282

integration of eqn. (2) when n - 3/2 does giveBond's law but, strictly speaking, Kick's andRittinger's laws do not result when n = 1 orn = 2 [8]. In fitting the general law toempirical batch grinding data [10, 11] (i.e.,Cmding best fit values for K and n in eqn. (2»,this second fonn is the most practical. Thereason for this is that in using the first formthe c.~~culation of the mean particle size, eqn.(3), requires the valu5 of n and K to beknown G priori. In this work, the second formof the law will be used and all representativesizes may be taken as per cent passing sizes,unless otherwise stated. To avoid inaccuracies,the exUema points of this general law (i.e.,n = 1 and n = 2) will be refered to as 'Kick-type' and 'Rittinger-type' laws respectively.

Which of the laws of comminution iscorrect? In other words is there a 'best' valuefor the exponent n in eqn. (2)? For manyy~ prior to the development of Bond's law(eqn. (lc», much was published on this ques-tion. The final outcome of these argumentswas the proposal that in the grinding of fineparticles (Ie. than l/J1n) a Rittinger-type lawwas applicable and for the crushing of coarseparticles a Kick-type law was valid [12].Bond, however, found that neither Kick's norRittinger's law of comminution was appli-cable to the practicable task of plant design.Hence, the development of the so-called thirdlaw of comminution [7], which is simplyderived from eqn. (2) by putting n equal tothe average of the exponents required for theRittinger- and Kick-type laws.

Hukki [4] notl~~ that the size ranges usedin the experimental testing of the various lawswere relatively narrow. To compensate forthis, experiments were performed throughoutthe entire range of sizes covered by cmshjngand grinding. In the conclusions to this work,Hukki [4] noted that (i) the net energyrequired in a comminution proce. increaseswith increasing fineness of the materialtreated; (ii) a log pk>t of the net energy-sizerelationship win be hyperbolic; (ill) thetheories of Kick, Bond and Rittinger aretangents to th. curve, having slopes 0, -1/2and -1, respectively, and (iv) the three com-minution laws, eqna. (la) - (lc), are onlyapproximations valid for reJatively narrowsize ranges.

The above points led Hukki [4] to theobservation that the exponent n in the

1---

~

J

idifferential form of energy-Size relationship,eqn. (2), is not a constant but a variabledependent on the representative particle sizex. Mathematically this can be expressed as

n = f(x)

so that eqn. (2) becomes

A}t.

to~cruimaon

dxdE--K ~ (4)

, ,

A new general form ,

To the author's knowledge, no fonn for thefunction f(x) in eqn. (4) has been proposed.On the development of a cortsistent, compre-hensive theory of comminution, the fonn off(x) may become self-evident. Such a theoryof comminution, however,: has eludedresearchers. The most direct JDeans of deter-mining a fonn for f(x) is by use of pragmatic~ents. For example, a P?ssible form fo{f(x) 11 \

log K log(ax of! b)(x)=2 + - - -~ (5)log x log x!

This function may appear somewhatcontrived, however, noting that

Em,

COIan<ranbyho.wiJdomaTh[1;etexthtqureibuba80ontho

--i,:

I...

X(1o1 a/lol x) = 0

will give on substitution of f(x) into eqn. (4)

d.x dxdE - -0 - - b - (6)X X2

Now, on integrating eqn. (6),

,

( Xl ) ( 1 1 )E = 0 In - + b - - - , (7)X2 X2 Xl ~

Therefore, the chosen form of f(x), eqn. (5),leads to a general comminution energy-sizereduction relationship which is a linear com-bination of the Kick- and Rittinger-type laws.Furthsmore, on replacing x in eqn. (6) by amean particle size defined by'

poea.debecaco01:ensqjlDU[a 1n(~) -; 1

.-: .will result in an expression of the same form

(7). In this case, however, th. expres-

a In ( ~ ) - ~%. % dP(x)

(8)

s.as eqn.sion may truly be viewed as a linear combina-tion of the full Kick and Rittinger laws.

As the particle size % becomes large, thefunction (%), given by eqn. (5), tends to 1.

wiaI:th

Alternatively, as % becomes small, f(%) tendsto 2. Therefore, for luge particles, i.e., in the

--J crushing range, the choice of f(%) approx-imates to Kick's equation. For small particles,on the oth« hand, i.e., the fme grindingrange, the choice of f(%) approximates toRittinger's equation. Hence, tI,t;! proposedform of the enefR'Y-siz~ reduction relation-ship, eqn. <.7), will exhibit the comminutionbehavior noted by Hukki [4], i.e., a log plotof the net energy-size relationship will behyperbolic with the theories of Kick, Bondand Rittinger tangents to the curve.

Empirical validationThe proposed 'combined' general law of

comminution, eqn. (7), will tend to Rittinger-and Kick-type laws at extl'emes of the sizennge, ths-eby exhibiting the behavior notedby HUti [4]. For intermediate size ranges,however, the accuracy of the combined lawwill require quantitative checking. This will bedone on u~ of batch grinding data for variousmaterials in various size ranges, cf. Table 1.The sources of these data are Smith and Lee[13], Celik [14], Iwasaki et al. [15], Freehet al. [16], and Bagga [17]. For details of theexperim('ntal conditions used in these studies,the reader is referred to the references inquestion. Each data set in Table 1 consists ofrepresentative size measures of the size distri-bution for each time interval used in thebatch grind. The parameter Pso refers to tht.80% weight passing size and Ploo is obtainedon extrapolating the straight-line portion ofthe log plot of the size distribution.

In implementing the combined law, allpossible pairs of feed and product sizes ineach data set will be used. For example, if adata set consists of 5 batch grinds, there willbe 10 po.ible pain of feed/product size thatcan be used in eqn. (7). The accUlacy of thecombined law on each data set is quantified,on noting that time of grind is proportional toenergy input, by calculating the sum ofsquares

.

.of

.. -1:(t - t..)2wbs-e t is the givm time of the batch lrin4and t.. is the estimated time calculated fromthe combined law for each feed/product sizepair % ./% 1. viz.,~

,~

283

(10)b(~ - ..!..)%2 %.

tn& . G 1n(~) +

The coefficients a and b in eqn. (10) are cal-culated for each data set such that the sum ofsquares in eqn. (9) is a minimum. As a furtherpoint of reference, the combined law, eqn.(10), is compared with the generalizedenergy-size reduction law, obtained onintegrating eqn. (2):

~)t -K ( -k-.It %2 (11)

where the coefficient K and the exponent nare calculated in a similar manner to the coef-ficients G and b on minimization of the sumof squares, eqn. (9). The required values of G,

b, K and n for each data set are given in Table2.

A tabulation of the values of tea' using thegeneralized law and the combined law, foreach data pair in the petroleum coke data setis given in Table 3. Comparison of the sum ofsquares, for each material data set, obtainedfrom eqna. (10) and (11) is made in Table 4.In all but three cases, the combined law givesa better fit to the experimental data. Further-more, in the cases where the generalized lawgives a better fit, the difference in the sum ofsquares is small.

From these results, it may be concludedthat in intermediate size ranges (2750 to60 /lm), the proposed combined law, eqns.(7) and (10), is a close approximation to theenergy-size reduction relationship. In general,this approximation is better than previouslyderived empiri~allaws. Furthermore, as thecombined law tends to Kick- and Rittinger-type laws at the extremes of the size range,this suggests that outside the intermediate sizerange the combined general law will still be are~nable approximation whereas otherempirical laws may break down.

Conclulion -Starting from the general equation

developed by Hukki [4], it has been demon-strated that in comminution a generalenergy-size reduction relationship which is alinear combination of Kick- and Rittinger-type laws can give a smooth and close fit toempirical data. Furthermore, this fit is

(9)

2M

TABLEt

Source Size meuurement Time of Ifind Sizetype (min) (IDD)

Spodumeae Smith and Lee [13) PI~ Feed0.7141.4292.8573.5714:2865.7147.143

10.00

Feed4.2867.143

10.00

Feed1.4292.85'14.2865.7147.1438.571

Beryl

Quartz

Rock salt

Petroleum coke

Quartz

-- ~

t.~..,.,,'~.'-

~~f.;;;TA f~I'.;

-If!fa

atl ~275014301140840730640485396275

~

!~.;Me ~:::

:. .

~

WeSmith and Lee [13) P1OO 2700660400285

ZTJ<»

MI

Smith and Lee [13) PIC» 2200960502338265210175

Smith and Lee [13] Pl00 Feed1.4292.8574.2865.71.7.143

1780800450302230180

~

Feed4.286

7.14310.00

14.29

Feed

1.4292.8574.286

7.143

0.331.003.006.00

15.0030.00 i

0.331.003.005.00

1~,-15.0030.00

Feed15.0020.0030.0060.00

Smith and Lee [13 ) Pl00 1520450285207144

Smith and Lee [13) P1OO 2560650330225136

Celik [14J p. 1150110083552018074

(14JCelik p. 80078059545025016070

Iwuati.taL (15) PIG 58518015011260

./

',"

TABLE 1 (continwd)

()ttawa land Freeb e' aL (16) p. Feed 5905.00 297

10.00 20015.00 14520.00 100

Menmann cement clinker Bana (17) p. 0.33 11501.00 1000

. 3.00 7505.00 ' 500

10.00 235

Western Kentucky coal Baua (17) p. 0.2 8001.00 6603.00 2956.00 145

.J OI;tawa sand

Menmann cement clinker

Western Kentucky coal

generally better than the fit obtained usingpreviously developed general laws.

The proposed combined law may have im-mediate UIe in computer simulationa, e.g., therecent grinding models develop«! by Cro.[18] and Voller et at [19]. For more generalUIeI, however, such as mill design (ct. Bond[20] and Rowlands and Kjos [21 I), furtherwork will be required, e.g., a relationshipbetween the Bond work index and the coeffi-cients a and b in eqn. (7) would have to be~arived. Finally, in any application of thecombined law of energy-size reduction, itshould be remembered that, like all previouatheGns of energy-size reduction, thedevelopment of the Jaw it based on purepragmatic reasoning.

.., "

~*'~ .

:~~c,. .",.;., _c

~:~'::;:""

~~J;,J;'~.,. .;"~ ,,~! .

p.

0.331.003.005.00

10.00

0.21.003.006.00

Bag,. [17)

Bagl. (17)

AcknowledgementsI would like to thank Dr. K. J. Reid, Dr.

K. A. Smith and Dr. M. Hebsur, respectivelyDirector. ASliltant Professor and ResearchAasociate at the Mineral Resourc. Center. forUleful dilcu8ion and guidance in this workand Dr. L. Auatin of the Pennsylvania StateUniversity for provKting experimental data. Iam also grateful for rmancial support through. Mineral ReIOurces Research Center post-doctoral feUoWlhip.

:. _..D_6Referencea1 G. E. Alar and J. H. Brown, Thc Canadian Mini",

GIld Mctallurrical BuUctin, 58 (1964) 871.2 L. G. Auain,lnd. E",. Chcm., 56 (1964) 18.3 C. C. Harris, Trana. 1M. Min. MctaL, 75 (1966)

C37.

-

Feed/product Time of lrind

Siae (IUD) Experimeat.al Genenlized law Combined law

1150 1100 0.67 0.20 0.251150 835 2.67 1.58 1.861150 520 5.67 4.50 4.951150 180 14.67 14.58 14.541150 74 29.67 28.94 29.161100' 835 2 1.38' 1.611100 520 5 4.29 4.701100 180 14 14.37 14.291100 74 29 28.74 28.91835 520 3 2.92 3.08835 180 12 12.99 12.68835 74 27 27.36 'l1.29520 180 9 10.07 9.59520 74 24 24.44 24.21180 7. 16 14.36 - 14.61

~.

TABLE 4

Comparimn of the law.

Material No. of feed/product paif8 Sum of Iquu.

Genenllaw Combined law

Spodumene 36 3.64 4.71Beryl 6 0.032 0.008Galena 21 0.567 0.361Microline 15 0.132 0.052Quartz 10 0.045 0.008Rock salt 10 0.068 0.032Petroleum coke 15 7.22 3.04Quartz 21 5.16 2.22Copper-nickel 10 5.08 5.39Ottawa sand 10 4.71 6.18Cement clinker 10 0.5045 0.469Coal 6 0.3076 0.268

4 R. T. Hukkj,AIME Trana., 220 (1961) 403.5 R. R. Van Rittineer, L.hrbuch der Aufbereitu,.,,-

iunde, Berlin, (1867).6 F. Kick, DOl Ge-.tz der proportiOMien Wider-

Itande uld .iM Anwendung, Leipzil (1885).7 F. C. Bond,AIAIE Trani., 193 (1962) 484.8 L. G. Auatin,Powder Tech1W>L, 7 (1973) 315.9 W. H. Walker, W. K. Len, W. H. McAdanw and

E. K. Gilliland, Principle, of Chemicol E,.,ineer-irt6, McGraw-Hili, New Yc.rk, 1937.

10 R. J. Charles, AIME Trana.. 208 (1957) 80.11 J. A. Holm_, Trana., Inlm. Claem. E"I78.. 36

(1957) 125.12 D. R. Walker and M. C. Shaw, AIME Trana., 199

(1954) 313.13 R. W. Smith and K. H. ~,AIME Trana., 241

(1968) 91.14 M. S. Celik, M.S. Thail, Pennlylvania State

Pou

~

Le

CM

A

CIZG

A

(I

stp.[:

k

.0nac~g

t.vbGCr.

,.,(

University, 1977.15 I. Iwasaki, A. S. Maliai and R. J. Lipp, Final

Report to MinnelOta Environmental QualityBoard Coppe,.-Nic~el Study, Mineral ReaourceaReleucb Center, University or Minnesota. April1978.

16 E. J. Freeb, W. E. Horst and R. C. Kellner, A/METrana., 238 (1967) 167.

17 P. B81P, M.S. Th_il, Pennsylvania State Univer-

r

lity, 1979. r-18 M. Cro., Powder Technol., 31 (1982) 233.19 V. R. Voller, K. A. Smith, M. CrO18 and K..I.

Reid, SME/AIME Fall Meeti,., 1981 pre-print,81-319.

20 F. C. Bond, Allil-Chambera Publication. 1962.21 C. A. Rowland. and D. M. Kjoa, in A. L. Mular

and R- B. Bhappu (Ed..), Mineral Proce..in, PlantDe,;p. SME/AIME, 1978. p. ~39.