P. Ftadziszewski, Mater. Chem. Phw., S (1990) 13.

5 E. Yigit, Int. J. Min. Process, 3 (1976) 365,

6 S. Tarasiewicz and P. Radziszewski, Ballmill Simulation Part I : A Kinetic

Ball Charge Model, SCS.

ANNEX I

Ball mill specifications

D mill

= 0.19 [ml P crit

= 72%

Lmill r 0.19 Cm1 'b

= 7500 [kg/m31

a = 0.026 [ml uS

q 0.4

Pfill q 20% 'k

= 0.3

liner : smooth