UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Studies on comminution. Jomoto, Kimitaka 1971

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1972_A7 J64.pdf [ 6.17MB ]
Metadata
JSON: 831-1.0081145.json
JSON-LD: 831-1.0081145-ld.json
RDF/XML (Pretty): 831-1.0081145-rdf.xml
RDF/JSON: 831-1.0081145-rdf.json
Turtle: 831-1.0081145-turtle.txt
N-Triples: 831-1.0081145-rdf-ntriples.txt
Original Record: 831-1.0081145-source.json
Full Text
831-1.0081145-fulltext.txt
Citation
831-1.0081145.ris

Full Text

STUDIES ON COMMINUTION  K i m i t a k a Jomoto BoAoSco  Hokkaido  U n i v e r s i t y , Japan,  1968  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF ' THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n t h e Department of MINERAL ENGINEERING •  We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e required standard  THE UNIVERSITY OF BRITISH COLUMBIA . December,  1971  In p r e s e n t i n g  t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r  an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y  available for reference  and  study.  I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying of t h i s f o r s c h o l a r l y purposes may by h i s r e p r e s e n t a t i v e s .  be  granted by  Mineral Engineering  The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada  Date  s h a l l not be  permission.  Department of  December  1971  Department or  I t i s understood t h a t copying or  of t h i s t h e s i s f o r f i n a n c i a l g a i n written  the Head of my  thesis  publication  allowed without  my  ABSTRACT The p h y s i c a l q u a n t i t y o f t h e square o f t e n s i l e s t r e n g t h o v e r Y o u n g s modulus has been p r o p o s e d as a 0  c r i t e r i o n o f comminution  o f r o c k by Oka and Majima.  was d e t e r m i n e d f o r f i v e d i f f e r e n t r o c k s , m e a s u r i n g s t r e n g t h and Young's modulus.  The v a l u e tensile  The t e s t i n g method used  t o d e t e r m i n e b o t h r o c k p r o p e r t i e s was a p o i n t - l o a d  tensile  s t r e n g t h t e s t and measurement o f p r o p a g a t i o n v e l o c i t y o f 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 o f the c r i t e r i o n  t h u s o b t a i n e d , t h e d r o p - w e i g h t i m p a c t t e s t and b a l l  mill  g r i n d i n g were c a r r i e d o u t u s i n g t h e i d e n t i c a l specimen 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 . experimental r e s u l t  used  The  was o b t a i n e d t h a t t h e square o f 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 t h a n t h e square o f t e n s i l e s t r e n g t h o v e r Young's modulus.  '  . • '  I n c h a p t e r 2, g r i n d i n g t e s t s were p e r f o r m e d w i t h a 12 i n c h b a t c h m i l l .  A t o r q u e meter was i n s t a l l e d t o e s t i m a t e  t h e energy c o n s u m p t i o n . n e c e s s a r y  to achieve s i z e  reduction.  From t h e t o r q u e measurement, t h e r e q u i r e d t o r q u e f o r t h e m i l l t u r n i n g remained a l m o s t c o n s t a n t r e g a r d l e s s o f t h e m a t e r i a l b e i n g ground. A new c o n c e p t , Energy I n d e x , was i n t r o d u c e d t o e v a l u a t e g r i n d i n g r e s u l t s and t o s t u d y g r i n d i n g p r o b l e m s .  F o r each r o c k  sample t e s t e d , t h e energy i n d e x was d e t e r m i n e d from t o r q u e measurements and t h e q u a n t i t y o f t h e p r o d u c t s .  I t was found  t h e energy i n d e x 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 i n d e x 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 SCOPE OF PRESENT WORK  —  5  '—  . 5  :  MATERIALS AND PREPARATION  6 11  EXPERIMENTAL METHODS 1  11  - Young's Modulus  2 - Tensile Strength  ;  _.  3 - Drop-Weight Impact T e s t  15  — —  EXPERIMENTAL RESULTS AND DISCUSSION 1  - Young's Modulus  2 - Tensile Strength  13  .  19 .  '  —.. 1 9  . .—  — —  :  3 - Drop-Weight Impact T e s t  24 28  o  4 - The Comparison o f S t /E and C r i t i c a l Height . 2 5 - The R e l a t i o n s h i p between S t /E and t h e xResults o f B a l l M i l l G r i n d i n g CONCLUSION — • — CHAPTER . 2 . .  36 40 ^  ENERGY INDEX IN,DRY TUMBLING- MILLING -  SCOPE OF PRESENT WORK MATERIALS AW  PREPARATION  ^5  - — — —  46  BALL MILL GRINDING TEST PROCEDURE  :  .  4b 51  EXPERIMENTAL RESULTS AND DISCUSSION 1  - Preliminary Test  2 - Torque Measurement  4 5  51  . •  —  55  - iii  Page  3 - Size D i s t r i b u t i o n k - Energy  59  Index  • CONCLUSION  —  SUMMARY  6^ ;  73  '  7^  :  SUGGESTIONS FOR FUTURE WORK  .— —  REFERENCES •  ;  APPENDIX  —  —  —  —  ?6 77 79  . LIST OF  FIGURES  Page  Figure 1  • Schematic i l l u s t r a t i o n of p u l s e for determining  Y o u n g s modulus 0  apparatus .  •—•—•—  11  2  C o m p r e s s i o n a l waves on t h e . o s c i l l o s c o p e s c r e e n  11-a  3  A p p a r a t u s 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 tester •—• Apparatus f o r d r o p - b a l l t e s t — ;  ka.  -  16 lb  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 n d e s i t e , . g r a n i t e , m a r b l e , s a n d s t o n e , and m a g n e t i t e 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 20  6  T e n s i l e strength of andesite, g r a n i t e , marble, s a n d s t o n e , and m a g n e t i t e :  25  7  Number o f i m p a c t s r e q u i r e d f o r f r a c t u r e o f ande.'. s i t e , g r a n i t e , m a r b l e , s a n d s t o n e , and m a g n e t i t e from v a r i o u s h e i g h t •— 29  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 andesite, g r a n i t e , marble, s a n d s t o n e , and m a g n e t i t e 32  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  10 11  2 2  R e l a t i o n s h i p between St  and  c r i t i c a l height  3? 37  2  R e l a t i o n s h i p between St /E and  energy index  kl  2  12  R e l a t i o n s h i p between St  13  T u m b l i n g m i l l and apparatus —  m-  I n f l u e n c e o f t h e 1 2 - i n c h m i l l speed on t h e g r o s s t o r q u e under t h e v a r i o u s l o a d i n g o f 1 inch b a l l s —— •  52  The e f f e c t o f b a l l l o a d i n g on t h e t o r q u e a t a spees o f 6k- rpm.  5^  and  energy index  M.  i t s torque measuring • —  ^9  :  •15 16  gross  The g r o s s t o r q u e f o r t h e m i l l f i l l e d w i t h without feed m a t e r i a l  and  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 e n e r g y 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 o f marble  60  •19a  Size d i s t r i b u t i o n s of various revolutions grinding  sample a f t e r 150  20  Rates o f f o r m a t i o n  21  Energy index as a f u n c t i o n o f product s i z e p a s s i n g 80 p e r c e n t •  69  22  C o r r e l a t i o n o f energy i n d e x t o Bond°s work index :  71  Energy index v e r s u s c r i t i c a l h e i g h t  71  23  o f mesh f r a c t i o n s ( m a r b l e )  60 66  - vi ' L I S T OF TABLES  Page  Table  7  1  Rock samples  —  2  Young" s modulus  3  E x p e r i m e n t a l r e s u l t s i n C h a p t e r 1.  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  5  Grinding conditions  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  '—•  •  •  --•  23  •  43  — - — — —  47 53  —  61 c-  Appendix ' Table 1  Measurement o f P r o p a g a t i o n V e l o c i t y o f P-Wave  2  Measurement o f T e n s i l e S t r e n g t h  .  83  3  E f f e c t of. B a l l Charge and M i l l Speed on. t h e Torque  86  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  87  •  . 80  - v i l ACKNOWLEGEMENTS The  author  i s indebted  t o P r o f e s s o r H.  Majima  o f the Department o f 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  H i s comments  and  suggestions  have h e l p e d  t o make p o s s i b l e t h i s  S i n c e r e g r a t i t u d e and  thesis.  t h a n k s a r e extended t o  D r . Y. F u j i n a k a and D r . H. Kiyama f o r t h e h e l p i n d e s i g n i n g t h e m i l l assembly used i n t h i s s t u d y and  f o r time w i l l i n g l y  g i v e n t o d i s c u s s i o n s on c o m m i n u t i o n . . A p p r e c i a t i o n and  thanks  a r e a l s o extended t o a l l members o f the Department o f M i n e r a l Engineering  f o r t h e i r assistance i n various f i e l d s during  this  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 t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a f e l l o w s h i p , t h e F r e d e r i c k Armand McDearmid S c h o l a r s h i p and B r a n c h , Department o f Energy and  from the Mines  Resources t o do t h e  necessary  work f o r t h i s t h e s i s , w h i l e s t u d y i n g a t t h e U n i v e r s i t y o f British  Columbia.  - 1 -  INTRODUCTION  ,  Comminution i s one o f t h e most i m p o r t a n t p r o c e s s e s w i t h i n t h e m i n i n g industry<>  unit  Numerous p a p e r s a r e  p u b l i s h e d on t h i s subject,. These a r e summarized i n t h e b i b l i ography " C r u s h i n g and G r i n d i n g ( 1 ) " a s w e l l as i n many r e v i e w 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 Important  year*  t o p i c s commonly d e a l t w i t h i n t h e s e s t u d i e s a r e :  (1) the. d e t e r m i n a t i o n o f p h y s i c a l c h a r a c t e r i s t i c s o f r o c k s or m i n e r a l s i n accordance w i t h c r u s h i n g o r g r i n d i n g c a p a b i l i t y , (2)  t h e e s t i m a t i o n o f energy f o r s i z e r e d u c t i o n , ( 3 ) t h e  e l u c i d a t i o n o f p a r a m e t e r s o f g r i n d i n g o p e r a t i o n s , (4-) t h e development o f g r i n d i n g t e c h n i q u e s < > 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  f o c u s e d on t h e d e t e r m i n a t i o n o f ease o f c o m m i n u t i o n o f m a t e r i a l s and on t h e e s t i m a t i o n o f e n e r g y c o n s u m p t i o n f o r s i z e r e d u c t i o n , . In s t u d y i n g these problems, a theory which s a t i s f i e s r e q u i r e m e n t s , h a s n o t been f o r m u l a t e d  0  practical  T h i s i s because t h e  g r i n d i n g system i s u s u a l l y c o m p l i c a t e d , and t h u s o n l y e m p i r i c a l s o l u t i o n s have b e e n o b t a i n e d t o t h e p r o b l e m s . Ease o f c o m m i n u t i o n o f m a t e r i a l s has b e e n t e n t a t i v e l y expressed  i n terms o f h a r d n e s s ,  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 , o r r e s i s t i v i t y t o a hammer t e s t o f materials  (2,3)«  However, none o f t h e s e c r i t e r i a ,  except  •  grindability  -  2  -  , can s a t i s f a c t o r i l y e x p r e s s t h e 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 a t l e a s t t h e f o l l o w i n g two  r e q u i r e m e n t s ; ( 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 c o m m i n u t i o n , and ( i i ) i t must be a b l e t o be  determined a c c u r a t e l y without  tedious  procedures. Recently  Oka  and  Majima ( 4 ) have s t u d i e d t h e  problem i n comminution a p p l y i n g the theory  energy  of e l a s t i c i t y  an i r r e g u l a r l y shaped r o c k p a r t i c l e , w h i c h i s s u b j e c t e d the a c t i o n of a p a i r of concentrated p a r t i c l e f r a c t u r e , i t was  loads.  to  For s i n g l e  found t h a t t h e s t r a i n energy  required  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 t h i r d order  o f p a r t i c l e s i z e and  s t r e n g t h , and modulus.  to  t o t h e second o r d e r o f  the tensile  i s i n v e r s e l y p r o p o r t i o n a l t o t h e v a l u e o f Young's  From t h i s f i n d i n g , t h e y s u g g e s t e d t h a t t h e r a t i o  t h e square o f t e n s i l e s t r e n g t h t o Young's modulus may  be  of  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 o r d e r t o d e t e r m i n e t h e i r c r i t e r i o n , measurements o f t e n s i l e s t r e n g t h as w e l l as Young's modulus a r e  required.  For the d e t e r m i n a t i o n  standard  o f b o t h r o c k p r o p e r t i e s , no  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 t h e r e s p o n s e o r r e s i s t a n c e of a m a t e r i a l to g r i n d i n g e f f o r t . The n u m e r i c a l e x p r e s s i o n o f g r i n d a b i l i t y has been p r o p o s e d as t h e r e s u l t o f e x p e r i m e n t a l work, s u c h as new s u r f a c e p r o d u c e d p e r u n i t o f p o t e n t i a l energy i n d r o p - w e i g h t t e s t , o r p r o d u c t w e i g h t p e r r e v o l u t i o n or the numbers o f r e v o l u t i o n s r e q u i r e d t o p r o d u c e t h e c o n s t a n t fineness i n m i l l grinding ( 3 ) 0  "  mentioned  requirements  l o a d , t e s t ( 5 ) was  3  "  f o r a c r i t e r i o n of comminution, p o i n t -  employed f o r d e t e r m i n g t e n s i l e s t r e n g t h ,  and measurement o f t h e p r o p a g a t i o n v e l o c i t y o f c o m p r e s s i o n a l waves was  done f o r d e t e r m i n i n g Y o u n g s m o d u l u s  The  other  r e a s o n f o r t h e c h o i c e o f t h e above t e s t methods was:  1) t o  6  0  use an i r r e g u l a r l y shaped specimen and t h e r e b y a v o i d weakening t h e specimen d u r i n g p r e p a r a t i o n , 2 ) t o 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 w h i c h o f t e n a p p e a r 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 o r weakness p l a n e s due t o m a c h i n i n g , and  3)  t h e same t e s t - p i e c e c a n be a p p l i e d f o r b o t h measurements, s i n c e t h e Young's modulus measurement i s a n o n - d e s t r u c t i v e test.  I t was  experiments  a l s o n e c e s s a r y t o c a r r y out  comminutiion  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 t h e g r i n d i n g p r o c e s s e s .  Drop-weight  i m p a c t 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 t h e y i n v o l v e d s i m p l e and complex e v e n t s ,  respectively. 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  t h e d r i v i n g power o f c o m m e r c i a l l y s i z e d t u m b l i n g m i l l d a t a based  on t h e l a b o r a t o r y t e s t measurementso  t h e method p r o p o s e d  from  Presently  by Bond ( 6 ) , w h i c h i s d e v e l o p e d  on t h e  b a s i s o f h i s c o n c e p t , "work i n d e x " , i s w i d e l y used f o r t h i s purpose.  U s i n g B o n d s h y p o t h e s i s , i t w i l l be seen t h a t t h e 9  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  for size  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 c o n s t a n t w h i c h i s e n t i t l e d "work i n d e x " i s p a r t i c u l a r l y dependent on  characteristics  _ 4 of the m a t e r i a l b e i n g ground. recognized  However, i t has been i m p l i c i t l y  f o r many y e a r s t h a t t h e r e may w e l l be a l a c k o f  p r o p o r t i o n a l i t y between energy c o n s u m p t i o n and g r i n d i n g efficiency i n m i l l s of d i f f e r e n t sizes.  F o r 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 c o r r e s p o n d i n g  values  d e t e r m i n e d from Bond's work i n d e x (•?)• As a consequence. Bond has p r o p o s e d t h e i n t r o d u c t i o n o f a s i z e f a c t o r v/hen r e l a t i n g t h e work i n d e x c a l c u l a t i o n s t o t h e o p e r a t i o n o f m i l l s l a r g e r than eight feet i n diameter(8)  0  We(17) have s u g g e s t e d  t h a t i t may be p o s s i b l e t o m o d i f y Bond's c o n c e p t o f v/ork i n d e x t o meet t h e - r e q u i r e m e n t 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 i n d e x w h i c h c a n c o l l e c t i v e l y evaluate  grinding efficiency. •A t u m b l i n g m i l l ' w h i c h was e q u i p p e d w i t h a t o r q u e  •pickup was c o n s t r u c t e d  f o r t h i s p u r p o s e , and some o f t h e  f a c t o r s a f f e c t i n g t h e 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 e x p e r i m e n t a l , r e s u l t s o b t a i n e d  are discussed  a new i n d e x c a l l e d a n "energy i n d e x " .  i n terms o f  The use o f t h e new  i n d e x 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 t h e e s t i m a t i o n o f t h e 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 b a s e d on l a b o r a t o r y measurements.  operation  -5 -  CHAPTER ONE  CRITERION OF COMMINUTION  SCOPE OF PRESENT WORK The p u r p o s e o f t h e s t u d i e s d i s c u s s e d i n t h i s c h a p t e r was t o examine e x p e r i m e n t a l l y t h e c r i t e r i o n o f c o m m i n i t i o n w h i c h has been p r o p o s e d b y Oka and Majima. 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 o f 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  I n order to obtain  The t e s t i n g method f o r  d e t e r m i n a t i o n o f 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 s t w h i c h c a n employ a n i r r e g u l a r l y .shaped s p e c i m e n .  tensile, For  Young's modulus, a measurement o f p r o p a g a t i o n v e l o c i t y o f a compressional modulus.  wave was c a r r i e d o u t , w h i c h y i e l d s a dynamic  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  including  one o r e sample, and c o n s i s t e d o f a n d e s i t e , g r a n i t e , m a r b l e , s a n d s t o n e , and m a g n e t i t e .  F o r experimental purposes f i f t e e n  specimens were t a k e n from each sample. F u r t h e r e x p e r i m e n t s were a l s o n e c e s s a r y t o 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 o f t h e most s i m p l e c a s e o f comminution e v e n t s , t h e d r o p - b a l l i m p a c t t e s t was u s e d .  The p u r p o s e o f t h i s t e s t i s  to  d e t e r m i n e t h e minimum h e i g h t o f 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, w h i c h i s termed t h e c r i t i c a l  height.  Thus measurements o f t h e c r i t i c a l h e i g h t p r o v i d e a c r i t e r i o n w h i c h e x p r e s s e s t h e ease o f comminution o f m a t e r i a l t o 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 o f r o c k m a t e r i a l s v a r y w i d e l y from one r o c k t o a n o t h e r .  T h e r e f o r e t h e t e s t samples  should include several 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 f r o m . e x p e r i m e n t a l r e s u l t s .  To  s a t i s f y t h i s r e q u i r e m e n t , f i v e d i f f e r e n t r o c k samples were examined: a n d e s i t e , . g r a n i t e , m a r b l e , s a n d s t o n e , and m a g n e t i t e . T a b l e 1 shows t h e 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 o f r o c k samples was d e t e r m i n e d  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  g i v e n v a l u e o f s p e c i f i c g r a v i t y i n T a b l e 1 i s a n average o f four to f i v e determinations. Coates and P a r s o n s ( 9 ) s u g g e s t e d t h a t t e n o r more t e s t - p i e c e s a r e r e q u i r e d t o determine the a c c e p t a b l e v a l u e o f t h e s t r e n g t h o f a r o c k as w e l l a s t h e o t h e r r o c k p r o p e r t i e s . Yamaguch ( 1 0 ) s u p p o r t e d , u s i n g m a t h e m a t i c a l  statistics  t o g e t h e r w i t h e x p e r i m e n t s , t h a t t h i s number o f t e s t p i e c e s r e q u i r e d a r e r e a s o n a b l e , even though a l l t h e t e s t p i e c e s a r e c u t from t h e same b l o c k o f r o c k . t h a n f i f t e e n specimens  A c c o r d i n g l y , more  f o r each o f f i v e r o c k . s a m p l e s  were  used f o r b o t h Young's modulus and t e n s i l e s t r e n g t h  ' .  measurement, and more t h a n t h i r t y t e s t p i e c e s o f each sample f o r the drop-weight  impact  test.  I t has o f t e n been o b s e r v e d t h a t many r o c k s , n o t o n l y s e d i m e n t a r y o r metamorphic r o c k s , b u t a l s o i g n e o u s 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 o f p h y s i c a l p r o p e r t i e s due t o  Table 1. Source  Rock  Andesite  Hope,  Rock Sarables Specific Gravity  Number o f Specimens Tested Tensile Young's Drop»Weight Strength Modulus Impact  BeCo  2»72  23  29  32  N o r t h Vancouver, «C o  2o71  15  15  30  Marble  Texada, B » C  2o73  18  23  38  Sandstone  Colorado, UoS oAo  2„67  15  15  31  Magnetite  Texadaj, B<,C.  4c63  15  20  33  Granite  B  0  -.. 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 o f Samples Andesite:  From Hope, B.C.  i s a p h ' a n i t i c and  the g r a i n s i z e o f b o t h f e l s i c and t h a n 0 . 1 mm  i n diameter.  i r r e g u l a r and/or a n h e d r a l .  The  v e r y dense?  mafic minerals i s  smaller  shape o f the q u a r t z c r y s t a l s i s .  The  quartz c r y s t a l impregnates  f r a c t u r e s composed o f t h i n v e i n l e t s r a n g i n g 0 . 5 t o 1 mm Granite;  and  5 0 p e r c e n t o f p o t a s h f e l d s p a r and  2 0 percent of mafic minerals.  The  plagioclase,,  " g r a n i t e " c o u l d be  q u a r t z m o n z o n i t e 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 and  p l a g i o c l a s e i s a l m o s t even.  f e l d s p a r , and and  The  p l a g i o c l a s e are 1 t o 3 mm,  Sandstone;  The  K-feldspar  and  biotite,  hornblende,  mm.  From Texada I s l a n d , B.C.  o f medium-grained c a l c i t e .  termed  g r a i n s i z e i s medium;. q u a r t z ,  o t h e r ferromagnesians are 0 . 2 to.1  Warble:  width.  c o n s i s t s o f about 3 0  From N o r t h V a n c o u v e r , B.C.  percent of quartz,  in  i s l i g h t gray;  g r a i n s i z e i s 1 to ' 3  consists mm..  From C o l o r a d o , U.S.A. has p a r a l l e l l a m i n a e .  layers vary thickness  from 1 mm  t o 2 cm,  s e p a r a b l e from o t h e r l a y e r s above and  and  below.  are The  visually rock c o n s i s t s ,  o f more than- 9 5 p e r c e n t rounded q u a r t z g r a i n s w h i c h a r e 0.1 0 . 2 mm  The  to  i n s i z e i n a porous' s i l i c a c e m e n t i n g 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 o f 9 7 p e r c e n t  m a g n e t i t e , and  the r e s t i s c h a l c o p y r i t e and  c h a l c o p y r i t e i s d i s s e m i n a t e d as s p e c k s and as i s o l a t e d v e i n l e t s , 0 ' . 5 t o 1 mm of magnetite v a r i e s widely  calcite.  The  c a l c i t e i s found  i n thickness.  from 0 . 5 t o 5. ram-  The  grain size  - 8 -  t h e 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 o r m i c r o 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 o r g r i n d i n g p r o c e s s e s ,  physical  p r o p e r t i e s o f r o c k s a r e i n g e n e r a l a v e r a g e d , and for  s p e c i a l care  t h e sample p r e p a r a t i o n w i t h r e s p e c t t o o r i e n t a t i o n was  not t a k e n i n t o a c c o u n t e x c e p t f o r sandstone« s a n d s t o n e sample was  The  c u t so t h a t i t c o u l d be s u b j e c t e d  compression perpendicular to i t s lamina.  T h i s was  o f t h e 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 measurement, and may  Colorado  generate a.considerable  g r i n d i n g t e s t r e s u l t s from t h o s e  of other  because  tensile  samples. Young's  However, t h e  immediate p u r p o s e i s t o f i n d a s u i t a b l e c r i t e r i o n  of  c o m m i n u t i o n w h i c h can be d e t e r m i n e d i n a l a b o r a t o r y technical tediousness.  Even though we  strength  deviation i n  I n t h i s s t u d y , t h e t e n s i l e s t r e n g t h and modulus o f r o c k samples a r e o f i n t e r e s t .  to  without  can d e t e r m i n e  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 t h e Young's modulus o f r o c k samples by u s i n g p r e c i s e l y shaped s p e c i m e n s , i t s h o u l d be n o t e d t h a t t h e p r e p a r a t i o n o f such specimens, i s n o t i n p a r t i c u l a r i t i s time-consuming. important  A l s o , i t i s not  t o d e t e r m i n e t h e r e a l t e n s i l e s t r e n g t h o r Young's  modulus f o r t h e p r e s e n t p u r p o s e .  To a v o i d t h e  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 t e s t and  simple,  difficulties strength  a dynamic measurement o f Young's modulus  was  employed w h i c h r e q u i r e d a s i m p l e p r e p a r a t i o n o f t e s t p i e c e s . . I t was  found t h r o u g h p r e l i m i n a r y t e s t s t h a t t h e 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 suitable  f o r t h e measurement  0  sphere  I r r e g u l a r l y shaped  was  specimens,  b o t h f a c e s o f which were c u t f o r c o n v e n i e n c e o f l o a d i n g , were used.  For the drop-weight impact t e s t , the t e s t  p i e c e , w a s p r e p a r e d t o be 6 cm s q u a r e and 3  i n thickness.  The s i z e and shape o f t h e specimens was a l m o s t t h e same i n all  samples.  -  10  -  EXPERIMENTAL METHODS The p r o c e d u r e s used f o r t h e 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 t h e method f o r d r o p - w e i g h t i m p a c t t e s t a r e a s follows<> 1  -  - Young"s Modulus The p u l s e v e l o c i t y measurement t e c h n i q u e  employed f o r d e t e r m i n g Young"s m o d u l u s arrangement  0  (13,1^)  was  A schematic  o f the apparatus is.shown i n F i g u r e 1  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 ,  0  I t consisted  Structural  B e h a v i o r Engg. L a b . ) , a n x - c u t 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.), (Type-5^-9,  T e k t r o n i x I n c . ) w i t h a. p r e a m p l i f i e r  an o s c i l l o s c o p e (Type-IA?,  T e k t r o n i x ) p and a p o r t a b l e c o m p r e s s i o n t e s t e r ( S o k k i s y a C o . ) . The p u l s e g e n e r a t o r used was c a p a b l e o f p r o d u c i n g a s q u a r e wave p u l s e w i t h 2 © 5 i 0 . 5 m i c r o 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 p e r s e c o n d .  The specimen  t o be t e s t e d was p l a c e d between a p a i r o f p l a t t e n s i n w h i c h t h e t r a n s m i t t e r and r e c e i v e r were mounted, and was l o a d e d by a h y d r a u l i c c o m p r e s s i o n t e s t e r f o r good m e c h a n i c a l contact„ . T h i s e n a b l e d one t o measure t h e t r a v e l - t i m e o f a c o m p r e s s i o n a l wave, t h a t i s , l o n g i t u d i n a l wave t h r o u g h t h e specimen the a c t i o n o f a n a r b i t r a r y loado  under  L o a d i n g p r e s s u r e was  i n c r 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 o b t a i n e d on t h e o s c i l l o s c o p e  6  .The l o a d was s e t a t 1 0 0 t o 2 0 0 k i l o g r a m s .  Then t h e t r a v e l - t i m e o f a c o m p r e s s i o n a l wave t h r o u g h a r o c k specimen was d e t e r m i n e d by m e a s u r i n g t h e d i s t a n c e between t h e  -  Figure 2 .  11a  -  A r r i v a l of Compressional  P u l s e a t t h e O r i g i n on an Screen  Oscilloscope  Compression T e s t e r  Oscilloscope  (Type-549) Seismic Timer-Pulse  Transmitter Platten  Trigger Input(A)  Generator  (Model-1007)  Trigger Input(B)  X-cut C r y s t a l Receiver Platten  High Gain D i f f . Amp. (Type-IA?)  Pulse Output  T r i g g e r Main P u l s e  U n i a x i a l Compression T e s t e r  F i g u r e 1.  Schematic I l l u s t r a t i o n o f P u l s e A p p a r a t u s f o r D e t e r m i n i n g Young's Modulus  1  -  -  12  s t a r t o f t h e sweep s i g n a l and t h e f i r s t a r r i v a l o f a c o m p r e s s i o n a l wave on t h e o s c i l l o s c o p e s c r e e n ( F i g u r e 2 ) . Thus t h e t r a v e l - t i m e o f t h e l o n g i t u d i n a l wave from t h e t r a n s m i t t e r t o r e c e i v e r t h r o u g h t h e s p e c i m e n was measured *  with 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 o f t h e wave m i g h t be a f f e c t e d t o some e x t e n t by t h e 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 o f t h e t r a n s m i t t e r , t h e r e c e i v e r and t h e p r e a m p l i f i e r , as. w e l l as by t h e mechanical c h a r a c t e r i s t i c s o f the contact t h e p l a t t e n s and a specimen„  surfaces  between  Therefore, the a c t u a l  travel-  t i m e was d e t e r m i n e d by. s u b t r a c t i n g t h e t r a v e l - t i m e o f t h e blank t e s t (that 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 t h e p i c k u p - p l a t t e n travel-timeo  d i r e c t l y ) from t h e t o t a l  The a c c u r a c y o f t r a v e l - t i m e measurement o b t a i n e d  by t h i s t e c h n i q u e was w i t h i n 0 . 0 5 m i c r o seconds when 5 m i c r o seconds o f f i x e d gauge and 1 / 1 0 0 v e r n i e r was u s e d . The v e l o c i t y o f c o m p r e s s i o n a l wave Vc i n a r o c k specimen i s d e r i v e d Ye = where L i s l e n g t h (second)o  from t h e f o l l o w i n g e q u a t i o n s — — dt  ( cm/sec )  —  —  (1)  o f a s p e c i m e n (cm) and d t i s t r a v e l - t i m e  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  t h e v a l u e s o f Vc i n t o t h e f o l l o w i n g e q u a t i o n : E = Vc  2  p  (l+v)(l-2v)  ( ) 2  1-v 2  where E i s Young's modulus (g/cm s e c ) , p i s t h e d e n s i t y o f 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 P o i s s o n ' s r a t i o  -  (dimensionless)o  13  -  Young's modulus e x p r e s s e d i n  engineering  • ' 2  u n i t , kg/cm,  , can be o b t a i n e d  by d i v i d i n g E by  l , 0 0 0 g ,  where g i s t h e a c c e l e r a t i o n o f g r a v i t y . The  p r e s e n t equipment was  not d e s i g n e d t o d e t e r m i n e  t h e v e l o c i t y o f s h e a r waves, s i n c e an x - c u t c r y s t a l p r o d u c e s a c o m p r e s s i o n a l p u l s e i n t h e specimen<= t h e r e f o r e , was  assumed t o l i e between  number m = 8) and reasonable value 2 - Tensile The  0  o  (m =  2 5 0  Poisson's 0  o  1 2 5  w h i c h was  ratio,  (Poisson°s considered  a  f o r common r o c k s .  Strength 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  i n t h i s s t u d y i s known as a p a i r o f p o i n t s l o a d 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)»  tensile 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 d e v e l o p e d f o r t h e use i r r e g u l a r l y shaped specimens was 3°  as shown i n F i g u r e  The  used f o r t h i s  a t t h e l o a d i n g end,  of  experiment  p l a t t e n which leads the  l o a d o n t o a specimen i s c y l i n d r i c a l , h a v i n g l w i t h a rounded edge.  o  used  concentrated  0 cm  diameter  A l o a d was  applied  t h r o u g h t h e c e n t r a l a x i s o f a specimen by means o f a p a i r o f p l a t t e n s u n t i l b r e a k a g e o c c u r r e d i n t h e specimen i n o r d e r t o measure t h e c r i t i c a l l o a d a t The  failure.  t e n s i l e s t r e n g t h o f a s p e c i m e n can be  from t h e f o l l o w i n g  calculated  formula: F  St = I* !1  =  .  .  0.9  5-  - ^ 7  d^  or •  —  (3)  - 14 -  Figure  3«  Apparatus f o r T e n s i l e  Measurement.  Strength  ( P o i n t - l o a d Compression T e s t e r )  -  15  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 o f l o a d ) ( c m ) The l o a d i n g r a t e was k e p t 1 0 0 0 t o 1 5 0 0 k g p e r m i n u t e through-, out t h e e x p e r i m e n t s .  3  -  Drop-Weight  Impact T e s t  A specimen p l a c e d on a s t e e l p l a t f o r m was  fractured  by a f r e e - f a l l i n g b a l l , u s i n g an a p p a r a t u s shown s c h e m a t i c a l y i n Figure 4 .  A manganese s t e e l b a l l o f 7 . 8 cm i n d i a m e t e r  w e i g h i n g 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  v a r i o u s h e i g h t s onto t h e t e s t p i e c e s .  The s t e e l  from  platform  was h e l d i n p l a c e by t h r e e 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 d i a m e t e r and 6 . 5 cm i n h e i g h t , on w h i c h t h r e e s t r a i n gauges were equipped f o r each p i l l a r .  The p u r p o s e o f t h e  1 1 ) t o e s t i m a t e t h e unconsumed energy f o r  s t r a i n gauges was  the f r a c t u r e o f s p e c i m e n , and 2 ) t o examine whether o r n o t t h e specimen f r a c t u r e a c t u a l l y , o c c u r s . The o u t p u t o f t h e s t r a i n gauges was  fed into a high  g a i n a m p l i f i e r t h r o u g h an i n t e g r a t i n g c i r c u i t w h i c h  integrated  t h e s t r a i n g e n e r a t e d by i m p a c t d u r i n g t h e i m p a c t d u r a t i o n . T h e r e f o r e , t h e a r e a A under t h e s t r a i n - t i m e c u r v e i s shown by t A =  •  *\ J  o  e (t) dt  -  16  -  To Winch  To Vacuum Pump  Vacuum H o l d e r Steel  Ball  Specimen Elevated Platform  Strain Gauge  H i g h G a i n Amp. /  • Recorder  Intergrating C i r c u i t Unit R i g i d Base  F i g u r e 3- • Schematic i l l u s t r a t i o n o f d r o p - b a l l impact t e s t e r  where  i s s t r a i n and t i s d u r a t i o n o f i m p a c t .  e  t i m e c u r v e i s r e c o r d e d on a c h a r t w i t h a h e a t pen.  The  strain-  sensitive  H o w e v e r , owing t o many m e c h a n i c a l c o n t a c t s "between  a t e s t - p i e c e and t h e p i l l a r s , t h e e x p e r i m e n t d i d n o t s u c c e e d i n obtaining the quantity with s u f f i c i e n t accuracy i n t h i s system. The d r o p - w e i g h t i m p a c t t e s t s from a p r e - s e t h e i g h t o n t o a specimen were r e p e a t e d u n t i l f r a c t u r e o f t h e specimen t o o k p l a c e , u n l e s s t h e f i r s t h i t caused f r a c t u r e o f t h e specimens  F o r each s e t o f b a l l h e i g h t s , 5 "to 8 specimens  were t e s t e d , t h e n t h e h e i g h t was i n c r e a s e d by 5 cm  intervals.  As d e s c r i b e d b e f o r e , two f a c e s o f a specimen were c u t p a r a l l e l and a p p r o x i m a t e l y 3 cm a p a r t .  However, i t was  found from t h e 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 o f 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 . T h e r e f o r e , c o r r e c t i o n has t o be a p p l i e d t o t h e e x p e r i m e n t a l results obtained.  F o r t h i s p u r p o s e , i t was assumed t h a t t h e  i m p a c t f o r c e from t h e d r o p p i n g b a l l i s p r o p o r t i o n a l t o t h e height a t which the b a l l  i s released, since the p o t e n t i a l  energy o f t h e b a l l i s p r o p o r t i o n a l t o t h e h e i g h t . equation  for this correction iss „ • , . 2(d-d) v nc = H • ( 1 + _ .—) d TT  where He = a d j u s t e d h e i g h t H  The  ,. . (4)  —  (cm)  = o r i g i n a l h e i g h t used i n t h e t e s t s  (cm)  * The d e r i v a t i o n o f t h e e q u a t i o n i s i n A p p e n d i x .  - 18 d = standard  thickness:  3»00  cm i n t h i s t e s t (cm)  d = t h i c k n e s s o f t h e specimen t e s t e d (cm) The h e i g h t o f t h e " b a l l has been a d j u s t e d as determined by E q u a t i o n cm t h i c k n e s s as s t a n d a r d . of  30  for- each specimen  ( 4 ) , t a k i n g t h e specimen o f 3 . 0 0 F o r example, t h e b a l l p o s i t i o n  cm f o r t h e t h i c k n e s s o f specimens o f 2.92,  3«00  and 3 . 0 9 cm i s e q u i v a l e n t t o t h o s e o f 3 1 . 7 . 3 0 . 0 , 28.9 cm respectively. F i g u r e 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 P r o c e d u r e , a w e l l - d e f i n e d o f the s i g n a l was  obtained  l o a d i n g range o f 100  image  on t h e o s c i l l o s c o p e i n t h e  t o 200  k g , v a r y i n g from one  t o another©  Therefore,  modulus, t h e  e f f e c t of loading pressure  specimen  p r i o r t o t h e measurements o f Young's  v e l o c i t y o f c o m p r e s s i o n a l waves was  on t h e  examined  e  propagation The  t e s t s were made by m e a s u r i n g t r a v e l - t i m e under t h e  preliminary various • o  l o a d i n g c o n d i t i o n s up t o a p p r o x i m a t e l y 1600  kg o r 9^«3  in stresso  of propagation  There was  a noticeable  increase  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 l o a d o t h a t the p r o p a g a t i o n v e l o c i t y through g r a n i t e was  highly  I t was  t i m e s t h a t a t 100  observed  specimens  dependent upon l o a d , w i t h the v e l o c i t y a t t o 1.12  kg/cm  kg b e i n g  1.07  kg l o a d .  andesite  sample showed t h e l e a s t e f f e c t o f l o a d i n g  1600  The pressure  on t h e v e l o c i t y , t h a t i s , t h e i n c r e a s e o f t h e v e l o c i t y a t 1600  kg was  l e s s t h a n 2 p e r c e n t o f t h a t a t 100 k g o  However, t h e r e  i s not  w i t h the i n c r e a s e v e l o c i t y was  significant difference i n travel-time  i n l o a d from 100  t o 200  k g where  measured f o r t h e d e t e r m i n a t i o n  the  o f Young's  moduluso E x p e r i m e n t a l r e s u l t s of the p r o p a g a t i o n v e l o c i t y measurements summarized i n F i g u r e t i m e v e r s u s t r a v e l distance» all  As  5* w h i c h p l o t s t r a v e l shown i n t h e s e f i g u r e s  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 z e r o  intercept,  T3  20  h  15  h  o o  CD  w o o  B 0) a •H  10  Eh  CD  5  2  Travel Figure  4  6  Distance 5.  (cm)  h  2  Travel  4  Distance  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 .-. <.  6  (cm)  Travel  Time  ( m i c r o second  oo  - IZ ~  )  Travel  Time  ( m i c r o second )  H"  CD  (-3  H-1-3  <; —•<;  CD. CD  8' f a • 2 »-3 p . SB Hca Cfa • 3 c f 3- © cs cf O O Cf ' •"' O O o P 3 ' CO 'o' ••.  •^d  3-o O 3  c f JKO'  ••;  O'  <  -  zz  -  -  23  s h o w i n g no d e l a y t i m e due  -  t o t h e c o n t a c t c o n d i t i o n between  t h e p l a t t e n and a s p e c i m e n .  The mean v a l u e and t h e s t a n d a r d  d e v i a t i o n o f t h e c o m p r e s s i o n a l wave v e l o c i t y t h r o u g h sample i s as f o l l o w s ! a n d e s i t e ; 6.01 + 0.10,'marble? magnetite;  5  = 10  +0.55,  + 0.26,  sandstone?  4 . 5 9 + 0 . 4 4 k i l o m e t e r per  3.86  each  granite; + 0.26,  3-81 and  second.  The p r o p a g a t i o n v e l o c i t y t h u s d e t e r m i n e d  may  differ  somehow from t h a t u s i n g a c y l i n d r i c a l s p e c i m e n w h i c h i s . commonly used f o r t h e measurements o f c o m p r e s s i o n a l shear  waves.  wave was  Since the f i r s t a r r i v a l of a  concerned  i n t h i s experiment,  . Rock  Density  g/cm ^  not a f f e c t  the  Young's Modulus  V e l o c i t y of P-Wave 10^  compressional  a shape o f t e s t p i e c e s  w h i c h composes a boundary c o n d i t i o n w i l l  T a b l e 2.  and/or  Young's Modulus m=8  m=6  0.125  0.167  10-  cm/sec  Andesite  2.72  6.01  Granite  2.71  3°  Marble  2.73  Sandstone Magnetite  5  m=4 0.250  kg/cm  9.67  905  8.35  3.87  3*75  3«35  5.10  6.99  6.76  6„04  2.67  3.86  3»91  3o79  3»38  4.75  4.59  9.85  9.53  8.51  81  m = P o i s s o n ' s number 1/m = P o i s s o n ' s r a t i o  - 2k experimental  results.  C o r r e l a t i o n e x p e r i m e n t s were c a r r i e d  out on c y l i n d r i c a l specimens h a v i n g 5 ° . 2 ' cm d i a m e t e r and i r r e g u l a r l y shaped specimens u s i n g m a g n e t i t e and m a r b l e o As c a n be s e e n i n F i g u r e 5 - ( c ) and ( e ) where a w h i t e  circle  d e n o t e s a c y l i n d r i c a l s p e c i m e n , t h e r e was no n o t i c e a b l e d i f f e r e n c e o b s e r v e d between shaped and unshaped s p e c i m e n s Then t h e Young's m o d u l i were computed from ( 2 ) with the d i f f e r e n t Poisson's  2 - Tensile  r a t i o , and l i s t e d  0  Equation i n Table 2 «  Strength  Figure  6 shows t h e r e s u l t s o f t h e p o i n t s l o a d  tensile  s t r e n g t h t e s t f o r a l l s a m p l e s , and i s t a b u l a t e d i n t h e Append! The  value  o f t e n s i l e s t r e n g t h computed from E q u a t i o n ( 3 )  i s t a k e n as o r d i n a t e , w h i l e t h e a b s c i s s a shows t h e d i a m e t e r o f an i n s c r i b e d s p h e r e o f a s p e c i m e n . According a l i t y constant  t o the recent report  involved i n Equation  ( 1 5 ) , the proportion-  ( 3 ) , that i s , k = 1 . ^ ,  i s r e l a t e d t o t h e r e l a t i v e shape d i m e n t i o n , and  Poisson°s r a t i o ,  e s p e c i a l l y t h e r a t i o o f t h e d i a m e t e r .of l o a d i n g p o i n t t o  t h a t o f t h e i n s c r i b e d sphere o f t h e specimen. 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 t h e o r e t i c a l l y by t h e r e s e a r c h e r s  0  A necessary  was a l s o  analyzed  The optimum v a l u e o f t h e  r a t i o . o f t h e s i z e o f l o a d i n g p o i n t t o t h a t o f t h e specimen i s 0 . 1 5 when t h e c o e f f i c i e n t k = 1 * ^ i n E q u a t i o n  (3).  D e c r e a s i n g t h i s r a t i o , t h e 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»  ? Tensile o o oo  o  ~a  ->•  CD ~5 CD  3 CD <-+ ro -s  o -+i 00  Strength  on o  IX)  ro .on O  i  i  O o  i  INS  (kg/cm")  GO  Q  oo o o  ®  . /  ®  -  o  -ti  ©  -a CD i — i O ZJ —00 3 o CD S H3 —'• cr ^ CD O CL 3  —  %  3>  /*.  ro 00  r+  a>  oo a •a r r a> ro 3 -s ro ro r+ ro o -s -h o oo -h  Strength  (kg/cm  Tensile  Strength  (kg/cm")  on O  O O  1  )  on O  1.  o o  i  /  oo  © ©  © In  -  ® h ©  •o  Marbl e  ro i — i o 3 00 3 O ro -5 CT -—- ro o CL 3  Tensile  200  200  (d)-Sandstone  (e)-Magnetite E  O  ®  ©  ^ 1 5 0  g  150  ©  -P  to  hi)  ®  100  ?  •p  +>  o>  .0)  1—1  100  0\  <H  CO Q>  10  50 EH  0  Z  3  5  7  Diameter of I n s c r i b e d Sphere o f Specimen (cm  50  3  ^  5  ?  Diameter o f I n s c r i b e d Sphere o f Specimen (cm) ngth o f ( d ) - S a n d s t o n e and ' ( e ) - M a g n e t i t e  -  27  -  I n t h e t e s t r e s u l t s shown i n F i g u r e 6, t h e c o n s t a n t r a t i o o f t h e s i z e o f l o a d i n g p o i n t t o t h a t o f specimens has not b e e n s a t i s f i e d , s i n c e t h e s i z e o f specimens was i n t h e r a n g e o f a p p r o x i m a t e l y 3 "to 6 cm, w h i l e t h a t o f l o a d i n g p o i n t was. k e p t c o n s t a n t loO cm*  T h e r e f o r e , t h e e f f e c t o f s i z e on  t h e t e n s i l e s t r e n g t h a p p e a r i n g i n F i g u r e 6 s h o u l d be i n d i r e c t . 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 strength w i t h the s p e c i m e n h a v i n g 5 cm d i a m e t e r , w h i c h f u l f i l l s a c o n s t a n t . r a t i o o f t h e p l a t t e n t o specimen s i z e , was d e t e r m i n e d  from t h e b e s t  f i t t i n g l i n e w h i c h was o b t a i n e d by t h e method o f l e a s t squares assuming a l i n e a r function,,  The c o r r e s p o n d i n g  linear  r e g r e s s i o n equations are; Andesite:  s t  (A)  =  330°^  Granite:  St(G)  Marble:  St(Mb) =  Sandstone:  s t  Magnetite:  s t  (S)  - 26 « x 0  = 128„S ~ 2 ? x 0  87°0  -  5*8 x '  = 1? °5 - 9 » 7 x 8  (Mg) = 1 3 3 - 7 - 1 0 . 1 x  where x i s s i z e o f s p e c i m e n s . The t e n s i l e s t r e n g t h o b t a i n e d 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 k g / c m  These t e n s i l e s t r e n g t h s a r e c l o s e l y averaged  2  f o r magnetite.  v a l u e s o f each  sample, s i n c e a 5 cm d i a m e t e r l o c a t e s a l m o s t a t t h e c e n t e r o f s i z e s o f t h e specimens t e s t e d o  -  28  3 - Drop-Weight Impact T e s t The r e s u l t s o f drop w e i g h t i m p a c t t e s t s a r e i l l u s t r a t e d i n F i g u r e ?. of  The o r d i n a t e e x p r e s s e s t h e h e i g h t  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 t h e p o t e n t i a l energy o f  the  d r o p p i n g b a l l d e t e r m i n e d by m u l t i p l y i n g t h e h e i g h t by  the  weight o f the drop b a l l o  On t h e a b s c i s s a , t h e number o f  drop t e s t s , r e q u i r e d t o g e t t h e f r a c t u r e o f a specimen i s giveno  .  •  We were p r i m a r i l y c o n c e r n e d w i t h t h e f r a c t u r e o f specimens by s i n g l e i m p a c t and t h i s w i l l be d i s c u s s e d i n :  terms o f c r i t i c a l h e i g h t .  The t e r m i n o l o g y o f c r i t i c a l  height  i s used f o r t h e h e i g h t below w h i c h f r a c t u r e c a n n o t t a k e p l a c e b y . s i n g l e impact.  However, t h e r e e x i s t s a c o n s i d e r a b l e  scatter  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 e x p e r i m e n t a l r e s u l t s . F u r t h e r m o r e one would e x p e c t such a s c a t t e r because i n most rocks there i s considerable difference i n stored s t r a i n  energy  w h i c h v a r i e s i n s t r e n g t h d e p e n d i n g on t h e d i r e c t i o n o f an a p p l i e d e x t e r n a l f o r c e (12)« s h o u l d be t r e a t e d  Therefore, the t e s t  statistically  results  0  The c o n t i n u o u s l i n e i n F i g u r e 8 d e p i c t s t h e r a t i o of  t h e number o f s p e c i m e n s  f r a c t u r e d by one drop from a  g i v e n h e i g h t range t o that,.of t o t a l specimens i n t h e same group.  The r a t i o t h u s d e f i n e d e x p r e s s e s t h e p r o b a b i l i t y o f  f r a c t u r e o f a specimen by s i n g l e i m p a c t a t a g i v e n h e i g h t l e v e l , and t h i s g i v e s t h e c r i t i c a l  height 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 o f repeated impact  ® 110  70  (a)Andesite  -  (b)Granite  i 100  -o @  90  —  its'  @  ®  . 60  "@ 0 d 9  (IF"  9  §>  ®  80  6 «  @ ©  (fp©  ®  0  ^  0  0  m  70  ®  ( 50  ©  ©  Q 0 ®  60  50  1 5  1 10  Number o f Impacts F i g u r e . 7«,  1 15  40  )  .  1  1  1  5  10  15  Number o f Impacts  Number o f Impacts R e q u i r e d f o r F r a c t u r e o f ( a ) A n d e s i t e . and ( b ) G r a n i t e from V a r i o u s H e i g h t  (d)Sandstone 60 \-  o  to CD  •H  50  80  o  h  •a  h  40  •H  .70 (J 6o  - i ®  ® ® • 01 o  i-J  30  .50  h  20  40  10  30 0  10 .Number o f Impacts Figure 7.  15  @  ©  ©  1  1  1  •5  10  15  Number o f Impacts  Number o f Impacts R e q u i r e d f o r F r a c t u r e o f ( c ) M a r b l e and (d)Sandstone from V a r i o u s H e i g h t  60 (e)Magnetite ©  50  € J  40  •  -©  30 u  ©  © © © ©  ©^  © ©  20  0  ©  J_ 5 io. Number o f Impacts F i g u r e 7.  15  Number o f Impacts R e q u i r e d f o r F r a c t u r e o f (e)Magnetite*from Various Height  o  l.o  0.5  P r o b a b i l i t y of Fracture F i g u r e 8.  o :.  •'  "  i.o  0.5  Probability  of  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 o r ( a ) - A n d e s i t e and ( b ) - G r a n i t e  Fracture Fracture  6o  90  50  80  40  70  6 o  E O  +>  si •30  •p 60 x w  r-l 20  .3 50 pq  (d)-Sandstone  (c)-Marble 40  10  _1  L  o  _l  1_  0.5 P r o b a b i l i t y of F i g u r e 8.  l.o Fracture  30,  X  l.o  0.5  P r o b a b i l i t y of  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 o r ( c ) - M a r b l e and ( d ) - S a n d s t o n e  Fracture  Fracture  60  iH  « 20  (e)-Magnetite  10  q  •  -  I  i  i  o •  i  i  _I  i  t  i  0.5  i  :  l.o  P r o b a b i l i t y of Fracture F i g u r e 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 Fracture f o r .(e). Magnetite  - 35 numbers. For example, the c r i t i c a l height f o r marble was determined  g r a p h i c a l l y as 4 5 . 0 cm with 100 percent  p r o b a b i l i t y and 3 8 . 0 cm with 50 percent p r o b a b i l i t y .  The  c r i t i c a l height corresponding to 100 percent p r o b a b i l i t y i s e s s e n t i a l l y an u n r e l i a b l e value f o r 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 t h i s study, the c r i t i c a l height corresponding to  50 percent p r o b a b i l i t y f o r the rock being tested was taken  as i t s representative c r i t i c a l height.  Among f i v e samples,  andesite consumes the l a r g e s t p o t e n t i a l energy, 19 kg-cm. This i s equivalent to 99 cm of c r i t i c a l height, i n d i c a t i n g 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 f o r others ares 68 cm f o r 57 cm f o r granite, 47 cm f o r magnetite, and 39 cm  sandstone, for marble.  Handling the t e s t r e s u l t s which record the repeated number, i t may be wise to apply the concept of p r o b a b i l i t y from a d i f f e r e n t point of view.  The r a t i o of  the number of specimens i n the same group at a given height l e v e l to the t o t a l number of impacts required f o r the fracture of these specimens i s adopted  f o r this  purpose.  This r a t i o i n i t s e l f i s the r e c i p r o c a l of the average number of  impacts required f o r the f r a c t u r e of a specimen at a given  height l e v e l , but i t also can be regarded as the p r o b a b i l i t y of  fracture of the specimen, per one impact at the given height.  For example, i f 36 impacts were required to f r a c t u r e 6 specimens  - 36 from the same height l e v e l , the p r o b a b i l i t y i s *>/36 =  1  /6.  The p r o b a b i l i t y calculated from t h i s d e f i n i t i o n i s also p l o t t e d i n Figure 8 by dashed l i n e s .  The  critical  heights with 50 percent p r o b a b i l i t y are obtained g r a p h i c a l l y as follows: 100 cm f o r andesite, 67 cm f o r sandstone, for  60.5  cm  g r a n i t e , k$ cm f o r magnetite, and 38 cm f o r marble. I t i s i n t e r e s t i n g to note that the difference of  c r i t i c a l height derived from both d e f i n i t i o n s i s only 1 to 2 percent at 50 percent p r o b a b i l i t y . determined  Thus, the c r i t i c a l heights  by both methods are equally u s e f u l to estimate the  resistivity  of rock samples to drop-weight  impact.  In t h i s  study, the c r i t i c a l height obtained from the former manner was  adopted.  4  The Comparison of S t / B * and C r i t i c a l 2  Using the. determined  Height  value of the t e n s i l e strength  as well as the Youn*s modulus of the rock samples, we c a l c u l a t e the value of S t / E f o r each sample. 2  St  2  can  The values of  and S t / E are tabulated together with other experimental 2  r e s u l t s i n Table 3 . of a new  In order to examine the a p p l i c a b i l i t y  comminution c r i t e r i o n to impact crushing events,  the values of S t / E are p l o t t e d i n Figure 9 against the 2  c r i t i c a l height at 50 percent p r o b a b i l i t i e s determined by the drop-weight  impact t e s t s .  As shown by the two dotted l i n e s ,  a r e l a t i o n s h i p i s recognized among andesite, magnetite,  St denotes t e n s i l e strength and E Young's modulus.  and  P o t e n t i a l Energy o  ( kg'w m )  o  o  —J—  Critical -3 P' OS CD \0  P 3  00 <+  U3  CD  l-»  ,—s.  O c+  X  P* O c+ 3 P' W  (-»  o JU Py«  Xw  CO CO H* c+ OS £ ^ CD c+ ©  *  ^  GO c+  ro  \• w  ° l ro  O  3  ro  Height  ( cm.)  m a r b l e , o r among s a n d s t o n e , g r a n i t e , m a g n e t i t e , and marble respectively.  However, i t i s d i f f i c u l t t o f i n d a r e l a t i o n s h i p  amongst a l l t h e s e . s a m p l e s . c r i t i c a l height.  Next S t was compared w i t h t h e 2  F i g u r e 10 shows t h e p l o t s o f S t  2  versus  c r i t i c a l h e i g h t f o r t h e s e samples and s u g g e s t s t h a t a l i n e a r relationship exists.  This observation  i n d i c a t e s that the  c r i t i c a l h e i g h t , w h i c h i s p r o p o r t i o n a l t o t h e energy r e q u i r e d t o f r a c t u r e r o c k specimens by i m p a c t , i s p r o p o r t i o n a l t o t h e square o f t h e t e n s i l e s t r e n g t h r a t h e r t h a n t o t h e square o f the t e n s i l e s t r e n g t h o v e r Young's modulus. To e x p l a i n t h e d i f f e r e n c e between e x p e r i m e n t a l  and  t h e o r e t i c a l r e s u l t s , a n e x a m i n a t i o n o f Oka and Maxima's t h e o r y i s necessary together results.  v/ith a d i s c u s s i o n o f t h e  The d e r i v a t i o n o f t h e t h e o r y  experimental  i s as f o l l o w s :  The energy w i s t h e p r o d u c t o f an a p p l i e d l o a d and the d i s p l a c e m e n t o f l o a d i n g p o i n t i n t h e d i r e c t i o n o f t h e l o a d . w = F • u  -.-  •  —  where F i s t h e l o a d and u i s d i s p l a c e m e n t . can be c a l c u l a t e d from e l a s t i c •u =  p _ x • E  (5)  Displacement u  theory. :  .  • (6)  They assumed t h a t t h e f r a c t u r e o f a n i r r e g u l a r r o c k  particle  i s o n l y p o s s i b l e when t h e t e n s i l e s t r e s s r e a c h e s t h e t e n s i l e s t r e n g t h o f t h e p a r t i c l e , where t h e 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 ) :  / . . ; . .  -  -  39  S u b s t i t u t i n g E q u a t i o n ( 3 ) and o b t a i n e d t h e energy  ( 6 ) i n t o Equation ( 1 ) , they  equations o.  •,-1.23  •  s t  2  3  ; E  ——(?)  x  E q u a t i o n ( 6 ) i s a f u n c t i o n o f Young's modulus., I n t h e d e r i v a t i o n o f E q u a t i o n ( 6 ) , i t i s q u e s t i o n a b l e v/hether t h e e l a s t i c theory' i s a p p l i c a b l e t o r o c k m a t e r i a l s , e s p e c i a l l y t o d e f o r m a t i o n , under t h e c r i t i c a l c o n d i t i o n n e a r f r a c t u r e , even though some m e c h a n i c a l p r o p e r t i e s o f r o c k s , w h i c h e x c l u d e t h e f r a c t u r e phenomena, c a n 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 t h e t h e o r y o f e l a s t i c i t y . t h a t Young's modulus  The r e a s o n b e i n g  by d e f i n i t i o n , h o l d s i t s p h y s i c a l  meaning i n t h e r a n g e 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 stress.  I t i s w e l l known t h a t t h e s t r a i n i s no  longer  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 .  In addition,  i n many c a s e s r o c k s have a l r e a d y been s u b j e c t e d t o s t r a i n i n a field,  and i f t h e r o c k c o n t a i n s much r e s i d u a l s t r a i n  energy  t h e s t r e s s s t r a i n c u r v e 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 o f t h e diagram  (16).  T h e r e f o r e , from t h i s p o i n t ,  t h e e l a s t i c t h e o r y i s not d i r e c t l y a p p l i c a b l e w i t h o u t t h i s c o n d i t i o n i n mind.  keeping  Thus, E q u a t i o n ( 6 ) can no l o n g e r be  a p p l i e d u s i n g t h e v a l u e o f Young's modulus o b t a i n e d i n t h e range o f 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  S t /E, o b t a i n e d i n t h i s e x p e r i m e n t theoretical, quantity.  i s d i f f e r e n t from  of  the  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 t h e l o a d F, w i t h o u t t h e  effect  - 4o o f Young's modulus.  -  2 T h e r e f o r e S t /E i n E q u a t i o n  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  be p o i n t e d out i n t h e e x p e r i m e n t a l method. i s the t a n g e n t  (7) would may  Young's modulus  o f t h e s t r e s s - s t r a i n c u r v e , but t h e  stress-strain  c u r v e i s n o t l i n e a r i n most c a s e s o f r o c k 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  o f Young's modulus a t which' t h e 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 / E obtained t h e o r e t i c a l l y . 2  o f Young's modulus, i n t h i s e x p e r i m e n t ,  was  value  o b t a i n e d from t h e  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. v a l u e o f Young's modulus used i n t h e o r e t i c a l  The  While,  the  consideration i s  n o t t h e v a l u e o f t h e dynamic, b u t o f t h e 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 m o d u l i has n o t b e e n difined.'.  T h i s m i g h t cause a d i s a g r e e m e n t between t h e o r y  experimental r e s u l t .  But i t might be adequate t o a p p l y  and the  dynamic modulus i n t h e e s t i m a t i o n o f 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 t h e impact t e s t o r g r i n d i n g t e s t . However, no c o m p l e t e e x p l a n a t i o n o f t h e d i s a g r e e m e n t has o b t a i n e d , and a f u r t h e r c o n s i d e r a t i o n o f t h i s m a t t e r  been  may  g i v e a s a t i s f a c t o r y answer t o t h e d i f f e r e n c e between t h e e x p e r i m e n t a l and t h e o r e t i c a l r e s u l t s . .  5-  The Mill  .  R e l a t i o n s h i p between S t / E and t h e R e s u l t s o f B a l l . 2  Grinding Based on t h e d r o p - w e i g h t t e s t s , t h e s q u a r e o f  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 as a c r i t e r i o n o f  30  30  Andesite  Andesite o -p 20  o  Granite K3  Magnetite  SlO c  1  CD  Sandstone  © Granite © / Magnetite ©  H3—  / Marble  Marble _!  o  I  Sandstone  .  I  l_  3 -2, S t V E ( x 1 0 kg/cm~ )  St 2  /  F i g u r e 1 1 . R e l a t i o n s h i p Between St /E and Energy Index.  2 , ( x  10  4  2 . 4 .  kg /cm  )  F i g u r e 1 2 . R e l a t i o n s h i p . Between St and E n e r g y Index..  - 42 -  comminution t h a n S t /E. However, t h e d r o p - w e i g h t i m p a c t t e s t r e p r e s e n t s a s i m p l e e v e n t i n comminution p r o c e s s .  Therefore, there 2  a r i s e s t h e q u e s t i o n o f whether t h e c r i t e r i o n S t i s a p p l i c a b l e t o t h e p r a c t i c a l comminution e v e n t s l i k e r o d o r . ball, m i l l  grinding. As d e s c r i b e d i n P a r t 2, t h e energy c o n s u m p t i o n o r  work r e q u i r e m e n t i n b a l l m i l l g r i n d i n g was e s t i m a t e d by means o f t h e energy i n d e x f o r t h e i d e n t i c a l samples used i n t h e t e s t s for the determination of S t / E .  The energy i n d e x r e p r e s e n t s  2  t h e e n e r g y c o n s u m p t i o n t o r e d u c e t h e -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 t h e l a b o r a t o r y b a l l m i l l under t h e g i v e n 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 p r o c e s s i n t h e b a t c h m i l l a t . a n e a r l y s t a g e w i l l be c o n t i n u o u s . I n F i g u r e 1 1 , and 12, t h e r e l a t i o n s h i p a r e t h e energy i n d e x v e r s u s S t / E and S t r e s p e c t i v e l y . 2  2  The f o r m e r  r e l a t i o n s h i p i s not recognized c l e a r l y i n s p i t e of considering t h e r a n g e s o f t h e q u a n t i t y o f S t / E , due t o t h e p o s s i b i l i t y 2  o f t h e change i n P o i s s o n s r a t i o . S i m i l a r l y as f o u n d i n t h e 2 ' p r e v i o u s s e c t i o n , S t i s more f a v o u r a b l e t o energy i n d i c e s , 9  •3c  e x c e p t t h e s a n d s t o n e sample,. T h i s r e s u l t may c o n f i r m t h e 2 e s t a b l i s h m e n t o f S t as t o a new c r i t e r i o n i n c o m m i n u t i o n O  p  r a t h e r t h a n S t / E , and S t  may be u s e f u l a s a c r i t e r i o n t o  e s t i m a t e t h e ease o f c o m m i n u t i o n o f r o c k m a t e r i a l s i n comminution i f t h e r e l a t i o n i s r e s e a r c h e d f o r each p a r t i c u l a r r o c k . * The anomalous b e h a v i o u r o f s a n d s t o n e sample i s mentioned i n P a r t 2.  Table 3 «  Rock  Tensile Strength /  St 4  9  kg/cm  c  Experimental Results  m=4  m=8  m=6  m=4  .0,1.25  0.250  0.125  0.167  0.250  0.167  10  xlO  5  kg/cm  Andesite  196  3.84  9.67  Granite  118  1.39  3.87  58  0.34  6.99  6.76  130  I.69  3-91.  3.79  89  0.79  Marble Sandstone Magnetite  St /E  Young's Modulus m=8 m=6  " 9 . 8 5  9.35 •  3.75  9.53  E = Young's modulus St = t e n s i l e s t r e n g t h m .= P o i s s o n ' s r a t i o l/m = P o i s s o n ' s r a t i o C r i t i c a l Heightj at 5 0 % probability ( B a l l w e i g h t ; 1 , 9 0 3 gms) Energy Index; 1 5 0 mesh 8 0 % p a s s i n g  :  2  2  10  -2  Critical Height  k g / cm  cm  EnergyIndex kwh/ton  8.35  3.97  4.11  4.60  99  26.7  3.35  3.60  3.71  4.16  57  9.7  6.04  0.48  O.50  4.32  4.46  5.00  68  0.80  O.83  0 . 9 3  ^7  3.38 8.51  '"  O.56  .  39 1  5.1 7.5  _ 44 -  CONCLUSION The q u a n t i t y , s q u a r e o f t e n s i l e s t r e n g t h o v e r Young's modulus,  d e r i v e d from t h e e l a s t i c t h e o r y  provided  a b a s i s o f c o m p a r i s o n f o r t h e comminution r e s i s t i v i t y or rock m a t e r i a l s . application  In c o n s i d e r a t i o n of the l i m i t of  of the theory of e l a s t i c i t y to the  c o n d i t i o n o f the rock f a i l u r e , the modified of t e n s i l e s t r e n g t h , appeares c l o s e l y  q u a n t i t y , square  related  h e i g h t and t h e b a l l m i l l g r i n d i n g r e s u l t s .  critical  to the  critical  The square o f  t e n s i l e s t r e n g t h i s p r o p o s e d t o be a new c r i t e r i o n t o estimate  t h e r e s i s t i v i t y o f r o c k s i n comminution, and  estimate  w i l l be e a s i l y c a r r i e d  out w i t h t h e use o f t h e ,  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 study.  this  ;  i n this  -  CHAPTER TWO .  45  -  ENERGY INDEX IN-DRY TUMBING MILLING  • SCOPE OF PRESENT WORK G r i n d i n g t e s t s were c a r r i e d o u t w i t h a 12' x'12 inch-millo  A t o r q u e meter was i n s t a l l e d between t h e d r i v i n g  s h a f t and t h e m i l l s h e l l .  T h i s made p o s s i b l e t h e d e t e r m i n a t i o n  o f t h e work r e q u i r e m e n t o f t h e m i l l d u r i n g o p e r a t i o n and t h e e s t i m a t i o n o f t h e energy c o n s u m p t i o n n e c e s s a r y f o r s i z e . reduction of material. F o r t h e purpose o f c o r r e l a t i n g t h e r e s u l t s w i t h t h e s e o b t a i n e d i n C h a p t e r 1, t h e m a t e r i a l ground was e s s e n t i a l l y t h e same a s t h a t used i n t h e p r e v i o u s e x p e r i m e n t s . One o f t h e o b j e c t s i n t h i s s t u d y was t o b r o a c h t h e p r o b l e m o f t h e e s t i m a t i o n o f energy c o n s u m p t i o n i n a c t u a l m i l l s by t h e r e s u l t s o f l a b o r a t o r y work. c o n c e p t , Energy I n d e x , was i n t r o d u c e d t o e v a l u a t e  A new grinding  r e s u l t s and a s s i s t i n t h e s t u d y o f g r i n d i n g p r o b l e m s .  The  dependency o f t h e e n e r g y i n d e x on e x p e r i m e n t a l p a r a m e t e r s was a l s o  investigated. Optimum g r i n d i n g c o n d i t i o n s were d e t e r m i n e d  e x p e r i m e n t a l l y and t h e s e c o n d i t i o n s were used t h r o u g h o u t the study.  - 46 MATERIALS AND  PREPARATION  The m a t e r i a l s used f o r t h e b a l l m i l l  grinding  e x p e r i m e n t s were e s s e n t i a l l y t h e same as t h o s e used i n the e x p e r i m e n t s r e f e r r e d t o 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 , m a r b l e , s a n d s t o n e , and m a g n e t i t e .  I n a d d i t i o n to  t h e s e 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 , w h i c h were p r o v i d e d from t h e Mines B r a n c h , Department o f E n e r g y ,  Mines  and R e s o u r c e s , Ottawa, were u s e d . L a r g e p i e c e s o f each r o c k sample were c r u s h e d t h r o u g h 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 s c r e e n e d d r y w i t h a Ro-Tap s c r e e n s h a k e r t o o b t a i n -6+8 f r a c t i o n s as f e e d .  mesh  T h i s was p r o v e d e x p e r i m e n t a l l y t o be a  s u i t a b l e s i z e f o r t h e 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 r e a s o n f o r t h e use o f s i n g l e s i z e f r a c t i o n s as a f e e d t o t h e g r i n d i n g m i l l , was t o o f f e r a b a s i s o f c o m p a r i s o n o f consumed e n e r g i e s f o r s i z e r e d u c t i o n o f d i f f e r e n t  samples.  Each amount o f m a t e r i a l t o be ground was o b t a i n e d by s p l i t t i n g t h e - 6 +8 mesh f r a c t i o n s from t h e same p o p u l a t i o n with a r i f f l e  sampler.  The a p p a r e n t s p e c i f i c g r a v i t y needed t o c a l c u l a t e the  volume o f f e e d was measured as f o l l o w s : - 6 +8 mesh  f r a c t i o n s were p l a c e d i n 1000  c c m e a s u r i n g c y l i n d e r and compacted  by s h a k i n g t o c o n t a i n a minimum amount o f v o i d s , and t h e n t h e a p p a r e n t s p e c i f i c g r a v i t y was d e t e r m i n e d ( s e e T a b l e *  T y l e r s t a n d a r d s c r e e n mesh  4).  -  Table 4 .  Sample  47  -  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 Specific Gravity  Apparent S p e c Grav. ( 6 / 8 mesh)  '•2,72  1.53  3 * ^ 2 . 8 8  Andesite Hope, B.C.  * Sample Charge kg  Granite N o r t h V a n c o u v e r , B.C.  2.71  1.28  Marble Texada, B.C.  2.73 '  .1.37  3«03  Sandstone C o l o r a d o , U.S.A.  2.67  1.20  2.71  Magnetite . Texada, B.C.  4 . 6 3  j :  2.31  5»20  :  Feldspar 2.58 ; P r o v i d e d by Mines B r a n c h  1.31  2.95  Limestone 2.75 ' P r o v i d e d by M i n e s B r a n c h  1.37  3*08  Quartz 2.65 P r o v i d e d by Mines B r a n c h  1.37  3 . 0 9  * The volume o f sample c h a r g e ( 2 2 5 0 c c ) i s o f t h e volume o f t h e b a l l v o i d s .  50  percent  - 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 a l w a y s  c o n d u c t e d d r y under f o l l o w i n g c o n d i t i o n s . ' . The shown i n F i g u r e 1 3 > was mild steel, bars.  The  a cylindrical  diameter-29.8  b a l l m i l l used ,  t u m b l i n g m i l l made o f  l e n g t h - 3 0 « 5 cm,  cm,  w i t h no  m i l l speed can be changed s t e p w i s e t o  v a r i o u s l e v e l s i n the range of  32  . .•  rpm.  to.100  lifter  several  In t h i s  mill  a s s e m b l y , a t o r q u e meter which:was c a p a b l e o f r e c o r d i n g maximum o f 3 0 k i l o g r a m - m e t e r was s h a f t and  the m i l l s h e l l  installed  between t h e  a driving  so as t o measure t h e r e q u i r e d . o p e r a t i n g  torque. : The  one  inch cast s t e e l b a l l s  w e i g h i n g 4?  were c h a r g e d as g r i n d i n g media t o occupy a p p a r e n t l y o f m i l l volume, w h i c h i s 1 0 6 a  m i l l was  m a i n t a i n e d a t 64 rpm  c r i t i c a l speed ratio  liters;  .  The  The  kilograms 5 0 percent  r o t a t i o n a l speed o f  which i s 7 9 percent of  material-to-void ratio,  the  the  t h a t i s , the  o f t h e volume o f m a t e r i a l t o t h a t o f t h e v o i d s i n t h e  b a l l c h a r g e , was stated.  kept constant  at 0 . 5 unless  otherwise  These g r i n d i n g c o n d i t i o n s a r e l i s t e d i n T a b l e 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 p e r f o r m e d f o r v a r i o u s  grinding  * The d e t a i l e d d e s c r i p t i o n s o f t h e c h a r a c t e r i s t i c s o f t h i s m i l l assembly i s p u b l i s h e d e l s e w h e r e ( 1 7 ) . ** S e l e c t i o n o f b o t h g r i n d i n g c o n d i t i o n s , s u c h as media c h a r g e and m i l l speed a r e d e s c r i b e d l a t e r i n t h e s e c t i o n o f preliminary testing.  - 49 -  Figure 1 3 .  Tumbling m i l l and i t s torque measuring  apparatus.  -  50  -  p e r i o d s , r a n g i n g from 1 0 t o 1 , 0 0 0 r e v o l u t i o n s , under t h e same g r i n d i n g c o n d i t i o n s and w i t h t h e c o r r e s p o n d i n g numbers o f 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 with.a r i f f l e  split  sampler i n o r d e r t o 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 a n a l y s i s . ,  The amount o f sample f o r  each s c r e e n i n g t e s t was a p p r o x i m a t e l y 180 .to 2 3 0 grams e x c e p t i n t h e case of. m a g n e t i t e .  The amount o f m a g n e t i t e s c r e e n e d  was 3 5 0 t o 400 grams s i n c e t h e a p p a r e n t s p e c i f i c g r a v i t y i s ' almost t w i c e t h a t o f the o t h e r samples.  The s c r e e n i n g ^  t e c h n i q u e used was t h e s t a n d a r d 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 o r d e r 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 t o e n s u r e e f f i c i e n t s c r e e n i n g . • F o r t h e s e p a r a t i o n o f c o a r s e r f r a c t i o n s a Ro-Tap s c r e e n s h a k e r was u s e d .  When t h e s c r e e n i n g l o s s exceeded 1 . 0 p e r c e n t  o f t o t a l w e i g h t , a n o t h e r 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 o u t on a sample t a k e n from t h e same l o t . Because o f a n i n s u f f i c i e n t amount o f f e e d f r a c t i o n s , a series of grinding t e s t s with feldspar, quartz, or limestone were p e r f o r m e d u s i n g one sample e a c h . ;  A f t e r one r u n o f t h e  g r i n d i n g t e s t , t h e sample was s i z e d , and t h e n r e t u r n e d t o t h e m i l l t o g e t h e r w i t h t h e r e s t o f t h e 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 p r o c e d u r e 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 t h e f o r m e r t e s t p r o c e d u r e because o f t h e u n a v o i d a b l e r e a r r a n g e m e n t o f m i l l c o n t e n t t o the steady s t a t e .  -  51  -  EXPERIMENTAL RESULTS AND 1  - Preliminary  DISCUSSION.  Test  The g r o s s t o r q u e r e q u i r e d t o d r i v e t h e b a l l w i t h o u t a r o c k sample charge was  mill  i n i t i a l l y i n v e s t i g a t e d under  t h e v a r i o u s c o n d i t i o n s o f l o a d o f media and r o t a t i o n a l speed of the m i l l . The dependency o f d r i v i n g t o r q u e on t h e l o a d o f 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 t h e  from z e r o l o a d t o 9 4 . 0 k g o f one equivalent  i n c h b a l l s which i s  t o 1 0 0 p e r c e n t o f i n t e r n a l m i l l volume.  l o a d i n g c o n d i t i o n , t h e e f f e c t o f m i l l speed was i n v e s t i g a t e d i n t h e range o f 3 ° . t o 1 0 0 rpm, 123  range  At  each  also  which i s 3 7 to.  p e r c e n t o f t h e c r i t i c a l speed o f t h e m i l l . F i g u r e 14 shows t h e r e s u l t s o f t h e s e t e s t s , and  i n d i c a t e s t h a t i n t h e p r a c t i c a l range o f b a l l c h a r g e , f o r 42 t o 48 p e r c e n t o f m i l l volume, t h e t o r q u e b e g i n s t o s h a r p l y a t around  7 0 rpm, w h i l e  is still  decrease  i n c r e a s i n g beyond  t h e c r i t i c a l speed when t h e b a l l charge i s l e s s t h a n about 3 3 percent.  I n t h e range o f common use f o r media c h a r g e ,  40 t o . 5 0 p e r c e n t , t h e maximum g r o s s t o r q u e was when t h e m i l l speed was speed.  observed  7 0 t o 8 0 percent of the  critical  The r e l a t i o n s h i p o f r e q u i r e d t o r q u e v e r s u s  o f b a l l i s p r e s e n t e d a t t h e speed o f 64 rpm  charge  i n Figure 1 5 »  The m i l l s h o u l d be r u n a t t h e p o i n t where t h e maximum torque i s required of m a t e r i a l s .  i n order t o get the e f f i c i e n t  comminution  On t h i s b a s i s , 5 0 p e r c e n t b a l l charge  and  Figure  Ik«,  I n f l u e n c e o f the 1 2 - i n c h m i l l speed on t h e g r o s s t o r q u e under the v a r i o u s l o a d i n g . o f 1 - i n c h b a l l s .  -  64 rpm  -  53  m i l l speed were c h o s e n as s t a n d a r d  c o n d i t i o n s throughout the experiments. c o n d i t i o n s are l i s t e d i n Table  Table 5 . Mill  Shell  grinding  These g r i n d i n g  5»  Grinding  Conditions  30.5 x 2 9 . 8 cm (12 x 11.75 Smooth-faced c y l i n d r i c a l  G r i n d i n g Method  Dry  Grinding  1 inch cast steel b a l l s , 5 0 percent m i l l charge  Mill  Media  Speed  in)  grinding  64 rpm,  47  kilograms  79 p e r c e n t of c r i t i c a l  speed  D r i v i n g Torque  2 2 5 kg-cm, l o a d e d m i l l w i t h 47 k g without 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  Power Requirement  148 w a t t s a t . 64 rpm 21 w a t t s f o r empty m i l l a t 64  rpm  ball,  ;  250  NO FEED  200  -©-  'XT  -o-  e o i bD ~  150  o E-i  10 S  Magnetite (5-20 kg)  100  Marble (;3°o8  kg)  50  EMPTY  100 G r i n d i n g .Media L o a d i n g .-' {fo o f m i l l volume) F i g u r e 1 5 . The e f f e c t o f b a l l l o a d i n g on t h e g r o s s t o r q u e a t a speed o f 64 rpm,  100  "  200  300  400  500  Number o f r e v o l u t i o n s F i g u r e 16. The g r o s s t o r q u e f o r t h e m i l l f i l l e d w i t h and. w i t h o u t f e e d m a t e r i a l .  •  '  -  55  -  •  2 - Torque Measurement The  t o r q u e was measured f o r each r u n o f t h e m i l l  to estimate  t h e energy^ c o n s u m p t i o n n e c e s s a r y t o a c h i e v e  reduction.  F i g u r e 16 p r e s e n t s  size  the gross torque readings f o r  m a g n e t i t e and marble w h i c h r e q u i r e d maximum and minimum t o r q u e r e s p e c t i v e l y among t h e samples t e s t e d as a f u n c t i o n o f m i l l revolutions.  The t o r q u e r e a d i n g s  sample were o b v i o u s l y  f o r t h e d r y g r i n d i n g o f each  l o w compared t o t h a t o f a m i l l  w i t h no f e e d . m a t e r i a l .  T h i s c o u l d be e x p l a i n e d by t h e  l u b r i c a t i n g e f f e c t due t o t h e f e e d m a t e r i a l r e d u c i n g 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 . Yang, e t . a l . obtained The  (18) r e p o r t e d  frictional  On t h e o t h e r  hand,  t h a t t h e l o w e s t t o r q u e was  f o r t h e m i l l t u r n i n g w i t h o u t any g r i n d i n g m a t e r i a l .  contradictory observation  might be c a u s e d by t h e use  o f t h e two d i f f e r e n t t y p e s o f m i l l s ,  t h a t i s , one a smooth-  f a c e d m i l l and t h e o t h e r w i t h l i f t e r b a r s . . was  turning  S i n c e Yang's work  done w i t h t h e m i l l e q u i p p e d w i t h e i g h t l i f t e r b a r s ,  slippage  o f t h e b a l l l o a d was l i t t l e a f f e c t e d by t h e l u b r i c a t i n g a c t i o n due  t o feed m a t e r i a l .  :  A f t e r t h e f i r s t 10 r e v o l u t i o n s , d u r i n g w h i c h t h e disturbance  o f t h e t o r q u e caused b y t h e s t a r t i n g a c t i o n o f t h e  m i l l s h e l l was o b s e r v e d , t h e r e q u i r e d t o r q u e l i n e a r l y f o r periods  of short duration.  increased  The t o t a l  revolutions  where t h e maximum t o r q u e was o b s e r v e d , were dependent upon each  * Energy i s c a l c u l a t e d f r o m t h e t o r q u e a s f o l l o w s ; e n e r g y ( w a t t s e c ) = 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 = t o r q u e (kg-cm)  - 56  -  r o c k sample and  r a n g e d from 80 t o 180  revolutions  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 t o r q u e d e c r e a s e d a s m a l l amount. Throughout the t o r q u e measurements, f o r a l l samples, b o t h t h e i n c r e a s e and t o 1,000  decrease of the torque,  r e v o l u t i o n s , was  limited  o f t h e average g r o s s t o r q u e .  f o r the  t o w i t h i n 3 "to 6 p e r c e n t  T h e r e f o r e i t c a n be  t h a t the r e q u i r e d torque i s almost constant. increase  period  regarded  However, t h e .  i n torque observed f o r the i n i t i a l p e r i o d of  may. s u g g e s t t h a t e f f i c i e n t g r i n d i n g o f f e e d m a t e r i a l take place during t h i s p e r i o d . by t h e r e s u l t s o b t a i n e d  This explanation  grinding can  i s supported  f r o m t h e s i z e a n a l y s i s w h i c h shows  t h a t t h e most r a p i d r a t e o f i n c r e a s e o f f i n e p r o d u c t s t o o k place during t h i s I t was t o r q u e due 10 p e r c e n t . that  time. a l s o observed t h a t the d i f f e r e n c e of  t o t h e d i f f e r e n t m a t e r i a l s ground was From t h i s r e s u l t , i t may  required  l e s s than  be p o s s i b l e t o s t a t e  t h e energy r e q u i r e d t o a c h i e v e s i z e r e d u c t i o n  m a t e r i a l s h a v i n g t h e same b u l k volume from one  of  s i z e to another  i s p r o p o r t i o n a l t o t h e number o f m i l l r e v o l u t i o n s and be p r i n c i p a l l y d e t e r m i n e d by g r i n d i n g t i m e .  Figure  might 17  shows t h e l i n e a r r e l a t i o n s h i p between expended energy e x p r e s s e d i n u n i t volume b a s i s and However, the a s s u m p t i o n o f c o n s t a n t  r e v o l u t i o n s o f the energy p e r  mill.  revolution  does not h o l d , f o r t h e energy u n i t on w e i g h t b a s i s , w h i c h i s e x p r e s s e d i n k i l o w a t t hour p e r t o n , u n l e s s the  materials  10  10  8  o <  Magnetite SandstoneGranite —  8  ft 6  W u  0)  05  w  Andesite' Feldspar Quartz Limestone Marble  M 0 ft  w  0)  Sandstone Granite — Marble \ Limestone/ Andesite Quartz . Feldspar  4  ft !>< W Magnetite  200  400  600  800  Number o f R e v o l u t i o n s F i g u r e 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  0  200 •Number o f  400  600  800  Revolutions  F i g u r e 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 - w e i g h t basis -  -  5 8  -  b e i n g ground have a p p r o x i m a t e l y t h e same a p p a r e n t gravity. required  Figure  specific  18 e x p r e s s e s t h e r e l a t i o n s h i p between  energy e x p r e s s e d i n u n i t w e i g h t 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  constant. These e x p e r i m e n t a l r e s u l t s i n d i c a t e t h a t t h e energy expended p e r r e v o l u t i o n o f t h e t u m b l i n g m i l l , f o r a g i v e n g r i n d i n g c o n d i t i o n , c a n be assumed as c o n s t a n t , regardless  of the materials  b e i n g g r o u n d , when expended  energy i s e x p r e s s e d on u n i t volume b a s i s .  -  3 - Size  59  -  Distribution The r e s u l t s o f s i z e a n a l y s e s o f g r i n d i n g  were p l o t t e d and l i s t e d  on Schumann's s i z e d i s t r i b u t i o n c u r v e ( 1 9 ) ,  i n Table 3 o f the  Appendix.  D i s t r i b u t i o n of material comminution  products  i n the f i n e s i z e s of a  p r o d u c t i s w e l l r e p r e s e n t e d by t h e  s i z e - d i s t r i b u t i o n equation: y =  100  following  ;  (—£-0 k  where y i s t h e c u m u l a t i v e p e r c e n t a g e o f m a t e r i a l  finer  t h a n s i z e x, e x p r e s s e d i n m i c r o n s , k i s t h e s i z e modulus, w h i c h i s t h e t h e o r e t i c a l maximum s i z e o r 1 0 0 p e r c e n t p a s s i n g s i z e i n t h e d i s t r i b u t i o n , <X i s t h e 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 o f the m a t e r i a l  and comminution  method.  F i g u r e 1 9 shows t h e Schumann p l o t s , o f w h i c h t h e o r d i n a t e and a b s c i s s a a r e t h e cummulative  percent passing a given  s i z e and t h e 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 p r e s e n t s t h e d i s t r i b u t i o n modulus f o r each c u r v e as a function  of the g r i n d i n g  The  following  p e r i o d s (see T a b l e 6 ) .  facts are generally  t h e d i s t r i b u t i o n modulus:  observed r e g a r d i n g  The d i s t r i b u t i o n modulus i s a l m o s t  e q u a l t o one r e g a r d l e s s o f m a t e r i a l s ,  i f the f r a c t u r e  is  caused s i m p l y by i m p a c t f o r c e , but because o f t h e complex a c t i o n s o f 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 grinding  p r o c e s s e s , t h e modulus t e n d s t o be l e s s t h a n  one  * T h i s p l o t o f y v e r s u s x e x p r e s s e d as t h e T y l e r mesh s i z e i s t h e same as l o g - l o g p l o t o f y v e r s u s x e x p r e s s e d as m i c r o n s .  -  60  -  100  C  •H  O 0)  CD >  •P  H O  100 48 .2.8 P a r t i c l e S i z e (mesh) Figure 19» Size D i s t r i b u t i o n s o f Marble. 200  100  50 0)  c •H  c o PL,  20.  h  10  CD > •H  -P ccj  5.  h  rH  I o  48 28 14 P a r t i c l e S i z e (mesh)  200  100  F i g u r e 1 9 - ( a ) S i z e D i s t r i b u t i o n s o f V a r i o u s Samples a f t e r 1 5 0 Revs G r i n d i n g .  Table 6 . Mill. Revolutions  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 *  Andesite  Granite  Marble  20  0.80  40  0.68  60  0.69  75  0.53  0.95  100  2  2  0„54  150  0.51  200  0.39  5  .* - 6 + 8 mesh f e e d  "  0.62  . •  O.89  0.53 . 0 . 8 8 0.78  -  0.41  500  1000  0 . 9 3  ,  250  800  0.87  2.52  0.64  0.89  125  300  Magnetite  0,82  .10  .  Sandstone  1.55  0.64  0.49  0.55  o.4o  0.79  0.38 1.07  -  (20*21).-  The  62  -  ,  ground m a t e r i a l s each have a  characteristic  modulus even though t h e g r i n d i n g c o n d i t i o n s a r e t h e same (21).  M o r e o v e r , the d i s t r i b u t i o n modulus i n wet  remains constant  d u r i n g s i z e r e d u c t i o n , b u t t h e s i z e modulus  i s s h i f t e d to a f i n e size  experimental  the  (23,24). results  obtained  i n this  were i n agreement w i t h a l l t h e s e o b s e r v a t i o n s c a s e o f t h e s a n d s t o n e sample. feed of s i n g l e  grinding  a s l i g h t tendency to decrease  i n size reduction, that i s , with  increase of g r i n d i n g time The  However, i n d r y  (20,22),  t h e d i s t r i b u t i o n modulus has w i t h an i n c r e a s e  grinding  except i n the  I t i s surprising  s i z e , - 6 + 8 mesh f r a c t i o n s  study  that f o r a  weighing 3 . 0 8 kg,  m a r b l e showed good Schuman's s i z e d i s t r i b u t i o n c u r v e c o v e r i n g a wide range from c o a r s e r t o f i n e r s i z e s , o n l y 10 revolutions grinding period modulus was  The  0  value  o f the  after  distribution  found t o d e c r e a s e 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  grinding periods  f o r a l l samples.  of  Fuerstenau ( 2 2 ) stated  t h a t the a p p a r e n t l o w e r i n g o f d i s t r i b u t i o n modulus i n . d r y g r i n d i n g must r e s u l t s  from t h e 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 f r o m g r i n d i n g media. the great  Therefore,  change o f t h e d i s t r i b u t i o n modulus i n t h i s  e x p e r i m e n t was fed i n t o the  dependent upon t h e l a r g e amount o f m a t e r i a l mill.  A possible explanation  of the unusual b e h a v i o r of  s a n d s t o n e on s i z e d i s t r i b u t i o n i s t h a t q u a r t z  grains  by t h e c e m e n t i n g m a t e r i a l s a r e e a s i l y l i b e r a t e d s t a g e i n g r i n d i n g and  cemented  at the  t h a t the a c t u a l g r i n d i n g process  early of  - 63 -. quartz  grains 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 c o m p l e t e d .  As can be r e a d o f f T a b l e .6  , t h e d i s t r i b u t i o n modulus o f  s a n d s t o n e changed s i g n i f i c a n t l y from 2 . 5 2 a t t h e g r i n d i n g period 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 e x p e c t t o o b t a i n t h e normal s i z e d i s t r i b u t i o n c u r v e o r modulus. The d i s t r i b u t i o n m o d u l i a t 1 5 0 r e v o l u t i o n s were estimated  from F i g u r e  1 9 , t o be 0 . 5 1 f o r a n d e s i t e ,  O.87 for  g r a n i t e , 0 . 6 2 f o r m a r b l e , 1 . 7 3 f o r s a n d s t o n e , 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 n o t o b s e r v e d between  the d i s t r i b u t i o n moduli materials obtained  thus obtained  i n the previous  and t h e r e s i s t i v i t y  Chapter,the c r i t i c a l  or the square o f the t e n s i l e s t r e n g t h .  T h i s c o u l d be  of height,  explained  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 n o t o n l y  on  t h e n a t u r e o f t h e m a t e r i a l , b u t a l s o g r e a t e l y on t h e g r i n d i n g c o n d i t i o n s o f t h e t e s t and on t h e comminution  device.  • -  ^ -  4 - E n e r g y Index • The e v a l u a t i o n o f t h e energy c o n s u m p t i o n o r work r e q u i r e m e n t f o r t h e g r i n d i n g m i l l o p e r a t i o n has been customarily  evaluated  on t h e b a s i s o f a n e n g i n e e r i n g  k i l o w a t t - h o u r p e r t o n o f o r e ground .  unit i n  Bond r e l a t e d t h e 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 t h e n e t e n e r g y n e c e s s a r y 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 t h e c o n c e p t o f Work Index ( 6 ) . Bond considered  t h a t t h e 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 t o t h e square r o o t o f t h e d i a m e t e r o f p r o d u c t particle.  However, C h a r l e s  ( 2 0 ) , as w e l l as o t h e r s  s t a t e d t h a t t h e r e l a t i o n o f energy i n p u t t o s i z e  (23,24)  reduction  may be d e r i v e d as a s p e c i a l c a s e and i s a p p l i c a b l e t o s p e c i f i c c a s e s i n t h e comminution s y s t e m .  A l s o , Smith (8) r e p o r t e d  t h e r e l a t i o n s h i p between a c t u a l h o r s e power and c a l c u l a t e d h o r s e power from' t h e Bond work i n d e x , u s i n g o p e r a t i n g obtained  from about t w e n t y cement p l a n t s .  data  That i s , a  r e a s o n a b l e c o r r e l a t i o n between b o t h h o r s e powers was o b s e r v e d f o r s m a l l e r and medium-sized m i l l s , whereas an o v e r - d e s i g n p r o b l e m c a n e x i s t f o r l a r g e r m i l l s more t h a n e i g h t f e e t i n diameter.  T h i s i n d i c a t e s a t l e a s t t h a t t h e work i n d e x i s  not o n l y a f u n c t i o n o f t h e n a t u r e o f m a t e r i a l g r o u n d , b u t a l s o p o s s i b l y t h a t o f the e f f i c i e n c y and/or c a p a c i t y o f a m i l l , as w e l l a s g r i n d i n g c o n d i t i o n s , s u g g e s t i n g t h e u n a p p r o p r i a t e n e s s o f Bond's d e f i n i t i o n o f work Therefore,  index.  i t may be w i s e t o reexamine t h e 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  reduction,  -  6 5  -  i n t r o d u c i n g a new i n d e x w h i c h i s somewhat d i f f e r e n t  from  work i n d e x . A t 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 o u t t o determine time.  the f i n e product o f the b a l l m i l l versus g r i n d i n g  The p r o g r e s s o f t h e p r o d u c t s o f v a r i o u s p a r t i c l e  size  a r e shown i n F i g u r e 2 0 , c o r r e s p o n d i n g t o g r i n d i n g t i m e o r m i l l revolution.  I t c a n be s e e n from t h i s f i g u r e f o r m a r b l e 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 c u r v e d p o r t i o n a f t e r 1 5 0 t o 2 0 0 r e v o l u t i o n s . The same t e n d e n c i e s were o b s e r v e d  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 o f b a t c h g r i n d i n g and i t may be caused by t h e l a c k o f new f e e d a n d / o r t h e a c c u m u l a t i o n o f f i n e r p r o d u c t s 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 t h e inside of the m i l l s h e l l .  T h e r e f o r e , i f one w i s h e s t o e s t i m a t e  t h e consumed energy t o p r o d u c e f i n e m a t e r i a l i n s i m u l a t e d continuous  o p e r a t i o n from t h a t o b t a i n e d i n a b a t c h m i l l , t h e  r e s u l t s o b t a i n e d from a l o n g e r g r i n d i n g p e r i o d t h a n 1 5 0 r e v o l u t i o n s o f m i l l r u n m i g h t be i n a d v i s a b l e , because t h e progress of f i n e r products i n continuous  o p e r a t i o n might  increase l i n e a r l y with respect 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» I n c o n n e c t i o n w i t h m i l l r e v o l u t i o n s , S m i t h and Lee ( 2 5 ) r e p o r t e d i n t h e i r c o m p a r i s o n 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 p r o d u c t s t o e s t i m a t e Bond g r i n d a b i l i t i e s from b a t c h  100  200  400 / Mill  600  800  1000  Revolutions  F i g u r e 20. Rates o f f o r m a t i o n o f mesh' f r a c t i o n s - M a r b l e -  1200  g r i n d a b i l i t l e s w i t h o u t any c i r c u l a t i n g l o a d , and 1 0 0 revolutions f o r coarser  ones.  The s l i g h t d i f f e r e n c e o f g r i n d i n g  r e v o l u t i o n s from o u r s t u d y m i g h t be caused by t h e 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 t h e amount o f f e e d . t e s t was b a s e d on Bond's s t a n d a r d g r i n d a b i l i t y t e s t  Their,  (26).and  the f e e d was 7 0 0 ml w h i c h was a b o u t one t h i r d o f our t e s t s for  a m i l l o f a l m o s t t h e same s i z e . As s t a t e d b e f o r e ,  the increase  of f i n e products  m i g h t be a l o n g t h e t a n g e n t o f i n i t i a l l i n e a r o b s e r v e d under. 1 5 0 r e v o l u t i o n s i n t h i s s t u d y .  increment And f o r t h e  comparison o f g r i n d i n g t e s t i n d i f f e r e n t rocks, t h i s  linear  p o r t i o n o f c u r v e may be t h e most s u i t a b l e f o r t h i s p u r p o s e , s i n c e one c a n a v o i d o t h e r c o m p l i c a t e d g r i n d i n g mechanisms w h i c h w i l l appear i n a l o n g e r  grinding  period.  Extrapolating the s t r a i g h t l i n e portion of desired s i z e , f o r example i n t h e case o f - 1 5 0 mesh ( s e e F i g u r e  20) ,  one c a n e s t i m a t e t h e n e c e s s a r y t o t a l m i l l r e v o l u t i o n s J t o achieve .80 percent o f cumulative weight o f the product, provided grinding.  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 As p r e v i o u s l y  shown i n F i 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  period,  so t h a t t h e energy c o n s u m p t i o n e x p r e s s e d i n k i l l o w a t t - h o u r p e r t o n , when 80 p e r c e n t o f t h e p r o d u c t p a s s e s I . 5 0 mesh o r any  o t h e r s i z e s , c a n be o b t a i n e d  by t h e f o l l o w i n g  equation.  -  68  -  = 1 . 7 x l O ~ ( T - To) N wg  (8)  7  where  W  = work i n k i l o w a t t - h o u r  per ton  T  = g r o s s t o r q u e 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 wg = w e i g h t o f f e e d i n a l a b o r a t o r y m i l l ,  increment i n kg  The h y p o t h e t i c a l e n e r g y r e q u i r e d t o a c h i e v e 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 " E n e r g y Index which i s not only a c h a r a c t e r i s t i c value be g r o u n d , b u t a l s o a f u n c t i o n o f f e e d and  other g r i n d i n g  f o r the m a t e r i a l t o  s i z e , product  size,  conditions.  Because o f t h e v a r i o u s g r i n d i n g mechanisms  incorporated  i n a t u m b l i n g m i l l and t h e v a r y i n g r e s i s t a n c e o f d i f f e r e n t m a t e r i a l s t o t h e v a r i o u s c o m m i n u t i o n mechanisms, i t i s questionable  whether o r n o t t h e energy i n d e x t h u s d e t e r m i n e d  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 p r o d u c t s i z e i s a p p l i c a b l e to other product s i z e s  Therefore, the  0  e n e r g y i n d i c e s o f a l l samples t e s t e d were d e t e r m i n e d . w i t h respect -200  t o varous s i z e s of products?  mesheso'  Figure  and  - 1 0 0 , - 1 5 0 ,  and  2 1 p r e s e n t s t h o s e energy i n d i c e s as a  f u n c t i o n of product s i z e passing paper.  - 6 5 ,  80 p e r c e n t ,  on l o g - l o g  The l i n e a r r e l a t i o n s h i p between t h e energy i n d e x  t h e p r o d u c t s i z e was f o u n d t o h o l d f o r a l l t h e samples  used o v e r t h e s i z e  -  -  69  40 — 30  \>"-®  20  V. c o •p \  •\ 10  Quartz  Civ'  \  —  x c  Andesite  \  .  \_  —  Limestone  '  Feldspar  —  I—I  \.  ft  Granite Magnetite ^  a N. 1  1  •  200  1  1  150  100  Marble .  Sandstone i  i  65  8 0 percent p a s s i n g product s i z e Figure'21.  (mesh)  Energy i n d e x as a f u n c t i o n o f p r o d u c t • *size p a s s i n g 80 p e r c e n t .  . . .  range s t u d i e d .  _  70  -  I t i s i n t e r e s t i n g to n o t i c e that the r a t i o of  t h e i n c r e a s e o f energy i n d e x w i t h t h e d e c r e a s e o f p r o d u c t s i z e i s not uniform f o r a l l samples. o f magnitude  Moreover, t h e o r d e r  i n energy i n d e x was changed as t h e p r o d u c t  s i z e was r e d u c e d , between t h e a n d e s i t e and m a r b l e  sample,  t h e f e l d s p a r and l i m e s t o n e sample, and t h e s a n d s t o n e and m a r b l e sample, r e s p e c t i v e l y .  T h e r e f o r e , t h e energy i n d e x  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 t h e power i n p u t e s t i m a t i o n f o r t h e o t h e r p r o d u c t s i z e , u n l e s s t h e s l o p e o f t h e s t r a i g h t l i n e i s known. The energy i n d e x was compared w i t h t h e Bond work i n d e x 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 w h i c h work i n d i c e s were d e t e r m i n e d by Mines B r a n c h , Department M i n e s and R e s o u r c e s , Ottawa.  o f Energy,  B o t h energy c o n c e p t s were based  on t h e a s s u m p t i o n o f 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 t h e r e s u l t and t h e r e l a t i o n s h i p o f b o t h c o n c e p t s 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  relationship  may b e • e x p r e s s e d :  -  Work Index = C x Energy Index where C i s c o n s t a n t .  The Bond work i n d e x i s i n t e n d e d f o r  c o m m e r c i a l m i l l s i n p r a c t i c e , whereas t h e energy i n d e x i s d i r e c t l y o b t a i n e d from t h e r e s u l t s o f 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 t h e d i f f e r e n c e e x p r e s s e d i n t h e c o e f f i c i e n t depends on t h e g r i n d i n g e f f i c i e n c y o f an a c t u a l m i l l and a l a b o r a t o r y mill.  I n t h i s c a s e , t h e i n c r e a s e o f f i n e p r o d u c t s 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 o f t h e l a b o r a t o r y  mill.  t o Bond,s Work Index.  Height.  - 72 -  Therefore, i f . the r e l a t i o n o f 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 o f 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 p r o d u c t s i z e , t h e work i n d e x i s e a s i l y d e t e r m i n e d by means o f t h e e n e r g y i n d e x o The r e l a t i o n s h i p between energy index, and t h e critical  h e i g h t o b t a i n e d i n C h a p t e r 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 t h e s i n g l e impact event i n -comminution r e p r e s e n t e d by means o f t h e d r o p - w e i g h t i m p a c t t e s t versus complex-events expressed b a l l m i l l  grinding.  The energy i n d i c e s were t a k e n a t - 1 0 0 , - 1 5 0 , and -200  mesh p a s s i n g 8 0 p e r c e n t t o compare w i t h t h e c r i t i c a l  height.  F i g u r e 2 3  quantities.  shows a l i n e a r r e l a t i o n s h i p between b o t h  The e x i s t a n c e o f a c l e a r r e l a t i o n s h i p o f s i n g l e  f r a c t u r e event t o b a l l m i l l g r i n d i n g s u g g e s t s t h e f o l l o w i n g : 1)  t h e c o m m i n u t i o n mechanism i n t h e b a l l m i l l m i g h t be i m p a c t  f r a c t u r e , i n great p a r t , a t the e a r l y stage of m i l l r u n . 2 ) the to  energy i n d e x w i l l  express the r e s i s t i v i t y o f m a t e r i a l s  c o m m i n u t i o n i n a p r a c t i c a l way.  CONCLUSION The  conclusions  of the i n v e s t i g a t i o n s I n P a r t  2  a r e as f o l l o w s : 1c  The  torque reading  f a c e d c y l i n d r i c a l m i l l was  f o r t h e d r y g r i n d i n g w i t h a smoothlow about 20 p e r c e n t compared with'  t h a t o f t h e m i l l t u r n i n g w i t h o u t any 2o  The  grinding material.  energy i n p u t , b a s e d on an e q u i - v o l u m e , t o a c h i e v e  t h e s i z e r e d u c t i o n o f any  m a t e r i a l c a n be d e t e r m i n e d 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  being  ground. 3„  A new  i n d e x "Energy I n d e x " w h i c h i s a f u n c t i o n o f  f e e d s i z e , p r o d u c t size,i ,and g r i n d i n g c o n d i t i o n s has k  p r o p o s e d , and  i t showed s a t i s f a c t o r y agreement w i t h t h e work  i n d e x 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 4.  been  a  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 i n d e x showed a l i n e a r r e l a t i o n s h i p w i t h t h e .  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  height.  -  .  74  • SUMMARY The a u t h o r i n t e n d e d t o 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 s t u d y : 1 ) an e x p e r i m e n t a l e x a m i n a t i o n o f a p o s s i b l e c r i t e r i o n f o r c o m m i n u t i o n , and 2)  investigation  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 t h e e s t i m a t i o n o f t h e d r i v i n g power o f t u m b l i n g m i l l s  from  t h e r e s u l t s o f l a b o r a t o r y work. I n p a r t 1, t h e q u a n t i t y S t / E p r o p o s e d by Oka 2  Majima was' e x p e r i m e n t a l l y examined  and  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 s a m p l e s .  For t h i s purpose,  t h e p o i n t - l o a d t e s t and t h e p r o p a g a t i o n v e l o c i t y measurement o f c o m p r e s s i o n a l 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 , t h e d r o p - w e i g h t i m p a c t t e s t and t h e b a l l m i l l g r i n d i n g t e s t was .performed, and t h e r e s u l t s were compared' w i t h S t /E and S t ~ , by means o f t h e c r i t i c a l h e i g h t . and t h e energy i n d e x .  I t was found t h a t , as a c r i t e r i o n o f p  comminution, t h e q u a n t i t y S t St /E. 2  And S t  2  i s more a p p l i c a b l e , t h a n t h a t o f  has been p r o p o s e d as a new c r i t e r i o n t o e x p r e s s  the r e s i s t i v i t y o f r o c k m a t e r i a l s t o  comminution.  I n P a r t 2, g r i n d i n g t e s t s were p e r f o r m e d t o a c h i e v e , t h e above p u r p o s e s w i t h t h e 12 i n c h b a l l m i l l e q u i p p e d w i t h a t o r q u e meter.  A c o n c e p t , Energy I n d e x , was  introduced to  e v a l u a t e g r i n d i n g r e s u l t s and t o s t u d y g r i n d i n g p r o b l e m s . Each r o c k sample t e s t e d was d e t e r m i n e d t h e energy i n d e x from t h e t o r q u e r e a d i n g s and t h e amount o f p r o d u c t s .  And i t was  - 75 found from t h e t o r q u e r e a d i n g s t h a t t h e energy i n p u t e x p r e s s e d on a n equi-volume  charge c a n be d e t e r m i n e d from t h e r e q u i r e d  g r i n d i n g t i m e whose v a l u e i s c h a r a c t e r i s t i c s o f t h e m a t e r i a l ground,,'  The r e l a t i o n s h i p o f b o t h c o n c e p t s . Bond work  and energy i n d e x was r e s e a r c h e d *  Index  -  -  76  SUGGESTIONS FOR FUTURE WORK On t h e b a s i s o f t h e r e s u l t s o b t a i n e d i n v e s t i g a t i o n described  from t h e  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 t h e e x p e r i m e n t a l and t h e o r e t i c a l r e s u l t s f o r t h e c r i t e r i o n o f comminution i s s t i l l unsolved.  Further  study o n . t h i s problem i s s t r o n g l y  recommended. . 2) I n o r d e r t o d e t e r m i n e • t h e energy i n d e x , t h e grinding period i s c r i t i c a l longer  i n t h e d r y method, s i n c e a  g r i n d i n g p e r i o d w i l l cause some o t h e r  grinding  mechanism o t h e r t h a n 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 t i m e o f 150 r e v o l u t i o n s w i l l be t o o s h o r t f o r a, tougher m a t e r i a l .  T h e r e f o r e , a v/et g r i n d i n g s t u d y s h o u l d be  a t t e m p t e d t o d e t e r m i n e t h e energy i n d e x , and t o s t u d y t h e r e l a t i o n s h i p between d r y and v/et  grinding.  3) . The maximum d r i v i n g t o r q u e was o b t a i n e d g r i n d media was c h a r g e d w i t h no f e e d .  when t h e  Therefore, t o eliminate  l u b r i c a t i n g a c t i o n due t o f e e d m a t e r i a l and/or s l i p p a g e o f g r i n d i n g media, t h e use o f a m i l l w i t h l i f t e r b a r s w i l l be more u s e f u l f o r t h e measurement o f r e q u i r e d e n e r g y f o r s i z e reduction.  -• 77 -  REFERENCES  1.  Department o f S c i e n t i f i c and I n d u s t r i a l R e s e a r c h , " C r u s h i n g and G r i n d i n g - A B i b l i o g r a p h y " , Her M a j e s t y ' s S t a t i o n e r y O f f i c e , London, ( 1 9 5 8 ) .  2.  C h e m i c a l E n g i n e e r s ' Handbook, 4 t h e d i t i o n , M c G r a w - H i l l .  3.  T a g g a r t , A.F., & Sons.  4.  Oka, Y., Majima, H., Vol.9,  No.2  Handbook o f M i n e r a l D r e s s i n g , John W i l e y Canadian M e t a l l u r g i c a l  (1970)  5.  H i r a m a t s u , Y., 0ka,Y., I n t . J . R o c k V 0 I . 3 (1966)  6.  Bond, F. C ,  AIMS. T r a n s . , V o l . 1 9 3 *  ?.  S m i t h , R.W.,  Mining Engineering,  Bond, F.  AIME,'Trans.,'(i960), Vol.  .8.  9.  C.,  Coates D.  Min.  Quarterly  F.,  P a r s o n s R. C ,  Sci., Vol.3,  (1966)  MechoMin.Sci., (1952)  April  (I96I) .217  I n t . J . Rock Mech. M i n .  10.  Yarriaguchi, U.,  11.  M c W I l l I a m s , J.R., T e s t i n g T e c h n i q u e s f o r Rock M e c h a n i c s , . 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. 13.  I n t . J . Rock Mech. M i n . S c i . V o l . 7  (1970)  Emery, C. L., Some A s p e c t s o f S t r a i n i n R o c k s j C o u r s e • Mineral Engineering 4 5 5 « O b e r t , L., D u v a l l , W.,I,, Rock M e c h a n i c s and t h e D e s i g n o f S t r u c t u r e s i n Rock, John W i l e y & Sons, I n c . (196?)  14.  B i r c h , F.,  15«  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.65,  No.4  J o u r n a l of Geophysical Research, Trans., (I960) V o l . 1 8 , No.191  (1969)  16. 17. 18.  Ernery, C. L., A p r i l , May.  Mine & Q u a r r y E n g i n e e r i n g , ( i 9 6 0 )  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 . Yang, D.C., Mempel, G., T r a n s . AIME., V o l . 2 3 8  Fuerstenau, (1967)  D.W.,  (1971)  Notes  -  78 -  19•  Schuhmann, R., J r . , M i n  20.  C h a r l e s , R. J . , AIME.Trans.  21. 22.  No.1189.  0  (1940)  T e c h n o l o g y , Tech. Pub. Vol.208„  (1957)  K i n a s e v i c h , R. S., F u e r s t e n a u , 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.  F u e r s t e n a u , 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 )  23.  M u l a r , A.S., AIME T r a n s . ,  24.  Schuhmann, R. J r . , AIME T r a n s V o l . 2 1 4 .  25.  26.  (1964)  • H u k k i , R. T.,  Vol'.223.  AIME T r a n s o ,  (1962)  Vol.220,  No.9o  (1959) (1959)  D e p t S c i e n t i f i c and I n d u s t r i a l R e s e a r c h , Warren Spring Laboratory, Mineral Processing Information 0  No.3>  (1962)  -  79  -  Appendix I  T a b l e 1, Measurement o f P r o p a g a t i o n V e l o c i t y o f P-Wave (Andesite) Length TravelV e l o c i t y of Sample Number Time P Wave x l o 5 cm/sec cm x 10~ sec 6  AAAAAAAAA-  1  5.83 6.43 6.05  2 3 4 5 •  4.6o. 5.33 4 . 3 8 6.16  6 7 8  5.52  9  A-10 A-11 A-12 A-13  '  A-14 . A-1.5  6.78  11.45  A-16  3.14  A-17 A-18  5»30  .  A-19 A-20  . .  A-22 A-23  4 . 9 3 4.01  °93 °99 .90 .06  6.30  6.95. 9.50 9.20 9 . 7 5 6.70  5.51 6.18  5 5 5 6  10.10 7.80'.' 8 . 8 5  6.19 3.69 5.61  6.18  6.05  9.70 10.85  6.48 6.00 6.35 5.51 6.00  9.35 "  5.92 6.34  9.75 8.70  6.33 6.34 6.01  9.55 5.23 8.90 8.10  5.95  6.08  6.76  •  5 . 9 4 5-78  3.29  6.05  3.^7 3.96 4 . 4 3  6.05 7.00 7.80  A-24  . 3 . 8 3  6.50  A-2 5 A-26  3.79  6.20 2.85  7  1.70  2.18  3.05 3.80  A-29 A-30  2.21 2 . 0 3  3.50  6.35 5.57. 5»746.31  3.40  5.97  •A-21  A-2  A-28  •  1.81  Mean v a l u e Standard d e v i a t i o n C o e f f i c i e n t of standsrd deviation  ... •  5.74 5 . 6 6 5.68 5o89 6.11  ..6.01 0.26  0.043  - 80 -  T a b l e 1 . Measurement o f P r o p a g a t i o n V e l o c i t y o f P-Wave Granite Sample Number  Length cm  TravelTime  Velocity of P Wave  x  x  10  sec  10  G-  1  5.44  14.25  3.82  G-  2  6.63  17.80  3.72  GG-  3  6.23  15.75  3.94  4  6.11  16.05  3.81  GG-  5  15.70  3.82  6  6.06  • 1 5 . 8 5  3«82  GG-  7  6.74  17.65  3.82  8  5.66  14.25  3-97  '6.00 .  '  G- •  9 •'  5.06  12.95  3.91  G-10  ,  5*93  15.95  3.72  4 . 7 7  12.30  3.88  4.71  12.75  3.69  G-13  5.79  15.15  3.82  G-14  4 . 9 3  1 3 . 6 5  G-15  5.48  14.70  G-11 G-12  ,  ,  Mean v a l u e Standard d e v i a t i o n C o e f f i c i e n t of standard deviation  -  cm,  3.61 3.73  3.81 0.10 0 . 0 2 6  - 81 -  Table' 1. Measurement o f P r o p a g a t i o n V e l o c i t y o f P-Wave Marble Sample Number  Length  TravelTime  cm Mb- 1 Mb- 2 Mb- 3 Mb- 4 Mb- 5 Mb- ' 6 Mb- 7 Mb- 8 Mb- 9 Mb-10  x  4.09 6.25 ,  - 6 10  Velocity of P Wave  sec  5  x 10  5.88  6 . 3 4 6.22  6.95 14.15 11.03 12.85  60O3  1 3 . 6 5  4.84 4.42  1 0 . 9 3 6.38  5.53 4 . 7 8  9 . 8 5 9.60  5.55 6.07  6.04 ' 3 . 0 5 5*47 5*33 6.43  .  . •'  4.42 5.75  5.82  5.25  11.05 11.98  Mb-12  2.91  6.20  4 . 3 8 4 . 6 9  Mb-13  3.97 5.12 4 . 2 6  8.05 1 0 . 9 8  ^.93 4 . 6 6  3.35  5.10 4.51 4 . 9 4  Mb-11  •  Mb-14 .  Mb-15  Mb-16  3.52  Mb-17  5.55 5.22  Mb-19  3.48  Mb-18--  .  7.80 11.23 10.30 6.20  Mean v a l u e Standard d e v i a t i o n C o e f f i c i e n t of standard deviation  MV-1 Mb"-2 Mb -3 0  Mb'-4  Mb'-5  5.41  7.18 4.426.53 6.56  10.85 14.30  7.80  12.55 1 3 . 6 5  .  5 . 0 7 5.61 5.10 0 . 5 5 0.108  4.98 5.02 5.67 5.20  4.81  ,  cm/sec  - 82 -  T a b l e 1. Measurement o f P r o p a g a t i o n V e l o c i t y o f P-Wave Sandstone '. Sample Number  Length  Travels Time  cm S- 1 • S- 2 s-  Ss-  SSSS-  x  4.89 5.00  5 6  5.22 4.88  7  6.56  8  4.19 4 . 6 3  9  4.64  S-10 S-11 S-12 S-1'3  •  S-14 • s-15  - 6  •.  sec  •  12.80  1 9 . 3 5 10.10  11.35  4.03  5«01  •13.25 15.45  3-78  4 . 2 3 5.03 6.98  MgMg-  8.95 11.20  13-40 '  4.82 4.90 '  4.15  11.85 .  4.16  7 8  .3.83  9  5.15 .4.47  Mg-10  6.19 6.14  Mg-11 Mg-12 Mg-13  4 . 6 5  Mg-14  4.07 5.^7  Mg-15  4.10  11.30 11.30  5-19  •  .  3.7^ 3«86 0.26  deviation .  4.07 5.30  9.10 .8.80 8.30  ^.73 4.61  11.50 9.60  4 . 6 6  9.55 9 . 7 5 12.90  0.067  ^ . 7 3 4 . 4 9 5.21  9.25  15.25 11.60  -  3.39 4.15  4.5?  2 3 4  w.{r-  .  4 . 1 0 4 . 1 6 4.05  .1  5 6  3-33 3.71  .  10.15 12.80  5.78  •  3 . 8 3  20.40 13.15  4 . 2 2  cm/  3.82  .  13.05  .  10 3.75 4.02  Mean v a l u e S t a n d a r d d e v i at i o n C o e f f i c i e n t o f Standard Magnetite MgMgMgMgMgMg-  5  x  16.20 12.05  6.09  4.85  3. • 4  10  Velocity of P Wave  3.97  4.48 '4.06 5.29 4 . 8 7 4.17 4 . 2 6  Mean v a l u e ^.59 ' 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  -1 Mg' 2 ' -0  Mg' - 3 Mg' - 4 Mg° - 5  . .-  6.87 6.69 . 6.25 6.17 7.35  15.75 15.0 5  4 . 3 6 4 . 4 5  15.75 13.15 15.10  3.97 4 . 6 9 4 . 8 7  Table 2 . Measurement o f T e n s i l e S t r e n g t h1 Granite  Andesite Specimen Number  ,  Thickness  Load a t Failure (kg)  (cm)  Tensile Strength (kg/cm Y  Specimen Number  Z  Thickness (cm)  Load a t Failure (kg)  Tensile Strength  1  5.44  3550  •  IO7  156  GG-  2  6.63  5550  .  113  6350  197  G-  3  6.20  5150  119  4.38  3300  15^  G-  4  •6.11  4750  113  5.52  5700  167  5  6.00  4500  112  A-10  3.69  3750  247  GG-  6  6..O6  5200  126  A-16  3.14  2500  226  A-17  5  5100  A-  1  5.87  7400  192  A-  3  6,05  6400  A-  5  5-36  A-  6  A-  8  30  7  6.74  5600  162  GG-  8  5.66  4000  214  c-  9  5.06  3550  G-10  5.93  4450 .  113 116  :  110 '  111  • 124  • A-18'  4.93  A-19  4.01  5150  A-20  3.29  3300  273  G-ll  4.77  2950  A-21  3.^7  2700  200  G-12  4,71  3200  A-22  3.96  3400  193  G-13  5.79  4000  106  A-23  4.43  269  G-14  *K93  3200  118  A-24  3.83  3050  185  G-15  5.48  2800  33  A-25  3-79  .4200  262  '  5850  .  5900  ..  286  .  •  128  Table 2 . Measurement o f T e n s i l e S t r e n g t h Marble Specimen Number Mb-  •  Sandstone Thickness  Load a t Failure (kg)  (cm)  Tensile Strength (kg/cm )  Specimen . Number  2  Thickness (cm)  Load a t . T e n s i l e F a i l u r e . Strength (kg/cm ) (kg) 2  S- 1  6.09  4500  108  S- 2  4.85  3300  125  4.89  2650  52  S- 3 S- 4  3100  2200  5^  S- 5  5.00 5.22  99 111  4100  134  6.04  2600  63  S-  4.88  3300  123  Mb- 8 Mb- 9  .5.47  1700  50  6.56  5950  123  5.83  .1500  40  4.19  3000  153  Mb-10 Mb-11  "6.43  2200  47 .  4.63  3400  5.25  2000  65  ' 4.64  3100  .141 .128  Mb-13 Mb-l 4  3.97 5.12  1050  60  4.22  2900  145  4600  153  Mb-15  4.26  5.19 ^.57  3300  146  Mb-l 6 Mb-l?  3.52  1050  75  S-14  3550  127  5.55  1950  56  S-15  5.01 5.78  5050  135  Mb-18  1850  61  Mb-l 9  5.22 3.48  900  67  Mb-20  4.98  1300  1  4.09  1050  Mb- 2  6.25  1900  Mb- 3  6.34  2550  57  Mb- 4  6.22  2250  Mb- 5 Mb- 6 '  6.03  '  56  1850  .  63  1300  •  . 63  '46  6  S- 7 • S- 8 V  S- 9 • S-10 S-11 . S-12 S-13  •  T a b l e 2 . Measurement o f T e n s i l e S t r e n g t h Magnetite Specimen .Thickness Number (cm) . Mg- 1 Mg-  2  Tensile. Strength (kg/cm )  Load a t Failure (kg)  2  4.23  1600  79  5-03  2300  81  Mg- 4 .  4.82  Mg-  5  4.9.0  2350  6  4.15  2100  4.16  1550  81"  3.83  1250  75  5.15  2350  79  1700  .77  . Mg.Mg-  7  Mg- 8 Mg- 9  .. 4,4?  Mg-10  Mg-l4  ,^  ..  2300  115 .  . 87 110  4.07  1750  Mg-16  3.43  1750  129  Mg-17 '. Mg-18  4.08  1950  105  4.1-6  1750  90  Mg-19  4.20  1400  70  Mg-20  4.79  2500  •  .  95:  100  I CO  I  Appendix I I  Table 3 . E f f e c t o f B a l l Charge and M i l l Speed on t h e Torque - B a l l Charge (kg) (Percentage of M i l l Volume)  Mill 32  47  .  55  58  Speed 62  ( r . p, m. )• 68  42,1  39.8  70  76  81  92  46.0  42.1  94.0  (100.0)  38.1  86.5  (  92.0)  57.5  79.0  (  83.8)  96.5  104.0  109.0  71.4  (  75.9)  1^5.5  1^9-5  1 5 3 . 5  '139.5  63.5  (  68.5)  184.0  185.5  144.0  99.1  55-5  (  58.9)  20.3.0  150.0  107.5  ^7.5  (  50  214.5  (  42.0)  211.0  31.^  (  33.^)  23.7  (  22.3)  15.8  (  14.8)  7.7  (  8.2)  '39-5  0  . (  0  ;4)  157.0  69.O  187.5 215.0  209.0  187.5  224.0  227.0  225,0  211.0  222,0  224.0  225.0  224.0  198,5  205.0  20?.0  .208.0  165.0.  I 6 8 . 5  170.0  172,5  126.5  )  -  84.3  85.O  29.1  33.0  .130.5  133.0  88.2  88.5  119.0  113.8  191.5  1 6 8 . 5  117.0  .  91.9  144.0 125.5  208.0  211.0 •176.0  136.5 •'91.2  38.3  76.6  213.0  214.5  179..0  179.5  •138.5 92.0  95.0  42,2  .  47.0  Appendix I I I T a b l e d . 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 75 (70  Mesh  /=  6  60.0  •8  21.0  10  .  7.7  :revs sec) Cum  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) 3 0 0 revs 800 revs 2 0 0 revs . 1 2 5 revs (750 (11? sec) sec) (281 sec) ( 1 8 8 1 sec) % % Cum % Cum % Cum % Cum  100.0  41.2  lOOoO  36.8  100.0  40.0  27.?  58.8  26.4  63.2  19.0  11.3  31.0  12.3  6.6  19.7  100.0  4,0  20.9  85.3  9.3  95.Q  36.7  1.5.7  64.4  10.0  85.7  7.5  24.4  12.3  48.7  10.5  75-7  1^.7  .  99.9  14  4.2  20  2.3  7.0  4.1  13.2  4.6  17.0  9.0  36.4  10.0  65.2  23.  1.4  4.7  2.4  9.1  2.8  12.3  6.0  2?.4  8.3  55.2  .35.  0.9  . 3->  1.6  6.6  4.3  21.4  43  0.5  2.5  1.0  5.0  65  0.4  1.9  0.7  100  0.2  1.5  0.5  150  0.2  1.3  200  1.1  1.1  11.2  '  "  1.9  9.6  1.2  7.7  2.8  17.1  0.9  6.5  2.2  14.2  30  0.7  5.5  1.5  12.0  0.4  2.7  0.5  4 . 8  1.2  10..  2.3  2.3  4.3  4.3  9.2  .  4.0  9.2  .  6.9 5.1  46.9 40.0 3^.9  5  3.5  30.4  2.5  26.9  24.4  24.4  Table 4 . Screen A n a l y s i s f o r t h e E a l l M i l l i n g ' . o f G r a n i t e Mesh  75 (70  revs sec)  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) revs 150 100 revs (141 sec) ( 9 4 sec)  # Curn 6  '  fo  •  Cum  19.7  100.0  23.7  80.3  17.8  87.0  10  14.6  56.6  13'.  69.2  14  10.7  42.0  12.0  8  .  13.0.  100.0  5  % Cum  % .  5.1  •  (234  revs sec)  %  % Cum  250  0.6  100.0  100.0  9o9  95.0  2.4  9 9 . ^  10.6  85.1  b. r>  97.0  55'.7'  12.3  74,5  7.9  92.7  .  84.8  20  8.5  31.3  10.8  43.7  13.0  62.2  11  28  6.1  22.8  8.4  32.9  11.2  4 9 . 2  13.7  72.9  35  4.8  16.7  7.0  24.5  9 . 8 .  38.0  13.4  59.2  48  3o3  11.9  5.0  17.5  7.1  28.2  10.6  45.8  4.1  12.5  5.9  21.1  4.3.  15.2  6.7  26.5  3»0  10.9  4.8  I 9 . 8  7.9  •7.9  •65  2.6  8.1  100  1.8  6.0  2.9  150  -.1.3  4.2  2.0  5.5  200  2.9  2.9  3.4  3.4  .  .  8.4  .  .9  8.7  15.0  35.2  15.0  Table 4 . S c r e e n A n a l y s i s f o r t h e B a l l M i l l i n g of Granite Total Grinding Revolutions ( G r i n d i n g Time) . 5 0 0 revs (469 sec)  Mesh  %  •  % Cum  6  1 0 0 0 revs (939 sec) .  %  % Cum  •  8  0.1  99.9 .'8  10  0.4  99  14  1.4  99.4  0.1  99.9  20  5.1  98.0  0.4  9 9 c 8  10.1  92.9  2.3  99°'+  14.1  82.8  28  35 48 65  „  1  2  « 9  12.7  68.7  .  6.9  97.1  9.9  9 0 , 2  55=8  13.3  80.3'  100  9.9  43.1  11.9  67.0  150  7.5  33.2  10.1  55-1  25.7  ^ 5 ° 0  45.0  200  25.7  Table 4. S c r e e n A n a l y s i s f o r t h e B a l l M i l l i n g o f M a r b l e 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) 1 0 revs 2 0 revs 40 r e v s 60 r e v s ( 9 sec) (19 sec) ( 3 8 sec) ( 5 6 sec) . . % % Gum % % Cum % % Cum ' % f5 Cum 100 o0 61.9 1 0 0 . 0 9 9 . 9 52.3 .37.1 99.9 25.5  Mesh  :  6  62.8  20.4  74.5  42.8  11.0  33.5  5.4.  13.1  10.1 7.8 6.0  38.1  19.2  47.6  4o9  20.1  7.0  28.4  "14  3.4  15.2  4 . 7  21.4  6.6  20  2.7  11.8  3.6  16.7  28  2.1  9.1  3.0  35  1.3  7.0  2.3  48  1.7  5.7  1.8  65  0.9  4.0  1.5  100  0.8  3.1  1.1  150  0.5  2.3  0.8,  200  1.8  2.6  8 10  I80O .  .  1.8  •  20.0 :  12.6  91.2  54.1  9.9  78.6  8.5  43.1  9 . 5  68.7  26.9  7o0  34.6  9 . 3  59.2  4 . 6  21.5  5.7  27*6  8.9  4 9 . 8  3.9  .16.9  4 . 9  21.9  8.4  40.9  2.9  13o0  3.7  17*0  6.7  32.6  2.4  10ol  3 o l  13»3  6.1  25o.9  10.2  4 . 8  19.7  3o6  14.9  9 . 3  1.7  7«7  2.4  3.4  1.3  6.0  1.7  7o8  2.6  4 . 7  4 . 7  6.1  6.1  4.5  1 0 0 revs ( 9 4 sec) % % Cum 8.8 100.0  .  11.3  11.3  Table 4, S c r e e n A n a l y s i s f o r t h e B a l l M i l l i n g o f M a r b l e Mesh  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) 1 0 0 0 revs 500 revs 2 0 0 revs (188 s e c ) ( 9 3 9 sec) ( 4 6 9 sec)  revs (141 sec)  150  % Cum  6  8.5  %  100.0  8  12.3  91*5  10  l l o l  14 .  10,5  1.7  f  Cum  %  % Cum  % Cum  100.0  3.5  98.3  0.1  99o9  79o2  5.1  94,8  0 . 2  99o8  .68.1  6.8  89.7  1.2  99o7  .  %  20  9.6  57.6  808  82.9  3o3  98.5  0,1  28  8o3  48.0  9»8  74.1  6,4  95o2  0 , 6  99*9  35  7.5  39.8  10.5  64.4  9 . 3  88.8  2.7  99o3  48  5.9  32.3  9.1  53.8  9»6  79o5  5*3  9 6 . 6  5  5«5  2 6 . 4  9-0  4 4 . 7  1 0 , 8  69.9  8.7  91o3  100  4 . 4  20.9  7c6  35<>7  9 . 4  59ol  9»6  82.6  150  3.5  16.5  6.1'  28.1  8,3  ^9»7  9 . 7  73 = 0  200  13.0  41,4  41,4  63*3  63o3  6  13o0  .  22.0  22.0  100.0  Table 4 . S c r e e n A n a l y s i s f o r t h e B a l l M i l l i n g o f Sandstone Mesh  •?5 (70  %. 6  11.4  8  "lO.O  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) 800 300 revs. revs. 150 revs. revs. (281 sec.) sec.) (750 (141 sec.) sec.) % Cum. % Cum. % Cum. % % Cum.. 99*9  1.0  100.0  0.1  100.0  88',  2.5  . 9 9 . 0  0.2  99.9  5  10  6.4  73.4  2.1  9.6.5  0.3  99.6  14  4.8  72,0  2.1  94.3-  0.3;  9 9 . 3  20  4 . 3  -67.2  2.4  92.3  0.5  99.0  28  4.0  63.O  2.7  89.9  35  5.5  59.0  4.4  87.1  48  7.3  7.7  82.8  '  53.5  98.5  0.1  .100.0  2.7  97.5  0.6  99.9  6.3  94.9  2,6  9.9.4  . 1 . 0  14.8  46.2  21.0  75.0  22.2  88.5  12.2  9 6 . 8  100  20.0  31.4  29.3  54.0  33.3  66.3  30.0  84.6  150  6.1  11.4  10.1  24.6  11.3  33.0  14.7  54.7  200  5.3  14.5  14.5  21  21.7  40.0  40.0  65  5.3  .7  Table 4,  Mesh . •  %  S c r e e n A n a l y s i s f o r the B a l l M i l l i n g o f M a g n e t i t e  7 5 revs. (70 sec.) % Cum.  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) 1 5 0 revs. 2 2 5 revs. 5 0 0 revs. (141 s e c ) (211 sec.) (469 s e c . ) % Cum. % % %'Cum. % Cum. 0.5  100.0  7.5  9 5 . 6  1.0  •99.5  75.9  •8.8  88.1  1.6  9 8 . 5  12..5  63.5  10.8  79-3  4.0  96.9  11.9  51.0  12.7  68.6  7.8  92.9  9.4  39.2  11.6  55.8  7.9  29.8  10.8  44.2  5.5  21.9  7°9.  33.5  25.8  100.0  10.2  100.0  8  21.4  72.9  13.8  89.8  10.  14.0  52.8  12.4  14  10.3  38.8  20  8.0  28.5  . 28  5.4  20.5  35  4.2  15.0  48  -  2.8  •  10.8  4.4  100.Q  6  2.3  ,8.0  4.5  16.4  6.9  25.6  100  1.6  5.7  3.3  11.9  5-1  18.6  1.50  1.2  4.1  2.4  .  3 = 7  200  2.9  2.9  6.2  65  •'-  8.6 6.2  9*9  11.6 .  1-3.7.' •  11.8  .  85.I 73.5 5.9.8.  11.4.  48.0  8.8  36.6  13.6  7.0  27.8  9c9  20.9  20.9  .  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,. L i m e s t o n e , and Quartz 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  ,  Feldspar 75 r e v s (70 sec) % io Cum  36.O  100.0  15»2  100,0  22.5  50.0  23.O  64,0  20.6  84.8  47.9. 32.6 22,8  9.7 5.5 3.2  27.5 .17.9 12.4  13o4  41 o0 27.6  16,7  64.2  13o3 10.2  ^7°5 34.2  .16.1  2.0  11.7  1.3  9.9  6.9 5-1  24.0 17«2  1.7"  8,7 6.5  7ol 5o0  3o4 2.6  1.3 1.0  4,9 3»6  0.9 0.7 0,6  3.5 2.4  1.8 •1.3  12,1 8.7 6.2 4.4  2,6  2.6  1.7  3.1  3.1  10  15.3 9.8 6.7  35 48 65 100 150 200  •  100,0  26,0 26.1  28  .' •Quartz 150 r e v s 75 r e v s (141 s e c ) (70 s e c ) % Cum% % Cum  50,0  6 8 14 20  Limestone 75 r e v s (70 sec) % % Cum  4.4 3»0 2.1  100,0 74.0  0.5 3,2  8.3 5c7  •9»2 7o2  3o7 2.8  "• 5=9 5.0  2,1  4.3. 3°7 3.2  1.5 1.0 0.8 1.7  19»3 13o6  .  -.95  -  A p p e n d i x IV D e r i v a t i o n o f E q u a t i o n 4 i n Page 17 Fracture  o f a specimen "by a f r e e - f a l l i n g b a l l was  assumed t o o c c u r o n l y when t h e t e n s i l e s t r e s s r e a c h e d t h e t e n s i l e s t r e n g t h o f a specimen.  As shown i n E q u a t i o n 3s  t h e 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 t o t h e square o f t h i c k n e s s o f a specimen. ' StoC - ~ d~  or  F oc S t - d  2  For a specimen h a v i n g s i z e o f d=d+Ad, where d i s standard  s i z e and a d i s i n c r e m e n t , t h e a p p l i e d f o r c e i s Foc(d + A d ) = d ( l + ~ 2  2  d  Then F might be p r o p o r t i o n a l t o t h e h e i g h t  of a b a l l position  H,. s i n c e t h e p o t e n t i a l energy o f t h e 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 H oc F oc d (1 +  2/Ad —-—) d  Hence t h e c o r r e c t e d h e i g h t He due t o t h e change o f t h i c k n e s s o f a specimen i s _2 ' d ( 1 + 2(d-oT)/ d ) He = H • p d TT  = H ( 1 + 2(d-d)'/ d )  -(4)  

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0081145/manifest

Comment

Related Items