A preliminary analysis of smooth roll crushers

International Journal of Mineral Processing, 6 (1980) 321--336 © Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands

321

A PRELIMINARY ANALYSIS OF SMOOTH ROLL CRUSHERS

L.G. AUSTIN, D.R. VAN ORDEN and J.W. PI~REZ The Pennsylvania State University, Department of Mineral Engineering, Mineral Processing Section, University Park, PA 16802 (U.S.A) (Received April 30, 1979; revised version accepted October 5, 1979)

ABSTRACT Austin, L.G., Van Orden, D.R. and P~rez, J.W., 1980. A preliminary analysis of smooth roll crushers. Int. J. Miner. Process., 6: 321--336. A brief review is given of the existing treatments of the fundamental description of this type of crusher. There does not appear to be any quantitative theoretical treatment of power requirement as a function of material properties. A model is developed to predict the size distribution resulting from any feed size distribution, based on the characteristic breakage parameters of a material between smooth rolls, and on the probability of sizes near the gap passing through with only a fraction broken. The characteristic parameters were measured for a coal and hard stone, and the validity of the model demonstrated.

INTRODUCTION T h e s m o o t h roll c r u s h e r is a simple a n d r o b u s t piece o f e q u i p m e n t which is widely u s e d f o r p r e p a r i n g c r u s h e d particles o f t h e o r d e r o f size o f 1 m m . It is also t h e s i m p l e s t f o r m of a f a m i l y o f roll crushers w h i c h include r i b b e d a n d t o o t h e d crushers. As p a r t o f a c o n t i n u i n g investigation o f t h e general field o f crushing a n d grinding (Austin, 1 9 7 1 ; A u s t i n et al., 1 9 7 6 a , b), we h a v e p e r f o r m e d an analysis o f the process design f a c t o r s involved in s m o o t h roll crushers. In this p a p e r we have s u m m a r i z e d existing i n f o r m a t i o n a n d deline a t e d areas o f f u t u r e research. T h e p a p e r also gives a m o d e l f o r p r e d i c t i n g t h e particle size d i s t r i b u t i o n s p r o d u c e d f r o m the roll crushing o f a given m a t e r i a l . T h e m o d e l is a p p l i e d t o d a t a f o r a b i t u m i n o u s coal and a s a n d s t o n e . T h e process design o f a simple roll c r u s h e r involves t h e following features: t h e r e q u i r e d m a x i m u m c a p a c i t y in t o n s per h o u r , t h e p o w e r n e e d e d , the c o m p a t i b i l i t y o f feed size to roll size a n d gap setting, c h o i c e o f roll speeds, a n d t h e size d i s t r i b u t i o n o f t h e c r u s h e d p r o d u c t . I t will be a s s u m e d in the analysis t h a t t h e materials t o be c r u s h e d are brittle a n d o f irregular shape, c o r r e s p o n d i n g t o m o s t r o c k s a n d coals.

322 THE MODE OF OPERATION The application of force to a particle in a smooth roll crusher is perhaps the simplest stress condition of any industrial size reduction equipment. A particle falling into the rolls is nipped and pulled further into the gap, thus subjecting it to simultaneous compression and shear (Fig. 1). Considering the idealized case of a spherical particle, the particle cannot be nipped if it is too large with respect to the roll diameter and gap setting. The particle is nipped when (Gaudin, 1939): tan(0/2) ~< p

(I)

where ~ is the coefficient of friction between the particle and the roll surface. 0 is the nip angle given by: cos(0/2) = ( d + x g ) / ( d + x )

(la)

where Xg is the gap setting, d the roll diameter and x, the particle diameter. In practice, the rolls become rough from wear, and the particles are n o t spherical, so that p is an apparent or effective coefficient of friction. For a given d, Xg, and p, eqs. 1 and l a define the sieve size Xmax at which particles are n o t readily nipped, so that mill capacity is less than expected and particles

~

.~

9C

xg ~

I

I

I

i

l

I

I

I

I

I

=

l

I

l

I

l

i

~

,

m N 6C

~ 5c 4C

~)~ Compression ,

~

30

06

20 I0 i

~/,~---~,,

~.~ . ~

~'~. ~

,o

i

. . . .

50 I~OLL

I

. . . .

I

,oo D/AMETER/GAP,

,~o

,

,

0 , 05 ,

200

d/×9

Fig. 1. Illustration of nip angle in rolls and crushing-shear forces.

Fig. 2. The values of maximum spherical particle size which are nipped by smooth rolls, as predicted by eqo 1.

323 build-up in the space above the gap. The use of feed sizes much larger than Xmax would lead to overflow from the rollers. Based on friction data reported by Brown and Pomeroy (1958), Xmax would vary from 35 mm for low rank coals to 17 mm for anthracite coals being crushed in a smooth roll crusher with a gap of 1.68 mm and roll diameter of 200 ram. There appears to be little systematic information on the influence of roll speed and material on the value of p. Figure 2 shows eq. 1 in graphical form. Assuming that the feed is in sizes less than Xmax, the m a x i m u m capacity per unit length of rolls is given by the ribbon of solid (Gaudin, 1939) which can be pulled through the rolls: (2)

M = F(1-e)pxguW

where ( 1 - e ) is the volume fraction of solids, p, their density, xg, the gap setting, W, the width of rolls, and u, the circumferential velocity of the rolls. The ribbon of solid is actually a ribbon of packed powder with a porosity of e, which is expected to be about 0.4. If the rolls are operating at different speeds it seems reasonable to use an arithmetic mean of the roll speeds. F is an empirical factor determined from test experience; ideally F would be 1 and in practice it varies from 0.2 to 0.9 for coals (Austin and McClung, 1979) depending on the strength of the coal and the roll diameter, with larger rolls and weaker coals giving higher values. The equation assumes of course that there is sufficient power to keep the rolls running at a constant speed even in maximum feed conditions. Brecker (1973) has studied the stresses produced on a particle nipped between rollers by measuring the force on the rollers with a semiconductor strain gage and oscilloscope. He assumed that the stress pattern can be derived by applying the simple static analysis of a sphere symmetrically loaded along the z-axis. The maximum tensile stress is predicted to be at the center of the sphere, where the principal stresses are given by: P OZ

--

O x

=

(42+15v)

nr 2 (14+10v) P Oy

--

21

(3)

nr 2 (28+20v)

where P is the force, r, the sphere radius, v, Poisson's ratio, and tensi!e stress is positive. He also assumed that failure would occur when ax - v (Oy+az) equals or exceeds a critical value Oe. For v = 0.25, this leads to: Oc = 0.373 P * / r 2

(4)

where P* is the failure load. For a cube the numerical factor was expected to be 0.313. The force on the rollers (both at the same speed) as a particle was crushed was measured for many particles and a median value of Oe determined. This was found to be close to the value determined by static compressive

324 tests, using eq. 4 in both cases to define ac, for a number of brittle materials of about 12 mesh in size. Equation 3 shows that tensile stresses are produced within the particle stressed by the rollers since Ox and ay are positive. Application of the Griffith crack theory (Griffith, 1920) states that fracture will initiate at microflaws occurring in these regions of tensile stress, leading to propagation of a crack at high velocity. As Austin and Klimpel (1964) have discussed, it is known that the initial crack will branch and rebranch into a tree of cracks, forming a suite of fragments. If one of the fragments is still too big to pass through the gap, it will be again nipped, pulled into the gap, restressed and fractured. The power necessary to rotate the rolls (excluding losses in the drive mechanisms) will depend on (a) the size and number of particles being simultaneously stressed in the rolls and (b) the mean magnitude of strain energy added to each of the collection of particles before fracture occurs, that is, the strength of the material. After fracture, the strain energy in each particle is converted primarily to heat (Austin and Klimpel, 1964). This type of size reduction device uses less power when run at low capacity, although energy efficiency (tons/kWh) may be lowest at low capacity due to constant losses in the drive train. There does not appear to be any existing quantitative theoretical treatment of power requirement as a function of material properties. The stress analysis is more complex when the roll surfaces are not moving at the same speed because of the non-symmetrical boundary conditions for shear. It is certain, however, that the non-uniform stress distribution induced by these conditions gives enhanced stress-concentration at microflaws, so that fracture occurs more readily. It is expected, therefore, that the use of different roll speeds will give reduced power, and hence, reduced specific crushing energy. However, too high a ratio will lead to poor nipping, so it is likely that there may be an optimum ratio of roll speeds. A MODEL FOR PRODUCT SIZE DISTRIBUTION The equations given above do not predict the size distribution of the product of roll crushing a given feed. In almost all uses of roll crushing, the production of excess fines is undesirable, but it is not possible to carry out consistent analysis of multistage versus single stage crushing unless product size distributions can be predicted. As a first step, we can assume that the stressing and fracture of any particle occurs almost independently of any other particle. Thus, breakage of particles of a given narrow size range will give a mean set of fragments irrespective of the number of the particles under consideration. Let bi.j be defined as follows: when particles in the sieve interval j (see Fig. 3) are passed through the crusher, the weight fraction which appears in size i is bi,j. It is convenient to accumulate values of bi,j from the b o t t o m size up to give the fraction less n

than the upper size of

i, B~j = ~ bk,j. k=i

Figure 4 shows a typical set of values

325 I00

I

I I

i

/

i

I

i

=

I

I

I ~

L

II

,

w N Z T

g c2_ /

~

Interv01 Number

h

I

I

200

I00

I

I

I

I

I=

=

500 I000 2000 SIEVE SIZE,/.L m

II

I

5000

IO~OOO

Fig. 3. D e f i n i t i o n o f x / 2 size intervals: the nth interval is the sub-sieve sink interval. tO0

,

,

,

, ~

518% I/4" ~-

0

o

I / 4 % 4 mesh 4 x 6

LU N

6

x 8

"

&

8

X12

"

v

u o

O O c]

FEED

v &

v

8 Q

.J IC

a

z

~

U

0

hi n

v A

Q

I...I

O

v

3 v

z~ '

I

02

i

I

Gap I

I

0.5 1.0 20 SIEVE S I Z E , r a m

I

50

iO

Fig. 4. The products from various feed sizes of coal crushed through a gap corresponding to 12 mesh between 200 m m diameter smooth rolls (2:1 roll speeds).

(see later). It can then be assumed that a feed size distribution consisting of a weight fraction of fl in size 1, f2 in size 2, etc., will give a product size distribution of: Pi = flbi, l + f 2 ~ ) i , 2 + . . . + f j b i j + . . . f i b i , i ,

n>~i>~j>~ l

(5)

326 or: i

Pi = ~j fj bi,j j=l

n/> i t> 1

,

(5a)

where Pi is the weight f r a c t i o n o f p r o d u c t in size i; clearly bi,y = 0 for i < j, since a small particle c a n n o t f r a c t u r e t o a large particle. T h e r e is evidence in the literature (Haultain, 1923) t h a t the values o f bi,y for a given set o f c o n d i t i o n s are d e p e n d e n t on the ratio of the feed particle size to the gap size (assuming t h a t feed sizes are m u c h smaller t h a n the roll d i a m e t e r ) , as illustrated in Fig. 5. If this is generally applicable, a single general m a t r i x b k,l * w o u l d be a sufficient description o f a given material and set of conditions.

I00~,i , l , i v ,=LIA N Z h-

N~0 z ~/

ed Size (mesh)

-

Gop Size (mesh)

8 x 12 ~ 12x16 0 16 x 2 0

v

=

o/ /

I

I

0.031

,

I

0.063

i

I

0.125

i

I

025

:

20

30 40

I

0.5

i

Gop I

1.0

I

20

RELATIVE SIZE , ×i/x 9

Fig. 5. Size distributions from various feed sizes erushed with a constant gap-to-feed-size setting (rolls 175 mm diameter). T h e relation b e t w e e n these values and the bi,i values for any given gap setting c o r r e s p o n d i n g t o a size interval n u m b e r ig is bi,j = b~,l where k = i - ig, l = ig - ] . Even so, the f o r m u l a t i o n is s o m e w h a t clumsy because the variation o f the total m a t r i x with material and c o n d i t i o n s has to be d e t e r m i n e d over all ranges o f p a r a m e t e r s o f interest. We have a t t e m p t e d t o r e d u c e the m a t r i x to f e w e r descriptive p a r a m e t e r s as follows. When material o f size j' is passed t h r o u g h the rolls, a f r a c t i o n 5j, i is left within the starting size range. Defining breakage as leading t o material smaller t h a n the starting size range, bJ,i can be c o n s i d e r e d as material which has by-

327 passed through the rolls: thus, a fraction ( 1 - s j ) by-passes without breakage and a fraction s i is broken. We will also assume that the first fracture leads to a set of p r i m a r y d a u g h t e r f r a g m e n t s denoted by the symbol bi,j. Daughterfragment material which is still larger than the gap setting can in turn by-pass t through the rolls or be selected for another fracture, with a fraction s i broken. s I does n o t necessarily equal si because a fragment of size i produced by fracture in the rolls might be expected to be in a better position to rebreak than t one dropped into the rolls, so that 1 >~s i >~ s i. PreCla~;sifter

Post Classifier

JBreakage I o;AX

°, Product

To Rebreakacje Pi si By- pass fi ( I -s i )

=.

Fig. 6. Schematic illustration of roll crusher model. fi =

feed fraction in size i;

si = fraction of fi broken; fi effective feed to mill; Pi = product fraction is size i leaving mill; s i = fraction ofp~ broken; Pi = product fraction in size i.

It is convenient to analyze this process of repeated breakage using Fig. 6. Let G be the mass of the sum of the f i s i components per unit mass of feed and T be the mass of thep~s~ components. Breakage of mass G + T of size distribution f[ produces a size distribution p~ given by: i-1 p~ =

~_j f ; b i , j , n >l i >

1

1--1

(6)

P'I = 0. At steady-state, the mass entering the breakage-action box is composed of new feed and rebreaking material:

(G +T)f} = fisi + (G +T)pl.sI Let f* = f[(G+ T) and p* =p~(G+T), then: f * : fisi +

pT ;

328 Multiplying both sides of eq. 6 by (G + T) and substituting from the above equation yields: i-1

p* = ~

bi,j(p?s ~ + fysj) , n >~i > 1

(7) p*= 0 The set of eq. 7 is computed sequentially, starting with size 1 to give p*, which is used to g e t p * , and so on. Since Y,Pi = 1: i n

(G+T) = ~P*'z i=1 and, then: #l

pl : p t i

j=l

p?

(8)

A mass balance on the final product may be written as: "The mass in size i of final product = the mass of size i leaving the pre-classifier + the mass of size i leaving the post-classifier", or:

Pi = f i ( 1 - s i ) + P i ( 1 - s i ) ( V + T) Substitution for G + T gives: n

Pi = f i ( 1 - s i ) + P i ( 1 - s i )

~-#P?

(9)

j=l Equations 7, 8 and 9 are used to simulate the roll crusher, via a computer program. In addition, fracture of coal in other size reduction devices is known to produce primary daughter-fragment distributions which can be normalized with respect to the breaking size (Austin and Luckie, 1972). If this applies, the fraction appearing in size 2 from fracture of size 1 is the same as that appearing in size 3 from fracture of size 2, etc. It is convenient then to use a vector bi~ to replace bi,j values. Making this assumption, eq. 7 becomes: i 1

P* = ~ b i - j ( p i s j + f j s j ) , j=l p*=

0

n>i>l

(7a)

329

This can be re-arranged to:

bi-, =

P~ - ~

E

bi-j(Pjs 1 +fjsj)

•

j=2 i>2

,s, ,

i>1

(10)

from which bi-j can be estimated when the product set Pi is from a single feed, fl = 1, if values of si and s~ are known. In addition, prior experience (Austin and Luckie, 1972) has shown that primary daughter-fragment distributions can be described by the parametric equation:

Bi-j = ¢

+ (1 - ¢) \ x ~ - ]

(11)

where the matrix of values is thus reduced to the three parameters q~, 7 and

(see Fig. 11 later): Bi_ j is again the cumulative form of bi-j. E X P E R I M E N T A L R E S U L T S A N D C A L C U L A T I O N OF P A R A M E T E R S

Feed fractions of coal in a x/~ sieve sequence were prepared by crushing and screening. Single fractions were fed into a laboratory smooth roll crusher with 200 mm diameter rolls, operated at roll speeds of 360 and 180 rpm, respectively. The feed was directed into a short central region of the roll over which the gap setting was reasonably uniform, using a vibrating feeder and chute. The inside of the receiving bin under the rolls was lined with smooth, taped cardboard to prevent loss of material. Weight losses were less than 1% and were treated as minus 38 pm (400 mesh) material. It was found that r

.4 _

0

G reenstone

z

th

(/1

50

50

w Q-

z

Z W w F-

I0

1.4 20 2.8 4.0 RELATIVE SIZE, xi/xg

5.7

w

Fig. 7. By-pass fractions for coal and O r e e n s t o n e crushed as in Fig. 4.

330 different feed rates to the rolls gave the same product-size distributions, providing the crushing zone was not choked. After feeding a single size interval, the fraction left within this size interval in the product is the by-pass fraction. The values of the by-pass fractions for the various feed sizes and materials are shown in Fig. 7. As would be expected, the fraction broken is 1 for sizes much bigger than the gap and close to zero for sizes smaller than the gap. The values plotted on log-probability paper are o o,

-

o

Belie Ayre South

I ;~z Q

5 20 40 6o

/ /

4o

95

5

99

I

999 99.9

~-

0.1 I

I

I

I I Jllll/

I.O

l

I Jlllll

O0

I00

SIEVE SIZE,ram

Fig. 8. Results of Fig. 7 plotted on log-probability paper.

shown in Fig. 8. Since the plots are straight lines the values can be approximately described by the log-normal form: 1 8i

-

~

x~i

J~ exp(-~2/2)d~ '

~

-

lnx - ln2 lno

(12)

where 2 is the median size and o is the standard deviation of the distribution. Figure 9 demonstrates that the fraction o f by-pass is a constant function of the ratio xi/xg under given conditions, because the same curve is obtained by holding the gap constant and varying xi or holding xi constant and varying the gap. Thus eq. 12 can be used in the form:

1 x~xg si - x/2~ _~ exp(-~2/2)d~'

l n ( x / x g ) - ln(M) ~ =

lno

(13)

Hence, M is the median and o is the standard deviation of the distribution calculated from: lno = ln(x/xg)(°841) - ln(x/xg)(°~sg) 2

(14)

331 \ \

0 xg - 12 mesh (I.70mm) 5mm)

o_ ,,=

05

0.5 z

o_

rn

0

I

I.O

I

1.4

I

t

~

2.0 2.8 40 RELATIVE S I Z E , x i / x Q

I

.5.7

1.0

Fig. 9. By-pass fractions obtained by varying breaking size at constant gap or breaking constant size with variable gap. TABLE I Characteristic parameters for the materials studied Belle Ayre South coal 7 q~ o M % ash % volatile matter ASTM rank Hardgrove grindability index

7 1.20 0.50 1.31 2.9

Greenstone 5 0.84 0.30 1.2 3.0

5.8 33.8 sub-bituminous C 58

w h e r e (X/Xg) (°'s4') and (X/Xg) (°''sg) are t h e sizes at which si is 0.841 and 0.159, respectively. T a b l e I s h o w s t h e p r o p e r t i e s o f t h e materials s t u d i e d a n d t h e p a r a m e t e r s o f the si curves. T h e p r o d u c t s o f 3 . 3 5 × 2.36 m m (6 × 8 m e s h ) feed passed t h r o u g h a gap setting o f 1.70 m m (12 m e s h ) were used in eq. 9 t o calculate bi-j, using the e x p e r i m e n t a l s3.3sx2.36 value a n d e s t i m a t i n g a set o f s I values via 1 >~ s~ >~si. This feed size was c h o s e n b e c a u s e it is close to t h e gap size, b u t n o t so close t h a t the particles are c r u s h e d in an a b n o r m a l m a n n e r . Based on m o t i o n p i c t u r e s m a d e b y H a u l t a i n (1923), G a u d i n ( 1 9 2 6 ) claims t h a t particles close in size t o t h e gap u n d e r g o f r a c t u r e originating f r o m transverse cracks t h r o u g h the particle f r o m roll t o roll, yielding t a b u l a r slabs. He p o s t u l a t e d t h a t breakage o c c u r s b y t w o m e c h a n i s m s : transverse cracks plus radiating cracks origin a t i n g f r o m t h e c o n t a c t p o i n t s o f particle t o rolls. T h e transverse cracks w o u l d

332

IO _ m --'

, , ~

i

i

~'~

i

n

i

i

~
i O

-~

a5

n~ u_ toO5

m ).5 z o_

<

O_

(D W o3

0

IO

1.4

P.O 2.8 4.0 RELATIVE S I Z E , x i / x g

5.7

Fig. 10. Estimates of second by-pass fractions 1 - s I used in the computations. I.U ~ -~-

[

I

I

I

I

I

]

l

I

I

I J

--

/ C

o

m-~

~

Greenstone

o Belle

Ayre

=

///~pe-I~

South

w W

i

OI

m F-

001

/I 1016

I

~

I

Q.Q31

I

I

I

O.OC:~ 0.125

RELATIVE

I

I

025

I

0 5

I

I.O

SIZE , x i / x j

Fig. 11. Cumulative primary daughter-fragment distributions for the materials studied.

presumably originate from the simple tensile stresses predicted by eq. 3, whereas the radiating cracks originate from tensile forces caused by local shear failure at the points of contact, as discussed by Sch6nert et al. (1977). The values of bi-i determined from eq. 10 are shown in Table II. The trial. and-error values of s~ used in the calculation are shown in Fig. 10, and the values of Bi-i are shown in Fig. 11.

333 T A B L E II

bi~ v ~ u e s

d e t e r m i n e d for the test m a t e r i ~ s for the first 10 ~ 2 i n t e ~ a l s

i -j

Belle A y r e S o u ~

Greenstone

1 2 3 4 5 6 7 8 9 10

0.0 0.6356 0.1484 0.0766 0.0485 0.0316 0.0206 0.0134 0.0088 0.0165

0.0 0.6845 0.1332 0.0543 0.0332 0.0256 0.0171 0.0132 0.0101 0.0288

V A L I D A T I O N OF T H E M O D E L

Before comparing experimental with model results it is instructive to determine the sensitivity of the computed size distributions to the values of 9, ~', ~ and sl-. The values of B i j are not very sensitive to 9, so the size distributions are not sensitive to/3: variation of ~ from 3 to 8 causes very little change. Figure 12 shows the sensitivity to ~', holding ¢ and s'i constant. Clearly, an incorrect ehoice for "y would give the wrong slope for the product size distribution. On the other hand, inereasing @ and s I alters the position of the size K)O

I

- t

I

I

I

I

m

I

I

I 0.1'25

i

[

I

fl~---'~

x

N ~A m Z "i"

~ ,0 Z

=

~

I.OO

1" hi

1.40

I

1

0.016

0.031

0.0~, RELATIVE

i

SIZE

I 0.25

, xi/x

L

I 0.5

K

1.0

I

Fig. 12. C o m p u t e d size distributions for a single feed size 1 (size 1 = 4 . 7 5 m m x 3 . 3 5 m m ) and a gap o f size 1 . 7 0 mm, s h o w i n g the e f f e c t o f varying the slope ~..

334 I00_[

I

i

I

~

I

i

I

i

I

1

I

i

J

[

~j~,-'r

x

Z w N Z I

Z W

-

~bo

7/Y o ]

'/I

/

i

0.016

I

i

J

OD31

i

0.063

0.125

RELATIVE

J

0.25

I

0.5

1.0

SIZE , x i / × I

Fig. 13. Computed size distributions for a single feed size 1 (size 1 = 4.75 m m × 3.35 ram) showing the effect of varying the intercept ~.

distribution to finer sizes but does not affect the slope of the curve, as shown in Fig. 13. Values of s i must be chosen to make the larger sizes match between experimental and model, s' values control the shape of the computed size distribution in the larger sizes and have been found to approximately follow r

t

Si = $i+1" 100_,

I

,

I

,

I

,

L~

® 3 / 8 " x 1/4" o I / 4 " × 4 mesh

~ NbJ

~

6

~ 8

~. 8 X 12

z

,,

/ ,

/Cf ~

" -4;

Predicted

]:

~

/®'/np/'p / ~ / []

/o

/

//

/

/

;~

/

c

~7

~, to

I

I

~/

0.016

0.031

I

I

I/

0D63 RELATIVE

I

0 125

I

I

025

l

I

05

I

0

SIZE , x i / x j

Fig. 14. Comparison of experimental size distributions with those predicted by the model: Belle Ayre South (Wyoming) coal, gap setting at 12 mesh.

335

Figure 14 shows the agreement between experiment and model using the B value parameters shown in Table I and the si, s~ values shown in Figs. 7 and 10. Agreement is reasonable. CONCLUSIONS

The size distributions obtained from crushing materials in a smooth roll crusher are dependent on the feed sizes compared to the gap size. A mathematical model has been formulated which expresses the produce size distribution as a function of feed size, a hypothesized normalized cumulative primary daughter-fragment distribution, and a vector of fractions selected for fracture (ranging from 1 for sizes much larger than the gap to almost zero for sizes smaller than the gap). Further, the primary daughter-fragment distribution has been expressed in parametric form, with the most important parameters being a slope 7 and an intercept ¢. The fraction of a ~/2 sieve interval of feed selected for breakage can also be described by dimensionless parameters M and a, which are the median and standard deviation of a log-probability distribution. The values of the descriptive parameters 7, ~, M and o vary from one material to another. The values of M possibly vary because different materials have different shapes (Austin et al., 1963). This study has delineated the following lines of research: (1) the values of effective coefficient of friction as a function of type of materials; (2) theoretical stress analysis of a sphere nipped between rollers, including compression and shear forces resulting from rolls rotating at different speeds; (3) estimation of power to run rolls as a function of material strength; (4) parametric description of the fraction selected for breakage and the primary daughter-fragment distribution as a function of coal type.

REFERENCES Austin, L.G., 1971. Powder Technol., 5:1--17. Austin, L.G. and Klimpel, R.R., 1964. Ind. Eng. Chem., 56: 1 8 - 2 9 . Austin, L.G. and Luckie, P.T., 1972. Powder Technol., 5: 267--277. Austin, L.G. and McClung, J.D., 1979. In: J. Leonard (Editor), Coal Preparation. Seeley W. Mudd Series, AIME, 7 : 1--34, Austin, L.G., Gardner, R.P, and Walker, P.L., 1963. Fuel, 42: 319--323. Austin, L.G., Luckie, P.T. and Von Seebach, H.M., 1976a. In: H. Rumpf and K. SchSnert (Editors), Proc. 4th Eur. Symp. Size Reduction, Nuremburg, Sept. 1975. DECHEMAMonogr. 79(A2): 519--538. Austin, L.G., Kimpel, R.R., Shoji, K., Bhatia, V.K., Jindal, V.K. and Savage, K., 1976b. Ind. Eng. Chem. Process Des. Dev., 15: 187--196. Brecker, J.N., 1973. Winter Annu. Meet. ASME, Detroit, Nov. 1973. Pap. 73-WA/Prod-14. Brown, J.H. and Pomeroy, C.D., 1958. In: W.H. Walton (Editor), Mechanical Properties of Non-Metallic Brittle Materials. Interscience, New York, N.Y., pp. 419--429.

336 Gaudin, A.M., 1926. Trans. AIME, 73: 253--310. Gaudin, A.M., 1939. Principles of Mineral Dressing. McGraw-Hill, New York, N.Y., pp. 41--43. Griffith, A.A., 1920. Philos. Trans. R. Soc. London, Ser. A, 221 : 163. Haultain, H.E.T., 1923. Trans. AIME, 69: 183. SchSnert, K., Von d. Ohe, W. and Rumpf, H., 1977. Rep. Inst. Mechanische Verfahrens technik, T.H. Karlsruhe.