CONFIRMATION OF THE THIRD THEORY

By Fred C. Bond

Since the Third Theory of Comminution was pre- sen ted eight years ago (I) i t h a s found increasing u s e in crushing and grinding problems. T h e pract ical util- ity of i t s w o k index equation i s quite generally ac- knowledged (2 ) . However, i t s theoretical ba s i s h a s been questioned in a t l e a s t three technical a r t ic les ( 3 ) ( 4 ) ('). The purpose of t h i s paper i s to present ex- perimental proof that i t i s scient if ical ly correct.

Part i c l e s under compressive s t r e s s a r e s trained and deformed. They absorb s train energy, and when t h i s locally exceeds the breaking strength, a crack tip forms. T h e surrounding strain energy flows t o the crack tip, which rapidly extends and sp l i t s the rock, releasing the s t ra in energy as heat. The initial energy flow cause s additional crack t ips in highly s trained areas. If the compression i s rapidly applied by im- pact , crack t ips may form before the strain energy h a s reached equilibrium in the part icle , thus decreas- ing the total work input required for breakage. T h e energy necessary t o break i s essen t ia l ly t he energy necessary to produce crack t ips , s i nce the energy necessary to extend the c racks to breakage i s already present a s strain energy in the deformed particles. After breakage nearly a l l of th i s energy appears a s heat .

The crack length cannot be measured directly. However, in p articles of regular and similar shape the crack tip length i s considered a s equal to the crack depth, or crack extension necessary to break, s o that the crack length equa ls the square root of one- half of the sur face area.

TheThi rdTheory s t a t e s that the useful work done in crushing and grinding i s directly proportional to the total length of the new cracks formed. I t can be con- firmed by showing that a constant work input produces a constant length of new c r acks when reducing the same material t o different product s i z e s . T h i s i s done in the present paper on a wide variety of material.

F. C. BOND is Consult ing Engineer, P roce s s - ing Machinery Dept., Allis-Chalmers Manufacturing Co., Milwaukee. TP 59B32. Manuscript, Nov. 5, 1958. AZME Trans., Vol. 21 7,1960. San Franc isco Meeting, February 1959. r 7

T h e cons tan t work input was supplied by one revolution of the 12" x 12" laboratory ball mill used in making indability t e s t s by the Allis-Chalmers

'Y method (I2 (I3). The new crack lengths produced per mill revolution were measured from a l l avai lable grindability t e s t r e su l t s a t 28, 35, 48, 65, and 100 mesh on fifteen different ores , and were found to re- main substant ial ly constant for each o r e a t a l l mesh s i z e s .

A new technique i s used for t h e measurement of crack lengths. Size distribution p lo ts of t h e mill feed and product a r e made by the Third Theory method (9) and the crack lengths are obtained from these p lo t s by the procedure described in the present paper. The energy input required to produce a unit length i s of fundamental importance in the s i z e reduction of brittle so l ids .

The crack length Cr i s expressed in centimeters per cubic centimeter of sol id material. I t bears a defi- n i te relationship to t h e external surface a r ea of the crushed o r ground solid.

A uniform shape must be assumed before the surface a r ea and crack length can be evaluated. In t h i s paper i t i s assumed that the relat ionship be- tween the surface a rea and the particle volume of a p a r t i c l e d m i c r o n s in diameter i s the same a s that of cubed-microns on a s ide. T h e external surface a r ea s of part icles with a cubical breakage probably agree approximately with t h i s rule, and correction factors can be applied when physical measurements of the surface a r e a s a re ava i lab le for comparison. However, the assumption of equivalent cubes ha s been found sat isfactory for most calculat ion purposes.

Assuming equivalent cubes, one cubic centimeter of par t ic les d microns in diameter will have a crack length Cr of J 30.000/d centimeters, and a surface a rea of 60,00O/d square centimeters. T h e spec i f ic crack length i s thus equal t o the square root of one- half the specif ic surface area.

Where Sa i s the surface area in square cent imeters per gram and Sg is the spec i f i c gravity of the ground solid, then

Determination of t h e c rack length of a crushed o r ground so l id r e q u i r e s (1) a s u i t a b l e method of p lo t t ing the s i z e dis t r ibut ion, (2) a c c e p t a n c e of a l imi t ing f i n e pa r t i c l e s i z e , or "grind limit", and (3) mathemat ical a n a l y s i s of the s i z e dis t r ibut ion plot. T h e s e th ree requirements a re developed below and in the appendix.

THIRD THEORY SIZE D l S T R I B U T I O N

When rock i s broken under compress ion the prod- u c t p a r t i c l e s formed ex tend through a large s i z e range. Some of the product p a r t i c l e s mus t b e c rushed to a sma l l s i z e and par t ia l ly rota ted out of thei r o r ig ina l posi t ion before the major c r a c k s can b e comple ted a n d the larger pa r t i c l e s sepa ra ted . T h i s d i s in teg ra t - ing ac t ion , together with the preferent ia l e x p o s u r e of the l a rge r p a r t i c l e s to s u c c e s s i v e breaking opera t ions , r e s u l t s in a s i z e dis t r ibut ion of the product h i c h i s the o b j e c t of in t ens ive s tudy.

T h e natural s i z e dis t r ibut ion of a homogeneous c rushed or g o u n d product undoubtedly f o l l o w s a mathemat ical law, a n d knowledge of t h i s l a w i s nec- e s s a r y before the work inpu t can b e complete ly eva l - uated in terms of pa r t i c l e s i z e produced. 'The power law advanced by ~ a u d i n , ( ? ) and Schuhmann

h a s been much used for th i s purpose. However, when it i s p lot ted on log- log paper wi th pa r t i c l e s i z e in microns a s a b s c i s s a and pe rcen t p a s s i n g a s ordi- n a t e , i t demons t rab ly d o e s not follow a s t r a i g h t l i n e , s o that i t s s l o p e ~ p r o b a b l y i s no t a co r rec t indica- tion of the s i z e dis t r ibut ion down to the gr ind limit.

'The Th i rd Theor , exponent ia l s i z e d i s t r i b ~ t i o n ( ~ ' of a homogeneous mater ia l apparent ly d o e s follow a s t r a igh t l ine , and i s u s e d in t h i s paper to e v a l u a t e the pa r t i c l e s i z e and work input. S ing le cyc le semi- logar i thmic p lo t t ing paper s u c h a s le t ter s i 7 e Il ietz- gen Vo. 340-L110 i s u s e d t o plot the s i z e dis t r ibut ion. T h e ve r t i ca l logar i thmic s c a l e ex tend ing from 1 0 % to 1006 i s u s e d a s the y s c a l e t o plot the percent cumu- l a t ive r e t a ined on , and t h e hor izontal l i nea r s c a l e , o r x s c a l e , r e p r e s e n t s the to ta l work input LVt in kilowatt- hours per ton divided by t h e work index W i . T h e 20% cumula t ive r e t a ined l i n e r e p r e s e n t s 80% p a s s i n g , and the ba lue of x a long th i s b a s e l ine i s d e s i g n a t e d as w, and e q u a l s W t / W i . T h e s i z e P in microns , or t h e s i z e which 80% of the mater ia l p a s s e s , i s found from

T h e v a l u e of P can be found more a c c u r a t e l y from t h i s type of p lo t than from the log-lop which i s cons t r i c t ed and curved in t h i s s i z e r ange . T h e n w i s uni ty a to ta l of o n e work index ki lowat t -hours per ton h a s been app l i ed tc, the rock, and P i s 100 microns .

Several back ing s h e e t s can be prepared to b e p lace ( l behind t h e le t ter s i z e paper . I tadia t - ing s t r a i g h t l i n e s are drawn on each s h e e t from t h e upper l e f t hand corner of the plot. Each l ine repre-

-

s e n t s a mesh s i z e in the ,: 2 s tandard s c r e e n s c a l c .

and c r o s s e s the b a s e l ine y = 20 a t the v a l u e of w found from Equa t ion (3).

Where P1 i s the s c r e e n open ing in microns . T h e l i n e s c r o s s w/2 a t y = 44.72%, w/4 a t y = 66.87%, a n d w/8 a t y = 81.78%.

A s e t of four di f ferent back ing s h e e t s i s conven- i en t , with di f ferent v a l u e s of the work input range x a s l i s t e d below in T a b l e I.

TABLE I

Racking i Range of x S e e t No. (Work Input) I U s e For

T h e v a l u e s with l e s s than 10% Cum. re t a ined o n can be plotted if n e c e s s a r y , by p lac ing a n addi t ional paper below the s h e e t cover ing 10% to 100% Cum., and e x t e n d i n g t h e mesh s i z e l i n e s from the back ing s h e e t s .

T h e exposure r a t io Er i s the e f fec t ive s lope of the plot ted s i z e dis t r ibut ion l ine , and e q u a l s t h e in- t e rcep t wq a t 100% Cum. re ta ined, d ivided by the va lue

( 1) (2 ) ( 3) (4 )

I . I C o u r s e g r i n d i n g p l o t in a s t r a i g h t l i n e of a i r la ter ia l w h i c h y i e l d e d cu rved l i n e s w h e n p l o t t e d by 1 2 o t h c r m e t h o d s . 10 I t i s morle on t w o - c y c l e s emi - log p a p e r t o c o v P r t h e r a n g e from 1 p c t t o 100 p c t C u m . r e t a i n e d on.

0 to0 .14 0 t o 0.70 0 to 1.40 0 to 3.50

Crushing Coarse Grinding Medium Grinding Fine Grinding

of w a t 20% retained. It var ies from 1 to 10 a s the amount of f ines present decreases . If a l l particle s i z e s present were equally exposed to s i z e reduction then the plotted distribution l ine would be vertical, xa would equal w, and the exposure ratio Er would equal unity. If the product cons is ted of part icles a l l of one s i z e the plotted distribution l ine would l i e above the mesh s i z e l ine , and Er would equal zero. In Fig. 1 Er equa ls 0.330, and P equals 6510 microns.

Where b i s the y intercept of the distribution l ine, then the exponential s i z e distribution equation i s

and log b = 21 .301 Er 1 - E r

When Er equals 0.500 each standard s c a l e s i eve s i z e fraction h a s absorbed the same amount of work input under the Third Theory, and the most nearly equivalent s lope of the log-log plot would be 1/2. In t h i s c a s e the constant a in Equation (4) equa ls 3.219/w, a n d b i n Equation (5) equa ls 500.

The value of Er t ends to increase with finer grinding in open circuit. T h e average of 50 lotted va lues on different materials gives approximately Er = 0.24 when P i s 10,000, Er = 0.33 when P i s 1000, and Er = 0.45 when P i s 100; with consider- able variation.

Curved s i z e distribution l i n e s resul t from natural or induced grain s i z e segregations. The plotted curves tend t o return t o the s traight tangent line, a s shown in Appendix A.

The previous paper on t h i s subject(') included a quantity cal led the exposure constant . T h i s e a s de- rived empirically to express a n approximate relation- sh ip between the work input and the exposure ratio. Since i t s publication the method was developed for calculat ing the crack length of a comminution prod- uct, which i s described in t h i s paper. The crack length calculation g ives much more accurate resul ts , and should supersede u s e of the empirical exposure constant.

G R I N D LIMIT

When a rock i s crushed or ground i t theoretically produces par t ic les of every possible s i z e between that of the feed and that of the unit l a t t i ce a t about 0.0005 micron. Flowever, the practical s i z e limit of p r t i c l e s produced by the ordinary grinding of rock now seems to l i e in the vicinity of 1/10 micron or 1000 Angstrom Units. I t i s believed that the regular la t t ice s tructure of c rys ta l s i s interrupted by zones of misalignment a t intervals in the order of 200 linear unit l a t t i ce groups. T h e s e constitute zones of weak- ness , a ~ d divide each rock crystal into mosaic blocks of a diameter approximately equal to the grind limit. Since the rock i s more difficult to break within a s ingle mosaic block than i t i s a long the mosaic boundary zones, crushing and grinding to coarse and intermediate s i z e s effectively ends a t the grind limit

s ize . A large increase in the fine grinding work in- put i s required to reduce rock appreciably below t h i s . s ize. In support of this reasoning i t i s observed that f ine f i laments of quartz or a sbe s to s , which presum- ably do not contain many regular mosaic boundaries, have a much higher specif ic tensi le s t rength than larger sec t ions with a mosaic structure.

The former grind limit of 0.700 micron was de- termined yea r s ago under the Rittinger Theory with the power law s i z e distribution.( ' ' ) When i t became possible to calculate crack lengths under the Third Theory i t was immediately apparent that the grind limit should be smaller. Many calculat ions were made from grindability t e s t s a t different mesh s i z e s on the same ore to try out different grind limits. I t was found that with a grind limit of 0.100 micron each revolution of the t e s t mill would p o d u c e the same crack length on the same ore ground a t different mesh s izes . The grind limit i s not precisely defined by these calcu- la t ions and probably var ies somewhat with different materials. However, it s e e m s to l i e within the l imits of 0.200 to 0.050 micron and one-tenth micron i s a sat isfactory average value. T h e grind limit t e s t cal- culations were too extensive to be included here.

I t i s assumed in this paper that a l l crushing and ordinary grinding effectively terminate a t the grind limit, and the work required i s calculated down to th i s s ize .

M A T H E M A T I C A L R E L A T I O N S H I P S

T h e Third Theory statement that the work input i s proportional to the crack length produced resu l t s in t he m r k index equation (6). When W represents the kilowatt-hours required to reduce a short ton from 80% pass ing F microns to 80% pass ing P microns, and W i i s the work index, then

Rh en the fourth The work

any three of the four quanti t ies are known can be found by transposing the equation.

input in joules, or watt-seconds, per gram i s 3.97 W.

T h e work index i s a crushing and grinding para- meter, which can be determined from plant operation or laboratory testing. Numerically, i t i s the Kwh re- quired to reduce a short ton from theoretically infi- ni te feed s i z e to 80% passing 100 microns. I t i s used to design crushing and grinding instal lat ions and to compare mechanical e f f ic ienc ies in exist ing plants.

T!le work index of any homogeneous material -

under homogeneous reduction conditions should con- tinue constant for a l l feed and product s i z e s . How- ever, natural or induced grain s i z e segregat ions fre- quently cause the work index to increase or decrease a s t he product s i z e becomes smaller. The same ef- fect i s produced by a se lec t ive action of the reduc- tion machine toward certain part icle s i z e s , which c a u s e s variat ions between the s l opes Er of the plotted s i z e distribution l i ne s of feed and product.

T h i s response of the work index to any se lec t ive change in crushing and grinding condit ions makes i t a valuable pract ical criterion of the actual work in- put required.

According to the Third Theory the specif ic work input var ies (a) a s the crack length produced, (b) a s the square root of the new surface a rea produced, and (c) a s t h e s i z e in microns 80% of the product p a s s e s to the exponent - 1/2, a s in Eq. (6), when the work index remains constant . T h e 80% passing s i z e P i s se lec ted a s a convenient and pract ical reference point. The Third Theory s i z e distribution plot s h o w s tha t the s i z e 100% p a s s e s i s an indeterminate extra- polation.

T h e three c r i t i cs of the Third Theory previously referred to ( 3 ) (4) ( 5 ) contend that var iat ions in the work index a t different product s i z e s indicate that the work input v a r i e s a s the product particle s i z e to a negative variable exponent which may be greater, equal to, or l e s s than -1,1'2 in individual c a s e s . If i t can be shown that the crack length produced by a constant work input remains constant a t different p o d u c t s i z e s , while the work index increases or de- c reases , the validity of the Third Theory i s confirmed.

I t i s shown in Appendix B that the total crack . .

length i s represented by the following equation:

173.2 Cr =- V f T

[ y l ( 1 - Er) t 6 9 . 9 2 E ~ G ] (7)

where y 1 i s l e s s than 100% cumulative retained on the grind limit of 1/10 micron, and G i s a finite inte- gral. The relat ionship between y l and G i s shown in Table B-I. Char t s can be prepared on log-log paper from T a b l e B-I1 by means of which the total crack length Cr can be found graphically for any value of the 80% passing s i z e P and the plotted s i z e distribu- tion s l o p e Er. With t h e s e char t s the crack leng ths of feed a n d product can he obtained immediately from the s i z e distribution plots, and the crack length per unit of work input can be rapidly calculated.

P R O O F T H A T C R A C K L E N G T H P R O D U C E D IS P R O P O R T I O N A L T O WORK I N P U T

T h e method described for plotting screen ana- l y s e s and measuring to ta l crack leng ths down to the grind limit of one-tenth micron supply the essen t ia l tools for t es t ing the Third Theory of Comminution. However, the variat ions between the breakage char- ac te r i s t i cs of different mater ials a re s o wide that t e s t s on o n e or two o r e s a re not suff icient to es tab- l i s t any theory. Conforming t e s t s on a wide variety of o res are necessary before a theory c a n be con- s idered a s proved.

Fortunately, such a s e r i e s of t e s t s ex i s t . Stand- ard closed circuit ball mill g i n d a b i l i t y t e s t s have

been made in the Allis-Chalmers Research Laboratory a t var ious mesh s i z e s on more than a thousand differ- en t mater ials , with publication of the resulting ne t grams per mill revolution and the work index values. ( I 2 ) ( I 3 ' Seventeen mate r ia l s were found in t h i s l i s t which had been tested by grinding to al l pass ing s i z e s of 28, 35, 48, 65, and 100 mesh, with a few ex- cept ions. T h e crack leng ths of minus 150 mesh and 200 mesh grindability products cannot be found accur- a tely un less the s i z e distribution ana lys i s of the products extends to s i z e s below 200 mesh.

Of the 17 s e t s of t e s t s ava i lab le one ( T e s t 1477A on Anaconda Copper ore) was rejected because of ob- vious inaccurac ies in some of the product screen ana- l y s e s ; and another ( T e s t 1036A on Quincy copper ore) was rejected because the mill feed w a s a blend of irregular fine s i z e s . T h e remaining 15 t e s t s were al l used to confirm the Third Theory with resu l t s l i s t ed in T a b l e s 11, 111, and IV.

The screen a n a l y s i s of each minus 6 mesh feed sample and each mesh undersize product sample was plotted by the Third Theory method, and the exposure ratio s t raight l ine w a s drawn to determine the s lope Er. In most c a s e s th i s line was tangent t o the plotted l ine a t t h e 80% pass ing (20% Cum. on) s ize .

If the plotted product l ine followed the s t raight tangent line the ore was des igna ted a s type I; if the plotted l ine curved to the right a t the finer s i z e s , in- dicat ing a natural grain s i z e deficiency, the ore was designated a s type 11; if the plotted line curved to the left a t the finer s i z e s , indicat ing a natural grain e x c e s s , the ore was designated a s type 111. T h e amount of departure of curves of type I1 a n d 111 from a straight line was indicated by the le t t e r s S, M, or L for small , medium, or large. If t h e plotted curve indicated a return to t h e s t raight tangent l ine a t 200 mesh or above, the le t ter R w a s added. None of the plots extended beyond 200 mesh.

T h e crack length Cr of each feed and product w a s found from the 80% pass ing s i z e P and the s lope E r by the c h a r t s prepared from T a b l e B-11. F ina l ly , the crack length in centimeters produced per revolution of t h e t e s t mill was computed from Equation (8) in Col. 10

Cm (Gross G/Rev.) (Crp - Crf) - -

Rev. - Specific Gravity

where Crp i s the crack length of each mesh undersize product and Crf i s the crack length of the new mill feed. Gross Grams/Rev. e q u a l s ne t Grams/Rev. di- vided by the fraction of the feed retained on the mesh s i z e tested. Since one mill revolution represen ts a constant work input, the constancy of the Cm/Rev. a t different mesh product s i z e s i s t h e confirmation of the Third Theory.

T a b l e I1 l i s t s the spec i f ic gravity and type of each ore t es ted .

r a b l e 111 l i s t s the resu l t s of each t e s t on each ore. The ores a r e arranged and numbered in the order of their work index variat ions. Ore No. 1 h a s the most rapidly increasing W i from 28 mesh to 1 0 0 mesh, ores in the middle of the l i s t have W i va lues approximately constant a t a l l mesh s i z e s , and ore No. 15 h a s the most rapidly decreas ing W i values. T h i s arrangement permits the comparison of other properties with the work index variat ions. Under the sys tem adopted by the c r i t i cs of the Third Theory ( 3 ) ( 4 ) ( 5 ) , the work done would vary a s the product par t ic le diameter to the exponent-> 1 / 2 for the No. 1 ore = -1/2 for the o res near the center of the Table , and -< 1 / 2 for the No. 15 ore.

Table IV l i s t s the averages from Table111 for e a c h mesh s i z e . The est imated va lues below 100 mesh were obtained by extrapolation. T h e y are useful for compari- son with individual bal l mill grindability resu l t s .

Comparison of Col. (10) with Col . (3) in Table 111 shows t h a t increases or d e c r e a s e s in the work index with finer grinding do not c a u s e changes in the crack length produced per mill revolution. T h e cent imeters of new crack length per revolution con- t inue as nearly constant a s the accuracy of th i s type of test ing al lows. T h e s e resu l t s a r e a definite con- firmation of the Third Theory of Comminution, and indicate that the work index variations resul t from individual heterogenei t ies of material or treatment.

Appendix C contains a more de ta i led d i scuss ion of the da ta in Table 111, together with a method for calculat ing the e f fec t of removing f ines from the feed, a n d for making grindability t e s t s with undersize feed and products.

Appendix D descr ibes a t e s t showing that an increase in the circulat ing load d e c r e a s e s the work index but does not decrease the crack length pro- duced per mill revolution.

TABLE II

No.

1

2

3

4

5

6

7

8

9

1 0

11

1 2

1 3

1 4

15

A-C Tes t No.

lOOOF

913A

1060A

lOOOB

1167A

504B

570A

910A

550B

730A

938A

lOOOG

lOOOA

1592

684A

Company

White P i n e (Copper Range) Sandstone

Morenci ( P h e l p s Dodge) Porphyry

Chino (Nevada Cons.) Porphyry

White P i n e (Copper Range) Sandstone Mixture - Hard & Soft

Reserve 'Mining Magnetic Taconite

Springs Mines, South Africa Quartz

L i t t l e Long L a c , Ontar io Quartz

Anaconda, Montana Si l ic ious

Benquet, P . I . Quartz

San L u i s , 'Mexico Quartz

Utah Copper, Arthur Mill Porphyry

White P i n e (Copper Range) Shale

White P i n e (Copper Range) Amygdaloid

Malartic, Quebec Quartz

Ajo (New Cornelia) Porphyry

Sp. Gr.

2.63

2.65

2.65

2.68

4.00

2.71

2.64

3.23

2.66

2.78

2.86

2.97

2.93

2.90

2.68

-6M Feed TY PC

11-L-R

111-L

11-L

111-L

111-L

I

11-U

I

11-M

I

11-S

11-L

11-S

11-M

11-L

Ore

Copper

Copper

Copper

Copper

Iron

Gold

Gold

Copper

Gold

Gold

Copper

Copper

Copper

Gold

Copper

T A B L E I11

(1) ( 2 ) ( 3 ) (4 ) ( 5 ) ( 6 ) (7 ) (8) ( 9 ) (10) (1 1) No. N e t F e e d Cm

Mesh Wi P E r G/Rev. % O n

~r ~ r 6 - K TYP e Rev.

(1) III-M I I I I

(2) I I I I I

(3) II-L II-L II-L I I

(4) III-S III-S III-S I I

(5) II-M II-S II-S II-S II-S

(6) II-S II-S II-S II-S Il-S

(7) II-L II-M 11-51 II-h1 I

(8) 11-L I I I I

144

Feed 28 7.2 35 10.1 48 12.0 6 5 14.8

100 16.0 Av. 12.02

Feed 28 9.3 35 10.1 48 10.6 6 5 10.7

100 11.7 A v . 10.48

Feed 28 9.3 35 10.7 48 9.8 65 10.9

100 12.4 Av. 10.62

Feed 2 8 12.0 35 10.6 48 10.9 65 11.8

100 13.0 r l v . 11.66

Feed 2 8 10.2 35 10.3 48 9.9 6 5 10.7

100 11.0 A v . 10.42

Feed 28 17.4 3 5 17.1 48 16.8 6 5 14.9

100 15.8 4 v . 16.40

Feed 28 17.0 3 5 16.7 48 16.6 6 5 16.2

100 15.4 A v . 16.38

Feed 2 8 11.4 35 12.5 48 12.1 6 5 11.4

100 12.0 A v . 11.88

TABLE I11 ( C o n t ' d . )

No. Net Feed Cm Mesh Wi P Er G GJP - Rev. K TYF G/Rev. %On Feed

28 16.6 3 5 16.8 48 15.2 6 5 15.4

Av. 16.00

Feed 2 8 16.9 35 15.2 48 15.0

100 16.4 .4v. 15.88

Feed 2 8 13.9 3 5 11.5 48 10.7 6 5 10.6

100 10.7 Av. 11.48

Feed 2 8 16.0 35 14.4 48 13.5 6 5 12.5

100 12.0 Av. 13.68

Feed 2 8 23.5 35 21.7 4 8 19.8 65 19.1

A v. 21.02

Feed 2 8 15.0 3 5 14.7 48 13.4 6 5 12.4

100 12.3 Av. 13.56

Feed 2 8 17.6 35 15.9 48 14.8 65 13.2

100 13.0 Av. 14.90

T A B L E IV - AVERAGES FROM TABLE 111

(2) (3) (4) (5) (6) (7) (8) (9) (10) (14) N e t F e e d E r

Mesh W i Grams % P (Calc.) Cr Cr ~m K Rev.

-

On (Calc.) Rev. F e e d - - 2121 .315 12.77 589 -

28 14.22 4.092 66.34 447.3 .290 23.00 487 17.75 20.97 35 13.89 3.119 72.30 325.8 .285 25.97 469 17.41 22.35 48 13.41 2.581 77.09 235.3 .262 28.51 4 38 17.13 22.83 6 5 13.19 2.070 80.91 163.9 .262 32.86 420 17.15 23.70

100 13.21 1.666 84.31 112.8 .265 38.68 411 17.23 24.38 Av. 13.59 - - - - - - 17.33 22.84

ESTIMATED

SUMMARY A N D C O N C L U S I O N S and a s b e s t o s , and the necess i ty of using empirical .. .

Extens ive evidence h a s been presented which Equation (C 3) to increase the plant work index va lues a t fine product s i z e s . T h e equat ions submitted permit

confirms the Third Theory of Comminution. All avai l - calculat ion of the crack lengths a t grind l imits other

able grindability t e s t s a t minus 28, 35, 48, 65 , and than 0.10 micron. Such calculat ions show that doubl- 100 mesh on 15 different ores were analyzed. T h e s e ing or halving th i s grind limit h a s comparatively l i t t le

t e s t s show that one revolution of the laboratory t e s t effect on the crack length values. The grind limit ball mill produced a constant new crack length on used i s not c r i t i ca l to confirmation of the Third Theory,

each ore, regardless of the mesh s i z e of the product and probably represents a n average value for different

and the work index variations a t e a c h mesh s i z e . . . T h e crack length in centimeters per c c of a

crushed or ground material containing i t s natural f ines equa ls by definition the square root of one-half of i t s surface a rea in s q . cm. per c c . T h e crack length i s found by char t s calculated from the Third Theory ex- ponential s i z e distribution plot ( 9 ) . Curves in the s i z e distribution l ines represent natural or induced grain s i z e segregat ions; the curves c a u s e variat ions in the work index, but they presumably osc i l l a te about the s t raight tangent l ine used in the crack length calcula- tion, and do not a f fec t the crack length produced.

- -

The crack length i s calculated down to the new g i n d limit of one-tenth micron. T h i s grind limit had been previously determined by essen t ia l ly the same method that w a s used to confirm the Third Theory, but from other t e s t da ta . T h u s the experimental evi- dence submitted confirms both the Third Theory and the grind limit.

T h e c a s e for the Third Theory would be fortified by a check determination of the grind limit us ing a different method. A direct check i s not avai lable a t present , but there i s considerable indirect evidence that a grind limit e x i s t s a t about this s i z e . T h i s in- c ludes analogy with recen t work on the s t ructure of meta l s , the frequent observation that there i s a pro- nounced change in the properties of s o l i d s a t the colloidal s i z e range, the increased spec i f ic tensi le strength of f ine filaments of such minerals a s quartz

mater ials . If the grind limit concept i s not allowed, the

correspondence of the f igures in Col. 1 0 of Table 111 must be otherwise explained, and no other explana- tion s e e m s avai lable .

T e s t s a t different circulating loads show that the work index d e c r e a s e s a s the circulat ing load increas- e s , while the crack length produced by a constant work input remains constant .

Knowledge of character is t ic va lues of the quantity Cr J P c a n be used to es t imate the probable s i z e d i s - tribution a t any 80% pass ing s i z e P. However, such es t imates wil l not include any s i z e segregat ion effects .

T h e laboratory work index i s calculated from the ne t grams made per mill revolution. However, th i s value i s affected by the configuration of the feed s i z e distribution both above and below the mesh s i z e t es t - ed. Any curvature resul t ing from natural grain s i z e e x c e s s e s or def iciencies , and any variation from the normal relat ionship between the feed and product tan- gent l ine s l o p e s , affect the ne t Grams/Rev. and the W i value. T h i s i s a l s o true in commercial plant mea- surements . If the work index i s to se rve a s a n accur- a t e measure of the work input required for reduction i t must necessar i ly vary in response to breakage hetero- genei t ies of the material.

Variat ions in the work index resul t from part ic le s i z e segra t ions and from variat ions in the eff ic iency

of reduction t o 80% p a s s i n g different s i z e s . They d o such a curve i s continued by a n a l y s i s a t increasingly not c a u s e variat ions in the total crack length pro- finer s i z e s , the curve will reverse i t s direction, re- duced down to the grind limit, a n d do not a f f e c t the turn to the s t raight l ine, and c ross i t . When the s t raight val idi ty of the Third Theory of comminution. l ine i s properly chosen the plotted curve will presum-

ably describe equivalent loops on both s i d e s of i t , R E F E R E N C E S and will follow i t upward to the grind limit, possible

with additional smaller osc i l l a t ions around it . The (1) F r e d C. Bond: The Third Theory of Comminution, s t raight l ine, cal led the tangent l ine, thus defines the AItlE Trans. , May 1952, Vol. 193, p 484. equivalent s lope of the curved s i z e distribution line (2) A. S. Kannewurf: ROCK PRODUCTS, may, 1957. of a crushed or ground material containing a l l of i t s (3) R. J . Char les : Energy-Size Reduction Relation- natural f ines . T h e curves a re caused by natural or

s h i p s in Comminution, AIME Trans . , Vol. 208, 8 0 (1957). induced grain s i z e s in the material, and affect the

work index va lues a s well a s the lotted s i z e distri- (4) J . A . Holmes: A Contribution to the Study of Com- bution a t those particle s i z e s . Ana lyses over a very

minution, Institution of Chemical Engineers , Eng- large range a re necessary to define t h e s e curves. land. T r a n s . Vol. 35, No. 2 (1957).

T h e three main types of s i z e distribution l ines (5) J o n a s Svensson and Jakob Murkes: An Empirical a re i l lustrated in F ig . A-1. T h e s e hypothet ical l ines Relat ionship between Work Input and Par t i c le Size differ in their curvature, but a l l represent materials Distribution before and a f te r Grinding. International with the same crack length C r of 32.6 c m i c c , and Vineral Dressing Congress , Stockholm, 1957.

(6) A. 0. Gates : An Experimental Invest igat ion in Rock with w = 0 . 5 8 0 , ~ ~ = 0.258, Er ~ 0 . 4 4 4 , a n d P = 297 Crushing performed a t Purdue University, AIME microns, or 80% p a s s i n g 48 mesh.

Trans . , Vol. 55, P. 875-909 (1916). In analyzing t h e s e lots the increasing constr ic- (7) A. 21. Gaudin: An Investigation of Crushing Phen- tion near the top of the chart should be remembered. A

omena, AIME Trans . 1926, Vol. 73 , pp 253-316. small loop near the top i s equivalent t o a larger one (8) R. Schuhmann: Pr inc ip les of Comminution, AIME below, and a curve near the bottom of the chart may

T P , 1189, Ju ly , 1940. appear to be almost a s t raight line. (9) Fred C . Bond: Comminution Exposure Cons tan t by Curve I in Fig. A-1 represents a material with

the Third Theory, MINING ENGINEERING, Decem- homogeneous breakage, and no natural or induced ber, 1957, AIME Trans . 1958. grain s i z e . I t lots in a s t raight l ine which cons t i tu tes

(10)A. F . Taggart : Handbook of Mineral Dressing. Sec. the tangent l ines of types I1 and 111. I t s work index 19, pp 145-150, with reference to original papers. continue 'Onstant a t mesh sizes. John Wiley and Sons, Inc., New York.

(11)Fred C. Bond and Walter L. Maxson: Grindability and Grinding Charac te r i s t i cs of Ores. AIME Trans . 4

Vol. 134, p 296 (1939). (12)Fred C. Bond: Standard Grindability T e s t s Tabu-

la ted. AIME Trans . Vol. 183, p. 313 (1949). (13)How to Determine Crusher and Grinding Mill S i z e s

. . . Accurately. Allis-Chalmers Bulletin 07R7945A, Milwaukee, Wilconsin.

(14)A Study of the Feas ib i l i ty of Hydraulic Transport of a T e x a s Ligni te , U.S.B.M. - R.I . 5404 (1958). Table 9, page 19.

APPENDIX A - GRAPHS

N A T U R A L GRAIN S I Z E S A N D C U R V E D DISTRIBUTION LINES

Computation of the Third Theory crack length requires a determination of the s l o p e Er or the plott- ed s i z e distribution l ine. If the plotted line is s t ra igh t represent ing a type I mater ial with homogen- e o u s breakage, it i s merely extended upward to the top of the chart , and the s lope Er i s found precisely. 7 1 , i , I\\\\ However, the plotted l ine may curve to the right L3 (type 11) or to the l e f t (type 111) a s i t i s t raced up the .<

..

chart . The meaning of these curves must be consider- * 0 10 e d , and the s l o p e of a s t raight l ine represent ing a n equiva len t crack length must be found.

T h i s i s made poss ib le by the discovery that when FIG. A-1 Three main types of s i z e distribution l ines .

'The c u r v e s in l i n e s I1 a n d 111 r e s u l t frorn loca l - i zed grain s i z e d e f i c i e n c i e s a n d e x c e s s e s . T h e s e may be induced gra in s i z e s r e su l t ing from a pa r t i cu la r reduct ion rliachine o r p r o c e s s . However, t hey more of ten r e p r e s e n t na tu ra l c h a r a c t e r i s t i c s of the rock or o the r ma te r i a l be ing broken, in which c a s e they a re c a l l e d na tu ra l gra in s i z e s . Na tu ra l gra in s i z e s c a n be d i s t i n g u i s h e d froni induced grain s i z e s by the i r p e r s i s t e n c e with d i f ferent reduct ion p r o c e s s e s .

A typ ica l na tu ra l grain s i z e d ~ r i a t e r i a l i s a l oose - ly cemen ted s a n d s t o n e which b reaks niore readi ly be- tween the cemen ted p a r t i c l e s than a c r o s s the individ- u a l g ra ins . I t s work index i n c r e a s e s a s the na tu ra l grain s i z e i s reacl led . I t s p lo t t ed s c r e e n a n a l y s i s a b o v e the na tu ra l gra in s i z e belongs t o type 11, and s h o w s a marked grain s i z e de f i c i ency . Curve I1 in F i g . A-1 s h o w s a maxirnurrl grain s i z e d e f i c i e n c y a t 150 rnesh. which d i s a p p e a r s where i t c r o s s e s the t an - gen t l ine a t 200 rnesh; the na tu ra l grain s i z e e x c e s s c e n t e r s a t a b o u t 325 m e s h , and i t r e tu rns t o the tan- gen t l i ne a t abou t 18 rnicrons. 4ny p o s s i b l e curvature a t s i z e s c o a r s e r than 65 rnesh i s no t showrl l ~ e c a u s e of t he e x p a n s i o n o r the cha r t nea r the bot tom, and the t angen t l i ne i s tlrawn through the p lo t t ed l i n e where i t c r o s s e s t h e b a s e l ine a t 20%. Cunl. r e t a ined on 48 rnesh. I t s work index shou ld i n c r e a s e from 100 mesh to 200 rnesh, s i n c e grinding of the hard individ- u a l s a n d s t o n e e r a i n s i n c r e a s e s .

Curve 111 r e p r e s e n t s a much c o a r s e r s a n d s t o n e with ;I natura l prain s i z e e x c e s s c e n t e r i n g a t 100 mest1 antl c r o s s i n g the tanqent l ine a t 200 rnes l~ . with a con lpensa t ins s i z e de r i c i ency cen te r ing a t abou t 37 rnicrons. \Yhen grouncl t o 80'; p a s s i n g '1.8 mesh any na tu ra l gra in s i z e d e f i c i e n c y a t t h i s s i z e i s imper- cep t ib l e , a n d the t a n g e n t 1 i n e . i ~ drawn t l~rougl i i t . I t s work index should i n c r e a s e frorn 28 mesh to 65 rnesh and l~rol ,ably d e c r e a s e s l igh t ly a t 200 mesh .

Ul'l~en i i material h a s a na tu ra l gra in s i z e i t s work index t e n d s t o i n c r e a s e wi th finer ?rinding a s the nat - ura l g a i n s i z e i s r eached , t o rerr~ain c o n s t a n t a t mesh s i z e s where the na tn ra l grain s i z e p e r s i s t s , and to d e c r e a s e so lnewha t a t rllesh s i z e s f iner t han the na t - ura l grain s i z e .

When a s c e n d i n g up the cha r t the minimum work index i s ob ta ined where the s i z e d i s t r ibu t ion l ine b e a r s to the r i ~ h t of t he t angen t l i ne d i rec t ion a t t he g r e a t e s t a n q l e , antl the maximum work index i s obta in- e( l where i t \ )ears to the lef t of the t angen t l i ne di- r ec t ion a t the g r e a t e s t ang le .

'The t angen t l ine which properly d e f i n e s the s l o p e Er i s u sua l ly drawn t angen t to the curved p lo t t ed l i n e a t the \ l a se l ine . Only in e x c e p t i o n a l c a s e s i s t he re s u f f i c i e n t cu rva tu re a t the b a s e l i ne to require chang- ing the s l o p e of the t angen t l i n e ; in t h e s e c a s e s :idJi- t ional s i z e a n a l y s e s of the mater ia l c a n a id in indi- c a t i n q the proper s l o p e .

F igure -2-2 g i v e s a n a c t u a l example of a l a rge type 11 cn rve . I t r e p r e s e n t s ~ n i n u s 1/:2 inch l ign i t e c o a l a f t e r be ing t r anspor t ed in a wet s lurry 7 1 . 5 ni i les through a ~ i ~ e l i n e . ( I 4 ) I t s h o w s a pronounced gra in

s i z e de f i c i ency presumably induced by s e l e c t i v e de- q a d a t i o n dur ing t r anspor t a t ion , which s t a r t e d a s c o a r s e a s 20 mesh a n d c e n t e r e d a t a b o u t 30 microns . Re low t h i s s i z e the p lot ted l i ne r e tu rns to the t an - g e n t l i ne , a n d c r o s s e s i t a t 6 niicrons. 'The s i z e ana ly - s i s e x t e n d s to 5 microns and s h o w s t h e s t a r t of the incluced grain s i z e e x c e s s below 6 microns . T h e c r a c k length Cr i s 17.6 c m / c c , with w = 0.220, x 2 =0.121, Er = 0.550, a n d the 80% p a s s i n g s i z e P e q u a l to 2065 microns .

FIG. A-2 Actual e x a r ~ l ~ l e of a large type I 1 curve.

'I'he 'l'liird Theory s i z e d is t r ibut ion e q u a t i o n s ( 9 ) s h o w t h a t where d i s a n y micron s i z e , then

10 Er 2 - l og y X = 6- L&- ( 1 - Er - 0.0699 r q (131)

l og b - l o g Y 10 l og b - 1.301

\&here y r e p r e s e n t s the pe rcen t cumulat ive we igh t r e t a ined on any q i n d l imi t s i z e i n microns G I then

log y = 2 ~7 - (2-1.301 Er) b P - L/% (1-Er) (113)

Equa t ion (B3) i s u s e d t o find the pe rcen t curnuln- t i ve r e t a ined on the grind lirnit G I , o r on any nlicron s i z e d which nlay be s u b s t i t u t e d for G I . Seven p lace l o g t a b l e s should be u s e d to e v a l u a t e Y accura t e ly .

,4ssurninK equivalent c u b e s , one cubic centimeter of par t i c les d microns in diameter wil l have a crack length Cr of d30 ,000/d cent imeters , a n d a surface area of 60,000.ld square cent imeters . I t follows from Equation (Bi) that

173.2 12.12 x Cr;- = Jb 2 - l o g y

Uy subst i tut ion in Equation (B2) and Equation (5) Cr i s found a s a function of y

The total crack length in a crushed or ground pro- duct extending down to the grind limit i s the summa- tion of the Cr va lues from y = 0 to y = yl , or

Vihere Cr i s defined by Equation (11) then

~ ~ 7 3 . = Cr (Total) =

A P P E N D I X 0 - CALCLTLA'TIONS CALCULATION S T E P S

T h e crack length calculat ion of a comminution product which contains i t s natural f ines i s made in the following s teps : (1) The s i z e distribution a s percent cumulative retain-

ed on i s plotted in the Third Theory exponential manner.(9) T h e product s i z e P and s lope E r a re obtained from the plot. If the plotted s i z e distri- bution line i s curved, the s lope line i s ordinarily drawn tangent to i t a t 20% Cum. retained.

(2) The grind limit GI in fract ions of a micron i s taken as 0.100, with 6 i equa l to 0.3162. The value of y l is found by a seven place table of logarithms from Equation (B11).

(3) The value of G for y l is found from the chart con- s t ructed from 'Table (B-I).

(4) T h e total crack length C r in cent imeters per c c i s found from Equation (R10).

(5) T h e total surface a rea above the grind limit i s found from Equation (B12) where Sg i s the specif ic gravity.

i' ( 2 - l o g ~ ) - - E r ( 2 - l o g ~ ) + 0 . 6 9 9 E r ST. Cm. 2 ( ~ r ) ' d~ Surface Area a ------ = - 2 - l o g y Gram S g (El21 (R7) (6) The crack energy C e in joules per c e n t i m e t e r b e -

(" tween the feed and product s i z e s is found from = y 1 ( 1 - E r ) + 1.610 E r dy ( n 8 ) Equation (B13) when C r p i s the total crack length 4.605 - logey of the product and Crf is the total crack length of the feed.

Khen ,4.605 - 1og.y = - loge$, , then the last C e = 3.97 it' Sg integral becomes Crp -Crf

s o that

173.2 Cr - 'Total ,/F [ y1 (1 - Er) + 69.92 E r G ] (1310)

'rile numerical value of the integral G i s finite for any value of l e s s than 100. I t i s considered as posi t ive to avoid confusion of s i g n s . I t s value f o r y 1' 99.00; retained on the grind limit w a s found by care- ful planimeter n leasure~nents to be 9.50. Values of G for other va lues of y l were found by other planimeter measure~nents , and adding or subtract ing a r e a s frorr~ 9.50. 'L'hese a re l i s ted in Table B-I

For accura te ca lcu la t ions of crack leng ths and sur face a r e a s the va lues of yl and G l i s ted in Table F3-I should be plotted to a large s c a l e on l inear paper.

The surface energy required S e in ergs per square centimeter of new surface a rea produced i s

-

(7) When Crf i s not avai lable the crack energy C e atlove the product s i z e can be found from Equation (015) . where W t equa ls W i t imes w

and 1.985 x lo7 W t Sg

S e = (crPj2

A useful approximate relat ionship between the ex- posure rat io E r and the s l o p e of the Schuhrrlann log- log plot i s given in Eq. (R17) below:

T A B L E B-I

The micron s i z e d a t any percent cumulative weight y retained o n d i s found from Eq. (B18 below: - 14.31 (log b - 1.301) (2 - log y

~ ' d = - w (log b - log rn) ) (B-18)

CRACK LENGTH CHARTS

Charts have been constructed s o tha t the crack length can be found graphically, and the laborious ca lcu la t ions described can be avoided. T h e value of Cr was calculated for each decimal uni t value of P from 80% p a s s i n g one micron to 80% pass ing one mil- lion microns, and for each decimal uni t value of the s lope Er from 0.1 to 0.9 inclusive; a total of 63 cal- culat ions. T h e s e are shown in Table R-11. Then these are plotted on single-cy cle log-log paper with C r on the vertical s c a l e and P on the horizontal s c a l e , s t raight l i n e s can be drawn for each decimal s lope value Er, and intermediate l ines c a n be inser ted using a logarithmic rule. Such a plot i s shown in outline in Fig. B-I. -

FIG. B-1 Crack length plotted graphically.

T A B L E R-11

NOTE: Eq. (B6) was integrated with the ass is tance o f Mr. Joseph Wegerer, formerly of the Allis-Chalmers Process ing Machinery Dept.

CRACK LENGTH VALUES FOR PLOTTING C r (cm/cc) * 1,000,000 P Microns 100

Er (Slope) C r 1

I

1,000 1 10,000 10

0.495 0.735 0.936 1.136 1.337 1.523 1.699

100,000

0.80 165.8 125.8 64.3 28.40

0.10 0.20 0.30 0.40 0.50 0.60 0.70

27.3 36.6 42.5 48.1 52.7 57.1 60.8

207.0 212.0 208.5 202.0 191.9 183.0 176.0

79.8 93.0

102.1 110.0 115.2 119.4 123.0

10.65 14.00 16.89 19.56 21.95 24.25 26.35

3.76 5.25 6.60 7.87 9.00

10.04 11.00

1.37 1.93 2.47 2.96 3.44 3.91 4.37

A s e t of s i x of t h e s e crack length char t s con- s t ructed from Table B-11 c a n be used t o find the value of the crack length Cr in cent imeters per c c of .solid over pract ical ly the ent i re range of crushing and grind- ing s i z e s directly from a Third Theory s i z e distribu- tion plot. T h e crack length i s a measure of the total work done in crushing and grinding. The sur face area of equivalent cubes in cubic cent imeters per gram down to the grind limit of 0.100 micron i s found by doubling the square of the crack length and dividing by the spec i f ic gravity.

APPENDIX C - COMMENTS

D A T A IN T A B L E S 111 and IV

No t e s t w a s made a t 6 5 mesh or ore No. 10, and no t e s t s were made a t 100 mesh on o r e s No. 9 and No. 13. T h e s e a re the only omissions in the Table. T h e da ta represent 72 grindability t e s t s on 15 ores , or about 500 complete grindability periods. No d a t a were omitted b e c a u s e of l ack of conformity.

In three c a s e s , those of o res No. 2, 4 and 5 , the s lope value E r of t h e s t a g e crushed minus 6 mesh bal l mill grindability feed w a s greater than that of the tangent a t t h e b a s e line. In a l l t h e s e c a s e s the plotted l ine abruptly changed direction to the right a t 20 mesh or coarser , instead of gradually curving to the right to indicate a typical type I1 natural p a i n s i z e deficiency. In t h e s e three c a s e s the s lope l ine E r of the feed was drawn through the intersect ion of the plot ted s ize distribution l ine with the base l ine paral le l to the direction of the plotted l ine a t s i z e s finer than 20 mesh. T h e grindability product l i n e s were a l l drawn tangent in the usual manner.

O r e s No. 4 and No. 5 a re known to cons i s t of a mixture of subs tan t ia l portions of harder and e a s i e r grinding mater ials , and the feed sample of ore No. 2 had a similar s i z e distribution plot. Mixtures of near- ly equal portions of relat ively hard and sof t mater ials , when crushed to such a s i z e that these mater ials a r e effectively segregated in different s i z e fract ions, may require a somewhat different s i z e a n a l y s i s treatment than that descr ibed for natural grain s i z e s .

Figure C-1 g ives a n example of the plotted curves. T h e type I1 s i z e dis t r ibut ion l i n e s and the s t ra igh t tangent l i n e s of t h e minus 6 mesh feed and the minus 28, 35, 48, and 6 5 mesh products of o re No. 11 are shown. The numerical va lues a r e l is ted in T a b l e Ill.

T h e most accura te determinations of the crack length produced per mil l revolution a re made a t 35, 48, and 6 5 mesh. T h e grindability t e s t s a t 28 mesh a r e somewhat l e s s accura te than those a t finer mesh s i z e s , s i n c e the length of each p i n d i n g period nec- e s s a r y to obtain the constant circulating load of 250% i s shor t , and the f i r s t few revolutions which dis tr i - bute the grinding charge may a f fec t t h e net grams of mesh undersize produced per revolution.

T h e products of the t e s t s a t minus 100 mesh have only t h e plotted points a t 150 and 200 mesh t o deter-

mine the s lope Er, which i s consequently not quite a s accura te a s t h o s e with more plotted points. An inspect ion of Col. (10) in Table 111 shows that the crack length produced per mill revolution i s some- what more errat ic in the 28 mesh and 100 mesh tes t s . However, the averages in T a b l e IV show consis tent r e s u l t s a t a l l mesh s i z e s .

In T a b l e 1V Col. (6), the average values of the 80% pass ing s i z e P in microns are l is ted. F o r the -6 mesh average mill feed P equa ls 6 3 8 of the opening of a 6 mesh s ieve , while the average value of the product s i z e P i s 77% of the sc reen openings P1 a t 2 5 0 8 circulating load.

T a b l e 111 shows that the crack length in centi- meters produced per mill revolution remains substan- tially cons tan t for each ore t es ted a t a l l product s i z e s from 2 8 mesh to 100 mesh, and Table 1V shows that the average for a l l 15 ores remains substant ial ly cons tan t a t a l l product s i z e s . T h i s s u s t a i n s theThird Theory s tatement that a constant amount of work in- put produces a constant new crack length, or that the new crack length produced i s proportional to the work input.

WORK I N D E X V A R I A T I O N S

In the Third Theory the work index i s a pract ical parameter which defines the work input necessary to

FIG. C-1 T y p e 11 s i z e distribution l i n e s and the straight tangent l i n e s of the minus 6 mesh feed and the minus 28. 35. 48. and 65 mesh products of ore 11. Numerical values are l i s t e d in Table 111.

reduce a shor t ton to 80% pass ing 100 microns, based upon the work input required over t h e s ize range tes t - ed. Natural grain s i z e s , varying reduction eff ic iencies , or any other conditions which a f fec t the e a s e of re- duction to 80% pass ing the given s i z e , affect the work index, which is much more variable than the crack length production.

The operat ing work index i s found by inser t ing plant d a t a in Equat ion (6). The work i n d e x i s deter- mined by laboratory bal l mill grindability t e s t s from the net grams of mesh undersize produced per revolu- tion. The following rev ised formula i s used:

wi = 44.5 / ( P I ) 0.23 0.82 x G b v x

where P1 i s the opening in microns of the mesh s i z e tested, and G b p i s the net grams of mesh undersize produced per mill revolution. T h i s equation g ives the work index a t average grinding efficiency of wet c losed circuit ball mi l l s 8 feet inside diameter. The efficiency varies a s the inside mill diameter to the exponent 0.20.

Column (3) in Table 111 l i s t s the work index of each grindability t e s t a s calculated from Equation (Cl ) . Since the 15 ores tested a re l i s ted in the order of increasing, s ta t ionary, and decreasing work index values, any cons i s ten t trend in theother va lues l i s ted should show some relat ionship t o the work index. One observed trend i s the decrease in the t re pared -6 mesh feed s l o p e Er and crack length Cr a s the t e s t numbers increase.

Following the n e t grarns per revolution a t equili- brium l is ted in Col. (4) are upward arrows, equa l s i g n s , and downward arrows. T h e s e indicate the direct ion which the net Grams/Rev. took during s u c c e s s i v e grinding periods in approaching equilibrium. ,An up- ward arrow shows that t h e net G/Rev. increased to equilibrium, a n equal sign shows that i t remained substant ial ly constant , and a downward arrow shows that it decreased to reach equilibrium. The general trend i s to decrease in the coarser s i z e s , p a s s through a no-change s i z e , and increase in the finer s i z e s , al- though there are many exceptions. T h e 72 t e s t s show- ed 23 decreasing, 21 equal, and 28 increasing va lues . At 23 mesh only one ore increased to equilibrium, and a t 100 mesh none decreased.

The gross grams produced per revolution i s ob- tained by dividing Col. (4) by Col. (5)/100.

The averages given d o not include the feed samples .

Col. (8) gives the crack length of each sample in cm c c , a s found from the Table 111 charts.

Col. (9) l i s t s the va lues of the parameter Cr 6 They decrease regularly from 6 mesh to 100 mesh, and form a nearly s t raight l ine when plotted aga ins t the mesh opening in microns on log-log paper. T h i s decrease i s a resu l t of the s l ight d e c r e a s e in the ex- posure ratio Er in Col . (7). T h i s parameter can be of

great va lue in predicting the tangent l ine s l o p e s of new reduction products.

The cons tan t cent imeters of crack length pro- duced per mill revolution in Col. (10) confirms the Third Theory and shows no relat ionship to the vary- ing work index values.

The value of K l is ted in Col. (11) shows a near- ly constant relationship of the work index to the crack length produced and feed and product s i z e s .

K = (Crp - Crf ) 6 Wi (Feed % On) T h e work index is thus nearly inversely propor-

t ional to the product of the cent imeters of new cracks per cc and the square root of the 80% pass ing s ize , divided by the percent weight of the feed retained on the mesh s i z e tested. T h e feed percent on and the 80% pass ing s i z e P ref lect natural or induced grain s i z e s which a f fec t the work index without affect ing the crack length produced.

Table IV s h o w s that the average va lues of K in Col. (11) increase s l ight ly with finer s i z e s . T h i s i s par t ly compensated by the s l i g h t decrease of the Wi values in Col. (3). T h e re la t ive invariability of K shows that the work index variat ions a re primarily the resul t of natural and induced grain s i z e deficien- c i e s and e x c e s s e s , within the framework of the Third Theory. The work index v a l u e s a re not consis tent ly proportional over the complete s i z e range to the prod- uct particle diameter to any constant exponent, s ince the exponent must vary a s the grain s i z e i s approached. The only exception i s the reduction of a homogeneous material, when the exponent i s -1/2 and the work index remains constant.

T h e crack energy and sur face energy inputs cal- culated from Equat ions B13, B14, B15, and B l 6 a re a l s o affected by grain s i z e segregat ions and reduc- tion efficiency variations, s i n c e these influence the work input W and the work index W i . The net work in- put to the grindability t e s t mill i s about 60 joules per revolution (12).

E Q U I V A L E N T SIZE O F S C A L P E D F E E D When a crushed or ground material h a s had part

of i t s natural f ines removed, or sca lped out, i t s re- s i s t a n c e to s i z e reduction per ton i s increased. T h i s can be expressed in terms of the increased 80% pass- s i z e PC of a ton of equivalent material containing i t s natural f ines . Where P i s the 80% p a s s i n g s i z e of the sca lped material, the equivalent PC i s the equivalent 80% pass ing s i z e per ton with f ines present , or the s i z e which will have the same crack length a s the s c a l p e d material with the natural s lope Er and para- meter Cr - of the unscalped material.

T h e value of PC can be found by making a Third Theory plot of the scalped material, finding the val- ue of Cr from the charts prepared from Table 1311, and then us ing the charts t o find the value PC which will have the natural s lope Er and natural Cr 16 value of the unscalped material, with t h e s a m e Cr value a s

the sca lped material. The equivalent 80% pass ing s i z e PC, or Fc when the material i s feed, i s used in the bas ic Third Theory equation (1) to calculate work input o r work index. T h e average value of Cr 6 in open circuit unscalped i s about 570, and the s l ope Er should be s e l ec t ed to conform to th i s value.

In c a s e s where the feed cons is t s of part icles which have been screened and vary in s i z e between c lo se limits, the f i rs t s t ep i s t o find the average micron diameter d of the s ized feed, o r the microns which 50% of the s ized feed would pa s s . The crack length of t h e s ized feed i s then found from Cr = d30,000/d. T h e equivalent feed s i z e F C i s found from F c = ( 5 7 0 / ~ r ) ' , f o ru se in Equation (1) .

CORRECTION F O R V E R Y FINE P R O D U C T

When the 80% pass ing s i z e P i s l e s s than 70 microns the work index Wi, a s determined from tes t s above that s i z e , i s multiplied by a factor f to account for the increased work done in producing sub-grind- limit part icles . The factor f i s found from the follow- ing empirical equation:

The smal l amount of material ordinarily present which i s finer than the grind limit when P > 70 probably cons i s t s principally of (1) tiny part icles which acted a s cement between the grains released by grinding, and ( 2 ) corners and edge fragments bro- ken from the mosaic blocks during grinding.

APPENDIX D - CIRCULATING LOADS

E F F E C T O F CIRCULATING LOADS

Grinding t e s t s a t 6 5 mesh were made on ore No. 15 in Table ID a t three different circulat ing loads in addition to the standard a t 250970. T h e resu l t s are l is ted in Table D-I.

A fair indication of the increased efficiency of grinding with increasing circulating loads i s given by the work index drop in Col. (3) . However, there i s no increase in the efficiency of new crack length production, a s shown in Col. (10). T h i s agrees with the Third Theory. An increased circulating load in- c r ea se s t he efficiency of grinding t o p a s s a certain s i z e , but it does not increase the production of new crack length or new surface area, except insofar a s it may ameliorate bal l coating or other deleter ious

conditions.

TARLE D-I

- - - -- - - - - - - - - - - - ---

No. 40 Circ. Net F e e d Cm TY pe Load Wi G/Rev. % On P Er Gr Cr "'7 KG K - - - - - - - -

(15) (Feed) 2270 ,248 11.1 529 11-L 150 13.5 1.83 82.2 159 ,227 31.8 401 17.2 23.5 11-L 250 13.2 2.02 82.2 173 .192 28.6 376 16.1 21.2 11-L 390 12.2 2.33 82.2 182 .189 27.9 376 17.7 22.5 11-L 815 12.0 2.39 82.2 183 .162 26.1 3 53 16.3 20.5