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UBC Theses and Dissertations

Studies on comminution. Jomoto, Kimitaka 1971

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STUDIES ON COMMINUTION Kimitaka Jomoto B o A o S c o Hokkaido U n i v e r s i t y , Japan, 1 9 6 8 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF ' THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n the Department of MINERAL ENGINEERING • We accept t h i s t h e s i s as conforming to the re q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA . December, 1971 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the Head of my Department or by h i s representatives. It i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of M i n e r a l Engineering The University of B r i t i s h Columbia Vancouver 8, Canada Date December 1971 ABSTRACT The p h y s i c a l q u a n t i t y of the square o f t e n s i l e s t r e n g t h over Young 0s modulus has been proposed as a c r i t e r i o n of comminution of rock by Oka and Majima. The value was determined f o r f i v e d i f f e r e n t r o c k s , measuring t e n s i l e s t r e n g t h and Young's modulus. The t e s t i n g method used to determine both rock p r o p e r t i e s was a p o i n t - l o a d t e n s i l e s t r e n g t h t e s t and measurement o f propagation v e l o c i t y of P-wave.' I n order to v e r i f y the a p p l i c a b i l i t y of the c r i t e r i o n thus o b t a i n e d , the drop-weight impact t e s t and b a l l m i l l g r i n d i n g were c a r r i e d out u s i n g the i d e n t i c a l specimen used f o r t e n s i l e s t r e n g t h and Young's modulus t e s t . The experimental r e s u l t was obtained t h a t the square of t e n s i l e s t r e n g t h i s more a p p l i c a b l e than the square of t e n s i l e s t r e n g t h over Young's modulus. ' . • ' In chapter 2, g r i n d i n g t e s t s were performed w i t h a 12 i n c h batch m i l l . A torque meter was i n s t a l l e d to estimate the energy consumption.necessary to achieve s i z e r e d u c t i o n . From the torque measurement, the r e q u i r e d torque f o r the m i l l t u r n i n g remained almost constant r e g a r d l e s s of the m a t e r i a l being ground. A new concept, Energy Index, was i n t r o d u c e d to evaluate g r i n d i n g r e s u l t s and t o study g r i n d i n g problems. For each rock sample t e s t e d , the energy index was determined from torque measurements and the q u a n t i t y of the p r o d u c t s . I t was found the energy index has a l i n e a r r e l a t i o n s h i p w i t h Bond work index under s e l e c t e d g r i n d i n g c o n d i t i o n . - i i -TABLE OF CONTENTS Page INTRODUCTION . CHAPTER 1 o CRITERION OF COMMINUTION '— 5 SCOPE OF PRESENT WORK — : . 5 MATERIALS AND PREPARATION 6 EXPERIMENTAL METHODS 1 1 1 - Young's Modulus 1 1 2 - T e n s i l e Strength ; _. . 1 3 3 - Drop-Weight Impact Test — — 1 5 EXPERIMENTAL RESULTS AND DISCUSSION ' 1 9 . 1 - Young's Modulus . —.. 1 9 2 - T e n s i l e Strength . — — — : 24 3 - Drop-Weight Impact Test 28 o 4 - The Comparison of St /E and C r i t i c a l Height . 36 2 5 - The R e l a t i o n s h i p between St /E and the xResults of B a l l M i l l G r i n d i n g 40 CONCLUSION — • — ^ CHAPTER . 2 . . ENERGY INDEX IN,DRY TUMBLING- MILLING - 4 5 SCOPE OF PRESENT WORK - — — ^ 5 MATERIALS AW PREPARATION — 46 BALL MILL GRINDING TEST PROCEDURE : . 4b EXPERIMENTAL RESULTS AND DISCUSSION 5 1 1 - P r e l i m i n a r y Test . 5 1 2 - Torque Measurement • — 5 5 - i i i -Page 3 - Si z e D i s t r i b u t i o n 5 9 k - Energy Index 6^ • CONCLUSION — ; ' 7 3 SUMMARY : 7 ^ SUGGESTIONS FOR FUTURE WORK . — — ?6 REFERENCES • ; 7 7 APPENDIX — — — — 7 9 . LIST OF FIGURES Page Figur e 1 • Schematic i l l u s t r a t i o n of p u l s e apparatus f o r determining Young 0 s modulus . •—•—•— 11 2 Compressional waves on t h e . o s c i l l o s c o p e screen 11-a 3 Apparatus f o r t e n s i l e s t r e n g t h measurement - 1^ • •W' Schematic i l l u s t r a t i o n of d r o p - b a l l impact t e s t e r •—• 16 ka. Apparatus f o r d r o p - b a l l t e s t — ; l b 5 T r a v e l - t i m e of compressional wave i n a n d e s i t e , . g r a n i t e , marble, sandstone, and magnetite as a f u n c t i o n of t r a v e l - d i s t a n c e 2 0 6 T e n s i l e s t r e n g t h of a n d e s i t e , g r a n i t e , marble, sandstone, and magnetite : 2 5 7 Number of impacts r e q u i r e d f o r f r a c t u r e of ande-.'. s i t e , g r a n i t e , marble, sandstone, and magnetite from v a r i o u s h e i g h t •— 2 9 8 C r i t i c a l h e i g h t of b a l l a t v a r i o u s p r o b a b i l i t i e s of f r a c t u r e f o r a n d e s i t e , g r a n i t e , marble, sandstone, and magnetite 3 2 2 9 R e l a t i o n s h i p between St /E and c r i t i c a l h e i g h t 3 ? 2 1 0 R e l a t i o n s h i p between St and c r i t i c a l h e i g h t 3 7 2 1 1 R e l a t i o n s h i p between St /E and energy index kl 2 1 2 R e l a t i o n s h i p between St and energy index M. 1 3 Tumbling m i l l and i t s torque measuring apparatus — • — ^ 9 m- I n f l u e n c e of the 1 2 - i n c h m i l l speed on the gross torque under the v a r i o u s l o a d i n g of 1 -i n c h b a l l s — — : • 5 2 • 1 5 The e f f e c t of b a l l l o a d i n g on the gross torque a t a spees of 6k- rpm. 5 ^ 1 6 The gross torque f o r the m i l l f i l l e d w i t h and without feed m a t e r i a l 5 ^ Page 17 R e l a t i o n s h i p between expended energy and m i l l r e v o l u t i o n s -volume b a s i s - • 57 18 R e l a t i o n s h i p between expended energy and m i l l r e v o l u t i o n s -weight b a s i s - — - — — — 57 19 S i z e d i s t r i b u t i o n s of marble 60 • 1 9 a S i z e d i s t r i b u t i o n s of v a r i o u s sample a f t e r 150 r e v o l u t i o n s g r i n d i n g 60 20 Rates of fo r m a t i o n of mesh f r a c t i o n s ( m a r b l e ) 66 2 1 Energy index as a f u n c t i o n of product s i z e p a s s i n g 80 percent • 69 2 2 C o r r e l a t i o n of energy index t o Bond°s work index : 71 2 3 Energy index versus c r i t i c a l h e i g h t 71 - v i -' LIST OF TABLES Page Table 1 Rock samples — 7 2 Young" s modulus '—• • • --• 2 3 3 Experimental r e s u l t s i n Chapter 1. • 4 3 4 M a t e r i a l s used f o r b a l l m i l l g r i n d i n g — - — — — 4 7 5 G r i n d i n g c o n d i t i o n s — 5 3 6 D i s t r i b u t i o n modulus from b a l l m i l l - t e s t s 61 Appendix ' c-Table 1 Measurement of Propagation V e l o c i t y of P-Wave . 80 2 Measurement of T e n s i l e Strength • . 8 3 3 E f f e c t of. B a l l Charge and M i l l Speed on. the Torque 8 6 4 Screen A n a l y s i s f o r the B a l l M i l l i n g 8 7 - v i l -ACKNOWLEGEMENTS The author i s indebted to P r o f e s s o r H. Majima of the Department of M i n e r a l E n g i n e e r i n g , who suggested t h a t the work f o r t h i s t h e s i s 'be u n d e r t a k e n His comments and suggestions have helped to make p o s s i b l e t h i s t h e s i s . Sincere g r a t i t u d e and thanks are extended to Dr. Y. F u j i n a k a and Dr. H. Kiyama f o r the h e l p i n d e s i g n i n g the m i l l assembly used i n t h i s study and f o r time w i l l i n g l y g i v e n to d i s c u s s i o n s on comminution. . A p p r e c i a t i o n and thanks are a l s o extended to a l l members of the Department of M i n e r a l E n g i n e e r i n g f o r t h e i r a s s i s t a n c e i n v a r i o u s f i e l d s d u r i n g t h i s study. I n a d d i t i o n the author a p p r e c i a t e s the funds r e c e i v e d under the U n i v e r s i t y of B r i t i s h Columbia f e l l o w s h i p , the F r e d e r i c k Armand McDearmid S c h o l a r s h i p and from the Mines Branch, Department of Energy and Resources to do the necessary work f o r t h i s thesis, w h i l e s t u d y i n g at the U n i v e r s i t y of B r i t i s h Columbia. - 1 -INTRODUCTION , Comminution i s one of the most important u n i t processes w i t h i n the mining industry<> Numerous papers are pu b l i s h e d on t h i s subject,. These are summarized i n the b i b l i -ography "Crushing and G r i n d i n g ( 1 ) " as w e l l as i n many review papers which appear i n e n g i n e e r i n g j o u r n a l s every year* Important t o p i c s commonly d e a l t w i t h i n these s t u d i e s a re: (1) the. d e t e r m i n a t i o n of p h y s i c a l c h a r a c t e r i s t i c s of rocks or m i n e r a l s i n accordance w i t h c r u s h i n g or g r i n d i n g c a p a b i l i t y , ( 2 ) the e s t i m a t i o n of energy f o r s i z e r e d u c t i o n , ( 3 ) the e l u c i d a t i o n of parameters o f g r i n d i n g o p e r a t i o n s , (4-) the development of g r i n d i n g techniques <> The author's i n t e r e s t i n g r i n d i n g problems i s p r e s e n t l y focused on the d e t e r m i n a t i o n of ease of comminution of m a t e r i a l s and on the e s t i m a t i o n of energy consumption f o r s i z e reduction,. In s t u d y i n g these problems, a theory which s a t i s f i e s p r a c t i c a l requirements, has not been f o r m u l a t e d 0 This i s because the g r i n d i n g system i s u s u a l l y complicated, and thus only e m p i r i c a l s o l u t i o n s have been obtained t o the problems. Ease of comminution of m a t e r i a l s has been t e n t a t i v e l y expressed i n terms of hardness, compressive s t r e n g t h , shear s t r e n g t h , g r i n d a b i l i t y , or r e s i s t i v i t y to a hammer t e s t of m a t e r i a l s ( 2 , 3 ) « However, none of these c r i t e r i a , except • - 2 -g r i n d a b i l i t y , can s a t i s f a c t o r i l y express the r e s i s t a n c e of m a t e r i a l s to comminution. In g e n e r a l , t h e r e f o r e , the c r i t e r i o n of comminution must meet at l e a s t the f o l l o w i n g two requirements; ( i ) i t must have p h y s i c a l meaning i n t i m a t e l y r e l a t e d t o comminution, and ( i i ) i t must be able to be determined a c c u r a t e l y without t e d i o u s procedures. Recently Oka and Majima ( 4 ) have s t u d i e d the energy problem i n comminution a p p l y i n g the theory o f e l a s t i c i t y to an i r r e g u l a r l y shaped rock p a r t i c l e , which i s subjected to the a c t i o n of a p a i r of concentrated l o a d s . For s i n g l e p a r t i c l e f r a c t u r e , i t was found t h a t the s t r a i n energy r e q u i r e d f o r s i z e r e d u c t i o n of a rock p a r t i c l e i s p r o p o r t i o n a l to the t h i r d order of p a r t i c l e s i z e and to the second order of t e n s i l e s t r e n g t h , and i s i n v e r s e l y p r o p o r t i o n a l to the value o f Young's modulus. From t h i s f i n d i n g , they suggested t h a t the r a t i o of the square of t e n s i l e s t r e n g t h t o Young's modulus may be the a p p r o p r i a t e - c r i t e r i o n of comminution. I n order t o determine t h e i r c r i t e r i o n , measurements of t e n s i l e s t r e n g t h as w e l l as Young's modulus are r e q u i r e d . For the d e t e r m i n a t i o n of both rock p r o p e r t i e s , no standard t e s t method has been e s t a b l i s h e d . In c o n s i d e r i n g the above *. G r i n d a b i l i t y i s b r o a d l y d e f i n e d as the response or r e s i s t a n c e of a m a t e r i a l t o g r i n d i n g e f f o r t . The numerical e x p r e s s i o n of g r i n d a b i l i t y has been proposed as the r e s u l t o f experimental work, such as new s u r f a c e produced per u n i t of p o t e n t i a l energy i n drop-weight t e s t , o r product weight per r e v o l u t i o n or the numbers of r e v o l u t i o n s r e q u i r e d to produce the constant f i n e n e s s i n m i l l g r i n d i n g ( 3 ) 0 " 3 " mentioned requirements f o r a c r i t e r i o n of comminution, p o i n t -load, t e s t ( 5 ) was employed f o r determing t e n s i l e s t r e n g t h , and measurement of the p r o p a g a t i o n v e l o c i t y of compressional waves was done f o r d e t e r m i n i n g Young 6s modulus 0 The other reason f o r the choice of the above t e s t methods was: 1) to use an i r r e g u l a r l y shaped specimen and thereby avoid weakening the specimen d u r i n g p r e p a r a t i o n , 2 ) to e l i m i n a t e p r o c e d u r a l e r r o r s which o f t e n appear i n d i r e c t t e s t i n g methods such as e c c e n t r i c l o a d i n g or weakness planes due to machining, and 3 ) the same t e s t - p i e c e can be a p p l i e d f o r both measurements, s i n c e the Young's modulus measurement i s a n o n - d e s t r u c t i v e t e s t . I t was a l s o necessary t o c a r r y out comminutiion experiments i n order to v e r i f y the c r i t e r i o n thus obtained as b e i n g a p p l i c a b l e t o the g r i n d i n g processes. Drop-weight impact t e s t s and b a l l - m i l l g r i n d i n g t e s t s were s e l e c t e d f o r these experiments as they i n v o l v e d simple and complex events, r e s p e c t i v e l y . I n v e s t i g a t o r s have o f t e n d i f f i c u l t y i n determining the d r i v i n g power of commercially s i z e d t umbling m i l l from data based on the l a b o r a t o r y t e s t measurementso P r e s e n t l y the method proposed by Bond ( 6 ) , which i s developed on the b a s i s of h i s concept, "work index", i s w i d e l y used f o r t h i s purpose. Using Bond 9s h y p o t h e s i s , i t w i l l be seen t h a t the d r i v i n g power i s p r o p o r t i o n a l t o r e q u i r e d energy f o r s i z e r e d u c t i o n , and t h a t p r o p o r t i o n a l i t y constant which i s e n t i t l e d "work index" i s p a r t i c u l a r l y dependent on c h a r a c t e r i s t i c s _ 4 -of the m a t e r i a l being ground. However, i t has been i m p l i c i t l y r ecognized f o r many years t h a t there may w e l l be a l a c k of p r o p o r t i o n a l i t y between energy consumption and g r i n d i n g e f f i c i e n c y i n m i l l s of d i f f e r e n t s i z e s . For example, i t i s •known t h a t a l a r g e d e v i a t i o n e x i s t s between a c t u a l power i n p u t f o r g r i n d i n g m i l l s i n cement i n d u s t r y and corresponding values determined from Bond's work index (•?)• As a consequence. Bond has proposed the i n t r o d u c t i o n of a s i z e f a c t o r v/hen r e l a t i n g the work index c a l c u l a t i o n s to the o p e r a t i o n o f m i l l s l a r g e r than e i g h t f e e t i n d i a m e t e r ( 8 ) 0 We(17) have suggested t h a t i t may be p o s s i b l e to modify Bond's concept of v/ork index to meet the-requirement by i n d u s t r y f o r i t s u n i v e r s a l a p p l i -c a t i o n by i n t r o d u c i n g a new index which can c o l l e c t i v e l y e valuate g r i n d i n g e f f i c i e n c y . •A tumbling m i l l ' w h i c h was equipped w i t h a torque •pickup was c o n s t r u c t e d f o r t h i s purpose, and some of the f a c t o r s a f f e c t i n g the g r i n d i n g e f f i c i e n c y were s t u d i e d . The experimental, r e s u l t s obtained are d i s c u s s e d i n terms of a new index c a l l e d an "energy index". The use of the new index w i l l e v e n t u a l l y o f f e r an improved method f o r the e s t i -mation of the power r e q u i r e d f o r p r a c t i c a l m i l l o p e r a t i o n based on l a b o r a t o r y measurements. -5 -CHAPTER ONE CRITERION OF COMMINUTION SCOPE OF PRESENT WORK The purpose of the s t u d i e s d i s c u s s e d i n t h i s chapter was to examine e x p e r i m e n t a l l y the c r i t e r i o n o f comminition which has been proposedby Oka and Majima. In order to o b t a i n the c r i t e r i o n q u a n t i t a t i v e l y , measurements of Young's.modulus and t e n s i l e s t r e n g t h were r e q u i r e d . The t e s t i n g method f o r the d e t e r m i n a t i o n of t e n s i l e s t r e n g t h was a p o i n t l o a d t e n s i l e , t e s t which can employ an i r r e g u l a r l y .shaped specimen. For Young's modulus, a measurement of prop a g a t i o n v e l o c i t y of a compressional wave was c a r r i e d out, which y i e l d s a dynamic modulus. The samples t e s t e d were f i v e d i f f e r e n t r o c k s i n c l u d i n g one ore sample, and c o n s i s t e d of a n d e s i t e , g r a n i t e , marble, sandstone, and magnetite. For experimental purposes f i f t e e n specimens were taken from each sample. F u r t h e r experiments were a l s o necessary to v e r i f y the a p p l i c a b i l i t y o f the c r i t e r i o n thus o b t a i n e d . As an example of the most simple case of comminution events, the d r o p - b a l l impact t e s t was used. The purpose of t h i s t e s t i s to determine the minimum height of b a l l p o s i t i o n r e q u i r e d to f r a c t u r e a sample, which i s termed the c r i t i c a l h e i g h t . Thus measurements of the c r i t i c a l h eight provide a c r i t e r i o n which expresses the ease o f comminution o f m a t e r i a l to be examined. - 6 -MATERIALS AMD PREPARATION ' • • The p h y s i c a l p r o p e r t i e s of rock m a t e r i a l s vary w i d e l y from one rock t o another. Therefore the t e s t samples should i n c l u d e s e v e r a l d i f f e r e n t types of rocks to o b t a i n a more u n i v e r s a l c o n c l u s i o n from.experimental r e s u l t s . To s a t i s f y t h i s requirement, f i v e d i f f e r e n t rock samples were examined: a n d e s i t e , . g r a n i t e , marble, sandstone, and magnetite. Table 1 shows the samples w i t h t h e i r o r i g i n and s p e c i f i c gravity.. The s p e c i f i c g r a v i t y of rock samples was determined f o r - 6 5 + 1 0 0 mesh p a r t i c l e s u s i n g a pycnometer method. The gi v e n value of s p e c i f i c g r a v i t y i n Table 1 i s an average of four t o f i v e d e t e r m i n a t i o n s . Coates and Parsons ( 9 ) suggested t h a t t e n or more t e s t - p i e c e s are r e q u i r e d to determine the acceptable value of the s t r e n g t h of a rock as w e l l as the other rock p r o p e r t i e s . Yamaguch ( 1 0 ) supported, u s i n g mathematical s t a t i s t i c s t o g e t h e r w i t h experiments, t h a t t h i s number of t e s t p i e c e s r e q u i r e d are reasonable, even though a l l the t e s t p i e c e s are cut from the same b l o c k of rock. A c c o r d i n g l y , more than f i f t e e n specimens f o r each of f i v e rock.samples were used f o r both Young's modulus and t e n s i l e s t r e n g t h ' . measurement, and more than t h i r t y t e s t p i e c e s of each sample f o r the drop-weight impact t e s t . I t has o f t e n been observed t h a t many r o c k s , not only sedimentary or metamorphic r o c k s , but a l s o igneous r o c k s , have a s i g n i f i c a n t d i f f e r e n c e of p h y s i c a l p r o p e r t i e s due t o Table 1. Rock Sarables Rock Source Number of Specimens Tested S p e c i f i c T e n s i l e Young's Drop»Weight G r a v i t y Strength Modulus Impact Andesite Granite Marble Sandstone Magnetite Hope, B e C o North Vancouver, B « C o Texada, B»C 0 Colorado, U o S o A o Texadaj, B<,C. 2 » 7 2 2 o 7 1 2 o 7 3 2 „ 6 7 4 c 6 3 2 3 1 5 18 1 5 1 5 2 9 1 5 2 3 1 5 2 0 3 2 3 0 3 8 3 1 3 3 -.. ya • -B r i e f P e t r o l o g i c a l D e s c r i p t i o n of Samples Ande s i t e : From Hope, B.C. i s aph'anitic and very dense? the g r a i n s i z e of both f e l s i c and mafic m i n e r a l s i s s m a l l e r than 0.1 mm i n diameter. The shape of the quartz c r y s t a l s i s . i r r e g u l a r and/or anh e d r a l . The quartz c r y s t a l impregnates f r a c t u r e s composed of t h i n v e i n l e t s r a n g i n g 0 . 5 to 1 mm i n w i d t h . G r a n i t e ; From North Vancouver, B.C. c o n s i s t s of about 3 0 percent of q u a r t z , 5 0 percent of potash f e l d s p a r and p l a g i o c l a s e , , and 2 0 percent of mafic m i n e r a l s . The " g r a n i t e " could be termed quartz monzonite p e t r o l o g i c a l l y s i n c e the r a t i o . o f the K - f e l d s p a r and p l a g i o c l a s e i s almost even. The g r a i n s i z e i s medium;. q u a r t z , f e l d s p a r , and p l a g i o c l a s e are 1 to 3 mm, and b i o t i t e , hornblende, and o t h e r ferromagnesians are 0 . 2 to.1 mm. Warble: From Texada I s l a n d , B.C. i s l i g h t gray; c o n s i s t s of medium-grained c a l c i t e . The g r a i n s i z e i s 1 to ' 3 mm.. Sandstone; From Colorado, U.S.A. has p a r a l l e l laminae. The l a y e r s vary t h i c k n e s s from 1 mm t o 2 cm, and are v i s u a l l y separable from other l a y e r s above and below. The rock c o n s i s t s , of more than- 9 5 percent rounded quartz g r a i n s which are 0.1 to 0 . 2 mm i n s i z e i n a porous' s i l i c a cementing m a t e r i a l . Magnetite: From Texada I s l a n d c o n s i s t s of 9 7 percent magnetite, and the r e s t i s c h a l c o p y r i t e and c a l c i t e . The c h a l c o p y r i t e i s disseminated as specks and c a l c i t e i s found as i s o l a t e d v e i n l e t s , 0 ' . 5 to 1 mm i n t h i c k n e s s . The g r a i n s i z e of magnetite v a r i e s w i d e l y from 0 . 5 to 5 . ram-- 8 -the d i f f e r e n t o r i e n t a t i o n s r e s u l t i n g from g e o l o g i c a l or micro s t r u c t u r e , a l t e r a t i o n , and other f a c t o r s ( 1 1 , 1 2 ) . However, i n c r u s h i n g or g r i n d i n g processes, p h y s i c a l p r o p e r t i e s of rocks are i n general averaged, and s p e c i a l care f o r the sample p r e p a r a t i o n w i t h r e s p e c t to o r i e n t a t i o n was not taken i n t o account except f o r sandstone« The Colorado sandstone sample was cut so t h a t i t could be subjected to compression p e r p e n d i c u l a r to i t s lamina. This was because of the d i f f i c u l t i e s o f sample p r e p a r a t i o n and t e n s i l e s t r e n g t h measurement, and may generate a.c o n s i d e r a b l e d e v i a t i o n i n g r i n d i n g t e s t r e s u l t s from those of other samples. I n t h i s study, the t e n s i l e s t r e n g t h and Young's modulus of rock samples are of i n t e r e s t . However, the immediate purpose i s to f i n d a s u i t a b l e c r i t e r i o n of comminution which can be determined i n a l a b o r a t o r y without t e c h n i c a l t e d i o u s n e s s . Even though we can determine the s o - c a l l e d t e n s i l e s t r e n g t h as w e l l as the Young's modulus of rock samples by u s i n g p r e c i s e l y shaped specimens, i t should be noted t h a t the p r e p a r a t i o n of such specimens, i s not simple, i n p a r t i c u l a r i t i s time-consuming. A l s o , i t i s not important to determine the r e a l t e n s i l e s t r e n g t h or Young's modulus f o r the present purpose. To a v o i d the d i f f i c u l t i e s of specimen p r e p a r a t i o n , a p o i n t - l o a d t e n s i l e s t r e n g t h t e s t and a dynamic measurement of Young's modulus was employed which r e q u i r e d a simple p r e p a r a t i o n of t e s t p i e c e s . . I t was found through p r e l i m i n a r y t e s t s t h a t the s i z e of - 9 -specimens e q u i v a l e n t t o a 3 "to 6 cm i n s c r i b e d sphere was s u i t a b l e f o r the measurement 0 I r r e g u l a r l y shaped specimens, both faces of which were cut f o r convenience of l o a d i n g , were used. For the drop-weight impact t e s t , the t e s t piece,was prepared t o be 6 cm square and 3 i n t h i c k n e s s . The s i z e and shape of the specimens was almost the same i n a l l samples. - 1 0 -EXPERIMENTAL METHODS The procedures used f o r the d e t e r m i n a t i o n o f Young's modulus and t e n s i l e s t r e n g t h and the method f o r drop-weight impact t e s t are as follows<> -1 - Young"s Modulus The p u l s e v e l o c i t y measurement technique ( 1 3 , 1 ^ ) was employed f o r determing Young"s modulus 0 A schematic arrangement of the apparatus is.shown i n F i g u r e 1 0 I t c o n s i s t e d of an e l e c t r o n i c p u l s e generator ( M o d e l - 1 0 0 7 , S t r u c t u r a l Behavior Engg. Lab.), an x-cut c r y s t a l - a p i e z o - e l a s t i c -t r a n s m i t t e r and r e c e i v e r ( S t y l e - B , S.B.E.L.), an o s c i l l o s c o p e (Type - 5 ^ - 9 , T e k t r o n i x Inc.) w i t h a. p r e a m p l i f i e r (Type-IA?, Tektronix)p and a p o r t a b l e compression t e s t e r ( S o k k i s y a Co.). The p u l s e generator used was capable of producing a square wave p u l s e w i t h 2 © 5 i 0 . 5 micro seconds w i d t h , a 1 5 0 v o l t s i n t e n s i t y , and 1 2 0 r e p e t i t i o n s per second. The specimen to be t e s t e d was placed between a p a i r of p l a t t e n s i n which the t r a n s m i t t e r and r e c e i v e r were mounted, and was loaded by a h y d r a u l i c compression t e s t e r f o r good mechanical contact„ . This enabled one t o measure the t r a v e l - t i m e of a compressional wave, t h a t i s , l o n g i t u d i n a l wave through the specimen under the a c t i o n of an a r b i t r a r y loado Loading pressure was incr e a s e d g r a d u a l l y u n t i l a c l e a r s i g n a l image was obtained on the o s c i l l o s c o p e 6 .The lo a d was set a t 1 0 0 to 2 0 0 k i l o g r a m s . Then the t r a v e l - t i m e of a compressional wave through a rock specimen was determined by measuring the d i s t a n c e between the - 1 1 a -F i g u r e 2 . A r r i v a l of Compressional Pulse a t the O r i g i n on an O s c i l l o s c o p e Screen Compression Tester Transmitter P l a t t e n X-cut C r y s t a l Receiver P l a t t e n O s c i l l o s c o p e (Type-549) Trigger Input(A) T r i g g e r Input(B) High Gain D i f f . Amp. (Type-IA?) Seismic Timer-Pulse Generator ( M o d e l - 1 0 0 7 ) Pulse Output Tr i g g e r Main Pulse 1 U n i a x i a l Compression Tester Figure 1. Schematic I l l u s t r a t i o n of Pulse Apparatus f o r Determining Young's Modulus - 1 2 -s t a r t of the sweep s i g n a l and the f i r s t a r r i v a l of a compressional wave on the o s c i l l o s c o p e screen ( F i g u r e 2 ) . Thus the t r a v e l - t i m e of the l o n g i t u d i n a l wave from the t r a n s m i t t e r t o r e c e i v e r through the specimen was measured * w i t h the o s c i l l o s c o p e o The t r a v e l - t i m e of the wave might be a f f e c t e d to some extent by the e l e c t r i c a l c h a r a c t e r i s t i c s of the t r a n s -m i t t e r , the r e c e i v e r and the p r e a m p l i f i e r , as. w e l l as by the mechanical c h a r a c t e r i s t i c s o f the conta c t s u r f a c e s between the p l a t t e n s and a specimen„ Therefore, the a c t u a l t r a v e l -time was determined by. s u b t r a c t i n g the t r a v e l - t i m e of the blank t e s t ( t h a t i s , the t r a v e l - t i m e through the t r a n s m i t t e r -p l a t t e n and the p i c k u p - p l a t t e n d i r e c t l y ) from the t o t a l t r a v e l - t i m e o The accuracy of t r a v e l - t i m e measurement obtained by t h i s technique was w i t h i n 0 . 0 5 micro seconds when 5 micro seconds of f i x e d gauge and 1 / 1 0 0 v e r n i e r was used. The v e l o c i t y of compressional wave Vc i n a rock specimen i s d e r i v e d from the f o l l o w i n g equations Ye = — — ( cm/sec ) — — ( 1 ) dt where L i s l e n g t h o f a specimen (cm) and dt i s t r a v e l - t i m e (second)o Then Young's modulus was c a l c u l a t e d by s u b s t i t u t i n g the v a l u e s of Vc i n t o the f o l l o w i n g equation: E = V c 2 p ( l + v ) ( l - 2 v ) ( 2) 1-v 2 where E i s Young's modulus (g/cm sec ), p i s the d e n s i t y of 3 m a t e r i a l b e i n g t e s t e d (g/cm ), and v i s Poisson's r a t i o - 1 3 -( d i m e n s i o n l e s s ) o Young's modulus expressed i n e n g i n e e r i n g • ' 2 u n i t , kg/cm, , can be obtained by d i v i d i n g E by l , 0 0 0 g , where g i s the a c c e l e r a t i o n of g r a v i t y . The present equipment was not designed to determine the v e l o c i t y of shear waves, s i n c e an x-cut c r y s t a l produces a compressional p u l s e i n the specimen<= Poisson's r a t i o , t h e r e f o r e , was assumed to l i e between 0 o 1 2 5 (Poisson°s number m = 8) and 0 o 2 5 0 (m = which was considered a reasonable value f o r common r o c k s . 2 - T e n s i l e Strength The method f o r d e t e r m i n i n g the t e n s i l e s t r e n g t h used i n t h i s study i s known as a p a i r of p o i n t s l o a d t e n s i l e s t r e n g t h t e s t f o r an i r r e g u l a r t e s t p i e c e ( 5 ) » A p a i r of p o i n t s l o a d i n g t e n s i l e t e s t e r developed f o r the use of i r r e g u l a r l y shaped specimens was used f o r t h i s experiment as shown i n F i g u r e 3 ° The p l a t t e n which leads the concentrated l o a d onto a specimen i s c y l i n d r i c a l , h a v i n g l o 0 cm diameter at the l o a d i n g end, w i t h a rounded edge. A l o a d was a p p l i e d through the c e n t r a l a x i s of a specimen by means of a p a i r of p l a t t e n s u n t i l breakage o c c u r r e d i n the specimen i n order to measure the c r i t i c a l l o a d a t f a i l u r e . The t e n s i l e s t r e n g t h of a specimen can be c a l c u l a t e d from the f o l l o w i n g formula: F St = I*1!- 5- or = 0 . 9 - ^ 7 • — ( 3 ) . . d^ - 14 -Figure 3 « Apparatus f o r T e n s i l e Strength Measurement. ( P o i n t - l o a d Compression Tester) - 1 5 -2 ' where St - t e n s i l e s t r e n g t h (kg/cm ) F = c r i t i c a l load at f a i l u r e (kg) 2 a or d = diameter of the i n s c r i b e d sphere of a specimen ( d i s t a n c e between p o i n t s of load)(cm) The l o a d i n g r a t e was kept 1 0 0 0 t o 1 5 0 0 kg per minute through-, out the experiments. 3 - Drop-Weight Impact Test A specimen placed on a s t e e l p l a t f o r m was f r a c t u r e d by a f r e e - f a l l i n g b a l l , u s i n g an apparatus shown s c h e m a t i c a l y i n F i g u r e 4 . A manganese s t e e l b a l l of 7 . 8 cm i n diameter weighing 1 , 9 0 3 gms, r e s t r a i n e d by vacuum, was r e l e a s e d from v a r i o u s h e i g h t s onto the t e s t p i e c e s . The s t e e l p l a t f o r m was h e l d i n p l a c e by three c y l i n d r i c a l s t e e l p i l l a r s , each 1 . 0 cm i n diameter and 6 . 5 cm i n h e i g h t , on which three s t r a i n gauges were equipped f o r each p i l l a r . The purpose of the s t r a i n gauges was 1 1 ) to estimate the unconsumed energy f o r the f r a c t u r e of specimen, and 2 ) to examine whether or not the specimen f r a c t u r e a c t u a l l y , o c curs. The output of the s t r a i n gauges was fed i n t o a h i g h g a i n a m p l i f i e r through an i n t e g r a t i n g c i r c u i t which i n t e g r a t e d the s t r a i n generated by impact d u r i n g the impact d u r a t i o n . Therefore, the area A under the s t r a i n - t i m e curve i s shown by t A = *\ e ( t ) dt • J o - 1 6 -To Winch To Vacuum Pump Vacuum Holder S t e e l B a l l S t r a i n Gauge Specimen E l e v a t e d P l a t f o r m High Gain Amp. / • Recorder I n t e r g r a t i n g C i r c u i t U n i t R i g i d Base Figure 3- • Schematic i l l u s t r a t i o n of d r o p - b a l l impact t e s t e r where e i s s t r a i n and t i s d u r a t i o n of impact. The s t r a i n -time curve i s recorded on a c h a r t w i t h a heat s e n s i t i v e pen. However, owing t o many mechanical c o n t a c t s "between a t e s t - p i e c e and the p i l l a r s , the experiment d i d not succeed i n o b t a i n i n g the q u a n t i t y w i t h s u f f i c i e n t accuracy i n t h i s system. The drop-weight impact t e s t s from a p r e - s e t h e i g h t onto a specimen were repeated u n t i l f r a c t u r e of the specimen took p l a c e , unless the f i r s t h i t caused f r a c t u r e of the specimens For each s e t of b a l l h e i g h t s , 5 "to 8 specimens were t e s t e d , then the h e i g h t was in c r e a s e d by 5 cm i n t e r v a l s . As d e s c r i b e d b e f o r e , two faces of a specimen were cut p a r a l l e l and approximately 3 cm a p a r t . However, i t was found from the p r e l i m i n a r y t e s t s t h a t a d e v i a t i o n of 1 2 mm i n t h i c k n e s s a f f e c t s s i g n i f i c a n t l y the drop-weight t e s t s . Therefore, c o r r e c t i o n has to be a p p l i e d t o the experimental r e s u l t s o b t a i n e d . For t h i s purpose, i t was assumed t h a t the impact f o r c e from the dropping b a l l i s p r o p o r t i o n a l to the height at which the b a l l i s r e l e a s e d , s i n c e the p o t e n t i a l energy of the b a l l i s p r o p o r t i o n a l to the h e i g h t . The equation f o r t h i s c o r r e c t i o n i s s „ • T T , . 2(d-d) v ,. . nc = H • ( 1 + _ .—) — (4) d where He = ad j u s t e d h e i g h t (cm) H = o r i g i n a l h e i g h t used i n the t e s t s (cm) * The d e r i v a t i o n of the equation i s i n Appendix. - 18 -d = standard t h i c k n e s s : 3 » 0 0 cm i n t h i s t e s t (cm) d = t h i c k n e s s of the specimen t e s t e d (cm) The height of the "ball has been adjusted for- each specimen as determined by Equation ( 4 ) , t a k i n g the specimen of 3 . 0 0 cm thickness as standard. For example, the b a l l p o s i t i o n of 3 0 cm f o r the t h i c k n e s s of specimens o f 2.92, 3 « 0 0 and 3 . 0 9 cm i s equivalent to those of 3 1 . 7 . 3 0 . 0 , 28.9 cm r e s p e c t i v e l y . Figure 4a. Apparatus f o r d r o p - b a l l t e s t . - 19 -EXPERIMENTAL RESULTS AND DISCUSSION 1 - Young?s Modulus -. As mentioned i n Procedure, a w e l l - d e f i n e d image of the s i g n a l was obtained on the o s c i l l o s c o p e i n the l o a d i n g range of 100 to 200 kg, v a r y i n g from one specimen to another© Therefore, p r i o r to the measurements of Young's modulus, the e f f e c t o f l o a d i n g pressure on the p ropagation v e l o c i t y of compressional waves was examined e The p r e l i m i n a r y t e s t s were made by measuring t r a v e l - t i m e under the v a r i o u s • o l o a d i n g c o n d i t i o n s up t o approximately 1600 kg or 9^«3 kg/cm i n s t r e s s o There was a n o t i c e a b l e i n c r e a s e of p r o p a g a t i o n v e l o c i t y i n a l l samples w i t h i n c r e a s i n g loado I t was observed t h a t the p ropagation v e l o c i t y through g r a n i t e specimens was h i g h l y dependent upon l o a d , w i t h the v e l o c i t y at 1600 kg b e i n g 1.07 to 1.12 times t h a t a t 100 kg l o a d . The a n d e s i t e sample showed the l e a s t e f f e c t of l o a d i n g pressure on the v e l o c i t y , t h a t i s , the i n c r e a s e o f the v e l o c i t y a t 1600 kg was l e s s than 2 percent of t h a t a t 100 k g o However, there i s not s i g n i f i c a n t d i f f e r e n c e i n t r a v e l - t i m e w i t h the i n c r e a s e i n l o a d from 100 to 200 kg where the v e l o c i t y was measured f o r the d e t e r m i n a t i o n of Young's moduluso Experimental r e s u l t s of the p ropagation v e l o c i t y measurements summarized i n F i g u r e 5* which p l o t s t r a v e l -time versus t r a v e l distance» As shown i n these f i g u r e s a l l l i n e s p l o t as a l i n e a r f u n c t i o n w i t h zero i n t e r c e p t , 2 4 6 T r a v e l Distance (cm) T3 o o CD w o o B 0) a •H E h CD 2 0 h 1 5 h 1 0 5 h 2 4 6 T r a v e l Distance (cm) Figure 5 . T r a v e l ' T i m e ' o f C o m p r e s s i o n a l Wave " i n ' ( a ) A n d e s i t e " a n d ^ ( b ) G r a n i t e ' . a s a F u n c t i o n " o f T r a v e l • D i s t a n c e .-. <. T r a v e l Time ( micro second ) oo - IZ ~ T r a v e l Time ( micro second ) H" CD (-3 H-1-3 <; —•<; CD. CD a 8 ' f • 2 »-3 p . SB H -ca Cfa • 3 -c f 3 - © cs c f O O Cf ' •"' O O o P 3 ' CO 'o' • • . •^d 3 - o O 3 c f J K -O ' ••; O' < - zz -- 2 3 -showing no delay time due to the contact c o n d i t i o n between t h e p l a t t e n and a specimen. The mean value and the standard d e v i a t i o n of the compressional wave v e l o c i t y through each sample i s as f o l l o w s ! a n d e s i t e ; 6.01 + 0.26, g r a n i t e ; 3-81 + 0.10,'marble? 5 = 10 + 0 . 5 5 , sandstone? 3.86 + 0.26, and magnetite; 4 . 5 9 + 0 . 4 4 k i l o m e t e r per second. The propagation v e l o c i t y thus determined may d i f f e r somehow from t h a t u s i n g a c y l i n d r i c a l specimen which i s . commonly used f o r the measurements of compressional and/or shear waves. Since the f i r s t a r r i v a l of a compressional wave was concerned i n t h i s experiment, a shape of t e s t p i e c e s which composes a boundary c o n d i t i o n w i l l not a f f e c t the Table 2. Young's Modulus . Rock D e n s i t y g/cm ^ V e l o c i t y of P-Wave 1 0 ^ cm/sec Young's Modulus m=8 m=6 m=4 0 . 1 2 5 0 . 1 6 7 0 . 2 5 0 1 0 - 5 kg/cm Andesite 2 . 7 2 6 . 0 1 9 . 6 7 905 8 . 3 5 G r a n i t e 2 . 7 1 3 ° 81 3 . 8 7 3 * 7 5 3 « 3 5 Marble 2 . 7 3 5 . 1 0 6 . 9 9 6 . 7 6 6„04 Sandstone 2 . 6 7 3 . 8 6 3 » 9 1 3 o 7 9 3 » 3 8 Magnetite 4 . 7 5 4.59 9 . 8 5 9.53 8 . 5 1 m = 1/m Poisson's = P o i s s o n ' number s r a t i o - 2k -experimental r e s u l t s . C o r r e l a t i o n experiments were c a r r i e d out on c y l i n d r i c a l specimens having 5 ° . 2 ' cm diameter and i r r e g u l a r l y shaped specimens u s i n g magnetite and marbleo As can be seen i n F i g u r e 5 - ( c ) and (e) where a white c i r c l e denotes a c y l i n d r i c a l specimen, there was no n o t i c e a b l e d i f f e r e n c e observed between shaped and unshaped specimens 0 Then the Young's moduli were computed from Equation ( 2 ) w i t h the d i f f e r e n t Poisson's r a t i o , and l i s t e d i n Table 2 « 2 - T e n s i l e Strength Figure 6 shows the r e s u l t s of the p o i n t s load t e n s i l e s t r e n g t h t e s t f o r a l l samples, and i s t a b u l a t e d i n the Append! The value of t e n s i l e s t r e n g t h computed from Equation ( 3 ) i s taken as o r d i n a t e , w h i l e the a b s c i s s a shows the diameter of an i n s c r i b e d sphere of a specimen. Accor d i n g t o the rec e n t r e p o r t ( 1 5 ) , the p r o p o r t i o n -a l i t y constant i n v o l v e d i n Equation ( 3 ) , t h a t i s , k = 1 . ^ , i s r e l a t e d to the r e l a t i v e shape dimention, Poisson°s r a t i o , and e s p e c i a l l y the r a t i o of the diameter .of l o a d i n g p o i n t to t h a t of the i n s c r i b e d sphere of the specimen. A necessary c o r r e c t i o n of the p r o p o r t i o n a l i t y constant was a l s o analyzed t h e o r e t i c a l l y by the r e s e a r c h e r s 0 The optimum value of the r a t i o . o f the s i z e of l o a d i n g p o i n t t o t h a t of the specimen i s 0 . 1 5 when the c o e f f i c i e n t k = 1 * ^ i n Equation ( 3 ) . Decreasing t h i s r a t i o , the t e n s i l e s t r e n g t h c a l c u l a t e d from Equation ( 3 ) may be overestimated» ? T e n s i l e S t r e n g t h ( k g / c m " ) o o oo o ~a ->• CD 3 ~5 CD CD <-+ ro o -s -+i o 0 0 - t i -a CD i — i O ZJ — 0 0 3 o CD S H3 —'• cr ^ CD O CL 3 INS GO on o IX) O o ro .on O oo o o i i i -Q . / ® ® © — % /*. 3> ro 00 r+ a> T e n s i l e S t r e n g t h ( k g / c m ) T e n s i l e S t r e n g t h ( k g / c m " ) oo a •a r r a> ro 3 -s ro ro r+ ro o -s -h o oo -h ro i — i o 3 00 3 O ro -5 CT -—- ro o CL 3 oo on O O O on O o o 1 1. / i -© © © In •o - ® h © Marbl e 2 0 0 O ^ 1 5 0 -P hi) ® 1 0 0 •p o> 1—1 <H CO Q> 5 0 0 (d)-Sandstone © ® Z 3 5 7 Diameter of I n s c r i b e d Sphere of Specimen (cm 2 0 0 E g 1 5 0 to ? 1 0 0 +> .0) 10 E H 5 0 (e)-Magnetite © 3 ^ 5 0\ ? Diameter o f I n s c r i b e d Sphere of Specimen (cm) ngth o f (d)-Sandstone and '(e)-Magnetite - 2 7 -In the t e s t r e s u l t s shown i n Figu r e 6, the constant r a t i o of the s i z e of l o a d i n g p o i n t to t h a t of specimens has not been s a t i s f i e d , s i n c e the s i z e of specimens was i n the range o f approximately 3 "to 6 cm, w h i l e t h a t of l o a d i n g p o i n t was. kept constant loO cm* Therefore, the e f f e c t of s i z e on the t e n s i l e s t r e n g t h appearing i n F i g u r e 6 should be indirect.o Taking t h i s i n t o c o n s i d e r a t i o n , the t e n s i l e s t r e n g t h w i t h the specimen having 5 cm diameter, which f u l f i l l s a c o n s t a n t . r a t i o of the p l a t t e n t o specimen s i z e , was determined from the best f i t t i n g l i n e which was obtained by the method of l e a s t squares assuming a l i n e a r function,, The corresponding l i n e a r r e g r e s s i o n equations are; An d e s i t e : s t ( A ) = 3 3 0 ° ^ - 2 6 0 « x G r a n i t e : St(G) = 128„S ~ 2 0 ? x Marble: S t ( M b ) = 8 7 ° 0 - 5*8 x' Sandstone: s t ( S ) = 1 ? 8 ° 5 - 9 » 7 x Magnetite: s t(Mg) = 1 3 3 - 7 - 1 0 . 1 x where x i s s i z e of specimens. The t e n s i l e s t r e n g t h obtained by s u b s t i t u t i n g 5 i n t o x f o r each sample i s : 1 9 6 f o r a n d e s i t e , 1 1 8 f o r g r a n i t e , 58 f o r marble, 1 3 0 f o r sandstone, and 8 9 kg/cm 2 f o r magnetite. These t e n s i l e s t r e n g t h s are c l o s e l y averaged values of each sample, s i n c e a 5 cm diameter l o c a t e s almost a t the c e n t e r of s i z e s of the specimens testedo - 28 3 - Drop-Weight Impact Test The r e s u l t s of drop weight impact t e s t s are i l l u s t r a t e d i n Fi g u r e ?. The o r d i n a t e expresses the height of b a l l p o s i t i o n o r a l t e r n a t i v e l y the p o t e n t i a l energy of the dropping b a l l determined by m u l t i p l y i n g the height by the weight of the drop b a l l o On the a b s c i s s a , the number of drop t e s t s , r e q u i r e d t o get the f r a c t u r e of a specimen i s g i v e n o . • We were p r i m a r i l y concerned w i t h the f r a c t u r e of specimens by s i n g l e impact and t h i s w i l l : be d i s c u s s e d i n terms of c r i t i c a l h e i g h t . The terminology of c r i t i c a l h e i g h t i s used f o r the height below which f r a c t u r e cannot take p l a c e b y . s i n g l e impact. However, there e x i s t s a c o n s i d e r a b l e s c a t t e r p o s s i b l y due t o i n h e r e n t f l a w s , i n experimental r e s u l t s . Furthermore one would expect such a s c a t t e r because i n most rocks there i s c o n s i d e r a b l e d i f f e r e n c e i n s t o r e d s t r a i n energy which v a r i e s i n s t r e n g t h depending on the d i r e c t i o n of an a p p l i e d e x t e r n a l f o r c e (12)« Therefore, the t e s t r e s u l t s should be t r e a t e d s t a t i s t i c a l l y 0 The continuous l i n e i n Figure 8 d e p i c t s the r a t i o of the number of specimens f r a c t u r e d by one drop from a g i v e n height range to that,.of t o t a l specimens i n the same group. The r a t i o thus d e f i n e d expresses the p r o b a b i l i t y of f r a c t u r e of a specimen by s i n g l e impact a t a g i v e n height l e v e l , and t h i s g i v e s the c r i t i c a l h eight i n simple p r o b a b i l i t y s t a t i s t i c s , e x c l u d i n g the i d e a of repeated impact 1 1 0 1 0 0 9 0 80 7 0 6 0 5 0 ® i ( a ) A n d esite - o — its' @ ® ® §> (fp-® Q ® 0 ® 1 1 1 7 0 6 . « 6 0 m 5 0 5 1 0 1 5 40 - (b)Granite @ @ © "@ 0 d (IF" 9 9 © 0 ^ 0 0 ( © © 1 1 1 ) . 5 1 0 1 5 Number o f Impacts Number of Impacts Fi g u r e . 7«, Number o f Impacts Required f o r F r a c t u r e of (a)Andesite . and ( b ) G r a n i t e from Various Height o to •H CD 60 \-50 h 40 h 30 h 20 10 0 10 .Number of Impacts 15 o •a •H i-J 80 .70 6o .50 40 30 (d)Sandstone ( - i J ® ® • 01 ® @ © © 1 1 1 o •5 10 Number of Impacts 1 5 Figure 7 . Number of Impacts Required f o r Fr a c t u r e of (c)Marble and (d)Sandstone from Various Height 60 50 4 0 30 20 (e)Magnetite © € J • -© © u © © © © © ©^ © © J_ 0 5 i o . Number of Impacts 15 F i g u r e 7 . Number o f Impacts Required f o r Fracture of (e)Magnetite*from Various Height o 0 . 5 l . o o •' " 0 . 5 i . o P r o b a b i l i t y of F r a c t u r e : . P r o b a b i l i t y of F r a c t u r e F i g u r e 8. C r i t i c a l Height o f B a l l a t Various P r o b a b i l i t i e s of F r a c t u r e f o r ( a ) - A n d e s i t e and ( b ) - G r a n i t e 6o 90 6 o +> si r-l 50 4 0 •30 20 10 (c)-Marble _1 L _ l 1_ o 0 . 5 l . o P r o b a b i l i t y of F r a c t u r e 80 70 E O •p 60 x w .3 50 pq 4 0 30, (d)-Sandstone X 0 . 5 l . o P r o b a b i l i t y o f F r a c t u r e F i g u r e 8. C r i t i c a l Height of B a l l at V a r i o u s P r o b a b i l i t i e s o f F r a c t u r e f o r (c)-Marble and (d)-Sandstone 60 iH « 20 -(e)-Magnetite 10 -q I i i i i _ I i t i i : • o • 0 . 5 l . o P r o b a b i l i t y of F r a c t u r e Figure 8. C r i t i c a l Height of B a l l at Various P r o b a b i l i t i e s of F r a c t u r e f o r .(e)-. Magnetite - 35 -numbers. For example, the c r i t i c a l height for marble was determined graphically as 4 5 . 0 cm with 100 percent probability and 3 8 .0 cm with 50 percent probability. The c r i t i c a l height corresponding to 100 percent probability i s essentially an unreliable value for rocks, with d i f f i c u l t y inherent i n i t s determination as expected from Figure 7» Therefore, i n this study, the c r i t i c a l height corresponding to 50 percent probability for the rock being tested was taken as i t s representative c r i t i c a l height. Among five samples, andesite consumes the largest potential energy, 19 kg-cm. This i s equivalent to 99 cm of c r i t i c a l height, indicating the highest r e s i s t i v i t y of comminution i n tested samples. The c r i t i c a l heights obtained for others ares 68 cm for sandstone, 57 cm for granite, 47 cm for magnetite, and 39 cm for marble. Handling the test results which record the repeated number, i t may be wise to apply the concept of probability from a different point of view. The ratio of the number of specimens in the same group at a given height level to the tot a l number of impacts required for the fracture of these specimens i s adopted for this purpose. This ratio i n i t s e l f i s the reciprocal of the average number of impacts required for the fracture of a specimen at a given height l e v e l , but i t also can be regarded as the probability of fracture of the specimen, per one impact at the given height. For example, i f 36 impacts were required to fracture 6 specimens - 36 -from the same height lev e l , the probability i s *>/36 = 1/6. The probability calculated from this definition i s also plotted i n Figure 8 by dashed l i n e s . The c r i t i c a l heights with 50 percent probability are obtained graphically as follows: 100 cm for andesite, 67 cm for sandstone, 60 .5 cm for granite, k$ cm for magnetite, and 38 cm for marble. It i s interesting to note that the difference of c r i t i c a l height derived from both definitions i s only 1 to 2 percent at 50 percent probability. Thus, the c r i t i c a l heights determined by both methods are equally useful to estimate the r e s i s t i v i t y of rock samples to drop-weight impact. In this study, the c r i t i c a l height obtained from the former manner was adopted. 4 The Comparison of St 2/B* and C r i t i c a l Height Using the. determined value of the tensile strength as well as the Youn*s modulus of the rock samples, we can calculate the value of St 2/E for each sample. The values of S t 2 and St 2/E are tabulated together with other experimental results i n Table 3 . In order to examine the applicability of a new comminution c r i t e r i o n to impact crushing events, the values of St 2/E are plotted i n Figure 9 against the c r i t i c a l height at 50 percent probabilities determined by the drop-weight impact tests. As shown by the two dotted lines, a relationship i s recognized among andesite, magnetite, and St denotes tensile strength and E Young's modulus. Potential Energy ( kg'w m ) o o —J— o C r i t i c a l Height ( cm.) -3 P' OS CD \0 00 <+ P U3 3 CD l-» ,—s. O c+ X P* O (-» c+ 3 ° l ro P' W o y JU P« X w CO CO O H* c+ 3 OS £ ro ^ CD c+ © * ^ GO c+ ro \ • w marble, or among sandstone, g r a n i t e , magnetite, and marble r e s p e c t i v e l y . However, i t i s d i f f i c u l t to f i n d a r e l a t i o n s h i p amongst a l l these.samples. Next S t 2 was compared w i t h the c r i t i c a l h e i g h t . Figure 10 shows the p l o t s of S t 2 versus c r i t i c a l h e i g h t f o r these samples and suggests t h a t a l i n e a r r e l a t i o n s h i p e x i s t s . This o b s e r v a t i o n i n d i c a t e s t h a t the c r i t i c a l h e i g h t , which i s p r o p o r t i o n a l to the energy r e q u i r e d to f r a c t u r e rock specimens by impact, i s p r o p o r t i o n a l t o the square o f the t e n s i l e s t r e n g t h r a t h e r than to the square of the t e n s i l e s t r e n g t h over Young's modulus. To e x p l a i n the d i f f e r e n c e between experimental and t h e o r e t i c a l r e s u l t s , an examination of Oka and Maxima's theory i s necessary together v/ith a d i s c u s s i o n of the experimental r e s u l t s . The d e r i v a t i o n of the theory i s as f o l l o w s : The energy w i s the product of an a p p l i e d l o a d and the displacement o f l o a d i n g p o i n t i n the d i r e c t i o n of the l o a d . w = F • u -.- • — ( 5 ) where F i s the l o a d and u i s displacement. Displacement u can be c a l c u l a t e d from e l a s t i c t h e o r y . p : • • u = _ . ( 6 ) x • E They assumed t h a t the f r a c t u r e of an i r r e g u l a r rock p a r t i c l e i s only p o s s i b l e when the t e n s i l e s t r e s s reaches the t e n s i l e s t r e n g t h of the p a r t i c l e , where the t e n s i l e s t r e n g t h i s g i v e n by E q u a t i o n ( 3 ) : / . . ; . . - 3 9 -S u b s t i t u t i n g Equation ( 3 ) and ( 6 ) i n t o Equation ( 1 ) , they obtained the energy equations o . 2 3 • , - 1 . 2 3 • s t ; x — — ( ? ) E Equation ( 6 ) i s a f u n c t i o n of Young's modulus., In the d e r i v a t i o n o f Equation ( 6 ) , i t i s questionable v/hether the e l a s t i c theory' i s a p p l i c a b l e to rock m a t e r i a l s , e s p e c i a l l y to deformation, under the c r i t i c a l c o n d i t i o n near f r a c t u r e , even though some mechanical p r o p e r t i e s of r o c k s , which exclude the f r a c t u r e phenomena, can be s a t i s f a c t o r i l y i n t e r p r e t e d by the theory of e l a s t i c i t y . The reason being t h a t Young's modulus by d e f i n i t i o n , holds i t s p h y s i c a l meaning i n the range where s t r a i n i s d i r e c t l y p r o p o r t i o n a l t o s t r e s s . I t i s w e l l known t h a t the s t r a i n i s no l o n g e r p r o p o r t i o n a l to the s t r e s s near f r a c t u r e c o n d i t i o n . I n a d d i t i o n , i n many cases ro c k s have a l r e a d y been subjected to s t r a i n i n a f i e l d , and i f the rock c o n t a i n s much r e s i d u a l s t r a i n energy the s t r e s s s t r a i n curve w i l l be no l o n g e r p r o p o r t i o n a l , even i n o r i g i n of the diagram ( 1 6 ) . Therefore, from t h i s p o i n t , the e l a s t i c theory i s not d i r e c t l y a p p l i c a b l e without keeping t h i s c o n d i t i o n i n mind. Thus, Equation ( 6 ) can no l o n g e r be a p p l i e d u s i n g the value of Young's modulus obtained i n the range of e l a s t i c i t y . This suggests t h a t the s i g n i f i c a n c e of St /E, obtained i n t h i s experiment i s d i f f e r e n t from the t h e o r e t i c a l , q u a n t i t y . I n t h i s i n s t a n c e , i t c o u l d be assumed t h a t the displacement of the l o a d i n g p o i n t u i n Equation ( 6 ) , i s d i r e c t l y p r o p o r t i o n a l t o the l o a d F, without the e f f e c t - 4o -2 of Young's modulus. Therefore St /E i n Equation (7) would be s u b s t i t u t e d f o r S t 2 . On the c o n t r a r y , the f o l l o w i n g may be p o i n t e d out i n the experimental method. Young's modulus i s the tangent of the s t r e s s - s t r a i n curve, but the s t r e s s - s t r a i n curve i s not l i n e a r i n most cases of rock m a t e r i a l s . Therefore, there remains u n c e r t a i n t y i n o b t a i n i n g a s a t i s f a c t o r y value of Young's modulus a t which' the s t r e s s l e v e l i s s u i t a b l e f o r the q u a n t i t y o f S t 2 / E obtained t h e o r e t i c a l l y . The value of Young's modulus, i n t h i s experiment, was obtained from the i n i t i a l tangent value i n the s t r e s s - s t r a i n curve. While, the value of Young's modulus used i n t h e o r e t i c a l c o n s i d e r a t i o n i s not the value of the dynamic, but of the s t a t i c modulus. The c o r r e l a t i o n between dynamic and s t a t i c moduli has not been difined.'. This might cause a disagreement between theory and experimental r e s u l t . But i t might be adequate to apply the dynamic modulus i n the e s t i m a t i o n of St /E, s i n c e dynamic f r a c t u r e i s i n v o l v e d i n the impact t e s t or g r i n d i n g t e s t . However, no complete e x p l a n a t i o n of the disagreement has been obtained, and a f u r t h e r c o n s i d e r a t i o n of t h i s matter may give a s a t i s f a c t o r y answer t o the d i f f e r e n c e between the experimental and t h e o r e t i c a l r e s u l t s . . . 5- The R e l a t i o n s h i p between S t 2 / E and the R e s u l t s of B a l l . M i l l G r i n d i n g Based on the drop-weight t e s t s , the square of t e n s i l e strength" i s more a p p l i c a b l e as a c r i t e r i o n of 3 0 o Andesite Magnetite Marble _! I . I l _ Granite K3 1 Sandstone H3— 3 -2, S t V E ( x 1 0 kg/cm~ ) 2 / Figure 1 1 . R e l a t i o n s h i p Between St /E and Energy Index. 3 0 o -p 2 0 o S l O CD c Andesite © G r a n i t e © / Magnetite © Sandstone / Marble 2 , 4 2 . 4 . St ( x 1 0 kg /cm ) Figure 1 2 . R e l a t i o n s h i p . Between St and Energy Index.. - 42 -comminution than St /E. However, the drop-weight impact t e s t r e p r e s e n t s a simple event i n comminution p r o c e s s . Therefore, there 2 a r i s e s the q u e s t i o n of whether the c r i t e r i o n St i s a p p l i c a b l e t o the p r a c t i c a l comminution events l i k e rod o r . b a l l , m i l l g r i n d i n g . As d e s c r i b e d i n P a r t 2, the energy consumption or work requirement i n b a l l m i l l g r i n d i n g was estimated by means of the energy index f o r the i d e n t i c a l samples used i n the t e s t s f o r the d e t e r m i n a t i o n of S t 2 / E . The energy index r e p r e s e n t s the energy consumption to reduce the -6 +8 mesh f r a c t i o n s t o d e s i r e d s i z e i n the l a b o r a t o r y b a l l m i l l under the given g r i n d i n g c o n d i t i o n s , assuming the i n i t i a l g r i n d i n g process i n the batch m i l l a t . a n e a r l y stage w i l l be continuous. I n F i g u r e 11, and 12, the r e l a t i o n s h i p are the energy index versus S t 2 / E and S t 2 r e s p e c t i v e l y . The former r e l a t i o n s h i p i s not re c o g n i z e d c l e a r l y i n s p i t e of c o n s i d e r i n g the ranges of the q u a n t i t y o f S t 2 / E , due t o the p o s s i b i l i t y of the change i n P o i s s o n 9 s r a t i o . S i m i l a r l y as found i n the 2 ' previous s e c t i o n , St i s more fa v o u r a b l e t o energy i n d i c e s , •3c except the sandstone sample,. This r e s u l t may c o n f i r m the 2 establishment of St as to a new c r i t e r i o n i n comminution O p r a t h e r than St /E, and St may be u s e f u l as a c r i t e r i o n to estimate the ease of comminution of rock m a t e r i a l s i n comminution i f the r e l a t i o n i s researched f o r each p a r t i c u l a r rock. * The anomalous behaviour of sandstone sample i s mentioned i n P a r t 2. Table 3 « Experimental Results Rock Te n s i l e Strength St / 9 4 kg/cm c xlO Young's Modulus m=8 m=6 m=4 . 0 , 1 . 2 5 0.167 0 . 2 5 0 1 0 5 kg/cm 2 S t 2 / E m=8 0 . 1 2 5 1 0 m=6 0 . 1 6 7 - 2 m=4 0 . 2 5 0 C r i t i c a l Height Energy-Index k g / cm cm kwh/ton Andesite 196 3.84 9 . 6 7 9 . 3 5 8 . 3 5 3 . 9 7 4 . 1 1 4 . 6 0 9 9 2 6 . 7 Granite 118 1 . 3 9 3 . 8 7 • 3 . 7 5 3 . 3 5 3 . 6 0 3 . 7 1 4 . 1 6 5 7 9 . 7 Marble 5 8 0 . 3 4 6 . 9 9 6 . 7 6 6.04 0.48 O . 5 0 '" O . 5 6 . 3 9 Sandstone 1 3 0 I . 6 9 3 - 9 1 . 3 . 7 9 : 3 . 3 8 4 . 3 2 4.46 5 . 0 0 6 8 1 5 . 1 Magnetite 8 9 0 . 7 9 " 9 . 8 5 9 . 5 3 8 . 5 1 0 . 8 0 O . 8 3 0 . 9 3 ^ 7 7 . 5 E = Young's modulus St = t e n s i l e s t r e n g t h m .= Poisson's r a t i o l/m = Poisson's r a t i o C r i t i c a l Height j at 5 0 % p r o b a b i l i t y ( B a l l weight; 1 , 9 0 3 gms) Energy Index; 1 5 0 mesh 8 0 % passing _ 44 -CONCLUSION The q u a n t i t y , square of t e n s i l e s t r e n g t h over Young's modulus, d e r i v e d from the e l a s t i c theory provided a b a s i s of comparison f o r the comminution r e s i s t i v i t y or rock m a t e r i a l s . I n c o n s i d e r a t i o n of the l i m i t of a p p l i c a t i o n of the theory of e l a s t i c i t y to the c r i t i c a l c o n d i t i o n of the rock f a i l u r e , the m o d i f i e d q u a n t i t y , square of t e n s i l e s t r e n g t h , appeares c l o s e l y r e l a t e d to the c r i t i c a l h e i g h t and the b a l l m i l l g r i n d i n g r e s u l t s . The square of t e n s i l e s t r e n g t h i s proposed t o be a new c r i t e r i o n t o estimate the r e s i s t i v i t y of rocks i n comminution, and t h i s estimate w i l l be e a s i l y c a r r i e d out w i t h the use of the , p o i n t l o a d t e n s i l e s t r e n g t h t e s t which.was employed i n t h i s study. ; - 4 5 -CHAPTER TWO . ENERGY INDEX IN-DRY TUMBING MILLING • SCOPE OF PRESENT WORK Gr i n d i n g t e s t s were c a r r i e d out w i t h a 12' x'12 i n c h - m i l l o A torque meter was i n s t a l l e d between the d r i v i n g s h a f t and the m i l l s h e l l . T h i s made p o s s i b l e the d e t e r m i n a t i o n of the work requirement of the m i l l d u r i n g o p e r a t i o n and the e s t i m a t i o n of the energy consumption necessary f o r s i z e . r e d u c t i o n of m a t e r i a l . For the purpose of c o r r e l a t i n g the r e s u l t s w i t h these obtained i n Chapter 1, the m a t e r i a l ground was e s s e n t i a l l y the same as t h a t used i n the p r e v i o u s experiments. One of the o b j e c t s i n t h i s study was to broach the problem of the e s t i m a t i o n of energy consumption i n a c t u a l m i l l s by the r e s u l t s of l a b o r a t o r y work. A new concept, Energy Index, was in t r o d u c e d t o evaluate g r i n d i n g r e s u l t s and a s s i s t i n the study of g r i n d i n g problems. The dependency of the energy index on experimental parameters was a l s o i n v e s t i g a t e d . Optimum g r i n d i n g c o n d i t i o n s were determined e x p e r i m e n t a l l y and these c o n d i t i o n s were used throughout the study. - 46 -MATERIALS AND PREPARATION The m a t e r i a l s used f o r the b a l l m i l l g r i n d i n g experiments were e s s e n t i a l l y the same as those used i n the experiments r e f e r r e d to i n P a r t 1. That i s , a n d e s i t e , g r a n i t e , marble, sandstone, and magnetite. I n a d d i t i o n to these samples, f e l d s p a r , q u a r t z , and l i m e s t o n e , which were provided from the Mines Branch, Department of Energy, Mines and Resources, Ottawa, were used. Large p i e c e s of each rock sample were crushed through a l a b o r a t o r y jaw, g y r a t o r y and cone c r u s h e r , and screened dry w i t h a Ro-Tap screenshaker t o o b t a i n -6+8 mesh f r a c t i o n s as feed. This was proved e x p e r i m e n t a l l y to be a s u i t a b l e s i z e f o r the g r i n d i n g t e s t w i t h a 12 i n c h m i l l . The reason f o r the use of s i n g l e s i z e f r a c t i o n s as a feed to the g r i n d i n g m i l l , was to o f f e r a b a s i s of comparison of consumed energies f o r s i z e r e d u c t i o n of d i f f e r e n t samples. Each amount of m a t e r i a l to be ground was obtained by s p l i t t i n g the - 6 +8 mesh f r a c t i o n s from the same p o p u l a t i o n w i t h a r i f f l e sampler. The apparent s p e c i f i c g r a v i t y needed to c a l c u l a t e the volume of feed was measured as f o l l o w s : - 6 +8 mesh f r a c t i o n s were placed i n 1000 cc measuring c y l i n d e r and compacted by shaking t o c o n t a i n a minimum amount of v o i d s , and then the apparent s p e c i f i c g r a v i t y was determined (see Table 4 ) . * T y l e r standard screen mesh - 4 7 -Table 4 . M a t e r i a l s used f o r B a l l M i l l . G r i n d i n g S p e c i f i c Apparent * Sample Sample G r a v i t y S p e c Grav. Charge ( 6 / 8 mesh) kg Andesite ' • 2 , 7 2 1 . 5 3 3 * ^ Hope, B.C. Gr a n i t e 2.7 1 1.28 2 . 8 8 North Vancouver, B.C. Marble 2 . 7 3 ' . 1 . 3 7 3«03 Texada, B.C. Sandstone 2 . 6 7 1 . 2 0 2 . 7 1 Colorado, U.S.A. j Magnetite 4 . 6 3 : 2 . 3 1 5 » 2 0 . Texada, B.C. : Feldspar 2 . 5 8 ; 1 . 3 1 2 . 9 5 Provided by Mines Branch Limestone 2 . 7 5 ' 1 . 3 7 3*08 Provided by Mines Branch Quartz 2 . 6 5 1 . 3 7 3 . 0 9 P r o v i d e d by Mines Branch * The volume of sample charge ( 2 2 5 0 cc) i s 5 0 percent of the volume of the b a l l v o i d s . - 48 -BALL MILL GRINDING TEST PROCEDURE The b a l l m i l l g r i n d i n g t e s t s were always . . • conducted dry under f o l l o w i n g conditions.'. The b a l l m i l l used , shown i n F i g u r e 1 3 > was a c y l i n d r i c a l tumbling m i l l made of m i l d s t e e l , d i a m e t e r - 2 9 . 8 cm, l e n g t h - 3 0 « 5 cm, w i t h no l i f t e r b a r s . The m i l l speed can be changed stepwise to s e v e r a l v a r i o u s l e v e l s i n the range of 3 2 t o . 1 0 0 rpm. In t h i s m i l l assembly, a torque meter which:was capable of r e c o r d i n g a maximum of 3 0 kilogram-meter was i n s t a l l e d between the d r i v i n g s h a f t and the m i l l s h e l l so as t o measure the r e q u i r e d .operating torque. : The one i n c h c a s t s t e e l b a l l s weighing 4? kilograms were charged as g r i n d i n g media to occupy a p p a r e n t l y 5 0 percent of m i l l volume, which i s 1 0 a 6 l i t e r s ; The r o t a t i o n a l speed of the m i l l was maintained a t 64 rpm which i s 7 9 percent of the c r i t i c a l speed . The m a t e r i a l - t o - v o i d r a t i o , t h a t i s , the r a t i o of the volume of m a t e r i a l t o t h a t of the voids i n the b a l l charge, was kept constant a t 0 . 5 u n l e s s otherwise s t a t e d . These g r i n d i n g c o n d i t i o n s are l i s t e d i n Table 5 i n the next s e c t i o n . The g r i n d i n g t e s t s were performed f o r v a r i o u s g r i n d i n g * The d e t a i l e d d e s c r i p t i o n s of the c h a r a c t e r i s t i c s of t h i s m i l l assembly i s p u b l i s h e d elsewhere ( 1 7 ) . ** S e l e c t i o n of both g r i n d i n g c o n d i t i o n s , such as media charge and m i l l speed are d e s c r i b e d l a t e r i n the s e c t i o n of p r e l i m i n a r y t e s t i n g . - 49 -Figure 1 3 . Tumbling mill and i t s torque measuring apparatus. - 5 0 -p e r i o d s , r a n g i n g from 1 0 to 1 , 0 0 0 r e v o l u t i o n s , under the same g r i n d i n g c o n d i t i o n s and w i t h the corresponding numbers of t e s t s f o r each sample 0 A f t e r each g r i n d i n g t e s t , ground m a t e r i a l was s p l i t w i t h . a r i f f l e sampler i n order to produce a r e p r e s e n t a t i v e sample f o r a s c r e e n i n g analysis., The amount o f sample f o r each s c r e e n i n g t e s t was approximately 180 .to 2 3 0 grams except i n the case of. magnetite. The amount of magnetite screened was 3 5 0 to 400 grams s i n c e the apparent s p e c i f i c g r a v i t y i s ' almost twice t h a t of the other samples. The screening^ technique used was the standard wet-and-dry method ( 3 ) « On • s c r e e n i n g , - 2 0 0 mesh f r a c t i o n s were f i r s t removed by wet s c r e e n i n g i n order t o a v o i d the misplacement o f p a r t i c l e s , p a r t i c u l a r l y f i n e r s i z e s , and to ensure e f f i c i e n t s c r e e n i n g . • For the s e p a r a t i o n of c o a r s e r f r a c t i o n s a Ro-Tap screen shaker was used. When the s c r e e n i n g l o s s exceeded 1 . 0 percent of t o t a l weight, another s c r e e n i n g a n a l y s i s was c a r r i e d out on a sample taken from the same l o t . Because of an i n s u f f i c i e n t amount of feed f r a c t i o n s , a s e r i e s of g r i n d i n g t e s t s w i t h f e l d s p a r , q u a r t z , o r limestone were performed u s i n g one sample ; each. A f t e r one run of the g r i n d i n g t e s t , the sample was s i z e d , and then r e t u r n e d to the m i l l t ogether w i t h the r e s t of the f r a c t i o n s t o be r e t e s t e d f o r the next d e s i r e d i n t e r v a l . T h i s procedure may cause a • s l i g h t l y d i f f e r e n t r e s u l t from the former t e s t procedure because of the unavoidable rearrangement of m i l l content t o the steady s t a t e . - 5 1 -EXPERIMENTAL RESULTS AND DISCUSSION. 1 - P r e l i m i n a r y Test The gross torque r e q u i r e d to d r i v e the b a l l m i l l without a rock sample charge was i n i t i a l l y i n v e s t i g a t e d under the v a r i o u s c o n d i t i o n s of l o a d of media and r o t a t i o n a l speed of the m i l l . The dependency of d r i v i n g torque on the l o a d of g r i n d i n g media was examined a t v a r i o u s l e v e l s i n the range from zero load to 9 4 . 0 kg of one i n c h b a l l s which i s e q u i v a l e n t to 1 0 0 percent of i n t e r n a l m i l l volume. At each l o a d i n g c o n d i t i o n , the e f f e c t o f m i l l speed was a l s o i n v e s t i g a t e d i n the range of 3 ° . t o 1 0 0 rpm, which i s 3 7 to. 1 2 3 percent of the c r i t i c a l speed of the m i l l . F i gure 14 shows the r e s u l t s of these t e s t s , and i n d i c a t e s t h a t i n the p r a c t i c a l range of b a l l charge, f o r 42 to 48 percent of m i l l volume, the torque begins to decrease s h a r p l y a t around 7 0 rpm, w h i l e i s s t i l l i n c r e a s i n g beyond the c r i t i c a l speed when the b a l l charge i s l e s s than about 3 3 p e r c e n t . I n the range of common use f o r media charge, 40 t o . 5 0 percent, the maximum gross torque was observed when the m i l l speed was 7 0 t o 8 0 percent of the c r i t i c a l speed. The r e l a t i o n s h i p of r e q u i r e d torque versus charge of b a l l i s presented a t the speed of 64 rpm i n F i g u r e 1 5 » The m i l l should be run a t the p o i n t where the maximum torque i s r e q u i r e d i n order to get the e f f i c i e n t comminution of m a t e r i a l s . On t h i s b a s i s , 5 0 percent b a l l charge and Figure Ik«, Influence of the 12-inch m i l l speed on the gross torque under the various l o a d i n g . o f 1-inch b a l l s . - 5 3 -64 rpm m i l l speed were chosen as standard g r i n d i n g c o n d i t i o n s throughout the experiments. These g r i n d i n g c o n d i t i o n s are l i s t e d i n Table 5» Table 5 . G r i n d i n g C o n d i t i o n s M i l l S h e l l G r i n d i n g Method G r i n d i n g Media M i l l Speed D r i v i n g Torque Power Requirement 3 0.5 x 2 9.8 cm (12 x 11.75 i n ) Smooth-faced c y l i n d r i c a l Dry g r i n d i n g 1 i n c h c a s t s t e e l b a l l s , 47 kilograms 5 0 percent m i l l charge 64 rpm, 79 percent of c r i t i c a l speed 2 2 5 kg-cm, loaded m i l l w i t h 47 kg b a l l , w ithout feed m a t e r i a l 3 2 kg-cm, w i t h empty m i l l s h e l l 148 watts at. 64 rpm 21 watts f o r empty m i l l a t 64 rpm ; 2 5 0 200 e o i bD ~ 1 5 0 o E-i 10 S 1 0 0 100 G r i n d i n g .Media Loading .-' {fo of m i l l volume) Figure 1 5 . The e f f e c t of b a l l l o a d i n g on the gross torque at a speed of 64 rpm, 5 0 NO FEED 'XT -©--o-Magnetite ( 5 - 2 0 kg) Marble (;3°o8 kg) EMPTY 100 " 200 3 0 0 400 Number of r e v o l u t i o n s 5 0 0 Figure 16. The gross torque f o r the m i l l f i l l e d w i t h and. without feed m a t e r i a l . • ' - 5 5 - • 2 - Torque Measurement The torque was measured f o r each run of the m i l l to estimate the energy^ consumption necessary to achieve s i z e r e d u c t i o n . F i g u r e 16 presents the gross torque readings f o r magnetite and marble which r e q u i r e d maximum and minimum torque r e s p e c t i v e l y among the samples t e s t e d as a f u n c t i o n of m i l l r e v o l u t i o n s . The torque readings f o r the dry g r i n d i n g of each sample were o b v i o u s l y low compared t o t h a t of a m i l l t u r n i n g w i t h no feed . m a t e r i a l . This c o u l d be e x p l a i n e d by the l u b r i c a t i n g e f f e c t due t o the feed m a t e r i a l r e d u c i n g f r i c t i o n a l r e s i s t a n c e between b a l l s and m i l l l i n i n g . On the other hand, Yang, e t . a l . (18) r e p o r t e d t h a t the lowest torque was obtained f o r the m i l l t u r n i n g without any g r i n d i n g m a t e r i a l . The c o n t r a d i c t o r y o b s e r v a t i o n might be caused by the use of the two d i f f e r e n t types of m i l l s , t h a t i s , one a smooth-faced m i l l and the other w i t h l i f t e r bars.. Since Yang's work was done w i t h the m i l l equipped w i t h e i g h t l i f t e r b a r s , s l i p p a g e of the b a l l l o a d was l i t t l e a f f e c t e d by the l u b r i c a t i n g a c t i o n due to feed m a t e r i a l . : A f t e r the f i r s t 10 r e v o l u t i o n s , d u r i n g which the dis t u r b a n c e of the torque caused by the s t a r t i n g a c t i o n of the m i l l s h e l l was observed, the r e q u i r e d torque i n c r e a s e d l i n e a r l y f o r p e r i o d s of s h o r t d u r a t i o n . The t o t a l r e v o l u t i o n s where the maximum torque was observed, were dependent upon each * Energy i s c a l c u l a t e d from the torque as f o l l o w s ; energy(watt sec) = 2 i c N . T x 9.8 x 1 0 ~ 3 where N = t o t a l m i l l r e v o l u t i o n s , T = torque (kg-cm) - 56 -rock sample and ranged from 80 to 180 r e v o l u t i o n s approximately. Then, a f t e r i n d i c a t i n g the maximum v a l u e , the torque decreased a s m a l l amount. Throughout the torque measurements, f o r a l l samples, both the i n c r e a s e and decrease of the torque, f o r the p e r i o d to 1,000 r e v o l u t i o n s , was l i m i t e d to w i t h i n 3 "to 6 percent of the average gross t o r q u e . Therefore i t can be regarded t h a t the r e q u i r e d torque i s almost constant. However, the . i n c r e a s e i n torque observed f o r the i n i t i a l p e r i o d of g r i n d i n g may. suggest t h a t e f f i c i e n t g r i n d i n g of feed m a t e r i a l can take p l a c e d u r i n g t h i s p e r i o d . This e x p l a n a t i o n i s supported by the r e s u l t s obtained from the s i z e a n a l y s i s which shows t h a t the most r a p i d r a t e of i n c r e a s e of f i n e products took pl a c e d u r i n g t h i s time. I t was a l s o observed t h a t the d i f f e r e n c e of r e q u i r e d torque due to the d i f f e r e n t m a t e r i a l s ground was l e s s than 10 p e r c e n t . From t h i s r e s u l t , i t may be p o s s i b l e to s t a t e t h a t the energy r e q u i r e d to achieve s i z e r e d u c t i o n of m a t e r i a l s having the same bulk volume from one s i z e to another i s p r o p o r t i o n a l to the number of m i l l r e v o l u t i o n s and might be p r i n c i p a l l y determined by g r i n d i n g time. F i g u r e 17 shows the l i n e a r r e l a t i o n s h i p between expended energy expressed i n u n i t volume b a s i s and r e v o l u t i o n s of the m i l l . However, the assumption of constant energy per r e v o l u t i o n does not h o l d , f o r the energy u n i t on weight b a s i s , which i s expressed i n k i l o w a t t hour per t o n , unless the m a t e r i a l s 10 W u 05 M 0 ft w 8 Magnetite Sandstone-Granite — Andesite' Feldspar Quartz Limestone Marble 10 o < 0) w 0) ft !>< W 200 400 600 800 Number of Revolutions Figure 1 ? . R e l a t i o n s h i p between expended energy and m i l l r e v o l u t i o n s - volume b a s i s 8 ft 6 4 Sandstone -Granite — Marble \ Limestone/ Andesite Quartz . Feldspar Magnetite 0 200 400 6 0 0 800 •Number of Revolutions Figure 18. R e l a t i o n s h i p between expended energy, and m i l l r e v o l u t i o n s - weight b a s i s -- 5 8 -b e i n g ground have approximately the same apparent s p e c i f i c g r a v i t y . Figure 18 expresses the r e l a t i o n s h i p between r e q u i r e d energy expressed i n u n i t weight b a s i s and m i l l r e v o l u t i o n s , and shows t h a t r e q u i r e d energy i s no l o n g e r c o n s t a n t . These experimental r e s u l t s i n d i c a t e t h a t the energy expended per r e v o l u t i o n of the tumbling m i l l , f o r a giv e n g r i n d i n g c o n d i t i o n , can be assumed as c o n s t a n t , r e g a r d l e s s of the m a t e r i a l s b e i n g ground, when expended energy i s expressed on u n i t volume b a s i s . - 5 9 -3 - S i z e D i s t r i b u t i o n The r e s u l t s of s i z e analyses of g r i n d i n g products were p l o t t e d on Schumann's s i z e d i s t r i b u t i o n curve ( 1 9 ) , and l i s t e d i n Table 3 of the Appendix. D i s t r i b u t i o n of m a t e r i a l i n the f i n e s i z e s of a comminution product i s w e l l represented by the f o l l o w i n g s i z e - d i s t r i b u t i o n equation: ; y = 1 0 0 (—£-0 k where y i s the cumulative percentage of m a t e r i a l f i n e r than s i z e x, expressed i n microns, k i s the s i z e modulus, which i s the t h e o r e t i c a l maximum s i z e o r 1 0 0 percent p a s s i n g s i z e i n the d i s t r i b u t i o n , <X i s the d i s t r i b u t i o n modulus, c h a r a c t e r i s t i c of the m a t e r i a l and comminution method. Figu r e 1 9 shows the Schumann p l o t s , of which the o r d i n a t e and a b s c i s s a are the cummulative percent p a s s i n g a g i v e n s i z e and the T y l e r mesh s i z e r e s p e c t i v e l y * . This a l s o presents the d i s t r i b u t i o n modulus f o r each curve as a f u n c t i o n of the g r i n d i n g p e r i o d s (see Table 6 ) . The f o l l o w i n g f a c t s are g e n e r a l l y observed r e g a r d i n g the d i s t r i b u t i o n modulus: The d i s t r i b u t i o n modulus i s almost equal to one r e g a r d l e s s of m a t e r i a l s , i f the f r a c t u r e i s caused simply by impact f o r c e , but because of the complex a c t i o n s of a b r a s i o n and c h i p p i n g as w e l l as impact i n m i l l g r i n d i n g processes, the modulus tends to be l e s s than one * This p l o t of y versus x expressed as the T y l e r mesh s i z e i s the same as l o g - l o g p l o t of y versus x expressed as microns. - 6 0 -1 0 0 C • H O 0) CD > •P H O 2 0 0 1 0 0 48 .2.8 P a r t i c l e S i z e (mesh) Fi g u r e 1 9» S i z e D i s t r i b u t i o n s o f Marble. 0) c • H c o PL, CD > • H -P ccj r H I o 1 0 0 5 0 2 0 . h 1 0 5 . h 2 0 0 1 0 0 48 28 14 P a r t i c l e S i z e (mesh) Figu r e 1 9 - ( a ) S i z e D i s t r i b u t i o n s of Various Samples a f t e r 1 5 0 Revs G r i n d i n g . Table 6 . D i s t r i b u t i o n modulus from b a l l m i l l - t e s t s * M i l l . Revolutions Andesite Granite Marble Sandstone Magnetite .10 0,82 2 0 0 . 8 0 4 0 0 . 6 8 6 0 0 . 6 9 7 5 0 . 5 3 0 . 9 5 2 . 5 2 0 . 9 3 1 0 0 0 . 8 9 0.64 1 2 5 0 „ 5 4 1 5 0 0 . 5 1 0 . 8 7 " 0 . 6 2 . • O . 8 9 2 0 0 0 . 3 9 0 . 5 3 . 2 2 5 , . 0 . 8 8 2 5 0 0 . 7 8 3 0 0 - 0.41 1 . 5 5 5 0 0 0.64 0 . 4 9 0 . 7 9 8 0 0 0 . 3 8 1 0 0 0 0 . 5 5 o . 4 o 1 . 0 7 .* - 6 + 8 mesh feed - 6 2 - , ( 2 0 * 2 1 ) . - The ground m a t e r i a l s each have a c h a r a c t e r i s t i c modulus even though the g r i n d i n g c o n d i t i o n s are the same ( 2 1 ) . Moreover, the d i s t r i b u t i o n modulus i n wet g r i n d i n g remains constant d u r i n g s i z e r e d u c t i o n , but the s i z e modulus i s s h i f t e d t o a f i n e s i z e ( 2 0 , 2 2 ) , However, i n dry g r i n d i n g the d i s t r i b u t i o n modulus has a s l i g h t tendency to decrease w i t h an i n c r e a s e i n s i z e r e d u c t i o n , t h a t i s , w i t h the i n c r e a s e of g r i n d i n g time ( 2 3 , 2 4 ) . The experimental r e s u l t s obtained i n t h i s study were i n agreement w i t h a l l these o b s e r v a t i o n s except i n the case of the sandstone sample. I t i s s u r p r i s i n g t h a t f o r a feed of s i n g l e s i z e , - 6 + 8 mesh f r a c t i o n s weighing 3 . 0 8 kg, marble showed good Schuman's s i z e d i s t r i b u t i o n curve c o v e r i n g a wide range from c o a r s e r to f i n e r s i z e s , o n l y a f t e r 1 0 r e v o l u t i o n s g r i n d i n g p e r i o d 0 The value o f the d i s t r i b u t i o n modulus was found to decrease s i g n i f i c a n t l y w i t h i n c r e a s e of g r i n d i n g p e r i o d s f o r a l l samples. Fuerstenau ( 2 2 ) s t a t e d t h a t the apparent l o w e r i n g of d i s t r i b u t i o n modulus i n . dry g r i n d i n g must r e s u l t s from the c o a r s e r p a r t i c l e s b e i n g • p r o t e c t e d by f i n e r p a r t i c l e s from g r i n d i n g media. Therefore, the great change of the d i s t r i b u t i o n modulus i n t h i s experiment was dependent upon the l a r g e amount of m a t e r i a l fed i n t o the m i l l . A p o s s i b l e e x p l a n a t i o n of the unusual behavior of sandstone on s i z e d i s t r i b u t i o n i s t h a t quartz g r a i n s cemented by the cementing m a t e r i a l s are e a s i l y l i b e r a t e d at the e a r l y stage i n g r i n d i n g and t h a t the a c t u a l g r i n d i n g process of - 63 -. quartz g r a i n s w i l l f o l l o w a f t e r l i b e r a t i o n has been completed. As can be read o f f Table .6 , the d i s t r i b u t i o n modulus of sandstone changed s i g n i f i c a n t l y from 2 . 5 2 a t the g r i n d i n g p e r i o d of 7 5 r e v o l u t i o n s to 1 . 0 7 at 8 0 0 r e v o l u t i o n s . Thus, on f u r t h e r g r i n d i n g one would expect to o b t a i n the normal s i z e d i s t r i b u t i o n curve or modulus. The d i s t r i b u t i o n moduli a t 1 5 0 r e v o l u t i o n s were estimated from Fig u r e 1 9 , t o be 0 . 5 1 f o r a n d e s i t e , O . 8 7 f o r g r a n i t e , 0.62 f o r marble, 1 . 7 3 f o r sandstone, and 0 . 8 9 f o r magnetite. A d i r e c t r e l a t i o n s h i p was not observed between the d i s t r i b u t i o n moduli thus obtained and the r e s i s t i v i t y of m a t e r i a l s obtained i n the p r e v i o u s Chapter,the c r i t i c a l h e i g h t , or the square of the t e n s i l e s t r e n g t h . This could be e x p l a i n e d by c o n s i d e r i n g t h a t d i s t r i b u t i o n modulus depends not only on the nature o f the m a t e r i a l , but a l s o g r e a t e l y on the g r i n d i n g c o n d i t i o n s of the t e s t and on the comminution d e v i c e . • - ^ -4 - Energy Index • The e v a l u a t i o n of the energy consumption or work requirement f o r the g r i n d i n g m i l l o p e r a t i o n has been c u s t o m a r i l y evaluated on the b a s i s of an e n g i n e e r i n g u n i t i n k i l o w a t t - h o u r per t o n of ore ground . Bond r e l a t e d the r e q u i r e d power f o r m i l l d r i v i n g w i t h the net energy necessary f o r s i z e r e d u c t i o n , i n t r o d u c i n g the concept of Work Index ( 6 ) . Bond considered t h a t the t o t a l work u s e f u l i n breakage i s i n v e r s e l y p r o p o r t i o n a l to the square r o o t of the diameter of product p a r t i c l e . However, Charles ( 2 0 ) , as w e l l as others ( 2 3,24) s t a t e d t h a t the r e l a t i o n of energy i n p u t t o s i z e r e d u c t i o n may be d e r i v e d as a s p e c i a l case and i s a p p l i c a b l e to s p e c i f i c cases i n the comminution system. A l s o , Smith (8) re p o r t e d the r e l a t i o n s h i p between a c t u a l horse power and c a l c u l a t e d horse power from' the Bond work index, u s i n g o p e r a t i n g data obtained from about twenty cement p l a n t s . That i s , a reasonable c o r r e l a t i o n between both horse powers was observed f o r s m a l l e r and medium-sized m i l l s , whereas an over-design problem can e x i s t f o r l a r g e r m i l l s more than e i g h t f e e t i n diameter. This i n d i c a t e s a t l e a s t t h a t the work index i s not on l y a f u n c t i o n of the nature of m a t e r i a l ground, but a l s o p o s s i b l y t h a t of the e f f i c i e n c y and/or c a p a c i t y of a m i l l , as w e l l as g r i n d i n g c o n d i t i o n s , suggesting the unappropriateness of Bond's d e f i n i t i o n of work index. Therefore, i t may be wise to reexamine the r e l a t i o n s h i p between d r i v i n g power and r e q u i r e d energy f o r s i z e r e d u c t i o n , - 6 5 -i n t r o d u c i n g a new index which i s somewhat d i f f e r e n t from work index. At f i r s t an i n v e s t i g a t i o n was c a r r i e d out to determine the f i n e product of the b a l l m i l l versus g r i n d i n g time. The progress of the products of v a r i o u s p a r t i c l e s i z e are shown i n Fig u r e 2 0 , corresponding to g r i n d i n g time or m i l l r e v o l u t i o n . I t can be seen from t h i s f i g u r e f o r marble t h a t the r a t e of p r o d u c t i o n of f i n e s i s a l i n e a r f u n c t i o n of time, f o l l o w e d by a curved p o r t i o n a f t e r 1 5 0 to 2 0 0 r e v o l u t i o n s . -The same tendencies were observed f o r a l l other samples. This o b s e r v a t i o n i s c h a r a c t e r i s t i c of batch g r i n d i n g and i t may be caused by the l a c k of new feed and/or the accumulation of f i n e r products c o a t i n g b a l l s , c o a r s e r p a r t i c l e s , and the i n s i d e of the m i l l s h e l l . Therefore, i f one wishes to estimate the consumed energy to produce f i n e m a t e r i a l i n simulated continuous o p e r a t i o n from t h a t obtained i n a batch m i l l , the r e s u l t s obtained from a l o n g e r g r i n d i n g p e r i o d than 1 5 0 r e v o l u t i o n s of m i l l run might be i n a d v i s a b l e , because the progress of f i n e r products i n continuous o p e r a t i o n might i n c r e a s e l i n e a r l y w i t h r e s p e c t to g r i n d i n g time, correspond-i n g to the f i n e r products a t 1 5 0 r e v o l u t i o n s i n the batch mill» In connection w i t h m i l l r e v o l u t i o n s , Smith and Lee ( 2 5 ) r e p o r t e d i n t h e i r comparison t e s t s w i t h Bond's g r i n d a b i l i t y t e s t t h a t 3 0 0 . t o 5 0 0 r e v o l u t i o n s i s recommended f o r f i n e r products t o estimate Bond g r i n d a b i l i t i e s from batch 100 200 400 / 6 0 0 800 1000 1200 M i l l Revolutions Figure 20. Rates of formation of mesh' f r a c t i o n s - Marble -g r i n d a b i l i t l e s without any c i r c u l a t i n g l o a d , and 1 0 0 r e v o l u t i o n s f o r co a r s e r ones. The s l i g h t d i f f e r e n c e of g r i n d i n g r e v o l u t i o n s from our study might be caused by the d i f f e r e n t g r i n d i n g c o n d i t i o n s , e s p e c i a l l y by the amount of fe e d . T h e i r , t e s t was based on Bond's standard g r i n d a b i l i t y t e s t ( 2 6 ) . a n d the feed was 7 0 0 ml which was about one t h i r d of our t e s t s f o r a m i l l of almost the same s i z e . As s t a t e d b e f o r e , the i n c r e a s e of f i n e products might be alo n g the tangent of i n i t i a l l i n e a r increment observed under. 1 5 0 r e v o l u t i o n s i n t h i s study. And f o r the comparison of g r i n d i n g t e s t i n d i f f e r e n t r o c k s , t h i s l i n e a r p o r t i o n of curve may be the most s u i t a b l e f o r t h i s purpose, since one can a v o i d other complicated g r i n d i n g mechanisms which w i l l appear i n a lo n g e r g r i n d i n g p e r i o d . E x t r a p o l a t i n g the s t r a i g h t l i n e p o r t i o n of d e s i r e d s i z e , f o r example i n the case of - 1 5 0 mesh (see Fi g u r e 2 0 ) , one can estimate the necessary t o t a l m i l l r e v o l u t i o n s J t o achieve . 8 0 percent of cumulative weight of the product, provi d e d the i n i t i a l g r i n d i n g r a t e can h o l d throughout the g r i n d i n g . As p r e v i o u s l y shown i n Fi g u r e 1 6 , the r e q u i r e d torque i s almost constant throughout the g r i n d i n g p e r i o d , so t h a t the energy consumption expressed i n k i l l o w a t t - h o u r per t o n , when 80 percent o f the product passes I . 5 0 mesh or any other s i z e s , can be obtained by the f o l l o w i n g equation. - 6 8 -= 1 . 7 x l O ~ 7 ( T - To) N wg (8) where W = work i n k i l o w a t t - h o u r per t o n T = gross torque i n kg-cm To - t o r q u e f o r empty m i l l t u r n i n g i n kg-cm . N = t o t a l r e v o l u t i o n s obtained h y p o t h e t i c a l l y from e x t r a p o l a t i n g the tangent of i n i t i a l increment wg = weight of feed i n a l a b o r a t o r y m i l l , i n kg The h y p o t h e t i c a l energy r e q u i r e d to achieve s i z e r e d u c t i o n w i t h i n i t i a l g r i n d i n g r a t e i s e n t i t l e "Energy Index which i s not only a c h a r a c t e r i s t i c value f o r the m a t e r i a l t o be ground, but a l s o a f u n c t i o n of feed s i z e , product s i z e , and other g r i n d i n g c o n d i t i o n s . Because of the v a r i o u s g r i n d i n g mechanisms i n c o r p o r a t e d i n a tumbling m i l l and the v a r y i n g r e s i s t a n c e of d i f f e r e n t m a t e r i a l s t o the v a r i o u s comminution mechanisms, i t i s questio n a b l e whether or not the energy index thus determined under f i x e d g r i n d i n g c o n d i t i o n s f o r a p a r t i c u l a r product s i z e i s a p p l i c a b l e to other product s i z e s 0 Therefore, the energy i n d i c e s of a l l samples t e s t e d were determined.with r e s p e c t t o varous s i z e s of products? - 6 5 , - 1 0 0 , - 1 5 0 , and - 2 0 0 mesheso' F i g u r e 2 1 p r e s e n t s those energy i n d i c e s as a f u n c t i o n of product s i z e p a s s i n g 80 p e r c e n t , on l o g - l o g paper. The l i n e a r r e l a t i o n s h i p between the energy index and the product s i z e was found t o h o l d f o r a l l the samples used over the s i z e - 6 9 -40 3 0 2 0 c o •p \ x c I—I ft 1 0 — \>"-® V. Andesite • \ C i v ' Quartz — \ \ . \_ Limestone — ' Feldspar — \. G r a n i t e Magnetite a ^ Marble . N. Sandstone 1 • 1 1 1 i i 2 0 0 1 5 0 1 0 0 6 5 8 0 percent p a s s i n g product s i z e (mesh) F i g u r e ' 2 1 . Energy index as a f u n c t i o n of product • *size p a s s i n g 80 percent. . . . _ 7 0 -range s t u d i e d . I t i s i n t e r e s t i n g to n o t i c e t h a t the r a t i o of the i n c r e a s e of energy index w i t h the decrease of product s i z e i s not uniform f o r a l l samples. Moreover, the order of magnitude i n energy index was changed as the product s i z e was reduced, between the a n d e s i t e and marble sample, the f e l d s p a r and limestone sample, and the sandstone and marble sample, r e s p e c t i v e l y . Therefore, the energy index determined f o r the p a r t i c u l a r product s i z e i s not a p p l i c a b l e t o the power i n p u t e s t i m a t i o n f o r the o t h e r product s i z e , u n less the slope of the s t r a i g h t l i n e i s known. The energy index was compared w i t h the Bond work index f o r f e l d s p a r , q u a r t z , and l i m e s t o n e , f o r which work i n d i c e s were determined by Mines Branch, Department of Energy, Mines and Resources, Ottawa. Both energy concepts were based on the assumption of 2 0 0 mesh p a s s i n g 1 0 0 p e r c e n t . Figure 2 2 shows' t h a t the r e s u l t and the r e l a t i o n s h i p of both concepts i s a l i n e a r f u n c t i o n w i t h zero i n t e r c e p t . The r e l a t i o n s h i p may be•expressed: -Work Index = C x Energy Index where C i s constant. The Bond work index i s intended f o r commercial m i l l s i n p r a c t i c e , whereas the energy index i s d i r e c t l y o btained from the r e s u l t s of l a b o r a t o r y m i l l g r i n d i n g . So t h a t the d i f f e r e n c e expressed i n the c o e f f i c i e n t depends on the g r i n d i n g e f f i c i e n c y of an a c t u a l m i l l and a l a b o r a t o r y m i l l . I n t h i s case, the i n c r e a s e of f i n e products I n an a c t u a l m i l l i s t w i c e (G = 0 . 5 ) t h a t of the l a b o r a t o r y m i l l . to Bond,s Work Index. Height. - 72 -Therefore, i f . the r e l a t i o n of both i n d i c e s i s obtained f o r each l a b o r a t o r y m i l l w i t h a g i v e n s e t of g r i n d i n g c o n d i t i o n s i n c l u d i n g product s i z e , the work index i s e a s i l y determined by means of the energy indexo The r e l a t i o n s h i p between energy index, and the c r i t i c a l h e i g h t obtained i n Chapter 1 was a l s o Investigated„ This r e l a t i o n s h i p c o n s i d e r s the s i n g l e impact event i n -comminution represented by means of the drop-weight impact t e s t versus complex-events expressed b a l l m i l l g r i n d i n g . The energy i n d i c e s were taken a t - 1 0 0 , - 1 5 0 , and - 2 0 0 mesh p a s s i n g 8 0 percent to compare w i t h the c r i t i c a l h e i g h t . F i g u r e 2 3 shows a l i n e a r r e l a t i o n s h i p between both q u a n t i t i e s . The e x i s t a n c e of a c l e a r r e l a t i o n s h i p of s i n g l e f r a c t u r e event to b a l l m i l l g r i n d i n g suggests the f o l l o w i n g : 1 ) the comminution mechanism i n the b a l l m i l l might be impact f r a c t u r e , i n great p a r t , at the e a r l y stage of m i l l r un. 2 ) the energy index w i l l express the r e s i s t i v i t y of m a t e r i a l s to comminution i n a p r a c t i c a l way. CONCLUSION The c o n c l u s i o n s of the i n v e s t i g a t i o n s I n P a r t 2 are as f o l l o w s : 1 c The torque r e a d i n g f o r the dry g r i n d i n g w i t h a smooth-faced c y l i n d r i c a l m i l l was low about 20 percent compared with' t h a t of the m i l l t u r n i n g without any g r i n d i n g m a t e r i a l . 2o The energy i n p u t , based on an equi-volume, to achieve the s i z e r e d u c t i o n of any m a t e r i a l can be determined from g r i n d i n g time,which i s a c h a r a c t e r i s t i c of the m a t e r i a l b e i n g ground. 3„ A new index "Energy Index" which i s a f u n c t i o n of feed s i z e , product size,i k,and g r i n d i n g c o n d i t i o n s has been proposed, and i t showed s a t i s f a c t o r y agreement w i t h the work index under s e l e c t e d g r i n d i n g c o n d i t i o n s a 4. Energy index i n c r e a s e s l i n e a r l y w i t h a decrease i n the p a r t i c l e s i z e of products on l o g - l o g paper. 5« Energy index showed a l i n e a r r e l a t i o n s h i p w i t h the . r e s u l t s of the drop-weight impact t e s t s expressed i n terms of the c r i t i c a l h e i g h t . - 7 4 . • SUMMARY The author intended to i n v e s t i g a t e two s u b j e c t s i n t h i s study: 1 ) an experimental examination of a p o s s i b l e c r i t e r i o n f o r comminution, and 2) i n v e s t i g a t i o n of s e v e r a l f a c t o r s which are necessary to e s t a b l i s h a b a s i s f o r the e s t i m a t i o n of the d r i v i n g power of tumbling m i l l s from the r e s u l t s of l a b o r a t o r y work. I n p a r t 1, the q u a n t i t y S t 2 / E proposed by Oka and Majima was' e x p e r i m e n t a l l y examined as a new c r i t e r i o n i n comminution w i t h f i v e d i f f e r e n t samples. For t h i s purpose, the p o i n t - l o a d t e s t and the propagation v e l o c i t y measurement of compressional wave was employed f o r the d e t e r m i n a t i o n of t e n s i l e s t r e n g t h and Young"s modulus, r e s p e c t i v e l y . For comminution t e s t s , the drop-weight impact t e s t and the b a l l m i l l g r i n d i n g t e s t was .performed, and the r e s u l t s were compared' w i t h St /E and S t ~ , by means of the c r i t i c a l h e i g h t . and the energy index. I t was found t h a t , as a c r i t e r i o n of p comminution, the q u a n t i t y St i s more a p p l i c a b l e , than t h a t of S t 2 / E . And S t 2 has been proposed as a new c r i t e r i o n to express the r e s i s t i v i t y of rock m a t e r i a l s to comminution. In P a r t 2, g r i n d i n g t e s t s were performed to achieve, the above purposes w i t h the 12 i n c h b a l l m i l l equipped w i t h a torque meter. A concept, Energy Index, was i n t r o d u c e d to evaluate g r i n d i n g r e s u l t s and to study g r i n d i n g problems. Each rock sample t e s t e d was determined the energy index from the torque readings and the amount of p r o d u c t s . And i t was - 75 -found from the torque readings t h a t the energy i n p u t expressed on an equi-volume charge can be determined from the r e q u i r e d g r i n d i n g time whose value i s c h a r a c t e r i s t i c s of the m a t e r i a l ground,,' The r e l a t i o n s h i p of both concepts. Bond work Index and energy index was researched* - 7 6 -SUGGESTIONS FOR FUTURE WORK On the b a s i s of the r e s u l t s obtained from the i n v e s t i g a t i o n d e s c r i b e d i n t h i s t h e s i s , the f o l l o w i n g i s recommended f o r f u r t h e r works 1) The d i f f e r e n c e between the experimental and t h e o r e t i c a l r e s u l t s f o r the c r i t e r i o n of comminution i s s t i l l unsolved. F u r t h e r study o n . t h i s problem i s s t r o n g l y recommended. . 2) I n order to determine • the energy index, the g r i n d i n g p e r i o d i s c r i t i c a l i n the dry method, s i n c e a long e r g r i n d i n g p e r i o d w i l l cause some other g r i n d i n g mechanism other than impact f o r s o f t e r m a t e r i a l s , w h i l e a g r i n d i n g time of 150 r e v o l u t i o n s w i l l be too s h o r t f o r a, tougher m a t e r i a l . Therefore, a v/et g r i n d i n g study should be attempted to determine the energy index, and t o study the r e l a t i o n s h i p between dry and v/et g r i n d i n g . 3) . The maximum d r i v i n g torque was obtained when the g r i n d media was charged w i t h no fe e d . Therefore, t o e l i m i n a t e l u b r i c a t i n g a c t i o n due to feed m a t e r i a l and/or s l i p p a g e of g r i n d i n g media, the use of a m i l l w i t h l i f t e r bars w i l l be more u s e f u l f o r the measurement o f r e q u i r e d energy f o r s i z e r e d u c t i o n . -• 77 -REFERENCES 1. Department of S c i e n t i f i c and I n d u s t r i a l Research, "Crushing and G r i n d i n g - A B i b l i o g r a p h y " , Her Majesty's S t a t i o n e r y O f f i c e , London, ( 1 9 5 8 ) . 2. Chemical Engineers' Handbook, 4 t h e d i t i o n , McGraw-Hill. 3 . Taggart, A.F., Handbook of M i n e r a l D r e s s i n g , John Wiley & Sons. 4 . Oka, Y., Majima, H., Canadian M e t a l l u r g i c a l Q u a r t e r l y V o l . 9 , No.2 (1970) 5 . Hiramatsu, Y., 0ka,Y., Int.J.Rock MechoMin.Sci., V 0 I . 3 (1966) 6 . Bond, F. C , AIMS. Trans., Vol.193* ( 1 9 5 2 ) ?. Smith, R.W., M i n i n g E n g i n e e r i n g , A p r i l ( I 9 6 I ) .8. Bond, F. C., A I M E , ' T r a n s . , ' ( i 9 6 0 ) , V o l . .217 9 . Coates D. F., Parsons R. C , I n t . J . Rock Mech. Min. Min. S c i . , V o l . 3 , (1966) 10. Yarriaguchi, U., I n t . J . Rock Mech. Min. S c i . Vol.7 (1970) 11. McWIllIams, J.R., T e s t i n g Techniques f o r Rock Mechanics,. ASTM S p e c i a l . t e c h n i c a l P u b l i c a t i o n , No.402. 12. Emery, C. L., Some Aspects of S t r a i n i n RocksjCourse Notes • M i n e r a l E n g i n e e r i n g 4 5 5 « 1 3 . Obert, L., D u v a l l , W.,I,, Rock Mechanics and the Design of S t r u c t u r e s i n Rock, John Wiley & Sons, I n c . ( 1 9 6 ? ) 14. B i r c h , F., J o u r n a l of Geophysical Research, Trans., V o l . 6 5 , No.4 (I960) 1 5« Oka, Y., Kiyama, H., Hiramatsu,Y., J o u r n a l of M a t e r i a l s S c i e n c e , Japan. Vol.18, No.191 ( 1 9 6 9 ) 1 6 . Ernery, C. L., Mine & Quarry E n g i n e e r i n g , A p r i l , May. ( i 9 6 0 ) 17. F u j i n a k a , Y«, Majima, H., Jomoto, K., Canadian M e t a l l u r g i c a l Q u a r t e r l y , Vol.10, N 0 3 . (1971) 18. Yang, D.C., Mempel, G., Fuerstenau, D.W., Trans. AIME., V o l . 2 3 8 (1967) - 7 8 -1 9 • Schuhmann, R., J r . , M i n 0 Technology, Tech. Pub. N o . 1 1 8 9 . ( 1 9 4 0 ) 2 0 . Charles, R. J . , AIME.Trans. Vol.208„ ( 1 9 5 7 ) 2 1 . K i n a s e v i c h , R. S., Fuerstenau, D.W., Canadian M e t a l l u r g i c a l Q u a r t e r l y , V o l . 3 , No.l. ( 1 9 6 4 ) 2 2 . Fuerstenau, D.W., S u l l i v a n , D.A.,Jr», AIME Trans• V o l . 2 2 0 , ( 1 9 6 2 ) 2 3 . Mular, A.S., AIME Trans., V o l ' . 2 2 3 . ( 1 9 6 2 ) 2 4 . Schuhmann, R. J r . , AIME T r a n s V o l . 2 1 4 . ( 1 9 5 9 ) 2 5 . • Hukki, R. T., AIME Trans o , V o l . 2 2 0 , No . 9 o ( 1 9 5 9 ) 2 6 . Dept 0 S c i e n t i f i c and I n d u s t r i a l Research, Warren Sp r i n g Laboratory, M i n e r a l P r o c e s s i n g Information No . 3 > ( 1 9 6 2 ) - 7 9 -Appendix I Table 1, Measurement of Propagation V e l o c i t y of P-Wave (Andesite) Sample Length T r a v e l - V e l o c i t y of Number Time P Wave cm x 1 0 ~ 6 s e c x l o 5 cm/sec A- 1 5 . 8 3 9 . 7 0 6 . 0 5 A- 2 6 . 4 3 1 0 . 8 5 5 ° 9 3 A- 3 6 . 0 5 1 0 . 1 0 5 ° 9 9 A- 4 4 . 6 o . 7 . 8 0 - 5 . 9 0 A- 5 • 5 . 3 3 '.' 8 . 8 5 6 . 0 6 A- 6 4 . 3 8 6 . 9 5 . 6 . 3 0 A- 7 6 . 1 6 9 . 5 0 6.48 A- 8 5 . 5 2 9 . 2 0 6.00 A- 9 6 . 1 9 9 . 7 5 6 . 3 5 A - 1 0 3 . 6 9 6 . 7 0 5 . 5 1 A - 1 1 5 . 6 1 9.35 " 6 . 0 0 A - 1 2 ' 6 . 7 8 1 1 . 4 5 5 . 9 2 A - 1 3 6.18 9 . 7 5 6 . 3 4 A-14 5 . 5 1 8 . 7 0 6 . 3 3 . A -1 .5 6 . 1 8 9 . 5 5 6 . 3 4 A - 1 6 3.14 5 . 2 3 6 . 0 1 A-17 5 » 3 0 8 . 9 0 5 . 9 5 A-18 4 . 9 3 8 . 1 0 6.08 A - 1 9 4 . 0 1 6 . 7 6 • 5 . 9 4 . A - 2 0 3 . 2 9 6 . 0 5 5 - 7 8 •A -21 3 . ^ 7 6 . 0 5 ... 5 . 7 4 . A - 2 2 3 . 9 6 7 . 0 0 • 5 . 6 6 . A - 2 3 4 . 4 3 7 . 8 0 5 . 6 8 A-24 . 3 . 8 3 6 . 5 0 5 o 8 9 A - 2 5 3 . 7 9 6 . 2 0 6 . 1 1 A - 2 6 • 1.81 2 . 8 5 6 . 3 5 A-2 7 1 . 7 0 3 . 0 5 5 . 5 7 . A-28 2.18 3 . 8 0 5 » 7 4 -A - 2 9 2 . 2 1 3 . 5 0 6 . 3 1 A - 3 0 2 . 0 3 3.40 5 . 9 7 Mean value ..6.01 Standard d e v i a t i o n 0 . 2 6 C o e f f i c i e n t of standsrd d e v i a t i o n 0.043 - 80 -Table 1 . Measurement of Propagation V e l o c i t y of P-Wave Gran i t e Sample Length T r a v e l - V e l o c i t y of Number Time P Wave cm x 1 0 sec x 1 0 cm, G- 1 5 . 4 4 1 4 . 2 5 3 . 8 2 G- 2 6 . 6 3 1 7 . 8 0 3 . 7 2 G- 3 6 . 2 3 1 5 . 7 5 3 .94 G- 4 6 . 1 1 1 6 . 0 5 3.81 G- 5 ' ' 6 . 0 0 . 1 5 . 7 0 3.82 G- 6 6 . 0 6 • 1 5 . 8 5 3 « 8 2 G- 7 6 . 7 4 1 7 . 6 5 3.82 G- 8 5 . 6 6 1 4 . 2 5 3 - 9 7 G- • 9 •' 5 . 0 6 1 2 . 9 5 3 . 9 1 G - 1 0 , 5 * 9 3 1 5 . 9 5 3 . 7 2 G - 1 1 4 . 7 7 1 2 . 3 0 3 . 8 8 G - 1 2 , , 4 . 7 1 1 2 . 7 5 3 . 6 9 G - 1 3 5 . 7 9 1 5 . 1 5 3 . 8 2 G-14 4 . 9 3 1 3 . 6 5 - 3 . 6 1 G - 1 5 5.48 1 4 . 7 0 3 . 7 3 Mean value Standard d e v i a t i o n C o e f f i c i e n t of standard 3 .81 0 . 1 0 0 . 0 2 6 d e v i a t i o n - 81 -Table' 1. Measurement of Propagation V e l o c i t y of P-Wave Marble Sample Length T r a v e l -Number Time - 6 cm x 1 0 sec Mb- 1 4 . 0 9 6 . 9 5 Mb- 2 6 . 2 5 1 4 . 1 5 . Mb- 3 , 6 . 3 4 1 1 . 0 3 Mb- 4 6 . 2 2 1 2 . 8 5 Mb- 5 6 0 O 3 1 3 . 6 5 Mb- ' 6 6.04 1 0 . 9 3 Mb- 7 ' 3 . 0 5 6 . 3 8 Mb- 8 5 * 4 7 9 . 8 5 Mb- 9 5 * 3 3 9 . 6 0 Mb - 1 0 6 . 4 3 1 1 . 0 5 Mb -11 • 5 . 2 5 1 1 . 9 8 Mb - 1 2 2 . 9 1 6 . 2 0 Mb-13 3 . 9 7 8 . 0 5 Mb-14 . 5 . 1 2 . 1 0 . 9 8 Mb - 1 5 4 . 2 6 3 . 3 5 Mb-16 3 . 5 2 7 .80 Mb - 1 7 5 . 5 5 1 1 . 2 3 Mb-18-- 5 . 2 2 1 0 . 3 0 Mb - 1 9 3.48 6 . 2 0 V e l o c i t y of P Wave 5 , x 1 0 cm/sec Mean value Standard d e v i a t i o n C o e f f i c i e n t of standard d e v i a t i o n M V - 1 Mb"-2 Mb 0 - 3 Mb ' - 4 Mb'-5 5.41 7.18 4.42-6 . 5 3 6 . 5 6 1 0.85 1 4 . 3 0 7.80 1 2 . 5 5 1 3 . 6 5 5 . 8 8 . 4.42 •' 5 . 7 5 4.84 4.42 5 . 5 3 4 . 7 8 5 . 5 5 6 . 0 7 5.82 . 4 . 3 8 4 . 6 9 ^ . 9 3 4 . 6 6 5 . 1 0 4 . 5 1 4 . 9 4 5 . 0 7 5 . 6 1 5 . 1 0 0 . 5 5 0 . 1 0 8 4.98 5 . 0 2 5 . 6 7 5 . 2 0 4.81 - 82 -Table 1. Measurement of Propagation V e l o c i t y of P-Wave Sandstone '. Sample Number Length T r a v e l s Time V e l o c i t y of P Wave cm - 6 •. x 1 0 sec 5 x 1 0 cm/ S- 1 • 6 . 0 9 1 6 . 2 0 3 . 7 5 S- 2 4 . 8 5 1 2 . 0 5 4 . 0 2 s - 3 . • 4 . 8 9 • 1 2 . 8 0 . 3.82 S- 4 5 . 0 0 1 3 . 0 5 3 . 8 3 s - 5 5 . 2 2 2 0.40 3 - 3 3 -S- 6 4 . 8 8 1 3 . 1 5 . 3 . 7 1 S- 7 6 . 5 6 1 9 . 3 5 3 . 3 9 S- 8 4 . 1 9 1 0 . 1 0 . 4 . 1 5 S- 9 4 . 6 3 1 1 . 3 0 4 . 1 0 S - 1 0 4.64 . 1 1 . 3 0 4 . 1 0 S - 1 1 • 4 . 2 2 1 0 . 1 5 4 . 1 6 S - 1 2 5 - 1 9 1 2 . 8 0 4 . 0 5 S - 1 ' 3 4 . 5 ? 1 1 . 3 5 4 . 0 3 S-14 • 5 « 0 1 • 1 3 . 2 5 3 - 7 8 s - 1 5 5 . 7 8 • 1 5 . 4 5 3 . 7 ^ Mean value Standard d e v i a C o e f f i c i e n t of t i o n Standard d e v i a t i o n 3 « 8 6 0 . 2 6 0 . 0 6 7 Magnetite Mg- .1 4 . 2 3 8 . 9 5 . ^ . 7 3 Mg- 2 5 . 0 3 1 1 . 2 0 4 . 4 9 Mg- 3 6 . 9 8 1 3-40 ' 5 . 2 1 Mg- 4 4.82 1 1 . 8 5 4 . 0 7 Mg- 5 • 4.90 9 . 2 5 5 . 3 0 Mg- 6 ' 4 . 1 5 . 9 . 1 0 3 . 9 7 w.{r- 7 4 . 1 6 . 8 . 8 0 ^ . 7 3 Mg- 8 . 3 . 8 3 8 . 3 0 4 . 6 1 Mg- 9 5 . 1 5 1 1 . 5 0 4.48 Mg-10 . 4 . 4 7 9 . 6 0 4 . 6 6 Mg-11 6.19 1 5 . 2 5 ' 4 . 0 6 Mg - 1 2 6.14 1 1 . 6 0 5 . 2 9 Mg - 1 3 4 . 6 5 . 9 . 5 5 4 . 8 7 Mg-14 4 . 0 7 9 . 7 5 4 . 1 7 Mg - 1 5 5 . ^ 7 1 2 . 9 0 4 . 2 6 Mean value ^ . 5 9 ' Standard d e v i a t i o n 0 . 4 4 C o e f f i c i e n t of standard d e v i a t i o n 0 . 0 9 6 ' - 0 - 1 6 . 8 7 1 5 . 7 5 4 . 3 6 Mg' - 2 6 . 6 9 . 1 5 . 0 5 4 . 4 5 Mg' - 3 . 6 . 2 5 1 5 . 7 5 3 . 9 7 Mg' - 4 . - 6 . 1 7 1 3 . 1 5 4 . 6 9 Mg° - 5 7 . 3 5 1 5 . 1 0 4 . 8 7 Table 2 . Measurement of T e n s i l e Strength 1 Andesite Granite Specimen Number Thickness (cm) Load at F a i l u r e (kg) T e n s i l e Strength (kg/cm ZY Specimen Number Thickness (cm) Load at F a i l u r e (kg) T e n s i l e Strength A- 1 5 . 8 7 7400 1 9 2 G- 1 5 . 4 4 3 5 5 0 • I O 7 A- 3 6 , 0 5 6400 1 5 6 G- 2 6 . 6 3 5 5 5 0 . 1 1 3 , A- 5 5 - 3 6 6 3 5 0 197 G- 3 6 . 2 0 5 1 5 0 1 1 9 A- 6 4 . 3 8 3 3 0 0 1 5 ^ G- 4 •6 . 1 1 4 7 5 0 1 1 3 A- 8 5 . 5 2 5 7 0 0 1 6 7 G- 5 6 . 0 0 4 5 0 0 1 1 2 A - 1 0 3.69 3 7 5 0 247 G- 6 6 . . O 6 5 2 0 0 1 2 6 A-16 3.14 2 5 0 0 2 2 6 G- 7 6 . 7 4 5 6 0 0 : 1 1 0 A-17 5 3 0 5 1 0 0 1 6 2 G- 8 5 . 6 6 4000 ' 1 1 1 • A-18' 4 . 9 3 ' 5 8 5 0 214 c- 9 5 . 0 6 3 5 5 0 • 124 A-19 4 . 0 1 5 1 5 0 .. 2 8 6 G - 1 0 5 . 9 3 4 4 5 0 . 1 1 3 A - 2 0 3 . 2 9 3 3 0 0 2 7 3 G - l l 4 . 7 7 2 9 5 0 1 1 6 A - 2 1 3 . ^ 7 2 7 0 0 2 0 0 G - 1 2 4 , 7 1 3 2 0 0 • 128 A - 2 2 3.96 3 4 0 0 1 9 3 G - 1 3 5 . 7 9 4000 1 0 6 A - 2 3 4 . 4 3 - 5 9 0 0 2 6 9 G-14 * K 9 3 3 2 0 0 118 A-24 3 . 8 3 . 3 0 5 0 1 8 5 G - 1 5 5.48 . 2800 3 3 A - 2 5 3 - 7 9 .4200 2 6 2 Table 2 . Measurement of T e n s i l e Strength Marble Sandstone Specimen Number Thickness (cm) Load at F a i l u r e (kg) T e n s i l e Strength (kg/cm 2) Specimen . Number Thickness (cm) Load at F a i l u r e (kg) . T e n s i l e . Strength (kg/cm 2) Mb- 1 4 . 0 9 1 0 5 0 5 6 S- 1 6 . 0 9 4 5 0 0 108 Mb- 2 6 . 2 5 1900 S- 2 4.85 3 3 0 0 1 2 5 Mb- 3 6 . 3 4 2 5 5 0 57 S- 3 4 . 8 9 2 6 5 0 99 Mb- 4 6.22 2 2 5 0 5 2 S- 4 5.00 3 1 0 0 111 Mb- 5 6 . 0 3 2 2 0 0 5^ S- 5 5.22 4100 1 3 4 • Mb- 6 ' 6.04 2 6 0 0 6 3 S- 6 4.88 3 3 0 0 1 2 3 Mb- 8 .5.47 1700 5 0 S- 7 6 . 5 6 5 9 5 0 1 2 3 Mb- 9 5.83 . 1 5 0 0 40 • S- 8 4.19 3 0 0 0 1 5 3 Mb-10 "6.43 2 2 0 0 47 . S- 9 4 . 6 3 3400 .141 Mb-11 5 . 2 5 2 0 0 0 6 5 V • S-10 ' 4.64 3 1 0 0 .128 Mb-13 3.97 - 1 0 5 0 6 0 S-11 4.22 2 9 0 0 145 Mb-l 4 5.12 1850 . 6 3 . S-12 5.19 4 6 0 0 1 5 3 • Mb - 1 5 4 . 2 6 ' 1 3 0 0 • . 6 3 S - 1 3 ^.57 3 3 0 0 146 Mb-l 6 3 . 5 2 1 0 5 0 75 S-14 5.01 3 5 5 0 127 Mb-l? 5.55 1 9 5 0 5 6 S - 1 5 5.78 5 0 5 0 1 3 5 Mb-18 5.22 1850 6 1 Mb-l 9 3.48 9 0 0 6 7 Mb-20 4.98 1 3 0 0 '46 Table 2 . Measurement of Te n s i l e Strength Magnetite Specimen Number .Thickness (cm) Load at F a i l u r e (kg) T e n s i l e . Strength (kg/cm 2) . Mg- 1 4 . 2 3 1 6 0 0 7 9 Mg- 2 5 - 0 3 2 3 0 0 81 Mg- 4 . 4.82 ,^ 2 3 0 0 1 1 5 Mg- 5 4.9.0 2 3 5 0 . . 8 7 . . Mg- 6 4 . 1 5 2 1 0 0 1 1 0 .Mg- 7 4 . 1 6 1 5 5 0 81" Mg- 8 3.83 1 2 5 0 7 5 Mg- 9 5 . 1 5 2 3 5 0 7 9 Mg - 1 0 .. 4,4? .. 1 7 0 0 . 7 7 Mg-l4 • 4.07 1 7 5 0 9 5 : Mg - 1 6 3.43 1 7 5 0 1 2 9 Mg-17 '. 4 . 0 8 1 9 5 0 1 0 5 Mg-18 4 . 1 -6 1 7 5 0 9 0 Mg -19 4 . 2 0 1400 7 0 Mg - 2 0 4 . 7 9 2 5 0 0 1 0 0 I CO I Appendix I I Table 3 . E f f e c t of B a l l Charge and M i l l Speed on the Torque - B a l l Charge (kg) (Percentage of M i l l Volume) M i l l Speed ( r . p, m. )• 3 2 4 7 . 5 5 5 8 6 2 68 7 0 7 6 8 1 9 2 9 4 . 0 ( 1 0 0 . 0 ) 3 8 . 1 3 9 . 8 42 , 1 4 2 . 1 4 6 . 0 8 6 . 5 ( 9 2 . 0 ) 5 7 . 5 - 69.O 7 6 . 6 8 8 . 5 7 9 . 0 ( 8 3 . 8 ) 9 6 . 5 104 . 0 1 0 9 . 0 1 1 9 . 0 1 1 3 . 8 7 1 . 4 ( 7 5 . 9 ) 1 ^ 5 . 5 1 ^ 9 - 5 1 5 3 . 5 ' 1 3 9 . 5 117 . 0 . 9 1 . 9 6 3 . 5 ( 6 8 . 5 ) 184 . 0 1 8 7 . 5 1 8 5 . 5 144 . 0 9 9 . 1 5 5 - 5 ( 5 8 . 9 ) 2 0 . 3 . 0 2 1 5 . 0 2 0 9 . 0 187 . 5 1 5 0 . 0 1 0 7 . 5 ^ 7 . 5 ( 5 0 ; 4 ) 214 . 5 224 . 0 2 2 7 . 0 2 2 5 , 0 2 1 1 . 0 1 9 1 . 5 1 6 8 . 5 1 4 4 . 0 ' 3 9 - 5 ( 4 2 . 0 ) 2 1 1 . 0 2 2 2 , 0 224 . 0 2 2 5 . 0 224 . 0 1 2 5 . 5 3 1 . ^ ( 3 3 . ^ ) 1 9 8 , 5 2 0 5 . 0 2 0 ? . 0 .208.0 2 0 8 . 0 2 1 1 . 0 2 1 3 . 0 214 . 5 2 3 . 7 ( 2 2 . 3 ) 1 5 7 . 0 1 6 5 . 0 . I 6 8 . 5 1 7 0 . 0 1 7 2 , 5 • 1 7 6 . 0 1 7 9 . . 0 179 . 5 1 5 . 8 ( 1 4 . 8 ) 1 2 6 . 5 . 1 3 0 . 5 1 3 3 . 0 1 3 6 . 5 • 1 3 8 . 5 7 . 7 ( 8 . 2 ) 8 4 . 3 85.O 8 8 . 2 • ' 9 1 . 2 9 2 . 0 9 5 . 0 0 . ( 0 ) 2 9 . 1 3 3 . 0 3 8 . 3 4 2 , 2 . 4 7 . 0 Appendix I I I Tabled. Screen A n a l y s i s f o r the B a l l M i l l i n g of Andesite T o t a l G r inding Revolutions ( G r i n d i n g Time) Mesh 7 5 : ( 7 0 /= revs sec) Cum 1 2 5 revs ( 1 1 ? sec) % % Cum 2 0 0 ( 1 8 8 revs . 1 sec) % Cum 3 0 0 (281 revs sec) % Cum 8 0 0 ( 7 5 0 revs sec) % Cum 6 6 0 . 0 1 0 0 . 0 41 .2 lOOoO 3 6 . 8 1 0 0 . 0 1 ^ . 7 1 0 0 . 0 4 , 0 . 9 9 . 9 •8 2 1 . 0 4 0 . 0 2 7 . ? 5 8 . 8 2 6 . 4 6 3 . 2 2 0 . 9 8 5 . 3 9 . 3 95.Q 1 0 . 7 . 7 19.0 1 1 . 3 3 1 . 0 1 2 . 3 3 6 . 7 1 . 5 . 7 64.4 1 0 . 0 8 5 . 7 14 4 . 2 1 1 . 2 6 . 6 1 9 . 7 7 . 5 24.4 1 2 . 3 48 .7 1 0 . 5 7 5 - 7 2 0 2 . 3 7 . 0 4 . 1 1 3 . 2 4 . 6 1 7 . 0 9 . 0 3 6 . 4 1 0 . 0 6 5 . 2 23. 1 . 4 4 . 7 2 . 4 9 . 1 2 . 8 1 2 . 3 6 . 0 2 ? . 4 8 . 3 5 5 . 2 . 3 5 . 0 . 9 . 3-> 1 . 6 6 . 6 ' 1 . 9 9 . 6 " 4 . 3 2 1 . 4 . 6 . 9 46.9 43 0 . 5 2 . 5 1 . 0 5 . 0 1 . 2 7 . 7 2 . 8 1 7 . 1 5 . 1 40.0 6 5 0 . 4 1 . 9 0 . 7 . 4 . 0 0 . 9 6 . 5 2 . 2 14 .2 3 ^ . 9 1 0 0 0 . 2 1 . 5 0 . 5 30 0 . 7 5 . 5 1 . 5 1 2 . 0 3 . 5 3 0 . 4 1 5 0 0 . 2 1 . 3 0 . 4 2 . 7 0 . 5 4 . 8 1 . 2 1 0 . . 5 2 . 5 2 6 . 9 2 0 0 1 . 1 1 . 1 2 . 3 2 . 3 4.3 4.3 9 . 2 9 . 2 24.4 24.4 Table 4 . Screen A n a l y s i s f o r the E a l l M i l l i n g ' . o f G r a n i t e T o t a l G r i n d i n g Revolutions ( G r i n d i n g Time) Mesh 7 5 revs 1 0 0 revs 1 5 0 revs 2 5 0 revs ( 7 0 sec) ( 9 4 sec) (141 sec) ( 2 3 4 sec) # Curn ' • fo Cum % % Cum % % Cum 6 1 9 . 7 1 0 0 . 0 1 3 . 0 . 1 0 0 . 0 . 5 . 1 • 1 0 0 . 0 0 . 6 1 0 0 . 0 8 . 2 3 . 7 8 0 . 3 1 7 . 8 8 7 . 0 9 o 9 9 5 . 0 2.4 9 9 . ^ 1 0 1 4 . 6 5 6 . 6 1 3 ' . 5 6 9 . 2 . 1 0 . 6 8 5 . 1 b. r> 9 7 . 0 14 1 0 . 7 4 2 . 0 1 2 . 0 5 5 ' . 7 ' 1 2 . 3 7 4 , 5 7 . 9 9 2 . 7 2 0 8 . 5 3 1 . 3 1 0 . 8 4 3 . 7 1 3 . 0 6 2 . 2 1 1 . 9 84.8 28 6 . 1 2 2 . 8 8 . 4 3 2 . 9 1 1 . 2 4 9 . 2 1 3 . 7 7 2 . 9 3 5 4 . 8 1 6 . 7 7 . 0 2 4 . 5 9 . 8 . 3 8 . 0 1 3 . 4 5 9 . 2 48 3 o 3 1 1 . 9 5 . 0 1 7 . 5 7 . 1 2 8 . 2 1 0 . 6 4 5 . 8 • 6 5 2 . 6 8 . 1 . 4 . 1 1 2 . 5 5 . 9 2 1 . 1 . 8 . 7 3 5 . 2 1 0 0 1 . 8 6 . 0 2 . 9 . 8 . 4 4 . 3 . 1 5 . 2 6 . 7 2 6 . 5 1 5 0 - . 1 . 3 4 . 2 2 . 0 5 . 5 3 » 0 1 0 . 9 4.8 I 9 . 8 2 0 0 2 . 9 2 . 9 3 . 4 3 . 4 7 . 9 • 7 . 9 1 5 . 0 1 5 . 0 Table 4 . Screen A n a l y s i s f o r the B a l l M i l l i n g of Granite T o t a l G r i n d i n g Revolutions ( G r i n d i n g Time) Mesh . 5 0 0 revs 1 0 0 0 revs ( 4 6 9 sec) ( 9 3 9 sec) % • % Cum . % % Cum 6 • 8 0 . 1 9 9 . 9 1 0 0 . 4 9 9 . ' 8 1 4 1 . 4 9 9 . 4 0 . 1 99 .9 2 0 5 . 1 9 8 . 0 0 . 4 9 9 c 8 2 8 „ 1 0 . 1 9 2 . 9 2 . 3 9 9 ° ' + 3 5 1 4 . 1 8 2 . 8 . 6 . 9 9 7 . 1 4 8 1 2 « 9 6 8 . 7 9 . 9 9 0 , 2 6 5 1 2 . 7 5 5 = 8 1 3 . 3 8 0 . 3 ' 1 0 0 9 . 9 4 3 . 1 1 1 . 9 6 7 . 0 1 5 0 7 . 5 3 3 . 2 1 0 . 1 5 5 - 1 2 0 0 2 5 . 7 2 5 . 7 ^ 5 ° 0 4 5 . 0 Table 4. Screen A n a l y s i s f o r the B a l l M i l l i n g of Marble T o t a l G r inding Revolutions ( G r i n d i n g Time) Mesh 1 0 revs 2 0 revs 40 revs 60 revs 1 0 0 revs ( 9 sec) ( 1 9 sec) ( 3 8 sec) ( 5 6 sec) . . ( 9 4 sec) % % Gum % % Cum : % % Cum ' % f5 Cum % % Cum 6 61.9 1 0 0 . 0 5 2 . 3 9 9 . 9 . 3 7 . 1 99.9 2 5 . 5 100 o0 8 . 8 100.0 8 I 8 0 O 3 8 . 1 1 9 . 2 4 7 . 6 2 0 . 0 6 2 . 8 2 0 . 4 7 4 . 5 . 1 2 . 6 9 1 . 2 1 0 . 4 o 9 2 0 . 1 7 . 0 2 8 . 4 : 9 . 3 4 2 . 8 11.0 5 4 . 1 9 . 9 7 8 . 6 "14 3 . 4 1 5 . 2 4 . 7 2 1 . 4 6 . 6 3 3 . 5 8 . 5 4 3 . 1 9 . 5 6 8 . 7 2 0 2 . 7 1 1 . 8 3 . 6 1 6 . 7 5 . 4 . 2 6 . 9 7 o 0 3 4 . 6 9 . 3 5 9 . 2 28 2 . 1 9.1 3 . 0 1 3 . 1 4 . 6 2 1 . 5 5 . 7 2 7 * 6 8 . 9 4 9 . 8 3 5 1 . 3 7 . 0 2 . 3 10.1 3 . 9 .16.9 4 . 9 2 1 . 9 8 . 4 4 0 . 9 48 1.7 5 . 7 1.8 7.8 2 . 9 1 3 o 0 3 . 7 1 7 * 0 6 . 7 3 2 . 6 6 5 0 . 9 4 . 0 1 . 5 6 . 0 2 . 4 1 0 o l 3 o l 1 3 » 3 6 . 1 2 5 o . 9 1 0 0 0 . 8 3 . 1 1 . 1 • 4 . 5 1.7 7 « 7 2 . 4 10.2 4 . 8 1 9 . 7 1 5 0 0 . 5 2 . 3 0 . 8 , 3 . 4 1 . 3 6 . 0 1.7 7 o 8 3 o 6 1 4 . 9 2 0 0 1 . 8 . 1 . 8 2 . 6 2 . 6 4 . 7 4 . 7 6 . 1 6 . 1 1 1 . 3 1 1 . 3 Table 4, Screen A n a l y s i s f o r the B a l l M i l l i n g of Marble T o t a l G r i n d i n g Revolutions ( G r i n d i n g Time) Mesh 1 5 0 (141 revs sec) % Cum 2 0 0 revs (188 sec) % f Cum 5 0 0 ( 4 6 9 % revs sec) % Cum 1 0 0 0 revs ( 9 3 9 sec) % % Cum 6 8 . 5 1 0 0 . 0 1 . 7 1 0 0 . 0 8 1 2 . 3 91 * 5 . 3 . 5 9 8 . 3 0 . 1 9 9 o 9 1 0 l l o l 7 9 o 2 5.1 9 4 , 8 0 . 2 9 9 o 8 14 . 1 0 , 5 . 6 8 . 1 6 . 8 8 9 . 7 1 . 2 9 9 o 7 2 0 9.6 57.6 8 0 8 82 . 9 3 o 3 9 8 . 5 0 , 1 1 0 0 . 0 28 8 o 3 4 8 . 0 9 » 8 7 4 . 1 6 , 4 9 5 o 2 0 , 6 9 9 * 9 3 5 7 . 5 3 9 . 8 1 0 . 5 6 4 . 4 9 . 3 8 8 . 8 2 . 7 9 9 o 3 48 5 . 9 3 2 . 3 9.1 5 3 . 8 9 » 6 7 9 o 5 5 * 3 9 6 . 6 6 5 5 « 5 2 6 . 4 . 9 - 0 4 4 . 7 1 0 , 8 6 9 . 9 8 . 7 9 1 o 3 1 0 0 4 . 4 2 0 . 9 7 c 6 35<>7 9 . 4 5 9 o l 9 » 6 8 2 . 6 1 5 0 3 . 5 16.5 6.1' 28.1 8 , 3 ^ 9 » 7 9 . 7 7 3 = 0 2 0 0 1 3 . 0 1 3 o 0 2 2 . 0 2 2 . 0 4 1 , 4 4 1 , 4 6 3 * 3 6 3 o 3 Table 4 . Screen A n a l y s i s f o r the B a l l M i l l i n g of Sandstone Mesh T o t a l G r i n d i n g Revolutions ( G r i n d i n g Time) • ? 5 r e v s . 1 5 0 r e v s . 3 0 0 r e v s . 8 0 0 r e v s . ( 7 0 sec.) (141 sec.) (281 sec.) ( 7 5 0 sec.) %. % Cum. % Cum. % Cum. % % Cum.. 6 1 1 . 4 9 9 * 9 1 . 0 1 0 0 . 0 0 . 1 1 0 0 . 0 8 " l O . O 8 8 ' , 5 2 . 5 . 9 9 . 0 0 . 2 9 9 . 9 1 0 6 . 4 7 3 . 4 2 . 1 9 . 6 . 5 0 . 3 9 9 . 6 14 4 . 8 7 2 , 0 2 . 1 9 4 . 3 - 0 . 3 ; 9 9 . 3 2 0 4 . 3 - 6 7 . 2 2 . 4 9 2 . 3 0 . 5 9 9 . 0 2 8 4 . 0 63.O 2 . 7 8 9 . 9 . 1 . 0 9 8 . 5 0 . 1 . 1 0 0 . 0 3 5 5 . 5 5 9 . 0 4 . 4 8 7 . 1 2 . 7 9 7 . 5 0 . 6 9 9 . 9 48 7 . 3 ' 5 3 . 5 7 . 7 8 2 . 8 6 . 3 9 4 . 9 2 , 6 9 . 9 . 4 6 5 1 4 . 8 4 6 . 2 2 1 . 0 7 5 . 0 2 2 . 2 8 8 . 5 1 2 . 2 9 6 . 8 1 0 0 2 0 . 0 3 1 . 4 2 9 . 3 5 4 . 0 3 3 . 3 6 6 . 3 3 0 . 0 8 4 . 6 1 5 0 6 . 1 1 1 . 4 1 0 . 1 2 4 . 6 1 1 . 3 3 3 . 0 1 4 . 7 5 4 . 7 2 0 0 5 . 3 5 . 3 1 4 . 5 1 4 . 5 2 1 . 7 2 1 . 7 4 0 . 0 4 0 . 0 Table 4, Screen A n a l y s i s f o r the B a l l M i l l i n g of Magnetite T o t a l G r i n d i n g R e v o l u t i o n s . ( G r i n d i n g Time)  Mesh . 7 5 r e v s . 1 5 0 r e v s . 2 2 5 r e v s . 5 0 0 r e v s . ( 7 0 sec.) (141 s e c ) ( 2 1 1 sec.) (469 sec.) • % % Cum. % Cum. % %'Cum. % % Cum. 6 2 5 . 8 1 0 0 . 0 1 0 . 2 1 0 0 . 0 4.4 1 0 0.Q 0 . 5 1 0 0 . 0 8 21.4 7 2 . 9 1 3 . 8 8 9 . 8 7 . 5 9 5 . 6 1 . 0 • 9 9 . 5 1 0 . 14 .0 5 2 . 8 1 2.4 7 5 . 9 • 8 . 8 8 8 . 1 1 . 6 9 8 . 5 14 1 0 . 3 3 8 . 8 1 2 . . 5 6 3 . 5 1 0 . 8 7 9 - 3 4 . 0 9 6 . 9 2 0 8 . 0 2 8 . 5 1 1 . 9 5 1 . 0 1 2 . 7 6 8 . 6 7 . 8 9 2 . 9 . 28 5.4 2 0 . 5 9.4 3 9 . 2 1 1 . 6 5 5 . 8 1 1 . 6 . 8 5 . I 3 5 4 . 2 1 5 . 0 7 . 9 2 9 . 8 1 0 . 8 44.2 . 1 - 3 . 7 . ' 7 3 . 5 48 - 2 . 8 • 1 0 . 8 5 . 5 2 1 . 9 7 ° 9 . 3 3 . 5 • 1 1 . 8 5 . 9 . 8 . 6 5 • '- 2 . 3 , 8 . 0 4 . 5 1 6.4 6 . 9 2 5 . 6 11.4. 48 .0 1 0 0 1 . 6 5 . 7 3 . 3 1 1 . 9 5 - 1 18 . 6 . 8 . 8 3 6 . 6 1 . 5 0 1 . 2 4 . 1 2.4 . 8 . 6 3 = 7 1 3 . 6 7 . 0 2 7 . 8 2 0 0 2 . 9 2 . 9 6 . 2 6 . 2 9 * 9 9 c 9 2 0 . 9 2 0 . 9 Table 4 , Screen A n a l y s i s f o r the B a l l M i l l i n g of Feldspar,. Limestone, and Quartz T o t a l Grinding Revolutions ( G r i n d i n g Time) Feldspar Limestone .' •- Quartz Mesh 75 revs 75 revs 75 revs 150 r e v s (70 sec) (70 sec) (70 sec) (141 sec) % io Cum % % Cum % Cum- % • % Cum , 6 26,0 100,0 50,0 100,0 36.O 100.0 15»2 100,0 8 26.1 74.0 22.5 50.0 23.O 64,0 2 0 . 6 84.8 10 15.3 4 7 . 9 . 9 . 7 27.5 13o4 41 o0 16 ,7 64.2 14 9.8 32.6 5 .5 .17.9 8.3 27 .6 13o3 ^7°5 20 6.7 22,8 3.2 12.4 5c7 19»3 10 .2 34.2 28 4 . 4 .16.1 2 .0 •9»2 3o7 13o6 6.9 24.0 35 3»0 11 . 7 1 . 3 7o2 2.8 9.9 5-1 17«2 48 2.1 8,7 0.9 "• 5=9 2,1 7 o l 3o4 12,1 65 1 .7" 6.5 0 .7 5.0 1 . 5 . 5o0 2 .6 8.7 100 1 . 3 4 ,9 0 ,6 4 . 3 . 1 . 0 3 . 5 1.8 6.2 150 1 .0 3»6 0.5 3°7 0.8 2 . 4 •1 .3 4 . 4 200 2,6 2.6 3,2 3.2 1.7 1 . 7 3.1 3.1 - . 9 5 -Appendix IV D e r i v a t i o n of Equation 4 i n Page 17 F r a c t u r e of a specimen "by a f r e e - f a l l i n g b a l l was assumed to occur only when the t e n s i l e s t r e s s reached the t e n s i l e s t r e n g t h of a specimen. As shown i n Equation 3s the t e n s i l e s t r e n g t h i s i n v e r s e l y p r o p o r t i o n a l to the square of t h i c k n e s s of a specimen. ' StoC - ~ or F oc S t - d 2 d~ For a specimen having s i z e of d=d+Ad, where d i s standard s i z e and ad i s increment, the a p p l i e d f o r c e i s Foc(d + A d ) 2 = d 2 ( l + ~ d Then F might be p r o p o r t i o n a l to the height o f a b a l l p o s i t i o n H,. s i n c e the p o t e n t i a l energy of the b a l l i s p r o p o r t i o n a l t o the h e i g h t . Therefore, -2 2/Ad H oc F oc d (1 + — - — ) d Hence the c o r r e c t e d h e i g h t He due to the change of t h i c k n e s s o f a specimen i s _2 T T ' d ( 1 + 2(d-oT)/ d ) He = H • p d = H ( 1 + 2(d-d)'/ d ) -(4) 

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