ELSEVIER Int. J. Miner. Process. 44-45 (1996) 521-537

lmEmnmonnt Joannut or rnmuuu PROCE5SlllE

Confined particle bed comminution under compressive loads

D.W. Fuerstenau a, 0. Gutsche b, P.C. Kapur ’

a University of California, Dept. of Materials Science and Mineral Engineering, Berkeley, CA, USA b Center for Particle Science and Technology, DuPont Co., Wilmington, DE, USA

’ Indian Institute of Technology, Dept. of Metallurgical Engineering, Kanpw, India

Abstmct

Eight minerals and ores were comminuted batchwise in a piston-die press under defined conditions that eliminated geometrical effects of the particle bed. The influence of bed pressure, material hardness and feed size on energy absorption, energy utilization and product size distributions was investigated. A set of compaction equations is proposed which provides explicit analytical expressions for relating pressure with piston travel distance, volume fraction of solids in the compressed bed and energy invested in the range of pressures studied. An energy utilization index is derived by coupling Rumpf’s similarity law of fracture mechanics with self-similar size distributions of ground particles. It is suggested that this index is a consistent and meaningful measure of the grindability of a solid, whether it is ground in the particle bed or crushed in the single-particle mode. The mean ratio of grindabilities in the two modes for eight minerals and ores is about 0.46, which is in reasonable agreement with alternate estimates. Grinding virtually comes to a halt at higher pressures. Using the limiting values of fines produced and a characteristic energy parameter, the cumulative amount of fines passing a given size can be correlated with the energy expended by a single equation which is valid for all passing sizes and solids tested.

1. Introduction

Densification and compaction of particulate solid mass under high compressive loads is encountered in natural phenomena - such as geotechnics and sedimentology - and in man-made processes. In the latter case, interestingly enough, compaction is employed for both size enlargement as well as for size reduction of particulate entities. In size reduction processes focus is primarily on the interrelated phenomena of energy absorp- tion, energy utilization, reduction ratio, grind limit and size distributions of the commin- uted product. In particle-bed comminution, unlike most conventional grinding mills,

0301-7!j16/96/$15.00 0 1996 Elsevier Science B.V. All rights reserved SSDI 0301-7516(95)00063-l

522 D.W. Fuerstemu et al./lnt. J. Miner. Process. 44-45 (1996) 521-537

energy is transferred directly to the charge mass and breakage occurs by very high stresses generated locally at the contact points between the particles of the tightly compressed bed (Fuerstenau et al., 1991). For this reason, among others, significantly enhanced energy efficiency is realized when a confined bed of particles is cornminuted under sufficiently high compressive loads. Large scale continuous grinding in the particle-bed mode is carried out in the newly invented choke-fed high-pressure roll mill (Schoenert, 1988).

There is considerable interest in elucidating the kinetic-cum- energetic related principles of particle-bed comminution. For detailed fundamental investigations of this grinding mode in the laboratory, the batch process in a piston-die press set-up has some advantages over the continuous process in a high-pressure roll mill. Substantially less feed sample is required in a piston-die than what is needed for a meaningful “run” on a laboratory-scale pressurized roll mill. The rate at which the bed is compressed in the former case can be fixed at a pre-assigned value or varied according to a pre-loaded program, whereas it is quite variable and inflexible in the latter case since it is determined in first instance by the geometry and speed of the rolls. A completely confined particle-bed mode of grinding is difficult to attain in the high-pressure roll mill because of the well known end effects that invariably result in some leakage of the feed. It is also far more convenient to comminute appropriately “designed” beds of hard-soft solids or coarse-fine mixtures in a piston-die arrangement, in order to isolate and study interparticle interactions and particulate environment effects in the compressed bed. In general, much higher grinding pressures can be attained in a piston-die press than with the high-pressure roll mill. In conclusion, the piston-die press set-up provides a convenient and versatile tool for the study and analysis of the absorption, dissipation and utilization of grinding energy, size spectra of the ground product and virtual cessation of further size reduction at high pressures. Based on our on-going investigations, this paper presents some results on these characteristic phenomena in the particle-bed mode of comminution.

2. Previous work

Although a number of investigators have studied particle-bed comminution in the piston-die press, undoubtedly the most detailed and significant work in this area is by Schoenert and coworkers. A brief summary of their conclusions that are pertinent to our work are described here. According to these authors, the container effect is eliminated provided the following inequalities are satisfied between the die diameter D, initial bed height h, and the maximum feed size X,,, (A ziz and Schoenert, 1980; Schoenert et al., 1990):

D/X,,,,, > 10; h/X,, > 6; h/D < l/3 (1) The densification path of the particulate bed depends on a large number of variables, including solid hardness, feed size, strain rate and presence or absence of water. The

D.W. Fuerstenuu et al./Int. J. Miner. Process. 44-45 (1996) 521-537 523

extent of compression can be related to the pressure p by an empirical exponential function (Schoenert et al., 1990)

8= 1 -exp[-(p/p,)n] (2)

where a. normalized compression number or ratio 8 is defined in terms of S,, and 6, the initial ,and instantaneous volume fractions solids in the bed, respectively, in the following manner:

f3=(S-6,)/(1--6,) (3)

The characteristic pressure pC is a measure of the solid bed resistance to compaction, and the exponent n is a parameter that depends on the densification path. Since the specific energy absorption E, can be approximately equated with the work of compres- sion per unit feed mass:

E, = A lS”“pds MO

where .A is the die cross-section area, A4 is charge mass, and s is piston travel distance which may be expressed in terms of the volume fraction solids as follows:

s = h[ ( 6 - &J/6] (5)

Based on the theoretical treatment by Weichert (1992), it is proposed that the fraction of a narrow size feed material broken F with the investment of E,,, compaction energy is given by (Schoenert and Mueller, 1990):

F/F,=l-exp[-(E,/E,,)P] (6)

where F, is the maximum fraction that can be broken, and Em, is a characteristic energy absorption parameter which depends on the particle size and its crushing strength. The exponent p is perceived as a parameter which accounts for both the crushing strength and the local distribution and intensity of the stress field created by virtue of the shape of the particles and their packing.

The size distribution of the comminuted product is described by a log-normal distribution function in which the upper limit of the size range is truncated at feed size X,. The distribution function is set up by employing the median size X5, in a series of transformations as under:

5=x/X,; 550 = X50/X, (7)

7=5/(1-5); %0=&o/(1-550) (8)

and

r= (1/o)ln(r1/7150); o= (1/2)ln(~~J77~6) (9)

If G(t; O,l> is the zero-mean and one-variance normal distribution function, the size distribution of the comminuted particles is given by (Schoenert and Mueller, 1990):

F’( x;E,,,) = G(r( L&,P)) (‘0)

524 D.W. Fuerstenau et al./Int. J. Miner. Process. 44-45 (1996) 521-537

A few comments on the results summarized above are in order. The integral term in Eq. 4, after substitutions of Eqs. 2, 3 and 5, cannot be evaluated analytically and therefore it is necessary to use numerical techniques. The upper boundary condition in the assumed relationship in Eq. 2 between the pressure and the normalized compaction ratio requires that 8= 1 as pressure tends to infinity. Evidently, zero porosity can be reached at ambient temperatures only if densification is accompanied by extensive visco-plastic flow of the solid. It is unlikely that a single and relatively simple equation can describe both the elastic and visco-plastic regimes as well as the overlapping intermediate stage, if any. In any case, the visco-plastic flow regime is not of immediate interest for particle-bed grinding in the range of pressures that are likely to be practically feasible. We also note that the truncated log-normal distribution function in Eq. 10 is not self-similar or self-preserving in the conventional sense, that is, the distribution function does not depend solely on a dimensionless normalized size x/X,, or ,$/tSO, whereas an overwhelming majority of cornminuted particle size distributions demonstrably do (Kapur, 1972, Kapur, 1987).

3. Experimental details

Experiments were designed to investigate the influence of bed pressure, feed size, feed composition and material properties on size reduction and compaction in compres- sion-loaded particle beds. Grinding tests were carried out in a piston-die press with steel dies of 65, 50, 40 and 30 mm diameter, loaded with an Instron Universal Testing Machine Model 1334. The piston travel rate was fixed at 1 cm per second. Eight minerals and ores, namely, barite, dolomite, galena, hematite, limestone, quartz, quartzite and taconite, were stage-crushed and sieved into narrow 2.4 x 3.3 and 1.7 X 2.4 mm size fractions. Other size fractions (4.7 X 6.7, 0.6 X 1 .O, 0.3 X 0.4 mm and 37 X 75 pm) of limestone and quartzite were also prepared for investigating the influence of feed size. The experimental procedure involved filling the die with sample, tapping and preloading the latter to about 0.2 MPa and measuring the initial bed height. The sample was then compression-loaded to a pre-selected bed pressure. At the end of the compaction cycle, the compressed bed was broken manually, the resulting aggregates immersed in water and then subjected to an ultra-sonic treatment for deagglomeration. The dispersed suspension was wet/dry sieved in the usual manner. Force-displacement data from the press were logged digitally and numerically integrated to obtain the work of compres- sion.

4. Compression equations and energy absorption

Fig. 1 shows typical curves of piston displacement, volume fraction of solids in the bed and specific energy absorbed as a function of pressure for 2.4 X 3.3 mm barite, galena, quartz and taconite. The displacement curves do not exhibit any definitive trend with the hardness of the solids. However, the compressibility of soft minerals in terms of’ the solids content of the compact is significantly greater than that of hard materials. At

D.W. Fuerstenuu et al./Int. J. Miner. Process. 44-45 (1996) 521-537 525

$ 250 _

f 200

t ei50

; 100

p50

0

0 Galena 0 Quartz v Tamnite

1 2345676 Piston Displacement, mm

Fig. 1. Piston displacement, solid content of the compact, and energy expended applied in the particle bed comminution of various minerals.

0.4 1 I 0 50 100 150 200 250 300

Bed Pressure, MPa 16

p 14

',

i;;

12

10

!6

o f 6

8 4

g2

0 0 50 100 150 200 250 300

Bed Pressure, MPa

as a function of pressure

250 MPa pressure, barite can be compacted to about 95 percent solids content as against about 85 percent for quartz and taconite, even though the energy absorption by the harder materials is considerably more.

Kawakita and Luedde (1970/71) listed fifteen equations of compaction that relate pressure with the solids content of the compact. Many of these empirical or semi-em- pirical relationships are specialized cases of the following general equation:

d19/dp = K( 1 - 6)“19~/,~ (11)

where ,K is a material and particle size specific parameter. The case where all three parameters are equal to unity is of particular interest for obtaining analytical expressions for compaction processes:

dl9/dp = 0( 1 - 6)/p (12)

526 D. W. Fuersienuu et al./ Int. J. Miner. Process. 44-45 (1996) 521-537

Integration of this equation with the boundary condition 8 = 6, when p + x yields a modified form of Tsuwa’s equation (Kawakita and Luedde, 1970/71):

e=ap/(l +up) (13)

in which the constant of integration a may be replaced by a characteristic pressure p, corresponding to a chosen compaction ratio of 0 = 19,, and hence:

e= (WPMl +WP,) (14)

where

C= e,/(i - e,) w

It is readily shown from Eq. 5 that the piston displacement is given by:

s= [W&l[(S- 4,)/%] (16)

where p is density of the solids. By substituting Eq. 3, this expression can be recast in terms of the compression ratio as follows:

s= E~/AP~,l[~~/(~~+~o)l (17) where A = 6, - 6, is the difference between the final and the initial compression ratios. By combining with Eq. 13 the piston displacement can also be represented as a function of pressure:

s = [MA/A& I[ P/( 4, + %P)] (18)

where a normalized pressure p is defined as:

P = CP/Pc (19)

1.0. , , , , , , , , ,

* a 0.9 - ??@ i&e 0.6 -

s 0.7 - fi e 0.6 -

QuartAte n 0.4 -

B V v 37X74pm

g 0.3 9 0 0.30x 0.42 mm P 0 0 0.59x 0.99

0.2 g 0 1.65 x 2.36 A 2.38 X 3.33 0 4.75X6.66

0 12 3 4 5 6 7 6 9 10 Dimensionless Pressure, P / PC

Fig. 2. Unique relationship between compaction ratio or number and a dimensionless normalized pressure for six size fractions of quartzite.

D.W. Fuerstenuu et al./lnt. J. Miner. Process. 44-45 (1996) 521-537 521

The specific energy absorbed in the particle-bed comminution process is given by substituting Eq. 18 into Eq. 4 and integrating:

E,, = [ AP,/P&?c] [ln( 1 + i&/&) - &P/( 6, + a,P)] or in terms of the compression ratio as follows:

K”=$$[ln( S;;l+_A(+&] (2’)

The compaction process is described by the set of Eqs. 14, 16-18, 20 and 21 which interrelates pressure applied, energy invested, piston displacement and compression ratio. According to Eq. 14, a dimensionless compaction number should be a unique function of the dimensionless pressure p/p, for a given value of the parameter c (c = 2,

250

200

150

100

50

0 1 0

'.Or

Dolomite 2.36 x 3.33 mm

1 2 3 4 5 6 7 8 9 10 Platen Dkplacement, mm

0.8

0.7

0.6

0.5

Dolomite 2.36x3.33 mm lQo.51 5&0.67

P,: 42.9 MPa

0 50 100 160 200 250 300

BedPressure, MPa 14 1 I p 12i Dolomtte T i 1o. 2.36x3.33mm

-model eq. 0 expeimental .

0 60 100 150 200 250 300 Bed Pressure. MPa

Fig. 3. Model equations and experimental data for piston displacement, solid content and energy invested as a function of pressure for medium hard dolomite.

528 D.W. Fuersrenuu er ul./lnt. J. Miner. Process. 44-45 (1996) 521-537

corresponding to 0, = 0.667 is convenient). Fig. 2 demonstrates this relationship for six size fractions of quartzite. Fig. 3 illustrates the extent of agreement between the model equations and the experimental data for piston displacement, solids content and energy invested as a function of pressure for 2.4 X 3.3 mm dolomite. Equally good agreement was obtained for the other minerals studied. The fit of the model equations to six size fractions of feed particles, namely limestone, is given in Fig. 4. Fig. 5 shows the variations of the three model parameters, namely, the initial and final solid contents and characteristic pressure, with feed size of limestone. In this instance and for quartzite, the initial and final solids contents increase almost linearly with increase in feed size. Even though, unlike the initial solids content, the final solids content does not have a clear cut physical meaning, but nevertheless its regular dependence on feed size suggests that it is not a completely arbitrary curve-fitting parameter. As expected, the characteristic pressure, which is a measure of the resistance to compaction and comminution, drops

250

P m50

0 0.30 I 0.42 mm

0 1 2 3 4 5 6 7 6 9 10 Piston Displacement, mm

0.9

E 0.8 :, CJ 8 07

; 0.6

0.5 Limestone

0.4 1 I I I I I I 0 50 100 150 200 250 300

Bed Preswre. MPa

9 12 - 0.59x0.99 7 2; 10

- -I .65 x 2.36 P - - 2.36 x 3.33

t 8 - 4.75 x 6.55 e- -

: 6 E

c 4 2 Limestone

Fig. 4. Comparison of the model with experimental data for six size fractions of limestone.

D.W. Fuerstenau et al./lni. J. Miner. Process. 44-4.5 (1996) 521-537 529

- loo p

^ a!

-60 0”

- 60 0.6 -

- 40

- 20

0.4 1 I . * . .**. I . . . . . . . . I . . . . . J 0

10 100 1000 loo00 Feed ParUcle Size, pm

Fig. 5. Dependence of initial and fmal solid contents and characteristic pressure on feed size in the particle bed comminution of limestone.

with increasing feed size, except that it reaches a more or less constant value in the case of quartzite for size fractions 1.7 X 2.4 mm and coarser.

5. Size distributions and energy utilization

As illustrated in Fig. 6, the size distribution curves of quartz particles generated in particle-bed comminution are self-similar when plotted against a dimensionless size that has been scaled by the median size. This means that a unique size distribution is associated with any given median size, and the size spectrum is driven forward on its trajectory by reduction in the median size only. As such, this scalar is a valid and consistent measure of the product fineness and moreover, its variation with energy input is also a valid and meaningful energy-size reduction relationship (Kapur, 1987). It turns out that by incorporating a theoretical property of self-similar distributions, one finds that the specific surface area is inversely proportional to the median size. From Rumpf’s similarity law of fracture mechanics (Rumpf, 1973), it is possible to derive the following energy-reduction ratio relationship (Gutsche et al., 1993):

X,-/X,,=j(X,)E,+e;(e= 1) (22) where j, a function of the feed size, is a convenient measure of the solid’s grindability or the energy utilization of the comminution process, which is independent of the quantum of energy expended and the extent of size reduction achieved. Fig. 7 presents the energy-reduction ratio plots for all eight materials examined. Fig. 8 shows the effect of feed size of limestone on the energy plots. Fig. 9 indicates that the grindability or energy utilization index of these two materials increases with feed size in the manner of a

530 D.W. Fuerstenau et al./lnt. J. Miner. Process. 44-45 (1996) 521-537

Fig. 6. Self. -similar size distributions of quartz comminuted in piston-die press.

0 A o A

A A

Quartz 1.65x2.36mm

A 2.55 J/g 0 5.93 0 12.84 0 23.23

1 1 . . . --.--’ - . - ..-..I . . . -.-J 10 100 1000 10000

Particle Size, pm

100

90 Quartz 1 60

1.65x2.36mm

A 2.55 J/g Go

70 0 5.33

60 0 12.e4 0 23.23

50

40

30

20

10

n

power law. The fact that the exponent value for limestone is nearly unity implies that the surface area produced is directly proportional to the energy input and as such, it provides an elegant demonstration of- Rittinger’s law in this instance. However, as seen by the plots given in this figure, this law does not hold for quartzite, as indeed it is also not applicable for many other substances.

Since the linear energy-reduction ratio relationship holds equally well when single particles are comminuted by compression, it is possible to compare energy utilization for

D.W. Fuerstenau et al./lnt. J. Miner. Process. 44-45 (1996) 521-537 531

Fig. 7. Energy-reduction ratio plots for eight solids.

comminution in these modes on a uniform and consistent basis. The two grindabilities for the eight materials are plotted in Fig. 10. The data lie approximately on a straight line witlh a slope of 0.46. In other words, on average energy utilization in the

14 ., ., ., , , ., , .

13 4.75 X6.68 mm

12

g:;

Y

2 3

z 7

53

5

P 5 4

3

2

1

0 2 4 6 3 10 12 14 16

%'=~cE-w, J/g

Fig. 8. Energy-reduction ratio plots for six size fractions of limestone.

532 D. W. Fuerstenuu et ul. /Int. J. Miner. Process. 44-45 (I 996) 521-537

0 Limestone

0.01 ’ ...a.,’ . .,.t**’ 8 ‘L.4 10 100 1000 10000

Feed Size, pm

Fig. 9. Power law relationship between energy utilization index or grindability and feed size of quartzite and limestone.

single-particle mode of comminution of 2.4 X 3.3 mm particles is about twice as much as in piston-die press grinding. This result is in conformity with previous conclusions based on surface area measurements on ground cement clinker (Schoenert, 1986).

7-

') 6- s

p 15

P 'E

4-

ii.

.j 3-

e.

if 2-

5sarlte G-G&M L-Limestone D-Dolomite Q-Quartz H-Hematite Qz-Quaftzi3e T-Taco&e

0

/

0 I 0.46

0 l- o#c$y , , . , . , . ( . , 0 12 3 4 5 6 7 8

Single-ParUclo Grindebility, Q/J

Fig. 10. Comparison of grindabilities in single-particle and confined particle-bed comminution modes.

D.W. Fuerstemu et al./Int. J. Miner. Process. 44-45 (1996) 521-537 533

6. Retardation phenomena in particle-bed grinding

Grinding virtually comes to a halt in tightly compressed bed at sufficiently high pressures. For further size reduction, it is necessary to loosen the bed, remove the fines and load the charge again. This near cessation of grinding is explicitly reflected in Eq. 6 for the probability of breakage of single-size particles in a confined bed. We mention in passing that in a simulation model proposed recently for high-pressure roll mill grinding (Fuerstenau et al., 1991), the expression for the fraction of solids broken out of the original feed size interval is:

F’ = 1 - exp[ -RPEA-Y/( 1 - y)] (23)

where y (0 I y < 1) is an energy dissipation exponent and ky is an energy normalized grinding rate parameter which depends on the material and the particle size. By incorporating the limiting breakage probability term, this expression can be made identical to Eq. 6.

Exaimination of the data on fines produced as a function of energy input reveals some interesling and revealing regularities. Fig. 11 gives plots of the cumulative amount of fines generated when 1.7 X 2.4 mm calcite is cornminuted in the piston-die press. The curves bend over much more than the traditional so-called zero-order production of fines plots observed with ball mill data, and the rate at which the curve for each size fraction moves towards its asymptote decreases with the mesh opening. It would seem that the presence of a sufficient amount of fines in the bed and the build up of an isostatic-like

-_ 0 4 6 12 16 20 24 26

Specific Energy, Jig

Fig. 11. Cumulative amount of the calcite tines produced as a function of energy expended in the piston-die press.

534 D.W. Fuerstenau et al./lnt. J. Miner. Process. 44-45 (1996) 521-537

0.9 -

0.6 -

0.7 - c

8 0.6- E : 0.5 -

-v -37 pm

Limeatone 2.S x 3.33 mm

140 MPa 1

0.01 ’ * ’ . ’ ’ ’ ’ ’ * ’ I 0.0 0.2 0.4 0.6 0.6 1.0

FmctIon ot Fines in the Mixture

Fig. 12. The fraction of 2.36 X 3.33 mm limestone particles broken at an applied pressure of 14.0 MPa when the feed particles are mixed with various proportions of limestone fines.

pressure field lead to a near cessation of particle breakage. This phenomenon is demonstrated more clearly in Fig. 12 which shows the breakage fraction of 2.4 X 3.3 mm limestone particles premixed with fines (prepared from the same material) in

0.9 -

0.6 -

0.7 -

0.6 -

a* 0.5 - G

0.4 -

0.3 -

0.2 -

0.1 -

1.65x2.36mm

A Calcite

DImensionless Energy, Es I Em

Fig. 13. Master plot of the fraction of maximum tines produced as a function of dimensionless energy for all passing sizes and materials tested.

D.W. Fuerstenuu et al./lnt. J. Miner. Process. 44-45 (1996) 521-537 535

W- 0

y8 70 - OFT

s OV

E w- V

= 0 50- = 9 40- Q

09 OOV

2 30- V

< V 20-

10 - 1.65 x 2.36 mm

10 100 1000 10000 Particle Size, pm

Fig. 14. Relationship between maximum fmes produced in the limit and passing size.

different proportions. Our analysis of the experimental data suggest that is possible to describe the observed trends by the following empirical equation:

(24)

where Qi is the amount of fines passing a mesh of size index i, QiB is the maximum amount of fines that can be produced (the asymptotic limit), and E,, is the energy input required to generate one-half the maximum amount of fines. Fig. 13 shows that a unique curve valid for all solids and all passing sizes can be generated by plotting the fraction of maximum fines produced as a function of a dimensionless energy, Em/E,,. More- over, the limiting values of fines can be correlated with the passing size by a single linear-log relationship, valid for both hard and soft minerals, as shown in Fig. 14. The half-value energy of size i is a measure of the energy required to produce half of Qp” and, thus, is a characteristic measure of grinding kinetics. Although not shown here, E,, also exhibits a definite decreasing trend with increasing cut size. From the master plots in Fig. 13 and the relationships given in Fig. 14 and for Es,,, it is possible to reproduce the size distributions of the cornminuted products. Fig. 15 shows the comparison of calculated and experimental size distributions for calcite and quartz.

7. Concluding remarks

We have attempted to present an alternate approach for analyzing and interpreting the phenomena of energy absorption, energy utilization and grinding retardation in the confined particle-bed comminution mode as carried out in a piston-die press. Availabil- ity of closed-form analytical expressions for the compaction process could form the starting point for a more realistic model of the high-pressure roll mill. A fairly complete

536 D.W. Fuersienau er ul./Inr. J. Miner. Process. 44-45 (1996) 521-537

0 26.63 1 - model eq.

EAiCle size, :r 10000

1.65 x 2.36 mm

10 100 1000 Particle Size, pm

loo00

Fig. 15. Size distributions of cornminuted calcite and quartz back-calculated from the master plot given in Fig. 13 and relationships shown in Fig. 14 and E,.

description of size distributions of the ground products is possible by fitting an appropriate statistical distribution function to the self-similar curve. Of particular significance is the energy-reduction ratio relationship which leads to a convenient and unambiguous measure of the energy utilization or solid grindability in the particle-bed grinding mode. The procedure described here can be used to build a data base for the grindabilities of minerals, ores and other brittle solids and of the effect of feed size on grindability. The same method can be employed to compare the energy efficiency of different grinding mills against a base line efficiency in single-particle comminution, provided of course the test mill exhibits linear energy-reduction ratio behavior. Quite general correlations exist for integrating and systematizing data relating to the retarda- tion phenomenon in particle-bed grinding, whose deeper understanding is perhaps crucial to possible further improvement of this mode of size reduction.

Acknowledgements

The authors wish to acknowledge the U.S. Bureau of Mines through a grant from the Generic Mineral Technology Center on Comminution (Grant No. G1135249) for the

D.W. Fuerstenau et al./lnt. J. Miner. Process. 44-45 (1996) 521-537 531

support of this research. Also PCK wishes to thank the Phoebe Apperson Hearst Foundation for partial support of his stay in Berkeley during the course of this research.

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